The generic extension map and modular standard modules

Johannes Droschl

Vol. 343 (2026), No. 1, 1–37

DOI: 10.2140/pjm.2026.343.1

Abstract

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 $C<!--FUNCTION\; APPLICATION-->$. We also compute the corresponding $\ell$-modular Rankin–Selberg $L$-functions and check that they agree with the $L$-functions of their $C<!--FUNCTION\; APPLICATION-->$-parameters constructed by Kurinczuk and Matringe.

Keywords

Rankin–Selberg $L$-function, modular representation theory, local Langlands correspondence

Mathematical Subject Classification

Primary: 11F70, 22E50

Milestones

Received: 11 March 2025

Revised: 10 April 2026

Accepted: 13 April 2026

Published: 29 April 2026