is a polynomial ring in one variable, graded so that the simple root
$\alpha\in\Sigma$ is in degree $2$. It follows that the objects in
$\mathcal{N}_{\mathrm{LV}}^{0}$ are graded $(R,R)$-bimodules. The standard
$(R,R)$-bimodules of Definition 6.2.1 are

$\displaystyle R_{id}\hskip 8.53581pt$ $\displaystyle(R\text{ as a
}(R,R)\text{-bimodule under the standard action}),\text{ and }$ $\displaystyle
R_{s}\hskip 11.38109pt$ $\displaystyle(R\text{ as a }(R,R)\text{-bimodule with
right action twisted by $s$}).$

We compute the action of $\mathbb{S}\text{Bim}=\langle
R,B_{s}\rangle_{\oplus,\ominus,(-)}$ on these generators.

(6.10) $\displaystyle
R_{id}\otimes_{R}B_{s}=R\otimes_{R}R\otimes_{R^{s}}R(1)=R\otimes_{R^{s}}R(1)=B_{s}$
(6.11) $\displaystyle
R_{s}\otimes_{R}B_{s}=R\otimes_{R}R\otimes_{R^{s}}R(1)=R_{s}\otimes_{R^{s}}R(1)=R\otimes_{R^{s}}R(1)=B_{s}$

As the square of $B_{s}$ is $B_{s}(1)\oplus B_{s}(-1)$ (equation (3.12)), we
see that

$\mathcal{N}_{\mathrm{LV}}^{0}=\langle
R_{\operatorname{id}},R_{s},B_{s}\rangle_{\oplus,\ominus,(-)}.$

(Compare the bimodule actions (6.10), (6.11), (3.12) to the convolution
actions (5.2), (5.3), (5.4).)

##  7\. The hypercohomology functor

The bridge between the geometric category $\mathcal{M}_{LV}^{0}$ and the
algebraic category $\mathcal{N}_{LV}^{0}$ is the $K$-equivariant
hypercohomology functor. As in (3.16), we have a functor

$\mathbb{H}_{K}^{\bullet}:D^{b}_{K}(X)\rightarrow H^{*}_{K}(X)\mathrm{-gmod},$

where $H^{*}_{K}(X)\mathrm{-gmod}$ is the category of graded modules over the
graded ring $H^{*}_{K}(X)$.

######  Lemma 7.0.1.

There is an isomorphism of rings

$H^{*}_{K}(X)\simeq P^{K}\otimes_{R^{W}}R,$

where $P^{K}$ is a $R^{W}$-module via the map $\phi$ in (6.8).

###### Proof.

We use standard facts about equivariant cohomology, proofs of which can be
found in [Bri98] in the case of a compact group. After a straightforward
reduction from complex reductive algebraic groups to compact groups, these
facts also hold in our setting. See also [Wys02, Prop 1.2.1].

Using three applications of the induction isomorphism in equivariant
cohomology [Bri98, §1, Rmk. 3] along with [Bri98, Prop. 1], we compute:

$\displaystyle H^{*}_{K}(G/B)$ $\displaystyle=H^{*}_{G}(G/B\times G/K)$
$\displaystyle=H^{*}_{G}(G\times_{B}G/K)$ $\displaystyle=H^{*}_{B}(G/K)$
$\displaystyle=H^{*}_{T}(G/K)$ $\displaystyle=H^{*}_{G}(G/K)\otimes_{R^{W}}R$
$\displaystyle=H^{*}_{K}(\mathrm{pt})\otimes_{R^{W}}R$
$\displaystyle=(H^{*}_{T_{K}}(\mathrm{pt}))^{W_{K}}\otimes_{R^{W}}R$
$\displaystyle=P^{K}\otimes_{R^{W}}R.$

∎

As in Section 3.6, the quotient map $P^{K}\otimes R\rightarrow
P^{K}\otimes_{R^{W}}R$ induces a fully faithful functor

(7.1) $P^{K}\otimes_{R^{W}}R\mathrm{-gmod}\rightarrow(P^{K},R)\mathrm{-gbim},$

and we identify $P^{K}\otimes_{R^{W}}R\mathrm{-gmod}$ with its image in
$(P^{K},R)\mathrm{-gbim}$. Hence we have a functor

(7.2)
$\mathbb{H}^{\bullet}_{K}:D^{b}_{K}(X)\rightarrow(P^{K},R)\mathrm{-gbim}.$

###  7.1. The essential image of the hypercohomology functor

######  Theorem 7.1.1.

The essential image of the functor

$\mathbb{H}^{\bullet}_{K}:\mathcal{M}_{LV}^{0}\rightarrow(P^{K},R)\mathrm{-gbim}$

defined in (7.2) is $\mathcal{N}_{LV}^{0}$.

By construction, $\mathcal{N}_{LV}^{0}$ is generated by objects of the form

$P_{x}\otimes_{R}B_{s_{1}}\otimes_{R}B_{s_{2}}\otimes\cdots\otimes_{R}B_{s_{n}}$

for $x\in{{}^{K}W^{\theta}}$ and $s_{i}\in S$. By Lemma 4.1.2,
$\mathcal{M}_{LV}^{0}$ is generated by objects of the form

$\bm{IC}_{\mathcal{Q}}\Theta_{s_{1}}\Theta_{s_{2}}\cdots\Theta_{s_{n}}$

for $\mathcal{Q}$ a closed $K$-orbit and $s_{i}\in S$. Hence to prove Theorem
7.1.1 it suffices to prove the following two lemmas.

######  Lemma 7.1.2.

For any closed $K$-orbit $\mathcal{Q}\subseteq X$,

$\mathbb{H}^{\bullet}_{K}(\bm{IC}_{\mathcal{Q}})=P_{x}$

for some $x\in{{}^{K}W^{\theta}}$, where $P_{x}$ is as in Definition 6.2.1.
Moreover, for every $x\in{{}^{K}W^{\theta}}$, there exists a closed $K$-orbit
$\mathcal{Q}\subseteq X$ such that
$\mathbb{H}_{K}(\bm{IC}_{\mathcal{Q}})=P_{x}$.

######  Lemma 7.1.3.

For every $s\in S$ and $\mathcal{F}\in\mathcal{M}_{LV}^{0}$,

$\mathbb{H}^{\bullet}_{K}(\mathcal{F}\Theta_{s})=\mathbb{H}^{\bullet}_{K}(\mathcal{F})\otimes_{R}B_{s}.$

######  Example 7.1.4.

$($Lemmas 7.1.2 and 7.1.3 for $SL_{2}(\mathbb{R}))$ In our running example
(Examples 3.2.4, 4.2.9, 5.1.4, 5.2.3, 6.2.6), the closed $K$-orbits are
precisely the $T$-fixed points in $X$: in the notation of (6.6),
$\mathcal{Q}_{0}=x_{id}$, and $\mathcal{Q}_{\infty}=x_{s}$. Hence the
$K$-equivariant hypercohomology of $\bm{IC}_{\mathcal{Q}_{0}}$ and
$\bm{IC}_{\mathcal{Q}_{\infty}}$ are given by the $T$-equivariant cohomology
of the corresponding $T$-fixed point:

$\displaystyle\mathbb{H}_{K}(\bm{IC}_{\mathcal{Q}_{0}})$
$\displaystyle=\operatorname{Hom}^{\bullet}_{T}(\mathbb{Q}_{x_{id}},\mathbb{Q}_{x_{id}})=H_{T}^{*}(x_{id}),\text{
and }$ $\displaystyle\mathbb{H}_{K}(\bm{IC}_{\mathcal{Q}_{\infty}})$
$\displaystyle=\operatorname{Hom}^{\bullet}_{T}(\mathbb{Q}_{x_{s}},\mathbb{Q}_{x_{s}})=H_{T}^{*}(x_{s}).$

For any $T$-fixed point $x_{w}$, $H^{*}_{T}(x_{w})\simeq R$ as left
$R$-modules via the Borel homomorphism. The right $R$-module structure on
$H^{*}_{T}(x_{w})$ is twisted by the parameter $w\in W$ specifying the
$T$-fixed point. (This can be seen by comparing first Chern classes of the
line bundles

$ET\times_{T}\mathbb{C}_{\chi}^{\times}\rightarrow ET\times_{T}x_{w}\hskip
2.84526pt\text{ and }\hskip
2.84526ptET\times_{T}wB\times_{B}\mathbb{C}_{\chi}^{\times}\rightarrow
ET\times_{T}x_{w}$

in $H^{*}(x_{w})$.)

With this computation, Lemma 7.1.3 can be seen for $SL_{2}(\mathbb{R})$ by
comparing the convolution actions (5.2), (5.3), and (5.4) with the bimodule
actions (6.10), (6.11) and (3.12).

###### Proof of Lemma 7.1.2.

Fix a closed $K$-orbit $\mathcal{Q}$. By Lemma 6.1.5, there exists
$w\in{{}^{K}W^{\theta}}$ such that the $T$-fixed point $x_{w}=wB/B$ lies in
$\mathcal{Q}$. Denote by $K_{w}:=\operatorname{stab}_{K}x_{w}$. Recall that
$K_{w}$ is a Borel subgroup in $K$ containing $T_{K}$ (see proof of Lemma
6.1.5) and $\mathcal{Q}\simeq K/K_{w}\simeq K\times_{K_{w}}x_{w}$. We have the
following isomorphisms of vector spaces:

(7.3)
$\mathbb{H}_{K}^{\bullet}(\bm{IC}_{\mathcal{Q}})\xrightarrow{\sim}H^{*}_{K}(\mathcal{Q})\xrightarrow{\sim}H^{*}_{K_{w}}(x_{w})\xrightarrow{\sim}H^{*}_{T_{K}}(x_{w}).$

The first is given by the adjunction $(i_{\mathcal{Q}}^{*},i_{\mathcal{Q}*})$,
the second is the induction isomorphism in equivariant cohomology [Bri98, §1,
Rmk. 3], and the third follows from the fact that $K_{w}/T_{K}$ is affine.

Recall that the $T_{K}$-equivariant cohomology of a point is isomorphic to
$P$. Explicitly, the isomorphism

(7.4) $P\xrightarrow{\sim}H^{*}_{T_{K}}(x_{w})$

is given by the Borel homomorphism, which is constructed as follows. Recall
that the ring $P$ is generated by $X(T_{K})_{\mathbb{Q}}$. From a character
$\chi\in X(T_{K})$, we construct a principal $\mathbb{C}^{\times}$-bundle

(7.5)
$ET_{K}\times_{T_{K}}\mathbb{C}^{\times}_{\chi}\xrightarrow{\pi_{\chi}}ET_{K}\times_{T_{K}}x_{w}\simeq
BT_{K},$

where $\mathbb{C}_{\chi}^{\times}$ is the $T_{K}$-space given by the action
$t\cdot z=\chi(t)z$ for $t\in T_{K}$ and $z\in\mathbb{C}^{\times}$. Because
$\pi_{\chi}$ is a $\mathbb{C}^{\times}$-bundle, it has a first Chern class
$c_{1}(\pi_{\chi})\in H^{2}(BT_{K})=H^{2}_{T_{K}}(x_{w})$. The homomorphism
(7.4) is defined by extending the assignment $\chi\mapsto c_{1}(\pi_{\chi})$
multiplicatively.

This establishes that $\mathbb{H}^{\bullet}_{K}(\bm{IC}_{\mathcal{Q}})\simeq
P$ as graded vector spaces. Moreover, all isomorphisms in (7.3) are clearly
compatible with the left $H_{K}^{*}(\mathrm{pt})$-module structures. As
$H_{K}^{*}(\mathrm{pt})\simeq P^{K}$ [Bri98, Prop. 1(i)], we see that
$\mathbb{H}^{\bullet}_{K}(\bm{IC}_{\mathcal{Q}})\simeq P$ as left
$P^{K}$-modules.

Each space in (7.3) also has a right $P$-action coming from the Borel
homomorphism. Specifically, from $\chi\in X(T_{K})$, we construct four
principal $\mathbb{C}^{\times}$-bundles:

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where the $B$-action on $\mathbb{C}^{\times}_{\chi}$ is defined by trivially
extending the $T_{K}$-action. By taking Chern classes, we obtain ring
homomorphisms $\varphi_{i}:\chi\mapsto c_{1}(\mu_{i}^{\chi})$:

${R}$${H^{*}_{T_{K}}(x_{w})}$${H^{*}_{K_{w}}(x_{w})}$${H^{*}_{K}(\mathcal{Q})}$${H^{*}_{K}(X)}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{3}}$$\scriptstyle{\varphi_{4}}$

The ring homomorphisms $\varphi_{i}$ give each space in (7.3) a right
$P$-module structure. Moreover, because the $\mathbb{C}^{\times}$-bundles
$\mu_{i}^{\chi}$ fit into the diagram

${EK\times_{T_{K}}wB\times_{B}\mathbb{C}^{\times}_{\chi}}$${EK\times_{T_{K}}wB\times_{B}\mathrm{pt}}$${EK\times_{K_{w}}wB\times_{B}\mathbb{C}^{\times}_{\chi}}$${EK\times_{K_{w}}wB\times_{B}\mathrm{pt}}$${EK\times_{K_{w}}KwB\times_{B}\mathbb{C}^{\times}_{\chi}}$${EK\times_{K_{w}}KwB\times_{B}\mathrm{pt}}$${EK\times_{K}KwB\times_{B}\mathbb{C}^{\times}_{\chi}}$${EK\times_{K}KwB\times_{B}\mathrm{pt}}$${EK\times_{K}G\times_{B}\mathbb{C}^{\times}_{\chi}}$${EK\times_{K}G\times_{B}\mathrm{pt}}$$\scriptstyle{\mu_{1}^{\chi}}$$\scriptstyle{\mu_{2}^{\chi}}$$\scriptstyle{\mu_{3}^{\chi}}$$\scriptstyle{\mu_{4}^{\chi}}$

in which every square is Cartesian, the isomorphisms in (7.3) are compatible
with these right $P$-actions. By precomposing with the map $\phi:R\rightarrow
P$ from (6.8), each of the vector spaces in (7.3) obtains a compatible right
$R$-module structure.

It remains to compute the right $R$-action on $H^{*}_{T_{K}}(x_{w})$ obtained
from $\varphi_{1}\circ\phi$. First note that the natural map

$H^{*}_{T}(x_{w})\rightarrow H^{*}_{T_{K}}(x_{w})$

obtained by post- and pre-composing $\phi$ with the Borel isomorphisms
$R\xleftarrow{\sim}H^{*}_{T}(x_{w})$ and
$P\xrightarrow{\sim}H^{*}_{T_{K}}(x_{w})$ sends

$c_{1}(\pi^{T}_{\lambda})\mapsto c_{1}(\pi_{\phi(\lambda)})$

for $\lambda\in X(T)$, where $\pi_{\lambda}^{T}$ is the
$\mathbb{C}^{\times}$-bundle
$ET\times_{T}\mathbb{C}_{\lambda}^{\times}\rightarrow ET\times_{T}x_{w}$ and
$\pi_{\phi(\lambda)}$ is as in (7.5). Moreover, if $\mu_{T}^{\lambda}$ is the
$\mathbb{C}^{\times}$-bundle

$ET\times_{T}wB\times_{B}\mathbb{C}^{\times}_{\lambda}\xrightarrow{\mu_{T}^{\lambda}}ET\times_{T}wB\times_{B}\mathrm{pt},$

then $\mu_{T}^{\lambda}\simeq\pi_{w\lambda}^{T}$ as
$\mathbb{C}^{\times}$-bundles, so in $H^{*}_{T}(x_{w})$, we have
$c_{1}(\mu_{T}^{\lambda})=wc_{1}(\pi_{\lambda}^{T})$.

Because the square

${ET\times_{T_{K}}wB\times\mathbb{C}^{\times}_{\phi(\lambda)}}$${ET\times_{T_{K}}wB\times_{B}\mathrm{pt}}$${ET\times_{T}wB\times_{B}\mathbb{C}^{\times}_{\lambda}}$${ET\times_{T}wB\times_{B}\mathrm{pt}}$$\scriptstyle{\mu^{\phi(\lambda)}}$$\scriptstyle{\mu_{T}^{\lambda}}$

is Cartesian, we have

$c_{1}(\mu^{\phi(\lambda)})=\phi(c_{1}(\mu_{T}^{\lambda}))=\phi(wc_{1}(\pi_{\lambda}^{T}))$

in $H^{*}_{T_{K}}(x_{w})$. Hence the right action of $g\in R$ on $p\in P\simeq
H^{*}_{T_{K}}(x_{w})$ is given by

$p\cdot g=p\phi(wg)=p\cdot_{w}g.$

This completes the proof of the first statement. The second statement follows
from the fact that every $T$-fixed point corresponding to $w\in{W^{\theta}}$
lies in some closed orbit $\mathcal{Q}$ (q.v. proof of Lemma 6.1.5). ∎

###### Proof of Lemma 7.1.3.

Set $Y:=G/P_{s}$ and denote by $C_{K}:=H^{*}_{K}(X)=P^{K}\otimes_{R^{W}}R$
(Lemma 7.0.1). The map
$\pi_{s}^{*}:\operatorname{End}^{\bullet}_{K}(\mathbb{Q}_{Y})\rightarrow\operatorname{End}^{\bullet}_{K}(\mathbb{Q}_{X})=C_{K}$
is injective with image $C_{K}^{s}=P^{K}\otimes_{R^{W}}R^{s}$. We identify
$H^{*}_{K}(Y)=\operatorname{End}_{K}^{\bullet}(\mathbb{Q}_{Y})$ with its image
under $\pi_{s}^{*}$. Consider the diagram of functors

${D^{b}_{K}(X)}$${C_{K}\mathrm{-gmod}}$${D^{b}_{K}(Y)}$${C^{s}_{K}\mathrm{-gmod}}$$\scriptstyle{\mathbb{H}^{\bullet}_{K}}$$\scriptstyle{\pi_{s*}}$$\scriptstyle{\mathrm{Res}^{s}}$$\scriptstyle{\pi_{s}^{*}}$$\scriptstyle{\mathbb{H}^{s}_{K}}$

where
$\mathbb{H}^{s}_{K}(-):=\operatorname{Hom}^{\bullet}_{D^{b}_{K}(Y)}(\mathbb{Q}_{Y},-)$
is the hypercohomology functor on $D^{b}_{K}(Y)$ and $\mathrm{Res}^{s}$ is the
functor given by restriction to $C^{s}_{K}\subseteq C_{K}$. The adjunction
$(\pi_{s}^{*},\pi_{s*})$ gives a natural isomorphism

(7.6) $\mathbb{H}_{K}^{s}\pi_{s*}\simeq\mathrm{Res}^{s}\mathbb{H}_{K}.$

Lemma 7.1.3 follows from the following proposition.

######  Proposition 7.1.5.

For $\mathcal{F}\in D^{b}_{K}(X)$ and $s\in S$,

$\mathbb{H}_{K}^{\bullet}(\pi_{s}^{*}\pi_{s*}\mathcal{F})\simeq
C_{K}\otimes_{C_{K}^{s}}\mathrm{Res}^{s}\mathbb{H}^{\bullet}_{K}(\mathcal{F})$

in $C_{K}\mathrm{-gmod}$.

Let $M\in C_{K}\mathrm{-gmod}$. Under the identification of
$C_{K}\mathrm{-gmod}$ with its image in $(P^{K},R)\mathrm{-gbim}$ (see 7.1),
the left $C_{K}$-module $C_{K}\otimes_{C_{K}^{s}}\mathrm{Res}^{s}M$
corresponds to the $(P^{K},R)$-bimodule $M\otimes_{R}B_{s}$, so Lemma 7.1.3
immediately follows from Proposition 7.1.5.

###### Proof of Proposition 7.1.5.

Our argument is exactly the argument in [Soe90, §3.2] with added equivariance.
We repeat the argument here for the benefit of readers less comfortable with
the German language.

For any $\mathcal{F}\in D^{b}_{K}(X)$, there is a canonical morphism of
$C_{K}$-modules

(7.7) $\displaystyle
C_{K}\otimes_{C_{K}^{s}}\mathbb{H}_{K}^{s}(\pi_{s*}\mathcal{F})$
$\displaystyle=\operatorname{Hom}_{K}^{\bullet}(\mathbb{Q}_{X},\pi_{s}^{*}\mathbb{Q}_{Y})\otimes_{\operatorname{End}^{\bullet}_{K}(\mathbb{Q}_{Y})}\operatorname{Hom}_{K}^{\bullet}(\mathbb{Q}_{Y},\pi_{s*}\mathcal{F})$
$\displaystyle\rightarrow\operatorname{Hom}^{\bullet}(\mathbb{Q}_{X},\pi_{s}^{*}\pi_{s*}\mathcal{F})=\mathbb{H}_{K}^{\bullet}(\pi_{s}^{*}\pi_{s*}\mathcal{F})$

given by $\varphi\otimes\psi\mapsto\pi_{s}^{*}\psi\circ\varphi$. We will show
that this canonical morphism is an isomorphism.

Because $\pi_{s}$ is smooth of relative dimension one, we have
$\pi_{s}^{*}=\pi_{s}^{!}[-2]$. Using this relation, along with the adjunction
$(\pi_{s!},\pi_{s}^{!})$ and the fact that $\pi_{s}$ is proper (hence
$\pi_{s!}=\pi_{s*}$), we compute

(7.8)
$\mathbb{H}_{K}^{\bullet}(\pi_{s}^{*}\pi_{s*}\mathcal{F})=\operatorname{Hom}^{\bullet}_{K}(\mathbb{Q}_{X},\pi_{s}^{!}\pi_{s*}\mathcal{F}[-2])=\operatorname{Hom}^{\bullet}_{K}(\pi_{s*}\mathbb{Q}_{X}[2],\pi_{s*}\mathcal{F}).$

We have canonical morphisms

$\mathbb{Q}_{Y}\rightarrow\pi_{s*}\pi_{s}^{*}\mathbb{Q}_{Y}=\pi_{s*}\mathbb{Q}_{X}=\pi_{s!}\pi_{s}^{!}\mathbb{Q}_{Y}[-2]\rightarrow\mathbb{Q}_{Y}[-2]$

given by the unit of the adjunction $(\pi_{s}^{*},\pi_{s*})$ and the counit of
the adjunction $(\pi_{s!},\pi_{s}^{!})$. Shifted, this is

$\mathbb{Q}_{Y}[2]\rightarrow\pi_{s*}\mathbb{Q}_{X}[2]\rightarrow\mathbb{Q}_{Y}.$

Because $Y$ has no negative cohomology groups, the composition is zero. By the
decomposition theorem, the sequence splits, and we conclude that

(7.9) $\pi_{s*}\mathbb{Q}_{X}[2]=\mathbb{Q}_{Y}\oplus\mathbb{Q}_{Y}[2].$

(Alternatively, (7.9) also follows from Lemma 4.2.4.) Using the splitting
(7.9) and the natural isomorphism (7.6), we continue the computation in (7.8).

$\displaystyle\operatorname{Hom}_{K}^{\bullet}(\pi_{s*}\mathbb{Q}_{X}[2],\pi_{s*}\mathcal{F})$
$\displaystyle=\operatorname{Hom}_{K}^{\bullet}(\mathbb{Q}_{Y}\oplus\mathbb{Q}_{Y}[2],\pi_{s*}\mathcal{F})$
$\displaystyle=\mathbb{H}_{K}^{s}(\pi_{s*}\mathcal{F})\oplus\mathbb{H}_{K}^{s}(\pi_{s*}\mathcal{F})[-2]$
$\displaystyle=(C_{K}^{s}\oplus
C_{K}^{s}[-2])\otimes_{C_{K}^{s}}\mathbb{H}^{s}_{K}(\pi_{s*}\mathcal{F})$
$\displaystyle=\operatorname{Hom}^{\bullet}_{K}(\mathbb{Q}_{Y},\mathbb{Q}_{Y}\oplus\mathbb{Q}_{Y}[-2])\otimes_{C_{K}^{s}}\mathbb{H}^{s}_{K}(\pi_{s*}\mathcal{F})$
$\displaystyle=\operatorname{Hom}^{\bullet}_{K}(\mathbb{Q}_{Y},\pi_{s*}\mathbb{Q}_{X})\otimes_{C_{K}^{s}}\mathbb{H}_{K}^{s}(\pi_{s*}\mathcal{F})$
$\displaystyle=\mathrm{Res}^{s}\mathbb{H}^{\bullet}_{K}(\mathbb{Q}_{X})\otimes_{C_{K}^{s}}\mathbb{H}_{K}^{s}(\pi_{s*}\mathcal{F})$
$\displaystyle=\mathrm{Res}^{s}C_{K}\otimes_{C_{K}^{s}}\mathbb{H}_{K}^{s}(\pi_{s*}\mathcal{F})$

This shows that the canonical morphism (7.7) is an isomorphism of
$C^{s}_{K}$-modules, hence it must be an isomorphicm of $C_{K}$-modules. The
proposition then follows from the natural isomorphism (7.6). ∎

This completes the proof of Lemma 7.1.3, and in turn, the proof of Theorem
7.1.1. ∎

######  Example 7.1.6.

$($Hypercohomology vanishes outside of $\mathcal{M}_{\mathrm{LV}}^{0}$ for
$SL_{2}(\mathbb{R}))$ Consider the $\bm{IC}$ sheaf corresponding to the Möbius
band local system $\mathcal{L}$ in Example 3.2.4, $\bm{IC}(X,\mathcal{L})$,
and its non-equivariant version, $\bm{IC}(X,\operatorname{For}\mathcal{L})$.
Recall that because $\mathcal{L}$ is clean,
$\bm{IC}(X,\operatorname{For}\mathcal{L})=j_{*}\operatorname{For}\mathcal{L}$,
where $j:\mathcal{O}\hookrightarrow X$ is inclusion of the open orbit. Hence
by equation (3.18),

$\mathbb{H}^{\bullet}(\bm{IC}(X,\operatorname{For}\mathcal{L}))=H^{*}(X;j_{*}\operatorname{For}\mathcal{L})=H^{*}(\mathbb{C}^{\times};\operatorname{For}\mathcal{L}).$

We showed (in two ways) in Example 5.2.4 that
$H^{*}(\mathbb{C}^{\times};\operatorname{For}\mathcal{L})=0$, so the ordinary
hypercohomology functor (and hence the equivariant hypercohomology functor)
vanishes on $\bm{IC}(X,\operatorname{For}\mathcal{L})$ (resp.
$\bm{IC}(X,\mathcal{L})$). This example illustrates that our techniques do not
necessarily capture the entire Lusztig–Vogan module in examples with non-
trivial (after forgetting equivariance) local systems.

Theorem 7.1.1 establishes that the hypercohomology functor

$\mathbb{H}^{\bullet}_{K}:\mathcal{M}_{LV}^{0}\rightarrow\mathcal{N}_{LV}^{0}$

is essentially surjective. In the following section, we show that it is also
fully faithful, and hence establishes an equivalence of categories

$\mathcal{M}_{LV}^{0}\simeq\mathcal{N}_{LV}^{0}.$

######  Remark 7.1.7.

This equivalence of categories appears (in slightly different language) as
Theorem 3.12 in [BV21].

###  7.2. Hypercohomology is fully faithful

######  Theorem 7.2.1.

The equivariant hypercohomology functor

$\mathbb{H}^{\bullet}_{K}:\mathcal{M}^{0}_{LV}\longrightarrow\mathcal{N}_{LV}^{0}$

is fully faithful.

To prove Theorem 7.2.1, we must show that for any
$\mathcal{F},\mathcal{G}\in\mathcal{M}_{LV}^{0}$, the map

(7.10)
$\operatorname{Hom}_{K}(\mathcal{F},\mathcal{G})\xrightarrow{\mathbb{H}_{K}^{\bullet}}\operatorname{Hom}_{(P^{K},R)\mathrm{-gbim}}(\mathbb{H}_{K}^{\bullet}(\mathcal{F}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))$

is bijective. We will do so in three steps:

  1. (1)

First, we establish that $\mathbb{H}^{\bullet}_{K}$ is compatible with certain
adjunctions, allowing us to reduce to the case where
$\mathcal{F}=\bm{IC}_{\mathcal{Q}}$. This is Lemma 7.2.4 and the preceding
material.

  2. (2)

Second, we establish that $\mathbb{H}_{K}^{\bullet}$ is faithful by showing
that the map (7.10) is injective. This is Theorem 7.2.5.

  3. (3)

Third, we establish that $\mathbb{H}_{K}^{\bullet}$ is full by showing that
the map (7.10) is surjective. This is Theorem 7.2.12.

######  Lemma 7.2.2.

For any $s\in S$, the functor
$\Theta_{s}:\mathcal{M}_{LV}^{0}\rightarrow\mathcal{M}_{LV}^{0}$ and the
functor
$-\otimes_{R}B_{s}:\mathcal{N}_{LV}^{0}\rightarrow\mathcal{N}_{LV}^{0}$ are
self-adjoint.

###### Proof.

For $\mathcal{F},\mathcal{G}\in D^{b}_{K}(X)$,

$\displaystyle\operatorname{Hom}(\mathcal{F},\pi_{s}^{*}\pi_{s*}\mathcal{G}[1])$
$\displaystyle=\operatorname{Hom}(\mathcal{F},\pi_{s}^{!}\pi_{s*}\mathcal{G}[-1])$
$\displaystyle=\operatorname{Hom}(\pi_{s!}\mathcal{F},\pi_{s*}\mathcal{G}[-1])$
$\displaystyle=\operatorname{Hom}(\pi_{s}^{*}\pi_{s*}\mathcal{F}[1],\mathcal{G}).$

This establishes the first claim. To see that $-\otimes_{R}B_{s}$ is self-
adjoint, note that the ring inclusion $R^{s}\hookrightarrow R$ is a degree $1$
Frobenius extension. (See [EMTW20, Ch. 8] for a detailed discussion of
Frobenius extensions.) Hence we have two adjoint pairs:

$(\operatorname{Ind}_{R^{s}}^{R},\operatorname{Res}_{R^{s}}^{R})\text{ and
}(\operatorname{Res}_{R^{s}}^{R}(2),\operatorname{Ind}_{R^{s}}^{R}).$

Each functor can be realized as tensor product:

$\operatorname{Ind}_{R^{s}}^{R}(-)=-\otimes_{R^{s}}R\text{ and
}\operatorname{Res}_{R^{s}}^{R}(-)=-\otimes_{R}R.$

It follows that for $M,N\in(P^{K},R)\mathrm{-gbim}$,

$\displaystyle\operatorname{Hom}(M,N\otimes_{R}B_{s})$
$\displaystyle=\operatorname{Hom}(M,N\otimes_{R}R\otimes_{R^{s}}R(1))$
$\displaystyle=\operatorname{Hom}(M\otimes_{R}R,N\otimes_{R}R(-1))$
$\displaystyle=\operatorname{Hom}(M\otimes_{R}R\otimes_{R^{s}}R(1),N)$
$\displaystyle=\operatorname{Hom}(M\otimes_{R}B_{s},N).$

∎

To show that the hypercohomology functor commutes with these adjunctions, we
will utilize the categorical module structure of $\mathcal{M}_{LV}^{0}$ and
$\mathcal{N}_{LV}^{0}$. Our arguments depend on some fundamental properties of
dual pairs in monoidal categories, which we recall in Appendix A. We encourage
any reader unfamiliar with these tools to read Appendix A before proceding.

######  Lemma 7.2.3.

For $s\in S$, the pair $({\bm{IC}}_{s},{\bm{IC}}_{s})$ is a dual pair in
$\mathcal{H}$, and the pair $(B_{s},B_{s})$ is a dual pair in
$\mathbb{S}\text{Bim}$.

###### Proof.

Denote by $\Theta^{B}_{s}:D^{b}_{B}(X)\rightarrow D^{b}_{B}(X)$ the push-pull
functor on $D^{b}_{B}(X)$:

$\Theta^{B}_{s}(\mathcal{F}):=\pi_{s}^{*}\pi_{s*}\mathcal{F}[1],$

where $\pi_{s}:X\rightarrow G/P_{s}$ is as in (3.4) and
${D^{b}_{B}(X)}$${D^{b}_{B}(G/P_{s})}$$\scriptstyle{\pi_{s*}}$$\scriptstyle{\pi_{s}^{*}}$
are the direct and inverse image functors on the $B$-equivariant derived
category. By analogous results to Lemma 7.2.2 and Lemma 4.1.2,
$\Theta^{B}_{s}$ is self-adjoint and can be realized as convolution with
$\bm{IC}_{s}$. Let $\eta:id\rightarrow\Theta^{B}_{s}\Theta^{B}_{s}$ (resp.
$\epsilon:\Theta^{B}_{s}\Theta^{B}_{s}\rightarrow id$) be the unit (resp.
counit) of the adjunction $(\Theta^{B}_{s},\Theta^{B}_{s})$ on $\mathcal{H}$.
Here $id$ is the identity functor on $\mathcal{H}$. We define co-evaluation
and evaluation morphisms in $\mathcal{H}$ by

$\displaystyle\mathbb{Q}_{B}\xrightarrow{\eta_{\mathbb{Q}_{B}}}\mathbb{Q}_{B}\Theta^{B}_{s}\Theta^{B}_{s}\simeq\mathbb{Q}_{B}*\bm{IC}_{s}*\bm{IC_{s}}\simeq\bm{IC}_{s}*\bm{IC}_{s},\text{
and }$
$\displaystyle\bm{IC}_{s}*\bm{IC}_{s}\simeq\mathbb{Q}_{B}*\bm{IC}_{s}*\bm{IC}_{s}\simeq\mathbb{Q}_{B}\Theta^{B}_{s}\Theta^{B}_{s}\xrightarrow{\epsilon_{\mathbb{Q}_{B}}}\mathbb{Q}_{B}.$

The zig-zag diagrams for the adjunction
$(\Theta^{B}_{s},\Theta^{B}_{s},\eta,\epsilon)$ guarantee that the diagrams
(A.1) and (A.2) commute.

Similarly, the functor
$-\otimes_{R}B_{s}:\mathbb{S}\text{Bim}\rightarrow\mathbb{S}\text{Bim}$ is
self-adjoint, and we denote by
$\eta^{\prime}:id\rightarrow-\otimes_{R}B_{s}\otimes_{R}B_{s}$ and
$\epsilon^{\prime}:-\otimes_{R}B_{s}\otimes_{R}B_{s}\rightarrow id$ the unit
and counit of the adjunction. We define co-evaluation and evaluation morphisms
in $\mathbb{S}\text{Bim}$ for the pair $(B_{s},B_{s})$ by

$\displaystyle
R\xrightarrow{\eta^{\prime}_{R}}R\otimes_{R}B_{s}\otimes_{R}B_{s}\simeq
B_{s}\otimes_{R}B_{s},\text{ and }$ $\displaystyle B_{s}\otimes_{R}B_{s}\simeq
R\otimes_{R}B_{s}\otimes_{R}B_{s}\xrightarrow{\eta^{\prime}_{R}}R.$

Again, the commutativity of the dual pair diagrams (A.1) and (A.2) follows
from the zig-zag diagrams for the adjunction
$(-\otimes_{R}B_{s},-\otimes_{R}B_{s},\eta^{\prime},\epsilon^{\prime})$. ∎

######  Lemma 7.2.4.

For every $s\in S$ and every pair of objects
$\mathcal{F},\mathcal{G}\in\mathcal{M}_{LV}^{0}$, the diagram

${\operatorname{Hom}(\mathcal{F},\mathcal{G}\Theta_{s})}$${\operatorname{Hom}(\mathcal{F}\Theta_{s},\mathcal{G})}$${\operatorname{Hom}(\mathbb{H}_{K}^{\bullet}(\mathcal{F}),\mathbb{H}_{K}^{\bullet}(\mathcal{G})\otimes_{R}B_{s})}$${\operatorname{Hom}(\mathbb{H}_{K}^{\bullet}(\mathcal{F})\otimes_{R}B_{s},\mathbb{H}_{K}^{\bullet}(\mathcal{G}))}$$\scriptstyle{\sim}$$\scriptstyle{\mathbb{H}^{\bullet}_{K}}$$\scriptstyle{\mathbb{H}_{K}^{\bullet}}$$\scriptstyle{\sim}$

commutes. Here the horizontal arrows are adjunction isomorphisms, and we are
using the isomorphism in Lemma 7.1.3 to replace
$\mathbb{H}_{K}^{\bullet}(\mathcal{G}\Theta_{s})$ with
$\mathbb{H}_{K}^{\bullet}(\mathcal{G})\otimes_{R}B_{s}$.

###### Proof.

It is enough to show that hypercohomology preserves the unit and counit of
each adjunction. In other words, if the unit and counit of the adjoint pair
${\mathcal{M}_{LV}^{0}}$${\mathcal{M}_{LV}^{0}}$$\scriptstyle{\Theta_{s}}$$\scriptstyle{\Theta_{s}}$
are $\eta:id\rightarrow\Theta_{s}\Theta_{s}$ and
$\epsilon:\Theta_{s}\Theta_{s}\rightarrow id$, it is enough to show that the
collections of morphisms

$\\{\mathbb{H}_{K}^{\bullet}(\eta_{\mathcal{F}})\\}_{\mathcal{F}\in\mathcal{M}_{LV}^{0}}\text{
and
}\\{\mathbb{H}_{K}^{\bullet}(\epsilon_{\mathcal{F}})\\}_{\mathcal{F}\in\mathcal{M}_{LV}^{0}}$

define the unit and counit of the adjunction of the adjunction
${\mathcal{N}_{LV}^{0}}$${\mathcal{N}_{LV}^{0}}$$\scriptstyle{-\otimes_{R}B_{s}}$$\scriptstyle{-\otimes_{R}B_{s}}$.

We begin by noting that because
$\mathbb{H}_{K}^{\bullet}:\mathcal{M}_{LV}^{0}\rightarrow\mathcal{N}_{LV}^{0}$
is essentially surjective (Theorem 7.1.1), every object
$M\in\mathcal{N}_{LV}^{0}$ is isomorphic to
$\mathbb{H}_{K}^{\bullet}(\mathcal{F})$ for some
$\mathcal{F}\in\mathcal{M}_{LV}^{0}$. Hence for each object in
$\mathcal{N}_{LV}^{0}$, there is a corresponding morphism in
$\\{\mathbb{H}_{K}^{\bullet}(\eta_{\mathcal{F}})\\}_{\mathcal{F}\in\mathcal{M}_{LV}^{0}}$
and in
$\\{\mathbb{H}_{K}^{\bullet}(\epsilon_{\mathcal{F}})\\}_{\mathcal{F}\in\mathcal{M}_{LV}^{0}}$.

Because $(\bm{IC}_{s},\bm{IC}_{s})$ and $(B_{s},B_{s})$ are dual pairs in
$\mathcal{H}$ and $\mathbb{S}\text{Bim}$, respectively (Lemma 7.2.3), we can
apply Lemma A.0.2 to $\mathcal{M}_{LV}^{0}$ and $\mathcal{N}_{LV}^{0}$. We
conclude that the unit and counit of the adjunction $(\Theta_{s},\Theta_{s})$
are

$\displaystyle\eta_{\mathcal{F}}:\mathcal{F}\xrightarrow{\mathcal{F}*\beta}\mathcal{F}*\bm{IC}_{s}*\bm{IC}_{s}\simeq\mathcal{F}\Theta_{s}\Theta_{s},\text{
and}$
$\displaystyle\epsilon_{\mathcal{G}}:\mathcal{G}\Theta_{s}\Theta_{s}\simeq\mathcal{G}*\bm{IC}_{s}*\bm{IC}_{s}\xrightarrow{\mathcal{G}*\mathcal{L}}\mathcal{G},$

where $\beta:\mathbbm{1}\rightarrow\bm{IC}_{s}*\bm{IC}_{s}$ and
$\mathcal{L}:\bm{IC}_{s}*\bm{IC}_{s}\rightarrow\mathbbm{1}$ are the co-
evaluation and evaluation morphisms for the dual pair
$(\bm{IC}_{s},\bm{IC}_{s})$. Similarly, the unit and counit for the adjoint
pair
${\mathcal{N}_{LV}^{0}}$${\mathcal{N}_{LV}^{0}}$$\scriptstyle{-\otimes_{R}B_{s}}$$\scriptstyle{-\otimes_{R}B_{s}}$
are

$\displaystyle\eta^{\prime}_{M}:M\xrightarrow{M\otimes_{R}\beta^{\prime}}M\otimes_{R}B_{s}\otimes_{R}B_{s},\text{
and }$
$\displaystyle\epsilon_{M}:N\otimes_{R}B_{s}\otimes_{R}B_{s}\xrightarrow{N\otimes_{R}\mathcal{L}^{\prime}}N,$

where $\beta^{\prime}:R\rightarrow B_{s}\otimes_{R}B_{s}$ and
$\mathcal{L}^{\prime}:B_{s}\otimes_{R}B_{s}\rightarrow R$ are the co-
evaluation and evaluation morphisms for the dual pair $(B_{s},B_{s})$.

By Lemma 7.1.3,

$\mathbb{H}_{K}^{\bullet}(\eta_{\mathcal{F}})=\mathbb{H}_{K}^{\bullet}(\mathcal{F})\otimes_{R}\beta^{\prime}=\eta^{\prime}_{\mathbb{H}_{K}^{\bullet}(\mathcal{F})}:\mathbb{H}_{K}^{\bullet}(\mathcal{F})\rightarrow\mathbb{H}_{K}^{\bullet}(\mathcal{F})\otimes_{R}B_{s}\otimes_{R}B_{s},$

and

$\mathbb{H}^{\bullet}_{K}(\epsilon_{\mathcal{F}})=\mathbb{H}_{K}^{\bullet}(\mathcal{F})\otimes\mathcal{L}^{\prime}=\epsilon^{\prime}_{\mathbb{H}^{\bullet}_{K}(\mathcal{F})}:\mathbb{H}_{K}^{\bullet}(\mathcal{F})\otimes_{R}B_{s}\otimes_{R}B_{s}\rightarrow\mathbb{H}_{K}^{\bullet}(\mathcal{F}).$

This proves the lemma. ∎

This completes the first step in our proof of Theorem 7.2.1. By Lemma 7.2.4,
proving Theorem 7.2.1 reduces to showing that the natural map

(7.11)
$\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})\xrightarrow{\mathbb{H}_{K}^{\bullet}}\operatorname{Hom}_{(P^{K},R)\mathrm{-gbim}}(\mathbb{H}^{\bullet}_{K}(\bm{IC}_{\mathcal{Q}}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))$

is bijective for any $\mathcal{G}\in\mathcal{M}_{LV}^{0}$ and closed $K$-orbit
$\mathcal{Q}$.

######  Theorem 7.2.5.

The map (7.11) is injective for any $\mathcal{G}\in\mathcal{M}_{LV}^{0}$ and
closed $K$-orbit $\mathcal{Q}$.

The proof of Theorem 7.2.5 proceeds as follows. First we show that for any
$\mathcal{G}\in D^{b}_{K}(X)$, the injectivity of (7.11) follows from the
injectivity of the canonical map

(7.12)
$\mathbb{H}_{K}^{\bullet}(i_{\mathcal{Q}!}i_{\mathcal{Q}}^{!}\mathcal{G})\rightarrow\mathbb{H}_{K}^{\bullet}(\mathcal{G})$

coming from the adjunction $(i_{\mathcal{Q}!},i_{\mathcal{Q}}^{!})$. This is
the content of Lemma 7.2.6. Then we show that (7.12) is injective for objects
$\mathcal{G}\in D^{b}_{K}(X)$ satisfying certain parity conditions (Lemma
7.2.10). Finally, we show that objects in $\mathcal{M}_{LV}^{0}$ satisfy the
necessary parity conditions (Corollary 7.2.11).

We start by recalling some basic facts in $D^{b}_{K}(X)$. Denote by
$i:=i_{\mathcal{Q}}:\mathcal{Q}\hookrightarrow X$, and let $\mathcal{G}\in
D^{b}_{K}(X)$ be arbitrary. Denote the hypercohomology functor on
$D^{b}_{K}(\mathcal{Q})$ by
$\mathbb{H}_{K}^{\mathcal{Q}}:=\operatorname{Hom}_{D^{b}_{K}(\mathcal{Q})}^{\bullet}(\mathbb{Q}_{\mathcal{Q}},-):D^{b}_{K}(\mathcal{Q})\rightarrow
H^{*}_{K}(\mathcal{Q})\mathrm{-gmod}$. The adjunction $(i^{*},i_{*})$ gives
isomorphisms

(7.13)
$\mathbb{H}_{K}^{\bullet}(\bm{IC}_{\mathcal{Q}})\simeq\mathbb{H}_{K}^{\mathcal{Q}}(\mathbb{Q}_{\mathcal{Q}})\simeq
H^{*}_{K}(\mathcal{Q}).$

The adjunction $(i_{!},i^{!})$ gives an injection

(7.14)
$\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})\simeq\operatorname{Hom}_{D_{K}^{b}(\mathcal{Q})}(\mathbb{Q}_{\mathcal{Q}},i^{!}\mathcal{G})\hookrightarrow\operatorname{Hom}^{\bullet}_{D^{b}_{K}(\mathcal{Q})}(\mathbb{Q}_{\mathcal{Q}},i^{!}\mathcal{G})=\mathbb{H}^{\mathcal{Q}}_{K}(i^{!}\mathcal{G}).$

Because $i^{*}\mathbb{Q}_{X}=\mathbb{Q}_{\mathcal{Q}}$, we have

(7.15)
$\operatorname{Hom}^{\bullet}_{D_{K}^{b}(\mathcal{Q})}(\mathbb{Q}_{\mathcal{Q}},i^{!}\mathcal{G})\simeq\operatorname{Hom}^{\bullet}_{K}(\mathbb{Q}_{X},i_{*}i^{!}\mathcal{G})\simeq\mathbb{H}_{K}^{\bullet}(i_{!}i^{!}\mathcal{G}).$

And finally, because $H^{*}_{K}(\mathcal{Q})$ is cyclic as a
$H^{*}_{K}(X)$-module (Lemma 7.1.2), there is a natural injecton

(7.16)
$\operatorname{Hom}_{K}(H^{*}_{K}(\mathcal{Q}),\mathbb{H}_{K}(\mathcal{G}))\hookrightarrow\mathbb{H}_{K}(\mathcal{G})$

given by $\varphi\mapsto\varphi(1)$.

For $M,N\in(P^{K},R)\mathrm{-gbim}$, denote by

$\operatorname{Hom}_{\mathrm{bim}}(M,N):=\operatorname{Hom}_{(P^{K},R)\mathrm{-gbim}}(M,N).$

######  Lemma 7.2.6.

The diagram

(7.17)
${\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})}$${\operatorname{Hom}_{\mathrm{bim}}(H^{*}_{K}(\mathcal{Q}),\mathbb{H}^{\bullet}_{K}(\mathcal{G}))}$${\mathbb{H}_{K}^{\mathcal{Q}}(i^{!}\mathcal{G})}$${\mathbb{H}^{\bullet}_{K}(\mathcal{G})}$${\mathbb{H}_{K}^{\bullet}(i_{!}i^{!}\mathcal{G})}$$\scriptstyle{a}$$\scriptstyle{\mathbb{H}_{K}^{\bullet}}$$\scriptstyle{d}$$\scriptstyle{b}$$\scriptstyle{\sim}$$\scriptstyle{c}$

obtained from (7.13)-(7.16) commutes for any $\mathcal{G}\in D^{b}_{K}(X)$.

###### Proof.

Denote by $\overset{*}{\eta}:\mathbbm{1}\rightarrow i_{*}i^{*}$ the unit of
the adjunction $(i^{*},i_{*})$, and by
$\overset{!}{\eta}:\mathbbm{1}\rightarrow i^{!}i_{!}$ (resp.
$\overset{!}{\epsilon}:i_{!}i^{!}\rightarrow\mathbbm{1}$) the unit (resp.
counit) of the adjunction $(i_{!},i^{!})$.

Let
$(\bm{IC}_{\mathcal{Q}}\xrightarrow{\varphi}\mathcal{G})\in\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})$.
We start by computing the image of $\varphi$ as we move counterclockwise
around the diagram. Under the injection $a$, $\varphi$ maps to the composition

$\mathbb{Q}_{\mathcal{Q}}\xrightarrow{\overset{!}{\eta}_{\mathbb{Q}_{\mathcal{Q}}}}i^{!}i_{!}\mathbb{Q}_{\mathcal{Q}}\xrightarrow{i^{!}\varphi}i^{!}\mathcal{G}.$

Applying the isomorphism $b$ to this and using the fact that
$\mathbb{Q}_{\mathcal{Q}}=i^{*}\mathbb{Q}_{X}$, we obtain

$\mathbb{Q}_{X}\xrightarrow{\overset{*}{\eta}_{\mathbb{Q}_{X}}}i_{*}i^{*}\mathbb{Q}_{X}\xrightarrow{i_{*}\overset{!}{\eta}_{\mathbb{Q}_{\mathcal{Q}}}}i_{*}i^{!}i_{!}\mathbb{Q}_{\mathcal{Q}}\xrightarrow{i_{*}i^{!}\varphi}i_{*}i^{!}\mathcal{G}.$

Finally, applying the map $c$ and using the fact that $i_{!}=i_{*}$, we obtain
the homomorphism

(7.18)
$\mathbb{Q}_{X}\xrightarrow{\overset{*}{\eta}_{\mathbb{Q}_{X}}}i_{!}i^{*}\mathbb{Q}_{X}\xrightarrow{i_{!}\overset{!}{\eta}_{\mathbb{Q}_{\mathcal{Q}}}}i_{!}i^{!}i_{!}\mathbb{Q}_{\mathcal{Q}}\xrightarrow{i_{!}i^{!}\varphi}i_{!}i^{!}\mathcal{G}\xrightarrow{\overset{!}{\epsilon}_{\mathcal{G}}}\mathcal{G}$

in $\mathbb{H}_{K}^{\bullet}(\mathcal{G})$.

