Cuspidal ${\ell }$-modular representations of $\operatorname {GL}_n({ F})$ distinguished by a Galois involution

Kurinczuk, Robert  ; Matringe, Nadir ; Sécherre, Vincent

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e48

Résumé

Let ${ F}/{ F}_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq 2$ with Galois automorphism $\sigma $, and let R be an algebraically closed field of characteristic $\ell \notin \{0,p\}$. We reduce the classification of $\operatorname {GL}_n({ F}_0)$-distinguished cuspidal R-representations of $\operatorname {GL}_n({ F})$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $\sigma $-self-dual cuspidal R-representation to be distinguished. Finally, we classify the distinguished cuspidal ${\overline {\mathbb {F}}_{\ell }}$-representations of $\operatorname {GL}_n({ F})$ having a distinguished cuspidal lift to ${\overline {\mathbb {Q}}_\ell }$.

Détail

DOI : 10.1017/fms.2024.140

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