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The generic extension map and modular standard modules

Johannes Droschl

TL;DR

This paper advances modular representation theory for GL_n over a non-archimedean field by proposing and comparing two natural candidates for modular standard modules: one obtained by reducing classical $ar{f Q}__$-based standard modules to $ar{f F}__$ via integral structures, and another built from the generic extension map on the cyclic quiver. It develops a modular Langlands-type classification through these constructions, establishes the compatibility of Rankin–Selberg L-factors with ${ m C}'$-parameters, and demonstrates a close alignment (and likely equality in many cases) between the two standard-module pictures. The paper shows that L-factors of modular standard modules satisfy $L(X,{ m S}_( rak m),{ m S}_{^{-1}}( rak n^c))=L(X,{ m C}'( rak m),{ m C}'( rak n))$, and provides a degeneration-based combinatorial framework via multisegments and Serre relations to control their structure. Together, these results lay groundwork for a modular Langlands program that mirrors the complex case while incorporating singular behavior in positive characteristic and modular reductions, with explicit links to ${ m C}$-parameters and Rankin–Selberg theory.

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 later 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 $\ell$-modular Rankin-Selberg $L$-function of both classes and check that they agree with the $L$-functions of their $\mathrm{C}$-parameters constructed by Kurinczuk and Matringe.

The generic extension map and modular standard modules

TL;DR

This paper advances modular representation theory for GL_n over a non-archimedean field by proposing and comparing two natural candidates for modular standard modules: one obtained by reducing classical -based standard modules to via integral structures, and another built from the generic extension map on the cyclic quiver. It develops a modular Langlands-type classification through these constructions, establishes the compatibility of Rankin–Selberg L-factors with -parameters, and demonstrates a close alignment (and likely equality in many cases) between the two standard-module pictures. The paper shows that L-factors of modular standard modules satisfy , and provides a degeneration-based combinatorial framework via multisegments and Serre relations to control their structure. Together, these results lay groundwork for a modular Langlands program that mirrors the complex case while incorporating singular behavior in positive characteristic and modular reductions, with explicit links to -parameters and Rankin–Selberg theory.

Abstract

In this paper we study two classes of -modular standard modules of the general linear group. The first class is obtained by reducing existing standard modules over to 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 later case we also manage to prove a modular version of the Langlands classification, similar to the work of Langlands and Zelevinsky over . We moreover compute the -modular Rankin-Selberg -function of both classes and check that they agree with the -functions of their -parameters constructed by Kurinczuk and Matringe.

Paper Structure

This paper contains 18 sections, 50 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Representations
  4. Cuspidal representations
  5. Multisegments
  6. Integral structures
  7. Generic extensions
  8. Degeneration order and Serre relations
  9. Whittaker models
  10. Derivatives
  11. Derivatives and Whittaker models
  12. Rankin Selberg L-factors
  13. Associative products and the map Sc
  14. Standard modules
  15. Standard modules over Ql
  16. ...and 3 more sections

Key Result

Theorem 1

The map $\mathcal{S}_\psi\colon \mathcal{M}ult_{{\overline{\mathbb{F}}_\ell}}^{ap}\rightarrow \mathfrak{Rep}_{W,\psi}$ satisfies property (1) and if $\mathfrak{m}\in \mathcal{M}ult_{{\overline{\mathbb{F}}_\ell},\square}^{ap}$, then $\langle\mathfrak{m}\rangle$ is a quotient of $\mathcal{S}_\psi(\mat

Theorems & Definitions (83)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 2.2.1: MinSec, ZelII
  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2: ZelII, DroMod
  • proof
  • Lemma 2.3.1: MinSec
  • proof
  • ...and 73 more