Abstract: Let X be a smooth projective curve over a finite field k, and let G be a reductive group. The unramified part of the theory of automorphic forms for the group G and the field k(X) studies functions on the k-points on the moduli space of G-bundles on X and the eigen-functions of the Hecke operators (to be reviewed in the talk!) acting there. The spectrum of the Hecke operators has continuous and discrete parts and it is described by the global Langlands conjectures (which in the case of functional fields are essentially proved by V.Lafforgue).
After recalling the above notions and constructions I will discuss what happens when k is replaced by a local field. The corresponding Hecke operators were essentially defined by myself and Kazhdan about 10 years ago, but the systematic study of eigen-functions has begun only recently. It was initiated several years ago by Langlands when k is archimedian and then Etingof, Frenkel and Kazhdan formulated a very precise conjecture describing the spectrum in terms of the dual group. Contrary to the classical case only discrete spectrum is expected to exist. I will discuss what is is known in the case when k is a local non-archimedian field (joint work in progress with D.Kazhdan).
Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode.