Automorphic vector bundles on the stack of $G$-zips

TL;DR

This paper investigates automorphic vector bundles on the stack of G-zips for a connected reductive group over a finite field, providing formulas for global sections and establishing an equivalence of categories.

Contribution

It introduces a general formula for global sections of automorphic vector bundles and establishes a categorical equivalence with admissible modules, advancing understanding of automorphic bundles on G-zips.

Findings

Formula for global sections in terms of Brylinski--Kostant filtration

Categorical equivalence between automorphic bundles and admissible modules

New insights into the structure of automorphic vector bundles on G-zips

Abstract

For a connected reductive group $G$ over a finite field, we study automorphic vector bundles on the stack of $G$-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski--Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of $G$-zips and a category of admissible modules with actions of a zero-dimensional algebraic subgroup a Levi subgroup and monodromy operators.