Coefficient systems and Jacquet modules

TL;DR

This paper demonstrates that irreducible supercuspidal representations of certain p-adic groups can be approximated by induced representations from compact subgroups, using coefficient systems and Jacquet modules.

Contribution

It establishes a near-induction property for supercuspidal representations of semisimple, simply connected groups of F-rank 1 over non-archimedean fields, extending understanding of their structure.

Findings

Supercuspidal representations contain specific compact subgroup representations.

Induced representations from these subgroups decompose into finite sums of supercuspidals.

The proof uses equivariant coefficient systems and Jacquet module analysis.

Abstract

Let F be a locally compact non-archimedean field and G the group of F-rational points of an algebraic group assumed to be defined over F, semisimple, simply connected and of F-rank 1. Let pi be a complex irreducible supercuspidal representation of G. We prove that pi is "nearly" induced in the following sense. There exist a maximal compact subgroup K of G and an irreducible smooth representation lamba of K such that pi contains lambda by restriction to K and such that the representation compactly induced from lambda to G is a finite direct sum of irreducible supercuspidal representations. The proof relies on the Schneider and Stuhler theory of equivariant coefficient systems and on a lemma on coefficient systems and Jacquet modules.