\mathcal{KL}(H)_{\mathscr{L}} & :=  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod_{\mathscr{L}}^{\mathcal{L}^+H}  &:=   \mathscr{L}_{\widehat{\on{Hecke}}} \on{-mod}( \mathcal{R}ep( \mathcal{L}^+ H )) \\
\mathcal{KL}(H)_{\mathscr{L}^{-1},\on{co}} & := \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod_{\mathscr{L}^{-1},\on{co}}^{\mathcal{L}^+H}&:=    \mathscr{L}_{\widehat{\on{Hecke}}}\on{-mod}^r( \mathcal{R}ep( \mathcal{L}^+ H )^\vee),
\end{eqnarray*}

where $\mathcal{R}ep( \mathcal{L}^+ H )$ (resp, $\mathcal{R}ep( \mathcal{L}^+ H )^\vee$) is equipped with the left (resp. right) $ [\mathcal{I}ndCoh^{!,\ren}( \widehat{\on{Hecke}}_H  ),\star]$-module structure described in Remark <ref> and Remark <ref>.

Several explanations are in order.

$\mathcal{KL}(H)_{\mathscr{L}} $ can also be defined as a geometric realization or totalization as follows. There is a functor

$$\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L},\Delta}: \Delta \to [\on{FactCat}_{\on{un}}]_{\on{lax-unital}} $$

that sends $[m]$ to $\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m )$ but sends a morphism $f:[m]\to [n]$ in $\Delta$ to the functor

$$  (p_f)^{!,\ren}(-)\overset{!}\otimes (p_{ f^{\gets}})^{!,\ren} ( \mathscr{L}_{ \widehat{\on{Hecke}} } ):  \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \to \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^n ), $$

where $f^{\gets}: [1]\to [m]$ is the map $f^{\gets}(0)=0, f^{\gets}(1)=f(0)$. The higher compatibilities are provided by the multiplicative structure[For instance, the composition laws are provided by

$$ p_{ (g\circ f)^\gets }^{!,\ren}(  \mathscr{L}_{ \widehat{\on{Hecke}} } ) \simeq p_{ g^\gets }^{!,\ren}(  \mathscr{L}_{ \widehat{\on{Hecke}} }) \overset{!}\otimes   p_{ (g\circ  f^\gets) }^{!,\ren}(  \mathscr{L}_{ \widehat{\on{Hecke}} } ) . $$
] of $\mathscr{L}$.
By passing to left adjoints, we also have a functor

$$\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L},\Delta^\op} : \Delta^\op \to \on{FactCat}_{\on{un}} $$

whose connecting functors are

$$   (p_f)^{\ren}_! [-  \overset{!}\otimes (p_{ f^{\gets}})^{!,\ren} ( \mathscr{L}_{ \widehat{\on{Hecke}} }^{-1}) ]:  \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^n ) \to \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ). $$

As in Remark <ref>, the totalization of the sections of $\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L}}$ satisfies the Bech-Chevalley condition, and the corresponding monad can be identified with the action of $ \mathscr{L}_{ \widehat{\on{Hecke}} }$ on $\mathcal{R}ep( \mathcal{L}^+ H )$. Therefore

$$ \mathcal{KL}(H)_{\mathscr{L}}  \simeq \lim \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L},\Delta} \simeq \colim   \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L},\Delta^\op}.$$

In particular, the sections of $\mathcal{KL}(H)_{\mathscr{L}}$ are compactly generated.

Passing to duality, the simplicial diagrams in Remark <ref> provides

$$\mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{ \mathscr{L}^{-1},\Delta }: \Delta \to [\on{FactCat}_{\on{un}}]_{\on{lax-unital}} $$

whose connecting functors are

$$ (p_{ f^{\gets}})^{!,\ren} (  \mathscr{L}_{ \widehat{\on{Hecke}} }^{-1} )  \overset{\on{act}}\otimes (p_f)_{\ren}^? ( - )   :  \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^m ) \to \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^n ),$$

where $\overset{\on{act}}\otimes$ means the $[\mathcal{I}nd\mathcal{C}oh^{!,\ren}(-), \overset{!}\otimes]$-action on $\mathcal{I}nd\mathcal{C}oh_{*,\ren}(-)$. By passing to left adjoints, we also have a functor

$$\mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L}^{-1},\Delta^\op}: \Delta^\op \to \on{FactCat}_{\on{un}} $$

whose connecting functors are

$$(p_f)_{*,\ren} [(p_{ f^{\gets}})^{!,\ren} (  \mathscr{L}_{ \widehat{\on{Hecke}} } )  \overset{\on{act}}\otimes - ]   :  \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^n ) \to \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^m ).$$

As before, we have

$$\mathcal{KL}(H)_{\mathscr{L}^{-1},\on{co}}  \simeq \lim \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L}^{-1},\Delta} \simeq \colim   \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L}^{-1},\Delta^\op}.$$

Similar to the non-twisted version, $\mathcal{KL}(H)_{\mathscr{L}}$ and $\mathcal{KL}(H)_{ \mathscr{L}^{-1},\on{co} }$ are dual to each other in the symmetric monoidal category $[\on{FactCat}_{\on{un}}]_{\on{lax-unital}}$.

One should think $\mathscr{L}$ as a factorization gerbe on $[\mathbb{B}\widehat{\mathcal{L}H}]_{\Ran_{\on{un,dR}}}$ equipped with a trivialization on $[\mathbb{B}\mathcal{L}^+H]_{\Ran_{\on{un,dR}}}$, and $\mathcal{KL}(H)_\mathscr{L}$ as the unital factorization category for the $\IndCoh^{!,\ren}$-theory on $[\mathbb{B}\widehat{\mathcal{L}H}]_{\Ran_{\on{un,dR}}}$ twisted by this gerbe.

Let $x\in X$ be a closed point. The group homomorphism $\widehat{\mathcal{L} H}_x \to \mathbb{BG}_m$ in Remark <ref> induces a central extension of $\widehat{\mathcal{L} H}_x$ that splits over ${\mathcal{L}^+ H}_x$. By <cit.>, this also corresponds to a weak $\widehat{\mathcal{L} H}_x$-action on $\on{Vect}$ whose restriction on ${\mathcal{L}^+ H}_x$ is trivialized. Let $\on{Vect}_{\mathscr{L}_x}$ be the correponding object in $\widehat{\mathcal{L}^+ H}_x\mod^{\on{weak}}$. It follows from Remark <ref> that $[\mathcal{KL}(H)_{\mathscr{L}}]_x$ can be identified with $(\on{Vect}_{\mathscr{L}_x}){^{  \widehat{\mathcal{L}H}_x,w}}$, which is then identified in <cit.> as $\widetilde{\mathcal{L}\mathfrak{h}_x}\on{-mod}^{\mathcal{L}^+ H_x}_{\mathscr{L}}$, where $(\widetilde{\mathcal{L}\mathfrak{h}_x})_{\mathscr{L}}$ is the corresponding central extension of $\mathcal{L}\mathfrak{h}_x$. Hence (when $H$ is reductive) $\mathcal{KL}(H)_{\mathscr{L}}$ is indeed a factorization version of the Kazhdan–Lusztig category.

§.§ The Tate Line Bundle

The goal of this subsection is to prove the following result:

Let $\mathscr{L}$ be a multiplicative factorization line bundle on $\widehat{\mathcal{L}H}_{\Ran_{\on{un,dR}}}$ that is trivial on $\mathcal{L}^+ H$. Then there exists a canonical equivalence

$$ \mathcal{KL}(H)_{ \mathscr{L} }  \simeq \mathcal{KL}(H)_{\mathscr{L}\otimes \mathscr{L}^{\on{Tate}},\on{co} }$$

in $\on{FactCat}_{\on{un}}$.

For any closed point $x\in X$, the combination of <cit.> and <cit.> provides an equivalence $ [\mathcal{KL}(H)_{ \mathscr{L} }]_x  \simeq [\mathcal{KL}(H)_{\mathscr{L}\otimes \mathscr{L}^{\on{Tate}} ,\on{ co}}]_x$. We will explain in Remark <ref> how to identify it with the restriction of our equivalence at $x$.

Step 1: In this step, for any $[m]$, we will construct $(m+1)$ different $[ \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ),\overset{!}\otimes]$-linear equivalences $\theta_i: \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \to \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^m )$.

For each $i: [0]\to [m]$, consider the corresponding projection $p_i: \widehat{\on{Hecke}}_H^m \to \mathbb{B} \mathcal{L}^+ H $. One can check directly that the following functors provide a self-duality on $\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m )$:

\[
\begin{aligned}
\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \otimes \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \xrightarrow{\otimes^!} \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \xrightarrow{  (p_i)^\ren_! } \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \mathbb{B} \mathcal{L}^+ H ) \to \\ \xrightarrow{?-\on{push}}  \mathcal{I}nd\mathcal{C}oh^{!,\ren}(\on{pt}) \simeq \mathbf{1} \\
\mathbf{1} \simeq  \mathcal{I}nd\mathcal{C}oh^{!,\ren}(\on{pt})  \xrightarrow{!-\on{pull}}  \mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_H^m ) \xrightarrow{ !-\on{push} }  \mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_H^m \times_{ (\mathbb{B} \mathcal{L}^+ H)  } \widehat{\on{Hecke}}_H^m ) \to\\
\xrightarrow{?\on{-push}} \mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_H^m \times \widehat{\on{Hecke}}_H^m ) \simeq \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \otimes \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ).
\end{aligned}
\]

Therefore for each $i: [0]\to [m]$, we obtain an equivalence

$$ \theta_i: \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \to \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^m ).$$

Note that the pairing functor in the above duality factors through the $!$-tensor product functor. It follows formally that $\theta_i$ has an $[ \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ),\overset{!}\otimes]$-linear structure.

The above construction is functorial in $[m]$ in the sense that for $f:[m]\to [n]$, we have a canonical commutative diagram

\xymatrix{
\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \ar[r]_-\simeq^-{\theta_i} \ar[d]^-{ (p_f)^{!,\ren}  }
& \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^m )
\ar[d]^-{ (p_f)^{?}_\ren  }  \\
\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^n ) \ar[r]_-\simeq^-{\theta_{f\circ i}} & \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^n ).

Step 2: In this step, we use the equivalences $\theta_i$ constructed in Step 1 to construct a multiplicative factorization line bundle $\mathscr{L}^{\on{diff}}$ on $\widehat{\mathcal{L}H}_{\Ran_{\on{un,dR}}}$ that is trivial on $\mathcal{L}^+ H$.

Consider the two $[\mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H ),\overset{!}\otimes]$-linear equivalences

$$ \theta_0,\theta_1: \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H ) \to \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H )$$

constructed in Step 1. There exists a unique unital factorization line bundle $\mathscr{L}^{\on{diff}}_{ \widehat{\on{Hecke}}_H}$ on $[\widehat{\on{Hecke}}_H ]_{\Ran_{\on{un,dR}}}$ such that

$$ \theta_0(-) \simeq  \theta_1(\mathscr{L}^{\on{diff}}_{ \widehat{\on{Hecke}}_H} \overset{!}\otimes - ) \simeq \mathscr{L}^{\on{diff}}_{ \widehat{\on{Hecke}}_H} \overset{\on{act}}\otimes \theta_1(-) $$

Let $\mathscr{L}^{\on{diff}}$ be its $!$-pullback along $\widehat{\mathcal{L}H}_{\Ran_{\on{un,dR}}}\to [\widehat{\on{Hecke}}_H ]_{\Ran_{\on{un,dR}}}$. Then the functoriality of $\theta_i$ described in Step 1 can upgrade $\mathscr{L}^{\on{diff}}$ to a multiplicative factorization line bundle on $\widehat{\mathcal{L}H}_{\Ran_{\on{un,dR}}}$ that is trivial on $\mathcal{L}^+ H$.

Step 3: It follows from construction that we have an equivalence

$$ \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L},\Delta} \simeq \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^\bullet )_{\mathscr{L} \otimes\mathscr{L}^{\on{diff}} ,\Delta}  $$

between cosimplicial objects in $[\on{FactCat}_{\on{un}}]_{\on{lax-unital}}$, whose value on any term is the equivalence

$$\theta_0: \mathcal{I}nd\mathcal{C}oh^{!,\ren}( \widehat{\on{Hecke}}_H^m ) \simeq \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\on{Hecke}}_H^m ) .$$

Hence we obtain an equivalence

$$\mathcal{KL}(H)_{ \mathscr{L} }  \simeq \mathcal{KL}^*(H)_{\mathscr{L}\otimes \mathscr{L}^{\on{diff}} }$$

in $[\on{FactCat}_{\on{un}}]_{\on{lax-unital}}$, which is automatically an equivalence in $\on{FactCat}_{\on{un}}$. Hence to finish the proof, we only need to show $\mathscr{L}^{\on{diff}} \simeq \mathscr{L}^{\on{Tate}}$.

Step 4: In this step, we construct two different $[\mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} ),\overset{!}\otimes ]$-linear equivalences

$$\mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} )\to \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\mathcal{L}H} ),$$

and we show that the difference of them is given by a unital factorization line bundle $\mathscr{L}^{\on{diff}'}$.

We claim the following functor is a perfect pairing, i.e., induces a self-duality on $\mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} )$:

\[
\begin{aligned}
\mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} ) \otimes \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} ) \xrightarrow{\otimes} \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} ) \xrightarrow{?\on{-push}} \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H}/ \mathcal{L}^+H ) \to \\ \xrightarrow{!\on{-push}} \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \on{pt} ) \simeq \mathbf{1}.
\end{aligned}
\]

To prove this, we only need to verify the factorization functor $\mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} )\to \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\mathcal{L}H} )$ induced by it is an equivalence. Consider the following lax-unital factorization functor

$$\mathbf{1} \xrightarrow{?\on{-pull}} \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \widehat{\mathcal{L}H} / \mathcal{L}^+H ) \xrightarrow{*\on{-pull}} \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\mathcal{L}H} )$$

and the corresponding unital factorization algebra $\mathcal{F}_0$. Unwinding the definitions, the aforementioned functor is given by

$$ -\overset{\on{act}}\otimes \mathcal{F}_0: \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} )\to \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\mathcal{L}H} ). $$

To prove its invertibility, we only need to show for any classical test scheme $s:S\to \Ran_{\on{un,dR}}$, the functor $-\overset{\on{act}}\otimes [\mathcal{F}_0]_s: \IndCoh^{!}(   \widehat{\mathcal{L}H}_s )\to \IndCoh_{*}(   \widehat{\mathcal{L}H}_s )$ is an equivalence. Let $[ \widehat{\mathcal{L}H}/\mathcal{L}^+H]_s \simeq \colim_\alpha Y_\alpha$ be a presentation of $[ \widehat{\mathcal{L}H}/\mathcal{L}^+H]_s$ as a classical ind-finite type ind-proper indscheme over $S$. Define $Z_\alpha:= Y_\alpha \times_{  \mathbb{B} \mathcal{L}^+H_s } S$ and $\pi_\alpha: Z_\alpha \to Y_\alpha$. Using Proposition <ref>, we only need to show the action of $\IndCoh^!(Z_\alpha)$ on $(\pi_\alpha)^*( \omega_{Y_\alpha})$ induces an equivalence $\IndCoh^!(Z_\alpha) \to \IndCoh_*(Z_\alpha)$. Write $Z_\alpha = \lim_k Y_\alpha^k$ such that $q_k:Y_\alpha^k \to Y_\alpha$ are smooth. Using Proposition <ref>, we only need to show the action of $\IndCoh(Y_\alpha^k)$ on $(q_k)^*(\omega_{Y_\alpha})$ induces an automorphism on $\IndCoh(Y_\alpha^k)$, but this follows from the fact that $q_k$ is smooth. This finishes the proof of the claim.

Let $\mathcal{F}_1$ be the unital factorization algebra that corresponds to

$$\mathbf{1} \xrightarrow{?\on{-pull}} \mathcal{I}nd\mathcal{C}oh_{*,\ren}( \mathcal{L}^+H \backslash\widehat{ \mathcal{L}H} ) \xrightarrow{*\on{-pull}} \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\mathcal{L}H} ).$$

By symmetry, the functor

$$ -\overset{\on{act}}\otimes \mathcal{F}_1: \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\mathcal{L}H} )\to \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\mathcal{L}H} ) $$

is also an equivalence. It follows that there exists a unique unital factorization line bundle $\mathscr{L}^{\on{diff}'}$ on $[\widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}$ such that $\mathcal{F}_0 \simeq \mathscr{L}^{\on{diff}'}\overset{\on{act}}\otimes \mathcal{F}_1 $.

Step 5: In this step, we show $\mathscr{L}^{\on{diff}'}$ has a canonical multiplicative structure, and $\mathscr{L}^{\on{diff}'}\simeq \mathscr{L}^{\on{diff}}$.

It is clear that $\mathscr{L}^{\on{diff}'}\boxtimes \mathscr{L}^{\on{diff}'}$ is the unique unital factorization line bundle on $[\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}$ such that $\mathcal{F}_0 \boxtimes \mathcal{F}_0 \simeq (\mathscr{L}^{\on{diff}'}\boxtimes \mathscr{L}^{\on{diff}'})\overset{\on{act}}\otimes (\mathcal{F}_1\boxtimes\mathcal{F}_1)$.

On the other hand, consider the composition

$$\on{mult}: [\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}} \xrightarrow{q} [\widehat{\mathcal{L}H}\times^{\mathcal{L}^+H} \widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}} \xrightarrow{\on{m}} [\widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}.$$

Using the base-change isomorphisms, we can obtain $q^*_\ren \circ m^{?}_\ren(  \mathcal{F}_i ) \simeq \mathcal{F}_i \boxtimes \mathcal{F}_i$. This implies $\mathcal{F}_0 \boxtimes \mathcal{F}_0 \simeq ( \on{mult}^{!,\ren}(   \mathscr{L}^{\on{diff}'}  )   ) \overset{\on{act}}\otimes (\mathcal{F}_1\boxtimes\mathcal{F}_1)$.

Therefore we have $\on{mult}^{!,\ren}(   \mathscr{L}^{\on{diff}'})\simeq \mathscr{L}^{\on{diff}'}\boxtimes \mathscr{L}^{\on{diff}'}$. Similarly we have $e^{!,\ren}(   \mathscr{L}^{\on{diff}'})\simeq \omega_{\on{pt}}$. One can work over $ [\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}$ to check these equivalences provide a multiplicative structure[By Remark <ref>, this is a property rather than additional structure.] on $\mathscr{L}^{\on{diff}'}$.

Now consider the projection $\pi: [\widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}} \to  [\widehat{\on{Hecke}}_H ]_{\Ran_{\on{un,dR}}}$. Using the base-change isomorphisms, we have $\pi^*_\ren \circ \theta_i(\omega) \simeq \mathcal{F}_i$. This implies $\pi^{!,\ren}( \mathscr{L}^{\on{diff}}_{ \widehat{\on{Hecke}}_H}  ) \simeq  \mathscr{L}^{\on{diff}'}$ as unital factorization line bundles. One can check that this equivalence is compatible with the multiplicative structures.

Step 6: In this last step, we show $\mathscr{L}^{\on{diff}} \simeq \mathscr{L}^{\on{Tate}}$.

