Localization for affine $\mathcal{W}$-algebras

First, these algebras exhibit Feigin–Frenkel duality: $\mathcal{W}_{\mathfrak{g},\kappa}\simeq\mathcal{W}_{\check{\mathfrak{g}},\check{\kappa}}$. Here $\check{\mathfrak{g}}$ is the Langlands dual Lie algebra, and $\check{\kappa}$ is the dual level (say, for $\kappa$ non-degenerate). This identification is expected to provide the interface for quantum local geometric Langlands; see [ff-duality], [bdh], [quantum-langlands-summary], and [whit].

For $\mathfrak{sl}_{2}$, the algebra $\mathcal{W}_{\kappa}$ is the completed enveloping algebra of a Virasoro algebra with central charge depending on $\kappa$. The Virasoro algebra is an infinite dimensional Lie algebra that has been extensively studied due to its appearance as a symmetry in conformal field theory and string theory.

There is a category $\mathscr{O}$ of lowest energy Virasoro representations. Remarkably, fairly general objects of this category appear in physical models. For example, highest weight modules in singular blocks appear already in free field theory, while continuous families of Verma modules appear in Liouville theory. For these reasons, lowest energy representations of the Virasoro algebra have received considerable study.

If $\mathfrak{g}$ has a simple factor of semisimple rank greater than one, $\mathcal{W}_{\kappa}$ is no longer associated with an infinite dimensional Lie algebra. However, there is a vertex algebra associated with $\mathcal{W}_{\kappa}$, which is conformal away from critical level. Therefore, $\mathcal{W}$-algebras provide fundamental examples of nonlinear symmetry algebras in conformal field theory.

Despite considerable interest in representation theory of $\mathcal{W}$-algebras, and folklore analogies between this subject and the geometry of affine flag variety, a direct connection between the two has not previously been established. Our main result, a localization theorem for affine $\mathcal{W}$-algebras, realizes this picture in a strong sense. Even for the Virasoro algebra, no geometric description of a category of its representations was previously known.

We first introduce the relevant Lie-theoretic data. Let $G$ be a semisimple, simply connected group¹¹1The results we prove straightforwardly extend, mutatis mutandis, to the case of any reductive group. with Lie algebra $\mathfrak{g}$, and $\kappa$ a noncritical level. Write $T$ for the abstract Cartan, $\mathfrak{t}$ for its Lie algebra, and $W_{\operatorname{f}}$ and $W$ for the finite and affine Weyl groups. Let $\lambda\in\mathfrak{t}^{*}$ be a weight that is regular of level $\kappa$. That is, we suppose it has trivial stabilizer under the level $\kappa$ dot action of the affine Weyl group, cf. Section 2.1.5.

We fix once and for all a Borel subgroup $B$ with unipotent radical $N$, so that $B/N\simeq T$. In addition, we fix a nondegenerate Whittaker character of conductor zero for the algebraic loop group $N_{F}$ of $N$

$$\psi:N_{F}\to{\mathbb{G}}_{a}.$$

The geometric side of our localization theorem is the DG category of $\kappa$-twisted, Whittaker equivariant, $\lambda$-monodromic $D$-modules on the enhanced affine flag variety. Below, we denote this category as

$$\mathsf{Whit}_{\kappa,\lambda\text{\textendash}mon}(\operatorname{Fl}),$$

cf. Section LABEL:ss:monodromy for our precise conventions.

The algebraic side of the localization theorem is the DG category of modules for $\mathcal{W}_{\kappa}$-algebra, cf. Section 2.3.3 for more detail. Below, we denote this category as

$$\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}.$$

Recall that the Verma modules $M_{\chi}$ for the $\mathcal{W}_{\kappa}$-algebra are parametrized by the central characters $\chi$ of the enveloping algebra $\mathscr{U}(\mathfrak{g})$. We identify the set of central characters with the set of closed points of $W_{\operatorname{f}}\backslash\mathfrak{t}^{*}$ via the Harish–Chandra homomorphism

$$\pi:\mathfrak{t}^{*}\rightarrow W_{\operatorname{f}}\backslash\mathfrak{t}^{*},$$

cf. Lemma LABEL:l:v2v for our precise normalizations.

With these notations in hand, we may state our main theorem.

