Representations of a reductive $p$-adic group in characteristic distinct from $p$

TL;DR

This paper studies irreducible cuspidal representations of reductive p-adic groups over fields of characteristic different from p, extending classification results to non-algebraically closed fields and analyzing supercuspidality conditions.

Contribution

It extends the classification of cuspidal types to non-algebraically closed fields and clarifies the supercuspidality criterion for induced representations.

Findings

Lists of cuspidal types are stable under automorphisms of the field.

Induced irreducible cuspidal representations are supercuspidal if and only if the originating type is supercuspidal.

Verification of classification and supercuspidality conditions for various groups.

Abstract

We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. When $C$ is algebraically closed, for many groups $G$, a list of cuspidal $C$-types $(J,\lambda)$ has been produced satisfying exhaustion, sometimes for a restricted kind of cuspidal representations, and often unicity. We verify that those lists verify Aut($C$)-stability and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main results concern supercuspidality. This notion makes sense for the representations $\lambda$ in the cuspidal $C$-types $(J,\lambda)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $\lambda$ is supercuspidal if and only $\lambda$ is supercuspidal.