Consider the autmorphism $\alpha=(\on{mult},\on{pr}_2)$ on $[\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}$. Using the base-change isomorphisms, we have $\alpha^*_\ren(  \mathcal{F}_1 \boxtimes  \mathcal{F}_1 )\simeq  \mathcal{F}_1 \boxtimes  \mathcal{F}_1$. On the other hand, we claim

\begin{equation} \label{eqn-proof-thm-tate-twist-1}
\alpha^*_\ren(  \mathcal{F}_0 \boxtimes  \mathcal{F}_0 )\simeq(\omega \boxtimes \mathscr{L}^{\on{Tate}})\overset{\on{act}}\otimes (\mathcal{F}_0\boxtimes\mathcal{F}_0).
\end{equation}

To simplify the notations, we prove the non-factorization version of the claim. The factorization version can be proved in the same way be replacing $\widehat{\mathcal{L}H}$ by $[\widehat{\mathcal{L}H}]_s$ for any classical test map $s:S\to \Ran_{ \on{un,dR} }$. We view $\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}$ as a group indscheme parameterized by its second factor and consider its closed sub-group scheme $\mathcal{K}$ containing those $(u,v)$ with $\on{Ad}_v u\in \mathcal{L}^+H$. Then we have a commutative diagram

\xymatrix{
\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H} \ar[r]^-{\beta} \ar[d]^-\alpha &
(\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H})/\mathcal{K} \ar[r]^-\gamma \ar[d]^-{\overline{\alpha}} &
\widehat{\mathcal{L}H} \ar[d] \\
\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H} \ar[r] &
\widehat{\mathcal{L}H}/\mathcal{L}^+H \times \widehat{\mathcal{L}}H/\mathcal{L}^+H \ar[r]^-{\on{pr}_2} &
\widehat{\mathcal{L}H}/\mathcal{L}^+H

such that the right square is Cartesian. Using the base-change isomorphisms, we obtain

\begin{equation} \label{eqn-proof-thm-tate-twist-2}
\alpha^*_\ren(  \mathcal{F}_0 \boxtimes  \mathcal{F}_0 )\simeq \beta^{*}_\ren\circ \gamma^{?}_\ren(\mathcal{F}_0).
\end{equation}

On the other hands, we can also view $\mathcal{K}':= \mathcal{L}^+H \times \widehat{\mathcal{L}H} $ as a closed sub-group scheme of $\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}$ parameterized by their second factors. Consider the chain $\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H} \xrightarrow{\beta'}
(\widehat{\mathcal{L}H} \times \widehat{\mathcal{L}H})/\mathcal{K}' \xrightarrow{\gamma'}
\widehat{\mathcal{L}H}$. We claim

\begin{equation} \label{eqn-proof-thm-tate-twist-3}
(\beta')^{*}_\ren\circ (\gamma')^{?}_\ren(  \mathscr{L}^{\on{Tate}}\overset{\on{act}}\otimes- ) \simeq (\beta)^{*}_\ren\circ (\gamma)^{?}_\ren(-).
\end{equation}

To prove this claim, consider the (classical) intersection $\mathcal{K}''$ of $\mathcal{K}$ and $\mathcal{K}'$ and the chains

$$ \mathbb{B}\mathcal{K}'' \xrightarrow{\phi} \mathbb{B}\mathcal{K} \xrightarrow{\xi}  \widehat{\mathcal{L}H},\; \mathbb{B}\mathcal{K}'' \xrightarrow{\phi'} \mathbb{B}\mathcal{K}' \xrightarrow{\xi'}   \widehat{\mathcal{L}H}.$$

Note that $\phi$ and $\phi'$ are of finite type and are nil-isomorphisms. Using Proposition <ref> and Proposition <ref>, one can deduce the following formula from its analogue in the finite type case:

\begin{equation} \label{eqn-proof-thm-tate-twist-4}
\phi^?_\ren(-) \simeq \phi^*_\ren( \xi^{!,\ren}( \mathcal{L})\overset{\on{act}}\otimes- ),\; (\phi')^?_\ren(-) \simeq (\phi')^*_\ren( (\xi')^{!,\ren}( \mathcal{L}')\overset{\on{act}}\otimes- )
\end{equation}

where $\mathcal{L}, \mathcal{L}'\in \IndCoh^{!,\ren}(  \widehat{\mathcal{L}H} )$ are (cohomologically shifted) line bundles whose restrictions along a classical test map $s:S\to \widehat{\mathcal{L}H}$ are respectively the determinant line bundles

$$\on{det}( \on{Lie}(\mathcal{K}|_s)/\on{Lie}(\mathcal{K}''|_s) [1]),\;\on{det}( \on{Lie}(\mathcal{K}'|_s)/\on{Lie}(\mathcal{K}''|_s) [1]).$$

By definition, $\mathcal{L}' \simeq \mathscr{L}^{\on{Tate}} \otimes \mathcal{L}$. Combining this with (<ref>), we obtain the desired equivalence (<ref>).

Now the desired equivalence (<ref>) follows from (<ref>) and (<ref>). Pluging in $\mathcal{F}_0 \simeq \mathscr{L}^{\on{diff}}\overset{\on{act}}\otimes \mathcal{F}_1 $, we obtain

\begin{eqnarray*}
(\omega \boxtimes \mathscr{L}^{\on{Tate}})\overset{\on{act}}\otimes (\mathcal{F}_0\boxtimes\mathcal{F}_0) & \simeq & \alpha^*_\ren(  \mathcal{F}_0 \boxtimes  \mathcal{F}_0 ) \\
& \simeq & \alpha^{!,\ren}(  \mathscr{L}^{\on{diff}}\boxtimes \mathscr{L}^{\on{diff}} )\overset{\on{act}} \otimes \alpha^*_\ren (  \mathcal{F}_1 \boxtimes  \mathcal{F}_1 ) \\
& \simeq & (  \mathscr{L}^{\on{diff}}\boxtimes (\mathscr{L}^{\on{diff}})^{\otimes 2} )\overset{\on{act}} \otimes (  \mathcal{F}_1 \boxtimes  \mathcal{F}_1 )  \\
& \simeq & (\omega \boxtimes \mathscr{L}^{\on{diff}})\overset{\on{act}}\otimes (\mathcal{F}_0\boxtimes\mathcal{F}_0).
\end{eqnarray*}

Since acting on $(\mathcal{F}_0\boxtimes\mathcal{F}_0)$ is an equivalence, we obtain $\omega \boxtimes \mathscr{L}^{\on{Tate}}\simeq \omega \boxtimes \mathscr{L}^{\on{diff}}$ and therefore $\mathscr{L}^{\on{Tate}}\simeq \mathscr{L}^{\on{diff}}$ as unital factorization line bundles. One can work over $ [\widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}\times \widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}$ to check this equivalence is compatible with the multiplicative structures. This finishes the proof of the theorem.

In this remark, we sketch how to identify the restriction of our equivalence in Theorem <ref> at a closed point $x\in X$ with the equivalence constructed in <cit.>.

We need to translate Raskin's construction to our setting. <cit.> says the monoidal category $A:=[\IndCoh^{!,\ren}( [\widehat{\on{Hecke}}_H ]_x ),\star]$ is rigid[This is related to the following fact, which we do not prove nor use in this paper. Consider the lax unital factorization functor:

$$ \lambda:  \mathcal{I}ndCoh^{!,\ren}( \widehat{\on{Hecke}}_H^1) \overset{e^{!,\ren}}\to  \mathcal{I}ndCoh^{!,\ren}( \widehat{\on{Hecke}}_H^0) \overset{?\on{-push}} \to \mathcal{I}ndCoh^{!,\ren}( \on{pt} ) \simeq \mathbf{1},$$

where $e: \widehat{\on{Hecke}}_H^0\to \widehat{\on{Hecke}}_H^1$ is the morphism corresponding to the unique map $[1]\to [0]$. Then $[ \mathcal{I}ndCoh^{!,\ren}( \widehat{\on{Hecke}}_H),\star ,\lambda]$ is a Frobenius algebra object in $[\on{FactCat}_{\on{un}}]_{\on{lax-unital}}$ (see <cit.>). A rigid monoidal DG category is in particular a Frobenius algebra object in $\on{DGCat}_{\on{cont}}$, and Raskin's construction actually only used this Frobenius algebra structure.

One can use this fact to obtain an equivalence $\mathcal{KL}(H)_{ \mathscr{L} }  \simeq \mathcal{KL}^*(H)_{\mathscr{L}\otimes \mathscr{L}^{\on{Serre}} }$, where $\mathscr{L}^{\on{Serre}}$ is a multiplicative factorization line bundle such that the Serre automorphism for the above Frobenius algebra (see <cit.>) is given by tensoring up with $\mathscr{L}^{\on{Serre}}$. However, it is harder to identify $\mathscr{L}^{\on{Serre}}$ with $\mathscr{L}^{\on{Tate}}$. For instance, Raskin proved (the non-factorization version) of this identification in <cit.> with the help of semi-infinite cohomologies.]. Consider the geometric left and right actions of $A$ on $M:=\on{Rep}( \mathcal{L}^+ H_x )$ defined respectively by (<ref>) and (<ref>), it is clear that these actions commute hence make $M$ an $(A,A)$-bimodule. One can verify that tensoring up with $[\mathscr{L}_{ \widehat{\on{Hecke}_H} }]_{x}$ is a monoidal automorphism on $A$. Using this automorphism, we can twist the $A$-actions on $\on{Rep}( \mathcal{L}^+ H_x )$ and obtain another $(A,A)$-bimodule $M_{\mathscr{L}}$.

In Raskin's language, the left $A$-module structure on $M_{\mathscr{L}}$ is the same as a weak $\widehat{\mathcal{L}H}_x$-action on $\on{Vect}$. Denote the corresponding weak $\widehat{\mathcal{L}H}_x$-module by $C_{\mathscr{L}}$. The invariance and coinvariance are defined respectively as $(C_{\mathscr{L}})^{\on{inv}}:=\on{Hom}_{A\mod^l}( M, M_{\mathscr{L}} )$ and $(C_{\mathscr{L}})_{\on{coinv}}:=M\otimes_A M_{\mathscr{L}}$. One can verify that the cosimplicial (resp. simplicial) diagram defining them satsifies the Beck-Chevalley conditions, which implies

$$(C_{\mathscr{L}})^{\on{inv}} \simeq [\mathcal{KL}(H)_{ \mathscr{L} }]_x ,\;(C_{\mathscr{L}})_{\on{coinv}}\simeq [ \mathcal{KL}^*(H)_{\mathscr{L}}]_x.$$

In <cit.>, Raskin used the general theory of rigid monoidal categories to obtain an equivalence

$$(C_{\mathscr{L}})^{\on{inv}}\simeq  (C_{\mathscr{L}}\otimes \chi_{\on{Serre}})_{\on{coinv}},$$

where $\chi_{\on{Serre}}$ is the weak $\widehat{\mathcal{L}H}_x$-module twisted from the trivial one by the Serre automorphism $\on{Se}_A$ (see <cit.>) on the rigid monoidal category $A$. Hence we only need to identify $\on{Se}_A$ with $[\mathscr{L}^{\on{Tate}}_{ \widehat{\on{Hecke}_H} }]_x \otimes^! (-)$ as symmetric monoidal functors. Recall the underlying endo-functor of $\on{Se}_A$ is the difference of two self-dualities on $A$ whose pairing functors are

$$A\otimes A \xrightarrow{\star} A \xrightarrow{\on{Maps}( \on{unit}_\star,- )} \on{Vect},\;A\otimes A \xrightarrow{\star^{\on{rev}}} A \xrightarrow{\on{Maps}( \on{unit}_\star,- )} \on{Vect}.$$

Using the base-change isomorphisms, we have $\on{Maps}( \on{unit}_\star, x\star y ) \simeq <\sigma^{!,\ren}(x),y>_1$, where $<-,->_1$ is the pairing in Step 1 of our proof of Theorem <ref> for the case $i=m=1$. Similarly we have $\on{Maps}( \on{unit}_\star, y\star x ) \simeq <\sigma^{!,\ren}(x),y>_0$. Since $\sigma^{!,\ren}$ is an involution, we obtain $\on{Se}_A\simeq \theta_1^{-1}\circ \theta_0$. Hence our proof of Theorem <ref> provides an equivalence $\on{Se}_A\simeq [\mathscr{L}^{\on{Tate}}_{ \widehat{\on{Hecke}_H} }]_x \otimes^! (-)$ as endo-functors. Now the monoidal structure on $\on{Se}_A$ would correspond to some multiplicative structure on $\mathscr{L}^{\on{Tate}}$. We only need to show it coincides with the standard one. But this follows from unwinding the definitions.

§.§ Variant: Factorization Module Category

We also need a factorization module version of the previous constructions. We introduce the following notations.

We use the following notations:

* $K\to H$ is a fixed closed subgroup of $H$.

* $x:\on{pt}\to \Ran_{\on{un}}$ is a fixed point of $\Ran_{\on{un}}$, which corresponds to a finite set $I_0\subset \on{Maps}(\on{pt},X)$, or equivalently, to closed points $x_i$ of $X$ labelled by $I_0$.

* $\Ran_{x,\on{un}}$ is the laft lax prestack defined in Section <ref>, and $\Ran_{x,\on{un,dR}}$ is its de-Rham lax prestack.

* $[\mathcal{L}^+ H]_{\Ran_{x,\on{un,dR}}}:= [\mathcal{L}^+ H]_{\Ran_{\on{un,dR}}} \times_{  \Ran_{\on{un,dR}} } \Ran_{x,\on{un,dR}}$, and similarly define $[\widehat{\mathcal{L} H}]_{\Ran_{x,\on{un,dR}}}$.

* $H_x:= \prod_{j\in J} H_{x_j}:= \prod_{j\in J} H$ and similarly define $K_x$.

* $[\mathcal{L}^+ H\times_{H_x} K_x ]_{ \Ran_{x,\on{un,dR}} }:= [\mathcal{L}^+ H]_{\Ran_{x,\on{un,dR}}} \times_{H_x} K_x$, which is a co-unital factorization module space of $\mathcal{L}^+ H_{\Ran_{\on{un,dR}}}$ supported on $x$.

* $[\widehat{\mathcal{L}H}\times_{ H_{x,\dR} } K_{x,\dR}  ]_{ \Ran_{x,\on{un,dR}}  }:=[\widehat{\mathcal{L} H}]_{\Ran_{x,\on{un,dR}}} \times_{H_{x,\dR}} K_{x,\dR} $ is the formal completion of $[\mathcal{L} H]_{\Ran_{x,\on{un,dR}}}$ along $[\mathcal{L}^+ H\times_{H_x} K_x ]_{ \Ran_{x,\on{un,dR}} }$, which is a corr-unital factorization module space of $[\mathcal{L} H]_{\Ran_{\on{un,dR}}}$ supported on $x$.

* $[ \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }}$ is the Cech nerve of $ [\mathbb{B}(\mathcal{L}^+ H\times_{H_x} K_x)]_{\Ran_{x,\on{un,dR}}} \to [\mathbb{B}(\widehat{\mathcal{L}H}\times_{ H_{x,\dR} } K_{x,\dR} )]_{\Ran_{x,\on{un,dR}}}$, which is a cosimplicial corr-unital factorization module space of $[\widehat{\on{Hecke}}_H^\bullet ]_{\Ran_{ \on{un,dR} }}$.

By definition, we have

$$ [ \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }} \simeq [ \widehat{\on{Hecke}}_{H}^\bullet  ]_{\Ran_{ x,\on{un,dR} }} \times_{ \mathbb{B}H_x^{\times (\bullet+1) }  }  \mathbb{B} K_x^{\times (\bullet+1) }.$$

As in the definitions of $\mathcal{KL}(H)_{\mathscr{L}}$ and $ \mathcal{KL}(H)_{\mathscr{L},\on{co}}$, we can define crystal of categories over $\Ran_{x,\on{un,dR}}$:

$$  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ,\;   \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}} ,$$

equipped respectively with unital factorization module structures for the unital factorization categories $\mathcal{KL}(H)_{\mathscr{L}}$ and $\mathcal{KL}(H)_{\mathscr{L},\on{co}}$.

We can also define a crystal of categories over $\Ran_{x,\on{un,dR}}$:

$$  \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }$$

equipped rwith unital factorization module structures for the unital factorization categories $\mathcal{R}ep( \mathcal{L}^+H )$.

Recall a map $y:\on{pt}\to \Ran_{x,\on{un}}$ corresponds to a finite subset $I\subset \on{Maps}(\on{pt},X)$ containing $I_0$. Then

$$[\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_H K}_{\mathscr{L}}  ]_{y} \simeq \bigotimes_{x_i\in I_0} \widetilde{\mathcal{L}\mathfrak{h}}_{x_i}\mod^{\mathcal{L}^+H_{x_i}\times_{H_{x_i}} K_{x_i}}_{\mathscr{L}} \bigotimes \bigotimes_{x_i\in I\setminus I_0} \on{KL}(H)_{\mathscr{L},x_i} .$$

§.§ Functoriality For the Group

Let $(K'\to H')$ be another closed embedding between finite type affine algebraic groups, and $(K'\to H')\to (K\to H)$ be group homomorphisms. Note that $\mathscr{L}$ induces a multiplicative factorization line bundle $\mathscr{L}'$ on $\mathcal{L}H'$.

Then the $!$-pullback functors induce lax unital morphisms

$$\on{res}: \mathcal{KL}(H)_{\mathscr{L}} \to \mathcal{KL}(H')_{\mathscr{L}'},  \;\on{res}: \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \to  \widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'}$$

compatible with the factorization structures, which we refer as the factorization restriction functors.

Dually, the $*$-pushforward functors lax unital morphisms

$$\on{C}^{\frac{\infty}{2}}_{\on{co}}:  \mathcal{KL}(H')_{\mathscr{L}',\on{co}} \to \mathcal{KL}(H)_{\mathscr{L},\on{co}}, \;\on{C}^{\frac{\infty}{2}}_{\on{co}}:  \widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}',\on{co}} \to \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}}$$

compatible with the factorization structures, which we call the co-version of the factorization semi-infinite cohomology functor.

Using (the factorization module version of) Theorem <ref>, we obtain a morphism

$$\on{C}^{\frac{\infty}{2}}:  \mathcal{KL}(H')_{\mathscr{L}'\otimes \mathscr{L}^{-\on{Tate}}_{H'}} \to \mathcal{KL}(H)_{\mathscr{L}\otimes \mathscr{L}^{-\on{Tate}}_{H}}, \;\on{C}^{\frac{\infty}{2}}:  \widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'\otimes \mathscr{L}^{-\on{Tate}}_{H'}} \to \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}\otimes \mathscr{L}^{-\on{Tate}}_{H}}$$

compatible with the factorization structures, which we call the the factorization semi-infinite cohomology functor.

§ LOCALIZATION AND LOCAL-GLOBAL CORRESPONDENCE

We keep Notation <ref>. Also,

* $X$ is assumed to be proper.

* The multiplicative factorization line bundle $\mathscr{L}$ is constructed from a central $\on{Lie}^*$-extension $\widetilde{\mathfrak{h}}_{\mathcal{D}}$ of $\mathfrak{h}_{\mathcal{D}}:=\mathfrak{h}\otimes \on{Diff}_X$ using the method in <cit.>.

* $\on{Bun}_H$ is the moduli stack of $H$-torsors on $X$.

* Let $(\on{Bun}_H\times\on{Ran}_{\on{un,dR}})^{H\on{-level}_{\infty}}$ be the prestack such that $(\on{Bun}_H\times\on{Ran}_{\on{un,dR}})^{H\on{-level}_{\infty}}(S)$ classifies

* a finite subset $I\subset \on{Maps}(S^{\on{cl,red}},X)$;

Let $\Gamma_I\to S\times X$ be the schema-theoretic sum of the graphs of the maps $S^{\on{cl,red}} \to X$;

* an $H$-torsor on $X\times S$ equipped with a trivialization on the adic multi-disk $\mathcal{D}_{\Gamma_I}'$.

* $\on{Bun}_H ^{K\on{-level}_x} \simeq \on{Bun}_H\times_{\mathbb{B}H_x } \mathbb{B}K_x$ is the moduli stack of $H$-torsors on $X$ equipped with $K$-structures at $x_j, j\in J$.

* $\mathscr{T}$ is the twisting on $\on{Bun}_H$ constructed from the central $\on{Lie}^*$-extension $\widetilde{\mathfrak{h}}_{\mathcal{D}}$ using the method in <cit.>.

* $\DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}$ is the DG category of $\mathscr{T}$-twisted D-modules on $\on{Bun}_H ^{K\on{-level}_x}$.

$$(\on{Bun}_H\times\on{Ran}_{x,\on{un,dR}})^{H\on{-level}_{\infty}}:=(\on{Bun}_H\times\on{Ran}_{\on{un,dR}})^{H\on{-level}_{\infty}}\times_{\on{Ran}_{\on{un,dR}} } \on{Ran}_{x,\on{un,dR}},$$

which is an $[\mathcal{L}^+ H\times_{H_x} K_x]_{\on{Ran}_{x,\on{un,dR}} }$-torsor on $\on{Bun}_H^{K\on{-level}_x}\times \on{Ran}_{x,\on{un,dR}}$. A standard re-gluing construction extend this $[\mathcal{L}^+ H\times_{H_x} K_x]_{\on{Ran}_{x,\on{un,dR}} }$-action to an $[\mathcal{L}H]_{\on{Ran}_{x,\on{un,dR}}}$-action. The restricted $[\widehat{\mathcal{L}H}\times_{H_{x,\dR}} K_{x,\dR}]_{\on{Ran}_{x,\on{un,dR}}}$-action preserves the projection to $(\on{Bun}_{H}^{K\on{-level}_x})_\dR\times\on{Ran}_{\on{un,dR}}$. Hence $\mathscr{T}$ induces a $\mathbb{G}_m$-gerbe on

$$(\on{Bun}_H\times\on{Ran}_{x,\on{un,dR}})^{H\on{-level}_{\infty}}/[\widehat{\mathcal{L}H}\times_{H_{x,\dR}} K_{x,\dR}]_{\on{Ran}_{x,\on{un,dR}}}$$

equipped with a trivialization on

$$(\on{Bun}_H\times\on{Ran}_{x,\on{un,dR}})^{H\on{-level}_{\infty}}/[\mathcal{L}^+H\times_{H_{x,\dR}} K_{x,\dR}]_{\on{Ran}_{x,\on{un,dR}}} \simeq  \on{Bun}_H\times_{\mathbb{B}H_{x,\dR} } \mathbb{B}K_{x,\dR} \times \on{Ran}_{x,\on{un,dR}} $$

Note that $\on{Bun}_H\times_{\mathbb{B}H_{x,\dR} } \mathbb{B}K_{x,\dR} $ is just the formal completion of $\on{Bun}_H$ along $\on{Bun}_H ^{K\on{-level}_x}$.

On the other hand, recall that the multiplicative factorization line bundle $\mathscr{L}$ induces a $\mathbb{G}_m$-gerbe on $[\mathbb{B}\widehat{\mathcal{L}H}]_{\on{Ran}_{\on{un,dR}}}$ equipped with a trivialization on $[\mathbb{B}\mathcal{L}^+H]_{\on{Ran}_{\on{un,dR}}}$. By restriction, we obtain another $\mathbb{G}_m$-gerbe on $(\on{Bun}_H\times\on{Ran}_{x,\on{un,dR}})^{H\on{-level}_{\infty}}/[\widehat{\mathcal{L}H}\times_{H_{x,\dR}} K_{x,\dR}]_{\on{Ran}_{x,\on{un,dR}}}$ equipped with a trivialization on $\on{Bun}_H\times_{\mathbb{B}H_{x,\dR} } \mathbb{B}K_{x,\dR} \times \on{Ran}_{x,\on{un,dR}} $.

It follows from the assumption that the above two $\mathbb{G}_m$-gerbes are canonical isomorphic.

Let $H'\to H$ be a group homomorphism. Then $\widetilde{\mathfrak{h}}_{\mathcal{D}}$ induces a central $\on{Lie}^*$-extension $\widetilde{\mathfrak{h}'}_{\mathcal{D}}$ of $\mathfrak{h}'_\mathcal{D}$. We often write $(\mathscr{T}', \mathscr{L}')$ for the corresponding twisting and factorization multiplicative line bundle.

Since $\on{Bun}_H $ is smooth, the dualizing complex $K\in \on{QCoh}(\on{Bun}_H )$ is invertible. Let $K[-\on{dim}( \on{Bun}_H )]$ be the corresponding line bundle, where $\on{dim}( \on{Bun}_H )$ is the locally constant function of dimension. Let $\mathscr{K}$ be the integral twisting on $\on{Bun}_H$ given by this line bundle. It follows from definitions that $-\mathscr{K}$ corresponds to the Tate central $\on{Lie}^*$-extension.

The following result summarizes what we need to know about chiral localization.

There are canonical unital morphisms

\begin{eqnarray*}
\on{Loc}:  &
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \to \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \boxtimes \mathbf{1},\\
\on{Loc}_{\on{co}}: &
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}} \to \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \boxtimes \mathbf{1}
\end{eqnarray*}

between crystal of categories on $\on{Ran}_{x,\on{un,dR}}$, where the RHS's are the constant crystals of categories. These morphisms are required to satisfy the following:

(1)(Compatibility with chiral homology)

$$[\on{Loc}]_{ x}: [ \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ]_x  \to \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}  $$

be the fiber of $\on{Loc}$ at $x:\on{pt}\to \on{Ran}_{x,\on{un,dR}}$. Then there is a canonical commutative diagram

\xymatrix{
[ \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ]_x  \ar[rr]^-{ [\on{Loc}]_{ x} }  \ar[d]^-{\on{ins}_{\on{unit}}}
& & \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \\
[ \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ]_{\on{Ran}_{x,\on{un,dR}}}  \ar[rr]^-{ [\on{Loc}]_{ \on{Ran}_{x,\on{un,dR}}  } }
& & \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \otimes \DMod(\on{Ran}_{x,\on{un}}) \ar[u]^-{\mathbf{Id}\otimes \on{C}^{\on{ch}}_{x,\on{un}} }

where the left vertical functor is the inserting unit functor (see Construction <ref> for its definition), and

$$\on{C}^{\on{ch}}_{x,\on{un}}:=p_!: \DMod(\on{Ran}_{x,\on{un}})\to \on{Vect}$$

is taking chiral homology, i.e., the $!$-pushforward[We remind that the well-definedness of this functor relies on the homological left cofinality of $\Ran_\dR \to \Ran_{\un, \dR}$; see <cit.> for more details.] functor along $p: \on{Ran}_{x,\on{un}}\to \on{pt}$. There is a similar canonical commutative diagram for $\on{Loc}_{\on{co}}$.

(2)(Compatibility with Tate twist) When $x$ corresponds to the empty subset $\emptyset\subset \on{Maps}( \on{pt},X )$, there is a canonical commutative diagram

\xymatrix{
\mathcal{KL}(H)_{\mathscr{L}\otimes \mathscr{L}^{\on{Tate}},\on{co} }
\ar[r]^-{ \on{Loc}_{\on{co}} }  \ar[d]^-\simeq
&  \DMod(\on{Bun}_H )_{\mathscr{T}-\mathscr{K}}  \boxtimes \mathbf{1}  \ar[d]^-{\simeq}_-{K\otimes - }
\\
\mathcal{KL}(H)_{ \mathscr{L} }
\ar[r]^-{ \on{Loc} }
&  \DMod(\on{Bun}_H )_{\mathscr{T}} \boxtimes \mathbf{1},

where the left vertical equivalence is due to Theorem <ref>, and the right equivalence is bacause the twisting $\mathscr{K}$ is integral (i.e., given by a shifted line bundle).

* We use

$$[\on{Loc}]_x^{\on{ch}}: [ \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ]_{\on{Ran}_{x,\on{un,dR}}} \to \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}   $$

to denote the composition $  (\mathbf{Id}\otimes \on{C}^{\on{ch}}_{x,\on{un}} )\circ   [\on{Loc}]_{ \on{Ran}_{x,\on{un,dR}}  } $.
For a unital factorization algebra $A$ in $ \mathcal{KL}(H)_{\mathscr{L}}$, we abuse notation and write

$$ [\on{Loc}]_x^{\on{ch}}: A\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  )\to \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}   $$

for the composition $[\on{Loc}]_x^{\on{ch}}\circ \oblv_{A}$.

(3)(Compatibility with restriction) Let $(K'\to H')\to (K\to H)$ be as in Construction <ref>. Then for any unital factorization algebra $A$ in $ \mathcal{KL}(H)_{\mathscr{L}}$, there is a canonical commutative diagram

\xymatrix{
A\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ) \ar[d]^-{\on{res}}
\ar[r]^-{[\on{Loc}]_x^{\on{ch}} }
& \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}  \ar[d]^-{!\on{-pull}} \\
\on{res}(A)\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'}  )
\ar[r]^-{[\on{Loc}]_x^{\on{ch}} }
& \DMod(\on{Bun}_{H'}^{K'\on{-level}_x})_{\mathscr{T}'},

where the left vertical functor is induced by the lax unital factorization functor $\on{res}$ in Construction <ref>.

(4)(Compatibility with semi-infinite cohomology) In the situation of (3), $K'=H',K=H$ and $H'\to H$ is a surjection with unipotent kernel. Then for any unital factorization algebra $A'$ in $ \mathcal{KL}(H')_{\mathscr{L}',\on{co}}$, there is a canonical commutative diagram

\xymatrix{
A'\on{-FactMod}_{\on{Ran}_{x,\on{un,dR}}}    \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}}  \ar[r]^-{[\on{Loc}]_x^{\on{ch}} }
& \DMod(\on{Bun}_{H'})_{\mathscr{T}'}  \ar[d]^-{*\on{-push}} \\
\on{C}^{\frac{\infty}{2}}_{\on{co}}(A')\on{-FactMod}_{\on{Ran}_{x,\on{un,dR}}}
\ar[r]^-{[\on{Loc}]_x^{\on{ch}} }
& \DMod(\on{Bun}_H)_{\mathscr{T}},

where the left vertical functor is induced by the lax unital factorization functor $\on{C}^{\frac{\infty}{2}}_{\on{co}}$ in Construction <ref>.

Consider the following correspondence

\[
\begin{aligned}
(\on{Bun}_H^{K\on{-level}_x})_\dR \times \on{Ran}_{x,\on{un,dR}} \gets  (\on{Bun}_H\times\on{Ran}_{x,\on{un,dR}})^{H\on{-level}_{\infty}}/[\widehat{\mathcal{L}H}\times_{H_{x,\dR}} K_{x,\dR}]_{\on{Ran}_{x,\on{un,dR}}} \to\\
\to [\mathbb{B}\widehat{\mathcal{L}H}\times_{ H_{x,\dR} } K_{x,\dR}  ]_{ \Ran_{x,\on{un,dR}}  }
\end{aligned}
\]

The morphism $\on{Loc}$ should be viewed as the pull-push along this correspondence for a suitable twisted renormalized $\IndCoh$-theory.

We did not claim $\on{Loc}$ or $\on{Loc}_{\on{co}}$ have any factorization structure. In fact, $\DMod(\on{Bun}_H )_{\mathscr{T}} \boxtimes \mathbf{1}$ is not even a weak factorization category.

§.§ Construction of the Localization Functors

We first review the global infinitesimal Hecke stacks constructed in <cit.> (see also <cit.>), which are global analogue of $\widehat{\on{Hecke}}_{H,K\on{-level}_x}^\bullet$.

Let $[\widehat{\on{Hecke}}_{H ,\on{glob}}^{K\on{-level}_x, \bullet}]_{\on{Ran}_{x,\on{un,dR}}}$ be the Cech nerve of the map

$$\on{Bun}_H^{K\on{-level}_x}\times \on{Ran}_{x,\on{un,dR}}\to (\on{Bun}_H\times\on{Ran}_{x,\on{un,dR}})^{H\on{-level}_{\infty}}/[\widehat{\mathcal{L}H}\times_{H_{x,\dR}} K_{x,\dR}]_{\on{Ran}_{x,\on{un,dR}}}.$$

We have isomorphisms $[(\widehat{\on{Hecke}}_{H ,\on{glob}}^{K\on{-level}_x, \bullet})_\dR]_{\on{Ran}_{x,\on{un,dR}}} \simeq (\on{Bun}_{H}^{K\on{-level}_x})_\dR\times\on{Ran}_{\on{un,dR}}$.

In the special case when $x:\on{pt}\to \on{Ran}_{\on{un}}$ corresponds to the empty set $\emptyset\subset \on{Maps}(\on{pt},X)$, we obtain $[\widehat{\on{Hecke}}_{H ,\on{glob}}^{\bullet}]_{\on{Ran}_{\on{un,dR}}}$. By definition, we have

$$ [ \widehat{\on{Hecke}}_{H,\on{glob}}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }} \simeq [ \widehat{\on{Hecke}}_{H,\on{glob}}^\bullet  ]_{\Ran_{ x,\on{un,dR} }} \times_{ \mathbb{B}H_x^{\times (\bullet+1) }  }  \mathbb{B} K_x^{\times (\bullet+1) }.$$

Each $[\widehat{\on{Hecke}}_{H ,\on{glob}}^{K\on{-level}_x, m}]_{ \on{Ran}_{x,\on{un,dR}}}$ is a unital lax prestack over $\on{Ran}_{x,\on{un,dR}}$. In particular, the connecting morphisms are strictly unital.

By Remark <ref> and Remark <ref>, we can assume $x:\on{pt} \to \on{Ran}_{\on{un}}$ corresponds to the empty set (i.e., we ignore the $K$-level structures).

Let $\phi:I\to J$ be an injective map between finite sets. To prove the claim, we only need to show the correspondence

$$ [\widehat{\on{Hecke}}_{H ,\on{glob}}^{ m}]_J \gets [\widehat{\on{Hecke}}_{H ,\on{glob}}^{ m}]_{\phi} \to [\widehat{\on{Hecke}}_{H ,\on{glob}}^{ m}]_{I}\times_{X^I} X^J $$

(that provides the corr-unital structure) has a degenerate right arm. Explicitly, $[\widehat{\on{Hecke}}_{H ,\on{glob}}^{ m}]_{\phi }(S)$ classifies

(i) Maps $x_j:S\to X$ labelled by $J$.

Let $\Gamma_J\to S\times X$ (resp. $\Gamma_I\to S\times X$) be the schema-theoretic sum of the graphs of all the $x_j$ (resp. of $x_{\phi(i)}$ for $i\in I$). Note that we have a closed embedding $\Gamma_I \to \Gamma_J$.

(ii) $H$-torsors $\mathcal{P}_0,\cdots,\mathcal{P}_m$ on $S\times X$ and isomorphisms

$$  [\mathcal{P}_0]|_{ (S\times X)\setminus \Gamma_I } \simeq \cdots \simeq [\mathcal{P}_m]|_{ (S\times X)\setminus \Gamma_I } $$

between their restrictions on $(S\times X)\setminus \Gamma_I $.

(iii) Isomorphisms

$$  [\mathcal{P}_0]\times_S S^{\on{cl,red}} \simeq \cdots \simeq [\mathcal{P}_m]\times_S S^{\on{cl,red}}, $$

where we view $[\mathcal{P}_k]\times_S S^{\on{cl,red}}$ as a $G$-torsor on $ S^{\on{cl,red}}\times X $.

By restriction, (ii) and (iii) provides two chains of isomorphisms

$$ [\mathcal{P}_0]|_{ (S\times X)\setminus \Gamma_I  } \times_S S^{\on{cl,red}} \simeq \cdots \simeq [\mathcal{P}_m]|_{(S\times X)\setminus \Gamma_I  } \times_S S^{\on{cl,red}}.$$

(iv) An isomorphism between the above two chains.

This makes the claim manifest because (ii)-(iv) do not depend on $J$.

It follows from Remark <ref> and the proof of Lemma <ref> that there are simplicial strictly co-unital maps

$$ [ \widehat{\on{Hecke}}_{H,\on{glob}}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }} \to [ \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }} .$$

Also, for any $[m]\to [n]$, the following commutative diagram is Cartesian:

\begin{equation} \label{eqn-global-local-hecke-cart}
\xymatrix{
[ \widehat{\on{Hecke}}_{H,\on{glob}}^{K\on{-level}_x,n}  ]_{\Ran_{ x,\on{un,dR} }} \ar[r] \ar[d]
& [ \widehat{\on{Hecke}}_{H,\on{glob}}^{K\on{-level}_x,m}  ]_{\Ran_{ x,\on{un,dR} }} \ar[d] \\
[ \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,n}  ]_{\Ran_{ x,\on{un,dR} }} \ar[r]
& [ \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,m}  ]_{\Ran_{ x,\on{un,dR} }} .
\end{equation}

For any quasi-compact open substack $U\subset \on{Bun}_H$, we write

$$[ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }}:=[ \widehat{\on{Hecke}}_{H,\on{glob}}^{K\on{-level}_x,\bullet}  ]_{\Ran_{ x,\on{un,dR} }} \times_{\on{Bun}_{H,\dR}} U_\dR.$$

As before, we also use the notations

$$[ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x}  ]_{\Ran_{ x,\on{un,dR} }}:= [ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,1}  ]_{\Ran_{ x,\on{un,dR} }},\; [U^{K\on{-level}_x}]_{\Ran_{ x,\on{un,dR} }}:= [ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,0}  ]_{\Ran_{ x,\on{un,dR} }}.$$

$[ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,m}  ]_{\Ran_{ x,\on{un,dR} }}$ is laft and satisfies (the $\Ran_{ x,\on{un,dR} }$-version of) $(\spadesuit)$ in Construction <ref>.

Follows from the Cartesian diagram (<ref>), Remark <ref>, Remark <ref> and the fact that $ U \times X^I \to \mathbb{B}[\mathcal{L}^+ H]_I $ is $*$-pullable.

$[ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,m}  ]_{\Ran_{ x,\on{un,dR} }}\to [ \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,m}  ]_{\Ran_{ x,\on{un,dR} }}$ satisfies (the $\Ran_{ x,\on{un,dR} }$-version of) the assumptions in Construction <ref>(b).

