${\mathcal {W}}$-algebras and Whittaker categories

Abstract

This article is concerned with Whittaker models in geometric representation theory, and gives applications to the study of affine ${\mathcal {W}}$-algebras. The main new innovation connects Whittaker models to invariants for compact-open subgroups of the loop group. This method, which has a counterpart for p-adic groups, settles a conjecture of Gaitsgory in the categorical setting. This method shows that Whittaker sheaves in geometric representation theory admit t-structures, as had previously been observed in some special cases. We then apply this method to the setting of affine ${\mathcal {W}}$-algebras. We study a new family of modules for affine ${\mathcal {W}}$-algebras, which can be thought of as analogues of certain tautological (“generalized vaccuum”) modules over the Kac-Moody algebra. Using the above t-structure, we obtain an affine analogue of Skryabin’s theorem that connects affine ${\mathcal {W}}$-algebras and Whittaker models. This theorem allows various geometric methods to be used to study affine ${\mathcal {W}}$-algebras. As one such application, we offer a new proof of one of Arakawa’s foundational results in the theory.

Appendices

Appendix A: Filtrations, Harish-Chandra modules, and semi-infinite cohomology

1.1 Overview

A.1.1. In this appendix, we give a slightly non-standard construction of the (quantum) Drinfeld-Sokolov reduction $\Psi :\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$, and discuss its compatibility with various filtrations. This material supports the calculations of Sect. 4.

A.1.2. The treatment we give here is quite lengthy, but this does not reflect the seriousness of the contents. There is a small number of ideas, and we summarize them here as themes to keep an eye towards:

• Recall that the Drinfeld-Sokolov functor is defined as $M \mapsto C^{\!\frac{\infty }{2}}({{\mathfrak {n}}}((t)),{{\mathfrak {n}}}[[t]],M \otimes (-\psi ))$, where $C^{\!\frac{\infty }{2}}$ is the semi-infinite cohomology functor. This functor should be thought of as “cohomology along ${{\mathfrak {n}}}[[t]]$ and homology along ${{\mathfrak {n}}}((t))/{{\mathfrak {n}}}[[t]]$,” although this does not quite make sense. Here we recall that cohomology is well-behaved⁷² for pro-finite dimensional Lie algebras, while homology is well-behaved for “discrete” Lie algebras (i.e., non-topologized ones).

  We give a slightly non-standard treatment of this semi-infinite cohomology functor, avoiding irrelevant Clifford algebras. Rather, we define:

  $$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {n}}}((t)),{{\mathfrak {n}}}[[t]],-) {:}{=}\underset{n \ge 0}{{\text {colim}}} \, C^{\bullet }({\text {Ad}}_{-n\check{\rho }(t)} {{\mathfrak {n}}}[[t]],-) [\dim ({\text {Ad}}_{-n\check{\rho }(t)} {{\mathfrak {n}}}[[t]]/{{\mathfrak {n}}}[[t]])]. \end{aligned}$$

  Here we recall that for finite-dimensional Lie algebras, Lie algebra cohomology and homology differ by a determinant twist and cohomological shift, so satisfy both covariant and contravariant functoriality (up to these twists and shifts) with respect to the Lie algebra. There is a relic of this for Lie algebra cohomology for profinite-dimensional Lie algebras, giving the structure morphisms in the above colimit. (So in fact, we should have included a twist by the line $\det ({\text {Ad}}_{-n\check{\rho }(t)} {{\mathfrak {n}}}[[t]]/{{\mathfrak {n}}}[[t]])$ to make the above structure maps canonical.)

  So much attention is paid to the construction of such morphisms.

• The other major theme is filtrations.

  The first question is given a (PBW) filtered module over a Lie algebra, how is its co/homology filtered? What is its associated graded?

  But there is a more subtle point in working with (finite or affine) ${\mathcal {W}}$-algebras: one would like the associated graded of a 1-dimensional module corresponding to a character $\psi$ of ${{\mathfrak {n}}}$ (or similarly for ${{\mathfrak {n}}}((t))$) to be the skyscraper sheaf at $\psi \in {{\mathfrak {n}}}^{\vee }$ in $\mathsf {QCoh}({{\mathfrak {n}}}^{\vee }) = {\text {Sym}}({{\mathfrak {n}}})\text {--}\mathsf {mod}= {\text {gr}}_{\bullet } U({{\mathfrak {n}}}) \text {--}\mathsf {mod}$. However, this is impossible: it is a graded module, so must be at the origin in ${{\mathfrak {n}}}^{\vee }$.

  The (standard) solution to this problem is the Kazhdan-Kostant method, which twists the filtration using $\check{\rho }:{{\mathbb {G}}}_m \rightarrow T$. This is the only canonical way of obtaining (non-derived) filtrations on ${\mathcal {W}}$-algebras.

  Finally, we emphasize the relationship between the PBW and Kazhdan-Kostant filtrations using bifiltrations; this material is used in Sect. 4 to settle a subtle homological algebra point regarding some Kazhdan-Kostant filtrations.

Each of these amount to completely elementary constructions and statements about Lie algebra co/homology, and its relation to Harish-Chandra conditions.

The reason this section is so long is rather out of a commitment to develop the theory with an emphasis on the categorical aspects. A primary reason for this is because, following [24], the derived category of ${{\mathfrak {g}}}[[t]]$ or ${{\mathfrak {g}}}((t))$-modules is subtle, and is better to understand through categories than (topological) algebras. One thing this requires, however, is some general formalism for working with filtrations on categories rather than on algebras and their modules.

This section also renders the theory in the $\mathsf {IndCoh}$ formalism of [42]. It has the advantage that it provides a robust framework for Lie algebra cohomology that does not rely on explicit formulae. But this accounts for some portion of the length: we have explained some elementary points in detail, in the hopes that this is instructive for understanding the formalism. We also hope that an $\mathsf {IndCoh}$ treatment gives the feeling why things are the way they are, and that they could never have been another way.

Finally, we advise the reader to look to Sect. 4.2.4, where we give another summary of what is actually needed from this section, which should also help identify what is most important here.

1.2 Filtrations

A.2.1. We begin with some abstract language about filtrations.

Definition A.2.1

Let ${\mathcal {C}}\in \mathsf {DGCat}_{cont}$ be given. A filtration on ${\mathcal {C}}$ is a datum of $\widetilde{{\mathcal {C}}} \in \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)\text {--}\mathsf {mod}$ plus an isomorphism:

$$\begin{aligned} {\mathcal {C}}\simeq \mathsf {Fil}\, {\mathcal {C}}\underset{\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)}{\otimes } \mathsf {QCoh}(({{\mathbb {A}}}_{\hbar }^1{\setminus } 0)/{{\mathbb {G}}}_m) = \underset{\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)}{\otimes } \mathsf {Vect}. \end{aligned}$$

Here ${{\mathbb {A}}}_{\hbar }^1$ is ${{\mathbb {A}}}^1$ with coordinate $\hbar$, and ${{\mathbb {G}}}_m$ acts by inverse⁷³ homotheties.

An object of $\mathsf {Fil}\, {\mathcal {C}}$ is called a filtered object of ${\mathcal {C}}$. Note that there is a canonical restriction functor $\mathsf {Fil}\, {\mathcal {C}}\rightarrow {\mathcal {C}}$. For ${\mathcal {F}}\in {\mathcal {C}}$, we refer to an extension $\widetilde{{\mathcal {F}}}$ of ${\mathcal {F}}$ to $\mathsf {Fil}\, {\mathcal {C}}$ as a filtration on ${\mathcal {F}}$.

We say a morphism $F:{\mathcal {C}}\rightarrow {\mathcal {D}}$ between filtered categories is filtered if we are given the data of $\mathsf {Fil}\, F: \mathsf {Fil}\, {\mathcal {C}}\rightarrow \mathsf {Fil}\, {\mathcal {D}}$ a $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$-linear functor.

In the language of [38], we would say $\mathsf {Fil}\, {\mathcal {C}}$ is a sheaf of categories over ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$ with fiber ${\mathcal {C}}$ at the open point of this stack.

Notation A.2.2

For a filtered category⁷⁴${\mathcal {C}}$ as above, we let ${\mathcal {C}}^{cl}$ denote the fiber of $\mathsf {Fil}\, {\mathcal {C}}$ at 0; it is called the associated semi-classical category. Note that ${{\mathbb {G}}}_m$ acts weakly on ${\mathcal {C}}^{cl}$. For ${\mathcal {F}}\in {\mathcal {C}}$ filtered, we let ${\text {gr}}_{\bullet } {\mathcal {F}}$ denote the induced object of ${\mathcal {C}}^{cl}$, obtained by taking the fiber at $\hbar = 0$. Note that ${\text {gr}}_{\bullet }({\mathcal {F}})$ comes from an object of ${\mathcal {C}}^{cl,{{\mathbb {G}}}_m,w}$, which we also denote by ${\text {gr}}_{\bullet }({\mathcal {F}})$. For a filtered functor $F: {\mathcal {C}}\rightarrow {\mathcal {D}}$, we obtain a functor $F^{cl}:{\mathcal {C}}^{cl} \rightarrow {\mathcal {D}}^{cl}$, which we refer to as the corresponding semi-classical functor.

Example A.2.3

$\mathsf {Vect}$ is canonically filtered, with $\mathsf {Fil}\, \mathsf {Vect}{:}{=}\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$.⁷⁵ In this case, a filtered object is a ${{\mathbb {G}}}_m$-representation with a degree 1 endomorphism. A ${{\mathbb {G}}}_m$-representation is the same as a ${{\mathbb {Z}}}$-graded vector space, and we suggestively denote the nth term of this graded vector space as $F_n(V)$. Then our degree 1 endomorphism is a sequence of maps $F_n(V) \rightarrow F_{n+1}(V)$.

The induced object of $\mathsf {Vect}$ is obtained by inverting this degree 1 endomorphism $\hbar$ and taking the degree 0 component: this is computed as the colimit $V {:}{=}{\text {colim}}_n F_n(V)$. (So in this formalism, filtrations are by definition exhaustive.) We compute ${\text {gr}}V$ by taking the cokernel of $\hbar$ acting on $\oplus _n F_nV$, which is $\oplus _n {\text {Coker}}(F_{n-1} V \rightarrow F_n V)$. In this example, we let ${\text {gr}}_n V$ denote the nth summand.

Example A.2.4

The above example generalizes to general ${\mathcal {C}}$ in place of $\mathsf {Vect}$: it has a canonical filtration defined by the category ${\mathcal {C}}\otimes \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$, and the above calculations render as is. We refer to this as the constant filtration on ${\mathcal {C}}$.

Example A.2.5

For A a filtered associative DG algebra, the Rees construction provides an algebra object $A_{\hbar } \in \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$ with generic fiber A. Then the category $\mathsf {Fil}\, A\text {--}\mathsf {mod}{:}{=}A_{\hbar }\text {--}\mathsf {mod}(\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m))$ provides a filtration on $A\text {--}\mathsf {mod}$. Note that $A\text {--}\mathsf {mod}^{cl}$ is the DG category of ${\text {gr}}_{\bullet } A$-modules. The induced functor $A\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$ is canonically filtered.

Example A.2.6

Suppose ${\mathcal {C}}$ carries a weak action of ${{\mathbb {G}}}_m$. Then we define:

$$\begin{aligned} \mathsf {Fil}\, {\mathcal {C}}{:}{=}{\mathcal {C}}^{{{\mathbb {G}}}_m,w} \underset{\mathsf {Rep}({{\mathbb {G}}}_m)}{\otimes } \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m). \end{aligned}$$

Here $\mathsf {Rep}({{\mathbb {G}}}_m)$ acts on each of these categories as on any weak ${{\mathbb {G}}}_m$-invariants. This tensor product is considered as acted on by $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$ in the obvious way. Note that this really does define a filtration on ${\mathcal {C}}$, and that there is a canonical functor ${\mathcal {C}}^{{{\mathbb {G}}}_m,w} \rightarrow \mathsf {Fil}\, {\mathcal {C}}$, given by exterior product with the structure sheaf of ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$.

Concretely, suppose that ${\mathcal {C}}= A\text {--}\mathsf {mod}$ for $A = \oplus _{i\in {{\mathbb {Z}}}} A^i$ a ${{\mathbb {Z}}}$-graded algebra. Note that A inherits a filtration from its grading: set $F_i A = \oplus _{j \le i} A^j$.⁷⁶ Then $\mathsf {Fil}\, A\text {--}\mathsf {mod}$ is the DG category of filtered modules over this filtered algebra. In this case, functor $A\text {--}\mathsf {mod}^{{{\mathbb {G}}}_m,w} \rightarrow \mathsf {Fil}\, A\text {--}\mathsf {mod}$ takes a graded module and creates a filtered one using the same construction as above.

A.2.2. We will use the following terminology in what follows: a filtered vector space $F_{\bullet } V$ is bounded from below if for all $i \ll 0$, $F_i V = 0$. Note that the functor ${\text {gr}}_{\bullet }:\mathsf {Fil}\, \mathsf {Vect}\rightarrow \mathsf {Vect}$ is conservative when restricted to the subcategory of filtered vector spaces with bounded below filtrations.⁷⁷

1.3 Finite-dimensional setting

A.3.1. Our formalism for semi-infinite cohomology will be built from the finite-dimensional setting, so we spend a while discussing this case.

We fix ${{\mathfrak {h}}}\in \mathsf {Vect}^{\heartsuit }$ a finite-dimensional Lie algebra.

A.3.2. The⁷⁸ PBW filtration on $U({{\mathfrak {h}}})$ defines a filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ with ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl} = \mathsf {QCoh}({{\mathfrak {h}}}^{\vee })$.

Here is a more geometric perspective. Let $\exp ({{\mathfrak {h}}})$ denote the formal group associated with ${{\mathfrak {h}}}$. Recall that ${{\mathfrak {h}}}\text {--}\mathsf {mod}= \mathsf {IndCoh}({{\mathbb {B}}}\exp ({{\mathfrak {h}}}))$ so that the forgetful functor corresponds to !-pullback to a point.

We have an induced family ${{\mathfrak {h}}}_{\hbar }$ of Lie algebras over ${{\mathbb {A}}}_{\hbar }^1$ given by ${{\mathfrak {h}}}\otimes {\mathcal {O}}_{{{\mathbb {A}}}_{\hbar }^1}$ with the bracket given by $\hbar$ times the bracket coming from ${{\mathfrak {h}}}$. This family is obviously ${{\mathbb {G}}}_m$-equivariant, so defines $\mathsf {Fil}\, {{\mathfrak {h}}}$ a Lie algebra over ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$. The generic fiber of $\mathsf {Fil}\, {{\mathfrak {h}}}$ is ${{\mathfrak {h}}}$, and special fiber is the vector space ${{\mathfrak {h}}}$ equipped with the abelian Lie algebra structure. We remark that the latter abelian Lie algebra has underlying formal group ${{\mathfrak {h}}}_0^{\wedge }$, that is, the formal completion at 0 of the vector space ${{\mathfrak {h}}}$; we use this notation at various points to emphasize the abelian nature.

Therefore, we obtain $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}{:}{=}\mathsf {IndCoh}({{\mathbb {B}}}\exp (\mathsf {Fil}\, {{\mathfrak {h}}}))$, which is a filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}$. Note that the “special fiber” ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl}$ is $\mathsf {IndCoh}({{\mathbb {B}}}{{\mathfrak {h}}}_0^{\wedge }) = {\text {Sym}}({{\mathfrak {h}}})\text {--}\mathsf {mod}= \mathsf {QCoh}({{\mathfrak {h}}}^{\vee })$.

Warning A.3.1

For obvious reasons, we prefer to think of $\mathsf {IndCoh}({{\mathbb {B}}}{{\mathfrak {h}}}_0^{\wedge })$ as $\mathsf {QCoh}({{\mathfrak {h}}}^{\vee })$, remembering that this category actually arises from the classifying space of a commutative formal group. Part of remembering this means internalizing that a duality occurs in this transition, so various pullbacks and pushforwards get swapped. For example, the fiber functor ${\text {Sym}}({{\mathfrak {h}}})\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$ (which corresponds to !-pullback ${\text {Spec}}(k) \rightarrow {{\mathbb {B}}}{{\mathfrak {h}}}_0^{\wedge }$) goes to global sections $\mathsf {QCoh}({{\mathfrak {h}}}^{\vee })\rightarrow \mathsf {Vect}$.

Example A.3.2

$U({{\mathfrak {h}}})$ is filtered, and its associated graded is the structure sheaf of ${{\mathfrak {h}}}^{\vee }$.

Warning A.3.3

The construction ${{\mathfrak {h}}}\mapsto \mathsf {Fil}\, {{\mathfrak {h}}}$ has some subtleties. We can rewrite its output: $\mathsf {Fil}\, {{\mathfrak {h}}}\in \mathsf {LieAlg}(\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m))$ is the same as a filtration on ${{\mathfrak {h}}}$ plus a Lie bracket on that filtered vector space extending the one on ${{\mathfrak {h}}}$. In our case, the filtration is $F_i {{\mathfrak {h}}}= 0$ for $i \le 0$ and $F_i {{\mathfrak {h}}}= {{\mathfrak {h}}}$ for $i>0$, so that the bracket $[-,-]:F_i {{\mathfrak {h}}}\otimes F_j {{\mathfrak {h}}}\rightarrow F_{i+j} {{\mathfrak {h}}}$ happens to factor through $F_{i+j-1}{{\mathfrak {h}}}$ (so that ${\text {gr}}_{\bullet } {{\mathfrak {h}}}$ is abelian, as we expect). In particular, ${{\mathfrak {h}}}= {\text {gr}}_1 {{\mathfrak {h}}}$.

This is slightly confusing for ${{\mathfrak {h}}}$ being abelian: since the generic and special fibers are the same, it is easy to wrongly confuse this filtration with the constant one.

A.3.3. Now recall that the functor:

$$\begin{aligned} C_{\bullet }({{\mathfrak {h}}},-):{{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}\end{aligned}$$

corresponds to the $\mathsf {IndCoh}$-pushforward functor for ${{\mathbb {B}}}\exp ({{\mathfrak {h}}})$. Therefore, we see that this functor is naturally filtered, i.e., it extends to give $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Fil}\, \mathsf {Vect}$. Note that the corresponding semi-classical functor $\mathsf {QCoh}({{\mathfrak {h}}}^{\vee }) \rightarrow \mathsf {Vect}$ is $*$-restriction to 0.

Remark A.3.4

Say $F_{\bullet } M \in \mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}$, and for simplicity let us assume M is in the heart of the t-structure and that $F_iM \hookrightarrow F_{i+1}M$. Recall that $C_{\bullet }({{\mathfrak {h}}},M)$ is computed by the complex:

$$\begin{aligned} \ldots \rightarrow \Lambda ^2 {{\mathfrak {h}}}\otimes M \rightarrow {{\mathfrak {h}}}\otimes M \rightarrow M \rightarrow 0 \rightarrow 0 \rightarrow \ldots \end{aligned}$$

with appropriate differentials. Then the resulting filtration on $C_{\bullet }({{\mathfrak {h}}},M)$ has $F_i$-term:

$$\begin{aligned} \ldots \rightarrow \Lambda ^2 {{\mathfrak {h}}}\otimes F_{i-2} M \rightarrow {{\mathfrak {h}}}\otimes F_{i-1} M \rightarrow F_i M \rightarrow 0 \rightarrow 0 \rightarrow \ldots \end{aligned}$$

A.3.4. Moreover, recall that $\omega _{{{\mathbb {B}}}\exp ({{\mathfrak {h}}})}$ is compact, as is its extension $\omega _{{{\mathbb {B}}}\exp (\mathsf {Fil}\, {{\mathfrak {h}}})}$.⁷⁹

Note that $\omega _{{{\mathbb {B}}}\exp ({{\mathfrak {h}}})}$ corresponds to the trivial representation in ${{\mathfrak {h}}}\text {--}\mathsf {mod}$. Therefore, the functor:

$$\begin{aligned} C^{\bullet }({{\mathfrak {h}}},-):{{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}\end{aligned}$$

of Lie algebra cohomology has a canonical filtered structure with graded $\mathsf {QCoh}({{\mathfrak {h}}}^{\vee }) \rightarrow \mathsf {Vect}$ given by the $(\mathsf {QCoh},!)$-pullback to $0 \in {{\mathfrak {h}}}^{\vee }$, i.e., the (continuous) right adjoint to the pushforward functor from 0.

