WINTER SCHOOL ON LOCAL GEOMETRIC LANGLANDS THEORY: PROGRAM

Transcription

1 WINTER SCHOOL ON LOCAL GEOMETRIC LANGLANDS THEORY: PROGRAM Contents 1. Day one GL-1: Introduction to quantum local geometric Langlands T-1: Tutorial on groups acting on categories Wh-1: The Whittaker model GL-2: The quantum parameter GL-3: The global quantum geometric theory 3 2. Day two Ch-1: The factorization category of D-modules on the affine Grassmannian T-2: Tutorial on factorization algebras and categories GL-4: Quantum Satake equivalence Wh-2: Proof of the equivalence of the two versions of the Whittaker model Ja-1: Jacquet functors for actions of loops groups on categories 7 3. Day three T-3: Tutorial on the functor of weak invariants GL-5: The functor of weak L(G)-invariants GL-6: Spectral side in the classical case FLE-1: Statement of the equivalence Wh-3: Statement of local quantum Langlands GL-7: Quantum Langlands for G being a torus Day four Ja-2: Jacquet functors and Langlands duality T-4: Tutorial on quantum groups T-5: Tutorial on factorization vs braided monoidal categories Ch-2: The factorization algebra Ω q (or now Nature encodes root data) Ja-3: Jacquet functor on Kac-Moody modules and the Kazhdan-Lusztig equivalence Wh-4: Duality for W-algebras Day five Wh-5: Jacquet functor on the Whittaker category FLE-2: Strategy of the proof of FLE via Jacquet functors Ja-4: Jacquet/Eisenstein functors in the global situation Ja-5: The semi-infinite IC sheaf FLE-3: Poor man s FLE: going to the small quantum group Ch-3: The Master chiral algebra 32 Date: November 30,

2 2 PROGRAM 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: the totality (technically, 2-category) of categories (technically, DG categories) equipped with an action of the loop group L(G) := G((t)) at a given level. We denote it by L(G)-mod. The quantum local geometric Langlands conjecture says that this 2-category is equivalent to the similarly defined category for the dual loop group L(Ǧ) at the dual level ˇ. I.e., we propose: Conjecture 1. There exists an equivalence of 2-categories L loc (1.1) L(G)-mod L(Ǧ)-mod ˇ. The equivalence (1.1) is supposed to be characterized (modulo the issues of Whittaker/Kac-Moody degeneracy) by the requirement that if C and Č correspond to each other under the above equivalence of 2-categories, then the Whittaker model of C, denoted Whit(C), should be equivalent as a category to the Kac-Moody model of Č, denoted KM(Č). The degenerate case should be controlled by the compatibility of the conjectural equivalence with the procedure of parabolic induction. This talk will be a general introduction to these ideas, and much of the rest of the workshops will be devoted to supplying details 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 a twist 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 then for loop groups, and consider examples. We will also review the theory of D-modules, de Rham prestacks, twistings, multiplicative twistings and their relation to central extensions of Lie algebras. Some background material on DG categories will be reviewed as well (compact generation, duality, tensor products) Wh-1: The Whittaker model. A key operation needed for the statement of the local quantum geometric Langlands conjecture is the 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 a locally compact group G, its Whittaker model is the space of vectors/functionals invariant against the maximal unipotent N against a non-degenerate character χ. Like in number theory, in our geometric setting, there are two possible operations: given C, we can consider the subcategory C L(N),χ of (L(N), χ)-invariant objects or the quotient category C L(N),χ of (L(N), χ)-coinvariants. These two operations are naturally dual to each other. However, it turns out that C L(N),χ and C L(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).

3 PROGRAM 3 A consequence of the equivalence of the two definitions of the functor Whit is that it is very wellbehaved from the algebro-analytic point of view (technically: commutes with both limits and colimits as well as duality) GL-2: The quantum parameter. Roughly speaking, the quantum parameter for the quantum geometric Langlands theory is what is called the Kac-Moody level (i.e., an Ad-invaraint bilinear form on g). However, this is literally the case only when G is semi-simple. If G contains has a central torus (and we will need to consider this case, because along with G we will consider all its Levi subgroups), an additional set of quantum parameters comes by considering extensions of z O X by ω X (here z is the center of g, and X is a curve equipped with a point x, so that k[[t]] is ÔX,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 even more fundamentally, we will explain that to a quantum parameter there corresponds a factorizable twisting on the factorizable affine Grassmannian Gr G, and also a mutiplicative factorizable twisting on the factorizable loop group L(G). This talk will also provide a background on the notions involved in the above definitions (the factorizable Grassmannian, loop group, etc). We will also relate quantum parameters to factorizable line bundles on Gr G that give rise to factorizable central extensions of L(G) (and thereby to integrable representations), and also to factorizable gerbes on Gr G, 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 Bun G on a global curve GL-3: The global quantum geometric theory. As in the classical theory of automorphic functions, the local Langlands theory is closely related to the global one. Given a quantum parameter and a global curve X, we can consider the corresponding category of twisted D-modules D-mod(Bun G). The caveat here is that there are two natural categories that one can consider, the!- and the *- ones, denoted D-mod(Bun G),! and D-mod(Bun G),, respectively. This has to do with the fact that Bun G is not quasi-compact, and, figuratively speaking, in D-mod(Bun G),! we generate objects using!-extensions from quasi-compact open substacks and in D-mod(Bun G),! we use *-extensions. The general principle, which we will see in other instances during the workshop, is that the choice of which version to consider depends on the sign of. For example, if G is simple, in which case is a multiple of the Killing form = c Kil, we use the!-version if (c + 1 ) 2 Q>0 and the *-version if (c + 1 ) 2 Q<0, and the two versions coincide if c k \ Q. (Here c = 1 is the critical value and 2 corresponds to the classical, as opposed to quantum, situation.) We will state the global (non-ramified) quantum geometric Langlands conjecture as follows (say, when is positive in the above sense): Conjecture 2. There exists an equivalence L glob (1.2) D-mod(Bun G),! D-mod(Bun Ǧ ) ˇ,.

4 4 PROGRAM Local-to-global functors. In order to state the local-to-global compatibility that is supposed to be satisfied by the conjectural equivalence (1.2), we will introduce several local-to-global functors. We introduce several functors that connect D-mod(Bun G) (both versions) with We start with (a naturally defined) functor then consider its Verdier conjugate Whit(Gr G) := Whit(D-mod(Gr G) ). Poinc : Whit(Gr G) D-mod(Bun G),, Poinc! : Whit(Gr G) D-mod(Bun G),!, and the right adjoint of Poinc! (same as dual of Poinc ) coeff : D-mod(Bun G),! Whit(Gr G). We will also introduce the Kazhdan-Lusztig category KL(G) := ĝ-mod L+ (G), (i.e., the category of Kac-Moody modules integrable with respect to L + (G)), and the localization functor Loc : KL(G) D-mod(Bun G),! (for positive), its Verdier conjugate and their respective right adjoints/duals Loc : KL(G) D-mod(Bun G),, Γ : D-mod(Bun G), KL(G) and Γ : D-mod(Bun G),! KL(G) Local-to-global compatibility. This equivalence (1.2) is supposed to be characterized (again, up to the issue of Whittaker/KM degeneracy) by the requirement that it makes the following diagrams commute: (1.3) In the above formula, KL(G) Loc D-mod(Bun G),! L Whit(Gr Ǧ )ˇ Poinc L glob D-mod(Bun Ǧ ) ˇ,. KL(G) L Whit(Gr Ǧ )ˇ is the Fundamental Local Equivalence for positive level, which would be one of the central themes in this workshop, see Talk FLE-1. (1.4) Passing to dual functors (and using the symmetry of the picture in G and Ǧ), we obtain a diagram where D-mod(Bun G),! coeff Whit(Gr G) L glob (L ˇ ) 1 D-mod(Bun Ǧ ) ˇ, Γ KL(Ǧ) ˇ, KL(G) L Whit(Gr Ǧ ) ˇ is the Fundamental Local Equivalence for negative level.

5 PROGRAM 5 By passing to left adjoints along the vertical arrows in (1.4), we obtain yet another commutative diagram: (1.5) D-mod(Bun G),! Poinc! L glob D-mod(Bun Ǧ ) ˇ, Loc Whit(Gr G) (L ˇ ) 1 KL(Ǧ) ˇ, (1.6) Juxtaposing diagrams (1.3) and (1.4) we obtain: KL(G) Loc D-mod(Bun G),! coeff Whit(Gr G) L L glob (L ˇ ) 1 which we call the Fundamental Commutative Diagram. Whit(Gr Ǧ )ˇ Poinc D-mod(Bun Ǧ ) ˇ, Γ KL(Ǧ) ˇ, Note that the commutativity of the outer square in (1.6) can be formulated unconditionally (we do not need L glob to state it): Conjecture 3. The square (1.7) commutes. coeff Loc KL(G) Whit(Gr G) L (L ˇ ) 1 Whit(Gr Ǧ )ˇ Γ Poinc KL(Ǧ) ˇ. In talk Wh-4 we will see that Conjecture 3 certain from another conjecture of local nature (see Conjecture 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: we want to compare two kinds of objects, each equipped with a rather complicated sounding structure an action of the corresponding loop group. The hope is that we can express this structure (on both sides) via another kind of structure, one that would 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 module category 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(Gr G) ) (resp., KL(G) ) that we already encountered in Talk GL-3.

6 6 PROGRAM Although both Whit(D-mod(Gr G) ) and KL(G) are still quite non-trivial, they are non-ramified in nature, and we can hope to achieve some sort of hands-on (i.e., close to combinatorial) description of both of them, so that we will able compare these categories (each on its side of Langlands correspondence) 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(Gr G) is naturally a factorization category, and if L(G) acts on C (at level ), then the same C has a natural structure of factorization module category over D-mod(Gr G). This defines a functor (2.1) L(G)-mod D-mod(Gr G) -mod Fact. 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 T-2: Tutorial on factorization algebras and categories. The notion of factorization category (and a related notion of factorization algebra in a given factorization 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 makes sense of them within the formalism of higher algebra. In process of doing so, we will review some of the 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 representations of 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 GL-4: Quantum Satake equivalence. One of the features of the local quantum geometric Langlands equivalence (1.1) is that if C L(G)-mod corresponds to Č L(Ǧ)-mod ˇ, then their categories of spherical objects, i.e., are equivalent as categories. C L+ (G) and ČL+ (Ǧ), As a formal consequence of this, one obtains that D-mod(Gr G) L(G)-mod corresponds to + D-mod(Gr Ǧ ) ˇ L (Ǧ)-mod ˇ under the equivalence (1.1): L loc (D-mod(Gr G) ) D-mod(Gr Ǧ ) ˇ. Applying the above principal one more time we deduce a conjectural equivalence between Sph G, := D-mod(Gr G) L+ (G) Conjecture 5. There exists an equivalence and Sph := Ǧ, ˇ D-mod(GrǦ L+ (Ǧ) ) ˇ : Sph G, Sph Ǧ, ˇ, as monoidal categories, compatible with the factorization structures.

7 PROGRAM 7 The above conjecture is the quantum counterpart of the geometric Satake equivalence. It plays the same role vis-à-vis Conjecture 1.2 as the usual geometric Satake does for the classical geometric Langlands (i.e., that (1.2) is compatible with the Sph G, -action on D-mod(Bun G) and the Sph - Ǧ, ˇ action on D-mod(Bun G) ˇ. Now, Conjecture 5 is very close to being a theorem. In fact, if is irrational, both sides are trivial (equivalent to Vect, the category of vector spaces). For rational, one shows that both sides are equivalent 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 (with the same result for Ǧ and ˇ). This group H had been first discovered by G. Lusztig as the recipient of the quantum Frobenius, see talk T 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 canonical equivalence C L(N),χ C L(N),χ. In this talk we will prove this statement. The reason for doing is not so much to give a proof in order to establish validity, but to introduce the ideas contained therein: what are the tools that allow us to prove something about substantially infinite-dimensional objects such as L(N)? The key observation is that one approximate each side, i.e., C L(N),χ and C L(N),χ, by adolescent Whittaker models, i.e., categories C In,χ, where I n is a certain family of subgroups of L(G) that tends to L(N) as n. The idea is that, unlike L(N), each I n is a group-scheme (rather than a group ind-scheme), and the operation C C In,χ is much more manageable. The subgroups and the corresponding categories C In,χ turn out to be very handy for other problems that involve the behavior of Whit(C) 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 between L(G)-mod and the corresponding category L(M)-mod, where M is the Levi quotient of a parabolic P on G. In order to simplify things, we will take P to be a Borel B, so that M is the abstract Cartan T. 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)-mod L(T )-mod : C J! (C) := C L(N) and C J (C) := C L(N). However, an analog of the functor that defines an equivalence C L(N),χ C L(N),χ, exists as a functor C L(N) C L(N) but is no longer an equivalence. Instead, we propose: Conjecture 6. The composition (2.2) C L(N) C C L(N ) is an equivalence. In the talk we will explain how this conjecture can be seen as a counterpart in our setting of Bernstein 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 the opposite Jacquet functor J. We will explicitly verify the following corollary of the above conjecture: namely, that the equivalence in question holds after we take L + (T )-invariants (which is the same as coinvariants) on both sides. In fact, we will show that the resulting equivalence is obtained by identifying both sides with C I, where

8 8 PROGRAM I L + (G) is the Iwahori subgroup (the caveat here is that since we use the Iwahori subgroup, for the moment, we can only prove this equivalence at a point, i.e., we do not yet know how to do the factorizable 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 categories equipped with an action of G. For C G-mod, we can consider the DG category C G -weak of weak G-invariants. This functor is representable: C G -weak Funct G-mod (g-mod, C), where g-mod G-mod is the category of modules over the Lie algebra, viewed as acted on by G via the adjoint action. Hence, C G -weak is acted on by the monoidal category HCh(G) := EndoFunct G-mod (g-mod) op. The monoidal category HCh(G) is that of Harish-Chandra bimodules. We will show that the resulting functor G-mod HCh(G)-mod, C C G -weak is an equivalence of 2-categories. As a formal consequence, one obtains a canonical equivalence (3.1) g-mod g-mod D-mod(G). HCh(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 KM(C) := C L(G) -weak. As in the finite-dimensional case, this functor is co-representable: KM(C) Funct L(G)-mod (ĝ-mod, C), where ĝ-mod is the category of modules over the Kac-Moody algebra attached to g at level. The functor KM gives rise to a functor (3.2) L(G)-mod HCh(L(G)) -mod, where HCh(L(G)) := EndoFunct L(G)-mod (ĝ-mod ) op.

9 PROGRAM Kac-Moody model as a factorization module. Let KL(G) be the Kazhdan-Lusztig category. We can think of it either as (ĝ-mod ) L+ (G) or as (D-mod(Gr G) ) L(G) -weak. Following up on ideas from Talk Ch-1, we will see that any module category over HCh(L(G)) -mod is automatically a factorization module category for KL(G). Thus, we obtain a functor (3.3) HCh(L(G)) -mod KL(G) -mod Fact, which is a cousin of the functor (2.1). We will propose: Conjecture 7. The functor (3.3) is fully faithful. Thus, we obtain that the operation C KM(C) defines a functor (3.4) L(G)-mod KL(G) -mod Fact. However, unlike the finite-dimensional situation, we encounter the issue of Kac-Moody degeneracy KM (non)-degeneracy. A feature of the infinite-dimensional situation is that, unless G is a torus, the functor (3.2) is no longer an equivalence (in fact, it fails to be conservative). I.e., for any C L(G)-mod we have a well-defined functor (3.5) ĝ-mod HCh(L(G)) KM(C) C. but it is not necessarily an equivalence. Conjecture 8. The functor (3.5) is fully faithful. We shall say that C is KM non-degenerate if the functor (3.5) is an equivalence. We will denote by (L(G)-mod ) KM -nondeg L(G)-mod the full subcategory that consists of KM non-degenerate objects. Thus, the functor KM defines an equivalence (3.6) (L(G)-mod ) KM -nondeg HCh(L(G)) -mod. Note that Conjecture 7 implies that the restriction of (3.4) to (L(G)-mod ) KM -nondeg is fully faithful 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 the quantum geometric Langlands story (one may think of it as letting tend to in a certain projective space to be explained in the talk). Specifically, we want we want to see how the phenomenon of KM-degeneracy plays out in this context. We will first consider the global situation. In this case, the KM non-degenerate part of D-mod(Bun G) is stipulated to be the category QCoh(LocSys G ). The entire D-mod(Bun G) is captured as a certain enlargement of QCoh(LocSys G ), denoted IndCoh nilp (LocSys G ) that consists of ind-coherent sheaves, with singular support belonging to the global nilpotent cone (the latter may be seen 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 of L(G)-mod is the 2-category DG categories over the ind-scheme Conn G(D ) of connections on the trivial G-bundle over the formal punctured disc D, equipped with a structure of (weak) equivariance with respect to L(G) that acts by gauge transformations.

10 10 PROGRAM 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 over QCoh(LocSys G (D )), where LocSys G (D ) := Conn G(D )/L(G). However, it is a priori not clear how to enlarge (L(G)-mod ) KM -nondeg defined in this way to an entire L(G)-mod. In this talk we will explain an idea that leads to the desired enlargement. We will consider a simplified situation, where instead of LocSys G (D ) we consider a finite-dimensional smooth algebraic stack Y, and we are interested in ways to enlarge the 2-category QCoh(Y)-mod. Namely, it will turn out that to every sub-lagrangian N T Y one can attach a certain 2-category, denote it QCoh(Y)-mod N, 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 LocSys G (D ) with N being the nilpotent cone. In a sense, this is a way to insert local Arthur parameters into the geometric Langlands story 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(Gr G) and KL(Ǧ)ˇ. However, things are a little more involved, in that the equivalence we are after depends on the sign of the level. Namely, let us assume that is positive. In this case, we propose: Conjecture 9. There exists a canonically defined equivalence of factorization categories (3.7) L : KL(G) Whit(Gr Ǧ ) ˇ. (We note that Whit(Gr Ǧ ) ˇ 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 module V λ KL(G) to the standard object j λ,! Whit(Gr Ǧ ) ˇ (here λ is a dominant weight of G and hence a dominant coweight of Ǧ). In addition, this equivalence is compatible with the action of Sph G, on KL(G) and the action of Sph G,ˇ on Whit(Gr Ǧ ) ˇ via quantum geometric Satake (see Talk GL-4). Now, we have canonical duality (to be explained in the talk). (KL(G) ) KL(G) Composing with (Whit(Gr Ǧ ) ˇ) Whit(Gr Ǧ )ˇ (here the right-hand side is naturally incarnated as coinvariants), from (3.7) we obtain an equivalence (3.8) L : KL(G) Whit(Gr Ǧ )ˇ. The functor L behaves quite differently from L. For example, it does not respect the t-structures, and it sends V λ KL(G) to the co-standard object j w0 (λ), Whit(Gr Ǧ ) ˇ. In addition, the functors L and L have a different-looking compatibility property with Jacquet functors (to be discussed in Talk FLE-2). One can precompose L with the Cartan involution on G and obtain a functor L τ : KL(G) Whit(Gr Ǧ )ˇ. The functors L and L τ glue into a family as varies (in particular, they are canonically the same when is irrational).

11 PROGRAM Wh-3: Statement of local quantum Langlands. The functor is co-representable by an object C Whit(C) Whit(L(G)) L(G)-mod, (we can think of Whit(L(G)) as obtained by applying the functor Whit to the category D-mod(L(G)) of -twisted D-modules on the loop group, acting on itself on the right). Denote Hecke-Whit(L(G)) := EndoFunct L(G)-mod (Whit(L(G)) ) op. The functor Whit can thus be seen as a functor (3.9) L(G)-mod Hecke-Whit(L(G)) -mod. As in Talk Ch-1, one shows that an object of Hecke-Whit(L(G)) -mod is naturally a factorization module for the factorization category Whit(Gr G). Thus, we obtain a functor (3.10) Hecke-Whit(L(G)) -mod Whit(Gr G) -mod Fact. As in the case of (2.1), we propose: Conjecture 10. The functor (3.10) is fully faithful. Thus, we obtain that Whit defines a functor (3.11) L(G)-mod Whit(Gr G) -mod Fact Precise statement of local quantum Langlands. Let be positive, and let us assume the Fundamental Local Equivalence, i.e., Conjecture 9. We propose: Conjecture 11. There exists an equivalence (1.1) that makes the following diagram commute KL(G) -mod Fact KM L(G)-mod Whit Whit(Gr G) -mod Fact L Whit(Gr Ǧ ) ˇ-mod Fact Whit L loc L(Ǧ)-mod ˇ KM ( L ) 1 KL(Ǧ)ˇ-modFact. There is a further compatibility that one requires from L loc to be discussed in talk Ja Several more concrete conjectures. that has to do with parabolic induction, Let us assume Conjectures 7 and 10. Then Conjecture 11 implies the following one: Conjecture 12. (a) With respect to the equivalence KL(G) -mod Fact Whit(Gr Ǧ ) ˇ-mod Fact, induced by (3.7), the essential images of the functors HCh(L(G)) -mod KL(G) -mod Fact and Hecke-Whit(L(Ǧ)) ˇ-mod Whit(GrǦ ) ˇ-mod Fact coincide.

12 12 PROGRAM (b) With respect to the equivalence KL(G) -mod Fact Whit(Gr Ǧ )ˇ-mod Fact, induced by (3.8), the essential images of the functors HCh(L(G)) -mod KL(G) -mod Fact and Hecke-Whit(L(Ǧ))ˇ-mod Whit(GrǦ )ˇ-mod Fact coincide. Conjecture 12 in turn implies: Conjecture 13. (a) There exist a canonical equivalence of monoidal categories HCh(L(G)) Hecke-Whit(L(Ǧ)) ˇ. (b) There exists a canonical equivalence of monoidal categories HCh(L(G)) Hecke-Whit(L(Ǧ))ˇ The local Fundamental Commutative Diagram. Assuming Conjecture 13, by passing to left adjoints along the vertical arrows in the upper portion of the above diagram, we obtain a local version of the Fundamental Commutative Diagram: (3.12) ĝ-mod HCh(L(G)) HCh(L(G)) -mod L(G)-mod Whit Hecke-Whit(L(G)) -mod L Hecke-Whit(L(Ǧ)) ˇ-mod L loc ( L ) 1 D-mod(L(Ǧ)) ˇ Hecke-Whit(L(Ǧ)) ˇ L(Ǧ)-mod ˇ KM HCh(L(Ǧ)) ˇ-mod. As in the global case, one can formulate a conjecture that the outer diagram in (3.12) commutes: Conjecture 14. The diagram Whit (ĝ-mod HCh(L(G)) HCh(L(G)) -mod ) Hecke-Whit(L(G)) -mod L Hecke-Whit(L(Ǧ)) ˇ-mod ( L ) 1 KM (D-mod(L(Ǧ)) ˇ ) Hecke-Whit(L(Ǧ)) ˇ HCh(L(Ǧ)) ˇ-mod. Note that the statement of Conjecture 14 depends on Conjecture 13, but not on the existence of the functor L loc ; in this sense it is more unconditional. In Talk Wh-4 we will show that (assuming Conjecture 13), the statement Conjecture 14 follows from another plausible statement, Conjecture Whittaker (non)-degeneracy. Much as KL, the functor (3.9) fails to be fully faithful. Namely, for C L(G)-mod, we have a canonically defined functor (3.13) Whit(L(G)) Hecke-Whit(L(G)) Whit(C) C, which is not, in general, an equivalence. Conjecture 15. The functor (3.13) is fully faithful.

13 PROGRAM 13 We shall say that C L(G)-mod is Whittaker non-degenerate if (3.13) is an equivalence. Let (L(G)-mod ) Whit -nondeg L(G)-mod denote the full subcategory consisting of Whittaker non-degenerate objects. Thus, the functor Whit defines an equivalence (3.14) (L(G)-mod ) Whit -nondeg Hecke-Whit(L(G)) -mod. We obtain that Conjecture 13 (combined with Conjectures 7 and 10) implies the following particular case of Conjecture 11: Conjecture 16. (a) There exists an equivalence that makes the diagram (L(G)-mod ) KL -nondeg -nondeg (L(Ǧ)-mod ˇ)Whit commute. (b) There exists an equivalence KL(G) -mod Fact KM KL -nondeg (L(G)-mod ) L Whit(Gr Ǧ ) ˇ-mod Fact Whit (L(Ǧ)-mod ˇ)Whit -nondeg (L(G)-mod ) KL -nondeg -nondeg (L(Ǧ)-modˇ)Whit that makes the diagram commute. KL(G) -mod Fact KM KL -nondeg (L(G)-mod ) L Whit(Gr Ǧ )ˇ-mod Fact Whit (L(Ǧ)-modˇ)Whit -nondeg One can formally deduce that the diagrams and commute as well. KL -nondeg (L(G)-mod ) Whit Whit(Gr G) -mod Fact KL -nondeg (L(G)-mod ) Whit Whit(Gr G) -mod Fact -nondeg (L(Ǧ)-mod ˇ)Whit KM ( L ) 1 KL(Ǧ)ˇ-modFact. -nondeg (L(Ǧ)-modˇ)Whit KM (L ) 1 KL(Ǧ) ˇ-modFact. Thus, we obtain that Conjecture 13 implies the appropriate non-degenerate cases of Conjecture 11.

14 14 PROGRAM 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 not been done yet). Essentially, the proof sould follow from the fact that L(T ) identifies with the Cartier dual of L(Ť ) via the Contou-Carrère symbol. This identification leads to the equivalences of monoidal categories in Conjecture 13. One deduces Conjecture 11 using the fact that the functor Whit of (3.9) does not do anything (is the identity functor), 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 the Cartan subgroup T as a tool to deduce the FLE for a reductive group G via Jacquet functors, see Talk FLE 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 the following diagram commute: (4.1) L(G)-mod J! L(Ǧ)-mod ˇ J L(T )-mod L(Ť )-mod ˇ. 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(Gr G) L(G)-mod Combining with (4.1) we obtain: Conjecture 17. We have an equivalence compatible with factorization. D-mod(Gr Ǧ ) ˇ L(Ǧ)-mod ˇ. (D-mod(Gr G) ) L(N) L+ (T ) (D-mod(Gr Ǧ ) ˇ) L( Ň) L + (Ť ), Under the equivalence of Conjecture 17, the standard object 0 in the LHS, i.e., the!-extension of the dualizing sheaf from the unit L(N)-orbit on Gr G, is supposed to go over to the object 0,co in the RHS, i.e., the image of δ 1,GrǦ D-mod(Gr Ǧ ) ˇ under the projection Classical version. D-mod(Gr Ǧ ) ˇ (D-mod(Gr Ǧ ) ˇ) L( Ň) L + (Ť ). In the talk, we will also explain the classical version of this statement (i.e., when is zero 1 ), and ˇ is. In this case, the right-hand side, i.e., (D-mod(Gr Ǧ ) ) L( Ň) L + (Ť ) degenerates to the (appropriately defined) category of ind-coherent sheaves on LocSys (D) Ť LocSys ˇB(D ) LocSys (D). Ǧ LocSysŤ (D ) LocSysǦ(D ) Note that when working over a point, the above fiber product identifies with ( Ň pt)/ǧ. ǧ 1 For us, zero=critical, but this does not affect D-mod(Gr G ) as a category, because the critical level is integral.

15 PROGRAM 15 The resulting equivalence is a theorem, due to [ABG]. D-mod(Gr G) L(N) L+ (T ) IndCoh( Ň pt /Ǧ) ǧ 4.2. T-4: Tutorial on quantum groups. First, we interpret the quantum parameter q as a quadratic form on the root lattice ˇΛ of our torus T with values in k. In fact, we need a little more: we need to refine q to a bilinear form (not necessarily symmetric). To this datum one attaches a braided monoidal category (in fact, a ribbon category), denoted Rep q (T ). This is the category of representations of the quantum torus. In what follows we will assume that the value of q on every simple root of G is non-trivial (this is a non-degeneracy condition) The positive part. Since Rep q (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 e i (here the index i runs over the set of vertices of the Dynkin diagram of G); denote it U q(n) free, equipped with the tautological Hopf algebra structure. Another is the co-free co-associative co-algebra on the same generators, denote it U q(n) co-free. We have a tautological map of Hopf algebras (4.2) U q(n) free U q(n) co-free, which is neither injective nor surjective. We note that the (graded) duals of U q(n) free and U q(n) co-free are the corresponding Hopf algebras U q(n ) co-free and U q(n ) free, respectively. And the dual of (4.2) is the corresponding tautological map U q(n ) free U q(n ) co-free. We introduce the Kac-DeConcini algebra U q(n) KD as the quotient of U q(n) free by the quantum Serre relations. One shows that (4.2) factors via U q(n) free U q(n) KD U q(n) co-free. We introduce Lusztig s algebra U q(n) Lus as the graded dual of U q(n ) KD (it is here that we use the non-degeneracy assumption on q). Thus, we obtain that (4.2) factors as U q(n) free U q(n) KD U q(n) Lus U q(n) co-free. Now the resulting map U q(n) KD U q(n) Lus is an isomorphism if and only if q is not a root of unity, i.e., if q(ˇα i) is non-torsion for every simple root ˇα i. If this is not the case, we denote by u q(n) the image of the above map: U q(n) KD u q(n) U q(n) Lus, this is the positive part of the small quantum group. Let Rep q (B + ) be the full subcategory in U q(n) Lus -mod consisting of objects on which the action of U q(n) Lus is locally nilpotent.

16 16 PROGRAM Representations of the entire quantum group. We now introduce the various versions of the category of representations of the entire quantum group. In every case, this will be a braided monoidal category. The construction will always proceed by first defining the corresponding abelian category, and then taking its derived category. The easiest category to introduce is Rep q (G) mixed. This is the (relative to Rep q (T )) Drinfeld s center of the category Rep q (B + ), i.e., Rep q (G) mixed = Z Dr,Repq (T )(Rep q (B + )). This is the category of representations of the mixed quantum group: one whose positive part is U q(n) Lus (and which is required to act locally nilpotently), and who negative part is U q(n ) KD. Now the theory bifurcates depending on whether or not q is a root of unity. Suppose first that it is not. Then we can think of Rep q (G) mixed as the quantum version of category O, and we define Rep q (G) (at the abelian level) as the full subcategory of Rep q (G) mixed consisting of objects on which the action of U q(n ) := U q(n ) Lus U q(n ) KD is also locally nilpotent. The restriction functor Rep q (G) Rep q (B + ) is fully faithful (even at the derived level), and we can therefore alternatively define Rep q (G) as the full subcategory of Rep q (B + ), consisting of objects on which the action of U q(n) can be extended to a locally nilpotent action of U q(n ). Suppose now that q is a root of unity, i.e., all q(ˇα i) are torsion. In this case, we introduce a whole array of categories. We let u q(g)-mod be the (relative to Rep q (T )) Drinfeld s center of the category u q(n)-mod, i.e., u q(g)-mod := Z Dr,Repq (T )(u q(n)-mod). This is the category of modules over the (graded) small quantum group. We introduce Rep q (G) 2 1 (at the abelian level) as the full subcategory in Repq (G) mixed, consisting of objects on which the action of U q(n ) KD factors through U q(n ) KD u q(n ). Finally, we introduce Rep q (G) (at the abelian level) to consist of objects of Rep q (G) 2 1, equipped with an extension of the u q(n )-action to a U q(n ) Lus -action. Thus, we have a sequence of restriction functors Rep q (G) mixed Rep q (G) 1 2 Repq (G) u q(g)-mod. One shows that the restriction functor Rep q (G) 2 1 Repq (G) is fully faithful (even at the derived level) The quantum Frobenius. Lusztig s quantum Frobenius defines a short exact sequence of Hopf algebras 1 u q(n) U q(n) Lus U(n H) 1, where n H is the maximal unipotent in the reductive group H of Talk GL-4, and U(n H) is its usual universal enveloping algebra, which can be viewed as an object of Rep q (T ) via Rep(T H) Rep q (T ). Dually, we have a short exact sequence 1 O N + U q(n ) KD u q(n ) 1. H This gives rise to the following pieces of structure: (i) An action of the monoidal category QCoh(O N + / Ad(B + )) (with respect to the point-wise tensor H product) on Rep q (G) mixed and an identification (at the derived level): (4.3) Rep q (G) mixed QCoh(O + / Ad(B N H )) H QCoh(pt /B H) Rep q (G) 1 2 ;

17 PROGRAM 17 (ii) An action of the monoidal category QCoh(pt /B H) Rep(B H) on Rep q (G) 1 2 (at the derived level): (4.4) Rep q (G) 2 1 Rep(T H) u q(g)-mod; Rep(B H ) and an identification (iii) An action of the monoidal category on Rep q (G) and an identification (at the derived level): Rep q (G) Combining (ii) and (iii), we obtain an identification Rep(B H) Rep q (G) 1 2. (4.5) Rep q (G) Rep(T H) u q(g)-mod T-5: Tutorial on factorization vs braided monoidal categories. Factorization categories exist also in the context of topology. When our curve X is A 1, they correspond to braided monoidal categories. An additional structure on a braided monoidal category (known is a ribbon structure) allows to associate to it a factorization category on any oriented 2-manifold. A Fact top (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 the symbol RH. Thus, one can attach to a ribbon category A an algebro-geometric factorization category Fact alg-geom (A) := RH ( Fact top (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 in question is the category Rep q (T ) the category of representations of the quantum torus. We will see that the resulting factorization category Fact alg-geom (Rep q (T )) identifies with D-mod(Gr Ť )ˇ (or, which is equivalent by the FLE for tori, with KL(T ) ), where q = exp(2πi ˇ)). In the process of doing so we will see that the q that appears in the quantum group is best interpreted as a factorization gerbe on Gr Ť Ch-2: The factorization algebra Ω q (or now Nature encodes root data). in it. Consider the (ribbon) braided monoidal category Rep q (T ) and the Hopf algebras U q(n) free U q(n) KD u q(n) U q(n) Lus U q(n) co-free Let A be one of these Hopf algebras. In the talk we will explain that the functor inv A of derived A-invariants applied to the augmentation module defines an E 2-algebra in Rep q (T ), to be denoted Inv A. Moreover, the Koszul duality functor inv A : A-mod nilp Inv(A)-mod (here A-mod nilp is the category of locally nilpotent A-modules) naturally lifts to a functor (4.6) Z Dr,Repq (T )(A-mod nilp ) Inv A -mod E 2. Applying the procedure from Talk T-5, we obtain that Inv A gives rise to a factorization algebra in Fact top (Rep q (T )), to be denoted Ω(A), so that Inv A -mod E 2 Ω(A)-mod Fact,

