I}\longrightarrow(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}.$

For any $n\geq 0$, let

$(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram},\leq n}_{J\to
I}\subset(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}_{J\to I}$

be the closed subscheme where the $1$-form is allowed to have poles of order
at most $1$ at $x$. Then we have

$(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}_{J\to
I}=\underset{n}{\operatorname{colim}}\
(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram},\leq n}_{J\to I},$

and the transition maps are regular closed embeddings. Moreover, the map

$(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram},\leq n}_{J\to
I}\longrightarrow(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}$

is a regular closed embedding for any $n\geq 0$, from which the claim follows.

The existence of $(\beta_{2}^{*})^{\operatorname{L}}$ follows from the fact
that
$\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}$
is the quotient of
$\operatorname{LS}_{H}(\mathring{D})^{x\operatorname{-ram},\log}_{X^{I}_{\operatorname{dR}}}$
by a certain Hecke groupoid, which is ind-proper because $H$ is reductive.

∎

As a consequence of the proposition, we can endow the object
$\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$ with a unital
structure and hence view it as an object of
$\mathsf{FactCat}^{\operatorname{lax-fact}}$. Explicitly, the structure
morphism attached to an injection $J\to I$ is given by

$(\beta^{*})^{\operatorname{L}}\alpha^{*}:\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X^{J}_{\operatorname{dR}}})\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}).$

As explained in §9.9 of [Ras16], the $1$-affineness of
$\operatorname{LS}_{H}(D)_{X^{I}_{\operatorname{dR}}}$ implies that
$\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$ factorizes
strictly, i.e. belongs to $\mathsf{FactCat}$.

It follows that

$(\iota^{*})^{\operatorname{L}}:\mathsf{QCoh}(\operatorname{LS}_{H}(D))\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$

is a morphism in $\mathsf{FactCat}$. Moreover, the pointwise tensor product on
$\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$ upgrades it
to an object of $\mathsf{ComAlg}(\mathsf{FactCat}_{\operatorname{lax-untl}})$.
The structure morphisms for this commutative algebra structure are generally
only lax unital with respect to factorization, i.e. they do not take place in
$\mathsf{FactCat}$. For instance, the morphism

$\operatorname{Vect}\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$

corresponding to the monoidal unit
$\mathcal{O}_{\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}}$ is not
strictly unital with respect to factorization, since the map

$(\iota^{*})^{\operatorname{L}}\mathcal{O}_{\operatorname{LS}_{H}(D)}\longrightarrow\mathcal{O}_{\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}}$

from the factorization unit to the monoidal unit is of course not an
isomorphism.

###  7.3. Local geometric class field theory in the de Rham setting

Let $\operatorname{Gr}_{T}^{\infty\cdot x}$ denote the corr-unital
factorization space classifying a point of $\operatorname{Gr}_{T}$ together
with a full level structure at $x$. In particular, the fiber of
$\operatorname{Gr}_{T}^{\infty\cdot x}$ at $x$ identifies with
$(\mathfrak{L}T)_{x}$, while its restriction to $X\setminus\\{x\\}$ identifies
with $\operatorname{Gr}_{T}$. We will write

$p:\operatorname{Gr}_{T}^{\infty\cdot x}\longrightarrow\operatorname{Gr}_{T}$

for the projection.

The commutative group structure on $\operatorname{Gr}_{T}^{\infty\cdot x}$
endows $\mathsf{D}(\operatorname{Gr}_{T}^{\infty\cdot x})$ with a convolution
symmetric monoidal structure, which makes it an object of
$\mathsf{ComAlg}(\mathsf{FactCat}_{\operatorname{lax-untl}})$. This
commutative algebra structure takes place in the lax unital setting,
corresponding to the fact that the natural morphism

$\delta_{1}\longrightarrow\omega_{\mathfrak{L}^{+}T}$

in $\mathsf{D}(\mathfrak{L}T)_{x}$ is not an isomorphism.

######  Theorem 7.3.1.

There is a canonical equivalence

$\mathbb{L}_{T}:\mathsf{D}(\operatorname{Gr}_{T}^{\infty\cdot
x})\tilde{\longrightarrow}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram}})$

in $\mathsf{ComAlg}(\mathsf{FactCat}_{\operatorname{lax-untl}})$ fitting into
a commutative square

${\mathsf{D}(\operatorname{Gr}_{T}^{\infty\cdot
x})}$${\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram}})}$${\mathsf{D}(\operatorname{Gr}_{T})}$${\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)).}$$\scriptstyle{\mathbb{L}_{T}}$$\scriptstyle{p_{\operatorname{dR},*}}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\sim}$

For any $n\geq 0$, let $\operatorname{Gr}_{T}^{n\cdot x}$ be the quotient of
$\operatorname{Gr}_{T}^{\infty\cdot x}$ classifying a point of
$\operatorname{Gr}_{T}$ together with a structure of level $n$ at $x$. To
prove the theorem, we will construct a system of isomorphisms

$\mathbb{L}_{T}^{(n)}:\mathsf{D}(\operatorname{Gr}_{T}^{n\cdot
x})\tilde{\longrightarrow}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n})$

in $\mathsf{ComAlg}(\mathsf{FactCat}_{\operatorname{lax-untl}})$, and obtain
$\mathbb{L}_{T}$ by passing to the limit.

###  7.4. Duality for quasicoherent sheaves on local systems

Fix a finite set $I$ and $n\geq 0$. Write

$\pi:\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}\longrightarrow X^{I}_{\operatorname{dR}}$

for the projection. We define a $\mathsf{D}(X^{I})$-linear functor

$\pi_{*,\operatorname{ren}}:\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})\longrightarrow\mathsf{D}(X^{I})$

as follows. Letting

$\rho:\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}}\longrightarrow X^{I}_{\operatorname{dR}}$

denote the projection, the usual direct image functor

$\rho_{*}:\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}})\longrightarrow\mathsf{D}(X^{I})$

is continuous and $\mathsf{D}(X^{I})$-linear by Corollary 3.12.2 of [Ras15c]
(the proof there is given at $x$ rather than over $X^{I}_{\operatorname{dR}}$,
but adapts to the latter setting). The map

$\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}$

realizes the latter as the quotient of the former by the formal group

$\mathcal{G}:=((\mathfrak{L}^{+}\check{T})^{x\operatorname{-ram},\leq
n}/\mathfrak{L}^{+}T)_{X^{I}_{\operatorname{dR}}}.$

The $\mathsf{D}(X^{I})$-module category $\mathsf{IndCoh}(\mathcal{G})$ admits
a natural symmetric monoidal structure with respect to convolution, which is
in fact rigid because $\mathcal{G}$ has finite-dimensional tangent space at
the identity. Since $\rho$ is equivariant for the action of $\mathcal{G}$, the
functor $\rho_{*}$ therefore factors through

$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}})\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}})_{\mathsf{IndCoh}(\mathcal{G})}\cong\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}),$

which yields the desired functor $\pi_{*,\operatorname{ren}}$.

######  Lemma 7.4.1.

The $\mathsf{D}(X^{I})$-linear pairing

$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})\underset{\mathsf{D}(X^{I})}{\otimes}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})\xrightarrow{\Delta^{*}}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})\xrightarrow{\pi_{*,\operatorname{ren}}}\mathsf{D}(X^{I})$

is perfect. This self-duality makes the following square commute:

${\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})}$${\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})^{\vee}}$${\mathsf{D}(X^{I})}$${\mathsf{D}(X^{I})^{\vee}.}$$\scriptstyle{\sim}$$\scriptstyle{\pi_{*,\operatorname{ren}}}$$\scriptstyle{(\pi^{*})^{\vee}}$$\scriptstyle{\sim}$

###### Proof.

Since $\rho_{*}$ is continuous and

$\Delta:\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}}\underset{X^{I}_{\operatorname{dR}}}{\times}\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}}$

is affine, it follows that

$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}})\underset{\mathsf{D}(X^{I})}{\otimes}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}})\xrightarrow{\Delta^{*}}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}})\xrightarrow{\rho_{*}}\mathsf{D}(X^{I})$

is a perfect pairing. Now the fact that the pairing in lemma is perfect
follows formally from the rigidity of $\mathsf{IndCoh}(\mathcal{G})$.

As for the commutativity of the square, note that $\rho_{*}$ and $\rho^{*}$
are dual by the construction of the self-duality on

$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\log,\leq
n}_{X^{I}_{\operatorname{dR}}}).$

The claim then follows formally from the construction of
$\pi_{*,\operatorname{ren}}$.

∎

###  7.5. Proof of Theorem 7.3.1

For any $n\geq 0$ and any finite set $I$, the Contou-Carrère pairing gives
rise to a bimultiplicative line bundle

$(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})_{\operatorname{dR}}\underset{X^{I}_{\operatorname{dR}}}{\times}\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{B}\mathbb{G}_{m}$
(7.5.1)

(cf. §6.3 of [RH23]). Since $\operatorname{Gr}_{T,X^{I}}^{n\cdot x}$ is an
ind-scheme locally of finite type, Verdier duality defines a perfect pairing

$\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})\underset{\mathsf{D}(X^{I})}{\otimes}\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})\longrightarrow\mathsf{D}(X^{I}).$

Thus the line bundle (7.5.1) determines an object of

$\mathsf{QCoh}((\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})_{\operatorname{dR}}\underset{X^{I}_{\operatorname{dR}}}{\times}\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})\tilde{\longrightarrow}\operatorname{Fun}_{\mathsf{D}(X^{I})}(\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}),\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})),$

which is the desired $\mathsf{D}(X^{I})$-linear functor

$\mathbb{L}_{T}^{(n)}:\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}).$

The bimultiplicativity of (7.5.1) implies that $\mathbb{L}_{T}^{(n)}$ is
symmetric monoidal, and the compatibility of (7.5.1) with factorization
implies that $\mathbb{L}_{T}^{(n)}$ defines a morphism in
$\mathsf{ComAlg}(\operatorname{Fact}_{\operatorname{lax-untl}})$ as $I$
varies.

On the other hand, by Lemma 7.4.1, the line bundle (7.5.1) determines a
$\mathsf{D}(X^{I})$-linear functor

$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}})\longrightarrow\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}).$

Define ${}^{\prime}\mathbb{L}_{T}^{(n)}$ to be the composition of this functor
with $\operatorname{inv}^{!}$, where

$\operatorname{inv}:\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}\longrightarrow\operatorname{Gr}_{T,X^{I}}^{n\cdot x}$

is inversion of the group law. We claim that ${}^{\prime}\mathbb{L}_{T}^{(n)}$
is inverse to $\mathbb{L}_{T}^{(n)}$.

Denote by $\mathcal{L}$ the line bundle (7.5.1), viewed as an object of

$\mathsf{QCoh}((\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})_{\operatorname{dR}}\underset{X^{I}_{\operatorname{dR}}}{\times}\operatorname{LS}_{\check{T}}(D)^{x\operatorname{-ram},\leq
n}_{X^{I}_{\operatorname{dR}}}).$

Write

$\upsilon:\operatorname{Gr}_{T,X^{I}}^{n\cdot x}\longrightarrow X^{I}$

for the projection.

######  Lemma 7.5.1.

There exist canonical isomorphisms

$(\upsilon_{\operatorname{dR},*}\otimes\operatorname{id})(\mathcal{L})\cong\delta_{1,X^{I}}\quad\text{and}\quad(\operatorname{id}\otimes\pi_{*,\operatorname{ren}})(\mathcal{L})\cong(\iota^{*})^{\operatorname{L}}\mathcal{O}_{\check{T},X^{I}}.$

Here $\mathcal{O}_{\check{T},X^{I}}$ denotes the image of $\omega_{X^{I}}$
under the functor

$\operatorname{coind}_{1}^{\check{T}}:\mathsf{D}(X^{I})\longrightarrow\mathsf{Rep}(\check{T})_{X^{I}_{\operatorname{dR}}}\cong\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D))_{X^{I}_{\operatorname{dR}}}.$

Before proving the lemma, let us explain how it implies that
$\mathbb{L}_{T}^{(n)}$ and ${}^{\prime}\mathbb{L}_{T}^{(n)}$ are mutually
inverse. Using the self-duality of
$\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot x})$, we see that the
composition

${}^{\prime}\mathbb{L}_{T}^{(n)}\circ\mathbb{L}_{T}^{(n)}:\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x})\longrightarrow\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot x})$

is given by a kernel $\mathcal{K}$ in

$\mathsf{D}(\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}\underset{X^{I}_{\operatorname{dR}}}{\times}\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}).$

The bimultiplicativity of (7.5.1) implies that

$\mathcal{K}\cong(\operatorname{id}\otimes\operatorname{inv}^{!})\operatorname{mult}^{!}(\upsilon_{\operatorname{dR},*}\otimes\operatorname{id})(\mathcal{L}),$

where

$\operatorname{mult}:\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}\underset{X^{I}_{\operatorname{dR}}}{\times}\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}\longrightarrow\operatorname{Gr}_{T,X^{I}}^{n\cdot x}$

denotes the multiplication map. The lemma therefore implies that

$\mathcal{K}\cong\Delta_{\operatorname{dR},*}\omega_{\operatorname{Gr}_{T,X^{I}}^{n\cdot
x}}$

is isomorphic to the kernel corresponding to the identity functor, and hence
${}^{\prime}\mathbb{L}_{T}^{(n)}\circ\mathbb{L}_{T}^{(n)}$ is isomorphic to
the identity functor. A symmetric argumentshows that
$\mathbb{L}_{T}^{(n)}\circ^{\prime}\mathbb{L}_{T}^{(n)}$ is isomorphic to the
identity.