Now we compute the image of $\varphi$ as we move clockwise around the diagram.
Under $\mathbb{H}_{K}^{\bullet}$, $\varphi$ maps to

$H^{*}_{K}(\mathcal{Q})\simeq\operatorname{Hom}_{D^{b}_{K}(\mathcal{Q})}^{\bullet}(i^{*}\mathbb{Q}_{X},\mathbb{Q}_{\mathcal{Q}})\xrightarrow{\sim}\operatorname{Hom}_{K}^{\bullet}(\mathbb{Q}_{X},i_{*}\mathbb{Q}_{\mathcal{Q}})\xrightarrow{\mathbb{H}_{K}^{\bullet}(\varphi)}\operatorname{Hom}_{K}^{\bullet}(\mathbb{Q}_{X},\mathcal{G})=\mathbb{H}^{\bullet}_{K}(\mathcal{G}),$

where the first arrow is the adjunction isomorphism sending

$(\mathbb{Q}_{\mathcal{Q}}=i^{*}\mathbb{Q}_{X}\xrightarrow{f}\mathbb{Q}_{\mathcal{Q}})\mapsto(\mathbb{Q}_{X}\xrightarrow{\overset{*}{\eta}_{\mathbb{Q}_{X}}}i_{*}i^{*}\mathbb{Q}_{X}\xrightarrow{i_{*}f}i_{*}\mathbb{Q}_{\mathcal{Q}})$

and the second arrow is post-composition with $\varphi$. Under $d$, this maps
to the image of the identity map:

(7.19)
$\mathbb{Q}_{X}\xrightarrow{\overset{*}{\eta}_{\mathbb{Q}_{X}}}i_{*}i^{*}\mathbb{Q}_{X}=i_{*}\mathbb{Q}_{\mathcal{Q}}\xrightarrow{\varphi}\mathcal{G}.$

The sequences (7.19) and (7.18) fit into the following diagram:

${\mathbb{Q}_{X}}$${i_{!}i^{*}\mathbb{Q}_{X}}$${i_{!}i^{!}i_{!}\mathbb{Q}_{\mathcal{Q}}}$${i_{!}i^{!}\mathcal{G}}$${\mathcal{G}}$${i_{*}\mathbb{Q}_{\mathcal{Q}}}$$\scriptstyle{\overset{*}{\eta}_{\mathbb{Q}_{X}}}$$\scriptstyle{i_{!}\overset{!}{\eta}_{\mathbb{Q}_{\mathcal{Q}}}}$$\scriptstyle{id}$$\scriptstyle{i_{!}i^{!}\varphi}$$\scriptstyle{\overset{!}{\epsilon}_{i_{!}\mathbb{Q}_{\mathcal{Q}}}}$$\scriptstyle{\overset{!}{\epsilon}_{\mathcal{G}}}$$\scriptstyle{\varphi}$

The zig-zag diagrams for the adjunction $(i_{!},i^{!})$ guarantee that the
left triangle commutes. The right triangle commuted by the naturality of
$\overset{!}{\epsilon}$. Hence the images agree, and the diagram (7.17)
commutes. ∎

Next we establish a condition on complexes $\mathcal{G}\in D^{b}_{K}(X)$ which
guarantees that the map (7.12) is injective. We will do this by applying
parity results from [JMW14]. To begin, we need to establish that our geometric
setting satisfies the necessary assumptions of loc. cit.

Let $\Lambda$ be the stratification of $X$ by $K$-orbits. We need to show that
the pair $(X,\Lambda)$ satisfies conditions (2.1) and (2.2) of [JMW14].
Because we are taking coefficients in a field, these assumptions reduce to
establishing the following lemma [JMW14, Rmk. 2.2].

######  Lemma 7.2.7.

For any $K$-orbit $\mathcal{O}$ in $X$ and
$\mathcal{L}\in\operatorname{Loc}_{K}(\mathcal{O})$,
$H^{n}_{K}(\mathcal{O};\mathcal{L})=0$ for $n$ odd.

###### Proof.

Let $x\in\mathcal{O}$, $K_{x}=\operatorname{stab}_{K}x$, and
$K_{x}^{\circ}\subseteq K_{x}$ the connected component of the identity. There
is a fibration

$\widetilde{\mathcal{O}}:=K/K_{x}^{\circ}\xrightarrow{\pi}\mathcal{O}\simeq
K/K_{x}$

given by $gK_{x}^{\circ}\mapsto g\cdot x$. The fibres of $\pi$ are isomorphic
to $K_{x}/K_{x}^{\circ}$, which has exponent $2$ [LV83]. The category
$\operatorname{Loc}_{K}(\mathcal{O})$ is equivalent to the category of
$\mathbb{C}[K_{x}/K_{x}^{\circ}]$-modules. In particular,
$\operatorname{Loc}_{K}(\mathcal{O})$ is semisimple because
$K_{x}/K_{x}^{\circ}$ is abelian. Under this equivalence,
$\pi_{*}\mathbb{Q}_{\widetilde{\mathcal{O}}}$ corresponds to the regular
representation. Hence all simple
$\mathcal{L}\in\operatorname{Loc}_{K}(\mathcal{O})$ appear as summands of
$\pi_{*}\mathbb{Q}_{\widetilde{\mathcal{O}}}$, so it is enough to show that
$H^{*}_{K}(\mathcal{O};\pi_{*}\mathbb{Q}_{\widetilde{\mathcal{O}}})$ vanishes
in odd degree.

Let $(K_{x}^{\circ})^{\mathrm{red}}$ be the maximal reductive quotient of
$K_{x}^{\circ}$. We have

$H^{*}_{K}(\mathcal{O};\pi_{*}\mathbb{Q}_{\widetilde{\mathcal{O}}})=H^{*}_{K}(\widetilde{\mathcal{O}})=H^{*}_{K_{x}^{\circ}}(\mathrm{pt})=H^{*}_{(K_{x}^{\circ})^{\mathrm{red}}}(\mathrm{pt})=\left(H^{*}_{T^{\prime}}(\mathrm{pt})\right)^{W^{\prime}},$

where $T^{\prime}$ is a maximal torus in $(K_{x}^{\circ})^{\mathrm{red}}$ and
$W^{\prime}$ is the Weyl group of $(K_{x}^{\circ})^{\mathrm{red}}$. It is a
classical result that $H_{T^{\prime}}^{*}(\mathrm{pt})$ is a polynomial ring
with generators in even degree (see Borel isomorphism in proof of Lemma
7.1.2), which establishes the result. ∎

Lemma 7.2.7 guarantees that we can apply results in [JMW14] in our setting.

######  Definition 7.2.8.

Let $\mathcal{G}\in D^{b}_{K}(X)$, and denote by
$\mathcal{H}^{i}(\mathcal{G})$ its cohomology sheaves. For an orbit
$\mathcal{O}$, denote by $k_{\mathcal{O}}:\mathcal{O}\rightarrow X$ the
locally closed inclusion map.

  1. (1)

The complex $\mathcal{G}$ is $*$-even (resp. $!$-even) if for all $K$-orbits
$\mathcal{O}$ in $X$, $\mathcal{H}^{n}(k_{\mathcal{O}}^{*}\mathcal{G})$ (resp.
$\mathcal{H}^{n}(k_{\mathcal{O}}^{!}\mathcal{G})$) vanishes for $n$ odd.

  2. (2)

The complex $\mathcal{G}$ is $*$-odd (resp. $!$-odd) if $\mathcal{G}[1]$ is
$*$-even (resp. $!$-even).

  3. (3)

The complex $\mathcal{G}$ is even (resp. odd) if it is both $*$-even (resp.
$*$-odd) and $!$-even (resp. $!$-odd).

  4. (4)

The complex $\mathcal{G}$ is parity if it splits into the direct sum of an
even and an odd complex.

######  Lemma 7.2.9.

[JMW14, Cor. 2.8] If $\mathcal{F}$ is $*$-even and $\mathcal{G}$ is $!$-odd,
then

$\operatorname{Hom}_{K}(\mathcal{F},\mathcal{G})=0.$

Using Lemma 7.2.9, we are able to prove our desired result. Fix a closed orbit
$\mathcal{Q}$ and set $i:=i_{\mathcal{Q}}:\mathcal{Q}\hookrightarrow X$.

######  Lemma 7.2.10.

If $\mathcal{G}\in D^{b}_{K}(X)$ is parity, then the natural map
$\mathbb{H}_{K}^{\bullet}(i_{!}i^{!}\mathcal{G})\rightarrow\mathbb{H}_{K}^{\bullet}(\mathcal{G})$
is injective.

###### Proof.

Let $U=X\smallsetminus\mathcal{Q}$. We have open and closed immersions

$U\xhookrightarrow{j}X\xhookleftarrow{i}\mathcal{Q},$

so for any $\mathcal{G}\in D^{b}_{K}(X)$, there is a distinguished triangle

(7.20) $i_{!}i^{!}\mathcal{G}\rightarrow\mathcal{G}\rightarrow
j_{*}j^{*}\mathcal{G}\xrightarrow{[1]}.$

Because the functor $\operatorname{Hom}_{K}(\mathbb{Q}_{X},-)$ is
cohomological, we obtain from (7.20) a long exact sequence

(7.21)
$\begin{split}\cdots\rightarrow\operatorname{Hom}^{0}_{K}(\mathbb{Q}_{X},i_{!}i^{!}\mathcal{G})&\rightarrow\operatorname{Hom}^{0}_{K}(\mathbb{Q}_{X},\mathcal{G})\\\
&\rightarrow\operatorname{Hom}^{0}(\mathbb{Q}_{X},j_{*}j^{*}\mathcal{G})\rightarrow\operatorname{Hom}^{1}(\mathbb{Q}_{X},i_{!}i^{!}\mathcal{G})\rightarrow\cdots\end{split}$

Assume that $\mathcal{G}$ is $!$-even. Then by Lemma 7.2.9,
$\operatorname{Hom}_{K}^{n}(\mathbb{Q}_{X},\mathcal{G})=\operatorname{Hom}_{K}(\mathbb{Q}_{X},\mathcal{G}[n])=0$
for $n$ odd, so every seventh term in (7.21) vanishes. Morover, we claim that
if $\mathcal{G}$ is $!$-even, then $j_{*}j^{*}\mathcal{G}$ is $!$-even as
well, so every 8th term in (7.21) also vanishes. To see this, first note that
$i^{!}j_{*}=0$, so $\mathcal{H}^{n}(i^{!}j_{*}j^{*}\mathcal{G})=0$ for all
$n$. Any other orbit $\mathcal{O}\neq\mathcal{Q}$ is contained in $U$, and the
inclusion map $k_{\mathcal{O}}$ factors

${\mathcal{O}}$${U}$${X.}$$\scriptstyle{\tilde{j}}$$\scriptstyle{k_{\mathcal{O}}}$$\scriptstyle{j}$

Hence

$k_{\mathcal{O}}^{!}j_{*}j^{*}\mathcal{G}=\tilde{j}^{!}j^{!}j_{*}j^{*}\mathcal{G}=\tilde{j}^{!}j^{*}j_{*}j^{*}\mathcal{G}=\tilde{j}^{!}j^{*}\mathcal{G}=k_{\mathcal{O}}^{!}\mathcal{G}.$

Here $j^{!}=j^{*}$ because $j$ is an open immersion. This shows that
$j_{*}j^{*}\mathcal{G}$ is also $!$-even, so the long exact sequence (7.21)
splits into exact sequences

(7.22)
$\begin{split}0\rightarrow\operatorname{Hom}^{n}_{K}(\mathbb{Q}_{X},i_{!}i^{!}\mathcal{G})&\xrightarrow{f^{n}}\operatorname{Hom}_{K}^{n}(\mathbb{Q}_{X},\mathcal{G})\\\
&\rightarrow\operatorname{Hom}^{n}_{K}(\mathbb{Q}_{X},j_{*}j^{*}\mathcal{G})\rightarrow\operatorname{Hom}_{K}^{n+1}(\mathbb{Q}_{X},i_{!}i^{!}\mathcal{G})\rightarrow
0\end{split}$

for $n$ even. In particular, for $!$-even $\mathcal{G}$, the map $f^{n}$ is
injective for all $n$ even (and zero for all $n$ odd). Analogously, if
$\mathcal{G}$ is $!$-odd, then (7.21) splits into (7.22) for $n$ odd, and
$f^{n}$ is injective for $n$ odd (and zero for $n$ even). Hence if
$\mathcal{G}$ is parity, $f^{n}$ is injective for all $n$. As the natural map
(7.12) is given by $\bigoplus_{n\in\mathbb{Z}}f^{n}$, this shows that if
$\mathcal{G}$ is parity, (7.12) is injective. ∎

Together, Lemma 7.2.10 and Lemma 7.2.6 imply that (7.11) is injective if
$\mathcal{G}$ is parity. Our final step is to show that every object in
$\mathcal{M}_{LV}^{0}$ is parity.

