For the second equivalence, note that it suffices to prove the corresponding
claim for non-unital factorization modules. Namely, by Proposition 3.8.4 and
the definition of the t-structure on unital modules, we have

$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}})^{\leq
0,\geq-m}=A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}\cap(A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}_{X^{I}_{\operatorname{dR}}})^{\leq 0,\geq-m}.$

As for the non-unital case, we use the equivalence

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}(A\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}},$

which commutes with the forgetful functors to
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ and in particular is t-exact.
The hypotheses imply that $A\otimes\omega_{X}[-1]$ and
$B\otimes\omega_{X}[-1]$ are connective in $\mathsf{D}(X)$, and that

$H^{i}(A\otimes\omega_{X}[-1])\tilde{\longrightarrow}H^{i}(B\otimes\omega_{X}[-1])$

for $i\geq-m$. Letting $\operatorname{Ran}_{I}$ denote the space of finite
subsets of $X$ equipped with an $I$-tuple of marked points, the map

$(\operatorname{Ran}\times\operatorname{Ran}_{I})_{\operatorname{disj}}\longrightarrow\operatorname{Ran}_{I}$

given by disjoint union is étale. It follows that for any $\mathcal{M}$ in
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}^{\leq 0}$, any $n\geq 1$, and any
nonempty finite set $I^{\prime}$, we have

$H^{i}(((A\otimes\omega_{X}[-1])^{\otimes
n}\overset{\operatorname{ch}}{\otimes}\mathcal{M})|^{!}_{X^{I^{\prime}\sqcup
I}})\tilde{\longrightarrow}H^{i}(((B\otimes\omega_{X}[-1])^{\otimes
n}\overset{\operatorname{ch}}{\otimes}\mathcal{M})|^{!}_{X^{I^{\prime}\sqcup
I}})$

for all $i\geq-m$. Here we are restricting along the natural map
$X^{I^{\prime}\sqcup I}\to\operatorname{Ran}_{I}$, and
$\overset{\operatorname{ch}}{\otimes}$ denotes the chiral action of
$\mathsf{Rep}(H)_{\operatorname{Ran}_{\operatorname{dR}}}$ on
$\mathsf{Rep}(H)_{\operatorname{Ran}_{I,\operatorname{dR}}}$ (cf. [FG12]
§7.2.4). Now the claim follows from Lemma 4.4.1.

∎

######  Proposition 4.4.4.

For any commutative algebra $A$ in $\mathsf{Rep}(H)^{\leq 0}$, the t-exact
functor $\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}$
restricts to an equivalence

$(A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}})^{\heartsuit}\tilde{\longrightarrow}(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}})^{\heartsuit}.$

###### Proof.

Consider the commutative square

${H^{0}(A)\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}}$${H^{0}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}}$${A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}}$${A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}}$$\scriptstyle{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}$$\scriptstyle{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}$

where the vertical functors, given by restriction of scalars along $A\to
H^{0}(A)$, are equivalences by Lemma 4.4.3. Thus we can assume that $A$ is
classical, in which case it is immediate from the definitions that
$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}$ is fully
faithful at the level of abelian categories. It remains to prove essential
surjectivity.

Recall the t-exact equivalence

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}(A\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}.$

Writing $i:Z_{I}\to X\times X^{I}$ and $p:Z_{I}\to X^{I}$ for the inclusion of
the incidence divisor and the projection as before, the structure map of a
chiral module $\mathcal{M}$ has the form

$j_{*}j^{*}((A\otimes\omega_{X}[-1])\boxtimes\mathcal{M})\longrightarrow
i_{\operatorname{dR},*}p^{!}\mathcal{M}.$

Objects in the essential image of

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}^{\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$

correspond under the above equivalence to those $\mathcal{M}$ such that the
composite map

$(A\otimes\omega_{X}[-1])\boxtimes\mathcal{M}\longrightarrow
j_{*}j^{*}((A\otimes\omega_{X}[-1])\boxtimes\mathcal{M})\longrightarrow
i_{\operatorname{dR},*}p^{!}\mathcal{M}$ (4.4.1)

vanishes. But notice that the functor $i^{*}_{\operatorname{dR}}$ is well-
defined on $(A\otimes\omega_{X}[-1])\boxtimes\mathcal{M}$, and moreover

$i^{*}_{\operatorname{dR}}((A\otimes\omega_{X}[-1])\boxtimes\mathcal{M})=A\otimes\mathcal{M}[1].$

Since $A$ is classical and the D-module underlying $\mathcal{M}$ belongs to
$\mathsf{D}(X^{I})^{\heartsuit}$, the map

$A\otimes\mathcal{M}[1]\longrightarrow\mathcal{M}$

corresponding to (4.4.1) vanishes. Thus

$A\text{\textendash}\mathsf{mod}^{\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}=A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}.$

The assertion for unital modules follows immediately, since

$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}=A\text{\textendash}\mathsf{mod}^{\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}\cap
A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$

as subcategories of

$A\text{\textendash}\mathsf{mod}^{\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}},$

and likewise for factorization modules.

∎

###  4.5. Local acyclity of the vacuum module

We continue to let $A$ denote a commutative algebra in $\mathsf{Rep}(H)^{\leq
0}$. For any finite set $I$, the functor

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$

is t-exact, and hence its (not necessarily continuous) right adjoint
$\operatorname{coind}^{\operatorname{fact}\to\operatorname{com}}$ is left
t-exact.

######  Proposition 4.5.1.

For any finite set $I$, the following conditions are equivalent:

  1. (i)

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

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$

preserves almost ULA objects;

  2. (ii)

the functor

$\operatorname{coind}^{\operatorname{fact}\to\operatorname{com}}:A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{+}\longrightarrow
A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{+}$

preserves filtered colimits bounded uniformly from below and is
$\mathsf{D}(X^{I})^{+}$-linear;

  3. (iii)

the vacuum module $\operatorname{Vac}_{A,X^{I}}$ is almost ULA in

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}.$

Moreover, these equivalent conditions hold for the pair $(A,H)$ if and only if
they hold for $(A,1)$, i.e. $A$ viewed as a commutative algebra in
$\operatorname{Vect}^{\leq 0}$.

###### Proof.

Since
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is ULA generated, Lemma 4.5.2 below implies that (i) is equivalent to (ii).

Condition (i) implies (iii) because the unit object in
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is ULA.

Consider the commutative square of $\mathsf{D}(X^{I})$-modules

${A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}}$${A\text{\textendash}\mathsf{mod}_{X^{I}_{\operatorname{dR}}}}$${A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}}$${A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}.}$$\scriptstyle{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}$$\scriptstyle{\operatorname{oblv}_{H}}$$\scriptstyle{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}$$\scriptstyle{\operatorname{oblv}_{H}}$

The horizontal functors are t-exact and conservative, and both admit
$\mathsf{D}(X^{I})$-linear and t-exact right adjoints, induced by the right
adjoint

$\operatorname{coind}_{H}:\mathsf{D}(X^{I})\longrightarrow\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$

of $\operatorname{oblv}_{H}$. In particular, the horizontal functors preserve
almost ULA objects. Applying Lemma 2.4.1, it follows immediately that (ii)
implies (i), and that (iii) is equivalent to (iv).

