WINTER SCHOOL ON LOCAL GEOMETRIC LANGLANDS THEORY: PROGRAM › ~Sergey.Lysenko › program_1.pdf ·...

WINTER SCHOOL ON LOCAL GEOMETRIC LANGLANDS THEORY:

PROGRAM

Contents

1. Day one 21.1. GL-1: Introduction to quantum local geometric Langlands 21.2. T-1: Tutorial on groups acting on categories 21.3. Wh-1: The Whittaker model 21.4. GL-2: The quantum parameter 31.5. GL-3: The global quantum geometric theory 32. Day two 52.1. Ch-1: The factorization category of D-modules on the affine Grassmannian 52.2. T-2: Tutorial on factorization algebras and categories 62.3. GL-4: Quantum Satake equivalence 62.4. Wh-2: Proof of the equivalence of the two versions of the Whittaker model 72.5. Ja-1: Jacquet functors for actions of loops groups on categories 73. Day three 83.1. T-3: Tutorial on the functor of weak invariants 83.2. GL-5: The functor of weak L(G)-invariants 83.3. GL-6: Spectral side in the classical case 93.4. FLE-1: Statement of the equivalence 103.5. Wh-3: Statement of local quantum Langlands 113.6. GL-7: Quantum Langlands for G being a torus 144. Day four 144.1. Ja-2: Jacquet functors and Langlands duality 144.2. T-4: Tutorial on quantum groups 154.3. T-5: Tutorial on factorization vs braided monoidal categories 174.4. Ch-2: The factorization algebra Ωq (or now Nature encodes root data) 174.5. Ja-3: Jacquet functor on Kac-Moody modules and the Kazhdan-Lusztig equivalence 204.6. Wh-4: Duality for W-algebras 225. Day five 245.1. Wh-5: Jacquet functor on the Whittaker category 245.2. FLE-2: Strategy of the proof of FLE via Jacquet functors 265.3. Ja-4: Jacquet/Eisenstein functors in the global situation 275.4. Ja-5: The semi-infinite IC sheaf 295.5. FLE-3: Poor man’s FLE: going to the small quantum group 305.6. Ch-3: The “Master” chiral algebra 32

Date: November 30, 2017.

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1. Day one

1.1. GL-1: Introduction to quantum local geometric Langlands.

In this talk, we will introduce the main object of study of the local geometric Langlands theory: thetotality (technically, 2-category) of categories (technically, DG categories) equipped with an action ofthe loop group L(G) := G((t)) at a given level κ. We denote it by L(G)-modκ.

The quantum local geometric Langlands conjecture says that this 2-category is equivalent to thesimilarly defined category for the dual loop group L(G) at the dual level −κ. I.e., we propose:

Conjecture 1. There exists an equivalence of 2-categories

(1.1) L(G)-modκLlocκ' L(G)-mod−κ.

The equivalence (1.1) is supposed to be characterized (modulo the issues of Whittaker/Kac-Moody

degeneracy) by the requirement that if C and C correspond to each other under the above equivalenceof 2-categories, then the Whittaker model of C, denoted Whit(C), should be equivalent as a category to

the Kac-Moody model of C, denoted KM(C).

The degenerate case should be controlled by the compatibility of the conjectural equivalence withthe procedure of parabolic induction.

This talk will be a general introduction to these ideas, and much of the rest of the workshops willbe devoted to supplying details.

1.2. T-1: Tutorial on groups acting on categories.

The key technical notion involved in the local geometric Langlands theory is that of DG category,equipped with an action of a group (or, more generally, a group ind-scheme).

In fact, there are two such notions: weak action and strong action, and the latter notion admits atwist corresponding to a central extension of the Lie algebra.

In this talk, we will introduce these notions, first in the case of finite-dimensional groups, and thenfor loop groups, and consider examples.

We will also review the theory of D-modules, de Rham prestacks, twistings, multiplicative twistingsand their relation to central extensions of Lie algebras. Some background material on DG categorieswill be reviewed as well (compact generation, duality, tensor products).

1.3. Wh-1: The Whittaker model.

A key operation needed for the statement of the local quantum geometric Langlands conjecture isthe passage from a category C equipped with an action of L(G) to its Whittaker model Whit(C).

This operation is modeled on the classical operation in number theory: given a representation of alocally compact group G, its Whittaker model is the space of vectors/functionals invariant against themaximal unipotent N against a non-degenerate character χ.

Like in number theory, in our geometric setting, there are two possible operations: given C, wecan consider the subcategory CL(N),χ of (L(N), χ)-invariant objects or the quotient category CL(N),χ

of (L(N), χ)-coinvariants. These two operations are naturally dual to each other.

However, it turns out that CL(N),χ and CL(N),χ are actually equiavalent (via a non-trivial procedure).The result is what we will call the Whittaker model of C, to be denoted Whit(C).

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A consequence of the equivalence of the two definitions of the functor Whit is that it is very well-behaved from the algebro-analytic point of view (technically: commutes with both limits and colimitsas well as duality).

1.4. GL-2: The quantum parameter.

Roughly speaking, the quantum parameter for the quantum geometric Langlands theory is what iscalled the Kac-Moody level (i.e., an Ad-invaraint bilinear form on g). However, this is literally the caseonly when G is semi-simple.

If G contains has a central torus (and we will need to consider this case, because along with G wewill consider all its Levi subgroups), an additional set of quantum parameters comes by consideringextensions of z⊗OX by ωX (here z is the center of g, and X is a curve equipped with a point x, so that

k[[t]] is OX,x).

In this talk we will recall the classical construction that associates to each quantum parameter a(factorizable) central extension of the (factorizable version of the) loop algebra L(g) := g((t)). But evenmore fundamentally, we will explain that to a quantum parameter there corresponds a factorizabletwisting on the factorizable affine Grassmannian GrG, and also a mutiplicative factorizable twisting onthe factorizable loop group L(G).

This talk will also provide a background on the notions involved in the above definitions (thefactorizable Grassmannian, loop group, etc).

We will also relate quantum parameters to factorizable line bundles on GrG that give rise to factor-izable central extensions of L(G) (and thereby to integrable representations), and also to factorizablegerbes on GrG, the latter being closely related to metaplectic parameters appearing in number theory.

Finally, we will explain how a quantum parameter gives rise to a twisting on BunG on a global curve.

1.5. GL-3: The global quantum geometric theory.

As in the classical theory of automorphic functions, the local Langlands theory is closely related tothe global one.

Given a quantum parameter κ and a global curve X, we can consider the corresponding category oftwisted D-modules D-mod(BunG)κ. The caveat here is that there are two natural categories that onecan consider, the !- and the *- ones, denoted D-mod(BunG)κ,! and D-mod(BunG)κ,∗, respectively. Thishas to do with the fact that BunG is not quasi-compact, and, figuratively speaking, in D-mod(BunG)κ,!we generate objects using !-extensions from quasi-compact open substacks and in D-mod(BunG)κ,! weuse *-extensions.

The general principle, which we will see in other instances during the workshop, is that the choiceof which version to consider depends on the sign of κ. For example, if G is simple, in which case κis a multiple of the Killing form κ = c · κKil, we use the !-version if (c + 1

2) ∈ Q>0 and the *-version

if (c + 12) ∈ Q<0, and the two versions coincide if c ∈ k \ Q. (Here c = − 1

2is the critical value and

corresponds to the classical, as opposed to quantum, situation.)

We will state the global (non-ramified) quantum geometric Langlands conjecture as follows (say,when κ is positive in the above sense):

Conjecture 2. There exists an equivalence

(1.2) D-mod(BunG)κ,!Lglobκ' D-mod(BunG)−κ,∗.

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1.5.1. Local-to-global functors.

In order to state the local-to-global compatibility that is supposed to be satisfied by the conjecturalequivalence (1.2), we will introduce several local-to-global functors.

We introduce several functors that connect D-mod(BunG)κ (both versions) with

Whit(GrG)−κ := Whit(D-mod(GrG)−κ).

We start with (a naturally defined) functor

Poinc∗ : Whit(GrG)−κ → D-mod(BunG)κ,∗,

then consider its Verdier conjugate

Poinc! : Whit(GrG)−κ → D-mod(BunG)κ,!,

and the right adjoint of Poinc! (same as dual of Poinc∗)

coeff : D-mod(BunG)κ,! →Whit(GrG)−κ.

We will also introduce the Kazhdan-Lusztig category

KL(G)κ := g-modL+(G)κ ,

(i.e., the category of Kac-Moody modules integrable with respect to L+(G)), and the localizationfunctor

Loc : KL(G)κ → D-mod(BunG)κ,!

(for κ positive), its Verdier conjugate

Loc : KL(G)−κ → D-mod(BunG)−κ,∗,

and their respective right adjoints/duals

Γ : D-mod(BunG)−κ,∗ → KL(G)−κ and Γ : D-mod(BunG)κ,! → KL(G)κ.

1.5.2. Local-to-global compatibility.

This equivalence (1.2) is supposed to be characterized (again, up to the issue of Whittaker/KMdegeneracy) by the requirement that it makes the following diagrams commute:

(1.3)

KL(G)κLκ−−−−−→ Whit(GrG)κ

Loc

y yPoinc∗

D-mod(BunG)κ,!Lglobκ−−−−−→ D-mod(BunG)−κ,∗.

In the above formula,

KL(G)κLκ' Whit(GrG)κ

is the Fundamental Local Equivalence for positive level, which would be one of the central themes inthis workshop, see Talk FLE-1.

Passing to dual functors (and using the symmetry of the picture in G and G), we obtain a diagram

(1.4)

D-mod(BunG)κ,!Lglobκ−−−−−→ D-mod(BunG)−κ,∗

coeff

y yΓ

Whit(GrG)−κ(L−κ)−1

−−−−−−→ KL(G)−κ,

where

KL(G)−κL−κ' Whit(GrG)−κ

is the Fundamental Local Equivalence for negative level.

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By passing to left adjoints along the vertical arrows in (1.4), we obtain yet another commutativediagram:

(1.5)


Poinc!

x xLoc


−−−−−−→ KL(G)−κ,

Juxtaposing diagrams (1.3) and (1.4) we obtain:

(1.6)


Loc

y yPoinc∗


coeff

y yΓ


−−−−−−→ KL(G)−κ,

which we call the Fundamental Commutative Diagram.

Note that the commutativity of the outer square in (1.6) can be formulated unconditionally (we donot need Lglob

κ to state it):

Conjecture 3. The square

(1.7)


coeff Loc

y yΓPoinc∗


−−−−−−→ KL(G)−κ.

commutes.

In talk Wh-4 we will see that Conjecture 3 certain from another conjecture of local nature (seeConjecture 21).

2. Day two

2.1. Ch-1: The factorization category of D-modules on the affine Grassmannian.

The statement of the local quantum geometric Langlands equivalence appears quite daunting: wewant to compare two kinds of objects, each equipped with a rather complicated sounding structure–anaction of the corresponding loop group.

The hope is that we can express this structure (on both sides) via another kind of structure, one thatwould be more amenable to Langlands-type comparison. This brings us to the idea of factorization.Namely, we will express the structure that we have at a point (i.e., an action of L(G) on a category C,or whatever it induces on Whit(C) or KM(C)) by interpreting it as a structure of factorization modulecategory for an appropriate factorization category.

The idea is that the acting agent, i.e., the factorization category in question, is a simpler object.For example, the acting agent for Whit(C) (resp., KM(C)) is the category Whit(D-mod(GrG)κ) (resp.,KL(G)−κ) that we already encountered in Talk GL-3.

