A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

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Author: Bangming Deng | Jie Du | Qiang Fu | 2012

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London Mathematical Society Lecture Note series: 401

A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory BANGMING DENG Beijing Normal University

JIE DU University of New South Wales, Sydney

QIANG FU Tongji University, Shanghai

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107608603 c B. Deng, J. Du and Q. Fu 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed andiboundiin the United Kingdom byithe MPGiBooksiGroup A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data ISBN 978-1-107-60860-3 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

We dedicate the book to our teachers: Peter Gabriel Shaoxue Liu Leonard Scott Jianpan Wang

2010 Mathematics Subject Classification. Primary 17B37, 20G43, 20C08; Secondary 16G20, 20G42, 16T20 Key words and phrases. affine Hecke algebra, affine quantum Schur algebra, cyclic quiver, Drinfeld double, loop algebra, quantum group, Schur–Weyl duality, Ringel–Hall algebra, simple representation

Abstract Over its one-hundred year history, the theory of Schur–Weyl duality and its quantum analogue have had and continue to have profound influences in several areas of mathematics such as Lie theory, representation theory, invariant theory, combinatorial theory, and so on. Recent new developments include, e.g., walled Brauer algebras and rational Schur algebras, quantum Schur superalgebras, and the integral Schur–Weyl duality for types other than A. This book takes an algebraic approach to the affine quantum Schur–Weyl theory. The book begins with a study of extended Ringel–Hall algebras associated with the cyclic quiver of n vertices and the Green–Xiao Hopf structure on their Drinfeld double—the double Ringel–Hall algebra. This algebra is presented in terms of Chevalley type and central generators and is proved to be isomorphic to the quantum loop algebra of the general linear Lie algebra. The rest of the book investigates the affine quantum Schur–Weyl duality on three levels. This includes • the affine quantum Schur–Weyl reciprocity; • the bridging role played by the affine quantum Schur algebra between the quantum loop algebra and the corresponding affine Hecke algebra; • Morita equivalence of certain representation categories; • the presentation of affine quantum Schur algebras; and • the realization conjecture for the double Ringel–Hall algebra which is proved to be true in the classical case. Connections with various existing works by Lusztig, Varagnolo– Vasserot, Schiffmann, Hubery, Chari–Pressley, Frenkel–Mukhin, and others are also discussed throughout the book.

Contents

Introduction

page 1

1

Preliminaries n and some notation 1.1 The loop algebra gl 1.2 Representations of cyclic quivers and Ringel–Hall algebras 1.3 The quantum loop algebra U( sln ) 1.4 Three types of generators and associated monomial bases 1.5 Hopf structure on extended Ringel–Hall algebras

9 9 12 17 20 25

2

Double Ringel–Hall algebras of cyclic quivers 2.1 Drinfeld doubles and the Hopf algebra D(n) 2.2 Schiffmann–Hubery generators 2.3 Presentation of D(n) 2.4 Some integral forms n ) 2.5 The quantum loop algebra U(gl 2.6 Semisimple generators and commutator formulas

31 31 37 41 45 49 55

3

Affine quantum Schur algebras and the Schur–Weyl reciprocity 3.1 Cyclic flags: the geometric definition 3.2 Affine Hecke algebras of type A: the algebraic definition 3.3 The tensor space interpretation 3.4 BLM bases and multiplication formulas 3.5 The D(n)-H(r )-bimodule structure on tensor spaces 3.6 A comparison with the Varagnolo–Vasserot action 3.7 Triangular decompositions of affine quantum Schur algebras 3.8 Affine quantum Schur–Weyl duality, I 3.9 Polynomial identity arising from semisimple generators 3.10 Appendix vii

62 63 68 74 77 79 88 95 102 106 115

viii

4

Contents

Representations of affine quantum Schur algebras 4.1 Affine quantum Schur–Weyl duality, II 4.2 Chari–Pressley category equivalence and classification 4.3 Classification of simple S(n, r )C -modules: the upward approach 4.4 Identification of simple S(n, r )C -modules: the n > r case (n) 4.5 Application: the set Sr 4.6 Classification of simple S(n, r )C -modules: the downward approach 4.7 Classification of simple U(n, r )C -modules

121 122 126

5

The presentation and realization problems 5.1 McGerty’s presentation for U(n, r) 5.2 Structure of affine quantum Schur algebras 5.3 Presentation of S(r, r ) 5.4 The realization conjecture 5.5 Lusztig’s transfer maps on semisimple generators

153 154 157 162 169 172

6

The classical (v = 1) case n ) 6.1 The universal enveloping algebra U(gl 6.2 More multiplication formulas in affine Schur algebras 6.3 Proof of Conjecture 5.4.2 at v = 1 6.4 Appendix: Proof of Proposition 6.2.3

179 179 185 190 194

Bibliography Index

201 205

132 136 141 143 150

Introduction

Quantum Schur–Weyl theory refers to a three-level duality relation. At Level I, it investigates a certain double centralizer property, the quantum Schur– Weyl reciprocity, associated with some bimodules of quantum gln and the Hecke algebra (of type A)—the tensor spaces of the natural representation of quantum gln (see [43], [21], [27]). This is the quantum version of the well-known Schur–Weyl reciprocity which was beautifully used in H. Weyl’s influential book [77]. The key ingredient of the reciprocity is a class of important finite dimensional endomorphism algebras, the quantum Schur algebras or q-Schur algebras, whose classical version was introduced by I. Schur over a hundred years ago (see [69], [70]). At Level II, it establishes a certain Morita equivalence between quantum Schur algebras and Hecke algebras. Thus, quantum Schur algebras are used to bridge representations of quantum gln and Hecke algebras. More precisely, they link polynomial representations of quantum gln with representations of Hecke algebras via the Morita equivalence. The third level of this duality relation is motivated by two simple questions associated with the structure of (associative) algebras. If an algebra is defined by generators and relations, the realization problem is to reconstruct the algebra as a vector space with hopefully explicit multiplication formulas on elements of a basis; while, if an algebra is defined in terms of a vector space such as an endomorphism algebra, it is natural to seek their generators and defining relations. As one of the important problems in quantum group theory, the realization problem is to construct a quantum group in terms of a vector space and certain multiplication rules on basis elements. This problem is crucial to understand their structure and representations (see [47, p. xiii] for a similar problem for Kac–Moody Lie algebras and [60] for a solution in the symmetrizable case). Though the Ringel–Hall algebra realization of the ±-part of quantum enveloping algebras associated with symmetrizable Cartan matrices was an important 1

2

Introduction

breakthrough in the early 1990s, especially for the introduction of the geometric approach to the theory, the same problem for the entire quantum groups is far from completion. However, Beilinson–Lusztig–MacPherson (BLM) [4] solved the problem for quantum gln by exploring further properties coming from the quantum Schur–Weyl reciprocity. On the other hand, as endomorphism algebras and as homomorphic images of quantum gln , it is natural to look for presentations for quantum Schur algebras via the presentation of quantum gln . This problem was first considered in [18] (see also [26]). Thus, as a particular feature in the type A theory, realizing quantum gln and presenting quantum Schur algebras form Level III of this duality relation. For a complete account of the quantum Schur–Weyl theory and further references, see Parts 3 and 5 of [12] (see also [17] for more applications). There are several developments in the establishment of an affine analogue of the quantum Schur–Weyl theory. Soon after BLM’s work, Ginzburg and Vasserot [32, 75] used a geometric and K -theoretic approach to investigate affine quantum Schur algebras1 as homomorphic images of quantum loop n ) of gln in the sense of Drinfeld’s new presentation [20], called algebra U(gl quantum affine gln (at level 0) in this book. This establishes at Level I the first centralizer property for the affine analogue of the quantum Schur–Weyl reciprocity. Six years later, investigations around affine quantum Schur algebras focused on their different definitions and, hence, different applications. For example, Lusztig [56] generalized the fundamental multiplication formulas [4, 3.4] for quantum Schur algebras to the affine case and showed that the “extended” quantum affine sln , U(n), does not map onto affine quantum Schur algebras; Varagnolo–Vasserot [73] investigated Ringel–Hall algebra actions on tensor spaces and described the geometrically defined affine quantum Schur algebras in terms of the endomorphism algebras of tensor spaces. Moreover, they proved that the tensor space definition coincides with Green’s definition [35] via q-permutation modules. Some progress on the second centralizer property has also been made recently by Pouchin [61]. The approaches used in these works are mainly geometric. However, like the non-affine case, there would be more favorable algebraic and combinatorial approaches. At Level II, representations at non-roots-of-unity of quantum affine sln and gln over the complex number field C, including classifications of finite dimensional simple modules, have been thoroughly investigated by Chari– Pressley [6, 7, 8], and Frenkel–Mukhin [28] in terms of Drinfeld polynomials. Moreover, an equivalence between the module category of the Hecke algebra 1 Perhaps they should be called quantum affine Schur algebras. Since our purpose is to establish

an affine analogue of the quantum Schur–Weyl theory, this terminology seems more appropriate to reflect this.

Introduction

3

H(r )C and a certain full subcategory of quantum affine sln (resp., gln ) has also been established algebraically by Chari–Pressley [9] (resp., geometrically by Ginzburg–Reshetikhin–Vasserot [31]) under the condition n > r (resp., n r ). Note that the approach in [31] uses intersection cohomology complexes. It would be interesting to know how affine quantum Schur algebras would play a role in these works. Much less progress has been made at Level III. When n > r , Doty–Green [18] and McGerty [58] have found a presentation for affine quantum Schur algebras, while the last two authors of this book have investigated the realization problem in [24], where they first developed an approach without using the stabilization property, a key property used in the BLM approach, and presented an ideal candidate for the realization of quantum affine gln . This book attempts to establish the affine quantum Schur–Weyl theory as a whole and is an outcome of algebraically understanding the works mentioned above. First, building on Schiffmann [67] and Hubery [40], our starting point is to present the double Ringel–Hall algebra D(n) of the cyclic quiver with n vertices in terms of Chevalley type generators together with infinitely many central generators. Thus, we obtain a central subalgebra Z(n) such that D(n) = U(n)Z(n) ∼ = U(n) ⊗ Z(n). We then establish an isomorphism between D(n) and Drinfeld’s quantum affine gln in the sense of [20]. In this way, we easily obtain an action on the tensor space which upon restriction coincides with the Ringel–Hall algebra action defined geometrically by Varagnolo–Vasserot [73] and commutes with the affine Hecke algebra action. Second, by a thorough investigation of a BLM type basis for affine quantum Schur algebras, we introduce certain triangular relations for the corresponding structure constants and, hence, a triangular decomposition for affine quantum Schur algebras. With this decomposition, we establish explicit algebra epimorphisms ξr = ξr,Q(v) from the double Ringel–Hall algebra D(n) to affine quantum Schur algebras S(n, r ) := S(n, r )Q(v) for all r 0. This algebraic construction has several nice applications, especially at Levels II and III. For example, the homomorphic image of commutator formulas for semisimple generators gives rise to a beautiful polynomial identity whose combinatorial proof remains mysterious. Like the quantum Schur algebra case, we will establish for n r a Morita equivalence between affine quantum Schur algebras S(n, r)F and affine Hecke algebras H(r )F of type A over a field F with a non-root-of-unity parameter. As a by-product, we prove that every simple S(n, r )F -module is finite dimensional. Thus, applying the classification of simple H(r )C modules by Zelevinsky [81] and Rogawski [66] yields a classification of simple

4

Introduction

S(n, r )C -modules. Hence, inflation via the epimorphisms ξr, C gives many n )-modules. We will also use ξ together finite dimensional simple UC (gl r,C with the action on tensor spaces and a result of Chari–Pressley to prove that n ) are all inflafinite dimensional simple polynomial representations of UC (gl tions of simple S(n, r )C -modules. In this way, we can see the bridging role played by affine quantum Schur algebras between representations of quantum affine gln and those of affine Hecke algebras. Moreover, we obtain a classification of simple S(n, r )C -modules in terms of Drinfeld polynomials and, when n > r , we identify them with those arising from simple H(r )C -modules. Our findings also show that, if we regard the category S(n, r )C -Mod of n )-modules, this category is S(n, r )C -modules as a full subcategory of UC (gl hi quite different from the category C ∩ C considered in [54, §6.2]. For example, the latter is completely reducible and simple objects are usually infinite dimensional, while S(n, r )C -Mod is not completely reducible and all simple objects are finite dimensional. As observed in [23, Rem. 9.4(2)] for quantum gl∞ and infinite quantum Schur algebras, this is another kind of phenomenon of infinite type in contrast to the finite type case. The discussion of the realization and presentation problems is also based on the algebra epimorphisms ξr and relies on the use of semisimple generators and indecomposable generators for D(n) which are crucial to understand the integral structure and multiplication formulas. We first use the new presentation for D(n) to give a decomposition for S(n, r ) = U(n, r )Z(n, r ) into a product of two subalgebras, where Z(n, r) is a central subalgebra and U(n, r ) is the homomorphic image of U(n), the extended quantum affine sln . By taking a close look at this structure, we manage to get a presentation for S(r, r ) for all r 1 and acknowledge that the presentation problem is very complicated in the n < r case. On the other hand, we formulate a realization conjecture suggested by the work [24] and prove the conjecture in the classical (v = 1) case. We remark that, unlike the geometric approach in which the ground ring must be a field or mostly the complex number field C, the algebraic, or rather, the representation-theoretic approach we use in this book works largely over a ring or mostly the integral Laurent polynomial ring Z[v, v −1 ]. We have organized the book as follows. In the first preliminary chapter, we introduce in §1.4 three different types of generators and their associated monomial bases for the Ringel–Hall algebras of cyclic quivers, and display in §1.5 the Green–Xiao Hopf structure on the extended version of these algebras. Chapter 2 introduces a new presentation using Chevalley generators n ) of gln . This is achieved by for Drinfeld’s quantum loop algebra U(gl

Introduction

5

constructing the presentation for the double Ringel–Hall algebra D(n) associated with cyclic quivers (Theorem 2.3.1), based on the work of Schiffmann and Hubery, and by lifting Beck’s algebra monomorphism from the quantum n ) to obtain an isomorphism sln with a Drinfeld–Jimbo presentation into U(gl between D(n) and U(gln ) (Theorem 2.5.3). Chapter 3 investigates the structure of affine quantum Schur algebras. We first recall the geometric definition by Ginzburg–Vasserot and Lusztig, the Hecke algebra definition by R. Green, and the tensor space definition by Varagnolo–Vasserot. Using the Chevalley generators of D(n), we easily obtain an action on the Q(v)-space with a basis indexed by Z and, hence, an action of D(n) on ⊗r (§3.5). We prove that this action commutes with the affine Hecke algebra action defined in [73]. Moreover, we show that the restriction of the action to the negative part of D(n) (i.e., to the corresponding Ringel–Hall algebra) coincides with the Ringel–Hall algebra action geometrically defined by Varagnolo–Vasserot (Theorem 3.6.3). As an application of this coincidence, the commutator formula associated with semisimple generators, arising from the skew-Hopf pairing, gives rise to a certain polynomial identity associated with a pair of elements λ, μ ∈ Nn (Corollary 3.9.6). The main result of the chapter is an elementary proof of the surjective homomorphism ξr from the double Ringel–Hall algebra D(n), i.e., the quantum loop n ), onto the affine quantum Schur algebra S(n, r ) (Theorem algebra U(gl 3.8.1). The approach we used is the establishment of a triangular decomposition of S(n, r ) (Theorem 3.7.7) through an analysis of the BLM type bases. In Chapter 4, we discuss the representation theory of affine quantum Schur algebras over C and its connection to polynomial representations of quantum affine gln and representations of affine Hecke algebras. We first establish a category equivalence between the module categories S(n, r )C -Mod and H(r )C -Mod for n r (Theorem 4.1.3). As an application, we will reinterpret Chari–Pressley’s category equivalence ([9, Th. 4.2]) between (level r ) representations of UC ( sln ) and those of affine Hecke algebras H(r )C , where n > r , in terms of representations of S(n, r )C (Proposition 4.2.1). We then develop two approaches to the classification of simple S(n, r )C -modules. In the socalled upward approach, we use the classification of simple H(r )C -modules of Zelevinsky and Rogawski to classify simple S(n, r )C -modules (Theorems 4.3.4 and 4.5.3), while in the downward approach, we determine the classification of simple S(n, r )C -modules (Theorem 4.6.8) in terms of simple polyn ). When n > r , we prove an identification nomial representations of UC (gl theorem (Theorem 4.4.2) for the two classifications. Finally, in §4.7, a classification of finite dimensional simple U(n, r )C -modules is also completed

6

Introduction

and its connections to finite dimensional simple UC ( sln )-modules and finite n )-modules are also discussed. dimensional simple (polynomial) UC (gl We move on to look at the presentation and realization problems in Chapter 5. We first observe S(n, r ) = U(n, r )Z(n, r), where U(n, r) and Z(n, r ) are homomorphic images of U(n) and Z(n), respectively, and that Z(n, r ) ⊆ U(n, r ) if and only if n > r . A presentation for U(n, r ) is given in [58] (see also [19] for the n > r case). Building on McGerty’s presentation, we first give a Drinfeld–Jimbo type presentation for the subalgebra U(n, r ) (Theorem 5.1.3). We then describe a presentation for the central subalgebra Z(n, r ) as a Laurent polynomial ring in one indeterminate over a polynomial ring in r − 1 indeterminates over Q(v). We manage to describe a presentation for S(r, r ) for all r 1 (Theorem 5.3.5) by adding an extra generator (and its inverse) together with an additional set of relations on top of the relations given in Theorem 5.1.3. What we will see from this case is that the presentation for S(n, r ) with r > n can be very complicated. We discuss the realization problem from §5.4 onwards. We first describe the modified BLM approach developed in [24]. With some supporting evidence, we then formulate the realization conjecture (Conjecture 5.4.2) as suggested in [24, 5.5(2)], and state its classical (v = 1) version. We end the chapter with a closer look at Lusztig’s transfer maps [57] by displaying some explicit formulas for their action on the semisimple generators for S(n, r ) (Corollary 5.5.2). These formulas also show that the homomorphism from U( sln ) to limS(n, n + m) induced by the transfer maps cannot either be extended to the ←−

double Ringel–Hall algebra D(n). (Lusztig already pointed out that it cannot be extended to U(n).) This somewhat justifies why a direct product is used in the realization conjecture. In the final Chapter 6, we prove the realization conjecture for the classical (v = 1) case. The key step in the proof is the establishment of more multiplication formulas (Proposition 6.2.3) between homogeneous indecomposable generators and an arbitrary BLM type basis element. As a by-product, we display a basis for the universal enveloping algebra of the loop algebra of gln (Theorem 6.3.4) together with explicit multiplication formulas between generators and arbitrary basis elements (Corollary 6.3.5). There are two appendices in §§3.10 and 6.4 which collect a number of lengthy calculations used in some proofs. Conjectures and problems. There are quite a few conjectures and problems throughout the book. The conjectures are mostly natural generalizations to the affine case, for example, the realization conjecture 5.4.2 and the conjectures in §3.8 on an integral form for double Ringel–Hall algebras and

Introduction

7

the second centralizer property in the affine quantum Schur–Weyl reciprocity. Some problems are designed to seek further solutions to certain questions such as “quantum Serre relations” for semisimple generators (Problem 2.6.4), the Affine Branching Rule (Problem 4.3.6), and further identification of simple modules from different classifications (Problem 4.6.11). There are also problems for seeking different proofs. Problems 3.4.3 and 6.4.2 form a key step towards the proof of the realization conjecture. Notational scheme. For most of the notation used throughout the book, if it involves a subscript or a superscript , it indicates that the same notation withouthas been used in the non-affine case, say, in [4], [12], [33], etc. Here the triangledepicts the cyclic Dynkin diagram of affine type A. For a ground ring Z and a Z-module (or a Z-algebra) A, we often use the notation AF := A ⊗ F to represent the object obtained by base change to a field F, which itself is a Z-module. In particular, if Z = Z[v, v −1 ], then we write A for AQ(v) . Acknowledgements. The main results of the book have been presented by the authors at the following conferences and workshops. We would like to thank the organizers for the opportunities of presenting our work. • Conference on Perspectives in Representation Theory, Cologne, September 2009; • International Workshop on Combinatorial and Geometric Approach to Representation Theory, Seoul National University, September 2009; • 2010 ICM Satellite Conference, Bangalore, August 2010; • 12th National Algebra Conference, Lanzhou, June 2010; • Southeastern Lie Theory Workshop: Finite and Algebraic Groups and Leonard Scott Day, Charlottesville, June 2011; • 55th Annual Meeting of the Australian Mathematical Society, Wollongong, September 2011. The research was partially supported by the Australian Research Council, the Natural Science Foundation of China, the 111 Program of China, the Program NCET, the Fok Ying Tung Education Foundation, the Fundamental Research Funds for the Central Universities of China, and the UNSW Goldstar Award. The first three and last two chapters were written while Deng and Fu were visiting the University of New South Wales at various times. The hospitality and support of UNSW are gratefully acknowledged. The second author would like to thank Alexander Kleshchev and Arun Ram for helpful comments on the Affine Branching Rule (4.3.5.1), and Vyjayanthi

8

Introduction

Chari for several discussions and explanations on the paper [9] and some related topics. He would also like to thank East China Normal University, and the Universities of Mainz, Virginia, and Auckland for their hospitality during his sabbatical leave in the second half of 2009. Finally, for all their help, encouragement, and infinite patience, we thank our wives and children: Wenlian Guo and Zhuoran Deng; Chunli Yu, Andy Du, and Jason Du; Shanshan Xia.

Bangming Deng Jie Du Qiang Fu Sydney 5 December 2011

1 Preliminaries

We start with the loop algebra of gln (C) and its interpretation in terms of matrix Lie algebras. We use the subalgebra of integer matrices of the latter to introduce several important index sets which will be used throughout the book. Ringel–Hall algebras H(n) associated with cyclic quivers (n) and their geometric construction are introduced in §1.2. In §1.3, we discuss the composition subalgebra C(n) of H(n) and relate it to the quantum loop algebra U( sln ). We then describe in §1.4 three types of generators for H(n), which consist of all simple modules together with, respectively, the Schiffmann–Hubery central elements, homogeneous semisimple modules, and homogeneous indecomposable modules, and their associated monomial bases (Corollaries 1.4.2 and 1.4.6). These generating sets will play different roles in what follows. Finally, extended Ringel–Hall algebras and their Hopf structure are discussed in §1.5.

n and some notation 1.1. The loop algebra gl For a positive integer n, let gln (C) be the complex general linear Lie algebra, and let n (C) := gln (C) ⊗ C[t, t −1 ] gl n (C) is spanned by E i, j ⊗ t m be the loop algebra of gln (C); see [47]. Thus, gl for all 1 i, j n, and m ∈ Z, where E i, j is the matrix (δk,i δ j,l )1k,l n . The (Lie) multiplication is the bracket product associated with the multiplication

(E i, j ⊗ t m )(E k,l ⊗ t m ) = δ j,k E i,l ⊗ t m+m . n (C) as a matrix Lie algebra. Let We may interpret the Lie algebra gl M, n (C) be the set of all Z×Z complex matrices A = (ai, j )i, j∈Z with ai, j ∈ C 9

10

1. Preliminaries

such that (a) ai, j = ai+n, j+n for i, j ∈ Z, and (b) for every i ∈ Z, the set { j ∈ Z | ai, j = 0} is finite. Clearly, conditions (a) and (b) imply that there are only finitely many non-zero entries in each column of A. For A, B ∈ M, n (C), let [ A, B] = AB − B A. Then (M, n (C), [ , ]) becomes a Lie algebra over C. Denote by Mn,• (C) the set of n × Z matrices A = (ai, j ) over C satisfying (b) with i ∈ [1, n] := {1, 2, . . . , n}. Then there is a bijection 1 : M, n (C) −→ Mn,• (C),

(ai, j )i, j∈Z −→ (ai, j )1i n, j∈Z .

(1.1.0.1)

For i, j ∈ Z, let Ei,j ∈ M, n (C) be the matrix (ek,l )k,l∈Z defined by 1, if k = i + sn, l = j + sn for some s ∈ Z; i, j ek,l = 0, otherwise. i, j

The set {E i,j |1 i n, j ∈ Z} is a C-basis of M, n (C). Since Ei,j+ln E p,q+kn = δ j, p E i,q+(l+k)n ,

for all i, j, p, q, l, k ∈ Z with 1 j, p n, it follows that the map l n (C), E M, n (C) −→ gl i, j+ln −→ E i, j ⊗ t , 1 i, j n, l ∈ Z

n (C) with is a Lie algebra isomorphism. We will identify the loop algebra gl M, n (C) in the sequel. n (Q) = M,n (Q) n := gl In Chapter 6, we will consider the loop algebra gl defined over Q and its universal enveloping algebra U(gln ) and triangular parts n )− , and U(gl n )0 . Here U(gl n )+ (resp., U(gl n )− , U(gl n )0 ) is the n )+ , U(gl U(gl n ) generated by E for all i < j (resp., E for all i > j, subalgebra of U(gl i, j i, j E i,i for all i). We will also relate these algebras in §6.1 with the specializations at v = 1 of the Ringel–Hall algebra H(n) and the double Ringel–Hall algebra D(n). We now introduce some notation which will be used throughout the book. Consider the subset M,n (Z) of M, n (C) consisting of matrices with integer entries. For each A ∈ M,n (Z), let ai, j i∈Z and co( A) = ai, j j∈Z . ro( A) = j∈Z

i∈Z

We obtain functions ro, co : M,n (Z) −→ Zn ,

n and some notation 1.1. The loop algebra gl

11

where Zn := {(λi )i∈Z | λi ∈ Z, λi = λi−n for i ∈ Z}. For λ = (λi )i∈Z ∈ Zn , A ∈ M,n (Z), and i 0 ∈ Z, let σ (λ) = λi and σ (A) = ai, j = i 0 +1i i 0 +n

i 0 +1ii 0 +n j∈Z

ai, j .

i 0 +1 ji 0 +n i∈Z

Clearly, both σ (λ) and σ ( A) are defined and independent of i 0 . We sometimes identify Zn with Zn via the following bijection 2 : Zn −→ Zn ,

λ −→ 2 (λ) = (λ1 , . . . , λn ).

(1.1.0.2)

For example, we define a “dot product” on Zn by λ μ := 2 (λ) 2 (μ) = n n i=1 λi μi , and define the order relation on Z by setting λ μ ⇐⇒ 2 (λ) 2 (μ) ⇐⇒ λi μi for all 1 i n.

(1.1.0.3)

Also, let ei ∈ Zn be defined by 2 (ei ) = ei = (0, . . . , 0, 1 , 0, . . . , 0). (i)

Let (n) :={A = (ai, j ) ∈ M,n (Z) | ai, j ∈ N} = M,n (N), Nn :={(λi )i∈Z ∈ Zn | λi 0}, and, for r 0, let (n, r ) :={A ∈ (n) | σ ( A) = r } and (n, r ) :={λ ∈ Nn | σ (λ) = r }. The set Mn (Z) can be naturally regarded as a subset of Mn,• (Z) by sending (ai, j )1i, j n to (ai, j )1i n, j∈Z , where ai, j = 0 if j ∈ Z\[1, n]. Thus, (the inverse of) 1 induces an embedding 1 : Mn (Z) −→ M,n (Z).

(1.1.0.4)

By removing the subscripts, we define similarly the subsets (n), (n, r ) of Mn (Z) and subset (n, r ) of Nn , etc. Note that 2 ((n, r )) = (n, r ). Let Z = Z[v, v −1 ], where v is an indeterminate, and let Q(v) be the fraction field of Z. For integers N , t with t 0, let

v N −i+1 − v −(N −i+1) N N ∈ Z and = 1. (1.1.0.5) = 0 v i − v −i t 1i t

If we put [m] = [N ]!

[t]! [N −t]!

v m −v −m v−v −1

=

m 1

and [N ]! := [1][2] · · · [N ], then

N t

=

for all 1 t N . Given a polynomial f ∈ Z and z ∈ C∗ := C\{0},

we sometimes write f z for f (z), e.g., [m]z , [m]!z , etc.

12

1. Preliminaries

When counting occurs, we often use N N := v t (N −t) t t to denote the Gaussian polynomials in v 2 . Also, for any Q(v)-algebra A and an invertible element X ∈ A , let t X v a−s+1 − X −1 v −a+s−1 X; a X; a = = 1, (1.1.0.6) and t 0 v s − v −s s=1

for all a, t ∈ Z with t 1.

1.2. Representations of cyclic quivers and Ringel–Hall algebras Let (n) (n 2) be 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. By Rep0(n) = Rep0F (n) we denote the category of ﬁnite dimensional nilpotent representations of (n) over F, i.e., representations V = (Vi , f i )i∈I of (n) such that all Vi are ﬁnite dimensional and the composition f n · · · f 2 f 1 : V1 → V1 is nilpotent. The vector dim V = (dimF Vi ) ∈ NI = Nn is called the dimension vector of V . (We shall sometimes identify NI with Nn under (1.1.0.2).) For each vertex i ∈ I , there is a one-dimensional representation Si in Rep0(n) satisfying (Si )i = F and (Si ) j = 0 for j = i. It is known that the Si form a complete set of simple objects in Rep0(n). Hence, each semisimple representation Sa in Rep0(n) is given by Sa = ⊕i∈I ai Si , where a = (a1 , . . . , an ) ∈ NI . A semisimple representation Sa is called sincere if a is sincere, namely, all ai are positive. In particular, the vector δ := (1, . . . , 1) ∈ Nn will often be used. Moreover, up to isomorphism, all indecomposable representations in Rep0(n) are given by Si [l] (i ∈ I and l 1) of length l with top Si . Thus, the isoclasses of representations in Rep0(n) are indexed by multisegments

1.2. Representations of cyclic quivers and Ringel–Hall algebras

13

π = i∈I, l 1 πi,l [i; l), where the representation M(π ) corresponding to π is defined by M(π ) = MF (π ) = πi,l Si [l]. 1i n,l 1

Since the set of all multisegments can be identified with the set + (n) = {A = (ai, j ) ∈ (n) | ai, j = 0 for i j } of all strictly upper triangular matrices via 3 : + (n) −→ ,

A = (ai, j )i, j∈Z −→

ai, j [i; j − i),

i< j,1i n

we will use + (n) to index the finite dimensional nilpotent representations. In particular, for any i, j ∈ Z with i < j, we have M i, j := M(E i,j ) = Si [ j − i ], and M i+n, j +n = M i, j . Thus, for any A = (ai, j ) ∈ + (n) and i 0 ∈ Z, M(A) = MF (A) = ai, j M i, j = 1i n,i < j

For A = (ai, j ) ∈ + (n), set

ai, j M i, j .

i 0 +1i i 0 +n,i < j

d(A) =

ai, j ( j − i ).

i< j,1i n

Then dimF M(A) = d(A). Moreover, for each λ = (λi ) ∈ NI , set Aλ = (ai, j ) n with ai, j = δ j,i +1 λi , i.e., Aλ = i=1 λi E i,i +1 . Then M(Aλ ) = λi Si =: Sλ (1.2.0.1) 1i n +

is semisimple. Also, for A ∈ (n), we write d(A) = dim M(A) ∈ ZI , the dimension vector of M(A). Hence, ZI is identified with the Grothendieck group of Rep0(n). A matrix A = (ai, j ) ∈ + (n) is called aperiodic if, for each l 1, there exists i ∈ Z such that ai,i+l = 0. Otherwise, A is called periodic. A nilpotent representation M(A) is called aperiodic (resp., periodic) if A is aperiodic (resp., periodic). It is well known that there exist Auslander–Reiten sequences in Rep0(n); see [1]. More precisely, for each i ∈ I and each l 1, there is an Auslander– Reiten sequence 0 −→ Si+1 [l] −→ Si [l + 1] ⊕ Si +1 [l − 1] −→ Si [l] −→ 0,

14

1. Preliminaries

where we set Si+1 [0] = 0 by convention. Si+1 [l] is called the Auslander– Reiten translate of Si [l], denoted by τ Si [l]. In this case, τ indeed defines an equivalence from Rep0(n) to itself, called the Auslander–Reiten translation. For each A = (ai, j ) ∈ + (n), we define τ (A) ∈ + (n) by M(τ (A)) = τ M(A). Thus, if we write τ (A) = (bi, j ) ∈ + (n), then bi, j = ai −1, j −1 for all i, j. We now introduce the degeneration order on + (n) and generic extensions of nilpotent representations. These notions play an important role in the study of bases for both the Ringel–Hall algebra H(n) of (n) and its composition subalgebra C(n); see, for example, [11, 13]. For two nilpotent representations M, N in Rep0(n) with dim M = dim N , define N dg M ⇐⇒ dimF Hom(X, N ) dimF Hom(X, M), for all X ∈ Rep0(n); (1.2.0.2) see [82]. This gives rise to a partial order on the set of isoclasses of representations in Rep0(n), called the degeneration order. Thus, it also induces a partial order on + (n) by letting A dg B ⇐⇒ M(A) dg M(B). By [62] and [11, §3], for any two nilpotent representations M and N , there exists a unique extension G (up to isomorphism) of M by N with minimal dim End(G). This representation G is called the generic extension of M by N and will be denoted by M ∗ N in the sequel. Moreover, for nilpotent representations M1 , M2 , M3 , (M1 ∗ M2 ) ∗ M3 ∼ = M1 ∗ (M2 ∗ M3 ). Also, taking generic extensions preserves the degeneration order. More precisely, if N1 dg M1 and N2 dg M2 , then N1 ∗ N2 dg M1 ∗ M2 . For A, B ∈ + (n), let A ∗ B ∈ + (n) be defined by M(A ∗ B) ∼ = M(A) ∗ M(B). As above, let Z = Z[v, v −1 ] be the Laurent polynomial ring in indeterminate v. By [65] and [36], for A, B1 , . . . , Bm ∈ + (n), there is a polynomial ϕ BA1 ,...,Bm ∈ Z[v 2 ] in v 2 , called the Hall polynomial, such that for any finite field F of q elements, ϕ BA1 ,...,Bm |v 2 =q (the evaluation of ϕ BA1 ,...,Bm at v 2 = q) M ( A)

equals the number FMFF(B1 ),...,MF (Bm ) of the filtrations 0 = Mm ⊆ Mm−1 ⊆ · · · ⊆ M1 ⊆ M0 = MF (A) such that Mt−1 /Mt ∼ = MF (Bt ) for all 1 t m. Moreover, for each A = (ai, j ) ∈ + (n), there is a polynomial a A = a A (v 2 ) ∈ Z in v 2 such that, for each finite field F with q elements, a A |v 2 =q = | Aut(MF (A))|; see, for example, [59, Cor. 2.1.1]. For later use, we give an

1.2. Representations of cyclic quivers and Ringel–Hall algebras

15

explicit formula for a A . Let m A denote the dimension of rad End(MF (A)), which is known to be independent of the field F. We also have

End(MF (A))/rad End(MF (A)) ∼ Mai, j (F), = 1i n, ai, j >0

where Mai, j (F) denotes the full matrix algebra of ai, j × ai, j matrices over F. Hence,

| Aut(MF (A))| = |F|m A |G L ai, j (F)|. 1i n, ai, j >0

Consequently, a A = v 2m A

(v 2ai, j −1)(v 2ai, j −v 2 ) · · · (v 2ai, j −v 2ai, j −2 ). (1.2.0.3)

1i n, ai, j >0

In particular, if z ∈ C∗ is not a root of unity, then a A |v 2 =z = 0. Let H(n) be the (generic) Ringel–Hall algebra of the cyclic quiver (n), which is by definition the free Z-module with basis {u A = u [M(A)] | A ∈ + (n)}. The multiplication is given by u A u B = v d( A),d(B) ϕ CA,B u C C∈+ (n)

for A, B ∈ + (n), where d(A), d(B) = dim Hom(M(A), M(B)) − dim Ext1 (M( A), M(B)) (1.2.0.4) is the Euler form associated with the cyclic quiver (n). If we write d(A) = (ai ) and d(B) = (bi ), then d( A), d(B) = ai bi − ai bi+1 . (1.2.0.5) i∈I

i∈I

Since both dimF End(MF (A)) and dimF MF (A) = d(A) are independent of the ground field, we put for each A ∈ + (n),

d A = dimF End(MF (A)) − dimF MF (A) and u A = vd A u A ;

(1.2.0.6)

cf. [13, (8.1)].1 As seen in [13], it is sometimes convenient to work with the PBW type basis { u A | A ∈ + (n)} of H(n). The degeneration order gives rise to the following “triangular relation” in H(n): for A1 , . . . , At ∈ + (n), u A1 · · · u At = v 1r<st d( Ar ),d( As ) ϕ AB1 ,..., At u B . (1.2.0.7) B dg A1 ∗···∗ At

1 There is another notation, called d , which will be defined in (3.1.3.2) for all A ∈ (n). A

16

1. Preliminaries

There is a natural NI -grading on H(n): H(n) = H(n)d ,

(1.2.0.8)

d∈N I

where H(n)d is spanned by all u A with d( A) = d. Moreover, we will frequently consider in the sequel the algebra H(n) = H(n) ⊗Z Q(v) obtained by base change to the fraction field Q(v). In order to relate Ringel–Hall algebras H(n) of cyclic quivers with affine quantum Schur algebras later on, we recall Lusztig’s geometric construction of H(n) (specializing v to a square root of a prime power) developed in [53, 54] (cf. also [55]). Let F = Fq be the finite field of q elements and d ∈ NI . Fix an I -graded F-vector space V = ⊕i ∈I Vi of dimension vector d, i.e., dimF Vi = di for all i ∈ I . Then each element in E V = ( fi ) ∈ HomF (Vi , Vi+1 ) | f n · · · f 1 is nilpotent i∈I

can be viewed as a nilpotent representation of (n) over F of dimension vector d. The group G V = i∈I G L(Vi ) acts on E V by conjugation. Then there is a bijection between the G V -orbits in E V and the isoclasses of nilpotent representations of (n) of dimension vector d. For each A ∈ + (n) with d(A) = d, we will denote by O A the orbit in E V corresponding to the isoclass of M(A). Define Hd = CG V (E V ) to be the vector space of G V -invariant functions from E V to C. Now let a, b ∈ Nn with d = a + b and fix I -graded F-vector spaces U = ⊕i ∈I Ui and W = ⊕i ∈I Wi with dimension vectors a and b, respectively. Let E be the set of triples (x, φ, ψ) such that (1) x ∈ E V , (2) the sequence φ

ψ

0 −→ W −→ V −→ U −→ 0 of I -graded spaces is exact, and (3) φ(W ) is stable by x. Let F be the set of pairs (x, W ), where x ∈ E V and W ⊆ V is an x-stable I -graded subspace of dimension vector b. Consider the diagram p1

p2

p3

EU × E W ←− E −→ F −→ E V , where p1 , p2 , p3 are projections defined in an obvious way. Given f ∈ CG U (EU ) = Ha and g ∈ CG W (E W ) = Hb , define the convolution product of f and g by 1

f g = q − 2 m(a,b) ( p3 )! h ∈ CG V (E V ) = Hd ,

1.3. The quantum loop algebra U( sln )

17

where h ∈ C(F) is the function such that p2∗ h = p1∗ ( f g) and m(a, b) = ai bi + ai bi+1 . 1i n

1i n

2 Consequently, H = d∈Nn Hd becomes an associative algebra over C. For each A ∈ + (n), let χO A be the characteristic function of O A and put

O A = q − 2 dim OA χO A . 1

By [13, 8.1]), there is an algebra isomorphism H(n) ⊗Z C −→ H, u A −→ v d( A)−d( A)d( A) χO A , for all A ∈ + (n), (1.2.0.9) 1 where C is viewed as a Z-module by specializing v to q 2 . In particular, this isomorphism takes u A to O A .

1.3. The quantum loop algebra U( sln ) As mentioned in the introduction, an important breakthrough for the structure of quantum groups associated with semisimple complex Lie algebras is Ringel’s Hall algebra realization of the ±-part of the quantum enveloping algebra associated with the same quiver; see [63, 64]. For the Ringel–Hall algebra H(n) associated with a cyclic quiver, it is known from [65] that a subalgebra, the composition algebra, is isomorphic to the ±-part of a quantum affine sln . We now describe this algebra and use it to display a certain monomial basis. The Z-subalgebra C(n) of H(n) generated by u [m Si ] (i ∈ I and m 1) is called the composition algebra of (n). By (1.2.0.8), C(n) inherits an NI -grading by dimension vectors: C(n) = C(n)d , d∈N I

where C(n)d = C(n) ∩ H(n)d . Let u i := u E = u [Si ] and define the i,i+1 divided power 1 m u i(m) = u ∈ C(n), [m]! i for i ∈ I and m 1. In fact, (m)

ui

= v m(m−1) u [m Si ] ∈ C(n) ⊂ H(n).

2 The algebra H defined also geometrically in [73, 3.2] has a multiplication opposite to the one

for H here.

18

1. Preliminaries

We now use the strong monomial basis property developed in [11, 13] to construct an explicit monomial basis of C(n). For each A = (ai, j ) ∈ + (n), define = A = max{ j − i | ai, j = 0}. In other words, is the Loewy length of the representation M(A). Suppose now A is aperiodic. Then there is i 1 ∈ [1, n] such that ai1 ,i1 + = 0, but ai1 +1,i1 +1+ = 0. If there are some ai1 +1, j = 0, we let p 1 satisfy ai1 +1,i1 +1+ p = 0 and ai1 +1, j = 0 for all j > i 1 + 1 + p; if ai1 +1, j = 0 for all j > i 1 + 1, let p = 0. Thus, > p. Now set t1 = ai1 ,i1 +1+ p + · · · + ai1 ,i1 + and define A1 = (bi, j ) ∈ + (n) by letting ⎧ ⎪ if i = i 1 , j i 1 + 1 + p; ⎪ ⎨0, bi, j = ai1 +1, j + ai1 , j , if i = i 1 + 1 < j, j i 1 + 1 + p; ⎪ ⎪ ⎩a , otherwise. i, j

Then, A1 is again aperiodic. Applying the above process to A1 , we get i 2 and t2 . Repeating the above process (ending with the zero matrix), we finally get two sequences i 1 , . . . , i m and t1 , . . . , tm . This gives a word w A = i 1t1 i 2t2 · · · i mtm , where i 1 , . . . , i m are viewed as elements in I = Z/nZ, and define the monomial (t ) (t )

(t )

u ( A) = u i11 u i22 · · · u imm ∈ C(n). The algorithm above can be easily modified to get a similar algorithm for quantum gln . We illustrate the algorithm with an example in this case. Example 1.3.1. If A =

1234 5 0 0 , then = 4, i = 1, 1 60 7

p = 1 and t1 = 2+3+4 = 9.

Here we ignore all zero entries on and below the diagonal for simplicity. Thus, A1 = A2 = A3 =

1000 7 3 4, 60 7 1000 7 0 0, 94 7 100 0 70 0 , 9 0 11

= 3, i 2 = 2, p = 1, and t2 = 3 + 4 = 7, = 2, i 3 = 3, p = 1, and t3 = 4, and = 1, i 4 = 4, p = 0, and t4 = 11.

1.3. The quantum loop algebra U( sln )

19

Now, for a matrix defining a semisimple representation, we have all = 1 and p = 0. So the remaining cases are i 5 = 3, t5 = 9; i 6 = 2, t6 = 7; and i 7 = 1, t7 = 1. Hence, (9) (7) (4) (11) (9) (7)

u (A) = u 1 u 2 u 3 u 4 u 3 u 2 u 1 . Proposition 1.3.2. The set {u (A) | A ∈ + (n) aperiodic} is a Z-basis of C(n). Proof. Let A ∈ + (n) be aperiodic and let w A = i 1t1 i 2t2 · · · i mtm be the corresponding word constructed as above. By [11, Th. 5.5], w A is distinguished, that is, ϕ AA1 ,...,Am = 1, where As = ts E is ,i s +1 for 1 s m. By [13, Th. 7.5(i)], the u (A) with A ∈ + (n) aperiodic form a Z-basis of C(n). We now define the quantum enveloping algebra of the loop algebra (the quantum loop algebra for short) of sln . Let C = C(n) = (ci, j )i, j ∈I be the n−1 , where I = Z/nZ. We always assume generalized Cartan matrix of type A that if n 3, then ci,i = 2, ci,i+1 = ci+1,i = −1 and ci, j = 0 otherwise. If n = 2, then c1,1 = c2,2 = 2 and c1,2 = c2,1 = −2. In other words, ⎛ ⎞ 2 −1 0 · · · 0 −1 ⎜−1 2 −1 · · · 0 0⎟ ⎜ ⎟ ⎜ ⎟ 0 −1 2 · · · 0 0 ⎜ ⎟ 2 −2 ⎟ (n 3). C= or C = ⎜ .. .. .. . . . ⎜ .. .. .. ⎟ −2 2 ⎜ . . . ⎟ ⎜ ⎟ ⎝0 0 0 · · · 2 −1⎠ −1 0 0 · · · −1 2 (1.3.2.1) The quantum group associated to C is denoted by U( sln ). Definition 1.3.3. Let n 2 and I = Z/nZ. The quantum loop algebra U( sln ) is the algebra over Q(υ) presented by generators i , K −1 , i ∈ I, E i , Fi , K i and relations, for i, j ∈ I , (QSL0) (QSL1) (QSL2) (QSL3) (QSL4)

1 K 2 · · · K n = 1; K i K j = K j K i , K i K −1 = 1; K i c i E j = υ i, j E j K i ; K i F j = υ −ci, j F j K i ; K

E i E j = E j E i , Fi F j = F j Fi if i = j ± 1; −K −1 K

i i (QSL5) E i F j − F j E i = δi, j υ−υ −1 ; (QSL6) E i2 E j − (υ + υ −1 )E i E j E i + E j E i2 = 0 if i = j ± 1 and n 3;

20

1. Preliminaries

(QSL7) Fi2 F j − (υ + υ −1 )Fi F j Fi + F j Fi2 = 0 if i = j ± 1 and n 3; (QSL6 ) E i3 E j − (v 2 + 1 + v −2 )E i2 E j E i + (v 2 + 1 + v −2 )E i E j E i2 − E j E i3 = 0 if i = j and n = 2; (QSL7 ) Fi3 F j − (v 2 + 1 + v −2 )Fi2 F j Fi + (v 2 + 1 + v −2 )Fi F j Fi2 − F j Fi3 = 0 if i = j and n = 2. For later use in representation theory, let UC ( sln ) be the quantum loop algebra defined by the same generators and relations (QSL0)–(QSL7) with v replaced by a non-root-of-unity z ∈ C∗ and Q(v) by C. A new presentation for U( sln ) and UC ( sln ), known as Drinfeld’s new presentation, will be discussed in §2.5. In this book, quantum affine sln always refers to the quantum loop (Hopf) algebra U( sln ).3 We will mainly work with U( sln ) or quantum groups defined over Q(v) and mention from time to time a parallel theory over C. Let U( sln )+ (resp., U( sln )− , U( sln )0 ) be the positive (resp., negative, zero) part of the quantum enveloping algebra U( sln ). In other words, U( sln )+ (resp., ±1 ), U( sln )− , U( sln )0 ) is a Q(v)-subalgebra generated by E i (resp., Fi , K i i ∈ I. Let C(n) = C(n) ⊗Z Q(v). Thus, C(n) identifies with the Q(v)-subalgebra H(n) generated by u i = u [Si ] for i ∈ I . Theorem 1.3.4. ([65]) There are Q(v)-algebra isomorphisms C(n) −→ U( sln )+ , u i −→ E i and C(n)op −→ U( sln )− , u i −→ Fi . By this theorem and the triangular decomposition U( sln ) = U( sln )+ ⊗ U( sln )0 ⊗ U( sln )− , the basis displayed in Proposition 1.3.2 gives rise to a monomial basis for U( sln ).

1.4. Three types of generators and associated monomial bases In this section, we display three distinct minimal sets of generators for H(n), each of which contains the generators {u i }i∈I for C(n). We also describe their associated monomial bases for H(n) in the respective generators. 3 If (QSL0) is dropped, it also defines a quantum affine sl with the central extension; see, n

e.g., [9].

1.4. Three types of generators and associated monomial bases

21

The first minimal set of generators contains simple modules and certain central elements. These generators are convenient for a presentation for the double Ringel–Hall algebras over Q(v) (or a specialization at a non-root-of-unity) associated to cyclic quivers (see Chapter 2). In [67] Schiffmann first described the structure of H(n) as a tensor product of C(n) and a polynomial algebra in infinitely many indeterminates. Later Hubery explicitly constructed these central elements in [39]. More precisely, for each m 1, let cm = (−1)m v −2nm (−1)dim End(M(A)) a A u A ∈ H(n), (1.4.0.1) A

where the sum is taken over all A ∈ + (n) such that d(A) = dim M(A) = mδ with δ = (1, . . . , 1) ∈ Nn , and soc M(A) is square-free, i.e., dim soc M(A) δ in the order defined in (1.1.0.3). Note that in this case, soc M(A) is squarefree if and only if top M(A) := M(A)/rad M(A) is square-free. The following result is proved in [67, 39]. Theorem 1.4.1. The elements cm are central in H(n). Moreover, there is a decomposition H(n) = C(n) ⊗Q(v) Q(v)[c1 , c2 , . . .], where Q(v)[c1 , c2 , . . .] is the polynomial algebra in cm for m 1. In particular, H(n) is generated by u i and cm for i ∈ I and m 1. We will call the central elements cm the Schiffmann–Hubery generators. Let A = (ai, j ) ∈ + (n). For each s 1, define m s = m s (A) = min{ai, j | j − i = s} and A = A − m j−i E i, j . 1i n, i < j

Then A is aperiodic. Moreover, for A, B ∈ + (n),

(1.4.1.1)

A = B ⇐⇒ A = B and m s (A) = m s (B), ∀ s 1. The next corollary is a direct consequence of Theorem 1.4.1 and Proposition 1.3.2. Corollary 1.4.2. The set

{u (A )

s 1

csm s (A) | A ∈ + (n)}

is a Q(v)-basis of H(n). Next, we look at the minimal set of generators consisting of simple modules and homogeneous semisimple modules. It is known from [73, Prop. 3.5]

22

1. Preliminaries

(or [13, Th. 5.2(i)]) that H(n) is also generated by u a = u [Sa ] for a ∈ NI ; see also (1.4.4.1) below. If a is not sincere, say ai = 0, then ua =

j∈I, j =i

v a j (1−a j ) ai−1 ai+1 u i−1 · · · u a11 u ann · · · u i+1 ∈ C(n). [a j ]!

(1.4.2.1)

Thus, H(n) is generated by u i and u a , for i ∈ I and sincere a ∈ NI . Indeed, this result can be strengthened as follows; see also [67, p. 421]. Proposition 1.4.3. The Ringel–Hall algebra H(n) is generated by u i and u mδ , for i ∈ I and m 1. Proof. Let H be the Q(v)-subalgebra generated by u i and u mδ for i ∈ I and m 1. To show H = H(n), it suffices to prove u a = u [Sa ] ∈ H for all a ∈ NI . Take an arbitrary a ∈ NI . We proceed by induction on σ (a) = i∈I ai to show u a ∈ H . If σ (a) = 0 or 1, then clearly u a ∈ H . Now let σ (a) > 1. If a is not sincere, then by (1.4.2.1), u a ∈ H . So we may assume a is sincere. The case where a1 = · · · = an is trivial. Suppose now there exists i ∈ I such that ai = ai+1 . Define a = (a j ), a = (a j ) ∈ NI by ai − 1, if j = i ; ai +1 − 1, if j = i + 1; aj = and a j = ai , otherwise, ai , otherwise. Then, in H(n), u i u a = v ai −ai +1 −1 (u X + v ai −1 [ai ]u a ) and u a u i+1 = v ai+1 −ai −1 (u X + v ai+1 −1 [ai+1 ]u a ), where X ∈ + (n) is given by M(X ) ∼ a j S j ⊕ (ai − 1)Si ⊕ (ai+1 − 1)Si+1 ⊕ Si [2]. = j =i,i+1

Therefore, u i u a − v 2ai −2ai +1 u a u i+1 = v ai −ai +1 −1 (v ai −1 [ai ] − v ai +1 −1 [ai+1 ])u a . The inequality ai = ai+1 implies v ai −1 [ai ] − v ai +1 −1 [ai +1 ] = 0. Thus, we obtain ua =

v ai −1 [a

v ai +1 −ai +1 v ai −ai+1 +1 u i u a − a −1 u a u i+1 . a −1 v i [ai ] − v ai +1 −1 [ai +1 ] i ] − v i +1 [ai+1 ]

Since σ (a ) = σ (a ) = σ (a) − 1, we have by the inductive hypothesis that both u a and u a belong to H . Hence, u a ∈ H . This finishes the proof.

1.4. Three types of generators and associated monomial bases

23

Remark 1.4.4. We will see that semisimple modules as generators are convenient for the description of Lusztig type integral forms. First, by [13, Th. 5.2(ii)], they generate the integral Ringel–Hall algebra H(n) over Z. Second, there are in §2.6 explicit commutator formulas between semisimple generators in the double Ringel–Hall algebra. Thus, a natural candidate for the Lusztig type form of quantum affine gln is proposed in §3.8. Finally, we introduce a set of generators for H(n) consisting of simple and homogeneous indecomposable modules in Rep0(n). Since indecomposable modules correspond to the simplest non-diagonal matrices, these generators are convenient for deriving explicit multiplication formulas; see §§3.4, 5.4, and 6.2. For each A ∈ + (n), consider the radical filtration of M(A) M( A) ⊇ rad M(A) ⊇ · · · ⊇ rad t−1 M(A) ⊇ rad t M(A) = 0, where t is the Loewy length of M(A). For 1 s t, we write rad s−1 M(A)/rad s M(A) = Sas ,

for some as ∈ NI .

Write m A = u a1 · · · u at . Applying (1.2.0.7) gives that mA = f (B)u B , B dg A

where f (B) ∈ Q(v) with f (A) = v −1

u A = f (A)

mA −

l<s al ,as

. In other words,

f (A)−1 f (B)u B .

B
Repeating the above construction for maximal B with B
where φ AB ∈ Q(v) with φ AA = f (A)−1 = 0. Proposition 1.4.5. For each s 1, fix an arbitrary i s ∈ Z+ . Then H(n) is generated by u i and u E for i ∈ I and s 1. i s ,i s +sn

Proof. For each m 1, let H(n)(m) be the Q(v)-subalgebra of H(n) generated by u i and u sδ for i ∈ I and 1 s m. By convention, we set H(n)(0) = C(n). Clearly, each H(n)(m) is also NI -graded. By Theorem 1.4.1, H(n)(m) is also generated by u i and cs , for i ∈ I and 1 s m. Moreover, H(n)(m−1) H(n)(m) for all m 1.

24

1. Preliminaries

We use induction on m to show the following Claim: H(n)(m) is generated by u i and u E

i s ,i s +sn

for i ∈ I and 1 s m.

Let m 1 and suppose the claim is true for H(n)(m−1) . Applying (1.4.4.1) to E = E i m ,i m +mn gives uE =

φ EB m B = φ EE m E +

B dg E

φ EB m B + φ C E mC ,

(1.4.5.1)

B
∼ where C = Amδ = i∈I m E i,i+1 , i.e., M(C) = Smδ and mC = u mδ . For each B with B dg E and B = C, the Loewy length of M(B) is strictly greater than 1. Using an argument similar to the proof of Proposition 1.4.3, we have m B ∈ H(n)(m−1) . We now prove that φ CE = 0. Suppose φ C E = 0. Then (m−1) u E ∈ H(n) . Furthermore, for each j ∈ I , there is 1 p n such that Ej, j+mn = τ p (E), where τ is the Auslander–Reiten translation defined in §1.2. Thus, applying τ p to (1.4.5.1) gives u E = u τ p (E ) = φ EE mτ p E + φ EB mτ p B . j, j +mn

B
Using a similar argument as above, we get u E ∈ H(n)(m−1) . By j, j+mn definition, we have H(n)(m−1) = H(n)d for all d ∈ NI with σ (d) < mn. d In particular, u E ∈ H(n)(m−1) for all i < j with j − i < mn. Consequently, i, j we get u Ei, j ∈ H(n)(m−1) for all i < j with j − i mn. By [37, Th. 3.1], each element in H(n) can be written as a linear combination of products of u A ’s with M( A) indecomposable. This together with the above (m−1) discussion implies that H(n)mδ = H(n)mδ . Thus, u mδ ∈ H(n)(m−1) . This contradicts the fact that H(n)(m−1) H(n)(m) . Therefore, φ CA = 0. We conclude from (1.4.5.1) that E u E − φC φ EB m B ∈ H(n)(m−1) , E u mδ = φ E m E + B
which shows the claim for H(n)(m) . This finishes the proof.

1.5. Hopf structure on extended Ringel–Hall algebras

25

By the proof of the above proposition, we see that for m 1, each of the following three sets {u i , cs | i ∈ I, 1 s m}, {u i , u sδ | i ∈ I, 1 s m}, and {u i , u E

i s ,i s +sn

| i ∈ I, 1 s m}

generates H(n)(m) . Hence, for each m 1, there are non-zero elements xm , ym ∈ Q(v) such that cm ≡ xm u mδ mod H(n)(m−1) and cm ≡ ym u E

i s ,i s +sn

mod H(n)(m−1) .

This together with Corollary 1.4.2 gives the following result. Corollary 1.4.6. The set ( A )

u (u sδ )m s ( A) | A ∈ + (n) s 1

is a Q(v)-basis of H(n), where A is defined in (1.4.1.1). For each s 1, choose i s ∈ Z+ . Then the set ( A )

u (u E )m s ( A) | A ∈ + (n) s 1

i s ,i s +sn

is also a Q(v)-basis of H(n).

1.5. Hopf structure on extended Ringel–Hall algebras It is known that generic Ringel–Hall algebras exist only for Dynkin or cyclic quivers. In the finite type case, these algebras give a realization for the ±-parts of quantum groups. It is natural to expect that this is also true for cyclic quivers. In other words, we look for a quantum group such that H(n) is isomorphic to its ±-part. We will see in Chapter 2 that this quantum group is the quantum loop algebra of gln in the sense of Drinfeld [20], which, in fact, is isomorphic to the so-called double Ringel–Hall algebra defined as the Drinfeld double of two Hopf algebras H(n)0 and H(n)0 together with a skew-Hopf pairing. In this section, we first introduce the pair H(n)0 and H(n)0 . We need some preparation. If we define the symmetrization of the Euler form (1.2.0.4) by (α, β) = α, β + β, α, then I together with ( , ) becomes a Cartan datum in the sense of [54, 1.1.1]. To a Cartan datum, there are associated root data in the sense of [54, 2.2.1]

26

1. Preliminaries

which play an important role in the theory of quantum groups. We shall fix the following root datum throughout the book. Definition 1.5.1. Let X = Zn , Y = Hom(X, Z), and let , rd : Y × X → Z be the natural perfect pairing. If we denote the standard basis of X by e1 , . . . , en and the dual basis by f 1 , . . . , f n , then f i , e j rd = δi, j . Thus, the embeddings I −→ Y, i −→ i˜ := f i − f i+1 and I −→ X, i −→ i = ei −ei+1 (1.5.1.1) with en+1 = e1 and f n+1 = f 1 define a root datum (Y, X, , rd , . . .). For notational simplicity, we shall identify both X and Y with ZI by setting ei = i = f i for all i ∈ I . Under this identification, the form , rd : ZI × ZI → Z becomes a symmetric bilinear form, which is different from the Euler forms , and its symmetrization ( , ). However, they are related as follows. Lemma 1.5.2. For a = ai i ∈ ZI , if we put a˜ = ai i˜ and a = ai i , then (1)

(a, b) = ˜a, b rd ;

(2)

a, b = a , brd , for all a, b ∈ ZI.

