Twisted geometric Satake equivalence

TL;DR

This paper extends the classical Satake equivalence to a twisted setting involving central extensions of loop groups, establishing a new tensor equivalence with a reductive group's representation category.

Contribution

It classifies central extensions of G(F) by the multiplicative group and constructs a tensor category of perverse sheaves that generalizes the Satake equivalence.

Findings

Classified central extensions of G(F) by the multiplicative group.

Established tensor equivalence with a reductive group's representation category.

Computed the root datum of the associated reductive group.

Abstract

We generalize the classical Satake equivalence as follows. Let k be an algebraically closed field, set O=k[[t]] and F=k((t)). For an almost simple algebraic group G we classify central extensions of G(F) by the multiplicative group. Any such extension E splits canonically over G(O). Consider the category of G(O)-biinvariant perverse sheaves on E with a given Gm-monodromy . We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group. We compute the root datum of this group.