2 Hecke Algebras with Unequal Parameters

These notes are an expanded version of the Aisenstadt lectures given at the CRM, Université de Montréal, in May/June 2002; they also include material from lectures given at MIT during the Fall of 1999 [L12]. I wish to thank Jacques Hurtubise for inviting me to give the Aisenstadt lectures. Hecke algebras arise as endomorphism algebras of representations of groups induced by representations of subgroups. In these notes we are mainly interested in a particular kind of Hecke algebras, which arise in the representation theory of reductive algebraic groups over finite or p-adic fields (see 0.3, 0.6). These Hecke algebras are specializations of certain algebras (known as Iwahori-Hecke algebras) which can be defined without reference to algebraic groups, namely by explicit generators and relations (see 3.2) in terms of a Coxeter group W (see 3.1) and a weight function L : W −→ Z (see 3.1), that is, a weighted Coxeter group. An Iwahori-Hecke algebra is completely specified by a weighted Coxeter graph, that is, the Coxeter graph of W (see 1.1) where for each vertex we specify the value of L at the corresponding simple reflection. A particularly simple kind of Iwahori-Hecke algebras corresponds to the case where the weight function is constant on the set of simple reflections (equal parameter case). In this case one has the theory of the ”new basis” [KL1] and cells [KL1], [L6], [L8]. The main goal of these notes is to try to extend as much as possible the theory of the new basis to the general case (of not necessarily equal parameters). We give a number of conjectures for what should happen in the general case and we present some evidence for these conjectures. We now review the contents of these notes. §1 introduces Coxeter groups following [Bo]. We also give a realization of the classical affine Weyl groups as periodic permutations of Z following an idea of [L4]. §2 contains some standard results on the partial order on a Coxeter group. In §3 we introduce the Iwahori-Hecke algebra attached to a weighted Coxeter group. Useful references for this are [Bo],[GP]. In §4 we define the bar operator following [KL1]. This is used in §5 to define the ”new basis” (cw) of an Iwahori-Hecke