Iwahori-Hecke Algebras and their Representation Theory. Lectures C.I.M.E., 1999

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris Subseries: Fondazione ...

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris Subseries: Fondazione C.I.M.E., Firenze Adviser: Pietro Zecca and European Mathematical Society

1804

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

I. Cherednik Ya. Markov R. Howe G. Lusztig

Iwahori-Hecke Algebras and their Representation Theory Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy June 28 - July 6, 1999 Editors: M. Welleda Baldoni Dan Barbasch

13

Authors

Editors

Ivan Cherednik Yavor Markov Department of Mathematics University of North Carolina Chapel Hill, NC 27599-3250 U.S.A

M. Welleda Baldoni Department of Mathematics University of Rome Tor Vergata Rome 00133 Italy

E-mail: [email protected] [email protected] Roger Howe Department of Mathematics Yale University New Haven, CT 06520-8283 U.S.A

E-mail: [email protected] Dan Barbasch Department of Mathematics 547 Malott Hall Cornell University Ithaca, NY 14853 U.S.A. E-mail: [email protected]

E-mail: [email protected] George Lusztig Department of Mathematics M.I.T. Cambridge, MA 02139 U.S.A E-mail: [email protected]

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): 20C08, 20C11, 19L47, 47B35 ISSN 0075-8434 ISBN 3-540-00224-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10903032

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Preface

This text consists of notes of three courses given during the CIME summer school which took place in 1999 in Martina Franca, Italy. The subject was Iwahori-Hecke algebras and their representation theory. The program consisted of several courses taught by senior faculty and some advanced lectures given by young researchers. The scheduled courses were G. Heckman, Representation theory of aﬃne Hecke algebras R. Howe, Aﬃne-like Hecke algebras and p-adic representations theory G. Lusztig Representations of aﬃne Hecke algebras I.Cherednik Hankel transform via double Hecke algebra The specialized lectures on more advanced topics were given by T. Haines, M. Nazarov, C. Kriloﬀ, U. Kulkharni, K. Maktouf, J. Kim and G. Papadoupoulo. The volume contains the notes of the courses by I. Cherednik, R. Howe, and G. Lusztig. G. Heckman was not able to provide notes for his course. We give some references later in the introduction. In the remainder of the introduction we give some background material on aﬃne Hecke algebras and extra references that complement the notes. Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were (probably) ﬁrst considered in the context of representation theory of ﬁnite groups of Lie type. For example for G = GL(2, Fq ), consider the question of decomposing the induced module I = IndG B [11] := {f : G −→ C : f (gb) = f (g)},

ι(g)f (x) := f (g −1 x),

where B is the subgroup of upper triangular matrices. One is naturally led to consider the algebra of intertwining operators of the module I. These are linear endomorphisms T : I −→ I satisfying T ◦ ι(g) = ι(g) ◦ T for all g ∈ G. This algebra of endomorphisms can be identiﬁed with the algebra of Bbiinvariant functions on G with multiplication structure given by convolution. In the case of GL(2, Fq ) it is generated (over C) by an element T satisfying the relation T 2 = (q − 1)T + q, and is called the ﬁnite Hecke algebra of type

VI

Preface

A1 . Its generalization to type An is of independent interest to knot theory because it is a quotient of the braid group. The Hecke algebras mentioned above play an important role in combinatorics and representation theory of GL(n) and the symmetric group. An account of their role in the theory of ﬁnite groups of Lie type is detailed in [Car] and the references therein. The above algebra is not what Hecke introduced in the context of automorphic forms. He deﬁned certain operators T (n) (n ∈ N) that act on automorphic forms for congruence groups and studied their eigenvalues and relation to Dirichlet series. An introduction to this theory can be found in [Se]. When one considers automorphic forms in the adelic setting ([Ge], [Bump]) the operators T (n) are very closely related to the previous example. One is led to consider the group of rational points G(F) for a local ﬁeld F with residual ﬁeld of characteristic p and a compact open subgroup K. The operators T (n) are K-biinvariant functions supported on certain cosets. This interpretation has led to a systematic study of the representation theory of groups over totally disconnected ﬁelds. The talks of Howe gave an overview of this research area, particularly the role of these algebras. The notes are in the article of R. Howe and C. Krillof Aﬃne-like Hecke algebras and p-adic representation theory The following is a brief list of the topics covered. – Structure of p-adic groups and their associated aﬃne Hecke algebras. – Bruhat and Iwahori-Bruhat decompositions from a geometric perspective. – Application of Hecke algebras in representations of the p-adic groups. Lusztig’s talks were focused on the special case when the compact open subgroup is an Iwahori subgroup. In this case very detailed knowledge of the representation theory is available due to his work partly joint with Kazhdan. His notes, G. Lusztig Notes on aﬃne Hecke algebras cover the following topics: – – – –

The aﬃne Hecke algebra H and equivariant K-theory Convolution Subregular case

A very diﬀerent ﬁeld where Hecke algebras play a role is in the area of special functions. MacDonald has made various conjectures about the existence of orthogonal polynomials in several variables attached to root systems. These polynomials are related to the spherical functions of real and p-adic groups. The conjectures have a natural interpretation in the context of representations of Iwahori-Hecke algebras. Much of the work in this area e.g. by Cherednik, Heckman and Opdam make essential use of this structure. Heckmann’s course provided an introduction to this area. We refer to the Bourbaki talks [He] and [M].

Preface

VII

In Cherednik’s notes the focus is on the advantages of the operator approach in the theory of Bessel functions and the classical Hankel transform. An account of these results can be found in the article I. Cherednik and Y. Markov, Hankel transform via double Hecke algebra: – – – – – – –

L-operator Hankel transform Dunkl operator Double H double prime Nonsymmetric eigenfunctions Inverse transform and Plancherel formula Truncated Bessel functions

We would like to thank the speakers and the participants for their scientiﬁc work which made the school a success. The summer school brought together young researchers from many different countries with senior faculty. Funds were provided by C.I.M.E., the National Science Foundation and the European Community. Welleda Baldoni and Dan Barbasch

References [Bo] A. Borel Admissible representations of a semisimple group over a local ﬁeld with vectors ﬁxed under an Iwahori subgroup, Invent. Math. 35, 1976, pp. 233259 [Bump] D. Bump Automorphic forms and representations, Cambridge Univ. Press [Car] R. Carter Groups of ﬁnite Lie type, conjugacy classes and complex characters, Wiley&Sons, 1985 [Ge] S. Gelbart Automorphic forms on adele groups, Annals of Math. St., Princeton University Press, vol. 83, Princeton, N.J. 1975 [He] G.J. Heckman Dunkl operators Seminaire Bourbaki, Asterisque 245, 1997, Exp. No. 828, pp. 223-246 [M] I.G. MacDonald Aﬃne Hecke algebras and orthogonal polynomials Seminaire Bourbaki, Asterisque 237 , 1996, Exp. No. 797, pp. 189–207 [Se] J-P. Serre Cours D’Arithmetique, Presses Universitaires de France, 1970

Table of Contents

Hankel transform via double Hecke algebra Ivan Cherednik and Yavor Markov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 L-operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hankel transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dunkl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nonsymmetric eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Master formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Double H double prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Algebraization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Inverse transform and Plancherel formula . . . . . . . . . . . . . . . . . . . . . . . 9 Finite-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Truncated Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 5 7 9 11 14 16 18 21 24

Aﬃne-like Hecke algebras and p-adic representation theory Roger Howe (Lecture Notes by Cathy Kriloﬀ ) . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structure of p-adic GL(V ) and Sp(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Structure of the Iwahori Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . 4 Results on representations of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Spherical Function Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 29 46 51 54 65 67

Notes on aﬃne Hecke algebras George Lusztig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The aﬃne Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 H and equivariant K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Subregular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Subregular case: type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Subregular case: types C, D, E, F, G . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 77 79 83 97 103

Hankel transform via double Hecke algebra Ivan Cherednik1 and Yavor Markov2 1

2

Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599 – 3250, USA [email protected] Partially supported by NSF grant DMS–9877048 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599 – 3250, USA [email protected]

This paper is a part of the course delivered by the ﬁrst author at UNC in 2000. The focus is on the advantages of the operator approach in the theory of Bessel functions and the classical Hankel transform. We start from scratch. The Bessel functions were a must for quite a few generations of mathematicians but not anymore. We mainly discuss the master formula expressing the Hankel transform of the product of the Bessel function by the Gaussian. By the operator approach, we mean the usage of the Dunkl operator and the H H , double H double prime, the rational degeneration of the double aﬃne Hecke algebra. This includes the transfer from the symmetric theory to the nonsymmetric one, which is the key tool of the recent development in the theory of spherical and hypergeometric functions. In the lectures, the Hankel transform was preceded by the standard Fourier transform, which is of course nonsymmetric, and the Harish-Chandra transform, which is entirely symmetric. We followed closely the notes of the lectures not yielding to the temptation of skipping elementary calculations. We do not discuss the history and generalizations. Let us give some references. The master formula is a particular case of that from [D]. Our proof is mainly borrowed from [C1] and [C2]. The nonsymmetric Hankel transform is due to C. Dunkl (see also [O,J]). We will see that it is equivalent to the symmetric one, as well as for the master formulas (see e.g. [L], Chapter 13.4.1, formula (9)). This is a special feature of the one-dimensional setup. Generally speaking, there is an implication nonsymmetric ⇒ symmetric, but not otherwise. We also study the truncated Bessel functions, which are necessary to treat negative half-integral k, when the eigenvalue of the Gaussian with respect to the Hankel transform is inﬁnity. They correspond to the ﬁnite-dimensional representations of the double H double prime, which are completely described in the paper. We did not ﬁnd proper references but it is unlikely that these functions never appeared before. They are very good to demonstrate the operator technique. We thank D. Kazhdan and A. Varchenko, who stimulated the paper a great deal, M. Duﬂo for useful discussion, and CIME for the kind invitation.

I. Cherednik et al.: LNM 1804, M.W. Baldoni and D. Barbasch (Eds.), pp. 1–25, 2003. c Springer-Verlag Berlin Heidelberg 2003

2

Ivan Cherednik and Yavor Markov

1 L-operator We begin with the classical operator L=(

∂ 2 2k ∂ ) + . ∂x x ∂x

Upon the conjugation: L = |x|−k H |x|k , H = (

∂ 2 k(1 − k) . ) + ∂x x2

(1)

Here k is a complex number. Both operators are symmetric = even. The ϕ-function is introduced as follows: Lϕλ (x, k) = 4λ2 ϕλ (x, k),

ϕλ (x, k) = ϕλ (−x.k),

ϕλ (0, k) = 1. (2)

We will mainly write ϕλ (x) instead of ϕλ (x, k). Since L is a DO of second order, the eigenvalue problem has a two-dimensional space of solutions. The even ones form a one-dimensional subspace and the normalization condition ﬁxes ϕλ uniquely. Indeed, the operator L preserves the space of even functions holomorphic at 0. The ϕλ can be of course constructed explicitly, without any references to the general theory of ODE. We look for a solution in the form ϕλ (x, k) = f (xλ, k). Set xλ = t. The resulting ODE is d2 f 1 df (t) + 2k (t) − 4f (t) = 0, dt2 t dt

a Bessel-type equation.

Its even normalized solution is given by the following series f (t, k) =

∞

t2m m! (k + 1/2) · · · (k − 1/2 + m) m=0

∞ 1 t2m = Γ (k + ) . 2 m=0 m!Γ (k + 1/2 + m)

(3)

So

1 1 f (t, k) = Γ (k + )t−k+ 2 Jk− 12 (2it). 2 The existence and convergence is for all t ∈ C subject to the constraint:

k = −1/2 + n, n ∈ Z+ .

(4)

The symmetry ϕλ (x, k) = ϕx (λ, k) plays a very important role in the theory. Here it is immediate. In the multi-dimensional setup, it is a theorem. Let us discuss other (nonsymmetric) solutions of (3) and (2). Looking for f in the form tα (1 + ct + . . .) in a neighborhood of t = 0, we get that the coeﬃcients of the expansion

Hankel transform via double Hecke algebra

f (t) = t1−2k

∞

3

cm t2m at t = 0

m=0

can be readily calculated from (3) and are well-deﬁned for all k. The convergence is easy to control. Generally speaking, such f are neither regular nor even. To be precise, we get even functions f regular at 0 when k = −1/2−n for an integer n ≥ 0, i.e. when (4) does not hold. These solutions cannot be normalized as above because they vanish at 0. Note that we do not need nonsymmetric f and the corresponding ϕλ (x) = f (xλ) in the paper. Only even normalized ϕ will be considered. The nonsymmetric ψ-functions discussed in the next sections are of diﬀerent nature. Lemma 1.1. (a) Let L◦ be the adjoint operator of L with respect to the C +∞ valued scalar product f, g0 = 2 0 f (x)g(x)dx. Then |x|−2k L◦ |x|2k = L. +∞ (b) Setting f, g = 2 0 f (x)g(x)x2k dx, the L is self-adjoint with respect to this scalar product, i.e. L(f ), g = f, L(g). ∂ ◦ Proof. First, the operator multiplication by x is self-adjoint. Second, ( ∂x ) = ∂ − ∂x via integration by parts. Finally,

d2 ◦ ∂ 2k ∂ ∂ 2k ( ))x2k ) + ( )◦ ( ))x2k = x−2k (( )2 − 2 dx ∂x x ∂x ∂x x ∂ ∂ + 2k(2k − 1)x2k−2 = x−2k (x2k ( )2 + 4kx2k−1 ∂x ∂x ∂ ∂ 2k ∂ − 2kx2k−1 − 2k(2k − 1)x2k−2 ) = ( )2 + = L. (5) ∂x ∂x x ∂x ∞ Therefore, L(f ), g = 2 0 L(f )gx2k dx = ◦ 2k 2 f L (x g) dx = 2 f x2k Lx−2k (x2k g) dx = f, L(g). (6) x−2k L◦ x2k = x−2k ((

R+

R+

Actually this calculation is not necessary if (1) is used. Indeed, H◦ = H.

2 Hankel transform Let us deﬁne the symmetric Hankel transform on the space of continuous functions f on R such that limx→∞ f (x)ecx = 0 for any c ∈ R. Provided (4), (Fk f )(λ) =

2 Γ (k + 1/2)

+∞

ϕλ (x, k)f (x)x2k dx.

(7)

0

The growth condition makes the transform well-deﬁned for all λ ∈ C, because ϕλ (x, k) ∼ Const(e2λx + e−2λx ) at x = ∞.

4

Ivan Cherednik and Yavor Markov

The latter is standard. We switch from F on functions to the transform of the operators: F(A)(F(f ) = F(A(f )). Remark that the Hankel transform of the function is very much diﬀerent from the transform of the corresponding multiplication operator. The key point of the operator technique is the following lemma. Lemma 2.1. Using the upper index to denote the variable (x or λ), (a)

F(Lx ) = 4λ2 ;

(b)

F(4x2 ) = Lλ ;

(c)

F(4x

∂ d ) = −4λ − 4 − 8k. ∂x dλ

Proof. Claim (a) is a direct consequence of Lemma 1.1 (b) with g(x) = ϕλ (x) : F(Lf ) = Lf, ϕλ = f, Lϕλ = 4λ2 f, ϕλ = 4λ2 F(f ) . Claim (b) results directly from the x ↔ λ symmetry of φ, namely, from the relation Lλ ϕλ (x) = 4x2 ψλ (x). Concerning (c), there are no reasons, generally speaking, to expect any simple Fourier transforms for the operators diﬀerent ∂ from L. However in this particular case: [Lx , x2 ] = 4x ∂x + 2 + 4k. Appling F ∂ 2 λ to both sides and using (a), (b), [4λ , L /4] = F(4x ∂x ) + 2 + 4k. Finally F(4x

d ∂ d ) = −4λ − 2 − 4k − 2 − 4k = −4λ − 4 − 8k ∂x dλ dλ

. Note that [x

∂ ∂ 2 , x ] = 2x2 , [x , Lx ] = −2Lx , ∂x ∂x

∂ because operators x ∂x , L are homogeneous of degree 2 and −2. So e = x2 , ∂ x + k + 1/2 = [e, f ] generate a representation of the f = −L /4, and h = x ∂x Lie algebra sl2 (C).

Theorem 2.1. (Master Formula) Assuming that Re k > − 12 , ∞ 2 1 2 2 2 ϕλ (x)ϕµ (x)e−x x2k dx = Γ (k + )eλ +µ ϕλ (µ), 2 0 ∞ 2 L 1 2 2 ϕλ (x) exp(− )(f (x))e−x x2k dx = Γ (k + )eλ f (λ), 4 2 0

(8)

provided the existence of exp(− L4 )(f (x)) and the integral in the second formula. 2

Proof. The left-hand side of the ﬁrst formula equals Γ (k +1/2)F(e−x ϕµ (x)). We set −x2 x2 ϕ− ϕµ (x), ϕ+ µ (x) = e µ (x) = e ϕµ (x). They are eigenfunctions of the operators

Hankel transform via double Hecke algebra 2

2

2

5

2

L− = e−x ◦ L ◦ ex , L+ = ex ◦ L ◦ e−x . ± with eigenvalue 2µ, To be more exact, ϕ± µ is a unique eigenfunction of L normalized by ϕ± (0) = 1. µ Express L− in terms of the operators from the previous lemma. 2 2 2 ∂ 2 ∂ 2 x2 ∂ ) e = e−x (ex ( )2 + 2(2x)ex + (2 + 4x2 )ex ) ∂x ∂x ∂x ∂ 2 ∂ 2 = ( ) + 4x + 2 + 4x , ∂x ∂x 2 2k ∂ 2 2 2 2k ∂ 2 2k 2k ∂ ex = e−x (ex + 2xex )= + 4k, e−x x ∂x x ∂x x x ∂x 2 ∂ 2k ∂ x2 ∂ L− = e−x (( )2 + )e = L + 4x + 2 + 4k + 4x2 . (9) ∂x x ∂x ∂x 2

e−x (

∂ − 2 − 4k + 4x2 . Now we may use Lemma 2.1: Analogously, L+ = L − 4x ∂x

F(Lx− ) =F(Lx ) + F(4x2 ) + F(4x =4λ2 + Lλ − 4λ Thus

∂ ) + F(2 + 4k) = ∂x

d − 4 − 8k + 2 + 4k = Lλ+ . dλ

(10)

x − x − − Lλ+ (Fϕ− µ ) = F(L− )(Fϕµ ) = F(L− ϕµ ) = 2µF(ϕµ ),

i.e. Fϕ− µ is an eigenfunction of L+ with the eigenvalue 2µ. Using the uniqueµ2 + ness, we conclude that F(ϕ− µ )(λ) = C(µ)e ϕµ (λ) for a constant C(µ). However the left-hand side of the master formula is λ ↔ µ symmetric as well as 2 λ2 +µ2 eµ ϕ+ ϕµ (λ). So C(µ) = C(λ) = C. Setting λ = 0 = µ, we get µ (λ) = e the desired. The second formula follows from the ﬁrst for f (x) = ϕµ (x, k). Move exp(µ2 ) to the left to see this. It is linear in terms of f (x) and holds for ﬁnite linear combinations of ϕ and inﬁnite ones provided the convergence. So it is valid for any reasonable f. We skip the detail.

3 Dunkl operator The above proof is straightforward. One needs the self-duality of the Hankel transform and the the commutator representation for x ∂/∂x. The self-duality holds in the general multi-dimensional theory. The second property is more special. Also our proof does not clarify why the master formula is so simple. There is a “one-line” proof of this important formula, which can be readily generalized. It involves the Dunkl operator: D=

∂ k − (s − 1), ∂x x

where s is the reﬂection s(f (x)) = f (−x).

(11)

6

Ivan Cherednik and Yavor Markov

The operator D is not local anymore, because s is a global operator apart from a neighborhood of x = 0. We are going to ﬁnd its eigenfunctions. Generally speaking, this may create problems since we cannot use the uniquness theorems from the theory of ODE. However everything is surprisingly smooth. Lemma 3.1. Considering x as the multiplication operator, s ◦ x = −x ◦ s,

s◦

∂ ∂ =− ◦ s, ∂x ∂x

(12)

(a) D2 = L upon the restriction to even functions, (b) s ◦ D ◦ s = −D and D2 ﬁxes the space of even functions. Proof. Indeed, (s ◦ x)(f (x)) = s(xf (x)) = −xf (−x) = (−x ◦ s)(f (x)). The ∂ ∂x is analogous. Then ∂ 2 ) − ∂x ∂ = ( )2 + ∂x

D2 = (

k ∂ (s − 1) − x ∂x k ∂ (s + 1) − x ∂x

∂ k (s − 1) + ∂x x ∂ k (s − 1) + ∂x x

k k (s − 1) (s − 1) x x k k (s − 1) (s − 1). x x

(13)

It is simple to calculate the ﬁnal formula but unnecessary. Applying (13) to symmetric (i.e. even) functions f (x), the two last terms will vanish, because (s − 1)(f (x)) = f (−x) − f (x) = 0. So (s + 1)(f (x)) = f (−x) + f (x) = 2f (x), ∂ 2 ∂ and D2 |even = ( ∂x ) + 2 xk ∂x = L. ∂ ∂ 2 s = − ∂x s = −dx, and s( xk (s − Claim (b) is obvious. Indeed, s2 = 1, s ∂x 1))s = − xk s(s2 − s) = − xk (s3 − s2 ) = − xk (s − 1). Thus s ◦ D ◦ s = −D. By the way, this implies that s ◦ D2 ◦ s = D2 , i.e. D2 commutes with s. So we do not need an explicit formula for D2 |even to see that it preserves even functions. +∞ Let us consider the standard scalar product f, g0 = −∞ f (x)g(x)dx. Here the functions are continuous C-valued continues on the real line R. One may add the complex conjugation to g but we will not do this. The scalar product is non-degenerate, so adjoint operators are well-deﬁned. We continue to use the notation H ◦ for the pairing f, g0 . Let us calculate the adjoint of D with respect to |x|2k . +∞ Proposition 3.1. Setting f, g = −∞ f (x)g(x)|x|2k dx, the Dunkl operator D is anti self-adjoint with respect to this scalar product, i.e. D(f ), g = −f, D(g). Equivalently, |x|−2k D◦ |x|2k = −D. ∂ ◦ ∂ Proof. Recall that x◦ = x and ( ∂x ) = − ∂x , where x is considered as the multiplication operator. Then s◦ = s : +∞ −∞ s(f ), g0 = f (−x)g(x)dx = f (t)g(−t)(−dt) = f, s(g)0 −∞

+∞

Hankel transform via double Hecke algebra

7

for t = −x. Hence, k ∂ − (s − 1))◦ |x|2k ∂x x ∂ k = |x|−2k (( )◦ − (s − 1)◦ ( )◦ )|x|2k ∂x x k ∂ − (s − 1) )|x|2k = |x|−2k (− ∂x x ∂ k = |x|−2k (− + (1 + s))|x|2k ∂x x k ∂ 2k = |x|−2k |x|2k (− ) + |x|−2k (− |x|2k ) + |x|−2k |x|2k (1 + s) ∂x x x ∂ k =− + (s − 1) = −D. (14) ∂x x

|x|−2k D◦ |x|2k = |x|−2k (

Finally, D(f ), g =

+∞

= −∞ +∞

+∞

−∞

D(f (x))g(x)|x|2k dx =

+∞

f (x)D◦ (|x|2k g(x))dx

(15)

−∞

f (x)|x|2k (|x|−2k D◦ |x|2k )(g(x))dx

=

−∞

f (x)(−D(g(x)))|x|2k dx = −f, D(g).

The proposition readily gives that |x|−2k L◦ |x|2k = L on even functions f. Indeed, L(f ), g = D2 (f ), g = f, D2 (g) = f, L(g), provided that g is even too. Recall that it was not diﬃcult to check this relation directly. In the multi-dimensional theory, this calculation is more involved and the usage of the (generalized) Dunkl operators makes perfect sense.

4 Nonsymmetric eigenfunctions Our next step will be a study of the eigenfunctions of the Dunkl operator: Dψλ (x, k) = 2λψλ (x, k),

ψλ (0, k) = 1.

(16)

We will use the shortcut notation f ι (x) = s(f (x)) = f (−x). Lemma 4.1. There exists a unique solution ψλ (x, k) of (16) which is analytic for all x ∈ R. It is represented in the form ψλ (x) = g(λx). provided that λ = 0. Without the normalization condition, ψ(x) is unique up to proportionality in R∗ for any λ. As λ = 0, it is given by the formula ψ0 = C1 + C2 x|x|−2k−1 , where C ∈ C are arbitrary constants.

8

Ivan Cherednik and Yavor Markov

Proof. Assuming that ψλ is a solution of (16), let ψλ0 =

1 1 (ψλ + ψλι ), ψλ1 = (ψλ − ψλι ), 2 2

be its even and odd parts. By Lemma 3.1 (b), Ds(ψλ (x)) = −sD(ψλ (x)) = −2λs(ϕλ (x)). Hence, (16) is equivalent to Dψλ0 = 2λψλ1 ; Dψλ1 = 2λψλ0

ψλ0 (0) = 1 ψλ1 (0) = 0.

(17)

Furthermore, D2 ψλ0 = 4λ2 ψλ0 . Since ψλ0 is even, Lψλ0 = 4λ2 ψλ due to Lemma 3.1. Therefore ψλ0 has to coincide with ϕλ from the ﬁrst section. This is true for all λ. If λ = 0, k 1 dϕλ 1 1 dψλ0 Dψλ0 = − (s − 1)ψλ0 = . (18) ψλ1 = 2λ 2λ dx x 2λ dx The last equality holds because ψλ0 = ϕλ is even. Finally, ψλ (x) = ϕλ (x) +

1 1 ϕ (x) = g(λx) for g = f + f , 2λ λ 2

(19)

where ϕλ (x) = f (λx), f is from (3), and f is the derivative. It is for λ = 0. Let us consider the case λ = 0. We have Dψλ0 = 0, ψλ0 (0) = 1 and Dψλ1 = 0, ψλ1 (0) = 0. Since ψλ0 is even, Dψλ0 = odd. So

0 dψλ dx

= 0. Thus ψλ0 (x) = 1. The ψλ1 is

dψλ1 (x) k dψλ1 (x) k 1 − (s − 1)ψλ1 (x) = + (ψλ (x) − ψλ1 (−x)) dx x dx x dψλ1 (x) 2k 1 + ψ (1). = dx x λ

Dψλ1 (x) =

Solving the resulting ODE, ψλ1 (x) = Cx|x|−2k−1 . In this proof, we used that Df (x) = f (x) on even functions and k 2k ∂ 2k (f (x) − f (−x)) = f (x) + f (x) = ( + )f (x) x x ∂x x on odd functions. By the way, it makes obvious the coincidence of L with ∂ ∂ + 2k D2 on even f . Indeed, D2 f (x) = ( ∂x x )( ∂x f (x)). For odd f, it is the ∂ ∂ ∂ 2 ( ∂x + 2k other way round: D f (x) = D(Df (x)) = ∂x (Df (x)) = ∂x x )(f (x)). In particular, Df (x) = f (x) +

D2 ψλ1 (x) =

∂ ∂ ∂ ∂ 2k 1 2k 1 ( + )ψ (x) = (( )2 + )ψ (x). ∂x ∂x x λ ∂x ∂x x λ

Hence ψλ1 (x) is also a solution of a second order diﬀerential equation. This equation is diﬀerent from that for ϕ, but not too diﬀerent. Comparing them we come to the important deﬁnition of the shift operator. We show the dek = ( ∂ )2 + ∂ 2k . pendence of L on k and set L ∂x

∂x x

Hankel transform via double Hecke algebra

9

k ◦ x = Lk+1 . Lemma 4.2. (a) x−1 ◦ L 1 2 1 k ψ = 4λ ψ . (b) L λ λ k ◦ x)(x−1 ψ 1 ) = 4λ2 (x−1 ψ 1 ). (c) (x−1 ◦ L λ λ Proof. The ﬁrst claim: ∂ ∂ 2 ∂ ∂ 2 ∂ ) ◦ x = x−1 (x( )2 + 2 ) = ( )2 + , ∂x ∂x ∂x ∂x x ∂x 2k ∂ ∂ 2k ∂ x−1 ◦ ◦ x = x−1 ◦ ◦ 2k = , ∂x x ∂x x ∂x k ◦ x = ( ∂ )2 + 2 ∂ + 2k ∂ = ( ∂ )2 + 2(k + 1) dx = Lk+1 . x−1 ◦ L ∂x x ∂x x ∂x ∂x x (20) x−1 ◦ (

k ψ 1 = D2 ψ 1 = 4λ2 ψ 1 due to (17). Claim (c) is a combination of (a) Then L λ λ λ and (b). Proposition 4.1. (Shift Formula) 2λ 1 1 ψλ (x, k) = ψ 0 (x, k + 1), i.e. x 1 + 2k λ

4λ2 1 dϕλ (x, k) = ϕλ (x, k + 1). x dx 1 + 2k (21)

Proof. Lemma 4.2 (c) implies that x−1 ψλ1 (x, k) = C(λ, k)ϕλ (x, k+1), because ϕλ (x, k + 1) is a unique even normalized solution of (2) for k + 1. Thanks −1 λ to (18) ψλ1 (x, k) = (2λ)−1 dϕ ϕλ (x, k) = C(λ, k)ϕλ (x, k + 1). dx (x, k). Thus x The constant C readily results from the expantion (3) of ϕλ (x, k). Explicitly: 0 = (Lk ϕλ − 4λ2 ϕλ )(0) ⇒ dϕλ 0 = (2k + 1)(x−1 )(0, k) − 4λ2 . dx

(22)

The shift formula can be of course checked directly without ψλ1 , a good exercise.

5 Master formula Let us deﬁne the nonsymmetric Hankel transform. We consider complexvalued C ∞ − functions f on R such that limx→∞ f (x)ecx = 0 for any c ∈ R and set +∞ 1 (Ff )(λ) = ψλ (x, k)f (x)|x|2k dx (23) Γ (k + 1/2) −∞ We assume that Re k > − 12 and always take ψ0 (x, k) = 1. Recall that the case λ = 0 is exceptional (Lemma 4.1 ): the dimension of the space of eigenfunctions is 2.

10

Ivan Cherednik and Yavor Markov

Let us compute the transforms of our main operators. Compare it with Lemma 2.1 . It is much more comfortable to deal with the operators of the ﬁrst order. The upper index denotes the variable. Lemma 5.1. F(Dx ) = −2λ;

(a)

(b)

F(2x) = Dλ ;

F(sx ) = sλ .

