# Higher localization and higher branching laws

Wen-Wei Li

###### Abstract

For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over
the complex numbers, the localization functor takes $\mathfrak{g}$-modules to
$D_{X}$-modules. We extend this construction to an equivariant and derived
setting using the formalism of h-complexes due to Beilinson–Ginzburg, and show
that the localizations of Harish-Chandra $(\mathfrak{g},K)$-modules onto
$X=H\backslash G$ have regular holonomic cohomologies when $H,K\subset G$ are
both spherical reductive subgroups. The relative Lie algebra homologies and
$\operatorname{Ext}$-branching spaces for $(\mathfrak{g},K)$-modules are
interpreted geometrically in terms of equivariant derived localizations. As
direct consequences, we show that they are finite-dimensional under the same
assumptions, and relate Euler–Poincaré characteristics to local index theorem;
this recovers parts of the recent results of M. Kitagawa. Examples and
discussions on the relation to Schwartz homologies are also included.

###### Contents

  1. 1 Introduction
    1. 1.1 Backgrounds
    2. 1.2 Higher localization
    3. 1.3 Higher branching
    4. 1.4 Twist by characters
    5. 1.5 On the analytic picture
    6. 1.6 Structure of this article
    7. 1.7 Conventions
  2. 2 Homological algebra over Harish-Chandra pairs
    1. 2.1 Review of dg-algebras and dg-modules
    2. 2.2 Harish-Chandra dg-algebras
    3. 2.3 Derived categories and functors
    4. 2.4 h-construction
    5. 2.5 Adjoint functors of oblivion
    6. 2.6 Invariants and co-invariants
  3. 3 $(\mathfrak{g},K)$-modules
    1. 3.1 Basic definitions
    2. 3.2 Standard resolutions
    3. 3.3 Bernstein–Lunts equivalence
    4. 3.4 A lemma on restriction
  4. 4 $D$-modules
    1. 4.1 Basic definitions
    2. 4.2 Inverse images
    3. 4.3 Comparison with the equivariant derived category
  5. 5 Localization and higher regularity
    1. 5.1 Derived localization functor
    2. 5.2 Localization and co-invariants
    3. 5.3 Statement of higher regularity
    4. 5.4 Criterion of regularity
    5. 5.5 End of the proof
  6. 6 Application to $\operatorname{Ext}$-branching
    1. 6.1 General setting of branching laws
    2. 6.2 Branching and localization
    3. 6.3 Some consequences of regularity
    4. 6.4 The monodromic setting
  7. 7 Comparison with the analytic picture
    1. 7.1 Schwartz homologies
    2. 7.2 Comparison map
    3. 7.3 Examples of comparison isomorphisms

##  1 Introduction

###  1.1 Backgrounds

We work over the complex numbers. Let $G$ be a connected reductive group with
Lie algebra $\mathfrak{g}$. The main theme of this work begins with the
functor

$\mathscr{D}_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot):\mathfrak{g}\text{-}\mathsf{Mod}\to\mathscr{D}_{X}\text{-}\mathsf{Mod}$

where $X$ is a smooth variety with right $G$-action (we say that $X$ is a
$G$-variety), with sheaf of algebraic differential operators
$\mathscr{D}_{X}$, and $\mathfrak{g}\text{-}\mathsf{Mod}$ (resp.
$\mathscr{D}_{X}\text{-}\mathsf{Mod}$) is the abelian category of
$\mathfrak{g}$-modules (resp. left $\mathscr{D}_{X}$-modules that are
$\mathscr{O}_{X}$-quasi-coherent). The tensor product is taken using the
homomorphism $j:U(\mathfrak{g})\to D_{X}:=\Gamma(X,\mathscr{D}_{X})$ induced
by $G$-action.

This functor is called _localization_. Its significance can be partly
explained by the fact that when $X$ is a homogeneous $G$-space, the fiber of
$\mathscr{D}_{X}\underset{U(\mathfrak{g})}{\otimes}V$ at $x\in X$ is
isomorphic to $V/\mathfrak{h}V$ where $H:=\mathrm{Stab}_{G}(x)$; therefore,
localization organizes _various_ spaces of co-invariants into a geometric
object over $X$.

The celebrated Beilinson–Bernstein localization is the special case when $X$
is the flag variety of $G$, but allowing some twists on $\mathscr{D}_{X}$. It
has tremendous consequences in the study of $\mathfrak{g}\text{-}\mathsf{Mod}$
and related structures. In this work we try to use localization in another
way. Namely, we start with a homogeneous $G$-space $X$ and study aspects of
representation theory of $G$ or its real form that are “relative” to $X$, by
means of the localization functors.

Localization has been utilized in a similar flavor in the “group case”
$G=H\times H$ and $X=H$ by D. Ben-Zvi and I. Ganev [6], who applied it to
study the asymptotics of matrix coefficients of admissible representations.
They also related this construction to Beilinson–Bernstein localization by
specialization at infinity. We remark that V. Ginzburg’s work [13] played an
important role there; specifically, it ensures that the localized
$D_{H}$-modules are regular holonomic and puts some control on its
characteristic variety — in fact, their irreducible constituents are character
$D_{H}$-modules.

A similar result of regularity is contained in [25] for localizations of
Harish-Chandra $(\mathfrak{g},K)$-modules to $X:=H\backslash G$, where
$H,K\subset G$ are reductive spherical subgroups. By a _Harish-Chandra
$(\mathfrak{g},K)$-module_, we mean a $(\mathfrak{g},K)$-module that is
finitely generated over $\mathfrak{g}$ and locally
$\mathcal{Z}(\mathfrak{g})$-finite, where $\mathcal{Z}(\mathfrak{g})$ is the
center of $U(\mathfrak{g})$. By sphericity of $H$ (ditto for $K$), we mean
that the homogeneous $G$-space $H\backslash G$ has an open Borel orbit. The
proof is based on a variant of Ginzburg’s criterion of regularity in [13].
Note that reductivity implies $X$ is affine, so we may work with
$D_{X}\text{-}\mathsf{Mod}$ instead of $\mathscr{D}_{X}\text{-}\mathsf{Mod}$.

Nonetheless, the usage of localization is made complicated by the following
two facts, at least.

  * •

Given an equivariant morphism $f:Y_{1}\to Y_{2}$ between smooth $G$-varieties,
there is no obvious way to relate the corresponding localizations via inverse
or direct images;

  * •

Localization is a right exact functor, but it is non-exact in many cases of
interest. This will be shown in Remark 5.5.9.

###  1.2 Higher localization

In the first part of this work, we generalize these ideas to the equivariant
and derived setting, i.e. to higher localizations.

First off, take a reductive subgroup $K\subset G$ and let
$(\mathfrak{g},K)\text{-}\mathsf{Mod}$ be the abelian category of
$(\mathfrak{g},K)$-modules. In representation theory, one is especially
interested in its subcategory of Harish-Chandra $(\mathfrak{g},K)$-modules.

Take an affine homogeneous space $X=H\backslash G$ and let
$(D_{X},K)\text{-}\mathsf{Mod}$ be the abelian category of $K$-equivariant
$D_{X}$-modules. Since $j:U(\mathfrak{g})\to D_{X}$ is equivariant, it is
easily seen that the localization lifts to the equivariant level:

$D_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot):(\mathfrak{g},K)\text{-}\mathsf{Mod}\to(D_{X},K)\text{-}\mathsf{Mod}.$

We wish to left-derive this right exact functor in order to obtain “higher
localizations”, and apply them to Harish-Chandra modules. Two issues arise
immediately.

  1. 1.

Homological algebra for $(\mathfrak{g},K)$-modules is done in the classical
way in [20]. On the other hand, the correct $K$-equivariant derived category
of $D_{X}$-modules requires less naive techniques when $K\neq\\{1\\}$, such as
that of Bernstein–Lunts [8] which replaces $X$ by various resolutions $P\to
X$, on which $K$ acts freely (see also [1]).

This kind of “type mismatch” makes it non-trivial to derive equivariant
localization. Nor is there any obvious way to define it as a compatible family
on resolutions.

  2. 2.

Another option is to consider $(\mathfrak{g},K)\text{-}\mathsf{Mod}\to
D_{X}\text{-}\mathsf{Mod}$ instead, in order to avoid equivariant derived
categories. This discards too many structures, thus impedes applications to
higher branching laws. Moreover, since nonzero projectives in
$(\mathfrak{g},K)\text{-}\mathsf{Mod}$ are never Harish-Chandra, it seems
difficult to prove regularity or holonomicity in cohomologies, unless one
addresses the first issue simultaneously.

For similar reasons, it seems unfeasible to define the derived localization
through realization functor [1, Theorem A.7.16].

Our approach to these problems is based on the formalism of _h-complexes_ ,
reviewed in §2. This theory is presented in [9], attributed to
Beilinson–Ginzburg and the earlier work of Duflo–Vergne [12]; for subsequent
developments, see [29, 27, 28, 19] and [5, Chapter 7]. Let
$A\in\\{U(\mathfrak{g}),D_{X}\\}$, so there is a natural homomorphism
$j:\mathfrak{k}\to A$ of Lie algebras (the Lie brackets in $A$ are
commutators). A weak $(A,K)$-module is a vector space $M$ with

  * •

a left $A$-module structure, say through
$\alpha:A\to\operatorname{End}_{\mathbb{C}}(M)$

  * •

an algebraic $K$-action, say through
$\rho:K\to\operatorname{Aut}_{\mathbb{C}}(M)$, compatibly with $A$,

but the $\mathfrak{k}$-actions $\mathop{}\\!\mathrm{d}\rho$ and $\alpha j$ on
$M$ are not required to agree; if they agree, we obtain an $(A,K)$-module. An
h-complex over $(A,K)$ is a complex $(C,d)$ of weak $(A,K)$-modules together
with a family of maps
$i_{\xi}\in\operatorname{End}^{-1}(C):=\operatorname{End}^{-1}_{\mathbb{C}}(C)$,
linear in $\xi\in\mathfrak{k}$, such that

  1. (i)

$ki_{\xi}k^{-1}=i_{\operatorname{Ad}(k)\xi}$ for all $k\in K$;

  2. (ii)

$i_{\xi}$ is $A$-linear for all $\xi\in\mathfrak{k}$;

  3. (iii)

$i_{\xi}i_{\eta}+i_{\eta}i_{\xi}=0$ for all $\xi,\eta\in\mathfrak{k}$;

  4. (iv)

$di_{\xi}+i_{\xi}d=(\mathop{}\\!\mathrm{d}\rho-\alpha j)(\xi)$.

The last homotopy condition implies that $\operatorname{H}^{n}(C)$ is an
$(A,K)$-module for all $n$. There are also translation functors $C\mapsto
C[1]$, mapping cones and $\operatorname{Hom}$-complexes for h-complexes. This
leads to the h-derived category ${}^{\mathrm{h}}\mathsf{D}(A,K)$: it is
triangulated and endowed with a $t$-structure whose heart is
$(A,K)\text{-}\mathsf{Mod}$. For every subgroup $T\subset K$ there are
functors of oblivion fitting into commutative diagrams up to isomorphism:

${{}^{\mathrm{h}}\mathsf{D}(A,K)}$${(A,K)\text{-}\mathsf{Mod}}$${{}^{\mathrm{h}}\mathsf{D}(A,T)}$${(A,T)\text{-}\mathsf{Mod}.}$$\scriptstyle{\operatorname{H}^{n}}$$\scriptstyle{\operatorname{H}^{n}}$

Therefore we obtain the bounded h-derived categories

${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K),\quad{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K).$

In [9, 27], these are shown to be equivalent to the usual bounded versions

$\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K),\quad\mathsf{D}^{\mathrm{b}}_{K}(X)$

of the derived category of $(\mathfrak{g},K)$-modules and the $K$-equivariant
derived category of $D_{X}$-modules, respectively; in particular, the
$\operatorname{Ext}$’s are the same. They proved the second case for non-
affine $X$ and non-reductive $K$ as well.

The best way to understand the h-construction is to do homological algebra
over _Harish-Chandra dg-algebras_ ; this is just an equivariant version of the
theory of dg-modules over dg-algebras (“dg” = differential graded), which is
nowadays standard; see [8] or [34]. Consequently, left and right h-derived
functors are defined, upon replacing projective (resp. injective) resolutions
by K-projective (resp. K-injective) resolutions. As we assumed $X$ is affine
and $K$ is reductive, there are enough K-projective bounded-above h-complexes
over $(D_{X},K)$.

We adopt this toolbox to deduce the following main theorem. Let

$\mathbf{Loc}_{X}=\mathbf{Loc}_{X,K}:{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)$

be the equivariant derived localization functor so obtained; see §5.1. An
advantage of the h-construction is that for all reductive subgroups $T$ of
$K$, one easily obtains commutative diagrams up to isomorphisms (Proposition
5.1.5):

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},T)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},T).}$$\scriptstyle{\mathbf{Loc}_{X,K}}$$\scriptstyle{\mathbf{Loc}_{X,T}}$

A closely related approach to $\mathbf{Loc}_{X,K}$ can be found in [5, §7.8],
as a part of the “Hecke patterns”.

######  Theorem 1.

Let $H$ and $K$ be spherical reductive subgroups of $G$. Let $V$ be a Harish-
Chandra $(\mathfrak{g},K)$-module, then the cohomologies of
$\mathbf{Loc}_{X}(V)$ are all regular holonomic as $D_{X}$-modules, and their
characteristic varieties are included in the nilpotent locus.

We refer to Theorem 5.3.4 and the results in that section for complete
statements. The proof is based on a criterion (Theorem 5.4.1) from [25]. One
must show that the cohomologies of $\mathbf{Loc}_{X}(V)$ are (R1) finitely
generated over $D_{X}$, (R2) carry $K$-equivariant structures, and (R3)
locally $\mathcal{Z}(\mathfrak{g})$-finite through

$\mathcal{Z}(\mathfrak{g})\subset U(\mathfrak{g})\xrightarrow{j}D_{X}.$

Note that (R2) is immediate in our setting. For the remaining conditions, one
passes to the case $K=\\{1\\}$ by commuting $\mathbf{Loc}_{X}$ and
$\operatorname{H}^{n}$ with oblivion. The hardest part is (R3): to prove it,
we need deep results of F. Knop [21] on the structure of the algebra
$D_{X}^{G}$ of invariant differential operators in the spherical case. This is
the content of Proposition 5.5.6.

We remark that $D_{X}^{G}$ also acts on the right of the functor
$D_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot)$, by letting $z\in D_{X}^{G}$
send $D\otimes w$ to $Dz\otimes w$. This induces a right $D_{X}^{G}$-action on
$\mathbf{Loc}_{X}$. As a by-product, Proposition 5.5.10 shows that $D_{X}^{G}$
acts locally finitely on the cohomologies of $\mathbf{Loc}_{X}(V)$, for all
Harish-Chandra $(\mathfrak{g},K)$-modules $V$.

###  1.3 Higher branching

We hope that the results on $\mathbf{Loc}_{X}$ will have some use in geometric
representation theory. However, this work is originally motivated by
_branching laws_ in harmonic analysis.

In the setting of $p$-adic groups $H\subset I$, the branching law studies the
spaces $\operatorname{Hom}_{H}(V|_{H},W)$ where $V$ (resp. $W$) is an
admissible representation of $I$ (resp. $H$). In particular, one is interested
in the dimension of the $\operatorname{Hom}$-space111We do not consider the
case of $\operatorname{Hom}_{H}(V,W|_{H})$ in this work.. The
$\operatorname{Ext}$-analogue or _higher branching law_ , proposed by D.
Prasad [30] for $p$-adic groups, considers the corresponding problem for
$\operatorname{Ext}^{n}_{H}(V|_{H},W)$ for general $n\geq 0$. One is also
interested in the Euler–Poincaré characteristic
$\sum_{n}(-1)^{n}\dim\operatorname{Ext}^{n}_{H}(V|_{H},W)$, well-defined as
long as $\dim\operatorname{Ext}^{n}_{H}(V|_{H},W)$ is finite and vanishes for
$n\gg 0$. By replacing $I$ by $H\times I$ in which $H$ embeds diagonally, the
problem can be reduced to the case $W=\mathbb{C}$, the trivial representation.

This work is motivated by higher branching laws in the Archimedean case. We
formulate it in the algebraic framework, namely by considering subgroups

${H}$${G}$${K^{H}}$${K.}$${\subset}$${\subset}$${\subset}$${\subset}$

where all groups are assumed to be complex reductive. We are led to study

$\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(V|_{H},\mathbb{C}),\quad
n\in\mathbb{Z}_{\geq 0}$

where $V$ is a $(\mathfrak{g},K)$-module and $V|_{H}$ denotes its restriction
to $(\mathfrak{h},K^{H})$. The $\operatorname{Ext}$ can be computed either in
the h-derived category or in the classical one.

Let $X=H\backslash G$ and let $x$ be the point $H\cdot 1$. The inclusion map
$i_{x}:\mathrm{pt}\to X$ is a morphism between $K^{H}$-varieties. By using the
relation between localization and co-invariants, and working systematically in
the h-derived categories, one obtains the following canonical isomorphisms.
For all $(\mathfrak{g},K)$-modules $V$ and all $n\in\mathbb{Z}$, we have

$\displaystyle\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(V|_{H},\mathbb{C})$
$\displaystyle\simeq\operatorname{Ext}^{n}_{D_{\mathrm{pt}},K^{H}}\left(i_{x}^{\bullet}(\mathbf{Loc}_{X}(V)),\mathbb{C}\right),$
(1.3.1) $\displaystyle\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H})$
$\displaystyle\simeq\operatorname{H}^{-n}\operatorname{L}\\!\left(\mathrm{coInv}^{\mathbb{C},K^{H}}_{\mathbb{C},\\{1\\}}\right)\left(i_{x}^{\bullet}\mathbf{Loc}_{X}(V)\right).$
(1.3.2)

For the complete statements, see Propositions 6.2.2 and 6.2.4. Here:

  * •

$\operatorname{H}_{n}(\mathfrak{h},K^{H};\cdot)$ are the relative Lie algebra
homologies of $(\mathfrak{h},K^{H})$-modules (see [20, p.157]), and note that

$\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(V|_{H},\mathbb{C})\simeq\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H})^{*};$

  * •

$\mathbf{Loc}_{X}(V):=\mathbf{Loc}_{X,K^{H}}(V)$, but it is also isomorphic to
the oblivion of $\mathbf{Loc}_{X,K}(V)$ via $K^{H}\subset K$;

  * •

$i_{x}^{\bullet}:{}^{\mathrm{h}}\mathsf{D}^{-}(D_{X},K^{H})\to{}^{\mathrm{h}}\mathsf{D}^{-}(D_{\mathrm{pt}},K^{H})$
is the h-version of the inverse image functor for $D$-modules;

  * •

$\operatorname{L}\\!\left(\mathrm{coInv}^{\mathbb{C},K^{H}}_{\mathbb{C},\\{1\\}}\right):{}^{\mathrm{h}}\mathsf{D}^{-}(\mathbb{C},K^{H})\to\mathsf{D}^{-}(\mathbb{C})$
is the left h-derived functor of taking co-invariants of h-complexes over
$(\mathbb{C},K^{H})$, see §2.6, and $\mathsf{D}(\mathbb{C})$ is the the
derived category of $\mathbb{C}=D_{\mathrm{pt}}$.