###### Proof of Lemma 7.5.1.

We begin by observing that $\mathbb{L}_{T}^{(n)}$ and
${}^{\prime}\mathbb{L}_{T}^{(n)}$ are mutually inverse over
$X\setminus\\{x\\}$, where they agree with the standard equivalence

$\mathsf{D}(\operatorname{Gr}_{T})\tilde{\longrightarrow}\mathsf{Rep}(\check{T})=\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(D))$

and its inverse respectively.

It follows that
$(\upsilon_{\operatorname{dR},*}\otimes\operatorname{id})(\mathcal{L})$
identifies with $\delta_{1}$ away from $x$. The factorization structure on
$(\upsilon_{\operatorname{dR},*}\otimes\operatorname{id})(\mathcal{L})$
therefore upgrades it to an object of

$\delta_{1}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{D}(\operatorname{Gr}_{T}^{n\cdot
x}))_{x}.$

Since $\delta_{1}$ is the unit object for the factorization structure on
$\mathsf{D}(\operatorname{Gr}_{T})$, restriction along the inclusion

$\operatorname{Gr}_{T,x}^{n\cdot
x}\longrightarrow\operatorname{Gr}_{T,X^{I}}^{n\cdot x}$

determines an equivalence

$\delta_{1}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{D}(\operatorname{Gr}_{T}^{n\cdot
x}))_{x}\cong\mathsf{D}(\operatorname{Gr}_{T,x}^{n\cdot x}).$

It therefore it suffices to show that

$(\upsilon_{\operatorname{dR},*}\otimes\operatorname{id})(\mathcal{L})|^{!}_{\operatorname{Gr}_{T,x}^{n\cdot
x}}\cong\delta_{1}.$

Using the fact that $\mathcal{O}_{\check{T}}$ is the unit for the
factorization structure on $\mathsf{Rep}(\check{T})$, a symmetric argument
reduces the construction of the other isomorphism to the claim that

$(\operatorname{id}\otimes\pi_{*,\operatorname{ren}})(\mathcal{L})|_{\operatorname{LS}_{\check{T}}(\mathring{D}_{x})^{\leq
n}}\cong(\iota^{*})^{\operatorname{L}}\mathcal{O}_{\check{T}}.$

Thus we have reduced to checking that $\mathbb{L}_{T}^{(n)}$ and
${}^{\prime}\mathbb{L}_{T}^{(n)}$ are mutually inverse over $x$. This is well-
known (cf. [RH23] §6.3): a choice of coordinate at $x$ allows one to decompose
$\operatorname{Gr}_{T,x}^{n\cdot x}$ and
$\operatorname{LS}_{\check{T}}(\mathring{D}_{x})^{\leq n}$ into direct factors
which can be analyzed explicitly.

∎

###  7.6. Proof of Theorem 6.6.1 for $G=T$

As explained in §6.13, it suffices to prove that $\operatorname{Sat}_{T,x}$ is
an equivalence. Let us reinterpret the monoidal functor

$\mathsf{D}(\mathcal{H}_{T,x})\xrightarrow{(\ref{presatake})}\mathcal{O}_{\check{T}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{T}\times\check{T}))_{x}.$

Being the fiber of the factorization category
$\mathsf{D}(\operatorname{Gr}_{T}^{\infty\cdot x})$ at $x$, the category
$\mathsf{D}(\mathfrak{L}T)_{x}$ admits commuting structures of
$\mathsf{D}(\mathfrak{L}T)_{x}$-module and factorization
$\mathsf{D}(\operatorname{Gr}_{T})$-module at $x$. In particular, it defines a
functor

$\mathsf{D}(\mathfrak{L}T)_{x}\text{\textendash}\mathsf{mod}\longrightarrow\mathsf{D}(\operatorname{Gr}_{T})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}.$

Similarly, the category
$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(\mathring{D}_{x}))$ admits
commuting structures of
$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(\mathring{D}_{x}))$-module and
factorization $\mathsf{Rep}(\check{T})$-module at $x$, hence it defines a
functor

$\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(\mathring{D}_{x}))\text{\textendash}\mathsf{mod}\longrightarrow\mathsf{Rep}(\check{T})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}.$

Taking the fiber of $\mathbb{L}_{T}$ at $x$, we obtain an equivalence

$\mathbbm{L}_{T,x}:\mathsf{D}(\mathfrak{L}T)_{x}\tilde{\longrightarrow}\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(\mathring{D}_{x}))$

of symmetric monoidal categories, which moreover respects the factorization
module structures for
$\mathsf{D}(\operatorname{Gr}_{T})\cong\mathsf{Rep}(\check{T})$. It therefore
gives rise to a commutative square

${\mathsf{D}(\mathfrak{L}T)_{x}\text{\textendash}\mathsf{mod}}$${\mathsf{D}(\operatorname{Gr}_{T})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}}$${\mathsf{QCoh}(\operatorname{LS}_{\check{T}}(\mathring{D}_{x}))\text{\textendash}\mathsf{mod}}$${\mathsf{Rep}(\check{T})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

in $\mathsf{DGCat}\text{\textendash}\mathsf{mod}$.

We have a monoidal equivalence

$\mathsf{D}(\mathcal{H}_{T,x})\tilde{\longrightarrow}\underline{\operatorname{End}}_{\mathsf{D}(\mathfrak{L}T)_{x}\text{\textendash}\mathsf{mod}}(\mathsf{D}(\operatorname{Gr}_{T,x})),$

where the right side means relative inner endomorphisms with respect to the
action of $\mathsf{DGCat}$ on
$\mathsf{D}(\mathfrak{L}T)_{x}\text{\textendash}\mathsf{mod}$. On the other
hand, there is a monoidal equivalence

$\mathcal{O}_{\check{T}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{T}\times\check{T}))_{x}\tilde{\longrightarrow}\underline{\operatorname{End}}_{\mathsf{Rep}(\check{T})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}}(\mathsf{Rep}(\check{T}))$

by [Ras16] §9.22.

Tracing through the constructions, we see that the previously constructed
monoidal functor

$\mathsf{D}(\mathcal{H}_{T,x})\longrightarrow\mathcal{O}_{\check{T}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{T}\times\check{T}))_{x}$

corresponds under the above identifications with the monoidal functor

$\underline{\operatorname{End}}_{\mathsf{D}(\mathfrak{L}T)_{x}\text{\textendash}\mathsf{mod}}(\mathsf{D}(\operatorname{Gr}_{T,x}))\longrightarrow\underline{\operatorname{End}}_{\mathsf{Rep}(\check{T})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}}(\mathsf{Rep}(\check{T}))$

induced by the upper-right circuit of the above commutative square. Using the
commutativity of the square (and hence Theorem 7.3.1), it follows that this
agrees with the monoidal functor induced by the lower-left circuit of the
square. But this is an equivalence by Theorem 9.13.1 in [Ras16] (in fact, the
theorem is proved after a series of reductions by establishing precisely this
equivalence).

This completes the proof of Theorem 6.6.1 for $G=T$.

##  8\. Factorization modules at a point

In this section, we will give an explicit description of the category of
factorization modules

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G)\otimes\mathsf{Rep}(M))_{x}$

at a fixed point $x\in X(k)$.

###  8.1. Lie-$!$ coalgebras

Recall that a _Lie-_ $!$ _coalgebra_ in $\mathsf{Rep}(H)$ is a Lie coalgebra
in the symmetric monoidal category $\mathsf{Rep}(H)\otimes\mathsf{D}(X)$. One
can give an alternative definition as follows. The category
$\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$ carries a non-
unital symmetric monoidal structure denoted by $\otimes^{*}$ (cf. [Ras15a]
§7.17). Note that the restriction functor

$\Delta^{!}:\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{X_{\operatorname{dR}}}\cong\mathsf{Rep}(H)\otimes\mathsf{D}(X)$

is carries $\otimes^{*}$ to $\otimes^{!}$, and in particular lifts to a
functor

$(\Delta^{!})^{\operatorname{enh}}:\mathsf{LieCoalg}^{\otimes^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}})\longrightarrow\mathsf{LieCoalg}^{\otimes^{!}}(\mathsf{Rep}(H)\otimes\mathsf{D}(X)).$

######  Proposition 8.1.1.

The functor $(\Delta^{!})^{\operatorname{enh}}$ restricts to an equivalence on
the full subcategory of
$\mathsf{LieCoalg}^{\otimes^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}})$
consisting of objects whose underlying D-module on $\operatorname{Ran}$ is
supported on the main diagonal.

###### Proof.

This follows from the observation that the left adjoint

$\Delta_{\operatorname{dR},*}:\mathsf{Rep}(H)\otimes\mathsf{D}(X)\longrightarrow\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$

of $\Delta^{!}$ is fully faithful and left-lax symmetric monoidal.

∎

There is another non-unital symmetric monoidal structure $\oplus^{*}$ on
$\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$, which is defined
in the same way as $\otimes^{*}$ but with $\boxtimes$ replaced by $\boxplus$.
Note that unlike $\otimes^{*}$, the operation $\oplus^{*}$ does not preserve
colimits in each variable. The restriction functor

$\Delta^{!}:\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{X_{\operatorname{dR}}}\cong\mathsf{Rep}(H)\otimes\mathsf{D}(X)$

carries $\oplus^{*}$ to $\oplus$, and hence lifts to a functor

$(\Delta^{!})^{\operatorname{enh},\oplus}:\mathsf{ComAlg}^{\oplus^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}})\longrightarrow\mathsf{ComAlg}^{\oplus}(\mathsf{Rep}(H)\otimes\mathsf{D}(X))\cong\mathsf{Rep}(H)\otimes\mathsf{D}(X).$

The following result first appeared in Rozenblyum’s thesis.

######  Proposition 8.1.2.

The functor $(\Delta^{!})^{\operatorname{enh},\oplus}$ admits a fully faithful
left adjoint

$\operatorname{Fact}^{\oplus}:\mathsf{Rep}(H)\otimes\mathsf{D}(X)\longrightarrow\mathsf{ComAlg}^{\oplus^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}).$

The essential image consists of those objects $\mathcal{M}$ such that the
structure map $\mathcal{M}\oplus^{*}\mathcal{M}\to\mathcal{M}$ induces an
equivalence

$(\mathcal{M}\boxplus\mathcal{M})|_{(\operatorname{Ran}_{\operatorname{dR}}\times\operatorname{Ran}_{\operatorname{dR}})_{\operatorname{disj}}}\tilde{\longrightarrow}(\operatorname{add}^{!}\mathcal{M})|_{(\operatorname{Ran}_{\operatorname{dR}}\times\operatorname{Ran}_{\operatorname{dR}})_{\operatorname{disj}}}.$

The category $\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$ is
also equipped with the “pointwise” tensor product $\otimes^{!}$. Note that for
any $\mathcal{M}$ in
$\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$, the functor
$(-)\otimes^{!}\mathcal{M}$ is lax symmetric monoidal with respect to
$\oplus^{*}$. In particular $\otimes^{!}$ naturally lifts to a symmetric
monoidal structure on
$\mathsf{ComAlg}^{\oplus^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}})$.
The functor $(\Delta^{!})^{\operatorname{enh},\oplus}$ is symmetric monoidal
with respect to $\otimes^{!}$, and hence $\operatorname{Fact}^{\oplus}$ is
left-lax symmetric monoidal. It therefore lifts to a functor

$\mathsf{LieCoalg}^{\otimes^{!}}(\mathsf{Rep}(H)\otimes\mathsf{D}(X))\longrightarrow\mathsf{LieCoalg}^{\otimes^{!}}(\mathsf{ComAlg}^{\oplus^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}})).$

### 8.2.

If $L$ is a Lie-$!$ coalgebra in $\mathsf{Rep}(H)$, we have seen that the
object $\Delta_{\operatorname{dR},*}L$ in
$\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$ has the structure
of Lie coalgebra with respect to $\otimes^{*}$. Given a finite set $I$, we can
consider $\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}$, where
$\operatorname{Ran}_{I}$ is the version of Ran space with marked points
indexed by $I$. This is acted on by
$\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$ with its
convolution product $\otimes^{*}$. In particular, we can define _Lie-_ $!$
_comodules_ for $L$ by

$L\text{\textendash}\mathsf{comod}^{!}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}:=\Delta_{\operatorname{dR},*}L\text{\textendash}\mathsf{comod}^{\otimes^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}})\times_{\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}}\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}.$

That is, an object of
$\Delta_{\operatorname{dR},*}L\text{\textendash}\mathsf{comod}^{\otimes^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}})$
is a Lie-$!$ comodule if its underlying object of $D(\operatorname{Ran}_{I})$
is supported on the main diagonal $X^{I}\to\operatorname{Ran}_{I}$.