As is typical in localization theory, the situation is especially nice when $\lambda$ is moreover antidominant of level $\kappa$. We refer to Section 2.1.5 for the standard definition of this notion, but emphasize that antidominant integral weights exist only for negative $\kappa$. We also recall from [whit] that there is a canonical construction of $t$-structures on DG categories of Whittaker equivariant $D$-modules. The construction is of semi-infinite nature, cf. Section 1.5.1.

As a consequence, we obtain the following calculation of characters of simple modules for $\mathcal{W}_{\kappa}$.

Corollary 1.3.10 and its extension to arbitrary blocks were originally proved by Arakawa in the seminal paper [ara]. In the antidominant case considered here, Corollary 1.3.10 provides a conceptual proof. Following the original methods [klic] of Kazhdan–Lusztig, various Kazhdan–Lusztig polynomials are well-known to calculate multiplicities in geometric settings. For example, in our setting, it is essentially standard that the Kazhdan–Lusztig polynomials considered here calculate multiplicities in $\mathsf{Whit}_{\kappa,\lambda\text{\textendash}mon}(\operatorname{Fl})^{\heartsuit}$. Therefore, we obtain Corollary 1.3.10 as a direct consequence of Theorem 1.3.7.

In the work of Arakawa, a Drinfeld-Sokolov reduction functor from $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ to $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}$ was shown to send Vermas to Vermas and simples to simples or zero, extending a conjecture of Frenkel–Kac–Wakimoto [fkw]. His results reduced the calculation of simple characters for $\mathcal{W}_{\kappa}$ to those for $\widehat{\mathfrak{g}}_{\kappa}$. These had been computed by Kashiwara–Tanisaki using localization onto the affine flag variety, as in the resolution of the original Kazhdan–Lusztig conjecture [ktl],[kt2],[kt]. Our work provides a direct geometric representation theoretic explanation for the relation between parabolic Kazhdan-Lusztig polynomials and $\mathcal{W}$-characters.

For finite $\mathcal{W}$-algebras, parallel results were proved by Ginzburg, Losev, and Webster [ginz], [los], [bw]. The expectation that something similar should hold in affine type was mentioned in work of Backelin–Kremnizer [krem].

For finite $\mathcal{W}$-algebras associated to a general nilpotent element $f$ of $\mathfrak{g}$, Dodd–Kremnizer realized representations with fixed central character in terms of asymptotic differential operators on the standard resolution of the intersection of the nilpotent cone with the Slodowy slice defined by $f$ [dk]. Arakawa–Kuwabara–Malikov then gave a chiralization of their construction to realize affine $\mathcal{W}$-algebras at critical level with fixed central character [akm].

For a principal nilpotent $f$, as in our results, the intersection of the Slodowy slice and the nilpotent cone is a point. In addition, we work at non-critical levels. Therefore, our localization results are disjoint from those of [akm].

For $G=SL_{n}$, Fredrickson–Neitzke observed a bijection between cells in a moduli spaces of wild Higgs bundles in genus zero and minimal models for the affine $\mathcal{W}$-algebra matching certain parameters [fn]. Our results do not speak to the phenomena they found.

The primary technical issue in proving Theorem 1.3.5 is its semi-infinite nature. As in [mys], semi-infinite phenomena present themselves when pairing abstract categorical methods and infinite-dimensional Lie groups. Below, we highlight some specific phenomena that appear in our present setting.

Our overall strategy is to prove Theorem 1.3.5 by passing to Whittaker equivariant objects in Kashiwara–Tanisaki’s localization theorem, which relates $D$-modules on the enhanced affine flag variety and Kac–Moody representations. However, there are no Whittaker equivariant objects in the abelian category of twisted $D$-modules on $\operatorname{Fl}$; this follows because $N_{F}$-orbits on $\operatorname{Fl}$ are infinite dimensional. Relatedly, there are no Whittaker equivariant objects in the abelian category of Kac–Moody representations.

However, for the corresponding DG categories of $D$-modules and Kac–Moody representations, there are robust categories of Whittaker equivariant objects. By the previous paragraph, such objects are necessarily concentrated in cohomological degree $-\infty$, i.e., these objects are in degrees $\leqslant-n$ for all integers $n$.²²2In particular, as we explain in more detail below, these DG categories are not the unbounded derived categories of the corresponding abelian categories. Thus, when we form global sections of a Whittaker $D$-module in Theorem 1.3.5, the underlying complex of vector spaces vanishes, but the corresponding object of $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}$ does not.

The connection between Whittaker equivariant objects in this setting and representations of $\mathcal{W}$-algebras was established by the second author in [whit]. In particular, one consequence of the main construction of [whit], which is crucial for the present work, is a systematic way to realize abelian categories of Whittaker equivariant objects, even though such objects lie in cohomological degree $-\infty$ when forgetting the equivariance.

The importance of considering categories of representations with objects in degree $-\infty$ in infinite dimensional algebras was first highlighted by Frenkel–Gaitsgory [fgl]. We follow them in sometimes referring to such methods as renormalization, and the resulting DG categories as renormalized.

Therefore, to handle the described issues of semi-infinite nature, and in particular to pass to Whittaker equivariant objects, we use a version of Kashiwara–Tanisaki localization which relates the renormalized categories of twisted $D$-modules and Kac–Moody representations and includes equivariance for categorical actions of loop groups. Such an enhancement is provided by the first author and J. Campbell in [ahc].

In the remainder of the introduction, we discuss some novel ingredients used to determine its essential image. On both sides of Theorem 1.3.5, the adolescent Whittaker filtration plays an essential role. This is a functorial filtration on Whittaker invariants of categorical loop group representations that was introduced in [whit].

On the geometric side, we show in Section 3 that the first term of the adolescent Whittaker filtration suffices to describe Whittaker $D$-modules on $\operatorname{Fl}$, i.e., we obtain a canonical equivalence between the Iwahori–Whittaker and Whittaker categories of $D$-modules on $\operatorname{Fl}$.

In the untwisted setting, this result was shown in [rcp]. However, here we deduce this result from a more general assertion. In effect,³³3We do not formulate the result in these terms. The assertion stated here follows from our Theorem 3.0.1 using the methods of [BZO]. we show that Whittaker invariants for a categorical representation generated by objects equivariant for the $n$th congruence subgroup of $\mathfrak{L}G$ are exhausted by the $n$th step in its adolescent Whittaker filtration, in line with the philosophy of [whit] Remark 1.22.2.

On the representation theoretic side, the adolescent Whittaker construction realizes subcategories of $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$ as certain categories of Harish-Chandra modules for $\widehat{\mathfrak{g}}_{\kappa}$.

We show that Category $\mathscr{O}$ for $\mathcal{W}_{\kappa}$ may be realized in the first step of the adolescent Whittaker filtration, i.e., as Iwahori–Whittaker modules for the Kac–Moody algebra. In Definition LABEL:d:catoiw, we provide an explicit description of the corresponding subcategory of Iwahori–Whittaker Kac–Moody representations; this subcategory is mapped isomorphically onto Category $\mathscr{O}$ for $\mathcal{W}_{\kappa}$ via Drinfeld–Sokolov reduction.

To our knowledge, the existence of such an Iwahori–Whittaker model for Category $\mathscr{O}$ was not anticipated prior to [whit]. As an application of this perspective on Category $\mathscr{O}$, we give a short new proof of the character formula for simple positive energy representations of $\mathcal{W}_{\kappa}$, originally obtained by Arakawa in the seminal paper [ara].

In order to obtain the above results, we give a new description of the adolescent Whittaker filtration of $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}$. Unlike the descriptions in [whit], which involved Kac–Moody algebras, our description is intrinsic to the $\mathcal{W}$-algebra, involving local nilpotency of certain explicit elements.

One important consequence of our description is that the abelian subcategories of $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$ provided by the adolescent Whittaker construction are closed under subobjects; this was not clear to the second author when writing [whit].

Having completed our analysis of the relevant steps in the adolescent Whittaker filtration, the determination of the essential image reduces to a problem on baby Whittaker categories. The latter question, which is now of finite-dimensional (i.e., no longer semi-infinite) nature, follows the pattern of standard arguments in geometric representation theory.