18 18 PROGRAM and thus (4.6) can be interpreted as (4.7) Z Dr,Repq (T )(A-mod nilp ) Ω(A)-mod Fact. We will see that since A is graded by the monoid ˇΛ pos, the object Ω(A) lives inside a full subcategory of Fact top (Rep q (T )) that can be described very explicitly. Namely, we will introduce a configuration space Conf(X, ˇΛ neg ) consisting of Λ neg -colored divisors. We will see that the datum of q gives rise to a factorization gerbe G q on Conf(X, ˇΛ neg ), and the corresponding subcategory of Fact top (Rep q (T )) identifies with Shv Gq (Conf(X, ˇΛ neg )). Thus, we obtain a sequence of factorization algebras in Shv Gq (Conf(X, ˇΛ neg )): (4.8) Ω co-free q Ω Lus q Ω small q Ω KD q Ω free q An explicit description of the Ω q algebras. A crucial feature of G q is that it is canonically trivial over the open subscheme Conf(X, ˇΛ neg j ) Conf(X, ˇΛ neg ). So that Shv Gq ( Conf(X, ˇΛ neg )) is the usual (non-twisted) category of sheaves. We will see that the factorization algebra Ω co-free q (resp., Ω(U q(n) free )) is the!- (resp., *-) extension of the sign local system on Conf(X, ˇΛ neg ). Another important observation is that the assignment A Ω(A) Shv Gq (Conf(X, ˇΛ neg )) sends Hopf algebras that are concentrated in cohomological degree 0 to perverse sheaves, and moreover it sends injections/surjections of Hopf algebras to injections/surjections of perverse sheaves. This immediately implies that the factorization algebra Ω small q is the!*-extension of the sign local system on Conf(X, ˇΛ neg ). The most important of these algebras, namely, Ω Lus q is a certain perverse sheaf that is squeezed between the!-extension and the!*-extension. It is a fact of crucial importance that Ω Lus q can be described explicitly by induction, and this will be the central idea of this talk How do these Ω q-algebras control representation theory? Our overall task (one on which we stake our hopes to prove the FLE) is to express the category Rep q (G) in factorization terms, ideally, as a the category of factorization modules over a factorization algebra. 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) inv Uq(n) Lus : Rep q(g) mixed Ω Lus q -mod Fact ; (4.10) inv Uq(n) Lus : Rep q (G) 1 2 Ω Lus q -mod Fact ; (4.11) inv Uq(n) Lus : Rep q (G) ΩLus q -mod Fact ; (4.12) inv uq(n) : u q(g)-mod Ω small q -mod Fact ;

19 PROGRAM 19 (4.13) inv uq(n) : Rep q (G) Ω small q -mod Fact. The functors (4.9) and (4.12) are equivalences (up to renormalization, i.e., up to tweaking homological algebra a bit to be explained in the talk). However, the other functors, the most important of which is (4.11), are not. Nonetheless, we will explain that the failure of the functor (4.11) to be an equivalence is controllable using a factorization algebra denoted Ω cl constructed from the commutative DG algebra C (n H), which is the cohomological Chevalley algebra of the Lie algebra n H (see Talk T-4). This will allow us to give a description of Rep q (G) in factorization terms. Essentially, what we will say will amount to saying that the forgetful functor Rep q (G) Rep q (G) Rep(B H) Rep q (G) 1 2 is fully faithful, while Rep q (G) 1 2 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 constructions on the de Rham side of Riemann-Hilbert. Note that the symmetric monoidal category Rep(T H) maps to the E 3-center of Rep q (T ). We consider Ω cl as a factorization algebra in Fact top (Rep(T H)), and the quantum Frobenius defines a map Ω cl Ω Lus q. We can consider Ω Lus q as a factorization algebra in the factorization category of commutative Ω cl -modules in Fact top (Rep(T H)), the later being equivalent to Fact top (Rep(B H)). Let (Ω Lus q -mod Fact ) Ω cl -com denote the resulting category of modules, i.e., the category of factorization Ω Lus q -modules in Fact top (Rep(B H)). We have an equivalence The functor inv Uq(n) Lus (Ω Lus q -mod Fact ) Ω cl -com Rep q (G) 1 2. naturally upgrades to a functor (4.14) Rep q (G) (Ω Lus q -mod Fact ) Ω cl -com, and further to a functor Rep q (G) Rep(B H) (Ω Lus q -mod Fact ) Ω cl -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) inv Uq (n) Lus Rep q (G) Rep(B H) Rep(T H) (Ω Lus q -mod Fact ) Ω cl -com Rep(B H ) Rep(B H ) Rep(T H) u q(g)-mod inv uq (n) Ω small q -mod Fact.