The assignment $I\mapsto
L\text{\textendash}\mathsf{comod}^{!}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
can be promoted to an object
$L\text{\textendash}\mathsf{comod}^{!}(\mathsf{Rep}(H))$ in
$\mathsf{FactCat}^{\operatorname{lax-fact}}$, similarly to the case of Lie-$*$
modules.

On the other hand, we have the object $\operatorname{Fact}^{\oplus}(L)$ in
$\mathsf{LieCoalg}^{\otimes^{!}}(\mathsf{ComAlg}^{\oplus^{*}}(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}))$.
For any finite set $I$, we obtain
$\operatorname{Fact}^{\oplus}(L)_{X^{I}_{\operatorname{dR}}}$, an object of
$\mathsf{LieCoalg}^{\otimes^{!}}(\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}})$.
Thus we can consider

$\operatorname{Fact}^{\oplus}(L)_{X^{I}_{\operatorname{dR}}}\text{\textendash}\mathsf{comod}(\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}),$

which is acted on by $\mathsf{D}(X^{I})$. Using the commutative
$\oplus^{*}$-algebra structure on $\operatorname{Fact}^{\oplus}(L)$, one can
assemble these into an object
$\operatorname{Fact}^{\oplus}(L)\text{\textendash}\mathsf{comod}(\mathsf{Rep}(H))$
of $\mathsf{FactCat}^{\operatorname{lax-fact}}$.

######  Proposition 8.2.1.

There is a canonical equivalence

$L\text{\textendash}\mathsf{comod}^{!}(\mathsf{Rep}(H))\tilde{\longrightarrow}\operatorname{Fact}^{\oplus}(L)\text{\textendash}\mathsf{comod}(\mathsf{Rep}(H))$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$, which commutes with the
forgetful functors to $\mathsf{Rep}(H)$.

###### Proof.

For any finite set $I$, we construct an equivalence

$L\text{\textendash}\mathsf{comod}^{!}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}\operatorname{Fact}^{\oplus}(L)_{X^{I}_{\operatorname{dR}}}\text{\textendash}\mathsf{comod}(\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}})$

which lifts the identity functor on
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$. The compatibility with
factorization structures will be manifest.

Since $\operatorname{Fact}^{\oplus}$ is oplax symmetric monoidal with respect
to $\otimes^{!}$, so is

$\displaystyle\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$
$\displaystyle\longrightarrow\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$
$\displaystyle\mathcal{M}$
$\displaystyle\mapsto\operatorname{Fact}^{\oplus}(\mathcal{M})_{X^{I}_{\operatorname{dR}}}.$

This defines an oplax action of $\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$ on
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$. On the other hand, we have the
oplax symmetric monoidal functor

$\Delta_{\operatorname{dR},*}:(\mathsf{Rep}(H)_{X_{\operatorname{dR}}},\otimes^{!})\longrightarrow(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}},\otimes^{*}).$

The action of
$(\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}},\otimes^{*})$ on
$\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}$ therefore
restricts to an oplax action of $\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$.
Thus it suffices to show that

$\Delta_{I,*}:\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}$

is a morphism of oplax $\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$-module
categories with respect to the above actions. This follows from the
observation that the
$\mathcal{M}\mapsto\operatorname{Fact}^{\oplus}(\mathcal{M})_{X^{I}_{\operatorname{dR}}}$
is isomorphic to the composition of

$\displaystyle\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$
$\displaystyle\longrightarrow\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}$
$\displaystyle\mathcal{M}$
$\displaystyle\mapsto\Delta_{\operatorname{dR},*}(\mathcal{M})\otimes^{*}\Delta_{I,*}\omega_{X^{I}}$

with

$\Delta_{I}^{!}:\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}.$

∎

###  8.3. Commutative Hopf algebras

Let $\mathbb{O}$ be a symmetric monoidal DG category. The functor

$\operatorname{triv}_{\mathsf{ComAlg}}:\mathbb{O}\longrightarrow\mathsf{ComAlg}^{\operatorname{aug}}(\mathbb{O})$

admits a left adjoint $\operatorname{coPrim}_{\mathsf{ComAlg}}$, and similarly

$\operatorname{triv}_{\mathsf{LieCoalg}}:\mathbb{O}\longrightarrow\mathsf{LieCoalg}(\mathbb{O})$

admits a right adjoint $\operatorname{Prim}_{\mathsf{LieCoalg}}$. We define

$\operatorname{coChev}:=[1]\circ\operatorname{coPrim}_{\mathsf{ComAlg}}\quad\text{and}\quad\operatorname{Chev}:=[-1]\circ\operatorname{Prim}_{\mathsf{LieCoalg}}.$

The Koszul duality between the commutative operad and the Lie co-operad means
that these functor lift to adjoint functors

$\operatorname{coChev}^{\operatorname{enh}}:\mathsf{ComAlg}^{\operatorname{aug}}(\mathbb{O})\rightleftarrows\mathsf{LieCoalg}(\mathbb{O}):\operatorname{Chev}^{\operatorname{enh}}.$

######  Proposition 8.3.1.

The adjoint functors

$\operatorname{B}_{\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}}:\mathsf{Grp}(\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}})\rightleftarrows\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}:\Omega_{\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}}$

are equivalences.

###### Proof.

This is proved in much the same way as Proposition 1.6.4 in Chapter 6 of
[GR20], the key difference being that the commutativity of the square

${\mathsf{Grp}(\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}})}$${\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}}$${\mathbb{O}^{\operatorname{op}}}$${\mathbb{O}^{\operatorname{op}}.}$$\scriptstyle{\operatorname{B}_{\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}}}$$\scriptstyle{\operatorname{oblv}}$$\scriptstyle{\operatorname{oblv}}$$\scriptstyle{[1]_{\mathbb{O}^{\operatorname{op}}}}$

is less obvious here. To prove this commutativity, we first rewrite

$\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}\cong\mathsf{LieAlg}(\mathbb{O}^{\operatorname{op}}).$

We remark that $\mathsf{LieAlg}(\mathbb{O}^{\operatorname{op}})$ does not
quite fit into the framework of _loc. cit._ , since the tensor product on
$\mathbb{O}^{\operatorname{op}}$ does not generally preserve colimits, but
nonetheless we can consider Lie algebras in this category. Given $L$ in
$\mathsf{Grp}(\mathsf{LieAlg}(\mathbb{O}^{\operatorname{op}}))$, we note that
the image of the simplicial object
$(\operatorname{B}_{\mathsf{LieAlg}(\mathbb{O}^{\operatorname{op}})}L)^{\bullet}$
under the forgetful functor

$\mathsf{Grp}(\mathsf{LieAlg}(\mathbb{O}^{\operatorname{op}}))\longrightarrow\mathsf{Grp}(\mathbb{O}^{\operatorname{op}})\cong\mathbb{O}^{\operatorname{op}}$

(the equivalence here is due to the stability of
$\mathbb{O}^{\operatorname{op}}$) is simply the Čech nerve of $L\to 0$. The
tensor product on $\mathbb{O}^{\operatorname{op}}$ certainly preserves this
particular sifted colimit in each variable, since the geometric realization of
the Čech nerve in $\mathbb{O}^{\operatorname{op}}$ computes the shift $L[1]$.
The claim now follows.

∎

The category of commutative Hopf algebras in $\mathbb{O}$ is defined by

$\mathsf{ComHopfAlg}(\mathbb{O}):=\mathsf{Grp}(\mathsf{ComAlg}^{\operatorname{aug}}(\mathbb{O})^{\operatorname{op}})^{\operatorname{op}}.$

By Proposition (8.3.1) and its proof, the functor

$(\operatorname{B}_{\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}})^{\operatorname{op}}\circ\mathsf{Grp}(\operatorname{coChev}^{\operatorname{enh},\operatorname{op}})^{\operatorname{op}}:\mathsf{ComHopfAlg}(\mathbb{O})\longrightarrow\mathsf{LieCoalg}(\mathbb{O})$

fits into a commutative square

${\mathsf{ComHopfAlg}(\mathbb{O})}$${\mathsf{LieCoalg}(\mathbb{O})}$${\mathsf{ComAlg}(\mathbb{O})}$${\mathbb{O},}$$\scriptstyle{\operatorname{oblv}}$$\scriptstyle{\operatorname{oblv}}$$\scriptstyle{\operatorname{coPrim}_{\mathsf{ComAlg}}}$

and hence we denote it by

$\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}:=(\operatorname{B}_{\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}})^{\operatorname{op}}\circ\mathsf{Grp}(\operatorname{coChev}^{\operatorname{enh},\operatorname{op}})^{\operatorname{op}}.$

Note that $\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}$
admits the right adjoint

$\operatorname{U}^{\operatorname{co}}:\mathsf{Grp}(\operatorname{Chev}^{\operatorname{enh},\operatorname{op}})^{\operatorname{op}}\circ(\Omega_{\mathsf{LieCoalg}(\mathbb{O})^{\operatorname{op}}})^{\operatorname{op}}:\mathsf{LieCoalg}(\mathbb{O})\longrightarrow\mathsf{ComHopfAlg}(\mathbb{O}).$

This choice of notation is justified by the following observation.

######  Proposition 8.3.2.

The composition

$\mathsf{LieCoalg}(\mathbb{O})\xrightarrow{\operatorname{U}^{\operatorname{co}}}\mathsf{ComHopfAlg}(\mathbb{O})\xrightarrow{\operatorname{oblv}}\mathsf{AssocCoalg}^{\operatorname{aug}}(\mathbb{O})$

is right adjoint to the functor

$\operatorname{cores}^{\mathsf{AssocCoalg}\to\mathsf{LieCoalg}}:\mathsf{AssocCoalg}^{\operatorname{aug}}(\mathbb{O})\longrightarrow\mathsf{LieCoalg}(\mathbb{O})$

of corestriction along the morphism of co-operads
$\emph{Assoc}^{*}\to\emph{Lie}^{*}$.

###### Proof.

The proof is analogous to that of Theorem 6.1.2 in Chapter 6 of [GR20].

∎

######  Corollary 8.3.2.1.

For any $A$ in $\mathsf{ComHopfAlg}(\mathbb{O})$, there is a canonical
morphism

$\operatorname{cores}^{\mathsf{AssocCoalg}\to\mathsf{LieCoalg}}(A)\longrightarrow\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}(A)$

in $\mathsf{LieCoalg}(\mathbb{O})$, with the same underlying morphism in
$\mathbb{O}$ as the unit map

$A\longrightarrow\operatorname{triv}_{\mathsf{ComAlg}}\operatorname{coPrim}_{\mathsf{ComAlg}}(A).$

In particular, the forgetful functor

$A\text{\textendash}\mathsf{comod}(\mathbb{O})\longrightarrow\mathbb{O}$

naturally lifts to a functor

$A\text{\textendash}\mathsf{comod}(\mathbb{O})\longrightarrow\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}(A)\text{\textendash}\mathsf{comod}(\mathbb{O}).$

###### Proof.

The morphism in question can be obtained as the composition

$\displaystyle\operatorname{cores}^{\mathsf{AssocCoalg}\to\mathsf{LieCoalg}}(A)$
$\displaystyle\longrightarrow\operatorname{cores}^{\mathsf{AssocCoalg}\to\mathsf{LieCoalg}}(\operatorname{U}^{\operatorname{co}}(\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}(A)))$
$\displaystyle\longrightarrow\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}(A).$

∎

###  8.4. Deformation theory

Let $\pi:Z\to Y$ be a morphism of prestacks equipped with a section
$\sigma:Y\to Z$ which is affine.