Fix a simply-connected semisimple algebraic group $G$ with Cartan and Borel subgroups $T\subset B$, and let $N$ denote the radical of $B$. Let $B^{-}$ be the corresponding opposed Borel. Let $\mathfrak{g},\mathfrak{b},\mathfrak{n}$ and $\mathfrak{t}$ denote the corresponding Lie algebras.

We write $\Lambda$ for the weight lattice, i.e., the characters of $T$, and $\check{\Lambda}$ for the coweight lattice, i.e., the cocharacters of $T$. Within $\check{\Lambda}$, we denote the coroots by $\check{\Phi}_{\operatorname{f}}$, and within them the simple coroots by

$$\check{\alpha}_{i},\quad\text{ for }i\in\mathscr{I}.$$

We will write $W_{\operatorname{f}}$ for the finite Weyl group. In addition to its linear action on the dual Cartan $\mathfrak{t}^{\vee}$, we will use its dot action. I.e., if we write $\rho$ for the half sum of positive roots in $\mathfrak{t}^{*}$, then the dot action of an element $w$ in $W_{\operatorname{f}}$ on $\lambda\in\mathfrak{t}^{*}$ is defined by

$$w\cdot\lambda:=w(\lambda+\rho)-\rho,$$

where the right-hand action is the linear one.

Recall that $\kappa$ denotes a level, i.e., an $\operatorname{Ad}$-invariant bilinear form on $\mathfrak{g}$. There are two particular levels that play distinguished roles. We will denote by $\kappa_{c}$ the critical level, i.e., minus one half times the Killing form. We denote by $\kappa_{b}$ the basic level. If $\mathfrak{g}$ is simple, $\kappa_{b}$ is the unique level for which the short coroots have squared length two. I.e., for a short coroot $\check{\check{\alpha}}$, one has

$$\kappa_{b}(\check{\alpha},\check{\alpha})=2.$$

For semisimple $\mathfrak{g}$, $\kappa_{b}$ is the unique level restricting to the basic form on each simple factor. For $G$ simple, we recall that $\kappa_{c}$ is necessarily a scalar multiple of ${\kappa_{b}}$; by definition, that scalar is minus the dual Coxeter number of $G$.

If $\mathfrak{g}$ is a simple Lie algebra, we call a level $\kappa$ noncritical if it does not equal $\kappa_{c}$. We call a level $\kappa$ positive if it lies in

$$\kappa_{c}+\mathbb{Q}^{\geqslant 0}\kappa_{b},$$

and $\kappa$ negative if it is not positive. If $\mathfrak{g}$ is a semisimple Lie algebra, write it as a sum of simple Lie algebras

$$\mathfrak{g}\simeq\bigoplus_{j\in\mathscr{J}}\mathfrak{g}_{j}$$

We say a level $\kappa$ for $\mathfrak{g}$ is noncritical if its restriction to each $\mathfrak{g}_{j}$ is noncritical. Similarly, we say $\kappa$ is positive if its restriction to each $\mathfrak{g}_{j}$ is positive, and $\kappa$ is negative if its restriction to each $\mathfrak{g}_{j}$ is negative.

Write $F:=\mathbb{C}(\!(t)\!)$ denote the field of Laurent series. Write $G_{F}$ for the algebraic loop group of $G$, and $\mathfrak{g}_{F}$ for its Lie algebra. Here and below, we will mean topological Lie algebras, so in this case

$${\mathfrak{g}}_{F}:=\mathfrak{g}\underset{\mathbb{C}}{\otimes}F.$$

Associated to our level $\kappa$ is the affine Lie algebra, given as a central extension

$$0\rightarrow\bigoplus_{j\in\mathscr{J}}\mathbb{C}\mathbf{c}_{j}\rightarrow\widehat{\mathfrak{g}}_{\kappa}\rightarrow{\mathfrak{g}}_{F}\rightarrow 0.$$

Explicitly, for elements $X$ and $Y$ of $\mathfrak{g}_{i}$ and $\mathfrak{g}_{j}$, for $i$ and $j$ in $\mathscr{J}$, and Laurent series $f$ and $g$ in $\mathbb{C}(\!(t)\!)$, the bracket is given by

$$[X\otimes f,Y\otimes g]\hskip 2.84526pt=\hskip 2.84526pt[X,Y]\otimes fg\hskip 2.84526pt+\hskip 2.84526pt\delta_{i,j}\cdot\kappa(X,Y)\cdot{\operatorname{Res}}\hskip 2.84526ptdfg\cdot\mathbf{c}_{i},$$

where $\operatorname{Res}$ denotes the residue, and $\delta_{i,j}$ the Kronecker delta function.