(c)

Proof. The ﬁrst formula is an immediate consequence of Proposition 3.1 (a) with g(x) = ψλ (x) : F(Df ) = Df, ψλ = −f, Dψλ = −2λf, ψλ = −2λF(f ). Claim (b) follows from the x ↔ λ symmetry. As to (c), use that ψλ (−x) = ψ−λ (x). Theorem 5.1. (Nonsymmetric Master Formula) ∞ 2 1 2 2 ψλ (x)ψµ (x)e−x |x|2k dx = Γ (k + )eλ +µ ψλ (µ), 2 −∞ ∞ 2 D2 1 2 ψλ (x) exp(− )(f (x)) e−x |x|2k dx = Γ (k + )eλ f (λ). 4 2 −∞

(24)

2

Proof. In the ﬁrst formula, the left-hand side equals Γ (k+1/2)F(e−x ψµ (x)). We set 2 2 ψµ− (x) = e−x ψµ (x), ψµ+ (x) = ex ψµ (x), 2

2

2

2

D− = e−x ◦ D ◦ ex , D+ = ex ◦ D ◦ e−x . The function ψµ± is an eigenfunction of D± with eigenvalue 2µ. The normalization ﬁxes it uniquely with the standard reservation about µ = 0. One gets: 2

D− = e−x (

2 ∂ ∂ k k − (s − 1))ex = + 2x − (s − 1) = D + 2x. ∂x x ∂x x

Correspondingly, D+ = D − 2x. Using Lemma 5.1 , x λ ) = F(Dx ) + F(2x) = −2λ + Dλ = D+ . F(D−

Therefore λ x x − (Fψµ− ) = F(D− )(Fψµ− ) = F(D− ψµ ) = 2µF(ψµ− ), D+

i.e. F(ψµ− ) is an eigenfunction of D+ with the eigenvalue 2µ, and F(ψµ− )(λ) = 2 C(µ)eµ ψµ+ (λ). Using the λ ↔ µ-symmetry, C(µ) = C(λ) = C and 2 1 C= e−x |x|2k dx = Γ (k + ). 2 R

Hankel transform via double Hecke algebra

11

Cf. the proof of the symmetric master formula. The second formula readily follows from the ﬁrst provided the existence 2 of the function exp(− D4 )(f (x)) and the corresponding integral. The latter function has to go to zero at x = ∞ faster than ecx for any c ∈ R. The symmetric master theorem is of course a particular case of (24). Indeed, we may replace ψλ (x) by 2ϕλ (x) = ψλ (x)+ψλ (−x) = ψλ (x)+ψ−λ (x) on the left-hand side. Then ψλ (µ) → ψλ (µ) + ψ−λ (µ) = ψλ (µ) + ψλ (−µ) = 2ϕλ (µ) 2

2

on the right-hand side. We use that the factor eλ +µ is even. Now we either repeat the same transfer for µ or simply symmetrize the integrand. It is more surprising that the nonsymmetric theorem can be deduced from the symmetric one. It is a special feature of the one-dimensional case. Generally speaking, there is no reason to expect such an implication. This may clarify why the nonsymmetric Hankel transform and ψ were of little importance in the classical theory of Bessel functions. They could be considered as a minor technical improvement of the symmetric theory. Now we have the opposite point of view. Let us deduce Theorem 5.1 from Theorem 2.1. We may assume that λ, µ = 0. Discarding the odd summands in the integrand, 2 2 ψλ ψµ e−x |x|2k dx = (ψλ0 + ψλ1 )(ψµ0 + ψµ1 )e−x |x|2k dx R R 2 0 0 1 1 −x2 = (ψλ ψµ + ψλ ψµ )e |x|2k dx = (ϕλ ϕµ + ψλ1 ψµ1 )e−x |x|2k dx. (25) R

R

The integral of ϕλ ϕµ is nothing else but (8). Let us use the shift formula to manage ψλ1 ψµ1 . See Proposition 4.1.

We get (ψλ1 ψµ1 )(x, k) =

4λµx2 (1+2k)2 (ϕλ ϕµ )(x, k

4λµ (1 + 2k)2 R 2 2 4λµ 3 = ϕλ (µ, k + 1)eλ +µ Γ (k + ) (1 + 2k)2 2 2 2 2λµ 1 = ϕλ (µ, k + 1)eλ +µ Γ (k + ) (1 + 2k) 2 2 2 1 = ψλ1 (µ, k)eλ +µ Γ (k + ). 2 This concludes the deduction. 2

(ψλ1 ψµ1 )(x, k)e−x |x|2k dx =

R

+ 1) and 2

(ϕλ ϕµ )(x, k + 1)e−x |x|2(k+1) dx

6 Double H double prime Let H H be the double degeneration of the double aﬃne Hecke algebra:

(26)

12

Ivan Cherednik and Yavor Markov

H H = ∂, x, s | sxs = −x,

s∂s = −∂,

[∂, x] = 1 + 2ks .

(27)

Its polynomial representation ρ : H H → End(P) in P = C[x] is as follows: ρ(x) = multiplication by x,

ρ(∂) = D,

ρ(s) = s,

where s is the reﬂection f → f ι , D is the Dunkl operator. The ﬁrst two of the deﬁning relations of H H are satisﬁed thanks to Lemma 3.1. As to the last, k ∂ k ∂ − (s − 1))x − x( − (s − 1)) ∂x x ∂x x ∂ k ∂ k =x + 1 − x (−s − 1) − x + x (s − 1) = 1 + 2ks. ∂x x ∂x x [D, x] = (

(28)

Theorem 6.1. cm,n, xm ∂ n s , where (a) Any nonzero ﬁnite linear combination H = m,n,

n, m ∈ Z+ , ∈ {0, 1}, acts as a nonzero operator in P. (b)Any element H ∈ H H can be uniquely expressed in the form H = m n x ∂ s . The representation ρ is faithful for any k. m,n,

Proof. Let H =

n,

fn ∂ n s =

N m=0

(gm ∂ m )(1 + s) +

M

(hm ∂ m )(1 − s) for

m=0

polynomials fm , gm , and hm . We may assume that and at least one of the leading coeﬃcients gN (x), hM (x) is nonzero. Then ρ(H) = L+ (1 + s) + L− (1 − s) ∂ N ∂ M ) +. . . and L− = hM (x)( ∂x ) +. . . for diﬀerential operators L+ = gN (x)( ∂x modulo diﬀerential operators of lower orders. Applying ρ(H) to even and odd functions, we get that ρ(H) = 0 implies that both L+ and L− have inﬁnite dimensional spaces of eigenfunctions. This is impossible. Claim (a) is veriﬁed. Concerning (b), any element H ∈ H H can be obviously expressed in the desired form. Such expression is unique and the representation ρ is faithful thanks to (a). The theorem is the key point of the representation theory of the double H. It is a variant of the so-called PBW theorem. There are not many algebras in mathematics and physics possessing this property. All have important applications. The double Hecke algebra is one of them. Next, we study the irreducibility of ρ. Lemma 6.1. The Dunkl operator D has only one eigenvalue in P, namely, λ = 0. If k = −1/2 − n for any n ∈ Z+ , then D has a unique (up to a constant) eigenfunction in P, the constant function 1. When k = −1/2 − n for n ∈ Z+ , the space of 0-eigenfunctions is C + Cx2n+1 .

Hankel transform via double Hecke algebra

13

Proof. Let p(x) ∈ P be an eigenfunction for D. Since D lowers the degree of any polynomial by 1, we have Dm+1 p = 0, where m = deg p. Therefore all eigenvalues of D are zero. Representing p as the sum p(x) = p0 (x) + p1 (x) ∂ 0 ∂ 1 1 of even p0 and odd p1 , Dp = ∂x p + ( ∂x p + 2k x p ) = 0. Both the even ∂ 0 p = 0 and and the odd parts of this expression have to be zero. Hence ∂x ∂ 1 2k 1 0 1 2l+1 al x , ∂x p + x p = 0. Therefore p =Const. Setting p (x) = (

2k 1 ∂ + )p (x) = al (2l + 1 + 2k)x2l = 0. ∂x x

If k = −1/2 − n for any n ∈ Z+ then al = 0 for any l, i.e p1 = 0. Otherwise k = −1/2 − n for a certain n ∈ Z+ and p1 (x) is proportional to x2n+1 . Theorem 6.2. (a) The representation ρ is irreducible if and only if k = −1/2 − n for any n ∈ Z+ . H (b) If k = −1/2 − n for n ∈ Z+ , then there exists a unique non-trivial H 2n+1 2n+1 )=x submodule of P, namely, (x P. Proof. Let {0} = V ⊂ P be a H H submodule of P. Let 0 = v ∈ V and m = deg v. Set P (m) = {p ∈ P | deg p ≤ m}. We have V (m) = V ∩P (m) = {0}. Then V (m) is D invariant. More exactly, D(V (m) ) ⊂ V (m−1) . Thus it contains an eigenfunction v0 of D. If k = −1/2 − n for n ∈ Z+ then Lemma 6.1 implies that v0 = 1. So 1 ∈ V and V ρ(xm )(1) = xm for any m ∈ N. This means that V = P and completes (a). If k = −1/2−n for n ∈ Z+ then Lemma 6.1 states that v0 = c1 +c2 x2n+1 ∈ V for constants c1 , c2 . If c1 = 0 then s(v0 ) + v0 = 2c1 ∈ V (the latter is sinvariant and V = P as above. If v0 = x2n+1 ∈ V then this argument gives that (x2n+1 ) = x2n+1 P ⊂ V. Moreover V cannot contain the polynomials of degree less than 2n + 1. Otherwise we can ﬁnd a 0-eigenvector of D in the space of such polynomials, which is impossible. Hence V = (x2n+1 ). The latter is invariant with respect to x and s. Its D-invariance readily follows from the formula D(xl ) = (l + (1 − (−1)l )k)xl−1 considered in the range l ≥ 2n + 1. We can reformulate the theorem as follows. The polynomial representation has a nontrivial (proper) H H -quotient if and only if k = −1/2−n for n ∈ Z+ . In the latter case, such quotient is unique, namely, V2n+1 = P/(x2n+1 ). Its dimension is 2n + 1. 0 Note that the subspace V2n+1 of V2n+1 generated by even polynomials ∂ is invariant with respect to the action of h = x ∂x + k + 1/2 and e = x2 , f = −L/4, satisfying the deﬁning relations of sl2 (C) (see Section 2). We get an irreducible representation of sl2 (C) of dimension n + 1.

14

Ivan Cherednik and Yavor Markov

7 Algebraization Let us use H H to formalize the previous considerations and to switch to the standard terminology of the representation theory. (a) Inner product. We call a representation V of H H pseudo-unitary if it possesses a non-degenerate C-bilinear form (u, w) such that (Hu, w) = (u, H v) for H ∈ H H for the anti-involution ∂ = −∂,

s = s,

x = x.

(29)

By anti-involution, we mean a C-automorphism satisfying (AB) = B A . We call such form -invariant. Formulas (29) are compatible with the deﬁning relations (27) of H H and therefore can be extended to the whole H H . This is straightforward. For instance, [x , ∂ ] = [x, −∂] = [∂, x] = 1 + 2ks = 1 + 2ks = [∂, x] . We add “pseudo” because the pairing, generally speaking, is not supposed to be positive and the functions can be complex-valued. The pairing f, g = R f (x)g(x)|x|2k dx gives an example, provided the existence of the integral. Taking real-valued functions, we make this inner product positive (no “pseudo”). Assuming that the functions are C ∞ , we need to examine the convergence at x = 0 and x = ∞. If Re (k) > −1/2 then it suﬃces to take regular f at x = 0. At inﬁnity, f (x)|x|k has to be of type 2 L1 (R). Polynomials times the Gaussian e−x are ﬁne. (b) Gaussians. A homomorphism γ : V → W for two H H -modules V, W is called a Gaussian if γH = τ (H)γ for the following automorphism τ of H H : τ (∂) = ∂ − 2x, τ (x) = x, τ (s) = s. (30) These formulas can be extended to an automorphism of H H . Indeed, τ (s)τ (∂)τ (s) = s(∂ + 2x)s = −∂ − 2x = −τ (∂), the same holds for x, and [τ (∂), τ (x)] = [d − 2x, x] = 1 + 2s = τ (1 + 2s). Note that 2 can be replaced by any constant α ∈ C in this deﬁnition. We get a family of automorphisms τα (∂) = ∂ − 2αx of H H . They lead to the following generalization of the master formula: 2 µ λ2 +µ2 1 ψλ (x)ψµ (x)e−αx |x|2k dx = ψλ ( )e α α−k Γ (k + ). α 2 R √ Here α > 0 to ensure the convergence. The substitution u = αx readily makes it equvalent to (24): use that ψλ (x) is a function of the xλ. One can 2 2 also follow the proof of the master formula employing e−αx Deαx = D+2αx.

Hankel transform via double Hecke algebra

15

If representations V, W are algebras of functions on the same set then γ can be assumed to be a function, to be more exact, the operator of multupli2 cation by a function. For instance, the multiplication by ex sends the poly2 nomial representation P to Pe±x . The latter is a H H -module too. Adding x2 all integral powers of e to P we make this multiplication an inner automorphism of the resulting algebra. However it is somewhat artiﬁcial. Alge2 braically, the resulting representation P[emx , m ∈ Z] is “too” reducible. An2 2 alytically, we mix together ex and e−x , functions with absolutely diﬀerent behaviour at inﬁnity. There are interesting examples of inner automorphisms τ, but they are ﬁnite-dimensional. (c) Hankel transform. Following Lemma 5.1, the operator Hankel transform is the following automorphism of H H : σ(∂) = −2x,

σ(s) = s,

σ(2x) = ∂.

(31)

These realations are obviously comapatible with the deﬁning relations of H H . Any homomorphism F : V → W of H H -modules inducing σ on H H can be 2 2 called a Hankel transform. The main example so far is F : Pe−x → Pe+x , where we identify x and λ in 23. Indeed, F ◦ D = −2x ◦ F, F ◦ 2x = D ◦ F, F ◦ s = s ◦ F upon this identiﬁcation. It is interesting to interpret the master formula from this viewpoint. It is 2 2 2 nothing else but the following identities for F = ex e∂ /4 ex : F∂ = −2xF,

Fs = sF,

F(2x) = ∂F.

(32)

This means that F is Hankel transform whenever it is well-deﬁned. Relations (32) can be deduced directly from the deﬁning relations. In the ﬁrst 2 2 place, check that [∂, x2 ] = 2x, [∂ 2 , x] = 2∂. Then get that [∂, ex ] = 2xex , 2 2 [e∂ /4 , x] = ∂ed /2 and use it as follows: 2

ex e∂

2

/4 x2

2

2

2

2

e 2x = ex e∂ 2

/4

2

2

2

2

(2x)ex = ex (∂ + 2x)e∂ ex = 2

2

2

(∂ − 2x + 2x)ex ed ex = ∂ex e∂ ex , 2

e−x e−∂

2

/4 −x2

e

2

2x = e−x e−∂ 2

2

2

/4

2

2

2

2

(2x)e−x = e−x (−∂ + 2x)e−∂ e−x =

2

2

2

2

(−∂ − 2x + 2x)ex ed )ex = −∂ex e∂ ex . The commutativity of F with s is obvious. Note the following braid identity which can be proved similarly: 2

F = ex e∂

2

/4 x2

e

2

2

= e∂ ex

/4 ∂ 2

e .

Actually we do need calculations from scratch because it suﬃces to use the nonsymmetric master formula Lemma 5.1 and the fact that P is a faithful representation. For instance, F(2x)F−1 and ∂ coincide in P. The previous consideration shows that the former is an element of H H . Hence they must coincide identically, i.e. in H H .

16

Ivan Cherednik and Yavor Markov

A good demonstration of the convenience of such an algebraization will be the case of negative half-integral k. Before switching to this case, let us conclude the “analytic” theory calculating the inverse Hankel transform.

8 Inverse transform and Plancherel formula Let Re k > −1/2. We use ψλ (x) from (16). +∞ 1 (Fre f )(λ) = ψλ (x)f (x)|x|2k dx, Γ (k + 1/2) −∞ +i∞ 1 ψx (−λ)g(λ)|λ|2k dλ. (Fim g)(x) = Γ (k + 1/2) −i∞

(33) (34)

The ﬁrst is nothing else but F from (23). We just show explicitely that the integration is real. Here we may consider C-valued functions f on R and g on C respectively such that f (x) = o(ecx ) at x = ∞ ∀c ∈ R and f ∈ g(λ) = o(eciλ ) at λ = i∞ ∀c ∈ R. Restricting ourselves with the polynomials times the Gaussian, 2

2

2

2

Fre : C[x]e−x → C[λ]eλ , Fim : C[λ]eλ → C[x]e−x . The ﬁrst map is an isomorphism. Let us discuss the latter. Let p(x) ∈ C[x]. Applying the master formula to pi (x) = p(ix), +∞ 2 2 1 ψλ (x)p(ix)e−x |x|2k dx = Fre (e−x p(ix)) Γ (k + 1/2) −∞ 2

= eλ exp((Dλ )2 /4)(p(iλ)). Since D ◦ I = iI ◦ D for I(f )(x) = f (ix), −(Du )2 (Dλ )2 p(iλ) = p(u) |u=iλ . 4 4 Now we replace λ → iλ, use that ψiλ (x) = ψλ (ix), and then integrate by substitution using z = ix. The resulting formula reads: +∞ 2 2 1 ψλ (ix)p(ix)e−x |x|2k dx = e−λ exp(−(Dλ )2 /4)(p(−λ)) Γ (k + 1/2) −∞

=

1 i Γ (k + 1/2)

(35) +i∞

−i∞ 2

2

ψλ (z)p(z)ez |z|2k dz. 2

Switching to λ, Fim (eλ p(λ)) = e−x exp(−(Dx )2 /4)(p(x)). We come to the inversion theorem.

(36)

Hankel transform via double Hecke algebra

17

2

Theorem 8.1. (Inversion Formula) Fim ◦ Fre = id in C[x]e−x , Fre ◦ 2 Fim = id in C[λ]eλ . 2

2

2

Proof. Fim ◦Fre (e−x p(x)) = e−x exp(−D2 /4) exp(D2 /4)(p(x)) = e−x p(x). The second formula is analogous. There is a simple “algebraic” proof based on the facts that the transform Fim ◦ Fre sends D → D, 2x → 2x, s → s. Thanks to the irreducibility of 2 C[x]e±x , we may apply the Schur lemma. However the spaces are inﬁnite dimensional, so a minor additional consideration is necessary. We will skip it because it practically coincides with that from the proof of the Plancherel formula. Provided Re k > − 12 , the inner products f, gre =

+∞

2k

f (x)g(x)|x| dx, −∞

f, gim

1 = i

+i∞

f (λ)g(−λ)|λ|2k dx

−i∞

(37) 2 2 are non-degenerate respectively in C[x]e−x and C[λ]eλ . It is obvious when R[x], R[λ] are considered instead of C[x], C[λ] and k ∈ R. Indeed, both forms become positive in this case. Concerning the second, use that g(−λ) = g(λ) for a real polynomial g. For complex-valued functions and k ∈ C, the claim requires proving. Let 2 2 us use the irreducibility of H H − modules C[x]e−x and C[x]ex , which is equivalent to the irreducibility of the polynomial representation P = C[x], which we already know for Re k > − 12 . Then the radical of the form . , . re 2 is a submodule of C[x]e−x . It does not coincide with the whole space, since √ 2 1 −x2 −x2 (38) e ,e re = e−2x |x|2k dx = ( 2)−2k−1 Γ (k + ). 2 R The same argument works for the imaginary integration. Theorem 8.2. (Plancherel Formula) 2 f, gre = f, gim for all f, g ∈ C[x]e−x , f = Fre (f ), g = Fre (g). 2

(39)

Proof. Setting P− = C[x]e−x , we need to check that . , . = . , . re coincides with f, g1 = f, gim for all f, g ∈ P− . In the ﬁrst place, Hf, g1 = f, H g1 for any H ∈ H H , i.e. this bilinear form is -invariant. Indeed, , gim = −2xf, gim Df, g1 = Df im = −f, Dg1 . = −f, −2(−x) g im = −f, Dg Similarly, 2xf, g1 = f, 2xg1 and s(f ), g1 = f, s(g)1 .

18

Ivan Cherednik and Yavor Markov 2

2

Setting . , . 2 = . , . − . , . 1 , we get e−x , e−x 2 = 0. Cf. 38. It is a

-invariant form as well. Let us demonstrate that it vanishes identically. 2 2 First, e−x , (D −2x)f = −(D +2x)e−x , f = 0, f = 0 for any f ∈ P− 2 due to the -invariance. So it is applicable to . , . 2 too. Second, Ce−x ∩ 2 2 2 (D − 2x)P− = ∅ because e−x , e−x = 0. Third, P− = Ce−x ⊕ (D − 2x)P− . 2 Really, the dimension of (D − 2x)Pn− for Pn = C[1, x, . . . , xn ]e−x is n + 1 since the kernel of the operator D −2x in P− is zero. However (D −2x)Pn− ⊂ P(n+1)− . Therefore (D − 2x)Pn− = P(n+1)− . Finally, 2

2

2

e−x , P− 2 = e−x , e−x + (D − 2x)P− 2 = 0 2

and e−x belongs to the radical of . , . 2 . Since the module P− is irreducible, the radical has to coincide with the whole P− . 2

Taking real k, the forms . , . re . , . im are positive on R[x]e−x and 2 R[λ]eλ . The Plancherel formula allows us to complete the function spaces extending the Fourier transforms Fre , Fim to the spaces of square integrable real-valued functions with respect to the “Bessel measure”: L2 (R, |x|2k dx) → L2 (iR, |λ|2k dλ) → L2 (R, |x|2k dx). The inversion and Plancherel formulas remain valid. Here we assume that k > − 12 . Let us discuss the case of negative halfintegers.

9 Finite-dimensional case Let k = −n − 12 for n ∈ Z+ . Then V2n+1 = P/(x2n+1 ) is an irreducible representation of H H . The elements of V2n+1 can be identiﬁed with polynomials of degree less than 2n + 1. Theorem 9.1. Finite-dimensional representations of H H exist only as k = −n − 1/2 for n ∈ Z+ . Given such k, the algebra H H has a unique ﬁnitedimensional irreducible representation up to isomorphisms, namely, V2n+1 . Proof. We will use that [h, x] = x, [h, ∂] = −∂ for h = (x∂ + ∂x)/2.

(40)

It readily follows from the deﬁning relations of H H . Actually (40) determines a super Lie algebra, which is osp(2, 1). One may use a general theory of this algebras. For our purpose, a reduction to sl2 is suﬃcient. ∂ Note that h is x ∂x + k + 1/2 in the polynomial representation. Since the latter is faithful, (40) is exactly the claim that x, D are homogeneous operators of degree ±1, which is obvious.

Hankel transform via double Hecke algebra

19

We will employ that e = x2 , f = −∂ 2 /4, and h satisfy the deﬁning relations of sl2 (C). Namely, [e, f ] = h because [∂ 2 , x2 ] = [∂ 2 , x]x + x[∂ 2 , x] = 2∂x + x(2∂), and the relations [h, e] = 2e, [h, f ] = −2f readily result from (40). Cf. Section 6. Let V be an irreducible ﬁnite-dimensional representation of H H . Then the 0 1 subspaces V , V of V formed respectively by s-invariant and s-anti-invariant vectors are preserved by h, e, and f. So they are sl2 (C)-modules. One gets ∂x = h + k + 1/2, x∂ = h − k − 1/2 in V 0 , and the other way round in V 1 . Let us check that k ∈ −1/2 − Z+ . All h-eigenvalues in V are integers thanks to the general theory of ﬁnite-dimensional representations of sl2 (C). We pick a nonzero h-eigenvector v ∈ V with the maximal possible eigenvalue m. Then m ∈ Z+ (the theory of sl2 ) and x(v) = 0 because the latter is an heigenvector with the eigenvalue m + 1. Hence ∂x(v) = 0, m + k + 1/2 = 0, and k = −1/2 − m. Let U 0 be a nonzero irreducible sl2 (C)−submodule of V 0 . The spectrum of h in U 0 is { −n, −n + 2, . . . , n − 2, n } for an integer n ≥ 0. Let vl = 0 be an h-eigenvector with the eigenvalue l. If e(v) = 0 then v = cvn for a constant c, and if f (v) = 0 then v = cv−n . Let us check that ∂x(vn ) = 0, x∂(v−n ) = 0, and ∂x(vl ) = 0 for l = n, x∂(vl ) = 0 for l = −n. Both operators, ∂x and x∂, obviously preserve U 0 : ∂x(vl ) = (l + k + 1/2)vl , x∂(vl ) = (l − k − 1/2)vl . Hence, ∂ 2 x2 (vl ) = ((∂x)2 + (1 − 2k)(∂x))(vl ) = (l + k + 1/2)(l − k + 3/2)vl . Setting l = n, we get that (n + k + 1/2)(n − k + 3/2) = 0 and k = −1/2 − n, because k < 0 and n − k + 3/2 > 0. Thus ∂x(vn ) = 0. The case of x∂ is analogous. The next claim is that x(vn ) = 0, ∂(v−n ) = 0. Indeed, x(v ) = 0 and ∂(v ) = 0 for v = x(vn ). Therefore 0 = [∂, x](v ) = (1 + 2ks)(v ) = (1 − 1 + n)v = nv . This means that either v = 0 or n = 0. In the latter case, v is proportional to v0 and therefore v = x(v0 ) = 0 as well. Similarly, ∂(v−n ) = 0. Now we use the formula

20

Ivan Cherednik and Yavor Markov

∂(x2 (vl )) = x(2 + x∂)(vl ) = (2 + l − k − 1/2)x(vl ) = (2 + l + n)x(vl ), and get that x(vl ) ∈ ∂(U 0 ) for any −n ≤ l ≤ n. Hence U = U 0 + ∂(U 0 ) is x-invariant. It is obviously ∂-invariant and s-invariant (⇐ ∂(V 0 ) ⊂ V 1 ). Also the sum is direct. Finally, U is a H H -module and has to coincide with V because the latter was assumed to be irreducible. The above formulas are suﬃcient to establish aH H - isomorphism U V2n+1 . Explicitly, the h-eigenvectors xi (v−n ) ∈ U will be identiﬁed with the monomials xi ∈ V2n+1 . Let us discuss the Hankel transform and related structures in the case of V2n+1 . We follow Section 7. (a) Form. To make “inner” we have to construct a non-degenerate bilinear pairing (u, v) on V2n+1 such that (Hu, v) = (u, H v). Here it is: ∀f, g ∈ V2n+1 set (f, g) = Res(f (x)g(x)x−2n−1 ), where Res( ai xi ) = a−1 . (41) The pairing is non-degenerate, because if f = axl + lower order terms, where a = 0 and 0 ≤ l ≤ 2n, then (f, x2n−l ) = a. We introduce a scalar product (f, g)0 = Res(f g) for polynomials in terms of x and x−1 . Denoting the conjugate of an operator A with respect to this pairing by A◦ , s◦ = −s, x◦ = x,

∂ ◦ ∂ = − , and x2n+1 D◦ x−2n−1 = −D. ∂x ∂x ◦

∂ ∂ = − ∂x , it follows from the The relation x◦ = x is obvious. Concerning ∂x −1 property Res(df /dx)=0 for any polynomial f (x, x ). The formula s◦ = −s results from Res(s(f (x)) = −Res(f (x)). Switching from ◦ to , we have

(sf, g) = Res(s(f )gx−2n−1 ) = −Res(f s(g)s(x−2n−1 )) = −(−1)2n+1 (f s(g)x−2n−1 ) = (f, s(g)). Finally, k k ∂ ∂ + (1 − s))◦ = − + (1 + s) , ∂x x ∂x x 2k k ∂ k ∂ − + (1 + s) = − + (−1 + s) = −D. =− ∂x x x ∂x x

D◦ = ( x−2k D◦ x2k

(42)

The ﬁrst equality on the second line holds because xk x2k = kx−2n−2 is an even function, and thus it commutes with the action of s. Finally (Df, g) = (Df, x2k g)0 = (f, x2k x−2k D◦ x2k (g))0 = −(f, x2k Dg)0 = −(f, Dg).

Hankel transform via double Hecke algebra

21

(b) Gaussian. The Gaussian does not exist in polynomials but of course ∞ 2 2 m can be introduced as a power series ex = m=0 (x ) /m! in the algebra of formal series C[[x]], a completion of the polynomial representation. 2 The by this series induces τ on H H . Its inverse is e−x = ∞ conjugation 2 m m=0 (−x ) /m!. The multiplication by the Gaussian does not preserve the space of polynomials but is well-deﬁned on V2n+1 because ∀f ∈ V2n+1 we have xm f = 0 for m ≥ 2n + 1. Finally, γ± =

2n

(±x2 )m /m!.

m=0

(c) Hankel transform. The operator D is nilpotent in V2n+1 because it lowers the degree of f ∈ V2n+1 by one. Therefore the operators exp(±D2 /4) ∈ C[[D]] are well-deﬁned in this representation as well as the Gaussians. It 2n suﬃces to take m=0 (±(D/2)2 )m /m!. Thus we may set 2

F = ex e

D2 4

2

ex

in V2n+1 .

(43)

Proposition 9.1. The map F is a Hankel transform on V2n+1 , i.e. F ◦ D = −F ◦ 2x, F ◦ 2x = D ◦ F, Fs = sF. These relations ﬁx it uniquely up to proportionality. Proof. We already know that F is a Hankel transform (the previous section).

is another one then the ratio FF

−1 commutes with x, D, and s because If F of the very deﬁnition. Since V2n+1 is irreducible (and ﬁnite dimensional) we

is proportional to F. get that F

10 Truncated Bessel functions Recall that D has only one eigenvalue in V2n+1 , namely, 0. Therefore we cannot deﬁne the ψλ as an eigenfunction of D in V2n+1 any longer. Instead, it will be introduced as the kernel of the Hankel transform. x λ Any linear operator A : V2n+1 → V2n+1 (the upper index indicates the variable) is a matrix. It means that A(f )(λ) = (f (x), α(x, λ)) = Res(f (x)α(x, λ)x−2n−1 ), where α(x, λ) =

2n l,m=0

cl,m xl λm =

2n

x2n−l A(xl ).

(44)

l=0

So here the kernel α(x, α) is uniquely deﬁned by A and vice versa. The truncated ψ-function is the kernel of F : F(f )(λ) = (f (x), ψλ (x)) = Res(f (x)ψλ (x)x−2n−1 ).

(45)

22

Ivan Cherednik and Yavor Markov

There is a somewhat diﬀerent approach. Let us use that the relations from Lemma 9.1 determine F uniquely up to proportionality. These relations are equivalent to the following properties of ψλ (x) : Dψλ (x) = 2λψλ (x) ψλ (x) = ψx (λ),

mod (x2n+1 , λ2n+1 ), ψλ (s(x)) = ψs(λ) (x).

Let us solve the ﬁrst equation. Setting ψλ (x) = l=2n,m=2n l=1,m=0

=

(46)

2n

l m l,m=0 cl,m x λ ,

1 cl,m (l + (1 − (−1)l )(−n − ))xl−1 λm 2

l=2n,m=2n−1

2cl,m xl λm+1

mod (x2n+1 , λ2n+1 ),

l=0,m=0

2 cl−1,m−1 for 2n ≥ l > 0, 2n ≥ m > 0, l + (1 − (−1)l )(−n − 1/2) where cl,0 = 0 = c2n,m for l > 0, m < 2n. (47) cl,m =

Using the x ↔ λ symmetry, we conclude that cl,0 = 0 = c0,l for nonzero l and cl,m = 0 for l = m. Thus ψλ (x) = gn (λx) for gn =

2n

cl tl , cl = cl,l ,

l=0

where the coeﬃcients are given by (47). fn (t) + (1/2)dfn /dt for the truncated Bessel function Finally, ngn (t) = 2m fn (t) = which is an even solution of the truncated Bessel m=0 c2m t equation (cf. Section 1): d2 f 1 df (t) + 2k (t) − 4f (t) = 0 dt2 t dt

mod (t2n ), k = −n − 1/2.

(48)

This equation is suﬃcient to determine the coeﬃcients of fn uniquely for any constant term c0 = c0,0 . They are given by the same formula (3) till c2n up to proportionality. This can be checked directly using explicit formulas which will be discussed next. Still c0 remains arbitrary. Recall that ψ was initially introduced as the kernel of F. So it comes with its own normalization. Let us calculate its c0 . One gets: 2

2

2

2

2

2

F(e−x ) = eλ exp(D2 /4)(eλ e−λ ) = eλ exp(D2 /4)(1) = eλ , so F(1 −

λ2 λ2n x2 x2n x4 + · · · + (−1)n )=1+ + ··· . + 1! 2! n! 1! n!

(49)

Hankel transform via double Hecke algebra

23

Here the transform of 1 is proportional to λ2n since the latter has to be λ . an eigenfunction of λ, i.e. a solution of the equation λF(1) = 0 in V2n+1 l λ l 2n−l Similarly, F(x ) = (D /2) F(1) is proportional to λ for 0 ≤ l ≤ 2n. Thus (49) leads to the relations F(x2m ) = (−1)m

m! λ2n−2m . (n − m)!

