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

LECTURES 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(GrGˇ)

%%K

KK KK KK KK

K Rep(Uq(G))

yyssssssssss

FS_q

Hereq∈C^×is not a root of unity, Rep(U_q(G)) is a certain category of representations of a quantum groupU_q(G), and Whit(GrGˇ) is the category of twisted Whittaker sheaves on the affine Grassmannian of the dual group ˇG. The intermediate category FS_q is the category of factorizable sheaves of Finkelberg and Schechtman. The goal of this talk is to give a conceptual understanding of the equivalence between Rep(U_q(G)) and FS_q using Koszul duality.

1. Quantum groups

1.1. Recall that Uq(G) is the Hopf algebra generated by Ei, Fi, and t ∈ T. Let Λ and ˇΛ denote the lattices of weights and coweights, respectively. Given ˇλ ∈ Λ, letˇ tλˇ = ˇλ(q)∈T. As usual, for any t∈T, we have the relation

tE_it⁻¹ =E_iα_i(t)

whereαi is the simple root corresponding to Ei. We also have EiFi =FiEi= t_diαˇi−t⁻¹_d

iαˇi

q^di−q^−di

where (α_i, α_i) = 2d_i. These generators also satisfy the rest of the quantum Serre relations.

The co-multiplication is given by

∆t=t⊗t

∆E_i=E_i⊗1 +t_diαˇi⊗E_i

∆F_i = 1⊗F_i+F_i⊗t_diαˇi

Date: February 5, 2008.

1

1.2. A representation ofUq(G) is a Λ-graded vector space (not nec. finite dim.) with an action of this algebra. An elementt∈T acts via

tv^λ=λ(t)v^λ

Let Uq(n⁺) denote the sub-algebra generated by the {E_i}. Define the subcategory O to be the representations on which U_q(n⁺) acts locally nilpotently. O is a braided monoidal category.

2. Factorizable Sheaves

LetX be a smooth complex curve andx₀ ∈X (e.g. X =A¹, x₀ = 0 ). Let Λ^pos⊂Λ denote the positive span of simple roots.

2.1. Given λ∈ −Λ^pos, let X^λ be the variety which classifies −Λ^pos-valued divisors of total weightλ, i.e. divisors of the form P

λ_ix_i, such that P

λ_i =λ. If λ=−P n_iα_i, then

X^λ =Y

i

X⁽ⁿⁱ⁾

whereX⁽ⁿⁱ⁾= Symⁿⁱ(X) denotes the ni-th symmetric power of the curve.

2.2. If λ ∈ Λ, let X_∞·x^λ ₀ denote the ind-scheme which classifies Λ-valued divisors of the formP

λixi, where P

λi =λ, and −λ_i∈Λ^pos forxi 6=x0. 2.3. If µ∈Λ, thenX_≤µx^λ

0 ⊂X_∞·x^λ ₀ classifies divisors of the form λ₀x₀+P

xi6=x0λ_ix_i withλ₀ ≤µ. Note that ifµ= 0, thenX_≤µx^λ

0 =X^λ.

2.4. Next we define a line bundle P^λ on X_∞·x^λ ₀. The fiber of P^λ atP λixi

O

i

ω_x^(λ_i^i,λi+2ρ).

(This was followed by a discussion of why this glues to a line bundle.) 2.5. By adding divisors, we get a map

X^λ1 ×X_∞·x^λ2 ₀

X_∞·x^λ1+λ₀²

Let (X^λ1×X_∞·x^λ2 ₀)disj ⊂X^λ1×X^λ2 denote the open subscheme consisting of disjoint divisors. Then we have thefactorization property:

P^λ1+λ2

(X^λ1×X∞·x^λ2 0)disj =P^λ1 P^λ2 Let

◦

X^λ ⊂X^λ denote the divisors of the form P

λixi where each λi is the negative of a simple root. ThenP^λ

◦

X^λ is trivial.

2.6. Next we define a basic q-twisted perverse sheaf Ω^λ on X^λ. Let

◦

Ω^λ = Ω^λ ◦

X^λ be thesign local system. Then we set

Ω^λ =j!∗

◦

Ω^λ wherej:

◦

X^λ →X^λis the inclusion map. These sheaves have the factorization property:

Ω^λ1+λ2

(X^λ1×X_∞·x^λ2

0)disj = Ω^λ1 Ω^λ2 2.7. The fibers of Ω^λ have the following property:

(Ω^λ)^P_λixi =O

i

(Ω^λi)_λixi Moreover,

(Ω^λ)_λx=

0 unlessλ=w(ρ)−ρ, w∈W C else

2.8. A factorizable sheaf (at x₀) is a collection of q-twisted perverse sheaves F^λ on X_∞·x^λ ₀ such that

F^λ1+λ2

(X^λ1×X_∞·x^λ2

0)disj = Ω^λ1 F^λ2 (plus associativity conditions).