Proof. Since all forms are bilinear, (1) follows from (i, j ) = i˜, j rd for all i, j ∈ I , and (2) from i, j = δi, j − δi+1, j = i , jrd . We may use the following commutative diagrams to describe the two relations: ZI × ZI (˜) × ( ) ZI × ZI

ZI × ZI

(, ) Z , rd

and

,

( ) × 1 ZI × ZI

Z. , rd

We also record the following fact which will be used below and in §2.1. Let F be a field. A Hopf algebra A over F is an F-vector space together with multiplication μA , unit ηA , comultiplication A , counit εA , and antipode σA which satisfy certain axioms; see, e.g., [72] and [12, §5.1]. Lemma 1.5.3. If A = (A , μ, η, , ε, σ ) is a Hopf algebra with multiplication μ, unit η, comultiplication , counit ε, and antipode σ , then A op = (A , μop , η, op , ε, σ ) is also a Hopf algebra. This is called the opposite Hopf algebra of A . Moreover, if σ is invertible, then both (A , μop , η, , ε, σ −1 ) and (A , μ, η, op , ε, σ −1 ) are also Hopf algebras, which are called semiopposite Hopf algebras.

1.5. Hopf structure on extended Ringel–Hall algebras

27

Let H(n)0 = H(n) ⊗Q(v) Q(v)[K 1±1 , . . . , K n±1 ].

(1.5.3.1)

Putting x = x ⊗1 and y = 1⊗ y for x ∈ H(n) and y ∈ Q(v)[K 1±1 , . . . , K n±1 ], + H(n)0 is a Q(v)-space with basis {u + A K α | α ∈ ZI, A ∈ (n)}. We are now ready to introduce the Ringel–Green–Xiao Hopf structure on H(n)0 . Let + (n)∗ := + (n)\{0}. Proposition 1.5.4. The Q(v)-space H(n)0 with basis {u + A K α | α ∈ ZI, A ∈ + (n)} becomes a Hopf algebra with the following algebra, coalgebra, and antipode structures. (a) Multiplication and unit (Ringel [64]): for all A, B ∈ + (n) and α, β ∈ ZI ,

+ u+ AuB =

v d( A),d(B) ϕ CA,B u + C,

C∈+ (n) d( A),α + Kα u+ u A Kα , A =v

K α K β = K α+β , and 1 = u+ 0 = K0. (b) Comultiplication and counit (Green [34]): for all C ∈ + (n) and α ∈ ZI , (u + C) =

v d( A),d(B)

A,B∈+ (n)

aAaB C + ϕ u ⊗ u+ A K d(B) , aC A,B B

(K α ) = K α ⊗ K α , ε(u + C ) = 0 (C = 0), and ε(K α ) = 1. α denotes ( K 1 )a1 · · · ( K n )an with K i = K i K −1 . Here, if α = (ai ), then K i+1 + (c) Antipode (Xiao [78]): for all C ∈ (n) and α ∈ ZI , σ (u + C ) = δC,0 +

(−1)m

m 1

aC 1 · · · aC m aC

+ (n) D∈ + (n)∗ C 1 ,...,C m ∈

× ϕCC1 ,...,Cm ϕCDm ,...,C1 u + D K −d(C) and σ (K α ) = K −α .

28

1. Preliminaries

Moreover, the inverse of σ is given by + σ −1 (u C ) = δC,0 +

m 1

(−1)m

+ (n) D∈ + (n)∗ C1 ,··· ,C m ∈

v2

i < j d(Ci ),d(C j )

aC 1 · · · aC m aC

−d(C) u + and × ϕCC1 ,...,Cm ϕCD1 ,...,Cm K D σ −1 (K α ) = K −α . Proof. The Hopf structure on H(n)0 is almost identical to the Hopf algebra α instead of K α H defined in the proof of [78, Prop. 4.8] except that we used K in the comultiplication and antipode. Thus, the comultiplication of H(n)0 defined here is opposite to that defined in [78, Th. 4.5], while the antipode is the inverse. Hence, by Lemma 1.5.3, H(n)0 is the Hopf algebra semi-opposite to a variant of the Hopf algebras considered in [78, loc cit]. (Of course, one can directly check by mimicking the proof of [78, Th. 4.5] that H(n)0 with the operations defined above satisfies the axioms of a Hopf algebra.) Remarks 1.5.5. (1) Because of Lemma 1.5.2, we are able to make the root datum used in the second relation in (a) invisible in the definition of H(n)0 . α in comultiplication and (2) Besides the modification of changing K α to K antipode, we also used the Euler form , (or rather the form , rd for the root datum given in Definition 1.5.1) instead of the symmetric Euler form ( , ) used in Xiao’s definition of H ([78, p. 129]) for the commutator formulas between 0 K α and u + A . This means that H is not isomorphic to H(n) . However, there 0 is a Hopf algebra homomorphism from H to H(n) by sending u + A K α to α whose image is the (Hopf) subalgebra H(n)0 generated by K i and u+ K A + + u A , for all i ∈ I and A ∈ (n). Note that it sends the central element 1 · · · K n = 1 in H(n)0 . K 1 · · · K n = 1 in H to K (3) The above modifications are necessary for compatibility with Lusztig’s construction for quantum groups in [54] and with the corresponding relations in affine quantum Schur algebras. + It is clear that the subalgebra of H(n)0 generated by u + A (A ∈ (n)) is isomorphic to H(n). The subalgebra generated by K α , α ∈ ZI , is isomorphic to the Laurent polynomial ring Q(v)[K 1±1 , . . . , K n±1 ].

Corollary 1.5.6. The Q(v)-space H(n)0 with basis {K α u − A | α ∈ ZI, A ∈ + (n)} becomes a Hopf algebra with the following algebra, coalgebra, and antipode structures.

1.5. Hopf structure on extended Ringel–Hall algebras

29

(a ) Multiplication and unit: for all A, B ∈ + (n) and α, β ∈ ZI , − − v d(B),d(A) ϕ CB,A u C , u− AuB = C∈+ (n) d(A),α u− Kα u− A Kα = v A,

K α K β = K α+β , and 1 = u− 0 = K0. (b ) Comultiplication and counit: for all C ∈ + (n) and α ∈ ZI , a AaB C − − )= v −d(B),d(A) ϕ A,B K −d(A) u − (u C B ⊗ uA, a C + A,B∈ (n)

(K α ) = K α ⊗ K α , − ε(u C ) = 0 (C = 0), and ε(K α ) = 1.

(c ) Antipode: for all C ∈ + (n) and α ∈ ZI , aC · · · aCm − ) = δC,0 + (−1)m v 2 i< j d(Ci ),d(C j ) 1 σ (u C aC + m 1

D∈ (n) + (n)∗ C1 ,...,Cm ∈

d(C) u − and × ϕCC1 ,...,Cm ϕCD1 ,...,Cm K D σ (K α ) = K −α . + Proof. Let H be the Q(v)-space with basis {K α (u − A ) | α ∈ ZI, A ∈ (n)} and define the following operations on H :

(a ) for all A, B ∈ + (n) and α, β ∈ ZI , − − (u − ) (u ) = v d(B),d(A) ϕ CB,A (u C ), A B C∈+ (n) −d(A),α (u − K α (u − A ) Kα = v A) , , K α K β = K α+β

and let 1 = (u − 0 ) = K0; + (b ) for all C ∈ (n) and α ∈ ZI , a AaB C − − ))= v d(A),d(B) ϕ A,B (u − ((u C B ) ⊗ K −d(B) (u A ) , a C + A,B∈ (n)

(K α ) ε((u − A) )

= K α ⊗ K α ,

= 0 (A = 0), and ε(K α ) = 1,

= K (K )−1 ; where K i i i+1

30

1. Preliminaries

(c ) for all C ∈ + (n) and α ∈ ZI , − σ ((u C ) ) = δC,0 + (−1)m m 1

v2

i< j d(Ci ),d(C j )

+ (n) D∈ + (n)∗ C 1 ,...,C m ∈

aC1 · · · aCm aC

× ϕCC1 ,...,Cm ϕCD1 ,...,Cm (u − D ) K d(C) and σ (K α ) = K −α .

By Lemma 1.5.3, if we replace the multiplication of H(n)0 by its opposite one and σ by σ −1 and keep other structure maps unchanged, then we obtain the semi-opposite Hopf algebra Hop of H(n)0 . It is clear that the Q(v)− linear isomorphism Hop → H taking u + A K α → (u A ) K −α preserves all the operations. Thus, H is a Hopf algebra with the operations (a )–(c ). Now, for α ∈ ZI and A ∈ + (n), set in H , −d( A),d( A) K α := K −α and u − K d(A) (u − A := v A) .

(1.5.6.1)

+ Then {K α u − A | α ∈ ZI, A ∈ (n)} is a new basis of H . It is easy to check that applying the operations (a )–(c ) to the basis elements K α u − A gives (a )– 0 (c ). Consequently, H(n) with the operations (a )–(c ) is a Hopf algebra.

The proof above shows that H(n)0 is the semi-opposite Hopf algebra of H(n)0 in the sense that multiplication and antipode are replaced by the opposite and inverse ones, respectively. Note that the inverse σ −1 of σ in H(n)0 is defined by − σ −1 (u C ) = δC,0 + (−1)m m 1

+ (n) D∈ + (n)∗ C1 ,...,Cm ∈

aC1 · · · aCm C ϕC1 ,...,Cm ϕCDm ,...,C1 u − D K d(C) . aC

+ By (a ), the subalgebra of H(n)0 generated by u − A ( A ∈ (n)) is isomorphic to H(n)op . Moreover, there is a Q(v)-vector space isomorphism

H(n)0 ∼ = Q(v)[K 1±1 , . . . , K n±1 ] ⊗Q(v) H(n)op .

(1.5.6.2)

Remark 1.5.7. If we put A = Z[(v m − 1)−1 ]m 1 , then (1.2.0.3) guarantees 0 0 that extended Ringel–Hall algebras H(n)A and H(n)A over A are welldefined. Thus, if v is specialized to z in a field (or a ring) F which is not a root 0 0 of unity, then extended Ringel–Hall algebras H(n) F := H(n)A ⊗ F and 0 H(n) F over F are defined.

2 Double Ringel–Hall algebras of cyclic quivers

A Drinfeld double refers to a construction of gluing two Hopf algebras via a skew-Hopf pairing between them to obtain a new Hopf algebra. We apply this construction in §2.1 to the extended Ringel–Hall algebras H(n)0 and H(n)0 discussed in §1.5 to obtain double Ringel–Hall algebras D(n) of cyclic quivers. The algebras D(n) possess a rich structure. First, by using the Schiffmann– Hubery generators and the connection with the quantum enveloping algebra associated with a Borcherds–Cartan matrix, we obtain a presentation for D(n) (Theorem 2.3.1). Second, by using Drinfeld’s new presentation for the quann ), we extend Beck’s (and Jing’s) embedding of quantum tum loop algebra U(gl n ) to obtain an isomorphism between D(n) and U(gl n ) affine sln into U(gl (Theorem 2.5.3). Finally, applying the skew-Hopf pairing to semisimple generators yields certain commutator relations (Theorem 2.6.3). Thus, we propose a possible presentation using semisimple generators; see Problem 2.6.4.

2.1. Drinfeld doubles and the Hopf algebra D(n) In this section, we first recall from [46] the notion of a skew-Hopf pairing and define the associated Drinfeld double. We then apply this general construction to obtain the Drinfeld double D(n) of the Ringel–Hall algebra H(n); see [78] for a general construction. Let A = (A , μA , ηA , A , εA , σA ) and B = (B, μB , ηB , B , εB , σB ) be two Hopf algebras over a field F. A skew-Hopf pairing of A and B is an F-bilinear form ψ : A × B → F satisfying: (HP1) ψ(1, b) = εB (b), ψ(a, 1) = εA (a), for all a ∈ A , b ∈ B; (HP2) ψ(a, bb ) = ψ(A (a), b ⊗ b ), for all a ∈ A , b, b ∈ B; 31

32

2. Double Ringel–Hall algebras of cyclic quivers

(HP3) ψ(aa , b) = ψ(a ⊗ a , B (b)), for all a, a ∈ A , b ∈ B; −1 (HP4) ψ(σA (a), b) = ψ(a, σB (b)), for all a ∈ A , b ∈ B, op

where ψ(a ⊗ a , b ⊗ b ) = ψ(a, b)ψ(a , b ), and B is defined by B (b) = b2 ⊗ b1 if B (b) = b1 ⊗ b2 . Note that we have assumed here that σB is invertible. Let A ∗ B be the free product of F-algebras A and B with identity. Then A ∗ B is the coproduct of A and B in the category of F-algebras. More precisely, for any fixed bases BA and BB for A , B, respectively, where both BA and BB contain the identity element, A ∗ B is the F-vector space spanned by the basis consisting of all words b1 b2 · · · bm (bi ∈ (BA \{1}) ∪ (BB \{1})) of any length m 0 such that bi bi +1 is not defined (in other words, bi , bi+1 are not in the same A or B) with multiplication given by “contracted juxtaposition” ⎧ , ⎪ b · · · bm b1 · · · bm if bm b1 is not defined; ⎪ ⎪ 1 ⎪ ⎨b · · · b if bm b1 is defined, 1 m−1 cb2 · · · bm , (b1 · · · bm ) ∗ (b1 · · · bm ) = ⎪ bm b1 = 0; ⎪ ⎪ ⎪ ⎩ 0, otherwise. Note that, since c = bm b1 = k λk ak , where all ak ∈ BA or all ak ∈ BB , is is a a linear combination of basis elements, the element b1 · · · bm−1 cb2 · · · bm linear combination of words and is defined inductively. Thus, 1 is replaced by the empty word. Let I = IA ,B be the ideal of A ∗ B generated by (b2 ∗ a2 )ψ(a1 , b1 ) − (a1 ∗ b1 )ψ(a2 , b2 ) (a ∈ A , b ∈ B), (2.1.0.1) where A (a) = a1 ⊗ a2 and B (b) = b1 ⊗ b2 . Moreover, I is also generated by b∗a− ψ(a1 , σB (b1 ))(a2 ∗ b2 )ψ(a3 , b3 ) (a ∈ A , b ∈ B), (2.1.0.2) op

op

(2) (2) where A (a) = a1 ⊗ a2 ⊗ a3 and B (b) = b1 ⊗ b2 ⊗ b3 (see, for example, [46, p. 72]). The Drinfeld double of the pair A and B is by definition the quotient algebra D(A , B) := A ∗ B/I. By (2.1.0.2), each element in D(A , B) can be expressed as a linear combination of elements of the form a ∗ b + I for a ∈ A and b ∈ B. Note that there is an F-vector space isomorphism

2.1. Drinfeld doubles and the Hopf algebra D(n)

33

D(A , B) −→ A ⊗F B, a ∗ b + I −→ a ⊗ b; see [71, Lem. 3.1]. For notational simplicity, we write a ∗ b + I as a ∗ b. Both A and B can be viewed as subalgebras of D(A , B) via a → a ∗ 1 and b → 1 ∗ b, respectively. Thus, if x and y lie in A or B, we write x y instead of x ∗ y. The algebra D(A , B) admits a Hopf algebra structure induced by those of A and B; see [46, 3.2.3]. More precisely, comultiplication, counit, and antipode in D(A , B) are defined by (a ∗ b) = (a1 ∗ b1 ) ⊗ (a2 ∗ b2 ), ε(a ∗ b) = εA (a)εB (b), and

(2.1.0.3)

σ (a ∗ b) = σB (b) ∗ σA (a), where a ∈ A , b ∈ B, A (a) = a1 ⊗ a2 , and B (b) = b1 ⊗ b2 . By modifying [71, Lem. 3.2], we obtain that the above ideal I can be generated by the elements in certain generating sets of A and B as described in the following result which will be used in §2.6. Lemma 2.1.1. Let A , B be Hopf algebras over F and let ψ : A × B → F be a skew-Hopf pairing. Assume that X A ⊆ A and X B ⊆ B are generating sets of A and B, respectively. If A (X A ) ⊆ spanF X A ⊗ spanF X A and B (X B ) ⊆ spanF X B ⊗ spanF X B , then the ideal I = IA ,B is generated by the following elements (b2 ∗ a2 )ψ(a1 , b1 ) − (a1 ∗ b1 )ψ(a2 , b2 ), for all a ∈ X A , b ∈ X B . (2.1.1.1) Proof. For a ∈ A and b ∈ B, put h a,b := (b2 ∗ a2 )ψ(a1 , b1 ) − (a1 ∗ b1 )ψ(a2 , b2 ). Let I be the ideal of A ∗ B generated by h a,b for all a ∈ X A , b ∈ X B , and set H = A ∗ B/I . We need to show I = I . Since I ⊆ I, it remains to show that for all a ∈ A and b ∈ B, h a,b ∈ I , or equivalently, h a,b = 0 in H. First, suppose a ∈ X A and b = y1 · · · yt with y j ∈ X B . We proceed by induction on t to show that h a,b ∈ I . Let b = y1 · · · yt−1 and b = yt . Write A (a) = a1 ⊗ a2 , A (a1 ) = a1,1 ⊗ a1,2 and A (a2 ) = a2,1 ⊗ a2,2 . The coassociativity of A implies that a1,1 ⊗a1,2 ⊗a2 = A (a1 )⊗a2 = a1 ⊗A (a2 ) = a1 ⊗a2,1 ⊗a2,2 .

34

2. Double Ringel–Hall algebras of cyclic quivers

Further, write B (b ) = b1 ⊗b2 and B (b ) = b1 ⊗b2 . Then B (b) = b1 b1 ⊗ b2 b2 . Thus, we obtain in H that (b2 b2 ∗ a2 )ψ(a1 , b1 b1 ) = b2 (b2 ∗ a2 )ψ(a1,1 , b1 )ψ(a1,2 , b1 ) = b2 (b2 ∗ a2,2 )ψ(a1 , b1 )ψ(a2,1 , b1 ) = (b2 ∗ a2,1 ) ∗ b1 ψ(a1 , b1 )ψ(a2,2 , b2 ) = = = =

(since a2 ∈ spanF X A and b ∈ X B ) (b2 ∗ a1,2 ) ∗ b1 ψ(a1,1 , b1 )ψ(a2 , b2 ) (a1,1 ∗ b1 )b1 ψ(a1,2 , b2 )ψ(a2 , b2 )

(by induction for a1 and b )

(a1 ∗ b1 )b1 ψ(a2,1 , b2 )ψ(a2,2 , b2 ) (a1 ∗ b1 b1 )ψ(a2 , b2 b2 ),

that is, h a,b ∈ I . Now suppose a = x1 · · · xs ∈ A with xi ∈ X A and b ∈ B. We proceed by induction on s. The case s = 1 has already been treated above. So assume s > 1. Let a = x1 · · · xs−1 and a = xs . Then (b2 ∗ a2 a2 )ψ(a1 a1 , b1 ) (2) (2) (2) = (b3 ∗ a2 )a2 ψ(a1 , b2 )ψ(a1 , b1 ) (2) (2) (2) = a1 ∗ (b2 ∗ a2 )ψ(a2 , b3 )ψ(a1 , b1 ) (by induction) (2) (2) (2) = a1 a1 ∗ b1 ψ(a2 , b3 )ψ(a2 , b2 ) (since a ∈ X A ) = a1 a1 ∗ b1 ψ(a2 a2 , b2 ), where (2) B (b) = proof.

b1(2) ⊗ b2(2) ⊗ b3(2) . Hence, h a,b ∈ I . This completes the

A skew-Hopf pairing can be passed on to opposite Hopf algebras (see Lemma 1.5.3). The following lemma can be checked directly. Lemma 2.1.2. Let A = (A , μA , ηA , A , εA , σA ), B = (B, μB , ηB , B , εB , σB ) be two Hopf algebras over a field F together with a skewHopf pairing ψ : A × B → F. Assume that σA and σB are both invertible. Then ψ is also a skew-Hopf pairing of the semi-opposite Hopf algeop op −1 −1 bras (A , μA , ηA , A , εA , σA ) and (B, μB , ηB , B , εB , σB ) (resp., op op −1 −1 (A , μA , ηA , A , εA , σA ) and (B, μB , ηB , B , εB , σB )).

2.1. Drinfeld doubles and the Hopf algebra D(n)

35

We end this section with the construction of the Drinfeld double associated with the Ringel–Hall algebras H(n)0 and H(n)0 introduced in §1.5. First, we need a skew-Hopf pairing. Applying Lemma 2.1.2 to [78, Prop. 5.3] yields the following result. For completeness, we sketch a proof. We introduce some notation which is used in the proof. For each α = i∈I ai i ∈ + ZI , write τ α = i ∈I ai−1 i . In particular, for each A ∈ (n), we have τ d(A) = d(τ (A)). Then, for α, β ∈ ZI , α = K α−τ α = K a1 · · · K nan . α, β = (α − τ α) β = −β, τ α and K 1 Proposition 2.1.3. The Q(v)-bilinear form ψ : H(n)0 × H(n)0 → Q(v) defined by − α β−d(A),d( A)+α+2d(A) −1 ψ(u + a A δ A,B , A Kα , Kβ u B ) = v

(2.1.3.1)

where α, β ∈ ZI and A, B ∈ + (n), is a skew-Hopf pairing. Proof. Condition (HP1) is obvious. We now check condition (HP2). Without − − loss of generality, we take a = u + A K α , b = K β u B , and b = K γ u C for + α, β, γ ∈ ZI and A, B, C ∈ (n). Then − − + − d(B),γ ψ(u + K β+γ u − A K α , K β u B K γ u C ) = ψ(u A K α , v B uC ) d(B),γ D = ψ(u + K β+γ v d(C),d(B) ϕC,B u− A Kα , v D) D

=v

x1

A ϕC,B a−1 A ,

where x1 = d(B), γ +d(C), d(B)+α(β +γ )−d(A), d(A)+α+2d(A). On the other hand, − − ψ((u + A K α ), K β u B ⊗ K γ u C ) a B aC − − = ψ( v d(B ),d(C ) ϕ BA ,C u C+ K α ⊗ u + B K d(C ) K α , K β u B ⊗ K γ u C ) a A B ,C a B aC − + − = v d(B ),d(C ) ϕ BA ,C ψ(u + C K α , K β u B )ψ(u B K d(C ) K α , K γ u C ) aA B ,C

A = v x2 ϕC,B a−1 A ,

where x2 = d(C), d(B) + α β − d(B), d(B) + α + (d(B) − τ d(B) + α) γ − d(C), d(C) + d(B) − τ d(B) + α + 2d(A). Here we have assumed A d(A) = d(B) + d(C) since ϕC,B = 0 otherwise. A direct calculation shows x1 = x2 . Hence, (HP2) holds.

36

2. Double Ringel–Hall algebras of cyclic quivers

Condition (HP3) can be checked similarly as (HP2). It remains to check − (HP4). Take a = u + A K α and b = K β u B . We may suppose A = 0 = B. Then − ψ(σ (u + A K α ), K β u B ) = (−1)m v y1 m 1

C 1 ,...,Cm ∈+ (n)∗

aC1 · · · aCm A ϕC1 ,...,Cm ϕCBm ,...,C1 aAaB

and −1 ψ(u + (K β u − A Kα, σ B )) m y2 = (−1) v m 1

C1 ,...,Cm ∈+ (n)∗

aC1 · · · aCm A ϕC1 ,...,Cm ϕCBm ,...,C1 , a AaB

where y1 = −d(B), α+(−α−d(A)+τ d( A))β −d(B), d(B)−α−d(A)+τ d(A) and y2 = d(A), d(B) − τ d(B) − β + α (d(B) − τ d(B) − β) − d( A), d(A) + α. Clearly, if d(A) = d(B), then ϕCA1 ,...,Cm ϕCBm ,...,C1 = 0. Hence, we may suppose d(A) = d(B). Then y1 = −α β − d(A), β + d(A), d(A) = y2 . − + −1 (K u − )), that is, (HP4) Therefore, ψ(σ (u + β B A K α ), K β u B ) = ψ(u A K α , σ holds.

Second, the Hopf algebras H(n)0 and H(n)0 together with the skewHopf pairing ψ give rise to the Drinfeld double 0 0 D (n) := D(H(n) , H(n) ).

Since as Q(v)-vector spaces, we have 0 0 ∼ 0 D ⊗ H(n)0 , (n) = D(H(n) , H(n) ) = H(n)

we sometimes write the elements in D (n) as linear combinations of a ⊗ b 0 0 for a ∈ H(n) and b ∈ H(n) . Moreover, it follows from (1.5.3.1) and (1.5.6.2) that there is a Q(v)-vector space isomorphism ±1 ±1 ∼ D (n) =H(n) ⊗Q(v) Q(v)[K 1 , . . . , K n ]

⊗Q(v) Q(v)[K 1±1 , . . . , K n±1 ] ⊗Q(v) H(n)op . Finally, we define the reduced Drinfeld double D(n) = D (n)/I ,

(2.1.3.2)

2.2. Schiffmann–Hubery generators

37

where I denotes the ideal generated by 1 ⊗ K α − K α ⊗ 1, for all α ∈ ZI . By the construction, I is indeed a Hopf ideal of D (n). Thus, D(n) is again a Hopf algebra. We call D(n) the double Ringel–Hall algebra of the cyclic quiver (n). Let D(n)+ (resp., D(n)− ) be the Q(v)-subalgebra of D(n) generated − + 0 by u + A (resp., u A ) for all A ∈ (n). Let D(n) be the Q(v)-subalgebra of D(n) generated by K α for all α ∈ ZI . Then D(n)+ ∼ = H(n), D(n)− ∼ = H(n)op , and D(n)0 ∼ = Q(v)[K 1±1 , . . . , K n±1 ].

(2.1.3.3)

Moreover, the multiplication map D(n)+ ⊗ D(n)0 ⊗ D(n)− −→ D(n) is an isomorphism of Q(v)-vector spaces. Also, we have D(n)0 := D(n)+ ⊗ D(n)0 ∼ = H(n)0 and D(n)0 := D(n)0 ⊗ D(n)− ∼ = H(n)0 . We will identify D(n)0 and D(n)0 with H(n)0 and H(n)0 , respectively, in the sequel. In particular, we may use the PBW type basis for H(n) to display a PBW type basis for D(n): + { u+ u− A K α B | A, B ∈ (n), α ∈ ZI }.

Remark 2.1.4. By specializing v to z ∈ C which is not a root of unity, (1.2.0.3) implies that the skew-Hopf pairing ψ given in (2.1.3.1) is welldefined over C. Hence the construction above works over C with H(n)0 , 0 etc., replaced by the corresponding specialization H(n)C , etc. (see Remark 1.5.7). Thus, we obtain the double Ringel–Hall algebra D,C (n) over C. In fact, if we use the ring A defined in Remark 1.5.7, the same reasoning shows that the skew-Hopf pairing in (2.1.3.1) is defined over A. Hence, we can form a double Ringel–Hall algebra D(n)A . Then D,C (n) ∼ = D(n)A ⊗ C and D(n) = D(n)A ⊗ Q(v).

2.2. Schiffmann–Hubery generators In this and the following sections, we will investigate the structure of D(n) by relating it with the quantum enveloping algebra of a generalized Kac–Moody algebra based on [67, 39]; see also [38, 14]. We first construct certain primitive central elements from the central elements of H(n) defined in (1.4.0.1). Recall that an element of a Hopf algebra with comultiplication is called primitive if

38

2. Double Ringel–Hall algebras of cyclic quivers (x) = x ⊗ 1 + 1 ⊗ x.

For m 1, let ± cm = (−1)m v −2nm

± (−1)dim End(M( A)) a A u ± A ∈ D(n) ,

(2.2.0.1)

A

where the sum is taken over all A ∈ + (n) such that d(A) = mδ and soc M(A) is square-free. We also define c0± = 1 by convention. By Theorem 1.4.1, the + and c− are central in D (n)+ and D (n)− , respectively. elements cm m ± u m be the generating funcFollowing [39, §4], let C ± (u) = 1 + m 1 cm ±} tions in indeterminate u associated with the sequence {cm m 1 and define ± elements xm by d 1 d ± X ± (u) = xm± u m−1 = log C ± (u) = ± C (u). du C (u) du m 1

Thus, for each m 1, ± xm± = mcm −

m−1

± xs± cm−s .

(2.2.0.2)

s=1

Recall from (1.2.0.6) the elements dim End(M( A))−dim M( A) ± u± u A ∈ D(n)± , for A ∈ + (n). A =v

In particular, for each 1 l n and m 1, u± E

l,l+mn

= v m−nm u ± E

l,l+mn

.

The following lemma can be deduced from [40, Lem. 12]. However, we provide here a proof for completeness. Lemma 2.2.1. For each m 1, xm± = v −nm (v m − v −m )

n l=1

u± E

l,l+mn

+ (v − v −1 )2 ym± ,

(2.2.1.1)

where ym± are Z-linear combinations of certain u± A such that d(A) = mδ and M(A) are decomposable. Proof. By (1.2.0.3), for each A ∈ + (n), a A ∈ Z is divisible by (v−v −1 )σ (A) , where σ (A) = 1i n, j ∈Z ai, j . Note that σ ( A) equals the number of indecomposable summands in M(A). Since a E = v 2(m−1) (v 2 − 1), for each l,l+mn 1 l n and m 1, it follows from (2.2.0.1) that ± cm ≡ v (1−n)m−1 (v − v −1 )

n l=1

u± E

l,l+mn

mod (v − v −1 )2 I± ,

2.2. Schiffmann–Hubery generators

39

where I± denote the Z-submodules of D(n)± spanned by all the u± A satisfying that M(A) are decomposable (i.e., σ (A) > 1). Since subrepresentations and quotient representations of each indecomposable representation of (n) are again indecomposable, it follows that, for each A ∈ + (n), ± ± ± ± u± u A ⊆ I± . A I ⊆ I and I

It suffices to show that xm±

≡v

−nm

m

(v − v

−m

)

n

u± E

l,l+mn

l=1

mod (v − v −1 )2 I± .

We proceed by induction on m. If m = 1, it is trivial since x1± = c1± . Let now m > 1. The inductive hypothesis together with (2.2.0.2) implies that xm± ≡ mv (1−n)m−1 (v − v −1 )

n

u± E

l,l+mn

l=1

−

m−1

n −ns s v (v − v −s ) u± E

s=1

n × v (1−n)(m−s)−1 (v − v −1 ) u± E

l,l+(m−s)n

l=1

l,l+sn

l=1

mod (v − v −1 )2 I± .

It is clear that for 1 l, l n, u± E

l,l+sn

u± E

l ,l +(m−s)n

≡ δl,l u± E

l,l+mn

mod I± .

We conclude that m−1 n xm± ≡ v (1−n)m m(1 − v −2 ) − (1 − v −2 )(1 − v −2s ) u± E s=1

≡ v −nm (v m − v −m )

n l=1

u± E

l=1

l,l+mn

l,l+mn

mod (v − v −1 )2 I± .

We further set z± m =

v nm x ± ∈ D(n)± , v m − v −m m

for m 1.

Applying (2.2.1.1) gives that z± m =

n l=1

u± E

l,l+mn

+

v nm (v − v −1 ) ± ym . [m]

(2.2.1.2)

40

2. Double Ringel–Hall algebras of cyclic quivers

− + − Hence, the elements z+ m (resp., zm ) are central in D(n) (resp., in D(n) ). Moreover, by Theorem 1.4.1, we have ± D(n)± = C(n)± ⊗ Q(v)[c1± , c2± , . . .] = C(n)± ⊗ Q(v)[z± 1 , z2 , . . .], (2.2.1.3) where C(n)± are the composition algebras generated by u i± , 1 i n. Let be the comultiplication on D(n) induced by Green’s comultiplication and σ be the antipode of D(n) induced by Xiao’s antipode; see Proposition 1.5.4 and Corollary 1.5.6. − Proposition 2.2.2. For each m 1, the elements z+ m and zm satisfy ± ± ± ± (z± m ) = zm ⊗ 1 + 1 ⊗ zm and σ (zm ) = −zm .

Moreover, for all i ∈ I and m, m 1, − + − + − ± [z+ m , zm ] = 0, [zm , u i ] = 0, [u i , zm ] = 0, and [zm , K i ] = 0.

In other words, z± m are central elements in D(n). Proof. By [39, Prop. 9], for each m 1, ± (cm )=

m

± cs± ⊗ cm−s .

s=0

Applying (2.2.0.2) implies that xm± are primitive; see [39, Cor. 10]. Hence, z± m are also primitive, that is, ± ± (z± m ) = zm ⊗ 1 + 1 ⊗ zm . ± Since D(n) is a Hopf algebra, we have μ(σ ⊗ 1)(z± m ) = ηε(zm ) = 0 for each m 1. Thus, ± ± ± 0 = μ(σ ⊗ 1)(z± m ⊗ 1 + 1 ⊗ zm ) = σ (zm ) + zm , ± i.e., σ (z± m ) = −zm . − − − + + Since (zm ) = z+ m ⊗ 1 + 1 ⊗ zm and (zm ) = zm ⊗ 1 + 1 ⊗ zm , applying − + (2.1.0.1) to zm and zm gives − − − − + + + ψ(z+ m , zm ) + zm ψ(1, zm ) + zm ψ(zm , 1) + zm zm ψ(1, 1)

− − − + + + − =z+ m zm ψ(1, 1) + zm ψ(zm , 1) + zm ψ(1, zm ) + ψ(zm , zm ). + + − + − It follows from ψ(1, 1) = 1 that z− m zm = zm zm , i.e., [zm , zm ] = 0. − + − + ± Similarly, we have [zm , u i ] = [u i , zm ] = [zm , K i ] = 0. The last assertion follows from (2.2.1.3).

2.3. Presentation of D(n)

41

By Theorem 1.3.4, there are isomorphisms ∼

∼

C(n)+ −→ U( sln )+ , u i+ −→ E i , and C(n)− −→ U( sln )− , u i− −→ Fi . Here we have applied the anti-involution U( sln )+ → U( sln )− , E i → Fi . By (2.2.1.3), there are decompositions D(n)+ = C(n)+ ⊗Q(v) Z(n)+ where Z(n)± m 1.

and D(n)− = C(n)− ⊗Q(v) Z(n)− , (2.2.2.1) ± := Q(v)[z± , z , . . .] are the polynomial algebras in z± m for 1 2

Remark 2.2.3. If z ∈ C is not a root of unity and H(n)C := H(n) ⊗ C is the C-algebra obtained by specializing v to z, then, by (1.2.0.3), we may use (2.2.0.2) to recursively define the central elements zm in H(n)C and, hence, ± elements z± m in D,C (n) ; see Remark 2.1.4. Thus, a C-basis similar to the one given in Corollary 1.4.2 can be constructed for D,C (n)± . In particular, we obtain decompositions ± ± D,C (n)± = C(n)± C ⊗ C[z1 , z2 , . . .],

and hence the C-algebra D,C (n) can be presented by generators − + u i+ , u i− , K i , K i−1 , z+ s , zs , i ∈ I, s ∈ Z and relations (QGL1)–(QGL8) as given in the last statement of Theorem 2.3.1 below.1

2.3. Presentation of D(n) Recall from (1.3.2.1) the Cartan matrix C = (ci, j ) of affine type An−1 and the quantum group U( sln ) associated with C as given in Definition 1.3.3. By sln )± under the isomorphism in Theorem 1.3.4, we identifying C(n)± with U( describe a presentation for D(n) as follows. Theorem 2.3.1. The double Ringel–Hall algebra D(n) of the cyclic quiver − −1 (n) is the Q(v)-algebra generated by Ei = u + i , Fi = u i , K i , K i , − + + z+ s , zs , i ∈ I, s ∈ Z with relations (i, j ∈ I and s, t ∈ Z ): (QGL1) K i K j = K j K i , K i K i−1 = 1; (QGL2) K i E j = v δi, j −δi, j +1 E j K i , K i F j = v −δi, j +δi, j+1 F j K i ; (QGL3) E i F j − F j E i = δi, j

i − K −1 K i , v−v −1

i = K i K −1 ; where K i+1

1 We will see in Proposition 2.4.5 that D (n) can be obtained as a specialization from the ,C Z-algebra D(n) by the base change Z → C, v → z. Thus, we can use the notation D(n)C

for this algebra.

42

2. Double Ringel–Hall algebras of cyclic quivers

(QGL4)

(−1)a

a+b=1−ci, j

(QGL5)

(−1)

a

a+b=1−ci, j

1 − ci, j a 1 − ci, j a

E ia E j E ib = 0, for i = j; Fia F j Fib = 0, for i = j;

+ + + − − − − + − − + (QGL6) z+ s zt = zt zs ,zs zt = zt zs , zs zt = zt zs ; + − − (QGL7) K i z+ s = zs K i , K i zs = zs K i ; + + − − − + + (QGL8) E i zs = zs E i , E i zs = zs E i , Fi z− s = zs Fi , and zs Fi = Fi zs .

Replacing Q(v) by C and v by a non-root-of-unity z ∈ C∗ , a similar result holds for D,C (n). We will prove this result by a “standardly” defined quantum enveloping algebra associated with a Borcherds–Cartan matrix. See Proposition 2.3.4 below for the proof. For each m ∈ Z+ , let Jm = [1, m] and set J∞ = [1, ∞) = Z+ . We then set m = ( Im = I ∪ Jm , for m ∈ Z+ ∪ {∞}. Define a matrix C ci, j )i, j ∈ Im by setting ci, j , if i, j ∈ I ; ci, j = 0, otherwise. m is a Borcherds–Cartan matrix which defines a generalized Kac–Moody C m we associate a quantum enveloping algebra algebra; see [5]. Thus, with C U(Cm ) as follows; see [48]. m be defined as above. The quantum Definition 2.3.2. Let m ∈ Z+ ∪{∞} and C m is the Q(v)-algebra presented enveloping algebra U(Cm ) associated with C by generators E i , Fi , K i , K i−1 , xs , ys , ks , k−1 s , i ∈ I, s ∈ Jm , and relations, for i, j ∈ I and s, t ∈ Jm , (R1) K i K j = K j K i , K i K i−1 = 1, ks kt = kt ks , ks k−1 s = 1, K i ks = ks K i ; (R2) K i E j = v δi, j −δi, j +1 E j K i , K i xs = xs K i , ks xt = xt ks , ks E i = E i ks ; (R3) K i F j = v −δi, j +δi, j+1 F j K i , K i ys = ys K i , ks yt = yt ks , ks Fi = Fi ks ; (R4) E i F j − F j E i = δi, j

i − K −1 K i , v−v −1

xs yt − yt xs = δs,t

ks −k−1 s , v−v −1

i = K i K −1 ; xs Fi = Fi xs , where K i +1 a 1 − ci, j (R5) (−1) E ia E j E ib = 0 for i = j ; a a+b=1−ci, j 1 − ci, j (R6) (−1)a Fia F j Fib = 0 for i = j; a a+b=1−ci, j

E i ys = ys E i ,

2.3. Presentation of D(n)

43

(R7) E i xs = xs E i , xs xt = xt xs ; (R8) Fi ys = ys Fi , ys yt = yt ys . m ) is a Hopf algebra with comultiplication , counit ε, and Moreover, U(C antipode σ defined by i + 1 ⊗ E i , (E i ) = E i ⊗ K (xs ) = xs ⊗ ks + 1 ⊗ xs , (K i±1 ) = K i±1 ⊗ K i±1 ,

−1 ⊗ Fi , (Fi ) = Fi ⊗ 1 + K i (ys ) = ys ⊗ 1 + k−1 s ⊗ ys , ±1 ±1 (k±1 s ) = ks ⊗ ks ;

ε(E i ) = ε(xs ) = 0 = ε(Fi ) = ε(ys ), ε(K i ) = ε(ks ) = 1; −1 , σ (Fi ) = − K i Fi , σ (K ±1 ) = K ∓1 , σ (E i ) = −E i K i

σ (xs ) = −xs k−1 s ,

i

i

∓1 σ (ys ) = −ks ys , and σ (k±1 s ) = ks ,

where i ∈ I and s ∈ Jm . Remark 2.3.3. The comultiplication here is opposite to the one given in [54, 3.1.10]. Consequently, the antipode is the inverse of the antipode given in [54, 3.3.1]. This change is necessary for its action on tensor space, commuting with the Hecke algebra action; cf. [12, §14.6]. m )+ (resp., m )− , m )0 ) be the Q(v)Let U+ = U(C U− = U(C U0 = U(C ±1 m ) generated by E i (resp., Fi , K ), for all i ∈ subalgebra of U(C I , and i ±1 xs (resp., ys , ks ), for all s ∈ Jm . Then, by [48, Th. 2.23], U(Cm ) admits a triangular decomposition m ) = U(C U+ ⊗ U0 ⊗ U− . m ) generated by E i , Fi , K i , K −1 (i ∈ I ) It is clear that the subalgebra of U(C i is the quantum enveloping algebra U( sln ) as defined in §1.3. Also, we denote m ) generated by E i , Fi , K i , K −1 (i ∈ I ). We by U(n) the subalgebra of U(C i will call U(n) the extended quantum affine sln .2 Now Theorem 2.3.1 follows immediately from the following result which describes the structure of D(n). It is a generalization of a result for the double Ringel–Hall algebra of a finite dimensional tame hereditary algebra in [38] to the cyclic quiver case. Its proof is based on Theorem 1.4.1. Proposition 2.3.4. There is a unique surjective Hopf algebra homomorphism ∞ ) → D(n) satisfying, for each i ∈ I and s ∈ J∞ , : U(C − ±1 − E i −→ u i+ , xs −→ z+ −→ K i± , k± s , Fi −→ u i , ys −→ zs , K i s −→ 1. 2 This is called quantum affine gl by Lusztig [56], which is an extension of quantum affine sl n n

by adding an extra generator to the 0-part. This extension is similar to the extension of quantum gln from quantum sln . In the literature, quantum affine sln refers also to the quantum loop algebra associated with sln ; see §2.5 below.

44

2. Double Ringel–Hall algebras of cyclic quivers

∞ ) generated by k±1 Moreover, Ker is the ideal of U(C s − 1, for all s ∈ J∞ . Proof. The existence and uniqueness of clearly follow from Definition 2.3.2, Theorem 1.3.4, Theorem 1.4.1, and Proposition 2.2.2. Further, induces surjective algebra homomorphisms ∞ )± −→ D(n)± . ± : U± = U(C By (R7), the multiplication map μ : U( sln )+ ⊗ Q(v)[x1 , x2 , . . .] −→ U+ is surjective. Hence, the composite + μ is surjective. By Theorem 1.4.1, + D(n)+ = C(n)+ ⊗Q(v) Q(v)[z+ 1 , z2 , . . .].

Since + μ(U( sln )+ ) = C(n)+ ∼ sln )+ , it follows that + μ is an iso= U( + morphism. Hence, is an isomorphism. Similarly, − is an isomorphism, too. ∞ ) generated by k±1 Let K be the ideal of U(C s −1 for s ∈ J∞ . It is clear that ∞ ) = K ⊆ Ker . The triangular decomposition U(C U+ ⊗ U0 ⊗ U− implies that + ∞ )/K = U(C U ⊗ Q(v)[K ±1 , . . . , K n±1 ] ⊗ U− + K /K. 1

Since D(n) = D(n)+ ⊗D(n)0 ⊗D(n)− = D(n)+ ⊗Q(v)[K 1±1, . . . , K n±1 ]⊗D(n)−,

we conclude that Ker = K, as required. Finally, it is straightforward to check that K is a Hopf ideal. Corollary 2.3.5. The double Ringel–Hall algebra D(n) is a Hopf algebra with comultiplication , counit ε, and antipode σ defined by i + 1 ⊗ E i , (E i ) = E i ⊗ K (K i±1 ) = K i±1 ⊗ K i±1 ,

−1 ⊗ Fi , (Fi ) = Fi ⊗ 1 + K i ± ± (z± s ) = zs ⊗ 1 + 1 ⊗ zs ;

ε(E i ) = ε(Fi ) = 0 = ε(z± ε(K i ) = 1; s ), −1 σ (E i ) = −E i K i , σ (Fi ) = − K i Fi , σ (K i±1 ) = K i∓1 , ± and σ (z± s ) = −zs ,

where i ∈ I and s ∈ J∞ . Remarks 2.3.6. (1) For notational simplicity, we sometimes continue to use u i± as generators of D(n). By Theorem 2.3.1, we see that there is a Q(v)algebra involution ς of D(n) satisfying ∓ K i±1 −→ K i∓1 , u i± −→ u i∓ , z± s −→ zs ,

2.4. Some integral forms

45

for i ∈ I and s 1. Thus, D(n) also admits a decomposition D(n) = D(n)− ⊗ D(n)0 ⊗ D(n)+ .

(2.3.6.1)

Moreover, two other types of generators for Ringel–Hall algebras discussed in §1.4 result in the corresponding generators for double Ringel–Hall algebras. Thus, we may speak of semisimple generators, etc. See more discussion in §2.6. (2) The subalgebra of D(n) generated by E i , Fi , K i , K i−1 (i ∈ I ) is isomorphic to U(n). We will always identify these two algebras and thus, view U(n) as a subalgebra of D(n). In particular, we have U(n) = C(n)− ⊗ D(n)0 ⊗ C(n)+ . − Moreover, if Z(n) = Q(v)[z+ m , zm ]m 1 denotes the central subalgebra − − + + of D(n) generated by . . . , z2 , z1 , z1 , z2 , . . ., then (2.3.6.1) together with (2.2.2.1) gives D(n) ∼ = U(n) ⊗ Z(n). − −1 (3) The subalgebra D(n) of D(n) generated by u + (A ∈ A , u A , Ki , Ki (n), i ∈ I ) is isomorphic to the reduced Drinfeld double of H(n)0 and H (n)0 ; see Remark 1.5.5(2). This subalgebra will be considered in §5.5 regarding a compatibility condition on the transfer maps. m ) can be naturally viewed as a subalgebra (4) For each m 1, U(C (m) of U(C∞ ). If we let D(n) be the subalgebra of D(n) generated by − for i ∈ I and s ∈ J , then the homomorphism E i , Fi , K i , K i−1 , z+ , z m s s in Proposition 2.3.4 induces a surjective Hopf algebra homomorphism m ) → D(n)(m) . Applying Proposition 1.4.3 shows that D(n)(m) is also U(C − generated by E i , Fi , K i , K i−1 , u + sδ , u sδ , for i ∈ I and s ∈ Jm . +

2.4. Some integral forms We introduce an integral form for the double Ringel–Hall algebra D(n) which is similar to those considered in [49, 42]. This form will be used in §6.1. See Remark 3.8.7(1) for a comparison with another possible form of Lusztig ∞ ) about the type. First, we observe the following commutator formula in U(C generators xm and ym . Proposition 2.4.1. For m, r, s 1, [xrm , ysm ]

=

min{r,s} ! i =1

i r " !s " km − k−1 m r−i i! ys−i m xm . i i v − v −1

(2.4.1.1)

46

2. Double Ringel–Hall algebras of cyclic quivers k −k−1

m m Proof. Since [xm , ym ] = v−v −1 = direct to check that, for s 1,

km ;0 1

± and [xm , k± m ] = 0 = [ym , km ], it is

km − k−1 m s−1 y . v − v −1 m We proceed by induction on r to prove (2.4.1.1). The case r = 1 is given as above. Now let r 1 and suppose (2.4.1.1) holds for r . If r < s, then [xm , ysm ] = s

[xrm+1 , ysm ] = xm [xrm , ysm ] + [xm , ysm ]xrm i r ! "! " r s km − k−1 km − k−1 m m s−1 r s−i r−i = i! x y x + s y x m m m i i v − v −1 v − v −1 m m i=1 i r ! "! " r−i r s km − k−1 km − k−1 m m s−i −1 = i! (s − i) ym + ys−i m xm xm −1 −1 i i v−v v−v i=1

km − k−1 m s−1 r y x v − v −1 m m ! " i r+1 r s km − k−1 m r+1−i = i! ys−i m xm i −1 i v − v −1 i=2 i r ! "! " r s km − k−1 km − k−1 m m s−1 r s−i r+1−i + i! y x + s y x m m i i v − v −1 v − v −1 m m +s

i=1

=

r+1 i=1

i r + 1 !s " km − k−1 m r +1−i i! ys−i . m xm i i v − v −1

The case r s can be treated similarly.

=U Z of U(C ∞ ) generated by K ±1 , K i ;0 , Consider the Z-subalgebra U i t (m) (m) k±1 = E im /[m]! and Fi = s , xs , ys together with the divided powers E i m ! Fi /[m] , where i ∈ I , t 0, and s, m 1. Further, we set + = , U − = , and U 0 = . U U+ ∩ U U− ∩ U U0 ∩ U (m)

+ (resp., U − ) is the Z-subalgebra of U(C ∞ ) generated by xs and E Then U i (m) (resp., ys and Fi ), for i ∈ I and s, m 1. Consequently, we obtain + = U ( − = U ( U sln )+ ⊗ Z[x1 , x2 , . . .] and U sln )− ⊗ Z[y1 , y2 , . . .],

where U ( sln )+ and U ( sln )− are the Z-subalgebras of U( sln ) generated by (m) (m) the divided powers E i and Fi , respectively. This implies in particular that + and U − are free Z-modules. both U 0 0 We now look at the structure ks ;0 of U . Let V be the Z-subalgebra generated K i ;0 ±1 ±1 by K i , , ks , 1 , for i ∈ I , t 0, and s 1. We first prove t

2.4. Some integral forms

47

that V 0 is Z-free. First, a result of Lusztig [52, 2.14] (see also [22] and [12, Th. 14.20]) shows that V 0 contains the Z-subalgebra V 1 generated by K i±1 and K it ;0 which has a Z-basis $ % K i ; 0 $$ a ∈ {0, 1}, t ∈ N, ∀i ∈ I , or i $ i ti i=1 #n−1 $ %

a K i ; 0 an K n ; 0 $$ i K = K · K a ∈ {0, 1}, t ∈ N, ∀i ∈ I . i i i $ i ti tn i=1 (2.4.1.2) ks ;0 Let V 2 be the Z-subalgebra of V 0 generated by k±1 , , for s 1, and let s 1 ks ; 0 ks ; 0 = [t]! . t t K=

#

n

a

Ki i

Then the formula [52, 2.3(g8)] with t = 1 becomes ks ; 0 ks ; 0 ks ; 0 t ks ; 0 v − [t]ks = . t 1 t t +1 Lemma 2.4.2. The set # k1 ; 0 km ; 0 δ M := k11 · · · kδmm ··· t1 tm

(2.4.1.3)

$ % $ $ m 1, δi ∈ {0, 1}, ti ∈ N $

forms a Z-basis of V 2 . Proof. Let W1 = spanZ M and # $ % km ; 0 $$ δ1 δm k1 ; 0 W2 = spanZ k1 · · · km ··· $ m 1, δi ∈ Z, ti ∈ N . t1 tm By (2.4.1.3), kst;0 ∈ V 2 , for s 1 and t 0. Thus W1 ⊆ W2 ⊆ V 2 . Clearly, ks ;0 2 k±1 s W2 ⊆ W2 . By (2.4.1.3) again, 1 W2 ⊆ W2 . Thus, V W2 ⊆ W2 . Since 2 2 1 ∈ W2 , V ⊆ W2 and, hence, V = W2 . For m 0, applying Lusztig’s formulas ks ; 0 t −1 m+1 ks ; 0 2t m ks ; 0 km+2 = v (v − v )k + v k and s s s t t +1 t ks ; 0 −t −1 −m ks ; 0 −2t −m+1 ks ; 0 k−m−1 = −v (v − v )k + v k s s s t t +1 t ks ;0 in [52, p. 278] yields k±m ∈ W1 , for any m 0. Hence, W2 ⊆ W1 and, s t therefore, W2 = W1 = V 2 .

48

2. Double Ringel–Hall algebras of cyclic quivers

0 is the Z-subalgebra of U(C ∞ ) genTheorem 2.4.3. The integral 0-part U K i ;0 ks ;0 ±1 ±1 0 is a free erated by K i , t , ks , 1 , for i ∈ I and s 1. Hence, U Z-module with basis {x y | x ∈ K, y ∈ M}. + V 0 U − . This is clearly a subset of U . Since, by (2.4.1.1), Proof. Let U = U 3 , and consequently, U is a subalgebra, it follows that U ⊆ U . Hence, U = U 0 0 0 = U ∩ U = V . The rest of the proof is clear. U We now look at an integral form of D(n). Definition 2.4.4. The integral form D(n) of D(n) is the Z-subalgebra gen + (m) − (m) , for i ∈ I and s, t, erated by K i±1 , K it ;0 , z+ and (u − s , zs , (u i ) i ) m 1. It is easy to see that the surjective Q(v)-algebra homomorphism ∞ ) −→ D(n) : U(C given in Proposition 2.3.4 induces a surjective Z-algebra homomorphism → D(n). We also set :U D(n) = D(n) ∩ D(n) for ∈ {+, −, 0}. + (resp., U − ). Hence, Then D(n)+ (resp., D(n)− ) is isomorphic to U + D(n)+ = C(n)+ ⊗Z Z[z+ 1 , z2 , . . .] and − D(n)− = C(n)− ⊗Z Z[z− 1 , z2 , . . .],

where C(n)+ ∼ sln )+ (resp., C(n)− ∼ sln )− ) is the Z-subalgebra of = U ( = U ( + (m) − (m) D(n) generated by the (u i ) (resp., (u i ) ). Moreover, as seen above, D(n)0 is a free Z-module with bases as given in (2.4.1.2). Hence, the multiplication map ∼

D(n)+ ⊗Z D(n)0 ⊗Z D(n)− −→ D(n)

(2.4.4.1)

is a Z-module isomorphism; see [12, Cor. 6.50]. In particular, D(n) is a free Z-module. The Z-algebra D(n) gives rise to a Z-form U(n) for U(n), the extended quantum affine sln : U(n) = D(n) ∩ U(n) = C(n)+ D(n)0 C(n)− .

(2.4.4.2)

We end this section with a description of the double Ringel–Hall algebras D,C (n) given in Remark 2.1.4 in terms of specialization. ks ;0 3 Of course, this fact can be proved directly by the relation [x , y ] = δ s t s,t 1 .

n ) 2.5. The quantum loop algebra U(gl

49

Proposition 2.4.5. If z ∈ C is a complex number which is not a root of unity, then the specialization D(n)C = D(n) ⊗ C at v = z is isomorphic to the C-algebra D,C (n). − Proof. If we assign to each u + A (resp., u A , K i ) degree d(A) = dim M(A) (resp., −d(A), 0), then D(n) admits a Z-grading D(n) = ⊕m∈Z D(n)m , and D(n) inherits a Z-grading D(n) = ⊕m∈Z D(n)m , where D(n)m = D(n) ∩ D(n)m . By the presentation of D,C (n) as given in Remark 2.1.4, it is clear that there is a graded algebra homomorphism D,C (n) → D(n)C . This map is injective since it is so on every triangular component. Now, a dimensional comparison on homogeneous components shows that it is an isomorphism.

Remark 2.4.6. Let H(n)+ be the Z-submodule of D(n) spanned by u + A for A ∈ + (n). Then H(n)+ is a Z-algebra which is isomorphic to H(n). We point out that in general, D(n)+ does not coincide with H(n)+ . For example, let n = 2. An easy calculation shows that + + + + −1 + z+ 1 = u 1 u 2 + u 2 u 1 − (v + v )u δ . + + + + + Since {u + 1 u 2 , u 2 u 1 , z1 } is a Z-basis for the homogeneous space D(2)δ , we + + conclude that u δ ∈ D(2) . If n = 3, then + + + + + + + + + z+ 1 =u 1 u 2 u 3 + u 2 u 3 u 1 + u 3 u 1 u 2 + + + + + + + + 2 −2 + − (v + v −1 )(u + 1 u 3 u 2 + u 2 u 1 u 3 + u 3 u 2 u 1 ) + (v + 1 + v )u δ . + Analogously, we have u + δ ∈ D(3) . + It seems difficult to prove directly that all central elements z+ m lie in H(n) . However, we will provide a proof of this fact in Corollary 3.7.5 via the integral quantum Schur algebras. Hence, D(n)+ is indeed a subalgebra of H(n)+ .

n ) 2.5. The quantum loop algebra U(gl In this section, we give an application of the presentation for D(n) defined in §2.3. More precisely, we will modify the construction in [40] to extend Beck’s algebra embedding of the quantum affine sln into the quantum loop algebra n ) defined by Drinfeld’s new presentation [20] to an explicit isomorphism U(gl n ). from D(n) to U(gl As discussed in §1.1, consider the loop algebra n = gln ⊗ Q[t, t −1 ], gl

50

2. Double Ringel–Hall algebras of cyclic quivers

which is generated by E i,i+1 ⊗ t s , E i +1,i ⊗ t s , and E j, j ⊗ t s , for s ∈ Z, 1 i n − 1, and 1 j n. We have the following quantum enveloping n ; see [20] (and also [28, 40]). algebra associated with gl n ) (or quantum affine Definition 2.5.1. (1) The quantum loop algebra U(gl ± gln ) is the Q(v)-algebra generated by xi,s (1 i < n, s ∈ Z), ki±1 and gi,t (1 i n, t ∈ Z\{0}) with the following relations: (QLA1) ki ki−1 = 1 = k−1 i ki , [ki , k j ] = 0; (QLA2) ki x±j,s = v ±(δi, j −δi, j+1 ) x±j,s ki , [ki , g j,s ] = 0; ⎧ ⎪0, if i = j, j + 1; ⎪ ⎨ ± [s] ± − js (QLA3) [gi,s , x j,t ] = ±v if i = j; s x j,s+t , ⎪ ⎪ ⎩∓v − js [s] x± , if i = j + 1; (QLA4) [gi,s , g j,t ] = 0;

s

φ+

j,s+t

−φ −

− i,s+t i,s+t (QLA5) [x+ ; i,s , x j,t ] = δi, j v−v −1 ± ± ± ± ± (QLA6) xi,s x j,t = x j,t xi,s , for |i − j| > 1, and [x± i,s+1 , x j,t ]v ±ci j = ± −[x±j,t+1 , xi,s ]v ±ci j ; ± ± ± ± ± (QLA7) [xi,s , [x j,t , x± i, p ]v ]v = −[xi, p , [x j,t , xi,s ]v ]v for |i − j | = 1, ± where [x, y]a = x y − ayx, and φi,s are defined via the generating functions in indeterminate u by ± ±s i± (u) := ki±1 exp ±(v − v −1 ) hi,±m u ±m = φi,±s u m 1

s 0

with ki = ki /ki +1 (kn+1 = k1 ) and hi,±m = v ±(i−1)m gi,±m − ±(i+1)m v gi+1,±m (1 i < n). n ) be the quantum loop algebra defined by the same generators (2) Let UC (gl and relations (QLA1)–(QLA7) with Q(v) replaced by C and v by z ∈ C∗ with z m = 1 for all m 1. Observe that, if we put i± (u) = ki±1 exp ±(v − v −1 ) m 1 gi,±m u ±m , then ± (uv i−1 ) i± (u) = ±i . i+1 (uv i+1 ) + − Also, φi,−s = φi,s = 0, for all s 1. Hence, (QLA5) becomes ⎧ ki − ki− ⎪ ⎪ v−v , if s + t = 0; ⎪ ⎨ + −1 φi,s+t + − [xi,s , xi,t ] = , if s + t > 0; v−v −1 ⎪ ⎪ − ⎪ −φ ⎩ i,s+t , if s + t < 0. v−v −1

n ) 2.5. The quantum loop algebra U(gl

51

By [2] (see also [44, Th. 2.2], [3, Th. 1], [28, §3.1], and [40, Th. 3]), there is a monomorphism (or Beck’s embedding) of Hopf algebras EB : U( sln ) → n ) such that U(gl + + − i −→ K ki , E i −→ xi,0 , Fi −→ x− i,0 (1 i < n), E n −→ εn , Fn −→ εn ,

where − − − εn+ = (−1)n [x− n−1,0 , · · · , [x3,0 , [x2,0 , x1,1 ]v −1 ]v −1 · · · ]v −1 kn and + + + + εn− = (−1)n k−1 n [· · · [[x1,−1 , x2,0 ]v , x3,0 ]v , · · · , xn−1,0 ]v .