######  Proposition 8.4.1.

The cotangent complex $T^{*}(Y/Z)$ relative to $\sigma$ has a natural
structure of Lie coalgebra in $\mathsf{QCoh}(Y)$, and $\sigma^{*}$ naturally
lifts to a functor

$(s^{*})^{\operatorname{enh}}:\mathsf{QCoh}(Z)\longrightarrow
T^{*}(Y/Z)\text{\textendash}\mathsf{comod}(\mathsf{QCoh}(Y)).$

###### Proof.

Let $\mathcal{G}:=Y\times_{Z}Y$, a group in prestacks affine over $Y$. Without
loss of generality, we can replace $Z$ by the classifying prestack
$\operatorname{B}_{Y}\mathcal{G}$. Let
$A\in\mathsf{ComHopfAlg}(\mathsf{QCoh}(Y)^{\leq 0})$ be the direct image of
$\mathcal{O}_{\mathcal{G}}$ under the projection $\mathcal{G}\to Y$, so we
have a canonical equivalence

$A\text{\textendash}\mathsf{comod}(\mathsf{QCoh}(Y))\tilde{\longrightarrow}\mathsf{QCoh}(\operatorname{B}_{Y}\mathcal{G})$

which fits into a commutative triangle

${A\text{\textendash}\mathsf{comod}(\mathsf{QCoh}(Y))}$${\mathsf{QCoh}(\operatorname{B}_{Y}\mathcal{G})}$${\mathsf{QCoh}(Y).}$$\scriptstyle{\operatorname{oblv}}$$\scriptstyle{s^{*}}$

Writing $e:Y\to\mathcal{G}$ for the unit section, it follows from the
definitions that we can identify

$\operatorname{coPrim}_{\mathsf{ComAlg}}(A)\cong
e^{*}T^{*}(\mathcal{G}/Y)\cong T^{*}(Y/B_{Y}\mathcal{G}).$

Using the functor
$\operatorname{coPrim}_{\mathsf{ComHopfAlg}}^{\operatorname{enh}}$ constructed
in the previous section, we obtain a Lie coalgebra structure on
$T^{*}(Y/B_{Y}\mathcal{G})$. The second claim follows from Corollary 8.3.2.1.

∎

### 8.5.

Now suppose that $\pi:Z\to Y$ is a morphism of corr-unital commutative
factorization spaces, again equipped with an affine section $\sigma:Y\to Z$.
We assume that for any injection of finite sets $J\to I$, the structure map

$Y_{I,J}\longrightarrow Y_{X^{I}_{\operatorname{dR}}}$

is a regular closed embedding, and likewise for $Z$. This allows us to equip
$\mathsf{QCoh}(Y)$ and $\mathsf{QCoh}(Z)$ with unital structures, and hence
upgrade them to objects of $\mathsf{FactCat}^{\operatorname{lax-fact}}$.
Namely, if

${Y_{I,J}}$${Y_{X^{J}_{\operatorname{dR}}}}$${Y_{X^{I}_{\operatorname{dR}}}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

is the correspondence defining the corr-unital structure on $Y$, then the
corresponding unital structure on $\mathsf{QCoh}(Y)$ is given by

$(\beta^{*})^{\operatorname{L}}\alpha^{*}:\mathsf{QCoh}(Y)_{X^{J}_{\operatorname{dR}}}\longrightarrow\mathsf{QCoh}(Y)_{X^{I}_{\operatorname{dR}}}.$

######  Lemma 8.5.1.

There is a canonical isomorphism of Lie coalgebras

$\operatorname{Fact}^{\oplus}(T^{*}(Y_{X_{\operatorname{dR}}}/Z_{X_{\operatorname{dR}}}))\tilde{\longrightarrow}T^{*}(Y_{\operatorname{Ran}_{\operatorname{dR}}}/Z_{\operatorname{Ran}_{\operatorname{dR}}})$

in $\mathsf{QCoh}(Y_{\operatorname{Ran}_{\operatorname{dR}}})$.

###### Proof.

The commutative factorization structures on $Y$ and $Z$ induce a structure of
commutative $\oplus^{*}$-algebra on
$T^{*}(Y_{\operatorname{Ran}_{\operatorname{dR}}}/Z_{\operatorname{Ran}_{\operatorname{dR}}})$,
which moreover lies in the essential image of $\operatorname{Fact}^{\oplus}$
by Proposition 8.1.2. It therefore suffices to produce an isomorphism of
Lie-$!$ coalgebras

$T^{*}(Y_{X_{\operatorname{dR}}}/Z_{X_{\operatorname{dR}}})\tilde{\longrightarrow}(\Delta^{!})^{\operatorname{enh}}T^{*}(Y_{\operatorname{Ran}_{\operatorname{dR}}}/Z_{\operatorname{Ran}_{\operatorname{dR}}}),$

which is immediate from the definitions.

∎

######  Proposition 8.5.2.

The functor $\sigma^{*}$ lifts to a morphism

$\mathsf{QCoh}(Z)\longrightarrow
T^{*}(Y_{X_{\operatorname{dR}}}/Z_{X_{\operatorname{dR}}})\text{\textendash}\mathsf{comod}^{!}(\mathsf{QCoh}(Y))$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$.

###### Proof.

This follows from Propositions 8.4.1 and 8.2.1 in view of Lemma 8.5.1.

∎

###  8.6. Deformations of local systems

Let us specialize to the situation

$\pi:\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}}{\times}\operatorname{LS}_{M}(D)\rightleftarrows\operatorname{LS}_{M}(D):\sigma.$

We begin with a geometric observation.

######  Proposition 8.6.1.

For any injection of finite sets $J\to I$, the corr-unital structure map

$\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}_{J\to
I}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}_{J\to
I}}{\times}\operatorname{LS}_{M}(D)_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}}{\times}\operatorname{LS}_{M}(D)_{X^{I}_{\operatorname{dR}}}$

is a regular closed embedding. Moreover, the section

$\sigma:\operatorname{LS}_{M}(D)_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}}{\times}\operatorname{LS}_{M}(D)_{X^{I}_{\operatorname{dR}}}$

is affine.

###### Proof.

Since $N_{P}$ is unipotent, the canonical map

$((\mathfrak{p}\otimes\Omega_{D}^{1})^{x\operatorname{-ram},\leq 1}_{J\to
I}\underset{(\mathfrak{m}\otimes\Omega_{D}^{1})^{x\operatorname{-ram},\leq
1}_{J\to
I}}{\times}(\mathfrak{m}\otimes\Omega_{D}^{1})_{X^{I}_{\operatorname{dR}}})/(\mathfrak{L}^{+}P)_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}_{J\to
I}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}_{J\to
I}}{\times}\operatorname{LS}_{M}(D)_{X^{I}_{\operatorname{dR}}}$

is an isomorphism. Both assertions follow readily from this presentation.

∎

The proposition allows us to endow

$\mathsf{QCoh}(\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}}{\times}\operatorname{LS}_{M}(D))$

with a unital structure, and hence view it as an object of
$\mathsf{FactCat}^{\operatorname{lax-fact}}$. As shown in §9.9 of [Ras16], the
$1$-affineness of $\operatorname{LS}_{P}(D)_{X^{I}_{\operatorname{dR}}}$
implies that the factorization structure is strict, i.e. this object belongs
to $\mathsf{FactCat}$.

The object $j_{!}j^{!}(\mathfrak{n}_{P}^{*}\otimes\omega_{X})$ inherits a
structure of Lie-$!$ coalgebra in $\mathsf{Rep}(M)$ from
$\mathfrak{n}_{P}^{*}\otimes\omega_{X}$. It is Verdier dual to the Lie-$*$
algebra $j_{*}j^{*}(\mathfrak{n}_{P}\otimes k_{X})$ which appears in §8 of
[Ras16].

######  Proposition 8.6.2.

The relative cotangent complex of

$\sigma:\operatorname{LS}_{M}(D)_{X_{\operatorname{dR}}}\longrightarrow\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}_{X_{\operatorname{dR}}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}_{X_{\operatorname{dR}}}}{\times}\operatorname{LS}_{M}(D)_{X_{\operatorname{dR}}}$

is canonically isomorphic to
$j_{!}j^{!}(\mathfrak{n}_{P}^{*}\otimes\omega_{X})$ as a Lie-$!$ coalgebra in
$\mathsf{Rep}(M)$.

###### Proof.

Let $U:=X\setminus\\{x\\}$. Since

$\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X_{\operatorname{dR}}}\underset{X_{\operatorname{dR}}}{\times}U_{\operatorname{dR}}=\operatorname{LS}_{H}(D)_{U_{\operatorname{dR}}}=\mathbb{B}H\times
U_{\operatorname{dR}},$

we have

$j^{!}T^{*}(\sigma)\cong
T^{*}(\operatorname{LS}_{M}(D)_{U_{\operatorname{dR}}}/\operatorname{LS}_{P}(D)_{U_{\operatorname{dR}}})\cong\mathfrak{n}_{P}^{*}\otimes\omega_{U}\cong
j^{!}(\mathfrak{n}_{P}^{*}\otimes\omega_{X}),$

from which we obtain a morphism of Lie-$!$ coalgebras

$j_{!}j^{!}(\mathfrak{n}_{P}^{*}\otimes\omega_{X})\longrightarrow
T^{*}(\sigma).$

It is an isomorphism over $U$ by construction, so it suffices to prove that

$i^{!}j_{!}j^{!}(\mathfrak{n}_{P}^{*}\otimes\omega_{X})\longrightarrow
i^{!}T^{*}(\sigma)$

is an isomorphism, where $i:\\{x\\}\to X$ is the inclusion.

Observe that

$i^{!}T^{*}(\sigma)\cong
T^{*}(\operatorname{LS}_{M}(D_{x})/\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))\cong\mathfrak{n}_{P}^{*}\otimes\operatorname{C}^{\operatorname{dR}}(\mathring{D}_{x}).$

By construction, the morphism in question is given by tensoring the canonical
isomorphism

$i^{!}j_{!}\omega_{U}\tilde{\longrightarrow}\operatorname{C}^{\operatorname{dR}}(\mathring{D}_{x})$

with $\mathfrak{n}_{P}^{*}$.

∎

######  Corollary 8.6.2.1.

The functor $\sigma^{*}$ lifts to a morphism

$\mathsf{QCoh}(\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}}{\times}\operatorname{LS}_{M}(D))\longrightarrow
j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))$

in $\mathsf{FactCat}$.

###### Proof.

Observe that

$\mathsf{QCoh}(\operatorname{LS}_{H}(D)_{X_{\operatorname{dR}}})=\mathsf{QCoh}(\mathbb{B}H\times
X_{\operatorname{dR}})=\mathsf{Rep}(H)\otimes\mathsf{D}(X)$

is canonically self-dual via the self-duality on $\mathsf{Rep}(H)$ and
$\mathsf{D}(X)$. The resulting equivalence

$\mathsf{Rep}(H)_{X}^{\operatorname{c},\operatorname{op}}\tilde{\longrightarrow}\mathsf{Rep}(H)_{X}^{\operatorname{c}}$

lifts to a contravariant equivalence $L\mapsto\mathbb{D}(L)$ between Lie
coalgebras in $\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$ (a.k.a. Lie-$!$
coalgebras in $\mathsf{Rep}(H)$) whose underlying object of
$\mathsf{Rep}(H)_{X_{\operatorname{dR}}}$ is compact and Lie-$*$ algebras in
$\mathsf{Rep}(H)$ satisfying the same condition. If $L$ is such a Lie-$*$
algebra, then there is a tautological equivalence

$L\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}\mathbb{D}(L)\text{\textendash}\mathsf{comod}^{!}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$

which upgrades to an isomorphism in $\mathsf{FactCat}^{\operatorname{lax-
fact}}$.

The fact that $j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))$ belongs to
$\mathsf{FactCat}$ (as opposed to merely $\mathsf{FactCat}^{\operatorname{lax-
fact}}$) follows from the existence of the monadic adjunction

$\operatorname{ind}:\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\rightleftarrows
j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M)):\operatorname{oblv}$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$ and
the strictness of the factorization structure on
$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))$.

∎

### 8.7.

Consider the canonical morphism of factorization spaces

$\iota:\operatorname{LS}_{P}(D)\longrightarrow\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}}{\times}\operatorname{LS}_{M}(D).$

Proposition 8.6.1 implies that $\iota^{*}$ admits a left adjoint
$(\iota^{*})^{\operatorname{L}}$.

