The generic extension map and modular standard modules

Abstract

In this paper we study two classes of $\ell$-modular standard modules of the general linear group. The first class is obtained by reducing existing standard modules over $\overline{\mathbb{Q}}_\ell$ to $\overline{\mathbb{F}}_\ell$ with respect to their natural integral structure. The second class is obtained by studying the generic extension map of the cyclical quiver, which was motivated by the construction of certain monomial bases of quantum algebras. In the latter case we also manage to prove a modular version of the Langlands classification, similar to the work of Langlands and Zelevinsky over $\mathbb{C}$. We moreover compute the corresponding $\ell$-modular Rankin-Selberg $L$-functions and check that they agree with the $L$-functions of their $\mathrm{C}$-parameters constructed by Kurinczuk and Matringe.