As shown in the proof of Proposition 4.3.2, the object $A_{X^{I}}$ is ULA,
hence almost ULA, in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$.
Thus (i) implies (iii), and for the same reason (ii) implies (iv).

To complete the proof, it suffices to show that (iv) implies (ii). We will
prove that the $\mathsf{IndCoh}(X^{I})$-linear functor

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}_{X^{I}}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}}$

preserves almost compact objects, which will follow if we can show that

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}_{X^{I}}^{\geq
m}\longrightarrow(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}})^{\geq
m}$

preserves compact objects for any $m\in\mathbb{Z}$. As the claim is local on
$X$, we can assume for simplicity that $X$ is affine. Then the DG category
$A\text{\textendash}\mathsf{mod}_{X^{I}}$ is compactly generated by the object
$A_{X^{I}}$, and hence the (non-stable) category
$A\text{\textendash}\mathsf{mod}_{X^{I}}^{\geq m}$ is compactly generated by
the objects $\tau^{\geq m}(A_{X^{I}}[-i])$ for $i\geq 0$. Since
$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}$ is t-exact,
we have

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}\tau^{\geq
m}(A_{X^{I}}[-i])=\tau^{\geq m}(\operatorname{Vac}_{A,X^{I}}),$

and the right side is compact in
$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}})^{\geq m}$ by
hypothesis.

∎

######  Lemma 4.5.2.

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are $\mathsf{D}(S)$-modules with
compatible t-structures, and that $F:\mathcal{C}\to\mathcal{D}$ is
$\mathsf{D}(S)$-linear and t-exact. In particular, the (not necessarily
continuous) right adjoint
$F^{\operatorname{R}}:\mathcal{D}\longrightarrow\mathcal{C}$ is left t-exact.
If

$F^{\operatorname{R}}:\mathcal{D}^{+}\longrightarrow\mathcal{C}^{+}$

preserves filtered colimits bounded uniformly from below and is
$\mathsf{D}(S)^{+}$-linear, then $F$ preserves almost ULA objects. If we
assume in addition that $\mathcal{C}$ is ULA generated, then the converse
holds.

###### Proof.

Cf. Proposition B.7.1 in [Ras21a] for the corresponding statement about ULA
objects. Assume that $F^{\operatorname{R}}$ preserves filtered colimits
bounded uniformly from below, which implies that $F$ preserves almost compact
objects. We have the commutative square of t-exact functors

${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{D}}$${\mathcal{C}}$${\mathcal{D},}$$\scriptstyle{\operatorname{id}\otimes
F}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}}}$$\scriptstyle{\operatorname{ind}_{\mathcal{D}}}$$\scriptstyle{F}$

where the vertical functors are conservative and have the continuous right
adjoints $\operatorname{oblv}_{\mathcal{C}}$ and
$\operatorname{oblv}_{\mathcal{D}}$. Thus Lemma 2.4.1 implies that the upper
horizontal functor preserves almost compact objects, from which it follows
that $F$ preserves almost ULA objects.

Now assume that $\mathcal{C}$ is ULA generated and that $F$ preserves almost
ULA objects. The latter assumption implies that for any ULA object $c$ in
$\mathcal{C}$, the object

$F(\operatorname{oblv}_{\mathcal{C}}(c))=\operatorname{oblv}_{\mathcal{D}}(F(c))$

is almost compact. Since
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ is generated as an
$\mathsf{IndCoh}(S)$-module by objects of the form
$\operatorname{oblv}_{\mathcal{C}}(c)$, where $c$ is ULA in $\mathcal{C}$, it
follows that

$\operatorname{id}\otimes
F:\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}\longrightarrow\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{D}$

preserves almost compact objects, and hence $F$ does as well by the above
commutative square. It follows immediately that

$F^{\operatorname{R}}:\mathcal{D}^{+}\longrightarrow\mathcal{C}^{+}$

preserves filtered colimits bounded uniformly from below. This functor is
automatically lax $\mathsf{D}(S)^{+}$-linear, so it remains to prove that this
structure is strict. Passing to right adjoints in the square above, we obtain
another commutative square

${\mathcal{D}^{+}}$${\mathcal{C}^{+}}$${(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{D})^{+}}$${(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{+},}$$\scriptstyle{F^{\operatorname{R}}}$$\scriptstyle{\operatorname{oblv}_{\mathcal{C}}}$$\scriptstyle{\operatorname{oblv}_{\mathcal{D}}}$$\scriptstyle{(\operatorname{id}\otimes
F)^{\operatorname{R}}}$

where the vertical functors are conservative and $\mathsf{D}(S)^{+}$-linear.
Thus it suffices to show that the lower horizontal functor is strictly
$\mathsf{IndCoh}(S)^{+}$-linear. The claim is local on $S$, which we can
therefore assume is affine. But then $\mathsf{IndCoh}(S)^{+}$ is generated by
the unit object under finite colimits and filtered colimits bounded uniformly
from below.

∎

### 4.6.

The following theorem plays a key technical role in this paper.

######  Theorem 4.6.1.

If $A$ is almost of finite type, then the equivalent conditions of Proposition
4.5.1 hold.

We also record the following important consequence of the theorem here.

######  Corollary 4.6.1.1.

If $A$ is of finite type and
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is coherent for every finite set $I$, then
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))$
satisfies the hypotheses of Theorem 3.9.2. In particular, under these
hypotheses
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is almost ULA generated for every finite set $I$.

###### Proof.

First, observe that the hypothesis that
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is coherent implies that it is almost ULA generated. This follows from
Proposition 4.1.2 and the $\mathsf{D}(X^{I})$-linear adjunction

$\operatorname{ind}_{A}:\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}\rightleftarrows
A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}:\operatorname{oblv}_{A}.$

Next, we claim that
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is coherent. By Lemma 2.4.7, it suffices to show that if $M$ is compact in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$,
then $M$ almost compact. By Proposition 4.4.4, we have
$M=\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}N$ for some
$N$ in
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$,
which is therefore almost compact since
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
was assumed coherent. Now the theorem implies that $M$ is almost compact.

Proposition 4.4.4 implies that the essential image of

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\geq
0}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\geq
0}$

generates the target under colimits, which implies that
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
is almost ULA generated.

Condition (ii) of Proposition 3.9.1 is satisfied by
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))$
because it holds for $\mathsf{Rep}(H)$, and the morphism

$\operatorname{oblv}_{A}:A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))\longrightarrow\mathsf{Rep}(H)$

in
$\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}})$
is conservative over $X^{I}_{\operatorname{dR}}$ for every finite set $I$. As
for condition (iii), note that Lemma 4.3.1 implies that it holds for
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))$ because $A$ was assumed to
be of finite type, and in particular eventually coconnective. Since

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))$

is a morphism in
$\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}})$,
the condition for
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))$
follows from Proposition 4.4.4. The remaining condition in Proposition 3.9.2
follows from the corresponding property of
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))$ in a similar fashion.

