Jordan decompositions of cocenters of reductive 𝑝-adic groups

He, Xuhua; Kim, Ju-Lee

doi:10.1090/ert/528

Jordan decompositions of cocenters of reductive $p$-adic groups
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by Xuhua He and Ju-Lee Kim

Represent. Theory 23 (2019), 294-324

DOI: https://doi.org/10.1090/ert/528

Published electronically: September 16, 2019

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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 $\overline {\mathcal {H}}^\mathrm {rig}_{r^+}$.

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