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Although both Whit(D-mod(GrG)κ) and KL(G)κ are still quite non-trivial, they are non-ramifiedin nature, and we can hope to achieve some sort of hands-on (i.e., close to combinatorial) descrip-tion of both of them, so that we will able compare these categories (each on its side of Langlandscorrespondence) directly. The latter will amount to the Fundamental Local Equivalence.

In this talk we will encounter the first instance of the passage “Action → Factorization”. Namely,we will observe that the category D-mod(GrG)κ is naturally a factorization category, and if L(G)acts on C (at level κ), then the same C has a natural structure of factorization module category overD-mod(GrG)κ. This defines a functor

(2.1) L(G)-modκ → D-mod(GrG)κ-modFact.

We will state the following conjecture and introduce some ideas that potentially lead to its proof:

Conjecture 4. The functor (2.1) is fully faithful.

2.2. T-2: Tutorial on factorization algebras and categories.

The notion of factorization category (and a related notion of factorization algebra in a given factor-ization category) is highly non-classical: it uses tools from higher algebra in an essential way.

In this talk we will first discuss the informal ways to introduce these notions, and then how to makessense of them within the formalism of higher algebra. In process of doing so, we will review some ofthe basic notions of higher category theory that would be useful for us during the rest of the workshop.

One of the constructions that we will review attaches to a symmetric monoidal category (resp.,commutative algebra) a factorization category (resp., factorization algebra).

A Fact(A), A ∈ ComAlg(A) Fact(A) ∈ Fact(A).

We will also review some constructions that will be ubiquitous in this workshop, such as the Kac-Moody factorization (a.k.a. chiral) algebra.

As in illustration of the ideas from Talk Ch-1, we will study the category of integrable representationsof L(G) (at a given integral level), and identify it with the category of factorization (a.k.a. chiral, VOA)modules for the corresponding integrable quotient of the Kac-Moody factorization algebra.

2.3. GL-4: Quantum Satake equivalence.

One of the features of the local quantum geometric Langlands equivalence (1.1) is that if C ∈L(G)-modκ corresponds to C ∈ L(G)-mod−κ, then their categories of spherical objects, i.e.,

CL+(G) and C

L+(G),

are equivalent as categories.

As a formal consequence of this, one obtains that D-mod(GrG)κ ∈ L(G)-modκ corresponds toD-mod(GrG)−κ ∈ L+(G)-mod−κ under the equivalence (1.1):

Llocκ (D-mod(GrG)κ) ' D-mod(GrG)−κ.

Applying the above principal one more time we deduce a conjectural equivalence between

SphG,κ := D-mod(GrG)L+(G)κ and SphG,−κ := D-mod(GrG)

L+(G)−κ :

Conjecture 5. There exists an equivalence

SphG,κ ' SphG,−κ,

as monoidal categories, compatible with the factorization structures.

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The above conjecture is the quantum counterpart of the geometric Satake equivalence. It playsthe same role vis-a-vis Conjecture 1.2 as the usual geometric Satake does for the classical geometricLanglands (i.e., that (1.2) is compatible with the SphG,κ-action on D-mod(BunG)κ and the SphG,−κ-action on D-mod(BunG)−κ.

Now, Conjecture 5 is very close to being a theorem. In fact, if κ is irrational, both sides are trivial(equivalent to Vect, the category of vector spaces). For κ rational, one shows that both sides areequivalent to (a twisted version of) the category of representations of a certain reductive group H,whose root datum can be read off that of G and κ (with the same result for G and −κ). This group Hhad been first discovered by G. Lusztig as the recipient of the quantum Frobenius, see talk T-5.

2.4. Wh-2: Proof of the equivalence of the two versions of the Whittaker model.

In Talk Wh-1, we stated that for a category C equipped with an action of L(G), there a canonicalequivalence

CL(N),χ → CL(N),χ.

In this talk we will prove this statement. The reason for doing is not so much to give a proof inorder to establish validity, but to introduce the ideas contained therein: what are the tools that allowus to prove something about substantially infinite-dimensional objects such as L(N)?

The key observation is that one approximate each side, i.e., CL(N),χ and CL(N),χ, by “adolescent

Whittaker models”, i.e., categories CIn,χ, where In is a certain family of subgroups of L(G) that “tendsto L(N)” as n 7→ ∞. The idea is that, unlike L(N), each In is a group-scheme (rather than a groupind-scheme), and the operation C 7→ CIn,χ is much more manageable.

The subgroups and the corresponding categories CIn,χ turn out to be very handy for other problemsthat involve the behavior of Whit(C).

2.5. Ja-1: Jacquet functors for actions of loops groups on categories.

As in the classical theory of representations of p-adic groups, we need to study the relation betweenL(G)-modκ and the corresponding category L(M)-modκ, where M is the Levi quotient of a parabolicP on G. In order to simplify things, we will take P to be a Borel B, so that M is the abstract CartanT . We choose a splitting T → B, so that we have a well-defined opposite Borel subgroup B−.

Like in the case of the Whittaker model, there are two functors L(G)-modκ ⇒ L(T )-modκ:

C 7→ J!(C) := CL(N) and C 7→ J∗(C) := CL(N).

However, an analog of the functor that defines an equivalence CL(N),χ → CL(N),χ, exists as a functor

CL(N) → CL(N) but is no longer an equivalence.

Instead, we propose:

Conjecture 6. The composition

(2.2) CL(N) → C→ CL(N−)

is an equivalence.

In the talk we will explain how this conjecture can be seen as a counterpart in our setting of Bern-stein’s 2nd adjointness theorem, which says that (in the context of representations of p-adic groups),parabolic induction, which is the right adjoint of the Jacquet functor J is also the left adjoint of theopposite Jacquet functor J−.

We will explicitly verify the following corollary of the above conjecture: namely, that the equivalencein question holds after we take L+(T )-invariants (which is the same as coinvariants) on both sides. Infact, we will show that the resulting equivalence is obtained by identifying both sides with CI , where

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I ⊂ L+(G) is the Iwahori subgroup (the caveat here is that since we use the Iwahori subgroup, forthe moment, we can only prove this equivalence at a point, i.e., we do not yet know how to do thefactorizable version).

3. Day three

3.1. T-3: Tutorial on the functor of weak invariants.

In this talk, which is a preparation for GL-5, we will focus on the finite-dimensional situation.Thus, let G be a finite-dimensional algebraic group, and let G-mod be the 2-category of DG categoriesequipped with an action of G.

For C ∈ G-mod, we can consider the DG category CG -weak of weak G-invariants. This functor isrepresentable:

CG -weak ' FunctG-mod(g-mod,C),

where g-mod ∈ G-mod is the category of modules over the Lie algebra, viewed as acted on by G viathe adjoint action.

Hence, CG -weak is acted on by the monoidal category HCh(G) := EndoFunctG-mod(g-mod)op. Themonoidal category HCh(G) is that of Harish-Chandra bimodules.

We will show that the resulting functor

G-mod→ HCh(G)-mod, C 7→ CG -weak

is an equivalence of 2-categories.

As a formal consequence, one obtains a canonical equivalence

(3.1) g-mod ⊗HCh(G)

g-mod ' D-mod(G).

3.2. GL-5: The functor of weak L(G)-invariants.

In talk Wh-1 we studied the operation

C→Whit(C).

In this talk we will study its Langlands dual counterpart. This is the operation

C 7→ KM(C) := CL(G) -weak.

As in the finite-dimensional case, this functor is co-representable:

KM(C) ' FunctL(G)-modκ(g-modκ,C),

where g-modκ is the category of modules over the Kac-Moody algebra attached to g at level κ.

The functor KM gives rise to a functor

(3.2) L(G)-modκ → HCh(L(G))κ-mod,

where

HCh(L(G))κ := EndoFunctL(G)-modκ(g-modκ)op.

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3.2.1. Kac-Moody model as a factorization module.

Let KL(G)κ be the Kazhdan-Lusztig category. We can think of it either as (g-modκ)L+(G) or as

(D-mod(GrG)−κ)L(G) -weak. Following up on ideas from Talk Ch-1, we will see that any module categoryover HCh(L(G))κ-mod is automatically a factorization module category for KL(G)−κ.

Thus, we obtain a functor

(3.3) HCh(L(G))κ-mod→ KL(G)−κ-modFact,

which is a cousin of the functor (2.1). We will propose:


Thus, we obtain that the operation C 7→ KM(C) defines a functor

(3.4) L(G)-modκ → KL(G)−κ-modFact.

However, unlike the finite-dimensional situation, we encounter the issue of Kac-Moody degeneracy.

3.2.2. KM (non)-degeneracy.

A feature of the infinite-dimensional situation is that, unless G is a torus, the functor (3.2) is nolonger an equivalence (in fact, it fails to be conservative). I.e., for any C ∈ L(G)-modκ we have awell-defined functor

(3.5) g-modκ ⊗HCh(L(G))κ

KM(C)→ C.

but it is not necessarily an equivalence.


We shall say that C is KM non-degenerate if the functor (3.5) is an equivalence. We will denote by

(L(G)-modκ)KM -nondeg ⊂ L(G)-modκ

the full subcategory that consists of KM non-degenerate objects. Thus, the functor KM defines anequivalence

(3.6) (L(G)-modκ)KM -nondeg ' HCh(L(G))κ-mod.

Note that Conjecture 7 implies that the restriction of (3.4) to (L(G)-modκ)KM -nondeg is fully faithful.

3.3. GL-6: Spectral side in the classical case.

In this talk we will make a detour to discuss the classical degeneration of one of the sides of thequantum geometric Langlands story (one may think of it as letting κ tend to ∞ in a certain projectivespace–to be explained in the talk). Specifically, we want we want to see how the phenomenon ofKM-degeneracy plays out in this context.

We will first consider the global situation. In this case, the KM non-degenerate part ofD-mod(BunG)∞ is stipulated to be the category QCoh(LocSysG). The entire D-mod(BunG)∞ iscaptured as a certain enlargement of QCoh(LocSysG), denoted IndCohnilp(LocSysG) that consists ofind-coherent sheaves, with singular support belonging to the global nilpotent cone (the latter may beseen as playing the role of global Arthur parameters in global geometric Langlands).

The local situation is more mysterious. We again stipulate that the KM non-degenerate part ofL(G)-mod∞ is the 2-category DG categories over the ind-scheme ConnG(D) of connections on thetrivial G-bundle over the formal punctured disc D, equipped with a structure of (weak) equivariancewith respect to L(G) that acts by gauge transformations.

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At least conjecturally, the functor of taking (weak) L(G)-invariants defines an equivalence from(L(G)-mod∞)KM -nondeg, defined as above, to the 2-category of DG categories that are modules overQCoh(LocSysG(D)), where

LocSysG(D) := ConnG(D)/L(G).

However, it is a priori not clear how to enlarge (L(G)-mod∞)KM -nondeg defined in this way to anentire L(G)-mod∞.

In this talk we will explain an idea that leads to the desired enlargement. We will consider asimplified situation, where instead of LocSysG(D) we consider a finite-dimensional smooth algebraicstack Y, and we are interested in ways to enlarge the 2-category QCoh(Y)-mod. Namely, it will turn outthat to every sub-Lagrangian N ⊂ T ∗Y one can attach a certain 2-category, denote it QCoh(Y)-modN,so that for N being the 0-section we recover the usual QCoh(Y)-mod.