∎

We remark that the hypothesis in the corollary holds if $\operatorname{Spec}A$
is a homogeneous space for $H$. Indeed, in that case we have

$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))\cong\mathsf{Rep}(H_{y})$

where $y\in(\operatorname{Spec}A)(k)$ is any geometric point and $H_{y}$ is
its stabilizer. On the other hand, if $H$ is the trivial group, then
$A\text{\textendash}\mathsf{mod}_{X^{I}_{\operatorname{dR}}}$ is usually not
coherent.

###  4.7. The case of a symmetric algebra

For the remainder of this section, we assume that $H$ is trivial unless
otherwise specified. As shown in Proposition 4.5.1, there no loss of
generality when proving Theorem 4.6.1 under this assumption.

Recall that the forgetful functor from chiral algebras to Lie-$*$ algebras on
$X$ admits a left adjoint $\operatorname{U}^{*\to\operatorname{ch}}$, called
the _chiral enveloping algebra_. We emphasize that in this paper, all chiral
algebras are non-unital, and hence our
$\operatorname{U}^{*\to\operatorname{ch}}$ is the augmentation ideal of the
unital version of the chiral enveloping algebra.

Below we prove the key special case of Theorem 4.6.1 where $A$ is a symmetric
algebra with finitely many generators. An important point is that in this
case, the chiral algebra corresponding to $A$ is also the chiral enveloping
algebra of a Lie-$*$ algebra.

######  Lemma 4.7.1.

Let $A:=\operatorname{Sym}V$ where $V$ is a complex of vector spaces. Write
$A^{+}:=\operatorname{Sym}^{>0}V$ for the augmentation ideal of $A$, viewed as
a non-unital commutative algebra. Then we have a canonical identification of
chiral algebras

$\operatorname{U}^{*\to\operatorname{ch}}(V\otimes\omega_{X}[-1])\tilde{\longrightarrow}A^{+}\otimes\omega_{X}[-1],$

where $V\otimes\omega_{X}[-1]$ has the abelian Lie-$*$ algebra structure and
$A^{+}\otimes\omega_{X}[-1]$ has the commutative chiral algebra structure
induced by the non-unital commutative algebra structure on $A^{+}$.

###### Proof.

Since $A^{+}\otimes\omega_{X}[-1]$ is commutative as a chiral algebra, the
underlying Lie-$*$ algebra

$\operatorname{oblv}^{\operatorname{ch}\to*}(A^{+}\otimes\omega_{X}[-1])$

is abelian, and we have a morphism of Lie-$*$ algebras

$V\otimes\omega_{X}[-1]\longrightarrow\operatorname{oblv}^{\operatorname{ch}\to*}(A^{+}\otimes\omega_{X}[-1]).$

By adjunction, we obtain a morphism of chiral algebras

$\operatorname{U}^{*\to\operatorname{ch}}(V\otimes\omega_{X}[-1])\longrightarrow
A^{+}\otimes\omega_{X}[-1],$

which we claim is an isomorphism. This can be seen by passing to the
associated graded of the PBW filtration on the chiral enveloping algebra, cf.
[FG12] Corollary 6.5.2.

∎

Recall that for a given Lie-$*$ algebra $L$, Lie-$*$ modules for $L$ assemble
into an object $L\text{\textendash}\mathsf{mod}^{*}$ in
$\mathsf{FactCat}^{\operatorname{lax-fact}}$. We remark that for any finite
set $I$, the unit object in
$L\text{\textendash}\mathsf{mod}^{*}_{X^{I}_{\operatorname{dR}}}$ is the
trivial module $\operatorname{triv}_{L}(\omega_{X^{I}})$.

Similarly, given a chiral algebra $B$, chiral modules for $B$ form an object
$B\text{\textendash}\mathsf{mod}^{\operatorname{ch}}$ in
$\mathsf{FactCat}^{\operatorname{lax-fact}}$. One way to see this is to use
the Koszul duality equivalence

$B\text{\textendash}\mathsf{mod}^{\operatorname{ch}}_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}\operatorname{C}^{\operatorname{ch}}(B)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}},$

where
$\operatorname{C}^{\operatorname{ch}}(B):=\operatorname{AddUnit}(\operatorname{C}^{\operatorname{ch}}_{+}(B))$
is obtained by adjoining a unit to the non-unital factorization algebra
$\operatorname{C}^{\operatorname{ch}}_{+}(B)$ Koszul dual to $B$ (here we also
used Proposition 3.8.2). This equivalence is $\mathsf{D}(X^{I})$-linear and
commutes with the forgetful functors to $\mathsf{D}(X^{I})$. We emphasize that
the unit object for the resulting factorization structure on
$B\text{\textendash}\mathsf{mod}^{\operatorname{ch}}$ corresponds to
$\operatorname{Vac}_{\operatorname{C}^{\operatorname{ch}}(B)}$, and in
particular the same object of $\mathsf{D}(X^{I})$ underlies both.

For any Lie-$*$ algebra $L$ on $X$, we have an adjunction

$\operatorname{ind}^{*\to\operatorname{ch}}:L\text{\textendash}\mathsf{mod}^{*}\rightleftarrows\operatorname{U}^{*\to\operatorname{ch}}(L)\text{\textendash}\mathsf{mod}^{\operatorname{ch}}:\operatorname{oblv}^{\operatorname{ch}\to*}$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$,
with $\operatorname{ind}^{*\to\operatorname{ch}}$ being strictly unital.

######  Lemma 4.7.2.

If $A=\operatorname{Sym}V$ where $V$ is a connective complex of vector spaces
with finite-dimensional total cohomology, then $\operatorname{Vac}_{A,X^{I}}$
is almost ULA in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}$.

###### Proof.

Let $A^{+}:=\operatorname{Sym}^{>0}V$ be as in Lemma 4.7.1, so by combining
the previous lemma and Koszul duality, we obtain a $\mathsf{D}(X^{I})$-linear
equivalence

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}(A^{+}\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}_{X^{I}_{\operatorname{dR}}},$

which is compatible with the forgetful functors to $\mathsf{D}(X^{I})$ and in
particular is t-exact. Now consider the $\mathsf{D}(X^{I})$-linear adjunction

$\operatorname{ind}^{*\to\operatorname{ch}}:(V\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{*}_{X^{I}_{\operatorname{dR}}}\rightleftarrows(A^{+}\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}_{X^{I}_{\operatorname{dR}}}:\operatorname{oblv}^{\operatorname{ch}\to*}.$

The functor $\operatorname{oblv}^{\operatorname{ch}\to*}$ intertwines the
forgetful functors to $\mathsf{D}(X^{I})$ and hence is t-exact. It follows
formally that $\operatorname{ind}^{*\to\operatorname{ch}}$ sends objects which
are almost ULA and eventually coconnective to almost ULA objects. Since
$\operatorname{ind}^{*\to\operatorname{ch}}$ is unital, it therefore suffices
to show that $\operatorname{triv}_{V\otimes\omega_{X}[-1]}(\omega_{X^{I}})$ is
almost ULA in
$(V\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{*}_{X^{I}_{\operatorname{dR}}}$
(of course it is eventually coconnective, being concentrated in cohomological
degree $-\\#I$).