(50)

For instance, F(x2n ) = (−1)n n!. This is exactly the coeﬃcient c0 above. We obtain that the normalization serving the truncated Hankel transform is ψλ (0) = −n!, c0 = gn (0) = fn (0) = −n!. (51) Formula (50) also results in F(x2m+1 ) = F(x(x2m )) = (D/2)F(x2m ) = (−1)m

m! λ2n−2m−1 . (n − m − 1)! (52)

Substituting, ψλ (x) =

n m=0

for fn (t) =

x2n−2m F(x2m ) +

n−1

1 x2n−2m−1 F(x2m+1 ) = fn (xλ) + fn (xλ) 2 m=0

n n (−1)m m! 2n−2m (−1)n−m (n − m)! 2m = t t . (n − m)! m! m=0 m=0

(53)

It is exactly the solution of (48) with the truncated normalization fn (0) = (−1)n n!. Truncated inversion. Concluding the consideration of the case k = −n − 12 for n ∈ Z+ , let us discuss the inversion. We have the following transformations and scalar products: F+ : C[x]/(x2n+1 ) → C[λ]/(λ2n+1 ),

F+ (f ) = Res(f (x)ψλ (x)x−2n−1 ),

F− : C[λ]/(λ2n+1 ) → C[x]/(x2n+1 ),

F− (f ) = Res(f (λ)ψx (−λ)λ−2n−1 ).

f, g+ = Res(f (x)g(x)x−2n−1 ),

f, g ∈ C[x]/(x2n+1 );

f, g− = Res(f (−λ)g(λ)λ−2n−1 ),

f, g ∈ C[λ]/(λ2n+1 ).

(54)

Here F+ (f ) = F(f ) = f in the notation above. The transform F− (f ) coincides with Fλ+ (f ) for even f (λ) and with −Fλ+ (f ) for odd f (λ). We can follow the “analytic” case and check that F− ◦ F+ commutes with D, x, s. Hence it is the multiplication by a constant thanks to the irreducibility of V2n+1 . The constant is F− ◦ F+ (1) and can be readily calculated. It is equally simple to calculate all F− ◦ F+ (xl ) using (50) and (52). For instance,

24

Ivan Cherednik and Yavor Markov

(−1)m m! 2n−2m λ = (n − m)! (−1)m m! (−1)n−m (n − m)! 2m = x = (−1)n x2m . (n − m)! m!

F− ◦ F+ (x2m ) = F−

Thus the truncated inversion reads: F− ◦ F+ = (−1)n id = F+ ◦ F− . Concerning the Plancherel formula, we may use the proportionality of the forms f, g+ and f, g− for f, g ∈ V2n+1 and their transforms f = F(f ), g = F(g). It results from the irreducibility of V2n+1 . A direct calculation is simple as well. Let f, f + = f, f =

2n

al a2n−l for f =

l=0 2n

g, g− =

2n l=0

(−1)l bl b2n−l for g = F(f ) =

l=0

al xl , 2n

bl λ l .

(55)

l=0

It is easy to check that bl b2n−l = (−1)l+n al a2n−1 . Indeed, using (50) and (52): b2m b2n−2m = (−1)m a2m

m! (n − m)! (−1)n−m a2n−2m (n − m)! m!

= (−1)n a2m a2n−2m , b2m+1 b2n−2m−1 = = (−1)m a2m+1

m! (n − m − 1)! (−1)n−m−1 a2n−2m−1 (n − m − 1)! m!

= (−1)n−1 a2m+1 a2n−2m−1 . We get the truncated Plancherel formula: f, g− = (−1)n f, g+ . The above consideration proves the coincidence for f = g, i.e. for the corresponding quadratic forms. It is of course suﬃcient.

References [C1]

Cherednik, I.: Diﬀerence Macdonald-Mehta conjectures. IMRN 10, 449–467 (1997).

Hankel transform via double Hecke algebra [C2] [D] [J] [L] [O]

25

Cherednik, I.: One-dimensional double Hecke algebras and Gaussians, CIME (2000). Dunkl, C.F.: Diﬀerential-diﬀerence operators associated to reﬂection groups, Trans. AMS. 311, 167–183 (1989). Jeu, M.F.E. de: The Dunkl transform, Invent. Math. 113, 147–162 (1993). Luke, J.: Integrals of Bessel functions, McGraw-Hill Book Company, New York-Toronto-London (1962). Opdam, E.M.: Dunkl operators, Bessel functions and the discriminant of a ﬁnite Coxeter group, Comp. Math. 85, 333–373 (1993).

Aﬃne-like Hecke algebras and p-adic representation theory Roger Howe (Lecture Notes by Cathy Kriloﬀ) Department of Mathematics Yale University New Haven, CT 06520-8283 [email protected]

These lectures begin with a discussion of the structure of p-adic groups (concentrating on GLn and Sp2n ) and their associated aﬃne Hecke algebras. The Bruhat and Iwahori-Bruhat decompositions are presented from a geometric perspective. Fundamental results and techniques from the representation theory of p-adic groups, stated without proof, are used to show the application of Hecke algebras in describing certain representations of the p-adic groups. The later part of the notes indicates why it is reasonable to believe that representations of Hecke algebras will in fact account for all representations of p-adic groups.

1 Introduction Let G be a locally compact totally disconnected group and K be any compact open subgroup. A representation (ρ, V ), consisting of a complex vector space, V and a homomorphism ρ : G → GL(V ), is smooth if the representation space V of G is a union of its K-ﬁxed vectors, written as V = V K . The K

subgroups K form a neighborhood basis of the identity element of G and as K gets smaller, V K gets bigger. If dim V K is ﬁnite for all K, then the representation with space V is admissible. The goal is to understand unitary irreducible smooth admissible representations of G using K. More speciﬁcally, denote by Cc∞ (G) the locally constant complex-valued functions on G with compact support. This forms an algebra under convolution and its representations are equivalent to the smooth representations of G. The Hecke algebra of G with respect to K, denoted H(K\G/K) or H(G//K), is the subalgebra of Cc∞ (G) consisting of K bi-invariant functions on G: H(G//K) = {f ∈ Cc∞ (G) : f (kgk ) = f (g) for all g ∈ G, k, k ∈ K}. This could also be written as Cc∞ (K\G/K) =

ag χKgK : ag ∈ C ,

I. Cherednik et al.: LNM 1804, M.W. Baldoni and D. Barbasch (Eds.), pp. 27–69, 2003. c Springer-Verlag Berlin Heidelberg 2003

28

Roger Howe (Lecture Notes by Cathy Kriloﬀ)

where χKgK is the characteristic function of the double coset KgK. Studying irreducible representations of H(G//K) allows us to understand irreducible representations of G with a K-ﬁxed vector. To explain this further, let us ﬁrst explore the relationship between G and K. Assume G is unimodular (i.e. left and right Haar measure are equal). Normalize Haar measure so that µG (K) = dg = 1. Then the characteristic K

function of K, χK , is idempotent in Cc∞ (G) and H(G//K) = χK ∗ Cc∞ (G) ∗ χK . Given this structure of H(G//K), we consider the following more general context. Let A be an associative algebra over C and let e ∈ A be idempotent so that eAe is a subalgebra of A. Then it is possible to connect representations of A and eAe. Let M(A) and M(eAe) be the categories of A-modules and eAe-modules. There exist restriction and induction functors r : M(A) → M(eAe) Y → eY

and

i : M(eAe) → M(A) Z → A ⊗ Z. eAe

These have the following properties: 1. r(i(Z)) = Z, 2. r is exact. Since r may annihilate some A-modules, it is not possible to recapture all representations of A from representations of eAe. However, if we let M(A, e) = {Y ∈ M(A) : Y = AeY } be the category of A-modules generated by e-ﬁxed vectors, and denote irreˆ then we have the following property. ducible A-modules by A, is bijective. 3. r : Aˆ ∩ M(A, e) → eAe Still there is the problem that M(A, e) may not be closed under taking submodules. However, the following are equivalent: 4. i(M(eAe)) = M(A, e), 5. M(A, e) is closed under taking submodules, 6. M(eAe) M(A, e). Thus, according to statement (iii) above, it is possible to understand at least those irreducible representations of G that have K-ﬁxed vectors by studying H(G//K). Proposition 1.1. H(G//K) has a basis consisting of fg = χKgK for g ∈ K\G/K.

Aﬃne-like Hecke algebras and p-adic representation theory

29

Proof. By deﬁnition, H(G//K) is generated by such functions. If a1 χKg1 K + · · · + a χKg K = 0, then evaluating at a representative of each double coset shows that a1 = · · · = a = 0 so such functions are also independent. Suppose that G is unimodular. Notice that if we write −1 KgK = ∪m Kg ∩ K) i=1 ki gK for ki ∈ K/(g n = ∪j=1 Kgkj for kj ∈ K/(gKg −1 ∩ K),

then in fact m = n = µ(KgK) since right and left Haar measure are equal. z Proposition 1.2. If fx ∗ fy = axy fz then the coeﬃcients azxy ∈ Z and z

thus µ(KxK)µ(KyK) =

azxy µ(KzK).

z

Proof. Writing KxK = ∪i ki xK yields χki xK = δki x ∗ χK . fx = χKxK = i

i

If f is left K-invariant, then δki x ∗ χK ∗ f = δki x ∗ (χK ∗ f ) = δki x ∗ f. fx ∗ f = i

i

i

Since fy is left K-invariant and can be written fy = fx ∗ fy =

i,j

δki x ∗ δk˜j y ∗ χK =

j

δk˜j y ∗ χK , this yields

azxy fz .

z

The ﬁrst sum can be written as an integral combination of characteristic functions of right K cosets. Partitioning the right cosets into double cosets yields a sum of characteristic functions of double cosets with integral coeﬃcients. The second statement follows directly from the ﬁrst. Corollary 1.1. If µ(KxK)µ(KyK) = µ(KxyK) then fx ∗ fy = fxy .

2 Structure of p-adic GL(V ) and Sp(V ) After some preliminaries, we indicate how diﬀerently double cosets can behave in a non-unimodular group. See Example 2.1.

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2.1 Preliminaries Let k be a p-adic ﬁeld, with ring of integers O, and maximal ideal P. Let π be ¯ a generator of P. Let k¯ = O/P be the residue class ﬁeld of k. Let q = #(k) ¯ Let p be the residual characteristic of k, that be the number of elements of k. ¯ Then q = pa , where a is the dimension of k¯ over is, the characteristic of k. the prime ﬁeld Fp of p elements. ¯ It is known that, for each Let k¯× denote the multiplicative group of k. element x ¯ of k¯× , there is a unique (q − 1)-th root of unity x in O ⊂ k, such that the image of x in k¯ is x ¯. Hence, by abuse of notation, we will henceforth understand by k¯× the group of (q − 1)-th roots of unity of k, or ¯ whichever makes sense in context. the multiplicative group of k, Let O× be the group of units in O. Note that O× = O \ P (set-theoretic diﬀerence). It is well-known that we can write k × = πO×

and

O× = k¯× (1 + P).

Here π is the cyclic group generated by π, k¯× is as above, and 1 + P = {1 + x : x ∈ P}. Example 2.1. Double cosets in a non-unimodular group. Let ab × G= : a ∈ k ,b ∈ k , 01 ab K= : a ∈ O× , b ∈ O , and 01 π0 z= . 01 In this case, a single arbitrarily many right cosets and left coset can equal a b −1 a πb vice versa. Since z z = , we know zK ⊆ Kz. Thus 01 0 1 1 x for l ≥ 0, Kz l K = Kz l = z l K, and 01 1x for l ≤ 0, Kz l K = z l K = , Kz l 01 where in both cases x ∈ O/π l O. Let V be a vector space of dimension n over k. By a lattice in V , we mean a compact, open O-module. Let Λ be a lattice in V . Then πΛ is also a lattice ¯ Choose a basis E ¯ = {¯ and Λ¯ = Λ/πΛ is a vector space over k. e1 , . . . , e¯m } for ¯ Lift each element e¯j of Λ¯ to an element ej of Λ. Λ. Let x be any element of Λ. If x ¯ is the image of x in Λ, then we may write

Aﬃne-like Hecke algebras and p-adic representation theory

x ¯=

m

β¯j e¯j ,

31

(1)

j=1

¯ If we lift the coeﬃcients β¯j to elements of O, where the β¯j are coeﬃcients in k. m then equation (1) says that x− βj ej = x1 belongs to πΛ. Replacing x with j=1

π −1 x1 in equation (1), and letting (β1 )j be the corresponding coeﬃcients, we m see that x − (βj + π(β1 )j )ej belongs to π 2 Λ. Continuing in this fashion, we j=1

see that the set E = {ej } spans Λ as an O-module, and in particular, spans V . It is easy to see that E must also be a linearly independent set, so it is a basis for V . Thus we may make the following statement. Proposition 2.1. Let Λ ⊂ V be a lattice in the vector space V over k. Let ¯ = {¯ ej } ⊂ Λ¯ = Λ/πΛ be the set of images of the ej E = {ej } ⊂ Λ, and let E ¯ If E ¯ is a basis for Λ, ¯ then E is an O-basis for Λ and a k-basis for V . in Λ. 2.2 Bruhat decomposition of GL(V ) We begin by stating some deﬁnitions that are needed for the two formulations of the Bruhat decomposition of G = GL(V ) discussed below. Let V be a vector space of dimension n over k. Deﬁnition 2.1. A set F = {U1 ⊂ U2 ⊂ · · · ⊂ Uk }, of nested subspaces Ui of V , is called a ﬂag in V . If dim Uj = j for 1 ≤ j ≤ k = n, then F is called complete. Deﬁnition 2.2. A line decomposition of V is a collection of lines ( = onedimensional subspaces) Lj of V , such that V is a direct sum: V = ⊕j Lj . Given a set, C = {Uα }, of subspaces of V , and a line decomposition V = ⊕Lj , the line decomposition is compatible with C if Uα = ⊕(Lj ∩ Uα ) for all α. A parabolic subgroup of GL(V ) can be viewed as the stabilizer of all subspaces in a ﬂag F, written as PF . In particular, a Borel subgroup B = PFo is the stabilizer of a complete ﬂag. Even more particularly, if we choose a basis E = {e1 , . . . , en } and denote by Vk the space spanned by the ﬁrst k elements of E, then the stabilizer of the complete ﬂag Fo = {V1 ⊂ V2 ⊂ · · · ⊂ Vn−1 ⊂ Vn } is the subgroup of upper triangular matrices with respect to the basis E. The notion of a line decomposition compatible with a ﬂag F leads easily to an analogous notion of a compatible basis for V . The previous statement says that if a compatible basis is chosen and the matrices in the stabilizer of a complete ﬂag are written in terms of this basis, then they are upper triangular. If Fo is a complete ﬂag with compatible basis F , set W = StabGL(V ) F . This is the Weyl group, which for GL(V ) is the symmetric group on n elements.

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Theorem 2.1 (Bruhat Decomposition). For any Borel subgroup, B, GL(V ) = BWB. This theorem has the following reﬁnements. Let A be the subgroup of GL(V ) for which the basis E is an eigenbasis, i.e., the stabilizer of all individual lines in the line decomposition corresponding to E. Let U be the unipotent radical of B (the set of elements of B which act trivially on the quotients Uj+1 /Uj ).

Theorem 2.2. GL(V ) = U W AU = U AW U = U w AW U w , for any w, w ∈ W , where U w = wU w−1 . U where the aﬃne Weyl group, W = This could also be written as G = U W W A is the semidirect product of W and A, or the stabilizer of the overall line decomposition of V . The Bruhat Decomposition also has the following equivalent more geometric reformulation. Theorem 2.3. G = BWB if and only if for any two ﬂags, F1 and F2 , there exists a line decomposition of V compatible with both F1 and F2 . Proof. First suppose that the geometric version holds. Fix an element g ∈ GL(V ) and a ﬂag F1 with B = StabGL(V ) F1 . Choose a compatible basis F1 for F1 and let W = StabGL(V ) F1 . Set g(F1 ) = F2 where F2 has compatible basis F2 = g(F1 ). Choose a basis E compatible with F1 and F2 and choose b1 ∈ B such that b1 (F1 ) = E. Since E is compatible with F1 and F2 and since −1 −1 b−1 1 (E) = F1 , F1 is a basis compatible with both b1 (F1 ) = F1 and b1 (F2 ) = −1 b1 g(F1 ) = F3 . This means we can ﬁnd w ∈ W so that w(F1 ) = F3 , i.e., so that the kth subspace in the ﬂag F3 is the span of the ﬁrst k elements in the reordered basis w(F1 ). Then w−1 b−1 1 g = b2 ∈ StabGL(V ) F1 = B, so g ∈ BWB. The idea is illustrated by the following commutative diagram. g

F1 −−−−→   b2

F2   −1 b1

w

F1 −−−−→ F3 Now suppose we are given any two ﬂags and expand them if necessary to complete ﬂags, F1 and F2 . As before, let B = StabGL(V ) F1 and let W = StabGL(V ) F1 where F1 is a compatible basis for F1 . Find g ∈ GL(V ) such that g(F1 ) = F2 and write g = b1 wb2 . We claim b1 (F1 ) = E is a basis compatible with both F1 and F2 . Since b1 ∈ StabGL(V ) F1 , E is compatible with F1 . Now consider b1 wb−1 1 (E). On the one hand, this equals b1 w(F1 ), which is some reordering of the vectors in E. But then that equals gb−1 2 (F1 ), −1 and since b−1 2 ∈ StabGL(V ) F1 , b2 (F1 ) is compatible with F1 and gets sent by g to a basis compatible with F2 . Thus, to establish the Bruhat Decomposition, we will prove the following result.

Aﬃne-like Hecke algebras and p-adic representation theory

33

Theorem 2.4 (Bruhat Decomposition, geometric version). Given any two ﬂags, F1 and F2 , there exists a line decomposition of V compatible with both F1 and F2 . Proof. Proceed by induction on the dimension of V , with the case of dimension one being trivial. Given any two ﬂags, expand them if necessary to complete ﬂags, F1 = {U1 ⊂ U2 ⊂ · · · ⊂ Un } and F2 = {W1 ⊂ W2 ⊂ · · · ⊂ Wn }, and let k1 be the smallest index such that U1 is contained in Wk1 . Let the ﬁrst line in the line decomposition be L1 = U1 . Let Z be any complement of Wk1 in V and set V1 = Wk1 −1 + Z. V1 is of dimension n−1 and since U1 +V1 = V , then Uj +V1 = V as well, for any j ≥ 1. This forces dim Uj ∩ V1 = dim Uj + dim V1 − dim (Uj + V1 ) = j − 1. Thus for 2 ≤ j ≤ n the Uj ∩ V1 form a complete ﬂag in V1 . Next, note that 1 ≤ j ≤ k1 − 1 Wj Wj ∩ V1 = Wk1 −1 + (Wj ∩ Z) k1 ≤ j ≤ n and therefore

j 1 ≤ j ≤ k1 − 1 dim (Wj ∩ V1 ) = j − 1 k1 ≤ j ≤ n

where the result is the same for both j = k1 − 1 and j = k. Thus the Wj ∩ V1 yield a complete ﬂag in V1 as well. By induction, there exists a line decompon sition V1 = Lj compatible with these two complete ﬂags. Using dimension j=2

arguments, it is possible to show that

n

Lj is a line decomposition of V com-

j=1

patible with the two original ﬂags. If we set σ(1) = k1 , then induction yields a permutation σ corresponding to the line decomposition. (For example, σ(2) would be the smallest index such that U2 ∩ V1 is contained in Wσ(2) ∩ V1 .) k Lσ−1 (j) . This shows how to reorder the lines so that Wk = j=1

2.3 Iwahori-Bruhat decomposition of GL(V ) Lattice Flags For p-adic groups, there is a lattice analog of ﬂags and an analog involving lattices of the Bruhat decomposition. Deﬁnition 2.3. A set L of lattices is called a lattice ﬂag if and only if (a) it is nested, i.e., totally ordered by inclusion, and (b) it is invariant under multiplication by k × .

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Roger Howe (Lecture Notes by Cathy Kriloﬀ)

Let L be a lattice ﬂag, and let Λo be a lattice in L. Since L is invariant under multiplication by scalars, if x = π a u is an element of k, where u ∈ O× is a unit, then xΛo = π a Λo , since uΛo = Λo . Thus, to satisfy (a), it is enough to know, for any lattice Λo ∈ L, that π ±1 Λo is also contained in L. Let Λ be any other element of L. For suﬃciently large positive exponents a, we will have π a Λ ⊂ Λo . Choose such an a to be as small as possible. By minimality we then have π a Λ ⊂ πΛo . Since L is totally ordered by inclusion, this implies that π a Λ ⊃ πΛo . Reduction modulo π attaches to π a Λ a subspace UΛ ⊂ Λo = Λo /πΛo . It is clear that π a Λ can be recovered from UΛ , as the unique lattice containing πΛo , and reducing modulo πΛo to UΛ . From π a Λ , all multiples of Λ can be recovered. Let Λ be a third lattice in L. Repeating the reasoning of the previous paragraph, we ﬁnd an exponent b such πΛo ⊂ π b Λ ⊂ Λo , and a subspace UΛ ⊂ Λo such that π b Λ /πΛo = UΛ . Appealing to the ordering with respect to inclusion of L, we see that the two subspaces UΛ and UΛ must also be related by inclusion, that is, one must contain the other. If we extend this analysis to all lattices of L, we see that Λ gives rise to a collection {UΛ˜ : Λ˜ ∈ L} of subspaces of Λo . Since an element of L can be recovered up to multiples from its associated subspace of Λo , the collection {UΛ˜ : Λ˜ ∈ L} determines L. The ordering with respect to inclusion of L translates to the same relation between the UΛ˜. Hence the UΛ˜ form a ﬂag in Λo . Thus, L determines and is determined by a ﬂag in Λo . Conversely, given a lattice Λo and a ﬂag {Uα } in Λo = Λo /πΛo , we can form the lattices Λα such that Λα ⊃ πΛo , and Λα /πΛo = Uα . Then taking all multiples π m Λα of these lattices, it is not hard to see that we obtain a lattice ﬂag. Thus, we conclude that the collection of all lattice ﬂags containing a given lattice Λo is in bijection with the collection of all ﬂags in the residue class vector space Λo . There is an obvious notion of maximal lattice ﬂag. It is clear that in the above correspondence between lattice ﬂags containing Λo and ﬂags in Λo , maximal lattice ﬂags correspond to maximal ﬂags in Λo . One consequence is that every lattice ﬂag can be extended to a maximal lattice ﬂag. Also, a lattice ﬂag is maximal if and only if the quotients, Λ /Λ , of consecutive elements Λ ⊂ Λ of the ﬂag are one-dimensional over the residue class ﬁeld k. Stabilizers of Lattices Let GL(V ) be the group of linear automorphisms of the vector space V . For a lattice Λ ⊂ V , let K(Λ) (= K when Λ is understood) be the subgroup of GL(V ) consisting of automorphisms of Λ. That is K(Λ) = {g ∈ GL(V ) : g(Λ) = Λ}. Proposition 2.2. There is a unique conjugacy class of maximal compact subgroups of GL(V ), consisting of the stabilizers K(Λ) of lattices Λ. In other words, every maximal compact subgroup of GL(V ) stabilizes a lattice, and all stabilizers of lattices are conjugate to each other in GL(V ).

Aﬃne-like Hecke algebras and p-adic representation theory

35

Proof. For any ﬁxed lattice Λ, the stabilizer K(Λ) is a compact and open subgroup of GL(V ). If Λ is another lattice, then we can ﬁnd g ∈ GL(V ) such that g(Λ) = Λ . This can be seen, for example, by choosing bases for Λ and for Λ , and letting g be the mapping which takes one basis to the other. Then K(Λ ) = g(K(Λ))g −1 , so K(Λ) and K(Λ ) are conjugate. Let H be any compact subgroup of GL(V ). Since K(Λ) is open, the intersection H ∩ K(Λ) has ﬁnite index in H. This means that the lattices {h(Λ) : h ∈ H} form a ﬁnite set. Hence the sum of such lattices will again be ˜ a lattice Λ˜ in V , and will evidently be stabilized by H. Hence, H ⊂ K(Λ). Consider the structure of K(Λ). Evidently, K(Λ) stabilizes all the multiples π a Λ of Λ. In particular, any h ∈ K(Λ) stabilizes πΛ, and therefore descends to deﬁne a mapping of the quotient Λ = Λ/πΛ. This gives us a homomorphism from K(Λ) to GL(Λ). By choosing a basis for Λ, we can see that this mapping is surjective, so that we have an exact sequence 1 → K1 (Λ) → K(Λ) → GL(Λ) → 1. An element y in K(Λ) = K is in K1 if and only if y(x) = x for all x in Λ. This means that y(x) − x belongs to πΛ. In other words, (1 − y)(Λ) ⊂ πΛ, or z = π −1 (y − 1) preserves Λ. Let End(Λ) denote the ring of linear maps of V which preserve Λ. It is a compact open subring of End(V ) - a lattice in End(V ). We have shown that K1 (Λ) = 1 + πEnd(Λ). We deﬁne the level m principal congruence subgroup of K(Λ) by Km (Λ) = 1 + π m End(Λ). This is a normal, open subgroup of K(Λ). As m → ∞, the groups Km run through a neighborhood basis of the identity in K. Also, the map y → 1 + y deﬁnes the ﬁrst of the following isomorphisms: Km (Λ)/Km+1 (Λ) π m End(Λ)/π m+1 End(Λ) End(Λ)/πEnd(Λ) End(Λ). From this, we see that the quotient groups Km /Km+1 are vector spaces over k, and in particular are abelian p-groups. As a by product of the description given above, we see that K(Λ) acts transitively on Λ − πΛ. Therefore, it will act transitively on π c Λ − π c+1 Λ for any integer c. This implies the following fact: Lemma 2.1. A lattice stabilized by K(Λ) has the form π c Λ for some integer c.

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Roger Howe (Lecture Notes by Cathy Kriloﬀ)

Recall now some generalities about pro-ﬁnite groups. Let H be a proﬁnite group, i.e., a compact, totally disconnected group. Since H is totally disconnected, every point has a neighborhood basis of open and closed sets. Thus, H has a neighborhood basis of the identity consisting of open and closed subsets Xα . For a given Xα , the stabilizer Hα of Xα under left multiplication will be open and closed, and contained in Xα itself. Since it is open, it is of ﬁnite index in H, whence it has only ﬁnitely many conjugates in H. Thus, the intersection of all these conjugate subgroups will again be open, and will be an open, normal subgroup of H, and again clearly contained in Xα . Thus, H allows a neighborhood basis of the identity element, consisting of open, normal subgroups. Call them Nα . This exhibits H as the inverse or projective limit of its ﬁnite quotients H/Nα : H = lim H/Nα . ←

Hence the term pro-ﬁnite. We can think of the order of H as being a product 2m2 3m3 5m5 · · · , where the exponents are either whole numbers or +∞. As Nα shrinks, an exponent in the order of H/Nα gets larger and larger, and either remains bounded, stabilizing at mp , or increases without bound, in which case mp = ∞. There is a Sylow theory for pro-ﬁnite groups. A Sylow subgroup Hp of H is a subgroup which projects to a Sylow subgroup in any ﬁnite quotient H/Nα . Proposition 2.3. 1. Sylow subgroups Hp of a pro-ﬁnite group H exist. 2. All Sylow p-subgroups of H are conjugate. 3. The order of Hp is pmp . We refer to a group in which only one exponent mp is positive as a pro-p group. Evidently, the Sylow subgroup Hp of H is a maximal pro-p subgroup of H. Return now to the stabilizer K(Λ) of a lattice. From our description above of the congruence subgroup K1 (Λ), we can see that it is a pro-p subgroup. It follows that all the exponents of K(Λ) are ﬁnite except for mp . Also the Sylow p-subgroups of K(Λ) will be the inverse images in K(Λ) of the Sylow psubgroups of GL(Λ). It is well known that these are the groups of “unipotent upper triangular matrices”. Precisely, if F is a maximal ﬂag in Λ, then the subgroup of GL(Λ) which stabilizes F and acts trivially on the quotients of successive elements of F is a Sylow p-subgroup. We will denote by J = J(Λ, F) the stabilizer in K(Λ) of F, and by J1 = J1 (Λ, F) the Sylow psubgroup of J. Then in fact, J is the normalizer in K of J1 , and J/J1 is × abelian and isomorphic to (k )n . As we have discussed above, there is naturally associated to the ﬂag F a maximal lattice ﬂag L. The stabilizer J(Λ, F) is equal to the stabilizer J(L). The subgroup J1 (Λ, F) = J1 (L) consists of elements which act trivially on

Aﬃne-like Hecke algebras and p-adic representation theory

37

the quotients of consecutive elements of L. We call J(L) an Iwahori subgroup of GL(V ). More generally, if M is any lattice ﬂag in V , then the stabilizer of (all the lattices of) M is called a parahoric subgroup of GL(V ). From our discussion of the relation between lattice ﬂags containing Λ and ﬂags in Λ, we know that if Λ belongs to M, then K1 (Λ) ⊂ J(M), and the quotient J(M)/K1 (Λ) is identiﬁed to the stabilizer of a ﬂag in Λ - a parabolic subgroup of GL(Λ). Combining the discussions of K(Λ) and of Sylow subgroups, we may make the following conclusion. Corollary 2.1. There is a unique conjugacy class of maximal compact pro-p subgroups of GL(V ), consisting of the J1 (L), for all maximal lattice ﬂags L. Iwahori-Bruhat Decomposition We now turn to the analog for lattice ﬂags of the Bruhat decomposition. Deﬁnition 2.4. Let L = {Λα } be a collection of lattices, and let V = ⊕j Lj be a line decomposition of V . We say that the line decomposition is compatible with L, if for each lattice Λα ∈ L, we have Λα = ⊕j (Λα ∩ Lj ). Proposition 2.4. Iwahori-Bruhat decomposition, geometric ver–sion Let L and M be two lattice ﬂags. Then there is a line decomposition V = ⊕j Lj compatible with both L and M. Proof. Select a lattice Λo of L. As we have seen, L is then associated to and determined by a ﬂag F(L) in Λo . We will construct another ﬂag in Λo using ˜ = (M ∩ Λo ) + πΛo . This is a lattice M. For each lattice M of M, set M between πΛo and Λo , so that it deﬁnes and is determined by a subspace U (M ) of Λo . For suﬃciently small elements of M, the subspace U (M ) will be zero, and for suﬃciently large elements of M, it will be all of Λo . Since the elements of M are totally ordered by inclusion the U (M ) will deﬁne a ﬂag in Λo . Call this ﬂag G(M). Since the quotient of successive elements of M is just a line over k, the ﬂag G(M) will be a maximal ﬂag. According to the Bruhat decomposition for GL(Λ), we can ﬁnd a line decomposition of Λo which is compatible with both ﬂags F(L), and G(M). Let {z j } be a basis for Λo selected from the lines of the line decomposition. Then, since the ﬂag F is deﬁned by lattices between Λo and πΛo , we may choose any elements {zj } lifting the z j to Λo , and these elements will be basis elements for lines, such that the resulting line decomposition of V will be compatible with L. On the other hand, each z j also spans the quotient of two spaces U (M1 ) and U (M2 ), where M1 ⊂ M2 are two successive elements of M. Thus, M1 is the largest element of M such that U (M1 ) does not contain z j , and M2 is the smallest element of M such that U (M2 ) does contain it. Thus, we may lift z j to an element zj belonging to M2 . We claim that if this is done, the line decomposition deﬁned by the zj is compatible also with M. Indeed, consider any lattice M in M. The multiples π c M of M deﬁne a subﬂag of G(M). According to our deﬁnitions, we have

38

Roger Howe (Lecture Notes by Cathy Kriloﬀ)

U (π c M )/U (π c+1 M ) (π c M ∩ Λo )/(π c M ∩ πΛo ) + (π c+1 M ∩ Λo ). An appropriate subset of the zj belong to π c M ∩ Λo , and reduce modulo πΛo to deﬁne a basis of the quotient. We observe that the formation of the quotients above is essentially symmetric in M and Λo . In particular, we have (π c M ∩ Λo )/((π c M ∩ πΛo ) + (π c+1 M ∩ Λo )) = π c [(M ∩ π −c Λo )/((M ∩ π −c+1 Λo ) + (πM ∩ π −c Λo ))]

(M ∩ π −c Λo )/((M ∩ π −c+1 Λo ) + (πM ∩ π −c Λo )).

