Examples of Hecke eigen-functions for moduli spaces of bundles over local non-archimedean field and an analog of Eisenstein series

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 archimedean 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-archimedean field $K$. In particular, I will talk about some version of the Eisenstein series operator which allows to construct a Hecke eigen-function over $K$ starting from a cuspidal Hecke eigen-function over finite field (joint work in progress with D.Kazhdan and A.Polishchuk).

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.