Going back to the loop group context, the idea is to apply this to Y replaced by LocSysG(D) with N

being the nilpotent cone. In a sense, this is a way to insert local Arthur parameters into the geometricLanglands story.

3.4. FLE-1: Statement of the equivalence.

As was already mentioned in Talk GL-3, the Fundamental Local Equivalence is an equivalence of(factorization) categories between Whit(GrG)κ and KL(G)κ.

However, things are a little more involved, in that the equivalence we are after depends on the signof the level. Namely, let us assume that κ is positive. In this case, we propose:

Conjecture 9. There exists a canonically defined equivalence of factorization categories

(3.7) L−κ : KL(G)−κ →Whit(GrG)−κ.

(We note that Whit(GrG)−κ that appears in the RHS is most naturally incarnated as invariants.)

This equivalence is supposed to respect the t-structures on both sides, and send the Weyl moduleVλ−κ ∈ KL(G)−κ to the standard object jλ,! ∈ Whit(GrG)−κ (here λ is a dominant weight of G and

hence a dominant coweight of G). In addition, this equivalence is compatible with the action of SphG,−κon KL(G)−κ and the action of SphG,κ on Whit(GrG)−κ via quantum geometric Satake (see Talk GL-4).

Now, we have canonical duality

(KL(G)−κ)∨ ' KL(G)κ

(to be explained in the talk).

Composing with

(Whit(GrG)−κ)∨ 'Whit(GrG)κ

(here the right-hand side is naturally incarnated as coinvariants), from (3.7) we obtain an equivalence

(3.8) Lκ : KL(G)κ →Whit(GrG)κ.

The functor Lκ behaves quite differently from L−κ. For example, it does not respect the t-structures,and it sends Vλκ ∈ KL(G)κ to the co-standard object j−w0(λ),∗ ∈Whit(GrG)−κ. In addition, the functors

L−κ and Lκ have a different-looking compatibility property with Jacquet functors (to be discussed inTalk FLE-2).

One can precompose Lκ with the Cartan involution on G and obtain a functor

Lτκ : KL(G)κ →Whit(GrG)κ.

The functors L−κ and Lτκ glue into a family as κ varies (in particular, they are canonically the samewhen κ is irrational).

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3.5. Wh-3: Statement of local quantum Langlands.

The functor

C 7→Whit(C)

is co-representable by an object

Whit(L(G))κ ∈ L(G)-modκ,

(we can think of Whit(L(G))κ as obtained by applying the functor Whit to the category D-mod(L(G))κof κ-twisted D-modules on the loop group, acting on itself on the right).

Denote

Hecke-Whit(L(G))κ := EndoFunctL(G)-modκ(Whit(L(G))κ)op.

The functor Whit can thus be seen as a functor

(3.9) L(G)-modκ → Hecke-Whit(L(G))κ-mod.

As in Talk Ch-1, one shows that an object of Hecke-Whit(L(G))κ-mod is naturally a factorizationmodule for the factorization category Whit(GrG)κ. Thus, we obtain a functor

(3.10) Hecke-Whit(L(G))κ-mod→Whit(GrG)κ-modFact.

As in the case of (2.1), we propose:


Thus, we obtain that Whit defines a functor

(3.11) L(G)-modκ →Whit(GrG)κ-modFact.

3.5.1. Precise statement of local quantum Langlands.

Let κ be positive, and let us assume the Fundamental Local Equivalence, i.e., Conjecture 9. Wepropose:

Conjecture 11. There exists an equivalence (1.1) that makes the following diagram commute

KL(G)−κ-modFact L−κ−−−−−→ Whit(GrG)−κ-modFact

KM

x xWhit

L(G)-modκLlocκ−−−−−→ L(G)-mod−κ

Whit

y yKM

Whit(GrG)κ-modFact (Lκ)−1

−−−−−→ KL(G)κ-modFact.

There is a further compatibility that one requires from Llocκ that has to do with parabolic induction,

to be discussed in talk Ja-2.

3.5.2. Several more concrete conjectures.

Let us assume Conjectures 7 and 10. Then Conjecture 11 implies the following one:

Conjecture 12.

(a) With respect to the equivalence

KL(G)−κ-modFact 'Whit(GrG)−κ-modFact,

induced by (3.7), the essential images of the functors

HCh(L(G))κ-mod→ KL(G)−κ-modFact and Hecke-Whit(L(G))−κ-mod→Whit(GrG)−κ-modFact

coincide.

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(b) With respect to the equivalence

KL(G)κ-modFact 'Whit(GrG)κ-modFact,

induced by (3.8), the essential images of the functors

HCh(L(G))−κ-mod→ KL(G)κ-modFact and Hecke-Whit(L(G))κ-mod→Whit(GrG)κ-modFact

coincide.

Conjecture 12 in turn implies:

Conjecture 13.

(a) There exist a canonical equivalence of monoidal categories

HCh(L(G))κ ' Hecke-Whit(L(G))−κ.

(b) There exists a canonical equivalence of monoidal categories

HCh(L(G))−κ ' Hecke-Whit(L(G))κ.

3.5.3. The local Fundamental Commutative Diagram.

Assuming Conjecture 13, by passing to left adjoints along the vertical arrows in the upper portionof the above diagram, we obtain a local version of the Fundamental Commutative Diagram:

(3.12)

HCh(L(G))κ-modL−κ−−−−−→ Hecke-Whit(L(G))−κ-mod

g-modκ ⊗HCh(L(G))κ

−y yD-mod(L(G))−κ ⊗

Hecke-Whit(L(G))−κ−

L(G)-modκLlocκ−−−−−→ L(G)-mod−κ

Whit

y yKM

Hecke-Whit(L(G))κ-mod(Lκ)−1

−−−−−→ HCh(L(G))−κ-mod.

As in the global case, one can formulate a conjecture that the outer diagram in (3.12) commutes:

Conjecture 14. The diagram

HCh(L(G))κ-modL−κ−−−−−→ Hecke-Whit(L(G))−κ-mod

Whit (g-modκ ⊗HCh(L(G))κ

−)y yKM (D-mod(L(G))−κ ⊗

Hecke-Whit(L(G))−κ−)

Hecke-Whit(L(G))κ-mod(Lκ)−1

−−−−−→ HCh(L(G))−κ-mod.

Note that the statement of Conjecture 14 depends on Conjecture 13, but not on the existence of thefunctor Lloc

κ ; in this sense it is more “unconditional”.

In Talk Wh-4 we will show that (assuming Conjecture 13), the statement Conjecture 14 follows fromanother plausible statement, Conjecture 22.

3.5.4. Whittaker (non)-degeneracy.

Much as KL, the functor (3.9) fails to be fully faithful. Namely, for C ∈ L(G)-modκ, we have acanonically defined functor

(3.13) Whit(L(G))κ ⊗Hecke-Whit(L(G))κ

Whit(C)→ C,

which is not, in general, an equivalence.


PROGRAM 13

We shall say that C ∈ L(G)-modκ is Whittaker non-degenerate if (3.13) is an equivalence. Let

(L(G)-modκ)Whit -nondeg ⊂ L(G)-modκ

denote the full subcategory consisting of Whittaker non-degenerate objects. Thus, the functor Whitdefines an equivalence

(3.14) (L(G)-modκ)Whit -nondeg ' Hecke-Whit(L(G))κ-mod.

We obtain that Conjecture 13 (combined with Conjectures 7 and 10) implies the following particularcase of Conjecture 11:

Conjecture 16.

(a) There exists an equivalence

(L(G)-modκ)KL -nondeg ' (L(G)-mod−κ)Whit -nondeg

that makes the diagram

KL(G)−κ-modFact L−κ−−−−−→ Whit(GrG)−κ-modFact

KM

x xWhit

(L(G)-modκ)KL -nondeg −−−−−→ (L(G)-mod−κ)Whit -nondeg

commute.

(b) There exists an equivalence

(L(G)-mod−κ)KL -nondeg ' (L(G)-modκ)Whit -nondeg

that makes the diagram

KL(G)κ-modFact Lκ−−−−−→ Whit(GrG)κ-modFact

KM

x xWhit

(L(G)-mod−κ)KL -nondeg −−−−−→ (L(G)-modκ)Whit -nondeg

commute.

One can formally deduce that the diagrams

(L(G)-modκ)KL -nondeg −−−−−→ (L(G)-mod−κ)Whit -nondeg

Whit

y yKM

Whit(GrG)κ-modFact (Lκ)−1

−−−−−→ KL(G)κ-modFact.

and

(L(G)-mod−κ)KL -nondeg −−−−−→ (L(G)-modκ)Whit -nondeg

Whit

y yKM

Whit(GrG)−κ-modFact (Lκ)−1

−−−−−→ KL(G)−κ-modFact.

commute as well.

Thus, we obtain that Conjecture 13 implies the appropriate non-degenerate cases of Conjecture 11.

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3.6. GL-7: Quantum Langlands for G being a torus.

Let G = T be a torus. In this case, Conjecture 11 is within easy reach (even though, this has notbeen done yet). Essentially, the proof sould follow from the fact that L(T ) identifies with the Cartierdual of L(T ) via the Contou-Carrere symbol.

This identification leads to the equivalences of monoidal categories in Conjecture 13. One deducesConjecture 11 using the fact that the functor Whit of (3.9) does not do anything (is the identityfunctor), while the functor KL of (3.2) is an equivalence.

The FLE in this case if also an easy (but important) observation. We will use the FLE for theCartan subgroup T as a tool to deduce the FLE for a reductive group G via Jacquet functors, see TalkFLE-2.

4. Day four

4.1. Ja-2: Jacquet functors and Langlands duality.

Jacquet functors are supposed to be compatible with the conjectural equivalence (1) by making thefollowing diagram commute:

(4.1)

L(G)-modκ −−−−−→ L(G)-mod−κ

J!

y yJ∗L(T )-modκ −−−−−→ L(T )-mod−κ.

Note that we use different Jacquet functors: in one case we use J! and in the other J∗.

Recall (see talk GL-4) that the equivalence (1) is supposed to send

D-mod(GrG)κ ∈ L(G)-modκ D-mod(GrG)−κ ∈ L(G)-mod−κ.

Combining with (4.1) we obtain:

Conjecture 17. We have an equivalence

(D-mod(GrG)κ)L(N)·L+(T ) ' (D-mod(GrG)−κ)L(N)·L+(T ),

compatible with factorization.

Under the equivalence of Conjecture 17, the standard object ∆0 in the LHS, i.e., the !-extension ofthe dualizing sheaf from the unit L(N)-orbit on GrG, is supposed to go over to the object ∆0,co in theRHS, i.e., the image of δ1,GrG

∈ D-mod(GrG)−κ under the projection

D-mod(GrG)−κ → (D-mod(GrG)−κ)L(N)·L+(T ).

4.1.1. Classical version.

In the talk, we will also explain the classical version of this statement (i.e., when κ is zero1), and κis ∞.

In this case, the right-hand side, i.e., (D-mod(GrG)∞)L(N)·L+(T ) degenerates to the (appropriately

defined) category of ind-coherent sheaves on

LocSysT (D) ×LocSysT (D)

LocSysB(D) ×LocSysG(D)

LocSysG(D).

Note that when working over a point, the above fiber product identifies with

( ˜N ×g

pt)/G.

1For us, zero=critical, but this does not affect D-mod(GrG)κ as a category, because the critical level is integral.

PROGRAM 15

The resulting equivalence

D-mod(GrG)L(N)·L+(T ) ' IndCoh( ˜N ×g

pt /G)

is a theorem, due to [ABG].

