“higher localization”.

The localization functor has the following symmetries. Let $D_{X}^{G}$ be the
algebra of $G$-invariant differential operators on $X$. For every $z\in
D_{X}^{G}$ and every h-complex $P$ over $(D_{X},K)$, we obtain an endomorphism
$\mathcal{R}_{z}$ of the h-complex $D_{X}\underset{U(\mathfrak{g})}{\otimes}P$
given by

$\displaystyle\mathcal{R}_{z}^{n}:D_{X}\underset{U(\mathfrak{g})}{\otimes}P^{n}$
$\displaystyle\to D_{X}\underset{U(\mathfrak{g})}{\otimes}P^{n}$
$\displaystyle D\otimes p$ $\displaystyle\mapsto Dz\otimes p.$

This is functorial in $P$ and passes to the h-derived category, giving rise to
a homomorphism of algebras

$\mathcal{R}:(D_{X}^{G})^{\mathrm{op}}\to\operatorname{End}_{\mathrm{functors}}(\mathbf{Loc}_{X}).$
(5.1.1)

In fact, the construction works over any $G$-variety $X$ in the sheafified
context. It is also compatible with oblivion, relative to the diagram in
Proposition 5.1.5.

###  5.2 Localization and co-invariants

Consider a homogeneous $G$-space $G=H\backslash G$, and let $x$ be the point
$H\cdot 1$ of $X$.

The arguments in §6 will rely crucially on the following fact relating
localizations and co-invariants. We do not assume $G$ is reductive or $X$ is
affine here.

######  Proposition 5.2.1.

For every $\mathfrak{g}$-module $V$, we have the canonical isomorphism

$\displaystyle V/\mathfrak{h}V$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{C}\underset{\mathscr{O}_{x}}{\otimes}\left(\mathscr{D}_{X}\underset{U(\mathfrak{g})}{\otimes}V\right)_{x}$
$\displaystyle v+\mathfrak{h}V$ $\displaystyle\mapsto 1\otimes(1\otimes v),$

where $\mathscr{O}_{x}$ is the local ring at $x$ and the $\otimes$ is relative
to the evaluation map at $x$.

###### Proof.

See [6, Lemma 2.2]. ∎

Let $K^{H}$ be a subgroup of $H$, acting trivially on $\mathbb{C}$. Several
observations are in order.

  * •

If $V$ is a weak $(\mathfrak{g},K^{H})$-module, the isomorphism in Proposition
5.2.1 is $K^{H}$-equivariant. Thus we obtain an isomorphism between two dg-
functors from ${}^{\mathrm{w}}\mathsf{C}(\mathfrak{g},K^{H})$ to
${}^{\mathrm{w}}\mathsf{C}(\mathbb{C},K^{H})$.

  * •

Furthermore, the functors $V\mapsto V/\mathfrak{h}V$ and
$V\mapsto\mathbb{C}\underset{\mathscr{O}_{x}}{\otimes}\left(\mathscr{D}_{X}\underset{U(\mathfrak{g})}{\otimes}V\right)_{x}$
both lift to the level of h-complexes

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

In fact, the dg-functor induced by $V\mapsto V/\mathfrak{h}V$ equals the
$\mathrm{coInv}^{U(\mathfrak{h}),K^{H}}_{\mathbb{C},K^{H}}$ from Proposition
2.6.1 (in the h-setting), where $\mathbb{C}$ is viewed as the quotient of
$U(\mathfrak{h})$ by augmentation ideal.

  * •

Since the data $i_{\xi}$ in an h-complex over $(\mathfrak{g},K^{H})$ are
$\mathfrak{g}$-linear, and the isomorphism in Proposition 5.2.1 is natural in
the $\mathfrak{g}$-module $V$, it also yields an isomorphism between these
functors on the level of h-complexes.

Observe that $\mathbb{C}=D_{\mathrm{pt}}$, and
$\mathbb{C}\underset{\mathscr{O}_{x}}{\otimes}(\cdot)_{x}$ equals the non-
derived inverse image of $\mathscr{D}_{X}$-modules via the inclusion
$i_{x}:\mathrm{pt}\to X$ of $X$. The next result is immediate.

######  Lemma 5.2.2.

Assume that $X$ is affine. Let $K^{H}$ be a subgroup of $H$. The following
diagram commutes up to a canonical isomorphism.

${{}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K^{H})}$${{}^{\mathrm{h}}\mathsf{C}(D_{X},K^{H})}$${{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})}$${{}^{\mathrm{h}}\mathsf{C}(D_{\mathrm{pt}},K^{H})}$$\scriptstyle{D_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot)}$$\scriptstyle{\mathbb{C}\underset{\mathscr{O}_{x}}{\otimes}(\cdot)_{x}}$$\scriptstyle{\mathrm{coInv}^{U(\mathfrak{h}),K^{H}}_{\mathbb{C},K^{H}}}$

When $K^{H}$ is reductive, one can take the left h-derived functor
$\operatorname{L}\\!\left(\mathrm{coInv}^{U(\mathfrak{h}),K^{H}}_{\mathbb{C},K^{H}}\right)$
on the bounded-above h-derived categories.

######  Proposition 5.2.3.

Assume that $X$ is affine. Let $K$ (resp. $K^{H}$) be a reductive subgroup of
$G$ (resp. $H$), and denote $\mathrm{oblv}^{K}_{K^{H}}$ the oblivion
${}^{\mathrm{h}}\mathsf{D}(D_{X},K)\to{}^{\mathrm{h}}\mathsf{D}(D_{X},K^{H})$.
There are canonical isomorphisms

$\displaystyle i_{x}^{\bullet}\mathbf{Loc}_{X,K^{H}}$
$\displaystyle\simeq\operatorname{L}\\!\left(\mathrm{coInv}^{U(\mathfrak{h}),K^{H}}_{\mathbb{C},K^{H}}\right),$
$\displaystyle
i_{x}^{\bullet}\left(\mathrm{oblv}^{K}_{K^{H}}\mathbf{Loc}_{X,K}(M)\right)$
$\displaystyle\simeq\operatorname{L}\\!\left(\mathrm{coInv}^{U(\mathfrak{h}),K^{H}}_{\mathbb{C},K^{H}}\right)(M|_{H}),$

where $M$ stands for an object of
${}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{g},K)$ and $M|_{H}$ denotes its image
in ${}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{h},K^{H})$.

###### Proof.

It suffices to prove the first isomorphism, since the second follows by
commuting $\mathbf{Loc}_{X}$ and oblivion using Proposition 5.1.5.

Note that the functors emitting from
${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K^{H})$ in the diagram of Lemma 5.2.2
preserve K-projectives, by Lemma 3.4.1. It remains to take left h-derived
functors in two ways, which give isomorphic results. ∎

Finally, $D_{X}^{G}$ acts on the right of
$i_{x}^{\bullet}\mathbf{Loc}_{X,K^{H}}$ through its action (5.1.1) on
$\mathbf{Loc}_{X,K^{H}}$. Let us relate it to co-invariants.

######  Proposition 5.2.4.

Let $z\in D_{X}^{G}$. Via Proposition 5.2.3, the $z$-action on
$i_{x}^{\bullet}\mathbf{Loc}_{X,K^{H}}$ arises from the following endomorphism
of the functor $\mathrm{coInv}^{U(\mathfrak{k}),K^{H}}_{\mathbb{C},K^{H}}$.
Near $x$, we can express $z$ as the image of some $f\otimes u$ under the
homomorphism

$\mathscr{O}_{X}\otimes U(\mathfrak{g})\to\mathscr{D}_{X},\quad f\otimes
u\mapsto fj(u)$

where $j:U(\mathfrak{g})\to D_{X}$ is the natural map. For every h-complex $P$
over $(\mathfrak{h},K^{H})$, we let $z$ act in each degree $n$ by

$\displaystyle P^{n}/\mathfrak{h}P^{n}$ $\displaystyle\to
P^{n}/\mathfrak{h}P^{n}$ $\displaystyle p+\mathfrak{h}P^{n}$
$\displaystyle\mapsto f(x)\cdot up+\mathfrak{h}P^{n}.$

###### Proof.

The action $\mathrm{id}\otimes\mathcal{R}_{z}^{n}$ on
$\mathbb{C}\underset{\mathscr{O}_{x}}{\otimes}D_{X}\underset{U(\mathfrak{g})}{\otimes}P^{n}$
depends only on the behavior of $z$ near $x$. Therefore it maps

$1\otimes 1\otimes p\mapsto 1\otimes fj(u)\otimes p=f(x)\otimes 1\otimes up.$

It remains to compare this with Proposition 5.2.1. ∎

###  5.3 Statement of higher regularity

The assumptions on $(\mathfrak{g},K)$ and $X$ from §5.1 remain in force.

######  Definition 5.3.1.

An object of ${}^{\mathrm{h}}\mathsf{D}(D_{X},K)$ is said to be _holonomic_
(resp.  _regular holonomic_) if all its cohomologies are holonomic (resp.
regular holonomic) as $D_{X}$-modules. This gives rise to a full triangulated
subcategory ${}^{\mathrm{h}}\mathsf{D}_{\mathrm{h}}(D_{X},K)$ (resp.
${}^{\mathrm{h}}\mathsf{D}_{\mathrm{rh}}(D_{X},K)$) of
${}^{\mathrm{h}}\mathsf{D}(D_{X},K)$, and one may impose boundedness
conditions as well.

Inside the equivariant derived category $\mathsf{D}^{\mathrm{b}}_{K}(X)$, we
also have $\mathsf{D}^{\mathrm{b}}_{K,\mathrm{h}}(X)$ (resp.
$\mathsf{D}^{\mathrm{b}}_{K,\mathrm{rh}}(X)$), the full triangulated
subcategory of objects with holonomic (resp. regular holonomic) cohomologies.
It matches ${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{h}}(D_{X},K)$
(resp. ${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{h}}(D_{X},K)$) under
Beilinson’s equivalence (Theorem 4.3.4).

For every $D_{X}$-module $L$, denote by $\mathrm{Ch}(L)\subset T^{*}X$ its
characteristic variety. We have the _moment map_

$\bm{\mu}:T^{*}X\to\mathfrak{g}^{*}.$

By choosing a base point, every homogeneous $G$-spaces take the form
$H\backslash G$, and $T^{*}(H\backslash
G)\simeq\mathfrak{h}^{\perp}\overset{H}{\times}G$. Then $\bm{\mu}$ maps
$[\lambda,g]$ to $\operatorname{Ad}^{*}(g^{-1})\lambda$ (co-adjoint action)
for all $\lambda\in\mathfrak{h}^{\perp}$ and $g\in G$.

Let $\mathcal{N}\subset\mathfrak{g}^{*}$ denote the nilpotent cone.

######  Definition 5.3.2.

Let ${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}+}(D_{X},K)$ be the
full subcategory of
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}}(D_{X},K)$ consisting of
objects $L$ such that

$\mathrm{Ch}\left(\operatorname{H}^{n}(L)\right)\subset\bm{\mu}^{-1}\left(\mathcal{N}\cap\mathfrak{k}^{\perp}\right)$

for all $n\in\mathbb{Z}$, where $\operatorname{H}^{n}(L)$ is viewed merely as
a $D_{X}$-module.

This is a full triangulated subcategory by standard properties of
characteristic varieties [15, §2.2]. Again, it has the counterpart
$\mathsf{D}^{\mathrm{b}}_{K,\mathrm{rh}+}(X)$ inside
$\mathsf{D}^{\mathrm{b}}_{K,\mathrm{rh}}(X)$.

######  Definition 5.3.3.

A normal $G$-variety is said to be _spherical_ if there is an open $B$-orbit
in $X$ for some (equivalently, any) Borel subgroup $B\subset G$. A subgroup
$H\subset G$ is said to be _spherical_ if the homogeneous $G$-space
$H\backslash G$ is spherical.

We are ready to state the higher regularity of localizations.

######  Theorem 5.3.4.

Let $G$ be a connected reductive group. Suppose that

  * •

$X$ is an affine spherical homogeneous $G$-space,

  * •

$K$ is a reductive spherical subgroup of $G$.

Let $V$ be a Harish-Chandra module over $(\mathfrak{g},K)$ (Definition 3.1.3),
then $\mathbf{Loc}_{X}(V)$ lies in
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}+}(D_{X},K)$.

The proof will be split into three chunks in §5.4, the hardest one among which
will be settled in §5.5.