Remark A.3.5

We retain the notation of Remark A.3.4. Recall that $C^{\bullet }({{\mathfrak {h}}},M)$ is computed by the complex:

$$\begin{aligned} \ldots \rightarrow 0 \rightarrow M \rightarrow {{\mathfrak {h}}}^{\vee } \otimes M \rightarrow \Lambda ^2 {{\mathfrak {h}}}^{\vee } \otimes M \rightarrow \ldots \end{aligned}$$

with appropriate differentials. Then the resulting filtration on $C^{\bullet }({{\mathfrak {h}}},M)$ has $F_i$-term:

$$\begin{aligned} \ldots \rightarrow 0 \rightarrow F_i M \rightarrow {{\mathfrak {h}}}^{\vee } \otimes F_{i+1} M \rightarrow \Lambda ^2 {{\mathfrak {h}}}^{\vee } \otimes F_{i+2} M \rightarrow \ldots \end{aligned}$$

Note that (unlike the case of Lie algebra homology) even if the filtration on M has $F_{-1}M = 0$, we may not have $F_{-1}C^{\bullet }({{\mathfrak {h}}},M) = 0$ (though $F_{-\dim {{\mathfrak {h}}}-1}C^{\bullet }({{\mathfrak {h}}},M)$ will be zero).

A.3.5. We now recall the following fact.

Lemma A.3.6

There is a canonical isomorphism of functors:

$$\begin{aligned} C^{\bullet }({{\mathfrak {h}}},(-)\otimes \det ({{\mathfrak {h}}})[\dim {{\mathfrak {h}}}]) \simeq C_{\bullet }({{\mathfrak {h}}},-). \end{aligned}$$

Moreover, suppose that we regard $\det ({{\mathfrak {h}}})[\dim {{\mathfrak {h}}}]$ as a filtered ${{\mathfrak {h}}}$-module with $F_{\dim {{\mathfrak {h}}}-i-1} \det ({{\mathfrak {h}}})[\dim {{\mathfrak {h}}}] = 0$ and $F_{\dim {{\mathfrak {h}}}+i}\det ({{\mathfrak {h}}})[\dim {{\mathfrak {h}}}] = \det ({{\mathfrak {h}}})[\dim {{\mathfrak {h}}}]$ for all $i \ge 0$. Then this isomorphism extends to an isomorphism of filtered functors, g inducing the usual isomorphism-up-to-twist-and-shift between $*$ and !-restriction to the point $0 \in {{\mathfrak {h}}}^{\vee }$.

Proof

Recall that for any $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$ (possibly non-filtered), $C^{\bullet }({{\mathfrak {h}}},M)$ has a canonical filtration $F_{\bullet }^{Chev} C^{\bullet }({{\mathfrak {h}}},M)$ (indexed by non-positive integers but bounded below) with ${\text {gr}}_{-i}^{Chev} C^{\bullet }({{\mathfrak {h}}},M) = M \otimes \Lambda ^i{{\mathfrak {h}}}^{\vee }[-i]$. Similarly, the functor of !-restriction along $0 \hookrightarrow {{\mathfrak {h}}}^{\vee }$ has a filtration with the same associated graded. One immediately sees that these “glue”: $C^{\bullet }({{\mathfrak {h}}},-)$ has a canonical filtration considered as a filtered functor.

In particular, we obtain a natural transformation of filtered functors ${{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$:

$$\begin{aligned} (-) \otimes \det ({{\mathfrak {h}}})^{\vee }[-\dim {{\mathfrak {h}}}] \rightarrow C^{\bullet }({{\mathfrak {h}}},-) \end{aligned}$$

where on the left hand side $\det ({{\mathfrak {h}}})^{\vee }[-\dim {{\mathfrak {h}}}]$ is equipped with the filtration with one jump in degree $-\dim {{\mathfrak {h}}}$. Evaluating this on $U({{\mathfrak {h}}})$, we claim that the composition:

$$\begin{aligned} \det ({{\mathfrak {h}}})^{\vee }[-\dim {{\mathfrak {h}}}] \rightarrow U({{\mathfrak {h}}}) \otimes \det ({{\mathfrak {h}}})^{\vee }[-\dim {{\mathfrak {h}}}] \rightarrow C^{\bullet }({{\mathfrak {h}}},U({{\mathfrak {h}}})) \end{aligned}$$

is an isomorphism of filtered complexes. Indeed, it suffices to check this at the associated graded level, where the claim is standard.

Then observe that using the (filtered) bimodule structure on $U({{\mathfrak {h}}})$, $C^{\bullet }({{\mathfrak {h}}},U({{\mathfrak {h}}}))$ can actually be considered as a filtered ${{\mathfrak {h}}}$-module. We claim that the above computation is true with this extra structure, considering $\det ({{\mathfrak {h}}})^{\vee }$ as an ${{\mathfrak {h}}}$-module in the obvious way.

Indeed, considering $U({{\mathfrak {h}}})$ as a filtered ${{\mathfrak {h}}}$-module via the right action, the morphism:

$$\begin{aligned} U({{\mathfrak {h}}}) \otimes \det ({{\mathfrak {h}}})^{\vee }[-\dim {{\mathfrak {h}}}] \rightarrow C^{\bullet }({{\mathfrak {h}}},U({{\mathfrak {h}}})) \end{aligned}$$

is a tautologically a morphism of filtered ${{\mathfrak {h}}}$-modules. Moreover, by a standard argument, this is also true if we consider $U({{\mathfrak {h}}})$ as a filtered ${{\mathfrak {h}}}$-module via the adjoint action. This shows the claim.

Now observe that by usual Morita theory, the datum of $C^{\bullet }({{\mathfrak {h}}},U({{\mathfrak {h}}}))$ as a filtered ${{\mathfrak {h}}}$-module completely is equivalent to the datum of the filtered functor $C^{\bullet }$. So the result follows from the observation that:

$$\begin{aligned} C^{\bullet }({{\mathfrak {h}}},U({{\mathfrak {h}}}) \otimes \det ({{\mathfrak {h}}})[\dim {{\mathfrak {h}}}]) = k = C_{\bullet }({{\mathfrak {h}}},U({{\mathfrak {h}}})) \end{aligned}$$

as filtered ${{\mathfrak {h}}}$-modules. $\square$

Remark A.3.7

From the perspective of usual complexes: for M a complex of ${{\mathfrak {h}}}$-modules, Lie algebra homology is $M \otimes \Lambda ^{\bullet } {{\mathfrak {h}}}$ with the appropriate grading and differential, while cohomology is $M \otimes \Lambda ^{\bullet } {{\mathfrak {h}}}^{\vee }$. Noting that $\Lambda ^i {{\mathfrak {h}}}= \det ({{\mathfrak {h}}}) \otimes \Lambda ^{\dim {{\mathfrak {h}}}- i} {{\mathfrak {h}}}^{\vee }$ and matching up the differentials and gradings then gives the claim.

Remark A.3.8

This isomorphism amounts to the calculation of $C^{\bullet }(U({{\mathfrak {h}}}))$ as a filtered complex acted on by ${{\mathfrak {h}}}$ (using the bimodule structure on $U({{\mathfrak {h}}})$). Since the claim is that the result is 1-dimensional in some cohomological degree with filtration jumping in one degree, the isomorphism above is easy to pin down uniquely.

A.3.6. We also observe that the above generalizes in the natural way to the setting of a morphism ${{\mathfrak {h}}}_1 \rightarrow {{\mathfrak {h}}}_2$ between (finite-dimensional, non-derived) Lie algebras. Here generalizing means that we obtain the previous discussion by considering the structure map ${{\mathfrak {h}}}\rightarrow 0$.

We draw attention to the following consequence of Lemma A.3.6 in this setting, which will play an important role in what follows. Let ${\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}:{{\mathfrak {h}}}_1\text {--}\mathsf {mod}\rightarrow {{\mathfrak {h}}}_2\text {--}\mathsf {mod}$ denote the left adjoint to the forgetful functor. Note that by realizing ${\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}$ as a $\mathsf {IndCoh}$-pushforward, we find that it is naturally filtered with semi-classical functor the quasi-coherent pullback along ${{\mathfrak {h}}}_2^{\vee } \rightarrow {{\mathfrak {h}}}_1^{\vee }$.

Corollary A.3.9

There is a canonical isomorphism of filtered functors:

$$\begin{aligned} C^{\bullet }\Big ({{\mathfrak {h}}}_2,{\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}\big ((-) \otimes \det ({{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1)[\dim {{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1]\big )\Big ) = C^{\bullet }({{\mathfrak {h}}}_1,-). \end{aligned}$$

Here $\det ({{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1)[\dim {{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1]$ is considered as trivially filtered with a single jump in degree ${{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1$.

Remark A.3.10

If ${{\mathfrak {h}}}_1 \rightarrow {{\mathfrak {h}}}_2$ is not injective, the quotient above should be replaced by the cone.

1.3.1 Harish-Chandra setting

Now suppose that $({{\mathfrak {h}}},K)$ is a finite-dimensional Harish-Chandra pair, so ${{\mathfrak {h}}}$ is as above, K is an affine algebraic group acting on ${{\mathfrak {h}}}$, and we are given a K-equivariant morphism ${\text {Lie}}(K) {=}{:}{{\mathfrak {k}}}\rightarrow {{\mathfrak {h}}}$.

Recall that a group always acts trivially on its classifying stack. Therefore, the induced action of K on ${{\mathbb {B}}}\exp ({{\mathfrak {h}}})$ factors through an action of $K_{dR}$ on ${{\mathbb {B}}}\exp ({{\mathfrak {h}}})$. Using somewhat strange notation, we let⁸⁰${{\mathbb {B}}}({{\mathfrak {h}}},K)$ denote the quotient ${{\mathbb {B}}}\exp ({{\mathfrak {h}}})/K_{dR}$. We let ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ denote $\mathsf {IndCoh}({{\mathbb {B}}}\exp ({{\mathfrak {h}}})/K_{dR})$, noting that this category is tautologically the K-equivariant category for the induced strong action of K on ${{\mathfrak {h}}}\text {--}\mathsf {mod}$.

Example A.3.11

For the Harish-Chandra pair $({{\mathfrak {k}}},K)$, we obtain ${{\mathbb {B}}}({{\mathfrak {k}}},K) = {{\mathbb {B}}}K$.

Example A.3.12

Note that the projection map ${\text {Spec}}(k) \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$ defines a group $({{\mathfrak {h}}},K)$ whose classifying stack is as notated. If ${{\mathfrak {h}}}= {\text {Lie}}(H)$ for $K \subseteq H$, then $({{\mathfrak {h}}},K)$ is the formal completion of K in H.

A.3.8. Now we recall that for any ind-affine nil-isomorphism $f:X \rightarrow Y$ of prestacks, [42] §IV.5.2 associates a deformation of this map, i.e., a prestack $Y_{\hbar }$ over ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$ and an ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$-morphism $X \times {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \rightarrow Y_{\hbar }$ giving f over the generic point.

For example, for ${\text {Spec}}(k) \rightarrow {{\mathbb {B}}}\exp ({{\mathfrak {h}}})$, we obtain the deformation ${{\mathbb {B}}}\mathsf {Fil}\, {{\mathfrak {h}}}$ corresponding to the PBW filtration.

For a Harish-Chandra datum as above, we obtain a deformation ${{\mathbb {B}}}\mathsf {Fil}\, ({{\mathfrak {h}}},K)$ associated with the map ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$. The special fiber of this deformation is $({{\mathbb {B}}}({{\mathfrak {h}}}/{{\mathfrak {k}}})_0^{\wedge } )/K$, where we note that $({{\mathfrak {h}}}/{{\mathfrak {k}}})_0^{\wedge }$ is⁸¹$\exp ({{\mathfrak {h}}}/{{\mathfrak {k}}})$ with ${{\mathfrak {h}}}/{{\mathfrak {k}}}$ considered as an abelian Lie algebra.

Therefore, we obtain a filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ with ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,cl} = \mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K)$.

Example A.3.13

When ${{\mathfrak {k}}}\xrightarrow {\simeq }{{\mathfrak {h}}}$, this is the constant filtration on $\mathsf {Rep}(K)$ (in the sense of Example A.2.4).

A.3.9. From now on, we assume ${{\mathfrak {k}}}\rightarrow {{\mathfrak {h}}}$ is injective.

A.3.10. $\mathsf {IndCoh}$ pushforward and pullback along the ind-affine nil-isomorphism ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$ induces a pair of adjoint functors:

$$\begin{aligned} {\text {ind}}= {\text {ind}}_{{{\mathfrak {k}}}}^{{{\mathfrak {h}}}}:\mathsf {Rep}(K) \rightleftarrows {{\mathfrak {h}}}\text {--}\mathsf {mod}^K:{\text {Oblv}}. \end{aligned}$$

These functors are evidently compatible with filtrations. At the semi-classical level, they induce the adjunction:

$$\begin{aligned} \mathsf {Rep}(K) \rightleftarrows \mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \end{aligned}$$

given by pullback and pushforward along the structure map $({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K \rightarrow {{\mathbb {B}}}K$.

In particular, we find that ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ admits a canonical t-structure for which ${\text {Oblv}}:{{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Rep}(K)$ is t-exact, and that the functor ${\text {ind}}$ is t-exact for this t-structure as well.

A.3.11. We focus on the case of Harish-Chandra cohomology.

Using the standard resolution of $\omega _{{{\mathbb {B}}}({{\mathfrak {h}}},K)}$ induced by the ind-affine nil-isomorphism ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$, we find that $\omega _{{{\mathbb {B}}}({{\mathfrak {h}}},K)}$ is compact. Moreover, this compactness remains true (for the same reason) for $\omega _{{{\mathbb {B}}}\mathsf {Fil}\, ({{\mathfrak {h}}},K)}$.

Therefore, mapping out of it defines a filtered functor:

$$\begin{aligned} C^{\bullet }({{\mathfrak {h}}},K,-):{{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Vect}. \end{aligned}$$

On associated, graded, this functor is given by taking !-restriction along $0/K \hookrightarrow ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K$, and then global sections (i.e., group cohomology for K).

Example A.3.14

If ${{\mathfrak {k}}}\xrightarrow {\simeq }{{\mathfrak {h}}}$, then $C^{\bullet }({{\mathfrak {k}}},K,-)$ computes group cohomology ${{\mathfrak {k}}}\text {--}\mathsf {mod}^K = \mathsf {Rep}(K) \rightarrow \mathsf {Vect}$. The filtration on this functor is the constant one.

Example A.3.15

If K is unipotent, then $C^{\bullet }({{\mathfrak {h}}},K,-)$ coincides with $C^{\bullet }({{\mathfrak {h}}},-)$ (composed with the forgetful functor ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow {{\mathfrak {h}}}\text {--}\mathsf {mod}$). However, the filtered structures are substantially affected by the presence of K, as can be seen e.g. by looking at the semi-classical level.

Remark A.3.16

We explain the above constructions using usual complexes. Suppose for simplicity that K is unipotent, so $C^{\bullet }({{\mathfrak {h}}},K,-) = C^{\bullet }({{\mathfrak {h}}},-)$ as a non-filtered functor. In the notation of Remark A.3.4, the filtration $F_{\bullet } M$ is a filtration in ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ if the $F_i M$ are K-submodules of M, i.e., if it induces a filtration on the image of $M \in {{\mathfrak {k}}}\text {--}\mathsf {mod}^K = \mathsf {Rep}(K)$.

In this case, we obtain a filtration on the cohomological Chevalley complex with ith term:

$$\begin{aligned}&\ldots \rightarrow 0 \rightarrow F_i M \rightarrow ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee } \otimes F_{i+1} M + {{\mathfrak {h}}}^{\vee } \otimes F_i M \\&\quad \rightarrow \Lambda ^2 ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee } \otimes F_{i+2} M + ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee } \wedge {{\mathfrak {h}}}^{\vee } \otimes F_{i+1} M + \Lambda ^2 {{\mathfrak {h}}}^{\vee } \otimes F_i M \rightarrow \ldots \end{aligned}$$

A.3.12. We now give versions of Corollary A.3.9 in this setting.

Lemma A.3.17

Let ${\text {ind}}$ denote the induction functor $\mathsf {Rep}(K) \rightarrow {{\mathfrak {h}}}\text {--}\mathsf {mod}^K$. There is a canonical isomorphism:

$$\begin{aligned} C^{\bullet }\Big ({{\mathfrak {h}}},K,{\text {ind}}\big ((-) \otimes \det ({{\mathfrak {h}}}/K)[\dim {{\mathfrak {h}}}/{{\mathfrak {k}}}]\big )\Big ) = C^{\bullet }(K,-):\mathsf {Rep}(K) \rightarrow \mathsf {Vect}\end{aligned}$$

of filtered functors. As before, $\det ({{\mathfrak {h}}}/{{\mathfrak {k}}})[\dim {{\mathfrak {h}}}/{{\mathfrak {k}}}])$ is filtered with a single jump in degree $\dim {{\mathfrak {h}}}/{{\mathfrak {k}}}$.

Proof

The proof is similar to Lemma A.3.6: we need to evaluate both sides on the regular representation⁸²${\mathcal {O}}_K$ and consider the result as a filtered K-module. For the right hand side, we obtain the trivial representation (in cohomological degree 0, with the filtration jumping only in degree 0).

Then one calculates $C^{\bullet }({{\mathfrak {h}}},K,{\text {ind}}{\mathcal {O}}_K )$ as $\det ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }[-\dim {{\mathfrak {h}}}/{{\mathfrak {k}}}]$ using standard filtrations as in Lemma A.3.6, which gives the claim. Here we note that for $C^{\bullet }({{\mathfrak {h}}},K,-)$ (considered merely as a filtered functor) the standard filtration has associated graded:

$$\begin{aligned} {\text {gr}}_{-i}C^{\bullet }({{\mathfrak {h}}},K,-) = C^{\bullet }(K, (-) \otimes \Lambda ^i ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }[-i]). \end{aligned}$$

$\square$

Suppose now that $({{\mathfrak {h}}}_1,K) \rightarrow ({{\mathfrak {h}}}_2,K)$ is a morphism of Harish-Chandra pairs. Note that $\mathsf {IndCoh}$ pushforward defines a functor ${\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}:{{\mathfrak {h}}}_1\text {--}\mathsf {mod}^K \rightarrow {{\mathfrak {h}}}_2\text {--}\mathsf {mod}^K$ compatible with forgetful functors and satisfying similar properties as before.

We have the following generalization of Lemma A.3.17.

Lemma A.3.18

There is a canonical isomorphism of filtered functors:

$$\begin{aligned}&C^{\bullet }\Big ({{\mathfrak {h}}}_2,K,{\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}\big ((-) \otimes \det ({{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1)[\dim {{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1]\big )\Big ) \\&\quad = C^{\bullet }({{\mathfrak {h}}}_1,K,-): {{\mathfrak {h}}}_1\text {--}\mathsf {mod}^K \rightarrow \mathsf {Vect}. \end{aligned}$$

As before, $\det ({{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1)[\dim {{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1]$ is filtered with a single jump in degree $\dim {{\mathfrak {h}}}_2/{{\mathfrak {h}}}_1$.

