Jordan decompositions of cocenters of reductive 𝑝-adic groups

Abstract

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