Shtukas for reductive groups and Langlands correspondence for function fields

TL;DR

This paper explores recent advances in the Langlands and geometric Langlands programs for function fields, focusing on the decomposition of automorphic forms via G-shtukas and the geometric Satake equivalence.

Contribution

It introduces a canonical decomposition of cuspidal automorphic forms for reductive groups over function fields using G-shtukas and geometric Satake.

Findings

Decomposition of automorphic forms indexed by Langlands parameters

Use of G-shtukas with multiple modifications in proofs

Application of geometric Satake equivalence

Abstract

We discuss recent developments in the Langlands program for function fields, and in the geometric Langlands program. In particular we explain a canonical decomposition of the space of cuspidal automorphic forms for any reductive group G over a function field, indexed by global Langlands parameters. The proof uses the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence.