Recent Developments in Lie Algebras, Groups and Representation Theory

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Recent Developments in Lie Algebras, Groups and Representation Theory

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Proceedings of Symposia in

PURE MATHEMATICS Volume 86

Recent Developments in Lie Algebras, Groups and Representation Theory 2009–2011 Southeastern Lie Theory Workshop Series Combinatorial Lie Theory and Applications October 9–11, 2009, North Carolina State University Homological Methods in Representation Theory May 22–24, 2010, University of Georgia Finite and Algebraic Groups June 1–4, 2011, University of Virginia

Kailash C. Misra Daniel K. Nakano Brian J. Parshall Editors

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SOCIETY

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ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ

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American Mathematical Society Providence, Rhode Island

2010 Mathematics Subject Classiﬁcation. Primary 17B37, 17B55, 17B56, 17B65, 17B67, 20C08, 20C11, 20G05, 20G42, 20G43.

Library of Congress Cataloging-in-Publication Data Recent developments in Lie algebras, groups and representation theory : Southeastern Lie Theory Workshop series 2009–2011 : Combinatorial Lie Theory and Applications, October 9–11, 2009, North Carolina State University : Homological Methods in Representation Theory, May 22–24, 2010, University of Georgia : Finite and Algebraic Groups, June 1–4, 2011, University of Virginia / Kailash C. Misra, Daniel K. Nakano, Brian J. Parshall, editors. pages cm. — (Proceedings of symposia in pure mathematics ; volume 86) Includes bibliographical references. ISBN 978-0-8218-6917-8 (alk. paper) 1. Lie algebras–Congresses. I. Misra, Kailash C. (1954–), editor of compilation. II. Nakano, Daniel K. (Daniel Ken), (1964–), editor of compilation. III. Parshall, Brian (1945–), editor of compilation. IV. Southeastern Lie Theory Workshop on Combinatorial Lie Theory and Applications (2009 : North Carolina State University). V. Southeastern Lie Theory Conference on Homological Methods in Representation Theory (2010 : University of Georgia). VI. Southeastern Lie Theory Workshop: Finite and Algebraic Groups (2011 : University of Virginia) QA252.3.R425 512482—dc23

2012 2012026116

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

17 16 15 14 13 12

North Carolina State University, October 2009

University of Georgia, May 2010

University of Virginia, June 2011

Contents Preface

ix

Perverse coherent sheaves on the nilpotent cone in good characteristic Pramod N. Achar

1

On the vanishing ranges for the cohomology of ﬁnite groups of Lie type II Christopher P. Bendel, Daniel K. Nakano, and Cornelius Pillen

25

Tilting modules for the current algebra of a simple Lie algebra Matthew Bennett and Vyjayanthi Chari

75

Endotrivial modules Jon F. Carlson

99

Super duality for general linear Lie superalgebras and applications Shun-Jen Cheng, Ngau Lam, and Weiqiang Wang

113

Structures and representations of aﬃne q-Schur algebras Jie Du

137

Multiplicative bases for the centres of the group algebra and Iwahori-Hecke algebra of the symmetric group Andrew Francis and Lenny Jones

159

Moonshine paths and a VOA existence proof of the Monster Robert L. Griess Jr.

165

Characteristic polynomials and ﬁxed spaces of semisimple elements Robert Guralnick and Gunter Malle

173

“Frobenius twists” in the representation theory of the symmetric group David J. Hemmer

187

The generalized Kac-Wakimoto conjecture and support varieties for the Lie superalgebra osp(m|2n) Jonathan Kujawa

201

An approach towards the Koll´ar-Peskine problem via the Instanton Moduli Space Shrawan Kumar

217

On the representations of disconnected reductive groups over Fq G. Lusztig

227

vii

viii

CONTENTS

Forced gradings in integral quasi-hereditary algebras with applications to quantum groups Brian J. Parshall and Leonard L. Scott

247

A semisimple series for q-Weyl and q-Specht modules Brian J. Parshall and Leonard L. Scott

277

Preface Lie theory represents a major area of mathematical research. Besides its increasing importance within mathematics (to geometry, combinatorics, ﬁnite and inﬁnite groups, etc.), it has important applications outside of mathematics (to physics, computer science, etc.). During the twentieth century, the theory of Lie algebras, both ﬁnite and inﬁnite dimensional, has been a major area of mathematical research with numerous applications. In particular, during the late 1970s and early 1980s, the representation theory of Kac-Moody Lie algebras (analogs of ﬁnite dimensional semisimple Lie algebras) generated intense interest. In part, the subject was driven by its interesting connections with such topics as combinatorics, group theory, number theory, partial diﬀerential equations, topology and with areas of physics such as conformal ﬁeld theory, statistical mechanics, and integrable systems. The representation theory of an important class of inﬁnite dimensional Lie algebras known as aﬃne Lie algebras led to the discovery of Vertex Operator Algebras (VOAs) in the 1980s. VOAs are precise algebraic counterparts to “chiral algebras” in two-dimensional conformal ﬁeld theory as formalized by Belavin, Polyakov, and Zamolodchikov. These algebras and their representations play important roles in a number of areas, including the representation theory of the Fischer-Griess Monster ﬁnite simple group and the connection with the phenomena of “monstrous moonshine,” the representation theory of the Virasoro algebra and aﬃne Lie algebras, and two-dimensional conformal ﬁeld theory. In 1985, the interaction of aﬃne Lie algebras with integrable systems led Drinfeld and Jimbo to introduce a new class of algebraic objects known as quantized universal enveloping algebras (also called quantum groups) associated with symmetrizable Kac-Moody Lie algebras. These are q-deformations of the universal enveloping algebras of the corresponding Kac-Moody Lie algebras, and, like universal enveloping algebras, they carry an important Hopf algebra structure. The abstract theory of integrable representations of quantum groups, developed by Lusztig, illustrates the similarity between quantum groups and Kac-Moody Lie algebras. The quantum groups associated with ﬁnite dimensional simple Lie algebras also have strong connections with the representations of aﬃne Lie algebras. The theory of canonical bases for quantum groups has provided deep insights into the representation theory of quantum groups. More recently, the theory of geometric crystals introduced by Berenstein and Kazhdan has opened new doors in representation theory. In particular, canonical bases at q = 0 (crystal bases) provide a beautiful combinatorial tool for studying the representations of quantum groups. The quantized universal enveloping algebra associated with an aﬃne Lie algebra is called a quantum aﬃne algebra. Quantum aﬃne algebras quickly became an interesting ix

x

PREFACE

and important topic of research, the representation theory of which parallels that of the corresponding aﬃne Lie algebras. But the theory is much deeper and richer than its classical counterpart, providing a clearer picture of connections with the other areas mentioned above. After the classiﬁcation of the ﬁnite simple groups (now complete), a full understanding of the representation theory of ﬁnite simple groups over ﬁelds k of arbitrary characteristic provides a major problem for the 21st century. The sporadic Fischer-Griess monster (mentioned above) gives one important example of a ﬁnite simple group closely related to Lie theory. Apart from the alternating groups and the 26 sporadic simple groups, the ﬁnite simple groups come in inﬁnite families closely related to the ﬁnite groups of rational points G(q) of simple algebraic groups G over algebraically closed ﬁelds k of positive characteristic p > 0. (The ﬁnite Ree and Suzuki groups are variations on this theme.) The representation theory of these ﬁnite groups of Lie type thus form a key area of investigation. One can consider a ﬁeld F , algebraically closed for simplicity, having characteristic , and investigate the category of F G(q)-representations. There are three cases to consider. First, in case = 0, take F = C, the complex numbers. This theory is the so-called ordinary representation theory of G(q). As a result of work of Deligne, Lusztig, and many other mathematicians over the past 35 years, the ordinary theory is quite well understood in comparison to the cases in which > 0. Second, if = p (the equal characteristic case), take F = k. By work of Steinberg, the irreducible kG(q)-modules all lift to irreducible rational representations of the algebraic group G. This fact has provided strong motivation for the study of the modular representation theory of the semisimple algebraic groups G over the past 30 years. For example, a famous conjecture due to Lusztig posits the characters of the irreducible representations when the characteristic p is large (bigger than the Coxeter number). For each type, this conjecture has been proved for p “large enough” by Andersen-Jantzen-Soergel. The proof follows a path from characteristic p to quantum groups at a root of unity to aﬃne Lie algebras and perverse sheaves. Thus, it ultimately involves the inﬁnite dimensional Lie theory discussed above. Although this approach fails to provide eﬀective bounds on the size of the prime p, a new avenue via a related combinatorial category has been recently investigated by Fiebig. As a result of Fiebig’s work, very large eﬀective bounds for Lusztig’s conjecture are now known. In addition, the determination of the characters for small p (i.e., less than the Coxeter number) remains largely uninvestigated. Third, when 0 < = p (the cross-characteristic case), much less is known in general. When G is a general linear group GLn (k), the determination of the decomposition numbers for the ﬁnite groups GLn (q) can be determined in terms of decomposition numbers for q-Schur algebras and then for quantum groups over ﬁelds of positive characteristic. This is the so-called Dipper-James theory. There are close connections with the representation theory of Hecke algebras and symmetric groups. In other types, much less is known; for example, the classiﬁcation of the irreducible representations is incomplete. A major problem for these other types would be to replace the quantum groups used for GLn (q) by some suitable structure. The modular representation theory has provided a crucial interface with the theory of ﬁnite dimensional algebras (especially, the theory of quasi-hereditary algebras introduced by Cline, Parshall and Scott). It seems likely that this direction

PREFACE

xi

will continue to prove fruitful. Another signiﬁcant feature of the modular representation theory of the ﬁnite groups of Lie type and the associated algebraic and quantum groups is the existence of a rich accompanying homological theory. Homological problems emerge immediately because of the failure of complete reducibility. In the equal characteristic case, the homological theory has been extensively developed, for the ﬁnite groups of Lie type, quantum enveloping algebras at roots of unity, restricted Lie algebras and inﬁnitesimal group schemes, as well as other settings. Geometric ideas enter via the theory of support varieties, which associate to each ﬁnite-dimensional module for a restricted Lie algebra (or ﬁnite group scheme) an algebraic variety. In the cross-characteristic case, much less in known about the cohomology. In the equal characteristic case, there is a considerable body of work involving the homological algebra of the inﬁnitesimal groups, and relations between the cohomology of G, its inﬁnitesimal subgroups, and its ﬁnite subgroups. Finally, we mention that the modular representation theory of general ﬁnite groups itself has a strong Lie-theoretic ﬂavor. In part, this is due to the famous Alperin conjecture, suggesting that the irreducible modular representations of general ﬁnite group should be classiﬁed in a “weight theoretic” way, much like irreducible modules for a complex semisimple Lie algebra are classiﬁed by their highest weights. Another notable conjecture, the Brou´e conjecture, has been recently veriﬁed for symmetric groups by Chuang and Rouquier using a the new method of “categoriﬁcation”. In 2009, the three editors established a network of Lie theorists in the southeastern region of the U.S. and proposed an annual regional workshop series of 3 to 4 days in Lie theory. The aim of these workshops was to bring together senior and junior researchers as well as graduate students to build and foster cohesive research groups in the region. With support from the National Science Foundation and the aﬃliated universities in the region, three successful workshops were held at North Carolina State University, the University of Georgia and the University of Virginia in 2009, 2010 and 2011 respectively. Each of these workshops was attended by over 70 participants. The workshops included expository talks by senior researchers and afternoon AIM style discussion sessions with a goal to educate graduate students and junior researchers in the early part of their study for research in diﬀerent aspects of Lie theory. In the third workshop at the University of Virginia, Professor Leonard Scott was honored on the eve of his retirement for his lifetime contributions to many of the aforementioned topics. The plenary speakers in the three workshops were invited to contribute to this proceedings. Most of the articles presented in this book are self-contained, and several survey articles, by Jon Carlson, Jie Du, Bob Griess, and David Hemmer are accessible to a wide audience of readers. The editors take this opportunity to acknowledge the conference participants, the contributors, and the editorial oﬃces of the American Mathematical Society for making this volume possible. Kailash C. Misra Daniel K. Nakano Brian J. Parshall

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Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Perverse coherent sheaves on the nilpotent cone in good characteristic Pramod N. Achar Abstract. In characteristic zero, Bezrukavnikov has shown that the category of perverse coherent sheaves on the nilpotent cone of a simply connected semisimple algebraic group is quasi-hereditary, and that it is derived-equivalent to the category of (ordinary) coherent sheaves. We prove that graded versions of these results also hold in good positive characteristic.

1. Introduction Let G be a simply connected semisimple algebraic group over an algebraically closed ﬁeld k of good characteristic. Let N denote the nilpotent variety in the Lie algebra of G. There is a “scaling” action of Gm on N that commutes with the G-action. Following [B1], we may consider the category of (G × Gm )-equivariant perverse coherent sheaves on N , denoted PCohG×Gm (N ). This category has some features in common with ordinary perverse sheaves, but it lives inside the derived category of (equivariant) coherent sheaves. In this note, we prove the following two homological facts about PCohG×Gm (N ). Theorem 1.1. The category PCohG×Gm (N ) is quasi-hereditary. ∼ Db CohG×Gm (N ). Theorem 1.2. We have Db PCohG×Gm (N ) = Theorem 1.1 means that the category contains a class of distinguished objects, called “standard” and “costandard” objects, that lead to a kind of Kazhdan–Lusztig theory. This result was proved in characteristic 0 in [B2]. (See also [A].) In fact, the proof given there “almost” works in positive characteristic as well; it is quite close to the proof given here. The same arguments also establish the corresponding result for PCohG (N ), where the Gm -action is forgotten. On the other hand, our proof of Theorem 1.2 makes use of the Gm -action in a crucial way (it means that various Ext-groups carry a grading which we exploit), so it cannot easily be forgotten. The proof is quite elementary: it relies only on general notions from homological algebra, and it is similar in spirit to the methods of [BGS]. Unfortunately, for the moment, these methods seem to be inadequate to prove the following natural analogue of Theorem 1.2. 2010 Mathematics Subject Classiﬁcation. Primary 20G05; secondary 14F05, 17B08. The author received support from NSF grant DMS-1001594. c c Mathematical 2012 American 0000 (copyright Society holder)

1

2

PRAMOD N. ACHAR

Conjecture 1.3. We have Db PCohG (N ) ∼ = Db CohG (N ). This conjecture is known to hold in characteristic 0 by [B4]. The proof given there involves relating CohG (N ) to perverse sheaves on the aﬃne ﬂag variety Fl for the Langlands dual group. It is likely (and perhaps already known to experts) that a similar approach using mixed perverse sheaves would allow one to bring in the Gm -action, leading to a characteristic-0 proof of Theorem 1.2 that is quite diﬀerent from the one given here. The reason for the restriction to characteristic 0 in [B4] is that the arguments there require the base ﬁeld k for G to coincide with the ﬁeld of coeﬃcients of sheaves on Fl. The sheaves in [B4], like nearly all constructible sheaves used in representation theory in the past thirty-ﬁve years, have their coeﬃcients in Q . But so-called modular perverse sheaves—perverse sheaves with coeﬃcients in a ﬁeld of positive characteristic—have recently begun to appear in a number of important applications [F, Ju, JMW, S]. It would be very interesting to develop a sheaftheoretic approach to Theorem 1.2 or Conjecture 1.3 in positive characteristic using modular perverse sheaves. In the present paper, the assumptions that G is simply connected and that k is of good characteristic are needed in order to invoke certain results from [BK] and [Ja]. In characteristic 0, it can be deduced from Theorems 1.1 and 1.2 that corresponding results hold for arbitrary connected reductive groups. In positive characteristic, however, isogenous groups need not have isomorphic nilpotent cones (see, e.g., [Ja, Remark 2.7]). In the latter case, the main theorems hold for groups with simply connected derived group, but not for arbitrary reductive groups. The paper is organized as follows. Sections 2 and 3 lay the homological-algebra foundations for the main results, starting with notation and deﬁnitions. The key result of that part of the paper is Theorem 3.15, which states that any quasiexceptional set satisfying certain axioms gives rise to a derived equivalence. In Section 4, we return to the setting of algebraic groups. Section 5 contains a number of technical lemmas on the so-called Andersen–Jantzen sheaves. The main theorems are proved in Section 6. Acknowledgments. While this project was underway, I beneﬁtted from numerous conversations with A. Henderson, S. Riche, and D. Treumann. I would also like to express my gratitude to the organizers of the Southeastern Lie Theory Workshop series for having given me the opportunity to participate in the May 2010 meeting. 2. Preliminaries on abelian and triangulated categories 2.1. Generalities. Fix an algebraically closed ﬁeld k. Throughout the paper, all abelian and triangulated categories will be k-linear and skeletally small (that is, the class of isomorphism classes of objects is assumed to be a set). Later, all schemes and algebraic groups will be deﬁned over k as well. For an abelian category A, we write Irr(A) for its set of isomorphism classes of simple objects. We say that A is a ﬁnite-length category if it is noetherian and artinian. Now, let T be a triangulated category. For objects X, Y ∈ T, we write Homi (X, Y ) = Hom(X, Y [i]). A full subcategory A ⊂ T is said to be admissible if it stable under extensions and direct summands, and if it satisﬁes the condition of [BBD, §1.2.5]. (Thus, our

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

3

use of the term “admissible” is slightly more restrictive than the deﬁnition used in [BBD].) If A ⊂ T is admissible, then it is automatically an abelian category, and every short exact sequence in A gives rise to a distinguished triangle in T. The heart of any t-structure on T is admissible. For the following fact, see [BBD, Remarque 3.1.17] or [BGS, Lemma 3.2.4]. Lemma 2.1. Let A be an admissible abelian subcategory of a triangulated category T. The natural map ExtiA (X, Y ) → HomiT (X, Y ) is an isomorphism for i = 0, 1. If it is an isomorphism for i = 0, 1, . . . , k, then it is injective for i = k + 1. Next, we recall the “∗” operation for objects of a triangulated category D. If X and Y are classes of objects in D, then we deﬁne there is a distinguished triangle . X ∗ Y = A ∈ D X → A → Y → with X ∈ X , Y ∈ Y By [BBD, Lemme 1.3.10], this operation is associative. In an abuse of notation, when X is a singleton {X}, we will often write X ∗ Y rather than {X} ∗ Y. Note that the zero object is a sort of “unit” for this operation. For instance, we have X ∗ Y ∗ 0 = X ∗ Y. Given a class X , X ∗ 0 is the class of all objects isomorphic to some object of X . 2.2. Tate twist. Many of our categories will be equipped with an automorphism known as a Tate twist, and denoted X → X1 . We will always assume that Tate twists are “faithful,” meaning that for any nonzero object X, we have X∼ = Xn

if and only if

n = 0.

A key example is the category Vectk of graded k-vector spaces, where the Tate twist is the “shift of grading” functor. For X ∈ Vectk , let Xn denote its nth graded component. Then Xm is the graded vector space given by (Xm )n = Xn−m . We regard k itself as an object of Vectk by placing it in degree 0. If X and Y are objects of an additive category equipped with a Tate twist, we let Hom(X, Y ) denote the graded vector space deﬁned by Hom(X, Y )n = Hom(X, Y −n ). Notations like Homi (−, −), Exti (−, −), and RHom(−, −) are deﬁned similarly. The following lemma is a graded analogue of [B2, Lemma 5]. Lemma 2.2. Let V be an object in D+ Vectk . (1) If there are integers n1 , . . . , nk such that 0 ∈ V n1 ∗ · · · ∗ V nk , then V = 0. (2) If there are integers n1 , . . . , nk > 0 such that k ∈ V ∗ V n1 ∗ · · · ∗ V nk , then H i (V ) = 0 for i < 0, and H 0 (V ) ∼ = k. For i > 0, H i (V ) is concentrated in strictly positive degrees.

4

PRAMOD N. ACHAR

Proof. (1) Suppose V = 0, and let m be the smallest integer such that H m (V ) = 0. Then, for any object X ∈ V n1 ∗ · · · ∗ V nk , it follows that the map H m (V n1 ) → H m (X) is injective. But if X = 0, this contradicts the assumption that H m (V ) = 0. (2) The argument given for part (1) shows that H i (V ) = 0 for i < 0, and that the map H 0 (V ) → H 0 (k) ∼ = k is injective. Let Y ∈ V n1 ∗ · · · ∗ V nk be such that there is a distinguished triangle V → k → Y →. If H 0 (V ) = 0, it would follow that H 0 (Y ) = 0, leading to a contradiction with the fact that H 0 (k) = 0, so it must be that H 0 (V ) ∼ = k. We then see from that distinguished triangle that H i (V ) ∼ = H i−1 (Y )

for all i ≥ 1.

Because all the ni are strictly positive, it follows from the fact that H 0 (V ) ∼ = k that H 0 (Y ) is concentrated in strictly positive degrees, and hence so is H 1 (V ). Thereafter, we proceed by induction on i: if H i (V ) is concentrated in strictly positive degrees, so is H i (Y ), and therefore so is H i+1 (V ). 2.3. Quasi-hereditary categories. Let S be a set equipped with a partial order ≤. Assume that every principal lower set is ﬁnite, i.e., that (2.1)

For all s ∈ S, the set {t ∈ S | t ≤ s} is ﬁnite.

Let A be a ﬁnite-length abelian category, and assume that one of the following holds: • “Ungraded case”: There is a ﬁxed bijection Irr(A) ∼ = S. • “Graded case”: A is equipped with a Tate twist, and there is a ﬁxed bijection Irr(A) ∼ = S × Z with the property that for a simple object L ∈ A, L corresponds to (s, n)

if and only if

L1 corresponds to (s, n + 1).

In the ungraded case, choose a representative simple object Σs for each s ∈ S, and let (≤s) A (resp. (<s) A) be the Serre subcategory of A generated by all simple objects Σt with t ≤ s (resp. t < s). In the graded case, let Σs denote a representative simple object corresponding to (s, 0) ∈ S × Z. In this case, (≤s) A (resp. (<s) A) denotes the Serre subcategory of A generated by all simple objects Σt n with t ≤ s (resp. t < s) and n ∈ Z. More generally, for any subset Ξ ⊂ S × Z, we let Ξ A denote the Serre subcategory of A generated by the Σt n with (t, n) ∈ Ξ. In the sequel, we will focus mostly on the graded case. With the above notation in place, the corresponding deﬁnitions and statements for the ungraded cases can usually be obtained simply by omitting Tate twists and by changing “Hom” and “Ext” to “Hom” and “Ext,” respectively. For instance, it is left to the reader to formulate the ungraded version of the following deﬁnition. Definition 2.3. A category A as above is said to be graded quasi-hereditary if for each s ∈ S, there is: (1) an object Δs and a surjective map φs : Δs Σs such that ker φs ∈ (<s) A

and

Hom(Δs , Σt ) = Ext1 (Δs , Σt ) = 0 if t > s.

(2) an object ∇s and an injective map ψ s : Σs → ∇s such that cok ψ s ∈ (<s) A

and

Hom(Σt , ∇s ) = Ext1 (Σt , ∇s ) = 0 if t > s.

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

5

Any object isomorphic to some Δs n is called a standard object, and any object isomorphic to some ∇s n is a costandard object. 2.4. Quasi-exceptional sets. We again let S be a set equipped with a partial order ≤ satisfying (2.1). Let D be a triangulated category, either equipped with a Tate twist (the “graded case”) or not (the “ungraded case”). As noted above, the deﬁnitions and lemmas below are usually stated only for the graded case. However, our ﬁrst deﬁnition comes with a caveat; see the remark below. Definition 2.4. A graded quasi-exceptional set in D is a collection of objects {∇s }s∈S such that the following conditions hold: (1) If s ≥ t, then Homi (∇s , ∇t ) = 0 for all i ∈ Z. (2) If i < 0, then Homi (∇s , ∇s ) = 0, and Hom(∇s , ∇s ) k. (3) If i > 0 and n ≥ 0, then Homi (∇s , ∇s n )) = 0. (4) The objects {∇s n | s ∈ S, n ∈ Z} generate D as a triangulated category. For s ∈ S, we denote by (<s) D the full triangulated subcategory of D generated by all ∇t n with t < s. Remark 2.5. An ungraded quasi-exceptional set is deﬁned with analogues of conditions (1), (2), and (4) above, but without condition (3). The omission of condition (3) makes the two cases substantially diﬀerent. In particular, the results of Section 3 apply only to the graded case. Definition 2.6. A graded quasi-exceptional set {∇s } in D is said to be dualizable if for each s ∈ S, there is an object Δs and a morphism ιs : Δs → ∇s such that: (1) The cone of ιs lies in (<s) D. (2) If s > t, Homi (Δs , ∇t ) = 0 for all i ∈ Z. The set {Δs } is known as the dual quasi-exceptional set. It follows from the second condition that Hom(Δs , X) = 0 for all X ∈ (<s) D. The proofs of the following two lemmas about a dual set are routine; we omit the details. Lemma 2.7. If {∇s } is a dualizable quasi-exceptional set, then the members of the dual set {Δs } are uniquely determined up to isomorphism. Lemma 2.8. Let {∇s } be a dualizable quasi-exceptional set, and let {Δs } be its dual set. Then: (1) If s ≤ t, then Homi (Δs , Δt ) = 0 for all i ∈ Z. (2) If i < 0, then Homi (Δs , Δs ) = 0, and Hom(Δs , Δs ) k. (3) If i > 0 and n ≥ 0, then Homi (Δs , Δs n ) = 0. (4) The objects {∇s n | s ∈ S, n ∈ Z} generate D as a triangulated category. Furthermore, for all i ∈ Z, there are natural isomorphisms ∼

∼

Homi (Δs , Δs ) → Homi (Δs , ∇s ) → Homi (∇s , ∇s ).

Definition 2.9. A dualizable quasi-exceptional set {∇ }s∈S with dual set {Δs }s∈S , is said to be abelianesque if we have s

Homi (∇s , ∇t ) = Homi (Δs , Δt ) = 0

for all i < 0.

The main technical result we need about quasi-exceptional sets is the following.

6

PRAMOD N. ACHAR

Theorem 2.10. Let D be a triangulated category with a Tate twist, and let {∇s }s∈S be an abelianesque dualizable quasi-exceptional set with dual set {Δs }s∈S . The categories D≤0 = {X ∈ D | Hom(X, ∇s n [d] = 0 for all n ∈ Z and all d < 0}, D≥0 = {X ∈ D | Hom(Δs n [d], X) = 0 for all n ∈ Z and all d > 0} constitute a bounded t-structure on D. In addition, its heart A = D≤0 ∩ D≥0 has the following properties: (1) A contains all Δs n and ∇s n . ∼ (2) There is a natural bijection Irr(A) → S × Z; the simple object Σs corresponding to (s, 0) ∈ S × Z is the image of the map ιs : Δs → ∇s . (3) A is a ﬁnite-length, graded quasi-hereditary category; the Δs n are the standard objects, and the ∇s n are the costandard objects. Proof sketch. A similar statement in the ungraded case, with the “abelianesque” condition omitted, is proved in [B2, Propositions 1 and 2]. In loc. cit., the standard and costandard objects in the heart are t H 0 (Δs ) and t H 0 (∇s ), where t 0 H (−) denotes cohomology with respect to the t-structure in the statement of the theorem. But the abelianesque condition clearly implies that the Δs and ∇s already lie in the heart of the t-structure, so the ungraded version of the theorem follows from the aforementioned results. The same arguments work in the graded case as well; cf. [B3, Proposition 4]. 2.5. Projective covers. We end this section with a result that lets us construct projectives in an abelian category starting from projectives in a Serre subcatgory. Its proof is similar to that of [BGS, Theorem 3.2.1]. Proposition 2.11. Let A be a ﬁnite-length abelian category. Let L ∈ A be a simple object with a projective cover M , and let R denote the kernel of M L. Let B ⊂ A be the Serre subcategory of objects that do not have L as a subquotient. Let L ∈ B be a simple object. Assume that the following conditions hold: (1) We have Hom(M, M ) k. (2) Inside B, L admits a projective cover P . (3) A and B are admissible subcategories of a triangulated category T, and (2.2)

Hom2T (M, R) = Hom2T (P , R) = 0.

Then L admits a projective cover P in A, arising in a short exact sequence (2.3)

0 → Ext1A (P , M )∗ ⊗ M → P → P → 0.

Proof. Let E = Ext1 (P , M ), and consider the identity map id : E → E as an element of Hom(E, E). Following this element through the chain of isomorphisms Hom(E, E) E ∗ ⊗ E E ∗ ⊗ Ext1 (P , M ) Ext1 (P , E ∗ ⊗ M ), we obtain a canonical element ν ∈ Ext1 (P , E ∗ ⊗M ). Form the short exact sequence corresponding to ν, and deﬁne P to be its middle term. We have thus constructed the sequence (2.3). We must now show that P is a projective cover of L . Because Hom(M, M ) k, we have natural isomorphisms Hom(E ∗ ⊗ M, M ) E ⊗ Hom(M, M ) E.

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

7

Consider now the following commutative diagram: −◦ν

Hom(E ∗ ⊗ M, M )

/ Ext1 (P , M )

E

/E

id

We see that the natural map Hom(E ∗ ⊗ M, M )) → Ext1 (P , M ) is an isomorphism, so from the long exact sequence associated to (2.3), we ﬁnd that the map Hom(P , M ) → Hom(P, M )

(2.4)

is an isomorphism as well, and that Ext1 (P, M ) → Ext1 (E ∗ ⊗ M, M ) is injective. But Ext1 (M, M ) = 0 because M is projective, so Ext1 (P, M ) = 0.

(2.5)

Before proceeding, we observe that R ∈ B. Otherwise, if R had a subquotient isomorphic to L, there would be a nonzero map M → R, since M is the projective cover of L. But the composition M → R → M yields a nonscalar element of Hom(M, M ), a contradiction. Next, for any object X ∈ B, we claim that Hom(E ∗ ⊗ M, X) = Ext1 (E ∗ ⊗ M, X) = 0. The former holds because L, the unique simple quotient of M , does not occur in any composition series for X, and the latter holds because M is projective. Then, from the long exact sequence obtained by applying Exti (·, X) to (2.3), we obtain the following isomorphisms for any X ∈ B: ∼

Hom(P , X) → Hom(P, X),

(2.6)

∼

Ext1 (P , X) → Ext1 (P, X). Since P is a projective object of B, the latter isomorphism actually implies Ext1 (P , X) = Ext1 (P, X) = 0.

(2.7)

Form the following commutative diagram with exact rows: Hom(P , R)

/ Hom(P , M )

/ Hom(P , L)

/ Ext1 (P , R)

Hom(P, R)

/ Hom(P, M )

/ Hom(P, L)

/ Ext1 (P, R)

The ﬁrst vertical map is an isomorphism by (2.6), and the second by (2.4). Both terms in the fourth column vanish by (2.7). Thus, the third vertical map is also an isomorphism. But Hom(P , L) = 0, so Hom(P, L) = 0 as well. Combining this with (2.6) and the fact that L is the unique simple quotient of P , we see that it is the unique simple quotient of P as well.

8

PRAMOD N. ACHAR

Moreover, from (2.7), we see that Ext1 (P, X) = 0 for all X ∈ B. To prove that P is a projective object of A, it remains only to show that Ext1 (P, L) = 0. Using Lemma 2.1, we may form the long exact sequence · · · → Ext1 (P, M ) → Ext1 (P, L) → Hom2T (P, R) → · · · . We saw in (2.5) that the ﬁrst term vanishes, and the assumption (2.2) implies that the last term does as well. Thus, Ext1 (P, L) = 0, as desired. 3. Derived equivalences from quasi-exceptional sets In this section, D will be a triangulated category equipped with a Tate twist and an abelianesque dualizable graded quasi-exceptional set {∇s }s∈S with dual set {Δs }, with S satisfying (2.1). Let A denote the heart of t-structure on D as in Theorem 2.10. Under mild assumptions, there is a natural t-exact functor of triangulated categories real : Db A → D, called a realization functor. For a construction of real in various settings, see [AR, Be, BBD]. The goal of this section is to prove that under an additional assumption (the “eﬀaceability property” of Section 3.2), this is an equivalence of categories. Below, Sections 3.1–3.3 contain a number preparatory results. The derived equivalence result, Theorem 3.15, is proved in Section 3.4. 3.1. Standard ﬁltrations and quasistandard objects. We begin with a number of technical lemmas on the existence and properties of certain objects which are ﬁltered by standard objects. Most of the results of this section are trivial in the case where the quasi-exceptional set is actually exceptional, meaning that Homi (∇s , ∇s ) = 0 for i > 0. Definition 3.1. Let X ∈ A. A ﬁltration 0 = X0 ⊂ X1 ⊂ · · · ⊂ Xk = X is called a standard ﬁltration if there are elements s1 , . . . , sk ∈ S and integers n1 , . . . , nk ∈ Z such that Xi /Xi−1 Δsi ni for each i. If X has such a ﬁltration with s1 = · · · = sk = s, X is said to be s-quasistandard. The notions of costandard ﬁltration and s-quasicostandard are deﬁned similarly. Definition 3.2. The standard order is the partial order Δ on S × Z given by (s, n) Δ (t, m)

if s < t, or else if s = t and n ≥ m.

Similarly, the costandard order ∇ is given by (s, n) ∇ (t, m)

if s < t, or else if s = t and n ≤ m.

A member of a subset Ξ ⊂ S × Z is said to be standard-maximal (resp. costandardmaximal ) if it is a maximal element of Ξ with respect to Δ (resp. ∇ ). A number of statements in this section, starting with the following lemma, contain both a “standard” part and a “costandard” part. In each instance, we will only prove the part pertaining to standard objects. It is, of course, a routine matter to adapt these arguments to the costandard case. For the maps φs : Δs → Σs and ψ s : Σs → ∇s as in Deﬁnition 2.3, we introduce the notation Rs = ker φs , Qs = cok ψ s .

9

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

Lemma 3.3. If (s, n) is standard-maximal in Ξ, then Δs n is a projective cover of Σs n in Ξ A. If (s, n) is costandard-maximal, then ∇s n is an injective hull of Σs n . Proof. We already know that Δs n has Σs n as its unique simple quotient, and that Ext1 (Δs n , Σt m ) = 0 whenever t ≥ s. To prove that Δs n is projective in Ξ A, it remains to show that Ext1 (Δs n , Σs m ) = 0

if m ≥ n.

Consider the short exact sequence 0 → Rs m → Δs m → Σs m → 0. Since Rs m ∈ A, we have Homi (Δs n , Rs m ) = 0 for all i ≥ 0. It follows that there is an isomorphism (<s)

∼

Ext1 (Δs n , Δs m ) → Ext1 (Δs n , Σs m ). When m ≥ n, Lemma 2.8(3) tells us that Ext1 (Δs n , Δs m ) = 0, as desired.

˜ (k) Δ s

with Proposition 3.4. For each k ≥ 0, there is an s-quasistandard object the following properties: ˜ (k) ˜ (k) (1) Hom(Δ s , Σs ) k, and Hom(Δs , Σt m ) = 0 if (t, m) = (s, 0). ˜ (k) (2) Ext1 (Δ s , Σt m ) = 0 if (t, m) Δ (s, −k). (k) ˜ s has a standard ﬁltration whose subquotients are various Δs m with (3) Δ −k ≤ m ≤ 0. ˜ s such that Similarly, there is an s-quasicostandard object ∇ (k) ˜ s ) k, and Hom(Σt m , ∇ ˜ s ) = 0 if (t, m) = (s, 0). (1) Hom(Σs , ∇ (k)

(k)

˜ s ) = 0 if (t, m) ∇ (s, k). (2) Ext1 (Σt m , ∇ (k) ˜ s has a costandard ﬁltration whose subquotients are various ∇s m (3) ∇ (k) with 0 ≤ m ≤ k. (0)

˜ s = Δs . For Proof. We proceed by induction on k. When k = 0, we set Δ this object, parts (1) and (2) are contained in Deﬁnition 2.3, and part (3) is trivial. ˜ (k−1) Suppose now that k > 0, and that we have already deﬁned Δ with the desired s properties. Let Ξ = {(t, m) | (t, m) Δ (s, −k)}, and let Ψ = Ξ (s, −k). ˜ (k) We will deﬁne Δ by invoking Proposition 2.11, but we must ﬁrst check that s ˜ (k−1) the hypotheses of that proposition are satisﬁed. Parts (1) and (2) say that Δ s Ψ is a projective cover of Σs in A. By Lemma 3.3, we have that Δs −k is a projective cover of Σs −k in Ξ A. Moreover, because Rs −k ∈ (<s) D, we know from Deﬁnition 2.6(2) that ˜ (k−1) , Rs −k ) = 0. Hom2 (Δs −k , Rs −k ) = Hom2 (Δ s

˜ (k−1) is s-quasistandard.) (The latter vanishing holds because Δ s (k) ˜ s be the projective cover of Σs in Ξ A obtained from Proposition 2.11. Let Δ ˜ (k) It is clear then that parts (1) and (2) of the proposition hold for Δ s . Moreover, we have an exact sequence ˜ (k−1) , Δs −k )∗ ⊗ Δs −k → Δ ˜ (k) → Δ ˜ (k−1) → 0 0 → Ext1 (Δ s

s

s

10

PRAMOD N. ACHAR

from which we can see that part (3) holds as well.

3.2. Eﬀaceability. For the remainder of Section 3, we assume that the quasiexceptional set {∇s } has the following additional property. Definition 3.5. An abelianesque quasi-exceptional set {∇s }s∈S is said to have the eﬀaceability property if the following two conditions hold: (1) For any morphism f : X[−d] → Δs where d > 0 and X is s-quasistandard, there is an object Y ∈ (≤s) A and an injective map g : Δs → Y such that g ◦ f = 0. (2) For any morphism f : ∇s → X[d] where d > 0 and X is s-quasicostandard, there is an object Y ∈ (≤s) A and a surjective map h : Y ∇s such that f ◦ h = 0. Lemma 3.6. For any morphism f : X[−d] → Δs where d > 0 and X is s-quasistandard, there is an s-quasistandard object Y and an injective map g : Δs → Y such that g ◦ f = 0. Moreover, every standard subquotient of Y /g(Δs ) is isomorphic to some Δs m with m > 0. Proof. Let g : Δs → Y be an embedding as in Deﬁnition 3.5. We must show how to replace this Y by a certain kind of s-quasistandard object. For now, we know only that Y ∈ (≤s) A. This means that Y /g(Δs ) has a ﬁltration with simple subquotients lying in (≤s) A. We may write: (3.1)

Y ∈ Δs ∗ Σt1 p1 ∗ Σt2 p2 ∗ · · · ∗ Σtk pk ,

with ti ≤ s for all i. From the distinguished triangle Δs → Σs → Rs [1] →, we have Σs m ∈ Δs m ∗ Rs [1]m . For each factor Σti pi in (3.1) with ti = s, let us replace it by Δs pi ∗ Rs [1]pi . We will then have (3.2)

Y ∈ Δs ∗ I1 ∗ · · · ∗ Il

⎧ ⎪ ⎨Δs m where each Ii is one of: Rs [1]m ⎪ ⎩ Σt m

for some m ∈ Z, for some m ∈ Z, or for some t < s and some m ∈ Z.

Note that each factor Ii that is not of the form Δs m belongs to (<s) D. Now, for I ∈ (<s) D, Hom(Δs m , I[1]) = 0 by Deﬁnition 2.6(2), so any distinguished triangle I → J → Δs m → splits. In other words, I ∗ Δs m contains only the isomorphism class of the direct sum I ⊕ Δs m , and in particular, we have I ∗ Δs m ⊂ Δs m ∗ I

if I ∈ (<s) D.

Using this fact, we can rearrange the expression (3.2) so that all factors of the form Δs m occur to the left of all factors in (<s) D. In other words, we may assume without loss of generality that (3.1) reads as follows: Y ∈ Δs ∗ Δs m1 ∗ · · · ∗ Δs mk ∗ Ik+1 ∗ · · · ∗ Il This means that there is a distinguished triangle (3.3)

Y →Y →I →

with Ii ∈ (<s) D for i ≥ k + 1.

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

11

with (3.4)

Y ∈ Δs ∗ Δs m1 ∗ · · · ∗ Δs mk

and

I ∈ Ik+1 ∗ · · · ∗ Il ⊂ (<s) D.

Recall that Hom(Δs [k]m , I) = Hom(Δs [k]m , I[1]) = 0 for any k, m ∈ Z, by Deﬁnition 2.6(2). It follows that Hom(X[−d], I) = Hom(X[−d]), I[1]) = 0. From the long exact sequence obtained by applying Hom(Δs , −) and Hom(X[−d], −) to (3.3), we obtain natural isomorphisms ∼

Hom(Δs , Y ) → Hom(Δs , Y )

and

∼

Hom(X[−d], Y ) → Hom(X[−d], Y ).

The ﬁrst of these shows that g : Δs → Y factors in a unique way through Y . Let g : Δs → Y be the induced map. Now, the fact that g ◦f = 0 means that g ◦f is in the kernel of Hom(X[−d], Y ) → Hom(X[−d], Y ), so the second isomorphism above shows that g ◦ f = 0. However, g does not necessarily have the other properties claimed in the proposition. To repair this, we use the “rearrangement” method again. By Lemma 2.8(3), we have Hom(Δs m , Δs [1]n ) = 0 if n > m, so it follows that Δs n ∗ Δs m ⊂ Δs m ∗ Δs n

if n > m.

Using this to rearrange terms in (3.4), we may ﬁrst assume without loss of generality that m1 ≤ m2 ≤ · · · ≤ mk , and then we may write Y ∈ Δs m1 ∗ · · · ∗ Δs mj ∗ Δs ∗ Δs mj+1 ∗ · · · ∗ Δs mk where m1 ≤ · · · ≤ mj ≤ 0 < mj+1 ≤ · · · ≤ mk . In other words, we have a distinguished triangle J → Y → Y → with J ∈ Δs m1 ∗ · · · ∗ Δs mj and Y ∈ Δs ∗ Δs mj+1 ∗ · · · ∗ Δs mk . Let g : Δs → Y be the natural map. Clearly, g ◦ f = 0. It remains only to check that g is injective. Recall that a map in A is injective if and only if its cone in D actually lies in the abelian category A (and in that case, the cone is the cokernel). By construction, the cone of g lies in Δs mj+1 ∗ · · · ∗ Δs mk ⊂ A, as desired. Proposition 3.7. Let k ≥ 0 and d ≥ 1. If n < k + d, then d s ˜s ˜ (k) Homd (Δ −n ) = 0. s n , Δs ) = Hom (∇ , ∇ (k)

Proof. We proceed by induction on d. Suppose ﬁrst that d = 1. Note that every composition factor of Δs −n is some Σt m with (t, m) Δ (s, −n). The assumption that n < k +1 means that (s, −n) Δ (s, −k), so in view of Lemma 2.1, the result follows from Propostion 3.4(2). ˜ (k) Now, suppose d > 1. Given f ∈ Homd (Δ s n , Δs ), choose an embedding g : Δs → Y as in Lemma 3.6, and let Z = Y /g(Δs ). Consider the exact sequence d ˜ (k) d ˜ (k) ˜ (k) Homd−1 (Δ s n , Z) → Hom (Δs n , Δs ) → Hom (Δs n , Y ).

For any m > 0, we have n − m < k + d − 1, so it follows from the inductive assumption that d−1 ˜ (k) ˜ (k) Homd−1 (Δ (Δs n − m , Δs ) = 0 if m > 0. s n , Δs m ) Hom Recall from Lemma 3.6 that the standard subquotients of Z are all various Δs m ˜ (k) with m > 0. It follows that Homd−1 (Δ s n , Z) = 0, so the map ˜ (k) n , Δs ) → Homd (Δ ˜ (k) n , Y ) Homd (Δ s s

12

PRAMOD N. ACHAR

is injective. The morphism f is in the kernel of this map (because g ◦ f = 0), so ˜ (k) f = 0. Thus, Homd (Δ s n , Δs ) = 0, as desired. Proposition 3.8. If t < s, or else if t = s and n − m < k + d, then d ˜ (k) ˜s Homd (Δ s n , Σt m ) = Hom (Σt n , ∇(k) m ) = 0.

˜ (k) Proof. If t < s, this follows from Deﬁnition 2.6(2) and the fact that Δ s n is s-quasistandard. If t = s, consider the exact sequence d ˜ (k) d+1 ˜ (k) ˜ (k) Homd (Δ (Δs n , Rs m ). s n , Δs m ) → Hom (Δs n , Σs m ) → Hom

˜ (k) The ﬁrst term vanishes by Proposition 3.7, and the last again because Δ s n is (k) d ˜ (<s) s-quasistandard and Rs m ∈ A. Therefore, Hom (Δs n , Σs m ) = 0. Corollary 3.9. If t = s, or else if t = s and n − m < k + d, then we have d t ˜ (k) ˜s Homd (Δ s n , ∇ m ) = Hom (Δs n , ∇(k) m ) = 0. 3.3. Subcategories associated to convex sets. The next step towards our theorem is to show that certain Serre subcategories of A with ﬁnitely many simple objects have enough projectives, and that these projectives have desirable Homvanishing properties in D. Definition 3.10. A subset Ξ ⊂ S × Z is said to be convex if the following two conditions hold: (1) For each (s, n) ∈ Ξ, we have Δs n ∈ Ξ A and ∇s n ∈ Ξ A. (2) For any s ∈ S, the set of integers {n ∈ Z | (s, n) ∈ Ξ} is either empty or an interval {a0 , a0 + 1, . . . , a0 + k}. Definition 3.11. Let Ξ ⊂ S × Z be a ﬁnite convex set. Let (s, n) ∈ Ξ, and let as = min{m ∈ Z | (s, m) ∈ Ξ}. Let ˜ (n−as ) . ΔΞ s = Δs The object ΔΞ s n is said to be Ξ-standard, or the Ξ-standard cover of Σs n . A Ξ-standard ﬁltration of an object X ∈ Ξ A is a ﬁltration each of whose subquotients is a Ξ-standard object. Lemma 3.12. Every ﬁnite subset of S × Z is contained in a ﬁnite convex set. Proof. Given a ﬁnite set Ξ ⊂ S × Z, let F0 (Ξ) ⊂ S be the set of s ∈ S such that one of the following conditions holds: • There is an n ∈ Z with (s, n) ∈ Ξ but either Δs n ∈ / Ξ A or ∇s n ∈ / Ξ A. • There are integers a < b < c such that (s, a), (s, c) ∈ Ξ but (s, b) ∈ / Ξ. Let F (Ξ) be the lower closure of F0 (Ξ), i.e., F (Ξ) = {s ∈ S | there is some s0 ∈ F0 (Ξ) such that s ≤ s0 }. It is obvious that F0 (Ξ) is ﬁnite, and then it follows from (2.1) that F (Ξ) is ﬁnite as well. Of course, Ξ is convex if and only if F (Ξ) is empty. We prove the lemma by induction on the size of F (Ξ). If F (Ξ) = ∅, let s ∈ F (Ξ) be a maximal element (with respect to ≤). Then s ∈ F0 (Ξ). Let us put a0 = min{n | (s, n) ∈ Ξ},

b0 = max{n | (s, n) ∈ Ξ}.

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

Finally, let

Ξ = Ξ ∪

13

Σt m occurs as a composition factor in . (t, m) some Δs r or ∇s r with a0 ≤ r ≤ b0

It is clear that s ∈ / F (Ξ ). Note that all the new pairs (t, m) ∈ Ξ Ξ have t ≤ s. It follows that every element s ∈ F0 (Ξ ) either belongs to F0 (Ξ) or is < s. Therefore, F (Ξ ) ⊂ F (Ξ) {s}, so by induction, Ξ , and therefore Ξ, is contained in a ﬁnite convex set. Lemma 3.13. If Ξ ⊂ S × Z is a ﬁnite convex set, and (s, n) ∈ Ξ, then ΔΞ s n belongs to Ξ A and has Σs n as its unique simple quotient. Moreover, for all d ≥ 1, we have (3.5)

Homd (ΔΞ s n , Σt m ) = 0 Hom

d

t (ΔΞ s n , ∇ m )

=0

if (t, m) ∈ Ξ and t ≤ s, for all (t, m) ∈ Ξ.

In particular, if (s, n) is a costandard-maximal element of Ξ, then ΔΞ s n is a projective cover of Σs n . Proof. Let as be as in Deﬁnition 3.11. Because Ξ is convex, the standard objects Δs as , Δs as + 1 , . . . , Δs n all belong to Ξ A, so it follows by Proposi˜ (n−as ) n does as well. We also already know that Σs n tion 3.4 that ΔΞ s n = Δs is the unique simple quotient of ΔΞ s n . Finally, note that if (s, m) ∈ Ξ, then as ≤ m, so n − m < (n − as ) + d. Therefore, (3.5) is an immediate consequence of Proposition 3.8 and Corollary 3.9. Proposition 3.14. Let Ξ ⊂ S × Z be a ﬁnite convex set. Every simple object Σs n ∈ Ξ A admits a projective cover P with a Ξ-standard ﬁltration. Moreover, we have (3.6)

Homd (P, X) = 0

for all d ≥ 1 and all X ∈ Ξ A.

Proof. We proceed by induction on the size of Ξ. If Ξ = ∅, there is nothing to prove. Otherwise, let (s, n) be a costandard-maximal element of Ξ, and let Ψ = Ξ {(s, n)}. Lemma 3.13 tells us that ΔΞ s n is a projective cover of Σs n satisfying (3.6). Next, let R denote the kernel of the map ΔΞ s n Σs n . R has a composition series consisting of Rs n and various Δs m with r ≤ m < n. We see thus that R is contained in Ψ A. Consider a pair (t, m) ∈ Ψ. We assume inductively that Σt m admits a projective cover P in Ψ A with a Ψ-standard ﬁltration and satisfying (3.6). In particular, we have Hom2 (P , R) = 0. We have already seen above that Hom2 (ΔΞ s n , R) = 0, so we may invoke Proposition 2.11 to obtain a projective cover P of Σt m in Ξ A. The key observation now is that every Ψ-standard object is also Ξ-standard. (This would not have been the case if we had instead deﬁned Ψ by deleting a standard-maximal element of Ξ.) Thus, we now see from the exact sequence (2.3) that P has a Ξ-standard ﬁltration. We must now establish (3.6). If (u, p) ∈ Ψ, we already know that Homd (P , Σu p ) = Homd (ΔΞ s n , Σu p ) = 0

14

PRAMOD N. ACHAR

for all d ≥ 1, so it follows that Homd (P, Σu p ) = 0 as well. It remains to show that Homd (P, Σs n ) = 0. Consider the exact sequence Homd−1 (P, Qs n ) → Homd (P, Σs n ) → Homd (P, ∇s n ). The ﬁrst term is already known to vanish because Qs n ∈ Ψ A, and the last term vanishes by Corollary 3.9 because P has Ξ-standard ﬁltration. Thus, we have Homd (P, Σs n ) = 0, as desired. 3.4. Main result. Given a set Ξ ⊂ S × Z, let Ξ D ⊂ D denote the full triangulated subcategory generated by Ξ A. In other words, Ξ D is full subcategory consisting of objects X all of whose cohomology objects H i (X) lie in Ξ A. Note that Ξ A is the heart of a bounded t-structure on Ξ D, so we have a realization functor real : Db (Ξ A) → Ξ D. Composition with the inclusion functor gives us a natural functor Db (Ξ A) → D. Theorem 3.15. For any convex set Ξ ⊂ S × Z, the natural functor Db (Ξ A) → D is fully faithful. In particular, the realization functor real : Db (A) → D is an equivalence of categories. Proof. We ﬁrst treat the case where Ξ is ﬁnite. For a ﬁxed object X ∈ Ξ A, {Extd (·, X)}d≥0 is a universal δ-functor, and Proposition 3.14 tells us that the δ-functor {Homd (·, X)}d≥0 is eﬀaceable. Since their 0th parts agree, there is a ∼ canonical isomorphism of functors Extd (·, X) → Homd (·, X). Therefore, the natural b Ξ functor D ( A) → D is full and faithful. Now, suppose Ξ is inﬁnite. For any ﬁnite convex subset Ψ ⊂ Ξ, consider the chain of maps g

h

ExtiΨ (X, Y ) → ExtiΞ (X, Y ) → Homi (X, Y ). We claim that g and h are both isomorphisms for all i. This is clearly the case for i = 0 and i = 1. Suppose, in fact, that it is known for i = 0, 1, . . . , d − 1. By Lemma 2.1, both g and h are injective for i = d. We know by the previous paragraph that the composition h ◦ g is an isomorphism for all i, so it follows that g and h are isomorphisms as well. We have shown that ExtdΞ (X, Y ) Homd (X, Y ) when X, Y ∈ Ψ A. But given any two objects X, Y ∈ Ξ A, we know from Lemma 3.12 that there exists a ﬁnite convex subset Ψ ⊂ Ξ such that X, Y ∈ Ψ A. It follows that ExtdΞ (X, Y ) Homd (X, Y ) for all X, Y ∈ Ξ A, so Db (Ξ A) → D is full and faithful. 4. Notation for semisimple groups 4.1. Representations and varieties. As noted in Section 2, we will work over a ﬁxed algebraically closed ﬁeld k. For an algebraic group H over k, let Rep(H) denote the category of rational representations of H, and Repf (H) ⊂ Rep(H) the subcategory of ﬁnite-dimensional representations. If H ⊂ K, we have the usual K induction and restriction functors indK H : Rep(H) → Rep(K) and resH : Rep(K) → K b Rep(H). We also use the derived functor RindH : D Rep(H) → Db Rep(K). (Of course, resK H is exact.)

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

15

For any variety X over k, we write k[X] for the ring of regular functions on X. If H is an algebraic group acting on X, we denote by CohH (X) the abelian category of H-equivariant coherent sheaves on X. For any F ∈ Db CohH (X), its derived global sections RΓ(F) may be regarded as an object of Db Rep(H). 4.2. Graded objects. Let Rep(H) be the category of graded rational Hrepresentations. This is, of course, equivalent to Rep(H × Gm ). Repf (H) is deﬁned similarly. For homogeneous components and Tate twists of objects of Rep(H), we retain the conventions introduced in Section 2.2 for Vectk . Consider km , the graded H-representation consisting of the trivial H-module concentrated in degree m. We can regard this as an (H × Gm )-equivariant sheaf on pt = Spec k. If X is a variety equipped with an action of H × Gm , we put OX m = a∗ km , where a : X → pt is the constant map. More generally, for any F ∈ D b CohH×Gm (X), we put L

Fm = F ⊗ OX m . We write U for any of the various functors that forget gradings. In particular, for coherent sheaves, we have U : Db CohH×Gm (X) → Db CohH (X). 4.3. Reductive groups. Throughout the rest of the paper, G will be a simply connected semisimple algebraic group over k, and the characteristic of k will be assumed to be good for G. Fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B. Let U ⊂ B be the unipotent radical, and let u be the Lie algebra of U . Recall that there is a T -equivariant isomorphism of varieties (4.1)

e : u → U.

Let Λ be the weight lattice of T . We will think of B as the “negative” Borel: we deﬁne Λ+ ⊂ Λ to be the set of dominant weights determined by declaring the weights of T on u to be the negative roots. Let W be the Weyl group, and let w0 ∈ W be the longest element. For any λ ∈ Λ, let dom(λ) denote the unique dominant weight in the W -orbit of λ. Let ≤ denote the usual partial order on Λ. That is, for μ, λ ∈ Λ, we say that μ ≤ λ if λ − μ is a nonnegative integer linear combination of positive roots. We also deﬁne a preorder on Λ as follows: μ λ if dom(μ) ≤ dom(λ). Obviously, ≤ and coincide on Λ+ . For any λ ∈ Λ, let kλ denote the 1-dimensional T -representation of weight λ. We may also regard this as a B-representation on which U acts trivially. For λ ∈ Λ+ , let L(λ), M (λ), and N (λ) denote the simple module, Weyl module, and dual Weyl module, respectively, of highest weight λ. These representations may sometimes be regarded as graded by placing them in degree 0. 4.4. Nilpotent cone and Springer resolution. Let N be the variety of nilpotent elements in the Lie algebra of G. We will also work with the ﬂag variety ˜ = G ×B u. All these varieties are acted B = G/B and the Springer resolution N on by G. Let Gm act on N by (z, x) → z 2 x, where z ∈ Gm and x ∈ N . This action commutes with the action of G. The same formula deﬁnes an action on u ˜ commuting with that of G. commuting with that of B, and so an action on N Finally, let Gm act trivially on B. The obvious projection maps, which we denote p π ˜ −→ B, N ←− N

16

PRAMOD N. ACHAR

are both (G × Gm )-equivariant. Our convention on the Gm -action means that the graded rings k[u] and k[N ] are concentrated in even, nonpositive degrees. We do not endow U with a Gm -action, so the map (4.1) only gives rise to an isomorphism ∼

k[U ] → U(k[u])

(4.2)

of ungraded T -representations. Any graded B-representation V gives rise to a locally free (G×Gm )-equivariant sheaf on B, denoted S (V ). Given a weight λ ∈ Λ, consider the object A(λ) = Rπ∗ p∗ S (kλ ). This is called the Andersen–Jantzen sheaf of weight λ. (It is known [KLT, Theorem 2] that A(λ) is actually a coherent sheaf, rather than a complex of sheaves, for λ dominant, but we will not use this fact.) For any λ ∈ Λ, let Dλ , Dλ ⊂ Db CohG×Gm (N ) be the full triangulated subcategories generated by all Tate twists of Andersen– Jantzen sheaves A(μ)n with μ λ or μ λ, respectively. Consider now the object in Db Rep(G) given by RΓ(A(λ)) ∼ = RΓ(p∗ p∗ S (kλ )). ∗ ∼ By the projection formula, we have p∗ p S (kλ ) = S (k[u] ⊗ kλ ), so RΓ(A(λ)) ∼ = RindG B (k[u] ⊗ kλ ).

(4.3)

The following lemma on representations of a Borel subgroup is certainly wellknown, but we include a proof for completeness. Lemma 4.1. Let μ ∈ Λ, and let U and V be rational B-representations such that all weights of U are ≤ μ but no weights of V are ≤ μ. Then RHom(U, V ) = 0. Proof. Let λ ∈ Λ. Note that indB T kλ is an injective B-module, since induction ∼ takes injective modules to injective modules. Since B ∼ = T U , we have indB T kλ = B B ∼ ind k U(k[u]) ⊗ k . The weights of k[u] are sums res k[U ] ⊗ kλ . By (4.2), resB λ T λ = T T of positive roots, so we see that all weights of indB T kλ are ≥ λ. The B-module V can be embedded in an injective module I 0 by taking a direct sum of copies of indB T kλ as λ varies over weights of V . It follows from the preceding paragraph that every weight of I is ≥ some weight of V . More generally, we can extend this an injective resolution (I n )n≥0 of V in which every weight of every term is ≥ some weight of V . Since no weight of V is ≤ μ, it follows that no weight of I n is ≤ μ either, so Hom(U, I n ) = 0 for all n. Thus, RHom(U, V ) = 0. 4.5. Perverse coherent sheaves. Recall that G acts on N with ﬁnitely many orbits, and that each orbit has even dimension. For an orbit C, let ηC be its generic point, and let iC : {ηC } → C be the inclusion map. An object F ∈ Db CohG (N ) is said to be a perverse coherent sheaf if the following two conditions hold: H i (i∗C F) = 0

for all i >

(i!C F)

for all i <

H

i

=0

1 2 1 2

codim C, codim C.

The second condition is equivalent to requiring that H i (i∗C DF) = 0

for all i >

1 2

codim C,

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

17

where D is the Serre–Grothendieck duality functor given by D = RHom(−, ON ). In fact, the functor D can be deﬁned using any equivariant dualizing complex [B1]. The fact that ON is a dualizing complex is equivalent to the fact that it is Gorenstein, cf. [BK, Theorem 5.3.2]. There is a choice of shifts and Tate twists here; our normalization agrees with the convention of [B3] but not with that of [B2]. The category PCohG (N ) of perverse coherent sheaves has a number of features in common with the more familiar perverse constructible sheaves. Key among these are that every object has ﬁnite length, and that the simple objects admit a characterization resembling that of intersection cohomology complexes. Simple objects are classiﬁed by pairs (C, V), where V is an irreducible G-equivariant vector bundle on C. The corresponding simple object will be denoted IC(C, V). The category PCohG×Gm (N ) is deﬁned in the same way as above. (Recall that the orbits of G × Gm coincide with those of G.) The forgetful functor U : Db CohG×Gm (N ) → Db CohG (N ) restricts to an exact functor U : PCohG×Gm (N ) → PCohG (N ) that takes simple objects of PCohG×Gm (N ) to simple objects of PCohG (N ). 5. Andersen–Jantzen sheaves and perverse coherent sheaves In this section, we prove a number of lemmas on Andersen–Jantzen sheaves. We work with (G × Gm )-equivariant sheaves throughout. For the most part, the Gm -action will play no essential role; nearly every statement in this section has an obvious G-equivariant analogue, with the same proof. The only exception to this is part (3) of Proposition 5.6, whose statement and proof involve imposing conditions on Tate twists. Many proofs in this section are closely modeled on those in [B2, Section 3], suitably modiﬁed to handle the diﬃculties that arise in positive characteristic. Lemma 5.1. For all λ ∈ Λ, we have DA(λ) ∼ = A(−λ). Proof. Recall that proper pushforward Rπ∗ commutes with Serre–Grothen˜ is given by D ˜ = RHom(−, π ! ON ). dieck duality, where the duality functor on N N It is a consequence of [BK, Lemma 3.4.2 and Lemma 5.1.1] that π ! ON ∼ = ON˜ , so ∗ ∗ ∗ ∼ ∼ D ˜ (p S (kλ )) = RHom(p S (kλ ), O ˜ ) = p S (k−λ ), N

N

and the lemma follows. Lemma 5.2. For all λ ∈ Λ, we have A(λ) ∈ PCoh all μ, λ ∈ Λ, we have Homi (A(μ), A(λ)) = 0 if i < 0.

G×Gm

(N ). In particular, for

Proof. Recall that the Springer resolution is semismall [Ja, Theorem 10.11]. This means that for any closed point x in an orbit C ⊂ N , we have dim π −1 (x) ≤ 1 2 codim C. Let UC be the union of the nilpotent orbits whose closure contains C. Thus, UC is an open G-stable subset of N , and C is the unique closed orbit therein. ˜C = π −1 (UC ). Every ﬁber of the proper map π : N ˜C → UC has dimension ≤ Let N 1 i ∗ ˜U = 0 2 codim C, so by [H, Corollary III.11.2], it follows that R π∗ (p S (kλ ))|N 1 ∗ for i > 2 codim C. Since iC is an exact functor (where iC : {ηC } → N is as in Section 4.5), it follows that H i (i∗C A(λ)) = 0 for i > 12 codim C. The same reasoning applies to A(−λ) ∼ = DA(λ), so A(λ) ∈ PCohG×Gm (N ).

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PRAMOD N. ACHAR

The last assertion of the lemma is just the general fact that Homi (X, Y ) always vanishes for i < 0 if X and Y are in the heart of some t-structure. Lemma 5.3. Let λ, μ ∈ Λ be two weights in the same W -orbit. If μ ≤ λ, then A(μ) ∈ Dλ ∗ A(λ)−2 , where is the length of the shortest element w ∈ W such that wλ = μ. Proof. The statement is trivial if μ = λ, so assume that μ < λ. It is easily seen by induction on that it suﬃces to prove this in the case where μ = sλ for some simple reﬂection s, say corresponding to the simple root α. Let n = α∨ , λ . Since sλ < λ, we have n > 0. Let Pα ⊂ G be the minimal parabolic subgroupcorresponding to α, and let pα : G/B → G/Pα be the projection map. Let ρ = 12 α, where the sum runs over all positive roots. Recall that G is assumed to be simply connected, so ρ lies in the Pα ∨ α weight lattice for G. Let Q = kρ−α ⊗ resP B indB kλ−ρ . Since α , λ − ρ = n − 1, Pα the weights of indB kλ−ρ are λ − ρ, λ − ρ − α, . . . , λ − ρ − (n − 1)α. Thus, the weights of Q are λ − α, λ − 2α, . . . , λ − nα = sλ. A standard fact relating induction, restriction, and tensor products tells us that (5.1)

Pα Pα α ∼ RindP B Q = RindB kρ−α ⊗ RindB kλ−ρ = 0, L

where the last equality follows from the fact that α∨ , ρ − α = −1. From the weights of Q, we see that there is a short exact sequence of B-modules 0 → ksλ → Q → K1 → 0, where the weights of K1 are λ. Applying Rπ∗ ◦ p∗ ◦ S , we see that (5.2)

A(sλ) ∈ Dλ ∗ Rπ∗ p∗ S (Q).

Similarly, there is a short exact sequence 0 → K2 → Q ⊗ kα → kλ → 0 where K2 has weights that are λ. We deduce that (5.3)

Rπ∗ p∗ S (Q ⊗ kα ) ∈ Dλ ∗ A(λ).

In view of (5.2) and (5.3), we see that the lemma will follow once we prove that (5.4) Rπ∗ p∗ S (Q ⊗ kα )−2 ∼ = Rπ∗ p∗ S (Q). Let uα be the Lie algebra of the unipotent radical of Pα , and consider its coordinate ring k[uα ]. It is the quotient of the graded ring k[u] by the ideal generated by α ∈ u∗ = (k[u])−2 . In other words, we have a short exact sequence 0 → k[u] ⊗ kα −2 → k[u] → k[uα ] → 0 of (B × Gm )-equivariant k[u]-modules, or equivalently of objects in CohB×Gm (u). A construction analogous to that of S then gives us a short exact sequence 0 → p∗ S (kα )−2 → ON˜ → i∗ ON˜α → 0 ˜ ). Here N ˜α = G ×B uα , and i : N ˜α → N ˜ is the inclusion map. in CohG×Gm (N Tensoring with p∗ S (Q), we see that (5.4) would follow if we knew that (5.5)

Rπ∗ (i∗ ON˜α ⊗ p∗ S (Q)) = 0.

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

19

Since N is an aﬃne variety, RΓ kills no nonzero object of Db CohG×Gm (N ), so it suﬃces to check that the object RΓ(Rπ∗ (i∗ ON˜α ⊗ p∗ S (Q))) ∼ = RΓ(Rp∗ (i∗ ON˜α ⊗ p∗ S (Q))) vanishes. By the projection formula. we have Rp∗ (i∗ ON˜α ⊗ p∗ S (Q)) ∼ = S (k[uα ] ⊗ Q), so to prove (5.4), we must check that RΓ(S (k[uα ] ⊗ Q)) = 0, or RindG B (k[uα ] ⊗ Q) = 0.

(5.6)

Pα L α ∼ But RindP B (k[uα ] ⊗ Q) = k[uα ] ⊗ RindB Q, so (5.6) follows from (5.1).

Lemma 5.4. For any μ ∈ Λ , we have Rπ∗ p S (M (μ)) ∼ = ON ⊗ M (μ). Moreover, there are weights ν1 , . . . , νk such that ∗

+

ON ⊗ M (μ) ∈ A(ν1 ) ∗ · · · ∗ A(νk ) ∗ A(μ)

(5.7)

where either νi μ or νi ∈ W μ but νi = μ for each i. As a consequence, ON ⊗ M (μ) ∈ PCohG×Gm (N ), and there is a surjective map M (μ) → A(μ). Proof. Since π ∗ (ON ⊗ M (μ)) ∼ = p∗ S (M (μ)), the projection formula implies ∗ L ∼ that Rπ∗ p S (M (μ)) = Rπ∗ ON˜ ⊗ (ON ⊗ M (μ)). But Rπ∗ ON˜ ∼ = ON by [BK, Theorem 5.3.2], so Rπ∗ p∗ S (M (μ)) ∼ = ON ⊗ M (μ). Next, there is a surjective map of B-representations resG B M (μ) → kμ . The kernel of this map has a ﬁltration whose subquotients are various kν1 , . . . , kνk , where either νi μ or νi ∈ W μ and ν < μ. Applying the functor Rπ∗ p∗ S , we see that (5.7) holds. It now follows from Lemma 5.2 that Rπ∗ p∗ S (M (μ)) ∈ PCohG×Gm (N ). In particular, there is a distinguished triangle K → ON ⊗ M (μ) → A(μ) → with K ∈ A(ν1 ) ∗ · · · ∗ A(νk ). Since all three terms belong to PCohG×Gm (N ), this is actually a short exact sequence in that category, and the map ON ⊗ M (μ) → A(μ) is surjective. Lemma 5.5. Let λ, μ ∈ Λ+ . (1) If λ ≤ μ, then RHom(ON ⊗ M (μ), A(λ)) = 0. (2) If λ = μ, then RHom(ON ⊗ M (μ), A(λ)) ∼ = k. Proof. We have RHom(ON ⊗ M (μ), A(λ)) ∼ = RHom(M (μ), RΓ(A(λ))) by adjunction. We will work with the latter object. Using (4.3), we have ∼ RHom (resG M (μ), k[u] ⊗ kλ ). RHom (M (μ), RΓ(A(λ))) = G

B

B

Of course, all weights of M (μ) are ≤ μ, and all weights of k[u]⊗kλ are ≥ λ. Part (1) then follows from Lemma 4.1. For part (2), let J ⊂ k[u] be the ideal spanned by all homogeneous elements of strictly negative degree. Thus, k[u] ⊗ kμ ∼ = kμ ⊕ (J ⊗ kμ ). Since all weights of J ⊗ kμ are > μ, Lemma 4.1 again tells us that RHom(resG B M (μ), J ⊗ kμ ) = 0. We conclude that RHomG (M (μ), RΓ(A(μ)n )) ∼ = RHomB (resG B M (μ), kμ n ) G ∼ = RHomG (M (μ), Rind kμ n ) ∼ = RHomG (M (μ), N (μ)n ), B

and this clearly vanishes for n = 0 and is 1-dimensional when n = 0. Proposition 5.6. Let λ ∈ Λ . We have: (1) If μ ∈ Λ and μ λ, then RHom(A(μ), A(λ)) = 0. +

20

PRAMOD N. ACHAR

(2) If i < 0, then Homi (A(λ), A(λ)) = 0, and Hom(A(λ), A(λ)) ∼ = k. (3) If i > 0 and n ≥ 0, then Homi (A(λ), A(λ)n ) = 0. (4) If μ ∈ Λ+ and λ = μ, then RHom(A(w0 μ), A(λ)) = 0. Proof. (1) Fix λ. In the proof, we will assume that μ is dominant and that λ ≤ μ, and we will show that RHom(A(wμ), A(λ)) = 0 for all w ∈ W . We proceed by induction with respect to . Assume that for all ν μ, we already know that RHom(A(ν), A(λ)) = 0. We know that RHom(ON ⊗ M (μ), A(λ)) = 0 by Lemma 5.5(1). On the other hand, applying RHom(−, A(λ)) to (5.7), we have RHom(ON ⊗ M (μ), A(λ)) ∈ RHom(A(μ), A(λ)) ∗ RHom(A(νk ), A(λ)) ∗ · · · ∗ RHom(A(ν1 ), A(λ)). All terms on the right-hand side with νi μ vanish by assumption and can be omitted. The remaining terms are those with νi ∈ W μ. By Lemma 5.3 and the inductive assumption, there is some integer n > 0 such that ∼ RHom(A(μ), A(λ))n RHom(A(νi ), A(λ)) = if νi ∈ W μ. Therefore, the expression above simpliﬁes to (5.8)

RHom(ON ⊗ M (μ), A(λ)) ∈

RHom(A(μ), A(λ)) ∗ RHom(A(μ), A(λ))n1 ∗ · · · ∗ RHom(A(μ), A(λ))nm . By Lemma 2.2(1), we conclude that RHom(A(μ), A(λ)) = 0. (2) The ﬁrst assertion of this part is contained in Lemma 5.2. The second assertion will be proved together with part (3) in the next paragraph. (3) This proof is similar to that of part (1). We know from Lemma 5.5(2) that RHom(ON ⊗ M (λ), A(λ)) ∼ = k. We may again carry out the calculations leading to (5.8), this time with μ = λ. Since n1 , . . . , nm > 0, Lemma 2.2(2) tells us that Hom(A(λ), A(λ)) ∼ = k, and that for i > 0, Homi (A(λ), A(λ)) is concentrated in strictly positive degrees. In other words, for n ≥ 0, Homi (A(λ), A(λ)n ) = 0. (4) If μ ≥ λ, then this is an instance of part (1). On the other hand, if μ > λ, then we apply Serre–Grothendieck duality and Lemma 5.1: RHom(A(w0 μ), A(λ)) ∼ = RHom(DA(λ), DA(w0 μ)) ∼ = RHom(A(−λ), A(−w0μ)). Now, −w0 μ and −w0 λ are both dominant, and −w0 μ > −w0 λ. In particular, we have −λ − w0 μ, so RHom(A(−λ), A(−w0μ)) = 0 by part (1) again. Lemma 5.7. Let C be the category of ﬁnitely-generated graded B-equivariant modules over the graded ring k[u]. Then Db C is generated as a triangulated category by objects of form k[u] ⊗ V n , where V is a ﬁnite-dimensional B-representation. Proof. In this proof, we will say that an object M ∈ C is free if it is a direct sum of objects of the form k[u] ⊗ V n . Let R be the functor which forgets the Baction (but retains the grading). Clearly, R takes free objects of C to free (graded) k[u]-modules. However, a module M ∈ C may have the property that R(M ) is a free module while M itself is not. Let us call a module M weakly free if R(M ) is free. It is easy to see that C has “enough” free objects, i.e., that every module is a quotient of a free module. Therefore, every module M has (possibly inﬁnite) resolution by free modules · · · → F1 → F0 → M → 0. Hilbert’s syzygy theorem, in the form found in, say, [CLO, Corollary 3.19], asserts that there is some n such

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

21

that the kernel of the map Fn → Fn−1 is free as a graded k[u]-module, i.e. weakly free. Thus, every module admits a ﬁnite resolution whose terms are either free or weakly free. It follows that Db C is generated by the weakly free modules. The lemma then follows from the following claim: Every weakly free module admits a ﬁnite ﬁltration whose subquotients are free modules. Let M be a weakly free module, and let m1 , . . . , mn be a set of homogeneous elements that constitute a basis for it as a free k[u]-module. Let N = max{deg mi }, and assume without loss of generality that m1 , . . . , mk have degree N and that mk+1 , . . . , mn have degree < N . Then m1 , . . . , mk must constitute a k-basis for the vector space MN . The k[u]-submodule M generated by m1 , . . . , mk is a free k[u]-module and a direct summand of R(M ). It is also stable under B and isomorphic to k[u] ⊗ MN as an object of C. In other words, M is a subobject of M in C; it is free, and the quotient M/M is weakly free. The claim then follows by induction on the rank of R(M ). ˜ ), we obtain the following Via the equivalences C ∼ = CohB×Gm (u) ∼ = CohG×Gm (N result. ˜ ) is generated as a triangulated category by Corollary 5.8. Db CohG×Gm (N ∗ the objects of the form p S (V )n , where V ranges over all ﬁnite-dimensional Brepresentations. Lemma 5.9. Db CohG×Gm (N ) is generated as a triangulated category by objects ˜ ). of the form Rπ∗ F, where F ∈ Db CohG×Gm (N Proof. Let D ⊂ Db CohG×Gm (N ) be the subcategory generated by objects ˜ ). Because PCohG×Gm (N ) is a ﬁnite-length category Rπ∗ F for F ∈ Db CohG×Gm (N that is the heart of a bounded t-structure, we have that the simple perverse coherent sheaves generate Db CohG×Gm (N ) as a triangulated category, so it suﬃces to show that the simple perverse coherent sheaves lie in D. Consider a simple perverse coherent sheaf IC(C, V), where C ⊂ N is a nilpotent orbit, and V is an irreducible G-equivariant vector bundle on C. Let Z = C C. We proceed by induction on C with respect to the closure partial order on nilpotent orbits. That is, we assume that IC(C , V ) ∈ D for all C ⊂ Z. The latter objects b CohG×Gm (N ) ⊂ Db CohG×Gm (N ) generate the full triangulated subcategory DZ consisting of objects whose support is contained in Z. Thus, our assumption implies b that DZ CohG×Gm (N ) ⊂ D. By[Ja, Proposition 5.9 and 8.8(II)], there is a parabolic subgroup P ⊃ B and a P -stable subspace v ⊂ u ∩ C such that the natural map q : G ×P v → C is a resolution of singularities of C. Consider the variety X = G ×B v. We have an ˜ , as well as an obvious smooth map h : X → G ×P v whose inclusion ˜ı : X → N ˜ be the inclusion map. ﬁbers are isomorphic to P/B. Let i : C → N G×Gm b (C) be an object such that i∗ G ∼ Let G ∈ D Coh = IC(C, V). (Because coherent pullback is not exact, some care must be taken to distinguish between ˜ Since RΓ(P/B, OP/B ) ∼ these two objects.) Let G˜ = (q ◦ h)∗ G, and let F = ˜ı∗ G. = k, it follows from the projection formula that the canonical adjunction morphism ∼ q ∗ G → Rh∗ h∗ (q ∗ G) is an isomorphism. Applying Rq∗ , we obtain an isomorphism ˜ Then, composing with G → Rq∗ q ∗ G, we get a morphism Rq∗ q ∗ G → R(q ◦ h)∗ G. ˜ G → R(q ◦ h)∗ G.

22

PRAMOD N. ACHAR

This map is at least an isomorphism over C, since q is an isomorphism over C. Thus, its cone K has support contained in Z. Applying i∗ , we have a distinguished triangle IC(C, V) → Rπ∗ F → i∗ K → . b Since Rπ∗ F ∈ D and i∗ K ∈ DZ CohG×Gm (N ) ⊂ D, we conclude that IC(C, V) ∈ D, as desired.

6. Proofs of the main results The results of Section 5 ﬁt the framework of Sections 2–3 and allow us to quickly deduce the main results. For λ ∈ Λ+ , let δλ denote the length of the shortest element w ∈ W such that wλ = w0 λ. We then put (6.1)

∇λ = A(λ)−δλ , Δλ = A(w0 λ)δλ

Proposition 6.1. The objects ∇λ constitute an abelianesque dualizable graded quasi-exceptional set in Db CohG×Gm (N ), and the Δλ form the dual set. Likewise, the objects U(Δλ ) constitute an abelianesque dualizable ungraded quasi-exceptional set in Db CohG (N ), and the U(Δλ ) form the dual set. Proof. Referring to Deﬁnition 2.4, we see that conditions (1)–(2) are proved in Proposition 5.6. To see that condition (4) holds, note that every graded ﬁnitedimensional B-representation arises by extensions among 1-dimensional representa˜ ), tions kλ n . By Corollary 5.8, the objects p∗ S (kλ n ) generate Db CohG×Gm (N and then by Lemma 5.9, the objects A(λ)n , where λ ∈ Λ and n ∈ Z, generate Db CohG×Gm (N ). The fact that it suﬃces to take the A(λ)n with λ dominant follows from Lemma 5.3 with an induction argument with respect to . Thus, the {A(λ)δλ } with λ ∈ Λ+ form a graded quasi-exceptional set. In fact, the aforementioned induction argument also shows that each Dλ is generated by the A(μ)n with μ ∈ Λ+ , μ < λ. So this category coincides with the one that would have been denoted (<λ) D in Section 2. By Lemma 5.3, there is a morphism A(w0 λ) → A(λ)−2δλ whose cone lies in Dλ . Combining this observation with Proposition 5.6(4), we see that the {A(w0 λ)δλ } forms a dual set. The fact that it is abelianesque is contained in Lemma 5.2. For the ungraded version, we omit part (3) of Deﬁnition 2.4. Since Proposition 5.6(3) was the only result of Section 5 without an ungraded analogue (see the remarks at the beginning of Section 5), the ungraded version of the present proposition also holds. Theorem 6.2. The categories PCohG×Gm (N ) and PCohG (N ) are quasi-hereditary, with standard and costandard objects as in (6.1). Proof. By Theorem 2.10, the objects Δλ and ∇λ determine a t-structure on each of Db CohG×Gm (N ) and Db CohG (N ) whose heart A is quasi-hereditary and in which those objects are standard and costandard, respectively. But it is easily seen from Lemma 5.2 and the deﬁnition given in Theorem 2.10 that every perverse coherent sheaf lies in A. The heart of one bounded t-structure cannot be properly contained in the heart of another bounded t-structure, so it must be that A coincides ˜ ) or PCohG (N ). with PCohG×Gm (N

PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC

23

∼

Theorem 6.3. The functor real : Db PCohG×Gm (N ) → Db CohG×Gm (N ) is an equivalence of categories. Proof. By Lemma 5.5(2), we have Homd (ON ⊗ M (λ), A(λ)) = 0 for all d > 0. It follows that if X ∈ PCohG×Gm (N ) is a λ-quasicostandard object, then Homd (ON ⊗ M (λ), X) = 0. By Lemma 5.4, we have a surjective map ON ⊗ M (λ) → A(λ). Thus, part (2) of Deﬁnition 3.5 holds. By Lemma 5.1, the Serre–Grothendieck duality functor exchanges standard and costandard objects, so part (1) of Deﬁnition 3.5 follows from part (2). By Theorem 3.15, the desired equivalence holds. References [A] [AR] [Be]

[BBD]

[BGS] [B1] [B2] [B3] [B4] [BK] [CLO] [F] [H] [Ja] [Ju] [JMW]

[KLT] [S]

P. Achar, On the equivariant K-theory of the nilpotent cone for the general linear group, Represent. Theory 8 (2004), 180–211. P. Achar and S. Riche, Koszul duality and semisimplicity of Frobenius, arXiv:1102.2820, submitted. A. Be˘ılinson, On the derived category of perverse sheaves, K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Mathematics, vol. 1289, Springer-Verlag, Berlin, 1987, pp. 27–41. A. Be˘ılinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Ast´ erisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171. A. Be˘ılinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527. R. Bezrukavnikov, Perverse coherent sheaves (after Deligne), arXiv:math.AG/0005152. R. Bezrukavnikov, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, Represent. Theory 7 (2003), 1–18. R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves, Invent. Math. 166 (2006), 327–357. R. Bezrukavnikov, Perverse sheaves on aﬃne ﬂags and nilpotent cone of the Langlands dual group, Israel J. Math. 170 (2009), 185–206. M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory, Progr. Math., vol. 231, Birkh¨ auser Boston, Boston, MA, 2005. D. A. Cox, J. Little, and D. O’Shea, Using algebraic geometry, 2nd ed., Graduate Texts in Mathematics, vol. 185, Springer, New York, 2005. P. Fiebig, Sheaves on aﬃne Schubert varieties, modular representations, and Lusztig’s conjecture, J. Amer. Math. Soc. 24 (2011), 133–181. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, no. 52, SpringerVerlag, New York, 1977. J. C. Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkh¨ auser Boston, Boston, MA, 2004, pp. 1–211. D. Juteau, Modular Springer correspondence and decomposition matrices, Ph.D. thesis, Universit´ e Paris 7, 2007. D. Juteau, C. Mautner, and G. Williamson, Perverse sheaves and modular representation theory, Geometric methods in representation theory II, S´emin. Congr., vol. 25, Soc. Math. France, 2010, pp. 313–350. S. Kumar, N. Lauritzen, and J. F. Thomsen, Frobenius splitting of cotangent bundles of ﬂag varieties, Invent. Math. 136 (1999), 603–621. W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (2000), 311–335.

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. E-mail address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 86, 0, XXXX 2012

On the Vanishing Ranges for the Cohomology of Finite Groups of Lie type II Christopher P. Bendel, Daniel K. Nakano, and Cornelius Pillen Abstract. The computation of the cohomology for ﬁnite groups of Lie type in the describing characteristic is a challenging and diﬃcult problem. In [BNP], the authors constructed an induction functor which takes modules over the ﬁnite group of Lie type, G(Fq ), to modules for the ambient algebraic group G. In particular this functor when applied to the trivial module yields a module with a natural G-ﬁltration. This ﬁltration was utilized in [BNP] to determine the ﬁrst non-trivial cohomology class when the underlying root system is of type An or Cn . In this paper the authors extend these results toward locating the ﬁrst non-trivial cohomology classes for the remaining ﬁnite groups of Lie type (i.e., the underlying root system is of type Bn , Cn , Dn , E6 , E7 , E8 , F4 , and G2 ) when the prime is larger than the Coxeter number.

1. Introduction 1.1. Let G be a simple algebraic group scheme over a ﬁeld k of prime characteristic p which is deﬁned and split over the prime ﬁeld Fp , and F : G → G denote the Frobenius map. The ﬁxed points of the rth iterate of the Frobenius map, denoted G(Fq ), is a ﬁnite Chevalley group where Fq denotes the ﬁnite ﬁeld with pr elements. An elusive problem of major interest has been to determine the cohomology ring H• (G(Fq ), k). Until recently, aside from small rank cases, it was not even known in which degree the ﬁrst non-trivial cohomology class occurs. This present paper is a sequel to [BNP] where we began investigating three related problems of increasing levels of diﬃculty: (1.1.1) Determining Vanishing Ranges: Finding D > 0 such that the cohomology group Hi (G(Fq ), k) = 0 for 0 < i < D. (1.1.2) Locating the First Non-Trivial Cohomology Class: Finding a D satisfying (1.1.1) such that HD (G(Fq ), k) = 0. A D satisfying this property will be called a sharp bound. (1.1.3) Determining the Least Non-Trivial Cohomology: For a sharp D as in (1.1.2) compute HD (G(Fq ), k). 2010 Mathematics Subject Classiﬁcation. Primary 20J06; Secondary 20G10. Research of the ﬁrst author was supported in part by NSF grant DMS-0400558. Research of the second author was supported in part by NSF grant DMS-1002135. c c 2012 American Mathematical Society XXXX

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Vanishing ranges (1.1.1) were found in earlier work of Quillen [Q], Friedlander [F] and Hiller [H]. Sharp bounds (1.1.2) were later found by Friedlander and Parshall for the Borel subgroup B(Fq ) of GLn (Fq ), and conjectured for the general linear group by Barbu [B]. A more detailed discussion of these results can be found in [BNP, Section 1.1]. In [BNP], for simple, simply connected G and primes p larger than the Coxeter number h, we proved that Hi (G(Fpr ), k) = 0 for 0 < i < r(p − 2). This provided an answer to (1.1.1) and improved on Hiller’s bounds [H]. For a group with underlying root system of type Cn , we demonstrated that D = r(p − 2) is in fact a sharp bound, answering (1.1.2). The ﬁrst non-vanishing cohomology, as in (1.1.3), was also determined. For type An , questions (1.1.2) and (1.1.3) were also answered, where the r > 1 cases required the prime to be at least twice the Coxeter number. Our methods also yielded a proof of Barbu’s Conjecture [B, Conjecture 4.11]. In this paper, we continue these investigations in two directions. First we consider the case when G is a group of adjoint type (as opposed to simply connected). For such G with p > h, when the root system is simply laced, one obtains a uniform sharp bound of r(2p − 3) answering (1.1.2) (cf. Corollary 3.3.1). The same uniform bound also holds for the adjoint versions of the twisted groups of types A, D, and E6 when p > h. We then consider the remaining types in the simply connected case. For G being simple, simply connected and having root system of type Dn with p > h, (1.1.2) and (1.1.3) are answered (cf. Theorem 4.5.2). For type En , (1.1.2) is answered for all primes p > h (with the exceptions of p = 17, 19 for type E6 ); cf. Theorems 5.1.3, 5.2.3, and 5.3.1. The calculations for the non-simply-laced groups are considerably more complicated. For type B we answer (1.1.2) when r = 1 and p > h, see Theorem 6.7.1. Some discussion of the situation for types G2 and F4 is given in Sections 7 and 8 respectively. For r = 1 and p > h, we ﬁnd improved answers to (1.1.1); cf. Theorem 7.5.1 and Theorem 8.1.1. Finding an answer to (1.1.2) and (1.1.3) continues to be elusive in these types although some further information towards answering these questions is obtained. 1.2. Notation. Throughout this paper, we will follow the notation and conventions given in the standard reference [Jan]. Let G be a simple, simply connected algebraic group scheme which is deﬁned and split over the ﬁnite ﬁeld Fp with p elements (except in Section 3.3 where G is assumed to be of adjoint type rather than simply connected). Throughout the paper let k be an algebraically closed ﬁeld of characteristic p. For r ≥ 1, let Gr := ker F r be the rth Frobenius kernel of G and G(Fq ) be the associated ﬁnite Chevalley group. Let T be a maximal split torus and Φ be the root system associated to (G, T ). The positive (resp. negative) roots are Φ+ (resp. Φ− ), and Δ is the set of simple roots. Let B be a Borel subgroup containing T corresponding to the negative roots and U be the unipotent radical of B. For a given root system of rank n, the simple roots will be denoted by α1 , α2 , . . . , αn (via the the Bourbaki ordering of simple roots). For type Bn , αn denotes the unique short simple root and for type Cn , αn denotes the unique long simple root. The highest (positive) root will be denoted α, ˜ and for root systems with multiple root lengths, the highest short root will be denoted α0 . Let W denote the Weyl group associated to Φ, and, for w ∈ W , let (w) denote the length of the element w (i.e., number of elements in a reduced expression for w).

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

27 3

Let E be the Euclidean space associated with Φ, and the inner product on E will be denoted by , . Let α∨ = 2α/α, α be the coroot corresponding to α ∈ Φ. The fundamental weights (basis dual to α1∨ , α2∨ , . . . , αn∨ ) will be denoted by ω1 , ω2 , . . . , ωn . Let X(T ) be the integral weight lattice spanned by these fundamental weights. The set of dominant integral weights is denoted by X(T )+ . For a weight λ ∈ X(T ), set λ∗ := −w0 λ where w0 is the longest word in the Weyl group W . By w · λ := w(λ + ρ) − ρ we mean the “dot” action of W on X(T ), with ρ being the half-sum of the positive roots. For α ∈ Δ, sα ∈ W denotes the reﬂection in the hyperplane determined by α. For a G-module M , let M (r) be the module obtained by composing the underlying representation for M with F r . Moreover, let M ∗ denote the dual module. 0 ∗ ∗ For λ ∈ X(T )+ , let H 0 (λ) := indG B λ be the induced module and V (λ) := H (λ ) be the Weyl module of highest weight λ. 2. General Strategy and Techniques 2.1. We will employ the basic strategy used in [BNP] in addressing (1.1.1)(1.1.3) which uses eﬀective techniques developed by the authors which relate Hi (G(Fq ), k) to extensions over G via an induction functor Gr (−). When Gr (−) is applied to the trivial module k, Gr (k) has a ﬁltration with factors of the form H 0 (λ) ⊗ H 0 (λ∗ )(r) [BNP, Proposition 2.4.1]. The G-cohomology of these factors can be analyzed by using the Lyndon-Hochschild-Serre (LHS) spectral sequence involving the Frobenius kernel Gr (cf. [BNP, Section 3]). In particular for r = 1, we can apply the results of Kumar-Lauritzen-Thomsen [KLT] to determine the dimension of a cohomology group Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ), which can in turn be used to determine Hi (G(Fpr ), k). The dimension formula involves the combinatorics of the well-studied Kostant Partition Function. This reduces the question of the vanishing of the ﬁnite group cohomology to a question involving the combinatorics of the underlying root system Φ. For root systems of types A and C the relevant root system combinatorics was analyzed in [BNP, Sections 5-6]. In the cases of the other root systems (B, D, E, F , G) the combinatorics is much more involved, and we handle these remaining cases in Sections 4-8. In this section, for the convenience of the reader, we restate the key results from [BNP] which will be used throughout this paper. 2.2. We ﬁrst record here a formula for −w · 0 that will be used at various times in the exposition [BNP, Observation 2.1]: Observation 2.2.1. If w ∈ W admits a reduced expression w = sβ1 sβ2 . . . sβm with βi ∈ Δ and m = (w), then −w · 0 = β1 + sβ1 (β2 ) + sβ1 sβ2 (β3 ) + · · · + sβ1 sβ2 . . . sβm−1 (βm ). Moreover, this is the unique way in which −w · 0 can be expressed as a sum of distinct positive roots. 2.3. The Induction Functor and Filtrations. Let Gr (k) := indG G(Fq ) (k). The functor Gr (−) is exact and one can use Frobenius reciprocity to relate extensions over G with extensions over G(Fq ) [BNP, Proposition 2.2]. Proposition 2.3.1. Let M, N be rational G-modules. Then, for all i ≥ 0, (M, N ) ∼ Exti = Exti (M, N ⊗ Gr (k)). G(Fq )

G

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

In order to make the desired computations of cohomology groups, we will make use of Proposition 2.3.1 (with M = k = N ). In addition, we will use a special ﬁltration on Gr (k) (cf. [BNP, Proposition 2.4.1]). Proposition 2.3.2. The induced module Gr (k) has a ﬁltration with factors of the form H 0 (λ) ⊗ H 0 (λ∗ )(r) with multiplicity one for each λ ∈ X(T )+ . 2.4. A Vanishing Criterion. The ﬁltration from Proposition 2.3.2 allows one to obtain a condition on G-cohomology which leads to vanishing of G(Fpr )cohomology (cf. [BNP, Corollary 2.6.1]). Proposition 2.4.1. Let m be the least positive integer such that there exists λ ∈ X(T )+ with Hm (G, H 0 (λ)⊗H 0 (λ∗ )(r) ) = 0. Then Hi (G(Fq ), k) ∼ = Hi (G, Gr (k)) = 0 for 0 < i < m. 2.5. Non-vanishing. While the identiﬁcation of an m satisfying Proposition 2.4.1 gives a vanishing range as in (1.1.1), it does not a priori follow that Hm (G(Fq ), k) = 0. The following theorem provides conditions which assist with addressing (1.1.2) or (1.1.3) [BNP, Theorem 2.8.1]. Theorem 2.5.1. Let m be the least positive integer such that there exists ν ∈ X(T )+ with Hm (G, H 0 (ν) ⊗ H 0 (ν ∗ )(r) ) = 0. Let λ ∈ X(T )+ be such that Hm (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0. Suppose Hm+1 (G, H 0 (ν) ⊗ H 0 (ν ∗ )(r) ) = 0 for all ν < λ that are linked to λ. Then (a) Hi (G(Fq ), k) = 0 for 0 < i < m; (b) Hm (G(Fq ), k) = 0; (c) if, in addition, Hm (G, H 0 (ν) ⊗ H 0 (ν ∗ )(r) ) = 0 for all ν ∈ X(T )+ with ν = λ, then Hm (G(Fq ), k) ∼ = Hm (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ). From the ﬁltration on Gr (k) in Proposition 2.3.2, Hi (G(Fq ), k) ∼ = Hi (G, Gr (k)) can be decomposed as a direct sum over linkage classes of dominant weights. For a ﬁxed linkage class L, let m be the least positive integer such that there exists ν ∈ L with Hm (G, H 0 (ν) ⊗ H 0 (ν ∗ )(r) ) = 0. Let λ ∈ L be such that Hm (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0. Suppose Hm+1 (G, H 0 (ν) ⊗ H 0 (ν ∗ )(r) ) = 0 for all ν < λ in L. Then analogous to Theorem 2.5.1, it follows that Hm (G(Fq ), k) = 0. See [BNP, Theorem 2.8.2]. 2.6. Reducing to G1 -cohomology. From Sections 2.4 and 2.5, the key to understanding the vanishing of Hi (G(Fpr ), k) is to understand Hi (G, H 0 (λ)⊗H 0 (λ∗ )(r) ) for all dominant weights λ. For r = 1, these groups can be related to G1 -cohomology groups (cf. [BNP, Lemma 3.1]). Lemma 2.6.1. Suppose p > h and let ν1 , ν2 ∈ X(T )+ . Then for all j j Hj (G, H 0 (ν1 ) ⊗ H 0 (ν2∗ )(1) ) ∼ = ExtG (V (ν2 )(1) , H 0 (ν1 )) ∼ = HomG (V (ν2 ), Hj (G1 , H 0 (ν1 ))(−1) ). We remark that the aforementioned lemma would hold for arbitrary rth-twists if it was known that the cohomology group Hj (Gr , H 0 (ν))(−r) admits a good ﬁltration, which is a long-standing conjecture of Donkin. For p > h, this is known for r = 1 by results of Andersen-Jantzen [AJ] and Kumar-Lauritzen-Thomsen [KLT]. In that case, the lemma is only needed when ν1 = ν2 . For arbitrary r we can

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often work inductively from the r = 1 case. This requires slightly more general Ext-computations and the possibility that ν1 = ν2 . 2.7. Dimensions for r = 1. From Lemma 2.6.1, for ν ∈ X(T )+ , the cohomology group Hi (G, H 0 (ν) ⊗ H 0 (ν ∗ )(1) ) can be identiﬁed with HomG (V (ν), Hi (G1 , H 0 (ν)(−1) ). It is well-known that, from block considerations, Hi (G1 , H 0 (ν)) = 0 unless ν = w · 0 + pμ for w ∈ W and μ ∈ X(T ). For p > h, from [AJ] and [KLT], we have i−(w) 2 (u∗ ) ⊗ μ) if ν = w · 0 + pμ indG 1 0 (−1) B (S (2.7.1) H (G1 , H (ν)) = 0 otherwise, where u = Lie(U ). Note also that, since p > h and ν is dominant, μ must also be dominant. For a dominant weight ν = pμ + w · 0, observe that from Lemma 2.6.1 and (2.7.1), we have Hi (G, H 0 (ν) ⊗ H 0 (ν ∗ )(1) ) ∼ = HomG (V (ν), Hi (G1 , H 0 (ν))(−1) ) i−(w) ∼ 2 (u∗ ) ⊗ μ)) = HomG (V (ν), indG B (S i−(w) ∼ = HomB (V (ν), S 2 (u∗ ) ⊗ μ).

Hence, if Hi (G, H 0 (ν) ⊗ H 0 (ν ∗ )(1) ) = 0, then ν − μ = (p − 1)μ + w · 0 must be a sum of (i − (w))/2 positive roots. For a weight ν and n ≥ 0, let Pn (ν) denote the dimension of the ν-weight space of S n (u∗ ). Equivalently, for n > 0, Pn (ν) denotes the number of times that ν can be expressed as a sum of exactly n positive roots, while P0 (0) = 1. The function Pn is often referred to as Kostant’s Partition Function. By using [AJ, 3.8], [KLT, Thm 2], Lemma 2.6.1, and (2.7.1), we can give an explicit formula for the dimension of Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) (cf. [BNP, Proposition 3.2.1, Corollary 3.5.1]). Proposition 2.7.1. Let p > h and λ = pμ + w · 0 ∈ X(T )+ . Then dim Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = (−1)(u) P i−(w) (u · λ − μ). u∈W

2

2.8. Degree Bounds. The following gives a fundamental constraint on nonzero i such that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = ExtiG (V (λ)(1) , H 0 (λ)) = 0. It is stated in a more general Ext-context as it will also be used in some inductive arguments for r > 1 (cf. [BNP, Proposition 3.4.1]). Proposition 2.8.1. Let p > h with γ1 , γ2 ∈ X(T )+ , both non-zero, such that γj = pδj + wj · 0 with δj ∈ X(T )+ and wj ∈ W for j = 1, 2. Assume ExtiG (V (γ2 )(1) , H 0 (γ1 )) = 0. (a) Let σ ∈ Φ+ . If Φ is of type G2 , assume that σ is a long root. Then pδ2 , σ ∨ − δ1 , σ ∨ + (w1 ) + w2 · 0, σ ∨ ≤ i. (b) If α ˜ denotes the longest root in Φ+ , then pδ2 , α ˜ ∨ − δ1 , α ˜ ∨ + (w1 ) − (w2 ) − 1 ≤ i. Equality requires that γ2 − δ1 = ((i − (w1 ))/2)α ˜ and −w2 · 0, α ˜ ∨ = (w2 ) + 1. (c) If γ1 = γ2 = pδ + w · 0, then i ≥ (p − 1)δ, α ˜ ∨ − 1.

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Proposition 2.8.1 can be generalized to the following (cf. [BNP, Proposition 4.3.1]). Proposition 2.8.2. Let p > h, 0 = λ ∈ X(T )+ and i ≥ 0. If Hi (G, H 0 (λ) ⊗ H (λ∗ )(r) ) = 0, then there exists a sequence of non-zero weights λ = γ0 , γ1 , . . . , γr−1 , γr = λ ∈ X(T )+ such that γj = pδj + uj · 0 for some uj ∈ W and nonzero δj ∈ X(T )+ . Moreover, for each 1 ≤ j ≤ r, there exists a nonnegative integer lj r l with ExtGj (V (γj )(1) , H 0 (γj−1 )) = 0 and j=1 lj = i. Furthermore, ⎞ ⎛ r ˜ ∨ ⎠ − r. (2.8.1) i ≥ ⎝ (p − 1)δj , α 0

j=1

˜ and that −uj ·0, α ˜∨ = Equality requires that pδj −δj−1 +uj ·0 = ((lj −(uj−1 ))/2)α (uj ) + 1 for all 1 ≤ j ≤ r. Note that the assumption Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0 in the proposition can be replaced by ExtkG/G1 (V (λ)(r) , Hl (G1 , H 0 (λ))) = 0, where k + l = i. In that case one arrives at the same conclusions with l1 = l. 3. Vanishing Ranges in the Simply Laced Case In this section we obtain some general vanishing information for those cases when the root system Φ is simply laced. In such cases, the longest root α ˜ and the longest short root α0 coincide. Following the discussion in Section 2, we want to consider when Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for i > 0 and λ ∈ X(T )+ . 3.1. To gain information on such cohomology groups, we will use Lemma 2.6.1 and (2.7.1). The following proposition will aid us in showing that certain cohomology groups are non-zero. Proposition 3.1.1. Let α ˜ denote the longest root of Φ and l be a nonnegative integer. Then HomG (V ((l + 1)α), ˜ indG (S l (u∗ ) ⊗ α)) ˜ ∼ = k. B

Proof. The claim follows from the diagram below and the fact that all modules in the commutative diagram below have a one-dimensional highest weight space with weight (l + 1)α. ˜ The ﬁrst embedding is a consequence of the fact that the module V (α) ˜ ⊗ · · · ⊗ V (α) ˜ has a Weyl module ﬁltration. The Weyl module V (α) ˜

(l+1) times

is isomorphic to the dual of the adjoint representation, g∗ . Clearly, g∗ maps onto u∗ and V (α) ˜ maps onto φ−α˜ (of weight α ˜ ) as B-modules. Hence, we obtain the two B-surjections in the ﬁrst line of the diagram. The remaining maps and the commutativity of the diagram arise via the universal property of the induction functor.

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V ((l + 1)α) ˜ → V (α) ˜ ⊗ · · · ⊗ V (α) ˜ u ∗ ⊗ · · · ⊗ u∗ ⊗ α ˜ S l (u∗ ) ⊗ α ˜

l times (l+1) times XX ↑ XX XXX XXX z indG (S l (u∗ ) ⊗ α ˜) B

3.2. For a G-module V and a dominant weight γ let [V ]γ denote the unique maximal summand of V whose composition factors have highest weights linked to γ. Lemma 3.2.1. Assume that the root system Φ of G is simply laced. Let α ˜ denote the longest root and deﬁne λ = pα ˜ + sα˜ · 0 = (p − h + 1)α. ˜ Then (a) for any non-zero dominant weight μ linked to zero we have Hi (G, H 0 (μ) ⊗ H 0 (μ∗ )(r) ) = 0 whenever i < r(2p − 3); (b) for any non-zero dominant weight μ linked to zero we have ExtkG/G1 (V (μ)(r) , Hl (G1 , H 0 (μ))) = 0 whenever k + l < r(2p − 3); H 0 (λ) if i = 2p − 3 i 0 (−1) ∼ ]0 = (c) [H (G1 , H (λ)) 0 if 0 < i < 2p − 3. Proof. We apply Proposition 2.8.2. Note that μ being linked to zero forces all the weights δj of Proposition 2.8.2 to be in the root lattice. This forces δj , α ˜ ∨ ≥ 2. Parts (a) and (b) now follow from equation (2.8.1) and the remark in Section 2.8. For part (c) we make use of Proposition 3.1.1 with l +1 = p−h+1 and conclude that (p−h) ∗ HomG (V (λ), indG (u ) ⊗ α ˜ )) ∼ = k. B (S Note that in the simply laced case (sα˜ ) = 2h − 3 which combined with (2.7.1) p−h ∗ (u ) ⊗ α)) ˜ ∼ yields indG = H2p−3 (G1 , H 0 (λ)(−1) ). B (S The weight λ is the smallest non-zero dominant weight in the zero linkage class. Any other non-zero weight μ in the linkage class will be of the form μ = λ + σ = (p − h + 1)α ˜ + σ, where σ is a non-zero sum of positive roots. Clearly μ cannot be a weight of S m (u∗ ) ⊗ α ˜ whenever m ≤ p − h. Hence, HomG (V (μ), Hi (G1 , H 0 (λ))(−1) ) ∼ = HomG (V (μ), indG B (S

i−(sα ˜) 2

(u∗ ) ⊗ α)) ˜ =0

for all 0 < i ≤ 2p − 3. Since H2p−3 (G1 , H 0 (λ))(−1) has a good ﬁltration one obtains [H2p−3 (G1 , H 0 (λ))(−1) ]0 ∼ = H 0 (λ). Part (b) now implies that [Hi (G1 , H 0 (λ))]0 = 0 whenever 0 < i < 2p − 3.

Theorem 3.2.2. Assume that the root system of G is simply laced. Then Hr(2p−3) (G(Fq ), k) = 0. Proof. Let μ be a weight in the zero linkage class. From Lemma 3.2.1(a), we ˜ know that Hi (G, H 0 (μ) ⊗ H 0 (μ∗ )(1) ) = 0 for i < r(2p − 3). Let λ = (p − h + 1)α. We next show by induction on r that Hr(2p−3) (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0. For r = 1, this follows from Lemma 2.6.1 and Lemma 3.2.1(c). For r > 1, we look at the Lyndon-Hochschild-Serre spectral sequence (r) E2k,l = ExtkG/G1 (V (λ)(r) , Hl (G1 , H 0 (λ))) ⇒ Extk+l , H 0 (λ)). G (V (λ)

32 8

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Lemma 3.2.1(b) implies that the E2k,l = 0 for k + l < r(2p − 3). Note that E2k,l = ExtkG (V (λ)(r−1) , Hl (G1 , H 0 (λ))(−1) ). Lemma 3.2.1(c) implies that E2k,l = 0 for l < 2p − 3, and, moreover, that H 0 (λ) is a summand of H2p−3 (G1 , H 0 (λ))(−1) . Hence, (cf. [BNP, Lemma 5.4]), (r−1)(2p−3),2p−3

E2

(r−1)(2p−3)

= ExtG

(V (λ)(r−1) , H2p−3 (G1 , H 0 (λ))(−1) )

(r−1)(2p−3)

has a summand isomorphic to ExtG (V (λ)(r−1) , H 0 (λ)). By induction, this (r−1)(2p−3),2p−3 Ext-group is non-zero, and hence E2 = 0 and transgresses to the E∞ page, which implies that r(2p−3)

ExtG

(V (λ)(r) , H 0 (λ)) = 0.

Since λ is the lowest non-zero dominant weight in the zero linkage class, the claim now follows by applying the argument given in Section 2.5 to the weight λ and the zero linkage class. 3.3. Finite groups of adjoint type. In this section we assume that G is simply laced and of adjoint type. The ﬁxed points of the rth iterated Frobenius map on G will again be denoted by G(Fq ). For example, if G is the adjoint group of type A then G(Fq ) is the projective general linear group P GLn (Fq ) with entries in the ﬁeld with q elements. Propositions 2.3.1 and 2.3.2 can also be applied to groups of adjoint type. Note that the root lattice and the weight lattice coincide in this case. Therefore all dominant weights of the form pδ + w · 0 are automatically in the zero linkage class. From Lemma 3.2.1, Theorem 3.2.2, and Proposition 2.4.1, one obtains the following corollary. Corollary 3.3.1. Assume that the root system Φ of G is simply laced and that G is of adjoint type. Then (a) Hi (G(Fq ), k) = 0 for 0 < i < r(2p − 3); (b) Hr(2p−3) (G(Fq ), k) = 0. Note that, for type An and q − 1 and n + 1 being relatively prime, the adjoint and the universal types of the ﬁnite groups coincide. In these cases the above claim was already observed in Theorems 6.5.1 and 6.14.1 of [BNP]. Remark 3.3.2. Let G be of type A, D, or E6 and of adjoint type. Let σ denote an automorphism of the Dynkin diagram of G. Then σ induces a group automorphism of G that commutes with the Frobenius morphism, which we will also denote by σ. Let Gσ (Fq ) be the ﬁnite group consisting of the ﬁxed points of σ composed with F . Note that σ ﬁxes the maximal root α. ˜ Therefore the discussion in this section also applies to the twisted groups Gσ (Fq ) of adjoint type. In particular, Corollary 3.3.1 holds for these groups as well. 4. Type Dn , n ≥ 4 Assume throughout this section that Φ is of type Dn , n ≥ 4, and that p > h = 2n − 2. Following Section 2, our goal is to ﬁnd the least i > 0 such that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )) = 0 for some λ.

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

33 9

4.1. Restrictions. Suppose that Hi (G, H 0 (λ)⊗H 0 (λ∗ )(1) ) = 0 for some i > 0 and λ = pμ + w · 0 with μ ∈ X(T )+ and w ∈ W . From Proposition 2.8.1(c), i ≥ (p − 1)μ, α ˜ ∨ − 1. For a fundamental dominant weight ωj , 1 if j = 1, n − 1, n ˜∨ = ωj , α 2 if 2 ≤ j ≤ n − 2. ˜ ∨ ≥ 2 and i ≥ 2p − 3. This Therefore, if μ = ω1 , ωn−1 , ωn , we will have μ, α reduces us to analyzing the cases when μ = ω1 , ωn−1 , ωn . 4.2. The case of ω1 . We consider ﬁrst the case that λ = pω1 + w · 0 and obtain the following restrictions. Lemma 4.2.1. Suppose Φ is of type Dn with n ≥ 4 and p > 2n − 2. Suppose λ = pω1 + w · 0 ∈ X(T )+ with w ∈ W . Then (a) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 2n; (b) if H2p−2n (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = pω1 − (2n − 2)ω1 = (p − 2n + 2)ω1 . Proof. Following the discussion in Section 2.7, λ − ω1 = (p − 1)ω1 + w · 0 must be a weight of S j (u∗ ) for j = i−(w) . Recall that ω1 = α1 + α2 + · · · + αn−2 + 2 1 1 2 αn−1 + 2 αn . Consider the decomposition of −w · 0 into a sum of (w) distinct positive roots (see Observation 2.2.1). Write (w) = a + b where a is the number of positive roots in this decomposition that contain α1 and b is the number of roots in this decomposition that do not contain α1 . Then λ − ω1 contains p − 1 − a copies of α1 . Since any root contains at most one copy of α1 , we have i − (w) = j ≥ p − 1 − a. 2 Replacing (w) by a + b and simplifying gives i ≥ 2p − 2 − a + b. The total number of positive roots containing an α1 is 2n − 2. Since we necessarily then have a ≤ 2n − 2, we can rewrite the above as i ≥ 2p − 2 − (2n − 2) + b = 2p − 2n + b ≥ 2p − 2n since b ≥ 0. This proves part (a). Furthermore, we see that i = 2p − 2n if and only if b = 0 and a = 2n − 2. In other words, when −w · 0 is expressed as a sum of distinct positive roots, it consists precisely of all 2n − 2 roots which contain an α1 . That is, −w · 0 = (2n − 2)ω1 , which gives part (b). 4.3. The case of ω1 continued. We will show in Proposition 4.3.2 that H2p−2n (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for λ = (p − 2n + 2)ω1 . To do this, we will make use of Proposition 2.7.1. We ﬁrst make some observations about relevant partition functions. Note that ω1 = 1 .

34 10

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

For Φ of type Dn , with n ≥ 4, and integers m, k, we set ⎧ (u) ⎪ ⎨ u∈W (−1) Pk (u · m1 ) P (m, k, n) := 1 ⎪ ⎩ 0

if m ≥ 1, k ≥ 0, if m = 0, k = 0, else.

Note that P (m, k, n) = dim HomG (V (m1 ), H 0 (G/B, S k (u∗ ))) = [ch H 0 (G/B, S k (u∗ )) : ch H 0 (m1 )], when m ≥ 0, k ≥ 0, n ≥ 4. Lemma 4.3.1. Suppose Φ is of type Dn with n ≥ 4 and m ≥ 0. (a) P (m, k, n) = 0 whenever k < m. (−1)(u) Pk (u · m1 − 1 ) = P (m − 1, k, n). (b) u∈W (u) (c) Pk (u · m1 + 1 ) = P (m + 1, k, n) − P (m + 1, k, n − 1), for u∈W (−1) n ≥ 5. (u) (d) Pk (u · m1 + 1 ) = P (m − 1, k − 2n + 2, n). u∈W (−1) (e) P (m, k, n) = P (m, k, n − 1) + P (m − 2, k − 2n + 2, n), for m ≥ 0, n ≥ 5. (f) P (m, m, n) = 1, for n ≥ 4 and m even. Proof. (a) The weight ω1 = 1 = 12 (α1 + α), ˜ written as a sum of simple roots, contains onecopy of α1 . Assume that 0 ≤ k < m and n ≥ 4. Recall that P (m, k, n) = u∈W (−1)(u) Pk (u · m1 ). Note that u · m1 = u(m1 ) + u · 0 is roots if and only if u(1 ) = 1 . Therefore, P (m, k, n) = a sum of positive(u) (−1) Pk (u · m1 ). If u(1 ) = 1 then u · m1 = m1 + u · 0. {u∈W |u(1 )=1 } In addition −u · 0, written as a sum of simple roots, contains no α1 . Hence, u · m1 , written as a sum of simple roots, contains exactly m copies of α1 . Each positive root of Φ contains at most one copy of α1 . Therefore at least m positive roots are needed to sum up to u · m1 . One concludes that Pk (u · m1 ) = 0 for k < m. (b) Again u · m1 − 1 is a sum of positive roots only if u(1 ) = 1 . Therefore,

(−1)(u) Pk (u · m1 − 1 ) =

(−1)(u) Pk (u · m1 − 1 )

{u∈W |u(1 )=1 }

u∈W

=

(−1)(u) Pk ((m − 1)1 + u · 0)

{u∈W |u(1 )=1 }

=

(−1)(u) Pk (u · (m − 1)1 )

{u∈W |u(1 )=1 }

=

P (m − 1, k, n).

(c) For the expression u · m1 + 1 to be a sum of positive roots one needs either u(1 ) = 1 or u(1 ) = 2 and u(2 ) = 1 . Set A = u∈W (−1)(u) Pk (u · m1 + 1 ).

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

Then

A=

35 11

(−1)(u) Pk (u · m1 + 1 )

{u∈W |u(1 )=1 }

+

(−1)(u) Pk (u · m1 + 1 )

{u∈W |u(1 )=2 ,u(2 )=1 }

=

(−1)(u) Pk ((m + 1)1 + u · 0)

{u∈W |u(1 )=1 }

+

(−1)(u)+1 Pk (sα1 m1 + u · 0 − α1 + 1 )

{u∈W |u(1 )=1 ,u(2 )=2 }

=

(−1)(u) Pk (u · (m + 1)1 )

{u∈W |u(1 )=1 }

−

(−1)(u) Pk (u · (m + 1)2 )

{u∈W |u(1 )=1 ,u(2 )=2 }

= P (m + 1, k, n) − P (m + 1, k, n − 1).

