Fundamental local equivalences in quantum geometric Langlands

TL;DR

This paper proves the conjectural fundamental local equivalence in quantum geometric Langlands, extending the Satake equivalence's role using Soergel module techniques to affine flag varieties.

Contribution

It establishes the fundamental local equivalence in quantum geometric Langlands and extends it to affine flag varieties, employing novel Soergel module methods.

Findings

Proves the fundamental local equivalence conjecture in quantum geometric Langlands.

Extends the equivalence to affine flag varieties.

Uses Soergel module techniques for the proof.

Abstract

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory--Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.

Equations

$${\widehat{\check{\mathfrak{g}}}}_{\check{\kappa }}\text{–}{\mathsf{mod}}^{{\mathfrak{L}}^{+}\check{G}}\text{and}{\widehat{\check{\mathfrak{g}}}}_{\check{\kappa }}\text{–}{\mathsf{mod}}^{\check{I}}.$$

$${\mathsf{Whit}}_{\kappa }^{\mathrm{sph}}\simeq {\widehat{\check{\mathfrak{g}}}}_{\check{\kappa }}\text{–}{\mathsf{mod}}^{{\mathfrak{L}}^{+}\check{G}}.$$

$${\mathsf{Whit}}_{\kappa }^{\mathrm{aff}}\simeq {\widehat{\check{\mathfrak{g}}}}_{\check{\kappa }}\text{–}{\mathsf{mod}}^{\check{I}}.$$

$${\widehat{\check{\mathfrak{g}}}}_{\infty }\text{–}{\mathsf{mod}}^{{\mathfrak{L}}^{+}\check{G}}=\mathsf{QCoh}(\check{\mathfrak{g}}[[t]]dt/{\mathfrak{L}}^{+}\check{G})=\mathsf{QCoh}(\mathbb{B}\check{G})=\mathsf{Rep}(\check{G}).$$

$$\mathsf{Rep}(\check{G})\to D({\mathrm{Gr}}_G{)}^{{\mathfrak{L}}^{+}G}\to \mathsf{Whit}({\mathrm{Gr}}_G)$$

$${\widehat{\check{\mathfrak{g}}}}_{crit}\text{–}{\mathsf{mod}}^{{\mathfrak{L}}^{+}\check{G}}\simeq \mathsf{QCoh}({\mathrm{Op}}_G^{\mathrm{unr}}).$$

$$Q\subset {\Lambda }_G.$$

$${\alpha }_i\in {\Lambda }_G\text{and}{\check{\alpha }}_i\in {\check{\Lambda }}_G.$$

$$\tau :\mathfrak{g}\stackrel{\sim }{\to }\mathfrak{g}.$$

$$[e_i,\tau (e_i)]=\check{{\alpha }_i},\text{for }i\in \mathcal{I}.$$

$$I:={\mathfrak{L}}^{+}G{\times }_{G}B\subset {\mathfrak{L}}^{+}G\subset \mathfrak{L}G.$$

$$T\simeq I/\mathring{I}.$$

$$\tilde{W}:=W_f⋉{\check{\Lambda }}_G.$$

$$W:=W_f⋉\check{Q}$$

$${\mathcal{C}}^{♡}:={\mathcal{C}}^{⩽0}\cap {\mathcal{C}}^{⩾0}.$$

$$D(\mathfrak{L}G)\text{–}\mathsf{mod}:=D(\mathfrak{L}G)\text{–}\mathsf{mod}({\mathsf{DGCat}}_{cont}).$$

$${\mathcal{C}}^H:={\mathsf{Hom}}_{D(H)\text{–}\mathsf{mod}}(\mathsf{Vect},\mathcal{C})\text{and}{\mathcal{C}}_H:=\mathsf{Vect}\underset{D(H)}{\otimes }\mathcal{C}.$$

$${\mathcal{C}}^{H,\chi }\text{and}{\mathcal{C}}_{H,\chi }.$$

$$\lambda \in \mathrm{Hom}(H,{\mathbb{G}}_m){\otimes }_{\mathbb{Z}}\mathbb{C}$$

$$\psi :H\to {\mathbb{G}}_a,$$

$$\mathcal{C}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{C}}_{\mathfrak{L}N,\psi }\simeq {\mathcal{C}}^{\mathfrak{L}N,\psi }.$$

$$D(X/H,\chi ):=D(X{)}_{H,\chi }.$$

$$\kappa :{\mathrm{Sym}}^2(\mathfrak{g})\to \mathbb{C}.$$

$$\underset{i\in \mathcal{I}}{\min }{\kappa }_b({\check{\alpha }}_i,{\check{\alpha }}_i)=2.$$

$$D_{\kappa }(\mathfrak{L}G)\text{–}\mathsf{mod}:=D_{\kappa }(\mathfrak{L}G)\text{–}\mathsf{mod}({\mathsf{DGCat}}_{cont}).$$

$$\mathcal{C}\in D_{\kappa }(\mathfrak{L}G)\text{–}\mathsf{mod},$$

$$D_{\kappa }(\mathfrak{L}N,\psi \backslash \mathfrak{L}G/I)$$

$$\mathfrak{Lg}=\mathfrak{g}\otimes \mathbb{C}((t))=\underset{n}{\lim }\mathfrak{g}\otimes \mathbb{C}((t))/t^n\mathbb{C}[[t]].$$

$$\begin{array}{c}{\xi }_1\otimes {\xi }_2\mapsto \mathrm{Res}\kappa ({\xi }_1,d{\xi }_2).\end{array}$$

Full text

1. 1 Introduction
2. 2 Preliminary material
3. 3 Fundamental local equivalences
4. A Proof in the general case

1. Introduction

1.1.

In the early days of the geometric Langlands program, experts observed that the fundamental objects of study deform over the space of levels $\kappa$ for the reductive group $G$. For example, if $G$ is simple, this is a 1-dimensional space. Moreover, levels admit duality as well: a level $\kappa$ for $G$ gives rise to a dual level111We include a translation by critical levels in the definition of $\check{\kappa}$, cf. Section 2.7 for our conventions on levels. $\check{\kappa}$ for the Langlands dual group $\check{G}$. This observation suggested the existence of a quantum geometric Langlands program, deforming the usual Langlands program.

The first triumph of this idea appeared in the work of Feigin–Frenkel [FF91], where they proved duality of affine $\mathcal{W}$-algebras: $\mathcal{W}_{G,\kappa}\simeq\mathcal{W}_{\check{G},\check{\kappa}}$. We emphasize that this result is quantum in its nature: the level $\kappa$ appears. For the critical level $\kappa=\kappa_{c}$, which corresponds to classical geometric Langlands, Beilinson–Drinfeld [BDdf] used Feigin–Frenkel duality to give a beautiful construction of Hecke eigensheaves for certain irreducible local systems.

1.2.

The major deficiency of the quantum geometric Langlands was understood immediately: the Satake equivalence is more degenerate.

For instance, in the classical case where $\kappa=\kappa_{c}$ is critical, the compatibility of global geometric Langlands with geometric Satake essentially characterizes the equivalence. Concretely, this means that for an irreducible $\check{G}$-local system $\sigma$ on a smooth, projective curve, one expects there to exist a canonical Hecke eigensheaf $\mathcal{A}_{\sigma}$ with eigenvalue $\sigma$.

This does not hold in the quantum setting. For instance, if $\kappa$ is irrational, then the Satake category is equivalent to $\mathsf{Vect}$ and the Hecke eigensheaf condition is vacuous. For rational $\kappa$, one hopes for a neutral gerbe of irreducible Hecke eigensheaves. This is known for a torus, but the gerbe is not at all canonically trivial, cf. [Lys15] Section 5.2. For general reductive $G$, we are not aware of any conjecture explicitly describing the relevant gerbe.

We remark that quantum geometric Langlands for rational $\kappa$ is closely tied to the theory of automorphic forms on metaplectic groups, cf. [GL18a]. It is well-known that Hecke eigenvalues form too coarse a decomposition in the metaplectic setting. Indeed, already in his first announcement [Hec36] of his eponymous operators, Hecke himself made this observation.

Es sei zum Schlauß noch erwähnt, daß für die Formen halbzahlinger Dimension, wie die einfachen Thetareihen und deren Potenzprodukte, sich eine ähnliche Theorie nicht aufbauen läßt. Da nämlich für diese die Zuordnung zu einer Stufe und zu einer Kongruenzgruppe bekanntlich nicht mehr so einfach wie bei ganzzahliger Dimension ist, so kann man die Operatoren $T_{m}$ nur für Quadratzahlen $m=n^{2}$ definieren, und man erhält so einen Zusammenhang nur zwischen den Koeffizienten $a(N)$ und $a(Nn^{2})$. (In conclusion, it should be mentioned that a similar theory cannot be developed for forms of half-integral weight, such as simple theta functions and monomials in them. Since the assignment of a level and a congruence group is not as easy as for integer dimensions, the operators $T_{m}$ can only be defined for square numbers $m=n^{2}$, and a relationship is obtained only between the coefficients $a(N)$ and $a(Nn^{2})$.)

In the study of metaplectic automorphic forms, one often repeatedly finds the guiding principle: it is more fruitful to study Whittaker coefficients than Hecke eigenvalues. Indeed, above Hecke exactly observes the gap that appears between the two in the metaplectic context, and that Whittaker coefficients provide the finer information. See [GPS80] for a classical application of these ideas.

We refer to [GGW18] for further discussion and more recent perspectives.

1.3.

In the geometric setting, Gaitsgory made significant advances in applying the above perspective. In [Gai08b] and [Gai16b], Gaitsgory, pursuing unpublished ideas he developed jointly with Lurie, formulated a series of conjectures regarding $\kappa$-twisted Whittaker $D$-modules on the affine Grassmannian $\operatorname{Gr}_{G}=\mathfrak{L}G/\mathfrak{L}^{+}G$ and the affine flag variety $\mathfrak{L}G/I$.

Let $\kappa$ be a nondegenerate level for $G$ and let $\check{\kappa}$ denote the dual level for $\check{G}$. Let $\mathsf{Whit}_{\kappa}^{\operatorname{sph}}$ denote the DG category of $\kappa$-twisted Whittaker $D$-modules on $\operatorname{Gr}_{G}$, and let $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$ denote the similar category for $\mathfrak{L}G/I$.

We write $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}$ for the affine Lie algebra associated to $\check{G}$ and $\check{\kappa}$, and let

[TABLE]

denote the DG categories of Harish–Chandra modules for the pairs $(\widehat{\check{\mathfrak{g}}}_{\check{\kappa}},\mathfrak{L}^{+}\check{G})$ and $(\widehat{\check{\mathfrak{g}}}_{\check{\kappa}},\check{I})$, respectively. We refer to Section 2 for further clarification on the notation.

Conjecture 1.3.1** **(Gaitsgory–Lurie,

[Gai08b] Conjecture 0.10).

There is an equivalence of DG categories

[TABLE]

Conjecture 1.3.2** (Gaitsgory, [Gai16b] Conjecture 3.11).**

There is an equivalence of DG categories

[TABLE]

We now state a preliminary version of our main result, and then discuss the hypothesis that appears in it.

Theorem**.**

Suppose that $\kappa$ is good in the sense of Section 3.4.4. Then Conjectures 1.3.1 and 1.3.2 are true.

1.3.3.

Briefly, a level $\kappa$ is good if and only if after restriction to every simple factor of $\mathfrak{g}$, $\kappa$ is either irrational, or a rational level whose denominator is coprime to the bad primes of the root system. We recall the latter always lie in $\{2,3,5\}$. For $G$ of type $A$, there are no bad primes, so every level is good in this case. For an explicit description for general $G$, see Figure 1.