The argument is similar to those that preceded it, so we omit it.

1.3.2 Group actions on filtered categories

We now put the above into a more conceptual framework whose perspective will be convenient at some points. We essentially rewrite the above using a version of the theory of group actions on categories.

Associated with the ind-affine nil-isomorphism, $K \rightarrow K_{dR}$ one has the deformation $\mathsf {Fil}\, K_{dR}$, which is a relative group⁸³prestack over ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$ with generic fiber $K_{dR}$ and special fiber $K \ltimes {{\mathbb {B}}}{{\mathfrak {k}}}_0^{\wedge }$. Note that because ${{\mathfrak {k}}}_0^{\wedge }$ is a commutative formal group, ${{\mathbb {B}}}{{\mathfrak {k}}}_0^{\wedge }$ actually is a commutative group prestack (acted on by K).

Observe that we have a fiber sequence of relative groups:

$$\begin{aligned} 1 \rightarrow \exp (\mathsf {Fil}\, {{\mathfrak {k}}}) \rightarrow \mathsf {Fil}\, K \rightarrow \mathsf {Fil}\, K_{dR} \rightarrow 1 \end{aligned}$$

(A.3.1)

where $\mathsf {Fil}\, K$ just means $K \times {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$.

Note that the usual convolution monoidal structure makes $\mathsf {IndCoh}(\mathsf {Fil}\, K_{dR}) \in \mathsf {Alg}(\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)\text {--}\mathsf {mod})$.

Definition A.3.19

Let ${\mathcal {C}}$ be a filtered category. A (strong) action of K on $\mathsf {Fil}\, {\mathcal {C}}$ is a $\mathsf {IndCoh}(\mathsf {Fil}\, K_{dR})$-module structure on $\mathsf {Fil}\, {\mathcal {C}}\in \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)\text {--}\mathsf {mod}$.⁸⁴

Warning A.3.20

There is a redundancy in the terminology: K acts on $\mathsf {Fil}\, {\mathcal {C}}$ could wrongly be understood to mean that $\mathsf {Fil}\, {\mathcal {C}}\in \mathsf {DGCat}_{cont}$ is a D(K)-module category. However, we are confident that there is no risk for confusion in our use of the above terminology: the meaning is completely determined by whether or not there is a $\mathsf {Fil}\,$ in front of what is acted upon.

We also note that there is no such risk for weak actions, i.e., if K acts on $\mathsf {Fil}\, {\mathcal {C}}$, then K acts weakly on $\mathsf {Fil}\, {\mathcal {C}}$ in the sense that $\mathsf {Fil}\, {\mathcal {C}}$ is a $\mathsf {QCoh}(K)$-module category.

Note that if K acts on $\mathsf {Fil}\, {\mathcal {C}}$, then K in particular acts on ${\mathcal {C}}$ (in the usual sense).

Moreover, ${\mathcal {C}}^{cl}$ receives an action of $\mathsf {IndCoh}(K \ltimes {{\mathbb {B}}}{{\mathfrak {k}}}_0^{\wedge })$. This is equivalent to saying that $\mathsf {QCoh}({{\mathfrak {k}}}^{\vee })$ (with the usual symmetric monoidal structure) acts on ${\mathcal {C}}^{cl}$, and K acts weakly on ${\mathcal {C}}^{cl}$, and these two actions are compatible under the adjoint action of K on ${{\mathfrak {k}}}^{\vee }$. In particular, ${\mathcal {C}}^{cl,K,w}$ may be thought of as a sheaf of categories on ${{\mathfrak {k}}}^{\vee }/K$.

Using the “trivial” action of K on $\mathsf {Fil}\, \mathsf {Vect}$ as a filtered category, we obtain an invariants and coinvariants formalism. By the same arguments as in [10] §2, the two are canonically identified.

In particular, we obtain a filtration on ${\mathcal {C}}^K$ with ${\mathcal {C}}^{K,cl} = ({\mathcal {C}}^{cl} \underset{\mathsf {QCoh}({{\mathfrak {k}}}^{\vee })}{\otimes } \mathsf {Vect})^{K,w}$, where $\mathsf {QCoh}({{\mathfrak {k}}}^{\vee })$ acts on $\mathsf {Vect}$ through the restriction to 0 functor. Geometrically, this means we take our sheaf of categories on ${{\mathfrak {k}}}^{\vee }/K$, restrict to 0/K, and then take global sections.

The adjoint functors ${\text {Oblv}}: {\mathcal {C}}^K \rightleftarrows {\mathcal {C}}: {\text {Av}}_*$ carry natural filtrations. semi-classically, ${\text {Oblv}}^{cl}$ is the functor:

$$\begin{aligned} ({\mathcal {C}}^{cl} \underset{\mathsf {QCoh}({{\mathfrak {k}}}^{\vee })}{\otimes } \mathsf {Vect})^{K,w} \rightarrow {\mathcal {C}}^{cl} \underset{\mathsf {QCoh}({{\mathfrak {k}}}^{\vee })}{\otimes } \mathsf {Vect}\rightarrow {\mathcal {C}}^{cl} \end{aligned}$$

where the first functor is forgetting and the second is $*$-pushforward along $0 \hookrightarrow {{\mathfrak {k}}}^{\vee }$. Of course, the semi-clclassical ${\text {Av}}_*^{cl}$ is the right adjoint to the functor we just described: !-pullback $0 \hookrightarrow {{\mathfrak {k}}}^{\vee }$ and then $*$-pushforward to ${{\mathbb {B}}}K$.

Example A.3.21

We can reformulate some of our earlier constructions in saying that K acts on $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}$ for a Harish-Chandra pair $({{\mathfrak {h}}},K)$. The associated filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ is our earlier one.

1.4 Derived categories

The following result will play an important role for us, but may be skipped at first pass.

Lemma A.4.1

(Bernstein-Lunts, [12]) ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ is the derived category of its heart. (Here we continue to assume ${{\mathfrak {k}}}\rightarrow {{\mathfrak {h}}}$ is injective.)

Proof

If ${{\mathfrak {k}}}\rightarrow {{\mathfrak {h}}}$ is an isomorphism, then ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K = \mathsf {Rep}(K)$, and this is a general property about algebraic stacks (see e.g. [32] Proposition 5.4.3).

In general, we have a t-exact forgetful functor ${\text {Oblv}}:{{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Rep}(K)$. Moreover, this functor admits a left adjoint ${\text {ind}}$, geometrically given by $\mathsf {IndCoh}$-pushforward.

Now observe that our hypothesis implies that the tangent complex of the morphism ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$ is concentrated in cohomological degree 0: it is ${{\mathfrak {h}}}/{{\mathfrak {k}}}$ considered as a K-representation. Therefore, the monad ${\text {Oblv}}\circ {\text {ind}}$ has a standard filtration with associated graded given by tensoring with ${\text {Sym}}({{\mathfrak {h}}}/{{\mathfrak {k}}})$, so in particular, this monad is t-exact. Since ${\text {Oblv}}$ is t-exact, we find that ${\text {ind}}$ is as well.

Therefore, we obtain a similar pair of adjoint functors between $D(\mathsf {Rep}(K)^{\heartsuit }) = \mathsf {Rep}(K)$ and $D({{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\heartsuit })$. Both forgetful functors ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Rep}(K)$ and $D({{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\heartsuit }) \rightarrow \mathsf {Rep}(K)$ are conservative and commute with colimits, so are monadic. Then we observe that the monads on $\mathsf {Rep}(K)$ are naturally identified, giving the claim.

$\square$

1.5 Kazhdan-Kostant twists

A.5.1. We now discuss how to render the standard solution to a standard problem in our framework.

We begin by describing the issue. Suppose ${{\mathfrak {g}}}$ and ${{\mathfrak {n}}}$ are as usual, and $\psi :{{\mathfrak {n}}}\rightarrow k$ is a non-degenerate character. We choose G-equivariant symmetric ${{\mathfrak {g}}}\simeq {{\mathfrak {g}}}^{\vee }$ and take $f \in {{\mathfrak {g}}}= {{\mathfrak {g}}}^{\vee }$ the principal nilpotent mapping to $\psi \in {{\mathfrak {n}}}^{\vee }$.

Recall that ${\mathcal {W}}^{fin}\text {--}\mathsf {mod}\simeq {{\mathfrak {g}}}\text {--}\mathsf {mod}^{N,\psi }$, and that ${\mathcal {W}}^{fin}$ is filtered with associated graded being the algebra of functions on $f+{{\mathfrak {b}}}/N$. In particular, $C^{\bullet }({{\mathfrak {n}}},(-\psi ) \otimes {\text {ind}}_{{{\mathfrak {n}}}}^{{{\mathfrak {g}}}}(\psi ))$ is filtered with associated graded being this algebra of functions.

However, this filtration is not induced by the obvious PBW filtration on ${\text {ind}}_{{{\mathfrak {n}}}}^{{{\mathfrak {g}}}}(\psi )$. Indeed, suppose more generally that M is any (PBW) filtered ${{\mathfrak {n}}}$-module. Then the induced filtration on $M \otimes -\psi$ has the same associated graded as M. (So in the case above, we will see the DG algebra $\Gamma ({{\mathfrak {b}}}/N,{\mathcal {O}}_{{{\mathfrak {b}}}/N})$ instead.)

The issue is with the filtration on the 1-dimensional representation $\psi$: jumping in a single degree, its associated graded is going to be the augmentation module of ${\text {Sym}}({{\mathfrak {n}}})$. In other words, its associated graded will be a ${{\mathbb {G}}}_m$-equivariant quasi-coherent sheaf on ${{\mathfrak {n}}}^{\vee }$, so it must be the skyscraper at the origin, though we would rather see the skyscraper at $\psi \in {{\mathfrak {n}}}^{\vee }$.

The solution⁸⁵ to this problem is to use $\check{\rho }$ to modify the filtration on $U({{\mathfrak {n}}})$, so that its associated graded is again ${\text {Sym}}({{\mathfrak {n}}})$ (but with a modified grading!), and the module $\psi$ is filtered with associated graded being the skyscraper at $\psi \in {{\mathfrak {n}}}^{\vee }$ instead. Namely, we set:

$$\begin{aligned} \begin{aligned} F_i^{KK} U({{\mathfrak {n}}}) = \underset{j}{\oplus } \, F_{i-j}^{PBW} U({{\mathfrak {n}}}) \cap U({{\mathfrak {n}}})^j \\ U({{\mathfrak {n}}})^j {:}{=}\{x \in U({{\mathfrak {n}}}) \mid {\text {Ad}}_{-\check{\rho }(\lambda )}(x) = \lambda ^j \cdot x\}. \end{aligned} \end{aligned}$$

We emphasize that the grading used here is induced by $-\check{\rho }: {{\mathbb {G}}}_m \rightarrow G^{ad}$, not $\check{\rho }$ itself (so ${{\mathfrak {n}}}$ is negatively graded). For example, for $i \in {{\mathcal {I}}}_G$, $e_i \in F_0^{KK} U({{\mathfrak {n}}})$, and for $\alpha$ a general root, $e_{\alpha } \in F_{1-(\check{\rho },\alpha )}^{KK} U({{\mathfrak {n}}})$.

One has a similar filtration on $U({{\mathfrak {g}}})$. Moreover, these filtrations naturally induce the “correct” filtration on ${\mathcal {W}}^{fin}$ (e.g., it is a filtration in the abelian category, not just the derived category).

Our present goal is to render the above ideas in the categorical framework. We will begin by discussing some generalities, and then apply these to the example of Harish-Chandra modules.

Warning A.5.1

We immediately see that Kazhdan-Kostant filtrations are typically unbounded from below and are not complete filtrations. This causes some technical problems (e.g., in Sect. 4), and requires care.

A.5.2. We begin by discussing how to twist a filtration by a grading to obtain a new filtration. Motivated by our particular concerns, we use the notation PBW to indicate an “old” filtration and KK to indicate a “new” filtration.

So suppose that $F_{\bullet }^{PBW} V \in \mathsf {Fil}\, \mathsf {Rep}({{\mathbb {G}}}_m)$, i.e., V is equipped with a grading $V = \oplus _j V^j$ and a filtration $F_{\bullet }^{PBW} V$ as a graded vector space, so we have a grading $F_i^{PBW} V = \oplus _j F_i^{PBW} V^j$ compatible with varying i in the natural sense.

Then we can twist the filtration $F_{\bullet }^{PBW}$ by the grading to obtain the filtration:

$$\begin{aligned} F_i^{KK} V = \underset{j}{\oplus } \, F_{i-j}^{PBW} V^j \end{aligned}$$

as before.

Here is a geometric interpretation. Recall that a filtered vector space is the same as a quasi-coherent sheaf on ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$. The data of a compatible filtration and grading as above is equivalent to a quasi-coherent sheaf on ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \times {{\mathbb {B}}}{{\mathbb {G}}}_m$, where we recover the underlying filtered vector space by pulling back along the first projection.

Formation of the KK filtration corresponds to pulling back along the graph of the projection ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \rightarrow {{\mathbb {B}}}{{\mathbb {G}}}_m$ instead.

In either perspective, we immediately find that:

$$\begin{aligned} {\text {gr}}_{i}^{KK} V = \underset{j}{\oplus } \, {\text {gr}}_{i-j}^{PBW} V^j. \end{aligned}$$

That is, ${\text {gr}}_{\bullet }^{KK} V = {\text {gr}}_{\bullet }^{PBW} V$ as vector spaces, although the gradings are different.

A.5.3. We now give a categorical version of the above.

Suppose ${\mathcal {C}}$ is a filtered category. We use the notation $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$ for the underlying $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$-module category, since we wish to construct another filtration on ${\mathcal {C}}$.

The extra data we need is an action of $\mathsf {QCoh}({{\mathbb {G}}}_m)$ (with its convolution monoidal structure) on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$ commuting with the $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$. In this case, we can again twist our filtration by this action in forming:

$$\begin{aligned} \mathsf {Fil}\, \!^{KK} {\mathcal {C}}{:}{=}\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}^{{{\mathbb {G}}}_m,w} \underset{\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m) \otimes \mathsf {Rep}({{\mathbb {G}}}_m)}{\otimes } \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m) \end{aligned}$$

where the action on the right term is induced by the symmetric monoidal functor:

$$\begin{aligned} \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m) \otimes \mathsf {Rep}({{\mathbb {G}}}_m) \rightarrow \mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m) \end{aligned}$$

of pullback along the graph of the structure map ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \rightarrow {{\mathbb {B}}}{{\mathbb {G}}}_m$.

More geometrically: we are given the datum of a sheaf of categories on ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \times {{\mathbb {B}}}{{\mathbb {G}}}_m$, and we observe that we can form two filtered categories from it, via pullback along the maps:

$$\begin{aligned} {\text {id}}\times p,\Gamma : {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \rightarrow {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \times {{\mathbb {B}}}{{\mathbb {G}}}_m \end{aligned}$$

where $p:{\text {Spec}}(k) \rightarrow {{\mathbb {B}}}{{\mathbb {G}}}_m$ is the tautological projection, and $\Gamma$ is the graph of the structure map as above. These two maps coincide over the open point ${{\mathbb {A}}}_{\hbar }^1{\setminus } 0/{{\mathbb {G}}}_m$, so are filtrations on the same category ${\mathcal {C}}$. By definition, the pullback along ${\text {id}}\times p$ defines the “PBW” filtration, and pullback along $\Gamma$ defines the KK filtration.

Remark A.5.2

There is a tautological functor $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}^{{{\mathbb {G}}}_m,w} \rightarrow \mathsf {Fil}\, \!^{KK} {\mathcal {C}}$.

Remark A.5.3

Note that ${\mathcal {C}}^{cl}$ is also the fiber at 0 of $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}$, but the weak ${{\mathbb {G}}}_m$-action is different: it is the diagonal action mixing the standard action of ${{\mathbb {G}}}_m$ on ${\mathcal {C}}^{cl}$ with the action coming from the weak ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$.

Example A.5.4

In Sect. A.5.2, it is tautological that the KK twisting construction is symmetric monoidal. So if A is an algebra with compatible filtration $F_{\bullet }^{PBW}$ and grading, we obtain a filtration $F_{\bullet }^{KK}$ on A (as an algebra). Therefore, we obtain two filtrations $\mathsf {Fil}\, \!^{PBW},\mathsf {Fil}\, \!^{KK}$ on $A\text {--}\mathsf {mod}$. Of course, the grading on A induces a weak ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} A$, and the general categorical construction above produces $\mathsf {Fil}\, \!^{KK} A$. The functor $\mathsf {Fil}\, \!^{PBW} A\text {--}\mathsf {mod}^{{{\mathbb {G}}}_m,w} \rightarrow \mathsf {Fil}\, \!^{KK} A\text {--}\mathsf {mod}$ corresponds to taking a graded and PBW filtered A-module and then applying the corresponding KK twist to obtain a KK filtered A-module.

A.5.4. The reader may skip this material for the time being, and return to it as necessary. Its purpose is closely tied to Warning A.5.1: Kazhdan-Kostant filtrations are typically incomplete and unbounded from below, even when a corresponding PBW filtration is bounded from below. To deal with this issue, we wish to yoke the two filtrations on our category.

Definition A.5.5

A bifiltration on ${\mathcal {C}}\in \mathsf {DGCat}_{cont}$ is a $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m)$-module category $\mathsf {BiFil}\,{\mathcal {C}}$ plus an isomorphism:

$$\begin{aligned}&{\mathcal {C}}\simeq \mathsf {BiFil}\,{\mathcal {C}}\underset{\mathsf {QCoh}({{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m)}{\otimes } \mathsf {QCoh}(({{\mathbb {A}}}_{\hbar _1}^1{\setminus } 0)/{{\mathbb {G}}}_m \times ({{\mathbb {A}}}_{\hbar _2}^1{\setminus } 0)/{{\mathbb {G}}}_m) \\&\quad = \mathsf {BiFil}\,{\mathcal {C}}\underset{\mathsf {QCoh}({{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m)}{\otimes } \mathsf {Vect}. \end{aligned}$$

A bifiltration on ${\mathcal {F}}$ in ${\mathcal {C}}$ is an object of $\mathsf {BiFil}\,{\mathcal {C}}$ restricting to ${\mathcal {F}}$.

Much of our earlier discussion generalies. E.g., we have an obvious notion of bifiltered vector spaces, and so on.

Note that a bifiltration on ${\mathcal {C}}$ indeed gives rise to two filtrations on ${\mathcal {C}}$, denoted $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$ and $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}$. These are respectively obtained by restricting $\mathsf {BiFil}\,{\mathcal {C}}$ to the loci:

$$\begin{aligned} \begin{aligned} {{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times ({{\mathbb {A}}}_{\hbar _2}^1{\setminus } 0)/{{\mathbb {G}}}_m = {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \\ ({{\mathbb {A}}}_{\hbar _1}^1{\setminus } 0)/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m = {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m. \end{aligned} \end{aligned}$$

Note that a bifiltration on ${\mathcal {F}}\in {\mathcal {C}}$ gives rise to PBW and KK filtrations on ${\mathcal {F}}$.

Remark A.5.6

Let ${\mathcal {C}}^{PBW\text {--}{cl}}$ and ${\mathcal {C}}^{KK\text {--}{cl}}$ denote the semi-classical categories associated with each of these filtrations. We claim that e.g. ${\mathcal {C}}^{PBW\text {--}{cl}}$ carries a natural KK filtration $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{PBW\text {--}{cl}}$; moreover, the weak ${{\mathbb {G}}}_m$-action on ${\mathcal {C}}^{PBW\text {--}{cl}}$ extends to one on $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{PBW\text {--}{cl}}$, and this action commutes with the action of $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$.