Note that $i_{x}^{\bullet}$ coincides with the inverse image functor under the
equivalence
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K^{H})\simeq\mathsf{D}^{\mathrm{b}}_{K^{H}}(X)$;
see Proposition 4.3.6. The same can also be said for
$\operatorname{L}\\!\left(\mathrm{coInv}^{\mathbb{C},K^{H}}_{\mathbb{C},\\{1\\}}\right)$,
but truncation is needed since this functor does not land in
$\mathsf{D}^{\mathrm{b}}(\mathbb{C})$ in general.

Actually, the deductions of (1.3.1) and (1.3.2) are routine once the basic
formalism of §§2.1—5.2 is set up. The h-formalism is needed only when
$K^{H}\neq\\{1\\}$.

The result below is a direct consequence of the earlier theorem on regularity.

######  Theorem 2.

Assume that $H,K\subset G$ are both spherical reductive subgroups. For all
Harish-Chandra $(\mathfrak{g},K)$-modules $V$ and all $n\in\mathbb{Z}$, there
are canonical isomorphisms

$\displaystyle\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(V|_{H},\mathbb{C})$
$\displaystyle\simeq\operatorname{Ext}^{n}_{D_{\mathrm{pt}},K^{H}}\left(i_{x}^{!}\mathbf{Loc}_{X}(V)[\dim
X],\mathbb{C}\right),$
$\displaystyle\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H})$
$\displaystyle\simeq\operatorname{H}^{-n+\dim
X}\operatorname{L}\\!\left(\mathrm{coInv}^{\mathbb{C},K^{H}}_{\mathbb{C},\\{1\\}}\right)\left(i_{x}^{!}\mathbf{Loc}_{X}(V)\right).$

When $K^{H}=\\{1\\}$ we also have

$\displaystyle\operatorname{Ext}^{n}_{\mathfrak{h}}(V|_{H},\mathbb{C})$
$\displaystyle\simeq\operatorname{Ext}^{n}_{D_{X}}\left(\mathbf{Loc}_{X}(V),i_{x,*}(\mathbb{C})[\dim
X]\right),$ $\displaystyle\operatorname{H}_{n}(\mathfrak{h};V|_{H})$
$\displaystyle\simeq\operatorname{H}^{-n-\dim
X}\left(i_{x}^{*}\mathbf{Loc}_{X}(V)\right).$

Here $i_{x}^{!}$, $i_{x}^{*}$ and $i_{x,*}$ are defined to match the
synonymous functors between the constructible equivariant derived categories
via Riemann–Hilbert correspondence.

All the complexes of $D$-modules above are bounded with regular holonomic
cohomologies. Consequently, all these vector spaces are finite-dimensional.

We refer to Theorem 6.3.2 and Proposition 6.3.6 for the complete statements.

The finiteness of $\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H})$ is covered
by a recent work of M. Kitagawa [18], who considers a broader setting of
branching laws and obtains uniform bounds. His approach is to interpret
relative Lie algebra cohomologies in terms of Zuckerman functors, and then
pass to homologies by Poincaré duality. The approach in this work relies on
standard results about constructible or regular holonomic equivariant derived
categories; although this is conceptually straightforward, the resulting bound
is less effective and involves more machinery.

In the special case $K^{H}=\\{1\\}$, we can connect the Euler–Poincaré
characteristic

$\mathrm{EP}_{\mathfrak{h}}(V|_{H},\mathbb{C}):=\sum_{n}(-1)^{n}\dim\operatorname{Ext}^{n}_{\mathfrak{h}}(V|_{H},\mathbb{C})$

to a celebrated topological counterpart, namely the local Euler–Poincaré
characteristic at $x$ of the _solution complex_ of
$\mathbf{Loc}_{X,\\{1\\}}(V)$. See Theorem 6.3.4 for the complete statement.

######  Theorem 3.

Retain the previous assumptions on $H,K\subset G$ and assume $K^{H}=\\{1\\}$.
Let $V$ be a Harish-Chandra $(\mathfrak{g},K)$-module, and set
$\mathcal{L}:=\mathbf{Loc}_{X,\\{1\\}}(V)$. Then
$\mathrm{EP}_{\mathfrak{h}}(V|_{H},\mathbb{C})$ equals the local
Euler–Poincaré characteristic

$\chi_{x}\left(\mathrm{Sol}_{X}(\mathcal{L})\right)$

of the solution complex of $\mathcal{L}$ at $x$, which is expressible in terms
of characteristic cycles of $\mathcal{L}$ and Euler obstructions by
Kashiwara’s local index theorem [15, Theorem 4.6.7].

We remark that the reductivity of $H$ and $K$ makes homological algebra over
$(D_{X},K)$ and $(\mathfrak{g},K)$ easier; it is also involved with the
criterion of regularity and the results of Knop in §§5.4—5.5. It is still
unclear whether our approach can extend to non-reductive subgroups.

It should be emphasized that the finiteness of $\operatorname{Ext}^{n}$ in the
spherical case is just a first example of the usage of higher localization,
and the computation of $\chi_{x}(\mathrm{Sol}_{X}(\mathcal{L}))$ requires
knowledge about characteristic cycles. To gain more applications in higher
branching laws, one needs a deeper understanding of $\mathbf{Loc}_{X}(V)$.
Regularity is merely the first step.

###  1.4 Twist by characters

In arithmetic applications, one often has to study branching laws in which the
trivial module $\mathbb{C}$ of $H$ is replaced by some character $\chi$, say
for $p$-adic groups. In the setting of $(\mathfrak{g},K)$-modules, one should
consider a $1$-dimensional $(\mathfrak{h},K^{H})$-module; by abusing notation,
we identify it with the underlying character $\chi:\mathfrak{h}\to\mathbb{C}$.
Denote by $\chi^{\vee}$ its contragredient $(\mathfrak{h},K^{H})$-module222In
the study of branching laws of real groups, many authors allow $\chi$ to be a
finite-dimensional module. It seems hard to incorporate this case into our
formalism..

Since $H$ is assumed to be reductive, $\chi$ factors through the Lie algebra
of some torus $S=H/\underline{H}$ and can be identified with an element of
$\mathfrak{s}^{*}$. The spaces

$\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(V|_{H},\chi),\quad\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H}\otimes\chi^{\vee})$

can still be interpreted via localization to $X=H\backslash G$, provided that
we work with twisted differential operators (TDO’s) on $X$.

Specifically, we have an $S$-torsor (on the left)

$\pi:\tilde{X}:=\underline{H}\backslash G\to H\backslash G=X$

that is also $G$-equivariant (on the right). This situation is already
considered by [9], where it is called an $S$-monodromic $G$-variety.

Let $\mathfrak{m}_{\chi}\subset\operatorname{Sym}(\mathfrak{s})$ be the
maximal ideal corresponding to $\chi$. The sheaf of $\chi$-twisted
differential operators on $X$ is

$\mathscr{D}_{X,\chi}:=(\pi_{*}\mathscr{D}_{\tilde{X}})^{S}\big{/}\mathfrak{m}_{\chi}(\pi_{*}\mathscr{D}_{\tilde{X}})^{S}.$

In the affine setting, $D_{X,\chi}=\Gamma(X,\mathscr{D}_{X,\chi})$ equals
$D_{\tilde{X}}^{S}/\mathfrak{m}_{\chi}D_{\tilde{X}}^{S}$. There is again a
$G$-equivariant homomorphism

$j:U(\mathfrak{g})\to D_{X,\chi},$

by which we define the $\chi$-twisted localization

$\mathbf{Loc}_{X,\chi}=\mathbf{Loc}_{X,K,\chi}:{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X,\chi},K).$

All the earlier results have monodromic counterparts in this context recorded
in §6.4. For example, Proposition 6.4.7 relates the fibers of
$D_{X,\chi}\underset{U(\mathfrak{g})}{\otimes}V$ to $(\mathfrak{h},\chi)$-co-
invariants of $V$, i.e. the quotient of $V$ by the span of all $\eta
v-\chi(\eta)v$ where $\eta\in\mathfrak{h}$ and $v\in V$. On the other hand,
the $K^{H}$-equivariant sheaf $\mathbb{C}$ on $\mathrm{pt}$ must be replaced
by the monodromic variant $\mathbb{C}_{\chi}$, and this gives rise to
monodromic versions of (1.3.1) and (1.3.2).

The key ingredient here is a monodromic version of the regularity Theorem
6.4.5 in the spherical case. One has to extend Knop’s results [21] to the
twisted setting of $D_{X,\chi}$. This is done in Lemma 6.4.4, by reducing to
the spherical homogeneous $S\times G$-space $\tilde{X}$.

We remark that the Riemann–Hilbert correspondence also holds in the twisted
(i.e. monodromic) and equivariant setting; one has to consider twisted
constructible sheaves, though. We refer to [17].

###  1.5 On the analytic picture

The discussions thus far focus on the algebraic setting. In the representation
theory of real groups, one often takes $G$ and $H$ to be real, and take $K$
(resp. $K^{H}$) corresponding to a maximal compact subgroup of $G$ (resp.
$H$). In studying
$\operatorname{Hom}_{\mathfrak{h},K^{H}}(V|_{H},\mathbb{C})$, one is more
interested in those element which extend continuously to the Casselman–Wallach
globalization $E$ of $V$. This is the same as the continuous linear
functionals on the space of co-invariants $E_{H}:=E\big{/}\sum_{h}(h-1)E$
endowed with the quotient topology.

For higher branching laws in the analytic setting, it is thus natural to
replace relative Lie algebra homologies by the _Schwartz homologies_

$\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H}),\quad n\in\mathbb{Z}$

defined by Y. Chen and B. Sun [3]; they are locally convex topological spaces,
not necessarily Hausdorff, and equal to $E_{H}$ when $n=0$. In fact, they
agree with the _smooth homologies_ constructed earlier by P. Blanc and D.
Wigner [10].

In §7.2 we will define the comparison maps

$c_{n}(E):\operatorname{H}_{n}\left(\mathfrak{h},K^{H};E^{K\text{-fini}}\right)\to\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H}),\quad
n\in\mathbb{Z}_{\geq 0}$

from the algebraic to the analytic picture, for every Casselman–Wallach
representation $E$ of $G$, where $E^{K\text{-fini}}$ is the
$(\mathfrak{g},K)$-module of $K$-finite vectors in $E$.

It is natural to ask if $c_{n}(E)$ is an isomorphism, at least when $H$ is a
reductive spherical subgroup of $G$ after extension of scalars to
$\mathbb{C}$. If it is indeed the case,
$\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H})$ will be finite-dimensional
since $\operatorname{H}_{n}(\mathfrak{h},K^{H};E^{K\text{-fini}})$ is; in
turn, this will imply $\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H})$ is
Hausdorff.

It seems too early to pass verdict on the question of comparison. Its validity
for $n=0$ is equivalent to the automatic continuity theorem, which is open
except when $H$ is a symmetric subgroup [32] or $H=G$; another (non-reductive)
example is when $H$ is maximal unipotent [14, 24]. Nonetheless, we will settle
two cases in affirmative.

  * •

In Example 7.3.2, we show that $c_{n}(E)$ is always an isomorphism when $E$ is
$K^{H}$-admissible in the sense of T. Kobayashi. This includes the case when
$H$ is symmetric, $H/K^{H}\to G/K$ is a holomorphic embedding of Hermitian
symmetric domains and $E$ is a unitary highest weight module. We refer to [23]
for an overview of admissible restrictions.

  * •

In Examples 7.3.5, 7.3.6, we prove that $c_{n}(E)$ is always an isomorphism
when $G=\operatorname{SL}(2)$ and $H$ is a reductive spherical subgroup. This
is done by an explicit computation with principal series, which suffices by a
general argument of Hecht–Taylor (Proposition 7.3.4). In fact, these
computations show that
$\operatorname{H}_{1}\left(\mathfrak{h},K^{H};E^{K\text{-fini}}\right)$ can
indeed be nonzero in this case.

In general, neither the finiteness nor Hausdorffness of
$\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H})$ is known for $n>0$, even when
$H$ is reductive and spherical. See [3] for related discussions.

###  1.6 Structure of this article

In §2 we collect the necessary definitions and backgrounds on Harish-Chandra
dg-algebras, explain the derived categories and derived functors in this
generality, and then proceed to the case of h-complexes and h-derived
categories. Our references on Harish-Chandra dg-algebras are [9, 29, 27, 28].

In §3, the formalism of h-complexes is applied to a pair $(\mathfrak{g},K)$,
and a generalized notion of Harish-Chandra modules is presented in Definition
3.1.3. The results are either taken from [9, 27], or are straightforward
analogues from the classical picture, cf. [20].

The §4.1 concerns the geometric counterpart. We review the definition of
h-derived category of $D$-modules, focusing on the affine case. Most of the
materials are from [9]. In addition, we show that Beilinson’s equivalence is
compatible with various operations on equivariant derived categories and their
h-analogues, especially for the case of inverse images.

In §5, we define the localization functor $\mathbf{Loc}_{X}$ in the
equivariant h-derived setting, relate its fibers to co-invariants, and prove
the main Theorem 5.3.4 about regularity.

In §6, we begin with the general algebraic framework of
$\operatorname{Ext}$-branching laws, relate both the $\operatorname{Ext}$
spaces and relative Lie algebra homologies to localizations, and draw some
consequences from regularity in the spherical case (Theorems 6.3.2, 6.3.4 and
Corollaries 6.3.3, 6.3.5). The monodromic setting discussed in §6.4 is
similar.

Finally, in §7 we review the Schwartz homologies and connect them to the
algebraic version via a family of maps $c_{n}(E)$. After several sundry
results, we present three cases in which the comparison maps $c_{n}(E)$ are
isomorphisms: admissible restriction (Example 7.3.2), the diagonal torus in
$\operatorname{SL}(2)$ (Example 7.3.5), and more general reductive spherical
subgroups of $\operatorname{SL}(2)$ (Example 7.3.6). This section is largely
independent of the previous ones.

###  1.7 Conventions

Unless otherwise specified (such as in §7), all vector spaces, varieties and
algebraic groups are defined over $\mathbb{C}$. Points of a variety are
assumed to be closed, and we write
$\mathrm{pt}:=\operatorname{Spec}\mathbb{C}$. By default, groups act on the
right of varieties.

We use the abbreviation $\otimes=\otimes_{\mathbb{C}}$. The dual of a vector
space $V$ is denoted by $V^{*}$; the invariants under a group $\Gamma$ is
denoted by $V^{\Gamma}$; the symmetric algebra is denoted by
$\operatorname{Sym}(V)$.

Differential operators on a smooth variety $X$ are assumed to be algebraic.
The structure sheaf of $X$ is denoted by $\mathscr{O}_{X}$. The algebra (resp.
sheaf of algebras) of differential operators on $X$ is denoted by $D_{X}$
(resp. $\mathscr{D}_{X}$).

The identity connected component of a Lie group or an algebraic group $G$ is
denoted by $G^{\circ}$; the opposite of $G$ is denoted by $G^{\mathrm{op}}$.
Subgroups are always assumed to be closed.

An affine group $K$ is said to be _reductive_ if $K^{\circ}$ is a connected
reductive group. Since we are in characteristic zero, this is equivalent to
the linear reductivity of $K$.

Lie algebras of Lie groups or algebraic groups are denoted by Gothic letters,
such as $\mathfrak{g}=\operatorname{Lie}G$. The universal enveloping algebra
of $\mathfrak{g}$ is denoted as $U(\mathfrak{g})$, and the center of
$U(\mathfrak{g})$ is denoted as $\mathcal{Z}(\mathfrak{g})$. The adjoint
action of a group is denoted by $\operatorname{Ad}$.

Categories and functors are all $\mathbb{C}$-linear. The opposite of a
category $\mathcal{C}$ is denoted by $\mathcal{C}^{\mathrm{op}}$. We neglect
all set-theoretic issues and employ only $1$-categories in this work, although
dg-enrichment still plays a vital role.

Complexes are assumed to be cochain complexes. Given such a complex $(C,d)$ or
simply $C$, we have the corresponding chain complex given by $C_{n}:=C^{-n}$.
We denote by $\mathsf{C}(\mathbb{C})$ (resp. $\mathsf{K}(\mathbb{C})$,
$\mathsf{D}(\mathbb{C})$) the category of complexes of $\mathbb{C}$-vector
spaces (resp. its homotopy category, its derived category), and put the
superscripts $+,-,\mathrm{b}$ to denote the full subcategories with
boundedness conditions. More generally, for an abelian category $\mathcal{A}$,
we have the categories $\mathsf{C}(\mathcal{A})$, $\mathsf{K}(\mathcal{A})$,
$\mathsf{D}(\mathcal{A})$ and so forth.

For an algebra $A$ (resp. Lie algebra $\mathfrak{g}$), we denote by
$A\text{-}\mathsf{Mod}$ (resp. $\mathfrak{g}\text{-}\mathsf{Mod}$) the abelian
category of left $A$-modules (resp. $\mathfrak{g}$-modules); the opposite
algebra of $A$ is denoted by $A^{\mathrm{op}}$. These notations will be
generalized to dg-modules over Harish-Chandra dg-algebras in §2.2.

