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

Cocenters of Hecke algebras $\mathcal{H}$ play an important role in studying mod $\mathrm{\ell }$ 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 ${\bar{\mathcal{H}}}_{r^{+}}^{\mathrm{r}\mathrm{i}\mathrm{g}}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\mathrm{\ell }\mathrm{\ne }p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an explicit basis of ${\bar{\mathcal{H}}}_{r^{+}}^{\mathrm{r}\mathrm{i}\mathrm{g}}$.