Parameters of Hecke algebras for Bernstein components of [formula omitted]-adic groups

Abstract

Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.Lusztig conjectured that the q-parameters are always integral powers of the cardinality of the residue field of F, and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of absolutely simple p-adic groups, and we prove it for most of those.