(d) We make use of the fact that ω1 = 1 is a minuscule weight and obtain: A = [( (−1)i ch H i (G/B, S k (u∗ ) ⊗ −1 )) : ch H 0 (m1 )] i≥0

= [ch H 0 (G/B, S k (u∗ ) ⊗ −1 )) : ch H 0 (m1 )] (by [KLT, Lemma 6]) = [ch H 0 (G/B, S k−2n+2 (u∗ ) ⊗ 1 )) : ch H 0 (m1 )] (u) = (−1) Pk−2n−2 (u · m1 − 1 ) (by [AJ, 3.8]) u∈W

=

P (m − 1, k − 2n + 2, n)

(by (b)).

Part (e) now follows directly from (c) and (d). (f) If n ≥ 5, it follows from (e) and (a) that P (m, m, n) = P (m, m, n − 1). So the claim holds if it holds for n = 4. If n = 4 and k = m, then (e) has to be replaced by P (m, m, 4) = (−1)(u) Pm (u · m2 ). {u∈W |u(1 )=1 ,u(2 )=2 }

Note that the both sides of the equation are zero unless m is even. For even m, a direct computation similar to that in the proof of [BNP, Lemma 6.11] shows that (−1)(u) Pm (u · m2 ) = 1. {u∈W |u(1 )=1 ,u(2 )=2 }

36 12

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Proposition 4.3.2. Suppose Φ is of type Dn with n ≥ 4. Assume that p > 2n − 2. Let λ = (p − 2n + 2)ω1 . Then H2p−2n (G, H 0 (λ) ⊗ H 0 (λ)(1) ) = k. Proof. From the previous discussion we know that λ = (p−2n+2)ω1 is of the form pω1 + w · 0 with (w) = 2n − 2. Set k = (i − l(w))/2. From Proposition 2.7.1 and Lemma 4.3.1(b), one concludes dim Hi (G, H 0 (λ) ⊗ H 0 (λ)(1) ) = [ch H 0 (G/B, S k (u∗ ) ⊗ ω1 ) : ch H 0 (λ)] = P (p − 2n + 1, k, n). By Lemma 4.3.1(a), this expression is zero unless k ≥ p−2n+1, and by Lemma 4.3.1(f) it follows that P (p − 2n + 1, p − 2n + 1, n) = 1. Replacing k by (i − 2n + 2)/2 and solving for i yields the claim. 4.4. The case of ωn−1 and ωn . We now consider the case that λ = pωn−1 + w · 0 or λ = pωn + w · 0 for w ∈ W with λ ∈ X(T )+ . Lemma 4.4.1. Suppose Φ is of type Dn with n ≥ 4 and p > 2n − 2. Suppose λ = pωn−1 + w · 0 ∈ X(T )+ or λ = pωn + w · 0 ∈ X(T )+ with w ∈ W , and Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for i = 0. Then (p − n)n ; (a) i ≥ 2 (b) if n ≥ 5, then i ≥ 2p − 2n + 2. Proof. We consider the case of ωn . By symmetry, the case of ωn−1 can be dealt with in a similar manner. Following the discussion in Section 2.7, λ − ωn = . Recall that (p − 1)ωn + w · 0 must be a weight of S j (u∗ ) for j = i−(w) 2 1 (n − 2) n (α1 + 2α2 + 3α3 + · · · + (n − 2)αn−2 ) + αn−1 + αn . 2 4 4 Consider the decomposition of −w · 0 into a sum of distinct positive roots (cf. Observation 2.2.1). Write (w) = a+b where a is the number of positive roots in this decomposition which contain αn and b is the number of roots in this decomposition − a copies of αn . Since any which do not contain αn . Then λ − ωn contains (p−1)n 4 root contains at most one copy of αn , we have ωn =

(p − 1)n i − (w) =j≥ − a. 2 4 Substituting (w) = a + b, rewriting, and simplifying, we get i≥

(p − 1)n − a + b. 2

The total number of positive roots containing αn is have a ≤ (n−1)n and b ≥ 0, we get 2

(n−1)n . 2

(p − 1)n (n − 1)n − +b 2 2 (p − n)n = +b 2 (p − n)n ≥ 2

i≥

Since we necessarily

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

37 13

which gives part (a). For part (b), assume that n ≥ 5. We want to show that (p − n)n ≥ 2p − 2n + 2. 2 This is equivalent to showing that (p − n)n ≥ 4p − 4n + 4. Consider the left hand side: (p − n)n = np − n2 = 4p + (n − 4)p − n2 . Hence the problem is reduced to showing that (n − 4)p − n2 ≥ −4n + 4 or (n − 4)p − n2 + 4n − 4 ≥ 0. Since p ≥ 2n − 1, we have (n − 4)p − n2 + 4n − 4 ≥ (n − 4)(2n − 1) − n2 + 4n − 4 = n2 − 5n = n(n − 5) ≥ 0 since n ≥ 5. Part (b) follows.

Note that if n = 4, (p − n)n 4(p − 4) = = 2p − 8 = 2p − 2n. 2 2 4.5. Summary for type D. The following two theorems summarize our ﬁndings when the root system is of type Dn . Theorem 4.5.1. Suppose Φ is of type Dn with n ≥ 4. Assume that p > 2n − 2. Then (a) Hi (G(Fp ), k) = 0 for0 < i < 2p − 2n; k if n ≥ 5 (b) H2p−2n (G(Fp ), k) = k ⊕ k ⊕ k if n = 4. Proof. Part (a) follows from Section 4.1, Lemma 4.2.1(a), Lemma 4.4.1, and Proposition 2.4.1. For part (b), when n ≥ 5, it follows from Section 4.1, Lemma 4.2.1, Proposition 4.3.2 and Lemma 4.4.1 that λ = (2p − 2n + 2)ω1 is the only dominant weight with H2p−2n (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0. Since λ is the lowest weight in its linkage class, the claim follows from Theorem 2.5.1. For n = 4 the symmetry of the root system yields H2p−2n (G, H 0 (λ) ⊗ H 0 (λ)(1) ) = k for the weights λ = (p − 6)ω1 , λ = (p − 6)ω3 and λ = (p − 6)ω4 , and those are the only weights with non-zero G-cohomology in degree 2p − 2n. Each weight is minimal in its own linkage class. The claim follows. Working inductively from the r = 1 case, we can obtain sharp vanishing bounds for arbitrary r. Theorem 4.5.2. Suppose Φ is of type Dn with n ≥ 4. Assume that p > 2n − 2. Then (a) Hi (G(Fq ), k) = 0 for 0 < i < r(2p − 2n); k if n ≥ 5 (b) Hr(2p−2n) (G(Fq ), k) = k ⊕ k ⊕ k if n = 4.

38 14

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Proof. For part (a), we need to show that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0 for 0 < i < r(2p − 2n) and λ ∈ X(T )+ . If that is true, then the claim follows from Proposition 2.4.1. For part (b), we require precise information on those dominant weights λ for which Hr(2p−2n) (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0. The root lattice ZΦ has four cosets within X(T ): ZΦ, {ω1 + ZΦ}, {ωn−1 + ZΦ}, and {ωn + ZΦ}. If λ is a weight in the root lattice claim (a) follows from Lemma 3.2.1(a). Furthermore, no such weights can contribute to cohomology in degree r(2p − 2n). Assume that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0 for some i > 0 and apply Proposition 2.8.2. Suppose ﬁrst that λ = pδ0 + u0 · 0 with δ0 = δr ∈ {ω1 + ZΦ}. For l each 1 ≤ j ≤ r, in order to have ExtGj (V (γj )(1) , H 0 (γj−1 )) = 0, then γj − δj−1 = lj −(uj−1 )

2 pδj + uj · 0 − δj−1 must be a weight in S (u∗ ). This implies that pδj − δj−1 must lie in the positive root lattice. Since δr ∈ {ω1 + ZΦ}, we necessarily have pδr ∈ {ω1 + ZΦ}. Since pδr − δr−1 ∈ ZΦ, it then follows that δr−1 ∈ {ω1 + ZΦ}. Inductively one concludes that δj ∈ {ω1 + ZΦ} for all j. For a weight γ, when expressed as a sum of simple roots, let N (γ) denote the number of copies of α1 that appear. Since a positive root contains at most one copy of α1 , we have

lj − (uj−1 ) ≥ pN (δj ) − N (−uj · 0) − N (δj−1 ). 2 From Observation 2.2.1, we know that −uj · 0 can be expressed uniquely as (uj ) distinct positive roots. Write (uj ) = aj + bj where aj denotes the number of roots containing an α1 and bj the number that do not. Then N (uj · 0) = −aj , and rewriting the above gives lj ≥ 2pN (δj ) − 2N (δj−1 ) − 2aj + aj−1 + bj−1 . Hence, summing over j gives i=

r j=1

lj ≥

r r r (2p − 2)N (δj ) − aj + bj . j=1

j=1

j=1

Recall that the δj are non-zero dominant weights. By the assumption on δj , N (δj ) ≥ 1. The total number of positive roots containing an α1 is 2n−2. Hence, aj ≤ 2n−2. With this, we get (4.5.1)

i ≥ r(2p − 2) − r(2n − 2) +

r j=1

bj = r(2p − 2n) +

r

bj ≥ r(2p − 2n),

j=1

since bj ≥ 0. This gives the necessary condition for part (a) for the coset {ω1 + ZΦ}. Before considering the remaining two cosets, towards addressing part (b), we consider when equality can hold in (4.5.1). As in Section 4.2 we see that equality holds in (4.5.1) if and only if N (δj ) = 1 and (uj ) = 2n − 2, which forces λ = γj = (p − 2n + 2)ω1 for all j. Moreover, one obtains from the discussion above and Proposition 4.3.2 for λ = (p − 2n + 2)ω1 that H 0 (λ) if i = 2p − 2n i 0 (−1) ∼ (4.5.2) [H (G1 , H (λ)) ]λ = 0 if 0 < i < 2p − 2n. Using the spectral sequence argument in the proof of Theorem 3.2.2 (see also the proof of [BNP, Lemma 5.4]), we can show that Hr(2p−2n) (G, H 0 (λ)⊗H 0 (λ∗ )(r) ) = k.

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

39 15

We prove this by induction on r with the r = 1 case being Theorem 4.5.1. Consider the Lyndon-Hochschild-Serre spectral sequence E2k,l = ExtkG/G1 (V (λ)(r) , Hl (G1 , H 0 (λ))) ∼ = Extk (V (λ)(r−1) , Hl (G1 , H 0 (λ))(−1) ) ⇒ Extk+l (V (λ)(r) , H 0 (λ)). G

G

From the remarks in Section 2.8 and the discussion above, E2k,l = 0 for k + l < r(2p − 2n). Furthermore, from (4.5.2), E2k,l = 0 for l < 2p − 2n. Finally, if E2k,l = 0 and k + l = r(2p − 2n), then, from the above conclusion that γj = λ for each j, (r−1)(2p−2n),2p−2n we must have l = 2p − 2n. Hence, the E2 -term survives to E∞ and is the only term to contribute in degree r(2p − 2n). Hence, by (4.5.2) and our inductive hypothesis, r(2p−2n)

ExtG

(r−1)(2p−2n) (V (λ)(r) , H 0 (λ)) ∼ (V (λ)(r−1) , H2p−2n (G1 , H 0 (λ))(−1) ) = ExtG (r−1)(2p−2n) ∼ (V (λ)(r−1) , H 0 (λ)) ∼ = k. = ExtG

To complete the proof of (a) we need to consider the case that λ = pδ0 + u0 · 0 with δ0 = δr ∈ {ωn−1 + ZΦ} ∪ {ωn + ZΦ}. As above, pδj + uj · 0 − δj−1 must lie in the positive root lattice. Since uj · 0 does, this implies that pδj − δj−1 must lie in the positive root lattice. When expressed as a sum of simple roots ωn−1 = 12 α1 + · · · (as does ωn ). Whereas, for 1 ≤ j ≤ n − 2, ωj = α1 + · · · . Since δ0 ∈ {ωn−1 + ZΦ} ∪ {ωn + ZΦ}, for pδ1 − δ0 to lie in the positive root lattice, when expressed as a sum of fundamental weights, pδ1 must contain an odd number of copies of ωn−1 and ωn in total. Since p is odd, this also holds for δ1 . Inductively, every δj has this property. We may assume therefore that each δj contains at least one copy of ωn or one copy of ωn−1 . Proceed as above, but let Nαn (γ) and Nαn−1 (γ) denote the number of copies of αn and αn−1 , respectively, appearing in γ. Set N (γ) = max{Nαn (γ), Nαn−1 (γ)}. Note that, for the weights γ that appear in what follows, both Nαn (γ) and Nαn−1 (γ) are nonnegative. Again, a positive root contains at most one copy of αn or αn−1 . Just as above, we get lj

≥

2pNαn (δj ) − 2Nαn (−uj · 0) − 2Nαn (δj−1 ) + (uj−1 )

and the corresponding dual statement for αn−1 . By choosing the appropriate root we obtain ≥ 2pN (δj ) − 2Nαn (−uj · 0) − 2Nαn (δj−1 ) + (uj−1 )

lj or lj

≥ 2pN (δj ) − 2Nαn−1 (−uj · 0) − 2Nαn−1 (δj−1 ) + (uj−1 ).

Either one will result in lj

≥

2pN (δj ) − 2N (−uj · 0) − 2N (δj−1 ) + (uj−1 ).

From earlier arguments we know that (uj−1 ) ≥ N (−uj−1 · 0). Hence lj

≥ 2pN (δj ) − 2N (−uj · 0) − 2N (δj−1 ) + N (−uj−1 · 0).

Summing over j, one obtains i=

r j=1

lj ≥

r j=1

[(2p − 2)N (δj ) − N (−uj · 0)] .

40 16

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Clearly, N (−uj · 0) ≤ N (−w0 · 0) = Substituting this gives

n(n−1) . 2

Moreover, we can say that N (δj ) ≥

n 4.

n rn(n − 1) i ≥ r(2p − 2) − 4 2 (p − 1)(n) n(n − 1) − =r 2 2 ≥ r(2p − 2n),

where the last inequality follows as in the proof of Lemma 4.4.1. Thus part (a) follows. For n ≥ 5, the last inequality is strict. Hence, λ = (p − 2n + 2)ω1 is the only dominant weight for which Hr(2p−2n) (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0. As in the proof of Theorem 4.5.1, since λ is minimal in its linkage class, part (b) follows. Similarly, for n = 4, by symmetry, part (b) follows. 5. Type E 5.1. Type E6 . Assume for this subsection that Φ is of type E6 with p > h = 12 (so p ≥ 13). The only dominant weights μ with μ, α ˜ ∨ < 2 are ω1 and ω6 . One concludes from Proposition 2.8.1 and Proposition 2.4.1 that Hi (G(Fp ), k) = 0 for all 0 < i < 2p − 3 unless there exists a weight λ of the form pω1 + w · 0 or of the form pω6 + w · 0 with Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for some 0 < i < 2p − 3. Lemma 5.1.1. Suppose Φ is of type E6 , p ≥ 13 and λ ∈ X(T )+ is of the form pω1 + w · 0 or pω6 + w · 0 with w ∈ W . Assume in addition that p = 13, 19. Then Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for all 0 < i < 2(p − 1). Proof. We prove the assertion for λ = pω1 + w · 0, w ∈ W. Let N denote the number of times that α1 appears in −w · 0 when written as a sum of simple roots. Note that all positive roots of Φ contain the simple root α1 at most once. This implies that N ≤ (w). Moreover, there are exactly 16 distinct positive roots containing α1 . Hence, N ≤ 16. Using ω1 = 1/3(4α1 + 3α2 + 5α3 + 6α4 + 4α5 + 2α6 ), we note that λ − ω1 , written as a sum of simple roots contains at least 4/3(p − 1) − N copies of α1 . From Section 2.7 we know that Hi (G, H 0 (λ)⊗H 0 (λ∗ )(1) ) = 0 and i > 0 imply that λ−ω1 is a sum of (i − (w))/2 many positive roots. Note that this can only happen if (p − 1) is divisible by 3. Again using the fact that α1 appears at most once in each positive root, one obtains the inequality: 4 i − (w) (p − 1) − N ≤ . 3 2 Solving for i yields 2 2 8 i ≥ (p − 1) − 2N + (w) ≥ 2(p − 1) + (p − 1) − N ≥ 2(p − 1) + (p − 1) − 16. 3 3 3 Note that equality holds if and only if N = (w) = 16. One obtains the desired claim i ≥ 2(p − 1) for all primes except those of the form p = 3t + 1 with 13 ≤ p ≤ 25, i.e., the primes p = 13 and p = 19. Theorem 5.1.2. Suppose Φ is of type E6 and p ≥ 13. (a) If p = 13, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 3; (ii) H2p−3 (G(Fp ), k) = 0.

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41 17

(b) If p = 13, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 16; (ii) H16 (G(Fp ), k) = 0. Proof. For p = 19, part (a) follows immediately from Lemma 5.1.1, Proposition 2.4.1, and Theorem 3.2.2. For the proof of part (b), set p = 13. Part (i) follows from the proof of Lemma 5.1.1. Let WI denote the subgroup of W generated by the simple reﬂections sα2 , ..., sα6 and let w denote the distinguished representative of the left coset w0 WI . Then (w) = 16 and −w · 0 equals the sum of all positive roots in Φ that contain α1 , which equals the weight 12ω1 . Let λ = pω1 + w · 0 = ω1 . Clearly, k∼ = HomG (V (λ), H 0 (λ)) ∼ = HomG (V (λ), indG (S 0 (u∗ ) ⊗ ω1 )) B

(16−(w))/2 ∗ ∼ (u ) ⊗ ω1 )) = HomG (V (λ), indG B (S 16 0 (−1) ∼ ) = HomG (V (λ), H (G1 , H (λ)) 16 0 0 ∗ (1) ∼ = H (G, H (λ) ⊗ H (λ ) ).

Since λ is the smallest dominant weight in its linkage class the assertion follows from the remarks in Section 2.5. For p = 19, part (a)(ii) follows from Theorem 3.2.2. It remains to show part (a)(i). If Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for i < 35 = 2p − 3 and λ ∈ X(T )+ , then λ = 19ω1 + w · 0 or λ = 19ω6 + w · 0 for some w ∈ W . From the proof of Lemma 5.1.1, one can see that i ≥ 32. Consider the case that λ = 19ω1 + w · 0. The ω6 case is dual and analogous. One can explicitly, with the aid of MAGMA, identify all w ∈ W such that λ ∈ X(T )+ and λ − ω1 lies in the positive root lattice. By considering the number of copies of α1 appearing in λ − ω1 (as in the proof of Lemma 5.1.1), one can identify the least k such that λ − ω1 is a weight in S k (u∗ ), and hence the least possible value of i. The three weights which can give a value of i < 35 are listed in the following table along with the minimum possible value of k. λ 7ω1 + ω4 7ω1 + ω2 7ω1

(w) 14 15 16

k i = 2k + (w) 10 34 9 33 8 32

For these weights, one can use MAGMA to explicitly compute (−1)(u) Pk (u · λ − ω1 ) u∈W

in order to apply Proposition 2.7.1. For λ = 7ω1 , one ﬁnds that in fact (−1)(u) P8 (u · λ − ω1 ) = 0 dim H32 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = u∈W

and

dim H34 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) =

(−1)(u) P9 (u · λ − ω1 ) = 0.

u∈W

So, for λ = 7ω1 , we have i ≥ 36. For λ = 7ω1 + ω2 one ﬁnds dim H33 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) =

u∈W

(−1)(u) P9 (u · λ − ω1 ) = 0.

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Therefore, i ≥ 35 in this case. Finally, for λ = 7ω1 + ω4 , one ﬁnds dim H34 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) =

(−1)(u) P10 (u · λ − ω1 ) = 0.

u∈W

Therefore, i ≥ 36 in this case and the claim follows.

We now consider the situation for arbitrary r. Sharp vanishing can be obtained for primes about twice the Coxeter number. Theorem 5.1.3. Suppose Φ is of type E6 and p ≥ 13. (a) If p = 13, 19 or p = 17 when r is even, then (i) Hi (G(Fq ), k) = 0 for 0 < i < r(2p − 3); (ii) Hr(2p−3) (G(Fq ), k) = 0. (b) If p = 13, then (i) Hi (G(Fq ), k) = 0 for 0 < i < 16r; (ii) H16r (G(Fq ), k) = 0. (c) If p = 17 and r is even, then (i) Hi (G(Fq ), k) = 0 for 0 < i < 27r; (ii) H31r (G(Fq ), k) = 0. (d) If p = 19, then (i) Hi (G(Fq ), k) = 0 for 0 < i < 32r; (ii) H35r (G(Fq ), k) = 0. Proof. The validity of parts (a)(ii), (c)(ii), and (d)(ii) follows from Theorem 3.2.2. For part (a)(i), we need to show that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0 for all dominant weights λ and all 0 < i < r(2p − 3). We argue along lines similar to that of the proof of Theorem 4.5.2. The root lattice ZΦ has three cosets within X(T ): ZΦ, {ω1 + ZΦ}, and {ω6 + ZΦ}. If λ is a weight in the root lattice, claim (a)(i) follows from Lemma 3.2.1(a). Assume that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0 for some i > 0 and apply Proposition 2.8.2. From above, we may assume that λ = pδ0 + u0 · 0 with δ0 = δr ∈ {ω1 + l ZΦ} ∪ {ω6 + ZΦ}. For each 1 ≤ j ≤ r, in order to have ExtGj (V (γj )(1) , H 0 (γj−1 )) = lj −(uj−1 )

2 0, then γj −δj−1 = pδj +uj ·0−δj−1 must be a weight in S (u∗ ). This implies that pδj −δj−1 must lie in the positive root lattice. Since δr ∈ {ω1 +ZΦ}∪{ω6 +ZΦ}, we necessarily have pδr ∈ {ω1 + ZΦ} ∪ {ω6 + ZΦ}. Since pδr − δr−1 ∈ ZΦ, it then follows that δr−1 ∈ {ω1 + ZΦ} ∪ {ω6 + ZΦ}. Inductively one concludes that δj ∈ {ω1 + ZΦ} ∪ {ω6 + ZΦ} for all j. Before continuing, we investigate this condition on δj a bit further. Recall that ω1 = 43 α1 + · · · + 23 α6 and ω6 = 23 α1 + · · · + 43 α6 . Suppose that δj ∈ {ω1 + ZΦ} and δj−1 ∈ {ω1 +ZΦ}. In order for pδj −δj−1 to lie in the root lattice, 43 p− 43 = 43 (p−1) would need to be an integer. In other words, p − 1 must be divisible by 3. The same argument holds if we assume that both δj and δj−1 lie in {ω6 + ZΦ}. On the other hand, suppose that δj ∈ {ω1 + ZΦ} and δj−1 ∈ {ω6 + ZΦ} (or vice versa). Then 43 p − 23 = 23 (2p − 1) (or 23 p − 43 = 23 (p − 2), respectively) must be an integer which implies that p − 2 is divisible by 3. Since p is a prime greater than three, either p − 1 is divisible by 3 or p − 2 is divisible by 3. Summarizing, if 3|(p − 1), then either each δj ∈ {ω1 + ZΦ} or each δj ∈ {ω6 + ZΦ}. We refer to this as the “consistent” case. Whereas, if 3|(p − 2), then we have an “alternating” situation

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

43 19

with the δj s alternately lying in {ω1 + ZΦ} or {ω6 + ZΦ}. Note further that since δ0 = δr , the alternating case can only occur if r is even. Consider ﬁrst the consistent case (when 3|(p − 1)). Suppose without loss of generality that each δj ∈ {ω1 + ZΦ}. For a weight γ, when expressed as a sum of simple roots, let N (γ) denote the number of copies of α1 that appear. Since a positive root contains at most one copy of α1 , we have lj − (uj−1 ) ≥ pN (δj ) − N (−uj · 0) − N (δj−1 ). 2 Rewriting this and using the fact that (see Observation 2.2.1) (uj−1 ) ≥ N (−uj−1 · 0) gives lj ≥ 2pN (δj ) − 2N (−uj · 0) − 2N (δj−1 ) + (uj−1 ) ≥ 2pN (δj ) − 2N (−uj · 0) − 2N (δj−1 ) + N (−uj−1 · 0). Therefore, i=

=

r j=1 r

lj ≥

r

(2pN (δj ) − 2N (−uj · 0) − 2N (δj−1 ) + N (−uj−1 · 0))

j=1

((2p − 2)N (δj ) − N (−uj · 0)).

j=1

There are only 16 positive roots which contain an α1 . Hence, Since N (δj ) ≥ 43 , we get r 4 4 (2p − 2) − 16 = r (2p − 2) − 16 = r 2p − 2 + i≥ 3 3 j=1

N (−uj · 0) ≤ 16. 1 (2p − 2) − 16 . 3

For p ≥ 25, we get i ≥ r(2p − 2) as desired. Note that for p = 17 and p = 23, 3 (p − 1), and so the only “bad” cases are p = 13 and p = 19. For p = 13, we conclude that i ≥ 16r, and for p = 19, we conclude that i ≥ 32r. Now consider the alternating case (which requires p − 2 being divisible by 3). Analogous to the proof of Theorem 4.5.2 for the type Dn case, for a weight γ, let Nα1 (γ) (or Nα6 (γ)) denote the coeﬃcient of α1 (or α6 ) when γ is expressed as a sum of simple roots. And then set N (γ) = max{Nα1 (γ), Nα6 (γ)} (where the max is considered only in cases where the quantities involved are nonnegative). Then we reach the same conclusion on i as above. In this case, p = 13 and p = 19 cannot occur. Moreover, p = 17 and p = 23 are potentially “bad.” However, for p = 23, since i is an integer, we still conclude that i ≥ r(2p − 3) as needed. For p = 17, we conclude that i ≥ 27r. That completes the proof of all parts except for part (b)(ii) with p = 13. This follows inductively from the r = 1 case by using the the spectral sequence argument as in the proofs of Theorem 3.2.2 and Theorem 4.5.2. For p = 17 when r is even and p = 19, the theorem does not give a sharp vanishing bound. 5.2. Type E7 . Assume for this subsection that Φ is of type E7 with p > h = 18 (so p ≥ 19.) The only dominant weight μ with μ, α ˜ < 2 is ω7 . Again we conclude from Proposition 2.8.1 and Proposition 2.4.1 that Hi (G(Fp ), k) = 0 for

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

all 0 < i < 2p − 3 unless there exists a weight λ of the form pω7 + w · 0 with Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for some 0 < i < 2p − 3. Lemma 5.2.1. Suppose Φ is of type E7 , p ≥ 19 and λ ∈ X(T )+ is of the form pω7 + w · 0 with w ∈ W . Assume in addition that p = 19, 23. Then Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for all 0 < i < 2(p − 1). Proof. Assume λ = pω7 + w · 0, w ∈ W. Let N denote the number of times that α7 appears in −w · 0 when written as a sum of simple roots. Note that all positive roots of Φ contain the simple root α7 at most once. This implies that N ≤ (w). Moreover, there are exactly 27 distinct positive roots containing α7 . Hence, N ≤ 27. When writing ω7 as a sum of simple roots the coeﬃcient for α7 is 3/2. Therefore λ − ω7 , written as a sum of simple roots contains at least 3/2(p − 1) − N copies of α7 . From Section 2.7, we know that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 and i > 0 imply that λ − ω1 is a sum of (i − (w))/2 many positive roots. Using the the fact that α7 appears at most once in each positive root, one obtains the inequality: i − (w) 3 (p − 1) − N ≤ . 2 2 Solving for i yields i ≥ 3(p − 1) − 2N + (w) ≥ 2(p − 1) + p − 1 − N ≥ 2(p − 1) + p − 1 − 27. Note that equality holds if and only if N = (w) = 27. Hence, i ≥ 2(p − 1) for all primes except for 18 < p ≤ 28, i.e., the primes p = 19 and p = 23. Theorem 5.2.2. Suppose Φ is of type E7 and p ≥ 19. (a) If p = 19, 23, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 3; (ii) H2p−3 (G(Fp ), k) = 0. (b) If p = 19, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 27; (ii) H27 (G(Fp ), k) = 0. (c) If p = 23, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 39; (ii) H39 (G(Fp ), k) = 0. Proof. Part (a) follows from Lemma 5.2.1, Proposition 2.4.1, and Theorem 3.2.2. For the proof of part (b), set p = 19. Part (i) follows from the proof of Lemma 5.2.1. Let WI denote the subgroup of W generated by the simple reﬂections sα1 , ..., sα6 and let w denote the distinguished representative of the left coset w0 WI . Then (w) = 27 and −w · 0 equals the sum of all positive roots in Φ that contain α7 , which equals the weight 18ω7 . Let λ = pω7 + w · 0 = ω7 . Using the same argument as for E6 , we obtain H27 (G1 , H 0 (λ))(−1) ∼ = H 0 (λ) and hence 27 0 0 ∗ (1) ∼ H (G, H (λ) ⊗ H (λ ) ) = k. Again λ is the smallest dominant weight in its linkage class and the assertion follows from the remarks in Section 2.5. For part (c), set p = 23. If Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for i < 43 = 2p − 3 and λ ∈ X(T )+ , then λ = 23ω7 + w · 0 for some w ∈ W . From the proof of Lemma 5.2.1, one can see that i ≥ 39. One can explicitly, with the aid of MAGMA, identify

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45 21

all w ∈ W such that λ ∈ X(T )+ and λ − ω7 lies in the positive root lattice. By considering the number of copies of α7 appearing in λ − ω7 (as in the proof of Lemma 5.2.1), one can identify the least k such that λ − ω7 is a weight in S k (u∗ ), and hence the least possible value of i. The four weights which can give a value of i < 43 are listed in the following table along with the minimum possible value of k. λ 5ω7 + ω4 5ω7 + ω3 5ω7 + ω1 5ω7

(w) 24 25 26 27

k 9 8 7 6

i = 2k + (w) 42 41 40 39

For these weights, one can use MAGMA to explicitly compute (−1)(u) Pk (u · λ − ω7 ) u∈W

in order to apply Proposition 2.7.1. For λ = 5ω7 , one ﬁnds that in fact (−1)(u) P6 (u · λ − ω7 ) = 1. dim H39 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = u∈W

Since there are no weights less than λ which can give cohomology in degree 40, H39 (G(Fp ), k) = 0. We next consider the situation for arbitrary r. Theorem 5.2.3. Suppose Φ is of type E7 and p ≥ 19. (a) If p = 19, 23, then (i) Hi (G(Fq ), k) = 0 for 0 < i < r(2p − 3); (ii) Hr(2p−3) (G(Fq ), k) = 0. (b) If p = 19, then (i) Hi (G(Fq ), k) = 0 for 0 < i < 27r; (ii) H27r (G(Fq ), k) = 0. (c) If p = 23, then (i) Hi (G(Fq ), k) = 0 for 0 < i < 39r; (ii) H39r (G(Fq ), k) = 0. Proof. The validity of part (a)(ii) follows from Theorem 3.2.2. For part (a)(i), we need to show that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(r) ) = 0 for all dominant weights λ and all 0 < i < r(2p − 3). An argument similar to that in the proof of Theorem 5.1.3 works here as well. The root lattice ZΦ has two cosets within X(T ), namely ZΦ and {ω7 + ZΦ}. If λ is a weight in the root lattice claim (a)(i) follows from Lemma 3.2.1(a). Consider then the case that λ ∈ {ω7 + ZΦ} and apply Proposition 2.8.2. As before, one ﬁnds that each δj ∈ {ω7 + ZΦ}. Furthermore, if we let N (γ) denote the coeﬃcient of α7 , when γ is expressed as a sum of simple roots, then we again conclude that r i≥ ((2p − 2)N (δj ) − N (−uj · 0)). j=1

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Here, since ω7 = α1 + · · · + 32 α7 , we have N (δj ) ≥ 32 . Furthermore, there are 27 positive roots which contain an α7 , and so N (−uj · 0) ≤ 27. Therefore, we get 3 (2p − 2) − 27 = r (2p − 3 + p − 27) . i≥r 2 For p ≥ 27, we have i ≥ r(2p − 3) as needed, which completes part (a). For p = 19 and p = 23, we conclude only that i ≥ 27r or i ≥ 39r, respectively, which gives parts (b)(i) and (c)(i). Parts (b)(ii) and (c)(ii) again follows inductively from the r = 1 case by the spectral sequence argument in Theorem 3.2.2 and Theorem 4.5.2. 5.3. Type E8 . Assume for this subsection that Φ is of type E8 with p > h = 30 (so p ≥ 31). Here the weight lattice and root lattice always coincide. From Corollary 3.3.1 we obtain the following. Theorem 5.3.1. Suppose Φ is of type E8 and p ≥ 31. Then (a) Hi (G(Fq ), k) = 0 for 0 < i < r(2p − 3); (b) Hr(2p−3) (G(Fq ), k) = 0. 6. Type Bn , n ≥ 3 Assume throughout this section that Φ is of type Bn , n ≥ 3, and that p > h = 2n. Note that type B2 is equivalent to type C2 which was discussed in [BNP]. However, for certain inductive arguments, at points we will allow n = 1, 2. Following Section 2, our goal is to ﬁnd the least i > 0 such that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for some λ. From Proposition 2.8.1, we know that i ≥ p − 2. 6.1. Restrictions. Suppose that Hi (G, H 0 (λ)⊗H 0 (λ∗ )(1) ) = 0 for some i > 0 and λ = pμ + w · 0 with μ ∈ X(T )+ and w ∈ W . In this case, the longest root α ˜ = ω2 . From Proposition 2.8.1, i ≥ (p − 1)μ, α ˜ ∨ − 1. For a fundamental dominant weight ωj , 1 if j = 1, n ∨ ˜ = ωj , α 2 if 2 ≤ j ≤ n − 1. ˜ ∨ ≥ 2 and i ≥ 2p − 3. Therefore, if μ = ω1 , ωn , we will have μ, α The following lemma shows that if n is suﬃciently large, and λ = pωn + w · 0, then one also has i ≥ 2p − 3. In fact strictly greater. Lemma 6.1.1. Suppose Φ is of type Bn with n ≥ 7 and p > 2n. Suppose λ = pωn + w · 0 ∈ X(T )+ with w ∈ W and Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0. Then i > 2p − 3. Proof. Following the discussion in Section 2.7, λ − ωn = (p − 1)ωn + w · 0 must be a weight of S j (u∗ ) for j = i−(w) . Recall that 2ωn = α1 + 2α2 + 3α3 + · · · + nαn . 2 Consider the decomposition of −w · 0 into a sum of distinct positive roots (cf. Observation 2.2.1). Write (w) = a + b + c where a is the number of positive roots in this decomposition which contain 2αn , b is the number of roots in this decomposition which contain αn , and c is the number of roots in this decomposition that do not contain αn . Then λ − ωn contains p−1 n − 2a − b 2

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

47 23

copies of αn . Since any root contains at most 2 copies of αn , we have i − (w) p−1 1 n − 2a − b . =j≥ 2 2 2 Replacing (w) by a + b + c and simplifying gives p−1 n − a + c. i≥ 2 The total number of positive roots which contain 2αn is n(n − 1)/2. Hence, a ≤ n(n − 1)/2 and c ≥ 0. So we have n n−1 p−n n2 p−1 n− n= n = 2p + −2 p− . (6.1.1) i≥ 2 2 2 2 2 Finally, using the assumption that p ≥ 2n + 1, one ﬁnds n n2 n2 7 i ≥ 2p + − 2 (2n + 1) − = 2p + − n − 2. 2 2 2 2 For n ≥ 7, we have i ≥ 2p − 2 as claimed.

For 3 ≤ n ≤ 6, the lemma is in fact false. These cases will be discussed in Sections 6.3 - 6.6. 6.2. The case of ω1 . In this section we investigate the case that λ = pω1 +w·0. Throughout this section Φ is of type Bn . In order to make use of some inductive arguments we allow n ≥ 1. We will frequently switch between the bases consisting of the simple roots {α1 , . . . , αn }, the fundamental weights {ω1 , . . . , ωn }, and the canonical basis {1 , . . . , n } of Rn . Following [Hum] we have αi = i − i+1 , for 1 ≤ i ≤ n − 1, and αn = n . Note that 1 = α0 is the maximal short root. The fundamental weights are ωj = 1 + · · · + j , for 1 ≤ j ≤ n − 1, and ωn = 1/2(1 + · · · + n ). In particular, ω1 = 1 . Definition 6.2.1. For Φ of type Bn , with n ≥ 1, we deﬁne ⎧ (u) ⎪ Pk (u · (m1 + (1 + · · · + j ))) ⎪ u∈W (−1) ⎪ ⎪ ⎪ ⎪ ⎨ P (m, k, j, n) := 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0

if m ≥ 0, k ≥ 0, and 1 ≤ j ≤ n, if m = −1, k = 0, and j = 1, else;

⎧ 0 k ∗ 0 ⎪ ⎨dim HomG (V (m1 + (1 + · · · + j )), H (G/B, S (u )) ⊗ H (1 )) T (m, k, j, n) := if m ≥ 0, k ≥ 0, and 1 ≤ j ≤ n, ⎪ ⎩ 0 else. Note that for p > 2n, P (m, k, j, n) = dim HomG (V (m1 + (1 + ... + j )), H 0 (G/B, S k (u∗ ))), which equals the multiplicity of H 0 (m1 + (1 + · · · + j )) in a good ﬁltration of H 0 (G/B, S k (u∗ )) (cf. [AJ, 3.8]). In particular, P (m, k, j, n) ≥ 0 for all m, k, j, and n.

48 24

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

Lemma 6.2.2. Suppose Φ is of type Bn with n ≥ 1 and p > 2n. If m ≥ 0, k ≥ 0, and 1 ≤ j ≤ n, then (u) (a) Pk (u · (m1 + (1 + ... + j )) − 1 ) = P (m − 1, k, j, n) + u∈W (−1) P (m, k, j − 1, n − 1); (u) (b) (−1) P k (u · m1 + 1 ) = P (m, k, 1, n) − P (m, k, 1, n − 1); u∈W (c) P (m, k, j, n) = 0 whenever k < m + 1; 2n (d) T (m, k, j, n) = 2n i=1 P (m − 1, k − i + 1, j, n) + i=1 P (m, k − i + 1, j − 1, n − 1) + P (m, k − n, j, n); (e) for n ≥ 2 and 1 ≤ j ≤ n − 1, T (m, k, j, n) ≥ P (m − 1, k, j, n) + P (m, k, j + 1, n) + P (m, k, j − 1, n); (f) T (m, k, n, n) ≥ P (m − 1, k, n, n) + P (m, k, n, n) + P (m, k, n − 1, n); (g) for l ≥ 0, P (2l, k, 1, n) = P (2l − 1, k − n, 1, n); (h) for l ≥ 0, u∈W (−1)(u) Pk (u · (2l + 1)1 + 1 ) = P (2l, k − n, 1, n). Proof. (a) P (m−1, k, j, n) = u∈W (−1)(u) Pk (u·((m−1)1 +(1 +· · ·+j ))). The expression u · ((m − 1)1 + (1 + · · · + j )) = u((m − 1)1 + (1 + · · · + j )) + u · 0 will be a sum of positive roots only if either u(1 ) = 1 or u(2 ) = 1 . If u is of the second type, then usα1 stabilizes 1 . Setting v = usα1 , one obtains (−1)(u) Pk (u · ((m − 1)1 + (1 + · · · + j ))) P (m − 1, k, j, n) = {u∈W |u(1 )=1 }

+ =

u∈W

−

(−1)(u) Pk (u · ((m − 1)1 + (1 + · · · + j )))

{u∈W |u(2 )=1 }

(−1)(u) Pk (u · (m1 + (1 + · · · + j )) − 1 )

(−1)(v) Pk (v · (m2 + (2 + · · · + j ))).

v∈W

The second term is just P (m, k, j − 1, n − 1) as claimed. Note that the formula also holds for m = 0. (b) The expression u∈W (−1)(u) Pk (u · m1 + 1 ) will be a sum of positive roots only if either u(1 ) = 1 or u(2 ) = 1 . Arguing as above one obtains (−1)(u) Pk (u · m1 + 1 ) = (−1)(u) Pk (u · m1 + 1 ) {u∈W |u(1 )=1 }

u∈W

+ =

u∈W

−

(−1)(u) Pk (u · m1 + 1 )

{u∈W |u(2 )=1 }

(−1)(u) Pk (u · (m1 + 1 ))

(−1)(v) Pk (v · (m2 + 2 )).

v∈W

The ﬁrst term is P (m, k, 1, n) and the second term is just P (m, k, 1, n − 1). (c) Assume that 0 ≤ k < m + 1. Part (a) implies that P (m, k, j, n) ≤ (−1)(u) Pk (u · ((m + 1)1 + (1 + · · · + j )) − 1 ). u∈W

Note that u · ((m + 1)1 + (1 + · · · + j )) − 1 is a sum of positive roots if and only if u(1 ) = 1 . If u(1 ) = 1 then u·((m+1)1 +(1 +· · ·+j ))−1 = (m+1)1 +u·(2 +

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

49 25

· · · + j ) and −u · (2 + · · · + j ), written as a sum of simple roots, contains no α1 . However, (m + 1)1 , written as a sum of simple roots, contains exactly m + 1 copies of α1 . Each positive root of Φ contains at most one copy of α1 . Therefore at least m+1 positive roots are needed to obtain the weight u·((m+1)1 +(1 +· · ·+j ))−1 . One concludes that Pk (u · ((m + 1)1 + (1 + · · · + j )) − 1 ) = 0 and the assertion follows. (d) For a simple root α, let Pα denote the minimal parabolic subgroup corresponding to α, and let uα denote the Lie algebra of the unipotent radical of Pα . From the short exact sequence 0 → α → u∗ → u∗α → 0 one obtains the Koszul resolution 0 → S k−1 (u∗ ) ⊗ α → S k (u∗ ) → S k (u∗α ) → 0. Tensoring with a weight μ yields 0 → S k−1 (u∗ ) ⊗ α ⊗ μ → S k (u∗ ) ⊗ μ → S k (u∗α ) ⊗ μ → 0. Induction from B to G yields the long exact sequence (6.2.1)

· · · → H i (G/B, S k−1 (u∗ ) ⊗ α ⊗ μ) → H i (G/B, S k (u∗ ) ⊗ μ) → H i (G/B, S k (u∗α ) ⊗ μ) → · · · .

We apply (6.2.1) with α = αj = j − j+1 and μ = −j , where 1 ≤ j ≤ n − 1, giving (6.2.2)

· · · → H i (G/B, S k−1 (u∗ ) ⊗ −j+1 ) → H i (G/B, S k (u∗ ) ⊗ −j ) → H i (G/B, S k (u∗α ) ⊗ −j ) → · · · .

Note that −j , αj∨ = −1 forces H i (Pα /B, −j ) = 0 for all i. The spectral sequence H r (G/Pα , S k (u∗α )) ⊗ H s (Pα /B, μ) ⇒ H r+s (G/B, S k (u∗α ) ⊗ μ) yields H i (G/B, S k (u∗α ) ⊗ −j ) = 0 for all i. Therefore, from (6.2.2), one obtains for 1 ≤ j ≤ n − 1 and i ≥ 0 (6.2.3) H i (G/B, S k−1 (u∗ ) ⊗ −j+1 ) ∼ = H i (G/B, S k (u∗ ) ⊗ −j ). Iterating this process yields (6.2.4)

H i (G/B, S k (u∗ ) ⊗ −i ) ∼ = H i (G/B, S k−n+i (u∗ ) ⊗ −n ).

Note that if k < n − i, the right hand side is identically zero, and the isomorphism still holds. Next we apply (6.2.1) with α = αn = n and μ = −n in order to obtain (6.2.5)

· · · → H i (G/B, S k−1 (u∗ )) → H i (G/B, S k (u∗ ) ⊗ −n ) → H i (G/B, S k (u∗α ) ⊗ −n ) → · · · .

Here −n , αn∨ = −2. Using the spectral sequence as above, one obtains H i (G/B, S k (u∗α ) ⊗ −n ) ∼ = H i−1 (G/Pα , S k (u∗α )) ⊗ H 1 (Pα /B, −n ) (6.2.6) ∼ = H i−1 (G/Pα , S k (u∗α )) ∼ = H i−1 (G/B, S k (u∗α )). Since H 0 (G/B, S k (u∗α ) ⊗ −n ) = 0, one obtains via by (6.2.5), (6.2.7) H 0 (G/B, S k−1 (u∗ )) ∼ = H 0 (G/B, S k (u∗ ) ⊗ −n ). From H i (G/B, S k (u∗ )) = 0 for i ≥ 1, using (6.2.5), one concludes

50 26

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

(6.2.8)

H i (G/B, S k (u∗ ) ⊗ −n ) ∼ = H i−1 (G/B, S k (u∗α )) for i ≥ 1.

Next, we apply (6.2.1) with α = αn = n and μ = 0 in order to obtain (6.2.9) · · · → H i (G/B, S k−1 (u∗ ) ⊗ n ) → H i (G/B, S k (u∗ )) → H i (G/B, S k (u∗α )) → · · · . From H i (G/B, S k (u∗ )) = 0 and H i (G/B, S k (u∗ ) ⊗ n ) = 0 for i ≥ 1 [KLT, 2.8], one concludes (6.2.10) 0 → H 0 (G/B, S k−1 (u∗ ) ⊗ n ) → H 0 (G/B, S k (u∗ )) → H 0 (G/B, S k (u∗α )) → 0,

and

(6.2.11)

H i (G/B, S k (u∗ ) ⊗ −n ) ∼ = H i−1 (G/B, S k (u∗α )) = 0 for i ≥ 2.

Similarly, apply (6.2.1) with α = αj = j−1 − j and μ = j , where 2 ≤ j ≤ n, to obtain · · · → H i (G/B, S k−1 (u∗ ) ⊗ j−1 ) → H i (G/B, S k (u∗ ) ⊗ j ) → H i (G/B, S k (u∗α ) ⊗ j ) → · · · .

As before this yields

(6.2.12)

H i (G/B, S k−1 (u∗ ) ⊗ j−1 ) ∼ = H i (G/B, S k (u∗ ) ⊗ j ).

Iterating this process results in

(6.2.13)

H i (G/B, S k−i+1 (u∗ ) ⊗ 1 ) ∼ = H i (G/B, S k (u∗ ) ⊗ i ).

Again, this isomorphism holds even if k < i − 1 (when the left side is identically zero). For any ﬁnite B-module M , we denote its Euler characteristic by

χ(M ) =

i≥0

(−1)i ch H i (G/B, M ).

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

51 27

From the above, one obtains ch(H 0 (G/B, S k (u∗ )) ⊗ H 0 (1 )) = χ(S k (u∗ ) ⊗ H 0 (1 )) n n χ(S k (u∗ ) ⊗ i ) + χ(S k (u∗ )) + χ(S k (u∗ ) ⊗ −i ) = i=1

=

n

i=1

χ(S k−i+1 (u∗ ) ⊗ 1 ) + χ(S k (u∗ ))

i=1

+

n

χ(S k−i+1 (u∗ ) ⊗ −n ) (by 6.2.4, 6.2.13)

i=1

=

n

ch H 0 (G/B, S k−i+1 (u∗ ) ⊗ 1 ) + ch H 0 (G/B, S k (u∗ )) (by [KLT, 2.8])

i=1

+

n

ch H 0 (G/B, S k−i+1 (u∗ ) ⊗ −n )

i=1

−

n

ch H 1 (G/B, S k−i+1 (u∗ ) ⊗ −n ) (by 6.2.8, 6.2.11)

i=1

=

n

ch H 0 (G/B, S k−i+1 (u∗ ) ⊗ 1 ) + ch H 0 (G/B, S k (u∗ ))

i=1

+

n

ch H 0 (G/B, S k−i (u∗ )) +

i=1

−

n

n

ch H 0 (G/B, S k−i−n+1 (u∗ ) ⊗ 1 )

i=1

ch H 0 (G/B, S k−i+1 (u∗ )) (by 6.2.7, 6.2.8, 6.2.10, 6.2.13)

i=1

=

2n

ch H 0 (G/B, S k−i+1 (u∗ ) ⊗ 1 ) + ch H 0 (G/B, S k−n (u∗ )).

i=1

The last equality yields T (m, k, j, n)

= =

dim HomG (V (m1 + (1 + · · · + j )), H 0 (G/B, S k (u∗ )) ⊗ H 0 (1 )) 2n

dim HomG (V (m1 + (1 + · · · + j )), H 0 (G/B, S k−i+1 (u∗ ) ⊗ 1 ))

i=1

+ dim HomG (V (m1 + (1 + · · · + j )), H 0 (G/B, S k−n (u∗ )). The assertion follows now from part (a). (e) A direct computation shows that ch(V (m1 + (1 + · · · + j )) ⊗ V (1 )) = ch V ((m − 1)1 + (1 + · · · + j )) + ch V (m1 + (1 + · · · + j+1 )) + ch V (m1 + (1 + · · · + j−1 )) + ch V ((m + 1)1 + (1 + · · · + j )) + ch V ((m − 1)1 + (1 + 2 ) + (1 + · · · + j )). It follows that T (m, k, j, n)

=

dim HomG (V (m1 + (1 + · · · + j )), H 0 (G/B, S k (u∗ )) ⊗ H 0 (1 ))

= dim HomG (V (m1 + (1 + · · · + j )) ⊗ V (1 )), H 0 (G/B, S k (u∗ ))) ≥ P (m − 1, k, j, n) + P (m, k, j + 1, n) + P (m, k, j − 1, n),

52 28

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

as claimed. Part (f) follows in similar fashion. (g) It is well-known that, for m ≥ 2, ch(H 0 (mω1 )) is equal to the diﬀerence of the mth and the (m − 2)nd symmetric power of the natural representation. The natural representation has one-dimensional weight spaces and includes the zero weight space. One concludes that the dimension of the zero weight space of the 2lth symmetric power equals the dimension of the zero weight space of the (2l +1)st symmetric power. The same is true for the pair H 0 (2lω1 ) and H 0 ((2l + 1)ω1 ). It follows from Kostant’s Theorem [Hum, 24.2] that P (2l − 1, k, 1, n) = P (2l, k, 1, n). (6.2.14) k≥0

k≥0

From (6.2.10) and (6.2.13) one obtains 0 → H 0 (G/B, S k−n (u∗ ) ⊗ 1 ) → H 0 (G/B, S k (u∗ )) → H 0 (G/B, S k (u∗αn )) → 0. All three modules have good ﬁltrations. Moreover, by part (i), dim HomG (V (2l1 + 1 ), H 0 (G/B, S k−n (u∗ ) ⊗ 1 )) = P (2l − 1, k − n, 1, n). Hence, for l ≥ 0 P (2l, k, 1, n) = P (2l − 1, k − n, 1, n) + dim HomG (V (2l1 + 1 ), H 0 (G/B, S k (u∗αn ))). Summing over all k ≥ 0 yields P (2l − 1, k, 1, n) = P (2l, k, 1, n) k≥0

k≥0

+

dim HomG (V (2l1 + 1 )), H 0 (G/B, S k (u∗αn ))).

k≥0

Comparing with (6.2.14) yields dim HomG (V (2l1 + 1 ), H 0 (G/B, S k (u∗αn ))) = 0, which forces P (2l, k, 1, n) = P (2l − 1, k − n, 1, n), for all k ≥ 0. (h) Following [AJ, 3.8], the multiplicity of ch V (2l1 + 1 ) in χ(S k (u∗ ) ⊗ −1 ) equals (−1)(u) Pk (u · ((2l + 1)1 ) + 1 ). u∈W

Moreover, by (6.2.11) and (6.2.4), χ(S k (u∗ ) ⊗ −1 ) = ch H 0 (G/B, S k (u∗ ) ⊗ −1 ) − ch H 1 (G/B, S k (u∗ ) ⊗ −1 )). In addition, from (6.2.8) and (6.2.4), the vanishing of HomG (V (2l1 + 1 ), H 0 (G/B, S k (u∗αn ))) forces the vanishing of HomG (V (2l1 + 1 ), H 1 (G/B, S k (u∗ ) ⊗ −1 )), for all k. Hence, (−1)(u) Pk (u·((2l+1)1 )+1 ) = dim HomG (V (2l1 +1 ), H 0 (G/B, S k (u∗ )⊗−1 )), u∈W

which equals P (2l, k − n, 1, n) by (6.2.4) and (6.2.7).

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

Proposition 6.2.3. Suppose Φ is of type m ≥ 0, k ≥ 0, and 1 ≤ j ≤ n, deﬁne ⎧ m + j+1 ⎪ 2 ⎪ ⎪ ⎨ m + 2j (6.2.15) t(m, j, n) = ⎪ m + 1 + 2n−j ⎪ 2 ⎪ ⎩ m + 1 + 2n−j−1 2

53 29

Bn with n ≥ 1 and p > 2n. For if if if if

j j j j

is is is is

odd and m is odd, even and m is even, even and m is odd, odd and m is even.

Then P (m, k, j, n) = 0 whenever k < t(m, j, n). Proof. By Lemma 6.2.2(c), P (m, k, j, n) = 0 if k < m + 1. We will prove the slightly stronger statement in the proposition inductively. To do so, we make some general observations. Deﬁne T (m, k, j, n) =

2n

P (m − 1, k − i + 1, j, n)

i=2 2n

+

P (m, k − i + 1, j − 1, n − 1) + P (m, k − n, j, n).

i=1

Observe that by Lemma 6.2.2(d), T (m, k, j, n) = T (m, k, j, n). Note further that if r is the smallest value of k for which T (m, k, j, n) = 0, then P (m − 1, r − 1, j, n) + P (m, r, j − 1, n − 1) + P (m, r − n, j, n) = 0. Suppose that P (m − 1, k, j, n) = 0 whenever k < t(m − 1, j, n) and that P (m, k, j − 1, n − 1) = 0 whenever k < t(m, j − 1, n − 1), then one could conclude that (6.2.16) T (m, k, j, n) = 0 whenever k < min{t(m − 1, j, n) + 1, t(m, j − 1, n − 1), m + 1 + n}. Moreover, parts (d) and (e) of Lemma 6.2.2 would imply that, for 2 ≤ j ≤ n − 1, (6.2.17)

P (m, k, j + 1, n) + P (m, k, j − 1, n) = 0 whenever T (m, k, j, n) = 0,

and from Lemma 6.2.2(f) (6.2.18)

P (m, k, n, n) + P (m, k, n − 1, n) = 0 whenever T (m, k, n, n) = 0.

In order to prove the proposition, we will use induction on n and or j. If n = 1 the claim follows from part (c) of Lemma 6.2.2. Moreover, parts (c) and (d) of the Lemma 6.2.2 imply that the claim holds for j = 1 and n ≥ 1. Step 1: Here we will show that P (m, k, j, n) = 0, whenever k < t(m, j, n) and m + j is odd. We will use induction on j. Assumption: P (m, k, l, n) = 0, whenever k < t(m, l, n), m + l is odd, and l ≤ j. Suppose that m + j + 1 is odd. Then m + j − 1 is also odd and the induction assumption implies that (6.2.16) holds. Together with (6.2.17) one obtains P (m, k, j+1, n) = 0 whenever k < min{t(m−1, j, n)+1, t(m, j−1, n−1), m+1+n}. It suﬃces therefore to verify that (6.2.19)

t(m, j + 1, n) ≤ min{t(m − 1, j, n) + 1, t(m, j − 1, n − 1), m + 1 + n}.

54 30

CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

From (6.2.15) it follows that = m + n − 2j m + 1 + 2n−(j+1)−1 2 t(m, j + 1, n) = m + 1 + 2n−(j+1) = m + n − j−1 2 2

if j is even and m is even, if j is odd and m is odd,

while

m+ t(m − 1, j, n) + 1 = m+ and

t(m, j−1, n−1) =

1 = m + n − 2j + 1 + 1 = m + n − j−1 2

2n−j + 2 2n−j−1 2

m+1+ m+1+

2n−2−(j−1)−1 2 2n−2−(j−1) = 2

= m + n − 2j m + n − j−1 2

if j is even and m is even, if j is odd and m is odd, if j is even and m is even, if j is odd and m is odd.

Inequality (6.2.19) indeed holds and Step 1 is complete. Step 2: Here we will show that P (m, k, n, n) = 0, whenever k < t(m, n, n) and m + n is even. Suppose that m + n is even. Step 1 implies that (6.2.16) holds for j = n. Together with (6.2.18) one obtains P (m, k, n, n) = 0 whenever k < min{t(m − 1, n, n) + 1, t(m, n − 1, n − 1), m + 1 + n}. It suﬃces therefore to verify that t(m, n, n) ≤ min{t(m − 1, n, n) + 1, t(m, n − 1, n − 1), m + 1 + n}. This can easily be done by looking at (6.2.15). It is left to the interested reader. Step 3: Here we will show that P (m, k, j, n) = 0 whenever k < t(m, j, n) and m + j is even. We use induction on n and on j. For j we work in decreasing order. The case j = n was settled above. Assumption: We assume that P (m, k, l, n − 1) = 0 whenever k < t(m, l, n − 1). In addition, we assume that P (m, k, l, n) = 0 whenever k < t(m, l, n), m + l is even, and l ≥ j. Suppose that m + j − 1 is even. The induction assumptions imply that (6.2.16) holds. By (6.2.17) one obtains P (m, k, j−1, n) = 0 whenever k < min{t(m−1, j, n)+1, t(m, j−1, n−1), m+1+n}. It suﬃces therefore to verify that (6.2.20)

t(m, j − 1, n) ≤ min{t(m − 1, j, n) + 1, t(m, j − 1, n − 1), m + 1 + n}.

From (6.2.15) one obtains:

m+ m+

j−1 2 j 2

m+ t(m − 1, j, n) + 1 = m+

j−1 2 j 2

t(m, j − 1, n) = while

and t(m, j − 1, n − 1) =

m+ m+

if j is odd and m is even, if j is even and m is odd. +1

j−1 2 j 2

if j is odd and m is even, if j is even and m is odd, if j is odd and m is even, if j is even and m is odd.

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

This proves inequality (6.2.20).

55 31

Theorem 6.2.4. Suppose Φ is of type Bn with n ≥ 2. Assume that p > 2n. Let λ = pω1 + w · 0 be a dominant weight. Then (a) Hi (G, H 0 (λ) ⊗ H 0 (λ)(1) ) = 0 for 0 < i < 2p − 2, whenever (w) is even; (b) Hi (G, H 0 (λ) ⊗ H 0 (λ)(1) ) = 0 for 0 < i < 2p − 3, whenever (w) is odd; (c) H2p−3 (G(Fp ), k) = 0. Proof. The set of dominant weights of the form λ = pω1 + w · 0, written in the -basis, are (p − (w) − 1)1 + (1 + · · · + (w)+1 ), with 0 ≤ (w) ≤ n − 1, and (p − (w) − 1)1 + (1 + · · · + 2n−(w) ), with n ≤ (w) ≤ 2n − 1.

Using Proposition 6.2.3 and Lemma 6.2.2(a), a direct computation shows that (−1)(u) Pk (u · ((p − (w) − 1)1 + (1 + · · · + (w)+1 )) − 1 ) = 0 whenever k < t,

u∈W

where

t=

and

(p − 1) − (p − 1) −

(w) 2 (w)+1 2

for 0 ≤ (w) ≤ n − 1 and (w) even, for 0 ≤ (w) ≤ n − 1 and (w) odd,

(−1)(u) Pk (u · ((p − (w) − 1)1 + (1 + .... + 2n−(w) )) − 1 ) = 0 whenever k < t,

u∈W

where

(p − 1) − t= (p − 1) −

(w) 2 (w)+1 2

for n ≤ (w) ≤ 2n − 1 and (w) even, for n ≤ (w) ≤ 2n − 1 and (w) odd.

Parts (a) and (b) follow from Proposition 2.7.1. Note that i = 2k + (w). Let λ be the lowest dominant weight of the form pω1 + w · 0. Then λ = (p − 2n + 1)1 and (w) = 2n − 1. We will show that (−1)(u) Pp−n−1 (u · ((p − 2n + 1)1 ) − 1 ) = 0. (6.2.21) u∈W

By Lemma 6.2.2(a) this is equivalent to showing that P (p − 2n − 1, p − n − 1, 1, n) is not zero. Lemma 6.2.2(b) and (h) imply that (6.2.22)

P (2l + 1, k, 1, n) = P (2l, k − n, 1, n) + P (2l + 1, k, 1, n − 1).

Note that (6.2.22) also holds for l = −1. Obviously P (2l − 1, 2l, 1, 1) = 1. It follows inductively from (6.2.22) that P (2l − 1, 2l, 1, n) = 0 for all n ≥ 0, l ≥ 0. From Lemma 6.2.2(g) one obtains now that P (2l, 2l + n, 1, n) = 0. Setting 2l = p − 2n − 1 yields P (p − 2n − 1, p − n − 1, 1, n) = 0. Hence, (6.2.21) holds. In Proposition 2.7.1, i = 2k − (w) = 2(p − n − 1) + 2n − 1 = 2p − 3 and one obtains H2p−3 (G, H 0 (λ) ⊗ H 0 (λ)(1) ) = 0. The weight λ is the lowest non-zero weight in its linkage class. Part (c) of the theorem follows now from the discussion after Theorem 2.5.1.

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6.3. Type B3 . Let Φ be of type B3 with p > h = 6 (so p ≥ 7). From the discussion in Section 6.1 and Theorem 6.2.4, in order to have Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 3, we must have λ = pω3 + w · 0 for w ∈ W . With the aid of MAGMA [BC, BCP] or other software, one can explicitly compute all w ·0 and determine which resulting λ are dominant. Further, λ − ω3 must be a weight i−(w) of S 2 (u∗ ). By direct computation, one can determine the least possible value of k for which λ − ω3 can be a weight of S k (u∗ ). The following table summarizes the weights which can give a value of i < 2p − 6. λ = pω3 + w · 0 (p − 6)ω3 + 2ω2 (p − 6)ω3 + ω1 (p − 6)ω3

(w) k i = 2k + (w) 3 p−5 2p − 7 5 p−6 2p − 7 6 p−7 2p − 8

Lemma 6.3.1. Suppose that Φ is of type B3 with p ≥ 7. Let λ = pμ + w · 0 ∈ X(T )+ with μ ∈ X(T )+ and w ∈ W . Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 8. If H2p−8 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 6)ω3 . H2p−8 (G, H 0 ((p − 6)ω3 ) ⊗ H 0 ((p − 6)ω3∗ )(1) ) = k. If H2p−7 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 6)ω3 + ω1 or λ = (p − 6)ω3 + 2ω2 . (e) H2p−8 (G(Fp ), k) = k.

(a) (b) (c) (d)

Proof. Parts (a), (b), and (d) follow from the discussion preceding the lemma. Part (c) follows from Proposition 2.7.1 and Lemma 6.3.3 below with m = p − 7. Since the weights in part (d) are larger than (p − 6)ω3 , by Theorem 2.5.1 and Theorem 6.2.4, we obtain part (e). Remark 6.3.2. The weights in part (d) also appear to give cohomology classes as veriﬁed for p = 7, 11, 13 by computer. For p = 7, λ = (p − 6)ω3 + ω1 gives a one-dimensional cohomology group. But for p = 11, 13, one gets a two-dimensional cohomology group. For all three primes, the weight λ = (p − 6)ω3 + 2ω2 gives a one-dimensional cohomology group. Lemma 6.3.3. Suppose that Φ is of type B3 . Let m ≥ 0 be an even integer. Then (−1)(u) Pm (u · ((m + 1)ω3 ) − ω3 ) = 1. u∈W

Proof. Let n be such that m = 2n. For n = 0, the claim readily follows, so we assume that n ≥ 1.1 We work with the epsilon basis for the root system. Then the positive roots are 1 , 2 , 3 , 1 + 2 , 1 + 3 , 2 + 3 , 1 − 2 , 1 − 3 , and 2 − 3 . Further ω3 = 12 (1 + 2 + 3 ). Relative to the basis, for any u ∈ W , u(i ) = ±j . That is, u permutes the i up to a sign. For u ∈ W , set xu := u·((m+1)ω3 )−ω3 . Using the fact that 2ρ = 51 +32 +3 , one ﬁnds that (6.3.1)

xu = u((n + 3)1 + (n + 2)2 + (n + 1)3 ) − 31 − 22 − 1 .

1 Indeed, for small values of n the claim can be veriﬁed by hand, and it has been veriﬁed for n ≤ 6 using MAGMA.

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By direct calculation, one can identify all u ∈ W for which xu is a positive root. There are twelve such elements which are summarized in the following table (using permutation notation) along with the parity of their lengths. An element marked with a superscript negative sign denotes the operation which consists of the given permutation of the i s followed by sending 3 to −3 . For example, let u = (123)− . Then u(1 ) = 2 , u(2 ) = −3 , and u(3 ) = 1 . u (u)

(1) even

(12) odd

(13) odd

(23) odd

(123) even

(132) even

(1)− odd

(12)− even

(13)− even

(23)− even

(123)− odd

(132)− odd

For these twelve u, using (6.3.1), one can explicitly compute xu . The values are summarized in the following table. Recall that m = 2n. For small values of n, some of these cannot be sums of positive roots. The necessary condition on n is given in the third column. u (1) (12) (13) (23) (123) (132) (1)− (12)− (13)− (23)− (123)− (132)−

xu := u · ((m + 1)ω3 ) − ω3 n1 + n2 + n3 (n − 1)1 + (n + 1)2 + n3 (n − 2)1 + n2 + (n + 2)3 n1 + (n − 1)2 + (n + 1)3 (n − 2)1 + (n + 1)2 + (n + 1)3 (n − 1)1 + (n − 1)2 + (n + 2)3 n1 + n2 − (n + 2)3 (n − 1)1 + (n + 1)2 − (n + 2)3 (n − 2)1 + n2 − (n + 4)3 n1 + (n − 1)2 − (n + 3)3 (n − 2)1 + (n + 1)2 − (n + 3)3 (n − 1)1 + (n − 1)2 − (n + 4)3

positive root sum n≥1 n≥1 n≥2 n≥1 n≥2 n≥1 n≥2 n≥2 n≥6 n≥4 n≥4 n≥6

We need to compute the appropriate alternating sum of the values of P2n (xu ) for these twelve values of u. We show below that there are four pairs of us for which the lengths have opposite parity and the values of P2n (xu ) are the same. Hence, those cancel each other out. We will further show that there is also a relationship between the remaining partition functions that will lead to the desired claim. To see these relationships, we make a few observations whose proofs are left to the interested reader. Observation 6.3.4. Let x = a1 1 +a2 2 +a3 3 with a1 +a2 +a3 = 3n. Suppose that x is expressed as a sum of 2n positive roots. (a) At least n of the roots have the form i +j (not necessarily all the same). (b) For any pair i, j ∈ {1, 2, 3} (with i = j), if ai + aj = 2n + c, then the root sum decomposition contains at least c copies of i + j . Observation 6.3.5. Let x = a1 1 + a2 2 − a3 3 with 1 ≤ a1 , a2 < a3 and a1 + a3 > 2n. Suppose that x is expressed as a sum of 2n positive roots. Then at least one of the roots is 1 − 3 and at least one is 2 − 3 . We now identify the four pairs (of opposite parity) where P2n (xu ) is the same. Case 1. (13) and (132).

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If n = 1, as noted above, x(13) = −1 + 2 + 33 cannot be expressed as a sum of positive roots. On the other hand, x(123) = 33 can be. However, it cannot be expressed as a sum of 2n = 2 positive roots. So we assume n ≥ 2. If x(13) is expressed as a sum of 2n positive roots, by Observation 6.3.4(b), at least one of the roots is 2 +3 (in fact, at least two). Hence, P2n (x(13) ) = P2n−1 (x(13) −(2 +3 )) = P2n−1 ((n − 2)1 + (n − 1)2 + (n + 1)3 ). Similarly, if x(132) is expressed as a sum of 2n positive roots, at least one of the roots is 1 + 3 (and one is 2 + 3 ). Hence, P2n (x(132) ) = P2n−1 ((n − 2)1 + (n − 1)2 + (n + 1)3 ) = P2n (x(13) ). For the remaining three pairs, if n is not suﬃciently large for xu to admit a positive root sum decomposition, then P2n (xu ) = 0 in both cases. So we assume in what follows that n is suﬃciently large to admit a root sum decomposition. Case 2. (1)− and (12)− . If x(1)− is expressed as a sum of 2n positive roots, by Observation 6.3.5, at least one of the roots is 1 − 3 . Hence, removing this root, P2n (x(1)− ) = P2n−1 ((n − 1)1 + n2 − (n + 1)3 ). Similarly, again by Observation 6.3.5, if x(12)− is expressed as a sum of 2n positive roots, then at least one of the roots is 2 − 3 . Hence, P2n (x(12)− ) = P2n−1 ((n − 1)1 + n2 − (n + 1)3 ) = P2n (x(1)− ). Case 3. (13)− and (132)− . Similar to Case 2, by removing an 2 − 3 for (13)− and removing an 1 − 3 for (132)− , one ﬁnds that P2n (x(12)− ) = P2n−1 ((n − 2)1 + (n − 1)2 − (n + 3)3 ) = P2n (x(132)− ). Case 4. (23)− and (123)− . Again, similar to Case 2 with a slight generalization of Observation 6.3.5, by removing two copies of 1 − 3 for (23)− and two copies of 2 − 3 for (123)− , one ﬁnds that P2n (x(23)− ) = P2n−2 ((n − 2)1 + (n − 1)2 − (n + 1)3 ) = P2n (x(123)− ). From Cases 1-4, we have that (6.3.2) (−1)(u) Pm (u·((m+1)ω3 )−ω3 ) = P2n (x(1) )−P2n (x(12) )−P2n (x(23) )+P2n (x(123) ). u∈W

We now deduce several relationships among the terms on the right hand side. If n = 1, only the ﬁrst three terms can be non-zero, and one can readily check that the claim holds. So we assume that n ≥ 2. Note that the following argument does still hold even when n = 1. Consider the identity element (1). Write P2n (x(1) ) = M1 + M2 + M3 where M1 denotes the number of root sum decompositions which contain an 1 + 2 , M2 denotes the number which contain an 1 + 3 but not an 1 + 2 , and M3 denotes the number which contain neither an 1 + 2 nor an 1 + 3 . For M1 , by assumption, the decomposition contains an 1 + 2 . Removing this root gives (6.3.3)

M1 = P2n−1 ((n − 1)1 + (n − 1)2 + n3 ).

For M2 , by assumption, the decomposition contains an 1 + 3 . Removing this root gives (6.3.4)

∗ M2 = P2n−1 ((n − 1)1 + n2 + (n − 1)3 ),

where P ∗ denotes the fact that we are only counting decompositions which contain no copies of 1 + 2 . By assumption, M3 is the number of root decompositions of n1 + n2 + n3 (into 2n positive roots) which do not contain an 1 + 2 nor

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an 1 + 3 . By Observation 6.3.4(a), any such decomposition contains at least n copies of 2 + 3 . Removing those leaves n1 which must be expressed as a sum of n positive roots without using 1 + 2 nor 1 + 3 . There is clearly only one such decomposition (using n copies of 1 ). Hence, M3 = 1. Consider now the word (12). Write P2n (x(12) ) = N1 + N2 where N1 denotes the number of root sum decompositions which contain at least one copy of 1 + 2 and N2 denotes the number which do not contain an 1 + 2 . In the ﬁrst case, by removing an 1 + 2 , we have N1 = P2n−1 ((n − 2)1 + n2 + n3 ).

(6.3.5)

In the second case (as well as the ﬁrst case), by Observation 6.3.4(b), any decomposition must include an 2 + 3 . Removing that, we see that ∗ N2 = P2n−1 ((n − 1)1 + n2 + (n − 1)3 ) = M2

from (6.3.4). From Observation 6.3.4(b), by removing an 1 + 3 , P2n (x(23) ) = P2n−1 ((n − 1)1 + (n − 1)2 + n3 ) = M1 , where the second equality follows from (6.3.3). From Observation 6.3.4(b), by removing an 2 + 3 , P2n (x(123) ) = P2n−1 ((n − 2)1 + n2 + n3 ) = N1 , where the second equality follows from (6.3.5)2 . From (6.3.2) and the preceding relationships, we have (−1)(u) Pm (u · ((m + 1)ω3 ) − ω3 ) u∈W

= P2n (x(1) ) − P2n (x(12) ) − P2n (x(23) ) + P2n (x(123) ) = M1 + M2 + M3 − N1 − N2 − P2n (x(23) ) + P2n (x(123) ) = M1 + M2 + 1 − N1 − M2 − M1 + N1 = 1 as claimed.

6.4. Type B4 . Let Φ be of type B4 with p > h = 8 (so p ≥ 11). As discussed in Section 6.3 for type B3 , in order to have Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 3, we must have λ = pω4 + w · 0 for w ∈ W . Again, by direct computation with MAGMA, the following table summarizes the weights which can give a value of i < 2p − 6. λ = pω4 + w · 0 (p − 8)ω4 + 2ω2 (p − 8)ω4 + ω1 (p − 8)ω4

(w) k i = 2k + (w) p−7 7 2p − 7 p−8 9 2p − 7 p − 9 10 2p − 8

Lemma 6.4.1. Suppose that Φ is of type B4 with p ≥ 11. Let λ = pμ + w · 0 ∈ X(T )+ with μ ∈ X(T )+ and w ∈ W . (a) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 8. 2 If

n = 1, x(123) cannot be expressed as a sum of positive roots. However, in that case, N1 is necessarily zero, and so we still have P2n (x(123) ) = N1 .

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(b) If H2p−8 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 8)ω4 . (c) H2p−8 (G, H 0 ((p − 8)ω4 ) ⊗ H 0 ((p − 8)ω4∗ )(1) ) = k. (d) If H2p−7 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 8)ω4 + ω1 or λ = (p − 8)ω4 + 2ω2 . (e) H2p−8 (G(Fp ), k) = k. Proof. Parts (a), (b), and (d) follow from the discussion preceding the lemma. Part (c) follows from Proposition 2.7.1 and Lemma 6.4.2 below with m = p − 9. Since the weights in part (d) are larger than (p − 8)ω4 , by Theorem 2.5.1 and Theorem 6.2.4, we obtain part (e). Lemma 6.4.2. Suppose that Φ is of type B4 . Let m ≥ 0 be an even integer. Then (−1)(u) Pm (u · ((m + 1)ω4 ) − ω4 ) = 1. u∈W

Proof. The arguments to follow are quite similar to those in the proof of Lemma 6.3.3. Let n be such that m = 2n. For n = 0, the claim readily follows, so we assume that n ≥ 1. As with the proof of Lemma 6.3.3, we work with the epsilon basis for the root system. Then the positive roots are 1 , 2 , 3 , 4 , 1 + 2 , 1 + 3 , 1 + 4 , 2 + 3 , 2 + 4 , 3 + 4 , 1 − 2 , 1 − 3 , 1 − 4 , 2 − 3 , 2 − 4 , and 3 − 4 . Further ω4 = 12 (1 + 2 + 3 + 4 ). Relative to the basis, for any u ∈ W , u(i ) = ±j . That is, u permutes the i up to a sign. For u ∈ W , let xu := u · ((m + 1)ω4 ) − ω4 . Using the fact that 2ρ = 71 + 52 + 33 + 4 , one ﬁnds that (6.4.1) xu = u((n + 4)1 + (n + 3)2 + (n + 2)3 + (n + 1)4 ) − 41 − 32 − 23 − 1 . By direct calculation, one ﬁnds that if u sends any i to −j (any j), then xu is either not expressible as a sum of positive roots or requires at least 2n + 1 roots to do so. Therefore, the only u that can contribute to the alternating sum under consideration are those u for which u(i ) = j . That is, u is simply one of the 24 permutations of the i s. Let u ∈ S4 ⊂ W . From (6.4.1), one ﬁnds that xu = a1 1 + a2 2 + a3 3 + a4 4 where a1 + a2 + a3 + a4 = 4n. Since the positive roots are of the form i , i + j , or i − j , for this to be expressed as a sum of 2n roots, each such root must be of the form i + j . That is the other two types of roots are not allowable. Similar to the arguments in the proof of Lemma 6.3.3, one can further see the following. Observation 6.4.3. Suppose that a1 1 + a2 2 + a3 3 + a4 4 is expressed as a sum of 2n positive roots where a1 + a2 + a3 + a4 = 4n. For any pair i, j ∈ {1, 2, 3, 4} (with i = j), if ai + aj = 2n + c, then the root sum decomposition contains at least c copies of i + j . Using Observation 6.4.3, by direct calculation, one can show that the 18 permutations u for which u(1 ) = 1 can be separated into 9 pairs of opposite parity having equal values of P2n (xu ). Hence the terms for those values of u cancel in the alternating sum. For example, consider the permutations (12) and (12)(43) of opposite parity. From (6.4.1), x(12) = (n − 1)1 + (n + 1)2 + n3 + n4 . By Observation 6.4.3, a decomposition of x(12) must contain at least one copy of 2 + 3 (as well as a copy of 2 + 4 ). Subtracting that root shows that P2n (x(12) ) = P2n−1 ((n − 1)1 + n2 + (n − 1)3 + n4 ).

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On the other hand, x(12)(34) = (n − 1)1 + (n + 1)2 + (n − 1)3 + (n + 1)4 . Here, x(12)(34) must contain a copy of 2 + 4 (in fact, at least two copies). Subtracting this root gives P2n (x(12)(34) ) = P2n−1 ((n − 1)1 + n2 + (n − 1)3 + n4 ) = P2n (x(12) ). The eight other pairings (which may not be unique) are (13) with (13)(24); (14) with (14)(23); (123) with (1243); (132) with (1342); (124) with (1234); (142) with (1432); (134) with (1324); and (143) with (1423). We leave the details to the interested reader. That leaves the six values of u for which u(1 ) = 1 : (1), (23), (24), (34), (234), and (243). However, as above, one can show that P2n (x(24) ) = P2n (x(243) ). So those terms cancel as well and we are reduced to (−1)(u) Pm (u·((m+1)ω4 )−ω4 ) = P2n (x(1) )−P2n (x(23) )−P2n (x(34) )+P2n (x(234) ). u∈W

From (6.4.1), we have x(1) = n1 + n2 + n3 + n4 . Write P2n (x(1) ) = M1 + M2 + M3 where M1 denotes the number of root sum decompositions which contain at least one copy of 2 + 3 , M2 denotes the number which contain no copies of 2 + 3 but contain at least one copy of 1 + 2 , and M3 denotes the number which contain neither an 2 + 3 nor an 1 + 2 . By assumption, subtracting a copy of 2 + 3 , we have (6.4.2)

M1 = P2n−1 (n1 + (n − 1)2 + (n − 1)3 + n4 ).

For M2 , subtracting a copy of 1 + 2 gives (6.4.3)

∗ ((n − 1)1 + (n − 1)2 + n3 + n4 ), M2 = P2n−1

where the P ∗ denotes the fact that the sum is only over those decompositions which do not contain a copy of 2 + 3 . For M3 , in order to get the n2 appearing in x(1) , there must be exactly n copies of 2 + 4 . But then the remaining n factors must all be 1 + 3 . In other words, M3 = 1. From (6.4.1), we have x(23) = n1 +(n−1)2 +(n+1)3 +n4 . Write P2n (x(23) ) = N1 + N2 where N1 denotes the number of root sum decompositions which contain at least one copy of 2 + 3 and N2 denotes the number which contain no copies of 2 + 3 . Subtracting a copy of 2 + 3 , we have (6.4.4)

N1 = P2n−1 (n1 + (n − 2)2 + n3 + n4 ).

For N2 , by Observation 6.4.3, any decomposition of x(23) contains at least one copy of 1 + 3 (as well as a copy of 3 + 4 ). Subtracting the 1 + 3 gives ∗ ((n − 1)1 + (n − 1)2 + n3 + n4 ) = M2 N2 = P2n−1

from (6.4.3). From (6.4.1), we have x(34) = n1 +n2 +(n−1)3 +(n+1)4 . From Observation 6.4.3, any decomposition of x(34) contains at least one copy of 2 + 4 (as well as a copy of 1 + 4 ). Subtracting the 2 + 4 gives P2n (x(34) ) = P2n−1 (n1 + (n − 1)2 + (n − 1)3 + n4 ) = M1 from (6.4.2). From (6.4.1), we have x(234) = n1 + (n − 2)2 + (n + 1)3 + (n + 1)4 . From Observation 6.4.3, any decomposition of x(234) contains at least one copy of 3 + 4

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(in fact, at least two copies). Subtracting this gives P2n (x(234) ) = P2n−1 (n1 + (n − 2)2 + n3 + n4 ) = N1 from (6.4.4). In summary, we have (−1)(u) Pm (u · ((m + 1)ω4 ) − ω4 ) u∈W

= P2n (x(1) ) − P2n (x(23) ) − P2n (x(34) ) + P2n (x(234) ) = M1 + M2 + M3 − N1 − N2 − P2n (x34 ) + P2n (x234 ) = M1 + M2 + 1 − N1 − M2 − M1 + N1 = 1 as claimed.

6.5. Type B5 . Let Φ be of type B5 with p > h = 10 (so p ≥ 11). As discussed in Section 6.3 for type B3 , in order to have Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p−3, we must have λ = pω5 +w·0 for w ∈ W . Speciﬁcally, substituting n = 5 into (6.1.1) gives p − 25 . (6.5.1) i ≥ 2p − 2 We obtain the following. Lemma 6.5.1. Suppose that Φ is of type B5 with p ≥ 11. Let λ = pω5 + w · 0 ∈ X(T )+ with w ∈ W . (a) If p = 17 or p ≥ 23, then Hi (G, H 0 (λ)⊗H 0 (λ∗ )(1) ) = 0 for 0 < i ≤ 2p−3. (b) Suppose p = 11. Then (i) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 7; (ii) if H2p−7 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 10)ω5 = ω5 ; (iii) H2p−7 (G, H 0 (ω5 ) ⊗ H 0 (ω5∗ )(1) ) = k; (iv) if H2p−6 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = ω1 + ω5 or λ = 2ω2 + ω5 ; (v) H2p−6 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = k for λ = ω1 + ω5 or λ = 2ω2 + ω5 ; (vi) H2p−7 (G(Fp ), k) = k. (c) Suppose p = 13. Then (i) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 5; (ii) if H2p−5 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 10)ω5 = 3ω5 ; (iii) H2p−5 (G, H 0 (3ω5 ) ⊗ H 0 (3ω5∗ )(1) ) = k; (iv) if H2p−4 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = ω1 + 3ω5 or λ = 2ω2 + 3ω5 ; (v) dim H2p−4 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 2 for λ = ω1 + 3ω5 ; (vi) H2p−4 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = k for λ = 2ω2 + 3ω5 ; (vii) H2p−5 (G(Fp ), k) = k. (d) Suppose p = 19. Then (i) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 3; (ii) if H2p−3 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 10)ω5 = 9ω5 ; (iii) dim H2p−3 (G, H 0 (9ω5 ) ⊗ H 0 (9ω5∗ )(1) ) = 15. Proof. For p ≥ 23, part (a) follows from (6.5.1). Parts (b)(i)-(v), (c)(i)-(vi), and (d) as well as part (a) for p = 17 follow by explicitly computing (with the aid of MAGMA) all possible w · 0, and then computing partition functions by hand or

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with the aid of MAGMA. For p = 11, since the weights in part (b)(iv) are larger than that in (b)(ii), part (b)(vi) follows from Theorem 2.5.1 and Theorem 6.2.4. Similarly, part (c)(vii) follows. 6.6. Type B6 . Let Φ be of type B6 with p > h = 12 (so p ≥ 13). As discussed in Section 6.3 for type B3 , in order to have Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 3, we must have λ = pω6 + w · 0 for w ∈ W . Recall the arguments in Section 6.1. Speciﬁcally, substituting n = 6 into (6.1.1) gives (6.6.1)

i ≥ 2p + (p − 18).

We obtain the following. Lemma 6.6.1. Suppose that Φ is of type B6 with p ≥ 13. Let λ = pω6 + w · 0 ∈ X(T )+ with w ∈ W . (a) If p ≥ 17, then Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i ≤ 2p − 3. (b) Suppose p = 13. Then (i) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 5; (ii) if H2p−5 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = (p − 12)ω6 = ω6 ; (iii) H2p−5 (G, H 0 (ω6 ) ⊗ H 0 (ω6∗ )(1) ) = k; (iv) if H2p−4 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, then λ = ω1 + ω6 or λ = 2ω2 + ω6 ; (v) H2p−4 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = k for λ = ω1 + ω6 or λ = 2ω2 + ω6 ; (vi) H2p−5 (G(Fp ), k) = k. Proof. Part (a) follows from (6.6.1). Parts (b)(i)-(v) follow by explicitly computing (with the aid of MAGMA) all possible w · 0, and then computing partition functions by hand or with the aid of MAGMA. Since the weights in part (b)(iv) are larger than that in part (b)(ii), part (b)(vi) follows from Theorem 2.5.1 and Theorem 6.2.4. 6.7. Summary for type B. Theorem 6.7.1. Suppose Φ is of type Bn with n ≥ 3. Assume that p > 2n. (a) If n ≥ 7 or p > 13 when n ∈ {5, 6}, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 3; (ii) H2p−3 (G(Fp ), k) = 0. (b) If n ∈ {5, 6} and p = 13, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 5; (ii) H2p−5 (G(Fp ), k) = k. (c) If n = 5 and p = 11, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 7; (ii) H2p−7 (G(Fp ), k) = k. (d) If n ∈ {3, 4}, then (i) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 8; (ii) H2p−8 (G(Fp ), k) = k. Proof. This follows from the discussion in Section 6.1, Theorem 6.2.4, Lemma 6.3.1, Lemma 6.4.1, Lemma 6.5.1, and Lemma 6.6.1.

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7. Type G2 Assume throughout this section that Φ is of type G2 and that p > h = 6 (so p ≥ 7). Following Section 2, our goal is to ﬁnd the least i > 0 such that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )) = 0 for some λ ∈ X(T )+ . 7.1. Restrictions. Suppose that Hi (G, H 0 (λ)⊗H 0 (λ∗ )(1) ) = 0 for some i > 0 and λ = pμ + w · 0 with μ ∈ X(T )+ and w ∈ W . From Proposition 2.8.1(c), i ≥ (p − 1)μ, α ˜ ∨ − 1. Consider the two fundamental weights ω1 and ω2 . Note that ω1 = α0 and ω2 = α. ˜ Furthermore, we have ω1 , α ˜ ∨ = 1 and ω2 , α ˜ ∨ = 2. ∨ Therefore, unless μ = ω1 = α0 , we have μ, α ˜ ≥ 2 and i ≥ 2p − 3. Suppose now that λ = pω1 + w · 0 for some w ∈ W . In order to have Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0, as discussed in Section 2.7, λ must be dominant and λ − ω1 must be a weight of S j (u∗ ) for some j. In other words, λ − ω1 must be expressible as a non-negative linear combination of positive roots. By direct calculation (by hand or with the aid of MAGMA), one can identify all possible λ. These are listed in the following table. As usual, si := sαi and e is the identity element. w e s1 s1 s2 s1 s2 s1 s1 s2 s1 s2 s1 s2 s1 s2 s1

(w) 0 1 2 3 4 5

λ = pω1 + w · 0 pω1 (p − 2)ω1 + ω2 (p − 5)ω1 + 2ω2 (p − 6)ω1 + 2ω2 (p − 6)ω1 + ω2 (p − 5)ω1

Note that each λ has the form λ = aω1 + bω2 for a ≥ 1 and 0 ≤ b ≤ 2. From Proposition 2.7.1, we know that for λ = pμ + w · 0, dim Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = (−1)(u) P i−(w) (u · λ − μ). u∈W

2

In the next section, we consider such partition functions. Since the prime p does not per se play a role in the partition function computations, we will work in a general setting. 7.2. Partitions I. Let λ = aω1 + bω2 for a ≥ 1 and 0 ≤ b ≤ 2. From the previous section, our goal is to make computations of (7.2.1) (−1)(u) Pk (u · λ − ω1 ). u∈W

In particular, we will identify the least value of k for which this sum is nonzero along with the value of the sum in that case. See Proposition 7.3.3 and Proposition 7.4.4. In order for Pk (u · λ − ω1 ) to be non-zero, u · λ − ω1 must lie in the positive (more precisely, non-negative) root lattice. By direct computation, one ﬁnds that there are only four elements u ∈ W for which this is true (under our assumptions on a and b above). This is summarized in the following table. The value of u·λ−ω1 is given in the root basis.

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

u e s1 s2 s1 s2

65 41

(u) u · λ − ω1 0 (2a + 3b − 2)α1 + (a + 2b − 1)α2 1 (a + 3b − 3)α1 + (a + 2b − 1)α2 1 (2a + 3b − 2)α1 + (a + b − 2)α2 2 (a − 6)α1 + (a + b − 2)α2

Note that in some of the cases a must be suﬃciently large in order for u · λ − ω1 to lie in the positive root lattice. Speciﬁcally, for s1 , one needs a ≥ 3 or b ≥ 1; for s2 , one needs a ≥ 2 or a ≥ 1 and b ≥ 1 or b ≥ 2; and for s1 s2 , one needs a ≥ 6. 7.3. Partitions II. As noted in Section 7.2, our goal is to ﬁnd the least value of k such that the sum (7.2.1) is non-zero. In this section, we notice some relationships among the partition functions which will allow us to identify a range under which the sum is zero. Lemma 7.3.1. Let λ = aω1 + bω2 with a ≥ 3 and 0 ≤ b ≤ 2. Suppose that k ≤ a + b − 2. Then Pk (λ − ω1 ) = Pk (s2 · λ − ω1 ). Proof. Recall the table in Section 7.2, and set γ1 := λ − ω1 = (2a + 3b − 2)α1 + (a + 2b − 1)α2 , γ2 := s2 · λ − ω1 = (2a + 3b − 2)α1 + (a + b − 2)α2 . Consider a decomposition of γ1 into k not necessarily distinct positive roots. Since a + 2b − 1 = (a + b − 2) + (b + 1) and k ≤ a + b − 2, at least b + 1 of those roots must contain 2α2 . However, the only root containing 2α2 is α ˜ = 3α1 + 2α2 . Hence, any decomposition of γ1 into k roots must contain at least b + 1 copies of α. ˜ Therefore (7.3.1) Pk (γ1 ) = Pk−b−1 (γ1 −(b+1)(3α1 +2α2 )) = Pk−b−1 ((2a−5)α1 +(a−3)α2 ). Now consider γ2 and the diﬀerence between the number of α1 s and α2 s appearing. Suppose γ2 = (mi α1 + ni α2 ) is expressed as a sum of k positive roots. Then (mi − ni ) = mi − ni = (2a + 3b − 2) − (a + b − 2) = a + 2b. Note that for each i, mi − ni ∈ {−1, 0, 1, 2}. Since k ≤ a + b − 2, for at least b + 1 values of i (in fact, at least b + 2 values), we must have mi − ni = 2. However, the only root where that occurs is 3α1 + α2 . Hence, any decomposition of γ2 into k roots must contain at least b + 1 copies of 3α1 + α2 . Therefore, Pk (γ2 ) = Pk−b−1 (γ2 − (b + 1)(3α1 + α2 )) = Pk ((2a − 5)α1 + (a − 3)α1 ). Combining this with (7.3.1) gives the claim.

Lemma 7.3.2. Let λ = aω1 + bω2 with a ≥ 6. Suppose that k ≤ a + b − 2. Then Pk (s1 · λ − ω1 ) = Pk (s1 s2 · λ − ω1 ). Proof. Recall the table in Section 7.2, and set γ3 := s1 · λ − ω1 = (a + 3b − 3)α1 + (a + 2b − 1)α2 , γ4 := s1 s2 · λ − ω1 = (a − 6)α1 + (a + b − 2)α2 . For γ3 , the same argument as in the preceding lemma gives (7.3.2) Pk (γ3 ) = Pk−b−1 (γ3 − (b + 1)(3α1 + 2α2 )) = Pk−b−1 ((a − 6)α1 + (a − 3)α2 ).

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Now consider γ4 and the diﬀerence between the number of α1 s and α2 s appearing. Suppose γ4 = (mi α1 + ni α2 ) is expressed as a sum of k positive roots. Then (mi − ni ) = mi − ni = (a − 6) − (a + b − 2) = −b − 4. Note that for each i, mi − ni ∈ {−1, 0, 1, 2}. For at least b + 1 values of i (in fact, at least b + 4 values), we must have mi − ni = −1. However, the only root where that occurs is α2 . Hence, any decomposition of γ4 into k roots must contain at least b + 1 copies of α2 . Therefore Pk (γ4 ) = Pk−b−1 (γ2 − (b + 1)α2 ) = Pk−b−1 ((a − 6)α1 + (a − 3)α2 ). Combining this with (7.3.2) gives the claim.

With the two aforementioned lemmas we can now prove the following proposition. Proposition 7.3.3. Let λ = aω1 + bω2 for a ≥ 1 and 0 ≤ b ≤ 2. For k ≤ a + b − 2, (−1)(u) Pk (u · λ − ω1 ) = 0. u∈W

Proof. From the discussion in Section 7.2, (−1)(u) Pk (u·λ−ω1 ) = Pk (λ−ω1 )−Pk (s1 ·λ−ω1 )−Pk (s2 ·λ−ω1 )+Pk (s1 s2 ·λ−ω1 ).

u∈W

For a ≥ 6, the claim follows from Lemma 7.3.1 and Lemma 7.3.2 above. For a < 6, one can see from the proof of Lemma 7.3.2 that the 2nd and fourth terms are zero. Hence, for 3 ≤ a ≤ 5, the result follows from Lemma 7.3.1. For 1 ≤ a ≤ 2, one can see from the proof of Lemma 7.3.1 that both the ﬁrst and third terms vanish, and so the result follows. When a is small, the claim could also be readily veriﬁed by hand. 7.4. Partitions III. Let λ = aω1 + bω2 for a ≥ 1 and 0 ≤ b ≤ 2. The goal of this section is to determine (−1)(u) Pa+b−1 (u · λ − ω1 ). u∈W

See Proposition 7.4.4. From the discussion in Section 7.2 we need to consider the following weights (with notation following Section 7.3): γ1 := λ − ω1 = (2a + 3b − 2)α1 + (a + 2b − 1)α2 , γ2 := s2 · λ − ω1 = (2a + 3b − 2)α1 + (a + b − 2)α2 , γ3 := s1 · λ − ω1 = (a + 3b − 3)α1 + (a + 2b − 1)α2 , γ4 := s1 s2 · λ − ω1 = (a − 6)α1 + (a + b − 2)α2 . More precisely, (7.4.1) (−1)(u) Pa+b−1 (u·λ−ω1 ) = Pa+b−1 (γ1 )−Pa+b−1 (γ2 )−Pa+b−1 (γ3 )+Pa+b−1 (γ4 ). u∈W

We ﬁrst make some reduction observations as done in the proofs in Section 7.3. Note that when a is small, some of the statements are trivially true since both sides

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67 43

are zero. But we include them here (and in the following statements) for simplicity of exposition. Observe also that the right hand side is independent of the value of b. Lemma 7.4.1. Let λ, γ1 , γ2 , γ3 , and γ4 be as above, and let k = a + b − 1. Then (a) (b) (c) (d)

Pk (γ1 ) = Pa−1 (2(a − 1)α1 + (a − 1)α2 ); Pk (γ2 ) = Pa−2 ((2a − 5)α1 + (a − 3)α2 ); Pk (γ3 ) = Pa−1 ((a − 3)α1 + (a − 1)α2 ); Pk (γ4 ) = Pa−2 ((a − 6)α1 + (a − 3)α2 ).

Proof. (a) Suppose that γ1 is decomposed as a sum of k positive roots. Similar to the argument in Lemma 7.3.1, since a + 2b − 1 = (a + b − 1) + b, at least b of those roots must (contain 2α2 and hence) be α ˜ = 3α1 + 2α2 . Hence, Pk (γ1 ) = Pk−b (γ1 − bα), ˜ and the claim follows. (b) Again, as in the proof of Lemma 7.3.1, since the diﬀerence in the number of α1 s and α2 s appearing in γ2 is a + 2b, if γ2 is expressed as k roots, then at least b + 1 of them must be 3α1 + α2 . Hence, Pk (γ2 ) = Pk−b−1 (γ2 − (b + 1)(3α1 + α2 )), and the claim follows. (c) As in part (a), we must have Pk (γ3 ) = Pk−b (γ3 − bα), ˜ and the claim follows. (d) As in part (b), similar to the proof of Lemma 7.3.2, we consider the diﬀerence in the number of α1 s and α2 s appearing in γ4 . Since this number is −b − 4, we can in particular assume that if γ4 is decomposed into k roots, then at least b + 1 of them are α2 . Hence, Pk (γ4 ) = Pk−b−1 (γ4 − (b + 1)α2 ), and the claim follows. With the aid of Lemma 7.4.1, we now observe that there are some relationships among the Pk (γi ). To this end, we introduce a bit of notation. Consider an arbitrary integer k ≥ 0 and weight γ = cα1 + dα2 for c, d ≥ 0. Any decomposition of γ into a sum of k positive roots is of one of two types: either the sum contains at least one copy of α ˜ or it does not contain any copies of α. ˜ Correspondingly, let Pk,α˜ (γ) and Pk,α˜ (γ) denote the number of such root sums. Then Pk (γ) = Pk,α˜ (γ) + Pk,α˜ (γ). Observe that Pk,α˜ (γ) = Pk−1 (γ − α ˜ ).

(7.4.2)

Lemma 7.4.2. Let λ, γ1 , γ2 , γ3 , and γ4 be as above, and let k = a + b − 1. Then (a) Pk (γ1 ) = Pk (γ2 ) + Pa−1,α˜ (2(a − 1)α1 + (a − 1)α2 ); (b) Pk (γ3 ) = Pk (γ4 ) + Pa−1,α˜ ((a − 3)α1 + (a − 1)α2 ). Proof. (a) We have Pk (γ1 ) = Pa−1 (2(a − 1)α1 + (a − 1)α2 )

(by Lemma 7.4.1(a))

= Pa−1,α˜ (2(a − 1)α1 + (a − 1)α2 ) + Pa−1,α˜ (2(a − 1)α1 + (a − 1)α2 ) = Pa−2 ((2a − 5)α1 + (a − 3)α2 ) + Pa−1,α˜ (2(a − 1)α1 + (a − 1)α2 )

(by (7.4.2))

= Pk (γ2 ) + Pa−1,α˜ (2(a − 1)α1 + (a − 1)α2 )

(by Lemma 7.4.1(b)).

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

(b) We have Pk (γ3 ) = Pa−1 ((a − 3)α1 + (a − 1)α2 )

(by Lemma 7.4.1(c))

= Pa−1,α˜ ((a − 3)α1 + (a − 1)α2 ) + Pa−1,α˜ ((a − 3)α1 + (a − 1)α2 ) = Pa−2 ((a − 6)α1 + (a − 3)α2 ) + Pa−1,α˜ ((a − 3)α1 + (a − 1)α2 )

(by (7.4.2))

= Pk (γ4 ) + Pa−1,α˜ ((a − 3)α1 + (a − 1)α2 )

(by Lemma 7.4.1(d)).

From (7.4.1), Lemma 7.4.1 and Lemma 7.4.2, we see that (7.4.3)

(−1)(u) Pa+b−1 (u · λ − ω1 ) = Pa−1,α˜ (2(a − 1)α1 + (a − 1)α2 )

u∈W

− Pa−1,α˜ ((a − 3)α1 + (a − 1)α2 ). Lemma 7.4.3. Let c ≥ 0. Then

c+1 , Pc,α˜ (2cα1 + cα2 ) − Pc,α˜ ((c − 2)α1 + cα2 ) = 3

where x denotes the least integer greater than or equal to x. Proof. Let η1 := 2cα1 + cα2 and η2 := (c − 2)α1 + cα2 . Observe ﬁrst that if c < 2, then Pc,α˜ (η2 ) = 0. On the other hand, we have P0,α˜ (0) = 1 and P1,α˜ (2α1 + α2 ) = 1, and so the claim holds for c < 2. Assume for the remainder of the proof that c ≥ 2. Observe that if ηi is expressed as a sum of c positive roots, none of which are α, ˜ then each root is necessarily of the form aα1 + α2 for a ∈ {0, 1, 2, 3}. So the question of possible decompositions involves looking only at the coeﬃcients of α1 . For nonnegative integers m, n, let Pm (n) denote the number of ways that n can be expressed as a sum of m integers n = n 1 + n2 + · · · + nm where ni ∈ {0, 1, 2, 3}. With this notation, Pc,α˜ (η1 ) = Pc (2c), Pc,α˜ (η2 ) = Pc (c − 2), and our goal is to compute Pc (2c) − Pc (c − 2) (when c ≥ 2). For m, n as above, let Sm (n) denote the set of such partitions of n into m integers. We ﬁrst show that there is an injection ϕ : Sc (c − 2) → Sc (2c). Let τ ∈ Sc (c − 2). Say τ : c − 2 = τ 1 + τ2 + · · · + τc , where τi ∈ {0, 1, 2, 3}. Let s denote the number of τi s which equal 3. The remaining c − s values must sum to c − 2 − 3s, and hence at most c − 2 − 3s of those terms can be non-zero. In other words, at least (c − s) − (c − 2 − 3s) = 2s + 2 of the remaining terms are zero. That is, we may assume that τ has the form: c − 2 = 3 + · · · + 3 + 0 + · · · + 0 +τ3s+3 + · · · + τc , s times

(2s+2) times

where, for (3s + 3) ≤ i ≤ c, τi ∈ {0, 1, 2}. Let ϕ(τ ) be the partition: 2c = 3 + · · · + 3 +2 + 2 + 0 + · · · + 0 +(τ3s+3 + 1) + (τ3s+4 + 1) + · · · + (τc + 1).

2s times

s times

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In words, the map ϕ leaves the initial s copies of 3 ﬁxed, sends s of the zeros to 3, sends two of the zeros to 2, leaves the other s zeros ﬁxed, and adds one to the unknown integers at the end. Note that those unknown integers are each at most 2, so adding one is allowable. One can also readily check that the new sum does indeed add up to 2c. It is clear that ϕ is an injection, but we will explicitly construct an inverse below. Observe that the resulting partition of 2c contains 2 at least twice. We claim that the image of ϕ is in fact precisely the subset X ⊂ Sc (2c) consisting of those partitions where 2 appears two or more times. Indeed, we can deﬁne a function ψ : X → Sc (c − 2) as follows. Let ξ ∈ X and s denote the number of times that zero appears in ξ. The remaining c − s values in ξ must sum to 2c. We know that at least two of those have value 2. The remaining c − s − 2 terms must sum to 2c − 4. Since 2(c − s − 2) = 2c − 4 − 2s, at least 2s of those terms must have value 3. In other words, ξ has the form: 2c = 0 + · · · + 0 +2 + 2 + 3 + · · · + 3 +ξ3s+3 + · · · + ξc , s times

2s times

where (for (3s + 3) ≤ i ≤ c) ξi ∈ {1, 2, 3}. Let ψ(ξ) be the partition: c − 2 = 0 + · · · + 0 + 3 + · · · + 3 +(ξ3s+3 − 1) + (ξ3s+4 − 1) + · · · + (ξc − 1). s times

(2s+2) times

In words, the map ψ leaves the zeros ﬁxed, sends the two 2s to zero, sends s copies of 3 to zero, leaves the other s copies of 3 ﬁxed, and subtracts one from each of the remaining integers. Clearly ψ is an inverse to φ. Hence, Pc (c − 2) = |X|. It remains to compute Pc (2c) − |X|. That is, we need to count the number of partitions 2c = n1 + n2 + · · · + nc , where ni ∈ {0, 1, 2, 3} but for which at most one value of ni = 2. Write c = 3m+t for m ≥ 0 and t < 3. Then it is a straightforward (but somewhat lengthy) computation to show that the number of such partitions is m + 1. This is left to the interested reader. The lemma follows. Applying Lemma 7.4.3 with c = a − 1, we obtain the following from (7.4.3). Proposition 7.4.4. Let λ = aω1 + bω2 for a ≥ 1 and 0 ≤ b ≤ 2. Then a . (−1)(u) Pa+b−1 (u · λ − ω1 ) = 3 u∈W

7.5. Vanishing Ranges. Suppose that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for some λ ∈ X(T )+ and i > 0. From the discussion in Section 7.1, we know that if i < 2p − 3, then λ must be of the form λ = pω1 + w · 0, and more precisely, that it must be one of the weights listed in Table 7.1. For each such λ, from Proposition 7.3.3 and Proposition 7.4.4, we can identify the least value of k such that (−1)(u) Pk (u · λ − ω1 ) = 0, u∈W

and moreover, identify the value of the sum. Further, from Proposition 2.7.1 (with k = (i − (w))/2)), we can then identify the least non-negative i with Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 along with the dimension of the cohomology group. This information is summarized in the following table. Here k and i are minimum

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

possible values, and dim gives the dimension of the cohomology group (equivalently the value of (7.2.1)). w

(w)

λ = pω1 + w · 0

e

0

pω1

p − 1 2p − 2

s1

1

(p − 2)ω1 + ω2

p − 2 2p − 3

s1 s2

2

(p − 5)ω1 + 2ω2

p − 4 2p − 6

s1 s2 s1

3

(p − 6)ω1 + 2ω2

p − 5 2p − 7

s1 s2 s1 s2

4

(p − 6)ω1 + ω2

p − 6 2p − 8

s1 s2 s1 s2 s1

5

(p − 5)ω1

p − 6 2p − 7

k

i

dim p 3

p 3

p 3

p 3

p 3

p 3

−1 −2 −2 −2 −2

Theorem 7.5.1. Suppose Φ is of type G2 and p ≥ 7. (a) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for i <2p − 8. p if λ = (p − 6)ω1 + ω2 3 −2 (b) dim H2p−8 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 else. (c) dim H2p−7 H 0 (λ) ⊗ H 0 (λ∗ )(1) ) (G, p if λ = (p − 5)ω1 or (p − 6)ω1 + 2ω2 3 −2 = 0 else. i (d) H (G(Fp ), k) = 0 for 0 < i < 2p − 8. Proof. Part (a) follows from Proposition 2.7.1 and Proposition 7.3.3. Parts (b) and (c) follow from the preceding table and discussion. Part (d) follows from part (a) and Proposition 2.4.1. One would like to apply Theorem 2.5.1 to conclude that H2p−8 (G(Fp ), k) = 0. However, the weight (p − 5)ω1 is less than and linked to the weight (p − 6)ω1 + ω2 and so the Theorem is not applicable. The non-zero cohomology from the weight (p − 5)ω1 in degree 2p − 7 could “cancel” some or all of the cohomology coming from the weight (p − 6)ω1 + ω2 . We refer the interested reader to [BNP, Section 2.7] for discussion of this interplay. In a similar manner, cohomology in degree 2p − 6 coming from the weight (p − 6)ω1 + ω2 could cancel that in degree 2p − 7 coming from the weight (p − 6)ω1 + 2ω2 . So it is not even possible to conclude that H2p−7 (G(Fp ), k) = 0. In summary, alternate methods are needed to determine the precise vanishing bound.

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8. Type F4 Assume throughout this section that Φ is of type F4 and that p > h = 12 (so p ≥ 13). Following the strategy laid out in Section 2, our goal is to ﬁnd the least i > 0 such that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for some λ ∈ X(T )+ . 8.1. Restrictions. Suppose that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for some i > 0 and λ = pμ + w · 0 with μ ∈ X(T )+ and w ∈ W . From Proposition 2.8.1, i ≥ (p − 1)μ, α ˜ ∨ − 1. For 1 ≤ i ≤ 3, we have ωi , α ˜ ∨ ≥ 2, while ω4 , α ˜ ∨ = 1. ∨ Therefore, unless μ = ω4 = α0 , we have μ, α ˜ ≥ 2 and i ≥ 2p − 3. Suppose now that λ = pω4 + w · 0 for some w ∈ W . With the aid of MAGMA, one can identify all w for which λ is in fact dominant. From Proposition 2.8.1(a), with σ = α0 and λ = pω4 + w · 0, since ω4 , α0∨ = 2, we have (8.1.1)

i ≥ 2(p − 1) + (w) + w · 0, α0∨ .

By checking all possible cases, one ﬁnds that (w) + w · 0, α0∨ ≥ −7. Combining this with (8.1.1), we conclude that i ≥ 2p − 9. From Proposition 2.4.1, we get the following. Theorem 8.1.1. Suppose Φ is of type F4 and p ≥ 13. Let λ ∈ X(T )+ . Then (a) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 9; (b) Hi (G(Fp ), k) = 0 for 0 < i < 2p − 9. 8.2. Based on the preceding discussion, the weights which could give Hi (G, H 0 (λ) ⊗H 0 (λ∗ )(1) ) = 0 for i ≤ 2p − 7 are summarized in the following table. λ = pω4 + w · 0 (p − 12)ω4 + ω2 (p − 12)ω4 + ω3 (p − 11)ω4 (p − 11)ω4 + 3ω1 (p − 12)ω4 + 2ω1 + ω3 (p − 12)ω4 + ω1 + ω2 (p − 11)ω4 + 2ω1 + ω2 (p − 12)ω4 + ω1 + ω2 + ω3 (p − 12)ω4 + 2ω2

(w) 13 14 15 10 11 12 9 10 11

w · 0, α0∨ −20 −21 −22 −16 −17 −18 −14 −15 −16

i 2p − 9 2p − 9 2p − 9 2p − 8 2p − 8 2p − 8 2p − 7 2p − 7 2p − 7

i−(w)

As seen in Section 2.7, λ − ω4 must be a weight of S 2 (u∗ ), and hence i is congruent to (w) mod 2. It follows that some of the above degree bounds are even higher. For example, consider λ = (p − 12)ω4 + ω3 = pω4 + w · 0. Since (w) = 14 but 2p − 9 is odd, the least value i could take would be 2p − 8. A similar situation holds for λ = (p − 12)ω4 + 2ω1 + ω3 and λ = (p − 12)ω4 + ω1 + ω2 + ω3 . Similarly, for the other weights in the above list, if the cohomology vanishes in the degree i listed, then the next possible non-vanishing degree is i + 2. We summarize this in the following lemma. Lemma 8.2.1. Suppose Φ is of type F4 , p ≥ 13 and λ ∈ X(T )+ . Suppose that Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0. Then (a) i ≥ 2p − 9; (b) if i = 2p − 9, then λ = (p − 12)ω4 + ω2 or (p − 11)ω4 ;

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CHRISTOPHER P. BENDEL, DANIEL K. NAKANO, AND CORNELIUS PILLEN

(c) if i = 2p − 8, then λ = (p − 12)ω4 + ω3 , (p − 11)ω4 + 3ω1 , or (p − 12)ω4 + ω1 + ω2 ; (d) if i = 2p − 7, then λ = (p − 12)ω4 + ω2 , (p − 11)ω4 , (p − 12)ω4 + 2ω1 + ω3 , (p − 11)ω4 + 2ω1 + ω2 , or (p − 12)ω4 + 2ω2 . 8.3. Conjectures. In principle, one could use Proposition 2.7.1 to compute the dimension of Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) in terms of partition functions for the weights in Lemma 8.2.1. For small p, one can use MAGMA to make this computation. For p = 13, 17, or 19, one ﬁnds that the two candidates in degree 2p − 9 have zero cohomology. They do give cohomology in degree 2p − 7. And in degree 2p − 8, the only one weight (of the three) which has cohomology is (p − 12)ω4 + ω3 . We make the following Conjecture 8.3.1. Suppose that Φ is of type F4 , p ≥ 13, and λ = pμ + w · 0 ∈ X(T )+ . Then (a) Hi (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for 0 < i < 2p − 8; (b) H2p−8 (G, H 0 (λ) ⊗ H 0 (λ∗ )(1) ) = 0 for λ = (p − 12)ω4 + ω3 . If part (a) of the conjecture holds, then Hi (G(Fp ), k) = 0 for 0 < i < 2p − 8 thus improving upon Theorem 8.1.1. However, even if part (b) of the conjecture also holds, it does not necessarily follow that H2p−8 (G(Fp ), k) = 0. Analogous to the situation for type G2 (cf. Section 7.5), cohomology in degree 2p − 7 from the weight (p − 11)ω4 could cancel out the cohomology in degree 2p − 8 from the weight (p − 12)ω4 + ω3 . Conjecture 8.3.1 is a special case of a more general conjecture on partition functions (known to hold for small values of m). Conjecture 8.3.1(a) would follow from parts (a) and (b) while Conjecture 8.3.1(b) would follow from part (c). Conjecture 8.3.2. Suppose that Φ is of type F4 and m ≥ 1. Then (−1)(u) Pm+1 (u · (mω4 + ω2 ) − ω4 ) = 0; (a) u∈W 0 if m is even, (u) (−1) Pm−1 (u · (mω4 ) − ω4 ) = (b) 1 if m is odd; u∈W (u) (c) (−1) Pm+1 (u · (mω4 + ω3 ) − ω4 ) = 1. u∈W

References [AJ] [B] [BNP] [BC] [BCP] [F] [FP]

H. H. Andersen, J. C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann., 269, (1984), 487–525. A. Barbu, On the range of non-vanishing p-torsion cohomology for GLn (Fp ), J. Algebra, 278, (2004), 456-472. C. P. Bendel, D. K. Nakano, C. Pillen, On the vanishing ranges for the cohomology of ﬁnite groups of Lie type, Int. Math. Res. Not. IMRN 2011; doi: 10.1093/imrn/rnr130. W. Bosma, J. Cannon, Handbook on Magma Functions, Sydney University, 1996. W. Bosma, J. Cannon, C. Playhoust, The Magma Algebra System I: The User Language, J. Symbolic Computation, 3/4 no. 24, (1997), 235-265. E. M. Friedlander, Computations of K-theories of ﬁnite ﬁelds, Topology, 15, (1976), 87109. E. M. Friedlander, B. J. Parshall, On the cohomology of algebraic and related ﬁnite groups, Invent. Math., 74, (1983), 85-118.

ON THE VANISHING RANGES FOR THE COHOMOLOGY OF FINITE GROUPS

[H] [Hum] [Jan] [KLT] [Q]

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H. L. Hiller, Cohomology of Chevalley groups over ﬁnite ﬁelds, J. Pure Appl. Alg., 16, (1980), 259-263. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, SpringerVerlag, 1972. J. C. Jantzen, Representations of Algebraic Groups, American Mathematical Society, Mathematical Surveys and Monographs, 107, 2003. S. Kumar, N. Lauritzen, J. Thomsen, Frobenius splitting of cotangent bundles of ﬂag varieties, Invent. Math., 136, (1999), 603-621. D. G. Quillen, On the cohomology and K-theory of ﬁnite ﬁelds, Ann. Math., 96, (1972), 551-586.

Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie WI 54751, USA E-mail address: [email protected] Department of Mathematics, University of Georgia, Athens GA 30602, USA E-mail address: [email protected] Department of Mathematics and Statistics, University of South Alabama, Mobile AL 36688, USA E-mail address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 86, 0, XXXX 2012

Tilting Modules for the Current Algebra of a Simple Lie Algebra Matthew Bennett and Vyjayanthi Chari

Abstract. The category of level zero representations of current and aﬃne Lie algebras shares many of the properties of other well–known categories which appear in Lie theory. In this paper we explore further similarities and develop a tilting theory for graded representations of the current algebra associated to a simple Lie algebra. The role of the standard and co–standard module is played by the ﬁnite–dimensional local Weyl module and the graded dual of the inﬁnite–dimensional global Weyl module respectively. We deﬁne the canonical ﬁltration of a graded module for the current algebra. In the case when g is of type sln+1 we show that the well–known necessary and suﬃcient homological condition for a canonical ﬁltration to be a good (or a ∇–ﬁltration) also holds in our situation. Finally, we construct the indecomposable tilting modules in our category and show that any tilting module is isomorphic to a direct sum of indecomposables.

Introduction The study of the representation theory of current algebras was largely motivated by its relationship to the representation theory of aﬃne and quantum aﬃne algebras associated to a simple Lie algebra g. However, it is also now of independent interest since it yields connections with problems arising in mathematical physics, for instance the X = M conjectures, see [1], [11], [18]. These connections arise from the fact that the current algebra is graded by the non–negative integers and that studying graded modules and their characters gives rise to interesting combinatorics. The work of [14] for instance, also relates certain graded characters to the Poincare polynomials of quiver varieties. The current Lie algebra is just the Lie algebra of polynomial maps from C → g and can be identiﬁed with the space g ⊗ C[t] with the obvious commutator. The Lie algebra and its universal enveloping algebra inherit a grading coming from the natural grading on C[t]. One is interested in the category I of Z–graded modules for g[t] with the restriction that the graded pieces are ﬁnite–dimensional. The simple objects in the category are just the graded shifts of the irreducible modules for g and so are parametrized by a set Λ consisting of pairs (λ, r), where λ is a dominant integral weight and r is an integer. However, the interest in this category 2010 Mathematics Subject Classiﬁcation. Primary 17B15. V.C. was partially supported by DMS-0901253. c c 2012 American Mathematical Society XXXX

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MATTHEW BENNETT AND VYJAYANTHI CHARI

stems from the fact that it has reducible but indecomposable objects. Many of these objects are either deﬁned in a way similar to, or play an analogous role to well–known constructions in Lie theory, say in the BGG category O associated to a simple Lie algebra or to representations of algebraic groups in characteristic p. Our work has some similarity with [16] although our set up is quite diﬀerent. In particular the grade zero piece of the algebra U(g[t]) is inﬁnite–dimensional. The category I contains the projective cover and the injective envelope of a simple object. Moreover, if we deﬁne a suitable partial order on Λ, then we can deﬁne the appropriate analog of the standard and costandard objects in I. An interesting feature in our case is that the standard object Δ(λ, r) is a ﬁnite–dimensional module called the local Weyl module which has been extensively studied (see [6], [10], [17], for instance). The co-standard object ∇(λ, r) however is inﬁnite–dimensional and is the (appropriately deﬁned) dual of the global Weyl module. Both modules lie in a nice subcategory of I which we call Ibdd and is the full subcategory consisting of objects whose weights are in a ﬁnite union of cones (as in O) and whose grades are bounded above. The main goal of this paper is to construct another family of non–isomorphic objects of Ibdd which are also indexed by Λ. These modules are denoted by T (λ, r) and have an inﬁnite ﬁltration in which the successive quotients are of the form Δ(μ, s) for (μ, s) ∈ Λ. The ﬁltration multiplicity of any given Δ(μ, s) is ﬁnite. We also show that these modules satisfy a nice homological property, namely that Ext1I (Δ(μ, s), T (λ, r)) = 0, (μ, s), (λ, r) ∈ Λ. In the case of algebraic groups (for instance, see [9],[15]) it is shown that the preceding condition is equivalent to the module having a ﬁltration by ∇(μ, s) and the module T (λ, r) is then called tilting. A crucial tool in that situation to proving this equivalence is to show that every module can be embedded into a module admitting a ∇–ﬁltration. In our case, we ﬁrst have to modify slightly the deﬁnition of the ∇–ﬁltration, but the more serious problem is to show that any object embeds into one which has a ∇–ﬁltration. If we restrict our attention to Ibdd then we are able to prove that any M embeds into an injective object of Ibdd . We show that if these injective objects admit a ∇–ﬁltration, then the modules T (λ, r) are tilting, and are all the indecomposable tilting objects in Ibdd . Moreover, we also prove that any tilting module in Ibdd is isomorphic to a direct sum of indecomposable tilting modules. In the case when g is of type sln+1 (see [3] for the n = 1 case and [2] for general n), it is shown that the injective envelope of a simple object does have ∇–ﬁltrations. In fact, it is also shown in those papers that the injective envelope a simple object in Ibdd (which is usually smaller) also has a ∇–ﬁltration. As a consequence, one sees that for sln+1 the modules T (λ, r) are indeed tilting modules. There are obviously a number of interesting questions one could ask about these modules which we will pursue elsewhere. Acknowledgements We are grateful to the referee for their careful reading of the paper and their comments, which led to a substantial improvement in the exposition. 1. Preliminaries 1.1. Throughout this paper we denote by C the ﬁeld of complex numbers and Z (resp. Z+ ) the set of integers (resp. nonnegative integers). For any Lie algebra a,

TILTING MODULES MODULES FOR THE CURRENT OF A SIMPLE LIE TILTING CURRENT ALGEBRA ALGEBRAOF LIE ALGEBRA ALGEBRA

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we denote by U(a) the universal enveloping algebra of a. Let t be an indeterminate and let a[t] = a ⊗ C[t] be the Lie algebra with commutator given by, [a ⊗ f, b ⊗ g] = [a, b] ⊗ f g, a, b ∈ a, f, g, ∈ C[t]. We identify a with the Lie subalgebra a ⊗ 1 of a[t]. The Lie algebra a[t] has a natural Z–grading given by the powers of t and this also induces a Z–grading on U(a[t]), and U(a[t])[s] = 0, s < 0, U(a[t])[0] = U(a). Each graded piece is a a–module under left and right multiplication by elements of a and hence also under the adjoint action of a. In particular, if dim a < ∞, then U(a[t])[r] is a free module for a (via left or right multiplication) of ﬁnite rank. 1.2. From now on, g denotes a ﬁnite–dimensional complex simple Lie algebra of rank n and h a ﬁxed Cartan subalgebra of g. Let I = {1, · · · , n} and ﬁx a set {αi : i ∈ I} of simple roots of g with respect to h and a set {ωi : i ∈ I} of fundamental weights. Let Q (resp. Q+ ) be the integer span (resp. Z+ –span) of {αi : i ∈ I} and P (resp. P + ) to be the Z (resp. Z+ ) span of {ωi : i ∈ I}. ± Let {x± be the Lie i , hi : i ∈ I} be a set of Chevalley generators of g and let n ± subalgebra of g generated by the elements xi , i ∈ I. We have, g = n− ⊕ h ⊕ n+ ,

U(g) = U(n− ) ⊗ U(h) ⊗ U(n+ ).

Given λ, μ ∈ h∗ , we say that λ ≥ μ iﬀ λ − μ ∈ Q+ . Let W be the Weyl group of g and let w0 ∈ W be the longest element of W . Given λ ∈ P + , let conv W λ ⊂ h∗ be the convex hull of the set W λ. 1.3. For any g-module M and μ ∈ h∗ , set Mμ = {m ∈ M : hm = μ(h)m, We say M is a weight module for g if

M=

h ∈ h}.

Mμ ,

μ∈h∗

and we set wt(M ) = {μ ∈ h∗ : Mμ = 0} . Any ﬁnite–dimensional g–module is a weight module. It is well-known that the set of isomorphism classes of irreducible ﬁnite-dimensional g-modules is in bijective correspondence with P + . For λ ∈ P + let V (λ) be a representative of the corresponding isomorphism class. It is generated by a vector vλ with deﬁning relations n+ vλ = 0,

λ(hi )+1 (x− vλ = 0, i )

hvλ = λ(h)vλ ,

h ∈ h, i ∈ I,

and note that wt V (λ) ⊂ conv W λ. The module V (0) is the trivial module for g and we shall write it as C. For a weight module with ﬁnite–dimensional weight spaces, the character of M is the element of the integral group ring Z[P ] deﬁned by, chg M = dimC Mμ e(μ), μ∈P

where e(μ) ∈ Z[P ] is the generator of the group ring corresponding to μ. The set {chg V (μ) : μ ∈ P + } is a linearly independent subset of Z[P ]. We say that M is a locally ﬁnite-dimensional g–module if it is isomorphic to a direct sum of ﬁnite–dimensional g–modules, in which case M is necessarily a weight module. Using Weyl’s theorem one knows that a locally ﬁnite-dimensional

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MATTHEW BENNETT AND VYJAYANTHI CHARI

g-module M is isomorphic to a direct sum of modules of the form V (λ), λ ∈ P + and hence wt M ⊂ P . Set (1.1)

+ + Mλn = M n ∩ Mλ ∼ = Homg (V (λ), M ).

+

M n = {m ∈ M : n+ m = 0}

1.4. Let I be the category whose objects are graded g[t]-modules V with ﬁnitedimensional graded components and where the morphisms are maps of graded g[t]modules. Thus an object V of I, is a Z–graded vector space V = ⊕s∈Z V [s] which admits a left action of g[t] satisfying (g ⊗ tr )V [s] ⊂ V [s + r],

s, r ∈ Z.

Clearly each graded piece is a g–module and hence objects of I are locally ﬁnite– dimensional g–modules. A morphism between two objects V , W of I is a degree zero map of graded g[t]–modules. Clearly I is closed under taking submodules, quotients and ﬁnite direct sums. For any r ∈ Z we let τr be the grade shifting operator. If V ∈ Ob I and μ ∈ P + , then + + + + V [r]nμ , V [r]nμ = Vμn ∩ V [r]. Vμn = r∈Z

The graded character of V ∈ Ob I is an element of the space of power series Z[P ][[u, u−1 ]], given by chgr V := chg (V [r])ur . r∈Z

Given V ∈ Ob I, the restricted dual V ∗ is the Z–graded vector space given by, V [−r]∗ , V ∗ [r] = V [−r]∗ . V∗ = r∈Z ∗

Moreover, V ∈ Ob I with the usual action: (xts )v ∗ (w) = −v ∗ (xts w), and (V ∗ )∗ ∼ = V as objects of I. Note that if V ∈ Ob I, then chg (V [r]∗ )u−r . chgr V ∗ := r∈Z

2. The main result 2.1. Let Ibdd be the full subcategory of I consisting of objects M satisfying the following two conditions: (i) there exists μ1 , · · · , μs ∈ P + (depending on M ) such that wt M ⊂

s

conv W μ ,

=1

(ii) there exists r ∈ Z (depending on M ) such that M [] = 0 if ≥ r. Notice that Ibdd is not closed under taking duals. Setting Λ = P + × Z, we now deﬁne three natural families of objects of Ibdd which are all indexed by Λ.

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2.2. Let ev0 : g[t] → g be the homomorphism of Lie algebras given by, ev0 (x ⊗ f ) = f (0)x. The kernel of ev0 is the graded ideal g ⊗ tC[t]. Any ﬁnite–dimensional g–module V can be regarded as an object of Ibdd by pulling back through ev0 and the pull back of V (λ) is denoted V (λ, 0). Set τr V (λ, 0) = V (λ, r) and let vλ,r ∈ V (λ, r) be the element corresponding to vλ . Given M ∈ Ob I denote by soc M the maximal semisimple submodule of M . The next proposition gives an explanation for restricting our study to Ibdd . Proposition. (i) Any irreducible object in I (or Ibdd ) is isomorphic to V (μ, r) for a unique element (μ, r) ∈ Λ. Moreover V (μ, r)∗ ∼ = V (−w0 μ, −r). (ii) Let M ∈ Ob Ibdd be non-trivial. Then soc M = 0 and we have V (λ, r)m(λ,r) , m(λ, r) = dim HomI (V (λ, r), M ). soc M ∼ = (λ,r)∈Λ

Proof. Part (i) is straightforward and a proof can be found in [5, Proposition 1.3]. For (ii), choose s ∈ Z such that M [s] = 0 and M [] = 0 for all > s. Since M [s] is a ﬁnite–dimensional g–module, there exists μ ∈ P + such that Homg (V (μ), M [s]) = 0. Since (g ⊗ tC[t])M [s] = 0, it follows that Homg[t] (V (μ, s), M ) = 0 proving that soc M = 0. The rest of (ii) is now immediate. 2.3. The next family we need are the local Weyl modules which were originally deﬁned in [8]. For the purposes of this paper, we shall denote them as Δ(λ, r), (λ, r) ∈ Λ. Thus, Δ(λ, r) is generated as a g[t]–module by an element wλ,r with relations: n+ [t]wλ,r = 0,

λ(hi )+1 (x− wλ,r = 0, i )

(h ⊗ t )wλ,r = δs,0 λ(h)wλ,r , s

where i ∈ I, h ∈ h and s ∈ Z+ . The following proposition summarizes the properties of Δ(λ, r) which are necessary for this paper (see for example [3]). Proposition. Let (λ, r) ∈ Λ. (i) The module Δ(λ, r) is indecomposable and ﬁnite–dimensional and hence an object of Ibdd . (ii) dim Δ(λ, r)λ = dim Δ(λ, r)[r]λ = 1, (iii) wt Δ(λ, r) ⊂ conv W λ, (iv) The module V (λ, r) is the unique irreducible quotient of Δ(λ, r). (v) {chgr Δ(λ, r) : (λ, r) ∈ Λ} is a linearly independent subset of Z[P ][u, u−1 ]. We denote by [Δ(λ, r) : V (μ, s)] the multiplicity of V (μ, s) in a Jordan–Holder series of Δ(λ, r).

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2.4. We now deﬁne the modules ∇(λ, r). These modules are usually deﬁned to be the dual of the modules Δ(λ, r), but in our situation the resulting modules would be too small. The correct deﬁnition is to take ∇(λ, r) to be the dual of the global Weyl modules W (λ, r). Here W (λ, r) is generated as a g[t]–module by an element wλ,r with relations: n+ [t]wλ,r = 0,

λ(hi )+1 (x− wλ,r = 0, i )

hwλ,r = λ(h)wλ,r , where i ∈ I and h ∈ h. It is not hard to see that the graded pieces of W (λ, r) are ﬁnite–dimensional. One notices ﬁrst that W (λ, r)[r] ∼ = V (λ), as g–modules and hence is ﬁnite–dimensional. Since W (λ, r)[s] ∼ = U(g ⊗ tC[t])[s − r]W (λ, r)[r], and dim U(g⊗tC[t])[s−r] < ∞, we see that W (λ, r) ∈ Ob I and that W (λ, r)[s] = 0 if s < r. Clearly the module Δ(λ, r) is a quotient of W (λ, r) and moreover V (λ, r) is the unique irreducible quotient of W (λ, r). It is known (see [4] or [8] ) that W (0, r) ∼ = C and that if λ = 0, the modules W (λ, r) are inﬁnite-dimensional and satisfy wt W (λ, r) ⊂ conv W λ. It follows from the preceding remarks, that if we set ∇(λ, r) = W (−w0 λ, −r)∗ , then ∇(λ, r) ∈ Ob Ibdd and soc ∇(λ, r) ∼ = V (λ, r). The next proposition summarizes the main results on ∇(λ, r) that are needed for this paper and is the dual version of the results on W (λ, r), proved for instance, in [3]. Proposition. Let (λ, r) ∈ Λ. The module ∇(λ, r) is an indecomposable object of Ibdd . dim ∇(λ, r)[r]λ = 1, and dim ∇(λ, r)[s] = 0 ⇐⇒ s ≤ r, wt ∇(λ, r) ⊂ conv W λ, Any submodule of ∇(λ, r) contains ∇(λ, r)[r]λ and the socle of ∇(λ, r) is V (λ, r). (v) {chgr ∇(λ, r) : (λ, r) ∈ Λ} is a linearly independent subset of Z[P ][[u, u−1 ]].

(i) (ii) (iii) (iv)

2.5. Definition. We say that M ∈ Ob I admits a Δ (resp. ∇)–ﬁltration if there exists an increasing family of submodules 0 ⊂ M0 ⊂ M1 ⊂ · · · , M= Mk , k

such that Mk /Mk−1 ∼ =

(λ,r)∈Λ

Δ(λ, r)mk (λ,r) ,

(resp., Mk /Mk−1 ∼ =

∇(λ, r)mk (λ,r) ),

(λ,r)∈Λ

for some choice of mk (λ, r) ∈ Z+ . If Mk = M for some k ≥ 0, then we say that M admits a ﬁnite Δ (resp. ∇)–ﬁltration. Finally,we say that M is tilting if M has both a Δ and a ∇–ﬁltration.

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Since dim M [r]λ < ∞ for all (λ, r) ∈ Λ, we see that if M has a Δ–ﬁltration (resp.∇–ﬁltration) Mk ⊂ Mk+1 , then mk (λ, r) = 0 for all but ﬁnitely many k. Since ⎞ ⎛ ⎝ chgr M = chgr Mk /Mk−1 = mk (λ, r)⎠ chgr Δ(λ, r), k≥0

(λ,r)∈Λ

k≥0

(where we understand that M−1 = 0) it follows from Proposition 2.3 that the ﬁltration multiplicity [M : Δ(λ, r)] = mk (λ, r), k≥0

is well -deﬁned and independent of the choice of the ﬁltration. An analogous statement holds for modules admitting a ∇–ﬁltration. Finally, we remark that if M is a direct sum of modules each of which admit a Δ (resp. ∇) ﬁltration then M also admits a Δ (resp. ∇) ﬁltration (one just takes the direct sum of the modules in the ﬁltration at each step). It follows that a direct sum of tilting modules is tilting. 2.6. The main goal of this paper is to understand tilting modules in Ibdd . In the case of algebraic groups (see [9], [15]) a crucial necessary result is to give a cohomological characterization of modules admitting a ∇–ﬁltration. The analogous result in our situation is to prove the following statement: An object M of Ibdd admits a ∇–ﬁltration iﬀ Ext1Ibdd ((Δ(λ, r), M ) = 0 for all (λ, r) ∈ Λ. It is not hard to see that the forward implication is true. The converse statement however requires one to prove that any object of Ibdd can be embedded in a module which admits a ∇–ﬁltration. At this point we can only prove the result for sln+1 and we explain the reason for these limitations in the next section. Summarizing, the ﬁrst main result that we shall prove in this paper is: Proposition. Let M ∈ Ob Ibdd . (i) If M admits a ∇–ﬁltration, then for all (λ, r) ∈ P + × Z, we have Ext1I (Δ(λ, r), M ) = 0. (ii) Let g be of type An , and assume that M ∈ Ibdd satisﬁes Ext1I (Δ(λ, r), M ) = 0 for all (λ, r) ∈ Λ. Then M admits a ∇–ﬁltration. 2.7. The second main result that we shall prove in this paper is the following: Theorem. (i) Given (λ, r) ∈ Λ, there exists an indecomposable module T (λ, r) ∈ Ob Ibdd which admits a Δ–ﬁltration and satisﬁes Ext1I (Δ(μ, s), T (λ, r)) = 0, (μ, s) ∈ Λ, T (λ, r)[r]λ = 1, wt T (λ, r) ⊂ conv W λ, and T (λ, r) ∼ = T (μ, s) iﬀ (λ, r) = (μ, s). (ii) If g is of type sln+1 , then T (λ, r) is tilting. Moreover any indecomposable tilting module in Ibdd is isomorphic to T (λ, r) for some (λ, r) ∈ Λ. Finally any tilting module in Ibdd is isomorphic to a direct sum of indecomposable tilting modules.

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MATTHEW BENNETT AND VYJAYANTHI CHARI

3. The canonical filtration and proof of Proposition 2.6 In this section we show that one can deﬁne in a canonical way a ﬁltration on any object of Ibdd such that the successive quotients embed into a direct sum of modules ∇(μ, s), (μ, s) ∈ Λ. To do this we need to understand the projective and injective objects of I although these are not objects of Ibdd . Using the canonical ﬁltration we get an upper bound for the character of any object of Ibdd . We then use this bound along with the BGG–reciprocity result proved in [2] and [3] to establish Proposition 2.6. 3.1. The category I contains the projective cover and the injective envelope of a simple object. For (λ, r) ∈ Λ, set P (λ, r) = U(g[t]) ⊗U(g) V (λ, r),

I(λ, r) = P (−w0 λ, −r)∗ .

Note that P (λ, r)[r] ∼ =g V (λ) ∼ =g I(λ, r)[r], P (λ, r)[s] = 0 = I(λ, −r)[−s],

s < r.

Clearly P (λ, r) is generated by the element pλ,r = 1 ⊗ vλ with deﬁning relations: λ(hi )+1 n+ pλ,r = 0, hpλ,r = λ(h)pλ,r , (x− pλ,r = 0. i )

The following was proved in [5, Proposition 2.1]. Proposition. For (λ, r) ∈ Λ, the object P (λ, r) is the projective cover in I of V (λ, r). Analogously, the object I(λ, r) is the injective envelope of V (λ, r) in I. Notice that P (λ, r)μ = 0 for inﬁnitely many μ ≥ λ and hence P (λ, r) (and also I(λ, r)) is not an object of Ibdd . However, we shall introduce quotients (resp. submodules) of these objects which do lie in Ibdd . 3.2. The object W (λ, r) which was deﬁned in Section 2.4 is the the maximal quotient (in I) of P (λ, r) such that wt W (λ, r) ⊂ λ − Q+ , or equivalently the maximal quotient whose weights are contained in conv W λ. Similarly, ∇(λ, r) is the maximal submodule of I(λ, r) whose weights are in the conv W λ. The following is now trivially proved. Lemma. For λ, μ ∈ P + with λ ≮ μ, we have Ext1I (W (λ, r), W (μ, s)) = 0 = Ext1I (∇(μ, r), ∇(λ, s)), for all r, s ∈ Z. 3.3. At this stage it is worth making the following remark. Deﬁne a partial order on P + × Z by: (λ, r) (μ, s) if either λ < μ or λ = μ and r ≤ s. Then it is not hard to see that, Δ(λ, r) is the maximal quotient of P (λ, r) such that Δ(λ, r)[s]μ = 0 =⇒ (μ, s) (λ, r). On the other hand, ∇(λ, r) is the maximal submodule of I(λ, r) satisfying, ∇(λ, r)[s]μ = 0 =⇒ (μ, s) (λ, r). Hence our choices of the Δ − −∇–modules is consistent with the ones usually made in the literature.

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3.4. Given Γ ⊂ P + , let I(Γ) be the full subcategory of I consisting of objects M such that wt M ⊂ conv W λ. λ∈Γ

The category Ibdd (Γ) is deﬁned similarly. Given M ∈ I, let MΓ be the maximal submodule of M that lies in I(Γ). We shall say that a subset Γ of P + is closed with respect to ≤ if λ ∈ Γ and μ ≤ λ implies μ ∈ Γ. Proposition. Let Γ ⊂ P + be closed with respect to ≤. (i) For (λ, r) ∈ Λ, we have that I(λ, r)Γ is an injective object of Ibdd (Γ) for (λ, r) ∈ Γ × Z. (ii) Suppose that M ∈ Ob Ibdd (Γ). There exists, an injective morphism ⊕m(λ,r) I(λ, r)Γ , m(λ, r) = dim HomI (V (λ, r), M ). M → (λ,r)∈Γ×Z

Proof. To prove (i), suppose that we have an injective morphism π : M → N and a morphism φ : M → I(λ, r)Γ of objects of of Ibdd and let ι : I(λ, r)Γ → I(λ, r) be the inclusion map. Since I(λ, r) is an injective object of I, it follows ˜ = ι.φ. Since immediately that we have a morphism φ˜ : N → I(λ, r) such that φ.π ˜ ) ∈ Ob Ibdd we see that φ(N ˜ ) ⊂ I(λ, r)Γ . Hence , φ.π ˜ = φ as required. φ(N To prove (ii) let (λ, r) ∈ Γ × Z and M ∈ Ob Ibdd (Γ) and let ιλ,r : V (λ, r) → I(λ, r)Γ be the canonical inclusion. Corresponding to any non–zero morphism ϕ : V (λ, r) → M , we have by using part (i) a morphism ϕ˜ : M → I(λ, r)Γ such that ϕ.ϕ ˜ = ιλ,r . If m(λ, r) > 0 it follows that by ﬁxing a basis for HomI (V (λ, r), M ) we have a morphism ⊕m(λ,r)

ϕλ,r : M → (I(λ, r)Γ )

.

Let 0 ∈ Z be such that M [] = 0 for all ≥ 0 which implies that m(λ, r) = 0 =⇒ r ≤ 0 . Since ϕλ,r M [s] = 0 =⇒ s ≤ r, it follows that we have a map m(λ,r) (I(λ, r)Γ ) , Φ:M →

m → {ϕλ,r (m)}(λ,r)∈Γ×Z .

(λ,r)∈Γ×Z

It remains to prove that Φ is injective. If ker Φ = 0 then we have soc(ker Φ) = 0 by Proposition 2.2. On the other hand, soc ker Φ ⊂ soc M and the restriction of Φ to soc M is injective by design. The proof of (ii) is complete. 3.5. From now on we ﬁx an enumeration λ0 , λ1 , · · · , λk , · · · of P + satisfying: λr − λs ∈ Q+ =⇒ r ≥ s. Given M ∈ Ibdd , deﬁne k(M ) ∈ Z+ to be minimal such that

k(M )

wt M ⊂

s=0

conv W λs .

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MATTHEW BENNETT AND VYJAYANTHI CHARI

For 0 ≤ s ≤ k(M ), let Ms be the maximal submodule of M whose weights lie in the union of the sets {conv W λr : r ≤ s}. Clearly

k(M )

Ms ⊂ Ms+1 ,

M=

Ms ,

s=0

Homg (V (λ, r), Ms+1 /Ms ) = 0 =⇒ λ = λs+1 .

(3.1)

We call the ﬁltration M0 ⊂ M1 ⊂ · · · ⊂ Mk(M ) = M the canonical ﬁltration of M . It follows from Proposition 3.4, and (3.1) that, Ms+1 /Ms embeds into a direct sum of modules of the form ∇(λs+1 , r), r ∈ Z. This gives, (3.2) chgr M =

chgr Ms /Ms−1 ≤

s≥0

i.e, [M : V (μ, )] ≤

dim HomI (V (λs , r), Ms /Ms−1 )ch∇(λs , r),

s≥0 r∈Z

dim HomI (V (λs , r), Ms /Ms−1 )[∇(λs , r) : V (μ, )],

s≥0 r∈Z

for all (μ, ) ∈ Λ. We claim that this is equivalent to, chgr M ≤ (3.3) dim HomI (Δ(λs , r), M )ch∇(λs , r). s≥0 r∈Z

The claim is obviously true if we prove that we have an isomorphism of vector spaces HomI (Δ(λs , r), M ) → HomI (V (λs , r), Ms /Ms−1 ). For this, observe that any non–zero map ϕ : Δ(λs , r) → M has its image in Ms . Moreover ϕ maps the unique maximal, proper submodule of Δ(λs , r) to Ms−1 and hence induces a non–zero map from V (λs , r) → Ms /Ms−1 . This proves that there is an injective linear map from Ψ : HomI (Δ(λs , r), M ) → HomI (V (λs , r), Ms /Ms−1 ). To see that Ψ is surjective, suppose that we have a non–zero map ψ : V (λs , r) → Ms /Ms−1 . Choose m ∈ Ms [r]λs such that ψ(vλs ,r ) = m ¯ where m ¯ is the image of m in Ms /Ms−1 . Since wt Ms ⊂ conv W λs it follows that n+ [t]m = 0. On the other hand since (h ⊗ tC[t])m ¯ = 0, we must have that (h ⊗ tC[t])m ∈ (Ms−1 )λs = 0. ˜ = ψ. Hence there exists a non–zero map ψ˜ : Δ(λs , r) → Ms → M such that Ψ(ψ) Finally, note that equality holds in (3.3) iﬀ the canonical ﬁltration is a ∇–ﬁltration. 3.6. The following result was proved in [3] when g is of type sl2 and in [2] when g is of type sln+1 . More precisely the dual of the following result was proved in these papers, i.e. it was shown that the projective objects had a canonical decreasing ﬁltration with successive quotients being the global Weyl modules W (μ, s). It is conjectured in [3] that the result is true in general. Theorem. Assume that g is of type sln+1 . Let Γ be a ﬁnite subset of P + . For all (λ, r) ∈ Γ × Z the canonical ﬁltration of I(λ, r)Γ is a ∇–ﬁltration. Moreover for all (μ, s) ∈ Λ, we have (3.4)

[I(λ, r)Γ : ∇(μ, s)] = [Δ(μ, s) : V (λ, r)] = dim HomI (Δ(μ, s), I(λ, r)Γ).

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85 11

3.7. We note the following consequence Proposition 3.4 and Theorem 3.6. Proposition. Assume that g is of type sln+1 and let M ∈ Ob Ibdd . Then M embeds into an object I(M ) of Ibdd which admits a ∇–ﬁltration. 3.8. To prove (ii) of Proposition 2.6, suppose that M ∈ Ob Ibdd satisﬁes Ext1I (Δ(λ, r), M ) = 0, (λ, r) ∈ Λ.

(3.5)

Assume also that we have an embedding 0 → M → I(M ) → Q → 0, where I(M ) ∈ Ob Ibdd has a ∇–ﬁltration, in which case Q ∈ Ob Ibdd . (In particular Proposition 3.7 shows that we can do this when g is of type sln+1 ). Applying HomI (Δ(λ, r), −) to the short exact sequence and using (3.5) gives, dim HomI (Δ(λ, r), I(M )) = dim HomI (Δ(λ, r), Q) + dim HomI (Δ(λ, r), M ), and hence, we have

(dim HomI (Δ(λs , r), I(M ))chgr ∇(λs , r) =

s≥0 r∈Z

dim HomI (Δ(λs , r), Q))chgr ∇(λs , r) +

s≥0 r∈Z

dim HomI (Δ(λs , r),

s≥0 r∈Z

M )chgr ∇(λs , r). Since I(M ) has a ∇–ﬁltration, it follows from (3.3) that the left hand side of the preceding equation is precisely chgr I(M ). On the other hand, since M, Q ∈ Ob Ibdd (3.3) shows that the right hand side of the preceding equation is bounded below by chgr M + chgr Q. Since chgr I(M ) = chgr M + chgr Q it follows that we must have dim HomI (Δ(λs , r), M )chgr ∇(λs , r), chgr M = s≥0 r∈Z

and similarly for chgr Q. Hence the canonical ﬁltration of M and Q are both ∇– ﬁltrations and Proposition 2.6(ii) is proved. 3.9. We need one more standard result (whose proof we include for convenience) to prove part (i) of Proposition 2.6, the main point is that it allows us to work with ﬁnite ∇–ﬁltrations. Lemma. Suppose that M ∈ Ob Ibdd has a (possibly inﬁnite) ∇–ﬁltration. Then M admits a ﬁnite ∇–ﬁltration 0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ Mk = M, Ms /Ms−1 ∼ ∇(λs , r)⊕[M :∇(λs ,r)] = r∈Z

where we recall that [M : ∇(λs , r)] < ∞ for all s and r. In particular if k is maximal such that Mλk = 0 then, there exists s ∈ Z and a surjective map M → ∇(λk , s) such that the kernel of this map also admits a ∇–ﬁltration.

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MATTHEW BENNETT AND VYJAYANTHI CHARI

Proof. Let N ⊂ N+1 be a ∇–ﬁltration of M and assume that λ ∈ P + is minimal such that [M : ∇(λ, r)] = 0 for some r ∈ Z. Using Lemma 3.2 and an induction on , we see that Ext1 (∇(λ, r), N ) = 0, ≥ 1, r ≥ Z. ˜ ⊂ N such that This implies that for each , we have N ˜ ∼ ˜ ∩ N−1 = 0, N ∇(λ, r)⊕m (λ,r) , N = r

N ∼ ∇(μ, s)⊕m (μ,s) . = ˜ N−1 ⊕ N (μ,s):μ=λ Deﬁne a ﬁltration M ⊂ M+1 , ≥ 1 of M by, ˜s , M = N−1 N s>

where we recall that N0 = 0. Then N M ∼ ∼ = = ˜ M−1 N−1 ⊕ N Since wt M ∩ P the proof.

+

∇(μ, s)⊕m (μ,s) .

(μ,s):μ=λ

is ﬁnite if M ∈ Ob Ibdd , an iteration of this argument completes

3.10. The following Lemma establishes Proposition 2.6(i). Lemma. For all (λ, r), (μ, s) ∈ Λ, Ext1I (Δ(λ, r), ∇(μ, s)) = 0. In particular if N ∈ Ob Ibdd has a ∇–ﬁltration then Ext1I (Δ(λ, r), N ) = 0. Proof. The proof is standard. Thus, suppose that we have a short exact sequence ι τ 0 → ∇(μ, s) → M → Δ(λ, r) → 0. Then Mλ = 0 and if μ λ we have (n+ [t])Mλ = 0 = (h ⊗ tC[t])Mλ . It follows from the deﬁning relations of Δ(λ, r) that if m ∈ M [r]λ is such that τ (m) = wλ,r , then U(g[t])m is a quotient of Δ(λ, r) via the map wλ,r → m and hence the sequence splits. If μ ≥ λ, then by taking duals we have a short exact sequence τ∗

ι∗

0 → Δ(λ, r)∗ → M ∗ → W (−w0 μ, −s) → 0. ∗ = 0 and using the deﬁning relations of Since −w0 μ ≥ −w0 λ we have n+ [t]M−w 0μ ∗ W (−w0 μ, −s) we see that ι splits. Suppose that N ∈ Ob Ibdd admits a ∇–ﬁltration and let p ∈ Z be such that N [s] = 0 if s > p. It follows from Lemma 3.9 that there exists k ∈ Z+ and a ﬁltration 0 ⊂ N0 ⊂ N1 ⊂ · · · ⊂ Nk = N such that ∇(λs , )m(λs ,) , Ns /Ns−1 ∼ = ≤p

TILTING MODULES MODULES FOR THE CURRENT OF A SIMPLE LIE TILTING CURRENT ALGEBRA ALGEBRAOF LIE ALGEBRA ALGEBRA

for some m(λs , ) ∈ Z+ . Since Ext1I (Δ(λ, r), Ns /Ns−1 ) →

87 13

(Ext1I (Δ(λ, r), ∇(λs , )⊕m(λs ,) )

≤p 1

it follows that Ext (Δ(λ, r), Ns /Ns−1 ) = 0. An obvious induction on s along with the fact that Nk = N proves the Lemma. 3.11. Proposition. Suppose that g is of type sln+1 . An object M of Ibdd has a ∇–ﬁltration iﬀ the canonical ﬁltration of M is a ∇–ﬁltration. Proof. Suppose that M has a ∇–ﬁltration. Then we have proved in Section 3.8 that chgr M = dim HomI (Δ(λ, r), M )chgr ∇(λ, r). (λ,r)∈Λ

Hence equality must hold in (3.3) which was written for the canonical ﬁltration. This proves that the canonical ﬁltration is a ∇–ﬁltration. The converse is obvious. 4. Modules with Δ–filtrations In our situation the fact that the dual of a Δ–module is not a ∇–module means that we have to also study properties of modules admitting a Δ–ﬁltration. We also need some results on the vanishing of Ext1I (Δ(λ, r), Δ(μ, s)) which will be used to construct the tilting modules in the next section. 4.1. Consider the projection map pr : U(g[t]) → U(h[t]) → 0 corresponding to the vector space decomposition,

U(g[t]) = U(h[t]) n− [t]U(g[t]) + U(g[t])n+ [t] . For i ∈ I and s ∈ Z+ , deﬁne elements Pi,s ∈ U(h[t]) recursively, by 1 (hi ⊗ tr )Pi,s−r . s r=1 s

Pi,0 = 1,

Pi,s = −

The following was proved in [12] (see [8]) for the current formulation: Lemma. For i ∈ I and s ≥ 1, we have, s − s s 2 pr((x+ i ⊗ t) (xi ) ) = (−1) (s!) Pi,s .

4.2. Proposition. (i) Let (λ, r) ∈ Λ and assume that N ∈ Ob I satisﬁes, N [s]λ = 0 if r < s ≤ r + 1 +

(4.1)

n i=1

Then, Ext1I (Δ(λ, r), N ) = 0.

λ(hi ).

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MATTHEW BENNETT AND VYJAYANTHI CHARI

(ii) If λ, μ ∈ P + and μ λ, we have Ext1I (Δ(λ, r), Δ(μ, )) = 0, for all r, ∈ Z, and Ext1I (Δ(λ, r), Δ(λ, r)) = 0, for all r ∈ Z, (iii) Given λ, μ ∈ P + there exists d(λ, μ) ∈ Z+ such that Ext1I (Δ(λ, r), Δ(μ, s)) = 0 =⇒ |r − s| ≤ d(λ, μ). Proof. Consider a short exact sequence, ι

τ

0 → N → M → Δ(λ, r) → 0, of objects of Ibdd . Since the sequence splits as g–modules, we can and do choose + m ∈ M [r]nλ such that τ (m) = wλ,r . Then τ ((hi ⊗ ts )m) = 0 for all s > 0 or equivalently (h ⊗ ts )m ∈ N [r + s] and so equation (4.1) gives n (hi ⊗ ts )m = 0, 0<s≤1+ λ(hi ). i=1

Taking s = 1 gives 2(x+ αi ⊗ t)m = [hi ⊗ t, xαi ]m = 0, k and repeating we ﬁnd that for all all i ∈ I and k ∈ Z+ we have (x+ αi ⊗ t )m = 0. Applying Lemma 4.1 we have s − s (x+ αi ⊗ t) (xαi ) m = Pi,s m = 0,

s > λ(hi ).

Since Pi,s is a polynomial in hi ⊗ tk , 1 ≤ k ≤ s, it follows by an obvious induction that (hi ⊗ ts )m = 0 for all i and s. Hence we have proved that m satisﬁes the deﬁning relations of Δ(λ, r) which means that τ splits. The proof of (ii) is similar and easier and we omit the details. Part (iii) is immediate from part (i) and the fact that Δ(μ, s) is ﬁnite–dimensional. 4.3. Proposition. Suppose that N ∈ Ob I is such that Ext1 (Δ(λ, r), N ) = 0 for all (λ, r) ∈ P + × Z. If M has a Δ–ﬁltration then Ext1 (M, N ) = 0. Proof. Consider a short exact sequence 0 → N → U → M → 0. Suppose that Mk ⊂ Mk+1 is a part of the Δ-ﬁltration of M and assume that Δ(μ, s)ms . Mk+1 /Mk ∼ = (μ,s)∈Λ

By assumption we have Ext1I (Mk+1 /Mk , N ) = 0. Let Uk ⊂ U be the preimage of Mk and note that Uk+1 /Uk ∼ = Mk+1 /Mk . Consider the short exact sequence 0 → N → Uk → Mk → 0. This sequence deﬁnes an element of Ext1 (Mk , N ). Since Mk has a ﬁnite Δ–ﬁltration it follows that Ext1 (Mk , N ) = 0. Hence the sequence splits and we have a map ϕk : Uk → N . We want to prove that ϕk+1 : Uk+1 → N can be chosen to extend ϕk . For this, applying Hom(−, N ) to 0 → Uk → Uk+1 → Uk+1 /Uk → 0,

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we get Hom(Uk+1 , N ) → Hom(Uk , N ) → 0, which shows that we can choose ϕk+1 to lift ϕk . Now deﬁning ϕ : U → N, ϕ(u) = ϕk (u), u ∈ Uk , we have the desired splitting of the original short exact sequence.

Together with Lemma 3.10 we now have, Corollary. Let M be a module with a Δ–ﬁltration. Then Ext1I (M, ∇(λ, r)) = 0, for all (λ, r) ∈ Λ. 4.4. Lemma. Let (λ, r), (μ, s) ∈ Λ. We have C, (λ, r) = (μ, s), HomI (Δ(λ, r), ∇(μ, s)) ∼ = 0 otherwise. Proof. Suppose that ϕ : Δ(λ, r) → ∇(μ, s) is non–zero. Then ϕ(wλ,r ) = 0 and hence we have λ ≤ μ and r ≤ s. Moreover since any submodule of ∇(μ, s) has non–zero socle it follows that ∇(μ, s)[s]μ must be in the image of ϕ which shows that μ ≤ λ and s ≥ r. 4.5. We end the section with a ﬁnal result needed to construct T (λ, r). It can be deduced from the fact (proved in [5]) that the space of extensions between irreducible objects of I is ﬁnite–dimensional, but we include a proof for convenience. Proposition. For (λ, r), (μ, s) ∈ Λ, we have dim Ext1I (Δ(λ, r), Δ(μ, s)) < ∞. Proof. Let π : P (λ, r) → Δ(λ, r) → 0 be the canonical projection which maps pλ,r to wλ,r . Apply Hom(−−, Δ(μ, s)) to the short exact sequence 0 → ker π → P (λ, r) → Δ(λ, r) → 0. Since P (λ, r) is a projective object of I, the result follows if we prove that dim HomI (ker π, Δ(μ, s)) < ∞.

(4.2)

Choose ∈ Z such that Δ(μ, s)[p] = 0 for all p > , in which case we have an injective map

ker π HomI (ker π, Δ(μ, s)) → dim HomI , Δ(μ, s) . p> ker π[] Since

dim

equation 4.2 is proved.

ker π p> ker π[]

=

dim ker π[p] < ∞,

p=r

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5. The modules T (λ, r) In this section we construct a family of indecomposable modules T (λ, r), (λ, r) ∈ Λ which admit a Δ–ﬁltration and satisfy Ext1I (Δ(μ, s), T (λ, r)) = 0, (μ, s) ∈ Λ. In the case when g is of type sln+1 , it follows that the modules T (λ, r) are tilting and we prove that any tilting module is a direct sum of copies of T (λ, r), (λ, r) ∈ Λ. The ideas we use are similar to the ones given in [15] but there are several diﬃculties to be overcome in our situation. 5.1. One of the ﬁrst diﬃculties we encounter in trying to use the algorithm given in [15] to construct T (λ, r) is that Λ cannot be enumerated by elements of Z+ . Hence our ﬁrst step is to show that we can ﬁnd a suitable subset (depending on (λ, r)) of Λ which can be so enumerated. This is done as follows. Recall that we have ﬁxed an enumeration λ0 , λ1 , · · · of P + such that λi ≤ λj =⇒ i ≤ j. Given k ≥ 0 deﬁne integers r0 ≥ r1 ≥ · · · ≥ rk recursively as follows. Set rk = 0 and for s < k, rs = max{r ≥ rs+1 : Δ(λs+1 , rs+1 )[r] = 0}. Set Sk = {(λs , r) : 0 ≤ s ≤ k, r ≤ rs }, and deﬁne ηk : Sk → Z+ by ηk (λs , rs − ) = k − s + (k + 1), s ≥ 0. It is easily checked that ηk deﬁnes a one to one map of sets. To see that it is surjective, given m ∈ Z+ write m = (k + 1) + k − s where 0 ≤ s ≤ k. Then, ηk (λs , rs − ) = m which proves that ηk is onto. As an example, consider the case when g = sl2 and λ = 4ω1 . If we identify P + with Z+ and use the obvious enumeration of Z+ , then 4ω1 = λ4 . By examining the graded characters of the local Weyl modules (see for example [6]) we ﬁnd r4 = 0, r3 = 4, r2 = 6, r1 = 7, r0 = 7. Then η4 (4ω, 0) = 0, η4 (3ω, 4) = 1, η4 (2ω, 6) = 2, η4 (ω, 7) = 3, η4 (0, 7) = 4, η4 (4ω, −1) = 5, etc. 5.2. From now on, we shall use the fact that Ext1I (M, N ) ∼ = Ext1I (τr M, τr N ) freely and without comment. The following result will play a crucial role in our construction. Proposition. Suppose that (λs , r), (λp , ) ∈ Sk for some 0 ≤ s, p ≤ k and assume that ηk (λs , r) < ηk (λp , ).Then, (i) (5.1)

Δ(λs , r)[] = 0, > r0 ≥ rs ≥ r.

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(ii) Δ(λp , )[r]λs = 0. (iii) Ext1I (Δ(λs , r), Δ(λp , )) = 0.

(5.2) (iv) If (μ, m) ∈ / Sk , then (5.3)

Ext1I (Δ(μ, m), Δ(λs , r) = 0.

Proof. Part (i) follows by noticing that λ0 = 0 and that Δ(λ0 , r) ∼ = τr C. Part (ii) is obvious if λs λp . If λs ≤ λp then rs ≥ rp and ηk (λs , r) < ηk (λp , ) =⇒ r + rp − > +rs . The deﬁnition of rs implies that Δ(λp , rp )[r + rp − ] = 0. Applying τ−rp proves (ii). To prove (iii), it suﬃces to show that (5.4)

Ext1I (Δ(λs , rs ), Δ(λp , − r + rs )) = 0.

If s ≥ p, this is immediate from Proposition 4.2(ii). If s < p, then rs ≥ rp and rp > − r + rs . Observing that rs ≥ max{m : Δ(λp , rp )[m] = 0} > max{m : Δ(λp , − r + rs )[m] = 0}, we see that (5.4) follows from Proposition 4.2(i) and part (iii) is proved. . The proof of (iv) is similar and we omit the details. 5.3. We need the following elementary result. Lemma. Suppose that M, N ∈ Ob I are such that 0 < dim Ext1I (M, N ) < ∞ and Ext1I (M, M ) = 0. Then, there exists U ∈ Ob I, d ∈ Z+ and a non–split short exact sequence 0 → N → U → M ⊕d → 0 so that Ext1I (M, U ) = 0. In what follows, we shall repeatedly use the fact that Proposition 4.5 implies that if N ∈ Ob I is ﬁnite–dimensional and admits a Δ–ﬁltration, then dim Ext1I (Δ(μ, r), N ) < ∞ for all (μ, r) ∈ Λ. 5.4. Fix k ∈ Z+ and given ∈ Z+ , denote by (μ , p ) ∈ Sk , the unique element such that ηk (μ , p ) = . Notice that (μ0 , p0 ) = (λk , 0). Deﬁne ﬁnite–dimensional objects {Ms : s ≥ 0} (depending on k) of I, recursively as follows. Set M0 = Δ(λk , 0). If Ext1I (Δ(μ1 , p1 ), M0 ) = 0, take M0 = M1 . Otherwise, let U ∈ Ob I be chosen as in Lemma 5.3 (with M = Δ(μ1 , p1 ) and N = M0 ). Clearly U is ﬁnite–dimensional and U [0]λ ∼ = (M0 )λ . k

k

Let M1 be the indecomposable summand of U which contains U [0]λk in which case we also have, Ext1I (Δ(μ1 , p1 ), M1 ) = 0.

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Since M0 is indecomposable and generated as a g[t]–module by (M0 )λk , it follows that we have an inclusion ι0 : M0 → M1 . Next, the fact that Δ(μ1 , p1 ) is indecomposable, implies that there exists d1 ∈ Z+ and a non–split short exact sequence of the form ι0 τ 0 → M0 → M1 → Δ(μ1 , p1 )⊕d1 → 0. Clearly, M1 is generated as a g[t]–module by the spaces M1 [0]λk and M1 [p1 ]μ1 , and Proposition 5.2 yields, (5.5)

Ext1I (Δ(μj , pj ), M1 ) = 0, j = 0, 1, Ext1I (Δ(μ, r), M1 ) = 0, (μ, r) ∈ / Sk .

Repeating this procedure and using Proposition 5.2 at each step we have the following proposition. Proposition. There exists a family Ms , s ≥ 0, of indecomposable ﬁnite– dimensional modules and injective morphisms ιs : Ms → Ms+1 of objects of I which have the following properties. (i) M0 = Δ(λk , 0), and for s ≥ 1, Ms /ιs−1 (Ms−1 ) ∼ = Δ(μs , ps )⊕ds , ds ∈ Z+ , dim Ms [0]λk = 1,

wt Ms ⊂ conv W λk .

(ii) Ext1I (Δ(μ , p ), Ms ) = 0, 0 ≤ ≤ s,

Ext1I (Δ(μ, r), Ms ) = 0, (μ, r) ∈ / Sk .

(iii) Ms [p] = 0, for all s ≥ 0,

(5.6)

p > r0 .

(iv) Ms is generated as a g[t]–module by the spaces {Ms [p ]μ : ≤ s}. Moreover, if we let ιr,s = ιs−1 · · · ιr : Mr → Ms , r < s, ιr,r = id, then (5.7)

Ms [p ]μ = ι,s (M [p ]μ ), s ≥ .

5.5. Let T (λk , 0) be the direct limit of {Ms , ιr,s : r, s ∈ Z+ , r ≤ s}. Since the maps ιr,s are injective morphisms it follows that the canonical morphism Ms → ˜ s of T (λk , 0) is injective and we have an isomorphism of Ms with a submodule M ˜s ⊂ M ˜ s+1 and T (λk , 0). Moreover, we have inclusions M ˜ s /M ˜ s−1 ∼ ˜ s, T (λk , 0) = M M = Ms /Ms−1 , s ≥ 0. s≥0

It follows from Proposition 5.4 that T (λk , 0) ∈ Ob Ibdd and that it has a Δ– ˜ s . Then, Propoﬁltration. From now on, by abuse of notation, we write Ms for M sition 5.4(iv) gives that (5.8) T (λk , 0)[p ]μ = M [p ]μ , Ms = U(g[t])T (λk , 0)[p ]μ . ≤s

To prove that T (λk , 0) is indecomposable, suppose that T (λk , 0) = U1 ⊕ U2 .

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Since dim T (λk , 0)[0]λk = 1, we may assume without loss of generality that T (λk , 0)[0]λk ⊂ U1 and hence M0 ⊂ U1 . Assume that we have proved by induction that Ms−1 ⊂ U1 . Since Ms is generated as a g[t]–module by the spaces {Ms [p ]μ : ≤ s}, it suﬃces to prove that Ms [ps ]μs ⊂ U1 . By (5.8), we have Ui [ps ]μs ⊂ Ms and hence Ms = (Ms−1 + U(g[t])U1 [ps ]μs ) U(g[t])U2 [ps ]μs . Since Ms is indecomposable by construction, it follows that U2 [ps ]μs = 0 and Ms ⊂ U1 which completes the inductive step. Proposition. Let (λk , 0) ∈ Λ. (i) There exists an indecomposable object T (λk , 0) in Ibdd which admits a ﬁltration by ﬁnite–dimensional modules (5.9) Ms = U(g[t])T (λk , 0)[p ]μ , s ≥ 0, ≤s

such that M0 ∼ = Δ(λk , 0) and the successive quotients are isomorphic to a ﬁnite–direct sum of Δ(μ, s), (μ, s) ∈ Sk . Moreover, wt T (λk , 0) ⊂ conv W λk , dim T (λk , 0)[0]λk = 1. (ii) For all (μ, s) ∈ Λ, we have Ext1I (Δ(μ, s)), T (λk , 0)) = 0,

(5.10)

(iii) Analogous statements hold for T (λk , r) = τr T (λk , 0) and moreover, ∼ T (λp , s) ⇐⇒ k = p, r = s. T (λk , r) = Proof. The proof of (i) is the discussion immediately preceding the proposition. To prove (ii), consider a short exact sequence 0 → T (λk , 0) → U → Δ(μ, s) → 0.

(5.11)

If μ λk , an argument identical to the one given in the proof of Lemma 3.10 proves that the short exact sequence in (5.11) must split. Suppose that μ ≤ λk . We claim that there exists r ∈ Z+ such that for all p ∈ Z Mr [] = Mp [], s ≤ ≤ s + 1 +

n

μ(hi ),

i=1

and Ext1I (Δ(μ, s), Mr ) = 0.

(5.12)

Since T (λk , 0) has ﬁnite–dimensional graded pieces there exists r˜ ∈ Z+ such that for all p ≥ r˜ we have Mr [] = Mp [], s ≤ ≤ s + 1 +

n

μ(hi ).

i=1

Set

r=

max{˜ r, s˜}, (μ, s) ∈ Sk , ηk (μ, s) = s˜, r˜, (μ, s) ∈ / Sk .

The claim now follows from Proposition 5.4(ii).

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Consider the short exact sequence 0 → Mr → T (λk , 0) → T (λk , 0)/Mr → 0. Applying HomI (Δ(μ, s), −) to the short exact sequence we get from Proposition 4.2(i) that Ext1I (Δ(μ, s), T (λk , 0)/Mr ) = 0. Using (5.12) we see that equation (5.10) is proved. Part (iii) is immediate.

The following is an analog of Fitting’s Lemma for the inﬁnite–dimensional modules T (λk , 0). Corollary. Let ψ : T (λk , 0) → T (λk , 0) be any morphism of objects of I. Then ψ(Ms ) ⊂ Ms for all s ≥ 0 and ψ is either an isomorphism or locally nilpotent, i.e., given m ∈ M , there exists ≥ 0 (depending on m) such that ψ (m) = 0. Proof. Since ψ preserves both weight spaces and graded components it follows from (5.9) that ψ(Ms ) ⊂ Ms for all s ≥ 0. Moreover, since Ms is indecomposable and ﬁnite–dimensional it follows from Fitting’s Lemma that the restriction of ψ to Ms , s ≥ 0 is either nilpotent or an isomorphism. If all the restrictions are isomorphisms then since T (λk , 0) is the union of Ms , s ≥ 0, it follows that ψ is an isomorphism. On the other hand, if the restriction of ψ to some Ms is nilpotent, then the restriction of ψ to all M , ≥ 0 is nilpotent which proves that ψ is locally nilpotent. 5.6. The following result is now a consequence of Proposition 2.6, Proposition 5.5(ii) and Theorem 3.6. Λ.

Proposition. If g is of type sln+1 , the objects T (λ, r) are tilting for all (λ, r) ∈

In the rest of the section we shall complete the proof of Theorem 2.7 by showing that any indecomposable tilting module is isomorphic to some T (λ, r) and that any tilting module in Ibdd is isomorphic to a direct sum of tilting modules. 5.7. Let T ∈ Ibdd be a ﬁxed tilting module. Using Proposition 2.6 and Corollary 4.3, we have (5.13)

Ext1I (T, ∇(λ, r)) = Ext1I (Δ(λ, r), T )) = 0, (λ, r) ∈ Λ.

Lemma. Suppose that T1 is any summand of T . Then T1 admits a ∇–ﬁltration and Ext1I (T1 , ∇(λ, r)) = 0, for all (λ, r) ∈ Z. Proof. Since Ext1 commutes with ﬁnite direct sums, we get Ext1I (T1 , ∇(λ, r)) = 0, Ext1I (Δ(λ, r), T1 )) = 0, (λ, r) ∈ Λ. Under the assumption that Proposition 2.6 is true, the second equality implies that T1 has a ∇-ﬁltration and the proof of the Lemma is complete.

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5.8. The preceding lemma illustrates one of the diﬃculties we face in our situation. Namely, we cannot directly conclude that T1 has a Δ–ﬁltration from the vanishing Ext–condition by using the dual of Proposition 2.6. However, we can prove, Proposition. Suppose that N ∈ Ibdd has a ∇–ﬁltration and satisﬁes Ext1I (N, ∇(λ, r)) = 0, for all (λ, r) ∈ Λ. There exists (μ, s) ∈ Λ such that T (μ, s) is a summand of N . Proof. Since N has a ∇–ﬁltration we can use Proposition 3.9 to choose (μ, s) ∈ Λ and surjective morphisms ϕ : N → ∇(μ, s) → 0, π : T (μ, s) → ∇(μ, s) → 0. We may also assume that ker ϕ and ker π have ﬁnite ∇–ﬁltrations. Let vμ,s be a non–zero element of ∇(μ, s)[s]μ and choose m ∈ N [s]μ and u ∈ T (μ, s)[s]μ so that ϕ(m) = vμ,s = π(u). Consider the short exact sequences 0 → ker ϕ → N → ∇(μ, s) → 0, and 0 → ker π → T (μ, s) → ∇(μ, s) → 0. Apply HomI (T (μ, s), −) to the ﬁrst sequence and HomI (N, −) to the second sequence. Since ker ϕ and ker π admit a ∇–ﬁltration, equation (5.13) and the hypothesis on N respectively, give, Ext1I (T (μ, s), ker ϕ) = 0

Ext1I (N, ker π) = 0.

Hence we have surjective maps HomI (T (μ, s), N ) → HomI (T (μ, s), ∇(μ, s)) → 0, HomI (N, T (μ, s)) → HomI (N, ∇(μ, s)) → 0. ˜ ∈ HomI (T (μ, s), N ) such that Choose ϕ˜ ∈ HomI (N, T (μ, s)) and π π.ϕ˜ = ϕ, ϕ.˜ π = π. This gives that π.ϕ.˜ ˜π=π Setting ψ = ϕ.˜ ˜ π , and using the fact that dim T (μ, s)[s]μ = 1, we see that ψ(u) = u. Corollary 5.5 proves that ψ is an isomorphism and hence π ˜ .ψ −1 is a splitting of ϕ˜ : N → T (μ, s) proving that T (μ, s) is a summand of N . Corollary. Any indecomposable tilting module is isomorphic to T (λ, r) for some (λ, r) ∈ Λ. Further if T ∈ Ob Ibdd is tilting there exists (λ, r) ∈ Λ such that T (λ, r) is isomorphic to a direct summand of T . Proof. Since T and T (λ, r) are tilting they satisfy (5.13) and the corollary follows.

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5.9. We emphasize again that the restriction in the following theorem stems only from the fact that Theorem 3.6 has not been proved outside the sln+1 –case. Theorem. Assume that g is of type sln+1 and let T ∈ Ob Ibdd . The following are equivalent. (i) T is tilting. (ii) Ext1I (Δ(λ, r), T ) = 0 = Ext1I (T, ∇(λ, r)), (λ, r) ∈ Λ. (iii) T is isomorphic to a direct sum of objects T (μ, s), (μ, s) ∈ Λ. Proof. Since the direct sum of tilting modules is tilting it follows that (iii) implies (i). Equation 5.13 proves that (i) implies (ii). We now prove that (ii) implies (iii). Proposition 2.6 implies that T has a ∇–ﬁltration. By Lemma 3.9 we can choose λ ∈ P + be maximal such that [T : ∇(λ, r)] = 0 for some r ∈ Z. Since T ∈ Ob Ibdd we can ﬁx a sequence r1 ≥ r2 ≥ · · · such that [T : ∇(λ, s)] = 0 =⇒ s = rj for some j ≥ 1. Note that either this sequence is ﬁnite, or an inﬁnite number of distinct integers occur. Again using Lemma 3.9 we have a surjective morphism T → ∇(λ, r1 ) and the proof of Proposition 5.8 gives that ∼ T (λ, r1 ) ⊕ T1 . T = Clearly T1 satisﬁes the same conditions as T and we can repeat this procedure. We ﬁnd that for j ≥ 1, there exists Tj ∈ Ob Ibdd such that T ∼ = Tj

j

T (λ, rs ).

s=1

Let πj : T → T (λ, rj ) be the canonical projection. Since T has ﬁnite–dimensional graded pieces and rj ≤ rj−1 and the modules T (λ, rj ) are all graded shifts of T (λ, 0), it follows that for any m ∈ T we have πj (m) = 0 for all but ﬁnitely many j. Hence we have a surjective map π:T → T (λ, rj ) → 0, and ker π = Tj . j≥1

j≥1

In particular, it follows that T ∼ = ker π

T (λ, rj )

ker π.

j≥1

Repeat the argument with ker π. Since (ker π)λ = 0, the argument stops eventually and the proof is complete. References [1] E. Ardonne and R. Kedem, Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas, J.Algebra 308 (2007), 270–294. [2] M. Bennett, A. Berenstein, V. Chari, A. Khoroshkin and S. Loktev, BGG reciprocity for the current algebra of sln+1 , in preparation. [3] M. Bennett, V. Chari, and N. Manning, BGG reciprocity for current algebras, to appear in Adv. Math. [4] V. Chari, G. Fourier and T. Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, 517–549. [5] V. Chari and J. Greenstein, Current algebras, highest weight categories and quivers, Adv. Math. 216 (2007), no. 2, 811–840.

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[6] V. Chari and S. Loktev, Weyl, Demazure and fusion modules for the current algebra of slr+1 , Adv. Math. 207 (2006), 928–960. [7] E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. [8] V. Chari and A. Pressley, Weyl modules for classical and quantum aﬃne algebras, Represent. Theory 5 (2001), 191–223 (electronic). [9] S. Donkin, Tilting modules for algebraic groups and ﬁnite dimensional algebras, A Handbook of Tilting Theory, London Math. Society Lect. Notes 332 (2007), 215–257. [10] G. Fourier and P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), no. 2, 566–593. [11] P. Di Francesco, and R. Kedem,Proof of the combinatorial Kirillov-Reshetikhin conjecture, arXiv:0710.4415. [12] H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480–551. [13] D. Happel, and C.M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399–443. [14] R. Kodera and K. Naoi, Loewy series of Weyl modules and the Poincar´ e polynomials of quiver varieties, arXiv:1103.4207. [15] O. Mathieu, Tilting modules and their applications, Adv. Studies in Pure Math. 26 (2000), 145–212. [16] V. Mazorchuk, Koszul duality for stratiﬁed algebras, II. Standardly stratiﬁed algebras, Jour. Aus. Math. Society, 89 (2010), 23–49. [17] K. Naoi, Weyl modules, Demazure modules and ﬁnite crystals for non-simply laced type, arXiv:1012.5480. [18] K. Naoi, Fusion products of Kirillov-Reshetikhin modules and the X = M conjecture, arXiv:1109.2450. Department of Mathematics, University of California, Riverside, CA 92521, ]U.S.A. E-mail address: [email protected] Department of Mathematics, University of California, Riverside, CA 92521, ]U.S.A. E-mail address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Endotrivial modules Jon F. Carlson Abstract. This paper is a survey of the current state of the eﬀort to classify endotrivial modules over the group algebras of ﬁnite groups.

1. Introduction and background. Endotrivial modules play an important role in the modular representation theory of ﬁnite groups and have been studied by many people since they were introduced by Dade more than 30 years ago. They serve as the building block of the endopermutation modules, and the classiﬁcation of endotrivial modules for p-groups was crucial for Bouc’s characterization of the Dade group [5]. In addition, the endotrivial modules form an important part of the Picard group of self-equivalence of the stable category of modules over a group algebra. Tensoring with an endotrivial module is a self-equivalence of Morita type on the stable category. Suppose that G is a ﬁnite group and k is a ﬁeld of characteristic p > 0. A ﬁnitely generated kG-module is an endotrivial module provided its endomorphism algebra Homk (M, M ) = Endk (M ) ∼ = k ⊕ proj . (Here ⊕ proj means the direct sum with some projective module.) Recall, that kG is a Hopf algebra, and the above isomorphism is an isomorphism of kG-modules. That is, in the stable category stmod(kG) of ﬁnitely generated kG-modules modulo projectives, Endk (M ) is isomorphic to the trivial module. A kG-module is an endopermutation module if Endk M is isomorphic to a permutation module in stmod(kG). Note that as kG-modules, Endk (M ) ∼ = M ∗ ⊗ M (where “⊗” means “⊗k ”). So we deﬁne the group T (G) to be the group with elements the equivalence classes [M ] of endotrivial modules M in the stable category ([M ⊕ proj] = [M ]), and with operation [M ] + [N ] = [M ⊗k N ]. Thus T (G) is an abelian group. Its identity is the class [k] of the trivial module k, and the inverse of the class [M ] of an endotrivial module M is the class [M ∗ ] of the k-dual of M . We should note that although we assume in this survey that our modules are ﬁnitely generated, we know that any endotrivial module, even of inﬁnite dimension, is the direct sum of a projective module and a ﬁnitely generated endotrivial module [2] 2000 Mathematics Subject Classiﬁcation. 20C20 . Partially supported by NSF grant No. 101102. 1 99

c 2012 American Mathematical Society

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JON F. CARLSON

A brief history of the subject starts with the work of Dade, published in 1978, which ﬁrst deﬁned endotrivial modules and classiﬁed them in the case that G is an abelian p-group [20, 21]. Dade showed that for p-nilpotent groups, the sources (in the sense of J. A. Green’s theory of vertices and sources) of the irreducible modules are endopermutation modules, which are built from endotrivial modules. Among other things, this means that if G is a p-nilpotent group, then any irreducible kGmodule is a direct summand of an induced module from an endopermutation module on a p-subgroup of G. This theorem was soon extended to p-solvable groups by Puig. Puig also demonstrated other roles of the endotrivial modules in block theory [31, 30]. For p-groups, the torsion free rank of the group of endotrivial modules was established by a theorem of Alperin in 1999 [1] and extended to arbitrary ﬁnite groups in [10]. A classiﬁcation of endotrivial modules for G a p-group was completed by the author and Jacques Th´evenaz [16, 17, 18] a few years later. Building on the classiﬁcation for p-groups, a classiﬁcation of endopermutation modules over p-groups by Bouc in 2005 [5]. Progress on classiﬁcations of endotrivial modules for other groups has been made by the author, Dave Hemmer, Nadia Mazza, Dan Nakano, Gabriel Navarro, Geoﬀ Robinson and Th´evenaz in various combinations. Some results for inﬁnitesimal group schemes have been added by the author and Nakano. As mentioned above, motivation for the latter work comes in part from the fact that if M is an endotrivial kG-module, then − ⊗ M : stmod(kG) → stmod(kG) is a self-equivalence of the stable category. So T (G) is a part of the Picard group of self-equivalences of the stable category - the self-equivalences of Morita type. We present some details of these results later in the paper. This papers is a composite of three talks that I gave in the spring of 2011 as surveys of recent work on endotrivial modules at the Maurice Auslander Distinguished Lectures in Woods Hole, in April, at the Leonard Scott Day which was connected to the Southeast Lie Theory meeting in Charlottesville in June, and at a meeting on Triangulated Categories and Applications at BIRS in Banﬀ, Canada in June. I was honored by the invitations and grateful to the organizers of each of these conferences for the chance to present this work. While a lot of the material was introduced in all three lectures, I tried to emphasize in each lecture aspects of the theory that would be of most interest to the audience at hand.

2. Basic results. Clearly, the trivial kG-module k is an endotrivial module. Moreover, if M is a one-dimensional kG-module, then M ⊗ M ∗ ∼ = k and hence, M is an endotrivial module. Some other well known examples require the following notations. If M is a kG-module, then Ω(M ) is the kernel of the projective cover P → M . So we have an exact sequence 0

/ Ω(M )

/P

/M

/ 0.

Inductively, we let Ωn (M ) = Ω(Ωn−1 (M )), for any n > 1. Similarly Ω−1 (M ) is the cokernel of the injective hull M → Q, and Ω−n (M ) = Ω−1 (Ω−n+1 (M )). Our basic examples, then, are the following.

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Examples: The trivial module k and its syzygies Ωn (k) are endotrivial modules. Examples: Any kG-module M of dimension 1, and all of its syzygies Ωn (M ) are endotrivial modules. The calculus that veriﬁes these example is the contained in the following two statements. (1) Ωm (M ) ⊗ Ωn (N ) ∼ and = Ωm+n (M ⊗ N ) ⊕ proj, m ∗ ∼ −m (2) (Ω (M )) = Ω (M ∗ ). Both of the statements can be derived from the self-injectivity of the group algebra kG and from the fact that for any kG-module M , M ⊗ P is projective whenever P is a projective kG-module. Everett Dade introduced the terms “endotrivial module” and “endopermutation module”, in his seminal papers [20]. In addition, he proved the following remarkable result. Theorem 2.1. If G is an abelian p-group having p-rank at least two, then T (G) ∼ = Z and is generated by [Ω(k)]. To understand the importance of the theorem for our investigation, we need to consider the following easy consequence of the Theorem of Chouinard (which in turn is a consequence of the Theorem of Quillen that we mention below). Proposition 2.2. A kG-module M is endotrivial if and only if its restriction ME to every elementary abelian p-subgroup E is an endotrivial module. We can give a brief sketch of the proof here. First note that the “only if” part is easy. For the “if”, we have an exact sequence 0

/ Ker

/ Homk (M, M )

tr

/k

/0

where tr is the trace map with kernel Ker. On restriction to any elementary abelian p-subgroup, the middle term is k ⊕ proj. Moreover, the identity homomorphism generates a submodule of Homk (M, M ) of dimension one, which has nonzero trace (since the dimension of the module is not divisible by p). Consequently, on restriction to any elementary abelian p-subgroup, the sequence splits, and hence, Ker is projective on restriction to every elementary abelian subgroup. Now by Chouinard’s Theorem (see one of the general references [4] or [19]), Ker is projective and hence also injective. So the sequence splits. What this means for an eﬀort at classiﬁcation of the endotrivial module can be seen from the restriction map: / T T (G) / T (G) res / T (E) /0 0 E where the product is over a complete set of representatives E of the maximal elementary abelian p-subgroups of G. For any such E, there is a restriction map T (G) −→ T (E) which takes that the class [M ] of an endotrivial module M to the class [ME ] of its restriction to a kE-module. Note that unless G has a cyclic or quaternion Sylow p-subgroup, T (E) ∼ = Z. Hence, the image of res is torsion free. If we wish to ﬁnd T (G), then we need to answer two questions

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• What is the image of the restriction map, res? and • What is the kernel, T T (G)? In later sections we survey what is knows about the answers to these questions. But ﬁrst we review the theory of support varieties. 3. A quick tour of support varieties. Suppose that G is a ﬁnite group or group scheme, and k is a ﬁeld of characteristic p > 0. By theorems of Evens-Golod-Venkov in the case of ﬁnite groups and Friedlander-Suslin in the case of ﬁnite group schemes, the cohomology ring H∗ (G, k) is a ﬁnitely generated k-algebra and for any ﬁnitely generated kG-module M , the cohomology H∗ (G, M ) is a ﬁnitely generated module over H∗ (G, k). It is an easy proof that the cohomology ring is graded-commutative and from these facts we deduce that its projectivized prime ideal spectrum VG (k) = Proj H∗ (G, k), is a ﬁnite dimensional projective variety. Now suppose that M is a ﬁnite dimensional kG-module. Let J(M ) be the annihilator in H∗ (G, k) of the cohomology module Ext∗kG (M, M ) ∼ = H∗ (G, Homk (M, M )). The (cohomological) support variety VG (M ) is the subvariety of VG (k) corresponding to J(M ) - that is, the set of prime ideals in H∗ (G, k) that contain J(M ). This variety has many properties that have proved to be useful. Some applications appear later in this paper. All of the results that we mention here can be found in one of the standard references [4, 19]. Theorem 3.1. Suppose that L, M and N are kG-modules. Then we have the following. (1) VG (M ) = ∅ if and only if M is a projective module. (2) If 0 → L → M → N → 0 is an exact sequence then, VG (M ) ⊆ VG (L) ∪ VG (N ) with equality if the sequence is split. (3) VG (M ⊗ N ) = VG (M ) ∩ VG (N ). (4) If VG (M ) = V1 ∪V2 with V1 ∩V2 = ∅, then there exist submodules M1 , M2 ⊆ M such that M ∼ = M1 ⊕ M2 with VG (M1 ) = V1 and VG (M2 ) = V2 . One more theorem has played an exceptionally important role in the subject. It is the theorem of Quillen. This was generalized to support varieties by AlperinEvens and independently by Avrunin. Theorem 3.2. Suppose that G is a ﬁnite group. Then • (Quillen) VG (k) = ∪ res∗G,E (VE (k)) where res∗G,E is the induced map on varieties coming from the restriction resG,E : H∗ (G, k) → H∗ (E, k), and where the union is over the collection of conjugacy classes of maximal elementary abelian subgroups of G. • (Alperin-Evens, Avrunin) If M is a ﬁnitely generated kG-module, then VG (M ) = ∪ res∗G,E (VE (M )). It follows that if we wish to compute the support variety of a module M we can do so by restricting to elementary abelian p-subgroups. At the level of elementary abelian p-subgroups, the support variety of M can be computed as the rank variety, introduced by the author. A generalization of the rank variety that works for all ﬁnite group schemes G was introduced by Friedlander and Julia Pevtsova a few years ago. This provides another sort of ”support variety” for modules over group schemes. It also provides a useful criterion for endotrivial modules.

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Remember that if x is a cyclic group of order p, then k x ∼ = k[t]/(tp ), by letting t = x − 1. A π-point is a ﬂat map αK : K[t]/(tp ) → KG for some extension K of k. It is additionally required that a π-point factor by ﬂat maps through some abelian unipotent subgroup scheme of G. The ﬂatness assures that KG is projective as a module over the algebra K[t]/(tp ). Two π-points αK and βL are equivalent if for ∗ all kG-modules M , αK (K ⊗ M ) is projective if and only if βL∗ (L ⊗ M ) is projective. ∗ By αK (K ⊗ M ), we mean the restriction of K ⊗ M to a K[t]/(tp )-module along αK . Let Π(G) be the set of all equivalence classes of π-points. Then Theorem 3.3. [23, 24]

Π(G) ∼ = VG (k) (Proj H∗ (G, k)).

If M is a kG-module and αK : K[t]/(tp ) → KG is a π-point, then the isomor∗ phism type of the restriction αK (K ⊗ M ) of M to a K[t]/(tp )-module is determined entirely by the Jordan canonical form of αK (t) on M . The Jordan type of αK (t) on M is a partition [p|p . . . p|p − 1| . . . |1], the list of the sizes of the Jordan blocks. Note that there is an ordering on partitions. With these notions, Friedlander, Pevtsova and Suslin prove the following rather surprising theorem. Theorem 3.4. [25] The collection of all classes [αK ], such that the Jordan type on M is less than maximal, is a closed set in Π(G). Moreover, αK has maximal Jordan type on M if and only if any other π-point in its equivalence class has maximal Jordan type. Friedlander, Pevtsova and Suslin used this to deﬁne a nonmaximal support variety for a ﬁnite dimensional kG-module M . This is the collection of equivalence classes of π-points whose Jordan type on M is less than maximal. A kG-module is said to have constant Jordan type if its nonmaximal support variety is empty, or, equivalently, any two π-points have the same Jordan type on M . What this has to do with endotrivial modules is the following. Theorem 3.5. [8] A kG-module M is an endotrivial module if and only if it has constant Jordan type of the form either [p | . . . | p | 1]

or

[ p | . . . | p | p − 1]

(all blocks have size p except one that has size 1 or p − 1). One of the important aspects of the theorem is that it holds for all ﬁnite group schemes. In the case of inﬁnitesimal group schemes, where Quillen’s theorem tells us little, this is one of the few techniques that is available. 4. Classifying endotrivial modules: an overview Classifying the endotrivial modules over G means ﬁnding the group T (G) and its generators. For ﬁnite p-groups, this was accomplished by the author and Th´evenaz a few years ago. Roughly speaking the theorems are the following. The cyclic case is easily proved knowing the indecomposable modules. The quaternion case had certainly been ﬁgured out by Dade in the early stages of the theory. Theorem 4.1. Suppose that G is a p-group of p-rank one. If G is cyclic of order pn then, T (G) ∼ = Z/2 if pn > 2 or T (G) = {0} if pn ≤ 2. If p = 2, and G is quaternion then, T (G) ∼ = Z/4 ⊕ Z/2, where the cyclic group of order 4 is generated by the class of Ω(k).

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Theorem 4.2. [17, 18] Suppose that G is a p-group having p-rank at least two. Then with a few exceptions (which we list below) T (G) ∼ = Z, generated by the class of Ω(k). If G is a p-group with p-rank at least 2, then the only situation in which T (G) has torsion is that where G is a semi-dihedral 2-group (p = 2). For this group, T (G) ∼ = Z ⊕ Z/2, the free part being generated by the class of Ω(k). So the question is the torsion free rank and the generators of T (G) in the case that G has p-rank at least 2. The torsion free rank was determined by Alperin [1], who determined a set of generators, though the proof that this is a complete set of generators only came later [6, 18]. We establish a number n = n(G) from the group structure as follows. First, if the maximal elementary abelian p-subgroups all have rank at least three, then n(G) = 1 and the torsion free part T F (G) of T (G) is cyclic (Z) generated by the class of Ω(k). Otherwise, let E1 , . . . , En be representatives of the conjugacy classes of elementary abelian subgroups of rank 2 such that (1) if the p-rank of G (the maximal rank of any elementary abelian p-subgroup) is 2), then E1 , . . . , En is a complete set of representatives of the conjugacy classes of maximal elementary abelian subgroups, or (2) if the p-rank of G is greater than 2, then E1 , . . . , En−1 is a complete set of representatives of the conjugacy classes of elementary abelian p-subgroups of p-rank two and En is contained in an elementary abelian subgroup of rank 3 or more. Then we have the theorem. Theorem 4.3. [1] Suppose that G is a p-group of p-rank at least 2, then the torsion free rank of T (G) is the number n = n(G) given above. We should note here that the group theory gives us a bound on the torsion free rank of T (G), that is independent of G. If p = 2, then the bound is 5 [7], while for odd p the torsion free rank is never greater than p + 1 [26]. Now suppose that G is any ﬁnite group. The work here has proceeded by attacking groups by their types. Roughly speaking almost all of the theorems look like the following. Theorem A.

Suppose that G is a ﬁnite group of type (ﬁll in here) . Then except for (ﬁll in here) , we have that T (G) = X(G) ⊕ (Image{res : T (G) → T (P )})

where P is the Sylow p-subgroup of G. Here X(G) is the group of linear characters of G, or one-dimensional kGmodules. An example of the theorem is the following. Note that for the symmetric groups, if p > 2, then X(G) ∼ = Z/2 is generated by the sign representation. When p = 2, X(G) = {0}. Theorem 4.4. [11, 9] Suppose that G = Sn the symmetric group on n letters for n ≥ p Then T (G) ∼ = Z ⊕ Z/2 except when (1) p = 2 and n = 2, 3, in which case T (G) = {0},

ENDOTRIVIAL MODULES

105 7

(2) p odd and p ≤ n < 2p, in which case T (G) = Z/2, (3) p odd and 2p ≤ n < 3p, in which case T (G) = Z ⊕ (Z/2)2 , (4) p2 ≤ n < p2 + p, in which case T (G) = (Z)2 ⊕ Z/2 for p > 2, while T (G) = (Z)2 if p = 2.

The ﬁrst two items in the theorem are driven by the fact that the Sylow psubgroup is cyclic. The third case is interesting because we get unexpected torsion endotrivial modules. This will be discussed more later. The last case come from the fact that G has maximal elementary abelian subgroups of rank 2, causing the torsion free rank of T (G) to be greater than one. This too will be discussed more later. Here is a brief summary of what is known. This is not meant to be complete but only to cover the highlights. Groups of Lie type. For G a group of Lie type in the deﬁning characteristic, the story is reasonably complete. This work appears in [10]. If the group has large Lie rank or if the ﬁeld deﬁning the group is large, the result is exactly what we expect: T (G) = X(G) ⊕ Z. Groups of type A1 are special and we discuss these below. The torsion subgroup of T (G) is trivial except when G has type A1 , twisted A2 and twisted B2 . The torsion free part of T (G) is cyclic generated by the class of Ω(k) for all groups of Lie rank at least three. Results are also available for the Borel and unipotent subgroups of G. Symmetric and alternating groups. The results for the symmetric group are outlined in Theorem 4.4. For alternating groups, there are similar theorems [9, 11]. Solvable and p-solvable groups. In [12], it is proved that if G is p-nilpotent, then T (G) ∼ = T (P ) ⊕ K(G), where P is a Sylow p-subgroup of G and K(G) is the kernel of the restriction T (G) −→ T (P ). In the case that G has p-rank at least 2, then it is further shown that K(G) = X(G). This last was also proved for solvable groups and it was conjectured that it should hold for for p-solvable. Moreover, it was prove that it must hold for p-solvable groups provided it held for p-nilpotent groups. This conjecture has now been settled in the aﬃrmative by Navarro and Robinson [29]. Groups with cyclic, quaternion or semi-dihedral Sylow subgroups. The endotrivial modules in the case that the Sylow p-subgroup is cyclic has been treated in the paper [28] by Mazza and Th´evenaz. When G has a quaternion Sylow 2subgroup (p = 2), the endotrivial group can be determined [13] with the help the Brauer-Suzuki Theorem that says that G has an odd-order normal subgroup H such that G/H has a central involution. For groups with semi-dihedral Sylow 2subgroups, the theorem [13] on endotrivial modules relies heavily on Erdmann’s analysis [22] of blocks with semi-dihedral defect group. The interesting thing here is that the exotic endotrivial modules occur at the bottom of their Auslander-Reiten components which have tree class D∞ . In all of the cases that T (P ) has torsion, the restriction map T (G) −→ T (P ) is surjective. Other special cases have been done by Mazza when the Sylow p-subgroup is normal [27], and by Robinson [32] in the case of simple endotrivial modules.

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JON F. CARLSON

5. The source of some of the torsion in T (G). One source of exceptions in Theorem A comes from groups that have a proper strongly p-embedded subgroup. This means basically that all of the fusions (conjugations) between p-elements takes place in the subgroup. An example is the group G = SL(2, pn ). A Sylow p-subgroup P of G has the form 1 b | b ∈ Fpn . P = 0 1 Its normalizer has the form N = NG (P ) =

a b 0 c

|

a, b, c ∈ Fpn , ac = 1 .

It is easy to check that the group has the property that if x ∈ G, x ∈ / N , then P ∩ xP x−1 = {1}. If M is a kN -module, then the Mackey formula says that (M ↑G x⊗M = N ) ↓N = (kG ⊗kN M ) ↓N = G/N

(x ⊗ M ) ↓N ∩xN x−1 ↑N = M ⊕ (proj).

N \G/N

So as a consequence, we have equivalences of the stable categories: stmod(kN )

stmod(kG).

given by induction and restriction. Thus, if M is a kN -module of dimension one, then the induced module M ↑G is an endotrivial kG-module: M ↑G ∼ = f (M ) ⊕ proj where f (M ) is the Green correspondent of M . In general, f (M ) does not have dimension one. Indeed, if G is the simple group P SL(2, pn ), then the trivial kN module kN is the only one-dimensional kN -module whose Green correspondent has dimension one. In a situation as above, the induction and restriction maps induce isomorphisms T (N ) T (G). For some time, it had been thought that perhaps, exotic torsion endotrivial modules occurred only for groups having strongly p-embedded subgroups. By “exotic” we mean an indecomposable endotrivial module M of dimension greater than one with the property that MP ∼ = k ⊕ proj. This, however, is not true. For an example we can return to the symmetric groups. Assume that p ≥ 3, and let G = Σ2p+b be the symmetric group on 2p + b letters for 0 ≤ b < p. Let H = Σp+b × Σp ⊆ G be the Young subgroup corresponding to the partition [p + b | p] of n = 2p + b. The Young module Y p+b,p , corresponding to ↑G , and it is identiﬁable as the one direct summand that H is a direct summand of kH is not in any such induced module coming from a Young subgroup corresponding to a larger partition. The result is Theorem 5.1. [10] The Young module Y p+b,p is an endotrivial module with the property that Y p+b,p ↓P ∼ = k ⊕ proj. Moreover, if b ≥ 1, then the Young subgroup H is not strongly p-embedded nor is it contained in a proper strongly p-embedded subgroup.

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6. The torsion free rank of T (G). The torsion free rank of T (P ) is greater than one when G has maximal elementary abelian p-subgroups of rank 2 and at least two classes of maximal elementary abelian subgroups. This was proved ﬁrst by Alperin in the case that G is a p-group. The same theorem (Theorem 4.3) holds, in general. But the reader should keep in mind that the torsion free rank n = n(G) may be smaller than n(P ). The proof of the general result is found in [10], but it is a generalization of an argument in [6]. A main point is that by a theorem of Alperin, also in [1], the group T (G) has a torsion free subgroup of maximal rank which is equal to the image of the restriction map n res : T (G) −→ T (Ei ) i=1

where E1 , . . . , En are as in the paragraph preceding Theorem 4.3. nSo the problem is to prove that the image of res has ﬁnite index in the the sum i=1 T (Ei ). Now remember that by Dade’s Theorem, T (Ei ) ∼ = Z and is generated by Ω(k). Hence to prove the extension of Theorem 4.3 to general ﬁnite groups, we need to show that for each i, there is an endotrivial kG-module M such that MEi ∼ = Ωm (k) ⊕ proj for ∼ some m = 0 and that MEj = k ⊕ proj for every j = i. The key to showing this fact lies in the theory of support varieties. We should emphasize that the method that we describe here only produces a subgroup of ﬁnite index in T (G). Getting the actual generators for the torsion free subgroup of the group of endotrivial modules has proved to be a very diﬃcult problem in the case that the torsion free rank of T (G) is two or more. It is one of the few problems left open in the studies [9, 10]. We speak more on this later. Suppose that G has a maximal elementary abelian subgroup E of rank 2, and at least two conjugacy classes of maximal elementary abelian subgroups. Choose subgroup E1 , . . . , En in the Sylow subgroup P of G, as in the paragraph coming just before Theorem 4.3. Note that E1 , . . . , En represent G-conjugacy classes of elementary abelian subgroups of rank 2. Let Z be the unique subgroup of order p in the center Z(P ) of the Sylow p-subgroup P of G. Note that Z(P ) must be cyclic by our assumptions. Quillen’s Theorem 3.2 tells us that VG (k) is a union of components VG (k) = W1 ∪ W2 ∪ · · · ∪ Wn , where for i = 1, . . . , n − 1, Wi = res∗G,Ei (VEi (k)). If G has p-rank 2, then Wn = res∗G,En (VEn (k)), otherwise it is the union of res∗G,E (VE (k)) where E runs through a collection representatives of the other classes of maximal elementary abelian subgroups. Note that each Wi is a folded version of VEi (k) ∼ = P1 . Moreover, because of the transitivity of the restriction map, we have that for any i and j, Wi ∩ Wj is the point res∗G,Z (VZ (k)). We choose an element ζ ∈ Hm (G, k) for some n such that resG,Z (ζ) is not nilpotent. We let ζ also denote a cocycle Ωm (k) −→ k so that we have an exact sequence 0

/ Lζ

/ Ωm (k)

ζ

/k

/0

where Lζ is the kernel. A rank variety calculation [19] conﬁrms that VG (Lζ ) = VG (ζ), the collection of all prime ideals that contain ζ. And we have that VG (Lζ ) =

n i=1

VG (ζ) ∩ Wi ,

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JON F. CARLSON

and moreover, VG (ζ) ∩ Wi ∩ Wj = ∅ by the choice of ζ. So for any i there are subvarieties V1 and V2 such that V1 ∩ V2 = ∅, VG (ζ) = V1 ∪ V2 where V1 contains VG (ζ) ∩ res∗G,Ei (VEi (k)), V2 contains VG (ζ) ∩ res∗G,E (VE (k)) for any maximal elementary abelian subgroup that is not conjugate to Ei . By Theroem 3.1, we have that Lζ ∼ = L1 ⊕ L2 where VG (Li ) = Vi for i = 1, 2 and a commutative diagram 0

0

L2

L2

0

/ Lζ

/ Ωm (k)

0

/ L1

/N

0

0

ζˆ

/k

/0

/k

/0

where N is the pushout of the lower right-hand square. Now we notice that (L2 )↓Ei is a projective kEi -module, and (L1 )↓E is projective for any maximal elementary abelian subgroup E not conjugate to Ei . It follows that N is an endotrivial module with the propeties that N↓Ei ∼ = Ωm (kE ) ⊕ proj and N↓E ∼ = kE ⊕ proj. for E not conjugate to a subgroup of Ei . This proves the theorem on the torsion-free rank. There is another construction of endotrivial module that comes from the work of Balmer and Favi [3]. It proceeds a follows. Assume that we have an open covering VG (k) = U1 ∪ U2 of the support variety. If Wi is the complement of Ui , let C(Ui ) denote the quotient category stmod(kG)/ stmod(kG)Wi where Wi is the collection of all closed subvarieties of Wi and stmod(kG)Wi is the full subcategory generated by modules whose support variety is in Wi . Then we consider the diagram M∈

/ C(U1 )

C = stmod(kG)

M2 ∈

C(U2 )

/ C(U1 ∩ U2 ) .

M1 M1 ∼ = M2

In this we have two objects M1 , M2 ∈ stmod(kG) satisfying the following. We suppose that there is an isomorphism σ : M1 → M2 in C(U1 ∩ U2 ). A gluing of M1 and M2 along the isomorphism σ is an object M of stmod(kG) which is locally isomorphic to M1 and M2 in a compatible way with σ: that is, M1 σ1 qq8 q q q q σ M MM MMM M σ2 & M2

∼ =

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109 11

commutes in C(U1 ∩U2 ). Such a gluing always exists in stmod(kG) and is unique up to isomorphism. Balmer and Favi noted that endotrival modules could be produced as gluings. But what can you get? Theorem 6.1. Consider the open complements Ui = VG (k) Wi of the above closed subsets Wi , for i = 1, 2. We have by assumption an open covering VG (k) = U1 ∪ U2 . With the above notations, the module N (in the above diagram) is the gluing of k and Ωm k along the isomorphism ζ −1 : k → Ωm k in C(U1 ∩ U2 ). 6k σ1 nnnn n n n nn ζ −1 N OO OOO O σ2 ' Ωm (k).

In fact, the gluing method produces the same collection of endotrivial modules as the cohomological push-out method described earlier in this section. Unfortunately, as noted earlier, this may not be the entire torsion-free part of T(G) 7. Endotrivial modules for inﬁnitesimal group schemes. In the last couple of years, Dan Nakano and the author have investigated the endotrivial modules for inﬁnitesimal subgroups of algebraic groups. An example of an inﬁnitesimal subgroup is the following. Suppose that G is an algebraic group deﬁned over a ﬁeld of characteristic p > 0. Let F : G → G be the Frobenius map, and let Gr be the scheme which is the kernel of F r . Then Gr is an inﬁnitesimal group scheme of height r. Inﬁnitesimal group schemes of height one are Morita equivalent to restricted p-Lie algebras. For inﬁnitesimal group schemes, the usual induction, restriction methods employed in the analysis of modules over ﬁnite groups is not available. As a result, we are left with a very basic question. Question 7.1. For G a general ﬁnite group scheme, is the group of endotrivial modules T (G) ﬁnitely generated? We have some evidence for this theorem in the form of the following. Theorem 7.2. [15] Suppose that G is a ﬁnite group scheme. For any positive integer n, there is at most a ﬁnite number of endotrivial kG-modules of dimension n. The proof of this uses the same idea of Dade [21] that was crucial for Puig’s proof [31] of ﬁnite generation of T (G) in the case that G is a ﬁnite group. In the case that G is a ﬁnite group, there is a theorem of the author (see [19], Theorem 8.2.11) saying that there is a bound on the dimension of any endotrivial module that whose restriction to every elementary abelian p-group has the form k ⊕ proj. This proves ﬁnite generation for endotrivial modules over ﬁnite groups. What is missing in the case of an arbitrary ﬁnite group scheme is an analog to Proposition 2.2 One implication of Theorem 7.2 is the following. Corollary 7.3. [15] If G is a connected reductive algebraic group, Then any endotrivial Gr -module is G-stable.

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JON F. CARLSON

For general ﬁnite group schemes, there are some fundamental question about liftings and extensions that are raised by the results. Definition 7.4. Assume that S is a normal subgroup scheme of H. An Smodule M stably lifts to H if there exists an H-module K such that K|S ∼ = M ⊕P where P is a projective S-module. Proposition 1. [15] The second syzygy Ω2 (k) := Ω2B1 (k) stably lifts to a B-module, but does not lift to a B-module. Theorem 7.5. For each n ∈ Z, ΩnGr (k) stably lifts to G. In general we are left with the question if G is a connected algebraic group, do the endotrivial Gr -modules stably lift to G-modules? There are a few cases where we have very speciﬁc information. An example is the following. Most of this is for unipotent inﬁnitesimal group schemes. Theorem 7.6. [14] Let U be the unipotent radical of a Borel subgroup of the semisimple algebraic group G of Lie rank at least 2. Then the inﬁnitesimal group scheme U1 has the property that T (U1 ) = Z and is generated by Ω(k), except in the case that p = 2, and G has type A2 . Assume that dim U ≥ 2 and T (Ur ) is generated by Ω(k). Let J be a proper subset of Δ. Then T ((PJ )r ) = X((PJ )r ) ⊕ Z, where PJ is the parabolic subgroup corresponding to J. Some investigation has also been made into when some of the canonical modules are endotrivial. The algebraic group of type A1 is exceptional. In this case we know T (G), that is, if G = SL2 , then T (G1 ) ∼ = Z ⊕ Z/2Z. Theorem 7.7. [14] Let G = SL2 . The Weyl module V (λ) (resp. H 0 (λ)) is endotrivial over Gr if and only if λ = npr or npr − 2 for n ≥ 0. However, this does not appear to happen very often. Theorem 7.8. [14] Suppose that p > 2 and that rank G ≥ 2. Then the only induced module H 0 (λ) (Weyl module V (λ) or (if p ≥ h) tilting module T (λ)) which is endotrivial over Gr is the trivial module. References [1] J. L. Alperin, A construction of endo-permutation modules, J. Group Theory 4 (2001), 3–10. [2] P. Balmer, D. J. Benson and J. F. Carlson, Gluing representations via idempotent modules and constructing endotrivial modules, J. Pure Appl. Algebra, 213(2009), 173 -193. [3] P. Balmer and G. Favi, Gluing techniques in triangular geometry, Q. J. Math. 58 (2007), 415–441. [4] D. J. Benson, Representations and Cohomology I, II, Cambridge Univ. Press, 1991. [5] S. Bouc, The Dade group of a p-group Invent. Math. 164 (2006), no. 1, 189231. [6] J. F. Carlson, Constructing endotrivial modules, J. Pure and Appl. Algebra, 206(2006), 83-110. [7] J. F. Carlson, Maximal elementary abelian subgroups of rank 2, J. Group Theory 10(2007), 5-14. [8] J. Carlson, E. Friedlander and J. Pevtsova, Modules of constant Jordan type, J. Reine Angew. Math., 2008(2008), 191-234.

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[9] J. Carlson, D. Hemmer and N. Mazza, The group of endotrivial modules for the symmetric and alternating groups, Proc. Edinburgh Math. Soc. (2) 53(2010), 83-95. [10] J. Carlson, N. Mazza and D. Nakano, Endotrivial modules for ﬁnite groups of Lie type, J. Reine Angew. Math., 595 (2006), 284–306. [11] J. Carlson, N. Mazza and D. Nakano, Endotrivial modules for the symmetric group, Proc. Edinburgh Math. Soc., 51 (2008), 1–22. [12] J. Carlson, N. Mazza and J. Th´ evenaz, Endotrivial modules for p-solvable groups, Trans. Amer. Math. Soc, 363(2011), 4979-4996. [13] J. Carlson, N. Mazza and J. Th´ evenaz, Endotrivial modules for groups with quaternion and semi-dihedral Sylow subgroups, J. Europ. Math. Soc. (to appear). [14] J.F. Carlson, D.K. Nakano, Endotrivial modules for ﬁnite group schemes, J. Reine Angew. Math., 653, (2011), 149–178. [15] J.F. Carlson, D.K. Nakano, Endotrivial modules for ﬁnite group schemes, II, Bulletin of the Inst. of Math. Academia Sinica (to appear). [16] J. F. Carlson and J. Th´ evenaz, Torsion endo-trivial modules, Algebras and Rep. Theory, 3 (2000), 303–335. [17] J. Carlson and J. Th´ evenaz, The classiﬁcation of torsion endo-trivial modules, Annals of Math. (2) 165(2005), 823-883. [18] J. Carlson and J. Th´ evenaz, The classiﬁcation of endo-trivial modules, Invent. Math. 158(2004), 389-411. [19] J. Carlson, L. Townsley, L. Valero-Elizondo and M. Zhang, Cohomology Rings of Finite Groups, Kluwer, Dordrecht, 2003. [20] E. C. Dade, Endo-permutation modules over p-groups, I, II, Ann. Math. 107 (1978), 459–494, 108 (1978), 317–346. [21] E. Dade, Algebraically rigid modules, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), pp. 195–215, Lecture Notes in Math., 832, Springer, Berlin, 1980. [22] K. Erdmann, Algebras and semidihedral defect groups. I, Proc. London Math. Soc. (3) 57 (1988), no. 1, 109–150. [23] E. Friedlander, J. Pevtsova, Representation-theoretic support spaces for ﬁnite group schemes, Amer. J. Math. 127 (2005), 379-420. [24] E. Friedlander, J. Pevtsova, Π-supports for modules for ﬁnite group schemes, Duke. Math. J. 139 (2007), 317–368. [25] E. Friedlander, J. Pevtsova, A. Suslin, Generic and Maximal Jordan types, Invent. Math. 168 (2007), 485–522. [26] N. Mazza, Connected components of the category of elementary abelian p-subgroups, J. Algebra, 320 (2008), 42–48. [27] N. Mazza, The group of endotrivial modules in the normal case, J. Pure Appl. Algebra 209 (2007), no. 2, 311–323. [28] N. Mazza, J. Th´evenaz, Endotrivial modules in the cyclic case, Archiv der Mathematik 89 (2007), no. 6, 497–503. [29] G. Navarro and G. Robinson, On endotrivial modules for p-solvable groups, preprint (2010). [30] L. Puig, Notes sur les p-alg` ebras de Dade, Preprint, 1988. [31] L. Puig, Aﬃrmative answer to a question of Feit, J. Algebra 131 (1990), 513–526. [32] G. Robinson, Endotrivial irreducible lattices, preprint (2011). Department of Mathematics, University of Georgia, Athens, Ga. 30602

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Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Super duality for general linear Lie superalgebras and applications Shun-Jen Cheng† , Ngau Lam†† , and Weiqiang Wang††† Abstract. We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as well as the identiﬁcation of the u-homology groups, under these category equivalences. As an application, we obtain a complete solution of the irreducible character problem for some new parabolic BGG categories of gl(m|n)-modules, including the full BGG category of gl(m|2)modules, in terms of type A Kazhdan-Lusztig polynomials.

1. Introduction Super duality is a powerful new approach developed in the past few years in the study of representation theory of Lie superalgebras. It provides a surprising direct link between various parabolic BGG categories of modules of Lie superalgebras and Lie algebras, and allows us to solve the fundamental irreducible character problem for Lie superalgebras in terms of Kazhdan-Lusztig polynomials for semisimple (or more generally Kac-Moody) Lie algebras. Super duality was ﬁrst formulated as conjectures for general linear Lie superalgebras in [CWZ, CW], and then formulated and established in much generality in [CL2, CLW1] recently. Our super duality solution of the irreducible character problem for Lie superalgebras ultimately depends on the solution [BB, BK] of Kazhdan-Lusztig conjecture for Lie algebras [KL, Deo]. We also refer to [BS] for a completely diﬀerent approach towards the very special version of super duality conjecture as formulated in [CWZ]. The goal of this paper is to formulate and establish a new form of super duality (which is an equivalence of categories) for the general linear Lie superalgebras. The super duality formalism typically takes advantage of a type A branch which many Dynkin diagrams possess. In this paper we will apply the super duality formalism simultaneously to the two ends of the (super type A) Dynkin diagrams. We in addition establish a correspondence of tilting modules under super duality, which was omitted in our previous paper [CLW1], besides the correspondences of standard and simple modules under super duality. We also show that super duality 2010 Mathematics Subject Classiﬁcation. Primary 17B10. † partially supported by an NSC-grant and an Academia Sinica Investigator grant. †† partially supported by an NSC-grant. ††† partially supported by an NSF grant. 1 113

c 2012 American Mathematical Society

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CHENG, LAM, AND WANG

is actually an equivalence of tensor categories. We remark that such equivalence of tensor categories holds as well in the settings of [CL2, CLW1] with suitable choice of Z2 -gradations though this issue was not addressed back then. The version of super duality in this paper allows us to settle the irreducible character problem in some new parabolic BGG categories of gl(m|n)-modules not covered in [CW, CL2, CLW1] in terms of parabolic Kazhdan-Lusztig polynomials of type A. Brundan [Br1] conjectured that the full BGG category of gl(m|n)-modules (of integer weights) categoriﬁes the Fock space V⊗m ⊗ (V∗ )⊗n , and he established a maximal parabolic version of this conjecture for the category of ﬁnite-dimensional gl(m|n)-modules, where V and V∗ denote the natural Uq (gl∞ )-module and its dual module, respectively. Various parabolic versions of Brundan’s conjecture have been formulated and settled since then via the super duality approach ([CW, CL2]). However, the irreducible character problem in the full BGG category for general m and n is still open (to which we will return elsewhere), except when m or n is at most 1. Our paper conﬁrms a variant of Brundan’s conjecture on the full BGG category of gl(m|n)-modules (with respect to a nonstandard Borel subalgebra of block type 1|m|1) for n = 2 or m = 2, and we will return to address the case for general m and n in another paper [CLW2]. However, in the full generality of [CLW2] it remains unclear if the Brundan-Kazhdan-Lusztig polynomials are actually (parabolic) Kazhdan-Lusztig polynomials, in contrast to the case treated here. (which is a central extension Recall the inﬁnite-dimensional Lie algebra gl ∞ with a central element denoted by K) has played a fundamental role in many ∞ -module M , then is called diﬀerent contexts. If K acts as a scalar on a gl the level of the module M . Let be a positive integer. A very special case of -modules of super duality states that the (semisimple) category of integrable gl ∞ positive level is equivalent to a suitable category of gl∞ -modules of negative level − (which is far from being obvious to be semisimple); A weak version of the super duality on the Grothendieck group level suﬃces to recover the character formulas established by completely diﬀerent methods in [KR] and [CL1, Remark 5.2]. Following Vogan [Vo], a computation of the u-homology groups is basically a computation of the (parabolic) Kazhdan-Lusztig polynomials. Super duality identiﬁes the corresponding u-homology groups with coeﬃcients in modules belonging to diﬀerent categories. This in particular allows us to recover easily the computation of u-homology groups with coeﬃcients in modules appearing in Howe duality decompositions (which were computed by completely diﬀerent techniques in [CK, CKW, HLT, LZ]). To keep the paper at a reasonable length, we have omitted several proofs leading towards super duality, when they are similar to the ones in [CL2, CLW1]. We also refer directly to [Br1, CW] for the Fock space formulation of some parabolic versions of Brundan’s conjecture. We shall use the following notations throughout this paper. The symbols Z, N, and Z+ stand for the sets of all, positive, and non-negative integers, respectively. For a superspace V = V¯0 ⊕ V¯1 and a homogeneous element v ∈ V , we use the notation |v| to denote the Z2 -degree of v. Finally all vector spaces, algebras, tensor products, et cetera, are over the ﬁeld of complex numbers C.

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2. Lie superalgebras associated to various Dynkin diagrams In this section, we introduce a general linear Lie superalgebra g of inﬁnite rank and its subalgebras g, g, g and g . We also formulate the ﬁnite-rank counterparts of these Lie superalgebras. 2.1. General linear Lie superalgebra. We ﬁx k ∈ Z+ . We consider the following ordered set I: ··· < −

(2.1)

1 3 1 3 < −1 < − < 0 < 1 < 2 <· · · < k < < 1 < < · · · , 2 2 2 2 k

For k > 0, set K = {1, . . . , k}, and for k = 0, set K = ∅. We deﬁne the following subsets of I: 1 I := K ∪ Z, I := K ∪ −Z+ ∪ ( + Z+ ), 2 1 1 I := K ∪ (− − Z+ ) ∪ N, I := K ∪ ( + Z). 2 2 Consider the inﬁnite-dimensional superspace V with ordered basis {vi |i ∈ I}. 0, if r ∈ Z ∪ K, and |vr | = ¯1, if r ∈ 12 + Z. With respect to this We declare |vr | = ¯ basis, a linear map on V may be identiﬁed with a complex matrix (ars )r,s∈I . The Lie superalgebra gl(V ) is the Lie subalgebra of linear transformations on V consisting of (ars ) with ars = 0 for all but ﬁnitely many ars ’s. Denote by Ers ∈ gl(V ) the elementary matrix with 1 at the rth row and sth column and zero elsewhere. Denote by g := gl(V ) ⊕ CK the central extension of gl(V ) by a one-dimensional center CK determined by the 2-cocycle α(A, B) := Str([J, A]B), where J =

A, B ∈ gl(V ),

r≤0 Err and Str denotes the supertrace. Observe that the cocycle α is a coboundary. Indeed, there is embedding ι : gl(V ) → g, deﬁned by sending A ∈ gl(V ) to A + Str(JA)K (cf. [CLW1, Section 2.5]). It is clear that ι(gl(V )) is an ideal of g and g is a direct sum of the ideals ι(gl(V )) and CK. For A ∈ gl(V ) we denote

:= ι(A) − Str(JA)K ∈ A g. rr ⊕ CK and the Let h and b denote the standard Cartan subalgebra ⊕r∈I CE h∗ by standard Borel subalgebra ⊕r≤s,r,s∈I CErs ⊕ CK, respectively. Deﬁne Λ0 ∈ Λ0 (K) = 1,

rr ) = 0, ∀r ∈ I. Λ0 (E

jj = δij for i, j ∈ I and i , K = 0. We set h∗ be determined by i , E Let i ∈ αr := r − r+ 12 ,

r∈

1 Z, 2

and

αj := j − j+1 ,

The totally ordered set I gives rise to a positive system for g + = { i − j | i < j, i, j ∈ I }, Δ

1 ≤ j ≤ k − 1.

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CHENG, LAM, AND WANG

whose associated fundamental system is given by { αr | r ∈ 12 Z \ {0} } ∪ {0 − 1 , k − 12 , α1 , α2 , · · · αk−1 }, if k = 0; Π= 1 { αr | r ∈ 2 Z }, if k = 0. Let K denote the following Dynkin diagram with fundamental system:

α1

α2

α3

αk−2

αk−1

···

for The following is a Dynkin diagram together with the fundamental system Π g,

where denotes an odd isotropic simple root:

D( g): ··· ··· K

− 1 α 1 α 1 α α − −1

−

2

0

k

1

2

2

1

When k = 0, the diagram above means that the middle three terms are replaced by the odd isotropic simple root 0 − 12 . The Dynkin diagrams in the rest of the paper are interpreted in the similar way for k = 0. Deﬁne 0 if r ∈ I; |r| = 1 if r ∈ I\I. Let τ be an automorphism of g of order 4 deﬁned by sr , rs ) := −(−1)|r|(|r|+|s|) E τ (E

(2.2)

τ (K) = −K.

rs 2.2. The subalgebras of g. The subalgebra of g generated by K and E with r, s ∈ I (respectively I, I , I ) is denoted by g (respectively g, g and g ). The Cartan subalgebras (respectively Borel subalgebras) of g, g, g and g induced from g are denoted by h, h, h and h (respectively b, b, b and b ) respectively. We set 1 βr := r − r+1 , ∀r ∈ Z. 2 Ln , Rn and Rn denote the following Dynkin diagrams Ln ,

For n ∈ Z+ , let

with ﬁxed fundamental systems:

···

Ln

β1−n β2−n β3−n β−2 β−1 L

n

β1 2

−n

β3 2

−n

β5 2

Rn

β1

β2

β3

Rn

β1

β3

β5

2

2

···

2

β

−n

···

···

−5 2

β

−3 2

βn−2

βn−1

β

n− 5 2

β

n− 3 2

Ln , Rn and Ln In the associated diagrams ,

the limit n → ∞, Dynkin of

L Rn are denoted by L , R ,

and R , respectively.

SUPER DUALITY AND APPLICATIONS

117 5

For m, n ∈ Z+ ∪ {∞}, let g(m, n) denote the subalgebra of g generated by Ers with −m < r, s < n + 1 and r, s ∈ I. Note that g(m, n) is a ﬁnite-dimensional subalgebra for m, n ∈ Z+ . Set g(m, n) = g ∩ g(m, n),

g(m, n) = g ∩ g(m, n), g (m, n) = g ∩ g(m, n).

g(m, n), g (m, n) = g ∩

Then h(m, n) = h∩ g(m, n) and b(m, n) = b∩ g(m, n) are the Cartan and Borel subalgebras of g(m, n), and we have four variants of Cartan and Borel subalgebras for the remaining four Lie superalgebras with self-explanatory notations. The Dynkin diagrams with given fundamental systems of g(m, n), g(m, n), g (m, n) and g (m, n) with respect to the Borel subalgebras b(m, n), b(m, n), b (m, n) and b (m, n), are given as follows: D(g(m, n)): Lm Rn K

k − 1 0 − ¯ 1

D(g(m, n)): L K Rn m

− 1

0 − ¯ k 2 1 L D(g (m, n)): Rn K

m 1 − ¯

k − 1 1 − 2

L D(g (m, n)): K Rn

m 1 − ¯

− 1 −

2

1

k

2

For m = n = ∞, these become the Dynkin diagrams and simple systems of g, g, g and g with respect to the Borel subalgebras b, b, b and b , respectively. The fundamental systems of g, g, g and g are denoted by Π, Π, Π and Π , respectively. These fundamental systems of g(m, n), g(m, n), g(m, n), g (m, n) and g (m, n) are (m, n), Π(m, n), Π(m, n), Π (m, n) and Π (m, n), respectively. denoted by Π 2.3. Levi subalgebras. We ﬁx an arbitrary subset Y0 of {αj | 1 ≤ j ≤ k − 1}. The set Y (respectively Y , Y , Y , Y ) denotes the union of Y0 with the subset of (respectively, of Π, Π, Π , Π ) consisting of elements of the form r − s , with Π r, s ∈ I ∩ 12 Z (respectively I ∩ 12 Z, I ∩ 12 Z, I ∩ 12 Z, I ∩ 12 Z) satisfying r < s ≤ 0 g or 0 < r < s. Let l (respectively l, l, l , l ) be the standard Levi subalgebra of (respectively g, g, g , g ) corresponding to the set Y (respectively Y , Y , Y , Y ). Let p = l + b (respectively p = l + b, p = l + b, p = l + b , p = l + b ) be the corresponding parabolic subalgebra of g (respectively g, g, g , g ) with nilradical u (respectively u, u, u , u ) and opposite nilradical u− (respectively u− , u− , u− , u− ). b ∩ l , b∩l ) Denote the standard Borel subalgebras b ∩ l (respectively b ∩ l, h∗ (respectively of l (respectively l, l , l ) by bl (respectively bl , bl , bl ). For μ ∈ ∗ ∗ h∗ , h , h∗ , h ), let L(l, μ) (respectively L(l, μ), L(l, μ), L(l , μ), L(l , μ)) denote the highest weight irreducible l-(respectively l-, l-, l -, l -)module of highest weight μ with respect to the standard Borel. We extend L(l, μ) to a p-module by letting u act trivially. Deﬁne as usual the parabolic Verma module Δ(μ) and its unique irreducible quotient L(μ) over g: Δ(μ) := Indgp L(l, μ),

Δ(μ) L(μ).

Similarly, we introduce the other four variants of parabolic Verma and irreducible quotient modules with self-explanatory notations.

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We shall also need the ﬁnite-rank counterparts of the above algebras. Let l(m, n) = l∩ g(m, n). Then l(m, n) is a Levi subalgebra of g(m, n). For μ ∈ h(m, n)∗ , let L(l(m, n), μ) denote the highest weight irreducible l(m, n)-module of highest (m,n) (μ) and weight μ. As above, we can deﬁne parabolic Verma g(m, n)-module Δ (m,n) (μ). In a completely parallel fashion, we have the its irreducible quotient L other four variants of Levi subalgebras, parabolic Verma modules, and so on, with self-explanatory notations. 2.4. The dominant weights. Given a partition μ = (μ1 , μ2 , . . .), let μ denote its conjugate partition. We also denote by θ(μ) the modiﬁed Frobenius coordinates of μ: θ(μ) := (θ(μ)1/2 , θ(μ)1 , θ(μ)3/2 , θ(μ)2 , . . .), where θ(μ)i−1/2 := μi − i + 1 , θ(μ)i := μi − i , i ∈ N. Here and below b := max{b, 0} for all b ∈ R. Let a, λ01 , . . . , λ0k ∈ C, λ− and λ+ be two partitions. Associated to the tuple λ = (a, λ01 , . . . , λ0k ; λ− , λ+ ), set Λ+ (λ) :=

k

λ0i i +

i=1 −

Λ (λ) := −

+

λ+ j j

Λ (λ) :=

λ0i i +

i=1

j∈N

λ− j −j+1

k

+ aΛ0 ,

−

Λ (λ) := −

j∈N

(λ+ )j j− 12 ,

j∈N

(λ− )j −j+ 12

+ aΛ0 .

j∈N

The tuple (a, λ01 , . . . , λ0k ; λ− , λ+ ) is said to satisfy a dominant condition if k λ0i i , α∨ ∈ Z+ ,

(2.3)

for all α ∈ Y0 ,

i=1

where α∨ denotes the coroot of α. Associated to such a dominant tuple and a ∈ C, we deﬁne the weights (which will be called dominant) := λ

k i=1

λ0i i −

θ(λ− )r −r+ 12 +

r∈ 12 N

λ := Λ− (λ) + Λ+ (λ) ∈ h∗ , −

λ := Λ (λ) + Λ+ (λ) ∈ h∗ ,

θ(λ+ )r r + aΛ0 ∈ h∗

r∈ 12 N +

∗

+

∗

λ := Λ− (λ) + Λ (λ) ∈ h ,

−

λ := Λ (λ) + Λ (λ) ∈ h .

The number a will be called the level of these weights. + + We denote by Pa+ (respectively Pa+ , P a , Pa+ , P a ) the set of all dominant (respectively λ, λ, λ , λ ) with a ﬁxed a ∈ C. Obviously, we weights of the form λ + + ↔λ↔λ↔ have bijective maps between Pa+ , Pa+ , P a , Pa+ and P a given by λ + + + λ ↔ λ for λ ∈ Pa . Finally, we let P = a∈C Pa . 3. Change of highest weights for diﬀerent Borel subalgebras In this section, using odd reﬂections, we will determine how a highest weight for a highest weight module changes from a standard Borel to another distinguished non-standard Borel subalgebra. We shall brieﬂy explain the eﬀect of an odd reﬂection on the highest weight of a highest weight irreducible module (cf., e.g., [PS, Lemma 1], [KW, Lemma

SUPER DUALITY AND APPLICATIONS

119 7

1.4]). Fix a Borel subalgebra B of a Lie superalgebra G with corresponding positive system Φ+ (B). Let α be an isotropic odd simple root and α∨ be its corresponding coroot. Applying the odd reﬂection with respect to α changes the Borel subalgebra B into a new Borel subalgebra Bα whose corresponding positive system is Φ+ (Bα ) = {−α} ∪ Φ+ (B) \ {α}. Lemma 3.1. Let λ be the highest weight with respect to B of an irreducible module. If λ, α∨ = 0, then the highest weight of this irreducible module with respect to Bα is λ − α. If λ, α∨ = 0, then the highest weight remains to be λ. Even though it is unclear how the structures of Verma modules are related via odd reﬂections, the relation of their characters is simply described as follows. Lemma 3.2. Let Δ(G, B, λ) and Δ(G, Bα , λ) denote the G-Verma modules of highest weight λ with respect to the Borel subalgebras B and Bα , respectively. Then we have chΔ(G, B, λ) = chΔ(G, Bα , λ − α). Proof. Follows from the identity Φ+ (Bα ) = {−α} ∪ Φ+ (B) \ {α}.

