$\delta_{Q}^{=\overline{w}}\circ\operatorname{oblv}^{U_{Q}({\mathbb{A}})}\circ\operatorname{CT}_{P\leftarrow
Q}^{\mathrm{enh}}\simeq\delta_{Q}^{=\overline{w}}\circ\operatorname{oblv}^{U_{Q}({\mathbb{A}})}\circ\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}\circ(p_{Q\to
P}^{\mathrm{enh}})^{!}|_{{\mathcal{I}}(G,P)}\to\delta_{Q}^{=\overline{w}}\circ(p_{Q\to
P}^{\mathrm{enh}})^{!}|_{{\mathcal{I}}(G,P)}$

induced by (18) is invertible. Now _imagine_ the adelic group
$U_{Q}({\mathbb{A}})$ exists in algebraic geometry and the analogy in Remark
2.3.6 can be made precise. For any object
${\mathcal{M}}_{P}\in{\mathcal{I}}(G,P)$, since it is
$U_{P}({\mathbb{A}})$-equivariant, so is $(p_{Q\to
P}^{\mathrm{enh}})^{!}({\mathcal{M}}_{P})$. Note that $U_{Q}/U_{P}\simeq
U_{Q}\cap M_{P}$. Hence

$\operatorname{oblv}^{U_{Q}({\mathbb{A}})}\circ\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}\circ(p_{Q\to
P}^{\mathrm{enh}})^{!}({\mathcal{M}}_{P})\simeq\operatorname{oblv}^{U_{Q}\cap
M_{P}({\mathbb{A}})}\circ\operatorname{Av}_{*}^{U_{Q}\cap
M_{P}({\mathbb{A}})}\circ(p_{Q\to P}^{\mathrm{enh}})^{!}({\mathcal{M}}_{P}).$

Hence we only need to show

$\delta_{Q}^{=\overline{w}}\to\delta_{Q}^{=\overline{w}}\circ\operatorname{oblv}_{*}^{U_{Q}\cap
M_{P}({\mathbb{A}})}\circ\operatorname{Av}_{*}^{U_{Q}\cap
M_{P}({\mathbb{A}})}$

is invertible. Note that the map

$\operatorname{Bun}_{R^{w}}^{(Q\cap
R^{w}){\mathrm{-gen}}}\simeq(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{R})^{=\overline{w}}\to\operatorname{Bun}_{L}$

factors through $\operatorname{Bun}_{L^{w}}^{(Q\cap L^{w}){\mathrm{-gen}}}$.
We can replace $\delta_{Q}^{=\overline{w}}$ by the composition

$\delta^{\prime}:\operatorname{DMod}(\operatorname{Bun}_{G}^{P{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}}^{(Q\cap
R^{w}){\mathrm{-gen}}})\xrightarrow{\mathrm{push}}\operatorname{DMod}(\operatorname{Bun}_{L^{w}}^{(Q\cap
L^{w}){\mathrm{-gen}}}).$

By Condition (ii), $U_{Q}\cap M_{P}\subseteq\ker(Q\cap R^{w}\to Q\cap L^{w})$,
hence $U_{Q}\cap M_{P}({\mathbb{A}})$ also acts on
$\operatorname{Bun}_{R^{w}}^{(Q\cap R^{w}){\mathrm{-gen}}}$ and stabilizes the
fibers of $\operatorname{Bun}_{R^{w}}^{(Q\cap
R^{w}){\mathrm{-gen}}}\to\operatorname{Bun}_{L^{w}}^{(Q\cap
L^{w}){\mathrm{-gen}}}$. This indicates

$\delta^{\prime}\simeq\delta^{\prime}\circ\operatorname{oblv}_{*}^{U_{Q}\cap
M_{P}({\mathbb{A}})}\circ\operatorname{Av}_{*}^{U_{Q}\cap
M_{P}({\mathbb{A}})}$

.

The above argument is non-rigorous at least for the following two reasons:

  * (1)

We did not define the pushforward functor along
$\operatorname{Bun}_{R^{w}}^{(Q\cap
R^{w}){\mathrm{-gen}}}\to\operatorname{Bun}_{L^{w}}^{(Q\cap
L^{w}){\mathrm{-gen}}}$, which is neither pseudo-proper nor represented by
algebraic stacks;

  * (2)

We did not define the adelic-actions and verify various properties of the
averaging functors.

Problem (1) is not essential. But Problem (2) seems out of reach with our
current techniques.

###  6.5. Recollection: the parabolic miraculous duality
${\mathcal{I}}(G,P)^{\vee}\simeq{\mathcal{I}}(G,P^{-})$

As explained in the beginning of this section, we need to use the duality
between ${\mathcal{I}}(G,P)$ and ${\mathcal{I}}(G,P^{-})$. The following
result is a global analogue of Theorem 3.1.2 and a generalization of the
miraculous duality on $\operatorname{Bun}_{G}$ (Theorem 6.1.2):

######  Theorem 6.5.1 ([Che21, Theorem E, Theorem 5.3.5]).

Let $P$ and $P^{-}$ be opposite parabolic subgroups of $G$ and $M:=P\cap
P^{-}$ be their Levi subgroup. Then the categories ${\mathcal{I}}(G,P)$ and
${\mathcal{I}}(G,P^{-})$ are canonically dual to each other. Moreover,

  * (1)

In the case $P=P^{-}=G$, we recover the miraculous duality on
$\operatorname{Bun}_{G}$;

  * (2)

Via this duality and the miraculous duality on $\operatorname{Bun}_{G}$, the
functors

$\operatorname{Eis}_{P\to
G}^{\mathrm{enh}}:{\mathcal{I}}(G,P)\to\operatorname{DMod}(\operatorname{Bun}_{G}),\;\;\operatorname{Eis}_{P^{-}\to
G}^{\mathrm{enh}}:{\mathcal{I}}(G,P^{-})\to\operatorname{DMod}(\operatorname{Bun}_{G})$

are conjugate to each other;

  * (3)

Via this duality and the miraculous duality on $\operatorname{Bun}_{M}$, the
functors

$\iota_{M,!}:\operatorname{DMod}(\operatorname{Bun}_{M})\to{\mathcal{I}}(G,P),\;\;\iota_{M,!}^{-}:\operatorname{DMod}(\operatorname{Bun}_{M})\to{\mathcal{I}}(G,P^{-})$

are conjugate to each other.

As in the local case (see §3.1), the above theorem formally implies:

######  Corollary 6.5.2.

Via the above dualities, we have:

  * (1)

The functors

$\displaystyle\operatorname{Eis}_{P\to
G}^{\mathrm{enh}}:{\mathcal{I}}(G,P)\to\operatorname{DMod}(\operatorname{Bun}_{G}),\;\;\operatorname{CT}_{G\leftarrow
P^{-}}^{\mathrm{enh}}:\operatorname{DMod}(\operatorname{Bun}_{G})\to{\mathcal{I}}(G,P^{-})$

are dual to each other;

  * (2)

The functors

$\displaystyle\iota_{M}^{!}:{\mathcal{I}}(G,P)\to\operatorname{DMod}(\operatorname{Bun}_{M}),\;\;\iota_{M,!}^{-}:\operatorname{DMod}(\operatorname{Bun}_{M})\to{\mathcal{I}}(G,P^{-})$

are dual to each other;

  * (3)

The functors

$\displaystyle\iota_{M}^{*}:{\mathcal{I}}(G,P)\to\operatorname{DMod}(\operatorname{Bun}_{M}),\;\;\iota_{M,*}^{-}:\operatorname{DMod}(\operatorname{Bun}_{M})\to{\mathcal{I}}(G,P^{-})$

are dual to each other.

######  Remark 6.5.3.

In fact, for any131313Except in this remark, we always use $Q$ to denote a
parabolic subgroup _smaller_ than $P$. We give up this convention here to be
compatible with [Che21]. $P\subseteq Q$ and $P^{-}\subseteq Q^{-}$, the
functors

$\displaystyle\operatorname{Eis}_{P\to
Q}^{\mathrm{enh}}:{\mathcal{I}}(G,P)\to{\mathcal{I}}(G,Q),\;\;\operatorname{CT}_{Q^{-}\leftarrow
P^{-}}^{\mathrm{enh}}:{\mathcal{I}}(G,Q^{-})\to{\mathcal{I}}(G,P^{-})$

are also dual to each other via the above dualities. This is implicit in
[Che21]. Let us explain the main ideas but treat the constructions in
_loc.cit._ as a blackbox.

We can assume $P$ and $Q$ are standard. In [Che21, §5.3] (see also [Che20a]),
the author considered a certain algebraic stack
$\overline{\operatorname{Bun}}_{G}$ equipped with a closed filtration
$(\overline{\operatorname{Bun}}_{G})_{\leq P}$ labelled by elements
$P\in\operatorname{Par}_{G}^{\mathrm{st}}$. There is a certain functor

${\mathbf{K}}:\operatorname{Par}_{G}^{\mathrm{st}}\to\operatorname{DMod}(\overline{\operatorname{Bun}}_{G})$

such that, for each $P\in\operatorname{Par}_{G}^{\mathrm{st}}$, the object
${\mathbf{K}}(P)$ is supported on the closed substack
$(\overline{\operatorname{Bun}}_{G})_{\leq P}$ of
$\overline{\operatorname{Bun}}_{G}$. There are maps

$\overline{\Delta}^{\mathrm{enh}}_{\leq
P}:(\overline{\operatorname{Bun}}_{G})_{\leq
P}\to\operatorname{Bun}_{G}^{P{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits}\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}$

functorial in $P$. The $!$-pushforward of ${\mathbf{K}}(P)$ along this map,
which we denote by ${\mathcal{K}}_{P}$, is contained in the full subcategory

${\mathcal{I}}(G,P)\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{I}}(G,P^{-})\simeq{\mathcal{I}}(G\mathbin{\mathop{\times}\displaylimits}G,P\mathbin{\mathop{\times}\displaylimits}P^{-})\subseteq\operatorname{DMod}(\operatorname{Bun}_{G}^{P{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits}\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}).$

Then [Che21, Theorem 5.3.5] says that ${\mathcal{K}}_{P}$ is the unit object
for a duality between ${\mathcal{I}}(G,P)$ and ${\mathcal{I}}(G,P^{-})$.

Hence, to show that $\operatorname{Eis}_{P\to Q}^{\mathrm{enh}}$ and
$\operatorname{CT}_{Q^{-}\leftarrow P^{-}}^{\mathrm{enh}}$ are mutually dual,
we only need to provide a canonical equivalence

$\operatorname{Eis}_{P\to
Q}^{\mathrm{enh}}\mathbin{\mathop{\otimes}\displaylimits}\operatorname{Id}({\mathcal{K}}_{P})\simeq\operatorname{Id}\mathbin{\mathop{\otimes}\displaylimits}\operatorname{CT}_{Q^{-}\leftarrow
P^{-}}^{\mathrm{enh}}({\mathcal{K}}_{Q})$

in
${\mathcal{I}}(G,Q)\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{I}}(G,P^{-})$.
This can be constructed as follows. Recall that the maps
$\overline{\Delta}^{\mathrm{enh}}_{\leq P}$ and the objects ${\mathbf{K}}(P)$
are functorial in $P$. Hence we obtain a morphism

$\operatorname{Eis}_{P\to
Q}^{\mathrm{enh}}\mathbin{\mathop{\otimes}\displaylimits}\operatorname{Eis}_{P^{-}\to
Q^{-}}^{\mathrm{enh}}({\mathcal{K}}_{P})\to{\mathcal{K}}_{Q},$

which induces a morphism

$\operatorname{Eis}_{P\to
Q}^{\mathrm{enh}}\mathbin{\mathop{\otimes}\displaylimits}\operatorname{Id}({\mathcal{K}}_{P})\to\operatorname{Id}\mathbin{\mathop{\otimes}\displaylimits}\operatorname{CT}_{Q^{-}\leftarrow
P^{-}}^{\mathrm{enh}}({\mathcal{K}}_{Q}).$

We only need to show that this morphism is invertible.

