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

Abstract

Let $G$ be a reductive group over a non-archimedean local field $F$. Consider an arbitrary Bernstein block $\mathrm{Rep}{\left(G\right)}^s$ 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 $\mathrm{Rep}{\left(G\right)}^s$. 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.