The image Im(EB ), known as the quantum loop algebra of sln , is the Q(v)± n ) generated by subalgebra of U(gl ki±1 , xi,s , and hi,t for 1 i < n, s, t ∈ Z with t = 0. Moreover, it can be proved (see, e.g., [28, §§2.1&3.1]) that the defining relations of this subalgebra are the relations (QLA5)–(QLA7) together with the relations: (QLA8) ki kj = k j ki , k1 k2 · · · kn = 1, [ ki , h j,s ] = 0, [hi,s , h j,t ] = 0; [s c ] ± ± ± ±c i, j (QLA9) ki x j,s = v x j,s ki , [hi,s , x j,t ] = ± si, j x±j,s+t , n−1 as given in (1.3.2.1). where (ci, j ) is the generalized Cartan matrix of type A This is Drinfeld’s new presentation for U(sln ). In the sequel, we will identify n ). U( sln ) with Im(EB ) and, thus, view U( sln ) as a subalgebra of U(gl In an entirely similar way we can identify UC ( sln ) defined in §1.3 with a n ) under the monomorphism EB,C .4 subalgebra of UC (gl Our next aim is to extend Beck’s embedding EB to a Hopf algebra isomorn ). For each s 1, define phism D(n) → U(gl θ±s = ∓

1 n ). (g1,±s + · · · + gn,±s ) ∈ U(gl [s]

(2.5.1.1)

n ). By It can be directly checked by the definition that the θs are central in U(gl [28, §3.3], they are all primitive, i.e., for all s ∈ Z\{0}, (θs ) = θs ⊗ 1 + 1 ⊗ θs . Remark 2.5.2. In [28, (4.6)], the authors introduced central elements Cs (s ∈ n ) which are defined by the generating function Z) in U(gl s 0

C±s u ±s =

n

i=1

gi,±m ±m exp ∓ u , [m] m>0

± 4 Let φ be the automorphism of U (gl s ± t a C n ) of the form xi,s → a xi,s , g j,t → a g j,t , and k j → k j . Then, for a = z n , φa ◦ EB,C is the isomorphism B−1 given in [28, Lem. 3.3].

52

2. Double Ringel–Hall algebras of cyclic quivers

that is,

C ±s u ±s = exp θ±m u ±m .

s 0

m>0

It follows that C0 = 1, C±1 = θ±1 and, for each s 1, Q(v)[θ±1 , . . . , θ±s ] = Q(v)[C±1 , . . . , C±s ] and θ±(s+1) ≡ C±(s+1) mod Q(v)[C±1 , . . . , C ±s ]. Now fix an integer s 1 and define n × n matrices X (±s) over Q(v) by ⎛ ⎞ 1 −v ±2s 0 ··· 0 0 ⎜ 0 v ±s −v ±3s · · · 0 0 ⎟ ⎜ ⎟ ⎜ 0 ±2s 0 v ··· 0 0 ⎟ ⎜ ⎟ (±s) X (±s) = (X i, j ) = ⎜ .. .. .. .. ⎟ .. ⎜ .. ⎟. . . . . . ⎟ ⎜ . ⎜ ⎟ ⎝ 0 0 0 · · · v ±(n−2)s −v ±ns ⎠ 1 1 1 1 1 ∓ [s] ∓ [s] ∓ [s] ··· ∓ [s] ∓ [s] By the definition of hi,±s and θ±s , hi,±s =

n

(±s)

X i, j g j,±s , for 1 i < n, and θ±s =

j =1

n

(±s)

X n, j g j,±s .

j =1

A direct calculation shows that det(X (±s) ) = ∓

±is 1 1 + v ±2s + · · · + v ±2(n−1)s v = 0 (n 2). [s] n−2 i=1

(±s)

We denote the inverse of X (±s) by Y (±s) = (Yi, j ). Thus, for each 1 i n, gi,±s =

n−1

(±s)

(±s)

Yi, j h j,±s + Yi,n θ±s .

j =1

n ) spanned by g1,±s , . . . , gn,±s coinTherefore, the Q(v)-subspace of U(gl n ) is cides with that spanned by h1,±s , . . . , hn−1,±s , θ±s . Consequently, U(gl also generated by ± k±1 i (1 i n), xi,t , hi,±s , θ±s (1 i < n, t ∈ Z, s 1).

n ) can be generated by Applying Beck’s embedding EB shows that U(gl ± ± k±1 i (1 i n), x j,0 (1 j < n), εn , θ±s (s 1).

Moreover, these generators satisfy relations similar to (QGL1)–(QGL8) in ±1 ± ± ± Theorem 2.3.1 in which K i±1 , u ±j , u ± n , zs are replaced by ki , x j,0 , εn , θ±s ,

n ) 2.5. The quantum loop algebra U(gl

53

respectively. Thus, we conclude that there is a surjective Q(v)-algebra homomorphism n ), EH : D(n) −→ U(gl ± ± K i±1 −→ ki±1 , u ±j −→ x±j,0 , u ± n −→ εn , zs −→ θ±s ,

(2.5.2.1)

for 1 i n, 1 j < n, and s 1. It is clear that EH is an extension of EB . The fact that the elements x±j,t and h j,±s (1 i < n, t ∈ Z, s 1) lie in Im(EB ) gives rise to the elements x±j,t = (EB )−1 (x±j,t ) and h j,±s = (EB )−1 (h j,±s ) in D(n) via the induced isomorphism EB : U( sln ) → Im(EB ). By Remark 2.3.6(2), D(n) is generated by ki±1 := K i±1 , x±j,t , h j,±s , z± s (1 i n, 1 j < n, t ∈ Z, s 1). Furthermore, for 1 i n and s 1, we define the elements gi,±s =

n−1

(±s) (±s) Yi, j h j,±s + Yi,n z± s ∈ D(n),

(2.5.2.2)

j =1

or equivalently, hi,±s = v ±(i −1)s gi,±s − v ±(i+1)s gi+1,±s (1 i < n, s 1) and 1 z± ( g1,±s + · · · + gn,±s ). s =∓ [s] This implies in particular that the set X := { k±1 x±j,t , gi,±s | 1 i n, 1 j < n, t ∈ Z, s 1} i , is also a generating set for D(n). Clearly, the x±j,t satisfy the relations (QLA5)–(QLA7). Since hi,±s together ±1 ± with k and x satisfy the relations (QLA8)–(QLA9) and the z± s are central i

j,t

elements, it follows from (2.5.2.2) that the gi,±s together with k±1 x±j,t sati and isfy the relations (QLA2)–(QLA4). In conclusion, the generators in X satisfy all the relations (QLA1)–(QLA7). Therefore, there is a surjective Q(v)-algebra homomorphism n ) −→ D(n) F : U(gl taking ki±1 → k±1 gi,±s , x±j,t → x±j,t . Obviously, both the i , gi,±s → composites EH F and FEH are the identity maps. This gives the following result.

54

2. Double Ringel–Hall algebras of cyclic quivers

Theorem 2.5.3. The surjective algebra homomorphism EH : D(n) → n ) given in (2.5.2.1) is a Hopf algebra isomorphism. In particular, U(gl n ). Theorem 2.3.1 gives another presentation for U(gl Proof. Clearly, EH is an algebra isomorphism. Since EB is a Hopf algebra embedding and the elements z± s and θ±s are primitive, it follows that EH is a bialgebra isomorphism. It is well known that if a bialgebra admits an antipode, then the antipode is unique; see, for example, [72, p.71]. This forces EH to be a Hopf algebra isomorphism. n )+ (resp., U(gl n )− ) be the subalgebra of U(gl n ) generated by Let U(gl − + − n )0 be εn , θs (resp., xi,0 , εn , θ−s ), for 1 i < n, s 1. Also, let U(gl n ) generated by the k±1 . The triangular decomposition the subalgebra of U(gl i n ). of D(n) given in (2.3.6.1) induces that of U(gl + xi,0 ,

Corollary 2.5.4. The multiplication map n )+ ⊗ U(gl n )0 ⊗ U(gl n )− −→ U(gl n ) U(gl is a Q(v)-space isomorphism. Remarks 2.5.5. (1) In [28], the authors introduced the Q(v)-subalgebra C of n ) generated by the central elements C±s as defined in Remark (2.5.2) and U(gl n ); see [28, (2.12)]. Indeed, considered the embedding of U( sln ) ⊗ C → U(gl under the isomorphism EH , C is identified with the central subalgebra Z(n) of D(n). n )+ ⊗ U(gl n )0 (2) In [40] a Hopf algebra isomorphism H(n)0 → U(gl −1 was established and, moreover, the elements EH−1 (x+j,−1 k−1 j ) and EH (gi,±s ) in D(n) were explicitly described. (3) The proof above can be easily modified to construct a C-algebra ison ), where the algebras are defined over morphism EH,C : D,C (n) → UC (gl C with respect to a non-root-of-unity z ∈ C∗ . This isomorphism will be used n ) with those of S(n, r )C in Chapter 4, in linking representations of UC (gl §§4.4–4.6. (4) With the above isomorphism, the double Ringel–Hall algebras D(n), D,C (n) will also be called a quantum affine gln . Moreover, the notation n ), UC (gl n ) throughout the book D(n), D,C (n) will not be changed to U(gl in order to emphasize the approach used in this book. (5) There is another geometric realization for the +-part of the quantum n ) in terms of the Hall algebra of a Serre subcategory of the loop algebra U(gl category of coherent sheaves over a weighted projective line; see [68, 4.3, 5.2].

2.6. Semisimple generators and commutator formulas

55

2.6. Semisimple generators and commutator formulas By [13, Th. 5.2], the Ringel–Hall algebra H(n) can be generated by semisimple modules over Z. Thus, semisimple generators would be crucial to the study of integral forms of D(n). In this section we derive commutator formulas between semisimple generators of D(n) (cf. [78, Prop. 5.5]). Recall from §1.2 that for the module M(A) associated with A ∈ + (n), we write d(A) = dim M(A) for the dimension vector of M(A). Define a subset + (n)ss of + (n) by setting + (n)ss = { A = (ai, j ) ∈ + (n) | ai, j = 0 for all j = i + 1}. In other words, A ∈ + (n)ss ⇐⇒ M(A) is semisimple. Then, by Propo− sition 1.4.3, D(n)+ (resp., D(n)− ) is generated by u + A (resp., u A ) for + + n ss ss all A ∈ (n) . We sometimes identify (n) with N via the map Nn → + (n)ss sending λ to A = Aλ with λi = ai,i+1 for all i ∈ Z. + ss − := span{u − | Lemma 2.6.1. Let X + := span{u + A | A ∈ (n) } and X A A ∈ + (n)ss }. Then

D(n) ∼ = H(n)0 ∗ H(n)0 /J , where J is the ideal of the free product H(n)0 ∗ H(n)0 generated by (1) (b2 ∗ a2 )ψ(a1 , b1 ) − (a1 ∗ b1 )ψ(a2 , b2 ) for all a ∈ X + , b ∈ X − , and (2) K α ∗ 1 − 1 ∗ K α for all α ∈ ZI, where (a) = a1 ⊗ a2 and (b) = b1 ⊗ b2 . Proof. For a ∈ H(n)0 and b ∈ H(n)0 , we write L(a, b) = (b2 ∗ a2 )ψ(a1 , b1 ) and R(a, b) = (a1 ∗ b1 )ψ(a2 , b2 ), where, for x = a, b, (x) = x1 ⊗ x2 . Define + ss X 0 := span{u + A K α | α ∈ ZI, A ∈ (n) } and + ss X 0 := span{K α u − A | α ∈ ZI, A ∈ (n) }.

Then X 0 (resp., X 0 ) generates H(n)0 (resp., H(n)0 ) and satisfies (X 0 ) ⊆ X 0 ⊗ X 0 (resp., (X 0 ) ⊆ X 0 ⊗ X 0 ). Thus, by Lemma 2.1.1, D(n) = H(n)0 ∗ H(n)0 / I, where I is the ideal of H(n)0 ∗ H(n)0 generated by

56

2. Double Ringel–Hall algebras of cyclic quivers

(1 ) L(a, b) − R(a, b) (a ∈ X 0 , b ∈ X 0 ), (2 ) K α ∗ 1 − 1 ∗ K α (α ∈ ZI ). Clearly, J ⊆ I. To show the reverse inclusion I ⊆ J , it suffices to prove that, + 0 , for u A K α ∈ X 0 and K β u − ∈ X B − + − L(u + A K α , K β u B ) ≡ R(u A K α , K β u B ) mod J .

By the definition of comultiplications in H(n)0 and H(n)0 , + (u + ) = f AA1 , A2 u + A A2 ⊗ u A1 K d( A2 ) and A1 ,A2 ∈+ (n)

(u − B) =

B1 ,B2 ∈+ (n)

−d(B1 ) u − ⊗ u − , g BB1 ,B2 K B2 B1

aA1 aA2 A aA ϕ A1 , A2

where f AA1 , A2 = v d( A1 ),d( A2 ) ϕ BB1 ,B2 . By the definition of J ,

and g BB1 ,B2 = v −d(B2 ),d(B1 )

− + − L(u + A , u B ) ≡ R(u A , u B ) mod J ,

where − L(u + A, uB) =

A1 , A2 ,B1 ,B2

− R(u + A, uB)=

A1 ,A2 ,B1 ,B2

(2.6.1.1)

+ f AA1 A2 g BB1 B2 u − B1 ∗ (u A1 K d(A2 ) ) − × ψ(u + and A2 , K −d(B1 ) u B2 ) − + − f AA1 A2 g BB1 B2 u + A2 ∗ ( K −d(B1 ) u B2 ) ψ(u A1 K d( A2 ) , u B1 ).

This together with Proposition 2.1.3 implies that − L(u + A Kα , Kβ u B ) + = f AA1 A2 g BB1 B2 (K β u − B1 ) ∗ (u A1 K d(A2 ) K α ) A1 ,A2 ,B1 ,B2

− × ψ(u + A K α , K β K −d(B1 ) u B2 ) 2 + ≡ v a K α+β f AA1 A2 g BB1 B2 u − B1 ∗ (u A1 K d( A2 ) ) A1 ,A2 ,B1 ,B2

≡ v a K α+β

aB1 aB2 aB

− × ψ(u + (by 2.6.1(2)) A2 , K −d(B1 ) u B2 ) − f AA1 A2 g BB1 B2 u + A2 ∗ ( K −d(B1 ) u B2 )

A1 ,A2 ,B1 ,B2

− × ψ(u + A1 K d( A2 ) , u B1 )

(by (2.6.1.1))

2.6. Semisimple generators and commutator formulas ≡

A1 , A2 ,B1 ,B2

57

− f AA1 A2 g BB1 B2 (u + A2 K α ) ∗ (K β K −d(B1 ) u B2 )

− × ψ(u + A1 K d( A2 ) K α , K β u B1 ) − = R(u + A K α , K β u B ) mod J ,

where a = α β + d(A1 ) + d(A2 ), α = α β + d( A), α, as desired. Recall the order relation on Zn defined in (1.1.0.3). Lemma 2.6.2. For α = (αi ), β = (βi ) ∈ Nn , let γ = γ (α, β) = (γi ) ∈ Nn be defined by γi = min{αi , βi+1 }. For each λ γ , define Cλ ∈ + (n) by M(Cλ ) = ((αi + βi − λi − λi−1 )Si ⊕ λi Si [2]). i∈I

Then uα uβ = v

i

αi (βi −βi +1 )

αi + βi − λi − λi−1 u Cλ . βi − λi−1

λγ i∈I

Proof. Clearly, each M(Cλ ) with λ γ is an extension of M(Aβ ) by M(Aα ), n n where Aα = i =1 αi E i,i+1 and Aβ = i =1 βi E i,i +1 . Conversely, each extension of M(Aβ ) by M(Aα ) is isomorphic to M(C λ ) for some λ γ . Hence, C C u α u β = v α,β ϕ Aαλ ,Aβ u Cλ = v i αi (βi −βi+1 ) ϕ Aαλ , Aβ u Cλ . λγ

λγ

The lemma then follows from the fact that

αi + βi − λi − λi−1 ϕ CAαλ ,Aβ = . βi − λi −1 i∈I

− Theorem 2.6.3. The algebra D(n) has generators u + A , K ν , u A (A ∈ + ss (n) , ν ∈ ZI ) which satisfy the following relations: for ν, ν ∈ ZI , A, B ∈ + (n)ss , − (1) K 0 = u + K ν K ν = K ν+ν ; 0 = u 0 = 1, + d( A),ν (2) K ν u A = v u + K , u − K = v d( A),ν K ν u − ν A A; A ν αi +βi −λ + + α (β −β ) i −λi −1 (3) u A u B = v i i i i +1 u Cλ , if A = Aα , λγ i ∈I βi −λi−1 B = Bβ , andγ = γ (α, β); αi +βi −λi −λi −1 − i βi (αi −αi +1 ) (4) u − u Cλ , if A = Aα , λγ i∈I AuB = v αi −λi −1 B = Bβ , and γ = γ (β, α);

58

2. Double Ringel–Hall algebras of cyclic quivers

(5) commutator relations: for all A, B ∈ + (n), v d(B),d(B)

A1 ,B1

A1 ,B1 d(B1 ),d(A)+d(B)−d(B1 ) + ϕ A,B v K d(B)−d(B1 ) u − B1 u A1

= v d(B),d( A)

A1 ,B1 d(B)−d(B1 ),d( A1 )+d(B),d(B1 ) ϕ A,B v A1 ,B1

d(B1 )−d(B) u + u − , ×K A1 B1 where A1 ,B1 ϕ A,B =

a A1 a B1 a AaB

a A aB A1 ,B1 ϕ A,B = 1 1 a AaB

A2 ∈+ (n)

A2 ∈+ (n)

v 2d( A2 ) a A2 ϕ AA1 ,A2 ϕ BB1 , A2 and (2.6.3.1)

v 2d( A2 ) a A2 ϕ AA2 ,A1 ϕ AB2 ,B1 .

Proof. Relations (1) and (2) follow from the definition, and (3) and (4) follow from Lemma 2.6.2. We now prove (5). As in the proof of Lemma 2.6.1, for A, B ∈ + (n), − L(u + A, uB) =

v d(A1 ),d(A2 )

A1 ,A2 ,B1 ,B2

=

a A1 a A2 aA

ϕ AA1 ,A2 v −d(B2 ),d(B1 )

a B1 a B2 aB

ϕ BB1 ,B2

+ + − × u− B1 ∗ (u A1 K d(A2 ) ) ψ(u A2 , K −d(B1 ) u B2 ) v d(A1 ),d(A2 )

A1 ,A2 ,B1 ,B2

a A1 a A2 aA

ϕ AA1 ,A2 v −d(B2 ),d(B1 )

a B1 a B2 aB

ϕ BB1 ,B2

−d(A ),d(A )+2d(A ) 1 + 2 2 2 × u− δ A2 ,B2 . B1 ∗ (u A1 K d(A2 ) ) v a A2

Hence, L(uA+ ,uB− ) ≡

v −d( A2 ),d(A)+d(B1 ),d( A2 )+2d( A2 )

A1 ,B1 , A2

a A1 a B1 a A 2 aA aB

d( A2 ) u − ) ∗ u + × (K B A1 1 −d(B),d( A) d(B1 ),d( A)+d(B)−d(B1 ) A1 ,B1 ≡v v ϕ A,B A1 ,B1

d(B)−d(B1 ) u − ∗ u + mod J . ×K B1 A1

ϕ AA1 ,A2 ϕ BB1 , A2

2.6. Semisimple generators and commutator formulas

59

By interchanging the running indices A1 and A2 , B1 and B2 ,

− R(u + A, uB) =

v d(A1 ),d(A2 )

a A1 a A 2 aA

A1 ,A2 ,B1 ,B2

≡

ϕ AA1 ,A2 v −d(B2 ),d(B1 )

a B1 a B2 aB

ϕ BB1 ,B2

− + − × u+ A2 ∗ ( K −d(B1 ) u B2 ) ψ(u A1 K d(A2 ) , u B1 ) v d(A1 ),d(A2 )−d(B),d(A1 )+2d(A1 )

A1 ,A2 ,B2

a A1 a A2 a B2 aA aB

ϕ AA1 ,A2 ϕ AB1 ,B2

−d(A1 ) u + ∗ u − ×K A2 B2 A1 ,B1 −d(B),d(B) d(B)−d(B1 ),d(A1 )+d(B),d(B1 ) ≡v v ϕ A,B A1 ,B1

−d(B)+d(B1 ) u + ∗ u − mod J . ×K A1 B1

This proves (5). Theorem 2.6.3 does not give a presentation for D(n) since the modules M(Cλ ) are not necessarily semisimple. It would be natural to raise the following question. Problem 2.6.4. Find the “quantum Serre relations” associated with semisimple generators to replace relations in Theorem 2.6.3(3)–(4), and prove that the relations given in Theorem 2.6.3 are defining relations for D(n). In this way, we obtain a presentation with semisimple generators for D(n). The relations in (5) above are usually called the commutator relations. We now derive a finer version of the commutator relations for semisimple generators. The next lemma follows directly from the definition of comultiplication in §1.5. Lemma 2.6.5. For A ∈ + (n), we have (2) (u + A) =

v

i > j d(A

A(1) , A(2) ,A(3) × u+ A(1)

(i ) ),d( A( j) )

ϕ AA(3) , A(2) , A(1)

a A(1) a A(2) a A(3) aA

(1) ⊗ u +(3) K d( A(1) )+d(A(2) ) ⊗ u+ K A(2) d( A ) A

and (2) (u − A) =

A(1) , A(2) , A(3)

v−

i < j d(A

(i) ),d( A( j) )

ϕ AA(3) , A(2) ,A(1)

a A(1) a A(2) a A(3) aA

−(d( A(2) )+d( A(3) )) u −(1) ⊗ K −d( A(3) ) u −(2) ⊗ u −(3) . ×K A A A

60

2. Double Ringel–Hall algebras of cyclic quivers

Proposition 2.6.6. Let X, Y ∈ + (n). Then, in D(n), + + − u− Y u X − u X uY a Aa B a B − X = v 1 ϕ A,B ϕ YA,B K −d( A) u + B u B a a X Y + A,B,B ∈ (n) A =0

+

v

2

+ (n) A,B,B ,C,C ∈ C =0 =C

v2

i< j d(X i ),d(X j )

a A a B aB X ϕ A,B,C ϕ YA,B ,C a X aY

(−1)m

m1 + (n)∗ X 1 ,...,X m ∈

d(C)−d( A) u + u − , a X 1 · · · a X m ϕ CX1 ,...,X m ϕ CX 1 ,...,X m K B B

where 1 = d(A), d(B) − d(Y ), d(A) + 2d(A) and 2 = d(A), d(B)+d(Y ), d(C)−d( A)−d(C), 2d(C)+d(B)+2d( A)+2d(C).

Proof. By [46, Lem. 3.2.2(iii)], for x ∈ H(n)0 and y ∈ H(n)0 , we have in D(n), yx = ψ(x1 , σ (y1 ))(x2 y2 )ψ(x3 , y3 ), where (2) (x) = x1 ⊗ x2 ⊗ x3 and (2) (y) = y1 ⊗ y2 ⊗ y3 . This together with Lemma 2.6.5 gives the required equality. The following result is a direct consequence of the above proposition (1) (m) together with the fact that for β = (βi ), β (1) = (βi ), . . . , β (m) = (βi ) n ∈ N, β ϕβ (1) ,...,β (m)

where

=

n

&& (1)

'' (m)

βi , . . . , βi

i =1

βi (1) (m) βi ,...,βi

βi

=

[[βi ]]! (1) (m) [[βi ]]! ...[[βi ]]!

=:

β (1) β , . . . , β (m)

,

and β = β (1) + · · · + β (m) .

Corollary 2.6.7. For λ, μ ∈ Nn , we have + u− μ uλ

− − u+ λ uμ

=

n α =0, α∈N αλ, αμ

0γ α

− xα,γ K 2γ −α u + λ−α u μ−α ,

(2.6.6.1)

2.6. Semisimple generators and commutator formulas

61

where xα,γ =v α,λ−α+μ,2γ −α+2γ ,α−γ −λ+2σ (α) aα−γ aλ−α aμ−α λ μ × · aλ aμ α − γ , λ − α, γ α − γ , μ − α, γ 2 (i) ( j) γ × (−1)m v 2 i< j γ ,γ aγ (1) · · · aγ (m) . γ (1) , . . . , γ (m) (i) m 1,γ =0, ∀i γ (1) +···+γ (m) =γ

This is the commutator formula for semisimple generators.

3 Affine quantum Schur algebras and the Schur–Weyl reciprocity

Like the quantum Schur algebra, the affine quantum Schur algebra has several equivalent definitions. We first present the geometric definition, given by Ginzburg–Vasserot and Lusztig, which uses cyclic flags and the convolution product. We then discuss the two Hecke algebra definitions given by R. Green and by Varagnolo–Vasserot. The former uses q-permutation modules, while the latter uses tensor spaces. Both versions are related by the Bernstein presentation for Hecke algebras of affine type A. In §3.4, we review the construction of BLM type bases for affine quantum Schur algebras and the multiplication formulas between simple generators and BLM basis elements (Theorem 3.4.2) developed by the last two authors [24]. Through the central element presentation for D(n) as given in Theorem 2.3.1, we introduce a D(n)-H(r )-bimodule structure on the tensor space in §3.5. This gives a homomorphism ξr from D(n) to S(n, r ). We then prove in §3.6 that the restriction of this bimodule action coincides with the H(n)op -H(r )bimodule structure defined by Varagnolo–Vasserot in [73]. Thus, we obtain an explicit description of the map ξr (Theorem 3.6.3). In §3.7, we develop a certain triangular relation (Proposition 3.7.3) among the structure constants relative to the BLM basis elements. With this relation, we display an integral PBW type basis and, hence, a (weak) triangular decomposition for an affine quantum Schur algebra (Theorem 3.7.7). Using the triangular decomposition, we easily establish the surjectivity of the homomorphism ξr from D(n) to S(n, r ) in §3.8 (Theorem 3.8.1). There are several important applications of this result which will be discussed in the next three chapters. As a first application, we end this chapter by establishing certain polynomial identities (Corollary 3.9.6) arising from the commutator formulas for semisimple generators discussed in §2.6. 62

3.1. Cyclic flags: the geometric definition

63

3.1. Cyclic flags: the geometric definition In this section we recall the geometric definition of affine quantum Schur algebras introduced by Ginzburg–Vasserot [32] and Lusztig [56]. Recall the notation (n, r), (n, r ), etc., introduced in §1.1. Let F be a field and fix an F[ε, ε−1 ]-free module V of rank r 1, where ε is an indeterminate. A lattice in V is, by definition, a free F[ε]-submodule L of V satisfying V = L ⊗F[ε] F[ε, ε −1 ]. For two lattices L , L of V , L + L is again a lattice. If, in addition, L ⊆ L, L/L is a finitely generated torsion F[ε]-module. Thus, as an F-vector space, L/L is finite dimensional. Let F = F,n be the set of all cyclic flags L = (L i )i∈Z of lattices of period n, where each L i is a lattice in V such that L i −1 ⊆ L i and L i−n = εL i , for all i ∈ Z. The group G of automorphisms of the F[ε, ε −1 ]-module V acts on F by g · L = (g(L i ))i∈Z for g ∈ G and L ∈ F. Thus, the map φ : F −→ (n, r ),

L −→ (dimF L i /L i−1 )i∈Z

induces a bijection between the set {F,λ }λ∈(n,r) of G-orbits in F and (n, r ). Similarly, let B = B,r be the set of all complete cyclic flags L = (L i )i∈Z of lattices, where each L i is a lattice in V such that L i −1 ⊆ L i , L i−r = εL i and dimF (L i /L i−1 ) = 1, for all i ∈ Z. The group G acts on F × F, F × B, and B × B by g · (L, L ) = (g · L, g · L ). For L = (L i )i ∈Z and L = (L i )i∈Z ∈ F, let X i, j := X i, j (L, L ) = L i−1 + L i ∩ L j . By lexicographically ordering the indices i, j, we obtain a filtration (X i, j ) of lattices of V . For i, j ∈ Z, let ai, j = dimF (X i, j / X i, j−1 ) = dimF

L i ∩ L j L i−1 ∩ L j + L i ∩ L j −1

.

By [56, 1.5] there is a bijection between the set of G-orbits in F× F and the matrix set (n, r ) by sending (L, L ) to A = (ai, j )i, j∈Z . Let O A ⊆ F × F be the G-orbit corresponding to the matrix A ∈ (n, r ). By [56, 1.7], for L, L ∈ F, (L, L ) ∈ O A ⇐⇒ (L , L) ∈ Ot A ,

(3.1.0.1)

where t A is the transpose of A. Similarly, putting ω = (. . . , 1, 1, . . .) ∈ (r, r ) and (r, r )ω = {A ∈ (r, r ) | ro(A) = co(A) = ω},

(3.1.0.2)

64

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

the G-orbits O A on B× B are indexed by the matrices A ∈ (r, r )ω , while the G-orbits O A on F × B are indexed by the set1 (nr , r)ω = {A ∈ (nr ) | ro(A) ∈ (n, r ), co(A) = ω}, where, like (n), (nr ) = {(ai, j )i, j ∈Z | ai, j ∈ N, ai, j = ai−n, j −r , ∀i, j ∈ Z,

n

ai, j ∈ N}.

i =1 j∈Z

(3.1.0.3) Clearly, with this notation, (r, r )ω = {A ∈ (rr ) | ro(A) = co(A) = ω}.

(3.1.0.4)

Assume now that F = Fq is the finite field of q elements and write F(q) for F and B(q) for B, etc. By regarding CF(q) and CB(q) as permutation G-modules, the endomorphism algebra S,q := EndCG (CF(q))op has a basis {e A,q } A∈(n,r) while H,q := EndCG (CB(q))op has a basis {e A,q } A∈(r,r)ω with the following multiplication: A ∈ n A, A , A ;q e A ,q , if co(A) = ro(A ); e A,q e A ,q = (3.1.0.5) 0, otherwise, where = (n, r ) (resp., = (r, r )ω ) and n A,A ,A ;q = |{L ∈ F(q) (resp., B(q)) | (L, L ) ∈ O A , (L , L ) ∈ O A }| (3.1.0.6) for any fixed (L, L ) ∈ O A . By [56, 1.8], there exists a polynomial p A,A , A ∈ Z in v 2 such that, for each finite field F with q elements, n A,A , A ;q = p A,A , A |v 2 =q . Thus, we have the following definition; see [56, 1.9]. Definition 3.1.1. The (generic) affine quantum Schur algebra S(n, r ) (resp., affine Hecke algebra H(r )) is the free Z-module with basis {e A | A ∈ (n, r )} (resp., {e A | A ∈ (r, r)ω }), and multiplication defined by A ∈ p A, A ,A e A , if co(A) = ro(A ); e A e A = (3.1.1.1) 0, otherwise. Both S(n, r ) and H(r ) are associative algebras over Z with an antiautomorphism e A → et A (see (3.1.3.4) below for a modified version). 1 The set is denoted by (n, r ) in [24].

3.1. Cyclic flags: the geometric definition

65

Alternatively, we can interpret affine quantum Schur algebras in terms of convolution algebras defined by G-invariant functions and the convolution product. Again, assume that F is the finite field of q elements, and for notational simplicity, let Y = F(q),

X = B(q),

and

G = G(q).

Define CG (Y × Y ), CG (Y × X ), and CG (X × X ) to be the C-span of the characteristic functions χO of the G-orbits O on Y × Y , Y × X , and X × X , respectively. With the convolution product χO (L, L )χO (L , L ), (3.1.1.2) (χO ∗ χO )(L, L ) = L ∈F

F

O

where O ⊂ F × and ⊂ F × F for various selections of F , F , and F , we obtain convolution algebras CG (Y × Y ) and CG (X × X ), and a CG (Y × Y )-CG (X × X )-bimodule CG (Y × X ). It is clear that S,q ∼ = √ CG (Y × Y ) and H,q ∼ = CG (X × X ), and specializing v to q gives an isomorphism S(n, r )C −→ CG (Y × Y ) (resp., H(r )C −→ CG (X × X )) (3.1.1.3) sending e A ⊗ 1 → χ A , where χ A denotes the characteristic function of the orbit O A . In the sequel, we shall identify S(n, r )C with CG (Y × Y ). Via the convolution product, CG (Y × X ) becomes a CG (Y × Y )CG (X × X )-bimodule. Thus, if we denote by T(n, r ) the generic form of CG (Y × X ), then T(n, r ) becomes an S(n, r )-H(r )-bimodule with a Z-basis {e A | A ∈ (nr , r)ω }. Remark 3.1.2. It is clear from the definition that this isomorphism continues √ √ to hold if C is replaced by the ring R = Z[ q, q −1 ]. In fact, we will frequently use the isomorphism S(n, r ) R ∼ = RG (Y × Y ) to derive formulas in S(n, r ) by doing computations in RG (Y × Y ); see, e.g., §§3.6–3.9 below. Observe that, for N n, F,n is naturally a subset of F,N , since every L = (L i ) ∈ F,n can be regarded as L = ( L i ) ∈ F,N , where, for all a ∈ Z, L i+a N = L n+an if n i N . Thus, if L i+a N = L i+an if 1 i n, and N = max{n, r }, then F,n ×F,n , F,n ×B,r , and B,r ×B,r can always be regarded as G-stable subsets of F,N × F,N , and the G-orbit O A containing = ( (L, L ) is the G-orbit O A containing ( L, L ), where A = (ai, j ) and A ai, j ) are related by, for all m ∈ Z, ak,l+mn , if 1 k, l n; (3.1.2.1) ak,l+m N = 0, if either n < k N or n < l N .

66

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Lemma 3.1.3. Let N = max{n, r}. By sending e A to e A, both S(n, r ) and H(r ) can be identified as (centralizer) subalgebras of S(N , r ), and T(n, r ) as a subbimodule of the S(n, r )-H(r )-bimodule S(N , r ). Proof. Define ω ∈ (N , r ) by setting (. . . , 1r , 1r , . . .), ω= (. . . , 1r , 0n−r , 1r , 0n−r , . . .),

if n r ; if n > r.

(3.1.3.1)

For λ ∈ (n, r ), let diag(λ) = (δi, j λi )i, j ∈Z ∈ (n, r ). If we embed (n, r ) into (N , r ) via the map μ → μ, where μ = (. . . , μ1 , . . . , μn , 0 N −n , μn+1 , . . . , μ2n , 0 N −n , . . .), ∼ and put e = μ∈(n,r) ediag( μ) and eω = ediag(ω) , then S(n, r ) = eS(N , r)e and H(r ) ∼ = eω S(N , r )eω , and T(n, r ) ∼ = eS(N , r )eω as S(n, r )-H(r )bimodules. Here all three isomorphisms send e A to e A. For A ∈ (n, r ), let [A] = v −d A e A ,

where

dA =

ai, j ak,l .

(3.1.3.2)

1in ik, j
(See [56, 4.1(b),4.3] for a geometric meaning of d A .) Then for λ ∈ (n, r ) and A ∈ (n, r ), we have [A], if λ = ro(A); [diag(λ)] · [A] = and 0, otherwise, [A], if λ = co(A); [A][diag(λ)] = 0, otherwise. (3.1.3.3) Moreover, by [56, 1.11], the Z-linear map τr : S(n, r ) −→ S(n, r ), [A] −→ [t A]

(3.1.3.4)

is an algebra anti-involution. We end this section with a close look at the basis {[A] | A ∈ (nr , r)ω } for T(n, r ) from its specialization CG (Y × X ). Let I(n, r ) = {i = (i k )k∈Z | i k ∈ Z, i k+r = i k + n for all k ∈ Z}.

(3.1.3.5)

We may identify the elements of I(n, r) with functions i : Z → Z satisfying i(s + r ) = i(s) + n for all s ∈ Z. (Note that a periodic function i can be identified with (i 1 , . . . , ir ) ∈ Zr , where i s = i(s) for all s. For more details, see (3.3.0.3) below.) Clearly, there is a bijection I(n, r ) −→ (nr , r)ω ,

i −→ Ai ,

(3.1.3.6)

3.1. Cyclic flags: the geometric definition

67

where Ai = (ak,l ) with ak,l = δk,il . Thus, the orbits of the diagonal action of G on Y × X are labeled by the elements of I(n, r), and Oi := O Ai is the orbit of the pair (Li , L∅ ), where the ith lattices of Li , L∅ are defined by Li,i = F v j and L∅,i = Fvj. (3.1.3.7) i( j )i

j i

Here v1 , . . . , vr is a fixed F[ε, ε −1 ]-basis of V and vi+r s = ε−s vi , for all s ∈ Z. This is because the difference set { j | i j k, j l}\({ j | i j k − 1, j l} ∪ { j | i j k, j l − 1}) {l}, if k = il ; = ∅, otherwise. Let di = d Ai (see (3.1.3.2)). If we set, for each L ∈ Y , Xi,L = {L ∈ X | (L, L ) ∈ Oi }, then, by [56, Lem. 4.3], di is the dimension of Xi,L (in the case where F is an algebraically closed field). More precisely, a direct calculation gives the following result. Lemma 3.1.4. For each i = (i j ) ∈ I(n, r ), let Inv(i) = {(s, t) ∈ Z2 | 1 s r, s < t, i s i t }. Then di = |Inv(i)|. Proof. Applying [56, Lem. 4.3] gives di = d Ai = δs,it δk,il = δs,it ( δk,il ) =

1tr sk, t
(

1t r s∈Z

δk,il ) = |Inv(i)|,

1t r i t k, t
as required. With the identification of (3.1.1.3), we have 1

[Ai ] = q − 2 di χi , where χi is the characteristic function of the orbit Oi .

s k, t
68

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

3.2. Affine Hecke algebras of type A: the algebraic definition We now follow [35, 73] to interpret affine quantum Schur algebras as endomorphism algebras of certain tensor spaces over the affine Hecke algebras associated with affine symmetric groups. Let S,r be the (extended) affine symmetric group consisting of all permutations w : Z → Z such that w(i + r ) = w(i) + r for i ∈ Z. This is the subset of all bijections in I(r, r ) which is defined in (3.1.3.5). Hence, Inv(w) is well-defined. More precisely, for any a ∈ Z, if we set Inv(w, a) = {(s, t) ∈ Z2 | a + 1 s a + r, s < t, w(s) > w(t)}, then Inv(w) = Inv(w, 0) and |Inv(w)| = |Inv(w, a)| for all a ∈ Z. There are several useful subgroups of S,r . The subgroup W of S,r con sisting of w ∈ S,r with ri=1 w(i) = ri=1 i is the affine Weyl group of type A with generators si (1 i r ) defined by setting

si ( j) =

⎧ ⎪ ⎪ j, ⎨

j − 1, ⎪ ⎪ ⎩ j + 1,

for j ≡ i, i + 1 mod r ; for j ≡ i + 1 mod r ; for j ≡ i mod r

(see [51, 3.6]). If S = {si }1i r , then (W, S) is a Coxeter system. For convenience of writing consecutive products of the form si si+1 si +2 · · · or si si−1 si−2 · · · , we set si +kr = si for all k ∈ Z. Observe that the cyclic subgroup ρ of S,r generated by the permutation2 ρ of Z sending j to j + 1, for all j , is in the complement of W . Observe also that s j+1 ρ = ρs j , for all j ∈ Z. 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 . We will identify A with Zr in the sequel. In particular, A is generated by ei = (0, . . . , 0, 1 , 0, . . . , 0) for 1 (i)

i r . Moreover, the subgroup of W generated by s1 , . . . , sr−1 is isomorphic to the symmetric group Sr . Recall the matrix set (r, r )ω defined in (3.1.0.2) which is clearly a group with matrix multiplication. The following result is well-known; see, e.g., [35, Prop. 1.1.3 & 1.1.5] for the last isomorphism.

2 The notation ρ will have a different use in §5.3.

3.2. Affine Hecke algebras of type A: the algebraic definition

69

Proposition 3.2.1. Maintain the notation above. There are group isomorphisms S,r ∼ = (r, r )ω ∼ = Sr Zr ∼ = ρ W. Proof. The first isomorphism is the restriction of the bijection defined in (3.1.3.6) for n = r ; see also (3.1.0.4). In particular, every w ∈ S,r is sent to the matrix Aw = (ak,l ) with ak,l = δk,w(l) . The second is easily seen from the fact that every w ∈ S,r can be written as xλ, where x ∈ Sr and λ ∈ Zr are defined uniquely by w(i) = λi r + x(i), for all 1 i, x(i ) r . If ek ∈ S,r (k ∈ [1, r ]) denotes the permutation ek (i ) = i for i = k, i ∈ [1, r ] and ek (k) = r + k, then ek = ρsr+k−2 · · · sk+1 sk , proving the last isomorphism.3 Let be the length function of the Coxeter system (W, S). Then (w) for w ∈ W is the length m such that w = si1 si2 · · · sim is a reduced expression. By [41], $$ w( j ) − w(i ) $$ $ $. (w) = $ $ r 1i < j r

Extend the length function to S,r by setting (ρ i w ) = (w ), for all i ∈ Z and w ∈ W . Since Inv(ρ i ) = ∅, induction on (w) shows that (w) = Inv(w),

for all w ∈ S,r .

(3.2.1.1)

The group S,r acts on the set I(n, r) given in (3.1.3.5) by place permutation: iw = (i w(k) )k∈Z ,

for i ∈ I(n, r ) and w ∈ S,r .

(3.2.1.2)

Clearly, every S,r -orbit has a unique representative in the fundamental set I(n, r )0 = {i ∈ I(n, r ) | 1 i 1 i 2 · · · ir n} = {iλ | λ ∈ (n, r)},

(3.2.1.3)

λ and where iλ = (i s )s∈Z ∈ I(n, r )0 if and only if i s = m, for all s ∈ Rm m ∈ Z, or equivalently, λ j = |{k ∈ Z | i k = j}|, for all j ∈ Z. Written in full, iλ is the sequence

(. . . , 0, . . . , 0, 1, . . . , 1, 2, . . . , 2, . . . , n, . . . , n , 1 + n, . . . , 1 + n , . . .). ( )* + ( )* + ( )* + ( )* + ( )* + λn

λ1

λ2

λn

λ1

3 If we put s , s , . . . , s on a circle clockwise and break the circle by removing s r 1 2 k−1 , then ρ −1 ek is obtained by flattening the circle into a line segment.

70

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Observe that every number of the form i +kn, for 1 i n, k ∈ Z, determines a constant subsequence (i j ) j∈R λ = (i + kn, . . . , i + kn) of i = iλ (of length i +kn λi ) indexed by the set λ Ri+kn = {λk,i−1 + 1, λk,i −1 + 2, . . . , λk,i−1 + λi = λk,i }, (3.2.1.4) , where λk,i−1 = kr + 1t i −1 λt . These sets form a partition j∈Z R λj of Z. Note that the fundamental set I(n, r )0 is obtained by shifting the one used in [73] by n. For λ ∈ (n, r ), let Sλ := S(λ1 ,...,λn ) be the corresponding standard Young subgroup of Sr (and hence, of S,r ), and let

D λ = {d | d ∈ S,r , (wd) = (w) + (d) for w ∈ Sλ }. By (3.2.1.1), one sees easily that d −1 ∈ D λ ⇐⇒ d(λk,i −1 + 1) < d(λk,i−1 + 2) < · · · < d(λk,i −1 + λi ), ∀1 i n, k ∈ Z ⇐⇒ d(λ0,i−1 + 1) < d(λ0,i−1 + 2) < · · · < d(λ0,i−1 + λi ), ∀1 i n. (3.2.1.5) The following result is the affine version of a well-known result for symmetric groups (see, e.g., [12, Th. 4.15, (9.1.4)]. It can be deduced from [73, 7.4]. For a proof, see [24, 9.2]). −1 Lemma 3.2.2. For λ, μ ∈ (n, r ), let D λ,μ = Dλ ∩ (Dμ ) . There is a bijective map

j : {(λ, w, μ) | w ∈ D λ,μ , λ, μ ∈ (n, r)} −→ (n, r )

(3.2.2.1)

sending (λ, w, μ) to A = (ak,l ), where, if iλ = (i a )a∈Z and iμ = ( ja )a∈Z , then μ

ak,l = |{t ∈ Z | i w(t) = k, jt = l}| = |Rkλ ∩ w Rl |

(3.2.2.2)

for all k, l ∈ Z. In particular, by certain appropriate embedding, restriction gives two bijections n j : {(λ, w, ω) | w ∈ D λ,μ , λ ∈ (n, r )} −→ (r , r)ω

(3.2.2.3)

and j : {(ω, w, ω) | w ∈ S,r } → (r, r )ω . μ

μ

We remark that, for w ∈ Sλ wSμ , the equality |Rkλ ∩ w Rl | = |Rkλ ∩ w Rl | holds. Hence, the matrix A is completely determined by λ, μ and the double coset Sλ wSμ , and is independent of the selection of the representative. Moreover, if j(λ, d, μ) = A, then j(μ, d −1 , λ) = t A, the transpose of A.

3.2. Affine Hecke algebras of type A: the algebraic definition

71

Corollary 3.2.3. For λ, μ ∈ (n, r ) and d ∈ D λ,μ with j(λ, d, μ) = A ∈ (i ) (n, r ), let ν be the composition of λi obtained by removing all zeros from row i of A. Then Sλ ∩ dSμ d −1 = Sν , where ν = (ν (1) , . . . , ν (n) ). μ

Proof. Let iμ = ( js )s∈Z ∈ I(n, r). Then js = k for all s ∈ Rk and k ∈ Z. μ μ Thus, l ∈ Riλ and d −1 (l) ∈ R j ⇐⇒ l ∈ Riλ ∩ d R j . For 1 i n, if Ii := (k, . . . , k )k∈Z = (. . . , 1, . . . , 1, 2, . . . , 2, . . . , n, . . . , n , . . .) ∈ Zλi , ( )* + ( )* + ( )* + ( )* + ai,k

ai,1

ai,2

ai,n

then, with the notation used in (3.2.1.4), (3.2.1.5) together with Lemma 3.2.2 implies Ii = ( jd −1 (λ0,i−1 +1) , . . . , jd −1 (λ0,i−1 +λi ) ). Hence, ( jd −1 (1) , . . . , jd −1 (r) ) = (I1 , . . . , In ). Since Sμ = StabS,r (Iμ ), it follows that Sλ ∩ dSμ d −1 = Sλ ∩ StabS,r (Iμ d −1 ) = StabSλ (Iμ d −1 ) = Sν . Corollary 3.2.4. There is a bijective map j∗ : I(n, r ) −→ {(λ, d, ω) | d ∈ D λ , λ ∈ (n, r )},

i −→ (λ, d, ω),

where i = iλ d. Moreover, if d + denotes a representative of Sλ d with maximal length, then di = |Inv(i)| = (d + ). Proof. Clearly, j∗ is the composition of the bijection given in (3.1.3.6) and the inverse of j given in (3.2.2.3). We now prove the last statement. Let L = {(s, t) ∈ Z2 | 1 s r, s < t} and j = iλ . Then Inv(i) = {(s, t) ∈ L | jd(s) jd(t) } = X 1 ∪ X 2

(a disjoint union),

where X 1 = {(s, t) ∈ L | jd(s) > jd(t) } and X 2 = {(s, t) ∈ L | jd(s) = jd(t) }. Since (s, t) ∈ X 2 if and only if d(s), d(t) ∈ Rkλ for some k ∈ Z, (3.2.1.5) forces d(s) < d(t). Hence, if S,r acts on Z2 diagonally, then X 2 = {(s, t) ∈ L | d(s) < d(t), d(s), d(t) ∈ Rkλ for some k ∈ Z} = d −1 {(s, t) ∈ Z2 | 1 d −1 (s) r, s < t, s, t ∈ Rkλ for some k ∈ Z}. Thus, |X 2 | is the length of the longest element w0,λ in Sλ by Lemma 3.10.1. Since, for (s, t) ∈ L\X 2 , jd(s) > jd(t) ⇐⇒ d(s) > d(t), applying (3.2.1.5) again yields Inv(d) = {(s, t) ∈ L | d(s) > d(t)} = {(s, t) ∈ L\X 2 | d(s) > d(t)} = X 1 . Consequently, di = |X 1 | + |X 2 | = (d) + (w0,λ ) = (d + ), as required. We record the following generalization (to affine symmetric groups) of a well-known result for Coxeter groups.

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3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

−1 Lemma 3.2.5. Let λ, μ ∈ (n, r ) and d ∈ D λ,μ . Then d Sλ d ∩ Sμ is a standard Young subgroup of Sμ . Moreover, each element w ∈ Sλ dSμ can be written uniquely as a product w = w1 dw2 with w1 ∈ Sλ and w2 ∈ D ν ∩ Sμ , −1 where ν ∈ (n, r ) is defined by Sν = d Sλ d ∩ Sμ , and the equality (w) = (w1 ) + (d) + (w2 ) holds.

Following [45], the (extended) affine Hecke algebra H(S,r ) over Z is defined to be the algebra generated by Tsi (1 i r ), Tρ±1 with the following relations: Ts2i = (v 2 − 1)Tsi + v 2 , Tsi Ts j = Ts j Tsi

(i − j ≡ ±1 mod r ),

Tsi Ts j Tsi = Ts j Tsi Ts j Tρ Tρ−1

=

Tρ−1 Tρ

(i − j ≡ ±1 mod r and r 3),

= 1, and

Tρ Tsi = Tsi +1 Tρ , where Tsr +1 = Ts1 . This algebra has a Z-basis {Tw }w∈S,r , where Tw = j

Tρ Tsi1 · · · Tsim if w = ρ j si1 · · · sim is reduced. The following result is well-known due to Iwahori–Matsumoto [41]. Recall the algebra H(r ) defined in §3.1 and the isomorphism given in (3.1.1.3). Lemma 3.2.6. There is a Z-algebra isomorphism H(S,r ) ∼ = H(r ) whose √ specialization of v to q gives a C-algebra isomorphism H(S,r ) ⊗ C ∼ = CG (X × X ). Thus, we will identify H(S,r ) with H(r) in the sequel. Let H(r ) = H(Sr ) be the subalgebra of H(r ) generated by Tsi (1 i < r ). Then H(r ) is the Hecke algebra of the symmetric group Sr . We finally set H(r) = H(r ) ⊗Z Q(v) and H(r ) = H(r) ⊗Z Q(v). For each λ ∈ (n, r ), let xλ = w∈Sλ Tw ∈ H(r ) and define SH (n, r ) := EndH(r) xν H(r ) . ν∈(n,r)

d H For λ, μ ∈ (n, r) and d ∈ D λ,μ , define φλ,μ ∈ S (n, r) as follows: d φλ,μ (xν h) = δμν Tw h, (3.2.6.1) w∈Sλ d Sμ d } forms a basis for where ν ∈ (n, r ) and h ∈ H(r ). Then the set {φλ,μ H S (n, r ).

3.2. Affine Hecke algebras of type A: the algebraic definition

73

Remarks 3.2.7. (1) We point out that, as a natural generalization of the q-Schur algebra given in [15, 16], the endomorphism algebra SH (n, r ) is d called the affine q-Schur algebra and the basis {φλ,μ } is the affine analogue of [16, 1.4]. (2) Let R be a commutative ring with 1 which is a Z-algebra. Then, by base change to R, a similar basis can be defined for S (n, r ; R) = EndH(r) R H

xλ H(r ) R .

λ∈(n,r)

As a result of this, the endomorphism algebra SH (n, r ) satisfies the base change property: SH (n, r ; R) ∼ = S H (n, r ) R . This property has already been √ √ −1 mentioned for R = Z[ q, q ] in Remark 3.1.2. Combining the base change property and [73, 7.4] gives the following result which extends the isomorphism given in Lemma 3.2.6 to affine quantum Schur algebras. Proposition 3.2.8. The bijection j given in Lemma 3.2.2 induces a Z-algebra isomorphism ∼

d h : S(n, r ) −→ SH (n, r ), e A −→ φλ,μ ,

for all A ∈ (n, r ) with A = j(λ, d, μ), where λ, μ ∈ (n, r ) and d ∈ D λ∈(n,r) x λ H(r ) as an S(n, r)-module via h, λ,μ . Moreover, regarding we obtain an S(n, r )-H(r )-bimodule isomorphism ∼

ev : T(n, r) −→

xλ H(r ), e A −→ xλ Td ,

λ∈(n,r)

for all A ∈ (nr , r)ω with A = j(λ, d, ω). Note that if we regard T(n, r ) as a subset of S(N , r) as in Lemma 3.1.3, the bimodule isomorphism is simply the evaluation map. Recall that, by removing the superscript , the notation Dλ,μ denotes the shortest (Sλ , Sμ )-coset representatives in Sr . If we identify (n, r ) with (n, r ) via (1.1.0.2), we obtain the following. d with λ, μ ∈ (n, r ) and Corollary 3.2.9. The subspace spanned by all φλ,μ d ∈ Dλ,μ is a subalgebra which is isomorphic to the quantum Schur algebra S(n, r ).

74

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Using the evaluation isomorphism, we now describe an explicit action of H(r ) on T(n, r). First, for λ ∈ (n, r ), d ∈ D λ , and 1 k r , ⎧ 2 ⎪ if dsk ∈ D ⎪ λ (then (dsk ) > (d)); ⎨v xλ Td , xλ Td · Tsk = xλ Tdsk , if (dsk ) > (d) and dsk ∈ D λ; ⎪ ⎪ ⎩v 2 x T + (v 2 − 1)x T , if (ds ) < (d)(then ds ∈ D). λ dsk

λ d

k

k

λ

(3.2.9.1) + Second, by Corollary 3.2.4, we obtain ev([Ai ]) = v −(d ) xλ Td if j∗ (i) = (λ, d, ω), where d + is a representative of Sλ d with maximal length. For w ∈ S,r , let w = v −(w) Tw . T Thus, for j = iλ and d as above, (3.2.9.1) becomes ⎧ jd ⎪ if dsk ∈ D ⎪ λ (then (dsk ) > (d)); ⎨v[A ], jd jds [A ]Tsk = [ A k ], if (dsk ) > (d) and dsk ∈ D λ; ⎪ ⎪ ⎩[Ajdsk ] + (v − v −1 )[Ajd ], if (ds ) < (d)(then ds ∈ D). k

k

λ

This together with Lemma 3.1.4 gives the first part of the following (and part (2) is clear from the definition). Proposition 3.2.10. Let i ∈ I(n, r ). (1) For any 1 k r , we have ⎧ i ⎪ if i k = i k+1 ; ⎪ ⎨v[A ], i is k [ A ]Tsk = [A ], if i k < i k+1 ; ⎪ ⎪ ⎩[Aisk ] + (v − v −1 )[Ai ], if i > i . k k+1 (2) [Ai ]Tρ = [Aiρ ], where ρ ∈ S,r is the permutation sending i to i + 1 for all i ∈ Z.

3.3. The tensor space interpretation We now interpret the right H(r)-module T(n, r ) in terms of the tensor space, following [73]. The Hecke algebra H(S,r ) admits the so-called Bernstein presentation which consists of generators e1 +···+e j −1 T e−1+···+e , X −1 (1 i r − 1, 1 j r ), Ti := Tsi , X j := T j j 1

3.3. The tensor space interpretation

75

and relations (Ti + 1)(Ti − v 2 ) = 0, Ti Ti +1 Ti = Ti+1 Ti Ti+1 , Ti T j = T j Ti (|i − j| > 1), X i X i−1 = 1 = X i−1 X i , X i X j = X j X i , Ti X i Ti = v 2 X i+1 , and X j Ti = Ti X j ( j = i, i + 1). Note that, for any dominant λ = (λi ) ∈ Zr (meaning λ1 · · · λr ), λ −1 . X λ := X 1 1 · · · X rλr = T λ

ρ = X −1 T −1 · · · T −1 since e1 = ρsr−1 · · · s2 s1 and In particular, Tρ = T 1 1 r−1 e1 . X 1−1 = T By definition, we have, for each 1 i r − 1, −1 Ti X i+1 = X i−1 Ti + (1 − v 2 )X i−1 , −1 X i+1 Ti = Ti X i−1 + (1 − v 2 )X i−1 ,

(3.3.0.1)

−1 −1 Ti−1 X i−1 = X i−1 + (1 − v −2 )X i+1 , and +1 Ti −1 −1 X i−1 Ti−1 = Ti−1 X i+1 + (1 − v −2 )X i+1 .

So X i Ti = Ti X i+1 + (1 − v 2 )X i+1 and Ti X i = X i +1 Ti + (1 − v 2 )X i+1 , etc. Then, for each a = (a1 , . . . , ar ) ∈ Zr , an inductive argument gives the formula X i +1 (X a − X asi ) X a Ti = Ti X asi + (1 − v 2 ) , (3.3.0.2) X i − X i+1 where asi = (a1 , . . . , ai−1 , ai+1 , ai , ai +2 , . . . , ar ). Let be the free Z-module with basis {ωi | i ∈ Z}. Consider the r -fold tensor space ⊗r and, for each i = (i 1 , . . . , ir ) ∈ Zr , write ωi = ωi1 ⊗ ωi2 ⊗ · · · ⊗ ωir = ωi1 ωi2 · · · ωir ∈

⊗r

.

We now follow [73] to define a right H(r )-module structure on ⊗r and to establish an H(r )-module isomorphism from T(n, r ) to ⊗r . Recall the set I(n, r ) defined in (3.1.3.5) and the action (3.2.1.2) of S,r on I(n, r ). If we identify I(n, r ) with Zr by the following bijection I(n, r ) −→ Zr , i −→ (i 1 , . . . , ir ),

(3.3.0.3)

then the action of S,r on I(n, r) induces an action on Zr . Also, the usual action of the place permutation of Sr on I (n, r), where I (n, r ) = {(i 1 , . . . , ir ) ∈ Zr | 1 i k n, ∀k},

76

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

is the restriction to Sr of the action of S,r on Zr (restricted to I (n, r )). We often identify I (n, r ) as a subset of I(n, r ), or I(n, r )0 as a subset of I (n, r ), depending on the context. By the Bernstein presentation for H(r ), Varagnolo–Vasserot extended in [73] the action of H(r ) on the finite tensor space ⊗r n , where n = span{ω1 , . . . , ωn }, given in [43] (see also [22]) to an action on ⊗r via the place permutation above. In other words, ⊗r admits a right H(r)-module structure defined by ⎧ ⎪ ωi · X t−1 = for all i ∈ Zr ; ⎪ ⎧ ωiet = ωi1 · · · ωit−1 ωit +n ωit+1 · · · ωir , ⎪ ⎪ ⎨ 2 ⎪ if i k = i k+1 ; ⎪ ⎨ v ωi , ⎪ ω · T = vωisk , if i k < i k+1 ; for all i ∈ I (n, r ), k i ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ vω + (v 2 − 1)ω , if i > i , isk

i

k

k+1

(3.3.0.4) where 1 k r − 1 and 1 t r . In general, for an arbitrary i ∈ Zr , there exist j ∈ I (n, r ) and a ∈ Zr satisfying ωj · X a = ωi . Then, by applying (3.3.0.2), we define ωi · Tk = (ωj · X a ) · Tk = ωj · (X a Tk ) X k+1 (X a − X ask ) = (ωj · Tk ) · X ask + (1 − v 2 )ωj · . X k − X k+1 Varagnolo–Vasserot have further established in [73, Lem. 8.3] an H(r )module isomorphism between T(n, r ) and ⊗r . This result justifies why the set I(n, r )0 defined in (3.2.1.3) is called a fundamental set. Recall from (3.1.3.6) the matrix Ai defined for every i ∈ I(n, r ). Proposition 3.3.1. There is a unique H(r )-module isomorphism g : T(n, r) −→

⊗r

such that [Ai ] −→ ωi for all i ∈ I(n, r )0 ,

which induces a Z-algebra isomorphism ∼

t : S(n, r ) −→ St(n, r ) := EndH(r) (

⊗r

).