For this, we recall that
$\operatorname{triv}_{V\otimes\omega_{X}[-1]}(\omega_{X^{I}})$ has the natural
nonnegative filtration whose $n^{\text{th}}$ associated graded piece is

$M_{n}:=\operatorname{ind}_{V\otimes\omega_{X}[-1]}(\operatorname{Sym}^{*,n}(V\otimes\omega_{X})\overset{*}{\otimes}(\Delta_{I})_{\operatorname{dR},*}\omega_{X^{I}}),$

where $\Delta_{I}:X^{I}\to\operatorname{Ran}_{I}$ is the main diagonal and

$\operatorname{ind}_{V\otimes\omega_{X}[-1]}:\mathsf{D}(\operatorname{Ran}_{I})\longrightarrow(V\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{*}(\mathsf{D}(\operatorname{Ran}_{I}))$

is the induction functor.

The lemma will follow if we prove these two assertions:

  1. (i)

For any $n\geq 0$, the object $M_{n}$ is ULA over $X^{I}$.

  2. (ii)

Given $m\in\mathbb{Z}$, for $n$ sufficiently large we have

$\operatorname{Hom}(M_{n},N)=0$

for any $N$ in
$((V\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{*}_{X^{I}_{\operatorname{dR}}})^{\geq
m}$.

For claim (i), note that $\operatorname{ind}_{V\otimes\omega_{X}[-1]}$
preserves ULA objects, being left adjoint to the $D(X^{I})$-linear functor
$\operatorname{oblv}_{V\otimes\omega_{X}[-1]}$. The object

$(V\otimes\omega_{X})^{\overset{*}{\otimes}n}\overset{*}{\otimes}(\Delta_{I})_{\operatorname{dR},*}\omega_{X^{I}}$

is the de Rham direct image of $V^{\otimes n}\otimes\omega_{X^{n}\times
X^{I}}$ along the proper map

$X^{n}\times
X^{I}\longrightarrow\operatorname{Ran}\times\operatorname{Ran}_{I}\longrightarrow\operatorname{Ran}_{I},$

hence is ULA over $X^{I}$. It follows that the object

$\operatorname{Sym}^{*,n}(V\otimes\omega_{X})\overset{*}{\otimes}(\Delta_{I})_{\operatorname{dR},*}\omega_{X^{I}}$

obtained by taking coinvariants for the symmetric group $\Sigma_{n}$ is also
ULA over $X^{I}$.

Finally, for (ii) it is enough to show that given $m\in\mathbb{Z}$, for $n$
sufficiently large we have

$\operatorname{Hom}_{\mathsf{D}(\operatorname{Ran}_{I})}((V\otimes\omega_{X})^{\overset{*}{\otimes}n}\overset{*}{\otimes}\Delta_{I,*}\omega_{X^{I}},(\Delta_{I})_{\operatorname{dR},*}N)=0$

for any $N\in\mathsf{D}(X^{I})^{\geq m}$ (since the same will therefore hold
for the $\Sigma_{n}$-coinvariants

$\operatorname{Sym}^{*,n}(V\otimes\omega_{X})\overset{*}{\otimes}(\Delta_{I})_{\operatorname{dR},*}\omega_{X^{I}}$

of the first object). Given nonempty finite $I^{\prime}$, define $Z$ to be the
fiber product

${Z}$${X^{n}\times X^{I}}$${X^{I^{\prime}\sqcup
I}}$${\operatorname{Ran}_{I}.}$

By base change, the object

$((V\otimes\omega_{X})^{\overset{*}{\otimes}n}\overset{*}{\otimes}(\Delta_{I})_{\operatorname{dR},*}\omega_{X^{I}})|^{!}_{X^{I^{\prime}\sqcup
I}}$

is the de Rham direct image of $V^{\otimes n}\otimes\omega_{X^{n}\times
X^{I}}$ along the finite map $Z\to X^{I^{\prime}\sqcup I}$, hence is
concentrated in cohomological degrees $\leq-n-\\#I$. Considering the same
cartesian square in the case $n=0$ shows that
$((\Delta_{I})_{\operatorname{dR},*}N)|^{!}_{X^{I^{\prime}\sqcup I}}$ is
concentrated in cohomological degrees $\geq-m$. The claim follows.

∎

###  4.8. Filtrations and the associated graded

Let $A$ be a commutative algebra in $\operatorname{Vect}^{\leq 0}$ equipped
with a nonnegative filtration. The functor

$\operatorname{gr}:A\text{\textendash}\mathsf{mod}^{\operatorname{fil}}\to\operatorname{gr}(A)\text{\textendash}\mathsf{mod}^{\operatorname{gr}}$

which takes the associated graded module is symmetric monoidal, as is the
forgetful functor

$\operatorname{oblv}^{\operatorname{fil}}:A\text{\textendash}\mathsf{mod}^{\operatorname{fil}}\longrightarrow
A\text{\textendash}\mathsf{mod}.$

In particular, both give rise to morphisms in $\mathsf{FactCat}$.

We will write

$A\text{\textendash}\mathsf{mod}^{\operatorname{fil}}_{\geq 0}\subset
A\text{\textendash}\mathsf{mod}^{\operatorname{fil}}$

for the full symmetric monoidal subcategory consisting of nonnegatively
filtered modules (not to be confused with a constraint on cohomological
degree, which we indicate with a superscript), and likewise for graded
modules.

We also observe that the category
$(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$ has a
grading by “total degree fiberwise over $X^{I}$,” characterized by the
condition that the natural conservative functor

$(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{Vect}^{\operatorname{gr}}\otimes
D(X^{I})$

respects this grading. We will write $M_{\operatorname{td}=i}$ for the graded
component of $M$ of total degree $i$.

######  Lemma 4.8.1.

Let $B$ be a nonnegatively graded connective commutative algebra, and suppose
we are given an inverse system

$\cdots\to M_{3}\to M_{2}\to M_{1}$

in
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$
such that for each $i\in\mathbb{Z}$, the inverse system

$\cdots\to(\operatorname{oblv}_{B}M_{3})_{\operatorname{td}=i}\to(\operatorname{oblv}_{B}M_{2})_{\operatorname{td}=i}\to(\operatorname{oblv}_{B}M_{1})_{\operatorname{td}=i}$

in $(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$
stabilizes. Then the forgetful functor

$\operatorname{oblv}_{B}:B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}\longrightarrow(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$

preserves the limit $\lim_{n}M_{n}$.