Consider the affine Cartan

$$\mathfrak{t}\oplus\bigoplus_{j\in\mathscr{J}}\mathbb{C}\mathbf{c}_{j}.$$

We may write its linear dual as

(2.1)

$$\mathfrak{t}^{*}\oplus\bigoplus_{j\in\mathscr{J}}\mathbb{C}\mathbf{c}_{j}^{*},$$

where $\mathbf{c}^{*}_{j}$ pairs with $\mathfrak{t}$ and the $\mathbf{c}_{i}$, for $i\neq j$, by zero and pairs with $\mathbf{c}_{j}$ by one. Let us denote the real affine coroots by $\check{\Phi}$, and the simple affine coroots by

$$\check{\alpha}_{i},\quad\text{ for }i\in\hat{\mathscr{I}}.$$

Explicitly, $\widehat{\mathscr{I}}:=\mathscr{I}\sqcup\mathscr{J}$. To write the corresponding elements of $\mathfrak{t}\oplus\mathbb{C}\mathbf{c}$, if we denote by $\check{\theta}_{j}$ the short dominant coroot of $\mathfrak{g}_{j}$, by $\kappa_{j}$ the restriction of $\kappa$ to $\mathfrak{g}_{j}$, and $\kappa_{b,j}$ the basic level for $\mathfrak{g}_{j}$, then they are given by

$$\check{\alpha}_{i},\quad\text{for }i\in\mathscr{I},\quad\quad\text{ and }\quad\quad\check{\alpha}_{j}:=-\check{\theta}_{j}+\frac{\kappa_{j}}{\kappa_{b,j}}\hskip 2.84526pt\mathbf{c}_{j},\quad\text{for }j\in\mathscr{J}.$$

We will write $W$ for the affine Weyl group. In addition to its linear action on the dual affine Cartan, we will use its dot action. I.e., if we write $\check{\rho}$ for the unique element of (2.1) satisfying

$$\langle\check{\rho},\check{\alpha}_{i}\rangle=1,\quad\text{for }i\in\widehat{\mathscr{I}},$$

then the dot action of $w$ in $W$ on an element $\lambda$ of (2.1) is given by

$$w\cdot\lambda=w(\lambda+\hat{\rho})-\hat{\rho}.$$

We may identify $\mathfrak{t}^{*}$ with the affine subspace

$$\mathfrak{t}^{*}+\sum_{j\in\mathscr{J}}\mathbf{c}_{j}^{*}.$$

This affine subspace is preserved by both the linear and dot actions of $W$, and we will only need these induced affine linear actions on $\mathfrak{t}^{*}$ in what follows.

Recall that a weight $\lambda$ of $\mathfrak{t}^{*}$ is antidominant if

$$\langle\lambda+\check{\rho},\check{\alpha}_{i}\rangle\notin\mathbb{Z}^{>0},\quad\text{for }i\in\widehat{\mathscr{I}},$$

and that $\lambda$ is dominant if

$$\langle\lambda+\check{\rho},\check{\alpha}_{i}\rangle\notin\mathbb{Z}^{<0},\quad\text{for }i\in\widehat{\mathscr{I}}.$$

If $\kappa$ is positive, then the dot orbit of $\lambda$ always contains a dominant weight, and if $\kappa$ is negative, then the dot orbit of $\lambda$ always contains an antidominant weight.

Associated to any weight $\lambda$ of $\mathfrak{t}^{*}$ is the subset of integral real affine coroots

$$\check{\Phi}_{\lambda}:=\{\check{\alpha}\in\check{\Phi}:\langle\check{\alpha},\lambda\rangle\in\mathbb{Z}\}.$$

The integral Weyl group $W_{\lambda}$ is the subgroup of $W$ generated by the reflections indexed by the integral coroots.

We recall that a weight $\lambda$ is said to be regular if its stabilizer in $W$ under the dot action is trivial. If $\kappa$ is not critical, this coincides with its stabilizer within $W_{\lambda}$ under the dot action.