Indeed, note that ${\mathcal {C}}^{PBW\text {--}{cl},{{\mathbb {G}}}_m,w}$ is the restriction to:

$$\begin{aligned} {{\mathbb {B}}}{{\mathbb {G}}}_m \times ({{\mathbb {A}}}_{\hbar _2}^1{\setminus } 0)/{{\mathbb {G}}}_m \subseteq {{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m \end{aligned}$$

so taking $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{PBW\text {--}{cl},{{\mathbb {G}}}_m,w}$ as the restriction to:

$$\begin{aligned} {{\mathbb {B}}}{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m \end{aligned}$$

(and applying de-equivariantization) gives the desired construction.

Of course, this works symmetrically in PBW and KK.

Example A.5.7

Suppose that we are in the setting of Sect. A.21, so ${\mathcal {C}}$ carries a single (PBW) filtration and a compatible weak ${{\mathbb {G}}}_m$-action. We claim that this data induces a bifiltration on ${\mathcal {C}}$ inducing the PBW and KK filtrations in the sense of Sect. A.21. For this, we note that we have the morphism:⁸⁶

$$\begin{aligned} \begin{aligned} {{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m \rightarrow {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \times {{\mathbb {B}}}{{\mathbb {G}}}_m \\ (s_1 \in {\mathcal {L}}_1, s_2\in {\mathcal {L}}_2) \mapsto (s_1 \otimes s_2 \in {\mathcal {L}}_1 \otimes {\mathcal {L}}_2, {\mathcal {L}}_2) \end{aligned} \end{aligned}$$

whose restriction to ${{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times ({{\mathbb {A}}}_{\hbar _2}^1{\setminus } 0)/{{\mathbb {G}}}_m$ is the ${\text {id}}\times p$ and whose restriction to $({{\mathbb {A}}}_{\hbar _1}^1{\setminus } 0)/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m = {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$ is $\Gamma$; so the operation of pullback (in the sheaf of categories language) along this morphism gives the desired structure.

Recall that ${\mathcal {C}}^{PBW\text {--}{cl}}$ is canonically isomorphic to ${\mathcal {C}}^{KK\text {--}{cl}}$ in this case, so we denote them each by ${\mathcal {C}}^{cl}$.

Recall that ${{\mathbb {G}}}_m \times {{\mathbb {G}}}_m$ weakly acts on ${\mathcal {C}}^{cl}$: one factor acts because this is always true for the semi-classical category, and the other factor acts because of the weak ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$. Then it is straightforward to verify that the KK filtration on ${\mathcal {C}}^{PBW\text {--}{cl}} = {\mathcal {C}}^{cl}$ is induced by taking the diagonal weak action of ${{\mathbb {G}}}_m$ on ${\mathcal {C}}^{cl}$ and applying Example A.2.6.

The situation with ${\mathcal {C}}^{KK\text {--}{cl}} = {\mathcal {C}}^{cl}$ is similar: its filtration is induced by the “canonical” weak ${{\mathbb {G}}}_m$-action on ${\mathcal {C}}^{cl}$, i.e., the one from the first ${{\mathbb {G}}}_m$-factor above (so is unrelated to the weak ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$).

A.5.5. We now begin to apply the above in the setting of Lie algebras and Harish-Chandra modules.

Before discussing Kazhdan-Kostant directly, let us discuss what we can obtain without the additional “grading” (i.e., weak ${{\mathbb {G}}}_m$-action). We use the language of Sect. A.3.13.

Suppose ${{\mathbb {G}}}_a$ acts on $\mathsf {Fil}\, {\mathcal {C}}$. Let $\psi$ denote the exponential (alias: Artin-Schreier) character sheaf on ${{\mathbb {G}}}_a$. Our problem is to construct a filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$.

First, note that (forgetting the filtrations) we can write ${\mathcal {C}}\mapsto {\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ in two steps: for $\widehat{{{\mathbb {G}}}}_a$ the formal completion of ${{\mathbb {G}}}_a$ at the origin, the group prestack ${{\mathbb {B}}}\widehat{{{\mathbb {G}}}}_a = {{\mathbb {G}}}_{dR}/{{\mathbb {G}}}_a$ acts on ${\mathcal {C}}^{{{\mathbb {G}}}_a,w}$. Note that $\mathsf {QCoh}({{\mathbb {B}}}\widehat{{{\mathbb {G}}}}_a) = \mathsf {QCoh}({{\mathbb {A}}}_{{\text {Lie}}}^1)$ with the convolution structure on the LHS corresponding to the tensor product structure on the RHS; here the subscript ${\text {Lie}}$ is used so we later remember the Lie-theoretic origins of this copy of the affine line. Therefore, ${\mathcal {C}}^{{{\mathbb {G}}}_a,w}$ fibers over ${{\mathbb {A}}}_{{\text {Lie}}}^1$, and we can take its fiber at $1 \in {{\mathbb {A}}}_{{\text {Lie}}}^1$, i.e., we can form:

$$\begin{aligned} {\mathcal {C}}^{{{\mathbb {G}}}_a,w} \underset{\mathsf {QCoh}({{\mathbb {A}}}_{{\text {Lie}}}^1)}{\otimes } \mathsf {Vect}\end{aligned}$$

using the restriction functor along $1 \hookrightarrow {{\mathbb {A}}}_{{\text {Lie}}}^1$. It is immediate to see that this tensor product is ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$.

In the filtered setting, let $\mathsf {Fil}\, \widehat{{{\mathbb {G}}}}_a$ denote the (commutative) relative formal group over ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$ defined by $\mathsf {Fil}\, {\text {Lie}}({{\mathbb {G}}}_a)$. As above, ${{\mathbb {B}}}\mathsf {Fil}\, \widehat{{{\mathbb {G}}}}_a$ acts on $\mathsf {Fil}\, {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$. Note that:

$$\begin{aligned} \mathsf {QCoh}({{\mathbb {B}}}\mathsf {Fil}\, \widehat{{{\mathbb {G}}}}_a) \simeq \mathsf {QCoh}(({{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar }^1)/{{\mathbb {G}}}_m) \in \mathsf {Alg}(\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)\text {--}\mathsf {mod}) \end{aligned}$$

where the left hand side is equipped with the convolution monoidal structure, and the right hand side is equipped with the tensor product monoidal structure. Moreover, since ${{\mathbb {A}}}_{{\text {Lie}}}^1$ occurs here as the coadjoint space ${\text {Lie}}({{\mathbb {G}}}_a)^{\vee }$, it is naturally equipped with the action of ${{\mathbb {G}}}_m$ by inverse homotheties; so both ${{\mathbb {A}}}^1$-factors are acted on in this way, and our graded algebra of functions is a polynomial algebra on the two degree 1 generators $\hbar$ and $x\hbar$ for $x\in {\text {Lie}}({{\mathbb {G}}}_a)$ the generator.

Warning A.5.8

The reader confused why we see $({{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar }^1)/{{\mathbb {G}}}_m$ and not ${{\mathbb {A}}}_{{\text {Lie}}}^1 \times ({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)$ should return to Warning A.3.3.

The upshot is that we obtain a filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ by taking $\mathsf {Fil}\, {\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ as $\mathsf {Fil}\, {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$ and restricting along the diagonal map:

$$\begin{aligned} {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \xrightarrow {x \mapsto (x,x)} ({{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar }^1)/{{\mathbb {G}}}_m. \end{aligned}$$

When there is risk for confusion, we refer to this as the PBW filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ and denote it by $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$.

Remark A.5.9

But in our hearts, we know that we would rather restrict along the non-existing map:

$$\begin{aligned} {{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m \xrightarrow {x \mapsto (1,x)} ({{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar }^1)/{{\mathbb {G}}}_m. \end{aligned}$$

Remark A.5.10

Note that ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi ,cl} = {\mathcal {C}}^{{{\mathbb {G}}}_a,cl}$ under this construction.

A.5.6. Let ${{\mathbb {G}}}_m$ act on the group ${{\mathbb {G}}}_a$ through inverse⁸⁷ homotheties. Since the morphism ${{\mathbb {G}}}_a \rightarrow {{\mathbb {G}}}_{a,dR}$ is ${{\mathbb {G}}}_m$-equivariant, ${{\mathbb {G}}}_m$ acts on $\mathsf {Fil}\, {{\mathbb {G}}}_{a,dR}$. For ${\mathcal {C}}$ “PBW” filtered, it makes sense to ask that a weak ${{\mathbb {G}}}_m$-action and a (strong) ${{\mathbb {G}}}_a$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$ be compatible (with the action of ${{\mathbb {G}}}_m$ on $\mathsf {Fil}\, {{\mathbb {G}}}_a$). Note that this in particular gives a weak ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$, so we may form $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$.

Observe that if we regard ${\text {Lie}}{{\mathbb {G}}}_a$ as a filtered Lie algebra through the PBW method of Sect. A.3.2 (remembering Warning A.3.3) and equip it with the above (degree $-1$) grading, then Kazhdan-Kostant twisting gives ${\text {Lie}}{{\mathbb {G}}}_a$ equipped with the constant filtration (jumping only in degree 0).

It follows formally that ${{\mathbb {B}}}\widehat{{{\mathbb {G}}}}_a$ (with no $\mathsf {Fil}\, \!$!) acts on $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$. Therefore, $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$ has an action of $\mathsf {QCoh}({{\mathbb {A}}}_{{\text {Lie}}}^1)$, and we may take its fiber at $1 \in {{\mathbb {A}}}_{{\text {Lie}}}^1$ (by approriately tensoring with $\mathsf {Vect}$). By definition, this is $\mathsf {Fil}\, \!^{KK} {\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$, the Kazhdan-Kostant filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$.

Recall that ${\mathcal {C}}^{cl}$ has commuting (because ${{\mathbb {G}}}_a$ is commutative) actions of $\mathsf {QCoh}({{\mathbb {G}}}_a)$ (under convolution) and $\mathsf {QCoh}({\text {Lie}}({{\mathbb {G}}}_a)^{\vee }) = \mathsf {QCoh}({{\mathbb {A}}}_{{\text {Lie}}}^1)$. One immediately finds that ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi ,KK\text {--}{cl}}$, the “semi-classical” category for the special fiber, is ${\mathcal {C}}^{cl,{{\mathbb {G}}}_a,w}|_{1\in {{\mathbb {A}}}_{{\text {Lie}}}^1}$, where the restriction notation means we form the appropriate tensor product.

Warning A.5.11

This Kazhdan-Kostant filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ is not obtained by applying the method of Sect. A.5.3 to the PBW filtration. Indeed, the semi-classical categories are different.

A.5.7. We now repeat the above to produce a bifiltration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ inducing the PBW and KK filtrations.

Note that ${\mathcal {C}}^{{{\mathbb {G}}}_a,w}$ carries a canonical bifiltration from Sect. A.5.4: it is induced by the weak ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}^{{{\mathbb {G}}}_a,w}$. Moreover, because ${\text {Lie}}{{\mathbb {G}}}_a$ is bifiltered by its PBW and KK filtrations by Sect. A.5.4, it follows that the corresponding bifiltered formal group acts on $\mathsf {BiFil}\,{\mathcal {C}}^{{{\mathbb {G}}}_a,w}$. Combining our analysis in the PBW and KK cases, we find that the action of $\mathsf {QCoh}({{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/{{\mathbb {G}}}_m)$ extends to an action of:

$$\begin{aligned} \mathsf {QCoh}\Big ( ({{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar _1}^1)/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/ {{\mathbb {G}}}_m \Big ) \end{aligned}$$

where the action on the first two factors is diagonal. So we obtain our desired bifiltration by setting $\mathsf {BiFil}\,{\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$ to be the restriction of $\mathsf {BiFil}\,{\mathcal {C}}^{{{\mathbb {G}}}_a,w}$ along the map:

$$\begin{aligned} {{\mathbb {A}}}_{\hbar _1}^1/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/ {{\mathbb {G}}}_m \xrightarrow {\Delta /{{\mathbb {G}}}_m \times {\text {id}}} ({{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar _1}^1)/{{\mathbb {G}}}_m \times {{\mathbb {A}}}_{\hbar _2}^1/ {{\mathbb {G}}}_m \end{aligned}$$

where $\Delta$ is the diagonal map ${{\mathbb {A}}}_{\hbar _1}^1 = {{\mathbb {A}}}^1 \rightarrow {{\mathbb {A}}}^1 \times {{\mathbb {A}}}^1 = {{\mathbb {A}}}_{{\text {Lie}}}^1 \times {{\mathbb {A}}}_{\hbar _1}^1$. By construction, it induces the PBW and KK filtrations on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi }$, with terminological conventions consistent with Sect. A.5.4.

Remark A.5.12

Recall from Remark A.5.6 that our bifiltration induces filtrations on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi ,PBW\text {--}{cl}}$ and ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi ,KK\text {--}{cl}}$ (in the notation of loc. cit.). It is straightforward to verify that the (KK) filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi ,PBW\text {--}{cl}} = {\mathcal {C}}^{cl,{{\mathbb {G}}}_a,w}|_{0 \in {{\mathbb {A}}}_{{\text {Lie}}}^1}$ is as in Example A.5.7, i.e., induced by the diagonal ${{\mathbb {G}}}_m$-action via Example A.2.6. The (PBW) filtration on ${\mathcal {C}}^{{{\mathbb {G}}}_a,\psi ,KK\text {--}{cl}} = {\mathcal {C}}^{cl,{{\mathbb {G}}}_a,w}|_{1 \in {{\mathbb {A}}}_{{\text {Lie}}}^1}$ is obtained from degenerating the character, i.e., it is the ${{\mathbb {G}}}_m$-equivariant sheaf of categories over ${{\mathbb {A}}}^1$ with fiber ${\mathcal {C}}^{cl,{{\mathbb {G}}}_a,w}|_{\lambda \in {{\mathbb {A}}}_{{\text {Lie}}}^1}$ at $\lambda \in {{\mathbb {A}}}^1$; of course, the ${{\mathbb {G}}}_m$-equivariance here comes from the ${{\mathbb {G}}}_m$-action on $\mathsf {Fil}\, \!^{PBW} {\mathcal {C}}$.

A.5.8. We now apply the above in the Harish-Chandra setting.

Suppose as before that $({{\mathfrak {h}}},K)$ is a Harish-Chandra pair with ${{\mathfrak {h}}}$ finite-dimensional and K an affine algebraic group. We suppose ${\text {Lie}}(K) \hookrightarrow {{\mathfrak {h}}}$ for simplicity. Suppose moreover that we are given a non-trivial character $\psi :K \rightarrow {{\mathbb {G}}}_a$; let $K^{\prime }$ denote the kernel. We also let $\psi$ denote the induced character ${{\mathfrak {k}}}\rightarrow k$, or the corresponding 1-dimensional ${{\mathfrak {k}}}$-module; similarly for $-\psi$.

Then since K acts on $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}$, ${{\mathbb {G}}}_a$ acts on $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K'}$, so by the above, we obtain a PBW filtration $\mathsf {Fil}\, \!^{PBW} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$ from Sect. A.5.5. We have ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi ,PBW\text {--}{cl}} = \mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K)$.

Suppose now that ${{\mathbb {G}}}_m$ acts on K; that the K-action on ${{\mathfrak {h}}}$ has been extended to ${{\mathbb {G}}}_m \ltimes K$; and that the character $\psi :K \rightarrow {{\mathbb {G}}}_a$ is ${{\mathbb {G}}}_m$-equivariant for the inverse homothety action of ${{\mathbb {G}}}_m$ on ${{\mathbb {G}}}_a$. Then $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K^{\prime }}$ carries an action of ${{\mathbb {G}}}_a = K/K^{\prime }$ and a weak action of ${{\mathbb {G}}}_m$, giving a datum as in Sect. A.5.6. Therefore, we obtain a KK filtration $\mathsf {Fil}\, \!^{KK} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$. We have:

$$\begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi ,KK\text {--}{cl}} = \mathsf {QCoh}(\psi + ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \end{aligned}$$

where $\psi + ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee } \subseteq {{\mathfrak {h}}}^{\vee }$ is the inverse image of $\psi$ under the map ${{\mathfrak {h}}}^{\vee } \rightarrow {{\mathfrak {k}}}^{\vee }$; this locus is closed under the K-action because $\psi$ is a character.

These two filtrations fit into a bifiltration by the general formalism.

Example A.5.13

For ${{\mathfrak {k}}}= {{\mathfrak {h}}}$, the PBW filtration on $\mathsf {Rep}(K) \overset{-\otimes \psi }{\simeq } {{\mathfrak {k}}}\text {--}\mathsf {mod}^{K,\psi }$ is the constant one (as we discussed before), and the KK filtration is induced from Example A.2.6 via the weak ${{\mathbb {G}}}_m$-action on $\mathsf {Rep}(K)$. In other words, we regard ${\mathcal {O}}_K$ as a graded coalgebra, so by loc. cit. it inherits a natural coalgebra filtration; then the KK filtration is obtained by considering filtered comodules. Note that the group cohomology functor $\mathsf {Rep}(K) \rightarrow \mathsf {Vect}$ is canonically bifiltered, e.g. because the trivial representation has a canonical⁸⁸ bifiltration.

It follows formally that the induction functor ${\text {ind}}_{{{\mathfrak {k}}}}^{{{\mathfrak {h}}}}:\mathsf {Rep}(K) = {{\mathfrak {k}}}\text {--}\mathsf {mod}^{K,\psi } \rightarrow {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$ is bifiltered. Applying this to the trivial representation with its canonical bifiltration, we see that the functor $C^{\bullet }({{\mathfrak {k}}},K,(-)\otimes -\psi ): {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi } \rightarrow \mathsf {Vect}$ is also naturally bifiltered. Its underlying semi-classical functors for the PBW and KK filtrations are the appropriate global sections functors:

$$\begin{aligned} \begin{aligned} \Gamma (({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K,-):\mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\\ \Gamma (\psi + ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K,-): \mathsf {QCoh}(\psi + ({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}. \end{aligned} \end{aligned}$$

Somewhat more generally,⁸⁹ suppose that the character $\psi$ is extended to ${{\mathfrak {h}}}$ and continues to satisfy the appropriate ${{\mathbb {G}}}_m$-equivariance.

Then for $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$, $M \otimes -\psi$ can be considered as an object of ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$, so it makes sense to take the Harish-Chandra cohomology:

$$\begin{aligned} C^{\bullet }({{\mathfrak {h}}},K,M \otimes -\psi ). \end{aligned}$$

This functor is bifiltered, with PBW and KK semi-classical versions given by:

$$\begin{aligned} \begin{aligned} \mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\\ \mathsf {QCoh}(\psi +({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\end{aligned} \end{aligned}$$

given by !-restriction to 0/K or $\psi /K$ followed by global sections on this stack (i.e., group cohomology for K).

Remark A.5.14

Let us describe what a KK filtration on an object of ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$ looks like concretely. Suppose $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi ,\heartsuit }$ observe that $M \otimes -\psi \in {{\mathfrak {k}}}\text {--}\mathsf {mod}^K = \mathsf {Rep}(K)$, i.e., the natural ${{\mathfrak {k}}}$-action integrates to the group. A sequence:

$$\begin{aligned} \ldots \subseteq F_i^{KK} M \subseteq F_{i+1}^{KK} M \subseteq \ldots \end{aligned}$$

is a KK filtration if:

• It is a filtration of M considered as a module over the KK-filtered algebra $U({{\mathfrak {h}}})$. In other words, if ${{\mathfrak {h}}}^j$ indicates the jth graded component of ${{\mathfrak {h}}}$, ${{\mathfrak {h}}}^j$ maps $F_i^{KK} M$ to $F_{i+j+1}^{KK} M$.