For n ∈ N, we introduce the following total orderings of the subsets of I: (3.1) 1 1 3 3 1 3 · · · ≺ − − n ≺ −1 − n ≺ − − n ≺ −n ≺ − n ≺ − n ≺ · · · ≺ − ≺ − 2 2 2 2 2 2 ≺ 1 − n ≺ · · · ≺ −2 ≺ −1 ≺ 0 ≺ 1 ≺ · · · ≺ k ≺ 1 ≺ 2 ≺ · · · ≺ n 3 1 3 1 ≺ ≺ ≺ ··· ≺ n + ≺ n + 1 ≺ n + ≺ n + 2 ≺ ··· , 2 2 2 2 (3.2) 3 1 1 3 3 1 · · · ≺ − − n ≺ −1 − n ≺ − − n ≺ −n ≺ − n ≺ − n ≺ · · · ≺ − ≺ − 2 2 2 2 2 2 3 1 1 ≺ 1 − n ≺ · · · ≺ −2 ≺ −1 ≺ 0 ≺ 1 ≺ · · · ≺ k ≺ ≺ ≺ · · · ≺ n − 2 2 2 1 3 ≺ 1 ≺ 2 ≺ ··· ≺ n ≺ n + ≺ n + 1 ≺ n + ≺ ··· , 2 2 (3.3) 3 1 · · · ≺ − − n ≺ −1 − n ≺ − − n ≺ −n ≺ 1 − n ≺ · · · ≺ −1 ≺ 0 2 2 1 3 3 1 ≺ − n ≺ − n ≺ ··· ≺ − ≺ − ≺ 1 ≺ ··· ≺ k ≺ 1 ≺ 2 ≺ ··· ≺ n 2 2 2 2 3 1 3 1 ≺ ≺ ≺ ··· ≺ n + ≺ n + 1 ≺ n + ≺ n + 2 ≺ ··· , 2 2 2 2 (3.4) 3 1 · · · ≺ − − n ≺ −1 − n ≺ − − n ≺ −n ≺ 1 − n ≺ · · · ≺ −1 ≺ 0 2 2 1 1 3 3 1 3 1 ≺ − n ≺ − n ≺ ··· ≺ − ≺ − ≺ 1 ≺ ··· ≺ k ≺ ≺ ≺ ··· ≺ n − 2 2 2 2 2 2 2 3 1 ≺ 1 ≺ 2 ≺ ··· ≺ n ≺ n + ≺ n + 1 ≺ n + ≺ ··· . 2 2 For any total ordering of I, there is a Borel subalgebra of g spanned by the Cartan rs such that r < s with respect to the subalgebra h and elements of the form E

120 8

CHENG, LAM, AND WANG

ordering. Conversely, any Borel subalgebra of g containing the Cartan subalgebra bs (n), bc (n), h determines a unique total ordering of I. Let bc (n) (respectively s b (n)) be the Borel subalgebras with respect to the ordering (3.1) (respectively (3.2), (3.3), (3.4)) of I. Two elements a and b in an ordered set are said to be adjacent if there is no element j in the set satisfying a < j < b or a > j > b. For an odd simple root of the g containing form r − s with r, s ∈ I in the root system of a Borel subalgebra B of the Cartan subalgebra h, the numbers r and s are adjacent with respect to the corresponding ordering. The ordering corresponding to the new Borel subalgebra obtained by applying the odd reﬂection with respect to r − s is the same as the ordering corresponding to the Borel subalgebra B except reversing the ordering of r and s. Remark 3.3. Any ordering preserving the orderings of positive integers, positive half integers, non-positive integers and negative half integers, and satisfying i < a < j for i ∈ − 12 Z+ , a ∈ K and j ∈ 12 N, can be obtained by a sequence of reversing the orderings of two adjacent indices r and s with r ∈ Z and s ∈ 12 + Z satisfying r, s > 0 or else r, s ≤ 0. Thus, the Borel subalgebra with respect to an ordering satisfying the conditions above can be obtained by applying a sequence of odd reﬂections to the standard Borel subalgebra b. Moreover, we can choose a sequence of odd reﬂections in such a way that it leaves the sets of roots of u and u− invariant. Hence, the Borel subalgebras bc (n), bs (n), bc (n) and bs (n) can be obtained from b by applying sequences of odd reﬂections leaving the sets of roots of u and u− invariant. Let us spell out precisely the sequence of odd reﬂections required to obtain the Borel subalgebra bs (n) from b leaving the sets of roots of u and u− invariant. This process can be easily modiﬁed for the remaining cases. Starting with the Dynkin diagram of g, we apply the following sequence 12 n(n−1) odd reﬂections. First we apply one odd reﬂection corresponding to 1 − 32 , then we apply two odd reﬂections corresponding to 2 − 52 and 1 − 52 . After that we apply three odd reﬂections corresponding to 3 − 72 , 2 − 72 , and 1 − 72 , et cetera, until ﬁnally we apply n−1 odd reﬂections corresponding to n−1 −n− 12 , n−2 −n− 12 , . . . , 1 −n− 12 . The corresponding fundamental system of the resulting new Borel subalgebra for g is listed as follows: ···

α−1

α

−1 2

0 − 1

K

k − 1 2

Rn

n− 1 2

− 1

β1

···

βn−1

αn

α

···

n+ 1 2

Now we apply the following sequence of 12 n(n−1) odd reﬂections to the Dynkin diagram above. First we apply one odd reﬂection corresponding to −1 −−1/2 , then we apply two odd reﬂections corresponding to −2 − −3/2 and −2 − −1/2 . After that we apply three odd reﬂections corresponding to −3 − −5/2 , −3 − −3/2 and −3 − −1/2 , et cetera, until ﬁnally we apply n − 1 odd reﬂections corresponding to 1−n − 3/2−n , 1−n − 5/2−n , . . . , 1−n − −1/2 . We obtain the Dynkin diagram of bs (n):

121 9

SUPER DUALITY AND APPLICATIONS

···

α−n

Ln

−1 2

Ln

− 1−n

0 − 1

K

k − 1

Rn

2

n− 1 2

− 1

Rn

αn

α

···

n+ 1 2

The following lemma is a variant of Lemma 3.1 in our setting. Lemma 3.4. Let α be an odd simple root of the form r − s , r, s ∈ I satisfying r < s ≤ 0 or 0 < r < s, in the root system of a Borel subalgebra B of l containing the Cartan subalgebra h. Let Bα denote the new Borel subalgebra obtained by applying the odd reﬂection with respect to α to the Borel subalgebra B. For μ ∈ h∗ , let v be a B-highest weight vector in L(l, μ). ss ) = 0, then L(l, μ) is an l-module of Bα -highest weight rr ) + μ(E (1) If μ(E α λ and v is a B -highest weight vector. ss ) = 0, then L(l, μ) is an l-module of Bα -highest weight rr ) + μ(E (2) If μ(E sr v is a Bα -highest weight vector. λ − α and E Using similar arguments as in the proofs of [CL2, Lemma 3.2] and [CL2, Corollary 3.3] (cf. [CLW1, Proposition 4.3]) together with Lemma 3.4, we have the following. Proposition 3.5. Let λ = (a, λ1 , . . . , λk ; λ− , λ+ ) ∈ Pa+ and n ∈ N. (i) Suppose that (λ− )1 ≤ n and (λ+ )1 ≤ n. Then the highest weight of L(l, λ) c with respect to the Borel subalgebra b (n) ∩ l is λ. Furthermore, Δ(λ) and are highest weight λ) L( g-modules of highest weight λ with respect to the Borel bc (n). (ii) Suppose that (λ− )1 ≤ n and λ+ 1 ≤ n. Then the highest weight of L(l, λ) s and λ) with respect to the Borel subalgebra b (n) ∩l is λ. Furthermore, Δ( L(λ) are highest weight g-modules of highest weight λ with respect to the Borel bs (n). + (iii) Suppose that λ− 1 ≤ n and (λ )1 ≤ n. Then the highest weight of L(l, λ) λ) with respect to the Borel subalgebra bc (n) ∩ l is λ . Furthermore, Δ( and L(λ) are highest weight g-modules of highest weight λ with respect to the Borel bc (n). + (iv) Suppose that λ− 1 ≤ n and λ1 ≤ n. Then the highest weight of L(l, λ) with s and respect to the Borel subalgebra b (n) ∩ l is λ . Furthermore, Δ(λ) are highest weight λ) L( g-modules of highest weight λ with respect to the s Borel b (n).

4. Super duality a , Oa , Oa , O and In this section, we ﬁrst introduce the module categories O a for the inﬁnite-rank Lie superalgebras g, g, g, g , and g , respectively. We also a → Oa , T : O a → Oa , T : O a → O and T : O a → O , introduce functors T : O a a and then show that they are equivalences of tensor categories which send parabolic Verma and simple modules to parabolic Verma and simple modules, respectively.

Oa

122 10

CHENG, LAM, AND WANG

be the category of

such that M

4.1. BGG categories. Let O g-modules M

is a semisimple h-module with ﬁnite-dimensional weight subspaces Mγ , γ ∈ h∗ , satisfying

decomposes over l into a direct sum of L(l, μ (i) M ) for μ ∈ P + .

) (ii) There exist ﬁnitely many weights λ1 , λ2 , . . . , λk ∈ P + (depending on M i −

, then γ ∈ λ such that if γ is a weight in M Z+ α, for some i. α∈Π

are (not necessarily even) The morphisms in O g-homomorphisms. We recall that a

1. Let O g and v ∈ M ϕ ∈ HomO (M , N ) means that ϕ(xv) = xϕ(v), for x ∈ consisting of the objects M such that Kv = av

∈O be the full subcategory of O

μ) and for all v ∈ M . We have O = a∈C Oa . The parabolic Verma modules Δ( μ) for μ ∈ P + lie in Oa , by Lemma 4.1 below. Analogously irreducible modules L( a we can deﬁne the other four variants O, O, O and O of the above categories for g, g, g , and g in self-expained notations, which contain the corresponding parabolic Verma and irreducible modules. := {μ ∈ Let Γ h∗ | μ(Err ) = 0, |r| 0}. For V = Vγ ∈ Oa , there is a Γ natural Z2 -gradation V = V¯0 V¯1 given by (4.1) V¯0 := and V¯1 := Vμ Vμ , ¯0 μ∈Γ

¯1 μ∈Γ

:= {μ ∈ Γ | where Γ r∈1/2+Z μ(Err ) ≡ (mod 2)} (cf. [CL2, Section 2.5]). Let We deﬁne O ¯0 to be the full subcategories Π denote the parity change functor on O. a a consisting of objects with Z2 -gradation given by (4.1) and deﬁne O ¯1 to be of O a a consisting of objects ΠM ¯0 . Note that the

with M

∈O the full subcategories of O a ¯0 and O ¯1 are of degree 0. ¯0 and O ¯1 are an abelian ¯ It is clear that O morphisms in O a a a a categories. let M

=M

¯0 + M

¯1 ∈ O,

¯0,μ = M

μ ∩ M

¯0 and M

¯1,μ = M

μ ∩ M

¯1 . Also we For M let

¯ :=

¯ :=

¯ :=

¯0,μ , M

¯1,μ ;

¯0,μ , M

¯1,μ .

¯ := M M M M M M 0 1 0 1 ¯0 μ∈Γ

¯1 μ∈Γ

¯1 μ∈Γ

¯0 μ∈Γ

:= M

¯ M

:= M

¯ M

.

¯ and M

¯ are submodules of M It is clear that M 0 1 0 1

Since h separates the Mμ ’s, we have Mμ = M¯0,μ M¯1,μ for all μ. Therefore M

= M

M

∈ O,

is even isomorphic to N ΠN

. Hence for every M M ¯ 0 for some N , N ∈ O . For N ∈ O, let pN : N −→ ΠN be the parity-reversing M ¯0 .

∈ O,

is isomorphic to N for some N ∈O isomorphism. Therefore for every M for M and

, N ∈ O, This implies that the kernel and the cokernel of ϕ belong to O ¯ 0

ϕ ∈ HomO (M , N ). Hence O is an abelian category. Also O and O have isomorphic ¯0 ,

, N ∈ O skeletons and hence they are equivalent categories. Note that for M

ϕ ∈ HomO (ΠM , N ) and ϕ ∈ HomO (M , ΠN ), we have ϕ = φ◦pΠM ◦φ and ϕ = pN

for some φ, φ ∈ HomO (M , N ). 1We remark that a diﬀerent deﬁnition of homomorphism is ϕ(xv) = (−1)|ϕ|·|x| xϕ(v). The

diﬀerence is inessential, as the map f → f ∗ given by f ∗ (v) := (−1)|f | f (v) provides a bijection between these two types of maps.

SUPER DUALITY AND APPLICATIONS ¯ 0

123 11

¯ 0

Analogously, Oa , Oa0 and Oa denote the respective full subcategories of Oa , Oa and Oa consisting of objects with Z2 -gradations given by (4.1) such that r are summed over half integers contained in the respective index sets; they are abelian and are equivalent to Oa , Oa and Oa , respectively. Using similar arguments as in the proof of [CLW1, Lemma 3.1], we have the following lemma. ¯

μ) and L( μ) Lemma 4.1. Let μ ∈ Pa+ . The restrictions to l of the g-modules Δ( + a. μ), L( μ) ∈ O decompose into direct sums of L(l, ν) for ν ∈ Pa . In particular, Δ( Analogous results hold for the other four variants of categories for g, g, g , and g .

4.2. The functors T , T , T and T . We shall introduce four functors T : a → Oa , T : O a → O and T : O a → O . We will describe in a → Oa , T : O O a a a → Oa , and the remaining three variants can be detail the construction of T : O treated similarly. By deﬁnition, g is naturally a subalgebra of g, l is a subalgebra of l, and h is a ∗ h∗ by regarding h as subalgebra of h. Furthermore, we may view h as a subspace of a direct summand of h with respect to the natural basis Err ’s. Given a semisimple

γ , we deﬁne

= ∗ M h-module M γ∈h

) :=

γ . T (M M γ∈h∗

) is an h-submodule of M

. It in addition M

= ∗ M

γ is an Note that T (M γ∈h l-module, then T (M

) is an l-submodule of M

. Furthermore, if M

= γ∈ h∗ Mγ is

) is a g-submodule of M

. a g-module, then T (M

The direct sum decomposition in M gives rise to the natural projection

TM : M −→ T (M )

is an l-module, then T which is an h–module homomorphism. If in addition M M

is a is also an l-module homomorphism. Furthermore, if M g-module, then TM

→ N is an is a g-module homomorphism. If f : M h-homomorphism, then the following induced map

) −→ T (N ) T [f] : T (M

→N is a l-homomorphism, is an h-module homomorphism. If in addition f : M

→ N is a then T [f] is an l-module homomorphism. Furthermore, if f : M g homomorphism, then T [f ] is a g-module homomorphism. Lemma 4.2. For λ ∈ Pa+ , we have = L(l, λ), = L(l, λ), T L(l, λ) T L(l, λ) = L(l , λ ), = L(l , λ ). T L(l, λ) T L(l, λ) = L(l, λ), and the other cases can be proved Proof. We will prove T L(l, λ) contains a ( by the same argument. By Proposition 3.5, L(l, λ) bs (n) ∩ l)-highest weight vector vλ of highest weight λ for n 0. The vector vλ clearly lies in , and it is a (b ∩ l)-singular vector since l ∩ b = l ∩ T L(l, λ) bs (n). Let w be

124 12

CHENG, LAM, AND WANG

. Taking n large enough, we can assume that w ∈ any weight vector in T L(l, λ) g(n, n) with respect to the Borel U ( g(n, n) ∩l)vλ . Note that the Dynkin diagram of subalgebra B = g(n, n) ∩ bs (n) is the following: Ln L

n 1 − 1−n − −

2

0

1

Rn Rn K

− 1 1 − 1 k

2

n−

2

Let N− be the opposite nilradical of B ∩ l. Then w ∈ U (N− )vλ , and hence we is a highest weight l-module with see that w ∈ U (l)vλ . This shows that T L(l, λ) respect to the Borel subalgebra l ∩ b of highest weight λ. is irreducible. In order to complete the proof we need to show that T L(l, λ) . Choose q 0 with q ≥ n such that Let w be a b ∩ l-singular vector in T L(l, λ) bs (q) w ∈ U ( g(q, q) ∩ l)vλ . Since the root vector corresponding to a simple root in either commutes with U ( g(q, q) ∩ l) or annihilates w, w is a bs (q)-singular vector and hence w is a multiple of vλ . Therefore T L(l, λ) is irreducible. Lemma 4.3. For μ, λ ∈ Pa+ , we have = L(l, μ) ⊗ L(l, λ), (i) T L(l, μ ) ⊗ L(l, λ) = L(l, μ) ⊗ L(l, λ), (ii) T L(l, μ ) ⊗ L(l, λ) = L(l , μ ) ⊗ L(l , λ ), ) ⊗ L(l, λ) (iii) T L(l, μ = L(l , μ ) ⊗ L(l , λ ). (iv) T L(l, μ ) ⊗ L(l, λ) = L(l, μ) ⊗ L(l, λ), and the other Proof. We will prove T L(l, μ ) ⊗ L(l, λ) cases can be proved by using the same argument. By Lemma 4.2, we have ⊇ L(l, μ) ⊗ L(l, λ). T L(l, μ ) ⊗ L(l, λ) The lemma now follows by a comparison of the characters on both sides of (i).

→ O, T : O → O and → O, T : O Proposition 4.4. The functors T : O → O are exact. T :O

then T (M

∈ O,

), Proof. In light of Lemma 4.2, it suﬃces to show that if M

T (M ), T (M ), and T (M ) lie in O, O, O , and O , respectively. We shall only

) ∈ O, as the argument for the other cases is similar. Also below show that T (M we will only prove the case k ≥ 1, as the case k = 0 is proved by slightly modifying the argument. and Π(R) to be the sets {α , . . . , α It is convenient to deﬁne Π(k), Π(L) 1 k−1 }, {α−1/2 , α−1 , . . .}, and {α1/2 , α1 , . . .}, respectively, for the remainder of the proof. Then there exists 1 ζ, 2 ζ, . . . ,r ζ, with 1 ζ, . . . ,r ζ ∈ P + such that

∈ O. Let M

any weight of M is bounded above by some i ζ. Ignoring the level recall that we have 0 − + j 0 0 0 + and j ζ − are partitions j ζ = (j ζ ,j ζ ,j ζ ), where ζ = (j ζ1 , . . . ,j ζk ), and j ζ + − with |j ζ | = kj and |j ζ | = lj . For each j ζ, let Pj be the following ﬁnite subset of h∗ : Pj := {(j ζ 0 , η, μ) | μ, η ∈ P with |μ| = kj and |η| = lj }.

) and P (M ) := r Pj . Set M := T (M j=1

SUPER DUALITY AND APPLICATIONS

125 13

We claim that given any weight ν of M , there exists γ ∈ P (M ) such that γ − ν ∈ Z+ Π. It suﬃces to prove the claim for ν ∈ P + . Since ν is also a weight of

, we have i ζ − ν ∈ Z+ Π, for some i. Thus M − ν = q(0 − ) + p( − 1/2 ) + κ− + κ+ + κ0 , 1 k 0 − + where p, q ∈ Z+ , κ ∈ α∈Π(k) Z+ α, κ ∈ β∈Π(L) Z+ β, and κ ∈ Z+ β. β∈Π(R) This implies that ν − is a partition of size li + q and ν + is a partition of ki + p, and hence there exists γ ∈ P (M ) such that ν + and ν − are obtained from the partitions γ + and γ − by adding p and q boxes, respectively. For every such a box of ν ± , we record the row number in which it was added to γ ± in the multisets J ± with |J + | = p and |J − | = q. Then we have ν = γ − 0κ − (k − j ) − (−j+1 − 1 ), iζ

j∈J +

j∈J −

and hence ν < γ. Thus, we conclude that M ∈ O.

4.3. A character formula. The following theorem can be regarded as a weak version of super duality which is to be established in Theorem 4.11. Using similar arguments as in the proof of [CLW1, Theorem 4.5], we have the following. a and λ ∈ P + . If M

∈O

is a highest weight Theorem 4.5. Let M g-module of a

highest weight λ, then T (M ), T (M ), T (M ) and T (M ) are highest weight modules over g, g, g and g of highest weights λ, λ, λ and λ , respectively. Furthermore, we have = Δ(λ) and T L( = L(λ); λ) λ) (i) T Δ( = Δ(λ) and T L( = L(λ); λ) λ) (ii) T Δ( = Δ (λ ) and T L( = L (λ ); λ) λ) (iii) T Δ( = Δ (λ ) and T L( = L (λ ). λ) λ) (iv) T Δ( By standard arguments, Theorem 4.5 implies the following character formula. Theorem 4.6. Let λ ∈ Pa+ , and write chL(λ) = μ∈Pa+ aμλ chΔ(μ), aμλ ∈ Z. Then = λ) θ ), + aμλ chΔ(μ (i) chL( μ∈Pa (ii) chL(λ) = μ∈Pa+ aμλ chΔ(μ). (iii) chL (λ ) = μ∈Pa+ aμλ chΔ (μ ), (iv) chL (λ ) = μ∈Pa+ aμλ chΔ (μ ). 4.4. Identiﬁcation of Kazhdan-Lusztig polynomials. For a module V over a Lie (super)algebra G and j ∈ Z+ , let Hj (G; V ) denote the jth homology group of G with coeﬃcient in V . For a precise deﬁnition of homology groups of Lie superalgebras with coeﬃcients in a module and a precise formula for the boundary operator we refer the reader to [Ta] or [CL2, Section 4]. a we denote by M = T (M

∈ O

) and M = T (M

), M = T (M

) and For M

M = T (M ). Using similar arguments as in the proof of [CLW1, Theorem 4.10], we have the following. Theorem 4.7. We have, for j ≥ 0,

)) ∼ (i) T (Hj ( u− ; M = Hj (u− ; M ), as l-modules.

126 14

CHENG, LAM, AND WANG

)) ∼ (ii) T (Hj ( u− ; M = Hj (u− ; M ), as l-modules.

(iii) T (Hj ( u− ; M )) ∼ = Hj (u− ; M ), as l -modules.

)) ∼ (iv) T (Hj ( u− ; M = Hj (u− ; M ), as l -modules. in Theorem 4.7 and using Theorem 4.5 we obtain the fol = L( λ) Setting M lowing. Corollary 4.8. For λ ∈ Pa+ and j ≥ 0, we have ∼ λ)) (i) T Hj ( u− ; L( = Hj (u− ; L(λ)), as l-modules. u− ; L(λ)) ∼ (ii) T Hj ( = Hj (u− ; L(λ)), as l-modules. ∼ λ)) (iii) T Hj ( u− ; L( = Hj (u− ; L (λ )), as l -modules. ∼ λ)) (iv) T Hj ( u− ; L( = Hj (u− ; L (λ )), as l -modules. a , Oa , We deﬁne the parabolic Kazhdan-Lusztig polynomials for the categories O + Oa , Oa and Oa associated to μ, λ ∈ Pa by letting ∞ μλ (q) := ), Hj u− ; L(λ) dimC Homl L(l, μ (−q)−j , μλ (q) := μλ (q) :=

n=0 ∞ j=0 ∞

dimC Homl L(l, μ), Hj u− ; L(λ) (−q)−j , dimC Homl L(l, μ), Hj u− ; L(λ) (−q)−j ,

j=0 ∞

μ λ (q) :=

μ λ (q) :=

j=0 ∞

dimC Homl L(l , μ ), Hj u− ; L (λ ) (−q)−j , dimC Homl L(l , μ ), Hj u− ; L (λ ) (−q)−j .

j=0

By Vogan’s homological interpretation of the Kazhdan-Lusztig polynomials [Vo, Conjecture 3.4] and the Kazhdan-Lusztig conjecture [KL] (proved in [BB, BK]), μλ (q) coincides with the original deﬁnition in the setting of semisimple Lie algebras and moreover μλ (1) = aμλ (cf. Theorem 4.6). The following reformulation of Corollary 4.8 is a generalization of Theorem 4.6. Theorem 4.9. For λ, μ ∈ Pa+ , we have the following identiﬁcation of Kazhdan a , Oa , Oa , O and O : Lusztig polynomials for the categories O a a (q) = μλ (q) = (q) = (q) = (q). μ λ

μλ

μ λ

μ λ

4.5. Super duality. Lemma 4.10. Let 0 −→ M −→ M −→M −→ 0 be an exact sequence of g (respectively g, g, g and g )-modules which are semisimple a (respectively Oa , Oa , over h (respectively h, h, h and h ) such that M , M ∈ O Oa and Oa ). Then M also belongs to Oa (respectively Oa , Oa , Oa and Oa ). Proof. The statement for Oa is clear. The statements for the other cases follow by arguments, for example, as for [CK, Theorems 3.1 and 3.2].

127 15

SUPER DUALITY AND APPLICATIONS

The arguments in [CLW1, Section 5] can be adapted to prove the following theorem. a → Oa , a → Oa , T : O Theorem 4.11 (Super Duality). The functors T : O a → O and T : O a → O are equivalences of categories. T :O a a

Remark 4.12. In an extreme yet interesting case, the Lie superalgebra D( g) in Section 2.1 (when k = 0) is associated to the following Dynkin diagram with all simple roots being odd; its subalgebra D(g) and D(g ) in Section 2.2 are Lie algebras associated to the following Dynkin diagrams and fundamental systems:

D( g): ··· ··· α

D(g ):

α

−1 2

α0

α1 2

α1

α3 2

β−3

β−2

β−1

β0

β1

β2

β3

β1

β3

β5

···

D(g):

α−1

−3 2

···

β

β

−7 2

−5 2

β

−3 2

β

−1 2

2

2

···

···

2

In the case when ∈ N, we let F be the (semisimple) subcategory of O of inte ∞ -modules of level . The corresponding subcategory of O is a suitable grable gl ∞ -modules of negative level −. By Theorem 4.11, U is equivasubcategory U of gl lent to F, and hence it is semisimple; moreover, we recover the character formulas for modules in U in [KR] and [CL1] that were obtained by completely diﬀerent methods. ¯ 0

¯0 , 4.6. Equivalence of tensor categories. Recall that any modules in O a ¯ 0

Oa , Oa0 and Oa are equipped natural Z2 -gradations given by (4.1). Note that

, the functors T , T , T and T preserve the Z2 -gradations of the modules. For M ¯ ¯ ¯ 0 0 0

N ∈ Oa , we have M ⊗ N ∈ Oa by [CK, Theorem 3.2] and hence Oa is a commutative tensor category (see, for example, [KS, Section 4.2]) with the trivial module as the ¯

¯ 0

¯ 0

unit object. Similarly, O, Oa , Oa0 and Oa are commutative tensor categories with the unit objects (cf. [CK, Theorem 3.1, 3.2]). By Lemma 4.3 and Theorem 4.11, we have the following theorem. ¯

¯

¯0 → O, T : O ¯0 → O0 , T : O ¯0 → O¯0 , and Theorem 4.13. The functors T : O ¯ 0 ¯0 → O are equivalences of tensor categories. T :O

∈ For M

¯0 O a

5. Tilting modules

= M

μ , we consider the restricted dual such that M μ∈h

, C) | f (M

γ ) = 0, for all but ﬁnitely many γ ∈

∨ := {f ∈ HomC (M h∗ }, M which is equipped with the usual Z2 -gradation and g-action. We twist the usual

∨ with the automorphism τ deﬁned in (2.2), and denote g-module structure on M

τ . The Z2 -gradation on M

τ equals the Z2 -gradation the resulting g-module by M

. It is easy to see deﬁned by (4.1). There are natural isomorphisms (M τ )τ = M τ τ = L( The restriction of τ to l is also denoted

) and L( λ) λ).

) = ch(M that ch(M

128 16

CHENG, LAM, AND WANG

¯0 for all

τ ∈ O by τ . Similarly, we can deﬁne L(l, μ )τ for μ ∈ Pa+ . Note that M a ¯ 0 since L(l, μ ¯0

∈O M )τ ∼ ) for all μ ∈ Pa+ . The contravariant functor from O = L(l, μ a a

to M

τ is also denoted by τ . Note that the functor to itself deﬁned by sending M τ is an isomorphism of categories. The restrictions of the automorphism τ of g to g, g, g and g are also denoted ¯ 0

¯ 0

by τ . Similarly, we can deﬁne M τ for an object M in Oa , Oa , Oa0 and Oa . We show that M τ belongs to the same category as M and (M τ )τ = M . Also we have ch(M τ ) = ch(M ) and Lτ = L for all irreducible highest weight modules L. We summarize the above discussion in the following. ¯

Proposition 5.1. We have T ◦ τ = τ ◦ T for T = T, T , T or T . An object M ∈ Oa is said to have a Verma ﬂag if it has a (possibly inﬁnite) increasing ﬁltration of g-modules: 0 = M0 ⊆ M1 ⊆ M2 ⊆ · · · such that M = ∪i≥0 Mi and each Mi /Mi−1 is either 0 or isomorphic to a parabolic Verma module Δ(λi ) for some λi ∈ P + . We deﬁne (M : Δ(μ)) for μ ∈ P + to be the number of subquotients of a Verma ﬂag of M that are isomorphic to Δ(μ). The ¯ ¯0 , O0 , O¯0 and notion of Verma ﬂag is deﬁned in a similar fashion in the categories O ¯ 0

O . A tilting module associated to λ ∈ P + in the category O is an indecomposable g-module U (λ) such that (1) U (λ) has a Verma ﬂag with Δ(λ) at the bottom, and (2) Ext1 (Δ(μ), U (λ)) = 0 for all μ ∈ P + . ¯0 ( μ) (respectively U (μ), U (μ ), U (μ )) in O For μ ∈ P + , the tilting module U ¯ 0

¯ 0

(respectively O , O0 , O ) are deﬁned in a similar fashion. ( From the proposition below, the tilting module U μ) (respectively U (λ), U (μ), ¯ ¯ 0 0 ¯ ¯ 0 U (μ ), U (μ )) belongs to O (respectively Oa , O , O0 , O ) for μ ∈ P + . ¯

a

a

a

a

a

there exists a unique (up to isomorphism) Proposition 5.2. For λ ∈ tilting module U (λ) associated to λ in Oa . There also exists a unique (up to iso (respectively U (λ), U (λ ), U (λ )) associated to (λ) morphism) tilting module U ¯ ¯ ¯0 (respectively O0 , O¯0 , O0 ). Moreover, we have (respectively λ, λ , λ ) in O λ a a a a = U (λ ) and T U (λ) = U (λ ) for = U (λ), T U (λ) = U (λ), T U (λ) T U (λ) Pa+ ,

λ ∈ Pa+ , and U τ = U for any tilting module U in either of these categories. Proof. The same proof for [CW, Theorem 3.14] ensures the existence of the tilting modules. From the construction of the tilting modules, we have U (λ)τ = U (λ) for λ ∈ Pa+ . The uniqueness of the tilting modules follows by adapting an argument of Soergel [So, Section 5] (also cf. [Br2]). As the category Oa may not have enough injective objects, we need to modify the proof therein as follows. First we remark that every element in End(U (λ)) for λ ∈ Pa+ is either locally nilpotent or an isomorphism, since U (λ) is indecomposable and all its weight space is ﬁnite dimensional and the usual argument via the Fitting decomposition works. Let C be an abelian category of g-modules containing Oa with enough injective objects such that every module in C is a semisimple h-module. Let U (λ) and U (λ) be two tilting modules associated to λ in Oa . Then Ext1O (Δ(μ), U (λ)) = 0

SUPER DUALITY AND APPLICATIONS

129 17

for all μ ∈ Pa+ . By Lemma 4.10, Ext1C (Δ(μ), U (λ)) = 0 for all μ ∈ Pa+ . Since U (λ)/Δ(λ) has a Verma ﬂag, we can apply [So, Lemma 5.10] and conclude that Ext1C (U (λ)/Δ(λ), U (λ)) = 0. Therefore we have 0 −→ Hom U (λ)/Δ(λ), U (λ) −→ Hom U (λ), U (λ) −→ Hom Δ(λ), U (λ) −→ 0. Hence there is ϕ ∈ Hom U (λ), U (λ) which restricts to the identity map on Δ(λ). Similarly, there is ϕ ∈ Hom U (λ), U (λ) which restricts to the identity map on Δ(λ). Therefore ϕ ◦ ϕ and ϕ ◦ ϕ are isomorphisms since they are not locally nilpotent. Hence U (λ) and U (λ) are isomorphic. By Theorem 4.13, Theorem 4.5 and the ﬁrst part of the proposition, tilting ¯ ¯ ¯0 , O0 , O¯0 and O0 . The last part modules exist and are unique in the categories O a a a a of the theorem also follows from Proposition 5.1, Theorem 4.5 and U (λ)τ = U (λ) for λ ∈ Pa+ . Remark 5.3. The argument above can be applied to show that there exists a unique tilting module associated to each dominant weight in the categories O, O, deﬁned in [CLW1, Section 3.2] as well. and O 6. Character formula for general linear Lie superalgebras In this section we use the super duality and truncation functors to obtain character formulas for irreducible modules of general linear Lie superalgebras of ﬁnite rank. 6.1. Module categories of ﬁnite-rank Lie superalgebras. We recall that the ﬁnite-rank Lie superalgebras g(m, n), g(m, n), g(m, n), g (m, n), and g (m, n) are deﬁned in Section 2.2 for m, n ∈ Z+ . In this subsection, we introduce certain module categories of these ﬁnite-rank Lie superalgebras, and relate them to the categories studied earlier via the truncation functors (see Proposition 6.2). Let O(m, n) be the full subcategory of g(m, n)-modules M such that M is a semisimple h(m, n)-module with ﬁnite-dimensional weight subspaces Mγ , γ ∈ h(m, n)∗ , satisfying (i) M decomposes over l(m, n) into a direct sum of L(l(m, n), μ) for μ ∈ P + . + (depending on M ) (ii) There exist ﬁnitely many weights λ1 , λ2 , . . . , λ k ∈ P such that if γ is a weight in M , then γ ∈ λi − α∈Π(m,n) Z+ α, for some i. Recall the central element K in the Lie superalgebra g(m, n). Let Oa (m, n) be the full subcategory of O(m, n) consisting of the objects M ∈ O(m, n) such that Kv = av for all v ∈ M . We certainly have O(m, n) = a∈C Oa (m, n). The parabolic Verma modules Δ(m,n) (μ) and irreducible modules L(m,n) (μ) for μ ∈ Pa+ ∩ h(m, n)∗ lie in Oa (m, n), by Proposition 6.2 below. The categories Oa (m, n) are abelian. Analogously we have four variants of the above categories with similar properties for Lie superalgebras g(m, n), g(m, n), g (m, n), and g (m, n) in self-explanatory notations. For V ∈ Oa (m, n), we deﬁne a natural Z2 -gradation V = V¯0 ⊕ V¯1 similar ¯0 (m, n) as before and ¯0 (m, n) of O to (4.1). This allows us to deﬁne a subcategory O a a analogously we have three variants of the above categories for Lie superalgebras g(m, n), g (m, n), and g (m, n).

130 18

CHENG, LAM, AND WANG

Let M be an h-, h-, h-, h - or h-module such that M = γ Mγ , where γ runs over γ ∈ i∈Z+ C−i + 1≤j≤k C¯j + j∈N Cj + CΛ0 . For m, n ∈ Z+ ∪ {∞}, we consider the truncated vector space Mν , tr(m,n) (M ) = ν

−m
1≤j≤k

C¯j +

Cj + CΛ0 . l, μ ) is an Lemma 6.1. Let m, n ∈ Z+ ∪ {∞} and μ ∈ Pa+ . Then, tr(m,n) L( l(m, n)-module, and l(m, n), μ jj = 0, ∀j ≤ −m, ∀j ≥ n + 1, L( ), if μ, E l, μ (6.1) tr(m,n) L( ) = 0, otherwise. where the sum is over ν ∈

C−i +

1≤j

Parallel results hold for the truncation of L(l, μ), L(l, μ), L (l , μ ), L (l , μ ). 0 → As a consequence of the lemma above, we obtain exact functors tr(m,n) : O a ¯

¯

¯

¯0 (m, n), tr(m,n) : Oa → Oa (m, n), tr(m,n) : O0 → O0 (m, n), tr(m,n) : O¯0 → O a a a a ¯ 0

¯ 0

Oa0 (m, n) and tr(m,n) : Oa → Oa (m, n), by sending M to tr(m,n) (M ). The following lemma can be proved by the same arguments as in [CLW1, Lemma 3.2] and [Don, Proposition 1.5]. ¯

Proposition 6.2. Let m, n ∈ Z+ ∪ {∞}, μ ∈ Pa+ and X = L, Δ, U . jj = 0, ∀j ≤ −m, j ≥ n + 1, (m,n) ( μ), if μ, E X (i) tr(m,n) X( μ) = 0, otherwise. jj = 0, ∀j ≤ −m, j ≥ n + 1, X(m,n) (μ), if μ, E (ii) tr(m,n) X(μ) = 0, otherwise. jj = 0, ∀j ≤ −m, j ≥ n + 1, X (m,n) (μ), if μ, E (iii) tr(m,n) X(μ) = 0, otherwise. jj = 0, ∀j ≤ −m, j ≥ n + 1, (μ ), if μ , E X(m,n) (iv) tr(m,n) X (μ ) = 0, otherwise. jj = 0, ∀j ≤ −m, j ≥ n + 1, X (m,n) (μ ), if μ , E (v) tr(m,n) X (μ ) = 0, otherwise. Remark 6.3. Let λ ∈ P + and m, n ∈ Z+ ∪ {∞}. Let d denote the boundary H• (u− ; L(λ)), λ)), operators for the complexes of the homology groups H• ( u− ; L( H• (u− ; L(λ)), H• (u− ; L(λ )), and H• (u− ; L(λ )). From the formula of d (see e.g. [CL2, (4.1)]) it is easy to see that the truncation functors are compatible with d, i.e., d ◦ tr(m,n) = tr(m,n) ◦ d. It follows that truncation functors tr(m,n) send these u-homology groups to the corresponding u-homology groups, and the corresponding module multiplicities of Levi subalgebras (or equivalently the corresponding Kazhdan-Lusztig polynomials) match under the truncation functors.

131 19

SUPER DUALITY AND APPLICATIONS

6.2. Character formulas for gl(k|2)-modules in the BGG category. Let λ ∈ P + . Then Proposition 6.2 and Theorem 4.6 together imply that, for any m, n ∈ Z+ , the character of the irreducible g (m|n)-module Lm,n (λ ) can be computed from knowledge of the irreducible character of the g-module L(λ). On the other hand chL(λ) is determined by the the coeﬃcients aμλ = μλ (1), where the polynomials μλ (q) are type A parabolic Kazhdan-Lusztig polynomials (see Section 4.4). Recalling the notations from Section 6.1 we note that g (1, 0) ∼ = gl(k|2) ⊕ CK (see Section 2.1), where gl(k|2) denotes the Lie subalgebra of linear transformations k on i=1 Cvi Cv− 12 Cv 12 . Recall that P + = a∈C Pa+ . Now choose Y0 = ∅ so

that l1,0 is simply the Cartan subalgebra of g (1, 0). Now for a ﬁxed a ∈ C, m, n ∈ N and λ0j ∈ Z we have λ = aΛ0 −

m−1

−i +

i=0

λ0j j +

j

n

l ∈ Pa+ .

l=1

(a, λ01 , . . . , λ0k ; λ− , λ+ ),

This corresponds to the dominant tuple and λ+ = (1n ). Thus, we have by Lemma 3.2 λ = aΛ0 − m− 12 + λ0j j + n 12 .

with λ− = (1m ),

j

k

Now let μ = p− 12 + j=1 λ0j j + q 12 be any integral weight of gl(k|2). Suppose we want to compute the character of the irreducible module with highest weight μ with respect to the Borel associated to the following fundamental system:

(6.2)

−1 2

− 1

α1

α2

···

αk−2

αk−1

k − 1 2

First, by tensoring with an integral multiple of the determinant representation, if necessary, we can assume that q = 0 and p ∈ Z. Now choose a = −p. Then the weight λ corresponding to the dominant tuple λ = (−p, λ01 , . . . , λ0k , ∅, ∅) restricts precisely to the weight μ on gl(k|2). It follows that the irreducible character of any integral highest weight with respect to the Borel associated to the fundamental system (6.2) are determined by the usual Kazhdan-Lusztig polynomials of type A Lie algebras. Now consider the standard Borel subalgebra of gl(k|2) corresponding to the following fundamental system: (6.3)

α1

α2

α3

···

αk−1 k − − 1 − 1 − 1 2 2 2

Note that the Dynkin diagram in (6.2) is related to the standard Dynkin diagram (6.3) by a sequence of odd reﬂections with respect to the following odd roots: (6.4)

− 12 − 1 , − 12 − 2 , . . . , − 12 − k .

Let us use Δ (γ) and L (γ) to denote the gl(k|2)-Verma and irreducible modules of highest weight γ with respect to the Borel associated to (6.3). Now suppose that

132 20

CHENG, LAM, AND WANG

we have for λ ∈ Pa+

chL (λ ) =

aμλ chΔ (μ),

μ

where Δ (μ) is the Verma module with respect to the Borel in (6.2). Let [λ ] be the highest weight obtained from λ by applying the sequence of odd reﬂections in (6.4) following the prescription of Lemma 3.4. Furthermore, we observe that

chΔ (μ) = chΔ (μ − k− 12 +

k

i ).

i=1

Thus, we conclude that

chL ([λ ] ) =

aμλ chΔ (μ +

μ

k

i − k− 12 ).

i=1

Therefore, the Kazhdan-Lusztig polynomials of type A also solve the character problem for irreducible highest weight modules with highest weights with respect to the standard Borel of gl(k|2). 6.3. A variant of Brundan’s conjecture. For notations on Fock spaces below, we will refer to [CWZ, CW] (which diﬀers somewhat from [Br1]). Let V with basis {vi }i∈Z denote the natural Uq (gl∞ )-module and let V∗ denote its dual module with basis {wi }i∈Z . The “Fock space” ∧m V∗ ⊗ V⊗k ⊗ ∧n V∗ admits a standard basis (using vi ’s and wi ’s). One can deﬁne a bar involution on ∧m V∗ ⊗ V⊗k ⊗ ∧n V∗ which is compatible with the bar involution on Uq (gl∞ ). This gives rise to the canonical basis and dual canonical basis on (a completion of) ∧m V∗ ⊗ V⊗k ⊗ ∧n V∗ (cf. [Lu, Br1]). We denote by Om|k|n the parabolic BGG category of gl(k|m + n)-modules of integral weights, which are semisimple with respect to the Levi subalgebra gl(m) ⊕ hk ⊕gl(n) (here hk denotes the Cartan in gl(k)), with respect to the Borel associated to the fundamental system below: 1 2

−m

− 3 2

−1 2

−m

···

−3 2

−

−1 2

− 1

k − 1

α1

α2

···

αk−2

αk−1

2

··· 1 − 3 2

2

n− 3 2

−

n− 1 2

We shall prove a variant of Brundan’s conjecture (see [Br1, Conjecture 4.32]) which can be informally formulated as follows. Note that the Borel subalgebra used here is not a standard one in contrast to Brundan’s original setting. Theorem 6.4. The category Om|k|n categoriﬁes the Uq (gl∞ )-module ∧m V∗ ⊗ V ⊗ ∧n V∗ , where the parabolic Verma, tilting and simple gl(k|m + n)-modules correspond to the standard, canonical, and dual canonical basis (at q = 1), respectively. ⊗k

The essence of Brundan’s conjecture is that the entries of the transition matrix between the (dual) canonical basis and the standard basis on ∧m V∗ ⊗ V⊗k ⊗ ∧n V∗ should be interpreted as the Kazhdan-Lusztig polynomials for the category Om|k|n . Indeed our approach is powerful enough to establish that these transition matrix entries coincide with the Poincare polynomials for the corresponding u-homology

SUPER DUALITY AND APPLICATIONS

133 21

groups deﬁned in Section 4.4. A precise formulation of Theorem 6.4 and its variants with self-contained proofs would take many pages as it would require us to repeat in our setting much of the notations and constructions in [Br1, CW] among other things. Let us instead sketch a proof of Theorem 6.4 below. The space of semi-inﬁnite q-wedges decomposes according to “sectors” labeled by Z ([KMS]). For a ∈ Z, let ∧∞+a V be the space of semi-inﬁnite q-wedges a ) of of sector a, which is isomorphic to the highest weight Uq (gl∞ )-module L(Λ ∞−a ∗ highest weight being the ath fundamental weight. Similarly the space ∧ V of a ). semi-inﬁnite wedges of “sector −a” is also isomorphic to the Uq (gl∞ )-module L(Λ ∞+a ∞−a ∗ ∼ The canonical isomorphism of Uq (gl∞ )-modules ∧ V=∧ V commutes with the bar involution. (This extends the observation made in [CWZ] for a = 0.) This induces an isomorphism of Uq (gl∞ )-modules ∧∞+a V ⊗ V⊗k ⊗ ∧∞ V ∼ = ∧∞−a V∗ ⊗ V⊗k ⊗ ∧∞ V∗ , which matches the standard, canonical, and dual canonical bases, respectively. Based on this together with Theorem 4.11 as well as a Fock space reformulation (cf. e.g. [CW, Theorem 4.14] for a special case; the general case is similar) of the Kazhdan-Lusztig conjecture for parabolic BGG categories in type A [BB, BK, KL], we establish a version of Theorem 6.4 for m = n = ∞. Theorem 6.4 for ﬁnite m and n follows from this version for m = n = ∞ and the compatibility of the (dual) canonical basis for varying m, n (compare [CWZ, Corollary 2.10], [CW, Corollary 2.6]). Remark 6.5. Note that Theorem 6.4 for m = n = 1 solves the irreducible character problem of the full BGG category of gl(k|2)-modules, which can also be viewed as a reformulation of the result in Section 6.2. On the other hand, it is easy to give a formula for the irreducible characters in the category Om|k|n for general m and n, in the spirit of Section 6.2, if one is willing to introduce messier notations. 7. Some more variations The formulation and construction in this paper have variations which unfortunately involve more complicated notations. In this section, we will explain one such variation and its implication on u-homology computation. For ﬁxed p, q ∈ Z+ , let F = {r ∈ 12 Z | − p < r ≤ 0 or 12 ≤ r ≤ q}. Set J = I ∪ F , J = I ∪ F , J = I ∪ F and J = I ∪ F . The subalgebra of g generated rs with r, s ∈ J (respectively J, J , J ) is denoted by G (respectively G, G and by E G ). Similar results in Section 3 and Section 4 are still valid if g (respectively g, g and g ) are replaced by G (respectively G, G and G ). It is worth pointing out + + that the sets Pa+ , P a , Pa+ and P a of all dominant weights are determined by the choice of the orderings preserving the orderings of positive integers, positive half integers, non-positive integers and negative half integers and satisfying i < b < j for i ∈ − 12 Z+ , b ∈ K and j ∈ 12 N. We do not carry out any details here. Let us illustrate by an example. Deﬁne orderings of J = I ∪ F , J = I ∪ F , J = I ∪ F and J = I ∪ F respectively by 1 3 1 · · · ≺ −2 ≺ −1 ≺ 0 ≺ −p + ≺ · · · ≺ − ≺ − ≺ 1 ≺ · · · ≺ k 2 2 2 3 1 1 ≺ ≺ ≺ ··· ≺ q − ≺ 1 ≺ 2 ≺ ··· , 2 2 2

134 22

CHENG, LAM, AND WANG

· · · ≺ −2 ≺ −1 ≺ 0 ≺ −p +

··· ≺ −

1 3 1 ≺ ··· ≺ − ≺ − ≺ 1 ≺ ··· ≺ k 2 2 2 1 3 ≺ 1 ≺ 2 ≺ ··· ≺ q ≺ ≺ ≺ ··· , 2 2

1 3 ≺ − ≺ −p + 1 ≺ · · · − 2 ≺ −1 ≺ 0 ≺ 1 ≺ · · · ≺ k 2 2 1 3 1 ≺ ≺ ≺ ··· ≺ q − ≺ 1 ≺ 2 ≺ ··· , 2 2 2

1 3 ≺ − ≺ −p + 1 ≺ · · · − 2 ≺ −1 ≺ 0 ≺ 1 ≺ · · · ≺ k 2 2 1 3 ≺ 1 ≺ 2 ≺ ··· ≺ q ≺ ≺ ≺ ··· . 2 2 The corresponding fundamental systems are indicated in the following Dynkin diagrams:

Lp G: R R K L q

1 − 1

1 − ¯ − 1

0 − 1 1 k −p − q− 2 2 2 2

Lp Rq

K L G: R

q − 1

1 − ¯ −

0 − 1 1 1 −p − k 2 2 2

Lp

G : R L K R q

1 − 1−p

− 1

1 − 1

−1 − ¯ k − q− 1 2 2 2

Lp Rq

L K G : R

1 − 1−p q − 1 − ··· ≺ −

−

−1 − ¯ 1

2

−

1

k

2

Let a, λ1 , . . . , λk ∈ C, λ and λ be two partitions. Associated to the tuple λ = (a, λ1 , . . . , λk ; λ− , λ+ ) satisfying the dominant condition (2.3), set Λ+ F (λ) := +

ΛF (λ) :=

k

+

λi i +

q

i=1

j=1

k

q

λi i +

i=1

Λ− F (λ) := −

p

(λ+ )j j− 12 +

−

ΛF (λ) := −

λ+ j − q j

j∈N

λ+ j j +

j=1

(λ+ )j − q j− 12 ,

j∈N

(λ− )j −j+ 12 −

j=1 p

λ− j − p −j+1 + aΛ0 ,

j∈N

λ− j −j+1 −

j=1

(λ− )j − p −j+ 12 + aΛ0 .

j∈N

Associated to such tuple, we deﬁne the weights λF := Λ− (λ) + Λ+ (λ), −

λF := Λ (λ) + Λ+ (λ), +

+

+

λF := Λ− (λ) + Λ (λ),

−

+

λF := Λ (λ) + Λ (λ).

The sets Pa+ , P a , Pa+ and P a of all dominant weights are the sets consisting of the weights of the form λF , λF , λF , λF , respectively. The main results on super duality in Sections 3, 4 and 5 aﬀord straightforward counterparts in this new setting.

SUPER DUALITY AND APPLICATIONS

135 23

Remark 7.1. The u-(co)homology formulas in the sense of Kostant and Enright [E] for unitarizable modules of general linear Lie superalgebras have been computed in [LZ] (cf. [HLT, Theorem 5.1, Theorem 4.4]), and they can be easily recovered via the super duality of this paper by an appropriate choice of λ and of the orderings deﬁned in the example above. Let us set k = p = 0 and choose λ + satisfying λ− 1 + λ1 ≤ a with a ∈ Z+ . Then we have Enright type u-(co)homology formulas for unitarizable modules L(G , λF ) (cf. [HLT, Theorem 4.4]). Using Theorem 4.7, its analogue for the irreducible modules L(G , λF ) together with Enright’s (co)homology formulas for L(G , λF ), we recover [LZ, Theorem 6.1].

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[LZ] [Lu] [PS] [So] [Ta]

[Vo]

CHENG, LAM, AND WANG

N. Lam and R.B. Zhang, u-cohomology formula for unitarizable modules over general linear superalgebras, J. Algebra 327 (2011), 50–70. G. Lusztig, Introduction to quantum groups, Progress in Math. 110, Birkh¨ auser, 1993. I. Penkov and V. Serganova, Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules, Ann. Inst. Fourier 39 (1989), 845–873. W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory (electronic) 2 (1998), 432–448. J. Tanaka, On homology and cohomology of Lie superalgebras with coeﬃcients in their ﬁnite-dimensional representations, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 3, 51–53. D. Vogan, Irreducible characters of semisimple Lie Groups II: The Kazhdan-Lusztig Conjectures, Duke Math. J. 46 (1979), 805–859. Institute of Mathematics, Academia Sinica, Taipei, Taiwan 11529 E-mail address: [email protected] Department of Mathematics, National Cheng-Kung University, Tainan, Taiwan 70101 E-mail address: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22904 E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Structures and Representations of Aﬃne q-Schur Algebras Jie Du Abstract. This paper provides a survey for the latest developments in the theory of aﬃne q-Schur algebras and Schur–Weyl duality between aﬃne quantum gln and aﬃne type A Hecke algebras. More precisely, we will establish, on the one side, an isomorphism between the double Ringel–Hall algebra D(n) of a cyclic quiver (n) and the quantum loop algebra of gln , and establish, on the other side, explicit epimorphisms from the double Ringel–Hall algebra to aﬃne q-Schur algebras S(n, r). Then, by two commuting actions on an aﬃne tensor space, we will establish, for n r, a category equivalence between representations of aﬃne q-Schur algebras and representations of aﬃne Hecke algebras. In this way, we describe two classiﬁcation theorems for simple representations of aﬃne q-Schur algebras over C when the parameter q is not a root of unity. Further structures of aﬃne q-Schur algebras will also be discussed as applications of the epimorphisms mentioned above.

1. Introduction It is well known that Schur algebras or their quantum analogue—q-Schur algebras—are a class of ﬁnite dimensional algebras which play an important role in the theory of Schur–Weyl duality. This theory links representations of the general or quantum linear group with those of the symmetric group or the associated Hecke algebra. Over its history of more than one hundred years, it continues to make profound inﬂuences in several areas of mathematics such as Lie theory, representation theory, invariant theory, combinatorics, etc. Recent new developments include, e.g., walled Brauer algebras and rational q-Schur algebras arising from quantum gln actions on mixed tensor spaces [11], quantum Schur superalgebras [16], quiver Schur algebras [37], and integral Schur–Weyl duality for type C [32], and so on. This paper provides a survey for the latest work [8] by Deng, Fu, and the author in the study of aﬃne q-Schur algebras and Schur–Weyl duality between aﬃne quantum gln and aﬃne type A Hecke algebras. The introduction of the aﬃne analogue of q-Schur algebras was given by Ginzburg–Vasserot [20] in 1993 as a generalization of the geometric approach to q-Schur algebras developed by Beilinson–Lusztig–MacPherson [2]. Later, in 1999, 2010 Mathematics Subject Classiﬁcation. Primary 17B37, 20G43, 20C08; Secondary 16G20, 16T20. Supported partially by Australian Research Council. c Mathematical 0000 (copyright Society holder) c 2012 American

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several other versions have appeared in [29], [38] (see also [22] for another deﬁnition). Independently, representations of aﬃne Hecke algebras H(r)C when the parameter q is not a root of unity have been investigated much earlier by Zelevinsky [42] in 1983 and Rogawski [35] in 1985. On the other hand, besides the standard representation theory of quantum groups (associated with symmetrizable Cartan matrices) as developed in [28], there is, in the aﬃne type case, a so-called pseudohighest weight module theory developed by Chari–Pressley around 1994 and its generalization to aﬃne quantum gln (or more precisely the quantum loop algebra of gln ) by Frenkel–Mukhin [17]. Though some connections under the assumption n ) (resp., UC (gl n )) and H(r)C n > r (resp., n r) between representations of UC (sl were observed in [6] (resp., [19]), it was unclear until the work [8] how the aﬃne qSchur algebra S(n, r)C plays a bridging role between aﬃne quantum gln and aﬃne Hecke algebras: S(n, r)C -Mod n )-Mod UC (gl

H(r)C -Mod

In this paper, we will present this bridging relation through a discussion on the structures and representations of aﬃne q-Schur algebras. n )-H(r)C -bimodule structure To build this bridging relation, we need a UC (gl ⊗r on the aﬃne tensor space ΩC . Using the geometric description of the Hall algebra H(n)C of a cyclic quiver (n), Varagnolo–Vasserot have already investigated an op H(n)C -H(r)C -bimodule structure on Ω⊗r C . Motivated by works of Schiﬀmann [36] and Hubery [24], we introduced the double Ringel–Hall algebra D,C (n) together with a Drinfeld–Jimbo type presentation involving Chevalley and (inﬁnitely many) central generators. Thus, the quantum aﬃne sln is the subalgebra generated by the Chevalley generators. By extending an isomorphism of Beck [1] between the quantum aﬃne sln of Drinfeld–Jimbo type and the quantum loop algebra of sln with Drinfeld’s new presentation [13], we obtain a Hopf algebra isomorphism between n ) (see Theorem 2.6). In this way, the natural action of D,C (n) D,C (n) and UC (gl on the aﬃne tensor space Ω⊗r C , which extends the action of Varagnolo–Vasserot, n ) on Ω⊗r and, hence, a UC (gl n )-H(r)C -bimodule gives rise to an action of UC (gl C ⊗r structure on ΩC . The double Ringel–Hall algebra has naturally a triangular decomposition with the ±-parts isomorphic to the Hall algebra. Thus, by an analysis of a spanning set of Beilinson–Lusztig–MacPherson (BLM) type for aﬃne q-Schur algebras, we managed to establish explicitly an epimorphism from the double Ringel–Hall algebra to aﬃne q-Schur algebras. In this way, the structures of aﬃne q-Schur algebras are nicely presented through its triangular decomposition, integral bases, central subalgebras, and generators/relations (for n r). These material will form the ﬁrst part of the paper. We will also mention a few conjectures at the end of Part 1 regarding Lusztig type forms, BLM type realization for the double Ringel–Hall algebra, and the second centralizer property in the aﬃne quantum Schur–Weyl reciprocity. Part 2 of the paper discusses the bridging relation between the representations. When the parameter is not a complex root of 1, two classiﬁcations for irreducible representations of aﬃne q-Schur algebras will be determined. As mentioned above, they arise from the corresponding classiﬁcations for aﬃne Hecke algebras H(r)C in

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n ) in terms of dominant terms of multisegments and for quantum loop algebras UC (gl Drinfeld polynomials. Moreover, we will identify them by establishing a bijection from multisegments of each length n and total length r to n-tuples of dominant Drinfeld polynomials of total degree r. It would be interesting to point out that research on further applications of the work is underway. For example, irreducible representations of aﬃne q-Schur algebras at a complex root of 1 have been classiﬁed by Fu.1 Also, Drinfeld polynomials associated with small representations L(λ)a of S(n, r)C (a ∈ C∗ ) have been computed in terms of row residue sequences of partition λ.2 Some Notations. Let Z = Z[υ, υ −1 ] be the ring of integral Laurent polynomials, let Q(υ) be the fraction ﬁeld of Z, and let C∗ = C\{0}. For integers N, t with t 0, let υ N −i+1 − υ −(N −i+1) N N = ∈ Z and = 1. i −i υ −υ t 0 1it m ]! N −υ −m ! = [t]![N and [N ] = m := [1][2] · · · [N ], then for If we put [m] = υυ−υ −1 1 t [N −t]! all 1 t N . We also write [m]q etc. to be the evaluation at q ∈ C. In Part 1, algebras are mostly deﬁned over Z or Q(υ). In particular, if an algebra is deﬁned over Z such as H(n), H(r), S(n, r) etc., then we will use the same letter in bold face to denote the corresponding algebra H(n), H(r), S(n, r) etc. obtained by based change to Q(υ). In Part 2, we will mainly consider representations over C for algebras H(r)C , S(n, r)C etc. which are obtained by specializing υ to a non-root of unity q ∈ C∗ . Acknowledgement. The paper was based on the two lectures given by the author at the Southeastern Lie Theory Workshop, Charlottesville, June 1–4, 2011. I would like to thank the organizers for the opportunity of attending this special conference which marks the retirement of Professor Leonard Scott. Leonard Scott was my postdoctoral advisor from 1988 to 1990, a crucial period in my life. Not only he taught me mathematics, but also his generosity, his great sense of humor, and his optimistic attitude towards life inﬂuenced me so much for so many years!

Part 1: The Structures Let = (n) denote the cyclic quiver n

1

2

3

n−2

n−1

with vertex set I = Z/nZ = {1, 2, . . . , n} and arrow set {i → i + 1 | i ∈ I}. Let F be a ﬁeld. In the category of ﬁnite-dimensional nilpotent representations of over F, the (one-dimensional) simple objects are denoted by Si , i ∈ I, and indecomposable 1Q. Fu, Aﬃne quantum Schur algebras at roots of unity, preprint 2012. 2J. Du and Q. Fu, Small representations for aﬃne q-Schur algebras, preprint (2012).

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objects by Si [l], for all i ∈ I and l 1. We will use the following matrix notation to denote the isomorphism classes (or isoclasses) of nilpotent representations. For i, j ∈ Z, let Ei,j = (ek,l )k,l∈Z be the Z × Z-matrix deﬁned by ek,l = δk,i+sn δl,j+sn (k, l, s ∈ Z). Let Θ(n) (resp., Θ+ (n)) be the N-span of {Ei,j |1 i − t n, j ∈ Z} (resp., {Ei,j |1 i n, j ∈ Z, i < j}. Let also Θ (n) = { A | A ∈ Θ+ (n)}. Thus, any A = (ai,j ) ∈ Θ(n) satisﬁes ai,j = ai+n,j+n for all i, j ∈ Z and each row has a ﬁnite sum. Let, for r 0 Θ(n, r) = {A ∈ Θ(n) | σ(A) = r}, Zn = {(λi )i∈Z ∈ ZZ | λi = λi+n , ∀i ∈ Z}, and Λ(n, r) = {λ ∈ Zn | λi 0 ∀i, σ(λ) = r},

n

where σ(A) = i=1 j∈Z ai,j and σ(λ) = λ1 + · · · + λn . Then, we may identify Λ(n, r) with the set Λ(n, r) of compositions of r into n parts. Note that taking sum of each row or column deﬁnes functions ro, co : Θ(n) → Zn . which map Θ(n,

r) onto Λ(n, r). For any A = 1in ai,j Ei,j ∈ Θ+ (n), deﬁne i<j

M (A) = MF (A) = ⊕ 1in ai,j Si [j − i]. i<j

Then {[M (A)] | A ∈ tions of .

Θ+ (n)}

is a complete set of isoclasses of nilpotent representa-

2. Double Ringel–Hall algebras and quantum loop algebras For F = Fq , let hC A,B be the number of submodules N of MFq (C) such that MFq (C)/N ∼ = MFq (A) and N ∼ = MFq (B). By a result of Ringel, there exist polyno2 C (υ ), called Hall polynomials, such that ϕC mials ϕC A,B A,B (q) = hA,B . Let H(n) be the (generic) Ringel–Hall algebra [33] associated with the cyclic quiver (n), which is by deﬁnition the free Z-module with basis {uA = u[M (A)] | A ∈ Θ+ (n)}. The multiplication is given by, for A, B ∈ Θ+ (n), uA uB = υ d(A),d(B) ϕC A,B uC C∈Θ+ (n)

where d(D) denotes the dimension vector of M (D) and d(A), d(B) = dim Hom(M (A), M (B)) − dim Ext1 (M (A), M (B)) is the Euler form associated with the cyclic quiver (n). If we write d(A) = (ai )i∈I and d(B) = (bi )i∈I , the dimension vectors of M (A) and M (B), respectively, then d(A), d(B) = ai bi − ai bi+1 . i∈I

i∈I

The Z-subalgebra C(n) of H(n) generated by u[mSi ] (i ∈ I and m 1) is called the composition algebra of (n) (see [34]). Theorem 2.1. Let C(n) = C(n) ⊗ Q(υ) and H(n) = H(n) ⊗ Q(υ). (1) ([34]) The Q(υ)-algebra C(n) is isomorphic to the ±-part of the quantum n. enveloping algebra of sl

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(2) ([36]) There are (algebraically independent) central elements cm , m 1, in H(n) such that H(n) = C(n) ⊗Q(υ) Q(v)[c1 , c2 , . . .]. Remark 2.2. The Ringel–Hall algebra H(n) has a grading H(n) = ⊕d∈Nn H(n)d by dimension vectors d. Each component H(n)d can be deﬁned geometrically in terms of invariant functions on a graded space ⊕i∈I Vi with d = (dim Vi )i under the conjugate action of i∈I GL(Vi ). This gives rise to a geometric deﬁnition of Hall algebras and, hence, results in many other applications. For example, Lusztig used this to introduce the canonical bases for quantum enveloping algebras. Let

0

= H(n) ⊗Q(υ) Q(υ)[K1±1 , . . . , Kn±n ] and

0

= Q(υ)[K1±1 , . . . , Kn±n ] ⊗Q(υ) H(n).

H(n) H(n)

Putting x = x ⊗ 1 and y = 1 ⊗ y for x ∈ H(n) and y ∈ Q(υ)[K1±1 , . . . , Kn±n ], + H(n)0 and H(n)0 are Q(υ)-spaces with bases {u+ A Kα | α ∈ ZI, A ∈ Θ (n)} − + and {Kα uA | α ∈ ZI, A ∈ Θ (n)}, respectively. By work of Green [21] and Xiao 0 0 [40], it is known that both H(n) and H(n) are Hopf algebras. Thus, one can build a larger Hopf algebra by using a so-called skew-Hopf pairing [26]. 0 0 By [40, Prop. 5.3], the Q(υ)-bilinear form ψ : H(n) × H(n) → Q(υ) deﬁned by − αβ−d(A),d(A)+α+2d(A) −1 ψ(u+ aA δA,B , A Kα , Kβ uB ) = υ

where α, β ∈ ZI and A, B ∈ Θ+ (n), is a skew-Hopf pairing. Here, d(A) is the dimension of M (A) and aA (q) = |AutMFq (A)|. 0 0 With ψ, we deﬁne I to be the ideal of the free product H(n) ∗ H(n) generated by (b2 ∗ a2 )ψ(a1 , b1 ) − (a1 ∗ b1 )ψ(a2 , b2 ) (a ∈ H(n)0 , b ∈ H(n)0 ), 3

where ΔH(n)0 (a) = a1 ⊗ a2 and ΔH(n)0 (b) = b1 ⊗ b2 . Deﬁne the Drinfeld 0 0 double by D ∗ H(n) /I and the double Ringle–Hall algebra by (n) := H(n) + − 0 D(n) = D (n)/J = H(n) · D(n) · H(n) ,

where J denotes the ideal generated by 1 ⊗ Kα − Kα ⊗ 1 for all α ∈ ZI, H(n)+ = − op H(n), H(n) = H(n) , and D(n)0 is the subalgebra generated by the Ki± . Remark 2.3. If q ∈ C∗ is not a root of unity, then aA (q) = 0. Thus, the Hopf pairing is well deﬁned over C with respect to such q. So the double Ringel–Hall algebra D,C (n) is deﬁned over C. The following result is built on the Hubery’s explicit description [23, 24] of Schiﬀmann’s central elements ci . Let (ci,j )i,j be the Cartan matrix of aﬃne type A. 3In fact, it is enough to take a, b to be all semisimple generators u+ , u− , respectively; see [8, a b

2.6.1].

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Theorem 2.4 ([8, 2.3.1,2.3.5]). The double Ringel–Hall algebra D(n) of the cyclic quiver (n) is the Hopf Q(υ)-algebra generated by Ei = u+ [Si ] , Fi = − −1 + − + u[Si ] , Ki , Ki , zs , zs , i ∈ I, s ∈ Z with relations (i, j ∈ I and s, t ∈ Z+ ): (QGL1) Ki Kj = Kj Ki , Ki Ki−1 = 1; (QGL2) Ki Ej = v δi,j −δi,j+1 Ej Ki , Ki Fj = v −δi,j +δi,j+1 Fj Ki ; −1 i −K K −1 i

(QGL3) Ei Fj − Fj Ei = δi,j υ−υ −1 , where Ki = Ki Ki+1 ; 1 − ci,j Eia Ej Eib = 0 for i = j; (QGL4) (−1)a a a+b=1−ci,j 1 − ci,j Fia Fj Fib = 0 for i = j; (QGL5) (−1)a a a+b=1−ci,j + + + − − − − + − z+ s zt = zt zs ,zs zt = zt zs , zs zt + + − − Ki zs = zs Ki , Ki zs = zs Ki ; + − − − Ei z+ s = zs Ei , Ei zs = zs Ei , Fi zs

+ (QGL6) = z− t zs ; (QGL7) + + (QGL8) = z− s Fi , and zs Fi = Fi zs . Moreover, the Hopf algebra structure is given by comultiplication Δ, counit ε, and antipode σ deﬁned by

i + 1 ⊗ Ei , Δ(Ei ) = Ei ⊗ K

−1 ⊗ Fi , Δ(Fi ) = Fi ⊗ 1 + K i

Δ(Ki±1 ) = Ki±1 ⊗ Ki±1 , ε(Ei ) = ε(Fi ) = 0, ε(Ki ) = 1;

i Fi , σ(K ±1 ) = K ∓1 ,

−1 , σ(Fi ) = −K σ(Ei ) = −Ei K i

i

i

± ± ε(z± Δ(z± s ) = zs ⊗ 1 + 1 ⊗ zs s ) = 0, ± ± and σ(zs ) = −zs ,

where i ∈ I and s ∈ Z+ . Replacing Q(υ) by C and υ by a non-root of unity q, a similar result holds for D,C (n). The subalgebra U(n) of D(n) generated by Ei , Fi , Ki , Ki−1 (i ∈ I) is the n ) generated by (extended) quantum aﬃne sln , which contains the subalgebra U(sl −1

Ei , Fi , Ki , Ki (i ∈ I). The latter is the quantum enveloping algebra of the loop algebra of sln . Motivated from the universal enveloping algebra of a loop algebra, Drinfeld introduced the following quantum version in [13]. ) (or quantum aﬃne gl ) Definition 2.5. (1) The quantum loop algebra U(gl n n ± and gi,t (1 i n, is the Q(υ)-algebra generated by xi,s (1 i < n, s ∈ Z), k±1 i t ∈ Z\{0}) with the following relations: (QLA1) ki k−1 = 1 = k−1 i i ki , [ki , kj ] = 0; ± ±(δi,j −δi,j+1 ) ± xj,s ki , [ki , gj,s ] = 0; (QLA2) ki xj,s = v ⎧ ⎪ 0 if i = j, j + 1, ⎨ [s] ± ± −is (QLA3) [gi,s , xj,t ] = ±v if i = j, s xi,s+t , ⎪ ⎩ (1−i)s [s] ± ∓v x , if i = j + 1; s i−1,s+t (QLA4) [gi,s , gj,t ] = 0; φ+

−φ−

− i,s+t i,s+t (QLA5) [x+ ; i,s , xj,t ] = δi,j v−v −1 ± ± ± ± ± ± ± (QLA6) xi,s xj,t = xj,t xi,s , for |i−j| > 1, and [x± i,s+1 , xj,t ]v ±cij = −[xj,t+1 , xi,s ]v ±cij ; ± ± ± ± ± (QLA7) [x± i,s , [xj,t , xi,p ]v ]v = −[xi,p , [xj,t , xi,s ]v ]v for |i − j| = 1,

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where [x, y]a = xy − ayx, and φ± i,s are deﬁned by the generating functions in indeterminate u: −1 ±s

±1 (2.1) Φ± ) hi,±m u±m ) = φ± . i (u) := ki exp(±(v − v i,±s u m1

s0

Here ki = ki /ki+1 (kn+1 = k1 ) and hi,m = v (i−1)m gi,m − v (i+1)m gi+1,m , for all 1 i < n. n ) be the quantum loop algebra deﬁned by the same generators (2) Let UC (gl and relations (QLA1)–(QLA7) with Q(υ) replaced by C and υ by a non-root of unity q ∈ C. Set, for each s 1, θ±s = ∓

1 (g1,s + · · · + gn,s ). [s]

n into The following result is a lifting of Beck’s embedding [1] from quantum sl n ) and Hubery’s embedding [24] of the extended Ringel–Hall algebras H(n)0 U(gl n ). into U(gl Theorem 2.6 ([8, 2.5.3]). There is a Hopf algebra isomorphism E : D(n) −→ n ) such that U(gl Ki±1 −→ k±1 i , En −→ ε+ n, where

Fj −→ x− j,0 (1 i n, 1 j < n),

Ej −→ x+ j,0 , Fn −→ ε− n,

z± s −→ θ±s (s 1),

− − − n −

ε+ n = (−1) [xn−1,0 , · · · , [x3,0 , [x2,0 , x1,1 ]υ −1 ]υ −1 · · · ]υ −1 kn and + + + + n −1 ε− n = (−1) kn [· · · [[x1,−1 , x2,0 ]υ , x3,0 ]υ , · · · , xn−1,0 ]υ .

Similarly, with υ replaced by a non-root of unity q ∈ C∗ , there is a C-algebra n ). isomorphism EC : D,C (n) → UC (gl This isomorphism is a key ingredient for the aﬃne q-Schur algebra S(n, r)C to n ) with those of the aﬃne link representations of the quantum loop algebra UC (gl Hecke algebra H(r)C of type A. For this purpose, we adjust E to the Hopf algebra n ) satisfying automorphism βC : D,C (n) −→ UC (gl (2.2)

Ki±1 −→ k±1 i ,

Ej −→ x+ j,0 ,

En −→ (−1)n υε+ n,

Fj −→ x− j,0 (1 i n, 1 j < n),

Fn −→ (−1)n υ −1 ε− n,

±s z± θ±s (s 1). s −→ ∓sυ

3. Aﬃne q-Schur algebras We now construct algebra epimorphisms from D(n) to aﬃne q-Schur algebras S(n, r) via the tensor spaces Ω⊗r . Let Ω be the free Z-module with basis {ωi | i ∈ Z} and let Ω = Ω ⊗Z Q(υ). − For i ∈ I and t ∈ Z+ , we deﬁne the actions of Ei , Fi , Ki±1 , z+ t , and zt on Ω by (3.1)

Ei · ωs = δi+1,¯s ωs−1 , z+ t · ωs = ωs−tn ,

Fi · ωs = δi,¯s ωs+1 , and

Ki±1 · ωs = υ ±δi,¯s ωs ,

z− t · ωs = ωs+tn .

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It is direct to check that Ω becomes a left D(n)-module. Hence, the Hopf structure on D(n) induces a D(n)-module structure on Ω⊗r . Thus, we obtain an algebra homomorphism ξ r : D(n) → End(Ω⊗r ).

(3.2)

Like the quantum gln case, the aﬃne Hecke algebra H(r) over Q(υ) (see deﬁnition below) acts on Ω⊗r and aﬃne q-Schur algebras are the endomorphism algebras of this H(r)-module. We now describe this action. Let S,r be the aﬃne symmetric group consisting of all permutations w : Z → Z such that w(i + r) = w(i) + r for i ∈ Z. There are several useful subgroups

r of S,r . r The subgroup W of S,r consisting of w ∈ S,r with i=1 w(i) = i=1 i is the aﬃne Weyl group of type A with generators si (1 i r) deﬁned by ⎧ ⎪ for j ≡ i, i + 1 mod r; ⎨j, si (j) = j − 1, for j ≡ i + 1 mod r; ⎪ ⎩ j + 1, for j ≡ i mod r. The cyclic subgroup ρ of S,r generated by the permutation ρ of Z sending j to j + 1, for all j, is in the complement of W . Observe that sj+1 ρ = ρsj , for all j ∈ Z and 1 j n with sr+1 = s1 . The subgroup A of S,r consisting of permutations y of Z satisfying y(i) ≡ i(mod r) is isomorphic to Zr via the map y → λ = (λ1 , λ2 , . . . , λr ) ∈ Zr , where y(i) = λi r + i, for all 1 i r. Thus, A is identiﬁed with Zr . In particular, A is generated by ei = (0, . . . , 0, 1 , 0, . . . , 0) for 1 i r. (i)

Moreover, the subgroup of W generated by s1 , . . . , sr−1 is isomorphic to the symmetric group Sr . These groups are related by the following group isomorphisms: S,r ∼ = ρ W ∼ = Sr Zr . The two isomorphisms give rise to two q-deformations of the group algebra of S,r . Following [25], the aﬃne Hecke algebra H(r) := H(S,r ) over Z is deﬁned to be the algebra generated by Tsi (1 i r), Tρ±1 with the following relations: Ts2i = (υ 2 − 1)Tsi + υ 2 , Tsi Tsj = Tsj Tsi T s i T s j T s i = Ts j T s i T s j Tρ Tρ−1

=

Tρ−1 Tρ

(i − j ≡ ±1 mod r),

(i − j ≡ ±1 mod r and r 3),

= 1, and Tρ Tsi = Tsi+1 Tρ .

This algebra has a Z-basis {Tw }w∈S,r , where Tw = Tρj Tsi1 · · · Tsim if w = ρj si1 · · · sim is reduced. Let H(r) = H(r) ⊗ Q(υ). Let H(r) = H(Sr ) denote the Hecke algebra over Z associated with the symmetric group Sr . Then, H(r) is isomorphic to the subalgebra generated by Tsi (1 i r − 1). The Hecke algebra H(r) admits also the so-called Bernstein presentation which consists of generators Ti := Tsi ,

Xj := T e1 +···+ej−1 T e−1 (1 i r − 1, 1 j r), 1 +···+ej

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and relations (Ti + 1)(Ti − υ 2 ) = 0, Ti Ti+1 Ti = Ti+1 Ti Ti+1 , Ti Tj = Tj Ti (|i − j| > 1), Xi Xi−1 = 1 = Xi−1 Xi , Xi Xj = Xj Xi , Ti Xi Ti = υ 2 Xi+1 , and Xj Ti = Ti Xj (j = i, i + 1). Use the Bernstein presentation, Varagnolo–Vasserot deﬁned an action of H(r) on the tensor space Ω⊗r in [38, 8.2(8)(9)] by ⎧ −1 for all i ∈ Zr ⎪ ⎪ωi · Xt = ⎧ωi1 · · · ωit−1 ωit +n ωit+1 · · · ωir , ⎪ ⎨ 2 ⎪ if ik = ik+1 ; ⎨ υ ωi , ⎪ ωi · Tk = vωisk , if ik < ik+1 ; for all i ∈ I(n, r), ⎪ ⎪ ⎪ ⎩ ⎩ vωisk + (υ 2 − 1)ωi , if ik+1 < ik , where 1 k r − 1, 1 t r, and I(n, r) = {(i1 , i2 , . . . , ir ) | 1 ij n, ∀j}. Let S(n, r) := EndH(r) (Ω⊗r ) and S(n, r) := EndH(r) (Ω⊗r ). Both algebras S(n, r) and S(n, r) are called aﬃne quantum Schur (or q-Schur) algebras over Z and Q(υ), respectively. Proposition 3.1 ([8, (3.5.4)]). The action of D(n) on Ω⊗r commutes with the H(r) action of Varagnolo–Vasserot. Hence, (3.2) induces algebra homomorphisms ξr : D(n) → EndH(r) (Ω⊗r ) = S(n, r) and ξr∨ : H(r) → EndS(n,r) (Ω⊗r )op . Similar statements hold for D,C (n) and H(r)C . Like the q-Schur algebra case, by specializing υ 2 to a prime power q, the algebras S(n, r)C and H(r)C and the tensor space Ω⊗r C have geometric descriptions in terms of varieties of ﬁltrations of lattices or cyclic ﬂags (see [20], [29], and [38]), where multiplication is a convolution product. This multiplication induces an S(n, r)C H(r)C -bimodule structure on Ω⊗r C (see [29, 1.10], [38, 7.4]). Remark 3.2. Using the geometric deﬁnition of Hall algebras mentioned in Remark 2.2, Varagnolo–Vasserot introduced in [38, Prop. 7.6] an algebra homoop − morphism from H(n)C to S(n, r)C . This induces a (geometric) action of H(n)C ⊗r on the tensor space ΩC . It is proved in [8, §3.6] that this action coincides with restriction of the action deﬁned by (3.1). Similar to q-Schur algebras, there is a standard (or deﬁning) basis {eA }A∈Θ(n,r) (notation of [29, 1.9]). For A ∈ Θ(n, r), let ai,j ak,l . (3.3) [A] = υ −dA eA , where dA = 1in ik,j
(See [29, 4.1(b),4.3] for a geometric meaning of dA .) 4. The Beilinson–Lusztig–MacPherson type construction Let Θ± (n) := {A ∈ Θ(n) | ai,i = 0 for all i}.

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Following [2], for A ∈ Θ± (n) and j ∈ Zn , deﬁne A(j, r) ∈ S(n, r) by

λj [A + diag(λ)], if σ(A) r; λ∈Λ(n,r−σ(A)) υ A(j, r) = 0, otherwise,

where λ · j = 1in λi ji . The set {A(j, r) | A ∈ Θ± (n), j ∈ Zn } forms a spanning set for S(n, r). Explicit bases can be extracted; see [14, Prop. 4.1] or [8, (3.4.1)]. The following four theorems in this and next sections summarize some structures of aﬃne q-Schur algebras. Theorem 4.1. (1)([8, Prop. 3.7.1]) There is a partial ordering on Θ± (n) such that, for any A = A+ + A− ∈ Θ± (n) with A+ ∈ Θ+ (n) and A− ∈ Θ− (n), gB,j,A;r B(j, r) (in S(n, r)). A+ (0, r)A− (0, r) = A(0, r) + ± B∈Θ (n) n B≺A,j∈Z

(2)([14, Th. 4.2]) There are explicit multiplication formulas for the products O(j, r)A(j , r),

Eh,h+1 (0, r)A(j, r), and Eh+1,h (0, r)A(j, r),

n where O stands for the zero matrix. More precisely,

for i ∈ Z, j, j ∈ Z and ± A = (ai,j ) ∈ Θ (n), if we put f (i) = f (i, A) = ji ah,j − j>i ah+1,j and

f (i) = f (i, A) = j
O(j, r)A(j , r) = υ jro(A) A(j + j , r),

A(j , r)O(j, r) = υ jco(A) A(j + j , r),

ah,i + 1 (A + Eh,i − Eh+1,i )(j + α h , r) 1 i
(0, r)A(j, r) = Eh,h+1

+

υ f (i)

i>h+1;ah+1,i 1

(A − Eh+1,h )(j + α h , r) − (A − Eh+1,h )(j + β h , r) 1 − υ −2 ah,h+1 + 1 (A + Eh,h+1 + υ f (h+1)+jh+1 )(j, r), 1

+ υ f (h)−jh −1

and Eh+1,h (0, r)A(j, r)

+

ah+1,i + 1 (A − Eh,i = υ + Eh+1,i )(j, r) 1 i
f (i)

i>h+1;ah,i 1

(A − Eh,h+1 )(j − α h , r) − (A − Eh,h+1 )(j + β h , r) 1 − υ −2 ah+1,h + 1 (A + Eh+1,h + υ f (h)+jh )(j, r). 1

+ υf

(h+1)−jh+1 −1

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(By convention, if an oﬀ-diagonal entry of A is negative, then A(j) = 0.) By the triangular relation in Theorem 4.1(1), we can establish a “triangular decomposition” for S(n, r), which is the key to proving the surjectivity of the map ξr in Proposition 3.1. Theorem 4.2 ([8, 3.7.7,3.8.1]). Let S(n, r)± = spanZ {A(0, r) | A ∈ Θ± (n)} and S(n, r)0 = spanZ {[diag(λ)] | λ ∈ Λ(n, r)}. Then S(n, r) = S(n, r)+ S(n, r)0 S(n, r)− and the set {A+ (0, r)[diag(λ(A))]A− (0, r) | A ∈ Θ(n, r)},

where λ(A) = (ai,i + j r. (3)([8, Th. 5.1.1]) The algebra U(n, r) can be presented by generators ei , fi , ki (i ∈ I) and relations: for all i, j ∈ I, (QS1) ki kj = kj ki ; (QS2) ki ej = υ δi,j −δi,j+1 ej ki , ki fj = υ −δi,j +δi,j+1 fj ki ; −1 ki ki − −1

(QS3) ei fj − fj ei = δi,j υ−υ −1 , where ki = ki ki+1 ; 1 − ci,j a b ei ej ei = 0 for i = j; (QS4) (−1)a a a+b=1−ci,j 1 − ci,j a b (QS5) fi fj fi = 0 for i = j; (−1)a a a+b=1−ci,j

(QS6) [ki ; r + 1]! = 0, k1 · · · kn = υ r .

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It seems to us that ﬁnding presentations for S(n, r) and n r is too complicated. We couldn’t get the general case except the following n = r case. Let ρ be the Q(υ)-linear isomorphism ρ : Ω⊗r −→ Ω⊗r , ρ(ωi1 ⊗ · · · ⊗ ωir ) = ωi1 −1 ⊗ · · · ⊗ ωir −1 .

+ Then, ρ = ξr ( a∈Nr ,σ(a)=r u a ) = eδ + eδ (i.e., the image of the sum of semisim

ple modules of dimension r) and ρ−1 = ξr ( a∈Nr u

−

± a ) = fδ + fδ , where u a = + − ai (ai −1) ± i υ ua , eδ = ξr (uδ ), and fδ = ξr (uδ ) with δ = (1, 1, . . . , 1). Theorem 5.2. The Q(υ)-algebra S(r, r) is generated by ei , fi , k±1 (1 i r) i and ρ±1 subject to the relations (QS1)–(QS6) above together with (QS0 )–(QS6 ) below: (QS0 ) ρρ−1 = ρ−1 ρ = 1, ρei ρ−1 = ei−1 , ρfi ρ−1 = fi−1 , ρki ρ−1 = ki−1 ; (QS1 ) (ki − υ)ρ = (ki − υ)e δ ; (QS2 ) ei ρ = ei e δ + υ+υ1 −1 e2i ei−1 · · · e1 er · · · ei+1 ; −1 (QS3 ) fi ρ = fi e δ + 1−υ1 −2 ei−1 · · · e1 er · · · ei+1 k−1 i (ki+1 − ki+1 ); −1 (QS4 ) (ki − υ)ρ = (ki − v)fδ ; (QS5 ) fi ρ−1 = fi f δ + υ+υ1 −1 f2i fi+1 · · · fr f1 · · · fi−1 ; −1 (QS6 ) ei ρ−1 = ei f δ + 1−υ1 −2 fi+1 · · · fr f1 · · · fi−1 k−1 i+1 (ki − ki ), where (with the notation of quantum divided powers) e δ = f δ =

n

i=1

a∈Nn σ(a)=r,ai =0

n

i=1

a∈Nn σ(a)=r,ai =0

(ei−1 )(ai−1 ) · · · (e1 )(a1 ) (en )(an ) · · · (ei+1 )(ai+1 )

(fi−1 )(ai−1 ) · · · (f1 )(a1 ) (fn )(an ) · · · (fi+1 )(ai+1 ) .

In particular, S(r, r) = U(r, r) ⊕

Q(υ)ρm .

0 =m∈Z

Note that (QS1 )–(QS6 ) may be derived from the multiplication formulas given in Theorem 4.1(2). 6. Some conjectures We end this part with three conjectures. The ﬁrst conjecture is a candidate of ). a Lusztig type integral Z-form for the quantum loop algebra U(gl n With the isomorphism between D(n) and gln ), we introduce the Z U( ±(m) , and u± subalgebra D(n)L of D(n) generated by Ki±1 , Kmi ;0 , ui a for all i ∈ I, n m > 0, and sincere a = (ai ) ∈ N (so all ai = 0). Here we used Hall algebra − notation with u+ i = Ei , ui = Fi and m Ki ; 0 Ki υ −s+1 − Ki−1 υ s−1 Ki ; 0 = = 1. and m 0 υ s − υ −s s=1 ±

±(m)

By [10, Th. 5.7], H(n) is generated by ui a Hopf subalgebra by [8, Prop. 3.5.6].

and u± a . This subalgebra is clearly

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If D(n)0 denotes the Z-subalgebra of D(n) generated by Ki±1 and +

−

i ∈ I and t > 0, then both H(n) D(n) and D(n) H(n) 0

0

Ki ;0 t

for

are Hopf subalgebras.

Conjecture 6.1. We conjecture that + −

(n) := H(n)+ D(n)0 H(n)− ∼ D(n)L = D = H(n) ⊗ D(n)0 ⊗ H(n) .

(n) is a subalgebra. Note that the restriction This is equivalent to saying that D

(n) is a surjective map onto S(n, r). of ξr to D ±(m) , and There is a smaller integral form D(n) generated by Ki±1 , Kmi ;0 , ui

z± m . This form is contained in D(n) ([8, 3.7.5]) and does not map onto S(n, r) ([8, 5.3.8]), but is useful for the reduction modulo υ = 1; see Chapter 6 in [8] for more details of this application. n ) over Z = C[υ, υ −1 ] In [17, 7.2], a so-called restricted integral form UZres (gl ki ;0 ±(m) ±1 is deﬁned as the Z -subalgebra generated by ki , m , xi,s , and gi,t /[t], for all n ) is a Hopf i ∈ I, m > 0, s ∈ Z, and t ∈ Z\{0}. However, it is not clear if UZres (gl subalgebra. The second conjecture is natural from the following observation. When r approaches to ∞, the algebra epimorphism ξr : D(n) → S(n, r) implies that “limr→∞ S(n, r) = D(n)”. Thus, quantum aﬃne gln can be realized as a limit of aﬃne q-Schur algebras. In the non-aﬃne case, this was established by Beilinson– Lusztig–MacPherson [2]. In fact, they discovered a stabilization property and con˙ the modiﬁed quantum gln ), ‘larger’ than the direct structed an algebra K (∼ = U, sum of all q-Schur algebras, whose completion (or limit) contains U(gln ) as a subalgebra. Thus, a new basis (called the BLM basis) for U(gln ) was introduced and multiplication formulas between generators and arbitrary basis element were given. However, in the aﬃne case, there is no stabilization property. So a modiﬁed approach was developed in [14] which we now describe below. For A ∈ Θ± (n) and j ∈ Zn , let A(j) = lim A(j, r) := υ λj [A + diag(λ)]. r→∞

λ∈Nn

(n) := ∈K r0 S(n, r). Let A(n) be the subspace of K(n) spanned by A(j) for all A ∈ Θ± (n) and j ∈ Zn . Since the structure constants (with respect to the BLM basis) appeared in the multiplication formulas given in Theorem 4.1(2) are independent of r, taking limits of both sides gives similar formulas in A(n). Thus, one sees easily that U(n), H(n)0 , H(n)0 as subalgebras are all in A(n). Clearly, A(j) =

r0 A(j, r)

(n) Conjecture 6.2 ([14, 5.2(2)]). The Q(υ)-space A(n) is a subalgebra of K which is isomorphic to D(n) or the quantum loop algebra U(gln ). The conjecture has been established in the classical (υ = 1) case. In this case, explicit multiplication formulas for Eh,h+mn [0]A[j] (m ∈ Z\{0}), where Eh,h+mn [0] are generators corresponding to homogeneous indecomposable modules, have been found; see [8, Ch. 6] for details (including notations). The last conjecture is natural from the Schur–Weyl duality.

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Conjecture 6.3. For n < r, the algebra homomorphisms ξr∨ : H(r) −→ EndS(n,r) (Ω⊗r )op ,

∨ op ξr,F : H(r)F −→ EndS(n,r)F (Ω⊗r F )

are surjective, where base change to an (inﬁnite) ﬁeld F is obtained by specializing υ to q ∈ F∗ . ∨ If q is a prime power, then Pouchin [31, Th. 8.1] has proved that ξr,C is surjective. Moreover, as proved in the non-aﬃne case [15], it would be interesting ∨ to know if the kernel of ξr,F can be described in terms of Kazhdan–Lusztig basis elements.

Part 2: The Representations As before, we assume that q ∈ C∗ is not a root of unity. Since S(n, r) and H(r) are deﬁned over Z, we may deﬁne S(n, r)F and H(r)F by base change to any ﬁeld or ring F. By presentations similar to the case over Q(υ), we may deﬁne the n ) over double Ringel–Hall algebras D,C (n) and the quantum loop algebras UC (gl C with respect to the parameter q; see 2.4 and 2.5. In this part, we investigate the n ), S(n, r)C , and H(r)C . relationship between representations of UC (gl 7. A category equivalence By Theorem 4.2, we have algebra epimorphism ξr,C : D,C (n) −→ S(n, r)C and algebra isomorphism (2.2) n ). βC : D,C (n) −→ UC (gl Hence, we obtain an algebra epimorphism (7.1)

n ) −→ S(n, r)C . ξr,C = ξr,C ◦ βC−1 : UC (gl

Like the non-aﬃne case, if n r, then there is an idempotent eω ∈ S(n, r)F such that H(r)F ∼ = eω S(n, r)F eω , where F is a commutative ring which is a Zalgebra. The ﬁrst part of the following result together with the epimorphism (7.1) shows that an aﬃne q-Schur algebra links representations of the quantum loop algebra n ) with those of the Hecke algebra H(r)C . UC (gl For an algebra A, let A-mod denote that category of ﬁnite dimensional left A-modules. Theorem 7.1. Let F be a ﬁeld and let q ∈ F∗ be a non-root of unity. (1) ([8, Th 4.1.3]) If n r, then there is a category equivalence F : H(r)F -mod −→ S(n, r)F -mod, M −→ Ω⊗r F ⊗H(r)C N. The inverse functor is the Schur functor G(M ) = eω M for all M ∈ S(n, r)F -mod. (2) ([8, Th 4.1.6]) Every simple S(n, r)F -module is ﬁnite dimensional. Note that part (2) is obtained by applying the ﬁrst part to the fact that every simple H(r)F -module is ﬁnite dimensional. By this theorem, we will determine in §8 and §11 simple S(n, r)C -modules through the classiﬁcation of simple H(r)C -modules by Zelevinsky [42] and Ro n )-modules by gawski [35], as well as the classiﬁcation of simple polynomial UC (gl Frenkel–Mukhin [17], which is built on Chari–Pressley [3, 5].

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Remark 7.2. For F = C, a direct category equivalence from H(r)C -mod to a n )-mod has been geometrically constructed in [19, Th. 6.8] full subcategory of UC (gl when n r. The simple objects in the full subcategory are described in terms of intersection cohomology complexes. 8. Simple S(n, r)C -modules arising from simple H(r)C -modules We ﬁrst recall a classiﬁcation of simple H(r)C -modules. Again, ﬁx q ∈ C∗ such that q m = 1 for all m 1. A segment s with center a ∈ C∗ and length k is by deﬁnition an ordered sequence s = (aq −k+1 , aq −k+3 , . . . , aq k−1 ) ∈ (C∗ )k . Write |s| = k and consider the set of all multisegments of total length r: Sr = {s = {s1 , . . . , sp } | r = Σi |si | =: |s|}.4 For s = {s1 , . . . , sp } ∈ Sr , let (a1 , a2 , . . . , ar ) = (s1 , . . . , sp ) ∈ (C∗ )r be the r-tuple obtained by juxtaposing the segments in s and let Js be the left ideal of H(r)C generated by Xj −aj for 1 j r. Then Ms = H(r)C /Js is a left H(r)C module which as an H(r)C -module is isomorphic to the regular representation of H(r)C . Let λ = (|s1 |, . . . , |sp |). Then, the element (−q 2 )−(w) Tw ∈ H(r)C yλ = w∈Sλ

generates the submodule H(r)C y¯λ of Ms which, as an H(r)C -module, is isomorphic to H(r)C yλ . Observe that (8.1) H(r)C yλ ∼ mμ,λ Eμ ), = Eλ ⊕ ( μ r,μλ

where Eν is the left cell module deﬁned by the Kazhdan–Lusztig’s C-basis [27] associated with the left cell containing w0,ν , the longest element of the Young subgroup Sν . Let Vs be the unique composition factor of the H(r)C -module H(r)C y¯λ such that the multiplicity of Eλ in Vs as an H(r)C -module is nonzero. We now can state the following classiﬁcation theorem due to Zelevinsky [42] and Rogawski [35]. The construction above follows [35]. Theorem 8.1. Let Irr(H(r)C ) be the set of isoclasses of all simple H(r)C modules. Then the correspondence s → [Vs ] deﬁnes a bijection from Sr to Irr(H(r)C ). Let Sr(n) = {s ∈ Sr | Ω⊗r C ⊗H(r)C Vs = 0}. (n)

Then Sr = Sr for all r n. We have the following classiﬁcation theorem; see [8, 4.3.4&4.5.3]. 4Here, possibly, s = s for some i = j. i j

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Theorem 8.2. The set {Ω⊗r C ⊗H(r)C Vs | s ∈ Sr } is a complete set of nonisomorphic simple S(n, r)C -modules. Moreover, we have5 (n)

(8.2)

Sr(n) = {s = {s1 , . . . , sp } ∈ Sr | p 1, |si | n ∀i}.

In particular, if n r, then the set {Ω⊗r C ⊗H(r)C Vs | s ∈ Sr } is a complete set of nonisomorphic simple S(n, r)C -modules. in (7.1), every Ω⊗r Thus, by ξr,C C ⊗H(r)C Vs is a simple UC (gln )-module. These simple representations reﬂect a certain structure of the quantum loop algebra n ) hidden in the theory of so-called pseudo-highest weight representations UC (gl [4, §12.2B].

9. Pseudo-highest weight representations Recall Deﬁnition 2.5 and the generating function (2.1). For 1 j n − 1 and n ) through the generating functions s ∈ Z, deﬁne the elements Pj,s ∈ UC (gl 1 n )[[u, u−1 ]]. Pj± (u) := exp − hj,±t (qu)±t = Pj,±s u±s ∈ UC (gl [t]q t1

s0

A comparison with (2.1) gives Φ± j (u)|υ=q

= k±1 j

Pj± (q −2 u) Pj± (u)

.

n ) through the For 1 i n and s ∈ Z, deﬁne the elements Qi,s ∈ UC (gl generating functions 1 ± ±t n )[[u, u−1 ]]. Qi (u) := exp − = gi,±t (qu) Qi,±s u±s ∈ UC (gl [t]q t1

s0

Note that Qi± (u) and Pj± (u) are related by Pj± (u) =

Qj± (uq j−1 ) ± Qj+1 (uq j+1 )

,

for all 1 j n − 1.

n )-module (resp., weight UC (gl n )-module) of type 1. Let V be a weight UC (sl Then V = ⊕μ∈Zn−1 Vμ (resp., V = ⊕λ∈Zn Vλ ) where kj w = q μj w, ∀1 j < n} Vμ = {v ∈ V | (resp., Vλ = {v ∈ V | kj w = q λj w, ∀1 j n}). Following [4, 12.2.4], a nonzero weight vector w ∈ V is called a pseudo-highest weight vector, if there exist some Pj,s ∈ C (resp. Qi,s ∈ C) such that x+ j,s w = 0,

Pj,s w = Pj,s w,

(resp., x+ j,s w = 0,

Qi,s w = Qi,s w).

for all s ∈ Z. The module V is called a pseudo-highest weight module if V = n )w (resp., V = UC (gl n )w) for some pseudo-highest weight vector w. UC (sl 5The proof of this result requires Theorem 10.1 below.

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Remark 9.1. There are integrable highest weight modules considered in [28, 6.2.3] and deﬁned in [28, 3.5.6] which are diﬀerent from pseudo-highest weight modules deﬁned here. This is seen as follows. ± 0 Let U± C be the subalgebra of UC (gln ) generated by xi,s for all i, s, and let UC ±1 n ) generated by all k n) = be the subalgebra of UC (gl and Qi,s . Then UC (gl i − 0 + ∼ − + 0 UC UC UC = UC ⊗ UC ⊗ UC by [17, Lem 7.4]. If we identify the new realization of n ) with the double Hall algebra D,C (n) under the isomorphism βC given in UC (gl − 0 − 0 + (2.2), then En = ε+ n ∈ UC UC and Fn = εn ∈ UC UC . If w0 is pseudo-highest weight −

vector, then U w0 = UC w0 . Now En w0 is not necessarily zero as En ∈ U− C kn (while Fn w0 = 0). However, a highest weight vector w0 always satisﬁes En w0 = 0. Since all Qi,s commute with the kj , each Vλ is a direct sum of generalized eigenspaces of the form Vλ,γ = {x ∈ Vλ | (Qi,s − γi,s )p x = 0 for some p (1 i n, s ∈ Z)},

±s where γ = (γi,s ) with γi,s ∈ C. Let Γ± j (u) = s0 γj,±s u . n )-module V (of type 1) is called a polynomial A ﬁnite dimensional weight UC (gl representation if, for every weight λ = (λ1 , . . . , λn ) ∈ Zn of V , the formal power ± series Γ± i (u) are polynomials in u of degree λi so that the zeroes of the functions + − Γi (u) and Γi (u) are the same. Following [17], an n-tuple of polynomials Q = (Q1 (u), . . . , Qn (u)) with constant terms 1 is called dominant if, for 1 i n − 1, the ratios Qi (q i−1 u)/Qi+1 (q i+1 u) are polynomials. Let Q(n) be the set of dominant n-tuples of polynomials. For Q = (Q1 (u), . . . , Qn (u)) ∈ Q(n), deﬁne Qi,s ∈ C (1 i n, s ∈ Z) by the following formula Q± Qi,±s u±s . i (u) = s0

−1 − ai u) ⇐⇒ f − (u) = 1im (1 − a−1 ). i u + n ) generated by x , Qi,s − Qi,s , and ki − q λi , Let I(Q) be the left ideal of UC (gl

Here f + (u) =

1im (1

j,s

for all 1 j n − 1, 1 i n and s ∈ Z, where λi = degQi (u), and deﬁne Verma type module n )/I(Q). M (Q) = UC (gl Then M (Q) has a unique simple quotient, denoted by L(Q). The polynomials Qi (u) are called Drinfeld polynomials associated with L(Q). Similarly, for an (n−1)-tuples P = (P1 (u), . . . , Pn−1 (u)) ∈ P(n) of polynomials with constant terms 1, deﬁne Pj,s ∈ C (1 j n − 1, s ∈ Z) as in Pj± (u) =

n ), Qi,s − Qi,s , and ki − Pj,±s u±s and let μj = degPj (u). Replacing UC (gl s0

n ), Pi,s − Pi,s , and ki − q μi , respectively, in the above construction q λi by UC (sl n )-module L(P). ¯ deﬁnes a simple UC (sl The polynomials Pi (u) are called Drinfeld ¯ polynomials associated with L(P).

Theorem 9.2. (1)([3, 4]) Let P(n) be the set of (n − 1)-tuples of polynomials ¯ with constant terms 1. The modules L(P) with P ∈ P(n) are all nonisomorphic ﬁnite dimensional simple UC (sln )-modules of type 1.

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n )-modules L(Q) with Q ∈ Q(n) are all nonisomorphic (2)([17]) The UC (gl n ). Moreover, ﬁnite dimensional simple polynomial representations of UC (gl ¯ L(Q)| ∼ = L(P) UC (sln )

where P = (P1 (u), . . . , Pn−1 (u)) with Pi (u) = Qi (q i−1 u)/Qi+1 (q i+1 u). We now use aﬃne q-Schur algebras S(n, r)C to link these simple pseudo n )-modules with the simple UC (gl n )-modules arising from simhighest weight UC (gl ple H(r)C -modules as given in Theorem 8.2. 10. Some Identiﬁcation Theorems By (7.1), every simple S(n, r)C -module of form Ω⊗r C ⊗H(r)C Vs given in Theorem n )-module. We now identify them as simple pseudo-highest 8.2 is a simple UC (gl n )-modules under the assumption n > r and compute the associated weight UC (gl Drinfeld polynomials Qs . Recall that Q(n) is the set of dominant n-tuples of polynomials. For r 1, let Q(n)r = Q = (Q1 (u), . . . , Qn (u)) ∈ Q(n) | r = deg Qi (u) . 1in

Let n > r. For s = {s1 , . . . , sp } ∈ Sr , write si = (ai q −νi +1 , ai q −νi +3 , . . . , ai q νi −1 ) ∈ (C∗ )νi , and deﬁne Qs = (Q1 (u), . . . , Qn (u)) by setting Qn (u) = 1 and, for 1 i n − 1, ± ± ± −i+1 )Pi+1 (uq −i+2 ) · · · Pn−1 (uq n−2i ), Q± i (u) = Pi (uq

where Pi± (u) =

±1 (1 − a±1 ). j u

1jp νj =i

If μi := deg Pi (u) = #{j ∈ [1, p] | νj = i} and λi := deg Qi (u) = #{j ∈ [1, p] | νj i}, then λ = (λ1 , . . . , λn−1 , λn ) is a partition of r and dual to (ν1 , . . . , νp ), and λi − λi+1 = μi for all 1 i < n. (n ) Note that, if s ∈ Sr (see (8.2)), where n < n, then Pn +1 (u) = · · · = Pn−1 (u) = 1. Thus, Qn +1 (u) = · · · = Qn (u) = 1. Thus, if n r, we also deﬁne (10.1)

Qs = (Q1 (u), . . . , Qn (u))

by dropping the 1’s. We have the following identiﬁcation theorem. Theorem 10.1 ([8, 4.4.2]). Maintain the notation above and let n > r. (1) The map s → Qs deﬁnes a bijection from Sr to Q(n)r . n )-module isomorphisms Ω⊗r ⊗H (r) Vs ∼ (2) There are UC (gl = L(Qs ) for all C C s ∈ Sr . Hence, the set {L(Q) | Q ∈ Q(n)r } forms a complete set of nonisomorphic simple S(n, r)C -modules.

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155 19

The condition n > r simply means S(n, r)C is a homomorphic image of quantum aﬃne sln . Thus, we may directly use the result [6, 7.6] together with a computation of the action of central elements to get the result. When n r, diﬀerent argument is required; see remarks after Theorem 11.4. We ﬁrst look at two examples which will be used to establish this result in next section. Example 10.2. Let a ∈ C∗ . (1) Assume r = 1 < n. Then Ma = H(1)C /H(1)C (X1 − a) is a simple H(1)C n )-module. If Q = (1−au, 1, . . . , 1), module and ΩC (a) := ΩC ⊗Ma is a simple UC (gl ∼ then the theorem above implies that ΩC (a) = L(Q). (2) For 1 i n − 1, deﬁne Qi,a ∈ Q(n) by setting Qn (u) = 1 and Qj (uq j−1 ) = (1 − au)δi,j Qj+1 (uq j+1 ) for 1 j n − 1. In other words, Qi,a = (1 − aq i−1 u, . . . , 1 − aq −i+3 u, 1 − aq −i+1 u , 1, . . . , 1). (ith component)

n )-module Li,a := L(Qi,a ) is also a Since n > i, by Theorem 10.1, the simple UC (gl simple S(n, i)C -module. The weight of the pseudo-highest weight vector of Li,a is λ(i, a) = (1i , 0n−i ). We end this section with another identiﬁcation for the 1-dimensional determinant representation of the aﬃne q-Schur algebra S(n, n)C . For any a ∈ C∗ , let s = (a, aq 2 , . . . , aq 2(n−1) ) which is regarded as a single segment, and let Deta := Ω⊗n C ⊗H(n)C Vs . Then, Deta is the submodule of Ms generated by y (n) and is a simple S(n, n)C module. Proposition 10.3 ([8, 4.6.5]). The simple S(n, n)C -module Deta has dimension 1 and Deta = spanC {ω1 ⊗ · · · ⊗ ωn ⊗ y (n) }. n )-module, Deta ∼ Moreover, as a UC (gl = L(Q), where Q = (Q1 (u), . . . , Qn (u)) ∈ Q(n) with Qi (u) = 1 − aq 2(n−i) u for all i = 1, 2, . . . , n. 11. A second classiﬁcation of simple S(n, r)C -modules By Example 10.2(1) together with the fact that Vs is a homomorphic image of some Ms , one can easily prove the following. n )-module. Lemma 11.1 ([8, 4.6.2]). Let V be a ﬁnite dimensional simple UC (gl Then the following conditions are equivalent: n )-module structure on V is inﬂated from an S(n, r)C -module (1) The UC (gl by the map ξr,C in (7.1); n )-module ΩC (a1 ) ⊗C · · · ⊗C ΩC (ar ) for (2) V is a quotient module of the UC (gl ∗ r some a ∈ (C ) ; (3) V is a quotient module of Ω⊗r C ; (4) V is a subquotient module of Ω⊗r C .

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Thus, if V is a simple S(n, r)C -module, then V is ﬁnite dimensional by Theorem 7.1(2). Hence, V is a quotient module of ΩC (a1 )⊗C · · ·⊗C ΩC (ar ) for some a ∈ (C∗ )r , ), by [17, by the lemma above. Since ΩC (a) is a polynomial representation of UC (gl n 4.3], the tensor product ΩC (a1 ) ⊗C · · · ⊗C ΩC (ar ) is a polynomial representation of n ). Hence, V is also a polynomial representation of UC (gl n ). We have proved UC (gl the following. Proposition 11.2 ([8, 4.6.4]). Every simple S(n, r)C -module is a polynomial n ). representation of UC (gl We now use Example 10.2(2) to show that every L(Q) with Q ∈ Q(n)r is isomorphic to the inﬂation of a simple S(n, r)C -module via (7.1). Let λ = (λ1 , . . . , λn ) with λj = deg(Qj (u)) and let Pi (u) =

Qi (uq i−1 ) , Qi+1 (uq i+1 )

1 i n − 1.

Write Qn (u) = (1 − b1 u) · · · (1 − bλn u) and (1 − ai,j u), Pi (u) = 1jμi

where μi = λi − λi+1 . Let V = L1 ⊗ · · · ⊗ Ln−1 ⊗ Detb1 ⊗ · · · ⊗ Detbλn where Li = Li,ai,1 ⊗ · · · ⊗ Li,ai,μi for 1 i n − 1 with Li,ai,j being an S(n, i)C module as deﬁned in Example 10.2(2), and Deta := Ω⊗n C ⊗H(n)C Vs , with s = (a, aq 2 , . . . , aq 2(n−1) ) is the 1-dimensional determinant representation of S(n, n)C (Proposition 10.3). Thus, V is a module for the algebra A = S(n, 1)⊗μ1 ⊗ · · · ⊗ S(n, n − 1)⊗μn−1 ⊗ S(n, n)⊗λn .

Since 1in−1 iμi + nλn = 1in λi = r, S(n, r)C is a subalgebra of A. Hence, restriction gives an S(n, r)C -module structure on V . Let wi,ai,k (resp. vj ) be a pseudo-highest weight vector of Li,ai,k (resp., Detbj ), and let w0 = w1 ⊗ w2 ⊗ · · · ⊗ wn−1 , where wi = wi,ai,1 ⊗ wi,ai,2 ⊗ · · · ⊗ wi,ai,μi , and v0 = v1 ⊗ · · · ⊗ vn . Since wi,ai,k has weight (1i , 0n−i ) and (1 − (ai,k q i−2j+1 u)±1 )wi,ai,k , if 1 j i; ± and Qj (u)wi,ai,k = wi,ai,k , if i < j n, Qj± (u)vi = (1 − (bi q 2(n−j) u)±1 )vi for 1 i λn , it follows from [17, Lem. 4.1] that Qj± (u)(w0 ⊗ v0 ) = Q± j (u)(w0 ⊗ v0 ). Moreover, the weight of w0 ⊗ v0 is λ = (μ1 , 0, . . . , 0) + (μ2 , μ2 , 0, . . . , 0) + (μn−1 , . . . , μn−1 , 0) + (λn , . . . , λn ). n )(w0 ⊗ v0 ) be the submodule of V generLet W = S(n, r)C (w0 ⊗ v0 ) = UC (gl ated by w0 ⊗ v0 . Then W is a pseudo-highest weight module whose pseudo-highest weight vector is a common eigenvector of ki and Qi,s with eigenvalues z λi and Qi,s , − respectively, where Qi,s are the coeﬃcients of Q+ i (u) or Qi (u). So the simple quotient module of W is isomorphic to L(Q) (cf. the construction in [17, Lem. 4.8]). Hence, we have proved the following.

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157 21

Proposition 11.3 ([8, 4.6.7]). Every L(Q) (Q ∈ Q(n)r ) is isomorphic to an inﬂation of a simple S(n, r)C -module by the map ξr,C . Combining the two propositions above yields the following classiﬁcation theorem. Theorem 11.4 ([8, 4.6.8]). For any n, r 1, the set L(Q) | Q ∈ Q(n)r is a complete set of nonisomorphic simple S(n, r)C -modules. It would be interesting to point out that the identiﬁcation in the n r case (see Theorem 10.1 for the n > r case) has recently been obtained by Deng–Du (see [18] n )for a diﬀerent proof). Thus, with the notation given in (10.1), we have UC (gl (n) ⊗r module isomorphisms ΩC ⊗H(r)C Vs ∼ = L(Qs ) for all s ∈ Sr . In particular, this (n) gives a bijection from Sr to Q(n)r in this case. References [1] J. Beck, Braid group action and quantum aﬃne algebras, Comm. Math. Phys. 165 (1994), 655–568. [2] A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GLn , Duke Math. J. 61 (1990), 655–677. [3] V. Chari and A. Pressley, Quantum aﬃne algebras, Comm. Math. Phys. 142 (1991), 261–283. [4] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. [5] V. Chari and A. Pressley, Quantum aﬃne algebras and their representations, Representations of groups (Banﬀ, AB, 1994), 59–78, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995. [6] V. Chari and A. Pressley, Quantum aﬃne algebras and aﬃne Hecke algebras, Paciﬁc J. Math. 174 (1996), 295–326. [7] B. Deng and J. Du, Identiﬁcation of simple representations for aﬃne q-Schur algebras, preprint. [8] B. Deng, J. Du and Q. Fu, A double Hall algebra approach to aﬃne quantum Schur–Weyl theory, LMS Lecture Notes Series (to appear). [9] B. Deng, J. Du, B. Parshall and J. Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs Volume 150, Amer. Math. Soc., Providence 2008. [10] B. Deng, J. Du and J. Xiao, Generic extensions and canonical bases for cyclic quivers, Can. J. Math. 59 (2007), 1260–1283. [11] R. Dipper, S. Doty and F. Stoll, Quantized mixed tensor space and Schur–Weyl duality I, II, arXiv:0806.0264v2, arXiv:0806.1227v2. [12] S. Doty and R. M. Green, Presenting aﬃne q-Schur algebras, Math. Z. 256 (2007), 311–345. [13] V. G. Drinfeld, A new realization of Yangians and quantized aﬃne alegbras, Soviet Math. Dokl. 32 (1988), 212–216. [14] J. Du and Q. Fu, A modiﬁed BLM approach to quantum aﬃne gln , Math. Z. 266 (2010), 747–781. [15] J. Du, B. Parshall and L. Scott, Quantum Weyl reciprocity and tilting modules, Commun. Math. Phys. 195 (1998), 321–352. [16] J. Du and H. Rui, Quantum Schur superalgebras and Kazhdan–Lusztig combinatorics, J. Pure Appl. Algebra 215 (2011) 2715–2737. ), Sel. math., New Ser. 8 (2002), [17] E. Frenkel and E. Mukhin, The Hopf algebra Rep Uq (gl ∞ 537–635. [18] Q. Fu, Aﬃne quantum Schur algebras and aﬃne Hecke algebras, preprint. [19] V. Ginzburg, N. Reshetikhin and E. Vasserot, Quantum groups and ﬂag varieties, Contemp. Math. 175 (1994), 101–130. [20] V. Ginzburg and E. Vasserot, Langlands reciprocity for aﬃne quantum groups of type An , Internat. Math. Res. Notices 1993, 67–85. [21] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), 361–377.

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[22] R. M. Green, The aﬃne q-Schur algebra, J. Algebra 215 (1999), 379–411. [23] A. Hubery, Symmetric functions and the center of the Ringel–Hall algebra of a cyclic quiver, Math. Z. 251 (2005), 705–719. ), preprint. [24] A. Hubery, Three presentations of the Hopf algebra Uv (gl n [25] V. F. R. Jones, A quotient of the aﬃne Hecke algebra in the Brauer algebra, Enseign. Math. 40 (1994), 313–344. [26] A. Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 29, Springer-Verlag, Berlin, 1995. [27] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke Algebras, Invent. Math. 53 (1979), 165–184. [28] G. Lusztig, Introduction to quantum groups, Progress in Math. 110, Birkh¨ auser, 1993. [29] G. Lusztig, Aperiodicity in quantum aﬃne gln , Asian J. Math. 3 (1999), 147–177. [30] K. McGerty, Generalized q-Schur algebras and quantum Frobenius, Adv. Math. 214 (2007), 116–131. [31] G. Pouchin, A geometric Schur–Weyl duality for quotients of aﬃne Hecke algebras, J. Algebra 321 (2009), 230–247. [32] J. Hu, BMW algebra, quantized coordinate algebra and type C Schur–Weyl duality, Represent. Theory 15 (2011), 1–62. [33] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–592. [34] C. M. Ringel, The composition algebra of a cyclic quiver, Proc. London Math. Soc. 66 (1993), 507–537. [35] J. D. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985), 443–465. [36] O. Schiﬀmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices (2000), 413–440. [37] C. Stroppel and B. Webster, Quiver Schur algebras and q-Fock space, preprint, arXiv:1110.1115v1. [38] M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297. [39] E. Vasserot, Aﬃne quantum groups and equivariant K-theory, Transf. Groups 3 (1998), 269– 299. [40] J. Xiao, Drinfeld double and Ringel-Green theory of Hall algebras, J. Algebra 190 (1997), 100–144. [41] D. Yang, On the aﬃne Schur algebra of type A. II, Algebr. Represent. Theory 12 (2009), 63–75. [42] A. V. Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible represer. 13 (1980), 165–210. netations of GLn , Ann. Sci. Ec. Norm. Sup. 4e S´ School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia. E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Multiplicative Bases for the Centres of the Group Algebra and Iwahori-Hecke Algebra of the Symmetric Group Andrew Francis and Lenny Jones Dedicated to Leonard Scott on the occasion of his retirement. Abstract. Let Hn be the Iwahori-Hecke algebra of the symmetric group Sn , and let Z(Hn ) denote its centre. Let B = {b1 , b2 , . . . , bt } be a basis for Z(Hn ) over R = Z[q, q −1 ]. Then B is called multiplicative if, for every i and j, there exists k such that bi bj = bk . In this article we prove that no multiplicative bases for Z(ZSn ) and Z(Hn ) when n ≥ 3. In addition, we prove that there exist exactly two multiplicative bases for Z(ZS2 ) and none for Z(H2 ).

