I want to tell you today about Dennis Gaitsgory’s recent work on quantum geometric Satake. If we had this blog six months ago, perhaps I would have written something about this then and it would have been “brand-new”. Anyway, six months old is still pretty new by math standards.
First, I should tell you a few works about regular geometric Satake (I had better resist writing classical geometric Satake) — one of my favorite subjects. I’ll approach it from the perspective of someone who studies representation theory of complex groups.
Let G be a complex reductive group. Then we have the semisimple category of finite dimensional representations of G. Geometric Satake is about constructing this category in a topological way. We take the Langlands dual group of of G (here is a definition) and then take its affine Grassmannian . The geometric Satake correspondence states that the category of equivariant perverse sheaves on Gr is equivalent to the representation category.
You might wonder why it is useful to realize the representation category in such a complicated way. Well, it does have some surprisingly useful consequences, such as the theory of MV cycles (see my thesis) or my recent work with Sabin Cautis. Actually, people in geometric Langlands usually view it the other way around — ie as a description of this category of perverse sheaves, but for the purposes of this post I think that my “backwards view” is best.
Quantum geometric Satake is about an attempt to do something similar for quantum groups. Namely to construct some category of perverse sheaves (or equivalently D-modules) using the affine Grassmannian which will realize the category of representations of a quantum group.
The affine Grassmannian has a natural line bundle coming from the central extension of and this leads to sheaves of twisted differential operators on Gr (much like one has twisted differential operators on the flag variety). The first attempts to construct a category involve looking at categories of twisted D-module on Gr which are equivariant — the amount of twisting is supposed to be the quantum paramenter. However, this does not work.
Gaitsgory’s idea (or maybe Jacob Lurie’s idea) is to start with the “Whittaker category” which is another version of geometric Satake. In this version, one considers perverse sheaves on Gr which are equivariant for the group , with respect to some fixed non-degenerate additive character . This theory was developed by Frenkel, Gaitsgory, and Vilonen and is motivated by the theory of Whittaker functions for p-adic groups. Surprisingly the above “quantum deformation” does work in this setting. Namely, Gaitsgory considers the category of twisted D-modules on Gr which are equivariant for with character . He then proves that this category is equivalent to the corresponding category of representations of the quantum group (again the amount of twisting equal the quantum parameter).
Now, there are two big caveats to make, both interesting. First, the orbits of are infinite dimensional, so this category perverse sheaves (or D-modules) is not a priori well-defined. There is a complicated trick for constructing this category of perverse sheaves which involves a compactification of the stack of N-bundles on a curve (I won’t go into that here).
Secondly, Gaitsgory doesn’t prove an equivalence with the category of representations of the quantum group, but rather with the category of factorizable sheaves which itself is equivalent to the category of representations of the quantum group, by the work of Bezrukavnikov, Finkelberg, Schechtman. This is itself an interesting topic, but unfortunately one for which I have neither energy or expertise to discuss right now!