When $Q=G$, this is exactly the content of the proof of [Che21, Theorem 5.3.5]
in [Che21, §6.13.2]. The proof there reduces the claim to [Che21, Goal 6.4.3],
and then to [Che21, Goal 6.4.17]. In the general case, we can proceed as in
_loc.cit._ and reduce it to [Che21, Goal 6.4.18].

######  Remark 6.5.4.

It is very possible that the dualities in Theorem 6.5.1 are compatible with
all the functorialities of ${\mathcal{I}}(-,-)$ in Proposition-Construction
4.1.5. Up to homotopy coherence, this follows from Corollary 6.5.2 and Remark
6.5.3. However, one needs more geometric inputs to give a homotopy coherent
proof, e.g., one needs to relate $\overline{\operatorname{Bun}}_{G}$ to
$\overline{\operatorname{Bun}}_{M}$.

######  Remark 6.5.5.

It is very possible that the dualities in Theorem 6.5.1 are compatible with
the $\operatorname{Sph}_{G,x}$-actions on ${\mathcal{I}}(G,P)$ and
${\mathcal{I}}(G,P^{-})$. This is _not_ obvious because there is no
$\operatorname{Sph}_{G,x}$-action on $\overline{\operatorname{Bun}}_{G}$.

###  6.6. The equivalence $\operatorname{Id}\to\beta_{G}\circ\beta_{G}^{L}$:
the dual Weyl filtration

In this subsection, we introduce the dual Weyl filtration on the functor

$\alpha_{P}:{\mathcal{I}}(G,P)\xrightarrow{\operatorname{Eis}_{P\to
G}^{\mathrm{enh}}}\operatorname{DMod}(\operatorname{Bun}_{G})\xrightarrow{\operatorname{CT}_{G\leftarrow
R}^{\mathrm{enh}}}{\mathcal{I}}(G,R)\xrightarrow{\iota_{L}^{!}}\operatorname{DMod}(\operatorname{Bun}_{L}).$

Via the dualities in Corollary 6.5.2, the above functor is dual to

$\operatorname{DMod}(\operatorname{Bun}_{L})\xrightarrow{\iota_{L,!}^{-}}{\mathcal{I}}(G,R^{-})\xrightarrow{\operatorname{Eis}_{R^{-}\to
G}^{\mathrm{enh}}}\operatorname{DMod}(\operatorname{Bun}_{G})\xrightarrow{\operatorname{CT}_{G\leftarrow
P^{-}}^{\mathrm{enh}}}{\mathcal{I}}(G,P^{-}).$

In other words, we have

$\alpha_{P}^{\vee}\simeq\operatorname{CT}_{G\leftarrow
P^{-}}^{\mathrm{enh}}\circ\operatorname{Eis}_{R^{-},!}.$

Consider the commutative diagram

$\textstyle{\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\operatorname{Bun}}_{R^{-}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Bun}_{G}.}$

By the base-change isomorphism, the functor $\operatorname{CT}_{G\leftarrow
P^{-}}^{\mathrm{enh}}\circ\operatorname{Eis}_{R^{-},!}$ is equivalent to the
composition of141414Here we do not need to distinguish the $!$-pushforward and
the $*$-pushforward along
$\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}}\to\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}$
because it is a disjoint union of proper maps.

$\displaystyle\epsilon_{P^{-}}:\operatorname{DMod}(\operatorname{Bun}_{L})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{-}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{-}})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})\xrightarrow{\mathrm{push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}})$

with

$\operatorname{Av}_{*}^{U_{P}^{-}({\mathbb{A}})}:\operatorname{DMod}(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}})\to{\mathcal{I}}(G,P^{-}).$

By definition, we have

$\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}}\simeq\mathbf{Maps}_{\mathrm{gen}}(X,G\backslash\overline{G/U_{R}^{-}}/L\leftarrow
P^{-}\backslash G/R^{-}).$

Similarly to Construction 6.4.1, we can define a filtration on this fiber
product. This filtration is still labelled by elements $\overline{w}\in
W_{G}/W_{R}$ because the definition of the poset $W_{G}/W_{R}$ is invariant
under the Cartan involution.

######  Construction 6.6.1.

For any $\overline{w}\in W_{G}/W_{R}$ and any representative $w\in W_{G}$ of
it, consider the closure $\overline{P^{-}wR^{-}}$ of $P^{-}wR^{-}$ in $G$.
Note that it only depends on $\overline{w}$. Define

$(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{\leq\overline{w}}:=\mathbf{Maps}_{\mathrm{gen}}(X,G\backslash\overline{G/U_{R}^{-}}/L\leftarrow
P^{-}\backslash\overline{P^{-}wR^{-}}/R^{-})$

Then we obtain a filtration of
$\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}}$
by the poset $W_{G}/W_{R}$. We also consider

$(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{=\overline{w}}:=\mathbf{Maps}_{\mathrm{gen}}(X,G\backslash\overline{G/U_{R}^{-}}/L\leftarrow
P^{-}\backslash P^{-}wR^{-}/R^{-})$

As in Remark 6.4.5, we have

(22)
$(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{=\overline{w}}\simeq\widetilde{\operatorname{Bun}}_{(R^{-})^{w}}^{P^{-}\cap(R^{-})^{w}{\mathrm{-gen}}},$

where $(R^{-})^{w}:=wR^{-}w^{-1}$, which is also the image of $R^{w}$ under
the Cartan involution.

######  Construction 6.6.2.

As in Construction 6.4.2 and Construction 6.4.4, we obtain a filtration on
$\epsilon_{P^{-}}$ such that

$\displaystyle\epsilon_{P^{-}}^{\leq\overline{w}}:\operatorname{DMod}(\operatorname{Bun}_{L})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{-}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{-}})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}((\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{\leq\overline{w}})\xrightarrow{\mathrm{push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}).$

We also have a similar description for $\epsilon_{P^{-}}^{=\overline{w}}$.
Namely,

  * •

If $P\in\operatorname{Par}_{G}^{\overline{w}}$, then

$\displaystyle\epsilon_{P^{-}}^{=\overline{w}}:\operatorname{DMod}(\operatorname{Bun}_{L})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{-}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{-}})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}((\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{=\overline{w}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}),$

which by (22) is equivalent to

$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L})\simeq\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{(R^{-})^{w}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{(R^{-})^{w}})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{(R^{-})^{w}}^{P^{-}\cap(R^{-})^{w}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}).$

  * •

If $P\notin\operatorname{Par}_{G}^{\overline{w}}$, then
$\epsilon_{P^{-}}^{=\overline{w}}\simeq 0$.

Composing with $\operatorname{Av}_{*}^{U_{P}^{-}({\mathbb{A}})}$, we obtain a
filtration on $\alpha_{P}^{\vee}$.

In view of Remark 6.5.3, the natural transformation (17) induces a natural
transformation

$\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\alpha_{Q}^{\vee}\simeq(\alpha_{Q}\circ\operatorname{CT}_{P\leftarrow
Q}^{\mathrm{enh}})^{\vee}\to(\alpha_{P})^{\vee}$

which, by construction, is given by

(23) $\displaystyle\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\alpha_{Q}^{\vee}\simeq\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\operatorname{CT}_{G\leftarrow
Q^{-}}^{\mathrm{enh}}\circ\operatorname{Eis}_{R^{-},!}\simeq\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\operatorname{CT}_{P^{-}\leftarrow
Q^{-}}^{\mathrm{enh}}\circ\operatorname{CT}_{G\leftarrow
P^{-}}^{\mathrm{enh}}\circ\operatorname{Eis}_{R^{-},!}\to$
$\displaystyle\to\operatorname{CT}_{G\leftarrow
P^{-}}^{\mathrm{enh}}\circ\operatorname{Eis}_{R^{-},!}\simeq(\alpha_{P})^{\vee}.$

######  Construction 6.6.3.

Similarly to Construction 6.4.3, the natural transformation (23) is compatible
with the filtrations.

In more detail, consider the map $p_{Q^{-}\to
P^{-}}^{\mathrm{enh}}:\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}}\to\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}$.
By definition

$\operatorname{oblv}^{U_{P}^{-}({\mathbb{A}})}\circ\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\simeq(p_{Q^{-}\to
P^{-}}^{\mathrm{enh}})_{!}\circ\operatorname{oblv}^{U_{Q}^{-}({\mathbb{A}})}.$

Hence we have a natural transformation

(24) $\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\operatorname{Av}_{*}^{U_{Q}^{-}({\mathbb{A}})}\to\operatorname{Av}_{*}^{U_{P}^{-}({\mathbb{A}})}\circ(p_{Q^{-}\to
P^{-}}^{\mathrm{enh}})_{!}$

between functors
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}})\to{\mathcal{I}}(G,P^{-})$.

On the other hand, the obvious commutative sqaure

$\textstyle{(\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{\leq\overline{w}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{\leq\overline{w}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}}$

induces a natural transformation

(25) $(p_{Q^{-}\to
P^{-}}^{\mathrm{enh}})_{!}\circ\epsilon_{Q^{-}}^{\leq\overline{w}}\to\epsilon_{P^{-}}^{\leq\overline{w}}.$

Combining with the previous paragraph, we obtain the desired natural
transformation

(26) $\displaystyle\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ(\alpha_{Q}^{\vee})^{\leq\overline{w}}\simeq\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\operatorname{Av}_{*}^{U_{Q}^{-}({\mathbb{A}})}\circ\epsilon_{Q^{-}}^{\leq\overline{w}}\to\operatorname{Av}_{*}^{U_{P}^{-}({\mathbb{A}})}\circ(p_{Q^{-}\to
P^{-}}^{\mathrm{enh}})_{!}\circ\epsilon_{Q^{-}}^{\leq\overline{w}}\to$
$\displaystyle\to\operatorname{Av}_{*}^{U_{P}^{-}({\mathbb{A}})}\circ\epsilon_{P^{-}}^{\leq\overline{w}}\simeq(\alpha_{P}^{\vee})^{\leq\overline{w}}$

######  Construction 6.6.4.

Passing back to the dual side, we obtain the _dual Weyl filtration_
$\alpha_{P}^{\prime\leq\overline{w}}$ on $\alpha_{P}$, where

$\alpha_{P}^{\prime\leq\overline{w}}:=((\alpha_{P}^{\vee})^{\leq\overline{w}})^{\vee}.$

The natural transformation (26) induces

(27) $\alpha_{P}^{\prime\leq\overline{w}}\circ\operatorname{CT}_{P\leftarrow
Q}^{\mathrm{enh}}\to\alpha_{Q}^{\prime\leq\overline{w}}.$

######  Remark 6.6.5.

We have not checked whether the dual Weyl filtration coincides with the Weyl
filtration, i.e.,
$\alpha_{P}^{\prime\leq\overline{w}}\simeq\alpha_{P}^{\leq\overline{w}}$.
Probably this can be proven by using the description of the unit object of the
duality between ${\mathcal{I}}(G,P)$ and ${\mathcal{I}}(G,P^{-})$ in [Che21].

###  6.7. The equivalence $\operatorname{Id}\to\beta_{G}\circ\beta_{G}^{L}$:
the cancellation lemma

In this subsection, we prove the cancellation lemma. The proof is very
technical and we believe future techniques can simplify it.