1. Introduction Quite a bit is known about the integral linear structure of the centres of Iwahori–Hecke algebras of ﬁnite Coxeter groups over the ring R = Z[q, q −1 ]. For instance, they are known to have a basis consisting of elements that specialize to conjugacy class sums [8]. Our focus here is on the Iwahori–Hecke algebra Hn of the symmetric group, where there is the suggestion of a multiplicative structure through a connection with symmetric polynomials of Jucys–Murphy elements [3, 11, 5]. This connection has yielded an alternative integral basis involving symmetric polynomials [6], but as yet no multiplicative relations. The ability to determine exactly how the product of elements of a basis for Z(Hn ), the centre of Hn , decomposes as a linear combination of basis elements would be particularly useful in any homomorphic context. For instance, the Brauer homomorphism plays an important role in the representation theory of the symmetric group, and has been generalized to Hn [10]. The fact that the codomain of this generalized Brauer homomorphism can be realized as certain products of elements from Z(Hn ) was instrumental in developing a Green correspondence for Hn [4]. However, as yet, an explicit determination of the structure constants has not been achieved [7]. Such a description would provide additional insight into the 2010 Mathematics Subject Classiﬁcation. Primary 20C08; Secondary 20B30, 11D61. Key words and phrases. Iwahori–Hecke algebra, centre, symmetric group. The ﬁrst author was supported by Australian Research Council Future Fellowship FT100100898. Both authors have beneﬁted greatly from the mentoring of Leonard Scott — in the case of the ﬁrst author as a postdoc mentor, and in the case of the second author as a PhD advisor. They would like to record their gratitude for his patient and wise tutelage. 1 159

c 2012 American Mathematical Society

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ANDREW FRANCIS AND LENNY JONES

representation theory of Hn . To this end, it is reasonable to search for bases for Z(Hn ) that have nice multiplicative properties. In this paper we address the question put by Jie Du (private communication) of whether Z(Hn ) might have a basis over R that is multiplicative (closed under the multiplication of the algebra). We answer this question in the negative for Z(Hn ) when n ≥ 3 by showing that no multiplicative bases exist for Z(ZSn ), the centre of the group algebra. We also prove that there exist exactly two multiplicative bases for Z(ZS2 ) and none for Z(H2 ). We point out that multiplicative bases do exist for Z(QSn ) and for Z(Hn ) over Q(q) for all n. For example, nested sums of central primitive orthogonal idempotents do the job. 2. Deﬁnitions and Notation Let Sn be the symmetric group on {1, . . . , n} with generating set of simple reﬂections S := {(i i + 1) | 1 ≤ i ≤ n − 1}. Let ZSn be the symmetric group algebra over the integers. The centre of ZSn , which we denote Z(ZSn ), has dimension equal to the number of partitions of n, and a Z-basis consisting of conjugacy class sums. Considered over the rational numbers, Z(QSn ) also contains a set of primitive orthogonal idempotents eλ in correspondence with the irreducible characters χλ for λ n. These idempotents form another basis for the centre of QSn , and can be expressed in terms of the irreducible characters as follows (see e.g. [1]): χλ (1) eλ = χλ (w)w. n! w∈Sn

Since the characters are constant on conjugacy classes, this expression can be rewritten in terms of class sums C: χλ (1) (2.1) eλ = χλ (wC )C n! C

where wC is a representative of the conjugacy class C. Remark 2.1. When the correspondence of a partition to its particular irreducible character, or conjugacy class is not needed, we simply use the notation χi , ei or Ci for an irreducible character, idempotent or conjugacy class, respectively. We also write wj for an element of the class Cj . Using Remark 2.1, we can rewrite (2.1) as χi (1) χi (wj )Cj . n! j=1 t

ei =

(2.2)

Definition 2.2. Let R = Z[q, q −1 ], where q is an indeterminate. The Iwahori– Hecke algebra Hn of Sn is the unital associative R-algebra with generators {Ti | 1 ≤ i ≤ n − 1}, where Ti corresponds to the simple reﬂection (i i + 1) ∈ S, subject to the relations T i T j = Tj T i Ti Ti+1 Ti = Ti+1 Ti Ti+1 Ti2

= q + (q − 1)Ti

if |i − j| ≥ 2 for 1 ≤ i ≤ n − 2 for 1 ≤ i ≤ n − 1.

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MULTIPLICATIVE BASES

The Iwahori–Hecke algebra Hn is a deformation of the symmetric group algebra ZSn [2]. In particular, the specialization of Hn at q = 1 is isomorphic to ZSn . It follows that any basis for Z(Hn ) over R specializes at q = 1 to a basis for Z(ZSn ). We denote the specialization of an element h ∈ Hn , the specialization of every element of a subset W of Hn , or the specialization of every entry in a matrix M , at q = 1 as h|q=1 , W |q=1 or M |q=1 , respectively. Because of the isomorphism Hn |q=1 ZSn , we think of the specialization of any element in Hn as being an element of ZSn . Definition 2.3. A basis B = {b1 , b2 , . . . , bt } for Z(Hn ) over R is called multiplicative if, for every i and j, there exists k such that bi bj = bk . Specialization at q = 1 gives analogous deﬁnitions for Z(ZSn ). 3. Multiplicative Bases The following preliminaries are needed to establish our main results. Lemma 3.1. Let {e1 , e2 , . . . , et } be a complete set of central primitive idempotents for Z(QSn ). Let B be a multiplicative basis for Z(ZSn ) and let b ∈ B. Then t b= εi ei , i=1

where εi ∈ {−1, 0, 1}. Proof. Consider the sequence b, b2 , b3 , b4 , . . .. Since B is multiplicative and ﬁnite, there exist distinct nonnegative integers k and m, and ˆb ∈ B, such that bk = ˆb = bm . Since {e1 , e2 , . . . , et } is a basis for Z(QSn ), we can write b = wi ∈ Q. Then t t wik ei = bk = ˆb = bm = wim ei , i=1

t i=1

wi ei , where

i=1

so that wik = wim for all i. Hence, for wi = 0, we have that wik−m = 1, which implies that wi = ±1 and the lemma is proven. The following theorem, which we state without proof, is due to W. Burnside [9]. Theorem 3.2 (Burnside). Let G be a ﬁnite group and let χ be a nonlinear irreducible character of G. Then there exists g ∈ G such that χ(g) = 0. We are now in a position to prove the following theorem. Theorem 3.3. There do not exist any multiplicative bases for either Z(Hn ) over R or for Z(ZSn ) when n ≥ 3. Proof. Since any basis for Z(Hn ) over R specializes at q = 1 to a basis for Z(ZSn ), it is enough to show that no multiplicative basis exists for Z(ZSn ). Let {e1 , e2 , . . . , et } be a complete set of central primitive idempotents for Z(QSn ). By way of contradiction, assume that B is a multiplicative basis for Z(ZSn ), and let

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b ∈ B. Considering b as an element of Z(QSn ), we have by Lemma 3.1 and (2.2) that t b= εi ei i=1

⎞ t χ (1) i = εi ⎝ χi (wj )Cj ⎠ n! i=1 j=1 t t χi (1)χi (wj ) Cj , εi = n! j=1 i=1 t

⎛

where εi ∈ {−1, 0, 1}. Since n ≥ 3, there exists at least one nonlinear irreducible character of Sn . Without loss of generality, let χt be nonlinear. Thus, by Theorem 3.2, there exists at least one conjugacy class, say Cs , such that χt (ws ) = 0. Then the absolute value of the coeﬃcient on Cs in b is

t−1

t−1

χ (1)χ (w )

1

i i s

εi εi χi (1)χi (ws )

=

n!

n! i=1

i=1

t−1 1 ≤ |χi (1)| |χi (ws )| n! i=1

1 2 χ (1) n! i=1 i t−1

≤

< 1. t−1 χi (1)χi (ws ) = 0. But then, the central Since b is integral, it follows that i=1 εi n! element Cs cannot be written as a linear combination of the elements of B, which contradicts the fact that B is a basis for Z(ZSn ). The case n = 2 requires a separate analysis which we give in the following theorem. Theorem 3.4. There exist exactly two multiplicative bases for Z(ZS2 ) and no multiplicative bases for Z(H2 ) over R. Proof. It is easy to see that the two bases {(1), (12)} and {(1), −(12)} for Z(ZS2 ) are multiplicative. To see that there are no others, let B = {b1 , b2 } be a multiplicative basis for Z(ZS2 ), where b1 = c1 (1) + c2 (12) and b2 = d1 (1) + d2 (12), with c1 , c2 , d1 , d2 ∈ Z. Since B is multiplicative, we have that b21 = bi and b22 = bj for some i and j. Suppose that b21 = b1 and b22 = b1 . Since b21 = b1 , equating coeﬃcients gives 2 c1 + c22 = c1 and 2c2 c2 = c2 , from which we conclude that c1 = 1 and c2 = 0. Thus b1 = (1). Then, since b22 = b1 , we have that either d1 = ±1 and d2 = 0 or d1 = 0 and d2 = ±1. The ﬁrst case here implies that b2 = (1), which is impossible since B is a basis and b1 = (1). The second case implies that b2 = ±(12), and so we get the two bases {(1), (12)} and {(1), −(12)}. The other cases are similar and they yield no new bases. To prove that no multiplicative basis for Z(H2 ) over R exists is a bit more tedious. Note that Z(H2 ) = H2 and that {1, T1 } is a basis for H2 over R. Suppose

MULTIPLICATIVE BASES

163 5

that B = {b1 , b2 } is a multiplicative basis for Z(H2 ) over R. Since B must specialize at q = 1 to a multiplicative basis for Z(H2 ), we will assume that b1 |q=1 = (1) and b2 |q=1 = (12). The case that b2 |q=1 = −(12) is similar. Then we can write b1 = f + gT1

and b2 = r + sT1 ,

where f, g, r, s ∈ R with f |q=1 = (1) = s|q=1 and g|q=1 = 0 = r|q=1 . Since B is multiplicative, we have that b21 = b1 or b21 = b2 . But b21 = f 2 + g 2 q + (2f g + g 2 (q − 1))T1 , and we see that b21 |q=1 = (1), which implies that b21 = b1 . Similarly, b22 = b1 . Equating coeﬃcients gives: (3.1)

f = f 2 + g 2 q = r 2 + s2 q

(3.2)

g = 2f g + g 2 (q − 1) = 2rs + s2 (q − 1).

If g ≡ 0, then b1 = f T1 and f 2 = f since b21 = b1 . Thus, f ≡ 1 and (3.3)

r 2 + s2 q = 1.

Since s|q=1 = 1, we have that s ≡ 0. Hence, from (3.2), we get 2r + s(q − 1) ≡ 0, so that r = −s(q − 1)/2. Substituting into (3.3) gives s2 = 4/(q + 1)2 ∈ R, which is a contradiction. Thus, g ≡ 0 and from (3.2) we have that f = (1 − g(q − 1))/2. Then substituting into (3.1) gives g 2 = 1/(q + 1)2 ∈ R, which completes the proof. 4. Comments and Conclusions One wonders if some minor adjustment of the idea of a multiplicative basis produces any integral bases for Z(Hn ) or for Z(ZSn ). Motivated by this speculation, we deﬁne the concept of a quasi-multiplicative basis. Definition 4.1. A basis B = {b1 , b2 , . . . , bt } for Z(Hn ) over R is called quasimultiplicative if, for every i and j, there exist k and some nonzero element fij ∈ R, such that bi bj = fij · bk . Specialization at q = 1 gives an analogous deﬁnition for Z(ZSn ). We see that if fij = 1 for all i and j in Deﬁnition 4.1, then B is a multiplicative basis. So Deﬁnition 4.1 is a generalization of Deﬁnition 2.3, and every multiplicative basis is a quasi-multiplicative basis. In a forthcoming paper, we investigate the existence of new quasi-multiplicative bases for Z(Hn ) and Z(ZSn ). References [1] Charles Curtis and Irving Reiner. Methods of representation theory, volume 1. Wiley, 1987. [2] Charles Curtis and Irving Reiner. Methods of representation theory, volume 2. Wiley, 1987. [3] Richard Dipper and Gordon James. Blocks and idempotents of Hecke algebras of general linear groups. Proc. London Math. Soc. (3), 54(1):57–82, 1987. [4] Jie Du. The Green correspondence for the representations of Hecke algebras of type Ar−1 . Trans. Amer. Math. Soc., 329(1):273–287, 1992. [5] Andrew Francis and John J. Graham. Centres of Hecke algebras: the Dipper-James conjecture. J. Algebra, 306(1):244–267, 2006. [6] Andrew Francis and Lenny Jones. A new integral basis for the centre of the Hecke algebra of type A. J. Algebra, 321(3):866–878, 2009. [7] Andrew Francis and Weiqiang Wang, The centers of Iwahori-Hecke algebras are ﬁltered, Representation Theory: Proceedings of the 4th International Conference on Representation Theory, Lhasa, Tibet, July 2007 (Jianpan Wang Zongzhu Lin, ed.), Contemporary Mathematics, vol. 478, American Mathematical Society, 2009, pp. 29–37.

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[8] Meinolf Geck and Rapha¨el Rouquier. Centers and simple modules for Iwahori-Hecke algebras. In Finite reductive groups (Luminy, 1994), pages 251–272. Birkh¨ auser Boston, Boston, MA, 1997. [9] I. Martin Isaacs. Character Theory of Finite Groups. AMS Chelsea Publishing, 2011. [10] Lenny K. Jones. Centers of generic Hecke algebras. PhD thesis, University of Virginia, 1987. [11] Andrew Mathas. Murphy operators and the centre of Iwahori-Hecke algebras of type A. J. Algebraic Combinatorics, 9:295–313, 1999. School of Computing, Engineering and Mathematics, University of Western Sydney, NSW 2751, Australia E-mail address, Andrew Francis: [email protected] Department of Mathematics, Shippensburg University, Pennsylvania, USA E-mail address, Lenny Jones: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Moonshine paths and a VOA existence proof of the Monster Robert L. Griess Jr.

1. Introduction This talk is a brief report on recent joint works by Ching Hung Lam and the speaker on developments in Monster and Moonshine theory [21, 22, 24]. Material from those articles is adapted in this report. The reader should consult those articles and the references in them for more details. We use the abbreviation VOA for vertex operator algebra [12] and the symbol M for the Monster [17]. First, we describe the theory of Moonshine Paths, which represent a relatively concrete way to pass from a node in the extended E8 -diagram to a conjugacy class of the Monster. This may help explain some of the mysterious coincidences found by McKay in the 1970s and by Glauberman and Norton in the late 1990s. Second, we describe a new existence proof of a group of Monster type using VOA theory. Simple current theory is a relatively new procedure developed by Miyamoto for building larger VOAs from a given VOA and a suitable module. Recently a VOA was created by Shimakura using simple currents. This VOA looks like the original Moonshine VOA. We determined that this VOA has an automorphism group of Monster type. Existing characterizations prove that a group of Monster type is unique and that this new VOA is isomorphic to the original Moonshine VOA of Frenkel-Lepowsky-Meurman. This existence proof [24] avoids most of the long technicalities in past existence proofs, at the cost of using VOA theory. Overall, this new proof represents noteworthy simpliﬁcation and shortening of basic existence theory. It is not really a “construction” but is rather an analysis of Aut(V ), where V is the previously constructed VOA of Shimakura [39]. We close with some general remarks on Monster-related proofs and future possibilities for improving Monster theory. 2. The extended E8 -diagram and dihedral groups in the Monster 8 , the In 1979, John McKay [31] noticed a remarkable correspondence between E extended E8 -diagram, and pairs of 2A-involutions in M, the Monster (the largest sporadic ﬁnite simple group). 2010 Mathematics Subject Classiﬁcation. Primary 20D08, 17B69. The author acknowledges support from United States NSA grant H98230-10-1-0201. 1 165

c 2012 American Mathematical Society

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3C ◦ (2.1) ◦ 1A

◦ 2A

◦ 3A

◦ 4A

◦ 5A

◦ 6A

◦ 4B

◦ 2B

There are 9 conjugacy classes of such pairs (x, y) [18], and the orders of the 8 products |xy|, for x = y, are the coeﬃcients of the highest root in the E8 -root system. Thus, the 9 nodes are labeled with 9 conjugacy classes of M. There is no obvious reason why there should be such a correspondence involving high-level theories from diﬀerent parts of the mathematical universe. In 2001, George Glauberman and Simon Norton [14] enriched this theory by adding details about the centralizers in the Monster of such pairs of involutions and relations involving the associated modular forms. Let (x, y) be such a pair and let 8 which is supported n(x, y) be its associated node. Let n (x, y) be the subgraph of E at the set of nodes complementary to {n(x, y)}. If (x, y) is a pair of 2A involutions and z is a 2B involution which commutes with x, y, Glauberman and Norton give a lot of detail about C(x, y, z). In particular, they explained how (in most cases) C(x, y, z) has a “new” relation to the extended E8 -diagram, namely that C(x, y, z)/O2 (C(x, y, z)) looks roughly like “half” of the Weyl group corresponding to the subdiagram n (x, y). The important and provocative McKay-Glauberman-Norton observations seemed like looking across a great foggy space, from one high mountain top to another. We want to realize their connections in a manner which is more down-to-earth, like walking along a path, making natural steps with familiar mathematical objects. These objects are lattices, vertex operator algebras, Lie algebras, Lie groups and ﬁnite groups.

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2.1. Compact Summary of Moonshine Path Strategy. This subsection contains a brief outline of how one may start with a node of the extended diagram 8 and move to a pair of 2A-involutions in the Monster, M. E 8 and the For simplicity, we describe two paths, one beginning with a node of E second one beginning with a pair of 2A-involutions in M. Each path ends with a subVOA generated by a pair of conformal vectors, for which theories on dihedral subVOAs give isomorphisms and enable us to splice the paths. Our Glauberman8 followed by the reverse of the above path Norton path consists of the path from E from M. The notation cvcc 21 means conformal vector of central charge 12 in a VOA. 8 : Path starting in E node → sublattice K of ﬁnite index in E8 → element r ∈ E8 (C) of order |E8 : K|, deﬁned by exponentiation → cvcc 12 e, f in VEE8 ≤ VE8 ⊕E8 → conjugacy of r to an element h in torus normalizer N (T) so h acts on the root lattice without eigenvalue 1 → a pair of EE8 lattices M, M < E83 and cvcc 21 eM , eM such that subV OAeM , eM ∼ = subV OAe, f → Niemeier lattice N with automorphism h so that N + (h ) and N+ (h ) are related to K; ﬁnd overlattice of N + (h ) ⊥ N+ (h )) isometric to Leech lattice. Path starting in M: distinct 2A-involutions x, y ∈ M → x, y correspond to unique cvcc 21 e , f (Miyamoto bijection) in V ; we may replace + x, y by conjugates to take e, f in VL (L means Leech lattice) At the endpoints of these two paths Existing results on dihedral subalgebras of VOAs prove that subV OAe , f ∼ = subV OAe, f if and only if n(x, y) is the node in the E8 procedure [27, 36].

3. Explanations and context The Moonshine Path theory oﬀers some context and explanations of the McKayGlauberman-Norton observations. 3.1. The leap. The leap from “Lie theory world” to the “discrete world” is perhaps the most dramatic transition in the path. This is where we regard an element of ﬁnite order playing two diﬀerent roles. See the points about conjugacy of r and h in the Compact Summary. On the VOA level, if L is an even lattice and VL denotes the lattice type VOA, Aut(VL ) is an algebraic group [9] of dimension at least rank(L), but Aut(VL+ ) is an algebraic group of dimension r/2, where r is the number of roots in L. Thus, one could have Aut(VJ+ ) ﬁnite and Aut(VK+ ) positive dimensional even if J and K are “nearby lattices”. This point may make the leap a bit less surprising.

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3.2. Why half a Weyl group? An ideal answer would be modeled like this. At a certain point on the path, one has a ﬁnite group F and an orthogonal representation F → O(n, C). Because of the change of lattice and associated VOA, there is a change of representation from the standard n-dimensional representation of the orthogonal group to a representation of the spin group on one of the half-spin modules. The half-spin module aﬀords a projective representation of SO(n, C), but not of O(n, C). The corresponding change in group representation theory is thereby restricted to an index 2 subgroup of F . This ideal answer is beautifully realized in the case of the 3C-node path [21], where F ∼ = Sym9 . In the 3C case, we use triality for D4 to ﬁnd a Leech lattice as exceptional overlattice of N + (h ) ⊥ N+ (h ), resulting in visible loss of half the Weyl group, going from n(x, y) to x, y (and then on to C(x, y, z)). Here, N + (h ) ∼ = √ 3E8 and N+ (h ) ∼ = A2 ⊗ E8 . A Moonshine path theory for the 5A-node has been presented in [22]. The ideal answer is not realized here, but see [22] for comments. There is analogy between the 3C and 5A cases because of the role played by integral lattices. The 5A-case required characterizations of two even lattices of two kinds: the ﬁrst has rank 8, discriminant group 54 [37] and the second has rank 16, discriminant group 54 [22]. The second lattice is not generally known and our characterization seems to be the ﬁrst one. Work on Moonshine paths for nodes 2A, 2B, 3A, 4A, 4B, 6A are in progress (the 1A case is easy). 4. The new VOA constructed with simple currents We deﬁne a ﬁnite group G to be of Monster type if it has an involution z whose centralizer CG (z) has the form 21+24 Co1 , is 2-constrained (i.e., satisﬁes z = CG (O2 (CG (z))) and z is conjugate to an element in CG (z) \ {z}. A short argument proves that such a G must be simple (e.g., see [17, 43]). The article [24] gives a new existence proof of a group of Monster type. In fact, a group of Monster type is unique up to isomorphism [18], so the group constructed in this article can be called “the” Monster, meaning the group constructed in [17]. To avoid specialized ﬁnite group theory in [24], we work with a group of Monster type and refer to [18] for uniqueness. Our basic strategy is described brieﬂy in the following paragraphs. It was inspired by the article of Miyamoto [35], which showed how to make eﬀective use of simple current modules and extensions. In [39], Shimakura gives a variation of Miyamoto’s construction. He takes + (VEE )3 and builds a candidate V for the Moonshine VOA using the theory of 8 simple current extensions. His construction of a VOA is more direct and shorter than Miyamoto who simultaneously builds a copy of M acting on his VOA. However, Shimakura’s method furnishes a useful large subgroup of Aut(V ) which is described below and is called J. + Using Aut((VEE )⊗3 ) ∼ = O + (10, 2) Sym3 , one can see in Aut(V ) a subgroup 8 5+10+20 ∼ (GL(5, 2) × Sym3 ) which leaves invariant a subVOA isomorphic to J = 2 + ⊗3 (VEE ) . 8 In [24], we name an involution z ∈ O2 (J) of trace 276 on V2 . We study the ﬁxed point subVOA, V z , and show that V z ∼ = VΛ+ (Λ is the Leech lattice) by use of simple current theory and characterizations.

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It is known that Aut(VΛ+ ) ∼ = 224 .Co1 . Therefore, CAut(V ) (z)/z embeds in 2 .Co1 . Module theory for VΛ+ is known well enough to show that possible extensions to a VOA are limited to two isomorphism types: one is VΛ (degree 1 term has dimension 24) and the other is the VOA V (with degree 1 term equal to 0). On each, 224 .Co1 acts by a projective representation. We conclude that CAut(V ) (z) ∼ = 21+24 Co1 . The fusion of z to CAut(V ) (z) \ {z} is seen in J. We conclude that Aut(V ) is a group of Monster type. In a sense, our existence proof is quite short. The hard group theory and caseby-case analysis of earlier proofs have essentially been eliminated by the use of VOA representation theory. + We refer to the overVOA of (VEE )3 described in [39] as a VOA of Moonshine 8 type, meaning a holomorphic VOA V = ⊕∞ n=0 Vn of central charge 24, so that V0 is 1-dimensional, V1 = 0 and the Monster acts as a group of automorphisms with faithful action on V2 . We mention that such a VOA is isomorphic to the standard Moonshine VOA constructed in [12], by [10, 28]. For the purpose of this article, it is not necessary to quote such characterizations. For background, we mention that the theory of simple current modules originated in the papers [13] and [38]. In [11, 30], certain simple current modules of a VOA are constructed using weight one semi-primary elements and extensions of a VOA by its simple current modules are also studied. The notion of simple current extension turns out to be a very powerful tool for constructing new VOAs from a known one [8, 26, 29, 30, 33, 35]. 24

4.1. About existence and uniqueness proofs of the Monster. The ﬁrst existence proof of the Monster was made in 1980, and published in [17]; see also [16]. A group C ∼ = 21+24 Co1 and a representation of degree 196883 was described. The hard part was to choose a C-invariant algebra structure, give an automorphism σ of it which did not come from C, then identify the group C, σ by proving ﬁniteness and proving that C is an involution centralizer in it. During the decade that followed [17], there were analyses, improvements and alternate viewpoints by Tits [40, 41, 42, 43] and Conway [4]. In the mid-80s, the theory of vertex algebras was developed. The Frenkel-Lepowsky-Meurman text [12] established the important construction of a Moonshine VOA and became a basic reference for VOA theory. The construction of the Monster done in [12] followed the lines of [17], but in a broader VOA setting. The articles [6, 7] constructed a VOA and gave a physics ﬁeld theory interpretation to aspects of [12, 17]. In 2004, Miyamoto [35], made signiﬁcant use of simple current extensions to give a new construction of a Moonshine VOA and of the Monster acting as automorphisms. An existence proof of the Monster was recently announced in [25], which uses theories of ﬁnite geometries and group amalgams. Uniqueness was ﬁrst proved in [18]. A diﬀerent uniqueness proof is indicated in [25]. We mention that the character table of the Monster [1] was determined in the late 1970s under the hypothesis that there exists an irreducible of dimension 196883 [15]. This hypothesis was not veriﬁed until [18] showed that a group of Monster type is unique and is therefore isomorphic to the group of [17] which plainly has an irreducible of dimension 196883.

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As far as we know, the ﬁrst proof that the Monster acts with 9 orbits on pairs of its 2A-involutions occurs in [18]. This property has been widely believed to be true since the late 1970s. 5. Comments on ﬁniteness proofs In the ﬁrst construction of M, the ﬁniteness proof was hard. The group was deﬁned as G := C, σ, where C ∼ / C. It was not obvious that = 21+24 Co1 and σ ∈ G was ﬁnite. A reduction modulo inﬁnitely many primes gave ﬁnite homomorphic images of G where characterizations from ﬁnite group theory applied. Tits had a better idea, to use the classiﬁcation of simple algebraic groups (if A is a ﬁnite dimensional algebra, Aut(A) is deﬁned by polynomial conditions, so is an algebraic group). For the algebra A deﬁned in [17], it is easy to show that if the connected component of the identity in Aut(A) has positive dimension, it is simple (because the subgroup G of Aut(A) constructed in [17] is irreducible without small index proper subgroups). Then routine arguments with highest weight modules and their tensors gives a contradiction. Algebraic group techniques can have wider application since the automorphism group of a ﬁnitely generated VOA is an algebraic group [9]. We let B be the 196884 dimensional algebra which occurs as the degree 2 piece of the Moonshine VOA, whose product is the natural ﬁrst VOA composition. It has A as its unique nontrivial composition factor. A second ﬁniteness technique involves eigenvalues of ad(e) (multiplication by e), where e is a type 2A idempotent in B. The set of all such e ﬁt on a sphere of radius |e|. The conﬁguration of eigenvalues of the ad(e) forces the distance between two such e to be bounded below, whence the set of such e is ﬁnite. Since this set spans B, Aut(B) is ﬁnite. Such arguments were used ﬁrst by Conway [4] and Meier-Neutsch [32]. 6. The future (Goal 1) We feel that there should be a relatively elementary determination of |Aut(V )| by its action on V . The ﬁrst published proof of the order of the Monster was [18]. It used a lot of specialized information about other ﬁnite simple groups. (Goal 2) There should be a way to teach just enough ﬁnite group theory, VOA theory and simple current theory to children (at least, to mathematical children) so that they can understand a construction of V and M in a short time. Ideally, nonexperts would be able to analyze these objects with relatively easy guidelines and not rely on a lot of specialized knowledge about many ﬁnite groups, as is common in current work with ﬁnite simple groups. This would make the 20 members of the Happy Family more accessible. The article [19] indicates the spirit we have in mind for (Goal 2). That article gives a uniqueness proof, by elementary arguments, for the Leech lattice in the style of building lattices from sublattices and gives a new way to analyze the isometry group, Co0 , which is relatively free of technicalities and special counting arguments. Here are more details. The logical order of steps in [19] diﬀers from earlier versions of this theory. We ﬁrst take a Leech lattice (any rank 24 even integral unimodular lattice without vectors of squared length 2; no explicit description is required) then deduce its uniqueness and properties of its automorphism group,

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the large Conway group. We use our uniqueness theorems to get transitivity on the sets of vectors of respective norms 4, 6, 8 and other conﬁgurations of vectors and sublattices without listing members of these sets or even knowing the order of the automorphism group, in contrast with the classic approaches. Next, we deduce existence and uniqueness of the Golay code, then existence and order of the Mathieu group, M24 . The older versions of this theory depend on existence and uniqueness of the Golay code to prove existence and uniqueness of the Leech lattice but ours does not. Simple observations of the Golay code then give immediate results about permutation representations of M24 . For example, we prove 5-transitivity of M24 without even knowing its order. Only minimal examination of particular codes is ever done, in contrast with older approaches [2, 3, 5, 20]. This logical sequence of ﬁrst obtaining the Conway group, then the Mathieu groups, is quite diﬀerent from those of earlier theories for these groups. References 1. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, ATLAS of ﬁnite groups. Clarendon Press, Oxford, 1985. 2. Conway, J. H. Three lectures on exceptional groups. Finite simple groups (Proc. Instructional Conf., Oxford, 1969), pp. 215247. Academic Press, London, 1971. MR0338152 (49 #2918) 3. Conway, John: A Group of Order 8, 315, 553, 613, 086, 720, 000. Bull. L.M.S. 1 (1969) 7988. 4. J.H. Conway, A simple construction for the Fischer-Griess monster group. Invent. Math. 79 (1985), no. 3, 513–540. 5. J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, 3rd Edition, Springer, New York, 1999. 6. L. Dolan, P. Goddard and P. Montague, Conformal ﬁeld theory, triality and the Monster group. Phys. Lett. B 236 (1990), no. 2, 165–72. 7. L. Dolan, P. Goddard and P. Montague, Conformal ﬁeld theory of twisted vertex operators. Nuclear Phys. B 338 (1990), no. 3, 529–601. 8. Chongying Dong, Robert L. Griess, Jr. and Gerald H¨ ohn, Framed vertex operator algebras, codes and the moonshine module, Comm. Math. Physics, 193, 1998, 407-448. 9. Chongying Dong, Robert L. Griess, Jr. , Automorphism groups of ﬁnitely generated vertex operator algebras, Michigan Math Journal, 50 (2002). 227-239. math.QA/0106051 10. Chongying Dong, Robert L. Griess, Jr. and Ching Hung Lam, Uniqueness results for the moonshine vertex operator algebra, American Journal of Mathematics, vol. 129, no. 2 (April, 2007), 583 - 609. 11. C. Dong, H. Li and G. Mason, Simple currents and extensions of vertex operator algebras, Comm. Math. Phys., 180(1996), 671-707. 12. I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster. Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988. 13. Fuchs, J., Gepner, D.: On the connection between WZW and free ﬁeld theories Nucl Phys B294, 30-42 (1998) 14. G. Glauberman and S. P. Norton, On McKay’s connection between the aﬃne E8 diagram and the Monster, Proceedings on Moonshine and related topics (Montreal, QC, 1999), 37–42, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001. 15. Robert L. Griess, Jr. , The structure of the “ Monster” simple group, Proceedings of the Conference on Finite Groups, Park City, Utah, February 1975, Academic Press, Inc.,1976, 113-118. 16. Robert L. Griess, Jr. , A construction of F1 as automorphisms of a 196,883-dimensional algebra, Proc. Natl. Acad. Sci., USA, 78, 1981, 689-691. 17. Robert L. Griess, Jr. , The friendly giant, Invent. Math., 69, 1982, 1-102. 18. Robert L. Griess, Jr. , Ulrich Meierfrankenfeld and Yoav Segev, A uniqueness proof for the Monster. Ann. of Math. (2) 130 (1989), no. 3, 567– 602.

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19. R.L. Griess Jr., Pieces of eight: semiselfdual lattices and a new foundation for the theory of Conway and Mathieu groups. Adv. Math. 148 (1999), no. 1, 75–104. 20. Robert L. Griess, Jr., Twelve Sporadic Groups, Springer Monographs in Mathematics, 1998, Springer-Verlag. 21. Robert L. Griess, Jr. and Ching Hung Lam, A moonshine path from E8 to the monster; Journal of Pure and Applied Algebra, 215 (2011) pp. 927-948. arxiv 11 oct 09). 22. Robert L. Griess, Jr. and Ching Hung Lam, A moonshine path for 5A and associated lattices of ranks 8 and 16, Journal of Algebra, vol 331, April 2011, 338-361; arXiv:1006.390 23. Robert L. Griess, Jr., ”An introduction to groups and lattices: ﬁnite groups and positive deﬁnite rational lattices”; published 2010 by Higher Education Press (in China) and published in 2011 by the International Press for the rest of the world. 24. Robert L. Griess, Jr. and Ching Hung Lam, A new existence proof of the Monster by VOA theory, about 31 pages; submitted; 8 March, 2011 arXiv:1103.1414v2 [math.QA] 25. Ivanov, A. A., The Monster Group and Majorana Involutions, Cambridge University Press, 2009. 26. C.H. Lam, N. Lam and H. Yamauchi, Extension of unitary Virasoro vertex operator algebra by a simple module. Internat. Math. Res. Notices 11 (2003), 577–611. 27. C. H. Lam, H. Yamada and H. Yamauchi, McKay’s observation and vertex operator algebras generated by 2 conformal vectors of central charge 1/2, Inter. Math. Res. Paper, No. 3 (2005), pp. 117-181. 28. C.H. Lam and H.Yamauchi, A characterization of the moonshine vertex operator algebra by means of Virasoro frames, Int. Math. Res. Not. IMRN 2007, no. 1, Art. ID rnm003, 10 pp. 29. C.H. Lam and H. Yamauchi, On the structure of framed vertex operator algebras and their pointwise frame stabilizers, Comm. Math. Phys. 277 (2008), no. 1, 237–285. 30. H.S. Li, Extension of Vertex Operator Algebras by a Self-Dual Simple Module, J. Algebra 187, 236–267(1997). 31. J. McKay, Graphs, singularities, and ﬁnite groups, Proc. Symp. Pure Mathe., Vol.37, Amer. Math. Soc., Providence, RI, 1980, 183-186. 32. MR1223664 (94g:17007) Meyer, Werner; Neutsch, Wolfram Associative subalgebras of the Griess algebra. J. Algebra 158 (1993), no. 1, 117. (Reviewer: Jonathan I. Hall), 17A99 (20D08) 33. M.Miyamoto, Binary codes and vertex operator(super) algebras, J. Algebra, 181, 207– 222(1996). 34. M.Miyamoto, The moonshine VOA and a tensor product of Ising models, in The Monster and Lie Algebras (Columbus, OH, 1996) (J. Ferrar and K. Harada, eds.), Walter de Gruyter, New York (1998), 99–110. 35. M.Miyamoto, A new construction of the moonshine vertex operator algebra over the real ﬁeld, Ann. Math. (2004). 36. S. Sakuma, 6-transposition property of τ -involutions of vertex operator algebras, Int. Math. Res. Not. IMRN 2007, no. 9, Art. ID rnm 030, 19 pp. 37. R. Scharlau and B. Hemkemeier, Mathematics of Computation, Volume 67, Number 222, April 1998, Pages 737749 S 0025-5718(98)00938-7 (an identical preprint appeared in December 1994 in the Bielefeld preprint series of SFB 343). 38. Schellekens, A N., Yankielowicz, S : Extended chiral algebras and modular invariant partition functions. Nucl Phys 327, 673-703 (1989) 39. H. Shimakura, An E8 approach the moonshine vertex operator algebra, to appear in J. London Math. Soc. 40. Tits, J.: R´ esum´ e de Cours, Annuaire du Coll6ge de France, 1982-1983, pp. 89-102 41. Tits, J.: Remarks on Griess’ construction of the Griess-Fischer sporadic group, I, II, IIt, IV. Mimeographed letters, December 1982 - July 1983 42. Tits, J.: Le Monstre (d’apr`es R. Griess, B. Fischer et al.), Seminaire Bourbaki, exposes n 620, novembre 1983, ti paraltre dans Ast6risque 43. J. Tits, On R. Griess’ “friendly giant”, Invent. Math. 78 (1984), no. 3, 491–499. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 USA E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Characteristic Polynomials and Fixed Spaces of Semisimple Elements Robert Guralnick and Gunter Malle Dedicated to Len Scott

Abstract. Answering a question of Frank Calegari, we extend some of our earlier results on dimension of ﬁxed point spaces of elements in irreducible linear groups. We consider characteristic polynomials rather than just ﬁxed spaces.

1. Introduction In [10], the authors answered a question of Peter Neumann and proved that if G is a nontrivial irreducible subgroup of GLn (k) = GL(V ) with k a ﬁeld, then there exists an element g ∈ G with dim CV (g) ≤ (1/3) dim V (where CV (g) denotes the ﬁxed space of g acting on V ). The example G = SO3 (k) with k not of characteristic 2 shows that 1/3 is best possible. Frank Calegari asked if one could ﬁnd g ∈ G such that the characteristic polynomial of g acting on V was of the form (T − 1)e f (T ) where f (1) = 0 and e < n/2. Calegari and Gee [2] are interested in the irreducibility of the Galois representations associated to self-dual cohomological automorphic forms for GLn , especially for small n. This result is easily seen to be true for ﬁnite G (or more generally compact G) in characteristic 0 by the orthogonality relations (see [12, Section 3] for this simple proof). For ﬁnite groups (or more generally for algebraic groups or if the characteristic is positive), the question reduces to ﬁnding a semisimple element g ∈ G with dim CV (g) < n/2 (recall an element in a linear group is semisimple if it is diagonalizable over the algebraic closure — for a ﬁnite group, this is equivalent to saying that char k does not divide the order of the element). The authors in [10, Thm. 1.3] proved a conjecture of Peter Neumann from 1966 about the minimum dimension of ﬁxed spaces. In particular, if G is ﬁnite and the characteristic does not divide |G|, this gives:

2010 Mathematics Subject Classiﬁcation. Primary 20C20. The ﬁrst author was partially supported by DMS 1001962. We thank Frank Calegari for his questions and comments. c Mathematical 0000 (copyright Society holder) c 2012 American

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Theorem 1.1. Let k be a ﬁeld of characteristic p ≥ 0 and G a nontrivial ﬁnite subgroup of GLn (k) = GL(V ) with p not dividing |G|. If V is an irreducible kG-module, then there exists a semisimple element g ∈ G with dim CV (g) ≤ (1/3) dim V . We note that this result depends upon the classiﬁcation of ﬁnite simple groups. However, the result had previously been unknown even for solvable groups. The conclusion of the theorem no longer holds in non-coprime characteristic. The simplest example is to take G = A.C the semidirect product of an elementary abelian group A of order 8 with the cyclic group of order 7 acting faithfully. Then G has an irreducible 7-dimensional representation over any ﬁeld of characteristic not 2. In particular, in characteristic 7, we see that every nontrivial semisimple element has a 3-dimensional ﬁxed space. More generally, if p = 2a − 1 is a Mersenne prime, let G = A.C with A elementary abelian of order p + 1 = 2a and C of order p acting faithfully on A. Then G has an irreducible representation of dimension p in characteristic p and every nontrivial semisimple element has a ﬁxed space of dimension (p − 1)/2. Moreover, taking direct products of this group with itself and the corresponding tensor product of representations, one gets examples of arbitrarily large dimension. Even in characteristic 0, the Example 6.5 in [10] shows that one can do no better than (1/9) dim V no matter how large the dimension of V . In this note, we will prove Calegari’s inequality and show that one can do better under various circumstances. If we consider connected algebraic groups, we can obtain similar results that do not depend upon the classiﬁcation of ﬁnite simple groups. Since a compact real Lie group has only semisimple elements and since the minimum value of dim CV (g) is attained on a nonempty open subset of G, we also obtain: Theorem 1.2. Let G = 1 be a connected compact real Lie subgroup of GLn (C) = GL(V ). Assume that CV (G) = 0. Then the average dimension (with respect to Haar measure) of CV (g) is at most (1/3) dim V . Let > 0. It was shown in [9, Thm. 6] that if G is compact and connected and V is irreducible, then in fact the average dimension of CV (g) is less than dim V as long as dim V is suﬃciently large. We have a similar result for Zariski dense subgroups of connected algebraic groups. Theorem 1.3. Let G = 1 be a subgroup of GLn (k) = GL(V ) with k an algebraically closed ﬁeld and char k = p ≥ 0. Assume that V is completely reducible, CV (G) = 0 and the Zariski closure of G is connected. Then the set of semisimple g ∈ G with dim CV (g) ≤ (1/3) dim V is open and Zariski dense in G. An easy consequence of Theorem 1.3 is: Corollary 1.4. Let G be an irreducible subgroup of GLn (k) = GL(V ) with char k = p ≥ 0. Assume that G is inﬁnite. Then there exists a semisimple g ∈ G such that dim CV (g) ≤ (1/3) dim V . Thus, we are reduced to considering ﬁnite groups. For the general case, we can show: Theorem 1.5. Let 1 = G ≤ GLn (k) = GL(V ) with char k = p ≥ 0. Assume that G acts irreducibly on V .

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(a) There exists a semisimple element g ∈ G with dim CV (g) < (1/2) dim V . (b) If p > n + 2 or p = 0, then there exists a semisimple element g ∈ G with dim CV (g) ≤ (1/3) dim V . (c) If p does not divide n, then there exists a semisimple element g ∈ G with dim CV (g) ≤ (3/8) dim V . (d) If n is prime and 2 is a multiplicative generator modulo n, then there exists a semisimple g ∈ G with dim CV (g) ≤ (1/3) dim V . In particular, this shows that if n ≤ 5, there exists a semisimple element g ∈ G with dim CV (g) ≤ 1. If n is prime, one can prove even stronger results: Theorem 1.6. Let G be a ﬁnite irreducible subgroup of GLn (k) with n an odd prime. Assume that char k = p > 2n − 3 (or p = 0). There exists a semisimple element x ∈ G with all eigenspaces of dimension at most 1. The paper is organized as follows. In the next section, we deal with algebraic groups and deduce the various results on algebraic and inﬁnite groups. We then consider various generation results about ﬁnite simple groups. In particular, in Section 3 we prove the following results that may be of independent interest: Theorem 1.7. Let G be a ﬁnite nonabelian simple group and p be a prime. Then unless (G, p) = (A5 , 5), there exist p -elements x, y, z ∈ G with xyz = 1 such that G = x, y. Theorem 1.8. Let G be a ﬁnite nonabelian simple group and p be a prime. Then there exist a pair of conjugate p -elements that generate G. Note that an immediate consequence of Theorem 1.7 and Scott’s Lemma [18] is: Corollary 1.9. Let G be a ﬁnite nonabelian simple subgroup of GLn (k) = GL(V ). Assume that CV (G) = CV ∗ (G) = 0. If G = A5 and char k = 5, assume further that V has no trivial composition factors. Then there exists a semisimple element g ∈ G with dim CV (g) ≤ (1/3) dim V . It is shown [10, Thm. 6.1] that in fact if > 0, G is a nonabelian ﬁnite simple group and V is an irreducible CG-module, then there exists g ∈ G with dim CV (g) < dim V as long as dim V is suﬃciently large (or equivalently |G| is suﬃciently large). This should be true for any algebraically closed ﬁeld. In Section 4, we consider ﬁnite groups and prove Theorem 1.5(a)–(c). We then consider representations of prime dimension, complete the proof of Theorem 1.5 and prove Theorem 1.6. In the ﬁnal section, we give some examples relating to the divisibility of characteristic polynomials of representations (a question asked by Calegari [2]). 2. Inﬁnite Groups In this section we prove Theorems 1.2, 1.3 and Corollary 1.4. These results do not require the classiﬁcation of ﬁnite simple groups. We ﬁrst prove Theorem 1.3. Let k be an algebraically closed ﬁeld of characteristic p ≥ 0. Let G be a subgroup of GLn (k) = GL(V ). We assume that G has no trivial composition factors on V and that Γ, the Zariski closure of G is connected.

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Our assumption is that G (equivalently Γ) acts completely reducibly on V . Thus, Γ is reductive. Note that for any e ≥ 0, {g ∈ Γ | dim CV (g) ≤ e} is an open subvariety of Γ. Moreover, the set of semisimple elements of Γ is also open. Let S be a rational irreducible kΓ-module. If follows by [7, Thm. 3.3] that the set of pairs of semisimple elements in Γ which generate an irreducible subgroup on S is a nontrivial open subvariety of Γ2 . Thus, the set of pairs of semisimple elements in Γ which have no ﬁxed points on V and whose product is semisimple is also an open nonempty subvariety of Γ2 . Thus, this set intersects G2 in a nonempty open subset of G2 . Choose x, y ∈ G such that x, y has no ﬁxed points on V with x, y and xy semisimple. By Scott’s Lemma [18], dim CV (x) + dim CV (y) + dim CV (xy) ≤ dim V and so some semisimple element g ∈ G satisﬁes dim CV (g) ≤ (1/3) dim V . This shows that the set of semisimple elements of Γ with dim CV (g) ≤ (1/3) dim V is an open dense subvariety of Γ. In particular, this set must intersect G whence Theorem 1.3 holds. Theorem 1.2 now follows immediately. We now prove Corollary 1.4. Arguing as in [10, Thm. 5.8], it suﬃces to work over an algebraically closed ﬁeld. Let Γ be the Zariski closure of G and Γ◦ the connected component of the identity in Γ. Note that Γ◦ = 1 as G is inﬁnite. Since V is irreducible for G, hence for Γ, Γ◦ acts completely reducibly without ﬁxed points on V . By Theorem 1.3, the set of semisimple g ∈ Γ◦ with dim CV (g) ≤ (1/3) dim V contains a dense open subset of Γ◦ and therefore intersects G ∩ Γ◦ non-trivially, as required. We close this section by showing that often one can do even better in the case of algebraic groups. There is a version of the following theorem for semisimple groups as well. Theorem 2.1. Let G be a simple simply connected algebraic group of rank r at least 2 over an algebraically closed ﬁeld k. Let V be a completely reducible rational kG-module with CV (G) = 0. Let g ∈ G be a regular semisimple element and assume that g 3 is not central (the latter can only fail if G = SL3 ). Then: (a) dim CV (g) ≤ (1/3) dim V , and (b) if g 2 is also regular, then any eigenspace of g has dimension at most (1/3) dim V . Proof. Let C be the conjugacy class of g. We ﬁrst prove (a). Let X be the variety of triples of elements all in C with product 1. By [11, Thms. 6.11, 6.15], this is an irreducible variety (of dimension 2 dim G − 3r) and the set of triples in X which generate a subgroup H such that each irreducible submodule of V remains irreducible for H (and non-isomorphic irreducibles remain non-isomorphic) is a dense open subvariety of X. Now (a) follows by Scott’s Lemma. If g 2 is regular semisimple, we consider the variety Y = {(x, y, z) ∈ C × C × C −2 | xyz = 1}. Precisely as above, we see that the subset of Y consisting of triples so that the subgroup they generate has the same collection of irreducibles as G is dense. Apply Scott’s Lemma to the elements (λ−1 x, λ−1 y, λ2 z) to conclude that the λ-eigenspace of at least one of them has dimension at most (1/3) dim V . Note that we do not need to assume that V is completely reducible in the previous result. The proof goes through verbatim as long as we assume that CV (G) = CV ∗ (G) = 0.

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3. Generation results The purpose of this section is the proof of the following generation results: Theorem 3.1. Let G be a ﬁnite non-abelian simple group, p a prime. Then we have: (a) G is generated by two conjugate p -elements, and (b) G is generated by three p -elements x, y, z with xyz = 1 unless (G, p) = (A5 , 5). Note that (A5 , 5) is a true exception to the conclusion of Theorem 3.1(b) since the largest element order of A5 prime to 5 is 3, and the triangle groups G(l, m, n) with l, m, n ≤ 3 are solvable. In [10, Thm. 1.1] we showed that any ﬁnite non-abelian simple group G is generated by a certain triple of conjugate elements with product 1. Thus the remaining task is to prove (a) and (b) for primes p dividing the common order of these elements. This will be shown in the subsequent propositions. Proposition 3.2. Theorem 3.1 holds for sporadic groups and the Tits group. Proof. For G a sporadic simple group both parts follow from [11, Table 4], where we exhibited a second generating systems (x, y, z) for G with product 1, with x ∼ y and the orders of x, y, z prime to those from [10]. Proposition 3.3. Theorem 3.1 holds for alternating groups. Proof. For G = An an alternating group, with n ≥ 11 odd, we produced in [10, Lemma 4.2] a generating system (x1 , y1 , z1 ) of n − 2-cycles with product 1. In [5, Cor. 17] there is given a generating system (x2 , y2 , z2 ) with x2 , y2 squares of an n − 1-cycle and o(z2 ) = n. For n ≥ 12 even, we gave a generating triple of n − 3-cycles in [10, Lemma 4.3], while [5, Cor. 14] gives a generating system with x2 , y2 both n − 1-cycles and o(z2 ) = n/2. This gives the claim unless p = 3 divides n. In the latter case, by [1, Cor. 2.2] for n > 12 there exists two n − 5-cycles with product of type (n − 2)(2), and these generate a transitive group since otherwise, the three elements would be in Sn−2 with the ﬁrst two being in An−2 . The n − 5 cycle guarantees the group generated is primitive and [20, Thm. 13.8] implies that they generate a 6-transitive group. The result now follows by [20, Thm. 51.1]. The alternating group A12 has a generating triple consisting of elements of order 11. For n = 5, 6, 7, 8, 9, 10 we gave generating triples of orders 5,5,7,7,7,7 respectively in [10, Lemma 4.4]. For n = 6, 7, 8, 9, 10 direct computation shows that there also exist generating triples of orders 4,5,15,15,15. Note that A5 is generated by two 3-cycles (with product of order 5). Proposition 3.4. Theorem 3.1 holds for exceptional simple groups of Lie type. Proof. For G of exceptional Lie type diﬀerent from 3D4 (q), we produced in [11, Thm. 2.2] a second generating system consisting of conjugate elements, while for 3D4 (q) in [11, Prop. 2.3] we gave a generating triple containing two conjugate elements. Moreover, in all cases, the element orders in these triples are coprime to those from [10]. Finally assume that G is of classical Lie type. For n ≥ 2, we let Φ∗n (q) denote the largest divisor of q n − 1 that is relatively prime to q m − 1 for all 1 ≤ m < n.

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Proposition 3.5. Theorem 3.1 holds for L2 (q), q ≥ 4. Proof. The groups L2 (q) with q ∈ {4, 5, 9} are isomorphic to alternating groups, for which the claim follows from Proposition 3.3. The group L2 (7) has generating triples consisting of elements of order 7, respectively of order 4. For q ≥ 8, q = 9, we gave in [10, Lemmas 3.14, 3.15] generating triples for L2 (q) of orders (q − 1)/d, where d = gcd(2, q − 1). Similarly, a direct calculation shows that there exist generating triples of order (q + 1)/d, which proves the claim. Definition 3.6. Let’s say that a pair (C, D) of conjugacy classes of a group G is generating if G = x, y for all (x, y) ∈ C × D. Note that by the result of Gow [6], if (C, D) is a generating pair for a ﬁnite group of Lie type consisting of classes of regular semisimple elements, then we ﬁnd generators (x, y) ∈ C × D with product in any given (noncentral) semisimple conjugacy class, for example in C −1 . Proposition 3.7. Theorem 3.1 holds for the groups Ln (q), n ≥ 3. Proof. Let G = Ln (q), n ≥ 3. Note that we may assume that (n, q) = (3, 2), (4, 2) as L3 (2) ∼ = L2 (7) and L4 (2) ∼ = A8 were already handled above. In [10, Prop. 3.13] we showed that G is generated by a triple of elements of order Φ∗n (q) when n is odd, respectively of order Φ∗n−1 (q) when n is even. For n = 4 and (n, q) = (6, 2), we showed in [11, Prop. 3.1 and 3.5] that there also exist generating triples of elements of orders coprime to the former ones, of the type stated in Theorem 3.1. The group L6 (2) is generated by a triple of elements of order 7. Now consider G = SL4 (q). Let C be a conjugacy class of regular semisimple elements of order (q 4 − 1)/(q − 1). According to [10, Lemma 2.3], the only maximal subgroups of SL4 (q) which might contain an element x ∈ C are the normalizer of 2 Ω− 4 (q), of Sp4 (q) or of GL2 (q ) ∩ SL4 (q). But in the ﬁrst two of these groups the centralizer order of elements of order q 2 + 1 is not divisible by (q 2 + 1)(q + 1) when q > 3. Thus, any x ∈ C lies in a unique maximal subgroup of G. Let C2 denote a class of regular semisimple elements in a maximal torus of order (q 2 − 1)(q − 1) of SL4 (q). Then this does not intersect the normalizer of GL2 (q 2 ) ∩ SL4 (q), so (C1 , C2 ) is a generating pair. By [6] there exist pairs with product in C1−1 , which must generate. Now pass to the quotient of SL4 (q) by its center. The group L4 (3) has a generating triple with elements of order 5. Proposition 3.8. Theorem 3.1 holds for the unitary groups Un (q), n ≥ 3, (n, q) = (4, 2). Proof. Let G = Un (q), n ≥ 3. The case n = 3 was already treated in [11, Prop. 3.1]. In [10, Prop. 3.11 and 3.12] we showed that G is generated by a triple of elements of order Φ∗2n (q) when n is odd, respectively of order Φ∗2n−2 (q) when n is even. For n ≥ 8 we showed in [11, Prop. 3.6] that there also exist generating triples of elements of orders coprime to the former ones, of the form required in Theorem 3.1. For the remaining n we argue in G = SUn (q) and then pass to the quotient by the center. For n = 7 let C1 contain elements of type 6+ and C2 elements of type 5− ⊕ 2+ , for n = 5 let C1 contain elements of type 4+ and C2 elements of type 3− ⊕ 2+ . Then any pair from C1 × C2 generates an irreducible subgroup, and by [10, Thm. 2.2] that can’t be proper, when (n, q) = (5, 2). We conclude using [6]. The group U5 (2) has generating triples of elements of order 5.

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If n = 4 or 6, the claim of Theorem 3.1(b) holds by [11, Prop. 3.6]. For Theorem 3.1(a), the group U6 (2) has a generating triple of elements of order 7. Else let’s take C1 a class of elements of order Φ∗2n−2 (q), C2 a class of elements of order Φ∗4 (q) when n = 4, Φ∗3 (q)Φ∗6 (q) when n = 6, and C3 one of C1 , C2 . Then the arguments in the proof of [10, Prop. 3.12] go through, with even better estimates, since the number of characters not vanishing on both C1 and C2 is smaller, and the same for the number of maximal subgroups containing elements from both classes. It follows that there exist generating triples with product 1 in C1 × C1 × C2 and in C1 × C2 × C2 . Since the orders in the two classes are relatively coprime, this gives the result. Proposition 3.9. Theorem 3.1 holds for the orthogonal groups O2n+1 (q), n ≥ 3. Proof. The group G = O2n+1 (q) possesses a generating triple of elements of order Φ∗2n (q), by [10, Prop. 3.7 and 3.8]. For n ≥ 7, we produced in [11, Prop. 7.9] a generating pair (C, D) of conjugacy classes containing regular semisimple elements of orders prime to Φ∗2n (q). For n = 4, 5, 6, let C contain elements of type (n − 1)− ⊕ 1− , D elements of type (n − 2)− ⊕ 2+ . Then the group generated by (x, y) ∈ C × D acts irreducibly or is contained in a 2n-dimensional orthogonal group. By consideration of suitable Zsigmondy primes, the latter can possibly only occur when q ≤ 3, which we exclude for the moment. Otherwise, an application of [11, Cor. 3.4] shows that (C, D) is a generating pair, and we conclude using [6]. The groups O9 (2) ∼ = S8 (2), O9 (3), O11 (2) ∼ = S10 (2), O11 (3), O13 (2) ∼ = S12 (2), O13 (3), possess generating triples with elements of orders 7, 13, 31, 41, 31, 61 respectively, Finally, for n = 3, we argued in [11, Prop. 3.8] that conjugacy classes of regular semisimple elements of types 3+ , 2− ⊕ 1+ form a generating pair in O7 (q) for q ≥ 5, and we produced generating triples of orders 7, 13 and 17 for O7 (2) ∼ = S6 (2), O7 (3), O7 (4) respectively. Proposition 3.10. Theorem 3.1 holds for the symplectic groups S2n (q), n ≥ 2, (n, q) = (2, 2). Proof. Note that we may assume that q is odd when n ≥ 3 by the result of Proposition 3.9. The group G = S2n (q) possesses a generating triple of elements of order Φ∗2n (q), resp. order 5 when (n, q) = (2, 3), by [10, Prop. 3.8]. For n ≥ 3, (n, q) = (4, 3), we found in [11, Prop. 7.8] a generating pair (C, D) of conjugacy classes containing regular semisimple elements of orders prime to Φ∗2n (q), and we may conclude as usual. The group S8 (3) has a generating triple with elements of order 13. For S4 (q) with q ≥ 3, the claim was already proved in [11, Prop. 3.1]. Proposition 3.11. Theorem 3.1 holds for the orthogonal groups O+ 2n (q), n ≥ 4. Proof. Let G = O+ 2n (q), n ≥ 4. In [10, Prop. 3.10] we showed that G is generated by a triple of elements of order dividing Φ∗2n−2 (q)(q + 1). For n = 4 we showed in [11, Prop. 3.10] that there also exist generating pairs of conjugate regular semisimple elements of orders coprime to the former ones, and we may conclude as before. For the n = 4 we may assume that q ≥ 3, since for O+ 8 (2) there exist generating triples of elements of order 9, and also of elements of order 7, by [10, Prop. 3.10] and [11, Prop. 3.10]. Let C1 contain regular semisimple elements of order (q 2 + 1)/d,

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with d = gcd(2, q − 1), in a maximal torus T of order (q 2 + 1)2 /d2 , and C2 the image of C1 under triality. Let (x, y) ∈ C1 × C2 . Then H := x, y contains T up to conjugation. By [16, Table I] the only maximal subgroup of O+ 8 (q) with this − 2 (q)×O (q)).2 . According to [6] there are (x, y) with product property is M = (O− 4 4 of order a Zsigmondy prime divisor of q 2 + q + 1, and hence not contained in M . This gives a triple as in Theorem 3.1(b). We now show Theorem 3.1(a) for n = 4 and q ≥ 3. Let C1 be a conjugacy class of elements of order Φ∗6 (q). Let C2 be a conjugacy class of regular semisimple elements of order (q 4 − 1)/4 (or q 4 − 1 if q is even) having precisely two non-trivial invariant subspaces. Let C3 be either C1 or C2 . Arguing as in [10, Prop. 3.10], one gets a lower bound for the number of triples (x, y, z) ∈ C1 × C2 × C3 with product 1 (the bound is actually much better than in [10] since there will be many fewer characters not vanishing on C1 and C2 ). Similarly, one gets an upper bound for the number of non-generating such triples (again the bound is much better than that given in [10] since there are many fewer maximal subgroups — for example, it is clear the group generated is irreducible). It follows that two elements of C1 or C2 will generate. Since the orders of the elements in C1 and C2 are relatively prime, the result follows. Proposition 3.12. Theorem 3.1 holds for the groups orthogonal O− 2n (q), n ≥ 4. Proof. The group G = O− 2n (q) possesses a generating triple of elements of order Φ∗2n (q), by [10, Prop. 3.6]. For (n, q) ∈ / {(4, 2), (4, 4), (5, 2), (6, 2)}, we produced in [11, Prop. 7.10] a generating pair (C, D) of conjugacy classes of G containing regular semisimple elements of orders prime to Φ∗2n (q). Now by [6] there exist triples in C × C × D, for example. Direct computation shows that the groups O− 8 (2), − − (4), O (2), O (2) possess generating triples with elements of orders 7, 13, 17, O− 8 10 12 31 respectively, which are again prime to Φ∗2n (q). 4. Finite Groups Here, we prove Theorem 1.5(a)–(c). By Corollary 1.4 it suﬃces to consider ﬁnite groups. Fix a ﬁeld k of characteristic char k = p. By Theorem 1.1, we may assume that p > 0. Assume that 1 = G ≤ GLn (k) = GL(V ) is irreducible on V . By extending scalars, we can reduce to the case that k is algebraically closed (V would at worst be a direct sum of Galois conjugates of a given irreducible module). Let N be a minimal normal subgroup of G. Thus, N acts completely reducibly on V and without ﬁxed points. We break up the argument depending upon the structure of N . Lemma 4.1. If |N | is odd, then N is an r-group for some odd prime r = p and there exists a p -element g ∈ N with dim CV (g) < (1/r) dim V ≤ (1/3) dim V . Proof. Since |N | is odd, N is solvable and since it is a minimal normal subgroup, it must be an r-group with r = p. Now apply [12, Cor. 1.3]. Lemma 4.2. If N is an elementary abelian 2-group, then p = 2 and (a) there exists an involution g ∈ N with dim CV (g) < (1/2) dim V ; (b) if p does not divide dim V , then there exists a p -element g ∈ G with dim CV (g) ≤ (1/3) dim V .

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Proof. Clearly, p = 2. By [12, Cor. 1.3], there exists g ∈ N with dim CV (g) < (1/2) dim V . So we may assume that p does not divide dim V . If N is central, thenany 1 = g ∈ N satisﬁes CV (g) = 0 and there is nothing to prove. Otherwise, V = Vi where the Vi are the distinct N -eigenspaces. Since V is irreducible, G permutes the Vi transitively. By [4, Thm. 1], there exists x ∈ G of prime power order r a having no ﬁxed points on the set of Vi . Since p does not divide dim V , r = p. If r is odd, then every orbit of x has size at least 3 whence dim CV (x) ≤ (1/3) dim V . So we may assume that r = 2. It follows as in the proof of [10, Thm. 5.8] that the average dimension of the ﬁxed point spaces of elements in the coset xN is at most (1/4) dim V . Since every element in xN is a 2-element, the result follows. The remaining case is when N is a direct product of t ≥ 1 isomorphic copies of a nonabelian simple group L. Let W be an irreducible kN -submodule of V . By reordering, we may write W = U1 ⊗ · · · ⊗ Ut where each Ui is an irreducible kL-module with Ui ∼ = k if and only if i > m for some m > 1. First we note that the proof of [10, Cor. 5.7] gives: Lemma 4.3. Let L be a simple group of Lie type over a ﬁnite ﬁeld of characteristic p and let E = L × · · · × L. Let k be an algebraically closed ﬁeld of characteristic p, V a completely reducible kE-module with CV (E) = 0. Then there exists a semisimple element x ∈ E with dim CV (x) ≤ (1/3) dim V . Actually, the proof in [10] does not work if L ∼ = L2 (5) with p = 5, or L ∼ = L2 (7) with p = 7. A more complicated proof can be given in these cases to show that the result is still true. The result also follows easily if L = L2 (q) using Theorem 2.1. Lemma 4.4. Let L be a nonabelian ﬁnite simple group and let E = L×· · ·×L (t copies). Let k be an algebraically closed ﬁeld of characteristic p, V = U1 ⊗ · · · ⊗ Ut a nontrivial irreducible kE-module with dim Ui = 3 for some i. There exists a p element x ∈ E (independent of V ) such that all eigenspaces of x have dimension at most (1/3) dim V . Proof. First assume that t = 1, so dim V = 3. By inspection, we can choose x of odd prime order with distinct eigenvalues (indeed unless p = 3, we can take x of order 3). Suppose that t > 1. Let y be the element of E with all coordinates equal to the x chosen above. Since x has all eigenspaces of dimension at most (1/3) dim Ui on Ui , the same is true on V . Lemma 4.5. Let L be a ﬁnite group with L = x, y and let E = L×· · ·×L. Let k be an algebraically closed ﬁeld of characteristic p, V = U1 ⊗ · · · ⊗ Ut a nontrivial irreducible kE-module with dim Ui ≥ 4 for some Ui . Let G be a diagonal copy of L in E. Let x1 , y1 and z1 be elements of G with each coordinate x, y or z = (xy)−1 respectively. Then dim CV (x1 ) + dim CV (y1 ) + dim CV (z1 ) ≤ (9/8) dim V . In particular, min{dim CV (x1 ), dim CV (y1 ), dim CV (z1 )} ≤ (3/8) dim V. Proof. The assumption that some Ui has dimension at least 4 gives that dim CV (G) ≤ (1/16) dim V . Indeed, assume that dim U1 ≥ 4. Then CV (G) ∼ = HomG (U1∗ , U2 ⊗ · · · ⊗ Ut ). Since dim U1 ≥ 4, dim CV (G) ≤ (1/4) dim(U2 ⊗ · · · ⊗ Ut ) ≤ (1/16) dim V.

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Similarly, dim CV ∗ (G) ≤ (1/16) dim V . By Scott’s Lemma [18] dim CV (x1 )+ dim CV (y1 ) + dim CV (z1 ) ≤ dim V + dim CV (G) + dim CV ∗ (G) ≤ (9/8) dim V.

The result follows.

Corollary 4.6. Let G be a ﬁnite group, k an algebraically closed ﬁeld of characteristic p and V a faithful irreducible kG-module. Let E = L × · · · × L be a minimal normal subgroup of G with L a nonabelian simple group. Then there exists a p -element x ∈ E with dim CV (x) ≤ (3/8) dim V . Proof. If L is of Lie type in characteristic p, the result follows by Lemma 4.3. So assume that this is not the case. Let W be any irreducible kE-submodule of V . Then W = U1 ⊗ · · · ⊗ Ut where each Ui is an irreducible kL-module. If dim Ui = 2, then p = 2 and L ∼ = SL2 (2f ) is of Lie type in characteristic 2, whence the result by Lemma 4.3. If dim Ui = 3 for some i, then this will be the case for every irreducible kEsubmodule (since any such is a twist of W ) and Lemma 4.4 applies. In the remaining case, dim Ui > 3 for every nontrivial Ui (and similarly for every twist of W ). By Theorem 1.7, we can choose p -elements x, y, z ∈ L which generate L and have product 1 aside from (L, p) = (A5 , 5). Note that A5 ∼ = L2 (5), so the latter is of Lie type in deﬁning characteristic, a case we already dealt with (alternatively, it would follow that each nontrivial Ui has dimension 5 and so if x is an element of E with each coordinate of order 3, dim CW (x) ≤ (9/25) dim V < (3/8) dim W ). It follows by Lemma 4.5 that there exists a p -element g ∈ E with dim CV (g) ≤ (3/8) dim V . We can now prove the ﬁrst three parts of Theorem 1.5. Proof of Theorem 1.5. Parts (a) and (c) follow from Lemmas 4.1, 4.2 and Corollary 4.6. Now assume that p > dim V + 2. By Lemma 4.1, we may assume that G has no odd order non-trivial normal subgroups. If O2 (G) = 1, Lemma 4.2 applies since p does not divide dim V . So we may assume that F (G) = 1. Let N = L × · · · × L be a minimal normal subgroup of G with L a nonabelian simple group. By [8, Thm. B], it follows that one of the following holds: (1) p does not divide |N |; (2) L is a ﬁnite group of Lie type in characteristic p; or (3) p = 11, N = J1 and dim V = 7. Thus by [10, Cor. 5.7] there exists x ∈ N with dim CV (x) ≤ (1/3) dim V . If x is a p -element, we are done. So we may assume that p divides the order of x (and so |N |). If N = J1 , p = 11 and dim V = 7, an element g of order 19 satisﬁes dim CV (g) = 1 < (1/3) dim V . If L has Lie type, then Lemma 4.3 yields a semisimple element g ∈ N with dim CV (g) ≤ (1/3) dim V . This completes the proof of (b). 5. Prime Degree Recall that a group is called quasi-simple if G is perfect and G/Z(G) is a nonabelian simple group.

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Let n be an odd prime. Let k be a ﬁeld with char k = p. Let G ≤ GLn (k) = GL(V ) be irreducible and ﬁnite. If G is not absolutely irreducible, then G must be cyclic and any generator has distinct eigenvalues on V (and is semisimple). So assume that G is absolutely irreducible and k is algebraically closed. Lemma 5.1. If p = n and the Sylow n-subgroup is not abelian, then G contains a subgroup A of order n such that V is a free kA-module. Proof. Let N be a minimal nonabelian n-subgroup of G. Then N acts irreducibly and is extraspecial, whence the result is clear. Lemma 5.2. Suppose that G acts imprimitively. (a) If n = p, then the Sylow n-subgroup S of G has order n and V is a free kS-module. (b) If n = p, then there exists an element x ∈ S with order a power of n having all eigenspaces of dimension at most 1. Proof. Since n is prime, G imprimitive implies that G permutes n one dimensional (linearly independent) subspaces. Thus, G surjects onto a transitive permutation group of degree n. In particular, n divides the order of G. Let N be the normal subgroup of G stabilizing each of the n one dimensional spaces. If p = n, then N has order prime to p, whence S has order n and the result follows. So assume that p = n. Let x ∈ G be an n-element with x not in N . Then xn is central in GL(V ) and its minimal polynomial is xn − a for some a ∈ k× , whence the result. Note that since n is prime if N is a minimal normal noncentral subgroup of G, then either N is an elementary abelian r-group for some prime r = p or N acts irreducibly. If N acts irreducibly, then either N is an n-group and p = n, whence N contains a (semisimple) element with distinct eigenvalues or N is quasi-simple. If Z(N ) = 1, then N contains a semsimple element x with CV (x) = 0. If N is simple, then Corollary 1.9 implies that there exists x ∈ N semisimple with dim CV (x) ≤ (1/3) dim V (if N = A5 , the result follows by inspection). If N is abelian and not a 2-group, then [12, Cor. 1.3] implies that there exists a (semisimple) x ∈ N with dim CV (x) < (1/3) dim V . Finally, if N is a 2-group and 2 is a multiplicative generator modulo n, then |N | = 2n−1 (because G permutes transitively the n eigenspaces of N and the smallest irreducible module of a cyclic group of order n in characteristic 2 has size 2n−1 ). It follows that there exists x ∈ N with −x a reﬂection, whence dim CV (x) = 1 ≤ (1/3) dim V . In particular, we have proved part (d) of Theorem 1.5. We next consider quasi-simple groups and ﬁrst show: Theorem 5.3. Let G be a ﬁnite quasi-simple group of Lie type in characteristic p. Let k be an algebraically closed ﬁeld of characteristic p. Suppose that V is a faithful irreducible kG-module of odd prime dimension n ≤ p. Then V is a twist of a restricted module and one of the following holds: (1) G = SL2 (q); (2) G = G2 (q) or 2 G2 (q) , n = 7; (3) G = Ωn (q) and V is a Frobenius twist of the natural module; or (4) G = SLn (q) or SUn (q) and V is a Frobenius twist of the natural module or its dual.

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Moreover, either there exists a semisimple element x ∈ G with all eigenspaces of dimension at most 1 on V or G = SL2 (p) and p ≤ 2n − 3. In all cases, there exists a semisimple element x ∈ G with dim CV (x) ≤ 1. Proof. For the ﬁrst part, it suﬃces to prove the result for algebraic groups. Let V = L(λ) where λ is the highest weight for V . By the Steinberg tensor product theorem and the fact that n is prime, V is a Frobenius twist of some restricted irreducible. So we may assume that λ is p-restricted. By [14], it follows that L(λ) is also the Weyl module whence the Weyl dimension formula holds for L(λ). Thus, it suﬃces to work in characteristic 0. The result in that case is due to Gabber, see [15, 1.6]. Aside from the ﬁrst case, any regular semisimple element of suﬃciently large order will have n distinct eigenvalues on V . Suppose that G = SL2 (q). Let x ∈ G have order q + 1. If two distinct weights for a restricted module can coincide on x, then 2 dim V − 2 ≥ q + 1. This can only occur if q = p ≤ 2n − 3. Moreover, no nontrivial weight vanishes on an element of order q+1. Since a restricted irreducible kSL2 (q)-module has distinct weights, the result follows. Theorem 5.4. Let n be an odd prime. Let G ≤ GL(V ) = GLn (k) be a ﬁnite irreducible quasi-simple group, where k is an algebraically closed ﬁeld of characteristic p. (a) If p > max{2n − 3, n + 2} or p = 0, then there exists a semisimple x ∈ G with all eigenvalues distinct. (b) If p > n + 2, there exists a semisimple element x ∈ G with dim CV (x) ≤ 1. Proof. If p divides |G|, then by [8, Thm. B], G is a ﬁnite group of Lie type in characteristic p (or G = J1 , p = 11 and n = 7, where we may take x of order 19) and Theorem 5.3 applies. If p does not divide |G|, then either p = 0 or V is the reduction of a characteristic 0 module. The list of possible groups and modules is given in [3, Thm. 1.2]. It is straightforward to see that the conclusion holds for these groups (most of the examples are related to Weil representations). Theorem 1.6 now follows from the previous results aside from the case n = 3 and char k = 5. In that case, any noncentral semisimple element of order greater than 2 has distinct eigenvalues. The next example shows that we do need some restriction on the characteristic. Example 5.5. Let k be an algebraically closed ﬁeld of positive characteristic p. Let G = SLp (k) = SL(V ). Then G acts by conjugation on W := End(V ). Since every semisimple g ∈ G is centralized by a maximal torus, we see that dim CW (g) ≥ p. Note that W is a uniserial module with two trivial composition factors and an irreducible composition factor V of dimension p2 − 2. Clearly, dim CV (g) ≥ dim CW (g) − 2 ≥ p − 2 for any semisimple element g of G (and since semisimple elements are Zariski dense, this is true for any g ∈ G). Note that dim V can be prime (eg, this is true for p = 5, 7, 13). The same holds for G(Fpa ) = SLp (pa ) for any a ≥ 1. 6. Characteristic Polynomials of Representations Let G be a group and V a ﬁnite dimensional kG-module with k a ﬁeld of characteristic p ≥ 0. Let chV denote the function from G to k[x] deﬁned by

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chV (g) = det(xI − g). Note that two modules have the same function if and only if their composition factors are the same. Our results on bounds for dim CV (g) for some g ∈ G can be phrased in asking: given a kG-module V what is the largest power of chk that divides chV ? Frank Calegari asked what one could say if V and W are two irreducible kGmodules and chV divides chW . Calegari and Gee [2] used this information to study Galois representations in very small dimensions. While we suspect that this does impose some constraints on the representations, we give some examples to show that it is not that rare (at least for groups of Lie type and algebraic groups in the natural characteristic). Example 6.1. Let G be a simple algebraic group over an algebraically closed ﬁeld of characteristic p > 0. Let V be an irreducible kG-module. By Steinberg’s tensor product theorem, V = V0 ⊗· · ·⊗Vm where Vi is a twist of a restricted module by the ith power of Frobenius. If 0 is a weight for some Vj , then clearly chV is a multiple of chVj , where Vj is the tensor product of all the Vi , i = j. The following example was shown to us by N. Wallach (in particular see [19]). Example 6.2. Let k be an algebraically closed ﬁeld of characteristic 0. Let G be a simple algebraic group over k. Let λ and μ be dominant weights with μ in the root lattice. Then chV (λ) divides chV (λ+μ) . It follows the same is true in positive characteristic p as long as p is suﬃciently large (depending upon λ and μ). Here are a few cases where one can compute this directly. We give one such case. Example 6.3. Let k be an algebraically closed ﬁeld of characteristic p ≥ 0. Let G = SLn (k) and let V = V (λ1 ) be the natural module. Then chV ((s+n)λ1 ) is a multiple of chV (sλ1 ) for p > s + n (or p = 0). References 1. E. Bertram, Even permutations as a product of two conjugate cycles. J. Combinatorial Theory Ser. A 12 (1972), 368–380. 2. F. Calegari, T. Gee, Irreducibility of automorphic Galois representations of GL(n), n at most 5. Preprint. 3. J. Dixon, A. Zalesskii, Finite primitive linear groups of prime degree. J. London Math. Soc. (2) 57 (1998), 126–134. 4. B. Fein, W. Kantor, M. Schacher, Relative Brauer groups. II. J. Reine Angew. Math. 328 (1981), 39–57. ´ lez-Diez, On Beauville structures on the groups Sn and An . Math. 5. Y. Fuertes, G. Gonza Z. 264 (2010), 959–968. 6. R. Gow, Commutators in ﬁnite simple groups of Lie type. Bull. London Math. Soc. 32 (2000), 311–315. 7. R. Guralnick, Some applications of subgroup structure to probabilistic generation and covers of curves. Pp. 301–320 in: Algebraic groups and their representations (Cambridge, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 517, Kluwer Acad. Publ., Dordrecht, 1998. 8. R. Guralnick, Small representations are completely reducible. J. Algebra 220 (1999), 531– 541. 9. R. Guralnick, M. Larsen, C. Manack, Low degree representations of simple Lie groups. Proc. Amer. Math. Soc., to appear. 10. R. Guralnick, G. Malle, Products of conjugacy classes and ﬁxed point spaces. J. Amer. Math. Soc. 25 (2012), 77–121.

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11. R. Guralnick, G. Malle, Simple groups admit Beauville structures. J. London Math. Soc., to appear. ´ ti, Average dimension of ﬁxed point spaces with applications. 12. R. Guralnick, A. Maro Advances Math. 226 (2011), 298–308. 13. G. Hiss, G. Malle, Low-dimensional representations of quasi-simple groups. LMS J. Comput. Math. 4 (2001), 22–63; Corrigenda: ibid. 5 (2002), 95–126. 14. J. Jantzen, Low-dimensional representations of reductive groups are semisimple. Pp. 255– 266 in: Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997. 15. N. Katz, Exponential sums and diﬀerential equations. Annals of Math. Studies 124, Princeton U. Press, Princeton, New Jersey, 1999. 16. P. Kleidman, The maximal subgroups of the ﬁnite 8-dimensional orthogonal groups P Ω+ 8 (q) and of their automorphism groups. J. Algebra 110 (1987), 173–242. ¨ beck, Small degree representations of ﬁnite Chevalley groups in deﬁning characteristic. 17. F. Lu LMS J. Comput. Math 4 (2001), 135–169. 18. L. Scott, Matrices and cohomology. Ann. of Math. 105 (1977), 473–492. 19. N. Wallach, Induced representations of Lie algebras. II. Proc. Amer. Math. Soc. 21 (1969), 161–166. 20. H. Wielandt, Finite permutation groups. Academic Press, New York–London, 1964. 3620 S. Vermont Ave, Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA. E-mail address: [email protected] FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany. E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

“Frobenius twists” in the representation theory of the symmetric group David J. Hemmer Abstract. For the general linear group GLn (k) over an algebraically closed ﬁeld k of characteristic p, there are two types of “twisting” operations that arise naturally on partitions. These are of the form λ → pλ and λ → λ + pr τ The ﬁrst comes from the Frobenius twist, and the second arises in various tensor product situations, often from tensoring with the Steinberg module. This paper surveys and adds to an intriguing series of seemingly unrelated symmetric group results where this partition combinatorics arises, but with no structural explanation for it. This includes cohomology of simple, Specht and Young modules, support varieties for Specht modules, homomorphisms between Specht modules, the Mullineux map, p-Kostka numbers and tensor products of Young modules.

1. Introduction Let k be an algebraically closed ﬁeld of characteristic p. An important construction in the representation theory of the general linear group G := GLn (k) is the Frobenius twist, which takes a G module M to the module M (1) . The action of G on M (1) is as on M except twisted by the Frobenius endomorphism F : G → G, which raises each matrix entry to the pth power. Probably the most important G modules are the Steinberg modules Str = L((pr − 1)ρ). For example the operation of “twist then tensor with Str ” plays a key role in the proof of Kempf’s vanishing theorem. In the last decade or so there have been a great variety of results and conjectures on the symmetric group Σd that “look like” they should come from doing a Frobenius twist or taking a tensor product with a Steinberg module, even though neither construction has any reasonable analogue in the world of kΣd modules. Regular twisting results involve partitions λ, pλ, p2 λ, etc... Results reminiscent of twisting then tensor with Str could relate λ with λ+pr τ. Both results we informally think of as twisting type theorems, keeping in mind again that there is no Frobenius twist for kΣd -modules. In this paper we survey the known results of this type, and add a couple more new results together with new examples, conjectures and a variety of open problems that remain. Particularly striking is the array of techniques that arise in the proofs of the various results. Twisting behavior seems to arise in many diﬀerent ways. It is probably naive to expect some kind of uniform “Frobenius twist” for symmetric 2010 Mathematics Subject Classiﬁcation. Primary 20C30. Research of the author was supported in part by NSA grant H98230-10-0192. 1 187

c 2012 American Mathematical Society

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groups that captures these diverse results. We will assume background information for kΣd representation theory as found in [22], and use the same notation. Let λ = (λ1 , λ2 , . . . , λn ) be a partition of d with at most n parts. These partitions correspond to dominant polynomial weights for G and many natural G modules are labeled by them; for example, irreducible modules L(λ), Weyl modules V (λ) and induced modules H0 (λ). The interested reader will ﬁnd Jantzen’s book [23] a deﬁnitive although likely unnecessary reference, as this paper will focus on the symmetric group. For a G module M , let M (r) denote the rth Frobenius twist [23, I.9.10] of M . A special case of the Steinberg Tensor Product Theorem [23, II.3.17] gives that: (1.1)

L(λ)(1) ∼ = L(pλ),

where pλ := (pλ1 , . . . , pλn ) pd. Equation 1.1 suggests the operation λ → pλ is quite natural for G, and it is not surprising to encounter theorems involving these “twisted” partitions. For example the isomorphism (1.2) H1 (G, L(λ)) ∼ = H1 (G, L(pλ)) is a special case of [4, Thm. 7.1] and is realized explicitly on the level of short exact sequences by applying the Frobenius twist. Let ρ = ρn denote the partition (n − 1, n − 2, . . . , 2, 1, 0). Another operation that arises frequently in the representation theory of G takes a partition λ to λ+pr τ for some other partition τ , where λ often involves the so-called Steinberg weight (pr − 1)ρ. For example [23, II.3.19]: (1.3) Hi ((pr − 1)ρ) ⊗ Hi (τ )(r) ∼ = Hi ((pr − 1)ρ + pr τ ). These results are also quite natural as Hi ((pr − 1)ρ) is the ubiquitous Steinberg module Str , which is simple and both projective and injective as a module for the Frobenius kernel Gr . Turning our attention to kΣd , we again ﬁnd modules labeled by partitions, this time by all partitions of d, not just those with at most n parts. For example we have Specht modules S λ , Young modules Y λ , irreducible modules Dλ for λ p-regular, etc. However there is no analogue of the Frobenius twist. Moreover pλ is a partition of pd, and so, for example, S λ and S pλ are modules for diﬀerent groups with no apparent connection. Nevertheless over the last ten years or so there have been numerous symmetric group results involving this kind of “twisting” of partitions, and the proofs use an impressive variety of diﬀerent techniques. Other results are reminiscent of (1.3), even though there is no natural analogue of the Steinberg module for Σd . 2. Schur subalgebras and the original “twist”. We believe the ﬁrst appearance of “twisting” type results for the symmetric group arose in the thesis of Henke, published in part in the paper [20]. (Although James’ computation of decomposition numbers for two-part partitions [21, 24.15] can be put in similar form). For example Henke proved: Theorem 2.1. Fix d and let λ = (d − k, k) be a p-regular partition. Then there is an a ≥ 1 such that there exists a strong submodule lattice isomorphism a between S (d−k,k) and S (d−k+cp ,k) for any c ≥ 1 such that cpa is even. Similar lattice isomorphisms exist for Young modules and permutation modules.

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More general results along the same line were obtain in [19], involving adding a large power of p to the ﬁrst part of a partition and obtaining equality of decomposition numbers and p-Kostka numbers. We should warn though that Section 4 of [19] has a gap. 1 The problem is that the set of weights considered, for example in Cor. 4.1 and following, is not an ideal or coideal. The proofs of these results involve constructing explicit isomorphisms between generalized Schur algebras in diﬀerent degrees. The numerical equalities obtained are then translated to the symmetric group setting.

3. Generic cohomology Only more recently have symmetric group results relating λ and pλ appeared. Many of these results are motivated by or suggestive of the famous generic cohomology theorem from [4], which we describe brieﬂy now. For G modules M1 and M2 , there is a natural map (3.1)

(r)

(r)

ExtiG (M1 , M2 ) → ExtiG (M1 , M2 )

induced by applying the Frobenius twist to the corresponding exact sequences of G modules. The map (3.1) is always an injection [23, II.10.14]. Thus, for ﬁxed i, we have a sequence of injective maps (3.2)

Hi (G, M ) → Hi (G, M (1) ) → Hi (G, M (2) ) → · · · .

When M is ﬁnite-dimensional, the sequence (3.2) is known [4] to stabilize, and the limit is called the generic cohomology Higen (G, M ) of M . Since G = GLn (k), the sequence (3.2) is known to stabilize immediately for i = 1, [4, Thm. 7.1] i.e. (3.3)

H1gen (G, M ) ∼ = H1 (G, M ).

Note that (1.2) is a special case. We call theorems relating cohomology of symmetric group modules U λ and U pλ “generic cohomology” or “stability” type theorems, where U can be S, M, D, Y . For example, a cohomology result relating S pλ and 2 S p λ would be called a generic cohomology type theorem, although we should warn that (1.1) does not hold for other natural G modules. For example H0 (λ)(1) is always a proper submodule of H0 (pλ).

4. Young modules For a partition λ d there is a corresponding Young subgroup Σλ ≤ Σd and the permutation module on the cosets is denoted M λ . The isomorphism classes of indecomposable summands of these permutation modules are also indexed by partitions of d and are called Young modules, denoted Y λ . These are important and well-studied modules. For example the set {Y λ | λ is p-restricted} is a complete set of projective indecomposable kΣd modules. Each Y λ is self-dual. Section 4.6 of [31] is a good basic reference for Young modules.

1 For

example p = 2, λ = (4, 3, 1), μ = (8), pd = 4 gives a counterexample to Cor. 4.5.

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4.1. p-Kostka numbers and tensor products. We have: Mλ ∼ =Yλ⊕

(4.1)

[M λ : Y μ ]Y μ

μ>λ

where > is the lexicographic order on partitions. The multiplicities [M λ : Y μ ] in (4.1) are known as p-Kostka numbers. As we will see there are results for Young r modules of both types, relating Y λ with Y pλ and with Y λ+p τ . In fact just considering p-Kostka numbers, both types of results arise. In [18], Henke determines completely the p-Kostka numbers when λ has two parts. She also obtains: Theorem 4.1. [18, Theorem 6.1] Let λ d and suppose λ1 ≥ d/2 and λ2 < pr . Then: r r [M λ+(ap ) : Y μ+(ap ) ] = [M λ : Y μ ] for every a ≥ 1 and μ d. Her proof uses the well-known multiplicity formula of Klyachko which gives a kind of recursion for p-Kostka numbers in terms of those for smaller partitions, where the assumptions in Theorem 4.1 ensures those decompositions are closely related. More recently in his 2011 thesis Gill proved a strengthened result: Theorem 4.2. [15, s Theorem 2.24] Let λ, μ d and a ≥ 1. Suppose μ has p-adic expansion μ = i=0 μ(i)pi . If pr > max(ps , λ2 ), then: r

[M λ+(ap

)

r

: Y μ+(ap ) ] = [M λ : Y μ ].

Gill’s techniques include an extensive analysis of Young vertices and the Brou´e correspondence for p-permutation modules. This approach to studying Young modules was pioneered by Erdmann in [10]. The twisting λ → pλ behaves very well with respect to the Young vertices, which Gill used to prove the following stability result on p-Kostka numbers under twisting: Theorem 4.3. [15, Theorem 2.21] Let λ, μ d. Then: [M λ : Y μ ] = [M pλ : Y pμ ]. We remark that an alternative proof of Theorem 4.3 could be given using the general linear group and an actual Frobenius twist. This is because p-Kostka numbers are equal to weight space multiplicities in simple GLn (k) modules. Namely: Proposition 4.4. [8, p. 55] The p-Kostka number [M λ : Y μ ] is the dimension of L(μ)λ , the λ weight space in the simple module L(μ). But there is no obvious way to use transfer the proof of Proposition 4.4 to give a short symmetric group proof of Theorem 4.3. 4.2. Young module cohomology. If one thinks of Y pλ as a twist of Y λ then the following theorem can be interpreted as a generic cohomology theorem for Young modules, valid in arbitrary degree:

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Theorem 4.5. [5, Theorem 13.2.1] Fix i > 0 and let p be arbitrary. Then there exists s(i) > 0 such that for any d and λ d we have Hi (Σpa d , Y p λ ) ∼ = Hi (Σpa+1 d , Y p a

a+1

λ

)

whenever a ≥ s(i). Here we have our ﬁrst example of what will be a common theme, a vector space isomorphism of cohomology but without any map realizing the isomorphism. Indeed the proof of Theorem 4.5 proceeds by using Schur functor techniques to translate the symmetric group cohomology problem to a representation theory problem for G. That in turn is solved using powerful algebraic topology techniques, and “reverse engineering” the answer gives the isomorphism; but one cannot trace back to ﬁnd a symmetric group proof or explicit realization of the isomorphism. For example if p = 2 and i = 1 we can compute s(i) = 1, i.e. [5, Theorem 12.4.1] gives: Ext1Σ2d (k, Y 2λ ) ∼ = Ext1Σ4d (k, Y 4λ ).

(4.2) This leads us to ask:

Problem 4.6. Can you prove (4.2) purely using symmetric group representation theory? Can you give an explicit map that takes a short exact sequence 0 → Y 2λ → U → k → 0 and produces the corresponding one for Y 4λ ? Recall that Hi (G, M ) ∼ = ExtiG (k, M ). This suggests a natural generalization of Theorem 4.5 would be to consider ExtiΣd (Y λ , Y μ ) ∼ = Hi (Σd , Y μ ⊗ Y λ ). Already the i = 0 case of this problem is extremely diﬃcult. Indeed knowing the dimension of HomkΣd (Y λ , Y μ ) for all λ, μ d is equivalent [5, Proposition 9.2.1] to knowing the decomposition matrix for the Schur algebra S(d, d), and thus contains more information than computing the decomposition matrix for the symmetric group, a notoriously intractable problem. While computing the actual dimensions is beyond reach, Gill managed to prove: Theorem 4.7. [14, Theorem 4.3] Let λ, μ d. Then: dimk HomkΣd (Y λ , Y μ ) ≤ dimk HomkΣpd (Y pλ , Y pμ ). Mackey’s theorem easily implies that Y λ ⊗Y μ is a direct sum of Young modules. The proof of Theorem 4.7 uses fact that dimk HomkΣd (Y λ , Y μ ) is the number of summands in Y λ ⊗ Y μ which have a trivial submodule. Since Y λ ⊆ M λ it is clear that HomkΣd (k, Y λ ) is at most one-dimensional. The λ for which it is nonzero are known (see [5, Proposition 12.1.1] for example), and these partitions are preserved under twisting. Then the following key theorem from [14] is used to complete the proof. Theorem 4.8. [14, Thoerem 3.6] Let λ, μ, τ d. Then: [Y λ ⊗ Y μ : Y τ ] = [Y pλ ⊗ Y pμ : Y pτ ]. This theorem is proved numerically by counting multiplicities, but looks like it should come from some explicit twist map! Several obvious questions arise:

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Problem 4.9. Theorem 4.7 implies a sequence of injections 2

2

0 → HomkΣd (Y λ , Y μ ) → HomkΣpd (Y pλ , Y pμ ) → HomkΣp2 d (Y p λ , Y p μ ) → · · · . Does this sequence stabilize? How many twists does it take to do so? Problem 4.10. Can one ﬁnd an explicit embedding of HomkΣd (Y λ , Y μ ) into HomkΣpd (Y pλ , Y pμ )? Problem 4.11. Can Theorem 4.7 be extended to ExtikΣd (Y λ , Y μ ) for i > 0, perhaps with more twists required as i grows in the spirit of Theorem 4.5? 5. Specht modules The Specht modules S λ are perhaps the most well-studied among all kΣd modules. They are a complete set of irreducible CΣd modules, but they are deﬁned over any ﬁeld and are not well understood over k. For example only quite recently was it proven which remain irreducible over k [11]. Computing the homomorphism space HomkΣd (S λ , S μ ) is an active area of research. Cohomology Hi (Σd , S λ ) was worked out in degree i = 0 more than thirty years ago in [21, 24.4], but the i = 1 case remains open. Recent results for Specht modules involve both twisting λ → pλ and λ → λ + pa τ . 5.1. Homomorphisms between Specht modules and decomposable Specht modules. There is quite a large literature on homomorphisms between Specht modules, for example Carter-Payne maps, row removal theorems, etc. When p > 2 it is known that S λ is indecomposable and HomkΣd (S λ , S λ ) ∼ = k. In 1980 r Murphy [33] analyzed the hook Specht modules S (d−r,1 ) in characteristic p = 2 and discovered they can have arbitrarily many indecomposable summands, so the dimension of HomkΣd (S λ , S λ ) can be arbitrarily large. Only in 2011 were such homomorphism spaces with dimension larger than one discovered in odd characteristic [6], [30]. Dodge’s examples are found in Rouquier blocks while Lyle ﬁnds explicitly examples with dimension two, then uses row and column removal theorems to get arbitrary dimension. In line with the theme of this paper we observe that the examples from [30, Theorem 1.2] are of the form: (5.1)

HomkΣd+3ap (S λ+a(p,p,p) , S μ+a(p,p,p) ).

Lyle suspects the spaces in 5.1 are all two-dimensional but does not prove this. If so it would give another example of the λ → λ + pr τ twisting. However she constructs the maps individually for each choice of a rather than, for example, by “twisting” the a = 1 case, so this is somewhat speculative at this point. We collect Murphy’s results below. Recall from [21, 6.7] that S λ ⊗sgn ∼ = (S λ )∗ . Since the sign representation is trivial in characteristic two, the assumption d ≥ 2r below does not impose a real restriction, all possible hooks are handled. Theorem 5.1. Let p = 2 and assume d ≥ 2r. r r (1) If d is even then S (d−r,1 ) is indecomposable and dim HomkΣd (S (d−r,1 ) , r S (d−r,1 ) ) = 1. r r (2) If d is odd and r is even then dim HomkΣd (S (d−r,1 ) , S (d−r,1 ) ) = r/2. r r (3) If d is odd and r is odd then dim HomkΣd (S (d−r,1 ) , S (d−r,1 ) ) = (r + 1)/2.

193 7

FROBENIUS TWISTS r

(4) If d is odd then S (d−r,1 ) is indecomposable if and only if d − r − 1 ≡ 0 mod 2L where 2L−1 ≤ r < 2L . From Theorem 5.1 we can extract the following twisting result: Proposition 5.2. Let p = 2 and d ≥ 2r. Then: r

r

r

r

(1) dim HomkΣd (S (d−r,1 ) , S (d−r,1 ) ) = dim HomkΣd (S (d−r+2,1 ) , S (d−r+2,1 ) ). r (2) Suppose 2L−1 ≤ r < 2L . Then S (d−r,1 ) is indecomposable if and only if L r S (d−r+2 ,1 ) is. Since a module is indecomposable if and only if its endomorphism algebra is local, we conclude from Proposition 5.2(2) that the vector space isomorphisms in Proposition 5.2(1) are not, in general, algebra isomorphisms. In [7], Dodge and Fayers discovered new inﬁnite series of decomposable Specht modules in characteristic two, the ﬁrst new examples since Murphy’s 1980 paper. Again in their series we see the twisting λ → λ + pr τ occurring. For example a special case of Theorem 3.1 in [7] is: Proposition 5.3. Let p = 2. Then S (4+4n,3,1,1) is decomposable for n ≥ 0. We can ask much more generally: Problem 5.4. Find general theorems relating HomkΣd (S λ , S μ ) with the space a a HomkΣd (S λ+(p ) , S μ+(p ) ). More generally, inspired by Lyle’s work, we could ask for homomorphism results relating λ + pr τ and μ + pr τ for more general τ. The work in [19] mentioned in Section 2 may be relevant here for Problem 5.4 As for comparing HomkΣd (S λ , S μ ) with HomkΣd (S pλ , S pμ ) in hopes of a result like Theorem 4.7, the following example suggests some caution. The computations were done in GAP4 [13] using code written by Matthew Fayers. Example 5.5. Let p = 3. Then: 6

dim HomkΣ9 (S (7,1,1) , S (3,1 ) ) dim HomkΣ27 (S dim HomkΣ81 (S

(21,3,3)

(63,9,9)

,S

,S

(9,36 )

(27,96 )

=

0

)

=

1

)

=

0.

5.2. Generic cohomology for Specht modules. We recently proved a generic cohomology type result for Specht modules. The proof proceeds by translating the problem to GLn (k) using the result of Kleshchev and Nakano [26, 6.3(b)]that: (5.2)

Hi (Σd , S λ ) ∼ = ExtiGLd (k) (H0 (d), H0 (λ), 0 ≤ i ≤ 2p − 4.

Using extensive knowledge on the structure of H0 (d) worked out by Doty [9] together with knowledge of cohomology for the Borel subgroup B and its Frobenius kernel Br , we applied the Lyndon-Hochschild-Serre spectral sequence to obtain: Theorem 5.6. Let p ≥ 3 and λ d. Then 2 H1 (Σpd , S pλ ) ∼ = H1 (Σp2 d , S p λ ).

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Theorem 5.6 can be interpreted as a generic cohomology theorem for Specht modules in degree one. We remark that the same result holds for H0 (Σd , S λ ), although somewhat trivially as HomkΣpd (k, S pλ ) is always zero unless λ = (d). It is natural to ask if this extends to a more general theorem like Theorem 4.5, that is: Problem 5.7. Fix i > 0. Is there a constant c(i) such that for any d with λ d and any a ≥ c(i) that Hi (Σpa d , S p λ ) ∼ = Hi (Σpa+1 d , S p a

a+1

λ

)?

As in the case of the Young module cohomology, the proof of Theorem 5.6 leaves one unable to produce an explicit map, so we can ask: Problem 5.8. Given an element 0 → S pλ → M → k → 0 in H1 (Σpd , S pλ ), 2 can one explicitly construct an extension of S p λ by k realizing the isomorphism in Theorem 5.6? We also proved a generic cohomology result of the other variety. Namely: Theorem 5.9. Let λ d and pr > d. Then: r

H1 (Σd , S λ ) = H1 (Σd+pr , S λ+(p ) ). Finally we ask for stronger results like that of Theorem 5.9. Problem 5.10. Let λ d and μ c. Can one ﬁnd more results that relate the r cohomology Hi (Σd , S λ ) and Hi (Σd+cpr , S λ+p μ )? We end this section by mentioning a diﬀerent sort of stability result that holds for H0 (Σd , S λ ) and, in all examples we have computed, also for H1 (Σd , S λ ). For an integer t let lp (t) be the least nonnegative integer satisfying t < p lp (t) . The following is an easy consequence of Theorem 24.4 in [21]: Lemma 5.11. Suppose λ = (λ1 , λ2 , . . . , λs ) d and suppose a ≡ −1 mod plp (λ1 ) . Then ∼ H0 (Σd+a , S (a,λ1 ,λ2 ,...,λs ) ). (5.3) H0 (Σd , S λ ) = This leads to the following Problem 5.12. Does the isomorphism in (5.3) hold for Hi for any other i > 0? 6. Complexity of symmetric group modules. Some computer calculations done by the VIGRE algebra group at the University of Georgia suggest that twisting of partitions may arise in determining the complexity of Specht modules. A thorough discussion of the complexity of modules can be found in [1, Ch. 5] Recall that an indecomposable module M has complexity the smallest c = c(M ) such that the dimensions in a minimal projective resolution are bounded by a polynomial of degree c − 1. The maximum possible complexity for M is the p-rank of the defect group of its block, which for the symmetric group is just the p-weight w of the block. Determining the complexity of various kΣd modules is an active area of research. The complexity c(Y λ ) was determined in [17, 3.3.2]. It is worth remarking

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195 9

that for λ d it follows immediately from that result that c(Y pλ ) = d, the maximum possible. Very little is known on the complexity of simple modules Dλ . The paper [17] gives an answer when Dλ is completely splittable. The preprint [29] show that simple modules in Rouquier blocks of weight w < p all have complexity w. In contrast to the “twisted” Young modules Y pλ having maximal possible complexity, it seems the situation for Specht modules S λ may be somewhat reversed. Say a partition λ is p × p if both λ and λ are of the form pτ . Equivalently if the Young diagram of λ is made up of p × p blocks. The UGA VIGRE Algebra Group made the following conjecture: Conjecture 6.1 (UGA VIGRE 2 ). Let S λ be in a block B of weight w. Then the complexity of S λ is w if and only if λ is not p × p. In [16] we proved that when λ is p × p then its complexity is not maximal, by ﬁnding a natural equivalent condition for p × p in terms of the abacus display for λ, and then looking at the branching behavior of S λ . The other (surely more diﬃcult!) direction of the conjecture remains open: Problem 6.2. Resolve the other direction of Conjecture 6.1. Problem 6.3. Suppose λ is p × p of weight w. Is the complexity of S λ equal to w − 1, or can it be less than w − 1? Problem 6.3 has been resolved only in the case λ = (p, p, . . . , p) p2 . In this case the support variety was computed explicitly by Lim [28], and its dimension (which equals the complexity) is indeed p − 1. The support variety for the Specht module S (3,3,3) in characteristic three provides a motivating example in Chapter 7 of the book [2] as a small-dimensional module with a very interesting support variety. Perhaps further twisting might lower the complexity even more: Problem 6.4. One can generalize the deﬁnition of p × p in several ways. For example one obvious generalization would be to require λ be p2 × p2 . Can one say anything interesting about these situations? Perhaps the complexity drops by 9 even more in this case? For example can one determine the complexity of S (9 ) in characteristic three? 7. Generic cohomology for simple modules and twists of the Mullineux map Recall that the irreducible modules for kΣd are labeled by p-regular partitions and are denoted Dλ . However there is another indexing, by p-restricted partitions and denoted Dμ . The latter is more natural in some ways, as Dμ is the image under the Schur functor of the irreducible G module L(μ). The labellings are related by: (7.1) Dλ ⊗ sgn ∼ = Dλ . It was a longstanding open problem to determine the partition m(λ) so that Dλ ⊗ sgn ∼ = Dm(λ) . This problem was ﬁnally solved by Kleshchev in [24]. A short time later Ford and Kleshchev [12] conﬁrmed that Kleshchev’s answer agreed with the original conjecture made by Mullineux in [32]. We will ﬁnd it useful to 2 This conjecture and some discussion can be found at http://www.math.uga.edu/ ~nakano/vigre/vigre.html

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use Mullineux’s original algorithm and also a diﬀerent (but of course equivalent) description given later by Xu in [34]. Both are nicely described in [3]. It follow from (7.1) that: Dλ ∼ = Dm(λ)

(7.2)

so one can easily arrive at a version of the Mullineux bijection, except on p-restricted partitions; namely λ → m(λ ) . 7.1. Generic cohomology for two-part irreducibles. Suppose one wanted a generic cohomology theorem for extensions between simple kΣd modules. The simple module L(λ) corresponds to Dλ , but pλ is never p-restricted so Dpλ does not exist. Strangely though something seems to be going on with the “wrong” upper notation. For example: Proposition 7.1. Assume p > 2. Let λ = (v, u) and μ = (s, r) be partitions of d. Then 2 2 Ext1kΣpd (Dpλ , Dpμ ) ∼ = Ext1kΣp2 d (Dp λ , Dp μ ). Proof. Assume u ≥ r without loss of generality. All the extensions between two-part simple modules were worked out by Kleshchev and Sheth in[27] (but see the Corrigendum [25]). In their notation we have pv − pu + 1 = 1 + i≥1 ai pi and the condition for the Ext group to be nonzero is that pu − pr = (p − ai )pi for some i such that ai > 0 and either ai+1 < p − 1 or u < pi+1 . This condition is clearly equivalent to the corresponding one for p2 v − p2 u + 1 and p2 u − p2 r. We remark that Proposition 7.1 requires the additional twist before the cohomology stabilizes. For example when p = 3 one can use Kleshchev-Sheth’s result to compute: Ext1Σ29 (D(20,9) , D(26,3) ) ∼ = 1 Ext (D(60,27) , D(78,9) ) ∼ =

k 0.

Σ87

Of course this suggests the following: Conjecture 7.2. Let λ, μ d. Then: Ext1kΣpd (Dpλ , Dpμ ) ∼ = Ext1kΣp2 d (Dp λ , Dp μ ). 2

2

Once again we have a vector space isomorphism in cohomology (Proposition 7.1) with no module homomorphisms realizing it! 7.2. Mullineux map and twists. The labelling of irreducibles Dμ by prestricted partitions seems more natural when comparing with GLn (k) (where actual Frobenius twists can occur). But pλ is never p-restricted, and we observed 2 above a relationship between Dpλ and Dp λ . Using (7.2) suggests some relationship between m(pλ) and m(p2 λ) . This led us to a strictly combinatorial question, namely is there any relation between twisting and the Mullineux map? And then of course given any such relation, is there a representation-theoretic interpretation? We have found a large class of partitions that have interesting behavior here. For example if λ = (λ1 , λ2 , . . . , λs ) d is a partition with distinct parts, deﬁne

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197 11

ˆ = (λp−1 , λp−1 , . . . , λp−1 ) (p − 1)d. λ s 1 2 Proposition 7.3. Suppose λ = (λ1 , λ2 , . . . , λs ) d has distinct parts. Then ˆ = (p − 1)λ. m(λ) Proof. Consider the p-regular version of Xu’s algorithm from [34] (nicely ˆ X using [3, Def 3.5] we obtain described in [3]). If we apply it to calculate λ j1 = j2 = · · · = jp−1 = s. At this point in the algorithm the ﬁrst column (consisting where λ ˆ and what remains is λ of s(p − 1) nodes) will have been removed from λ, denotes λ with its ﬁrst column removed. One can now apply induction using [3, Proposition 3.6(2 )] or just continue with the algorithm to obtain (p − 1)λ. Corollary 7.4. Let λ d have distinct parts. Then m(pμ) = pm(μ) ˆ for both μ = (p − 1)λ and μ = λ. Proof. By Proposition 7.3 we have: m((p)(p − 1)λ) = pλ ˆ = pλ =

pm((p − 1)λ)

and ˆ m(pλ)

= m(pλ) = p(p − 1)λ ˆ = pm(λ).

The appearance above of p(p − 1)λ for λ having distinct parts is reminiscent of the twist of the Steinberg weight p(p − 1)ρ from the GLn (k) theory, although we have no representation-theoretic interpretation at this time. The examples in Corollary 7.4 are not the only ones where m(pμ) = pm(μ) although other examples seem to be rare. For example if p = 5 and d = 20 then Corollary 7.4 yields all μ 20 where m(5μ) = 5m(μ). On the other hand for p = 5 and λ = (3, 3) we have m(15, 15) = (10, 10, 10) = 5m(3, 3), although λ does not have distinct parts.. This brings us to: Problem 7.5. Classify all λ such that m(pλ) = pm(λ) and give a representationtheoretic interpretation of the answer. The subset of such λ arising in Corollary 7.4 is certainly closed under twisting, as are all other examples we have computed, so we conjecture: Conjecture 7.6. Suppose m(pλ) = pm(λ). Then m(p2 λ) = pm(pλ). More generally we can ask: Problem 7.7. Classify all λ such that m(pλ) = pτ for some τ.

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For example when p = 5 and λ = (6, 6, 4) we have: m(30, 30, 20) = (203 , 54 ) = 5m(10, 3, 3). Even for partitions without such a nice relationship, there still seems to be some intriguing behavior among the m(pa λ) for various a. For example we will now derive a sort of Steinberg tensor product theorem for Mullineux maps. First deﬁne τn = m(1n ), so the trivial kΣn module is Dτn . It is well known that τn = (p − 1, p − 1, p − 1, · · · , p − 1, a). Lemma 7.8. Suppose λ = (λ1 , λ2 , . . . , λs ) is a partition with distinct parts. Then: m(pλ) = τpλ1 + τpλ2 + · · · + τpλs . For example if λ = (4, 2, 1) and p = 5 then m(5λ) = (12, 9, 6, 4, 4) = (45 ) + (4, 4, 2) + (4, 1) = τ20 + τ10 + τ5 . Proof. The proof is by induction on s where the case s = 1 is just the deﬁnition of τn . We will use the original algorithm of Mullineux, described and proved in [12, p. 272]. Since the parts of λ are distinct, then all the rim p-hooks removed in determining the Mullineux symbol Gp (pλ) are horizontal. Thus we determine the Mullineux symbol Gp (m(pλ)) = (7.3) sp s(p − 1)

··· ···

sp s(p − 1)

(s − 1)p (s − 1)(p − 1)

··· ···

(s − 1)p (s − 1)(p − 1)

··· ···

p p−1

··· ···

p p−1

where the ﬁrst column in (7.3) occurs λs times, the second occurs λs−1 − λs times, the third λs−2 − λs−1 , etc. So the last column occurs λ1 − λ2 times. Notice sp then that removing the columns we obtain the Mullineux symbol for s(p − 1) p(λ1 − λs , λ2 − λs , . . . , λs−1 − λs ). So we apply induction and the result follows. Using the result above and drawing a diagram with the appropriate τpλi and τp2 λi , the following corollary is immediate: Corollary 7.9. Suppose λ has distinct parts. Then: ˆ m(p2 λ) − m(pλ) = pλ. Using Proposition 7.3 one can rewrite this as: m(p2 λ) − m(pλ) = pm((p − 1)λ). For arbitrary λ with repeated parts we conjecture a weaker form of stability after multiple “twists:” Conjecture 7.10. Let λ d. Then there exist 1 ≤ a < b such that: m(pb λ) = m(pa λ) + pa τ. As an example of Conjecture 7.10 needing multiple twists consider the following: Example 7.11. Let p = 7 and λ = (292 , 24, 42 , 33 , 2, 1). Then: m(75 λ) − m(7λ) = 7(1238405 , 96005 , 54004 , 38405 , 8006 , 4006 ). If 1 ≤ x < y < 5 then m(7y λ) − m(7x λ) is not of the form 7τ , i.e. this stability really requires at least ﬁve twists.

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Finally we remark that the results above on the Mullineux map did not depend on p being prime, and any possible representation-theoretic interpretations might hold true for the Hecke algebra of type A at an eth root of unity. It seems ﬁtting to close with a completely general (and quite possibly absurd) question: Problem 7.12. Can one say anything interesting about how the structure of the principal block B0 (kΣpd ) is reﬂected inside B0 (kΣp2 d )? For example Theorem 4.7 is a statement about Cartan invariants. References 1. D. J. Benson, Representations and cohomology. II, second ed., Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1998, Cohomology of groups and modules. MR1634407 (99f:20001b) 2. D.J. Benson, Representations of elementary abelian p-groups, preprint, 2010. 3. Christine Bessenrodt, Jørn B. Olsson, and Maozhi Xu, On properties of the Mullineux map with an application to Schur modules, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 443–459. MR1684242 (2000c:20024) 4. Edward Cline, Brian Parshall, Leonard Scott, and Wilberd van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), no. 2, 143–163. MR0439856 (55 #12737) 5. Fred R. Cohen, David J. Hemmer, and Daniel K. Nakano, On the cohomology of Young modules for the symmetric group, Advances in Mathematics 224 (2010), 1419–1461. 6. Craig J. Dodge, Large dimension homomorphism spaces between Specht modules for symmetric groups, J. Pure Appl. Algebra 215 (2011), no. 12, 2949–2956. MR2811577 7. Craig J. Dodge and Matthew Fayers, Some new decomposable Specht modules, preprint, 2011. 8. Stephen Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), no. 1, 39–60. 9. Stephen R. Doty, The submodule structure of certain Weyl modules for groups of type An , J. Algebra 95 (1985), no. 2, 373–383. MR801273 (86j:20035) 10. Karin Erdmann, Young modules for symmetric groups, J. Aust. Math. Soc. 71 (2001), no. 2, 201–210, Special issue on group theory. MR1847190 (2002f:20015) 11. Matthew Fayers, Irreducible Specht modules for Hecke algebras of type A., Advances in Mathematics 193 (2005), 438–452. 12. Ben Ford and Alexander S. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 (1997), no. 2, 267–308. MR1477629 (98k:20015a) 13. The GAP Group, Gap – groups, algorithms, and programming, version 4.4.12, 2008. 14. Christopher Gill, Tensor product of young modules, Vol. 366 (September 2012), pp. 12–34, http://dx.doi:10.1016/j.jalgebra.2012.05.014. 15. Christopher C. Gill, Tensor products, trivial source modules and related algebras., Ph.D. thesis, University of Oxford, 2011. 16. David J. Hemmer, The complexity of certain Specht modules for the symmetric group, J. Algebraic Combin. 30 (2009), no. 4, 421–427. MR2563134 17. David J. Hemmer and Daniel K. Nakano, Support varieties for modules over symmetric groups, J. Algebra 254 (2002), no. 2, 422–440. 18. Anne Henke, On p-Kostka numbers and Young modules, European J. Combin. 26 (2005), no. 6, 923–942. MR2143202 (2006e:20021) 19. Anne Henke and Steﬀen Koenig, Relating polynomial GL(n)-representations in diﬀerent degrees, J. Reine Angew. Math. 551 (2002), 219–235. MR1932179 (2003k:20066) 20. Anne E. Henke, Schur subalgebras and an application to the symmetric group, J. Algebra 233 (2000), no. 1, 342–362. MR1793600 (2001m:20063) 21. Gordon James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, no. 682, Springer-Verlag, 1978. 22. Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. 23. Jens C. Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, 2003.

