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 Π 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 EndG(Π)-Mod can be treated as the module category of a twisted affine Hecke algebra.We draw two major consequences. Firstly, we show that the equivalence of categories between Rep(G)s and EndG(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.