######  Corollary 5.3.5.

Under the assumptions of Theorem 5.3.4, let
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{HC}}(\mathfrak{g},K)$ be the
full triangulated subcategory of
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)$ consisting of objects
whose cohomologies are Harish-Chandra $(\mathfrak{g},K)$-modules. Then
$\mathbf{Loc}_{X}$ restricts to

${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{HC}}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}+}(D_{X},K).$

###### Proof.

Since $\mathbf{Loc}_{X}$ is triangulated, one can truncate and shift to reduce
to the case of Harish-Chandra $(\mathfrak{g},K)$-modules. ∎

We remark that in view of Theorems 3.3.1 and 4.3.4, the result above can be
rephrased in terms of “usual” derived categories as

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

However, the construction of $\mathbf{Loc}_{X}$ passes through h-derived
categories.

######  Remark 5.3.6.

Take $X=H\backslash G$ for some subgroup $H\subset G$. By Matsushima’s
criterion [31, Theorem 3.8], $H\backslash G$ is affine if and only if $H$ is
reductive. The conditions in Theorem 5.3.4 are thus symmetric in $H$ and $K$:
both are required to be spherical and reductive. One can also view
$\mathbf{Loc}_{H\backslash G}(V)$ as objects of the derived category of the
stack $H\backslash G/K$.

######  Example 5.3.7.

In order to appreciate the property of characteristic varieties in Definition
5.3.2, let us take

  * •

$H$: a connected reductive group, embedded diagonally in $G:=H\times H$,

  * •

$K:=H$ as a subgroup of $G$,

  * •

$X:=H$ with $G$ acting by $x(h_{1},h_{2})=h_{2}^{-1}xh_{1}$.

Note that $X\simeq H\backslash G$ by choosing $1$ as the base point, and
$H\backslash G/K$ becomes the adjoint quotient stack $\frac{H}{H}$.

The $(\mathfrak{g},K)$-modules in this case give rise to Harish-Chandra
bimodules. They are closely related to harmonic analysis on $H(\mathbb{C})$.

All conditions in Theorem 5.3.4 are met in this case. Given a Harish-Chandra
$(\mathfrak{g},K)$-module $V$ and $n\in\mathbb{Z}$, the irreducible
constituents of the $\operatorname{Ad}$-equivariant $D_{H}$-module
$\operatorname{H}^{n}\mathbf{Loc}_{H}(V)$ are actually _character
$D_{H}$-modules_, i.e. regular holonomic $D_{H}$-modules corresponding to
_character sheaves_ via Riemann–Hilbert (see [26, §2.1], and also [13]).
Indeed, this follows from the condition on
$\mathrm{Ch}\left(\operatorname{H}^{n}\mathbf{Loc}_{H}(V)\right)$ and [26,
Theorem 4.4].

###  5.4 Criterion of regularity

Let $K\subset G$ and $X$ be as in Theorem 5.3.4. Our proof of Theorem 5.3.4 is
based on the result below from [25], which is a variant of Ginzburg’s [13,
Corollary 8.9.1].

######  Theorem 5.4.1.

Suppose that $M$ is a $D_{X}$-module with the following properties:

  1. (R1)

$M$ is finitely generated over $D_{X}$;

  2. (R2)

$M$ carries a structure of $K$-equivariant $D_{X}$-module;

  3. (R3)

$M$ is locally $\mathcal{Z}(\mathfrak{g})$-finite, where
$\mathcal{Z}(\mathfrak{g})$ acts through the homomorphism
$j:U(\mathfrak{g})\to D_{X}$ of algebras.

Then $M$ is regular holonomic, and
$\mathrm{Ch}(M)\subset\bm{\mu}^{-1}(\mathcal{N}\cap\mathfrak{k}^{\perp})$.

###### Proof.

This is an instance of [25, Proposition 3.4, Corollary 5.7]. ∎

We can now depict the strategy for proving Theorem 5.3.4.

###### Proof of Theorem 5.3.4.

Let $V$ be a Harish-Chandra $(\mathfrak{g},K)$-module. Set
$\mathcal{L}:=\mathbf{Loc}_{X,\\{1\\}}(V)$, which is also the image of
$\mathbf{Loc}_{X,K}(V)$ under
${}^{\mathrm{h}}\mathsf{D}(D_{X},K)\to\mathsf{D}(X)$ by Proposition 5.1.5. We
shall verify (R1) — (R3) for the cohomologies of $\mathcal{L}$.

First, we identify $\mathcal{L}$ with
$D_{X}\underset{U(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}V$. Since $V$
is finitely generated over $\mathfrak{g}$ and $U(\mathfrak{g})$ is left (and
right) Noetherian, there is a free resolution over $\mathfrak{g}$:

$\cdots\to U(\mathfrak{g})^{\oplus n_{1}}\to U(\mathfrak{g})^{\oplus n_{0}}\to
V\to 0,\quad n_{i}\in\mathbb{Z}_{\geq 0}.$

Hence $D_{X}\underset{U(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}V$ is
represented by the complex

$\cdots\to D_{X}^{\oplus n_{i+1}}\to D_{X}^{\oplus n_{i}}\to\cdots.$

Since $D_{X}$ is left (and right) Noetherian by [15, Proposition 1.4.6], the
cohomologies of the complex above are finitely generated $D_{X}$-modules. This
verifies (R1).

As for (R2), from (2.5.2) we see that $\operatorname{H}^{n}(\mathcal{L})$ is
the image of $\operatorname{H}^{n}(\mathbf{Loc}_{X,K}(V))$, for all $n$.

What remains is (R3); this will be settled by Proposition 5.5.6. ∎

Note that the case of zeroth cohomology, i.e. the non-derived
$D_{X}\underset{U(\mathfrak{g})}{\otimes}V$, has been addressed in [25,
Example 5.5 (iii)].

###  5.5 End of the proof

Hereafter, $G$ is a connected reductive group, and $X$ is a smooth affine
$G$-variety.

We say a left module $M$ over a commutative algebra $A$ is _locally
$A$-finite_ if it it is the union of finite-dimensional $A$-submodules.

######  Lemma 5.5.1.

Suppose $A$ is a finitely generated commutative algebra. For every short exact
sequence $0\to N^{\prime}\to N\to N^{\prime\prime}\to 0$ of $A$-modules, we
have: $N$ is locally finite if and only if $N^{\prime}$ and $N^{\prime\prime}$
are both locally finite.

###### Proof.

It suffices to explain the “if” part. Let $x\in N$. Its image in
$N^{\prime\prime}$ is annihilated by an ideal $I$ of finite codimension (as
vector subspace), hence $Ix\subset N^{\prime}$. Since $A$ is Noetherian, $I$
is finitely generated, hence there exist ideals $I_{1},\ldots,I_{k}$ of finite
codimension such that $I_{1}\cdots I_{k}Ix=0$. However $I_{1}\cdots I_{k}I$ is
also of finite codimension, by the structure of $A$. ∎

######  Lemma 5.5.2.

Let $M$ be a $D_{X}$-module, thus also a $U(\mathfrak{g})$-submodule via the
homomorphism $j:U(\mathfrak{g})\to D_{X}$. Let $M^{\natural}\subset M$ be a
$\mathfrak{g}$-submodule. Then the map

${\alpha:D_{X}\otimes M^{\natural}}$${M}$${P\otimes m}$${Pm}$

is a homomorphism of $\mathfrak{g}$-modules, if we let $\theta\in\mathfrak{g}$
act on $P\in D_{X}$ by $\theta\odot P:=[j(\theta),P]$.

###### Proof.

Omit $j$ to simplify notation. Given $\theta\in\mathfrak{g}$, we have

$\displaystyle\theta\left(\alpha(P\otimes m)\right)$
$\displaystyle=\theta(Pm)=(\theta P)m-(P\theta)m+P(\theta m)$
$\displaystyle=[\theta,P]m+P(\theta m)$
$\displaystyle=\alpha\left((\theta\odot P)\otimes
m\right)+\alpha(P\otimes\theta m)$ $\displaystyle=\alpha\left(\theta(P\otimes
m)\right)$

by the standard definition of the $\mathfrak{g}$-module structure on
$D_{X}\otimes M$. ∎

######  Lemma 5.5.3.

Let $M$ be a $D_{X}$-module as before, and let $M^{\flat}\subset M$ be a
$\mathcal{Z}(\mathfrak{g})$-submodule that is locally finite, and generates
$M$ over $D_{X}$. Then $M$ is locally $\mathcal{Z}(\mathfrak{g})$-finite.

###### Proof.

Take the $\mathfrak{g}$-submodule $M^{\natural}:=U(\mathfrak{g})M^{\flat}$ of
$M$. Note that $M^{\natural}$ is locally $\mathcal{Z}(\mathfrak{g})$-finite,
and generates $M$ over $D_{X}$. Note that $G$ acts algebraically on $D_{X}$ by
$P\xmapsto{g}gPg^{-1}$; this is indeed clear, and is included in Proposition
5.1.1. Its derivative is the action $\odot$ in Lemma 5.5.2, hence $D_{X}$ is a
union of finite-dimensional $\mathfrak{g}$-submodules.

Consider now the $\mathfrak{g}$-linear surjection $\alpha:D_{X}\otimes
M^{\natural}\twoheadrightarrow M$ from Lemma 5.5.2. By a theorem of Kostant
[20, Theorem 7.133] (and taking $\varinjlim$), the $\mathfrak{g}$-module
$D_{X}\otimes M^{\natural}$ is seen to be locally
$\mathcal{Z}(\mathfrak{g})$-finite. Hence so is $M$. ∎

Note that in Lemma 5.5.3, if $M^{\flat}$ has infinitesimal character $\chi$,
then $M$ as a $\mathcal{Z}(\mathfrak{g})$-module is supported on
$\chi+\\{\text{weights}\\}$. Here we describe
$\operatorname{Spec}\mathcal{Z}(\mathfrak{g})$ via Harish-Chandra’s
isomorphism.

For every homomorphism of algebras
$\chi:\mathcal{Z}(\mathfrak{g})\to\mathbb{C}$, we have the ideal

$\mathfrak{m}_{\chi}:=\operatorname{ker}(\chi)\subset\mathcal{Z}(\mathfrak{g}),$

and we say a $\mathfrak{g}$-module has _infinitesimal character_ $\chi$ if
$\mathcal{Z}(\mathfrak{g})$ acts through $\chi$. For example, the
$\mathfrak{g}$-module below has infinitesimal character $\chi$:

$M_{\chi}:=U(\mathfrak{g})/\mathfrak{m}_{\chi}U(\mathfrak{g}).$ (5.5.1)

For all $\mathfrak{g}$-module $V$, define the $D_{X}$-modules

$\operatorname{Tor}^{U(\mathfrak{g})}_{n}\left(D_{X},V\right):=\operatorname{H}^{-n}\left(D_{X}\underset{U(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}V\right),\quad
n\in\mathbb{Z}.$

######  Definition 5.5.4.

Following [21, pp.254–255], we define $\mathcal{Z}(X)$ to be the center of
$D_{X}^{G}$. It is known to be the whole $D_{X}^{G}$ when $X$ is spherical,
see loc. cit.

Note that $j:U(\mathfrak{g})\to D_{X}$ restricts to
$\mathcal{Z}(\mathfrak{g})\to\mathcal{Z}(X)$.

######  Lemma 5.5.5.

Given $\chi$, the $D_{X}$-modules
$\operatorname{Tor}^{U(\mathfrak{g})}_{n}\left(D_{X},M_{\chi}\right)$ are
locally $\mathcal{Z}(\mathfrak{g})$-finite for all $n\in\mathbb{Z}$, where
$M_{\chi}$ is as in (5.5.1).

###### Proof.

This relies critically on some results of B. Kostant and F. Knop. We will
perform change-of-rings through the commutative diagram

${D_{X}}$${\mathcal{Z}(X)}$${U(\mathfrak{g})}$${\mathcal{Z}(\mathfrak{g})}$free$\scriptstyle{j}$$\scriptstyle{j}$free

of algebras; here “free” means free as left and right modules. Indeed,

  * •

the freeness of $U(\mathfrak{g})$ over $\mathcal{Z}(\mathfrak{g})$ is due to
Kostant — see [20, Theorem 7.114],

  * •

the freeness of $D_{X}$ over $\mathcal{Z}(X)$ is due to Knop [21, Theorem 9.5
(c)].

