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 $p$. 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.

45 pages This is the final version

References in corpus (5)

• Comparison of Bushnell-Kutzko and Yu's constructions of supercuspidal representations
• Representations of a $p$-adic group in characteristic $p$
• On Representations of Reductive $p$--adic Groups over $\mathbb{Q}$--algebras
• Semisimple characters for inner forms II: Quaternionic inner forms of classical groups
• Modulo $\ell$-representations of $p$-adic groups $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(F)$

Cited by in corpus (3)

• A note on the admissibility of smooth simple $RG$-modules
• Representations of $S{L}_2(F)$
• Averaging functors in Fargues' program for GL_n