### Acknowledgement

The author would like to thank Yangyang Chen, Jan Frahm and Masatoshi Kitagawa
for useful conversations and their expertise. This work is supported by
NSFC-11922101.

##  2 Homological algebra over Harish-Chandra pairs

###  2.1 Review of dg-algebras and dg-modules

Below is a short review about dg-modules over dg-algebras. These materials are
standard, and detailed expositions can be found in [8, Part II] or [34].

First of all, we make $\mathsf{C}(\mathbb{C})$ into a symmetric monoidal
category by using Koszul’s sign rule, namely: the tensor product of complexes
$M$ and $N$ are given by $(M\otimes N)^{n}=\bigoplus_{i+j=n}M^{i}\otimes
N^{j}$, with $d(m\otimes n)=dm\otimes n+(-1)^{i}m\otimes dn$, and the braiding
$M\otimes N\stackrel{{\scriptstyle\sim}}{{\rightarrow}}N\otimes M$ is
$m\otimes n\mapsto(-1)^{ij}n\otimes m$ on $M^{i}\otimes N^{j}$.

######  Definition 2.1.1.

If a complex $A$ carries the structure of an algebra333Also known as a
_monoid_ in some textbooks. in the symmetric monoidal category
$\mathsf{C}(\mathbb{C})$, then $A$ is said to be a _dg-algebra_.

Multiplication in a dg-algebra is given by a morphism $A\otimes A\to A$ in
$\mathsf{C}(\mathbb{C})$. In concrete terms, this means that there are linear
maps $A^{i}\otimes A^{j}\to A^{i+j}$ for all $i,j\in\mathbb{Z}$, satisfying

$d(xy)=dx\cdot y+(-1)^{i}x\cdot dy$

and are associative and unital with respect to some element in $A^{0}$.

It is customary to identify the complex $A$ with $\bigoplus_{n}A^{n}$ and view
$d$ as an endomorphism of that vector space. Note that dg-algebras in degree
zero are simply algebras.

######  Definition 2.1.2.

By the abstract formalism of algebras in symmetric monoidal categories, it
makes sense to define left (resp. right) $A$-modules $M$, with scalar
multiplication given by morphisms $A\otimes M\to M$ (resp. $M\otimes A\to M$),
which we call left (resp. right) _dg-modules_ over $A$. Homomorphisms between
dg-modules are defined as morphisms between complexes that respect scalar
multiplications.

In concrete terms, a left dg-module over $A$ is a complex $M$ together with
linear maps $A^{i}\otimes M^{j}\to M^{i+j}$ for all $i,j\in\mathbb{Z}$,
satisfying

$d(am)=da\cdot m+(-1)^{i}a\cdot dm$

and are associative and unital. Again, it is customary to identify $M$ with
the vector space $\bigoplus_{n}M^{n}$, so that $M$ is also a left $A$-module
in the non-dg sense. Right dg-modules over $A$ are described similarly, and
one can also define bimodules.

When $A=\mathbb{C}$ (in degree zero), left and right dg-modules are nothing
but complexes.

We now review the tensor product of dg-modules; see [34, Definition 3.3.23]
for details.

######  Definition 2.1.3.

Let $M$ be a right dg-module and $N$ be a left dg-module over a dg-algebra
$A$. Let us form the vector space $M\underset{A}{\otimes}N$ in the non-dg
sense; the kernel of the natural linear map

$\bigoplus_{n}(M\otimes
N)^{n}\simeq(\bigoplus_{i}M^{i})\otimes(\bigoplus_{j}M^{j})\to
M\underset{A}{\otimes}N$

is readily seen to be a subcomplex of $M\otimes N$. In this way, we see that
$M\underset{A}{\otimes}N$ comes from a complex, which will also be denoted as
$M\underset{A}{\otimes}N$.

If $M$ (resp. $N$) is a dg-bimodule over $(R,A)$ (resp. over $(A,S)$), where
$R$ and $S$ are dg-algebras, then $M\underset{A}{\otimes}N$ inherits a natural
$(R,S)$-bimodule structure.

The _opposite dg-algebra_ $A^{\text{op}}$ of $A$ is the same complex with the
multiplication $x\overset{\mathrm{op}}{\cdot}y:=(-1)^{ij}yx$ for all $x\in
A^{i}$ and $y\in A^{j}$. One passes between left dg-modules over $A$ and right
dg-modules over $A^{\mathrm{op}}$ by the rule $ma:=(-1)^{ij}am$, for $a\in
A^{i}$ and $m\in M^{j}$. Henceforth, we will mainly consider left dg-modules.

######  Definition 2.1.4.

A _dg-category_ is a category $\mathcal{C}$ enriched over the symmetric
monoidal category $\mathsf{C}(\mathbb{C})$. In other words, to each pair
$(X,Y)$ of objects is assigned the $\operatorname{Hom}$-complex
$\operatorname{Hom}^{\bullet}_{\mathcal{C}}(X,Y)\in\mathsf{C}(\mathbb{C})$,
and there are morphisms of composition in $\mathsf{C}(\mathbb{C})$ of the form

$\operatorname{Hom}^{\bullet}_{\mathcal{C}}(Y,Z)\otimes\operatorname{Hom}^{\bullet}_{\mathcal{C}}(X,Y)\to\operatorname{Hom}^{\bullet}_{\mathcal{C}}(X,Z),$

subject to strict associativty and unit laws. The non-dg version
$\operatorname{Hom}_{\mathcal{C}}(X,Y)$ can be recovered by taking
$0$-cocycles in $\operatorname{Hom}^{\bullet}_{\mathcal{C}}(X,Y)$. We also
write
$\operatorname{End}^{\bullet}_{\mathcal{C}}(X):=\operatorname{Hom}^{\bullet}_{\mathcal{C}}(X,X)$.

A _dg-functor_ $\mathcal{C}\to\mathcal{D}$ between dg-categories is a functor
whose action on $\operatorname{Hom}$ sets upgrades to
$\operatorname{Hom}$-complexes.

For example, for any abelian category $\mathcal{A}$, the category
$\mathsf{C}(\mathcal{A})$ is naturally a dg-category. In the case of
$\mathsf{C}(\mathbb{C})$, the $\operatorname{Hom}$-complexes will be denoted
as $\operatorname{Hom}^{\bullet}$. Let $M$ and $N$ be complexes, typical
elements in $\operatorname{Hom}^{n}(M,N)$ take the form
$(f^{k})_{k\in\mathbb{Z}}$ with $f^{k}:M^{k}\to N^{k+n}$.

######  Definition 2.1.5.

Let $\mathcal{C}$ be a dg-category. Its opposite $\mathcal{C}^{\mathrm{op}}$
is the dg-category with the same objects, and
$\operatorname{Hom}_{\mathcal{C}^{\mathrm{op}}}^{\bullet}(X,Y):=\operatorname{Hom}_{\mathcal{C}}^{\bullet}(Y,X)$,
with morphisms of composition

$\operatorname{Hom}_{\mathcal{C}^{\mathrm{op}}}^{p}(Y,Z)\otimes\operatorname{Hom}_{\mathcal{C}^{\mathrm{op}}}^{q}(X,Y)\to\operatorname{Hom}_{\mathcal{C}^{\mathrm{op}}}^{p+q}(X,Z)$

defined as $(-1)^{pq}$ times the reversed composition in $\mathcal{C}$.

We now extend the definition of $\operatorname{Hom}$-complexes to dg-modules.

######  Definition 2.1.6.

Let $M$ and $N$ be left dg-modules over a dg-algebra $A$. Define
$\operatorname{Hom}^{\bullet}_{A}(M,N)$ as the subcomplex of
$\operatorname{Hom}^{\bullet}(M,N)$ given by

$\operatorname{Hom}^{n}_{A}(M,N):=\left\\{\begin{array}[]{r|l}(f^{k})_{k\in\mathbb{Z}}\in\operatorname{Hom}^{n}(M,N)&\forall
i,j,\;\forall a\in A^{i},\;m\in M^{j}\\\
&f^{i+j}(am)=(-1)^{ni}af^{j}(m)\end{array}\right\\}.$

If $M,N,L$ are left dg-modules over $A$, we have the morphism of composition

$\operatorname{Hom}^{\bullet}_{A}(N,L)\otimes\operatorname{Hom}^{\bullet}_{A}(M,N)\to\operatorname{Hom}^{\bullet}_{A}(M,L)$

in $\mathsf{C}(\mathbb{C})$. We write
$\operatorname{End}^{\bullet}_{A}(M):=\operatorname{Hom}^{\bullet}_{A}(M,M)$.

######  Remark 2.1.7.

For every complex $M$, composition makes $\operatorname{End}^{\bullet}(M)$
into a dg-algebra. It is routine to check (see eg. [34, Proposition 3.3.17])
that given a dg-algebra $A$, to promote $M$ into a left dg-module over $A$
amounts to prescribing a homomorphism of dg-algebras

$A\to\operatorname{End}^{\bullet}(M).$

######  Definition 2.1.8.

For every dg-algebra $A$, we denote the abelian category of left dg-modules
over $A$ by $A\text{-}\mathsf{dgMod}$ in order to emphasize that its objects
are complexes.

######  Proposition 2.1.9.

The construction of $\operatorname{Hom}^{\bullet}_{A}$ upgrades
$A\text{-}\mathsf{dgMod}$ to a dg-category.

###### Proof.

The $0$-cocycles in $\operatorname{Hom}^{\bullet}_{A}(M,N)$ are nothing but
morphisms between dg-modules. ∎

For any complex $M$, its translate $M[1]$ is given by $M[1]^{n}=M^{n+1}$ and
$d_{M[1]}^{n}=-d_{M}^{n+1}$. The translation functor on
$A\text{-}\mathsf{dgMod}$ is given as follows (see [34, Definition 4.1.9]).

######  Definition 2.1.10.

Let $M$ be a left dg-module over a dg-algebra $A$. We turn $M[1]$ into a left
dg-module over $A$ by setting the scalar multiplication $\cdot$ of $A$ on
$M[1]$ to be

$a\cdot m=(-1)^{i}am,\quad a\in A^{i},\quad m\in M[1]^{j}=M^{j+1}.$

###  2.2 Harish-Chandra dg-algebras

To begin with, let $K$ be an affine algebraic group. Set
$O_{K}:=\Gamma(K,\mathscr{O}_{K})$. An abstract representation $V$ of $K$,
possibly infinite-dimensional, is said to be _algebraic_ if it arises from a
$O_{K}$-comodule structure on $V$; in other words, $V$ is the union of finite-
dimensional subrepresentations $V_{i}$ on which $K$ acts through a
homomorphism $\sigma_{i}:K\to\operatorname{GL}(V_{i})$ of algebraic groups. In
general, we say $v\in V$ is $K$-algebraic if it belongs to some algebraic
subrepresentation, and define

$V^{K\text{-alg}}:=\\{v\in V:\text{algebraic}\\}.$

This is the maximal $K$-algebraic subrepresentation of $V$. The assignment
$V\mapsto V^{K\text{-alg}}$ is functorial, and one readily checks that

$V^{K\text{-alg}}=V^{K^{\circ}\text{-alg}}.$

The class of algebraic representations is closed under subquotients, direct
sums and tensor products. If $\sigma:K\to\operatorname{Aut}_{\mathbb{C}}(V)$
gives rise to an algebraic representation, then its derivative at identity

$\mathop{}\\!\mathrm{d}\sigma:\mathfrak{k}\to\operatorname{End}_{\mathbb{C}}(V)$

makes sense: all reduces to the finite-dimensional case.

######  Definition 2.2.1.

We say that $K$ acts algebraically on a complex $M$, if $M^{n}$ is an
algebraic representation of $K$ and $d^{n}:M^{n}\to M^{n+1}$ is
$K$-equivariant, for each $n$. In particular, this induces a homomorphism
$K\to\operatorname{Aut}_{\mathsf{C}(\mathbb{C})}(M)$ of abstract groups.

######  Definition 2.2.2 ([9, 1.9.1]).

By a _Harish-Chandra dg-algebra_ we mean a quadruplet

$(A,K,\sigma,j),$

or more succinctly a pair $(A,K)$, where:

  * •

$A$ is a dg-algebra and $K$ is as above,

  * •

$K$ acts algebraically on $A$, and induces a homomorphism
$\sigma:K\to\operatorname{Aut}_{\mathsf{dga}}(A)$, where $\mathsf{dga}$ is the
category of dg-algebras;

  * •

$j:\mathfrak{k}\to A^{0}\subset A$ is a $K$-equivariant linear map satisfying

$\displaystyle d\circ j=0,$ $\displaystyle
j([\xi_{1},\xi_{2}])=[j(\xi_{1}),j(\xi_{2})]:=j(\xi_{1})j(\xi_{2})-j(\xi_{2})j(\xi_{1})$

for all $\xi_{1},\xi_{2}\in\mathfrak{k}$;

  * •

furthermore, we impose the condition that for all $\xi\in\mathfrak{k}$,

$(\mathop{}\\!\mathrm{d}\sigma)(\xi)=[j(\xi),\cdot]\;\in\operatorname{End}_{\mathbb{C}}(A).$

A morphism $(A,K)\to(A^{\prime},K^{\prime})$ between Harish-Chandra dg-
algebras consists of a homomorphism $\varphi:K\to K^{\prime}$ between
algebraic groups together with a $\varphi$-equivariant homomorphism $\psi:A\to
A^{\prime}$ between dg-algebras, such that

${\mathfrak{k}}$${A}$${\mathfrak{k}^{\prime}}$${A^{\prime}}$$\scriptstyle{j}$$\scriptstyle{\operatorname{Lie}\varphi}$$\scriptstyle{\psi}$$\scriptstyle{j^{\prime}}$

commutes.

######  Definition 2.2.3.

Let $(A,K)$ be a Harish-Chandra dg-algebra. A (left) dg-module over $(A,K)$ is
a left dg-module $M$ over $A$, say given by a homomorphism
$\alpha:A\to\operatorname{End}^{\bullet}(M)$ between dg-algebras, together
with a group homomorphism
$\rho:K\to\operatorname{Aut}_{\mathsf{C}(\mathbb{C})}(M)$ making $K$ act
algebraically on $M$, subject to the following compatibilities:

  * •

$\alpha(\sigma(k)a)=\rho(k)\alpha(a)\rho(k)^{-1}$ in
$\operatorname{End}^{\bullet}(M)$, for all $k\in K$ and $a\in A$;

  * •

$\alpha\circ j=\mathop{}\\!\mathrm{d}\rho$ as linear maps
$\mathfrak{k}\to\operatorname{End}_{\mathsf{C}(\mathbb{C})}(M)$.

Homomorphisms $M\to N$ between dg-modules over $(A,K)$ are $K$-equivariant
homomorphisms in $\operatorname{Hom}_{A}(M,N)$. The abelian category so
obtained is denoted by $(A,K)\text{-}\mathsf{dgMod}$.

When $K=\\{1\\}$, this reverts to the dg-category $A\text{-}\mathsf{dgMod}$
reviewed in §2.1.

Every morphism $(\varphi,\psi):(A,K)\to(A^{\prime},K^{\prime})$ between
Harish-Chandra dg-algebras induces a pullback functor

$(\varphi,\psi)^{*}:(A^{\prime},K^{\prime})\text{-}\mathsf{dgMod}\to(A,K)\text{-}\mathsf{dgMod}.$
(2.2.1)

This can be naturally upgraded to a dg-functor between dg-categories.

######  Remark 2.2.4.

When $A$ is simply a $\mathbb{C}$-algebra, we can define $(A,K)$-modules as
dg-modules over $(A,K)$ in degree zero, and the resulting abelian category
will be denoted by $(A,K)\text{-}\mathsf{Mod}$.

###  2.3 Derived categories and functors

The basic reference for what follows is [9, 1.9]. Let $(A,K)$ be a Harish-
Chandra dg-algebra. Many operations on $A\text{-}\mathsf{dgMod}$ carry over to
$(A,K)\text{-}\mathsf{dgMod}$ by imposing $K$-equivariance. Below are some key
examples.

Translation functor

Let $M$ be a dg-module over $(A,K)$. We make $M[1]$ into a dg-module over
$(A,K)$ using the dg-module structure over $A$ from Definition 2.1.10, and
leaving the $K$-action intact.

The $\operatorname{Hom}$-complex

Let $M$, $N$ be dg-modules over $(A,K)$. Define
$\operatorname{Hom}^{\bullet}_{A,K}(M,N)$ to be the subcomplex of
$\operatorname{Hom}^{\bullet}_{A}(M,N)$ (see Definition 2.1.6) given by

$\operatorname{Hom}^{n}_{A,K}(M,N)=\left\\{(f^{k})_{k\in\mathbb{Z}}\in\operatorname{Hom}^{n}_{A}(M,N):\forall
k,\;f^{k}\;\text{is $K$-equivariant}\right\\}.$

Morphisms between dg-modules over $(A,K)$ are nothing but $0$-cocycles in
$\operatorname{Hom}^{\bullet}_{A,K}$. This turns $(A,K)\text{-}\mathsf{dgMod}$
into a dg-category.

Homotopy category

Two morphisms $f,g\in\operatorname{Hom}_{A,K}(M,N)$ are said to be homotopic
if $f-g$ is a coboundary in $\operatorname{Hom}^{0}_{A,K}(M,N)$. The homotopy
category $\mathsf{K}(A,K)$ of $(A,K)\text{-}\mathsf{dgMod}$ has the same
objects as $(A,K)\text{-}\mathsf{dgMod}$, whilst

$\operatorname{Hom}_{\mathsf{K}(A,K)}(M,N):=\operatorname{H}^{0}\operatorname{Hom}^{\bullet}_{A,K}(M,N)$

for all objects $M$ and $N$. The translation functor $M\mapsto M[1]$ passes to
$\mathsf{K}(A,K)$, and we have
$\operatorname{Hom}_{\mathsf{K}(A,K)}(M,N[n])\simeq\operatorname{H}^{n}\operatorname{Hom}^{\bullet}_{A,K}(M,N)$.

Acyclic objects

We say $M$ is acyclic if it is acyclic as an object of
$\mathsf{C}(\mathbb{C})$.

Quasi-isomorphisms

If $f\in\operatorname{Hom}_{A,K}(M,N)$ induces isomorphisms
$\operatorname{H}^{n}(M)\to\operatorname{H}^{n}(N)$ for every $n$, we say $f$
is a quasi-isomorphism. This property depends only on the image of $f$ in the
homotopy category.

Mapping cones

Let $f\in\operatorname{Hom}_{A,K}(M,N)$. The mapping cone $\mathrm{Cone}(f)$
taken in $\mathsf{C}(\mathbb{C})$ is made into a dg-module over $(A,K)$ by
recalling that $\mathrm{Cone}(f)$ is $M[1]\oplus N$ as graded vector spaces,
and we let $A$ and $K$ act on $M[1]$ and $N$ in the way prescribed before. The
natural morphisms

$N\to\mathrm{Cone}(f)\to M[1]$

are morphisms in $(A,K)\text{-}\mathsf{dgMod}$. Note that this is a
$K$-equivariant version of [34, Definition 4.2.1].

The constructions above make $\mathsf{K}(A,K)$ into a triangulated category:
the distinguished triangles are the ones isomorphic to

$M\xrightarrow{f}N\to\mathrm{Cone}(f)\xrightarrow{+1}$

given before, where $f:M\to N$ is any morphism in
$(A,K)\text{-}\mathsf{dgMod}$.

######  Definition 2.3.1.

Let $\mathsf{D}(A,K)$ be the Verdier quotient of $\mathsf{K}(A,K)$ by acyclic
objects; equivalently, it is obtained from $\mathsf{K}(A,K)$ by inverting
quasi-isomorphisms. We call $\mathsf{D}(A,K)$ the _derived category_ of
$(A,K)$. It inherits the triangulated structure from $\mathsf{K}(A,K)$.

When $K=\\{1\\}$, the above reduces to the well-known derived category of dg-
modules: see [8, §10] or [34, Chapter 7].

In order to better understand $\mathsf{D}(A,K)$, one has to adapt the notion
of K-projective and K-injective complexes into our context. Cf. [34, §§10.1 —
10.2].

######  Definition 2.3.2.

Let $M$ be a dg-module over $(A,K)$.

  * •

We say $M$ is _K-injective_ if

$Q\;\text{is acyclic}\implies\operatorname{Hom}^{\bullet}_{A,K}(Q,M)\;\text{is
acyclic};$

equivalently,

$Q\;\text{is acyclic}\implies\operatorname{Hom}_{\mathsf{K}(A,K)}(Q,M)=0.$

  * •

We say $M$ is _K-projective_ if

$Q\;\text{is acyclic}\implies\operatorname{Hom}^{\bullet}_{A,K}(M,Q)\;\text{is
acyclic};$

equivalently,

$Q\;\text{is acyclic}\implies\operatorname{Hom}_{\mathsf{K}(A,K)}(M,Q)=0.$

Being K-injective (resp. K-projective) is thus a property within
$\mathsf{K}(A,K)$. If $M$ is K-injective (resp. K-projective), then

$\displaystyle\operatorname{Hom}_{\mathsf{K}(A,K)}(X,M)\simeq\operatorname{Hom}_{\mathsf{D}(A,K)}(X,M)$
$\displaystyle(\text{resp.}\;\operatorname{Hom}_{\mathsf{K}(A,K)}(M,X)\simeq\operatorname{Hom}_{\mathsf{D}(A,K)}(M,X))$

for all dg-module $X$ over $(A,K)$. Cf. [34, Theorems 10.1.13 and 10.2.9].

We now move to $t$-structure. We say $A$ is _non-positively graded_ if
$A^{n}=0$ for all $n>0$.

######  Proposition 2.3.3.

Suppose that $A$ is non-positively graded. For every dg-module $M$ over
$(A,K)$, the “smart truncation” $\tau^{<0}M$ in $\mathsf{C}(\mathbb{C})$ is
actually a dg-submodule of $M$ over $(A,K)$, thus so is $\tilde{\tau}^{\geq
0}M:=M/\tau^{<0}M$. We have the canonical distinguished triangle

$\tau^{<0}M\to M\to\tilde{\tau}^{\geq 0}M\xrightarrow{+1}$

in $\mathsf{K}(A,K)$ as well as in $\mathsf{D}(A,K)$.

###### Proof.

Routine, see [9, 1.9.4 Lemma]. ∎

Using the fact above, we equip $\mathsf{D}(A,K)$ with a $t$-structure and
define its full triangulated subcategories $\mathsf{D}^{\star}(A,K)$ where
$\star\in\\{+,-,\mathrm{b}\\}$. They are characterized by
$\operatorname{H}^{n}=0$ for $n\ll 0$, $n\gg 0$ and $|n|\gg 0$ respectively.

We also denote by $\mathsf{K}^{\star}(A,K)$ (where
$\star\in\\{+,-,\mathrm{b}\\}$) the full triangulated subcategories of
$\mathsf{K}(A,K)$ consisting of bounded below, bounded above, and bounded
complexes respectively, so that $\mathsf{D}^{\star}(A,K)$ is equivalent to the
corresponding Verdier quotients.

Now comes the definition of derived functors in an abstract context. More
general formulations exist and can be found in [34, Chapter 8], for example.

######  Definition 2.3.4.

Let $(A,K)$ (resp. $(A^{\prime},K^{\prime})$) be Harish-Chandra dg-algebras,
and let $\mathsf{K}$ (resp. $\mathsf{K}^{\prime}$) be a full triangulated
subcategory of $(A,K)\text{-}\mathsf{dgMod}$ (resp.
$(A^{\prime},K^{\prime})\text{-}\mathsf{dgMod}$). Denote the functor to
Verdier quotients modulo acyclic objects as $Q:\mathsf{K}\to\mathsf{D}$ and
$Q^{\prime}:\mathsf{K}^{\prime}\to\mathsf{D}^{\prime}$ respectively.

Consider a triangulated functor $F:\mathsf{K}\to\mathsf{K}^{\prime}$. A left
(resp. right) _derived functor_ of $F$ is defined to be a triangulated functor
$\operatorname{L}\\!F$ (resp. $\operatorname{R}\\!F$) from $\mathsf{D}$ to
$\mathsf{D}^{\prime}$ which is the right (resp. left) Kan extension of
$Q^{\prime}F$ along $Q:\mathsf{K}\to\mathsf{D}$.

Whenever they exist, $\operatorname{L}\\!F$ and $\operatorname{R}\\!F$ fit
into 2-cells:

${\mathsf{K}}$${\mathsf{K}^{\prime}}$${\mathsf{D}}$${\mathsf{D}^{\prime}}$$\scriptstyle{F}$$\scriptstyle{Q}$$\scriptstyle{Q^{\prime}}$$\scriptstyle{\operatorname{L}\\!F}$
${\mathsf{K}}$${\mathsf{K}^{\prime}}$${\mathsf{D}}$${\mathcal{D}^{\prime}}$$\scriptstyle{F}$$\scriptstyle{Q}$$\scriptstyle{Q^{\prime}}$$\scriptstyle{\operatorname{R}\\!F}$
(2.3.1)

The morphisms $\Rightarrow$ between functors are part of the data of derived
functors. The universal property of Kan extension amounts to asserting that
every 2-cell of the left (resp. right) form as (2.3.1) uniquely “retracts to”
(resp. is uniquely “inflated from”) the 2-cell of $\operatorname{L}\\!F$
(resp. $\operatorname{R}\\!F$).

######  Example 2.3.5.

The easiest example is the case when $F$ is _exact_ , i.e. when $F$ preserves
acyclicity (equivalently, preserves quasi-isomorphisms). For such functors,
$\operatorname{L}\\!F=\operatorname{R}\\!F$ exists and is simply induced from
the universal property of Verdier quotients; furthermore, the $\Rightarrow$ in
(2.3.1) are isomorphisms.

The pull-back functors $(\varphi,\psi)^{*}$ from (2.2.1) are exact, since they
preserve the underlying complexes.

For general $F$, the left (resp. right) derived functor can be accessed from
K-injective (resp. K-projective) resolutions as in the usual setting, if they
exist.

######  Definition 2.3.6.

Let $M$ be a dg-module over $(A,K)$. A K-injective (resp. K-projective)
_resolution_ of $M$ is a quasi-isomorphism $M\to I$ (resp. $P\to M$) in
$(A,K)\text{-}\mathsf{dgMod}$ where $I$ is K-injective (resp. $P$ is
K-projective).

Let $\mathsf{K}$ be a full triangulated subcategory of $\mathsf{K}(A,K)$. If
every object $M$ of $\mathsf{K}$ admits a K-injective (resp. K-projective)
resolution within $\mathsf{K}$, then $\mathsf{K}$ is said to have enough
K-injectives (resp. K-projectives).

######  Theorem 2.3.7.

Assume that $(A,K)$ be non-positively graded. Then $\mathsf{K}^{+}(A,K)$ has
enough K-injectives; if we assume moreover that $K$ is reductive, then
$\mathsf{K}^{-}(A,K)$ has enough K-projectives.

###### Proof.

The first part follows from [9, 1.15.3]. The second part follows from [29,
§5.6]. ∎

The following is a special instance of [34, Theorems 10.1.20 and 10.2.15].

######  Proposition 2.3.8.

Consider the situation of Definition 2.3.4.

  * •

If $\mathsf{K}$ has enough K-injectives, then $\operatorname{R}\\!F$ exists,
and $(\operatorname{R}\\!F)(M)\simeq Q^{\prime}F(I)$ if $M\to I$ is a
K-injective resolution within $\mathsf{K}$.

  * •

If $\mathsf{K}$ has enough K-projectives, then $\operatorname{L}\\!F$ exists,
and $(\operatorname{L}\\!F)(M)\simeq Q^{\prime}F(P)$ if $P\to M$ is a
K-projective resolution within $\mathsf{K}$.

As a result, one can define
$\operatorname{RHom}_{A,K}(X,Y)\in\mathsf{D}(\mathbb{C})$ for $X$ bounded
above and $Y$ bounded below by the familiar recipe:

  * •

either as the derived functor of $\operatorname{Hom}^{\bullet}_{A,K}(X,\cdot)$
from $(A,K)\text{-}\mathsf{dgMod}^{+}$ to $\mathsf{C}(\mathbb{C})$,

  * •

or as the derived functor of $\operatorname{Hom}^{\bullet}_{A,K}(\cdot,Y)$
from $(A,K)\text{-}\mathsf{dgMod}^{-}$ to
$\mathsf{C}(\mathbb{C})^{\mathrm{op}}$.

The boundedness conditions are removable if the relevant K-injective or
K-projective resolutions exist. In any case, for all $n\in\mathbb{Z}$ we have

$\operatorname{H}^{n}\operatorname{RHom}_{A,K}(X,Y)\simeq\operatorname{Hom}_{\mathsf{D}(A,K)}(X,Y[n])=:\operatorname{Ext}^{n}_{A,K}(X,Y).$
(2.3.2)

Given functors
$\mathsf{K}\xrightarrow{F}\mathsf{K}^{\prime}\xrightarrow{G}\mathsf{K}^{\prime\prime}$,
the universal property furnishes canonical morphisms

$\operatorname{R}\\!(GF)\to(\operatorname{R}\\!G)(\operatorname{R}\\!F),\quad(\operatorname{L}\\!G)(\operatorname{L}\\!F)\to\operatorname{L}\\!(GF),$

provided that all these derived functors exist. We record the following
standard result (see eg. [34, Remark 8.4.31]) which is also immediate from
Proposition 2.3.8. Recall that a functor is said to be exact if it preserves
acyclicity.

######  Corollary 2.3.9.

Keep the notations of Definition 2.3.4 and consider functors
$\mathsf{K}\xrightarrow{F}\mathsf{K}^{\prime}\xrightarrow{G}\mathsf{K}^{\prime\prime}$.

  * •

Suppose $\mathsf{K}$ and $\mathsf{K}^{\prime}$ have enough K-injectives. If
either $F$ preserves K-injectives or $G$ is exact, then
$\operatorname{R}\\!(GF)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\operatorname{R}\\!G)(\operatorname{R}\\!F)$.

  * •

Suppose $\mathsf{K}$ and $\mathsf{K}^{\prime}$ have enough K-projectives. If
either $F$ preserves K-projectives or $G$ is exact, then
$(\operatorname{L}\\!G)(\operatorname{L}\\!F)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{L}\\!(GF)$.

The standard tool to ensure the preservation of K-injectives or K-projectives
is adjunction.

######  Proposition 2.3.10.

Let $\mathcal{C}$ and $\mathcal{C}^{\prime}$ be dg-categories, each is of the
form $(A,K)\text{-}\mathsf{dgMod}$ for some $(A,K)$. Suppose that

${F:\mathcal{C}}$${\mathcal{D}:G}$

is a pair of adjoint functors, and both $F$ and $G$ upgrade to dg-functors
(see the discussions after (2.2.1)). Then

  * •

$F$ preserves K-projectives if $G$ is exact;

  * •

$G$ preserves K-injectives if $F$ is exact.

###### Proof.

Adjunction is described by morphisms $\eta:\mathrm{id}_{\mathcal{C}}\to GF$
and $\epsilon:FG\to\mathrm{id}_{\mathcal{C}^{\prime}}$ satisfying the
triangular equalities. As $F$ and $G$ upgrade to dg-functors,
$(\eta,\epsilon)$ induces not only functorial bijections

$\operatorname{Hom}_{\mathcal{C}^{\prime}}(FX,Y)\simeq\operatorname{Hom}_{\mathcal{C}}(X,GY),$

but also functorial isomorphisms of complexes

$\operatorname{Hom}^{\bullet}_{\mathcal{C}^{\prime}}(FX,Y)\simeq\operatorname{Hom}^{\bullet}_{\mathcal{C}}(X,GY),$

i.e. a dg-adjunction. The assertions follow at once. ∎

###  2.4 h-construction

Throughout this subsection, we consider a Harish-Chandra dg-algebra $(A,K)$
with $A$ concentrated at degree zero, i.e. $A$ is an algebra coming with
compatible homomorphisms

$j:\mathfrak{k}\to A,\quad\sigma:K\to\operatorname{Aut}_{\mathsf{alg}}(A).$

Recall from Remark 2.2.4 that the $(A,K)$-modules are simply left $A$-modules
equipped with compatible $K$-actions. The dg-category of complexes of
$(A,K)$-modules is denoted by $\mathsf{C}(A,K)$.

######  Definition 2.4.1.

With the conventions above, a _weak $(A,K)$-module_ is a $\mathbb{C}$-vector
space $M$ equipped with homomorphisms

$\alpha:A\to\operatorname{End}_{\mathbb{C}}(M),\quad\rho:K\to\operatorname{Aut}_{\mathbb{C}}(M),$

making $M$ into a left $A$-module and an algebraic representation of $K$
respectively, such that

$\alpha(\sigma(k)a)=\rho(k)\alpha(a)\rho(k)^{-1}$

holds in $\operatorname{End}_{\mathbb{C}}(M)$ for all $k\in K$ and $a\in A$,
i.e. $\alpha$ is $K$-equivariant.

Weak $(A,K)$-modules form an abelian category. Given a weak $(A,K)$-module
$M$, define

$w:=\mathop{}\\!\mathrm{d}\rho-\alpha\circ
j:\mathfrak{k}\to\operatorname{End}_{\mathbb{C}}(M).$ (2.4.1)

This map is clearly $K$-equivariant. It is also $A$-linear: indeed,

$\displaystyle[\alpha(j(\xi)),\alpha(a)]$
$\displaystyle=\alpha\left([j(\xi),a]\right)$
$\displaystyle=\alpha\left(\mathop{}\\!\mathrm{d}\sigma(\xi)\cdot
a\right)=[\mathop{}\\!\mathrm{d}\rho(\xi),\alpha(a)]$

for all $\xi\in\mathfrak{k}$ and $a\in A$; the last equality follows from the
$K$-equivariance of $\alpha$. This entails $[w(\xi),\alpha(a)]=0$.

Therefore, a weak $(A,K)$-module is an $(A,K)$-module if and only if $w=0$.

######  Definition 2.4.2.

Denote by ${}^{\mathrm{w}}\mathsf{C}(A,K)$ the category of complexes of weak
$(A,K)$-modules. The $\operatorname{Hom}$-complex in this dg-category is
denoted by ${}^{\mathrm{w}}\operatorname{Hom}^{\bullet}$.

Our main reference for the h-construction below is [9]; the idea is attributed
to Duflo–Vergne [12] and Beilinson–Ginzburg in loc. cit.

######  Definition 2.4.3 ([9, 1.5]).

An _h-complex_ over $(A,K)$ is a complex $C=(C,d)$ of weak $(A,K)$-modules
together with a linear map

$i:\mathfrak{k}\to\operatorname{End}^{-1}(C):=\operatorname{End}^{-1}_{\mathbb{C}}(C),\quad\xi\mapsto
i_{\xi},$

such that

  1. (i)

$ki_{\xi}k^{-1}=i_{\operatorname{Ad}(k)\xi}$ for all $k\in K$;

  2. (ii)

$i_{\xi}$ is $A$-linear;

  3. (iii)

$i_{\xi}i_{\eta}+i_{\eta}i_{\xi}=0$ for all $\xi,\eta\in\mathfrak{k}$;

  4. (iv)

$di_{\xi}+i_{\xi}d=w(\xi)$ (recall (2.4.1)).

The morphisms between h-complexes are morphisms between complexes of weak
$(A,K)$-modules that commute with all $i_{\xi}$.

The homotopy condition (iv) implies that the cohomologies of an h-complex are
actually $(A,K)$-modules.

We remark that h-complexes are called _equivariant complexes_ in [29, 27, 28,
19].

######  Definition 2.4.4.

Denote by ${}^{\mathrm{h}}\mathsf{C}(A,K)$ the category of h-complexes over
$(A,K)$.