######  Proposition 6.7.1 (Cancellation lemma for the dual Weyl filtration).

For fixed $\overline{w}\in W_{G}/W_{R}$ and any representative $w$ of it, let
$Q,P\in\operatorname{Par}_{G}^{\overline{w}}$ with $Q\subseteq P$ satisfy the
following conditions:

  * (i)

$Q\cap R^{w}=P\cap R^{w}$;

  * (ii)

$U_{Q}\cap M_{P}\subseteq U_{R}^{w}$, where $U_{Q}$ (resp. $U_{R}$) is the
unipotent radical of $Q$ (resp. $R$), $M_{P}$ is the Levi subgroup of $P$ and
$U_{R}^{w}:=wU_{R}w^{-1}$.

Then (20) induces an equivalence

$\alpha_{Q}^{\prime=\overline{w}}\circ\operatorname{CT}_{P\leftarrow
Q}^{\mathrm{enh}}\simeq\alpha_{P}^{\prime=\overline{w}}$

between functors
${\mathcal{I}}(G,P)\to\operatorname{DMod}(\operatorname{Bun}_{L})$.

######  Remark 6.7.2.

In Corollary 6.8.2, we will show Condition (ii) actually follows from
Condition (i) and the assumption $Q\subseteq
P\in\operatorname{Par}_{G}^{\overline{w}}$.

###### Proof.

By duality, we need to show that the natural transformation

$\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ(\alpha_{Q}^{\vee})^{=\overline{w}}\to(\alpha_{P}^{\vee})^{=\overline{w}}$

is an equivalence.

Using the Cartan involution on $G$, Condition (i) and (ii) are equivalent to

  * (i-)

$Q^{-}\cap(R^{-})^{w}=P^{-}\cap(R^{-})^{w}$;

  * (ii-)

$U_{Q}^{-}\cap M_{P}\subseteq(U_{R}^{-})^{w}$.

Hence, by (22), we have

$(\operatorname{Bun}_{G}^{P^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{=\overline{w}}\simeq(\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\widetilde{\operatorname{Bun}}_{R^{-}})^{=\overline{w}}.$

It follows that (25) induces an equivalence

$(p_{Q^{-}\to
P^{-}}^{\mathrm{enh}})_{!}\circ\epsilon_{Q^{-}}^{=\overline{w}}\simeq\epsilon_{P^{-}}^{=\overline{w}}.$

It remains to show the natural transformation

$\operatorname{Eis}_{Q^{-}\to
P^{-}}^{\mathrm{enh}}\circ\operatorname{Av}_{*}^{U_{Q}^{-}({\mathbb{A}})}\circ\epsilon_{Q^{-}}^{=\overline{w}}\to\operatorname{Av}_{*}^{U_{P}^{-}({\mathbb{A}})}\circ(p_{Q^{-}\to
P^{-}}^{\mathrm{enh}})_{!}\circ\epsilon_{Q^{-}}^{=\overline{w}}$

induced by (24) is an equivalence. To simplify the notations, we apply the
Cartan involution and show that

$\operatorname{Eis}_{Q\to
P}^{\mathrm{enh}}\circ\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}\circ\epsilon_{Q}^{=\overline{w}}\to\operatorname{Av}_{*}^{U_{P}({\mathbb{A}})}\circ(p_{Q\to
P}^{\mathrm{enh}})_{!}\circ\epsilon_{Q}^{=\overline{w}}$

is an equivalence.

Similarly to Remark 6.4.8, our strategy is to show that the images of the
functor $\epsilon_{Q}^{=\overline{w}}$ are “$U_{Q}\cap
M_{P}({\mathbb{A}})$-equivariant” objects, and then show the natural
transformation (24) sends any such object to an invertible morphism.

For this, we first need to define such “$U_{Q}\cap
M_{P}({\mathbb{A}})$-equivariant objects”. Consider the functor

(28)
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{Q})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}}).$

######  Definition 6.7.3.

Define
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}\subseteq\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})$
to be the full subcategory sitting in the following Cartesian diagram

${\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}}$${\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})}$${\operatorname{DMod}(\operatorname{Bun}_{M_{Q}})}$${\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}}).}$$\scriptstyle{\subseteq}$${\lrcorner}$$\scriptstyle{(\ref{eqn-
proof-
cancellation-1})}$$\scriptstyle{\subseteq}$$\scriptstyle{*\mathrm{-pull}}$

######  Remark 6.7.4.

The essential image of the functor
$\operatorname{DMod}(\operatorname{Bun}_{M_{Q}})\to\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$
is stable under limits. To see this, we can replace the $*$-pullback functor
by the $!$-pullback functor and show the latter has a left adjoint. By the
contraction principle (see e.g. [DG15, Appendix C]), the left adjoint, i.e.,
the $!$-pushforward functor along
$\operatorname{Bun}_{Q/U_{P}}\to\operatorname{Bun}_{M_{Q}}$ is indeed well-
defined and given by the $!$-pullback functor along
$\operatorname{Bun}_{M_{Q}}\to\operatorname{Bun}_{Q/U_{P}}$.

Using the base-change isomorphisms, it is easy to see that
${\mathcal{I}}(G,Q)\subseteq\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}$.

Then to finish the proof of the cancellation lemma (Porposition 6.7.1), it is
enough to prove the following two lemmas:

######  Lemma 6.7.5.

For any arrow $Q\subseteq P$ in $\operatorname{Par}^{\mathrm{st}}_{G}$, the
natural transformation

$\textstyle{\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}}$$\scriptstyle{(p_{Q\to
P}^{\mathrm{enh}})_{!}}$$\textstyle{{\mathcal{I}}(G,Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Eis}_{Q\to
P}^{\mathrm{enh}}}$$\textstyle{\operatorname{DMod}(\operatorname{Bun}_{G}^{P{\mathrm{-gen}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Av}_{*}^{U_{P}({\mathbb{A}})}}$$\textstyle{{\mathcal{I}}(G,P)}$

is invertible when restricted to
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}$.

######  Lemma 6.7.6.

Let $\overline{w}\in W_{G}/W_{R}$ and $Q\subseteq P$ be an arrow in
$\operatorname{Par}_{G}^{\overline{w}}$. If $U_{Q}\cap M_{P}\subseteq
U_{R}^{w}$, then the functor

$\displaystyle\epsilon_{Q}^{=\overline{w}}:\operatorname{DMod}(\operatorname{Bun}_{L})\simeq\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}}^{Q\cap
R^{w}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}}).$

factors through
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}$.

###### Proof of Lemma 6.7.5..

It suffices to prove the assertion after composing the natural transformation
in question with the forgetful functor

$\iota_{M_{P}}^{!}:{\mathcal{I}}(G,P)\to\operatorname{DMod}(\operatorname{Bun}_{M_{P}}).$

Let
${\mathcal{F}}\in\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}$. We have a canonical arrow
$\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}({\mathcal{F}})\to{\mathcal{F}}$,
where, abusing notation, we view both ${\mathcal{F}}$ and
$\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}({\mathcal{F}})$ as objects in
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})$.

Now note that the functor
$\iota_{M_{P}}^{!}\circ\operatorname{Av}_{*}^{U_{P}({\mathbb{A}})}$ is
equivalent to

$\operatorname{DMod}(\operatorname{Bun}_{G}^{P{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{P})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{P}})$

because they have the same left adjoint (Lemma 2.3.7). Then, unwinding the
definitions, the assertion of the lemma boils down to proving that the functor

$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{P{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{P})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{P}})$

sends
$\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}({\mathcal{F}})\to{\mathcal{F}}$
to an invertible morphism.

To this end, observe that

$\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}^{P{\mathrm{-gen}}}}}\operatorname{Bun}_{P}\simeq\operatorname{Bun}_{P}^{Q{\mathrm{-gen}}};$

thus, using proper base-change and Remark 2.3.18, the above composition is
equivalent to

$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{P}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{P})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{P}}).$

Since $\operatorname{Bun}_{P}^{Q{\mathrm{-gen}}}\to\operatorname{Bun}_{P}$ and
$\operatorname{Bun}_{M_{P}}^{Q/U_{P}{\mathrm{-gen}}}\to\operatorname{Bun}_{M_{P}}$
are pseudo-proper, this is equivalent to

$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{P}^{Q{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{P}}^{Q/U_{P}{\mathrm{-gen}}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{P}}).$

Then it suffices to prove that

$\theta:\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{P}^{Q{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{P}}^{Q/U_{P}{\mathrm{-gen}}})$

sends
$\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}({\mathcal{F}})\to{\mathcal{F}}$
to an invertible morphism.

Using base-change, it is easy to check that $\theta$ sends
$\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})^{U_{Q}\cap
M_{P}({\mathbb{A}})}$ into
${\mathcal{I}}(P,Q)\subseteq\operatorname{DMod}(\operatorname{Bun}_{M_{P}}^{Q/U_{P}{\mathrm{-gen}}})$.
Hence we just need to show that, for any ${\mathcal{K}}\in{\mathcal{I}}(P,Q)$,
the morphism

$\operatorname{Maps}({\mathcal{K}},\theta(\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}({\mathcal{F}})))\to\operatorname{Maps}({\mathcal{K}},\theta({\mathcal{F}}))$

is invertible. As shown in the proof of Proposition-Construction 4.1.5, the
left adjoint $\theta^{L}$ of $\theta$ sends ${\mathcal{I}}(P,Q)$ to
${\mathcal{I}}(G,Q)$, and, as such, it is just the functor
$\operatorname{Eis}:{\mathcal{I}}(P,Q)\to{\mathcal{I}}(G,Q)$. Hence we just
need to check that the natural arrow

$\operatorname{Maps}(\operatorname{Eis}({\mathcal{K}}),\operatorname{Av}_{*}^{U_{Q}({\mathbb{A}})}({\mathcal{F}}))\to\operatorname{Maps}(\operatorname{Eis}({\mathcal{K}}),{\mathcal{F}})$

is an isomorphism: this is obvious because $\operatorname{Eis}({\mathcal{K}})$
is contained in ${\mathcal{I}}(G,Q)$.

∎[Lemma 6.7.5]

To prove Lemma 6.7.6, we first need to perform some preliminary
simplifications. By definition, we need to show that

$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}}^{Q\cap
R^{w}{\mathrm{-gen}}})\to$
$\displaystyle\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{Q})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$

factors through the fully faithful pullback functor
$\operatorname{DMod}(\operatorname{Bun}_{M_{Q}})\to\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$.
We write

$\widetilde{\operatorname{Bun}}_{R^{w}}^{Q\cap
R^{w}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}^{Q{\mathrm{-gen}}}}}\operatorname{Bun}_{Q}=:(\widetilde{\operatorname{Bun}}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}},$

and similarly when $\widetilde{\operatorname{Bun}}_{R^{w}}$ is replaced by
$\operatorname{Bun}_{R^{w}}$. By the base-change isomorphism, the above
displayed functor is equivalent to

(29)
$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}((\widetilde{\operatorname{Bun}}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}}).$

Now we use the defect stratification on
$\widetilde{\operatorname{Bun}}_{R^{w}}$ to reduce to a similar statement
about $\operatorname{Bun}_{R^{w}}$. In other words, we prove the following
result.

######  Lemma 6.7.7.

For $\overline{w}\in W_{G}/W_{R}$, let $Q\subseteq P$ be an arrow in
$\operatorname{Par}_{G}^{\overline{w}}$ such that $U_{Q}\cap M_{P}\subseteq
U_{R}^{w}$. Assume that the functor

(30)
$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}((\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$

factors through the pullback functor
$\operatorname{DMod}(\operatorname{Bun}_{M_{Q}})\to\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$.
Then so does the functor (29).