######  Proposition 8.7.1.

The functor $(\iota^{*})^{\operatorname{L}}$ fits into a commutative square

${\mathsf{Rep}(P)}$${\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))}$${\mathsf{QCoh}(\operatorname{LS}_{P}(D)^{x\operatorname{-ram}}\underset{\operatorname{LS}_{M}(D)^{x\operatorname{-ram}}}{\times}\operatorname{LS}_{M}(D))}$${j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))}$$\scriptstyle{(\iota^{*})^{\operatorname{L}}}$$\scriptstyle{\operatorname{res}^{N_{P}}_{\mathfrak{n}_{P}}}$$\scriptstyle{(s^{*})^{\operatorname{enh}}}$

in $\mathsf{FactCat}$, where the right vertical functor is given by induction
along the morphism of Lie-$*$ algebras $\mathfrak{n}_{P}\otimes k_{X}\to
j_{*}j^{*}(\mathfrak{n}_{P}\otimes k_{X})$.

###### Proof.

The square in question tautologically commutes if the vertical functors are
replaced with their right adjoints. Thus we must show that the resulting
natural transformation

$\operatorname{ind}_{\mathfrak{n}_{P}\otimes
k_{X}}^{j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})}\operatorname{res}_{\mathfrak{n}_{P}}^{N_{P}}\longrightarrow(s^{*})^{\operatorname{enh}}(\iota^{*})^{\operatorname{L}}$

is an isomorphism. For any finite set $I$, both composite functors are
$\mathsf{Rep}(M)_{X^{I}_{\operatorname{dR}}}$-linear, and since
$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$ is generated as a
$\mathsf{Rep}(M)_{X^{I}_{\operatorname{dR}}}$-module by
$\operatorname{triv}_{P}(\omega_{X^{I}})$, it suffices to check that this
natural transformation is an isomorphism after evaluating on this object.
Applying the conservative functor

$\operatorname{oblv}:j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(M)_{X^{I}_{\operatorname{dR}}},$

we have reduced to proving that

$\operatorname{ind}_{\mathfrak{n}_{P}\otimes
k_{X}}^{j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})}\operatorname{triv}_{\mathfrak{n}_{P}}(\omega_{X^{I}})\longrightarrow
s^{*}(\iota^{*})^{\operatorname{L}}\operatorname{triv}_{P}(\omega_{X^{I}})$

is an isomorphism in $\mathsf{Rep}(M)_{X^{I}_{\operatorname{dR}}}$. Using
factorization, we we can assume that $I$ is a singleton. Over
$U=X\setminus\\{x\\}$, then this morphism is simply the identity map on
$\operatorname{triv}_{M}(\omega_{U_{\operatorname{dR}}})$, so it suffices to
check that the induced map on the fiber at $x$ is an isomorphism.

In particular, we have reduced to proving the commutativity of the square in
question at the point $x$. Then we are in the realm of finite-dimensional
deformation theory, and it can be identified with the commutative square

${\mathsf{QCoh}(\operatorname{LS}_{P}(D_{x}))}$${\mathsf{QCoh}(\operatorname{LS}_{P}(D_{x})^{\wedge}_{\operatorname{LS}_{M}(D_{x})})}$${\mathsf{QCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))}$${\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}),}$$\scriptstyle{(\iota^{*})^{\operatorname{L}}}$$\scriptstyle{(\widehat{\iota}^{*})^{\operatorname{L}}}$

where the horizontal morphisms are given by inverse image and

$\widehat{\iota}:\operatorname{LS}_{P}(D_{x})^{\wedge}_{\operatorname{LS}_{M}(D_{x})}\longrightarrow(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}$

is obtained from $\iota$ by formal completion along
$\operatorname{LS}_{M}(D_{x})$. All of the stacks appearing here are formally
smooth, and for such stacks we have the equivalence

$\Upsilon_{Y}:\mathsf{QCoh}(Y)\tilde{\longrightarrow}\mathsf{IndCoh}(Y),$

which intertwines inverse image for $\mathsf{QCoh}$ with $!$-inverse image for
$\mathsf{IndCoh}$. Since the maps $\iota$ and $\widehat{\iota}$ are proper,
the above square can be identified with

${\mathsf{IndCoh}(\operatorname{LS}_{P}(D_{x}))}$${\mathsf{IndCoh}(\operatorname{LS}_{P}(D_{x})^{\wedge}_{\operatorname{LS}_{M}(D_{x})})}$${\mathsf{IndCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))}$${\mathsf{IndCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}),}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\widehat{\iota}_{*}}$

which commutes by base change for the standard functors on $\mathsf{IndCoh}$.

∎

In particular, it follows that $(s^{*})^{\operatorname{enh}}$ induces a
morphism

$\mathsf{QCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))\longrightarrow
j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))_{x}$ (8.7.1)

of factorization $\mathsf{Rep}(P)$-module categories at $x$.

######  Proposition 8.7.2.

The morphism (8.7.1) restricts to an equivalence on the full subcategory

$\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}),$

embedded via the left adjoint of the restriction functor

$\mathsf{QCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))\longrightarrow\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}).$

###### Proof.

Identify (8.7.1) with the composition

$\displaystyle\mathsf{QCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))$
$\displaystyle\stackrel{{\scriptstyle\Upsilon}}{{\tilde{\longrightarrow}}}\mathsf{IndCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))$
$\displaystyle\longrightarrow\mathsf{IndCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})})$

as in the proof of Proposition 8.7.1.

∎

###  8.8. Factorization modules for $\Upsilon(\mathfrak{n}_{P})$

The lax factorization category
$\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$ makes
$\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))$ a factorization
$\mathsf{Rep}(H)$-module at $x$. The action of
$\mathsf{QCoh}(\operatorname{LS}_{\Gamma}(\mathring{D}_{x}))$ on itself by
tensor product of quasicoherent sheaves commutes with the factorization
$\mathsf{Rep}(H)$-module structure. This yields a functor

$\displaystyle\Phi_{H}:\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))\text{\textendash}\mathsf{mod}$
$\displaystyle\longrightarrow\mathsf{Rep}(H)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}$

which commutes with the forgetful functors to $\mathsf{DGCat}$. On the level
of objects, this means that there is a natural factorization
$\mathsf{Rep}(H)$-module structure on any $\mathcal{C}$ in
$\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))\text{\textendash}\mathsf{mod}$
coming from the factorization module structure on the second factor of

$\mathcal{C}\otimes_{\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))}\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))=\mathcal{C}.$

The restriction functor

$\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{H}(D_{x})})$

admits a fully faithful
$\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))$-linear left adjoint.
It follows that the restriction of scalars functor

$\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{H}(D_{x})})\text{\textendash}\mathsf{mod}\longrightarrow\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x}))\text{\textendash}\mathsf{mod}$

is fully faithful.

######  Theorem 8.8.1.

The restriction of $\Phi_{H}$ to the full subcategory
$\mathsf{QCoh}(\operatorname{LS}_{H}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{H}(D_{x})})\text{\textendash}\mathsf{mod}$
is fully faithful.

###### Proof.

This is Theorem 9.13.1 of [Ras16].

∎

######  Proposition 8.8.2.

There is a canonical isomorphism

$\Phi_{P}(\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}))\tilde{\longrightarrow}\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))_{x}$

in $\mathsf{Rep}(P)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}$,
where
$\mathsf{QCoh}(\operatorname{LS}_{P}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{P}(D_{x})})$
acts on

$\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})})$

via restriction along the composition

$\displaystyle(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}$
$\displaystyle\longrightarrow(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{P}(D_{x})}$
$\displaystyle\longrightarrow\operatorname{LS}_{P}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{P}(D_{x})},$

and the factorization $\mathsf{Rep}(P)$-module structure on
$\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))_{x}$
arises from the morphism

$\mathsf{Rep}(P)\longrightarrow\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\xrightarrow{\operatorname{ind}^{*\to\operatorname{fact}}}\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))_{x}$

in $\mathsf{FactCat}$.

###### Proof.

Lemma 8.16.1 of [Ras16] says that

$\operatorname{ind}^{*\to\operatorname{ch}}:j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))_{x}\tilde{\longrightarrow}\operatorname{U}^{\operatorname{ch}}(j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X}))\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(M))_{x}$

is an equivalence. Identifying

$j^{*}\operatorname{U}^{\operatorname{ch}}(j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X}))\tilde{\longrightarrow}j^{*}\operatorname{U}^{\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X})$

as chiral algebras on $X\setminus\\{x\\}$ and applying Proposition 5.1.1, we
obtain an equivalence

$j_{*}j^{*}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{*}(\mathsf{Rep}(M))_{x}\tilde{\longrightarrow}\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))_{x}.$

By construction, this equivalence respects the factorization
$\mathsf{Rep}(P)$-module structures.

We can apply $\Phi_{P}$ to the
$\mathsf{QCoh}(\operatorname{LS}_{P}(\mathring{D}_{x}))$-linear functor

$(\iota_{x}^{*})^{\operatorname{L}}:\mathsf{Rep}(P)=\mathsf{QCoh}(\operatorname{LS}_{P}(D_{x}))\longrightarrow\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})})$

to obtain a morphism in
$\mathsf{Rep}(P)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}$.
Inspecting the constructions, we see that taking the fiber at $x$ of the
morphism $(\iota^{*})^{\operatorname{L}}$ in
$\mathsf{FactCat}^{\operatorname{lax-fact}}$ from Proposition 8.7.1 yields the
same morphism. Now apply Proposition 8.7.2.

∎

###  8.9. Spectral Hecke category at a point

Finally, we deduce the promised description of factorization
$\Upsilon(\mathfrak{n},\mathcal{O}_{G})$-modules at $x$.

######  Theorem 8.9.1.

There is a canonical equivalence

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{x}\tilde{\longrightarrow}\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{G\times
M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{G\times
M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}).$

###### Proof.

As explained in [Ras16] §§9.22-24, for any factorization
$\mathsf{Rep}(G)$-module category $\mathcal{C}$ at $x$, we can identify

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G)\otimes\mathcal{C})_{x}\tilde{\longrightarrow}\operatorname{Fun}_{\mathsf{Rep}(G)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}}(\mathsf{Rep}(G),\mathcal{C}).$

Here we view the diagonal bimodule $\mathcal{O}_{G}$ as a factorization
algebra in $\mathsf{Rep}(G\times G)$, and $\mathsf{Rep}(G)\otimes\mathcal{C}$
as a factorization $\mathsf{Rep}(G\times G)$-module at $x$. Moreover, if
$\mathcal{C}$ lies in the essential image of

$\mathsf{QCoh}(\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})})\text{\textendash}\mathsf{mod}$

under $\Phi_{G}$, it follows that

$\displaystyle\mathsf{Rep}(G)\underset{\mathsf{QCoh}(\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})})}{\otimes}\mathcal{C}$
$\displaystyle\tilde{\longrightarrow}\operatorname{Fun}_{\mathsf{QCoh}(\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})})}(\mathsf{Rep}(G),\mathcal{C})$
$\displaystyle\tilde{\longrightarrow}\operatorname{Fun}_{\mathsf{Rep}(G)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{x}}(\mathsf{Rep}(G),\mathcal{C})$
$\displaystyle\tilde{\longrightarrow}\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G)\otimes\mathcal{C})_{x},$

where we used Theorem 8.8.1 for the second equivalence. The first equivalence
uses the observation that $\mathsf{Rep}(G)$ is dualizable and self-dual as an
object of
$\mathsf{QCoh}(\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})})\text{\textendash}\mathsf{mod}$,
cf. _loc. cit._ §9.18.

We now apply the above in the case

$\mathcal{C}=\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}),$

viewed as a
$\mathsf{QCoh}(\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})})$-module
via inverse image along the projection

$\operatorname{LS}_{P}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{P}(D_{x})}\longrightarrow\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})}.$

Note that by Lemma 3.6.1, we have an equivalence

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G)\otimes\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M)))_{x}\tilde{\longrightarrow}\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{x}.$

Applying Proposition 8.8.2 and the equivalence in the previous paragraph
identifies the right hand side with

$\mathsf{Rep}(G)\underset{\mathsf{QCoh}(\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})})}{\otimes}\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}).$

The morphism

$\operatorname{LS}_{G}(D_{x})\longrightarrow\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})},$

being isomorphic to $\\{0\\}/G\to\widehat{\mathfrak{g}}_{0}/G$, is affine,
which implies that the above tensor product identifies with

$\mathsf{QCoh}(\operatorname{LS}_{G}(D_{x})\underset{\operatorname{LS}_{G}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{G}(D_{x})}}{\times}(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x}))^{\wedge}_{\operatorname{LS}_{M}(D_{x})}).$

Finally, using the identification

$\operatorname{LS}_{H}(\mathring{D}_{x})^{\wedge}_{\operatorname{LS}_{H}(D_{x})}\tilde{\longrightarrow}\widehat{\mathfrak{h}}_{0}/H,$

it is easy to see that the canonical map from the above fiber product to

$\operatorname{LS}_{G}(D_{x})\underset{\operatorname{LS}_{G}(\mathring{D}_{x})}{\times}\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{M}(D_{x})=\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{G\times
M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{G\times M}(D_{x})$

factors through an isomorphism with the formal completion of the latter along
$\operatorname{LS}_{M}(D_{x})$.

