FACTORIZABLE SHEAVES AND QUANTUM GROUPSLECTURES BY DENNIS GAITSGORY, 1/24/08 AND 1/28/08

The idea is to formulate a kind of Langlands duality for quantum groups (and later,a quantum geometric Langlands conjecture). To this end, we consider the following
diagram of equivalences:
Whit(GrǦ )

KKK
KKK
KKK
K%

Rep(Uq (G))

FSqs
sssss
sssy sHere q ∈ C× is not a root of unity, Rep(Uq (G)) is a certain category of representationsof a quantum group Uq (G), and Whit(GrǦ ) is the category of twisted Whittaker sheaves
on the affine Grassmannian of the dual group Ǧ. The intermediate category FSq is thecategory of factorizable sheaves of Finkelberg and Schechtman. The goal of this talkis to give a conceptual understanding of the equivalence between Rep(Uq (G)) and FSq
using Koszul duality.
1. Quantum groups
1.1. Recall that Uq (G) is the Hopf algebra generated by Ei , Fi , and t ∈ T . Let Λand Λ̌ denote the lattices of weights and coweights, respectively. Given λ̌ ∈ Λ̌, lettλ̌ = λ̌(q) ∈ T . As usual, for any t ∈ T , we have the relationtEi t−1 = Ei αi (t)where αi is the simple root corresponding to Ei . We also have
E i Fi = Fi E i =tdi α̌i − t−1
di α̌iq di − q −diwhere (αi , αi ) = 2di . These generators also satisfy the rest of the quantum Serrerelations.
The co-multiplication is given by∆t = t ⊗ t∆Ei = Ei ⊗ 1 + tdi α̌i ⊗ Ei
∆Fi = 1 ⊗ Fi + Fi ⊗ tdi α̌iDate: February 5, 2008.
12LECTURES BY DENNIS GAITSGORY, 1/24/08 AND 1/28/08

1.2. A representation of Uq (G) is a Λ-graded vector space (not nec. finite dim.) withan action of this algebra. An element t ∈ T acts viatv λ = λ(t)v λ
Let Uq (n+ ) denote the sub-algebra generated by the {Ei }. Define the subcategory
O to be the representations on which Uq (n+ ) acts locally nilpotently. O is a braided
monoidal category.
2. Factorizable Sheaves
Let X be a smooth complex curve and x0 ∈ X (e.g. X = A1 , x0 = 0 ). Let Λpos ⊂ Λdenote the positive span of simple roots.
pos2.1. Given λ ∈ −Λpos , let X λ be the variety
which classifies
PP −Λ -valued divisors
P oftotal weight λ, i.e. divisors of the form
λi xi , such that
λi = λ. If λ = − ni αi ,thenY
Xλ =X (ni )in(n)where X i = Sym i (X) denotes the ni -th symmetric power of the curve.
λ2.2. If λP∈ Λ, let X∞·x
denote the ind-scheme which classifies Λ-valued divisors of0P
the form
λi xi , whereλi = λ, and −λi ∈ Λpos for xi 6= x0 .
Pλλ2.3. If µ ∈ Λ, then X≤µx
⊂ X∞·xclassifies divisors of the form λ0 x0 + xi 6=x0 λi xi00λwith λ0 ≤ µ. Note that if µ = 0, then X≤µx
= X λ.
02.4.

λNext we define a line bundle Pλ on X∞·x
. The fiber of Pλ at0O
ωx(λi i ,λi +2ρ) .

P

λi xii(This was followed by a discussion of why this glues to a line bundle.)
2.5.

By adding divisors, we get a mapλ2X λ1 × X∞·x0

λ1 +λ2X∞·x0λ2 )Let (X λ1 × X∞·x⊂ X λ1 × X λ2 denote the open subscheme consisting of disjoint0 disjdivisors. Then we have the factorization property:

Pλ1 +λ2 (X λ1 ×X λ2∞·x0 )disj◦

Let X λ ⊂ X λ denote the divisors of the formof a simple root. Then Pλ ◦ λ is trivial.
X= Pλ1  Pλ2
P

λi xi where each λi is the negativeFACTORIZABLE SHEAVES AND QUANTUM GROUPS

3◦

2.6. Next we define a basic q-twisted perverse sheaf Ωλ on X λ . Let Ωλ = Ωλ ◦ λ beXthe sign local system. Then we set◦

Ωλ = j!∗ Ωλ◦where j : X λ → X λ is the inclusion map. These sheaves have the factorization property:Ωλ1 +λ2 (X λ1 ×X λ2∞·x0 )disj

2.7.

= Ωλ1  Ωλ2

The fibers of Ωλ have the following property:O(Ωλ )P λi xi =(Ωλi )λi xiiMoreover,λ(Ω )λx =0C

unless λ = w(ρ) − ρ, w ∈ W
else

2.8. A factorizable sheaf (at x0 ) is a collection of q-twisted perverse sheaves Fλ onλX∞·xsuch that0Fλ1 +λ2 (X λ1 ×X λ2 )= Ωλ1  Fλ2∞·x0 disj(plus associativity conditions).
2.9. Let FS denote the category of factorizable sheaves at x0 . Then FS ' O as abeliancategories. For example, given the following diagram
λ(X=µx)0 disjj2/ Xλ=µx0

j1/ Xλ≤µx0

we define∇µ = (j1 )∗ (j2 )!∗ (sign)∆µ = (j1 )! (j2 )!∗ (sign)
Lµ = (j1 )!∗ (j2 )!∗ (sign)
There are all examples of factorizable sheaves at x0 , corresponding to the Verma,
co-Verma, and irreducible representations, respectively.
2.10.

