Smooth representations of reductive p -ADIC groups: structure theory via types

Abstract

Here, F denotes a non-Archimedean local field (with finite residue field) and G the group of F-points of a connected reductive algebraic group defined over F. Let R(G) denote the category of smooth, complex representations of G. Let B(G) be the set of pairs (L, σ), where L is an F-Levi subgroup of G and σ is an irreducible supercuspidal representation of L, taken modulo the equivalence relation generated by twisting with unramified quasi characters and G-conjugacy. To s ∈ B(G), one can attach a full (abelian) sub-category Rs(G) of R(G); the theory of the Bernstein centre shows that R(G) is the direct product of these Rs(G). The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix s ∈ B(G). Let K be a compact open subgroup of G and ρ an irreducible smooth representation of K. The pair K, ρ is an s-type if it has the following property: an irreducible representation π of G contains ρ if and only if π in Rs(G). Let H(G, ρ) be the Hecke algebra of compactly supported ρ-spherical functions on G; if (K,ρ) is an s-type, then the category Rs(G) is canonically equivalent to the category H(G, ρ)-Mod of H(G, ρ-modules. Let M be a Levi subgroup of G; there is a canonical map B(M) to B(G). Take t ∈ B(M) with image s ∈ B(G). The choice of a parabolic subgroup of G with Levi component M gives functors of parabolic induction and Jacquet restriction connecting Rt(M) with Rs(G). We assume given a t-type (KM,ρM) in M; the paper concerns a general method of constructing from this data an s-type (K,ρ) in G. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism H(M,ρM) → H(G,ρ). The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing H(G,ρ) explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types. 1991 Mathematics Subject Classification: 22E50.