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

Abstract

We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. In all known cases, such a representation is the compactly induced representation ${\text{ind}}_J^G\lambda$ from a smooth $C$-representation $\lambda$ of a compact modulo centre subgroup $J$ of $G$. When $C$ is algebraically closed, for many groups $G$, a list of pairs $(J,\lambda )$ has been produced, such that any irreducible cuspidal $C$-representation of $G$ has the form ${\text{ind}}_J^G\lambda$, for a pair $(J,\lambda )$ unique up to conjugation. We verify that those lists are stable under the action of field automorphisms of $C$, and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main result concerns supercuspidality. This notion makes sense for the irreducible cuspidal $C$-representations of $G$, but also for the representations $\lambda$ above, which involve representations of finite reductive groups. In most cases we prove that ${\text{ind}}_J^G\lambda$ is supercuspidal if and only if $\lambda$ is supercuspidal.