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

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 $(J,λ)$ 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 $λ$ in the cuspidal $C$-types $(J,λ)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $λ$ is supercuspidal if and only $λ$ is supercuspidal.

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