• Consider ${\mathcal {O}}_K$ as a filtered coalgebra using the ${{\mathbb {G}}}_m$-action on it and Example A.2.6. Then the coaction map $(M \otimes -\psi ) \rightarrow (M\otimes -\psi ) \otimes {\mathcal {O}}_K$ should be filtered.

If K is connected, this is equivalent to asking that ${{\mathfrak {k}}}^j$ acting on $M \otimes -\psi$ takes $F_i^{KK} M \otimes -\psi$ to $F_{i+j}^{KK} M \otimes -\psi$.

Suppose now that $\psi$ is ${{\mathbb {G}}}_m$-equivariantly extended to ${{\mathfrak {h}}}$. We also suppose that K is unipotent, so $C^{\bullet }({{\mathfrak {h}}},K,-)$ coincides with $C^{\bullet }({{\mathfrak {h}}},-)$ as a non-filtered functor. Then the KK filtration on $C^{\bullet }({{\mathfrak {h}}},K,M \otimes -\psi )$ is similar to the filtration from Remark A.3.16; its ith term is:

1.6 Compact Lie algebras

A.6.1. We now begin to move to an infinite dimensional setting. Let ${{\mathfrak {h}}}$ be a profinite-dimensional Lie algebra, so ${{\mathfrak {h}}}= {\text {lim}}_i {{\mathfrak {h}}}/{{\mathfrak {h}}}_i$ is a filtered limit of finite-dimensional Lie algebras ${{\mathfrak {h}}}/{{\mathfrak {h}}}_i \in \mathsf {LieAlg}(\mathsf {Vect}^{\heartsuit })$, and with all structure maps being surjective. Of course, ${{\mathfrak {h}}}_i \subseteq {{\mathfrak {h}}}$ indicates the corresponding normal open Lie subalgebra, which is of the requisite type.

Following [24] §22–23, we define:

$$\begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}{:}{=}\underset{i}{{\text {colim}}} \, {{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}\in \mathsf {DGCat}_{cont} \end{aligned}$$

where our structure functors are forgetful functors. By our assumptions, for each structure map ${{\mathfrak {h}}}/{{\mathfrak {h}}}_i \twoheadrightarrow {{\mathfrak {h}}}/{{\mathfrak {h}}}_j$, the induced functor:

$$\begin{aligned} {\text {Oblv}}: {{\mathfrak {h}}}/{{\mathfrak {h}}}_j\text {--}\mathsf {mod}\rightarrow {{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}\end{aligned}$$

has a continuous right adjoint: it is Lie algebra cohomology with respect to the ${{\mathfrak {h}}}_j/{{\mathfrak {h}}}_i$. Therefore, the above colimit is also the limit under these right adjoint functors.

By Lemma 5.2.5, ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ has a canonical t-structure compatible with filtered colimits with heart the abelian category of discrete⁹⁰${{\mathfrak {h}}}$-modules. Moreover, if our indexing category is countable, ${{\mathfrak {h}}}\text {--}\mathsf {mod}^+$ is the (bounded below) derived category of this abelian category.

We see that ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ is compactly generated, and that it has a canonical trivial representation $k \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$ that is compact. Therefore, we have a continuous functor $C^{\bullet }({{\mathfrak {h}}},-): {{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$, which is defined as the complex of maps from the trivial representation.

Remark A.6.1

Note that the t-structure on ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ is not necessarily left complete. Indeed, suppose ${{\mathfrak {h}}}$ is abelian and infinite-dimensional. Then⁹¹${\text {Ext}}_{{{\mathfrak {h}}}\text {--}\mathsf {mod}}^{\bullet }(k,k) = \Lambda ^{\bullet } {{\mathfrak {h}}}^{\vee }$, so there are non-zero maps $k \rightarrow k[n]$ for each $n \ge 0$. If the t-structure were left complete, we would have $\oplus _{n\ge 0} k[n] \xrightarrow {\simeq }\prod _{n \ge 0} k[n]$ (proof: consider the Postnikov tower for the LHS). But this is impossible: we would have constructed a map $k \rightarrow \oplus _{n \ge 0} k[n]$ that would not factor through any finite direct sum, contradicting the compactness of k.

In fact, the t-structure is not even left separated. Here is one explicit way to see this. Then ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ is canonically self-dual in the sense of [35]: indeed, each ${{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}$ has a canonical Serre self-duality, and we tautologically have:⁹²

$$\begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{\vee } = \underset{{\text {Oblv}}^{\vee }}{{\text {lim}}} {{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}\end{aligned}$$

where the notation indicates the limit under the functors dual to the forgetful functors; these are given by coinvariants with respect to the kernels, which differ from the invariants by a shift and tensoring with a 1-dimensional representation, according to Lemma A.3.6. This readily implies the claim: we should replace Serre self-duality on each ${{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}$ by its composition with the functor of shifting by $\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}_i$ and tensoring with the determinant of the adjoint representation.

It follows that we have an equivalence ${{\mathbb {D}}}:\mathsf {Pro}({{\mathfrak {h}}}\text {--}\mathsf {mod}^c)^{op} \simeq {{\mathfrak {h}}}\text {--}\mathsf {mod}$, where ${{\mathfrak {h}}}\text {--}\mathsf {mod}^c$ indicates the subcategory of compact objects. The objects $U({{\mathfrak {h}}}/{{\mathfrak {h}}}_i)$ are compact in ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ (and even generate), and form a filtered projective system in the obvious way; we denote this pro-object by $U({{\mathfrak {h}}})$. The object ${{\mathbb {D}}}U({{\mathfrak {h}}}) \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$ is then obviously non-zero, but lives in $\cap _n {{\mathfrak {h}}}\text {--}\mathsf {mod}^{\le -n}$ because ${{\mathbb {D}}}U({{\mathfrak {h}}}) = {\text {colim}}_i {{\mathbb {D}}}U({{\mathfrak {h}}}/{{\mathfrak {h}}}_i)$, and ${{\mathbb {D}}}U({{\mathfrak {h}}}/{{\mathfrak {h}}}_i)$ lies in cohomological degree $-\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}_i$.

Remark A.6.2

Note that ${{\mathfrak {h}}}^{\vee }$ is a Lie coalgebra, so there is a general formalism of taking comodules over it. It is straightforward to show that ${{\mathfrak {h}}}^{\vee }\text {--}\mathsf {comod}$ is the left completion of ${{\mathfrak {h}}}\text {--}\mathsf {mod}$.

Warning A.6.3

For $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}^+$, $C^{\bullet }({{\mathfrak {h}}},M)$ is computed by a standard complex, but this is not true for general $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$. To make this precise, note that for any $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$, one can form a semi-cosimplicial diagram and the canonical morphism:

This map is an equivalence for $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}^+$, but not for general M. Indeed, considering the right hand side as a functor in the variable M, it is easy to see that it will not commute with colimits.

A.7. We have a canonical filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}$. Indeed, this follows immediately from the fact that the functors ${\text {Oblv}}: {{\mathfrak {h}}}/{{\mathfrak {h}}}_j \text {--}\mathsf {mod}\rightarrow {{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}$ are filtered.

We have:

$$\begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl} \simeq \underset{i}{{\text {colim}}} \, \mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {h}}}_i)^{\vee }) \end{aligned}$$

where the colimit is under $*$-pushforward functors along the closed embeddings $\alpha _{i,j}:({{\mathfrak {h}}}/{{\mathfrak {h}}}_j)^{\vee } \hookrightarrow ({{\mathfrak {h}}}/{{\mathfrak {h}}}_i)^{\vee }$.

We claim that ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl}$ is canonically isomorphic to⁹³$\mathsf {IndCoh}({{\mathfrak {h}}}^{\vee })$, where ${{\mathfrak {h}}}^{\vee }$ is considered as an indscheme. Indeed, in the standard $\mathsf {IndCoh}$ notation from [36], we have commutative diagrams:

with vertical arrows equivalences; of course, these commutative diagrams have the requisite compatibilities of higher category theory. Therefore, ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl}$ is equivalent to this colimit; since the functors $\alpha _{i,j,*}^{\mathsf {IndCoh}}$ admit the continuous right adjoints $\alpha _{i,j}^!$, we obtain the claim.

As before, the functor $C^{\bullet }({{\mathfrak {h}}},-):{{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$ is filtered, and with semi-classical functor $\mathsf {IndCoh}({{\mathfrak {h}}}^{\vee }) \rightarrow \mathsf {Vect}$ given as !-restriction to $0 \in {{\mathfrak {h}}}^{\vee }$.

1.6.1 Actions of group schemes on filtered categories

It is straightforward to generalize the above definitions to the Harish-Chandra setting and to compute the outputs. But it is quite clarifying in this setting to generalize the language of Sect. A.3.13, so we do so.

A.7.2. Suppose K is an affine group scheme; we write K as a filtered limit ${\text {lim}}_i K/K_i$ for $K_i \subseteq K$ a normal subgroup scheme with $K/K_i$ an affine algebraic group. Recall that an action of an algebraic group on a filtered category induced a weak action the semi-classical category. As a warm-up, we begin our discussion there.

Definition A.7.1

A weak action of K on a category ${\mathcal {C}}\in \mathsf {DGCat}_{cont}$ a $\mathsf {QCoh}(K)$-module structure on ${\mathcal {C}}$, where $\mathsf {QCoh}(K)$ is given the convolution monoidal structure.

A renormalized (weak)⁹⁴action of K is an object of ${\text {lim}}_i \mathsf {QCoh}(K/K_i)\text {--}\mathsf {mod}$, where the structure functors $\mathsf {QCoh}(K/K_i)\text {--}\mathsf {mod}\rightarrow \mathsf {QCoh}(K/K_j)\text {--}\mathsf {mod}$ are given by weak invariants with respect to $K_j/K_i$.

Notation A.7.2

We say a renormalized action of K is on ${\mathcal {C}}\in \mathsf {DGCat}_{cont}$ if the compatible system of $\mathsf {QCoh}(K/K_i)$-module categories is denoted by ${\mathcal {C}}^{K_i\text {--}ren}$ and ${\mathcal {C}}= {\text {colim}}\, {\mathcal {C}}^{K_i\text {--}ren}$. Note that this is necessarily a co/limit situation, i.e., all structure functors admit right adjoints.

Example A.7.3

Define $\mathsf {Rep}(K)$ as ${\text {colim}}\, \mathsf {Rep}(K/K_i) \in \mathsf {DGCat}_{cont}$. (This category should not be confused with $\mathsf {QCoh}({{\mathbb {B}}}K)$: they have t-structures and coincide on bounded below derived categories, but $\mathsf {Rep}(K)$ is always compactly generated while $\mathsf {QCoh}({{\mathbb {B}}}K)$ may not be; rather, the latter category is the left completion of the former.)

Clearly this definition makes sense for any affine group scheme. Moreover, $\mathsf {Rep}(K_i)$ is weakly acted upon by $K/K_i$. So setting $\mathsf {Vect}^{K_i\text {--}ren} {:}{=}\mathsf {Rep}(K_i)$, we obtain a renormalized action of K on $\mathsf {Vect}$.

Remark A.7.4

Note that the trivial representation in $\mathsf {Rep}(K)$ is compact, so we have a continuous group cohomology functor $C^{\bullet }(K,-):\mathsf {Rep}(K) \rightarrow \mathsf {Vect}$.

Remark A.7.5

Note that for any ${\mathcal {C}}$ with a renormalized action of K, $\mathsf {Rep}(K)$ acts on ${\mathcal {C}}^{K\text {--}ren}$: indeed, writing ${\mathcal {C}}^{K\text {--}ren} = ({\mathcal {C}}^{K_i\text {--}ren})^{K/K_i,w}$, we find $\mathsf {Rep}(K_i)$ acts, so taking the colimit over i gives the claim.

In fact, since the colimit defining $\mathsf {Rep}(K)$ is under (symmetric) monoidal functors, and since:

$$\begin{aligned} \mathsf {QCoh}(K/K_i)\text {--}\mathsf {mod}\xrightarrow {(-)^{K/K_i,w}} \mathsf {Rep}(K/K_i)\text {--}\mathsf {mod}\end{aligned}$$

is an equivalence (by 1-affineness of ${{\mathbb {B}}}K/K_i$), we find that the functor ${\mathcal {C}}\mapsto {\mathcal {C}}^{K\text {--}ren}$ is actually an equivalence between $\mathsf {Rep}(K)\text {--}\mathsf {mod}$ and the (2-)category of categories with a renormalized K-action.

Remark A.7.6

Suppose that X is a quasi-compact quasi-separated classical⁹⁵ scheme with a K-action. By Noetherian approximation, we can write $X = {\text {lim}}_i X^i$ under affine morphisms and $K = {\text {lim}}_i K/K_i$ as above such that $X^i$ is finite type and $K/K_i$ acts on $X^i$, with these actions being compatible in the natural sense as we vary i. Finally, we assume that all structure morphisms among the $X^i$ are flat.

In this case, we obtain a renormalized action of K on $\mathsf {QCoh}(X)$ by setting:

$$\begin{aligned} \mathsf {QCoh}(X)^{K\text {--}ren} \,{:}{=}\, \underset{i}{{\text {colim}}} \, \mathsf {QCoh}(X^i/(K/K_i)). \end{aligned}$$

To emphasize a stacky perspective, we sometimes use the notation:

$$\begin{aligned} \mathsf {QCoh}^{ren}(X/K) = \mathsf {QCoh}(X)^{K\text {--}ren}. \end{aligned}$$

All structure maps here are affine, so this is a co/limit situation. We similarly have $\mathsf {IndCoh}(X)^{K\text {--}ren}$; the flatness of our structure maps implies that this is also a co/limit. (All of this is invariant under choices of presentations as limits.)

Note that for $X = {\text {Spec}}(k)$, we recover $\mathsf {Rep}(K)$ by this construction.

More generally, let $X = {\text {colim}}_j X_j$ be an indscheme, with the $X_j$ schemes satisfying the above hypotheses. Moreover, we assume the closed embeddings among the $X_j$ are finitely presented and eventually coconnective (e.g. regular). Note that $\mathsf {QCoh}(X) {:}{=}{\text {lim}}\, \mathsf {QCoh}(X_j)$ is a co/limit, and similarly for $\mathsf {IndCoh}(X)$.⁹⁶ Clearly each of these categories has a canonical renormalized action of K.

A.7.3. We now discuss the filtered setting.

Definition A.7.7

A (strong) action of K on a filtered category is an object of:

$$\begin{aligned} \underset{i}{{\text {lim}}} \, \mathsf {IndCoh}(\mathsf {Fil}\, (K/K_i)_{dR})\text {--}\mathsf {mod}(\mathsf {QCoh}({{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m)\text {--}\mathsf {mod}) \end{aligned}$$

i.e., a compatible system of filtered categories acted on by the $K/K_i$.

As before, we say this action is on $\mathsf {Fil}\, {\mathcal {C}}$ if our compatible system is denoted $\mathsf {Fil}\, {\mathcal {C}}^{K_i}$ and $\mathsf {Fil}\, {\mathcal {C}}= {\text {colim}}_i \mathsf {Fil}\, {\mathcal {C}}^{K_i}$. Again, this is a co/limit.

If K has a prounipotent tail, then ${\mathcal {C}}$ inherits a (strong) action of K with all notation compatible, i.e., the generic fiber ${\mathcal {C}}^{K_i}$ of $\mathsf {Fil}\, {\mathcal {C}}^{K_i}$ is the $K_i$-invariants for this action.

Remark A.7.8

At the semi-classical level, we obtain an object of:

$$\begin{aligned} \underset{i}{{\text {lim}}} \, \mathsf {QCoh}(({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }/(K/K_i))\text {--}\mathsf {mod}{=}{:}\mathsf {ShvCat}^{ren}({{\mathfrak {k}}}^{\vee }/K). \end{aligned}$$

Here we use the sheaf of categories notation because, by [41],

$$\begin{aligned} \mathsf {ShvCat}({{\mathfrak {k}}}^{\vee }) \,{:}{=}\, \underset{i}{{\text {lim}}} \, \mathsf {QCoh}(({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee })\text {--}\mathsf {mod}\ne \mathsf {QCoh}({{\mathfrak {k}}}^{\vee })\text {--}\mathsf {mod}. \end{aligned}$$

(Although the RHS is a full subcategory of the LHS.) We use the superscript ren because of the relationship to our notion of a renormalized action of K on a category.

In the above setup, we use the notation ${\mathcal {C}}^{cl}|_{({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }}^{K_i\text {--}ren}$ to indicate the corresponding object of $\mathsf {QCoh}(({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }/(K/K_i))\text {--}\mathsf {mod}$. With this notation, we are encouraging the reader to imagine ${\mathcal {C}}^{cl}$ as sitting over ${{\mathfrak {k}}}^{\vee }$ and equipped with a compatible weak (or better: renormalized) action of K.

Then observe that the filtration on ${\mathcal {C}}^K$ has ${\mathcal {C}}^{K,cl} = {\mathcal {C}}^{cl}|_0^{K\text {--}ren}$, and more generally, ${\mathcal {C}}^{K_i,cl} = {\mathcal {C}}^{cl}|_{({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }}^{K_i\text {--}ren}$.

Remark A.7.9

Note that in the above setting, we may reformulate our semi-classical data in saying that we have a compatible system of categories ${\mathcal {C}}^{cl}|_{({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }}$ equipped with renormalized K-actions and $\mathsf {QCoh}(({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee })$-module category structures and satisfying the natural compatibilities.

We let ${\mathcal {C}}^{cl}$ denote the limit of this diagram. Note that this is actually a co/limit situation. Then ${\mathcal {C}}^{cl}$ can be thought of as the global sections of the sheaf of categories on ${{\mathfrak {k}}}^{\vee }$ that our datum induced. Note that ${\mathcal {C}}$ itself actually is a filtered category, with ${\mathcal {C}}^{cl}$ as its semi-classical version.

Note that the place to be careful about making mistakes in distinguishing sheaves of categories from module categories is that we may have:

$$\begin{aligned} {\mathcal {C}}^{cl}|_{({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }} \ne {\mathcal {C}}^{cl} \underset{\mathsf {QCoh}({{\mathfrak {k}}}^{\vee })}{\otimes } \mathsf {QCoh}(({{\mathfrak {k}}}/{{\mathfrak {k}}}_i)^{\vee }). \end{aligned}$$

A.7.4. Now suppose that we are in a Harish-Chandra setting: we assume we are given a projective system of Harish-Chandra data $({{\mathfrak {h}}}/{{\mathfrak {h}}}_i,K/K_i)$ with ${{\mathfrak {k}}}/{{\mathfrak {k}}}_i \rightarrow {{\mathfrak {h}}}/{{\mathfrak {h}}}_i$ injective. Our two projective systems ${{\mathfrak {h}}}/{{\mathfrak {h}}}_i$ and $K/K_i$ are assumed to satisfy our earlier (e.g., finiteness) hypotheses.

We then set:

$$\begin{aligned} \mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}^K \,{:}{=}\, \underset{i}{{\text {colim}}} \, \mathsf {Fil}\, {{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}^{K/K_i} \in \mathsf {DGCat}_{cont}. \end{aligned}$$

Note that this is a co/limit situation; the right adjoint to the forgetful functor:

$$\begin{aligned} \mathsf {Fil}\, {{\mathfrak {h}}}/{{\mathfrak {h}}}_j\text {--}\mathsf {mod}^{K/K_j} \rightarrow \mathsf {Fil}\, {{\mathfrak {h}}}/{{\mathfrak {h}}}_i\text {--}\mathsf {mod}^{K/K_i} \end{aligned}$$

is given by (the filtered version of) Harish-Chandra cohomology with respect to $({{\mathfrak {h}}}_j/{{\mathfrak {h}}}_i,K_j/K_i)$.

Note that this construction makes sense for each $K_i$ in place of K. Moreover, $K/K_i$ acts on $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K_i}$, and we have natural compatibilities as we take invariants. Therefore, the above data defines an action of K on $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}$.

