DAY I, TALK 5: GLOBAL QUANTUM THEORY (GL-3)

Transcription

1 DAY I, TALK 5: GLOBAL QUANTUM THEORY (GL-3) D. ARINKIN Contents 1. Global Langlands Correspondence Formulation of the global correspondence Classical limit Aside: Class field theory Normalization 2 2. Local-to-global: Whittaker D-modules Description via Gr Description via Drinfeld compactification 4 3. Local-to-global: Representations of the Kac-Moody algebra The Kazhdan-Lusztig category The localization functor D-modules on Bun G 6 4. All Together Now 8 References 8 1. Global Langlands Correspondence 1.1. Formulation of the global correspondence. Fix all the usual objects: k is a field of characteristic zero, X is a (projective smooth connected) curve, G is a reductive group, and Ǧ is its Langlands dual. Roughly speaking (to be corrected later) the quantum geometric Langlands correspondence is an equivalence of categories κ (1.1) D-mod(Bun G ) κ D-mod(BunǦ) ˇκ. Here κ and ˇκ are twistings, so the categories on the two sides of the correspondence are basically modules over certain TDO rings on the corresponding stacks. The relation between twists κ and ˇκ is easiest to explain when G is semisimple. Then the twisting on Bun G is given by a G-invariant form on g, or, equivalently, a W -invariant form on t. If the form is non-degenerate (the non-degeneracy of the twisting is required for (1.1)), it induces a W -invariant form on ť; this is what ˇκ is. In other words, κ and ˇκ are supposed to be mutual inverses. Remark 1. It is better to include the critical twist in κ and ˇκ. In other words, the value κ = 0 (which we do not allow in (1.1)!) should correspond to the TDO acting on ω 1/2 Bun G, not on O BunG (equivalently, κ is shifted by half of the Killing form). Date: January 15,

2 2 D. ARINKIN This can sometimes be ignored, because ω 1/2 actually exists as a line bundle on Bun G, and so the corresponding category D-mod(Bun G ) ω 1/2 is equivalent to D-mod(Bun G ). However, the equivalence is given by twisting by the line bundle ω 1/2, so it messes up some constructions: for instance, the free D-module becomes the module induced from a line bundle Classical limit. Recall that the (hopefully) more familiar classical global geometric Langlands conjecture is supposed to be (roughly speaking) an equivalence (1.2) QCoh(LocSys G ) D-mod(BunǦ). This is in fact the limit of (1.1) as κ (and so ˇκ 0): LocSys G is the twisted cotangent bundle to Bun G, so the twisted differential operators on Bun G can be viewed as a quantization of the sheaf O LocSysG, and so QCoh(LocSys G ) is basically D-mod(Bun G ) κ for κ. Note that unlike (1.2), the correspondence (1.1) is symmetric! (Partly because we are slightly simplifying things...) 1.3. Aside: Class field theory. The classical limit case explains why κ and ˇκ have to be mutual inverses (and at any rate, the most natural way to pass from t to ť involves inversion). However, why is there the minus sign ( ˇκ on the RHS)? This is just the way things are, and one way to see it is to check the case when G = T is a torus. This will probably be explained later (talk GL-7), but a short story is as follows: Let A be an abelian variety, and A its dual. (In the case we are interested in, A = Bun G, A = BunǦ, or, more precisely, the abelian variety part of these stacks.) The Fourier-Mukai transform is an equivalence F : QCoh(A) QCoh(A ). It admits an upgrade: the Polishchuk-Rothstein transform, which can be stated as follows (in the non-degenerate case): Theorem 2. Let λ be a non-degenerate twisting on A. There is a corresponding non-degenerate twisting λ on A such that F upgrades to an equivalence of categories F λ : D-mod(A) λ D-mod(A ) λ. (Basically, Theorem 2 claims that two monads on the two sides of the Fourier- Mukai transform match.) Now we can see that if λ is positive, λ must be negative. This is in fact a feature of the Fourier-Mukai transform: ample bundles get sent to anti-ample bundles (and so must the corresponding TDO s, in order for F λ to be an upgrade of F) Normalization. Let us now write some axioms that hopefully would fix the correspondence κ. The main property of the classical Langlands correspondence 0 is the Hecke eigenproperty: there is a big algebra (monoidal category) of operators (functors) acting on D-mod(BunǦ), and the correspondence is supposed to intertwine them with some natural functors on QCoh(LocSys G ). However, this approach does not work in the quantum case: there are essentially no Hecke operators acting on D-mod(Bun G ) κ for irrational κ.

3 GLOBAL 3 Exercise 1. Make this statement precise (it is about twisted G(O)-equivariant modules on the affine Grassmannian) and prove it. (Or go to talk GL-4.) There is a different approach to normalizing 0 that can be extended to the quantum case. Namely, the correspondence should send the structure sheaf O LocSysG to the first Fourier coefficient D BunǦ-modules; more generally, all the tautological vector bundles on O LocSysG correspond to the non-degenerate Fourier coefficient D BunǦ-modules. Remarks 3. (1) The condition on tautological bundles can be obtained from the condition on O BunǦ by the action of the Hecke functors, however, the functors are not explicitly relevant for the correspondence. (2) The conditions can also be stated in the adjoint way: for instance, the first Fourier coefficient functor on D BunǦ corresponds to the functor of global sections Γ(LocSys G, ). (3) The conditions do not quite determine : one has to worry about the constant term/eisenstein series functors on the automorphic side and about the difference between QCoh and IndCoh on the spectral side. The quantum version of this compatibility can be expressed as the following commutative diagram: L κ (1.3) KL(G) κ Whit(GrǦ)ˇκ Loc D-mod(Bun G ) κ Poinc κ D-mod(BunǦ) ˇκ. Here KL is the Kazhdan-Lusztig category of representations of the Kac-Moody algebra, Loc is the localization functor (which in some sense constructs tautological D-modules on Bun G ), Whit is the category of Whittaker sheaves on the affine Grassmannian ( non-degenerate local Fourier coefficients ), and the Poincaré functor Poinc is basically the push-forward under the map GrǦ BunǦ. The top arrow of the diagram is the Fundamental Local Equivalence that matches Whittaker D-modules with tautological sheaves; it is important because it is a concrete local (=hopefully computable) object that normalizes something global and complicated. Remarks 4. (1) In fact, the diagram (1.3) requires fixing a point, or, even better, considering the relative version over X. In fact, it is even better to work over X n (i.e., fix n points), or over X Ran (i.e., take factorization structure into account). If all of these structures are included, (1.3) is supposed to determine κ, modulo something degenerate. Life is particularly easy when κ is irrational, in which case everything is non-degenerate. (2) The diagram (1.3) is not entirely correct: there are actually two versions of the objects/functors in it. This is related to the difference between and! version of D-modules, which we discuss below. (3) There is a discrepancy of twists on the right-hand side. This is essentially just a matter of conventions. (4) We can also take adjoint functors and rewrite the diagram with vertical arrows going up:

4 4 D. ARINKIN (1.4) KL(G) κ L κ Whit(GrǦ)ˇκ Γ D-mod(Bun G ) κ coeff κ D-mod(BunǦ) ˇκ. (5) A side remark: why is it that the Satake category (G(O)-equivariant D- modules on Gr) deforms to something that is generically trivial (so useless for normalization) while the category of the Whittaker D-modules on Gr has a meaninful description? One reason is that the Satake category has nontrivial derived data (irreducible objects have higher Ext s). This is similar to the following fact: if F is a coherent sheaf on a variety M, and the derived fiber F m for m M is non-zero and sits in a single cohomological degree, then F is locally free near m. (Which is a version of the Nakayama Lemma.) 2. Local-to-global: Whittaker D-modules Consider the Whittaker category and the functors that relate it to D-modules on BunǦ. For simplicity, we consider the version of this construction with a single fixed point x X. Let us write simply Gr, Bun, etc for GrǦ, BunǦ, and so on. The functors and categories considered have essentially geometric nature (correspond to spaces and maps between them), and the twisting plays very little role, so we will omit it below Description via Gr. The category Whit is defined as the full subcategory of sheaves on Gr spanned by sheaves that are twisted-equivariant under the action of L(N) (with respect to a non-degenerate character χ). We have a natural map µ : Gr Bun (which basically modifies the trivial Ǧ-bundle at x according to the point of Grassmannian) and Poinc is the pushforward with respect to µ. In fact, we have two functors: Poinc! = µ! and Poinc = µ. The functor Poinc! admits a right adjoint, coeff = µ!. However, defining them carefully requires caution, because Gr is an ind-space; moreover, the orbits of L(N) on Gr are infinitedimensional, all non-zero objects of Whit(Gr) are really ind-supported. Moreover, in fact, the two functors land in different versions ( and!) of the category of D- modules on Bun; more on this later Description via Drinfeld compactification. When we study the functors relating Whit(Gr) with D-mod(Bun), we can replace Gr by a more finitedimensional object defined using the Drinfeld compactification. Here is the construction for Ǧ = SL(2) (for general description, see Dennis s paper). Fix the line bundle l := ω 1/2 on X. Denote by M the moduli stack of collections consisting of an SL(2)-bundle E and a rational map ι : l E that is required to be non-zero and needs to be regular on X {x} (so it can have any kind of singularity at x, but only zeros elsewhere). Clearly, M is an ind-stack equipped with a forgetful map µ : M Bun.

