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THE TAMELY RAMIFIED FUNDAMENTAL LOCAL EQUIVALENCE

AT INTEGRAL LEVEL

JUSTIN CAMPBELL AND GURBIR DHILLON

Abstract. Let G be an almost simple algebraic group with Langlands dual G, and fix anoncritical integral level κ for G, with dual level κ for G. We prove an equivalence betweenκ-twisted Whittaker D-modules on the affine flag variety of G and affine Category O forG at level −κ, as conjectured by Gaitsgory. To do so, we prove an affine version ofMilicic-Soergel’s Whittaker localization of blocks of Category O.

1. Introduction

Let G be an almost simple complex algebraic group with Langlands dual G. The Funda-mental Local Equivalence is a conjectural identification, proposed by Gaitsgory and Lurie,between the category of twisted Whittaker D-modules on the affine Grassmannian for Gand the category of representations at the dual level of the affine Lie algebra of G, in-tegrable for its arc group [9]. This conjecture provides a deformation of the GeometricSatake equivalence to all Kac-Moody levels, and is expected to play a similarly fundamen-tal role in the quantum geometric Langlands program as the Satake equivalence in theusual Langlands correspondence.

The Fundamental Local Equivalence is a conjecture of unramified nature, in that theD-modules live on the quotient of loop group of G by its arc group, and the representationsof the affine Lie algebra of G are integrable for its arc group. Gaitsgory has conjectured atamely ramified Fundamental Local Equivalence, wherein one replaces arc subgroups withIwahori subgroups [11]. In this paper we prove the latter conjecture under an integralityassumption on the level.

2. Statement of Results

To formulate the Fundamental Local Equivalence precisely, we introduce some notation.We begin with the twisted Whittaker D-modules on the affine flag variety. Write g for theLie algebra of G, LG for its algebraic loop group, and FlG for its affine flag variety. Thereis a canonical embedding

(g∗ ⊗ g∗)G −→ Pic(FlG)⊗Z C,which sends the Killing form to the line bundle coming from the Tate extension of LG. Ac-cordingly, to an invariant bilinear form κ one may associate the corresponding cocompletedg-category of twisted D-modules Dκ(FlG).

Date: Spring 2018.

1

2 JUSTIN CAMPBELL AND GURBIR DHILLON

To form its Whittaker subcategory, for a Borel B of G write N for its unipotent radical.For a nondegenerate character ψ of LN of conductor zero, we will take the associated co-complete dg-category of (LN,ψ)-equivariant twisted D-modules. Writing I for the Iwahorisubgroup of LG corresponding to B, we will write this category as

Dκ(I\LG/LN,ψ) ' D−κ(FlG)LN,ψ.

We refer the reader to Section 4 of [5] for the definition of the category of invariants for agroup ind-scheme such as LN .

We next describe the Langlands dual side, i.e. Iwahori-integrable modules for theaffinization of the dual Lie algebra. Write g for the Lie algebra of G, and LG for itsloop group. For an invariant bilinear form κ on g, consider the affine Lie algebra gκ, andits renormalized dg-category of representations gκ -mod. Let I be the Iwahori subgroup ofLG corresponding to I, and consider the corresponding category of I-equivariant objects

gκ -modI .

Finally, recall that one may identify Kac-Moody levels, i.e. the lines of invariant bilinearforms, for G and G. Namely, writing Wf for their Weyl group, we have:

(g∗ ⊗ g∗)G ' (h∗ ⊗ h∗)Wf ' (h⊗ h)Wf ' (g∗ ⊗ g∗)G.

We will follow the convenient practice in the subject of incorporating into the above equiv-alence a shift by the critical levels κc for G and κc for G, as we spell out more carefully inSection 7. For a level κ for G, write κ for the corresponding level of G. We may now statethe tamely ramified Fundamental Local Equivalence, as conjectured by Gaitsgory.

Conjecture 2.1. [11] For any nonzero κ, there is an equivalence:

Dκ(I\LG/LN,ψ) ' gκ -modI . (2.2)

In this paper we prove Conjecture 2.1 under an integrality assumption on the level.Recall the basic level κb for g, which gives the short coroots of g squared length two.

Theorem 2.3. Suppose that κ is a nonzero integral multiple of κb. Then Conjecture 6.4is true for κ.

To our knowledge, Theorem 2.3 gives the first known cases of Conjecture 2.1. We nowbriefly indicate the structure of the argument. A basic pattern in proving equivalencesin the Langlands program is that, given the combinatorial nature of Langlands duality,one often shows that both sides of the proposed equivalence admit the same combinatorialdescription. The Category O of an affine Lie algebra is known to admit such a Coxeter-theoretic description, via Kazhdan-Lusztig theory and Soergel modules, and we will usethis to provide the desired combinatorial control of both sides of (2.2).

It is quite plausible that the Kac-Moody side of (2.2) should be controlled by Cate-gory O, provided one handles issues of cocompletion, compactness, and renormalizationappropriately. Accordingly, we show in Theorem 5.15 that for negative κ it is roughly theind-completion of the bounded derived category of O. While this would be false for positiveκ, we may instead reduce to the negative level case via Kac-Moody duality.

THE TAMELY RAMIFIED FUNDAMENTAL LOCAL EQUIVALENCE AT INTEGRAL LEVEL 3

The Whittaker side of (2.2) looks superficially more distant from Category O than theKac-Moody side. For semisimple Lie algebras, it is known that integral blocks of itsCategory O may be realized as holonomic partial Whittaker sheaves on its flag variety[12], [16], [17], [22]. However, a generalization to affine type is not straightforward, dueto the unavailability of a center for noncritical affine algebras. Indeed, even the specialcase of identifying a regular block of Category O at a negative integral level with Schubert-constructible perverse sheaves on the affine flag variety has not appeared explicitly in theliterature.

Nonetheless, based on ongoing work of the second named author on the geometric rep-resentation theory of affine W-algebras, we were led to suspect that the relation betweenintegral blocks of O and partial Whittaker sheaves should persist in affine type, and indeedit does.

To state this precisely, suppose G is simply connected. Write I for the prounipotentradical of I, and let χ be an additive character of I. One may form the category of partialWhittaker sheaves on the affine flag variety

D(I\LG/I, χ).

Next, we introduce the relevant Kac–Moody representations. Let µ be an integral antidom-inant weight for the affinization of g, and consider the block Oµ of Category O containingthe simple module with highest weight µ. Suppose that χ and µ are compatible, in thatthe affine simple roots on which χ is nonzero coincide with the affine simple roots whoseassociated reflections stabilize µ. Then we have

Theorem 2.4. There is a canonical up to scalars equivalence between the compact objectsin the heart of the Whittaker category and the corresponding block of affine Category O

D(I\LG/I, χ)♥,c ' Oµ. (2.5)

Let us mention in passing that Theorem 2.4 has nice applications to (i) relations betweentranslation functors and nearby cycles conjectured by the first named author, new alreadyin finite type, (ii) localization of singular blocks of affine Category O on affine partial flagvarieties, and (iii) a relation between Drinfeld–Sokolov reduction and translation functorsconjectured by the second named author. These will be explained elsewhere.

To prove Theorem 2.3, our assumption on the level κ is precisely that the twist onthe Whittaker side is trivial. Moreover, by a theorem of Raskin, one may identify the(LN,ψ)-and (I ,Adt−ρ ψ)-invariants of FlG, after which we may use Theorem 2.4 to identifythe Whittaker sheaves coming from each connected component of FlG with Kac-Moodyrepresentations. It is reasonable to suppose a more involved variant of this argument couldapply to the general case of Conjecture 2.1, and this is the subject of ongoing work of thesecond named author.

Finally, as we explain in Conjecture 7.14, the Coxeter-theoretic combinatorics we utilizemay be understood as relatively lossless projections of a local geometric Shimura corre-spondence, a perspective which to our knowledge is new even in finite type.


Organization of the paper. Sections 3-6 are devoted to a proof of Theorem 2.4. Namely,in Section 3, we give an argument for the case of nondegenerate Whittaker characters infinite type which makes the Hecke equivariance manifest. In Section 4, we discuss compactgenerators for the Whittaker category, and analyze the action of the affine Hecke categoryon the vacuum object. In Section 5, we provide a similar analysis for the Iwahori equivariantcategory of gκ -mod. In Section 6, we use the results of Sections 4 and 5 to bootstrap theresult of Section 3 to a proof of Theorem 2.4. Finally, in Section 7, we prove Theorem 2.3.

Conventions regarding dg-categories. In this paper we understand “dg-category”to mean “C-linear stable (∞, 1)-category.” We denote by DGCat the (∞, 1)-category ofidempotent-complete dg-categories with exact C-linear functors, and write DGCatcont forthe (∞, 1)-category of cocomplete (presentable) dg-categories with colimit-preserving func-tors. All limits and colimits are to be understood in the sense of (∞, 1)-categories.

We occasionally make use of the Lurie tensor product on DGCatcont. Recall that for twoobjects C and D, the dg-category C⊗D is characterized by the universal property that aC-linear colimit-preserving functor C⊗D→ E is equivalent to a functor C×D→ E whichis C-linear and preserves colimits in each variable separately.

Acknowledgments. It is a pleasure to thank Roman Bezrukavnikov, Dan Bump, MichaelFinkelberg, Dennis Gaitsgory, Sam Raskin, Ben Webster, David Yang, and Zhiwei Yun formany helpful conversations. Part of this work was done when the second named authorvisited the first author named at Caltech in Fall 2018, and later when both visited MSRIduring Spring 2019. We are grateful to both institutions for their hospitality and pleasantworking conditions.

3. The categorical sign representation

Let L be a finite dimensional reductive group with Lie algebra l with fixed triangulardecomposition l = n− ⊕ h ⊕ n. Write B for the Borel of L with Lie algebra h ⊕ n, and Nfor its unipotent radical. Write αi, αi, i ∈ I, for the simple roots and coroots of l.

Recall that the Hecke algebra associated to L is a Z[q±1/2] algebra with standard gen-erators Ti, i ∈ I, satisfying a braid relation and the quadratic relation

(Ti + 1)(Ti − q) = 0, i ∈ I.

Accordingly, it has a sign representation Csgn, on which each Ti acts as -1. In this section,we explain that there are two equivalent incarnations of the categorical sign representationVectsgn of the categorical Hecke algebra D(B\L/B). Namely, we will see it roughly as amaximally singular block of Category O or as nondegenerate Whittaker modules on theflag variety.

Let us introduce the Whittaker side. Let N− denote the connected subgroup of L withLie algebra n−, and recall its one parameter subgroups Ni, i ∈ I, whose Lie algebras arethe negative simple root spaces. Fix a nondegenerate additive character ψ : N− → C, i.e.one whose restriction to each Ni is nonzero. We may accordingly form the category of


Whittaker D-modules on the flag variety of L, i.e.

D(B\L/N−, ψ) := D(B\L)N−,ψ,

which is a full subcategory of D(B\L) because N− is unipotent. This carries an action ofD(B\L/B) via left convolution. As a category, it is equivalent to Vect, with generator Wψ

its unique indecomposable object which lies in the heart D(B\L)♥.Let us describe the singular side. Write Z for the center of the universal enveloping

algebra of l, and for a central character χ : Z → C write l -modχ for the category of lmodules with generalized central character χ. I.e., this is the full subcategory of l -modconsisting of objects whose cohomology groups are all set theoretically supported on χwhen restricted from l to Z. For any χ, l -modχ carries a canonical action of D(L), and

hence its B equivariant category l -modBχ carries a canonical action of D(B\L/B). Recall

the Harish–Chandra isomorphism Z ' Sym hWf , and the resulting identification of centralcharacters with dot orbits of Wf on h∨. Let χ be maximally singular, i.e. correspondingto a singleton orbit which we by abuse of notation still denote by χ. Writing ρ for the halfsum of the positive roots, we may take χ = −ρ, and if l is semisimple this is the uniquechoice. Then l -modBχ as a category is canonically equivalent to Vect, with generator the

unique Verma module Mχ := indlbCχ in the block.

Theorem 3.1. For χ maximally singular, there is a t-exact equivalence of D(B\L/B)modules:

D(B\L/N−, ψ) ' l -modBχ .

This theorem is originally due to Milicic-Soergel [17] via a different argument, wherethey use the language of Harish-Chandra modules rather than the Hecke category.

Proof. We will produce a D(B\L/B)-equivariant map from D(B\L/N−, ψ) to l -modBχ ,and check it sends our generators to one another. To produce the desired functor, we willprovide a D(L)-equivariant functor

F : D(L/N−, ψ) −→ l -modχ

and then take B-invariants. By Corollary 2.4.6 in [5], the category D(L/N−, ψ) can alsobe realized as Whittaker coinvariants, i.e. for any D(L)-module C we have a canonicalequivalence

HomD(L) -mod(D(L/N−, ψ),C) ' CN−,ψ. (3.2)

To define F , it therefore suffices to specify an object of l -modN−,ψ

χ , which we construct

as follows. Consider the universal Whittaker module M(ψ) := Indln− Cψ, which lies in

l -modN−,ψ. Recall that, as with any central character, the forgetful functor from l -modχ

to l -mod admits a continuous right adjoint:

Oblvχ : l -modχ � l -mod : Ri!χ.


These functors restrict to an adjunction between the Whittaker subcategories, and we setM(ψ, χ) := Ri!χM(ψ).

It remains to argue that the resulting functor FB : D(B\L/N−, ψ) → l -modBχ is an

equivalence. It suffices to show that FB sends Wψ to Mχ, up to a cohomological shift.Writing δe for the delta D-module on the identity of L, by the construction of (3.2) wehave F (AvN−,ψ,∗ δe) 'M(ψ, χ). It follows that

FB(AvB,∗AvN−,ψ,∗ δe) ' AvB,∗ F (AvN−,ψ,∗ δe) ' AvB,∗M(ψ, χ).

To calculate the latter, note that

Homl -modBχ(Mχ,AvB,∗M(ψ, χ)) ' Homl -modχ(Mχ, Ri

!χM(ψ))

' Homl -mod(Mχ,M(ψ))

' Homb -mod(Cχ,Resbl M(ψ)) ' C[−dimB],

where in the last step we use that M(ψ) is a free rank one U(b)-module. Noting thatAvB,∗AvN−,ψ,∗ δe 'Wψ[−dimN ], it follows that the functor associated to M(ψ, χ)[dim h]produces the desired t-exact equivalence. �

4. Partial Whittaker sheaves

In this section, we study partial Whittaker sheaves on the affine flag variety. We wouldlike to argue that (i) these form a highest weight category, and (ii) all the (co)standardobjects can be reconstructed from the one with the smallest support via the action of theaffine Hecke algebra.

However, since for (i) we are not working with an abelian category of holonomic objects,but rather a renormalized unbounded dg derived category of arbitrary objects, what wewill actually prove is that it can be reconstructed from the compact objects in the heartof the t-structure, which indeed form a highest weight category.

4.1. Whittaker sheaves on ind-schemes. The argument for (i) holds rather generally.So for the moment, let X be any finite type algebraic variety equipped with an actionof a prounipotent group scheme U with only finitely many orbits. Accordingly, X has astratification

X =⊔λ∈Λ

Cλ

by the orbits Cλ, λ ∈ Λ, and by prounipotence each orbit is an affine space. For anyadditive character χ : U → Ga, consider the Whittaker category D(X)U,χ, which is a fullsubcategory of D(X) by prounipotence of U .

Proposition 4.1. Fix a stratum Cλ. Then D(Cλ)U,χ is nonzero if and only if χ vanisheson the stabilizer of c for any point c ∈ Cλ. If D(Cλ)U,χ is nonzero, then it admits a t-exactequivalence with Vect, and compact objects in D(Cλ)U,χ are lisse.


Proof. This statement is well-known, but due to its importance for us we remind the readerof the proof. Choose c ∈ Cλ and denote its stabilizer in U by Uc. Taking the fiber at cinduces an equivalence

D(Cλ)U,χ ' VectUc,χ,

where the action of Uc on Vect is the trivial one. Since Uc is connected, if χ|Uc is nonzerothen VectUc,χ = 0. On the other hand, since Uc is prounipotent, if χ|Uc = 0 then we have anequivalence VectUc,χ ' Vect. In the latter case, the resulting equivalence Vect ' D(Cλ)U,χ

can be normalized to send C to χ! exp[1− dimCλ], where χ : Cλ → Ga is the smooth mapinduced by χ and exp denotes the exponential D-module on Ga, placed in cohomologicaldegree zero. �

Let Λ ⊂ Λ index the strata Cλ with D(Cλ)U,χ nonzero. For each λ ∈ Λ, write jλ,!, jλ,∗for the ! and ∗ extensions, respectively, of the indecomposable object in D(Cλ)U,χ,♥. Notethat Λ carries a natural partial order coming from orbit closures, i.e. for λ, ν ∈ Λ, we haveλ 6 ν if and only if Cλ lies in the closure of Cν .

Theorem 4.2. Write A ⊂ D(X)U,χ,♥ for the full subcategory consisting of coherent D-modules. Then

(1) A is a highest weight category with standard objects jλ,! and costandard objectsjλ,∗, λ ∈ Λ.

(2) The tautological map A → D(X)U,χ,♥ induces a fully faithful embedding Db(A) →D(X)U,χ.

(3) The above embedding exhibits D(X)U,χ as the ind-completion of Db(A). I.e., D(X)U,χ

is compactly generated with compact objects given by the essential image of Db(A).

Proof. For (1), we observe that since U acts on X with finitely many orbits, any object ofA is automatically holonomic. In particular A has finite length, and its simple objects arethe intermediate extensions jλ,!∗ for λ ∈ Λ.

Checking the remaining conditions of A being highest weight, as enumerated in Section3.2 of [4], are straightforward, except for possibly the vanishing of Ext2

A(jλ,!, jν,∗) for λ, ν ∈Λ. However, this injects into H2 RHomD(X)U,χ(jλ,!, jν,∗), and it is straightforward to seethat

RHomD(X)U,χ(jλ,!, jν,∗) ' RHomD(X)(jλ,!, jν,∗) '

{C λ = ν,

0 λ 6= ν.(4.3)

For (2), one can use the presentation of Db(A) as the homotopy category of boundedcomplexes of tilting objects (as in [2]) and Equation (4.3). Equivalently, one can use thatthe jλ,!, λ ∈ Λ generate Db(A), as do the jλ,∗, λ ∈ Λ.

For (3), it suffices to check that the jλ,!, λ ∈ Λ, generate D(X)U,χ,♥. Suppose M is an

object of D(X)U,χ,♥ such that RHom(jλ,!,M) ' 0, λ ∈ Λ. Taking a Cλ open in X, it followsby adjunction from Proposition 4.1 that the restriction of M to Cλ vanishes. Hence M ispushed forward from the complement of Cλ, and we finish by induction on the number ofstrata. �


Suppose now X is an ind-scheme presented as a filtered colimit lim−→αXα of finite type

schemes under closed embeddings. In this case, we have

D(X) = lim−→D(Xα),

where the colimit is taken in DGCatcont with the transition functors given by direct image.Suppose that X carries an action of a prounipotent group U such that each Xα is U -stable. Recall from [5] Section 4 that for a category C acted on by D(U), there is acanonical equivalence between invariants and coinvariants CU ' CU , and similarly fortwisted (co)invariants. In particular, taking U -invariants commutes with colimits, andhence

D(X)U,χ = (lim−→D(Xα))U,χ ' lim−→(D(Xα)U,χ).

Suppose that U acts on each Xα with only finitely many orbits. In this case, let Λαindex the orbits on Xα supporting Whittaker sheaves, and set Λ := lim−→Λα. Λ carries apartial order induced from those of the Λα, i.e. the closure relation on strata, and for anyλ ∈ Λ the collection {ν ∈ Λ : ν 6 λ} is finite. For each λ ∈ Λ, consider as before the !- and∗- extensions jλ,!, jλ,∗, λ ∈ Λ, and further consider the indecomposable tilting object Tλ.

Proposition 4.4. For each α, write Aα for the subcategory of D(Xα)U,χ,♥ defined inTheorem 4.2. Then for each α → β, the map Db(Aα) → Db(Aβ) is fully faithful, and thetautological functor

lim−→Db(Aα) −→ D(X)U,χ

exhibits the latter as the ind-completion of the former, where the colimit is taken in DGCat.In particular, D(X)U,χ,c is equivalent to the bounded homotopy category of tilting objectsTλ, λ ∈ Λ.

Proof. That Db(Aα) → Db(Aβ) is fully faithful can be seen by (i) considering Aα as a‘closed’ highest weight subcategory of Aβ, or (ii) via the Kashiwara lemma and Theorem4.2(2).

To see the compact generation claim, given a diagram of compactly generated dg-categories Cα such that the transition maps iα,β preserve compactness, its colimit taken inDGCatcont is canonically equivalent to the ind-completion of the colimit of the compactobjects lim−→Ccα taken in DGCat. The claim about tiltings follows from the explicit form of

colimits in DGCat, cf. [21], and the analogous claim for each Aα. �

4.2. Whittaker sheaves on the affine flag variety. In this subsection, g = n−⊕ h⊕ nwill denote a Kac-Moody algebra of finite or untwisted affine type, though everythingshould apply mutatis mutandis to the general case. Write G for the corresponding group,B for its Borel subgroup with Lie algebra h⊕ n, and B\G for its flag variety. Denote thesimple roots by

αi, i ∈ I.

For a subset of I of finite type, write l for the corresponding Levi subalgebra, L for thecorresponding Levi subgroup, and BL for its standard Borel. Write W for the Weyl groupof G, WL for the Weyl group of l, and denote the longest element of the latter by wL◦ .


As before, write N for the prounipotent radical of B, and consider its conjugate

N−L := wL◦NwL◦ .

An additive character ψ : N−L → Ga is at most nonzero on the simple root one-parameter

subgroups wL◦NαiwL◦ , i ∈ I. Fix a ψ which is nonzero precisely on the negative simple roots

of L. We will be concerned with the partial Whittaker category

D(B\G/N−L , ψ) := D(B\G)N−L ,ψ.

Recall that the orbits of N−L on B\G are given by the finite dimensional Schubert cells

Cw := BwN−L , w ∈W.

From Proposition 4.1, it follows that Cw supports a Whittaker sheaf if and only if w is ofminimal length in its right WL-coset. Let us write WL for the set of such minimal lengthcoset representatives, and write

jψw,!, jψw,∗, w ∈WL,

for the corresponding standard and costandard objects. Note that jψe,! ' jψe,∗, where e

denotes the identity element of W , and write δψ for this simple object.The Hecke algebra D(B\G/B) acts on the partial Whittaker category by left convolution

D(B\G/B)⊗D(B\G/N−L , ψ) −→ D(B\G/N−L , ψ), M ⊗N −→M ?N.

For w ∈W , write jw,!, jw,∗ for the the respective extensions of the constant D-module fromBwB.

Proposition 4.5. For any w ∈WL we have:

(1) jw,! ? δψ ' jψw,!.

(2) jw,∗ ? δψ ' jψw,∗.

(3) Under the equivalences of (1) and (2), applying − ? δψ to the canonical map

jw,! −→ jw,∗

yields a nonzero map jψw,! → jψw,∗.

Proof. By the assumption that w ∈WL, the relevant convolution morphism between Schu-bert cells is an isomorphism, cf. [1] for more details in an ostensibly special case. �

5. Affine Category O and Harish–Chandra modules

In this section, first we perform the comparison between Category O and Harish–Chandramodules for the Iwahori. Second, we show the linkage principle is a decomposition as affineHecke algebra modules. Note this assertion is non-trivial in affine type due to the absenceof central characters.


5.1. Compact generators for Harish–Chandra modules. Let κ be any non-criticallevel, and gκ the corresponding affine Lie algebra. Write gκ -mod for its renormalizedcocomplete dg-category of representations, which carries an action of Dκ(LG). For acompact open subgroup H of LG, we may accordingly form the invariants category

gκ -modH .

For our application, we are interested in gκ -modI , for I the Iwahori subgroup of LG. Inthe next subsections we will use the following assertion

Proposition 5.1. Write Λ for the character lattice of I. Then the Verma modules

Mλ, λ ∈ Λ,

are compact generators of gκ -modI .

The reader happy to take Proposition 5.1 on faith may wish to skip the remainderof this subsection. We will deduce Proposition 5.1 from a more general presentation ofgκ -modH , stated in unpublished notes of Gaitsgory [10]. Consider the abelian category ofHarish–Chandra modules. Within its unbounded derived dg-category, take the pretrian-gulated envelope of the representations induced from finite-dimensional representations ofH. Denote its ind-completion by

(gκ, H) -mod .

Theorem 5.2. There is a canonical equivalence

(gκ, H) -mod ' gκ -modH .

In the remainder of this subsection, we give a proof of Theorem 5.2, largely followingthe argument sketched in loc. cit. Write H as an inverse limit

H = lim←−i

Hi, i ∈ Z>0,

where the Hi are finite dimensional and each morphism is surjective with unipotent kernel.Its renormalized category of representations is the colimit in DGCatcont of the categoriesof representations of the Hi under inflation:

Rep(H) = lim−→Rep(Hi).

Denote the unbounded derived category of its abelian category of representations byRep(H)naıve, and note that there is a tautological map

Rep(H) −→ Rep(H)naıve. (5.3)

Lemma 5.4. Consider the pretriangulated envelope of the finite-dimensional representa-tions of H taken in Rep(H)naıve. Then the morphism (5.3) identifies its ind-completionwith Rep(H).


Proof. Note that for each i, Rep(Hi) is compactly generated by its finite dimensionalrepresentations. Since the inflation maps preserve compactness, it follows that Rep(H) iscompactly generated by the finite dimensional representations of H. It remains to checkthat (5.3) is fully faithful on such objects, for which we give two arguments. First, thisfollows from using the cobar complex and the identification OH ' lim−→OHi .

Second, we can use the following result of Raskin.

Theorem 5.5. Let Ci, i ∈ I, be a filtered diagram of cocomplete dg-categories equipped witht-structures. Assume for each i→ j that the corresponding morphism

Ci −→ Cj

is t-exact and admits a continuous right adjoint, and set

C := lim−→i

Ci.

Then there is a unique t-structure on C which is compatible with filtered colimits and suchthat each insertion Ci → C is t-exact. Moreover suppose that for each i ∈ I, the canonicalmap

D(C♥i )+ −→ C+i

is an equivalence. Then the same is true for C+ provided that I is countable.

In our situation, it is straightforward to see that (5.3) is t-exact and restricts to anequivalence

Rep(H)♥ −→ Rep(H)naıve,♥.

By Raskin’s theorem, it follows that (5.3) is an equivalence on bounded below objects, andin particular fully faithful on the compact generators of Rep(H). �

Writing h for the Lie algebra of H, and writing hi for the Lie algebra of Hi, we have

h = lim←−i

hi.

For each i, write hi -mod for the cocomplete dg-category of representations of hi. Therenormalized category of smooth representations of h is the colimit of those of the hi underinflation:

h -mod = lim−→ hi -mod .

Denote the unbounded derived category of the the abelian category of smooth representa-tions of h by h -modnaıve. As before, there is a tautological functor

h -mod −→ h -modnaıve . (5.6)

Lemma 5.7. Consider the pretriangulated envelope of the finitely generated smooth repre-sentations of h in h -modnaıve. Then the morphism (5.6) identifies its ind-completion withh -mod.

Proof. Similar to the second argument of Lemma 5.4. �


Note that each hi -mod carries an action of D(H), and hence so does h -mod. In partic-ular, we can form the invariant category

h -modH .

Lemma 5.8. There is a canonical equivalence

h -modH ' Rep(H).

Proof. Since taking invariants for a group scheme commutes with colimits, we have

h -modH ' lim−→(hi -modH).

Fix an i, and write Ui for the kernel of the projection H → Hi. Since Ui acts trivially onhi -mod and is prounipotent, we have

hi -modH ' (hi -modUi)Hi ' hi -modHi ' Rep(Hi).

Under the above equivalences, the transition maps identify with inflation, hence

h -modH ' lim−→(hi -modH) ' lim−→Rep(Hi) ' Rep(H),

as desired. �

Recall that h -mod and gκ -mod carry t-structures such that their bounded below partscoincide with those of the unrenormalized categories. In particular, there is an adjunction

Ind : h -mod+ � gκ -mod+ : Oblv . (5.9)

Lemma 5.10. The adjunction (5.9) induces an adjunction

Ind : h -mod� gκ -mod : Oblv (5.11)

such that Oblv is conservative.

Proof. To obtain the full adjunction, one may consider the composite

h -modc −→ h -mod+ −→ gκ -mod+ −→ gκ -mod

and ind-extend. To see that Oblv is conservative, note that h contains a basis uj of open

subalgebras about 0 in gκ. Applying Ind to Indhui C therefore yields a collection of compact

generators for gκ -mod, which is equivalent to the conservativity of its right adjoint. �

We will show in the appendix that

Lemma 5.12. Both functors in (5.11) carry a canonical datum of D(H)-equivariance.

By the lemma, (5.11) induces an adjunction

Ind : h -modH � gκ -modH : Oblv .

Via Lemma 5.8, we may rewrite this as

Ind : Rep(H)� gκ -modH : Oblv . (5.13)


Finally, let us turn to Harish–Chandra modules. Write (gκ, H) -modnaıve for the un-

bounded derived category of the abelian category (gκ, H) -mod♥. There is an adjunction

Ind : Rep(H)naıve � (gκ, H) -modnaıve : Oblv .

By the definition of the renormalization of Harish-Chandra modules, (gκ, H) -modc is thepretriangulated envelope of the image of Rep(H)c. As Oblv is t-exact, it follows that(gκ, H) -modc consists of almost perfect objects in (gκ, H) -modc, i.e. if an object of(gκ, H) -modc belongs to (gκ, H) -mod≥−n, then it is compact in (gκ, H) -mod≥−n. Inparticular, we have a canonical identification

(gκ, H) -mod+ ' (gκ, H) -modnaıve,+ .

By an argument to similar to Lemma 5.10, we obtain an adjunction

Ind : Rep(H)� (gκ, H) -mod : Oblv (5.14)

wherein Oblv is conservative. Now observe that the monads on Rep(H) coming from(5.11), (5.14) canonically identify. Hence Theorem 5.2 follows by Barr-Beck-Lurie.

5.2. Category O and Harish–Chandra modules. Fix a negative level κ, i.e.

κ /∈ κc + Q>0κb.

Consider the corresponding affine Lie algebra gκ and its usual abelian category O. Re-call that Λ denotes the character lattice of the Iwahori, and within O consider the fullsubcategory OΛ of objects whose weights lie in Λ. Note that there is a tautological map

OΛ −→ gκ -modI,♥ .

We will now explain how to reconstruct the entire renormalized derived category of Harish-Chandra modules from this embedding.

For λ ∈ Λ, recall the Verma module Mλ, its simple quotient Lλ, the dual Verma Aλ,and the indecomposable tilting module Tλ. This induces a partial order on Λ, where λ 6 νif Lλ is a subquotient of Mν . Introduce the basic closed sets

Λ6ν = {λ ∈ Λ : λ 6 ν}, ν ∈ Λ.

A subset Λ′ ⊂ Λ is said to be closed if it is a union of basic closed sets. Finally, by ourassumption on the level, each Λ6ν is finite, hence we can write Λ as an ascending union

Λ0 ⊂ Λ1 ⊂ Λ2 · · · , Λ =⋃i>0

Λi,

where each Λi is finite and closed.

Theorem 5.15. For each i, write OΛi for the Serre subcategory of OΛ consisting of finitesuccessive extensions of the Lλ, λ ∈ Λi. Then

(1) OΛi is a highest weight category with standard objects Mλ and costandard objectsAλ, λ ∈ Λi.

(2) For each i, the tautological map DbOΛi → gκ -modI is fully faithful.


(3) The induced map

lim−→i

DbOΛi −→ gκ -modI ,

where the colimit is taken in DGCat, exhibits the latter as the ind-completion ofthe former.

(4) gκ -modI,c is canonically equivalent to bounded homotopy category of tilting modulesTλ, λ ∈ Λ.

Proof. Assertion (1) is well known. For assertion (2), it suffices to check that for µ, ν ∈ Λwe have

RHom(gκ,I) -mod(Mµ, Aν) ∼=

{C µ = ν,

0 µ 6= ν

This is also well known, but we recall the calculation. If we write U for the prounipo-tent radical of the Iwahori, and for λ ∈ Λ we write Cλ for the corresponding characterrepresentation of the torus T , we have

RHom(gκ,I) -mod(Mµ, Aν) ' RHomRep(I)(Cµ, Aν)

' RHomRep(T )(Cµ, AhUν ) ' RHomRep(T )(Cµ,Cν).

For (3), note that by point (2) the map out of the colimit is fully faithful. Moreover, notethatDb(OΛi) coincides with the pretriangulated envelope of theMλ, λ ∈ Λi. By Proposition

5.1, it follows that the map has essential image the compact objects of gκ -modI , as desired.Finally, (4) follows from (3). �

5.3. The Linkage Principle. In this section, κ continues to be any negative level. Thehighest weights of Category O are parametrized by the dual Cartan h∗.

We will need the linkage principle, i.e. the decomposition of O into blocks, which wenow briefly review. Recall the loop algebra

Lg = g⊗CC((z)).

There is a unique invariant bilinear form κb on g for which the short coroots have squaredlength two. Associated to κb is the central extension of Lg

0 −→ Cc −→ gκb −→ Lg −→ 0,

with Lie bracket given by the formula

[X ⊗ f, Y ⊗ g] = [X,Y ]⊗ fg + κb(X,Y ) Res fdg c, X, Y ∈ g, f, g,∈ C((z)).

The affine coroots determine linear functionals on the extended dual Cartan h∗⊕Cc∨. Wewill only need the real affine coroots, which we denote by Φ. Associated to this is a linearaction of the affine Weyl group W on the extended dual Cartan h∗⊕Cc∨. Label the simple

affine coroots αi, i ∈ I, and note there is a unique element ρ in h∗ ⊕ Cc∨ satisfying

〈ρ, αi〉 = 1, i ∈ I.


Associated to this is the ρ-shifted dot action of W . Writing κ as kκb, k ∈ C, the highestweights for gκ identify with the affine hyperplane

h∗κ := h∗ + kc∨.

The linear and dot actions of W preserve h∗κ. For λ ∈ h∗κ, there is an associated collectionof real coroots

Φλ := {α ∈ Φ : 〈λ+ ρ, α〉 ∈ Z}.Write Wλ for the corresponding integral Weyl group, i.e. the subgroup of W generated bythe reflections sα, α ∈ Φλ. With this, we can state the well known

Theorem 5.16. (Linkage principle) For λ ∈ h∗κ, the block Oλ of O for gκ containing Lλhas highest weights Wλ · λ.

Let us apply this to OΛ. Note that the integral weights have a common integral Weylgroup, i.e.

Proposition 5.17. For any λ, λ′ ∈ Λ, we have Φλ = Φλ′.

Proof. Write Φf for the coroots of g. There is a standard enumeration of the affine coroots

as α(n), for a choice of finite coroot α ∈ Φf and integer n ∈ Z. One may calculate that

〈λ+ ρ, α(n)〉 = 〈λ+ ρ, α〉+n

2(κ− κc)(α, α). (5.18)

Now note that if λ ∈ Λ, the first summand on the right hand side of (5.18) is an integer. �

Let us write Wκ for the integral Weyl group arising from Proposition 5.17. Write Of.l.Λ

for the full subcategory of OΛ consisting of finite length objects, and similarly Of.l.λ for a

block therein. Theorem 5.16 yields a decomposition of abelian categories

Of.l.Λ =

⊕λ∈Wκ\Λ

Of.l.λ . (5.19)

We would like to bootstrap this to an analogous decomposition for Harish-Chandra mod-ules. Accordingly, for λ ∈ Wκ\Λ, write gκ -modIλ for the full cocomplete subcategory of

gκ -modI with compact generators

Mµ, µ ∈Wκ · λ.

Proposition 5.20. There is a decomposition in DGCatcont:

gκ -modI =⊕

λ∈Wκ\Λ

gκ -modIλ . (5.21)

Proof. Recall the Λi, i > 0, from Theorem 5.15. For each i and λ ∈Wκ\Λ, set

Oλ,i := Oλ ∩ OΛi .

For fixed i, it follows from Equation 5.19 that we have a decomposition in DGCat

DbOΛi =⊕

λ∈Wκ\Λ

DbOλ,i.


Taking a colimit over i in DGCat and ind-completing, we are therefore done by Theorem5.15(3). �

Corollary 5.22. For arbitrary κ′, there is a decomposition in DGCatcont:

gκ′ -modI =⊕

λ∈Wκ′\Λ

gκ′ -modIλ .

Proof. The decomposition at positive rational levels follows from the decomposition at neg-ative rational levels by Kac-Moody duality. More carefully, recall semi-infinite cohomologygives an LG-equivariant perfect pairing

C∞2

+∗(g2κc , L+g,−⊗−) : gκ′ -mod⊗ g−κ′+2κc -mod −→ Vect .

This induces a perfect pairing on I-equivariant categories. The associated contravariantequivalence on compact objects interchanges Mλ at level κ′ with M−λ−2ρ[dimN ] at level−κ′ + 2κc. �

5.4. Action of the affine Hecke algebra. In this section, we take G to be simplyconnected and κ to be negative integral, i.e.

κ ∈ κc + Z<0κb.

In this situation, there is a canonical monoidal equivalence Dκ(LG) ' D(LG), and hencean action of the affine Hecke algebra H := D(I\LG/I) on gκ -modI . Recall that λ ∈ Λ isantidominant if

〈λ+ ρ, αi〉 6= 1, 2, 3, . . . , i ∈ I.

Each orbit of Wκ on Λ contains a unique antidominant weight, hence we may identify Wκ\Λwith such weights. For an antidominant weight λ, its stabilizer Wλ in W is a parabolicsubgroup, and we denote by W λ the set of minimal length coset representatives for W/Wλ.The goal of this subsection is to prove the following two theorems.

Theorem 5.23. The direct sum decomposition (5.21) is one of affine Hecke algebra mod-ules. I.e., for each λ ∈ Λ antidominant, the action of H preserves gκ -modIλ.

To state the second theorem, for λ antidominant, write δλ for Mλ ' Lλ. Write

jw,!, jw,∗, w ∈W,for the corresponding standard and costandard objects in D(I\LG/I), cf. the discussionpreceding Proposition 4.5.

Theorem 5.24. Fix an antidominant weight λ ∈ Λ, and w ∈W λ.

(1) jw,! ? δλ 'Mw·λ.

(2) jw,∗ ? δλ ' Aw·λ

(3) Under the equivalences of (1) and (2), applying − ? δλ to the canonical map

jw,! −→ jw,∗

yields a nonzero map Mw·λ → Aw·λ.


5.4.1. The regular case. We first handle those λ which are regular, i.e. have trivial stabilizerin W , following the affine localization theorem due to Kashiwara-Tanisaki [14], Beilinson-Drinfeld [3], and Frenkel-Gaitsgory [8].

Write I for the prounipotent radical of I, and consider the enhanced flag variety LG/I.This carries a right action of T , and hence we can consider the monodromic category

Dκ(LG/I)T,w,λ. We have a global sections functor

Dκ(LG/I)T,w,λOblvR−−−−→ IndCoh(LG/I)T,w,λ

ΓIndCoh

−−−−−→ VectT,w(−)hT−−−−→ Vect . (5.25)

Here IndCoh(LG/I) denotes the dg-category of ind-coherent sheaves on LG/I, and ΓIndCoh

is the functor of ind-coherent global sections. We remark that IndCoh(LG/I) is canonically

equivalent to QCoh(LG/I), but the usual global sections functor Γ on the latter is notcontinuous. However, ΓIndCoh can be characterized as the unique continuous functor whichagrees with Γ on complexes supported on (finite-dimensional) subschemes of LG/I.

The composition (5.25) canonically lifts to a strongly LG-equivariant functor

ΓT : Dκ(LG/I)T,w,λ −→ gκ -mod, (5.26)

cf. Proposition A.4 from the appendix.

Theorem 5.27. After taking I-invariants, ΓT gives an equivalence

ΓT : Dκ(I\LG/I)T,w,λ ' gκ -modIλ . (5.28)

Proof. For the Schubert cell

j : IwI −→ LG, w ∈W,

consider the standard and costandard objects j◦w,!, j◦w,∗. As shown in Theorem 4.2 and

Proposition 4.4, the j◦w,!, w ∈ W, are compact generators, as are the j◦w,∗, w ∈ W . We willuse the following:

Theorem 5.29. (Kashiwara–Tanisaki) For any w ∈W , we have

(1) ΓT j◦w,! 'Mw·λ,

(2) ΓT j◦w,∗ ' Aw·λ,(3) ΓT sends a nonzero map jw,! → jw,∗ to a nonzero map Mw·λ → Aw·λ.

We are therefore done by Theorems 5.15 and 5.16, as both categories are identified withthe ind-completion of the same bounded homotopy category of tiltings. �

We may now deduce Theorems 5.23 and 5.24 in the regular case.

Proof of Theorem 5.23 for regular λ. Consider any D(LG)-representations C,D equippedwith an equivariant map C→ D. After taking I-invariants, we obtain a map of H modulesCI → DI . In particular, the essential image of CI is preserved by the action of the affineHecke algebra. Taking C,D as in Theorem 5.27 gives the claim. �


Proof of Theorem 5.24 for regular λ. By Theorem 5.27, it suffices to address the analogousclaim in D(I\LG/I), where it is clear. �

5.5. The singular case.

Proof of Theorem 5.23 for singular λ. For any λ ∈ Λ, not necessarily antidominant, con-sider the global sections functor as in (5.28). Note that it sends jw,∗ to an object in degreezero with the same Jordan-Holder content as Aw·λ. It follows that the subcategory ofgκ -modI generated under colimits by essential image of ΓT coincides with gκ -modIλ. �

For λ singular antidominant, one can show that Theorem 5.29 holds for w ∈W λ, whichimplies Theorem 5.24. However, we will instead deduce Theorem 5.24 from the regularcase via translation functors.

For any two levels κ1, κ2, the usual tensor product of representations gives a map

gκ1 -modnaıve⊗ gκ2 -modnaıve −→ gκ1+κ2 -modnaıve .

Restricting to compact objects of the renormalized derived categories, we obtain a map

gκ1 -modc⊗c gκ2 -modc −→ gκ1+κ2 -mod+,

where⊗c denotes the tensor product on pre-triangulated idempotent-complete dg-categories.Ind-extending, we obtain a map

gκ1 -mod⊗ gκ2 -mod −→ gκ1+κ2 -mod . (5.30)

Notice that the left hand side of (5.30) carries a diagonal an action of Dκ1+κ2(LG).

Lemma 5.31. The functor (5.30) carries a canonical Dκ1+κ2(LG)-equivariant structure.

Suppose that κ1 is integral. By passing to integrable objects, we obtain an equivariantfunctor

gκ1 -modLG⊗ gκ2 -mod −→ gκ1+κ2 -mod .

If we further pick an object M of gκ2 -modLG, this induces an equivariant functor

M⊗− : gκ1 -mod −→ gκ1+κ2 -mod .

We may further take I-invariants to obtain a Hecke-equivariant functor

M⊗− : gκ1 -modI −→ gκ1+κ2 -modI .

Suppose that κ2 is also integral. If we pick λ, µ ∈ Λ, we may include and project on thecorresponding blocks to form the composition

λMν : gκ1 -modIν −→ gκ1 -modI −→ gκ1+κ2 -modI −→ gκ1+κ2 -modIλ .

If κ2 is also integral, by Theorem 5.23 the composite is a Hecke equivariant functor.Now take κ negative integral, and λ a singular antidominant weight. Recall that a weight

π is called dominant if〈π, αi〉 ∈ 0, 1, 2, . . . , i ∈ I.

We may write λ = ν+π, where π is dominant and ν is regular antidominant. We will take

λMν to be the usual translation functor T, for which we need to check the following.


Lemma 5.32. For π dominant, Lπ carries a canonical LG-equivariant structure.

Proof. This follows from the argument of Lemma 5.26 and affine Borel-Weil-Bott. �

Proof of Theorem 5.24 for singular λ. Having already proved the regular case, it sufficesto know for w ∈W λ that

TMw·ν 'Mw·λ TAw·ν ' Aw·λ TLw·ν ' Lw·λ.The first and third are explicitly stated by Kashiwara–Tanisaki in [15], and the middlefollows from recalling that translation commutes with the standard duality D on O. �

6. Affine Milicic-Soergel equivalence

Let G be simply connected, and fix a standard parahoric P = LUP in LG. Let κ be anegative integral level, and λ an antidominant weight at level κ whose stabilizer in W isWL.

Theorem 6.1. There is a canonical t-exact and H-equivariant equivalence

D(I\LG/◦I−L , ψ) ' gκ -modIλ . (6.2)

Proof. Recall the objects δψ and δλ. We claim it suffices to produce an H-equivariantfunctor in either direction which interchanges δψ and δλ. Suppose we have such a functorF. By assumption, WL = W λ, and hence by Proposition 4.5 and Theorem 5.24 we have

that F interchanges Mw·λ and jψw,! and Aw·λ and jψw,∗ for any w ∈WL. Moreover, it induces

an isomorphism

RHom(My·λ, Aw·λ) ' RHom(jψy,!, jψw,∗), y, w ∈WL, (6.3)

since these vanish unless y = w, in which case they are one dimensional and identified byProposition 4.5(3) and Theorem 5.24(3). It then follows that F is an equivalence, e.g. byProposition 4.4 and Theorem 5.15(4).

As in Theorem 3.1, to construct such an F we will produce an LG-equivariant functor

D(LG/◦I−L , ψ) −→ gκ -mod .

This is equivalent to specifying an object of gκ -mod◦I−L ,ψ, which will be the parabolic

induction of the corresponding object for L.More precisely, write BL for the Borel of L contained in I, and write ML

λ for the corre-

sponding Verma module for l of highest weight λ. Write N−L for the unipotent radical ofthe opposite Borel of L whose Lie algebra contains h, and note that ψ restricts to a non-degenerate character of N−L . In the proof of Theorem 3.1, we produced an object M(ψ, χ)

of l -modN−L ,ψ such that, up to a cohomological shift, we had

AvBL,∗M(ψ, χ) 'MLλ .

Write p for the Lie algebra of P , and take the parabolic induction

M := Indgκp Inflp

l M(ψ, χ).


As in the proof of Theorem 3.1, it remains to calculate the I-average of M , and check thatit agrees with δλ up to a cohomological shift. The short exact sequence of group schemes

1 −→ UP −→ I −→ BL −→ 1

yields a canonical isomorphism of functors

AvI,∗ ' AvBL,∗ ◦AvUP ,∗ .

Since M is by construction UP equivariant, by the prounipotence of UP we have

AvUP ,∗M 'M.

As inflation and induction are both canonically BL equivariant functors, cf. the appendix,we have

AvBL,∗ Indgκp Inflp

l M(ψ, χ) ' Indgκp Inflp

l AvBL,∗M(ψ, χ) ' Indgκp Inflp

l MLλ 'Mλ,

as desired. �

Finally, let us note that one can switch between I−L - and I-invariants. Namely, if C isa strong representation of LG, g an element of LG(C), and H an open subgroup of LG,then there is a canonical identification

δg ?− : CH ' CgHg−1. (6.4)

In particular, if we write ψL for an additive character of I which is nonzero precisely onthe simple positive roots of L, and we choose a lift of wL◦ to L, we obtain

Corollary 6.5. There is a canonical t-exact and H-equivariant equivalence

D(I\LG/◦I, ψL) ' gκ -modIλ .

7. The fundamental local equivalence

Let g be a simple Lie algebra, and g its Langlands dual. Recall the notion of dual levels

κ ∈ Sym2(g∗)g, κ ∈ Sym2(g∗)g.

It will be helpful for us to be more concrete. Write κb(g) for the basic level of g, and κb(g)for the basic level of g. There is a corresponding trivialization of the lines of invariantforms

C ' Sym2(g∗)g, C ' Sym2(g∗)g,

interchanging 1 ∈ C and the basic levels. Denote the common lacing number of g and gby r. Finally, write h∨ for the dual Coxeter number of g, and Lh∨ for the dual Coxeternumber of g. Then explicitly the dual of level −h∨ + k is

− Lh∨ − 1

rk, k ∈ C. (7.1)


7.1. The extended affine Weyl group and Dynkin diagram automorphisms. LetG be an almost simple group with Lie algebra g. Then it is well known that its fundamentalgroup acts via Dynkin diagram automorphisms on gκ [13]. It will be convenient to use thefollowing construction of that action, which is probably known.

Associated to the fixed triangular decomposition of g is a maximal torus T of G. WritingΛG for its coweight lattice, one forms the extended affine Weyl group

W ext := Wf n ΛG.

The inclusion of the coroot latice into ΛG gives a short exact sequence

1 −→W −→W ext −→ π1(G) −→ 1.

Let us write Dy(gκ) for the automorphism group of the Dynkin diagram of gκ, which weidentify with the corresponding automorphisms of gκ. The desired homomorphism Dy fitsinto an exact sequence

1 −→W −→W ext Dy−−→ Dy(gκ).

The construction is as follows. Consider the normalizer NG(T ), and form its loop groupLNG(T ). Its component group identifies with W ext, and its neutral component with thatof LT . For any C-point w of LNG(T ), we may restrict its adjoint action to the standardIwahori

Iw−→ Adw I.

The latter group is another Iwahori subgroup containing T , and hence may be conjugatedby some y in LNGsT back to I, where Gs denotes the simply connected cover of G. Thecomposition

Iw−→ Adw I

y−→ I

when restricted to the abelian quotient of the prounipotent radical yields the desired ele-ment of Dy(gκ). It is straightforward to check that is independent of the choice of y, andthat this action factors through the component group W ext, provided that one remembersthat the C-points of L+T and of the neutral component of LT coincide.

In the following subsection we will use the following observation and accompanyingnotation. For w an element of W ext, write Dyw for the image of w under Dy. Then forany standard Levi L of LG, its image DywL is again a standard Levi.

7.2. The Whittaker side of the FLE. We will now rewrite the Whittaker category onthe affine flag variety in terms of Kac-Moody representations. As in Corollary 6.5, for astandard Levi L of LGs, write ψL for any additive character of I which is nonzero preciselyon the positive simple roots of L. Note that all such are interchanged under the action ofT as in Equation (6.4). By a slight abuse of notation, we will often write G instead of thestandard Levi Gs.

Fix an auxiliary negative integral level κ′ such that there exists a regular antidominantweight at level κ′. At such a level, for each standard Levi L we may choose an antidominantweight µL whose stabilizer in W is WL. Finally, write Is for the Iwahori of LGs.


Proposition 7.2. There is an H-equivariant isomorphism

D(I\LG/LN,ψ) '⊕

λ∈π1(G)

gκ′ -modIsµDy(−λ−ρ)G(7.3)

Proof. As shown by Raskin [18], the Whittaker category coincides with the baby Whittakercategory, i.e.

D(I\LG/LN,ψ) ' D(I\LG/Adt−ρ I ,Adt−ρ ψG).

We will now decompose the latter as a D(FlGs)-module over the connected components ofFlG:

D(I\LG/Adt−ρ I ,Adt−ρ ψG) '⊕

λ∈π1(G)

D(Adtλ Is\LGs/Adt−ρ I ,Adt−ρ ψG)

'⊕

λ∈π1(G)

D(Is\LGs/Adt−λ−ρ I ,Adt−λ−ρ ψG)

'⊕

λ∈π1(G)

D(Is\LGs/I, ψDy−λ−ρG)

Applying Corollary 6.5 to the summands in the last expression yields the proposition. �

7.3. The Kac–Moody side of the FLE. We are interested in studying affine CategoryO for g at a level

κ = −Lh∨ +1

rk, k ∈ Z<0. (7.4)

We will write W for the affine Weyl group of g, and continue to write W for the affineWeyl group of g. Recall that the weight lattice for the Iwahori I of LG identifies with ΛG,and that all such weights have a common integral Weyl group Wκ, cf. Proposition 5.17.

Proposition 7.5. For a level of the form (7.4), there is a canonical isomorphism of Coxetergroups

Wκ 'W.

Proof. After composing with the automorphism of hk given by translation by ρ, the actionof W on hk identifies it with Wf n 1

rk Ql, where Wf is the finite Weyl group acting linearly,

and Ql is the long coroot lattice acting by translations. The form κb for g induces aWf invariant inner product on h for which the long coroots have squared length two.Accordingly, for a coroot α, write αl for its long multiple, i.e.

αl =

{α, κb(α, α) = 2,

rα, κb(α, α) = 2r

For any λ ∈ h, let us write tλ for translation by λ. Finally, recall the enumeration of realaffine roots

Φ ' Φf × Z.


With this notation, the affine reflection for α(n), α ∈ Φf , n ∈ Z, is given by

sα(n)= t−

n2rk

αlsαtn

2rkαl . (7.6)

As discussed in Proposition 5.17, for α(n) to lie in Wκ is equivalent to

− nκb(α, α)

2rk∈ Z. (7.7)

Recall that κ(α, α) is 2 or 2r, depending on whether α is short or long. If we write Φs

for the short coroots, and Φl for the long coroots, then it follows from (7.7) that Wκ isgenerated by the reflections corresponding to

Φs × rkZ t Φl × kZ. (7.8)

Combining this with Equation (7.6), it follows that Wκ is identified with Wf n Q, where Qdenotes the coroot lattice. Applying κb for g identifies this with Wf n Ql, the semidirectproduct of Wf and the long root lattice, acting on h∗, i.e. W ext. Moreover, it is straight-forward to see these identifications intertwine the set of simple reflections correspondingto the walls of the negative alcove.

�

Recall that the simple generators of Wκ correspond to the positive elements of (7.6)which cannot be written as a nontrivial sum of other positive elements of (7.6), cf. [15]. Ifwe consider the finite simple roots αi, i ∈ I, along with the highest root θ, we obtain thefollowing.

Corollary 7.9. The simple reflections of Wκ are given by

sαi,(0), i ∈ I, and s−θ,(−k) = t−

θ2 sθt

θ2 .

It follows from Proposition 7.5 that the block decomposition of O is given by the dotaction of Wf n Q on ΛG. We now determine the antidominant weights under this action.

As notation, let us call a coweight µ ∈ ΛG negative minuscule if for every positive root α,we have

−1 6 〈α, µ〉 6 0.

Lemma 7.10. A coweight λ ∈ ΛG is antidominant with respect to Wκ if and only if λ+ ρis negative minuscule.

Proof. By Corollary 7.9, it follows that λ is antidominant if and only if

〈αi, λ+ ρ〉 6 0, i ∈ I, and −1 6 〈θ, λ+ ρ〉.Since θ is the highest root, the claim follows. �

Let us index the simple reflections of Wκ by I, the affine simple roots of g, as in Corollary7.9. If λ+ρ is the negative minuscule coweight ωi, for some i ∈ I, it follows that the stabilizerof λ is the parabolic subgroup of Wκ generated by the simple reflections corresponding to

I \ i.


7.4. Combining the two sides. To compare the two sides, it remains to examine thesummands arising in the Whittaker category as in Proposition 7.2, which were indexed byπ1(G). To account for the ρ-shift in Lemma 7.11, write Gad for the adjoint form G, andΛGad

for the coweight lattice. Recall that under the identification

π1(Gad) ' ΛGad/Q,

we may associate to each element λ in π1(Gad) a unique coset representative in Λ which isminuscule.

Lemma 7.11. For λ ∈ π1(Gad), write ωi for its unique minuscule representative. Then

Dy−λG is the standard Levi of LGs corresponding to I \ i.

This lemma essentially appears with slightly different language and conventions in [13],but we include a somewhat different proof for the convenience of the reader.

Proof. We must show that Dy−ωi sends the affine node 0 to i. This is equivalent to Dyωisending i to 0.