######  Corollary 7.2.11.

All complexes in $\mathcal{M}_{LV}^{0}$ are parity.

###### Proof.

Let $k_{\mathcal{O}}:\mathcal{O}\rightarrow X$ be inclusion. Because the
functors $k_{\mathcal{O}}^{*}$ are exact for all $K$-orbits $\mathcal{O}$,
Remark 4.4.4 implies that
$\mathcal{H}^{n}(k_{\mathcal{O}^{\prime}}^{*}\bm{IC}(\overline{\mathcal{O}},\mathcal{L}))=0$
for any $\mathcal{O}^{\prime}\in K\backslash X$ and $n$ odd. Hence
$\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$ is $*$-even. Because
$\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$ is self-dual, it must also be
$!$-even [JMW14, Rmk. 2.5(3)]. By the decomposition theorem,
$\pi_{s}^{*}\pi_{s*}\bm{IC}(\overline{\mathcal{O}},\mathcal{L})$ is a sum of
shifts of $\bm{IC}$ sheaves. Because $\mathcal{M}_{LV}^{0}$ is generated by
the action of push-pull functors, any object in $\mathcal{M}_{LV}^{0}$
decomposes into a sum of shifts of $\bm{IC}$ sheaves. Hence any object in
$\mathcal{M}_{LV}^{0}$ can be split into a sum of an even complex and an odd
complex. ∎

This completes the second step in the proof of Theorem 7.2.1. We have now
established that the functor $\mathbb{H}_{K}^{\bullet}$ is faithful on
$\mathcal{M}_{LV}^{0}$. It remains to show that $\mathbb{H}_{K}^{\bullet}$ is
full.

######  Theorem 7.2.12.

The map (7.11) is surjective for $\mathcal{G}\in\mathcal{M}_{LV}^{0}$.

The structure of the proof of Theorem 7.2.12 is as follows. By Lemma 7.2.6 and
Theorem 7.2.5, for $\mathcal{G}\in\mathcal{M}_{LV}^{0}$ we have injections

(7.23)
${\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})}$${\mathbb{H}_{K}^{\bullet}(\mathcal{G})}$${\operatorname{Hom}_{\mathrm{bim}}(\mathbb{H}_{K}^{\bullet}(\bm{IC}_{\mathcal{Q}}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))\hskip
28.45274pt}$$\scriptstyle{F}$$\scriptstyle{G}$

given by $F=c\circ b\circ a$ and $G=d\circ\mathbb{H}_{K}^{\bullet}$ in the
pentagon diagram (7.17). Moreover, because $\mathcal{G}$ is parity (Corollary
7.2.11), the natural restriction map (7.25)

$\mathbb{H}_{K}^{\bullet}(\mathcal{G})\xrightarrow{\operatorname{res}}\mathbb{H}_{T_{K}}^{\bullet}(\operatorname{res}\mathcal{G})$

and the localization map

(7.24)
$\mathbb{H}_{T_{K}}^{\bullet}(\operatorname{res}\mathcal{G})\xrightarrow{i_{T_{K}}^{*}}\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G}),$

where $i_{T_{K}}:X^{T_{K}}\hookrightarrow X$ is the inclusion of $T_{K}$-fixed
points, are also injective. This is shown in Lemma 7.2.13 and Lemma 7.2.14. In
Lemma 7.2.16 we show that the image of
$\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})$ and the image of
$\operatorname{Hom}_{\mathrm{bim}}(\mathbb{H}_{K}^{\bullet}(\bm{IC}_{\mathcal{Q}}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))$
in $\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$
agree under this series of injections. Because the diagram (7.17) in Lemma
7.2.6 commutes and (7.11) is injective (Theorem 7.2.5), this implies the
surjectivity of (7.11).

Let $\operatorname{res}:D^{b}_{K}(X)\rightarrow D^{b}_{T_{K}}(X)$ be the
restriction functor sending
$\mathcal{G}=(\mathcal{G}_{X},\overline{\mathcal{G}},\beta)$ to
$\operatorname{res}\mathcal{G}:=(\mathcal{G}_{X},f^{*}\overline{\mathcal{G}},\beta)$,
where $f$ is the projection

$X\times_{T_{K}}EK\xrightarrow{f}X\times_{K}EK.$

Denote by
$\mathbb{H}_{K}^{\bullet}(\mathcal{G})\xrightarrow{\operatorname{res}}\mathbb{H}_{T_{K}}^{\bullet}(\mathcal{G})$
the induced map on hypercohomology:

(7.25)
$\varphi=(\varphi_{X},\overline{\varphi})\xmapsto{\operatorname{res}}\operatorname{res}\varphi:=(\varphi_{X},f^{*}\overline{\varphi}).$

######  Lemma 7.2.13.

If $\mathcal{G}\in D^{b}_{K}(X)$ is parity, then
$\mathbb{H}_{K}^{\bullet}(\mathcal{G})\xrightarrow{\operatorname{res}}\mathbb{H}^{\bullet}_{T_{K}}(\operatorname{res}\mathcal{G})$
is injective.

###### Proof.

Let $\mathcal{G}=(\mathcal{G}_{X},\overline{\mathcal{G}},\beta)\in
D^{b}_{K}(X)$. The Leray spectral sequence for
$\operatorname{res}\mathcal{G}=(\mathcal{G}_{X},f^{*}\overline{\mathcal{G}},\beta)\in
D^{b}_{T_{K}}(X)$ associated to the fibration

$K/{T_{K}}\rightarrow X\times_{T_{K}}EK\xrightarrow{f}X\times_{K}EK$

is

$E^{pq}_{2}=H^{p}(X\times_{K}EK;R^{q}f_{*}f^{*}\overline{\mathcal{G}})\implies
H^{p+q}(X\times_{T_{K}}EK;f^{*}\overline{\mathcal{G}}).$

Because $f$ is a fibration with fiber $K/{T_{K}}$, we have

$R^{q}f_{*}f^{*}\overline{\mathcal{G}}\simeq\overline{\mathcal{G}}\otimes
H^{q}(K/{T_{K}}).$

Hence
$\mathbb{H}^{\bullet}_{T_{K}}(\operatorname{res}\mathcal{G})=H^{*}(X\times_{T}EK;f^{*}\overline{\mathcal{G}})$
can be computed with the spectral sequence

(7.26) $H^{p}(X\times_{K}EK;\overline{\mathcal{G}})\otimes
H^{q}(K/{T_{K}})\implies
H^{p+q}(X\times_{T_{K}}EK;f^{*}\overline{\mathcal{G}}).$

Let $B_{K}=K\cap B$, so $K/B_{K}$ is the flag variety of $K$. We have a
fibration

$B_{K}/{T_{K}}\rightarrow K/{T_{K}}\rightarrow K/B_{K}$

with affine fibres, so

$H^{*}(K/{T_{K}})=H^{*}(K/B_{K})\simeq P/P_{+}^{K},$

where $P_{+}\subseteq P$ is the augmentation ideal and
$P_{+}^{K}:=P_{+}^{W_{K}}$ is the $W_{K}$ invariants in $P_{+}$. In
particular, $H^{*}(K/{T_{K}})$ vanishes in odd degrees.

By Lemma 7.2.9, if $\mathcal{G}$ is $!$-even (resp. $!$-odd), then
$\mathbb{H}_{K}^{\bullet}(\mathcal{G})=H^{*}(X\times_{K}EK;\overline{\mathcal{G}})$
vanishes in odd (resp. even) degree. Hence if $\mathcal{G}$ is $!$-parity
(i.e., either $!$-even or $!$-odd), the spectral sequence (7.26) has a
chessboard pattern, hence degenerates on the $E_{2}$ page. In this case,

(7.27) $\mathbb{H}_{T_{K}}^{\bullet}(\operatorname{res}\mathcal{G})\simeq
P\otimes_{P^{K}}\mathbb{H}_{K}^{\bullet}(\mathcal{G}).$

If $\mathcal{G}$ is parity, then $\mathcal{G}$ is the direct sum of a $!$-even
and a $!$-odd complex, hence (7.27) also holds. This implies the result. ∎

Next we will show that the localization map (7.24) is also injective.

######  Lemma 7.2.14.

Let $\mathcal{G}\in\mathcal{M}_{LV}^{0}$ and
$i_{T_{K}}:X^{T_{K}}\hookrightarrow X$ be the inclusion of $T_{K}$-fixed
points. The localization map
$\mathbb{H}_{T_{K}}^{\bullet}(\operatorname{res}\mathcal{G})\xrightarrow{i_{T_{K}}^{*}}\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$
is injective.

###### Proof.

Let $\operatorname{For}:D^{b}_{K}(X)\rightarrow D^{b}(X)$ be the forgetful
functor, as in (3.13). Clearly $\operatorname{For}$ factors through
$D^{b}_{K}(X)\xrightarrow{\operatorname{res}}D^{b}_{T_{K}}(X)$.

By the localization theorem in equivariant hypercohomology [GKM98, Thm.
1.6.2], $i_{T_{K}}^{*}$ is injective if $\operatorname{res}\mathcal{G}$ is
equivariantly formal; that is, if the spectral sequence for its equivariant
cohomology

(7.28)
$E_{2}^{pq}=H^{p}_{T_{K}}(\mathrm{pt})\otimes\mathbb{H}^{\bullet}(\mathcal{G})\implies\mathbb{H}^{p+q}_{T_{K}}(\operatorname{res}\mathcal{G})$

degenerates at the $E_{2}$ page. Here
$\mathbb{H}^{\bullet}(\mathcal{G})=\operatorname{Hom}^{\bullet}_{D^{b}_{c}(X)}(\mathbb{Q}_{X},\operatorname{For}\mathcal{G})$
as in (3.13). Because $H^{*}_{T_{K}}(\mathrm{pt})$ vanishes in odd degree, to
show that $\operatorname{res}\mathcal{G}$ is equivariantly formal, it is
enough to show that $\mathbb{H}^{\bullet}(\mathcal{G})$ vanishes in all
degrees of one parity (as this implies the spectral sequence (7.28) has a
checkerboard pattern, hence degenerates at $E_{2}$).

Let $C:=H^{*}(X)=R/R_{+}^{W}$ be the cohomology ring of $X$. Note that $C$ is
a graded ring that vanishes in odd degree. Since
$\mathcal{G}\in\mathcal{M}_{LV}^{0}$, $\mathcal{G}$ is a direct summand of a
finite direct sum of shifts of objects of the form
$\bm{IC}_{\mathcal{Q}}*\bm{IC}_{s_{1}}*\cdots*\bm{IC}_{s_{n}}$ for some closed
orbit $\mathcal{Q}$ and sequence of simple reflections $(s_{1},\ldots,s_{n})$.
If
$\mathbb{H}^{\bullet}(\bm{IC}_{\mathcal{Q}}*\bm{IC}_{s_{1}}*\cdots*\bm{IC}_{s_{n}})$
vanishes in odd degree for all closed orbits $\mathcal{Q}$ and all subsets of
simple roots, then $\mathbb{H}^{\bullet}(\mathcal{G})$ does as well, so
without loss of generality, we may assume that
$\mathcal{G}=\bm{IC}_{\mathcal{Q}}*\bm{IC}_{s_{1}}*\cdots*\bm{IC}_{s_{n}}$. By
[Soe90, Lem. 13, Kor. 2], we have

$\mathbb{H}^{\bullet}(\mathcal{G})=C\otimes_{C^{s_{n}}}\cdots\otimes_{C^{s_{2}}}C\otimes_{C^{s_{1}}}H^{*}(\mathcal{Q})[\dim
Q].$

As $\mathcal{Q}\simeq K/B_{K}$ is a flag variety, $H^{*}(\mathcal{Q})$
vanishes in odd degrees, hence $\mathbb{H}^{\bullet}(\mathcal{G})$ vanishes in
one parity. ∎

Our final step is to compare the images of
$\operatorname{Hom}_{\mathrm{bim}}(H^{*}_{K}(\mathcal{Q}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))$
and $\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})$ in
$\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$. To
do so, we will use the $R$-bimodule structure of
$\mathbb{H}^{\bullet}_{T_{K}}(i^{*}_{T_{K}}\operatorname{res}\mathcal{G})$.

