The Tannakian Formalism and the Langlands Conjectures

TL;DR

This paper establishes a connection between the Tannakian formalism and the Langlands conjectures by characterizing homomorphisms between Grothendieck semirings of reductive groups and their relation to group homomorphisms.

Contribution

It proves that certain semiring homomorphisms correspond to actual group homomorphisms, linking Tannakian formalism with Langlands conjectures.

Findings

Homomorphisms from Grothendieck semirings correspond to group homomorphisms.

Irreducible representations are preserved under these homomorphisms.

Connects Tannakian formalism with Langlands conjectures.

Abstract

Let H be a connected reductive group over an algebraically closed field of characteristic zero, and let G be an abstract group. In this note we show that every homomorphism from the Grothendieck semiring of H to that of G which maps irreducible representations to irreducibles, comes from a group homomorphism from G to H. We also connect this result with the Langlands conjectures.