GAAG seminar - Symmetries of geometric Eisenstein series

I will report on work in progress, joint with Justin Campbell. For a reductive group G, we prove a factorizable Koszul duality result for the spherical category associated to the affine Grassmannian of G. As a consequence, we construct an action of the Langlands dual Lie algebra ǧ on the intersection cohomology of the moduli of quasimaps from a curve X into the flag variety G/B, similar to one introduced by Feigin, Finkelberg, Kuznetsov, and Mirkovíc. Along the way we develop some infinite-dimensional deformation theory using techniques inspired by recent work of Clausen and Scholze on “solid” geometry.