Square integrable representations and Hecke eigenvalues over a local non-archimedian field

Abstract

Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\bpo$ of $G$-bundles on ${\mathbb P}^1$ endowed with a trivialization at $0$ and $\infty$. This space is a representation of $G(F)\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $π$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of $π$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.