1.4. Related works

As emphasized by Gaitsgory in the initial papers [Gai08b] and [Gai16b], Conjectures 1.3.1 and 1.3.2 provide quantum analogues of theorems in classical local geometric Langlands, i.e. for $\kappa$ or $\check{\kappa}$ at the critical level. We presently review these statements.

1.4.1.

For $\kappa=\kappa_{c}$ critical, Conjecture 1.3.1 is known from work of Frenkel–Gaitsgory–Vilonen, and is a variant of the geometric Satake equivalence.

In this case, heuristically we have $\check{\kappa}=\infty$; to interpret this carefully, we refer to [Zha17] for details. Standard arguments then give:

[TABLE]

Here the action of $\mathfrak{L}^{+}\check{G}$ on $\check{\mathfrak{g}}[[t]]dt$ is the gauge action.

Then by [FGV01], the composition:

[TABLE]

is an equivalence, where the first functor is the geometric Satake functor [MV07] and the second functor is given by convolution on the unit object of the right hand side.

We emphasize that unlike with the Satake equivalence, the equivalence $\mathsf{Rep}(\check{G})\xrightarrow{\sim}\mathsf{Whit}(\operatorname{Gr}_{G})$ is an equivalence of derived categories, not merely abelian categories. This amounts to the cleanness property of spherical Whittaker sheaves from [FGV01] and the geometric Casselman–Shalika formula from loc. cit.

Remark 1.4.2*.*

That (1.3) is an equivalence is special to integral levels. That is, at non-integral levels, the spherical Hecke category produces only a small part of the spherical Whittaker category. This failure, especially at rational levels, is part of our interest in Theorem Theorem.

1.4.3.

For $\kappa=\kappa_{c}$ critical, a version of Conjecture 1.3.2 is the main result of [AB09a]. This deep work of Arkhipov–Bezrukavnikov was one of the most significant breakthroughs in geometric Langlands, and underlies seemingly countless advances in the area since.

1.4.4.

By the above discussion, for $\kappa=\kappa_{c}$, Conjecture 1.3.1 is a sort of variant of the geometric Satake theorem. One of Gaitsgory’s key insights is that in quantum geometric Langlands, Hecke operators play a diminished role, while Conjecture 1.3.1 plays the fundamental role that Satake plays in the classical theory.

1.4.5.

For $\check{\kappa}=\check{\kappa}_{c}$ critical, Conjecture 1.3.1 is the main result of [FG09c], where Frenkel–Gaitsgory construct an equivalence

[TABLE]

Here the right hand side is the indscheme of unramified (or monodromy-free) opers for $G$; this category is the $\kappa\to\infty$ limit of $\mathsf{Whit}_{\kappa}(\operatorname{Gr}_{G})$ as above, and is defined in loc. cit. For the centrality of this result in the geometric Langlands program, see [Gai15] Section 11.

Conjecture 1.3.2 in this case is a folklore extension of the main results of [FG09a] and [FG], but whose complete proof is not recorded in the literature.

1.4.6.

Finally, let us discuss the previously known cases of Conjectures 1.3.1 and 1.3.2. In the original paper [Gai08b], Gaitsgory proved Conjecture 1.3.1 for $\kappa$ irrational, and in fact a stronger version of it, as we presently describe below. As far as we are aware, no other cases of the conjectures have been obtained.

1.5. Factorization

In fact, Gaitsgory conjectured more, related to the factorization of the Beilinson–Drinfeld affine Grassmannian.

In Conjecture 1.3.1, he conjectured an equivalence of factorization categories, cf. [Gai08b] and [Ras15a]. Similarly, in Conjecture 1.3.2, it is expected that the equivalence should be one of factorization modules for the (conjecturally equivalent) factorization categories appearing in Conjecture 1.3.1.

When $\kappa=\kappa_{c}$, these goals are implicit in the original work. In the spherical case, this is spelled out in [Ras15b] Theorem 6.36.1. In the Iwahori case, a weak version of the compatibility with factorization module structures was shown in [Ras16a] Theorem 10.8.1.

1.6. The role of this paper

In the decade since their formulation, Gaitsgory has been advancing an ambitious program to establish the fundamental local equivalences. We refer to [ABC*+*18] for an overview of this project; [Gai08b], [BG08], and [BG08] for early work on it; and [Gai16a], [Gai17], [Gai18a], [Gai18b], [Gai19], [GL19], and [GL18b] for some of his recent advances in this project.

Gaitsgory’s program, though still incomplete, represents a new paradigm for Kac-Moody algebras, quantum groups, and quantum geometric Langlands. It is full of lovely, innovative constructions and numerous breakthroughs. It is also quite sophisticated, as seems always to be the case when working with factorization algebras.

Our work is not intended to supersede the eventual conclusion of Gaitsgory’s project. Rather, we regard the equivalences of Conjectures 1.3.1 and 1.3.2 (i.e., forgetting factorization) to be interesting results in geometric representation theory and geometric Langlands.

For example, as discussed above, the $\kappa=\kappa_{c}$ analogues of our results include the geometric Casselman–Shalika formula [FGV01] and the deep work of Arkhipov–Bezrukavnikov [AB09b]. In fact, as we hope to explain elsewhere, the geometric part of our study of $\mathsf{Whit}_{\kappa}^{\operatorname{sph}}$, suitably adapted to the function-field setting, should imply some new function-theoretic results on the metaplectic Casselman–Shalika formula.

Moreover, while our present results are expected to be interesting outcomes of Gaitsgory’s methods, we find it desirable to have a more direct argument.

1.7. Methods

Our techniques are remarkably elementary in comparison to the above work of Gaitsgory or e.g. [AB09b]. Our main input is classical methods developed by Soergel and his school.

1.7.1.

In his initial work [Soe90], Soergel showed that a block of Category $\mathcal{O}$ for $\mathfrak{g}$ can be reconstructed from the Weyl group of $G$. Fiebig [Fie06] extended this work to Kac–Moody algebras. As a consequence of Fiebig’s work, the category $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}\text{\textendash}\mathsf{mod}^{\check{I}}$ can be completely recovered from the combinatorial datum of the root datum of $\check{G}$ and the level $\check{\kappa}$.

To prove Conjecture 1.3.2, we provide a similar Coxeter-theoretic description of $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$. We do this by relating $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$ to $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{I}$,222In finite type, an analogous result appears in Milicic–Soergel [MS97]. Their techniques are not available in the affine setting, so our methods differ. We use the perspective of loop group actions on categories to study Kac–Moody representations. We convolve by an explicit object, constructed from $\mathcal{W}$-algebras. In contrast, in the finite-type setting, [MS97] relies on good properties of Harish–Chandra bimodules with generalized central characters. The theory of affine Harish–Chandra bimodules is in its infancy and is much more difficult than in finite type. As we hope to explain elsewhere, our methods are sufficient to establish similar properties of a suitable category of Harish–Chandra bimodules in affine type. However, it should also be possible to prove this equivalence directly by a Soergel module argument, cf. Remark 3.6.4. which allows us to apply Fiebig’s results directly to $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$.

We then prove Conjecture 1.3.2 by matching Langlands dual combinatorics. Here we draw the reader’s attention to Theorem 3.5.6, which is a combinatorial shadow of quantum Langlands duality.

It is striking that these fundamental conjectures of Gaitsgory have been open for over a decade, but admit a solution that almost could have been given at the time.

Remark 1.7.2*.*

In fact, Theorem 3.5.6, combined with the description of twisted Hecke categories as Soergel bimodules, obtained in finite type recently in [LY19], should yield quantum Langlands duality for affine Hecke categories. Therefore, Soergel’s methods, as applied in our paper, should suffice to prove the local quantum geometric Langlands correspondence for categorical representations generated by Iwahori invariant vectors.

Remark 1.7.3*.*

Because of our reliance on [Fie06], our construction is a little non-canonical. Indeed, in loc. cit., there is a choice of projective cover of simple objects. With that said, hewing closer to the Koszul dual picture as in [LY19] would provide canonical equivalences.

Remark 1.7.4*.*

After completing this paper, we learned of the thesis of Chris Dodd [Dod11], which reproves the results of Arkhipov–Bezrukavnikov [AB09b] by a Soergel module argument. Our argument may be thought of as a quantum deformation of his approach. We thank Roman Bezrukavnikov for bringing this to our attention.

1.8. Comparison

In short, we relate Langlands dual categories using Fiebig’s combinatorial description of blocks of affine Category $\mathcal{O}$.

Gaitsgory’s program compares these categories via a factorization algebra $\Omega_{q}$ (and some of its cousins), which may also be constructed directly from the root datum of $G$, cf. [Gai08a], [ABC*+*18] and [Gai19].

It would be quite interesting to find a direct relationship between these two perspectives.

Remark 1.8.1*.*

We highlight one point of departure in our perspective as compared to Gaitsgory’s. At negative levels, our equivalence is $t$-exact by construction and matches highest weight structures. This was previously anticipated in the spherical case by Gaitsgory, but was ambiguous in the affine case. After we told him about our results, he found an argument showing that a similar property must hold for the equivalence he is working on.

In our approach, these properties are key in deducing the parahoric version of the theorem from the Iwahori version.

Acknowledgments. We would like to thank D. Ben-Zvi, R. Bezrukavnikov, A. Braverman, D. Bump, D. Gaitsgory, S. Kumar, S. Lysenko, I. Mirkovic, W. Wang, B. Webster, D. Yang, and Z. Yun for interest, encouragement and helpful discussions.

Part of this work was carried out at MSRI, where S.R. was in residence. In addition, this research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Research and Innovation. We thank all these institutions for their hospitality.

2. Preliminary material

In this section, we collect standard definitions and notation. We invite the reader to skip to the next section and refer back as needed.

2.1. Notation for groups

Let $G$ be a reductive group over $\mathbb{C}$.333Our arguments apply more generally for any split reductive group over a field $k$ of characteristic [math]. That is, the cohomology that appears is purely de Rham, never étale or Betti.

2.1.1.

We fix once and for all a pinning $(T,B,\psi)$ of $G$. That is, we fix $T\subset B\subset G$ with $T$ (resp. $B$) a Cartan (resp. Borel) subgroup of $G$. In addition, for $N$ the unipotent radical of $B$, we fix a nondegenerate character $\psi:N\to\mathbb{G}_{a}$.

2.1.2.

Given the above data, there is a canonically defined Langlands dual group $\check{G}$, which also comes with a pinning. In particular, we have Cartan and Borel subgroups $\check{T}\subset\check{B}\subset\check{G}$. Again, $\check{N}$ denotes the unipotent radical of $\check{B}$.

2.1.3.

The data $T\subset B\subset G$ determines a Borel $B^{-}$ opposite to $B$, so $B^{-}\cap B=T$. We denote its radical by $N^{-}$. The same applies for $\check{G}$.

2.1.4.

We denote the appropriate Lie algebras by $\mathfrak{g},\mathfrak{b},\mathfrak{n},\mathfrak{t},\mathfrak{b}^{-},\mathfrak{n}^{-},\check{\mathfrak{g}},\check{\mathfrak{b}},\check{\mathfrak{n}},\check{\mathfrak{t}},\check{\mathfrak{b}}^{-}$, and $\check{\mathfrak{n}}^{-}$.

2.1.5.

We write $\check{\Lambda}_{G}$ for the lattice of coweights of $G$, i.e., the cocharacter lattice of $T$, and $\Lambda_{G}$ for the lattice of weights of $G$, i.e., the character lattice of $T$. We denote the root lattice, i.e., the integral span of the roots by

[TABLE]

In other words, $Q=\Lambda_{G^{ad}}$ for $G^{ad}$ the adjoint group of $G$. Similarly, we let $\check{Q}\subset\check{\Lambda}_{\check{G}}$ denote the coroot lattice, and one has $\check{Q}=\Lambda_{\check{G}^{ad}}$.