Using the $\mathcal{Z}(\mathfrak{g})$-flatness of $U(\mathfrak{g})$, inside
$\mathsf{D}(\mathfrak{g}\text{-}\mathsf{Mod})$ we have

$M_{\chi}\simeq
U(\mathfrak{g})\underset{\mathcal{Z}(\mathfrak{g})}{\otimes}\frac{\mathcal{Z}(\mathfrak{g})}{\mathfrak{m}_{\chi}}\simeq
U(\mathfrak{g})\underset{\mathcal{Z}(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}\frac{\mathcal{Z}(\mathfrak{g})}{\mathfrak{m}_{\chi}}.$

Set $N_{\chi}:=\mathcal{Z}(\mathfrak{g})/\mathfrak{m}_{\chi}$. Since change-
of-ring preserves K-projectives, performing
$\underset{}{\overset{\mathrm{L}}{\otimes}}$ in stages leads to

$\displaystyle
D_{X}\underset{U(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}M_{\chi}$
$\displaystyle\simeq
D_{X}\underset{U(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}\left(U(\mathfrak{g})\underset{\mathcal{Z}(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}N_{\chi}\right)$
$\displaystyle\simeq
D_{X}\underset{\mathcal{Z}(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}N_{\chi}$
$\displaystyle\simeq
D_{X}\underset{\mathcal{Z}(X)}{\overset{\mathrm{L}}{\otimes}}\left(\mathcal{Z}(X)\underset{\mathcal{Z}(\mathfrak{g})}{\overset{\mathrm{L}}{\otimes}}N_{\chi}\right)\;\quad\text{in}\;\mathsf{D}(D_{X}\text{-}\mathsf{Mod}).$

Hence by the $\mathcal{Z}(X)$-flatness of $D_{X}$, for all $n\geq 0$ we have
in $D_{X}\text{-}\mathsf{Mod}$

$\operatorname{Tor}^{U(\mathfrak{g})}_{n}\left(D_{X},M_{\chi}\right)\simeq
D_{X}\underset{\mathcal{Z}(X)}{\otimes}\operatorname{Tor}^{\mathcal{Z}(\mathfrak{g})}_{n}\left(\mathcal{Z}(X),N_{\chi}\right).$
(5.5.2)

We contend that for all $n\geq 0$,

$\mathcal{Z}(\mathfrak{g})\;\text{acts
on}\;\operatorname{Tor}^{\mathcal{Z}(\mathfrak{g})}_{n}\left(\mathcal{Z}(X),N_{\chi}\right)\;\text{through}\;\chi.$
(5.5.3)

Here $\mathcal{Z}(\mathfrak{g})$ acts through
$j:\mathcal{Z}(\mathfrak{g})\to\mathcal{Z}(X)$. The action of
$z\in\mathcal{Z}(\mathfrak{g})$ is given as follows. Take a projective
resolution $\cdots\to Q_{1}\to Q_{0}\to N_{\chi}\to 0$, then there exist
$f_{i}\in\operatorname{End}_{\mathcal{Z}(\mathfrak{g})}(Q_{i})$ making

${\cdots}$${Q_{1}}$${Q_{0}}$${N_{\chi}}$${0}$${\cdots}$${Q_{1}}$${Q_{0}}$${N_{\chi}}$${0}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{0}}$$\scriptstyle{z}$

commutative; different choices of $(f_{i})_{i\geq 0}$ are related by
$\mathcal{Z}(\mathfrak{g})$-linear homotopies. Take
$\mathcal{Z}(X)\underset{\mathcal{Z}(\mathfrak{g})}{\otimes}(\cdot)$ on the
whole diagram, then the induced action on $\operatorname{H}_{n}$ is the
desired one; the commutativity of $\mathcal{Z}(X)$ is crucial here.

  * •

Naturally, one can take $f_{i}=z$.

  * •

On the other hand, one can also take $f_{i}=\chi(z)\mathrm{id}$. This proves
(5.5.3).

By (5.5.2),
$\operatorname{Tor}^{U(\mathfrak{g})}_{n}\left(D_{X},M_{\chi}\right)$ is
generated by
$M^{\flat}:=1\otimes\operatorname{Tor}^{\mathcal{Z}(\mathfrak{g})}_{n}\left(\mathcal{Z}(X),N_{\chi}\right)$
as a $D_{X}$-module. By combining (5.5.3) and Lemma 5.5.3, the local
$\mathcal{Z}(\mathfrak{g})$-finiteness follows. ∎

######  Proposition 5.5.6.

Let $V$ be a $\mathfrak{g}$-module that is finitely generated and locally
$\mathcal{Z}(\mathfrak{g})$-finite. Then
$\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},V)$ is locally
$\mathcal{Z}(\mathfrak{g})$-finite for all $n\in\mathbb{Z}$.

###### Proof.

Using the assumptions on $V$, one produces a filtration
$0=V_{0}\subset\cdots\subset V_{n}=V$ such that each subquotient has an
infinitesimal character. By the long exact sequence (of $D_{X}$-modules) for
$\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},\cdot)$ and Lemma 5.5.1, we
are thus reduced to the case that $V$ has infinitesimal character $\chi$.

We argue by dimension shifting. The case $n<0$ is trivial. By taking
generators of $V$ and using (5.5.1), there is a short exact sequence of
$\mathfrak{g}$-modules

$0\to W\to M_{\chi}^{\oplus I}\to V\to 0$

where $I$ is some (small) set; all modules have infinitesimal character
$\chi$. In the long exact sequence we have the piece

$\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},M_{\chi}^{\oplus
I})\to\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},V)\to\operatorname{Tor}^{U(\mathfrak{g})}_{n-1}(D_{X},W).$

The third term is locally $\mathcal{Z}(\mathfrak{g})$-finite by recursion; so
is the first term by Lemma 5.5.5 and the fact that
$\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},\cdot)$ commutes with direct
sums. This completes the proof. ∎

######  Remark 5.5.7.

When $V$ has infinitesimal character $\chi$, the support of
$\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},V)$ as a
$\mathcal{Z}(\mathfrak{g})$-module can be loosely controlled as discussed
after Lemma 5.5.3.

We record a by-product of the proofs above, which implies that
$D_{X}\underset{U(\mathfrak{g})}{\otimes}(\cdot)$ is usually non-exact.

######  Proposition 5.5.8.

Define $M_{\chi}$ by (5.5.1) for all $\chi$. We have

$\left[\forall\chi,\;\operatorname{Tor}^{U(\mathfrak{g})}_{1}(D_{X},M_{\chi})=0\right]\iff\mathcal{Z}(X)\;\text{is
flat over}\;\mathcal{Z}(\mathfrak{g}).$

###### Proof.

Let $N_{\chi}:=\mathcal{Z}(\mathfrak{g})/\mathfrak{m}_{\chi}$. By the local
criterion of flatness, $\mathcal{Z}(X)$ is flat over
$\mathcal{Z}(\mathfrak{g})$ if and only if
$\operatorname{Tor}_{1}^{\mathcal{Z}(\mathfrak{g})}(\mathcal{Z}(X),N_{\chi})=0$
for all $\chi$. In turn, this is equivalent to
$\operatorname{Tor}^{U(\mathfrak{g})}_{1}(D_{X},M_{\chi})=0$ by (5.5.2)
together with the freeness of $D_{X}$ over $\mathcal{Z}(X)$. ∎

######  Remark 5.5.9.

In view of the structure theorem in [21] and the notation therein,
$\mathcal{Z}(X)$ is flat over $\mathcal{Z}(\mathfrak{g})$ if and only if the
natural map between categorical quotients

$(\rho+\mathfrak{a}_{X}^{*})\sslash W_{X}\to\mathfrak{t}^{*}\sslash W$

is flat, which rarely holds; a necessary condition is that
$\mathrm{rank}(X)=\mathrm{rank}(G)$. Therefore $D_{X}$ is rarely flat over
$U(\mathfrak{g})$, by Proposition 5.5.8.

However, flatness of $\mathcal{Z}(X)$ does hold when $X$ is the affine closure
of $U\backslash G$ where $U$ is a maximal unipotent subgroup: this is a
combination of [21, Lemma 6.4] and Pittie–Steinberg theorem. Note that
$\mathcal{Z}(Y)$ is defined for all normal $G$-varieties $Y$ in loc. cit., and
it is an equivariant-birational invariant of $Y$.

The same arguments also lead to the following fact about the right
$\mathcal{Z}(X)$-action on $\mathbf{Loc}_{X}$, which might be of independent
interest.

######  Proposition 5.5.10.

Under the assumptions of Proposition 5.5.6,
$\operatorname{Tor}_{n}^{U(\mathfrak{g})}(D_{X},V)$ is locally
$\mathcal{Z}(X)$-finite under the action (5.1.1), for all $n\in\mathbb{Z}$.

Furthermore, if $V$ has infinitesimal character $\chi$, then the support of
$\operatorname{Tor}_{n}^{U(\mathfrak{g})}(D_{X},V)$ as a
$\mathcal{Z}(X)$-module is a subset of the fiber of $\chi$ under
$\operatorname{Spec}(\mathcal{Z}(X))\to\operatorname{Spec}(\mathcal{Z}(\mathfrak{g}))$,
which is finite.

###### Proof.

First, Lemma 5.5.1 applies to $\mathcal{Z}(X)$-modules, by describing the
structure of $\mathcal{Z}(X)$ using Knop’s theorem.

As before, one reduces to the case when $V$ has infinitesimal character
$\chi$, and then to the case $V=M_{\chi}$. Let $\mathcal{Z}(\mathfrak{g})$ act
on $\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},M_{\chi})$ via

$\mathcal{Z}(\mathfrak{g})\to\mathcal{Z}(X)\xrightarrow{\eqref{eqn:Z-action}}\operatorname{End}_{D_{X}}\left(\operatorname{Tor}^{U(\mathfrak{g})}_{n}(D_{X},M_{\chi})\right).$

By inspecting the proof of Lemma 5.5.5, we see that the action above is simply
$\chi$. We conclude by the fact that $\mathcal{Z}(X)$ is a finite
$\mathcal{Z}(\mathfrak{g})$-module; see [21, p.254]. ∎

##  6 Application to $\operatorname{Ext}$-branching

###  6.1 General setting of branching laws

Suppose we are given a pair $(\mathfrak{i},K^{I})$ (see §3.1) and its subpair
$(\mathfrak{h},K^{H})$, where $\mathfrak{h}\subset\mathfrak{i}$ and
$K^{H}\subset K^{I}$; see §3.4 for the precise conditions on subpairs. This
induces the diagonal embedding of pairs

$(\mathfrak{h},K^{H})\to(\mathfrak{i}\times\mathfrak{h},K^{I}\times K^{H}).$

On the first component it is $U(\mathfrak{h})\to U(\mathfrak{i})\otimes
U(\mathfrak{h})$, given by $\theta\mapsto\theta\otimes 1+1\otimes\theta$ for
all $\theta\in\mathfrak{h}$. On the second component it is $k\mapsto(k,k)$ for
all $k\in K^{H}$.

Consider h-complexes $M$ over $(\mathfrak{i},K^{I})$ and $N$ over
$(\mathfrak{h},K^{H})$. Taking tensor product yields an h-complex over
$(\mathfrak{i}\times\mathfrak{h},K^{I}\times K^{H})$, which we denote as
$M\boxtimes N$.

######  Definition 6.1.1.

Denote by $M|_{H}$ the restriction of $M$ to an h-complex over
$(\mathfrak{h},K^{H})$. Define $(M\boxtimes N)|_{H}$ similarly via the
diagonal embedding. Ditto for the functors induced on derived categories.

By the general formalism of §2.3, we can define $\operatorname{RHom}$ between
h-complexes over these pairs; they will be denoted as
$\operatorname{RHom}_{\mathfrak{h},K^{H}}$ and so on.

######  Proposition 6.1.2.

Assume $K^{H}$ and $K^{I}$ are reductive. Let $M$ (resp. $N$) be an h-complex
over $(\mathfrak{i},K^{I})$ (resp. $(\mathfrak{h},K^{H})$). Denote by
$N^{\vee}$ the contragredient of $N$, as in Definition 3.1.2. There is a
canonical isomorphism in $\mathsf{D}(\mathbb{C})$

$\operatorname{RHom}_{\mathfrak{h},K^{H}}\left(M|_{H},N^{\vee}\right)\simeq\operatorname{RHom}_{\mathfrak{h},K^{H}}\left((M\boxtimes
N)|_{H},\mathbb{C}\right).$

The same isomorphism also holds for complexes of
$(\mathfrak{i},K^{I})$-modules and $(\mathfrak{h},K^{H})$-modules, with the
corresponding $\operatorname{RHom}$.