Thus, for each zj , there is an appropriate exponent cj such that the elements π −cj zj , will be a basis for M , and when reduced modulo πM will provide a basis compatible with the ﬂag in M deﬁned by the multiples of Λo . Since this is true for any lattice in M, this means that the elements zj deﬁne a line decomposition compatible with M. Thus the proposition is proved. Remark 2.1. 1. As in the case of the ordinary Bruhat Decomposition, this geometric version of the Iwahori-Bruhat Decomposition has a group theoretic formulation. As above, let J(L) = J be the stabilizer of the maximal lattice ﬂag L. Let Λo be a lattice in L, and let K(Λo ) be the stabilizer of Λo . Let V = ⊕j Lj be a line decomposition of V compatible with L. Let A be the group of transformations which stabilize all the lines Lj , and ˜ = AW be the aﬃne Weyl group of transformations which stabilize let W the collection {Lj } (but which may permute the Lj among themselves). Then the group-theoretic version of the Iwahori-Bruhat Decomposition is ˜ J(L). GL(V ) = J(L)W 2. Here we are abusing language by calling AW the “aﬃne Weyl group”. Correctly speaking, the aﬃne Weyl group is AW/A0 , where A0 = A∩J(L) is the maximal compact subgroup of A. 3. There is also a related description of the double cosets of the maximal compact subgroup K(Λo ): GL(V ) = K(Λo )AK(Λo ). This is sometimes called the Cartan decomposition. It follows easily from the Iwahori-Bruhat decomposition. It also exists in a slightly ﬁner version, in which A is replaced by a subsemigroup A0+ (see Section 3), which forms a fundamental domain for the action of W on A. 2.4 Bruhat decomposition of Sp(V ) Let V be a vector space of dimension 2n equipped with a symplectic form , . This allows us to deﬁne a map, denoted by : End(V ) → End(V ), such that T u, v = u, T v. This map has the following properties:

Aﬃne-like Hecke algebras and p-adic representation theory

39

1. is linear, 2. (T ) = T , 3. (ST ) = T S . The symplectic group can be deﬁned as Sp(V ) = {g ∈ GL(V ) | gu, gv = u, v = u, g gv or g g = I}, with Lie algebra sp(V ) = {T ∈ End(V ) | T = −T }. There is the usual notion of duality with respect to the symplectic form. For a subspace U of V , deﬁne U ⊥ = {v ∈ V | v, u = 0 for all u ∈ U }. Then (U ⊥ )⊥ = U and dim U + dim U ⊥ = dim V . We say U is isotropic if , |U = 0, which is equivalent to U ⊂ U ⊥ . If all subspaces in a ﬂag are isotropic then we say the ﬂag is isotropic. If a ﬂag has the form F = {U1 ⊂ U2 ⊂ · · · ⊂ U }, where Ui⊥ = Uj then the ﬂag is called self-dual. Notice that then in fact j = − i + 1. There is a bijection between isotropic ﬂags and self-dual ﬂags, obtained by adding to the isotropic ﬂag the orthogonal subspaces of the subspaces in the ﬂag. We will refer to a self-dual ﬂag with = 2n − 1 (so Ui⊥ = U2n−i ) as complete. Such ﬂags have the form F = {U1 ⊂ ⊥ ⊥ ⊂ Un−2 ⊂ · · · ⊂ U1⊥ }, and are also called symplectic U2 ⊂ · · · ⊂ Un ⊂ Un−1 ﬂags. Let E = {ej , fj : 1 ≤ j ≤ n} be a symplectic basis for V . This means that ei , ej = 0 = fi , fj , and ei , fj = δij , where δij is Kronecker’s delta. Let B be the stabilizer of the complete symplectic ﬂag associated to E, and let W denote the Weyl group, the stabilizer of the basis E modulo signs (meaning, that the set ±E is stable under W , and W acts by permutations on the set of pairs {±ej }.{±fj }. Theorem 2.5 (Bruhat Decomposition). For any Borel subgroup, B, Sp(V ) = BWB. Once again there are various reﬁnements. Let U be the unipotent radical of B (the set of elements of B which act trivially on the quotients Uj+1 /Uj ). denote the Let A be the subgroup that ﬁxes every line kei or kfi and let W = WA aﬃne Weyl group, the stabilizer of the set of lines {kei , kfi }. Then W (but this is no longer a semidirect product as it was in the case of GL(V )). U . Theorem 2.6. Sp(V ) = BW U = U W There is again an equivalent geometric reformulation. Let F1 and F2 be two complete, self-dual ﬂags in V . We know from the Bruhat theory for GL(V )

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that we can ﬁnd a line decomposition of V compatible with both F1 and F2 . However, this does us little good in understanding the structure of Sp(V ), because Sp(V ) does not act transitively on the set of line decompositions of V . In the context of Sp(V ), we need to capitalize on the self-dual structure of the Fj to show that we can ﬁnd a line decomposition compatible with the symplectic structure. Let V = ⊕j Lj be a line decomposition of V . We will call it symplectic if, for each index j, there is a unique index j such that Lj is paired non-trivially with Lj . That is, Lj is orthogonal to all but one of the other Lk . Then the planes Lj ⊕ Lj decompose V into an orthogonal direct sum of non-degenerate planes. Equivalently, we can ﬁnd basis elements for the lines Lj such that they form a symplectic basis for V . Theorem 2.7 (Bruhat Decomposition, geometric version). Given any two self-dual ﬂags, F1 and F2 , there exists a symplectic line decomposition of V compatible with both F1 and F2 . Proof. Fix a symplectic form on V . Proceed by induction on the dimension of V , with the case of dimension two being straightforward. Given two self-dual ﬂags, expand them if necessary to obtain complete ﬂags, F1 = {U1 ⊂ U2 ⊂ · · · ⊂ U2n−1 } and F2 = {W1 ⊂ W2 ⊂ · · · ⊂ W2n−1 }, where Uj⊥ = U2n−j and Wj⊥ = W2n−j . Deﬁne k1 = σ(1) as the smallest index such that U1 is contained in Wk1 . Then taking complements, W2n−k1 ⊂ U2n−1 and 2n − k1 is the largest such index. Choose a line L ⊂ W2n−k1 +1 complimentary to W2n−k1 . Notice that this means Wk1 −1 ⊂ L⊥ and k1 − 1 is the largest such index. Since U1 and L are paired nontrivially under the symplectic form and since U1 is isotropic, P = U1 ⊕ L is a hyperbolic plane and P ⊥ is a (2n − 2)dimensional subspace of V . We claim that the Uj ∩ P ⊥ and the Wj ∩ P ⊥ form symplectic ﬂags in P ⊥ . First we show they are self dual, i.e., that (Uj ∩ P ⊥ )⊥ ∩ P ⊥ = U2n−j ∩ P ⊥

(2)

(U2n−j + P ) ∩ P ⊥ = U2n−j ∩ P ⊥ ,

(3)

or equivalently, that

and similarly for Wj . For simplicity of notation, set 2n − j = i. For the Ui case, choose x = u + ∈ Ui + P , with u ∈ Ui and ∈ L. (Note that U1 ⊂ Ui , so that Ui + P = Ui + L.) Suppose x ∈ P ⊥ . Then u + ⊥ U1 , but u ∈ Ui ⊂ U2n−1 = U1⊥ , so ⊥ U1 . Since L and U1 are paired nontrivially, this forces = 0 and x = u ∈ Ui . Thus (Ui + P ) ∩ P ⊥ ⊂ Ui ∩ P ⊥ and (Ui + P ) ∩ P ⊥ = Ui ∩ P ⊥ . Similarly, take x = w + u1 + ∈ Wi + P . We want to show that if x ∈ P ⊥ then x ∈ Wi . There are four slightly diﬀerent cases depending on the relative positions of i, k1 and 2n − k1 + 1. Set

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a = min{k1 − 1, 2n − k1 } b = max{k1 , 2n − k1 + 1}. Case 1, i ≤ a: If x ∈ P ⊥ , then w + u1 + ⊥ U1 and since w ∈ W2n−k1 ⊂ and u1 ∈ U1 ⊂ U1⊥ , we see = 0. But then x = w + u1 is also in L⊥ and w ∈ Wk1 −1 ⊂ L⊥ implies u1 = 0 so x ∈ Wi . Case 2, k1 ≤ i ≤ 2n − k1 : Since U1 ⊂ Wi , we can write x = w + . If x ∈ P ⊥ then w + ⊥ U1 , but w ∈ W2n−k1 ⊂ U1⊥ , so = 0. Case 3, 2n − k1 + 1 ≤ i ≤ k1 − 1: Since L ⊂ Wi , we can write x = w + u1 . If x ∈ P ⊥ , then w + u1 ⊥ L, but w ∈ Wk1 −1 ⊂ L⊥ , so u1 = 0. Case 4, i ≥ b: Since U1 ⊂ Wi and L ⊂ Wi , P ⊥ ⊂ Wi so certainly (Wi + P ) ∩ P ⊥ ⊂ Wi ∩ P ⊥ . To see that both Ui ∩ P ⊥ and Wi ∩ P ⊥ form complete ﬂags, notice that

U1⊥

Ui = (Ui ∩ P ) ⊕ (Ui ∩ P ⊥ ) Wi = (Wi ∩ P ) ⊕ (Wi ∩ P ⊥ ). These decompositions are in fact equivalent to the equalities (Ui + P ) ∩ P ⊥ ⊂ Ui ∩ P ⊥ (Wi + P ) ∩ P ⊥ ⊂ Wi ∩ P ⊥ , as can be shown directly. The decompositions make it clear that for 1 ≤ i ≤ 2n − 1, dim (Ui ∩ P ⊥ ) = i − 1, and   if i ≤ a i ⊥ dim (Wi ∩ P ) = i − 1 if a < i ≤ b   i − 2 if b < i. Now by induction, ﬁnd a symplectic line decomposition P ⊥ =

2n

Lj

j=3

compatible with these two ﬂags. Setting L1 = U1 and L2 = L, it is pos2n Lj is a symplectic line sible to show, using dimension arguments, that j=1

decomposition of V compatible with the two original symplectic ﬂags. 2.5 Iwahori-Bruhat decomposition of Sp(V ) Self-dual Lattice Flags Let Λ be a lattice in the symplectic space V . The dual lattice Λ∗ is deﬁned by Λ∗ = {y ∈ V : y, x ∈ O, for every x ∈ Λ}. We have deﬁned a lattice ﬂag in Section 2.3. We will call a lattice ﬂag L self-dual if, for every lattice Λ in L, the dual lattice Λ∗ is also in L.

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Example 2.2. Let {ej , fj } be a symplectic basis for V . If Λo is the lattice spanned by this basis, then it is easy to see that Λ∗o = Λo . We say that Λo is self dual. For 1 ≤ a ≤ n (where dim V = 2n), let Λa be the lattice spanned by {ej , 1 ≤ j ≤ n} ∪ {fj , a < j ≤ n} ∪ {πfj , 1 ≤ j ≤ a}. Let Λn+a be the lattice spanned by {ej , 1 ≤ j ≤ n − a} ∪ {πej , n − a < j ≤ n} ∪ {πfj , 1 ≤ j ≤ n}. Then set Λb+2nc = π c Λb for 0 ≤ b < 2n, and c ∈ Z. The reader can check that the Λm form a complete lattice ﬂag, which is self-dual. Precisely, Λ∗m = Λ−m . We will investigate the structure of self-dual lattice ﬂags in V . We will ﬁnd in particular that any complete self-dual lattice ﬂag is symplectically equivalent to the example just given. The set of all lattices in V itself forms a lattice, in the sense of ordered sets, with the operations of intersection and sum as the lattice operations. We may observe that a lattice ﬂag is trivially closed under these two operations: since its lattices are totally ordered by inclusion, the intersection of any two lattices in the ﬂag is the smaller one, and the sum is the larger one. Lattice ﬂags have another property: they are scalable, in the sense that if Λ is in a lattice ﬂag, then so is sΛ for any scalar s. To study the structure of self-dual lattice ﬂags, we will begin by studying scalable self-dual lattices of lattices. We will call a lattice Λ almost self dual if Λ ⊂ Λ∗ ⊂ π −1 Λ. Proposition 2.5. Any scalable self-dual lattice lattice contains an almost self-dual lattice. Proof. Let Λ be any lattice in V . We will show that by taking multiples, duals, intersections and sums, we can construct beginning with Λ an almost self-dual lattice. We will assume the standard result from the “theory of elementary divisors”, that we can ﬁnd a basis {ej , fj : 1 ≤ j ≤ n} for Λ, such that ej , f = δj π cj . Although the basis is not unique, the integers cj , if arranged in decreasing order, are well-deﬁned. We will call the cj the exponents of Λ. Given such a basis for Λ, it is then easy to see that Λ∗ has a basis −cj {π ej , π −cj fj }. From this, the exponents of Λ∗ are seen to be −cj . Also, the exponents for π d Λ are cj + 2d. We note that a lattice is almost self-dual if and only if its exponents are all either 0 or 1. If all the exponents of Λ are the same, then we can scale Λ by an appropriate power of π, to make every exponent of the rescaled lattice either 0 or 1, according as to whether the original exponent value is even or odd. The diﬀerence between the largest and smallest exponent of Λ is a measure of how far from self-dual a scalar multiple of Λ can be. We can reduce the “exponent spread” of Λ as follows. Arrange the exponents in descending order; c1 ≥ c2 ≥ c3 ≥ · · · ≥ cn . Now scale Λ by the appropriate power of π to arrange that −1 ≤ c1 + cn ≤ 2. Now form Λ ∩ Λ∗ . It exponents will be |cj |,

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and these will have maximum spread max{|c1 |, |cn |}, which is roughly half of the spread of Λ itself. It is not hard to see that by repeating this process, we can reduce the exponent spread to not more than 2, and the exponents themselves to lie in the range from 0 to 2. If the exponents are all 0 or 1, then we have an almost self-dual lattice. So we only have to worry if the values of the exponents include 2. If the lattice Λ has exponents 0, 2, and perhaps 1, we form Λ = Λ + πΛ∗ . We can compute that Λ has exponents only 0 and perhaps 1. Thus Λ is the desired almost self-dual lattice. Now consider a self-dual lattice ﬂag L. If Λ is any lattice in L, then the multiples of Λ and of Λ∗ form a self-dual subﬂag of L. Thus, some multiple of Λ or of Λ∗ must be almost self-dual. Conversely, given an almost self dual lattice Λ, one can check that the multiples of Λ and of Λ∗ do constitute a self-dual lattice ﬂag. Another way of formulating the fact that a self-dual lattice ﬂag must contain an almost self-dual lattice, is to observe that every lattice Λ in the ﬂag must have exponents of only two values, say c and c − 1; for as we have noted above, if the exponents of Λ are {cj }, then the exponents of Λ∗ are {−cj }, and the exponents of π d Λ are {cj + 2d}. If Λ is almost self-dual, then its exponents are 0 and 1, and so every multiple of Λ has exponents with only two values which diﬀer by 1. Conversely, if Λ is a lattice with only two values for exponents, with the values diﬀering by 1, then some multiple of Λ or of its dual will be self-dual, so that Λ does generate a self-dual lattice ﬂag. Now consider a self-dual lattice ﬂag L which is complete. We claim that L must contain a self-dual lattice. We know from the discussion so far that L must contain at least an almost self-dual lattice. Let Λ be one. Then, since L is complete, it contains a lattice Λ such that Λ ⊂ Λ , and Λ /Λ is a line over k. The exponents of Λ must be the same as those of Λ, except for one, which is one less than a corresponding exponent of Λ. (This can be seen, for example, by observing that the volume of Λ can be expressed in terms of its exponents.) The exponents of Λ are all either 0 or 1; if Λ is not itself self-dual, then 1 certainly occurs. Since Λ must have exponents with only two values, it cannot have -1 as an exponent, so its exponents again must have values 0 and 1, with one more 0 and one less 1 than the exponents of Λ. Hence, we see that by looking for a maximal almost-self dual lattice in L, we will ﬁnd a lattice which is actually self-dual. Still considering the self-dual, maximal lattice ﬂag L, let Λo be a self-dual lattice in L. Since L is totally ordered by inclusion, we may label the lattices in L by integers, in a manner such that Λm+1 is the largest element of L strictly contained in Λm . This labeling is clearly unique. Then the lattices Λm for 0 ≤ m ≤ 2n are contained between Λo and πΛo = Λ2n . They therefore deﬁne a ﬂag in Λ = Λ/πΛ. The condition that each Λm should have at most two exponent values translates into the condition that the ﬂag U m = {Λm /πΛo } should be a self-dual ﬂag in Λo . Thus, we see that L is indeed isometric to the example given above of a self-dual complete lattice ﬂag.

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Iwahori-Bruhat Decomposition Now consider two complete, self-dual lattice ﬂags. Since these are in particular lattice ﬂags, we know that we can ﬁnd a line decomposition of V which is compatible with both of them. However, as in the case of ordinary ﬂags, we can do better. Proposition 2.6. Iwahori-Bruhat decomposition, geometric ver–sion For any two complete, self-dual lattice ﬂags in V , there is a symplectic line decomposition compatible with both ﬂags. Proof. We label the lattices in L and M by the integers, in the standard way described above: Λo is the self-dual element of L. Consecutive elements of L are labeled by consecutive integers, with positive integers labeling sublattices of Λo . The labeling of M is similar. Consider the lattice Λo in L. It is self-dual. The k vector space Λo = Λo /πΛo inherits a symplectic form by reducing modulo P the restriction to Λo of the form , on V . For a lattice Ma of M, consider the subspace U (Ma ) of Λo deﬁned by (Ma ∩ Λo ) + πΛo . Consider the orthogonal complement of U (Ma ) with respect to the symplectic form on Λo . It can be lifted to a lattice contained between Λo and πΛo . As such, it will consist of elements y ∈ Λo such that x, y lies in P for all x in Ma ∩ Λo . This means that π −1 y belongs to ((Ma ∩ Λo ) + πΛo )∗ . We compute that ((Ma ∩ Λo ) + πΛo )∗ = (Ma ∩ Λo )∗ ∩ (πΛo )∗ = (Ma∗ + Λ∗o ) ∩ π −1 Λo = (M−a ∩ π −1 Λo ) + Λo = π −1 ((πM−a ∩ Λo ) + πΛo ). It follows that U (Ma )⊥ = U (πM−a ) = U (M−a+2n ). Here the ⊥ is understood to refer to Λo . Thus, we see that the ﬂag in Λo deﬁned by M will be a self-dual ﬂag. The ﬂag deﬁned by L is likewise self-dual. According to the ordinary Bruhat decomposition for Sp(Λo ), we can ﬁnd a symplectic basis for Λo compatible with both ﬂags. According to the Iwahori-Bruhat decomposition for GL(V ), we can lift this basis to a basis of Λo such that the corresponding line decomposition is compatible with both lattice ﬂags. It remains to show that we can lift the basis in such a way that we get a symplectic line decomposition - that is, we can lift the symplectic basis for Λo to a symplectic basis for Λo that is also compatible with both lattice ﬂags. As in the earlier arguments, we proceed by induction on dim V . Let a1 be the smallest index such that Ma1 ⊂ πΛo . That is, Ma1 ⊂ πΛo but Ma1 −1 ⊂ πΛo . Then, according to the duality computation above, M−a1 ⊃ π −1 Λo , but M−a1 +1 ⊃ π −1 Λo . It follows that we can choose e1 ∈ Ma1 −1 and f1 ∈ πM−a1 ∩ Λo , such that e1 , f1 = 1. We note that e1 and f1 are both in Λo , and that, according to the proof of the Bruhat Decomposition for vector spaces, we can choose them compatible with the ﬂag deﬁned by the Λk also. That means that, if P is the plane in Λo spanned by e1 and f1 (taken modulo

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πΛo ), then each subspace of Λo deﬁned by Λk for 0 ≤ k ≤ 2n, is the direct ⊥ sum of its intersections with P and with P . From this, it follows that, if P is the plane in V spanned by e1 and f1 , then each lattice Λk satisﬁes Λk = (Λk ∩ P ) ⊕ (Λk ∩ P ⊥ ). (Here of course P ⊥ refers to orthogonal complement in V rather than in Λo .) In brief, the decomposition V = P ⊕ P ⊥ is compatible with the lattice ﬂag L. We claim that the decomposition V = P ⊕ P ⊥ is also compatible with the lattice ﬂag M. To see this, consider the lattice Mo , the self-dual member of M, and consider the ﬂag in M o deﬁned by M. We may write a1 − 1 = 2n + b1 − 1 with 0 < b1 ≤ 2n. Then −a1 + 2n = −2n + 2n − b1 , and 0 ≤ 2n − b1 < 2n. With this notation, we see that π − e1 is in Mo , and indeed, is in Mb1 −1 . Similarly, π f1 is in Mo , or more precisely, is in M2n−b1 . From this, we see that the images in M o of π − e1 and π f1 are compatible with the ﬂag deﬁned by the Mk . Since π − e1 and π f1 obviously span the same plane P as do e1 and f1 , we can argue as in the case of the ﬂag L, that the decomposition V = P ⊕ P ⊥ is compatible with M, as desired. Thanks to this compatibility, we can now ﬁnish the argument by induction on dim V , in parallel with the argument for the Bruhat Decomposition for Sp(V ). Parahoric Subgroups Given L a self-dual lattice ﬂag in V , we denote the stabilizer in Sp(V ) of all the lattices of L by KL . The groups KL are referred to as parahoric subgroups. If L is complete, we also write KL = J. This is the Iwahori subgroup of Sp(V ). If K ⊂ Sp(V ) is any compact subgroup, then as we have seen in the discussion of GL(V ), K will preserve some lattice Λ in V . It will then also preserve all lattices obtained from L by the operations of scalar multiplication, duality, intersection and sum. In other words, it will preserve the self-dual lattice of lattices generated by L. Hence, according to Proposition 2.5, it will preserve some almost self-dual lattice, and likewise the self-dual lattice ﬂag it generates. Hence, any compact subgroup of Sp(V ) is contained in a parahoric subgroup. Among the parahoric subgroups, the smallest ones are of course the stabilizers of the members of a complete self dual ﬂag - the Iwahori subgroups. The largest are the stabilizers of a single almost self-dual lattice (and hence, of all elements of the self-dual lattice ﬂag it generates). Unlike the case of GL(V ), these groups are not all conjugate inside Sp(V ). However, despite this, they all have the same Sylow p-subgroup. More precisely, we can show that the Sylow p-subgroup of the Iwahori group is the Sylow p-subgroup of any parahoric (up to conjugacy). Indeed, let H be a compact pro-p subgroup of Sp(V ). Then H preserves some self-dual lattice ﬂag L. We may suppose that L is maximal with respect to being H-stable. We claim then that L is complete. If not, let Λ ∈ L be a maximal almost self-dual lattice in L. If Λ is not itself self-dual, then Λ∗ /Λ is a k vector space, and H acts on Λ∗ /Λ via a ﬁnite quotient H, which is of course a p-group. Further, Λ∗ /Λ inherits

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a symplectic form, and H will preserve this form. Since H is a p-group, it will preserve some line L1 ⊂ Λ∗ /Λ, and also its orthogonal subspace. The line and its orthogonal subspace lift to a nested pair of mutually dual, Hinvariant lattices strictly contained between Λ and Λ∗ . Also, the smaller of the two lattices will be almost self-dual. These lattices and their multiples can be adjoined to L to make a larger H-invariant lattice ﬂag. Hence, maximality of L implies that Λ must have been self-dual. Then H acts on Λ = Λ/πΛ, and will preserve a maximal, self-dual ﬂag there, since it acts via a p-group. This ﬂag then determines a complete self-dual lattice ﬂag, which will evidently be invariant by H. Thus, H is contained in the pro-p part of an Iwahori subgroup, as claimed.

3 Structure of the Iwahori Hecke Algebra 3.1 H(G//K) First we study H(G//K), where K = K(Λ) ⊂ GL(V ) is the stabilizer of a lattice in V , or, in the case of Sp(V ), K = K(Λ) ∩ Sp(V ), where Λ is a selfdual lattice. In the case of GLn , let A◦ be the group of all diagonal matrices whose entries are powers of π. Let A◦+ be the subsemigroup of A◦ deﬁned by   m    π 1        π m2   ◦+ A =  : m1 ≤ m2 ≤ . . . ≤ mn .  . ..           mn π In the case of Sp2n , ﬁrst choose a symplectic basis, and order it so that the matrix of the symplectic form is 0 −In , In 0 where In is the n × n identity matrix. Then set a 0 ◦ ◦ A = : a ∈ A for GLn , 0 a−1 and let A◦+ be the subsemigroup for which a belongs to A◦+ for GLn . The Weyl group acts on the diagonal torus by conjugation. This action permutes the diagonal entries, and it preserves the subgroup A◦ It is not diﬃcult to verify that the semigroup A◦+ contains a unique element in each W orbit in A◦ . The group K contains a set of representatives for W . As noted at the end of Section 2.3, this implies that we have the decompositions G = KA◦+ K = JA◦ K. From the ﬁrst of these decompositions, we deduce the following result.

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Proposition 3.1. H(G//K) is commutative. Proof. The proof is based on the simple and classical statement, due to Gel’fand, that if an algebra A allows an anti-automorphism which is the identity, then A is commutative. Here we start with the transpose map sending g to g t for g ∈ GLn , which preserves K and A◦+ . This induces an antiautomorphism of the convolution algebra Cc∞ (G) and is the identity upon restriction to H. A similar argument works for Sp2n as long as we use coordinates with respect to a symplectic basis. Thus irreducible representations of K occur at most once in any representation of G. The operators originally constructed by Hecke were elements of H(G//K) where G = SL2 or G = PGL2 . 3.2 H(G//J ) The extended aﬃne Weyl group For G = GLn , the Weyl group W is the group of permutations, Sn , generated by the n − 1 transpositions sj for j = 1, . . . , n − 1 where sj interchanges ej and ej+1 . The extended aﬃne Weyl ◦ , is almost an aﬃne Coxeter group generated by reﬂections. Genergroup, W ◦ include the s1 , . . . , sn−1 together with two additional generators, ators for W s0 and t, where     0 π −1 01  1   0 1         .. .. ..  s0 =  t=  . . . .        1 0 1 π 0 π 0 The group generated by the sj , 1 ≤ j ≤ n − 1, is the usual Weyl group of permuation matrices. If we adjoin s0 , we get a group containing all diagonal matrices of determinant one (whose diagonal entries are powers of π). Adding t gives us a group containing all diagonal matrices (whose diagonal entries are powers of π). The choice of t is based on the fact that tsj t−1 = sj−1 so t normalizes JGLn , where   × O O ... O  .   P . . . . . . ..  . JGLn =    . .  .. . . O P . . . P O× ◦ include the usual Coxeter relations, s2 = 1 (but now The relations in W i with 0 ≤ i ≤ n − 1) and (si sj )mij = 1, where for GLn , mi,i+1 = 3 and mij = 2 if |i − j| > 1 with |i − j| computed modulo n. But we also have the relations tsj t−1 = sj−1 .

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◦ For G = Sp2n , JSp2n = JGLn ∩ Sp2n and the generators of the group W can most easily be described by listing only the symplectic basis vectors that they change in some way (here 1 ≤ j ≤ n − 1): s0 : e1 → πf1 f1 → −π

−1

sj : ej ↔ ej+1 e1

fj ↔ fj+1

sn : en → fn fn → −en .

◦ , the length of a word is deﬁned to be the For both of these versions of W minimum number of generators of type sj (i.e. excluding any occurrences of the generator t in the case of GLn ). This is used so that µ(JwJ) = q (w) . Iwahori-Matsumoto presentation The relations for H(G//J) consist of two types. To describe the ﬁrst, in place of (si sj )mij = 1, we use the braid relation si sj si · · · = sj si sj · · · , where each side is a reduced expression for the longest word in W . This avoids the use of inverses. Since l(si sj si · · · ) = l(sj si sj · · · ) = mij and l(uv) = l(u) + l(v) implies µ(JuvJ) = µ(JuJ)µ(JvJ), we know µ(Jsi sj si · · · J) = µ(Jsi J)µ(Jsj J)µ(Jsi J) · · · = q mij and µ(Jsj si sj · · · J) = µ(Jsj J)µ(Jsi J)µ(Jsj J) · · · = q mij . Thus the braid relations hold in H(G//J): fsi fsj fsi · · · = fsi sj si ··· = fsj si sj ··· = fsj fsi fsj · · · . Secondly, the quadratic relations s2i = 1 become fs2i = (q − 1)fsi + qf1 . Theorem 3.1. H(G//J) is the algebra generated by the appropriately indexed fsi (and ft when G = GLn ) subject to the following relations. 1. fs2i = (q − 1)fsi + qf1 2. fu fv = fuv if (u) + (v) = (uv) 3. ft fsi = fsi+1 ft when G = GLn Notice that the third relation uses that ft−1 = ft−1 , which follows from the fact that t is by deﬁnition of length zero. This presentation shows that the structure of H(G//J) is similar to that ◦ where of a Coxeter group and allows us to view H as a deformation of W the operators have eigenvalues q and −1 rather than 1 and −1. However, ◦ , which is it obscures the abelian subgroup of diagonal matrices inside W often useful. This abelian subgroup is revealed by an anternative presentation developed by J. Bernstein and A. Zelevinsky.

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49

Conversion to the Bernstein-Zelevinsky presentation We now show how, in the case of GL(V ), to convert from the Iwahori-Matsumoto presentation given above to the Bernstein-Zelevinsky presentation. This presentation is often used, in part because it makes clear the presence of a large abelian ◦. subalgebra inside H(G//J) corresponding to the diagonal subgroup of W ◦ for GLn , set In the aﬃne Weyl group W   −1 π   ..   .   −1   π ,  ak =   1     . ..   1 where the ﬁrst k entries along the diagonal are π −1 . These elements with ◦ and it is easy to 1 ≤ k ≤ n generate a free semigroup of rank n inside W check that (ak ) = k(n − k). Notice that  −1  π   ..   .   −1   π     1   and sk ak sk =  −1  π     1     . ..   1  −2  π   ..   .   −2   π   −1   π   = ak−1 ak+1 , ak sk ak sk =  −1  π     1     . .  .  1 where each of these has the ﬁrst k − 1 entries equal to π −2 . Note also that sk commutes with aj for j = k and therefore in particular with ak−1 and ak+1 . If we rewrite the above identity as (ak sk )ak = ◦ in the equation is the sum ak−1 ak+1 sk , then the length of the element of W of the lengths of the factors, so the identity fak sk fak = fak−1 fak+1 fsk ,

(4)

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Roger Howe (Lecture Notes by Cathy Kriloﬀ)

is valid in the Hecke algebra H(GLn //J). Since (ak sk ) = (ak ) − 1, the identity fak = fak sk fsk also holds in H(GLn //J). Now we use this together with the quadratic relation to see that fak fsk = fak sk fs2k = fak sk ((q − 1)fsk + q) = (q − 1)fak sk fsk + qfak sk = (q − 1)fak + qfak sk . Solving this for fak sk and substituting into the identity 4 yields ( ' 2 1 1 f f f − 1 − a s a k k k q q fak = fak−1 fak+1 fsk . Since the elements on the right commute, reorder them to fak+1 fsk fak−1 and on the left and fa−1 on the right. (Notice then multiply throughout by fa−1 k k−1 that fak−1 is indeed invertible. More generally, for any w in the aﬃne Weyl group, fw is invertible, since it is a product of generators of H(G//J), and the generators are obviously invertible, by the quadratic relations.) This yields ( ' −1 −1 −1 −1 1 1 f (f f ) − 1 − (5) s a k ak−1 q k q (fak fak−1 ) = fak fak+1 fsk = fak+1 fak fsk . In equation 5, replace fw by q 1

Tk = q − 2 fsk ,

yk = q −

−(w) 2

n−2k+1 2

Then 5 becomes n−2k+1 1 12 2 yk q q Tk q

−

'

fw and rename by setting

fak fa−1 , k−1

q−1 q

( q

and yk+1 = q −

n−2k+1 2

Combining powers of q, dividing by q

n−2k 2

yk = q

n−2k−1 2

n−2k−1 2

fak+1 fa−1 . k

1

yk+1 q 2 Tk .

