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Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980’s, Kawanaka introduced generalised Gelfand–Graev representations of the finite group $$ G\left({\mathbbm{F}}_q\right) $$ G F q , assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka’s construction can be carried out when we drop the assumptions on q. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig’s concept of special unipotent classes of G in terms of weighted Dynkin diagrams.

H.G. Jacob's elegant approach to the rational canonical, or Frobenius normal form of a linear map is presented here in pure matrix language, thereby avoiding the abstract machinery and prerequisites in the original paper. Related algorithmic aspects and an efficient implementation in the computer algebra system GAP are also discussed.

The article “Generalised Gelfand-Graev Represenations in Bad Characteristic?,” written by Meinolf Geck, was originally published Online First without Open Access.

This is a preliminary version of the first chapter of a book project on the character theory of finite groups of Lie type. It provides the foundations from the general theory of reductive algebraic groups over a finite field.

Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach., 20 pages; minor corrections and additions

Let G be a finite group of Lie type and St k be the Steinberg representation of G, defined over a field k. We are interested in the case where k has prime characteristic l and St k is reducible. Tinberg has shown that the socle of St k is always simple. We give a new proof of this result in terms of the Hecke algebra of G with respect to a Borel subgroup and show how to identify the simple socle of St k among the principal series representations of G. Furthermore, we determine the composition length of St k when G = GL n ( q ) or G is a finite classical group and l is a so-called linear prime.

Let $G(q)$ be a finite group of Lie type over a field with $q$ elements, where $q$ is a prime power. The Green functions of $G(q)$, as defined by Deligne and Lusztig, are known in \textit{almost} all cases by work of Beynon--Spaltenstein, Lusztig und Shoji. Open cases exist for groups of exceptional type ${^2\!E}_6$, $E_7$, $E_8$ in small characteristics. We propose a general method for dealing with these cases, which procedes by a reduction to the case where $q$ is a prime and then uses computer algebra techniques. In this way, all open cases in type ${^2\!E}_6$, $E_7$ are solved, as well as at least one particular open case in type $E_8$., 29 pages; added one case in type E8

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality., 16 pages

According to an old result of Sch\"utzenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups. We show that Sch\"utzenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz' conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types; combined with previous work, this leaves type $E_8$ as the only remaining open case., Comment: 33 pages

Let W be a finite Coxeter group. It is well known that the number of involutions in W is equal to the sum of the degrees of the irreducible characters of W . Following a suggestion of Lusztig, we show that this equality is compatible with the decomposition of W into Kazhdan–Lusztig cells. The proof uses a generalisation of the Frobenius–Schur indicator to symmetric algebras, which may be of independent interest.

Let W W be a finite Weyl group and σ \sigma a non-trivial graph automorphism of W W . We show a remarkable relation between the σ \sigma -twisted involution module for W W and the Frobenius–Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig and Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius–Schur indicators for “unipotent characters” of twisted dihedral groups.

Lie-theoretic structures of type E8 (e.g., Lie groups and algebras, Iwahori–Hecke algebras and Kazhdan–Lusztig cells, …) are considered to serve as a “gold standard” when it comes to judging the effectiveness of a general algorithm for solving a computational problem in this area. Here, we address a problem that occurred in our previous work on decomposition numbers of Iwahori–Hecke algebras, namely, the computation of invariant bilinear forms on so-called W-graph representations. We present a new algorithmic solution which makes it possible to produce and effectively use the main results in further applications.

James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split $BN$-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$. Here, we present a variation of this idea which leads to a new, and quite elementary construction of~$\mathfrak{g}$ itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of $\mathfrak{g}$., 14 pages, minor corrections

Let W be a finite Coxeter group and L be a weight function on W in the sense of Lusztig. We have recently introduced a pre-order relation ⪯ L on the set of irreducible characters of W which extends Lusztig’s definition of “families” and which, conjecturally, corresponds to the ordering given by Kazhdan–Lusztig cells. Here, we give an explicit description of ⪯ L for W of type B n and any L. (All other cases are known from previous work.) This crucially relies on some new combinatorial constructions around Lusztig’s “symbols”. Combined with previous work, we deduce general compatibility results between ⪯ L and Lusztig’s a-function, valid for any W,L.

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on these families. Following an idea of Spaltenstein, we show that $\leq_{\cLR}$ can be characterised (and effectively computed) in terms of standard operations in the character ring of $W$. If, moreover, $W$ is the Weyl group of an algebraic group $G$, then $\leq_{\cLR}$ can be interpreted, via the Springer correspondence, in terms of the closure relation among the "special" unipotent classes of $G$.

By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided that the Kazhdan--Lusztig basis of $H$ satisfies certain deep properties. If $W$ is crystallographic and $H$ is a one-parameter algebra, then these properties are known to hold thanks to a geometric interpretation. In this paper, we develop some new general methods for verifying these properties, and we do verify them for two-parameter algebras of type $I_2(m)$ and $F_4$ (where no geometric interpretation is available in general). Combined with previous work by Alvis, Bonnaf\'e, DuCloux, Iancu and the author, we can then extend Lusztig's construction of an explicit isomorphism to all types of $W$, without any restriction on the parameters of $H$., Comment: final version; some minor corrections, including change of title (old title: "Remarks on Iwahori--Hecke algebras with unequal parameters"). To appear in "Pure and Applied Mathematics Quaterly"

In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' conjecture for Iwahori–Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are:•the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham–Lehrer, and•the explicit determination of W-graphs for the irreducible (generic) representations of Hecke algebras of type E7 and E8 by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's MeatAxe and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.