In particular, g is an S(n, r )-H(r)-bimodule isomorphism. Moreover, spe√ cializing v to q yields a CG (Y × Y )-CG (X × X )-bimodule isomorphism over C from CG (Y × X ) to ∀i ∈ I(n, r )0 .

⊗r

C

:=

⊗r

1

⊗ C sending [Ai ] = q − 2 di χi to ωi ,

Proof. The first assertion follows from [24, Lem. 9.5]. By regarding ⊗r as an S(n, r )-module via t, g induces an S(n, r )-H(r )-bimodule isomorphism. The last assertion follows from the isomorphism (3.1.1.3) and the definition that T(n, r ) is the generic form of CG (Y × X ).

3.4. BLM bases and multiplication formulas

77

si , for 1 i r −1, on the Remark 3.3.2. As seen above, since the action of T basis elements ωi for i ∈ I (n, r ) follows the same rules as the action on [Ai ] given in Proposition 3.2.10(1), it follows that g([Ai ]) = ωi for all i ∈ I (n, r ). sr are different. Hence, if we identify T(n, r ) with However, the actions of T ⊗r under g, then {[Ai ]} i∈I(n,r ) and {[ωi ]}i∈I(n,r) form two different bases with the subset {[Ai ] = ωi }i∈I (n,r) in common. We now identify CG (Y × X ) with √ q gives isomorphisms EndH(r )C (

⊗r

C

⊗r

C

. Consequently, specializing v to

)∼ = EndH(r)C (CG (Y × X )) ∼ = CG (Y × Y ) ∼ = S(n, r )C .

These algebras will be identified in the sequel. Also, let = ⊗Z Q(v), i.e., is a Q(v)-vector space with basis {ωi | i ∈ Z}. Then the right action of H(r) on ⊗r extends to a right action of H(r ) = H(r ) ⊗Z Q(v) on ⊗r . Hence, we have the Q(v)-algebra isomorphism S(n, r ) := S(n, r ) ⊗Z Q(v) ∼ = EndH(r ) (⊗r ). We will make the identifications S(n, r) = EndH(r) ( EndH(r ) (⊗r ).

⊗r )

and S(n, r ) =

3.4. BLM bases and multiplication formulas We now follow [4] (cf. [73]) to define BLM bases for the affine quantum Schur algebra S(n, r ) as discussed in [24]. Let ± (n) = {A ∈ (n) | ai,i = 0 for all i}

(3.4.0.1)

be the set of 0-diagonal matrices in (n). For A ∈ ± (n) and j ∈ Zn , define A(j, r ) ∈ S(n, r ) by λj if σ (A) r ; λ∈(n,r−σ (A)) v [A + diag(λ)], A(j, r ) = (3.4.0.2) 0, otherwise, where λ · j = 1i n λi ji . For the convenience of later use, we extend the definition to matrices in Mn,(Z) by setting A(j, r ) = 0 if some off-diagonal entries of A are negative. The elements A(j, r ) are the affine version of the elements defined in [4, 5.2] and have been defined in terms of characteristic functions in [73, 7.6]. The following result is the affine analogue of [25, 6.6(2)]; see [24, Prop. 4.1].

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3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Proposition 3.4.1. For a fixed 1 i 0 n, the set B,i0 ,r := { A(j, r ) | A ∈ ± (n), j ∈ Nn , ji0 = 0, σ (j) + σ ( A) r } forms a Q(v)-basis for S(n, r ). In particular, the set B,r := {A(j, r ) | A ∈ ± (n), j ∈ Nn , σ ( A) r } is a spanning set for S(n, r ). We call B,i0 ,r a BLM basis of S(n, r ) and call B,r the BLM spanning set. As in the finite case, one would expect that there is a basis B for quantum affine gln such that its image in S(n, r ) is B,r ∪ {0} for every r 0. See §5.4 for a conjecture. The affine analogue of the multiplication formulas given in [4, 5.3] has also been established in [24]. As seen in [24, Th. 4.2], these formulas are crucial to a modified approach to the realization problem for quantum affine sln . We shall also see in Chapter 5 that they are useful in finding a presentation for the affine quantum Schur algebras S(r, r ). Let αi = ei − e i+1 , β i = −ei − ei+1 n ∈ Z. Theorem 3.4.2. Assume 1 h n. For i ∈ Z, j, j ∈ Zn , and A ∈ ± (n), if we put f (i ) = f (i, A) = j i ah, j − j >i ah+1, j and f (i) = f (i, A) = j
(3.4.2.1)

where 0 stands for the zero matrix, E h,h+1 (0, r) A(j, r ) =

+

v f (i )

i
v f (i)

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

+ v f (h)− jh −1

ah,i + 1 − E (A + E h,i h+1,i )(j + α h , r ) 1

ah,i + 1 − E ( A + E h,i h+1,i )(j, r ) 1

( A − E h+1,h )(j + α h , r ) − ( A − E h+1,h )(j + β h , r )

+ v f (h+1)+ jh+1

1 − v −2

ah,h+1 + 1 (A + E h,h+1 )(j, r ), 1

(3.4.2.2)

3.5. The D(n)-H(r )-bimodule structure on tensor spaces

79

and

E h+1,h (0, r )A(j, r) =

vf

(i)

i

+

v

f (i )

i>h+1;ah,i 1

+vf +vf

(h+1)− j h+1 −1

(h)+ j h

ah+1,i + 1 (A − E h,i + E h+1,i )(j, r ) 1

ah+1,i + 1 (A − E h,i + E h+1,i )(j − α h , r) 1

(A − E h,h+1 )(j − α h , r ) − (A − E h,h+1 )(j + β h , r)

ah+1,h 1

1 − v −2 +1 (A + E h+1,h )(j, r ).

(3.4.2.3)

According to Proposition 1.4.5, the double Ringel–Hall algebra D(n) has generators corresponding to simple modules and homogeneous indecomposable modules. It would be natural to raise the following question. In fact, we will see in §5.4 that the solution to this problem is the key to prove the realization conjecture 5.4.2; cf. Problem 6.4.2. Problem 3.4.3. Find multiplication formulas for E h,h+sn (0, r )A(j, r ) for all s = 0 in Z.

3.5. The D(n)-H(r)-bimodule structure on tensor spaces As in §3.3, let be the Q(v)-vector space with basis {ωs | s ∈ Z}. Our idea is to use the presentation of D(n) to define the natural representation of D(n). This induces a left D(n)-module structure on the tensor space ⊗r which commutes with the right action of the affine Hecke algebra H(r ) and, hence, an algebra homomorphism ξr : D(n) → S(n, r ). We then partially establish the affine analogue of the quantum Schur–Weyl duality in §3.8. For i ∈ I and m ∈ Z+ , we define the actions of K i±1 , E i , Fi , and z± m on by E i · ωs = δi+1,¯s ωs−1 , K i±1 · ωs = v ±δi,¯s ωs , and

Fi · ωs = δi,¯s ωs+1 , z± m · ωs = ωs∓mn .

(3.5.0.1)

Lemma 3.5.1. With the action defined as above, becomes a left D(n)module. Proof. For i ∈ I and m ∈ Z+ , we denote by κi±1 , φi+ , φi− , and ψm± the Q(v)linear transformations of induced by the actions of K i±1 , E i , Fi , and z± m on defined above, respectively. The assertion follows from the fact that κi±1 , φi+ ,

80

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

φi− , and ψm± satisfy the relations similar to (QGL1)–(QGL8) in Theorem 2.3.1. This is because those relations involving ψm± are clear, while the others (involving only κi±1 , φi+ , and φi− ) follow from the natural representation of the extended quantum affine sln on (see, e.g., [24, (9.5.1)]). Let ξ1 : D(n) → EndQ(v) () be the algebra homomorphism associated with the D(n)-module . Proposition 3.5.2. (1) The D(n)-module is indecomposable. (2) For each non-zero element a ∈ Q(v), the subspace Va spanned by ωi − aωi+n , for all i ∈ Z, is a submodule of . Moreover, ∼ = Va , and the quotient module (a) := /Va is simple. Proof. (1) It is easy to check that the map π : → , ωi → ωi +n is a D(n)-module homomorphism and so is π −1 . Take an arbitrary f ∈ EndD(n) () and write f (ω0 ) = s∈Z bs ωs , where all but finitely many bs ∈ Q(v) are zero. Then, for each i ∈ I , E i f (ω0 ) = E i ( bs ωs ) = δi +1,¯s bs ωs−1 = bi+1+tn ωi+tn . s∈Z

s∈Z

t∈Z

On the other hand, E i f (ω0 ) = f (E i ω0 ) = 0 if i = n − 1. Hence, bi+1+tn = 0 unless i = n − 1 and f (ω0 ) = t∈Z btn ωtn . Thus, for each 1 s < n, f (ωs ) = f (Fs−1 · · · F1 Fn ω0 ) = Fs−1 · · · F1 Fn f (ω0 ) = btn ωtn+s . t∈Z

For any k = s ± mn ∈ Z with 0 s < n and m > 0, ∓ f (ωk ) = f (z∓ btn ωtn+k . m ωs ) = zm f (ωs ) =

t∈Z

Hence, f = t∈Z btn We conclude that EndD(n) () = Q(v)[π, π −1 ] is a Laurent polynomial algebra over Q(v), which implies that is indecomposable. (2) Clearly, Va is a submodule of , and the map → Va , ωi → ωi − aωi+n gives the required D(n)-module isomorphism. Also, the quotient module (a) = /Va has a basis {ω¯ i = ωi + Va | i ∈ I } with the D(n)-module structure given by πt.

E i · ω¯ j = δi+1, j ω¯ j−1 , K i±1 · ω¯ j = v ±δi, j ω¯ j , and

Fi · ω¯ j = δi, j ω¯ j+1 , z± ¯ j = a ±m ω¯ j , m ·ω

for all i, j ∈ I and m 1. It is obvious that (a) is a simple D(n)-module.

3.5. The D(n)-H(r )-bimodule structure on tensor spaces

81

There will be a different construction for (a) in §4.6 through representations of the Hecke algebra H(1). As shown in §2.1, there is a PBW type basis for D(n). The action of these basis elements on the basis {ωs | s ∈ Z} of can be described as follows. Proposition 3.5.3. For each 0 = A ∈ + (n) and s ∈ Z, we have u+ δ A,Et,s ωt and u− δ A,Es,t ωt , A · ωs = A · ωs = t<s

s
dA ± where, as in (1.2.0.6), u± A = v u A with d A = dim End(M(A)) − dim M(A). In particular, if A = 0 and M(A) is decomposable, then u+ u− A ·ωs = 0 = A ·ωs .

Proof. Define a Q(v)-linear map φ = φ + : D(n)+ → EndQ(v) (), u+ A → φ A by setting φ0 = id and, for A = 0, φ A : −→ , ωs −→ δ A,Et,s ωt . t<s

Define φ − : D(n)− → EndQ(v) () similarly. We only prove the first equality. The second one can be proved analogously. We first show that φ is an algebra homomorphism. For i < j in Z, set M i, j = M(E i, j ) = Si [ j − i] as in §1.2 and write j − i = an + b for a 0 and 0 b < n. Then it is easy to check that # a, if b = 0; dim End(M i, j ) = (3.5.3.1) a + 1, if b = 0. We claim that, for l < s < t, dim End(M l,t ) = dim End(M l,s ) + dim End(M s,t ) + dim M l,s , dim M s,t (3.5.3.2) (cf. proof of [13, Lem. 8.2]). Indeed, write s − l = a1 n + b1 and t − s = a2 n + b2 , for a1 , a2 0 and 0 b1 , b2 < n. Then t −l = (a1 +a2 )n +b1 +b2 . If b1 = 0 or b2 = 0, the equality follows from (3.5.3.1) since dim M l,s , dim M s,t = 0 by (1.2.0.5). Suppose now b1 = 0 and b2 = 0. Then dim End(M l,s ) + dim End(M s,t ) = a1 + a2 + 2 and # dim Endk (M ) = l,t

a1 + a2 + 1, a1 + a2 + 2,

if b1 + b2 n; if b1 + b2 > n.

82

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Let d = (di ) = dim M l,s . Then

#

dim M l,s , dim M s,t = dt−1 − ds−1 =

−1, 0,

if b1 + b2 n; if b1 + b2 > n.

Therefore, the equality (3.5.3.2) holds. For A, B ∈ + (n), we have

d A +d B + + u+ u+ u A u B = v dim M( A),dim M(B)+d A +d B A B =v

v −dC ϕ CA,B u C+ .

C

Thus, to prove that φ is an algebra homomorphism, it suffices to show that for A, B ∈ + (n), −d φ A φ B = v dim M(A),dim M(B)+d A +d B v C ϕ CA,B φC . C

By definition, we have, for s ∈ Z,

φ A φ B (ωs ) = 0 = v dim M( A),dim M(B)+d A +d B

v −dC ϕ CA,B φC (ωs )

C

unless B and A =

= E s,t and A = E l,s for some l El,s for some l < s < t. Then

< s < t. Now, suppose that B = E s,t

φ A φ B (ωs ) = ωl and

v −dC ϕ CA,B φC (ωs ) = v −d D ωl ,

C

where D =

E l,t

D = 1). From (3.5.3.2) it follows that (and ϕ A,B −d φ A φ B (ωs ) = ωl = v dim M(A),dim M(B)+d A +d B v C ϕ CA,B φC (ωs ). C

Hence, φ is an algebra homomorphism. To complete the proof, it remains to show that φ coincides with the restric+ tion of ξ1 to D(n)+ . Since D(n)+ is generated by u + i and zm , for i ∈ I and + m ∈ Z , we need to check that + + + φ(u + i ) = ξ1 (u i ) and φ(zm ) = ξ1 (zm ).

The equality φ(u i+ ) = ξ1 (u i+ ) is trivial. For each s ∈ Z, we have by definition + ξ1 (z+ m )(ωs ) = zm · ωs = ωs−mn .

On the other hand, by (2.2.1.2), z+ m =

n l=1

u+ E

l,l+mn

+

v nm (v − v −1 ) + ym , [m]

3.5. The D(n)-H(r )-bimodule structure on tensor spaces

83

where ym+ is a linear combination of certain u+ A with M(A) decomposable. + Hence, φ(ym ) = 0. Thus, φ(z+ m )(ωs )

=

n l=1

φ( u+ )(ωs ) E l,l+mn

=

n

δ E

l,l+mn ,E b,s

ωb = ωs−mn .

l=1 b<s

+ + (Note that E a,b = E a+n,b+n for a, b ∈ Z.) Hence, φ(zm ) = ξ1 (zm ). This completes the proof.

Remark 3.5.4. The above proposition implies that the action of D(n)− on induced from φ − coincides with the action given in [73, Lem. 8.1] which is defined geometrically through the algebra homomorphism ζ1− : D(n)− → S(n, 1); see (3.6.2.1) below. Now, for each r 1, the Hopf algebra structure of D(n) induces a left D(n)-module structure on the tensor space ⊗r which has a Q(v)-basis {ωi | i ∈ Zr }. Since z± m are primitive elements, we have by (3.5.0.1) that, for each m 1 and ωi = ωi1 ⊗ · · · ⊗ ωir ∈ ⊗r , z± m · ωi =

r

ωi1 ⊗ · · · ⊗ ωis−1 ⊗ ωis ∓mn ⊗ ωis+1 ⊗ · · · ⊗ ωir .

(3.5.4.1)

s=1

Recall from §3.3 that H(r ) has a right action on ⊗r . The following result says that the left action of D(n) and the right action of H(r) on ⊗r commute. Proposition 3.5.5. For each r 1, the actions of D(n) and H(r ) on ⊗r commute. In other words, the tensor space ⊗r is a D(n)-H(r )-bimodule. Proof. Since D(n) is generated by the set + D := {K i±1 , E i , Fi , z± m | i ∈ I, m ∈ Z }

and H(r ) is generated by the set H := {Tk , X s±1 | 1 k < r, 1 s r}, it suffices to show that u · ωi · h = u · ωi · h

(3.5.5.1)

for all u ∈ D, h ∈ H , and i ∈ Zr . It is easy to check from the definition that (3.5.5.1) holds for u = K i±1 and arbitrary h ∈ H (resp., for h = X s±1 and arbitrary u ∈ D).

84

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Furthermore, by (3.5.4.1), for m 1 and i ∈ Zr , ±m z± Xs . m · ωi = ωi ·

(3.5.5.2)

1s r

This implies that, for any h ∈ H , ±m ±m (z± · ω ) · h = ω · X · h = (ωi · h) · X s = z± i i m s m · (ωi · h)

1s r

1s r

since 1s r X s±m are central elements in H(r ). Consequently, it remains to prove that (3.5.5.1) holds for i ∈ Zr , u = E i , Fi (i ∈ I ), and h = Tk (1 k < r ). Clearly, the subalgebra of D(n) generated by K i±1 , E j , and F j (i ∈ I and j ∈ I \{n}) is isomorphic to the quantum enveloping algebra Uv (gln ) of gln , and the subalgebra of H(r ) generated by Tk (1 k < r ) is isomorphic to the Hecke algebra H(r ) of Sr . Thus, by applying the result on quantum Schur algebras (see, e.g., [12, Lem. 14.23]), we obtain that (3.5.5.1) holds for u = E i , Fi (i ∈ I \{n}), h = Tk (1 k < r ), and i ∈ I (n, r ) = {(i 1 , . . . , ir ) ∈ Zr | 1 i s n, ∀s}. Suppose now i = (i 1 , . . . , ir ) ∈ I (n, r). Then, by definition, E n · ωi = δi j ,1 v g(i, j ) ωi−e j , 1 j r

where g(i, j ) = |{s | j < s r, i s = n}| − |{s | j < s r, i s = 1}| and e j = (δs, j )1s r ∈ Zr . In the following we show case by case that (E n · ωi ) · Tk = E n · (ωi · Tk ) for 1 k < r . Case i k = i k+1 . In this case, (E n · ωi ) · Tk = v 2 δi j ,1 v g(i, j ) ωi−e j + δik ,1 v g(i,k) ωi−ek · Tk 1 j r j =k,k+1

+ v g(i,k+1) ωi−ek+1 · Tk .

(3.5.5.3)

If i k = i k+1 = 1, then g(i, k) = g(i, k + 1) − 1. Hence, δik ,1 v g(i,k) ωi−ek · Tk + v g(i,k+1) ωi−ek+1 · Tk = δik ,1 v g(i,k) ωi+(n−1)ek · (X k Tk ) + v g(i,k+1) ωi+(n−1)ek+1 · (X k+1 Tk ) = δik ,1 v g(i,k)+2 ωi+(n−1)ek · (Tk−1 X k+1 ) + v g(i,k+1) ωi+(n−1)ek+1 · (Tk X k + (v 2 − 1)X k+1 ) = δik ,1 v g(i,k)+1 ωi−ek+1 + v g(i,k+1) (vωi−ek + (v 2 − 1)ωi−ek+1 ) = δik ,1 v 2 v g(i,k) ωi−ek + v g(i,k+1) ωi−ek+1 .

3.5. The D(n)-H(r )-bimodule structure on tensor spaces

85

Applying (3.5.5.3) gives that (E n · ωi ) · Tk = v 2 δi j ,1 v g(i, j ) ωi−e j = v 2 E n · ωi = E n · (ωi · Tk ). 1 j r

Case i k < i k+1 . In particular, i k+1 = 1. By definition, we have (E n · ωi ) · Tk = v g(i, j) ωi−e j · Tk =

1 j r, i j =1

v g(i, j) ωi+(n−1)e j · (X j Tk )

1 j r, i j =1

=

v g(i, j ) ωi+(n−1)e j · (Tk X j )+ δik ,1 v g(i,k) ωi+(n−1)ek · (v 2 Tk−1 X k+1 )

1 jr, i j =1 j =k,k+1

=

v g(i, j)+1 ωisk −e j + δik ,1 v g(i,k) v 1−δik+1 ,n ωisk −ek+1 .

1 jr, i j =1 j =k,k+1

Since g(isk , k + 1) = g(i, k) − δik+1 ,n and g(isk , j ) = g(i, j ) for j = k, k + 1, we get that (E n · ωi ) · Tk = v v g(isk , j ) ωisk −e j + vδik ,1 v g(isk ,k+1) ωisk −ek+1 1 jr, i j =1 j =k,k+1

= v E n · ωisk = E n · (ωi · Tk ). Case i k > i k+1 . In this case, i k = 1. It follows from the definition that (E n · ωi ) · Tk = v g(i, j ) ωi+(n−1)e j · (X j Tk ) =

1 j r, i j =1

v

g(i, j)

ωi+(n−1)e j · (Tk X j )

1 jr, i j =1 j =k,k+1

=

+ δik+1 ,1 v g(i,k+1) ωi+(n−1)ek+1 · (Tk X k + (v 2 − 1)X k+1 ) v g(i, j) (vωisk −e j + (v 2 − 1)ωi−e j ) 1 jr, i j =1 j =k,k+1

+ δik+1 ,1 v g(i,k+1) (v 1+δik ,n ωisk −ek + (v 2 − 1)ωi−ek+1 ) =v v g(i, j ) ωisk −e j + δik+1 ,1 v g(i,k+1)+δik ,n ωisk −ek 1 j r, i j =1 j =k,k+1

+ (v 2 − 1)

1 j r, i j =1

v g(i, j ) ωi−e j .

86

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

Since g(isk , k) = g(i, k + 1) − δik ,n and g(isk , j) = g(i, j) for j = k, k + 1, we have (E n · ωi ) · Tk = v v g(isk , j) ωisk −e j + (v 2 − 1)E n · ωi 1 jr (isk ) j =1

= v E n · ωisk + (v 2 − 1)E n · ωi = E n · (ωi · Tk ). Similarly, we have for i ∈ I (n, r ) and 1 k < r , (Fn · ωi ) · Tk = Fn · (ωi · Tk ). We conclude that (3.5.5.1) holds for i ∈ I (n, r ), u = E i , Fi , and h = Tk , where i ∈ I and 1 k < r . In general, for an arbitrary j ∈ Zr , write ωj = ωi · (X 1t1 · · · X rtr ) with i ∈ I (n, r ) for some t1 , . . . , tr ∈ Z. Let X be the subalgebra of H(r ) generated by X 1±1 , . . . , X r±1 . Then, by (3.3.0.2), X 1t1 · · · X rtr Tk = Tk x+y for some x, y ∈ X . By the above discussion, we infer that for u = E i or Fi (i ∈ I ), (u · ωj ) · Tk = (u · ωi ) · (X 1t1 · · · X rtr ) · Tk = (u · ωi ) · (Tk x + y) = u · (ωi · Tk ) · x + u · (ωi · y) = u · (ωi · Tk ) · x + u · (ωi · y) = u · ωi · (Tk x + y) = u · (ωj · Tk ). The proof is completed. For each r 1, the D(n)-H(r )-bimodule structure on ⊗r induces Q(v)algebra homomorphisms ξr :D(n) −→ EndH(r ) (⊗r ) and

ξr∨ :H(r ) −→ EndD(n) (⊗r )op .

(3.5.5.4)

Remark 3.5.6. By Remark 3.5.4, the PBW type basis action for the Ringel– Hall algebra D(n)− described in Proposition 3.5.3 induces via comultiplication an action on the tensor space ⊗r . Varagnolo and Vasserot [73, Lem. 8.2] have outlined a proof of the fact that this Ringel–Hall algebra action commutes with the action of H(r ). Proposition 3.5.3 and Remark 3.5.4 show that this Ringel–Hall algebra action coincides with the restriction of the double Ringel– Hall algebra action above, while the proof in Proposition 3.5.5 is a complete and more natural proof via the natural representation of D(n). See §3.6 for an explicit description of the map ξr and a further comparison with the work of Varagnolo and Vasserot. We now describe the action of semisimple generators of D(n) on ⊗r . Recall that, for each a = (ai ) ∈ Nn = Nn , ± Sa = ⊕i∈I ai Si , u ± u± a = u [Sa ] , and a =v

1i n ai (ai −1)

u± [Sa ] .

3.5. The D(n)-H(r )-bimodule structure on tensor spaces

87

The following result is a direct consequence of the definition of the comultiplications in Proposition 1.5.4(b) and Corollary 1.5.6(b ). See [73, 8.3] for the second formula. Proposition 3.5.7. For a ∈ Nn , we have (r −1) ( u+ a)= (s) (t) (1) ⊗ · · · ⊗ (1) v s>t a ,a u+ ⊗ u+ K u+ K (r−1) a(1) a(2) a a(r) a +···+a a=a(1) +···+a(r)

and (r −1) ( u− a)= (s) (t) (2) (r) ⊗ v s>t a ,a u− K u− K u− . (r) ⊗ · · · ⊗ a(1) −(a +···+a ) a(r−1) −a a(r ) a=a(1) +···+a(r)

This proposition together with Proposition 3.5.3 gives the following corollary. Corollary 3.5.8. For a = (ai ) ∈ Nn and i = (i 1 , . . . , ir ) ∈ Zr , we have m (m −1)ei s ,ei t ωi −m ⊗ · · · ⊗ ωi −m u+ v s>t t s r r and a · ωi = 1 1 m i ∈{0,1}, ∀i a=m 1 e i 1 −1 +···+m r eir −1

u− a · ωi =

v

s>t

m s (m t −1)ei s ,ei t

ωi1 +m 1 ⊗ · · · ⊗ ωir +m r .

m i ∈{0,1}, ∀i a=m 1 e i 1 +···+m r eir

In particular, if σ (a) =

i∈I

ai > r , then u+ u− a · ωi = 0 = a · ωi .

Corollary 3.5.9. If n > r , then ξr (D(n)) = ξr (U(n)). − ±1 + − Proof. By Proposition 1.4.3, D(n) is generated by u + i , u i , K i , u sδ , u sδ (i ∈ I , s ∈ Z+ ). If n > r , then dim Ssδ = sn > r . By Corollary 3.5.8, − ⊗r u+ sδ · ωi = 0 = u sδ · ωi , for all ωi ∈ . − Hence, ξr (u + sδ ) = 0 = ξr (u sδ ) for all s 1, and ξr (D(n)) is generated by the + − ±1 images of u i , u i , K i . This proves the equality.

Remark 3.5.10. We remark that, if z ∈ C is not a root of unity and D,C (n) is the specialized double Ringel–Hall algebra considered in Proposition 2.4.5, then there is a D,C (n)-H(r )C -bimodule action on the complex space ⊗r C which gives rise to algebra homomorphisms op ξr,C : D,C (n) −→ S(n, r )C and ξr,∨C : H(r )C −→ EndD,C (n) ( ⊗r C ) . (3.5.10.1)

88

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

3.6. A comparison with the Varagnolo–Vasserot action In [73], Varagnolo–Vasserot defined a Hall algebra action on the tensor space via the action on (Remark 3.5.4) and the comultiplication , and proved in [73, 8.3] that this action agrees with the affine quantum Schur algebra action via the algebra homomorphism ζr− described in Proposition 3.6.1(1) below. We will define the algebra homomorphism ζr+ in 3.6.1(2) opposite to ζr− and show that ξr in (3.5.5.4) is their extension. In other words, ξr coincides with ζr+ and ζr− upon restriction (Theorem 3.6.3). We first follow [73] to define an algebra homomorphism from the Ringel– Hall algebra H(n) of the cyclic quiver (n) to the affine quantum Schur algebra S(n, r ). This definition relies on an important relation between cyclic flags and representations of cyclic quivers. Recall from (3.1.1.3) the geometric characterization of affine quantum Schur algebras and the flag varieties Y = F(q), X = B(q), and G = G(q) over the finite field F = Fq . Let L = (L i )i ∈Z , L = (L i )i∈Z ∈ Y satisfy L ⊆ L i.e., L i ⊆ L i for all i ∈ Z. By [32, §9] and [56, 5.1], we can view L/L as a nilpotent representation V = (Vi , f i ) of (n) over F such that Vi = L i /L i and f i is induced by the inclusion L i ⊆ L i +1 for each i ∈ I . Here we identify L n+1 /L n+1 with L 1 /L 1 via the multiplication by ε. Further, for L ⊆ L , we define the integers a(L, L ) = dimF (L i /L i )(dimF (L i+1 /L i ) − dimF (L i /L i )) and 1i n

c(L , L) =

dimF (L i+1 /L i+1 )(dimF (L i+1 /L i ) − dimF (L i /L i )).

1i n

(3.6.0.1) Recall also from (1.2.0.9) the geometric characterization H of the Ringel– 1 Hall algebra H(n). Thus, by specializing v to q 2 , there is a C-algebra isomorphism H(n) ⊗Z C −→ H, u A −→ O A = q − 2 dim OA χO A , 1

where O A is the G V -orbit in the representation variety E V corresponding to the isoclass of M(A). By identifying S(n, r ) ⊗Z C with CG (Y × Y ), H(n) ⊗Z C with H (and hence, H(n)op ⊗Z C with Hop ), and recalling the elements A(j, r ) ∈ S(n, r ), for each A ∈ ± (n) and j ∈ Zn , defined in (3.4.0.2), [73, Prop. 7.6] can now be stated as the first part of the following (cf. footnote 2 of Chapter 1). Proposition 3.6.1. (1) There is a Z-algebra homomorphism ζr− : H(n)op −→ S(n, r ), u A −→ t A(0, r ),

for all A ∈ + (n)

3.6. A comparison with the Varagnolo–Vasserot action

89

such that the induced map ζr− ⊗ idC : Hop −→ CG (Y × Y ) is given by 1 q − 2 a(L,L ) f (L /L), if L ⊆ L ; − (ζr ⊗ idC )( f )(L, L ) = (3.6.1.1) 0, otherwise. (2) Dually, there is a Z-algebra homomorphism ζr+ : H(n) −→ S(n, r ), u A −→ A(0, r ), for all A ∈ + (n) with the induced map ζr+ ⊗ idC : H −→ CG (Y × Y ) given by 1 q − 2 c(L,L ) f (L/L ), if L ⊆ L; + (ζr ⊗ idC )( f )(L, L ) = 0, otherwise.

(3.6.1.2)

Proof. Statement (1) is given in [73, 7.6]. We only need to prove (2). We first observe from the proof of [56, Lem. 1.11] that 2 2 1 1 t dA − d A = ai, j − ai, j . 2 2 1i n

Let

ζr+

1 j n

j ∈Z

i∈Z

be the composition of the algebra homomorphisms (ζr− )op

τr

H(n) −→ S(n, r )op −→ S(n, r ), where τr is the anti-involution on S(n, r ) given in (3.1.3.4). Thus, τr induces a map τr ⊗ id : CG (Y ×Y ) → CG (Y × Y ). Applying this to the characteristic function χ A of the orbit O A in Y × Y , noting (3.1.0.1), yields 1

1

(τr ⊗ id)(χ A )(L, L ) = q 2 (d A −dt A ) χt A (L, L ) = q 2 (d A −dt A ) χ A (L , L) which is non-zero if and only if (L , L) ∈ O A . This implies ai, j = dimF

L i ∩L j . ∩L +L ∩L L i−1 j j−1 i

j∈Z ai, j

Hence, by the proof of [56, 1.5(a)], dimF (L i /L i −1 ) = and dimF (L j /L j−1 ) = i∈Z ai, j . Putting

b(L, L ) =

1 ( ((dimF (L i /L i −1 ))2 − (dimF (L i /L i −1 ))2 )), 2 1i n

1i n

1 2 b(L,L )

we obtain (τr ⊗ id)(χ A )(L, L ) = q χ A (L , L). Hence, 1 (τr ⊗ id)(g)(L, L ) = q 2 b L,L ) g(L , L), for g ∈ CG (Y × Y . Taking g = (ζr− ⊗ id)( f ) with f ∈ H, and applying (3.6.1.1) gives 1 q 2 (b(L,L )−a(L ,L)) f (L/L ), if L ⊆ L; − op (τr ⊗id)((ζr ) ⊗id)( f )(L, L ) = 0, otherwise. (3.6.1.3)

90

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

For L ⊆ L, we have dimF (L i+1 /L i )2 1i n

=

1i n

=2

(dimF (L i +1 /L i+1 ) + dimF (L i +1 /L i ) − dimF (L i /L i ))2

(dimF (L i /L i ))2 +

1i n

+2

1i n

−2

(dimF (L i+1 /L i ))2

1i n dimF (L i+1 /L i+1 ) dimF (L i +1 /L i ) dimF (L i+1 /L i+1 ) dimF (L i /L i )

1i n

−2

dimF (L i+1 /L i ) dimF (L i /L i ).

1i n

Hence, b(L, L ) − a(L , L) = −c(L, L ) and (3.6.1.2) now follows from (3.6.1.3). For notational simplicity, we write A(j, r ) for A(j, r ) ⊗ 1 in S(n, r ) ⊗Z C. 1 By taking f to be q − 2 dim O A χO A in (3.6.1.1) and (3.6.1.2), we obtain the following. Corollary 3.6.2. For A ∈ + (n) and (L, L ) ∈ Y × Y , we have 1 q − 2 (c(L,L )+dim OA ) , if L ⊆ L and L/L ∈ O A ; A(0, r )(L, L ) = 0, otherwise and t

A(0, r )(L, L ) =

q − 2 (a(L,L )+dim OA ) ,

if L ⊆ L and L /L ∈ O A ;

0,

otherwise.

1

We now identify H(n) as D(n)+ and H(n)op as D(n)− via (2.1.3.3). Then ζr± induce Q(v)-algebra homomorphisms ζr± : D(n)± −→ S(n, r ).

(3.6.2.1)

Theorem 3.6.3. For every r 0, the map ξr : D(n) → S(n, r ) defined in (3.5.5.4) is the (unique) algebra homomorphism satisfying + + ξr (K 1 1 · · · K n n ) = 0(j, r ), ξr ( u+ u A ) = A(0, r ), and A ) = ζr ( j

j

− − ξr ( u− u A ) = t A(0, r ), A ) = ζr (

3.6. A comparison with the Varagnolo–Vasserot action

91

for all j = ( j1 , . . . , jn ) ∈ Zn and A ∈ + (n). In particular, we have ξr |D(n)± = ζr± . Proof. Since D(n) is generated by K i±1 , 1 i n, together with semisimn n ple generators u ± i =1 λi E i,i +1 . By Proposition 3.6.1, Aλ , λ ∈ N, where Aλ = it suffices to prove t (1) ξr (K i ) = 0(ei , r ); (2) ξr ( u− u+ Aλ ) = ( Aλ )(0, r); and (3) ξr ( Aλ ) = Aλ (0, r ).

To prove them, we apply Propositions 3.3.1 and 3.5.5 to compare the actions of both sides on ωi = [ Ai ], for all i ∈ I(n, r )0 . Suppose i = iμ . By (3.5.0.1), K i · ωi = v μi ωi . Since ro(Ai ) = μ, (3.1.3.4) implies 0(ei , r )[Ai ] = v μi [Ai ]. Hence, 0(ei , r) · ωi = K i · ωi , proving (1). The proof of (2) is given in [73, 8.3]. We now prove (3) for completeness. So we need to show that n Aλ (0, r ) · ωi = u+ λ · ωi , for all i ∈ I(n, r)0 and λ ∈ N.

(3.6.3.1)

The equality is trivial if σ (λ) > r as both sides are 0. We now assume σ (λ) r and prove the equality by writing both sides as a linear combination of the basis {[Ai ]}i∈I(n,r) ; cf. Remark 3.3.2. By Proposition 3.3.1, the left-hand side of (3.6.3.1) becomes Aλ (0, r ) · ωi = Aλ (0, r )[Ai ]. We now compute this by regarding Aλ (0, r )[Ai ] as the convolu1 tion product q − 2 di Aλ (0, r ) ∗ χi ; see Remark 3.1.2 and compare [73, 8.3]. By the definition (3.1.1.2), for j ∈ I(n, r ) and λ ∈ Nn , ( Aλ (0, r ) ∗ χi )(Lj , L∅ ) =

(Aλ (0, r ))(Lj , L)χi (L, L∅ )

L∈Y

=

(Aλ (0, r ))(Lj , L),

L∈Y (L,L∅ )∈Oi

where Lj is defined in (3.1.3.7), and Oi = O Ai is the orbit containing (Li , L∅ ) (see also (3.1.3.7) for the definition of L∅ ). For L = (L i ) ∈ Y and (L, L∅ ) ∈ Oi , there is g ∈ G such that (L, L∅ ) = g(Li , L∅ ). In other words, L = gLi and L∅ = gL∅ . The fact that i ∈ I(n, r )0 implies that for each t ∈ Z, there exists lt ∈ Z such that Li,t = L∅,lt . (More precisely, if i = iμ and t = s + kn with 1 s n, then lt = μ1 + · · · + μs + kr .) Thus, L t = g(Li,t ) = g(L∅,lt ) = L∅,lt = Li,t for any t ∈ Z. This implies that L = Li . Hence, by Corollary 3.6.2, 1

(Aλ (0, r) ∗ χi )(Lj , L∅ ) = (Aλ (0, r ))(Lj , Li ) = q − 2 (c(j,i)+dim OAλ )

92

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

if and only if Li ⊆ Lj , Lj /Li ∼ = M(Aλ ), where c(j, i) = c(Lj , Li ). The latter is equivalent by definition to the conditions {k ∈ Z | i k t} ⊆ {k ∈ Z | jk t} ⊆ {k ∈ Z | i k t + 1} and dimF Lj,t /Li,t = λt , for all t ∈ Z. Hence, Li ⊆ Lj , Lj /Li ∼ = M(Aλ ) ⇐⇒i t − 1 jt i t and λt = |j−1 (t) ∩ i−1 (t + 1)| for all t ∈ Z ⇐⇒j = i − m and λ = m 1 ei1 −1 + · · · + m r eir −1 , for some m s ∈ {0, 1}, since, for all s ∈ Z, m s = 1 ⇐⇒ js = i s − 1 ⇐⇒ s ∈ ∪t∈Z (j−1 (t) ∩ i−1 (t + 1)). r is uniquely determined by (m , m , . . . , m ) ∈ Zr and e ∈ Zr Here, m ∈ Z 1 2 r i corresponds to ei = (δk,i )1k r under (1.1.0.2). (Note that i − m ∈ I(n, r ) if i ∈ I(n, r ).) Since dim O Aλ = 0 (see, e.g., [12, (1.6.2)]), we obtain, for i ∈ I(n, r)0 and λ ∈ Nn , 1 1 Aλ (0, r )[ Ai ] = q − 2 di Aλ (0, r ) ∗ χi = q 2 (−c(i−m,i)−di ) χi−m , m∈M

m i ∈ {0, 1}, λ = m 1 ei1 −1 + · · · + m r eir −1 }. Hence, Aλ (0, r ) · ωi = v −c(i−m,i)−di +di−m [Ai−m ]. (3.6.3.2)

where M = {m ∈

Zr

|

m∈M

We now calculate the right-hand side of (3.6.3.1). By Corollary 3.5.8, + u+ · ω = u · (ω ⊗ · · · ⊗ ω ) = v −c (i,m) ωi1 −m 1 ⊗ · · · ⊗ ωir −m r , i i i r 1 λ λ m∈M

where c (i, m) :=

t<s

m t (1 − m s )e i s , ei t =

m t (1 − m s )δit ,is − d(i, m),

1t<s r

with d(i, m) = 1t<s r m t (1 − m s )δi t ,i s +1 . To make a comparison with (3.6.3.2), we need to write ωi−m := ωi1 −m 1 ⊗ · · · ⊗ ωir −m r as a linear combination of the basis {[Aj ]}j∈I(n,r) . For i ∈ I(n, r )0 , order the set {t ∈ Z | 1 t r, i t = 1, m t = 1} = {t1 , t2 , . . . , ta } by t1 < t2 < · · · < ta . Then, by definition, ωi−m X t−1 X t−1 · · · X t−1 = ωj , a 1 2 where j = i − m + n(et1 + et2 + · · · + eta ) ∈ I (n, r ). By Proposition 3.3.1 (see also Remark 3.3.2), ωj = [Aj ]. Thus, ωi−m = ωj X ta X ta−1 · · · X t1 = [ Aj ]X ta X ta−1 · · · X t1 .

3.6. A comparison with the Varagnolo–Vasserot action

93

ρ −1 = T r−1 · · · T 2 T 1 X 1 , it follows that Since T −1 · · · T −1 T −1 Xk = T k r−2 r−1 Tρ −1 T1 T2 · · · Tk−1 , t−1 = T ta − (v − v −1 ), for 1 k r . Now, by Proposition 3.2.10 and noting T a v −1 [ Ajsta ], if n = jta = jta +1 ; t−1 = [ Aj ] T a [Ajsta ], if n = jta > jta +1 . Thus, repeated application of Proposition 3.2.10 gives −b jsta ···sr−1 ρ −1 t−1 · · · T −1 T −1 [Aj ]T ], r−2 r−1 Tρ −1 = v [A a

where b = |{s ∈ Z | ta < s r, i s = n, m s = 0}|. Since 1 i 1 · · · ir n, it follows that b = |{s ∈ Z | 1 s r, i s = n, m s = 0}|. If = jsta · · · sr−1 , j = jsta · · · sr−1 ρ −1 , then jr = n and so j1 = j0 = 0 < jk for all 2 k r . The last inequality is seen from the fact that { jk }2k r = { jk }1k r−1 contains only positive integers. Applying Proposition 3.2.10 again yields j

1 · · · T ta −1 = [ Aj s1 ···sta −1 ] = [Aj−neta ]. [Aj ]T

Hence, [Aj ]X ta = v −b [ Aj−neta ]. Continuing this argument, we obtain eventually ωi−m = v −ab [Ai−m ], where ab = |{(s, t) ∈ Z2 | 1 s, t r, i s = n, m s = 0, i t = 1, m t = 1}| = m t (1 − m s )δis ,n δit ,1 = m t (1 − m s )δis ,n δit ,1 = d(i, m), 1s,t r

1t<s r

since 1 t < s r and i t = i s + 1 imply i s = n and i t = 1. Therefore, we obtain u+ v −c (i,m)−d(i,m) [Ai−m ]. λ · ωi = m∈M

Comparing this with (3.6.3.2), it remains to prove that d(i, m) + c (i, m) = c(i − m, i) + di − di−m . The number c(i − m, i) = c(Li−m , Li ) is defined in (3.6.0.1). Since dim Li,i +1 /Li,i = |i−1 (i + 1)| = δis ,i +1 and λi =

1s r

1s r

m s δis −1,i ,

for i ∈ Z,

94

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

it follows that c(i − m, i) =

1i n

=

1s

=

λi +1 (|i−1 (i + 1)| − λi ) =

1s,t r

m t (1 − m s )δit −1,is +

1t<s r

m t (1 − m s )δit −1,is

m t (1 − m s )δit −1,is .

m t (1 − m s )δit −1,is (1 − δis ,n ) + d(i, m).

1s
Hence, c(i − m, i) − c (i, m) = 2d(i, m) + −

m t (1 − m s )δit −1,is (1 − δis ,n )

1s
m t (1 − m s )δis ,it .

1t<s r

On the other hand, for any i ∈ I(n, r ), di = |Inv(i)| by Lemma 3.1.4. Observe that the set I :={(s, t) ∈ L | i s i t + 1} ∪ {(s, t) ∈ L | i s = i t , m s = m t or m s = 0, m t = 1} is the intersection Inv(i − m) ∩ Inv(i), where L = {(s, t) ∈ Z2 | 1 s r, s < t}. It turns out that Inv(i − m)\I = {(s, t) ∈ L | i s + 1 = i t , m s = 0, m t = 1} and Inv(i)\I = {(s, t) ∈ L | i s = i t , m s = 1, m t = 0}. Hence, di−m − di = m t (1 − m s )δit ,is +1 − m s (1 − m t )δit ,is . 1sr t∈Z, s
1sr t∈Z, s
Since m a+r = m a , i a+r = i a + n, for all a ∈ Z, and i ∈ I(n, r )0 , it follows that i s = i t , for all 1 s r, t > r , and i t = n + 1 ⇐⇒ i t = 1 for some 1 t r with t = t + r. Thus, the above sums can be rewritten as di−m − di = m t (1 − m s )δit ,is +1 (1 − δis ,n ) 1s
+

m t (1 − m s )δit ,1 δis ,n −

1t<s r

= c(i − m, i) − c (i, m) − d(i, m), as required.

1s
m s (1 − m t )δit ,is

3.7. Triangular decompositions of affine quantum Schur algebras

95

3.7. Triangular decompositions of affine quantum Schur algebras In this section we study the “triangular parts” of S(n, r ). We first show that S(n, r ) admits a triangular relation for certain structure constants relative to the BLM basis. This relation allows us to produce an integral basis from which we obtain a triangular decomposition. We will give an application of this decomposition in the next section by proving that the algebra homomorphism ξr : D(n) → S(n, r ) defined in §3.5 is surjective. Keep the notation in the previous sections. We will continue to use the convolution product to derive properties or formulas via the isomorphism mentioned in Remark 3.1.2. Thus, when the convolution product ∗ is used, we automatically mean that the affine quantum Schur algebra is the algebra √ √ √ S(n, r ) R with R = Z[ q, q −1 ] obtained by specializing v to q, for a prime power q, and is identified with the convolution algebra RG (Y × Y ). The following results are taken from [56]. Lemma 3.7.1. Let A = (ai, j ) ∈ (n, r ) and let (L, L ) ∈ O A , where L = (L i )i∈Z and L = (L i )i∈Z . (1) A is upper triangular if and only if L i ⊆ L i for all i. (2) A is lower triangular if and only if L i ⊆ L i for all i. (3) dim(L i /L i−1 ) = k∈Z ai,k and dim(L i /L i−1 ) = k∈Z ak,i . L j Li (4) dim L ∩L = s i,t j as,t and dim L ∩L = s i,t j as,t . i

j −1

i−1

j

For A ∈ M,n (Z), let ⎧ as,t , ⎪ ⎨ s i,t j σi, j (A) = ⎪ as,t , ⎩ s i,t j

if i < j; if i > j.

For any fixed x0 ∈ Z and i < j, it is easy to see that there are two bijective maps {(b, s, t) ∈ Z3 | s − bn i < j t − bn, s ∈ [x0 + 1, x0 + n]} −→ {(s, t) ∈ Z2 | s i, t j}, (b, s, t) −→ (s − bn, t − bn)

96

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

and {(b, s, t) ∈ Z3 | t − bn i < j s − bn, s ∈ [x0 + 1, x0 + n]} −→ {(s, t) ∈ Z2 | s j, t i }, (b, s, t) −→ (s − bn, t − bn). Thus, we obtain an alternative interpretation of σi, j ( A): ⎧ ⎪ as,t |{b ∈ Z | s − bn i < j t − bn}|, ⎪ ⎪ ⎨ x0 +1s x0 +n s
s>t

if i < j ; if i > j .

0

In particular, σi, j (A) < ∞. Further, for A, B ∈ M,n (Z), define (see [24, §6]) B A ⇐⇒ σi, j (B) σi, j (A), for all i = j.

(3.7.1.1)

We put B ≺ A if B A and, for some pair (i, j) with i = j, σi, j (B) < σi, j (A). It is shown in [24, Th. 6.2] that, if A, B ∈ + (n) satisfy d(A) = d(B), then B dg A ⇐⇒ B A. In other words, the ordering is an extension of the degeneration order defined in (1.2.0.2). For A ∈ (n), write A = A+ + A0 + A− = A± + A0 , where A+ ∈ + (n), − A ∈ − (n), A± = A+ + A− , and A0 is a diagonal matrix. Here − (n) := {tB | B ∈ + (n)}. Lemma 3.7.2. Let A ∈ ± (n) with σ (A) r . (1) For μ ∈ (n, r − σ (A+ )), ν ∈ (n, r − σ (A− )), if ([A+ + diag(μ)] ∗ [ A− + diag(ν)])(L, L ) = 0, where (L, L ) ∈ O B for some B ∈ (n), then B A. (2) For λ ∈ (n, r − σ ( A)), if (L, L ) ∈ O A+diag(λ) , then {L ∈ F | (L, L ) ∈ O A+ +diag(μ) , (L , L ) ∈ O A− +diag(ν) } μ∈ (n,r−σ ( A+ )) ν∈ (n,r−σ ( A− ))

= {L ∩ L }.

3.7. Triangular decompositions of affine quantum Schur algebras

97

Proof. (1) Since ([A+ + diag(μ)] ∗ [ A− + diag(ν)])(L, L ) = 0, there exists L ∈ F such that (L, L ) ∈ O A+ +diag(λ) and (L , L ) ∈ O A− +diag(μ) . Also, (L, L ) ∈ O B . Hence, by Lemma 3.7.1(1)&(2), L ⊆ L and L ⊆ L , and by Lemma 3.7.1(4), ⎧ ⎪ Li ⎪ dim , if i < j ; ⎨ L ∩L i j−1 σi, j (A) = and Lj ⎪ ⎪ dim , if i > j , ⎩ L ∩L i−1

j

i−1

j

⎧ ⎪ Li ⎪ , if i < j ; ⎨dim L i ∩L j−1 σi, j (B) = L j ⎪ ⎪ , if i > j . ⎩dim L ∩L Therefore,

L

⊆

L ∩ L

and

⎧ L i ∩L j−1 ⎪ ⎪ 0, ⎨dim L i ∩L j−1 σi, j (A) − σi, j (B) = L i−1 ∩L ⎪ ⎪ ⎩dim L ∩L j 0, i−1

j

if i < j; if i > j.

Consequently, B A. (2) If L ∈ F satisfies (L, L ) ∈ O A+ +diag(μ) and (L , L ) ∈ O A− +diag(ν) , for some μ ∈ (n, r − σ (A+ ) and ν ∈ (n, r − σ (A− ), then L ⊆ L ∩ L as seen above. Thus, for all i ∈ Z, L i ∩ L i Li dim = dim(L i /L i ) − dim L i L i ∩ L i Li Li = dim − dim L i ∩ L i L i ∩ L i = as,t − as,t = 0, s i
s i
by Lemma 3.7.1(4) again. Hence, L = L ∩ L , proving the assertion. Proposition 3.7.3. Let A ∈ ± (n). Then the following triangular relation relative to ≺ holds: A+ (0, r )A− (0, r ) = A(0, r ) + f C, A [C] (in S(n, r )) C∈(n,r) C ≺A

= A(0, r ) +

± (n) B∈ n B≺A,j∈Z

g B,j, A;r B(j, r ) (in S(n, r )),

98

3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

where f C, A ∈ Z, g B,j,A;r ∈ Q(v). Proof. Since the elements B(j, r ) span S(n, r ) by Proposition 3.4.1, the second equality follows from the first one. We now prove the first equality. Let r ± = σ (A), r + = σ ( A+ ), and r − = σ ( A− ). There is nothing to prove if r ± > r . Assume now r ± r . By (3.1.1.1), A+ (0, r ) A− (0, r ) = [A+ + diag(μ)][A− + diag(ν)] =

C∈(n,r ) μ,ν

=

μ∈(n,r−r + ) ν∈(n,r−r − )

v −d A+ +diag(μ) −d A− +diag(ν) p A+ +diag(μ),A− +diag(ν),C v dC [C]

f C, A [C],

C∈(n,r )

where f C,A =

v dC −d A+ +diag(μ) −d A− +diag(ν) p A+ +diag(μ),A− +diag(ν),C ∈ Z.

μ∈ (n,r−r + ) ν∈ (n,r−r − )

If f C,A = 0, then p A+ +diag(μ),A− +diag(ν),C = 0 for some μ, ν as above. Thus, by definition, there is a finite field F of q elements such that p A+ +diag(μ),A− +diag(ν),C |v 2 =q = n A+ +diag(μ),A− +diag(ν),C;q = 0, where n A+ +diag(μ),A− +diag(ν),C;q is defined in (3.1.0.6). By Lemma 3.7.2(1), we conclude C A. We need to prove that f C,A = 1, for all C = A + diag(λ) with λ ∈ (n, r − r ± ). First, Lemma 3.7.2(2) implies that there exist unique μ, ν such that n A+ +diag(μ),A− +diag(ν),A+diag(λ);q = 1. Thus, f A+diag(λ),A = v d A+diag(λ) −d A+ +diag(μ) −d A− +diag(ν) .

(3.7.3.1)

Second, since d A+diag(λ) = d A+ +diag(μ) + d A− +diag(ν) by a direct computation (see Lemma 3.10.2), we obtain f A+diag(λ),A = 1. Finally, since is a partial order on ± (n) by [24, Lem. 6.1], we conclude that A+ (0, r )A− (0, r ) = A(0, r) + g, where g is a Z-linear combination of [C] with C ∈ (n, r) and C ≺ A. We now use the triangular relation to establish a triangular decomposition for the Z-algebra S(n, r ). Consider the following Z-submodules of S(n, r ) S(n, r )+ = spanZ {A(0, r ) | A ∈ + (n)},

S(n, r )− = spanZ {A(0, r ) | A ∈ − (n)}, and S(n, r ) = spanZ {[diag(λ)] | λ ∈ (n, r )}. 0

(3.7.3.2)

3.7. Triangular decompositions of affine quantum Schur algebras

99

As homomorphic images of Ringel–Hall algebras (see Proposition 3.6.1), both S(n, r )+ and S(n, r )− are Z-subalgebras of S(n, r ). It can be directly checked that S(n, r)0 is also a Z-subalgebra of S(n, r). Moreover, S(n, r )0 is isomorphic to the 0-part of the (non-affine) quantum Schur algebra. We will call the subalgebras S(n, r )− , S(n, r )0 , and S(n, r )+ the negative, zero, and positive parts of S(n, r ), respectively. The next result shows that all three X ;a subalgebras are free as Z-modules. Recall the notation t introduced in (1.1.0.6). Proposition 3.7.4. Let ki = 0(ei , r ), 1 i n. (1) The elements A(0, r ) (resp., t A(0, r )), for A ∈ + (n) with σ (A) r , form a Z-basis of S(n, r )+ (resp., S(n, r )− ). In particular, ξr maps H(n)± onto S(n, r )± . (2) For λ ∈ (n, r ), k1 ; 0 k2 ; 0 kn ; 0 lλ := ··· = [diag(λ)]. λ1 λ2 λn In particular, the set {lλ | λ ∈ (n, r)} forms a Z-basis of S(n, r )0 . Proof. Assertion (1) follows from the definition of S(n, r )± and Proposition 3.4.1. To see (2), since ki [diag(μ)] = v νi [diag(ν)][diag(μ)] =

ν∈(n,r) v μi [diag(μ)], by (3.1.3.4), it follows that kit;0 [diag(μ)] = μt i [diag(μ)]. Hence, for λ ∈ (n, r ), μ1 μn lλ = lλ [diag(μ)] = ··· [diag(μ)] = [diag(λ)], λ1 λn μ∈(n,r)

μ∈(n,r )

as desired.4 As in §2.3, let D(n)± (resp., H(n)± ) be the Q(v)-submodules (resp., Z+ submodules) of D(n) spanned by u ± A , for A ∈ (n). They are respectively the Q(v)-subalgebras and Z-subalgebras of D(n). The above proposition together with the results in §3.6 gives the following result; see Remark 2.4.6. + − − Corollary 3.7.5. For each s 1, we have z+ s ∈ H(n) and zs ∈ H(n) . + − − Proof. We only prove z+ s ∈ H(n) . The proof for zs ∈ H(n) is similar. By Proposition 3.6.1 and Theorem 3.6.3, the restriction of ξr : D(n) → S(n, r ) gives an algebra homomorphism

ζr+ : D(n)+ −→ S(n, r ) = EndH(r) (⊗r ) 4 In the literature, l is denoted by k or 1 . We modified the notation in order to introduce its λ λ λ preimage Lλ in D(n); see (5.1.1.1).

100 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity + taking u+ A → A(0, r) for A ∈ (n). Write z+ f A u+ s = A, A∈+ (n) ⊗r ) ⊆ ⊗r . where all but finitely many f A ∈ Q(v) are zero. By (3.5.4.1), z+ s ( + + ⊗r Hence, ζr (zs ) ∈ EndH(r) ( ) = S(n, r ); see Proposition 3.3.1. In other words, ζr+ (z+ ) = f A(0, r ) = f A A(0, r ) ∈ S(n, r )+ . A s A∈+ (n)

A∈+ (n), σ (A)r

By Proposition 3.7.4(1), {A(0, r) | A ∈ + (n), σ (A) r } is a Z-basis for S(n, r )+ . Hence, f A ∈ Z, for all A ∈ + (n) with σ ( A) r . Since r can be chosen to be an arbitrary positive integer, it follows that f A ∈ Z, for all + A ∈ + (n). We conclude that z+ s ∈ H(n) . We now patch the three bases to obtain a basis for S(n, r ). For A ∈ (n), define (cf. [4]) ( j − i)( j − i + 1) (i − j)(i − j + 1) ||A|| = ai, j + ai, j . 2 2 1in 1i n i< j

i> j

Lemma 3.7.6. For A ∈ (n), the equality || A|| = σi, j (A) + σi, j (A) 1i n i< j

1in i> j

holds. In particular, if A, B ∈ (n) satisfy A ≺ B, then || A|| < ||B||. Proof. By definition, we have σi, j (A) = as,t = as,t |{(i, j ) | s i < j t}| 1i n si< jt

1in i< j

=

1sn s
(t − s)(t − s + 1) as,t 2 1sn

and

s

σi, j (A) =

1i n t j
1in i> j

=

as,t =

as,t |{(i, j ) | t j < i s}|

1sn s>t

(s − t)(s − t + 1) as,t . 2 1sn s>t

The assertion follows from the definition of ||A||.

