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

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 ${\overline{\mathcal{H}}}_{r^{+}}^{\mathrm{rig}}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell \ne p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an explicit basis of ${\overline{\mathcal{H}}}_{r^{+}}^{\mathrm{rig}}$.

29 pages

References in corpus (2)

• Tame tori in p-adic groups and good semisimple elements
• Cocenter of $p$-adic groups, II: induction map