2.1.6.

We let $\mathscr{I}$ denote the set of nodes of the Dynkin diagram of $G$. For $i\in\mathscr{I}$, the corresponding simple roots and coroots are denoted

[TABLE]

2.1.7.

Our choice of pinning defines a standard involutive anti-homomorphism444One sometimes finds this involution called the Chevalley involution, or the Cartan involution, but the latter terminology is potentially misleading.

[TABLE]

More precisely, for $i\in\mathscr{I}$, let $e_{i}\in\mathfrak{n}$ be the unique vector of weight $\alpha_{i}$ such that $\psi(e_{i})=1$. Then $\tau$ is the unique involution such that $\tau([x,y])=-[\tau(x),\tau(y)]$, $\tau|_{\mathfrak{t}}=\text{id}_{\mathfrak{t}}$, and

[TABLE]

We observe that $\tau$ lifts to an involutive anti-homomorphism on $G$, which we also denote by $\tau$.

2.2. Loops and arcs

2.2.1.

For any affine variety $Z$ of finite type, we let $\mathfrak{L}Z$ denote its algebraic loop space and $\mathfrak{L}^{+}Z$ denote its algebraic arc space. The former is an ind-scheme, while the latter is an affine scheme. There is a canonical evaluation map $\mathfrak{L}^{+}Z\to Z$ given by evaluation of a jet at the origin.

For $Z=H$ an affine algebraic group, $\mathfrak{L}H$ is a group ind-scheme, with $\mathfrak{L}^{+}H\subset\mathfrak{L}H$ a group subscheme.

2.2.2.

The Borel subgroup $B\subset G$ defines the Iwahori subgroup

[TABLE]

Dually, we a preferred Iwahori subgroup $\check{I}\subset\mathfrak{L}\check{G}$. We denote the prounipotent radical of $I$ by $\mathring{I}$ and note the canonical isomorphism

[TABLE]

2.3. Weyl groups

The combinatorics of affine Weyl groups plays an important role in this paper. We recall notation and fundamental constructions below.

2.3.1.

We let $W_{f}$ denote the Weyl group of $G$, and remind that $W_{f}$ is also the Weyl group of $\check{G}$.

2.3.2.

The extended affine Weyl group of $G$ is the semidirect product

[TABLE]

The subgroup of $\widetilde{W}$ given by

[TABLE]

is the affine Weyl group, where we remind that $\check{Q}$ was the coroot lattice.

2.3.3.

Let $S_{f}\subset W_{f}$ denote the set of simple reflections $s_{i}$ for $i\in\mathscr{I}$. Let $S\subset W$ denote the union of $S_{f}$ with the set of simple affine reflections as in Section 7 of [Kac90]. We remind that the simple affine reflections are indexed by simple factors of $G$.

The pairs $(W_{f},S_{f})$ and $(W,S)$ are Coxeter systems, i.e., Coxeter groups with preferred choices of simple reflections. We remind that the Bruhat order and length function on $W$ each extend in a standard way to $\widetilde{W}$.

2.4. Categories

We repeatedly work with DG categories and their symmetries. In this setting, we use the following conventions.

2.4.1.

We let $\mathsf{DGCat}_{cont}$ denote the symmetric monoidal $\infty$-category of cocomplete DG categories and continuous DG functors as defined in [GR17] Section I.1.10. We denote the binary product underlying the symmetric monoidal structure by $-\otimes-$; this is the Lurie tensor product of [Lur17].

For simplicity, we sometimes refer to $\infty$-categories as categories, and similarly for DG categories.

2.4.2.

Given a $t$-structure on a DG category $\mathscr{C}$, we let $\mathscr{C}^{\leqslant 0}$ and $\mathscr{C}^{\geqslant 0}$ denote the subcategories of connective and coconnective objects. That is, we use cohomological indexing notations.

We denote the heart of such a $t$-structure by

[TABLE]

2.5. $D$-modules

We make essential use of categories of $D$-modules on ind-pro-finite-type schemes such as $\mathfrak{L}G$, as developed in [Ber17] and [Ras15c]. For an indscheme $X$, we denote by $D(X)$ what is in loc. cit. denoted by $D^{*}(X)$.

2.5.1.

By functoriality, $D(\mathfrak{L}G)$ carries a canonical convolution monoidal structure. We denote the corresponding ($\infty$-)category of DG categories equipped with an action of $D(\mathfrak{L}G)$ by

[TABLE]

We use similar notation for other group (ind-)schemes such as $\check{I}$ and $\mathfrak{L}N$.

2.6. Invariants and coinvariants

2.6.1.

Given a group ind-scheme $H$, and a $D(H)$-module $\mathscr{C}$, we denote its categories of invariants and coinvariants respectively by

[TABLE]

See [Ber17] for further discussion.

2.6.2.

Similarly, for a multiplicative $D$-module $\chi$ on $H$, we denote the corresponding categories of twisted invariants and twisted coinvariants by

[TABLE]

Our multiplicative $D$-modules will be obtained by one of the following two procedures.

2.6.3.

First, any character

[TABLE]

determines a character $D$-module $``t^{\lambda}"$ of $H$. In this case, we denote twisted invariants by $\mathscr{C}^{H,\lambda}$. We apply this construction particularly for $H=T$ and $H=I$.

2.6.4.

Similarly, given an additive character

[TABLE]

we obtain a character $D$-module $``e^{\psi}"$ on $H$. Here we denote twisted invariants by $\mathscr{C}^{H,\psi}$. We apply this constructions for $H=\mathfrak{L}N$.

2.6.5.

Suppose $H$ is an affine group scheme with pro-unipotent radical $H^{u}$. We suppose that $H/H^{u}$ is finite type.

By [Ber17], the canonical forgetful map $\mathscr{C}^{H}\rightarrow\mathscr{C}$ admits a continuous right adjoint $\operatorname{Av}_{*}^{H}$. Moreover, there is a canonical equivalence $\mathscr{C}_{H}\simeq\mathscr{C}^{H}$ fitting into a commutative diagram

[TABLE]

The same applies in the presence of a multiplicative $D$-module $\chi$.

2.6.6.

Now suppose $\mathscr{C}\in D(\mathfrak{L}G)\text{\textendash}\mathsf{mod}$. Let $\psi:\mathfrak{L}N\to\mathbb{G}_{a}$ denote the Whittaker character of $\mathfrak{L}N$.

In this case, [Ras16b] Theorem 2.1.1 provides a canonical equivalence

[TABLE]

We highlight that this case is more subtle than that of an affine group scheme considered above.

2.6.7.

As a final piece of notation, for $X$ an ind-scheme with an action of a group ind-scheme $H$, we use the notation

[TABLE]

By [Ras15c] Proposition 6.7.1, this notation is unambiguous.

In the setting of either (2.1) or (2.2), we remark that these coinvariants coincide with invariants.

2.7. Levels

Recall that a level $\kappa$ for $G$ is a $G$-invariant symmetric bilinear form

[TABLE]

2.7.1.

We let $\kappa_{\mathfrak{g},c}$ denote the critical level for $G$, i.e., $-\frac{1}{2}$ times the Killing form of $G$. Where $G$ is unambiguous, we simply write $\kappa_{c}$.

2.7.2.

Suppose $G$ is simple. A level $\kappa$ is rational if $\kappa$ is a rational multiple of the Killing form and irrational otherwise. We say $\kappa$ is positive if $\kappa-\kappa_{c}$ is a positive rational multiple of the Killing form. We say a level $\kappa$ is negative if $\kappa$ is not positive or critical. In particular, any irrational level is negative.

For general reductive $G$, we say a level $\kappa$ is rational, irrational, positive, or negative if its restrictions to each simple factor are so.

2.7.3.

A level $\kappa$ is nondegenerate if $\kappa-\kappa_{c}$ is nondegenerate as a bilinear form. For such $\kappa$, the dual level $\check{\kappa}$ for $\check{G}$ is the unique nondegenerate level such that the restriction of $\check{\kappa}-\check{\kappa}_{\check{\mathfrak{g}},c}$ to $\mathfrak{t}^{*}$ and the restriction of $\kappa-\kappa_{\mathfrak{g},c}$ to ${\mathfrak{t}}$ are dual symmetric bilinear forms.

2.7.4.

For a simple Lie algebra $\mathfrak{g}$, the basic level $\kappa_{\mathfrak{g},b}=\kappa_{b}$ is the unique positive level such that the short coroots have squared length two, i.e.

[TABLE]

2.8. Twisted $D$-modules

Given a level $\kappa$, there is a canonical monoidal DG category of twisted $D$-modules, $D_{\kappa}(\mathfrak{L}G)$, see for example [Ras16b] Section 1.29. We again use the notation

[TABLE]

2.8.1.

We recall that the multiplicative twisting defined by $\kappa$ is canonically trivialized on $\mathfrak{L}^{+}G$ and $\mathfrak{L}N$. In particular, for

[TABLE]

we can make sense of invariants and coinvariants of $\mathfrak{L}^{+}G$ and $\mathfrak{L}N$ with coefficients in $\mathscr{C}$. The same applies for any subgroup, in particular for $I\subset\mathfrak{L}^{+}G$. Also, the same applies with a twisting, e.g. in the Whittaker setup for $\mathfrak{L}N$. We remark that the identification (2.2) of Whittaker invariants and coinvariants [Ras16b] was proved more generally for $D_{\kappa}(\mathfrak{L}G)\text{\textendash}\mathsf{mod}$.

Example 2.8.2*.*

$D_{\kappa}(\mathfrak{L}G)$ carries commuting actions of $D(\mathfrak{L}N)$ and $D(I)$, so following the convention of Section 2.6.7, we use the notation

[TABLE]

for the appropriate invariants = coinvariants category.

2.9. Affine Lie algebras

2.9.1.

Let $\mathfrak{L}\mathfrak{g}$ denote the Lie algebra of $\mathfrak{L}G$, considered with its natural inverse limit topology, i.e.

[TABLE]

Given a level $\kappa$, one obtains a continuous 2-cocycle $\mathfrak{L}\mathfrak{g}\otimes\mathfrak{L}\mathfrak{g}\to\mathbb{C}$ given by

[TABLE]

Here $d$ is the exterior derivative and $\operatorname{Res}$ is the residue. We denote the corresponding central extension by

[TABLE]

2.9.2.

We denote by $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$ the abelian category of smooth representations of $\widehat{\mathfrak{g}}_{\kappa}$ on which the central element $\mathbf{1}$ acts via the identity.

We let $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ denote the DG category introduced by Frenkel–Gaitsgory in Sections 22 and 23 of [FG09b]. This DG category is compactly generated and carries a canonical $t$-structure with heart $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$. However, there are some non-zero objects in

[TABLE]

so $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ is not the derived category of $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$.

2.9.3.

In Sections 10 and 11 of [Ras19], a $D_{\kappa}(\mathfrak{L}G)$-module structure on $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ was constructed, enhancing previous constructions of Beilinson–Drinfeld and Frenkel–Gaitsgory, cf. [BDdf] Section 7 and [FG06] Section 22.

2.9.4.

Let $H$ be a sub-group scheme of $\mathfrak{L}^{+}G$ of finite codimension, and consider the corresponding category of equivariant objects

[TABLE]

By [Ras16b] Lemma A.35.1, the bounded below category

[TABLE]

canonically identifies with bounded below derived category of its heart $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{H,\heartsuit}$, which are Harish–Chandra modules for the pair $(\widehat{\mathfrak{g}}_{\kappa},H)$. Moreover, $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{H}$ is compactly generated by inductions of finite dimensional $H$-modules.

2.9.5.

Let $\kappa$ be a level. We abuse notation in letting $\kappa$ also denote the map $\kappa:\mathfrak{t}\to\mathfrak{t}^{*}$.

The dot action of $\widetilde{W}$ on $\mathfrak{t}^{*}$ is defined by having $W_{f}$ act through the usual dot action, and $\check{\Lambda}_{G}$ act through translations via $\kappa-\kappa_{c}$. I.e., writing $\rho$ for the half sum of the positive roots, for any $\lambda\in\mathfrak{t}^{*}$ we have

[TABLE]

2.9.6.

To discuss integral Weyl groups, we need some more standard facts about this dot action. Recall that the affine real coroots of $\widehat{\mathfrak{g}}_{\kappa}$ are a subset of $\mathfrak{t}\oplus\mathbb{C}\mathbf{1}$. In particular, to a such a coroot $\check{\alpha}$, we may associate its classical part $\check{\alpha}_{cl}$, i.e. its projection to $\mathfrak{t}$. This is a coroot of $\mathfrak{g}$, and in particular we may associate a classical root $\alpha_{cl}\in\mathfrak{t}^{*}$.

2.9.7.

Via a standard construction, $\check{\alpha}$ acts as an affine linear functional on $\mathfrak{t}^{*}$, which we denote by $\langle\check{\alpha},-\rangle$, in such a way that, writing $s_{\check{\alpha}}$ for the associated reflection in $\widetilde{W}$, one has

[TABLE]

Briefly, this arises via restricting a linear action of $\widetilde{W}$ on $(\mathfrak{t}\oplus\mathbb{C}\mathbf{1})^{*}$ to the affine hyperplane of functionals whose pairing with $\mathbf{1}$ is 1. We refer the reader to Sections 3.1 and 3.4 of [Dhi19], where this is reviewed in greater detail.

3. Fundamental local equivalences

In this section, we suppose $G$ is simple of adjoint type and $\kappa$ is a negative level for $G$. Under these assumptions, we prove the main theorem, i.e., the fundamental local equivalence for good $\kappa$. In Appendix A, we deduce the same result for general $G$ and general good $\kappa$ from this case.

3.1. Overview of the argument

The proof of the main theorem requires fine arguments involving combinatorics of affine Lie algebras. To help the reader understand what follows, we begin with an overview of the main ideas. This inherently requires referring to concepts that have not been introduced yet, so the reader may safely skip this material and refer back as necessary.

We omit some technical considerations at this point in the discussion. For example, we do not carefully distinguish here between abelian and derived categories.

3.1.1.

Suppose $\lambda\in\mathfrak{t}^{*}$ is a weight of $\mathfrak{g}$. Let us denote the block of Category $\mathscr{O}$ for $\widehat{\mathfrak{g}}_{\kappa}$ containing the Verma module $M_{\lambda}$ by

[TABLE]

In Section 3.2.3, we recall that $\lambda$ determines a subgroup $W_{\lambda}\subset W$, its integral Weyl group.555In spite of the notation, $W_{\lambda}$ depends also on $\kappa$. This is a Coxeter group, i.e. comes equipped with a set of simple reflections.

We make important use of Theorem 3.5.2, which is due to Fiebig [Fie06], following earlier work of Soergel [Soe90]. This result asserts that if $\lambda$ is antidominant666In the sense of affine Kac–Moody algebras. In particular, the definition depends on $\kappa$. $\mathscr{O}_{\kappa,\lambda}$ is determined as a category by the data of (i) the Coxeter group $W_{\lambda}$ along with (ii) the subgroup

[TABLE]

stabilizing $\lambda$ under the dot action of $W$ on $\mathfrak{t}^{*}$, cf. Section 2.9.5.

Remark 3.1.2*.*

For us, the most important case is when $\lambda$ is integral. Here we write $W_{\mathfrak{g},\kappa}$ in place of $W_{\lambda}$. In this case, $W_{\mathfrak{g},\kappa}$ contains the finite Weyl group $W_{f}$. If $\kappa$ is irrational, the two are equal, and the simple reflections in $W_{\mathfrak{g},\kappa}=W_{f}$ are the usual ones determined by our fixed Borel. If $\kappa$ is rational, there is one additional simple reflection; we provide an explicit formula for it in Lemma 3.4.3.

3.1.3.

In Theorem 3.2.7, we find a block decomposition for the Whittaker category $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$ on the affine flag variety for $G$. These blocks are indexed by $W_{\mathfrak{g},\kappa}$-orbits in $\check{\Lambda}_{G}=\Lambda_{\check{G}}$.

In Theorem 3.3.3, we show that up to varying $\kappa$ by an integral translate (which does not affect the Whittaker category), its neutral block is equivalent to an integral block of Category $\mathscr{O}$ for $\widehat{\mathfrak{g}}_{-\kappa}$. We generalize this to general blocks at good777In our actual exposition, the definition of good level is essentially rigged so such a result holds. The content is rather in Proposition 3.4.5, which provides a concrete description of good levels. levels in Corollary 3.4.8. These identifications preserve the natural highest weight structures on both sides. From now on, we assume that $\kappa$ is good.

3.1.4.

For each block of $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$, we explicitly compute the combinatorial datum (3.1) of the corresponding block of Category $\mathscr{O}$ for the corresponding Kac–Moody algebra888Again, this Kac–Moody algebra is essentially $\widehat{\mathfrak{g}}_{-\kappa}$, except that we may need to replace $-\kappa$ by an integral translate. and provide a form of Langlands duality for this datum.

Essentially by construction, for any such block the corresponding integral Weyl group is $W_{\mathfrak{g},\kappa}$, with simple reflections as indicated above. In Theorem 3.5.6, we construct an isomorphism $W_{\mathfrak{g},\kappa}\simeq W_{\check{\mathfrak{g}},\check{\kappa}}$ preserving simple reflections, and that is the identity on $W_{f}$.999We are not aware of the identification of Theorem 3.5.6 having appeared previously in the literature.

Note that for any block of $\mathscr{O}$ for the Kac–Moody algebra, the corresponding integral Weyl group canonically acts on the set of isomorphism classes of simple objects in this block. Therefore, the above considerations provide an action of $W_{\check{\mathfrak{g}},\check{\kappa}}$ on the set of isomorphism classes of simple objects in $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$, which is canonically identified with $\check{\Lambda}_{G}=\Lambda_{\check{G}}$. Again, by construction, this action coincides with the dot action of Section 2.9.5, but for $\check{G}$ rather than $G$.

In Corollary 3.2.11 and in the proof of Theorem 3.6.1, we check that simple objects of $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$ corresponding to antidominant weights of $\check{G}$ are also standard objects for the highest weight structure on this category. These observations amount to matching the data (3.1) with that of affine Category $\mathscr{O}$ for $\check{G}$, completing the proof of Theorem Theorem in the Iwahori case.

3.1.5.

In Section 3.7, under the hypotheses of this section, we deduce the parahoric version of Theorem Theorem from the Iwahori version. In particular, this includes the spherical version of the theorem.

We do this by identifying the parahoric categories as full101010This fully faithfulness only holds at the abelian categorical level. subcategories of the corresponding Iwahori categories and then identifying the essential images under the isomorphism of Theorem 3.6.1.

3.2. Block decomposition for the Whittaker category

We begin by decomposing the Whittaker category into blocks.

3.2.1.

To do so, it will be useful to simultaneously consider the case of twisted Whittaker categories. Thus, we fix $\lambda\in\mathfrak{t}^{*}$, and consider

[TABLE]

While these DG categories depends on both $\lambda$ and $\kappa$, we will study them with fixed $\kappa$ and varying $\lambda$, and for this reason suppress $\kappa$ for ease of notation.

3.2.2.

For indexing reasons, it will be convenient to rewrite this category as follows. Consider the automorphism of $\mathfrak{L}G$ given by

[TABLE]

where $\tau$ is as in Section 2.1.7. This induces an equivalence

[TABLE]

where the $\psi$ on the right-hand side of (3.2) denotes a nondegenerate character of $\mathfrak{L}N^{-}$ of conductor one. In the following discussion, we will think of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ via the latter expression.

3.2.3.

It will be important for us to consider $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ as a module for the following version of the affine Hecke category.

Let $W_{\lambda}$ denote the integral Weyl group of $\lambda$.111111Later in the paper, the twist will be fixed to $\lambda=0$, but the group and level will vary. We will accordingly denote this integral Weyl group by $W_{\mathfrak{g},-\kappa}$ instead. We hope this does not cause confusion, and will reintroduce this change in notation when it first occurs. Recall this is the subgroup of $W$ generated the reflections $s_{\check{\alpha}}$ corresponding to affine coroots $\check{\alpha}$ satisfying

[TABLE]

where the pairing is via the level $-\kappa$ action as in Section 2.9.7. For $w\in W_{\lambda}$, write $j_{w,!*}$ for the intermediate extension of the simple object of

[TABLE]

We define $\mathscr{H}_{\lambda}$ to be the full subcategory

[TABLE]

generated under colimits and shifts by the objects

[TABLE]

By Section 4 of [LY19],121212While loc. cit. is written ostensibly over a finite field and in the finite type setting, the arguments we cite from it straightforwardly adapt to the present situation. $\mathscr{H}_{\lambda}$ is closed under convolution, so admits a unique monoidal DG structure for which its embedding into $\eqref{e:heckealg}$ is monoidal. In addition, loc. cit. shows that $\mathscr{H}$ is the neutral block of $\eqref{e:heckealg}$, i.e., it is the minimal direct summand containing the identity element.

3.2.4.

Recall that the Iwasawa decomposition provides $\mathfrak{L}G$ with a stratification by the double cosets

[TABLE]

Let $\widetilde{W}^{f}\subset\widetilde{W}$ denote the subset of elements of minimal length in their left $W_{f}$-cosets. We now describe the affine Whittaker category for a single stratum.

Lemma 3.2.5**.**

For $w\notin\widetilde{W}^{f}$, the affine Whittaker category on $Iw\mathfrak{L}N^{-}$ vanishes, i.e. we have

[TABLE]

For $w\in\widetilde{W}^{f}$, $!$-restriction to any closed point gives an equivalence

[TABLE]

Proof.

The twistings corresponding to both $\kappa$ and $\lambda$ are trivializable on a double coset. Moreover, $\psi$ is trivial on the stabilizer in $\mathfrak{L}N^{-}$ of the point $w\in I\backslash\mathfrak{L}G$ if and only if $w\in\widetilde{W}^{f}$, which implies the desired identities. ∎

For $w\in\widetilde{W}^{f}$ we denote the corresponding standard, simple, and costandard objects of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ by

[TABLE]

Explicitly, under the identification with $\mathsf{Vect}$ above, they correspond to the relevant extensions of $\mathbb{C}[-\ell(ww_{\circ})]\in\mathsf{Vect}$, where $\ell$ denotes the length function on $\widetilde{W}$, and $w_{\circ}$ denotes the longest element of $W_{f}$.

3.2.6.

We next obtain the block decomposition of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$. To state it, for any double coset

[TABLE]

we write $\mathsf{Whit}_{\lambda,y}^{\operatorname{aff}}$ for the full subcategory of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ generated under colimits and shifts by the objects

[TABLE]

Theorem 3.2.7**.**

Each $\mathsf{Whit}_{\lambda,y}^{\operatorname{aff}}$ is preserved by the action of $\mathscr{H}_{\lambda}$, and the direct sum of inclusions yields an $\mathscr{H}_{\lambda}$-equivariant equivalence

[TABLE]

Proof.

For $w\in\widetilde{W}$ consider the corresponding standard, simple, and costandard objects

[TABLE]

We use $-\overset{I}{\star}-$ to denote the convolution functor

[TABLE]

and the induced functor

[TABLE]

For any $w\in\widetilde{W}$, if we by abuse of notation denote its image in $\widetilde{W}^{f}\xrightarrow{\sim}\widetilde{W}/W_{f}$ again by $w$, we claim there exist equivalences

[TABLE]

Note that in (3.6), the object $j_{e,*}^{\psi}$ belongs to $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$, whereas $j_{w,!}^{\psi}$ and $j_{w,*}^{\psi}$ belong to $\mathsf{Whit}_{w\cdot\lambda}^{\operatorname{aff}}$.

We break the proof (3.6) into several cases. First, suppose $w$ lies in $\widetilde{W}^{f}$. Then the second identity in (3.6) follows from the observation that the convolution map

[TABLE]

is an isomorphism. The first identity for such $w$ then follows by the cleanness of $j_{e,*}^{\psi}$ and the ind-properness of the multiplication map

[TABLE]

Next, suppose $w=s$ is a simple reflection of $W_{f}$. The image of the convolution map as in (3.7) is contained in the locally closed sub-indscheme of $\mathfrak{L}G$ corresponding to the strata $I\mathfrak{L}N^{-}\cup Is\mathfrak{L}N^{-}$. As the only stratum of its closure which supports Whittaker sheaves is $I\mathfrak{L}N^{-}$, it is enough to compute the $!$-restrictions of our convolutions to the identity

[TABLE]

Let us write $\iota$ for the involution $g\mapsto g^{-1}$ on $\mathfrak{L}G$. By base change we may compute the $!$-fibre as

[TABLE]

as desired. A similar calculation on $\mathbb{P}^{1}$ yields for the first identity in (3.6) for $w=s$.

Finally, for general $w\in\widetilde{W}$, we write $w=w^{f}w_{f}$ for $w^{f}\in\widetilde{W}^{f}$ and $w_{f}\in W_{f}$. Choosing a reduced expression for $w_{f}$, and the corresponding factorizations of $j_{w,!}$ and $j_{w,*}$, we are reduced to the cases considered above for twists in $\widetilde{W}\cdot\lambda$. Given (3.6), the assertions of the theorem follow by the same arguments as in [LY19] Propositions 4.7 and 4.9. ∎

3.2.8.

Having obtained the block decomposition of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$, we now record some properties of each block as an $\mathscr{H}_{\lambda}$ module. We begin with some relevant combinatorics.

Recall that $W_{\lambda}$ is a Coxeter group. Namely, write $\check{\Phi}^{+}$ for the positive real coroots, and consider the subset

[TABLE]

With this, a reflection $s_{\check{\alpha}}$ of $W_{\lambda}$ is simple if and only if $\check{\alpha}$ is not expressible as a sum of two other elements of $\check{\Phi}^{+}_{\lambda}$.

Lemma 3.2.9**.**

Each double coset $W_{\lambda}y{W}_{f}$ contains a unique element minimal with respect to the Bruhat order. In particular, each $W_{\lambda}$ orbit on $\widetilde{W}/W_{f}$ contains a unique element $yW_{f}$ minimal with respect to the Bruhat order. Moreover, its stabilizer

[TABLE]

is a parabolic subgroup of $W_{\lambda}$.

Proof.

Let $y$ be any element of minimal length in $W_{\lambda}yW_{f}$. We first claim that $y\leqslant w_{\lambda}y$ for every $w_{\lambda}\in W_{\lambda}$. Indeed, for any reflection $s_{\check{\alpha}}$ of $W_{\lambda}$ with corresponding positive coroot $\check{\alpha}$, it follows from the minimal length of $y$ that

[TABLE]

i.e., that $y^{-1}(\check{\alpha})>0$. The claimed minimality now follows by induction on the length of element of $W_{\lambda}$ with respect to its Coxeter generators.

We next show $y\leqslant w_{\lambda}yw_{f}$, for any $w_{f}$ in $W_{f}$. However, it is clear that $y\leqslant ys$, for any simple reflection $s$ of $W_{f}$. This implies that for any $w\in\widetilde{W}$, $y\leqslant w$ if and only if $y\leqslant ws_{j}$. Applying this and a straightforward induction on the length to $w=w_{\lambda}y$ yields the claim.

It remains to show that (3.8) is a parabolic subgroup of $W_{\lambda}$. To see this, consider $W_{y^{-1}\lambda}=y^{-1}W_{\lambda}y$. The positivity property (3.9) of $y$ implies that conjugation by $y^{-1}$ defines an isomorphism of Coxeter systems

[TABLE]

i.e. exchanges their simple reflections. It is therefore enough to see that $W_{f}\cap W_{y^{-1}\lambda}$ is a parabolic subgroup of $W_{y^{-1}\lambda}$, which is straightforward, cf. the proof of Lemma 8.8 of [Dhi19]. ∎

3.2.10.

Combining Lemma 3.2.9 and Theorem 3.2.7 we obtain:

Corollary 3.2.11**.**

For a double coset $W_{\lambda}yW_{f}$ with minimal element $y$, the corresponding object of $\mathsf{Whit}_{\lambda,y}^{\operatorname{aff}}$ is clean, i.e.,

[TABLE]

In the following proposition, we collect some properties of the action of $\mathscr{H}_{\lambda}$ on $j_{y,!}^{\psi}$. To state them, let us denote the set of elements of $W_{\lambda}$ of minimal length in their left cosets with respect to the parabolic subgroup (3.8) by

[TABLE]

We additionally recall that $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ carries a canonical $t$-structure. Namely, an object $\mathscr{M}$ is coconnective, i.e., lies in $\mathsf{Whit}_{\lambda}^{\operatorname{aff},\geqslant 0}$, if and only if

[TABLE]

Such a $t$-structure may be seen directly via gluing of $t$-structures stratum by stratum, and coincides with the general construction of Section 5.2 and Appendix B of [Ras16b], cf. loc. cit. Remark B.7.1.

Proposition 3.2.12**.**

For $y$ as in Corollary 3.2.11, and any $w$ in $W_{\lambda},$ there are equivalences

[TABLE]

Moreover, if $w$ lies in $W_{\lambda}^{f}$, then convolution with $j_{y,*}^{\psi}$ yields an isomorphism of lines

[TABLE]

Proof.

Both assertions follow from (3.6) by standard arguments. Namely, the proof of [LY19] Proposition 5.2 yields (3.10). It follows from (3.10) that convolution with $j_{y,*}^{\psi}$ is $t$-exact, which readily implies (3.11). ∎

Remark 3.2.13*.*

One can show that the clean objects $j_{y,!*}^{\psi}$ constructed above are the only clean extensions in $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$. Since we do not use this fact, we only sketch the proof. Namely, write $\leqslant$ for the Bruhat order on $W_{\lambda}^{f}$. One can in fact show that for any $w,x\in W_{\lambda}^{f}$ the space of intertwining operators

[TABLE]

vanishes unless $w\leqslant x$, in which case it is one dimensional and consists of embeddings. This may be deduced from (3.10) and the analogous classification of intertwining operators between standard objects of $\mathscr{H}_{\lambda}$. The latter may be proved directly or deduced via localization from the Kac–Kazhdan theorem on homomorphisms between Verma modules.

3.3. Whittaker–singular duality

We now use the above noted properties of the $\mathscr{H}_{\lambda}$-action to produce, under necessary hypotheses, equivalences between the neutral block of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ and a block of Category $\mathscr{O}$ for $\widehat{\mathfrak{g}}_{-\kappa}$.

3.3.1.

Consider the category of $(I,\lambda)$-integrable modules for $\widehat{\mathfrak{g}}_{-\kappa}$, which we denote by

[TABLE]

Recall this is compactly generated by the Verma modules $M_{\theta},\text{ for }\theta\in\lambda+\Lambda_{G},$ and note the integral Weyl group of any such $\theta$ is $W_{\lambda}$.

3.3.2.

We identify the desired blocks by matching their combinatorics to that of $\mathsf{Whit}_{e,\lambda}^{\operatorname{aff}}$ as follows. Suppose $M_{\Theta}$ is an irreducible Verma module in (3.12) such that the stabilizer of $\Theta$ under the dot action of $W_{\lambda}$ is given by

[TABLE]

Consider the corresponding block of (3.12), i.e. the full subcategory compactly generated by $M_{\theta}$, for $\theta\in W_{\lambda}\cdot\Theta$, and denote it by

[TABLE]

Theorem 3.3.3**.**

There is a canonical $\mathscr{H}_{\lambda}$-equivariant and $t$-exact equivalence

[TABLE]

Remark 3.3.4*.*

For emphasis: by definition of $\Theta$, the integral Weyl group of this block is $W_{\lambda}$, and the stabilizer of $\Theta$ in $W_{\lambda}$ is $W_{\lambda}\cap W_{f}$.

Remark 3.3.5*.*

The minus sign in front of the level is an artifact of Section 3.2.2. As we will discuss in greater detail in Section 3.4.1, we potentially need to apply an integral translation to $\kappa\rightsquigarrow\kappa+\kappa^{\prime}$ before finding $\Theta$ as above; in practice, this translation in particular assures that of making $-\kappa-\kappa^{\prime}$ is negative.

Example 3.3.6*.*

Theorem 3.3.3 is always applicable when the twist $\lambda$ is trivial. Namely, by an integral translation, we may assume $-\kappa$ is sufficiently negative, in which case we may take $\Theta$ to be $-\rho$.

Proof of Theorem 3.3.3.

To produce an $\mathscr{H}_{\lambda}$-equivariant functor, we will construct a $D_{-\kappa}(\mathfrak{L}G)$-equivariant functor and then pass to $(I,\lambda)$-equivariant objects. After further projecting onto a block of (3.12), we will show this is the sought-for equivalence.

We begin by constructing an $D_{-\kappa}(\mathfrak{L}G)$-equivariant functor

[TABLE]

To do so, recall that for any $D_{-\kappa}(\mathfrak{L}G)$-module $\mathscr{C}$, one has a canonical equivalence

[TABLE]

Explicitly, if we write $\operatorname{ins}\delta$ for the insertion of the delta function at the identity of $\mathfrak{L}G$ into Whittaker coinvariants, the equivalence (3.14) is given by evaluation at $\operatorname{ins}\delta$. Applying this to $\mathscr{C}\simeq\widehat{\mathfrak{g}}_{-\kappa}\text{\textendash}\mathsf{mod}$, we obtain via the affine Skryabin equivalence, cf. [Ras16b],

[TABLE]

Therefore, to produce $\mathsf{F}$ we must specify a module for the $\mathcal{W}_{-\kappa}$-algebra.

We do so as follows. Write $\operatorname{Zhu}(\mathcal{W}_{-\kappa})$ for the Zhu algebra of $\mathcal{W}_{-\kappa}$, and $Z\mathfrak{g}$ for the center of the universal enveloping algebra of $\mathfrak{g}$. Let us normalize the identification

[TABLE]

as in [Dhi19]. Writing $\chi(\Theta)$ for the character of $Z\mathfrak{g}$ corresponding to $\Theta$, i.e. via its action on Verma module for $\mathfrak{g}$ with highest weight $\Theta$, consider the associated local cohomology $Z\mathfrak{g}$-module $Ri_{\chi(\Theta)}^{!}Z\mathfrak{g}$. We will take $\mathsf{F}$ to be associated to the corresponding $\mathcal{W}_{-\kappa}$-module

[TABLE]

where $\operatorname{pind}$ denotes the standard induction from Zhu algebra modules to vertex algebra modules, i.e., the left adjoint to the functor of taking singular vectors.