Every h-complex is a complex of weak $(A,K)$-modules, whence the functor
${}^{\mathrm{h}}\mathsf{C}(A,K)\to{}^{\mathrm{w}}\mathsf{C}(A,K)$. We upgrade
${}^{\mathrm{h}}\mathsf{C}(A,K)$ into a dg-category by defining
${}^{\mathrm{h}}\operatorname{Hom}^{\bullet}(C_{1},C_{2})$ as the subcomplex
of ${}^{\mathrm{w}}\operatorname{Hom}^{\bullet}(C_{1},C_{2})$:

${}^{\mathrm{h}}\operatorname{Hom}^{n}(C_{1},C_{2}):=\left\\{(f^{l})_{l\in\mathbb{Z}}\in{}^{\mathrm{w}}\operatorname{Hom}^{n}(C_{1},C_{2}):\forall
l,\xi,\;f^{l-1}i_{\xi}=(-1)^{n}i_{\xi}f^{l}\right\\}.$

As usual, we say a morphism (resp. an object) of
${}^{\mathrm{h}}\mathsf{C}(A,K)$ is a quasi-isomorphism (resp. acyclic) if it
is so as a complex. The following definition thus makes sense.

######  Definition 2.4.5.

Using ${}^{\mathrm{h}}\operatorname{Hom}^{\bullet}$, we define the homotopy
category ${}^{\mathrm{h}}\mathsf{K}(A,K)$ of ${}^{\mathrm{h}}\mathsf{C}(A,K)$,
and the _h-derived category_ ${}^{\mathrm{h}}\mathsf{D}(A,K)$ is defined as
its Verdier quotient by acyclic complexes, or equivalently by inverting quasi-
isomorphisms.

The K-injective and K-projective h-complexes are also defined in this way. The
derived functors in this setting are also called _h-derived functors_.

For an h-complex $C$, we make $C[1]$ into an h-complex by setting

$i^{\prime}:\mathfrak{k}\to\operatorname{End}^{-1}(C[1]),\quad\xi\mapsto
i^{\prime}_{\xi}:=-i_{\xi}.$

Using this, mapping cones can be defined for any morphism $f:C_{1}\to C_{2}$
in ${}^{\mathrm{h}}\mathsf{C}(A,K)$. In this way,
${}^{\mathrm{h}}\mathsf{K}(A,K)$ and ${}^{\mathrm{h}}\mathsf{D}(A,K)$ become
triangulated categories.

For every h-complex $C$, the “smart truncation” $\tau^{<0}C$ as a subcomplex
of weak $(A,K)$-modules is an h-complex, and so is $\tilde{\tau}^{\geq
0}C:=C/\tau^{<0}C$. Consequently, ${}^{\mathrm{h}}\mathsf{D}(A,K)$ is endowed
with a $t$-structure, whose heart is exactly the category
$(A,K)\text{-}\mathsf{Mod}$ of $(A,K)$-modules.

We wish to compare the h-derived category with the naive one. There is an
evident dg-functor

$\mathsf{C}((A,K)\text{-}\mathsf{Mod})\to{}^{\mathrm{h}}\mathsf{C}(A,K)$
(2.4.2)

by turning complexes over $(A,K)\text{-}\mathsf{Mod}$ into h-complexes with
$i_{\xi}=0$. It induces a functor between homotopy categories and preserves
acyclicity.

Recall that the derived category $\mathsf{D}((A,K)\text{-}\mathsf{Mod})$ of
$(A,K)\text{-}\mathsf{Mod}$ is a triangulated category with $t$-structure
whose heart is $(A,K)\text{-}\mathsf{Mod}$. For
$\star\in\\{\;,+,-,\mathrm{b}\\}$, from (2.4.2) we obtain a functor

$\alpha:\mathsf{D}^{\star}((A,K)\text{-}\mathsf{Mod})\to{}^{\mathrm{h}}\mathsf{D}^{\star}(A,K).$
(2.4.3)

It is obviously $t$-exact.

All the assertion above can be checked by hand. Nonetheless, the formalism of
Harish-Chandra dg-algebras provides a conceptually more satisfactory approach
to the h-construction. We present a summary below after some preparations.

First, suppose that $A_{i}$ are dg-algebras for $i=1,2$. By the general theory
of algebras in a symmetric monoidal category, the tensor product $A_{1}\otimes
A_{2}$ of complexes underlies a dg-algebra.

Secondly, consider the dg Lie-algebra

$\overline{\mathfrak{k}}:=\left[\mathfrak{k}\xrightarrow{\mathrm{id}}\mathfrak{k}\right],\quad\text{degrees:}\;-1,0,$

whose Lie bracket on
$\overline{\mathfrak{k}}^{0}\otimes\overline{\mathfrak{k}}^{0}$ and
$\overline{\mathfrak{k}}^{0}\otimes\overline{\mathfrak{k}}^{-1}$ equals the
$[\cdot,\cdot]$ for $\mathfrak{k}$, and is zero otherwise. Its universal
enveloping dg-algebra is

$U(\overline{\mathfrak{k}})=\underbracket{\bigwedge^{\bullet}\mathfrak{k}}_{\deg\leq
0}\otimes\underbracket{U(\mathfrak{k})}_{\deg=0}.$

The differential $d$ is induced from the complex $\overline{\mathfrak{k}}$;
explicit formulas will be given in §3.2. We let $K$ act on
$U(\overline{\mathfrak{k}})$ by adjoint actions on both $\otimes$-slots.

######  Theorem 2.4.6.

We have the following equivalences of dg-categories.

  1. (i)

Take $B=A$. Then

$(B,K)\text{-}\mathsf{dgMod}\simeq\mathsf{C}((A,K)\text{-}\mathsf{Mod}).$

  2. (ii)

Take $B=U(\mathfrak{k})\otimes A$ with diagonal $K$-action, where
$U(\mathfrak{k})$ is viewed as a dg-algebra concentrated at degree zero.
Define $j_{B}:\mathfrak{k}\to B$ by

$j_{B}(\xi)=\xi\otimes 1+1\otimes j(\xi).$

This makes $(B,K)$ into a Harish-Chandra dg-algebra, and

$(B,K)\text{-}\mathsf{dgMod}\simeq{}^{\mathrm{w}}\mathsf{C}(A,K).$

Specifically, the action of $\xi\in\mathfrak{k}\subset U(\mathfrak{k})$
corresponds to $w(\xi)$; see (2.4.1).

  3. (iii)

Take $B=U(\overline{\mathfrak{k}})\otimes A$ with diagonal $K$-action. Define
$j_{B}:\mathfrak{k}\to B$ by

$j_{B}(\xi)=j^{\natural}(\xi)\otimes 1+1\otimes j(\xi)$

where $j^{\natural}$ is the embedding $\mathfrak{k}\hookrightarrow 1\otimes
U(\mathfrak{k})\subset U(\overline{\mathfrak{k}})$. This makes $(B,K)$ into a
Harish-Chandra dg-algebra, and

$(B,K)\text{-}\mathsf{dgMod}\simeq{}^{\mathrm{h}}\mathsf{C}(A,K).$

Specifically, the action of
$\xi\in\mathfrak{k}\subset\bigwedge^{\bullet}\mathfrak{k}$ corresponds to
$i_{\xi}$.

All these equivalences are identity on the underlying complexes; they preserve
acyclicity, quasi-isomorphisms and homotopies.

###### Proof.

The case (i) is trivial. Case (iii) is [9, 1.11.1], and (ii) is explained in
[28, p.2201]. ∎

######  Remark 2.4.7.

The dg-algebra $B$ above is always non-positively graded. Hence Theorem 2.3.7
provides enough K-injectives (resp. K-projectives, assuming $K$ reductive) for
the bounded below (resp. bounded above) homotopy categories.

######  Remark 2.4.8.

The h-version of $\operatorname{Ext}$ functors, denoted by
${}^{\mathrm{h}}\operatorname{Ext}_{A,K}$, can be defined through the
equivalence in Theorem 2.4.6 (iii). As the familiar $\operatorname{Ext}$
functor, it can be computed in terms of
${}^{\mathrm{h}}\operatorname{Hom}^{\bullet}$ and K-injective or K-projective
resolutions of h-complexes.

In view of these identifications, the dg-functor (2.4.2) corresponds to the
pullback induced by the $K$-equivariant homomorphism

$\epsilon\otimes\mathrm{id}:U(\overline{\mathfrak{k}})\otimes
A\to\mathbb{C}\otimes A\simeq A$

of dg-algebras, where $\epsilon:U(\overline{\mathfrak{k}})\to\mathbb{C}$ is
the augmentation homomorphism. On the other hand, pull-back along the
inclusion $U(\mathfrak{k})\otimes A\to U(\overline{\mathfrak{k}})\otimes A$
amounts to forgetting the datum $i$ in h-complexes.

###  2.5 Adjoint functors of oblivion

Consider Harish-Chandra dg-algebras $(A,K,\sigma,j)$,
$(A^{\prime},K^{\prime},\sigma^{\prime},j^{\prime})$ and a morphism
$(A,K)\to(A^{\prime},K^{\prime})$. In most of the scenarios, the maps $A\to
A^{\prime}$ and $K\to K^{\prime}$ will be inclusions. For this reason, the
corresponding exact functor from (2.2.1) will be named as _oblivion_ instead
of pullback. Denote it by

$\mathrm{oblv}:(A^{\prime},K^{\prime})\text{-}\mathsf{dgMod}\to(A,K)\text{-}\mathsf{dgMod}.$

We begin with the change of dg-algebras, i.e. the case $K^{\prime}=K$.
Consider a $K$-equivariant homomorphism $\varphi:A\to A^{\prime}$ between dg-
algebras such that $\varphi j=j^{\prime}$. Two constructions on a dg-module
$M$ over $(A,K)$ will be needed.

  * •

Definition 2.1.3 affords the dg-module $A^{\prime}\underset{A}{\otimes}M$ over
$A^{\prime}$. Let $K$ acts diagonally on it; this action is algebraic.

  * •

Since $A^{\prime}$ is a dg-bimodule over $(A,A^{\prime})$, the
$\operatorname{Hom}$-complex $\operatorname{Hom}^{\bullet}_{A}(A^{\prime},M)$
is actually a left dg-module over $A^{\prime}$. It carries the standard
$K$-action

$f\xmapsto{k\in K}\underbracket{k}_{M}\circ
f\circ\underbracket{k^{-1}}_{A^{\prime}},$ (2.5.1)

which respects differentials and $A^{\prime}$-action. Now take
$\operatorname{Hom}^{\bullet}_{A}(A^{\prime},M)^{K\text{-alg}}$.

We contend that they yield dg-modules over $(A^{\prime},K)$.

######  Lemma 2.5.1.

Both constructions above yield dg-functors

$(A,K)\text{-}\mathsf{dgMod}\to(A^{\prime},K)\text{-}\mathsf{dgMod}.$

###### Proof.

We first check that $A^{\prime}\underset{A}{\otimes}M$ is a dg-module over
$(A^{\prime},K)$. Writing the actions by $K$, $A$ and $A^{\prime}$ as left
multiplication, we have

$\displaystyle(ka^{\prime}_{1})\cdot(a^{\prime}_{2}\otimes m)$
$\displaystyle=(ka^{\prime}_{1})a^{\prime}_{2}\otimes m$
$\displaystyle=k\cdot\left((a^{\prime}_{1}\cdot k^{-1}a^{\prime}_{2})\otimes
k^{-1}m\right)$ $\displaystyle=k\cdot a^{\prime}_{1}\cdot
k^{-1}\cdot(a^{\prime}_{2}\otimes m)$

for all $a^{\prime}_{1},a^{\prime}_{2}\in A^{\prime}$, $k\in K$ and $m\in M$.
This is the first condition in Definition 2.2.3.

As for the second condition, taking derivative of the diagonal $K$-action
yields

$\displaystyle\mathop{}\\!\mathrm{d}\rho(\xi)(a^{\prime}\otimes m)$
$\displaystyle=(\mathop{}\\!\mathrm{d}\sigma^{\prime}(\xi)a^{\prime})\otimes
m+a^{\prime}\otimes(\mathop{}\\!\mathrm{d}\rho(\xi)m)$
$\displaystyle=[j^{\prime}(\xi),a^{\prime}]\otimes
m+a^{\prime}\otimes(j(\xi)m)$
$\displaystyle=\left([j^{\prime}(\xi),a^{\prime}]+a^{\prime}j^{\prime}(\xi)\right)\otimes
m$ $\displaystyle=(j^{\prime}(\xi)a^{\prime})\otimes
m=j^{\prime}(\xi)\cdot(a^{\prime}\otimes m)$

for all $\xi\in\mathfrak{k}$ since $j^{\prime}=\varphi j$, and we are done.

Similarly, one readily checks that
$\operatorname{Hom}^{\bullet}_{A}(A^{\prime},M)^{K\text{-alg}}$ satisfies
these two conditions.

Standard facts from the theory of dg-modules (see eg. [34, §9.1]) show that
they are dg-functors if we forget $K$-actions. Our case follows by imposing
$K$-equivariance. ∎

######  Proposition 2.5.2.

The dg-functors from Lemma 2.5.1 fit into adjunctions

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###### Proof.

Consider the first adjunction. It is a standard fact that

$\operatorname{Hom}_{A^{\prime}}(A^{\prime}\underset{A}{\otimes}M,M^{\prime})\simeq\operatorname{Hom}_{A}(M,\mathrm{oblv}(M^{\prime}))\;\;\text{canonically.}$

Adding $K$-equivariance yields the desired adjunction for
$\operatorname{Hom}_{A^{\prime},K}$. The second adjunction can be deduced in a
similar way, by using standard adjunctions for dg-modules. ∎

######  Corollary 2.5.3.

The functor $A^{\prime}\underset{A}{\otimes}(\cdot)$ preserves K-projectives
and $\operatorname{Hom}^{\bullet}_{A}(A^{\prime},\cdot)^{K\text{-alg}}$
preserves K-injectives.

###### Proof.

Oblivion is exact, so the assertions follow from Proposition 2.3.10. ∎

Next, we describe the adjoint functors of oblivion in h-construction
(Definition 2.4.3).

######  Proposition 2.5.4.

Suppose that $A$ and $A^{\prime}$ are both in degree zero in the circumstance
of Proposition 2.5.2. In what follows, $M$ stands for an arbitrary h-complex
over $(A,K)$, say with $i^{M}_{\xi}\in\operatorname{End}^{-1}(M)$ in its data
($\xi\in\mathfrak{k}$).

  1. (i)

The left adjoint of

$\mathrm{oblv}:{}^{\mathrm{h}}\mathsf{C}(A^{\prime},K)\to{}^{\mathrm{h}}\mathsf{C}(A,K)$

is the dg-functor

$A^{\prime}\underset{A}{\otimes}(\cdot):{}^{\mathrm{h}}\mathsf{C}(A,K)\to{}^{\mathrm{h}}\mathsf{C}(A^{\prime},K),$

where $K$ acts diagonally on $A^{\prime}\underset{A}{\otimes}M$, and the
degree $-1$ endomorphisms $i^{\otimes}_{\xi}$ of
$A^{\prime}\underset{A}{\otimes}M$ are given by

$i^{\otimes}_{\xi}(a^{\prime}\otimes m)=a^{\prime}\otimes i^{M}_{\xi}(m).$

  2. (ii)

The right adjoint of $\mathrm{oblv}$ is the dg-functor

$\operatorname{Hom}^{\bullet}_{A}(A^{\prime},\cdot)^{K\text{-alg}}:{}^{\mathrm{h}}\mathsf{C}(A,K)\to{}^{\mathrm{h}}\mathsf{C}(A^{\prime},K),$

where $K$ acts à la (2.5.1) on
$\operatorname{Hom}^{\bullet}_{A}(A^{\prime},M)$ and the degree $-1$
endomorphisms $i^{\operatorname{Hom}}_{\xi}$ of
$\operatorname{Hom}^{\bullet}_{A}(A^{\prime},M)$ are given by

$(i_{\xi}^{\operatorname{Hom}}f)(a^{\prime})=i^{M}_{\xi}f(a^{\prime}),\quad
f\in\operatorname{Hom}^{n}_{A}(A^{\prime},M).$

###### Proof.

This is essentially a combination of Theorem 2.4.6 and Proposition 2.5.2,
since

$\displaystyle(U(\overline{\mathfrak{k}})\otimes
A^{\prime})\underset{U(\overline{\mathfrak{k}})\otimes A}{\otimes}M$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\leftarrow}}A^{\prime}\underset{A}{\otimes}M,$
$\displaystyle\operatorname{Hom}^{\bullet}_{U(\overline{\mathfrak{k}})\otimes
A}(U(\overline{\mathfrak{k}})\otimes A^{\prime},M)$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Hom}^{\bullet}_{A}(A^{\prime},M)$

as dg-modules over $A^{\prime}$, and these isomorphisms are $K$-equivariant.
It remains to identify $i_{\xi}^{\operatorname{Hom}}$ and $i_{\xi}^{\otimes}$,
and this is routine. ∎

######  Remark 2.5.5.

The functor
$\operatorname{Hom}^{\bullet}_{A}(A^{\prime},\cdot)^{K\text{-alg}}$
specializes to the co-induction in [9, 1.12] in the special case
$A=\mathbb{C}$.

Finally, we consider change of group, i.e. the case $A=A^{\prime}$. For the
sake of simplicity, we only treat the right adjoint of oblivion in the case of
h-construction.

######  Theorem 2.5.6 (P. Pandžić).

Suppose that $A$ is concentrated at degree zero. Let $T\to K$ be a
homomorphism of affine groups and assume $T$ is reductive. Then there is a
pair of adjoint functors

${\mathrm{oblv}:{}^{\mathrm{h}}\mathsf{C}(A,K)}$${{}^{\mathrm{h}}\mathsf{C}(A,T):\Gamma^{\mathrm{eq}}_{K,T}.}$

Moreover, $\Gamma^{\mathrm{eq}}_{K,T}$ upgrades to a dg-functor and induces a
$t$-exact functor between homotopy categories. The functor
$\Gamma^{\mathrm{eq}}_{K,T}$ is exact (i.e. preserves acyclicity).

###### Proof.

This is [28, Theorems 3.1.2 — 3.1.5]. The assertion that
$\Gamma^{\mathrm{eq}}_{K,T}$ upgrades to a dg-functor is contained in the
cited proofs. ∎

In [28], the functors $\Gamma^{\mathrm{eq}}_{K,T}$ are called _equivariant
Zuckerman functors_. A comparison with the classical Zuckerman functors is
given in [28, Theorem 3.3.2].