###### Proof.

By Proposition 3.8.4, the inclusion

$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}\longrightarrow
B\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$

preserves limits. Applying the equivalence

$B\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}(B\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}},$

it suffices to check the corresponding claim in the category of non-unital
chiral modules. The forgetful functor

$(B\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}((\operatorname{Vect}^{\operatorname{gr}})_{\operatorname{Ran}_{I,\operatorname{dR}}})\longrightarrow(\operatorname{Vect}^{\operatorname{gr}})_{\operatorname{Ran}_{I,\operatorname{dR}}}$

preserves limits, so it is enough to prove that the limit
$\lim_{n}\operatorname{oblv}_{B}M_{n}$ taken in
$(\operatorname{Vect}^{\operatorname{gr}})_{\operatorname{Ran}_{I,\operatorname{dR}}}$
belongs to the full subcategory
$(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$. But
this is clear, since for each graded component of fixed total degree the limit
stabilizes and hence belongs to
$(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$.

∎

######  Corollary 4.8.1.1.

With $B$ as in the lemma, suppose we are given a sequence of objects
$Q_{1},Q_{2},\cdots$ in
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$
such that for each $i\geq 1$, the object $\operatorname{oblv}_{B}Q_{i}$ in
$(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$ is
concentrated in total degree $\geq i$. Then the map

$\bigoplus_{i\geq 1}Q_{i}\longrightarrow\prod_{i\geq 1}Q_{i}$

in
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$
is an isomorphism.

###### Proof.

Applying the lemma to the inverse system with terms

$\prod_{i=1}^{n}Q_{i},$

we see that the product

$\prod_{i\geq 1}Q_{i}$

is preserved by $\operatorname{oblv}_{B}$. Now the corollary follows from the
observation that the map

$\bigoplus_{i\geq 1}\operatorname{oblv}_{B}Q_{i}\longrightarrow\prod_{i\geq
1}\operatorname{oblv}_{B}Q_{i}$

in $\operatorname{Vect}^{\operatorname{gr}}_{X^{I}_{\operatorname{dR}}}$ is an
isomorphism, since in each graded component of fixed total degree, all but
finitely many factors in the product vanish.

∎

######  Lemma 4.8.2.

Fix a nonnegatively filtered connective commutative algebra $A$. If the
conditions of Proposition 4.5.1 hold for $\operatorname{gr}(A)$, then they
hold for $\operatorname{oblv}^{\operatorname{fil}}A$.

###### Proof.

_Step 1:_ With $B$ as in Lemma 4.8.1, fix $m\in\mathbb{Z}$ and suppose that
$P$ is a compact object in
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}^{\geq
m}$. We claim that there exists $r\geq 0$ such that for any object $Q$ of
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}^{\geq
m}$ with $\operatorname{oblv}_{B}Q$ concentrated in total degree $\geq r$, we
have

$\operatorname{Hom}_{B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}}(P,Q)=0.$

Otherwise we could choose a sequence $Q_{1},Q_{2},\cdots$ in
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}^{\geq
m}$ such that $\operatorname{oblv}_{B}Q_{i}$ is concentrated in total degree
$\geq i$ and

$\operatorname{Hom}_{B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}}(P,Q_{i})\neq
0$

for any $i\geq 1$. By Corollary 4.8.1.1, this contradicts the compactness of
$P$ in
$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}^{\geq
m}$.

_Step 2:_ By Lemma 3.4.2, the functor

$\operatorname{gr}:(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}\longrightarrow(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}$

admits a right adjoint which is $\mathsf{D}(X^{I})$-linear and compatible with
factorization. In particular, this lifts to a $\mathsf{D}(X^{I})$-linear
adjunction

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}\rightleftarrows\operatorname{gr}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}.$

We define a functorial _descending_ filtration on any $M$ in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}$
as follows. Let $F_{0}:=M$, and for any $n\geq 0$ define

$F_{n+1}:=\operatorname{fib}(F_{n}\to\operatorname{gr}F_{n})$

(here we are suppressing from the notation the right adjoint of
$\operatorname{gr}$, which commutes with the forgetful functors to
$\mathsf{D}(X^{I})$). We claim that if $M$ is concentrated in total degree
$\geq 0$ then

$(\operatorname{oblv}_{A}M)_{\operatorname{td}=0}\longrightarrow(\operatorname{oblv}_{A}\operatorname{gr}M)_{\operatorname{td}=0}$

is an isomorphism. It will then follow inductively that $F_{n}$ is
concentrated in total degree $\geq n$ for any $n\geq 0$, and hence that

$M\tilde{\longrightarrow}\lim_{n}M/F_{n}$

by Lemma 4.8.1.

Recall that $X^{I}$ is stratified with strata $Z(p)$ indexed by surjections
$p:I\to J$. Using the Cousin filtration, it is enough to check the claim over
each $Z(p)$. There we are working in the multifiltered category
$(\operatorname{Vect}^{\operatorname{fil}})^{\otimes
J}\otimes\mathsf{D}(Z(p))$ with its standard grading by total degree, and the
claim is evident.

_Step 3:_ Let $V:=\operatorname{ind}_{X^{I}}(\operatorname{Vac}_{A,X^{I}})$,
an object of
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}$,
so the lemma will follow if we prove that
$\operatorname{oblv}^{\operatorname{fil}}V$ is almost compact in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}$.
Fixing $m\in\mathbb{Z}$, we must show that

$\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}}(\operatorname{oblv}^{\operatorname{fil}}V,N)$

preserves filtered colimits when viewed as a functor of $N$ in
$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}})^{\geq
m}$. Note that the functor

$\operatorname{oblv}^{\operatorname{fil}}:\operatorname{Vect}^{\operatorname{fil}}_{\geq
0}\longrightarrow\operatorname{Vect}$

admits a symmetric monoidal right adjoint
$\operatorname{triv}^{\operatorname{fil}}_{\geq 0}$, which equips a vector
space with the trivial nonnegative filtration. In particular, this lifts to a
$\mathsf{D}(X^{I})$-linear adjunction

$\operatorname{oblv}^{\operatorname{fil}}:A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}}_{\geq
0})_{X^{I}_{\operatorname{dR}}}\rightleftarrows
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}:\operatorname{triv}^{\operatorname{fil}}_{\geq
0}.$

Thus we have

$\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}}(\operatorname{oblv}^{\operatorname{fil}}V,N)=\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,\operatorname{triv}^{\operatorname{fil}}_{\geq
0}N).$

Fix $r$ as in Step 1 for $B:=\operatorname{gr}(A)$ and $P:=\tau^{\geq
m}\operatorname{gr}(V)$. Recalling the descending filtration on
$M:=\operatorname{triv}^{\operatorname{fil}}_{\geq 0}N$ constructed in Step 2,
we claim that

$\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,F_{r})=0.$

Since $F_{r}=\lim_{k}F_{r}/F_{r+k}$, it suffices to show that

$\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,F_{n}/F_{n+1})=0$

for any $n\geq r$. By construction we have

$\displaystyle\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,F_{n}/F_{n+1})$
$\displaystyle=\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,\operatorname{gr}F_{n})$
$\displaystyle=\operatorname{Hom}_{\operatorname{gr}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}}(\operatorname{gr}(V),\operatorname{gr}F_{n}).$

Note that $M=\operatorname{triv}^{\operatorname{fil}}_{\geq 0}N$ implies that
$\operatorname{gr}F_{n}$ is concentrated in cohomological degrees $\geq m$ for
any $n\geq 0$ (this can be checked over each stratum $Z(p)$, where it becomes
an obvious statement about the trivial nonnegative multifiltration). Now the
claim follows from Step 1.