Write P = LUP for the standard Levi of G corresponding to I \ i. Since ωi is minuscule,if we expand θ as a sum of simple roots, the coefficient of αi is one. It follows that the Liealgebra of UP is simple as an L-module, and in particular that the longest element wL◦ ofthe Weyl group of L exchanges θ and αi. From this observation, it is straightforward tosee that Dyωi may be given, after picking lifts of w◦ and wL◦ to G, as the composition

LGsAd

tωi−−−−→ LGsAd

wL◦−−−−→ LGsAdw◦−−−→ LGs.

In particular, this sends the finite simple root αi to the affine root α0. �

Having assembled all the necessary ingredients, we are ready to prove our main theorem.

Proof of Theorem 2.3. Recall that κ is negative. On the Whittaker side, Proposition 7.2yields the decomposition

D(I\LG/LN,ψ) '⊕

λ∈π1(G)

gκ′ -modIsµDy(−λ−ρ)G(7.12)

On the Kac–Moody side, Subsection 7.3 and Proposition 5.20 yields

ˆgκ -modI '⊕

λ∈π1(G)

ˆgκ -modI−λ−ρ (7.13)

For fixed λ, by Theorem 5.15 the compact objects in the heart of the corresponding sum-mands of (7.12), (7.13) are blocks of affine Category O for g and g, respectively. Underthe identification of their integral Weyl groups given in Proposition 7.5, the stabilizers oftheir antidominant representatives coincide by Lemma 7.11. Soergel’s theory, as proved forKac–Moody algebras by Fiebig [7], yields a t-exact equivalence between the correspondingblocks. Applying Theorem 5.15, this induces an equivalence between the full blocks ofHarish-Chandra modules, as desired.


Finally, the case of positive κ follows from the negative case by dualizing. Namely, thestandard self duality of D(FlG) carries a canonical datum of LG equivariance. By theidentification of Whittaker invariants and coinvariants, due to Raskin [19], this induces anidentification of the Whittaker categories, as explained in [6]. On the Kac-Moody side, onemay similarly use Kac-Moody duality, and the canonical identification of I-invariants andcoinvariants. �

As a final remark, let us give an alternative perspective on the appearance of whatamounts to a Soergel modules argument in the proof of Theorem 2.3. A general expectationof local quantum geometric Langlands is the exchanging of I-invariants and I-invariants.Applying this to the endomorphisms of the corepresenting objects, namely Dκ(FlG) andDκ(FlG), we obtain the following conjecture, due to Gaitsgory.

Conjecture 7.14. There is a canonical monoidal equivalence

Dκ(I\LG/I) ' Dκ(I\LG/I).

Appendix A. Equivariance data for some standard functors in (Geometric)Representation Theory

In what follows, by a group we mean either (i) an algebraic group of finite type, (ii) anquasicompact affine group scheme of profinite type with a prounipotent radical of finitecodimension, or (iii) a group ind-scheme G containing an open subgroup K of type (ii) forwhich G/K is ind-proper, and in particular ind-finite type. Such a group has a Lie algebrag which is (i) discrete, (ii) linearly compact, or (iii) a Tate vector space, respectively.

In these cases one can make sense of the algebra object D∗(G) in DGCatcont, the coalge-bra object D!(G), the notion of a weak action of G and the functors of weak (co)invariants.Further, one can consider the renormalized category of Lie algebra representations g -mod,and the canonical identification

g -mod ' D!(G)G,w,

where one takes weak invariants with respect to the right action of G, and retains a stronglevel κ via left multiplication and the identification

(D!(G)G,w)∨ ' D∗(G)G,w.

Moreover, for a multiplicative twisting T on G, consider the corresponding extension gκof g. One can form the algebra object D∗κ(G) of twisted D-modules, its dual coalgebraobject D!

κ(G), the renormalized category of Lie algebra representations gκ -mod, and theidentification

gκ -mod ' D!κ(G)G,w.


A.1. Restriction and induction.

Proposition A.1. Let π : H → G be a group morphism, T a multiplicative twisting onG, and hκ → gκ the corresponding morphism of centrally extended Lie algebras. Then thenatural restriction map

Res : gκ -mod −→ hκ -mod

carries a canonical datum of Dκ(H)-equivariance.

Proof. The restriction functor identifies with the composite

D!κ(G)G,w

Oblv−−−→ D!κ(G)H,w

π!

−→ D!κ(H)H,w

and the two appearing functors carry natural H-equivariant structures. �

Corollary A.2. Let T,T′ be two multiplicative twistings, and gκ, gκ′ the correspondingcentral extensions of g. The tensor product of representations

gκ -mod⊗ gκ′ -mod −→ gκ+κ′ -mod

carries a canonical datum of D∗κ+κ′(G)-equivariance.

Proof. One has a natural identification of renormalized derived categories

gκ -mod⊗ gκ′ -mod ' (g⊕ g)κ+κ′ -mod,

where as usual (g ⊕ g)κ+κ′ denotes the quotient of gκ ⊕ gκ′ identifying the two canonicalcentral elements. Under this identification, the above functor corresponds to restrictionalong the diagonal embedding G→ G×G. �

Proposition A.3. Let ι : H → G be an open embedding of a subgroup scheme, i.e. G/His of ind-finite type. Then the induction functor

Ind : hκ -mod −→ gκ -mod

carries a canonical datum of H-equivariance.

Proof. By the assumption on ι, the morphismH → G is of finite presentation. In particular,ι! admits a left adjoint

ι∗ : D!κ(H)→ D!

κ(G).

Recall for any category C with a weak action of G, the forgetful functor from G-invariantsto H-invariants admits a continuous right adjoint

Oblv : CG,w � CH,w : Avw,∗ .

With this notation, the induction functor identifies with

D!κ(H)H,w

ι∗−→ D!κ(G)H,w

Avw,∗−−−−→ D!κ(G)G,w,

and the two appearing functors carry natural H-equivariant structures. �

One should be able to show that the equivariance datum of the above Proposition coin-cides with the a priori op-lax datum induced by adjunction from the equivariance datumfor restriction.


A.2. Global sections on the enhanced flag variety. Let us write Fl for the enhancedaffine flag variety. Following Beilinson–Drinfeld, by its formal smoothness, for any level κone has the global algebra of differential operators Γ(Fl, Dκ) which contains a copy of g−κ.Moreover, ind-coherent global sections defines a functor

ΓIndCoh : Dκ(Fl)→ Vect .

Moreover, these sections carry the structure of discrete right modules for Γ(Fl, Dκ), and inparticular left modules for gκ. Let us denote this enhanced functor by

ΓBD : Dκ(Fl)→ gκ -mod .

It is expected that ΓBD carries a canonical datum of strong equivariance.We now show enough consequences of this to suffice for the purposes of this paper.

Proposition A.4. Suppose C carries an action of Dκ(LG), and one is given a weakly equi-variant functor F : C→ Vect. Then F canonically factorizes through a strongly equivariantfunctor F:

CF−→ gκ -mod

Oblv−−−→ Vect .

Proof. Raskin in [20] shows that one has a right adjoint to

Oblv : Dκ(LG) -mod→ LG -modweak .

It therefore remains to apply the construction of loc. cit. to Vect. Namely, letting H runover compact open subgroups of LG, we have

OblvR(Vect) ' lim−→H

VectLGH ,w ' gκ -mod,

as desired. �

From the Proposition, we obtain an a priori different lift of ΓIndCoh to an equivariantfunctor

Γ : Dκ(Fl)→ gκ -mod .

While we expect that Γ ' ΓBD, we will only show

Proposition A.5. The functors Γ,ΓBD coincide after restriction to hearts, i.e. one hasa natural isomorphism of functors

Γ,ΓBD : Dκ(Fl)♥ → gκ -mod♥

which induces the identity natural transformation between

Oblv ◦Γ,Oblv ◦ΓBD : Dκ(Fl)♥ → Vect♥ .

Proof. Because we are working with the abelian category of Lie algebra representations,we may reduce to the analogous assertion for each sl2 corresponding to the affine simple

roots αi, i ∈ I.For fixed αi, by considering the Schubert decomposition one sees that Fl has a filtration

by SL2 stable subvarieties which are rationally smooth. We are therefore reduced to theanalogous assertion for a linear algebraic group acting on a smooth algebraic variety, whereit is straightforward. �


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