5 GLOBAL 5 We now define a version of the Whittaker category Whit(M). It is going to be a full subcategory Whit(M) D-mod(Bun) cut out by the following twistedequivariance condition. Given a point y X {x}, put M (y) := {(E, ι) : ι(y) 0}. The vector space of polar parts ω(k y )/ω(o y ) acts on M (y), and we define Whit(M (y) ) D-mod(M (y) ) to be the full subcategory of sheaves that are twisted-equivariant for the character res (exp) on ω(k y )/ω(o y ). Exercise 2. (1) Any F Whit(M (y) ) is in fact supported on collections (E, ι) where ι has no zeros (but it can have a pole at x). (This is equivalent to the following statement: if D is a divisor supported away from y, and the map res : ω(k y )/ω(o y ) factors through ω(k y )/ω(o y ) H 1 (X, ω(d)), then D 0.) (2) Given y, y X {x}, the equivariance conditions on M (y) and M (y ) are compatible on M (y) M (y ). 3. Local-to-global: Representations of the Kac-Moody algebra 3.1. The Kazhdan-Lusztig category. The twisting κ defines the Kac-Moody central extension 0 k ĝ κ g((t)) 0. Basically, it is given by the cocycle (f, g) res(κ(f, dg)). The Kazhdan-Lusztig category KL κ is the category of g κ -modules on which the central character acts tautologically, and the action of g[[t]] is integrated to a representation of the positive loop group L + (G). (The condition makes sense because the Kac-Moody extension splits over g[[t]].) The last condition is precisely equivariance with respect to the strong action of L + (G), so we can write KL κ = (ĝ mod) L+ (G) κ. For any V Rep(L + (G)) c (where c denotes compact objects, that is, finitedimensional representations), the induced module Ind(V ) := Indĝκ k g[[t]] (V ) is integrable. The objects {Ind(V ) : V Rep(L + (G)) c } form a system of compact generators for KL κ (in fact, it is enough to consider V Rep(G) c ) The localization functor. The localization functor Loc : KL κ D-mod(Bun G ) κ is most easily described on objects Ind(V ): V defines a tautological vector bundle V on Bun G, and Loc(Ind(V )) = D κ OBunG V (the induced D κ -module). For more general W KL κ, we can describe Loc(W ) via the following steps:

6 6 D. ARINKIN (1) The action of L + (G) on W gives a infinite-dimensional vector bundle W on Bun G ; (2) The action of ĝ κ on W induces, at every point E Bun G, the action of the Kac-Moody central extension 0 k (ĝ E ) κ (g E ) K x 0 on W E. (3) The extension splits over the discrete Lie subalgebra on the fiber W E, and we put Γ(X {x}, g E ) (g E ) K x, Loc(W ) E = (W E ) Γ(X {x},ge ). (4) The rest of the action defines the structure of a D κ -module on Loc(W ). However, this description has a surprising feature: it exists in two versions (you heard this before), but we are not free to choose between them: the choice is determined by the sign of κ (now this, you have not heard)! 3.3. D-modules on Bun G. The first two subsections can be skipped if you are not interested in examples Elementary exercise. Let us start with the following exercise. Let D = k x, d dx be the ring of differential operators on A1. Put M := D/D(x d dx λ). By construction, M is a coherent (and in fact holonomic) D-module. What module is it? It is easy to see that M is a local system on A 1 {0} with regular singularities at 0 and at, and residues λ and λ, respectively. So the question is to understand M as the extension of this local system. Here is the answer: { j (M A1 {0}), λ Z 0 M = j! (M A1 {0}), λ Z <0. Here j : A 1 {0} A 1. (If λ Z, both answers coincide.) Question 5. Hold on, aren t and! Verdier dual to each other? Why are they not behaving in the same way? Exercise 3. What happens on A n? Now the module has one generator annihilated by d x i dx i λ, so it is no longer holonomic. The question is to find out when it is obtained by (resp.!) extension from A n 0. You can solve this by brute force by using the blowup and reducing to the previous example, or you can be clever and just compute its! (resp. ) fiber at the origin to see when it vanishes. Answer: { j (M A M = n {0}), λ Z 0 j! (M A1 {0}), λ Z n.