The space $\mathbb{H}_{T_{K}}^{\bullet}(\operatorname{res}\mathcal{G})$ is
naturally a module over $H^{*}_{T_{K}}(X)=P\otimes_{R^{W}}R$, so it also has
the structure of an $(P,R)$-bimodule. Via $i_{T_{K}}^{*}$,
$\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$
inherits a $P\otimes_{R^{W}}R$-module structure from
$\mathbb{H}_{T_{K}}^{\bullet}(\operatorname{res}\mathcal{G})$: for
$\varphi\in\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$
and $f\otimes g\in P\otimes_{R^{W}}R$,

$f\otimes g\cdot\varphi=\varphi\circ i_{T_{K}}^{*}(f\otimes g).$

With respect to this action, $i_{T_{K}}^{*}$ is a homomorphism of
$P\otimes_{R^{W}}R$-modules.

Recall that the $T_{K}$-fixed points in $X$ align with the $T$ fixed points,
so they are isolated and in bijection with $W$ (Lemma 6.1.4). Denote the
skyscraper sheaf on $x_{w}=wB/B\in X^{T_{K}}$ by $\mathbb{Q}_{w}$. Analogously
to the notation defined in (3.10) for raising $R$-bimodules to polynomial
powers, we set the following notation for objects in derived categories: for
every $q=\sum q_{i}v^{i}\in\mathbb{Z}[v^{\pm 1}]$ and $\mathcal{G}$ a complex
in a derived category, we denote by

$\mathcal{G}^{\oplus q}:=\bigoplus\mathcal{G}[i]^{\oplus q_{i}}.$

######  Lemma 7.2.15.

As graded $(P,R)$-bimodules,

$\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})\simeq\bigoplus_{w\in
W}(P_{w})^{\oplus q_{w}}$

where $q_{w}\in\mathbb{Z}[v^{\pm 1}]$ are given by the decomposition
$i_{T_{K}}^{*}\operatorname{res}\mathcal{G}=\bigoplus_{w\in
W}(\mathbb{Q}_{w})^{\oplus q_{w}}$ and $P_{w}\in(P,R)\mathrm{-gbim}$ is the
standard module corresponding to $w\in W$ (given by the obvious adaptation to
Definition 6.2.1).

###### Proof.

To begin, note that since $\mathbb{Q}_{X^{T_{K}}}=\bigoplus_{w\in
W}\mathbb{Q}_{w}$, we clearly have

$\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})=\operatorname{Hom}_{T_{K}}^{\bullet}(\bigoplus_{w\in
W}\mathbb{Q}_{w},\bigoplus_{w\in W}(\mathbb{Q}_{w})^{\oplus
q_{w}})=\bigoplus_{w\in W}(P_{w})^{\oplus q_{w}}$

as vector spaces, so the content of the lemma is that the $R$-bimodule
structures align.

The $(P,R)$-bimodule structure on
$\mathbb{H}_{T_{K}}^{\bullet}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$
comes from the algebra homomorphism

(7.29)
$P\otimes_{R^{W}}R=H^{*}_{T_{K}}(X)\xrightarrow{i_{T_{K}}^{*}}H^{*}_{T_{K}}(X^{T_{K}})=\bigoplus_{w\in
W}P_{w},$

so to prove the lemma, we need to determine the image of an element $f\otimes
g\in P\otimes_{R^{W}}R$ under $i_{T_{K}}^{*}$. The algebra $P\otimes_{R^{W}}R$
is generated by $P\otimes 1$ and $1\otimes X(T)_{\mathbb{Q}}$, so it suffices
to determine the image of $p\otimes 1$ and $1\otimes\chi$ for $p\in R$ and
$\chi\in X(T)_{\mathbb{Q}}$.

The map (7.29) is compatible with the left
$P=H^{*}_{T_{K}}(\mathrm{pt})$-module structures on either side because the
commutativity of the the diagram

${X}$${X^{T_{K}}}$${\mathrm{pt}}$$\scriptstyle{\pi}$$\scriptstyle{i_{T_{K}}}$$\scriptstyle{\pi_{T_{K}}}$

implies the commutativity of the diagram

${H^{*}_{T_{K}}(X)}$${H^{*}_{T_{K}}(X^{T_{K}})}$${H^{*}_{T_{K}}(\mathrm{pt})}$$\scriptstyle{i_{T_{K}}^{*}}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi_{T_{K}}^{*}}$

Hence

$i_{T_{K}}^{*}(p\otimes 1)=p\cdot i_{T_{K}}^{*}(1\otimes
1)=p\cdot(1,\ldots,1)=(p,\ldots,p).$

It remains to determine the image of $1\otimes\chi$. First, note that we have
a natural ring homomorphism $P\rightarrow H_{T_{K}}^{*}(X^{T_{K}})$ given by
the Borel homomorphism. The construction is similar to that in the proof of
Lemma 7.1.2: for $\chi\in X(T_{K})$ and $w\in W$, construct
$\mathbb{C}^{\times}$-bundles

$\displaystyle ET_{K}\times_{T_{K}}G\times_{B}\mathbb{C}^{\times}_{\chi}$
$\displaystyle\xrightarrow{\mu^{\chi}}ET_{K}\times_{T_{K}}G\times_{B}\mathrm{pt}\simeq
ET_{K}\times_{T_{K}}X$ $\displaystyle
ET_{K}\times_{T_{K}}wB\times_{B}\mathbb{C}^{\times}_{\chi}$
$\displaystyle\xrightarrow{\mu_{w}^{\chi}}ET_{K}\times_{T_{K}}wB\times_{B}\mathrm{pt}\simeq
BT_{K},$

where $\chi$ is inflated trivially to a character of $B$. By taking first
Chern classses, we obtain ring homomorphisms

$\displaystyle P$
$\displaystyle\xrightarrow{\varphi}H^{*}_{T_{K}}(X);\chi\mapsto
c_{1}(\mu^{\chi}),\text{ and }$ $\displaystyle P$
$\displaystyle\xrightarrow{\varphi_{w}}H^{*}_{T_{K}}(x_{w});\chi\mapsto
c_{1}(\mu_{w}^{\chi}).$

Because $\mu^{\chi}$ and $\mu^{\chi}_{w}$ fit into the Cartesian square

${ET_{K}\times_{T_{K}}G\times_{B}\mathbb{C}^{\times}_{\chi}}$${ET_{K}\times_{T_{K}}G\times_{B}\mathrm{pt}\simeq
ET_{K}\times_{T_{K}}X}$${ET_{K}\times_{T_{K}}wB\times_{B}\mathbb{C}^{\times}_{\chi}}$${ET_{K}\times_{T_{K}}wB\times_{B}\mathrm{pt}\simeq
BT_{K},}$$\scriptstyle{\mu^{\chi}}$$\scriptstyle{\mu_{w}^{\chi}}$

the diagram

${H^{*}_{T_{K}}(X)}$${P}$${\displaystyle{\bigoplus_{w\in
W}H^{*}_{T_{K}}(x_{w})}}$$\scriptstyle{i^{*}_{T_{K}}}$$\scriptstyle{\varphi}$$\scriptstyle{\displaystyle{\bigoplus_{w\in
W}\varphi_{w}}}$

commutes. This gives a natural ring homomorphism $\bigoplus_{w\in
W}\varphi_{w}:P\rightarrow H^{*}_{T_{K}}(X^{T_{K}})$. The image of
$1\otimes\chi\in P\otimes_{R^{W}}R$ is then given by pre-composing this ring
homomorphism with $\phi:R\rightarrow P$:

$1\otimes\chi\mapsto\bigoplus_{w\in W}1\otimes\varphi_{w}\circ\phi(\chi).$

In the proof of Lemma 7.1.2, we showed that the right action of $g\in R$ on
$p\in P\simeq H^{*}_{T_{K}}(x_{w})$ obtained from $\varphi_{w}\circ\phi$ is
given by

$p\cdot g=p\varphi(wg),$

so (7.29) sends

$1\otimes\chi\mapsto(\phi(wx))_{w\in W}.$

Hence for $f\otimes g\in P\otimes_{R^{W}}R$,

$i_{T_{K}}^{*}:f\otimes g\mapsto(f\phi(wg))_{w\in W}.$

This completes the proof. ∎

Using the $(P,R)$-bimodule structure of
$\mathbb{H}^{\bullet}_{T_{K}}(i_{T_{K}}^{*}\operatorname{res}\mathcal{G})$, we
can now show that the images of $i_{T_{K}}^{*}\circ\operatorname{res}\circ F$
and $i_{T_{K}}^{*}\circ\operatorname{res}\circ G$ agree. As before, we set
$i:=i_{\mathcal{Q}}:\mathcal{Q}\hookrightarrow X$.

######  Lemma 7.2.16.

Let $F,G,\operatorname{res},i_{T_{K}}^{*}$ be as in (7.23), (7.25), and
(7.24). Then the images of $i_{T_{K}}^{*}\circ\operatorname{res}\circ F$ and
$i_{T_{K}}\circ\operatorname{res}\circ G$ agree.

###### Proof.

In the proof of Lemma 7.2.6, we showed that the image of $F$ in
$\mathbb{H}_{K}^{\bullet}(\mathcal{G})$ is

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\psi\in\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})\right\\},$

where $\overset{*}{\eta}:\mathbbm{1}\rightarrow i_{*}i^{*}$ is the unit of the
adjunction $(i^{*},i_{*})$.

We can also identify the image of $G$ in
$\mathbb{H}_{K}^{\bullet}(\mathcal{G})$. As elements of $\operatorname{im}G$
are of the form $\psi(1)$ for some
$\psi\in\operatorname{Hom}_{\mathrm{bim}}(H^{*}_{K}(\mathcal{Q}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))$,
and $H^{*}_{K}(\mathcal{Q})\simeq P_{v}$ as $P^{K}\otimes_{R^{W}}R$-modules
for some $v\in{{}^{K}W^{\theta}}$ (Lemma 7.1.2), any element
$g=\psi(1)\in\operatorname{im}G$ has the property that for any $p\in P^{K}$
and $r\in\varphi^{-1}(p)$,

(7.30) $\displaystyle p\otimes 1\cdot g=p\otimes 1\cdot\psi(1)$
$\displaystyle=\psi(p\otimes 1\cdot 1)$ $\displaystyle=\psi(p)$
$\displaystyle=\psi(1\otimes v^{-1}r\cdot 1)$ $\displaystyle=1\otimes
v^{-1}r\cdot\psi(1)=1\otimes v^{-1}r\cdot g.$

Moreover, if $g\in\mathbb{H}_{K}^{\bullet}(\mathcal{G})$ satisfies (7.30),
there exists
$\psi\in\operatorname{Hom}_{\mathrm{bim}}(H^{*}_{K}(\mathcal{Q}),\mathbb{H}_{K}^{\bullet}(\mathcal{G}))$
defined by $\psi(1)=g$ such that $G(\psi)=g$, so

$\operatorname{im}G=\left\\{\varphi\in\mathbb{H}_{K}^{\bullet}(\mathcal{G})\mid(p\otimes
1-1\otimes v^{-1}r)\cdot\varphi=0\text{ for all }p\in P^{K}\text{ and
}r\in\phi^{-1}(p)\right\\}.$

We will show that under $i_{T_{K}}^{*}\circ\operatorname{res}$,
$\operatorname{im}F$ and $\operatorname{im}G$ map to the same subset of
$\mathbb{H}_{T_{K}}^{\bullet}(i^{*}_{T_{K}}\operatorname{res}\mathcal{G})$.
Let $\varphi\in\operatorname{im}F$. Under $\operatorname{res}$ and
$i^{*}_{T_{K}}$, $\varphi$ maps to

$\mathbb{Q}_{X^{T_{K}}}=\bigoplus_{w\in
W}\mathbb{Q}_{w}\xrightarrow{i_{T_{K}}^{*}\operatorname{res}\varphi}i_{T_{K}}^{*}\operatorname{res}\mathcal{G}=\bigoplus_{w\in
W}(\mathbb{Q}_{w})^{\oplus q_{w}}.$

Because $\varphi$ factors through $\bm{IC}_{\mathcal{Q}}$,
$i_{T_{K}}^{*}\operatorname{res}\varphi$ must factor through
$i_{T_{K}}^{*}\bm{IC}_{\mathcal{Q}}$. The $T_{K}$-fixed points in
$\mathcal{Q}$ correspond to Weyl group elements in the same $W_{K}$-coset as
$v$ (Lemma 6.1.5), so

$i_{T_{K}}^{*}\bm{IC}_{\mathcal{Q}}=\bigoplus_{w\in W_{K}v}\mathbb{Q}_{w}.$

Hence $i_{T_{K}}^{*}\operatorname{res}\varphi$ vanishes outside of $W_{K}v$,
and we have an inclusion

$\operatorname{im}i_{T_{K}}^{*}\circ\operatorname{res}\circ
F\subseteq\operatorname{im}i_{T_{K}}^{*}\bigcap\bigoplus_{w\in
W_{K}v}(R_{w})^{\oplus q_{w}}.$

We claim that this is actually an equality

(7.31) $\operatorname{im}i_{T_{K}}^{*}\circ\operatorname{res}\circ
F=\operatorname{im}i_{T_{K}}^{*}\bigcap\bigoplus_{w\in W_{K}v}(R_{w})^{\oplus
q_{w}}.$

Indeed, recall that if we denote by $i$ the inclusion of $\mathcal{Q}$ into
$X$ and $j$ the inclusion of the complement, we have

$\operatorname{Hom}_{K}(\bm{IC}_{\mathcal{Q}},\mathcal{G})\simeq\operatorname{Hom}_{D^{b}_{K}(\mathcal{Q})}(\mathbb{Q}_{\mathcal{Q}},i^{!}\mathcal{G})$

(see (7.14)) and the short exact sequence

(7.32)
$0\rightarrow\mathbb{H}^{\bullet}_{T_{K}}(i_{!}i^{!}\mathcal{G})\rightarrow\mathbb{H}^{\bullet}_{T_{K}}(\mathcal{G})\rightarrow\mathbb{H}_{T_{K}}^{\bullet}(j_{*}j^{*}\mathcal{G})\rightarrow
0$

given by standard parity sheaf arguments. Applying the localization theorem to
every term in (7.32) implies that an element in
$\mathbb{H}^{\bullet}_{T_{K}}(\mathcal{G})$ (represented by a tuple
$(f_{w})_{w\in W}$) is in the image of
$\mathbb{H}_{T_{K}}^{\bullet}(i_{!}i^{!}\mathcal{G})$ in (7.32) if and only if
$f_{w}=0$ for all $w$ outside of $W_{K}v$, so (7.31) holds.