, and rearranging produces 1

1

Tk yk − yk+1 Tk = (q 2 − q − 2 )yk . Using the action sk (yk ) = yk+1 , this can be rewritten in the form of the Bernstein-Zelevinsky relation 1

1

Tk yk − sk (yk )Tk = (q 2 − q − 2 )

sk (yk ) − yk . sk (yk )yk−1 − 1

(6)

Note also that Tk yj = yj Tk for j = k, k + 1 since then sk commutes with aj and aj−1 as observed above. In addition, the yj generate an abelian subalgebra and the Tk satisfy the relations Ti T j = Tj T i Tk Tk+1 Tk = Tk+1 Tk Tk+1

if |i − j| > 1 for 1 ≤ k ≤ n − 1.

Also, the relation in 6 in fact holds for any Laurent polynomial in the yj .

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4 Results on representations of G 4.1 Fundamental techniques For more details, see the references for further reading. Parabolic induction The primary method of constructing representations of p-adic groups is parabolic induction. Begin with a parabolic subgroup P of the group G and decompose it as P = MP UP where MP is a Levi component and UP is the unipotent radical of P . Let σ be a (smooth admissible) representation of MP on the space Y and lift σ to P by taking it to be trivial on UP . The induced representation IndG P (σ), called a principal series, consists of the space {f : G → Y : f (pg) = σ(p)f (g) for all p ∈ P, g ∈ G} and the action IndG P (σ)(g)f (h) = f (hg). It is possible to prove the following facts. 1. If σ is admissible then IndG P (σ) is admissible. G ∗ ∗ 2. (IndG σ) = Ind (δ σ ) where δP is the modular function of P deﬁned P P P using p-adic absolute value by δP (m) = | det Ad(m|uP )|−1 p 1/2

3. If σ is unitary then IndG P (δP σ) is unitary. In the special case when the parabolic subgroup is the Borel subgroup, then the Levi component is a torus and the decomposition is P = B = AU . Since A is abelian, take ψ to be a character trivial on the compact part of the torus, i.e. a character of A/A0 A0 where A0 = A ∩ K. Then the induced representation IndG B (ψ) is called an unramiﬁed minimal principal series, or a spherical principal series. Since G = KB, each spherical principal series contains a unique K-ﬁxed vector. (In fact we will see below that the process of forming spherical principal series will yield all representations with K-ﬁxed vectors.) These induced representations are highly important in the study of automorphic forms since any given representation of an adele group over a global ﬁeld factors into a tensor product of representations of groups over local ﬁelds, and almost all factors are spherical principal series. Jacquet modules Let ρ be a smooth (admissible) representation of G. If ρ can be realized as a principal series from some parabolic, then it must be induced from a representation trivial on UP , the unipotent radical of the parabolic. So we look for invariants for UP . Set Y (UP ) = {ρ(u)y − y : y ∈ Y, u ∈ UP }. This records all nontrivial action of UP . The quotient YUP = Y /Y (UP ) is the maximal quotient of Y on which UP acts trivially. It is called a Jacquet module. Proposition 4.1. Y (UP ) is an MP -submodule of Y . The proof involves a simple calculation. We denote the action of MP on YUP by ρU .

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Roger Howe (Lecture Notes by Cathy Kriloﬀ)

Frobenius Reciprocity Proposition 4.2. Let σ be a smooth representation of M and let ρ be a smooth representation of G. Then HomG (ρ, IndG P (σ)) HomMP (ρU , σ). So if YUP = {0} then ρ is a constituent of the principal series representation arising from P . Let JU denote the quotient map from Y (admissible as a G-module, but not as an M -module) to YUP (which is admissible as an M module). Proposition 4.3. Let UP− be the unipotent subgroup of G opposite to UP . Suppose H is a compact subgroup of G that can be factored as H = (H ∩ P UP− )(H ∩ M )(H ∩ UP ). Then JU : Y H → YUH∩M is surjective. P Notice that such a factorization is often possible, for example whenever H is a principal congruence subgroup. In fact, for all suﬃciently small compact subgroups of M it is possible to ﬁnd an H as in the proposition. Then when P Y H is ﬁnite dimensional, YUH∩M will also be ﬁnite dimensional and thus if P ρ is admissible, so is ρUP . This means that it is possible to check whether a representation occurs as part of a principal series by checking if some Jacquet module is nonzero. If all Jacquet modules are zero (i.e. YUP (ρ) = 0 for all P ) then we say ρ is cuspidal, meaning it has compactly supported matrix coeﬃcients. In particular, they are square-integrable (a.k.a. discrete series). Such representations are characteristic of p-adic group representation theory. Harish-Chandra philosophy of cusp forms The above background leads to the philosophy of cusp forms, which involves a two-step program to classify representations of G. 1. Classify cuspidal representations. 2. Decompose all principal series IndG P (σ) where σ is cuspidal. Completing this program will guarantee that all representations have been found, but will involve redundancy since a representation could appear in more than one induced representation. However, such cases are very limited, as demonstrated in the following proposition. First, notice that for ﬁxed M , it is possible to choose U and P = M U so that ρ appears as a submodule G of IndG P (σ) since JU is exact. Thus by ρ ∈ IndP (σ) it is meant that ρ is a G submodule of IndP (σ). if ρ ∈ IndG (σ) and ρ ∈ IndG (σ ), then Proposition 4.4. For ρ ∈ G, P P (MP , σ) is conjugate to (MP , σ ), meaning that MP is conjugate to MP in such a way that σ is sent to σ .

Aﬃne-like Hecke algebras and p-adic representation theory

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4.2 Connection of Jacquet functor to H(G//J ) Recall the semigroup A◦+ introduced in Section 3.1. This subsemigroup of the torus contains one representative of each conjugacy class in A◦ . Under conjugation, A◦+ stretches UP+ and shrinks UP− (upper and lower triangular matrices respectively) in the sense that for a ∈ A◦+ , a(J ∩ UP+ )a−1 ⊃ J ∪ UP+ and a(J ∩ UP− )a−1 ⊂ J ∪ UP− . This implies that (ab) = (a) + (b) for all a, b ∈ A◦+ and therefore that fa ∗ fb = fab . Thus {fa : a ∈ A◦+ } A◦+ is a semigroup. Let H be a subgroup of G satisfying the properties in Proposition 4.3 and properties analogous to those above with respect to A◦+ . Then for 1 χHaH (normalized to be idempotent when a = 1), JU (ρ(fa )y) = fa = µ(H) 1 δP (a)ρUP (faM )JU (y), where faM = µM (H∩M ) χ(H∩M )a(H∩M ) . Thus the action of an abelian subsemigroup in H(G//J) transfers to an action of a subsemigroup corresponding to M . Since the fa form a commutative semigroup, we can write YH = Y H,λ ◦+ λ∈A H,λ

where the Y are generalized eigenspaces with λ = (λ(a1 ), λ(a2 ), . . . ) for the canonical generators ak deﬁned as in 3.2. Pλ Corollary 4.1. JU maps Y H,λ to YUH∩M,δ and is surjective. P

The proof involves considering the diﬀerence between G and M in terms of A◦+ . If elements of M are written as block diagonal matrices and the index k ends a block, then ak becomes central in M and therefore faMk must be invertible. So JU acting on Y H,λ is either an isomorphism or zero, depending on whether λ(ak ) = 0 or λ(ak ) = 0, for ak in the center of M . 4.3 Category equivalence The consequence for H(G//J) is the following category equivalence. First, recall that for H = J, all fa are invertible, as noted in Section 3.2. Therefore for the generalized eigenspace Y J,λ , there are no zero components in λ. This implies JUB (Y J ) is injective and therefore an isomorphism. Corollary 4.2 (Borel). then ρJ = {0} if and only if ρ is a submodule of the principal 1. If ρ ∈ G, series.

54

Roger Howe (Lecture Notes by Cathy Kriloﬀ)

2. M(G, eJ ) M(H(G//J)) where the ﬁrst is the category of G-modules generated by J-ﬁxed vectors and the second is the category of H(G//J)modules. Proof. 1. Given a vector v in ρJ , it is sent by the Jacquet functor JU to a vector that transforms according to an admissible representation of A. Hence it can be decomposed as a sum of generalized eigenvectors for A, with each generalized eigenvector associated to a character of A. Since JU (v) will be A ∩ J = A0 invariant, all those characters must be trivial on A0 , i.e., unramiﬁed. Thus, the associated induced representations, into which ρ embeds by Proposition 4.2, are spherical principal series. 2. As noted in Section 1, the main point is to show that M(G, eJ ) is closed under taking submodules. Suppose Y is generated by Y J , but Z ⊂ Y has Z J = {0}. Then Z can be mapped non-trivially into some IndG P (σ), for appropriate parabolic P and supercuspidal representation σ. This mapping is found by ﬁrst taking the Jacquet functor to P , and then projecting to the component that transforms according to σ, and ﬁnally using Frobenius Reciprocity (Proposition 4.2). Since JU is exact, and likewise projection to a supercuspidal component, there must also be a non-trivial J map from Y to IndG guarantees P (σ).The fact that Y is generated by Y G J that (IndP σ) = {0}. According to the remark at the beginning of this section, this means that σ must have a non-trivial Jacquet module with respect to a Borel subgroup, and with respect to unramiﬁed characters of the maximal torus. According to Proposition 4.4, this means that P is already a Borel, and σ an unramiﬁed character - that is, Z maps to a spherical principal series, contrary to the assumption that it did not have any J-ﬁxed vectors. Thus H(G//J) provides a precise understanding of the spherical principal series of the group G, in other words about the algebraic structure of G. We have not addressed the unitary structure or the Plancherel formula. It is in fact also possible to detect which representations of G are unitary by use of H(G//J). This is due to Barbasch and Moy.

5 Spherical Function Algebras There are many other representations of G besides spherical principal series, and although the general representation is far from completely known, we will now indicate how to begin to extend the above result. We will look especially at the minimal principal series, that is, representations induced from an arbitrary character of the Borel subgroup. As a ﬁrst strategy one might think to look at algebras H(G//K) as K is made smaller and smaller. However these algebras become more and more complicated and there is a great deal of redundancy since any J-ﬁxed vector

Aﬃne-like Hecke algebras and p-adic representation theory

55

is also a K-ﬁxed vector for K ⊂ J. We would like to ﬁlter out those representations we already know and understand. A way of doing this is to look at non-trivial representations of compact groups. This leads to the study of spherical function algebras. 5.1 Structure Let K be an open compact subgroup of a locally compact group G and let ρ : K → GL(V ) be an irreducible representation of K. Cc∞ (G; End(V )) is the algebra of matrix valued functions on G under convolution and is isomorphic to Cc∞ (G) ⊗ End(V ). Deﬁne the following subalgebra of Cc∞ (G; End(V )): H(G//K; ρ) = {f : G → End(V ) | f (k1 gk2 ) = ρ(k1 )f (g)ρ(k2 )}. This is called a spherical function algebra, or also, a Hecke algebra. Normalize and extend ρ in the following way, ρ(k)/µG (K) g = k ∈ K eρ (g) = 0 g∈ / K. Then eρ is idempotent as an element of Cc∞ (G; End(V )) and we have H(G//K; ρ) = eρ ∗ Cc∞ (G; End(V )) ∗ eρ . Just as H(G//K) determines the irreducible representations of G which contain K-ﬁxed vectors, H(G//K; ρ) describes those irreducible representations of G which contain ρ∗ on restriction to K. Support issues Although double coset algebras have a natural basis consisting of the χKgK , the situation is more complicated with spherical function algebras, both because there may be more than one element supported on a given coset, and because some cosets may support no elements of the algebra. In order to characterize those cosets which do support an element of H(G//K; ρ), ﬁrst notice that the mapping f → f (g) deﬁnes an isomorphism H(G//K; ρ)|KgK HomKg (ρ, Ad∗ g(ρ)), where Kg = K ∩ gKg −1 , Adg(K) = gKg −1 , and for h ∈ Adg(K), Ad∗ g(ρ)(h) = ρ(g −1 hg).

56

Roger Howe (Lecture Notes by Cathy Kriloﬀ)

Elements of HomKg (ρ, Ad∗ g(ρ)) are intertwining maps between the two representations, and there exists an element in H(G//K; ρ) with support on KgK if there exist nontrivial intertwining operators, i.e. if HomKg (ρ, Ad∗ g(ρ)) = {0}. We say that g intertwines ρ, or g is in the support of H(G//K; ρ). More generally, suppose G is a locally compact group and K1 and K2 are two compact open subgroups of G. Given irreducible representations ρj of Kj , on spaces Vj , j = 1, 2, we say that g ∈ G intertwines ρ1 and ρ2 if there exists a non-zero T in Hom(V2 , V1 ), such that ρ1 (k)T = T ρ2 (g −1 kg), where k ∈ K1 ∩ gK2 g −1 . 5.2 Lie lattices and characters The simplest way to determine whether g is in the support of certain spherical function algebras is to use the exponential map to reduce the computation to looking at the Lie algebra of G. Suppose G is a p-adic group with p suﬃciently large and suppose G has Lie algebra g ⊂ End(X). Recall that P is the maximal ideal of the ring O of integers in our base ﬁeld k. Let P j , for j ∈ Z be the j-th power of P—it is the set of all elements zπ j , with z ∈ O. We deﬁne the valuation function ordk on k by setting ordk (x) = j

if x ∈ P j − P j+1 = π j O× .

Observe that ordk (xy) = ordk (x) + ordk (y). A sequence {xn } in k will converge to zero if and only if ordk (xn ) goes to +∞. Consider the exponential mapping, deﬁned by the usual power series: exp(x) =

∞ xn . n! n=0

This will converge if and only if the individual terms goto zero, which means xn = n ordk (x) − that their valuations should go to ∞. We have ordk n! ordk (n!). We can calculate that n n n + 2 + 3 + ... ordk (a) = ordk (p) ordk (n!) = p p p a=1 n . ≤ ordk (p) p−1 n x ordk (p) Hence ordk ≥ n ordk (x) − . Thus exp(x) makes sense pron! p−1 ordk (p) viding that ordk (x) > . In particular, if p is suﬃciently large relative p−1 to the degree of k over the p-adic numbers Qp , the function exp is deﬁned on the whole prime ideal P. n

Aﬃne-like Hecke algebras and p-adic representation theory

Set

 P  ..  J =.  P

57

 O ... O . . . . ..  . . . . O ... P

If p is suﬃciently large and the dimension of the base ﬁeld over Qp can be bounded, then the exponential map is deﬁned on J and exp(J) is a maximal pro-p subgroup of GL(X). Consider a Lie lattice, Λ, inside J, i.e. a lattice such that [Λ, Λ] ⊂ Λ. The exponential of a Lie lattice need not be a group, but a mild condition will guarantee that it is. Deﬁnition 5.1. A lattice Λ is elementarily exponentiable or e.e. if adp−1 (y)(Λ) ⊂ p2 Λ for all y ∈ Λ. If the Lie lattice Λ is elementarily exponentiable, then exp(Λ) is a group and therefore acts on the Lie algebra g via the adjoint action. Furthermore, ˆ given a character ψ ∈ Λ, Ad(exp Λ)(ψ) = ψ

if and only if

[Λ, Λ] ⊂ ker ψ.

Either of these equivalent conditions then implies that ψ ◦ log is a character of exp Λ, so this provides a useful way to construct characters. For semisimple groups these results generalize as follows. Let g be a semisimple Lie algebra with Killing form κ. Suppose Θ is an additive character of the base ﬁeld k such that O ⊂ KerΘ and π −1 O is not contained in ker Θ. Given a lattice Λ ⊂ g with Λ¯ g/Λ⊥ , where Λ⊥ is the annihilator of Λ with respect to κ, deﬁne ψx (y) = Θ(κ(x, y)) for x ∈ g and y ∈ Λ. (Note that it is not necessary to use the Killing form here. Any invariant bilinear form with no factors of p in its normalization would work.) Proposition 5.1. Given g ∈ G and two e.e. lattices Λ1 and Λ2 in g with respective characters ψz1 and ψz2 , g intertwines ψz1 ◦ log on exp Λ1 with ⊥ ψz2 ◦ log on exp Λ2 if and only if (z1 + Λ⊥ 1 ) ∩ Adg(z2 + Λ2 ) = ∅. (Here we assume ψzj ◦ log is a character on exp Λj for j = 1, 2.) This proposition provides a geometric condition for intertwining. Kirillov bicharacter Let Λ be an e.e. lattice. For x, y ∈ Λ, deﬁne ψz ([x, y]) = Θ(κ(z, [x, y])) = Θ(κ([z, x], y)) = Bψz (x, y). This is the Kirillov bicharacter, a skew symmetric bilinear form that deﬁnes a character in x and y. Then ψz ◦log is a character if the bicharacter is trivial, i.e. if ψz ([x, y]) = 1.

58

Roger Howe (Lecture Notes by Cathy Kriloﬀ)

Example 5.1 (sl2 ). Choose a basis {h, e+ , e− } for sl2 such that [h, e+ ] = 2e+ [h, e− ] = −2e− [e+ , e− ] = h. If Λo equals the usual Cartan decomposition, Λo = to ⊕ gα,o ⊕ g−α,o = Oh ⊕ Oe+ ⊕ Oe− , then set Λ = π Λo . Take > 1 and ψ = ψah where a ∈ π − O× . We now compute the Kirillov bicharacter. There is no contribution to Bψ except from commutators of elements in gα,o and g−α,o since each of the subalgebras in the Cartan decomposition is commutative. Thus it suﬃces to compute, for b, c ∈ O, Bψ (be+ , ce− ) = Θ(κ(ah, [be+ , ce− ])) = Θ(κ([ah, be+ ], ce− )) = Θ(2abcκ(e+ , e− )) = Θ(8abc). Set Λk,r,s = tk ⊕ gα,r ⊕ g−α,s where tk = π k to , gα,r = π r gα,o and g−α,s = π g−α,o . This is a lattice in sl2 . Then Bψ = 1 on Λk,r,s if and only if r +s ≥ . Take r + s = so that ψ˜ = ψ ◦ log is a character. Set L = Lk,r,s = exp(Λk,r,s ) and take k < so that Bψ is not trivial on all of Λk,r,s . s

5.3 Harish-Chandra homomorphism for SL2 Let ψ˜ and L be as above. Let A=

b 0 × : b ∈ k 0 b−1

be thediagonal torus in SL2 . Here and below, we will denote the matrix b 0 simply by the element b of k × appearing in its (1, 1)-entry. For each 0 b−1 b in A, it is easy to see that there is a function fb of the spherical function ˜ such that fb is supported on LbL, and fb (1 b2 ) = algebra H(SL2 //L, ψ) ˜ ψ(1 2 ). ˜ = LAL. Proposition 5.2. 1. Supp(H(SL2 //L, ψ)) 2. Setting f˜b = fb |A , the normalized restriction map τ : fb →

(µSL2 (L)µSL2 (LBL))1/2 ˜ fb µA (L ∩ A)

˜ and H(A//(L ∩ A), ψ). ˜ is an isomorphism between H(SL2 //L, ψ)

Aﬃne-like Hecke algebras and p-adic representation theory

59

Remark 5.1. ˜ is rather easy to understand. Note that, The Hecke algebra H(A//(L ∩ A), ψ) since A is commutative, double cosets are equal to left or right cosets. Thus, (L∩A)b(L∩A) = b(L∩A). Denote by fˆb the characteristic function of b(L∩A). Since ψ˜ can be extended from L∩A to a character of all of A, the map f˜b → fˆb ˜ to C ∞ (A/(L ∩ A)), which is deﬁnes an isomorphism from H(A//(L ∩ A), ψ) c just the convolution algebra of a discrete abelian group. µ 2 (L) The map τ is in fact an L2 isometry, up to the factor µASL(L∩A) . This means that the Plancherel measure of the series of representations of SL2 associated ˜ is essentially the same as the Plancherel measure for to H(A//(L ∩ A), ψ) A/(L ∩ A) (which is just Haar measure on the Pontrjagin dual of A/(L ∩ A)). Proof (Proof of Proposition 5.2). For simplicity in the following argument, we will assume that Haar measures have been normalized so that µSL2 (L) = 1 = µA (L ∩ A). This does not aﬀect the argument in any essential way, but does simplfy some formulas, in which factors involving these two measures would have to appear, absent such a normalization. We ﬁrst assume statement (i) and prove statement (ii). Write A = A+ ∪ − A , where b 0 −1 A+ = : b ∈ O 0 b−1 and −

A =

b 0 :b∈O 0 b−1

If we write b = π u, where u is a unit in O, we have that Ad(b)(Λk,r,s ) = Λk,r+2 ,s−2 . Thus A+ is the semigroup of elements of A that stretch gα , and shrink g−α , and A− is the semigroup of elements that do the opposite. In particular, if b1 and b2 are any two elements of A+ , then µSL2 (Lb1 L)µSL2 (Lb2 L) = µSL2 (Lb1 b2 L).

(7)

Given this, a very general argument lets us show that fb1 ∗fb2 = fb1 b2 . Indeed, let Lb1 L = ∪j j b1 L, and Lb2 L = ∪k k b2 L be decompositions of the respective double cosets of L into right cosets. We may assume that one of the j is the identity, and likewise for the k . Then, under the condition 7, we know from the proof of Proposition 1.2, that ∪j,k j b1 k b2 L is a decomposition of Lb1 b2 L into disjoint right L cosets. We can write fb1 = j ψ(j )δ j ∗δb1 ∗f1 , and similarly fb2 = k ψ(k )δ k ∗ ˜ In analogy with the δb2 ∗ f1 . Here f1 is the identity element of H(SL2 //L, ψ). argument of Proposition 1.2, we can write fb1 ∗ fb2 = j,k ψ(j k )δ j ∗ δb1 ∗ δ k ∗ δb2 ∗ f1 . As noted, thanks to condition 7 we know that the various terms in this sum are supported on disjoint right L cosets, so the only term which contributes to fb1 ∗ fb2 (b1 b2 ) is the term δb1 ∗ δb2 ∗ f1 . Hence fb1 ∗ fb2 (b1 b2 ) = fb1 b2 (b1 b2 ), so the two functions must be equal.

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Remark 5.2. Set U+ =

1x :x∈k 01

and U − =

10 :y∈k . y1

These are the root subgroups relative to the torus A. We also need integral versions of them: 1x 10 + r − s and Us = Ur = :x∈P :y∈P . 01 y1 Thus, Ur+= exp(gα,r ), and Us− = exp(g−α,s ). π w 0 If b = , with w a unit, and < 0 (so that b is in A+ ), then 0 π− w−1 we can write 10 bL. Lk,r,s bLk,r,s = LbL = y 1 s s−2 y∈P /P

1x 10 = ux and = uy . Then given b1 and b2 , with notation parallel 01 y1 to that for b, we see that Lb1 Lb2 L = u y b1 u z b2 L Set

y∈P s /P s−21 z∈P s /P s−22

=

y∈P s /P s−21 z∈P s /P s−22

=

uy uπ21 z b1 b2 L

y∈P s /P s−21 z∈P s /P s−22

=

1 0 b b L y + π −2 1 z 1 1 2

uy˜b1 b2 L.

y˜∈P s /P s−2(1 +2 )

Thus, the decomposition of the product double coset into right cosets can be done very explicitly and cleanly in this situation. We continue the proof of Proposition 5.2. Observe that, as a semigroup, A π 0 . Thus to ﬁnish showing is generated by A+ and by the element b1 = 0 π −1 that τ is a homomorphism, it is enough to show that fb1 ∗ fb−1 = µ(Lb1 L)f1 , 1 (−1 ' fb−1 f 1 . As in the remark above, or in other words, that µ(Lb11L)1/2 = µ(Lb1bL) 1/2 we can write δuy ∗ δb−1 ∗ f1 and fb1 = f1 ∗ δb1 ∗ δuy . fb−1 = 1

y∈P s /P s+2

1

y∈P s /P s+2

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Thus, the convolution of the two is fb1 ∗ fb−1 = f1 ∗ δb1 ∗ δuy ∗ δuz ∗ δb−1 ∗ f1 1

1

y,z∈P s /P s+2

=

f1 ∗ δb1 ∗ δuy+z ∗ δb−1 ∗ f1

y,z∈P s /P s+2

=# (P s /P s+2 )

1

f1 ∗ δb1 ∗ δuy˜ ∗ δb−1 ∗ f1 1

y˜∈P s /P s+2

= #(P s−2 /P s )

f1 ∗ δuy˜ ∗ f1 .

y˜∈P s /P s+2

Now the support criterion tells us that in this sum, only the term corresponding to y = 0 survives. Thanks to our normalization of Haar measure, this term is just f1 . Since also #(P s−2 /P s ) = µ(Lb1 L), this gives us the relation we wanted. Finally, let us demonstrate the support criterion, statement (i) of Proposition 5.2. This involves a fairly simple calculation plus an appeal to Proposition 5.1. The basis of the calculation is contained in the following lemma. ⊥ Lemma 5.1. Ad(L)(ah + (Λ⊥ k,r,s ∩ t)) = ah + Λk,r,s .

Proof. The proof is an easy exercise using Hensel’s Lemma. ˜ Then Proposition 5.2 says that we Suppose that g ∈ Supp(H(SL2 //L, ψ)). ⊥ can ﬁnd an element z ∈ Ad(g)(ah + Λ⊥ k,r,s ) ∩ (ah + Λk,r,s ). ¿From Lemma 5.1, we can write z = Ad(h1 )(y), with y ∈ ah + (Λ⊥ k,r,s ∩ t). Similarly, we can also write z = Ad(g)(Ad(hl2 ))(y ) for some 2 in L and y in ah+(Λ⊥ k,r,s ∩t). Comparing these two expressions for z, we see that Ad(h1 )(y) = Ad(g)Ad(h2 )(y ), or Ad(h−1 1 gh2 )(y ) = y. Notice that, because of the requirement that > k, the set ah+Λ⊥ k,r,s does not contain the zero element of t. This also implies that, since ah + Λ⊥ k,r,s is “convex”, in the sense that it contains the average of any two of its elements, if it contains an element y, it does not contain −y. But the only elements of t which are conjugate to some y in t are y itself and −y. Thus, we conclude that g = h−1 1 gh2 centralizes y. Since t consists simply of multiples of y, this means that g centralizes t, and hence belongs to A, or in other words g ∈ LAL, as desired. This concludes the proof of Proposition 5.2. Remark 5.3. The key point that makes Proposition 5.2 work is that we chose r and s to make r + s = , which is the largest possible value such that the Kirillov form vanishes on Λk,r,s , so that we can use ψah ◦ log to deﬁne a character on L = exp(Λk,r,s . The fact that r + s = is what makes Lemma 5.3.2 true, and this is in turn is what makes part i) of Proposition 5.2 work. Further, as we saw in the proof, the second part of Proposition 5.2, which is what we really

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want to know, depends crucially on the ﬁrst part. We note again that there are many possible pairs r, s that will work; the essential condition is on their sum. Let Ao = A+ ∩ A− be the maximal compact subgroup of A. It is easy to see that Ao normalizes L, so that the product Ao L will be a group. Further, the character ψah of L, extends to Ao L. The extensions are naturally identiﬁed with the extensions from A ∩ L to Ao of ψ|A∩L . Denote one such extension by Ψ . Then easy arguments show that H(A//Ao , Ψ ) Cc (A/Ao ) Cc (k × /O) Cc (Z). This may be regarded as the simplest possible example of an aﬃne Hecke algebra. Furthermore, an easy corollary of Proposition 5.2 is that H(SL2 //Ao L, Ψ ) H(A//Ao , Ψ ). Thus, Proposition 5.2 is a prototype for more general Hecke algebra isomorphisms. We note that the spherical function algebra H(SL2 //Ao L, Ψ ) controls the non-spherical principal series of SL2 corresponding to all characters of A ˜ is isomorwhich restrict to Ψ on Ao . Also, one can show that H(SL2 //L, ψ) phic to the direct sum of the algebras H(SL2 //Ao L, Ψ ) where Ψ runs over all ˜ controls possible extensions of ψah from A ∩ L to Ao . Thus, H(SL2 //L, ψ) the union of a collection of principal series representations of SL2 . For completeness in the case of SL2 , one should also analyze spherical function × algebras H(SL2 //J, φ), where φ is a character of J/J1 Ao /A1 k . Here −1 one can show that if φ = φ , then H(SL2 //J, φ) Cc (Z). In the case of ˜ SL ) is the the sign character sgn = sgn−1 , one has H(SL2 //J, sgn) Cc (W 2 group algebra of the aﬃne Weyl group of SL2 , which is the inﬁnite dihedral group. 5.4 General minimal principal series The example described in detail in Section 5.3 is not so diﬀerent from the general case, at least not when p is suﬃciently large. Let g be a split semisimple Lie algebra, and let t ⊂ g be a split Cartan subalgebra. Write g=t⊕ gα α

for the root space decomposition of g with respect to t. The gα are the root spaces - the eigenspaces for the action adt of t on g by commutator. Let to ⊂ t be the lattice in t consisting of elements x such that the eigenvalues of ad(x) are integers. We can ﬁnd a Lie lattice Λ ⊂ g, containing to . Under the assumption that the residual characteristic p is large, we will have Λ = to ⊕ Λ ∩ gα α

for any lattice containing to and invariant under adto .

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63

Recall that the Killing form κ( , ) on g is deﬁned by the formula κ(x, y) = tr(adx, ady). An element x of a Lie lattice such as Λ will of course satisfy adx(Λ) ⊂ Λ. It follows that κ(x, y) must also be integral. Thus, letting Λ∗ denote the lattice dual to Λ by the Killing form, we will have Λ ⊂ Λ∗ . Thus, if we have a Lie lattice in g such that Λ∗ = Λ, it must be a maximal Lie lattice. Chevalley’s description of the generators and relations for a simple Lie algebra in terms of roots shows that when p is large (actually, it does not have to be very large for this), self-dual Lie lattices do exist in any semi-simple Lie algebra g. For g = sln , the stabilizer of any lattice is a self-dual Lie lattice, and for g = sp2n , the stabilizer of any self-dual lattice is a self-dual Lie lattice. For any ﬁxed root α, set tα = [gα , g−α ]. This is a line in t. The span sα = tα ⊕ gα ⊕ g−α is a Lie algebra, isomorphic with sl2 . We have a decomposition g = sα ⊕ s⊥ α, where the ⊥ is with respectto κ. Suppose that Λ = to + α Λ ∪ gα is a self-dual Lie lattice in g. Then the intersection sα ∩ Λ is a self-dual Lie lattice in sα , and Λ = sα ∩ Λ ⊕ s⊥ α ∩ Λ. Set gα,o = gα ∩ Λ and tα,o = tα ∩ Λ. The fact that Λ is self-dual implies that [gα,o , gβ,o ] = gα+β,o whenever α+β is a root. Similarly, [gα,o , g−α,o ] = tα,o . As in Section 5.3, for an integer , set Λ = π Λ, and t = π to , and gα, = π gα,o and tα, = π tα,o . Let ψ be a character of t1 . We can represent ψ in the form ψ(x) = χ(κ(x, yψ )), where χ is a character of (the additive group of) k, such that O ⊂ ker χ, but π −1 O ⊂ ker χ. By the depth of ψ, denoted dψ , we mean the smallest such that t ⊂ ker ψ. This is also equal to the smallest such that yψ ∈ t− . Similarly deﬁne dψ,α to be the smallest such that tα, ⊂ ker ψ. Notice that dψ = maxα dψ,α The notion of depth extend in a straightforward way from characters of t1 to characters of tk . Lemma 5.2. dψ,α+β ≤ max{dψ,α , dψ,β } The proof uses that tα+β, ⊂ tα, + tβ, , which can be shown by considering rank 2 algebras. In fact for the classical cases, only sl3 and sp4 need to be considered. Corollary 5.1. For any ﬁxed d0 , {β | dψ,β ≤ d0 } span a Levi component Mψ,d0 .

