Jordan decompositions of cocenters of reductive P-adic groups

Abstract

Cocenters of Hecke algebras H play an important role in studying mod ℓ or ℂ harmonic analysis on connected p-adic reductive groups. On the other hand, the depth r Hecke algebra Hr+ is well suited to study depth r smooth representations. In this paper, we study depth r rigid cocenters Hr+rig of a connected reductive p-adic group over rings of characteristic zero or ℓ ≠ p. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth r rigid cocenter, hence find an explicit basis of Hr+rig.