Passing to $(I,\lambda)$-equivariant objects, we obtain a $D_{-\kappa}(I,\lambda\backslash\mathfrak{L}G/I,-\lambda)$-equivariant functor

[TABLE]

Let us determine $\mathsf{F}(j_{e,*}^{\psi})$. To do so, write $\mathbb{C}_{\psi}$ for the one dimensional representation of $\mathfrak{n}^{-}$ associated to the additive character $\psi$ of $\mathfrak{L}N^{-}$. As we will explain in more detail below, we then have

[TABLE]

To see the first identification (3.17), one may use that (3.16) is an iterated extension of Verma modules, and in particular arises from the first step of the adolescent Whittaker filtration of $\mathcal{W}_{-\kappa}\operatorname{-mod}$, cf. [Ras16b].

For the second identification (3.18), if we write $K_{1}$ for the first congruence subgroup of $\mathfrak{L}^{+}G$, one has a corresponding factorization

[TABLE]

The second identification then follows from the $K_{1}$-integrability of a parabolically induced module, the prounipotence of $K_{1}$, cf. Theorem 4.3.2 of [Ber17], and the $D(G)$-equivariance of $\operatorname{pind}_{\mathfrak{g}}^{\widehat{\mathfrak{g}}_{-\kappa}}$.

For the third identification (3.19), for any $\xi\in\lambda+\Lambda_{G}$, let us write $M_{\xi,\mathfrak{g}}$ for the corresponding Verma module for $\mathfrak{g}$. One has by adjunction

[TABLE]

The latter vanishes unless $\xi\in W_{f}\cdot\Theta$, in which case we continue

[TABLE]

where in the last step one uses the canonical identification

[TABLE]

By our assumption on $\Theta$, each $M_{\xi,\mathfrak{g}}$, for $\xi\in(W_{f}\cdot\Theta)\cap\lambda+\Lambda_{G}$, generates a block of $\mathfrak{g}\text{\textendash}\mathsf{mod}^{B,\lambda}$ equivalent to $\mathsf{Vect}$, hence we have shown

[TABLE]

The identity (3.19) then follows by applying $\operatorname{pind}_{\mathfrak{g}}^{\widehat{\mathfrak{g}}_{-\kappa}}$.

The projection of the sum of Verma modules (3.19) onto $\widehat{\mathfrak{g}}_{-\kappa}{\text{\textendash}\mathsf{mod}}^{I,\lambda}_{\Theta}$ picks out the summand

[TABLE]

cf. Lemma 8.7 of [Dhi19]. Accordingly, we consider the composition

[TABLE]

where the latter map is projection onto the block. Note that while the blocks of (3.12) are in general not preserved by $D_{-\kappa}(I,\lambda\backslash\mathfrak{L}G/I,-\lambda)$, they are preserved by $\mathscr{H}_{\lambda}$. In particular, by construction the composition (3.20), which we denote by $\Psi$, is an $\mathscr{H}_{\lambda}$-equivariant functor which sends $j_{e,*}^{\psi}$ to $M_{\Theta}$.

It remains to see that $\Psi$ is an equivalence. By (3.10) and $\mathscr{H}_{\lambda}$-equivariance, we obtain identifications

[TABLE]

where the last identity is a standard consequence of localization. Similarly, if for $\theta\in\lambda+\Lambda$ we denote the contragredient dual of $M_{\theta}$ by $A_{\theta}$, we obtain identifications

[TABLE]

It is therefore enough to check, for any $y,w\in W_{\lambda}$, the map

[TABLE]

is an equivalence. This again follows from (3.11), as desired. ∎

3.3.7.

While Theorem 3.2.7 realizes the neutral block of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$, this may be applied to other blocks as follows. Fix a double coset

[TABLE]

with associated minimal element $y$ and block $\mathsf{Whit}_{\lambda,y}^{\operatorname{aff}}$.

Proposition 3.3.8**.**

Convolution with the clean object

[TABLE]

yields a $t$-exact equivalence

[TABLE]

Proof.

The argument for [LY19] Proposition 5.2 applies mutatis mutandis. ∎

3.4. Application of Whittaker-singular duality and the classification of

good levels

3.4.1.

We would like to relate an arbitrary block $\mathsf{Whit}_{\lambda,y}^{\operatorname{aff}}$ of $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ to Kac–Moody representations. Via Proposition 3.3.8, we should apply Theorem 3.2.7 to $\mathsf{Whit}_{y^{-1}\cdot\lambda,e}^{\operatorname{aff}}$. Therefore, we would like to produce a suitable Verma module in

[TABLE]

It will useful to expand the collection of available Verma modules. For example, if $-\kappa$ is positive rational and the twist $y^{-1}\cdot\lambda$ is trivial, there are no irreducible Verma modules in (3.21). To address this, note that for any integral level $\kappa^{\prime}$ for $G$ one has a tautological $t$-exact equivalence

[TABLE]

We may further increase our supply of integral levels and characters as follows. Write $G_{s}$ for the simply connected form of $G$, and $I_{s}$ for the Iwahori subgroup of its loop group. Then the tautological embedding

[TABLE]

induces an equivalence of neutral blocks.

3.4.2.

To analyze when we may find the desired Verma modules, we will need some basic properties of the relevant integral Weyl group. So, for an arbitrary level $\kappa_{\circ}$, let us denote by $W_{\mathfrak{g},\kappa_{\circ}}$ the integral Weyl group of $0\in\mathfrak{t}^{*}$ at level $\kappa_{\circ}$, i.e. what was denoted by $W_{0}$ in the notation of 3.2.3.

To describe the simple reflections in $W_{\mathfrak{g},\kappa_{\circ}}$, recall the canonical identification of $W$ with the semidirect product

[TABLE]

For any finite coroot $\check{\alpha}$ we denote by $s_{\check{\alpha}}$ the corresponding reflection in $W_{f}$, and for an element $\check{\lambda}$ of the coroot lattice $\check{Q}$ we write $t^{\check{\lambda}}$ for the corresponding translation in $W$. In addition, let us write $\check{\theta}_{s}$ for the short dominant coroot, $\check{\theta}_{l}$ for the long dominant coroot, and $r$ for the lacing number of $\mathfrak{g}$.

Lemma 3.4.3**.**

For any $\kappa_{\circ}$ and integral level $\kappa^{\prime}$, there are canonical identifications

[TABLE]

intertwining their inclusions into $W$. If $\kappa_{\circ}$ is irrational, then $W_{\mathfrak{g},\kappa_{\circ}}\simeq W_{f}$, i.e., they coincide as subgroups of $W$. If $\kappa_{\circ}$ is rational, write it as

[TABLE]

where $h^{\vee}$ is the dual Coxeter number, $p$ and $q$ are coprime integers, and $\kappa_{b}$ is the basic level. Then $W_{\mathfrak{g},\kappa_{\circ}}$ has simple reflections given by the simple reflections of $W_{f}$ and the additional reflection

[TABLE]

Proof.

Recall the standard enumeration of the affine real coroots as

[TABLE]

where $\check{\Phi}_{f}$ denotes the finite coroots. With this enumeration, the element $\check{\alpha}_{n}\in\mathfrak{t}\oplus\mathbb{C}\mathbf{1}$, for $\check{\alpha}\in\check{\Phi}_{f}$ and $n\in\mathbb{Z}$, is given by

[TABLE]

In particular $\check{\alpha}_{n}$ belongs to $W_{\mathfrak{g},\kappa_{\circ}}$ if and only if

[TABLE]

is an integer, which straightforwardly implies the claims of the lemma. ∎

3.4.4.

Having explicitly identified the Coxeter generators of the integral Weyl group, we will now obtain for most levels highest weights with prescribed stabilizers within it. Let us formulate this problem precisely. If we write $\Lambda_{G_{s}}$ for the weight lattice, and recall that $I_{s}$ denotes the Iwahori subgroup of $\mathfrak{L}G_{s}$, then

[TABLE]

is compactly generated by the Verma modules $M_{\lambda}$, for $\lambda\in\Lambda_{G_{s}}$. In particular, this category has highest weights consisting of the weight lattice. Let us say a level $\kappa_{\circ}$ is good if for any finite parabolic subgroup $W_{\circ}$ of $W_{\mathfrak{g},\kappa_{\circ}}$ there exists an integral level $\kappa^{\prime}$ and a simple Verma module

[TABLE]

whose highest weight has stabilizer $W_{\circ}$. Let us classify the good levels.

Proposition 3.4.5**.**

Every irrational level is good. A rational level

[TABLE]

where $p$ and $q$ are coprime integers, is good if and only if $q$ is coprime to the number $n(\mathfrak{g})$ associated to $\mathfrak{g}$ in Figure 1.

Proof.

For $\kappa_{\circ}$ irrational, the claim is clear as $W_{\mathfrak{g},\kappa_{\circ}}\simeq W_{f}$. For $\kappa_{\circ}$ rational, which we may take to be negative, a weight $\lambda\in\Lambda_{G_{s}}$ is antidominant if and only if

[TABLE]

as follows from (3.23). Let us write $\omega_{i},$ for $i\in\mathscr{I}$, for the fundamental weights, and write the dominant coroots as sums of simple coroots

[TABLE]

Recall the standard correspondence between finite parabolic subgroups $W_{\circ}$ of $W_{\mathfrak{g},\kappa_{\circ}}$ and nonempty faces of the above alcove, which associates to a face the stabilizer of any interior point. It follows that, after the transformation $\lambda\mapsto-\lambda-\rho$, we are looking for points of $\Lambda_{G_{s}}$ within the alcove with vertices at

[TABLE]

Recalling that we are free to replace $p$ by any element of $p+q\mathbb{Z}$, it is straightforward to see that we can find points of $\Lambda_{G_{s}}$ in the interior of every face of the alcove if and only if for each $i\in\mathscr{I}$ one has

[TABLE]

To see this, note these conditions are tautologically equivalent to being able to realize each vertex of the alcove as a point of $\Lambda_{G_{s}}$, hence they are necessary. To see they are sufficient, suppose they are satisfied. Via this assumption, for any positive integer $N$ we may replace $p$ with an element of $p+q\mathbb{Z}$ so that

[TABLE]

In particular, we may assume that $N$ is greater than the number of vertices of the alcove, i.e. $N>\lvert I\rvert+1$. In this case, every face of the alcove contains an interior point expressible as a convex combination of the vertices with coefficients in $\frac{1}{N}\mathbb{Z}$. As such a convex combination is a point of $\Lambda_{G_{s}}$, we are done.

Finally, recalling the $n_{i}$ and $m_{i}$ for each type, cf. Plates I-IX of [Bou02], yields the entries of Figure 1. Namely, if $n_{i}$ is prime, $p\in(n_{i},q)$ if and only if $(n_{i},q)=1$, and if $n_{i}$ is composite, then each of its prime divisors occurs as another $n_{i^{\prime}}$. ∎

3.4.6.

In what follows, we will be most concerned with the Whittaker category on $\mathfrak{L}G/I$, i.e., with $\mathsf{Whit}_{\lambda}^{\operatorname{aff}}$ for $\lambda=0$. In this case, we replace the notation $\lambda$ by the level $\kappa$.

In other words, $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}\coloneqq\mathsf{Whit}_{0}^{\operatorname{aff}}$. We similarly denote the summands $\mathsf{Whit}_{0,y}^{\operatorname{aff}}$ of this category considered in Theorem 3.2.7 by $\mathsf{Whit}_{\kappa,y}^{\operatorname{aff}}$.

3.4.7.

Let us obtain, for good levels $\kappa_{\circ}$, the Kac–Moody realization of blocks of the Whittaker category. Fix a double coset, whose minimal length element we denote by $y$, in

[TABLE]

Corollary 3.4.8**.**

If $\kappa_{\circ}$ is good, then for any $y$ as above the corresponding block $\mathsf{Whit}_{\kappa,y}^{\operatorname{aff}}$ admits an equivalence with a block of $\widehat{\mathfrak{g}}_{\kappa_{\circ}+\kappa^{\prime}}\text{\textendash}\mathsf{mod}^{I,y^{-1}\cdot 0},$ for some integral level $\kappa^{\prime}$.

Proof.

By Proposition 3.3.8 and Theorem 3.2.7, it is enough to produce a simple Verma module in

[TABLE]

whose (antidominant) highest weight $\Theta$ has stabilizer under the dot action of $W$ given by

[TABLE]

By (3.9), it is equivalent to produce a simple Verma module in

[TABLE]

whose highest weight $y\cdot\Theta$ has stabilizer given by

[TABLE]

The latter is provided by Proposition 3.4.5, as desired. ∎

Remark 3.4.9*.*

As in Remark 3.3.4, and at the risk of redundancy, we emphasize: by construction, the integral Weyl group of the above block is identified as a Coxeter system with $W_{\mathfrak{g},\kappa}$, and the stabilizer of $\Theta$ in $W_{\mathfrak{g},\kappa}$ is $W_{\mathfrak{g},\kappa_{\circ}}\cap yW_{f}y^{-1}$.

Remark 3.4.10*.*

We recall that our running assumption in this section is that $\kappa$ is negative, as will be important for the arguments when we get to the fundamental local equivalences. However, everything until this point, in particular our analysis of the twisted Whittaker categories, applies to an arbitrary level $\kappa$.

3.5. From $\widehat{\mathfrak{g}}$-modules to $\widehat{\check{\mathfrak{g}}}$-modules

3.5.1.

To relate blocks of Category $\mathscr{O}$ for $\widehat{\mathfrak{g}}_{-\kappa}$ and $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}$, we would like to use the following result of Fiebig.

Let $\mathfrak{k}$ be an affine Lie algebra with Cartan subalgebra $\mathfrak{h}$. Fix $\alpha\in\mathfrak{h}^{*}$ such that the Verma module $M_{\alpha}$ is simple, and write $\mathscr{O}_{\alpha}$ for the corresponding block of Category $\mathscr{O}$ for $\mathfrak{k}$. Let us write $W_{\alpha}$ for the integral Weyl group of $\alpha$, and $W_{\alpha}^{\circ}$ for its subgroup stabilizing $\alpha$ under the dot action.

Theorem 3.5.2** ([Fie06] Thm. 4.1).**

As an abelian highest weight category, $\mathscr{O}_{\alpha}$ is determined up to equivalence by the Coxeter system $W_{\alpha}$ along with its subgroup $W_{\alpha}^{\circ}$.

Remark 3.5.3*.*

Theorem 3.5.2, as written in loc. cit., applies to symmetrizable Kac–Moody algebras, and in particular affine Kac–Moody algebras. Recall the latter consists of a Laurent polynomial version of the affine Lie algebra along with a degree operator $t\partial_{t}$. One may apply it to the present situation as follows. At any noncritical level, Category $\mathscr{O}$ for the affine Lie algebra canonically embeds as a Serre subcategory of Category $\mathscr{O}$ for the affine Kac–Moody algebra. Namely, one sets $t\partial_{t}$ to act by the semisimple part of $-L_{0}$, where $L_{0}$ is the Segal–Sugawara energy operator.

3.5.4.

To apply Theorem 3.5.2 in our situation, we must relate $W_{-\kappa,\mathfrak{g}}$ and $W_{\check{\kappa},\check{\mathfrak{g}}}$.

To do so, fix a level $\kappa_{\circ}$ for $\mathfrak{g}$. Let us write $\check{Q}$ for the coroot lattice and $Q$ for the root lattice of $\mathfrak{g}$. Associated to $\kappa_{\circ}$ is a map

[TABLE]

where $\kappa_{\mathfrak{g},c}$ denotes the critical level for $\mathfrak{g}$. In particular we may consider the sublattice of $\check{Q}$ given by

[TABLE]

Suppose that $\kappa_{\circ}$ is noncritical. Recall that, if we write $\check{\kappa}_{\check{\mathfrak{g}},c}$ for the critical level for $\check{\mathfrak{g}}$, then by definition, the dual nondegenerate bilinear form to $\kappa_{\circ}-\kappa_{\mathfrak{g},c}$ is $\check{\kappa}_{\circ}-\check{\kappa}_{\check{\mathfrak{g}},c}$. It follows we have a canonical identification

[TABLE]

3.5.5.

With this, we may canonically identify the integral Weyl groups on the opposite sides of quantum Langlands duality.

Theorem 3.5.6**.**

For any level $\kappa_{\circ}$, under the identification $W\simeq W_{f}\ltimes\check{Q}$ one has

[TABLE]

Moreover, for a noncritical level $\kappa_{\circ}$, there is a canonical isomorphism of Coxeter systems

[TABLE]

Remark 3.5.7*.*

As $G$ and $\check{G}$ in general have different affine Weyl groups, there is, perhaps, something surprising about Theorem 3.5.6.

Proof of Theorem 3.5.6.

We begin with (3.26). Recall from the proof of Lemma 3.4.3 that for a finite coroot $\check{\alpha}$ and integer $n$, the affine coroot $\check{\alpha}_{n}$, belongs to $W_{\mathfrak{g},\kappa_{\circ}}$ if and only if

[TABLE]

is an integer. This integrality condition may be rewritten as

[TABLE]

which in turn is equivalent to $n\check{\alpha}\in\check{Q}_{\kappa_{\circ}}.$ As the affine reflection in $W$ corresponding to $\check{\alpha}_{n}$ is explicitly given by $s_{\check{\alpha}}t^{n\check{\alpha}},$ it follows that we have an inclusion

[TABLE]

and that the left-hand side includes the translations $t^{n\check{\alpha}},$ for $n\check{\alpha}$ as above. To see that (3.28) is an equality, it suffices to see that $\check{Q}_{\kappa_{\circ}}$ lies in the left-hand side. But if we write an element $\check{\lambda}$ of $\check{Q}$ as a linear combination

[TABLE]

we have that $\kappa_{\circ}(\lambda)$ lies in $Q$ if and only if $\kappa_{\circ}(n_{i}\check{\alpha}_{i})$ lies in $Q$, for all $i\in\mathscr{I}$. Considering the affine coroots $(\check{\alpha}_{i})_{n_{i}}$ for $i\in\mathscr{I}$ and composing the translational parts of their reflections yields the desired equality.

Let us use the equality (3.26) to prove (3.27). Namely, via (3.22), we may assume that $\kappa_{\circ}$ is positive. Under this assumption, we will show the composite identification

[TABLE]

is an isomorphism of Coxeter systems. That is, we claim the sets of simple reflections are exchanged under the identification

[TABLE]

If $\kappa_{\circ}$ is irrational, this is clear, as both sides are $W_{f}$. Otherwise, let us write the level as

[TABLE]

where $p,q$ are positive coprime integers. Recall the affine simple reflection $s_{0,\kappa_{\circ}}$ from Lemma 3.4.3. Applying (3.29) to it, we obtain, writing $\theta_{s}$ for the short dominant root and $\theta_{l}$ for the long dominant root,

[TABLE]

To finish, recall that $\check{\kappa}_{\circ}$ is given by

[TABLE]

where $h^{\vee}_{\check{\mathfrak{g}}}$ and $\check{\kappa}_{\check{\mathfrak{g}},b}$ are the dual Coxeter number and basic level for $\check{\mathfrak{g}}$, respectively. Comparing the analog of (3.23) for $\check{\mathfrak{g}}$ to (3.31) shows the intertwining of the affine simple reflections by (3.29), as desired. ∎

3.5.8.

We may apply these as follows. Suppose $\kappa$ and $\check{\kappa}$ are dual levels, and $\kappa^{\prime}$ is an integral level for $G_{s}$. Suppose we are given a $y\in\widetilde{W}$ of minimal length in $W_{\mathfrak{g},-\kappa+\kappa^{\prime}}y$, and a simple Verma module

[TABLE]

Suppose we are further given a simple Verma module $M_{\check{\nu}}$ in $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}\mathsf{-mod}^{\check{I}_{s}}$, such that the stabilizers of $\mu$ and $\check{\nu}$ are identified via

[TABLE]

Corollary 3.5.9**.**

In the above situation, there is a $t$-exact equivalence

[TABLE]

Proof.

For either category, which we temporarily denote by $\mathscr{C}$, and by $\mathscr{C}^{\heartsuit,c}$ the finite length objects in its heart, the canonical map

[TABLE]

realizes the latter as the ind-completion of the former. To see this, note that since the blocks contain a simple Verma module, $\mathscr{C}^{\heartsuit,c}$ indeed consists of compact objects, and the fully faithfulness may be checked from Verma to dual Verma modules. Either $\mathscr{C}^{\heartsuit,c}$ is a block of Category $\mathscr{O}$ for the corresponding affine algebra, whence we are done by the assumptions on $\mu$ and $\check{\nu}$, the identification of Coxeter systems (3.32), and Theorem 3.5.2. ∎

3.6. The tamely ramified fundamental local equivalence

Recall we have assumed that $G$ is simple of adjoint type and that $\kappa$ is negative.

Theorem 3.6.1**.**

For $\kappa$ good, there is a $t$-exact equivalence

[TABLE]

Proof.

Recall our identification from 3.2.4 of the isomorphism classes of simple objects in the left-hand side of (3.33) with the coset space

[TABLE]

We showed in Lemma 3.2.9 that each orbit of $W_{\mathfrak{g},-\kappa}$ on the latter contains a minimal element $y$ with respect to the Bruhat order. In Corollary 3.4.8 we showed the corresponding block $\mathsf{Whit}_{\kappa,y}^{\operatorname{aff}}$ of the left-hand side of (3.33) identifies with a block of twisted $I_{s}$-integrable modules for $\widehat{\mathfrak{g}}_{-\kappa+\kappa_{\circ}}$, for an integral level $\kappa^{\prime}$. As discussed in the proof of Corollary 3.4.8, its integral Weyl group identifies as a Coxeter system with $W_{\mathfrak{g},-\kappa}$, and with this identification the stabilizer of the highest weight of a simple Verma module is given by

[TABLE]

On the other hand, if we denote by $\check{\Lambda}_{G}$ the cocharacter lattice of $T$, i.e. the character lattice of $\check{T}$, recall that $\widetilde{W}$ is explicitly

[TABLE]

Consider its action on $\check{\Lambda}_{G}$, where $W_{f}$ acts by the dot action and an element $\check{\lambda}$ in $\check{\Lambda}_{G}$ acts as translation by minus $\check{\lambda}$. Acting on $-\check{\rho}$ yields a $\widetilde{W}$-equivariant identification

[TABLE]

With this, recalling that $\kappa$ is negative, the restriction of (3.35) along the composition

[TABLE]

yields the standard dot action of $W_{\check{\mathfrak{g}},\check{\kappa}}$ on $\check{\Lambda}_{G}$. Moreover, under the equivalence (3.35), for any reflection $s_{\alpha}$ in $W_{\check{\mathfrak{g}},\check{\kappa}}$ and element $x$ of $\widetilde{W}$, one has that

[TABLE]

where the latter $\leqslant$ denotes the standard partial order on $\check{\Lambda}_{G}$. To see this note that, if we write $\Phi^{+}$ for the positive real coroots of $\check{\mathfrak{g}}$ and $\Phi_{f}$ for the finite coroots of $\check{\mathfrak{g}}$, both sides of (3.36) are equivalent to

[TABLE]

Using (3.36), we may describe the block decomposition of

[TABLE]

as follows. In its usual formulation, due to Deodhar–Gabber–Kac [DGK82], blocks correspond to $W_{\check{\mathfrak{g}},\check{\kappa}}$ dot orbits on $\check{\Lambda}_{G}$, and each contains a unique simple Verma module. Under the identification of its highest weights with $\widetilde{W}/W_{f}$ via (3.35), each orbit of $W_{\check{\mathfrak{g}},\check{\kappa}}$ contains a minimal element $y$ with respect to the Bruhat order. By (3.36), the corresponding Verma module is antidominant, i.e. simple, and has stabilizer

[TABLE]

Comparing Equations (3.34) and (3.37), we are done by Corollary 3.5.9. ∎

3.6.2.

