Appears in collection : Summer School 2021: Enumerative Geometry, Physics and Representation Theory

I am going to explain a series of conjectures due to D.Gaiotto which provide a geometric realization of categories of representations of certain quantum super-groups (such as U_q(gl(M|N)) via the affine Grassmannian of certain (purely even) algebraic groups. These conjectures generalize both the well-known geometric Satake equivalence and the so called Fundamental Local Equivalence of Gaitsgory and Lurie (which will be recalled in the talk).

In the 2nd part of the talk I will explain a recent proof of this conjecture for U_q(N|N-1) (for generic q), based on a joint work with Finkelberg and Travkin.