Tensor product of C-injective modules

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness     

On Weyl structures

We characterize the relation between the geometrical properties of Weyl manifolds and the algebraic properties of the Weyl algebras (§1) and the deformation algebras associated to two conformal Weyl connections (§2). The last section is devoted to the study of the Weyl-Lyra algebras associated to a conformal Weyl connection and a conformal semisymmetric connection.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

Hecke-Clifford Algebras and Spin Hecke Algebras I: The Classical Affine Type

Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners and describe the centres. We further develop connections of these algebras with the usual degenerate (i.e., graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.     

Derivations of generalized Weyl algebras

A class of the associative and Lie algebras A[D] = A F[D] of Weyl type are studied, where A is a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] is the polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A such that A is D-simple. The derivations of these associative and Lie algebras are precisely determined.     

A super Frobenius formula for the characters of Iwahori–Hecke algebras

In this article, we establish a super Frobenius formula for the characters of Iwahori–Hecke algebras. We define Hall–Littlewood supersymmetric functions in a standard manner to make supersymmetric functions from symmetric functions, and give some properties of supersymmetric functions. Based on Schur–Weyl reciprocity between Iwahori–Hecke algebras and the general quantum super algebras, which was obtained in Mitsuhashi [H. Mitsuhashi, Schur–Weyl reciprocity between the quantum superalgebra and the Iwahori–Hecke algebra, Algeb. Represent. Theor. 9 (2006), pp. 309–322.], we derive that the Hall–Littlewood supersymmetric functions, up to constant, generates the values of the irreducible characters of Iwahori–Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case that was obtained in Ram [A. Ram, A Frobenius formula for the characters of the Hecke algebra, Invent. Math. 106 (1991), pp. 461–488.] and the classical super case.     

Lusztig's isomorphism theorem for cellular algebras

In this paper, we first introduce a notion of semisimple system with parameters, then we establish Lusztig's isomorphism theorem for any cellular semisimple system with parameters. As an application, we obtain Lusztig's isomorphism theorem for Ariki-Koike algebras, cyclotomic q-Schur algebras and Birman-Murakami-Wenzl algebras. Second, using the results for certain Ariki-Koike algebras, we prove an analogue of Lusztig's isomorphism theorem for the cyclotomic Hecke algebras of type G(p,p,n) (which are not known to be cellular in general). These generalize earlier results of [G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981) 490-498.] on such isomorphisms for Iwahori-Hecke algebras associated to finite Weyl groups.     

A categorical approach to Weyl modules

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.     

Local and global methods in representations of Hecke algebras

This paper aims at developing a “local-global” approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the “good” prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.     

Cellular Structure of q-Brauer Algebras

Let R be a commutative noetherian domain. The q-Brauer algebras over R are shown to be cellular algebras in the sense of Graham and Lehrer. In particular, they are iterated inflations of Hecke algebras of type A. When R is a field of arbitrary characteristic, we determine for which parameters the q-Brauer algebras are quasi-hereditary. Then, using the general theory of cellular algebras we parametrize all irreducible representations of q-Brauer algebras.     

Hecke algebras from groups acting on trees and HNN extensions

We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal $C^1$-completion. The paper unifies algebraic and analytic approaches by focusing on the common geometric thread. The results have implications for the general theory of totally disconnected locally compact groups.     

Explicit irreducible representations of the¶Iwahori–Hecke algebra of Type F4

A general method for computing irreducible representations of Weyl groups and Iwahori–Hecke algebras was introduced by the first author in [10]. In that paper the representations of the algebras of types A _n , B _n , D _n and G _n were computed and it is the purpose of this paper to extend these computations to F ₄. The main goal here is to compute irreducible representations of the Iwahori–Hecke algebra of type F ₄ by only using information in the character table of the Weyl group. Received: Received: 30 July 1998     

REPRESENTATIONS OF AFFINE HECKEALGE BRAS OF TYPE G2

Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.     

On Schur's Q-functions and the Primitive Idempotents of a Commutative Hecke Algebra

Let B _n denote the centralizer of a fixed-point free involution in the symmetric group S _2n . Each of the four one-dimensional representations of B _n induces a multiplicity-free representation of S _2n , and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of S _n.     

The irreducible characters of the alternating Hecke algebras

This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and we show that these character values determine the irreducible characters completely. As an application, we determine a splitting field for the alternating Hecke algebras in the semisimple case.     

Affine and degenerate affine BMW algebras: actions on tensor space

The affine and degenerate affine Birman–Murakami–Wenzl (BMW) algebras arise naturally in the context of Schur–Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients of the affine and degenerate affine BMW algebras. In this paper, we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements—the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.     

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov (e.g., to Chow groups, Grothendieck’s \(K_0\) , connective \(K\) -theory, elliptic cohomology, and algebraic cobordism). The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings, respectively. We also introduce a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra, respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.     

A constructive proof of the Peter‐Weyl theorem

We present a new and constructive proof of the Peter‐Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*‐algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural continuation in the line of Bishop's work on locally compact, but Abelian, groups [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)