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Roger Howe (Lecture Notes by Cathy Kriloﬀ)

The Levi component Mψ,d0 in the Corollary consists of all directions where the character ψ is trivial. In general the roots where a particular character has depth less than or equal to a speciﬁed level, say j, forms a Levi subalgebra and every Levi subalgebra can be described in this way. If such an Mψ,j = t then we say ψ is nondegenerate at level j. Choose an ordering for the roots. Set J = t1 ⊕ gα,o ⊕ gα,1 . α>0

α<0

Then J is the Lie lattice corresponding to an Iwahori subgroup of G. Given any character ψ of t1 , we may extend ψ to a character of J by declaring ψ to be trivial on any root space: ψ(x) = 1, for all x in gα ∩ J. Deﬁne a function on J × J by the formula Bψ (x, y) = ψ([x, y]) for x and y in J. This is the Kirillov bicharacter associated to ψ. Analogous to the sl2 case, consider the Kirillov bicharacter associated to ψ and construct a Lie lattice Λψ maximal with respect to the condition Bψ |Λψ = 1. (Bψ breaks into a sum of bicharacters on copies of sl2 .) For each α such that dψ,α > j, choose aα , a−α ≥ 0 so that aα + a−α = dψ,α . If dψ,α = j, set aα = a−α = j. We also require that Λψ = tk ⊕ α gα,aα be a Lie lattice (aα+β ≤ aα + aβ ). There do exist such possibilities )for aα* . For dψ,α for example, take aα = 0 for α > 0 and a−α = dψ,α , or take aα = 2 * ) dψ,α + 1 α > 0 and a−α = dψ,α − aα = . 2 Extend ψ to a character of Λψ by declaring ψ to be trivial on gα,aα . Then ψ˜ = ψ ◦ log is a character of Lψ = exp Λψ . Let Mψ,1 = Mψ be as in Corollary 5.1. ˜ then g ∈ Lψ Mψ Lψ . Lemma 5.3. If g ∈ Supp(H(G//Lψ , ψ)), ˜ 1 k2 ) for k1 , k2 ∈ Lψ , and fm to Let m ∈ Mψ and deﬁne fm (k1 mk2 ) = ψ(k have value 0 oﬀ of Lψ mLψ . Then set f˜m = fm |Mψ . Theorem 5.1. With L˜ψ = Lψ ∩ Mψ , the map τ : fm →

µG (Lψ )µG (Lψ mLψ ) µMψ (L˜ψ )µMψ (L˜ψ mL˜ψ )

1/2 f˜m

is an isomorphism of algebras and is, up to a constant multiple, an L2 isometry. The proof is nearly the same as for G = SL2 above.

Aﬃne-like Hecke algebras and p-adic representation theory

65

6 Consequences The discussion in Section 5.4 implies the minimal principal series can be ˆ This is done by eliminating analyzed by reduction to H(G//J, φ) for φ ∈ J. the co-root directions where the character has positive depth on the torus and focusing on where the depth is zero. Such algebras have been analyzed by A. Roche and shown to be aﬃne Hecke algebras. In fact these ideas can be extended much further. Recall that Λo = to ⊕ α gα,o is a lattice corresponding to a good maximal parahoric subgroup. Also recall that J = to ⊕ gα,o ⊕ g−α,1 α>0

plays the role of the Iwahori lattice. Choose a Lie lattice Λˇ such that J ⊂ Λˇ ⊂ Λo to play the role of the parahoric lattice. Let Γ be a ﬁnite subgroup of AutG with |Γ | < p (although for many of the results, any cardinality relatively prime to p is suﬃcient). Then g = gΓ ⊕ h and we deﬁne the projection map prΓ : g → gΓ by 1 γ(x). prΓ (x) = |Γ | γ∈Γ

Notice that [gΓ , h] ⊂ h. Suppose that Λˇ and t are stable under the action of Γ . Then γ(gα ) = gγ(α) and γ(gα,(α) ) = gγ(α),(γ(α)) , where (α) is 0 or 1 as appropriate. ˇk (the isomorphism holds since the order Now take ψ ∈ (ˆtk )Γ tΓk → Λ of Γ is relatively prime to p). Then the depths will be constant on the orbits of γ, i.e. dψ,α = dψ,γ(α) . Construct Λψ to be Γ invariant. If −α and α are in the same orbit under Γ then to do this we must adjust the α and −α root spaces simultaneously which may not be consistent with the requirement that aα + a−α = dψ,α , so this may not quite be possible. The best possible may only be aα + a−α = dψ,α ± 1. If so, deﬁne Λ ψ using dψ,α − 1 and Λψ using dψ,α + 1 (so Λψ ⊂ Λ ψ ) as well as L ψ = exp Λ ψ and Lψ = exp Λψ . Then 1. 2.

ψ˜ = ψ ◦ log is a character of Lψ . ψ˜ is AdL ψ invariant.

3. There exists a unique character ρψ ∈ L ψ such that ρψ |L = mψ˜ where ψ + +1/2 +L + + ψ+ m = dim ρψ = + + . + Lψ + Modulo the kernel of ψ˜ we have the Heisenberg group and we take representations of it. We get a similar result to that before for ﬁxed points, namely

66

Roger Howe (Lecture Notes by Cathy Kriloﬀ)

Γ (Lψ )Γ = Lψ ∩ GΓ = exp ΛΓ ψ and Lψ = (Lψ ) ⊕ (Lψ ∩ h) with a similar result for the case of Lψ .

Proposition 6.1. Supp(H(GΓ /(L ψ )Γ , ρΓψ )) = (L ψ )Γ MψΓ (L ψ )Γ It is an open question whether there is an analogous Hecke algebra isomorphism associated with this support result. If so, then this may complete 80-90% of the theory. This was done in some cases for G = GLn Howe and Moy, but has not been pushed further. Example 6.1. Let G be a reductive group over k. Let k˜ be a Galois extension ˜ k) over k which splits some or possibly all of the tori in Gk . Let Γ = Gal(k, ˜ which acts on Gk˜ and the Lie algebra gk˜ = gk ⊗ k. Lemma 6.1. If t ⊂ gk and t ⊗ k˜ is split, then we can ﬁnd a Γ -stable Lie lattice Λˇ for t ⊗ k˜ ⊂ gk˜ . This has been veriﬁed when G is classical. Then M Γ is the centralizer of a torus and can be compact. In the extreme case where Γ is nondegenerate, M Γ is a torus and again might be compact. If M Γ is compact then the corresponding spherical function algebra is ﬁnite dimensional, and the representation induced from such ψ will be a ﬁnite sum of irreducible supercuspidal representations. This case of the construction yields all known supercuspidal representations. Supercuspidal representations which correspond to characters of compact tori were constructed by Morris The above construction extends Morris’s work, and explains the origin of other supercuspidal representations that do not correspond to characters of compact tori. Example 6.2. If G = Sp2n then M Γ is a product of unitary groups, copies of GLk , and possibly one copy of Sp2m . This yields supercuspidal representations corresponding to characters of a non-maximal torus whose centralizer is a product of compact unitary groups. At the moment, this kind of construction accounts for all known examples of supercuspidal representations, at least in the regime of large residual characteristic. For the full story, what we have done here must be further reﬁned, by extending ρΓψ to a cuspidal representation on a parahoric subgroup in M Γ . For Sp2n , this more reﬁned construction yields supercuspidal representations when the centralizer of a torus is contains no GLk factors. One can ask whether the Hecke algebras that arise in this way are all generalized aﬃne Hecke algebras. The answer is yes in many cases. For the case of GLn this was settled by Howe-Moy and Bushnell-Kutzko (with no restriction on characters). For algebras associated to level zero, this was done by Morris. A. Roche has obtained results for principal series, and many new examples have been provided by J. Kim.

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These notes represent an adaptation and summary or synthesis of a range of papers. Rather than give precise references, we list below for each section a selection of papers which serve either as background or development of the content of the notes. It is a pleasure to thank Cathy Kriloﬀ for shouldering the main burden in preparing the text.

References 2. Structure of p-adic GL(V ) and Sp(V ) [BT] F. Bruhat and J. Tits, Groupes r´eductifs sur un corps local I. Inst. Hautes ´ Etudes Sci. Publ. Math. 42 (1972), 1–251.

3. Structure of the Iwahori Hecke Algebra [BZ] [Bo]

[Ca] [Ch] [H4] [KL] [L1] [L2] [Ro]

J. Bernstein and A. Zelevinsky, Representations of the group GL(n, F ) where F is a non-archimedean local ﬁeld. Russian Math. Surveys 31 (1976), 1–68. A. Borel, Admissible representations of a semisimple group over a local ﬁeld with vectors ﬁxed under an Iwahori subgroup. Invent. Math. 35 (1976), 233– 259. W. Casselman, The unramiﬁed principal series of p-adic groups. I. The spherical function. Compositio Math. 40 (1980), 387–406. I. Cherednik, A new interpretation of Gelfand-Tsetlin bases. Duke Math. J. 54 (1987), 563–577. R. Howe, Hecke algebras and p-adic GLn . Contemp. Math. 177 (1994), 65– 100. D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke Algebras. Invent. Math. 87 (1987), 153–215. G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups. Trans. Amer. Math. Soc. 277 (1983), 623–653. —, Aﬃne Hecke algebras and their graded version. J. Amer. Math. Soc. 2 (1989), 599–635. J. Rogawski, On modules over the Hecke algebra of a p-adic group. Invent. Math. 79 (1985), 443–465.

4. Results on Representations of G [BM1] D. Barbasch and A. Moy, A unitarity criterion for p-adic groups. Invent. Math. 98 (1989), 19–37. [BM2] —, Reduction to real inﬁnitesimal character in aﬃne Hecke algebras. J. Amer. Math. Soc. 6 (1993), no. 3, 611–635. [BM3] —, Whittaker Models with Iwahori Fixed Vector. Contemporary Math. 177 (1994), ams [BM4] —, Unitary Spherical Spectrum for p-adic classical groups. Acta Appl. Math. 44, Kluwer Academic Press, 1996.

68 [C]

[Ca]

Roger Howe (Lecture Notes by Cathy Kriloﬀ) P. Cartier, Representations of p-adic groups: a survey. Automorphic Forms, Representations and L-functions, A. Borel and W. Casselman (eds.), Proc. Symp. Pure Math. XXXIII, Parts 1 & 2, American Mathematical Society, Providence, RI, 1979, 111–155. W. Casselman, The unramiﬁed principal series of p-adic groups I. The spherical function. Compositio Math. 40 (1980), 387–406.

5. Spherical Function Algebras [BK1] C. Bushnell and P. Kutzko, The Admissible Dual of GL(N ) via Compact Open Subgroups, Ann. of Math. Stud., vol. 129, Princeton Univ. Press, 1993. [H1] R. Howe, Kirillov theory for compact p-adic groups. Paciﬁc J. Math. 73 (1977), 365–381. [H4] —, Hecke algebras and p-adic GLn . Contemp. Math. 177 (1994), 65–100. [R] A. Roche, Types and Hecke Algebras for principal series representations of ´ split reductive p-adic groups. Ann. Sci. Ecole Norm. Sup. 31 (1998), 361– 413.

6. Consequences J. Adler, Reﬁned anisotropic K-types and supercuspidal representations. Paciﬁc J. Math. 185 (1998), 1–32. [As] C. Asmuth, Weil representations of symplectic p-adic groups. Amer. J. Math. 101 (1979), 885–908. [BK1] C. Bushnell and P. Kutzko, The Admissible Dual of GL(N ) via Compact Open Subgroups, Ann. of Math. Stud., vol. 129, Princeton Univ. Press, 1993. [BK2] —, Smooth representations of reductive p-adic groups: Structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. [H2] R. Howe, Some qualitative results on the representation theory of Gln over a p-adic ﬁeld. Paciﬁc J. Math. 73 (1977), 479–538. [H3] —, Tamely ramiﬁed supercuspidal representations of Gln . Paciﬁc J. Math. 73 (1977), 437–560. [H4] —, Hecke algebras and p-adic GLn . Contemp. Math. 177 (1994), 65–100. [HM1] R. Howe (with A. Moy), Harish-Chandra Homomorphisms for p-adic Groups, CBMS Regional Conf. Ser. in Math., vol. 59, Amer. Math. Soc., Providence, RI, 1985. [HM2] R. Howe and A. Moy, Hecke algebra isomorphisms for GL(n) over a p-adic ﬁeld. J. Alg. 131 (1990), 388–424. [K] J.-L. Kim, Hecke algebras of classical groups over p-adic ﬁelds and supercuspidal representations. Amer. J. Math. 121 (1999), no. 5, 967–1029. [L3] G. Lusztig, Classiﬁcation of unipotent representations of simple p-adic groups. Internat. Math. Res. Notices 11 (1995), 517–589. [M1] L. Morris, Some tamely ramiﬁed supercuspidal representations of symplectic groups. Proc. London Math. Soc. 63 (1991), 519–551. [M2] —, Tamely ramiﬁed supercuspidal representations of classical groups II. Rep´ Norm. Sup. 25 (1992), 233–274. resentation theory. Ann. Sci. Ec. [M3] —, Tamely ramiﬁed intertwining algebras. Invent. Math. 114 (1994), 1–54. [A]

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[MP1] A. Moy and G. Prasad, Unreﬁned minimal K-types for p-adic groups. Invent. Math. 116 (1994), 393–408. [MP2] —, Jacquet Functors and unramiﬁed minimal K-types. Comment. Math. Helv. 71 (1996), 98–121. [Wa] J.-L. Waldspurger, Alg´ebras de Hecke et induites de repr´esentations cuspidales, pour GL(N ). J. Reine Angew. Math. 370 (1986), 127–191. [Y] J.-K. Yu, Tame construction of supercuspidal representations. preprint, 1998.

Notes on aﬃne Hecke algebras George Lusztig Department of Mathematics, M.I.T., Cambridge, MA 02139 [email protected] Supported in part by the National Science Foundation

Introduction Aﬃne Hecke algebras play an important role in the study of representations of reductive groups over a p-adic ﬁeld. They also encode critical information about the characters of modular representations of semisimple algebraic groups in positive characteristic and of quantum groups at roots of 1. One approach to aﬃne Hecke algebras is via equivariant K-theory. In this approach, the parameter of the algebra is interpreted as the standard generator of the representation ring of C∗ . In these lectures, the K-theoretic approach to the aﬃne Hecke algebras is emphasized. We will use this approach to give an exposition of the classiﬁcation of the simple modules of the aﬃne Hecke algebra. We will also give a survey of the results of [L6] on bases in equivariant K-theory (subregular case) and will extend these results to the not necessarily simply laced case. Contents 1. The aﬃne Hecke algebra 2. H and equivariant K-theory 3. Convolution 4. Subregular case 5. Subregular case: type A 6. Subregular case: types C, D, E, F, G

1 The aﬃne Hecke algebra 1.1 Let X, Y be two ﬁnitely generated free abelian groups with a given perfect αi |i ∈ I} ⊂ Y ) pairing , : X × Y → Z. Assume that {αi |i ∈ I} ⊂ X (resp. {ˇ are the simple roots (resp. simple coroots) of a root datum of ﬁnite type in X, Y . For i ∈ I let si : X → X be the reﬂection si (x) = x − x, α ˇ i αi and let W be the subgroup of GL(X) generated by si , i ∈ I (the Weyl group, a Coxeter group). Let A = Z[v, v −1 ] where v is an indeterminate. Let A[X] be the group algebra of X with coeﬃcients in A. The basis element of A[X] corresponding to x is denoted by [x]. Let H be the A-algebra with 1 deﬁned by the generators

I. Cherednik et al.: LNM 1804, M.W. Baldoni and D. Barbasch (Eds.), pp. 71–103, 2003. c Springer-Verlag Berlin Heidelberg 2003

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T˜i (i ∈ I) and θx (x ∈ X) and the relations (a) (T˜i + v −1 )(T˜i − v) = 0 for any i ∈ I; (b) T˜i T˜j T˜i · · · = T˜j T˜i T˜j . . . (µ factors in both products), for any i = j in I with si sj of order µ in W ; (c) θx θx = θx+x for any x, x ∈ X; (d) θ0 = 1; (e) θx T˜i − T˜i θsi (x) = (v − v −1 )θ [x]−[si (x)] 1−[−αi ]

for any i ∈ I, x ∈ X. The fraction in (e) is apriori an element of the quotient ﬁeld of A[X] but it actually belongs to A[X]; for p = x∈X cx [x] ∈ A[X] with cx ∈ A we write θp = x∈X cx θx ∈ H. 1.2 More generally, assume that we are given functions λ : I → N, λ∗ : {i ∈ I|ˇ αi ∈ 2Y } → N ˇ i = αi , α ˇ i = −1. such that λ(i) = λ(i ) whenever i, i ∈ I satisfy αi , α ∗ To λ, λ we can attach the A-algebra H with 1 deﬁned by the generators T˜i (i ∈ I) and θx (x ∈ X) and the relations 1.1(b)-(d) together with (a) (T˜i + v −λ(i) )(T˜i − v λ(i) ) = 0 for any i ∈ I; (b) θx T˜i − T˜i θsi (x) = (v λ(i) − v −λ(i) )θ [x]−[si (x)] 1−[−αi ]

for any i ∈ I, x ∈ X with α ˇi ∈ / 2Y ; ∗ ∗ (c) θx T˜i − T˜i θsi (x) = ((v λ(i) − v −λ(i) ) + (v λ (i) − v −λ (i) )θ−αi )θ [x]−[si (x)] 1−[−2αi ]

for any i ∈ I, x ∈ X with α ˇ i ∈ 2Y . The fractions are interpreted as in 1.1. In the ”special case” where λ(i) = 1 for all i and λ∗ (i) = 1 for all i such that λ∗ (i) is deﬁned, we recover the algebra in 1.1. 1.3 H is called an aﬃne Hecke algebra. It is another incarnation of the Hecke algebra of an (extended) aﬃne Weyl group introduced by Iwahori and Matsumoto [IM] in which the parameters attached to the various (aﬃne) simple reﬂections are allowed to depend on the simple reﬂection. The fact that the Hecke algebra in the Iwahori-Matsumoto presentation can be described in the form above has been stated by Bernstein (in the ”special case” 1.1); the proof was given in [L1] in the ”special case” and in [L3] in general. The existence of the two incarnations of H reﬂects (in the ”special case” 1.1) the fact that an algebraic group over a p-adic ﬁeld is at the same time an inﬁnite dimensional group over a ﬁnite ﬁeld. A strategy for classifying irreducible representations of a semisimple split p-adic group, say, with non-zero vectors ﬁxed by an Iwahori group, has been to convert the problem to that of the classiﬁcation of irreducible representations of an aﬃne Hecke algebra in the

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√

Iwahori-Matsumoto presentation (with v specialized to a power of p), then to pass to the presentation in 1.1 which is particularly well suited for the study of representations. 1.4 For w ∈ W we set

T˜w = T˜i1 T˜i2 . . . T˜in ∈ H

where i1 , i2 , . . . , in is any sequence in S such that w = si1 si2 . . . sin and such that n is minimum possible; this is known to be independent of the choice of sequence. Recall that n is the length, l(w) of w. H is free as an A-module. Indeed, the elements T˜w θx (with w ∈ W, x ∈ X) form an A-basis of H; the elements θx T˜w (with w ∈ W, x ∈ X) form another A-basis of H. (See [[L3] 3.4].) 1.5 Let w0 be the longest element of W . Let χ → χ be the involutive antiautomorphism of the A-algebra H deﬁned by T˜i → T˜i for i ∈ I and ˜ T˜w−1 θ → θ for x ∈ X. T −x w0 (x) w0 0

2 H and equivariant K-theory 2.1 The idea that the representations of H (in the ”special case” 1.1) should be geometrically understood in terms of equivariant K-theory, the parameter v being interpreted as the standard generator of the representation ring of C∗ , was formulated in [L2], where the principal series representations of H were treated from this point of view. Subsequently, this idea has been developed in [KL], [GI]. In this section we want to explain the method of [KL] to construct representations of H. We shall need some K-theoretical preparation. 2.2 All algebraic varieties are assumed to be over C. If M is a linear algebraic group we say that Z is an M -variety if M acts on Z and Z can be imbedded as a closed M -stable subvariety of a smooth algebraic variety with an algebraic action of M . For such Z, let CohM (Z) be the category of coherent M -equivariant sheaves on X and let V ecM (Z) be the full subcategory of CohM (Z) whose objects are locally free coherent sheaves. (We identify objects of V ecM (Z) with the corresponding M -equivariant vector bundles on

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Z.) Let KM (Z) be the Grothendieck group of the category CohM (Z). This is naturally an RM -module where RM = KM (point) is the Grothendieck ring of ﬁnite dimensional representations of M . Let Z be another M -variety and let f : Z → Z be an M -equivariant morphism. If f is smooth, then the inverse image f ∗ : KM (Z ) → KM (Z) is well deﬁned; if f is proper, the the direct image f∗ : KM (Z) → KM (Z ) is well deﬁned (it is deﬁned using an alternating sum of higher direct images). If Z is a closed M -stable subvariety of Z, let CohM (Z; Z ) be the full subcategory of CohM (Z) whose objects are the objects of CohM (Z) whose support is contained in Z . Let KM (Z; Z ) be the Grothendieck group of the cate∼ gory CohM (Z; Z ). We have an obvious isomorphism KM (Z )−→KM (Z; Z ). Let φ : E1 → E0 be a morphism in V ecM (Z) and let Z be a closed subvariety of Z such that φ is an isomorphism over Z −Z . Let F ∈ CohM (Z). Let K1 , K0 be the kernel and cokernel of the morphism 1⊗φ : F ⊗E1 → F ⊗E0 in CohM (Z). Then K1 , K0 ∈ CohM (Z; Z ). Thus, K1 , K0 give rise to elements ˜ 0 ∈ KM (Z; Z ). Then F → K ˜0 − K ˜ 1 is a well deﬁned homomorphism ˜ 1, K K γφ : KM (Z) → KM (Z; Z ) = KM (Z ). 2.3 Let G be a connected reductive algbraic group over C. Let g be the Lie algebra of G and let gn be the variety of nilpotent elements in g. Let B be the variety of all Borel subalgebras of g. For any parabolic subalgebra p of g we denote by np the nil-radical of p. A parabolic subalgebra p of g is said to be almost minimal if the variety of Borel subalgebras contained in p is 1-dimensional. Let I be a ﬁnite set indexing the G-orbits on the set of almost minimal parabolic subalgebras (for the adjoint action). A parabolic subalgebra in the G-orbit indexed by i is said to have type i. Let Pi be the variety of all parabolic subalgebras of type i. Let πi : B → Pi be the morphism deﬁned by πi (b) = p where b ∈ B, p ∈ Pi , b ⊂ p. Let X be the set of isomorphism classes of algebraic G-equivariant line bundles on B where G acts on B by the adjoint action. Then X is a ﬁnitely generated free abelian group under the operation given by tensor product of line bundles. For each i ∈ I, let Li ∈ X be the tangent bundle along the ﬁbres of πi : B → Pi . Given i ∈ I and L ∈ X, we deﬁne an integer m by the requirement that the Euler characteristic of any ﬁbre of πi (a projective line) with coeﬃcients in the restriction of L to that ﬁbre (regarded as a coherent sheaf) is m + 1. We set m = α ˇ i (L) ∈ Z. Then α ˇ i : X → Z is a group homomorphism. We will often write the tensor product of two line bundles L, L as LL and the dual of L as L−1 . Let X be a free abelian group (in additive notation) with a given isomor∼ phism X −→X denoted by x → Lx . (Thus, Lx Lx = Lx+x for x, x ∈ X.) Let

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α ˇ

i αi ∈ X be deﬁned by Lαi = Li . The composition X → X−→Z is denoted ˇ i ∈ Y ) are as in again by α ˇ i . Let Y = Hom(X, Z). Then (X, Y, αi ∈ X, α 1.1 and the most general (X, Y, αi ∈ X, α ˇ i ∈ Y ) in 1.1 is thus obtained. We deﬁne W, H as in 1.1.

2.4 Let G = G × C∗ . We regard B and g as G-varieties with G-action (g, λ) : b → Ad(g)b and (g, λ) : y → λ−2 Ad(g)y. Then gn is a G-stable subvariety of g. Let M be a closed subgroup of G. Then B, gn can be regarded as M -varieties. We denote by v the element pr2 M −→C∗ of RM . Then for any M -variety U , we can regard v as a line bundle in V ecM (U ), by taking the inverse image of v ∈ KM (point) under U → point. For an integer n we deﬁne v n ∈ V ecM (U ) to be the n-th tensor power of v. Let e ∈ gn ; assume that M is contained in the stabilizer of e in G. Then Be = {b ∈ B|e ∈ b} is an M -stable subvariety of B. Let i ∈ I. Let Be,i = {p ∈ Pi |e ∈ p},

Be,i = {b ∈ B|e ∈ πi (b)}.

Deﬁne a : Be → Be,i , b : Be,i → Be,i by b → πi (b). Then a is proper and b is 1 smooth (a P -bundle). Let L be the line bundle on Be,i whose ﬁbre at b is p/b where p = πi (b) ∈ Pi . This is restriction of the G-equivariant (hence G-equivariant, with trivial C∗ -action) line bundle Li on B hence is an M -equivariant line bundle. It has a canonical section whose value at b is the image of e ∈ πi (b) in πi (b)/b. Since the sections of L are the same as the sections of v −2 L, we obtain a section of the line bundle v −2 L, which is in fact M -invariant. This section of v −2 L vanishes exactly over Be . It deﬁnes a map of line bundles C → v −2 L; taking duals and tensoring by v −1 we ﬁnd a map of line bundles φ : vL−1 → v −1 which is an isomorphism outside Be . By the construction in 2.2, φ gives rise to a homomorphism γφ : KM (Be,i ) → KM (Be ). We deﬁne a homomorphism ci : KM (Be ) → KM (Be ) as the composition b∗

γφ

∗ KM (Be )−→K M (Be,i )−→KM (Be,i )−→KM (Be ).

a

Next, let x ∈ X. We regard Lx as an object of V ecG (B) with C∗ acting trivially. We can also regard Lx as an object of V ecM (Be ) by restriction from B. Let θx : KM (Be ) → KM (Be ) (a) be the homomorphism deﬁned by F → Lx ⊗ F.

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Let (s, c) ∈ G be an element in the stabilizer of e and let M be the smallest diagonalizable subgroup of G containing (s, c). Now (s, c) deﬁnes a ring homomorphism h : RM → C (it attaches to an M -module the trace of (s, c) on that M -module). This makes C into an RM -module and we can form C ⊗RM KM (Be ). The operators ci , θx : KM (Be ) → KM (Be ) are RM -linear hence they induce operators C ⊗RM KM (Be ) → C ⊗RM KM (Be ) denoted again by ci , θx . In the remainder of this section and in the next section we assume that G has simply connected derived group. Proposition 2.5 There is a unique H-module structure on C ⊗RM KM (Be ) such that v n ∈ A acts as multiplication by h(v n ), v − T˜i acts as ci and θx is induced by the map 2.4(a). This is proved in [KL] except that there we use equivariant topological Khomology instead of the Grothendieck group KM (). But in our case the two theories coincide, thanks to [DLP]. 2.6 Let A be the subgroup of G consisting of all g ∈ G such that Ad(g)e = e and gs = sg. This group acts naturally on Be and its action commutes with the action of M . For any g ∈ A and any F ∈ CohM (Be ), we have (g −1 )∗ F ∈ CohM (Be ). This deﬁnes an action of A on KM (Be ) which is RM linear hence deﬁnes an action of A on C ⊗RM KM (Be ). For any irreducible C-representation ρ of A which is trivial on the identity component A0 , we consider Eρ = HomA (ρ, C⊗RM KM (Be )). The H-module structure on C⊗RM KM (Be ) is compatible with the A-module structure, hence it induces an Hmodule structure on Eρ . Proposition 2.7 Assume that c ∈ C∗ is not a root of 1. (a) We have Eρ = 0 if and only if ρ appears in the complex homology space of {b ∈ Be |Ad(s)b = b} regarded as an A-module in a natural way. ¯ρ . (b) If Eρ = 0 then Eρ has a unique simple quotient H-module E (c) Consider the set of G-conjugacy classes of triples (e, s, ρ) where e ∈ gn , s is a semisimple element of G such that Ad(s)e = c2 e and ρ is an irreducible representation of A/A0 (as in 2.6) satisfying (a). On the other hand, consider the set of isomorphism classes of simple H-modules (over C) in which v acts ¯ρ . as multiplication by c. These two sets are in bijection under (e, s, ρ) → E The comments after the statement of 2.5 apply here as well. 2.8 Let e, h, f be an sl2 -triple of g. Let ζ : SL2 → G be the homomorphism of algebraic groups whose tangent map at 1 carries

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-

0 ( 00 10 ) to e, ( 01 00 ) to f , 0 −1 to h. Let C be a maximal torus of {g ∈ G|Ad(g)e = e, Ad(g)h = h, Ad(g)f = f }. Let H = C × C∗ . We identify H with a closed subgroup of G via , 0 , c). (g, c) → (gζ 0c c−1

Then Be is an H-stable subvariety of B. Proposition 2.9 There is a unique H-module structure on KH (Be ) such that v n ∈ A acts as multiplication by v n ∈ RH , v − T˜i acts as ci and θx acts as in 2.4(a). This follows by applying 2.5 for all (s, c) ∈ H and using [[L5], 1.14]. 2.10 Let i ∈ I and let R be a closed H-stable subvariety of Be . We say that R is i-saturated if the following holds: b ∈ R, b ∈ Be , πi (b) = πi (b ) =⇒ b ∈ R. If this condition is satisﬁed, then the deﬁnition of T˜i makes sense for R instead of Be and yields an RH -linear map T˜i : KH (R) → KH (R). This is compatible with T˜i : KH (Be ) → KH (Be ) under the direct image map KH (R) → KH (Be ) induced by the inclusion R ⊂ Be . Hence the image of this map is is stable under T˜i : KH (Be ) → KH (Be ).