7 GLOBAL Twisted D-modules on A n / G m. But what does it have to do with the subject? Consider the stack X := A n / G m and its open substack U := (A n {0})/ G m P n 1. Let j : U X be the open embedding. λ k gives a natural twisting on X (which is actually pulled back from B(G m )). Just to normalize things: let O(1) be the tautological line bundle on X (so O(1) U O P n 1(1)). Let D λ be the TDO ring acting on O(λ). The module M described above is the pullback p D λ under the chart p : A n X. (The pullback of the twisting is trivial, so p D λ is an untwisted D-module.) So we proved the following claim: Claim 6. (1) j (D λ U ) = D λ if λ Z 0 ; (2) j! (D λ U ) = D λ if λ Z n. Note that the pushforward j! requires some comments, because we are applying it to a non-holonomic module. However, there is the following general claim: Claim 7. For any λ, the pushforward functor j : D-mod(U) λ D-mod(X) λ preserves coherence. Therefore, the pushforward functor j! : D-mod(U) λ D-mod(X) λ is well-defined and preserves coherence as well. Note that preservation of coherence is unusual for pushforward under open embedding: it is a very stacky phenomenon D-modules vs!-d-modules. Still, what does it have to do with the subject? It turns out that the open embedding j : U X is a model for how the stack Bun G looks far, far away. This all has to do with non-quasicompactness of Bun G. For simplicity, assume G = SL 2 : Let Bun 0 Bun G be the open substack of semistable bundles. Let Bun k Bun G be the stack of bundles E that fit into an exact sequence (3.1) 0 l E l 1 0, deg l = k > 0. (So l E is in fact the Harder-Narasimhan filtration.) Bun k is locally closed, and Bun k := i k Bun i is open and quasi-compact. The point is, that the geometry of Bun k is non-trivial for small k, but if k 0, it is much simpler, because (3.1) splits. Moreover, it is not hard to describe the smooth neighborhood of Bun k in Bun k (basically, by choosing l 1 E and looking at how it deforms). The description includes the above example as the most essential piece. (Details are in Drinfeld-Gaitsgory.) This leads to the following theorem: Theorem 8. The stack Bun admits a filtration by open quasi-compact stacks Bun = k Bun k such that the embedding j k : Bun k Bun k+1 has the following properties for k 0: (1) j k, : D-mod(Bun k ) κ D-mod(Bun k+1 ) κ preserves coherence (and therefore, j k,! is well-defined and preserves coherence as well).

8 8 D. ARINKIN (2) The natural map D κ ( Bun k+1 ) j k, (D κ ( Bun k )) is an isomorphism unless κ is positive rational on one of the simple factors. (3) The dual map D κ ( Bun k+1 ) j k,! (D κ ( Bun k )) is an isomorphism unless κ is negative rational on one of the simple factors. Moreover, the theorem extends to D-modules induced from tautological bundles. We now see that there are two essentially different ways to define the category of twisted D-modules on Bun G : D-mod! (Bun G ) κ = lim{d-mod(bun k G ), j k,!} D-mod (Bun G ) κ = lim{d-mod(bun k G ), j k, }. Both categories are compactly generated. The Verdier duality shows that D-mod! (Bun G ) κ and D-mod (Bun G ) κ are dual to each other. Moreover, we see that the natural target of the functor Loc is D-mod! (Bun G ) κ for positive κ and it is D-mod! (Bun G ) κ for negative κ. (Here positive and negative should be understood in the same way as in Theorem All Together Now Now is the time to put it all together. This gives the following corrected form of the diagram (1.3) (suppose κ is positive): Lκ (4.1) KL(G) κ Whit(GrǦ)ˇκ Loc D-mod! (Bun G ) κ Poinc κ D-mod (BunǦ) ˇκ. L κ (4.2) KL(G) κ Whit(GrǦ) ˇκ Loc D-mod (Bun G ) κ Poinc! κ D-mod! (BunǦ)ˇκ. Here we are using different notation (L and L) for the Fundamental Local Equivalence for negative and positive, because the functors are essentially different. However, the two diagrams are Verdier conjugate. References [1] D. Gaitsgory. Quantum Langlands correspondence. arxiv: [2] D. Gaitsgory. Outline of the proof of the geometric Langlands conjecture for GL(2). Astérisque, no. 370 (2015): [3] D. Gaitsgory, Twisted Whittaker model and factorizable sheaves. Selecta Mathematica, New Series 13, no. 4 (2008): [4] A. Stoyanovsky, On quantization of the geometric Langlands correspondence. arxiv:math/ v5.

9 GLOBAL 9 [5] A. Stoyanovsky, Quantum Langlands duality and conformal field theory. arxiv:math/