###### Proof.

Recall that $\widetilde{\operatorname{Bun}}_{R^{w}}$ has a defect
stratification such that the disjoint union of the strata is of the form
$\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H$,
where $H$ is certain Hecke ind-stack for $L^{w}$-torsors. Let

$\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H\to\widetilde{\operatorname{Bun}}_{R^{w}}$

be the locally closed embedding for this stratification. Then the composition

$\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H\to\widetilde{\operatorname{Bun}}_{R^{w}}\to\operatorname{Bun}_{G}$

is given by

$\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H\xrightarrow{\mathrm{pr}_{1}}\operatorname{Bun}_{R^{w}}\to\operatorname{Bun}_{G}.$

By the same method as in [BG02, Theorem 6.2.10], the image of the composition

$\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H)$

is contained in the image of the pullback functor
$\operatorname{DMod}(H)\to\operatorname{DMod}(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H)$.
Let ${\mathcal{F}}$ be any object in
$\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}})$ such that its
$!$-restriction to
$\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H$
is an object pulled back from $H$. Then we only need to show the functor

$\phi:\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}((\widetilde{\operatorname{Bun}}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$

sends ${\mathcal{F}}$ to an object pulled back from
$\operatorname{Bun}_{M_{Q}}$.

We can assume ${\mathcal{F}}$ is supported on a single connected component
$\widetilde{\operatorname{Bun}}_{R^{w},\lambda}$ of
$\widetilde{\operatorname{Bun}}_{R^{w}}$. Write this connected component as a
union $\cup U_{\alpha}$ of open substacks such that each $U_{\alpha}$ is the
union of _finitely_ many defect strata. Then
${\mathcal{F}}\simeq\lim_{\alpha}{\mathcal{F}}_{\alpha}$, where
${\mathcal{F}}_{\alpha}$ is the $*$-extension of
${\mathcal{F}}|_{U_{\alpha}}$.

We claim each $\phi({\mathcal{F}}_{\alpha})$ is an object pulled back from
$\operatorname{Bun}_{M_{Q}}$. To prove the claim, we can replace
${\mathcal{F}}_{\alpha}$ by the $*$-extension of ${\mathcal{F}}|^{!}_{Y}$,
where $Y$ is some defect stratum of $\widetilde{\operatorname{Bun}}_{R^{w}}$.
By definition, such $*$-extension is contained in the image of the functor

$\varphi:\operatorname{DMod}(H)\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{L^{w}}}}H)\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{w}}).$

By the base-change isomorphism, the composition $\phi\circ\varphi$ is
equivalent to

$\operatorname{DMod}(H)\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{(\ref{eqn-
proof-lem-proof-
cancellation-3-0})}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}}).$

Hence the assumption on the functor (30) implies the claim.

By the above claim and Remark 6.7.4, the object
$\lim_{\alpha}\phi({\mathcal{F}}_{\alpha})$ is pulled back from
$\operatorname{Bun}_{M_{Q}}$. It remains to show
$\phi(\lim_{\alpha}{\mathcal{F}}_{\alpha})\to\lim_{\alpha}\phi({\mathcal{F}}_{\alpha})$
is an isomorphism. Recall that each connected component of
$\operatorname{Bun}_{Q}$ is quasi-compact over $\operatorname{Bun}_{G}$, hence
each connected component of

$(\widetilde{\operatorname{Bun}}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}}$

is quasi-compact over $\widetilde{\operatorname{Bun}}_{R^{w}}$. Thus, for a
fixed such connected component, the inverse images of
$U_{\alpha}\subseteq\widetilde{\operatorname{Bun}}_{R^{w}}$ in it stabilize
for large enough $U_{\alpha}$. This makes the desired claim manifest.

∎[Lemma 6.7.7]

Next, let us prove that the assumption of Lemma 6.7.7 holds true. This will
conclude the proof of Lemma 6.7.6. By definition, we have

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}}\simeq\mathbf{Maps}_{\mathrm{gen}}(X,Q\backslash
G/R^{w}\leftarrow{\mathbb{B}}(Q\cap R^{w})),$

which, up to non-reduced structures, it is isomorphic to

$\mathbf{Maps}_{\mathrm{gen}}(X,Q\backslash\overline{QR^{w}}/R^{w}\leftarrow{\mathbb{B}}(Q\cap
R^{w})).$

For any finite collection $\underline{x}=\big{\\{}x_{1},\dots,x_{n}\big{\\}}$
of closed points of $X$, let

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near
}\underline{x}}\subseteq\mathbf{Maps}_{\mathrm{gen}}(X,Q\backslash\overline{QR^{w}}/R^{w}\leftarrow{\mathbb{B}}(Q\cap
R^{w}))$

be the open substack classifying those maps $X\to
Q\backslash\overline{QR^{w}}/R^{w}$ that send the points $x_{i}$, and
therefore a Zariski neighborhood of them, into ${\mathbb{B}}(Q\cap
R^{w})\subseteq Q\backslash\overline{QR^{w}}/R^{w}$. By definition, when
$\underline{x}$ varies, these open substacks form a cover.

The following result says we can replace Lemma 6.7.6 to a similar statement
concerning these open substacks.

######  Lemma 6.7.8.

Let $\overline{w}\in W_{G}/W_{R}$ and $Q\subseteq P$ be an arrow in
$\operatorname{Par}_{G}^{\overline{w}}$ such that $U_{Q}\cap M_{P}\subseteq
U_{R}^{w}$. For any finite collection
$\underline{x}=\big{\\{}x_{1},\dots,x_{n}\big{\\}}$ of closed points of $X$,
suppose that the functor

(31)
$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}((\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near
}\underline{x}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}}).$

factors through the pullback functor
$\operatorname{DMod}(\operatorname{Bun}_{M_{Q}})\to\operatorname{DMod}(\operatorname{Bun}_{Q/U_{P}})$.
Then so does the functor (30).

###### Proof.

The lemma does _not_ follow immediately from Zariski descent and Remark 6.7.4
because

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}}$

can not be covered by finitely many

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near }\underline{x}}.$

However, this problem is not essential because

(32)
$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}}\to\operatorname{Bun}_{Q/U_{P}}$

is quasi-compact when restricted to each connected component of the source.

In more detail, recall $\operatorname{Bun}_{R^{w}}\to\operatorname{Bun}_{G}$
is quasi-compact when restricted to each connected component of the source,
and $\operatorname{Bun}_{Q}\to\operatorname{Bun}_{Q/U_{P}}$ is quasi-compact.
Hence the last claim of the above paragraph is true. To prove the lemma, we
can replace $\operatorname{Bun}_{M_{Q}}$ by any fixed quasi-compact open
substack $V\subseteq\operatorname{Bun}_{M_{Q}}$ and replace the map (32) by

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{M_{Q}}}}V\to\operatorname{Bun}_{Q/U_{P}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{M_{Q}}}}V.$

Recall that $\operatorname{Bun}_{Q/U_{P}}$ is quasi-compact over
$\operatorname{Bun}_{M_{Q}}$. Hence the target of the above map is quasi-
compact and so is each connected component of the source. Now each such
connected component can be covered by finitely many

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near
}\underline{x}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{M_{Q}}}}V,$

and we are done by Zariski descent and Remark 6.7.4.

∎[Lemma 6.7.8]

###### Proof of Lemma 6.7.6..

We just need to show that the assumption of Lemma 6.7.8 is valid. Write

$1\to K_{1}\to G_{1}\to H_{1}\to 1,\;\;1\to K_{2}\to G_{2}\to H_{2}\to 1$

for the short exact sequences

$1\to(Q\cap U_{R}^{w})\to Q\cap R^{w}\to(Q\cap R^{w})/(Q\cap U_{R}^{w})\to
1,\;\;1\to U_{Q}/U_{P}\to Q/U_{P}\to Q/U_{Q}\to 1.$

We claim both sequences have canonical splittings. For the second sequence,
the splitting is provided by the isomorphism

$M_{Q}\to Q\to Q/U_{P}\to Q/U_{Q}.$

For the first one, we only need to show that

$Q\cap L^{w}\to Q\cap R^{w}\to(Q\cap R^{w})/(Q\cap U_{R}^{w})$

is an isomorphism. Note that both $Q$ and $L^{w}$ contain the Cartan subgroup
$T$. Hence, the weight decomposition of $\operatorname{Lie}(G)$ implies the
above map induces an isomorphism of Lie algebras. On the other hand, choose
${\mathbb{G}}_{m}\to T$ such that the adjoint ${\mathbb{G}}_{m}$-action on
$R^{w}$ contracts it onto $L^{w}$. The restricted action on the closed
subscheme $Q\cap R^{w}$ must contract it onto $Q\cap L^{w}$. In particular,
$Q\cap L^{w}$ has non-empty intersection with any connected component of
$Q\cap R^{w}$. This finishes the proof of the claim.

Write $K_{0}:=U_{Q}\cap M_{P}$. By assumption, $K_{0}\subseteq K_{1}$, and the
composition $K_{0}\to G_{1}\to G_{2}$ induces an isomorphism $K_{0}\to K_{2}$.
Note that $K_{0},K_{1},K_{2}$ are unipotent.

Consider the fiber product

$\operatorname{Hecke}_{G_{2},\underline{x}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Hecke}_{H_{2},\underline{x}}}}\operatorname{Bun}_{H_{2}}$

studied in the proof of Lemma 2.4.1. Informally, it classifies Hecke
modifications of $G_{2}$-torsors at $\underline{x}$ that induce trivial Hecke
modifications of the induced $H_{2}$-torsors. As in the proof of Lemma 2.4.1,
an object ${\mathcal{F}}\in\operatorname{DMod}(\operatorname{Bun}_{G_{2}})$ is
pulled back from $\operatorname{Bun}_{H_{2}}$ iff the pullbacks of
${\mathcal{F}}$ along the two maps

$\operatorname{Bun}_{G_{2}}\xleftarrow{h^{l}}\operatorname{Hecke}_{G_{2},\underline{x}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Hecke}_{H_{2},\underline{x}}}}\operatorname{Bun}_{H_{2}}\xrightarrow{h^{r}}\operatorname{Bun}_{G_{2}}$

are isomorphic.

Now we reformulate the above condition using loop group actions. Consider the
$G_{2}({\mathcal{O}}_{\underline{x}})$-torsor $Z_{2}$ on
$\operatorname{Bun}_{G_{2}}$ that classifies $G_{2}$-torsors equipped with a
trivialization on $\operatorname{Spec}{\mathcal{O}}_{\underline{x}}$. The
standard re-gluing construction entends the
$G_{2}({\mathcal{O}}_{\underline{x}})$-action on $Z_{2}$ to a
$G_{2}({\mathcal{K}}_{\underline{x}})$-action. Note that we have

$\operatorname{DMod}(\operatorname{Bun}_{G_{2}})\simeq\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}.$

It follows the above condition is equivalent to
${\mathcal{F}}\in\operatorname{DMod}(Z_{2})^{K_{2}({\mathcal{K}}_{\underline{x}})H_{2}({\mathcal{O}}_{\underline{x}})}$.

