Hecke algebras and symplectic reflection algebras

The current article is a short survey on the theory of Hecke algebras, and in particular Kazhdan-Lusztig theory, and on the theory of symplectic reflection algebras, and in particular rational Cherednik algebras. The emphasis is on the connections between Hecke algebras and rational Cherednik algebras that could allow us to obtain a generalised Kazhdan-Lusztig theory, or at least its applications, for all complex reflection groups.Comment: This survey was written for the Proceedings of the MSRI Introductory Workshop in Noncommutative Algebraic Geometry and Representation Theory (2013

James' Conjecture for Hecke algebras of exceptional type, I

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 $E_7$ and $E_8$ 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 {\sf MeatAxe} and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.Comment: 24 pages; corrected some misprints, added Remark 4.1

Analogues of Weyl's formula for reduced enveloping algebras

In this note we study simple modules for a reduced enveloping algebra U_chi(g) in the critical case when chi element of g^* is ``nilpotent''. Some dimension formulas computed by Jantzen suggest modified versions of Weyl's dimension formula, based on certain reflecting hyperplanes for the affine Weyl group which might be associated to Kazhdan--Lusztig cells.Comment: AMS-LaTeX, 10 pages, 2 figure

Hecke algebras with unequal parameters and Vogan's left cell invariants

In 1979, Vogan introduced a generalised $\\tau$ -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.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1405.573

How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system

For each finite, irreducible Coxeter system $(W,S)$, Lusztig has associated a set of "unipotent characters" \Uch(W). There is also a notion of a "Fourier transform" on the space of functions \Uch(W) \to \RR, due to Lusztig for Weyl groups and to Brou\'e, Lusztig, and Malle in the remaining cases. This paper concerns a certain $W$-representation $\varrho_{W}$ in the vector space generated by the involutions of $W$. Our main result is to show that the irreducible multiplicities of $\varrho_W$ are given by the Fourier transform of a unique function \epsilon : \Uch(W) \to \{-1,0,1\}, which for various reasons serves naturally as a heuristic definition of the Frobenius-Schur indicator on \Uch(W). The formula we obtain for $\epsilon$ extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which $W$ is a Weyl group. We include in addition a succinct description of the irreducible decomposition of $\varrho_W$ derived by Kottwitz when $(W,S)$ is classical, and prove that $\varrho_{W}$ defines a Gelfand model if and only if $(W,S)$ has type $A_n$, $H_3$, or $I_2(m)$ with $m$ odd. We show finally that a conjecture of Kottwitz connecting the decomposition of $\varrho_W$ to the left cells of $W$ holds in all non-crystallographic types, and observe that a weaker form of Kottwitz's conjecture holds in general. In giving these results, we carefully survey the construction and notable properties of the set \Uch(W) and its attached Fourier transform.Comment: 38 pages, 4 tables; v2, v3, v4: some corrections and additional reference