Penghui Li

We recall the Betti Geometric Langlands Conjecture proposed by Ben-Zvi-Nadler. In genus 1 case, we use an uniformization method to calculate the ellitpic character sheaves in terms of Lusztig's character sheaves. And then construct a functor from the category of elliptic character sheaves (the semistable part of automorphic category) to the spectral category in the Conjecture. The functor is fully-faithful if and only if certain conjecture of Hecke categories holds. We prove the analogous conjecture for Weyl groups. The construction uses three previous results: Ben-Zvi-Nadler's identification of character sheaves as trace of Hecke category, Bezrukavnikov's Langlands duality for affine Hecke category, and Ben-Zvi-Nadler-Preygel's gluing of the spectral categories (in genus 1). This is a joint work with D. Nadler.