Similarly, we can define a $G_{1}({\mathcal{O}}_{\underline{x}})$-torsor
$Z_{1}$ on
$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near }\underline{x}}$ and extend the
$G_{1}({\mathcal{O}}_{\underline{x}})$-action on $Z_{1}$ to a
$G_{1}({\mathcal{K}}_{\underline{x}})$-action. Note that the
$K_{1}({\mathcal{K}}_{\underline{x}})$-action on the source of

$Z_{1}\to
Z_{1}/G_{1}({\mathcal{K}}_{\underline{x}})\simeq(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near }\underline{x}}\to\operatorname{Bun}_{L^{w}}$

stabilizes the fibers (because $K_{1}$ is contained in the kernel of $Q\cap
R^{w}\to L^{w}$). Hence the image of the $!$-pullback functor

$\operatorname{DMod}(\operatorname{Bun}_{L^{w}})\to\operatorname{DMod}((\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near
}\underline{x}})\simeq\operatorname{DMod}(Z_{1})^{G_{1}({\mathcal{O}}_{\underline{x}})}$

is contained in the full subcategory

$\operatorname{DMod}(Z_{1})^{K_{1}({\mathcal{K}}_{\underline{x}})H_{1}({\mathcal{O}}_{\underline{x}})}\subseteq\operatorname{DMod}(Z_{1})^{G_{1}({\mathcal{O}}_{\underline{x}})}.$

Also note that there is a canonical map $Z_{1}\to Z_{2}$ intertwining the
$G_{i}({\mathcal{K}}_{\underline{x}})$-actions and compatible with the map

$(\operatorname{Bun}_{R^{w}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{Q})^{Q\cap
R^{w}\textnormal{ near }\underline{x}}\to\operatorname{Bun}_{M_{Q}}.$

It remains to show the functor

(33)
$\operatorname{DMod}(Z_{1})^{G_{1}({\mathcal{O}}_{\underline{x}})}\simeq\operatorname{DMod}(Z_{1}/G_{1}({\mathcal{O}}_{\underline{x}}))\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(Z_{2}/G_{2}({\mathcal{O}}_{\underline{x}}))\simeq\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}$

sends
$K_{1}({\mathcal{K}}_{\underline{x}})H_{1}({\mathcal{O}}_{\underline{x}})$-equivariant
objects to
$K_{2}({\mathcal{K}}_{\underline{x}})H_{2}({\mathcal{O}}_{\underline{x}})$-equivariant
objects. This does _not_ follow immediately from $K_{1}\supseteq K_{0}\simeq
K_{2}$ because $K_{0}$ is not a normal subgroup of $G_{1}$ hence we cannot
define the notion of
$K_{0}({\mathcal{K}}_{\underline{x}})H_{1}({\mathcal{O}}_{\underline{x}})$-equivariant
objects.

Nevertheless, this can still be formally proved as follows.

We first replace the relevant ind-group schemes by group schemes. This step is
standard. Namely, write
$K_{2}({\mathcal{K}}_{\underline{x}})\simeq\bigcup_{\alpha}V_{2}^{\alpha}$ as
union of group schemes stabilized by the adjoint action of
$H_{2}({\mathcal{O}}_{\underline{x}})$. For example, $V_{2}^{\alpha}$ can be
chosen to be $hK_{2}({\mathcal{O}}_{\underline{x}})h^{-1}$ for certain central
elements $h$ in $H_{2}({\mathcal{O}}_{\underline{x}})$. We can assume that
$V_{2}^{\alpha}$ contains $K_{2}({\mathcal{O}}_{\underline{x}})$. For a fixed
such $V_{2}:=V_{2}^{\alpha}$, let $V_{0}$ be the corresponding subgroup of
$K_{0}({\mathcal{K}}_{x})$. Let $V_{1}\subseteq
K_{1}({\mathcal{K}}_{\underline{x}})$ be a sub-group scheme containing $V_{0}$
and stabilized by $H_{1}({\mathcal{O}}_{\underline{x}})$. We only need to show
that the functor (33) sends
$V_{1}H_{1}({\mathcal{O}}_{\underline{x}})$-equivariant objects to
$V_{2}H_{2}({\mathcal{O}}_{\underline{x}})$-equivariant objects.

As in the proof of Lemma 6.7.8, we replace $\operatorname{Bun}_{M_{Q}}$ by a
fixed quasi-compact open substack and therefore assume
$Z_{i}/G_{i}({\mathcal{O}}_{\underline{x}})$ are quasi-compact algebraic
stacks. In particular, we have Verdier self-duality on them. Passing to
duality, we just need to show that the functor

$\displaystyle\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\simeq\operatorname{DMod}(Z_{2}/G_{2}({\mathcal{O}}_{\underline{x}}))\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(Z_{1}/G_{1}({\mathcal{O}}_{\underline{x}}))\simeq$
$\displaystyle\simeq\operatorname{DMod}(Z_{1})^{G_{1}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{1})^{V_{1}H_{1}({\mathcal{O}}_{\underline{x}})}$

sends the kernel of

$\operatorname{Av}_{*}:\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\to\operatorname{DMod}(Z_{2})^{V_{2}H_{2}({\mathcal{O}}_{\underline{x}})}$

to $0$. Let ${\mathcal{C}}$ be this kernel. We have a commutative square

$\textstyle{\operatorname{DMod}(Z_{1})^{G_{1}({\mathcal{O}}_{\underline{x}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Av}_{*}}$$\scriptstyle{\operatorname{oblv}}$$\textstyle{\operatorname{DMod}(Z_{1})^{V_{1}H_{1}({\mathcal{O}}_{\underline{x}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{oblv}}$$\textstyle{\operatorname{DMod}(Z_{1})^{K_{1}({\mathcal{O}}_{\underline{x}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Av}_{*}}$$\textstyle{\operatorname{DMod}(Z_{1})^{V_{1}},}$

whose vertical arrows are conservative. Hence we just need to check that

$\displaystyle\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\simeq\operatorname{DMod}(Z_{2}/G_{2}({\mathcal{O}}_{\underline{x}}))\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(Z_{1}/K_{1}({\mathcal{O}}_{\underline{x}}))\simeq$
$\displaystyle\simeq\operatorname{DMod}(Z_{1})^{K_{1}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{1})^{V_{1}}$

sends ${\mathcal{C}}$ to $0$. Since
$\operatorname{DMod}(Z_{1})^{K_{1}({\mathcal{O}}_{\underline{x}})}\subseteq\operatorname{DMod}(Z_{1})^{K_{0}({\mathcal{O}}_{\underline{x}})}$,
the above functor is equivalent to

$\displaystyle\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\simeq\operatorname{DMod}(Z_{2}/G_{2}({\mathcal{O}}_{\underline{x}}))\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(Z_{1}/K_{0}({\mathcal{O}}_{\underline{x}}))\simeq$
$\displaystyle\simeq\operatorname{DMod}(Z_{1})^{K_{0}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{1})^{V_{1}},$

which in turn is equivalent to

$\displaystyle\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{oblv}}\operatorname{DMod}(Z_{2})^{K_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(Z_{1})^{K_{0}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{1})^{V_{0}}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{1})^{V_{1}}.$

Thus, it suffices to prove that

$\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{oblv}}\operatorname{DMod}(Z_{2})^{K_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(Z_{1})^{K_{0}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{1})^{V_{0}}$

sends ${\mathcal{C}}$ to $0$. Note that the $!$-pullback functor
$\operatorname{DMod}(Z_{2})\to\operatorname{DMod}(Z_{1})$ is compatible with
the $K_{2}({\mathcal{O}}_{\underline{x}})\simeq
K_{0}({\mathcal{O}}_{\underline{x}})$-actions. By [Ras16, Corollary 2.17.10],
the above composition is equivalent to

$\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{oblv}}\operatorname{DMod}(Z_{2})^{K_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{2})^{V_{2}}\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(Z_{1})^{V_{0}}.$

Then we are done because the composition of the first two functors is
equivalent to

$\operatorname{DMod}(Z_{2})^{G_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{Av}_{*}}\operatorname{DMod}(Z_{2})^{V_{2}H_{2}({\mathcal{O}}_{\underline{x}})}\xrightarrow{\operatorname{oblv}}\operatorname{DMod}(Z_{2})^{V_{2}}.$

∎[Lemma 6.7.6]

∎[Proposition 6.7.1]

###  6.8. The equivalence $\operatorname{Id}\to\beta_{G}\circ\beta_{G}^{L}$:
finishing the proof

In this subsection, we use the cancellation lemma (Proposition 6.7.1) to
finish the proof.

We first give the following convenient description of
$\operatorname{Par}_{G}^{\overline{w}}$.

######  Lemma 6.8.1.

For $P\in\operatorname{Par}_{G}^{\mathrm{st}}\setminus\\{G\\}$, we have

$P\in\operatorname{Par}_{G}^{\overline{w}}\;\;\;\text{iff}\;\;\;B\cap
M_{P}\subseteq R^{w},$

where $B$ is the fixed Borel subgroup, $M_{P}$ is the Levi subgroup of $P$,
$w\in W_{G}$ is any representative of $\overline{w}$ and $R^{w}:=wRw^{-1}$.

###### Proof.

By definition of the Bruhat order, $P\in\operatorname{Par}_{G}^{\overline{w}}$
iff $BwR$ is closed in $PwR$ iff $BR^{w}$ is closed in $PR^{w}$ iff
$U_{P}\backslash BR^{w}$ is closed in $U_{P}\backslash PR^{w}$.

Note that $U_{P}\backslash PR^{w}\simeq
U_{P}\backslash(P\mathbin{\mathop{\times}\displaylimits}R^{w})/(P\cap
R^{w})\simeq(M_{P}\mathbin{\mathop{\times}\displaylimits}R^{w})/(P\cap R^{w})$
and $U_{P}\backslash BR^{w}\simeq(B\cap
M_{P}\mathbin{\mathop{\times}\displaylimits}R^{w})/(B\cap R^{w})$. Hence the
above condition is equivalent to the map

(34) $(B\cap M_{P}\mathbin{\mathop{\times}\displaylimits}R^{w})/(B\cap
R^{w})\to(M_{P}\mathbin{\mathop{\times}\displaylimits}R^{w})/(P\cap R^{w})$

being a closed embedding.

Note that
$M_{P}R^{w}=(M_{P}\mathbin{\mathop{\times}\displaylimits}R^{w})/(M_{P}\cap
R^{w})\to(M_{P}\mathbin{\mathop{\times}\displaylimits}R^{w})/(P\cap R^{w})$ is
a $U_{P}\cap R^{w}$-torsor. Hence (34) is a closed embedding iff so is its
base-change along this map. This base-change is $(B\cap M_{P})R^{w}\to
M_{P}R^{w}$. It is a closed embedding iff $(B\cap M_{P})R^{w}/R^{w}\to
M_{P}R^{w}/R^{w}$ is a closed embedding. This map can be rewritten as

(35) $B\cap M_{P}/(B\cap R^{w}\cap M_{P})\to M_{P}/(R^{w}\cap M_{P}).$

Note that $B\cap M_{P}$ is a Borel subgroup of $M_{P}$. We claim $R^{w}\cap
M_{P}$ is a parabolic subgroup of $M_{P}$. To see this, first note that
$R^{w}$ contains the Cartan subgroup $T$. We can choose a homomorphism
${\mathbb{G}}_{m}\to T$ such that the adjoint ${\mathbb{G}}_{m}$-action on $G$
has attractor equal to $R^{w}$. Then $R^{w}\cap M_{P}$ is the attractor of the
restricted ${\mathbb{G}}_{m}$-action on $M_{P}$. This proves the claim.

Using the above claim, we see (35) is a closed embedding iff $(B\cap
M_{P})(R^{w}\cap M_{P})$ is closed in $M_{P}$ iff $B\cap M_{P}$ is contained
in $R^{w}\cap M_{P}$ iff $B\cap M_{P}$ is contained in $R^{w}$.

∎[Lemma 6.8.1]

######  Corollary 6.8.2.

In the statement of Proposition 6.7.1, Condition (ii) follows from Condition
(i). In other words, for $P,Q\in\operatorname{Par}_{G}^{\overline{w}}$ such
that $P\cap R^{w}=Q\cap R^{w}$, we have $U_{Q}\cap M_{P}\subseteq U_{R}^{w}$.

###### Proof.

Note that $U_{Q}\cap M_{P}\subseteq B\cap M_{P}$. Hence by Lemma 6.8.1,
$U_{Q}\cap M_{P}\subseteq R^{w}$.