Follows from the Cartesian diagram (<ref>), and the fact that $[U^{K\on{-level}_x}]_{\Ran_{ x,\on{un,dR} }}$ is QCA over $\Ran_{ \on{un,dR} }$.

$$  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \simeq \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{R}ep( \mathcal{L}^+ H\times_{H_x} K_x )),$$

where $\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } $ is the line bundle in $\mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{K\on{-level}_x})$ defined as the pullback of $\mathscr{L}_{ \widehat{\on{Hecke}}_H }$. Recall we also have

$$  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \simeq \lim \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta} \simeq \colim \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op}, $$

where the limit is taken in $[\on{CrysCat}(\on{Ran}_{x,\on{un,dR}})]_{\on{lax-unital}}$, while the colimit is taken in $\on{CrysCat}(\on{Ran}_{x,\on{un,dR}})$. Also these diagrams are “twisted” by $\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} }$ in a way similar to Remark <ref>.

For any quasi-compact open substack $U\subset \on{Bun}_H$, the Cartesian squares (<ref>) provide an $ [\mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{K\on{-level}_x}),\star]$-action on $ \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})$ (viewed as objects in $[\on{CrysCat}(\on{Ran}_{x,\on{un,dR}})]_{\on{lax-unital}}$). Consider[Since $\widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}$ is laft, we do not need to use the renormalized $\IndCoh$-theories.]

$$  \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ).$$

Similar to Remark <ref>, this crystal of categories can be written as a geometric realization or totalization as follows. Consider the functor

$$\mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta}: \Delta \to [\on{CrysCat}(\on{Ran}_{x,\on{un,dR}})]_{\on{lax-unital}} $$

that sends $[m]$ to $\mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,m}  )$, and sends a morphism $f:[m]\to [n]$ in $\Delta$ to the $!$-pullback functor twisted by the line bundle on $ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x}$ that is the pullback of $\mathscr{L}_{ \widehat{\on{Hecke}} }$ along $ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x} \to \widehat{\on{Hecke}}_{H}$. Passing to left adjoints, we also have a functor

$$\mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op}: \Delta^\op \to \on{CrysCat}(\on{Ran}_{x,\on{un,dR}}) $$

connected by twisted pushforward functors (which are simultaneously $!$-pushforward and $*$-pushforward functors). Then we have

\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[r] \ar[d]^-\simeq &
\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  )\ar[d]^-\simeq \\
\colim \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op} \ar[r]^-{!\on{-pull}}\ar[d]^-\simeq
\colim \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op}\ar[d]^-\simeq \\
\lim \mathcal{I}nd\mathcal{C}oh^{!,\ren}(   \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta} \ar[r]^-{!\on{-pull}}
\lim \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta},

where the limit is taken in $[\on{CrysCat}(\on{Ran}_{x,\on{un,dR}})]_{\on{lax-unital}}$, while the colimit is taken in $\on{CrysCat}(\on{Ran}_{x,\on{un,dR}})$. Note that the second horizontal morphism is well-defined because of the Cartesian diagram (<ref>). Since the horizontal morphisms are induced by $!$-pullback functors along co-unital maps, they are unital morphisms between crystals of categories.

Similarly, we have

\begin{equation}
\label{eq:coloc-construction}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}}  \ar[d]^-\simeq &
\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  )\ar[d]^-\simeq \ar[l] \\
\colim \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op} \ar[d]^-\simeq
\colim \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op}\ar[d]^-\simeq  \ar[l]_-{*\on{-push}}\\
\lim \mathcal{I}nd\mathcal{C}oh_{*,\ren}(   \widehat{\on{Hecke}}_{H}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta}
\lim \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta} \ar[l]_-{*\on{-push}}.
\end{equation}

By Lemma <ref>, we can pass to left adjoints along the horizontal direction and obtain a unital morphism

$$  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}} \to \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ).$$

We have a unital morphism

\begin{equation*}
\begin{aligned}
\on{ind}:\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  )\simeq  \colim \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op} \to\\
\to  \IndCoh( (U^{K\on{-level}_x})_\dR)_{\mathscr{T}}\boxtimes \mathbf{1} \simeq \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1}
\end{aligned}
\end{equation*}

induced by twisted pushforward functors. Here we used the fact that $\mathscr{L}$ and $\mathscr{T}$ are compatible (see Remark <ref>). Namely, by restriction along the augementation $ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}\to (U^{K\on{-level}_x})_\dR$, the twisting $\mathscr{T}$ induces a cosimplicial $\mathbb{G}_m$-gerbe on $ \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}$ equipped with a trivialization of its 0-term. Then the compatibility statement implies $ \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op} $ is the simplicial diagram twisted by this gerbe.

Passing to right adjoints, we obtain a lax-unital morphism

$$\on{oblv}: \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1} \to \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  )$$

induced by twisted $!$-pullback functors.

Recall we have $\DMod(\on{Bun}_H^{K\on{-level}_x })_{\mathscr{T}} \simeq \lim_{U} \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}$, where the connecting functors are given by pullbacks. Consider the following compositions of unital morphisms:

\begin{eqnarray*}
\on{Loc}_U: \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \xrightarrow{!\on{-pull}} \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \xrightarrow{\on{push}} \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1}  \\
\on{Loc}_{U,\on{co}}: \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}} \xrightarrow{*\on{-pull}} \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \xrightarrow{\on{push}} \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1}.
\end{eqnarray*}

They are functorial in $U$, i.e., compatible with the pullback functors $\DMod( U_1^{K\on{-level}_x} )\to \DMod( U_2^{K\on{-level}_x} )$. Passing to limits, we obtain unital morphisms

\begin{eqnarray*}
\on{Loc}: \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \to \DMod( \on{Bun}_H^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1}  \\
\on{Loc}_{\on{co}}: \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L},\on{co}} \to \DMod( \on{Bun}_H^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1}.
\end{eqnarray*}

§.§ Compatibility with Chiral Homology

Since $\on{Loc}$ is unital, we have a commutative square

\xymatrix{
[ \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ]_x  \ar[rr]^-{ [\on{Loc}]_{ x} }  \ar[d]^-{\on{ins}_{\on{unit}}}
& & \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}
\ar[d]^-{\on{ins}_{\on{unit}}}\\
[ \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ]_{\on{Ran}_{x,\on{un,dR}}}  \ar[rr]^-{ [\on{Loc}]_{ \on{Ran}_{x,\on{un,dR}}  } }
& & \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \otimes \DMod(\on{Ran}_{x,\on{un}})

Note that the right vertical functor sends $\mathcal{F}$ to $\mathcal{F}\boxtimes \omega_{ \on{Ran}_{x,\on{un}} }$. Recall $\on{Ran}_{x,\on{un}} $ is homologically contractible (see <cit.>). Hence $\on{C}^{\on{ch}}_{x,\on{un}}( \omega_{ \on{Ran}_{x,\on{un}} } ) \simeq \omega_{\on{pt}}$. This proves Theorem <ref>(1).

§.§ Compatibility with Tate Twist

Unwinding the definitions, we only need to construct the following commutative diagram for cosimplicial crystals of categories:

\xymatrix{
\mathcal{I}nd\mathcal{C}oh_{*,\ren}(\widehat{\on{Hecke}}_{H}^{\bullet}  )_{\mathscr{L}\otimes \mathscr{L}^{\on{Tate}},\Delta}
\ar[r]^-{*\on{-pull}}   &
\mathcal{I}nd\mathcal{C}oh(\widehat{\on{Hecke}}_{H,U}^{\bullet}  )_{\mathscr{L}\otimes \mathscr{L}^{\on{Tate}},\Delta}
\\
\mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{\bullet}  )_{\mathscr{L},\Delta}  \ar[r]^-{!\on{-pull}} \ar[u]^-\simeq_-{\theta_0}  &
\mathcal{I}nd\mathcal{C}oh(\widehat{\on{Hecke}}_{H,U}^{\bullet}  )_{\mathscr{L},\Delta}   \ar[u]^-\simeq_-{\vartheta_0}

such that

* The horizontal morphisms were defined in Construction <ref>.

* The left vertical equivalences were defined in Step 3 of the proof of Theorem <ref>. Namely, for each $[m]$, we use the unique $\mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{m}  )$-linear equivalence $\theta_0^{[m]}$ such that

$$\mathcal{I}nd\mathcal{C}oh(\on{pt})\xrightarrow{!\on{-pull}}  \mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{0}  ) \xrightarrow{\theta_0^{[m]}}  \mathcal{I}nd\mathcal{C}oh_{*,\ren}(\widehat{\on{Hecke}}_{H}^{m}  ) $$

is equivalent to

$$ \mathcal{I}nd\mathcal{C}oh(\on{pt})\xrightarrow{*\on{-pull}}  \mathcal{I}nd\mathcal{C}oh_{*,\ren}(\widehat{\on{Hecke}}_{H}^{0}  ) \xrightarrow{(p_0)_\ren^? }  \mathcal{I}nd\mathcal{C}oh_{*,\ren}(\widehat{\on{Hecke}}_{H}^{m}  ) .$$

* For each $[m]$, the right vertical equivalence $\vartheta_0^{[m]}$ is tensoring up with
$ p_0^!\circ \Upsilon ( K^{-1}  )$.

We only need to show $\vartheta_0^{[m]}$ is the unique $\mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{m}  )$-linear equivalence such that

\xymatrix{
\mathcal{I}nd\mathcal{C}oh_{*,\ren}(\widehat{\on{Hecke}}_{H}^{m}  )
\ar[r]^-{*\on{-pull}}   &
\mathcal{I}nd\mathcal{C}oh(\widehat{\on{Hecke}}_{H,U}^{m}  )
\\
\mathcal{I}nd\mathcal{C}oh^{!,\ren}(\widehat{\on{Hecke}}_{H}^{m}  )  \ar[r]^-{!\on{-pull}} \ar[u]^-\simeq_-{\theta_0}  &
\mathcal{I}nd\mathcal{C}oh(\widehat{\on{Hecke}}_{H,U}^{m}  )   \ar[u]^-\simeq_-{\vartheta_0^{[m]}}

commutes. Using the base-change isomorphisms, we only need to prove the case $m=0$, which then follows from the fact that $\vartheta_0^{[0]} =  \Upsilon ( K^{-1}  )\otimes -$ is the unique $\mathcal{I}nd\mathcal{C}oh(U )$-linear equivalence such that

$$\mathcal{I}nd\mathcal{C}oh(\on{pt})\xrightarrow{!\on{-pull}}  \mathcal{I}nd\mathcal{C}oh(U   ) \xrightarrow{\vartheta_0^{[0]}} \mathcal{I}nd\mathcal{C}oh(U  )$$

is equivalent to the $*$-pullback functor.

§.§ Compatibility with Propogation

Our strategy for the proof of Theorem <ref>(3-4) is to reduce to the case when the factorization algebra $A$ (resp. $A'$) is the unit factorization algebra. To achieve this reduction, we need to use the tool of propogation of factorization modules, and prove $\on{Loc}$ and $\on{Loc}_{\on{co}}$ are compatible with propogations. We first explain what these propogations mean.

The propagation method was initially introduced by Beilinson and Drinfeld in <cit.>. We emphasize that this technique only exists in the unital setting.

Consider the laft lax prestack

\[\on{Arrow}( \on{Ran}_{x,\on{un}} ) := S \mapsto \tx{Fun}([2], \Ran_{x, \un}(S)).\]

We have two projections

$$p_s,p_t: \on{Arrow}( \on{Ran}_{x,\on{un}} )\to \on{Ran}_{x,\on{un}} $$

“remembering” respectively the source and the target. Note that we have a $2$-morphism $\mathfrak{i}:p_s\to p_t$. We equip $\on{Arrow}( \on{Ran}_{x,\on{un}} )$ with the factorization module space structure (for the factorization space $\on{Ran}_{\on{un}}$) such that $p_t$ is compatible with the factorization module structures but $p_s$ is not. For a map $y:\on{pt}\to \on{Ran}_{\on{un}}$ corresponding to a finite subset $J$ with $I_t\cap J = \emptyset$, the map

$$(\on{Ran}_{\on{un}} \times \on{Arrow}( \on{Ran}_{x,\on{un}} ))_{\on{disj}} \to \on{Arrow}( \on{Ran}_{x,\on{un}} )$$

sends $(y,z)$ to the point corresponding to $I_0\subset I_s \subset J\sqcup I_t$.

Explicitly, a map $z:\on{pt}\to \on{Arrow}( \on{Ran}_{x,\on{un}} )$ corresponds to a chain $I_0\subset I_s\subset I_t$ of finite subsets of $\on{Maps}(\on{pt},X)$ such that $I_0\subset I_s$ (resp. $I_0\subset I_t$) corresponds to the composition $\on{pt}\to \on{Arrow}( \on{Ran}_{x,\on{un}} ) \xrightarrow{p_s} \on{Ran}_{x,\on{un}}$ (resp. $\on{pt}\to \on{Arrow}( \on{Ran}_{x,\on{un}} ) \xrightarrow{p_t} \on{Ran}_{x,\on{un}}$).

[Propogation of factorization modules]

Let $V\to W$ be a morphism between laft prestacks. For any unital factorization category $\mathcal{A}$ over $W\times \Ran_{\on{un,dR}}$ (or equivalently, a $W$-family of unital factorization categories over $\Ran_{\on{un,dR}}$), and for any factorization $\mathcal{A}$-module $\mathcal{C}$ over $V\times \on{Ran}_{x,\on{un,dR}}$, the (co)restriction $\mathbf{cores}_{p_{t,\dR}}(\mathcal{C})$, which is a crystal of categories over $V\times  \on{Arrow}( \on{Ran}_{x,\on{un}} )_\dR$, inherits a factorization $\mathcal{A}$-module structure. Moreover, for any unital factorization algebra $A$ in $\mathcal{A}$, there is a canonical functor

$$ \on{prop}_A: A\on{-FactMod}_{\on{un}}( \mathcal{C} ) \to A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) $$

such that the following diagram commutes

\xymatrix{
A\on{-FactMod}_{\on{un}}( \mathcal{C} ) \ar[d]^-{\on{prop}_A} \ar[r]^-{\oblv}
& \mathbf{\Gamma}( \on{Id}_\dR, \mathcal{C} ) \ar[d]^-{\on{res}} \\
A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) \ar[r]^-{\oblv}
\mathbf{\Gamma}( p_{t,\dR}, \mathcal{C} ),

where the bottom functor is the restriction functor for sections of $\mathcal{C}$. More generally, for any morphism $f: A \to B$ between unital factorization algebras, we can use restriction of factorization modules (as defined in Proposition <ref>) to define the composition

$$ \on{prop}_{A}: B\on{-FactMod}_{\on{un}}( \mathcal{C} ) \xrightarrow{\on{prop}_B} B\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) \xrightarrow{\on{res}_f}  A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) $$

and have a natural transformation

\begin{equation} \label{eqn-nt-prop-res-oblv}
\xymatrix{
B\on{-FactMod}_{\on{un}}( \mathcal{C} ) \ar[d]^-{\on{prop}_{A}} \ar[r]^-{\oblv}
& \mathbf{\Gamma}( \on{Id}_\dR, \mathcal{C} ) \ar[d]^-{\on{res}} \\
A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) )\ar@{=>}[ru] \ar[r]^-{\oblv}
\mathbf{\Gamma}( p_{t,\dR}, \mathcal{C} ),
\end{equation}

In the special case when $A=\on{unit}$ is the unit factorization algebra of $\mathcal{A}$, we have the following commutative diagram

\xymatrix{
B\on{-FactMod}_{\on{un}}( \mathcal{C} ) \ar[d]^-{\on{prop}_{\on{unit}}} \ar[r]^-{\oblv}
& \mathbf{\Gamma}( \on{Id}_\dR, \mathcal{C} ) \ar[d]^-{\on{ins}_{\on{unit}}}\\
\on{unit}\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) \ar[r]^-{\oblv}
\mathbf{\Gamma}( p_{t,\dR}, \mathcal{C} ),

where $\on{ins}_{\on{unit}}$ is the functor of unit insertion

$$ \on{ins}_{\on{unit}}: \mathbf{\Gamma}( \on{Id}_\dR, \mathcal{C} )\xrightarrow{\on{res}}\mathbf{\Gamma}( p_{s,\dR}, \mathcal{C} ) \xrightarrow{\mathfrak{i}_!}  \mathbf{\Gamma}( p_{t,\dR}, \mathcal{C} ).$$

Digression: what is propagation?
Let us explain the name. Suppose $A$ is a unital factorization algebra and $N$ is its, such that its fiber at a $\mb{C}$-point

\[y := [\{x\} \subseteq \{x, x_1, \ldots, x_n\}] \in \on{Arrow}( \on{Ran}_{x,\on{un}} )(\mb{C})\]

is given by some vector space $N_0 \tensor A_0^{\tensor (n - 1)}$. The map $y$ induces a map

\[\left(\Ran_{\un, y} := S \mapsto \{\alpha: S \to \Ran_{\un} \mid (x, x_1, \ldots, x_n) \subseteq \alpha\}\right) \to \on{Arrow}( \on{Ran}_{x,\on{un}} )\]

Then the pullback of $\tx{prop}_A(N)$ along this map produces a unital $A$-factorization module on the former, which should be thought of as consisting of $(N_0, A_0, \ldots, A_0)$ supported at $(x, x_1, \ldots, x_n)$ respectively. Thus, it can be seen as “propagating” the algebra $A$ to other points on the curve.

Another perspective is as follows. As explained in <cit.>, via étale hyperdescent the category of $A$-unital factorization modules in $C$ itself form a weak unital factorization category under chiral fusion, and the underlying sheaf of $\tx{prop}_A(N)$ can be seen as observing $A$ as an element $A^\tx{enh} \in \tx{FactAlg}_{\un}(A\tx{-FactMod}_\un(\mc{C}))$ and $N$ as an element $N^\tx{enh} \in A^\tx{enh}\tx{-FactMod}_\un(A\tx{-FactMod}_\un(\mc{C}))$.
This is, of course, evocative of the “enhancement” equivalences

\[\tx{AlgEnh}: \mc{O}\tx{-Alg}(\mc{C})_{A /} \simeq \mc{O}\tx{-Alg}(A\mod^{\mc{O}}(\mc{C}))\]

\[\tx{ModEnh}: A\mod^{\mc{O}}(\mc{C}) \simeq \tx{AlgEnh}(A)\mod^{\mc{O}}(A\mod^{\mc{O}}(\mc{C}))\]

which holds for any coherent $\infty$-operad $\mc{O}$, any (weakly) $\mc{O}$-monoidal category $\mc{C}$ and any $\mc{O}$-algebra $A \in \mc{C}$, as established in <cit.> and <cit.>.