4.2. T-4: Tutorial on quantum groups.

First, we interpret the quantum parameter q as a quadratic form on the root lattice Λ of our torus Twith values in k∗. In fact, we need a little more: we need to refine q to a bilinear form (not necessarilysymmetric). To this datum one attaches a braided monoidal category (in fact, a ribbon category),denoted Repq(T ). This is the category of representations of the quantum torus. In what follows we willassume that the value of q on every simple root of G is non-trivial (this is a non-degeneracy condition).

4.2.1. The positive part.

Since Repq(T ) is a braided monoidal category, it makes sense to talk about Hopf algebras in it.We will consider several of them. One is the free associative algebra on the generators ei (here theindex i runs over the set of vertices of the Dynkin diagram of G); denote it Uq(n)free, equipped withthe tautological Hopf algebra structure. Another is the co-free co-associative co-algebra on the samegenerators, denote it Uq(n)co-free. We have a tautological map of Hopf algebras

(4.2) Uq(n)free → Uq(n)co-free,

which is neither injective nor surjective. We note that the (graded) duals of Uq(n)free and Uq(n)co-free

are the corresponding Hopf algebras Uq(n−)co-free and Uq(n

−)free, respectively. And the dual of (4.2)is the corresponding tautological map Uq(n

−)free → Uq(n−)co-free.

We introduce the Kac-DeConcini algebra Uq(n)KD as the quotient of Uq(n)free by the quantum Serrerelations. One shows that (4.2) factors via

Uq(n)free Uq(n)KD → Uq(n)co-free.

We introduce Lusztig’s algebra Uq(n)Lus as the graded dual of Uq(n−)KD (it is here that we use the

non-degeneracy assumption on q). Thus, we obtain that (4.2) factors as

Uq(n)free Uq(n)KD → Uq(n)Lus → Uq(n)co-free.

Now the resulting map Uq(n)KD → Uq(n)Lus is an isomorphism if and only if q is “not a root ofunity”, i.e., if q(αi) is non-torsion for every simple root αi. If this is not the case, we denote by uq(n)the image of the above map:

Uq(n)KD uq(n) → Uq(n)Lus,

this is the positive part of the small quantum group.

Let Repq(B+) be the full subcategory in Uq(n)Lus-mod consisting of objects on which the action of

Uq(n)Lus is locally nilpotent.

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4.2.2. Representations of the entire quantum group.

We now introduce the various versions of the category of representations of the entire quantumgroup. In every case, this will be a braided monoidal category. The construction will always proceedby first defining the corresponding abelian category, and then taking its derived category.

The easiest category to introduce is Repq(G)mixed. This is the (relative to Repq(T )) Drinfeld’s center

of the category Repq(B+), i.e.,

Repq(G)mixed = ZDr,Repq(T )(Repq(B+)).

This is the category of representations of the “mixed” quantum group: one whose positive part isUq(n)Lus (and which is required to act locally nilpotently), and who negative part is Uq(n

−)KD.

Now the theory bifurcates depending on whether or not q is a root of unity. Suppose first thatit is not. Then we can think of Repq(G)mixed as the quantum version of category O, and we define

Repq(G) (at the abelian level) as the full subcategory of Repq(G)mixed consisting of objects on which

the action of Uq(n−) := Uq(n

−)Lus ' Uq(n−)KD is also locally nilpotent. The restriction functor

Repq(G) → Repq(B+) is fully faithful (even at the derived level), and we can therefore alternatively

define Repq(G) as the full subcategory of Repq(B+), consisting of objects on which the action of Uq(n)

can be extended to a locally nilpotent action of Uq(n−).

Suppose now that q is a root of unity, i.e., all q(αi) are torsion. In this case, we introduce a whole

array of categories. We let•uq(g)-mod be the (relative to Repq(T )) Drinfeld’s center of the category

uq(n)-mod, i.e.,•uq(g)-mod := ZDr,Repq(T )(uq(n)-mod).

This is the category of modules over the (graded) small quantum group.

We introduce Repq(G)12 (at the abelian level) as the full subcategory in Repq(G)mixed, consisting of

objects on which the action of Uq(n−)KD factors through Uq(n

−)KD uq(n−).

Finally, we introduce Repq(G) (at the abelian level) to consist of objects of Repq(G)12 , equipped

with an extension of the uq(n−)-action to a Uq(n

−)Lus-action. Thus, we have a sequence of restrictionfunctors

Repq(G)mixed → Repq(G)12 → Repq(G)→ •

uq(g)-mod.

One shows that the restriction functor Repq(G)12 → Repq(G) is fully faithful (even at the derived

level).

4.2.3. The quantum Frobenius.

Lusztig’s quantum Frobenius defines a short exact sequence of Hopf algebras

1→ uq(n)→ Uq(n)Lus → U(nH)→ 1,

where nH is the maximal unipotent in the reductive group H of Talk GL-4, and U(nH) is its usualuniversal enveloping algebra, which can be viewed as an object of Repq(T ) via Rep(TH)→ Repq(T ).

Dually, we have a short exact sequence

1→ ON+H→ Uq(n

−)KD → uq(n−)→ 1.

This gives rise to the following pieces of structure:

(i) An action of the monoidal category QCoh(ON+H/Ad(B+)) (with respect to the point-wise tensor

product) on Repq(G)mixed and an identification (at the derived level):

(4.3) Repq(G)mixed ⊗QCoh(O

N+H

/Ad(BH ))QCoh(pt /BH) ' Repq(G)

12 ;

PROGRAM 17

(ii) An action of the monoidal category QCoh(pt /BH) ' Rep(BH) on Repq(G)12 and an identification

(at the derived level):

(4.4) Repq(G)12 ⊗

Rep(BH )Rep(TH) ' •uq(g)-mod;

(iii) An action of the monoidal category Rep(H) on Repq(G) and an identification (at the derived level):

Repq(G) ⊗Rep(H)

Rep(BH) ' Repq(G)12 .

Combining (ii) and (iii), we obtain an identification

(4.5) Repq(G) ⊗Rep(H)

Rep(TH) ' •uq(g)-mod.

4.3. T-5: Tutorial on factorization vs braided monoidal categories.

Factorization categories exist also in the context of topology. When our curve X is A1, they corre-spond to braided monoidal categories. An additional structure on a braided monoidal category (knownis a ribbon structure) allows to associate to it a factorization category on any oriented 2-manifold.

A 7→ Facttop(A).

Now, we have a Riemann-Hilbert functor that associates to any topological factorization category(satisfying some finiteness conditions) an algebro-geometric one. We will denote this assignment by thesymbol RH. Thus, one can attach to a ribbon category A an algebro-geometric factorization category

Factalg-geom(A) := RH(Facttop(A)

).

For A symmetric monoidal, we recover the construction A Fact(A) from Talk T-2.

We will consider in detail a particular example of this situation, where the ribbon category inquestion is the category Repq(T )–the category of representations of the quantum torus. We will see

that the resulting factorization category Factalg-geom(Repq(T )) identifies with D-mod(GrT )κ (or, whichis equivalent by the FLE for tori, with KL(T )κ), where q = exp(2πi · κ)).

In the process of doing so we will see that the “q” that appears in the quantum group is bestinterpreted as a factorization gerbe on GrT .

4.4. Ch-2: The factorization algebra Ωq (or now Nature encodes root data).

Consider the (ribbon) braided monoidal category Repq(T ) and the Hopf algebras

Uq(n)free Uq(n)KD uq(n) → Uq(n)Lus → Uq(n)co-free

in it.

Let A be one of these Hopf algebras. In the talk we will explain that the functor invA of derivedA-invariants applied to the augmentation module defines an E2-algebra in Repq(T ), to be denotedInvA. Moreover, the Koszul duality functor

invA : A-modnilp → Inv(A)-mod

(here A-modnilp is the category of locally nilpotent A-modules) naturally lifts to a functor

(4.6) ZDr,Repq(T )(A-modnilp)→ InvA -modE2 .

Applying the procedure from Talk T-5, we obtain that InvA gives rise to a factorization algebra inFacttop(Repq(T )), to be denoted Ω(A), so that

InvA -modE2 ' Ω(A)-modFact,

18 PROGRAM

and thus (4.6) can be interpreted as

(4.7) ZDr,Repq(T )(A-modnilp)→ Ω(A)-modFact.

We will see that since A is graded by the monoid Λpos, the object Ω(A) lives inside a full subcategoryof Facttop(Repq(T )) that can be described very explicitly. Namely, we will introduce a configuration

space Conf(X, Λneg) consisting of Λneg-colored divisors. We will see that the datum of q gives riseto a factorization gerbe Gq on Conf(X, Λneg), and the corresponding subcategory of Facttop(Repq(T ))identifies with

ShvGq (Conf(X, Λneg)).

Thus, we obtain a sequence of factorization algebras in ShvGq (Conf(X, Λneg)):

(4.8) Ωco-freeq → ΩLus

q → Ωsmallq → ΩKD

q → Ωfreeq .

4.4.1. An explicit description of the Ωq algebras.

A crucial feature of Gq is that it is canonically trivial over the open subscheme

Conf(X, Λneg)

j−→ Conf(X, Λneg).

So that ShvGq (

Conf(X, Λneg)) is the usual (non-twisted) category of sheaves.

We will see that the factorization algebra Ωco-freeq (resp., Ω(Uq(n)free)) is the !- (resp., *-) extension

of the sign local system on

Conf(X, Λneg).

Another important observation is that the assignment

A Ω(A) ∈ ShvGq (Conf(X, Λneg))

sends Hopf algebras that are concentrated in cohomological degree 0 to perverse sheaves, and moreoverit sends injections/surjections of Hopf algebras to injections/surjections of perverse sheaves. Thisimmediately implies that the factorization algebra Ωsmall

q is the !*-extension of the sign local system on

Conf(X, Λneg).

The most important of these algebras, namely, ΩLusq is a certain perverse sheaf that is squeezed

between the !-extension and the !*-extension. It is a fact of crucial importance that ΩLusq can be

described explicitly by induction, and this will be the central idea of this talk.

4.4.2. How do these Ωq-algebras control representation theory?

Our overall task (one on which we stake our hopes to prove the FLE) is to express the categoryRepq(G) in “factorization terms”, ideally, as a the category of factorization modules over a factorizationalgebra. We will not be able to do quite that, and in the talk we will explain what it is that we can do.

Using (4.7), we will consider the following several functors:

(4.9) invUq(n)Lus : Repq(G)mixed → ΩLusq -modFact;

(4.10) invUq(n)Lus : Repq(G)12 → ΩLus

q -modFact;

(4.11) invUq(n)Lus : Repq(G)→ ΩLusq -modFact;

(4.12) invuq(n) :•uq(g)-mod→ Ωsmall

q -modFact;

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(4.13) invuq(n) : Repq(G)→ Ωsmallq -modFact.

The functors (4.9) and (4.12) are equivalences (up to renormalization, i.e., up to tweaking homo-logical algebra a bit–to be explained in the talk). However, the other functors, the most important ofwhich is (4.11), are not.

Nonetheless, we will explain that the failure of the functor (4.11) to be an equivalence is controllableusing a factorization algebra denoted Ωcl constructed from the commutative DG algebra C·(nH), whichis the cohomological Chevalley algebra of the Lie algebra nH (see Talk T-4). This will allow us to givea description of Repq(G) in “factorization terms”.