_Step 4:_ Continuing the notation of Step 3, we have

$\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}}}(\operatorname{oblv}^{\operatorname{fil}}V,N)=\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,M/F_{r}).$

Thus it suffices to show that for any $n\geq 0$,

$\operatorname{Hom}_{A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{fil}})_{X^{I}_{\operatorname{dR}}}}(V,F_{n}/F_{n+1})=\operatorname{Hom}_{\operatorname{gr}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}}(\operatorname{gr}(V),\operatorname{gr}F_{n})$

preserves filtered colimits when viewed as a functor of $N$ in
$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}})^{\geq
m}$. Since $\operatorname{gr}F_{n}$ belongs to
$\operatorname{gr}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\operatorname{Vect}^{\operatorname{gr}})_{X^{I}_{\operatorname{dR}}}^{\geq
m}$ by Step 3, we are done by the hypothesis on $\operatorname{gr}(A)$.

∎

###  4.9. Finitely presented algebras

A connective commutative algebra $A$ is called $0$_-finitely presented_ if it
is isomorphic to a classical finitely generated polynomial algebra. For $n\geq
1$, we say that $A$ is $n$_-finitely presented_ if

$A\cong B\underset{\operatorname{Sym}(V[n-1])}{\otimes}k$

where $B$ is $(n-1)$-finitely presented and $V$ is a finite-dimensional
classical vector space equipped with a $k$-linear map $V[n-1]\to B$. If $A$ is
$n$-finitely presented for some $n\geq 0$, we say that $A$ is _finitely
presented_.

For example, an algebra $A$ is $1$-finitely presented if and only if its
spectrum is a derived global complete intersection.

######  Lemma 4.9.1.

The conditions of Proposition 4.5.1 hold for $A$ finitely presented.

###### Proof.

We prove by induction on $n$ that the conditions hold for any $A$ of the form
$B\otimes\operatorname{Sym}V$, where $B$ is $n$-finitely presented and $V$ is
a connective complex of vector spaces with finite-dimensional total cohomology
(for varying $n$, this statement is equivalent to the assertion of the lemma).
The case $n=0$ is precisely Lemma 4.7.2. In general, we have

$B\cong C\underset{\operatorname{Sym}(W[n-1])}{\otimes}k$

where $C$ is $(n-1)$-finitely presented and $W$ is finite-dimensional and
classical. Thus $B$ admits a nonnegative filtration with associated graded
algebra

$\operatorname{gr}B\cong C\otimes\operatorname{Sym}(W[n]).$

Consequently $A$ admits a nonnegative filtration with associated graded

$\operatorname{gr}A\cong
C\otimes\operatorname{Sym}(W[n])\otimes\operatorname{Sym}V\cong
C\otimes\operatorname{Sym}(W[n]\oplus V).$

Theorem 4.6.1 holds for $\operatorname{gr}A$ by the inductive hypothesis,
whence the theorem holds for $A$ by Lemma 4.8.2.

∎

###  4.10. Algebras almost of finite type

We will use the following approximation lemma to prove Theorem 4.6.1.

######  Lemma 4.10.1.

If $A$ is connective and almost of finite type, there exists a directed system

$B_{0}\longrightarrow B_{1}\longrightarrow B_{2}\longrightarrow\cdots$

of connective commutative algebras mapping to $A$ such that

  1. (1)

$B_{n}$ is $n$-finitely presented for every $n\geq 0$,

  2. (2)

$H^{-i}(B_{n})\tilde{\to}H^{-i}(A)$ for $i<n$,

  3. (3)

$H^{-n}(B_{n})\to H^{-n}(A)$ is surjective.

In particular $\operatorname{colim}_{n}B_{n}\tilde{\to}A$.

Finally, we can prove the theorem.

###### Proof of Theorem 4.6.1.

Choose a filtration $\operatorname{colim}_{n}B_{n}\tilde{\to}A$ as in Lemma
4.10.1. Fix $m\geq 0$. Lemma 4.4.3 implies that for $n>m$, the restriction of
scalars induces an equivalence

$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}})^{\leq
0,\geq-m}\tilde{\longrightarrow}(B_{n}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}_{\operatorname{dR}}})^{\leq
0,\geq-m}.$

Moreover, by Lemmas 4.3.1 and 4.4.2, this equivalence sends

$\tau^{\geq-m}(\operatorname{Vac}_{A,X^{I}}[-\\#I])\mapsto\tau^{\geq-m}(\operatorname{Vac}_{B_{n},X^{I}}[-\\#I]).$

The inclusion

$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}})^{\leq
0,\geq-m}\longrightarrow(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}})^{\geq-m}$

preserves compact objects because the t-structure on
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}}$ is compatible
with filtered colimits. Since $\operatorname{Vac}_{B_{n},X^{I}}$ is almost ULA
by Lemma 4.9.1, it follows that

$\tau^{\geq-m}(\operatorname{oblv}_{X^{I}}\operatorname{Vac}_{A,X^{I}}[-\\#I])$

is compact in
$(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}_{X^{I}})^{\geq-m}$.
Thus $\operatorname{Vac}_{A,X^{I}}$ is almost ULA as desired.

∎

##  5\. Spectral Hecke categories

In this section, we construct the factorization categories which appear on the
spectral (a.k.a. Langlands dual) side. These arise as renormalizations of
certain categories of factorization modules.

###  5.1. Chiral induction

Recall that the Lie bracket on $\mathfrak{n}_{P}$ induces the structure of
Lie-$*$ algebra on the constant sheaf $\mathfrak{n}_{P}\otimes k_{X}$. The
category of Lie-$*$ modules

$(\mathfrak{n}_{P}\otimes k_{X})\text{-mod}^{*}(\mathsf{Rep}(M))$

forms an object in $\mathsf{FactCat}^{\operatorname{lax-fact}}$. Lemma 7.7.1
in [Ras21a] says that $(\mathfrak{n}_{P}\otimes
k_{X})\text{-mod}^{*}(\mathsf{Rep}(M))$ is equivalent to the factorization
category attached to the symmetric monoidal category
$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))$, and in
particular belongs to $\mathsf{FactCat}$. In what follows we will pass freely
between the two.

Following [Ras21a], we write

$\Upsilon(\mathfrak{n}_{P},-):\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\longrightarrow\mathsf{Rep}(M)$

for the morphism in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ defined as
the composition

$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\xrightarrow{\operatorname{ind}^{*\to\operatorname{ch}}}\operatorname{U}^{*\to\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X})\text{-mod}^{\operatorname{ch}}(\mathsf{Rep}(M))\xrightarrow{\operatorname{oblv}}\mathsf{Rep}(M)$

(this is the chiral analogue of the $\mathbb{E}_{2}$-monoidal functor denoted
by $\operatorname{Chev}_{\Upsilon}$ in the introduction).