3.7. Triangular decompositions of affine quantum Schur algebras 101 For A ∈ (n) and i ∈ Z, define the “hook sum” σi ( A) = ai,i + (ai, j + a j,i ). j
It is easy to see that σ (A) = σ (A) = (σi ( A))i∈Z ∈ Nn

1i n

and

σi (A). Let p A = A+ (0, r )lσ ( A) A− (0, r ). (3.7.6.1)

We now describe a PBW type basis for S(n, r ). Theorem 3.7.7. Keep the notation introduced above. There exist g B,A ∈ Z such that p A = [A] + g B,A [B]. B∈(n,r),B≺A

Moreover, the set Pr := {p A | A ∈ (n, r )} forms a Z-basis for S(n, r). In particular, we obtain a (weak) triangular decomposition: S(n, r ) = S(n, r )+ S(n, r )0 S(n, r )− . Proof. By the notational convention right above Lemma 3.7.2, if A ∈ (n, r ), then σi (A± ) is the i th component of co(A+ ) + ro(A− ) and lσ ( A) A− (0, r ) = [diag(σ (A))] A− (0, r ) = A− (0, r )[diag(σ (A) + co(A− ) − ro(A− ))]. On the other hand, A+ (0, r )A− (0, r ) = A± (0, r ) + g, where g is a Z-linear combination of [B] with B ∈ (n, r ) and B ≺ A, by Proposition 3.7.3. Thus, p A = A+ (0, r )lσ ( A) A− (0, r ) = A+ (0, r )A− (0, r )[diag(σ (A) + co(A− ) − ro(A− ))] = A± (0, r )[diag(σ (A) + co(A− ) − ro(A− ))] + g = [ A± + diag(σ ( A) − (co(A+ ) + ro(A− ))] + g = [ A] + g , where g is the Z-linear combination of [B] with B ∈ (n, r ) and B ≺ A. Thus, the set Pr is linearly independent. To see that it spans, we can apply Lemma 3.7.6 and an induction on ||A|| to show that [A] is a Z-linear combination of p B with B ∈ (n, r ). Hence, Pr forms a Z-basis for S(n, r). The last assertion follows from Proposition 3.7.4.

102 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

3.8. Affine quantum Schur–Weyl duality, I We now use the triangular decomposition given in Theorem 3.7.7 to partially establish an affine analogue of the quantum Schur–Weyl reciprocity. As in Remark 2.3.6(4), for each m 0, let D(n)(m) (resp., D(n)(0) ) − ±1 + − denote the subalgebra of D(n) generated by u + i , u i , K i , zs , zs (resp., + − ±1 u i , u i , K i ) for i ∈ I , 1 s m (resp., for i ∈ I , m = 0). Thus, D(n)(0) = U(n) as defined in Remark 2.3.6(2). Theorem 3.8.1. Let n, r be two positive integers with n 2. (1) The algebra homomorphism ξr : D(n) → S(n, r) is surjective. (2) If we write r = mn + m 0 with m 0 and 0 m 0 < n, then ξr induces a surjective algebra homomorphism D(n)(m) → S(n, r ). In particular, if n > r , then ξr induces a surjective algebra homomorphism U(n) → S(n, r ). (3) If n r , then the map ξr∨ : H(r ) → EndD(n) (⊗r )op given in (3.5.5.4) is an isomorphism. Proof. (1) As in Remark 2.4.6, let H(n)+ (resp., H(n)− ) be the Z− + subalgebra of D(n) generated by u + A (resp., u A ) for all A ∈

(n), and ±1 let D(n)0 be the Z-subalgebra of D(n) generated by K i and K it ;0 , for i ∈ I and t > 0 (see (2.4.1.2) for bases of D(n)0 ), and set (n) := H(n)+ D(n)0 H(n)− ∼ D = H(n)+ ⊗ D(n)0 ⊗ H(n)− . (3.8.1.1) By Theorem 3.6.3 and Proposition 3.7.4, ξr maps H(n)ε onto S(n, r )ε for (n) onto S(n, r ) by ε = +, − and D(n)0 onto S(n, r )0 . Hence, ξr maps D Theorem 3.7.7. Taking a base change to Q(v) gives the required surjectivity. − ±1 + − (2) By Proposition 1.4.3, D(n) is generated by u + i , u i , K i , u sδ , u sδ (i ∈ I , s ∈ Z+ ). For each s 1, the semisimple module Ssδ has dimension sn. Thus, if s > m 1, then dim Ssδ > r . Moreover, in this case, we have by Corollary 3.5.8 that, for each ωi ∈ ⊗r , − u+ sδ · ωi = 0 = u sδ · ωi . − In other words, ξr (u + sδ ) = 0 = ξr (u sδ ) whenever s > m and m 1. The asser− tion follows from the fact that D(n)(m) is generated by u i+ , u i− , K i±1 , u + sδ , u sδ (i ∈ I , 0 s m); see Remark 2.3.6(4). The last assertion follows from Corollary 3.5.9. (3) By (1), we have EndD(n) (⊗r ) = EndS(n,r) (⊗r ). On the other hand, by Lemma 3.1.3 and Proposition 3.3.1, H(r ) ∼ = eω S(n, r )eω and ⊗r ∼ =

3.8. Affine quantum Schur–Weyl duality, I

103

S(n, r )eω (see the proof of Lemma 3.1.3). Hence, EndS(n,r) (⊗r ) ∼ = EndS(n,r) (S(n, r )eω ) ∼ = (eω S(n, r )eω )op . This implies that ξr∨ is an isomorphism. Combining the above theorem with Corollary 3.5.8 yields the following result which can also be derived from [56, §4.1, Th 8.2].5 Corollary 3.8.2. Suppose n > r . Then ξr : D(n) → S(n, r) induces a surjective Z-algebra homomorphism θr : U(n) −→ S(n, r ), where U(n) is the Z-subalgebra of D(n) generated by K i±1 , and (u i− )(m) for i ∈ I and t, m 1 (see §2.3).

K i ;0 + (m) , (u i ) t

Proof. By Proposition 3.7.4, ξr : D(n) → S(n, r ) induces surjective Zalgebra homomorphisms ξr,+Z : H(n)+ −→ S(n, r )+ and ξr,−Z : H(n)− −→ S(n, r )− . By [13, Th. 5.2], H(n)+ is generated by (u i+ )(m) and u + a , for i ∈ I , m 1, and sincere a ∈ Nn . Since n > r , it follows from Corollary 3.5.8 that + n ξr,+Z (u + a ) = 0, for all sincere a ∈ N . Thus, ξr,Z gives rise to a surjective Z-algebra homomorphism θr+ : C(n)+ −→ S(n, r )+ , (m) . where C(n)+ is the Z-subalgebra of D(n) generated by the (u + i ) Similarly, we obtain a surjective Z-algebra homomorphism

θr− : C(n)− −→ S(n, r )− , (m) . By where C(n)− is the Z-subalgebra of D(n) generated by the (u − i ) (2.4.4.2),

U(n) = C(n)+ D(n)0 C(n)− . The assertion then follows from the triangular decomposition of S(n, r ) given in Theorem 3.7.7. If z ∈ C∗ is not a root of unity, and D,C (n) is the double Ringel–Hall algebra over C with parameter z considered in Remark 2.1.4, then we have algebra homomorphisms ξr,C and ξr,∨C as given in (3.5.10.1). The proof of the theorem above gives the following. 5 Lusztig constructed a canonical basis in [56, §4.1] for S (n, r ) and proved that those canonical basis elements labeled by aperiodic matrices form a basis for U(n, r ) := ξr (U(n)).

104 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity Corollary 3.8.3. The C-algebra homomorphism ξr,C : D,C (n) −→ S(n, r )C is surjective. If n r , ξr,∨C : H(r )C −→ EndD,C (n) ( isomorphism.

⊗r op )

C

is an

The above corollary together with [61, Th. 8.1] shows the following Schur– Weyl reciprocity in the affine quantum case. √ Corollary 3.8.4. Let q be a prime power. By specializing v to q, the ⊗r D,C (n)-H(r )C -bimodule C induces algebra homomorphisms ξr,C : D,C (n) −→ EndC (

⊗r

C

) and ξr,∨C : H(r )C −→ EndC (

⊗r op

C

)

such that Im (ξr,C ) = EndH(r)C (

⊗r C )

= S(n, r)C and Im (ξr,∨C ) = EndS(n,r)C (

⊗r op C ) .

∼ U(gl n ), the quantum Remarks 3.8.5. (1) As established in §2.5, D(n) = n ) → loop algebra. Hence, ξr induces a surjective algebra homomorphism U(gl S(n, r ). Similarly, ξr,C induces an algebra epimorphism UC (gln ) → n ) is the quantum loop algebra over C defined in S(n, r )C . Here UC (gl Definition 2.5.1, which is isomorphic to D,C (n) by Remark 2.5.5(3); cf. Theorem 2.5.3. It would be interesting to find explicit formulas for the action n ) on the tensor space ⊗r . of generators of U(gl (2) In [75, Th. 2], Vasserot has also constructed a surjective map #z from n ) to the K -theoretic construction K G (Z)z of S(n, r )C . It would also UC (gl be interesting to know if #z is equivalent to the epimorphism ξr,C , namely, if ∼ n ) → gr ◦ #z = f ◦ ξr,C under the isomorphisms f : UC (gl D,C (n) and ∼ G gr : K (Z )z → S(n, r )C (see [32, (9.4)]). (3) By the epimorphisms ξr and ξr,C , different types of generators for double Ringel–Hall algebras (see Remark 2.3.6(1)) give rise to corresponding generators for affine quantum Schur algebras. Thus, we may speak of semisimple generators, homogeneous indecomposable generators, etc., for S(n, r ). See §§5.4, 6.2. We end this section with a few conjectures. In the proof of the surjectivity of ξr in Theorem 3.8.1(1), we proved that the (n) defined in (3.8.1.1) maps onto the restriction of ξr to the free Z-module D (integral) algebra S(n, r). Since H(n)± are generated by semisimple generators (see [13, Th. 5.2(ii)]), it is natural to expect that the commutator formulas (n). Naturally, if the following conjecture given in Corollary 2.6.7 hold in D (n) an integral form of Lusztig type for D(n). was true, we would call D

3.8. Affine quantum Schur–Weyl duality, I

105

(n) is a subalgebra of D(n). Conjecture 3.8.6. The Z-module D Since both H(n)+ D(n)0 and D(n)0 H(n)− are subalgebras, this conjecture is equivalent to proving that all coefficients c(λ, μ, α) := 0 0γ α x α,γ K 2γ −α appearing in Corollary 2.6.7, are in D(n) . We make some comparisons with the integral form D(n) for D(n) introduced in Definition 2.4.4 and the restricted integral form discussed in [28, §7.2]. Remarks 3.8.7. (1) Since the integral composition algebra C(n)± is a Zsubalgebra of the integral Ringel–Hall algebra H(n)± which also contains the central generators z± m by Corollary 3.7.5, it follows that (n) ⊂ D(n). D(n) ⊂ D

(3.8.7.1)

However, we will see in Remark 5.3.8 that the restriction to D(n) of the homomorphism ξr in general does not map onto the integral affine quantum Schur (n), and we cannot use this integral form algebra S(n, r ). Thus, D(n) = D to get the Schur–Weyl theory at the roots of unity. n ) is the C[v, v −1 ]-subalgebra of (2) The restricted integral form Uvres (gl

n ) generated by divided powers (x± )(m) , k± , ki ;0 , and gi,m (see [28, U(gl i,s

i

t

[m]

n ) under the isomorphism EH given in §7.2]). If we identify D(n) with U(gl res (2.5.2.1) (see Theorem 2.5.3), (2.5.1.1) implies that z± m = θ±m ∈ Uv (gln ) n ). However, it is for all m 1. Thus, D(n) ⊗Z C is a subalgebra of Uvres (gl res not clear if Uv (gln ) is a subalgebra of D(n) ⊗Z C. Also, if we assume the (n) are Hopf subalgebras. But, as pointed conjecture, then both D(n) and D res n ) is a Hopf subalgebra. out in [28], it is not known if Uv (gl op The surjective homomorphism ξr,∨C : H(r )C → EndS(n,r)C ( ⊗r C ) for √ v = q was established by a geometric method. We do not know in general if the surjectivity holds over Q(v). Since both ξr : D(n) → S(n, r) and ξr,C : D,C (n) → S(n, r )C are surjective, the following conjecture gives the affine Schur–Weyl reciprocity over Q(v) and C for a non-root-of-unity specialization.

Conjecture 3.8.8. For n < r , the algebra homomorphisms ξr∨ : H(r ) −→ EndS(n,r) (⊗r )op and ξr,∨C : H(r)C −→ EndS(n,r)C (

⊗r op C )

are surjective, where a base change to C is obtained by specializing v to a non-root-of-unity z ∈ C.

106 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity With the truth of Conjecture 3.8.6, specializing v to any element in a field (n) ⊗ F F (of any characteristic) results in a surjective homomorphism D → S(n, r ) ⊗ F. It is natural to further expect that the affine Schur–Weyl reciprocity holds at roots of unity. Conjecture 3.8.9. The affine quantum Schur–Weyl reciprocity over any field F holds.

3.9. Polynomial identity arising from semisimple generators In this last section we will give an application of our theory. We use the commutator formulas in Theorem 2.6.3(5) to derive a certain polynomial identity which seems to be interesting in its own right. For λ ∈ Nn , set as in (1.2.0.1) Sλ =

n

λi Si and Aλ =

i =1

n

+ λi E i,i+1 ∈ (n).

i=1

Then, d(Aλ ) = dim Sλ = σ (λ) and M(Aλ ) = Sλ . Furthermore, for λ, α, β ∈ λ to be the Hall polynomial ϕ Aλ Nn , set ϕα,β Aα , Aβ and λ λ1 λ2 λn = ··· α α1 α2 αn

(cf. (2.6.6.1)).

Recall from (1.1.0.3) that, for λ = (λi ), μ = (μi ) ∈ Nn , λ μ means λi μi for all i. Recall also from (1.2.0.6) and (1.2.0.3) the number d A and the polynomial a A . For semisimple modules, we have the following easy formulas. Lemma 3.9.1. Let λ, α ∈ Nn satisfy α λ. Then λ λ ϕα,λ−α = , d Aλ = (λ2i −λi ), and aλ := a Aλ = α 1i n

λ λ In particular, ϕλ−α,α = ϕα,λ−α = v2

n

i =1

αi (αi −λi )

1i n 0sλi −1

aλ aα aλ−α .

We will also use the abbreviation for the elements a A aB A1 ,B1 ϕ A,B = 1 1 v 2d( A2 ) a A2 ϕ AA1 ,A2 ϕ BB1 , A2 and a AaB + A2 ∈ (n)

a A aB A1 ,B1 ϕ A,B = 1 1 a AaB

A2 ∈+ (n)

v 2d( A2 ) a A2 ϕ AA2 ,A1 ϕ AB2 ,B1 ,

(v 2λi −v 2s ).

3.9. Polynomial identity arising from semisimple generators

107

defined in (2.6.3.1) by setting, for λ, μ, α, β ∈ Nn , A ,A A ,A . α,β α,β ϕλ,μ = ϕ Aλα,Aμβ and ϕλ,μ = ϕ Aλα,Aμβ . Proposition 3.9.2. For λ, μ ∈ Nn and A, B ∈ + (n), if ϕ AA,B or ϕ AA,B is λ ,Aμ λ , Aμ non-zero, then there exist α, β ∈ Nn such that A = Aα , B = Aβ , λ − α = μ − β 0, and . α,β α,β ϕλ,μ = ϕλ,μ =

1 aλ−α

v 2σ (λ−α)+

1i n

2(αi (αi −λi )+βi (βi −μi ))

.

Aμ Aλ Proof. If either ϕ AA,B = 0 or ϕ AA,B = 0, then ϕ A,C ϕ B,C = 0 or λ , Aμ λ , Aμ A

Aλ μ ϕC,A ϕC,B = 0 for some C. In either case, both M(A) and M(B), as submodules or quotient modules of a semisimple module, are semisimple, and λ − d(A) = d(C) = μ − d(B). Write A = Aα , B = Aβ for some α, β ∈ Nn . Then λ − α = μ − β and so aλ−α = aμ−β . By the last assertion of Lemma 3.9.1, one sees immediately that aα aβ λ μ A,B A,B ϕ Aλ , Aμ = ϕ Aλ ,Aμ = aλ−α v 2σ (λ−α) aλ aμ α β 1 2σ (λ−α)+1in 2(αi (αi −λi )+βi (βi −μi )) = v , aλ−α

as required. Recall the surjective homomorphism ξr : D(n) → S(n, r ) as explicitly described in Theorem 3.6.3. For A, B ∈ + (n), let X A,B := ξr (L A,B ) and Y A,B := ξr (R A,B ), where L A,B (resp., R A,B ) are the LHS (resp., RHS) of the commutator relation given in Theorem 2.6.3(5). Then X A,B = Y A,B . In fact, since these commutator formulas continue to hold in D,C (n), where v is specialized to a non-root-of-unity in C (see Remark 2.1.4), X A,B = Y A,B holds in S(n, r )C . In particular, for each prime power q, by specializing v to √ q, we will view both X A,B and Y A,B as elements in the convolution algebra q CG (Y × Y ) ∼ = S(n, r )C . In this case, denote X A,B and Y A,B by X A,B and q q q Y A,B , respectively. Thus, we have X A,B = Y A,B , where A ,B √ √ d(B1 ),d( A)+d(B)−d(B1 )−d A −d B q 1 1 1 1 X A,B = q d(B),d(B) ϕ A,B (q) q A1 ,B1

q

Y A,B

× kd(B)−d(B1 ) ∗ (t B1 )(0, r ) ∗ A1 (0, r ) and √ √ d(B)−d(B1 ),d(A1 )+d(B),d(B1 )−d A −d B A1 ,B1 1 1 = q d(B),d(A) ϕ A,B (q) q A1 ,B1

× kd(B1 )−d(B) ∗ A1 (0, r ) ∗ (t B1 )(0, r ).

(3.9.2.1)

108 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity a −a Here ka = in=1 ki i ki+1i , for any a = (ai ) ∈ Nn , with ki = ξr (K i ) = 0(ei , r ), for any i ∈ I . q q In the following we are going to use the equality X A,B = Y A,B to derive some interesting polynomial identities. In the rest of this section, we fix the finite field F with q elements. As in §3.1, let Y = F(q) be the set of all cyclic flags L = (L i )i∈Z of lattices in a fixed F[ε, ε−1 ]-free module V of rank r 1. Lemma 3.9.3. For a ∈ Nn and (L, L ) ∈ Y × Y , 1 q 2 a,λ , if L = L and (L, L) ∈ Odiag(λ) for λ ∈ (n, r ); ka (L, L ) = 0, otherwise. Proof. Let λ ∈ (n, r ). By Lemma 3.7.1(1)&(2), (L, L ) ∈ Odiag(λ) if and only if L = L and λi = dim(L i /L i−1 ) for all i. Thus, if ka (L, L ) = 0, then L = L . Now we assume L = L and (L, L) ∈ Odiag(λ) . Since ka = 1 (a −a )μ i −1 i 1in i 2 [diag(μ)] and [diag(μ)] = χOdiag(μ) , it follows μ∈(n,r ) q 1 1 (a −a that ka (L, L) = q 2 1i n i i −1 )λi = q 2 a,λ .

q N −i+1 −1 For integers N , t with t 0, let Nt = . For α, β ∈ Nn q i −1 q

1i t

and (L, L ) ∈ Y × Y , consider the subsets of Y :

X (α, β, L, L ) := {L ∈ Y | L, L ⊆ L , L /L ∼ = Sβ , L /L Y (α, β, L, L ) := {L ∈ Y | L ⊆ L, L , L/L ∼ = Sα , L /L

∼ = Sα } and ∼ = Sβ }.

Here, L, L ⊆ L and L ⊆ L, L are the short form of L ⊆ L , L ⊆ L and L ⊆ L, L ⊆ L , respectively. Lemma 3.9.4. For α, β ∈ Y and (L, L ) ∈ Y × Y , if X (α, β, L, L ) = ∅ or Y (α, β, L, L ) = ∅, then L i + L i ⊆ L i+1 ∩ L i+1 and βi − dim((L i + L i )/L i ) = αi − dim((L i + L i )/L i ) 0, for all i ∈ Z. Moreover, in this case,

|X (α, β, L, L )| =

1i n

|Y (α, β, L, L )| =

1i n

&& &&

dim(L i+1 ∩ L i+1 /(L i + L i )) αi − dim((L i + L i )/L i ) dim(L i ∩ L i /(L i−1 + L i−1 )) αi − dim((L i + L i )/L i )

and q

''

. q

= (L i /L i , f i ) and maps f i , f i induced

L /L

Proof. If, as representations of (n), both (L i /L i , f i ) are semisimple, then the linear

''

L /L = from the

3.9. Polynomial identity arising from semisimple generators

109

for inclusion L i ⊆ L i+1 are zero maps. This forces L i ⊆ L i+1 and L i ⊆ L i+1 all i. Hence,

X (α, β, L, L ) = ∅ ⇐⇒ ∃ L ∈ X (α, β, L, L ) L t + L t ⊆ L t ⊆ L t+1 ∩ L t+1 , for t ∈ Z; ⇐⇒ dim(L t /L t ) = βt , dim(L t /L t ) = αt , for t ∈ Z ⎧ ⎪ for t ∈ Z; ⎪ ⎨ L t + L t ⊆ L t ⊆ L t+1 ∩ L t+1 , ⇐⇒ dim(L t /(L t + L t )) = βt − dim((L t + L t )/L t ), for t ∈ Z; ⎪ ⎪ ⎩dim(L /(L + L )) = α − dim((L + L )/L ), for t ∈ Z, t t t t t t t and Y (α, β, L, L ) = ∅ ⇐⇒ ∃ L ∈ Y (α, β, L, L ) L t ⊆ L t ⊆ L t+1 , dim(L t /L t ) = αt , for t ∈ Z; ⇐⇒ L t ⊆ L t ⊆ L t+1 , dim(L t /L t ) = βt , for t ∈ Z L t−1 ⊆ L t ⊆ L t , dim(L t /L t ) = αt , for t ∈ Z; ⇐⇒ L t−1 ⊆ L t ⊆ L t , dim(L t /L t ) = βt , for t ∈ Z ⎧ ⎪ for t ∈ Z; ⎪ ⎨ L t−1 + L t−1 ⊆ L t ⊆ L t ∩ L t , ⇐⇒ dim(L t ∩ L t /L t ) = αt − dim((L t + L t )/L t ), for t ∈ Z; ⎪ ⎪ ⎩dim(L ∩ L /L ) = β − dim((L + L )/L ), for t ∈ Z. t t t t t t t The rest of the proof is clear. For any λ = (λi )i∈Z , z = (z i )i∈Z ∈ Nn , define the polynomials in v 2 over Z: ν 2 −νi 2 i Pλ,z (v 2 ) = v 1in 2 +(λi −νi )(zi −νi−1 ) 0νλ ν∈Nn

n

z i+1 2 νi ! λi × (v − 1) [[νi ]] νi νi i=1

and (v 2 ) = Pλ,z

v2

ν 2 −νi

1in

i

2

+(λi −νi )(z i+1 −νi+1 )

0νλ ν∈Nn

×

n

i=1

νi

(v − 1) [[νi ]] 2

!

λi νi

zi νi

.

110 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

We now prove that these polynomials occur naturally in the coefficients of X A,B , Y A,B for A, B ∈ + (n)ss when they are written as a linear combination of eC , C ∈ (n, r ). Theorem 3.9.5. For x, y ∈ (n, r ), (L, L ) ∈ F,x (q) × F,y (q), and λ, μ ∈ Nn , let λ := λ − γ and μ := μ − δ, where γ = (dim(L i + L i )/L i )i∈Z and δ = (dim(L i + L i )/L i )i∈Z . If X λ,μ (L, L ) := X Aλ , Aμ (L, L ) = 0 or Yλ,μ (L, L ) := Y Aλ ,Aμ (L, L ) = 0, then L i + L ⊆ L i +1 ∩ L for all i ∈ Z and λ= μ 0. Moreover, putting q

q

q

q

i +1

i

z = (z i ) with z i = dim(L i ∩ L i /(L i−1 + L i−1 ))), for all i ∈ Z, we have in this case:

1 1 2 q X λ,μ (L, L ) = q 2 ( fλ + 1in (λi +μi +λi −λi )) · Pλ,z (q) and s −1 q 1in 1s λi

Yλ,μ (L, L ) = q q

where fλ =

1 2 λ + 1in (λi +μi +λi −λi )) 2 ( f

1in 1s λi

1i n

qs

1 · P (q), − 1 λ,z

λi (λi −1 − λi − yi )− xi+1 (δi + λi ) + μ, μ + δ + λ, λ.

Proof. We need to compute the value of (3.9.2.1) at (L, L ) for A = Aλ , B = Aμ (hence, A1 = Aα and B1 = Aβ ). By Corollary 3.6.2 and Lemma 3.9.3, and noting dim O Aν = 0 for ν ∈ Nn , ( kμ−β ∗ t Aβ (0, r ) ∗ Aα (0, r ))(L, L ) = ( kμ−β ∗ t Aβ (0, r ))(L, L )Aα (0, r)(L , L ) L ∈Y

=

1

q 2 (μ−β,x−a(L,L )−c(L ,L )) ,

L ∈Y ,L,L ⊆L L /L∼ =Sβ ,L /L ∼ =Sα

where a(L, L ), c(L , L ) are defined in (3.6.0.1). Thus, if X λ,μ (L, L ) = 0 q or Yλ,μ (L, L ) = 0, then some ϕ AA,B = ϕ AA,B = 0. So applying the first λ , Aμ λ , Aμ assertion of Proposition 3.9.2 yields 1 (μ,μ+β,μ−β+β,λ−d A −d Aα ) q α,β β X λ,μ (L, L ) = ϕλ,μ (q)q 2 q

n α,β∈N λ−α=μ−β0

×

L ∈Y ,L,L ⊆L L /L∼ =Sβ ,L /L ∼ =Sα

1

q 2 (μ−β,x−a(L,L )−c(L ,L )) .

3.9. Polynomial identity arising from semisimple generators

111

Similarly,

Yλ,μ (L, L ) = q

1

α,β

ϕλ,μ (q)q 2

n α,β∈N λ−α=μ−β0

×

(μ,λ+μ−β,α+μ,β−d A −d Aα )

β

1

q 2 (β−μ,x−a(L ,L

)−c(L,L ))

.

L ∈Y ,L ⊆L,L L/L ∼ =Sβ =Sα ,L /L ∼

If L ∈ Y satisfies L, L ⊆ L , L /L ∼ = Sβ , and L /L ∼ = Sα , then a(L, L ) + c(L , L ) = βi (xi+1 − βi ) + αi+1 (yi+1 − αi ) =: ♥. 1i n

1i n

Likewise, if L ∈ Y satisfies L ⊆ L, L , L/L ∼ = Sα , and L /L ∼ = Sβ , then L i ⊆ L i+1 and L i ⊆ L i+1 for all i. Thus, (L i+1 /L i )/(L i /L i ) ∼ /L i and (L i+1 /L i )/(L i+1 /L i ) ∼ , = L i+1 = L i+1 /L i+1

and so /L i ) = xi+1 − αi+1 + αi . dim(L i+1 /L ) = y Similarly, dim(L i+1 i+1 − βi+1 + βi . Hence, i

a(L , L ) + c(L, L ) = βi (dim(L i+1 /L i ) − βi ) + αi+1 (dim(L i+1 /L i ) − αi ) 1i n

=

1i n

βi (xi+1 − αi+1 + αi − βi ) +

1i n

=

βi (xi+1 − βi ) +

1i n

αi+1 (yi+1 − βi+1 + βi − αi )

1i n

αi+1 (yi+1 − αi ) = ♥,

1i n

and consequently, X λ,μ (L, L ) = q

α,β

n α,β∈N λ−α=μ−β0

1

ϕλ,μ (q)q 2

(−d Aα−d A −♥) β

1

× q 2 (μ,μ+β,μ−β+β,λ+μ−β,x) |X (α, β, L, L )| and Yλ,μ (L, L ) = q

α,β

1

ϕλ,μ (q)q 2

(−d Aα−d A −♥) β

n α,β∈N λ−α=μ−β0 1

× q 2 (μ,λ+μ−β,α+μ,β+β−μ,x) |Y (α, β, L, L )|.

112 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity Thus, X λ,μ (L, L ) = 0 or Yλ,μ (L, L ) = 0 implies some X (α, β, L, L ) = ∅ or Y (α, β, L, L ) = ∅. Hence, by Lemma 3.9.4, L i + L i ⊆ L i+1 ∩ L i+1 for all i and β − δ = α − γ 0. The latter together with λ − α = μ − β 0 implies q

q

λ− μ = (λ − μ) − (γ − δ) = (α − β) − (γ − δ) = 0 and λ = λ − γ α − γ 0. So we have proved the first assertion. q q It remains to simplify X λ,μ (L, L ) and Yλ,μ (L, L ) under the assumption that L i + L i ⊆ L i+1 ∩ L i+1 for all i and λ= μ 0. First, for z i = dim(L i ∩ L i /(L i−1 + L i−1 )), Lemma 3.9.4 gives

z i+1 |X (α, β, L, L )| = . αi − γi q 1i n

Second, for α, β ∈ Nn satisfying λ − α = μ − β 0, Lemma 3.9.1 and Proposition 3.9.2 imply 1

α,β

ϕλ,μ (q)q 2

(−d Aα −d A −♥) β

=

1i n (λi −αi +αi (αi −λi )+βi (βi −μi ))

q ×q

=q

1 2

| Aut(Sλ−α )|

1 2

(βi −βi2 +αi −αi2 )−βi (xi +1 −βi )−αi (yi −αi −1 )

1in

1

q2 1in (λi +μi )

1i n

αi (αi +αi −1 −2λi −yi )+βi (2βi −2μi −x i+1 )

| Aut(Sλ−α )|

,

since 2λi − αi + βi = λi + μi for all i . Hence, X λ,μ (L, L ) q

=q

1 2

1in (λi +μi )

1

q2

1in

| Aut(Sλ−α )|

n α,β∈N λ−α=μ−β0

×q

αi (αi +αi −1 −2λi −yi )+βi (2βi −2μi −x i+1 )

1 2 (μ,μ+β,μ−β+β,λ+μ−β,x)

z i+1 . αi − γi q

1i n

Here we have implicitly assumed α γ (or equivalently, β δ). Setting ν = α − γ = β − δ gives 1

X λ,μ (L, L ) = q 2 q

1i n (λi +μi )

n ν∈N 0νλ−γ

1

z i +1 q 2 fν , | Aut(Sλ−γ −ν )| νi q

1i n

3.9. Polynomial identity arising from semisimple generators

113

where (γi + νi )(γi + νi + γi−1 + νi−1 − 2λi − yi )

fν =

1i n

+ (δi + νi )(2(γi + νi − λi ) − xi +1 )

(3.9.5.1)

+ μ, μ + δ + ν, λ − (γ + ν) + δ + ν, λ + λ − (γ + ν), x, for ν ∈ Nn with 0 ν λ. Similarly, Yλ,μ (L, L ) q

=q

1 2

1

1i n (λi +μi )

q2

1i n

| Aut(Sλ−α )|

n α,β∈N λ−α=μ−β0

×q

αi (αi +αi −1 −2λi −yi )+βi (2βi −2μi −x i+1 )

1 2 (μ,λ+μ−β,α+μ,β+β−μ,x)

1i n

1

=q 2

1i n (λi +μi )

n ν∈N 0νλ−γ

zi αi − γi

q

1

z i q 2 gν , | Aut(Sλ−γ −ν )| νi q

1i n

where (γi + νi )(γi + νi + γi−1 + νi−1 − 2λi − yi )

gν =

1i n

+ (δi + νi )(2(γi + νi − λi ) − xi +1 )

(3.9.5.2)

+ μ, λ + λ − γ − ν, γ + ν + μ, δ + ν + γ + ν − λ, x. Simplifying fλ − f ν and gλ − gν and substituting, we obtain, by Lemma 3.10.3 in the appendix, X λ,μ (L, L ) q

q 2 1i n (λi +μi )+ 2 fλ q 1in (λi zi −νi (zi +λi +λi+1 −νi −νi +1 )) z i+1 = | Aut(Sλ−ν )| νi q 1

1

1i n

0ν λ n ν∈N

=

0ν λ n ν∈N

hν q

1i n (λi z i −νi z i −νi λi+1 )

1i n 1sνi

(q zi+1 −s+1 − 1)

114 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

and Yλ,μ (L, , L ) q

1

1

q 2 1in (λi +μi )+ 2 fλ q 1i n (λi zi+1 −νi (zi +1 +λi +λi −1 −νi −νi+1 )) z i = | Aut(Sλ−ν )| νi q 1i n

0ν λ n ν∈N

=

1in (λi z i +1 −νi z i+1 −νi λi −1 )

hν q

0ν λ n ν∈N

(q zi −s+1 − 1),

1in 1sνi

where q−

hν =

1i n νi (λi −νi −νi+1 )

| Aut(Sλ−ν )|

qs

1i n 1sνi

1 . −1

Further simplification by Lemma 3.9.1 gives hν =

q−

q

=q

1in νi (λi −νi −νi +1 )

1 1in 2 (λi −νi −1)(λi −νi )

1 2 1in 2 (λi −λi )

1i n 1s λi

qs

1i n 1s λi −νi

1 q s q −1

1 1 − 1 1i n q s − 1 1sνi

νi2 −νi 1in ( 2

+νi νi +1 )

λi . νi q

1i n

Thus, q

1

2 1in (λi +μi +λi −λi ))

X λ,μ (L, L ) = q 2 ( fλ +

1 · Qλ,z and qs − 1

1in 1s λi

Yλ,μ (L, L ) = q 2 ( fλ + 1

q

2 1in (λi +μi +λi −λi ))

1in 1s λi

where, for pν,λ (q) = q Qλ,z = Qλ,z =

0ν λ ν∈Nn

0ν λ ν∈Nn

1in (

pν,λ (q) · q pν,λ (q) · q

νi2 −νi 2

+νi νi +1 )

1 · Qλ,z , −1

λi νi

1i n

1i n (λi z i −νi z i −νi λi+1 )

qs

q

,

(q zi+1 −s+1 − 1) and

1i n 1sνi

1i n (λi z i+1 −νi z i+1 −νi +1 λi )

1i n 1sνi

= Q . Finally, it is clear to see that Pλ,z = Qλ,z and Pλ,z λ,z

(q zi −s+1 − 1).

3.10. Appendix

115

The fact that ξr is an algebra homomorphism immediately gives the following polynomial identity. Corollary 3.9.6. For any λ = (λi )i∈Z , z = (z i )i∈Z ∈ Nn , Pλ,z (v 2 ) = (v 2 ). Pλ,z Proof. If z = 0, then the equality holds trivially. Now suppose z = 0 and set n r = i=1 z i . Let F be a finite field with q elements. By Theorems 3.6.3 and q q 2.6.3(5), we have X λ,μ (L, L ) = Yλ,μ (L, L ) for all λ, μ and (L, L ). Take L = (L i )i∈Z = L ∈ Y (thus, λ=λ=μ= μ) such that dim L i /L i−1 = z i , q q for all i ∈ Z. Applying Theorem 3.9.5 to the equality X λ,λ (L, L) = Yλ,λ (L, L) (q). gives the equality Pλ,z (q) = Pλ,z (v 2 ) Remark 3.9.7. We point out that the polynomial identity Pλ,z (v 2 ) = Pλ,z ± is equivalent to the fact that the algebra homomorphisms ζr : D(n)± → S(n, r ) constructed by Varagnolo and Vasserot [73] (see (3.6.2.1)), which have easy extensions ζr0 : D(n)0 → S(n, r ) and ζr0 : D(n)0 → S(n, r ), can be extended to an algebra homomorphism D(n) → S(n, r). In fact, there is an obvious linear extension ξr of ζr0 and ζr0 which is an algebra homomorphism if and only if ξr preserves the commutator relations in Theorem 2.6.3(5) on semisimple generators (see Lemma 2.6.1). This is the (q) for every prime power q and λ, z. case by Theorem 3.9.5 if Pλ,z (q) = Pλ,z : (n, r ) → Z[v 2 ] For every λ ∈ Nn and r 0, define functions f λ,r , f λ,r 2 2 such that f λ,r (z) = Pλ,z (v ) and f λ,r (z) = Pλ,z (v ). The polynomial identity shows that the two functions are identical.

Problem 3.9.8. Give a direct (or combinatorial) proof of the identity (v 2 ), or equivalently, f Pλ,z (v 2 ) = Pλ,z λ,r = f λ,r .

3.10. Appendix In this appendix, we prove a few lemmas which have been used in the previous sections. The first one reflects some affine phenomenon for the length of the longest element w0,λ of Sλ . This is used in the proof of Corollary 3.2.4. Lemma 3.10.1. For λ ∈ (n, r) and d ∈ D λ , let Y = {(s, t) ∈ Z2 | 1 d −1 (s) r, s < t, s, t ∈ Rkλ for some k ∈ Z} and Z = {(s, t) ∈ Z2 | 1 s r, s < t, s, t ∈ Rkλ for some 1 k n}. Then |Y | = |Z | = (w0,λ ).

116 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity Proof. For a ∈ Z let Ya = {b ∈ Z | a < b, a, b ∈ Rkλ for some k ∈ Z}. We first claim |Ya1 | = |Ya2 | whenever a1 ≡ a2 mod r . Indeed, write a2 = a1 + cr λ and assume a1 ∈ Rkλ (see (3.2.1.4)). Then a2 ∈ Rk+cn . By definition, λ Ya2 = {b ∈ Z | a2 < b, b ∈ Rk+cn } = {b ∈ Z | a1 < b − cr, b − cr ∈ Rkλ }.

Hence, there is a bijection from Ya2 to Ya1 defined by sending b to b − cr , proving the claim. Since the remainders of d(i ) when divided by r are all distinct and Y = {(d(s), j ) | j ∈ Yd(s) }, 1s r

it follows from the claim that |Y | = |Yd(s) | = |Ys | 1s r

=

1s r

|{t ∈ Z | s < t, s, t ∈ Rkλ for some k ∈ Z}|

1s r

= |Z | = (w0,λ ), as desired. The next lemma is used in the proof of Proposition 3.7.3. Lemma 3.10.2. Keep the notation A, A+ , A− , λ, μ, ν used in (3.7.3.1). We have d A+diag(λ) = d A+ +diag(μ) + d A− +diag(ν) . Proof. Recall from Lemma 3.7.2(2) that μ, ν are uniquely determined by the conditions (L, L ) ∈ O A+ +diag(μ) and (L , L ) ∈ O A− +diag(ν) , whenever (L, L ) ∈ O A+diag(λ) and L = L ∩ L . By Lemma 3.7.1(3), for i ∈ Z, λi + ai,k = dim(L i /L i −1 ) = μi + ai,k and k,k =i

λi +

k,i
ak,i =

dim(L i /L i−1 )

k,k =i

Thus, μi = λi + for such μ, ν,

k
d A+diag(λ) =

ai,k and νi = λi + 1i n ik; j
ai, j ak,l +

= νi + k
1in ki
ak,i .

k,k>i

ak,i for all i. Moreover, since

λi ak,l +

j
λk ai, j ,

3.10. Appendix

d A+ +diag(μ) =

1i n i k; j < l i < j; k < l

=

ai, j ak,l +

ai, j ak,l +

λi ak,l +

1i n ki
1i n i k; j < l i > j; k > l

=

μi ak,l

1i n ki
1i n i k; j < l i < j; k < l

d A− +diag(ν) =

ai, j ak,l +

117

ai, j ak,l , and

1i n k i j

νk ai, j

1i n j
ai, j ak,l +

1i n i k; j < l i > j; k > l

λk ai, j +

1i n j
ai, j ak,l ,

1i n j
it follows that d A+diag(λ) − d A+ +diag(μ) − d A− +diag(ν) = ai, j ak,l − ai, j ak,l − 1i n ik; j
1i n i k, j < l i < j, k < l

−

1i n k i j

=

ai, j ak,l −

ai, j ak,l

1i n j

ai, j ak,l −

1i n i k, j < l i > j, k < l

ai, j ak,l −

1i n k i j

ai, j ak,l

1i n i k; j < l i > j; k > l

ai, j ak,l

1i n j
= 0, as required. Let f ν , gν be the numbers defined as in (3.9.5.1) and (3.9.5.2). The following lemma establishes a certain relationship between these numbers and is used in the proof of Theorem 3.9.5. Lemma 3.10.3. Maintain the notation in the proof of Theorem 3.9.5. For any 0 ν λ, we have fλ = gλ , fλ − f ν = −2 gλ − gν = −2

1i n

1i n

λi z i +

2νi (z i + λi + λi+1 − νi − νi +1 ), and

1i n

λi z i +1 +

1i n

2νi (z i+1 + λi + λi −1 − νi − νi+1 ).

118 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity Proof. Since λ=λ−γ =μ−δ = μ, it follows that λi ) + μ, λ + μ, δ + λ λi (λi−1 − λi − yi ) − xi+1 (δi + gλ = 1i n

=

λi ) + λ + δ, λ + μ, μ λi (λi−1 − λi − yi ) − xi+1 (δi +

1i n

= fλ . By definition, for 0 ν λ = λ − γ, ( λi + γi )( λi−1 + γi−1 − (γi + λi ) − yi ) fλ − f ν = 1i n

−

(δi + μi )xi+1 + δ + μ, γ + λ

1i n

−

(γi + νi )(−γi + νi + γi−1 + νi−1 − 2 λi − yi )

1i n

−

(δi + νi )(2(νi − λi ) − xi+1 )

1i n

− δ + ν, λ − ν − δ + ν, γ + λ − λ − ν, x λi γi−1 − λi2 − λi yi + γi μi λi = λi−1 + λi−1 − xi+1 1i n

+

− γi νi−1 − 3νi2 − νi γi−1 − νi νi−1

(4νi λi + νi yi − 2δi νi + 2δi λi + νi xi+1 )

1i n

+ μ − ν, γ + λ − δ + ν, λ − ν − λ − ν, x = f ν + f ν , where f ν =

λi−1 + λi−1 − xi+1 λi λi γi−1 − λi2 − λi yi + γi μi + δi λi 1i n

λi + λi+1 ) + + μi μi γi − μi γi+1 − μi

λi+1 − xi λi + xi+1 λi (δi

1i n

and f ν =

− γi νi − γi νi−1 − 2νi2 − νi γi−1 − νi νi−1 + 2νi λi + νi yi

1i n

− δi νi + νi γi+1 + 2νi (−δi νi+1 − νi νi+1 + xi νi ). λi+1 + 1i n

3.10. Appendix

119

Since μ = λ and γi−1 + γi − yi − xi + δi−1 + δi = −2zi for all i ∈ Z, we have λi (γi−1 − yi + δi + γi − γi+1 − xi ) + λi−1 + λi+1 γi δi f ν = 1i n

1i n

=

1i n

λi (γi−1 + γi − yi − xi + δi−1 + δi )

1i n

= −2

λi z i

1i n

and f ν =

=

νi − (γi−1 + γi − yi − xi + δi−1 + δi )

1i n

− 2νi − νi−1 + 2 λi + 2 λi+1 − νi+1

2νi (z i + λi + λi+1 − νi − νi+1 ).

1i n

Consequently, for 0 ν λ, we obtain that fλ − f ν = −2 λi z i + 2νi (z i + λi + λi+1 − νi − νi+1 ). 1i n

1i n

Since μ = λ, for 0 ν λ, we have gλ − gν = ( λi + γi )(γi−1 + λi−1 − γi − λi − yi ) 1i n

−

(δi + λi )xi+1 + δ + λ, δ + λ

1i n

−

(γi + νi )(−γi + γi−1 + νi + νi−1 − 2 λi − yi )

1i n

−

(δi + νi )(2(ν − λi ) − xi+1 )

1i n

− λ − ν, γ + ν − δ + λ, δ + ν − ν − λ, x = gν + gν , where gν =

λi γi − λi yi + γi λi γi−1 − λi−1 − 2xi+1 λi 1i n

+ 3δi λi γi+1 + xi λi − δi λi+1 + λi

120 3. Affine quantum Schur algebras and the Schur–Weyl reciprocity

and gν =

λi + νi yi − 3δi νi γi νi − γi νi−1 − 2νi2 − νi γi−1 − νi νi−1 + 2νi

1i n

+ 2νi xi+1 + δi νi+1 + 2 λi νi+1 − νi γi+1 − νi νi+1 − xi νi .

It is easy to check that for all i ∈ Z, γi−1 − γi + 2γi+1 + xi − yi − 2xi+1 + 3δi − δi−1 = −2z i+1 . Therefore, gν =

λi (γi−1 − γi + 2γi+1 + xi − yi − 2xi+1 + 3δi − δi−1 )

1i n

= −2

λi z i+1

1i n

and gν =

=

−(γi−1 − γi + 2γi+1 + xi − yi − 2xi+1 + 3δi − δi−1 )

1i n

− 2νi − νi−1 − νi+1 + 2 λi + 2 λi−1

2νi (z i+1 + λi + λi−1 − νi − νi+1 ).

1i n

Thus, for 0 ν λ, gλ − gν = −2 2νi (z i+1 + λi + λi−1 − νi − νi+1 ). λi z i+1 + 1i n

1i n

4 Representations of affine quantum Schur algebras

The quantum Schur–Weyl duality links representations of quantum gln with those of Hecke algebras of symmetric groups. More precisely, there are category equivalences between level r representations of quantum gln and representations of the Hecke algebra of Sr . This relationship provides in turn two approaches to representations of quantum Schur algebras. First, one uses the polynomial representation theory of quantum gln to determine representations of quantum Schur algebras. This is a downward approach. Second, representations of quantum Schur algebras can also be determined by those of Hecke algebras. We call this an upward approach. In this chapter, we will investigate the affine versions of the two approaches. When the parameter v is specialized to a non-root-of-unity z ∈ C, classification of finite dimensional simple (or irreducible) polynomial representations n ) was completed by Frenkel–Mukhin [28], of the quantum loop algebra UC (gl based on Chari–Pressley’s classification of finite dimensional simple representations of the quantum loop algebra UC ( sln ) [6, 7]. On the other hand, Zelevinsky [81] and Rogawski [66] classified simple representations of the Hecke algebra H(r )C . Moreover, Chari–Pressley have also established a category equivalence between the module category H(r )C -mod and a certain full subcategory of UC ( sln )-mod when n > r without using affine quantum Schur algebras. We will first establish in §4.1 a category equivalence between the categories S(n, r )F -Mod and H(r )F -Mod when n r (Theorem 4.1.3). As an immediate application of this equivalence, we prove that every simple representation in S(n, r )F -Mod is finite dimensional (Theorem 4.1.6). We will briefly review the classification theorems of Chari–Pressley and Frenkel–Mukhin in §4.2 and of Rogawski in §4.3. We will then use the category equivalence to classify simple representations of affine quantum Schur algebras via Rogawski’s classification of simple representations of H(r )C (Theorem 4.3.4). 121

122

4. Representations of affine quantum Schur algebras

From §4.4 onwards, we will investigate the downward approach. Using n ) → S(n, r )C , every simple the surjective homomorphism ξr, C : UC (gl n )-module. Our question is how S(n, r )C -module is inflated to a simple UC (gl to identify those inflated simple modules. We use a two-step approach. First, motivated by a result of Chari–Pressley, we identify in the n > r case the simple S(n, r )C -modules arising from simple H(r )C -modules in terms of n ) (Theorem 4.4.2). Then, using simple polynomial representations of UC (gl this identification, we will classify all simple S(n, r)C -modules through simn ) (Theorem 4.6.8). Thus, all finite ple polynomial representations of UC (gl n ) are inflations of dimensional simple polynomial representations of UC (gl simple representations of affine quantum Schur algebras. In this way, like the classical theory, affine quantum Schur algebras play a bridging role between polynomial representations of quantum affine gln and those of affine Hecke algebras of Sr . We end the chapter by presenting a classification of finite dimensional simple U(n, r )C -modules (Theorem 4.7.5). Throughout the chapter, all representations are defined over a ground field F with a fixed specialization Z → F by sending v to a non-root-of-unity z ∈ F. From §4.2 onwards, F will be the complex number field C. The algebras n ) are defined in §2.5 with Drinfeld’s new presentation. For UC ( sln ) and UC (gl any algebra A considered in this chapter, the notation A -mod represents the category of all finite dimensional left A -modules. We also use A -Mod to denote the category of all left A -modules.

4.1. Affine quantum Schur–Weyl duality, II In this section we establish the affine version of the category equivalences mentioned in the introduction above. As a by-product, we prove that every irreducible representation over the affine quantum Schur algebra is finite dimensional. Recall from §2.3 (see footnote 2 there) the extended affine sln , U(n), generated by E i , Fi , K i± subject to relations (QGL1)–(QGL5) given in Definition 2.3.1. By dropping the subscripts from U(n), S(n, r), and H(r ), we obtain the notation U(n), S(n, r ), and H(r ) for quantum gln , quantum Schur algebras, and Hecke algebras of type A. The algebras U(n), S(n, r ), and H(r ) will be naturally viewed as subalgebras of U(n), S(n, r ), and H(r ), respecn ). tively. Moreover, we may also regard U(n) as a subalgebra of D(n) = U(gl Let U(n) be the Z-form of U(n) defined in (2.4.4.2) and let U (n), S(n, r ), and H(r ) denote the corresponding Z-free subalgebras of U(n), S(n, r ), and H(r ).

4.1. Affine quantum Schur–Weyl duality, II

123

The tensor space ⊗r over Z is a U(n)-H(r )-bimodule via the actions (3.5.0.1) and (3.3.0.4), where the affine H(r )-action is a natural extension from the H(r )-action on ⊗r spanned n , where n is the free Z-submodule of ⊗r by the elements ωi (1 i n); see (3.3.0.4). Thus, n is a U (n)-H(r )subbimodule of ⊗r by restriction. Lemma 4.1.1. There is a U (n)-H(r )-bimodule isomorphism ⊗r n

∼

⊗H(r) H(r ) −→

⊗r

, x ⊗ h −→ xh.

Proof. Clearly, there is a U (n)-H(r )-bimodule homomorphism ⊗r ⊗r , x ⊗ h −→ xh. n ⊗H(r) H(r ) −→ a1 ar {Tw X 1 · · · X r | w ∈ Sr , ai ∈ Z, 1 i r } forms

ϕ:

Since the set for H(r ), it follows that the set

a Z-basis

X := {ωi ⊗ X 1a1 · · · X rar | i ∈ I (n, r ), ai ∈ Z, 1 i r } forms a Z-basis for ϕ(ωi ⊗

a X 11

⊗r n

⊗H(r) H(r ). Furthermore, by (3.3.0.4), a

· · · X rar ) = ωi X 1 1 · · · X rar = ωi−n(a1 ,a2 ,...,ar ) ,

for all (a1 , a2 , . . . , ar ) ∈ Zr and i ∈ I (n, r ). Thus, the bijection (3.3.0.3) from I(n, r ) to Zr = I (n, r ) + nZr implies that ϕ(X ) = {ωi | i ∈ I(n, r )} forms a Z-basis for ⊗r . Hence, ϕ is a U (n)-H(r )-bimodule isomorphism. For N = max{n, r }, let ω ∈ (N , r ) be defined as in (3.1.3.1). Thus, if ω = (ωi ), then ωi = 1 for 1 i r , and ωi = 0 if n > r and r < i n. Let eω = ediag(ω) . Corollary 4.1.2. If n r , then there is an S(n, r )-H(r )-bimodule isomorphism S(n, r )eω ∼ = S(n, r )eω ⊗H(r) H(r ). Proof. By the proof of Lemma 3.1.3, there is an S(n, r )-H(r)-bimodule isomorphism S(n, r )eω ∼ = T(n, r). By Proposition 3.3.1, this isomorphism extends to a bimodule isomorphism S(n, r )eω ∼ = ⊗r (cf. the proof of Theorem 3.8.1(3)). Now, the required result follows from the lemma above. We are now ready to establish a category equivalence. For the rest of this section, we assume F is a (large enough) field which is a Z-algebra such that the image z ∈ F of v is not a root of unity. Thus, both H(r )F and S(n, r )F are semisimple F-algebras, and every finite dimensional U (n)F -module is completely reducible. The following result generalizes the category equivalence between S(n, r )F -mod and H(r )F -mod (n r) to the affine case.

124

4. Representations of affine quantum Schur algebras

Theorem 4.1.3. Assume that n r . The categories S(n, r )F -Mod and H(r )F -Mod are equivalent. It also induces a category equivalence between S(n, r )F -mod and H(r )F -mod. Hence, in this case, the algebras S(n, r )F and H(r )F are Morita equivalent. Proof. The S(n, r )F -H(r )F -bimodule

⊗r

F

induces a functor

F : H(r )F -Mod −→ S(n, r)F -Mod, L −→

⊗r

F

⊗H(r )F L .

(4.1.3.1)

Since n r , there is a functor, the Schur functor, G : S(n, r )F -Mod −→ H(r )F -Mod, M −→ eω M. Here we have identified eω S(n, r )F eω with H(r )F . We need to prove that there are natural isomorphisms G ◦ F ∼ = idH(r)F -Mod , which is clear (see, e.g., [33, (6.2d)]), and F ◦ G ∼ = idS(n,r)F -Mod . By the semisimplicity, for any S(n, r )F -module M, there is a left S(n, r )F module isomorphism f : S(n, r )F eω ⊗H(r)F eω M ∼ =M

(4.1.3.2)

defined by f (x ⊗ m) = xm for any x ∈ S(n, r )F eω and m ∈ eω M. By Corollary 4.1.2, (4.1.3.2) induces a left S(n, r )F -module isomorphism g : S(n, r )F eω ⊗H(r)F eω M ∼ =M satisfying g(x ⊗ m) = xm, for any x ∈ S(n, r )F eω and m ∈ eω M. We now claim that g is an S(n, r )F -module isomorphism. Indeed, by identifying S(n, r )F with SH (n, r ) under the isomorphism given in Proposition d 1 3.2.8 and considering the basis {φλ,μ }, we have eω = φω,ω and 1 S(n, r )F eω = HomH(r)F (H(r )F , xλ H(r )F ) = φλ,ω H(r )F , λ∈(n,r)

λ∈(n,r)

1 ∈ S(n, r ) . Since g(φ 1 ⊗ m) = φ 1 m, for all λ ∈ (n, r ) and where φλ,ω F λ,ω λ,ω m ∈ eω M, it follows that, for any λ ∈ (n, r ), h ∈ H(r )F , and m ∈ eω M, 1 1 1 1 g(φλ,ω h ⊗ m) = g(φλ,ω ⊗ hm) = φλ,ω (hm) = (φλ,ω h)m.

Hence, g(x ⊗ m) = xm, for all x ∈ S(n, r )F eω and m ∈ eω M. Thus, g is an S(n, r )F -module isomorphism. Therefore, for any M ∈ S(n, r )F -Mod, there is a natural isomorphism F ◦ G(M) ∼ = S(n, r )F eω ⊗H(r)F eω M ∼ = M, proving F ◦ G ∼ = idS(n,r)F -Mod .

4.1. Affine quantum Schur–Weyl duality, II

125

The last assertion follows from the fact that, if N is a finite dimensional H(r )F -module, then Lemma 4.1.1 implies that F(N ) = ⊗r F ⊗H(r)F N is also finite dimensional. Remarks 4.1.4. (1) For the case F = C, a direct category equivalence from n )-mod has been established H(r )C -mod to a full subcategory of UC (gl in [31, Th. 6.8] when n r . The construction there is geometric, using intersection cohomology complexes. (2) See [76] for a similar equivalence in the context of representations of p-adic groups. See also [74] for a connection with representations of double affine Hecke algebras of type A. Following [9, 2.5], a finite dimensional U (n)F -module M is said to be of level r if every irreducible component of M is isomorphic to an irreducible component of ⊗r n,F . In other words, a U (n)F -module M has level r if and only if it is an S(n, r )F -module. We will generalize this definition to the affine case; see Corollary 4.6.9 and Remark 4.6.10 below. Note that, since levels are defined for modules of quantum sln in [9, 2.5], the condition n > r there is necessary. Recall that a U(n)F -module M is called of type 1, if it is a direct sum M = ⊕λ∈Zn Mλ of its weight spaces Mλ which have the form Mλ = {x ∈ M | K i x = z λi x}. In other words, a U(n)F -module M is of type 1 if it is of type 1 as a U (n)F module. Theorem 4.1.3 immediately implies the following category equivalence due to Chari–Pressley [9]. However, a different functor is used in [9, Th. 4.2]. We will make a comparison of the two functors in the next section. Corollary 4.1.5. If n > r , then the functor F induces a category equivalence between the category of finite dimensional U(n)F -modules of type 1 which are of level r when restricted to U (n)F -modules and the category of finite dimensional H(r )F -modules. Proof. By Corollary 3.5.9 (see also Lemma 5.2.1 below), the condition n > r implies S(n, r )F = U(n, r )F is a homomorphic image of U(n)F . Thus, level r representations considered are precisely S(n, r )F -modules. Now the result follows immediately from the theorem above. If N n, then there is a natural injective map : (n) −→ (N ),

= ( A = (ai, j ) −→ A ai, j ),

126

4. Representations of affine quantum Schur algebras

is defined in (3.1.2.1). Similarly, there is an injective map where A : Zn −→ ZN , λ −→ λ,

(4.1.5.1)

where λi = λi for 1 i n and λi = 0 for n + 1 i N . See the proof of Lemma 3.1.3. These two maps induce naturally by Lemma 3.1.3 an injective algebra homomorphism (not sending 1 to 1) for A ∈ (n, r ). ιr = ιn,N,r : S(n, r ) −→ S(N , r), [A] −→ [ A] In other words, we may identify S(n, r ) as the centralizer subalgebra 1 = eS(N , r )e of S(N , r ), where e = λ∈(n,r) φλ, λ∈(n,r) [diag(λ)] ∈ λ S(N , r ). Now, the fact that every simple module of an affine Hecke algebra is finite dimensional implies immediately that the same is true for affine quantum Schur algebras. Theorem 4.1.6. Every simple S(n, r)F -module is finite dimensional. Proof. If n r , then the algebras S(n, r )F and H(r )F are Morita equivalent. The assertion follows from the fact that every simple H(r )F -module is finite dimensional. If n < r , then, by taking N = r , S(n, r )F ∼ = eS(r, r )F e, where e = 1 . It follows that each simple S(n, r )F -module is isomorphic to λ∈(n,r) φ λ, λ eL for a simple S(r, r )F -module L; see, for example, [33, 6.2(g)]. As shown above, all simple S(r, r )F -modules are finite dimensional. Hence, so are all simple S(n, r )F -modules.

4.2. Chari–Pressley category equivalence and classification We first prove that the functor F defined in (4.1.3.1) coincides with the functor F defined in [9, Th. 4.2]. Then, we describe the Chari–Pressley classification of finite dimensional simple UC ( sln )-modules and its generalization to n ) for later use. UC (gl From now on, we assume the ground field F = C, the complex number field, which is a Z-module where v is mapped to z with z m = 1 for all m 1. Following [9], let E θ and Fθ be the operators on n,C defined respectively by E θ · ωi = δi,n ω1 and Fθ · ωi = δi,1 ωn .

(4.2.0.1)

4.2. Chari–Pressley category equivalence and classification

127

θ = K 1 · · · K n−1 = K −1 = K −1 K 1 .1 For a left H(r ) F -module M, Let K 0 0 define F(M) =

⊗r n,C

⊗H(r)C M.