Essentially, what we will say will amount to saying that the forgetful functor

Repq(G)→ Repq(G) ⊗Rep(H)

Rep(BH) ' Repq(G)12

is fully faithful, while Repq(G)12 can be algorithmically expressed via Repq(G)mixed by means of (4.3).

But we want to express all of this in factorization terms, in order to later perform analogous construc-tions on the de Rham side of Riemann-Hilbert.

Note that the symmetric monoidal category Rep(TH) maps to the E3-center of Repq(T ). We consider

Ωcl as a factorization algebra in Facttop(Rep(TH)), and the quantum Frobenius defines a map

Ωcl → ΩLusq .

We can consider ΩLusq as a factorization algebra in the factorization category of commutative Ωcl-modules

in Facttop(Rep(TH)), the later being equivalent to Facttop(Rep(BH)).

Let

(ΩLusq -modFact)Ωcl -com

denote the resulting category of modules, i.e., the category of factorization ΩLusq -modules in

Facttop(Rep(BH)).

We have an equivalence

(ΩLusq -modFact)Ωcl -com ' Repq(G)

12 .

The functor invUq(n)Lus naturally upgrades to a functor

(4.14) Repq(G)→ (ΩLusq -modFact)Ωcl -com,

and further to a functor

Repq(G) ⊗Rep(H)

Rep(BH)→ (ΩLusq -modFact)Ωcl -com,

and the latter is an equivalence. Thus, we the functor (4.14) is fully faithful as desired.

Note also that we have a commutative diagram(4.15)

Repq(G) ⊗Rep(H)

Rep(BH) ⊗Rep(BH )

Rep(TH)inv

Uq(n)Lus

−−−−−−−−→ (ΩLusq -modFact)Ωcl -com ⊗

Rep(BH )Rep(TH)

∼y y∼

•uq(g)-mod

invuq(n)

−−−−−−→ Ωsmallq -modFact.

20 PROGRAM

4.5. Ja-3: Jacquet functor on Kac-Moody modules and the Kazhdan-Lusztig equivalence.

Our approach to the proof of the FLE is to construct functors from each side to the correspondingcategory for T , and (try to) express the categories for G via these functors. In this talk, we will performthis procedure for the KL side.

Predictably, it is here that the sign of the level κ will be play the most significant role, as thebehavior of the category KL(G)κ depends on the level.

Notational convention: From now on we will assume that κ is positive. Negative levels will bedenoted by −κ.

4.5.1. Positive level case.

We construct the functor

(4.16) jKLκ,∗ : KL(G)κ → KL(T )κ

to be the composition

(4.17) KL(G)κ → (D-mod(GrG)κ)L(N)·L+(T ) ⊗KL(G)κconvolution−→

→ (g-modκ)L(N)·L+(T ) → (t-modκ)L+(T ) = KL(T )κ.

In the above formula, the first arrow, i.e.,

KL(G)κ → (D-mod(GrG)κ)L(N)·L+(T ) ⊗KL(G)κ,

is given by tensoring with the object ∆0,co ∈ (D-mod(GrG)κ)L(N)·L+(T ), see Talk J-2.

The third arrow, i.e.,

(4.18) J∗(g-modκ)L+(T ) := (g-modκ)L(N)·L+(T ) → (t-modκ)L+(T )

is the functor C∞2 (L(n),−) =: C

∞2∗ (L(n),−) of semi-infinite cohomology with respect to L(n) := n((t)).

4.5.2. Negative level case.

The situation at the negative level is trickier. We construct the functor

(4.19) jKL−κ,! : KL(G)−κ → KL(T )−κ

to be the composition

(4.20) KL(G)−κ → (D-mod(GrG)−κ)L(N)·L+(T ) ⊗KL(G)−κconvolution−→

→ (g-mod−κ)L(N)·L+(T ) → (t-mod−κ)L+(T ) ' KL(T )−κ.

Here the first arrow, i.e.,

KL(G)−κ → (D-mod(GrG)−κ)L(N)·L+(T ) ⊗KL(G)−κ

is given by tensoring with ∆0 ∈ (D-mod(GrG)−κ)L(N)·L+(T ), see Talk J-2.

The third arrow is a non-standard functor

C∞2

! (L(n),−) : J!(g-mod−κ)L+(T ) := (g-mod−κ)L(N)·L+(T ) → (t-mod−κ)L

+(T ).

In the talk we will describe the functor C∞2

! (L(n),−) via the equivalence

(g-mod−κ)L(N)·L+(T ) ' (g-mod−κ)I ,

and also its incarnation on the other side of Riemann-Hilbert.

PROGRAM 21

4.5.3. Factorization description.

One can regard the following conjecture as a way to describe the category (g-modκ)L(N)·L+(T ) (resp.,

(g-mod−κ)L(N)·L+(T )) essentially combinatorially.

Denote by ΩKM,Lusκ (resp., ΩKM,Lus

−κ ) the factorization algebra in KL(T )κ (resp., KL(T )−κ) equal to

jKLκ,∗(V0

κ) (resp., jKL−κ,!(V0

−κ)), where V0 denotes the vacuum module at the given level. We propose:

Conjecture 18.

(a) The functor C∞2∗ (L(n),−) defines an equivalence

(g-modκ)L(N)·L+(T ) → ΩKM,Lusκ -modFact.

(b) The functor C∞2

! (L(n),−) defines an equivalence

(g-mod−κ)L(N)·L+(T ) → ΩKM,Lus−κ -modFact.

Parallel to Talk Ch-2, the composite functors

KL(G)κ := g-modL+(G)κ → (g-modκ)L(N)·L+(T ) → ΩKM,Lus

κ -modFact

and

KL(G)−κ := g-modL+(G)−κ → (g-mod−κ)L(N)·L+(T ) → ΩKM,Lus

−κ -modFact

are not equivalences. But the failure to be such is controllable, using the same factorization algebraΩcl, by a procedure outlined at the end of Talk Ch-2. We will return to this in Talks FLE-2 and FLE-3.

4.5.4. Relation to the Kazhdan-Lusztig equivalence.

We propose:

Conjecture 19. The factorization algebra ΩLusq (resp., ΩLus

q−1) corresponds under Riemann-Hilbert (see

talk T-5) to ΩKM,Lusκ (resp., ΩKM,Lus

−κ ).

Assuming this conjecture, we obtain that Conjecture 18 can be reformulated as follows:

Conjecture 20.

(a) The category Repq(G)mixed corresponds under Riemann-Hilbert to

J∗(g-modκ)L+(T ) := (g-modκ)L(N)·L+(T ).

(b) The category Repq−1(G)mixed corresponds under Riemann-Hilbert to

J!(g-mod−κ)L+(T ) := (g-mod−κ)L(N)·L+(T ).

In the talk we will explain that the usual Kazhdan-Lusztig equivalence may be interpreted as a state-ment that the category Repq−1(G) corresponds under Riemann-Hilbert to KL(G)κ (this equivalence iscompatible with the t-structures and send Weyl modules to Weyl modules).

Applying duality, we obtain that Repq(G) corresponds under Riemann-Hilbert to KL(G)κ. However,

the latter equivalence is not t-exact and sends the dual Weyl module with highest weight λ to the Weylmodule with highest weight −w0(λ).

The equivalences of Conjecture 20 are supposed to make the following diagrams commute

Repq(G) −−−−−→ Repq(G)mixedy yKL(G)κ −−−−−→ (g-modκ)L(N)·L+(T )

22 PROGRAM

andRepq−1(G) −−−−−→ Repq−1(G)mixedy yKL(G)−κ −−−−−→ (g-mod−κ)L(N)·L+(T ),

where the lower horizontal arrows are the functors appearing in formulas (4.17) and (4.20), respectively.

4.6. Wh-4: Duality for W-algebras.

The upper portion of the diagram in Conjecture 11 implies that that the equivalence Llocκ of (1) is

supposed to send

g-modκ ∈ L(G)-modκ (D-mod(L(G))−κ)L(N),χ ∈ L(G)-mod−κ,

where (L(N), χ)-invariants are taken with respect the action of L(N) on l(G) by right multliplication.

Applying to this the lower portion of the diagram in Conjecture 11 we obtain an equivalence

Whit(g-modκ) ' KM(

(D-mod(L(G))−κ)L(N),χ).

However, we have (tautologically, to be explained in the talk):

KM(

(D-mod(L(G))−κ)L(N),χ)'Whit(g-modκ).

Hence, we obtain an equivalence of factorization categories

(4.21) Whit(g-modκ) 'Whit(g-modκ).

The equivalence (4.21) has been deduced from the conjectural equivalence (1). However, fortunately,(4.21) is actually a theorem. Namely, one introduces the W-algebra Wg,κ as the BRST reduction ofthe Kac-Moody chiral algebra at level κ.

There is a (tautologically) defined functor

(4.22) Whit(g-modκ)→Wg,κ-modFact,

and it was recently proved by S. Raskin that this functor is an equivalence.

Given this, the sought-for equivalence, (4.21) is a consequence of the isomorphism of chiral algebras

(4.23) Wg,κ 'Wg,κ,

due to Feigin and Frenkel.

One actually has to be a little more careful here: the category Wg,κ-modFact that appears in (4.22)is a renormalized version of the naive category of factorization Wg,κ-modules. So one needs also to seethat there renormalizations match up on the two sides of (4.23).

4.6.1. Compatibility with FLE.

Note now that the category Whit(g-modκ) comes equipped with a tautological functor

Whit(D-mod(L(G))κ)⊗ g-modκ →Whit(g-modκ).

Pre-composing with

Whit(D-mod(GrG)κ)⊗KL(G)κ →Whit(D-mod(L(G))κ)⊗ g-modκ,

we obtain a functor

(4.24) Whit(D-mod(GrG)κ)⊗KL(G)κ →Whit(g-modκ).

We propose:

PROGRAM 23

Conjecture 21.

(a) The diagram

Whit(D-mod(GrG)κ)⊗KL(G)κ(Lκ)−1⊗Lκ−−−−−−−→ KL(G)κ ⊗Whit(D-mod(GrG)κ)y y

Whit(g-modκ) Whit(g-modκ)

∼y y∼

Wg,κ-modFact −−−−−→ Wg,κ-modFact

commutes.

(b) The diagram

Whit(D-mod(GrG)−κ)⊗KL(G)−κ(L−κ)−1⊗L−κ−−−−−−−−−→ KL(G)−κ ⊗Whit(D-mod(GrG)−κ)y y

Whit(g-mod−κ) Whit(g-mod−κ)

∼y y∼

Wg,−κ-modFact ∼−−−−−→ Wg,−κ-modFact

commutes.

Conjecture 21 does not appear very far-fetched. Namely, one observes that the images of

Whit(D-mod(GrG)κ)→Whit(D-mod(L(G))κ)→Whit(D-mod(L(G))κ)⊗ g-modκ →Whit(g-modκ)

and

KL(G)κ → g-modκ →Whit(D-mod(L(G))κ)⊗ g-modκ →Whit(g-modκ)

“commutatively fuse with each other”. This is supposed to reduce the statement of Conjecture 21 tothe commutativity of the diagram

KL(G)κLκ−−−−−→ Whit(D-mod(GrG)κ)y y

Whit(g-modκ)∼−−−−−→ Whit(g-modκ),

whereas the latter should be amenable to explicit analysis.

4.6.2. The pairing.

Denote

Wg,κ =: W := Wg,κ.