Since $U_{Q}\cap M_{P}$ and $U_{R}^{w}$ are connected, we only need to show
that $\operatorname{Lie}(U_{Q}\cap
M_{P})\subseteq\operatorname{Lie}(U_{R}^{w})$. Since $P,Q,R^{w}$ contain the
Cartan subgroup $T$, by the weight decomposition of $\operatorname{Lie}(G)$,
we just need to show that $\operatorname{Lie}(U_{Q}\cap
M_{P})\cap\operatorname{Lie}(L^{w})=0$. Applying the Cartan involution, this
amounts to $\operatorname{Lie}(U_{Q}^{-}\cap
M_{P})\cap\operatorname{Lie}(L^{w})=0$.

Note that $\operatorname{Lie}(U_{Q}^{-}\cap
M_{P})\subseteq\operatorname{Lie}(P)$ but $\operatorname{Lie}(U_{Q}^{-}\cap
M_{P})\cap\operatorname{Lie}(Q)=0$. Hence, by $P\cap R^{w}=Q\cap R^{w}$, we
have $\operatorname{Lie}(U_{Q}^{-}\cap M_{P})\cap\operatorname{Lie}(R^{w})=0$.
In particular, $\operatorname{Lie}(U_{Q}^{-}\cap
M_{P})\cap\operatorname{Lie}(L^{w})=0$ as desired.

∎[Corollary 6.8.2]

Then we give a purely combinatorial sufficient condition for the assumption of
the cancellation lemma. For convenience, we introduce the following
definition:

######  Definition 6.8.3.

For any positive simple root $\check{\alpha}$ of $G$ and standard parabolics
$Q\subseteq P\in\operatorname{Par}_{G}^{\mathrm{st}}$, we write
_$P=Q\langle\check{\alpha}\rangle$_ if the Dynkin diagram of $M_{P}$ is
obtained from that of $M_{Q}$ by adding the vertex corresponding to
$\check{\alpha}$.

######  Lemma 6.8.4.

If $\overline{w}\neq\overline{w}_{\circ}\in W_{G}/W_{R}$, then there exists a
negative simple root $-\check{\alpha}$ which is not a weight of
$\operatorname{Lie}(R^{w})$. Moreover, for $Q\subseteq
P\in\operatorname{Par}_{G}^{\mathrm{st}}\setminus\\{G\\}$ such that
$P=Q\langle\check{\alpha}\rangle$, we have

  * (1)

$P\in\operatorname{Par}_{G}^{\overline{w}}$ iff
$Q\in\operatorname{Par}_{G}^{\overline{w}}$;

  * (2)

If $P,Q\in\operatorname{Par}_{G}^{\overline{w}}$, then $P\cap R^{w}=Q\cap
R^{w}$.

###### Proof.

Suppose $\operatorname{Lie}(R^{w})$ has all negative simple roots as weights.
Then $N^{-}\subseteq R^{w}$ hence $R^{w}$ has transversal intersection with
$B^{-}$, which contradicts $\overline{w}\neq\overline{w}_{\circ}$.

Let $-\check{\alpha}$ be a negative simple root which is not a weight of
$\operatorname{Lie}(R^{w})$. Then $\check{\alpha}$ must be a weight of
$\operatorname{Lie}(R^{w})$.

For (1), by Lemma 6.8.1, we only need to show that $B\cap M_{P}\subseteq
R^{w}$ iff $B\cap M_{Q}\subseteq R^{w}$. The “only if” part is obvious. The
“if” part follows from the above paragraph because $\operatorname{Lie}(B\cap
M_{P})$ is generated by $\operatorname{Lie}(B\cap M_{Q})$ and the weight space
of $\check{\alpha}$.

For (2), suppose $P\cap R^{w}\neq Q\cap R^{w}$. Then there exists a negative
root $-\check{\lambda}$ which is a weight of $\operatorname{Lie}(P\cap R^{w})$
but not of $\operatorname{Lie}(Q\cap R^{w})$. Since
$P=Q\langle\check{\alpha}\rangle$, we must have
$-\check{\lambda}=-n\check{\alpha}-\check{\mu}$, where $n>0$ and $\check{\mu}$
is a non-negative linear sum of positive simple roots. We can assume there is
no root strictly smaller than $\check{\lambda}$ satisfying the same
assumption.

Note that $\check{\mu}\neq 0$, otherwise $-\check{\alpha}$ would be a weight
of $\operatorname{Lie}(R^{w})$ which contradicts our assumption. It follows
that $\check{\lambda}$ is not simple: let us write
$\check{\lambda}=\check{\lambda}_{1}+\check{\lambda}_{2}$ for two strictly
positive roots $\check{\lambda}_{1},\check{\lambda}_{2}$. Since
$-\check{\lambda}$ is a weight of $\operatorname{Lie}(P)$, so are
$\check{\lambda}_{1}$ and $-\check{\lambda}_{1}$. In other words,
$\pm\check{\lambda}_{1}$ are both weights of $\operatorname{Lie}(M_{P})$.
Hence, by the assumption $P\in\operatorname{Par}_{G}^{\overline{w}}$ and Lemma
6.8.1, $\check{\lambda}_{1}$ is a weight of $\operatorname{Lie}(R^{w})$.
Consequently, $-\check{\lambda}_{2}=(-\check{\lambda})+\check{\lambda}_{1}$ is
also a weight of $\operatorname{Lie}(P\cap R^{w})$. By the minimality
assumption on $\check{\lambda}$, we see that $-\check{\lambda}_{2}$ must be a
weight of $\operatorname{Lie}(Q\cap R^{w})$. Write $\check{\lambda}_{2}$ as a
sum of positive simple roots. Since $P=Q\langle\check{\alpha}\rangle$, this
sum has no $\check{\alpha}$ term. By symmetry, the sum for
$\check{\lambda}_{1}$ has no $\check{\alpha}$ term either. But
$\check{\lambda}=\check{\lambda}_{1}+\check{\lambda}_{2}=n\check{\alpha}+\check{\mu}$
contains $\check{\alpha}$ terms. A contradiction!

∎[Lemma 6.8.4]

Now we are ready to finish the proof of Theorem 5.1.4.

###### Proof of Theorem 5.1.4..

Recall our Goal 6.3.1: we need to show that

(36)
$\iota_{L}^{!}({\mathcal{M}}_{R})\to\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}({\mathcal{M}}_{P})$

is invertible, where
$\alpha_{P}=\operatorname{CT}_{R,*}\circ\operatorname{Eis}_{P\to
G}^{\mathrm{enh}}$,
${\mathcal{M}}_{P}\in{\mathcal{I}}(G,P)^{G\mathrm{-atemp}}$ and
$\operatorname{CT}^{\mathrm{enh}}_{P\leftarrow
Q}({\mathcal{M}}_{Q})\simeq{\mathcal{M}}_{P}$ for $Q\subseteq P$.

Using the dual Weyl filtration on $\alpha_{P}$ (Construction 6.6.4), we obtain
a filtration on the RHS by $W_{G}/W_{R}$, whose $\leq\overline{w}$-term is

$\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime\leq\overline{w}}({\mathcal{M}}_{P}).$

To conclude, it suffices to prove the following two lemmas.

######  Lemma 6.8.5.

We have:

  * (1)

The morphism (36) factors through an equivalence

$\iota_{L}^{!}({\mathcal{M}}_{R})\to\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime=\overline{1}}({\mathcal{M}}_{P});$

  * (2)

For $\overline{w}\in W_{G}/W_{R}$, if
$\overline{w}\neq\overline{1},\overline{w}_{\circ}$, then

$\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime=\overline{w}}({\mathcal{M}}_{P})\simeq 0.$

######  Lemma 6.8.6.

For $\overline{w}=\overline{w}_{\circ}$,

$\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime=\overline{w}_{\circ}}({\mathcal{M}}_{P})\simeq 0.$

We write the above results as two separate lemmas because they are proved
using different ingredients. We will prove the former using the cancellation
lemma, while prove the latter using the Deligne–Lusztig duality and Theorem
3.0.2.

###### Proof of Lemma 6.8.5.

In this proof, we do not need the assumption that ${\mathcal{M}}_{P}$ is
$G$-anti-tempered.

We assume $\overline{w}\neq\overline{w}_{\circ}$. Let $-\check{\alpha}$ be as
in Lemma 6.8.4, i.e., a negative simple root which is not a weight of
$\operatorname{Lie}(R^{w})$. Let
$\operatorname{Par}^{-\check{\alpha}\notin}_{G}\subseteq\operatorname{Par}_{G}^{\mathrm{st}}\setminus\\{G\\}$
be the sub-poset containing those $Q$ for which $-\check{\alpha}$ is not a
weight of $\operatorname{Lie}(Q)$. Then there is a retraction

$\operatorname{Par}_{G}^{\mathrm{st}}\setminus\\{G\\}\to\operatorname{Par}^{-\check{\alpha}\notin}_{G}$

sending both $Q\langle\alpha\rangle$ and $Q$ to $Q$.

We claim that the functor

$\operatorname{Par}_{G}^{\mathrm{st}}\setminus\\{G\\}\to\operatorname{DMod}(\operatorname{Bun}_{L}),\;P\mapsto\alpha_{P}^{\prime=\overline{w}}({\mathcal{M}}_{P})$

factors through the above retraction. As we now explain, this follows easily
from the cancellation lemma. We just need to show that the arrow $Q\to
Q\langle\alpha\rangle=:P$ induces an isomorphism

$\alpha_{Q}^{\prime=\overline{w}}({\mathcal{M}}_{Q})\to\alpha_{P}^{\prime=\overline{w}}({\mathcal{M}}_{P}).$

Suppose that $Q\notin\operatorname{Par}_{G}^{\underline{w}}$: then
$P\notin\operatorname{Par}_{G}^{\underline{w}}$ by Lemma 6.8.4(1), which
implies that the source and the target are both $0$ (see Construction 6.6.2).
Now suppose that $Q\in\operatorname{Par}_{G}^{=\underline{w}}$: then Lemma
6.8.4(1) implies that $P\in\operatorname{Par}_{G}^{=\underline{w}}$, too. By
Corollary 6.8.2 and Lemma 6.8.4(2), the assumptions of the cancellation lemma
(Proposition 6.7.1) are satisfied. Hence the desired morphism is an
isomorphism.

By the claim, we have

$\mathbin{\mathop{\operatorname{colim}}\displaylimits_{Q\in\operatorname{Par}^{-\check{\alpha}\notin}_{G}}\,}\alpha_{Q}^{\prime=\overline{w}}({\mathcal{M}}_{Q})\simeq\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime=\overline{w}}({\mathcal{M}}_{P}).$

Note that $\operatorname{Par}^{-\check{\alpha}\notin}_{G}$ has a final object:
the maximal standard parabolic subgroup $S$ such that the Dynkin diagram of
$M_{S}$ is obtained from that of $G$ by removing the vertex corresponding to
$\check{\alpha}$. Consequently,

$\alpha_{S}^{\prime=\overline{w}}({\mathcal{M}}_{S})\simeq\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime=\overline{w}}({\mathcal{M}}_{P}).$

When $\underline{w}\neq\underline{1}$, we claim
$S\notin\operatorname{Par}_{G}^{=\overline{w}}$. Of course, this implies (2)
because in that case $\alpha_{S}^{\prime=\overline{w}}\simeq 0$ (see
Construction 6.6.2). To prove the claim, by Lemma 6.8.1, we only need to show
that $B\cap M_{S}$ is not contained in $R^{w}$. Suppose the contrary, then
$\operatorname{Lie}(R^{w})$ contains all the positive simple roots of $M_{S}$.
On the other hand, by definition, $\operatorname{Lie}(R^{w})$ contains
$\check{\alpha}$. Hence $R^{w}$ is a standard parabolic subgroup. Then $B\cap
M_{S}\subseteq R^{w}$ implies $R^{w}=S$ because both $R^{w}$ and $S$ are
maximal proper standard parabolic subgroups, and neither of them contains
$-\check{\alpha}$ as a weight. But this forces $\underline{w}=\underline{1}$,
because $S$ is standard. This finishes the proof of (2).

