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The purpose of Chapter 1 is to collect some basic results about algebraic groups (with proofs where appropriate) which will be needed for the discussion of characters and applications in later chapters. In particular, one of our aims is to arrive at the point where we can give a precise definition of a `series of finite groups of Lie type' ${G(q)}$, Indexed by a parameter $q$. We also introduce a number of tools which will be helpful in the discussion of examples. For a reader familiar with the basic notions about algebraic groups, root data and Frobenius maps, it might just be sufficient to browse through this chapter on a first reading, in order to see some of our notation. A central role is played by the 'isogeny theorem' which is illustrated with numerous examples, including a quite thorough discussion of Frobenius and Steinberg maps. The final section discusses in some detail the first applications to the character theory of finite groups of Lie type: the `Multiplicity--Freeness' Theorem.
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.
We present the ordinary Harish-Chandra theory for finite groups with a BN-pair in arbitrary non-defining characteristic and the relation to Hecke algebras. We then introduce Lusztig induction for finite reductive groups, explain its basic properties, use it to define and investigate the duality operation on the character ring and the Steinberg character. In the final section we explain the d-Harish-Chandra theories for finite reductive groups which play a fundamenhtal role in modular representation theory of finite reductive groups.
After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig‘s result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.
Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich and vast area of mathematics. It incorporates tools and methods from algebraic geometry, topology, combinatorics and computer algebra, and has since evolved substantially. With this book, the authors meet the need for a contemporary treatment, complementing in core areas the well-established books of Carter and Digne–Michel. Focusing on applications in finite group theory, the authors gather previously scattered results and allow the reader to get to grips with the large body of literature available on the subject, covering topics such as regular embeddings, the Jordan decomposition of characters, d-Harish–Chandra theory and Lusztig induction for unipotent characters. Requiring only a modest background in algebraic geometry, this useful reference is suitable for beginning graduate students as well as researchers.
We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older CHEVIE system, for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan–Lusztig cells and W-graphs, which works efficiently for all finite groups of rank ≤8 (except E8). We also discuss the computation of Lusztig’s leading coefficients of character values and distinguished involutions (which works for E8 as well). Our experiments suggest a re-definition of Lusztig’s ‘special’ representations which, conjecturally, should also apply to the unequal parameter case. Supplementary materials are available with this article.
Let H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.
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 type Bn, 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 algebra Hn in terms of the Dipper-James-Murphy basis of Hn.
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.
We consider the Dipper–James q-Schur algebra [Sscr ]q(n, r)k, defined over a field k and
with parameter q ≠ 0. An understanding of the representation theory of this algebra
is of considerable interest in the representation theory of finite groups of Lie type and
quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]q(n, r)k is a
semisimple algebra if q is not a root of unity. Much more interesting is the case when
[Sscr ]q(n, r)k is not semisimple. Then we have a corresponding decomposition matrix
which records the multiplicities of the simple modules in certain ‘standard modules’
(or ‘Weyl modules’). A major unsolved problem is the explicit determination of these
decomposition matrices.
We consider the Dipper–James q-Schur algebra [Sscr ]q(n, r)k, defined over a field k and
with parameter q ≠ 0. An understanding of the representation theory of this algebra
is of considerable interest in the representation theory of finite groups of Lie type and
quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]q(n, r)k is a
semisimple algebra if q is not a root of unity. Much more interesting is the case when
[Sscr ]q(n, r)k is not semisimple. Then we have a corresponding decomposition matrix
which records the multiplicities of the simple modules in certain ‘standard modules’
(or ‘Weyl modules’). A major unsolved problem is the explicit determination of these
decomposition matrices.
In this paper, we consider Kawanaka's generalized Gelfand-Greav characters associated with the various unipotent classes of a finite group of Lie type $G(q)$ (where $q$ is a power of a prime~$p$). Under the assumption that $p$ is large enough, Lusztig [{\em Advances in Math.} 94 (1992) 139-179] has expressed these characters in terms of characteristic functions of certain character sheaves on $G$, where the resulting formulae contain as unknown quantities certain fourth roots of unity. It is one purpose of this paper to determine these roots of unity explicitly, in the case where the centre of $G$ is connected. We then use these results to study the restriction of character sheaves to the unipotent variety of~$G$. Our motivation was to find a more conceptual approach to the properties listed by Lusztig at the end of the introduction of [{\em J. Algebra} 104 (1986) 146-194].