Endomorphism algebras and Hecke algebras for reductive p-adic 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 Πof 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 (Π)-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 (Π)-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.

New in second version: - compatibility of the constructions with parabolic induction is discusssed - the presentation of the arguments for temperedness is improved - a paragraph about a smaller progenerator of Rep(G)^s, and how that works out in the non-cuspidal case. In 2023 a problem with the part on temperedness in paragraph 9.1 surfaced. This is repaired in an appendix to version 3