∎

######  Corollary 8.9.1.1.

The monad on $\mathsf{Rep}(P)$ induced by the adjunction

$\mathsf{Rep}(P)\rightleftarrows\mathsf{Sph}^{\operatorname{spec}}_{G,P,x},$

where the left adjoint is (5.6.2), is isomorphic to the monad given by the
associative algebra $\operatorname{Sym}((\mathfrak{g}/\mathfrak{n}_{P})[-2])$.

###### Proof.

Observe that the equivalence of Theorem 8.9.1 restricts to an equivalence

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{x}\tilde{\longrightarrow}\mathsf{QCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{G\times
M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{G\times
M}(D_{x}))^{\wedge}_{\operatorname{LS}_{P}(D_{x})}).$

We claim that this equivalence is t-exact with respect to the usual
t-structure on quasicoherent sheaves: recall that

$(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{G\times
M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{G\times
M}(D_{x}))^{\wedge}_{\operatorname{LS}_{P}(D_{x})}\cong(\mathfrak{n}_{P}\underset{\mathfrak{g}}{\times}\\{0\\})/P,$

and in particular this is an algebraic stack of finite type. The t-structure
on

$\mathsf{QCoh}((\mathfrak{n}_{P}\underset{\mathfrak{g}}{\times}\\{0\\})/P)$

is uniquely characterized by the fact that direct image along

$\\{0\\}/P\longrightarrow(\mathfrak{n}_{P}\underset{\mathfrak{g}}{\times}\\{0\\})/P$

is t-exact, whence the claim.

It follows that the above equivalence preserves coherence, and hence
renormalizes to

$\mathsf{Sph}_{G,P,x}^{\operatorname{spec}}\tilde{\longrightarrow}\mathsf{IndCoh}((\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{G\times
M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{G\times
M}(D_{x}))^{\wedge}_{\operatorname{LS}_{P}(D_{x})}).$

The functor (5.6.2) corresponds to direct image of ind-coherent sheaves along

$\iota:\operatorname{LS}_{P}(D_{x})\longrightarrow(\operatorname{LS}_{P}(\mathring{D}_{x})\underset{\operatorname{LS}_{G\times
M}(\mathring{D}_{x})}{\times}\operatorname{LS}_{G\times
M}(D_{x}))^{\wedge}_{\operatorname{LS}_{P}(D_{x})},$

where we identify

$\mathsf{IndCoh}(\operatorname{LS}_{P}(D_{x}))\cong\mathsf{QCoh}(\operatorname{LS}_{P}(D_{x}))\cong\mathsf{Rep}(P)$

using the t-exact functor $\Psi$. After a choice of formal coordinate at $x$,
this becomes the map

$\\{0\\}/P\longrightarrow(\mathfrak{n}_{P}\underset{\mathfrak{g}}{\times}\\{0\\})/P.$

If we identify

$\Gamma(\mathfrak{n}_{P}\underset{\mathfrak{g}}{\times}\\{0\\},\mathcal{O})=\operatorname{Sym}((\mathfrak{g}/\mathfrak{n}_{P})^{*}[1]),$

then we have

$\iota^{!}\iota_{*}V=\operatorname{Hom}_{\operatorname{Sym}((\mathfrak{g}/\mathfrak{n}_{P})^{*}[1])}(k,V)=\operatorname{Sym}((\mathfrak{g}/\mathfrak{n}_{P})[-2])\otimes
V$

for any $V$ in $\mathsf{Rep}(P)$, i.e. the monad $\iota^{!}\iota_{*}$ is given
by the algebra $\operatorname{Sym}((\mathfrak{g}/\mathfrak{n}_{P})[-2])$
Koszul dual to $\operatorname{Sym}((\mathfrak{g}/\mathfrak{n}_{P})^{*}[1])$,
as desired.

∎

##  9\. The equivalence for $P=G$

In this section, we complete the proof of the factorizable derived Satake
equivalence, as formulated in Theorem 6.6.1.

###  9.1. The derived Satake transform as a morphism of monads

By the construction of the morphism (6.6.1), we have a commutative triangle

${\mathsf{Rep}(\check{G})}$${\mathsf{D}(\mathcal{H}_{G})}$${\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{G}))}$$\scriptstyle{\operatorname{Sat}_{G}^{\operatorname{naive}}}$$\scriptstyle{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}$$\scriptstyle{(\ref{presatake})}$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$. After renormalization, we
obtain a commutative triangle

${\mathsf{Rep}(\check{G})}$${\mathsf{Sph}_{G}}$${\mathsf{Sph}_{\check{G}}^{\operatorname{spec}}}$$\scriptstyle{\operatorname{Sat}_{G}}$

in $\mathsf{FactCat}$. By Lemma 6.3.1 and Theorem 4.6.1 respectively, the
diagonal functors preserve ULA objects over each $X^{I}$, hence admit
factorizable right adjoints. Moreover, their right adjoints are conservative
and hence monadic. The above triangle therefore corresponds to a morphism

$\Psi_{G}\longrightarrow\Psi_{\check{G}}^{\operatorname{spec}}$ (9.1.1)

between the corresponding monads on $\mathsf{Rep}(\check{G})$ in
$\mathsf{FactCat}_{\operatorname{lax-untl}}$, which we will prove is an
isomorphism. As previously explained, Theorem 6.6.1 will follow if we prove
that the morphism of monads

$\Psi_{G,x}\longrightarrow\Psi_{\check{G},x}^{\operatorname{spec}}$

is an isomorphism over each $x\in X(k)$.

###  9.2. Explicit description of the monads at a point

We will need the following lemma.

######  Lemma 9.2.1.

Let $H$ be an affine group scheme and suppose that $\mathsf{Rep}(H)$ acts on a
DG category $\mathcal{C}$. For any objects $c,d$ in $\mathcal{C}$, we have a
canonical isomorphism

$\underline{\operatorname{Hom}}_{\mathsf{Rep}(H)}(c,d)\tilde{\longrightarrow}\operatorname{Hom}_{\mathcal{C}}(c,\mathcal{O}_{H}\star
d)$

in $\mathsf{Rep}(H)$, where the $H$-action on the right side arises by
functoriality from the $H$-action on $\mathcal{O}_{H}$ by right translations.

Now we deduce an explicit description of the monad $\Psi_{G,x}$ from a well-
known equivariant cohomology calculation originally due to Ginzburg.

######  Proposition 9.2.2.

The monad

$\Psi_{G,x}:\mathsf{Rep}(\check{G})\longrightarrow\mathsf{Sph}_{G,x}\longrightarrow\mathsf{Rep}(\check{G})$

is isomorphic to the monad given by the associative algebra
$\operatorname{Sym}(\check{\mathfrak{g}}[-2])$.

###### Proof.

The functor

$\mathsf{Rep}(\check{G})\longrightarrow\mathsf{Sph}_{G,x}$

obtained by renormalizing $\operatorname{Sat}_{G,x}^{\operatorname{naive}}$ is
monoidal, and in particular can be viewed as a
$\mathsf{Rep}(\check{G})$-linear functor. It follows that $\Psi_{G,x}$ is
given by the associative algebra
$\underline{\operatorname{End}}_{\mathsf{Rep}(\check{G})}(\delta_{\mathfrak{L}^{+}G})$.
Lemma 9.2.1 yields a $\check{G}$-equivariant isomorphism

$\underline{\operatorname{End}}_{\mathsf{Rep}(\check{G})}(\delta_{\mathfrak{L}^{+}G})\tilde{\longrightarrow}\operatorname{Hom}_{\mathsf{Sph}_{G,x}}(\delta_{\mathfrak{L}^{+}G},\mathcal{O}_{\check{G}}\star\delta_{\mathfrak{L}^{+}G}).$

Note that this is an isomorphism of associative algebras in
$\mathsf{Rep}(\check{G})$, where the algebra structure on the right side is
induced by the algebra structure on $\mathcal{O}_{\check{G}}$ and composition
of morphisms in $\mathsf{Sph}_{G,x}$.

Then apply Theorem 7.6.1 from [ABG04], which supplies an isomorphism of
associative algebras

$\operatorname{Hom}_{\mathsf{Sph}_{G,x}}(\delta_{1},\mathcal{O}_{\check{G}}\star\delta_{\mathfrak{L}^{+}G})\tilde{\longrightarrow}\operatorname{Sym}(\check{\mathfrak{g}}[-2]).$

∎

Next, we show that the monad $\Psi_{\check{G},x}^{\operatorname{spec}}$ is
abstractly isomorphic to the same associative algebra as $\Psi_{G}$.

######  Proposition 9.2.3.

The monad

$\Psi_{\check{G},x}^{\operatorname{spec}}:\mathsf{Rep}(\check{G})\longrightarrow\mathsf{Sph}_{\check{G},x}^{\operatorname{spec}}\longrightarrow\mathsf{Rep}(\check{G})$

is isomorphic to the monad given by the associative algebra
$\operatorname{Sym}(\check{\mathfrak{g}}[-2])$.

###### Proof.

This is Corollary 8.9.1.1, applied in the case $\check{P}=\check{G}$.

∎

###  9.3. Another reduction step

We claim that if $\operatorname{Sat}_{G,x}$ induces an isomorphism

$\operatorname{End}_{\mathsf{Sph}_{G,x}}(\delta_{\mathfrak{L}^{+}G})\tilde{\longrightarrow}\operatorname{End}_{\mathsf{Sph}_{\check{G},x}^{\operatorname{spec}}}(\operatorname{Vac}_{\mathcal{O}_{\check{G}}})$

on endomorphisms of the unit objects, then it is an equivalence.

We have previously shown that if the morphism of monads (9.1.1) is an
isomorphism over $x$, then $\operatorname{Sat}_{G,x}$ is an equivalence.
Propositions 9.2.2 and 9.2.3 show that both of these monads are abstractly
isomorphic to the associative algebra
$\operatorname{Sym}(\check{\mathfrak{g}}[-2])$ in $\mathsf{Rep}(\check{G})$,
so we must prove that the endomorphism

$\varphi_{G}:\operatorname{Sym}(\check{\mathfrak{g}}[-2])\longrightarrow\operatorname{Sym}(\check{\mathfrak{g}}[-2])$

corresponding to (9.1.1) is an isomorphism.

It suffices to show that $\varphi_{G}$ is an isomorphism at the level of
cohomology. We view the cohomology of the associative DG algebra
$\operatorname{Sym}(\check{\mathfrak{g}}[-2])$ as a classical graded
commutative algebra, so we need only prove that $\varphi_{G}$ restricts to an
isomorphism on the generators $\check{\mathfrak{g}}$. On the other hand, the
subspace

$H^{2}(\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}})\subset\check{\mathfrak{g}}$

generates the adjoint representation, so it is enough to show that
$\varphi_{G}$ restricts to an isomorphism on the algebra of invariants
$\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}$.

From the proof of Proposition 9.2.2 we obtain a commutative square

${\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}}$${\operatorname{Sym}(\check{\mathfrak{g}}[-2])}$${\operatorname{End}_{\mathsf{Sph}_{G,x}}(\delta_{\mathfrak{L}^{+}G})}$${\underline{\operatorname{End}}_{\mathsf{Rep}(\check{G})}(\delta_{\mathfrak{L}^{+}G}),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the left vertical isomorphism is the composition

$\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}\tilde{\longrightarrow}\operatorname{Sym}(\mathfrak{g}[-2])^{G}\tilde{\longrightarrow}\operatorname{C}_{\operatorname{dR}}(\mathbb{B}G)\tilde{\longrightarrow}\operatorname{End}_{\mathsf{Sph}_{G,x}}(\delta_{\mathfrak{L}^{+}G}).$

Similarly, Proposition 9.2.3 yields a commutative square

${\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}}$${\operatorname{Sym}(\check{\mathfrak{g}}[-2])}$${\operatorname{End}_{\mathsf{Sph}_{\check{G},x}^{\operatorname{spec}}}(\operatorname{Vac}_{\mathcal{O}_{\check{G}}})}$${\underline{\operatorname{End}}_{\mathsf{Rep}(\check{G})}(\operatorname{Vac}_{\mathcal{O}_{\check{G}}}).}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

The claim now follows.

