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

Robert Kurinczuk, Nadir Matringe, Vincent Sécherre

Abstract

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of ${\rm GL}_n(F)$ which are ${\rm GL}_n(F_0)$-distinguished, that is, which carry a non-zero ${\rm GL}_n(F_0)$-invariant linear form. In the case when $\ell\neq2$, an $\ell$-modular cuspidal representation of ${\rm GL}_n(F)$ is ${\rm GL}_n(F_0)$-distinguished if and only if it lifts to a ${\rm GL}_n(F_0)$-distinguished cuspidal $\ell$-adic representation, whereas when $\ell=2$, it is ${\rm GL}_n(F_0)$-distinguished if and only if it is conjugate-self-dual.

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

Abstract

Let be a quadratic extension of non-Archimedean locally compact fields with residual characteristic , and be a prime number different from . We classify those -modular cuspidal irreducible representations of which are -distinguished, that is, which carry a non-zero -invariant linear form. In the case when , an -modular cuspidal representation of is -distinguished if and only if it lifts to a -distinguished cuspidal -adic representation, whereas when , it is -distinguished if and only if it is conjugate-self-dual.

Paper Structure

This paper contains 7 sections, 131 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The first main result ($\ell\neq2$)
  4. Rankin--Selberg factors over non-Archimedean fields
  5. Gamma factors over finite fields
  6. Gamma factors and distinction
  7. The second main result ($\ell=2$)

Theorems & Definitions (43)

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