Denote by $\Upsilon(\mathfrak{n}_{P})$ the factorization algebra in
$\mathsf{Rep}(M)$ obtained by applying $\Upsilon(\mathfrak{n}_{P},-)$ to the
unit object in
$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))$. In
particular, applying (3.5.1) to $\Upsilon(\mathfrak{n}_{P},-)$ yields a
morphism

$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\longrightarrow\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))$
(5.1.1)

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

######  Proposition 5.1.1.

We have a canonical identification

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

of factorization algebras in $\mathsf{Rep}(M)$, i.e.
$\Upsilon(\mathfrak{n}_{P})$ is the factorization algebra corresponding to the
chiral enveloping algebra of the Lie-$*$ algebra $\mathfrak{n}_{P}\otimes
k_{X}$. Moreover, we have a commutative triangle

${\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))}$${\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))}$${\operatorname{U}^{*\to\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(M))}$$\scriptstyle{(\ref{indups})}$$\scriptstyle{\operatorname{ind}^{*\to\operatorname{ch}}}$$\scriptstyle{\sim}$

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

###### Proof.

Since the functor $\operatorname{ind}^{*\to\operatorname{ch}}$ is unital, the
factorization algebra $\Upsilon(\mathfrak{n}_{P})$ is by definition the object
of $\mathsf{Rep}(M)$ underlying the unit in

$\operatorname{U}^{*\to\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(M)).$

As explained in §4.7, the latter is
$\operatorname{C}^{\operatorname{ch}}(\operatorname{U}^{*\to\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X}))$, whence the first assertion of the proposition.

As for the commutative triangle, we construct the horizontal equivalence as
the composition

$\displaystyle\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))$
$\displaystyle\tilde{\longrightarrow}\operatorname{C}^{\operatorname{ch}}_{+}(\operatorname{U}^{*\to\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X}))\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(M))$
$\displaystyle\tilde{\longrightarrow}\operatorname{U}^{*\to\operatorname{ch}}(\mathfrak{n}_{P}\otimes
k_{X})\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(M)).$

The commutativity of the triangle is then immediate from the construction of
(5.1.1.

∎

Invoking the identifications from Proposition 5.1.1, we have an adjunction

$\operatorname{ind}^{*\to\operatorname{ch}}:\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\rightleftarrows\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M)):\operatorname{oblv}^{\operatorname{ch}\to*}$
(5.1.2)

in $\mathsf{FactCat}_{\operatorname{lax-untl}}^{\operatorname{lax-fact}}$.
Note that the right adjoint is compatible with the forgetful functors to
$\mathsf{Rep}(M)$, and in particular is conservative. As in [Ras21a] Corollary
7.8.1, this implies that
$\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))$
belongs to $\mathsf{FactCat}$, i.e. factorizes strictly.

### 5.2.

Note that the symmetric monoidal functor

$\operatorname{res}^{N_{P}}_{\mathfrak{n}_{P}}:\mathsf{Rep}(P)\longrightarrow\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))$

is fully faithful because $\mathfrak{n}_{P}$ is nilpotent.

######  Lemma 5.2.1.

For any finite set $I$, the functor

$\operatorname{res}^{N_{P}}_{\mathfrak{n}_{P}}:\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$

is fully faithful.

###### Proof.

Since the symmetric monoidal category $\mathsf{Rep}(P)$ is rigid and the unit
object in $\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))$ is
compact, Lemma 3.4.2 says that corresponding morphism in $\mathsf{FactCat}$
admits a right adjoint in $\mathsf{FactCat}_{\operatorname{lax-untl}}$. The
unit of this adjunction is an isomorphism because it is compatible with
factorization and is an isomorphism over $X$.

∎

In what follows, we will freely identify
$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$ with its essential image in
$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$.

######  Proposition 5.2.2.

The monad on
$\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$
arising from the adjunction (5.1.2) preserves the full subcategory
$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$ for any finite set $I$.

We will apply a general lemma, which we now prepare to state. Let
$\mathcal{C}$ be a commutative lax factorization category, i.e. an object of
$\mathsf{ComAlg}(\mathsf{FactCat}^{\operatorname{lax-fact}})$, and $L$ a
Lie-$*$ algebra in $\mathcal{C}$. Then we have adjoint functors

$\operatorname{ind}^{*\to\operatorname{ch}}:L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\rightleftarrows\operatorname{U}^{*\to\operatorname{ch}}(L)\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}:\operatorname{oblv}^{\operatorname{ch}\to*}$

for any finite set $I$.

Recall that $L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})$ has a canonical
structure of commutative lax factorization category compatible with the
forgetful functor

$\operatorname{oblv}_{L}:L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})\longrightarrow\mathcal{C}.$

In particular, for any finite set $I$ the category

$L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}$

is equipped with a symmetric monoidal structure lifting the one on
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$, which will also be denoted by
$\otimes^{!}$.

Viewing $L$ with the adjoint action as an object of the symmetric monoidal
category
$L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})_{X_{\operatorname{dR}}}$,
the symmetric algebra $\operatorname{Sym}^{!}(L[1])$ is a commutative algebra
in this category and hence gives rise to a commutative factorization algebra
in $L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})$.

######  Lemma 5.2.3.

The monad
$\operatorname{oblv}^{\operatorname{ch}\to*}\operatorname{ind}^{*\to\operatorname{ch}}$
on
$L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}$
admits a canonical filtration with associated graded isomorphic to

$\operatorname{Sym}^{!}(L[1])_{X^{I}}\overset{!}{\otimes}(-).$

###### Proof.

First, we record the apparently weaker assertion that the functor

$\operatorname{oblv}_{\operatorname{U}^{*\to\operatorname{ch}}(L)}\operatorname{ind}^{*\to\operatorname{ch}}:L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathcal{C}_{X^{I}_{\operatorname{dR}}}$

is filtered with associated graded

$\operatorname{oblv}_{L}(\operatorname{Sym}^{!}(L[1])_{X^{I}}\overset{!}{\otimes}(-)).$

This is a variant of Corollary 6.5.2 in [FG12] and is proved in the same way.

