BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Braverman (University of Toronto) DTSTART:20211117T210000Z DTEND:20211117T220000Z DTSTAMP:20260423T035406Z UID:MITLie/43 DESCRIPTION:Title: Examples of Hecke eigen-functions for moduli spaces of bundles over local non-archimedean field and an analog of Eisenstein series\nby Alexander Braverman (University of Toronto) as part of MIT Lie groups seminar\n\nLe cture held in 2-142.\n\nAbstract\nLet 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 $k(X)$ s tudies 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 t alk!) acting there. The spectrum of the Hecke operators has continuous and discrete parts and it is described by the global Langlands conjectures (w hich in the case of functional fields are essentially proved by V.Lafforgu e).\n\nAfter recalling the above notions and constructions I will discuss what happens when $k$ is replaced by a local field. The corresponding Heck e 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 a nd then Etingof\, Frenkel and Kazhdan formulated a very precise conjecture describing the spectrum in terms of the dual group. Contrary to the class ical 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 oper ator which allows to construct a Hecke eigen-function over $K$ starting fr om a cuspidal Hecke eigen-function over finite field (joint work in progr ess with D.Kazhdan and A.Polishchuk).\n LOCATION:https://researchseminars.org/talk/MITLie/43/ END:VEVENT END:VCALENDAR