In the special case when $\mathcal{A}$ and $\mathcal{C}$ are the constant crystals, i.e., $\mathcal{A} = \IndCoh(W)\boxtimes \mathbf{1}$ and $\mathcal{C}=\IndCoh(V)\boxtimes \mathbf{1}$, the functor

$$ \on{res}: \mathbf{\Gamma}( \on{Id}_\dR, \mathcal{C} )  \to \mathbf{\Gamma}( \on{p}_{t,\dR}, \mathcal{C} ) $$

is just

$$p_t^!: \IndCoh(V)\otimes \DMod( \Ran_{x,\on{un}} ) \to \IndCoh(V)\otimes \DMod(\on{Arrow}( \Ran_{x,\on{un}}) ).$$

Hence we obtain a natural transformation

\begin{equation} \label{eqn-prop-A'-nt}
\xymatrix{
B\on{-FactMod}_{\on{un}}( \IndCoh(V) \boxtimes \mathbf{1} ) \ar[d]^-{\on{prop}_{A}} \ar[r]^-{\oblv}
& \IndCoh(V)\otimes \DMod( \Ran_{x,\on{un}} )  \\
A\on{-FactMod}_{\on{un}}( \IndCoh(V) \boxtimes\mathbf{cores}_{p_{t,\dR}}(\mathbf{1}) )\ar@{=>}[ru] \ar[r]^-{\oblv}
\IndCoh(V)\otimes \DMod(\on{Arrow}( \Ran_{x,\on{un}}) ) \ar[u]^-{p_{t,!}},
\end{equation}

The diagram (<ref>) does not commute in general. But we have the following results:

The diagram (<ref>) commutes after taking chiral homology, i.e., after composing with

$$\on{C}_{x,\on{un}}^{\on{ch}}: \IndCoh(V)\otimes \DMod( \Ran_{x,\on{un}} ) \to \IndCoh(V).$$

Recall $\on{C}_{x,\on{un}}^{\on{ch}}$ is given by $!$-pushforward along $p:  \Ran_{x,\on{un}}\to \on{pt}$. In the case when $B = A$, the following stronger result is true:

The composition

$$ p_! \circ p_{s,!}\circ p_t^!  \simeq p_! \circ p_{t,!}\circ p_t^! \to p_!  $$

in $\on{Funct}( \DMod( \Ran_{x,\on{un}} ) ) \to \on{Vect}$ is an equivalence. Moreover, the natural transformation

$$ p_{s,!}\circ p_t^! \to p^!\circ p_! $$

is an equivalence.

Before we prove, let us explain the names.
For a set of unital $A$-factorization modules $N_1, \ldots, N_k$ inserted at distinct points $x_1, \ldots, x_k \in X(\mb{C})$, let us write the chiral homology as

\[\langle N_1, \ldots, N_k \rangle^{A}_{x_1, \ldots, x_k};\]

Given a unital factorization $A$-module $M$ supported at $x$, the equality between fibers of $p_{s,!}\circ p_t^!(M)$ and $p^!\circ p_!(M)$ at some $(x, x_1, \ldots, x_n)$, given by the propagation lemma for chiral homology, translates to

\[\langle M, A, \ldots, A \rangle^{A}_{x, x_1, \ldots, x_n} \simeq \langle M \rangle^A_{x},\]

which says that when computing $A$-chiral homology, extra copies of $A$ would not affect the result. The propagation-restriction lemma, on the other hand, roughly states that if $M$ is a $B$-factorization module and $A \to B$ is a map between unital factorization algebras, then we have

\[\left\langle\langle M \rangle_x^A, \langle M, B \rangle_{x, x_1}^A, \langle M, B, B \rangle_{x, x_1, x_2}^A, \ldots \right\rangle \simeq \langle M \rangle^B_x\]

where the LHS now ranges over all of $\Ran_{x, \un}$.

Consider the pullback diagram

\[\xymatrix{
\Ran_{x, \un} \times_{\{x\}} \Ran_{x, \un} \ar^{q_2}[r] \ar^{q_1}[d] & \Ran_{x, \un} \ar^{p}[d] \\
\Ran_{x, \un} \ar^{p}[r] & \{x\}

We claim that $(q_1)_!$ is well-defined, and the natural transformation $(q_1)_! \circ (q_2)^! \Rightarrow p^! \circ p_!$ is an isomorphism. Indeed, this can be checked by by first establishing the corresponding facts for the following diagram

\[\xymatrix{
\Ran_{x, \un} \times_{R} \Ran_{x} \ar[r] \ar[d] & \Ran_{x} \ar[d] \\
\Ran_{x, \un} \ar[r] & \{x\}

using the fact that $\Ran_x$ is pseudo-proper, then use the universal homological left cofinality of $\Ran_{x} \to \Ran_{x, \un}$, provable as in <cit.>. There is a map of lax prestacks

\[\iota: \tx{Arrow}(\Ran_{x, \un}) \to \Ran_{x, \un} \times_{\{x\}} \Ran_{x, \un} \hspace{1em} (I_0 \subseteq I_s \subseteq I_t) \mapsto ((I_0 \subseteq I_s), (I_0 \subseteq I_t))\]

\[\tx{such that} \hspace{1em} q_2 \circ \iota \simeq p_t \hspace{1em} \tx{and} \hspace{1em} q_1 \circ \iota \simeq p_s.\]

It suffices to establish that $\iota$ is universally homologically left cofinal. As in <cit.>, it suffices to check that for every affine $S$, every $S$-point

\[((I_0 \subseteq I_1), (I_0 \subseteq I_2)) \in \Ran_{x, \un} \times_{\{x\}} \Ran_{x, \un}(S),\]

and every $t: T \to S$ where $T$ is affine, the category

\[\{J \in \Ran(T), K \in \Ran(T), \alpha: I_0|_t \subseteq J \subseteq K~\tx{such that}~I_1|_t \subseteq J, I_2|_t \subseteq K\}\]

is a contractible category; but this category has an initial element given by $(I_1|_t, I_1|_t \cup I_2|_t)$.

We first prove the special case $A \simeq \tx{unit}_\mc{A}$, for $\eta_B: \tx{unit}_\mc{A} \to B$ the canonical map attached to any factorization algebra $B$. Let $\iota_s: \Ran_{x, \un} \to \tx{Arrow}(\Ran_{x, \un})$ be the inclusion map $(I_0 \subseteq I_1) \mapsto (I_0 \subseteq I_1 \subseteq I_1)$. It follows from Proposition <ref> and Proposition <ref> that

\[\tx{res}_{\eta_B}: B\tx{-FactMod}_\un(\mc{C}) \to \tx{unit}_\mc{A}\tx{-FactMod}_\un(\mc{C})\]

is isomorphic to the functor on $\mc{C}$ induced by

\[p_s^! \circ \iota_{s}^!: \DMod(\tx{Arrow}(\Ran_{x, \un})) \to \DMod(\tx{Arrow}(\Ran_{x, \un})),\]

So it suffices to establish $p_! \circ (p_s)_! \circ p_s^! \circ \iota_s^! \circ p_t^! \simeq p_! \circ (p_s)_! \circ p_s^! \simeq p_!$, which follows from the homological contractibility of $p_s$.

Each $f: A \to B$ comes with a natural map $\eta_B \to f$, which by functoriality of $\tx{res}_f$ induces a natural transformation

\[\alpha: (\tx{id} \times p_!) \circ \tx{prop}_{\tx{unit}_\mc{A}} \Rightarrow (\tx{id} \times p_!) \circ \tx{prop}_{A};\]

unfolding the construction of $\tx{res}_f$ we see that $\alpha$ is induced by the following natural transformation of functors $\DMod(\tx{Arrow}(\Ran_{x, \un})) \to \Vect$:

\[\alpha_{\tx{univ}}: p_! \circ (p_s)_! \circ (i \circ p_s)^! \Rightarrow p_! \circ (p_s)_!\]

which is in turn induced the natural transformation $i \circ p_s \Rightarrow \tx{id}$ (here $i: \Ran_{x, \un} \to \tx{Arrow}(\Ran_{x, \un})$ is the map $I_0 \subseteq I_1 \mapsto I_0 \subseteq I_1 \subseteq I_1$). Thus it suffices to show that $\alpha_{\tx{univ}}$ is an equivalence. We get this by observing that $i \circ p_s$ is universally homologically left cofinal, which follows from the same argument as in the previous lemma.

One can also define $2\on{Chain}( \on{Ran}_{x,\on{un}} )$ classifying $I_0\subset I_1\subset I_2 \subset I_3$ and propogation functor

$$\on{prop}_A: A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) \to A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{3,\dR}}(\mathcal{C}) ),$$

where $p_3: 2\on{Chain}( \on{Ran}_{x,\on{un}} )\to  \on{Ran}_{x,\on{un}}$ “remembers” $I_0\subset I_3$. One can similarly prove variants of the lemmas.

We apply Construction <ref> to the factorization category $\mathcal{A}:=\mathcal{KL}(H)_{\mathscr{L}}$ and its factorization module category $\mathcal{C}:=  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}$. Using the fact that $\on{Loc}$ is a morphism between crystals of categories, the natural transformation (<ref>) induces a natural transformation

\begin{equation} \label{eqn-nt-loc-prop}
\xymatrix{
B\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} ) \ar[d]^-{\on{prop}_{A}} \ar[r]^-{\on{Loc}}
& \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \otimes \DMod( \on{Ran}_{x,\on{un,dR}} )  \\
A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} ) )\ar@{=>}[ru]\ar[r]^-{\on{Loc}}
\DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \otimes \DMod(\on{Arrow}(\on{Ran}_{x,\on{un,dR}}) ) \ar[u]^-{p_{t,!}}.
\end{equation}

The following is the main result of this subsection.

The diagram (<ref>) commutes after taking chiral homology, i.e., after composing with

$$ \on{C}_{x,\on{un}}^{\on{ch}}: \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \otimes \DMod( \on{Ran}_{x,\on{un,dR}} ) \to  \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}.$$

The similar claim for $\on{Loc}_{\on{co}}$ is also true.

The lemma does not follow immediately from Propogation-restriction Lemma for Chiral homology (see Lemma <ref>) because $\on{Loc}$ has no factorization structure. See Warning <ref>.

The rest of this subsection is devoted to the proof of the lemma. We only prove the claim for $\on{Loc}$ because the proof for $\on{Loc}_{\on{co}}$ is similar. Our strategy is to reduce to problem to a similar claim about another morphism

$$ \on{Loc}^+: \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } \to  \DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H}\boxtimes \mathbf{1}$$

which is equipped with factorization structures such that we can deduce the claim from Propogation-restriction Lemma for Chiral homology (see Lemma <ref>). We first explain the construction of $\on{Loc}^+$.

By assumption, the twisting $\mathscr{T}$ on $\on{Bun}_H ^{K\on{-level}_x}$ is the restriction of a twisting on $\on{Bun}_H$. Hence we have an adjoint pair

$$\on{ind}: \DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H} \adjoint \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} : \on{oblv}, $$

$$\DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H} \simeq \IndCoh( \on{Bun}_H\times_{ \mathbb{B}H_{x,\dR} }  \mathbb{B}K_{x,\dR} ) $$

is the category of relative D-modules.

Let $U$ be a quasi-compact open substack of $\on{Bun}_H$. Consider the Cech nerve $U^{K\on{-level}_x,\bullet}$ of the projection

U^{K\on{-level}_x} = U\times_{ \mathbb{B}H_{x} }  \mathbb{B}K_{x} \to
U\times_{ \mathbb{B}H_{x,\dR} }  \mathbb{B}K_{x,\dR}. $$

It is easy to see the colimit of this simplicial laft prestack is just $U\times_{ \mathbb{B}H_{x,\dR} }  \mathbb{B}K_{x,\dR}$. Hence

$$ \DMod(U ^{K\on{-level}_x})_{/U}  \simeq \lim_{!\on{-pull}} \IndCoh( U^{K\on{-level}_x,\bullet} ) \simeq \colim_{!\on{-push}} \IndCoh( U^{K\on{-level}_x,\bullet} ) $$

We have a simplicial map $ U^{K\on{-level}_x,\bullet} \times \Ran_{x,\on{un,dR}} \to \widehat{ \on{Hecke} }_{H,U}^{K\on{-level}_x,\bullet} $. Since the gerbe in Remark <ref> is equipped with a trivialization on $U\times_{ \mathbb{B}H_{x,\dR} }  \mathbb{B}K_{x,\dR}$, the twisted $!$-pullback functors induce a cosimplicial lax-unital morphism

$$  \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta} \to  \IndCoh( U^{K\on{-level}_x,\bullet}) \boxtimes \mathbf{1},$$

and the twisted $!$-pushforward functors induce a simplicial unital morphism

$$\IndCoh( U^{K\on{-level}_x,\bullet}) \boxtimes \mathbf{1}\to  \mathcal{I}nd\mathcal{C}oh(   \widehat{\on{Hecke}}_{H,U}^{K\on{-level}_x,\bullet}  )_{\mathscr{L},\Delta^\op}.$$

Passing to limits and colimits, we obtain an adjoint pair

$$\on{ind}:  \DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}\adjoint \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) :\on{oblv} . $$

By construction, the composition

$$  \DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}\adjoint \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \adjoint  \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1} $$

is just the the adjoint pair between categories of relative D-modules and (twisted) absolute D-modules.

Similar to Construction <ref>, we can construct a unital morphism

\begin{equation} \label{eqn-loc-relative-crystal}
\on{Loc}_U^+: \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } \to  \DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}
\end{equation}

by writting both sides as simplicial colimits (or simplicial limits) and using $!$-pullback functors to connect them. Moreover, when $x$ corresponds to the empty set, the unital morphism

\begin{equation} \label{eqn-loc-relative-crystal-alg}
\on{Loc}_U^+: \mathcal{R}ep(\mathcal L^+ H) \to  \IndCoh(U )\boxtimes \mathbf{1}
\end{equation}

can be upgraded to a unital factorization functor, where the LHS is a factorization category over $\on{Ran}_{\on{un,dR}}$, while the RHS is over $U\times \on{Ran}_{\on{un,dR}}$ (or equivalently, a $U$-family of factorization categories). Also, the morphism (<ref>) can be upgraded to a morphism between factorization module categories intertwining (<ref>), where the RHS is viewed as a factorization module category over $ (U ^{K\on{-level}_x})_\dR \times_{U_\dR} U \times \Ran_{x,\on{un,dR}}$.

It follows we have a commutative diagram

\begin{equation} \label{eqn-loc-full-sph}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[r] \ar[d]^-{\on{oblv}} &
\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \ar[d]^-{\on{oblv}} \\
\mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } \ar[r]^-{\on{Loc}_U^+} &
\DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}.
\end{equation}

Moreover, using the base-change isomorphisms, one can show that this square is left adjointable along the vertical direction, i.e., it induces a commutative diagram

\begin{equation} \label{eqn-loc-full-sph-2}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[r]  &
\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \\
\mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } \ar[r]^-{\on{Loc}_U^+}\ar[u]^-{\on{ind}} &
\DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}\ar[u]^-{\on{ind}},
\end{equation}

which then induces a commutative diagram

\begin{equation} \label{eqn-loc-full-sph-3}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[r]^-{\on{Loc}_U}  &
\DMod(U ^{K\on{-level}_x})\boxtimes \mathbf{1} \\
\mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } \ar[r]^-{\on{Loc}_U^+}\ar[u]^-{\on{ind}} &
\DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}\ar[u]^-{\on{ind}}.
\end{equation}

Passing to right adjoints along vertical direction (and allowing $U$ to change), we obtain a 2-morphism

\begin{equation} \label{eqn-natural-transformation-loc-oblv}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[r]^-{\on{Loc}} \ar[d]^-{\on{oblv}} &
\DMod( \on{Bun}_H^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1}\ar[d]^-{\on{oblv}} \\
\mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } \ar[r]^-{\on{Loc}^+} \ar@{=>}[ru]&
\DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H}\boxtimes \mathbf{1}.
\end{equation}

This 2-morphism is not invertible in general. However, we are going to prove the following result and use it to deduce Lemma <ref>.

Up to chiral homology, the digram (<ref>) commutes for factorization objects. More precisly, for any unital factorization algebra $A$ in $ \mathcal{KL}(H)_{\mathscr{L}}$, the natural transformation

\begin{equation} \label{eqn-lem-loc-compatible-oblv}
\xymatrix{
A\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} ) \ar[d]^-{\on{oblv}}
\ar[r]^-{ \on{Loc} }
& \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}} \otimes \DMod( \on{Ran}_{x,\on{un,dR}}) \ar[d]^-{\on{oblv}} \\
\on{oblv}(A)\on{-FactMod}_{\on{un}}( \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }  )
\ar[r]^-{\on{Loc}^+ } \ar@{=>}[ur]
& \DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H}  \otimes \DMod( \on{Ran}_{x,\on{un,dR}}) ,
\end{equation}

is invertible after composing with

$$ \on{C}^{\on{ch}}_{x,\on{un}}: \DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H}  \otimes \DMod( \on{Ran}_{x,\on{un,dR}}) \to \DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H}.$$

The similar claim for $\on{Loc}_{\on{co}}$ is also true.

We also need the following variant, which can be proved similarly[One needs to use the construction in Remark <ref>].

Lemma <ref> remains valid if we

* replace $\on{Ran}_{x,\on{un,dR}}$ by $\on{Arrow}(\on{Ran}_{x,\on{un,dR}})$;

* replace $\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}$ and $\mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }  $ by their (co)restriction along $p_{t,\dR}$;

* replcae the chiral homology functor $\on{C}^{\on{ch}}_{x,\on{un}}$ by $p_{s,!}$.

We first deduce Lemma <ref> from Lemma <ref>.

Proof of Lemma <ref>: Note that we have an obvious commutative diagram

\xymatrix{
B\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} ) \ar[d]^-{\on{prop}_{A}} \ar[r]^-{\on{oblv}}
& \on{oblv}(B)\on{-FactMod}_{\on{un}}( \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x } ) \ar[d]^-{\on{prop}_{\on{oblv}(A)}}  \\
A\on{-FactMod}_{\on{un}}( \mathbf{cores}_{p_{t,\dR}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} ) ) \ar[r]^-{\on{oblv}}
\on{oblv}(A)\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(  \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }) ) .

Hence by Lemma <ref> and Variant <ref>, we can reduce the problem to a similar claim about $\on{Loc}^+$. But then it follows immediately from Propogation-restriction Lemma for Chiral homology (see Lemma <ref>) because $\on{Loc}^+$ has a factorization structure.

The rest of this subsection is devoted to the proof of Lemma <ref>. The strategy is to use Propogation-restriction Lemma for Chiral homology (see Lemma <ref>) again to reduce to the case when $A$ is the unit factorization algebra.

To avoid overboxes, we use the following shorthands:

\begin{equation*}
\begin{aligned}
\mathcal{A}^+:= \mathcal{R}ep(\mathcal{L}^+H),\;\mathcal{A}:=\mathcal{KL}(H)_{\mathscr{L}},\;
\mathcal{C}^+:= \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x },\; \mathcal{C}:=  \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} ,\\ D^+:= \DMod(U ^{K\on{-level}_x})_{/U} ,\; \mathcal{D}:=\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ),\; D':=  \DMod(U ^{K\on{-level}_x})_{\mathscr{T}}.
\end{aligned}
\end{equation*}

We write $\mathbb{V}$ for the unit factorization algebra of $\mathcal{A}$.

