Moduli spaces of principal F-bundles

TL;DR

This paper constructs and studies moduli spaces of principal F-bundles over curves, generalizing previous moduli of F-sheaves, with the aim of connecting their cohomology to the Langlands correspondence.

Contribution

It introduces a new class of moduli spaces of principal F-bundles associated to reductive groups and representations, extending prior work on F-sheaves.

Findings

Defined properties of the moduli spaces of F-bundles.

Established their relation to Langlands correspondence.

Generalized existing moduli spaces of F-sheaves.

Abstract

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) $F$-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group $G$, and an irreducible algebraic representation $\ov{\om}$ of $(\check{G})^n/Z(\check{G})$. Our spaces generalize moduli spaces of $F$-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case $G=GL_r$ and $\ov{\om}$ is the tensor product of the standard representation and its dual. The importance of the moduli spaces of $F$-bundles is due to the belief that Langlands correspondence should be realized in their cohomology.