###### Proof.

We will only sketch the case of h-complexes; the other case is similar and
well-known.

Take a K-projective resolution $P\to M$ in
${}^{\mathrm{h}}\mathsf{C}(\mathfrak{i},K^{I})$. Then so is $P|_{H}\to M|_{H}$
in ${}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})$ by Lemma 3.4.1. We claim
that $(P\boxtimes N)|_{H}\to(M\boxtimes N)|_{H}$ is a K-projective resolution
as well.

Since $((\cdot)\boxtimes N)|_{H}\simeq(\cdot)|_{H}\otimes N$ is exact, the
above is indeed a quasi-isomorphism. By [27, Lemma 3.2], $(\cdot)|_{H}\otimes
N$ has a right adjoint dg-functor given by internal $\operatorname{Hom}$ (see
§3.1)

$\operatorname{Hom}^{\bullet}_{\mathbb{C}}(N,(\cdot)|_{H})^{K_{H}\text{-alg}}.$

As seen in the proof of Lemma 3.4.1, this right adjoint is exact, thus the
claim follows.

All in all,
$\operatorname{RHom}_{\mathfrak{h},K^{H}}\left(M|_{H},N^{\vee}\right)$ is
represented by the complex

${}^{\mathrm{h}}\operatorname{Hom}^{\bullet}_{\mathfrak{h},K^{H}}\left(P|_{H},N^{\vee}\right)\simeq{}^{\mathrm{h}}\operatorname{Hom}^{\bullet}_{\mathfrak{h},K^{H}}\left(P|_{H}\otimes
N,\mathbb{C}\right)$

where the aforementioned dg-adjunction is applied once again. By the claim,
the right-hand side represents
$\operatorname{RHom}_{\mathfrak{h},K^{H}}\left((M\boxtimes
N)|_{H},\mathbb{C}\right)$. ∎

Cf. [30, Proposition 2.6] for the case of $p$-adic groups.

Under the assumptions of reductivity, the study of general
$\operatorname{Ext}$-branching

${}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},N^{\vee})$

for various $M$, $N$ thus reduces to the special case
${}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})$,
once we replace $\mathfrak{i}$ (resp. $K^{I}$) by
$\mathfrak{g}:=\mathfrak{i}\times\mathfrak{h}$ (resp. $K:=K^{I}\times K^{H}$).
This recipe is certainly well-known, at least in degree $n=0$ and for
complexes of $(\mathfrak{g},K)$-modules instead of h-complexes.

We record another easy fact about
$\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(\cdot,\mathbb{C})$.

######  Proposition 6.1.3 ([20, Corollary 3.2]).

Consider a subpair $(\mathfrak{h},K^{H})$ of $(\mathfrak{g},K)$ and assume
$K^{H}$ and $K$ are reductive. Let $V$ be a $(\mathfrak{g},K)$-module. There
are canonical isomorphisms

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

for each $n\in\mathbb{Z}$, where
$\operatorname{H}_{n}\left(\mathfrak{h},K^{H};\cdot\right)$ is the relative
Lie algebra homology.

Note that
${}^{\mathrm{h}}\operatorname{Ext}_{\mathfrak{h},K^{H}}\simeq\operatorname{Ext}_{\mathfrak{h},K^{H}}$
by Corollary 3.3.2. The upcoming results about $\operatorname{Ext}^{n}$ can
all be viewed as assertions about
$\operatorname{H}^{n}\left(\mathfrak{h},K^{H};V\right)^{*}$.

###  6.2 Branching and localization

Consider a connected reductive group $G$ and its subgroups

${H}$${G}$${K^{H}}$${K.}$${\subset}$${\subset}$${\subset}$${\subset}$ (6.2.1)

We assume that $H$, $K$ and $K^{H}$ are all reductive.

######  Lemma 6.2.1.

The following diagram commutes up to canonical isomorphism

${{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})}$${{}^{\mathrm{h}}\mathsf{C}(\mathbb{C},K^{H})}$${\mathsf{C}(\mathbb{C})^{\mathrm{op}}}$$\scriptstyle{\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},K^{H}}}$$\scriptstyle{\operatorname{Hom}_{1}}$$\scriptstyle{\operatorname{Hom}_{2}}$

where

$\displaystyle\operatorname{Hom}_{1}$
$\displaystyle:={}^{\mathrm{h}}\operatorname{Hom}^{\bullet}_{\mathfrak{h},K^{H}}(\cdot,\mathbb{C}),$
$\displaystyle\operatorname{Hom}_{2}$
$\displaystyle:={}^{\mathrm{h}}\operatorname{Hom}^{\bullet}_{\mathbb{C},K^{H}}(\cdot,\mathbb{C}),$

and all actions upon $\mathbb{C}$ are defined to be trivial.

###### Proof.

All functors in view are dg-functors, and it amounts to establishing a
canonical isomorphism in $\mathsf{C}(\mathbb{C})$

${}^{\mathrm{h}}\operatorname{Hom}^{\bullet}_{\mathbb{C},K^{H}}\left(\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},K^{H}}(N),\mathbb{C}\right)\simeq{}^{\mathrm{h}}\operatorname{Hom}^{\bullet}_{\mathfrak{h},K^{H}}(N,\mathbb{C})$

where $N$ is any h-complex over $(\mathfrak{h},K^{H})$. Indeed, this follows
from the adjunction between co-invariants and inflation (Proposition 2.6.1);
the adjunction extends to the dg-level, cf. the proof of Proposition 2.3.10. ∎

Consider the affine homogeneous $G$-space

$X:=H\backslash G,\quad x:=H\cdot 1.$

Thus $x$ is a $K^{H}$-fixed point in $X$, and we have the corresponding
$K^{H}$-equivariant morphism

$i_{x}:\mathrm{pt}\to X.$

Note that $D_{\mathrm{pt}}=\mathbb{C}$, with trivial $K^{H}$-action.

To save space, we will omit the oblivion
${}^{\mathrm{h}}\mathsf{D}(D_{X},K)\to{}^{\mathrm{h}}\mathsf{D}(D_{X},K^{H})$
in the statements below, and adopt the notation $M\mapsto M|_{H}$ of
Definition 6.1.1 on the level of h-derived categories.

######  Proposition 6.2.2.

For all objects $M$ of ${}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{g},K)$, there
are canonical isomorphisms in $\mathsf{D}^{+}(\mathbb{C})$:

$\operatorname{RHom}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})\simeq\operatorname{RHom}_{D_{\mathrm{pt}},K^{H}}\left(i_{x}^{\bullet}(\mathbf{Loc}_{X}(M)),\mathbb{C}\right).$

Here
$i_{x}^{\bullet}:{}^{\mathrm{h}}\mathsf{D}^{-}(D_{X},K^{H})\to{}^{\mathrm{h}}\mathsf{D}^{-}(D_{\mathrm{pt}},K^{H})$
is the inverse image functor introduced in §4.2.

###### Proof.

The functor
${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})$
preserves K-projectives by Lemma 3.4.1, hence the left-hand side is the left
h-derived functor of the composition of

${}^{\mathrm{h}}\mathsf{C}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})\xrightarrow{\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},K^{H}}}{}^{\mathrm{h}}\mathsf{C}(\mathbb{C},K^{H})\xrightarrow{\operatorname{Hom}_{2}}\mathsf{C}(\mathbb{C})^{\mathrm{op}},$

in the notation of Lemma 6.2.1.

Since $\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},K^{H}}$ also preserves
K-projectives by Proposition 2.6.2, one can take left h-derived functors in
stages. Using Proposition 5.2.3, the result is isomorphic to the composition
of

$i_{x}^{\bullet}\left(\mathbf{Loc}_{X}\right):{}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{-}(\mathbb{C},K^{H})$

with
$\operatorname{L}\\!(\operatorname{Hom}_{2}):{}^{\mathrm{h}}\mathsf{D}^{-}(\mathbb{C},K^{H})\to\mathsf{D}^{+}(\mathbb{C})^{\mathrm{op}}$.
However, $\operatorname{L}\\!(\operatorname{Hom}_{2})$ is just
$\operatorname{RHom}_{D_{\mathrm{pt}},K^{H}}(\cdot,\mathbb{C})$. ∎

######  Corollary 6.2.3.

For all objects $M$ of $\mathsf{D}^{\mathrm{b}}(\mathfrak{g},K)$ and all
$n\in\mathbb{Z}$, there are canonical isomorphisms:

$\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})\simeq\operatorname{Ext}^{n}_{\mathsf{D}^{\mathrm{b}}_{K^{H}}(\mathrm{pt})}\left(i_{x}^{\bullet}(\varepsilon\mathbf{Loc}_{X}(M)),\mathbb{C}\right).$

Here $\varepsilon$ is the equivalence in Theorem 4.3.4.

###### Proof.

Combine the Bernstein–Lunts equivalence (Corollary 3.3.2) with Beilinson’s
equivalence $\varepsilon$ (Corollary 4.3.5), noting that $\varepsilon$ is
compatible with inverse images (Proposition 4.3.6). ∎

One can also interpret the relative Lie algebra homologies in terms of
localization.

######  Proposition 6.2.4.

Let $V$ be a $(\mathfrak{g},K)$-module. There are canonical isomorphisms

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

.

###### Proof.

The arguments from Proposition 6.2.2 shows that the right-hand side is the
$\operatorname{H}^{-n}$ of the composition

${}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{h},K^{H})\xrightarrow{\operatorname{L}\\!\left(\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},K^{H}}\right)}{}^{\mathrm{h}}\mathsf{D}^{-}(\mathbb{C},K^{H})\xrightarrow{\operatorname{L}\\!\left(\mathrm{coInv}^{\mathbb{C},K^{H}}_{\mathbb{C},\\{1\\}}\right)}\mathsf{D}^{-}(\mathbb{C}).$

Taking co-invariants preserves K-projectives. By the transitivity (2.6.1), the
composition above folds into

${}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{h},K^{H})\xrightarrow{\operatorname{L}\\!\left(\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},\\{1\\}}\right)}\mathsf{D}^{-}(\mathbb{C}).$

It remains to show that after taking $\operatorname{H}^{-n}$, the second arrow
gives $\operatorname{H}_{n}(\mathfrak{h},K^{H};W)$ when applied to any
$(\mathfrak{h},K^{H})$-module $W$.

Use transitivity to break
$\mathrm{coInv}^{\mathfrak{h},K^{H}}_{\mathbb{C},\\{1\\}}$ into two stages

${}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})\xrightarrow{\mathrm{coInv}_{1}}\mathsf{C}(\mathfrak{h},K^{H})\xrightarrow{\mathrm{coInv}_{2}}\mathsf{C}(\mathbb{C}),$

cf. Example 2.6.3 for $\mathrm{coInv}_{1}$. The left derived functors are
decomposed accordingly. Now take the standard resolution $W\otimes
N\mathfrak{h}\to W$ in ${}^{\mathrm{h}}\mathsf{C}(\mathfrak{h},K^{H})$ (see
§3.2). Unwinding the construction of $\mathrm{coInv}_{1}$, one obtains

$\mathrm{coInv}_{1}(W\otimes N\mathfrak{g})\simeq
W\otimes\mathrm{coInv}_{1}(N\mathfrak{g})$

in $\mathsf{C}(\mathfrak{h},K^{H})$. It is shown in [28, Proposition 3.2.7]
that $\mathrm{coInv}_{1}(N\mathfrak{g})$ is the standard resolution of
$\mathbb{C}$ in $\mathsf{C}(\mathfrak{h},K^{H})$, i.e. the relative standard
complex. Hence $\mathrm{coInv}_{2}(W\otimes\mathrm{coInv}_{1}(N\mathfrak{g}))$
represents the relative Lie algebra homologies of $W$, as desired. ∎

###  6.3 Some consequences of regularity

Throughout this section, we let $G$ be a connected reductive group, $X$ be a
spherical affine homogeneous $G$-space, and $K$ be a spherical reductive
subgroup of $G$. We consider a point $x$ of $X$ and a reductive subgroup
$K^{H}$ of $K$ that stabilizes $x$. In particular,
$i_{x}:\mathrm{pt}\hookrightarrow X$ is $K^{H}$-equivariant.

By taking $H:=\mathrm{Stab}_{G}(x)$ to identify $X$ with $H\backslash G$, the
situation is like (6.2.1); the extra assumption here is that $H$ and $K$ are
both spherical.

For every smooth $K^{H}$-variety $Y$, let
$\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{cons}}(Y)$ denote the bounded
$K^{H}$-equivariant derived category of constructible sheaves defined by
Bernstein–Lunts [8], equipped with perverse $t$-structure. Assuming $Y$ is
affine, we have equivalences

${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}}(D_{Y},K^{H})\simeq\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{rh}}(Y)\simeq\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{cons}}(Y).$

  * •

The first one is Beilinson’s equivalence (Theorem 4.3.4).

  * •

The second one is the equivariant Riemann–Hilbert correspondence; see [8, 4.2]
or [17, Theorem 4.6.2].

######  Definition 6.3.1.

Let
${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}}(D_{X},K^{H})}$${{}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}}(D_{\mathrm{pt}},K^{H})}$$\scriptstyle{i_{x}^{*},i_{x}^{!}}$$\scriptstyle{i_{x,*}}$
be the functors that correspond to the synonymous functors between
$\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{cons}}(X)$ and
$\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{cons}}(\mathrm{pt})$.

Therefore we have

$\displaystyle i_{x}^{!}$ $\displaystyle=i_{x}^{\bullet}[-\dim X],$
$\displaystyle i_{x}^{*}$ $\displaystyle=i_{x}^{!}[2\dim
X]=i_{x}^{\bullet}[\dim X]\quad\text{when}\;K^{H}=\\{1\\},$

and $i_{x}^{*}$ is left adjoint to $i_{x,*}$. For the relation between
$i_{x}^{!}$ and $i_{x}^{\bullet}$, see eg. [15, Theorem 7.1.1]. For the
relation between the $i_{x}^{*}$ and $i_{x}^{!}$ in the non-equivariant case,
see eg. [8, p.8].

######  Theorem 6.3.2.

Let $M$ be an object of the category
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{HC}}(\mathfrak{g},K)$
introduced in Corollary 5.3.5. There is a canonical isomorphism

$\operatorname{RHom}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})\simeq\operatorname{RHom}_{D_{\mathrm{pt}},K^{H}}\left(i_{x}^{!}\mathbf{Loc}_{X}(M)[\dim
X],\mathbb{C}\right).$

When $K^{H}=\\{1\\}$, this is also isomorphic to

$\operatorname{RHom}_{D_{\mathrm{pt}}}\left(i_{x}^{*}\mathbf{Loc}_{X}(M)[-\dim
X],\mathbb{C}\right)\simeq\operatorname{RHom}_{D_{X}}\left(\mathbf{Loc}_{X}(M),i_{x,*}(\mathbb{C})[\dim
X]\right).$

All these complexes of $D$-modules are bounded with regular holonomic
cohomologies.

###### Proof.

By Corollary 5.3.5, $\mathbf{Loc}_{X}(M)$ is in
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}}(D_{X},K)$. It remains to
apply Proposition 6.2.2 and the relations recalled above. ∎

Note that by [15, Examples 1.5.23 and 1.6.4], $i_{x,*}(\mathbb{C})$ is the
$D_{X}$-module generated by the Dirac measure at $x$.

In order to apply Theorem 6.3.2 to concrete problems, one needs a deeper,
quantitative understanding of $\mathbf{Loc}_{X}(M)$. We only give some crude
applications below, which bypasses this issue.

######  Corollary 6.3.3.

Let $M$ be as in Theorem 6.3.2. Then
${}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})$
is finite-dimensional for all $n\in\mathbb{Z}$. It vanishes for $|n|\gg 0$.

###### Proof.

To prove the vanishing for $|n|\gg 0$, simply take the standard resolution of
$M$.

To prove the finiteness of
${}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})$,
it suffices to show that for every object $N$ of
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}}(D_{\mathrm{pt}},K^{H})$,
we have

$\dim{}^{\mathrm{h}}\operatorname{Ext}^{n}_{D_{\mathrm{pt}},K^{H}}\left(N,\mathbb{C}\right)<+\infty.$

There are at least two ways to see this. (i) Working in
$\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{cons}}(\mathrm{pt})$, use the
constructibility of $\mathrm{R}\mathscr{H}\mathit{om}$ (see [1, Desideratum
6.4.1]) together with the functor $\mathrm{Inv}^{\leq m}_{K^{H},*}$ of
truncated invariants introduced in [1, Proposition 6.6.7] ($m\gg 0$) to reach
$\operatorname{Ext}^{n}$. (ii) Use $m$-acyclic resolutions and holonomicity to
access $\operatorname{Ext}^{n}$: see [9, Proposition 2.13.1] together with the
proof of [9, Theorem 2.13]. ∎

In particular, the Euler–Poincaré characteristic for branching laws

$\mathrm{EP}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C}):=\sum_{n}(-1)^{n}\dim{}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},\mathbb{C})$
(6.3.1)

is well-defined, for all object $M$ of
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{HC}}(\mathfrak{g},K)$.

######  Theorem 6.3.4.

Take $K^{H}=\\{1\\}$ in the formalism above. Let $M$ be in
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{HC}}(\mathfrak{g},K)$ and set
$\mathcal{L}:=\mathbf{Loc}_{X,\\{1\\}}(M)$. Define the solution complex
$\mathrm{Sol}_{X}(\mathcal{L})\in\mathsf{D}^{\mathrm{b}}_{\mathrm{cons}}(X)$
of $\mathcal{L}$ by [15, p.118]. Then

$\operatorname{RHom}_{\mathfrak{h}}(M|_{H},\mathbb{C})\simeq
i_{x}^{*}\mathrm{Sol}_{X}(\mathcal{L}).$

Consequently, the $\mathrm{EP}_{\mathfrak{h},\\{1\\}}(M|_{H},\mathbb{C})$ in
(6.3.1) equals the local Euler–Poincaré characteristic

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

of $\mathrm{Sol}_{X}(\mathcal{L})$ at $x$, which can be expressed in terms of
characteristic cycles and Euler obstructions by Kashiwara’s local index
theorem [15, Theorem 4.6.7].

###### Proof.

By [15, Proposition 4.7.4] we have

$\mathrm{Sol}_{X}(\mathcal{L})[\dim
X]\simeq\mathrm{DR}_{X}\left(\mathbb{D}_{X}\mathcal{L}\right)$

canonically, where $\mathrm{DR}_{X}$ (resp. $\mathbb{D}_{X}$) denotes the non-
equivariant de Rham functor (resp. duality endo-functor) for $X$. Since
$\mathrm{DR}_{\mathrm{pt}}=\mathrm{id}$, the Riemann–Hilbert correspondence
leads to

$i_{x}^{*}\mathrm{Sol}_{X}(\mathcal{L})\simeq\left(\mathbb{D}_{\mathrm{pt}}i_{x}^{!}\mathcal{L}\right)[-\dim
X]\simeq\mathbb{D}_{\mathrm{pt}}\left(i_{x}^{!}\mathcal{L}[\dim X]\right).$

Now put $K^{H}=\\{1\\}$ in Theorem 6.3.2 to infer that
$i_{x}^{*}\mathrm{Sol}_{X}(\mathcal{L})\simeq\operatorname{RHom}_{\mathfrak{h}}(M|_{H},\mathbb{C})$.
The remaining assertions follow at once. ∎

The isomorphism in Theorem 6.3.4 generalizes [25, Proposition 10.2].

We now specialize to the case when $M$ is concentrated in degree zero.

######  Corollary 6.3.5.

For every Harish-Chandra $(\mathfrak{g},K)$-module $V$ and $n\geq 0$, the
dimension of

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

is finite. It also equals
$\dim\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H})$.

###### Proof.

The isomorphism in the first part is Corollary 3.3.2. The second part follows
from Corollary 6.1.3. ∎

The finiteness in Corollary 6.3.5 is also a consequence of a more general
result of M. Kitagawa [18, Fact 4.7]; a uniform bound is given in [18,
Corollary 7.17]. In the case $K^{H}=\\{1\\}$, the finiteness of
$\dim\operatorname{H}_{n}(\mathfrak{h};V|_{H})$ is shown in [2, Proposition
4.2.2] when $K$ is a symmetric subgroup in good position relative to $H$.

Likewise, Proposition 6.2.4 and Corollary 5.3.5 lead to the following
statement for relative Lie algebra homologies.

######  Proposition 6.3.6.

Let $V$ be a Harish-Chandra $(\mathfrak{g},K)$-module. There are canonical
isomorphisms

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

for all $n\in\mathbb{Z}$; when $K^{H}=\\{1\\}$, it is also isomorphic to

$\operatorname{H}^{-n-\dim X}\left(i_{x}^{*}\mathbf{Loc}_{X}(V)\right).$

To better understand the effect of $\operatorname{H}^{-n+\dim
X}\operatorname{L}\\!\left(\mathrm{coInv}^{\mathbb{C},K^{H}}_{\mathbb{C},\\{1\\}}\right)$,
we remark that on the constructible side, it matches the composition

$\mathsf{D}^{\mathrm{b}}_{K^{H},\mathrm{cons}}(\mathrm{pt})\xrightarrow{\mathrm{Inv}^{\geq
m}_{K^{H},!}}{}^{\mathrm{p}}\mathsf{D}^{-}_{\mathrm{cons}}(\mathrm{pt})^{\geq
m}\xrightarrow{{}^{\mathrm{p}}\operatorname{H}^{-n\pm\dim
X}}\left\\{\text{f.d.}\;\mathbb{C}\text{-vector spaces}\right\\}$

for $m\leq-n+\dim X$, where $\mathrm{Inv}^{\geq m}_{K^{H},!}$ stands for the
functor of truncated co-invariants in [1, Proposition 6.6.7]. This can be
deduced from Proposition 4.3.7 and adjunction. For a detailed discussion, see
[1, §6.6].

From this, the finiteness of
$\dim\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H})$ can be deduced directly
from Proposition 6.3.6.

###  6.4 The monodromic setting

Let $G$ and $X=H\backslash G$ be as in §6.3, so $H$ is reductive. In addition,
we fix a character of Lie algebra

$\chi:\mathfrak{h}\to\mathbb{C}.$

It must factor through the Lie algebra of a torus quotient, say
$S=H/\underline{H}$ where $\underline{H}\lhd H$. Accordingly, there is a
morphism of $G$-varieties

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

Note that $S$ acts on the left of $\tilde{X}$, commuting with $G$ and makes
$\pi$ into an $S$-torsor. Since $S$ is a torus, $\pi$ is even Zariski-locally
trivial. Also note that $\tilde{X}$ is affine since $X$ is. In [9, 2.2], $\pi$
is said to make $X$ into an _$S$ -monodromic $G$-variety_.

The character $\chi$ corresponds to an element of $\mathfrak{s}^{*}$. Let
$\mathfrak{m}_{\chi}\subset\operatorname{Sym}(\mathfrak{s})$ be the
corresponding maximal ideal; its image in
$\widetilde{\mathscr{D}}_{X}:=(\pi_{*}\mathscr{D}_{\tilde{X}})^{S}$ is
central. Recall the following notion from [4, p.24] (see also [25]).

######  Definition 6.4.1.

Set
$\mathscr{D}_{X,\chi}:=\widetilde{\mathscr{D}}_{X}/\mathfrak{m}_{\chi}\widetilde{\mathscr{D}}_{X}$
be the sheaf of algebras of twisted differential operators (TDO’s) attached to
$\chi$. Put $D_{X,\chi}:=\Gamma(X,\mathscr{D}_{X,\chi})$.

By trivializing $\pi$ locally over $X$, we see that $\mathscr{D}_{X,\chi}$ is
a locally trivial sheaf of TDO’s. Taking $\chi=0$ reverts to the untwisted
version $\mathscr{D}_{X}$.

######  Remark 6.4.2.

For later use, we remark that taking inverse images realizes an equivalence
from $\mathscr{D}_{X,\chi}\text{-}\mathsf{Mod}$ to the category of
$(S,\chi)$-monodromic $\mathscr{D}_{\tilde{X}}$-modules, the latter being a
twisted version of $S$-equivariance; see [25, (2.3) and Proposition 2.8] or
[4, 1.8.10 Lemma] for details. A further inverse image via
$G\to\underline{H}\backslash G=\tilde{X}$ brings us to
$(H^{\mathrm{op}},\chi)$-monodromic $\mathscr{D}_{G}$-modules.

The actions of $G$ and $S$ commute, hence we obtain $G$-equivariant
homomorphisms
$U(\mathfrak{g})\to\widetilde{\mathscr{D}}_{X}\to\mathscr{D}_{X,\chi}$. Since
$X$ is affine, we may and will work with $D_{X,\chi}$-modules instead. The
previous constructions now become

$D_{X,\chi}\simeq D_{\tilde{X}}^{S}/\mathfrak{m}_{\chi}D_{\tilde{X}}^{S}.$

We obtain a natural $G$-equivariant homomorphism $j:U(\mathfrak{g})\to
D_{X,\chi}$.

Given a reductive subgroup $K\subset G$, there is also a monodromic version of
Beilinson’s equivalence (Theorem 3.3.1), identifying
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}(D_{X,\chi},K)$ with the monodromic
equivariant derived category $\mathsf{D}^{\mathrm{b}}_{K,\chi}(X)$ of
$D$-modules; see [9, 2.14].

######  Remark 6.4.3.

Since we fixed $\chi$, there is no need to consider
$(\tilde{\mathscr{D}}_{X},F|K)$-modules as in [9, 2.5] where $F:=K\times S$.
This corresponds to taking $I=\mathfrak{m}_{\chi}$ in the formalism of [9,
2.3].

In the context of TDO’s arising from monodromic structures, the notions of
holonomicity and regularity can be found in [17, §7.14]444The author is
indebted to Masatoshi Kitagawa for clarifications on this point.. They reduce
to the untwisted theory by taking inverse images via $\pi$. We define
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}+}(D_{X,\chi},K)$
accordingly, cf. Definition 5.3.2.

The derived monodromic localization functor

$\mathbf{Loc}_{X,\chi}=\mathbf{Loc}_{X,K,\chi}:{}^{\mathrm{h}}\mathsf{D}(\mathfrak{g},K)\to{}^{\mathrm{h}}\mathsf{D}(D_{X,\chi},K)$
(6.4.1)

can still be defined, with amplitude in $[-\dim G,0]$.

The following construction is needed for the monodromic counterpart of Theorem
5.3.4. Define the commutative algebra $\mathcal{Z}(\tilde{X})$ for the
$S\times G$-variety $\tilde{X}$ by applying Definition 5.5.4. There is a
homomorphism

$\operatorname{Sym}(\mathfrak{s})\otimes\mathcal{Z}(\mathfrak{g})\simeq\mathcal{Z}(\mathfrak{s}\times\mathfrak{g})\to\mathcal{Z}(\tilde{X})\subset
D_{\tilde{X}}^{S\times G}\subset D_{\tilde{X}}^{S}$

of algebras. Define

$\mathcal{Z}_{\chi}(X):=\mathcal{Z}(\tilde{X})/\mathfrak{m}_{\chi}\mathcal{Z}(\tilde{X}).$

We obtain homomorphisms

$\mathcal{Z}(\mathfrak{g})\to\mathcal{Z}_{\chi}(X)\to D_{X,\chi}^{G}\subset
D_{X,\chi}.$

Thus $\mathcal{Z}_{\chi}(X)$ acts on the right of $\mathbf{Loc}_{X,\chi}$
through $\mathcal{Z}_{\chi}(X)\to D_{X,\chi}^{G}$, as in the non-twisted case
(5.1.1).

######  Lemma 6.4.4.

The algebra $\mathcal{Z}_{\chi}(X)$ is commutative. Moreover, $D_{X,\chi}$ is
projective over $\mathcal{Z}_{\chi}(X)$ and $\mathcal{Z}_{\chi}(X)$ is
finitely generated over $\mathcal{Z}(\mathfrak{g})$, both as left and right
modules.

###### Proof.

We know that $\mathcal{Z}(\tilde{X})$ is commutative, hence so is
$\mathcal{Z}_{\chi}(X)$.

Next, the projectivity and finite generation as left and right modules hold on
the level of

$\mathcal{Z}(\mathfrak{s}\times\mathfrak{g})\to\mathcal{Z}(\tilde{X})\hookrightarrow
D_{\tilde{X}}$

by [21]. Note that $\mathcal{Z}(\tilde{X})$ is actually included in
$D_{\tilde{X}}^{S}$. We claim that $D_{\tilde{X}}^{S}$ is also projective as
left and right $\mathcal{Z}(\tilde{X})$-modules.

Since $S$ is reductive, acting algebraically on $D_{\tilde{X}}$ and trivially
on $\mathcal{Z}(\tilde{X})$, one can realize $D_{\tilde{X}}^{S}$ canonically
as a direct $\mathcal{Z}(\tilde{X})$-summand of $D_{\tilde{X}}$, thus the
claim follows at once. This can also be seen by identifying $S$-action on
$D_{\tilde{X}}$ with a grading by the cocharacter lattice of $S$.

By the claim, we may tensor the homomorphisms

$\mathcal{Z}(\mathfrak{s}\times\mathfrak{g})\to\mathcal{Z}(\tilde{X})\hookrightarrow
D_{\tilde{X}}^{S}$

with $\operatorname{Sym}(\mathfrak{s})/\mathfrak{m}_{\chi}$ over
$\operatorname{Sym}(\mathfrak{s})$ on the left or right. The required
projectivity and finite generation persist. ∎

######  Theorem 6.4.5.

Suppose that $X$ is affine and spherical, $K$ is a reductive spherical
subgroup of $G$, and $V$ is a Harish-Chandra $(\mathfrak{g},K)$-module. Then
$\mathbf{Loc}_{X,\chi}(V)$ lies in
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{rh}+}(D_{X,\chi},K)$.

###### Proof.

Recall the strategy in the non-monodromic case in §§5.4–5.5. The key criterion
from [25] applies in the monodromic setting. In fact, after ascending to $G$
by Remark 6.4.2, it boils down to showing that a finitely generated,
$(H^{\mathrm{op}}\times K,\chi\otimes\mathrm{triv})$-monodromic and locally
$\mathcal{Z}(\mathfrak{g}\times\mathfrak{g})$-finite $D_{G}$-module is regular
holonomic, and this is indeed covered by [25, Theorem 5.6]. Note that $\chi$
is required to be trivial on the nilpotent radical of $\mathfrak{h}$ in loc.
cit., but $H$ is reductive here.

The other ingredients in the proof carry over, by using Lemma 6.4.4. The bound
on the characteristic varieties are obtained in the same way as [25,
Proposition 3.4]. ∎

Likewise, Corollary 5.3.5 and Proposition 5.5.10 also admit monodromic
versions.

Given the base-point $x$ of $X$, the inclusion map $i_{x}$ gives rise to an
inclusion of $S$-monodromic $K^{H}$-varieties, given by the Cartesian square

${\pi^{-1}(x)}$${\tilde{X}}$${\mathrm{pt}}$${X.}$$\scriptstyle{\pi}$$\scriptstyle{i_{x}}$

Next, suppose that

  * •

$K^{H}$ is a reductive subgroup of $K$ that fixes $x$,

  * •

a character $K^{H}\to\mathbb{G}_{\mathrm{m}}$ is given, whose derivative
coincides with $\chi|_{\mathfrak{k}^{H}}$.

Therefore $\chi$ can be viewed as a $(\mathfrak{h},K^{H})$-module.

Note that
$D_{\mathrm{pt},\chi}\simeq\operatorname{Sym}(\mathfrak{s})/\mathfrak{m}_{\chi}$.
Define the $K^{H}$-equivariant $D_{\mathrm{pt},\chi}$-module
$\mathbb{C}_{\chi}$ by letting $\mathfrak{s}$ (resp. $K^{H}$) act on
$\mathbb{C}$ through $\chi$ (resp. the given
$K^{H}\to\mathbb{G}_{\mathrm{m}}$). Below is the monodromic counterpart of
Proposition 6.2.2.

######  Proposition 6.4.6.

For all objects $M$ of ${}^{\mathrm{h}}\mathsf{D}^{-}(\mathfrak{g},K)$, there
are canonical isomorphisms in $\mathsf{D}^{+}(\mathbb{C})$:

$\operatorname{RHom}_{\mathfrak{h},K^{H}}(M|_{H},\chi)\simeq\operatorname{RHom}_{D_{\mathrm{pt},\chi},K^{H}}\left(i_{x}^{\bullet}(\mathbf{Loc}_{X,\chi}(M)),\mathbb{C}_{\chi}\right)$

where $i_{x}^{\bullet}$ is the h-derived inverse image for $K^{H}$-equivariant
monodromic $D$-modules.

###### Proof.

Same as the non-monodromic case. It suffices to replace $\mathbb{C}$ by
$\mathbb{C}_{\chi}$, then apply the following analog of Proposition 5.2.1. ∎

######  Proposition 6.4.7.

For every $\mathfrak{g}$-module $V$, its space of $(\mathfrak{h},\chi)$-co-
invariants is isomorphic to
$\mathbb{C}\underset{\mathscr{O}_{X,x}}{\otimes}\left(\mathscr{D}_{X,\chi}\underset{U(\mathfrak{g})}{\otimes}V\right)$,
by sending the image of $v\in V$ to $1\otimes(1\otimes v)$.

###### Proof.

Same as the one given in [6, Lemma 2.2]. It suffices to replace the map
$\mathscr{O}_{X}\otimes\mathfrak{g}\to\mathscr{D}_{X}$ by
$\mathscr{O}_{X}\otimes\mathfrak{g}\to\mathscr{D}_{X,\chi}$. ∎

Below is the monodromic counterpart of Theorem 6.3.2. Define the functors
$i_{x}^{!}$, $i_{x}^{*}$ and $i_{x,*}$ in the same manner as in §6.3.

######  Theorem 6.4.8.

Let $M$ be an object of the category
${}^{\mathrm{h}}\mathsf{D}^{\mathrm{b}}_{\mathrm{HC}}(\mathfrak{g},K)$. There
are canonical isomorphisms

$\operatorname{RHom}_{\mathfrak{h},K^{H}}(M|_{H},\chi)\simeq\operatorname{RHom}_{D_{\mathrm{pt},\chi},K^{H}}\left(i_{x}^{!}\mathbf{Loc}_{X,\chi}(M)[\dim
X],\mathbb{C}_{\chi}\right).$

When $K^{H}=\\{1\\}$, this is also isomorphic to

$\operatorname{RHom}_{D_{\mathrm{pt},\chi}}\left(i_{x}^{*}\mathbf{Loc}_{X,\chi}(M)[-\dim
X],\mathbb{C}_{\chi}\right)\\\
\simeq\operatorname{RHom}_{D_{X,\chi}}\left(\mathbf{Loc}_{X,\chi}(M),i_{x,*}(\mathbb{C}_{\chi})[\dim
X]\right).$

All these complexes of $D$-modules are bounded with regular holonomic
cohomologies.

###### Proof.

Since the Riemann–Hilbert correspondence has a monodromic version [16, Theorem
3.16.2], and similarly in the equivariant case, one can repeat the proof of
Theorem 6.3.2. Caution: the monodromic Riemann—Hilbert correspondence lands in
the derived category of “twisted sheaves”. ∎

Accordingly, we deduce the consequences about:

  * •

${}^{\mathrm{h}}\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(M|_{H},\chi)$ (cf.
Corollary 6.3.3),

  * •

$\operatorname{Ext}^{n}_{\mathfrak{h},K^{H}}(V|_{H},\chi)$,
$\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H}\otimes\chi^{\vee})$ (cf.
Corollary 6.3.5), where $V$ is a Harish-Chandra $(\mathfrak{g},K)$-module and
$\chi^{\vee}$ means the contragredient of $\chi$.

The finiteness of
$\dim\operatorname{H}_{n}(\mathfrak{h},K^{H};V|_{H}\otimes\chi^{\vee})$ is
still covered by the work of M. Kitagawa [18, Fact 4.7].

Finally, the Proposition 6.3.6 also has a monodromic analogue:

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

When $K^{H}=\\{1\\}$, it is isomorphic to $\operatorname{H}^{-n-\dim
X}\left(i_{x}^{*}\mathbf{Loc}_{X,\chi}(V)\right)$.

##  7 Comparison with the analytic picture

###  7.1 Schwartz homologies

In this subsection, we work on the analytic side of representation theory.
Therefore, we take

  * •

$G$: an almost linear Nash group,

  * •

$K$: a maximal compact subgroup of $G$.

They are both Lie groups, and we will consider continuous representations of
them.