To conclude, we record some properties of the above equivalence, which will be used in Proposition A.2.2 below.

Proposition 3.6.3**.**

For $\kappa$ good, the obtained equivalence

[TABLE]

interchanges, for any $\check{\lambda}$ in $\check{\Lambda}_{G}$, the (co)standard and simple objects, i.e.,

[TABLE]

Proposition 3.6.3 immediately implies, via the constructions of the appendix, a similar statement for general reductive groups $G$ at negative level.

Remark 3.6.4*.*

Having obtained the tamely ramified fundamental local equivalence for good levels, let us outline a variant of the proof which may be desirable.

Presently, we relate blocks of $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$ and $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}\text{\textendash}\mathsf{mod}^{\check{I}}$ by relating the former to $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{I}$ and applying Fiebig’s results. However, it should be possible to adapt the arguments of Bezrukavnikov–Yun on $\mathbb{V}$-functors provided in Section 4 and 5 of [BY13] to $\mathsf{Whit}_{\kappa}^{\operatorname{aff}}$, and thereby identify each block with a category of (possibly singular) Soergel modules. Comparing this with the similar identification provided by Fiebig, and matching the combinatorics exactly as in the proof of Theorem 3.6.1 should yield the desired equivalence.

This would remove the assumption of goodness on $\kappa$, and such a description of the Whittaker category should be equally applicable in other sheaf-theoretic contexts, e.g. metaplectic Whittaker sheaves over function fields.

3.7. Parahoric fundamental local equivalences

3.7.1.

Recall the canonical bijection between the simple roots of $\mathfrak{g}$ and $\check{\mathfrak{g}}$, which were indexed by $\mathscr{I}$. In particular, for a standard parahoric subgroup

[TABLE]

which corresponds to a subset $\mathscr{J}$ of $\mathscr{I}$, we may associate a dual parahoric

[TABLE]

3.7.2.

Let us obtain a parahoric extension of the tamely ramified fundamental local equivalence. In particular, we continue to assume that $\kappa$ is negative and $G$ is simple of adjoint type.

Theorem 3.7.3**.**

For $\kappa$ good, there is an equivalence

[TABLE]

Proof.

It is enough to produce an equivalence, which we denote provisionally by a dotted line, fitting into commutative diagram

[TABLE]

where we normalize the pull-back $\pi^{!*}$ associated to $\pi:\mathfrak{L}G/I\rightarrow\mathfrak{L}G/P$ to be $t$-exact. Here, as in the proof of Corollary 3.5.9, the superscripts $\heartsuit,c$ denote compact objects in the heart, which in the present situation is equivalent to finite length objects in the heart. Noting that both vertical arrows in (3.39) are full embeddings of Serre subcategories, it is enough to see that the essential images of the simple objects under are intertwined by (3.33). To see this claim, recall we denoted the subset of simple roots corresponding to the dual parahorics $P$ and $\check{P}$ by $\mathscr{J}$. With this, the simple objects on the Whittaker side lying in the essential image are intermediate extensions from the orbits

[TABLE]

where $x$ satisfies the three conditions

[TABLE]

The simple objects on the Kac–Moody side lying in the essential image have highest weights $\check{\lambda}$ satisfying

[TABLE]

Write $\check{\lambda}=x\cdot-\check{\rho}$, for $x\in\widetilde{W}$ acting as in (3.35). We may assume $x$ is of maximal length in its left $W_{f}$ coset, in which case (3.41) is equivalent to $x$ satisfying the three conditions

[TABLE]

We finish by noting that (3.40) and (3.42) describe the same subset of $\widetilde{W}$, namely the unique elements of maximal length in double cosets $W_{\mathscr{J}}xW_{f}$, such that the double coset satisfies

[TABLE]

where $\check{\Phi}_{f}$ denotes the finite coroots and $\check{\Phi}_{\mathscr{J}}$ the coroots of the Levi associated to $\mathscr{J}$. ∎

3.7.4.

We next observe that the analog of Proposition 3.6.3 again holds in the parahoric setting. As notation for the highest weights of $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}\text{\textendash}\mathsf{mod}^{\check{P}}$, let us write

[TABLE]

Proposition 3.7.5**.**

For $\kappa$ good, the obtained equivalence

[TABLE]

interchanges, for $\check{\lambda}$ in $\check{\Lambda}^{+}_{G,P}$, the corresponding (co)standard and simple objects.

Proof.

For simple objects this is contained in the proof of Theorem 3.7.3. For the remaining claims, note that the standard and costandard objects of the parahoric categories may be obtained from the standard and costandard objects of the Iwahori categories by appling the left and right adjoints of the vertical arrows in (3.39), respectively. Hence the claim follows from Proposition 3.6.3. ∎

Via the reductions in the appendix, a similar statement applies for general reductive $G$ at negative level.

3.7.6.

We finish with two remarks.

Remark 3.7.7*.*

Applying Theorem 3.7.3 in the maximal case, i.e., for the parahorics given by the arc groups $\mathfrak{L}^{+}G$ and $\mathfrak{L}^{+}\check{G}$, we obtain the spherical fundamental local equivalence for good levels, namely

[TABLE]

Remark 3.7.8*.*

We would like to record here the expectation that, for dual parahorics $P$ and $\check{P}$ as above, local quantum Langlands duality exchanges the operations of taking $P$ and $\check{P}$ invariants. For the Iwahori and arc subgroups, this already appears in the literature [Cam18], [Gai18c]. However, while one has a canonical bijection between the affine simple roots for $\mathfrak{g}$ and $\check{\mathfrak{g}}$, we do not expect an interchanging of the corresponding invariants of more general parahoric subgroups. Indeed, already the analog of Theorem 3.7.3 will typically fail, unless $\check{\kappa}$ is integral and $\mathfrak{g}$ is simply laced.

Appendix A Proof in the general case

In this appendix, we spell out how to deduce the general case of the conjectures from the case of $G$ of adjoint type and $\kappa$ a negative level. While we write the reductions in the tamely ramified case, they apply mutatis mutandis in the parahoric cases as well.

A.1. Good levels for general $G$

Recall the notion of a good level for a simple group, cf. Section 3.4.4 and Proposition 3.4.5. Let us a say a level $\kappa$ for a reductive group $G$ is good if it is good after restriction to each simple factor of $G$.

The following reductions in fact show the general case of the conjectures for reductive $G$ and $\kappa$ good reduce to the case of simple $G$ of adjoint type and $\kappa$ a good negative level. In particular, via Theorems 3.6.1 and 3.7.3, we obtain the fundamental local equivalences for general $G$ at good levels.

A.2. Finite isogenies

Suppose we are given a finite isogeny of pinned connected reductive groups

[TABLE]

The morphism $i$ yields a closed embedding of affine flag varieties, and hence a fully faithful embedding

[TABLE]

Consider the Langlands dual isogeny of connected reductive groups

[TABLE]

Associated to $\check{\iota}$ is a fully faithful restriction map

[TABLE]

To see the claimed fully faithfulness, one may use the following lemma.

Lemma A.2.1**.**

Suppose one is a given a fibre sequence of quasi-compact affine group schemes

[TABLE]

where $K$ is of finite type, the prounipotent radical $H^{u}$ is of finite codimension in $H$, and $pt/K$ is homologically contractible, i.e.,

[TABLE]

Then for any $D(Q)$-module $\mathscr{C}$, the restriction map $\mathscr{C}^{Q}\rightarrow\mathscr{C}^{H}$ is fully faithful.

Proof.

For $\mathscr{C}\simeq D(Q)$, this is exactly the assumption of homological contractibility. The case of general $\mathscr{C}$ follows from taking its cobar resolution as a $D(Q)$-module and using the commutation of $Q$ and $H$ invariants with colimits. ∎

We may apply the lemma to $\check{I}_{2}\rightarrow\check{I}_{1}$, as the kernel identifies with the kernel of $\check{\iota}$. Combining these assertions, we obtain:

Proposition A.2.2**.**

Suppose that one has an equivalence of the form (1.2) for $(G_{2},\kappa)$ and $(\check{G}_{2},\check{\kappa})$. Assume it exchanges the full subcategory generated under shifts and colimits by the Whittaker sheaves

[TABLE]

Then it induces an equivalence of the form (1.2) for $(G_{1},\kappa)$ and $(\check{G}_{1},\check{\kappa})$, fitting into a commutative diagram

[TABLE]

A.3. Products

By the preceding subsection, we may replace our group, after passing to a finite quotient thereof, by the product of a semisimple group of adjoint type and a torus. We next reduce to the case of a single factor.

Suppose that $G$ factors as a product of pinned connected reductive groups $G\simeq G_{1}\times G_{2}.$ Associated to this is a tensor product decomposition

[TABLE]

On the Langlands dual side, we obtain a decomposition $\check{G}\simeq\check{G}_{1}\times\check{G}_{2}$, and a similar tensor product decomposition

[TABLE]

In particular, to provide an equivalence as in (1.2) for $G_{1}\times G_{2}$, it is enough to do so for $G_{1}$ and $G_{2}$ separately.

A.4. Tori

Let us dispose of the torus factor of $G$. Given dual tori $T$ and $\check{T}$, it is clear that both sides of (1.2) canonically identify as

[TABLE]

corresponding to the components of the affine Grassmannian of $T$ and the Fock modules for $\widehat{\check{\mathfrak{t}}}_{\check{\kappa}}$, respectively.

A.5. Positive level

We have reduced the conjecture to $G$ simple of adjoint type, and a $\kappa$ arbitrary. We now reduce to $\kappa$ negative via cohomological duality.

For a connected reductive group $G$, there is a canonical $D_{\kappa}(\mathfrak{L}G)$-equivariant duality

[TABLE]

induced by Verdier duality. This yields a duality of the Whittaker invariants, cf. Section 4 of [Dhi19], i.e.,

[TABLE]

On the Kac–Moody side, recall that semi-infinite cohomology (defined with respect to any compact open subalgebra) induces an $D_{\kappa}(\mathfrak{L}G)$-equivariant duality

[TABLE]

cf. Section 9 of [Ras19].Passing to $\check{I}$ invariants, we obtain a duality

[TABLE]

Accordingly, an equivalence as in (1.2) at level $\kappa$ follows by duality from such an equivalence at level $-\kappa$.

We now check the compatibility of the above with the assumption of Proposition A.2.2 concerning essential images. For dual categories $\mathscr{C}$ and $\mathscr{C}^{\vee}$, let us write $\mathscr{C}^{c}$ and $(\mathscr{C}^{\vee})^{c}$ for their full subcategories of compact objects, and denote their induced identification by

[TABLE]

Lemma A.5.1**.**

Fix any $\check{\lambda}$ in $\check{\Lambda}_{G}$, and write $\rho$ and $\check{\rho}$ for the half sum of the positive roots and coroots of $G$, respectively. With respect to the duality datum (A.3), we have

[TABLE]

With respect to the duality datum (A.4), normalized with respect to the Lie algebra of the Iwahori $\check{I}$, we have

[TABLE]

The proof of the lemma is straightforward, cf. Lemma 9.8 of [Dhi19] for the assertion regarding Verma modules.