###  9.4. Conclusion of the proof

Finally, to prove that $\operatorname{Sat}_{G,x}$ induces an isomorphism on
endomorphisms of the unit objects, we reduce to the case $G=T$ using
$\operatorname{Sat}_{G,B}$ as an intermediary. Consider the commutative
diagram in $\mathsf{FactCat}$

${\mathsf{Sph}_{G}}$${\mathsf{Sph}_{\check{G}}^{\operatorname{spec}}}$${\mathsf{Sph}_{G,B}}$${\mathsf{Sph}_{\check{G},\check{B}}^{\operatorname{spec}}}$${\mathsf{Sph}_{T}}$${\mathsf{Sph}_{\check{T}}^{\operatorname{spec}},}$$\scriptstyle{\operatorname{Sat}_{G}}$$\scriptstyle{\operatorname{Sat}_{G,B}}$$\scriptstyle{\operatorname{Sat}_{T}}$

where $\mathsf{Sph}_{G}\to\mathsf{Sph}_{G,B}$ is the unique morphism of
$\mathsf{Sph}_{G}$-modules in $\mathsf{FactCat}$ (uniqueness is a consequence
of unitality), and similarly for the other vertical functors.

Taking the fiber at $x$ and passing to endomorphisms of the unit objects, we
obtain a commutative diagram of associative algebras

${\operatorname{End}_{\mathsf{Sph}_{G,x}}(\delta_{\mathfrak{L}^{+}G})}$${\operatorname{End}_{\mathsf{Sph}_{\check{G},x}^{\operatorname{spec}}}(\operatorname{Vac}_{\mathcal{O}_{\check{G}}})}$${\operatorname{End}_{\mathsf{Sph}_{G,B,x}}(\Delta^{0})}$${\operatorname{End}_{\mathsf{Sph}_{\check{G},\check{B},x}^{\operatorname{spec}}}(\operatorname{Vac}_{\Upsilon(\check{\mathfrak{n}},\mathcal{O}_{\check{G}})})}$${\operatorname{End}_{\mathsf{Sph}_{T,x}}(\delta_{\mathfrak{L}^{+}T})}$${\operatorname{End}_{\mathsf{Sph}_{\check{T},x}^{\operatorname{spec}}}(\operatorname{Vac}_{\mathcal{O}_{\check{T}}}).}$

Let us describe the algebras in the middle row: first, we have

$\operatorname{Sym}(\check{\mathfrak{t}}[-2])\tilde{\longrightarrow}\operatorname{Sym}(\mathfrak{t}^{*}[-2])\tilde{\longrightarrow}\operatorname{C}_{\operatorname{dR}}(\mathbb{B}T)\tilde{\longrightarrow}\operatorname{End}_{\mathsf{Sph}_{G,B,x}}(\Delta^{0}).$

By Corollary 8.9.1.1, we have an equivalence

$\mathsf{Sph}_{\check{G},\check{B},x}^{\operatorname{spec}}\tilde{\longrightarrow}\operatorname{Sym}((\check{\mathfrak{g}}/\check{\mathfrak{n}})[-2])\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{B}))$

which sends

$\operatorname{Vac}_{\Upsilon(\check{\mathfrak{n}},\mathcal{O}_{\check{G}})}\mapsto\operatorname{Sym}((\check{\mathfrak{g}}/\check{\mathfrak{n}})[-2]).$

In particular, we have isomorphisms

$\displaystyle\operatorname{End}_{\mathsf{Sph}_{\check{G},\check{B},x}^{\operatorname{spec}}}(\operatorname{Vac}_{\Upsilon(\check{\mathfrak{n}},\mathcal{O}_{\check{G}})})$
$\displaystyle\tilde{\longrightarrow}\operatorname{End}_{\operatorname{Sym}((\check{\mathfrak{g}}/\check{\mathfrak{n}})[-2])\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{B}))}(\operatorname{Sym}((\check{\mathfrak{g}}/\check{\mathfrak{n}})[-2]))$
$\displaystyle\tilde{\longrightarrow}\operatorname{Sym}((\check{\mathfrak{g}}/\check{\mathfrak{n}})[-2])^{\check{B}}=\operatorname{Sym}(\check{\mathfrak{t}}[-2]).$

To summarize, under these identifications the above commutative diagram of
associative algebras becomes

${\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}}$${\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}}$${\operatorname{Sym}(\check{\mathfrak{t}}[-2])}$${\operatorname{Sym}(\check{\mathfrak{t}}[-2])}$${\operatorname{Sym}(\check{\mathfrak{t}}[-2])}$${\operatorname{Sym}(\check{\mathfrak{t}}[-2]).}$$\scriptstyle{\varphi_{G}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\varphi_{T}}$$\scriptstyle{\operatorname{id}}$

Here the downward vertical arrows are given by the Chevalley homomorphism

$\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}\tilde{\longrightarrow}\operatorname{Sym}(\check{\mathfrak{t}}[-2])^{W}\longrightarrow\operatorname{Sym}(\check{\mathfrak{t}}[-2]).$

We have previously proved that Theorem 6.6.1 holds in the case $G=T$, which
implies that $\varphi_{T}$ is an isomorphism. It follows that

$\varphi_{G}:\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}\longrightarrow\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}$

is injective at the level of cohomology, hence an isomorphism, since
$\operatorname{Sym}(\check{\mathfrak{g}}[-2])^{\check{G}}$ has finite-
dimensional cohomologies.

##  10\. The equivalence for a proper parabolic

In this section, we deduce Theorem 6.12.3 from Theorem 6.6.1 and the main
theorem of [Ras16].

###  10.1. Recovering spherical objects from Whittaker invariants

Consider the pairing in $\mathsf{FactCat}$ defined as the composite

$\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)\tilde{\longrightarrow}\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\otimes\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)\longrightarrow\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi},$

where the first functor is the equivalence of Theorem 6.5.1 and the second
functor is given by convolution. By Proposition 3.4.1, this corresponds to a
morphism

$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)\longrightarrow\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi}.$
(10.1.1)

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$. By construction, this
morphism is equivariant for the right action of $\mathfrak{L}G$.

######  Proposition 10.1.1.

The morphism (10.1.1) factors through the inclusion

$\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}\longrightarrow\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi},$

and moreover sends the unit $\delta_{\mathfrak{L}^{+}G}$ to the image of the
regular bimodule $\mathcal{O}_{\check{G}}$ under the functor

$\displaystyle\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G})$
$\displaystyle\tilde{\longrightarrow}\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}$
$\displaystyle\xrightarrow{\operatorname{oblv}_{\mathfrak{L}^{+}G}}\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}.$

###### Proof.

For first assertion, note that (10.1.1) can also be obtained from the
composite

$\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)\longrightarrow\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)\xrightarrow{\operatorname{Av}_{!}^{\mathfrak{L}N^{-},\psi}}\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi},$

where the first morphism is the action of $\mathsf{Rep}(\check{G})$ coming
from $\operatorname{Sat}_{G}^{\operatorname{naive}}$.

For the second claim, apply the functor of $\mathfrak{L}^{+}G$-invariants to
(10.1.1) to obtain a morphism

$\mathsf{D}(\mathcal{H}_{G})\longrightarrow\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\tilde{\longrightarrow}\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}).$

By construction, this agrees with the morphism of associative algebras (6.5.1)
and in particular preserves the monoidal unit. The claim follows.

∎

Observe that for a fixed $x\in X(k)$, the category

$\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}$

has a canonical structure of factorization module at $x$ for the factorization
category

$\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\cong\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}),$

compatible with the functor

$\operatorname{oblv}_{\mathfrak{L}^{+}G}:\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}N^{-},\psi}\longrightarrow\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}.$

The previous proposition implies that (10.1.1) induces a morphism

$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x},$

equivariant for the right action of $(\mathfrak{L}G)_{x}$.

If $\mathcal{C}$ is a left $\mathsf{D}(\mathfrak{L}G)_{x}$-module, we obtain a
functor

$\mathcal{C}^{\mathfrak{L}^{+}G}\tilde{\longrightarrow}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\underset{\mathsf{D}(\mathfrak{L}G)_{x}}{\otimes}\mathcal{C}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}\underset{\mathsf{D}(\mathfrak{L}G)_{x}}{\otimes}\mathcal{C}.$

Note that since the $\mathsf{D}(\mathfrak{L}G)_{x}$-module and factorization
$\mathsf{Rep}(\check{G})$-module structures on the category

$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}$

commute, the category

$\mathcal{C}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}\cong\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}\underset{\mathsf{D}(\mathfrak{L}G)_{x}}{\otimes}\mathcal{C}$

automatically inherits a factorization $\mathsf{Rep}(\check{G})$-module
structure, and hence we can identify

$\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}\underset{\mathsf{D}(\mathfrak{L}G)_{x}}{\otimes}\mathcal{C}\tilde{\longrightarrow}\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathcal{C}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}.$

We will show that the resulting functor

$\mathcal{C}^{\mathfrak{L}^{+}G}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathcal{C}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}.$
(10.1.2)

is “nearly” an equivalence.

###  10.2. Tempered geometric Satake equivalence

Given a $\mathsf{D}(\mathfrak{L}G)_{x}$-module category $\mathcal{C}$, we
define the _anti-tempered_ subcategory of $\mathcal{C}^{\mathfrak{L}^{+}G}$ by

$\mathcal{C}^{\mathfrak{L}^{+}G,\operatorname{anti-
temp}}:=\ker(\mathcal{C}^{\mathfrak{L}^{+}G}\xrightarrow{\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}}\mathcal{C}^{\mathfrak{L}N^{-},\psi}).$

The _tempered_ subcategory
$\mathcal{C}^{\mathfrak{L}^{+}G,\operatorname{temp}}$ is defined to be the
right complement of $\mathcal{C}^{\mathfrak{L}^{+}G,\operatorname{anti-
temp}}$. The inclusion

$\mathcal{C}^{\mathfrak{L}^{+}G,\operatorname{temp}}\longrightarrow\mathcal{C}^{\mathfrak{L}^{+}G}$

admits a right adjoint.

######  Proposition 10.2.1.

The composite functor

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}G,\operatorname{temp}}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}G}=\mathsf{D}(\mathcal{H}_{G})_{x}\xrightarrow{(\ref{presatake})}\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}))_{x}$

is an equivalence.

###### Proof.

Recall that (6.6.1) is t-exact, and that by Theorem 6.6.1 it becomes an
equivalence after renormalization. By Corollary 2.3.7 of [Ber21], the
subcategory

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}G,\operatorname{anti-
temp}}\subset\mathsf{D}(\mathcal{H}_{G})_{x}$

consists precisely of the infinitely connective objects, i.e. those with
vanishing cohomology objects in all degrees. It follows that the functor

$\mathsf{D}(\mathcal{H}_{G})_{x}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}G,\operatorname{temp}}$

right adjoint to the inclusion realizes the latter as the left completion of
the t-structure on the former. On the other hand, the proof of Proposition
8.9.1.1 shows that we have a t-exact equivalence

$\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}))_{x}\cong\mathsf{QCoh}((\\{0\\}\underset{\check{\mathfrak{g}}}{\times}\\{0\\})/\check{G}),$

and in particular the t-structure on the source is left complete. Thus

$\mathsf{Sph}_{\check{G},x}^{\operatorname{spec}}=\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}))_{x}^{\operatorname{ren}}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}))_{x}$

realizes the target as the left completion of the source. It follows that the
functor in the proposition is obtained by applying left completion to the
equivalence of Theorem 6.6.1, hence is an equivalence.

∎

### 10.3.

We will need a couple of lemmas.

######  Lemma 10.3.1.

The convolution functor

$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}_{x}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\longrightarrow\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi}$

is an equivalence onto

$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}.$

###### Proof.

This functor is fully faithful because $\mathcal{H}_{G,x}$ is ind-proper (see
Theorem 3.1.5 of [CD21]), so it suffices to show that the image generates the
target under colimits. Tensor the functor

$\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}:\mathsf{D}(\mathcal{H}_{G})_{x}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}_{x}$

with $\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}$ over
$\mathsf{D}(\mathcal{H}_{G})_{x}$ to obtain a functor

$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}_{x}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}.$

Its composition with the functor in question identifies with

$\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}:\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\longrightarrow\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi},$

whose image generates the accessible subcategory by definition.

∎

######  Lemma 10.3.2.

The functor

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}=\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G}$

is an equivalence onto
$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}$.

###### Proof.

Recall that the inclusion

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G}$

admits a right adjoint, which is automatically
$\mathsf{D}(\mathcal{H}_{G})_{x}$-linear because
$\mathsf{D}(\mathcal{H}_{G})_{x}$ is compactly generated by left and right
dualizable objects. It follows that the functor in question is fully faithful.