Following B. Feigin, let us note that Conjecture 21 can also be expressed as the existence of a pairing

P : KL(G)κ ⊗KL(G)κ →W-modFact

that can be defined either as

KL(G)κ ⊗KL(G)κId⊗Lκ' KL(G)κ ⊗Whit(D-mod(L(G))κ)→Wg,κ-modFact 'W-modFact

or as

KL(G)κ ⊗KL(G)κLκ⊗Id' Whit(D-mod(L(G))κ)→ KL(G)κ →Wg,κ-modFact 'W-modFact.

In particular, the category W-modFact can be viewed as a factorization module category overKL(G)κ ⊗ KL(G)κ. It is clear from the construction that the factorization structures with respect

24 PROGRAM

to KL(G)κ and KL(G)κ separately come from actions of HCh(L(G))−κ and HCh(L(G))−κ, respec-tively. However, it is not immediately clear that these actions commute with one another (unless weassume Conjecture 3.3 or Conjecture 13). So we propose:

Conjecture 22. The structure on W-modFact of factorization module category over KL(G)κ⊗KL(G)κcomes from an action of the monoidal category

HCh(L(G))−κ ⊗HCh(L(G))−κ.

A similar discussion applies to κ replaced by −κ (and L replaced by L).

4.6.3. Relation to the global Fundamental Commutative Diagram.

We now return to Conjecture 3. We can interpret it as an isomorphism between two functors

KL(G)κ ⊗KL(G)κ → Vect .

We claim that this isomorphism follows from Conjecture 21(a). Namely, one can show that each ofthese functors identifies with

KL(G)κ ⊗KL(G)κP→W-modFact → Vect,

where the second arrow is the functor of chiral homology (a.k.a. conformal blocks) with respect to W.

4.6.4. Relation to the local Fundamental Commutative Diagram.

Let us now return to Conjecture 14 (so we are assuming Conjecture 13). We claim that Conjecture 22implies Conjecture 14.

Namely, by unwinding the definitions, one shows that the two functors

HCh(L(G))κ ⇒ HCh(L(G))−κ

are given by

− ⊗HCh(L(G))−κ

W-modFact.

5. Day five

5.1. Wh-5: Jacquet functor on the Whittaker category.

In this talk we will study “the most infinite-dimensional” category that arises from geometry, namely

D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)κ ,

where L(N−) ·L+(T )-invariants are taken with respect to left multiplication, and (L(N), χ)-invariantsare taken with respect to right multiplication.

From Conjecture 11 (combined with Conjecture 6) we obtain:

Conjecture 23.

(a) There exists a canonical equivalence

(5.1) (g-modκ)L(N−)·L+(T ) ' D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)−κ .

(b) There exists a canonical equivalence

(5.2) (g-mod−κ)L(N)·L+(T ) ' D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)κ .

PROGRAM 25

5.1.1. The restriction functor.

Recall that the categories

(g-modκ)L(N−)·L+(T ) and (g-mod−κ)L(N)·L+(T )

came equipped with functors to KL(T )κ and KL(T )−κ denoted, C∞2∗ (L(n−),−) and C

∞2

! (L(n),−),respectively.

We will now describe a functor

Res! : D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)κ → D-mod(L(T ))L

+(T )κ ' D-mod(GrT )−κ,

that corresponds to the above functors under the equivalences of Conjecture 23. We remark here that,

just as the category D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)κ itself, the functor Res! is geometric in nature

(is not specific to D-modules) and so “does not feel the sign of the level”; i.e., the same constructionworks for κ replaced by −κ.

Namely, let us describe the λ component of Res for λ ∈ Λ, thought of as a connected component ofGrT . This functor is obtained by applying the forgetful functor

D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)κ → D-mod(L(G))L

+(T )κ ,

followed by a functor D-mod(L(G))L+(T )κ → Vect, obtained by pairing with a particular object Sλ ∈

D-mod(L(G))L+(T )−κ .

The object Sλ, in its turn is the image under the forgetful functor

D-mod(L(G))(L(N)·L+(T ))×L(N−)−κ → D-mod(L(G))

L+(T )−κ

to the object equal to !-averaging with respect to L(N) × L(N−) of the D-module of distributions ontλ · L+(T ) ⊂ L(T ).

5.1.2. Factorization description.

We have a naturally defined functor

(5.3) Whit(GrG)κ := D-mod(L(G))(L(N),χ)×L+(G)κ ' D-mod(L(G))

L+(G)×(L(N),χ)−κ →

→ D-mod(GrG)L(N−)·L+(T )−κ ⊗D-mod(L(G))

L+(G)×(L(N),χ)−κ → D-mod(L(G))

(L(N−)·L+(T ))×(L(N),χ)−κ ,

where the last arrow is given by convolution and the preceding arrow by tensoring with the object

∆0 ∈ D-mod(GrG)L(N−)·L+(T )−κ .

Denote the composite functor

Whit(GrG)κ → D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)−κ

Res!−→ D-mod(GrT )κ

by jWhitκ,! .

Denote by ΩWhit,Lusκ the factorization algebra in D-mod(GrT )κ, equal to the image under jWhit

κ,! ofthe vacuum object of D-mod(GrG)κ.

Parallel to Conjecture 18, we have:

Conjecture 24. The functor Res! defines an equivalence

D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)−κ → ΩWhit,Lus

κ -modFact.

26 PROGRAM

5.1.3. The classical case.

Conjecture 24 is meaningful when κ = 0. In this case the factorization algebra ΩWhit,Lusκ is what

we previously denoted by Ωcl (in the classical limit, the group H is G); it is constructed out of thecommutative DG algebra C·(n).

We obtain:

Conjecture 25. The functor Res! defines an equivalence

D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ) → Ωcl-modFact.

An approximation to this conjecture was established in S. Raskin’s thesis. At a point (i.e., thenon-factorizable version), Conjecture 25 is equivalent to

Whit(D-mod(FlG)) ' QCoh(n/AdB),

which is a theorem of Arkhipov-Bezrukavnikov.

5.2. FLE-2: Strategy of the proof of FLE via Jacquet functors.

Recall that in Talk Wh-5 we defined the functor

jWhitκ,! : Whit(GrG)κ → D-mod(GrT )κ.

In this talk we will give a more direct geometric description of this functor.

It is not true that the corresponding functor

jWhitκ,! : Whit(GrG)κ → ΩWhit,Lus

κ -modFact

is an equivalence. However, we expect that its failure to be an equivalence is controllable in terms ofthe factorization algebra Ωcl that has already appeared in Talk Ch-2.

Namely, we introduce the category (ΩWhit,Lusκ -modFact)Ωcl -com, and we upgrade jWhit

κ,! to a functor

(5.4) Whit(GrG)κ ⊗Rep(H)

Rep(BH)→ (ΩWhit,Lusκ -modFact)Ωcl -com.

We propose:

Conjecture 26. The functor (5.4) is an equivalence.

In fact, Conjecture 26 is very closed to being a theorem; we will return to this in Talk FLE-3.

Now, we have a parallel story on the KL side: we introduce the categories

(ΩKM,Lusκ -modFact)Ωcl -com and (ΩKM,Lus

−κ -modFact)Ωcl -com

and we refine the functors jWhitκ,∗ and jWhit

−κ,! to functors

(5.5) KL(G)κ ⊗Rep(H)

Rep(BH)→ (ΩKM,Lusκ -modFact)Ωcl -com

and

(5.6) KL(G)−κ ⊗Rep(H)

Rep(BH)→ (ΩKM,Lus−κ -modFact)Ωcl -com,

respectively.

We propose:

Conjecture 27.

(a) The functor (5.5) is an equivalence.

(b) The functor (5.6) is an equivalence.

Now, the strategy of the proof of FLE is the following. Note that a prerequisite to Conjecture 23 is:

PROGRAM 27

Conjecture 28. Let κ be positive.

(a) With respect to the equivalence Lκ : KL(T )κ ' D-mod(GrT )κ, the factorization algebra2 Ω−,KM,Lusκ

corresponds to ΩWhit,Lusκ .

(b) With respect to the equivalence L−κ : KL(T )−κ ' D-mod(GrT )−κ, the factorization algebra ΩKM,Lus−κ

corresponds to ΩWhit,Lus−κ .

Assuming this conjecture, we obtain an equivalence

(5.7) Ω−,KM,Lusκ -modFact ' ΩWhit,Lus

κ -modFact.

Thus, in order to prove the FLE it remains to show that with respect to the above equivalence, theessential images of the (fully faithful) functors

KL(G)κ → KL(G)κ ⊗Rep(H)

Rep(B−H)→ (Ω−,KM,Lusκ -modFact)Ω−,cl -com

and

Whit(GrG)κ →Whit(GrG)κ ⊗Rep(H)

Rep(BH)→ (ΩWhit,Lusκ -modFact)Ωcl -com

generate the same subcategories.

And similarly, for

KL(G)−κ → KL(G)−κ ⊗Rep(H)

Rep(BH)→ (ΩKM,Lusκ -modFact)Ωcl -com

and

Whit(GrG)−κ →Whit(GrG)−κ ⊗Rep(H)

Rep(BH)→ (ΩWhit,Lus−κ -modFact)Ωcl -com.

5.3. Ja-4: Jacquet/Eisenstein functors in the global situation.

For a parabolic P with Levi quotient M , the operation of *-pull and !-push along

BunG ← BunP → BunM

defines a functor

Eis! : D-mod(BunM )κ,! → D-mod(BunG)κ,!.

Dually, the operation of of *-pull and !-push along the same diagram defines a functor

Eis∗ : D-mod(BunM )−κ,∗ → D-mod(BunG)−κ,∗.

The global quantum Langlands equivalence (1.2) is supposed to be compatible with these functorsby making the following diagram commute:

D-mod(BunG)κ,! −−−−−→ D-mod(BunG)−κ,∗

Eis!

x xEis∗

D-mod(BunM )κ,! −−−−−→ D-mod(BunM )−κ,∗.

We also have the diagram of dual functors

(5.8)


CT∗

y yCT?

D-mod(BunM )κ,! −−−−−→ D-mod(BunM )−κ,∗,

where CT? is the non-standard Constant Term functor, defined as the dual of Eis!.

2The superscript “-” in the next formula indicates that we perform the construction with respect to B− rather thanB.

28 PROGRAM

In addition to the functor CT∗, there is also the functor

CT! : D-mod(BunG)κ,! → D-mod(BunM )κ,!.

given by *-pull and !-push. An important observation is that we have a canonical isomorphism

(5.9) CT∗ ' CT−! ,

where the superscript “-” means that we are considering this functor for the opposite parabolic.

5.3.1. Local-to-global compatibility.

For simplicity, we take P = B and so M = T . We will explain that we have the following commu-tative diagrams:

KL(G)κj−,KLκ,∗−−−−−→ KL(T )κ

Loc

y yLoc

D-mod(BunG)κ,!CT−∗−−−−−→ D-mod(BunT )κ,!

and

KL(G)−κjKL−κ,!−−−−−→ KL(T )−κ

Loc

y yLoc

D-mod(BunG)−κ,∗CT?−−−−−→ D-mod(BunT )−κ,∗

and also

Whit(GrG)−κjWhit−κ,!−−−−−→ Whit(GrT )−κ

Poinc!

y yPoinc!

D-mod(BunG)κ,!CT−

!−−−−−→ D-mod(BunT )κ,!and

Whit(GrG)κjWhitκ,!−−−−−→ Whit(GrT )κ

Poinc∗

y yPoinc∗

D-mod(BunG)−κ,∗CT?−−−−−→ D-mod(BunT )−κ,∗.