In this paper, we study the Kazhdan–Lusztig cells of a Coxeter group W W in a “relative” setting, with respect to a parabolic subgroup W I ⊆ W W_I \subseteq W . This relies on a factorization of the Kazhdan–Lusztig basis { C w } \{\mathbf {C}_w\} of the corresponding (multi-parameter) Iwahori–Hecke algebra with respect to W I W_I . We obtain two applications to the “asymptotic case” in type B n B_n , as introduced by Bonnafé and Iancu: we show that { C w } \{\mathbf {C}_w\} is a “cellular basis” in the sense of Graham and Lehrer, and we construct an analogue of Lusztig’s canonical isomorphism from the Iwahori–Hecke algebra to the group algebra of the underlying Weyl group of type B n B_n .

Let $H$ be the Iwahori–Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. We establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan–Lusztig basis and Lusztig's results on the $a$-function.

In this paper, we study Lusztig’s a-function for a Coxeter group with unequal parameters. We determine that function explicitly in the “asymptotic case” in typeBn, where the left cells have been determined in terms of a generalized Robinson-Schensted correspondence by Bonnafé and the second author. As a consequence, we can show that all of Lusztig’s conjectural properties (P1)–(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the “leading matrix coefficients” introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori-Hecke algebraHnin terms of the Dipper-James-Murphy basis ofHn.

Following Lusztig, we consider a Coxeter group W together with a weight function L. This gives rise to the pre-order relation ⩽ L and the corresponding partition of W into left cells. We introduce an equivalence relation on weight functions such that, in particular, ⩽ L is constant on equivalent classes. We shall work this out explicitly for W of type F 4 and check that several of Lusztig's conjectures concerning left cells with unequal parameters hold in this case, even for those parameters which do not admit a geometric interpretation. The proofs involve some explicit computations using CHEVIE .

We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

Barbasch and Vogan showed that the Kazhdan–Lusztig cells of a finite Weyl group are compatible with parabolic subgroups. Their proof uses the known bridge between the theory of cells and the theory of primitive ideals. In this paper, an elementary, self-contained proof of this result is provided, which works for arbitrary Coxeter groups and Lusztig's general definition of cells (involving Iwahori–Hecke algebras with unequal parameters). The argument is based on a recent paper by Howlett and Yin.

In this paper we are concerned with the problem of determining the character values and Schur indices of a finite group of Lie type overFq{\mathbb F}_q. We show that (under some conditions onqq) these values lie in the ring of algebraic integers generated by(1+±q)/2(1+ \sqrt {\pm q})/2and roots of unity of order prime toqq. Furthermore, we determine the Schur indices for some of the (nonrational) unipotent characters in exceptional groups. Our results, combined with previous results due to Gow, Ohmori and Lusztig, imply that there are only 6 cases left where the Schur index of a cuspidal unipotent character remains unknown. Our methods rely, in an essential way, on Lusztig’s theory of character sheaves.

Lusztig's classification of the unipotent characters of a finite Chevalley or Steinberg group involves a certain non-abelian Fourier transformation. We construct analogous transformations for the Suzuki and Ree groups, based on a set of axioms derived from Lusztig's theory of character sheaves. We also determine Fourier matrices for the “spetses” (in the sense of Broue, Michel, and the second author) associated with twisted dihedral groups. This completes the determination of Fourier matrices for all “spetses” associated with finite Coxeter groups. We end by collecting common properties of these Fourier matrices and the eigenvalues of Frobenius of character sheaves and unipotent characters.

In 1979, Vogan introduced a generalised τ-invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.

Following Lusztig, we investigate constructible characters, leading coefficients and left cells for a finite Coxeter group W W in the case of unequal parameters. We obtain explicit results for W W of type F 4 F_4 , B n B_n and I 2 ( m ) I_2(m) ( m m even) which support Lusztig’s conjecture that known results about left cells in the equal parameter case should remain valid in the case of unequal parameters.

We apply Lusztig’s theory of cells and asymptotic algebras to the Iwahori–Hecke algebra of a finite Weyl group extended by a group of graph automorphisms. This yields general results about splitting fields (extending earlier results by Digne–Michel) and decomposition matrices (generalizing earlier results by the author). Our main application is to establish an explicit formula for the number of simple modules in type D n D_n (except in characteristic 2 2 ), using the known results about type B n B_n due to Dipper, James, and Murphy and Ariki and Mathas.

We consider the symmetric group Sn (regarded as a Coxeter group) and the corresponding Iwahori-Hecke algebra Hq(Sn). In 1975, Starkey introduced the character table of Hq(Sn) and gave an elegant rule for determining that table, in terms of the ordinary character table of Sn. We give a new simple proof of that rule, whose main ingredients are the exterior powers of the reflection representation of Sn and, otherwise, only general properties of Coxeter groups.