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24. A. S. Kleshchev, Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. (2) 54 (1996), no. 1, 25–38. MR1395065 (96m:20019c) 25. A. S. Kleshchev and J. Sheth, Corrigendum: “On extensions of simple modules over symmetric and algebraic groups” [J. Algebra 221 (1999), no. 2, 705–722; MR1728406 (2001f:20091)], J. Algebra 238 (2001), no. 2, 843–844. MR1823787 26. Alexander S. Kleshchev and Daniel K. Nakano, On comparing the cohomology of general linear and symmetric groups, Paciﬁc J. Math. 201 (2001), no. 2, 339–355. 27. Alexander S. Kleshchev and J. Sheth, On extensions of simple modules over symmetric and algebraic groups, J. Algebra 221 (1999), no. 2, 705–722. 28. Kay Jin Lim, The varieties for some Specht modules, J. Algebra 321 (2009), no. 8, 2287–2301. MR2501521 (2010c:20008) 29. Kay Jin Lim and Kai Meng Tan, The complexity of some simple modules of the symmetric groups, preprint (2011). 30. Sin´ ead Lyle, Large dimensional homomorphism spaces between Weyl modules and Specht modules, preprint (2011). 31. Stuart Martin, Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol. 112, Cambridge University Press, Cambridge, 1993. 32. G. Mullineux, Bijections of p-regular partitions and p-modular irreducibles of the symmetric groups, J. London Math. Soc. (2) 20 (1979), no. 1, 60–66. MR545202 (80j:20016) 33. Gwendolen Murphy, On decomposability of some Specht modules for symmetric groups, J. Algebra 66 (1980), no. 1, 156–168. MR591250 (81m:20019) 34. Maozhi Xu, On Mullineux’ conjecture in the representation theory of symmetric groups, Comm. Algebra 25 (1997), no. 6, 1797–1803. MR1446130 (98c:20027) Department of Mathematics, University at Buffalo, SUNY, 244 Mathematics Building, Buffalo, NY 14260, USA E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 0, XXXX 2012

The generalized Kac-Wakimoto conjecture and support varieties for the Lie superalgebra osp(m|2n) Jonathan Kujawa Abstract. Atypicality is a fundamental combinatorial invariant for simple supermodules of a basic Lie superalgebra. Boe, Nakano, and the author gave a conjectural geometric interpretation of atypicality via support varieties. Inspired by low dimensional topology, Geer, Patureau-Mirand, and the author gave a generalization of the Kac-Wakimoto atypicality conjecture. We prove both of these conjectures for the Lie superalgebra osp(m|2n).

1. Introduction 1.1. Let g = g¯0 ⊕ g¯1 be a basic classical Lie superalgebra over the complex numbers. An important category of g-supermodules is the category F of ﬁnite dimensional integrable g-supermodules. Starting with the work of Kac [12, 13], the category F has been the object of investigation for more than 30 years by numerous researchers. Of particular interest is the simple supermodules in F. Most eﬀorts have focused on obtaining character formulas (to mention only a few of the more prominent papers in the area, see [5, 6, 11, 16]). Recently two new lines of investigation have developed. The category F shares a number of features with the modular representations of ﬁnite groups and, more generally, ﬁnite group schemes. For example, F is not semisimple, has enough projectives, and usually projectives and injectives in F coincide. Motived by the successful use of cohomology and support varieties in the ﬁnite group scheme setting, the authors of [3] began an investigation of F using analogous tools. They conjectured that support varieties provide a geometric interpretation of the combinatorial invariant known as atypicality. Namely, given a supermodule M in F, let V(g,g¯0 ) (M ) denote the support variety associated to M as in [3]. Given a simple g-supermodule L(λ) of highest weight λ, let atyp(λ) denote the atypicality of λ. Precise deﬁnitions can be found in the body of this paper. The following “atypicality conjecture” is given in [3, Conjecture 7.2.1]1 .

2010 Mathematics Subject Classiﬁcation. Primary: 17B56, 17B10. Research of the author was partially supported by NSF grant DMS-0734226 and NSA grant H98230-11-1-0127. 1 More accurately, there the support variety of the detecting algebra is used but conjecturally the dimension of that variety coincides with the one used here. c c 2012 American Mathematical Society XXXX

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Conjecture 1.1. Let g be a basic classical Lie superalgebra and let L(λ) be a simple supermodule in F. Then, dim V(g,g¯0 ) (L(λ)) = atyp(λ). This conjecture was proven for gl(m|n) in [4]. In a diﬀerent direction, the authors of [9] were motivated by questions in low dimensional topology to introduce modiﬁed trace and dimension functions for F. Given a supermodule M = M¯0 ⊕ M¯1 in F the appropriate analogue of dimension is superdimension: sdim(M ) := dim M¯0 − dim M¯1 . Plainly the superdimension can equal zero. Let def(g) denote the maximum possible value of atyp(λ) as λ ranges over the highest weights of simple supermodules in F. The following conjecture of Kac and Wakimoto [14, Conjecture 3.1] makes precise when the superdimension of a simple supermodule vanishes. Conjecture 1.2. Let L(λ) be a simple g-supermodule in F, then atyp(λ) = def(g) if and only if sdim(L(λ)) = 0. The authors of [9] introduce modiﬁed dimension functions for F and prove that they are a natural replacement for the superdimension when the superdimension vanishes. In particular, they provide the following generalization of the Kac-Wakimoto conjecture [9, Conjecture 6.3.2]. Given a supermodule M , let IM denote the full subcategory of all supermodules which appear as a direct summand of M ⊗ X for some supermodule X in F. Conjecture 1.3. Let g be a basic classical Lie superalgebra and let L(λ) be a simple g-supermodule. Then L(λ) admits an ambidextrous trace and we can deﬁne a modiﬁed dimension function dL(λ) on IL(λ) . If L(μ) is another simple supermodule with atyp(μ) ≤ atyp(λ), then L(μ) is an object of IL(λ) , and atyp(μ) = atyp(λ) if and only if dL(λ) (L(μ)) = 0. If L(λ) is the trivial supermodule, then IL(λ) = F, atyp(λ) = def(g), and dL(λ) = sdim. In this way the generalized Kac-Wakimoto conjecture specializes to the ordinary Kac-Wakimoto conjecture. Recently Serganova proved the ordinary KacWakimoto conjecture for gl(m|n) and osp(m|2n) and the generalized Kac-Wakimoto conjecture for gl(m|n) [15]. 1.2. In the present paper we consider the case when g equals gl(m|n) or osp(m|2n). That is, the Lie superalgebras of type ABCD in the Kac classiﬁcation [12]. Taken together these are the inﬁnite families of basic classical Lie superalgebras in the Kac classiﬁcation. The results we prove are new for osp(m|2n) but, as we described above, are known for gl(m|n) by the work of Serganova [15] and Boe, Kujawa, and Nakano [4]. However the proofs work equally well for gl(m|n) so we include them. Perhaps the most interesting case which still remains is the type Q Lie superalgebras. Although not basic, they have a notion of atypicality and the geometric and topological viewpoints apply. In Section 3 we prove the generalized Kac-Wakimoto conjecture for F. It is worth remarking that this has the following purely representation theoretic corollary. Let L(λ) and L(μ) be simple g-supermodules with atyp(λ) = atyp(μ). Then

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there is a supermodule X in F such that L(λ) is a direct summand of L(μ) ⊗ X. This in turn implies the support variety of all simple supermodules of the same atypicality coincide (see Theorem 4.1). Similarly the complexity of the simple supermodules of the same atypicality coincide. The recent calculation of complexity for the simple supermodules for gl(m|n) in [1] depends crucially on this result. In Section 4 we compute the support varieties of the simple g-supermodules in F. We show that if L(λ) is a simple g-supermodule, then V(g,g¯0 ) (L(λ)) ∼ = Aatyp(λ) . In particular this veriﬁes Conjecture 1.1. Note that the support variety is canonically deﬁned for any object of F. Thus the above result justiﬁes the deﬁnition of the atypicality of a general supermodule via atyp(M ) = dim V(g,g¯0 ) (M ). 1.3. Acknowledgements. The author would like to recognize his collaborations with Boe and Nakano, and Geer and Patureau-Mirand. The results of this paper are a direct outgrowth of that work. The author is thankful for many stimulating conversations. The author would also like to acknowledge Serganova for helpful discussions. In particular, several key ingredients used here were developed by Serganova, Duﬂo-Serganova, and Gruson-Serganova. Finally, the author would like to recognize the stimulating environment provided by the Southeast Lie Theory conference series.

2. Preliminaries 2.1. All vector spaces will be over the complex numbers, C, and ﬁnite dimensional unless otherwise stated. In most cases the vector spaces will have a Z2 -grading, V = V¯0 ⊕ V¯1 , and we will write v¯ ∈ Z2 for the degree of a homogeneous element v ∈ V . We call an element v ∈ V even (resp. odd ) if v¯ = ¯0 (resp. v¯ = ¯1). Let g = g¯0 ⊕ g¯1 denote one of the Lie superalgebras gl(m|n), osp(2m|2n), and osp(2m + 1|2n) as deﬁned in [12]. In each case g¯0 is reductive as a Lie algebra and so g is classical in the sense of [3]. Furthermore, in each case we may deﬁne a bilinear form ( , ) : g ⊗ g → C by (x, y) = str(xy), where str is the supertrace. This deﬁnes a nondegenerate, supersymmetric, invariant, even bilinear form and so by deﬁnition g is basic. Fix a choice of Cartan subalgebra h ⊂ g¯0 as in [11]. The bilinear form on g induces a bilinear form on the dual of the Cartan subalgebra, h∗ , which we again denote by ( , ). In particular, we may choose a basis for h∗ , ε1 , . . . , εm , δ1 , . . . , δn , on which (εi , εj ) = δi,j ,

(εi , δj ) = 0,

(δi , δj ) = −δi,j .

With respect to our choice of Cartan subalgebra we have a decomposition of g into root spaces. Each root space is one dimensional and spanned by a homogenous vector. Consequently, we may deﬁne the parity of a root to be the parity of the corresponding root space. We write Φ (resp. Φ¯0 and Φ¯1 ) for the set of roots (resp. set of even and odd roots).

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We can explicitly describe the root systems as follows. If g = gl(m|n), then the roots are Φ¯0 = {εi − εj | i = j} ∪ {δi − δj | i = j} , Φ¯1 = {±(εi − δj )} . If g = osp(2m|2n), then the roots are Φ¯0 = {±εi ± εj | i = j} ∪ {±δi ± δj | i = j} ∪ {2δi } , Φ¯1 = {±εi ± δj } . If g = osp(2m + 1|2n), then the roots are Φ¯0 = {±εi ± εj | i = j} ∪ {±δi ± δj | i = j} ∪ {±εi } ∪ {2δi } , Φ¯1 = {±εi ± δj } . In each case the subscripts on the epsilons are from among 1, . . . , m and the subscripts on the deltas are from among 1, . . . , n. We call a ﬁnite dimensional g-supermodule integrable if all its weights lie in the Z-span of ε1 , . . . , εm , δ1 , . . . , δn . Let F = F(g) denote the category of all integrable ﬁnite dimensional g-supermodules and all (not necessarily grading preserving) gsupermodule homomorphisms. We should remark that our F is a full subcategory of the category F(g, g¯0 ) considered in [3]. However, the projective cover in F(g, g¯0 ) of any object of F lies in F. This implies that projective resolutions, cohomology, support varieties, etc. in the two categories coincide. By g-supermodule we will always mean an object in F unless otherwise stated. Let F¯0 denote the category of all ﬁnite dimensional integrable g-supermodules and all grading preserving gsupermodule homomorphisms. In [15] Serganova considers a full subcategory of F¯0 . However, the parity change functor allows us to apply her results without loss to F¯0 and F. We ﬁx the same choice of Borel subalgebra b containing h as in [11] and deﬁne ρ to be the half sum of the positive even roots minus the half sum of the positive odd roots. See just before [11, Corollary 3] for a list of the simple roots and the ρ corresponding to this choice of b. Then the simple objects of F are parameterized by highest weight with respect to our choice of h and b. We write L(λ) for the simple supermodule of highest weight λ ∈ h∗ . By deﬁnition, we call λ ∈ h∗ a dominant integral weight if it is the highest weight of some simple supermodule in F. For an explicit description of the dominant integral highest weights with respect to these choices, see [11, Corollary 3]. The maximal number of pairwise orthogonal isotropic roots with respect to the bilinear form on h∗ is the defect of g. We write def(g) for the defect of g. In our case def(gl(m|n)) = def(osp(2m|2n)) = def(osp(2m + 1|2n)) = min(m, n). The atypicality of the simple supermodule L(λ) is deﬁned to be the maximal number of pairwise orthogonal isotropic roots which are also orthogonal to λ + ρ. We write atyp(λ) for this number. We write (2.1) A(λ) = α1 , . . . , αatyp(λ) , for a ﬁxed choice of such roots. Although the set A(λ) is not unique, it is known that the size of the set is well deﬁned and, furthermore, does not depend on our choice

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of Cartan or Borel subalgebras. Consequently, it makes sense to write atyp(L) for a simple supermodule L. 2.2. Given a g-supermodule M = M¯0 ⊕ M¯1 the superdimension of M is given by sdim(M ) = dim(M¯0 ) − dim(M¯1 ). Note that by deﬁnition we have atyp(L) ≤ def(g) for any simple g-supermodule L. The following conjecture of Kac and Wakimoto [14, Conjecture 3.1] makes precise when we in fact have equality. Conjecture 2.1. Let L be a simple g-supermodule in F, then atyp(L) = def(g) if and only if sdim(L) = 0. While investigating generalized trace and dimension functions on nonsemisimple tensor categories, Geer, Patureau-Mirand and the author gave a generalized Kac-Wakimoto conjecture [9, Conjecture 6.3.2] (see Conjecture 2.3). Serganova recently proved the ordinary Kac-Wakimoto conjecture for gl(m|n) and osp(m|2n) and the generalized Kac-Wakimoto conjecture for gl(m|n) [15]. Our ﬁrst goal is to prove the generalized Kac-Wakimoto conjecture for osp(m|2n). 2.3. In [9] generalized trace and dimension functions on ribbon categories were introduced. We only provide the deﬁnitions and results we need and refer the reader to loc. cit. for additional details. In order to be mathematically correct we work in F¯0 for the remainder of this section and Section 3 so as to have a ribbon category. However simple arguments using the parity change functor show the results also hold in F. We leave this to the interested reader. For any object V in F¯0 , let V ∗ denote the dual supermodule. For any V, W in F¯0 , let V ⊗ W denote the tensor product supermodule (where the tensor product is over C). For any object V in F¯0 , let evV : V ∗ ⊗ V → C be the evaluation morphism given by f ⊗ x → f (x) and let coevV : C → V ⊗ V ∗ be the coevaluation morphism given by 1 → ni=1 vi ⊗ fi , where v1 , . . . , vn is a homogeneous basis for V and where fi ∈ V ∗ is deﬁned by fi (vj ) = δi,j . Deﬁne the graded “ﬂip” map cV,W : V ⊗ W → W ⊗ V by v ⊗ w → (−1)v¯·w¯ w ⊗ v. Finally, for all V in F¯0 we set the “twist map” θV : V → V to be the identity. The above data makes F¯0 into a ribbon category. For short we write evV = evV ◦cV,V ∗ and coevV = cV,V ∗ ◦ coevV . Fix a pair of objects V and W in F¯0 and an endomorphism f of V ⊗ W . To such objects and morphisms we use the ribbon category structure to deﬁne the following morphisms: trL (f ) = (evV ⊗ IdW ) ◦ (IdV ∗ ⊗f ) ◦ (coevV ⊗ IdW ) ∈ EndF¯0 (W ), and

trR (f ) = (IdV ⊗ evW ) ◦ (f ⊗ IdW ∗ ) ◦ (IdV ⊗ coevW ) ∈ EndF¯0 (V ). Given an object J in F¯0 , the ideal IJ is the full subcategory of all objects which appear as direct summands of J ⊗ X for some object X in F¯0 . More precisely, M is an object of IJ if and only if there is an object X in F¯0 and morphisms α : M → J ⊗ X and β : J ⊗ X → M with β ◦ α = IdM . For example, if P is a projective supermodule in F¯0 , then IP is precisely the full subcategory of projective objects. For short we denote this particular ideal by Proj. If IJ is an ideal in F¯0 then a trace on IJ is a family of linear functions t = {tV : EndF¯0 (V ) → C}

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where V runs over all objects of IJ and such that following two conditions hold. (1) If U ∈ IJ and W is an object of F¯0 , then for any f ∈ EndF¯0 (U ⊗ W ) we have (2.2)

tU⊗W (f ) = tU (trR (f )) . (2) If U, V ∈ IJ then for any morphisms f : V → U and g : U → V in F¯0 we have

(2.3)

tV (g ◦ f ) = tU (f ◦ g).

For V an object of F¯0 , we say a linear function t : EndF¯0 (V ) → C is an ambidextrous trace on V if for all f ∈ EndF¯0 (V ⊗ V ) we have t(trL (f )) = t(trR (f )). For short we call a supermodule ambidextrous if it is simple and if it admits a nonzero ambidextrous trace. The following theorem summarizes several results from [9, Section 3.3] as they apply here. Theorem 2.2. Let L be a simple g-supermodule. If IL admits a trace then the map tL is an ambidextrous trace on L. Conversely, an ambidextrous trace on L extends uniquely to a trace on IL . Furthermore, the trace on IL and the ambidextrous trace on L are unique up to multiplication by an element of C. Given a trace on IJ , {tV }V ∈IJ , we deﬁne the modiﬁed dimension function on objects of IJ , dJ : Ob(IJ ) → C, by taking the modiﬁed trace of the identity morphism: dJ (V ) = tV (IdV ). We can now state the generalized Kac-Wakimoto conjecture [9, Conjecture 6.3.2]. Conjecture 2.3. Let g be a basic classical Lie superalgebra and let J be a simple g-supermodule. Then J is ambidextrous and if L is another simple supermodule with atyp(L) ≤ atyp(J), then L is an object of IJ and atyp(L) = atyp(J) if and only if dJ (L) = 0. This conjecture was proven for gl(m|n) by Serganova in [15]. 3. Generalized Kac-Wakimoto Conjecture 3.1. We ﬁrst prove that every simple g-supermodule in F¯0 is ambidextrous. To do so we use the ﬁbre functor introduced by Duﬂo and Serganova [7] and further developed by Serganova [15]. We ﬁrst summarize the results of theirs which we require. Let G¯0 denote the connected reductive algebraic group with Lie algebra g¯0 . Let X = {x ∈ g¯1 | [x, x] = 0} . Given an element x ∈ X , the G¯0 -orbit of x contains elements of the form x1 +· · ·+xk , where xi lies in the root space gαi and α1 , . . . , αk are pairwise orthogonal, isotropic roots. It is straightforward to see that the number k depends only on the orbit and so it makes sense to deﬁne the rank of x to be k. We write rank(x) for this

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number. By deﬁnition the rank of x is among 0, 1, . . . , def(g). Using the root space decomposition of g it is not diﬃcult to see that every possible value is achieved. Given an x ∈ X , let Centg (x) denote the centralizer of x in g and set gx = Centg (x)/[x, g]. Note that gx is a Lie superalgebra and if y ∈ X with rank(x) = rank(y), then gx ∼ = gy . Furthermore, if rank(x) = k, then we have: • if g is gl(m|n), then gx is isomorphic to gl(m − k|n − k); • if g is osp(2m + 1|2n), then gx is isomorphic to osp(2(m − k) + 1|2(n − k)); • if g is osp(2m|2n), then gx is isomorphic to osp(2(m − k)|2(n − k); If x ∈ X , then in the enveloping superalgebra of g we have 0 = [x, x] = 2x2 . Hence for any supermodule M in F¯0 the linear map M → M given by the action of x squares to zero. That is, it makes sense to deﬁne Mx = Ker(x)/ Im(x). Note that Mx is naturally a gx -supermodule and the assignment M → Mx deﬁnes a functor from F¯0 to F¯0 (gx ) which is called the ﬁbre functor. We write f → fx for the functor’s action on a morphism f . Note that the ﬁbre functor is a functor of ribbon categories. 3.2. Recall that Proj is the ideal of all projective objects. By [8, Theorem 4.8.2] the ideal Proj is known to admit a nontrivial trace. We write tp = {tpV | V ∈ Proj} for this trace. We call a simple supermodule typical if it has atypicality zero. By [13, Theorem 1] if a simple supermodule is typical, then it is projective. Furthermore if T is a typical simple supermodule, since Proj admits a nontrivial trace and IT = Proj, it follows from Theorem 2.2 that tpT (IdT ) = 0. Theorem 3.1. Let g denote gl(m|n), osp(2m + 1|2n), or osp(2m|2n). Then every simple g-supermodule in F¯0 is ambidextrous. Proof. Let L be a simple supermodule in F¯0 of atypicality k. Fix x ∈ X with rank(x) = k. For any g-supermodule M let ϕx : Endg (M ) → Endgx (Mx ) denote the algebra map induced by the ﬁbre functor via ϕx (f ) = fx . By [15, Corollary 2.2] Lx is a direct sum of typical supermodules and so is projective. More generally, if M is an object of IL , then it is a direct summand of L ⊗ Y for some supermodule Y . Applying the ﬁbre functor we see that Mx is a direct summand of Lx ⊗ Yx and so is projective. Consequently it makes sense for any M in IL to deﬁne a map, tM , by the composition (3.1)

tM := tpMx ◦ϕx : EndF¯0 (M ) → C.

Since the ﬁbre functor is a functor of ribbon categories it is straightforward to verify that t = {tM | M ∈ IL } is a (possibly trivial) trace on the ideal IL . We now prove t is nontrivial. First we assume g is either gl(m|n) or osp(2m + 1|2n). By [15, Corollary 2.2], since L is a simple g-supermodule of atypicality k, we have (3.2) Lx ∼ = T ⊗ Cx (L)

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as gx -supermodules, where T is a typical simple gx -supermodule and Cx (L) is a superspace with trivial gx -action. Using (3.2) we compute tL (IdL ): tL (IdL ) = tpLx (IdL,x ) = tpLx (IdLx ) = tpT (trR (IdLx )) = tpT (IdT ) sdim (Cx (L)) . The ﬁrst equality is by the deﬁnition of t, the second by the deﬁnition of the ﬁbre functor, the third is by (2.3) and (3.2), and the last is by direct calculation. Furthermore, since T is a typical simple supermodule we have tpT (IdT ) = 0 and by [15, Theorem 2.3] we have sdim (Cx (L)) = 0. Therefore tL is nontrivial and so t is a nontrivial trace on IL and L is ambidextrous. The case when g = osp(2m|2n) is argued similarly. The only diﬀerence is that by [15, Corollary 2.2] we instead have Lx ∼ = T ⊗ Cx (L) ⊕ T σ ⊗ Cx (L) as gx -supermodules. Here T is a typical simple gx -supermodule, T σ is the typical simple supermodule obtained by twisting T by the involution σ : gx → gx given just before [15, Corollary 2.2], and Cx (L) and Cx (L) are superspaces with trivial gx -action. Using linearity and calculating as before, we have (3.3)

tL (IdL ) = tpT (IdT ) sdim (Cx (L)) + tpT σ (IdT σ ) sdim (Cx (L)) .

We now claim that tpT (IdT ) = tpT σ (IdT σ ). Deﬁne an endofunctor of F¯0 by twisting by σ: on objects the functor is given by M → M σ and is the identity on morphisms. Twisting by σ is a functor of ribbon categories and takes Proj to itself. Thus we may deﬁne a new family of maps tσ = {tσV | V ∈ Proj} on Proj by precomposing by this functor: tσM (f ) = tpM σ (f ). The fact that twisting by σ is a functor of ribbon categories implies that tσ is a trace on Proj. Using this new trace we can rewrite our claim as tpT (IdT ) = tσT (IdT ). Thus to prove our claim it suﬃces to prove that the traces tp and tσ coincide. That is, that tpV = tσV for all V in Proj. By the explicit description of σ given in [15, Section 2] there exist typical simple supermodules U for which U σ = U and for such a supermodule it is immediate that tpU = tσU . However, IU = Proj and so by Theorem 2.2 a trace on Proj is completely determined by tpU . That is, since tpU = tσU , we in fact have that tp and tσ coincide on all of Proj. In particular, tpT (IdT ) = tσT (IdT ) and so tpT (IdT ) = tpT σ (IdT σ ). Returning to (3.3), we obtain (3.4)

tL (IdL ) = tpT (IdT ) [sdim (Cx (L)) + sdim (Cx (L))] .

By [15, Theorem 2.3] we have sdim (Cx (L) ⊕ Cx (L)) = sdim (Cx (L)) + sdim (Cx (L)) = 0. Furthermore T is a simple object in Proj and so tpT (IdT ) = 0. Combining these observations with (3.4) we see that tL is nontrivial. That is, t deﬁnes a nontrivial trace on IL and L is ambidextrous. We remark that the ambidextrous trace on L given in the proof may depend on the choice of x. However, since L is simple any two traces diﬀer only by a scalar multiple. We also remark that our reduction to the typical case is inspired by Serganova’s analogous approach for gl(m|n) given in [15]. However, Serganova

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used a diﬀerent argument to prove the nontriviality of the trace on IL . Her approach uses the explicit description of the trace on typical supermodules given via supercharacters in [10]. 3.3. When k = 0, we set S0 to be a typical simple supermodule. Then IS0 = Proj and it contains every typical simple supermodule and has a nontrivial trace by [8, Theorem 4.8.2]. By [15, Lemma 6.3], for each 0 < k ≤ def(g), there exists a simple g-supermodule, Sk , of atypicality k such that every simple supermodule of atypicality k lies in ISk . By the previous theorem Sk is ambidextrous and, in particular, we may ﬁx an x ∈ X of rank k which deﬁnes a nontrivial trace on ISk . In either case we denote the trace on ISk by t = {tV | V ∈ ISk } and the corresponding dimension function by dSk . Proposition 3.2. Let 0 ≤ k ≤ def(g) and let Sk be the simple g-supermodule given above. Let L be a simple supermodule of atypicality k. Then dSk (L) = 0 and IL = IS k . Proof. Since L lies in ISk , we have IL ⊆ ISk . By [9, Theorem 4.2.1] the nonvanishing of dSk (L) implies that the ideals are equal. Thus it suﬃces to compute dSk (L). If k = 0, then this is a consquence of Theorem 2.2 and the fact that IL = Proj. If k > 0, then by the previous theorem L is ambidextrous and using the element x ﬁxed above we also have a nontrivial trace on IL . We denote this trace by t = {tV | V ∈ IL }. Recall in particular that tL (IdL ) = 0. Using the deﬁnition of t and t we have dSk (L) = tL (IdL ) = tpLx (IdL,x ) = tL (IdL ) = 0. It is worth making explicit the following representation theoretic interpretation of the previous result. Corollary 3.3. Let g denote gl(m|n) or osp(m|2n). Let L1 and L2 be two simple supermodules in F¯0 with the same atypicality. Then there are supermodules X1 and X2 in F¯0 such that L2 is a direct summand of L1 ⊗ X1 and L1 is a direct summand of L2 ⊗ X2 . By the previous theorem for each 0 ≤ k ≤ def(g), the ideal generated by a simple supermodule of atypicality k is independent of the choice of simple supermodule. Consequently, we write Ik for the ideal generated by a simple of atypicality k. In particular, I0 = Proj (as typical supermodules are projective) and Idef(g) = F¯0 (as the trivial supermodule has atypicality equal to the defect and generates the entire category). Furthermore it is not diﬃcult to see using the translation functors of [11, Sections 5-6] that for each atypicality k = 1, . . . , def(g), there is a simple supermodule of atypicality k, L, and simple supermodule of atypicality k − 1, L , so that L is an object in IL . Thus we have I0 ⊆ I1 ⊆ · · · ⊆ Idef(g) . Given x ∈ X of rank k, we write t for the trace on Ik deﬁned by (3.1) and d for the corresponding modiﬁed dimension function. Proposition 3.4. Let 0 < k ≤ def(g) and let L be a simple supermodule of atypicality strictly less than k. Then L is an object of Ik and d(L) = 0 and IL Ik .

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Proof. The fact that L is an object of Ik follows from the discussion preceeding the proposition. We now compute d(L) using the deﬁnition of t on Ik . Since L has atypicality strictly less than k we know by [15, Theorem 2.1] that Lx = 0. It is then immediate that d(L) = 0. Since L is an object of Ik , we have IL ⊆ Ik . However, by [9, Theorem 4.2.1] the vanishing of the modiﬁed dimension implies that the inclusion is strict. Combining the above results we have Conjecture 2.3. We also have the following description of the ideals deﬁned by simple objects. Theorem 3.5. If Ik denotes the ideal deﬁned by a simple supermodule of atypicality k in F¯0 , then Ik is independent of this choice. Furthermore, these ideals form the following chain of inclusions Proj = I0 I1 I2 · · · Idef(g) = F¯0 . 4. Support Varieties 4.1. Given a classical Lie superalgebra a and an object M in F(a), let V(a,a¯0 ) (M ) denote the support variety of M as deﬁned in [3] and let cF (a) (M ) the complexity of M in F(a) (i.e. the rate of growth of a minimal projective resolution of M in F). As an application of the generalized Kac-Wakimoto conjecture we see that for a simple supermodule these depend only on atypicality. Theorem 4.1. Let g denote gl(m|n) or osp(m|2n) and let L1 and L2 be two simple objects of F(g) of the same atypicality. Let a ⊆ g denote a subalgebra of g which is itself a classical Lie superalgebra. Then (4.1)

V(a,a¯0 ) (L1 ) = V(a,a¯0 ) (L2 )

(4.2)

cF (a) (L1 ) = cF (a) (L2 )

Proof. By Corollary 3.3 there is a g-supermodule X such that L1 is a direct summand of L2 ⊗ X. By the basic properties of support varieties [4, Equations (4.6.3) and (4.6.4)] this implies V(a,a¯0 ) (L1 ) ⊆ V(a,a¯0 ) (L2 ⊗ X) ⊆ V(a,a¯0 ) (L2 ) ∩ V(a,a¯0 ) (X) ⊆ V(a,a¯0 ) (L2 ). However, this argument is symmetric in L1 and L2 and so we have the equality of support varieties. To prove equality of complexity, we observe that the argument used for gl(m|n) in the proof of [1, Theorem 8.1.1] applies verbatim with the exception that references to [15, Corollary 6.7] should be replaced with references to the generalized Kac-Wakimoto conjecture. 4.2. Using Theorem 4.1 and the line of argument for gl(m|n) used in [4], we now compute the support varieties for the simple supermodules of F. If L(λ) is typical, then it is projective by [13, Theorem 1] and the support variety is trivial. Theorem 4.3 immediately follows. Consequently we assume atyp(L(λ)) > 0 in what follows. Given 0 < k ≤ def(g), let gk be the subalgebra of g deﬁned as follows: • for g = gl(m|n), gk = gl(k|k), • for g = osp(2m + 1|2n), gk = osp(2k + 1|2k), • for g = osp(2m|2n), gk = osp(2k|2k).

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We identify gk as a subalgebra of g as follows. For gl(m|n), gk is the subalgebra isomorphic to gl(k|k) whose roots lie in the intersection of Φ with the R-span of εm−k+1 , . . . , εm , δ1 , . . . , δk . Similarly, for osp(2m|2n) and osp(2m + 1|2n), gk is the subalgebra whose roots like in the intersection of Φ with the R-span of εm−k+1 , . . . , εm , δn−k+1 , . . . , δn . In particular, note that gk has defect k. Let Z = Z(U (g)) denote the center of the universal enveloping superalgebra of g. Given a simple g-supermodule L(λ) of highest weight λ we may use Schur’s lemma to deﬁne an algebra homomorphism χλ : Z → C by the equation zv = χλ (z)v for all z ∈ Z and all v ∈ L(λ). Using these central characters we have a decomposition of F into blocks F= F χ, where the direct sum runs over all algebra homomorphisms χ : Z → C. It is known that all simple supermodules in F χ have the same atypicality and so it makes sense to refer to this as the atypicality of the block. In particular, the principal block of F(gk ) has atypicality k. Gruson and Serganova prove that every block2 of F(g) of atypicality k is equivalent to the principle block of F(gk ). We will need to study the functor which gives this equivalence. Let l denote the subalgebra gk + h ⊆ g. Fix a choice of h ⊂ h so that h is a central subalgebra of l and l = gk ⊕ h . Given λ ∈ h∗ , let λ ∈ (h )∗ denote the map obtained by restricting λ to h . Given a dominant weight μ, Gruson and Serganova call μ stable if A(μ) (where A(μ) is as in (2.1)) is a subset of the roots for l and (μ + ρ, β) > 0 for all β ∈ Φ+ which are not roots of l. Say λ and μ are stable ¯ 0 dominant integral weights and χμ = χλ . Then we have by [11, Section 3] and references therein that μ can be written as w(λ + ρ + i ni αi ) − ρ where w ∈ W , the Weyl group of g¯0 , the sum is over the elements of A(λ), and ni ∈ C for all i. From this it follows that λ = μ . If μ is a stable dominant integral weight, then is straightforward to verify that on L(μ) the Gruson-Serganova functor given in [11, Section 5] coincides with the functor Resμ : F(g) → F(gk ) given by (4.3)

Resμ (N ) = {n ∈ N | h n = μ (h )n for all h ∈ h } .

Note that this is indeed a gk -supermodule as h commutes with gk . Let N be an object of F χμ such that for every composition factor L(γ) of N , the weight γ is stable. An induction on composition series length using that γ = μ shows that the Gruson-Serganova functor coincides with Resμ on N . 4.3. The inclusion gk → g induces a map in relative cohomology, res : H• (g, g¯0 ; M ) → H• (gk , gk,¯0 ; M ), for any M in F(g). Note that this coincides with the map induced by the restriction functor, Res : F(g) → F(gk ). We then have the following commutative diagram.

(4.4)

Ig (M ) → H• (g, g¯0 ; C) ⏐ ⏐ resC

m1

−−−−→

H• (g, g¯0 ; M ⊗ M ∗ ) ⏐ ⏐res

m2

Igk (M ) → H• (gk , gk,¯0 ; C) −−−−→ H• (gk , gk,¯0 ; M ⊗ M ∗ ) 2 More precisely, for osp(2m|2n) half the blocks of atypicality k are equivalent to the principle block of osp(2k + 2|2k). See [11, Section 5] for details. However, by Theorem 4.1 we may safely assume that our simple supermodule does not lie in one of these blocks.

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Here the horizontal maps are those induced by the exact functor −⊗M , and Ig (M ) (resp. Igk (M )) is the kernel of this map. Recall that this is the ideal which deﬁnes V(g,g¯0 ) (M ) (resp. V(gk ,gk,¯0 ) (M )). For clarity in our notation we shall capitalize the names of functors and call the induced maps in cohomology by the same name but in lower case. For example, in (4.4) resC denotes the map induced by the restriction functor Res (with coeﬃcients in the trivial supermodule). Let J denote the kernel of resC . Fix d ≥ 0 so that J is generated by elements of degree no more than d. Such a d exists because H• (g, g¯0 ; C) is a Noetherian ring (indeed by [3, Theorem 4.1.1] it is a polynomial ring). Now choose a dominant integral weight λ and let P• → L(λ) be a ﬁxed projective resolution of L(λ) in F(g). We set Γ to be the set of highest weights of the composition factors of P0 , . . . , Pd . Applying the algorithm given in the proof of [7, Lemma 12] we may choose λ so that γ is stable for all γ ∈ Γ. Let us write e ⊆ g and ˜e ⊆ gk for the detecting subalgebras as deﬁned in [3, Section 4]. We may assume that ˜e¯1 ⊆ e¯1 . To see this, we see that one can choose a set Ω as in [3, Table 2] to obtain an explicit basis for e¯1 for which ˜e¯1 ⊆ e¯1 . The following proposition records certain properties of (4.4) and is straightforward generalization of [4, Proposition 4.7.3]. For completeness we include the proof. Proposition 4.2. Let J denote the kernel of the map resC and ﬁx d ≥ 0 so that J is generated by elements of degree no more than d. Then the following statements about (4.4) hold true. (a) The map resC is a surjective algebra homomorphism. (b) Let M = L(λ) be a simple supermodule in F of atypicality k. Then the map m2 is injective. (c) Assume M = L(λ) be a simple supermodule in F so that the elements of the set Γ deﬁned above are stable. Then J ⊆ Ig (M ). Proof. One proves (a) as follows. By [3, Theorem 3.3.1(a)] there are ﬁnite pseu for which restriction induces isomorphisms doreﬂection groups W and W • • ∗ W H (g, g¯0 ; C)l → S(e¯1 ) and H (gk , gk,¯0 ; C) → S(˜e∗¯1 )W . From the identiﬁcation e¯1 ⊆ e¯1 , one has the canonical algebra homomorphism given by restriction of functions

ρ : S(e∗¯1 )W → S(˜e∗¯1 )W

and the explicit description of e and e allows one to verify that this map is surjective. As all maps are induced by restrictions, one has the following commutative diagram. H• (gl(m|n), gl(m|n)¯0 ; C) −−− −→ S(e∗¯1 )W ⏐ ⏐ ⏐ ⏐ρ resC

H• (gl(k|k), gl(k|k)¯0 ; C) −−− −→ S(˜e∗¯1 )W

Therefore, the map resC is surjective. To prove (b) one argues as follows. We ﬁrst assume that λ is stable. We then have the decomposition (4.5)

L(λ) = Resλ (L(λ)) ⊕ Gλ (L(λ))

THE GENERALIZED KAC-WAKIMOTO CONJECTURE

as gk -supermodules, where (4.6)

Gλ (L(λ)) =

213 13

{x ∈ L(λ) | hx = ν(h)x for all h ∈ h }.

∗

ν∈(h ) ν=λ

Now since Resλ (L(λ)) coincides with the output of the Gruson-Serganova equivalence it is a simple gk -supermodule in the principle block of F(gk ). That is, it is a simple supermodule of the same atypicality as the trivial gk -supermodule. By Theorem 4.1 this implies the following equality. The remaining inclusions follow by the basic properties of support varieties: V(gk ,gk,¯0 ) (C) = V(gk ,gk,¯0 ) (Resλ (L(λ))) ⊆ V(gk ,gk,¯0 ) (L(λ)) ⊆ V(gk ,gk,¯0 ) (C) . Therefore we have V(gk ,gk,¯0 ) (L(λ)) = V(gk ,gk,¯0 ) (C) .

(4.7)

Applying Theorem 4.1 again, it follows that (4.7) holds for arbitrary L(λ) when λ has atypicality k. However H• (gk , gk,¯0 ; C) is a polynomial ring and so has no nonzero nilpotent elements. This along with (4.7) implies that Igk (L(λ)) = (0) and the injectivity of m2 follows. We now prove (c). By our assumption on Γ the functor Resλ coincides with the Gruson-Serganova equivalence on the ﬁrst d degrees of cohomology and so resλ deﬁnes an isomorphism in cohomology in those degrees. Let Resl : F(g) → F(l) be the restriction functor and Pλ : F(l) → F(gk ) be the functor given by projection onto the λ weight space with respect to the action of h , then Resλ = Pλ ◦ Resl . Since resλ is injective for i = 0, . . . , d, resl must also be injective in these degrees. The fact that l = gk ⊕ h for a central abelian subalgebra h implies that the restriction functor F(l) → F(gk ) induces an injective map on cohomology. Composing the Resl with this functor yields the restriction functor g → gk and, hence, res is injective. From this and the commutativity of the diagram (4.4), it follows that the generators of J and hence J itself lies in Ig (M ). 4.4. We can now compute the support varieties of the simple supermodules. Let e ⊆ g be the detecting subalgebra of g. Let W be the ﬁnite pseduoreﬂection groups given by [3, Theorem 3.3.1(a)]. For any g-supermodule, M , the inclusion e → g induces a map of support varieties res∗ : V(e,e¯0 ) (M ) → V(g,g¯0 ) (M )

(4.8) with image (4.9)

res∗ V(e,e¯0 ) (M ) ∼ = V(e,e¯0 ) (M )/W.

The proof of the following theorem closely parallels the analogous result in [4]. We include the proof for completeness. Theorem 4.3. Let g be gl(m|n) or osp(m|2n). Let L(λ) be a simple g-supermodule of atypicality k. Let e ⊆ gk be the detecting subalgebra of gk chosen so that e¯1 ⊆ e¯1 . Then, (a) (4.10)

res∗ ( e¯1 ) = res∗ V(e,e¯0 ) (L(λ)) = V(g,g¯0 ) (L(λ)) ∼ = Ak .

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(b) (4.11)

V(e,e¯0 ) (L(λ)) = W · e¯1 . In particular, V(e,e¯0 ) (L(λ)) is the union of ﬁnitely many k-dimensional subspaces.

Proof. One proves (a) as follows. By Theorem 4.1 we may compute the support variety of any simple supermodule of atypicality k. We choose L(λ) so that the statements of Proposition 4.2 hold true. By Proposition 4.2(c) we have that Ker(resC ) ⊆ Ig (L(λ)). On the other hand, it follows by the commutativity of (4.4) and the injectivity of m2 (Proposition 4.2(b)) that Ig (L(λ)) ⊆ Ker(resC ). Therefore, Ig (L(λ)) = Ker(resC ). Using the surjectivity of resC and the description of H• (gk , gk,¯0 ; C)) as a polynomial ring in k variables, we have V(g,g0¯ ) (L(λ)) ∼ = MaxSpec (H• (g, g¯0 ; C)/ Ker(resC )) ∼ = MaxSpec H• (gk , gk,¯0 ; C) ∼ = Ak . Now consider V(e,e¯0 ) (L(λ)). Recall that ˜e¯1 ⊆ e¯1 . Since L(λ) is stable it follows that L(λ) contains a simple gk -supermodule in the principle block for gk as a direct summand (namely Resλ L(λ)). Since all modules in the same block have the same atypicality, it follows by Theorem 4.1 that V(˜e,˜e¯0 ) (L(λ)) = V(˜e,˜e¯0 ) (C). By the rank variety description of e support varieties it must be that for any x ∈ ˜e¯1 , L(λ) is not projective as an x-supermodule. Here x denotes the Lie subsuperalgebra generated by x. This statement is equally true if we view x as an element of e¯1 . Thus, we have e¯1 ⊆ V(e,e¯0 ) (L(λ)). Therefore by (4.9) one has, (4.12)

res∗ ( e¯1 ) ⊆ res∗ (V(e,e¯0 ) (L(λ))) ⊆ V(g,g¯0 ) (L(λ)) ∼ = Ak .

However, by (4.9) the map res∗ is ﬁnite-to-one so res∗ ( e¯1 ) is a k-dimensional closed subset of Ak . However Ak is a k-dimensional irreducible variety. Therefore res∗ ( e¯1 ) = Ak and all the containments in (4.12) must be equalities. This proves (a). To prove (b) we simply use the fact that the ﬁbers of the map res∗ are precisely the orbits of the ﬁnite group W. The above theorem immediately implies the validity of the atypicality conjecture for gl(m|n) and osp(m|2n). Corollary 4.4. Let g denote gl(m|n) and osp(m|2n) and let L(λ) be a simple g-supermodule of atypicality k. Then dim V(e,e¯0 ) (L(λ)) = dim V(g,g¯0 ) (L(λ)) = atyp(L(λ)). References [1] B. B. Boe, J. R. Kujawa, D. K. Nakano, Complexity for modules over the classical Lie superalgebra gl(m|n), Compositio Math., to appear. , Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. [2] Not., (2011), Issue 3, 696-724. , Cohomology and support varieties for Lie superalgebras, Trans. Amer. Math. Soc. [3] 362 (2010), no. 12, 6551–6590. , Cohomology and Support Varieties for Lie Superalgebras II, Proc. London Math. [4] Soc. 98 (2009), no. 1, 19–44. [5] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2003), no. 1, 185–231.

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[6] S.-J. Cheng, N. Lam, W. Wang, Weiqiang, Super duality and irreducible characters of orthosymplectic Lie superalgebras, Invent. Math. 183 (2011), no. 1, 189–224. [7] M. Duﬂo, V. Serganova, On associated variety for Lie superalgebras, arXiv:math/0507198, (2005). [8] N. Geer, J. R. Kujawa, B. Patureau-Mirand, Ambidextrous objects and trace functions for nonsemisimple categories, Proc. Amer. Math. Soc., to appear. , Generalized trace and modiﬁed dimension functions on ribbon categories, Selecta [9] Math., Volume 17, Issue 2 (2011), 453–504 [10] N. Geer, B. Patureau-Mirand, An invariant supertrace for the category of representations of Lie superalgebras, Paciﬁc J. Math. 238 (2008), no.2, 331–348. [11] C. Gruson, V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852–892. [12] V. G. Kac, Lie superalgebras, Adv. in Math. 26 (1977), no. 1, 8-96. , Characters of typical representations of classical Lie superalgebras, Comm. Algebra [13] 5 (1977), no. 8, 889-897. [14] V. Kac, M. Wakimoto, Integrable highest weight modules over aﬃne superalgebras and number theory, Progr. Math., 123, Birkh¨ auser Boston, Boston, MA, 1994, 415–456. [15] V. Serganova, On superdimension of an irreducible representation of a basic classical lie superalgebra, Supersymmetry in Mathematics and Physics, Springer LNM, editors S. Ferrara, R. Fioresi , V.S. Varadarajan (2011), to appear. , Characters of irreducible representations of simple Lie superalgebras, Proceedings [16] of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 583–593. Mathematics Department, University of Oklahoma, Norman, OK 73019 E-mail address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 86, 0, XXXX 2012

An approach towards the Koll´ ar-Peskine problem via the Instanton Moduli Space Shrawan Kumar 1. Introduction Koll´ar and Peskine (cf. [BC, page 278]) asked the following question on complete intersections over the ﬁeld C of complex numbers. In this note, the ﬁeld C is taken as the base ﬁeld. By a variety, we mean a complex quasiprojective (reduced) (but not necessarily irreducible) variety. Question 1.1. Let Ct ⊂ P3 be a family of smooth curves parameterized by the formal disc D := spec R, where R is the formal power series ring C[[t]] in one variable. Assume that the general member of the family is a complete intersection. Then, is the special member C0 also a complete intersection? By using a construction due to Serre, the above problem is equivalent to the following (cf. [Ku]). Question 1.2. Let Vt be a family of rank two vector bundles on P3 . Assume that the general member of the family is a direct sum of line bundles. Then, is the special member V0 also a direct sum of line bundles? Let us consider the following slightly weaker version of the above question. Question 1.3. Let Vt be a family of rank two vector bundles on P3 . Assume that the general member of the family is a trivial vector bundle. Then, is the special member V0 also a trivial vector bundle? In the next section, we show that the above question is equivalent to a question on the nonexistence of algebraic maps from P3 to the inﬁnite Grassmannian X associated to the aﬃne SL(2). Speciﬁcally, we have the following result (cf. Theorem 2.5): Theorem 1.4. Let X be any irreducible projective variety. Then, the following two conditions are equivalent: (a) Any rank-2 vector bundle F on X × D with trivial determinant, such that F|X×D∗ is trivial, is itself trivial. (b) There exists no nonconstant morphism X → X . Thus, Question 1.3 is equivalent to the following question (cf. Question 2.6): Question 1.5. Does there exist no nonconstant morphism P3 → X ? c c 2012 American Mathematical Society XXXX

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Let Mord∗ (P1 , X ) denote the set of base point preserving morphisms from P1 → X of degree d. It is a complex algebraic variety. As we show in Section 3, any morphism φ : P3 → X of degree d, preserving the base points, canonically induces a morphism φˆ : C3 \{0} → Mord∗ (P1 , X ). Let Md be the set of isomorphism classes of rank two vector bundles V over P2 with trivial determinant and with second Chern class d together with a trivialization of V|P1 . Then, Md has a natural variety structure, which will be referred to by the Donaldson moduli space. Donaldson showed that there is a natural diﬀeomorphism between Md and the instanton moduli space Id of Yang-Mills d-instantons over the ﬂat R4 with group SU(2) modulo based gauge equivalence. As shown by Atiyah, there is a natural embedding i : Mord∗ (P1 , X ) → Md as an open subset (cf. Proposition 3.1). Thus, the morphism φˆ gives rise to a morphism (still denoted by) φˆ : C3 \{0} → Md . Deﬁne an action of C∗ on C3 \{0} by homothecy and on P2 via: z · [λ, μ, ν] = [z −1 λ, μ, ν]. This gives rise to an action of C∗ on Md via the pull-back of bundles. Then, the embedding φˆ is C∗ -equivariant (cf. Theorem 3.2). We would like to make the following conjecture (cf. Conjecture 3.3). Conjecture 1.6. For any d > 0, there does not exist any C∗ -equivariant morphism fˆ : C3 \{0} → Md . Assuming the validity of the above conjecture 1.6, we get that there is no nonconstant morphism φ : P3 → X . Thus, by Theorem 1.4, assuming the validity of the above Conjecture 1.6, any rank-2 bundle F on P3 × D with trivial determinant, such that F|P3 ×D∗ is trivial, is itself trivial (cf. Corollary 3.4). As a generalization of the above, we would like to make the following conjecture (cf. Conjecture 3.5). Conjecture 1.7. For any n ≥ 2, let Xn be the inﬁnite Grassmannian associated to the group G = SL(n), i.e., Xn := SL(n, K)/ SL(n, R), where K is the quotient ﬁeld of R. Then, there does not exist any nonconstant morphism φ : Pn+1 → Xn . Finally, in Section 4, we recall an explicit construction of the moduli space Md via the monad construction and show that the C∗ -action on Md takes a relatively simple form (cf. Lemma 4.3). Acknowledgements. It is my pleasure to thank N. Mohan Kumar, who brought to my attention the Koll´ ar-Peskine problem and with whom I had several very helpful conversations/correspondences. I also thank J. Koll´ar for a correspondence. This work was partially supported by the NSF grant DMS-0901239.

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2. Koll´ ar-Peskine problem and inﬁnite Grassmannian For more details on the following construction of the inﬁnite Grassmannians, see [K, Chapter 13]. Set G = SL(2, K), P = SL(2, R), where K := C[[t]][t−1 ] denotes the ring of Laurent series in one variable and R is the subring C[[t]] of power series. The ring homomorphism R → C, t → 0, gives rise to a group homomorphism π : P → SL(2, C). Deﬁne B = π −1 (B), where B ⊂ SL(2, C) is the Borel subgroup consisting of the upper triangular matrices. For any d ≥ 0, deﬁne n d t 0 P/P ⊂ G/P. B Xd = n=0 0 t−n Then, Xd admits a natural structure of a projective variety and ∪d≥0 Xd = G/P. Moreover, Xd is irreducible (of dimension d), and Xd → Xd+1 is a closed embedding. In particular, X := G/P is a projective ind-variety. For any integer d ≥ 0, consider the set Ld of R-submodules L ⊂ K ⊗C V such that td Lo ⊂ L ⊂ t−d Lo , and dimC (L/td Lo ) = 2d, where V := C2 and Lo := R ⊗ V . Let L := ∪d≥0 Ld . Any element of L is called an R-lattice in K ⊗C V . The group SL(2, K) acts canonically on K ⊗C V . Recall the following from [K, Lemma 13.2.14]. Lemma 2.1. The map g SL(2, R) → gLo (for g ∈ SL(2, K)) induces a bijection β : X → L. Let X be any irreducible projective variety and let F be a rank two vector bundle on X × D with trivial determinant, where D := spec R. Fix a trivialization of the determinant of F. Assume that F|X×D∗ is trivial, D∗ being the punctured formal disc D∗ := spec K. Fix a compatible trivialization τ of F|X×D∗ (compatible with the trivialization of the determinant of F). For any x ∈ X, H 0 (x × D, F) → H 0 (x × D∗ , F) K ⊗C V. Thus, Lx := H 0 (x × D, F) → K ⊗C V. It can be seen that Lx is an R-lattice in K ⊗C V . Moreover, the map x → Lx provides a morphism φF (τ ) : X → X under the identiﬁcation of Lemma 2.1 (depending upon the trivialization τ ). If we choose a diﬀerent compatible trivialization τ of the bundle F|X×D∗ , it is easy to see that the morphism φF (τ ) diﬀers from φF (τ ) by the left multiplication of an element g ∈ G, i.e., φF (τ )(x) = gφF (τ )(x), for all x ∈ X. (To prove this, observe that any morphism X × D ∗ → SL(2, C) is constant in the X-variable since X is an irreducible projective variety by assumption.) Set [φF ] as the equivalence class of the map φF (τ ) : X → X (for some compatible trivialization τ ), where two maps X → X are called equivalent if they diﬀer by left multiplication by an element of G. Thus, [φF ] does not depend upon the choice of the compatible trivialization τ of F|X×D∗ .

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Lemma 2.2. The bundle F is trivial on X × D if and only if the map [φF ] is a constant map. Proof. If F is trivial on X × D, then [φF ] is clearly a constant map. Conversely, assume that [φF ] is a constant map. Choose a compatible trivialization τ of F|X×D∗ so that ∀x ∈ X. Lx = Lo , Let φ := φF (τ ). Take a basis {e1 , e2 } of V . This gives rise to unique sections σ1 (x), σ2 (x) ∈ H 0 (x × D, F) corresponding to the elements 1 ⊗ e1 and 1 ⊗ e2 respectively under the map φ. Let s1 , s2 ∈ H 0 (X × D∗ , F) be everywhere linearly independent sections such that σi (x)|x×D∗ = si|x×D∗ , for all x ∈ X. It suﬃces to show that σ1 (x), σ2 (x) are linearly independent at 0 as well. Take a small open subset U ⊂ X so that the bundle F|U×D is trivial. Fix a compatible trivialization τ of F|U×D . Then, the sections σi can be thought of as maps U × σ

i D −→ V which are linearly independent over any point of U × D∗ . From this it is easy to see that σi are linearly independent over any point of U × D since the transition matrix over U × D∗ with respect to the two trivializations τ and τ of F|U ×D∗ has determinant 1. Covering X by such small open subsets U , the lemma is proved.

As above, a bundle F gives rise to a morphism φF : X → X (unique up to the left multiplication by an element of G). Conversely, any morphism φ : X → X gives rise to a bundle F. Before we can prove this, we need the following result. Let V := P1 × V → P1 be the trivial rank-2 vector bundle over P1 , where V is the two dimensional complex vector space C2 . For any g ∈ G, deﬁne a rank-2 locally free sheaf Vg on P1 as the sheaf associated to the following presheaf: For any Zariski open subset U ⊂ P1 , set Vg (U ) = H 0 (U, V), if 0 ∈ / U, and Vg (U ) = {σ ∈ H 0 (U \ {0}, V) : (σ)0 ∈ g(R ⊗C V )}, if 0 ∈ U, where (σ)0 denotes the germ of the rational section σ at 0 viewed canonically as an element of K ⊗C V . With this notation, we have the following result from [KNR, Proposition 2.8]. (In fact, we only give a particular case of loc. cit. for G = SL(2, C) and for the curve C = P1 , which is suﬃcient for our purposes.) Proposition 2.3. There is a rank-2 algebraic vector bundle U on X × P1 satisfying the following: (1) The bundle U is of trivial determinant, (2) The bundle U is trivial restricted to X × (P1 \ {0}), (3) For any x = gP ∈ X (for g ∈ G), the restriction U|x×P1 is isomorphic with the locally free sheaf Vg as above. Lemma 2.4. For any morphism φ : X → X , there exists a rank two vector bundle Fφ on X × D with trivial determinant (explicitly constructed in the proof ) such that Fφ |X×D∗ is trivial and such that the associated morphism [φFφ ] = [φ]. Proof. As in Proposition 2.3, consider the vector bundle U on X × P1 of rank two. Let Uφ be the pull-back of the family U to X × P1 via the morphism φ × Id. Let Fφ be the restriction of Uφ to X × D, where D is the formal disc around 0.

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Then, by the properties (1)-(2) of Proposition 2.3, the bundle Fφ satisﬁes the ﬁrst two properties of the lemma. Finally, by the property (3) of Proposition 2.3 and the deﬁnition of the map [φFφ ], it is easy to see that [φFφ ] = [φ]. Combining Lemmas 2.2 and 2.4, we get the following theorem: Theorem 2.5. Let X be any irreducible projective variety. Then, the following two conditions are equivalent: (a) Any rank-2 vector bundle F on X × D with trivial determinant, such that F|X×D∗ is trivial, is itself trivial. (b) There exists no nonconstant morphism X → X . By virtue of the above theorem, an aﬃrmative answer of Question 1.3 is equivalent to an aﬃrmative answer of the following question. Observe that under the assumptions of Question 1.3, the family Vt , thought of as a rank-2 vector bundle V on P3 × D, has trivial determinant by virtue of [H, Exercise 12.6(b), Chap. III]. Also, V|P3 ×D∗ is trivial by the semicontinuity theorem (cf. [H, §12, Chap. III]). Question 2.6. Does there exist no nonconstant morphism P3 → X ? Definition 2.7. Recall (cf. [K, Proposition 13.2.19 and its proof]) that the singular homology H2 (X , Z) Z and it has a canonical generator given by the Schubert cycle of complex dimension 1. For any morphism φ : P3 → X , deﬁne its degree to be the integer d = dφ such that the induced map in homology φ∗ : H2 (P3 , Z) → H2 (X , Z) induced by φ is given via multiplication by d. Since the pull-back of the ample generator of Pic X H 2 (X , Z) (which is globally generated) is a globally generated line bundle on P3 , d ≥ 0 and d = 0 if and only if φ is a constant map. For any rank-2 bundle F on P3 × D with trivial determinant such that F|X×D∗ is trivial, we deﬁne its deformation index d(F) = d[φF ] . Proposition 2.8. For any morphism φ : P3 → X , dφ is divisible by 6. Equivalently, for any F as in the above deﬁnition, d(F) is divisible by 6. Proof. Consider the induced algebra homomorphism in cohomology: φ∗ : H ∗ (X , Z) → H ∗ (P3 , Z), induced by φ. By the deﬁnition, the induced map at H 2 is multiplication by dφ . Moreover, by [K, Exercise 11.3.E.4], for any i ≥ 0, H 2i (X , Z) is a free Z-module of rank 1 generated by the Schubert class i . Moreover, i+j

i+j .

i · j = i In particular, 6 3 = 31 . From this the proposition follows.

3. Koll´ ar-Peskine problem and the instanton moduli space Take any morphism φ : P3 → X , with degree d = dφ . Assume that φ([0, 0, 0, 1]) is the base point xo := 1 · P ∈ X . Deﬁne the map π : C3 \{0} × P1 −→ P3 , (x, [λ, μ]) −→ [λx, μ],

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for x ∈ C3 \{0} and [λ, μ] ∈ P1 . There is an action of C∗ on C3 \{0} × P1 by 1 z · x, [λ, μ] = zx, [ λ, μ] , for z ∈ C∗ . z Then, π factors through the C∗ -orbits. Consider the composite morphism φ¯ = φ ◦ π : C3 \{0} × P1 → X . ¯ 0) = xo for any x ∈ C3 \{0}, where 0 ∈ P1 is the base point [0, 1]. Observe that φ(x, d 1 Let Mor∗ (P , X ) denote the set of base point preserving morphisms from P1 → X of degree d (taking 0 to xo ). Then, by Grothendieck’s result on Hilbert schemes, Mord∗ (P1 , X ) acquires the structutre of a complex algebraic variety. Moreover, the map φˆ : C3 \{0} → Mord∗ (P1 , X ), ¯ is a morphism. canonically induced from φ, Let us consider the embedding P1 → P2 , [λ, μ] → [λ, μ, 0]. Fix d ≥ 0 and let Md be the set of isomorphism classes of rank two vector bundles V over P2 with trivial determinant and with second Chern class d together with a trivialization of V|P1 . The isomorphism is required to preserve the trivialization of V over P1 . Then, Md has a natural variety structure. Moreover, any bundle V ∈ Md is semistable. (By [OSS, Chapter I, Lemma 3.2.2], V is trivial on generic lines ⊂ P2 . Thus, by [OSS, Chapter II, Lemma 2.2.1], V is semistable.) We will refer to Md as the Donaldson moduli space. Donaldson [D] showed that there is a natural diﬀeomorphism between Md and the instanton moduli space Id of Yang-Mills d-instantons over the ﬂat R4 with group SU(2) modulo based gauge equivalence. Deﬁne an action of C∗ on Mord∗ (P1 , X ) via: (z · γ)[λ, μ] = γ[zλ, μ],

(1)

for z ∈ C∗ , γ ∈ and [λ, μ] ∈ P1 . Also, deﬁne an action of C∗ on P2 via: Mord∗ (P1 , X )

z · [λ, μ, ν] = [z −1 λ, μ, ν].

(2)

This gives rise to an action of C∗ on Md via the pull-back of bundles, i.e., for V ∈ Md , [X] ∈ P2 , the ﬁber of z · V over [X] is given by: (z · V)[X] = Vz·[X] .

(3)

(Observe that P → P is stable under C∗ and hence the trivialization of V|P1 pulls back to a trivialization.) Recall the following result from [A, § 2]. 1

2

Proposition 3.1. There is a natural embedding i : Mord∗ (P1 , X ) → Md as an open subset. Moreover, i is C∗ -equivariant with respect to the C∗ actions as in equations (1) and (3). The following result summarizes the above discussion. Theorem 3.2. To any morphism φ : P3 → X of degree d preserving the base points, there is a canonically associated C∗ -equivariant morphism (deﬁned above) φˆ : C3 \{0} → Md , where C∗ acts on C3 \{0} via the multiplication.

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Moreover, φ is constant (i.e., d = 0) iﬀ φˆ is constant. We would like to make the following conjecture. Conjecture 3.3. For any d > 0, there does not exist any C∗ -equivariant morphism fˆ : C3 \{0} → Md . Assuming the validity of the above conjecture, we get the following. Corollary 3.4. Assuming the validity of Conjecture 3.3, there does not exist any nonconstant morphism φ : P3 → X . Thus, by Theorem 2.5, assuming the validity of Conjecture 3.3, any rank-2 bundle F on P3 × D with trivial determinant, such that F|P3 ×D∗ is trivial, is itself trivial. As a generalization of the above corollary, I would like to make the following conjecture. Conjecture 3.5. For any n ≥ 2, let Xn be the inﬁnite Grassmannian associated to the group G = SL(n), i.e., Xn := SL(n, K)/ SL(n, R). Then, there does not exist any nonconstant morphism φ : Pn+1 → Xn . Remark 3.6. An interesting aspect of this approach is that Question 1.3 involving an arbitrary family of (not necessarily semistable) vector bundles on P3 is reduced to a question about the Donaldson moduli space Md consisting of rank two semistable bundles on P2 . 4. Monad construction of Md This section recalls an explicit construction of the moduli space Md via the monad construction. We refer to [OSS, §§3,4, Chap. II] for more details on the monad construction (see also [B] and [Hu]). Fix an integer d ≥ 0. Let H, K, L be complex vector spaces of dimensions d, 2d+2, d respectively. By monad one means linear maps parameterized by Z ∈ C3 , depending linearly on Z: A

B

H →Z K →Z L, such that the composite BZ ◦ AZ = 0, for all Z ∈ C3 . The monad is said to be nondegenerate if for all Z ∈ C3 \ {0}, BZ is surjective and AZ is injective. In this case, we get a vector bundle on P2 with ﬁber at the line [Z] the vector space E(A, B) := Ker BZ / Im AZ . Then, any rank-2 bundle on P2 with the second Chern class d, which is trivial on some line, is isomorphic with E(A, B), for some monad (A, B). Moreover, such a monad (A, B) is unique up to the action of GL(H) × GL(K) × GL(L). Let [λ, μ, ν] be the homogeneous coordinates on P2 . If we only consider bundles on P2 trivial on the ﬁxed line ν = 0, the condition on the corresponding monad is that the composite Bλ Aμ = −Bμ Aλ is an isomorphism, where (for Z = (λ, μ, ν)) AZ := Aλ λ + Aμ μ + Aν ν, and BZ := Bλ λ + Bμ μ + Bν ν. In the following, t denotes the transpose, Id×d denotes the identity matrix of size d × d, 0d×d denotes the zero matrix of size d × d and α, β, a and b are matrices of indicated sizes. For such bundles (i.e., rank-2 bundles on P2 with the second Chern

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class d, which are trivial on the line ν = 0), using the action of GL(H) × GL(K) × GL(L), one can choose bases for H, K, L so that the maps are given as follows: t t t t t Aλ = Id×d , 0d×d , 0d×2 , Aμ = 0d×d , Id×d , 0d×2 , Aν = αd×d , βd×d , ad×2 , Bλ = 0d×d , Id×d , 0d×2 , Bμ = −Id×d , 0d×d , 0d×2 , Bν = −βd×d , αd×d , bd×2 , and the following condition is satisﬁed: Bν Aν = 0, which is equivalent to the condition [α, β] + bat = 0. The restriction of the bundle E(A, B) to the line ν = 0 has a standard frame given by the last 2 basis vectors of K C2d+2 . For any d ≥ 0, let Sˆd be the closed subvariety of matrices (α, β, a, b) such that α, β are d × d matrices and a, b are d × 2 matrices and they satisfy: (1) [α, β] + bat = 0. Let Sd be the open subset of Sˆd satisfying, in addition, the following condition: t (2) For all λ, μ ∈ C, αt + λId×d , β t + μId×d , a is injective and −(β + μId×d ), α + λId×d , b is surjective. We recall the following result due to Barth from [D, Proposition 1]. Theorem 4.1. For any d ≥ 0, the variety Md is isomorphic with the quotient of the variety Sd by the action of GL(d) under: g · (α, β, a, b) = (gαg −1 , gβg −1 , (g −1 )t a, gb), for g ∈ GL(d), and (α, β, a, b) ∈ Sd . Remark 4.2. The aﬃne variety Sˆd is stable under the above action of GL(d). Moreover, the open subset of stable points of Sˆd (under the GL(d)-action) is precisely equal to Sd (cf. [D, Lemma on page 458 and its proof]). Lemma 4.3. Under the above isomorphism of the variety Md with the quotient of Sd by GL(d), the action of C∗ transports to the action: z · (α, β, a, b) = (zα, β, za, b), for z ∈ C∗ , (α, β, a, b) ∈ Sd . Proof. The C∗ -action on Md via the pull-back corresponds to the bundle: Ker(−μId×d − νβ, z −1 λId×d + να, νb) Ker(z −1 λBλ + μBμ + νBν ) = −1 Im(z λAλ + μAμ + νAν ) Im (z −1 λId×d + ναt , μId×d + νβ t , νa)t =

Ker(−μId×d − νβ, z −1 λId×d + να, νb) . Im (λId×d + zναt , zμId×d + zνβ t , zνa)t

Changing the basis in C2d+2 = Cd × Cd × C2 in the second factor to {zej }1≤j≤d , where {ej } is the original basis, we get that the last term in the above equation is equal to Ker(−μId×d − νβ, λId×d + zνα, νb) . Im (λId×d + zναt , μId×d + νβ t , zνa)t This proves the lemma.

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References [A] M.F. Atiyah. Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), 437–451. [AJ] M.F. Atiyah and J.D.S. Jones. Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), 97–118. [BC] E. Ballico and C. Ciliberto. Algebraic curves and projective geometry, Lecture Notes in Mathematics vol. 1389, Springer-Verlag (1989). [B] W. Barth. Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63–91. [BH] W. Barth and K. Hulek. Monads and Moduli of vector bundles, Manuscripta Math. 25 (1978), 323–347. [D] S.K. Donaldson. Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–460. [GG] W.L. Gan and V. Ginzburg. Almost-commuting variety, D-modules, and Cherednik algebras, Inter. Math. Res. Papers vol. 2006 (2006), 1–54. [H] R. Hartshorne. Algebraic Geometry, Springer-Verlag (1977). [Hu] K. Hulek. On the classiﬁcation of stable rank-r vector bundles over the projective plane, In: Proc. Nice Conference on Vector Bundles and Diﬀerential Equations, A. Hirschowitz (editor), Birkh¨ auser, Boston (1980), pp. 113–144. [Ku] N.M. Kumar. Smooth degeneration of complete intersection curves in positive characteristic, Invent. Math. 104 (1991), 313–319. [K] S. Kumar. Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics vol. 204, Birkh¨ auser, Boston (2002). [KNR] S. Kumar, M.S. Narasimhan and A. Ramanathan. Inﬁnite Grassmannians and moduli spaces of G-bundles, Math. Annalen 300 (1994), 41–75. [OSS] C. Okonek, M. Schneider and H. Spindler. Vector Bundles on Complex Projective Spaces, Progress in Mathematics vol. 3, Birkh¨ auser, Boston (1980). [P1] J. Le Potier. Fibr´ es stables de rang 2 sur P2 (C), Math. Annalen 241 (1979), 217–256. [P2] J. Le Potier. Sur le groupe de Picard de l’espace de modules des ﬁbr´es stables sur P2 , Ann. ´ Norm. Sup. 13 (1981), 141–155. Scient. Ec. ´ Norm. Sup. [PT] J. Le Potier and A. Tikhomirov. Sur le morphisme de Barth, Ann. Scient. Ec. 34 (2001), 573–629. [S] S. A. Stromme. Ample divisors on ﬁne moduli spaces on the projective plane, Math. Z. 187 (1984), 405–423.

Address: Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA (email: [email protected])

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Proceedings of Symposia in Pure Mathematics Volume 86, 0, XXXX 2012

On the representations of disconnected reductive groups over Fq G. Lusztig Introduction Let G be a connected reductive algebraic group deﬁned over a ﬁnite ﬁeld Fq . One of the main tools in the study of representations of the ﬁnite group G(Fq ) over a ﬁeld of characteristic zero is the use of certain varieties Xw (see [DL1]) on which G(Fq ) acts (here w is a Weyl group element). Now let σ : G → G be a quasisemisimple automorphism of G and let m ≥ 1 be an integer such that σ m = 1. ˆ of G with the cyclic group of order m with Consider the semidirect product G generator σ; this is naturally an algebraic group deﬁned over Fq . Now the ﬁnite ˆ q ) acts naturally on the disjoint union w Xw and from this one can group G(F ˆ q ). For example, this again derive information about the representations of G(F observation has been used by the author in his proof of the ﬁniteness of the number ˆ (see [Sp, I, of unipotent G-conjugacy classes in the connected component Gσ of G ˆ q ) has 4.1]); the connection between the varieties Xw and the representations of G(F been systematically investigated by Digne and Michel [DM] and by Malle [Ma]. In this paper we try to extend some results on unipotent representations established ˆ q ). One of the key steps in the description [L2] of the for G(Fq ) in [L2] to G(F set of unipotent representations of G(Fq ) is the deﬁnition of a partition of that set into subsets indexed by the two-sided cells [KL1] of the Weyl group such that certain explicit Q-linear combinations of virtual representations of G(Fq ) given by the alternating sum of the cohomologies of the various Xw are linear combinations of unipotent representations corresponding to a ﬁxed two-sided cell. A conjectural ˆ q ) was formulated by Malle in [Ma] extension of this statement to the case of G(F and proved in this paper (see 2.4(ii)). Our proof is a generalization of that in [L2]; the main new ingredient is the use of the generalization given in [L1] to certain Hecke algebras with unequal parameter of the polynomials Py,w deﬁned in [KL1] and of their geometric interpretation stated in [L1] (and generalizing that in [KL2]) which is proved in this paper. It turns out that these generalized polynomials appear naturally in the study of the varieties Xw in connection with a ˆ q ) and they provide the necessary tools to prove the nontrivial G(Fq )-coset of G(F above conjecture. 2010 Mathematics Subject Classiﬁcation. Primary 20G99. Supported in part by the National Science Foundation grant DMS-0758262. c c 2012 American Mathematical Society XXXX

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G. LUSZTIG

Notation. Let k be an algebraic closure of the ﬁnite ﬁeld Fp with p elements. ¯l If q is a power of p we denote by Fq the subﬁeld of k with q elements. Let Q be an algebraic closure of the ﬁeld of l-adic numbers (l is a ﬁxed prime number = p). All algebraic varieties in this paper are over k. For an algebraic variety Y of pure dimension let Hi (Y ) (resp. Hic (Y )) be the i-th hypercohomology space (resp. hypercohomology with compact support) of Y with coeﬃcients in the intersection ¯ l ); let H i (Y ) = H i (Y, Q ¯ l ). If K is a complex of l-adic cohomology complex IC(Y, Q c c i sheaves on an algebraic variety Y we denote by H K the i-th cohomology sheaf of K and by Hyi K the stalk of Hi K at y ∈ Y . The cardinal of a ﬁnite set S is denoted by |S|. If S is a set and f : S → S is a map we set S f = {s ∈ S; f (s) = s}. We set A = Z[v, v −1 ] where v is an indeterminate. 1. Preliminaries 1.1. Let G be a connected reductive algebraic group over k. Let F : G → G be the Frobenius map relative to an Fq -rational structure on G (q is a power of p). Let B be the variety of Borel subgroups of G. Note that F induces an endomorphism B → F (B) of B. Let W be the Weyl group of G viewed as an indexing set for the G-orbits on B × B (simultaneus conjugation); for w ∈ W let Ow be the Gorbit corresponding to w. We regard W as a Coxeter group with set of simple reﬂections {si ; i ∈ I} in the standard way; let l : W → N be the corresponding length function. For any I ⊂ I let sI be the longest element of the subgroup of W generated by {si ; i ∈ I }. Let ≤ be the standard partial order on W . Now F induces an automorphism δ : W → W (compatible with the length function) by the requirement that w ∈ W, (B, B ) ∈ Ow implies (F (B), F (B )) ∈ Oδ(w) . Let sgn : W → {±1} be the homomorphism w → (−1)l(w) . 1.2. Let IrrW be a set of representatives for the isomorphism classes of irreducible representations of W over Q. For E ∈ IrrW let ME be the set of linear maps of ﬁnite order Δ : E → E such that Δ(w(e)) = δ(w)(Δ(e)) for any w ∈ W , e ∈ E. Then |ME | is 0 or 2. Let Irrδ W = {E ∈ IrrW ; |ME | = 2}. For E ∈ IrrW we deﬁne E † ∈ IrrW by E † ∼ = E ⊗ sgn. Let H be the Hecke algebra over A of W with respect to the weight function w → l(w) on W . Thus H has an A-basis (Tw )w∈W and we have Tw Tw = Tww if w, w ∈ W , l(ww ) = l(w)+l(w ); moreover we have (Tsi −v 2 )(Tsi +1) = 0 for i ∈ I. Note that Q[W ] = Q⊗A H where Q is viewed as an A-algebra via v → 1; w ∈ Q[W ] corresponds to 1 ⊗ Tw . Let J be the ring with Z-basis (tw )w∈W deﬁned as in [L3, 18.3] in terms of (W, l) and let Φ : H → A ⊗ J be the A-algebra homomorphism deﬁned in [L3, 18.9]. After applying Q⊗A to Φ we obtain an algebra isomorphism ∼ ΦQ : Q[W ] → Q ⊗ J. Via this isomorphism any E ∈ IrrW becomes a simple Q ⊗ Jmodule denoted by E♠ . Let E ∈ Irrδ W , Δ ∈ ME . We deﬁne Δ : E♠ → E♠ by Δ(ξ) = ΦQ (Δ(Φ−1 Q (ξ))); we have tδ(w) (Δ(ξ)) = Δ(tw (ξ)) for any w ∈ W, ξ ∈ E♠ . ∼ After applying Q(v)⊗A to Φ we obtain an algebra isomorphism ΦQ(v) : Q(v)⊗H → Q(v) ⊗ J. Via this isomorphism the simple Q(v) ⊗ J-module Q(v) ⊗Q E♠ (E as above) becomes a simple Q(v) ⊗ H-module denoted by Ev2 and the isomorphism 1 ⊗ Δ : Q(v) ⊗Q E♠ → Q(v) ⊗Q E♠ becomes an isomorphism Δ : Ev2 → Ev2 such that Tδ(w) (Δ(ξ)) = Δ(Tw (ξ)) for any w ∈ W, ξ ∈ Ev2 . We assume that for each E ∈ Irrδ W we have choosen an element Δ ∈ ME .

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

229 3

Let a : W → N be the function deﬁned (in terms of l : W → N) in [L2, (5.27.1)] or equivalently as in [L3, 13.6]. Let E → aE be the function IrrW → N deﬁned in [L2, (4.1.1)]. For E ∈ IrrW we set aE = aE † . For w ∈ W we have

v −l(w) tr(ΔTw , Ev2 ) = cw,Δ,E v aE + lower powers of v where cw,Δ,E ∈ Z (see [L2, (5.1.23)]); by an argument similar to that in [L3, 20.10], † ) where Δ : E † → E † is Δ ⊗ 1 : E ⊗ sgn → E ⊗ sgn. we have cw,Δ,E = tr(Δ , E♠ Let RW be the vector space of formal linear combinations E∈Irrδ W rE E, rE ∈ Q. For w ∈ W we set cw,Δ,E E ∈ RW . Aw = E∈Irrδ W

(Compare [L2, (5.11.6)].) 1.3. Let w ∈ W . Following [DL1] we set Xw = {B ∈ B; (B , F (B )) ∈ Ow }. Let

¯ w = {B ∈ B; (B , F (B )) ∈ ∪z∈W ;z≤w Oz } X

be the closure of Xw in B. For x ∈ GF , Ad(x) : B → B (conjugation by x) leaves ¯ w stable and induces linear automorphisms Ad(x)∗ of Hi (X ¯ w ), Hci (Xw ). Then Xw , X ∗−1 i ¯ i F makes H (Xw ), Hc (Xw ) into G -modules. Let E be a set of reprex → Ad(x) sentatives for the isomorphism classes of irreducible representations of GF which i i ¯ appear in Hc (Xw ) for some w ∈ W, i ∈ N or equivalently in H (Xw ) for some w ∈ W, i ∈ N (thus E is the set of unipotent representations of GF .) Let [E]Z be the Grothendieck group of the category of representations of GF which are ﬁnite sums of unipotent representations. Then [E] := Q ⊗ [E]Z has a basis {ρ; ρ ∈ E}. For f ∈ [E] let (ρ : f ) ∈ Q be the coeﬃcient of ρ ∈ E in f . For w ∈ W let Rw = i (−1)i Hci (Xw ) viewed as an element of [E]. As in [L2, 3.7], for any E ∈ Irrδ W we deﬁne tr(Δw, E)Rw ∈ [E]. RE = |W |−1

w∈W

For any ξ = E∈Irrδ W rE E ∈ RW we set Rξ = lar, for w ∈ W , RAw is deﬁned.

E∈Irrδ W

rE RE ∈ [E]. In particu-

1.4. Let ≤LR be the preorder on W deﬁned in [KL1] and let ∼LR be the corresponding equivalence relation on W (the equivalence classes are the two-sided cells of W ; they form a set C). For w, w ∈ W we write w
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Using the theorem and 1.4(a) we see that for c ∈ C, w ∈ c we have (a) RAw ∈ ρ∈E;ρ=c∗ Qρ. The following result is contained in [L2, 6.15]. Theorem 1.6. Let w ∈ W . We have ¯ w ) = (−1)l(w)−a(w) RA + (a) Hl(w)−a(w) (X w

nw ,w RAw ∈ [E]

w ∈W ;w
where nw ,w ∈ Q. ¯ w )ρ the ρ-isotypic component For w ∈ W, i ∈ N and ρ ∈ E we denote by Hi (X F i ¯ of the G -module H (Xw ) and we set ¯ w ) = ¯ w )ρ Hl(w)−a(w) (X Hl(w)−a(w) (X ρ;ρ=c∗

where c ∈ C is deﬁned by w ∈ c. Let c ∈ C. Using the theorem and 1.5(a) we see that if w ∈ c then ¯ w ) is a Q-linear combination of ρ ∈ E such that c∗ ≤LR ρ. (b) Hl(w)−a(w) (X Hence projecting the equality (a) onto the subspace generated by the ρ ∈ E such that ρ = c∗ we see that for any w ∈ c we have ¯ w ) = (−1)l(w)−a(w) RA . Hl(w)−a(w) (X w

(c)

(This projection maps each RAw with w
(−1)l(w)−a(w) RAw is an actual GF -module.

(d)

¯ w ))ρ = 0 for some w ∈ ρ∗ . Corollary 1.7. Let ρ ∈ E. We have Hl(w)−a(w) (X l(w )−a(w ) ¯ ∗ If c ∈ C, w ∈ c and H (Xw )ρ = 0 then ρ ≤LR c .

Let c = ρ. We can ﬁnd E ∈ Irrδ W such that (ρ : RE ) = 0, E c. By [L2, 5.13(ii)], E is a Q-linear combination of elements Ax (x ∈ c∗ ). Hence RE is a Q-linear combination of elements RAx (x ∈ c∗ ). It follows that (ρ : Ax ) = 0 for ¯ x ))ρ = 0. The last sentence in some x ∈ c∗ . Hence by 1.6(c) we have Hl(x)−a(x) (X the corollary follows from 1.6(b). 2. The main results 2.1. We preserve the setup of 1.1. In addition, we ﬁx an automorphism σ : G → G such that for some (B, B ∗ ) ∈ OsI we have σ(B) = B, σ(B ∗ ) = B ∗ and such that σF = F σ : G → G. We also ﬁx an integer m ≥ 1 such that σ m = 1. Now σ induces a (length preserving) automorphism of W denoted again by σ; thus, for w ∈ W, (B1 , B2 ) ∈ Ow we have (σ(B1 ), σ(B2 )) ∈ Oσ(w) ; moreover σδ = δσ : W → W . For i ∈ I we have σ(si ) = sσ(i) where σ : I → I is a ˆ be the semidirect product of G with the cyclic group of order m bijection. Let G ˆ we have σgσ −1 = σ(g) for all g ∈ G. Note that with generator σ so that in G ˆ is naturally an aﬃne algebraic group with identity component G. We extend G ˆ → G ˆ (denoted again by F ) by σ i g → σ i F (g) for F to a homomorphism G ˆ i ∈ [0, m − 1], g ∈ G. This is the Frobenius map for an Fq -rational structure on G. F F ˆ Note that G is the semidirect product of G with the cyclic group of order m with generator σ.

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231 5

Let d ≥ 1 be an integer with the following property: there exists (B, B ∗ ) ∈ OsI such that σ(B) = B, σ(B ∗ ) = B ∗ , F d (B) = B, F d (B ∗ ) = B ∗ , F d acts on d B ∩ B ∗ as t → tq (clearly, such d exists). Let r ≥ 1 be a multiple of d and let r F = F σ : G → G, a Frobenius map relative to an Fq -rational structure on G where q = q r . Note that w ∈ W, (B1 , B2 ) ∈ Ow implies (F (B1 ), F (B2 )) ∈ Oσ(w) . ˙ = {w ∈ W ; σ(w) = w}. It is known that W ˙ is itself a Weyl group with Let W ˙ ˙ →N standard generators sω where ω runs over the set I of σ-orbits on I. Let l˙ : W ˙ ˙ ˙ be the length function of W ; thus l(sω ) = 1 for any ω ∈ I. The restriction l|W˙ of l ˙ is a weight function (in the sense of [L3, 3.1]) on the Coxeter group W ˙ . We to W ˙ l(w) ˙ deﬁne sgn ˙ : W → {±1} by sgn(w) ˙ = (−1) . ˙ be a set of representatives for the isomorphism classes of irre2.2. Let IrrW ˙ over Q. Now δ : W → W restricts to an automorphism ducible representations of W ˙ ˙ and the weight function of W (denoted again by δ) preserving the set {sω , ω ∈ I} ˙ l|W˙ . For E ∈ IrrW let ME be the set of linear maps of ﬁnite order Δ : E → E ˙ , e ∈ E. Then |ME | is 0 or 2. such that Δ(w(e)) = δ(w)(Δ(e)) for any w ∈ W ˙ we deﬁne E † ∈ IrrW ˙ by ˙ = {E ∈ IrrW ˙ ; |ME | = 2}. For E ∈ IrrW Let Irrδ W † ∼ E = E ⊗ sgn. ˙ ˙ with respect to the weight function Let H˙ be the Hecke algebra over A of W ˙ , l|W˙ . Thus H˙ has an A-basis (T˙w )w∈W˙ and we have T˙w T˙w = T˙ww if w, w ∈ W ˙l(ww ) = l(w)+ ˙ ˙l(w ); moreover we have (T˙s −v 2l(sω ) )(T˙s +1) = 0 for ω ∈ I. ˙ Note ω ω ˙ ] = Q ⊗A H˙ where Q is viewed as an A-algebra via v → 1; w ∈ Q[W ] that Q[W corresponds to 1 ⊗ T˙w . Let¯: A → A be the ring involution such that v → v −1 . Let¯: H˙ → H˙ be the ˙ ˙ ring homomorphism such that aT˙w = a ¯T˙w−1 −1 for w ∈ W , a ∈ A. For any x, y ∈ W ˙ we deﬁne a polynomial Rx,y (X) ∈ Z[X], (X is an indeterminate) by the equality R˙ x,y (v 2 )v 2l(x) T˙x−1 T˙y = −1 . ˙ x∈W

(See [KL1, (2.0.a)], [L3, 4.3].) Note that R˙ x,y = 0 unless x ≤ y. It follows that R˙ x,y (v 2 )v 2l(x) T˙xs . (a) T˙y T˙s = I

I

˙ ;x≤y x∈W

˙ there is a unique element C˙ w ∈ H˙ such that For any w ∈ W C˙ w = sgn(yw) ˙ P˙ y,w (v −2 )v l(w)−2l(y) T˙y ˙ ;y≤w y∈W

=

sgn(yw) ˙ P˙ y,w (v 2 )v −l(w)+2l(y) T˙y−1 −1

˙ ;y≤w y∈W

where P˙ y,w (X) ∈ Z[X] has degree ≤ (l(w) − l(y) − 1)/2 if y < w and P˙ w,w (X) = 1. ˙ we have from the deﬁnitions (See [L3, §5]; compare [KL1].) For y ≤ w in W v 2l(y) R˙ y,z (v 2 )P˙ z,w (v 2 ). (b) v 2l(w) P˙ y,w (v −2 ) = ˙ ;y≤z≤w z∈W

232 6

G. LUSZTIG

˙ a polynomial Q˙ y,w (X) ∈ Z[X] We set P˙ y,w = 0 if y ≤ w. We deﬁne for y, w in W ˙ by the requirement that for any y, w in W , (c)

sgn(zw) ˙ P˙y,z (X)Q˙ z,w (X) is 1 if y = w and is 0 if y = w.

˙ z∈W

˙ and Let J˙ be the ring with Z-basis (t˙w )w∈W˙ deﬁned as in [L3, 18.3] in terms of W ˙ ˙ the weight function l|W˙ and let Φ : H → A ⊗ J be the A-algebra homomorphism deﬁned in [L3, 18.9]. (The deﬁnitions and results of [L3] that were just quoted are applicable in view of [L3, §16].) After applying Q⊗A to Φ we obtain an algebra ∼ ˙ ]→ ˙ Via this isomorphism any E ∈ IrrW ˙ becomes Q ⊗ J. isomorphism ΦQ : Q[W ˙ a simple Q ⊗ J-module denoted by E♠ . Let E ∈ Irrδ W , Δ ∈ ME . We deﬁne ˙ ˙ Δ : E♠ → E♠ by Δ(ξ) = ΦQ (Δ(Φ−1 Q (ξ))); we have tδ(w) (Δ(ξ)) = Δ(tw (ξ)) for any ˙ w ∈ W , ξ ∈ E♠ . After applying Q(v)⊗A to Φ we obtain an algebra isomorphism ∼ ˙ Via this isomorphism the simple Q(v)⊗ J-module ˙ ΦQ(v) : Q(v)⊗A H˙ → Q(v)⊗Z J. ˙ Q(v) ⊗Q E♠ (E as above) becomes a simple Q(v) ⊗A H-module denoted by Ev2 and the isomorphism 1 ⊗ Δ : Q(v) ⊗Q E♠ → Q(v) ⊗Q E♠ becomes an isomorphism ˙ , ξ ∈ Ev 2 . Δ : Ev2 → Ev2 such that T˙δ(w) (Δ(ξ)) = Δ(T˙w (ξ)) for any w ∈ W ˙ We assume that for each E ∈ Irrδ W we have chosen an element Δ ∈ ME .

¯ w are stable under φ := F d : B → B ˙ . Now Xw and X 2.3. Let w ∈ W (since φ commutes with F and it acts trivially on W ) and under σ : B → B (since σ commutes with F and σ(w) = w); hence they are F -stable and there are ¯ w ) and Hci (Xw ). Hence Xw , X ¯w induced automorphisms φ∗ , (F r )∗ , σ ∗ , F ∗ of Hi (X F i i F ˆ are stable under the G -action σ g : B → σ (Ad(g)B) (i ∈ [0, m − 1], g ∈ G ) ¯ w ) and Hci (Xw ) by σ i g → (σ ∗ )−1 Ad(g −1 )∗ . Let E˙ be a set ˆ F acts on Hi (X and G ˆF of representatives for the isomorphism classes of irreducible representations of G whose restriction to GF is a direct sum of unipotent representations. Let E0 be ˆ F -module; let E˙0 be the set of the set of all ρ ∈ E such that ρ extends to a G ˙ ˜Z be the Grothendieck group of all ρ˙ ∈ E˙ such that ρ| ˙ GF is irreducible. Let [E] F ˆ the category of representations of G whose restriction to GF are ﬁnite sums of ˙ ˜ := Q ⊗ [E] ˙ ˜Z has a basis {ρ; ˙ For ρ˙ ∈ E, ˙ unipotent representations. Then [E] ˙ ρ˙ ∈ E}. ˙ ˜ we denote by (ρ˙ : f )˜ the coeﬃcient of ρ˙ in f . Let [E] ˙ be the quotient of f ∈ [E] ˙ ˜ by the subspace consisting of all elements whose character restricted to GF σ [E] ˙ tr(xσ, h) makes sense for any x ∈ GF and we can is zero. Note that for h ∈ [E], ˙ deﬁne for ρ˙ ∈ E,

(ρ˙ : h) = |GF |−1

tr((xσ)−1 , ρ)tr(xσ, ˙ h).

F

x∈G

˙ ˜, h is its image in [E] ˙ and ρ˙ ∈ E, ˙ then We show that if f ∈ [E] (a)

(ρ˙ : h) =

(ρ˙ ⊗ χ : f )˜χ(σ)−1

χ∈L

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

233 7

ˆ F , trivial on GF . Indeed, where L is the set of one dimensional characters of G (ρ˙ ⊗ χ : f )˜χ(σ)−1 χ∈L

= |GF |−1 m−1

tr(xσ j , ρ˙ ⊗ χ)tr((xσ j )−1 , f )χ(σ)−1

χ∈L x∈GF ,j∈[0,m−1]

= |GF |−1 m−1

= |GF |−1

j −1 tr(xσ j , ρ)tr((xσ ˙ ) , f)

x∈GF ,j∈[0,m−1]

χ(σ j−1 )

χ∈L

−1 tr(xσ, ρ)tr((xσ) ˙ , f ) = (ρ˙ : h),

F

x∈G

as desired. ˙ ρ be the (one dimensional) subspace of [E] ˙ generated by For any ρ ∈ E0 let [E] ˙ ˙ GF . From the deﬁnitions we have a direct sum the elements ρ˙ ∈ E0 such that ρ = ρ| decomposition ˙ = ⊕ρ∈E [E] ˙ ρ. [E] 0 ˙ ρ . We have ˙ c = ⊕ρ∈E ;ρ∗ =c [E] For any c ∈ C we set [E] 0

˙ = ⊕c∈C [E] ˙ c. [E] The following result is clear from the deﬁnitions. ˙ be such that (ρ˙ : h) = 0 for any ρ˙ ∈ E. ˙ Then h = 0. (b) Let h ∈ [E] ˙ For w ∈ W let R˙ w = (−1)i Hci (Xw ) i

˙ we set ˙ For any E ∈ Irrδ W viewed as an element of [E]. ˙ |−1 ˙ tr(Δw, E)R˙ w ∈ [E]. R˙ E = |W ˙ w∈W

˙ and any i ∈ Z we deﬁne tr(Δh, Ev2 ; i) ∈ Z by For any ξ ∈ H˙ any E ∈ Irrδ W tr(Δh, Ev2 ; i)v i . tr(Δh, Ev2 ) = i

Part (i) of the following theorem is a partial generalization of [L2, 3.8(ii)]; part (ii) (a partial generalization of [L2, 4.23], see also 1.5) provides an aﬃrmative answer to a conjecture of G. Malle [Ma] (which he proved in the case where rank(G) ≤ 6). ˙ the following equality holds in Q[v] ⊗ [E]: ˙ Theorem 2.4. (i) For any w ∈ W ¯ w )v i = (a) (−1)i Hi (X tr(Δ P˙ y,w (v 2 )T˙y , Ev2 )R˙ E . i

˙ E∈Irrδ W

˙ y∈W

Equivalently, for any i ∈ Z we have ¯w) = tr(Δ P˙ y,w (v 2 )T˙y , Ev2 ; i)R˙ E . (−1)i Hi (X ˙ E∈Irrδ W

˙ y∈W

˙ c. ˙ . There exists c ∈ C such that R˙ E ∈ [E] (ii) Let E ∈ Irrδ W The proof of (i) (which has much in common with that of [L2, 3.8(ii)]) is given in 2.15. The proof of (ii) is given in 2.19 where the two-sided cell c in (ii) is explicitly described in terms of E (see 2.19(c)).

234 8

G. LUSZTIG

¯ l ; for any integer n we set q n/2 = 2.5. We ﬁx a square root q 1/2 of q in Q n/2 nr/2 ˙ ¯ ˙ ¯ l is viewed as an A-algebra via (q ) ,q =q . Let Hq = Ql ⊗A H where Q 1/2 v → q . ¯ l . For any w ∈ W ˙ we deLet F be the vector space of functions B F → Q ˙ ˙ ﬁne a linear map Tw : F → F by Tw (f ) = f for f ∈ F where f (B) = F ˙ B ∈B F ;(B,B )∈Ow f (B ) for B ∈ B . In this way F becomes a Hq -module. Note ˙ ˙ ˙ that the linear maps Tw : F → F (w ∈ W ) are linearly independent. For w, w ∈ W ˙ ˙ ˙ we have Tw Tw = w ∈W˙ Nw,w ,w Tw where 1/2 n

Nw,w ,w = |{B ∈ B F ; (B1 , B) ∈ Ow , (B, B2 ) ∈ Ow }| F for any (B1 , B2 ) ∈ Ow . Using this and 2.2(a) we see that Ny,sI ,z T˙z = R˙ x,y (q)q l(x) T˙xsI ˙ z∈W

˙ x∈W

˙ as linear maps F → F. Using the linear independence of the linear for any y ∈ W ˙ maps Tz we deduce (a) R˙ x,y (q)q l(x) = Ny,s ,xs I

I

˙ . for any x, y in W ˙ we deﬁne Bw ∈ B by the 2.6. Now let B, B ∗ be as in 2.1. For any w ∈ W ∗ conditions (B, Bw ) ∈ Ow , (Bw , B ) ∈ Ow−1 sI . Note that Bw is ﬁxed by F r : B → B ˙ let and by σ : B → B hence by F : B → B. For z ∈ W Bz = {B ∈ B; (B, B ) ∈ Oz }, B¯z = {B ∈ B; (B, B ) ∈ ∪z ∈W ;z ≤z Oz }, Az = {B ∈ B; (B , BzsI ) ∈ OsI }; z note that Bz , B¯z , A are stable under F r : B → B and under σ : B → B hence under F : B → B. ˙ . Let Kw = IC(B¯w , Q ¯ l ) be the intersection cohomology complex of Let w ∈ W ˙ such that y ≤ w, σ induces an isomorphism of local systems B¯w . For any y ∈ W ∼ ∼ i i Kw → HB Kw σ ∗ (Hi Kw |By ) → Hi Kw |By ; this induces an automorphism σ ∗ : HB y y since By ∈ By is ﬁxed by σ. The trace of this automorphism is denoted by ni,y,w,σ . The following generalization of [KL2, 4.3] was stated without proof in [L1, (8.1)]; its proof (which uses [KL2, 4.2]) is a generalization of that of [KL2, 4.3]. ˙ such that y ≤ w we have Theorem 2.7. For any y, w ∈ W P˙ y,w (X) = n2i,y,w,σ X i . i∈N

˙ we have |(Bz ∩ Ay )F | = Nz,s ,ys ; using From the deﬁnitions, for z, y ∈ W I I 2.5(a), it follows that (a) |(Bz ∩ Ay )F | = R˙ y,z (q)q l(y) . ¯ l ) of Note that Kw |B¯w ∩Ay is the intersection cohomology complex IC(B¯w ∩ Ay , Q B¯w ∩Ay since B¯w ∩Ay is open in B¯w . Now the Grothendieck-Lefschetz trace formula for F : B¯w ∩ Ay → B¯w ∩ Ay gives: i (−1)i tr(F ∗ , Hic (B¯w ∩ Ay )) = (−1)i tr(F ∗ , HB Kw ). i

B ∈(B¯w ∩Ay )F

i

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

235 9

˙ ) we obtain By specifying z ∈ W such that B ∈ OzF (so that z is necessarily in W (−1)i tr(F ∗ , Hic (B¯w ∩ Ay )) i

=

˙ ;z≤w B ∈(Bz ∩Ay )F z∈W

i (−1)i tr(F ∗ , HB Kw ).

i

i y F and Note that tr(F ∗ , HB Kw ) is independent of B when B runs through (Bz ∩A ) F i even when B runs through Bz (since the local system H Kw |Bz is B-equivariant for the obvious transitive B-action on Bz with connected isotropy groups); moreover we have Bz ∈ BzF and we deduce that (−1)i tr(F ∗ , Hic (B¯w ∩ Ay )) i

=

|(Bz ∩ Ay )F |

i (−1)i tr(F ∗ , HB Kw ); z i

˙ ;z≤w z∈W

using (a) and the fact that R˙ y,z (X) = 0 unless y ≤ z, we deduce that i (−1)i tr(F ∗ , Hic (B¯w ∩ Ay )) = (−1)i tr(F ∗ , HB Kw ). R˙ y,z (q)q l(y) z i

i

˙ ;y≤z≤w z∈W

By Poincar´e duality for intersection cohomology we have (−1)i tr(F ∗ , Hic (B¯w ∩ Ay )) = q l(w) (−1)i tr(F ∗−1 , Hi (B¯w ∩ Ay )) i

i

since B¯w ∩ A is of pure dimension l(w) (or is empty). By [KL2, 1.5, 4.5] we have y

i Kw ) tr(F ∗−1 , Hi (B¯w ∩ Ay )) = tr(F ∗−1 , HB y

hence q l(w)

i (−1)i tr(F ∗−1 , HB Kw ) y

i

=

R˙ y,z (q)q l(y)

i (−1)i tr(F ∗ , HB Kw ). z i

˙ ;y≤z≤w z∈W

i i Kw is zero if i is odd while if i is even, (F r )∗ acts on HB Kw By [KL2, 4.2], HB z z i/2 ∗ i i/2 as q times a unipotent transformation hence F acts on HBz Kw as q times a unipotent transformation times the action of σ ∗ (which commutes with the unipotent transformation, since σF r = F r σ). Thus we have i q l(w) q −i/2 tr(σ ∗−1 , HB Kw ) y i∈2N

=

R˙ y,z (q)q l(y)

˙ ;y≤z≤w z∈W

i q i/2 tr(σ ∗ , HB Kw ). z

i∈2N

˙ such that z ≤ w we set: Now for any z ∈ W i tr(σ ∗ , HB Kw )X i/2 ∈ R[X], P˜z,w (X) = z i∈2N

(X) P˜z,w

=

i∈2N

i tr(σ ∗−1 , HB Kw )X i/2 ∈ R[X] z

236 10

G. LUSZTIG

¯ l generated by the m-th roots of 1. By the deﬁnition of where R is the subring of Q ˜ intersection cohomology, Pz,w , P˜z,w are polynomials in X of degree ≤ (l(w) − l(z) − 1)/2 (if z < w) and are equal to 1 if z = w. We have (q −1 ) = q l(y) R˙ y,z (q)P˜z,w (q). q l(w) P˜y,w ˙ ;y≤z≤w z∈W

Since here q is an arbitrary power of q d it follows that (X −1 ) = X l(y) R˙ y,z (X)P˜z,w (X). (b) X l(w) P˜y,w ˙ ;y≤z≤w z∈W

Note also that

X l(w) P˙ y,w (X −1 ) =

(c)

X l(y) R˙ y,z (X)P˙ z,w (X).

˙ ;y≤z≤w z∈W

We show by induction on l(w) − l(y) that (d) P˜y,w (X) = P˜ (X) = P˙ y,w (X) y,w

˙ such that y ≤ w. If l(y) = l(w) we have y = w and all three terms in for any y ∈ W (d) are 1. Now assume that y < w and that the result is known when y is replaced ˙ where y < z ≤ w. Substracting (c) from (b) we then ﬁnd (using that by z ∈ W ˙ Ry,y = 1): (X −1 ) − P˙ y,w (X −1 )) = X l(y) (P˜y,w (X) − P˙ y,w (X)). X l(w) (P˜y,w

Thus (e) X (l(w)−l(y)/2 (P˜y,w (X −1 ) − P˙ y,w (X −1 )) = X (−l(w)+l(y))/2 (P˜y,w (X) − P˙ y,w (X)). −n/2 ; the right hand side of (e) The left hand side of (e) belongs to n>0 RX n/2 belongs to n>0 RX . Hence both sides must be zero. This proves (d). The theorem is proved.

˙ let Lw = IC(X ¯w, Q ¯ l ) be the intersection complex of X ¯ w (a 2.8. For w ∈ W F ¯ variety of pure dimension l(w)). For any x ∈ G and any B ∈ Xw such that i i Ad(x)F (B ) = B we have an induced isomorphism F ∗ Ad(x)∗ : HB Lw → HB Lw . We show: Proposition 2.9. In the setup of 2.8 we have ¯ w )) (−1)i tr(F ∗ Ad(x)∗ , Hi (X i

=

P˙ y,w (q)

˙ ;y≤w y∈W

(−1)i tr(F ∗ Ad(x)∗ , Hci (Xy )). i

By the Grothendieck-Lefschetz trace formula we have ¯ w )) (−1)i tr(F ∗ Ad(x)∗ , Hi (X i

(a)

=

¯ w ;Ad(x)F (B )=B B ∈X

i

i (−1)i tr(F ∗ Ad(x)∗ , HB Lw ).

¯w → X ¯ w is a Frobenius map relative to (Note that Ad(x)F = Ad(x)σF r : X ¯w an Fq -rational structure since Ad(x)σ is an automorphism of ﬁnite order of X

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

237 11

¯w → X ¯ w .) In the sum over B in (a) commuting with the Frobenius map F r : X ˙ . (Indeed we can specify y ∈ W such that B ∈ Xy ; we have necessarily y ∈ W assume that B ∈ Xy . Then Ad(x)F (B ) ∈ Xσ(y) ; hence if Ad(x)F (B ) = B then σ(y) = y.) Thus the right hand side of (a) becomes i (−1)i tr(F ∗ Ad(x)∗ , HB Lw ) ˙ ;y≤w B ∈Xy ;Ad(x)F (B )=B y∈W

i

˙ such that y ≤ w we have and it is enough to show that for any y ∈ W i (−1)i tr(F ∗ Ad(x)∗ , HB Lw ) B ∈Xy ;Ad(x)F (B )=B

i

= P˙y,w (q) (−1)i tr(F ∗ Ad(x)∗ , H i (Xy )). i

By the Grothendieck-Lefschetz trace formula we have (−1)i tr(F ∗ Ad(x)∗ , H i (Xy )) = |{B ∈ Xy ; Ad(x)F (B ) = B }|. (b) i

Thus it is enough to show that for any B ∈ Xy such that Ad(x)F (B ) = B we have i ˙ (−1)i tr(F ∗ Ad(x)∗ , HB Lw ) = Py,w (q). i

Let w˙ ∈ G be such that w(B ˙ ∩B ∗ )w˙ −1 = B ∩B ∗ and (B, wB ˙ w˙ −1 ) ∈ Ow . Let Gw = −1 ¯ ¯ w = ∪z∈W ;z≤w Gz B wB, ˙ Gw = {g ∈ G; g F (g) ∈ Gw }. Let Gw = ∪z∈W ;z≤w Gz , G ˜ w (resp. L ˜ w ) be the intersection cohomology be the closures of Gw , Gw in G. Let K ¯ w (resp. G ¯ w ) with coeﬃcients in Q ¯ l . Deﬁne α : G ¯ w → X ¯ w by complex of G −1 ¯ → G ¯ by Φ(g) = xF (g); g → gBg (a principal B-bundle). Deﬁne Φ : G w w ¯ w and α(Φ(g)) = note that Φ is a Frobenius map for an Fq -rational structure on G −1 ¯ . Hence α (B ) is Φ-stable. Since α−1 (B ) is a Ad(x)F (α(g)) for any g ∈ G w homogeneous B-space with a compatible Fq -rational structure given by Φ we can i i ˜ ﬁnd g ∈ α−1 (B ) such that Φ(g ) = g . We have canonically HB Lw = Hg Lw ∗ ∗ i ∗ i ˜ Φ and tr(F Ad(x) , HB Lw ) = tr(Φ , Hg Lw ); moreover we have g ∈ (Gy ) . Deﬁne ¯ w by g → g −1 F (g) (an ´etale covering). Deﬁne Φ : G ¯w → G ¯ w by ¯ w → G L : G ¯ w . We have canonically g → F (g); note that L(Φ(g)) = Φ (L(g)) for any g ∈ G ∗ i ˜ ∗ i Φ ˜ w = Hi K ˜ ˜ Hgi L L(g ) w and tr(Φ , Hg Lw ) = tr(Φ , HL(g) Kw ); moreover, L(g) ∈ Gy . ¯ w → B¯w by g → gBg −1 (a principal B-bundle). Note that β(Φ (g)) = Deﬁne β : G i ˜ ¯ w . We have canonically Hi K F (β(g)) for any g ∈ G L(g ) w = Hβ(L(g )) Kw and ˜ w ) = tr(F ∗ , Hi Kw ); moreover, β(L(g)) ∈ ByF . By the proof of tr(Φ∗ , Hi K L(g)

β(L(g))

2.7 we have i i tr(F ∗ , Hβ(L(g)) Kw ) = tr(F ∗ , HB Kw ) = ni,y,w,σ q i/2 . y

It remains to use the equality P˙ y,w (q) =

(−1)i ni,y,w,σ q i/2

i∈N

which follows from 2.7. The proposition is proved.

238 12

G. LUSZTIG

˙ . The equality in Proposition 2.9 can be rewitten as 2.10. Let w ∈ W ¯ w )) (−1)i tr((φ∗ )r σ ∗ Ad(x)∗ , Hi (X i

=

P˙ y,w (q0r )

˙ ;y≤w y∈W

(−1)i tr((φ∗ )r σ ∗ Ad(x)∗ , Hci (Xy )) i

for r ∈ {1, 2, 3, where q0 = (q )d . Here we can make formally r tend to zero and we obtain ¯ w )) (−1)i tr(σ ∗ Ad(x)∗ , Hi (X i

(a)

=

˙ ;y≤w y∈W

P˙ y,w (1)

(−1)i tr(σ ∗ Ad(x)∗ , Hci (Xy )). i

2.11. If x ∈ GF we have F (x) ∈ GF since F F = F F ; moreover we have F be the semidirect product of GF with the F rm (x) = x since F rm = F m . Let G cyclic group of order rm with generator θ in which we have θzθ −1 = F (z) for all z ∈ GF . For any z ∈ GF we deﬁne a linear map z : F → F (F as in 2.2) by z(f ) = f (f ∈ F) where f (B1 ) = f (z −1 B1 z) for B1 ∈ B F . Now z → z makes F into a GF -module. We deﬁne a linear map θ : F → F by f → f (f ∈ F) where f (B1 ) = f (F −1 (B1 )) for B1 ∈ B F (this is well deﬁned since F F = F F ). This F -module structure linear map together with the GF -module structure deﬁne a G on F. Following Shintani, for each x ∈ GF we deﬁne (noncanonically) an element x ˆ ∈ GF as follows. We write x = aF (a−1 ) with a ∈ G and we set x ˆ = a−1 F (a). F Note that x ˆ ∈ G (since F F = F F ) hence x ˆ : F → F is well deﬁned. We have the following result (compare [L2, 2.10]):

˙ and any x ∈ GF we have (with notation Proposition 2.12. For any w ∈ W of 2.11): (−1)i tr(F ∗ x∗ , Hci (Xw )) = tr(θ −1 x ˆ−1 T˙w−1 , F). i

By 2.9(b) with y replaced by w, the left hand side of the equality above is equal to |{B ∈ Xw ; Ad(x)F (B ) = B }| hence to |{B ∈ B; (B , F (B )) ∈ Ow , aF (a−1 )F (B )F (a)a−1 = B }|. Setting B1 = a−1 B a, we see that the last number is equal to |{B1 ∈ B; (aB1 a−1 , F (a)F (B1 )F (a−1 )) ∈ Ow ; F (B1 ) = B1 }| = |{B1 ∈ B F ; (B1 , x ˆF (B1 )ˆ x−1 ∈ Ow }|. ¯ l such that fB (B ) is For any B1 ∈ B F we denote by fB1 the function B F → Q 1 ˙ 1 if B = B1 and 0 if B = B1 . Let f1 = Tw−1 (fB1 ). We have f1 (B ) = 1 if / Ow−1 . Let f2 = x ˆ−1 (f1 ). We have (B , B1 ) ∈ Ow−1 , f1 (B ) = 0 if (B , B1 ) ∈ −1 −1 f2 (B ) = 1 if (ˆ xB x ˆ , B1 ) ∈ Ow−1 , f2 (B ) = 0 if (ˆ xB x ˆ , B1 ) ∈ / Ow−1 . Let f3 = θ −1 (f2 ). We have f3 (B ) = 1 if (ˆ xF (B )ˆ x−1 , B1 ) ∈ Ow−1 , f2 (B ) = 0 if

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

239 13

(ˆ xF x ˆ−1 (B ), B1 ) ∈ / Ow−1 . We see that ˆ−1 T˙w−1 , F) = |{B1 ∈ B F ; (ˆ xF (B1 )ˆ x−1 , B1 ) ∈ Ow−1 }| tr(θ −1 x = |{B1 ∈ B F ; (B1 , x ˆF (B1 )ˆ x−1 ) ∈ Ow }|. This completes the proof. ˙ , x ∈ GF . Combining 2.9 and 2.12 we obtain 2.13. Let w ∈ W ¯ w )) = (−1)i tr(F ∗ Ad(x)∗ , Hi (X ˆ−1 T˙y−1 , F). P˙ y,w (q)tr(θ −1 x (a) i

˙ ;y≤w y∈W

¯ w ) → Hi (X ¯ w ) (notation of 2.10) By [L2, 2.20, 3.8], for ρ ∈ E, i ∈ N, φ∗ : Hi (X i ¯ leaves stable H (Xw )ρ (notation of 1.6) and acts on it as λρ q id/2 U where U is a ¯ w )ρ and λρ is a root of 1 depending only on ρ unipotent transformation of Hi (X ∗ ¯ w )ρ to Hi (X ¯ w )ρ where ρ = ρ, while (not on w, i); also if ρ ∈ E − E0 , σ maps Hi (X ∗ i ¯ ∗−1 ¯ w )ρ into if ρ ∈ E0 then σ leaves stable H (Xw )ρ and the σ action makes Hi (X ˆ F -module. It follows that aG ¯ w )) = ¯ w )ρ ). (b) (−1)i tr(F ∗ x∗ , Hi (X (−1)i λr/d q i/2 tr(σ ∗ x∗ , Hi (X ρ

i

i

ρ∈E0

We can ﬁnd d ≥ 1 such that λdρ = 1 for all ρ ∈ E. From now on we assume that r is a multiple of dd . Then (b) becomes ¯ w )) = ¯ w ))˜tr((xσ)−1 , ρ). (c) (−1)i tr(F ∗ x∗ , Hi (X (−1)i q i/2 (ρ˙ : Hi (X ˙ i

i

ρ∈ ˙ E˙0

¯ l ⊗A to Φ in 2.2 (where Q ¯ l is viewed as an A-algebra 2.14. After applying Q ∼ ¯ ˙ ˙ via v → q 1/2 ) we obtain an algebra isomorphism Φq : H˙ q → Q l ⊗ J. If E ∈ IrrW ˙ ¯ ¯ then the simple Ql ⊗ J-module Ql ⊗ E♠ (see 2.2) can be viewed via Φq as a simple ˙ we deﬁne Δ : Eq → Eq to correspond H˙ q -module denoted by Eq . If E ∈ Irrδ W ¯ l ⊗ E♠ → Q ¯ l ⊗ E♠ ; we have T˙δ(w) (Δ(ξ)) = Δ(T˙w (ξ)) for any under Φq to 1 ⊗ Δ : Q ˙ w ∈ W , ξ ∈ Eq . ˙ let FE be the Eq -isotypic component of the Hq -module F; thus For E ∈ IrrW we have F = ⊕E∈IrrW˙ FE . Note that each FE is a GF -submodule of F. From the ˙ we have T˙w θ −1 = θ −1 T˙δ(w) : F → F. It follows that deﬁnitions, for any w ∈ W θ −1 : F → F permutes the summands FE of F according to the permutation of ˙ induced by δ : W → W . In particular only the summands corresponding to IrrW ˙ , we have ˙ are mapped into themselves. We see that for z ∈ GF , y ∈ W E ∈ Irrδ W tr(θ −1 z −1 T˙y−1 , FE ). tr(θ −1 z −1 T˙y−1 , F) = ˙ E∈Irrδ W

˙ we set VE = HomH (Eq , F); this is an irreducible GF -module (the For E ∈ IrrW q F G -module structure comes from that of F); we have a canonical isomorphism of ∼ ˙ , the (Hq , GF )-modules Eq ⊗ VE → FE . Via this isomorphism, assuming E ∈ Irrδ W −1 map θ : FE → FE becomes an isomorphism XE : Eq ⊗ VE → Eq ⊗ VE necessarily of the form XE = XE ⊗ XE where XE : Eq → Eq , XE : VE → VE are isomorphisms ˙ ˙ F (z) : VE → VE for any such that Tw XE = XE Tδ(w) : Eq → Eq and zXE = XE F z ∈ G . Thus XE acts on Eq as a nonzero constant times Δ−1 : Eq → Eq ; we may assume that the constant is 1 (by absorbing it into XE ) so that XE = Δ−1 .

