The Tannakian formalism and the Langlands conjectures

Abstract

Let $H$ be a connected reductive group over an algebraically closed field of characteristic zero, and let $\Gamma$ be an abstract group. In this note, we show that every homomorphism of Grothendieck semirings $\varphi :K_0^{+}\left[H\right]\to K_0^{+}\left[\Gamma \right]$, which maps irreducible representations to irreducible, comes from a group homomorphism $\rho :\Gamma \to H\left(K\right)$. We also connect this result with the Langlands conjectures.