Note that ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,cl} = \mathsf {IndCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K)$: the calculation is the same as the one we gave for ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl}$.

We have a natural filtered Harish-Chandra cohomology functor ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Vect}$ with expected semi-classical version given by !-restriction to 0/K followed by group cohomology.

A.7.5. Now suppose in the above setting that we have compatible ${{\mathbb {G}}}_m$-actions on each $K/K_i$ and ${{\mathfrak {h}}}/{{\mathfrak {h}}}_i$. Suppose moreover that we are given a ${{\mathbb {G}}}_m$-equivariant character $\psi :K \rightarrow {{\mathbb {G}}}_a$ for the action of ${{\mathbb {G}}}_m$ on ${{\mathbb {G}}}_a$ by inverse homotheties.

The construction of Sect. A.5.8 applies as is, giving a Kazhdan-Kostant filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$ fitting into a bifiltration with the PBW filtration. It has similar properties to the finite-dimensional version, up to the various differences we saw above between the finite and profinite-dimensional settings.

A.7.6. Now suppose that we are given ${{\mathfrak {h}}}^0 \subseteq {{\mathfrak {h}}}$ an open subalgebra, so ${{\mathfrak {h}}}/{{\mathfrak {h}}}^0$ is finite-dimensional. We assume the pair $({{\mathfrak {h}}}^0,K)$ satisfies the profinite-dimensional Harish-Chandra conditions as above, so ${{\mathfrak {k}}}\subseteq {{\mathfrak {h}}}^0 \subseteq {{\mathfrak {h}}}$ and ${{\mathfrak {h}}}^0$ is a K-submodule of ${{\mathfrak {h}}}$.

We have the following version of Corollary A.3.9 and Lemma A.3.18.

Lemma A.7.10

• (1) The forgetful functor ${{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow {{\mathfrak {h}}}^0\text {--}\mathsf {mod}$ admits a left adjoint ${\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}}$ as a filtered functor. The induced semi-classical functor $({\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}})^{cl}$ is the $(\mathsf {IndCoh},*)$ pullback functor:

  $$\begin{aligned} \mathsf {IndCoh}({{\mathfrak {h}}}^{0,\vee }) \rightarrow \mathsf {IndCoh}({{\mathfrak {h}}}^{\vee }) \end{aligned}$$

  i.e., the left adjoint to the $\mathsf {IndCoh}$-pushforward along the projection ${{\mathfrak {h}}}^{\vee } \rightarrow {{\mathfrak {h}}}^{0,\vee }$.

  There is a canonical isomorphism of filtered functors ${{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}$:

  $$\begin{aligned} C^{\bullet }\Big ({{\mathfrak {h}}},{\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}}\big ((-) \otimes \det ({{\mathfrak {h}}}/{{\mathfrak {h}}}^0)[\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}^0]\big )\Big ) = C^{\bullet }({{\mathfrak {h}}}^0,-) \end{aligned}$$

  where $\det ({{\mathfrak {h}}}/{{\mathfrak {h}}}^0)[\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}^0])$ is filtered with a single jump in degree $\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}^0$.

• (2) The functor ${\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}}$ is a morphism of filtered categories acted on by K. The induced functor ${\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}}: {{\mathfrak {h}}}^0\text {--}\mathsf {mod}^K \rightarrow {{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ has semi-classical version:

  $$\begin{aligned} \mathsf {IndCoh}^{ren}(({{\mathfrak {h}}}^0/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {IndCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \end{aligned}$$

  again given by $(\mathsf {IndCoh},*)$-pullback.

  There is a canonical isomorphism of filtered functors ${{\mathfrak {h}}}^0\text {--}\mathsf {mod}^K \rightarrow \mathsf {Vect}$:

  $$\begin{aligned} C^{\bullet }\Big ({{\mathfrak {h}}},K,{\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}}\big ((-) \otimes \det ({{\mathfrak {h}}}/{{\mathfrak {h}}}^0)[\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}^0]\big )\Big ) = C^{\bullet }({{\mathfrak {h}}}^0,K,-). \end{aligned}$$

• (3) Suppose now that we are given the extra data of Sect. A.7.5, and suppose that we are given a character $\psi :{{\mathfrak {h}}}\rightarrow k$ extending the same-named character on ${{\mathfrak {k}}}$. Then there is a canonical isomorphism of bifiltered functors ${{\mathfrak {h}}}^0\text {--}\mathsf {mod}^{K,\psi } \rightarrow \mathsf {Vect}$:

  $$\begin{aligned} C^{\bullet }\Big ({{\mathfrak {h}}},K,{\text {ind}}_{{{\mathfrak {h}}}^0}^{{{\mathfrak {h}}}}\big ((-) \otimes (-\psi ) \otimes \det ({{\mathfrak {h}}}/{{\mathfrak {h}}}^0)[\dim {{\mathfrak {h}}}/{{\mathfrak {h}}}^0]\big )\Big ) = C^{\bullet }({{\mathfrak {h}}}^0,K, - \otimes (-\psi )). \end{aligned}$$

  Each of these functors has semi-classical version:

  $$\begin{aligned} \mathsf {IndCoh}^{ren}(\psi +({{\mathfrak {h}}}^0/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\end{aligned}$$

  given by !-restriction to $\psi /K$ followed by $\Gamma ^{\mathsf {IndCoh}}$ (i.e., group cohomology with respect to K).

Proof

These results follow immediately from their finite-dimensional counterparts by passing to the limit. $\square$

1.7 Tate setting

A.8.1. Our treatment here follows [37] at some points.

Suppose ${{\mathfrak {h}}}\in \mathsf {Pro}(\mathsf {Vect}^{\heartsuit })$ is a Tate Lie algebra, i.e., ${{\mathfrak {h}}}$ is a limit under surjective maps of (possibly infinite dimensional) vector spaces, has a continuous Lie bracket, and an open profinite dimensional Lie subalgebra.⁹⁷

Suppose moreover that we are given a Harish-Chandra datum $({{\mathfrak {h}}},K)$ with K a group scheme and ${{\mathfrak {k}}}\hookrightarrow {{\mathfrak {h}}}$ an open⁹⁸ subalgebra. We are going to define a filtered category ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ acted on by K.

First, observe that the group prestack $({{\mathfrak {h}}},K)$ from Sect. A.3.7 still makes sense, receiving a canonical ind-affine nil-isomorphism ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$. The definition from loc. cit. does not make sense as is: de Rham spaces and formal completions are best avoided in infinite type. Instead, we assume ${{\mathfrak {h}}}= {\text {Lie}}(H)$ for a group indscheme H with $K \subseteq H$ a compact open subgroup;⁹⁹ then $({{\mathfrak {h}}},K)$ is the formal completion of K in H. In general, one can appeal e.g. to [7] 7.11.2 (v) for the construction.

We form the simplicial diagram:

given by applying the Cech construction to the morphism ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$; note that the geometric realization of this diagram is ${{\mathbb {B}}}({{\mathfrak {h}}},K)$. Moreover, note that each term in the simplicial diagram is of the form “an ind-finite type indscheme modulo an action of K.” Therefore, $\mathsf {IndCoh}^{ren}$ makes sense for each term of this diagram. We define ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ as the totalization:

Since all the morphisms in our simplicial diagram are ind-affine nil-isomorphisms, this is a co/limit. The Beck-Chevalley formalism easily implies that the forgetful functor ${\text {Oblv}}:{{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Rep}(K)$ is monadic, and in particular, admits a left adjoint ${\text {ind}}_{{{\mathfrak {k}}}}^{{{\mathfrak {h}}}}$.

It is easy to see that ${\text {ind}}_{{{\mathfrak {k}}}}^{{{\mathfrak {h}}}} \circ {\text {Oblv}}$ is conservative and t-exact, so ${\text {Oblv}}$ is monadic and ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$ has a canonical t-structure with ${\text {Oblv}}$ and ${\text {ind}}_{{{\mathfrak {k}}}}^{{{\mathfrak {h}}}}$ both t-exact. We have:

Lemma A.8.1

(Bernstein-Lunts, [12]) If $K = {\text {lim}}_i K/K_i$ is a countable inverse limit, ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,+}$ is the bounded below derived category of the heart of its t-structure.

Indeed, $\mathsf {Rep}(K)^+ = D^+(\mathsf {Rep}(K)^{\heartsuit })$ by Lemma 5.2.5 (which is where the countability hypothesis enters), and then the same argument as in the finite-dimensional Lemma A.4.1 applies.

A.8.2. Now note that the deformations defined in [42] §IV.5.2 makes sense in the infinite type setup and are well-behaved in our setup. Applying this to ${{\mathbb {B}}}K \rightarrow {{\mathbb {B}}}({{\mathfrak {h}}},K)$, we obtain a prestack $\mathsf {Fil}\, {{\mathbb {B}}}({{\mathfrak {h}}},K)$ over ${{\mathbb {A}}}_{\hbar }^1/{{\mathbb {G}}}_m$ with special fiber $({{\mathbb {B}}}({{\mathfrak {h}}}/{{\mathfrak {k}}})_0^{\wedge })/K$.

We can form the above Cech construction along this deformation and imitate the above construction to obtain a filtration on ${{\mathfrak {h}}}\text {--}\mathsf {mod}^K$. We claim that ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,cl}$ is canonically isomorphic to $\mathsf {QCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K)$, with $({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }$ the continuous dual considered as an affine scheme. Indeed, we need to compute:

Here it makes sense to replace ${{\mathfrak {h}}}/{{\mathfrak {k}}}$ by any K-representation V. Since any K-representation is the union of its finite-dimensional representations, we find that the above is $\mathsf {IndCoh}^{ren}({{\mathbb {B}}}({{\mathfrak {h}}}/{{\mathfrak {k}}})_0/K)$ (where the renormalization makes sense because ${{\mathbb {B}}}({{\mathfrak {h}}}/{{\mathfrak {k}}})_0$ is the appropriate colimit of the classifying stacks acted on by K corresponding to finite-dimensional subrepresentations of ${{\mathfrak {h}}}/{{\mathfrak {k}}}$). Clearly $\mathsf {IndCoh}^{ren}({{\mathbb {B}}}({{\mathfrak {h}}}/{{\mathfrak {k}}})_0^{\wedge }/K) \simeq \mathsf {QCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K)$ as desired.

Finally, note that if $K = {\text {lim}}_i K/K_i$ as before, then the above construction makes sense for each of the compact open normal subgroup schemes $K_i \subseteq K$. Moreover, $K_i/K_j$-invariants for $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K_j}$ are easily seen to give $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K_i}$.

This is exactly the data to define the filtered category ${{\mathfrak {h}}}\text {--}\mathsf {mod}$ acted on by K. Note that:

$$\begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{cl} = \mathsf {IndCoh}({{\mathfrak {h}}}^{\vee }) \,{:}{=}\, \underset{i}{{\text {colim}}} \, \mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {k}}}_i)^{\vee }) \in \mathsf {DGCat}_{cont} \end{aligned}$$

with the colimit being under $*$-pushforward functors; we label this colimit as $\mathsf {IndCoh}$ for the same reason as in Sect. A.7.

More precisely, recall that our semi-classical data is a renormalized sheaf of categories on ${{\mathfrak {k}}}^{\vee }/K$. This sheaf of categories is described in the same way as in Sect. A.7.4.

A.8.3. Next, we note that the above makes sense even if ${{\mathfrak {k}}}$ is not an open subalgebra.

More precisely, and with apologies for the notation change, choose $H_0$ a group scheme and a Harish-Chandra datum $(H_0,{{\mathfrak {h}}})$ with ${{\mathfrak {h}}}_0 \subseteq {{\mathfrak {h}}}$ open. Then suppose that K is a group subscheme of $H_0$, with no hypothesis that it be compact open (e.g., K could be trivial).

Then as above, we have an action of $H_0$ on $\mathsf {Fil}\, {{\mathfrak {h}}}\text {--}\mathsf {mod}$. We claim that given any action of $H_0$ on $\mathsf {Fil}\, {\mathcal {C}}$, we can restrict to obtain an action of K on $\mathsf {Fil}\, {\mathcal {C}}$.

Indeed, if $H_0 = {\text {lim}}_i \, H_0/H_i$ for $H_i$ a normal subgroup scheme, note that $H_i K$ is a compact open subgroup scheme of H; we then set:

$$\begin{aligned} \mathsf {Fil}\, {\mathcal {C}}^K \,{:}{=}\, \underset{i}{{\text {colim}}} \, \mathsf {Fil}\, {\mathcal {C}}^{H_i K} \in \mathsf {DGCat}_{cont}. \end{aligned}$$

We remark that this is a co/limit. Replacing K by a compact open subgroup (of K), we obtain the requisite data.

1.7.1 Semi-infinite cohomology

We now make a more stringent assumption on ${{\mathfrak {h}}}$: suppose that it is a union¹⁰⁰ of open pro-finite dimensional subalgebras ${{\mathfrak {h}}}= {\text {colim}}_i \, {{\mathfrak {h}}}_i$. We fix an initial index “0” and let ${{\mathfrak {h}}}_0$ denote the corresponding open subalgebra.

We assume that for every ${{\mathfrak {h}}}_i \subseteq {{\mathfrak {h}}}_j$, the action of ${{\mathfrak {h}}}_i$ on $\det ({{\mathfrak {h}}}_j/{{\mathfrak {h}}}_i)$ is trivial; e.g., this is automatically the case if ${{\mathfrak {h}}}$ is ind-pronilpotent. For later use, we observe that in this case:

$$\begin{aligned} {\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}(M \otimes \det ({{\mathfrak {h}}}_j/{{\mathfrak {h}}}_i)) = {\text {ind}}_{{{\mathfrak {h}}}_1}^{{{\mathfrak {h}}}_2}(M) \otimes \det ({{\mathfrak {h}}}_j/{{\mathfrak {h}}}_i) \end{aligned}$$

(A.8.2)

since we are just tensoring by a line. (This line is essentially just a placeholder, ensuring the canonicity of various isomorphisms.)

In this case, we obtain a semi-infinite cohomology functor:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,-): {{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}\end{aligned}$$

defined as follows.

Note that any of the compact open subalgebras ${{\mathfrak {h}}}_i$, we have a forgetful functor ${{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow {{\mathfrak {h}}}_i\text {--}\mathsf {mod}$, which is conservative and admits the left adjoint ${\text {ind}}_{{{\mathfrak {h}}}_i}^{{{\mathfrak {h}}}}$. Then we claim that the induced map:

$$\begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}\rightarrow \underset{i}{{\text {lim}}} \, {{\mathfrak {h}}}_i\text {--}\mathsf {mod}\end{aligned}$$

is an equivalence. Indeed, both sides are clearly monadic over ${{\mathfrak {h}}}_0\text {--}\mathsf {mod}$. This is a co/limit, so we also obtain:

$$\begin{aligned} \underset{i}{{\text {colim}}} \, {{\mathfrak {h}}}_i\text {--}\mathsf {mod}\xrightarrow {\simeq }{{\mathfrak {h}}}\text {--}\mathsf {mod}\in \mathsf {DGCat}_{cont} \end{aligned}$$

where we are using the induction functors on the left hand side.

We then define a functor:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,{\text {ind}}_{{{\mathfrak {h}}}_i}^{{{\mathfrak {h}}}}(-)): {{\mathfrak {h}}}_i\text {--}\mathsf {mod}\rightarrow \mathsf {Vect}\end{aligned}$$

as:

$$\begin{aligned} C^{\bullet }({{\mathfrak {h}}}_i,(-) \otimes \det ({{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0)[\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0]). \end{aligned}$$

By Lemma A.7.10 (1) and (A.8.1), for ${{\mathfrak {h}}}_i \subseteq {{\mathfrak {h}}}_j \subseteq {{\mathfrak {h}}}$, we have canonical isomorphisms:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,{\text {ind}}_{{{\mathfrak {h}}}_i}^{{{\mathfrak {h}}}}(-)) \simeq C^{\!\frac{\infty }{2}}\big ({{\mathfrak {h}}},{{\mathfrak {h}}}_0,{\text {ind}}_{{{\mathfrak {h}}}_j}^{{{\mathfrak {h}}}_i} {\text {ind}}_{{{\mathfrak {h}}}_i}^{{{\mathfrak {h}}}_j}(-)\big ). \end{aligned}$$

These are compatible as we vary indices, so we obtain the desired functor $C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,-)$.

Notation A.8.2

This functor depends only in a mild way on ${{\mathfrak {h}}}_0$, but it is convenient for our purposes to keep it in the notation.

Remark A.8.3

Here is another perspective. Note that by the general co/lim formalism for a filtered diagram, any $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$ can be written as ${\text {colim}}_i \, {\text {ind}}_{{{\mathfrak {h}}}_i}^{{{\mathfrak {h}}}} M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}$, i.e., we forget down to ${{\mathfrak {h}}}_i$ and then induce. So we find:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,M) = \underset{i}{{\text {colim}}} \, C^{\bullet }({{\mathfrak {h}}}_i,M \otimes \det ({{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0)[\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0]). \end{aligned}$$

Applying this formula to $M \in {{\mathfrak {h}}}\text {--}\mathsf {mod}^{\heartsuit }$ (or a bounded below chain complex of such objects) and computing $C^{\bullet }({{\mathfrak {h}}}_i,-)$ by the standard resolution, one recovers the usual complex computing semi-infinite cohomology in this case; i.e., this perspective recovers the classical one.

The above analysis applies just as well in the filtered setting, so we obtain a canonical filtration on $C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,-)$. The semi-classical functor:

$$\begin{aligned} \mathsf {IndCoh}({{\mathfrak {h}}}^{\vee }) \rightarrow \mathsf {Vect}\end{aligned}$$

is given by !-restricting¹⁰¹ to obtain an object of $\mathsf {QCoh}(({{\mathfrak {h}}}/{{\mathfrak {h}}}_0)^{\vee })$; and noting that this is $\mathsf {QCoh}$ of an affine scheme, we then take $*$-restriction to $0 \in ({{\mathfrak {h}}}/{{\mathfrak {h}}}_0)^{\vee }$. Indeed, by Lemma A.7.10, the functor $C^{\bullet }({{\mathfrak {h}}}_i,(-) \otimes \det ({{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0)[\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0])$ semi-classically gives the functor:

$$\begin{aligned} \mathsf {IndCoh}({{\mathfrak {h}}}_i^{\vee }) \rightarrow \mathsf {Vect}\end{aligned}$$

that is the composition of !-restricting to $({{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0)^{\vee }$ and then $*$-restricting to 0; passing to the colimit in i gives the claim.

Remark A.8.4

From the perspective of Remark A.8.3, we have:

$$\begin{aligned} F_i C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,M) = \underset{i}{{\text {colim}}} \, F_{i-\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0} C^{\bullet }({{\mathfrak {h}}}_i,M \otimes \det ({{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0)[\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0]). \end{aligned}$$

(The shift in indexing reflects the repeatedly emphasized fact that our determinant lines are considered as filtered with a jump in degree $\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0$.)