We will study how these diagrams, when juxtaposed with (5.8) interact with the diagrams (1.5) and(1.3) (i.e., the Fundamental Commutative diagram (1.6)).

5.3.2. More general Eisenstein and Constant Term functors.

We will show that there exist functors

Eis!,ext : D-mod(BunT )κ,! ⊗ (D-mod(GrG)−κ)L(N)·L+(T ) → D-mod(BunG)κ,!

and

Eis∗,ext : D-mod(BunT )−κ,∗ ⊗ (D-mod(GrG)κ)L+(T )

L(N) → D-mod(BunG)−κ,∗.

They are supposed to make the following diagram commute


Eis!,ext

x xEis∗,ext

D-mod(BunT )κ,! ⊗ (D-mod(GrG)−κ)L(N)·L+(T ) −−−−−→ D-mod(BunT )−κ,∗ ⊗ (D-mod(GrG)κ)L+(T )

L(N),

where

(D-mod(GrG)−κ)L(N)·L+(T ) ' (D-mod(GrG)κ)L+(T )

L(N)

PROGRAM 29

is the equivalence of Conjecture 17.

Dually, we have the corresponding constant term functors:

D-mod(BunG)κ,! ⊗ (D-mod(GrG)κ)L+(T )

L(N) −−−−−→ D-mod(BunG)−κ,∗ ⊗ (D-mod(GrG)−κ)L(N)·L+(T )

CT∗,ext

y CT?,ext

yD-mod(BunT )κ,! −−−−−→ D-mod(BunT )−κ,∗.

For example, the non-standard functor CT? appearing in diagram (5.8) is CT?,ext(−,∆0).

Parallel to (5.9) we also have the functor

CT!,ext : D-mod(BunG)κ,! ⊗ (D-mod(GrG)κ)L(N)·L+(T ) → D-mod(BunT )κ,!

and a commutative diagram

D-mod(BunG)κ,! ⊗ (D-mod(GrG)κ)L(N−)·L+(T ) −−−−−→ D-mod(BunG)κ,! ⊗ (D-mod(GrG)κ)L+(T )

L(N)

CT−!,ext

y CT∗,ext

yD-mod(BunT )κ,!

Id−−−−−→ D-mod(BunT )κ,!,

where the top horizontal arrow is induced by the functor (2.2).

We will study the interaction of this diagram and the Fundamental Commutative Diagram (1.6) viathe local-to-global compatibilities that arise from the conjectural local commutative diagrams

(D-mod(GrG)κ)L(N−)·L+(T ) ⊗KL(G)κ∼−−−−−→ (D-mod(GrG)−κ)L(N−)·L+(T ) ⊗Whit(GrG)κ

convolution

y yconvolution

(g-modκ)L(N−)·L+(T ) −−−−−→ D-mod(L(G))(L(N−)×L+(T ))×(L(N),χ)−κ

C∞2∗ (L(n−),−)

y yRes!

KL(T )κ∼−−−−−→ D-mod(GrT )κ

and

(D-mod(GrG)−κ)L(N)·L+(T ) ⊗KL(G)−κ∼−−−−−→ (D-mod(GrG)κ)L(N−)·L+(T ) ⊗Whit(GrG)−κ

convolution

y yconvolution

(g-mod−κ)L(N)·L+(T ) −−−−−→ D-mod(L(G))(L(N−)×L+(T ))×(L(N),χ)κ

C∞2

!(L(n),−)

y yRes

KL(T )−κ∼−−−−−→ D-mod(GrT )−κ.

5.4. Ja-5: The semi-infinite IC sheaf.

In this talk we will introduce a particular object (in fact, a factorization algebra)

IC∞2 ∈ (D-mod(GrG)κ)L(N)·L+(T ).

We will denote by IC∞2,− its counterpart in (D-mod(GrG)κ)L(N−)·L+(T ), and let us denote by IC

∞2

co

(resp., IC∞2,−

co ) the projection of IC∞2,− (resp., IC

∞2 ) under

(D-mod(GrG)κ)L(N−)·L+(T ) → (D-mod(GrG)κ)L(N)·L+(T )

and

(D-mod(GrG)κ)L(N)·L+(T ) → (D-mod(GrG)κ)L(N−)·L+(T ),

30 PROGRAM

respectively.

5.4.1. The role of IC∞2 in the global setting.

Consider the functor

D-mod(BunT )κ,! → D-mod(BunG)κ,!

equal to

Eis!,ext(−, IC∞2 ).

In the talk we will explain that this is the functor of compactified Eisenstein series, denoted elsewherein the literature by Eis!∗.

5.5. FLE-3: Poor man’s FLE: going to the small quantum group.

5.5.1. The role of IC∞2 for the Kazhdan-Lusztig category.

Let us denote by jKLκ,!∗ the functor

KL(G)κ → KL(T )κ

given by

KL(G)κIC∞2

co ⊗−−→ (D-mod(GrG)κ)L(N)·L+(T ) ⊗KL(G)κconvolution−→

→ (g-modκ)L(N)·L+(T )

C∞2∗ (L(n),−)→ KL(T )κ.

Similarly, let jKL−κ,!∗ be the functor

KL(G)−κ → KL(T )−κ

given by

KL(G)−κIC∞2 ⊗−−→ (D-mod(GrG)−κ)L(N)·L+(T ) ⊗KL(G)κ

convolution−→

→ (g-mod−κ)L(N)·L+(T ) C∞2

!(L(n),−)→ KL(T )−κ.

One can show that the functors factor through the de-equivariantizations

KL(G)κ ⊗Rep(H)

Rep(TH)→ KL(T )κ and KL(G)−κ ⊗Rep(H)

Rep(TH)→ KL(T )−κ.

Let ΩKM,smallκ (resp., ΩKM,small

−κ ) denote the image of the vacuum module under the functor jKLκ,!∗

(resp., jKL−κ,!∗).

Conjecture 29.

(a) The resulting functor

jKLκ,!∗ : KL(G)κ ⊗

Rep(H)Rep(TH)→ ΩKM,small

κ -modFact

is an equivalence.

(b) The resulting functor

jKL−κ,!∗ : KL(G)−κ ⊗

Rep(H)Rep(TH)→ ΩKM,small

−κ -modFact

is an equivalence.

PROGRAM 31

5.5.2. Quantum group interpretation.

We propose:

Conjecture 30. The factorization algebra Ωsmallq (resp., Ωsmall

q−1 ) goes over under Riemann-Hilbert to

ΩKM,smallκ (resp., ΩKM,small

−κ ).

Conjecture 31.

(a) With respect to the Kazhdan-Lusztig equivalence Repq(G) ' KL(G)κ, the functor invuq(n) corre-

sponds to jKLκ,!∗.

(b) With respect to the Kazhdan-Lusztig equivalence Repq−1(G) ' KL(G)−κ, the functor invuq(n)

corresponds to jKL−κ,!∗.

Note that Conjecture 31 makes Conjecture 29 very plausible. Indeed, on the Betti side of Riemann,we have equivalences

Repq(G) ⊗Rep(H)

Rep(TH) ' •uq(g)-modinvuq(n)

−→ Ωsmallq -modFact.

5.5.3. The role of IC∞2 for the Whittaker category.

Let us denote by jWhitκ,!∗ the functor

Whit(GrG)κ → D-mod(GrT )κ

given by

Whit(GrG)κIC∞2,−

−→ D-mod(GrG)L(N−)·L+(T )−κ ⊗D-mod(L(G))

L+(G)×(L(N),χ)−κ →

→ D-mod(L(G))(L(N−)·L+(T ))×(L(N),χ)−κ

Res!−→ D-mod(GrT )κ

One can show that jWhitκ,!∗ factors the de-equivariantization

Whit(GrG)κ ⊗Rep(H)

Rep(TH)→ D-mod(GrT )κ.

Let us denote by ΩWhit,smallκ the image of the vacuum object in Whit(GrG)κ under jWhit

κ,!∗ .

We propose:

Conjecture 32. The resulting functor

jWhitκ,!∗ : Whit(GrG)κ ⊗

Rep(H)Rep(TH)→ ΩWhit,small

κ -modFact

is an equivalence.

The above conjecture is close to being a theorem: it is the subject of work in progress of Gaitsgory-Lysenko.

5.5.4. Comparison with the Langlands dual side.

We propose:

Conjecture 33. Under the equivalence

(D-mod(GrG)κ)L(N)·L+(T ) ' (D-mod(GrG)−κ)L(N)·L+(T ),

the object IC∞2 goes over to IC

∞2

co .

In addition, we propose:

32 PROGRAM

Conjecture 34.

(a) With respect to the equivalence Lκ : KL(T )κ ' D-mod(GrT )κ, the factorization algebra Ω−,KM,smallκ

corresponds to ΩWhit,smallκ .

(b) With respect to the equivalence L−κ : KL(T )−κ ' D-mod(GrT )−κ, the factorization algebra

ΩKM,small−κ corresponds to ΩWhit,small

−κ .

Assuming the above two conjectures, we obtain that we are supposed to have commutative diagrams

KL(G)κ ⊗Rep(H)

Rep(TH)Lκ−−−−−→ Whit(GrG)κ ⊗

Rep(H)Rep(TH)

j−,KLκ,!∗

y yjWhitκ,!∗

Ω−,KM,smallκ -modFact ∼−−−−−→ ΩWhit,small

κ -modFact

and

KL(G)−κ ⊗Rep(H)

Rep(TH)L−κ−−−−−→ Whit(GrG)−κ ⊗

Rep(H)Rep(TH)

jKL−κ,!∗

y yjWhit−κ,!∗

ΩKM,small−κ -modFact ∼−−−−−→ ΩWhit,small

−κ -modFact.

Note, however, that if we assume Conjectures 31 and 33, we obtain unconditional equivalences

KL(G)κ ⊗Rep(H)

Rep(TH) 'Whit(GrG)κ ⊗Rep(H)

Rep(TH)

and

KL(G)−κ ⊗Rep(H)

Rep(TH) 'Whit(GrG)−κ ⊗Rep(H)

Rep(TH).

This is what we call “poor man’s FLE”: instead of the original FLE we prove its de-equivariantizedversion.

5.5.5. Restoring the original FLE.

Recall from Talk FLE-2 that our strategy of proof of the FLE relies on Conjectures 27 and 26. Thepoint is that these conjectures follow from Conjectures 32 and 29. Let us explain this for the Whittakercategory, the KL case being analogous. In fact, we claim that we have an analog of diagram (4.15):


Rep(BH) ⊗Rep(BH )

Rep(TH)jWhitκ,!−−−−−→ (ΩWhit,Lus

κ -modFact)Ωcl -com ⊗Rep(BH )

Rep(TH)

∼y y∼


Rep(TH)jWhitκ,!∗−−−−−→ ΩWhit,small

κ -modFact.

Now, Conjecture 32 says the bottom horizontal arrow in the above diagram is an equivalence. Hence,so is the top horizontal arrow. But this, in turn, implies that the original functor


Rep(BH)jWhitκ,!−→ (ΩWhit,Lus

κ -modFact)Ωcl -com

is an equivalence.

5.6. Ch-3: The “Master” chiral algebra.

PROGRAM 33

5.6.1. The chiral algebra of differential operators.

Consider the (factorization) category D-mod(L(G))κ; the operation of taking global sections definesa functor

Γ : D-mod(L(G))κ → g-modκ ⊗ g-mod−κ.