######  Corollary 2.5.7.

If $T$ is reductive, then
$\mathrm{oblv}:{}^{\mathrm{h}}\mathsf{C}(A,K)\to{}^{\mathrm{h}}\mathsf{C}(A,T)$
preserves K-projectives.

###### Proof.

Apply Proposition 2.3.10. ∎

Finally, we record the easy observation that $\mathrm{oblv}$ induces functors
between h-derived categories for all $T\to K$ (by exactness), and the diagrams

${{}^{\mathrm{h}}\mathsf{C}(A,K)}$${(A,K)\text{-}\mathsf{Mod}}$${{}^{\mathrm{h}}\mathsf{C}(A,T)}$${(A,T)\text{-}\mathsf{Mod}}$$\scriptstyle{\operatorname{H}^{n}}$$\scriptstyle{\mathrm{oblv}}$$\scriptstyle{\operatorname{H}^{n}}$
${{}^{\mathrm{h}}\mathsf{D}(A,K)}$${(A,K)\text{-}\mathsf{Mod}}$${{}^{\mathrm{h}}\mathsf{D}(A,T)}$${(A,T)\text{-}\mathsf{Mod}}$$\scriptstyle{\operatorname{H}^{n}}$$\scriptstyle{\mathrm{oblv}}$$\scriptstyle{\operatorname{H}^{n}}$
(2.5.2)

commute for all $n\in\mathbb{Z}$.

###  2.6 Invariants and co-invariants

Consider a Harish-Chandra dg-algebra $(A,K)$ together with

  * •

$\mathfrak{a}\subsetneq A$: a $K$-invariant dg-ideal;

  * •

$N\lhd K$: a subgroup acting trivially on $A/\mathfrak{a}$.

From this we obtain a morphism $(A,K)\to(A/\mathfrak{a},K/N)$ between Harish-
Chandra dg-algebras. The pull-back dg-functor in this case is called
_inflation_

$\mathrm{Infl}^{A,K}_{A/\mathfrak{a},K/N}:(A/\mathfrak{a},K/N)\text{-}\mathsf{dgMod}\to(A,K)\text{-}\mathsf{dgMod}.$

######  Proposition 2.6.1.

There exists a left (resp. right) adjoint dg-functor
$\mathrm{coInv}^{A,K}_{A/\mathfrak{a},K/N}$ (resp.
$\mathrm{Inv}^{A,K}_{A/\mathfrak{a},K/N}$) of
$\mathrm{Infl}^{A,K}_{A/\mathfrak{a},K/N}$.

###### Proof.

The construction is standard: $\mathrm{Inv}^{A,K}_{A/\mathfrak{a},K/N}$ (resp.
$\mathrm{coInv}^{A,K}_{A/\mathfrak{a},K/N}$) takes the maximal quotient dg-
module (resp. dg-submodule) on which $\mathfrak{a}$ and $N$ act trivially. ∎

The transitivity
$\mathrm{Infl}^{A,K}_{A/\mathfrak{a},K/N}=\mathrm{Infl}^{A,K}_{A/\mathfrak{a},K}\mathrm{Infl}^{A/\mathfrak{a},K}_{A/\mathfrak{a},K/N}$
implies

$\displaystyle\mathrm{Inv}^{A,K}_{A/\mathfrak{a},K/N}$
$\displaystyle\simeq\mathrm{Inv}^{A/\mathfrak{a},K}_{A/\mathfrak{a},K/N}\mathrm{Inv}^{A,K}_{A/\mathfrak{a},K},$
(2.6.1) $\displaystyle\mathrm{coInv}^{A,K}_{A/\mathfrak{a},K/N}$
$\displaystyle\simeq\mathrm{coInv}^{A/\mathfrak{a},K}_{A/\mathfrak{a},K/N}\mathrm{coInv}^{A,K}_{A/\mathfrak{a},K}.$

Note that the left and right adjoint of
$\mathrm{Infl}^{A,K}_{A/\mathfrak{a},K}$ have been given in §2.5. The right
adjoint of $\mathrm{Infl}^{A,K}_{A,K/N}$ is given in Theorem 2.5.6 when $K$ is
reductive.

######  Proposition 2.6.2.

The functors $\mathrm{coInv}^{A,K}_{A/\mathfrak{a},K/N}$ (resp.
$\mathrm{Inv}^{A,K}_{A/\mathfrak{a},K/N}$) preserves K-projectives (resp.
K-injectives).

###### Proof.

This follows from the exactness of inflation and Proposition 2.3.10. ∎

In general, $\mathrm{Inv}$ (resp. $\mathrm{coInv}$) is not exact, and one has
to consider its right (resp. left) h-derived functor, provided that
$(A,K)\text{-}\mathsf{dgMod}$ or some suitable subcategory has enough
K-injectives (resp. K-projectives). For these h-derived functors, the
adjunction to $\mathrm{Infl}^{A,K}_{A/\mathfrak{a},K/N}$ still holds on the
level of derived categories.

######  Example 2.6.3.

Let us specialize to the h-construction. Assume $A$ is in degree zero,
realized as the quotient of $B:=U(\overline{\mathfrak{k}})\otimes A$ modulo
the augmentation ideal (Theorem 2.4.6). The inflation

$\mathsf{C}(A,K)=(A,K)\text{-}\mathsf{dgMod}\to(B,K)\text{-}\mathsf{dgMod}\simeq{}^{\mathrm{h}}\mathsf{C}(A,K)$

is the obvious inclusion. The functor of invariants (resp. co-invariants)
extracts
$\bigcap_{\xi}(\operatorname{ker}(i_{\xi})\cap\operatorname{ker}(w(\xi)))$
(resp. takes the quotient modulo
$\sum_{\xi}(\operatorname{im}(i_{\xi})+\operatorname{im}(w(\xi))$).

The left h-derived functor $\operatorname{L}\\!(\mathrm{coInv})$ in this
setting played a major role in [27].

##  3 $(\mathfrak{g},K)$-modules

###  3.1 Basic definitions

Following [9], we consider the following data

  * •

$\mathfrak{g}$: finite-dimensional Lie algebra,

  * •

$K$: affine algebraic group,

  * •

$\operatorname{Ad}:K\to\operatorname{Aut}_{\text{Lie alg.}}(\mathfrak{g})$: a
homomorphism of algebraic groups,

  * •

$\iota:\mathfrak{k}\to\mathfrak{g}$: inclusion of Lie algebras,

subject to the conditions below

  * •

$\iota$ is $K$-equivariant,

  * •

$(\mathop{}\\!\mathrm{d}\operatorname{Ad}(\xi))(x)=[\iota(\xi),x]$ for all
$\xi\in\mathfrak{k}$ and $x\in\mathfrak{g}$.

Note that no grading is put on $U(\mathfrak{g})$. As $\iota$ extends to an
inclusion $U(\mathfrak{k})\to U(\mathfrak{g})$ and $\operatorname{Ad}$ extends
to an algebraic action on $U(\mathfrak{g})$, the following is obvious.

######  Proposition 3.1.1.

The quadruplet $(U(\mathfrak{g}),K,\operatorname{Ad},\iota)$ is a Harsh-
Chandra dg-algebra in the sense of Definition 2.2.2, concentrated at degree
zero.

We shall write $(\mathfrak{g},K)$ instead of $(U(\mathfrak{g}),K)$. Thus we
obtain the notions of $(\mathfrak{g},K)$-modules, weak
$(\mathfrak{g},K)$-modules, and h-complexes over $(\mathfrak{g},K)$ by the
general formalism in §2.4.

The constructions below for tensor products and internal
$\operatorname{Hom}$’s are taken from [27, p.80]. Let $M$ and $N$ be objects
of ${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)$, say with data $i^{M}_{\xi}$
and $i^{N}_{\xi}$ for all $\xi\in\mathfrak{k}$. We make $M\otimes N$ into an
h-complex by letting

$\displaystyle\eta(x\otimes y)$ $\displaystyle=\eta x\otimes y+x\otimes\eta
y,\quad\eta\in\mathfrak{g},$ $\displaystyle k(x\otimes y)$
$\displaystyle=kx\otimes ky,\quad k\in K,$ $\displaystyle i^{M\otimes
N}_{\xi}(x\otimes y)$ $\displaystyle=i^{M}_{\xi}(x)\otimes y+(-1)^{p}x\otimes
i^{N}_{\xi}(y),\quad\xi\in\mathfrak{k},$

where $x\in M^{p}$ and $y\in N^{q}$, for all $(p,q)\in\mathbb{Z}^{2}$.

Let $\mathfrak{g}$ and $K$ act on the $\operatorname{Hom}$-complex
$\operatorname{Hom}^{\bullet}_{\mathbb{C}}(M,N)$ via

$\displaystyle(\eta f)(m)=\eta(f(m))-f(\eta m),\quad\eta\in\mathfrak{g},$
$\displaystyle(kf)(m)=k(f(k^{-1}m)),\quad k\in K,$

where $f\in\operatorname{Hom}^{n}_{\mathbb{C}}(M,N)$, $n\in\mathbb{Z}$. We
take $\operatorname{Hom}^{\bullet}_{\mathbb{C}}(M,N)^{K\text{-alg}}$ and
define

$(i^{\operatorname{Hom}}_{\xi}f)(m)=i^{N}_{\xi}f(m)-(-1)^{n}f(i^{M}_{\xi}m),\quad
f\in\operatorname{Hom}^{n}_{\mathbb{C}}(M,N)^{K\text{-alg}},\quad\xi\in\mathfrak{k}.$

######  Definition 3.1.2.

The h-complex constructed above is called the _internal $\operatorname{Hom}$_
of $M$ and $N$. By taking $N=\mathbb{C}$, one arrives at the notion of
_contragredient_ h-complexes $M^{\vee}$ over $(\mathfrak{g},K)$.

The next notion applies only to $(\mathfrak{g},K)$-modules. Recall that a
$\mathcal{Z}(\mathfrak{g})$-module is called _locally
$\mathcal{Z}(\mathfrak{g})$-finite_ if it is a union of finite-dimensional
submodules.

######  Definition 3.1.3.

A $(\mathfrak{g},K)$-module $M$ is said to be a _Harish-Chandra module_ if the
following two conditions hold.

  1. (i)

$M$ is finitely generated as a $\mathfrak{g}$-module;

  2. (ii)

$M$ is locally $\mathcal{Z}(\mathfrak{g})$-finite.

Harish-Chandra modules form a Serre subcategory of
$(\mathfrak{g},K)\text{-}\mathsf{Mod}$. Cf. Lemma 5.5.1.

######  Remark 3.1.4.

In the applications to Lie groups, $K$ is often taken to be a compact Lie
group instead, and the algebraicity of the $K$-action on $M$ translates into
local $K$-finiteness; see eg. [20, p.45 and (1.64)].

From §5 onward, we will specialize to the case where
$\mathfrak{g}=\operatorname{Lie}G$ for some connected reductive group $G$ and
$K$ is a reductive subgroup of $G$; the maps $\iota$ and $\operatorname{Ad}$
will then be the obvious ones. If $K$ is a symmetric subgroup, then a
$(\mathfrak{g},K)$-module $M$ is Harish-Chandra if and only if it is finitely
generated over $\mathfrak{g}$ and admissible; see [20, Corollary 7.223] and
[33, 3.4.1 Theorem] after taking suitable real forms.

###  3.2 Standard resolutions

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. The standard complex
$N\mathfrak{g}$ is given by

$\cdots\to U(\mathfrak{g})\otimes\bigwedge^{2}\mathfrak{g}\to
U(\mathfrak{g})\otimes\bigwedge^{1}\mathfrak{g}\to
U(\mathfrak{g})\otimes\bigwedge^{0}\mathfrak{g}$

in degrees $\ldots,-2,-1,0$ and zero elsewhere. The differentials

$\partial_{n}:U(\mathfrak{g})\otimes\bigwedge^{n}\mathfrak{g}\to
U(\mathfrak{g})\otimes\bigwedge^{n-1}\mathfrak{g}$

for $n\geq 1$ are

$\displaystyle
u\otimes(\xi_{1}\otimes\cdots\otimes\xi_{n})\mapsto\sum_{i=1}^{n}(-1)^{i+1}u\xi_{i}\otimes(\xi_{1}\wedge\cdots\widehat{\xi_{i}}\cdots\wedge\xi_{n})$
$\displaystyle+\sum_{p<q}(-1)^{p+q}u\otimes([\xi_{p},\xi_{q}]\wedge\cdots\widehat{\xi_{p}}\cdots\widehat{\xi_{q}}\cdots)$

where $\widehat{\cdots}$ means terms to be neglected.

Let $U(\mathfrak{g})$ act on each
$U(\mathfrak{g})\otimes\bigwedge^{n}\mathfrak{g}$ by left multiplication, and
take the augmentation homomorphism

$\epsilon:U(\mathfrak{g})=U(\mathfrak{g})\otimes\bigwedge^{0}\mathfrak{g}\to\mathbb{C}.$

It is well known [20, Theorem 2.122] that
$\epsilon:N\mathfrak{g}\to\mathbb{C}$ furnishes a K-projective resolution of
the trivial $\mathfrak{g}$-module $\mathbb{C}$.

We now consider the data $(\mathfrak{g},K)$ of §3.1. View $\mathbb{C}$ as the
trivial $(\mathfrak{g},K)$-module.

######  Theorem 3.2.1 (P. Pandžić [28, Theorem 3.2.6]).

Let $K$ act diagonally on each term of $N\mathfrak{g}$, and define

$\displaystyle i_{\xi}:U(\mathfrak{g})\otimes\bigwedge^{n}\mathfrak{g}$
$\displaystyle\to U(\mathfrak{g})\otimes\bigwedge^{n+1}\mathfrak{g}$
$\displaystyle u\otimes\lambda$
$\displaystyle\mapsto-u\otimes(\xi\wedge\lambda)$

for all $\xi\in\mathfrak{k}$, $u\in U(\mathfrak{g})$,
$\lambda\in\bigwedge^{n}\mathfrak{g}$ and $n\geq 0$. Then $N\mathfrak{g}$ is
an h-complex over $(\mathfrak{g},K)$.

If $K$ is reductive, then $\epsilon:N\mathfrak{g}\to\mathbb{C}$ is a
K-projective resolution in the sense of h-complexes.

The following result is essentially a formal consequence of Theorem 3.2.1.

######  Theorem 3.2.2 (P. Pandžić [27, Theorem 3.5]).

Assume that $K$ is reductive. For every h-complex $M$ over $(\mathfrak{g},K)$,
the morphism $\mathrm{id}\otimes\epsilon:M\otimes N\mathfrak{g}\to M$ is a
K-projective resolution in the h-sense.

Therefore, the homotopy category
${}^{\mathrm{h}}\mathsf{K}^{\star}(\mathfrak{g},K)$ of h-complexes have enough
K-projectives for $\star\in\\{\;,+,-,\mathrm{b}\\}$ as long as $K$ is
reductive.

The upshot is that there are canonical K-projective resolutions $M\otimes
N\mathfrak{g}\to M$ for all $M$ in ${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)$
when $K$ is reductive. Moreover, these are also K-projective resolutions in
$\mathsf{C}(\mathfrak{g}\text{-}\mathsf{Mod})$; this agrees with the assertion
of Corollary 2.5.7 (take $T=\\{1\\}$).

###  3.3 Bernstein–Lunts equivalence

Let $(\mathfrak{g},K)$ be as in §3.1. Let

$\mathsf{C}(\mathfrak{g},K),\quad\mathsf{K}(\mathfrak{g},K),\quad\mathsf{D}(\mathfrak{g},K)$

be the category of complexes of $(\mathfrak{g},K)$-modules, its homotopy
category and derived category, respectively. Recall that they have the
h-avatars

${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K),\quad{}^{\mathrm{h}}\mathsf{K}(\mathfrak{g},K),\quad{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},K)$

and so on for the variants with superscripts $+,-,\mathrm{b}$ to denote full
subcategories with boundedness conditions. The comparison functor (2.4.3)
becomes

$\alpha:\mathsf{D}^{\star}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{\star}(\mathfrak{g},K),\quad\star\in\\{\;,+,-,\mathrm{b}\\}.$
(3.3.1)

The following equivalences are due to Bernstein–Lunts [9] and P. Pandžić [27].

######  Theorem 3.3.1.

The functor
$\alpha:\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)$
is an equivalence. If $K$ is reductive, then so is
$\alpha:\mathsf{D}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},K)$.
In both cases, the functors are triangulated and preserve $t$-structures.

###### Proof.

The first part is [9, Theorem 1.3] and the second is [27, Theorem 1.1].
Moreover, we have noted after (2.4.3) that $\alpha$ is triangulated and
preserves $t$-structures. ∎

Denote the $\operatorname{Ext}$ functors for $\mathsf{C}(\mathfrak{g},K)$ and
${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)$ by
$\operatorname{Ext}^{\bullet}_{\mathfrak{g},K}$ and
${}^{\mathrm{h}}\operatorname{Ext}^{\bullet}_{\mathfrak{g},K}$, respectively.
See Remark 2.4.8.

######  Corollary 3.3.2.

There are canonical isomorphisms

$\operatorname{Ext}^{n}_{\mathfrak{g},K}(M,N)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{g},K}(\alpha
M,\alpha N)$

for all objects $M,N$ of $\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)$ and all
$n\in\mathbb{Z}$, compatibly with long exact sequences for
$\operatorname{Ext}$. If $K$ is reductive, the same holds for all objects of
$\mathsf{D}(\mathfrak{g},K)$.

###### Proof.

For the left-hand side we have
$\operatorname{Ext}^{n}_{\mathfrak{g},K}(M,N)\simeq\operatorname{Hom}_{\mathsf{D}(\mathfrak{g},K)}(M,N[n])$,
and similarly for the right-hand side. ∎

In view of these results, we will often omit $\alpha$ from the formulas, and
denote the $\operatorname{Ext}$ functor for both
$\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)$ and
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)$ by
$\operatorname{Ext}^{\bullet}_{\mathfrak{g},K}$.