For $\widehat{\mathfrak{g}}_{\kappa}$ the affine algebra as above, we let $\mathcal{W}_{\kappa}$ denote the $\mathcal{W}$-algebra associated to $\widehat{\mathfrak{g}}_{\kappa}$ and a principal nilpotent element $f$ of $\mathfrak{g}$. I.e., $\mathcal{W}_{\kappa}$ is the vertex algebra obtained as the quantum Drinfeld-Sokolov reduction of the vacuum representation of $\widehat{\mathfrak{g}}_{\kappa}$, cf. [aran] and Chapter 15 of [fbz]. For $\kappa$ noncritical, $\mathcal{W}_{\kappa}$ contains a canonical conformal vector. Throughout this paper, unless otherwise specified, we assume $\kappa$ is noncritical.

We recall that the Zhu algebra of $\mathcal{W}_{\kappa}$ identifies with the center of the universal enveloping algebra of $\mathfrak{g}$. We refer to Lemma LABEL:l:v2v for our precise normalization of this isomorphism. In particular, for a central character $\chi$, we denote by $M_{\chi}$ the corresponding Verma module for $\mathcal{W}_{\kappa}$, and by ${L}_{\chi}$ its unique simple quotient.

For a DG category $\mathscr{C}$ equipped with a $t$-structure, we write $\mathscr{C}^{\heartsuit}$ for the abelian category of objects lying in its heart, and $\mathscr{C}^{+}$ for its full subcategory consisting of bounded below, i.e., eventually coconnective, objects.

We denote the abelian category of smooth modules for $\widehat{\mathfrak{g}}_{\kappa}$ on which each $\mathbf{c}_{j}$, for $j\in\mathscr{J}$, acts via the identity by

(2.2)

$$\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}.$$

We denote by $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ the DG category of smooth modules for $\widehat{\mathfrak{g}}_{\kappa}$, which is a renormalization of the unbounded DG derived category of $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$ introduced by Frenkel–Gaitsgory [fg]. Explicitly, within the bounded derived category of (2.2), consider the subcategory generated under cones and shifts from modules induced from trivial representations of compact open subalgebras of $\widehat{\mathfrak{g}}_{\kappa}$; by definition, $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ is the ind-completion of this category. The category $\widehat{\mathfrak{g}}_{\kappa}\text{\textendash}\mathsf{mod}$ carries a $t$-structure with heart (2.2); its bounded below part identifies canonically with the bounded below derived category of (2.2). Further details may be found in Sections 22 and 23 of [fg] as well as the very readable Section 2 of [gn].

We denote the abelian category of modules, in the sense of vertex algebras, over $\mathcal{W}_{\kappa}$ by

(2.3)

$$\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}.$$

We denote by $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}$ the DG category of modules for $\mathcal{W}_{\kappa}$. This is a renormalization of the unbounded DG derived category of $\mathcal{W}_{\kappa}\text{\textendash}\mathsf{mod}^{\heartsuit}$ introduced in [whit]. As in the Kac–Moody case, it may be constructed as the ind-completion of the subcategory generated by an explicit collection of modules within the bounded derived category of (2.3), cf. Section 4 of loc. cit. and Section 4 of the present paper.

We denote by $\mathsf{DGCat}$ the $(\infty,1)$-category of DG categories and DG functors between them. We let $\mathsf{DGCat}_{cont}\subset\mathsf{DGCat}$ denote the $1$-full subcategory of of cocomplete DG categories and continuous DG functors between them. We refer to [gaitsroz] for background material on the $\infty$-categorical perspective on DG categories.

For a general treatment of $D$-modules on infinite dimensional varieties, we refer the reader to [rdm] and [beraldo]. However, with the exception of ($\kappa$-twisted) $D$-modules on the loop group, we will only deal with the DG category of $D$-modules on ind-schemes $X$ of ind-finite type.

Let us summarize the relevant aspects of the theory in this simpler case. For an indscheme $X$ written⁴⁴4Of course, for cardinality reasons, not every ind-scheme can be written as such a union. However, this is satisfied for our examples of interest, so for concreteness, we assume it. as an ascending union

$$Z_{0}\rightarrow Z_{1}\rightarrow Z_{2}\rightarrow\cdots$$

of schemes of finite type along closed embeddings, the category $D(X)$ of $D$-modules on $X$ is the colimit in $\mathsf{DGCat}_{cont}$ of the corresponding categories for the $Z_{i}$ along $*$-pushforwards, i.e.,

(2.4)

$$D(X)\simeq\operatorname{colim}D(Z_{0})\rightarrow D(Z_{1})\rightarrow D(Z_{2})\rightarrow\cdots.$$

In particular, by the $t$-exactness of these pushforwards, $D(X)$ has a natural $t$-structure. Moreover, its bounded below part $D(X)^{+}$ identifies with the bounded below derived category of its heart $D(X)^{\heartsuit}$, cf. Lemma 5.4.3 of [whit]. Finally, we should note that $D(X)$ is compactly generated, with

$$D(X)^{c}\simeq\operatorname{colim}D(Z_{0})^{c}\rightarrow D(Z_{1})^{c}\rightarrow D(Z_{2})^{c}\rightarrow\cdots,$$

where the superscript $c$ denotes compact objects and the appearing colimit is taken in $\mathsf{DGCat}$. Plainly, a compact object of $D(X)$ is simply a bounded complex of $D$-modules with coherent cohomology $*$-extended from some $Z_{n}$, and $\operatorname{Hom}$ between two such objects may be computed in any $Z_{n}$ containing both their supports.

We recall that $\kappa$ defines categories of twisted $D$-modules the loop group $G_{F}$ and its descendants. More precisely, there is a monoidal DG category $D_{\kappa}(G_{F})$ defined in [whit] Section 1.30 and [mys]; in the latter source, this category is denoted $D_{\kappa}^{*}(G(K))$. This twisting is canonically trivialized on the compact open subgroup $G_{O}$ of regular maps from the disc to $G$, and below we use the restriction of this trivialization to $I\subset G_{O}$. Similarly, the twisting is canonically trivialized on the group ind-scheme $N_{F}\subset G_{F}$ of loops into $N$.

Throughout this paper, we make essential use of techniques from categorical representation theory. We review below some basic features we employ; the reader may wish to consult the user-friendly [beraldo] or the foundational paper [mys] as further references. We also refer to [paris-notes] for some useful lecture notes on the subject.

Given a group ind-scheme $H$, by functoriality the multiplication on $H$ endows the category $D(H)$ with a monoidal structure given by convolution. We denote the associated $(\infty-)$category of DG categories equipped with an action of $D(H)$ by

$$D(H)\text{\textendash}\mathsf{mod}:=D(H)\text{\textendash}\mathsf{mod}(\mathsf{DGCat}_{cont}).$$

Given a $D(H)$-module $\mathscr{C}$, one may form its categories of invariants and coinvariants, respectively given by

$$\mathscr{C}^{H}:=\mathsf{Hom}_{D(H)\text{\textendash}\mathsf{mod}}(\mathsf{Vect},\mathscr{C})\quad\text{and}\quad\mathscr{C}_{H}:=\mathsf{Vect}\underset{D(H)}{\otimes}\mathscr{C}.$$

There are tautological ‘forgetting and averaging’ adjunctions

$$\operatorname{Oblv}:\mathscr{C}^{H}\rightleftarrows\mathscr{C}:\operatorname{Av}_{H,*}$$

$$\operatorname{ins}^{L}:\mathscr{C}_{H}\rightleftarrows\mathscr{C}:\operatorname{ins}.$$

If $H$ is a group scheme whose pro-unipotent radical is of finite codimension, then there is a canonical identification of invariants and coinvariants. Namely, the map $\mathscr{C}_{H}\rightarrow\mathscr{C}^{H}$ induced by $\operatorname{Av}_{H,*}$ is an equivalence. We will use these equivalences implicitly throughout.

