Speaker:  Alexander Braverman (University of Toronto) 

Title:  Hecke operators over local fields, Calogero-Moser varieties and lifting

Abstract:  Let G be a split reductive group over a finite field k.  A well-known result of Prasad says that the space of (complex-valued) functions with finite support on the moduli space of G-bundles on P¹ over k endowed with Borel structures at 0 and infinity is naturally isomorphic to the regular bi-module over the Iwahori-Hecke algebra of G over the field k((t)).  We shall present  a generalization of this result in two directions:  first, we replace Borel structures at 0 and infinity with a full trivialization at those points, and also we replace the finite field k with a local non-archimedian field F. 

We shall also discuss two types of applications of this generalization:

     1) We show how the relativistic Calogero-Moser variety of G arises as the
         spectrum of Hecke operators over F in a certain special case

     2) We give a conjectural explicit description of the character of the
         lifting of a cuspidal representation of G(F) to the group G(E) for any
         finite extension E of F.

This is a joint work in progress with D. Kazhdan.