###  3.4 A lemma on restriction

Here we consider two pairs $(\mathfrak{g},K)$ and $(\mathfrak{h},T)$ as in
§3.1, assuming:

  * •

$\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$;

  * •

$T$ is a subgroup of $K$;

  * •

the maps $K\to\operatorname{Aut}(\mathfrak{g})$,
$T\to\operatorname{Aut}(\mathfrak{h})$, $\mathfrak{k}\to\mathfrak{g}$ and
$\mathfrak{t}\to\mathfrak{h}$ are compatible with the inclusions above.

Therefore we obtain a morphism of pairs
$(U(\mathfrak{h}),T)\to(U(\mathfrak{g}),K)$ in the sense of §2.2, and
similarly for the Harish-Chandra dg-algebras obtained by h-construction.

Below is the h-counterpart of a standard result [20, Proposition 2.57 (c)]
about restriction (or oblivion); the proof is also similar to the one in loc.
cit.

######  Lemma 3.4.1.

Suppose that $T$ is reductive. Then the functor
${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},T)$
is exact and preserves K-projectives.

###### Proof.

Exactness is known. As for the preservation of K-projectives, we break the
restriction into

${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},T)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},T).$

The first step preserves K-projectives by Corollary 2.5.7. In view of
Proposition 2.3.10, it suffices to show that
${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},T)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},T)$
has a right adjoint which is exact and upgrades to a dg-functor.

The desired right adjoint dg-functor is given by Proposition 2.5.4 (ii),
namely

$\operatorname{Hom}^{\bullet}_{U(\mathfrak{h})}(U(\mathfrak{g}),\cdot)^{T\text{-alg}}:{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},T)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},T).$

It remains to show that
$\operatorname{Hom}^{\bullet}_{U(\mathfrak{h})}(U(\mathfrak{g}),M)^{T\text{-alg}}$
is acyclic for every acyclic h-complex $M$ over $(\mathfrak{h},T)$.

By reductivity, we may choose an $T^{\circ}$-invariant subspace
$\mathfrak{q}\subset\mathfrak{g}$ under $\operatorname{Ad}$, such that
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{q}$. Next, apply [20, Lemma 2.56]
(with $L=\\{1\\}$ and swapping left and right) to obtain an isomorphism

$U(\mathfrak{h})\otimes\operatorname{Sym}(\mathfrak{q})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}U(\mathfrak{g})\quad\text{as
$\mathfrak{h}$-modules.}$

In loc. cit. the map is $u\otimes v\mapsto u\sigma(v)$ where
$\sigma:\operatorname{Sym}(\mathfrak{q})\to U(\mathfrak{g})$ is
symmetrization. Hence it is also $T^{\circ}$-equivariant.

Hence there is a $T^{\circ}$-equivariant (relative to (2.5.1)) isomorphism in
$\mathsf{C}(\mathbb{C})$:

$\displaystyle\operatorname{Hom}^{\bullet}_{U(\mathfrak{h})}(U(\mathfrak{g}),M)^{T^{\circ}\text{-alg}}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Hom}^{\bullet}_{\mathbb{C}}(\operatorname{Sym}(\mathfrak{q}),M)^{T^{\circ}\text{-alg}}$
$\displaystyle\varphi$
$\displaystyle\mapsto\varphi|_{1\otimes\operatorname{Sym}(\mathfrak{q})}.$

Consider any algebraic representation $W$ of $T^{\circ}$ and a short exact
sequence $0\to V^{\prime}\to V\to V^{\prime\prime}\to 0$ of algebraic
representations of $T^{\circ}$. Taking a compact real form of $T^{\circ}$ and
passing from algebraic to locally $T^{\circ}(\mathbb{R})$-finite modules, by
[20, Proposition 1.18b] the short exact sequence splits. As a consequence,

$0\to\operatorname{Hom}_{\mathbb{C}}(W,V^{\prime})^{T^{\circ}\text{-alg}}\to\operatorname{Hom}_{\mathbb{C}}(W,V)^{T^{\circ}\text{-alg}}\to\operatorname{Hom}_{\mathbb{C}}(W,V^{\prime\prime})^{T^{\circ}\text{-alg}}\to
0$

is acyclic in $\mathsf{C}(\mathbb{C})$.

From this fact,
$\operatorname{Hom}^{\bullet}_{\mathbb{C}}(\operatorname{Sym}(\mathfrak{q}),M)^{T^{\circ}\text{-alg}}$
and
$\operatorname{Hom}^{\bullet}_{U(\mathfrak{h})}(U(\mathfrak{g}),M)^{T^{\circ}\text{-alg}}$
are seen to be acyclic. It remains to recall that
$(\cdots)^{T\text{-alg}}\simeq(\cdots)^{T^{\circ}\text{-alg}}$. ∎

##  4 $D$-modules

###  4.1 Basic definitions

Let $K$ be an affine algebraic group. A smooth variety $X$ is said to be a
_$K$ -variety_ if $K$ acts algebraically on the right of $X$. Morphisms of
$K$-varieties are defined to be $K$-equivariant morphisms of varieties.

Let $X$ be a smooth variety. Unless otherwise specified, $D_{X}$-modules
(resp. $\mathscr{D}_{X}$-modules) will mean left modules (resp.
$\mathscr{O}_{X}$-quasi-coherent left modules). We denote by

$\mathsf{D}^{\star}(X),\quad\star\in\\{\;,+,-,\mathrm{b}\\}$

the derived category of $\mathscr{D}_{X}$-modules with boundedness condition
$\star$; for affine $X$, they are just
$\mathsf{D}^{\star}(D_{X}\text{-}\mathsf{Mod})$.

If $X$ is a smooth affine $K$-variety, then we obtain the homomorphism

$j:U(\mathfrak{k})\to D_{X}$

of algebras, mapping $\xi\in\mathfrak{k}$ to the vector field on $X$ generated
by $\xi$. On the other hand, $K$ acts on $D_{X}$ in the following way. The
right $K$-action on $X$ induces a left $K$-action on regular functions; let
$P\in D_{X}$ and $k\in K$, then ${}^{k}P\in D_{X}$ is the operator

$\varphi\mapsto k(P(k^{-1}\varphi))$

for all regular function $\varphi$. It belongs to $D_{X}$: in fact ${}^{k}P$
is just the transport of structure by $k$ acted on $P$. Denote by
$\sigma:K\to\operatorname{Aut}(D_{X})$ the resulting homomorphism.

######  Proposition 4.1.1.

Give a smooth affine $K$-variety $X$, the quadruplet $(D_{X},K,\sigma,j)$ is a
Harsh-Chandra dg-algebra in the sense of Definition 2.2.2, concentrated at
degree zero.

###### Proof.

This is stated in [9, 2.1] (see the references therein). See also [25]. ∎

Fix a smooth affine $K$-variety $X$ hereafter. We obtain the notions of
$(D_{X},K)$-modules, weak $(D_{X},K)$-modules, and h-complexes over
$(D_{X},K)$ by the general formalism in §2.4. In particular, we obtain the
categories

${}^{\mathrm{h}}\mathsf{C}(D_{X},K),\quad{}^{\mathrm{h}}\mathsf{K}(D_{X},K),\quad{}^{\mathrm{h}}\mathsf{D}(D_{X},K)$

as well as the versions with boundedness conditions.

In comparison with the standard terminologies, we have:

  * •

$(D_{X},K)$-modules are exactly the (strongly) _$K$ -equivariant
$D_{X}$-modules_;

  * •

weak $(D_{X},K)$-modules are exactly the _weakly $K$-equivariant
$D_{X}$-modules_.

See also the explanations after [25, Definition 2.2].

For all subgroup $T$ of $K$ and $n\in\mathbb{Z}$, it is clear that

${{}^{\mathrm{h}}\mathsf{D}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{D}(D_{X},T)}$${(D_{X},K)\text{-}\mathsf{Mod}}$${(D_{X},T)\text{-}\mathsf{Mod}}$$\scriptstyle{\operatorname{H}^{n}}$$\scriptstyle{\operatorname{H}^{n}}$

commutes, where the horizontal arrows are the evident ones, and the upper one
is $t$-exact triangulated. Taking $T=\\{1\\}$, the second column becomes
$\mathsf{D}(X)\xrightarrow{\operatorname{H}^{n}}D_{X}\text{-}\mathsf{Mod}$.

######  Remark 4.1.2.

If $x\in X$ is fixed by $K$, we can also talk about $(D_{X,x},K)$-modules,
etc., where $D_{X,x}$ is the stalk of $D_{X}$ at $x$. Taking stalks yield the
dg-functors

${}^{\mathrm{h}}\mathsf{C}(D_{X},K)\to{}^{\mathrm{h}}\mathsf{C}(D_{X,x},K)$

and so forth. In fact it is $D_{X,x}\underset{D_{X}}{\otimes}(\cdot)$; it
preserves K-projectives by Corollary 2.5.3.

######  Remark 4.1.3.

The assumption that $X$ is affine is artificial here. It is clear that the
whole formalism can be sheafified. For any smooth $K$-variety $X$, there is a
theory of $(\mathscr{D}_{X},K)$-modules, weak $(\mathscr{D}_{X},K)$-modules,
h-complexes over $(\mathscr{D}_{X},K)$ and the corresponding derived
categories. See [9, 2.16.1], or [19] for partial flag varieties. Nevertheless,
this level of generality is needed in this work.

Several constructions from the theory of $D$-modules generalize to the
h-setting. For example, the same ideas from §3.2 upgrades the _Spencer
complex_ [15, (1.5.6)]

$\cdots\to\mathscr{D}_{X}\underset{\mathscr{O}_{X}}{\otimes}\bigwedge^{2}\mathscr{T}_{X}\to\mathscr{D}_{X}\underset{\mathscr{O}_{X}}{\otimes}\bigwedge^{1}\mathscr{T}_{X}\to\mathscr{D}_{X}\underset{\mathscr{O}_{X}}{\otimes}\bigwedge^{0}\mathscr{T}_{X}$

into an h-complex over $(\mathscr{D}_{X},K)$, where $\mathscr{T}_{X}$ stands
for the tangent sheaf of $X$. Denote this h-complex as $NX$, then the morphism
$\mathscr{D}_{X}\to\mathscr{O}_{X}$ given by $D\mapsto D(1)$ yields a quasi-
isomorphism $NX\to\mathscr{O}_{X}$ in
${}^{\mathrm{h}}\mathsf{C}(\mathscr{D}_{X},K)$.

###  4.2 Inverse images

There are two basic operations on derived categories of $D$-modules: inverse
image (suitably shifted) and de Rham push-forward. The goal here is to define
inverse images for h-complexes on smooth affine $K$-varieties. We do not
consider de Rham push-forward in this work.

For every morphism $f:X\to Y$ between smooth varieties, we set

$\mathrm{rd}(f):=\dim X-\dim Y.$

Assuming that there are enough flat $\mathscr{D}_{Y}$-modules (eg. when $Y$ is
quasi-projective), the basic theory of $D$-modules provides a functor

$f^{\bullet}=f^{!}[-\mathrm{rd}(f)]:\mathsf{D}^{-}(Y)\to\mathsf{D}^{-}(X).$

  * •

The $f^{\bullet}$ above is the inverse image functor denoted by
$\operatorname{L}\\!f^{*}$ in [15, p.33], the left derived functor of
$\mathscr{D}_{X\to Y}\underset{f^{-1}\mathscr{D}_{Y}}{\otimes}f^{-1}(\cdot)$;

  * •

The $f^{!}$ is denoted by $f^{\dagger}$ in loc. cit., and $f^{!}$ corresponds
to the $!$-pullback under Riemann–Hilbert correspondence when applied to
regular holonomic complexes.

The effect of $\mathscr{D}_{X\to
Y}\underset{f^{-1}\mathscr{D}_{Y}}{\otimes}f^{-1}(\cdot)$ is to take the
pullback of a $\mathscr{D}_{Y}$-module as a quasi-coherent sheaf, and then
equip it with a natural $\mathscr{D}_{X}$-module structure.

If $f$ is a morphism between $K$-varieties, then $\mathscr{D}_{X\to Y}$ is
$K$-equivariant, and this functor can be lifted to the level of h-complexes,
cf. Proposition 2.5.4 (i). The point is to take its left h-derived functor.
This splits into two cases.

Smoooth inverse image

Suppose $f$ is smooth, then $\mathscr{D}_{X\to
Y}\underset{f^{-1}\mathscr{D}_{Y}}{\otimes}f^{-1}(\cdot)$ preserves acyclic
objects, hence it induces a $t$-exact functor

$f^{\bullet}:{}^{\mathrm{h}}\mathsf{D}(D_{Y},K)\to{}^{\mathrm{h}}\mathsf{D}(D_{X},K)$

as well as $f^{!}:=f^{\bullet}[\mathrm{rd}(f)]$.

General inverse image

For general $f$, we shall assume $K$ is reductive. Theorem 2.3.7 ensures that
${}^{\mathrm{h}}\mathsf{C}(D_{Y},K)$ has enough K-projectives. Hence we may
define the left h-derived functor

$f^{\bullet}:{}^{\mathrm{h}}\mathsf{D}^{-}(D_{Y},K)\to{}^{\mathrm{h}}\mathsf{D}^{-}(D_{X},K)$

and $f^{!}:=f^{\bullet}[\mathrm{rd}(f)]$. Note that $f^{\bullet}$ can be
computed from K-flat resolutions, and K-projective implies K-flat (cf. [34,
Proposition 10.3.4]).

Clearly, these two definitions agree in overlapping cases. The following
results are trivial for smooth inverse images, thus we state them only for the
general ones.

######  Lemma 4.2.1.

Suppose $K$ is reductive. Consider morphisms
$X\xrightarrow{f}Y\xrightarrow{g}Z$ of smooth affine $K$-varieties. There is a
canonical isomorphism $(gf)^{\bullet}\simeq f^{\bullet}g^{\bullet}$.

###### Proof.

Analogous to the non-equivariant version; see the proof of [15, Propositionp
1.5.11]. It boils down to standard properties of derived tensor products, but
for our setting of h-complexes, the dg-counterparts are needed (cf. [34,
§12.3]). ∎

######  Lemma 4.2.2.

Suppose $K$ is reductive. Let $f:X\to Y$ be a morphism between smooth affine
$K$-varieties. Then $f^{\bullet}$ restricts to
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{Y},K)\to{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)$.

###### Proof.

It suffices to show that $f^{\bullet}$ has bounded amplitude. The functor of
forgetting $K$-equivariance

${}^{\mathrm{h}}\mathsf{C}(D_{Y},K)\to{}^{\mathrm{h}}\mathsf{C}(D_{Y},\\{1\\})=\mathsf{C}(D_{Y}\text{-}\mathsf{Mod})$

is exact and preserves K-projectives by Corollary 2.5.7. Ditto for $Y$
replaced by $X$. Therefore the left square of the diagram

${{}^{\mathrm{h}}\mathsf{D}^{-}(D_{Y},K)}$${{}^{\mathrm{h}}\mathsf{D}^{-}(D_{X},K)}$${(D_{X},K)\text{-}\mathsf{Mod}}$${\mathsf{D}^{-}(Y)}$${\mathsf{D}^{-}(X)}$${D_{X}\text{-}\mathsf{Mod}}$$\scriptstyle{f^{\bullet}}$$\scriptstyle{\operatorname{H}^{n}}$$\scriptstyle{f^{\bullet}}$$\scriptstyle{\operatorname{H}^{n}}$

commutes up to a canonical isomorphism — to see this, compute left h-derived
functors via Corollary 2.3.9. So does the right square by (2.5.2). The
amplitude of the upper $f^{\bullet}$ can thus be bounded by that of the lower
one, which is finite by [15, Corollary 1.4.20]. ∎

The non-affine case is not needed in this work, and is left to the interested
reader.

###  4.3 Comparison with the equivariant derived category

To begin with, suppose $X$ is a smooth $K$-variety.

######  Definition 4.3.1.

Let $\mathsf{D}^{\mathrm{b}}_{K}(X)$ be the _bounded equivariant derived
category_ of $\mathscr{D}_{X}$-modules under $K$-action defined by Bernstein
and Lunts [8, 4.2]. It is a triangulated category endowed with a
$t$-structure, whose heart is the category of $K$-equivariant
$\mathscr{D}_{X}$-modules.

For details about the definition of $\mathsf{D}^{\mathrm{b}}_{K}(X)$, we refer
to [8], [9, 2.11], or to the more systematic [1, Chapter 6] in the setting of
constructible sheaves. Below is only a sketch.

######  Definition 4.3.2.

A _resolution_ of a $K$-variety $X$ is a pair $(P,p)$ where $P$ is a
$K$-torsor and $p:P\to X$ is a smooth affine $K$-equivariant morphism. We also
set $\overline{P}:=P/K$. Morphisms $f:(Q,q)\to(P,p)$ are defined as
commutative diagrams

${Q}$${P}$${X}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{p}$

where $f$ is a smooth $K$-equivariant morphism. From $f$ we obtain a smooth
morphism $\overline{f}:\overline{Q}\to\overline{P}$.

######  Example 4.3.3.

The _trivial resolution_ of $X$ is $P:=X\times K$ and $p:=\mathrm{pr}_{1}$,
where $K$ acts on $P$ by $(x,h)k=(xk,k^{-1}h)$. Then
$X\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\overline{P}$ by mapping $x\in
X$ to the orbit of $(x,1)$.

For the construction of sufficiently many resolutions of $X$ (namely,
$n$-acyclic ones [1, Definition 6.1.18] for every $n\in\mathbb{Z}_{\geq 0}$),
we refer to [1, §6.1].

  * •

The objects in $\mathsf{D}^{\mathrm{b}}_{K}(X)$ are collections $M$ of objects
$M_{P}$ in $\mathsf{D}^{\mathrm{b}}(\overline{P})$ for every resolution
$(P,p)$, together with isomorphisms

$\alpha_{f}:\overline{f}^{\bullet}M_{P}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M_{Q}$

for each morphism $f:(Q,q)\to(P,p)$, compatibly with compositions in the sense
that

$\alpha_{g}\circ\overline{g}^{\bullet}(\alpha_{f})=\alpha_{fg}.$

  * •

The morphisms $M\to N$ in $\mathsf{D}^{\mathrm{b}}_{K}(X)$ are collections of
morphisms $\varphi_{P}:M_{P}\to N_{P}$ compatibly with various $\alpha_{f}$.