A.8.5. Now observe that the above all makes sense in the Harish-Chandra setting as well. Indeed, if we have the Harish-Chandra datum $({{\mathfrak {h}}},K)$ with K a group scheme, then note that ${{\mathfrak {k}}}\subseteq {{\mathfrak {h}}}_i$ for $i\gg 0$, so the formula:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,K,-) \,{:}{=}\, \underset{i}{{\text {colim}}} \, C^{\bullet }({{\mathfrak {h}}}_i,K,M \otimes \det ({{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0)[\dim {{\mathfrak {h}}}_i/{{\mathfrak {h}}}_0]) \end{aligned}$$

makes sense by cofinality and defines a filtered functor:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,K,-):{{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Vect}. \end{aligned}$$

If ${{\mathfrak {h}}}_0 = {{\mathfrak {k}}}$, then the corresponding semi-classical functor:

$$\begin{aligned} \mathsf {IndCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\end{aligned}$$

is given by !-restriction to 0/K followed by group cohomology with respect to K. If ${{\mathfrak {h}}}_0 \subseteq {{\mathfrak {k}}}$ with $\det ({{\mathfrak {k}}}/{{\mathfrak {h}}}_0)$ a trivial ${{\mathfrak {h}}}_0$-representation, then we can twist to reduce to this case, i.e., the functor:

$$\begin{aligned} C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {h}}}_0,K, (-) \otimes \det ({{\mathfrak {k}}}/{{\mathfrak {h}}}_0)^{\vee }[-\dim {{\mathfrak {k}}}/{{\mathfrak {h}}}_0]): {{\mathfrak {h}}}\text {--}\mathsf {mod}^K \rightarrow \mathsf {Vect}. \end{aligned}$$

will have the semi-classical functor described above.

Notation A.8.5

When ${{\mathfrak {h}}}_0 = {{\mathfrak {k}}}$, we use the notation $C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},K,-)$ in place of $C^{\!\frac{\infty }{2}}({{\mathfrak {h}}},{{\mathfrak {k}}},K,-)$.

1.7.2 KK version

Suppose now that $({{\mathfrak {h}}},K)$ is equipped with a ${{\mathbb {G}}}_m$-action as before. Suppose moreover that K is equipped with a character $\psi :K \rightarrow {{\mathbb {G}}}_a$ that is ${{\mathbb {G}}}_m$-equivariant for the inverse homothety action on the target.

Then ${{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi }$ makes sense, and inherits PBW and KK filtrations fitting into a bifiltration as before; this is immediate from our earlier constructions and the general KK formalism.

We have:

$$\begin{aligned} \begin{aligned} {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi ,PBW\text {--}{cl}} = \mathsf {IndCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \\ {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi ,KK\text {--}{cl}} = \mathsf {IndCoh}^{ren}(\psi +({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \end{aligned} \end{aligned}$$

as before. The KK filtration on the former and the PBW filtration on the latter are as in Remark A.5.12.

Remark A.8.6

Suppose now that ${{\mathfrak {h}}}$ satisfies the hypotheses of Sect. A.8.4, and let ${{\mathfrak {h}}}_0$ be as in loc. cit. Suppose that $\psi :{{\mathfrak {k}}}\rightarrow k$ is extended to a character $\psi :{{\mathfrak {h}}}\rightarrow k$, so the functor:

$$\begin{aligned} C^{\!\frac{\infty }{2}}\big ({{\mathfrak {h}}},{{\mathfrak {h}}}_0,K,(-) \otimes -\psi \big ): {{\mathfrak {h}}}\text {--}\mathsf {mod}^{K,\psi } \rightarrow \mathsf {Vect}\end{aligned}$$

makes sense. Note that if the subalgebras ${{\mathfrak {h}}}_i \subseteq {{\mathfrak {h}}}$ are all graded subalgebras (with respect to the grading on ${{\mathfrak {h}}}$ induced by the ${{\mathbb {G}}}_m$-action), and $\psi :{{\mathfrak {h}}}\rightarrow k$ is graded for the degree $-1$ grading on the target, then $C^{\!\frac{\infty }{2}}\big ({{\mathfrak {h}}},{{\mathfrak {h}}}_0,K,(-) \otimes -\psi \big )$ is naturally bifiltered. If ${{\mathfrak {k}}}= {{\mathfrak {h}}}_0$, then the induced semi-classical functors:

$$\begin{aligned} \begin{aligned} \mathsf {QCoh}^{ren}(({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\\ \mathsf {QCoh}^{ren}(\psi +({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }/K) \rightarrow \mathsf {Vect}\end{aligned} \end{aligned}$$

are given by $*$-restriction to 0/K and $\psi /K$ followed by global sections, i.e., group cohomology with respect to K. Here we note that our hypothesis means ${{\mathfrak {k}}}$ is open in ${{\mathfrak {h}}}$, so $({{\mathfrak {h}}}/{{\mathfrak {k}}})^{\vee }$ is a scheme, not an indscheme; so our definition of $\mathsf {IndCoh}^{ren}$ in this infinite type setting means that it tautologically coincides with $\mathsf {QCoh}^{ren}$.

1.7.3 Central extensions

Finally, we explain the straightforward extension of the above to central extensions of ${{\mathfrak {h}}}$.

A.8.8. Suppose that in the above notation, we are given $\widehat{{{\mathfrak {h}}}}$ a Tate Lie algebra and a central extension:

$$\begin{aligned} 0 \rightarrow k \rightarrow \widehat{{{\mathfrak {h}}}} \rightarrow {{\mathfrak {h}}}\rightarrow 0 \end{aligned}$$

of ${{\mathfrak {h}}}$ by the abelian Lie algebra k.

We suppose that K is as in Sect. A.8, so ${{\mathfrak {k}}}\subseteq {{\mathfrak {h}}}$ is an open subalgebra. We suppose moreover that we are given a Harish-Chandra datum $(\widehat{{{\mathfrak {h}}}},K \times {{\mathbb {G}}}_m)$ compatible in the sense that the projections:

$$\begin{aligned} \begin{aligned} \widehat{{{\mathfrak {h}}}} \rightarrow {{\mathfrak {h}}}\\ K \times {{\mathbb {G}}}_m \xrightarrow {p_1} K \end{aligned} \end{aligned}$$

induce a morphism of Harish-Chandra data. Moreover, we assume that the structural morphism $k \xrightarrow {x \mapsto (0,x)} {{\mathfrak {k}}}\times k = {\text {Lie}}(K \times {{\mathbb {G}}}_m) \rightarrow \widehat{{{\mathfrak {h}}}}$ is the given embedding of k into $\widehat{{{\mathfrak {h}}}}$; and that the ${{\mathbb {G}}}_m \subseteq K \times {{\mathbb {G}}}_m$-action on $\widehat{{{\mathfrak {h}}}}$ is trivial.

Remark A.8.7

Note that the extension $\widehat{{{\mathfrak {h}}}}$ is canonically split over ${{\mathfrak {k}}}$.

Remark A.8.8

Of course, everything that follows generalizes to the case where the central ${{\mathbb {G}}}_m$ is replaced by a torus. We mention this because it is necessary for the setting of affine Kac-Moody algebras for non-simple ${{\mathfrak {g}}}$, as in Sect. 1.8.2.

A.8.9. We want to form the DG category $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}$, which morally is the DG category of $\widehat{{{\mathfrak {h}}}}$-modules on which $1 \in k \subseteq \widehat{{{\mathfrak {h}}}}$ acts by the identity. For this, we will construct an action of $\mathsf {QCoh}({{\mathbb {A}}}^1)$ on $\widehat{{{\mathfrak {h}}}}\text {--}\mathsf {mod}$; it then makes sense to take the fiber at $1 \in {{\mathbb {A}}}^1$ (by tensoring with the restriction to 1 functor $\mathsf {QCoh}({{\mathbb {A}}}^1) \rightarrow \mathsf {Vect}$).

Indeed, note that our Harish-Chandra datum induces an action of ${{\mathbb {G}}}_{m,dR}$ on ${{\mathbb {B}}}(\widehat{{{\mathfrak {h}}}},K)$: ${{\mathbb {G}}}_m$ acts because it acts on ${{\mathfrak {h}}}$ commuting with K, and the action of the formal group is trivial because our Harish-Chandra data was extended to $(\widehat{{{\mathfrak {h}}}},K \times {{\mathbb {G}}}_m)$. Moreover, the underlying ${{\mathbb {G}}}_m$-action is canonically trivial, because ${{\mathbb {G}}}_m$ acts trivially on $\widehat{{{\mathfrak {h}}}}$. Therefore, we obtain an action of ${{\mathbb {B}}}\widehat{{{\mathbb {G}}}}_m = {{\mathbb {G}}}_{m,dR}/{{\mathbb {G}}}_m$ on ${{\mathbb {B}}}(\widehat{{{\mathfrak {h}}}},K)$.¹⁰²

Then observe that $\mathsf {QCoh}({{\mathbb {B}}}\widehat{{{\mathbb {G}}}}_m) \simeq \mathsf {QCoh}({{\mathbb {A}}}^1)$, with the convolution monoidal structure on the left hand side corresponding to the tensor product on the right hand side, so we obtain the desired action on $\widehat{{{\mathfrak {h}}}}\text {--}\mathsf {mod}$ by definition of this category, and therefore the definition of the category $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}$.

Example A.8.9

A splitting of the Lie algebra morphism $\widehat{{{\mathfrak {h}}}} \rightarrow {{\mathfrak {h}}}$ gives an identification $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}\simeq {{\mathfrak {h}}}\text {--}\mathsf {mod}$ compatible with all extra structures.

A.8.10. There is a filtered version of the above, quite similar to Sect. A.5.5. We use the ${{\mathbb {G}}}_m$ action on $\mathsf {Fil}\, \widehat{{{\mathfrak {h}}}}\text {--}\mathsf {mod}$ as above, finding that $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}$ is filtered with semi-classical category $\mathsf {IndCoh}({{\mathfrak {h}}}^{\vee })$. So the situation is not sensitive to the central extension.

The rest of the usual package generalizes as is to this setting. We have an action of K on $\mathsf {Fil}\, \widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}$, so obtain a filtration on $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}^K$. If we have compatible ${{\mathbb {G}}}_m$-actions on $\widehat{{{\mathfrak {h}}}}$ and K with $k \subseteq \widehat{{{\mathfrak {h}}}}$ acted on trivially, we obtain a bifiltration on $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}^K$; if K is equipped with an appropriately ${{\mathbb {G}}}_m$-equivariant additive character, we obtain a bifiltration on $\widehat{{{\mathfrak {h}}}}_1\text {--}\mathsf {mod}^{K,\psi }$ as well. The semi-classical categories are as expected, i.e., the same as if we worked with ${{\mathfrak {h}}}$ instead of its central extension, and the various restriction and induction functors satisfy the standard functoriality properties at the semi-classical level.

Appendix B: Proof of Lemma 5.2.1

1.1 Overview

We give two proofs of this result: a geometric one based in the theory of D-modules, and a representation theoretic one.

The former approach, which was sketched after the statement of Lemma 5.2.1, is more versatile and conceptual. But for technical reasons, we only know how to apply this method for n sufficiently large.¹⁰³

The second one is more ad hoc. The idea is that we can compute the associated graded of this functor using (the proof of) Theorem 3.1.1 and verify exactness here. However, the problems with unboundedness of Kazhdan-Kostant filtrations come in here, and we use some tricks to circumvent this.

Remark B.1.1

There is a homology between the two approaches: $(\check{\rho },\alpha _{max})$ is involved in the technical issues on both sides. Perhaps this hints at a more systematic solution.

1.2 Geometric approach

B.2.1. We begin with the D-module approach. Since ${\mathcal {C}}= \widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}$ and its Harish-Chandra variants are fairly general examples of categories acted on by a group, we introduce some axiomatics about the relationship between such group actions and t-structures. We then establish general results about ${\text {Av}}_*$ and ${\text {Av}}_!$, and deduce Lemma 5.2.1 from here.

1.2.1 Axiomatics

Fix H an affine algebraic group and ${\mathcal {C}}$ a DG category acted on weakly by H.

Suppose ${\mathcal {C}}$ is equipped with a t-structure compatible with filtered colimits. Note that $\mathsf {QCoh}(H) \otimes {\mathcal {C}}$ inherits a t-structure: $(\mathsf {QCoh}(H) \otimes {\mathcal {C}})^{\le 0}$ is generated under colimits by objects ${\mathcal {O}}_H \boxtimes {\mathcal {F}}$ for ${\mathcal {F}}\in {\mathcal {C}}^{\le 0}$.

Lemma B.2.1

The following conditions are equivalent:

• (1) The functor ${\text {Oblv}}\circ {\text {Av}}_*^w: {\mathcal {C}}\rightarrow {\mathcal {C}}$ is t-exact.

• (2) The functor ${\text {act}}: \mathsf {QCoh}(H) \otimes {\mathcal {C}}\rightarrow {\mathcal {C}}$ is t-exact.

• (3) The functor ${\text {coact}}: {\mathcal {C}}\rightarrow \mathsf {QCoh}(H) \otimes {\mathcal {C}}$ is t-exact.

• (4) ${\mathcal {C}}^{H,w}$ admits a t-structure such that ${\text {Oblv}}$ and ${\text {Av}}_*^w$ are t-exact.

• (5) The $\mathsf {QCoh}(H)$-linear equivalence:

  $$\begin{aligned} \mathsf {QCoh}(H) \otimes {\mathcal {C}}\rightarrow \mathsf {QCoh}(H) \otimes {\mathcal {C}}\end{aligned}$$

  (B.2.1)

  induced by ${\text {coact}}: {\mathcal {C}}\rightarrow \mathsf {QCoh}(H) \otimes {\mathcal {C}}$ is t-exact.

Proof

Note that we have a functor $p_2^*:{\mathcal {C}}\xrightarrow {{\mathcal {F}}\mapsto {\mathcal {O}}_H \boxtimes {\mathcal {F}}} \mathsf {QCoh}(H) \otimes {\mathcal {C}}$, which admits the conservative right adjoint $p_{2,*}$. We claim $p_2^*$ and $p_{2,*}$ are t-exact. Indeed, $p_2^*$ is tautologically right t-exact, so $p_{2,*}$ is left t-exact. But from the definition of the t-structure, we see $p_{2,*}$ is right t-exact as well, so t-exact. Then since $p_{2,*}p_2^* = {\mathcal {O}}_H \otimes -$ is t-exact, we obtain the t-exactness of $p_2^*$ as well.

We will deduce the other conditions from (1). Since e.g. (4) obviously implies it, this suffices.

Recall from the Beck-Chevalley formalism that we have:

$$\begin{aligned} p_{2,*} {\text {coact}}= {\text {act}}p_2^* = {\text {Oblv}}{\text {Av}}_*^w. \end{aligned}$$

Since $p_{2,*}$ is t-exact and conservative, we see that (1) implies (3).

We now deduce (2) from (1); by the above, we assume (3) as well. Note that t-exactness of ${\text {coact}}$ implies that its right adjoint ${\text {act}}$ is left t-exact. Since $p_2^*({\mathcal {C}}^{\le 0})$ generates $(\mathsf {QCoh}(H) \otimes {\mathcal {C}})^{\le 0}$, it suffices to show ${\text {act}}p_2^*$ is right t-exact, but this is clear since ${\text {Oblv}}{\text {Av}}_*^w$ is t-exact by assumption.

For (4), observe that ${\mathcal {C}}^{H,w}$ is the limit of a cosimplicial diagram with t-exact structure maps in the underlying semi-cosimplicial diagram (by (3)). This implies the existence of a t-structure with ${\text {Oblv}}:{\mathcal {C}}^{H,w} \rightarrow {\mathcal {C}}$ t-exact. To see ${\text {Av}}_*^w$ is t-exact, it suffices to see that ${\text {Oblv}}{\text {Av}}_*^w$ is, but this is given.

Finally, note that the equivalence (B.2.1) intertwines the functors $p_2^*$ and ${\text {coact}}$. Therefore, it suffices to see that ${\text {coact}}({\mathcal {C}}^{\le 0})$ generates $(\mathsf {QCoh}(H) \otimes {\mathcal {C}})^{\le 0}$ under colimits. But this follows because ${\text {act}}$ is t-exact and conservative.

$\square$

If these equivalent conditions are satisfied, we say the t-structure is compatible with the weak action of H.

B.2.3. Now suppose that H acts strongly on ${\mathcal {C}}$.

We say that the action is compatible with the t-structure if it is compatible for the weak action. It is equivalent to say that:

$$\begin{aligned} {\text {coact}}[-\dim H]:{\mathcal {C}}\rightarrow D(H) \otimes {\mathcal {C}}\end{aligned}$$

is t-exact. As in the weak setting, ${\mathcal {C}}^H$ inherits a t-structure with ${\text {Oblv}}:{\mathcal {C}}^H \rightarrow {\mathcal {C}}$ being t-exact.

Lemma B.2.2

In the above setting, the functor ${\text {Av}}_*:{\mathcal {C}}\rightarrow {\mathcal {C}}^H$ has cohomological amplitude $[0,\dim H]$.

More generally, for $K \subseteq H$ with H/K affine, the functor ${\text {Av}}_*:{\mathcal {C}}^K \rightarrow {\mathcal {C}}^H$ has cohomological amplitude $[0,\dim H/K]$.

Proof

${\text {Av}}_*$ is left t-exact because it is right adjoint to a t-exact functor.

For the upper bound on the amplitude, note that weak averaging from ${\mathcal {C}}^{K,w} \rightarrow {\mathcal {C}}^{H,w}$ is t-exact because H/K is affine. Observe that weak averaging is given by convolution with $D_{H/K}$, $*$-averaging is given by convolution with the constant D-module $k_{H/K}$, and then use the de Rham resolution of $k_{H/K}$, which consists of free D-modules in degrees $[0,\dim H/K]$, to complete the argument. $\square$

1.2.2 !-averaging

We now want a version of the above for !-averaging. It is essentially the same, but slightly more subtle because !-averaging may not be defined.

Moreover, the proof of Lemma B.2.2 in the case where H/K was affine used the fact that de Rham cohomology on an affine scheme is right t-exact. The corresponding fact for compactly supported de Rham cohomology is harder to show (for non-holonomic D-modules), and is the main theorem of [51].

B.2.5. Suppose in the above setting that are given $K_1,K_2$ two subgroups of H, and characters $\psi _i:K_i \rightarrow {{\mathbb {G}}}_a$ that coincide on $K_1 \cap K_2$. Suppose that for every ${\mathcal {C}}\in \mathsf {DGCat}_{cont}$ acted on by H, the functor ${\text {Av}}_!^{\psi _2}:{\mathcal {C}}^{K_1,\psi _1} \rightarrow {\mathcal {C}}^{K_2,\psi _2}$ given by restricting to the intersection $K_1 \cap K_2$ and then !-averaging is defined functorially in ${\mathcal {C}}$.

Lemma B.2.3

Suppose ${\mathcal {C}}$ is acted on by H, and equipped with a t-structure compatible with the t-structure. Suppose moreover that the t-structure on ${\mathcal {C}}$ is compactly generated, i.e., ${\mathcal {C}}^{\le 0}$ is compactly generated (in the sense of general category theory).

Then under the above hypotheses, if $K_2/K_1 \cap K_2$ is affine, then ${\text {Av}}_!^{\psi _2}:{\mathcal {C}}^{K_1,\psi _1} \rightarrow {\mathcal {C}}^{K_2,\psi _2}$ has cohomological amplitude ${[-\dim K_2/K_1 \cap K_2,0]}$.

We need the following result, which appeared already as [39] Lemma 4.1.3. We include the proof for the reader’s convenience.

Lemma B.2.4

Let ${\mathcal {C}}\in \mathsf {DGCat}_{cont}$ be equipped with a compactly generated t-structure. Let $F:{\mathcal {D}}_1 \rightarrow {\mathcal {D}}_2 \in \mathsf {DGCat}_{cont}$ be given, and suppose that the categories ${\mathcal {D}}_i$ are equipped with t-structures, and that F is left t-exact. Then:

$$\begin{aligned} {\text {id}}_{{\mathcal {C}}} \otimes F: {\mathcal {C}}\otimes {\mathcal {D}}_1 \rightarrow {\mathcal {C}}\otimes {\mathcal {D}}_2 \end{aligned}$$

is left t-exact.