Let $E$ be a smooth Fréchet representation of $G$ of moderate growth. In [11],
Y. Chen and B. Sun defined the _Schwartz homologies_
$\operatorname{H}^{\mathcal{S}}_{n}(G;E)$ for $n\in\mathbb{Z}_{\geq 0}$. These
are locally convex topological vector spaces which are possibly non-Hausdorff,
functorial in $E$, and satisfy various desirable properties such as long exact
sequences and Shapiro’s lemma. In particular,

$\operatorname{H}^{\mathcal{S}}_{0}(G;E)=E_{G}:=E\big{/}\sum_{g\in
G}(g-\mathrm{id})(E)\;+\;\text{quotient topology}.$

We refer to [11] for all the details and terminologies.

Let $\mathfrak{g}$ be the complexified Lie algebra of $G$. Every smooth
representation $E$ of $G$ gives rise to an $(\mathfrak{g},K)$-module in a
generalized sense: it is a vector space equipped with compatible structures of
$\mathfrak{g}$-module and $K$-module; unlike §3.1, the $K$-action here is not
assumed to be algebraic or locally finite, but only smooth.

Let $\mathfrak{p}:=\mathfrak{g}/\mathfrak{k}$, on which $K$ acts; we may and
do choose a decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ as
$K$-modules, and denote by $\mathcal{P}:\mathfrak{g}\to\mathfrak{p}$ the
corresponding projection.

######  Definition 7.1.1.

For an $(\mathfrak{g},K)$-module $E$, the relative Lie algebra homologies of
$E$ are

$\operatorname{H}_{n}(\mathfrak{g},K;E):=\operatorname{H}_{n}\left(C(\mathfrak{g},K;E)\right)$

where the _standard complex_ $C(\mathfrak{g},K;E)$ for $E$ is defined by

$C_{n}(\mathfrak{g},K;E):=(\bigwedge^{n}\mathfrak{p})\underset{K}{\otimes}E,\quad
n\in\mathbb{Z}_{\geq 0}.$

Here we make the left $K$-module $\bigwedge^{n}\mathfrak{p}$ into a right one
by $uk:=k^{-1}u$, so $\otimes_{K}$ is simply the tensor product over the
abstract group algebra $\mathbb{C}[K]$. Alternatively,
$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E$ is also the space of co-
invariants (algebraically) of $\bigwedge^{n}\mathfrak{p}\otimes E$ under the
diagonal $K$-action.

The boundary maps in the complex

$\partial_{n}:C_{n}(\mathfrak{g},K;E)\to C_{n-1}(\mathfrak{g},K;E)$

for $n\geq 1$ are the usual ones (cf. [20, (2.126b)]):

$\partial_{n}((\xi_{1}\wedge\cdots\wedge\xi_{n})\otimes
v)=\sum_{i=1}^{n}(-1)^{i}(\cdots\wedge\widehat{\xi_{i}}\wedge\cdots)\otimes\xi_{i}v\\\
+\sum_{p<q}(-1)^{p+q}(\mathcal{P}[\xi_{p},\xi_{q}]\wedge\cdots\widehat{\xi_{p}}\cdots\widehat{\xi_{q}}\cdots)\otimes
v.$

Now let $E$ be a smooth Fréchet representation of $G$ of moderate growth. Then
each $C_{n}(\mathfrak{g},K;E)$ gets topologized, each $\partial_{n}$ is
continuous, and the homologies are endowed with quotient topologies. In [11,
Theorem 7.7] it is proved that

$\operatorname{H}^{\mathcal{S}}_{n}(G;E)\simeq\operatorname{H}_{n}(\mathfrak{g},K;E),\quad
n\in\mathbb{Z}_{\geq 0}$ (7.1.1)

canonically and topologically.

######  Definition 7.1.2.

Let $E$ be a smooth Fréchet representation of $G$ of moderate growth. Denote
by $E^{K\text{-fini}}$ the subspace of $K$-finite vectors in $E$.

Therefore $E^{K\text{-fini}}$ is an $(\mathfrak{g},K)$-module in the usual
algebraic sense. By functoriality and (7.1.1), we obtain canonical linear maps

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

######  Proposition 7.1.3.

Let $E$ be a smooth Fréchet representation of $G$ of moderate growth. For
every $n$, we have

$\operatorname{H}_{n}\left(\mathfrak{g},K;E^{K\text{-fini}}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{H}^{\mathcal{S}}_{n}(G;E)$

as vector spaces.

###### Proof.

Let $\mathrm{Irr}(K)$ be the set of isomorphism classes of finite-dimensional
irreducible representations of $K$. Let $\mathcal{M}(K)$ be the unital algebra
of bounded complex Borel measures on $K$ under convolution. Let $R(K)$ be the
non-unital subalgebra comprised of left $K$-finite (equivalently, right
$K$-finite) elements, which is also spanned by the matrix coefficients from
$\mathrm{Irr}(K)$ once we fix a Haar measure. See [20, I.2].

By continuity and completeness, $\mathcal{M}(K)$ acts on $E$, thus so does
$R(K)$.

For each $\gamma\in\mathrm{Irr}(K)$, define

$\chi_{\gamma}:=\dim\gamma\cdot\Theta_{\gamma^{\vee}}\cdot\frac{\mathop{}\\!\mathrm{d}k}{\mathrm{mes}(K)}\;\in
R(K),$

where $\Theta_{\gamma^{\vee}}$ is the character of the contragredient
$\gamma^{\vee}$; see [20, (1.23)]. This is an idempotent in $R(K)$ which is
$K$-invariant under conjugation.

For every finite subset $\Gamma$ of $\mathrm{Irr}(K)$, we define
$\chi_{\Gamma}:=\sum_{\gamma\in\Gamma}\chi_{\gamma}$. This is still a
$K$-invariant idempotent. When applied to locally $K$-finite representations,
its effect is to project to the $\Gamma$-isotypic part.

The evident maps $C_{n}(\mathfrak{g},K;E^{K\text{-fini}})\to
C_{n}(\mathfrak{g},K;E)$ give a morphism between complexes. It suffices to
show it is an isomorphism in each degree $n$.

First, we prove that $C_{n}(\mathfrak{g},K;E^{K\text{-fini}})\to
C_{n}(\mathfrak{g},K;E)$ is surjective. Take $\Gamma$ that contains all
irreducible constituents of $\bigwedge^{n}\mathfrak{p}$. Let $r\mapsto
r^{\prime}$ be the anti-involution of $R(K)$ induced by $k\mapsto k^{-1}$. For
every $\omega\otimes v\in\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E$, we
claim that

$\omega\otimes v=\chi_{\Gamma}\omega\otimes
v=\omega\otimes\chi^{\prime}_{\Gamma}v.$

Indeed, the first equality follows from the choice of $\Gamma$. As for the
second one, in the proof of [11, Theorem 7.7], a “strong projective
resolution” (45) of $E$ is shown to be topologically isomorphic to
$C(\mathfrak{g},K;E)$ after taking $(\cdot)_{G}$. It is part of the theory of
Schwartz homologies, especially [11, Theorem 5.9], that taking $(\cdot)_{G}$
of such a resolution produces Fréchet spaces in each degree. Hence the
algebraic co-invariants in the formation of
$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}v$ are actually topological,
i.e.

$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E=(\bigwedge^{n}\mathfrak{p}\otimes
E)\bigg{/}\overline{\langle k\omega\otimes kv=\omega\otimes v:k\in
K,\ldots\rangle}.$

This implies $\chi_{\Gamma}\omega\otimes
v=\omega\otimes\chi^{\prime}_{\Gamma}v$ in
$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E$, by approximating
$\chi_{\Gamma}$ by Dirac measures. Surjectivity follows since
$\chi_{\Gamma}E=E^{\Gamma}$, the $\Gamma$-isotypic part of
$E^{K\text{-fini}}$.

Let us show injectivity. By general properties of $\otimes$ over
$\mathbb{C}[K]$, we have

$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}(E^{K\text{-fini}})=\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}\varinjlim_{\Gamma}E^{\Gamma}\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\varinjlim_{\Gamma}\left(\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E^{\Gamma}\right)$

where the $\varinjlim$ over finite subsets $\Gamma\subset\mathrm{Irr}(K)$ is
filtered. The map from the leftmost term to
$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E$ is determined by the
compatible family of evident maps

$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E^{\Gamma}\to\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E.$

Fixing $\Gamma$, the map above is a split injection since $\chi_{\Gamma}$ is a
$K$-invariant idempotent, so

$E=\chi_{\Gamma}E\oplus(\mathrm{id}-\chi_{\Gamma})E=E_{\Gamma}\oplus(\mathrm{id}-\chi_{\Gamma})E$

as $\mathbb{C}[K]$-modules. As the filtered $\varinjlim$ of injections is
still injective, this shows the injectivity of
$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}(E^{K\text{-fini}})\to\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E$.

Note that the same method shows that
$\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E^{\Gamma}\to\bigwedge^{n}\mathfrak{p}\underset{K}{\otimes}E^{\Gamma^{\prime}}$
is split injective whenever $\Gamma\subset\Gamma^{\prime}$. ∎

###  7.2 Comparison map

We now consider almost linear Nash groups

${H}$${G}$${K^{H}}$${K.}$${\subset}$${\subset}$${\subset}$${\subset}$ (7.2.1)

analogously to the algebraic setting of (6.2.1), the assumptions now being

  * •

$G$ is reductive,

  * •

$K$ (resp. $K^{H}$) is a maximal compact subgroup of $G$ (resp. $H$); this
implies $K^{H}=H\cap K$.

We consider _Casselman–Wallach representations_ of $G$, i.e. smooth admissible
Fréchet representation of moderate growth; taking $K$-finite vectors
establishes an equivalence between this category and the category of Harish-
Chandra $(\mathfrak{g},K)$-modules. See [7].

For a smooth Fréchet representation $E$ of $G$ of moderate growth, its
restriction $E|_{H}$ to $H$ is still smooth Fréchet of moderate growth, by
unraveling definitions. However, admissibility is usually lost.

In view of (7.1.1) applied to $H$ and $K^{H}$, the inclusion
$E^{K\text{-fini}}\subset E$ induces natural linear maps

$c_{n}(E):\operatorname{H}_{n}\left(\mathfrak{h},K^{H};E^{K\text{-fini}}\right)\to\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H}),\quad
n\in\mathbb{Z}_{\geq 0}$ (7.2.2)

between vector spaces.

######  Definition 7.2.1.

We call (7.2.2) the _comparison maps_ for the Casselman–Wallach representation
$E$ relative to the data (7.2.1).

The following question is thus natural.

For what data (7.2.1), representations $E$ and $n$ is $c_{n}(E)$ an
isomorphism?

######  Remark 7.2.2.

In view of Proposition 7.1.3, the map $c_{n}(E)$ is the same as the

$\operatorname{H}_{n}\left(\mathfrak{h},K^{H};E^{K\text{-fini}}\right)\to\operatorname{H}_{n}\left(\mathfrak{h},K^{H};E^{K^{H}\text{-fini}}\right)$

induced by $E^{K\text{-fini}}\subset E^{K^{H}\text{-fini}}$.

For such questions, it is usual to focus on the case when $H$ is a real
spherical subgroup of $G$. An affirmative answer for all $E$ will have strong
consequences. For example, in degree $n=0$ it amounts to _automatic
continuity_ for invariant linear functionals (cf. [2, Theorem A]), which are
hard-core results available only in sporadic cases, for example when (i) $n=0$
and $H\subset G$ is a symmetric subgroup [32], or (ii) when $n$ is arbitrary
and $H\subset G$ is a maximal unipotent subgroup [14, 24].

In order to apply the earlier results, let us assume further that

  * •

all the groups $G$, $K$, $H$, $K^{H}$ arise from the algebraic setting
(6.2.1), namely they arise from $\mathbb{R}$-points of affine groups;

  * •

$H$ is a reductive spherical subgroup of $G$, when viewed as complex groups.

Since $V:=E^{K\text{-fini}}$ is a Harish-Chandra $(\mathfrak{g},K)$-module, by
Corollary 6.3.5 or results of [18], the source of $c_{n}(E)$ is finite-
dimensional. Under these assumptions, whether $c_{n}(E)$ is an isomorphism or
not is still wide open. In §7.3, we will consider some cases in which
$c_{n}(E)$ turn out to be isomorphisms for all $n$.