For the description of the following structures of
$\mathsf{D}^{\mathrm{b}}_{K}(X)$, see also see [1, pp.286–287]; note that one
works with constructible sheaves in loc. cit. and the smooth inverse images
differ by a shift by $\mathrm{rd}(f)$.

Forgetting equivariance

The functor
$\mathbf{oblv}:\mathsf{D}^{\mathrm{b}}_{K}(X)\to\mathsf{D}^{\mathrm{b}}(X)$
sends an object $M$ to $M_{X\times K}$, by taking the trivial resolution.

Translation functor

Define $M[1]$ to be the collection $(M_{P}[1])_{(P,p)}$.

Distinguished triangles

A triangle $M_{1}\to M_{2}\to M_{3}\xrightarrow{+1}$ is said to be
distinguished if $M_{1,P}\to M_{2,P}\to M_{3,P}\xrightarrow{+1}$ is, for every
resolution $(P,p)$.

Bounded $t$-structure

For any bounded interval $I$, an object $M$ lies in $\mathsf{D}^{I}_{K}(X)$ if
and only if $\mathbf{oblv}(M)$ lies in $\mathsf{D}^{I}(X)$.

More generally, let $T\subset K$ be a subgroup. There is a $t$-exact functor

$\mathbf{oblv}^{K}_{T}:\mathsf{D}^{\mathrm{b}}_{K}(X)\to\mathsf{D}^{\mathrm{b}}_{T}(X)$

whose definition à la [1, Definition 6.5.2] is sketched below.

Any resolution $(P,p)$ can be viewed as a resolution of $X$ as a $T$-variety.
Let $\mathsf{D}^{\mathrm{b}}_{T\subset K}(X)$ be the category defined in the
same way as $\mathsf{D}^{\mathrm{b}}_{T}(X)$, but allowing only the
resolutions so obtained. This gives rise to a functor

$\mathbf{oblv}^{K}_{T\subset
K}:\mathsf{D}^{\mathrm{b}}_{K}(X)\to\mathsf{D}^{\mathrm{b}}_{T\subset K}(X).$

On the other hand, [1, Lemmas 6.4.7, 6.4.8] realizes
$\mathsf{D}^{\mathrm{b}}_{T}(X)$ as a full subcategory of
$\mathsf{D}^{\mathrm{b}}_{T\subset K}(X)$. All these constructions being
triangulated and $t$-exact, one can verify that $\mathbf{oblv}^{K}_{T\subset
K}$ lands in $\mathsf{D}^{\mathrm{b}}_{T}(X)$ by checking on hearts, as in [1,
p.292]. This yields $\mathbf{oblv}^{K}_{T}$.

Moreover, it is proved in loc. cit. that
$\mathsf{D}^{\mathrm{b}}_{\\{1\\}}(X)\simeq\mathsf{D}^{\mathrm{b}}(X)$ and
$\mathbf{oblv}^{K}_{\\{1\\}}\simeq\mathbf{oblv}$.

######  Theorem 4.3.4 (A. Beilinson, see [9, Theorem 2.13]).

Let $X$ be a smooth affine $K$-variety. There is an equivalence between
triangulated categories

$\varepsilon:{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)\to\mathsf{D}^{\mathrm{b}}_{K}(X)$

with the following properties.

  1. (i)

For every subgroup $T\subset K$, the diagram

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${\mathsf{D}^{\mathrm{b}}_{K}(X)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},T)}$${\mathsf{D}^{\mathrm{b}}_{T}(X)}$$\scriptstyle{\varepsilon}$$\scriptstyle{\mathbf{oblv}^{K}_{T}}$$\scriptstyle{\varepsilon}$

commutes up to isomorphism; in particular, $\varepsilon$ commutes with
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)\to\mathsf{D}^{\mathrm{b}}(X)$
and $\mathbf{oblv}$.

  2. (ii)

It is $t$-exact.

  3. (iii)

It induces identity on hearts, which are the category of equivariant
$D_{X}$-modules on both sides.

###### Proof.

The equivalence is stated in loc. cit.; the case here is simpler since $X$ is
affine and we do not consider monodromic structures yet. Let us sketch the
definition of $\varepsilon$ now. Given a resolution $(P,p)$, consider the
functors

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{P},K)}$${\mathsf{D}^{\mathrm{b}}((D_{P},K)\text{-}\mathsf{Mod})}$${\mathsf{D}^{\mathrm{b}}(\overline{P}).}$$\scriptstyle{p^{\bullet}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(4.3.1)

The middle equivalence is the functor $\alpha$ in the step 2 of proof of [9,
Theorem 2.13]. The rightmost equivalence comes from
$D_{\overline{P}}\text{-}\mathsf{Mod}\simeq(D_{P},K)\text{-}\mathsf{Mod}$, as
$P\to\overline{P}$ is a $K$-torsor; it is also the functor in the step 1 of
the cited proof.

Suppose that an object $M$ of
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)$ is given. When $(P,p)$
varies, we obtain a compatible collection of objects of
$\mathsf{D}^{\mathrm{b}}(\overline{P})$. Hence (4.3.1) yields the functor
$\varepsilon$. It is triangulated since all the functors in (4.3.1) are.

Consider the property (i). Given $T\subset K$, define $\varepsilon_{T\subset
K}:{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},T)\to\mathsf{D}^{\mathrm{b}}_{T\subset
K}(X)$ by the same recipe (4.3.1), but allowing only resolutions restricted
from $K$. By the earlier review on $\mathbf{oblv}^{K}_{T}$, it suffices to
commute

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${\mathsf{D}^{\mathrm{b}}_{K}(X)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},T)}$${\mathsf{D}^{\mathrm{b}}_{T\subset
K}(X)}$$\scriptstyle{\varepsilon}$$\scriptstyle{\mathbf{oblv}^{K}_{T\subset
K}}$$\scriptstyle{\varepsilon_{T\subset K}}$

up to isomorphism. But this is immediate by construction.

As $\varepsilon$ is seen to commute with
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)\to\mathsf{D}^{\mathrm{b}}(X)$
and $\mathbf{oblv}$, it is also $t$-exact, whence (ii).

Consider (iii). If $M$ is an equivariant $D_{X}$-module, then its image under
(4.3.1) is the descent to $\overline{P}$ of its inverse image in
$(D_{P},K)\text{-}\mathsf{Mod}$. When $(P,p)$ varies, such a collection also
gives rise to an object in the heart of $\mathsf{D}^{\mathrm{b}}_{K}(X)$ that
corresponds to $M$; for details, see [1, Theorem 6.4.10] and its proof. Beware
of the shifts in the constructible setting considered in [1]. ∎

######  Corollary 4.3.5.

Let $X$ be a smooth affine $K$-variety. There are canonical isomorphisms

${}^{\mathrm{h}}\operatorname{Ext}^{n}_{D_{X},K}(M,N)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Ext}^{n}_{\mathsf{D}^{\mathrm{b}}_{K}(X)}(\varepsilon
M,\varepsilon N)$

for all objects $M,N$ of ${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)$ and
all $n\in\mathbb{Z}$, compatibly with long exact sequences for
$\operatorname{Ext}$.

###### Proof.

Analogous to Corollary 3.3.2. ∎

The inverse image functor is also defined on equivariant derived categories,
denoted by the same symbol $f^{\bullet}$. They match the h-counterpart under
$\varepsilon$ by the next result.

######  Proposition 4.3.6.

Assume that $K$ is reductive. Let $f:X\to Y$ be a morphism between smooth
affine $K$-varieties. The following diagram commutes up to a canonical
isomorphism:

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{Y},K)}$${\mathsf{D}^{\mathrm{b}}_{K}(Y)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${\mathsf{D}^{\mathrm{b}}_{K}(X).}$$\scriptstyle{\varepsilon}$$\scriptstyle{f^{\bullet}}$$\scriptstyle{f^{\bullet}}$$\scriptstyle{\varepsilon}$

###### Proof.

Given a resolution $q:Q\to Y$, we form the Cartesian square

${P}$${Q}$${X}$${Y}$$\scriptstyle{\tilde{f}}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{f}$

then $p$ is also a resolution. Let $\overline{f}:\overline{P}\to\overline{Q}$
denote the induced morphism. Then

$\mathrm{rd}(p)=\mathrm{rd}(q),\quad\mathrm{rd}(\overline{f})=\mathrm{rd}(f)=\mathrm{rd}(\tilde{f}).$

Recall the construction (4.3.1) of $\varepsilon$, and consider the following
diagram

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{Y},K)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{Q},K)}$${\mathsf{D}^{\mathrm{b}}((D_{Q},K)\text{-}\mathsf{Mod})}$${\mathsf{D}^{\mathrm{b}}(\overline{Q})}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{P},K)}$${\mathsf{D}^{\mathrm{b}}((D_{P},K)\text{-}\mathsf{Mod})}$${\mathsf{D}^{\mathrm{b}}(\overline{P}).}$$\scriptstyle{f^{\bullet}}$$\scriptstyle{q^{\bullet}}$$\scriptstyle{\tilde{f}^{\bullet}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\overline{f}^{\bullet}}$$\scriptstyle{p^{\bullet}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

The left square commutes by Lemma 4.2.1. As for the right rectangle, consider
a bounded complex $\overline{M}$ of $D_{\overline{Q}}$-modules. We may and do
take a quasi-isomorphism $\overline{F}\to\overline{M}$ such that
$\overline{F}$ is K-flat when viewed over
$O_{\overline{Q}}:=\Gamma(\overline{Q},\mathscr{O}_{\overline{Q}})$. Then
their inverse images as complexes of $(D_{Q},K)$-modules $F\to M$ is still a
quasi-isomorphism, and $F$ is K-flat over $O_{Q}$ since

$F\underset{O_{Q}}{\otimes}(\cdot)\simeq\overline{F}\underset{O_{\overline{Q}}}{\otimes}O_{Q}\underset{O_{Q}}{\otimes}(\cdot)\simeq\overline{F}\underset{O_{\overline{Q}}}{\otimes}(\cdot).$

The image of $F$ in ${}^{\mathrm{h}}\mathsf{C}(D_{Q},K)$ can be used to
compute $\tilde{f}^{\bullet}$, whilst $\overline{F}$ can be used to compute
$\overline{f}^{\bullet}$. This concludes the commutativity since

${Q}$${\overline{Q}}$${P}$${\overline{P}}$$\scriptstyle{f}$$\scriptstyle{\overline{f}}$
commutes.

When $(Q,q)$ varies, the resulting $(P,p)$ determine objects of
$\mathsf{D}^{\mathrm{b}}_{K}(X)$ by [1, Lemma 6.4.8]. The assignment

$\left(M_{Q},\ldots\right)_{(Q,q)}\mapsto\left(\overline{f}^{\bullet}M_{Q},\ldots\right)_{(P,p)},$

where $M_{Q}$ are objects of $\mathsf{D}^{\mathrm{b}}(\overline{Q})$, is
exactly the recipe of Bernstein–Lunts [8] for defining $f^{\bullet}$ or its
shift $f^{!}$ on equivariant derived categories; see [1, §6.5]. This completes
the proof. ∎

Finally, suppose that $N\lhd K$ and $X$ is a smooth affine $K/N$-variety.
There is an inflation functor

$\mathrm{Infl}^{K}_{K/N}:\mathsf{D}^{\mathrm{b}}_{K/N}(X)\to\mathsf{D}^{\mathrm{b}}_{K}(X).$

The construction is based on the fact that if $p:P\to X$ is a resolution as
$K$-varieties, then so is its quotient $p^{\prime}:P^{\prime}:=P/N\to X$ as
$K/N$-varieties, and
$\overline{P}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\overline{P^{\prime}}$.
It is $t$-exact.

######  Proposition 4.3.7.

The diagram

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K/N)}$${\mathsf{D}^{\mathrm{b}}_{K/N}(X)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${\mathsf{D}^{\mathrm{b}}_{K}(X)}$$\scriptstyle{\mathrm{Infl}^{K}_{K/N}}$$\scriptstyle{\varepsilon}$$\scriptstyle{\mathrm{Infl}^{K}_{K/N}}$$\scriptstyle{\varepsilon}$

commutes up to isomorphism, where the
$\mathrm{Infl}^{K}_{K/N}:=\mathrm{Infl}^{D_{X},K}_{D_{X},K/N}$ on the left is
defined in §2.6.

###### Proof.

Given a resolution $(P,p)$ as $K$-varieties, define $(P^{\prime},p^{\prime})$
as above. By the construction (4.3.1) of $\varepsilon$, it suffices to check
the commutativity of

${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K/N)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{P^{\prime}},K/N)}$${\mathsf{D}^{\mathrm{b}}((D_{P^{\prime}},K/N)\text{-}\mathsf{Mod})}$${\mathsf{D}^{\mathrm{b}}(\overline{P^{\prime}})}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{P},K)}$${\mathsf{D}^{\mathrm{b}}((D_{P},K)\text{-}\mathsf{Mod})}$${\mathsf{D}^{\mathrm{b}}(\overline{P})}$$\scriptstyle{(p^{\prime})^{\bullet}}$$\scriptstyle{\mathrm{Infl}^{K}_{K/N}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{p^{\bullet}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

up to isomorphism, where the vertical arrows except the first one are inverse
images via $P\to P^{\prime}$ and
$\overline{P}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\overline{P^{\prime}}$.
This is clear. ∎

##  5 Localization and higher regularity

###  5.1 Derived localization functor

Let $G$ be a connected reductive group, and let $K\subset G$ be a reductive
subgroup. Let $X$ be an affine smooth $G$-variety.

From this we obtain a pair $(\mathfrak{g},K)$ as in §3.1: the homomorphism
$\operatorname{Ad}:K\to\operatorname{Aut}(\mathfrak{g})$ is just the adjoint
action, and $\iota:\mathfrak{k}\to\mathfrak{g}$ is the inclusion. Recall that
we write $(\mathfrak{g},K)$ instead of $(U(\mathfrak{g}),K)$ for the pairs.

On the other hand, we also obtain $(D_{X},G)$ and its subpair $(D_{X},K)$. The
constructions in §4.1 are applicable.

######  Proposition 5.1.1.

The map $j:U(\mathfrak{g})\to D_{X}$ induced from the $G$-action on $X$
induces a morphism $(\mathfrak{g},K)\to(D_{X},K)$ of pairs in the sense of
§2.2.

###### Proof.

The map $j$ is $K$-equivariant since $K\subset G$. The requirement that
$\mathfrak{k}\to U(\mathfrak{g})\xrightarrow{j}D_{X}$ composes to
$j|_{\mathfrak{k}}:\mathfrak{k}\to D_{X}$ is trivial. ∎

Therefore we have the change-of-algebra dg-functor in Proposition 2.5.4 (i):

$D_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot):{}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{C}(D_{X},K).$

Being left adjoint to oblivion, it preserves K-projectives.

######  Definition 5.1.2.

Denote by
$\mathbf{Loc}_{X}=\mathbf{Loc}_{X,K}:{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}(D_{X},K)$
the h-derived functor of $D_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot)$,
called the _h-derived localization functor_.

Since $K$ is reductive, the definition makes sense:
${}^{\mathrm{h}}\mathsf{K}(\mathfrak{g},K)$ has enough K-projectives, namely
the ones from standard resolutions. The functor $\mathbf{Loc}_{X}$ is right
$t$-exact.

######  Remark 5.1.3.

For general smooth $G$-variety, one can also define $\mathbf{Loc}_{X}$ with
the sheafified pair $(\mathscr{D}_{X},K)$. The Beilinson–Bernstein
localization corresponds to the case when $X=\mathcal{B}$ is the flag variety,
allowing twisted differential operators, and considering only
$\operatorname{H}^{0}\mathbf{Loc}_{\mathcal{B}}$.

######  Proposition 5.1.4.

The triangulated functor $\mathbf{Loc}_{X}$ has amplitude in $[-\dim G,0]$. In
particular, it restricts to

$\mathbf{Loc}_{X}:{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X},K).$

###### Proof.

Use standard resolutions (Theorem 3.2.2). ∎

######  Proposition 5.1.5.

Let $T$ be a reductive subgroup of $K$. The diagram below commutes up to
canonical isomorphism:

${{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},K)}$${{}^{\mathrm{h}}\mathsf{D}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},T)}$${{}^{\mathrm{h}}\mathsf{D}(D_{X},T).}$$\scriptstyle{\mathbf{Loc}_{X,K}}$$\scriptstyle{\mathbf{Loc}_{X,T}}$

In particular, the same holds for

${{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},K)}$${{}^{\mathrm{h}}\mathsf{D}(D_{X},K)}$${\mathsf{D}(\mathfrak{g}\text{-}\mathsf{Mod})}$${\mathsf{D}(X).}$$\scriptstyle{\mathbf{Loc}_{X,K}}$$\scriptstyle{D_{X}\underset{U(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}(\cdot)}$

###### Proof.

The starting point is the commutativity of

${{}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)}$${{}^{\mathrm{h}}\mathsf{C}(D_{X},K)}$${{}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},T)}$${{}^{\mathrm{h}}\mathsf{C}(D_{X},T).}$

The vertical arrows are exact and preserve K-projectives by Corollary 2.5.7.
Taking left h-derived functors via Corollary 2.3.9 gives the desired result. ∎

######  Proposition 5.1.6.

Let $V$ be a $(\mathfrak{g},K)$-module. There is a canonical isomorphism of
$K$-equivariant $D_{X}$-modules

$\operatorname{H}^{0}\mathbf{Loc}_{X}(V)\simeq
D_{X}\underset{U(\mathfrak{g})}{\otimes}V;$

on the right-hand side, $D_{X}$ acts by left multiplication and $K$ acts
diagonally.

###### Proof.

As $K$ is reductive, one can take the standard resolution $P:=V\otimes
N\mathfrak{g}\to V$. Now take the h-complex
$D_{X}\underset{U(\mathfrak{g})}{\otimes}P$. Its $\operatorname{H}^{0}$ yields
$D_{X}\underset{U(\mathfrak{g})}{\otimes}V$. ∎

In view of this result, $\mathbf{Loc}_{X}$ or its cohomologies can be seen as