When $\underline{w}=\underline{1}$, we have $S=R$. Hence to prove (1), we need
to show the natural transformation $\iota_{L}^{!}\to\alpha_{R}$ factors
through an equivalence $\iota_{L}^{!}\simeq\alpha_{R}^{\prime=\overline{1}}$.

Recall (see Construction 6.6.2) that $(\alpha_{R}^{\vee})^{=\overline{1}}$ is
equivalent to

$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{R^{-}})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{R^{-}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{R^{-}{\mathrm{-gen}}})\xrightarrow{\operatorname{Av}_{*}}{\mathcal{I}}(G,R^{-}),$

which by definition is equivalent to
$\iota_{L,!}^{-}:\operatorname{DMod}(\operatorname{Bun}_{L})\to{\mathcal{I}}(G,R^{-})$.
Passing to duality, we see
$\alpha_{R}^{\prime=\overline{1}}\simeq\iota_{L}^{!}:{\mathcal{I}}(G,R)\to\operatorname{DMod}(\operatorname{Bun}_{L})$
(see Corollary 6.5.2). It follows from the construction that
$\iota_{L}^{!}\to\alpha_{R}$ indeed factors through this equivalence. This
finishes the proof of (1).

∎[Lemma 6.8.5]

It remains to prove Lemma 6.8.6. We first need some preparations. Let
$S:=(R^{-})^{w_{\circ}}:=w_{\circ}R^{-}w_{\circ}^{-1}$. Note that $S$ is a
maximal standard (proper) parabolic subgroup of $G$. Let $-\check{\beta}$ be
the unique negative simple root which is not a weight of
$\operatorname{Lie}(S)$.

Recall
$\operatorname{Par}_{S}^{\mathrm{st}}\subseteq\operatorname{Par}_{G}^{\mathrm{st}}$
is the full subcategory of standard parabolic subgroups $P$ contained in $S$.
We claim

$\operatorname{Par}_{S}^{\mathrm{st}}=\operatorname{Par}_{G}^{=\overline{w}_{\circ}}.$

By Lemma 6.8.1, we only need to show that $B\cap M_{P}\subseteq S^{-}$ iff
$P\subseteq S$, but this is obvious.

Let $[1]$ be the 1-simplex viewed as a category, i.e., it contains two objects
$0$ and $1$, and a unique non-invertible arrow $0\to 1$. Then we have an
equivalence

(37)
$(\operatorname{Par}_{S}^{\mathrm{st}}\setminus\\{S\\}\mathbin{\mathop{\times}\displaylimits}[1])\bigsqcup_{\operatorname{Par}_{S}^{\mathrm{st}}\setminus\\{S\\}\mathbin{\mathop{\times}\displaylimits}\\{0\\}}\operatorname{Par}_{S}^{\mathrm{st}}\to\operatorname{Par}_{G}^{\mathrm{st}}\setminus\\{G\\}$

which sends

  * •

$(Q,0)\in(\operatorname{Par}_{S}^{\mathrm{st}}\setminus\\{S\\}\mathbin{\mathop{\times}\displaylimits}[1])$
to $Q$;

  * •

$(Q,1)\in(\operatorname{Par}_{S}^{\mathrm{st}}\setminus\\{S\\}\mathbin{\mathop{\times}\displaylimits}[1])$
to $Q\langle\check{\beta}\rangle$;

  * •

$P\in\operatorname{Par}_{S}^{\mathrm{st}}$ to $P$.

By Construction 6.6.2, if
$P\notin\operatorname{Par}_{G}^{=\overline{w}_{\circ}}=\operatorname{Par}_{S}^{\mathrm{st}}$,
then $\alpha_{P}^{\prime=\overline{w}_{\circ}}\simeq 0$. Hence the equivalence
(37) implies

$\mathbin{\mathop{\operatorname{colim}}\displaylimits_{P\in\operatorname{Par}_{G}^{\mathrm{st}},P\neq
G}\,}\alpha_{P}^{\prime=\overline{w}_{\circ}}({\mathcal{M}}_{P})\simeq\operatorname{coFib}(\mathbin{\mathop{\operatorname{colim}}\displaylimits_{Q\in\operatorname{Par}_{S}^{\mathrm{st}},Q\neq
S}\,}\alpha_{Q}^{\prime=\overline{w}_{\circ}}({\mathcal{M}}_{Q})\to\alpha_{S}^{\prime=\overline{w}_{\circ}}({\mathcal{M}}_{S}))$

By assumption, we have
${\mathcal{M}}_{Q}\simeq\operatorname{CT}^{\mathrm{enh}}_{S\leftarrow
Q}({\mathcal{M}}_{S})$. Hence it remains to prove that

(38)
$\operatorname{coFib}(\mathbin{\mathop{\operatorname{colim}}\displaylimits_{Q\in\operatorname{Par}_{S}^{\mathrm{st}},Q\neq
S}\,}\alpha_{Q}^{\prime=\overline{w}_{\circ}}\circ\operatorname{CT}^{\mathrm{enh}}_{S\leftarrow
Q}\to\alpha_{S}^{\prime=\overline{w}_{\circ}})$

sends ${\mathcal{I}}(G,S)^{G\mathrm{-atemp}}$ to $0$.

By Construction 6.6.4, the functor (38) is the dual functor of

(39)
$\operatorname{coFib}(\mathbin{\mathop{\operatorname{colim}}\displaylimits_{Q\in\operatorname{Par}_{S}^{\mathrm{st}},Q\neq
S}\,}\operatorname{Eis}^{\mathrm{enh}}_{Q^{-}\to
S^{-}}\circ(\alpha_{Q}^{\vee})^{=\overline{w}_{\circ}}\to(\alpha_{S}^{\vee})^{=\overline{w}_{\circ}}),$

where each $(\alpha_{Q}^{\vee})^{=\overline{w}_{\circ}}$ is equivalent to (see
Construction 6.6.2)

(40)
$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L})\simeq\operatorname{DMod}(\operatorname{Bun}_{M_{S}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{S})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S}^{Q^{-}\cap
S{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}})\xrightarrow{\operatorname{Av}_{*}}{\mathcal{I}}(G,Q^{-}).$

We claim the following diagram is Cartesian:

${\widetilde{\operatorname{Bun}}_{S}^{Q^{-}\cap
S{\mathrm{-gen}}}}$${\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}}}$${\widetilde{\operatorname{Bun}}_{S}^{S^{-}\cap
S{\mathrm{-gen}}}}$${\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}}.}$${\lrcorner}$

The claim follows from

${\mathbb{B}}(Q^{-}\cap S)\simeq{\mathbb{B}}(S^{-}\cap
S)\mathbin{\mathop{\times}\displaylimits_{{\mathbb{B}}S^{-}}}{\mathbb{B}}Q^{-},$

which in turn is a consequence of $(S^{-}\cap S)Q^{-}=S^{-}$.

Applying the base-change isomorphism to this diagram, we see that
$(\alpha_{Q}^{\vee})^{=\overline{w}_{\circ}}$ is equivalent to

$\displaystyle\operatorname{DMod}(\operatorname{Bun}_{L})\simeq\operatorname{DMod}(\operatorname{Bun}_{M_{S}})\xrightarrow{*\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{S})\xrightarrow{!\mathrm{-push}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S})\to$
$\displaystyle\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S}^{M_{S}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}})\xrightarrow{\operatorname{Av}_{*}}{\mathcal{I}}(G,Q^{-}).$

By Definition 2.3.16, we have a commutative diagram

$\textstyle{\operatorname{DMod}(\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{!\mathrm{-pull}}$$\scriptstyle{\operatorname{Av}_{*}}$$\textstyle{\operatorname{DMod}(\operatorname{Bun}_{G}^{Q^{-}{\mathrm{-gen}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Av}_{*}}$$\textstyle{{\mathcal{I}}(G,S^{-})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{CT}^{\mathrm{enh}}_{S^{-}\leftarrow
Q^{-}}}$$\textstyle{{\mathcal{I}}(G,Q^{-}).}$

We obtain

$(\alpha_{Q}^{\vee})^{=\overline{w}_{\circ}}\simeq\operatorname{CT}^{\mathrm{enh}}_{S^{-}\leftarrow
Q^{-}}\circ(\alpha_{S}^{\vee})^{=\overline{w}_{\circ}}$

and can thus rewrite (39) as

$\operatorname{coFib}(\mathbin{\mathop{\operatorname{colim}}\displaylimits_{Q\in\operatorname{Par}_{S}^{\mathrm{st}},Q\neq
S}\,}\operatorname{Eis}^{\mathrm{enh}}_{Q^{-}\to
S^{-}}\circ\operatorname{CT}^{\mathrm{enh}}_{S^{-}\leftarrow
Q^{-}}\to\operatorname{Id})\circ(\alpha_{S}^{\vee})^{=\overline{w}_{\circ}}.$

Now consider the endo-functor of ${\mathcal{I}}(G,S^{-})$ defined by

(41)
$\mathrm{DL}_{S^{-}}:=\operatorname{coFib}(\mathbin{\mathop{\operatorname{colim}}\displaylimits_{Q\in\operatorname{Par}_{S}^{\mathrm{st}},Q\neq
S}\,}\operatorname{Eis}^{\mathrm{enh}}_{Q^{-}\to
S^{-}}\circ\operatorname{CT}^{\mathrm{enh}}_{S^{-}\leftarrow
Q^{-}}\to\operatorname{Id}).$

We have reduced Lemma 6.8.6 (and therefore the entire proof) to the following
goal.

######  Goal 6.8.7.

For any ${\mathcal{F}}\in\operatorname{DMod}(\operatorname{Bun}_{L})$ and
${\mathcal{M}}\in{\mathcal{I}}(G,S)^{G\mathrm{-atemp}}$, we have

$\langle\mathrm{DL}_{S^{-}}\circ(\alpha_{S}^{\vee})^{=\overline{w}_{\circ}}({\mathcal{F}}),{\mathcal{M}}\rangle\simeq
0,$

where $\langle-,-\rangle$ is the pairing functor between
${\mathcal{I}}(G,S^{-})$ and ${\mathcal{I}}(G,S)$.

We first deduce this from the following lemmas.

######  Lemma 6.8.8.

Let
${\mathcal{I}}(G,S^{-})^{\iota_{*}{\mathrm{-gen}}}\subseteq{\mathcal{I}}(G,S^{-})$
be the full subcategory generated under colimits by the image of
$\iota_{M_{S},*}^{-}:\operatorname{DMod}(\operatorname{Bun}_{M_{S}})\to{\mathcal{I}}(G,S^{-})$
(see Definition 2.3.13). Then the image of
$(\alpha_{S}^{\vee})^{=\overline{w}_{\circ}}$ is contained in
${\mathcal{I}}(G,S^{-})^{\iota_{*}{\mathrm{-gen}}}$.

######  Lemma 6.8.9.

The image of the functor

${\mathcal{I}}(G,S)\xrightarrow{\mathrm{DL}_{S}}{\mathcal{I}}(G,S)\xrightarrow{\iota_{M_{S}}^{*}}\operatorname{DMod}(\operatorname{Bun}_{M_{S}})$

is contained in
$\operatorname{DMod}(\operatorname{Bun}_{M_{S}})^{*{\mathrm{-gen}}}$.