We will also need the twists of the above by a character. The data of a multiplicative $D$-module $\chi$ on $H$,⁵⁵5Strictly speaking, in this generality this should be regarded as an object of the dual category $D^{!}(H)$., is equivalent to an action of $D(H)$ on $\mathsf{Vect}$ which we denote by $\mathsf{Vect}_{\chi}$. One has associated categories of twisted invariants and coinvariants

$$\mathscr{C}^{H,\chi}:=\mathsf{Hom}_{D(H)\text{\textendash}\mathsf{mod}}(\mathsf{Vect}_{\chi},\mathscr{C})\quad\text{and}\quad\mathscr{C}_{H,\chi}:=\mathsf{Vect}_{\chi}\underset{D(H)}{\otimes}\mathscr{C}.$$

These fit into adjunctions as above, and under the same hypotheses on $H$ one may canonically identify twisted invariants and coinvariants.

We will need two examples of nontrivial multiplicative $D$-modules. First, given an element

$$\lambda\in\operatorname{Hom}(H,\mathbb{G}_{m})\underset{\mathbb{Z}}{\otimes}\mathbb{C}$$

one has an associated multiplicative $D$-module $``t^{\lambda}"$ on $H$. We denote the corresponding twisted invariants or coinvariants by a superscript or subscript $(H,\lambda)$, respectively. In what follows, this will principally be applied to Iwahori subgroups of $G_{F}$.

Second, given an element $\psi\in\operatorname{Hom}(H,\mathbb{G}_{a})$ one has an associated multiplicative $D$-module $``e^{\psi}"$ on $H$. We denote the corresponding twisted invariants and coinvariants by a superscript or subscript of $(H,\psi)$, respectively. This will principally be applied to $N_{F}$ and related prounipotent subgroups of $G_{F}$.

Applying the general constructions of the previous section, to a $D_{\kappa}(G_{F})$-module $\mathscr{C}$ one may attach its Whittaker invariants and coinvariants

$$\mathscr{C}^{N_{F},\psi}\quad\text{and}\quad\mathscr{C}_{N_{F},\psi}.$$

A principal result of [whit] is that these may be canonically identified. This is non-trivial because $N_{F}$ is not a group scheme, but rather a group ind-scheme. We employ this identification throughout. Unless an argument is biased toward one of these perspectives, we refer to them both as the Whittaker model of $\mathscr{C}$, and we denote this category by $\mathsf{Whit}(\mathscr{C})$.

Refining the equivalence between invariants and coinvariants above, [whit] constructed a canonical filtration of any Whittaker model by full subcategories

$$\mathsf{Whit}^{\leqslant 1}(\mathscr{C})\subset\mathsf{Whit}^{\leqslant 2}(\mathscr{C})\subset\cdots\quad\quad\underset{n}{\operatorname{colim}}\,\mathsf{Whit}^{\leqslant n}(\mathscr{C})\simeq\mathsf{Whit}(\mathscr{C}).$$

As we presently review, these may be identified with the twisted invariants for certain compact open subgroups $\mathring{I}_{n}$ of $G_{F}$.

Fix a positive integer $n$. Consider the $n^{th}$ congruence subgroup $K_{n}$ of $G_{O}$, and write $G_{n-1}$ for the quotient, and similarly consider $N_{n-1}$. One may form the subgroup of $G_{O}$ consisting of arcs which until $n$th order lie in $N$, i.e.

$$\mathring{J}_{n}:=G_{O}\underset{G_{n-1}}{\times}N_{n-1}.$$

One then obtains $\mathring{I}_{n}$ by conjugation, namely

$$\mathring{I}_{n}:=\operatorname{Ad}_{t^{-n\check{\rho}}}\mathring{J}_{n}.$$

Each $\mathring{I}_{n}$ admits a unique additive character $\psi$ which is (i) trivial on $\mathring{I}_{n}\cap B^{-}_{F}$ and (ii) agrees with the Whittaker character on $\mathring{I}_{n}\cap N_{F}$. There are canonical identifications

$$\mathsf{Whit}^{\leqslant n}(\mathscr{C})\simeq\mathscr{C}^{\mathring{I}_{n},\psi},$$

and the transition functors in the above filtration are given by averaging, cf. Section 2 of [whit] for more details.

The first case step in the filtration will be of particular importance to us. Note that $\mathring{I}_{1}$ is the prounipotent radical of an Iwahori subgroup. For ease of notation, we often denote them by $\mathring{I}$ and $I$, respectively. In what follows, we sometimes refer to the corresponding invariants as the baby Whittaker model.