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Quantum geometric Satake *August 8, 2007*

*Posted by Joel Kamnitzer in geometric Langlands, quantum groups.*

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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!

## Comments

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Does this require/forbid you to be at a root of unity?

If it doesn’t forbid them, are we likely to see a Jared Anderson-like rule for computing fusion coefficients come from this?

So far the result is proven only in the non-root of unity case, although it is conjectured to hold at a root of unity as well.

I’m not sure exactly how fusion would work though. The problem is that this category of perverse sheaves is not obviously a tensor category — the analog in terms of functions is considering functions on G((t)) which are left invariant by G[[t]] and right equivariant by N((t)) and you can’t convolve such functions. However, Dennis has a replacement for this convolution structure, namely a fusion structure (that’s fusion in terms of working with the diagonal X in X x X (X a curve), not to be confused the fusion of representations). Recall that in the usual Satake case there is both fusion and convolution. However the “Jared Anderson rule” uses the convolution structure in the usual case, so I’m not sure what the analog will be here. I’ll keep an eye out for it.

I’m curious in what way the “Jared Anderson rule” uses

convolution, rather than fusion — don’t they define the same operation on sheaves? is it the explicit form of convolution that’s used?

(the fact that there are both of them

is supposed to explain, from a topological field theory point of view, why the Satake category is symmetric and not just braided, or E_3 rather than E_2.. I guess when we pass to quantum groups we need to drop back down to braided, so need to lose convolution)

I have tried to confuse the fusion of representations with that

on the Grassmannian — after all they’re both aspects of the same

conformal field theory operator product expansion or

topological field theory pair-of-pants product..

however I don’t understand

how to describe the integrable loop group representations in

a setting where there’s also this geometric fusion that you mention

(they don’t live on the right affine Grassmannian I think) – though

I was told Teleman has a picture for all of this.. anyone?

Recall that in the usual Satake case there is both fusion and convolution.Huh?

Maybe I’m confused about the word “usual”. Are we losing geometric, but keeping quantum? Because if we’re losing quantum but keeping geometric, then I don’t understand what fusion is.

If convolution corresponds to tensor product (as in geometric Satake), then something’s going to be weird since tensor product of quantum group reps adds levels. And level corresponds to which root of unity, right? So convolution will be some correspondence between three different categories.

I hope I’m making some sense here.

To answer Allen’s question (and David’s first one at the same time), in geometric Satake (which is what I meant above by usual) there are two ways of defining the tensor structure on the category of G[[t]] equivariant perverse sheaves.

The first, called convolution, is to look at convolution of the affine Grassmannian with itself, namely . It is a Gr bundle over Gr which also maps to Gr. Under this description, tensor product multiplicities become components of fibres of the map to Gr (once you restrict to the convolution of one over another ). This is where Jared’s formula comes from.

The second way to define the tensor structure is called fusion. As David mentions above, the two definitions are equivalent. The fusion structure (which is unrelated (at least not directly related – see the last paragraph of David’s email) to the fusion of quantum group reps) involves working with the Beilinson-Drinfeld Grassmannian over a product of curves. In this formulation, I don’t think that it is possible to see what varieties give you tensor product multiplicities.

Thanks Joel!

Regarding Allen’s question: unless I’m confusing notions,

the tensor product of quantum group representations doesn’t add levels, or change q (eg root of unity) — we have a family

of tensor categories (or Hopf algebras)

labeled by q, ie tensor product is q by q. I think what Allen is referring to is that under the equivalence between quantum group representations and loop group representations, tensor product goes

not to tensor product (which adds levels) but to fusion of representations. When we can realize these representations geometrically (as D-modules) on the affine Grassmannian (eg at critical or negative level, but not at positive level where the integrable reps live) then this fusion is realized by the B-D fusion or factorization picture that Joel explains, if I understand correctly. (I think this might be close

to the idea of factorizable sheaves, which maybe we can get Joel to explain — please???)

Joel – is it correct to think that in the fusion picture the multiplicities would be given in terms of the fibers of the (nonalgebraic) collapsing map from a nearby fiber in the fusion family to the Grassmannian? ie it would be given topologically, but not naturally algebraically?

Thanks David — of course I was mixing up quantum group tensoring (“=” affine group fusion) with affine group tensoring. (Though now I’m very vaguely curious how a quantum group person can view affine group tensoring, mixing roots of unity.)

David – I believe that you are correct that in the fusion picture the multiplicities can be given by the fibres of a collapsing map from a nearby fibre. I’ve never thought about trying to understand those fibres systematically — it may be harder than understanding the fibres of the convolution morphism since as you say, this is just a topological picture. So to answer Allen’s original question, perhaps you can get varieties which record tensor product coefficients for quantum groups at a root of unity.

Collapsing maps aren’t algebraic — they’re only well-defined up to homotopy (or perhaps stratified symplectomorphism). In particular, the fibers aren’t going to be varieties.

Example: {xy=1} has a collapsing to {xy=0}, shrinking the {|x|=1} neck to the singular point, and away from there a symplectomorphism. (Ask for it to be U(1)-equivariant and to preserve the real structure, and it becomes unique.) The fiber is S^1.

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