20 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 corresponding category for T, and (try to) express the categories for G via these functors. In this talk, we will perform this procedure for the KL side. Predictably, it is here that the sign of the level will be play the most significant role, as the behavior of the category KL(G) depends on the level. Notational convention: From now on we will assume that is positive. denoted by Positive level case. We construct the functor (4.16) j KL, : KL(G) KL(T ) to be the composition convolution (4.17) KL(G) (D-mod(Gr G) ) L(N) L + (T ) KL(G) In the above formula, the first arrow, i.e., Negative levels will be (ĝ-mod ) L(N) L + (T ) ( t-mod ) L + (T ) = KL(T ). KL(G) (D-mod(Gr G) ) L(N) L + (T ) KL(G), is given by tensoring with the object 0,co (D-mod(Gr G) ) L(N) L + (T ), see Talk J-2. The third arrow, i.e., (4.18) J (ĝ-mod ) L + (T ) := (ĝ-mod ) L(N) L + (T ) ( t-mod ) L + (T ) is the functor C 2 (L(n), ) =: C Negative level case. (L(n), ) of semi-infinite cohomology with respect to L(n) := n((t)). The situation at the negative level is trickier. We construct the functor (4.19) j KL,! : KL(G) KL(T ) to be the composition (4.20) KL(G) (D-mod(Gr G) ) L(N) L+ (T ) convolution KL(G) Here the first arrow, i.e., (ĝ-mod ) L(N) L+ (T ) ( t-mod ) L+ (T ) KL(T ). KL(G) (D-mod(Gr G) ) L(N) L+ (T ) KL(G) is given by tensoring with 0 (D-mod(Gr G) ) L(N) L+ (T ), see Talk J-2. The third arrow is a non-standard functor C 2! (L(n), ) : J! (ĝ-mod ) L+ (T ) := (ĝ-mod ) L(N) L+ (T ) ( t-mod ) L+ (T ). In the talk we will describe the functor C 2! (L(n), ) via the equivalence (ĝ-mod ) L(N) L+ (T ) (ĝ-mod ) I, and also its incarnation on the other side of Riemann-Hilbert.

21 PROGRAM Factorization description. One can regard the following conjecture as a way to describe the category (ĝ-mod ) L(N) L + (T ) (resp., (ĝ-mod ) L(N) L+ (T ) ) essentially combinatorially. Denote by Ω KM,Lus (resp., Ω KM,Lus ) the factorization algebra in KL(T ) (resp., KL(T ) ) equal to j KL, (V 0 ) (resp., j KL,!(V 0 )), where V 0 denotes the vacuum module at the given level. We propose: Conjecture 18. (a) The functor C 2 (L(n), ) defines an equivalence (ĝ-mod ) L(N) L + (T ) Ω KM,Lus -mod Fact. (b) The functor C 2! (L(n), ) defines an equivalence and Parallel to Talk Ch-2, the composite functors KL(G) := ĝ-mod L+ (G) KL(G) := ĝ-mod L+ (G) (ĝ-mod ) L(N) L+ (T ) Ω KM,Lus -mod Fact. (ĝ-mod ) L(N) L + (T ) Ω KM,Lus -mod Fact (ĝ-mod ) L(N) L+ (T ) Ω KM,Lus -mod Fact are not equivalences. But the failure to be such is controllable, using the same factorization algebra Ω cl, by a procedure outlined at the end of Talk Ch-2. We will return to this in Talks FLE-2 and FLE Relation to the Kazhdan-Lusztig equivalence. We propose: Conjecture 19. The factorization algebra Ω Lus q talk T-5) to Ω KM,Lus (resp., Ω KM,Lus ). (resp., Ω Lus q 1 ) corresponds under Riemann-Hilbert (see Assuming this conjecture, we obtain that Conjecture 18 can be reformulated as follows: Conjecture 20. (a) The category Rep q (G) mixed corresponds under Riemann-Hilbert to J (ĝ-mod ) L + (T ) := (ĝ-mod ) L(N) L + (T ). (b) The category Rep q 1(G) mixed corresponds under Riemann-Hilbert to J! (ĝ-mod ) L+ (T ) := (ĝ-mod ) L(N) L+ (T ). In the talk we will explain that the usual Kazhdan-Lusztig equivalence may be interpreted as a statement that the category Rep q 1(G) corresponds under Riemann-Hilbert to KL(G) (this equivalence is compatible with the t-structures and send Weyl modules to Weyl modules). Applying duality, we obtain that Rep q (G) corresponds under Riemann-Hilbert to KL(G). However, the latter equivalence is not t-exact and sends the dual Weyl module with highest weight ˇλ to the Weyl module with highest weight w 0(λ). The equivalences of Conjecture 20 are supposed to make the following diagrams commute Rep q (G) Rep q (G) mixed KL(G) (ĝ-mod ) L(N) L + (T )

22 22 PROGRAM and Rep q 1(G) Rep q 1(G) mixed KL(G) (ĝ-mod ) L(N) L+ (T ), where the lower horizontal arrows are the functors appearing in formulas (4.17) and (4.20), respectively Wh-4: Duality for W-algebras. The upper portion of the diagram in Conjecture 11 implies that that the equivalence L loc supposed to send ĝ-mod L(G)-mod (D-mod(L(Ǧ)) ˇ)L(Ň),χ L(Ǧ)-mod ˇ, of (1) is where (L(N), χ)-invariants are taken with respect the action of L(Ň) on l(ǧ) by right multliplication. Applying to this the lower portion of the diagram in Conjecture 11 we obtain an equivalence ( Whit(ĝ-mod ) KM (D-mod(L(Ǧ)) ˇ)L(Ň),χ). However, we have (tautologically, to be explained in the talk): ( KM (D-mod(L(Ǧ)) ˇ)L(Ň),χ) Whit( ǧ-modˇ). Hence, we obtain an equivalence of factorization categories (4.21) Whit(ĝ-mod ) Whit( ǧ-modˇ). 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 W g, as the BRST reduction of the Kac-Moody chiral algebra at level. There is a (tautologically) defined functor (4.22) Whit(ĝ-mod ) W g,-mod Fact, 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) W g, W ǧ,ˇ, due to Feigin and Frenkel. One actually has to be a little more careful here: the category W g,-mod Fact that appears in (4.22) is a renormalized version of the naive category of factorization W g,-modules. So one needs also to see that there renormalizations match up on the two sides of (4.23) Compatibility with FLE. Note now that the category Whit(ĝ-mod ) comes equipped with a tautological functor Pre-composing with we obtain a functor Whit(D-mod(L(G)) ) ĝ-mod Whit(ĝ-mod ). Whit(D-mod(Gr G) ) KL(G) Whit(D-mod(L(G)) ) ĝ-mod, (4.24) Whit(D-mod(Gr G) ) KL(G) Whit(ĝ-mod ). We propose: