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

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