###### Proof of Lemma 6.8.6.

By Goal 6.8.7 and Lemma 6.8.8, it suffices to prove that

$\langle\mathrm{DL}_{S^{-}}\circ\iota_{M_{S},*}^{-}({\mathcal{K}}),{\mathcal{M}}\rangle\simeq
0,$

for any ${\mathcal{K}}\in\operatorname{DMod}(\operatorname{Bun}_{M_{S}})$ and
${\mathcal{M}}\in{\mathcal{I}}(G,S)^{G\mathrm{-atemp}}$. By Corollary 6.5.2
and Remark 6.5.3, the functor $\mathrm{DL}_{S^{-}}\circ\iota_{M_{S},*}^{-}$ is
dual to $\iota_{M_{S}}^{*}\circ\mathrm{DL}_{S}$. Hence we need to prove that

$\iota_{M_{S}}^{*}\circ\mathrm{DL}_{S}({\mathcal{M}})\simeq 0.$

By Lemma 6.8.9,

$\iota_{M_{S}}^{*}\circ\mathrm{DL}_{S}({\mathcal{M}})\in\operatorname{DMod}(\operatorname{Bun}_{M_{S}})^{*{\mathrm{-gen}}}$

and therefore is $M_{S}$-tempered ([Ber21b, Theorem B]).

On the other hand, the functor $\mathrm{DL}_{S}$ is
$\operatorname{Sph}_{G,x}$-linear, hence $\mathrm{DL}_{S}({\mathcal{M}})$ is
$G$-anti-tempered because so is ${\mathcal{M}}$. Theorem 3.0.2(4) implies that
$\iota_{M_{S}}^{*}\circ\mathrm{DL}_{S}({\mathcal{M}})$ is also $M_{S}$-anti-
tempered.

Combining the above two paragraphs, we see
$\iota_{M_{S}}^{*}\circ\mathrm{DL}_{S}({\mathcal{M}})\simeq 0$ as desired.

∎[Lemma 6.8.6]

It remains to prove Lemma 6.8.8 and Lemma 6.8.9.

###### Proof of Lemma 6.8.8.

By (40), the functor $(\alpha_{S}^{\vee})^{=\overline{w}_{\circ}}$ factors
through

$\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S}^{M_{S}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}})\xrightarrow{\operatorname{Av}_{*}}{\mathcal{I}}(G,S^{-}).$

We claim that the image of above functor already lands in
${\mathcal{I}}(G,S^{-})^{\iota_{*}{\mathrm{-gen}}}$. Let
$\lambda\in\Lambda_{G,S}$ and consider the corresponding connected component
$\widetilde{\operatorname{Bun}}_{S,\lambda}^{M_{S}{\mathrm{-gen}}}$. By
Corollary 2.3.14, we only need to show that

$\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S,\lambda}^{M_{S}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}})\xrightarrow{\operatorname{Av}_{*}}{\mathcal{I}}(G,S^{-})\xrightarrow{(\iota_{M_{S}}^{-,\mu})^{!}}\operatorname{DMod}(\operatorname{Bun}_{M_{S},\mu})$

is $0$ unless $\mu\geq\lambda$. Here we flip the sign of the inequality in the
statement of Corollary 2.3.14 because we apply it to $S^{-}$.

By definition, the above composition is given by

$\operatorname{DMod}(\widetilde{\operatorname{Bun}}_{S,\lambda}^{M_{S}{\mathrm{-gen}}})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}})\xrightarrow{!\mathrm{-pull}}\operatorname{DMod}(\operatorname{Bun}_{S^{-},\mu})\xrightarrow{*\mathrm{-push}}\operatorname{DMod}(\operatorname{Bun}_{M_{S},\mu}).$

By the base-change isomorphism, it suffices to check that

$\widetilde{\operatorname{Bun}}_{S,\lambda}^{M_{S}{\mathrm{-gen}}}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}^{S^{-}{\mathrm{-gen}}}}}\operatorname{Bun}_{S^{-},\mu}=\emptyset$

unless $\mu\geq\lambda$. Then we are done: this fiber product is just

$(\widetilde{\operatorname{Bun}}_{S,\lambda}\mathbin{\mathop{\times}\displaylimits_{\operatorname{Bun}_{G}}}\operatorname{Bun}_{S^{-},\mu})^{M_{S}{\mathrm{-gen}}},$

i.e. the parabolic Zastava space defined in [BFGM02, §2], and the desired
inequality was proved there.

∎[Lemma 6.8.8]

###### Proof of Lemma 6.8.9.

We will show that the image of

(42)
$\operatorname{DMod}(\operatorname{Bun}_{M_{S}})\xrightarrow{\iota_{M_{S},!}}{\mathcal{I}}(G,S)\xrightarrow{\mathrm{DL}_{S}}{\mathcal{I}}(G,S)\xrightarrow{\iota_{M_{S}}^{*}}\operatorname{DMod}(\operatorname{Bun}_{M_{S}})$

is contained in
$\operatorname{DMod}(\operatorname{Bun}_{M_{S}})^{*{\mathrm{-gen}}}$.

Consider the functor
$\mathrm{DL}_{M_{S}}:\operatorname{DMod}(\operatorname{Bun}_{M_{S}})\to\operatorname{DMod}(\operatorname{Bun}_{M_{S}})$
in Theorem 6.1.4 (applied to the reductive group $M_{S}$). Using notations in
§4.1, it is a certain colimit of functors

${\mathcal{I}}(S,S)\xrightarrow{\operatorname{CT}}{\mathcal{I}}(S,Q)\xrightarrow{\operatorname{Eis}}{\mathcal{I}}(S,S).$

On the other hand, by definition (see (41), the functor $\mathrm{DL}_{S}$ is a
similar colimit of functors

${\mathcal{I}}(G,S)\xrightarrow{\operatorname{CT}}{\mathcal{I}}(G,Q)\xrightarrow{\operatorname{Eis}}{\mathcal{I}}(G,S).$

By the base-change isomorphism for ${\mathcal{I}}(-,-)$ (Lemma 4.5.1), it is
easy to see that

$\iota_{M_{S}}^{!}\circ\mathrm{DL}_{S}\simeq\mathrm{DL}_{M_{S}}\circ\iota_{M_{S}}^{!}.$

Passing to dualities (Corollary 6.5.2 and Remark 6.5.3), we obtain that

$\mathrm{DL}_{S}\circ\iota_{M_{S},!}\simeq\iota_{M_{S},!}\circ\mathrm{DL}_{M_{S}}.$

Hence the functor (42) is equivalent to $\mathrm{DL}_{M_{S}}$. By Theorem
6.1.4, it suffices to show that the image of the functor $\operatorname{Ps-
Id}_{M_{S},*}$ is contained in
$\operatorname{DMod}(\operatorname{Bun}_{M_{S}})^{*{\mathrm{-gen}}}$. But this
is known, see [DG13, §4.4.3].

∎[Lemma 6.8.9]

∎[Theorem 5.1.4]

##  Appendix A Lax $\operatorname{Sph}_{G}$-linear functors are strict

In this appendix, we prove the following result:

######  Proposition A.1.1.

Any _continuous_ (left or right)-lax $\operatorname{Sph}_{G}$-linear is
strictly $\operatorname{Sph}_{G}$-linear.

######  Definition A.1.2.

Following [GR19, Chapter 1, Definition 9.1.2], a stable presentable monoidal
category ${\mathcal{A}}$ is _rigid_ if the following conditions hold:

  * (i)

The right adjoint
$m^{R}:{\mathcal{A}}\to{\mathcal{A}}\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{A}}$
of the multiplication functor $m$ is continuous;

  * (ii)

The functor $m^{R}$, which a priori is right-lax
$({\mathcal{A}},{\mathcal{A}}^{\mathrm{rev}})$-linear, is strictly linear;

  * (iii)

The unit object $1_{\mathcal{A}}$ is compact.

######  Proposition A.1.3 ([GR19, Chaper 1, Lemma 9.3.6]).

For a presentable monoidal category ${\mathcal{A}}$ satisfying Condition (i)
and (ii) in Definition A.1.2, any continuous lax ${\mathcal{A}}$-linear
functor is strictly linear151515Warning: the proof in _loc.cit._ contains
_lots_ of typos..

∎[Proposition A.1.3]

######  Remark A.1.4.

In fact, [GR19] stated the above results for rigid monoidal categories.
However, the proofs there did not use Condition (iii), nor the stable
assumption on ${\mathcal{A}}$.

######  Remark A.1.5.

$\operatorname{Sph}_{G}$ is _not_ rigid because its unit object is not
compact. Indeed, the unit object is the pushforward of $\omega$ along
${\mathbb{B}}G({\mathcal{O}})\to G({\mathcal{O}})\backslash
G({\mathcal{K}})/G({\mathcal{O}})$ and
$\omega\in\operatorname{DMod}({\mathbb{B}}G({\mathcal{O}}))\simeq\operatorname{DMod}({\mathbb{B}}G)$
is not compact (see [DG13, §7.1.4]).

Nevertheless, we have:

######  Lemma A.1.6.

The monoidal category $\operatorname{Sph}_{G}$ satisfies Condition (i) and
(ii) in Definition A.1.2.

###### Proof.

Recall the multiplication functor $m$ is given by $!$-pull-$*$-push along the
following correspondence

$G({\mathcal{O}})\backslash
G({\mathcal{K}})/G({\mathcal{O}})\xleftarrow{u}G({\mathcal{O}})\backslash
G({\mathcal{K}})\mathbin{\mathop{\times}\displaylimits^{G({\mathcal{O}})}}G({\mathcal{K}})/G({\mathcal{O}})\xrightarrow{v}G({\mathcal{O}})\backslash
G({\mathcal{K}})/G({\mathcal{O}})\mathbin{\mathop{\times}\displaylimits}G({\mathcal{O}})\backslash
G({\mathcal{K}})/G({\mathcal{O}}),$

where $u$ is ind-proper and $v$ is pro-smooth and in particular placid (see
[Ras15, §4.10]). It follows that $m\simeq u_{*}\circ v^{!}\simeq u_{!}\circ
v^{!}$ has a continuous right adjoint $m^{R}\simeq v_{*,\mathrm{ren}}\circ
u^{!}$, where $v_{*,\mathrm{ren}}$ is the renormalized pushforward functor
defined in _loc.cit._. Hence we verified Condition (i). Also, Condition (ii)
follows from the base-change isomorphisms between such renormalized functors
and usual $!$-pullback functors (see [Ras15, Proposition. 4.11.1]).

∎[Lemma A.1.6]

_Proof of Proposition A.1.1._ Now Proposition A.1.1 follows from Lemma A.1.6
and Proposition A.1.3.

∎[Proposition A.1.1]

We also need the following technical lemma.

######  Lemma A.1.7 ([GR19, Chapter 1, Lemma 9.3.2].).

For a presentable monoidal category ${\mathcal{A}}$ satisfying Condition (i)
and (ii) in Definition A.1.2 and any left ${\mathcal{A}}$-module
${\mathcal{M}}$, the action functor
$a:{\mathcal{A}}\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{M}}\to{\mathcal{M}}$
has a continuous right adjoint $a^{R}$, which is canonically isomorphic to

${\mathcal{M}}\xrightarrow{1_{\mathcal{A}}}{\mathcal{A}}\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{M}}\xrightarrow{m^{R}\mathbin{\mathop{\otimes}\displaylimits}\operatorname{Id}}{\mathcal{A}}\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{A}}\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{M}}\xrightarrow{\operatorname{Id}\mathbin{\mathop{\otimes}\displaylimits}a}{\mathcal{A}}\mathbin{\mathop{\otimes}\displaylimits}{\mathcal{M}}.$

∎[Lemma A.1.7]

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