Modular representations of GL(n) distinguished by GL(n-1) over a p-adic field

TL;DR

This paper classifies irreducible smooth representations of GL(n) over a p-adic field that have nonzero invariants under GL(n-1), revealing differences from the complex case when the residue field size relates to the characteristic.

Contribution

It provides a complete classification of distinguished mod $ ext{ell}$ representations of GL(n) over p-adic fields for certain residue field conditions, extending understanding beyond the complex case.

Findings

Classifies all irreducible smooth mod $ ext{ell}$ representations of GL(n) with GL(n-1)-invariants.

Shows the dimension of invariant linear forms can be 2 in some cases, unlike the complex case.

Highlights differences in representation theory depending on residue field characteristics.

Abstract

Let $\F$ be a non-Archimedean locally compact field, $q$ be the cardinality of its residue field, and $\R$ be an algebraically closed field of characteristic $\ell$ not dividing $q$.We classify all irredu\-cible smooth $\R$-representations of $\GL\_n(\F)$ having a nonzero $\GL\_{n-1}(\F)$-inva\-riant linear form, when $q$ is not congruent to $1$ mod $\ell$.Partial results in the case when $q$ is $1$ mod $\ell$ show that, unlike the complex case, the space of $\GL\_{n-1}(\F)$-invariant linear forms has dimension $2$ for certain irreducible representations.