Hecke algebras for the 1st congruence subgroup and bundles on ${\mathbb{P}}^1$ I: the case of finite field

Let $G$ be a split reductive group over a finite field $k$. In this note we study the space $V$ of finitely supported functions on the set of isomorphism classes $G$-bundles on the projective line ${\mathbb{P}}^1$ endowed with a trivialization at $0$ and $\mathrm{\infty }$. We show that $V$ is naturally isomorphic to the regular bimodule over the Hecke algebra $A$ of the group $G(k((t)))$ with respect to the first congruence subgroup. As a byproduct we show that Hecke operators at points different from $0$ and $\mathrm{\infty }$ to generate the “stable center” of $A$. We provide an expression of the character of the lifting of an irreducible cuspidal representation of $GL(N,k)$ to $GL(N,k^{\mathrm{\prime }})$ where $k^{\mathrm{\prime }}$ is a finite extension of $k$ in terms of these generators. In a subsequent publication we plan to develop analogous constructions in the case when $k$ is replaced by a local non-archimedian field.