Jordan Decompositions of cocenters of reductive $p$-adic groups

TL;DR

This paper establishes a Jordan decomposition for depth r rigid cocenters of reductive p-adic groups, providing an explicit basis and advancing understanding of harmonic analysis on these groups.

Contribution

It introduces a Jordan decomposition for depth r rigid cocenters, offering a new explicit basis under mild hypotheses, enhancing the structural understanding of these cocenters.

Findings

Established Jordan decomposition of depth r rigid cocenters.

Provided an explicit basis for the cocenters.

Applicable to rings of characteristic zero or  eq p.

Abstract

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