240 14

G. LUSZTIG

rm m rm Since XE = 1 and (XE ) = 1 we see that (XE ) = 1 so that the GF -module structure on VE extends to a GF -module structure in which θ −1 acts as XE . We F ˙ see that for z ∈ G , y ∈ W , we have tr(θ −1 z −1 , VE )tr(Δ−1 T˙y−1 , Eq ). tr(θ −1 z −1 T˙y−1 , F) = ˙ E∈Irrδ W

We now substitute this (with z = x ˆ) into 2.13(a), taking into account 2.13(c) and using that tr(Δ−1 T˙y−1 , Eq ) = tr(T˙y Δ, Eq ) = tr(ΔT˙y , Eq ) we obtain ¯ w ))˜tr((xσ)−1 , ρ) (−1)i q i/2 (ρ˙ : Hi (X ˙ i

=

(a)

ρ∈ ˙ E˙0

P˙ y,w (q)

˙ y∈W

tr(θ −1 x ˆ−1 , VE )tr(ΔT˙y , Eq )

˙ E∈Irrδ W

˙ , x ∈ GF . for any w ∈ W ˙ and 2.15. Multiplying both sides of 2.14(a) by sgn(wu) ˙ Q˙ w,u (q) with u ∈ W ˙ we obtain summing over all w ∈ W ¯ w ))˜sgn(wu) (−1)i q i/2 (ρ˙ : Hi (X ˙ Q˙ w,u (q)tr((xσ)−1 , ρ) ˙ i

=

˙ ρ∈ w∈W ˙ E˙0

tr(θ −1 x ˆ−1 , VE )tr(ΔT˙u , Eq )

˙ E∈Irrδ W

˙ , x ∈ GF . Multiplying both sides of the last equality by for any u ∈ W |GF |−1 tr(xσ, ρ˙ ) with ρ˙ ∈ E˙0 and summing over all x ∈ GF we obtain ¯ w ))˜sgn(wu) (−1)i q i/2 (ρ˙ ⊗ χ : Hi (X ˙ Q˙ w,u (q)χ(σ)−1 i

(a)

˙ χ∈L w∈W F −1

= |G |

F

x∈G

tr(θ −1 x ˆ−1 , VE )tr(ΔT˙u , Eq )tr(xσ, ρ˙ )

˙ E∈Irrδ W

˙ , ρ˙ ∈ E˙0 . Here L is as in 2.3(a). We have used that, if ρ, for any u ∈ W ˙ ρ˙ ∈ E˙0 , F −1 −1 −1 , ρ)tr(xσ, ˙ ρ˙ ) is χ(σ) if ρ˙ = ρ˙ ⊗ χ for some χ ∈ L then |G | x∈GF tr((xσ) ˙ , and is 0, otherwise. Next we note that for E, E ∈ Irrδ W q l(u) tr(ΔT˙u , Eq )tr(ΔT˙u , Eq ) ˙ u∈W

is equal to 0 if E = E and is equal to D(E)(q) for some polynomial D(E)(X) ∈ Q[X] if E = E ; moreover, D(E)(q) is a nonzero rational number. ˙ Multiplying both sides of (a) by D(E )(q)−1 q l(u) tr(ΔT˙u , Eq ) with E ∈ Irrδ W ˙ we obtain and summing over all u ∈ W (−1)i q l(u) D(E )(q)−1 i

(b)

˙ χ∈L u∈W ˙ w∈W

¯ w ))˜sgn(wu) × tr(ΔT˙u , Eq )q i/2 (ρ˙ ⊗ χ : Hi (X ˙ Q˙ w,u (q)χ(σ)−1 = |GF |−1 tr(θ −1 x ˆ−1 , VE )tr(xσ, ρ˙ ) x∈GF

˙ . for any ρ˙ ∈ E˙0 , E ∈ Irrδ W

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

241 15

Let h(r) be the common value of the two sides of (b). We now show that (for ﬁxed ρ˙ , E ) h(r) has the following properties: (i) |GF |h(r) is a cyclotomic integer; (ii) when r varies (as a multiple of dd ), h(r) is in a ﬁxed (cyclotomic) subﬁeld ¯ of Ql of ﬁnite degree over Q; (iii) all complex conjugates of h(r) have absolute value ≤ 1. Now (i) is obvious from the right hand side of (b) since the character values of a ˆ F ) are cyclotomic integers. Also (ii) is obvious from ﬁnite group (namely GF and G the left hand side of (b); we use that there are only ﬁnitely many χ(σ) (roots of 1) and that tr(ΔT˙u−1 , Eq ) ∈ Q(q 1/2 ). To prove (iii) we set: tr(θ −1 x ˆ−1 , VE )tr(ˆ xθ, VE ) e(r) = |GF |−1 x∈GF

e (r) = |GF |−1

tr(σ −1 x−1 , ρ˙ )tr(xσ, ρ˙ ).

F

x∈G

By the Cauchy-Schwarz inequality, the absolute value square of h(r) (in the form given by the right hand side of (b)) is ≤ than e(r)e (r). Hence it is enough to show that e(r) = e (r) = 1. Now e (r) = 1 by the orthogonality relations for irreducible ˆ F which remain irreducible when restricted to GF . It remains to characters of G show that e(r) = 1. Setting x = aF (a−1 ), a ∈ G in the deﬁnition of e(r) we have e(r) = |GF |−1 |GF |−1 tr((a−1 F (a)θ)−1 , VE )tr(a−1 F (a)θ, VE ). a∈G; F (aF (a−1 ))=aF (a−1 )

(We use that, if a is in the sum and b ∈ GF then tr(a−1 F (a)θ, VE ) = tr(b−1 a−1 F (a)F (b)θ, VE ). It is enough to show that if z ∈ GF then tr(zθ, VE ) = tr(b−1 zF (b)θ, VE ) which follows from F (b)θ = θb.) Setting z = a−1 F (a) ∈ GF we obtain e(r) = |GF |−1 tr((zθ)−1 , VE )tr(zθ, VE ) z∈GF

and this equals 1 by the orthogonality relations for irreducible characters of GF F which remain irreducible when restricted to G . This proves (iii). If we combine the various imbeddings of the number ﬁeld in (ii) into C we get an imbedding of that ﬁeld into some Cn . By (i),(iii) the set {h(r)} is carried by that imbedding into a discrete and bounded subset of Cn . Hence the set {h(r)} is ﬁnite. This holds for any ρ˙ , E which are in ﬁnite number. We see that there exists an inﬁnite subset I of {dd , 2dd , 3dd , . . . } such that for any ρ˙ , E , the expression (b) is a constant cρ˙ ,E when r runs through I. Thus we have (−1)i q l(u) D(E )(q)−1 tr(ΔT˙u , Eq ) i

(c)

×q

˙ χ∈L u∈W ˙ w∈W i/2

¯ w ))˜sgn(wu) (ρ˙ ⊗ χ : Hi (X ˙ Q˙ w,u (q)χ(σ)−1 = cρ˙ ,E

˙ , r ∈ I. for any ρ˙ ∈ E˙0 , E ∈ Irrδ W

242 16

G. LUSZTIG

We multiply the two sides of 2.14(a) by |GF |−1 tr(xσ, ρ˙ ) with ρ˙ ∈ E˙0 and sum over all x ∈ GF ; we obtain ¯ w ))˜χ(σ)−1 (−1)i q i/2 (ρ˙ ⊗ χ : Hi (X

i

=

χ∈L

P˙ y,w (q)

˙ y∈W

|GF |−1 tr(xσ, ρ˙ )tr(θ −1 x ˆ−1 , VE )tr(ΔT˙y , Eq )

˙ x∈GF E∈Irrδ W

that is ¯ w ))˜χ(σ)−1 = (−1)i q i/2 (ρ˙ ⊗ χ : Hi (X P˙ y,w (q) i

˙ y∈W

χ∈L

cρ˙ ,E tr(ΔT˙y , Eq )

˙ E∈Irrδ W

˙ , ρ˙ ∈ E˙0 , r ∈ I. Since r runs through an inﬁnite set we have an for any w ∈ W equality of polynomials in v: (d) ¯ w ))˜χ(σ)−1 v i = (−1)i (ρ˙ ⊗ χ : Hi (X cρ˙ ,E tr(ΔT˙y , Ev2 ) P˙ y,w (v 2 ) i

˙ y∈W

χ∈L

˙ E∈Irrδ W

˙ , ρ˙ ∈ E˙0 . Similarly, since (c) holds for r running through an inﬁnite for any w ∈ W set, we have an equality of polynomials in v: (−1)i v 2l(u) D(E )(v 2 )−1 i

˙ χ∈L u∈W ˙ w∈W

¯ w ))˜sgn(wu) × tr(ΔT˙u , Ev 2 )(ρ˙ ⊗ χ : Hi (X ˙ Q˙ w,u (v 2 )χ(σ)−1 v i = cρ˙ ,E ˙ . Setting v = 1 in the last equality and using the for any ρ˙ ∈ E˙0 , E ∈ Irrδ W ˙ | we obtain identity D(E )(1) = |W ˙ |−1 cρ˙ ,E = (−1)i |W i

˙ χ∈L u∈W ˙ w∈W

¯ w ))˜sgn(wu) × tr(Δu, E )(ρ˙ ⊗ χ : Hi (X ˙ Q˙ w,u (1)χ(σ)−1 .

(e)

By 2.3(a) we have for ﬁxed ρ˙ : ¯ w ))˜χ(σ)−1 = ¯ w )) (−1)i (ρ˙ ⊗ χ : Hi (X (−1)i (ρ˙ : Hi (X i

i

χ∈L

which by 2.10(a) is equal to

P˙ y,w (1)

˙ ;y≤w y∈W

(−1)i (ρ˙ : Hci (Xy )). i

Substituting this into (e) we obtain ˙ |−1 tr(Δu, E )sgn(wu) cρ˙ ,E = |W ˙ Q˙ w,u (1) ˙ u∈W ˙ w∈W F −1

× |G |

˙ ;y≤w y∈W

P˙ y,w (1)

(−1)i (ρ˙ : Hci (Xy )). i

Using the deﬁnition of Q˙ w,u (1) we deduce ˙ |−1 tr(Δu, E )|GF |−1 (ρ˙ : R˙ u ) = (ρ˙ : R˙ E ) cρ˙ ,E = |W ˙ u∈W

REPRESENTATIONS OF DISCONNECTED REDUCTIVE GROUPS OVER Fq

243 17

˙ . Substituting this into (d) and using the equality for any ρ˙ ∈ E˙0 , E ∈ Irrδ W

¯ w ))χ(σ)−1 = (ρ˙ : Hi (X ¯ w )) (ρ˙ ⊗ χ : Hi (X

χ∈L

(see 2.3(a)), we obtain i

¯ w ))v i = (−1)i (ρ˙ : Hi (X

P˙ y,w (v 2 )

˙ y∈W

(ρ˙ : R˙ E )tr(ΔT˙y , Ev2 )

˙ E∈Irrδ W

˙ , ρ˙ ∈ E˙0 . The previous equality also holds for ρ˙ ∈ E˙ − E˙0 (in that for any w ∈ W case both sides are zero). This proves 2.4(a) in view of 2.3(b). ˙ (with the weight function l| ˙ ) deﬁned in ˙ LR be the preorder on W 2.16. Let ≤ W [L3, 8.1] where it is denoted by ≤LR and let ∼ ˙ LR be the corresponding equivalence ˙ (the equivalence classes are the two-sided cells of W ˙ with the weight relation on W ˙ ˙ ˙ ˙ LR w function above; they form a set C). For w, w ∈ W we write w
where c˙w,Δ,E ∈ Z; by an argument similar to that in [L3, 20.10], we have c˙w,Δ,E = † ) where Δ : E † → E † is Δ ⊗ 1 : E ⊗ sgn ˙ → E ⊗ sgn. ˙ tr(Δ , E♠ ˙ Let RW˙ be the vector space of formal linear combinations ˙ rE E, E∈Irrδ W ˙ we set rE ∈ Q. For w ∈ W ˙w = c˙w,Δ,E E ∈ R˙ W˙ . A ˙ E∈Irrδ W

From the deﬁnitions we see that ˙ w is a linear combination of E ∈ Irrδ W ˙ w ∈ c then A ˙ such that (a) If c ∈ C, ∗ cE = c . ˙ then Conversely, if E ∈ Irrδ W ˙ x (x ∈ c∗ ). (b) E is a Q-linear combination of elements A E The proof is along the lines of that of [L2, 5.13(ii)] (which is the analogous result ˙ ). for W instead of W ˙ In For any ξ = E∈Irrδ W˙ rE E ∈ R˙ W˙ we set R˙ ξ = E∈Irrδ W˙ rE R˙ E ∈ [E]. ˙ ˙ ˙ particular, for w ∈ W , RA˙ w ∈ [E] is deﬁned.

244 18

G. LUSZTIG

˙ . We have Proposition 2.17. Let w ∈ W ¯ w ) = (−1)l(w)+a(w) Hl(w)+a(w) (X ¯w ) (−1)l(w)−a(w) Hl(w)−a(w) (X ˙ (a) nw ,w R˙ A˙ w ∈ [E] = R˙ A˙ w + ˙ ,w < ˙ LR w w ∈W

for certain rational numbers nw ,w . The ﬁrst equality follows from the Lefschetz hard theorem in intersection coho¯ w which has pure dimension l(w). In the rest of the proof we mology applied to X concentrate on the second equality. Using the second part of 2.4 with i = l(w)+a(w) we see that it is enough to show that tr(Δ P˙ y,w (v 2 )T˙y , Ev2 ; l(w) + a(w))E ˙ E∈Irrδ W

(b)

˙w+ =A

˙ y∈W

˙ ,w < ˙ w ∈W

A˙ w LR w

(equality in R˙ W˙ ). Let c ∈ C˙ be such that w ∈ c. From the deﬁnitions we see that, ˙ , the operator ˙ P˙ y,w (v 2 )T˙y : Ev2 → Ev2 is zero unless c∗ ≤ ˙ for E ∈ Irrδ W E LR c. y∈W Thus the sum in the left hand side of (b) can be restricted to the E such that ˙ we have ˙ LR c. Next we show that for any E ∈ Irrδ W c∗E ≤ (b) tr(Δ P˙ y,w (v 2 )T˙y , Ev2 ) = c˙w,Δ,E v l(w)+aE + lower powers of v. ˙ y∈W

˙ , y < w we (Using the deﬁnition of c˙w,Δ,E , it is enough to show that for any y ∈ W have P˙ y,w (v 2 )tr(ΔT˙y , Ev2 ) ∈ v l(w)+aE −1 Z[v −1 ]; since tr(ΔT˙y , Ev2 ) ∈ v l(y)+aE Z[v −1 ], it is enough to note that P˙ y,w (v 2 ) ∈ v l(w)−l(y)−1 Z[v −1 ].) It follows that any E ˙ LR c and which appears with nonzero coeﬃcient in the ﬁrst sum in (b) satisﬁes c∗E ≤ ∗ that the contribution to that sum of the E such cE = c (hence aE = a(w), see [L3, 20.6(c)]) is c˙ E = A˙ w w,Δ,E

˙ ;c∗ =c E∈Irrδ W E

˙ LR c then E is a linear (see 2.16(a)). It remains to show that if E satisﬁes c∗E < ˙ ˙ ˙ combination of elements Aw with w ∈ W , w

LR c

˙c, [E]

˙ ≥c = ⊕c ∈C;c≤ c [E] ˙ c . [E] LR

By [L3, 23.5] any c ∈ C˙ is contained in a unique two-sided cell c! ∈ C; moreover, the ˙ = c). Also (c∗ )! = (c! )∗ . (Indeed, if map C˙ → C, c → c! is injective (we have c! ∩ W ! ∗ w ∈ c so that w ∈ c , we have wsI ∈ c so that wsI ∈ c∗ and wsI ∈ (c∗ )! ; but we have also wsI ∈ c! sI = (c! )∗ . Hence the desired equality.) We show: ˙ ≤c! . (a) If c ∈ C˙ and w ∈ c then R˙ A˙ w ∈ [E] ˙ , w < ˙ LR w. By We can assume that this is true if w is replaced by w with w ∈ W

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¯ w ) is a sum of representations isomorphic to ρ 1.7, the GF -module Hl(w)−a(w) (X ∗ ! l(w)−a(w) ¯ ˙ ≤c! . Using now 2.17 we see that such that ρ ≤LR c . Hence H (Xw ) ∈ [E] ˙ ≤c! nw ,w R˙ ˙ ∈ [E] (b) R˙ ˙ + ˙ ˙

Aw

w ∈W ,w
Aw

with nw ,w ∈ Q. By the induction hypothesis for any w in the last sum (with ˙ we have R˙ ˙ ∈ [E] ˙ ≤c ! . By arguments in [L3, §16] (see especially w ∈ c , c ∈ C) Aw ˙ LR w we deduce that w ≤LR w that is c! ≤ c! . Hence 16.6, 16.13(a)), from w ≤ ≤c! ≤c! ˙ ˙ ˙ ˙ ≤c! . Introducing this in (b) we see that R˙ ˙ ∈ [E] ⊂ [E] . Thus RA˙ w ∈ [E] Aw ˙ ≤c! . This proves (a). [E] We show: ˙ ≤(c∗E )! . ˙ then R˙ E ∈ [E] (c) If E ∈ Irrδ W This follows from (a) using 2.16(b). 2.19. Recall the operation of “duality” [DL2, DL3] D : [E] → [E] which is a linear map such that D(ρ) = ρ˜ where ρ → ρ˜ is an involution of E and such that D(Rw ) = sgn(w)Rw for any w ∈ W . It follows that if E ∈ Irrδ W then D(RE ) = ±RE † . Hence if ρ ∈ E, then ρ ˜ = ρ∗ . ˙ → [E] ˙ with The operator D is generalized in [DM, §3] to a linear map D˙ : [E] ˙ ρ) the following properties: (i) if ρ˙ ∈ E˙0 and ρ| ˙ GF = ρ ∈ E0 then D( ˙ is up to scalar ˙ R˙ w ) = sgn(w) ˙ R˙ w for any of the form ρ˙ ∈ E˙0 where ρ˙ |GF = ρ˜ (as above); (ii) D( ˙ ˙ ˙ ˙ ˙ w ∈ W . It follows that if E ∈ Irrδ W then D(RE ) = ±RE † . We show: ˙ E] ˙ ≤c ) = [E] ˙ ≥c∗ . (a) if c ∈ C then D([ It is enough to show that, if ρ ∈ E0 , ρ˜ ∈ E0 are as above, then ρ∗ ≤LR c implies ρ ˜∗ ≥LR c∗ , that is ρ ≥LR c∗ . But this follows from the fact ∗ reverses the preorder ≤LR . Applying D˙ to 2.18(c) with E replaced by E (as above) and using (a) we obtain ˙ ≥((c∗E )! )∗ . We have ((c∗ )! )∗ = (c! )∗ = (c∗ )! . Hence R˙ E ∈ [E] E E E ˙ ≥(c∗E )! . (b) R˙ E ∈ [E] ˙ ≤(c∗E )! ∩ [E] ˙ ≥(c∗E )! . Hence From (b) and 2.18(b) we deduce that R˙ E ∈ [E] ˙ (c∗E )! . (c) R˙ E ∈ [E] This proves 2.4(ii). From (c) we deduce, using 2.16(a), that for any c ∈ C˙ and any w ∈ c we have ˙ c! . (d) R˙ A˙ w ∈ [E] ˙ w ∈ c. Let πc! : [E] ˙ → [E] ˙ c! be the canonical projection. We 2.20. Let c ∈ C, apply πc! to the equality between the ﬁrst and third member of 2.17(a) and we use 2.19(d). We obtain ¯ w )) = R˙ ˙ . (−1)l(w)−a(w) πc! (Hl(w)−a(w) (X Aw ˙ , w < ˙ LR w then πc! (R˙ A˙ ) = 0. (Indeed, let c ∈ C˙ be We have used that if w ∈ W w such that w ∈ c . We have c = c. Using 2.19(d) for w instead of w it is enough to show that c! = c! ; this follows from the injectivity of the map c → c! .) Note that ¯ w ) (see 1.6) is a G ˆ F -submodule of Hl(w)−a(w) (X ¯ w ); moreover from Hl(w)−a(w) (X l(w)−a(w) ¯ ¯ w ) (equality in [E]). ˙ (Xw )) = Hl(w)−a(w) (X the deﬁnitions we have πc! (H Thus we have the following generalization of 1.6(c): (a)

¯ w ) = R˙ ˙ . (−1)l(w)−a(w) Hl(w)−a(w) (X Aw

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In particular, the following generalization of 1.6(d) holds: (b)

ˆ F -module. (−1)l(w)−a(w) R˙ A˙ w can be represented by an actual G

2.21. Assume now that G is an even special orthogonal group over Fq and that ˆ is an even full orthogonal σ is conjugation by a reﬂection deﬁned over Fq so that G group over Fq . Using the results of this paper (especially 2.20(b)) one can show ˙ then there exists c ∈ C such that that if E ∈ Irrδ W 2n R˙ E = ρ˜ ρ∈E;ρ=c

where n is deﬁned by |{ρ ∈ E; ρ = c}| = 22n and for each ρ in the sum, ρ˜ denotes a certain extension of ρ to a GF -module. The proof will be given elsewhere. References [DL1] P.Deligne and G.Lusztig, Representations of reductive groups over ﬁnite ﬁelds, Ann.Math. 103 (1976), 103-161. [DL2] P.Deligne and G.Lusztig, Duality for representations of a reductive group over a ﬁnite ﬁeld, J.Alg. 74 (1982), 284-291. [DL3] P.Deligne and G.Lusztig, Duality for representations of a reductive group over a ﬁnite ﬁeld, II, J.Alg. 81 (1983), 540-549. ´ [DM] F.Digne and J.Michel, Groupes r´ eductifs nonconnexes, Ann.Ec.Norm.Sup. 27 (1994), 345406. [KL1] D.Kazhdan and G.Lusztig, Representations of Coxeter groups and Hecke algebras , Inv.Math. 53 (1979), 165-184. [KL2] D.Kazhdan and G.Lusztig, Schubert varieties and Poincar´ e duality, Proc.Symp.Pure Math., vol. 36, Amer.Math.Soc., 1980, pp. 185-203. [L1] G.Lusztig, Left cells in Weyl groups, Lie groups representations, LNM 1024, Springer Verlag, 1983, pp. 99-111. [L2] G.Lusztig, Characters of reductive groups over a ﬁnite ﬁeld, Ann.Math.Studies 107, Princeton U.Press, 1984, 384p. [L3] G.Lusztig, Hecke algebras with unequal parameters, CRM Monograph Ser.18, Amer. Math. Soc., 2003, 136p. [Ma] G.Malle, Generalized Deligne-Lusztig characters, J.Alg. 159 (1993), 64-97. [Sp] N.Spaltenstein, Classes unipotentes et sous-groupes de Borel, LNM 946, Springer Verlag, 1982. Department of Mathematics, M.I.T., Cambridge, Massachusetts 02139

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

Forced Gradings in Integral Quasi-hereditary Algebras with Applications to Quantum Groups Brian J. Parshall and Leonard L. Scott Abstract. Let O be a discrete valuation ring with fraction ﬁeld K and residue over O provides a bridge between the ﬁeld k. A quasi-hereditary algebra A K := K ⊗ A over representation theory of the quasi-hereditary algebra A over k. In one the ﬁeld K and the quasi-hereditary algebra Ak := k ⊗O A K –mod is a full subcategory of the category of modimportant example, A k –mod is a full subcategory ules for a quantum enveloping algebra while A of the category of modules for a reductive group in positive characteristic. This paper considers ﬁrst the question of when the positively graded algebra n n+1 := gr A AK ) is quasi-hereditary. A main ren≥0 (A ∩ rad AK )/(A ∩ rad sult gives suﬃcient conditions that gr A be quasi-hereditary. The main require ment is that each graded module gr Δ(λ) arising from a A-standard (Weyl) module Δ(λ) have an irreducible head. An additional hypothesis requires that K be quasi-hereditary, a property recently proved [16] the graded algebra gr A to hold in some important cases involving quantum enveloping algebras. In the arises from regular dominant weights for a quantum enveloping case where A algebra at a primitive pth root of unity for a prime p > 2h − 2 (where h is the is quasi-hereditary. Coxeter number), a second main result shows that gr A The proof of this diﬃcult result involves interesting new methods involving integral quasi-hereditary algebras. It also depends on previous work [16] of the authors, including a continuation of the methods there involving tightly graded subalgebras, and a development of a quantum deformation theory over O = Z(p) (ζ), where ζ is a pth root of unity. This (new) deformation theory, given in Appendix II (§7), extends work of Andersen-Jantzen-Soergel [2], and is worthy of attention in its own right. As we point out, the present paper provides an essential step in our work on p-ﬁltrations of Weyl modules for reductive algebraic groups over ﬁelds of positive characteristic.

1. Introduction Quasi-hereditary algebras are certain ﬁnite dimensional algebras over a ﬁeld which arise naturally in Lie-theoretic representation theory, in both geometric and algebraic contexts. For example, if G is a reductive algebraic group in positive characteristic, the category of ﬁnite dimensional rational G-modules generated by the irreducible modules having highest weights in a ﬁnite saturated set of dominant 2000 Mathematics Subject Classiﬁcation. Primary 17B55, 20G; Secondary 17B50. Research supported in part by the National Science Foundation DMS-1001900. c Mathematical c 0000 (copyright Society holder) 2012 American

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BRIAN J. PARSHALL AND LEONARD L. SCOTT

weights is equivalent to the module category A–mod for a quasi-hereditary algebra A; see [3]. In a geometric vein, certain categories of perverse sheaves are similarly equivalent to A–mod for a quasi-hereditary algebra A; see [14]. On the other hand, quasi-hereditary algebras are deﬁned abstractly, so they are objects of interest to ﬁnite dimensional algebraists; see [19], for example. In [16], the authors studied naturally occurring hypotheses which, given a quasi-hereditary algebra A, guarantee a quasi-hereditary structure on the positively graded algebra gr A := radi A/ radi+1 A. For general quasi-hereditary algebras A, it is quite unexpected that gr A should also be quasi-hereditary, though [16] demonstrated that this does occur in several situations of interest in the theory of quantum and algebraic groups. over a commutative ring O In [4], the notion of an integral quasi-herediary A was introduced. See §2 below for a summary. In this paper, O is taken to be a DVR with fraction ﬁeld K and residue ﬁeld k = O/(π). We take up the question over O, the positively graded of when, for the integral quasi-hereditary algebra A algebra A ∩ radi A K := gr A i+1 ∩ rad K A A i≥0

is an integral quasi-hereditary algebra over O. The (integral) quasi-heredity of implies, for example, that the K-algebra A K := K ⊗O A and the k-algebra A A = Ak := A ⊗O k are both quasi-hereditary. We will always assume that A is associated with a ﬁxed weight poset Λ indexing its irreducible modules, over provides a link between the either K or k, and is “split” (see §2). The algebra A K and the k-algebra A. representation theory of the K-algebra A is quasi-hereditary over O, then gr A K is also quasi-hereditary Of course, if gr A (with the same weight poset Λ as A, and with standard modules gr Δ(λ) K )—already K is a diﬃcult property to prove. We simply assume, as a starting point, that gr A quasi-hereditary. In fact, in the applications we have in mind, this assumption will is integral be valid. The ﬁrst main theorem, Theorem 4.17, establishes that gr A and standard modules gr Δ(λ)) quasi-hereditary (with the same poset as A if and 1 only if each gr Δ(λ), λ ∈ Λ, has a simple head. Here Δ(λ) ∩ radi Δ(λ) K gr Δ(λ) := . i+1 Δ(λ)K i≥0 Δ(λ) ∩ rad with standard ﬁltrations Corollary 4.15 gives conditions under which A-modules M with standard ﬁltrations. give rise to gr A-modules gr M In §4, we show that the simple head hypothesis required in Theorem 4.17 holds, are satisﬁed.2 provided certain conditions involving an integral subalgebra a of A The conditions are stated in Conditions 5.1 and the hypotheses of Theorem 5.3. The latter theorem is based on the (new) notion of a tight lattice (see Deﬁntion 2.2 just below). All this is pulled together in Corollary 5.4 and Theorem 5.5. 1These gr-constructions make sense for any integral domain and lattices for algebras over them, and, as far as we know, have not been previously studied. 2The use of such a special subalgebra has been previously explored by us in [16], and this section is a natural extension of those results.

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In §5, we consider the case in which the integral quasi-hereditary algebra A arises as a suitable “regular weight” homomorphic image of a quantum enveloping algebra at a root of unity. (The regular blocks of the famous q-Schur algebras provide examples of such algebras.) The ﬁrst main result here is Theorem 6.1 which veriﬁes that Conditions 5.1 hold in case p > h (the Coxeter number) and A is associated to a ﬁnite ideal of regular weights. Then the main result in this section, is a quasi-hereditary algebra given in Theorem 6.3, shows that, when p ≥ 2h−2, gr A with standard modules gr Δ(λ). Its proof completes the veriﬁcation process begun in Theorem 6.1 by checking that the hypotheses of Theorem 5.3 hold, so that all the results of §4 are available. A key step in the proof of Theorem 6.1, namely, the veriﬁcation of Condition 5.1(5), is postponed to Appendix II (§6), entitled “Quantum deformation theory over O.” The appendix is inspired by, and heavily uses, the deformation theory in the paper [2], in which Andersen-Jantzen-Soergel showed that the (known to be true) Lusztig conjecture for quantum groups at a pth root of unity implies the characteristic p version of the conjecture for primes p 0, depending on the root system. The paper [2] presents two parallel deformation theories, the ﬁrst in positive characteristic for restricted Lie algebra representations, and the second for the small quantum group at a root of unity ζ. In our appendix, we extend the results of [2] to include a third deformation theory, for a natural integral form of the small quantum group over an appropriate DVR O ⊆ Q(ζ) with ζ a pth root of unity. We require that p be a prime > h. The main consequence of this theory is the demonstration of a positive grading on the O-integral form which base changes to the grading on the small (“regular block”) quantum group over Q(ζ) studied in [2, §17, §18]. See Theorem 8.1, which can also be regarded as a main result of this paper. This is just what is needed to complete the veriﬁcation of Condition 5.1(5), as required in the proof of Theorem 6.1(c). Observe that [2, §18] is quoted earlier in checking Condition 5.1(1) (in “case 2” of [2]). One area where the results of this paper are very useful is the study of ﬁltra and for A = A k . For example, an important tions of standard modules both for A question concerns the existence of p-ﬁltrations of Weyl modules Δ(λ) (even in the non-regular case) for a reductive group G, i. e., ﬁltrations with sections of the form L(μ0 ) ⊗ Δ(μ1 )(1) with μ0 a p-restricted dominant weight and μ1 an arbitrary dominant weight. See [1], though the terminology goes back earlier to an 1990 MSRI lecture of Donkin, and the concept already appears in the work of Jantzen [9]. Of course, anything that can be said about ﬁltrations of standard modules for gr A and gr A can be phrased as statements about ﬁltrations of standard modules for and A. The additional structure aﬀorded by the gradings makes such ﬁltraA tion assertions even more interesting. Indeed, [17], which uses our results here to present new results on p-ﬁltrations of Weyl modules for semisimple groups from this perspective, using tools unavailable in the ungraded (or non-integral) setting. We also expect to use [17] and the Lusztig modular character conjecture to improve the Koszulity theory given in [16]. One unexpected feature of our results in the present paper is that the Lusztig modular conjecture is not required. We do use the root of unity quantum analog of the conjecture, but that is known to be a theorem for p > h [21].

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2. Some preliminary notation/results Let (K, O, k) is a p-local system, i. e., O is a discrete valuation ring with maximal ideal m = (π), fraction ﬁeld K, and residue ﬁeld k = O/m.3 The ﬁeld k is allowed to have arbitrary characteristic; in our applications it will have positive characteristic p. Let AO be an O-algebra, which will always be assumed to be an O-free module of ﬁnite rank. Base change to K and k deﬁnes algebras AK := K ⊗O AO and Ak := k ⊗O AO = AO /πAO over K and k, respectively. It will be often convenient and denote Ak simply by A. More generally, if M is an A to denote AO by A ∼ module, write MK for K ⊗O M and M for Mk = k ⊗O M = M /π M ; sometimes, is said to be a A-lattice for M k . The A-module M if it is O-ﬁnite and we write M torsion free. (This deﬁnition applies when O is any integral domain; see Reiner [18, of a ﬁnite dimensional A K -module N pp. 44, 129]. Any O-ﬁnite A-submodule M is a A-lattice, and is said to be a full lattice in N if also K M = N , in which case ∼ be the category of O-ﬁnite A-modules. The category K ⊗O M = N . Let A-mod A-mod contains all A-lattices, and also contains A–mod as a full subcategory. It will often be very convenient, for a non-negative integer n, to denote the n ∩ radn A by K )n , the nth power of K of A Here radn A K = (rad A ideal A rad A. K . Of the radical (= maximal nilpotent ideal) of the ﬁnite dimensional algebra A n n n course, if n ≥ 1, rad A is a nilpotent ideal in AK and (rad A)K = rad AK . Deﬁne A ∩ radn A K n n+1 A/ := (2.1) gr A = rad rad A. n+1 ∩ rad K A A n≥0

n≥0

K = is a ﬁnite O-free module and (gr A) K∼ Necessarily the graded algebra gr A = gr A n n+1 ⊕n≥0 rad AK / rad AK . A similar notation will be used for modules and lattices. For any A-lattice n n n M and non-negative integer n, let rad M = M ∩ rad MK (where rad MK = K ), and set K )M (radn A (2.2)

:= gr M

M ∩ radn M K n n+1 / . = rad M rad M n+1 ∩ rad K M M

n≥0

n≥0

is a graded gr A-lattice, Then gr M and )K ∼ K = ⊕n≥0 radn M K / radn+1 M K . (gr M = gr M There are one-to-one correspondences of isomorphism classes of irreducible modules: K ) ↔ Irr(gr A K ) ↔ Irr(A K ). (2.3) Irr((gr A) Hence, ∩ rad A K is an O-pure nilpotent ideal in A. Also, A k ) ↔ Irr(gr A) ↔ Irr((A/( k )) ↔ Irr(A) ↔ Irr(A). (2.4) Irr((gr A) radA) The relation between the sets Irr(AK ) and Irr(A) may be subtle (or non-existent); in which the however, in the next section, we consider a family of O-algebras A 3In the text, given a ring or algebra R, an R-module M is ﬁnite (or R-ﬁnite) if it is ﬁnitely generated as an R-module.

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irreducible A-modules correspond bijectively, up to isomorphism, to the irreducible K -modules. Further, there is an evident common indexing set Λ. If λ ∈ Λ, we will A K -(resp., A-) module. K (λ) (resp., L(λ)) be the corresponding irreducible A let L The following general lemma which will be needed in §5. Lemma 2.1. Suppose aK ⊆ bK ⊆ AK are ﬁnite dimensional K-algebras. Assume that every irreducible AK -module is absolutely irreducible and a direct sum of irreducible bK -modules, each of which is absolutely irreducible over aK . Then (a) bK ∩ rad AK = rad bK , and aK ∩ rad AK = aK ∩ rad bK = rad aK . (b) The algebras aK and bK (as well as AK ) are split semisimple. Any Wedderburn complement aK,0 of aK is contained in a Wedderburn complement bK,0 of bK , and every Wedderburn complement bK,0 of bK is contained in a Wedderburn complement AK,0 of AK . Proof. We ﬁrst prove (a). Since bK ∩ rad AK is a nilpotent ideal in bK , we have bK ∩rad AK ⊆ rad bK . Since rad bK kills every irreducible AK -module, we also have rad bK ⊆ rad AK , and so rad bK ⊆ bK ∩ rad AK . Thus, bK ∩ rad AK = rad bK . A similar argument shows that aK ∩ rad AK = rad aK . Since aK and bK are subalgebras of AK , all their irreducible modules can be found as composition factors of irreducible AK -modules. Thus, all irreducible aK -modules are absolutely irreducible, and all irreducible bK -modules are are absolutely irreducible. In particular, aK / rad aK and bK / rad bK are split semisimple algebras, as is AK / rad AK . By part (a) above, we have aK / rad aK ⊆ bK / rad bK ⊆ AK / rad AK . Now (b) follows from standard separability arguments (vanishing of Hochschild 1- and 2-cohomology). The following general deﬁnition makes sense when O is any integral domain, though we will apply it only in our DVR context. Definition 2.2. Let b be an O-ﬁnite and torsion free O-algebra, and let M be a b-lattice. Call M is tight if (2.5)

r r , = ( rad M b)M rad

∀r ∈ N.

, the left-hand side always contains the rightObserve that, for any b-lattice M = hand side in (2.5), and that equality holds for M b or any ﬁnite projective b-module, which are, thus, tight lattices for b. Finally, this section concludes with the following well-known elementary lemma which will often be used without comment. Part (a) is [18, Thm. 4.0]. Part (b) gives one of the many characterizations of purity in the case of a DVR. We leave the easy proof to the reader. be a submodule of an Lemma 2.3. Let R be an integral domain, and let N R-ﬁnite, torsion free module M . (That is, M is an R-lattice in the sense of [18, p. 44] Then is pure in M (meaning that if M /N is torsion free) if and only if N = (a) N M ∩ NK , where K is the fraction ﬁeld of R; is pure in M if and only (b) if R = O is a DVR with fraction ﬁeld K, then N k obtained by base-change is injective. k → M if the natural map N

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3. Integral quasi-hereditary algebras We maintain the general notation from above, unless otherwise noted. As is a “split” quasi-hereditary algebra (QHA) over O in the sense of [4, sume that A Defn. (3.2)]. This deﬁnition requires that there exists a sequence 0 = J0 ⊂ J1 ⊂ of ideals in A such that Ji /Ji−1 is an heredity ideal in A/ Ji−1 for · · · ⊂ Jt = A 4 0 < i ≤ t. K –mod and A–mod are highest weight categories (HWCs) [3] It follows that A with the same poset Λ indexing the (isomorphism classes of) irreducible modules K (λ) (resp., L(λ)) for A K (resp., A). Indeed, by [7, Prop. 2.1.1], the category L A-mod is an O-ﬁnite highest weight category in the sense of [7, Defn. 2.1].5 The is “split” implies that the irreducible modules for A K and A are assumption that A are absolutely irreducible. Also, the category A-mod has standard modules Δ(λ), K -mod (resp., λ ∈ Λ, which are A-lattices with the property that the HWCs A k (λ) := A–mod) has standard objects {ΔK (λ) := Δ(λ)K }λ∈Λ (resp., {Δ(λ) = Δ Δ(λ)k }λ∈Λ ). Similarly, A-mod has costandard objects ∇(λ), λ ∈ Λ, etc. Each irreducible A-module L(λ) has a (unique, up to isomorphism) projective cover P(λ) (i. e., a in A-mod; see [7, Prop. 2.3] or [4, p. 157]6, and P (λ) has a Δ-ﬁltration ﬁltration with sections isomorphic to Δ(μ), μ ∈ Λ). In this ﬁltration, Δ(λ) appears as the top section, and all other sections Δ(μ) have μ > λ. Clearly, P(λ) is a A-lattice, and a direct summand of A (viewed as a module over itself). Moreover, decomposes a direct sum of various copies of P(λ) (each the (left) A-module A appearing dim L(λ) times). Observe that gr P(λ) is the projective cover of L(λ) in either gr A-mod (un graded module category) or in gr A-grmod (graded module category). This claim follows from dimension considerations, since (gr P (λ))0 , as a nonzero quotient of ∼ 0∼ P(λ), has L(λ) as a homomorphic image. (Note head A = head A.) = head (gr A) While P(λ)k is the projective cover P (λ) in A–mod of L(λ), it is not generally K (λ) in A K –mod. true that P (λ)K = P(λ)K is the projective cover PK (λ) of L However, we do have (3.1) P (λ)K ∼ PK (μ)⊕mλ,μ , = PK (λ) ⊕ μ>λ

for non-negative integers mλ,μ . This follows since ΔK (λ) is a quotient of P(λ)K , and since all other standard modules ΔK (μ) appearing in any ΔK -ﬁltration of P(λ)K have weights μ > λ. Applying the functor gr to (3.1) gives a similar decomposition 4An ideal J is heredity provided that (i) A/ J is O-free; (ii) J2 = J; (iii) J is a projective left is semisimple over O. The split case, used here, requires (over A-module; and (iv) E = EndA (J) DVRs, such as O) the stronger, but easier to state, condition that E is a direct sum of full matrix algebras Mn (O) over O. 5See also [7, Prop. 2.1.2, Rem./Defn. 2.1.3], which explains the essential equivalence of the O-ﬁnite highest weight category notion (in the ﬁnite weight poset case) with that of a split QHA (over a commutative Noetherian ring O, which is a DVR here). If A-mod is a HWC with poset Λ, we often informally say that A is a QHA with poset Λ. 6The arguments in [7] and [4] do not require that O be complete, or make any use of a completion of O.

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of gr (P(λ)K ) = (gr P(λ))K , although we, as yet, know little about gr ΔK -ﬁltrations of gr (P(λ))K or of PK (λ). This issue will be addressed in Hypothesis 4.7 below. 4. A main result This section approaches the question of determining conditions on an integral quasi-hereditary algebra A˜ over O which guarantee that the graded algebra gr A as deﬁned in (2.1) is quasi-hereditary. The ﬁrst step is to ﬁnd some properties and gr A might “obviously” share, at least in common situations. To this end A we introduce a general framework of “weight algebras”. In this framework we can replace the DVR O by a general commutative ring R. If p ∈ SpecR and if M is an R-module, let Mp denoted the Rp -module obtained by localizing M at p. Finally set M (p) := Mp /pMp , a vector space over the residue ﬁeld R(p) := Rp /pRp . Let B be an R-ﬁnite algebra. We call B a weight algebra, if there is a set {eν | ν ∈ X} of orthogonal idempotents in B, summing to 1, such that, for each p ∈ Spec R, some subset Xp of X bijectively indexes the irreducible Bp -modules in such a way that, if ν ∈ Xp corresponds to the irreducible Bp -module Lp (ν), then eν Lp (ν) = 0. Somewhat loosely, we refer to X as the set of “weights” of B. If M is a B-module and ν ∈ X, we write Mν := eν M , the ν-weight space of M . In this paper, we only consider weight algebras B as above for which the sets Xp can be uniformly chosen to be the same subset Λ of X. In this case, we say B is Λ-uniform. The weights in X\Λ then becomes largely irrelevant. In particular, we note the following. Proposition 4.1. Suppose that B is a Λ-uniform weight algebra with weight set X. Then B is Morita equivalent to a Λ-uniform weight algebra with weight set Λ. Proof. Put P = because of λ∈Λ Beλ . Then P is a projective generator ∼ the non-vanishing conditions eλ Lp (λ) = 0 above. Thus, B := λ,μ∈Λ eλ Beμ = op (EndB P ) is R-ﬁnite as a module and is Morita equivalent to B. The idempotents eλ , λ ∈ Λ, all belong to B , and sum to the identity in B . It is easily checked that these idempotents, indexed by Λ, give B the structure of a Λ-uniform weight algebra with weight set Λ. Some useful concepts can be deﬁned in the generality of Λ-uniform weight algebras B. Let Γ is a nonempty subset of Λ. Let N be a ﬁnite B-module. Let NΓ be the largest quotient module for which, given ν ∈ Λ, eν NΓ = 0 implies ν ∈ Γ. Explicitly, NΓ = N/ ν∈Λ\Γ Beν N . Considering B as a left module over itself, it follows that BΓ is an R-algebra, and NΓ is a BΓ -module. The notation agrees with is a split QHA over O the terminology at the end of Example 2.1. In case B = A Γ is an integral QHA with with weight poset Λ, then, in case Γ is an ideal in Λ, A indexed by poset Γ and standard modules Δ(γ), γ ∈ Γ (the standard modules of A elements in Γ). The following deﬁnition introduces an even stronger property for weight algebras which will hold in our applications. The deﬁnition is somewhat redundant, in that conditions (1) and (2) on the set Λ of weights for B imply already that B is Λ-uniform, and give a “natural” indexing of irreducible modules. Definition 4.2. Let B a Λ-uniform weight algebra. Then B is Λ-standard if the following conditions hold:

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(1) dim Lp (λ)λ = 1 for all λ ∈ Λ and all p ∈ Spec R. (Here the dimension is computed over the residue ﬁeld R(p) of Rp .) (2) There is a poset ordering ≤ on Λ such that, for all p ∈ Spec R, and all λ, μ ∈ Λ, if Lp (λ)μ = 0, then μ ≤ λ. When R = O, then Spec O = {m, 0}, with O(0) = K and O(m) = k. The following lemma, which applies for any Noetherian integral domain R, is one of the main points of the “weight algebra” development of this section. The proof is essentially obvious, noting that the ideal B ∩ rad BK and its gr B counterpart are both nilpotent ideals. The Noetherian hypotheses on R insures that gr B is ﬁnite B-module. Lemma 4.3. Assume that B is a weight algebra over a Noetherian integral n B rad domain R with fraction ﬁeld K, and put gr B = n≥0 . If {eμ | μ ∈ X} is n+1 rad B a set of idempotents giving B the structure of a weight algebra over R, the “same” set {eμ | u ∈ X} ⊆ (gr B)0 ⊆ gr B gives gr B the structure of a weight algebra. If Λ is a subset of X, then B is Λ-standard if and only if gr B is Λ-standard. be a Now return to the situation in which R = O is a DVR as above. Let A is an A-lattice. Λ-standard weight algebra over O, and suppose that N If λ ∈ Λ, (λ) denote the A K -submodule of N K generated by all μ-weight vectors in let N K λ is called λ-primitive7 in N if the NK with μ ∈ Λ and μ > λ. An element v ∈ N 8 image of Ov in N /N ∩ NK (λ) is pure and nonzero. (Equivalently, v has nonzero /N ∩N (λ))k .) image in (N K λ is strongly λ-primitive if v represents a homogeneous We say that v ∈ N /N ∩N (λ) is pure and element [v] in gr N such that the image of O[v] in gr N K /N ∩N (λ))k .) If v is nonzero. (Equivalently, [v] has nonzero image in (gr N K strongly λ-primitive, then it is λ-primitive; see the proposition below. Necessarily, v and, of course, [v] are non-zero. If v is strongly λ-primitive, we also say that [v] is strongly λ-primitive. is said to be primitive (resp., strongly primitive) if it is An element v ∈ N λ-primitive (resp., strongly λ-primitive) for some λ. The astute reader will observe or any gr A-lattice, that “primitive” can be deﬁned for gr N and that every strongly primitive element of gr N is both primitive and homogeneous. For some wellbehaved lattices, the notions coincide; that is, the primitive homogeneous elements are strongly primitive. See Remark 4.18(c). It is useful to use the technical of gr N translation of the “strongly primitive” notion in (b) of the result below. be a Λ-standard weight algebra over O and let N be Proposition 4.4. Let A λ . a A-lattice. Let λ ∈ Λ and v ∈ N ∩ ∩N (λ) and Ov + (N (a) The element v is λ-primitive if and only if v ∈ N K NK (λ)) is O-pure. 7This notion is inspired by Xi [22], but is stronger, even for A K , than the deﬁnition of λ ⊆ N K,λ is primitive in our sense, then λ must be “primitive” used there. Notice, if v ∈ N (λ))μ = 0. K /N maximal among μ ∈ Λ such that (N K 8This equivalence and its “strongly λ-primitive” analogue below require that O be a DVR, though in most of the discussion, including Proposition 4.4 below, it is only necessary that O be an integral domain.

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(b) The element v is strongly λ-primitive if and only if there is a non-negative n (λ)), and Ov + N ∩ , v ∈ N ∩ (radi+1 N K + N integer i such that v ∈ rad N K i+1 (rad NK + NK (λ)) is O-pure in N . (c) If v is strongly λ-primitive, it is λ-primitive. λ is λ-primitive if and only if the image of Ov in Proof. The element v ∈ N ∩N (λ), and the N /N ∩ NK (λ) is nonzero and pure, i. e., if and only if v ∈ N K quotient ∩N (λ)) N Ov + (N K / ∩N (λ) ∩N (λ) N N K K is torsion free. Equivalently, /(Ov + (N ∩N K (λ)) N ∩N (λ)) is pure in N . This proves is torsion free, which just means that Ov + (N K (a). λ represents a homogeneous element We now prove (b). The element v ∈ N /N ∩N (λ)) if and only if [v] of grade i in gr N with nonzero image in gr (N K i i+1 ∩ rad N (λ) for some (uniquely determined) i. K and v ∈ rad K + N v ∈N N K /(N ∩N (λ))) is equivalent to the purity of Purity of the image of O[v] in (gr (N K /(N ∩N (λ)))i for this integer i. The is equivalent to the purity of O[v] in (gr (N K ∩ (N (λ) + radi+1 N . This proves (b). K ) in N the sum Ov + (N K Finally, to see (c), assume that v is strongly λ-primitive, and choose the index ∩ (N (λ) + radi+1 N K ), and i as in the proof of (b) above. Thus, v ∈ F := (N K . Put E := (N ∩ (N (λ)) ⊆ F . Then Ov ∩ F = 0, so that the Ov + F is pure in N K quotient ∼ = F/E (Ov + F)/(Ov + E) = F/(F ∩ (Ov + E)) /(Ov + E) is torsion free, which implies, by (a), is torsion free. It follows that N that v is λ-primitive. This proves (c). generated by all of its strongly primitive elements The gr A-submodule of gr N Since strongly is denoted gr N (or by gr AN to emphasize the dependence9 on A). primitive elements in gr N are, by deﬁnition, homogeneous, the gr A-module gr N is a graded submodule of gr N . is a split integral QHA over O with poset Λ. We call A Now suppose that A toral quasi-hereditary (TQHA) if it is a Λ-standard weight algebra over O, using the given poset structure on Λ. Many integral QHAs come naturally equipped with such a structure. In any case, it can usually be assumed, by passing to a Morita equivalent algebra. In particular, we note the following. is a split integral QHA over O with poset Proposition 4.5. Suppose that A over O. In addition, the algebras Λ. Then A is Morita equivalent to a TQHA B and gr B K , respectively. gr A and gr AK are Morita equivalent to the algebras gr B 9The set of strongly primitive elements of gr N does not depend on A, so long as N is an A

by an O-pure ideal acting trivially lattice. (So, for example, we could use instead a quotient of A .) However, the gr A-submodule on N generated by these strong primitive elements might depend and, thus, on A. (The dependence may sometimes be eliminated with strong hypotheses on gr A, ; see Remark 4.18(b).) on N

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Proof. Let P(λ) be the PIM (projective indecomposable module) associated to λ ∈ Λ, and put P := λ∈Λ P(λ). Clearly, P is a ﬁnite projective generator for so that E := (End (P))op is Morita equivalent to A. If we take P (λ) = Ae λ for A, A identiﬁes with the algebra orthogonal idempotents {eλ }λ∈Λ , then E λ,μ∈Λ eλ Aeμ . With this identiﬁcation, {eλ }λ∈Λ ⊆ E is a set of orthogonal idempotents summing It is easily checked that these idempotents give to the identity e := eλ of E. → E = eAe the structure of a Λ-standard weight algebra. (Use the equivalence M eM from A-mod to E-mod.) = P is a projective generator for A, we have Ae A = A. This implies Since Ae that K e)n = e radn A K e, n ∈ N, (e rad A from which the remaining Morita equivalences can be deduced.

be a TQHA over O. For λ ∈ Λ, the standard A-module Lemma 4.6. Let A Δ(λ) is generated by its λ-weight space Δ(λ)λ := eλ Δ(λ) = Δ(λ) ∩ ΔK (λ)λ . Proof. The head of Δ(λ) is isomorphic to L(λ). Also, Δ(λ) λ is O-free of rank 1. The lemma follows from Nakayama’s lemma. We will sometimes need to assume one or both of the following following hypotheses. is a TQHA over O with poset Λ. Hypotheses 4.7. (1) A K is a QHA with the same weight poset Λ as A K . (2) The graded K-algebra gr A K -mod of not necessarily For each λ ∈ Λ, the standard modules in the HWC gr A K (λ). K -modules are the modules gr Δ graded gr A Lemma 4.8. Assume that Hypothesis 4.7(1) holds. Let P be projective in A-mod. Then gr P = gr P. Proof. First, it is useful to note that every ﬁnite projective A-module P has the property that the positive grade terms of gr P are contained in rad(gr P), the and intersection of all maximal submodules. This follows from the case of P = A the nilpotence of the ideal of positive grade terms. In particular, P and P have the same head. See also the discussion above (2.4). Now to prove the lemma, it suﬃces to treat the case P = P(λ), λ ∈ Λ. The map P(λ)λ → Δ(λ) λ is surjective. So Lemma 4.6 implies there exists an element v ∈ P(λ)λ whose image in Δ(λ) λ is a generator for Δ(λ). The image of v in L(λ) = headΔ(λ) is nonzero, as is the image of [v] in L(λ) = head gr P (λ). Thus, [v] generates gr P(λ) by Nakayama. It is, thus, suﬃcient to show that v is strongly λ primitive. However, the image of v in P(λ)/P(λ)∩(rad P(λ)K + P(λ)†K (λ)) is an Agenerator of the latter nonzero module, hence generates an O-pure submodule. Assuming Hypothesis 4.7(1), Δ(λ) is generated by a strongly λ-primitive el ement vλ ∈ Δ(λ) and there are no μ-primitive elements in Δ(λ) for μ = λ. λ Thus, gr Δ(λ) is generated by [vλ ] ∈ gr Δ(λ), but we do not, in general, know that

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gr Δ(λ) = gr Δ(λ), although this does hold if λ is maximal in Λ, by Lemma 4.8.10 is a pure A-submodule , the map In favorable cases—see Lemma 4.10—if R of N → gr N /R, though we do not have inde → gr N /R induces a surjection gr N gr N → gr N /R. (The best we pendent conditions which guarantee surjectivity for gr N have is Corollary 4.16, whose proof uses the surjection gr N → gr N /R of Lemma 4.11, and requires additional hypotheses. An immediate consequence of the discussion above is the following result. Lemma 4.9. Assume Hypothesis 4.7(1). For λ ∈ Λ, gr Δ(λ) has a simple head if and only if gr Δ(λ) = gr Δ(λ). The next two results are technical results needed in Lemma 4.12. is an A-lattice, is Lemma 4.10. Assume Hypothesis 4.7(1). Suppose N and R an O-pure A-submodule. λ ∈ Λ, maximal with respect to Nλ = 0, such that the following conditions hold. ⊆A K N λ . (1) R (or, equivalently, in N λ ), for all i ∈ N. K )λ is pure in N (2) Rλ + (N ∩ radi N j λ ⊆ rad N λ = K . Then N (3) Let jλ be the largest index j such that N jλ 11 N λ ). λ (rad R → gr N /R induces a surjection gr N /R). → gr (N Then the map gr N /R is the Proof. We ﬁrst prove that each strongly primitive element in gr N image of a strongly primitive element in gr N . /R) i be such a strongly primitive element, represented (through Let [v] ∈ (gr N . We take [v] to be μ-strongly primitive for some abuse of notation) by v ∈ N K ), the O-module K + R ∩ (radi N some μ ∈ Λ. Thus, for some grade i, v ∈ N i+1 Ov + N ∩ (rad NK + RK + N (μ)K ) is pure, and does not collapse to the right λ = 0, there μ = 0. Since λ is maximal with N hand summand. Without loss, v ∈ N are three cases to consider. K ⊆ A K N (μ)K . Thus, K ⊆ N Case 1: λ > μ Using condition (1), we have that R i+1 i+1 K + N (μ)K ) = N ∩ (rad (μ)K ). K + R K + N ∩ (rad N N N Thus, the image of v ∈ (gr N )i is μ-strongly primitive, and maps to [v] in /R) i. (gr N (μ)K )λ = 0, so that, using condition (1), we Case 2: μ < λ and μ = λ. Then (N get ∩ (radi+1 N K + N (μ)K ))μ = (N ∩ (radi+1 N (μ)K )μ K + R K + N N ∩ (radi+1 N (μ)K ) is pure and does not collapse to K + N so that, again, Ov + N )i is strongly primitive, the right-hand summand. Again, the image of v in (gr N and maps to [v] in (gr N /R). 10We cannot just reduce to the maximal case without changing the algebra gr A that is acting on gr Δ(λ). This is a subtle point, since the physical graded O-module structure of the latter gr A-module remains the same, if we attempt such a reduction. However, the submodule changes generated by a given vector might change, as gr A 11It can be shown that (3) =⇒ (2).

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(μ) = N (λ) = 0 by the maximality of λ. For each j ∈ N, Case 3: μ = λ. Here N i λ + ( K + radj N by condition (2) )λ is a lattice in (R K )λ , and is pure in N R rad N . Hence, j

∩ (radj N K ))λ = R λ + ( K + R )λ , (N rad N

j ∈ N.

λ ⊆ R λ + (N ∩ rad K ) by condition (3). We claim that i > jλ . If not, then N N Thus, λ + N ∩ radi+1 N λ + N ∩ radi+1 N K ) = R K , Ov + (R i+1

/R, taking j = i in the previous contradicting the strong primitivity of [v] in gr M display. λ and, of Thus, we may assume that i > jλ . Notice [v] = [v ] if v ∈ v + R i+1 ∩NK ) remains unchanged if v replaces v. course, the expression Ov + (R + rad λ ∩ radjλ +1 N K . Using condition (3), we may now assume that v ∈ N +N ∩ radi+1 N K ) ∩ (Ov + R K )λ λ ∩ radjλ +1 N (N λ ∩ radjλ +1 N λ ∩ radi+1 N K + N K = Ov + R ∩ radi+1 N K ). = Ov + (N )i is strongly λ-primitive, and maps to to [v] ∈ (gr N /R) i. So the image of v in (gr N This completes the proof in all cases that [v] is the image of a strongly primitive . element of gr N in gr N /R is contained in gr N /R. It remains to prove that the image of gr N For this it is suﬃcient to show that the image in gr N /R of a strongly primitive is either strongly primitive or zero. element in gr N K represents a strongly primitive element [u] in μ ∩ radi N Suppose that u ∈ N )i , and suppose that the image of [u] in (gr N /R) i is not zero. There are again (gr N three cases, as above, depending on the relationship of μ to λ. If μ = λ, we may use the formulas developed in Cases 1 and 2 above to prove /R) is strongly primitive. If μ = λ and i > jλ , condition (3) the image of [u] in gr (N /R is zero, as is, essentially argued in the discussion implies the image of [u] in gr N of Case 3. This completes the proof of the lemma. A prestandard module of weight λ ∈ Λ is a graded gr A-module D(λ) which is a λ. submodule of gr Δ(λ), and which satisﬁes the condition that (gr Δ(λ))λ = D(λ) is This implies that D(λ) is a full gr A-lattice in gr ΔK (λ). Equivalently, D(λ) a graded gr A-module satisfying gr Δ(λ) ⊆ D(λ) ⊆ gr Δ(λ). For example, both and gr Δ(λ) are prestandard modules of weight λ. The notation gr Δ(λ) gr Δ(λ) J is a natural quasi-hereditary quotients Thus, if A = A/ depends on the algebra A. , associated with the poset ideal Γ containing λ, then gr Δ(λ) is a module for gr A but the analogue gr A Δ(λ) of gr Δ(λ) constructed for A , rather than A, may → gr A may not be surjective). However, if be larger (because the map gr A and is also a gr A -module, then D(λ) ⊆ Δ(λ) is prestandard with respect to gr A, it remains prestandard and still satisﬁes the required sandwich property. We have gr Δ(λ) ⊆ gr A Δ(λ) ⊆ D(λ) ⊆ gr A Δ(λ) = gr Δ(λ).

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by (gr Δ(λ)) This is because gr A Δ(λ) is generated over gr A λ . In particular, gr A Δ(λ) is itself prestandard. Later, we introduce much stronger hypotheses which guarantee that gr Δ(λ) = gr Δ(λ), so that all of this structure collapses. But, for now, we will keep track of it. is a graded gr A Lemma 4.11. Assume Hypothesis 4.7(1). Suppose that E module satisfying n n ⊆ i )(si ) i )(si ) ⊆ E gr Δ(λ gr Δ(λ i=1

i=1

i) for λ1 , · · · , λn ∈ Λ, s1 , · · · , sn ∈ Z. Then there are prestandard modules D(λ with sections Fi /Fi−1 ∼ has a ﬁltration 0 = F0 ⊆ F1 ⊆ · · · ⊆ Fn = E such that E = 12 D(λi )(si ) with λi and si as above, i = 1, · · · , n. (It may be that D(λi ) is distinct j ), even if λi = λj .) from D(λ Proof. The proof is an easy induction, using projections, on the number n of summands. Now we are ready to prove the main lemma. For convenience, we assume both parts of Hypotheses 4.7. be an A-lattice Lemma 4.12. Assume Hypothesis 4.7. Let N which has a Δ ﬁltration. Assume also that gr NK = (gr N )K has a gr ΔK -ﬁltration as a graded K -module. Then gr N has a graded ﬁltration with sections graded gr A-modules gr A which have the form D(λ)(s), λ ∈ Λ and s ∈ N, for prestandard modules D(λ). (Diﬀerent sections can be associated to non-isomorphic prestandard modules of the K . (In same weight λ.) Applying K ⊗O − gives a graded gr ΔK -ﬁltration of gr N particular, gr N is a full lattice in gr NK .) Moreover, we may choose the ﬁltration 0 = F0 ⊆ F1 ⊆ · · · ⊆ Fr = gr N )(s ), for so that , if λ < λ belong to Λ and Fi /Fi−1 ∼ Fj /Fj−1 ∼ = D(λ)(s), = D(λ some i, j, s, s , then i > j. has a Δ-ﬁltration, such Proof. Since N it contains a pure submodule M is a direct sum of copies of Δ(λ), that M for some λ ∈ Λ such that all weights of irreducible sections of N /M are smaller than λ or not comparable to it. The /M also has a Δ-ﬁltration. module N Put M ∩ (rad A K )n N K = . gr # M ∩ (rad A K )n+1 N K M n≥0

→ gr N of graded gr A-modules, There is an obvious inclusion map ι : gr # M and the image Im ι is the kernel of the natural map gr N → gr N /M . (We do is a full lattice in an not claim this latter map is surjective.) The module gr # M # K -module gr MK , as discussed in [16] (without the analogously deﬁned graded gr A K of graded K subscript), and there is an analogous inclusion ιK : gr # MK → gr N K /M K ). AK -modules, with Im ιK the kernel of the natural map gr NK → gr (N 12A diﬀerent ordering of the weights λ , · · · , λ could conceivably result in diﬀerent pren 1

i ). standard modules D(λ

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is a direct sum of d copies of Δ(λ) Let d = d(λ) = dim NK,λ . Thus, M by K . Also, gr N K must contain a direct sum of d copies of the construction of M K -modules. (N K K (λ), viewed, for the moment, as ungraded gr A the modules gr Δ will also contain a graded version of the direct sum; see below.) Since NK /MK has zero λ-weight space, this direct sum must be contained in Im ιK . By dimension considerations, it must equal Im ιK . Taking gradings into account, we have a decomposition d K (λ)(mi ) K ∼ gr # M gr Δ = i=1

for some non-negative integers m1 ≤ m2 ≤ · · · ≤ md . As usual, the notation K (λ) (viz., gr Δ K (λ)(0)) K (λ)(m) indicates that the usual grade 0 head of gr Δ gr Δ is given grade m (m ∈ Z). λ as a direct sum Write M λ = λ,n M M n≥0

λ,n is an O-module chosen as a complement to M λ ∩ (rad A K )n+1 N K in where M n Mλ ∩ (rad AK ) NK . The latter O-module is pure in M and is the full λ-weight ∩ (rad A K )n N λ,n in (gr # M K . The image gr # M n,λ of M )n is the full space in M # λ-weight space (gr M )n,λ . The image is 0 unless n = mi for some i. In fact, the λ,n . We have number of mi equal to n is the rank of M ⎧ =A M λ = ⎨M n≥0 AMλ,n ⎩A M λ,n ∼ λ,n . ⊗O M = Δ(λ) λ,n is regarded as an O-module only. Here, in the tensor product, M λ,n = 0. Thus, Recall that m1 is the smallest integer n with M ⎧ K ⊆ (rad A K )m1 N ⎨M ⎩M ⊆ (rad A K )m1 +1 N K . The two displays above show that the two naturally isomorphic O-submodules and gr # M )m ⊆ gr N are generated as O-modules by λ,m ⊆ (gr N λ,m ⊆ N M 1 1 1 and gr N , respectively. Put R = A M λ,m ∼ strongly primitive elements of N 1 = # as Δ(λ) ⊗O Mλ,m1 , and RK = K R. Form the graded module gr R → gr N , and note (gr # R) λ ⊆ gr N identiﬁes naturally with M λ,m1 . constructed for M # # We may identify gr R with a submodule of gr M . As such, (4.1)

λ,m λ,m = gr # M λ,m ∼ gr N gr # R 1 1 = 1

K has a graded submodule consisthas rank dim MK,λ,m1 . Consequently, since gr N K (λ)(m1 ), the image of gr # R K ing of a direct sum of dim MK,λ,m1 -copies of gr Δ # K -isomorphism gr R K ∼ must be that submodule. Thus, there is a graded gr A = K (λ)(m1 ) ⊗K M K,λ,m . gr Δ 1 K ∩(rad A K )n N K , The isomorphism (4.1) has implications for the ﬁltration terms R # n ≥ 0, whose successive quotients deﬁne the grades of gr R. In fact, it is easy to

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see (4.2)

K ∩ (rad A K )m1 +s N K = R K , K )s R (rad A

s ≥ 0.

K ∩ (rad A K )n N K )s R K must be some (rad A K is an easy (The proof that each R K )n N K ∩ (rad A K = 0. downward induction on n, starting the largest n for which R m1 +1 The precise indexing can then be made from the fact that RK ∩ (rad AK ) NK K .) is the largest intersection not equal to R are deﬁned by successive quotients of the Since the grade n terms of gr # R K , (4.2) gives ∩ (rad A K )n N ﬁltration terms R ∼ ∼ gr # R = (gr R)(m 1 ) = (gr Δ(λ))(m1 ) ⊗O Mλ,m1 in gr A-grmod. /R satK , we next check that N To complete the proof by induction on dim N isﬁes the hypotheses of the lemma. By construction, N /M has a ﬁltration with /R is visibly an ungraded direct sum of copies sections Δ(ν), ν ∈ Λ\{λ}, while M K /M K ) must, by construction, have a graded ﬁltration of Δ(λ). Similarly, gr (N K (μ)(s), μ ∈ Λ\{λ}, s ≥ 0. The kernel of the map with sections modules gr Δ K /R K ) → gr (N K /M K ) is the cokernel of the map gr # R K → gr # M K , which gr (N K (λ)(t), t ≥ 0. Thus, N /R K -modules gr Δ is, visibly, a direct sum of graded gr A satisﬁes the hypotheses of the lemma. /R) has a graded ﬁltration with sections various modules By induction, gr (N gr Δ(ν)(s), ν ∈ Λ, s ∈ N, here, and in the statement of the lemma. Applying K /R K . It is K ⊗O −, these become sections of a graded gr ΔK -ﬁltration of gr N easily checked that the hypotheses of Lemma 4.11 apply to N and R, so that the ∩ gr # R, which → gr N /R is surjective. The latter map has kernel gr N map gr N contains gr Mλ,m1 . Thus, # # ∼ gr Δ(λ)(m 1 ) ⊗O Mλ,m1 ⊆ gr N ∩ gr R ⊆ gr R = gr Δ(λ)(m1 ) ⊗O Mλ,m1 ,

∼ noting the isomorphism gr Δ(λ)(m 1 ) ⊗O Mλ,m1 = (gr A)gr Mλ,m1 . The lemma # = gr N ∩ gr R, and induction. now follows by Lemma 4.11, applied to E Remark 4.13. (a) Character arguments in the (common) Grothendieck group K -mod show that [N K : ΔK (λ)] = [gr N K : gr ΔK (λ)] for N K -mod and gr A for A as in Lemma 4.12. Also, [NK : ΔK (λ)] is obviously the same as [N : Δ(λ)] and K : gr ΔK (λ)] counts the number of (collective occurrences) of prestandard [gr N , viewed as an ungraded gr A modules D(λ) in any prestandard ﬁltration of gr N module (a full lattice in gr NK ). (b) Assuming Hypothesis 4.7, any PIM P(λ), λ ∈ Λ, may be used for N in Lemma 4.12. Here gr P (λ) = gr P (λ) by Lemma 4.8. It has a prestandard ﬁltration by Lemma 4.12, and the proof of Lemma 4.12 shows that the top term of the ﬁltration may be taken to be gr Δ(λ). Also, the kernel of the map from μ > λ. This P (λ) = P (λ) onto gr Δ(λ) is ﬁltered by prestandard modules D(μ), is also guaranteed by part (a) above of this remark. The following proposition is, in some sense, a corollary of the proof of Lemma 4.12, though additional argument is required.

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BRIAN J. PARSHALL AND LEONARD L. SCOTT

be an A-lattice Proposition 4.14. Let N satisfying the hypothesis of Lemma 4.12 (which includes Hypotheses 4.7). Assume also that gr Δ(ν) has a simple head, K (ν)] = 0. Then gr N = gr N and all the K : Δ for each ν ∈ Λ satisfying [N prestandard modules in the ﬁltration of Lemma 4.12 are “standard” (i. e., if a ∼ shifted copy of D(ν) occurs in the ﬁltration, then D(ν) = gr Δ(ν)). Before giving the proof, we record the following immediate corollary (of the proposition and Lemma 4.12, using the Morita equivalence of Proposition 4.5). is a split QHA over O with poset Λ, and Corollary 4.15. Suppose that A K is a QHA algebra with the same poset Λ. Let N be a A-lattice that gr A with a Δ-ﬁltration, and suppose gr Δ(ν) has a simple head for each ν ∈ Λ satisfying K (ν)] = 0. Then gr N has a ﬁltration with sections gr Δ(λ), K : Δ λ ∈ Λ. [N Proof of Propostion 4.14. According to Lemma 4.8, the head of a given = gr Δ(λ). Since any module D(λ) is sandwiched gr Δ(λ) guarantees that gr Δ(λ) between gr Δ(λ) and gr Δ(λ), we must have D(λ) = gr Δ(λ). We now continue with the notation and proof of Lemma 4.12. The displayed ∩ gr # R = gr # R. inclusions in the last paragraph are now equalities, and so gr N # That is, gr R ⊆ gr N . is the kernel of the map gr N → gr N /R. This map sends We recall that gr # R gr N onto gr N /R, according to the paragraph quoted above. The paragraph /R satisﬁes the hypotheses before that, in the proof of Lemma 4.12, notes that N . In particular, N /R has a Δ-ﬁltration. of the lemma required on N This is part of a Δ-ﬁltration of N , since R is a direct sum of standard modules by construction (see its introduction earlier in the proof of the lemma). /R satisﬁes the hypothesis of the current proposition, and we It follows that N ) that gr N /R = gr N /R. can assume inductively (using induction on the rank of N and gr N have the same image and kernel, so must be equal. This But now gr N proves the proposition. As a further corollary of Lemma 4.12, Proposition 4.14, and their proofs, we have N , Λ satisfy the hypotheses of the preceding corollary, Corollary 4.16. Let A, Γ ∼ )Γ as gr A-lattices. and let Γ ⊆ Λ be a poset ideal. Then gr N In particular, = (gr N → gr N Γ is surjective. the natural map gr N Proof. Again, we may assume the hypothesis of Proposition 4.14, using Morita equivalence. Now, we simply continue the discussion, in the proof of the latter result, choosing λ to belong to Γ. (If Γ = ∅, there is nothing to prove.) Proceeding , it can be assumed that by induction on the rank of N /R) Γ∼ /R) Γ. gr (N = (gr N is a direct sum of copies of Δ(λ), Since R it is contained in the kernel of the ∼ /R) Γ∼ Γ . Also, since surjection N → NΓ . Thus, (N /R)Γ = NΓ , and so gr (N = gr N # gr R is a direct sum of shifted copies of gr Δ(λ), each of which has head which is is contained in the kernel of the surjection gr N → (gr N )Γ . a shift of L(λ), gr # R # Of course, gr R is the kernel of the map gr N → gr N /R. The latter map is

FORCED GRADINGS IN INTEGRAL QUASI-HEREDITARY ALGEBRAS

263 17

surjective, as Lemma 4.12 and Proposition 4.14 (and their proofs) show. Thus, /R) Γ . The corollary now follows. )Γ ∼ (gr N = (gr N An analogue of Corollary 4.16 holds for QHAs over ﬁelds (without any integral hypothesis or conclusion). See Appendix I. We are now ready to establish a main theorem of this paper. is a split QHA over O with poset Λ, and gr A K Theorem 4.17. Suppose that A is a QHA over K with the same poset Λ. Further, suppose that gr Δ(λ) has a simple head, for all λ ∈ Λ. Then gr A is a split QHA over O with poset Λ and standard objects gr Δ(λ), λ ∈ Λ. Proof. Using Proposition 4.5 to pass to a Morita equivalent algebra, we can assume that Hypothesis 4.7 holds. Let P be any O-ﬁnite projective A-module. Then N := P satisﬁes the hypotheses of Proposition 4.14. Now apply Remark 4.13(b) to P = P(λ), for any given λ ∈ Λ. This provides an exact sequence 0 → Ω → gr P(λ) → gr Δ(λ) → 0 in which Ω has a ﬁltration with sections gr Δ(ν), ν > λ. In particular, ⎧ ⎨ Homgr A(Ω, L(μ)) = 0, ⎩Hom

(Ω, gr Δ(μ)) gr A

unless μ > λ. Thus (4.3)

⎧ ⎨

= 0,

L(μ)) = Ext1gr A(gr Δ(λ),

⎩Ext1

gr A

0,

(gr Δ(λ), gr Δ(μ)) =

0,

is a gr A-module unless λ < μ (λ, μ ∈ Λ). In particular, if M such that the 1 composition factors L(μ) of M satisfy ν > λ, then Ext (gr Δ(λ), M ) = 0.13 gr A

viewed as a left module over itself in Proposition 4.14. We Next, take P = A, can rearrange the (ungraded version of the) ﬁltration guaranteed by Proposition 4.14 so that all copies of a given gr Δ(λ) occur contiguously. Removing all redundant ﬁltration indices j and those for which Fj+1 /Fj and Fj /Fj−1 are isomorphic to the same module gr Δ(λ), gives a new ﬁltration 0 ⊆ Fj1 ⊆ Fj2 ⊆ · · · ⊆ Fjs = gr A, i )⊕mi , i = 1, · · · , s, for with 1 = j1 < j2 < · · · < js = r and Fji+1 /Fji ∼ = Δ(λ suitable elements λi ∈ Λ and positive integers mi . We may assume that λi = λi if i = i , and also that λi < λi if i < i . 13There are many ways to prove this. First, note the hypothesis on M =M /π M holds for

/π a M , with a > 0, and, thus, on M /π a M . The required any of its homomorphic images π a−1 M a is replaced by vanishing holds if M is replaced by M /π M , so it suﬃces to show it holds when M . For large enough a, the latter module is torsion free. So we may assume to start that M πa M K satisfy μ > λ since the analogous property K (μ) of M is torsion free. All composition factors L K is a QHA with poset Λ. Thus, over k holds. Thus, Ext1 (gr ΔK (λ), MK ) = 0, since gr A gr AK

Ext1

gr A

) is torsion free. However, multiplication by π on this Ext1 -group is surjective, (gr Δ(λ), M

since Ext1

gr A

) = 0. So the Ext1 -group itself vanishes. (gr Δ(λ), M

264 18

BRIAN J. PARSHALL AND LEONARD L. SCOTT

On the other Now take Ji = Fji . By deﬁnition, Ji is a left ideal in gr A. We claim that Ji x ⊆ Ji . Right multiplication by x deﬁnes an hand, let x ∈ gr A. x → gr A, and it suﬃces to show that the induced map Ji → endomorphism gr A → (gr A)/ Ji is the zero map. If this map is nonzero, tensoring with K deﬁnes gr A K )/JiK . However, JiK is ﬁltered by standard modules a nonzero map JiK → (gr A K )/JiK gr ΔK (λj ), j = 1, · · · , s for some non-negative integer s = si , while (gr A is ﬁltered by modules gr ΔK (μ) with no μ greater than, or equal to, any λj . Hence, K )/JiK ) = 0, a contradiction. Therefore, Ji x ⊆ Ji as claimed, Homgr AK (JiK , (gr A and Ji is an ideal in gr A. is a ﬁnite O-subalgebra of Next, for λ ∈ Λ, End (gr Δ(λ)) gr A

∼ Endgr A(gr Δ(λ)) K = Endgr A K (gr ΔK (λ)) = K, ∼ so Endgr A(Δ(λ)) = O. Thus, i )⊕mi ) ∼ Endgr A(Ji /Ji−1 ) ∼ = Endgr A(Δ(λ = Mmi (O). k Fix i, and put Γi = {γ ∈ Λ | γ ≤ λi , for some i > i}. Then Jik ⊆ B := (gr A) and B/Jik is the largest quotient algebra of B whose composition factors L(ν) 2 is a B/Jik -module, its composition factors L(γ) satisfy satisfy ν ∈ Γi . Since Jik /Jik 2 γ ∈ Γi . On the other hand, Jik /Jik is a homomorphic image of Jik , which is ﬁltered 2 with sections gr Δ(λ), λ ∈ Γi , each of which has head L(λ). So, if Jik /Jik = 0, it 2 must have a composition factor L(λ), λ ∈ Γi . Therefore, Jik = Jik , so Ji = Ji2 by Nakayama’s lemma. Ji−1 -projective, it suﬃces to show that Finally, to show Ji /Ji−1 is (gr A)/ gr Δ(λi ) is (gr A)/Ji−1 -projective. But this follows from the sentence immediately following (4.3). is a split QHA over O, as claimed. This construction also In conclusion, gr A shows that gr Δ(λ) is standard. Remark 4.18. (a) We could give an alternative version of part of the above and Λ satisfying proof by appealing to Corollary 4.16. In particular, we note, for A the hypothesis of Theorem 4.17, and any poset ideal Γ ⊆ Λ, that Γ ) ∼ Γ. gr (A = (gr A) → gr A Γ is surjective. So, if N is an A Γ -lattice, (b) Consequently, the map gr A with A and Γ as above, there is a physical equality of O-modules = gr N . gr AN A Γ

Λ, N satisfy (c) In a similar spirit, another consequence of Corollary 4.16: If A, the hypothesis of the corollary, then the strongly primitive elements of gr N are the . This follows by taking Γ = {γ | γ > λ} for same as the primitive elements of gr N any given λ ∈ Λ, and using the isomorphism /N ∩N (λ) = gr N )Γ ∼ /gr N ∩ (gr N )K (λ). Γ ∼ gr N = (gr N = gr N K

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265 19

5. A special case In this section, unless otherwise noted, we continue the notation and assume both the conditions of Hypotheses 4.7 of the previous section. In particular, the is a split QHA with standard objects Δ(λ), algebra A λ ∈ Λ. The irreducible modules and PIMS of the QHA AK are denoted LK (λ) and PK (λ), respectively. We sometimes write ΔK (λ) (which is isomorphic to Δ(λ) K ) for its standard modules. K is also a QHA under Hypothesis 4.7, speciﬁcally item (2). It The algebra gr A ∼ has standard objects (gr Δ(λ)) K = gr Δ(λ)K = gr ΔK (λ), λ ∈ Λ. has a pure subalgebra In addition, we assume, for the rest of this section, that A a and a Wedderburn complement AK,0 of AK . For use in the results below, we record the following conditons. Conditions 5.1. (1) aK has a tight grading aK = aK,0 ⊕ aK,1 ⊕ · · · 14. K = (rad K = A K (rad (2) rad A aK ) A aK ). (3) For λ ∈ Λ, ΔK (λ) has a graded aK -structure, and is generated as an aK -module by ΔK (λ)0 . K,0 -stable. Also, A K,0 contains aK,0 (deﬁned in (1)) (4) In (3), ΔK (λ)0 is A and all idempotents eλ , λ ∈ Λ. (5) a has a positive grading a = ⊕r≥0 ar such that K ar = aK,r , the rth grade of aK in (1), for each r ∈ N. Observe that Conditions (5.1)(1) & (5) can be made independently of the QHA In this spirit, we have the following result. algebra A. Proposition 5.2. Let a be an algebra which is free and ﬁnite over the DVR O and satisﬁes Conditions 5.1(1) & (5). Then the following statements hold: (a) For each r ∈ N, we have ar = ai = aK . There a∩ aK,r and i≥r a ∩ radr ∼ is an isomorphism a → gr a of graded O-algebras sending x ∈ ar to its image [x] in r r+1 a/rad a = (gr a)r . rad is an (b) If M a-lattice, the following statements are equivalent: is tight. (See Deﬁnition 2.2.) (i) M r M = rad for each r ∈ N. (ii) ai M i≥r is generated by (gr M )0 . (iii) The gr a-lattice gr M O-direct summand of a, we have Proof. We begin with (a). Since ar is an that ar = ai = aK , since a∩ ar,K = a∩ aK,r . Similarly, we get i≥r a ∩ radr aK = (rad aK,i )r = aK,i = ( ai )K . aK ) r = ( radr i≥1

i≥r

i≥r

In particular, there is a well-deﬁned O-linear graded map sending x ∈ ar to its image [x] ∈ a ∩ radr aK / aK , and the map is clearly a ∩ radr+1 multiplicative. Thus, it is a graded homomorphism of O-algebras. The equations i≥r ai = aK a ∩ radr r+1 ai = aK show that the map is an isomorphism. Part (a) is and i≥r+1 a ∩ rad now proved. 14This means that aK,n with aK is positively graded, aK = n≥0 aK,0 semisimple and with

aK generated by aK,1 ; see [4, §4]. Equivalently, aK ∼ aK . = gr

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BRIAN J. PARSHALL AND LEONARD L. SCOTT

We now prove (b). The equivalence (i) ⇐⇒ (ii) is immediate from the sum formula for a ∩ radr aK in part (a). Now suppose that (ii) holds. Then, using the isomorphism of (a), )0 = [ /(M ∩ rad M K ) = ( ∩ radr+i M + M ∩ radr+1 M K )/M K , (gr a)r (gr M ar ]M ar M . But (ii) the second equality holding by deﬁnition of the action of gr a on gr M gives that +M ∩ radr+1 M K = + = =M ∩ radr M K . ai M ai M ar M ar M i≥r+1

i≥r

)0 = gr M r , and (iii) holds. Thus, (gr a)r (gr M )0 = gr M r for all r ∈ N. This gives that If (iii) holds, then (gr a)r (gr M ∩ radr M +M ∩ radr+1 M K = M K ar M calculating with (gr a)r = [ ar ], as above. Downward induction on r proves (ii). (Note that (ii) holds for suﬃciently large r, with both sides equal to 0.) The proposition is proved. K -submodule EK (λ) of PK (λ) by the short exact For λ ∈ Λ, deﬁne the A sequence (5.1)

0 → EK (λ) → PK (λ) → ΔK (λ) → 0.

be a QHA over O with weight poset Λ, which satisﬁes Theorem 5.3. Let A Hypothesis 4.7 and Conditions 5.1(1)—(5). Let λ ∈ Λ be such that the PIM PK (λ) K -mod contains a full A-lattice P(λ)† with the following properties: in A † † (i) P (λ) = Av, where v ∈ P (λ)λ ; (ii) P(λ)† = P(λ)†0 ⊕ P(λ)† ∩ rad PK (λ), where P (λ)†0 is an O-direct summand K,0 -submodule of K P (λ)† = PK (λ); of P(λ)† such that K P(λ)†0 + EK (λ) is an A † (iii) P (λ) |a is tight. Then Δ(λ)| a is tight. † Proof. Since LK (λ) ∼ = PK (λ)/ rad PK (λ), (ii) implies K P(λ)0 +EK (λ)/EK (λ) ∼ = LK (λ)|AK,0 as an AK,0 -module. Now let φ : PK (λ) → ΔK (λ) be a surjection. Then φ(rad PK (λ)) = rad ΔK (λ), so that φ(K P (λ)†0 ) is nonzero and isomorphic to K,0 LK (λ)|AK,0 . Since the latter module appears with multiplicity one as an A † composition factor of ΔK (λ), we must have φ(K P (λ) ) = ΔK (λ)0 , the AK,0 submodule of ΔK (λ) in Conditions (5.1)(3) & (4). (Observe that aK,i ΔK (λ)0 = 0 ΔK (λ)0 ∩ rad ΔK (λ) = ΔK (λ)0 ∩ i≥1

by Conditions (5.1)(1), (2), (3).) Since P(λ)†0 is a complement in P(λ)† to P (λ)† ∩rad PK (λ) = P(λ)† ∩rad P (λ)†K , which is i≥1 ai P (λ)† , by the assumed tightness (iii) and Proposition 5.2, we have ai P(λ) = P(λ)†0 + ai P(λ)†0 + ai P(λ)† ). P(λ)† = P(λ)0 + i≥1

i≥1

i≥1

FORCED GRADINGS IN INTEGRAL QUASI-HEREDITARY ALGEBRAS

267 21

Further iterating this equation (or just applying Nakayama’s lemma), we get that ai P(λ)†0 . P(λ)† = P(λ)†0 + i≥1

Thus, φ(P(λ)† ) = φ(P (λ)†0 ) +

ai φ(P(λ)†0 ).

i≥1

Since Kφ(P(λ)†0 ) = φ(K P(λ)†0 ) = ΔK (λ)0 , the aK -graded structure of Condition (3) shows the above sum is direct, even inside the right-hand side sum. That is, ai φ(P(λ)† ) = ai φ(P(λ)† ). 0

i≥1

0

i≥1

We claim that the a-lattice φ(P (λ)† )|a is tight. By Propositon 5.2, it is suﬃcient to check the equality ai φ(P(λ)† ) = φ(P(λ)† ) ∩ radr φ(P (λ))†K , ∀r ∈ N. i≥r

ai = For r = 0, this is trivial, since i≥0 a, and aφ(P (λ)† ) = φ(P(λ)† ). For r ≥ 1, † ai P(λ) = i≥r ai P(λ)0 . Consequently, we may argue as above that i≥r ai φ(P(λ)† ) = ai φ(P(λ)†0 ). i≥r

i≥r

Multiplying by K, the left-hand side gives (rad aK )r φ(K P(λ)† ) = radr φ(P(λ)† )K . The right-hand side, however, is an O-direct summand of φ(P(λ)† ), as shown by the direct sum decomposiiton above. Hence, it is equal to the intersection with φ(P(λ)† ) of its product with K, i. e., it is equal to φ(P(λ)† ) ∩ radr φ(P(λ)†K ), a-lattice. Since φ(P(λ)† ) = Aφ(v), with v and φ(v) and so φ(P(λ)† ) is tight as an † ∼ being λ-weight vectors, we have φ(P (λ) ) = Δ(λ), proving the theorem. be a QHA over O with weight poset Λ. If λ ∈ Λ Corollary 5.4. Let A satisﬁes the hypothesis of Theorem 5.3, then gr Δ(λ) has a simple head. Explicitly, ∼ head(gr Δ(λ)) The assertion still holds if λ belongs to = L(λ), the head of Δ(λ). Γ -module. an ideal Γ, and gr Δ(λ) is regarded as a gr A Proof. By Theorem 5.2, Δ(λ) is tight as a a-lattice. Next, note the action By of gr a on gr Δ(λ) factors naturally through the graded map gr a → gr A. generates gr Δ(λ) as a gr a -module, and thus as a Propostion 5.2(c), (gr Δ(λ) 0 gr A-module. But (gr Δ(λ))0 has a simple gr A-head, since it has a simple A-head (which, of course, is also the head L(λ) of Δ(λ)). Now we can use Theorem 4.17, applied to ideals in Λ, to obtain some integral Γ . The assertion below regarding N follows from Corollary QHAs of the form gr A 4.15.

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BRIAN J. PARSHALL AND LEONARD L. SCOTT

be a QHA over O with weight poset Λ. If Γ is an ideal in Theorem 5.5. Let A Γ is a QHA Λ such that the hypothesis of Theorem 5.3 holds for all λ ∈ Γ, then gr A (over O) with weight poset Γ, and standard modules gr Δ(λ), λ ∈ Γ. In addition, if has a gr Δ-ﬁltration is a A Γ -lattice with a Δ-ﬁltration, then the gr A-lattice gr N N (a ﬁltration with sections gr Δ(μ), μ ∈ Γ. 6. Quantum enveloping algebras For the rest of this paper, let Φ be a (classical) irreducible root system. Let C = (ci,j ) be the r ×r Cartan matrix of Φ; thus, ci,j = (αi∨ , αj ) if Π = {α1 , · · · , αr } is a ﬁxed simple set of roots of Φ. Let Φ+ be the corresponding set of positive roots, and let α0 ∈ Φ+ be the maximal short root. For α ∈ Φ, let (6.1)

dα := (α, α)/(α0 , α0 ) ∈ {1, 2, 3}.

Next, let p > 2 be a prime integer. If Φ has type G2 , then assume p > 3. (Later additional assumptions will be made on p.) Thus, p does not divide any of the integers dα deﬁned above in (6.1). In the polynomial ring Z[v], let m = (p, v −1) and put A = Z[v]m . Setting O = A/(φp (v)), with φp (v) = v p−1 + · · · + 1, the pth cyclotomic polynomial, (K, O, Fp ) is a p-modular system, where K = Q(ζ). Here ζ := v+(φp (v)) is a primitive pth root of unity. Under the quotient map π : O → Fp , we have π(ζ) = 1. Later it will be important to observe that, given α ∈ Φ, (6.2)

ζ dα − 1 = uα (ζ − 1),

for some unit uα ∈ O.

dα −1

In fact, we can take uα = ζ +· · ·+ζ +1 ∈ O, which is a unit since π(uα ) = dα ·1 in Fp by comments just above. Let U be the quantum enveloping algebra over the function ﬁeld Q(v), with generators Eα , Fα , Kα , Kα−1 , α ∈ Π, satisfying the usual relations. For β ∈ Φ, write β = α∈Π nα α (for integers nα ) and set (6.3) Kβ := Kαnα ∈ U . α∈Π Let UA be the A-subalgebra generated by the divided powers Eα = Eαi /[i]!α , (i) := O ⊗A U and Fα = Fαi /[i]!α and the elements Kα Kα−1 , α ∈ Π, i ∈ N.15 Set U A ζ . Uζ = K ⊗O U ζ The category Uζ -mod of ﬁnite dimensional integrable, type 1 Uζ -modules has irreducible modules Lζ (λ) indexed by the poset X + of dominant weights [11, §5.1]. In fact, it is a highest weight category (in the sense of [3]) with standard (resp., costandard) modules denoted Δζ (λ) (resp., ∇ζ (λ)), λ ∈ X + . Given a ﬁnite ideal Λ in the poset X + , let Uζ -mod[Λ] be the full subcategory of Uζ -mod consisting of objects whose composition factors have the form Lζ (λ), λ ∈ Λ. Let I be the of U -mod[Λ], and set A =U := U /I. Then A is an integral annihilator in U ζ ζ ζ,Λ ζ QHA in the sense above. See [5], [7] for more details. We can repeat the previous paragraph, replacing Λ by a ﬁnite non-empty ideal of the poset of p-regular dominant weights.16 We wish to verify that the Hypothesis (i)

15Here [i]! means that the polynomial [i]! is to be evaluated at v dα . α 16A weight λ ∈ X is p-regular if (λ + ρ, α∨ ) ≡ 0 mod p for all roots α. (Here ρ is the Weyl

weight.) So we assume that (1) Λ ⊆ X + consists of p-regular weights, and (2) if μ ∈ X + is p-regular and μ ≤ λ, then μ ∈ Λ.

FORCED GRADINGS IN INTEGRAL QUASI-HEREDITARY ALGEBRAS

269 23

=A Λ for Λ a suitably large ideal of p-regular weights. 4.7 holds for the algebra A Recall from [16, §8 ] that Λ is fat provided, given any p-restricted, p-regular weight λ, we have 2(p − 1)ρ + w0 λ ∈ Λ. This implies, in particular, that Λ contains all p-restricted, p-regular dominant weights. For α ∈ Π, Kαp − 1 is central in Uζ . Consider the O-Hopf algebra ζ := U ζ /Kαp − 1 | α ∈ Π. U ζ,K . (Of couse, the Kαp − 1 ∈ I.) Following Lusztig [13, 6.5(b)] and Let Uζ := U [12, Thm. 8.3.4], let u ζ and uζ = u ζ,K be the small quantum enveloping algebras. ζ (resp., Uζ ). In addition, u Thus, u ζ (resp., uζ ) is a (normal) subalgebra of U ζ is a dim g . Also, uζ admits a triangular decomposition free O-module of rank p + 0 u− ζ ⊗ uζ ⊗ uζ −→ uζ , mult

− + − 0 0 where u+ ζ , uζ , uζ are certain subalgebras of the algebras Uζ , Uζ , Uζ deﬁned in [12], [13]. Finally, let uζ is deﬁned to be the product of the p-regular blocks in the algebra ζ = u ζ ∩ uζ . It is easy to see that u ζ is a direct factor of u ζ , and k ⊗ u ζ uζ . Deﬁne u 17 is the product of all regular blocks of k ⊗ u ζ .

=U ζ,Λ for a ﬁnite, non-empty ideal Λ of p-regular domTheorem 6.1. Let A inant weights. (a) Hypothesis 4.7 holds provided p > h. is a pure (b) If, in addition, Λ is a fat ideal, then the image a of u ζ in A subalgebra such that Conditions (5.1)(1)–(4) hold. (c) Also, for p > h and Λ fat, Condition 5.1(5) holds for the grading in Condition 5.1(1). Proof. We begin by showing that Hypothesis 4.7(1) holds. We have already is an integral QHA. We need to check that A contains an approremarked that A ζ -module A is priate set of idempotents. We recall from [7, Cor. 3.3] that the left U 0 a direct sum of weight spaces, i. e., , spaces upon which Uζ acts in a uniform way ζ -module A determined by μ. Let X = XΛ be the (integral) weights μ for which U has a nonzero μ-weight space. All weights in X are p-regular, and Λ ⊆ X. Let C be is commutative. Since A is a direct sum of rank 1 C of U 0 . Thus, C the image in A ζ 0 modules (on which Uζ acts with some weight λ ∈ X), the algebra CK has a faithful K is (split) semisimple and a K . Hence, C completely reducible module, namely, A direct sum of copies of K, one copy for each weight. Similarly, the image C of C in Ak = A is split semisimple, and its irreducible (1-dimensional) modules are also

is the completion of O, to A , where O indexed by the set X. If we base change A O

⊆ A lift to idempotents in O

C ⊆A , preserving orthogonality. idempotents in C O

⊆ KO

C. In this way, we get a set of |X| orthogonal primitive idempotents in OC 17Write u = u ⊕ us , where us is the product of all singular blocks of u . Then all ζ ζ ζ ζ ζ composition factors of any “reduction mod p” of (a full u ζ -lattice in) a uζ -module, such as k ⊗ ( uζ ∩ uζ ), have p-regular weights. A similar statement holds for usζ , using p-singular weights. Now uζ ∩ uζ ⊕ u ζ ∩ usζ ) is a homomorphic image of the lattices u ζ / uζ ∩ uζ and u ζ / uζ ∩ usζ in usζ u ζ /( and uζ , respectively. This common quotient must be zero, giving the claimed properties of u ζ .

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already has a complete set of |X| orthogonal primitive idempotents in the But K C ⊆ KO

C. By the uniqueness of complete sets of central (commutative) algebra K C primitive idempotents, the two sets of |X| idempotents we have constructed are actually lie in the same. Hence, the idempotents {eμ | μ ∈ X} constructed for K C

K C ∩ OC = C ⊆ A. This set of idempotents, together with its subset indexed by the structure of a TQHA. Thus, Hypothesis 4.7(1) holds. Λ, gives A Property (2) holds for p > h by [16, Thm. 8.4] (with p playing the role of e). This completes the proof of (a). Now we prove (b). Since Λ is fat, u ζ,K = uζ maps injectively onto its image K . The same holds for u aK in A ζ since u ζ is contained in uζ , so, in particular, ∼ ζ maps isomorphically onto the image of k ⊗ a u ζ = a. Similarly, using fatness, k ⊗ u in k ⊗ A. The latter image is not a priori isomorphic to k ⊗ a, but follows here, a. Now Lemma 2.3 implies that a is pure in A. because u ζ ∼ = In [2, §§18.17–18.21], it is proved that uζ is a Koszul algebra, a property which implies that uζ has a tight grading (i. e., a grading making it isomorphic to gr uζ ), provided that p > h and the Lusztig character conjecture holds in the quantum enveloping algebra case (which is true [21]). Thus, Condition 5.1(1) holds. Condition 5.1(2) follows from left-right symmetry and [16, Lem. 8.3], and Condition 5.1(3) follows from [16, Thm. 6.4]. In fact, choose a larger ideal Λ containing Λ and consisting of p-regular weights, such that the projective cover P (λ) of L(λ) in UΛ is projective for aK . The ﬁnal condition of (4) follows from Lemma 2.1. This proves (b). The proof of (c) requires further results from [2], and it is given in §7, Appendix II. Remark 6.2. As remarked in the proof, the graded algebra uζ is isomorphic K aK ⊆ gr A to gr uζ as graded algebras. Also, Conditions 5.1(2) show that the gr (whether Λ is fat or not). Thus, the natural map u ζ → a induces a graded map ζ in the u ζ → gr A, under the hypotheses of the theorem (using the grading on u proof of part (c)). The map is an injection when Λ is fat. Theorem 6.3. Assume that p > 2h−2 is an odd prime and consider the algebra =U ζ,Γ , where Γ is an non-empty ideal of p-regular dominant weights. Then gr A A is a QHA over O, with standard modules gr Δ(λ), λ ∈ Γ. Proof. Our goal is to verify the hypotheses of Theorem 5.5 (which are the hypotheses of Theorem 5.3). Conditions 5.1 have already been dealt with in Theorem 6.1, leaving checking hypotheses (i)—(iii) in the statement of Theorem 5.3: Fix λ ∈ Γ. In the category of Uζ -modules, we have18 (6.4)

PK (λ) ∼ = QK (λ0 ) ⊗ LK (λ1 )[1] .

Enlarging Γ to a larger poset Λ of weights (with Γ an ideal in Λ), we can insure K -module. We assume this has been that PK (λ) in the display makes sense as an A done. 18If M is a U (g )-module, then M [1] denotes the U -module obtained by “pulling M back” K ζ through the Frobenius morphism Uζ → U (gK ). If M = LK (λ) is the irreducible gK -module of highest weight λ, then M [1] ∼ = LK (pλ).

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271 25

0 ), a PIM in a certain highest Assuming p ≥ 2h − 2, there is an A-module Q(λ A, with weight category, lifting a corresponding truncated projective module for A/π 19 0 )/π Q(λ 0 ) is 0 )K ∼ ak -projective, and Q(λ = QK (λ0 ). (See [7].) Also, Q(λ0 ) := Q(λ a-projective. Thus, Q(λ0 ) can be given an a-grading, so Q(λ0 ), which is O-free, is as 0) ∼ 0 ) ⊆ aK ⊗a LK (λ0 ), Q(λ a ⊗a0 L(λ = K,0 0 )/Q(λ 0 )∩ 0 ) is a full a-lattice in LK (λ0 ) isomorphic to the projection Q(λ where L(λ ∼ ∼ Q(λ0 )K in LK (λ0 ) = Q(λ0 )K / rad Q(λ0 )K = PK (λ0 )/ rad PK (λ0 ). Fix a nonzero λ-weight vector v = v0 ⊗ v1 ∈ QK (λ0 )0 ⊗ LK (λ1 )[1] , with v0 ∈ QK (λ0 )0 = (A ⊗A0 LK (λ0 ))0 ∼ = LK (λ0 ). Put = A0 ⊗A0 LK (λ0 ) ∼ (6.5)

0) ⊗ L min (λ1 )[1] , P(λ)† = Q(λ

min (λ1 )[1] is the full U ζ -lattice in LK (λ1 )[1] generated by v1 . (Of course, this where L ζ -modules to make sense of the tensor lattice, as well as Q(λ0 ), must be viewed as U † product.) In general, P(λ) is not a projective A-module, though it is projective ∼ for a. Its scalar extension P (λ)K = PK (λ) is AK -projective. The projectivity of P(λ)† |a implies that it is tight. (See the observation following Deﬁnition 2.2.) This establishes part (iii) of the hypothesis of Theorem 5.3. since adding P(λ)† = Q(λ 0 )≥1 ⊗ L(λ 1 )[1] to both sides of Also, P(λ)† = Av, ≥1 gives P(λ)† , as may be checked in Av 0) ⊗ L min (λ1 )[1] . (P(λ)† + rad PK (λ))/ rad PK (λ) ∼ = L(λ (Notice that the latter module is a homomorphic image of Δ(λ). This follows from a result of Z. Lin, c.f., [5, Prop. 1.9].) This establishes hypothesis (i) of Theorem 5.3. K,0 v. To establish hypothesis (ii) of Theorem 5.3, observe that K P(λ)† ⊆ A ∼ Modulo EK (λ), AK,0 v generates an AK,0 -submodule of PK (λ)/EK (λ) = ΔK (λ). K,0 -module The latter module is completely reducible as an AK,0 -module, and the A generated by its one-dimensional λ-weight space is isomorphic to LK (λ)|AK,0 . Thus, K,0 v + EK (λ)/EK (λ) have the same dimension and K P(λ0 ) + EK (λ)/EK (λ) and A are thus equal (in view of the above containment). This establishes hypothesis (ii). 7. Appendix I We prove the following result. Lemma 7.1. Suppose that Γ is a poset ideal in a ﬁnite poset Λ, and that B is a QHA over a ﬁeld K with weight poset Λ. Assume that gr B is also a QHA with weight poset Λ, where we identify irreducible modules for gr B and B through B/ rad B. Then gr (P (γ)Γ ) is a PIM for (gr B)Γ , for each γ ∈ Γ. Here P (γ) is the PIM for B associated to γ. In particular, for λ ∈ Λ, gr Δ(λ) is the standard module for gr B associated to λ. 19Reference [7] discusses integral lifting of truncated projective modules. Q(λ ) is a PIM in 0 the category of G-modules whose composition factors L(ν) all satisfy ν ≤ 2(p − 1)ρ + w0 λ0 . This assumes p ≥ 2h − 2; see [10].

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Proof. If Γ = Λ, there is nothing to prove. Otherwise, let μ ∈ Λ\Γ be maximal in Λ, and put m = [P (γ) : L(μ)]. Thus, P (γ) contains a submodule D∼ = Δ(μ)⊕m , and P (γ)/D ∼ = P (γ)Λ , where Λ = Λ\{μ}. ∼ We have D/ rad D = L(μ)⊕m , and gr # D/gr # rad D is a graded version of D/ rad D, as may be checked by counting composition factors. Write gr # D/gr # rad D = L(μ)(i)⊕mi , i∈N

with m = mi , mi ≥ 0. Since μ is maximal, gr Δ(μ) is projective both as a graded or ungraded gr Bmodule, and its shifts (gr Δ(μ))(i) are also projective. Also, the head of (gr Δ(μ))(i) is L(μ)(i). Consequently, there is a graded module map φ : D := Δ(μ)(i)⊕mi → gr # D, i∈N

giving a surjection when composed with gr # D → gr # D/gr # rad D. By the assumption that gr B is a QHA, the ungraded projective gr B-module gr P (γ) has a ﬁltration by standard modules, which may be assumed to have bottom term D , a direct sum of standard modules. Consequently, if (gr # D) denotes the ungrqaded gr B-modules obtained by forgetting the grading on gr # D, we have D ⊆ Image φ ⊆ (gr # D) , where φ is the ungraded version of the map φ. (Of course, the underlying image is the same as for the graded version. A similar remark applies for (gr # D) . Since dim D = m dim Δ(μ) = dim D = dim gr # D, it follows that equality holds in the inclusions (gr P (γ))Λ = gr P (γ)/D ∼ = gr (P (γ)/D) = gr (P (γ)Λ ) as ungraded gr A-modules. (Recall the graded isomorphism gr P (γ)/gr # D ∼ = (gr P (γ)/D).) The lemma now follows by induction on Λ\Γ. As a corollary of the proof, we have the following result. In the statement, we maintain the notation and hypotheses of the above lemma. The corollary is parallel to Corollary 4.16. Corollary 7.2. Let M be a (ﬁnite dimensional) B-module such that M and gr M have standard ﬁltrations. Then (gr M )Γ = gr (MΓ ). Proof. This is obtained by essentially the same argument as above, using induction on Λ\Γ. 8. Appendix II: Quantum deformation theory over O The goal of this appendix is provide a proof of Theorem 6.1(c). The proof will require some preparation and relies on results in [2]. In the process, we extend (when p > h is a prime) the quantum deformation theory of [2] to a version over O = Z(p) [ζ], where ζ is a pth root of unity. We let Φ be an irreducible root system as in §5, and let ζ be a primitive pth root of unity. Assume that p is an odd prime, p = 3 in case Φ has type G2 . We also use other notation of §5 above, e. g., dα = (α, α)/(α0 , α0 ). Notations of [2] will

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273 27

be used as they arise. For instance, U denotes the quantum algebra deﬁned in [2, §1.3] over Q(ζ), a homomorphic image of the DeConcini-Kac quantum algebra U2 . Thus, U = U − U 0 U + is the quantum enveloping algebra deﬁned on [2, p. 15]. Here U + , U − are ﬁnite dimensional, while U 0 is the algebra of Laurent polynomials in Kα , α ∈ Π. Let Kα − 1 S := O d ,α ∈ Π . ζ α −1 This is a polynomial algebra over O, an O-form of the polynomial algebra K [Kα , α ∈ Π]. Observe that, for any α ∈ Φ+ (not just in Π), the expression Hα :=

Kα − 1 ζ dα − 1

makes sense, using the deﬁnition of Kα as a product given on [2, p. 47] (see also (6.3)). A recursive argument, applying the identity xy − 1 = (x − 1)y + (y − 1) to the numerator of Hα , shows that Hα belongs to S . (Note that the denominator may be adjusted to have the correct dα by multiplying by a unit in O, since all the numbers dα are relatively prime to p.) We now make the following additional observations for α ∈ Φ+ , using the notation [Kα ; j], 0 ≤ j < p on [2, p. 48], and given implicitly below: = Kα − 1 ∈ S , for (1) Kα ∈ S . (This is obvious, since (ζ dα − 1) ζKdαα−1 −1 α ∈ Φ+ .) (2) Let S be the completion (or just localization) with respect to the augmentation ideal of S (i. e., the kernel of the augmentation map S → O. Then Kα−1 ∈ S . (3) [Kα ; 0] ∈ Kα−1 S , since [Kα ; 0] =

Kα−1 Kα + 1 Kα − 1 · , · ζ −dα ζ dα + 1 ζ dα − 1

and p is odd (which implies ζ dα + 1 is a unit in O). (4) [Kα ; j] ∈ Kα−1 S , 1 ≤ j < p, since [Kα ; j] = Kα ζ dα j − ζ −dα j Kα−1 ζ dα − ζ dα (Kα − 1)ζ dα j + ζ −dα j (1 − Kα−1 ) ζ dα j − ζ −dα j + dα ζ dα − ζ dα ζ − ζ −dα dα j − ζ −dα j 1 Kα − 1 ˙ dα j + ζ −dα j K −1 ) + ζ · (ζ = . α ζ dα − 1 (ζ dα + 1)ζ −dα ζ dα − ζ −dα =

(5) [Kα ; j]−1 ∈ S (or the corresponding localization of S ), as follows from (2) and (4), 1 ≤ j < p. dα r (6) log Kα ∈ S , since (ζ r−1) ∈ O for all integers r ≥ 1. As mentioned above, the proofs in this section will require the results and methods of [2]. In particular, we follow the notation of that paper closely. Consider the algebras S and BS [2, p. 180] (expanded on [2, p. 179, bottom] to allow direct sums of compositions of wall-crossing functors; indeed, we should take BS := End# K(Wa ,S) (QI (S))—see [2, p. 219]—for suﬃciently large I). The algebra

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S is a symmetric algebra on the integral root lattice associated to Φ (the latter denoted R in [2]). The algebra BS is Z-graded, compatibly with a grading on S in which every root has grade 2. It gives rise, by base change, to various algebras BA , for any commutative S-algebra A. Taking A = Q(ζ) yields an algebra isomorphic to the ungraded endomorphism algebra of the “Y -projective generator” of a certain category CK (Ω), K = Q(ζ), with Ω any regular orbit of the aﬃne Weyl group Wa ∼ = Wp on integral weights. This result requires restrictions on p: it must be a positive odd integer > h, not divisible by 3 if Φ is of type G2 . (In [2], Φ is allowed to be decomposable.) Examination of this ungraded endomorphism algebra shows op that BK is Morita equivalent to the block algebra of the small quantum group uζ associated to the orbit Ω . See [2, Thm. 16.18], which is essentially already proved in this “Case 2” by the Claim [2, §14.13(4)]. When p > h is a prime, we can deﬁne the above Morita equivalence over a DVR O with fraction ﬁeld K, namely, O := Z(p) [ζ] = Z[ζ](p,ζ−1) . This is not done in [2], but, with S and S above in hand, it is possible to modify the arguments there to establish this Morita equivalence. Here is an outline: We wish to transport many of the results stated in [2] for the algebra U over K := Q(ζ) in “Case 2” to an O-form − 0 + UO of U . Like U , the form UO has a triangular decomposition UO = UO UO UO − + + with UO , UO generated over O by the Fα , Eα , α ∈ Φ , and ±1 0 := O KO , Hα , α ∈ Π , UO the localization of S above with respect to the multiplicatively closed set generated by the Kα ’s. Using (3) and (4) above, and standard identities (e.g., Kac’s identity [6, Lem. 5.27]), we ﬁnd that UO is an O-algebra, an O-form of U . It also inherits the comultiplication on U , deﬁned in [2, §7.1]. Observe Δ(

Kα − 1 Kα − 1 Kα − 1 )= d ⊗ Kα + 1 ⊗ d . d α α ζ −1 ζ −1 ζ α −1

0 Deﬁne BO to be the O-algebra obtained by localizing UO at the multiplicatively + closed set generated by all [Kα ; j], α ∈ Φ , 0 ≤ j < p. Then BO is an O-form of the K-algebra B deﬁned in [2, p. 48, bottom]. Put Hα = [Kα ; 0] as at the beginning of [2, §8]. The results of [2, §8.6] hold (with similar proofs) if B is replaced by BO . In particular, they hold for the BO -algebra A = S , which has other properties we need, such as those required on [2, p. 103, top]. Thus, the functor V, given in [2, p. 86], may be deﬁned on the category CA (Ω), using for the eβ (λ), on [2, p. 86], generators of the Ext1 -groups described in terms of A in [2, §§8.6, 8.7]. The orbit Ω on [2, p. 86] need not be regular, and this ﬂexibility is important for discussing translation to, from, and through the walls. The target category of V is a certain “combinatorial category” K(Ω, A) on which “combinatorial translation functors” are deﬁned. The “combinatorial” category K(Ω, A) = K(Ω) is deﬁned by analogy with [2, §9.4,§14.4]. (If Ω is regular, K(Ω, A) is isomorphic to K(Wa , A) used above.) These translation functors are shown in [2, Prop., §10.11] to commute with the module theoretic translation functors. (Commutativity there implies commutativity in our case.) 0 Since S is a unique factorization domain, so are UO and BO also unique factorization domains. In particular, the critical intersection property A= Aβ β∈Φ+

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holds for A above, since it is ﬂat over BO . See [2, Lem. 9.1] and its proof. The −1 K −Kα ∈ BO with β = α ∈ Φ+ , algebra Aβ is obtained by inverting all Hα = ζ dαα−ζ −d α β . Now [2, Prop. 9.4] essentially holds as stated,20 and and setting Aβ = A ⊗BO BO with the same proof, for any A ﬂat over BO , and, in particular, for A = S . This gives a fully faithful functor

VΩ : FCA (Ω) → K(Ω), with FC A (Ω) denoting the category of A-ﬂat UO -modules, which are a direct sum of weight spaces (“X-graded” in the sense of [2]) with all weights in Ω, and satisfying the conditions of [2, §2.3] with U replaced by UO . The symmetric algebra S = S(ZΦ) is written in [2, §14.3] using the symbol hα to denote a root α ∈ Φ. It is then given two diﬀerent interpretations, as log Kα and as dα Hα in “Case 1” and “Case 2,” respectively. We can essentially handle both interpretations in our set-up at the same time: Fix hα := log Kα ∈ S as in (6) above. Then dα Hα diﬀer from hα only by multiplication by a unit in S , and a similar comparison may be made to the generators Hα of S . In particular, S is isomorphic to the completion of S ⊗Z O with respect to its augmentation ideal. So A = S is ﬂat over S, giving the nice base-change property of [2, Lem. 14.8] for passing from objects and morphisms of K(Ω, S) to K(Ω, A). Together with the isomorphism VΩ above, this implies that BS ⊗S A is the endomorphism algebra of a projective generator of CA (Ω). Base changing this generator from A to O shows, as in [2, 14.13(4)], that BS ⊗S O is the endomorphism algebra of a projective generator for a regular block of u ζ -mod. See [2, §16.9] for a more complete treatment over K. The algebra BS ⊗S O retains the Z-grading of BS and is compatible with the grading of B ⊗S K. The latter grading is shown to be positive and even Koszul in [2, §§18.17–8.21]. (The hypothesis there on Lusztig conjecture holds for the quantum case when p > h.) The positive grading on BS ⊗S O transports to a positive grading on u ζ , with the required properties in (c). (The algebra u ζ is Morita equivalent to op a product of copies of the algebra (BS ⊗S O) , so it has the form eM e, where M is a full matrix algebra over a ﬁnite product of copies of the algebra (BS ⊗S O)op , and e ∈ M is an idempotent such that M e is a projective generator of M -mod. Then gr eM e ∼ = e(gr M )e ∼ = eM e, the latter isomorphism coming from a similar one for (BS ⊗S O)op ), completing the proof.

As a corollary of the proof, we have the following new result, independent of Theorem 6.1 and worthy of attention in its own right. The notations uζ and u ζ may be found above Theorem 6.1. They refer to the “regular” part of the small quantum group uζ and of its integral form, respectively. Theorem 8.1. Assume that p > h is a prime. Then the algebra u ζ over O has a positive grading which base changes to the Koszul grading on uζ obtained in [2, §§17-18 and p. 231]. 20In checking the analogues of [2, §9.4] as well as the previous results in §8 mentioned above, a good strategy is to read through these results and, when references are made to previous sections in [2], to look back at those references and check that they hold in our context.

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References 1. H. Andersen, p-Filtrations and the Steinberg module, J. Algebra 244 (2001), 664–683. 2. H. Andersen, J. Jantzen, W. Soergel, Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p, Ast´ erique 220 (1994). 3. E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 85–99. 4. E. Cline, B. Parshall and L. Scott, Integral and graded quasi-hereditary algebras, I, J. Algebra 131 (1990), 126–160. 5. E. Cline, B. Parshall, and L. Scott, Reduced standard modules and cohomology, Trans. Amer. Math. Soc. 361 (2009), 5223-5261. 6. B. Deng, J. Du, B. Parshall, and J.-P Wang, Finite Dimensional Algebras and Quantum Groups, Math. Surveys and Monographs 150, Amer. Math. Soc. (2008). 7. J. Du and L. Scott, Lusztig conjectures, old and new, I, J. reine Angew. math. 455 (1994), 141–182. 8. J. Du, B. Parshall, and L. Scott, Quantum Weyl reciprocity and tilting modules, Comm. Math. Physics 195 (1998), 321–352. 9. J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine und Angew. Math. 317 (1980), 157-199. 10. J.C. Jantzen, Representations of algebraic groups, 2nd ed. American Mathematical Society (2003). 11. J. .C Jantzen, Lectures on Quantum Groups, Grad. Studies in Math., 6, Amer. Math. Soc., 1996. 12. G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. A.M.S. 3, 257–296. 13. G. Lusztig, Quantum groups at roots of 1, Geo. Dedicata 35 (1990), 89–114. 14. B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Lecture Notes in Mathematics Series, Carleton University, 3 (1988), 1–105. 15. B. Parshall and L. Scott, Quantum Weyl reciprocity for cohomology, Proc. London Math. Soc. 90 (2005), 655-688. 16. B. Parshall and L. Scott, A new approach to the Koszul property in representation theory using graded subalgebras, J. Inst. Math. Jussieu, doi:10.1017/S1474748012000679 (2012) (see also arXiv:0910.0633). 17. B. Parshall and L. Scott, On p-ﬁltrations of Weyl modules, to appear. 18. I Reiner, Maximal Orders, Academic Press (1975). 19. C. Ringel, The category of good modules over a quasi-hereditary algebra has almost-split sequences, Math. Zeit. 208 (1991), 209–225. 20. L. Scott, Semistandard ﬁltrations and highest weight categories, Mich. Math. J. 58 (2009), 339–360. 21. T. Tanisaki, Character formulas of Kazhdan-Lusztig type, Representations of ﬁnite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun. 40, Amer. Math. Soc., Providence, RI (2004) 261–276. 22. N. Xi, Maximal and primitive elements in Weyl modules for type A2 , J. Algebra 215 (1999), 735–756. Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 86, 2012

A Semisimple Series for q-Weyl and q-Specht Modules Brian J. Parshall and Leonard L. Scott Abstract. In [42], the authors studied the radical ﬁltration of a Weyl mod◦

ule Δζ (λ) for quantum enveloping algebras Uζ (g) associated to a ﬁnite dimen√ ◦ sional complex semisimple Lie algebra g. There ζ 2 = e 1 and λ was, initially, required to be e-regular. Some additional restrictions on e were required— e. g., e > h, the Coxeter number, and e odd. Translation to a facet gave an explicit semisimple series for all quantum Weyl modules with singular, as well as regular, weights. That is, the sections of the ﬁltration are explicit semisimple modules with computable multiplicities of irreducible constituents. However, in the singular case, the ﬁltration conceivably might not be the radical ﬁltration. This paper shows how a similar semisimple series result can be ◦ obtained for all positive integers e in case g has type A, and for all positive integers e ≥ 3 in type D. One application describes semisimple series (with computable multiplicities) on q-Specht modules. We also discuss an analogue for Weyl modules for classical Schur algebras and Specht modules for symmetric group algebras in positive characteristic p. Here we assume the James Conjecture and a version of the Bipartite Conjecture. In an appendix, the authors present new results relating various partial orders (e. g., the ↑ and Bruhat-Chevalley orders) which are used in the paper.

