Character tables of commutative hecke algebras associated with exceptional weyl groups

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups when they are commutative. In the case when Weyl groups are of classical type, they are already known in [D.1] and [D.2]. 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 exceptional Weyl groups.     

Three addition theorems for someq-Krawtchouk polynomials

Theq-Krawtchouk polynomials are the spherical functions for three different Chevalley groups over a finite field. Using techniques of Dunkl to decompose the irreducible representations with respect to a maximal parabolic subgroup, we derive three addition theorems. The associated polynomials are related to affine matrix groups.During the preparation of this paper the author was partially supported by NSF grant MCS78-02410.     

Standard bases for affine parabolic modules and nonsymmetric Macdonald polynomials

We establish a connection between (degenerate) nonsymmetric Macdonald polynomials and standard bases and dual standard bases of maximal parabolic modules of affine Hecke algebras. Along the way we prove a (weak) polynomiality result for coefficients of symmetric and nonsymmetric Macdonald polynomials.     

Character Tables of Parabolic Subgroups of the Chevalley Groups of Type G 2

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G ₂(q), where q is a power of a prime p > 3.     

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.     

Chevalley群中Levi子群在抛物子群中的扩群

本文以根系及Ｗｅｙｌ群为基础,以换位子公式为工具,对任意域上任意类型Ｃｈｅｖａｌｌｅｙ群,针对极大抛物子群情形,确定了Ｌｅｖｉ子群在抛物子群中的所有扩群．     

Basic reductions in the description of normal subgroups

Classification of subgroups in a Chevalley group G(Φ, R) over a commutative ring R, normalized by the elementary subgroup E(Φ, R), is well known. However, for exceptional groups, in the available literature neither the parabolic reduction nor the level reduction can be found. This is due to the fact that the Abe-Suzuki-Vaserstein proof relied on localization and reduction modulo the Jacobson radical. Recently, for the groups of types E ₆, E ₇, and F ₄, the first-named author, M. Gavrilovich, and S. Nikolenko have proposed an even more straightforward geometric approach to the proof of structure theorems, similar to that used for exceptional cases. In the present paper, we give still simpler proofs of two key auxiliary results of the geometric approach. First, we carry through the parabolic reduction in full generality: for all parabolic subgroups of all Chevalley groups of rank ≥ 2. At that we succeeded in avoiding any reference to the structure of internal Chevalley modules, or explicit calculations of the centralizers of unipotent elements. Second, we prove the level reduction, also for the most general situation of double levels, which arise for multiply-laced root systems. Bibliography: 64 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 30–52.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

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.     

Splitting fields for extended complex reflection groups and Hecke algebras

Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k. Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer. Also, we generalize the corresponding result for cyclotomic Hecke algebras to Hecke algebras attached to extended finite complex reflection groups.     

Calculations in exceptional groups over rings

In the present paper, we discuss a major project whose goal is to develop theoretical background and working algorithms for calculations in exceptional Chevalley groups over commutative rings. We recall some basic facts concerning calculations in groups over fields, and indicate complications arising in the ring case. Elementary calculations as such are no longer conclusive. We describe the basics of calculations with elements of exceptional groups in their minimal representations, which allow one to reduce calculations in the group itself to calculations in subgroups of smaller rank. For all practical purposes, such calculations are much more efficient than localization methods. Bibliography: 147 titles.     

Semisimple Types in GLn

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.     

Infinitesimal Hecke algebras

In this Note, we define infinitesimal analogues of the Iwahori–Hecke algebras associated with finite Coxeter groups. These are reductive Lie algebras for which we announce several decomposition results. These decompositions yield irreducibility results for representations of the corresponding (pure) generalized braid groups deduced from Hecke algebra representations through tensor constructions. To cite this article: I. Marin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A note on the Hochschild homology and cyclic homology of a topological algebra

In this note we use a topological version of Hochschild homology and cyclic homology of a commutative algebra, introduced by P. Seibt in [Se₂], to show, that periodic homology can be used to calculate the relative algebraic de Rham cohomology of a morphism of affine Q-schemes of finite type as defined in [Ha], chapt. III, §4.     

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     

A lattice isomorphism theorem for cluster groups of mutation-Dynkin type An

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type $A_n$ case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group.     

On the Induction of Kazhdan-Lusztig Cells

Barbasch and Vogan showed that the Kazhdan–Lusztig cellsof a finite Weyl group are compatible with parabolic subgroups.Their proof uses the known bridge between the theory of cellsand the theory of primitive ideals. In this paper, an elementary,self-contained proof of this result is provided, which worksfor arbitrary Coxeter groups and Lusztig's general definitionof cells (involving Iwahori–Hecke algebras with unequalparameters). The argument is based on a recent paper by Howlettand Yin. 2000 Mathematics Subject Classification 20C08.