Vol. 4, No. 2, 2022

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

Guy Henniart and Marie-France Vignéras

Vol. 4 (2022), No. 2, 249–305

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 ind JGλ from a smooth C-representation λ of a compact modulo centre subgroup J of G. When C is algebraically closed, for many groups G, a list of pairs (J,λ) has been produced, such that any irreducible cuspidal C-representation of G has the form ind JGλ, for a pair (J,λ) 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 λ above, which involve representations of finite reductive groups. In most cases we prove that ind JGλ is supercuspidal if and only if λ is supercuspidal.

modular representations of reductive $p$-adic groups, cuspidal types, supercuspidal modular representations
Mathematical Subject Classification
Primary: 11F55, 22E50
Received: 16 November 2020
Revised: 14 August 2021
Accepted: 31 August 2021
Published: 24 August 2022
Guy Henniart
Laboratoire de Mathématiques d’Orsay
Univ. Paris–Sud
Marie-France Vignéras
Institut de Mathématiques de Jussieu–Paris Rive Gauche
Université de Paris 7