Next we repeat this construction for n points. We defineXnλXnPP
as the ind-scheme which classifies (x10 , . . . , xn0 , λi xi ) where
λi = λ, and λi is negλλative away from x10 , . . . , xn0 . Therefore, X∞·x
isthefiberoverx
0 of X1 .
0
2.11. Let FSn denote the category of factorizable sheaves on Xnλ . For example, FS1
is the category of local systems on S with coefficients in FS. Also, FS2 / FS1 is thecategory local systems on X × X − ∆(X) with coefficients in FS × FS.

4LECTURES BY DENNIS GAITSGORY, 1/24/08 AND 1/28/08

3. Koszul duality
In this section we state the main theorem/construction and explain how it relates toKoszul duality. From now on X = C.
3.1. Let Λ ⊃ Λpos be a lattice containing a semi-group of positive elements. Let A bea Λ-graded Hopf algebra. Suppose that A0 = k and Aµ is finite dimensional.
Note that Uq (n+ ) is not a Hopf algebra in the usual category of Λ-graded vector
spaces. However, it is a Hopf algebra in the category of Λ-graded vector spaces equippedwith a different braiding:Cµ ⊗ C νq (µ,ν)/ Cν ⊗ Cµ .

3.2. Theorem. To a Hopf algebra A one attaches canonically a system of (not twisted!)
perverse sheaves ΩλA on X λ with the factorization property. Moreover:(1) i∗λx (ΩλA ) = (TorA (k, k))λ(2) This construction yields an equivalence of categories between these Hopf algebras and systems of perverse sheaves on X λ with the factorization property.
(3) There is a canonical equivalence of categories between(A]A∗op )-modules on which A>0 acts locally nilpotently
andfactorizable sheaves with respect to ΩA
3.3.

The dual sheaf D(ΩA ) = ΩA∗ is also factorizable. Thereforei!λx (ΩA ) = (ExtA∗ (k, k))λMoreover, the Ωλ from above corresponds to ΩUq (n− ) .
3.4. Let A be an augmented Λpos -graded associative algebra. Let B = k ⊗A k thoughtof as a DG co-algebra via the bar construction. Koszul duality yields an equivalence ofcategories
D(A-modules on which A>0 acts locally nilpotently ) ' D(B-comodules)
M 7→ TorA (k, M )The quasi-inverse to this functor is given by
N 7→ ExtB (k, N )3.5. Let us now discuss factorizable sheaves in dimension 1. Let B be a DG co-algebra.
Let Ran(R) denote Ran space of R. It is a topological space whose points are finite
non-empty collections of points of R. We define a complex of sheaves ΩB on Ran(R).
(ΩB ){x1 ,...,xn } = B ⊗ . . . ⊗ B
Since R is one-dimensional and oriented (S 1 would work too), it suffices to define(ΩB ){x} → (ΩB ){x1 ,x2 }We take this map to be the co-multiplication B → B ⊗ B.
We have the following:

FACTORIZABLE SHEAVES AND QUANTUM GROUPS

5H ∗ (Ran(S 1 ), ΩB ) = H∗ (B) = the Hochschild homology of B
(Beilinson made a comment that one could guess the S 1 -equivariant cohomology...)HS∗ 1 (Ran(S 1 ), ΩB ) = the cyclic homology of B ??
3.6.

We have a map
Ran(R) × Ran(R) → Ran(R)◦given by taking the union of finite subsets. Let (Ran)disj ⊂ Ran(R) × Ran(R) denotethe open subset of pairs of disjoint points. Then ΩB has a factorization property onRan(R).
3.7. Let x0 ∈ R. Then Ranx0 (R) is the space of finite subsets that contain x0 . Let M
be a bi-comodule over B. We define a sheaf ΩB,M on Ranx0 (R). We let(ΩB,M ){x0 ,...,xn } = M ⊗ B ⊗ . . . ⊗ B
as before, the structure maps are sufficient to define a sheaf:
M →B⊗MM →M ⊗B
Moreover, we haveH ∗ (Ranx0 (S 1 ), ΩB,M ) = H∗ (B, M )3.8.

Suppose B is augmented. ThenHc∗ (Ran(R), ΩB ) = ExtB (k, k) ' A

If M is a left B-comodule, then ΩB,M is a sheaf on Ranx0 (R≤x0 ). Furthermore,H ∗ (Ranx0 (R≤x0 ), ΩB,M ) = ExtB (k, M )3.9.

For each n, we have a diagram of DG co-algebras:
B = k ⊗A k/ k ⊗An kO∼(k ⊗A k)⊗nwhere the vertical arrow is a quasi-isomorphism. Such a structure is called an E2co-algebra.
3.10.

If B is an E2 co-algebra, then ΩB is a factorizable complex on Ran(R2 ).

6

3.11.

LECTURES BY DENNIS GAITSGORY, 1/24/08 AND 1/28/08

On the other hand, suppose we have such an ΩB . Then∗(ΩB )
Hc∗ (ΩB Ran(R) ) = A = HRan(iR)which implies that ΩB is a perverse sheaf. Now let I1 , I2 be two disjoint open intervals
in R. We have a map
Ran(I1 ) × Ran(I2 ) → Ran(R)which givesHc∗ (Ran(I1 ), ΩB ) ⊗ Hc∗ (Ran(I2 ), ΩB ) → Hc (Ran(R), ΩB )Since each open interval is homeomorphic to R, this yield a multiplication map A⊗A →∗A. Similarly, using HRan(iR), we get a co-multiplication A → A ⊗ A.
(Here Drinfeld made a comment that this picture is what originally led him to define
quantum groups).