Proof

Let ${\mathcal {F}}\in {\mathcal {C}}^{\le 0}$ be compact. Let ${{\mathbb {D}}}{\mathcal {F}}:{\mathcal {C}}\rightarrow \mathsf {Vect}$ denote the corresponding continuous functor $\underline{{\text {Hom}}}_{{\mathcal {C}}}({\mathcal {F}},-)$. Note that ${{\mathbb {D}}}{\mathcal {F}}$ is left t-exact because ${\mathcal {F}}\in {\mathcal {C}}^{\le 0}$.

We have induced functors:

$$\begin{aligned} {{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_i}: {\mathcal {C}}\otimes {\mathcal {D}}_i \rightarrow \mathsf {Vect}\otimes {\mathcal {D}}_i = {\mathcal {D}}_i. \end{aligned}$$

The main observation is that ${\mathcal {G}}\in {\mathcal {C}}\otimes {\mathcal {D}}_i$ lies in $({\mathcal {C}}\otimes {\mathcal {D}}_i)^{\ge 0}$ if and only if:

$$\begin{aligned} {{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_i} ({\mathcal {G}}) \in {\mathcal {D}}_i^{\ge 0} \end{aligned}$$

for all ${\mathcal {F}}$ as above. Indeed, for ${\mathcal {H}}\in {\mathcal {D}}_i$, the (possibly non-continuous) composite functor:

$$\begin{aligned} {\mathcal {C}}\otimes {\mathcal {D}}_i \xrightarrow {{{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_i}} {\mathcal {D}}_i \xrightarrow {\underline{{\text {Hom}}}_{{\mathcal {D}}_i}({\mathcal {H}},-)} \mathsf {Vect}\end{aligned}$$

coincides with $\underline{{\text {Hom}}}_{{\mathcal {C}}\otimes {\mathcal {D}}_i}({\mathcal {F}}\boxtimes {\mathcal {H}},-)$, as follows by observing that it is the right adjoint to the functor $k \mapsto {\mathcal {F}}\boxtimes {\mathcal {H}}$. Taking ${\mathcal {H}}\in {\mathcal {D}}_i^{\le 0}$, this immediately implies the observation.

Therefore, we need to show that for ${\mathcal {G}}\in ({\mathcal {C}}\otimes {\mathcal {D}}_1)^{\ge 0}$, we have:

$$\begin{aligned} ({{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_2}) \circ ({\text {id}}_{{\mathcal {C}}} \otimes F) ({\mathcal {G}}) \in {\mathcal {D}}_2^{\ge 0} \end{aligned}$$

for all compact ${\mathcal {F}}\in {\mathcal {C}}^{\le 0}$. By functoriality, we have:

$$\begin{aligned} ({{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_2}) \circ ({\text {id}}_{{\mathcal {C}}} \otimes F) ({\mathcal {G}}) = F \circ ({{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_1})({\mathcal {G}}). \end{aligned}$$

Because $({{\mathbb {D}}}{\mathcal {F}}\otimes {\text {id}}_{{\mathcal {D}}_1})({\mathcal {G}}) \in {\mathcal {D}}_1^{\ge 0}$ by the above, we obtain the claim from left t-exactness of F. $\square$

Proof of Lemma B.2.3

The functor ${\text {Av}}_!^{\psi _2}$ is right t-exact because it is a left adjoint to a t-exact functor. So it remains to show the other bound.

First, suppose that ${\mathcal {C}}= D(H)$. Then $D(H)^{K_1,\psi _1}$ is compactly generated by coherent D-modules. Therefore, for the t-structure on $D(H)^{K_1,\psi _1}$, compact objects are closed under truncations. So it suffices to show that every compact object of $D(H)^{K_1,\psi _1,\ge 0}$ maps to $D(H)^{K_2,\psi _2,\ge -\dim K_2/K_1 \cap K_2}$.

This follows immediately from the fact that !-pushforward is left t-exact on coherent objects, which is Theorem 3.3.1 of [51]. (The cohomological shift by $\dim K_2/K_1 \cap K_2$ arises because !-averaging is !-convolution with a dualizing D-module.)

For general ${\mathcal {C}}$, we use the commutative diagram:

The horizontal arrows are obviously conservative and t-exact up to shift (by assumption on the action on ${\mathcal {C}}$), while the right vertical arrow has the correct amplitude by the above and Lemma B.2.4. This immediately implies the same for the left vertical arrow. $\square$

Remark B.2.5

In the case ${\mathcal {C}}= {{\mathfrak {h}}}\text {--}\mathsf {mod}$, the argument given amounts to using the Beilinson-Bernstein localization functor to pass from the Lie algebra to D-modules.

Remark B.2.6

The above works just as well when H is a group scheme and the $K_i$ are compact open subgroup schemes: indeed, there is a normal compact open subgroup scheme of H contained in the $K_i$, reducing the problem to the finite-dimensional version. But it is not clear how to show the lemma for H being the loop group, since ${\text {coact}}$ is no longer t-exact up to shift (it maps into infinitely connective objects).

Remark B.2.7

The above works just as well in the setting of twisted D-modules.

1.2.3 Geometric proof of Lemma 5.2.1 for n large enough

We will show Lemma 5.2.1 for $n \ge (\check{\rho },\alpha _{max})$ (alias: the Coxeter number of G minus 1).

The right exactness is immediately given by Lemma B.2.2. The issue in applying Lemma B.2.3 is that we need ${\text {Av}}_!$ to be defined and functorial for subgroups of a group scheme, not a group indscheme such as G(K).

But in the given range of n, $\mathring{I}{}_n$ and $\mathring{I}{}_{n+1}$ are both contained in ${\text {Ad}}_{-(n+1)\check{\rho }(t)} G(O)$. Since the existence and functoriality of ${\text {Av}}_!$ is really about convolution identities, this means that for any category strongly acted on by ${\text {Ad}}_{-(n+1)\check{\rho }(t)} G(O)$, we can !-average from $(\mathring{I}{}_n,\psi )$ to $(\mathring{I}{}_{n+1},\psi )$, and this !-averaging coincides with the $*$-averaging up to the shift by $2\Delta$ from Theorem 2.3.1. Now Lemma B.2.3 applies and gives the desired left t-exactness for $m = n+1$, which evidently suffices.

Remark B.2.8

If for ${\mathcal {C}}$ we had D-modules on a reasonable indscheme X acted on by G(K) (or the $\kappa$-twisted version of this notion), then we could apply [51] directly, without needing the general Lemma B.2.3. That is, we would not need any restrictions on n.

1.3 Representation theoretic approach

We now indicate a representation theoretic approach to treat Lemma 5.2.1 for all n.

Proof of Lemma 5.2.1

Step 1.

Note that by the general formalism from “Appendix A”, $\iota _{n,m,*}:\mathsf {Whit}^{\le n}(\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}) \rightarrow \mathsf {Whit}^{\le m}(\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod})$ is filtered for the KK filtration with associated semi-classical functor:

$$\begin{aligned} \mathsf {QCoh}^{ren}(f+{\text {Lie}}\mathring{I}{}_n^{\perp }/\mathring{I}{}_n) \rightarrow \mathsf {QCoh}^{ren}(f+{\text {Lie}}\mathring{I}{}_m^{\perp }/\mathring{I}{}_m) \end{aligned}$$

given by push/pull along the correspondence:

(B.2.1)

up to cohomological shift and a determinant twist. The main observation is that this functor is t-exact. (The “up to cohomological shift” is compatible with the shift by $(m-n)\Delta$ in Lemma 5.2.1.)

Indeed, the pushforward in this correspondence is obviously t-exact because the map is affine. It remains to see that the left leg of the correspondence is flat (and in fact, smooth).

This follows from the explicit description of both sides from the proof of Theorem 3.1.1. Indeed, first say $n>0$ for simplicity. Then both sides are classifying stacks over $f+t^{-n}{\text {Ad}}_{-n\check{\rho }(t)}{{\mathfrak {b}}}^e[[t]]$ by loc. cit. Moreover, the relevant group schemes are congruence subgroups of jets into the group scheme of regular centralizers. We then obtain the claim from the smoothness of that group scheme.

If $n = 0$ and $m>n$, then the relevant map $f+{{\mathfrak {g}}}[[t]] \cap {\text {Lie}}\mathring{I}{}_m^{\perp }/G(O) \cap \mathring{I}{}_m \rightarrow {{\mathfrak {g}}}[[t]]/G(O)$ factors through ${{\mathfrak {g}}}^{reg}(O)/G(O)$, which is the classifying stack over $f+{{\mathfrak {b}}}^e[[t]]$ of jets into regular centralizers. So the same analysis applies.

Step 2. To show $\iota _{n,m,*}[(m-n)\Delta ]$ is t-exact, it suffices to show that it is left t-exact, since Lemma B.2.2 implies the right t-exactness. For this, it suffices to show that it suffices to show that for ${\mathcal {F}}\in \widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}^{\mathring{I}{}_n,\psi ,\heartsuit }$, $\iota _{n,m,*}({\mathcal {F}})[(m-n)\Delta ]$ is also in cohomological degree 0.

For $n>0$, it suffices to take ${\mathcal {F}}$ to be a quotient of ${\text {ind}}_{\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi )$. Indeed, such quotients generate the abelian category under extensions and filtered colimits. Similarly, for $n = 0$, it suffices to take ${\mathcal {F}}$ to be a quotient of a Weyl module (i.e., a quotient of ${{\mathbb {V}}}_{\kappa }^{\lambda } {\text {ind}}_{{{\mathfrak {g}}}[[t]]}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(V^{\lambda })$ for $\lambda \in \Lambda ^+$ a dominant coweight, where $V^{\lambda }$ is the highest weight representation of G, and is acted on by ${{\mathfrak {g}}}[[t]]$ through the quotient ${{\mathfrak {g}}}$).

Here is a wrong conclusion to the argument, which we correct in what follows. The modules ${\text {ind}}_{\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi )$ (resp. ${{\mathbb {V}}}_{\kappa }^{\lambda }$) have KK filtrations, so the quotient ${\mathcal {F}}$ inherits one as well. Therefore, $\iota _{n,m,*}({\mathcal {F}})$ has a canonical filtration. By Step 1, ${\text {gr}}_{\bullet } \iota _{n,m,*}({\mathcal {F}})[(m-n)\Delta ]$ is concentrated in cohomological degree 0.

However, because the KK filtration on ${\text {ind}}_{\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi )$ is not bounded below, it is not clear that the filtration on $\Psi ({\mathcal {F}})$ is bounded below in this case (and probably it is not). That is, the argument from the proof of Theorem 4.2.1 does not adapt well to this setting. So we give a different method below, which essentially uses different bookkeeping to avoid this issue.

Step 3. Of course, it suffices to treat the case where G is not a torus, so we assume this in one follows. We first additionally suppose that $n > 0$.

Let $h \in {{\mathbb {Q}}}^{>1}$ be a rational number (greater than 1) to be specified later. This choice defines a grading on the Kac-Moody algebra with degrees lying in $h{{\mathbb {Z}}}\subseteq {{\mathbb {Q}}}$ as follows. Note that the Kac-Moody algebra has canonical $L_0{:}{=}t\partial _t$ and $\check{\rho }$-gradings. Consider it as equipped with the grading $-h\check{\rho }-(h-1)L_0$ (so e.g., $t^i e_{\alpha }$ has degree $-h(\check{\rho },\alpha )-(h-1)i$).

The subalgebras ${\text {Lie}}\mathring{I}{}_n,{\text {Lie}}\mathring{I}{}_m$ are obviously graded. Moreover, the character $\psi :{\text {Lie}}\mathring{I}{}_n \rightarrow k$ vanishes on homogeneous components apart from degree $-1$, so we can use the KK formalism from “Appendix A”. Note that there is no problem in using fractional indices, though our filtration will be graded similarly. (Clearing denominators, it is the same as renormalizing the PBW filtration to have the same associated graded, but with jumps only at multiples of the denominator of h.) Let us refer to this as the KK’ filtration on $\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}$, etc. Note that if $h = 1$, this is recovering the usual KK filtration.

A straightforward calculation, which is performed in the next step, shows that we can take h so that the induced KK’ filtration on ${\text {ind}}_{{\text {Lie}}\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi )$ to be bounded from below (it is essential that $n>0$ here).

Of course, the same boundedness occurs for the induced KK’ filtration on ${\mathcal {F}}$, any quotient of our induced module.

It is straightforward to see that the induced KK’ filtration on:

$$\begin{aligned} C^{\bullet }({\text {Lie}}\mathring{I}{}_m,\mathring{I}{}_n \cap \mathring{I}{}_m, {\mathcal {F}}\otimes -\psi _{\mathring{I}{}_m}) \end{aligned}$$

is then bounded from below as well. First, observe that (for any $h>1$) there is a compact open subalgebra of ${\text {Lie}}\mathring{I}{}_n$ on which the $-h\check{\rho }-(h-1)L_0$-degrees are negative. It follows that the degrees on $\Lambda ^i \big ({\text {Lie}}\mathring{I}{}_n\big )^{\vee }$ are bounded from below independently of i, since a compact open subalgebra has finite codimension. This shows that the induced filtration on:

$$\begin{aligned} C^{\bullet }(\mathring{I}{}_n,{\mathcal {F}}\otimes -\psi ) \end{aligned}$$

is bounded from below, or similarly for $\mathring{I}{}_n\cap \mathring{I}{}_m$. The Harish-Chandra cohomology appearing above differs from the latter group cohomology by tensoring with the exterior algebra of ${\text {Ad}}_{-m\check{\rho }(t){{\mathfrak {n}}}[[t]]}/{\text {Ad}}_{-n\check{\rho }(t)}{{\mathfrak {n}}}[[t]]$, so the result follows.

This Chevalley complex computes:

$$\begin{aligned} \underline{{\text {Hom}}}_{\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}^{\mathring{I}{}_m}}( {\text {ind}}_{{\text {Lie}}\mathring{I}{}_m}^{\widehat{{{\mathfrak {g}}}}_{\kappa }} \psi , \iota _{n,m,*}({\mathcal {F}})) ) \end{aligned}$$

by definition of $\iota _{n,m,*}$ as $*$-averaging. To compute the associated graded, one takes ${\text {gr}}_{\bullet }^{KK'}({\mathcal {F}}) \in \mathsf {QCoh}(f+{\text {Lie}}\mathring{I}{}_n^{\perp }/\mathring{I}{}_n)^{\heartsuit }$, applies pull-push along the correspondence (B.2.1), applies the cohomological shift by $(m-n)\Delta$ and the determinant twist, and then applies global sections on the stack $f+{\text {Lie}}\mathring{I}{}_m^{\perp }/\mathring{I}{}_m$.

The upshot is that the resulting object of $\mathsf {Vect}$ is in cohomological degrees $\ge (m-n)\Delta$ by the exactness of our pull-push operation and because of the cohomological shift. This means the same is true for the Chevalley complex above. Because ${\text {ind}}_{{\text {Lie}}\mathring{I}{}_m}^{\widehat{{{\mathfrak {g}}}}_{\kappa }} \psi$ generates $\mathsf {Whit}^{\le m}(\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod})^{\le 0}$ under colimits, we finally obtain that $\iota _{n,m,*}({\mathcal {F}})$ is in cohomological degrees $\ge (m-n)\Delta$, hence is in degree $(m-n)\Delta$, as was desired.

Step 4. It remains to define h and check the desired boundedness. For this, let $\alpha _{max}$ denote the highest root, and take:

$$\begin{aligned} h {:}{=}\frac{n(\check{\rho },\alpha _{max})}{1+(n-1)(\check{\rho },\alpha _{max})}. \end{aligned}$$

(E.g., for $n = 1$, h is one less than the Coxeter number of G.)

We want to see that KK’ filtration on ${\text {ind}}_{{\text {Lie}}\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi )$ is bounded below: in fact, we will see $F_{-1}^{KK'} {\text {ind}}_{{\text {Lie}}\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi ) = 0$. It suffices to show that the non-zero graded degrees on:

$$\begin{aligned} {\text {gr}}_{\bullet }^{KK'} {\text {ind}}_{{\text {Lie}}\mathring{I}{}_n}^{\widehat{{{\mathfrak {g}}}}_{\kappa }}(\psi ) = {\text {Sym}}^{\bullet }({{\mathfrak {g}}}((t))/{\text {Lie}}\mathring{I}{}_n) \end{aligned}$$

are $\ge 0$. Note that in the notation from the proof of Lemma 5.3.2, this associated graded is an algebra generated by elements $\frac{e_{\alpha }}{t^r}$ ($r\ge n(\check{\rho },\alpha )+1$) and $\frac{f_{\beta }}{t^r}$ ($r\ge -n(\check{\rho },\beta )-n+1$), which have gradings:

$$\begin{aligned} \begin{aligned} -h(\check{\rho },\alpha )+(h-1)r+1 \\ h(\check{\rho },\beta )+(h-1)r+1. \end{aligned} \end{aligned}$$

We need to show that these numbers are each $\ge 0$ for $\alpha$ (resp. $\beta$) a positive root (resp. or zero) and r in the appropriate range.

Regarding the “$\alpha$ inequality,” note that:

$$\begin{aligned} h \ge \frac{n(\check{\rho },\alpha )}{1+(n-1)(\check{\rho },\alpha )}. \end{aligned}$$

(B.2.2)

Then the bound on r means the KK’ degree of $\frac{e_{\alpha }}{t^r}$ is:

$$\begin{aligned}&-h(\check{\rho },\alpha )+(h-1)r + 1 \ge -h(\check{\rho },\alpha ) + (h-1)(n(\check{\rho },\alpha )+1) + 1 \\&\quad = h\big (1+(n-1)(\check{\rho },\alpha )\big )-n(\check{\rho },\alpha ) \end{aligned}$$

which is non-negative by (B.2.2).

For the second inequality, first note that:¹⁰⁴

$$\begin{aligned} (n-1) h = \frac{n(n-1)}{\frac{1}{(\check{\rho },\alpha _{max})}+n-1} < \frac{n(n-1)}{n-1} = n. \end{aligned}$$

(B.2.3)

Then the bound on r gives the degree of $\frac{f_{\beta }}{t^r}$ as:

$$\begin{aligned}&h(\check{\rho },\beta ) + (h-1) r + 1 \ge h(\check{\rho },\beta ) + (h-1) \big (-n(\check{\rho },\beta )-n+1\big ) +1 \\&\quad = (\check{\rho },\beta )\big (n-(n-1)h\big ) \end{aligned}$$

which is non-negative by (B.2.3) (recall our normalization that $\beta$ is 0 or a positive root).

Step 5. Finally, we treat the case $n = 0$. Here are three arguments.

Observe that (e.g. by Theorems 5.1.1 and 5.3.4), it suffices to show that $\Psi :\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}^{G(O)} \rightarrow \mathsf {Vect}$ is t-exact.

First, this result can be found in the literature: at non-critical level, this is [26] Proposition 2 plus the Sugawara construction, and at critical level this is [25] Theorem 3.2.

Second, one can organize the above differently: [26] uses Arakawa exactness in an essential way, and our generalization Corollary 7.2.4 of it, which removes the use of the extended affine Kac-Moody algebra, allows one to use the Frenkel–Gaitsgory method directly.

Finally, note that any object of $\widehat{{{\mathfrak {g}}}}_{\kappa }\text {--}\mathsf {mod}^{G(O)}$ has a $\check{\rho }$-grading, and morphisms preserve these gradings. Therefore, ${\mathcal {F}}$ (${:}{=}$ a quotient of ${{\mathbb {V}}}_{\kappa }^{\lambda }$) has canonical $\check{\rho }$-gradings, and also inherits PBW and KK filtrations from ${{\mathbb {V}}}_{\kappa }^{\lambda }$. These satisfy the usual compatibility in the KK formalism. Therefore, we can apply the method from Theorem 4.2.1 to obtain the desired result.

$\square$