Consider the unital morphism $\on{Loc}_U:\mathcal{C} \to \mathcal{D} \xrightarrow{\on{ind}}  D'\boxtimes \mathbf{1}$. By Construction <ref>, it induces the following commutative diagram

\xymatrix{
A\on{-FactMod}_{\on{un}}(\mathcal{C}) \ar[d]^-{\on{prop}_{\mathbb{V}}}
\ar[rr]^-{[\on{Loc}_U]_{\on{Id}_{\dR}}} & &
D'\otimes \DMod( \on{Ran}_{x,\on{un}} ) \ar[d]^-{\on{ins}_{\on{unit}}\simeq p_s^!}
\\	\mathbb{V}\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) \ar[rr]^-{[\on{Loc}_U]_{p_{t,\dR}}} & &
D'\otimes \DMod( \on{Arrow}_{ \on{Ran}_{x,\on{un}}} ).

It is easy to see $p_s$ is homologically contractible, i.e., $p_s^!$ is fully faithful. Hence we have a commutative diagram

\begin{equation} \label{eqn-prop-unit-loc-chiral-homology}
\xymatrix{
A\on{-FactMod}_{\on{un}}(\mathcal{C}) \ar[d]^-{\on{prop}_{\mathbb{V}}}
\ar[rr]^-{[\on{Loc}_U]_{\on{Id}_{\dR}}} & &
D'\otimes \DMod( \on{Ran}_{x,\on{un}} )
\\	\mathbb{V}\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ) \ar[rr]^-{[\on{Loc}_U]_{p_{t,\dR}}} & &
D'\otimes \DMod( \on{Arrow}_{ \on{Ran}_{x,\on{un}}} ) \ar[u]^-{p_{s,!}}.
\end{equation}

On the other hand, the factorization structures on $\on{Loc}_{U}^+: \mathcal{C}^+ \to D^+\boxtimes \mathbf{1}$ induces a commutative diagram

\xymatrix{
\on{oblv}(A)\on{-FactMod}_{\on{un}}(\mathcal{C}^+) \ar[d]^-{\on{prop}_{\oblv(\mathbb{V})}}
\ar[rr]^-{[\on{Loc}^+_U]_{\on{Id}_{\dR}}} & &
\on{Loc}^+_U\circ \on{oblv}(A)\on{-FactMod}_{\on{un}}( D^+\boxtimes \mathbf{1} )  \ar[d]^-{\on{prop}_{\on{Loc}^+_U\circ \on{oblv}(\mathbb{V})}}
\\	\oblv(\mathbb{V})\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}^+) ) \ar[rr]^-{[\on{Loc}^+_U]_{p_{t,\dR}}} & &
\on{Loc}^+_U\circ \on{oblv}(\mathbb{V})\on{-FactMod}_{\on{un}}( D^+\boxtimes \mathbf{cores}_{p_{t,\dR}}(\mathbf{1}) ).

By Propogation-restriction Lemma for Chiral homology (see Lemma <ref>), the natural transformation

\xymatrix{
\on{Loc}^+_U\circ \on{oblv}(A)\on{-FactMod}_{\on{un}}( D^+\boxtimes \mathbf{1} )  \ar[d]^-{\on{prop}_{\on{Loc}^+_U\circ \on{oblv}(\mathbb{V})}} \ar[r]^-{\on{oblv}}
& D^+ \otimes \DMod( \Ran_{x,\on{un}})
\\
\on{Loc}^+_U\circ \on{oblv}(\mathbb{V})\on{-FactMod}_{\on{un}}( D^+\boxtimes \mathbf{cores}_{p_{t,\dR}}(\mathbf{1}) ) \ar[r]^-{\on{oblv}} \ar@{=>}[ru]
& D^+ \otimes \DMod( \on{Arrow}(\Ran_{x,\on{un}}))  \ar[u]^-{p_{t,!}}.

becomes invertible after taking chiral homology. Hence

\xymatrix{
\on{oblv}(A)\on{-FactMod}_{\on{un}}(\mathcal{C}^+) \ar[d]^-{\on{prop}_{\oblv(\mathbb{V})}}
\ar[rr]^-{[\on{Loc}^+_U]_{\on{Id}_{\dR}}} & &
D^+ \otimes \DMod( \Ran_{x,\on{un}})
\\	\oblv(\mathbb{V})\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}^+) ) \ar[rr]^-{[\on{Loc}^+_U]_{p_{t,\dR}}}  \ar@{=>}[rru] & &
D^+ \otimes \DMod( \on{Arrow}(\Ran_{x,\on{un}})) \ar[u]^-{p_{t,!}}

becomes invertible after taking chiral homology.

Finally, similar to (<ref>), we have a natural transformation

\begin{equation} \label{eqn-lem-loc-compatible-oblv-prop}
\xymatrix{
\mathbb{V}\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) )\ar[d]^-{\on{oblv}}
\ar[rr]^-{p_{s,!}\circ [\on{Loc}_U]_{p_{t,\dR}} }
& & D'\otimes \DMod( \on{Ran}_{x,\on{un}} )   \ar[d]^-{\on{oblv}} \\
\oblv(\mathbb{V})\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}^+) )
\ar[rr]^-{p_{s,!}\circ [\on{Loc}^+_U]_{p_{t,\dR}} }  \ar@{=>}[urr]
& & D^+\otimes \DMod( \on{Ran}_{x,\on{un}} ).
\end{equation}

Proof of Lemma <ref>:
Combining the above three constructions, we have reduced Lemma <ref> to the following result.

The natural transformation (<ref>) is invertible.

$$ \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}) \xrightarrow{\on{ins}_{\on{unit}}} \mathbb{V}\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) )$$

is an equivalence. Also recall the image of

$$ \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}^+) \xrightarrow{\on{ind}} \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}) $$

generates the target category because its right adjoint is conservative. Hence we only need to show (<ref>) is invertible after precomposing with

$$ \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}^+) \xrightarrow{\on{ind}} \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}) \xrightarrow{\on{ins}_{\on{unit}}} \mathbb{V}\on{-FactMod}_{\on{un}}(\mathbf{cores}_{p_{t,\dR}}(\mathcal{C}) ). $$

In other words, we only need to show

\xymatrix{
\mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}) \ar[d]^-{\on{oblv}}
\ar[rr]^-{p_{s,!}\circ [\on{Loc}_U]_{p_{t,\dR}}}
& & D'\otimes \DMod( \on{Ran}_{x,\on{un}} )   \ar[d]^-{\on{oblv}} \\
\mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}^+)
\ar[rr]_-{p_{s,!}\circ [\on{Loc}^+_U]_{p_{t,\dR}}} \ar@{=>}[urr]
& & D^+\otimes \DMod( \on{Ran}_{x,\on{un}} ) ,

is invertible after precomposing with

$$ \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}^+) \xrightarrow{\on{ind}} \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}) \xrightarrow{\on{ins}_{\on{unit}}} \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}).$$

Since $\on{ind}:\mathcal{C}^+ \to \mathcal{C} $ is unital, the above functor is equivalent to

$$\mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}^+) \xrightarrow{\on{ins}_{\on{unit}}} \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}^+) \xrightarrow{\on{ind}} \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C})

Note that we have a commutative diagram:

\xymatrix{
\mathbf{\Gamma}(\on{Id}_\dR,\mathcal{C}^+)
\ar[r]^-{\on{ins}_{\on{unit}}}  \ar[d]^-{ \on{Loc}^+ }
& \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}^+)\ar[r]^-{\on{ind}}  \ar[d]^-{ \on{Loc}^+ }
& \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}) \ar[r]^-{\on{oblv}}  \ar[d]^-{ \on{Loc} }
&  \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{C}^+) \ar[d]^-{ \on{Loc}^+ } \\
\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1}) \ar[r]^-{\on{ins}_{\on{unit}}}
& \mathbf{\Gamma}(\on{p}_{t,\dR},D^+\boxtimes \mathbf{1})  \ar[r]^-{\on{ind}}
& \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{D})  \ar[r]^-{\on{oblv}}
&  \mathbf{\Gamma}(\on{p}_{t,\dR},D^+\boxtimes \mathbf{1})  ,

where the left square is because $\on{Loc}^+:\mathcal{C}^+\to D^+\boxtimes \mathbf{1} $ is unital, and the right two squares follow from (<ref>) and (<ref>) (by taking $\mathbf{\Gamma}(\on{p}_{t,\dR},-)$). Hence we only need to show the natural transformation from

\begin{equation*}
\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1}) \xrightarrow{\on{ins}_{\on{unit}}}
\mathbf{\Gamma}(\on{p}_{t,\dR},D^+\boxtimes \mathbf{1})  \xrightarrow{\on{ind}}  \mathbf{\Gamma}(\on{p}_{t,\dR},D'\boxtimes \mathbf{1})  \xrightarrow{\on{oblv}}
\mathbf{\Gamma}(\on{p}_{t,\dR},D^+\boxtimes \mathbf{1}) \xrightarrow{p_{s,!}} \mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1})
\end{equation*}

$$ 	\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1}) \xrightarrow{\on{ins}_{\on{unit}}}
\mathbf{\Gamma}(\on{p}_{t,\dR},D^+\boxtimes \mathbf{1})  \xrightarrow{\on{ind}}  \mathbf{\Gamma}(\on{p}_{t,\dR},\mathcal{D})  \xrightarrow{\on{oblv}}
\mathbf{\Gamma}(\on{p}_{t,\dR},D^+\boxtimes \mathbf{1}) \xrightarrow{p_{s,!}} \mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1}) $$

is invertible.

Note that we also have a natural transformation from

\begin{equation*}
\mathbf{\Gamma}(x,D^+\boxtimes \mathbf{1}) \xrightarrow{\on{ins}_{\on{unit}}}
\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1})  \xrightarrow{\on{ind}}  \mathbf{\Gamma}(\on{Id}_\dR,D'\boxtimes \mathbf{1})  \xrightarrow{\on{oblv}}
\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1}) \xrightarrow{p_{s,!}} \mathbf{\Gamma}(x,D^+\boxtimes \mathbf{1})
\end{equation*}

$$ \mathbf{\Gamma}(x,D^+\boxtimes \mathbf{1}) \xrightarrow{\on{ins}_{\on{unit}}}
\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1})  \xrightarrow{\on{ind}}  \mathbf{\Gamma}(\on{Id}_\dR,\mathcal{D})  \xrightarrow{\on{oblv}}
\mathbf{\Gamma}(\on{Id}_\dR,D^+\boxtimes \mathbf{1}) \xrightarrow{p_{s,!}} \mathbf{\Gamma}(x,D^+\boxtimes \mathbf{1}) .$$

By the propogation lemma for chiral homology (see Lemma <ref>), the natural transformation in the last paragraph can be obtained from this one by tensoring up with $\DMod(\on{Ran}_{x,\on{un,dR}})$. Hence we only need to show this natural transformation is invertible.

In other words, we only need to show the chiral homology of the monad

$$ \DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}\to \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  )  \to \DMod(U ^{K\on{-level}_x})_{/U}\boxtimes \mathbf{1}$$

is equivalent to the monad

$$ \DMod(U ^{K\on{-level}_x})_{/U} \to \DMod(U ^{K\on{-level}_x} )_{\mathscr{T}} \to \DMod(U ^{K\on{-level}_x})_{/U}. $$

When $\mathscr{T}$ is trivial and $x$ corresponds to the empty subset, this is the main theorem of <cit.>. The general case can be proved using the same method[Details will be provided in the second author's thesis.].

§.§ Compatibility with Restriction

In this and the following subsection, $U$ (resp. $V$) is a quasi-compact open substack of $\on{Bun}_H$ (resp. $\on{Bun}_{H'}$) such that the image of $V$ under the map $\on{Bun}_{H'} \to \on{Bun}_{H}$ is contained in $U$.

We first construct a natural transformation unerlying the desired commutative square.

By Construction <ref>, we have a commutative diagram induced by twisted $!$-pullback functors

\xymatrix{
\DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1} \ar[r]^-{\on{oblv}} \ar[d]^-{!\on{-pull}}
& \mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \ar[d]^-{!\on{-pull}}\\
\DMod( V^{K'\on{-level}_x} )_{\mathscr{T}'}\boxtimes \mathbf{1} \ar[r]^-{\on{oblv}}
& \mathscr{L}'_{\widehat{\on{Hecke}}_{H'}^{K'\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( V^{K'\on{-level}_x})  ).

Passing to left adjoints, we obtain a $2$-morphism

\xymatrix{
\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \ar[r]^-{\on{ind}} \ar[d]^-{!\on{-pull}}
& \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1} \ar[d]^-{!\on{-pull}}\\
\mathscr{L}'_{\widehat{\on{Hecke}}_{H'}^{K'\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( V^{K'\on{-level}_x})  )
\ar[r]^-{\on{ind}} \ar@{=>}[ur]
&\DMod( V^{K'\on{-level}_x} )_{\mathscr{T}'}\boxtimes \mathbf{1} .

On the other hands, by Construction <ref> we have a commutative diagram induced by twisted $!$-pullback functors:

\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[r]^-{!\on{-pull}} \ar[d]^-{\on{res}}  &
\mathscr{L}_{\widehat{\on{Hecke}}_H^{K\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U^{K\on{-level}_x})  ) \ar[d]^-{!\on{-pull}}  \\
\widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'} \ar[r]^-{!\on{-pull}}  &
\mathscr{L}'_{\widehat{\on{Hecke}}_{H'}^{K'\on{-level}_x} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( V^{K'\on{-level}_x})  ).

Hence we obtain a $2$-morphism

\begin{equation} \label{eqn-natural-transformation-loc-restriction}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  \ar[r]^-{\on{Loc}_U}  \ar[d]^-{\on{res}}
& \DMod( U^{K\on{-level}_x} )_{\mathscr{T}}\boxtimes \mathbf{1} \ar[d]^-{!\on{-pull}}\\
\widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'}
\ar[r]^-{\on{Loc}_V} \ar@{=>}[ur]
&\DMod( V^{K'\on{-level}_x} )_{\mathscr{T}'}\boxtimes \mathbf{1}.
\end{equation}

This $2$-morphism is not invertible in general. However, we are going to prove the following result, which implies Theorem <ref>(3):

Up to chiral homology, the digram (<ref>) commutes for factorization objects. More precisly, for any unital factorization algebra $A$ in $ \mathcal{KL}(H)_{\mathscr{L}}$, the natural transformation

\begin{equation} \label{eqn-lem-loc-compatible-!-pull}
\xymatrix{
A\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}}  ) \ar[d]^-{\on{res}}
\ar[r]^-{[\on{Loc}]_x^{\on{ch}} }
& \DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}  \ar[d]^-{!\on{-pull}} \\
\on{res}(A)\on{-FactMod}_{\on{un}}( \widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'}   )
\ar[r]^-{[\on{Loc}]_x^{\on{ch}} } \ar@{=>}[ur]
& \DMod(\on{Bun}_{H'}^{K'\on{-level}_x})_{\mathscr{T}'},
\end{equation}

induced by it is invertible.

$$\on{oblv}:\DMod(\on{Bun}_{H'} ^{K'\on{-level}_x})_{\mathscr{T}'}\to \DMod(\on{Bun}_{H'} ^{K'\on{-level}_x})_{/\on{Bun}_{H'}}$$

is conservative, we only need to show (<ref>) becomes invertible after composing with this functor. But this follows from  Lemma <ref> (applied to both $(K\to H)$ and $(K'\to H)$) and the following obvious commutative diagrams (where all the arrows are induced by twisted $!$-pullback functors):

\begin{equation*}
\xymatrix{
\widetilde{\mathcal{L}\mathfrak{h}}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }_{\mathscr{L}} \ar[d]^-{\on{res}} \ar[r]^-{\on{oblv}}
& \mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }  \ar[d]^-{\on{res}}
\DMod(\on{Bun}_H ^{K\on{-level}_x})_{\mathscr{T}}  \ar[r]^-{\on{oblv}} \ar[d]^-{!\on{-pull}} &
\DMod(\on{Bun}_H ^{K\on{-level}_x})_{/\on{Bun}_H} \ar[d]^-{!\on{-pull}} \\
\widetilde{\mathcal{L}\mathfrak{h}'}\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }_{\mathscr{L}'} \ar[r]^-{\on{oblv}}
& \mathcal{L}^+\mathfrak{h}'\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x },
\DMod(\on{Bun}_{H'} ^{K'\on{-level}_x})_{\mathscr{T}'}  \ar[r]^-{\on{oblv}} &
\DMod(\on{Bun}_{H'} ^{K'\on{-level}_x})_{/\on{Bun}_{H'}},
\end{equation*}

\xymatrix{
\mathcal{L}^+\mathfrak{h}\on{-}mod^{\mathcal{L}^+H\times_{H_x} K_x }  \ar[d]^-{\on{res}}  \ar[r]^-{\on{Loc}_U^+}
\DMod(U ^{K\on{-level}_x})_{/U} \boxtimes\mathbf{1} \ar[d]^-{!\on{-pull}} \\
\mathcal{L}^+\mathfrak{h}'\on{-}mod^{\mathcal{L}^+H'\times_{H'_x} K'_x }
\ar[r]^-{\on{Loc}_{U'}^+}
\DMod( (U') ^{K'\on{-level}_x})_{/U'}\boxtimes\mathbf{1} .

§.§ Compatibility with Semi-infinite Cohomology

We first construct a natural transformation unerlying the desired commutative square.

It follows from Construction <ref> and Construction <ref> that we have a commutative diagram induced by twisted $*$-pushforward functors

\xymatrix{
\mathcal{KL}(H')_{\mathscr{L}',\on{co}}   \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}}
& \mathscr{L}'_{\widehat{\on{Hecke}}_{H'} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( V ) ) \ar[l] \ar[d]^-{*\on{-push}} \ar[r] &
\DMod(V)_{\mathscr{T}'}\boxtimes 1 \ar[d]^-{*\on{-push}} \\
\mathcal{KL}(H)_{\mathscr{L},\on{co}}
& \mathscr{L}_{\widehat{\on{Hecke}}_{H} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U ) )  \ar[l] \ar[r] &
\DMod(U)_{\mathscr{T}}\boxtimes 1.

Passing to left adjoints, we obtain

\xymatrix{
\mathcal{KL}(H')_{\mathscr{L}',\on{co}}   \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}} \ar[r]
& \mathscr{L}'_{\widehat{\on{Hecke}}_{H'} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( V ) )  \ar[d]^-{*\on{-push}} \ar[r] &
\DMod(V)_{\mathscr{T}'}\boxtimes 1 \ar[d]^-{*\on{-push}} \\
\mathcal{KL}(H)_{\mathscr{L},\on{co}}  \ar[r] \ar@{=>}[ru]
& \mathscr{L}_{\widehat{\on{Hecke}}_{H} } \on{-mod}( \mathcal{I}nd\mathcal{C}oh( U ) )  \ar[r] &
\DMod(U)_{\mathscr{T}}\boxtimes 1.

