Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian

The geometric Satake equivalence of Ginzburg and Mirković-
Vilonen, for a complex reductive group $G$, is a realization of the tensor category of representations of its Langlands dual group ${}^L G$ as a category of ``spherical'' perverse sheaves on the affine grassmannian $\operatorname {Gr}_G = G(\mathbb{C}(\mspace {-3.5mu}(t)\mspace {-3.5mu})/G(\mathbb{C}[\mspace {-2mu}[t]\mspace {-2mu}])$. Since its original statement it has been generalized in two directions: first, by Gaitsgory, to the Beilinson-Drinfeld or factorizable grassmannian, which for a smooth complex curve $X$ is a collection of spaces over the powers $X^n$ whose general fiber is isomorphic to $\operatorname {Gr}_G^n$ but with the factors ``fusing'' as they approach points with equal coordinates, allowing a more natural description of the structures and properties even of the Mirković-Vilonen equivalence. The second generalization, due recently to Finkelberg-Lysenko, considers perverse sheaves twisted in a suitable sense by a root of unity, and obtains the category of representations of a group other than the Langlands dual. This latter result can be considered as part of ``Langlands duality for quantum groups''.

In this work we obtain a result simultaneously generalizing all of the above. We consider the general notion of twisting by a gerbe and define the natural class of ``factorizable'' gerbes by which one can twist in the context of the Satake equivalence. These gerbes are almost entirely described by the quadratic forms on the weight lattice of $G$. We show that a suitable formalism exists such that the methods of Mirković-Vilonen can be applied directly in this general context virtually without change and obtain a Satake equivalence for twisted perverse sheaves. In addition, we present new proofs of the properties of their structure as an abelian tensor category.