######  Remark 7.2.3.

A smooth Fréchet representation $F$ of $H$ of moderate growth is said to be
_homologically finite_ (resp.  _homologically separated_) if
$\operatorname{H}^{\mathcal{S}}_{n}(H;F)$ is finite-dimensional (resp.
Hausdorff) for all $n$; homological finiteness implies homological separation.
See [3] for an overview. With all the previous assumptions (algebraicity,
sphericity), it is still unknown whether $E|_{H}$ is homologically finite or
not, except for specific choices of $H$ or $E$, or for $n=0$. If $c_{n}(E)$ is
surjective, then $E|_{H}$ will be homologically finite. In particular,
homological finiteness will hold in all the examples of §7.3.

######  Remark 7.2.4.

In parallel with the monodromic setting considered in §6.4, one should also
consider $\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H}\otimes\chi^{-1})$ when
$\chi$ is a smooth character of $H$ factoring through the unipotent radical of
$H$, and modify (7.2.2) accordingly.

###  7.3 Examples of comparison isomorphisms

Consider the groups in (7.2.1); we assume that they are all affine groups in
what follows.

######  Proposition 7.3.1.

Let $E$ be a Casselman–Wallach representation of $G$. If
$E^{K^{H}\text{-fini}}=E^{K\text{-fini}}$, then the comparison map $c_{n}(E)$
is an isomorphisms for all $n$.

###### Proof.

Immediate from Remark 7.2.2. ∎

######  Example 7.3.2.

Suppose $E$ is of the form $E=B^{\infty}$, where $B$ is an irreducible unitary
representation of $G$ and $B^{\infty}$ denotes its smooth part. We say $B$ is
_$K^{H}$ -admissible_ if $B|_{K^{H}}$ decomposes discretely into a Hilbert
direct sum of irreducibles with finite multiplicities. By [22, Proposition
1.6], this implies that $B^{K\text{-fini}}$ is _discretely decomposable_ as an
$(\mathfrak{h},K^{H})$-module. The notions of admissibility and discrete
decomposability under restriction have been extensively studied by T.
Kobayashi and his collaborators; see [22] or [23, §4.1] for an overview.

We claim that when $B$ is $K^{H}$-admissible, $c_{n}(E)$ is an isomorphism for
all $n$.

Indeed, by [22, Proposition 1.6], the $K^{H}$-admissibility implies

$B^{K\text{-fini}}=(B|_{H})^{\infty,K^{H}\text{-fini}};$

the left-hand side is $E^{K\text{-fini}}$ whilst the right-hand side contains
$B^{\infty,K^{H\text{-fini}}}=E^{K^{H}\text{-fini}}$. Hence Proposition 7.3.1
can be applied.

Discrete decomposability is a rare phenomenon, so the scope of the condition
above is limited. Nonetheless, it covers some interesting families: suppose
$H\subset G$ is a symmetric subgroup, and $H/K^{H}\to G/K$ is a holomorphic
embedding of Hermitian symmetric domains, then every unitary highest weight
module (eg. holomorphic discrete series) is $K^{H}$-admissible, hence
discretely decomposable over $(\mathfrak{h},K^{H})$. See [22, Fact 5.4].

######  Example 7.3.3.

The $K^{H}$-admissibility mentioned above always holds if $H=K$.

Before stating the next example, we record a standard reduction due to
Hecht–Taylor.

######  Proposition 7.3.4.

If $c_{n}(E)$ is an isomorphism for all $n$ and all principal series
representations $E$, then the same is true for all Casselman–Wallach
representations $E$.

###### Proof.

In view of the isomorphisms of abstract vector spaces

$\operatorname{H}_{n}\left(\mathfrak{h},K^{H};E^{K^{H}\text{-fini}}\right)\simeq\operatorname{H}_{n}\left(\mathfrak{h},K^{H};E\right)\simeq\operatorname{H}^{\mathcal{S}}_{n}(H;E|_{H})$

of Proposition 7.1.3, one can reiterate the proof of [14, Proposition 3],
which proceeds by resolving the $(\mathfrak{g},K)$-module $E^{K\text{-fini}}$
into a complex of principal series via Casselman’s Subrepresentation Theorem,
and analyzing the corresponding map between two double complexes, each
computing $(\mathfrak{h},K^{H})$-homologies in the vertical direction.

The point here is that
$(\bigwedge^{p}\mathfrak{p})\underset{K^{H}}{\otimes}(\cdot)$ is an exact
functor from locally $K^{H}$-finite representations to vector spaces, for
every $p\in\mathbb{Z}$. Indeed, taking $K^{H}$-co-invariants is an exact
functor on such representations, as follows from [20, Proposition 1.18]. In
[14] one took $\otimes$ instead of $\otimes_{K^{H}}$, but the exactness is all
one needs to analyze the spectral sequence $E_{2}^{q,-p}$ therein. ∎

######  Example 7.3.5.

Take $G=\operatorname{SL}(2)$, $K=\operatorname{SO}(2)$,
$H\simeq\mathbb{G}_{\mathrm{m}}$ being the diagonal torus, and $K^{H}=K\cap
H$; note that $H$ is a symmetric subgroup of $G$, and $K^{H}=\mu_{2}:=\\{\pm
1\\}$. We also take $U$ to be the upper-triangular unipotent subgroup, and
identify all these groups with their $\mathbb{R}$-points. By identifying
$\mathbb{R}^{2}$ with row vectors and $U\backslash G$ with
$\mathbb{R}^{2}\smallsetminus\\{0\\}$, the principal series can be described
as follows, on the level of Casselman–Wallach representations.

Let $\lambda\in\mathbb{C}$ and $\epsilon\in\\{0,1\\}$. Define

$V^{\epsilon}_{\lambda}:=\left\\{\begin{array}[]{r|l}f:\mathbb{R}^{2}\smallsetminus\\{0\\}\to\mathbb{C}&\forall
t\in\mathbb{R}^{\times},\;x\in\mathbb{R}^{2}\smallsetminus\\{0\\}\\\
\text{smooth}&f(tx)=\mathrm{sgn}(t)^{\epsilon}|t|^{\lambda}f(x)\end{array}\right\\}.$

It is topologized in the standard way, and $G$ acts by $(gf)(x)=f(xg)$.

In particular, $\bigl{(}\begin{smallmatrix}a&\\\
&a^{-1}\end{smallmatrix}\bigr{)}$ maps $f$ to $(x_{1},x_{2})\mapsto
f(ax_{1},a^{-1}x_{2})$, and $-1\in K^{H}$ maps $f$ to $(-1)^{\epsilon}f$. We
also remark that $V^{\epsilon}_{\lambda}$ is the normalized parabolic
induction of

$\begin{pmatrix}a&*\\\
&a^{-1}\end{pmatrix}\mapsto\mathrm{sgn}(a)^{\epsilon}|a|^{-\lambda-1},\quad
a\in\mathbb{R}^{\times}.$

Let $\theta:=\bigl{(}\begin{smallmatrix}1&\\\ &-1\end{smallmatrix}\bigr{)}$,
which is $K^{H}$-invariant and generates $\mathfrak{h}$. The
$\theta$-invariants in $V^{\epsilon}_{\lambda}$ are precisely the functions
$f$ which are locally constant on the real curve $x_{1}x_{2}=c$, for each
$c\in\mathbb{R}$.

Clearly, $(V^{\epsilon}_{\lambda})^{K^{H}\text{-fini}}=V^{\epsilon}_{\lambda}$
is strictly larger than $(V^{\epsilon}_{\lambda})^{K\text{-fini}}$.

For $L\in\\{K^{H},K\\}$, the complex computing
$\operatorname{H}_{\bullet}(\mathfrak{h},K^{H};(V^{\epsilon}_{\lambda})^{L\text{-fini}})$
is

$\displaystyle\mathbb{C}\theta\underset{K^{H}}{\otimes}(V^{\epsilon}_{\lambda})^{L\text{-fini}}$
$\displaystyle\to\mathbb{C}\underset{K^{H}}{\otimes}(V^{\epsilon}_{\lambda})^{L\text{-fini}},\quad\deg=-1,0,$
(7.3.1) $\displaystyle\theta\otimes v$ $\displaystyle\mapsto 1\otimes\theta
v.$

When $\epsilon=0$, the $\operatorname{H}_{1}$ is
$\\{f\in(V^{0}_{\lambda})^{L\text{-fini}}:\theta f=0\\}$. On each open
quadrant, $f$ takes the form $c|x_{1}x_{2}|^{\lambda/2}$ for some
$c\in\mathbb{C}$. In order that they glue to a smooth function on
$\mathbb{R}^{2}\smallsetminus\\{0\\}$, the constants $c$ must coincide and we
must have $\lambda\in 4\mathbb{Z}_{\geq 0}$, in which case
$f(x_{1}x_{2})=c(x_{1}x_{2})^{\lambda/2}$ is polynomial, hence $K$-finite.
Therefore

$\operatorname{H}_{1}\left(\mathfrak{h},K^{H};(V^{0}_{\lambda})^{L\text{-fini}}\right)=\begin{cases}0,&\text{if}\;\lambda\notin
4\mathbb{Z}_{\geq 0}\\\
\mathbb{C}(x_{1}x_{2})^{\lambda/2},&\text{if}\;\lambda\in 4\mathbb{Z}_{\geq
0}.\end{cases}$

When $\epsilon=1$, we have
$\mathbb{C}\theta\underset{K^{H}}{\otimes}(V^{1}_{\lambda})^{L\text{-fini}}=\mathbb{C}\underset{K^{H}}{\otimes}(V^{1}_{\lambda})^{L\text{-fini}}=0$,
thus

$\operatorname{H}_{n}\left(\mathfrak{h},K^{H};(V^{1}_{\lambda})^{L\text{-fini}}\right)=0,\quad
n\in\mathbb{Z}.$

These computations show that $c_{1}(V_{\lambda}^{\epsilon})$ is always an
isomorphism. The same also holds for $c_{0}(V_{\lambda}^{\epsilon})$, by
either invoking the automatic continuity theorem for $H\subset G$, or a direct
computation as before. By Proposition 7.3.4, it follows that $c_{n}(E)$ is an
isomorphism for all Casselman–Wallach representations $E$ and all $n$.

The example above shows that, even in the simplest case, it may happen
simultaneously that

  * •

$E^{K^{H}\text{-fini}}\supset E^{K\text{-fini}}$ strictly;

  * •

$c_{n}(E)$ is an isomorphism for all $n$;

  * •

some higher homologies of $E|_{H}$ are nonzero.

In particular, the condition in Proposition 7.3.1 is sufficient but not
necessary in general.

######  Example 7.3.6.

The proper reductive spherical subgroups $H$ of $G=\operatorname{SL}(2)$,
identified with their $\mathbb{R}$-points, are

$\begin{pmatrix}*&\\\ &*\end{pmatrix},\quad\left\langle\begin{pmatrix}*&\\\
&*\end{pmatrix},\begin{pmatrix}&1\\\
-1&\end{pmatrix}\right\rangle,\quad\operatorname{SO}(2)$

up to conjugacy. Take $K=\operatorname{SO}(2)$, so that $K^{H}:=K\cap H$ is
maximal compact in $H$ in each case.

In Example 7.3.5 it is shown that $c_{n}(E)$ is an isomorphism in the first
case, for all $E$ and $n$. We will settle the remaining cases below.

For the second case (the normalizer of the diagonal torus), retain the
notations from Example 7.3.5 and consider the principal series
$V_{\lambda}^{\epsilon}$, which suffices. One has
$K^{H}\simeq\mathbb{Z}/4\mathbb{Z}$ whose generators act on
$\theta\in\mathfrak{h}$ by $-1$. For $L\in\\{K^{H},K\\}$ we may decompose
$(V_{\lambda}^{\epsilon})^{L\text{-fini}}$ into $K^{H}$-isotypic components.
The complex (7.3.1) becomes

$(V_{\lambda}^{\epsilon})^{L\text{-fini}}[\rho]\to(V_{\lambda}^{\epsilon})^{L\text{-fini}}[\mathrm{triv}],\quad
v\mapsto\theta v,$

where $\rho$ (resp. $\mathrm{triv}$) denotes the character of $K^{H}$ mapping
the generators to $-1$ (resp. $1$), and $[\cdots]$ are the isotypic
components. As in Example 7.3.5, the homologies turn out to be independent of
$L$. Hence $c_{n}(E)$ is an isomorphism for all $E$ and $n$.

Finally, the third case $H=\operatorname{SO}(2)$ is covered by Example 7.3.3.

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