3 Convolution 3.1 In this section we describe an alternative method to deﬁne an H-module structure on K M (Be ) (as in 2.4) which follows Ginzburg [GI] (with a diﬀerent normalization, as in [L4]) and is based on a construction (convolution) which ﬁrst appeared (in a non-equivariant setting) in [KT]. 3.2 As a preparation, consider a linear algebraic group M , a smooth M -variety Z and two closed M -stable subsets Z , Z of Z. Then we have an RM -bilinear intersection product (or ”Tor product”) KM (Z ) × KM (Z ) → KM (Z ∩ Z )

(a)

denoted by F , F → F ⊗L F . It is deﬁned as follows. Start with F ∈ CohM (Z ), F ∈ CohM (Z ). We regard F , F as sheaves on Z, zero outside Z , Z respectively. We can ﬁnd a ﬁnite resolution of F by

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sheaves in V ecM (Z); the same applies to Z . We take the tensor product of these two resolutions and we get a complex of sheaves in V ecM (Z) which is exact outside Z ∩ Z . The cohomology sheaves of this complex belong to CohM (Z; Z ∩ Z ) hence they deﬁne elements of KM (Z; Z ∩ Z ) = KM (Z ∩ Z ). The alternating sum of these elements is, by deﬁnition, F ⊗L F . 3.3 We preserve the setup of 2.3. Let i ∈ I. Let ¯i = {(b, b ) ∈ B × B|πi (b) = πi (b )}. O (a closed subvariety of B × B). Let Z¯i be the set of all (y, b, b ) ∈ gn × B × B ¯i and y ∈ np where p = πi (b) = πi (b ). such that (b, b ) ∈ O ˇ i (x ) = α ˇ i (x ) = −1 and x + x = −αi . We choose x , x ∈ X such that α ¯i is independent of the choice Then the restriction of Lx Lx from B×B to O ¯i ). The inverse image of x , x . This restriction, denoted Li , belongs to V ecG (O ¯ ¯ of Li under the obvious map Zi → Oi is again denoted by Li ; it belongs to V ecG (Z¯i ). Let Λ = {(y, b) ∈ gn × B|y ∈ b} (a smooth G-variety hence an M -variety). Let M, e, Be be as in 2.4. Then Be may be regarded as a subvariety of Λ by b → (e, b). We regard Z¯i , Λ×Be as closed M -stable subvarieties of Λ×Λ in an obvious way. Their intersection is Z¯i ∩ (Be × Be ). Then we have a Tor-product (see 3.2): ⊗L : KM (Z¯i ) × KM (Λ × Be ) → KM (Z¯i ∩ (Be × Be )). The second projection pr2 : Λ × Be → Be is smooth and the ﬁrst projection pr1 : Z¯i ∩ (Be × Be ) → Be is proper. Hence F → pr1∗ ((v −1 Li ) ⊗L (pr2∗ F )) is a well deﬁned RM -linear map c1 : KM (Be ) → KM (Be ). Proposition 3.4 There is a unique H-module structure on KM (Be ) such that v n ∈ A acts as multiplication by v n ∈ RM , −v −1 − T˜i acts as ci and θx acts as in 2.4(a). See [[L4], Sec. 7, 8 and 10.1]. Proposition 3.5 If M is as in 2.9, then the H-module structures on KM (Be ) described in 2.9 and 3.4 coincide.

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We must only show that the action of T˜i in the two module structures on KH (Be ) coincide. Both H-module structures can be expressed in terms of equivariant topological K-homology. Now in [[KL], Sec. 5] there is an argument by which one can get information about the action of T˜i on a Khomology space of Be from information about the action of T˜i on a Khomology space of B. (In that reference we use this to show that the relations of H are satisﬁed.) But the same argument shows that it is enough to verify the coincidence of the two actions of T˜i on a K-homology space of B. In this case we use the explicit formulas [[KL], 3.10],[[L4], 7.23]. This completes the proof. 3.6 In the setup of 3.3 we denote by Be (i) the closed subset of Be consisting of those b such that e ∈ np where p = πi (b). The map pr1 : Z¯i ∩ (Be × Be ) → Be has image contained in Be (i). Hence the image of c1 : KM (Be ) → KM (Be ) is contained in the image of KM (Be (i)) → KM (Be ) (direct image map induced by the inclusion Be (i) ⊂ Be ). 3.7 We will need another K-theoretic construction: Serre-Grothendieck duality. For any smooth connected M -variety Z we denote by ΩZ ∈ V ecM (Z) the line bundle of top exterior diﬀerential forms on Z. Let Z be an M -variety. We deﬁne a group homomorphism (a) DZ : KM (Z) → KM (Z) as follows. We choose an M -equivariant closed imbedding of Z into a smooth ˜ Let F ∈ CohM (Z). We can regard F as a sheaf on Z, ˜ zero M -variety Z. p p+1 → . . . in outside Z. We can ﬁnd a complex of sheaves · · · → F → F CohM (Z) such that F p are zero for p > 0 and for |p| large, and the p-th cohomology sheaf is zero except in degree 0 where it is F. This gives rise to a complex of sheaves · · · → F˜ p → F˜ p+1 → . . . in V ecM (Z) where for any connected component Z˜j of Z˜ of dimension n we have F˜ p |Z˜j = Hom(F −n−p |Z˜j , ΩZ˜j ). ˜ Z) hence they The cohomology sheaves of this complex belong to CohM (Z; ˜ give rise to elements of KM (Z; Z) = KM (Z). Taking the alternating sum of these elements, we obtain an element of KM (Z) which is, by deﬁnition, DZ (F). This deﬁnes (a); it is well deﬁned, independent of the choices.

4 Subregular case 4.1 In the remainder of this paper we assume that g is simple and that G is simply connected. We ﬁx e, h, f as in 2.8 and assume that e is subregular,

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that is, Be (see 2.4) is pure of dimension 1. Let C and H = C × C∗ ⊂ G be as in 2.8. Let Λe = {(y, b) ∈ gn × B|[y − e, f ] = 0, y ∈ b}. Then Λe is an H-stable subvariety of gn × B. The subvariety {e} × Be = Be of Λe is H-stable. It is known that Λe is smooth, connected of dimension 2. In this section we deﬁne a subset B± Be ⊂ KH (Be ) following [L4]. (This can be deﬁned also for arbitrary e ∈ gn , see [L5].) 4.2 Let J be a set that indexes the irreducible components of Be . Let Vj be the irreducible component of Be indexed by j ∈ J. It is isomorphic to P 1 . We can regard J as the set of vertices of a graph (called also J) in which j, j are joined if Vj ∩ Vj = ∅. (Then this intersection is a single point pj,j .) It is known that J is a Coxeter graph of type A, D or E. For j ∈ J, Vj is a projective line; in fact it is a ﬁbre of πi : B → Pi for a well deﬁned i ∈ I. Then j → i is a map ω : J → I. This map is surjective and its ﬁbres are the orbits of a cyclic group Z/dZ acting on J by graph automorphisms. If G is of type A, D, E, then J = I and d = 1; if G is of type Cn , (n ≥ 3), then J is of type Dn+1 and d = 2; if G is of type Bn (n ≥ 2), then J is of type A2n−1 and d = 2; if G is of type F4 , then J is of type E6 and d = 2; if G is of type G2 , then J is of type D4 and d = 3. A Coxeter graph has a canonical involution (called opposition); it is induced by conjugation by the longest element in the corresponding Weyl group. 4.3 An involution of the Lie algebra g is said to be an opposition if, on some Cartan subalgebra of g, is multiplication by −1. We can ﬁnd (cf. [L5]) an opposition : g → g such that (a) (e) = −e, (h) = h, (f ) = −f ; (b) = −1 on the Lie algebra of C; (c) induces the opposition involution of the Coxeter graph J. (Note that, by (a), induces an involution : Be → Be and this in turn induces an involution of the graph J, so that (c) makes sense.) If G is of type D, E, we have C = {1} hence condition (b) can be omitted; also, condition (c) is automatically satisﬁed. In this case, is uniquely determined. If G is of type Cn , (n ≥ 3) or F4 or G2 then we have C = {1} hence condition (b) can be omitted. In this case, is uniquely determined by (a),(c).

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If G is of type A or Bn (n ≥ 2) then dim C = 1. In this case, condition (c) is automatically satisﬁed (in the presence of (a),(b)); moreover is uniquely determined up to conjugation by Ad(g) for some g ∈ C. Let ξ → ξ † be the involution of RH induced by the automorphism of H = C × C∗ given by (g, c) → (g −1 , c). The involution of Be is compatible with the H-action in the following way: ((g, c)b) = (g, c)† (b) for (g, c) ∈ H, b ∈ Be . If F ∈ CohH (Be ), then the inverse image of F under is again naturally an object of CohH (Be ); we denote it by ∗ F. This deﬁnes an involution ∗ of KH (Be ) which is RH -semilinear with respect to the involution † : RH → RH . 4.4 We consider the RH -bilinear pairing KH (Be ) × KH (Λe ) → RH given by (F : F ) = π∗ (F ⊗L F ). Here the Tor-product is relative to the smooth variety Λe and its closed subvarieties Be , Λe with intersection Be ; π is the map from Be to the point. Let ν = l(w0 ). We now deﬁne a pairing (||) : KH (Be ) × KH (Λe ) → RH by (F ||F ) = (−v)ν−2 (∗ T˜w0 (F ) : F )† where ∗ acts as in 4.3 and T˜w0 ∈ H acts as in 2.9 or 3.4. This pairing is RH -linear in the ﬁrst variable and RH -semilinear (as in 4.3) in the second variable. We deﬁne β˜ : KH (Be ) → KH (Be ) by ˜ ) = (−v)−ν ∗ T˜−1 DB (F ) β(F w0 e where ∗ acts as in 4.3, T˜w0 ∈ H acts as in 2.9 or 3.4 and DBe : KH (Be ) → KH (Be ) is as in 3.7. By [[L4], 12.10], β˜ is an involution. Let Cˆ be the group of characters of C. We have Cˆ ⊂ RC ⊂ RC [v, v −1 ] = RC×C∗ = RH . Hence for τ ∈ Cˆ and F ∈ KH (Be ), the product τ F ∈ KH (Be ) is well deﬁned. From the deﬁnition, we see that ˜ F ) = τ β(F ˜ ) β(τ ˆ F ∈ KH (Be ). for τ ∈ C, 4.5 Let k : Be → Λe be the inclusion. As in [[L5], 5.11] we deﬁne −1 ˜ ]}. B± Be = {F ∈ KH (Be )|β(F ) = F, (F ||k∗ F ) ∈ 1 + RC [v

A priori we only have (F ||k∗ F ) ∈ RH = RC×C∗ = RC [v, v −1 ].

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4.6 If q is an H-ﬁxed point of Vj and k : {q} → Vj , k : {q} → Be are the inclusions, we shall sometimes denote k∗ (C) ∈ KH (Vj ) and k∗ (C) ∈ KH (Be ) again by q. Also, if O ∈ V ecH (Vj ) we shall denote again by O the object of CohH (Be ) obtained by extending O to Be by 0 outside Vj . 4.7 As in [[L6], 3.4] we see that the RH -module KH (Be ) is free. As a basis we may take the elements Oj−1 (j ∈ J) and q; here Oj−1 is any H-equivariant line bundle on Vj such that Vj has Euler characteristic 0 with coeﬃcients in Oj−1 and q is an H-ﬁxed point in Be . From the deﬁnition of the action of T˜i in 2.9 we see that, if i = ω(j) and p is an H-ﬁxed point in Vj then (a) T˜i (Oj−1 ) = vOj−1 and (b) T˜i (p) = −v −1 p + bOj−1 for some b ∈ RH such that b → 0 under the obvious ring homomorphism RH → R{1} = Z. In particular, {ξ ∈ KH (Vj )|(T˜i − v)ξ = 0} = RH Oj−1 . From 3.6 we see that (T˜i + v −1 )KH (Be ) is contained in the image of KH (∪j∈J;ω(i)=j Vj ) → KH (Be ) (direct image map). Moreover (T˜i + v −1 )KH (Be ) must be annihilated by T˜i − v. Note that ∪j∈J;ω(i)=j Vj is an i-saturated subvariety of Be (a disjoint union of projective lines), see 2.10. It follows that (T˜i + v −1 )KH (Be ) ⊂ ⊕j∈J;i=ω(j) RH Oj−1 .

(c)

If i ∈ I, j ∈ J are such that i = ω(j), then T˜i (Oj−1 ) = −v −1 Oj−1 + cj Oj−1 with cj ∈ RH ; j

here j runs over the elements of J that are joined with j and satisfy ω(j ) = i. (We use (c) and the fact that Vj ∪ (∪J Vj ) is i-saturated.) If i ∈ I and p is an H-ﬁxed point in Be − ∪j∈J;i=ω(j) Vj then, using (c) and the fact that {p} is i-saturated, we see that (e) T˜i (p) = −v −1 p.

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5 Subregular case: type A 5.1 In this section we assume that G = SL(E) where E is a C-vector space with basis 1 , 2 , . . . , n (n ≥ 3). We set 0 = 0. We may assume that e, h, f : E → E are the linear maps e k = (n − 1 − k) k+1 for k ∈ [1, n − 1], e n = 0, h k = (−n + 2k) k for k ∈ [1, n − 1], h n = 0, f k = (k − 1) k−1 for k ∈ [1, n − 1], f n = 0. We identify B with the variety of complete ﬂags in E in an obvious way. For k ∈ [1, n − 1], the subspaces Ek = C n−k + · · · + C n−1 ,

Ek = C n−k+1 + · · · + C n

are k-dimensional and e-stable. We set E0 = 0, En = E. Note that e = 0 on E2 /E0 , E3 /E1 , . . . , En /En−2 . Hence for k ∈ [1, n − 1], the set Vk of complete ﬂags in E of the form ⊂ . . . ⊂ En−1 } {E1 ⊂ . . . ⊂ Ek−1 ⊂ Wk ⊂ Ek+1

(with variable Wk of dimension k) is a projective line contained in Be . We have Be = ∪k∈[1,n−1] Vk . We may take I = J = [1, n − 1]. We may assume that C is identiﬁed with C∗ and is such that the imbedding of H = C × C∗ = C∗ × C∗ in G = SL(E) is given by the following action of C∗ × C∗ on E: (c , c) : k → c−1 c−n+2k k for k ∈ [1, n − 1],

(c , c) : n → cn−1 n .

This is compatible with the action of e in the following way: (c , c)(ex) = c2 e(c , c)(x) for x ∈ E. In particular, if ξ ∈ Be , then (c , c)ξ ∈ Be . Each subspace Ek , Ek is stable under this action. The ﬁxed point set of the H-action on Be consists of the n points: ), pk−1,k = (E1 ⊂ . . . ⊂ Ek−1 ⊂ Ek ⊂ . . . ⊂ En−1

k ∈ [1, n].

For k ∈ [2, n − 1] we have Vk−1 ∩ Vk = {pk−1,k }. Hence in this case, the deﬁnition of pk−1,k given above agrees with the deﬁnition given in 4.2. For k ∈ [1, n], let Lk−1,k be the line bundle on the ﬂag manifold whose ﬁbre at D∗ = (0 = D0 ⊂ D1 ⊂ D2 ⊂ . . . ⊂ Dn−1 ⊂ Dn = E) is Dk /Dk−1 . This is SL(V )-equivariant (even GL(V )-equivariant). We have L0,1 ⊗ L1,2 ⊗ . . . ⊗ Ln−1,n = C as SL(V )-equivariant line bundles. α ˇ k maps Lk−1,k → −1,

Lk,k+1 → 1,

Ll−1,l → 0, if l = k, k + 1.

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5.2 The ﬁbre of Lk−1,k at pl−1,l is (as a representation of H): Ek /Ek−1 = v −1 v n−2k if k < l; Ek /Ek−1 = v n−1 if k = l; = v −1 v n−2k+2 if k > l. Ek /Ek−1 Here v (resp, v) is the character C∗ × C∗ → C∗ given by the ﬁrst (resp. second) projection. 5.3 For k ∈ [1, n − 1], the simple root αk is Lk = L−1 k−1,k ⊗ Lk,k+1 ; this is the line bundle on the ﬂag manifold whose ﬁbre at D∗ is Hom(Dk /Dk−1 , Dk+1 /Dk ). Note that Lk |Vk is the tangent bundle of Vk . From 5.2 we see that the ﬁbre of Lk at pk,k+1 is v n v −n+2k , the ﬁbre of Lk at pk−1,k is v −n v n−2k . 5.4

For k ∈ [1, n − 1] let O = Okb ,b;a ,a be the H-equivariant line bundle on Vk whose ﬁbre at pk−1,k is v b v b and whose ﬁbre at pk,k+1 is v a v a . Here a − b = mn, a − b = m(−n + 2k) where m + 1 is the Euler characteristic of Vk with coeﬃcients in O. If j : {pk−1,k } → Vk , j : {pk.k+1 } → Vk are the inclusions, we have exact sequences 0 → Okn,−n+2k;0,0 → Ok0,0;0,0 → j∗ (C) → 0, 0 → Ok0,0;−n,n−2k → Ok0,0;0,0 → j∗ (C) → 0 in CohH (Vk ). Hence pk−1,k = Ok0,0;0,0 − Okn,−n+2k;0,0 ,

pk,k+1 = Ok0,0;0,0 − Ok0,0;−n,n−2k

in KH (Vk ) and in KH (Be ). We have

Okb ,b;a ,a + Oka ,a;b ,b = v a v a + v b v b in KH (Vk ) and in KH (Be ), whenever a − b = mn, a − b = m(−n + 2k) for some m.

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5.5 From 5.2, 5.4, we see that the restriction of Lk−1,k to Vl is Ol−1,n−2k;−1,n−2k if k ∈ [1, l − 1], Oln−1,0;−1,n−2k if k = l, Ol−1,n−2k+2;n−1,0 if k = l + 1, Ol−1,n−2k+2;−1,n−2k+2 if k ∈ [l + 2, n]. Lemma 5.6 Let k ∈ [1, n − 1]. We have T˜k (Ok0,0;−n,n−2k ) = vOk0,0;−n,n−2k in KH (Vk ) or in KH (Be ). This is a special case of 4.7(a). Lemma 5.7 Let k ∈ [1, n − 1] and let ξ ∈ KH (Be ). We have Lk−1,k (T˜k ξ) = (T˜k + v −1 − v)(Lk,k+1 ξ). Here Lk−1,k acts as by tensor product. This is a special case of the relation 1.1(e) which holds in the H-module KH (Be ). Lemma 5.8 (a) T˜1 (p01 ) = −v −1 p01 − O1n,−n+1;0,−1 + O10,1;−n,n−1 . (b) T˜k (p01 ) = −v −1 p01 for k ≥ 2. We prove (a). By 4.7(b), we have T˜1 (p01 ) = −v −1 p01 + bO10,0;−n,n−2 for some b ∈ RH . Let ζ = O10,0;−n,n−2 . By 5.7, we have L0,1 T˜1 p01 = (T˜1 + v −1 − v)(L0,1 p01 ) = v −1 v n−2 (T˜1 + v −1 − v)p01 , L0,1 (−v −1 p01 + bO10,0;−n,n−2 ) = = v −1 v n−2 (−v −1 p01 + bO10,0;−n,n−2 + (v −1 − v)p01 ), −v −1 v n−1 p01 + bO1n−1,0;−1,n−2 O10,0;−n,n−2 = −v −1 v n−1 p01

mod ζ.

The last equality takes place in KH (V1 ). Now O1n−1,0;−1,n−2 O10,0;−n,n−2 = O1n−1,0;−1,n−2 (−O1−n,n−2;0,0 + v −nv n−2 + 1) = = −O1−1,n−2;−1,n−2

mod ζ = −v −1 v n−2 p01

mod ζ.

Hence −v −1 v n−1 p01 − bv −1 v n−2 p01 = −v −1 v n−1 p01

mod ζ,

−v −1 v n−1 − bv −1 v n−2 = −v −1 v n−1 . Thus, b = −v n v −n+1 + v; this proves (a). Now (b) follows from 4.7(e).

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Lemma 5.9 Assume that k ∈ [2, n − 1]. We have (in KH (Be )): 0,0;−n,n−2k+2 0,0;−n,n−2k+2 = −v −1 Ok−1 − vOk0,0;−n,n−2k (a) T˜k Ok−1 (b) T˜k−1 O0,0;−n,n−2k = −v −1 O0,0;−n,n−2k − v −1 O0,0;−n,n−2k+2 . k

k

k−1

We prove (a). Let p˜ = pk−1,k . By 4.7(d), we have 0,0;−n,n−2k+2 0,0;−n,n−2k+2 = −v −1 Ok−1 + cOk0,0;−n,n−2k T˜k Ok−1 for some c ∈ RH . We have 0,0;−n,n−2k+2 0,0;−n,n−2k+2 Lk−1,k T˜k Ok−1 = (T˜k + v −1 − v)Lk,k+1 Ok−1 0,0;−n,n−2k+2 Lk−1,k (−v −1 Ok−1 + cOk0,0;−n,n−2k ) = = (T˜k + v −1 − v)v −1 v n−2k O0,0;−n,n−2k+2

=v

k−1 0,0;−n,n−2k+2 (−v Ok−1 0,0;−n,n−2k+2 v)Ok−1 )

−1 n−2k

v

+ (v −1 −

−1

+ cOk0,0;−n,n−2k

0,0;−n,n−2k+2 + cOk0,0;−n,n−2k ) = Lk−1,k (−v −1 Ok−1 = (T˜k + v −1 − v)v −1 v n−2k O0,0;−n,n−2k+2

=v

k−1 0,0;−n,n−2k+2 (−v Ok−1 0,0;−n,n−2k+2 v)Ok−1 ),

−1 n−2k

v

+ (v −1 −

−1

+ cOk0,0;−n,n−2k

−1,n−2k+2;n−1,0 0,0;−n,n−2k+2 Ok−1 + cOkn−1,0;−1,n−2k Ok0,0;−n,n−2k = − v −1 Ok−1 0,0;−n,n−2k+2 ). = v −1 v n−2k (cOk0,0;−n,n−2k − vOk−1

Here the products are taken in KH (Vk−1 ) or KH (Vk ) (note that Vk−1 , Vk are smooth). We have −1,n−2k+2;n−1,0 0,0;−n,n−2k+2 0,0;0,0 Ok−1 = v −1 v n−2k+2 Ok−1 Ok−1 0,0;−n,n−2k+2 = v −1 v n−2k+2 (˜ p + Ok−1 ),

Okn−1,0;−1,n−2k ⊗ Ok0,0;−n,n−2k = = Okn−1,0;−1,n−2k (−Ok−n,n−2k;0,0 + v −n v n−2k + 1) = −Ok−1,n−2k;−1,n−2k + (v −n v n−2k + 1)Okn−1,0;−1,n−2k = −v −1 v n−2k (˜ p + Okn,−n+2k;0,0 ) + (v −n v n−2k + 1)Okn−1,0;−1,n−2k . We get 0,0;−n,n−2k+2 − v −1 v n−2k+1 (˜ p + Ok−1 )

p + Okn,−n+2k;0,0 ) + c(v −n v n−2k + 1)Okn−1,0;−1,n−2k − cv −1 v n−2k (˜ 0,0;−n,n−2k+2 = v −1 v n−2k (cOk0,0;−n,n−2k − vOk−1 ).

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The coeﬃcient of p˜ gives −v −1 v n−2k+1 − cv −1 v n−2k = 0 hence c = −v. This proves (a). The proof of (b) is similar. Lemma 5.10 Assume that k, k ∈ [1, n − 1], k = k, k ± 1. Then T˜k (Ok0,0;−n,n−2k ) = −v −1 Ok0,0;−n,n−2k . This is a special case of 4.7(d). 5.11 For k ∈ [1, n − 1] we set Ok = Ok0,−n+k;−n,−k ∈ KH (Be ). We rewrite some of the earlier identities as follows. Here k, l are such that both sides make sense. T˜k Ok−1 = −v −1 Ok−1 − Ok ; T˜k−1 Ok = −v −1 Ok − Ok−1 ; T˜l (Ok ) = −v −1 Ok if l = k − 1, k, k + 1; T˜k (Ok ) = vOk ; pk−1,k = pk,k+1 + (−v n v k + v n−k )Ok ; T˜k (p01 ) = −v −1 p01 for k ≥ 2; T˜1 (p01 ) = −v −1 p01 + (−v n + v n )O1 . 5.12 From 5.11 we see that

p01 = pn−1,n +

(−v n v k + v n−k )Ok .

k∈[1,n−1]

For k ∈ [1, n − 1], let Ak = (v n − v n )(v n−k − v −n+k )/(v n − v −n ), Ak = (−v n v n + 1)(v k − v −k )/(v n − v −n ). Then Ak = Ak + (−v n v k + v n−k ) hence we can set Ak Ok = pn−1,n + ξ = p01 + k∈[1,n−1]

Ak Ok .

k∈[1,n−1]

The equations (v + v −1 )A1 = A2 + v n − v n , (v + v −1 )A3 = A2 + A4 , . . . ,

(v + v −1 )A2 = A1 + A3 , (v + v −1 )An−1 = An−2

can be also expressed in the form T˜k ξ = −v −1 ξ for all k ∈ [1, n − 1].

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Lemma 5.13 For k ∈ [1, n − 1] we have Ok = −(−1)ν v ν−n On−k , (a) T˜w−1 0 (b) T˜w0 Ok = −(−1)ν v −ν+n On−k . (Compare [[L6], 4.5].) Let H0 be the A-subalgebra of H generated by {T˜k |k ∈ [1, n−1]}. Let M be the RH -submodule of KH (Be ) with basis {Ok |k ∈ [1, n− 1]}. Then M is an H0 -submodule of KH (Be ). Since {m ∈ M|T˜k m = vm} = RH Ok } and T˜w0 T˜k T˜w−1 = T˜n−k it follows that T˜w0 (RH Ok ) = RH On−k . 0 ˜ Hence Tw0 Ok = bk On−k where bk ∈ RH . Note that bk is invertible in RH since T˜w0 : M → M is an isomorphism. If k ∈ [2, n] we have T˜k Ok−1 = −v −1 Ok−1 − Ok , hence T˜w0 T˜k Ok−1 = −v −1 T˜w0 Ok−1 − T˜w0 Ok , T˜n−k T˜w0 Ok−1 = bk−1 T˜n−k Ok−1 = −v −1 bk−1 On−k+1 − bk On−k , T˜n−k Ok−1 = −v −1 On−k+1 − bk b−1 k−1 On−k . = 1. Thus, b is independent of k. Thus there exists It follows that bk b−1 k k−1 an invertible element v c v c ∈ RH with = ±1, c, c ∈ Z such that T˜w0 Ok = v c v c On−k for all k. The determinant (over RH ) of T˜w0 : M → M is on the one hand equal to ±v c(n−1) v c (n−1) and on the other hand is equal to the ν-th power of the determinant of T˜k : M → M where k ∈ [1, n − 1], that is to ((−1)n v −n+3 )ν . Thus, ±v c(n−1) v c (n−1) = ((−1)n v −n+3 )ν . It follows that c = 0 and c = (−n + 3)n/2. This proves (b) up to sign. To determine the sign, we specialize v = 1. Under this specialization M becomes the reﬂection representation of W tensor the sign representation. In this representation w0 acts as Ok → −(−1)ν On−k . This proves (b). Now (a) follows from (b). The lemma is proved. Lemma 5.14 pn−1,n = (−v)ν p01 − (−1)νk∈[1,n−1] v ν (v n−k − v −n+k )Ok . (a) T˜w−1 0 (b) T˜w pn−1,n = (−v)−ν p01 − (−1)ν v n v −ν+n (v n−k − v −n+k )Ok . 0

k∈[1,n−1]

We prove (a). Let ξ be as in 5.12. Clearly, T˜w−1 (ξ) = (−v)ν ξ, or equivalently 0 (pn−1,n + Ak Ok ) = (−v)ν (p01 + Ak Ok ). T˜w−1 0 k

k∈[1,n−1]

Hence T˜w−1 (pn−1,n ) − 0

An−k (−1)ν v ν−n Ok = (−v)ν p01 + (−v)ν

k

T˜w−1 (pn−1,n ) = (−v)ν p01 + (−1)ν 0

Ak Ok ,

k

(v ν−n An−k + v ν Ak )Ok .

k

We have

v −n An−k + Ak = −(v n−k − v −n+k ).

This proves (a). We prove (b). Clearly, T˜w0 (ξ) = (−v)−ν ξ or equivalently

Notes on aﬃne Hecke algebras

T˜w0 (pn−1,n +

Ak Ok ) = (−v)−ν (p01 +

k

89

Ak Ok ).

k

Hence T˜w0 (pn−1,n ) −

An−k (−1)ν v −ν+n Ok = (−v)−ν p01 + (−v)−ν

k

Ak Ok ,

k

T˜w0 (pn−1,n ) = (−v)−ν p01 + (−1)ν

(v −ν+n An−k + v −ν Ak )Ok .

k

We have

An−k + v −n Ak = −v n (v n−k − v −n+k ).

The lemma is proved. Lemma 5.15 For k ∈ [1, n − 1] we have ∗ Ok = v n On−k . n,−k;0,−n+k as objects of It is enough to show that ∗ Ok0,−n+k;−n,−k = On−k ∗ 0,−n+k;−n,−k at pn−k−1,n−k = V ecH (Vn−k ). By deﬁnition, the ﬁbre of Ok (pk,k+1 ) is v n v −k and its ﬁbre at pn−k,n−k+1 = (pk−1,k ) is v −n+k . These n,−k;0,−n+k at the corresponding points. The are the same as the ﬁbres of On−k lemma follows.

Lemma 5.16 For k ∈ [1, n − 1] we have DBe (Ok ) = −v n v n Ok . We have ΩVk = Okn,−n+2k;−n,n−2k . Hence DBe (Ok ) = DBe (Ok0,−n+k;−n,−k ) = −Ok0,n−k;n,k ΩVk = −Ok0,n−k;n,k Okn,−n+2k;−n,n−2k = −Okn,k;0,n−k = −v n v n Ok . 5.17 We set On = p01 −

v n−k Ok = pn−1,n −

k∈[1,n−1]

v n v k Ok .

k∈[1,n−1]

Lemma 5.18 For k ∈ [1, n], we have (−v)−ν T˜w−1 ∗ DBe (Ok ) = Ok . 0 Assume ﬁrst that k ∈ [1, n − 1]. We have (−v)−ν T˜w−1 ∗ DBe (Ok ) = (−v)−ν T˜w−1 ∗ (−v n v n Ok ) = 0 0 − (−v)−ν v n T˜w−1 On−k 0

= (−v)−ν v n (−1)ν v ν−n On−k .

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Next we note that DBe (p01 ) = p01 and ∗ (p01 ) = pn−1,n . We now consider the case k = n using the already known case k ∈ [1, n − 1]:

∗ DBe (On ) = (−v)−ν T˜w−1 ωD(p01 − (−v)−ν T˜w−1 0 0 v −n+k Ok

k∈[1,n−1]

= p01 −

k∈[1,n−1]

(pn−1,n ) − = (−v)−ν T˜w−1 0 (v

n−k

= p01 −

− v −n+k )Ok −

k∈[1,n−1]

v n−k Ok )

v −n+k Ok

k∈[1,n−1]

v

n−k

Ok = On .

k∈[1,n−1]

The lemma is proved. Lemma 5.19 We have

T˜w0 On = (−v)−ν p01 + (−v)−ν v n −ν

= (−v)

−ν

pn−1,n + (−v)

v k Ok

k∈[1,n−1]

v n−k Ok .

k∈[1,n−1]

We have T˜w0 On = T˜w0 (pn−1,n −

v n v k Ok )

k∈[1,n−1] −ν

= (−v)

p01 − (−1)

+ (−1)ν v −ν+n

ν

v n v −ν+n (v n−k − v −n+k )Ok

k∈[1,n−1]

v n v n−k Ok .

k∈[1,n−1]

The lemma follows. 5.20 Consider a hermitian form (, ) on KH (Be ) with values in RH (linear in the ﬁrst variable, antilinear in the second with respect to the ring involution † : RH → RH (that is, v † = v −1 , v † = v) such that (χξ, ξ ) = (ξ, χ ξ ) and (ξ, ξ ) = (ξ , ξ)† for χ ∈ H, ξ, ξ ∈ KH (Be ). Lemma 5.21 There exists c ∈ RH such that (a) (Ok , Ok ) = c if k, k in [1, n − 1] are consecutive; (b) (Ok , Ok ) = −(v + v −1 )c for all k ∈ [1, n − 1];

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(c) (Ok , Ok ) = 0 if k = k in [1, n − 1] are not consecutive; (d) (p01 , Ok ) = 0 if k ∈ [2, n − 1]; (e) (p01 , O1 ) = −c(−v n + v n ); (f ) (p01 , p01 ) = −cv −1 (v n − v n )(v −n − v n ). (a)-(d) are proved as in [[L6], 5.1]. We have (T˜1 p01 , O1 ) = (p01 , T˜1 O1 ) hence (−v −1 p01 + (−v n + v n )O1 , O1 ) = (p01 , vO1 ), (v + v −1 )(p01 , O1 ) = −c(−v n + v n )(v + v −1 ) and (e) follows. We write θL instead of θx where L = Lx , x ∈ X. From the deﬁnitions we ˜−1 ˜ have θL −1 = Tw0 θLn−1,n Tw0 in H. Hence 01

(θL−1 p01 , O1 ) = (p01 , T˜w−1 θLn−1,n T˜w0 O1 ) = 0 01

θLn−1,n On−1 ). = −(−1)ν v −ν+n (p01 , T˜w−1 0 −1,−n+2;n−1,0 hence Recall that Ln−1,n |Vn−1 = On−1 −1,−n+2;n−1,0 0,−1;−n,−n+1 −1,−n+1;−1,−n+1 On−1 = On−1 θLn−1,n On−1 = On−1 0,0;0,0 0,0;−n,−n+2 = v −1 v −n+1 On−1 = v −1 v −n+1 (pn−1,n + On−1 )

= v −1 v −n+1 (pn−1,n + vOn−1 ). We have θL−1 p01 = v −n+1 p01 . Thus, 01

v −n+1 (p01 , O1 ) = −(−1)ν v −ν+n (p01 , T˜w−1 (v −1 v −n+1 pn−1,n + v −1 v −n+2 On−1 )) 0 = −(−1)ν v −ν+n (p01 , v −1 v −n+1 ((−v)ν p01 − − (−1)ν v ν (v n−k − v −n+k )Ok ) − (−1)ν v −1 v −n+2 v ν−n O1 ) k∈[1,n−1]

= −v v(p01 , p01 ) + v v n (p01 , O1 ). Thus, v −n+1 (p01 , O1 ) = −v v(p01 , p01 ) + v v n (p01 , O1 ). Here we substitute (p01 , O1 ) by the expression in (e); (f) follows. The lemma is proved. 5.22 We now take (ξ, ξ ) = (ξ||k∗ ξ ) (notation of 5.4, 5.5). This satisﬁes the conditions in 5.20 (see [L4]) hence 5.21 is applicable. Lemma 5.23 For this (, ) we have c = −v −1 .