Then F(M) is equipped with the U (n)C -module structure induced by that on ⊗r n,C . By [9, Th. 4.2], F (M) becomes a U(n)C -module via the action: E 0 (w ⊗ m) =

r

(Y j+ w) ⊗ (X j m) and

j=1

F0 (w ⊗ m) =

r

(Y j− w) ⊗ (X −1 j m),

j=1 ⊗r n,C ,

where w ∈ are defined by

m ∈ M and the operators Y j± ∈ EndC (

⊗r n,C )

(1 j r )

−1 )⊗(r− j ) = 1⊗( j −1) ⊗ Fθ ⊗ ( K 0 )⊗(r− j) and Y j+ = 1⊗( j−1) ⊗ Fθ ⊗ ( K θ θ )⊗( j −1) ⊗ E θ ⊗ 1⊗(r − j) = ( K −1 )⊗( j −1) ⊗ E θ ⊗ 1⊗(r− j ) . Y j− = ( K 0 Hence, we obtain a functor F : H(r )C -mod −→ U(n)C -mod. When n > r , Chari–Pressley used this functor to establish a category equivalence between H(r )C -mod and the full subcategory of U(n)C -mod consisting of finite dimensional U(n)C -modules which are of level r when restricted to U (n)C -modules; see Corollary 4.1.5. On the other hand, the algebra homomorphism ξr,C : U(n)C → S(n, r)C (cf. Remark 3.5.10) gives an inflation functor ϒ : S(n, r )C -mod −→ U(n)C -mod. Thus, we obtain the functor ϒF = ϒ ◦ F : H(r )C -mod −→ U(n)C -mod. Proposition 4.2.1. There is a natural isomorphism of functors ϕ : F ϒF. In other words, for any H(r ) F -modules M, M and homomorphism f : M → M , there is a U(n)C -module isomorphism ∼

ϕ M : F(M) −→ ϒF(M),

w ⊗ m −→ w ⊗ m,

1 The K and E , F below should be regarded as K , E , F , respectively, if the index set n n n 0 0 0

I = Z/nZ is identified as {1, 2, . . . , n}.

128

4. Representations of affine quantum Schur algebras

for all w ∈ ⊗r n,C and m ∈ M such that ϒF( f )ϕ M = ϕ M F ( f ). Here ⊗ = ⊗H(r )C and ⊗ = ⊗H(r)C .

Proof. We first recall from Remark 3.5.10 that the representation U(n)C → EndC ( ⊗r C ) factors through the algebra homomorphism ξr,C : U(n)C → S(n, r )C . Thus, by Lemma 4.1.1, there is a U (n)C -module isomorphism ϕ = ϕ M : F (M) −→ ϒF(M) w ⊗ m −→ w ⊗ m

⊗r n,C ,

(for all w ∈

m ∈ M).

It remains to prove that ϕ(E 0 (ωi ⊗ m)) = E 0 (ωi ⊗ m) and ϕ(F0 (ωi ⊗ m)) = F0 (ωi ⊗ m), for all i ∈ I (n, r ) and m ∈ M. By the definition above, for i ∈ I (n, r ) and m ∈ M, ϕ(E 0 (ωi ⊗ m)) = ϕ =

r

(Y j+ ωi ) ⊗ (X j m)

j=1 r Y j+ ωi j =1

⊗ X j m =

r

(4.2.1.1) (Y j+ ωi )X j

⊗ m,

j=1

where, by (4.2.0.1) and (3.3.0.4), (Y j+ ωi )X j = v =v

j+1kr (δi k ,n −δi k ,1 )

δi j ,1 (ωi1 · · · ωi j −1 ωn ωi j +1 · · · ωir )X j

j+1kr (δi k ,n −δi k ,1 )

δi j ,1 ωi1 · · · ωi j −1 ω0 ωi j+1 · · · ωir . (4.2.1.2)

⊗r

On the other hand, E 0 acts on Corollary 2.3.5. Since (r−1) (E 0 ) =

C

r

via the comultiplication as described in ⊗(r− j )

1⊗( j −1) ⊗ E 0 ⊗ K 0

,

j=1

this together with (3.5.0.1) gives E 0 ωi =

r

v

j+1kr (δi k ,n −δi k ,1 )

δi j ,1 ωi1 · · · ωi j −1 ω0 ωi j+1 · · · ωir

j=1

=

r

(Y j+ ωi )X j .

j=1

(Note that, for 1 i, j n, the values δi, j are unchanged regardless of viewing i, j as elements in Z or in I = Z/nZ.) Hence, by (4.2.1.1), E 0 ϕ(ωi ⊗ m) = E 0 ωi ⊗ m = ϕ(E 0 (ωi ⊗ m)).

4.2. Chari–Pressley category equivalence and classification

129

Similarly, we can prove that for i ∈ I (n, r ) and m ∈ M, F0 ϕ(ωi ⊗ m) r = v 1k j −1 (δik ,1 −δik ,n ) δi j ,n ωi1 · · · ωi j−1 ωn+1 ωi j +1 · · · ωir ⊗ m j=1

= ϕ(F0 (ωi ⊗ m)), as required. The commutativity relation ϒF( f )◦ϕ M = ϕ M ◦F ( f ) is clear. We now recall another theorem of Chari–Pressley which classifies finite dimensional simple UC ( sln )-modules and its generalization by Frenkel– Mukhin to the classification of finite dimensional irreducible polynomial n ). representations of UC (gl For 1 j n − 1 and s ∈ Z, define the elements P j,s ∈ UC ( sln ) through the generating functions 1 ±t P± (u) := exp − h (zu) j,±t j [t]z t 1 = P j,±s u ±s ∈ UC ( sln )[[u, u −1 ]]. s 0

Here we may view P ± j (u) as formal power series (at 0 or ∞) with coefficients in UC (sln ). Note that these elements are related to the elements φ ± j,s used in the definition of UC ( sln ) (see Definition 2.5.1(2)) through the formula ±j (u) = k±1 j

−2 P± j (z u)

P± j (u)

.

(4.2.1.3)

(For a proof, see, e.g., [10, p. 291].)2 For any polynomial f (u) = 1i m (1 − ai u) ∈ C[u] with constant term 1 and ai ∈ C∗ , define f ± (u) (so that f + (u) = f (u)) as follows:

f ± (u) = (1 − ai±1 u ±1 ). (4.2.1.4) 1i m

−1 + Note that f − (u) = ( m i=1 (−ai u) ) f (u). Let V be a finite dimensional representation of UC ( sln ) of type 1. Then V = ⊕λ∈Zn−1 Vλ , where Vλ = {x ∈ V | ki x = z λi x, 1 i n − 1}, 2 If P ± (u) denote the elements defined in line 2 above [28, (4.1)], then P ± (u) = P ± (uz). We j j j

have corrected a typo by removing the + sign in exp(∓ ...). If the + sign is there, then the − ± −1 case in i± (u) = ki±1 P ± j (uz )/P j (uz) as given in [28, (4.1)] is no longer true. Also, (4.2.1.3) shows that + sign is unnecessary.

130

4. Representations of affine quantum Schur algebras

and, since all Pi,s commute with the k j , each Vλ is a direct sum of generalized eigenspaces of the form Vλ,γ = {x ∈ Vλ | (Pi,s − γi,s ) p x = 0 for some p (1 i n − 1, s ∈ Z)},

(4.2.1.5)

±s where γ = (γi,s ) with γi,s ∈ C. If we put & ± = s 0 γ j,±s u , then j (u) by [29, Prop. 1], there exist polynomials f i (u) = 1 j m i (1 − a j,i u) and gi (u) = 1 j n i (1 − b j,i u) in C[u] such that

&i± (u) =

f i± (u)

gi± (u)

and

λi = m i − n i .

(4.2.1.6)

Following [7, 12.2.4], a non-zero (μ-weight) vector w ∈ V is called a pseudo-highest weight vector if there exist some P j,s ∈ C such that x+j,s w = 0,

P j,s w = P j,s w,

and k j w = z μ j w,

for all 1 j n − 1 and s ∈ Z. The module V is called a pseudo-highest weight module3 if V = UC ( sln )w for some pseudo-highest weight vector w. For notational simplicity, the expressions P j,s w = P j,s w, for all s ∈ Z, will be written as a single expression ± −1 P± j (u)w = P j (u)w ∈ V [[u, u ]],

where P j± (u) =

P j,±s u ±s .

(4.2.1.7)

s 0

Let P(n) be the set of (n − 1)-tuples of polynomials with constant terms 1. For P = (P1 (u), . . . , Pn−1 (u)) ∈ P(n), define P j,s ∈ C, for 1 j n−1 and s ∈ Z, as in P j± (u) = s 0 P j,±s u ±s , where P j± (u) is defined by (4.2.1.4). Let I¯(P) be the left ideal of UC ( sln ) generated by x+j,s , P j,s − P j,s , and μ k j − z j , for 1 j n − 1 and s ∈ Z, where μ j = degP j (u), and define the “Verma module” ¯ M(P) = UC ( sln )/ I¯(P). ¯ ¯ Then M(P) has a unique simple quotient, denoted by L(P). The polynomials ¯ Pi (u) are called Drinfeld polynomials associated with L(P). The following result is due to Chari–Pressley (see [6, 7] or [8, pp.7–8]). ¯ Theorem 4.2.2. The modules L(P) with P ∈ P(n) are all non-isomorphic finite dimensional simple UC (sln )-modules of type 1. 3 There are simple highest weight integrable modules considered in [54, 6.2.3] (defined in λ

[54, 3.5.6]) which, in general, are infinite dimensional.

4.2. Chari–Pressley category equivalence and classification

131

n ) be the algebra over C generated by x± (1 i < n, s ∈ Z), Let UC (gl i,s ki±1 , and gi,t (1 i n, t ∈ Z\{0}) with relations similar to (QLA1)–(QLA7) as defined in §2.5. Chari–Pressley’s classification is easily generalized to that n )-modules as follows; see [28, §4]. of simple UC (gl 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 (zu) = Qi,±s u ±s ∈ UC (gl [t]z t 1

s 0

Since hi,m = z (i −1)m gi,m − z (i +1)m gi+1,m , it follows from the definitions that P± j (u) =

j −1 ) Q± j (uz j+1 ) Q± j +1 (uz

,

for 1 j n − 1. n ), a non-zero As in the case of UC ( sln ), for a representation V of UC (gl (λ-weight) vector w ∈ V is called a pseudo-highest weight vector if there exist some Q i,s ∈ C such that x+j,s w = 0,

Qi,s w = Q i,s w,

and ki w = z λi w,

(4.2.2.1)

for all 1 i n, 1 j n − 1, and s ∈ Z. The module V is called n )w for some pseudo-highest a pseudo-highest weight module if V = UC (gl weight vector w. Associated to the sequence (Q i,s )s∈Z , define two formal power series by Q± Q i,±s u ±s . (4.2.2.2) i (u) = s 0

We also write the short form Qi± (u)w = Q i± (u)w for the relations Qi,s w = Q i,s w (s ∈ Z). n )-module V is called a polynomial represenA finite dimensional UC (gl tation if the restriction of V to UC (gln ) is a polynomial representation of type 1 and, for every weight λ = (λ1 , . . . , λn ) ∈ Nn of V , the formal power series &i± (u) associated to the eigenvalues (γi,s )s∈Z defining the generalized eigenspaces Vλ,γ as given in (4.2.1.5), where Pi,s is replaced by Qi,s and n −1 by n, are polynomials in u ± of degree λi so that the zeros of the functions &i+ (u) and &i− (u) are the same. Following [28], an n-tuple of polynomials Q = (Q 1 (u), . . . , Q n (u)) with constant terms 1 is called dominant if, for each 1 i n − 1, the ratio Q i (uz i−1 )/Q i +1 (uz i+1 ) is a polynomial. Let Q(n) be the set of dominant ntuples of polynomials.

132

4. Representations of affine quantum Schur algebras

For Q = (Q 1 (u), . . . , Q n (u)) ∈ Q(n), define Q i,s ∈ C, for 1 i n and s ∈ Z, by the following formula Q± Q i,±s u ±s , i (u) = s 0

n ) where Q i± (u) is defined using (4.2.1.4). Let I (Q) be the left ideal of UC (gl generated by x+j,s , Qi,s − Q i,s , and ki − z λi , for 1 j n − 1, 1 i n, and s ∈ Z, where λi = degQ i (u), and define n )/I (Q). M(Q) = UC (gl Then M(Q) has a unique simple quotient, denoted by L(Q). The polynomials Q i (u) are called Drinfeld polynomials associated with L(Q). n )-modules L(Q) with Q ∈ Q(n) are Theorem 4.2.3. ([28]) The UC (gl all non-isomorphic finite dimensional simple polynomial representations of n ). Moreover, UC (gl ¯ L(Q)| ∼ = L(P), UC (sln )

where P = (P1 (u), . . . , Pn−1 (u)) with Pi (u) = Q i (uz i −1 )/Q i +1 (uz i+1 ).

4.3. Classification of simple S(n, r)C -modules: the upward approach We first recall the classification of irreducible representations of H(r )C or equivalently, simple H(r )C -modules. For a = (a1 , . . . , ar ) ∈ (C∗ )r , let Ma = H(r )C /Ja , where Ja is the left ideal of H(r )C generated by X j − a j for 1 j r . Then Ma is an H(r )C module of dimension r! and, regarded as an H(r )C -module by restriction, Ma is isomorphic to the regular representation of H(r )C . Applying the functor F to Ma yields an S(n, r )C -module ⊗r C ⊗H(r)C Ma and hence, a D,C (n)-module inflated by the homomorphism ξr,C given in (3.5.10.1). The following result, which will be used in the next section, tells how the central generators z± t of D,C (n) defined in (2.2.1.2) act on this module. Lemma 4.3.1. Let a = (a1 , . . . , ar ) ∈ (C∗ )r and w ∈ ±t we have z± t w = 1s r as w for t 1.

⊗r

C

⊗H(r)C Ma . Then

Proof. We may assume w = ωi ⊗ h¯ for some i ∈ I(n, r ) and h¯ ∈ Ma . Since for each t 1, 1s r X s±t is a central element in H(r )C , it follows from

4.3. Classification of simple S(n, r )C -modules

133

(3.5.5.2) that z± t w = ωi

X s±t ⊗ h¯ = ωi ⊗ h

1s r

=

as±t ωi

1s r

⊗ h 1¯ =

X s±t 1¯

1s r

as±t w.

1s r

A segment s with center a ∈ C∗ is by definition an ordered sequence s = (az −k+1 , az −k+3 , . . . , az k−1 ) ∈ (C∗ )k . Here k is called the length of the segment, denoted by |s|. If s = {s1 , . . . , s p } is an unordered collection of segments, define ℘ (s) to be the partition associated with the sequence (|s1 |, . . . , |s p |). That is, ℘ (s) = (|si1 |, . . . , |si p |) with |si1 | · · · |si p |, where |si1 |, . . . , |si p | is a permutation of |s1 |, . . . , |s p |. We also call |s| := σ (℘ (s)) the length of s. Let Sr be the set of unordered collections of segments s with |s| = r . Then Sr = ∪λ∈+ (r) Sr,λ , where Sr,λ = {s ∈ Sr | ℘ (s) = λ} and + (r ) is the set of partitions of r . For s = {s1 , . . . , s p } ∈ Sr,λ , let a(s) = (s1 , . . . , s p ) ∈ (C∗ )r be the r -tuple obtained by juxtaposing the segments in s. Then the element yλ = (−z 2 )−(w) Tw ∈ H(r )C w∈Sλ

generates the submodule H(r )C y¯λ of Ma(s) which, as an H(r )C -module, is isomorphic to H(r )C yλ . However, H(r )C yλ ∼ m μ,λ E μ ) = Sλ ⊕ ( m μ,λ Sμ ), (4.3.1.1) = Eλ ⊕ ( μ&r,μλ

μ&r,μλ

where Sν := E ν is the left cell module defined by the Kazhdan–Lusztig C-basis [50] associated with the left cell containing w0,ν . The notation Sν indicates that E ν is the Specht module associated with the dual partition ν of ν (i.e., isomorphic to H(r )C yν Twν xν in the notation of [15, p.25]). Let Vs be the unique composition factor of the H(r )C -module H(r )C y¯λ such that the multiplicity of Sλ in Vs as an H(r)C -module is non-zero. We can now state the following classification theorem due to Zelevinsky [81] and Rogawski [66]. The construction above follows [66].

134

4. Representations of affine quantum Schur algebras

Theorem 4.3.2. 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 ). We record the following general result. Lemma 4.3.3. Let S and H be two algebras over a field F. If V is an S-H bimodule, M is a left H -module, and e ∈ S is an idempotent element, then eV is an eSe-H -bimodule and there is an eSe-module isomorphism (eV ) ⊗ H M ∼ = e(V ⊗ H M) (ew ⊗ m −→ e(w ⊗ m)). Proof. There is a natural map α : (eV ) ⊗ H M → V ⊗ H M defined by sending ew ⊗ m to ew ⊗ m. The map α induces a surjective map α¯ : (eV ) ⊗ H M → e(V ⊗ H M). On the other hand, there is a surjective right H -module homomorphism from V to eV defined by sending w to ew, for w ∈ V . This map induces a natural surjective map β : V ⊗ H M → (eV ) ⊗ H M defined by sending w ⊗ m to ew ⊗ m, for w ∈ V and m ∈ M. By restriction, we get a map β¯ : e(V ⊗ H M) → (eV ) ⊗ H M. Since α¯ β¯ = id and β¯ α¯ = id, the assertion follows. Let Sr(n) = {s ∈ Sr | (n)

Then Sr

⊗r

C

⊗H(r)C Vs = 0}.

= Sr for all r n. We have the following classification theorem.

Theorem 4.3.4. The set {

⊗r

C

⊗H(r )C Vs | s ∈ Sr(n) }

is a complete set of non-isomorphic simple S(n, r )C -modules. In particular, if n r , then the set { ⊗r C ⊗H(r )C Vs | s ∈ Sr } is a complete set of nonisomorphic simple S(n, r )C -modules. Proof. If n r, then the assertion follows from Theorem 4.1.3. Now we assume n < r . As in the proof of Theorem 4.1.6, we choose N = r and regard (n, r ) as a subset of (r, r ) via the map μ → μ given in (4.1.5.1). Then S(n, r )C identifies a centralizer subalgebra eS(r, r)C e of S(r, r )C , where 1 . e = λ∈(n,r ) [diag( λ)] = λ∈(n,r) φλ, λ Recall from Proposition 3.3.1 that the tensor space ⊗r identifies T(n, r ). Thus, T(r, r )C is the tensor space on which S(r, r )C acts. Hence, the set

4.3. Classification of simple S(n, r )C -modules

135

{T(r, r )C ⊗H(r)C Vs | s ∈ Sr } forms a complete set of non-isomorphic simple S(r, r)C -modules. By [33, 6.2(g)] or a direct argument, the set {e(T(r, r)C ⊗H(r)C Vs ) | s ∈ Sr }\{0} forms a complete set of non-isomorphic simple S(n, r)C -modules. Since ∼ eT(r, r )C ∼ e x H (r ) x = μ C = μ H(r )C μ∈ (r,r)

∼ =

μ∈(n,r)

xμ H(r )C ∼ =

μ∈(n,r)

⊗r

C

,

by Lemma 4.3.3, there is an S(n, r )C -module isomorphism e(T(r, r )C ⊗H(r)C Vs ) ∼ = (eT(r, r)C ) ⊗H(r)C Vs ∼ =

⊗r

C

⊗H(r)C Vs ,

proving the n < r case. From the proof above, one sees easily that, for + (n, r ) = + (r )∩(n, r ), -

Sr,λ ⊆ Sr(n) .

λ∈+ (n,r)

We will prove in §4.5 that this inclusion is an equality and, thus, complete the classification of simple S(n, r )C -modules when n < r in the upward approach. (n) The next example shows that Sr = Sr does occur. Example 4.3.5. For any a ∈ C∗ , there is an algebra homomorphism eva : H(r )C → H(r )C , called the evaluation map (see [9, 5.1]), such that (1) eva (Ti ) = Ti , 1 i r − 1; −2( j−1) (2) eva (X j ) = az T j−1 · · · T2 T1 T1 T2 · · · T j−1 , 1 j r . Thus, every simple H(r )C -module Sλ is also a simple H(r )C -module (Sλ )a . If n < r , then those s such that Vs is isomorphic to (S(1r ) )a , where ℘ (s) = (r ), (n) are not in Sr . We end this section with a brief discussion of a certain branching rule. Since the parameter z for the specialization Z → C, v → z is not a root of unity, the Hecke algebra H(r )C is semisimple. Thus, as an H(r )C -module, Vs is semisimple. By (4.3.1.1), we can decompose the H(r )C -module Vs |H(r)C = [Vs : Sμ ]Sμ . (4.3.5.1) μ&r,μ℘ (s)

136

4. Representations of affine quantum Schur algebras

Here the multiplicity [Vs : Sμ ] m μ,℘ (s) , the multiplicity of Sμ in H(r )C y℘ (s) , and [Vs : S℘ (s) ] = 1. We will call the description of the nonzero multiplicities in (4.3.5.1) the affine-to-finite Branching Rule or simply the affine Branching Rule. Problem 4.3.6. Describe the affine Branching Rule. In other words, find a necessary and sufficient condition for the multiplicities [Vs : Sμ ] given in (4.3.5.1) to be non-zero. There is a similar branching rule for S(n, r )C to S(n, r )C . We will make a comparison between them in §4.5.

4.4. Identification of simple S(n, r)C -modules: the n > r case We now use the downward approach to identify the simple S(n, r )C -modules when n r , and we will complete another classification in §4.6. More precisely, we will prove that each simple S(n, r )C -module is an irreducible n ) in Proposition 4.6.4 and prove that polynomial representation of UC (gl n ) is a simple S(n, r )C each irreducible polynomial representation of UC (gl module for some r in Proposition 4.6.7. Combining Propositions 4.6.4 with 4.6.7, we can classify the simple modules for S(n, r )C . Recall the C-algebra D,C (n) defined in Remark 2.1.4 and the Hopf Cn ) discussed in Theorem 2.5.3 algebra isomorphism EH,C : D,C (n) → UC (gl and Remark 2.5.5(3). In order to make use of the results in [9], we need to adjust this isomorphism so that its restriction to UC ( sln ) agrees with the one given in [9, p.318]. By the presentation given in Theorem 2.3.1, it is easy to see that there is a Hopf algebra automorphism g of D,C (n) satisfying g(K i±1 ) = K i±1 ,

g(u i± ) = u ± i , (1 i < n),

n ±1 ± g(u ± n ) = (−1) z u n ,

±s ± g(z± s ) = ∓sz zs (s 1).

(The scalar (−1)n z ±1 is for the adjustment, while the scalar ∓sz ±s becomes apparent in the computation right above (4.4.2.3).) Combining the two gives the following. Proposition 4.4.1. There is a Hopf C-algebra isomorphism n ) f = EH,C ◦ g : D,C (n) −→ UC (gl such that

4.4. Identification of simple S(n, r )C -modules: the n > r case K i±1 −→ ki±1 ,

137

± u± i −→ xi,0 (1 i < n),

n ±1 ± u± n −→ (−1) z εn ,

±s z± s −→ ∓sz θ±s (s 1).

Now the C-algebra epimorphism ξr,C : D,C (n) → S(n, r )C described in Corollary 3.8.3 (and Theorem 3.8.1) together with f gives a C-algebra epimorphism n ) −→ S(n, r )C . ξr, C := ξr,C ◦ f −1 : UC (gl

(4.4.1.1)

n )-module via Thus, every S(n, r )C -module will be inflated into a UC (gl this homomorphism. In particular, every simple S(n, r )C -module given in n )-module. We now identify them for the Theorem 4.3.4 is a simple UC (gl n ) n > r case in terms of the irreducible polynomial representations of UC (gl described in Theorem 4.2.3. Recall from §4.2 that Q(n) is the set of dominant n-tuples of polynomials. For r 1, let Q(n)r = Q = (Q 1 (u), . . . , Q n (u)) ∈ Q(n) | r = deg Q i (u) . 1i n

Assume n > r. For s = {s1 , . . . , s p } ∈ Sr with si = (ai z −μi +1 , ai z −μi +3 , . . . , ai z μi −1 ) ∈ (C∗ )μi , define Qs = (Q 1 (u), . . . , Q n (u)) by setting recursively 1, Q i (u) = Pi (uz −i+1 )Pi+1 (uz −i +2 ) · · · Pn−1 (uz n−2i ), where Pi (u) = 1 j p (1 − a j u).

(4.4.1.2)

if i = n; if n − 1 i 1,

μ j =i

Since every μ j r, Pr+1 (u) = · · · = Pn−1 (u) = 1. Hence, every Qs has the form (Q 1 (u), . . . , Q r (u), 1, . . . , 1). Moreover, putting νi := degPi (u) = #{ j ∈ [1, p] | μ j = i} and λi := degQ i (u) = #{ j ∈ [1, p] | μ j i} gives a partition λ = (λ1 , . . . , λn−1 , λn ) of r dual to (μ1 , . . . , μ p ), and λi − λi+1 = νi for all 1 i < n. From the definition, s consists of νi segments of length i with centers determined by the roots of Pi (u) for all i.

138

4. Representations of affine quantum Schur algebras

We now have the following identification theorem. Theorem 4.4.2. Maintain the notation above and let n > r . The map n )-module s → Qs defines a bijection from Sr to Q(n)r , and induces UC (gl ⊗r isomorphisms C ⊗H(r)C Vs ∼ = L(Qs ), for all s ∈ Sr . Hence, the set {L(Q) | Q ∈ Q(n)r } forms a complete set of non-isomorphic simple S(n, r )C -modules. Proof. By the algebra homomorphism ξr, C given in (4.4.1.1), every S(n, r )C n )-module. Let [M] denote the isoclass of M. module M is regarded as a UC (gl By Theorem 4.3.4, it suffices to prove that ∼ (1) ⊗r C ⊗H(r)C Vs = L(Qs ), and (2) {[L(Q)] | Q ∈ Q(n)r } = {[ ⊗r C ⊗H(r )C Vs ] | s ∈ Sr }. We first prove (1). Let s = {s1 , . . . , s p } ∈ Sr be an unordered collection of segments with si = (ai z −μi +1 , ai z −μi +3 , . . . , ai z μi −1 ) ∈ (C∗ )μi .

Thus, r = 1i p μi . Let a = a(s) = (s1 , . . . , s p ) ∈ (C∗ )r be the sequence obtained by juxtaposing the segments in s. Then the simple S(n, r )C -module ⊗r C ⊗H(r )C Vs becomes a simple UC (gln )-module via (4.4.1.1). As a sim ple UC (sln )-module, this module is isomorphic to the Chari–Pressley module F (Vs ) by Proposition 4.2.1. Applying [9, 7.6] yields a UC ( sln )-module iso∼ ¯ morphism ⊗r ⊗ V L(P), where P = (P (u), . . . , Pn−1 (u)) 1 H(r)C s = C with

±1 Pi± (u) = (1 − a ±1 1 i n − 1. (4.4.2.1) j u ), 1 j p μ j =i

n )-module, by [28, Lem. 4.2], if w0 ∈ ⊗r ⊗H (r ) Vs As a simple UC (gl C C is the pseudo-highest weight vector of weight λ = (λ1 , . . . , λn ), then λ is a partition of r , since λ is also the highest weight of a simple submodule of + the S(n, r )C -module ⊗r C ⊗H(r)C Vs , and there exist Q i (u) ∈ C[[u]] and − −1 Q i (u) ∈ C[[u ]], 1 i n, such that Qi± (u)w0 = Q i± (u)w0 , P j± (u)

=

Q ±j (uz j −1 ) Q ±j +1 (uz j+1 )

± P± j (u)w0 = P j (u)w0 ,

K i w0 = z λi w0 ,

, and degPi (u) = λi − λi+1 . (4.4.2.2)

We now prove that Q ± i (u) are polynomials of degree λi .

4.4. Identification of simple S(n, r )C -modules: the n > r case

139

∓t Since f (z± t ) = ∓t z θ±t , for all t 1, as in Proposition 4.4.1, it follows from (2.5.1.1) and Lemma 4.3.1 that

t z ±t gi,±t w0 = z± (ai z −μi +2k−1 )±t w0 . t w0 = [t]z 1i p 1i n

1kμi

Thus,

Q i± (u)w0 =

1i n

Qi± (u)w0

1i n

1 = exp − gi,±t (uz)±t w0 [t]z t 1 1i n 1 = exp − (ai uz 2k−1−μi )±t w0 t 1i p =

1i p 1kμi

=

1kμi

exp

−

t 1

1 t 1

t

(ai uz 2k−1−μi )±t w0

±1 1 − ai uz 2k−1−μi w0 ,

1i p 1kμi

as −

1 2k−1−μi )±t t 1 t (ai uz

1i n

Q± i (u)

= log 1 − (ai uz 2k−1−μi )±1 . Hence,

2k−1−μi ±1 = 1 − ai uz .

(4.4.2.3)

1i p 1kμi

On the other hand, by (4.4.2.2), ± 2(n−i) Q i± (u) = Pi± (uz −i+1 )Pi±+1 (uz −i+2 ) · · · Pn−1 (uz n−2i )Q ± ) n (uz (4.4.2.4) for all 1 i n, and by (4.4.2.1),

Pi± (uz −i+1 )Pi± (uz −i +3 ) · · · Pi± (uz i −1 )

= (1 − (a j uz −μ j +1 )±1 )(1 − (a j uz −μ j +3 )±1 ) · · · (1 − (a j uz μ j −1 )±1 ) 1 j p μ j =i

=

1 j p, μ j =i 1kμ j

(1 − (a j uz 2k−1−μ j )±1 ).

140

4. Representations of affine quantum Schur algebras

Thus,

Pi± (uz −i+1 )Pi± (uz −i+3 ) · · · Pi± (uz i −1 )

1i n−1

±1 1 − a j uz 2k−1−μ j .

=

1 j p 1kμ j

Hence,

1i n

=

Q i± (u)

Pi± (uz −i +1 )Pi± (uz −i+3 ) · · ·

1i n−1

±1 = 1 − ai uz 2k−1−μi

Pi± (uz i−1 )

2l Q± n (uz )

0l n−1

2l Q± n (uz ).

0l n−1

1i p 1kμi

Combining this with (4.4.2.3) yields

2l Q± n (uz ) = 1. 0l n−1

So we have

exp

1 gn,±t z ±2lt (uz)±t w0 [t]z t 1 0l n−1

Qn± (uz 2l )w0 = w0 .

−

=

0l n−1

It follows that

1 − gn,±t [t]z t 1

z

±2lt

(uz) w0 = 0. ±t

0l n−1

This forces gn,±t w0 = 0, for all t 1. Consequently, ± Q± n (u)w0 = Qn (u)w0 = w0 , ± and hence, Q ± n (u) = 1. We conclude by (4.4.2.4) that all Q i (u) are polynomials with constant term 1 and Qs = (Q 1 (u), . . . , Q n (u)) ∈ Q(n). Moreover,

1i n

deg Q i (u) =

1 j n−1

j deg P j (u) =

1 j p

μj = r =

n i =1

λi .

(n)

4.5. Application: the set Sr

141

This forces λn = 0 and consequently, deg Q i (u) = λi , for all 1 i n. Therefore, ⊗r C ⊗H(r)C Vs is a simple polynomial representation of UC (gln ) ∼ and ⊗r ⊗ V L(Q ), proving (1). s H(r)C s = C We now prove (2). For any Q = (Q 1 (u), . . . , Q n (u)) ∈ Q(n) such that r = 1i n λi with λi = degQ i (u), we now prove that L(Q) is a simple S(n, r )C -module. Since the polynomials P j (u) =

Q j (uz j−1 ) (1 j n − 1) Q j+1 (uz j+1 )

have constant term 1 and deg P j (u) = λ j − λ j+1 =: ν j , it follows that λ ∈ + (n, r ) is a partition with at most n parts. So n > r implies λn = 0. Moreover, we may write, for 1 i n − 1, Pi (u) = (1 − aν1 +···+νi −1 +1 u)(1 − aν1 +···+νi −1 +2 u) · · · (1 − aν1 +···+νi−1 +νi u), where a −1 ∈ C, 1 j p = i νi , are the roots of Pi (u). Let j s = {s1 , . . . , s p }, where si = (ai z −μi +1 , ai z −μi +3 , . . . , ai z μi −1 ) and (μ1 , . . . , μ p ) = (1ν1 , . . . , (n − 1)νn−1 ), and let a = (s1 , . . . , s p ). Since μj = iνi = λi = r, 1 j p

1i n−1

1i n

(C∗ )r .

we have a ∈ By the first part of the proof, we see Q = Qs and, hence, ⊗r ∼ C ⊗H(r )C Vs = L(Q) as UC (gln )-modules. In other words, L(Q) is a simple S(n, r )C -module.

4.5. Application: the set Sr(n) (n)

We can apply Theorem 4.4.2 to determine the index set Sr used in the Classification Theorem 4.3.4. Recall from Proposition 3.7.4 that the idempotents lλ = [diag(λ)], for λ ∈ (n, r ), which form a basis for S(n, r )0C . Note also that S(n, r )0C = S(n, r )0C and (n, r ) is identified with (n, r) under the map 2 in (1.1.0.2). Lemma 4.5.1. If L(Q) is a simple S(n, r )C -module with pseudo-highest weight λ and lμ L(Q) = 0, then λ μ. n )-module, then lμ L(Q) = L(Q)μ is its Proof. If we regard L(Q) as a UC (gl μ-weight space. However, if w0 is a pseudo-highest weight vector in L(Q)λ ,

142

4. Representations of affine quantum Schur algebras

n ) = UC (gl n )− UC (gl n )0 UC (gl n )+ then the triangular decomposition4 UC (gl n )− w0 . Here UC (gl n )± (resp., ([28, Lem. 7.4]) implies L(Q) = UC (gl n )0 ) are the subalgebras generated by all x± (resp., k j , g j,t ), for all UC (gl i,s 1 i n−1, s ∈ Z (resp., 1 j n, t ∈ Z\{0}). Our assertion follows from − the fact that every xi,s w0 has weight λ; see (QLA2) in Definition 2.5.1. For Q ∈ Q(n)r , let deg(Q) = (deg(Q 1 (u)), · · · , deg(Q n (u))). As usual, regard (n, r ) as a subset of (N , r) as in (4.1.5.1). Lemma 4.5.2. Assume N > r n. Let e = μ∈(n,r) lμ . Then the set {eL(Q) | Q ∈ Q(N )r , deg(Q) ∈ (n, r)} forms a complete set of non-isomorphic simple S(n, r )C -modules. Proof. By Theorem 4.4.2 and [33, 6.2(g)], the set {eL(Q) = 0 | Q ∈ Q(N )r } forms a complete set of non-isomorphic simple S(n, r )C -modules. Thus, it is enough to prove that, for Q ∈ Q(N )r , eL(Q) = 0 if and only if deg(Q) ∈ (n, r ), i.e., the last N − n parts of deg(Q) are all zero. Let Q ∈ Q(N )r and λ = deg(Q). If λ ∈ (n, r), then e(lλ L(Q)) = lλ L(Q) = 0 and, hence, eL(Q) = 0. Conversely, assume eL(Q) = 0. Since 1 = α∈(N ,r) lα , eL(Q) = e lα L(Q) = lα L(Q). α∈(N,r)

α∈(n,r)

This together with the fact that eL(Q) = 0 implies that there exists α ∈ (n, r ) such that lα L(Q) = 0. Since lα L(Q) = 0, by Lemma 4.5.1, α λ and, hence, r = 1i n αi 1i n λi r , forcing λ ∈ (n, r ). Theorem 4.5.3. We have Sr(n) = {s = {s1 , . . . , s p } ∈ Sr | p 1, |si | n, ∀i }. Proof. We choose N > 0 such that N > max{r, n}. Then, by Theorem 4.4.2, {L(Qs ) | s ∈ Sr } = {L(Q) | Q ∈ Q(N )r } is a complete set of non-isomorphic simple S(N, r )C -modules. Let e = μ∈(n,r ) lμ . By the proof of Theorem 4.3.4 we have eL(Qs ) ∼ = ⊗r C ⊗H(r)C Vs for s ∈ Sr . Thus, by Lemma 4.5.2, the set {

⊗r

C

⊗H(r)C Vs | s ∈ Sr , deg(Qs ) ∈ (n, r )}

forms a complete set of non-isomorphic simple S(n, r )C -modules. Now, write Qs = (Q 1 (u), . . . , Q N (u)). (Then Q r +1 (u) = · · · = Q N (u) = 1.) The condition deg(Qs ) ∈ (n, r ), where s = {s1 , . . . , s p } ∈ Sr , is equivalent to 4 This triangular decomposition is different from the one given in Corollary 2.5.4.

4.6. Classification of simple S(n, r )C -modules

143

Q n+1 (u) = · · · = Q N (u) = 1, which means Pn+1 (u) = · · · = PN −1 (u) = 1. By (4.4.1.2), this condition holds if and only if each segment si in s has length |si | n for 1 i p. + Since { ⊗r n,C ⊗H(r)C Sμ }μ∈ (n,r) forms a complete set of simple S(n, r )C ⊗r modules, we may speak of multiplicities [ ⊗r C ⊗H(r)C Vs : n,C ⊗H(r)C Sμ ] for

(n)

all s ∈ Sr . An immediate consequence of the theorem above is the following multiplicity identity. Recall that λ is the dual partition of λ. Corollary 4.5.4. The partition ℘ (s) associated with s ∈ Sr(n) gives rise to a (n) well-defined surjective map ℘ : Sr → + (n, r ), s → ℘ (s) . Moreover, (n) + for s ∈ Sr and μ ∈ (n, r ), we have [

⊗r

C

⊗H(r)C Vs :

⊗r n,C

⊗H(r)C Sμ ] = [Vs : Sμ ].

Proof. By Lemma 4.1.1 and (4.3.1.1), we have S(n, r )C -module isomorphisms ⊗r ∼ ⊗r ∼ m μ ( ⊗r C ⊗H(r)C Vs = n,C ⊗H(r)C Vs = n,C ⊗H(r)C Sμ ), μ&r,μ℘ (s)

where m μ = [Vs : Sμ ]. Now, + (n, r ). Hence, in this case, [

⊗r

C

⊗H(r)C Vs :

⊗r n,C

⊗r n,C

⊗H(r)C Sμ = 0 if and only if μ ∈

⊗H(r)C Sμ ] = m μ = [Vs : Sμ ],

as desired.

4.6. Classification of simple S(n, r )C -modules: the downward approach We now complete the classification of simple S(n, r)C -modules by removing the condition n > r in Theorem 4.4.2. We will continue to use the downward approach with a strategy different from that in the previous section. Throughn ) with D,C (n) via the isomorphism f out this section, we will identify UC (gl n )-module in Proposition 4.4.1 and regard every S(n, r )C -module as a UC (gl via the algebra homomorphism ξr,C : UC (gln ) → S(n, r )C ; see (4.4.1.1). n )-module via ξ Consider the UC (gl 1,C C (a)

:= F(Ma ) =

C

⊗H(1)C Ma

for a ∈ C∗ . By Proposition 4.2.1, C (a) ∼ = n,C ⊗H(1)C Ma ∼ = n,C as S(n, 1)C -modules. Hence, dim C (a) = n. See Proposition 3.5.2 for a different construction of C (a).

144

4. Representations of affine quantum Schur algebras

n )-module since dimC Ma = 1 By Theorem 4.1.3, C (a) is a simple UC (gl and Ma = Vs with s = (a) ∈ S1 . Since n > 1, by Theorem 4.4.2, C (a)

∼ = L(Q) with Q 1 (u) = 1 − au and Q i (u) = 1, for 2 i n. (4.6.0.1)

n )-modules C (a) are very useful, and we will prove that every The UC (gl simple S(n, r )C -module is a quotient module of C (a1 ) ⊗C · · · ⊗C C (ar ), for some a ∈ (C∗ )r , in Corollary 4.6.2. For i ∈ Z, let ωi = ωi ⊗ 1¯ ∈ C (a). n )-module Lemma 4.6.1. For any a = (a1 , . . . , ar ) ∈ (C∗ )r , there is a UC (gl isomorphism ϕ:

C (a1 ) ⊗C

· · · ⊗C

C (ar )

−→

⊗r

C

⊗H(r )C Ma

¯ for i ∈ I(n, r ), where defined by sending ωi to ωi ⊗ 1, ωi = ωi1 ⊗ · · · ⊗ ωir . Moreover, as an S(n, r )C -module, C (a1 ) ⊗C · · · ⊗C C (ar ) is isomorphic ∗ r to the finite tensor space ⊗r n,C , for all a ∈ (C ) . Proof. The set { ωi | 1 i n} forms a basis of

C (a).

Hence, the set { ωi | i ∈ I (n, r)}

forms a basis of C (a1 ) ⊗C · · · ⊗C C (ar ). Similarly by Proposition 4.2.1, we have ⊗r

C

⊗H(r)C Ma ∼ =

⊗r n,C

⊗H(r)C Ma ∼ =

⊗r n,C

(4.6.1.1)

as S(n, r )C -modules. So the set {ωi ⊗ 1¯ | i ∈ I (n, r )} forms a basis of ϕ:

⊗r

C

⊗H(r)C Ma . Hence, there is a linear isomorphism

C (a1 ) ⊗C

· · · ⊗C

C (ar )

−→

⊗r

C

⊗H(r )C Ma

¯ for all i ∈ I (n, r ). defined by sending ωi to ωi ⊗ 1, Now we assume i ∈ I(n, r ). We write i = j + nt with j ∈ I (n, r ) and t ∈ Zr . Then ϕ( ωi ) = ϕ((a1−t1 ω j1 ) ⊗ · · · ⊗ (ar−tr ω jr )) = a1−t1 a2−t2 · · · ar−tr ωj ⊗ 1¯ = ωj X 1−t1 X 2−t2 · · · X r−tr ⊗ 1¯ ¯ = ωi ⊗ 1.

4.6. Classification of simple S(n, r )C -modules

145

n )-module isomorphism. The last assertion is It follows easily that ϕ is a UC (gl clear from (4.6.1.1). n )-module. Then Corollary 4.6.2. Let V be a finite dimensional simple UC (gl the following conditions are equivalent: (1) V can be regarded as an S(n, r )C -module via ξr, C ; n )-module C (a1 ) ⊗C · · · ⊗C (2) V is a quotient module of the UC (gl ∗ r for some a ∈ (C ) ; (3) V is a quotient module of ⊗r C ; (4) V is a subquotient module of ⊗r C .

C (ar )

Proof. By the lemma above, the map ⊗r

C

−→

C (a1 ) ⊗C

· · · ⊗C

C (ar ),

ωi −→ ωi

n )-module epimorphism (say, induced by the natural H(r)C -module is a UC (gl epimorphism H(r )C → Ma ). Hence, (2) implies (3). Certainly, (3) implies (4). Since ⊗r C is an S(n, r )C -module, (4) implies (1). If V can be regarded as an S(n, r )C -module via ξr, C , then, by Theorem 4.3.4, V ∼ = ⊗r C ⊗H(r)C Vs for some s. Since Vs is a homomorphic image of some Ma (see [9, 3.4], say), it follows that V is a homomorphic image of ⊗r C ⊗H(r)C Ma , which is, by Lemma 4.6.1, isomorphic to C (a1 ) ⊗C · · · ⊗C C (ar ). Hence, V is a homomorphic image of C (a1 ) ⊗C · · · ⊗C C (ar ), proving (2). n ), S(n, r )C , etc., under consideration Remark 4.6.3. The algebras UC (gl are all defined over C with parameter z which is not a root of unity. As an S(n, r )C -module, the finite dimensional tensor space ⊗r n,C is a semisimple module (i.e., is completely reducible). In the affine case, however, the infinite dimensional tensor space ⊗r C is not completely reducible. The fact that every simple S(n, r )C -module is a homomorphic image of ⊗r C does reflect a certain degree of the complete reducibility. Using Corollary 4.6.2, we can prove the first key result for the classification theorem. Proposition 4.6.4. Every simple S(n, r )C -module is a polynomial represenn ). tation of UC (gl Proof. Let V be a simple S(n, r )C -module. Then V is finite dimensional by Theorem 4.1.6. Thus, V is a quotient module of C (a1 ) ⊗C · · · ⊗C C (ar ) for some a ∈ (C∗ )r , by Corollary 4.6.2. Now, (4.6.0.1) implies that C (a) is n ). So, by [28, 4.3], the tensor product a polynomial representation of UC (gl

146

4. Representations of affine quantum Schur algebras

n ). Hence, · · · ⊗C C (ar ) is a polynomial representation of UC (gl n ). V is also a polynomial representation of UC (gl C (a1 ) ⊗C

n )-module Deta defined in the following lemma plays For a ∈ C∗ , the UC (gl n ) as the quantum determithe same role in the representation theory of UC (gl nant in the representation theory of U (n)C = U (gln )C . One can easily see that by restriction Deta is isomorphic to the quantum determinant for U (n)C . Lemma 4.6.5. Fix a ∈ C∗ . Let a = s = (a, az 2 , . . . , az 2(n−1) ) regarded as a single segment, and let Deta :=

⊗n

C

⊗H(n)C Va ,

where Va = Vs is the submodule of Ma generated by y (n) . Then dim Deta = 1 and Deta = spanC {ω1 ⊗ · · · ⊗ ωn ⊗ y (n) }. n )-module, Deta ∼ Moreover, as a UC (gl = L(Q), where Q = (Q 1 (u), . . . , Q n (u)) ∈ Q(n) with Q i (u) = 1 − az 2(n−i ) u, for all i = 1, 2, . . . , n. Proof. Recall the notation used in §4.3. We have Ma ∼ = H(n)C and, by Theorem 4.3.2, Va = H(n)C y (n) = Cy (n) , since Ti y(n) = −y(n) , for all 1 i n − 1. By Theorem 4.1.3, Deta is a simple S(n, n)C -module. By Proposition 4.6.4, Deta ∼ = L(Q), for some Q ∈ Q(n). Since ⊗n ωi H(n)C , n,C = i∈I(n,n)0

by Proposition 4.2.1, Deta ∼ =

⊗n n,C

⊗H(n)C Va ∼ =

ωi H(n)C ⊗H(n)C Va .

i∈I(n,n)0

Hence, Deta = spanC {ωi ⊗ y (n) | i ∈ I(n, n)0 }. If i k = i k+1 in an i ∈ I(n, n)0 for some 1 k n − 1, then z 2 ωi ⊗ y (n) = ωi Tk ⊗ y (n) = ωi ⊗ Tk y (n) = −ωi ⊗ y (n) . This forces ωi ⊗ y (n) = 0 as z is not a root of unity. The only i ∈ I(n, n)0 with i k = i k+1 , for 1 k n − 1, is i = (1, 2, . . . , n). Hence, Deta = C ω1 ⊗ · · · ⊗ ωn ⊗ y (n) . Since K i w0 = zw0 , where w0 = ω1 ⊗ · · · ⊗ ωn ⊗ y (n) , it follows that deg Q i (u) = 1, for all 1 i n. On the other hand, since Pi (u) =

Q i (uz i −1 ) Q i +1 (uz i+1 )

4.6. Classification of simple S(n, r )C -modules

147

is a polynomial, we must have Pi (u) = 1, for all 1 i n − 1. Suppose Q n (u) = 1 − bu, for some b ∈ C∗ . Then Q i (u) = 1 − bz 2(n−i ) u for all i. Thus, as in the proof of Theorem 4.4.2, Lemma 4.3.1 implies t z ±t gi,±t w0 = z± (az 2(k−1) )±t w0 , t w0 = [t]z 1i n

1k n

for all t 1 and, hence,

±1 1 − bz 2(n−i) u w0 = Q i± (u)w0 1i n

=

1i n

2(k−1) ±1 ± Qi (u)w0 = 1 − az u w0 .

1i n

1k n

Equating the coefficients of u forces a = b. Lemma 4.6.6. Let V be a simple S(n, k)C -module and W be a simple S(n, l)C -module. Then V ⊗ W is an S(n, k + l)C -module. n )-module Proof. By Corollary 4.6.2, V is a quotient module of the UC (gl ∗ )k and W is a quotient mod(a ) ⊗ · · · ⊗ (a ) for some a ∈ (C C 1 C C C k ule of C (b1 ) ⊗C · · · ⊗C C (bl ) for some b ∈ (C∗ )l . Thus, V ⊗ W is a n )-module quotient module of the UC (gl C (a1 ) ⊗C

· · · ⊗C

C (ak ) ⊗C

C (b1 ) ⊗C

· · · ⊗C

C (bl ),

⊗(k+l)

which is isomorphic to C ⊗H(r)C M(a,b) , by Lemma 4.6.1. Hence, as a quotient module of an S(n, k + l)C -module, V ⊗ W is an S(n, k + l)C module. We remark that it is possible to embed H(k)C ⊗ H(l)C into H(k + l)C as a subalgebra (see, e.g., [9, 3.2]) and, hence, S(n, k + l)C is embedded as a subalgebra in S(n, k)C ⊗ S(n, l)C . Thus, by restriction, the S(n, k)C ⊗ S(n, l)C -module V ⊗ W is an S(n, k + l)C -module. See the map * in §5.5 or [56, 1.2] for a geometric construction of the embedding. For 1 i n − 1 and a ∈ C∗ , define Qi,a ∈ Q(n) by setting Q n (u) = 1 and Q j (uz j −1 ) = (1 − au)δi, j , Q j +1 (uz j +1 ) for 1 j n − 1. In other words, Qi,a = (1 − az i−1 u, . . . , 1 − az −i +3 u, 1 − az −i +1 u , 1, . . . , 1). (i)

n )-module Since n > i, by Theorem 4.4.2, the simple UC (gl L i,a := L(Qi,a )

148

4. Representations of affine quantum Schur algebras

is also a simple S(n, i)C -module. The weight of a pseudo-highest weight vector of L i,a is λ(i, a) = (1i , 0n−i ). Now we can prove the second key result for the classification theorem. Proposition 4.6.7. If Q = (Q i (u)) ∈ Q(n) with r = 1i n deg(Q i (u)), then L(Q) is (isomorphic to) a simple S(n, r )C -module. Proof. Let λ = (λ1 , . . . , λn ) with λ j = deg(Q j (u)) and let Pi (u) =

Q i (uz i−1 ) , Q i+1 (uz i +1 )

1 i n − 1.

Write Q n (u) = (1 − b1 u) · · · (1 − bλn u) and

Pi (u) = (1 − ai, j u), 1 j μi

where μi = λi − λi +1 . Let V = L 1 ⊗ · · · ⊗ L n−1 ⊗ Detb1 ⊗ · · · ⊗ Detbλn , where L i = L i,ai,1 ⊗ · · · ⊗ L i,ai,μi for 1 i n − 1. Let wi,ai,k (resp., v j ) be a pseudo-highest weight vector of L i,ai,k (resp., Detb j ), and let w0 = w1 ⊗ w2 ⊗ · · · ⊗ wn−1 and v0 = v1 ⊗ · · · ⊗ vn , where wi = wi,ai,1 ⊗ wi,ai,2 ⊗ · · · ⊗ wi,ai,μi , for 1 i n − 1. Since wi,ai,k has weight (1i , 0n−i ) and (1 − (ai,k z i−2 j+1 u)±1 )wi,ai,k , if 1 j i; Q± and j (u)wi,ai,k = wi,ai,k , if i < j n, 2(n− j ) ±1 Q± u) )vi for 1 i λn , j (u)vi = (1 − (bi z

it follows from [28, Lem. 4.1] that ± Q± j (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 ). Let W be the submodule of V generated 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 Q i,s , respectively,

4.6. Classification of simple S(n, r )C -modules

149

where Q i,s are the coefficients of Q i± (u). So the simple quotient module of W is isomorphic to L(Q) (cf. the construction in [28, Lem. 4.8]). Since 1i n−1 iμi + nλn = 1i n λi = r , by Lemma 4.6.6, V is an S(n, r )C -module. Hence, L(Q) has an S(n, r )C -module structure. Now using Propositions 4.6.4 and 4.6.7 we can prove the following classification theorem. Theorem 4.6.8. For any n, r 1, the set L(Q) | Q ∈ Q(n)r is a complete set of non-isomorphic simple S(n, r )C -modules. Proof. By Proposition 4.6.7, the set L(Q) | Q ∈ Q(n)r consists of nonisomorphic simple S(n, r )C -modules. It remains to prove that every simple S(n, r )C -module is isomorphic to L(Q) for some Q ∈ Q(n)r . n )-module Let V be a simple S(n, r )C -module. Then V ∼ = L(Q) as a UC (gl for some Q ∈ Q(n) by Proposition 4.6.4. Let l = 1i n deg Q i (u). Then, by Proposition 4.6.7, L(Q) is an S(n, l)C -module. Thus, by restriction, V is a module for the q-Schur algebra S(n, r )C and V is also a module for the q-Schur algebra S(n, l)C . Hence, r = l. Corollary 4.6.9. Let V be a finite dimensional irreducible polynomial repren ). Then V can be regarded as an S(n, r )C -module via ξ sentation of UC (gl r,C if and only if V is of level r as a U (n)C -module. Proof. If V can be regarded as an S(n, r)C -module via ξr, C , then we may view V as an S(n, r )C -module by restriction and, hence, V is of level r as a U(n)C -module. Conversely, suppose that V is of level r as a U (n)C -module and V = L(Q) for some Q ∈ Q(n). Then V is an S(n, r )C -module by Theorem 4.6.8, where r = 1i n degQ i (u). Hence, V is of level r as a U (n)C -module. So r = r and V is an S(n, r )C -module. Remark 4.6.10. It is reasonable to make the following definition. A finite n )-module is said to be of level r if it is an S(n, r )C dimensional UC (gl n )-module of level r , then its commodule via ξr,C . Thus, if V is a UC (gl position factors are all homomorphic images of ⊗r C . It would be interesting to know if the converse is also true. It is natural to make a comparison between the Classification Theorems 4.3.4 and 4.6.8 and to raise the following problem. Problem 4.6.11. Generalize the Identification Theorem 4.4.2 to the case where n r .

150

4. Representations of affine quantum Schur algebras

4.7. Classification of simple U(n, r )C -modules The homomorphic image U(n, r )C of the extended affine quantum sln , U(n)C , is a proper subalgebra of S(n, r )C when n r . In other words, by n ) → S(n, r )C restriction, the surjective algebra homomorphism ξr, C : UC (gl induces a surjective algebra homomorphism ξr,C : U(n)C → U(n, r)C . In this section, we will classify finite dimensional simple U(n, r )C -modules. Let P ∈ P(n) and λ ∈ + (n, r ) be such that λi − λi+1 = degPi (u), for 1 i n − 1. Define ¯ M(P, λ) = U(n)C / I¯(P, λ), where I¯(P, λ) is the left ideal of U(n)C generated by x+ i,s , Pi,s − Pi,s , and k j − z λ j for 1 i n −1, s ∈ Z, and 1 j n, where Pi,s are defined using ¯ (4.2.1.7). The U(n)C -module M(P, λ) has a unique simple quotient U(n)C ¯ module, which is denoted by L(P, λ). Lemma 4.7.1. For Q ∈ Q(n), let λ = (λ1 , . . . , λn ) with λi = deg(Q i (u)) for 1 i n and P = (P1 (u), . . . , Pn−1 (u)) ∈ P(n) be such that P j (u) =

Q j (uz j −1 ) , Q j +1 (uz j+1 )

¯ for 1 j n − 1. Then L(P, λ) ∼ = L(Q)|U(n)C . n ) Proof. Let w0 ∈ L(Q) be a pseudo-highest weight vector. Since UC (gl ± is generated by U(n)C and the central elements zt , for t 1, every simn )-module is a simple U(n)C -module by restriction. In particular, ple UC (gl L(Q)|U(n)C is simple. So we have L(Q) = U(n)C w0 . Hence, there is a ¯ surjective U(n)C -module homomorphism ϕ : M(P, λ) → L(Q) defined ¯ by sending u¯ to uw0 , for u ∈ U(n)C . Thus, L(Q) ∼ λ)/Ker ϕ = M(P, ¯ as U(n)C -modules. Since L(Q)|U(n)C is simple and M(P, λ) has a unique ¯ ¯ simple quotient L(P, λ), we have L(P, λ) ∼ = L(Q) as U(n)C -modules. Corollary 4.7.2. Let P ∈ P(n) and λ ∈ + (n, r ) with λi − λi +1 = degPi (u) ¯ for 1 i n − 1. Then L(P, λ) is a U(n, r )C -module via ξr, C and ∼ ¯ ¯ L(P, λ)|UC (sl n ) = L(P). Proof. Let Q n (u) = 1 + u λn . Using the formula P j (u) =

Q j (uz j −1 ) , Q j +1 (uz j+1 )

we define the polynomials Q i (u), for 1 i n − 1. Then we have Q = (Q 1 (u), . . . , Q n (u)) ∈ Q(n) and λi = degQ i (u), for 1 i n. By

4.7. Classification of simple U(n, r )C -modules

151

¯ Theorem 4.6.8, L(Q) is an S(n, r )C -module. So, by Lemma 4.7.1, L(P, λ) ∼ = ¯ L(Q)|U(n)C is a U(n, r )C -module. Hence, L(P, λ)|UC (sl is simple since the n ) algebra homomorphism ξr, C : UC ( sln ) → U(n, r )C is surjective. Since U(n, r )C contains S(n, r )C as a subalgebra, it follows that, if λ ∈ ¯ + (n, r ), then L(P, λ) is an S(n, r )C -module and ¯ ¯ L(P, λ) = L(P, λ)μ , (4.7.2.1) μλ μ∈(n,r )

¯ ¯ where L(P, λ)μ denotes the weight space of L(P, λ) as a U (n)C -module. ∼ L(P ¯ ¯ ∗ , λ∗ ), Lemma 4.7.3. For partitions λ, λ∗ and P, P∗ ∈ P(n), if L(P, λ) = then P = P∗ and λ = λ∗ . In particular, for dominant polynomials Q, Q∗ ∈ Q(n), L(Q)|U(n)C ∼ = L(Q∗ )|U(n)C if and only if deg Q i (u) = deg Q ∗i (u) and j−1 Q j (uz )/Q j +1 (uz j+1 ) = Q ∗j (uz j−1 )/Q ∗j+1 (uz j +1 ), for all 1 i n and 1 j n − 1. ∼ L(P ¯ ¯ ∗ , λ∗ ), it follows Proof. By (4.7.2.1) we have λ = λ∗ . Since L(P, λ) = ∼ ∼ ¯ ∗ , λ∗ )| ∼ ¯ ¯ from Corollary 4.7.2 that L(P) λ)|UC (sl = L(P, n ) = L(P UC (sln ) = ¯L(P∗ ). Therefore, P = P∗ . Lemma 4.7.4. Let V be a finite dimensional simple U(n, r )C -module. Then there exist P = (P1 (u), . . . , Pn−1 (u)) ∈ P(n) and λ ∈ + (n, r ) with λi − ¯ λi+1 = degPi (u), for all 1 i n − 1, such that V ∼ λ). = L(P, Proof. Since ξr, C : UC ( sln ) → U(n, r )C is surjective, V is a simple UC ( sln )module. Let w0 be a pseudo-highest weight vector satisfying μi x+ i,s w0 = 0, Pi,s w0 = Pi,s w0 , and ki w0 = z w0 ,

for all 1 i n − 1 and s ∈ Z, where μi = degPi (u). Using the idempotent decomposition 1 = ν∈(n,r) lν , ν∈(n,r) lν w0 = w0 = 0 implies that there exists λ ∈ (n, r ) such that lλ w0 = 0. It is clear that lλ w0 is also a pseudo-highest weight vector satisfying + xi,s lλ w0 = 0, Pi,s lλ w0 = Pi,s lλ w0 , and ki lλ w0 = z μi lλ w0 .

On the other hand, ki lλ w0 = z λi lλ w0 , for 1 i n. Thus, k i lλ w 0 = λ −λ i i+1 z lλ w0 . So λi − λi+1 = μi , for 1 i n − 1. Hence, there is a ¯ surjective U(n)C -module homomorphism ϕ : M(P, λ) → V defined by sending u¯ to ulλ w0 , for all u ∈ U(n)C . This surjection induces a U(n)C -module ¯ isomorphism V ∼ λ). = L(P, Altogether this gives the following classification theorem.