1. Introduction In the modular representation theory of a reductive group G (or a quantum ◦ enveloping algebra Uζ (g), with ζ 2 a primitive eth root of 1), the general failure of complete reducibility has given rise, in the past 40 years, to a rich cohomology ◦ theory for both G and Uζ (g). See [27] for a compilation of many results. The related question of better understanding important ﬁltrations of certain modules, e. g., Weyl modules, also has attracted considerable attention. See, for example, [28, pp. 445, 455], [1, §8], [45], [42] on ﬁltrations with semisimple sections as well as [29, §3], [17], [2], [15, §6], and [44] for the somewhat analogous p-ﬁltrations. Interesting ﬁltrations can take many forms, but a basic ﬁltration for any ﬁnite dimensional module M is its radical ﬁltration M ⊇ rad M ⊇ rad2 M ⊇ · · · . In this case, the sections radi M/ radi+1 M are, of course, semisimple (i. e., completely reducible), so that {radi M } is an example of a “semisimple series,” mentioned in 2010 Mathematics Subject Classiﬁcation. Primary 17B55, 20G; Secondary 17B50. Research supported in part by the National Science Foundation DMS1001900. c Mathematical 0000 (copyright Society holder) c 2012 American

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the title of this paper. In recent work, the authors [42] succeeded in calculating the multiplicities of the irreducible constituents for the radical series sections in the quantum Weyl modules associated to regular weights. It was required that ◦ e > h, the Coxeter number of g. In addition, e was required to be odd (and there were some other mild conditions on e, depending on the root system). For such “large” e, we also could describe the sections in a semisimple series for quantum Weyl modules with singular highest weights (but we were unable to show the series was the radical series, though this seems likely to be the usual case). Our methods also were applicable for Weyl modules in suﬃciently large positive characteristics having highest weights in the Jantzen region. This paper completes part of this project by giving, for types A and D, an explicit semisimple series for quantum Weyl modules for all positive integers e, except that in type D2m+1 it is required e ≥ 3. Explicit formulas for the multiplicities of the irreducible modules for each semisimple section are also obtained. In particular, in type A, our previous results are extended to all small e and all even e. Interestingly, these previous results, given in [42] for e odd and > h, play a key role here in obtaining the results for e even and/or small. Extensions of these results to other types would be possible provided there were improvements in the Kazhdan-Lusztig correspondence as quoted in [50, p. 273]. This paper is organized so as to make such extensions easy to obtain once such improvements are known. The goals of the previous paragraph for small e are achieved in several steps. First, by Kazhdan-Lusztig [35], there is a category equivalence of the (appropriate) quantum enveloping algebra module category with the module category of an aﬃne (untwisted) Lie algebra, consisting of modules of level −e − g (where g is the dual Coxeter number).1 Second, working entirely within the representation theory of the aﬃne Lie algebra, there is an equivalence to a module category of level −e − g, where e is suitably large. Here we make good use of the fact that category equivalences at the aﬃne Lie algebra level provides the ﬂexibility to treat small/even values of e. Now using [35] again, we move to the quantum enveloping algebra situation where the results of [42] are applicable. The approach is nontrivial and takes up §§3–7. It requires the interaction of several highest weight categories of Lie algebra modules (some of them new) and exact functors between them. In particular, we treat (various versions of) categories of g = [g, g]-modules ◦ which are integrable in the direction of g, and we also study their associated standard and costandard modules. Section 7 contains several contributions to further understanding these categories; see, for instance, Theorem 7.3 which both mirrors and uses the ﬁltration results of [42, Thm. 8.4, Cor. 8.5], and whose proof requires the combinatorial equivalences obtained by Fiebig [23, Thm. 11]. All of this work ◦ is done when g is an arbitrary complex semisimple Lie algebra. Much of what we need for the quantum case (in particular, the entire e odd case) could be done by working with the translation functor theory we provide, which gives many categorial equivalences without the need to construct inverses at a Verma ﬂag level, as in [22], or to construct explicit combinatorial deformations, as in [23]. However, the latter theory of Fiebig is theoretically very satisfying and has many additional practical advantages. In particular, it allows us, in our quantum situation, to deal with the e even case. 1If g is the aﬃne Lie algebra, we really work with modules for g = [g, g].

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One huge advantage of our extension of the results of [42] to small e is that the results can be used to obtain, in type A and working with the q-Schur algebras Sq (n, r) with q = ζ 2 , semisimple series and multiplicity formulas for the Specht modules of the Hecke algebras Hq (r). Small e results are required because the contravariant Schur functor from Sq (n, r)-mod → mod-Hq (r), taking Weyl modules to Specht modules, is only exact when r ≤ n. On the other hand, the treatment of meaningful cases (i. e., Hq (r) not semisimple) requires e ≤ r, so that e ≤ n = h. Thus, e is “small” in the sense of this paper. (Also, except when e = r = h, we have e < h, and all weights are singular.) Another application is to Weyl modules for classical Schur algebras S(n, r) in characteristic p > 0. The weights λ are required to be viewed as partitions of a positive integer r satisfying r < p2 . Also, we assume (the deﬁning characteristic version of) the James conjecture [26] and a Schur algebra version of the Bipartite Conjecture [21]; see §8.3. With these assumptions, we show that both Weyl modules and corresponding Specht modules have explicit semisimple series, with multiplicities of irreducible modules explicitly given in terms of inverse KazhdanLusztig polynomials. Returning to the quantum case, there is an interesting overlap, in type A with e = 3, between our results and methods and those of Peng Shan [47]. Her focus is on the Jantzen ﬁltration and ours is on a semisimple series. In the case of regular weights, the sections of the Jantzen ﬁltration are semisimple; in fact, Theorem 7.5(a), together with the multiplicities given in [47], imply, in the regular weight case, that the Jantzen ﬁltration is the radical ﬁltration. Semisimplicity of the sections of the Jantzen ﬁltration remains unknown for singular weights. However, semisimplicity is likely, since the section multiplicities in [47] agree with those for the semisimple series studied in this paper.2 The extension in Theorem 8.1.1 below of the multiplicity results of [44, Thm. 9.1] (which involve q-Schur algebras, so are in type A) to eliminate any restriction on e was announced by us in Remark 9.2 in arXiv: 0910.06733 v.2 (August 2010). In §9, Appendix I, we provide (apparently new) equivalences in the aﬃne case between ↑-style orders [27], [31], and the Bruhat-Chevalley orders. The proofs in this section are all combinatorial. The results are used in our proofs here, and Theorem 9.6 has also been used in [24] to complete an argument in [4], relevant to the Koszulity of some of the algebras A we consider in the regular case. See footnote 7. In Theorem 7.3, for example, we prove only that gr AΓ is Koszul, not the stronger property that AΓ is Koszul. Although the Koszulity of gr AΓ is all that is needed in the semisimple series results in this paper, it is still interesting to know about the Koszulity of AΓ , as argued in footnote 7. For a (non-Lie theoretic) example when gr A is Koszul, but A is not Koszul, see [11].

2Finally, there is an additional overlap of the results of this paper in type A with work of Shan-Varagnolo-Vasserot [48]. Their paper implies in type A that the algebras we study here are actually Koszul, thus, isomorphic to the algebra gr A that we study. See also [42, Ft. 1] for the possibility of extending the results to other types, at least for regular weights. However, it is the Koszulity of gr A, rather than that of A, which is needed the applications in [43] (in the regular case, but for all types) and in [44].

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2. Notation: Lie algebras The following notation is standard, mostly following [30], [32], and [33] with cosmetic diﬀerences. (For example, our root system is denoted Φ rather than Δ. ◦

The classical ﬁnite root system is denoted Φ. If V is a complex vector space and V ∗ is its dual, the natural pairing V ∗ × V → C between V ∗ and V is usually denoted φ, v = φ(v). Finite notation: ◦

(1) g is ﬁnite dimensional, complex, simple Lie algebra, with Cartan subalge◦

◦

◦

bra h, Borel subalgebra b ⊃ h. ◦

◦

◦

◦

◦

(2) Φ, Π = {α1 , · · · , αr }, Φ+ : roots of h in g, simple roots determined by b, ◦

◦

positive roots determined by Π. These are subsets of h∗ , which is identiﬁed ◦

◦

◦

with h using the restriction of the Killing form on g to h normalized so ◦

that the induced form on h∗ satisﬁes (θl , θl ) = 2 if θl is the maximal root ◦ ◦ in Φ. Let ρ = 12 α be the Weyl weight. ◦ + ◦∨

∨

◦

α∈Φ

◦

(3) Φ = {α | α ∈ Φ}: coroot system of Φ, identiﬁes with {α∨ = 2α/(α, α) | α ∈ ◦

Φ}. ◦

◦

(4) W = sα2 , · · · , sαr : Weyl group of g, generated by fundamental reﬂec◦

tions sαi : E → E, where E = h∗ R is the Euclidean space associated to the ◦

Killing form on h. (5) 1 , · · · , r : fundamental dominant weights; thus, (i , αj∨ ) = δi,j , 1 ≤ i, j ≤ r. ◦ ◦ r r (6) P , P + : the weight lattice i=1 Zi , and the set i=1 Ni of dominant weights. ◦

◦+

◦

◦

(7) Q, Q : the root lattice for g and the positive root lattice of g. ◦ (8) θs and θl : the maximal short and long roots of g. ◦ (9) h, g: Coxeter and dual Coxeter numbers; thus h − 1 = (ρ, θs∨ ) and g − 1 = ◦ ∨ ρ, θl , i = 1, · · · , r. (10) D = (θl , θl )/(θs , θs ) ∈ {1, 2, 3}. Aﬃne notation: ◦

◦

(1) g := (C[t, t−1 ]⊗ g)⊕Cc⊕Cd: aﬃne Lie algebra attached to g, with central element c. ◦ (2) h = h ⊕ Cc ⊕ Cd be the “Cartan subalgebra” of g. Following [50, p. 268], consider χ, δ ∈ h∗ deﬁned by ⎧ ⎪ ⎨χ(h) = δ(h) = 0; χ(c) = δ(d) = 1; ⎪ ⎩ χ(d) = δ(c) = 0. Thus, (2.1)

◦∗

h∗ := h ⊕ Cχ ⊕ Cδ.

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◦

Here h∗ identiﬁes with a subspace of h∗ by making it vanish on d and c. (3) Φim: = {nδ | 0 = n ∈ Z}, the imaginary roots. ◦

(4) Φre := {jδ + α, | j ∈ Z, α ∈ Φ}, the real roots. (5) Φ∨ = {α∨ | α ∈ Φre } ⊂ h, aﬃne roots. (6) Φ = Φre ∪ Φim , the root system of g.

◦

◦

(7) Φ+ = {jδ | j ∈ Z+ } ∪ {jδ + α | j ∈ Z+ , α ∈ Φ} ∪ Φ+ : positive roots. If α ∈ Φ, gα ⊂ g is the α-root space. The algebra g has Borel subalgebra b := h, gα , α ∈ Φ+ . Let α0 = δ − θl , so that Π := {α0 , α1 , · · · , αr } is ◦ the set of simple roots for g. Then α0∨ = c − θl∨ . Put ρ = ρ + gχ. Thus, ρ(c) = g. (8) Q = Zα0 ⊕ · · · ⊕ Zαr , Q+ = Nα0 ⊕ · · · ⊕ Nαr : root lattice, positive root lattice. (9) W := sα | α ∈ Φre = sα | α ∈ Π : Weyl group of g.3 (10) For λ ∈ h∗ , Φ(λ) := {α ∈ Φre | λ + ρ, α∨ ∈ Z}. (11) C := {λ ∈ h∗ | (λ + ρ)(c) = λ + ρ, c = 0}, the non-critical region. (12) C − := {λ ∈ C | λ + ρ, α∨ ≤ 0, for all α ∈ Φ+ (λ) = Φ(λ) ∩ Φ+ }, as − introduced in [33]. Also, Crat is the set of λ ∈ C − such that λ, αi∨ Z, i = 1, · · · , r, and λ(c) ∈ Q. (13) Denote the non-degenerate, invariant, symmetric bilinear form on g given ◦ in [10, §16.1] by (x, y), x, y ∈ g. Its restriction to g agrees with the normalized Killing form mentioned above. (14) ≤ denotes the dominance order on h∗ : thus, λ ≥ μ if and only if λ − μ ∈ Q+ . This should not be confused with various other partial orders, e. g., ≤λ (see Remark 5.2 (b)), or ≤e (see below (7.1). (15) For λ ∈ C − , let W (λ) be the subgroup of W generated by the sα , α ∈ Φ(λ). Also, let W0 (λ) be the subgroup of W (λ) generated by those α satisfying λ + ρ, α∨ ) = 0 for all α ∈ Φ(λ). 3. Module categories Following [33], let O = O(g) be the category of g-modules M which are weight modules for h having ﬁnite dimensional weight spaces Mλ , λ ∈ h∗ , and which have the property that, given ξ ∈ h∗ , the weight space Mξ+σ = 0 for only ﬁnitely many σ ∈ Q+ .4 Any λ ∈ h∗ deﬁnes a one-dimensional module (still denoted λ) for the universal enveloping algebra U (b) of b. Let M (λ) := U (g) ⊗U(b) λ be the Verma module for g of highest weight λ. It has a unique irreducible quotient module L(λ). Both M (λ) and L(λ) belong to O. In fact, the irreducible modules L(λ), λ ∈ h∗ , are all the irreducible objects in O, and they are non-isomorphic for distinct λ. ◦

3This deﬁnition result in a natural identiﬁcation W ∼ W Q. Some other authors would = ◦

instead take the Weyl group associated to g to be W Q∨ , where Q∨ is the root lattice of the dual ◦

root system (equivalently, the coroot lattice associated to Φ. These Coxeter groups are isomorphic; see [39, p. 400]. 4This category, used in [33], is quite close to the original category, denoted O in [30]. Indeed, the latter category is contained in O, and any object M in O in which [M : L(λ)] = 0 (in the sense of [30, p. 151]) implies λ ∈ C is a direct sum of objects (i. e., block projections) in the category O, as follows from [33, Prop. 3.3]. Following [33], the symbol “O” means something diﬀerent in this paper.

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The category O has a contravariant, exact duality M → M , obtained by taking to be the linear dual Mλ∗ of the λ-weight space of M , and making g act on M through the Chevalley anti-involution (sending ei , fi , h to ei , fi , −h, respectively). Then M ∼ = M , M ∈ O. Also, M and M have the same weight space dimensions (i. e., they have the same formal character). Clearly, L(λ) ∼ = L(λ). In particular, the dual Verma module M (λ) := M (λ) has socle L(λ). (See [39, p. 158] for similar notation of duality on a slightly diﬀerent category. See also [33, pp. 26– 28].) For k ∈ Q, let Ok be the full subcategory of O consisting of all modules for which the central element c acts as multiplication by the scalar k. For example, M (λ), L(λ) ∈ Ok for k := λ(c) ∈ Q. In addition, the duality M → M restricts to a duality on Ok . Put ◦ g := [g, g] = (C[t, t−1 ] ⊗ g) ⊕ Cc, Mλ

◦

the derived subalgebra of g. Let h := h ⊕ Cc, so that h∗ = h∗ ⊕ Cχ. Next, deﬁne Mres (g) to be the full subcategory of U (g)-modules M with the property that, given any v ∈ M , gα v = 0 for all but a ﬁnite number of positive roots α. The objects in Mres (g) are not required to be weight modules; that is, Mres (g) is not a subcategory of O. Replacing g by g gives a similar category Mres ( g) of res g-modules. For k ∈ Q, let Mres (g) be the full subcategory of M (g) consisting k of modules upon which c acts by multiplication by k. A subcategory Mres g) of k ( Mres ( g) is deﬁned similarly. The Casimir operator Ω, deﬁned in [30, §2.5], belongs to a completion Uc (g) of U (g) [30, p. 229].5 In particular, Ω deﬁnes a locally ﬁnite operator on each object in Mres (g), commuting with the action of g. For a ∈ C, let Mres k,a (g) denote the res res full subcategory of Mk (g) consisting of objects M in Mk (g) upon which Ω − a res acts locally nilpotently, and let Mres,d k,a (g) be the full subcategory of Mk,a (g) having objects on which the element d ∈ g has a semisimple action (in addition to the local nilpotence of Ω − a). Proposition 3.1. (Kac-Polo) For k = −g (the dual Coxeter number) and a ∈ C, there is a full embedding Fk,a : Mres g) −→ Mres k ( k (g) ∼

g) −→ Mres,d of abelian categories, inducing an equivalence of Mres k ( k,a (g). Moreover, the inverse of the equivalence is given by restriction. ∼ g), Fk,a (M ) ∈ Mres,d That is, for each M ∈ Mres g = M k ( k,a (g) and Fk,a (M )| (naturally). Also, any object in Mres,d k,a (g) is isomorphic to Fk,a (M ), for some M ∈ res g). Mk ( Proof. A brief outline of the proof may be found in Soergel [49, pp. 446– 447]. We ﬁll in some details. First, the algebra Uc ( g) injects naturally into Uc (g), since tensor induction takes Mres ( g) into Mres (g). If T0 ∈ Uc ( g) denotes the (0th) Sugawara operator, the discussion in [30, p. 228–229] shows that the equation T0 = −2(c + g)d + Ω 5[30] only deﬁnes U ( c g), but the same deﬁnition works for g.

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holds in Uc (g). By [30, Lem. 12.8], the equation [T0 , x] = [−2(c + g)d, x], for x ∈ g, holds in Uc ( g). If we consider the corresponding equation of operators on an object M ∈ Mres g), we may replace c + g by k + g = 0. Letting d act as the operator k ( T0 −a gives an action of g on M . Equivalently, τ x − xτ acts as [d, x] on M τ := −2(k+g) for each x ∈ g ⊆ Uc ( g). The operator T0 is locally ﬁnite on M [30, p. 229], as is τ . For any complex number , and any positive integer n, let Mn, := {m ∈ M | (τ − )n m = 0}. If x ∈ g is a γ-eigenvector for ad d, we easily ﬁnd, by induction on n, that xM,n ⊆ Mγ+,n . (Alternatively, see [10, Prop. 2.7].) Let τs be the semisimple part of the locally ﬁnite operator τ on M . The operator τs acts as multiplication by on M,n , and by γ + on Mγ+,n . For m ∈ M,n and x as above, we have τs (xm) = (γ + )xm = γxm + xm = [d, x]m + xτs m and so τs x − xτs acts as [d, x] on M . This, letting d act as τs , gives an action of g on M , extending that of g. (Note that g is spanned by the eigenvectors of ad d.) The constructed g-module belongs to Mres k (g) and d acts semisimply (as τs ). The equation Ω−a τ =d+ −2(k + g) shows that Ω − a acts as a nonzero scalar multiple of the locally nilpotent part of τ . So it is itself locally nilpotent. Finally, the assignment of M to the constructed g-module is clearly functorial providing a functor Fk,a : Mres g) → Mres k ( k,a (g) with res,d Fk,a (M )|g = M . The construction shows that Fk,a (M ) ∈ Mk,a , and, clearly, any ∼ object N ∈ Mres,d g ). (Note that d must act as the semisimple k,a satisﬁes N = Fk,a (N | T0 −a Ω−a part of −2(k+g) and −2(k+g) must act as the locally nilpotent part.) This completes the proof. We will assume for the rest of this paper, unless otherwise explicitly stated to the contrary, that k is a rational number with k + g < 0. We next deﬁne below, for such a k ∈ Q, a category Ok of g-modules. The deﬁnition is taken from [50], adapted from [35]. In Corollary 3.2, Ok is shown to be equivalent to a category of g-modules, and is more fully integrated into the g-module theory in §5. Given any g-module M and positive integer n, let M (n) be the subspace of ◦ all m ∈ M such that x1 · · · xn m = 0 for any choice of x1 , · · · , xn ∈ tC[t] ⊗ g. Now deﬁne Ok to be the full subcategory of g-modules M such that (a) c acts as multiplication by k; (b) each M (n) is ﬁnite dimensional; and (c) M = n≥1 M (n). ◦

Since each M (n) is evidently a g-submodule of M |g◦ , (c) implies that M is a locally ◦

g, in ﬁnite, hence semisimple, g-module. Then, by (a), M is a weight module for the sense that it decomposes into weight spaces for h. In addition, Ok is a full subcategory of Mres g). k ( In [35, Defn. 2.15, Thm. 3.2]6, it is shown that all objects in Ok have ﬁnite length. The irreducible modules involved are all generated by a highest weight vector having weight λ satisfying λ, αi∨ ≥ 0, for i = 1, · · · , r and λ(c) = k. These 6In this reference, the authors deﬁne a category O which turns out to be O for k = κ − g. κ k The discussion is given only for the simply laced root system case, but this restriction is not necessary [38].

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irreducible modules are non-isomorphic for distinct λ above. Conversely, any gmodule with a ﬁnite composition series having irreducible quotients of this form belong to Ok . At the level of g-modules deﬁne O+ = O+ (g) to be the full subcategory of O consisting of all objects M such that [M : L(μ)] = 0 implies μ, αi∨ ∈ N for i = 1, · · · , r. Let O+,ﬁnite be the full category of O+ consisting of objects which have ﬁnite length. Similarly, for any k ∈ Q with k < −g, let Ok = Ok (g) be the full subcategory of O consisting of all objects upon which c acts by multiplication by k, and let + O+ k be the full subcategory of O consisting of all objects in both Ok and O . If + + a ∈ C, let Ok,a be the full subcategory of Ok consisting of all objects upon which and O+,ﬁnite be the full Ω acts with generalized eigenvalue a. Finally, let O+,ﬁnite k k,a + subcategories of O+ and O , respectively, consisting of objects of ﬁnite length.7 k k,a Corollary 3.2. Suppose k ∈ Q, k < −g. Then the restriction to g of any belongs to O . Conversely, if M ∈ O , then F M belongs to object in O+,ﬁnite k k k,a k,a . These two functors are mutually inverse, up to a natural isomorphism, O+,ﬁnite k,a and provide an equivalence ∼

−→ Ok , O+,ﬁnite k,a ◦∗

◦

Given any weight μ ∈ h∗ = h ⊕ Cχ ⊕ Cδ, deﬁne μ to be the projection of μ ◦∗ ◦ into h , and, for k ∈ Q, put μk = μ + kχ ∈ ( h)∗ . Also, if k = −g, put μk,a := μk + bδ,

(3.1) ◦

where b =

◦ ◦

a−(μ+2ρ,μ) 2(k+g)

◦

◦

depends on k and μ, as well as a. Note that μ| ◦ = μ and h

μ|h = μk if and only if μ has level k. Also, μ = μk,a if and only if μ has level k and the Casimir operator Ω acts with eigenvalue a on L(μ), the irreducible g-module of high weight μ. (This is an easy calculation from [10, Prop. 11.36]. See also [30, ◦∗ p. 229].) Since we regard ( h)∗ and h as contained in h∗ , μk and μk,a are deﬁned for μ in these spaces as well. As a corollary of this discussion, we have the following. Proposition 3.3. Suppose μ, μ ∈ C and μ = w · μ for some element w ∈ W . (Here w · μ := w(μ + ρ) − ρ is the usual dot action of W .) Then, if μ = μ , we have μ|h = μ |h . Proof. The levels of μ, μ have the value k = (μ + ρ, δ) = (w(μ + ρ), δ) = (μ + ρ, δ), after noting that w(δ) = δ. By [10, Prop. 11.36], Ω acts on L(μ) and L(μ ) by multiplication by

a = (μ + ρ, μ + ρ) − (ρ, ρ) = (w(μ + ρ), w(μ + ρ)) − (ρ, ρ) = (μ + ρ, μ + ρ) − (ρ, ρ). 7Any indecomposable object of O+ , or of O , or any object in a single block of O , already k k k

has ﬁnite length. The argument is given below.

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◦

If μ|h = μ |h , then μ = μ and (by above) ◦ k,a

μ = μk,a = μ

◦ ,k,a

=μ

= μ ,

as required. 4. Weyl groups and linkage classes

Maintain the above notation. For α ∈ Φre , form the reﬂection sα : h∗ → h∗ , x → x − x, α∨ α. The Weyl group W = sα | α ∈ Φre for g is a Coxeter group with fundamental reﬂection set S = {sα | α ∈ Π}, where Π is the set of simple roots of g. For λ ∈ h∗ , Φ(λ) := {α ∈ Φre | λ + ρ, αi∨ ∈ Z} is a subsystem of Φre , in the sense of [33, §2]. The subgroup W (λ) := sα | α ∈ Φ(λ) ≤ W is a Coxeter group with fundamental reﬂection set S(λ) := {sα | α ∈ Π(λ)}, where Π(λ) consists of those α ∈ Φ+ (λ) such that sα (Φ+ (λ)\{α}) = Φ+ (λ)\{α}. Here Φ+ (λ) := Φ+ ∩ Φ(λ). Let Φ0 (λ) := {α ∈ Φ(λ) | λ + ρ, α∨ = 0. Also, let + + Π0 (λ) := {α ∈ Φ+ 0 (λ) | sα (Φ0 (λ)\{α}) = Φ0 (λ)\{α}.

Then W0 (λ) := {w ∈ W (λ) | w · λ = λ} is a Coxeter group with a fundamental reﬂections set S0 (λ) := {sα | α ∈ Π0 (λ)}; see [33, §2]. Let

[λ] := W (λ) · λ; . [λ]+ = {μ ∈ [λ] | μ, αi∨ ∈ N, 1 ≤ i ≤ r} An element λ ∈ C − is called regular if W0 (λ) = {1}. Let O[λ] be the full subcategory of O consisting of modules M such that [M : L(μ)] = 0 implies μ ∈ [λ]. Let λ ∈ C − . Then λ is the smallest element in [λ], in the sense that W (λ) · λ ⊆ λ + Q+ . In particular, all objects of O[λ] have ﬁnite length (as can be deduced from the deﬁntion of O and [λ]). As shown in [33, Prop. 3.1], using work of Kac-Kazhdan [31], if μ ∈ [λ], the Verma module M (μ) belongs to O[λ]. Also, M (μ) belongs to O[λ]. If M ∈ O is indecomposable with [M : L(μ)] = 0, for some μ ∈ [λ], then M ∈ O[λ] [33, Prop. 3.2]. Or, if μ ∈ [λ] and if M is an indecomposable module linked to L(μ) via a chain of indecomposable modules Mi in O, i = 0, · · · , n, and irreducible modules L(μi ) satisfying M = M0 and [Mi : L(μi )] = 0 = [Mi+1 : L(μi )], and μ = μn , then M ∈ O[λ]. Moreover, if M = L(ν) with ν ∈ [λ], there is such a chain, using [33, Prop. 31] (which quotes [31]). That is, O[λ] is the “block” of O associated to L(μ), and the irreducible modules L(ν), ν ∈ [λ], constitute a linkage class. The inclusion i∗ : O+ → O admits a right adjoint i! and a left adjoint i∗ . Explicitly, given M ∈ O, M+ = i! M (resp., M + = i∗ M ) is the largest submodule (resp., quotient module) lying in O+ . For λ ∈ C − , O+ [λ] will denote the full subcategory of O[λ] consisting of all M such that [M : L(μ)] = 0 implies μ ∈ [λ]+ .

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Consider the parabolic subalgebra ◦

p := (C[t] ⊗ g) ⊕ Cc ⊕ Cd of g. Its Levi factor is denoted ◦

◦

l := g ⊕ Cc ⊕ Cd = g + h. ◦

Any λ ∈ h∗ determines an irreducible l-module s(λ) whose restriction to g is ir◦

reducible. Writing λ = λ + kχ + bδ, k, b ∈ C, s(λ) is ﬁnite dimensional if and ◦

◦

only if λ ∈ X + (and in this case its restriction to g is then the ﬁnite dimensional ◦

◦

irreducible module V (λ) of highest weight λ). ◦

Proposition 4.1. Let μ = μ + kχ + bδ ∈ h∗ , with k < −g (the dual Coxeter ◦

◦

number), and μ dominant on h. Then

M (μ)+ ∼ = U (g) ⊗U(p) s(μ); M (μ)+ ∼ = HomU(p) (U (g), s(μ)). Also, (M (μ)+ ) ∼ = M ∗ (μ)+ . ◦

◦

Proof. Each subspace tn ⊗ g ⊂ g ⊂ U (g), n ∈ Z, is a g-submodule of U (g) ◦ under the adjoint action, isomorphic to the adjoint module g. Obviously, U (g)⊗U(p) s(μ), as a left g-module under multiplication, is the homomorphic image of a direct ◦ ◦ ◦ sum of modules (tn1 ⊗ g) ⊗ (tn2 ⊗ g) ⊗ · · · ⊗ (tnm ⊗ g) ⊗ s(μ), n1 , · · · , nm ∈ Z≤0 , ◦ m ∈ N. All these tensor products are ﬁnite dimensional g-modules. Thus, if L(ν) is a g-composition factor of U (g) ⊗U(p) s(μ), its highest weight space must generate ◦

a ﬁnite dimensional g-module. In particular, (ν, αi∨ ) ∈ N, for i = 1, · · · n. Let U (l) ⊗U(l∩b) μ be the Verma module for l with highest weight μ. It inﬂates naturally to p as U (p) ⊗U(b) μ. There is an exact sequence (4.1)

0 −→ N −→ U (p) ⊗U(b) μ −→ s(μ) −→ 0 ◦

of p-modules. From the classical theory of g-Verma modules, the highest weight of any composition factor Y of N has the property that (, αi∨ ) < 0 for some i. The section U (g) ⊗U(p) Y of U (g) ⊗U(p) N has an irreducible head with the same highest weight . It follows now, by tensor inducing the exact sequence (4.1) of p-modules, that U (g) ⊗U(p) s(μ) is the largest quotient of M (μ) = U (g) ⊗U(b) μ with all composition factors in O+ . That is, M (μ)+ ∼ = U (g) ⊗U (p) s(μ). A similar argument establishes the assertion for M (μ)+ and the ﬁnal assertion is obvious. 5. Highest weight categories Throughout this section, ﬁx λ ∈ C − . The set [λ] = W (λ) · λ is a poset, putting r + μ ≤ ν if and only if ν − μ ∈ Q := i=0 Nαi .8 The set [λ] has a unique minimal element, namely, λ. If ν ∈ [λ], then {μ ∈ [λ] | μ ≤ ν} is a ﬁnite poset ideal in [λ]. If Γ ⊆ [λ], let O[Γ] be the full subcategory of O consisting of objects with 8If Γ is a poset ideal in a poset Λ, i. e., if ν ≤ γ ∈ Γ =⇒ ν ∈ Γ, we write Γ Λ. We will also consider other partial orders on [λ] in this section.

A SEMISIMPLE SERIES FOR q-WEYL AND q-SPECHT MODULES

poset

Δ(ν)

∇(ν)

O[Γ]

Γ [λ]

M (ν)

M (ν)

O+ [Γ]

Γ [λ]+

M (ν)+

M (ν)+

Category

O∞ [λ]

[λ]

M (ν)

M (ν)

O+,∞ [λ]

[λ]+

M (ν)+

M ∗ (ν)+

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Table 1. Various categories

have “composition factors” (in the sense of [30, p. 151]) L(γ), γ ∈ Γ. It will often convenient (in §§6,7) to denote O[Γ] by OΓ [λ] if λ needs to be mentioned. We will consider the following categories of g-modules. In each case, the irreducible modules are indexed, up to isomorphism, by a poset ideal Γ in [λ] or in [λ]+ . In case Γ ⊆ [λ]+ , we let O+ [Γ] = OΓ,+ [λ] be the full subcategory of O+ consisting of modules composition factors L(ν), ν ∈ Γ. Also, for ν ∈ Γ, there are given two modules Δ(ν) and ∇(ν) in the category. An object X in O∞ [λ] (resp., O+,∞ [λ]) is by deﬁnition a directed union {X Γ }Γ of modules X Γ ∈ OΓ [λ] ranging over ﬁnite Γ [λ] (resp., Γ+ [λ]+ ). Theorem 5.1. For λ ∈ C − , each of the categories listed in Table 1 below is a highest weight category (in the sense of [9]) with standard (Weyl) modules Δ(ν), costandard modules ∇(ν) and indicated poset. In particular, each of these categories has enough injective objects. If Γ is a ﬁnite poset ideal, then objects in the categories O[Γ] and O+ [Γ] have ﬁnite composition series. Proof. The fact that O[Γ] is a highest weight category follows from the dual of the deﬁnition [9] and the fact that projectives in suitable “truncated” categories have Verma module ﬁltrations [46, Lem. 10]. The latter reference does not use an arbitrary poset ideal Γ, but every such Γ is contained in one of theirs, which is suﬃcient, see [22, Lem. 2.3] for the case when Γ is any ideal generated by a single element. In [46], the authors also treat the parabolic case, and generalized Verma modules. Thus, it follows similarly that O+ [Γ] is a highest weight category. If Γ ≤ Γ are two poset ideals, then the injective hull IΓ (ν) of any given irre ducible L(ν) in O[Γ] embeds in the injective hull IΓ (ν) in O[Γ ]. Taking a directed union over the chain of poset ideals Γn , n ∈ N, with Γn ⊆ [λ] gives an injective hull in O∞ [λ] (since Γn [λ]). It follows easily from [9] that O∞ [λ] is a highest weight category. Similarly, O+,∞ [λ] is also a highest weight category. The last assertion is clear from the deﬁnition of O. Remark 5.2. (a) The category O[λ], λ ∈ C − , is the union Γ O[Γ] over all ﬁnite poset ideals Γ ⊂ [λ] (with respect to ≤). The category O[λ] satisﬁes most of the axioms in [9] for a highest weight category, though not all. (There are not enough injective objects.) However, each full subcategory O[Γ] does have enough injective and projective objects, and is a highest weight category. As we have seen, the categories O[Γ] can be used to formally complete O[λ] to a highest weight category O∞ [λ]. Some authors, speaking more informally, simply call such categories (like O[λ]) a highest weight category.

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Similar remarks apply to O+ [λ], deﬁned to be O+ ∩ O[λ], as well as many of the other categories we have introduced. (b) For λ ∈ C − , there is another poset structure on [λ] = W (λ) · λ. Explicitly, given μ ∈ W (λ) · λ, write μ = wμ · λ where wμ ∈ W (λ) is the unique element w ∈ W (λ) of minimal length such that w · λ = μ.9 Then, for μ, ν ∈ [λ], put μ ≤λ ν if and only if wμ ≤λ wν in the Bruhat-Chevalley order ≤λ on the Coxeter group − , the above W (λ). We will show in Remark 5.6(a) that, at least when λ ∈ Crat categories are highest weight categories using ≤λ (or its restriction to [λ]+ ) with the same standard and costandard objects. ∼

Definition 5.3. For μ, ν ∈ h∗ , put ν ≤ μ provided that the following two conditions hold: (1) μ, ν have the same level k = −g, i. e., k = μ(c) = ν(c) = −g; (2) ν k,a ≤ μk,a for some a ∈ C. (See (3.1.) ◦

Condition (2) does not actually depend on a. Indeed, writing μ = μ + kχ, ◦ ◦ ◦ ν = ν + kχ, then μk,a = μ + kχ + bδ and ν k,a = ν + kχ + b δ, then ⎧ ◦ ◦ ◦ ρ,μ) ⎪ ⎨b = a−(μ+2 k+g ⎪ ⎩b =

◦

◦ ◦

a−(ν+2ρ,ν) . k+g

Thus, ν k,a ≤ μk,a if and only if (1) b − b ∈ N; ◦

◦

◦+

(2) μ − ν + (b − b )θl ∈ Q The parameter a drops out of b − b which depends only on μ, ν and k. Thus, any ∼ a may be used in deﬁning ν ≤ μ := μ| h, for For λ ∈ C − , we temporarily write [λ]∼ for the collection of all μ μ ∈ [λ]. As a consequence of the discussion, we have: Proposition 5.4. The restriction map [λ] → [λ]∼ is a poset isomorphism, if [λ] is given the poset structure ≤ deﬁned in the ﬁrst paragraph of this section), and ∼

if [λ]∼ is given its poset structure via ≤ . Proof. The bijectivity of restriction has already been established in Proposition 3.3. As noted in its proof, an inverse on [λ]∼ of restriction is provided by ∼

μ → μ k,a = μk,a , where λ = λk,a . As the deﬁnition of ≤ shows, this inverse is order preserving, as is the restriction map itself. We introduce some further categories, obtained by restricting to g all the categories listed in Table 1, as well as O[λ] and O+ [λ], decorating the resulting strict Each of these image category with a “tilde” (i. e., changing O to O). g-categories has an associated poset, given in Table 1 for the corresponding g-category. (We can view the posets as abstract sets, useful for labeling irreducible, standard, and ∼

costandard modules. As such, there is no need to pass to version using ([λ]∼ , ≤ ) in view of the poset isomorphism in Proposition 5.4. Keeping the Table 1 version eases our notational burden.) Thus, in the proposition below, we use the ≤ partial order 9See [33, Introduction]. Or consult Appendix I below.

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on [λ] and [λ]+ , whether or not we are dealing with categories of g or g-modules. We will extend the proposition to the partial order ≤λ in Remark 5.2(b), as well as to additional partial orders ≤nat discussed there. For use below, deﬁne (5.1)

− Crat := {λ ∈ C − | λ, αi∨ ∈ Z, for each i = 1, · · · , r and λ(c) ∈ Q}.

− In particular, each λ ∈ Crat has level a rational number k and k < −g since − λ ∈ C . Thus, we can write (−k + g) = e/m, with e and m relatively prime positive integers. Assuming D divides m, we ﬁnd that (m/D)δ − θs ∈ Φ+ (λ), so ◦

◦

that λ + ρ, ((m/D)δ − θs )∨ ≤ 0. This implies that 0 ≥ λ + ρ, θs∨ ) ≥ −e, a familiar ◦

◦

condition from alcove geometry. Together with the condition that λ ∈ P , it is − suﬃcient to guarantee that λ ∈ h∗ satisfying λ(c) = k belongs to Crat belongs to − C . For a larger picture, see Proposition 7.1 below. Finally, note that, for λ ∈ C − − with λ(c) ∈ Q, [λ]+ = ∅ unless λ ∈ Crat . (Use the fact that μ ∈ [λ] implies that μ ∈ λ + Q.) − Proposition 5.5. Let λ ∈ Crat and let Γ [λ] (resp., Γ+ [λ]+ ) be ﬁnite. The + + ∞ +,∞ [λ]+ are all highest weight categories categories O[Γ], O [Γ ], O [λ], and O + + with weight posets Γ, Γ , [λ] and [λ] , respectively. Each of these categories is equivalent to its counterpart for g-modules, as is each of the categories O[λ] and + O [λ]. The functors providing these equivalences are, in each case, given by the restriction functor of g-modules to g-modules, and by applying Fk,a to objects in, Γ [λ], with a determined by λ. There is a natural common extension say, a category O ∞ ∞ [λ], which is also inverse to the on O of the functors Fk,a to the functors Fk,a restriction functor.

Proof. Let Mres,∞ ( g), Mres,d,∞ (g) denote the categories of g, g modules, rek k res,d spectively, which are directed unions of objects in Mres ( g ), M (g), respectively. k k ∞ Then the functors Fk,a extend in an obvious way to functors Fk,a on the direct union ∞ categories, giving equivalences inverse to restriction. In particular, Fk,a provides, ∞ +,∞ [λ] equivalences inverse to restriction to the versions without the O [λ] and O “tilde.” The remaining equivalences are obvious. Each of the g-modules categories above has irreducible, standard and costandard modules. These modules will be denoted by placing a “tilde” over their g (μ) and M (μ) are the irreducible, standard and counterparts. Thus, L(μ), M (μ)+ , M (μ)+ are costandard modules for O[λ], if μ ∈ [λ]. If μ ∈ [λ]+ , then M + [λ]+ . All of these modules are the the standard and costandard modules for O restrictions to g of their g-counterparts. Remark 5.6. (a) Let E be an abstract highest weight category [9] with weight poset (Λ, ≤), and with costandard modules ∇(λ), λ ∈ Λ. Assume there also exist standard modules Δ(λ), λ ∈ Λ, and assume that Δ(λ) and ∇(λ) have the same composition factors (with multiplicities). There is a natural partial order ≤nat , at least when Δ(λ) and ∇(λ) have the same composition factors (which holds for all the categories above). More precisely, ≤nat is the partial order generated by the requirement that μ ≤nat ν when [Δ(ν) : L(μ)] = 0. Let Λnat = (Λ, ≤nat ) denote this new poset. Then E is also a highest weight category with respect to Λnat with the same standard and costandard modules. If ν ≤nat μ, then clearly ν ≤ μ.

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Now suppose that is a partial order on the set Λ such that, for each μ, ν ∈ Λ, μ ≤nat =⇒ μ ν. Then it is easily seen that E is a highest weight category with respect to (Λ, ), with the same standard and costandard modules. Moreover, if Γ is a -ideal in Λ, it is also a ≤nat -ideal in Λ. − Returning to the situation of Proposition 5.5, where λ ∈ Crat , there is also, in addition to the partial orders ≤nat and ≤ on [λ], the partial order ≤λ discussed in Remark 5.2(b) and a further partial order ↑ (discussed in §9, Appendix I). The orders ≤λ and ↑ are shown to the same on [λ] in Proposition 9.1 below. Also, just using λ ∈ C − , μ ≤nat ν implies in an obvious way, using the remarks above Proposition 9.1 that μ ↑ ν, and, thus, now using Proposition 9.1, μ ≤λ ν. In turn, μ ≤λ ν implies μ ≤ ν, when λ ∈ C − . We summarize this discussion as follows. (5.2)

μ ≤nat ν =⇒ μ ≤λ ν =⇒ μ ≤ ν (μ, ν ∈ [λ], λ ∈ C − ).

− There is a version of these implications for [λ]+ , when λ ∈ Crat .

(5.3)

μ ≤nat ν =⇒ μ ≤λ ν =⇒ μ ≤ ν (μ, ν ∈ [λ]+ ).

The meaning of ≤nat in (5.3) is not quite the same as it is in (5.2) since the ≤nat for [λ]+ is computed with respect to diﬀerent standard modules. However, if μ ≤nat ν in (5.3), then μ ≤nat ν in (5.2). This implies the validity of (5.3). We will mostly be using (5.3), so we have preferred not to use separate notations for the two orders denoted ≤nat . (Finally, it is interesting to observe that in (5.2) we have ≤nat =≤λ , though we will not need this fact.) The main conclusion to be drawn is that Proposition 5.5 holds as written, if the order ≤ is replaced by ≤λ or by ≤nat , although it must be understood that the meaning of ≤nat varies between [λ] and [λ]+ . A similar observation holds regarding Theorem 5.1. All these conclusions, including the inequalities (7.2) and (7.3) hold if ≤λ is replaced by the (restriction of the) Bruhat-Chevalley order on W , though we will not use this fact. (Apply [34, Lem. 2.2.11(iii)] with w = 1 there. This shows that two elements of W (λ) related by ≤λ are similarly related under the Bruhat-Chevalley order on W .) k, O + and O +,ﬁnite , just as the (strict) images (b) We can also deﬁne categories O k k +,ﬁnite of under the restriction functor of the corresponding categories Ok , O+ k and O g-modules. Each such strict image is a full subcategory of g-modules, by Proposition +,ﬁnite ∼ 3.1. (All of their objects belong to Mres g).) Also, O = Ok . But it is not true k ( k that any of these new categories is equivalent to the category of g-modules from which its name is derived, since, in particular, no generalized eigenvector of the +,ﬁnite , the category Ok Casimir operator Ω has been speciﬁed. Thus, while Ok ∼ =O k +,ﬁnite is not equivalent to Ok . Instead, Ok is equivalent to O+,ﬁnite as Corollary 3.2 k,a is equivalent to a direct sum of copies of Ok .) This shows. (It is true that O+,ﬁnite k is not an issue, however, with the g-categories in Theorem 5.1, since for example the generalized eigenvalue of Ω on objects of O∞ [λ] is determined by λ. − . He In [50], Tanisaki described the group W (λ) explicitly for any λ ∈ Crat also describes the dot action of W (λ) on [λ] in more explicit terms. Implicit in his discussion is a description of Φ(λ), together with a set of fundamental roots (which can in any event be easily calculated). We return to this in §7.1.

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291 15

6. Translation functors h∗ relative to the decomLet P be a ﬁxed Z-lattice h∗ whose projection onto ◦

◦

◦

◦

position (2.1) is P + Zχ, where P is the weight lattice of Φ in h∗ . Let P + be the set of all λ ∈ P such that λ, αi∨ ∈ N for all i = 0, · · · , r. Given λ, μ ∈ C − with μ − λ ∈ W P + , Kashiwara-Tanisaki [33] deﬁne an exact translation functor Tμλ : O[λ] −→ O[μ]. The deﬁnition is the familiar one, taking a “block” projection, after tensoring with an irreducible module L(γ), γ ∈ P + ∩ W (μ − λ). We summarize the key properties they prove in the following proposition.10 Proposition 6.1. ([33, Props. 3.6, 3.8]) Assume λ, μ ∈ C − satisfy μ − λ ∈ W P + and Φ0 (λ) ⊆ Φ0 (μ). (a) For any w ∈ W (λ), Tμλ M (w · λ) ∼ = M (w · μ). (b) Also, Tμλ L(w

· λ)) =

+ + L(w · μ), if w(Φ+ 0 (μ)\Φ0 (λ)) ⊆ Φ (λ); 0, otherwise.

Remark 6.2. Recall that if λ ∈ [λ] we let wλ ∈ W (λ) be the shortest element + + w ∈ W (λ) such that w · λ = λ . The condition w(Φ+ 0 (μ)\Φ0 (λ)) ⊆ Φ (λ) in Proposition 6.1(b) is equivalent to wλ = wμ where λ = w · λ and μ = w · μ. In case λ is regular, the condition is that w = wμ . We add some additional useful properties of Tμλ after extending these functors in two ways. First, the “block” projection deﬁnition of translation extends easily to O∞ [λ]. We use the same notation Tμλ for this extension, so that now we have the functor Tμλ : O∞ [λ] −→ O∞ [μ]. ∼ ∞ [μ] deﬁne Second, Tμλ and the natural equivalences O[λ] = O∞ [λ] and O∞ [μ] ∼ =O a composite functor ∞ [λ] −→ O ∞ [μ]. Tμλ : O Therefore, there is a commutative diagram Tμλ

O∞ [λ] ↓∼

−→ O∞ [μ] ↓∼

∞ [λ] O

μ ∞ [μ] −→ O

Tλ

where the vertical maps are just restriction of functors. We now list these additional properties of Tμλ . Proposition 6.3. Assume λ, μ ∈ C − satisfy the properties Φ0 (λ) ⊆ Φ0 (μ) and μ − λ ∈ W P + of Proposition 6.1. Then the following statements hold. (w · μ). (w · λ) ∼ (a) Tμλ M (w · λ) ∼ = M (w · μ) and Tμλ M =M 10Observe that the condition μ − λ ∈ W P + implies that W (μ) = W (λ).

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(b) If Φ0 (λ) = Φ0 (μ), then Tμλ and Tμλ give equivalences of categories ⎧ ∼ ⎨Tμλ : O∞ [λ] −→ O∞ [μ] ∼ ∞ ⎩Tλ : O ∞ [λ] −→ O [μ]. μ

(c) Again assume that Φ0 (λ) = Φ0 (μ). If Γ ⊆ [λ], set Γ = {w · μ | w · λ ∈ Γ}. Then Γ is a poset ideal in ([λ], ≤λ ) if and only if Γ is a poset ideal in ([μ], ≤μ ). In this case, the posets Γ and Γ are isomorphic by the evident map w · λ → w · μ, and the functors Tμλ and Tμλ induce (by restriction) category equivalences ⎧ ∼ ⎨Tμλ : OΓ [λ] −→ OΓ [μ] ∼ Γ ⎩Tλ : O Γ [λ] −→ O [μ]. μ

Proof. We ﬁrst prove (a). Because L(μ) ∼ = L(μ), the duality on O[μ] preserves blocks in the category O[λ]. It thus follows by construction that the translation functor Tμλ commutes with duality. Now apply Theorem 6.1(a). Next, we consider (c). First, the assertions concerning Γ and Γ follow from Remark 6.2. Because the exact functor Tμλ takes standard (resp., costandard) modules Δ(w · λ) (resp. ∇(w · λ)), w · λ ∈ Γ, in O[λ] to standard (resp., costandard) modules Δ(w · μ) (resp., ∇(w · μ)) in O[μ], the comparison theorem [40, Thm. 5.8] implies it is an equivalence of categories. A similar argument applies to Tμλ . This proves (c). Finally, (b) follows from (c), writing [λ] as a (directed) union of ﬁnite ideals Γ. − Lemma 6.4. Let λ, μ ∈ Crat satisfying the conditions μ−λ ∈ W P + and Φ0 (λ) ⊆ Φ0 (μ). Let w ∈ W (λ) = W (μ). (a) If w · μ ∈ [μ]+ , then w · λ ∈ [λ]+ . Also, when w = wμ , for μ ∈ [μ]+ , then w = wλ for λ = w · μ. (b) Assume λ := w · λ ∈ [λ]+ . Put μ = w · μ. If wλ = wμ , then μ ∈ [μ]+ . ◦

Proof. To prove (a), it must be shown ﬁrst that if α ∈ Π, then w · λ, α∨ ≥ 0. ◦

By hypothesis, 0 ≤ w·μ, α∨ = w(μ+ρ), α∨ )−1 for any simple root α ∈ Π. Hence, (μ + ρ, w−1 (α)∨ ≥ 1. Since μ ∈ C − , this forces w−1 (α) < 0. Now suppose that ◦

w·λ, α∨ < 0, for some α ∈ Π ⊆ Φ0 (λ). Then, arguing as before, λ+ρ, w−1 (α)∨ ≤ 0. If λ + ρ, w−1 (α)∨ < 0, then, since λ ∈ C − , w−1 (α) > 0, which is impossible. Thus, w−1 (α) ∈ Φ0 (λ) ⊆ Φ0 (μ), again impossible. Next, for the last assertion of (a), notice that w = wμ means that wμ is of minimal length in its coset wW0 (μ), so it is certainly of minimal length in wW0 (λ) ⊆ wW0 (μ), giving w = wλ . Now (a) is completely established. Now we prove (b). We can assume that w = wμ = wλ . Assume, for some ◦

α ∈ Π, w·μ, α∨ < 0. In other words, w(μ+ρ), α∨ < 1, i. e., μ+ρ, w−1 (α)∨ ≤ 0. But w · λ, α∨ ≥ 0, so that λ + ρ, w−1 (α)∨ ) ≥ 1. Since λ ∈ C − , this means that w−1 (α) < 0, so that l(sα w) < l(w). Thus, sα w · μ = w · μ. Since μ ∈ C − , we have μ + ρ, w−1 (α)∨ ≥ 0. Combining the two previous paragraphs, μ + ρ, w−1 (α)∨ = 0. However, this means that sα w(μ + ρ) = μ + ρ, or equivalently sα w · μ = μ = w · μ, contradicting the assumption that w = wμ since l(sα w) < l(w).

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− Proposition 6.5. Let λ, μ ∈ Crat (see (5.1) satisfy μ − λ ∈ W P + and Φ0 (λ) ⊆ λ Φ0 (μ). Then the functor Tμ takes objects in O+ [λ] to objects in O+ [μ]. Also, Tμλ + [λ] to objects in O + [μ]. If τ ∈ [μ]+ , then L(wτ · λ) is the unique maps objects in O + irreducible module L(ν) ∈ O [λ] for which Tμλ L(ν) = L(τ ). A similar statement + [λ]. ) and O holds for L(τ

Proof. We wish to show that L := Tμλ L(λ ) ∈ O+ [μ]. If L = 0, then it is L(μ ) where μ = w · μ and wλ = wμ by Remark 6.2. Thus, μ ∈ [μ]+ by Lemma 6.4(b), as desired. This proves the ﬁrst assertion of the proposition. The second assertion concerning Tμλ is an easy consequence. The ﬁnal assertion follows from Lemma 6.4(a) and Remark 6.2.

We also obtain − satisfy μ − λ ∈ W P + and Φ0 (λ) ⊆ Φ0 (μ). For Lemma 6.6. Let λ, μ ∈ Crat + w ∈ W (λ) with w · λ ∈ [λ] ,

Tμλ M (w · λ)+ ∼ = M (w · μ)+ λ + (w · μ)+ . (w · λ) ∼ Tμ M =M

(w · μ)+ = 0. (If w · μ ∈ [λ]+ , then M (w · μ)+ = 0 and M Proof. By Propositions 6.1 and 6.5, Tμλ M (w·λ)+ is a quotient of M (w·μ) and belongs to O+ [μ]. Thus, Tμλ M (w·λ)+ is a quotient of M (μ)+ . We can compare their ◦

characters using Proposition 6.1 and Weyl’s character formula (applied for g on the ◦

◦

irreducible modules V (w · λ), V (w · μ)). This shows (after inducing s(w ·λ), s(w ·μ) from p to g) that Tμλ M (w ·λ)+ and M (w ·μ)+ have the same character. Hence, they (w · μ)+ , (w · λ)+ ∼ are isomorphic. A similar argument applies to show Tμλ M =M proving the lemma. Now the following analogue of Proposition 6.5 follows as in the proof of the latter. − Proposition 6.7. Let λ, μ ∈ Crat satisfy the conditions (of Proposition 6.1) μ − λ ∈ W P + and Φ0 (λ) ⊆ Φ0 (μ). Then the following statements hold. ∗ (w · μ)+ . ∗ (w · λ)+ ∼ (a) Tμλ M ∗ (w · λ)+ ∼ = M ∗ (w · μ)+ and Tμλ M =M + Also, when w = wμ , for μ ∈ [μ] , then w = wλ for λ = w · μ. (b) If Φ0 (λ) = Φ0 (μ), then Tμλ and Tμλ give equivalences of categories ⎧ ∼ ⎨Tμλ : O+,∞ [λ] −→ O+,∞ [μ] ∼ +,∞ ⎩Tλ : O +,∞ [λ] −→ O [μ]. μ

(c) Again assume that Φ0 (λ) = Φ0 (μ). If Γ ⊆ [λ]+ , set Γ = {w · μ | w · λ ∈ Γ}. Then Γ is a poset ideal in ([λ]+ , ≤λ ) if and only if Γ is a poset ideal in ([μ]+ , ≤μ ). In this case, the posets Γ and Γ are isomorphic by the evident map w · λ → w · μ, and the functors Tμλ and Tμλ induce (by restriction) category equivalences ⎧ ∼ ⎨Tμλ : OΓ,+ [λ] −→ OΓ ,+ [μ] ∼ Γ ,+ ⎩Tλ : O Γ,+ [λ] −→ O [μ]. μ

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7. Quantum enveloping algebras and category equivalences ◦

We continue to work with the indecomposable (classical) root system Φ, and we let be a positive integer. Recall D = (θl , θl )/(θs , θs ) ∈ {1, 2, 3}. Let

, if is odd; (7.1) e := /2, if is even. ◦

We will consider the aﬃne Weyl group We associated to W and the integer e. This group is discussed in great detail in Jantzen’s textbook [27, II,§6], where the integer e is denoted p (but is not required to be a prime). There is a natural ◦

◦

◦

dot action of We := W eQ on the set of integer weights P ⊆ h∗+ , given by ◦ ◦ ◦ ◦ w · μ = w(μ + ρ) − ρ for w ∈ We . In turn this induces a faithful (non-linear) action ◦

of We on the Euclidean space E associated to the root system Π (see notation list ◦

in §2). The action without the · is the usual linear action of W , together with ◦

eQ acting by translation. The fundamental reﬂections s0 , s1 , · · · , sr for We consist ◦

of the usual reﬂections si associated to fundamental roots αi ∈ Π, i = 1, · · · , r, ◦

together with the reﬂection s0 in the aﬃne hyperplane {x ∈ h∗ | x + ρ, θs∨ = −e }. In this way, We becomes a Coxeter group. We will usually denote its length function by le and let ≤e denote the Bruhat-Chevalley partial order on We .11 The following proposition is an easy calculation, similar to those given in [50, p. 269]. The ﬁrst observations of this kind are likely those of [36]. We state it ◦

only for (D, e) = 1. (In particular, this condition holds when Φ is simply laced since D = 1 in that case.) A somewhat similar result holds without the assumption (D, e) = 1, though the group We must be modiﬁed; see [50, Lem. 6.3]. Proposition 7.1. (a) Let be a positive integer and let e be as in (7.1). Let − λ ∈ Crat with λ(c) = k, and assume that −(k + g) = /2D and that (D, e) = 1. ∼ There is an isomorphism φ : We −→ W (λ) sending s0 , · · · , sr to the fundamental reﬂections of the Coxeter group W (λ) deﬁned by 2δ − θs , α1 , · · · , αr ∈ Φ(λ) if is odd, and to the fundamental reﬂections of W (λ) deﬁned by δ − θs , α1 , · · · , αr if ◦

◦

is even. In both cases, if μ ∈ P , then ⎧ ◦ ◦ ◦ ⎨φ (w) · (μ + kχ) = w · μ + kχ, w ∈ W ;

(7.2) ◦ ◦ ◦ ⎩φ (eγ) · (μ + kχ) = μ + eγ + kχ mod Z δ, γ ∈ Q.

Consequently, if μ ∈ h∗ with μ(c) = k and w ∈ We , then (7.3)

◦

φ (w) · μ = w · μ + kχ

mod Cδ.

(b) More generally, if −(k + g) = e/m for some positive integers e, m with ∼ (m, e) = 1 and D|m, then there is an isomorphism φe,λ : We → W (λ), where φe,λ (s0 ) = s(m/D)δ−θs and, for i = 1, · · · , r, φe,λ (sαi ) is equal to the fundamental 11We can also regard W has a Coxeter group with fundamental reﬂections s , s , · · · , s , e r 0 1 ◦

taking s0 to be the reﬂection in the aﬃne hyperplane {x ∈ h∗ | x + ρ, θs∨ = +e }. In this case, let le and ≤e be the length and Bruhat-Chevalley order on We . See the Appendix I for much more discussion.

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reﬂection deﬁned by α1 , · · · , αr . The roots (m/D)δ−θs , α1 , · · · , αr are the (positive) fundamental roots in Φ(λ). Each equation in (7.2) and (7.3) holds as written, but with φ replaced by φe,λ . Remark 7.2. The maps φe,λ and φ agree when /2D = e/m, which is our major interest in this paper (the “quantum case”, at least when (D, e) = 1.) The exact description of φ (w) · μ above (or of φe,λ (w) · μ) is ◦

◦

◦

φ (w) · μ = (w · μ)k,a = (w · μ + kχ)k,a = w · μ + kχ + bδ, where b is chosen so that the Casimir operator Ω acts on L(φ (w) · μ) with the same ◦

◦

◦

μ+2ρ,w·μ) action as on L(μ). That is, a = (μ + 2ρ, μ) and b = a−(w·2(k+g) . This reader is cautioned that the projections onto Cδ for μ and for φ (w) · μ will generally be ◦

diﬀerent. In particular, if w = eγ, γ ∈ Q, then φ (eγ) acts as a translation by eγ mod Cδ on the elements μ of level k in h∗ . That is, φ (eγ) · μ = μ + eγ mod Cδ. However, it is not true in general that φ (eγ) · μ = μ + eγ exactly, even if γ is replaced on the right by any ﬁxed element of γ + Zδ. One consequence of having to work mod Cδ with level k weights is that the ◦

◦

meaning of dominance orders in the correspondence between P and P +kχ mod Cδ is lost. However, the Bruhat-Chevalley order is preserved. See the §9, Appendix I for a discussion of the Bruhat-Chevalley orders relative to the often used partial orders ↑ of strong linkage. ◦

Let ζ ∈ C be a primitive th root of unity and set q = ζ 2 . Let Uζ = Uζ (Φ) ◦

be the (Lusztig) quantum enveloping algebra at ζ for the root system Φ. Let Q be the category of type 1, integrable, ﬁnite dimensional Uζ -modules. According to Tanisaki [50, Thm. 7.1], summarizing work of Kazhdan-Lusztig [35] and Lusztig [38], there is a category equivalence (7.4)

∼

F : O− /2D−g −→ Q

between the category O− /2D−g of g-modules and the category Q . This holds for ◦

all positive integers , when Φ has type A or D, but restrictions are required in the other cases; see [50, Thm. 7.1, Rem. 7.2]—note that Rem. 7.2(a) should be replaced by r > h, the Coxeter number. In the notation of the previous section, letting k := −/2D − g, Ok has a block decomposition ﬁnite [λ]+ . O − λ∈Crat ,λ(c)=k

+ bδ]+ identiﬁes Observe that the “ﬁnite” can be omitted here, and, generally, O[λ + ; indeed, these are identical subcategories of the category of g-modules. with O[λ] ◦ + For μ ∈ [λ] , F L(μ) = Lζ (μ). Also, as noted in [50, Thm. 7], Kazhdan-Lusztig ◦ ◦ (μ) ∼ proved that F M = Δζ (μ). (Here μ ≡ μ + kχ mod Cδ.)

− , and let Γ be a ﬁnite ideal in [λ]+ with respect to ≤λ (see ReLet λ ∈ Crat Γ,+ [λ] mark 5.2(b)). Recall that every object M in the highest weight category O Γ,+ has a ﬁnite composition series. Let P be a projective generator for O [λ]. If Γ,+ [λ] ∼ AΓ := End(P ) , then O = AΓ -mod. It is convenient to choose P so that the Γ,+ radn AΓ / radn+1 AΓ modules in O [λ] are actually AΓ -modules. Put gr AΓ =

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be the positively graded algebra obtained from AΓ using its radical ﬁltration.12 Let Γ,+ [λ] be the category gr AΓ -grmod of graded gr AΓ -modules. gr O − be regular, and let Γ be a ﬁnite ideal in the poset Theorem 7.3. Let λ ∈ Crat ([λ] , ≤λ ). Γ,+ [λ] = gr AΓ -grmod is a highest weight category with (a) The category gr O poset Γ (or Γnat ) and standard objects gr M (μ)+ , μ ∈ Γ. Γ,+ [λ] has a graded (b) The algebra gr AΓ is a Koszul algebra. Also, gr O Kazhdan-Lusztig theory with respect to the length function on Γ deﬁned by the Coxeter length. (c) For μ = x · λ ∈ [λ]+ , form the radical ﬁltration M (μ)+ = F 0 (μ) F 1 (μ) · · · F m (μ) = 0 of M (μ)+ . For ν = y · λ ∈ [λ]+ , [F i (μ)/F i+1 (μ) : L(ν)] is the coeﬃcient of tl(x)−l(y)−i in the inverse Kazhdan-Lusztig polynomial Qy,x for the Coxeter group W (λ). +

− Proof. Suppose that λ ∈ Crat is regular and that k = λ(c). Write −(k + g) = e/m, where (e, m) = 1. ∼

Case 1: D divides m. Proposition 7.1(b) gives an isomorphism φe,λ : We → W (λ) of Coxeter groups, matching up indicated sets of fundamental reﬂections. For a ◦

rational number k , put λ = λ + k χ, so that k = λ (c). We can choose k so − that λ ∈ Crat is regular. We can also choose k so that −(k + g) = /2D for a ◦

postive integer not divisible by 2 or 3 (if Φ has type G2 ). Deﬁning e as in (7.1) (using for and e for e), we have (D, e ) = 1. Proposition 7.1(a) then gives an ∼ isomorphism φ : We → W (λ ), again matching up fundamental reﬂections. Thus, there is an isomorphism W (λ) → W (λ ) preserving fundamental reﬂections. Since λ and λ are both regular, the equality W0 (λ) = W0 (λ ) is trivial. Therefore, by ∼ → O[λ ]. (See Proposition 5.5.) [23, Thm. 11], there is a category equivalence O[λ] Since the orders ≤λ and ≤λ obviously correspond, standard modules correspond (implicit in [23]). Similar comments apply to costandard modules. The sets [λ]+ and [λ ]+ are easily characterized (when λ, λ are regular) in terms of representing elements w · λ, w · λ by requiring w to be of maximal length in ◦ ∼ O+ [λ ], as W w ⊆ W (λ) and similarly for w . Thus, we get an equivalence O+ [λ] = + [λ ]. (See Proposition 5.5.) Since the partial + [λ] ∼ well as an equivalence of O =O orders correspond, so do their ideals. Let Γ ≤ [λ ]+ correspond to Γ ≤ [λ]+ . Γ ,+ [λ ]. Γ,+ [λ] ∼ Clearly, O =O Adjusting k further we can assume that F is a category equivalence. In particular, we can assume that is odd, not divisible by 3 in type G2 , and > h. ◦

If Γ is an ideal in the ≤λ partial order, then Γ is an ideal in the ↑e -order, by ◦

◦

Appendix I, Theorem 9.6. Therefore, Γ is an ideal in the natural order ≤nat on P . ◦ ◦ ◦ ◦ ◦ Γ ,+ [λ ]) = Q [Γ ]. (As is well-known, when μ ≤nat ν, then μ ↑e ν.) We have F (O ∼

◦

Γ,+ [λ] → Q [Γ ] Combining this with the previous paragraph gives an equivalence O 12The algebra A is only determined up to Morita equivalence. However, if A and B are Γ two Morita equivalent ﬁnite dimensional algebras, then gr A is Morita equivalent to gr B, and, in addition, these two graded algebras have equivalent graded module categories. See Appendix II.

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of highest weight categories. Notice that by Proposition 6.7(c), the posets Γ and Γ are isomorphic. Now apply [42, Cor. 8.5]. 13 ◦

Case 2: D does not divide m: In this case, W (λ) ∼ = We∨ = W Q∨ , which is the ∨ aﬃne Weyl group for the dual root system Φ . (See [50, Lem. 6.2].) By [23, Γ,+ [λ] is equivalent to a similar category, but replacing g by the Thm. 11] again, O aﬃne Lie algebra associated to the dual root system. Hence, Case 2 reduces to Case 1. (Notice that [23, Thm. 11] does not require the underlying Lie algebras to be the same!) ◦

− Theorem 7.4. Let λ = λ + kχ + bδ ∈ Crat . For μ ∈ [λ]+ , M (μ)+ has a + 0 1 m ﬁltration M (μ) = F (μ) ⊇ F (μ) ⊇ · · · ⊇ F (μ) = 0 in which each section F i (μ)/F i+1 (μ) is a semisimple g-module, and such that, given any ν ∈ [λ]+ , the i i+1 multiplicity [F (μ)/F (μ) : L(ν)] is the coeﬃcient of tl(wμ )−l(wν )−i in the inverse Kazhdan-Lusztig polynomial Qwν ,wμ associated to W (λ). If λ is regular, then the ﬁltration F • (μ) is the radical ﬁltration of M (μ)+ .

Proof. We prove this result in the “quantum case” (discussed in the proof of Theorem 7.3), speciﬁcally, in which −(k + g) = /2D with (D, e) = 1. We leave the “non-quantum case” to the reader. The case in which λ is regular is handled in the previous theorem, so assume that λ is not regular. For a suﬃciently large positive ◦ integer n, the weight λ := −2ρ + (k − n)χ + bδ is regular (so Φ0 (λ ) = ∅ ⊆ Φ0 (λ)), ◦

◦

◦

◦

− . Also, λ − λ = λ + 2ρ + nχ ∈ W (P + + nχ) ⊆ W P + . In addition, and it lies in Crat Φ(λ) = Φ(λ), so W (λ) = W (λ ). Let μ ∈ [λ]+ , so that μ = wμ · λ, where wμ ∈ W (λ) has minimal length among all w ∈ W (λ) for which μ = w · λ. Thus, by Lemma 6.4, μ := wμ · λ ∈ [λ ]+ , and so Tλλ M (μ )+ = M (μ)+ . Put F i (μ) = Tλλ radi M (μ )+ . Then F • (μ) is a ﬁltration of M (μ)+ . Since the functor Tλλ is exact we have

F i (μ)/F i+1 (μ) ∼ = Tλλ (radi M (μ )+ / radi+1 M (μ )+ )

is semisimple. To determine the multiplicity [F i (μ)/F i+1 (μ) : L(ν)], we can assume that ν ∈ [λ]+ . Write ν = wν · λ. Since [radi M (μ)+ / radi+1 M (μ)+ : L(wλ ·λ)] = [radi M (μ )+ / radi+1 M (μ )+ : L(wν ·λ )], which is the coeﬃcient of tl(wμ )−l(wν )−i in the inverse Kazhdan-Lusztig polynomial Qwν ,wλ of W (λ), the result follows. Now we consider an analogous result for the quantum enveloping algebras Uζ of type An or Dn . Let C := {x ∈ E | − < x + ρ, α0∨ ,

x + ρ, α∨ < 0, ∀α ∈ Π}

be the anti-dominant chamber. Given a dominant weight ν, let wν ∈ We have minimal length so that wν−1 ν ∈ C − . 13Results claimed in [4, §9.5 (para. 1), Lem. 9.10.5, Thm. 9.10.2], together with the order compatibility result given in Theorem 9.6 in Appendix I, imply that AΓ itself is Koszul. (According to [24, Appendix], a result like our Theorem 9.6 is required in [4, Lem. 9.10.5] to make its proof work. In turn, this lemma is required for [4, Thm. 9.10.1], a main result.) However, we only need that gr AΓ is Koszul for the results below in the singular weight case, and no better result is obtained in these cases by knowing AΓ is Koszul here. (Koszulity is not generally preserved under exact functors.) See also footnote 2 regarding the recent preprint [48].

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Theorem 7.5. Assume that Φ has type A or D. Also, for type D2n+1 , it is required that e ≥ 3.14 ◦

(a) Assume that μ, ν ∈ P + are dominant weights which are We -conjugate. Then the standard module Δζ (μ) has ﬁltration Δζ (μ) = F 0 (μ) ⊇ · · · ⊇ F m (μ) = 0 by Uζ -submodules with each section F i (μ)/F i+1 (μ) a semisimple Uζ -module. Further, the multiplicity [F i (μ)/F i+1 (μ) : Lζ (ν)] can be taken to be the coeﬃcient of tl(wμ )−l(wν )−i in the inverse Kazhdan-Lusztig polynomial Qwν ,wμ associated to W (λ). If μ is regular, then the ﬁltration F • (μ) is the radical ﬁltration of Δζ (μ). (b) Assume that e ≥ h. Let Γ be a ﬁnite ideal of e-regular weights. Let BΓ be the ﬁnite dimensional algebra whose module category identiﬁes with the category of Uζ -modules having highest weights in Γ. Then the algebra gr BΓ is a Koszul algebra. Also, the category gr BΓ -grmod has a graded Kazhdan-Lusztig theory with respect to the length function (deﬁned on We -orbits in Γ by the Coxeter length.) Proof. This follows from Theorem 7.4, since for types A and D as indicated, the functor F is an equivalence of categories, preserving standard modules. The argument above, traced through from the proof of Theorem 7.3, gives the additional result that, under the hypotheses of Theorem 7.5(b), there is an isomorphism ExtnBΓ (L, L ) ∼ = Extngr BΓ (L, L ), valid for all n ≥ 0 and for all irre ducible BΓ -modules L, L (which are naturally irreducible gr BΓ -modules); see [42, Cor. 8.5(a)]. Consequently, the homological algebra of BΓ in Theorem 7.5(b) is very close to that of gr BΓ . Of course, BΓ is even isomorphic to gr BΓ , if we grant the Koszulity of AΓ argued in footnote 7. 8. Applications In this section, we reinterpret Theorem 7.5 for the q-Schur algebras and then pass to a similar result for Specht modules for Hecke algebras. Then we brieﬂy raise some open questions. Finally, we obtain some similar results for classical Schur algebras in positive characteristic, involving the James conjecture and the bipartite conjecture. = 8.1. q-Schur and Hecke algebras. Given a Coxeter system (W, S), let H −1 H(W ) be the Hecke algebra over Z = Z[q, q ] (Laurent polynomials in a variable q) with basis {τw | w ∈ W } and deﬁning relations

τsw , if l(sw) = l(w) + 1 τs τw = for s ∈ S, w ∈ W . qτw + (q − 1)τsw , otherwise →H be the Z-algebra involution deﬁned by Ψ(τw ) = (−q)l(w) τ −1 Let Ψ : H w−1 . If is a H-module, Ψ denotes the module obtained by making H act through M then M Ψ. For example, let Sr be the symmetric group of degree r, and let S = {(1, 2), · · · , ) simply (r − 1, r)}. Then (Sr , S) is a Coxeter system. In this case, denote H(W + by H, or H(r) if r needs to be mentioned. Let Λ(n, r) (resp., Λ (n, r)) be the 14There is no other restriction on the positive integer e. In case e is odd, the arguments in

Theorems 7.3 and 7.4 can be rearranged to treat all quantum cases, using translation functors alone, without recourse to [23]. Of course, use of [23] not only handles the e even case, but also − in Theorems 7.3 and 7.4. allows a treatment for aﬃne Lie algebras of all weights in Crat

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set of compositions (resp., partitions) λ of r into n parts (resp., with at most n parts); let Λ(r) = Λ(r, r) and Λ+ (r) = Λ+ (r, r)). For λ ∈ Λ(n, r), let Tλ be the deﬁned by λ, and T(n, r) := right “permutation” module for H λ∈Λ(n,r) Tλ . The (integral) q-Schur algebra (of bidegree (n, r)) is the endomorphism algebra (8.1.1)

Sq (n, r) := EndH (T(n, r)).

Given any commutative Z-algebra K (e. g., a ﬁeld), let Sq (n, r)K )(or just Sq (n, r) r) ⊗ K—it has a description similar to if K is clear) denote the K-algebra S(n, K and T (n, r) = T(n, r)K , respectively. (8.1.1), replacing H and T (n, r) by H = H From now on assume that K contains Q(ζ), where ζ is a primitive th root of 1. Put q = ζ 2 , a primitive eth root of 1, in the notation of the previous section. (No restriction is placed on e, except as otherwise noted.) The triple (Sq (n, r), T, H) satisﬁes the “ATR” set-up prosyletized in [12]. In particular, given M ∈ mod–H (right modules), put M := HomH (M, T ) ∈ Sq (n, r)–mod, and, given N ∈ Sq (n, r)–mod, let N := HomSq (n,r) (N, T ). In this way, there is a contravariant functor M → M (resp., N → N ) from mod–H to Sq (n, r)–mod (resp., Sq (n, r)–mod to mod–H). The convenience of denoting them by the same symbol overcomes the annoyance of denoting them by the same symbol! If Uζ is the (Lusztig) quantum enveloping algebra of type An−1 over K, there is a surjective homomorphism Uζ Sq (n, r). In this way, Sq (n, r)-mod is embedded in Uζ -mod. In addition, Sq (n, r)–mod is a highest weight category with poset (Λ+ (n, r), ) deﬁned by the dominance order on partitions. Irreducible modules Lq (λ), standard modules Δq (λ), and costandard modules ∇q (λ) are all indexed ¯ where by Λ+ (n, r). When regarded as Uζ -modules, Lq (λ) gets relabeled as Lζ (λ), ◦

¯ ∈ P + is deﬁned as follows: write λ = (λ1 , · · · , λn ), λ1 ≥ · · · ≥ λn ≥ 0, and put λ ¯ λ = a1 1 + · · · + ar−1 n−1 with ai := λi − λi+1 . (In this expression, we label the simple roots for sln in the usual way, as in [5].) Each λ ∈ Λ+ (r), thus determines ¯ ∈ C − (the wλ ∈ Sr which has minimal length among all w satisfying w−1 · λ anti-dominant chamber for Uζ ). In particular, for λ ∈ Λ+ (n, r), we have ⎧ ¯ ∼ ⎨Δq (λ) = S λ, (8.1.2) ⎩∇ (λ) ¯ ∼ = SλΨ . q In this expression, λ denotes the partition conjugate to λ ∈ Λ+ (r). In addition, the irreducible H-modules are indexed by the set Λ+ (r)row-reg of (row) e-regular ¯ is e-restricted partitions (i. e., no row is repeated e-times). If λ ∈ Λ+ (r), then λ (i. e., it has all coeﬃcients of fundamental dominant weights positive and < e) if and only if λ is e-regular. Then for λ ∈ Λ+ res (r) (the e-restricted partitions), ⎧ ⎨DλΨ , λ ∈ Λ+ res (r); (8.1.3) L(λ) ∼ = ⎩0, otherwise. Theorem 8.1.1. Assume that K is a ﬁeld containing Q(ζ). (a) For λ ∈ Λ+ (n, r), the q-Weyl module Δq (λ) for the q-Schur algebra Sq (n, r) has a ﬁltration Δq (λ) = F 0 (λ) ⊇ F 1 (λ) ⊇ · · · ⊇ F m (λ) = 0 with semisimple sections F i (λ)/F i+1 (λ) in which, given ν ∈ Λ+ (r), the multiplicity of Lq (ν) in

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F i (λ)/F i+1 (λ) is the coeﬃcient of tl(wλ¯ )−l(wν¯ )−i in the inverse Kazhdan-Lusztig polynomial Qwν¯ ,wλ¯ associated to the aﬃne Weyl group We of type An−1 . (b) For λ ∈ Λ+ (r), the q-Specht module S λ for the Hecke algebra H has a ﬁltration 0 = G0 (λ) ⊆ G1 (λ) ⊆ · · · ⊆ Gm (λ) = S λ with semisimple sections Gi+1 (λ)/Gi (λ) in which, given ν ∈ Λ+ res (r), the multiplicity of the irreducible Hmodule DνΨ in the section Gi+1 (λ)/Gi (λ) is the coeﬃcient of tl(wλ¯ )−l(wν¯ )−i in the inverse Kazhdan-Lusztig polynomial Qwν¯ ,wλ¯ associated to the aﬃne Weyl group We of type Ar−1 . Proof. (a) is merely a translation into the language of q-Schur algebras of Theorem 7.5(a). As for (b), we can take n = r. We ﬁrst observe T := T (r, r) ∼ = Sq (n, r)f for an idempotent f ∈ Sq (r, r) [41, p. 664], and so T is projective. In addition, T is a tilting module for Sq (r, r) and is therefore self-dual. See [18, Thm. 8.4]. Thus, T is also an injective Sq (r, r)-module and so the “diamond functor” (−) = HomSq (r,r) (−, T ) : Sq (r, r)–mod −→ mod–H is exact. Hence, (a) implies (b), putting Gi (λ) = F m−i (λ) .

8.2. Open questions. We raise some open questions. Question 8.2.1. Given λ ∈ Λ+ (r), when is it true that the ﬁltration described in the proof of Theorem 8.1.1(b) is the socle ﬁltration of S λ ? One should at least ¯ is regular in the sense of alcove assume that λ is restricted, and the case where λ geometry is already interesting. Question 8.2.2. When is there a positive grading on H (with grade 0 semisimple) such that for each λ ∈ Λ+ (r) there is a graded H-module structure on S λ , so that the multiplicities of irreducible H0 ∼ = H/ rad H-modules in each grade are as predicted by Theorem 8.1.1(b)? The same question may be asked for the quotient algebras H(n, r) deﬁned in [18] and for the H(n, r)-modules S λ , λ ∈ Λ+ (n, r). Question 8.2.3. In [7], a Z-grading on Specht modules is given with respect to a Z-grading of the Hecke algebra. Since this grading is not, in general, a positive grading with the grade 0 term a semisimple algebra, individual grades of a given graded module are not necessarily semisimple modules. Nevertheless, it appears from the form of the graded multiplicities in [7], together with [51], that these multiplicities are the same coeﬃcients which appear in our Theorem 8.1.1(b). The question, therefore, arises as to when it is possible to “regrade” the Hecke algebra H (shifting grades of projective indecomposable summands and passing to an endomorphism algebra) to achieve a positively graded algebra with grade 0 term semisimple in such a way that the induced regradings of the Specht modules agree with our ﬁltration sections as in Question 8.2.2. When this is possible, it answers Question 8.2.2 in a very speciﬁc way. Question 8.2.4. For λ ∈ Λ+ (n, r), when is the ﬁltration for Δq (λ) described in Theorem 8.1.1(a) given by the radical series? The same question can be asked in all types; [42, Thm. 8.4(c)] gives a positive answer for regular highest weights. Lin [37, Rem. 2.9(1)] suggests a positive answer in the singular case, at least for generic weights.

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Question 8.2.5. When is there a positive grading on Sq (n, r) (with grade 0 semisimple) and on the standard modules Δq (n, r) so that the grade i section multiplicities are predicted by those in Theorem 8.1.1(a) (for all i)? From the general theory of graded quasi-hereditary algebras [42], if Sq (n, r) has a positive grading, its standard modules will automatically be graded. Question 8.2.6. In [3], Ariki gives a Z-grading on Sq (n, r) and the standard modules under mild restrictions on e. One can ask when some regrading process in this case serves to give a positive question in Question 8.2.5 above. When n ≥ r, [3] computes the multiplicities of graded irreducible modules in his graded standard modules giving an answer involving (inverse) Kazhdan-Lusztig polynomials. Is there some positive regrading of the grading in [3] possible so that the multiplicities in each grade agree with those in Theorem 8.1.1(a)? Question 8.2.7. When is gr Sq (n, r) a quasi-hereditary algebra? When is it Koszul? One can also ask when gr Sq (n, r) has a Kazhdan-Lusztig theory in the the sense of [11], though it should be stated that the same question is open for singular blocks of Sq (n, r) itself. Of course, all the above questions for q-Schur algebras can be asked in other types, i. e., for generalized q-Schur algebras. 8.3. Positive characteristic. Now assume that k is an algebraically closed ﬁeld of positive characteristic p. For positive integers n, r, let S(n, r) = S1 (n, r) be the classical Schur algebra over k of bidegree (n, r); see [20] for a detailed discussion in this special case. The irreducible S(n, r)-modules L(λ) are indexed by partitions λ ∈ Λ+ (n, r). √ Form the PID Z = Z(p) [ζ], where q := ζ 2 = p 1. (Here Z(p) denotes the ring of integers localized at the prime ideal (p).) The q-Schur algebra Sq (n, r), taken over Q(ζ), has a standard integral Z -form Sq (n, r) such that S(n, r) = Sq (n, r) ⊗Z k. For λ ∈ Λ+ (n, r), choose a S(n, r) -stable Z -lattice Lq (λ) in Lq (λ), and let Lq (λ) = Lq (λ) ⊗Z k be the S(n, r)-module obtained by base change to k. The following is a special case of a conjecture of James [26]. Conjecture 8.3.1. (James Conjecture, deﬁning characteristic case) If p2 > r, Lq (λ) ∼ = L(λ) for all λ ∈ Λ+ (n, r). The (full) James conjecture has been veriﬁed for all r ≤ 10 [26].15 It is also known that the James conjecture holds, for a ﬁxed n and all r, provided that p is suﬃciently large.The conjecture is trivial unless p ≤ r, so that r must grow with p for the conclusion to be substantive. ¯ be the dominant weight for SLn determined by λ. Let For λ ∈ Λ+ (n, r), let λ ¯ ∈ C − , the wλ¯ be the unique element x in the aﬃne Weyl group Wp such that x · λ anti-dominant chamber. (See the discussion two paragraphs above Theorem 8.1.1.) In the same spirit, but motivated by [21], [8, Thm. 6.3], and, especially, the notion of an abstract Kazhdan-Lusztig theory given in [11], we conjecture the following. Conjecture 8.3.2. (Schur Algebra Bipartite Conjecture, explicit form) For p2 > r, the Ext1 -quiver of S(n, r) is a bipartite graph. Explicitly, the decomposition E = {λ | l(wλ¯ ) ∈ 2Z},

O = {λ | l(wλ¯ ) ∈ 2Z + 1 }

15An advantage of using the James conjecture (over the Lusztig conjecture) is that it does not require that p ≥ n.

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of Λ+ (n, r) is compatible with the bipartite Ext1 -quiver. The ﬁrst sentence in Conjecture 8.3.2 means that Λ+ (n, r) decomposes into a disjoint union Λ+ (n, r) = E ∪ O (the “even” and the “odd” partitions) such that Ext1S(n,r) (L(λ), L(ν)) = 0, whenever λ, ν ∈ Λ+ (n, r) are either both in E or both in O. The second sentence provides explicit E and O. Theorem 8.3.3. Consider the Schur algebra S(n, r) in positive characteristic p. Assume that p2 > r and that the James Conjecture 8.3.1 and the Bipartite Conjecture 8.3.2 are true. (a) For λ ∈ Λ+ (n, r), let {F i (λ)} be the semisimple series of Δq (λ), q = ζ 2 = √ p 1, given in Theorem 8.1.1. Let L be the Sq (n, r) -stable Z -lattice in Δq (λ) generated by a highest weight vector.16 Set F i (λ) = F i (λ) ∩ L and F i (λ) = F i (λ) ⊗Z k. Then {F i (λ)} is a semisimple series in Δ(λ) = Δq (λ). Furthermore, for μ ∈ Λ+ (n, r), (8.3.1)

[F i (λ)/F i+1 (λ) : L(μ)] = [F i (λ)/F i+1 (λ) : Lq (μ)].

In particular, this multiplicity is given by a coeﬃcient in an inverse KazhdanLusztig polynomial, as in Theorem 8.1.1. (b) For λ ∈ Λ+ (r), the Specht module S λ for Sr has a ﬁltration 0 = G0 (λ) ⊆ 1 G (λ) ⊆ · · · ⊆ Gm (λ), in which, given μ ∈ Λ+ res , the multiplicity of the irreducible Sr -module DμΨ in the section Gi+1 (λ)/Gi (λ) is the coeﬃcient of tl(wλ¯ )−l(wμ¯ )−i in the inverse Kazhdan-Lusztig polynomial Qwμ¯ ,wλ¯ associated to the aﬃne Weyl group We of type Ar−1 . Proof. Since Conjecture 8.3.1 is assumed to hold, a section F i (λ)/F i+1 (λ) reduces mod p to an S(n, r)-module whose composition factor multiplicities are given by (8.3.1). To check that {F i (λ)} is a semisimple series Δ(λ), it suﬃces to check that these sections are semisimple. But, if L(τ ) and L(σ) both appear in a composition series in F i (λ)/F i+1 (λ), then the coeﬃcients of tl(wλ¯ )−l(wτ¯ )−i in Qwλ¯ ,wτ¯ and of tl(wλ¯ )−l(wσ¯ )−i in Qwλ¯ ,wσ¯ are nonzero. However, these are polynomials in q = t2 , so that l(wλ¯ ) − l(wτ¯ ) − i ≡ l(wλ¯ ) − l(wσ¯ ) − i

mod 2.

It follows that wτ¯ and wσ¯ have the same parity, so that F i (λ)/F i+1 (λ) is semisimple by Conjecture 8.3.2, as required for (a). Part (b) is proved in the same way as Theorem 8.1.1(b). Remark 8.3.4. James also conjectured [26] a version of Conjecture 8.3.1 for q-Schur algebras, with q an eth root of 1 and ep > r. It is at least reasonable to ask, as a question, if the analog of Conjecture 8.3.2 holds under this assumption. Positive answers (to this question, and to this version of the James conjecture), would have consequences for the ﬁnite general linear groups G in a non-deﬁning characteristic p. See [14, §9], which shows the group algebra of G has a large quotient which is a sum of tensor products of q-Schur algebras at various roots of unity q. Another reference is [6]. For a diﬀerent (though possibly related) use of Weyl module ﬁltrations for (general) ﬁnite Chevalley groups, see [28] and [45]. 16We make this choice for simplicity and to insure that Δ(λ) = L = Δ (λ) . However, with q

any choice of lattice L, L has the same sections.

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9. Appendix I ∗

For λ ∈ h , Coxeter groups W (λ) and W0 (λ) have been deﬁned at the beginning of §4. Moreover, speciﬁc fundamental sets of reﬂections Π(λ) and Π0 (λ) ⊆ Π(λ), respectively, were deﬁned. When λ ∈ C − , [33, Lem. 2.13] implies that W0 (λ) is a parabolic subgroup of W . In particular, this means that the Bruhat-Chevalley partial order ≤λ on W (λ) restricts to the Bruhat-Chevalley order on W0 (λ). Similarly, the length function lλ on W (λ) restricts to the length function on W0 (λ). If ν, μ ∈ C, write ν ↑ μ if either ν = μ, or, if there are positive real roots β1 , · · · , βm (allowing repetition) such that the corresponding reﬂections sβi satisfy (9.1)

ν = sβm sβm−1 · · · sβ1 · μ ≤ · · · ≤ sβ1 · μ ≤ μ.

(Observe that this forces each βi ∈ Φ(ν) = Φ(μ).) According to a result of Kac-Kazhdan (see [33, Thm. 3.1]), ν ↑ μ if and only if [M (μ) : L(ν)] = 0. In this section, we will give many further equivalences − in case γ, μ ∈ [λ]+ for some λ ∈ Crat . The proposition below gives the ﬁrst of these. Eventually, we will relate the ↑ order to a similar order well-known in alcove geometry, as well as to various Bruhat-Chevalley orders. See Theorem 9.6 and 9.7 below, both of which appear to be new. Proposition 9.1. Suppose that λ ∈ C − and that μ, ν ∈ [λ]. Then μ ↑ ν implies − μ ≤λ ν. If λ ∈ Crat , then μ ↑ ν if and only if μ ≤λ ν. Proof. Write ν = y · λ and μ = w · λ, where y, w ∈ W (λ) have minimal length (and are uniquely determined, since W0 (λ) is parabolic in W (λ)). Suppose that ν ↑ μ and let sβ1 , · · · , sβm be as in (9.1). We can assume that all the inequalities in (9.1) are strict, i. e., putting μ0 = μ and, for 1 ≤ i ≤ m, μi = sβi · μi−1 then μi−1 − μi = ni βi for some ni ∈ Z+ . In particular, ν = λ and βi ∈ Φ+ (λ), for each i = 1, · · · , m. Take a reduced expression w = t1 · · · tu , where tj = sγj ∈ S(λ). (Actually, u = 0 since y·λ > λ.) Let β ∈ Φ+ (λ) be so that sβ w·λ < w·λ; for example, we could take β = β1 in (9.1). Then w(λ + ρ), β ∨ = λ + ρ, w−1 (β)∨ > 0. Since λ ∈ C − , this means that w−1 (β) < 0. Thus, if j ≥ 1 is maximal with tj−1 · · · t1 (β) > 0, we obtain that β = t1 · · · tj−1 (γj ), for some j, 1 ≤ j ≤ u. Let w1 ∈ W (λ) be of minimal lλ -length with w1 ·λ = sβ w·λ. That is, w1 is a distinguished member of the tj · · · tu <λ w. left-coset sβ wW0 (λ) in W (λ). Thus, w1 ≤λ sβ w, while sβ w = t1 · · · So w1 <λ w. Continuing this way, eventually gives y <λ w, as desired. − Next, assuming λ ∈ Crat , we start from the assumption y <λ w and show that ν ↑ μ using induction on lλ (w) − lλ (y). For most of the proof, we will just use the fact that λ ∈ C − . But near the end, it is useful to know that the fundamental roots of Φ(λ) are linearly independent, in order to quote [25, §5]. Independence follows − from λ ∈ Crat , using [50, Lem. 6.3]. By [25, Prop., p. 122], there exists x ∈ W (λ) with y ≤λ x <λ w and lλ (w) − lλ (x) = 1. Put x = w v, where v ∈ W0 (λ) and w ∈ xW0 (λ) is the element of shortest length in this left coset. Then y = x v where x ≤λ x and v ≤λ v. The minimality of y implies that y = x ≤λ w . Possibly, y = w , but, in any case, induction implies that ν ↑ w ·λ = x·λ. However, x may be obtained from a reduced expression w = t1 · · · tu (as in the second paragraph of the proof) by removing one of the fundamental (with respect to W (λ)) reﬂections tj . Put β = t1 · · · tj+1 (γj ).

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Then β ∈ Φ(λ) and w−1 (β) = −tu · · · tj+1 (αj ). By [25, §5.6, Ex. 1], w−1 (β) < 0 and β > 0. So sβ w · λ ≤ w · λ. Thus, x · λ ↑ w · λ = μ, so x · λ ↑ μ, as desired. − In the following result, we use the fact that, if λ ∈ Crat , then in addition to ◦

W0 (λ), W is a parabolic subgroup of W (λ). − , μ ∈ [λ]+ . Then there is a distinguished (i.e., Lemma 9.2. Suppose λ ∈ Crat ◦

minimal length) (W , W0 (λ))-double coset representative d ∈ W (λ) with μ = w0 d · λ. The element d is unique; write d = d(μ). Proof. Choose an element d ∈ W (λ) of minimal length with d · λ = w0 · μ, ◦+

or, equivalently, μ = w0 d · λ. Since μ ∈ [λ]+ , sα · μ < μ for all α ∈ Φ , whereas ◦

◦

sα · (d · λ) = sα · (w0 · μ) = sα (w0 (μ + ρ)) − ρ = sα (w0 (μ + ρ)) − ρ > w0 · μ = d · λ. ◦+

Since λ ∈ C − , d−1 (α) > 0 for all α ∈ Φ . ◦

For all w ∈ W , we will show (9.2)

lλ (wd) = lλ (w) + lλ (d), ◦

so that d is a distinguished right coset representative for W in W (λ). We proceed ◦

by induction on lλ (w). Thus, let w = sγi · · · sγ1 be a reduced expression for w ∈ W , and assume that (9.2) holds. ◦

For γi+1 ∈ Π, suppose sγi+1 sγi · · · sγ1 is a reduced expression of sγi+1 w. Equivalently, w−1 (γi+1 ) > 0. Thus, d−1 w−1 (γi+1 ) > 0 by the ﬁrst paragraph of the proof. By [34, Prop. 2.1.9(ii)], we obtain that lλ (sγi+1 wd) = 1 + lλ (w) + lλ (d), as required. It follows d has the form d w where d is a distinguished double coset repre◦

sentation of W dW0 (λ) and w ∈ W0 (λ). Minimality of lλ (d) among x ∈ W (λ) for which x · λ = μ implies that d = d , as desired. In general, w0 d will not be the element ∈ W (λ) of minimal length with · λ = μ. We write = (μ) for such an element and note d(μ) ≤λ (μ) ≤λ w0 d(μ). The element (μ) was denoted wμ in Remark 5.2(b). In particular condition (2) below is equivalent to μ ≤λ ν by deﬁnition. − . Then the following are Proposition 9.3. Suppose μ, ν ∈ [λ]+ with λ ∈ Crat equivalent: (1) d(μ) ≤λ d(ν); (2) (μ) ≤λ (ν); (3) w0 d(μ) ≤λ w0 d(ν).

Proof. Note w0 d(μ) = (μ)w (μ) for some w (μ) ∈ W (λ). By deﬁnition, (μ) is the element of minimal lλ -length in (μ)W0 (λ). Thus, if (3) holds, then (μ) ≤λ w0 d(μ) ≤λ w0 d(ν) = (ν)w (ν), and it follows that (μ) <λ (ν), which is (2). Assuming (2), d(μ) ≤λ (μ) ≤λ (ν) = ◦

w0 d(ν)w (ν)−1 = w1 d(ν)w2 with w1 ∈ W , w2 ∈ W0 (ν) and lλ (w1 d(ν)w2 ) = lλ (w1 ) + lλ (d(ν)) + lλ (w2 ). Then d(μ) = w1 d(ν) w2 with w1 ≤λ w1 , d(ν) ≤λ d(ν),

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w2 ∈ w, and lλ (w1 ) + lλ (d(ν) ) + lλ (w2 ) = lλ (d(μ)). Minimality of lλ (d(μ)) gives d(μ) = d(ν) ≤λ d(ν), which is (1). Clearly, (1) implies (3), and the proposition is proved. There is a further partial order and equivalence to add to the list. For y, w ∈ W (λ), write y ≤λ w, if w0 yw0 ≤λ w0 ww0 , and put lλ (y) = lλ (w0 yw0 ). In aﬃne geometry, these operations amount to a change in generating fundamental reﬂections from the walls of the standard anti-dominant alcove to the walls of the standard dominant alcove. − Deﬁne f (μ), for μ ∈ [λ]+ and λ ∈ Crat , to be the element f ∈ W (λ) with minimal lλ -length satisfying f · (w0 · λ) = μ. − Proposition 9.4. Suppose μ ∈ [λ]+ , λ ∈ Crat . Then f (μ) = w0 d(μ)w0 . More+ over, if also ν ∈ [λ] , then

f (μ) ≤λ f (ν) ⇐⇒ d(μ) ≤λ d(ν). Proof. By deﬁnition, lλ (w0 f w0 ) = lλ (f ). Thus, for f = f (μ), the element w0 f w0 is the (unique) element f of minimal length lλ (f ) satisfying w0 f · λ = μ. However, w0 d(μ) · λ = μ, so w0 f W0 (λ) = w0 d(μ)W0 (λ). Thus, lλ (d(μ)) ≤ lλ (f ), since d(μ) is distinguished. Therefore, d(μ) = f = w0 f (μ)w0 . So f (μ) = w0 d(μ)w0 . This last assertion is now obvious, and the proof is complete. Finally, we want to compare the order ≤λ with the strong linkage order, in the sense of [27, II, §6.7]. Sections 6.1–6.11 of the latter reference apply for any positive integer p, which we take to be e. For brevity, we refer the reader to those sections for details on the alcove notation used below. “Alcoves” are regarded as certain ◦

open subsets of RP , and the closure of an alcove is then a fundamental domain for ◦

◦

the “dot” action of the aﬃne Weyl group We = W eQ. The “standard alcove” C satisﬁes ◦ 0 < x + ρ, α∨ < e, for all x ∈ C, and this inequality deﬁnes C. For any alcove C1 , there are unique integers nα = nα (C), for each α ∈ Φ+ , deﬁned by ◦

nα e < x + ρ, α∨ < (nα + 1)e and the function d(C1 ) is deﬁned as α∈Φ+ nα . Allowing the right hand inequality “< (nα + 1)e” above to be the weaker “≤ (nα + 1)e, deﬁnes the elements of the

1 of C1 . upper closure C ◦

There is a “dot” action of We on alcoves, agreeing with its “dot” action on P , which is generated by reﬂections in the walls of the closure C of C. (This is not the same notation as in Proposition 7.1, where the walls of the (closure of the) antidominant chamber w0 ·C were used.) For y ∈ We , write le (y) for the length of y with respect to this set of generating reﬂections, and y ≤e w when y, w ∈ We and y ≤ w in the Bruhat-Chevalley order with respect to these generating reﬂections. (Thus, y ≤e w if and only if w0 yw0 ≤e w0 ww0 if and only if φe,λ (w0 yw0 ) ≤λ φe,λ (w0 ww0 ) if and only if φe,λ (y) ≤λ φe,λ (w), using the map φe,λ : We → W (λ) in the context ◦

of Proposition 7.1.) If μ ∈ P + , deﬁne fe (μ) to be the unique element f ∈ We with

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le (f ) minimal satisfying f · x = μ for some x ∈ C, the closure of C. Equivalently, μ ∈ f · C [27, II, 6.11]. From separating hyperplane considerations, le (f ) = d(f · C).

(9.3)

Using this identity, the following lemma is mostly an easy exercise. ◦

◦

◦

Lemma 9.5. Let ξ ∈ P + ∩ Q, i. e., a dominant weight of g lying in the root ◦

◦

lattice, and let z ∈ We correspond to eξ ∈ eQ (i. e., z · x = x + eξ, for all x ∈ RP ). ◦

Then, for any μ ∈ P + ,

⎧ ⎨ fe (z · μ) = zfe (μ) ⎩l (zf (μ)) = e e

le (z) + le (fe (μ)).

1 + eξ = C Proof. Note that C 1 + eξ, just by the deﬁnition of the upper closure. Thus, fe (z · μ) = zfe (μ). The length additivity is an easy calculations with the identity (9.3) and is left to the reader. Now we can prove our main result on strong linkage in the sense of [27, II, 6.1–6.11]. To avoid conﬂict with our notation on h∗ , we use ↑e to denote the ↑ ◦

order in [27, II, Ch. 6]. That is, if μ, ν ∈ P , write μ ↑e ν to mean that μ = ν or ◦

there exists a chain μ = μ0 ≤ μ1 ≤ · · · ≤ μm = ν in P and reﬂections s1 , · · · , sm (not necessarily fundamental) in We such that si · μi = μi+1 , for i = 1, · · · , m. The ◦

order ≤ used is the usual dominance order on P . ◦

Theorem 9.6. Let μ, ν ∈ P + lie in the same orbit of We under the dot action. Then μ ↑e ν ⇐⇒ fe (μ) ≤e fe (ν) where fe (μ) and fe (ν) are the elements of We described above. Proof. First, note the general Coxeter group fact that if u, w, z ∈ We , and, if the lengths of zy and of zw are obtained by adding the length of z to that of y and to that of w, respectively, then y ≤e w ⇐⇒ zy ≤e zw. The implication that y ≤e w =⇒ zy ≤e zw is obvious and the reverse implication reduces immediately to the case le (z) = 1, where it is obvious. (If zy ≤e zw, then y ≤e zy ≤e w, or else, zy = zw where w ≤e w, whence y = w ≤e w.) This fact, together with the preceding lemma, allows us to replace y = f (μ) and ◦

w = f (ν) by zy, zw with z ∈ We corresponding to an element eξ with ξ ∈ Q and ◦

also dominant (i. e., in P + ). At the same time, we can replace μ, ν by μ + eξ, ν + eξ, respectively. Obviously μ ↑e ν ⇐⇒ μ + eξ ↑e ν + eξ. This equivalence holds for ◦

arbitrary weights μ, ν ∈ P and thus may also be applied to intermediate instances of ↑e . So starting from μ ↑e ν and making such an adjustment, we may assume all elements μ0 ≤ μ1 ≤ · · · ≤ μm in the deﬁning chain are dominant, and also le (f (μi )) = le (fe (μi−1 )) + 1, for i = 1, · · · , m. (See the argument in [27, II, 6.10].)

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This implies that μi−1 is obtained from μi by striking out a simple reﬂection, and so, in particular, fe (μi−1 ) <e fe (μi ) for each i = 1, · · · , m, and fe (μ) ≤e f (ν). ◦

Next, suppose that fe (μ) ≤e fe (ν) with μ, ν ∈ P + in the same We -orbit under the dot action. We want to show that μ ↑e ν. By construction, μ, ν belong to fe (μ) · C and fe (ν) · C, respectively. This is equivalent to fe (μ) · C ↑e fe (ν) · C, adapting the notation of [27, II, 6.11(3)]. As in [27], an alcove C1 is called dominant ◦+

if nα (C1 ) ≥ 0 for all α ∈ Φ . Both f (μ) · C and f (ν) · C are dominant. So, it suﬃces to prove that y1 · C ↑e y2 · C, whenever y1 ≤e y2 and y1 · C, y2 · C are both dominant (y1 , y2 ∈ We ). For any y ∈ We , y · C is dominant if and only if y is of minimal le - length in ◦

the right coset W y. We now proceed by induction on the diﬀerence le (y2 ) − le (y1 ). Without loss of generality, we can assume this diﬀerence is positive. By [25, Prop., p. 122], there exists y ∈ We with y1 ≤e y <e y2 and le (y) + 1 = le (y2 ). Thus, y = sy2 , where s is ◦

◦

an (aﬃne) reﬂection in the hyperplane H = Hα,n = {x ∈ RP | x + ρ, α∨ = ne}. The hyperplane H must be one of those separating the dominant alcove y2 · C from C. See [25, Thm., p. 93; Thm. 5.8, Ex. 1, p. 117]. Since y2 · C is dominant, it follows that n > 0 for the parameter n in H = Hα,n . ◦

Now [27, II, 6.8 Prop.] can be applied, to give a (unique) w ∈ W with wy · C dominant and wy · C ↑ y2 · C. ([27] actually proves much more.) Then y1 ≤e ◦

y, and so y1 ≤e wy, since wy is of minimal length in the coset W y. (We use the factorization y = w−1 (wy) with le (y) = le (w−1 ) + le (wy). Thus, a reduced expression of y1 can be obtained from one for y by omitting suitable reﬂections ◦

from w−1 , and from wy. However, y1 is minimal in W y1 , so y1 ≤e wy.) We have le (y1 ) ≤ le (wy) ≤ le (y) < le (y2 ), so y1 · C ↑ wy · C by induction. It follows that y1 · C ↑ y2 · C, and the theorem is proved. We can now give the promised relation of the order ≤λ with the strong linkage ◦

− order ↑e . The order ≤λ is on W (λ), λ ∈ Crat , whereas ↑e is on P . We have already ◦

compared ↑e on P with ≤e on We in Theorem 9.6. It remains now only to compare ≤e on We with ≤λ on W (λ). Proposition 7.1 and its discussion easily give part (i) of the theorem below, and part (ii) then summarizes results already proved in this section. − Theorem 9.7. Let λ ∈ Crat with λ(c) = e/m, where e, m are relatively prime positive integers, and m is divisible by D. Then ◦

(i) f (μ) = φe,λ (fe (μ)) for each μ ∈ [λ]+ ; (ii) if μ, ν ∈ [λ]+ , then the following statements are equivalent: ◦ ◦ (1) μ ↑e ν); (2) (3) (4) (5) (6)

◦

◦

fe (μ) ≤e fe (ν); f (μ) ≤λ f (ν); d(μ) ≤λ d(ν); μ ≤λ ν; μ ↑λ ν.

308 32

BRIAN J. PARSHALL AND LEONARD L. SCOTT

10. Appendix, II: Graded algebras and Morita equivalence We show that if A is a ﬁnite dimensional algebra which is Morita equivalent to another ﬁnite dimensional algebra B, then the graded algebras gr A and gr B obtained from the radical ﬁltrations of A and B are also Morita equivalent. Also, the corresponding categories of ﬁnite dimensional graded modules categories are equivalent. Let A be a ﬁnite dimensional algebra over a ﬁeld. Let P be any (ﬁnite dimensional) projective A-module. Any A-map x : P → P which doesn’t send P to rad P can be multiplied by another A-map P → P to get the identity on an irreducible summand of P/ rad P . So the original map x can’t be in the radical of End P . On the other hand, if the original map x does send P into rad P , so does any element in the ideal N that x generates in End P , So a power of the ideal N is zero. Consequently, x must be in rad End P . We have now characterized the radical rad End P of End P as the space of all maps x : P → P with xP contained in rad P . Next, let’s add the condition that P is a projective generator. Then rad P is clearly the span of images xP of elements x in End P such that xP is contained in rad P . That is, rad P = (rad End P )P . It follows inductively that radr P = (rad End P )r P . (Assume this isomorphism for r − 1 and multiply both sides by rad A.) Thus, when P is viewed as a left A = End P -module, and we make it into a gr A -module—call it gr P —by using the radical series of A , we get a module natural identiﬁcation of gr P with gr P as a vector space. If we let A be the opposite algebra of A , then the vector space gr P provides a gr A, gr A -bimodule through this identiﬁcation. However, Morita theory tells us that, if we similarly regard P as an (A, A )bimodule, it provides a Morita context. In particular, P is a projective generator as right A -module or left A -module. Thus gr P is a projective generator as a left gr A or right gr A -module. We have thus recaptured the Morita context provided by P for A and A by one provided by gr P for gr A and gr A , and so the latter algebras are Morita equivalent. Moreover, since the bimodule providing this equivalence is graded, we obtain an equivalence between the categories of graded gr A-modules and graded gr A -modules. In particular, gr A is Koszul and only if gr A is Koszul. References 1. H. Andersen and M. Kaneda, Semisimple series of modules for the ﬁrst Frobenius kernel in a reductive algebraic group, Proc. London Math. Soc. 59 (1989), 74–89. 2. H. Andersen, p-Filtrations and the Steinberg module, J. Algebra 244 (2001), 664–683. 3. S. Ariki, Graded q-Schur algebras, arXiv:0903.3454 (2009). 4. S. Arkhipov, R. Bezrukavinkov, and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), 595–678. 5. N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4 —6, Springer (2002). 6. J. Brundan, R. Dipper, and A. Kleshchev, Quantum linear groups and representations of GLn (Fq ), Mem. Amer. Math. Soc. 149 (2001), no. 706, viii+112 pp. 7. J. Brundan, A. Kleshchev, and W. Wang, Graded Specht modules, arXiv:0901.0218 (2009). 8. J. Chuang and K. M. Tan, Filtrations of Rouquier blocks of symmetric groups and Schur algebras, Proc. London Math. Soc. 86 (2003), 685–706. 9. E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 85–99. 10. R. Carter, Lie algebras of ﬁnite and aﬃne type, Cambridge Press (2005).

A SEMISIMPLE SERIES FOR q-WEYL AND q-SPECHT MODULES

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11. E. Cline, B. Parshall, and L. Scott, Abstract Kazhdan-Lusztig theories, Tˆ ohoku Math. J. 45 (1993), 511-534. 12. E. Cline, B. Parshall, and L. Scott, Stratifying endomorphism algebras, Memoirs Amer. Math. Soc. 591 (1996), 119+iv pages. 13. E. Cline, B. Parshall, and L. Scott, Graded and non-graded Kazhdan-Lusztig theories, in Algebraic Groups and Lie Groups, ed. by G. I. Lehrer, Camb. Univ. Press (1997), 105–125. 14. E. Cline, B. Parshall, and L. Scott, Generic and q-rational representation theory, Publ. RIMS (Kyoto), 35 (1999), 31–90. 15. E. Cline, B. Parshall, and L. Scott, Reduced standard modules and cohomology, Trans. Amer. Math. Soc. 361 (2009), 5223–5261. 16. B. Deng, J. Du, B. Parshall, and J.-p. Wang, Finite Dimensional Algebras and Quantum Groups, Math. Surveys and Monographs 150, Amer. Math. Soc., 2008. 17. S. Donkin, Lecture at MSRI, November 14, 1990 (a conjecture recorded in [2, p. 664–665]). 18. J. Du, B. Parshall, and L. Scott, Quantum Weyl reciprocity and tilting modules, Comm. Math. Physics 195 (1998), 321–352. 19. J. Du, A noted on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), 363-372. 20. J. A. Green (with K. Erdmann and M. Schocker), Polynomial Representations of GLn : with an Appendix on Schensted Correspondence and Littelmann Paths, Lecture Notes in Math. 830, Springer (2007). 21. M. Fayers, and K. M. Tan, The ordinary quiver of a weight three block of the symmetric groups is bipartite, Adv. in Math. 209 (2007) 69-98. 22. P. Fiebig, Centers and translation functors for the category O over Kac-Moody algebras, Math. Zeit. 243 (2003), 689-717. 23. P. Fiebig, The combinatorics of the category O over symmetrizable Kac-Moody Lie algebras, Trans. Groups 11 (2006), 29–49. 24. T. Hodge, K. Paramasamy, and L. Scott, Remarks on the ABG induction theorem, in preparation. 25. J. Humphreys, Reﬂection groups and Coxeter groups, Cambridge U. Press, 1990. 26. G. James, The decomposition matrices of GLn (q) for n ≤ 10, Proc. London Math. Soc. 60 (1990), 225265. 27. J. C. Jantzen, Representations of algebraic groups, 2nd ed., Math. Surveys and Monographs 107, Amer. Math. Soc., 2003. 28. J. C. Jantzen, Filtrierungen der Darstellungen in der Hauptserie endlicher Chevalley-Gruppen, Proc. London Math. Soc. 49 (1984), 445-482. 29. J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine und Angew. Math. 317 (1980), 157-199. 30. V. Kac, Inﬁnite dimensional Lie algebras, Cambridge, 1990. 31. V. Kac and D. Kazhdan, Structures of representations with highest weight of inﬁnite dimensional Lie algebras, Adv. Math. 34 (1979), 97–108. 32. M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for aﬃne Lie algebras with negative level, Duke Math. J. 77 (1995), 21–62. 33. M. Kashiwara and T. Tanisaki, Characters of irreducible modules with non-critical highest weights over aﬃne Lie algebras, Proc. of the International Conference on Representation Theory, East China Normal University, Shanghai (2000), Springer, 275–296. 34. M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras, III—the positive rational case, Asian J. Math. 2 (1998), 779–832. 35. D. Kazhdan and G. Lusztig, Tensor structures arising from aﬃne Lie algebras, I, II, III, IV, J. Amer. Math. Soc., 6 (1993), 905–947, 949–1011, 7 335–381, 383–453. 36. S. Kumar, Toward a proof of Lusztig’s conjecture concerning negative level representations of aﬃne Lie algebras, J. Algebra 164 (1994), 115–527. 37. Z. Lin, Comparisons of extensions of modules for algebraic groups and quantum groups, Proc. Symp. Pure Math. 63 (1998), 355–367. 38. G. Lusztig, Monodromic systems on aﬃne ﬂag manifolds, Proc. Royal Soc. Series A, 445 (1994), 231–246; Errata, 450 (1995), 731–732. 39. R. Moody and A. Pianzola, Lie algebras with triangular decomposition, Wiley (1995). 40. B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Proc. Ottawa Workshop in Algebra (1987), 1–105.

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41. B. Parshall and L. Scott, Quantum Weyl reciprocity for cohomology, Proc. London Math. Soc. 90 (2005), 655–688. 42. B. Parshall and L. Scott, A new approach to the Koszul duality property in representation theory using graded subalgebras, J. Math. Inst. Jussieu, in press (2011). 43. B. Parshall and L. Scott, Forced grading in integral quasi-hereditary algebras with applications to quantum enveloping algebras, to appear. 44. B. Parshall and L. Scott, On p-ﬁltrations of Weyl modules, in preparation. 45. C. Pillen, Loewy series for principal series representations of ﬁnite Chevalley groups, J. Algebra 189 (1997), 101–124. 46. A. Rocha-Caridi and N. Wallach, Projective modules over graded Lie algebras, I, Math. Z. 180 (1982) 151–177. 47. P. Shan, Graded decomposition matrices of v-Schur algebras via Jantzen ﬁltration, arXiv:1006.1545v1 (2010). 48. P. Shan, M. Varagnolo, and E. Vasseot, Koszul duality of aﬃne Kac-Moody algebras and cyclotomic DAHA, arXiv: 1107.0146 (2011). 49. W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, J. Representation Theory 2 (1998), 432–448. 50. T. Tanisaki, Character formulas of Kazhdan-Lusztig type, in Representations of ﬁnite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun. 40, Amer. Math. Soc., Providence, RI (2004) 261–276. 51. M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebras, Duke J. Math. 100 (1999), 267–297. Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: [email protected]

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