To see that the image of this functor is
$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}$, it
suffices to show that that the kernel of the right adjoint

$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}$

is equal to
$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G,\operatorname{anti-temp}}$.
This follows from the commutative square

${\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}}$${\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}}$${\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}N^{-},\psi}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}}$${\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}.}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{Av}_{!}^{\mathfrak{L}N^{-},\psi}\otimes\operatorname{id}}$$\scriptstyle{\operatorname{Av}_{!}^{\mathfrak{L}N^{-},\psi}}$$\scriptstyle{\sim}$

∎

Restrict (10.1.2) to the tempered subcategory to obtain a functor

$\mathcal{C}^{\mathfrak{L}^{+}G,\operatorname{temp}}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathcal{C}^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}.$
(10.3.1)

######  Proposition 10.3.3.

The functor (10.3.1) is an equivalence for any
$\mathsf{D}(\mathfrak{L}G)_{x}$-module category $\mathcal{C}$.

###### Proof.

First, we can immediately reduce to the universal case
$\mathcal{C}=\mathsf{D}(\mathfrak{L}G)_{x}$. Namely, we have

$\mathcal{C}^{\mathfrak{L}^{+}G,\operatorname{temp}}\cong\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\underset{\mathsf{D}(\mathfrak{L}G)_{x}}{\otimes}\mathcal{C},$

and the functor in question is given by tensoring

$\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}$
(10.3.2)

with $\mathcal{C}$ over $\mathsf{D}(\mathfrak{L}G)_{x}$.

Next, we reduce to the case
$\mathcal{C}=\mathsf{D}(\operatorname{Gr}_{G})_{x}$. First, observe that

$\displaystyle\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes$
$\displaystyle\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi})_{x}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}$
$\displaystyle\tilde{\longrightarrow}\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\underset{\mathsf{D}(\mathcal{H}_{G})}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G))_{x}$
$\displaystyle\tilde{\longrightarrow}\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}G)^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x},$

where the first equivalence is because the right
$\mathsf{D}(\mathcal{H}_{G})_{x}$-module and factorization
$\mathsf{Rep}(\check{G})$-module structures on

$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}_{x}$

commute, and the second equivalence uses Lemma 10.3.1. On the other hand, we
have

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\underset{\mathsf{D}(\mathcal{H}_{G})_{x}}{\otimes}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}\tilde{\longrightarrow}\mathsf{D}(\mathfrak{L}G)_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}$

by the previous lemma. Thus if we can establish that (10.3.1) is an
equivalence for $\mathcal{C}=\mathsf{D}(\operatorname{Gr}_{G})_{x}$, then the
case $\mathcal{C}=\mathsf{D}(\mathfrak{L}G)_{x}$ will follow by tensoring with
$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{x}$ over
$\mathsf{D}(\mathcal{H}_{G})_{x}$.

To see that

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}$

is an equivalence, observe that

$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}=\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}\tilde{\longrightarrow}\mathsf{Rep}(\check{G})$

by Theorem 6.5.1. The resulting functor

$\mathsf{D}(\operatorname{Gr}_{G})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{G}))_{x}$

is an equivalence by Proposition 10.2.1.

∎

###  10.4. Proof of Theorem 6.12.3

As shown in §6.13, it is enough to prove that $\operatorname{Sat}_{G,P,x}$ is
an equivalence for a fixed $x\in X(k)$.

Apply Proposition 10.3.3 in the case
$\mathcal{C}=(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}$,
which yields the equivalence

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\tilde{\longrightarrow}\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}})_{x}.$

Theorem 6.8.1 supplies an isomorphism

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}\tilde{\longrightarrow}\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))$

in $\mathsf{FactCat}$, and the commutative triangle in the proof of Theorem
6.12.1 yields a commutative triangle

${\mathsf{Rep}(\check{G})}$${(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}}$${\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M})).}$$\scriptstyle{\Upsilon(\check{\mathfrak{n}}_{\check{P}},-)}$$\scriptstyle{\sim}$

Here the left diagonal morphism is the composite

$\mathsf{Rep}(\check{G})\longrightarrow\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\xrightarrow{\operatorname{Av}_{!}^{\mathfrak{L}N^{-},\psi}}(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}},$

where the first morphism is given by the acting on the unit via
$\operatorname{Sat}_{G}^{\operatorname{naive}}$. We therefore obtain an
equivalence

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\tilde{\longrightarrow}\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\Upsilon(\check{\mathfrak{n}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M})))_{x},$

and by definition the latter category is equivalent to

$\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{M}))_{x}.$

Equip the category

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}$

with the t-structure characterized by the requirement that the functor

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G}\longrightarrow(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}$
(10.4.1)

right adjoint to the inclusion is left t-exact. In fact, this functor is
t-exact: its kernel

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{anti-
temp}}$

equals the kernel of $\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}$ by
definition, and by Proposition 6.11.1 the latter consists of infinitely
connective objects, i.e. those with vanishing cohomology objects in all
degrees. In particular this kernel is stable under truncation functors, which
implies that (10.4.1) is t-exact as claimed.

Combining the above equivalences, we obtain

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}\tilde{\longrightarrow}\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{M}))_{x}.$
(10.4.2)

Tracing through the constructions, we see that this is the restriction of
(6.12.2) to the tempered subcategory. We claim that the triangle

${(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G}}$${(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}}}$${\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{M}))_{x}}$$\scriptstyle{(\ref{tempquotps})}$$\scriptstyle{(\ref{psfunc1})}$$\scriptstyle{(\ref{temppsequiv})}$

commutes, which amounts to the assertion that (6.12.2) vanishes on

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{anti-
temp}}.$

As noted above, this subcategory consists of infinite connective objects, and
the t-structure on

$\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{G})\otimes\mathsf{Rep}(\check{M}))_{x}\cong\mathsf{QCoh}((\check{\mathfrak{n}}_{\check{P}}\underset{\check{\mathfrak{g}}}{\times}\\{0\\})/\check{P})$

is left complete, so the claim follows.

It therefore suffices to show that (10.4.1) restricts to an equivalence on
eventually coconnective objects, or equivalently on coconnective objects. The
restriction of this functor to coconnective objects admits the left adjoint

$((\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G,\operatorname{temp}})^{\geq
0}\longrightarrow(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G}\xrightarrow{\tau^{\geq
0}}((\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G})^{\geq
0},$

which is fully faithful because (10.4.1) is t-exact. So it is enough to show
that (10.4.1) is conservative on eventually coconnective objects. But

$\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}:(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}^{+}G}\longrightarrow(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\mathfrak{L}N^{-},\psi}$

factors through (10.4.1), and the former is conservative on eventually
coconnective objects by Proposition 6.11.1.

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  * [BG08] Alexander Braverman and Dennis Gaitsgory.  Deformations of Local Systems and Eisenstein Series.  Geometric and Functional Analysis, 2008.
  * [BL18] Roman Bezrukavnikov and Ivan Losev.  On dimension groth of modular irreducible representations of semisimple Lie algebras.  326:59–89, 2018.
  * [BZFN10] David Ben-Zvi, John Francis, and David Nadler.  Integral transforms and Drinfeld centers in derived algebraic geometry.  Journal of the American Mathematical Society, 23(4):909–966, 2010\.
  * [BZNP17] David Ben-Zvi, David Nadler, and Anatoly Preygel.  Integral transforms for coherent sheaves.  Journal of the European Mathematical Society, 19(12):3763–3812, 2017.
  * [CD21] Justin Campbell and Gurbir Dhillon.  Affine Harish-Chandra bimodules and Steinberg-Whittaker localization.  arXiv preprint arXiv:2108.02806, 2021.
  * [CF21a] Lin Chen and Yuchen Fu.  An extension of the Kazhdan-Lusztig equivalence.  arXiv preprint arXiv:2111.14606, 2021.
  * [CF21b] Lin Chen and Yuchen Fu.  An extension of the Kazhdan-Lusztig equivalence.  arXiv preprint arXiv:2111.14606, 2021.
  * [CGR23] Lin Chen, Dennis Gaitsgory, and Sam Raskin.  The geometric Langlands functor II: Equivalence on the Eisenstein part.  2023\.
  * [Che20] Lin Chen.  Nearby cycles on Drinfeld-Gaitsgory-Vinberg Interpolation Grassmannian and long intertwining functor.  arXiv preprint arXiv:2008.09349, 2020.
  * [FG12] John Francis and Dennis Gaitsgory.  Chiral Koszul duality.  Selecta Mathematica, 18(1):27–87, 2012.
  * [FGV01] Edward Frenkel, Dennis Gaitsgory, and Kari Vilonen.  Whittaker patterns in the geometry of moduli spaces of bundles on curves.  Annals of Mathematics, 153(3):699–748, 2001.
  * [FR22] Joakim Færgeman and Sam Raskin.  Non-vanishing of geometric Whittaker coefficients for reductive groups.  arXiv preprint arXiv:2207.02955, 2022.
  * [FR23] Joakim Færgeman and Sam Raskin.  The Arinkin–Gaitsgory temperedness conjecture.  Bulletin of the London Mathematical Society, 55(3):1419–1446, 2023\.
  * [Gai11] Dennis Gaitsgory.  What acts on geometric Eisenstein series.  2011\.  Available at https://people.mpim-bonn.mpg.de/gaitsgde/GL/WhatActs.pdf.
  * [Gai13] Dennis Gaitsgory.  Ind-coherent sheaves.  Moscow Mathematical Journal, 13(3):399–528, 2013.
  * [Gai14] Dennis Gaitsgory.  Sheaves of categories and the notion of 1-affineness.  Available at people.math.harvard.edu/~gaitsgde/GL/shvsofcat.pdf, 2014.
  * [Gai15] Dennis Gaitsgory.  Outline of the proof of the geometric Langlands conjecture for GL(2).  Astérisque, 2015.
  * [Gai19] Dennis Gaitsgory.  The semi-infinite intersection cohomology sheaf.  2019\.  Available at people.math.harvard.edu/~gaitsgde/GL/semiinfIC.pdf.
  * [Gai22] Dennis Gaitsgory.  The semi-infinite intersection cohomology sheaf-II: the Ran space version.  In Representation Theory and Algebraic Geometry, pages 151–265. Springer, 2022.
  * [GL19] Dennis Gaitsgory and Sergey Lysenko.  Metaplectic Whittaker category and quantum groups: the” small” FLE.  arXiv preprint arXiv:1903.02279, 2019.
  * [GR19] Dennis Gaitsgory and Nick Rozenblyum.  A study in derived algebraic geometry: Volume I: correspondences and duality, volume 221.  American Mathematical Society, 2019.
  * [GR20] Dennis Gaitsgory and Nick Rozenblyum.  A study in derived algebraic geometry: Volume II: deformations, Lie theory and formal geometry, volume 221.  American Mathematical Society, 2020.
  * [Lur17a] Jacob Lurie.  Higher algebra.  Available at http://www.math.ias.edu/~lurie/papers/HA.pdf, 2017\.
  * [Lur17b] Jacob Lurie.  Higher topos theory.  Available at http://www.math.ias.edu/~lurie/papers/HTT.pdf, 2017\.
  * [Noc23] Guglielmo Nocera.  A model for the $\mathbb{E}_{3}$ fusion-convolution product of constructible sheaves on the affine Grassmannian, 2023.
  * [Ras15a] Sam Raskin.  Chiral categories.  Available at gauss.math.yale.edu/~sr2532/chiralcats.pdf, 2015.
  * [Ras15b] Sam Raskin.  D-modules on infinite-dimensional varieties.  https://gauss.math.yale.edu/ sr2532/dmod.pdf, 2015.
  * [Ras15c] Sam Raskin.  On the notion of spectral decomposition in local geometric Langlands.  2015\.  Available at gauss.math.yale.edu/~sr2532/locsys.pdf.
  * [Ras16] Sam Raskin.  Chiral principal series categories II: the factorizable Whittaker category.  2016\.  Available at https://gauss.math.yale.edu/~sr2532/cpsii.pdf.
  * [Ras19] Sam Raskin.  Homological methods in semi-infinite contexts.  Available at https://gauss.math.yale.edu/~sr2532/topalg.pdf, 2019\.
  * [Ras21a] Sam Raskin.  Chiral principal series categories I: finite dimensional calculations.  Advances in Mathematics, 388:107856, 2021.
  * [Ras21b] Sam Raskin.  $\mathcal{W}$-algebras and Whittaker categories.  Selecta Mathematica, 27(3):1–114, 2021.
  * [RH23] Sam Raskin and Justin Hilburn.  Tate’s thesis in the de Rham setting.  Journal of the American Mathematical Society, 36:917–1001, 2023\.
  * [Yan21] Ruotao Yang.  Twisted Whittaker category on affine flags and category of representations of mixed quantum group.  arXiv preprint arXiv:2104.09848, 2021.
  * [YR22] David Yang and Sam Raskin.  Affine Beilinson-Bernstein localization at the critical level.  arXiv preprint arXiv:2203.13885, 2022.