152

4. Representations of affine quantum Schur algebras

Theorem 4.7.5. The set ¯ { L(P, λ) | P ∈ P(n), λ ∈ + (n, r ), λi − λi+1 = degPi (u), ∀1 i < n} is a complete set of non-isomorphic finite dimensional simple U(n, r )C modules. Define an equivalence relation ∼ on Q(n) be setting, for Q, Q ∈ Q(n), Q ∼ Q ⇐⇒ degQ i (u) = degQ i (u), 1 i n, and Q j (uz j −1 ) Q j (uz j −1 ) = , 1 j n − 1. Q j +1 (uz j+1 ) Q j +1 (uz j+1 ) Corollary 4.7.6. Let r = Q(n)r / ∼ denote the set of equivalence classes and choose a representative Qπ ∈ π for every π ∈ r . Then the set {L(Qπ )|U(n)C : π ∈ r } is a complete set of non-isomorphic finite dimensional simple U(n, r )C modules. Moreover, if n > r , then r = Q(n)r . Proof. The last assertion follows from the fact that, if n > r , then U(n, r )C = S(n, r )C . As seen in Theorem 4.2.3, there is a rougher equivalence relation ∼ on Q(n) defined by setting for Q, Q ∈ Q(n), Q j (uz j−1 ) Q j (uz j −1 ) Q ∼ Q ⇐⇒ = , for 1 j n − 1, Q j +1 (uz j+1 ) Q j +1 (uz j +1 )

such that the equivalence classes are in one-to-one correspondence to simple UC ( sln )-modules.

5 The presentation and realization problems

As seen in Chapters 2 and 3, the double Ringel–Hall algebra D(n) is presented by generators and relations, while the affine quantum Schur algebra S(n, r ) is defined as an endomorphism algebra which is a vector space with an explicitly defined multiplication. Now the algebra epimorphism from D(n) to S(n, r) raises two natural questions: how to present affine quantum Schur algebras S(n, r ) in terms of generators and relations, and how to realize the double Ringel–Hall algebra D(n) in terms of a vector space together with an explicitly defined multiplication. In this chapter and the next, we will tackle these problems. Since D(n) ∼ = U(n) ⊗ Z(n) by Remark 2.3.6(2), it follows that S(n, r ) = U(n, r)Z(n, r ), where U(n, r ) (resp., Z(n, r )) is the homomorphic image of the quantum group U(n), the extended quantum affine sln , (resp., the central subalgebra Z(n)) under the map ξr . We first review in §5.1 a presentation of McGerty for U(n, r ). This is a proper subalgebra of S(n, r ) if n r . As a natural affine analogue of the presentation given by Doty–Giaquinto [18], we will modify McGerty’s presentation to obtain a Drinfeld–Jimbo type presentation for U(n, r ) (Theorem 5.1.3). We then determine the structure of the central subalgebra Z(n, r ) of S(n, r ) (Proposition 5.2.3). However, it is almost impossible to combine the two to give a presentation for S(n, r ). In §5.3, we will use the multiplication formulas given in §3.4 to derive some extra relations for an extra generator required for presenting S(r, r) for all r 1 (Theorem 5.3.5). In particular, we will then easily see why the Hopf algebra U considered in [35, 3.1.1]1 maps onto affine quantum Schur algebras when n r ; see Remark 5.3.2. Strictly speaking, U cannot be regarded as a quantum enveloping algebra since it does not have a triangular decomposition. 1 This algebra n ) in [35]. U is denoted by U (gl

153

154

5. The presentation and realization problems

From §5.4 onwards, we will discuss the realization problem. We first formulate a realization conjecture in §5.4, as suggested in [24, 5.2(2)], and its classical (v = 1) version. In the last section, we show that Lusztig’s transfer maps are not compatible with the map ξr for the double Ringel–Hall algebra D(n) considered in Remark 2.3.6(3). This justifies why we cannot have a realization in terms of an inverse limit of the transfer maps. We will then establish the conjecture for the classical case in the next chapter.

5.1. McGerty’s presentation for U(n, r ) The presentation problem for S(n, r ) when n > r is relatively easy. In this case, S(n, r) = U(n, r ) is a homomorphic image of U(n). By using McGerty’s presentation for U(n, r ), we obtain a new presentation for U(n, r ) similar to that for quantum Schur algebras given in [18] (cf. [26]). In particular, this gives Doty–Green’s result [19] for S(n, r ) with n > r (removing the condition n 3 required there). Let ξr : D(n) → S(n, r ) be the surjective homomorphism defined in (3.5.5.4). For each r 1, define U(n, r ) := ξr (U(n)). Clearly, U(n, r ) is generated by the elements ei := ξr (E i ) = E i,i+1 (0, r ), fi := ξr (Fi ) = E i+1,i (0, r ), and

ki := ξr (K i ) = 0(ei , r ),

for all i ∈ I . n−1 as in Let C = (ci, j ) denote the generalized Cartan matrix of type A (1.3.2.1). Recall the elements lλ , λ ∈ (n, r ), defined in Proposition 3.7.4(2). The following result is taken from [58, Prop. 6.4 & Lem. 6.6]. Theorem 5.1.1. As a Q(v)-algebra, U(n, r ) is generated by ei , fi , lλ

(i ∈ I, λ ∈ (n, r ))

subject to the following relations (1) lλ lμ = δλ,μ lλ , λ∈(n,r) lλ = 1; lλ+ei −e ei , if λ + ei − e i+1 ∈ (n, r ); i+1 (2) ei lλ = 0, otherwise; lλ−ei +e fi , if λ − ei + ei+1 ∈ (n, r ); i +1 (3) fi lλ = 0, otherwise;

5.1. McGerty’s presentation for U(n, r )

155

(4) ei f j − f j ei = δi, j λ∈(n,r) [λi − λi+1 ]lλ ; a 1 − ci, j (5) (−1) eai e j ebi = 0 for i = j; a a+b=1−ci, j a 1 − ci, j (6) (−1) fia f j fbi = 0 for i = j. a a+b=1−ci, j

This theorem is the affine version of Theorem 3.4 in [18]. Naturally, one expects the affinization of Theorem 3.1 in [18] for a Drinfeld–Jimbo type presentation. In fact, this is an easy consequence of the following result for the Laurent polynomial algebra U0 := Q(v)[K 1±1 , . . . , K n±1 ] = U(n)0 , whose proof can be found in [18, Prop. 8.2, 8.3] and [26, 4.5, 4.6] in the context of quantum gln and quantum Schur algebras (though the result itself was not explicitly stated there). For any μ ∈ Nn and 1 i n, let n

Ki ; 0 Lμ = , v μ = (v μ1 , . . . , v μn ), and μi (5.1.1.1) i =1 [K i ;r + 1]! = (K i − 1)(K i − v) · · · (K i − vr ). If we regard Lμ as a function from U0 to Q(v), then Lλ (v μ ) = δλ,μ , for all λ, μ ∈ (n, r ). Lemma 5.1.2. The ideals Ir and Jr of U0 generated by the sets Ir = {1 − +λ∈(n,r ) Lλ } ∪ {Lλ Lμ − δλ,μ Lλ | λ, μ ∈ (n, r )} ∪ {K i Lλ − v λi Lλ | 1 i n, λ ∈ (n, r )} and Jr = {κ := K 1 · · · K n − vr , [K i ; r + 1]! | 1 i n}, are the same. Proof. For completeness, we provide here a direct proof. Consider the algebra epimorphism ϕ : U0 −→ U0 /K 1 − v μ1 , . . . , K n − v μn , f −→ ( f (v μ ))μ∈(n,r) . μ∈(n,r)

It is clear from the relations Lλ (v μ ) = δλ,μ that Ir = Ker ϕ. Applying the Chinese Remainder Theorem yields2 2 The ideal J appearing in the proof of [12, Lem. 13.36] should be

/

x 1 − v μ1 , . . . , x n − v μn ,

0μ1 ,...,μn r

while the quotient algebra R/J should have dimension (r + 1)n .

156

5. The presentation and realization problems Ir =

/

K 1 − v μ1 , . . . , K n − v μn

μ∈(n,r)

/

=

(κ + K 1 − v μ1 , . . . , K n − v μn )

0μ1 ,...,μn r

= κ +

/

K 1 − v μ1 , . . . , K n − v μn

0μ1 ,...,μn r

= κ + [K 1 ; r + 1]! , . . . , [K n ; r + 1]! = Jr , since κ ∈ K 1 − v μ1 , . . . , K n − v μn ⇐⇒ μ1 + · · · + μn = r . The above lemma together with Theorem 5.1.1 gives the following result, which, as mentioned above, was proved in [19] under the assumption that n 3 and n > r . Theorem 5.1.3. The algebra U(n, r ) is generated by the elements ei , fi , ki (i ∈ I = Z/nZ) subject to the relations: (QS1) ki k j = k j ki ; (QS2) ki e j = v δi, j −δi, j +1 e j ki , ki f j = v −δi, j +δi, j+1 f j ki ; −1 k − k

−1 i i (QS3) ei f j − f j ei = δi, j v−v −1 , wher e ki = ki ki+1 ; 1 − ci, j a b (QS4) (−1)a ei e j ei = 0, for i = j; a a+b=1−ci, j 1 − ci, j a b (QS5) (−1)a fi f j fi = 0, for i = j; a a+b=1−ci, j

(QS6) [ki ; r + 1]! = 0, k1 · · · kn = vr . Proof. First, by the lemma, the ideal Ir of U(n) generated by Ir is the same as the ideal Jr generated by Jr . Second, if ξr∗ : U(n) → S(n, r ) denotes the restriction of ξr to U(n), then it is clear that Ir ⊆ Ker ξr∗ (see the proof of Proposition 3.7.4). Thus, we obtain an epimorphism ϕ : U(n)/Ir −→ U(n)/Ker ξr∗ ∼ = U(n, r ) satisfying ϕ(E i + Ir ) = ei , ϕ(Fi + Ir ) = fi , and ϕ(Lλ + Ir ) = lλ for i ∈ I and λ ∈ (n, r ). On the other hand, it is straightforward to check that all relations given in Theorem 5.1.1 hold in U(n)/Ir (see, e.g., the proof of [12, Lem 13.40]). Thus, applying Theorem 5.1.1 yields a natural algebra homomorphism

5.2. Structure of affine quantum Schur algebras

157

ψ : U(n, r ) −→ U(n)/Ir satisfying ei → E i + Ir , fi → Fi + Ir , and lλ → Lλ + Ir . Therefore, ϕ has to be an isomorphism, forcing Ker ξr∗ = Ir = Jr . Remarks 5.1.4. (1) It is possible to replace the relations in (QS6) by the relations [k1 ; μ1 ]! [k2 ; μ2 ]! · · · [kn ; μn ]! = 0, for all μ ∈ Nn with σ (μ) = r + 1, −1 and replace kn used in (QS1)–(QS5) by kn := vr k−1 1 · · · kn−1 . The new presentation uses only 3n − 1 generators. For more details, see [26] or [12, §13.10]. It is interesting to point out that the homomorphic image U(∞, r) of U(gl∞ ), which is a proper subalgebra of the infinite quantum Schur algebra S(∞, r ), for all r 1, has only a presentation of this type; see [24, 4.7,5.4]. (2) If Ir denotes the ideal of D(n) generated by Ir , then

D(n)/ Ir ∼ = U(n, r ) ⊗ Z(n). Thus, adding (QS6) to relations (QGL1)–(QGL8) in Theorem 2.3.1 gives a presentation for this algebra. The relations (QS1)–(QS6) in Theorem 5.1.3 will form part of the relations in a presentation for S(r, r ), r 1; see §5.3.

5.2. Structure of affine quantum Schur algebras When n r , U(n, r ) is a proper subalgebra of S(n, r ). The next two sections are devoted to the study of the structure of affine quantum Schur algebras S(n, r ) in this case. We will first see in this section a general structure of S(n, r ) inherited from D(n) and give an explicit presentation for S(r, r ) in §5.3. We first endow S(n, r ) with a Z-grading through the surjective algebra homomorphism ξr : D(n) −→ S(n, r) = EndH(r ) (⊗r ). − If we assign to each u + A (resp., u A , K i ) the degree d(A) = dim M(A) (resp., −d(A), 0), then D(n) admits a Z-grading D(n) = ⊕m∈Z D(n)m . By definition, we have, for each x ∈ D(n)m and ωi = ωi1 ⊗ · · · ⊗ ωir ∈ ⊗r ,

x · ωi =

l p=1

a p ωj( p) ,

158

5. The presentation and realization problems

where a p ∈ Q(v) and j( p) = ( j1, p , . . . , jr, p ) ∈ Zr satisfy rs=1 js, p = r s=1 i s − m, for all 1 p l. Thus, letting S(n, r )m = ξr (D(n)m ) gives a decomposition S(n, r ) = S(n, r )m m∈Z

of S(n, r ), i.e., S(n, r ) is Z-graded, too. − By Remark 2.3.6(2), there is a central subalgebra Z(n) = Q(v)[z+ m , zm ]m 1 of D(n) such that D(n) = U(n)⊗Q(v) Z(n). This gives another subalgebra of S(n, r ) Z(n, r ) := ξr (Z(n)) such that S(n, r ) = U(n, r )Z(n, r). In other words, ξr induces a surjective algebra homomorphism U(n, r ) ⊗ Z(n) −→ S(n, r), x ⊗ y −→ xξr (y). Clearly, Z(n, r ) is contained in the center of S(n, r ). By [56, Th. 7.10 & 8.4] (see also Corollary 3.5.9), we have the following result. Lemma 5.2.1. The equality U(n, r ) = S(n, r ) holds if and only if n > r . In other words, Z(n, r ) ⊆ U(n, r ) if and only if n > r . − Moreover, for each m 1, ξr (z+ m ) ∈ S(n, r )mn and ξr (zm ) ∈ S(n, r )−mn , and the Z-grading of S(n, r ) induces a Z-grading U(n, r ) = U(n, r )m m∈Z

of U(n, r), where U(n, r )m = U(n, r) ∩ S(n, r )m . We are now going to determine the structure of both Z(n, r) and U(n, r ). For all 1 s r , define commuting Q(v)-linear maps φs : ⊗r −→ ⊗r , ωi −→ ωi−nes = ωi1 ⊗ · · · ⊗ ωis −n ⊗ · · · ⊗ ωir and set, for each m 1, pm =

r

φsm

and qm =

s=1

By (3.5.4.1), pm = Moreover,

ξr (z+ m)

and qm =

r

φs−m .

s=1

ξr (z− m ).

Thus, they both lie in S(n, r ).

Z(n, r ) = Q(v)[pm , qm ]m 1 .

5.2. Structure of affine quantum Schur algebras

159

Remark 5.2.2. Recall from (3.5.5.4) the algebra homomorphism ξr∨ : H(r ) −→ EndQ(v) (⊗r )op . It is easy to see from the definition that, for each m 1, pm = ξr∨ (X 1m + · · · + X rm ) and qm = ξr∨ (X 1−m + · · · + X r−m ). It is well known that, for each m ∈ Z, the element X 1m + · · · + X rm is central in H(r ). Now let σ1 , . . . , σr (resp., τ1 , . . . , τr ) denote the elementary symmetric polynomials in φ1 , . . . , φr (resp., φ1−1 , . . . , φr−1 ), i.e., for 1 s r, σs = φt1 · · · φts (resp., τs = φt−1 · · · φt−1 ). s 1 1t1 <···
1t1 <···
Then σs , τs ∈ Z(n, r ) and Z(n, r ) = Q(v)[σ1 , . . . , σr , τ1 , . . . , τr ]. Since τr = σr−1 and τs = σr−s τr for each 1 s < r, this implies that Z(n, r ) = Q(v)[σ1 , . . . , σr , σr−1 ]. Proposition 5.2.3. The set λ

r −1 λr X := {σ1λ1 · · · σr−1 σr | λ1 , . . . , λr−1 ∈ N, λr ∈ Z}

forms a Q(v)-basis of Z(n, r ). In other words, Z(n, r ) is a (Laurent) polynomial algebra in σ1 , . . . , σr , σr−1 . Proof. It is obvious that Z(n, r ) is spanned by X . It remains to show that X is linearly independent. For i = (i 1 , . . . , ir ), j = ( j1 , . . . , jr ) ∈ Zr , we define the lexicographic order i lex j if i = j or there exists 1 < s r such that ir = jr , . . . , i s = js , i s−1 < js−1 . Clearly, this gives a linear ordering on Zr . For each λ = (λ1 , . . . , λr−1 , λr ) ∈ Nr−1 × Z, define λ

r−1 λr σ λ = σ1λ1 · · · σr−1 σr .

160

5. The presentation and realization problems

Then, by definition, for i = (i 1 , . . . , ir ) ∈ Zr , we obtain that3 σ λ (ωi ) = ωi∗λ + bj ωj , i∗λ
where all but finitely many bj ∈ N are zero and i ∗ λ = i 1 − λr n, i 2 − (λr−1 + λr )n, . . . , ir − (λ1 + · · · + λr )n . Note that, for λ, μ ∈ Nr−1 × Z, i ∗ λ = i ∗ μ ⇐⇒ λ = μ. Now suppose m

(t)

at σ λ

= 0,

t=1

where at ∈ Q(v) and λ(t) ∈ Nr−1 × Z, for 1 t m. Fix an i ∈ Zr . Without loss of generality, we may suppose i ∗ λ(1)
x

where is a linear combination of ωj with i ∗ λ(1)
j

n−1 {kj = k11 · · · kn−1 | j = ( j1 , . . . , jn−1 ) ∈ (Nn−1 )r },

where (Nn−1 )r = {j = ( j1 , . . . , jn−1 ) ∈ Nn−1 | j1 + · · · + jn−1 r }. Let, further, Z(n, r ) be the subalgebra of S(n, r ) generated by Z(n, r ) and S(n, r )0 . 3 The linear maps σ and σ λ should not be confused with the sum function σ defined on i n in §1.1. M,n (Z) and Z

5.2. Structure of affine quantum Schur algebras

161

Proposition 5.2.5. The multiplication map Z(n, r ) ⊗Q(v) S(n, r )0 −→ Z(n, r ) is a Q(v)-algebra isomorphism. Proof. It remains to show that the set {σ λ kj | λ ∈ Nn−1 × Z, j ∈ (Nn−1 )r } is linearly independent. By definition, for each i ∈ Zr , kj (ωi ) = v f (i,j) ωi , where f (i, j) ∈ Z is determined by i and j. Thus, by the proof of Proposition 5.2.3, we infer that, for λ ∈ Nn−1 × Z, j ∈ (Nn−1 )r , and i ∈ Zr , σ λ kj (ωi ) = v f (i,j) ωi∗λ + cj ωj , i∗λ
where all but finitely many cj ∈ Q(v) are zero. Since i ∗ λ = i ∗ μ if and only if λ = μ, it suffices to show that, for each fixed λ, {σ λ kj | j ∈ (Nn−1 )r } is a linearly independent set. This follows from the fact that {kj | j ∈ (Nn−1 )r } is a linearly independent set. By the discussion above, the center of the positive part S(n, r )+ of S(n, r ) contains the polynomial algebra Q(v)[σ1 , . . . , σr ] := Z(n, r )+ , and the center of the negative part S(n, r )− contains the polynomial algebra Q(v)[τ1 , . . . , τr ] := Z(n, r )− . Moreover, S(n, r )+ = U(n, r )+ · Z(n, r )+ and S(n, r )− = U(n, r )− · Z(n, r)− . For each m 0, set ± S(n, r )± m = S(n, r ) ∩ S(n, r )±m

U(n, r )± m

and

±

= U(n, r ) ∩ U(n, r )±m .

Then we obtain an N-grading on S(n, r )± and U(n, r)± : ± S(n, r )± = S(n, r )± U(n, r)± m and U(n, r ) = m. m 0

m 0

− Note that, for m 1, pm ∈ S(n, r )+ mn and qm ∈ S(n, r )mn and, for 1 s + − r , σs ∈ S(n, r )sn and τs ∈ S(n, r)sn .

Proposition 5.2.6. For each m 0, ± dim S(n, r )± m /U(n, r )m

= |{A ∈ + (n) | A is periodic with σ (A) r, d(A) = m}|.

162

5. The presentation and realization problems

Proof. We only prove the assertion for the “+” case; the “−” case is similar. By [13, §8], the composition subalgebra C(n)+ of D(n)+ has a basis4 where E + A

+ {E + A | A ∈ (n) is aperiodic}, A u + with all B periodic and η A ∈ Q(v). = u+ B
Furthermore, by [11, §6], for A, B ∈ + (n),

B dg A =⇒ σ (B) σ (A). Thus, if A ∈ + (n) is aperiodic with σ (A) > r , then by (3.4.0.2), ξr (E + ) = A(0, r ) + η BA B(0, r ) = 0. A B
Hence, for each m 0, the set + Xm := {ξr (E + A ) | A ∈ (n) is aperiodic with σ (A) r, d(A) = m}

is a spanning set for U(n, r )+ m . By Proposition 3.7.4(1), the set { A(0, r ) | A ∈ + (n), σ ( A) r, d(A) = m} is a basis of S(n, r)+ m . Hence, Xm is a linearly independent set and, thus, is a basis of U(n, r)+ . This gives the desired assertion. m Corollary 5.2.7. For each r 2, S(r, r )+ = U(r, r )+ ⊕ Q(v)σ1t and S(r, r )− = U(r, r )− ⊕ Q(v)τ1t . t 1

t 1

Proof. By Proposition 5.2.6, + dim S(r, r )+ m /U(r, r )m

=

1, 0,

if m = 0 and m ≡ 0 mod r , otherwise.

+ Since σ1 = p1 = ξr (z+ 1 ), it follows from Theorem 3.8.1(2) that S(r, r ) is generated by ei (1 i r ) and σ1 . This implies the first decomposition. The second one can be proved similarly.

5.3. Presentation of S(r, r) In this section we give a presentation for S(r, r ) by describing explicitly a complement of U(r, r ) in S(r, r ). 4 The basis was used to give an elementary construction for the canonical basis of C (n)±

in [13].

5.3. Presentation of S(r, r )

163

We first consider the general case and recall the surjective algebra homomorphism ξr : D(n) → S(n, r ). For each a = (ai ) ∈ Nn , we write in D(n), ± u± u± a = u [Sa ] and a =v

i

ai (ai −1) ± ua

(cf. §1.4).

By (1.4.2.1), if a is not sincere, say ai = 0, then

=

a j (a j −1) ± ua ± (ai −1 ) (a1 ) ± (an ) (u i −1 ) · · · (u ± (u n ) · · · (u i±+1 )(ai +1 ) 1)

u a± = v

j

∈ U(n).

Thus, if a is not sincere, then both ea := ξr ( u+ u− a ) and fa := ξr ( a ) lie in U(n, r ). Now consider the following element ρ in EndQ(v) (⊗r ): ρ : ⊗r −→ ⊗r , ωi −→ ωi−e1 −···−er , i.e., ρ(ωi1 ⊗· · ·⊗ωir ) = ωi1 −1 ⊗· · ·⊗ωir −1 . It is clear that ρ r = σr ∈ S(n, r ). Lemma 5.3.1. Suppose n r . Then ρ= ea and ρ −1 = a∈Nn , σ (a)=r

fa .

a∈Nn , σ (a)=r

In particular, if n > r , then ρ, ρ −1 ∈ U(n, r ). Proof. For each a ∈ Nn , let Ya denote the set of the sequences j = ( j1 , . . . , jr ) satisfying the condition that, for each 1 i n, ai = |{1 s r | js = i}|. By Corollary 3.5.8, for each ωi1 ⊗ · · · ⊗ ωir ∈ ⊗r , u+ δ j1 ,i1 −1 · · · δ jr ,ir −1 ωi1 −1 ⊗ · · · ⊗ ωir −1 , a · (ωi 1 ⊗ · · · ⊗ ωir ) = j∈Ya

where, for m ∈ Z, m denotes its residue class in Z/nZ. Therefore, u+ a · (ωi 1 ⊗ · · · ⊗ ωir ) = ωi 1 −1 ⊗ · · · ⊗ ωir −1 , a∈Nn , σ (a)=r

that is, the first equality holds. The second one can be proved similarly. The above lemma implies that eδ := ρ − eδ and fδ := ρ −1 − fδ

(5.3.1.1)

164

5. The presentation and realization problems

are in U(r, r ), where δ = (1, . . . , 1). More precisely, eδ =

fδ =

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 ) and

(fi−1 )(ai −1 ) · · · (f1 )(a1 ) (fn )(an ) · · · (fi+1 )(ai +1 ) . (5.3.1.2)

From now on, we assume n = r. By §1.4 and Theorem 3.8.1(2), S(r, r ) is generated by ei , fi , ki (1 i n = r ), and eδ , fδ . It follows from (5.3.1.1) and (5.3.1.2) that S(r, r ) is also generated by the ei , fi , ki , and ρ, ρ −1 . Since S(r, r )r+ = U(r, r)r+ ⊕ Q(v)ρ = U(r, r)r+ ⊕ Q(v)σ1 and S(r, r )r− = U(r, r)r− ⊕ Q(v)ρ −1 = U(r, r )r− ⊕ Q(v)τ1 , it follows from Corollary 5.2.7 that S(r, r )+ = U(r, r )+ ⊕

Q(v)ρ t and

t 1

−

−

S(r, r ) = U(r, r ) ⊕

Q(v)ρ −t .

(5.3.1.3)

t 1

Remark 5.3.2. As a consequence of the above discussion, we obtain [35, Th. 3.4.8] which states that (for n r with n 3) there is a surjective algebra homomorphism αr : U → S(n, r ) taking R to ρ −1 , where U is a Hopf algebra obtained from U(n) by adding primitive elements R, R −1 with R R −1 = R −1 R = 1. It seems to us that U is not a quantum group in a strict sense since it does not admit a triangular decomposition. Proposition 5.3.3. For each 1 i r , we have in S(r, r ), (1) ki eδ = veδ ; (2) ei eδ = v+v1 −1 e2i ei−1 · · · e1 er · · · ei+1 ;

−1 (3) fi eδ = 1−v1 −2 ei −1 · · · e1 er · · · ei+1 k−1 i (ki+1 − ki+1 ); (4) ki fδ = vfδ ; (5) fi fδ = v+v1 −1 f2i fi +1 · · · fr f1 · · · fi−1 ;

(6) ei fδ =

1 f · · · fr f1 · · · fi −1 ki−1 +1 (ki 1−v −2 i+1

− k−1 i ).

Proof. All these relations can be deduced from the multiplication formulas in Theorem 3.4.2. However, we provide here a direct proof for (1) and (2). The relations (4) and (5) can be proved in a similar manner.

5.3. Presentation of S(r, r )

165

By definition, we have in D(r ), + + 2 u i+ u + δ = u [M] + (v + 1)u δ+ei ,

where M = S1 ⊕ · · · ⊕ Si−1 ⊕ Si [2] ⊕ Si+2 ⊕ · · · ⊕ Sr . On the other hand, + + + + + 2 (u i+ )2 u + i−1 · · · u 1 u r · · · u i+1 = v(v + 1)u [2Si ] u [S1 ⊕···Si−1 ⊕Si +1 ⊕···⊕Sr ] + = (v + v −1 )(u + [M] + u δ+ei ).

Since Sδ+ei is semisimple of dimension r + 1, it follows that ξr (u + δ+ei ) = 0. Hence, + + ei eδ = ξr (u + i u δ ) = ξr (u [M] ) =

=

1 + + + ξr ((u i+ )2 u + i−1 · · · u 1 u r · · · u i+1 ) v + v −1

1 e2 ei−1 · · · e1 er · · · ei +1 , v + v −1 i

which gives the relation (2). The relation (1) follows from the fact that for each ωi1 ⊗ · · · ⊗ ωir ∈ ⊗r , u δ · (ωi1 ⊗ · · · ⊗ ωir ) = δ j1 ,i1 −1 · · · δ jr ,ir −1 ωi1 −1 ⊗ · · · ⊗ ωir −1 =

ωi 1 −1 ⊗ · · · ⊗ ωir −1 , 0,

j∈Yδ

if i 1 , . . . , i r are pairwise distinct; otherwise.

Note that, if i 1 , . . . , i r are pairwise distinct, then K i · (ωi1 ⊗ · · · ⊗ ωir ) = vωi1 ⊗ · · · ⊗ ωir . The above proposition together with ρ = eδ + eδ and ρ −1 = fδ + fδ gives the following result. Corollary 5.3.4. For each 1 i r , we have in S(r, r ), (QS1 ) (ki − v)ρ = (ki − v)eδ ; (QS2 ) ei ρ = ei eδ + v+v1 −1 ei2 ei −1 · · · e1 er · · · ei +1 ;

−1 (QS3 ) fi ρ = fi eδ + 1−v1 −2 ei−1 · · · e1 er · · · ei+1 k−1 i (ki+1 − ki+1 ); (QS4 ) (ki − v)ρ −1 = (ki − v)fδ ; (QS5 ) fi ρ −1 = fi fδ + v+v1 −1 f2i fi +1 · · · fr f1 · · · fi−1 ;

(QS6 ) ei ρ −1 = ei fδ +

1 f · · · fr f1 · · · fi−1 k−1 i+1 (ki 1−v −2 i+1

where eδ and fδ are given in (5.3.1.2).

− k−1 i ),

166

5. The presentation and realization problems

Theorem 5.3.5. The Q(v)-algebra S(r, r ) is generated by ei , fi , ki±1 (1 i r ) and ρ ±1 subject to the relations (QS0 )

ρρ −1 = ρ −1 ρ = 1, ρei ρ −1 = ei −1 , ρfi ρ −1 = fi−1 , ρki ρ −1 = ki −1

together with the relations (QS1 )–(QS6 ) and the relations (QS1)–(QS6) in Theorem 5.1.3. In particular, S(r, r ) = U(r, r ) ⊕ Q(v)ρ m . 0 =m∈Z

Proof. Let S be the Q(v)-algebra generated by xi , yi , zi±1 (1 i r ) and η±1 with the relations (QS1)–(QS6) and (QS0 )–(QS6 ) (here we replace ±1 ei , fi , ki±1 , and ρ ±1 by xi , yi , z±1 i , and η , respectively). Thus, there is a surjective Q(v)-algebra homomorphism ϒ : S xi

−→ S(r, r ), −→ ei , yi −→ fi , zi± −→ ki± , η±1 −→ ρ ±1 .

Let U be the Q(v)-subalgebra of S generated by xi , yi , and zi±1 (1 i r ). Then ϒ induces a surjective homomorphism ϒ1 : U → U(r, r ). Since the relations (QS1)–(QS6) are the defining relations for U(r, r ), there is also a natural surjective homomorphism : U(r, r) → U . Clearly, the compositions ϒ1 and ϒ1 are identity maps. Thus, both ϒ1 and are isomorphisms. It is clear that for each 0 = t ∈ Z, ρ t lies in S(r, r )r t . We claim that ρ t does not lie in U(r, r )r t . Otherwise, applying relations (QS0 )–(QS6 ) would give that ρ = ρ t ρ −t+1 ∈ U(r, r )r t ρ −t+1 ⊆ U(r, r )r . This implies S(r, r ) = U(r, r ), which contradicts Lemma 5.2.1. Hence, ρ t ∈ U(r, r )r t for all 0 = t ∈ Z. For each m ∈ Z, choose a Q(v)-basis Bm of U(r, r )m . Then the set B := ∪m∈Z Bm forms a basis of U(r, r ). Moreover, by the discussion above, the set B ∪ {ρ t | 0 = t ∈ Z} is linearly independent in S(r, r ). On the other hand, by the definition of S , the set {(c) | c ∈ B} ∪ {ηm | 0 = m ∈ Z} is a spanning set of S . Since ϒ((c)) = c and ϒ(ηm ) = ρ m , it follows that the above spanning set is a basis of S . Therefore, ϒ is an isomorphism. This finishes the proof. Remark 5.3.6. We can also present S(r, r ) by using generators ei , fi , ki±1 (1 i r ), eδ and fδ , but the relation between eδ and fδ is not clear. Furthermore, if we use generators σ1 and τ1 instead of ρ and ρ −1 , the relations

5.3. Presentation of S(r, r )

167

would be much more complicated; see the example below. From the relations obtained in the case n = r , it seems very hard to get the relations for the general case n < r. Example 5.3.7. We consider the special case n = r = 2. In this case, U(2, 2) = ξ2 (U(2)) is generated by ei , fi , ki for i = 1, 2, and Z(2, 2) = Q(v)[σ1 , σ2 , σ2−1 ]. Moreover, ρ ∈ EndQ(v) (⊗2 ) is defined by ρ(ωs ⊗ ωt ) = ωs−1 ⊗ ωt−1 and satisfies ρ 2 = σ2 . A direct calculation shows that ρ=

1 (e2 + e1 e2 + e2 e1 + e22 − σ1 ). v + v −1 1

By (5.3.1.3), we obtain that S(2, 2)+ = U(2, 2)+ ⊕

Q(v)ρ m = U(2, 2)+ ⊕

m 1

Q(v)σ1m .

m 1

Similarly, ρ −1 =

1 (f2 + f1 f2 + f2 f1 + f22 − τ1 ) v + v −1 1

and S(2, 2)− = U(2, 2)− ⊕

Q(v)ρ −m = U(2, 2)− ⊕

m 1

Q(v)τ1m .

m 1

Hence, σ1 τ1 ∈ U(2, 2) and S(2, 2) = U(2, 2) ⊕ Q(v)ρ m = U(2, 2) ⊕ Q(v)σ1m ⊕ Q(v)τ1m . 0 =m∈Z

m 1

Furthermore, the following relations hold in S(2, 2): σ1 e1 = e1 e2 e1 , σ1 e2 = e2 e1 e2 , τ1 f1 = f1 f2 f1 , τ1 f2 = f2 f1 f2 , σ1 f1 = e1 e2 f1 + f1 e2 e1 , σ1 f2 = e2 e1 f2 + f2 e1 e2 , τ1 e1 = f1 f2 e1 + e1 f2 f1 , τ1 e2 = f2 f1 e2 + e2 f1 f2 ,

(5.3.7.1)

σ1 (ki − v) = (e1 e2 + e2 e1 )(ki − v), i = 1, 2, and τ1 (ki − v) = (f1 f2 + f2 f1 )(ki − v), i = 1, 2. Remark 5.3.8. Recall from §2.3 that D(n) is the Z-subalgebra of D(n) ±1 K i ;0 + (m) − generated by K i , , z+ , and (u i− )(m) for i ∈ I and s , zs , (u i ) t s, t, m 1. By Proposition 3.6.1 and Corollary 3.7.5, all ξr (z± s ) lie in S(n, r ). Thus, the Q(v)-algebra homomorphism ξr : D(n) → S(n, r ) induces a Zalgebra homomorphism ξr,Z : D(n) → S(n, r). By Corollary 3.8.2, ξr,Z

168

5. The presentation and realization problems

is surjective in case n > r . However, it is in general not surjective as shown below. Let n = r = 2. By Theorem 5.3.5, S(2, 2) = U(2, 2) ⊕ Q(v)ρ m , 0 =m∈Z

where ρ = a∈N2 , σ (a)=2 ea and ρ −1 = a∈N2 , σ (a)=2 fa ; see Lemma 5.3.1. −1 lie By definition, ea = ξ2 ( u+ u− a ) and fa = ξ2 ( a ). It follows that ρ and ρ in S(2, 2). Using an argument similar to the proof of Corollary 3.8.2, we can (m) show that S(2, 2) can be generated by e(m) (i = 1, 2, m 1), lλ (λ ∈ i , fi −1 (2, 2)), ρ, and ρ . Thus, we obtain that S(2, 2) = U(2, 2) ⊕ Zρ m , 0 =m∈Z (m)

(m)

where U(2, 2) is the Z-subalgebra of S(2, 2) generated by ei , fi , lλ for i = 1, 2, m 1, and λ ∈ (2, 2). On the other hand, the image Im ξ for ξ := ξ2,Z : D(2) → S(2, 2) is ki ;0 (m) + − the Z-subalgebra of S(2, 2) generated by k±1 i , t , ξ(zs ), ξ(zs ), ei , and (m) − fi for i = 1, 2 and s, t, m 1. Since σ1 = ξ(z+ 1 ) and τ1 = ξ(z1 ), we have by the example above that −1 2 2 ξ(z+ 1 ) = −(v + v )ρ + (e1 + e1 e2 + e2 e1 + e2 )

and −1 −1 ξ(z− + (f21 + f1 f2 + f2 f1 + f22 ). 1 ) = −(v + v )ρ

Furthermore, for each s 1, (s−1)/2 s ± (ξ(z± ))s − t=1 ξ(z )ρ ±2t , ± 1 ξ(zs ) = s/2 s t ± s−2t±2t ± s (ξ(z1 )) − t=1 t ξ(zs−2t )ρ ,

if s is odd; if s is even.

By (5.3.7.1), all the elements ei ρ ±1 , fi ρ ±1 , ρ ±1 ei , ρ ±1 ei lie in U(2, 2) for i = 1, 2. Thus, an inductive argument implies that for each s 1, s s −s ±s ξ(z± mod U(2, 2). s ) ≡ (−1) (v + v )ρ

From (5.3.7.1) it also follows that ki ρ ±1 ≡ vρ ±1 ≡ ρ ±1 ki mod U(2, 2) for i = 1, 2. We conclude that neither of ρ and ρ −1 lies in Im ξ . Therefore, the Z-algebra homomorphism ξ = ξ2,Z : D(2) → S(2, 2) is not surjective.

5.4. The realization conjecture

169

5.4. The realization conjecture We now look at the realization problem for quantum affine gln . In the nonaffine case, Beilinson–Lusztig–MacPherson [4] provided a construction for quantum gln via quantum Schur algebras. In order to generalize the BLM approach to the affine case, a modified BLM approach has been introduced in [24]. On the one hand, this approach produces a realization of quantum gln which serves as a “cut-down” version of the original BLM realization. On the other hand, most of the constructions in this approach can be generalized to the affine case. The conjecture proposed below is a natural outcome from this consideration; see [24, 5.2]. Let K(n) be the Z-algebra which has Z-basis {[A]} A∈(n) and multiplication defined by [ A] · [B] = 0 if co(A) = ro(B), and [A] · [B] as given in S(n, r ) if co(A) = ro(B) and r = σ (A), where co(A) (resp., ro(A)) is the column (resp., row) sum vector associated with A (see §1.1). This algebra has no identity but infinitely many idempotents [diag(λ)] for all λ ∈ Nn . Moreover, K(n) ∼ = ⊕r 0 S(n, r ). (Note that S(n, 0) = Z with a basis labeled by the zero matrix.) (n) be the vector space of all formal Let K(n) = K(n)Q(v) and let K (possibly infinite) Q(v)-linear combinations A∈(n) β A [A] which have the following properties: ∀ x ∈ Nn , the sets {A ∈ (n) | β A = 0, ro( A) = x} and { A ∈ (n) | β A = 0, co(A) = x} are finite. In other words, for λ, μ ∈ Nn , both β A [diag(λ)] · [ A] and A∈(n)

(5.4.0.1)

β A [A] · [diag(μ)]

A∈(n)

(n) by setting are finite sums. Thus, there is a well-defined multiplication on K α A [ A] · β B [B] := α A β B [A][B]. A∈(n)

B∈(n)

A,B∈(n)

(n). This algebra has an This defines an associative algebra structure on K identity element λ∈Nn [diag(λ)], the sum of all [D] with D a diagonal matrix in (n), and contains K(n) as a natural subalgebra without identity. Note (n) is isomorphic to the direct product algebra that K r 0 S(n, r), and that the anti-involutions τr given in (3.1.3.4) induce the algebra anti-involution (n) −→ K (n), τ := τr : K β A [A] −→ β A [t A]. r

A∈(n)

A∈(n)

(5.4.0.2)

170

5. The presentation and realization problems

(n) by For A ∈ ± (n) and j ∈ Zn , define A(j) ∈ K A(j) = v λj [A + diag(λ)], λ∈Nn

where λ j = 1i n λi ji . Clearly, A(j) = r 0 A(j, r ). (n) spanned by A(j) for all A ∈ ± (n) and Let A(n) be the subspace of K n j ∈ Z. Since the structure constants (with respect to the BLM basis) appearing in the multiplication formulas given in Theorem 3.4.2 are independent of r , we immediately obtain the following similar formulas in A(n). Here, again for the simplicity of the statement, we extend the definition of A(j) to all the matrices in Mn,(Z) by setting A(j) = 0 if some off-diagonal entries of A are negative. Theorem 5.4.1. Maintain the notation used in Theorem 3.4.2. The following multiplication formulas hold in A(n): 0(j)A(j ) = v jro( A) A(j + j ), A(j )0(j) = v jco( A) A(j + j ), where 0 stands for the zero matrix,

E h,h+1 (0)A(j) =

v f (i)

i

+

v

f (i)

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

+ v f (h)− jh −1

vf v

(i )

f (i)

i >h+1;ah,i 1

+vf

(h+1)− j h+1 −1

(h)+ j h

1 − v −2

ah,h+1 + 1 (A + E h,h+1 )(j), 1

i

+vf

ah,i + 1 (A + E h,i − E h+1,i )(j) 1

and

+

ah,i + 1 (A + E h,i − E h+1,i )(j + α h) 1

( A − E h+1,h )(j + α h ) − ( A − E h+1,h )(j + β h )

+ v f (h+1)+ jh+1

E h+1,h (0) A(j) =

(5.4.1.1)

(5.4.1.2)

ah+1,i + 1 (A − E h,i + E h+1,i )(j) 1

ah+1,i + 1 (A − E h,i + E h+1,i )(j − α h) 1

(A − E h,h+1 )(j − α h ) − (A − E h,h+1 )(j + β h )

ah+1,h 1

1 − v −2 +1 (A + E h+1,h )(j).

(5.4.1.3)

5.4. The realization conjecture

171

There is a parallel construction in the non-affine case as discussed in [24, §8]. By removing the sub/superscripts , we obtain the corresponding objects and multiplication formulas in this (non-affine) case. Since the quantum gln , denoted by U(n), is generated by E i , Fi , and K ±1 j (1 i n −1, 1 j n), the corresponding multiplication formulas define an algebra isomorphism U(n) → A(n) sending E h to E h,h+1 (0), Fh to E h+1,h (0), and K j to 0(e j ). In this way, we obtain a realization of the quantum gln . This is the so-called modified approach introduced in [24]. In this approach, the algebra K(n) as a direct sum of all quantum Schur algebras S(n, r ) is just a homomorphic image of the BLM algebra K constructed in [4], which is isomorphic to the modified ˙ quantum group U(n). However, since we avoid using the stabilization property (see [4, §4]) required in the construction of K, it can be generalized to obtain the affine construction above. The Q(v)-space A(n) is a natural candidate for a realization of D(n) ∼ = n ). Since D(n) has generators other than simple generators, the formuU(gl las in Theorem 5.4.1 are not sufficient to show that A(n) is a subalgebra. However, as seen in [24, 5.4 &7.5], these formulas are sufficient to embed the subalgebras U(n), H(n)0 and H(n)0 into A(n). Thus, it is natural to formulate the following conjecture. Conjecture 5.4.2. ([24, 5.5(2)]) The Q(v)-space A(n) is a subalgebra of (n) which is isomorphic to D(n) or the quantum loop algebra U(gl n ). K As seen in §1.4, there are three types of extra generators for D(n). We expect to derive more multiplication formulas between these extra generators and the BLM basis elements in S(n, r ) (see, e.g., Problem 6.4.2) and, hence, to prove the conjecture. As a first test, we will establish the conjecture for the classical (v = 1) case in the next chapter. We end this section with a formulation of the conjecture in this case. (n)Q Let Z → Q be the specialization by sending v to 1, and let K be the vector space of all formal (possibly infinite) Q-linear combinations A∈(n) β A [ A]1 satisfying (5.4.0.1). Here [A]1 denotes the image of [A] in S(n, r )Q . For any r > 0, A ∈ ± (n), and j ∈ Nn , define in S(n, r )Q (cf. [30, (3.0.3)]) j if σ (A) r ; λ∈(n,r −σ (A)) λ [A + diag(λ)]1 , A[j, r ] = (5.4.2.1) 0, otherwise, j where λj = in=1 λi i . (These elements play a role similar to the elements A(j, r ) for affine quantum Schur algebras. See §6.2 below for more discussion of the elements.) Let

172

5. The presentation and realization problems

A[j] =

∞

(n)Q . A[j, r ] ∈ K

(5.4.2.2)

r =0

Then the classical version of Conjecture 5.4.2 claims that the Q-span A,Q [n] (n)Q which is isomorphic to the algebra of all A[j] is a subalgebra of K D(n)Q := D(n)Q /K i − 1 | 1 i n, where D(n) is the integral form defined in Definition 2.4.4. We will prove in §6.1 that the specialized algebra D(n)Q is isomorphic to the universal n ) over Q introduced in §1.1. enveloping algebra U(gl

5.5. Lusztig’s transfer maps on semisimple generators In [57] Lusztig defined an algebra homomorphism φr,r−n : S(n, r) −→ S(n, r − n), for r n, called the transfer map. In this section, we describe the images of A(0, r ) and t A(0, r ) under φr,r −n for all A = Aλ = n i=1 λi E i,i +1 ∈ (n). Throughout this section, for a prime power q, S(n, r)C = S(n, r ) ⊗Z C √ always denotes the specialization of S(n, r ) at v = q. In other words, S(n, r )C is a C-vector space with a basis {e A = e A ⊗ 1 | A ∈ (n, r )}; see Definition 3.1.1. We first recall from [57, §1] the definition of φr,r−n . As in §3.1, let V be a free F[ε, ε −1 ]-module of rank r and let F = F,n (V ) be the set of all cyclic flags L = (L i )i ∈Z , where each L i is a lattice in V such that L i−1 ⊆ L i and L i−n = εL i for all i ∈ Z. If F is the finite field with q elements, then we use the notation Y = F(q) for F, and S(n, r )C is identified with CG (Y × Y ), where G = G V is the group of automorphisms of the F[ε, ε−1 ]-module V . Given L = (L i ), L = ( L i ) ∈ F with L ⊆ L, i.e., L i ⊆ L i for all i ∈ Z, L/ L can be viewed as a nilpotent representation of the cyclic quiver (n); see §3.6. As in (1.2.0.1), for each λ = (λi )i ∈Z ∈ Nn , let Sλ = ⊕1i n λi Si , where Si denotes the simple representation of (n) corresponding to the vertex i, and + set Aλ = 1i n λi E i,i+1 ∈ (n). For A ∈ ± (n) and j ∈ Zn , let A(j, r) ∈ S(n, r ) be the BLM basis elements defined as in (3.4.0.2). We also view A(j, r ) as an element in S(n, r )C √ via specializing v to q. By Corollary 3.6.2, for all λ ∈ Nn and L = (L i ), L = ( L i ) ∈ F, we have v −c(L,L) , if L ⊆ L and L/ L∼ = Sλ ; Aλ (0, r )(L, L) = (5.5.0.1) 0, otherwise,

5.5. Lusztig’s transfer maps on semisimple generators

where c(L, L) =

dim(L i+1 / L i+1 ) dim( L i+1 / L i ) − dim(L i / Li )

1i n

=

173

dim(L i / L i ) dim(L i /L i−1 ) − dim(L i / Li ) .

1i n

Moreover, 0(λ, r )(L, L) =

#

v λμ , 0,

if L = L; otherwise,

(5.5.0.2)

where μ = (μi ) ∈ Nn with μi = dim L i /L i−1 . Now let r, r , r 0 satisfy r = r + r . Let V be a direct summand of V of rank r . Set V = V /V . Define π : F,n (V ) → F,n (V ), L → L (resp., π : F,n (V ) → F,n (V ), L → L ) by setting L i = (L i + V )/ V (resp., L i = L i ∩ V ), for all i ∈ Z. Thus, for each i ∈ Z, there is an exact sequence 0 −→ L i /L i−1 −→ L i /L i−1 −→ L i /L i −1 −→ 0 of F-vector spaces. Let F be the finite field with q elements. Following [57, 1.2], let * : S(n, r )C → S(n, r )C ⊗ S(n, r )C be the map defined by5 *( f )(L , L , L , L ) = f (L, L) L∈F,n (V )

for f ∈ S(n, r )C , where L , L ∈ F,n (V ), L , L ∈ F,n (V ), and L is a fixed element in F,n (V ) satisfying π (L) = L and π (L) = L , and the sum L) = L and π ( L) = L . is taken over all L ∈ F,n (V ) such that π ( For two lattices L , L in V , define (L : L ) = dimF (L/ L) − dimF (L / L), where L is a lattice contained in L ∩ L . For L = (L i ), L = (L i ) ∈ F,n (V ), set n (L : L ) = (L i : L i ). i =1

We finally define a C-linear isomorphism ξ : S(n, r )C → S(n, r )C by

ξ( f )(L, L ) = q (L:L )/2 f (L, L ) 5 The map * is denoted by in [57].

174

5. The presentation and realization problems

for f ∈ S(n, r )C and L, L ∈ F,n (V ). Then ξ is an algebra isomorphism; see [57, 1.7]. As before, let δ = (δi ) ∈ Nn with all δi = 1. For each A = (ai, j ) ∈ (n, n), if ro(A) = δ or co(A) = δ, then we set sgn A = 0. Suppose now ro(A) = co(A) = δ. Then there is a unique permutation w : Z → Z such that ai, j = δw( j ),i . In this case, we set sgn A = (−1)Inv(w) , where Inv(w) = |{(i, j ) ∈ Z2 | 1 i n, i < j, w(i) > w( j )}| (cf. §3.2). By [57, 1.8], there is an algebra homomorphism χ : S(n, n)C → C such that χ (e A ) = sgn A . In particular, for 1 i n and λ ∈ Zn , (λδ)/2 χ (E . i,i+1 (0, n)) = 0, χ (E i+1,i (0, n)) = 0, and χ (0(λ, n)) = q q

Suppose r n and let φr,r−n denote the composition ξ ⊗χ

*

S(n, r )C −→ S(n, r − n)C ⊗ S(n, n)C −→ S(n, r − n)C ⊗ C = S(n, r − n)C . Lusztig [57] showed that, for each pair A ∈ (n, r ) and B ∈ (n, r − n), there is a uniquely determined polynomial f A,B (v, v −1 ) ∈ Z = Z[v, v −1 ] such that for each finite field F of q elements, q φr,r −n (q −d A /2 e A ) = f A,B (q 1/2 , q −1/2 )q −d B /2 e B . B∈(n,r−n)

This gives a Q(v)-algebra homomorphism φr,r−n : S(n, r ) → S(n, r − n) defined by setting φr,r−n ([A]) = f A,B (v, v −1 )[B], B∈(n,r−n)

which is called the transfer map. Moreover, φr,r−n takes E i,i+1 (0, r ) −→ E i,i +1 (0, r − n),

E i+1,i (0, r ) −→ E i+1,i (0, r − n), and

0(λ, r ) −→ v λδ 0(λ, r − n), for i ∈ Z and λ ∈ Zn . By the definition, for each i ∈ I , we have φr,r−n ξr (K i ) = v 0(ei, n − r ) = 0(e i , n − r ) = ξr−n (K i ). Thus, φr,r−n ξr = ξr −n . However, if we view U( sln ) as a subalgebra of D(n) i for i ∈ I and denote the restriction of ξr (resp., ξr−n ) generated by E i , Fi , K to U( sln ) still by ξr (resp., ξr−n ), then φr,r−n ξr = ξr−n , i.e., the following triangle commutes

5.5. Lusztig’s transfer maps on semisimple generators ξr U( sln )

175

S(n, r ) φr,r−n

ξr −n

S(n, r − n)

As a result, there is an induced algebra homomorphism U( sln ) → limS(n, n + m); see [57, 3.4]. As seen above (a fact pointed out by Lusztig ←− in [57]), if U( sln ) is replaced by U(n), then the diagram above is not commutative. Hence, the homomorphism U( sln ) → limS(n, n + m) cannot be ←−

extended to U(n), nor to D(n). However, it is natural to ask if this homomorphism can be extended to the double Ringel–Hall algebra D(n) introduced in Remark 2.3.6(3), which has the same 0-part as U( sln ). We now show below that this is not the case. It is known from §3.7 that S(n, r) is generated by the Aλ (0, r ), t Aλ (0, r ) and 0(ei, r ) for λ ∈ Nn and i ∈ I . In the following we describe the images of Aλ (0, r) and t Aλ (0, r ) under φr,r−n . As in §2.1, for λ = (λi ) ∈ Zn , define τ λ ∈ Zn by setting (τ λ)i = λi−1 , for all i ∈ Z. Proposition 5.5.1. Keep the notation above. Let F be the finite field with q elements. For each λ = (λi ) ∈ Nn , we have *(Aλ (0, r)) = q (μτ ν−τ μν)/2 Aμ (0, r )0(ν, r ) ⊗ Aν (0, r )0(−μ, r ). n μ,ν∈N μ+ν=λ

Dually, we have *(t Aλ (0, r )) =

q (μτ ν−τ μν)/2 ·t Aμ (0, r )0(τ ν, r )⊗t Aν (0, r )0(−τ μ, r ).

n μ,ν∈N μ+ν=λ

Proof. We only prove the first formula. The second one can be proved similarly. For μ + ν = λ with μ, ν ∈ Nn , we write #μ,ν = Aμ (0, r )0(ν, r ) ⊗ Aν (0, r )0(−μ, r ). Thus, it suffices to show that for all fixed L , L ∈ F,n (V ) and L , L ∈ F,n (V ), *(Aλ (0, r ))(L , L , L , L ) = q (μτ ν−τ μν)/2 #μ,ν (L , L , L , L ). n μ,ν∈N μ+ν=λ

176

5. The presentation and realization problems

By the definition and (5.5.0.1), we obtain that *(Aλ (0, r ))(L , L , L , L ) = Aλ (0, r )(L, L) =q

(−

L λ (dim L /L −λ ))/2 i i i−1 i 1i n

,

where denotes the cardinality of the set L := { L ∈ F,n (V ) | L ⊆ L, L/ L∼ L) = L , π ( L) = L }. = Sλ , π ( Now let L ∈ L . For each i ∈ Z, consider the projection θi : L i −→ L i = (L i + V )/V . Thus, θi−1 ( Li ) = L i + L i ∩ V = L i + L i and L i /θi−1 ( Li ) ∼ L i . This = L i / implies that dim θi−1 ( L i )/ L i = dim L i / L i − dim L i / L i = dim L i / L i . The semisimplicity of L/ L ∼ L i . Hence, we have the = Sλ shows L i−1 ⊆ inclusions L i−1 + L i ⊆ L i ⊆ θi−1 ( Li ) = L i + L i ⊆ L i . Then θi−1 ( L i )/(L i−1 + L i ) = L i /(L i−1 + L i ) + (L i−1 + L i )/(L i −1 + L i ) and dim θi−1 ( L i )/(L i−1 + L i ) = dim L i /(L i−1 + L i ) + dim L i / L i . The inequality dim(L i −1 + L i )/(L i −1 + L i ) dim L i / L i gives that dim(L i−1 + L i )/(L i−1 + L i ) = dim L i / L i and θi−1 ( L i )/(L i−1 + L i ) = L i /(L i−1 + L i ) ⊕ (L i−1 + L i )/(L i −1 + L i ). Thus, L i /(L i −1 + L i ) is a complement of (L i−1 + L i )/(L i−1 + L i ). Consequently,

= i , 1i n

where i is the number of subspaces in θi−1 ( L i )/(L i−1 + L i ) which are complementary to (L i−1 + L i )/(L i −1 + L i ). Further, = = =

dim θi−1 ( L i )/(L i−1 + L i ) = dim L i /(L i−1 + L i ) − dim L i /θi−1 ( Li ) dim L i /(L i−1 + L i ) + dim(L i −1 + L i )/(L i−1 + L i ) − dim L i / L i dim L i /L i−1 − dim(L i−1 + L i )/L i−1 + dim L i / L i − dim L i / L i dim L i /L i−1 + dim L i / L i − dim L i / Li .

5.5. Lusztig’s transfer maps on semisimple generators

177

Therefore,

i = q dim L i / L i (dim L i /L i−1 −dim L i / L i ) . We finally get that *(Aλ (0, r ))(L , L , L , L ) = q a/2 , where a= −λi (dim L i /L i−1 −λi )+2 dim L i / L i (dim L i /L i−1 −dim L i / L i ) . 1i n

On the other hand, by (5.5.0.1) and (5.5.0.2), #μ,ν (L , L , L , L ) = 0 ⇐⇒ L ⊆ L , L ⊆ L , L / L ∼ L ∼ = Sμ , L / = Sν μ ∼ i ⇐⇒ L i ⊆ L i , L i ⊆ L i , L i / L i = F , L i / L i ∼ = Fνi , ∀ i ∈ Z. Moreover, if this is the case, then #μ,ν (L , L , L , L ) =Aμ (0, r )(L , L )0(ν, r )( L , L )Aν (0, r )(L , L )0(−μ, r )( L , L ) =q b/2 , where b = −c(L , L ) − c(L , L ) +

(νi dim L i / L i−1 − μi dim L i / L i−1 ).

1i n

Since

c(L , L ) = 1i n μi (dim L i /L i−1 − μi ), c(L , L ) = 1i n νi (dim L i /L i−1 − νi ), dim L i / L i −1 = dim L i /L i−1 + μi−1 − μi , and dim L i / L i−1 = dim L i /L i−1 + νi−1 − νi ,

it follows that b= −μi dim L i /L i −1 − νi dim L i /L i−1 + μ2i + νi2 1i n

+ νi L i /L i−1 − μi dim L i /L i−1 + νi μi−1 − μi νi−1

=a + τ μ ν − μ τ ν. Note that λi = μi + νi and dim L i /L i −1 = dim L i /L i−1 + dim L i /L i−1 , for all i ∈ Z. Therefore, *(Aλ (0, r ))(L , L , L , L ) = q (μτ ν−τ μν)/2 #μ,ν (L , L , L , L ), μ+ν=λ

which finishes the proof.

178

5. The presentation and realization problems

Corollary 5.5.2. Suppose r n. If λ = (λi ) ∈ Nn satisfies λi 1 for all i ∈ Z, then φr,r −n (Aλ (0, r )) = Aλ (0, r − n) + Aλ−δ (0, r − n) · 0(δ, r − n) and φr,r −n (t Aλ (0, r )) = t Aλ (0, r − n) + t Aλ−δ (0, r − n) · 0(δ, r − n). Proof. Write r = r − n. Fix a finite field F with q elements. Then applying the above proposition gives *( Aλ (0, r )) = q (μτ ν−τ μν)/2 Aμ (0, r )0(ν, r ) ⊗ Aν (0, n)0(−μ, n). n μ,ν∈N μ+ν=λ

If σ (ν) > n, then Aν (0, n) = 0. If 1 σ (ν) n and ν = δ, then χ (Aν (0, n)) = 0. Thus, q

φr,r (Aλ (0, r )) = (ξ ⊗ χ )*(Aλ (0, r )) =ξ(Aλ (0, r ))χ (0(−λ, n)) + ξ(Aλ−δ (0, r )0(δ, r ))χ (Aδ (0, n)0(−λ + δ, n)) =Aλ (0, r ) + Aλ−δ (0, r )0(δ, r ). The second formula can be proved similarly. By the above corollary, if λ = (λi ) ∈ Nn satisfies λi 1, for all i ∈ Z and σ (λ) r , then φr,r −n ξr ( Aλ (0, r )) = ξr −n ( Aλ (0, r )) and φr,r −n ξr (t Aλ (0, r )) = ξr −n (t Aλ (0, r )). This shows that the transfer maps φr,r −n are not compatible with the homomorphisms ξr : D(n) → S(n, r ).