We note that, in general, this functor fails to be conservative. Let δL+(G) ∈ D-mod(L(G))κ be thevacuum object. Denote

D(G)κ := Γ(L(G), δL+(G)) ∈ g-modκ ⊗ g-mod−κ.

This is the chiral algebra of differential operators on G at level κ. Since δL+(G) is equivariant with

respect to L+(G)× L+(G), the object Dκ naturally lifts to an object KL(G)κ ⊗KL(G)−κ.

The functor Γ naturally gives rise to a functor

(5.10) Γ : D-mod(L(G))κ → D(G)κ-modFact.

The functor (5.10) is t-exact, and defines an equivalence at the level of hearts. But, in general, itfails to be an equivalence (it even fails to be conservative).

We propose the following conjecture that says that D(G)κ-modFact is “the closest KM non-degenerateapproximation” to D-mod(L(G))κ:

Conjecture 35. We have an equivalence

g-modκ ⊗HCh(L(G))κ

g-mod−κ ' D(G)κ-modFact.

As a formal consequence we obtain:

(5.11) g-mod−κ ⊗L(G)

D(G)κ-modFact ' g-mod−κ.

5.6.2. Construction of the Master chiral algebra.

Let us assume the FLE and Conjecture 21. Recall the functor

P : KL(G)κ ⊗KL(G)κ →W-modFact.

We denote by

M−κ,−κ ∈ KL(G)−κ ⊗W-modFact ⊗KL(G)−κ

the factorization algebra equal to

(IdKL(G)−κ ⊗P ⊗ IdKL(G)−κ)(D(G)−κ ⊗D(G)κ).

5.6.3. The Master chiral algebra and the local conjecture.

We can consider the category of factorization M−κ,−κ-modules in g-mod−κ⊗W-modFact⊗g-mod−κ,denoted M−κ,−κ-modFact, as acted on by L(G) at level −κ and L(G) at level −κ.

The operation

(5.12) C 7→ C ⊗L(G)

M−κ,−κ-modFact

defines a functor

L(G)-modκ → L(G)-mod−κ.

It follows from (5.11) (see the long manipulation below) that (5.12) factors as

L(G)-modκ (L(G)-modκ)KM-nondeg → (L(G)-mod−κ)KM-nondeg → L(G)-mod−κ.

Let Mlocκ denote the resulting functor

(5.13) (L(G)-modκ)KM-nondeg → (L(G)-mod−κ)KM-nondeg.

34 PROGRAM

We will show that it has the features expected from the composition

(L(G)-modκ)KM-nondeg → L(G)-modκLlocκ−→ L(G)-mod−κ (L(G)-mod−κ)KM-nondeg.

Namely, the local Fundamental Commutative Diagram reads as

HCh(L(G))κ-mod −−−−−→ Hecke-Whit(L(G))−κ-mody(L(G)-modκ)KM-nondeg

yyL(G)-modκ

Llocκ−−−−−→ L(G)-mod−κyy (L(G)-mod−κ)KM-nondegy

Hecke-Whit(L(G))κ-mod −−−−−→ HCh(L(G))−κ-mod

We claim that the resulting diagram

HCh(L(G))κ-mod −−−−−→ Hecke-Whit(L(G))−κ-mod −−−−−→ L(G)-mod−κy y(L(G)-modκ)KM-nondeg Mloc

κ−−−−−→ (L(G)-mod−κ)KM-nondegy yL(G)-modκ −−−−−→ Hecke-Whit(L(G))κ-mod −−−−−→ HCh(L(G))−κ-mod

does commute. Namely, we claim that the composition

(5.14) HCh(L(G))κ-mod→ (L(G)-modκ)KM-nondeg Mlocκ−→

→ (L(G)-mod−κ)KM-nondeg → HCh(L(G))−κ-mod

Is given by

C 7→ C ⊗HCh(L(G))−κ

W-modFact,

where we regard W-modFact as a category acted on by HCh(L(G))−κ ⊗HCh(L(G))−κ, see Talk Wh-4.

This assertion is equivalent to the assertion that

g-modκ ⊗L(G)

M−κ,−κ-modFact ⊗L(G)

g-modκ 'W-modFact.

PROGRAM 35

We have:

g-modκ ⊗L(G)

M−κ,−κ-modFact ⊗L(G)

g-modκ '

'M−κ,−κ-modFact

((g-modκ ⊗

L(G)g-mod−κ)⊗W-modFact ⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)=

= (D(G)−κ ⊗D(G)κ)-modFact

((g-modκ ⊗

L(G)g-mod−κ)⊗W-modFact ⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)'

D(G)κ-modFact

(D(G)−κ-modFact((g-modκ ⊗

L(G)g-mod−κ)⊗W-modFact)⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)

D(G)κ-modFact

(D(G)−κ-modFact((g-modκ ⊗

L(G)g-mod−κ ⊗ g-modκ)L(N),χ)⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)

D(G)κ-modFact

((D(G)−κ-modFact(g-modκ ⊗

L(G)g-mod−κ ⊗ g-modκ))L(N),χ ⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)(5.11)' D(G)κ-modFact

(g-modL(N),χ

κ ⊗ (g-mod−κ ⊗L(G)

g-modκ)

)'

' D(G)κ-modFact

(W-modFact ⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)'

' D(G)κ-modFact

(g-modL(N),χκ ⊗ (g-mod−κ ⊗

L(G)

g-modκ)

)'

' D(G)κ-modFact

((g-modκ ⊗ (g-mod−κ ⊗

L(G)

g-modκ))L(N),χ

)'

'

(D(G)κ-modFact(g-modκ ⊗ (g-mod−κ ⊗

L(G)

g-modκ))

)L(N),χ (5.11)

' g-modL(N),χκ 'W-modFact.

In the above manipulation, pulling the functor of (L(N), χ) (resp., (L(N), χ)) invariants out ofparenthesis is justified due to the fact this operation commutes with limits and colimits.

5.6.4. The Master chiral algebra and the global conjecture.

Consider the object

Mglob−κ,−κ := (Loc⊗oblvW ⊗ Loc)(M−κ,−κ) ∈ D-mod(BunG)−κ,∗ ⊗D-mod(BunG)−κ,∗,

where oblvW denotes the forgetful functor W-modFact → Vect.

We use Mglob−κ,−κ as a kernel to define a functor

(5.15) D-mod(BunG)κ,! → D-mod(BunG)−κ,∗.

We will now explain the connection between Mglobκ and the conjectural equivalence (1.2).

Let

D-mod(BunG)KM-nondegκ,! ⊂ D-mod(BunG)κ,! and D-mod(BunG)KM-nondeg

−κ,∗ ⊂ D-mod(BunG)−κ,∗

be the full subcategories generated by the essential image of the corresponding functors

Loc : KL(G)κ → D-mod(BunG)κ,! and Loc : KL(G)−κ → D-mod(BunG)−κ,∗.

We propose

36 PROGRAM

Conjecture 36. The functors

Loc : KL(G)κ → D-mod(BunG)KM-nondegκ,! and Loc : KL(G)−κ → D-mod(BunG)KM-nondeg

−κ,∗

are Verdier quotients.

This conjecture is equivalent to the following one:

Conjecture 37.

(a) The endofunctor of D-mod(BunG)κ,! defined by

(Loc⊗Loc)(D(G)−κ) ∈ D-mod(BunG)−κ,∗ ⊗D-mod(BunG)κ,!

identifies with the colocalization functor with respect to the full subcategory

D-mod(BunG)KM-nondegκ,! ⊂ D-mod(BunG)κ,!.

(b) The endofunctor of D-mod(BunG)−κ,∗ defined by

(Loc⊗Loc)(D(G)κ) ∈ D-mod(BunG)κ,! ⊗D-mod(BunG)−κ,∗

identifies with the colocalization functor with respect to the full subcategory

D-mod(BunG)KM-nondeg−κ,∗ ⊂ D-mod(BunG)−κ,∗.

It follows that the functor (5.15) factors as

D-mod(BunG)κ,! → D-mod(BunG)KM-nondegκ,! → D-mod(BunG)KM-nondeg

−κ,∗ → D-mod(BunG)−κ,∗.

Let Mglobκ denote the resulting functor

(5.16) D-mod(BunG)KM-nondegκ,! → D-mod(BunG)KM-nondeg

−κ,∗ .

We will now show that it has the properties expected from the composite

D-mod(BunG)KM-nondegκ,! → D-mod(BunG)κ,!

Lglobκ−→ D-mod(BunG)−κ,∗ D-mod(BunG)KM-nondeg

−κ,∗ .

Namely, let us write the Fundamental Commutative Diagram (1.6) as


Loc

yD-mod(BunG)KM-nondeg

κ,!

yPoinc∗yD-mod(BunG)κ,!

Lglobκ−−−−−→ D-mod(BunG)−κ,∗y

coeff

y D-mod(BunG)KM-nondeg−κ,∗y


−−−−−−→ KL(G)−κ.

PROGRAM 37

We claim that the diagram


Poinc∗−−−−−→ D-mod(BunG)−κ,∗

Loc

y yD-mod(BunG)KM-nondeg

κ,!

Mglobκ−−−−−→ D-mod(BunG)KM-nondeg

−κ,∗y yD-mod(BunG)κ,!

coeff−−−−−→ Whit(GrG)−κ(L−κ)−1

−−−−−−→ KL(G)−κ

does commute.

Namely, we claim that the composite functor

KL(G)κLoc−→ D-mod(BunG)KM-nondeg

κ,!

Mglobκ−→ D-mod(BunG)KM-nondeg

−κ,∗ → KL(G)−κ

is such that the resulting pairingKL(G)κ ⊗KL(G)κ → Vect

is given by

(5.17) KL(G)κ ⊗KL(G)κP→W-modFact ConfW−→ Vect,

where the last arrow is the functor of factorization homology with respect to W, see Talk Wh-4.

Indeed, let us note that Conjecture 37 implies that for M ∈ KL(G)κ

(5.18) (〈−,−〉 ⊗ Id) (Loc⊗Loc⊗Loc)(M⊗D(G)−κ) ' Loc(M),

where 〈−,−〉 denotes the Verdier duality pairing

D-mod(BunG)κ,! ⊗D-mod(BunG)−κ,∗ → Vect .

Hence, for M′ ∈ KL(G)κ and M′′ ∈ KL(G)κ, the value of the pairing (5.17) is

((〈−,−〉) (Loc⊗Loc))⊗ (ConfW P )⊗ ((〈−,−〉) (Loc⊗Loc)) (M⊗D(G)−κ ⊗D(G)κ ⊗M′) '

' ((〈−,−〉) (Loc⊗Loc))⊗(

(〈−,−〉) (Loc⊗(Poinc∗ Lκ)))⊗((〈−,−〉) (Loc⊗Loc)) (M⊗D(G)−κ⊗D(G)κ⊗M′) '

(5.18)' ((〈−,−〉)⊗ (〈−,−〉))

(Loc⊗(Poinc∗ Lκ)⊗ (Loc⊗Loc)

)(M⊗D(G)κ ⊗M

′) '

' ((〈−,−〉)⊗ (〈−,−〉)) (

(Poinc∗ Lκ)⊗ Loc)⊗ (Loc⊗Loc))

(M⊗D(G)κ ⊗M′) '

(5.18)' (〈−,−〉)

((Poinc∗ Lκ)⊗ Loc

)(M⊗M

′) ' (ConfW P )(M⊗M′).

WINTER SCHOOL ON LOCAL GEOMETRIC LANGLANDS THEORY: PROGRAM › ~Sergey.Lysenko › program_1.pdf ·...

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