The outer square is just

\begin{equation} \label{eqn-natural-transformation-loc-coh}
\xymatrix{
\mathcal{KL}(H')_{\mathscr{L}',\on{co}}   \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}} \ar[r]^-{\on{Loc}_{V,\on{co}}}
\DMod(V)_{\mathscr{T}'}\boxtimes 1 \ar[d]^-{*\on{-push}} \\
\mathcal{KL}(H)_{\mathscr{L},\on{co}}   \ar[r]^-{\on{Loc}_{U,\on{co}}} \ar@{=>}[ru]
\DMod(U)_{\mathscr{T}}\boxtimes 1.
\end{equation}

This $2$-morphism is not invertible in general. However, we are going to prove the following result, which implies Theorem <ref>(4):

Up to chiral homology, the digram (<ref>) commutes for factorization objects. More precisly, for any unital factorization algebra $A'$ in $ \mathcal{KL}(H')_{\mathscr{L}',\on{co}}$, the natural transformation

\begin{equation*}
\xymatrix{
A'\on{-FactMod}_{\on{Ran}_{x,\on{un,dR}}}   \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}} \ar[r]^-{[\on{Loc}]^{\on{ch}}_{x,\on{co}}}
\DMod(\on{Bun}_{H'})_{\mathscr{T}'} \ar[d]^-{*\on{-push}} \\
\on{C}^{\frac{\infty}{2}}_{\on{co}}(A')\on{-FactMod}_{\on{Ran}_{x,\on{un,dR}}}  \ar[r]^-{[\on{Loc}]^{\on{ch}}_{x,\on{co}}}\ar@{=>}[ru]
\DMod(\on{Bun}_H)_{\mathscr{T}}.
\end{equation*}

induced by it is invertible.

Let $\mathbb{V}'_{\on{co}}$ be the unit factorization algebra of $ \mathcal{KL}(H')_{\mathscr{L}',\on{co}}$. By Lemma <ref>, we can reduce the lemma to the following result.

The natural transformation

\begin{equation} \label{eqn-lem-loc-compatible-*-push-prop}
\xymatrix{
\mathbb{V}'_{\on{co}}\on{-FactMod}_{ \on{Arrow}(\on{Ran}_{x,\on{un,dR}})}   \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}} \ar[r]^-{\on{Loc}_{\on{co}}}
\DMod(\on{Bun}_{H'})_{\mathscr{T}'} \otimes \DMod( \on{Arrow}(\on{Ran}_{x,\on{un,dR}}) ) \ar[d]^-{*\on{-push}} \\
\on{C}^{\frac{\infty}{2}}_{\on{co}}(\mathbb{V}'_{\on{co}})\on{-FactMod}_{ \on{Arrow}(\on{Ran}_{x,\on{un,dR}})}   \ar[r]^-{\on{Loc}_{\on{co}}}\ar@{=>}[ru]
\DMod(\on{Bun}_H)_{\mathscr{T}} \otimes \DMod( \on{Arrow}(\on{Ran}_{x,\on{un,dR}}) ).
\end{equation}

is invertible after composing with

$$  p_{s,!}:  \DMod( \on{Arrow}(\on{Ran}_{x,\on{un,dR}}) )\to  \DMod( \on{Ran}_{x,\on{un,dR}}). $$

In the proof of the lemma, we use the following shorthands:

* $\pi: \on{Ran}_{x,\on{un}} \to \on{Ran}_{\on{un}}$ is the obvious preojection;

* $\mathcal{R}ep:= \mathcal{R}ep(\mathcal{L}^+ H),\; \mathcal{R}ep':= \mathcal{R}ep(\mathcal{L}^+ H'),\; \mathcal{KL}_{\on{co}}:= \mathcal{KL}(H)_{\mathscr{L},\on{co}},\; \mathcal{KL}'_{\on{co}}:= \mathcal{KL}(H')_{\mathscr{L}',\on{co}}$ are the unital factorization categories over $\on{Ran}_{\on{un}}$;

* $\mathcal{I}nd\mathcal{C}oh: \IndCoh(\on{Bun}_H) \boxtimes \mathbf{1},\;\mathcal{I}nd\mathcal{C}oh': \IndCoh(\on{Bun}_H') \boxtimes \mathbf{1} $
are the crystal of categories over $\on{Ran}_{\on{un}}$, similarly for $\mathcal{DM}od$ and $\mathcal{DM}od'$.

Similar to the proof of Lemma <ref>, we only need to show (<ref>) is invertible after composing with

$$ \mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep') \xrightarrow{\on{ind}} \mathbf{\Gamma}(\pi_\dR, \mathcal{KL}')   \xrightarrow{\on{ins}_{\on{unit}}} \mathbb{V}'_{\on{co}}\on{-FactMod}_{ \on{Arrow}(\on{Ran}_{x,\on{un,dR}})}. $$

We have commutative squares

\xymatrix{
\mathcal{R}ep' \ar[r]^-{\on{Loc}^+_{\on{co}}} \ar[d]^-{\on{ind}}
\mathcal{I}nd\mathcal{C}oh'  \ar[d]^-{\on{ind}} &
\mathcal{R}ep \ar[r]^-{\on{Loc}^+_{\on{co}}} \ar[d]^-{\on{ind}}
\mathcal{I}nd\mathcal{C}oh   \ar[d]^-{\on{ind}}
& \mathcal{R}ep' \ar[r]^-{\on{ind}} \ar[d]^-{\on{inv}}
\mathcal{KL}'_{\on{co}}    \ar[d]^-{\on{C}^{\frac{\infty}{2}}_{\on{co}}} \\
\mathcal{KL}'_{\on{co}}   \ar[r]^-{\on{Loc}_{\on{co}}}
\mathcal{D}mod' , &
\mathcal{KL}_{\on{co}}   \ar[r]^-{\on{Loc}_{\on{co}}}
\mathcal{D}mod ,
& \mathcal{R}ep \ar[r]^-{\on{ind}}
\mathcal{KL}_{\on{co}},

where the last one is induced by twisted $*$-pushforward morphisms, and the other two squares are just (the co-version of) (<ref>). Note that the horizontal morphisms are strictly unital. Hence we only need to show

\begin{equation} \label{eqn-proof-lem-loc-compatible-*-push-prop-1}
\xymatrix{
\mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep')  \ar[d]^-{\on{inv}} \ar[r]^-{\on{Loc}^+_{\on{co}}} &
\mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{I}nd\mathcal{C}oh')  \ar[d]^-{*\on{-push}}
\\
\mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep) \ar[r]^-{\on{Loc}^+_{\on{co}}} \ar@{=>}[ru] &
\mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{I}nd\mathcal{C}oh)
\end{equation}

is invertible after precomposing with

$$ \on{ins}_{\on{unit}}:\mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep')\to  \mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep')$$

and composing with

$$p_{s,!} :\mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{I}nd\mathcal{C}oh)\to\mathbf{\Gamma}(\pi_\dR, \mathcal{I}nd\mathcal{C}oh).$$

Since $H'\to H$ is a surjection with unipotent kernel, the image of the restriction functor

$$ \on{res}:  \mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep) \to \mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep')$$

generates the target. Hence we can further precompose with this functor. Note that
$$\mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep)  \xrightarrow{\on{res}} \mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep') \xrightarrow{ \on{ins}_{\on{unit}}}  \mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep')$$

is equivalent to

$$\mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep) \xrightarrow{ \on{ins}_{\on{unit}}}  \mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep) \xrightarrow{\on{res}}  \mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep')$$

because $\on{res}: \mathcal{R}ep\to \mathcal{R}ep'$ is strictly unital.

Recall the pointwise-tensor products induce a symmetric monoidal structure on $\mathcal{R}ep$, and the monoidal unit of $\mathbf{\Gamma}( \on{Id}_\dR, \mathcal{R}ep)$ (which is the trivial representation) is also the factorization unit. Note that all the relevant prestacks in the above statement

$$ [\mathbb{B}\mathcal{L}^+H]_{\on{Ran}_{\on{un}}} ,\; [\mathbb{B}\mathcal{L}^+H']_{\on{Ran}_{\on{un}}}, \on{Bun}_H \times \on{Ran}_{\on{un}},\; \on{Bun}_{H'} \times \on{Ran}_{\on{un}}$$

are defined over $[\mathbb{B}\mathcal{L}^+H]_{\on{Ran}_{\on{un}}}$. Using the base-change isomorphisms and their corollary, the projection formulae, we can reduce to show (<ref>) is invertible when precomposing with $\on{res} \circ \on{ins}_{\on{unit}}$, composing with $p_{s,!}$, and evaluating at the monoidal unit of $\mathbf{\Gamma}(\pi_\dR, \mathcal{R}ep)$. In other words, we only need to show (<ref>) is invertible when evaluating at the monoidal unit of $ \mathbf{\Gamma}((\pi\circ p_t)_\dR, \mathcal{R}ep')$ and composing with $p_{s,!}$.

Using Propogation Lemma for Chiral homology (see Lemma <ref>), we can further reduce to the following lemma.

The Beck-Chevalley natural transformation

\xymatrix{
[\mathcal{R}ep(\mathcal{L}^+ H')]_{\on{Ran}_{\on{un,dR}}} \ar[r]^-{ \on{Loc}^+_{\on{co}}} \ar[d]^-{\on{inv}} &
\IndCoh(\on{Bun}_{H'})\otimes \DMod(\on{Ran}_{\on{un}} )\ar[d]^-{*\on{-push}} \\
[\mathcal{R}ep(\mathcal{L}^+ H)]_{\on{Ran}_{\on{un,dR}}} \ar[r]^-{ \on{Loc}^+_{\on{co}}} \ar@{=>}[ru] &
\IndCoh(\on{Bun}_{H})\otimes \DMod(\on{Ran}_{\on{un}} )

is invertible when evaluating at the monoidal unit and taking chiral homology, i.e., composing with

$$ \on{C}_{\on{un}}^{\on{ch}}: \IndCoh(\on{Bun}_{H})\otimes \DMod(\on{Ran}_{\on{un}} )\to  \IndCoh(\on{Bun}_{H}). $$

The claim remains correct if we replace $\on{Loc}^+_{\on{co}}$ by $\on{Loc}$ and the $*$-pushforward functor by the $?$-pushforward functor.

Recall $\on{Loc}^+_{\on{co}}$ is defined as the $*$-pullback functor for $\mathcal{I}nd\mathcal{C}oh_{*,\ren}$-theory along

$$ \on{Bun}_H \times \Ran_{\on{un,dR}} \to [\mathbb{B} \mathcal{L}^+ H)]_{\Ran_{\on{un,dR}}} $$

while $\on{Loc}^+$ is defined as the $!$-pullback functor for $\mathcal{I}nd\mathcal{C}oh^{!,\ren}$-theory along the same map. It follows that

$$\on{Loc}^+ \simeq  K_{\on{Bun}_H}\otimes \on{Loc}^+_{\on{co}}.$$

Then it is clear that the claims for $\on{Loc}$ and $\on{Loc}_{\on{co}}$ are equivalent. We will freely switch between these two versions.

Let $q: \on{Bun}_{H'} \to \on{Bun}_{H}$ be the projection. Let $I:=\on{inv}(\on{unit})\in \mathcal{R}ep(\mathcal{L}^+ H)$ be the unital factorization algebra. We want to show

$$\on{Loc}^+ (I) \to  q_?(\omega_{\on{Bun}_{H'} }) \boxtimes \omega_{ \on{Ran}_{\on{un}} } $$

is sent to an isomorphism by $\on{C}_{\on{un}}^{\on{ch}}$.

We need to use the theory of commutative factorization categories and commutative factorization algebras. We first review what these mean.

Recall the factorization structure on $\mathcal{R}ep(\mathcal{L}^+ H)$ is induced by the factorization structure on $[\mathcal{L}^+ H]_{\on{Ran}_{\on{un,dR}}}$. Moreover, the latter factorization structure on $[\mathcal{L}^+ H]_{\on{Ran}_{\on{un,dR}}}$ can be upgraded to a cocommutative one. Roughly speaking, this means the isomorphism

$$[\mathcal{L}^+ H]_{ (\on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}})_{\on{disj}} }  \simeq ([\mathcal{L}^+ H]_{ \on{Ran}_{\on{un,dR}}} \times [\mathcal{L}^+ H]_{ \on{Ran}_{\on{un,dR}}} )_{\on{disj}}$$

can be extended to a morphism

$$ [\mathcal{L}^+ H]_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}} }  \to [\mathcal{L}^+ H]_{ \on{Ran}_{\on{un,dR}}} \times [\mathcal{L}^+ H]_{ \on{Ran}_{\on{un,dR}}}  $$

defined over $\on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}$ subject to higher compatibilities (which are actually structures). This implies the factorization structure on $\mathcal{R}ep(\mathcal{L}^+ H)$ can be upgraded to a commutative one. This means the equivalence

$$   (\mathcal{R}ep(\mathcal{L}^+ H) \boxtimes \mathcal{R}ep(\mathcal{L}^+ H))|_{ (\on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}})_{\on{disj}} } \simeq \mathcal{R}ep(\mathcal{L}^+ H) |_{ (\on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}})_{\on{disj}} } $$

can be extended to a morphism

$$ \on{mult}:  (\mathcal{R}ep(\mathcal{L}^+ H) \boxtimes \mathcal{R}ep(\mathcal{L}^+ H))|_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}} } \to \mathcal{R}ep(\mathcal{L}^+ H) |_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}} $$

between crystal of categories on $\on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}$ subject to higher compatibilities.

The factorization functor $\on{inv}: \mathcal{R}ep(\mathcal{L}^+ H')\to \mathcal{R}ep(\mathcal{L}^+ H)$
can be upgrated to a right-lax commutative factorization functor. Roughly speaking, this means we have a $2$-morphism

\xymatrix{
(\mathcal{R}ep(\mathcal{L}^+ H') \boxtimes \mathcal{R}ep(\mathcal{L}^+ H'))|_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}} }
\ar[r]^-{\on{mult}} \ar[d]^-{ \on{inv}\boxtimes \on{inv} } &
\mathcal{R}ep(\mathcal{L}^+ H') |_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}} \ar[d]^-{ \on{inv} }\\
(\mathcal{R}ep(\mathcal{L}^+ H) \boxtimes \mathcal{R}ep(\mathcal{L}^+ H))|_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}} }
\ar[r]^-{\on{mult}} \ar@{=>}[ru] &
\mathcal{R}ep(\mathcal{L}^+ H) |_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}}

subject to higher compatibilities. Namely, in the $\mathcal{I}nd\mathcal{C}oh^{!,\ren}$-theory, the horizontal morphisms are induced by $!$-pullback functors, the vertical morphisms are induced by $?$-pushforward fucntors (which are the right adjoints of $!$-pullback functors), and the $2$-morphism is induced by the Bech-Chevalley natural transformations. It follows that $I=\on{inv}(\on{unit})$ can be upgraded to a commutative (unital) factorization algebra. This means we have a morphism

$$ \on{mult}( I\boxtimes I ) \to I|_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}} $$

in $[\mathcal{R}ep(\mathcal{L}^+ H)]_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}}}$ subject to higher compatibilities.

Also, the commutative factorization algebra structure on $I$ can upgrade $\on{Loc}^+(I)$ to a commutative factorization algebra inside $\IndCoh(\on{Bun}_H)\boxtimes \mathbf{1}$, which is a $\on{Bun}_H$-family of commutative factorization categories. This means we have a canonical morphism

$$ \on{Loc}^+(I) \otimes^!_{\on{Bun}_H} \on{Loc}^+(I) \to \on{Loc}^+(I)|_{ \on{Ran}_{\on{un,dR}}\times\on{Ran}_{\on{un,dR}} } $$

subject to higher compatibilities.

It is clear that $q_?(\omega_{\on{Bun}_{H'} }) \boxtimes \omega_{ \on{Ran}_{\on{un}} }$ is also a commutative factorization algebra inside $\IndCoh(\on{Bun}_H)\boxtimes \mathbf{1}$ and the map

\begin{equation}\label{eqn-proof-lem-conformal-block-chiral-homology-1}
\on{Loc}^+ (I) \to  q_?(\omega_{\on{Bun}_{H'} }) \boxtimes \omega_{ \on{Ran}_{\on{un}} }
\end{equation}

is a morphism between commutative factorization algebras.

By Proposition <ref>, the restriction functor

$$  [\mathcal{R}ep(\mathcal{L}^+ H)]_{ \on{Ran}_{\on{un,dR}} } \to [\mathcal{R}ep(\mathcal{L}^+ H) ]_{X_\dR} $$

induces an equivalence

$$ \on{CommFactAlg}_{\on{un}}(\mathcal{R}ep(\mathcal{L}^+ H)  ) \simeq \on{CommAlg}_{\on{un}}( [\mathcal{R}ep(\mathcal{L}^+ H) ]_{X_\dR} ).$$

Similarly we have

$$ \Delta^!: \on{CommFactAlg}_{\on{un}}(\IndCoh(\on{Bun}_H)\boxtimes \mathbf{1}  ) \simeq \on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H\times X_\dR) ).$$

Also, the symmtric monoidal functor

$$\on{pr}_1^!:\IndCoh(\on{Bun}_H)\to  \IndCoh(\on{Bun}_H\times X_\dR) $$
induces a functor

$$\on{pr}_1^{!,\on{CommAlg}}: \on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H) ) \to \on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H\times X_\dR) ) $$

such that the corresponding functor

$$ \on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H) ) \to  \on{CommFactAlg}_{\on{un}}(\IndCoh(\on{Bun}_H)\boxtimes \mathbf{1}  ) $$

sends $\mathcal{M}$ to $\mathcal{M}\boxtimes \omega_{\on{Ran}_{\on{un}}}$. Passing to left adjoints, we obtain a commutative diagram

\xymatrix{
\on{CommFactAlg}_{\on{un}}(\IndCoh(\on{Bun}_H)\boxtimes \mathbf{1}  ) \ar[r]_-{\Delta^!}^-\simeq\ar[d]^-{ \on{C}^{\on{ch}}_{\on{un}} }  &
\on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H\times X_\dR) )\ar[ld]^-{ ( \on{pr}_1^{!,\on{CommAlg}})^L } \\
\on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H) ).

Hence we only need to show (<ref>) is sent to an isomorphism by $( \on{pr}_1^{!,\on{CommAlg}})^L\circ \Delta^!$.

Unwinding the definitions, we have reduce the problem to the following claim. Consider the Beck-Chevalley natural transformation

\xymatrix{
[\mathcal{R}ep(\mathcal{L}^+ H')]_{X_\dR} \ar[r]^-{ \on{Loc}^+_{\on{co}}} \ar[d]^-{\on{inv}} &
\IndCoh(\on{Bun}_{H'} \times X_\dR)\ar[d]^-{*\on{-push}} \\
[\mathcal{R}ep(\mathcal{L}^+ H)]_{X_\dR} \ar[r]^-{ \on{Loc}^+_{\on{co}}} \ar@{=>}[ru] &
\IndCoh(\on{Bun}_{H}\times X_\dR )

and its value on the monoidal unit, which is a morphism in $\on{CommAlg}_{\on{un}}( \IndCoh(\on{Bun}_H\times X_\dR) )$. We only need to show this morphism is sent to an isomorphism by $( \on{pr}_1^{!,\on{CommAlg}})^L$. This claim can be rewritten using $\on{QCoh}$-theory as follows. The value of[For a symmetric monoidal DG category $\mathcal{C}$ equipped with a compatible $t$-structure, the category $\on{CommAlg}_{\on{un}}^{\on{cc}}(\mathcal{C})$ is the full subcategory of $\on{CommAlg}_{\on{un}}(\mathcal{C})$ containing those objects $A$ that are coconnective, i.e., such that $A\in \mathcal{C}^{\ge 0}$ and $H^0(A)\in \mathcal{C}^{\heartsuit}$ is the monoidal unit.]

