Modular Representations of ${\rm GL}{(n)}$ Distinguished by ${\rm GL}{(n-1)}$ 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 l not dividing q⁠. We classify all irreducible smooth R-representations of GLn(F) having a non-zero GLn−1(F)-invariant linear form, when q is not congruent to 1 mod l⁠. Partial results in the case when q is 1 mod l show that, unlike the complex case, the space of GLn−1(F)-invariant linear forms has dimension 2 for certain irreducible representations..