Supercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involution
| Autor: | Vincent Sécherre |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Involution (mathematics)
Pure mathematics Mathematics::Number Theory 01 natural sciences 11F85 0103 physical sciences modular representation FOS: Mathematics 0101 mathematics Representation Theory (math.RT) Mathematics::Representation Theory p-adic reductive group Mathematics 11F70 Algebra and Number Theory [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] Cuspidal representation 010102 general mathematics cuspidal representation 2010 MSC: 22E50 11F70 11F85 16. Peace & justice Galois involution distinguished representation 22E50 22E50 11F70 11F85 010307 mathematical physics Mathematics - Representation Theory |
| Zdroj: | Algebra Number Theory 13, no. 7 (2019), 1677-1733 Algebra & Number Theory Algebra & Number Theory, Mathematical Sciences Publishers 2019, 13 (7), pp.1677-1733. ⟨10.2140/ant.2019.13.1677⟩ |
| ISSN: | 1937-0652 |
| DOI: | 10.2140/ant.2019.13.1677⟩ |
| Popis: | Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $\sigma$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $\pi$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $\pi^{\sigma}\simeq\pi^{\vee}$ (such a representation is said to be $\sigma$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times}$. When $\pi$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $\pi$ to be ${\rm GL}_n(F_0)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $\omega$ the non-trivial $R$-character of $F_0^{\times}$ trivial on $F/F_0$-norms, we prove that any $\sigma$-selfdual supercuspidal $R$-representation is either distinguished or $\omega$-distinguished, but not both. In the modular case, that is when $\ell>0$, we give examples of $\sigma$-selfdual cuspidal non-supercuspidal representations which are not distinguished nor $\omega$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan-Kable-Tandon. Comment: 56 pages |
| Databáze: | OpenAIRE |
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