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

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.