\xymatrix{
\on{CommAlg}_{\on{un}}^{\on{cc}}(\on{QCoh}( [\mathbb{B}\mathcal{L}^+ H']_{X_\dR} )) \ar[r]^-{ *\on{-pull}} \ar[d]^-{*\on{-push}} &
\on{CommAlg}_{\on{un}}^{\on{cc}}(\QCoh(\on{Bun}_{H'} \times X_\dR))\ar[d]^-{*\on{-push}} \\
\on{CommAlg}_{\on{un}}^{\on{cc}}(\on{QCoh}( [\mathbb{B}\mathcal{L}^+ H]_{X_\dR} )) \ar[r]^-{ *\on{-pull}} \ar@{=>}[ru] &
\on{CommAlg}_{\on{un}}^{\on{cc}}(\QCoh(\on{Bun}_{H}\times X_\dR ))

at the monoidal unit is sent to an isomorphism by

$$ \on{CommAlg}_{\on{un}}^{\on{cc}}(\QCoh(\on{Bun}_{H}\times X_\dR )) \subset \on{CommAlg}_{\on{un}}(\QCoh(\on{Bun}_{H}\times X_\dR )) \xrightarrow{ (\on{pr}_1^{*,\on{CommAlg}})^L }  \on{CommAlg}_{\on{un}}(\QCoh(\on{Bun}_{H})). $$

Let $O_1 \to O_2$ be the value of the above natural transformation at the monoidal unit. Note that

$$(\on{pr}_1^{*,\on{CommAlg}})^L(O_2) \simeq q_*( O_{\on{Bun}_{H'}} ),$$

where $q: \on{Bun}_{H'} \to \on{Bun}_H$ is the projection map. Indeed, this can be seen by translating back to the chiral homology description for $(\on{pr}_1^{*,\on{CommAlg}})^L$. In particular, $(\on{pr}_1^{*,\on{CommAlg}})^L(O_2)$ is coconnective.

We claim $(\on{pr}_1^{*,\on{CommAlg}})^L(O_1)$ is also coconnective. We first finish the proof assuming this claim. We only need to show $O_1\to O_2$ is sent to an isomorphism by the partially defined left adjoint

$$L:=(\on{pr}_1^{*,\on{CommAlg}^{\on{cc}}})^L:\on{CommAlg}_{\on{un}}^{\on{cc}}(\QCoh(\on{Bun}_{H}\times X_\dR )) \to \on{CommAlg}_{\on{un}}^{\on{cc}}(\QCoh(\on{Bun}_{H} )).$$

Using the assumption that $H'\to H$ is surjective with unipotent kernel, one can check that the maps

$$  [\mathbb{B}\mathcal{L}^+ H']_{X_\dR}\to [\mathbb{B}\mathcal{L}^+ H]_{X_\dR},\;\on{Bun}_{H'}\to \on{Bun}_{H} $$

are co-affine[A map $Y\to Z$ is co-affine if for any affine test scheme $S\to Z$, the fiber product $S\times_Z Y$ is co-affine in the sense of <cit.>.]. Hence we have

$$  [\mathbb{B}\mathcal{L}^+ H']_{X_\dR}\times_{[\mathbb{B}\mathcal{L}^+ H]_{X_\dR}  }  (\on{Bun}_{H}\times X_\dR) \simeq \on{cSpec}(O_1)  ,\; \on{Bun}_{H'}\times X_\dR \simeq \on{cSpec}(O_2),$$

$$\on{cSpec}: \on{CommAlg}_{\on{un}}^{\on{cc}}(\QCoh(\on{Bun}_{H}\times X_\dR )) \to \on{PreStk}_{ / \on{Bun}_{H}\times X_\dR}

is the (relative) co-affine spectrum functor (see <cit.>), it is charaterized by the functorial equivalences

$$ \on{Maps}_{\on{Bun}_{H}\times X_\dR}( \on{Spec}(A), \on{cSpec}(O) ) \simeq \on{Maps}_{ \on{CommAlg}_{\on{un}}(\QCoh(\on{Bun}_{H}\times X_\dR )) }(O,A) $$

for any connective $A\in \on{CommAlg}_{\on{un}}^{\on{cn}}(\QCoh(\on{Bun}_{H}\times X_\dR ))$. Consider the Weyl retriction functor

$$W: \on{PreStk}_{ / \on{Bun}_{H}\times X_\dR} \to \on{PreStk}_{ / \on{Bun}_{H}}.$$

For any affine test scheme $Y$ over $\on{Bun}_H$, we have

\begin{eqnarray*}
& & \on{Maps}_{\on{Bun}_H}( Y,W([\mathbb{B}\mathcal{L}^+ H']_{X_\dR}\times_{[\mathbb{B}\mathcal{L}^+ H]_{X_\dR}  }  (\on{Bun}_{H}\times X_\dR) )  ) \\
&\simeq & \on{Maps}_{\on{Bun}_H\times X_\dR}( Y\times X_\dR,[\mathbb{B}\mathcal{L}^+ H']_{X_\dR}\times_{[\mathbb{B}\mathcal{L}^+ H]_{X_\dR}  }  (\on{Bun}_{H}\times X_\dR) )   \\
&\simeq & \on{Maps}_{[\mathbb{B}\mathcal{L}^+ H]_{X_\dR} }( Y\times X_\dR,[\mathbb{B}\mathcal{L}^+ H']_{X_\dR}  ) \\
& \simeq & \on{Maps}_{\mathbb{B}H\times X }( Y\times X,\mathbb{B}H'\times X )\\
& \simeq &  \on{Maps}_{\on{Bun}_H}( Y, \on{Bun}_{H'}),
\end{eqnarray*}

where we use the fact that the Weyl restriction functor $\on{PreStk}_{/X} \to \on{PreStk}_{/X_\dR}$ sends $\mathbb{B}H\times X$ to $[\mathbb{B}\mathcal{L}^+ H]_{X_\dR}$. Hence

$$W([\mathbb{B}\mathcal{L}^+ H']_{X_\dR}\times_{[\mathbb{B}\mathcal{L}^+ H]_{X_\dR}  }  (\on{Bun}_{H}\times X_\dR) ) \simeq \on{Bun}_{H'}.$$

Since $\on{cSpec}$ is fully faithful (see <cit.>), we obtain

$$L(O_1) \simeq q_*( O_{\on{Bun}_{H'}} ) \simeq L(O_2)$$

as desired.

It remains to show $(\on{pr}_1^{*,\on{CommAlg}})^L(O_1)$ is coconnective. Let $\mathfrak{n}$ be the Lie algebra of the kernel $H'\to H$. Recall we have a $\on{Lie}^*$-algebra $\mathfrak{n}_D:= \mathfrak{n}\otimes D_X \in \DMod(X)$. Similarly, we have a $\on{Bun}_H$-family of $\on{Lie}^*$-algebras $\widetilde{\mathfrak{n}_D}\in \on{QCoh}(\on{Bun}_H) \otimes \DMod(X)$ such that its restriction at a $H$-torsor $\mathcal{P}\in \on{Bun}_H$ is the $\on{Lie}^*$-algebra $(\mathfrak{n}_D)_{\mathcal{P}}$ twisted by $\mathcal{P}$ (defined via the adjoint action of $H$ on $\mathfrak{n}$). Let $C(\widetilde{\mathfrak{n}_D} )\in \on{QCoh}(\on{Bun}_H) \otimes \DMod(X)$ be the $\on{Bun}_H$-family of commutative algebras that is Koszul dual to $\widetilde{\mathfrak{n}_D} $ (see Proposition <ref>). By definition, $O_1\simeq C(\widetilde{\mathfrak{n}_D} )$. By <cit.>, we have

$$(\on{pr}_1^{*,\on{CommAlg}})^L(C(\widetilde{\mathfrak{n}_D} )) \simeq C \circ \Gamma(X_\dR, \widetilde{\mathfrak{n}_D}),  $$

where $\Gamma(X_\dR, \widetilde{\mathfrak{n}_D}) \in \on{QCoh}(\on{Bun}_H)$ is a $\on{Bun}_H$-family of $\on{Lie}$-algebras, i.e., a Lie-algebra object in $\on{QCoh}(\on{Bun}_H)$ and $C \circ \Gamma(X_\dR, \widetilde{\mathfrak{n}_D})$ is its Koszul dual. Then we are done because $C \circ \Gamma(X_\dR, \widetilde{\mathfrak{n}_D})$ is clearly coconnective.

§ TORIC FUNDAMENTAL LOCAL EQUIVALENCE

The starting point is the following classical result due to <cit.>:

For any classical Noetherian scheme $S_0$, we have an isomorphism of $S$-group schemes

\[\underline{\Hom}_{\tx{Grp}/S_0}(\mc{L} \mb{G}_m, \underline{\mb{G}_m}) \simeq \mc{L} \mb{G}_m\]

where $\underline{\Hom}_{\tx{Grp}/S_0}$ denotes the relative group homomorphism prestack, $\mc{L} \mb{G}_m$ is the relative loop space over $S_0$, and $\underline{\mb{G}_m})$ is the constant group scheme.

In particular this gives a map of group prestacks known as the Contou-Carrère symbol

\[\langle \bullet, \bullet \rangle: \mc{L} \mb{G}_m \times_{S_0} \mc{L} \mb{G}_m \to \underline{\mb{G}_m}.\]

At one point it can be described as follows (<cit.>). The determinant line bundle on $\mc{L} \mb{G}_m$ determines a central extension $\widetilde{\mc{L} \mb{G}_m}$, and we have, for any test affine scheme $A$,

\[\langle f, g \rangle^{-1} = \tilde{f} \cdot \tilde{g} \cdot \tilde{f}^{-1} \cdot \tilde{g}^{-1} \in \mb{G}_m(A)\]

for any $f, g \in \mb{G}_m(K)(A)$ and $\tilde{f}, \tilde{g}$ any lifting to the central extension. Since the central extension canonically splits on $\mc{L}^+ \mb{G}_m$, the pairing is canonically trivial on $\mc{L}^+ \mb{G}_m$ and thus induces another group homomorphism

\[\mc{L}^+ \mb{G}_m \times_{S_0} (\mc{L} \mb{G}_m / \mc{L}^+ \mb{G}_m) \to \mb{G}_m;\]

In fact, this pairing is perfect in the following sense (<cit.>):

We have an isomorphism of $S_0$-group schemes

\[\underline{\Hom}_{\tx{Grp}/S_0}(\mc{L}^+ \mb{G}_m, \underline{\mb{G}_m}) \simeq \mc{L}\mb{G}_m / \mc{L}^+ \mb{G}_m.\]

An important property of this symbol is that it is bi-multiplicative, i.e. for every $S$-point $x, y, z \in \mc{L}\mb{G}_m(S)$ we have:

\[\langle x y, z \rangle = \langle x, z \rangle \langle y, z \rangle, \langle x, y z \rangle = \langle x, y \rangle \langle x, z \rangle.\]

As observed in <cit.>, the Contou-Carrère symbol is furthermore compatible with the flat connection on the loop group and is factorizable because the determinant line bundle is; therefore we have a (non-unital) factorizable version of the symbol:

\[[\mc{L}^+ \mb{G}_m]_{\Ran_\dR} \times_{\Ran_\dR} [\tx{Gr}_{\mb{G}_m}]_{\Ran_\dR} \to \mb{G}_m \times \Ran_\dR\]

Delooping on the first variable, we have a pairing between $\tx{Gr}_{\mb{G}_m}$ and its 1-dual, i.e. a map

\[[\mb{B} \mc{L}^+ \mb{G}_m]_{\Ran_\dR} \times_{\Ran_\dR} [\tx{Gr}_{\mb{G}_m}]_{\Ran_\dR} \to \mb{B} \mb{G}_m \times \Ran_\dR\]

which is explicitly describable as follows. An $S$-point of the LHS is given by

\[(x, \mc{P}_1, \mc{P}_2, \alpha)\]

where $x = (x_i)_{i \in I}$ is an $S$-point of $\Ran_\dR$, $\mc{P}_1$, $\mc{P}_2$ two line bundles on the adic disk $\mc{D}'_{\Gamma_x}$, and $\alpha$ a trivialization of $\mc{P}_2$ on $\mc{D}^\circ_{\Gamma_x}$. Then the pairing sends it to

\[(x, \det(\Gamma(\mc{D}'_{\Gamma_x}, \mc{P}_1), \Gamma(\mc{D}'_{\Gamma_x}, \mc{P}_1 \tensor \mc{P}_2))^{-1} \tensor \det(\Gamma(\mc{D}'_{\Gamma_x}, \mc{O}), \Gamma(\mc{D}'_{\Gamma_x}, \mc{P}_2)))\]

where $\Gamma(\mc{D}'_{\Gamma_x}, \mc{P}_1)$ sends $\mc{P}_1$ to the corresponding lattice in the Tate vector space $\Gamma(\mc{D}^\circ_{\Gamma_x}, \mc{P}_1)$, which we identify with $\Gamma(\mc{D}^\circ_{\Gamma_x}, \mc{P}_1 \tensor \mc{P}_2)$ via $\alpha$; $\det$ stands for the determinant line bundle.

The following was shown in <cit.>:

This pairing is again perfect. More precisely, we have an isomorphism of counital factorization spaces

\[\underline{\Hom}_{\tx{Grp}/X^I}([\mb{B} \mc{L}^+ \mb{G}_m]_{X^I}, \underline{\mb{G}_m}) \simeq [\tx{Gr}_{\mb{G}_m}]_{X^I}\]

after base-changing along $X^I \to \Ran_\dR$.

The counital structure mentioned in the proposition above comes from a corr-unital structure of the pairing, given by a map of corr-unital factorization spaces

\[[\mb{B}\mc{L}^+ \mb{G}_m]_{\Ran_{\un, \dR}} \times_{\Ran_{\un, \dR}} \tx{Gr}_{\mb{G}_m, \Ran_{\un, \dR}} \to \mb{B} \mb{G}_m \times \Ran_{\un, \dR}\]

here the fiber product is taken as corr-unital factorization spaces (note this is really necessary: $\mb{B}\mc{L}^+ \mb{G}_m$ is co-unital but $\tx{Gr}_{\mb{G}_m}$ is unital). Unfolding definition, we see the corresponding unital structure, for any fixed injection $\alpha: I \to J$ of finite sets, is given by

\[\xymatrix{
& [\mb{B} \mc{L}^+ \mb{G}_m]_{X_J} \times_{X^J} X^J \times_{X^I} [\tx{Gr}_{\mb{G}_m}]_{X^I} \ar[ld] \ar[rd] \\
([\mb{B}\mc{L}^+ \mb{G}_m]_{X^I} \times_{X^I} [\tx{Gr}_{\mb{G}_m}]_{X^I}) \times_{X^I} X^J & & [\mb{B}\mc{L}^+ \mb{G}_m]_{X^J} \times_{X^J} [\tx{Gr}_{\mb{G}_m}]_{X^J}

the leftward map is given by restriction while the rightward one uses Beauville-Laszlo. Then the two maps to $\mb{B} \mb{G}_m$ (via either leg) canonically give the same result, so we have a well-defined corr-unital map.

Using the pairing $\coweightLat \tensor_{\mb{Z}} \weightLat \to \mb{Z}$, the above discussion immediately generalizes to give a pairing

\[[\mb{B}\mc{L}^+ T]_{\Ran_{\un, \dR}} \times_{\Ran_{\un, \dR}} \tx{Gr}_{\check{T}, \Ran_{\un, \dR}} \to \mb{B} \mb{G}_m \times \Ran_{\un, \dR}\]

uniquely determined by the requirement that, if $\mc{P}_1, \mc{P}_2$ are line bundles, $\lambda: \mb{G}_m \to T, \cmu: \mb{G}_m \to \check{T}$, then the pairing on the induced $T$ (resp. $\check{T}$) bundles should be the pairing of $\mc{P}_1$ and $\mc{P}_2$ raise to the $\lambda(\cmu)$-th power.

§.§ Levelwise Cartier Duality

Recall that we have a placid presentation $\mc{L}^+T_{X^I} \simeq \limit_i \mc{L}^{(n)}T_{X^I}$ by the $n$th jet schemes, where each $\mc{L}^{(n)}T_{X^I}$ is a smooth affine $X^I$-group scheme. Consider the Cartier dual

\[Z_n := \underline{\Hom}_{\tx{Grp}/X^I}(\mc{L}^{(n)}T_{X^I}, \underline{\mb{G}_m})\]

The goal of this subsection is to prove the following statement:

Each $Z_n$ is an ind-flat ind-finite group ind-scheme, and for each $n < m$, the map $Z_n \to Z_m$ is an ind-closed embedding, and we have $\tx{Gr}_{\check{T}, X^I} \simeq \colimit_n Z_n$. In other words, this provides an ind-flat placid presentation of $\tx{Gr}_{\check{T}, X^I}$.

An ind-flat placid presentation of $\tx{Gr}_{G, X^I}$ is expected to exist for all reductive $G$. However, we are currently unaware of such a construction.

One readily reduces to the case of $T = \mb{G}_m$. Let $r: (\mc{L}^{(n)} \mb{G}_m)_{X^I} \to X^I$ be the projection, and set

\[\mc{O}_n := r_*(\mc{O}) \in \QCoh^{\heartsuit}(X^I);\]

it is a Hopf algebra object in said category. It is clearly flat as an $\mc{O}_{x^I}$-module, and is in fact locally free (<cit.>), so we reduce to an immediate check on geometric fibers.

More explicitly, let $\Gamma_n$ be the scheme-theoretic sum of $n$th neighborhood of the graphs in $X^I \times X$, and let $p: \Gamma_n \to X^I$ be the projection map. If we consider $\mb{G}_m \simeq \mb{A}^2 \times_{\mb{A}^1} \{1\}$ as embedded into $\mb{A}^2$ via $x \mapsto (x, x^{-1})$, then we can write

\begin{equation}
\label{eq:sym-expression}
\mc{O}_n \simeq \tx{Sym}_{\mc{O}_{X^I}}(p_*(\mc{V}_2^\vee \tensor K)) \tensor_{\tx{Sym}_{\mc{O}_{X^I}}(p_*\tensor K)} \mc{O}_{X^I}
\end{equation}

where $K := p^!(\mc{O}_{X^I})$ is the relative dualizing sheaf, and $\mc{V}_r^\vee := \underline{\Hom}_{\mc{O}_{\Gamma_n}}(\mc{O}_{\Gamma_n}^{\oplus r}, \mc{O}_{\Gamma_n}) \in \QCoh^\heartsuit(\Gamma_n)$.

The algebra map $\tx{Sym}_{\mc{O}_{X^I}}(p_* K) \to \mc{O}_{X^I}$ is induced by the trace map $p_* p^!(\mc{O}) \to \mc{O}$, and the algebra map

\[\tx{Sym}_{\mc{O}_{X^I}}(p_* K) \to \tx{Sym}_{\mc{O}_{X^I}}(p_*(\mc{V}_2^\vee \tensor K))\]

is induced by a sheaf map