Abstract: Let G, G^L be Langlands dual reductive groups. The geometric Satake equivalence is the wonderful fact that one can realize the monoidal category of G^L-representations as G[[t]]-equivariant perverse sheaves on the affine Grassmannian of G.
The affine Grassmannian is a basic example of a semi-infinite partial flag manifold Fl_P, associated to the parabolic P = G. For a general parabolic P, we will explain how G[[t]]-equivariant perverse sheaves on Fl_P are equivalent to Rep(P^L). Moreover, one can obtain not only Rep(P^L) but Rep(M^L), where M^L is the Levi quotient of P^L, by considering G[[t]]-equivariant perverse sheaves which are factorization modules for the semi-infinite intersection cohomology sheaf on Fl_P.
Time permitting, we will indicate the derived equivalences these abelian statements fit into, and how relaxing G[[t]]-equivariance to smaller parahorics yields realizations of the `regular blocks' for various quantum groups, including small and mixed quantum groups, as well as analogues with positive characteristic coefficients.
The contents of this talk builds on work and conjectures of many people, notably Arkhipov, Bezrukavnikov, Braverman, Feigin, Finkelberg, Frenkel, Gaitsgory, Lusztig, Mirkovic, and Raskin, and is the subject of works in progress with Campbell, Chen, Lysenko, and Achar--Riche.