Parameters of Hecke algebras for Bernstein components of p-adic groups

TL;DR

This paper investigates the parameters of affine Hecke algebras associated with Bernstein components of p-adic groups, computing them in various cases and confirming conjectures about their nature and origin.

Contribution

The paper computes the q-parameters of affine Hecke algebras for many Bernstein blocks and proves Lusztig's conjecture for most simple p-adic groups.

Findings

Computed q-parameters for principal series of quasi-split groups

Confirmed Lusztig's conjecture for most simple p-adic groups

Reduced conjecture to simple group cases and proved it

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 q_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 simple p-adic groups, and we prove it for most of those.