A comparison of endomorphism algebras

Abstract

Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G(F), we have two kinds of progenerators: the compactly induced representation indKG(F)(ρ) of a type (K,ρ), and the parabolically induced representation IPG(ΠM) of a progenerator ΠM of a Bernstein block for a Levi subgroup M of G. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra EndG(F)(indKG(F)(ρ)) for a depth-zero type (K,ρ) in [20] with the description of the endomorphism algebra EndG(F)(IPG(ΠM)) in [33], that are described in terms of affine Hecke algebras.