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We have (O2 , O1 ) = (−v)ν−2 (O2 : T˜w0 ∗ (O1 )) = (−v)ν−2 (O2 : T˜w0 v n On−1 ) = (−v)ν−2 v n (O2 : −(−1)ν v −ν+n O1 ) = −v n v n−2 (O2 : O1 ). The ﬁbre of O2 = O20,−n+2;−n,−2 at p12 is v −n+2 and the ﬁbre of O1 = O10,−n+1;−n,−1 at p12 is v −n v −1 . Since n ≥ 3, V2 , V1 intersect transversally in Λe at p12 , so that (O2 : O1 ) = v −n+2 (v −n v −1 ) = v −n v −n+1 . Hence (O2 , O1 ) = −v n v n−2 v −n v −n+1 = −v −1 . On the other hand, (O2 , O1 ) = c. The lemma is proved. Lemma 5.24 We have (a) (Ok , Ok ) = −v −1 if k, k in [1, n − 1] are consecutive; (b) (Ok , Ok ) = 1 + v −2 for all k ∈ [1, n − 1]; (c) (Ok , Ok ) = 0 if k = k in [1, n − 1] are not consecutive; (d) (p01 , Ok ) = 0 if k ∈ [2, n − 1]; (e) (p01 , O1 ) = v −1 (−v n + v n ); (f ) (p01 , p01 ) = v −2 (v n − v n )(v −n − v n ); (g) (On , O1 ) = −v −1 v n ; (h) (On , On−1 ) = −v −1 ; (i) (On , Ok ) = 0 for k ∈ [2, n − 2]; (j) (On , On ) = 1 + v −2 . (a)-(f) follow from 5.21, 5.23. We prove (g). We have (On , O1 ) = (p01 − v n−k Ok , O1 ) k∈[1,n−1]

= (p01 , O1 ) − v n−1 (O1 , O1 ) − v n−2 (O2 , O1 ) = v −1 (−v n + v n ) − v n−1 (1 + v −2 ) + v n−2 v −1 = −v −1 v n . We prove (h). We have (On , On−1 ) = (p01 −

v n−k Ok , On−1 )

k∈[1,n−1]

= (p01 , On−1 ) − v(On−1 , On−1 ) − v 2 (On−2 , On−1 ) = −v(1 + v −2 ) + v 2 v −1 = −v −1 . We prove (i). For k ∈ [2, n − 2] we have (On , Ok ) = (p01 − v n−k Ok , Ok ) k ∈[1,n−1]

= −v n−k (1 + v −2 ) + v n−k−1 v −1 + v n−k+1 v −1 = 0.

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We prove (j). We have (On , On ) = (p01 − v n−k Ok , p01 − v n−k Ok ) k

k

= v −2 (v n − v n )(v −n − v n ) − v −1 v n−1 (−v n + v n ) − v −1 v n−1 (−v −n + v n ) + v 2k (1 + v −2 ) − 2 v 2k+1 v −1 = 1 + v −2 . k∈[1,n−1]

k∈[1,n−2]

The lemma is proved. Proposition 5.25 B± Be is the signed A-basis of KH (Be ) consists of the elements ±v s Ok (k ∈ [1, n], s ∈ Z). Any element ξ in the list above satisﬁes (ξ, ξ) = 1 + v −2 by 5.24 and ˜ β(ξ) = ξ by 5.18. Hence ξ ∈ B± Be . Conversely, let ξ ∈ B± . Let ∂ : RH → A be the group homomorphism Be given by v s v t → v t if s = 0 and v s v t → 0 if s = 0. Now the elements Ok (k ∈ [1, n − 1]) and p01 form an RH -basis of KH (Be ). Using the deﬁnition of On (see 5.17) we deduce that the elements Ok (k ∈ [1, n]) form an RH -basis of KH (Be ). Thus, we have ξ = s∈Z,k∈[1,n] cs,k v s Ok where cs,k ∈ A are 0 for all but ﬁnitely many (s, k). We can ﬁnd an integer t0 such that for all s, k we have cs,k ∈ cs,k v t0 1 + v t0 −1 Z[v −1 ]) with cs,k ∈ Z for all s, k and cs,k = 0 for some s, k. By 5.24, if (s, k) = (s , k ), then ∂(v s Ok , v s Ok ) ∈ v −1 Z[v −1 ]. It follows that ∂(ξ , xi ) ∈ 1 + v −1 Z[v −1 ] = v 2t0 = ( s,k cs,k 2 )v 2t0 + v 2t0 −1 Z[v −1 ] Since, from our assumption, we have ∂(ξ , xi ) ∈ 1 + v −1 Z[v −1 ], it follows 2 that t0 = 0 and s,k cs,k = 1. Hence there is a unique value of (s, k) for which cs,k = ±1 and cs,k = 0 for all other values of (s, k). In other words, (a) there is a unique value of (s, k) for which cs,k ∈ ±1 + v −1 Z[v −1 ] and cs,k ∈ v −1 Z[v −1 ] for all other values of (s, k). ˜ Since β(ξ ) = ξ , we have s s s∈Z,k∈[1,n] cs,k v Ok = s∈Z,k∈[1,n] cs,k v Ok where¯: A → A is the ring involution given by v t → v −t . Hence (b) cs,k = cs,k for all s, k. Combining (a),(b) we see that there is a unique value of (s, k) for which cs,k ∈ ±1 and cs,k = 0 for all other values of (s, k). The proposition follows. 5.26 Let S be the algebra of polynomials in two variables x1 , x2 with coeﬃcients in C. Then Γ = {r ∈ C∗ |rn = 1} acts on S by algebra automorphisms s : x1 → sx1 , s : x2 → s−1 x2 . Let I be the ideal of S generated by the

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elements x1 x2 , xn1 , xn2 . This ideal is Γ -invariant. Let S˜ = S/I. Then S˜ has a basis 1, x1 , x21 , . . . , xn−1 , x2 , x22 , . . . , xn−1 . 1 2 On S we have an action of H = C∗ × C∗ given by (c , c) : xs1 xt2 → cs−t cs+t xs1 xt2 . This commutes with the Γ -action. Each basis element spans a line that is H-stable and Γ -stable. We can number the irreducible characters of Γ as ρk (k ∈ [0, n − 1]) so that ρ0 is the unit representation and the ρk -isotypic component of S˜ is C1 if k = 0 and is Cxk1 ⊕ Cxn−k if k ∈ [1, n − 1]. 2 We get an induced H-action on the algebraic variety H consisting of all ideals J in S such that J is Γ -stable and S/J ∼ = RegΓ as a Γ -module ; here, RegΓ is the regular representation of Γ . Note that H is a naturally a closed subvariety of a Hilbert scheme. Let H0 be the subvariety of H considting of all J ∈ H such that I ⊂ J . We may identify in an obvious way H0 with the variety consisting of all ideals of S˜ which are Γ -stable and isomorphic to RegΓ − C as a virtual Γ -module. Assume that k ∈ [1, n−1]. For any (a, b) ∈ C2 −{0}, the subspace spanned by xn−1 , xn−2 , . . . , xn−k+1 , axk1 + bxn−k , xk+1 , xk+2 , . . . , x1n−1 (a) 2 2 2 2 1 1 is a two sided ideal in S˜ which deﬁnes a point of H0 . When a, b vary, we get a subvariety Πk of H0 isomorphic to P 1 . We have ∪k Πk = H0 . For k ∈ [1, n], let pk−1,k be the subspace spanned by xn−1 , xn−2 , . . . , xn−k+1 , xk1 , xk+1 , xk+2 , . . . , x1n−1 . 2 2 2 1 1 Note that pk−1,k ∈ Πk , pk,k+1 ∈ Πk . ) = v −n v n−2k as The tangent space to Πk at pk−1,k is Hom(Cxk1 , Cxn−k 2 an H-module. The tangent space to Πk at pk,k+1 is Hom(Cxn−k , Cxk1 ) = v n v −n+2k as 2 an H-module. ∼ According to [IN] and [B] there exists an isomorphism H−→Λe which ∼ 0 ∼ restricts to an isomorphism H −→Be and to isomorphisms Πk −→Vk for all k ∈ [1, n − 1]. These isomorphisms can be assumed to be compatible with the H-actions. We use them to identify H = Λe , H0 = Be , Πk = Vk . Then pk−1,k as just deﬁned is the same as pk−1,k deﬁned in 5.1. 5.27 For l ∈ [0, n − 1] let E l be the vector bundle over H whose ﬁbre at J is HomΓ (ρl , S/J ). The ﬁbre of E l at the point 5.26(a) is HomΓ (ρl , Cxl1 ) if l < k, HomΓ (ρl , (Cxk1 + Cxn−k )/C(axk1 + bxn−k )) if k = l, 2 2

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HomΓ (ρl , Cxn−l 2 ) if l > k. Note that E 0 = C. Assume that l ∈ [1, n − 1]. If k = l, the restriction E l |Πk is a trivial line bundle (if we forget the H-equivariant structure). n−l l The ﬁbre of E l at pl−1,l is (Cxl1 + Cxn−l (hence is 2 )/Cx1 = Cx2 −n+l n−l v ). v n−l The ﬁbre of E l at pl,l+1 is (Cxl1 + Cxn−l = Cxl1 (hence is v l v l ). 2 )/Cx2 l Hence the restriction E |Πl is Ol−n+l,n−l,l,l = v −n+l Ol0,n−l;n,l = v −n+l (Ol )∗ . Note also that the ﬁbre of E l at p01 is Cx2n−l hence it is v −n+l v n−l . 5.28 Let E l be the line bundle dual to E l . Note that E 0 = C. Assume that l ∈ [1, n − 1]. If k = l, the restriction E l |Πk is a trivial line bundle (if we forget the H-equivariant structure). The restriction E l |Πl is v n−l Ol . The ﬁbre of E l at p01 is v n−l v −n+l . Lemma 5.29 Let l ∈ [1, n − 1]. We have (a) (Ol ||E l ) = v −n+l v −2 ; (b) (Ok , E l ) = 0 if k ∈ [1, n], k = l. Assume ﬁrst that k ∈ [1, n − 1]. We have (Ok ||E l ) = (−v)ν−2 (∗ T˜w0 Ok : E l )† = −(−v)ν−2 (−1)ν v −ν+n v −n (Ok : E l )† = −v −n v n−2 (Ok : E l )† . Now (Ok : E l ) is the alternating sum of the cohomologies of Πk with coeﬃcients in the line bundle Ok ⊗ E l |Πk . If l = k, then this is zero. If l = k then Ok ⊗ E k |Πk = v n−k Ok ⊗ Ok . We have Ok ⊗ Ok = Ok0,−n+k;−n,−k (−Ok−n,−k;0,−n+k + v −n v −k + v −n+k ) = −Ok−n,−n;−n,−n + (v −n v −k + v −n+k )Ok and this contributes −v −n v −n to the Euler characteristic. Hence (c) (Ok : E l ) = 0 if k = l, (Ol : E l ) = v n−l (−v −n v −n ) = −v −l v −n . Thus (a) and (b) (with k ∈ [1, n − 1]) follow. We have

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(On ||E l ) = (−v)ν−2 (∗ T˜w0 On : E l )†

= (−v)ν−2 (∗ ((−v)−ν pn−1,n + (−v)−ν ν−2

= (−v)

−ν

((−v)

−ν

p01 + (−v)

v n−k Ok ) : E l )†

k∈[1,n−1]

v n v n−k On−k : E l )†

k∈[1,n−1] −2

= (−v)

−2

= (−v)

n l

l †

(p01 + v v Ol : E )

(v n−l v −n+l − −v n v j v −j v −n )† = 0.

(We have used 5.19 and (c).) The lemma is proved. Lemma 5.30 We have (a) (Ok ||C) = 0 for all k ∈ [1, n − 1]; (b) (On ||C) = v −2 . For any k ∈ [1, n − 1] we have (c) (Ok : C) = 0. since the cohomology of P 1 with coeﬃcients in a line bundle with Euler characteristic 0 is 0. We deduce (Ok ||C) = (−v)ν−2 (∗ T˜w0 Ok : C)† = −(−v)ν−2 (∗ (−1)ν v −ν+n On−k : C)† = −v n−2 v n (Ok : C)† = 0, which proves (a). Using the equalities (p01 : C) = (pn−1,n , C) = 1 and (c) we deduce (On ||C) = (−v)ν−2 (∗ T˜w0 On : C)† = (−v)ν−2 (∗ ((−v)−ν p01 + (−v)−ν v n

v k Ok ) : C)†

k

= v −2 (pn−1,n : C)† = v −2 . The lemma is proved. 5.31 For k ∈ [1, n] we deﬁne ek ∈ V ecH (H0 ) = V ecH (Λe ) by ek = v −n+k v 2 E k if k ∈ [1, n − 1], en = v 2 E 0 = v 2 C. We have (Ok ||ek ) = δk,k for all k, k ∈ [1, n]. In particular, we see that the pairing (||) is non-singular and that {ek |k ∈ [1, n]} is an RH -basis of KH (Λe ). Arguing now as in [[L6], 7.9], we see that ±v s Ok (k ∈ [1, n], s ∈ Z) coincides with the set B± Λe deﬁned in [[L5], 5.11] (a signed basis of KH (Λe )).

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6 Subregular case: types C, D, E, F, G 6.1 In this section we assume that G is of type C, D, E, F or G. In this case we have C = {1} hence H = C∗ . We want to describe the set B± Be . The case where G is of type D or E is discussed in [L6]; here we shall review the results in [L6] including at the same time G of type C, F, G. We have J = {j01 , j11 , . . . , ja11 } ∪ {j02 , j12 , . . . , ja22 } ∪ {j03 , j13 , . . . , ja33 } (a disjoint union except for j01 = j02 = j03 ) where a1 , a2 , a3 are ≥ 1, u j, j are joined in J precisely when {j, j } = {jtu , jt+1 } with u ∈ {1, 2, 3}, 0 ≤ t < au . We denote j01 = j02 = j03 by j0 . u . Note that For u ∈ {1, 2, 3}, 0 ≤ t < au , we write put,t+1 instead of pjtu ,jt+1 1 p0,1 , p20,1 , p30,1 are distinct points of Vj0 , but they are the same as elements of KC∗ or KC∗ (Be ) which are denoted by p. The ﬁxed point set of the H = C∗ -action of Be has connected components µj (j ∈ J) where µj = Vj if j = j0 ; µj = {put,t+1 } if j = jtu with u ∈ {1, 2, 3}, 0 < t < au , µj = {q u } if j = jtu with u ∈ {1, 2, 3}, t = au . Thus, if j = jtu with u ∈ {1, 2, 3}, t = au , there are two H-ﬁxed points on Vj , namely put−1,t and q u . 6.2 There is a unique homomorphism n0 : X → Z such that n0 (αi ) = −2 if i = ω(j0 ), n0 (αi ) = 0 if i = ω(j0 ). For j ∈ J we deﬁne a homomorphism nj : X → Z by nj0 = n0 and ni (x) = n0 (siu1 siu2 . . . sit1 (x)) where j = jtu , u ∈ {1, 2, 3}, 0 < t ≤ au and iur = ω(jru ) for r ∈ [1, t]. If x ∈ X, we regard Lx ∈ V ecG (B) as an object of V ecC∗ (B) by restriction via C∗ = H ⊂ G. In particular we obtain a C∗ -action on the ﬁbre of Lx at a C∗ -ﬁxed point on Be . (a) Let j ∈ J, x ∈ X and let b ∈ µj . Then C∗ acts on the ﬁbre of Lx through the character v nj (x) . This is proved as in [[L6], 3.2]. 6.3 Fix j ∈ J and m ∈ Z. There is, up to isomorphism, a unique C∗ -equivariant line bundle Ojm on Vj such that (a) the Euler characteristic of Vj with coeﬃcients in Ojm is m + 1; (b) if j = jtu , then C∗ acts on the ﬁbre of Ojm at a point of µj by v tm . If j = j0 and f : {pu0,1 } → Vj is the inclusion, we have

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p = Oj0 − Oj−1 ∈ KC∗ (Vj ).

6.4 We regard Ojm as an object of CohC∗ (Be ) (zero outside Vj ); this object is denoted again by Ojm . As in [L6], we see that KC∗ (Be ) is the A-module with generators Ojm (j ∈ J, m ∈ Z) and relations 0 Oj0tu − v −t Oj−1 − v t+1 Oj−1 u = Oj u u t t+1 t+1 for u ∈ {1, 2, 3}, 0 ≤ t < au , Ojm+1 + Ojm−1 = (v t + v −t )Ojm for j = jtu , 0 ≤ t ≤ au , m ∈ Z. It follows that (a) an A-basis of KC∗ (Be ) is given by Oj−1 (j ∈ J) and p = Oj00 − Oj−1 . 0 6.5 α ˇ (x)

For x ∈ X, the restriction of Lx to Vj is v s Oj i αi (x) (with j = jtu ). s = nj (x) − tˇ

where i = ω(j) and

6.6 As in [[L6], 3.6], we have (a) θx p = v n0 (x) p; ˇ i (x) = 1, then θx Ojm = v nj (x)−t Ojm+1 and (b) if j = jtu , i = ω(j) and α m−1 θx−αi Ojm = v nj (x)−t Oj . u Lemma 6.7 Let j = jtu , j = jt−1 with u ∈ {1, 2, 3}, 0 < t ≤ au . Let p˜ = put−1,t . Let i = ω(j), i = ω(j ). We have (a) T˜i p˜ = −v −1 p˜ + (v t−1 − v −t+1 )Oj−1 ; (b) T˜i p˜ = −v −1 p˜ + (v t − v −t )Oj−1 .

The proof (based on 4.7(b) and 6.6) is along the same lines as that of [[L6], 3.9]. (The argument in [[L6], 3.9] can be simpliﬁed since here we know in advance that the coeﬃcient of p˜ in (a) and (b) is −v −1 (by 4.7(b)). 6.8 L et i0 = ω(j0 ). The following is a special case of 6.7: (a) T˜i0 p = −v −1 p + (v − v −1 )Oj−1 . 0 Lemma 6.9 Let i ∈ I be such that i = ω(j0 ). Then (a) T˜i p = −v −1 p.

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This follows from 4.7(e) except in the special case where ∪j;ω(j)=i Vj contains all three points pu01 . In this special case, G is be of type G2 and i is uniquely determined. For j = j0 , Vj is i-saturated and contains one of the points pu01 hence T˜i p = −v −1 p + aj Oj−1 where aj ∈ A. (We use 4.7(c).) Thus aj Oj−1 takes the same value for three diﬀerent j. It follows that aj = 0 and (a) is proved. Lemma 6.10 If i ∈ I, j ∈ J are such that i = ω(j), then (a) T˜i (Oj−1 ) = vOj−1 . This is a special case of 4.7(a). Lemma 6.11 Assume that i ∈ I, j ∈ J are such that i = ω(j). Then (a) T˜i (Oj−1 ) = −v −1 Oj−1 − j ∈J Oj−1 where J consists of all j ∈ J that are joined with j and satisfy ω(j ) = i. u If J = ∅, then (a) follows from 4.7(d). Assume now that j = jtu , j = jt−1 with 0 < t ≤ au and i = ω(j ). Then (a) is proved in the same way as [[L6], 3.11(a)] (using 4.7(d) and 6.7); again the proof in [[L6], 3.11] is simpliﬁed u by the use of 4.7. Assume next that j = jt−1 , j = jtu with 1 < t ≤ au and i = ω(j ). Then (a) is proved in the same way as [[L6], 3.11(b)]. It remains to consider the case where j = j0 , i = ω(j1u ). We use 4.7(d). By symmetry, the coeﬃcients cj in 4.7(d) are independent of j . Thus, ) = −v −1 Oj−1 + c j ∈J Oj−1 with c ∈ A. T˜i (Oj−1 0 0 Setting f = j ∈J Oj−1 , we have (b) T˜i (Oj−1 ) = −v −1 Oj−1 + cf . 0 0 From the earlier part of the proof we deduce (c) T˜i0 f = −v −1 f − |J |Oj−1 0 where i0 = ω(j0 ). Using (b),(c) we can express the braid group relation . . . T˜i0 T˜i T˜i0 = . . . T˜i T˜i0 T˜i on the A-submodule spanned by f, Oj0 as an equality of two explicit 2 × 2 matrices with entries in A. This equality gives us c = −1 if |J | is 1 or 2 and (c + 1)(3c + 1) = 0 if |J | = 3. In the last case we must also have c = −1 since c ∈ A. This proves (a).

Lemma 6.12 Let j → j ∗ be the opposition involution of the Coxeter graph J. Let h be the Coxeter number of G. This is the same as the Coxeter number of the graph J. We have (a) T˜w0 (Oj−1 ) = −(−v)−ν+h Oj−1 ∗ ; −1 −1 ν−h −1 ˜ Oj ∗ . (b) Tw0 (Oj ) = −(−v) When G is of type D, E, this is proved in [[L6], 4.5]. Assume now that G is of type Cn (n ≥ 3), F4 or G2 . Let σ be a generator of the cyclic group Z/dZ which acts on J (d ∈ {2, 3}) as in 4.2. Let H0 be the A-subalgebra of H generated by {T˜i |i ∈ I}. Let M be the A-submodule of KH (Be ) with basis {Oj−1 |j ∈ J}. Then M is an H0 submodule of KH (Be ).

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Let M be the A-submodule of M generated by the elements Oj−1 with −1 σ(j) = j and Oj−1 + Oσ(j) with σ(j) = j; let M be the A-submodule of M −1 , Oj−1 − Oσ−1 generated by the elements Oj−1 − Oσ(j) −1 (j) with σ(j) = j. Then M , M are H-submodules and their direct sum is M (after extending scalars to Q(v)). Now M is irreducible as an H-module (if scalars are extended to an algebraic closure of Q(v)); the same holds for M except for type G2 . Now T˜w0 is central in H0 (since in our case w0 is central in W .) Hence T˜w0 acts on M as multiplication by b ∈ A. Similarly, (in type C, F ), T˜w0 acts on M as multiplication by b ∈ A. Now b can be determined as in [[L6], 4.5] since for v = 1, M becomes the reﬂection representation of W . We see that b = −(−v)−ν+h . For G of type Cn , M has rank 1 and any T˜i such that |ω −1 (i)| = 1 acts as −v −1 on it, while the T˜i such that ω −1 (i) = 2 acts as v on it. Hence T˜w0 2 acts on it as (−v −1 )n −n v n = v −ν+h . For G of type F4 , M has rank 2 and any T˜i such that |ω −1 (i)| = 1 acts as −v −1 on it, while any T˜i such that |ω −1 (i)| = 1 acts with eigenvalues −v −1 , v. Hence the determinant of T˜w0 on M is v −24 . Hence T˜w0 acts on M as ±v −12 . If we specialize v to 1, w0 acts as 1. Hence T˜w0 acts on M as v −12 . For G of type G2 , M has rank 2 and one checks easily that T˜w0 acts on it as −1. Thus, the action on T˜w0 on M , M is explicitly described, hence its action on M is determined and (a) follows. Now (b) follows from (a). 6.13 We consider the system of equations with unknowns Aj (j ∈ J): (v + v −1 )Aj = j ∈J;j,j joined Aj , if j ∈ J − {j0 }, (v + v −1 )Aj = j ∈J;j,j joined Aj − (v − v −1 ), if j = j0 . This system has a unique solution with Aj ∈ Q(v) for all j ∈ J. It satisﬁes ¯ −Bj B Aj = vh/2j−v−h/2 where Bj ∈ vZ[v] for all j ∈ J and ¯: A → A is as in 5.25. (See [[L6], 4.1, 4.2].) Note that h is even in our case. The polynomials Bj are given explicitly in [[L6], 1.10]. (p) = (−v)ν p+(−v)ν (1+v −h ) j∈J Aj Oj−1 ; Lemma 6.14 We have (a) T˜w−1 0 (b) T˜w0 (p) = (−v)−ν p + (−v)−ν (1 + v h ) j∈J Aj Oj−1 . Let ξ = p + j∈J Aj Oj−1 . Using 6.8-6.11, the deﬁnition of Aj and the fact that Aj is constant on any Z/dZ orbit in J, we see that T˜i ξ = −v −1 ξ for all ˜ ξ = (−v)−ν ξ or equivalently i ∈ I. It follows 0 that Tw ˜ Tw0 (p + j∈J Aj Oj−1 ) = (−v)−ν (p + j∈J Aj Oj−1 ). Using 6.12 and the equality Aj ∗ = Aj we deduce T˜w0 (p) − j∈J Aj (−v)−ν+h Oj−1 = (−v)−ν (p + j∈J Aj Oj−1 ) and (b) follows. Next note that T˜w−1 ξ = (−v)ν ξ or equivalently 0

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(p + j∈J Aj Oj−1 ) = (−v)ν (p + j∈J Aj Oj−1 ). T˜w−1 0 Using again 6.12 we deduceν−h −1 T˜w−1 (p) − Oj = (−v)ν (p + j∈J Aj Oj−1 ) j∈J Aj (−v) 0 and (a) follows. Lemma 6.15 Let p = p − j∈J Bj v −h/2 Oj−1 . We have T˜w0 (p) = (−v)−ν (p + j∈J v h/2 Bj Oj−1 ). This follows immediately from 6.12,6.13, 6.14. (Compare [[L6], 4.7].) 6.16 Let G be a connected semisimple simply connected algebraic group over C whose Coxeter graph is J as in 2.2. The analogues for G of the various objects attached to G will be denoted by the same symbol underlined. Thus, I, g, W , w0 , ν, h, H, T˜w0 ∈ H are deﬁned. We ﬁx an sl2 -triple e, h, f in g such that e is subregular in g. Let ˜ k be the Be , Λe , , J be the analogues for G of Be , Λe , , J. Let (:), (||), β, ˜ k (k as in 4.5). analogues for G of (:), (||), β, Note that I = J and h = h. Moreover, according to [S] we can ﬁnd a ∼ C∗ -equivariant isomorphism Λe −→Λe which carries Be onto Be and is such that the induced map J → J is the identity map. (We have J = I = J.) Under this isomorphism k corresponds to k. Using these isomorphisms we may identify KC∗ (Be ) = KC∗ (Be ) and KC∗ (Λe ) = KC∗ (Λe ). It is then clear that (:) = (:) and DBe = DBe . It is also clear that, for j ∈ J = J, the element Oj−1 deﬁned in terms of G is the same as that deﬁned in terms of G. From the ∗ −1 deﬁnition, we have ∗ Oj−1 = Oj−1 = Oj−1 . Also, p deﬁned in terms ∗ , Oj of G is the same as p deﬁned in terms of G. Moreover, ∗ (p) = p, ∗ (p) = p. Since the elements Oj−1 together with p form an A-basis of KC∗ (Be ), it follows that ∗ = ∗ on KC∗ (Be ). Comparing 6.12, 6.14 with the analogous formulas for G we see that T˜w0 = (−v)−ν+ν T˜w0 : KC∗ (Be ) → KC∗ (Be ). Using this we see that (F ||F ) = (−v)ν−2 (∗ T˜w0 F : F ) = (−v)ν−2 (∗ T˜w0 F : F ) = (F ||F ), ˜ β˜ = (−v)−ν ∗ T˜w−1 DBe = (−v)−ν ∗ T˜w−1 DBe = β. 0 0 ± It follows that B± Be deﬁned in terms of G is the same as BBe deﬁned in terms of G. Since the latter is known from [[L6], 6.4], we see that the following holds (for G):

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Proposition 6.17 B± Be is the signed A-basis of KC∗ (Be ) consisting of ± the elements v −h/2 Oj−1 (j ∈ J) and p. Here p is deﬁne in terms of G. Proposition 6.18 There exist C∗ -equivariant vector bundles E j (j ∈ J) on Λe such that ∗ (a) v 2 E j (j ∈ J), v 2 C form an A-basis of KC∗ (Λe ) dual to the basis v −h/2 Oj−1 (j ∈ J), p of KC∗ (Be ) with respect to (||). (b) B± Λe (see [[L5], 5.11]) is the signed A-basis of KC∗ (Λe ) consisting of ∗ ± the elements v 2 E j (j ∈ J), v 2 C. The arguments above reduce this to the case where G is replaced by G, which is known by [[L6], 7.7, 7.9]. 6.19 Let z be the centralizer of f in g. This is an H-stable subspace of g. The element t≥0 (−1)t z(t) ∈ RH = A (where z(t) is the t-th exterior power of z) is divisible in A by (1 − v −2 )r (where r is the rank of g); see [[L5], 3.2]. The quotient is denoted by ∇e ∈ A. If ξ ∈ KC∗ (Λe ), there is a unique element m(ξ) ∈ KB∗ (Be ) such that k∗ (m(ξ)) = ∇e ξ with k as in 4.5 (see [[L5], 3.5]). ∗ Let J˜ = J {0}. For j ∈ J˜ we deﬁne bj ∈ KC∗ (Λe ) by bj = v 2 E j if 2 j ∈ J, b0 = v C. We want to compute the inner products (m(bj )||bj ) ∈ A ˜ (These products are of interest in connection with the for any j, j ∈ J. representation theory of Lie algebras in characteristic p; see [L7].) Let P = w∈W v −2l(w) ∈ A. Let m : KC∗ (Λe ) → KC∗ (Be ), P, ∇e be the analogues for G of m : KC∗ (Λe ) → KC∗ (Be ), P, ∇e . The analogue of bj for G is bj itself (see 6.17, 6.18). Proposition 6.20 (m(bj )||bj ) =

P P (m(bj )||bj ).

(Note that (m(bj )||bj ) is explicitly computed in [L7].) e From the deﬁnitions we have (m(bj )||bj ) = ∇ ∇e (m(bj )||bj ). It remains to observe that P e (a) ∇ ∇e = P . Indeed, by computation, we see that both sides of (a) are equal to (1 − v −2 )(1 − v −2(n+1) )−1 (for type Cn ), (1 − v −2 )2 (1 − v −10 )−1 (1 − v −18 )−1 (for type F4 ), (1 − v −2 )2 (1 − v −8 )−2 (for type G2 ); for types D, E both sides of (a) are clearly 1.

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[B]