Endomorphism algebras and Hecke algebras for reductive p-adic groups

TL;DR

This paper demonstrates that Bernstein blocks of smooth representations of reductive p-adic groups are closely related to twisted affine Hecke algebras, leading to classification results and preservation of temperedness.

Contribution

It establishes an almost Morita equivalence between Bernstein blocks and twisted affine Hecke algebras, providing new classification tools and proving a version of the ABPS conjecture.

Findings

Bernstein blocks are 'almost' Morita equivalent to twisted affine Hecke algebras.

The equivalence preserves temperedness of finite length representations.

Provides a classification of irreducible representations in terms of associated complex tori and finite groups.

Abstract

Let G be a reductive p-adic group and let Rep(G)^s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)^s, by analysing the algebra of G-endomorphisms of a progenerator \Pi of that category. We show that Rep(G)^s is "almost" Morita equivalent with a (twisted) affine Hecke algebra. This statement is made precise in several ways, most importantly with a family of (twisted) graded algebras. It entails that, as far as finite length representations are concerned, Rep(G)^s and End_G (\Pi)-Mod can be treated as the module category of a twisted affine Hecke algebra. We draw two consequences. Firstly, we show that the equivalence of categories between Rep(G)^s and End_G (\Pi)-Mod preserves temperedness of finite length representations. Secondly, we provide a classification of the irreducible representations in Rep(G)^s, in terms of the complex torus and the finite group canonically associated to Rep(G)^s. This proves a version of the ABPS conjecture and enables us to express the set of irreducible $G$-representations in terms of the supercuspidal representations of the Levi subgroups of $G$. Our methods are independent of the existence of types, and apply in complete generality.