2.9. Let FS denote the category of factorizable sheaves atx₀. Then FS'Oas abelian categories. For example, given the following diagram

(X_=µx^λ ₀)disj

j2 //X_=µx^λ ₀ ^{j1 //}X_≤µx^λ

0

we define

∇_µ= (j1)∗(j2)!∗(sign)

∆µ= (j1)!(j2)!∗(sign) L_µ= (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 fornpoints. We define X_n^λ

Xⁿ

as the ind-scheme which classifies (x¹₀, . . . , xⁿ₀,P

λixi) where P

λi =λ, and λi is neg- ative away fromx¹₀, . . . , xⁿ₀. Therefore,X_∞·x^λ ₀ is the fiber overx0 of X₁^λ.

2.11. Let FSn denote the category of factorizable sheaves on X_n^λ. For example, FS1

is the category of local systems on S with coefficients in FS. Also, FS₂/FS₁ is the category local systems onX×X−∆(X) with coefficients in FS×FS.

3. Koszul duality

In this section we state the main theorem/construction and explain how it relates to Koszul duality. From now onX=C.

3.1. Let Λ⊃Λ^pos be a lattice containing a semi-group of positive elements. Let Abe a Λ-graded Hopf algebra. Suppose thatA₀ =k andA_µ 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 equipped with adifferent braiding:

C^µ⊗C^ν

q^(µ,ν)//C^ν ⊗C^µ.

3.2. Theorem. To a Hopf algebraAone attaches canonically a system of (not twisted!) perverse sheaves Ω^λ_Aon X^λ with the factorization property. Moreover:

(1) i^∗_λx(Ω^λ_A) = (Tor_A(k, k))^λ

(2) This construction yields an equivalence of categories between these Hopf alge- bras 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 whichA^>0 acts locally nilpotently and

factorizable sheaves with respect to Ω_A 3.3. The dual sheaf D(ΩA) = ΩA^∗ is also factorizable. Therefore

i^!_λx(Ω_A) = (Ext_A^∗(k, k))^λ Moreover, the Ω^λ from above corresponds to Ω_Uq(n⁻₎.

3.4. LetAbe an augmented Λ^pos-graded associative algebra. LetB =k⊗_Akthought of as a DG co-algebra via the bar construction. Koszul duality yields an equivalence of categories

D(A-modules on whichA^>0 acts locally nilpotently )'D(B-comodules) M 7→Tor_A(k, M)

The quasi-inverse to this functor is given by

N 7→Ext_B(k, N)

3.5. Let us now discuss factorizable sheaves in dimension 1. LetBbe 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 ofR. We define a complex of sheaves Ω_B onRan(R).

(Ω_B)_{x1,...,xn} =B⊗. . .⊗B

SinceR is one-dimensional and oriented (S¹ would work too), it suffices to define (Ω_B)_{x} →(Ω_B)_{x1,x2}

We take this map to be the co-multiplicationB →B⊗B.

We have the following:

H^∗(Ran(S¹),Ω_B) =H∗(B) = the Hochschild homology of B

(Beilinson made a comment that one could guess theS¹-equivariant cohomology...) H_S^∗1(Ran(S¹),ΩB) = the cyclic homology ofB ??

3.6. We have a map

Ran(R)×Ran(R)→Ran(R) given by taking the union of finite subsets. Let (

R◦an)_disj ⊂Ran(R)×Ran(R) denote the open subset of pairs of disjoint points. Then ΩB has a factorization property on Ran(R).

3.7. Letx0∈R. ThenRanx0(R) is the space of finite subsets that containx0. LetM be a bi-comodule overB. We define a sheaf Ω_B,M onRan_x0(R). We let

(ΩB,M){x₀,...,xn} =M⊗B⊗. . .⊗B as before, the structure maps are sufficient to define a sheaf:

M →B⊗M M →M⊗B Moreover, we have

H^∗(Ran_x0(S¹),Ω_B,M) =H∗(B, M) 3.8. Suppose B is augmented. Then

H_c^∗(Ran(R),ΩB) = ExtB(k, k)'A

IfM is a left B-comodule, then Ω_B,M is a sheaf onRan_x0(R^≤x0). Furthermore, H^∗(Ran_x0(R^≤x0),Ω_B,M) = Ext_B(k, M)

3.9. For each n, we have a diagram of DG co-algebras:

B=k⊗_Ak ^//k⊗_Aⁿk

(k⊗_Ak)^⊗n

∼

OO

where the vertical arrow is a quasi-isomorphism. Such a structure is called an E₂ co-algebra.

3.10. IfB is anE₂ co-algebra, then Ω_B is a factorizable complex onRan(R²).

3.11. On the other hand, suppose we have such an ΩB. Then H_c^∗(ΩB

Ran(R)) =A=H_R^∗_an(iR)(ΩB)

which implies that Ω_B is a perverse sheaf. Now letI₁, I₂ be two disjoint open intervals inR. We have a map

Ran(I₁)×Ran(I₂)→Ran(R) which gives

H_c^∗(Ran(I₁),Ω_B)⊗H_c^∗(Ran(I₂),Ω_B)→H_c(Ran(R),Ω_B)

Since each open interval is homeomorphic toR, this yield a multiplication mapA⊗A→ A. Similarly, usingH_Ran(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).