Now let $\varphi\in\operatorname{im}G$. As both $\operatorname{res}$ and
$i_{T_{K}}^{*}$ are compatible with $(P^{K},R)$-bimodule structures, we have
that $(p\otimes 1-1\otimes v^{-1}r)\in P^{K}\otimes_{R^{W}}R$ annihilates
$i_{T_{K}}^{*}\operatorname{res}\varphi=(p_{w}^{q_{w}})_{w\in
W}\in\bigoplus_{w\in W}(P_{w})^{q_{w}}$ for all $p\in P^{K}$ and
$r\in\phi^{-1}(p)$. By Lemma 7.2.15, for $p\in P^{K}$ and
$r\in\varphi^{-1}(p)$,

(7.33) $\displaystyle p\otimes 1\cdot i_{T}^{*}\operatorname{res}\varphi$
$\displaystyle=(pp_{w}^{q_{w}})_{w\in W},\text{ and }$ (7.34) $\displaystyle
1\otimes{v^{-1}r}\cdot i_{T_{K}}^{*}\operatorname{res}\varphi$
$\displaystyle=(\phi(wv^{-1}r)p_{w}^{q_{w}})_{w\in W}.$

The actions (7.33) and (7.34) agree if and only if $w\in W_{K}v$. Hence

(7.35) $\operatorname{im}i_{T}^{*}\circ\operatorname{res}\circ
G=\operatorname{im}i_{T}^{*}\bigcap\bigoplus_{w\in W_{K}v}(P_{w})^{\oplus
q_{w}}.$

As (7.31) and (7.35) align, we are done. ∎

This completes the proof of Theorem 7.2.1, and establishes an equivalence of
categories between $\mathcal{M}_{LV}^{0}$ and $\mathcal{N}_{LV}^{0}$. By
Theorem 3.5.2, we have established the main result of our paper.

######  Corollary 7.2.17.

Let $\mathcal{N}_{LV}^{0}$ be the category in Definition 6.2.5 and
$M_{LV}^{0}$ the $\bm{H}$-module in Theorem 5.1.3. As right $\bm{H}$-modules,

$[\mathcal{N}_{LV}^{0}]_{\oplus}\simeq M_{LV}^{0}.$

######  Example 7.2.18.

$($An alternate basis of $M_{LV}$ for $SL_{2}(\mathbb{R}))$ The formulas in
Example 6.2.6 describe the Lusztig–Vogan action on an alternate basis of
$M_{LV}$ (the basis corresponding to $\bm{IC}$ sheaves). In particular, by
passing to the split Grothendieck group in equations (6.10), (6.11) and
(3.12), we obtain the following equations in $M_{LV}$:

(7.36) $\displaystyle\mathbb{Q}_{\mathcal{Q}_{0}}\cdot b_{s}$
$\displaystyle=m_{s},$ (7.37)
$\displaystyle\mathbb{Q}_{\mathcal{Q}_{\infty}}\cdot b_{s}$
$\displaystyle=m_{s},$ (7.38) $\displaystyle m_{s}\cdot b_{s}$
$\displaystyle=vm_{s}+v^{-1}m_{s},$

where
$m_{s}=v\mathbb{Q}_{\mathcal{Q}_{0}}+v\mathbb{Q}_{\mathcal{Q}_{\infty}}+\mathbb{Q}_{\mathcal{O}}\in
M_{LV}$ corresponds to $[B_{s}]\in[\mathcal{N}_{\mathrm{LV}}^{0}]_{\oplus}$
(resp.
$[\bm{IC}(X,\mathbb{Q}_{\mathcal{O}})]\in[\mathcal{M}_{\mathrm{LV}}^{0}]_{\oplus}$)
under the isomorphism $[\mathcal{N}_{\mathrm{LV}}^{0}]_{\oplus}\simeq
M_{\mathrm{LV}}$ (resp. $[\mathcal{M}_{\mathrm{LV}}^{0}]_{\oplus}\simeq
M_{\mathrm{LV}}$). We note that in this example, the formulas describing the
Lusztig–Vogan action are considerably cleaner in this basis. (Compare to
equations (3.7).)

##  Appendix A Dual pairs in monoidal categories

In this appendix we recall the definition of a dual pair in a monoidal
category and describe a procedure for constructing adjunctions from dual
pairs. These results provide the categorical framework for the proof of Lemma
7.2.4.

Recall that a monoidal category consists of the data of a category
$\mathcal{A}$, along with a bifunctor
$\otimes:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$, a unit object
$\mathbbm{1}$, and natural isomorphisms

$\displaystyle\alpha_{X,Y,Z}:(X\otimes Y)\otimes Z$
$\displaystyle\xrightarrow{\sim}X\otimes(Y\otimes Z)\hskip
8.53581pt\text{(associator)},$ $\displaystyle\lambda_{X}:\mathbbm{1}\otimes X$
$\displaystyle\xrightarrow{\sim}X\hskip 8.53581pt\text{(left unitor), and }$
$\displaystyle\rho_{X}:X\otimes\mathbbm{1}$
$\displaystyle\xrightarrow{\sim}X\hskip 8.53581pt\text{(right unitor)}$

satisfying the pentagon and unit axiom [EGNO16, Ch. 1].

######  Definition A.0.1.

A dual pair in a monoidal category
$(\mathcal{A},\otimes,\alpha,\mathbbm{1},\rho,\lambda)$ is a pair $(X,X^{*})$
of objects in $\mathcal{A}$ such that there exist morphisms

$\beta:\mathbbm{1}\rightarrow X\otimes X^{*}\text{ (co-evaluation)},\hskip
8.53581pt\gamma:X^{*}\otimes X\rightarrow\mathbbm{1}\text{ (evaluation)}$

making the diagrams

(A.1) $\mathbbm{1}\otimes X$$(X\otimes X^{*})\otimes X$$X\otimes(X^{*}\otimes
X)$$X\otimes\mathbbm{1}$$X$$\scriptstyle\beta\otimes
X$$\scriptstyle\alpha_{X,X^{*},X}$$\scriptstyle
X\otimes\gamma$$\scriptstyle\rho_{X}$$\scriptstyle\lambda_{X}$

and

(A.2) $X^{*}\otimes\mathbbm{1}$$X^{*}\otimes(X\otimes X^{*})$$(X^{*}\otimes
X)\otimes X^{*}$$\mathbbm{1}\otimes X^{*}$$X^{*}$$\scriptstyle
X^{*}\otimes\beta$$\scriptstyle\alpha_{X^{*},X,X^{*}}$$\scriptstyle\gamma\otimes
X^{*}$$\scriptstyle\lambda_{X^{*}}$$\scriptstyle\rho_{X^{*}}$

commute.

Let $(\mathcal{A},\otimes,\alpha,\mathbbm{1},\rho,\lambda)$ be a monoidal
category. Recall that a right $\mathcal{A}$-module consists of the data of a
category $\mathcal{M}$, equipped with a bifunctor
$\otimes:\mathcal{M}\times\mathcal{A}\rightarrow\mathcal{M}$ and natural
isomorphisms

$\displaystyle\alpha_{M,X,Y}:(M\otimes X)\otimes Y$
$\displaystyle\xrightarrow{\sim}M\otimes(X\otimes Y)\text{ (associator)}$
$\displaystyle\rho_{M}:M\otimes\mathbbm{1}$
$\displaystyle\xrightarrow{\sim}M\text{ (right unitor) }$

satisfying the pentagon axiom and the unit axoim [EGNO16, Ch. 7].

######  Lemma A.0.2.

Let $(X,X^{*},\beta:\mathbbm{1}\rightarrow X\otimes X^{*},\gamma:X^{*}\otimes
X\rightarrow\mathbbm{1})$ be a dual pair in a monoidal category
$(\mathcal{A},\otimes,\alpha,\mathbbm{1},\rho,\lambda)$, and let $\mathcal{M}$
be a right $\mathcal{A}$-module. Define functors

${\mathcal{M}}$${\mathcal{M}}$$\scriptstyle{-\otimes X}$$\scriptstyle{-\otimes
X^{*}}$

and morphisms

$\displaystyle\eta_{M}:M\xrightarrow{\rho_{M}^{-1}}M\otimes\mathbbm{1}\xrightarrow{M\otimes\beta}M\otimes
X\otimes X^{*}$ $\displaystyle\epsilon_{M}:M\otimes X^{*}\otimes
X\xrightarrow{M\otimes\gamma}M\otimes\mathbbm{1}\xrightarrow{\rho_{M}}M$

for all objects $M$ in $\mathcal{M}$. Then $(-\otimes X,-\otimes X^{*})$ is an
adjoint pair of functors, with unit $\eta:id\rightarrow-\otimes X\otimes
X^{*}$ and counit $\epsilon:-\otimes X^{*}\otimes X\rightarrow id$.

###### Proof.

Naturality of $\rho$ guarantees that the collection of morphisms
$\\{\eta_{M}\\}_{M\in\mathcal{M}}$ and $\\{\epsilon_{M}\\}_{M\in\mathcal{M}}$
form natural transformations. To see that the zig-zag diagrams commute, we
factor them as follows (ignoring associativity isomorphisms):

(A.3) ${M\otimes X}$${M\otimes\mathbbm{1}\otimes X}$${M\otimes X\otimes
X^{*}\otimes X}$${M\otimes X}$${M\otimes X\otimes\mathbbm{1}}$${M\otimes
X}$$\scriptstyle{\rho_{M}^{-1}\otimes
X}$$\scriptstyle{id}$$\scriptstyle{M\otimes\beta\otimes
X}$$\scriptstyle{M\otimes\lambda_{X}}$$\scriptstyle{M\otimes
X\otimes\gamma}$$\scriptstyle{id}$$\scriptstyle{M\otimes\rho_{X}}$$\scriptstyle{\rho_{M\otimes
X}}$ (A.4) ${M\otimes X^{*}}$${M\otimes X^{*}\otimes\mathbbm{1}}$${M\otimes
X^{*}\otimes X\otimes X^{*}}$${M\otimes X^{*}}$${M\otimes\mathbbm{1}\otimes
X^{*}}$${M\otimes X^{*}}$$\scriptstyle{\rho_{M\otimes
X^{*}}^{-1}}$$\scriptstyle{id}$$\scriptstyle{M\otimes
X^{*}\otimes\beta}$$\scriptstyle{M\otimes\gamma\otimes
X^{*}}$$\scriptstyle{id}$$\scriptstyle{\rho_{M}\otimes X^{*}}$

In (A.3), the commutativity of the top left triangle follows from the unit
axiom in $\mathcal{M}$, commutativity of the top right square follows from the
first dual pair axiom (A.1), and commutativity of the bottom right triangle
follows from the pentagon axiom and unit axiom of $\mathcal{M}$, as in
[EGNO16, Prop. 2.2.4].

In (A.4), commutativity of the top left triangle follows from the pentagon
axiom and unit axiom in $\mathcal{M}$, as in [EGNO16, Prop. 2.2.4],
commutativity of the top right square follows from the second dual pair axiom
(A.2), and commutativity of the bottom right triangle follows from the unit
axiom in $\mathcal{M}$. ∎

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