Let X be as a smooth projective curve over a finite field k.
It is well-known that the space of unramified automorphic forms for the
global field k(X) and a reductive group G can be identified with the
space of functions on the set of k-points of the moduli space of
G-bundles on X. This space is acted on by a large commutative algebra of
Hecke operators and the Langlands conjecture (essentially known in this
case) describes the set of their common eigen-values.
Langlands and then Etingof, Frenkel and Kazhdan initiated the study of a
similar problem when k is replaced by a local field F. I shall describe
some recent progress in this area concentrating on the case when F is
non-archimedian.
We shall discuss the following 1) exact definition of the space on which
the Hecke operators act (here the difficulty is related to the fact that
Bun_G is a stack rather than a scheme; this difficulty is resolved by a
general algebro-geometric construction) 2) why the cuspidal (but not
Eisenstein) spectrum over k embeds into that over F. 3) The appearance
of automorphic forms for groups of the form G(k[t]/t^2) in this story
and their relationship with the Hitchin system.
Based on a series of joint works with D.Kazhdan and A.Polishchuk. | 