6 The classical (v = 1) case

n ) be the universal enveloping algebra of the loop algebra gl n (Q) as Let U(gl mentioned in §1.1. We will establish, on the one hand, a surjective homomorn ) to the affine Schur algebra S(n, r )Q via the natural action phism from U(gl n ) is of gln (Q) on the Q-space Q , and prove, on the other hand, that U(gl isomorphic to the specialization D(n)Q of the integral double Ringel–Hall algebra D(n) at v = 1 and K i = 1. In this way, we obtain a surjective algebra homomorphism ηr : D(n)Q → S(n, r )Q which is regarded as the classical (v = 1) version of the surjective homomorphism ξr : D(n) → S(n, r ). We then prove the classical version of Conjecture 5.4.2 via the homomorphisms ηr . A crucial step to establish the conjecture in this case is the extension of the multiplication formulas given in §3.4 to formulas between homogeneous indecomposable generators and arbitrary BLM basis elements. This is done in §6.2. The conjecture in the classical case is proved in §6.3. In order to distinguish the specializations at non-roots-of-unity v = z ∈ C considered in previous chapters from the specialization at v = 1, we will particularly consider the specialization Z → Q by sending v to 1 throughout the chapter. Thus, H(r)Q identifies the group algebra QS,r , Q is the Qspace with basis {ωi }i ∈Z , and S(n, r )Q identifies the classical affine Schur algebra. In other words, S(n, r )Q ∼ = EndQS,r ( ⊗r Q ).

n ) 6.1. The universal enveloping algebra U (gl n (Q) has a basis {E }1i n, j ∈Z . Recall from §1.1 that the loop algebra gl i, j Thus, the natural action of these basis elements on Q defined by ωi+tn , if k = j + tn; E i, j ωk = (6.1.0.1) 0, otherwise 179

6. The classical (v = 1) case

180

n )-module structure on Q and, hence, on the r -fold tensor gives rise to a U(gl ⊗r product Q . Thus, we obtain an algebra homomorphism n ) −→ S(n, r )Q . ηr : U(gl

(6.1.0.2)

The fact that ηr is surjective has already been established in [79, Th. 6.6(i)] by a coordinate algebra approach. We now present a different proof by identifying the above action with the Hall algebra action as discussed in §§3.5–3.6. Thus, a more explicit description for ηr is obtained. n ) as the specialization D(n)Q of the integral form First, we interpret U(gl D(n) given in Definition 2.4.4 at v = 1 and 1. Let U(n) be the all K i = (m) (m) , for Z-subalgebra of D(n) generated by K i±1 , K it ;0 , (u + ) and (u − i i ) 1 i n and t, m 1. Then D(n) ∼ = U(n) ⊗Z Z[z± m ]m 1 ; see (2.4.4.2). By specializing v to 1, Q is regarded as a Z-module. Consider the Q-algebra D(n)Q := D(n)Q /K i − 1 | 1 i n (6.1.0.3) ∼ = (D(n) ⊗Z Q[v, v −1 ])/v − 1, K i − 1 | 1 i n. If x ∈ D(n), then x¯ denotes the image in D(n)Q . Since [m]v=1 = m 1, K i ;0 D(n)Q is generated by u i± (1 i n), z± (t 0). Note m (m 1), t that D(n)Q inherits a Hopf algebra structure from D(n)Q . Ln = gl Ln (Q) = Let sln (Q) = sln (Q) ⊗ Q[t, t −1 ] and set gl sln (Q) ⊕ QE , L (Q). Then the set where E = E is in the center of gl 1i n

n

i,i

X :={E i,j+ln | 1 i, j n, i = j, l ∈ Z} ∪ {E i,i | 1 i n} ∪ {E i,i+ln − E i+1,i+1+ln | 1 i n − 1, l ∈ Z, l = 0}

Ln . Let forms a Q-basis for gl zm =

E h,h+mn

for m = 0.

1h n

n = gl n (Q). By the Then X ∪ {zm | m ∈ Z, m = 0} forms a Q-basis for gl PBW theorem for universal enveloping algebras, n ) ∼ L ) ⊗ Q[zm ]m∈Z\{0} , U(gl = U(gl n

(6.1.0.4)

) is the enveloping algebra of gl . Note that the zm are central where U(gl n n elements in U(gln ). E E (E −1)···(E −t+1) n ). Let D(n)± = U(n)± ⊗ Let ti,i := i,i i,i t! i,i ∈ U(gl ± Z[zm ]m 1 be the ±-part of D(n) (see (2.4.4.1)) and let D(n)± Q = D(n)± ⊗ Q. L

L

n ) 6.1. The universal enveloping algebra U (gl

181

n ) Theorem 6.1.1. There is a Hopf algebra isomorphism φ : D(n)Q → U (gl K i ;0 E + , u − to E , and z± to defined by sending to ti,i , u i to E i,i s i +1 i+1,i t ∼ D(n)± , the z±s for 1 i n and s, t 1. In particular, D(n)± = homomorphic image of D(n)± in D(n)Q .

Q

Q

Thus, the bar on the elements of D(n)± Q can be dropped. Proof. By [52, 6.7] and [54], specializing v to 1 induces an algebra isomorphism L ) φ1 : U(n)Q := U(n)Q /K i − 1 | 1 i n −→ U(gl n K i ;0 E + , u − → E , for all 1 defined by taking t → ti,i , u i → E i,i+1 i i+1,i i n and t 1. Since D(n) ∼ = U(n) ⊗ Z[z± m ]m 1 is an algebra isomorphism, it follows that D(n)Q ∼ ] = U(n)Q ⊗ Q[z± m m 1 . Thus, the isomorphism φ1 together with (6.1.0.4) gives the required algebra isomorphism n ), φ = φ1 ⊗ φ2 : D(n)Q −→ U(gl ± where φ2 : Q[z± m ]m 1 → Q[zm ]m∈Z\{0} is the isomorphism sending zm to z±m for all m 1. The compatibility of Hopf structures is clear since u i± and z± m are primitive elements. ± Finally, choose a Z-basis {y ± j | j ∈ J } of U(n) . Then

Y

±

# =

y± j

k

(zi± )ai

$ % $ $ j ∈ J, k 1, ai ∈ N, ∀i $

(6.1.1.1)

i=1

is a basis for D(n)± . So base change gives a basis YQ± = {y = y ⊗ 1} y∈Y ±

± ± ± ± for D(n)± Q . Let YQ be the image of YQ in D(n)Q . Since φ(YQ ), as part of a n ), is linearly independent, it follows that the bar map D(n)± → basis for U(gl Q

D(n)± Q is injective, proving the last assertion.

The integral form D(n) given in (2.4.4.1) acts on the free Z-module , thanks to (3.5.4.1). This induces an action of D(n)Q on Q with K i acting trivially and, hence, an action of D(n)Q on Q . Thus, we obtain an action of D(n)Q on ⊗r Q which gives an algebra homomorphism ξ¯r : D(n)Q −→ S(n, r )Q . This homomorphism can also be regarded as the reduction modulo v = 1 and K i = 1 of the restriction ξr |D(n) : D(n) → S(n, r ) of the map in (3.5.5.4).

6. The classical (v = 1) case

182

Though the map ξr |D(n) is not surjective in general by Example 5.3.8, we will see that the map ξ¯r is surjective. Assuming [79, Th. 6.6(i)], this can be seen from the following compatibility condition. Proposition 6.1.2. For any r 0, we have ηr ◦ φ = ξ¯r . ⊗r Proof. The action of u i+ (resp., u − i ) on Q is the same as the action of E i,i+1 ± (resp., E i+1,i ) for all i ∈ I and, by (3.5.4.1), the action of zm is the same as the action of z±m for all m 1. This proves the compatibility condition.

To establish the surjection directly, we want to understand the structure of D(n)Q . Recall that the idea used in the proof of Theorem 3.8.1 is to apply Proposition 3.7.4(1) to get the surjection on every triangular part of the affine Schur algebra. Thus, if we can prove that D(n)Q contains Ringel–Hall alge± bras H(n)± Q := H(n) ⊗ Q (at v = 1), then the surjection follows. Note that this fact cannot be seen directly from the definition of D(n)Q .

Lemma 6.1.3. ([59, Cor. 4.1.1]) For 1 k t, let Ak = (ai,(k)j ), B = (bi, j ) ∈ + (n). (1) If σ (B) tk=1 σ (Ak ) and B = tk=1 Ak , then (v 2 − 1)|ϕ AB1 , A2 ,...,At . t (2) If B = k=1 Ak , then $

$ B bi, j ! 2 $ (v − 1)$ ϕ A1 ,A2 ,...,At − . (1) (2) (t) 1in, j ∈Z ai, j !ai, j ! · · · ai, j ! i< j

Proposition 6.1.4. (1) The Ringel–Hall algebra H(n)± Q at v = 1 is generated ± by u for all i < j. E i, j

n )+ (resp., U(gl n )− ) denotes the subalgebra of U(gl n ) generated (2) If U (gl by E i, j (resp., E j,i ) for all i < j, then there are algebra isomorphisms f ± : n )± → H(n)± taking E → u + (resp., E → u − ), for all i < j. U(gl i, j j,i Q Ei, j

Ei, j

(3) The Ringel–Hall algebras H(n)± Q are subalgebras of D(n)Q so that − D(n)Q = H(n)+ Q ⊗ (0-part) ⊗ H(n)Q .

(4) We have f ± = φ −1 |U (gl

± n)

n )± → H(n)± . : U(gl Q

Proof. It suffices to prove the + case. (1) Let H be the subalgebra generated by all u E . By induction on σ (B) i, j

and applying Lemma 6.1.3, we can easily prove that every u B ∈ H, for all B ∈ + (n). Hence, H(n)Q ⊆ H.

n ) 6.1. The universal enveloping algebra U (gl

183

+ (2) Let gl n be the Lie subalgebra of gln generated by E i, j (i < j). Then n )+ is isomorphic to the enveloping algebra of gl + + U(gl n and in U(gln ) , [E i,j , E k,l ] = δ j¯,k¯ E i,l+ j−k − δl¯,i¯ E k, j+l−i , for i, j, k, l ∈ Z.

On the other hand, by Lemma 6.1.3, for i < j and k < l, we have in H(n)+ Q, u + u + − u + u + = δ j¯,k¯ u + E i, j

E k,l

Ek,l

E i, j

Ei,l+ j −k

− δl,¯ i¯ u +

E k, j+l−i

.

(6.1.4.1)

n )+ → H(n)+ such that Thus, there is an algebra homomorphism f + : U(gl Q f + (E i,j ) = u + for i < j. Ei, j

L+

Let = {(i, j) | 1 i n, j ∈ Z, i < j}. By Lemma 6.1.3 again, we + see, for any A = (i, j )∈L+ ai, j E i, j ∈ (n),

+ ai, j (u ) = ai, j ! u + ϕ B (1)u + A + B, E i, j

(i, j)∈L+

(i, j)∈L+

B∈+ (n), σ (B)<σ (A)

where ϕ B ∈ Z and the products are taken with respect to a fixed total order on L+ . Hence, the set $ #

% $ + + ai, j $ (u ) $ A = (ai, j ) ∈ (n) (i, j )∈L+

Ei, j

+ is a Q-basis of H(n)+ Q . Thus, f sends a PBW-basis of U(gln ) to a basis of + H(n)Q . Hence, f + is an isomorphism. (3) By Remark 2.4.6 (see also Corollary 3.7.5), D(n)+ is a subalgebra of H(n)+ . Thus, base change induces an algebra homomorphism ιQ : + + D(n)+ Q → H(n)Q , sending the basis YQ given in (6.1.1.1) to its image n + + YQ+ . By (2.2.1.2), z+ in H(n)+ m = i=1 u E Q equals f (zm ). Hence,

i,i+mn

n )+ = U(gl n )+ ⊗ Q[zm ]m 1 . This proves ( f + )−1 (YQ+ ) is a basis for U(gl that ιQ is an isomorphism, and consequently, by Theorem 6.1.1, D(n)+ ∼ = L

Q

H(n)+ Q.

(4) Now, by (2.2.1.2) again, z+ m =

m 1,

n

+ l=1 u E

l,l+mn

in D(n)Q . Hence, for

+ ( f + )−1 (z+ m ) = zm = φ(zm ).

Also, ( f + )−1 (u i+ ) = φ(u i+ ). Therefore, ( f + )−1 = φ|H(n)+ . Q

0(ei , r )

Let ki = ∈ S(n, r), 1 i n, be as considered in Proposition 3.7.4, and let L = {(i, j) | 1 i n, j ∈ Z}. Then the set {E i,j }(i, j)∈L

6. The classical (v = 1) case

184

n (Q) and, hence, generates U(gl n ). We have almost proved forms a basis for gl the following result, which is independent of [79, Th. 6.6(i)]. The subscripts 1 indicate the elements in S(n, r )Q obtained from elements in S(n, r ) by specializing v to 1. Theorem 6.1.5. The map ηr (or ξ¯r ) is surjective. Moreover, we have ki ; 0 ηr E i,i )= and ηr (E j,k ) = E j,k (0, r )1 , for all i, j = k. 1 1 Proof. By Theorem 3.6.3, we have ξ¯r K it ;0 = kit;0 1 , ξ¯r (u + A ) = A(0, r )1 , t A(0, r ) . Applying the isomorphism φ given in Theorem and ξ¯r (u − ) = 1 A 6.1.1 yields the corresponding formulas for ηr . Now, by Proposition 6.1.2, the surjectivity of ηr follows from the surjectivity of ξ¯r since ξ¯r maps every triangular part of D(n)Q onto the corresponding part of S(n, r )Q = − 0 S(n, r )+ Q S(n, r )Q S(n, r)Q as given in Theorem 3.7.7. E Remark 6.1.6. We now give a direct proof for the formula ηr ti,i = kit;0 1 . For λ ∈ (n, r ), let iλ be defined as in the proof of Proposition 3.3.1. Since, (ω w) = (E ω )w = λ ω w = ( ki ;0 ω )w (cf. for all w ∈ S,r , E i,i iλ i iλ i,i iλ 1 1 iλ ) = ki ;0 . Hence, (6.1.0.1)), we obtain ηr (E i,i 1 1 ηr

E i,i t

=

t−1 1

t! s=0

=

(λi − s)[diag(λ)]1

λ∈(n,r)

λ∈(n,r)

λi ki ; 0 [diag(λ)]1 = . t t 1

The map ξ¯r : D(n)Q → S(n, r )Q is the classical (v = 1) version of the map ξr : D(n) → S(n, r ). In the next two sections, we will describe explicitly the image of the map

(n)Q , ξ¯ = ξ¯r : D(n)Q −→ K r 0

or equivalently, the map η=

n ) −→ K (n)Q . ηr : U(gl

(6.1.6.1)

r 0

(n)Q defined at the end of §5.4 with the Here we have identified the algebra K direct product r 0 S(n, r )Q .

6.2. More multiplication formulas in affine Schur algebras

185

6.2. More multiplication formulas in affine Schur algebras In order to prove Conjecture 5.4.2 in the classical case, we use the elements A[j, r ] for A ∈ ± (n), j ∈ Nn defined in (5.4.2.1): j if σ (A) r ; λ∈(n,r−σ (A)) λ [A + diag(λ)]1 , A[j, r ] = 0, otherwise. These elements cannot be obtained by specializing v to 1 from the elements A(j, r ) defined in (3.4.0.2). However, we have A[0, r ] = A(0, r )1 for A ∈ ± (n) (assuming 00 = 1). We also point out another difference when σ (A) = r . In this case, A[j, r ] = δ0,j [A]1 while A(j, r )1 = [A]1 for all j ∈ Nn . We will first show that, for a given r > 0, the set { A[j, r ]} A∈± (n),j∈Nn spans the affine Schur algebra S(n, r)Q . Then we derive some multiplication formulas between A[j, r ] and certain generators corresponding to simple and homogeneous indecomposable representations of the cyclic quiver (n). We will leave the proof of the conjecture to the next section. The following result is the classical counterpart of [24, Prop. 4.1]. Its proof is similar to the proof there; cf. [30, 4.3,4.2]. Proposition 6.2.1. For any fixed 1 i 0 n, the set {A[j, r ] | A ∈ ± (n), j ∈ Nn , ji0 = 0, σ (A) + σ (j) r} is a basis for S(n, r )Q . In particular, the set {A[j, r ] | A ∈ ± (n), j ∈ Nn , σ ( A) r} forms a spanning set of S(n, r )Q . The first two of the following multiplication formulas in affine Schur algebras are a natural generalization of the multiplication formulas for Schur algebras, given in [30, Prop. 3.1], which are the classical version of the quantum formulas in [4, Lem. 5.3]. The third formula is new and is the key to the proof of Conjecture 5.4.2 in the classical case. It would be interesting to find the corresponding formula for affine quantum Schur algebras. For the simplicity of the statement in the next result, we also set A[j, r ] = 0 if some off-diagonal entries of A are negative. Theorem 6.2.2. Assume 1 h, t n, j = ( jk ) ∈ Nn , and A = (ai, j ) ∈ ± (n). The following multiplication formulas hold in S(n, r )Q : (1) 0[e t , r ]A[j, r] = A[j + et , r ] + s∈Z at,s A[j, r ];

6. The classical (v = 1) case

186 (2) for ε ∈ {1, −1},

E h,h+ε [0, r ]A[j, r ] =

(ah,i + 1)(A + E h,i − E h+ε,i )[j, r ]

ah+ε,i 1 ∀i =h,h+ε

i jh + (−1) (A − E h+ε,h )[j + (1 − i )eh , r ] i 0i jh jh+ε + (ah,h+ε + 1) (A + E h,h+ε )[j − i eh+ε , r ]; i 0i jh+ε

(3) for m ∈ Z\{0},

E h,h+mn [0, r] A[j, r ] =

(ah,s+mn + 1)(A + E h,s+mn − E h,s )[0, r ]

s ∈{h,h−mn} ah,s 1

+

(ah,h+mn + 1)

0t jh

+

0t jh

(−1)

t

jh t

jh t

(A + E h,h+mn )[j − t e h , r]

(A − E h,h−mn )[j + (1 − t)e h , r ].

It is natural to compare Theorem 6.2.2(1)–(2) with [24, (4.2.1-3)] or Theorem 3.4.2 which generalize the corresponding ones for quantum Schur algebras. They are not obtained from the quantum counterpart by specializing v to 1. For example, the second sum in the right-hand side of (2) above is slightly different from the quantum version. In the case where σ (A) = r + 1 and ah+ε,h 1, the left-hand side of Theorem 6.2.2(2) is zero. By the remark at the beginning of this section, the right-hand side is also zero since j + (1 − i)e h = 0, for all 0 i jh . The proof of the following result will be given at the end of the chapter as an appendix; see §6.4. It should be pointed out that the first formula can also be obtained from [56, 3.5] by specializing v to 1, while the second formula is the key to the proof of part (3) of the theorem above. Proposition 6.2.3. Let 1 h n, B = (bi, j ) ∈ (n, r ), and λ = ro(B). (1) If ε ∈ {1, −1} and λ e h+ε , then [E h,h+ε +diag(λ−eh+ε )]1 [B]1 =

(bh,i +1)[B+E h,i −E h+ε,i ]1 .

i∈Z,bh+ε,i 1

(Here, by convention, 1 h < n for ε = 1 and 1 < h n for ε = −1.)

6.2. More multiplication formulas in affine Schur algebras

187

(2) If m ∈ Z\{0} and λ e h , then [E h,h+mn +diag(λ−e h )]1 [B]1 =

(bh,s+mn +1)[B+E h,s+mn −E h,s ]1 .

s∈Z bh,s 1

We now use these formulas to prove the theorem. Proof. The proof of formula (1) is straightforward. Since, for any A ∈ (n, r ) and λ ∈ (n, r ), [A]1 , if λ = ro(A); [diag(λ)]1 [A]1 = 0, otherwise, it follows that

0[e t , r ]A[j, r ] = =

λ∈(n,r−1)

μ∈(n,r−σ (A))

(

at, j + μt )μj [A + diag(μ)]1

μ∈(n,r−σ (A)) j∈Z

=

μj [A + diag(μ)]1

λt μj [A + diag(μ)]1 (where λ = ro(A) + μ)

μ∈(n,r−σ (A))

=

λt [diag(λ)]1

μ

j+et

μ∈(n,r−σ (A))

[A + diag(μ)]1 +

at, j A[j, r] = RHS.

j ∈Z

We now prove formula (2). For convenience, we set [B]1 = 0 if one of the entries of B is negative. Since [A]1 [B]1 = 0 whenever co( A) = ro(B), we have E h,h+ε [0, r] A[j, r ] = μj [E h,h+ε + diag(μ + ro(A) − eh+ε )]1 [A + diag(μ)]1 . μ∈(n,r−σ (A))

By Proposition 6.2.3(1), [E h,h+ε + diag(μ + ro( A) − eh+ε )]1 [A + diag(μ)]1 = (ah,i + 1)[A + E h,i − E h+ε,i + diag(μ)]1 i =h,h+ε ah+ε,i 1

+ (μh + 1)[A − E h+ε,h + diag(μ + eh )]1

+ (ah,h+ε + 1)[A + E h,h+ε + diag(μ − e h+ε )]1 .

6. The classical (v = 1) case

188

Thus, E h,h+ε [0, r ]A[j, r ] =

(ah,i + 1)(A + E h,i − E h+ε,i )[j, r ] + Yh + Yh+ε ,

i =h,h+ε ah+ε,i 1

where

Yh =

μj (μh + 1)[A − E h+ε,h + diag(μ + eh )]1

μ∈(n,r−σ (A))

=

j μi i (μh + 1 − 1) jh (μh + 1)

μ∈(n,r−σ (A)) i =h

× [A − E h+ε,h + diag(μ + eh )]1 jh = (−1)i (A − E h+ε,h )[j + (1 − i)eh , r ], i 0i jh

and

Yh+ε =

μj (ah,h+ε + 1)[A + E h,h+ε + diag(μ − e h+ε )]1

μ∈(n,r−σ (A))

= (ah,h+ε + 1)

j μi i (μh+ε − 1 + 1) jh+ε

μ∈(n,r−σ (A)) i =h+ε

= (ah,h+ε + 1)

× [ A + E h,h+ε + diag(μ − e h+ε )]1 jh+ε μj−i eh+ε i

0i jh+ε

= (ah,h+ε + 1)

0i jh+ε

μ∈(n,r−σ (A)−1)

× [ A + E h,h+ε + diag(μ)]1 jh+ε ( A + E h,h+ε )[j − i eh+ε , r ]. i

Substituting gives (2). Finally, we prove formula (3). The proof is similar to that of (2). First, with the same reasoning, E h,h+mn [0, r ]A[j, r] = μj [E h,h+mn + diag(μ) + r o(A) − e h ]1 · [A + diag(μ)]1 . μ∈ (n,r−σ (A))

Applying Proposition 6.2.3(2) yields [E h,h+mn + diag(μ) + r o(A) − e h ]1 · [A + diag(μ)]1 = (ah,s+mn + 1)[A + E h,s+mn − E h,s + diag(μ)]1 s ∈{h,h−mn}

6.2. More multiplication formulas in affine Schur algebras

189

+ (ah,h+mn + 1)[( A + E h,h+mn ) + diag(μ − e h )]1

+ (μh + 1)[( A − E h,h−mn ) + diag(μ + e h )]1 .

Thus, [0, r ] A[j, r ] = E h,h+mn (ah,s+mn + 1)(A + E h,s+mn − E h,s )[0, r ] + X1 + X2 , s ∈{h,h−mn}

where

X1 = (ah,h+mn + 1)

μj [(A + E h,h+mn ) + diag(μ − e h )]1

μ∈ (n,r −σ (A))

μ∈ (n,r −σ (A))

s =h 1sn

= (ah,h+mn + 1)

j

μss (μh − 1 + 1) jh

× [(A + E h,h+mn ) + diag(μ − e h )]1 jh j−t e h (μ − e h) t (n,r −σ ( A))

= (ah,h+mn + 1)

μ∈ 0t jh

= (ah,h+mn

× [A + E h,h+mn + diag(μ − e h )]1 jh + 1) )[j − t e (A + E h,h+mn h , r] t 0t jh

and X2 =

μ∈ (n,r −σ (A))

=

μj (μh + 1)[( A − E h,h−mn ) + diag(μ + e h )]1

j

μss (μh + 1 − 1) jh (μh + 1)

μ∈ (n,r −σ (A)) s =h

=

× [( A − E h,h−mn ) + diag(μ + e h )]1 jh j+(1−t)e h (μ + e (−1)t h) t (n,r−σ ( A))

μ∈ 0t jh

× [(A − E h,h−mn ) + diag(μ + e h )]1 jh (A − E h,h−mn )[j + (1 − t)e = (−1)t h , r ], t 0t jh

proving (3). This completes the proof of the theorem.

6. The classical (v = 1) case

190

6.3. Proof of Conjecture 5.4.2 at v = 1 We now use Theorem 6.2.2 to prove Conjecture 5.4.2 in the classical case. Recall from the proof of Proposition 6.1.4 that the specialized Ringel–Hall algebra H(n)Q is generated by u i, j := u E for all i < j ∈ Z. As seen from i, j Proposition 1.4.5, the Ringel–Hall algebra H(n) over Q(v) can be generated by the elements associated with simple and homogeneous indecomposable representations of (n). We first prove that this is also true for H(n)Q . Lemma 6.3.1. The H(n)Q is generated by the elements u i = u i,i+1 and u i,i+mn , for i ∈ Z and m 1. In particular, the subalgebra S(n, r )+ Q + (resp., S(n, r )− ) spanned by A[0, r ] = A(0, r ) , for all A ∈ (n) (resp., 1 Q [0, r ] and E A ∈ − (n)), can be generated by E h,h [0, r ], for all h,h+mn 1 h n, h − h = −1, and m 1, (resp., h − h = 1 and m −1). Proof. Let K be the subalgebra of H(n)Q generated by the elements u i,i+1 and u i,i+mn , for i ∈ Z and m 1. It is enough to prove u i, j ∈ K, for all i < j. Write j − i = mn + k, where m ∈ Z and 1 k n. If k = n, then u i, j ∈ K. Now assume 1 k < n. We apply induction on k. If k = 1, then by (6.1.4.1), u i,mn+i+1 = u i,i +1 u i +1,mn+1+i − u i +1,mn+i+1 u i,i +1 ∈ K. Now suppose k > 1 and u i, j ∈ K, for all i < j with j − i = mn + k − 1. Then by (6.1.4.1) and the inductive hypothesis, u i,mn+k+i = u i,i +1 u i +1,mn+k+i − u i+1,mn+k+i u i,i+1 ∈ K, proving the first assertion. The last assertion follows from Proposition 3.6.1. (n)Q be the vector space of all formal (possibly infinite) As in §5.4, let K Q-linear combinations A∈(n) β A [A]1 satisfying (5.4.0.1). Recall from (5.4.2.2) that, for A ∈ ± (n) and j ∈ Nn , A[j] =

∞

(n)Q . A[j, r ] ∈ K

r =0

Furthermore, let and be the orders on Zn and (n) defined in (1.1.0.3) and (3.7.1.1), respectively. Proposition 6.3.2. For A ∈ ± (n) and j ∈ Nn , we have j A+ [0]0[j]A− [0] = A[j] + f A,j A[j ] + j <j n j ∈N

j

B,j

B,j

f A,j B[j ],

± (n) B∈ n B≺A, j ∈N

where f A,j , f A,j ∈ Q and A = A+ + A− with A+ , t(A− ) ∈ + (n).

6.3. Proof of Conjecture 5.4.2 at v = 1

191

Proof. For each r 0, by Lemma 6.3.1, we may write A+ [0, r ] as a linear combination of monomials in E h,h+1 [0, r ] and E h,h+mn [0, r ]. By Theorem B,j

6.2.2, there exist f A,j ∈ Q (independent of r ) such that

A+ [0, r ]0[j, r] A− [0, r ] =

B,j

f A,j B[j , r ],

(6.3.2.1)

± (n) B∈ n j ∈N

for all r 0. On the other hand, using an argument similar to the second display for the computation of p A in the proof of Theorem 3.7.7 yields A+ [0, r ]0[j, r ]A− [0, r ] =

λj A+ [0, r ][diag(λ)]1 A− [0, r ]

λ∈(n,r)

=

λj [A + diag(λ − σ (A))]1 + g,

λ∈(n,r ) λσ (A)

where σ (A) = co(A+ ) + ro(A− ) and g is a Q-linear combination of [B]1 with B ∈ (n, r ) and B ≺ A. Since λj = (λ − σ (A) + σ (A))j = (λ − σ (A))j +

j <j n j ∈N

j

f A,j (λ − σ (A))j ,

j

where f A,j ∈ Z are independent of λ and r , it follows that A+ [0, r ]0[j, r ]A− [0, r] = A[j, r ] +

j <j n j ∈N

j

f A,j A[j , r ] + g.

Combining this with (6.3.2.1) proves the assertion. For integers a −1 and l 1, let Za,l = {a + i | i = 1, 2, . . . , l} and (Za,l )n = {λ ∈ Nn | λi ∈ Za,l , ∀i }. Lemma 6.3.3. For fixed integers a −1 and n, l 1, if we order (Za,l )n lexicographically and form an l n × l n matrix Bn = (λμ )λ,μ∈(Za,l )n , where μ μ μ λμ = λ1 1 λ2 2 · · · λn n , then det(Bn ) = 0. Proof. Write (Za,l )n = {a1 , a2 , . . . , al n } with ai
6. The classical (v = 1) case

192

⎛ (a + 1)(a+1) Bn−1 ⎜(a + 2)(a+1) Bn−1 ⎜ Bn = ⎜ .. ⎝ .

(a + 1)a+2 Bn−1 (a + 2)(a+2) Bn−1 .. .

(a + l)(a+1) Bn−1

(a + l)(a+2) Bn−1

··· ··· .. . ···

⎞ (a + 1)a+l Bn−1 (a + 2)(a+l) Bn−1 ⎟ ⎟ ⎟. .. ⎠ . (a + l)(a+l) Bn−1

Thus, det(Bn )

⎛

(a + 1)Bn−1 (a + 2)Bn−1 .. .

Bn−1 l ⎜

n−1 ⎜ Bn−1 = (a + i )(a+1)l det ⎜ . ⎝ .. i=1

(a + l)Bn−1

Bn−1

··· ··· .. . ···

⎞ (a + 1)l−1 Bn−1 (a + 2)l−1 Bn−1 ⎟ ⎟ ⎟ .. ⎠ . (a + l)l−1 Bn−1

l

n−1 = (a +i)(a+1)l i=1

⎛

Bn−1 ⎜ Bn−1 ⎜ × det⎜ . ⎝ .. Bn−1 =

⎞ 0 ((a +2)l−1 −(a +2)l−2 (a +1))Bn−1 ⎟ ⎟ ⎟ .. ⎠ .

··· ··· .. .

0 (2−1)Bn−1 .. . (l −1)Bn−1

···

((a +l)l−1 −(a +l)l−2 (a +1))Bn−1

l l

n−1 n−1 (a + i )(a+1)l det(Bn−1 ) ( j − 1)l i=1

j =2

⎛

Bn−1 ⎜ .. × det ⎝ . Bn−1

··· .. . ···

(a

+ 2)l−2 B .. .

n−1

⎞ ⎟ ⎠.

(a + l)l−2 Bn−1

Hence, det(Bn ) = det(Bn−1 )l

l

n−1 (a + i)(a+1)l

(i − j)l

n−1

= 0,

1 j
i=1

by induction. Note that, in the proof above, if a = −1, then l

Zl := Z−1,l = {0, 1, . . . , l − 1}

and so the product i=1 (a + i)(a+1)l = 1. (n)Q As introduced at the end of §5.4, let A,Q [n] be the subspace of K ± n spanned by the elements A[j] for all A ∈ (n) and j ∈ N. The following n ). Let theorem gives a realization of the universal enveloping algebra U(gl −1 n n (Zl ) = 2 ((Zl ) ), where 2 is defined in (1.1.0.2). n−1

6.3. Proof of Conjecture 5.4.2 at v = 1

193

(n)Q with Q-basis Theorem 6.3.4. The Q-space A,Q [n] is a subalgebra of K B = {A[j] | A ∈ ± (n), j ∈ Nn }. Moreover, the map η := r 0 ηr defined in (6.1.6.1) is injective and induces η

n ) ∼ a Q-algebra isomorphism U(gl = A,Q [n]. Proof. We first prove the linear independence of B. Suppose f A,j A[j] = 0 A∈± (n), j∈Nn

for some f A,j ∈ Q. Then 0=

± (n) A∈ n j∈N

f A,j A[j] =

r 0

± (n) A∈ λ∈ (n,r −σ ( A))

λj f A,j [A + diag(λ)].

j∈Nn

Thus, j∈Nn λj f A,j = 0 for all A ∈ ± (n), λ ∈ (n, r − σ (A)), and r σ (A). So, when A is arbitrarily fixed, there is a finite subset J of Nn satisfying f A,j = 0 for all j ∈ J . Choose 1 such that J is a subset of (Zl )n and set f A,j = 0 if j ∈ (Zl )n \J . Since ∪r σ (A) (n, r − σ (A)) = Nn contains (Zl )n , it follows that j∈(Zl )n λj f A,j = 0 for all λ ∈ (Zl )n . Applying Lemma 6.3.3 gives f A,j = 0 for all j. Hence, B forms a basis for A,Q [n]. n ) = U(gl n )+ U (gl n )0 U(gl n )− , by identifying U(gl n )± with Since U(gl ± H(n)Q , the set + j1 jn − n {u + A (E 1,1 ) · · · (E n,n ) u B | A, B ∈ (n), j ∈ N}

n ). Now Theorem 6.1.5 implies forms a basis of U (gl j1 jn − η(u + (E 1,1 ) · · · (E n,n ) u t A− ) = A+ [0]0[j]A− [0]. A+

By Proposition 6.3.2, the set { A+ [0]0[j]A− [0] | A ∈ ± (n), j ∈ Nn } forms another basis for A,Q [n]. Hence, η is injective and A,Q [n] is exactly the image of η. This completes the proof of the theorem. This theorem together with Theorem 6.2.2 implies immediately the follown ). ing multiplication formulas in U (gl n ) of the loop algebra Corollary 6.3.5. The universal enveloping algebra U(gl ± n gln (Q) has a basis { A[j] | A ∈ (n), j ∈ N} which satisfies the following multiplication formulas: for 1 h, t n, j = ( jk ) ∈ Nn , A = (ai, j ) ∈ ± (n), ε ∈ {1, −1}, and m ∈ Z\{0},

6. The classical (v = 1) case

194

0[e t ]A[j] =A[j + et ] +

E h,h+ε [0]A[j] =

at,s A[j],

s∈Z

(ah,i + 1)(A + E h,i − E h+ε,i )[j]

ah+ε,i 1 ∀i =h,h+ε

jh ( A − E h+ε,h )[j + (1 − i)eh ] i 0i jh jh+ε + (ah,h+ε + 1) (A + E h,h+ε )[j − i eh+ε ], i +

(−1)i

0i jh+ε

and

E h,h+mn [0]A[j] =

(ah,s+mn + 1)( A + E h,s+mn − E h,s )[0]

s ∈{h,h−mn} ah,s 1

+

(ah,h+mn + 1)

0t jh

+

0t jh

(−1)

t

jh t

jh t

(A + E h,h+mn )[j − t e h]

( A − E h,h−mn )[j + (1 − t)e h ].

Remark 6.3.6. There should be applications of these multiplication formun )Z generated by the las. For example, one may define the Z-subalgebra U(gl divided powers of E h,h+ε [0] together with E h,h+mn [0] for all 1 h n, ε ∈ {1, −1} and m ∈ Z\{0}. This should serve as the Kostant Z-form of n ). U(gl

6.4. Appendix: Proof of Proposition 6.2.3 For a finite subset X ⊆ S,r , define X = x∈X x ∈ QS,r . If A = j(λ, d, μ) is the matrix corresponding to the double coset Sλ dSμ for λ, μ ∈ (n, r ), d d ∈ D λ,μ , then the element [A]1 ∈ S(n, r )Q is the map φλ,μ (at v = 1) as defined in (3.2.6.1). Note also that, if v = 1, then xλ = Sλ . Thus, [ A]1 (Sν ) = 0 for ν = μ and [A]1 (Sμ ) = Sλ dSμ = Sλ dD ν ∩ Sμ by Lemma 3.2.5,

where ν is the composition defined by Sν = d −1 Sλ d ∩ Sμ (see Corollary 3.2.3 for a precise description of ν). Lemma 3.2.5 implies also that for λ, μ ∈ (n, r ) and w ∈ S,r , Sλ wSμ = |w−1 Sλ w ∩ Sμ |Sλ wSμ . This fact will be frequently used in the proofs below.

(6.4.0.1)

6.4. Appendix: Proof of Proposition 6.2.3

195

Recall also from (3.2.1.4) the sets Riλ+kn = {λk,i −1 + 1, λk,i−1 + 2, . . . , λk,i −1 + λi } associated with λ ∈ (n, r ), where λk,i −1 := kr + i−1 j=1 λ j . P ROPOSITION 6.2.3(1). Let A = (ai, j ) = j(λ, d, μ) ∈ (n, r ) with ro(A) = λ. If ε ∈ {1, −1} and λ e h+ε , then [E (ah,i + 1)[A + E h,h+ε + diag(λ − eh+ε )]1 [ A]1 = h,i − E h+ε,i ]1 , i∈Z, ah+ε,i 1

where 1 h, h + ε n. Proof. Observe that ro(E h,h+ε ) = eh and co(E h,h+ε ) = eh+ε . Thus, applying (3.2.2.2) yields j(λ + αh,ε , 1, λ) = E h,h+ε + diag(λ − eh+ε ), where αh,ε = eh − eh+ε . In other words, the matrix E h,h+ε + diag(λ − eh+ε ) is defined by the double coset Sλ−α Sλ . Hence, putting L = [1, n] × Z, we h,ε have by Corollary 3.2.3 that [E h,h+ε + diag(λ − eh+ε )]1 [A]1 (Sμ ) = [E h,h+ε + diag(λ − eh+ε )]1 (Sλ dSμ )

1 = S Sλ dSμ as,t ! λ+α h,ε s,t∈L

=

s,t∈L

where

γ = γ (ε) =

1 as,t !

Sλ+α D γ ∩ Sλ dSμ , h,ε

(λ1 , . . . , λh , 1, λh+1 − 1, λh+2 , . . . , λn ),

if ε = 1;

(λ1 , . . . , λh−1 − 1, 1, λh , λh+1 , . . . , λn ),

if ε = −1.

λ For i ∈ Rh+ε = {λ0,h+ε + 1, λ0,h+ε + 2, . . . , λ0,h+ε + λ0,h+ε+1 }, define yε,i ∈ Sλ by setting 1···λ λ + 1 ··· i − 1 i i + 1 · · · λ0,h+1 · · · r y1,i = 1 · · · λ0,h λ0,h + 2 · · · i λ + 1 i + 1 ···λ ···r 0,h

and y−1,i =

0,h

1 · · · λ0,h−2 · · · i − 1 1 · · · λ0,h−2 · · · i − 1

0,h

i λ0,h−1

i +1 i

··· ···

0,h+1

λ0,h−1 λ0,h−1 − 1

λ0,h−1 + 1 · · · r λ0,h−1 + 1 · · · r

.

6. The classical (v = 1) case

196

λ Then D γ ∩ Sλ = {yε,i | i ∈ Rh+ε }. Hence,

[E h,h+ε + diag(λ − eh+ε )]1 [A]1 (Sμ ) =

s,t∈L

1 as,t !

Sλ+α yε,i dSμ .

λ i∈Rh+ε

h,ε

(ε,i )

Let B (ε,i) = (bs,t ) ∈ (n, r ) be the matrix associated with λ + αh,ε , μ and the double coset Sλ+α yε,i dSμ . Since h,ε

λ+α −1 yε,i (Rs h,ε )

=

⎧ λ ⎪ ⎪ ⎨ Rs ,

if 1 s n but s = h, h + ε;

Rhλ ∪ {i}, ⎪ ⎪ ⎩ R λ \{i }, h+ε

if s = h; if s = h + ε,

λ , it follows that, for i ∈ Rh+ε (ε,i)

bs,t

λ+α

μ

−1 = |d −1 yε,i Rs h,ε ∩ Rt | ⎧ ⎪ if 1 s n but s = h, h + ε; ⎪ ⎨as,t , μ −1 = ah,t + |{d (i) ∩ Rt }|, if s = h; ⎪ ⎪ μ ⎩a −1 h+ε,t − |{d (i) ∩ Rt }|, if s = h + ε. μ

If ti ∈ Z is the unique integer such that d −1 (i ) ∈ Rti (and ah+ε,ti 1), then (ε,i) bs,t

=

⎧ ⎪ ⎪ ⎨as,t ,

if 1 s n, s = h, h + ε or t = ti ;

ah,t + 1, if s = h, t = ti ; ⎪ ⎪ ⎩a h+ε,t − 1, if s = h + ε, t = ti .

− E λ This implies that B (ε,i ) = A + E h,t h+ε,ti , for all i ∈ Rh+ε . By Corollary i 3.2.3 again, [E h,h+ε + diag(λ − eh+ε )]1 [A]1 (Sμ )

1 (ε,i ) = bs,t !Sλ+α yε,i dSμ h,ε as,t ! λ s,t∈L

=

i∈Rh+ε s,t∈L

(ε,i)

bs,t ! (ε,i ) [B ]1 (Sμ ) as,t

λ s,t∈L i∈Rh+ε

=

ah,t + 1 i [B (ε,i) ]1 (Sμ ). a h+ε,t i λ

i∈Rh+ε

6.4. Appendix: Proof of Proposition 6.2.3

197

Finally, [E h,h+ε + diag(λ − eh+ε )]1 [A]1 ah,t + 1 λ = |{i ∈ Z | i ∈ Rh+ε , t = ti }| [A + E h,t − E h+ε,t ]1 ah+ε,t t∈Z,ah+ε,t 1 = (ah,t + 1)[A + E h,t − E h+ε,t ]1 , t∈Z,ah+ε,t 1 μ

λ , t = t }| = |d −1 R λ as |{i ∈ Z | i ∈ Rh+ε i h+ε ∩ Rt | = ah+ε,t .

We need some preparation before proving Proposition 6.2.3(2). We follow the notation used in §3.2. Thus, for 1 i r , ei = (0, . . . , 0, 1 , 0, . . . , 0) ∈ (i)

S,r is the permutation sending i to i + r and j to j, for all 1 j r with j = i, and ei = ρsr+i −2 · · · sr+1 sr · · · si +1 si as seen in the proof of Proposition 3.2.1. Note that s j +1 ρ = ρs j for all j ∈ Z. To λ ∈ (n, r), h ∈ [1, n] with λh = 0, and m > 0, we associate the element u λm,h =

1 · · · λ0,h−1 λ0,h−1 + 1 λ0,h−1 + 2 1 · · · λ0,h−1 λ0,h−1 + 2 λ0,h−1 + 3

··· ···

λ0,h − 1 λ0,h

in S,r and let u λ−m,h = (u λm,h )−1 . Then 1···λ λ0,h−1 + 1 λ0,h−1 + 2 u λ−m,h = 1 · · · λ0,h−1 −mr +λ λ +1 0,h−1

0,h

0,h−1

λ0,h mr +λ0,h−1 +1

··· ···

λ0,h λ0,h − 1

λ0,h + 1 · · · r λ0,h + 1 · · · r

λ0,h + 1 · · · r λ0,h + 1 · · · r

.

For any i ∈ Rhλ , removing those simple reflections s j from ei indexed by the numbers i, i + 1, . . . , λ0,h − 1 and r + λ0,h−1 , r + λ0,h−1 + 1, . . . , r + i − 2 yields the shortest representative ρsr+λ0,h−1 −1 · · · sλ0,h +1 sλ0,h of the double coset Sλ ei Sλ . Clearly, u λ1,h = ρsr+λ0,h−1 −1 · · · sλ0,h +1 sλ0,h . This observation has the following generalization. Lemma 6.4.1. Maintain the notation introduced above. Suppose λ (n, r ), m ∈ Z\{0}, and 1 h n. (1) u λm,h is the shortest element in Sλ eim Sλ for all i ∈ Rhλ ; (2) j(λ, u λm,h , λ) = E h,h−mn + diag(λ − e h ); λ λ −1 (3) (u m,h ) Sλ u m,h ∩ Sλ = Sν , where (λ1 , . . . , λh−1 , λh − 1, 1, λh+1 , . . . , λn ), if m > 0; ν= (λ1 , . . . , λh−1 , 1, λh − 1, λh+1 , . . . , λn ), if m < 0.

∈

6. The classical (v = 1) case

198

Proof. For m > 0, it is straightforward to check that eim = si−1 · · · sλ0,h−1 +2 sλ0,h−1 +1 · u λm,h · sλ0,h −1 · · · si+1 si . Thus, Sλ eim Sλ = Sλ u λm,h Sλ . Now, by definition, u λm,h ( j) < u λm,h ( j + 1), for any j with s j ∈ Sλ and m ∈ Z\{0}. Hence, by (3.2.1.5), u λm,h ∈ D λ,λ , proving = e (1). The assertion (2) follows from Lemma 3.2.2, noting that e h h−mn and |Rsλ ∩ u λm,h Rtλ | = E h,h−mn + diag(λ − e h ) s,t . The assertion (3) is a consequence of (2) and Corollary 3.2.3. P ROPOSITION 6.2.3(2). Let h ∈ [1, n] and A = (ai, j ) ∈ (n, r ) with ro(A) = λ. If m ∈ Z\{0} and λ e h , then [E h,h+mn +diag(λ−e (ah,s+mn +1)[ A+E h,s+mn −E h,s ]1 . h )]1 [A]1 = s∈Z, ah,s 1

Proof. As above, let λ0,h = λ1 + · · · + λh

(and λ0,0 = 0).

Assume μ = co( A) and d ∈ D λ,μ such that j(λ, d, μ) = A. By Lemma 6.4.1 and Corollary 3.2.3 (and (6.4.0.1)), [E h,h+mn + diag(λ − e h )]1 [A]1 (Sμ )

= Sλ (u λm,h )−1 Sλ · d · D α ∩ Sμ

(noting λh > 0)

1 Sλ (u λm,h )−1 Sλ · Sλ dSμ |Sλ | 1 1 = Sλ (u λm,h )−1 Sλ · Sλ · d · Sμ |Sλ | 1sn as,t ! =

=

1sn t∈Z

=

1sn t∈Z

t∈Z

1 as,t !

Sλ (u λm,h )−1 Sλ · d · Sμ

1 Sλ · (u λm,h )−1 · D β ∩ Sλ · d · Sμ , as,t !

where Sα = d −1 Sλ d ∩ Sμ and Sβ = u λm,h Sλ (u λm,h )−1 ∩ Sλ with β=

(λ1 , . . . , λh−1 , 1, λh − 1, λh+1 , . . . , λn ), if m > 0; (λ1 , . . . , λh−1 , λh − 1, 1, λh+1 , . . . , λn ), if m < 0.

6.4. Appendix: Proof of Proposition 6.2.3

199

We now compute D β ∩ Sλ . For m = 0 and λ0,h−1 + 1 i λ0,h (i.e., λ i ∈ Rh ), define wm,i ∈ Sλ as follows. If m > 0, then wm,λ0,h−1 +1 = 1 and, for λ0,h−1 + 1 < i, wm,i :=

1 · · · λ0,h−1 1 · · · λ0,h−1

λ0,h−1 + 1 λ0,h−1 + 2

··· ···

i −1 i

i +1 i +1

i λ0,h−1 + 1

· · · λ0,h · · · r · · · λ0,h · · · r

.

If m < 0, then wm,λ0,h = 1 and, for i < λ0,h , wm,i :=

1 · · · λ0,h−1 · · · i − 1 1 · · · λ0,h−1 · · · i − 1

i λ0,h

i +1 i

··· ···

λ0,h λ0,h − 1

λ0,h + 1 · · · r λ0,h + 1 · · · r

.

It is also clear that D β ∩ Sλ = {wm,i | λ0,h−1 + 1 i λ0,h } for all m ∈ Z\{0}. Hence, [E h,h+mn +diag(λ−e h )]1 [A]1 (Sμ ) =

1

a ! 1sn s,t t∈Z

Sλ ·(u λm,h )−1 wm,i d ·Sμ .

i∈Rhλ

(m,i) Let B (m,i) = (bs,t ) ∈ (n, r ) be the matrix associated with λ, μ and the λ double coset Sλ (u m,h )−1 wm,i dSμ , where m ∈ Z\{0} and i ∈ Rhλ . Since

−1 λ wm,i u m,h (Rsλ )

=

Rsλ ,

if 1 s n but s = h;

(Rhλ \{i}) ∪ {mr

+ i},

if s = h,

it follows that, for i ∈ Rhλ , μ

(m,i) −1 λ bs,t = |d −1 wm,i u m,h Rsλ ∩ Rt | as,t , if 1 s n but s = h; = μ μ −1 −1 as,t − |{{d (i)} ∩ Rt }| + |{d (mr + i)} ∩ Rt |, if s = h. μ

If ti ∈ Z is the unique integer such that d −1 (i) ∈ Rti (and thus, ah,ti 1), μ then d −1 (mr + i) ∈ Rmn+ti and (m,i) bh,t

=

⎧ ⎪ ⎪ ⎨ah,t ,

a − 1, ⎪ h,t ⎪ ⎩a + 1, h,t

if t ∈ {ti , mn + ti }; if t = ti ; if t = mn + ti .

6. The classical (v = 1) case

200

for all i ∈ R λ . Thus, applying This implies that B (m,i ) = A + E h,mn+t − E h,t h i i Corollary 3.2.3 again yields [E h,h+mn + diag(λ − e h )]1 [A]1 (Sμ )

1 (m,i ) = bs,t !Sλ (u λm,h )−1 wm,i dSμ as,t ! λ 1sn 1sn i ∈Rh

t∈Z

t∈Z

(m,i) bs,t ! (m,i) = [B ]1 (Sμ ) as,t ! λ 1sn i∈Rh

=

t∈Z

i∈Rhλ ,ah,ti 1

ah,mn+ti + 1 (m,i) [B ]1 (Sμ ). ah,ti

Therefore, [E h,h+mn + diag(λ − e h )]1 [A]1 ah,mn+t + 1 = |{i ∈ Z | i ∈ Rhλ , t = ti }| [A + E h,mn+t − E h,t ]1 a h,t t∈Z ah,t 1

=

(ah,t+mn + 1)[A + E h,t+mn − E h,t ]1 ,

t∈Z ah,t 1

μ

since ah,t = |d −1 Rhλ ∩ Rt | = |{i ∈ Z | i ∈ Rhλ , t = ti }| by Lemma 3.2.2. Proposition 6.2.3(2) is the key to the establishment of the multiplication formulas in Theorem 6.2.2, which in turn play a decisive role in the proof of the conjecture in the classical case (see Proposition 6.3.2). It would be natural to raise the following question as a finer version of Problem 3.4.3. Problem 6.4.2. Find the quantum version of the multiplication formulas given in Proposition 6.2.3(2) for affine quantum Schur algebras.

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Index

A(j, r ), BLM basis element, 77 A[j, r ], BLM basis element for affine Schur algebra, 185 Aλ , matrix defining the semisimple representation Sλ , 13 C = C(n) , Cartan matrix of a cyclic quiver, 19 I , index set Z/nZ = {1, 2, . . . , n} of simple generators, 12 M( A), representation of (n) associated with A, 13 M ∗ N , generic extension, 14 R λj , j ∈ Z, partition of Z, 70 Sλ , semisimple representation, 13 Si [l], indecomposable representation of (n), 12 H(n), Ringel–Hall algebra of (n), 15 (n, r ), set of compositions of r into n parts, 11 + (n, r ), set of partitions in (n, r ), 135 + (r ), set of partitions of r , 133 , free Z-module with basis {ωi | i ∈ Z}, 75 F = F,n , set of cyclic flags of period n, 63 F,λ , G-orbit in F, 63 H(r ), affine Hecke algebra, 64 n with (n, r ), set of sequences λ ∈ Z σ (λ) = r , 11 (n), set of Z × Z-matrices over N with n periodicity, 11 − (n), set of strictly lower triangular matrices in (n), 96 + (n), set of strictly upper triangular matrices in (n), 13

+ (n)∗ , subset of non-zero matrices in + (n), 27 ± (n), set of 0-diagonal matrices in (n), 77 (n, r ), set of matrices A ∈ (n) with σ ( A) = r , 11 (n)Q , completion of K(n)Q , 171 K n , set of infinite sequences of integers with n Z periodicity, 11 K(n), algebra isomorphic to ⊕r 0 S(n, r ), 169 S,r , affine symmetric group, 68 U(n), extended quantum affine sln , 43 U(n, r ), homomorphic image of U(n), 154 C(n)± , subalgebras of D(n), ∼ = C(n), 40 d( A), dimension vector of M( A), 13 lλ , idempotent [diag(λ)], 99, 151, 154 σ (A), sequence of hook sums, 101 co(A), column sum vector, 10, 169 C(n), composition algebra of (n), 17 D(n), double Ringel–Hall algebra, 37 D(n), integral form of D(n), 48 D(n)0 , 0-part of D(n), 48 D(n)± , ±-part of D(n), 48 Sλ , Young subgroup associated with λ, 70 aA , order of Aut(M(A)), 14 d( A), dimension of M( A), 13 j, bijection between double cosets and (n, r ), 70 , , Euler form, 15 n or Zn , 11 , order on Z dg , degeneration order, 14 C∗ , set of non-zero complex numbers, 11 ν , dual partition of ν, 133

205

206

Index

, order on M,n (Z) or (n), 96 ro(A), row sum vector, 10, 169 ∼ n ), 54, EH,C , isomorphism D,C (n) → UC (gl 136 ∼ n ), 54 EH , isomorphism D(n) → U(gl n , U (gln ), universal enveloping algebra of gl 179 Z = Z[v, v −1 ], Laurent polynomial ring in indeterminate v, 11, 14, 64, 122 X , set of complete cyclic flags B,r (q), 65 Y , set of cyclic flags F,n (q), 65 σ (A), sum of n consecutive rows of A, 11 σ (λ), sum of n consecutive components of λ, 11 t A, transpose of A, 63 m , C ∞ , Borcherds–Cartan matrices, 42 C α , K i , 27, 35, 42 K (n), candidate of Lusztig form, 102 D (n), 49, 102 H(n)+ , +-part of D (n), 102 H(n)− , −-part of D (n), cyclic quiver with n vertices, 12 p A , PBW type basis element for S(n, r ), 101 ϕ BA ,...,B , Hall polynomial, 14 m 1 ξr , epimorphism D(n) → S(n, r ), 86, 102 ξr∨ , homomorphism H(r ) → EndD(n) (⊗r ), 86 ξr,C , epimorphism D,C (n) → S(n, r )C , 87, 137 n ) → S(n, r )C , ξr, C , epimorphism UC (gl 137 D (n), D (n) with reduced 0-part, 45 d A , as in [A] := v−d A e A , 66

∼ over Q(v), S(n, r ), 77 the ±, 0-part, S(n, r )± , S(n, r )0 , 98 the Hecke algebra definition, SH (n, r ), 72 the tensor space definition, St (n, r ), 76 affine symmetric group, 68 affine Weyl group of type A, W , 68 aperiodic matrix, 13

affine Branching Rule, 136 affine Hecke algebra ∼ at a prime power q, H,q , 64 ∼ at a prime power q, CG (X × X ), 65 ∼ of S,r , H(S,r ), 72 ∼ over Z, H(r ), 64 ∼ over Q(v), H(r ), 72 affine quantum Schur algebra ∼ at a prime power q, S,q , 64 ∼ at a prime power q, CG (Y × Y ), 65 ∼ over Z, S(n, r ), 64

Euler form, 15 symmetric ∼, ( , ), 25 extended quantum affine sln , 43, 153 extended Ringel–Hall algebra, 27, 28

d A , as in u˜ A := v d A u A , 15 u A = u [M(A)] , basis elements of H(n), 15 u a , semisimple generator, 22 u i , simple generators for H(n), 17 u i± , simple generators for D(n), 40 w0,λ , longest element of Sλ , 71 Z(n), central subalgebra, 45, 158

Bernstein presentation, 74 BLM basis, 78 BLM spanning set, 78 Cartan datum, 25 Cartan matrix, 19 ∼ of a cyclic quiver, 19 m , C ∞ , 42 Borcherds–∼, C central elements z+ m of D(n), 40, 99, 183 central elements cm of H(n), 21 central subalgebra of D(n), 45, 153, 158 commutator formula, 61 commutator relation, 59 composition algebra, 17 convolution product, 65 cyclic flag, 63 cyclic quiver, 12 defining relations (QGL1)–(QGL8), 42 (QLA1)–(QLA7), 50 (QS1 )–(QS6 ), (QS0 ), 165 (QS1)–(QS6), 156 (QSL0)–(QSL7), (QSL6 ), (QSL7 ), 20 (R1)–(R8), 43 degeneration order, 14, 96 dimension vector, 12 double Ringel–Hall algebra D (n), ∼ with reduced 0-part, 45 D(n), ∼, 37 Drinfeld double, 32, 36 reduced ∼, 36 Drinfeld polynomials, 130, 132

Gaussian polynomial, 12 generators for S(n, r ), 104 indecomposable ∼, 179 semisimple ∼, 172, 175 generators for H(n) homogeneous indecomposable ∼, 23

Index

homogeneous semisimple ∼, 21 Schiffmann–Hubery ∼, 21 generators for D(n), 45 semisimple ∼, 55, 61, 86 indecomposable representation of (n), 12 integral form ∼ of D(n), D(n), 48 (n), 102 candidate of Lusztig form, D length function, 69 loop algebra of gln , 9, 49, 179 Morita equivalence, 124 nilpotent representation, 12 aperiodic ∼, 13 periodic ∼, 13 opposite Hopf algebra, 27 PBW type basis, 15, 37, 81 PBW type basis for S(n, r ), 101 periodic matrix, 13 polynomial identity, 115 polynomial representation, 131 primitive element, 37 pseudo-highest weight module, 130, 131 pseudo-highest weight vector, 130, 131 quantum affine sln , 20 extended ∼, U(n), 43 quantum affine gln , 50

207

quantum loop algebra n ), 50 ∼ of gln , U(gl ∼ of sln , U( sln ), 20, 51 realization conjecture, 171 Ringel–Hall algebra double ∼, D(n), 37 extended ∼, H(n)0 , 28 extended ∼, H(n)0 , 27 root datum, 26 semi-opposite Hopf algebra, 27 semisimple representation of (n), Sa , 12 simple representation ∼ of H(r )C , Vs , 133 ∼ of S(n, r )C , 134, 149 n ), L(Q), 132 ∼ of UC (gl n ), L(P), ¯ ∼ of UC (sl 130 ∼ of (n), Si , 12 ¯ ∼ of U(n)C , L(P, λ), 150 sincere, 12 skew-Hopf pairing, 31 symmetrization, 25 tensor space T(n, r ), 65 ∼ over Q(v), ⊗r , 83 ∼ over Z, ⊗r , 75 transfer map, 172 triangular decomposition of S(n, r ), 101 triangular relation, 97

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