Now we will deduce the lemma. Observe that the adjoint action canonically
upgrades $L$ to a Lie-$*$ algebra in the commutative lax factorization
category $L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})$. It follows
immediately that $\operatorname{U}^{*\to\operatorname{ch}}(L)$ upgrades to a
chiral algebra in $L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})$.
Moreover, the morphism

$\operatorname{oblv}_{L}:L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})\longrightarrow\mathcal{C}$

of commutative lax factorization categories induces isomorphisms

$L\text{\textendash}\mathsf{mod}^{*}(L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C}))\tilde{\longrightarrow}L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})$

and

$\operatorname{U}^{*\to\operatorname{ch}}(L)\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C}))\tilde{\longrightarrow}\operatorname{U}^{*\to\operatorname{ch}}(L)\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathcal{C}).$

These fit into a commutative square

${L\text{\textendash}\mathsf{mod}^{*}(L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C}))}$${\operatorname{U}^{*\to\operatorname{ch}}(L)\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C}))}$${L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})}$${\operatorname{U}^{*\to\operatorname{ch}}(L)\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathcal{C})}$$\scriptstyle{\operatorname{ind}^{*\to\operatorname{ch}}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{ind}^{*\to\operatorname{ch}}}$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$, so the lemma follows from the
weaker assertion above applied to $L$ as a Lie-$*$ algebra in
$L\text{\textendash}\mathsf{mod}^{*}(\mathcal{C})$.

∎

###### Proof of Proposition 5.2.2.

Since $\mathfrak{n}\otimes k_{X}$ belongs to
$\mathsf{Rep}(P)_{X_{\operatorname{dR}}}$, it follows that

$\operatorname{Sym}^{!}(\mathfrak{n}\otimes k_{X}[1])_{X^{I}}$

belongs to $\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$. Now apply Lemma
5.2.3 to deduce the proposition.

∎

### 5.3.

For any finite set $I$, define

$\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}\subset\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$

to be the full subcategory generated by the image of
$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$ under
$\operatorname{ind}^{*\to\operatorname{ch}}$. In view of Proposition 5.2.2, it
follows that these assemble into an object of $\mathsf{FactCat}$, and we have
a monadic adjunction

$\operatorname{ind}^{*\to\operatorname{ch}}:\mathsf{Rep}(P)\rightleftarrows\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(M)):\operatorname{oblv}^{\operatorname{ch}\to*}$
(5.3.1)

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ with the left adjoint
belonging to $\mathsf{FactCat}$.

We equip

$\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$

with the unique t-structure such that
$\operatorname{oblv}^{\operatorname{ch}\to*}$ is left t-exact. Equivalently,
the connective objects are generated by the image of $\mathsf{Rep}(P)^{\leq
0}$ under $\operatorname{ind}^{*\to\operatorname{ch}}$. Proposition 7.11.1(2)
of [Ras21a] says that $\operatorname{ind}^{*\to\operatorname{ch}}$ is actually
t-exact.

######  Proposition 5.3.1.

For any finite set $I$, the t-structure on

$\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$

is coherent.

We will apply the following general lemma.

######  Lemma 5.3.2.

Let $F:\mathcal{C}\to\mathcal{D}$ be a t-exact functor whose restriction

$F:\mathcal{C}^{\heartsuit}\longrightarrow\mathcal{D}^{\heartsuit}$

is fully faithful. Assume also that the right adjoint $F^{\operatorname{R}}$
is continuous and conservative. If the t-structure on $\mathcal{C}$ is
coherent, then so is the t-structure on $\mathcal{D}$.

###### Proof.

Standard arguments show that any compact object in $\mathcal{D}^{\heartsuit}$
admits a finite filtration with subquotients of the form $F(c)$, where $c$ is
compact in $\mathcal{C}^{\heartsuit}$. If the t-structure on $\mathcal{C}$ is
coherent, then any compact object in $\mathcal{C}^{\heartsuit}$ is almost
compact in $\mathcal{C}$. Since $F$ preserves almost compact objects and any
extension of almost compact objects is almost compact, the lemma follows.

∎

###### Proof of Proposition 5.3.1.

Lemma 7.16.1 of [Ras21a] says that

$\operatorname{ind}^{*\to\operatorname{ch}}:\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}\longrightarrow\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(M))_{X^{I}_{\operatorname{dR}}}$

is fully faithful on the heart, and the t-structure
$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$ is coherent by Proposition
4.1.2. Thus the hypotheses of Lemma 5.3.2 apply to the adjunction (5.3.1).

∎

###  5.4. The spectral Chevalley functor as a factorization algebra

We will abuse notation slightly by also writing $\Upsilon(\mathfrak{n}_{P},-)$
for the composition

$\mathsf{Rep}(G)\xrightarrow{\operatorname{res}^{G}_{P}}\mathsf{Rep}(P)\xrightarrow{\operatorname{res}^{N_{P}}_{\mathfrak{n}_{P}}}\mathfrak{n}_{P}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(M))\xrightarrow{\Upsilon(\mathfrak{n}_{P},-)}\mathsf{Rep}(M).$

This is a morphism in $\mathsf{FactCat}_{\operatorname{lax-untl}}$, and hence
can be viewed as a factorization algebra in
$\underline{\operatorname{Hom}}_{\mathsf{FactCat}_{\operatorname{lax-
untl}}}(\mathsf{Rep}(G),\mathsf{Rep}(M))$. But $\mathsf{Rep}(G)$ is
canonically self-dual in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ by
Proposition 3.4.1, so we can identify

$\underline{\operatorname{Hom}}_{\mathsf{FactCat}_{\operatorname{lax-
untl}}}(\mathsf{Rep}(G),\mathsf{Rep}(M))\cong\mathsf{Rep}(G)\otimes\mathsf{Rep}(M)\cong\mathsf{Rep}(G\times
M).$

We will write $\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})$ for the
factorization algebra in $\mathsf{Rep}(G\times M)$ corresponding to
$\Upsilon(\mathfrak{n}_{P},-)$. As the notation suggests, this factorization
algebra can also be obtained by applying the morphism

$\operatorname{id}_{\mathsf{Rep}(G)}\otimes\Upsilon(\mathfrak{n}_{P},-):\mathsf{Rep}(G)\otimes\mathsf{Rep}(G)\longrightarrow\mathsf{Rep}(G)\otimes\mathsf{Rep}(M)\cong\mathsf{Rep}(G\times
M)$

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ to $\mathcal{O}_{G}$, viewed
as a commutative factorization algebra in
$\mathsf{Rep}(G)\otimes\mathsf{Rep}(G)$.

By construction, there is a canonical morphism of factorization algebras

$\operatorname{unit}_{\mathsf{Rep}(G)}\boxtimes\Upsilon(\mathfrak{n}_{P})\longrightarrow\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})$

in $\mathsf{Rep}(G\times M)$. By Lemma 3.6.1, this induces an equivalence

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))\tilde{\longrightarrow}\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G)\otimes\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))).$
(5.4.1)

By definition, the factorization algebra
$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})$ is the image of
$\operatorname{res}^{G\times G}_{G\times P}$ under the functor

$\operatorname{id}_{\mathsf{Rep}(G)}\otimes(\Upsilon(\mathfrak{n}_{P},-)\circ\operatorname{res}^{N_{P}}_{\mathfrak{n}_{P}}):\mathsf{Rep}(G\times
P)\longrightarrow\mathsf{Rep}(G)\otimes\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M))),$

and in particular belongs to the full subcategory

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

We define

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

to be the full subcategory corresponding to

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

under the equivalence (5.4.1).

The adjunction (5.3.1) induces an adjunction

$\mathsf{Rep}(G\times
P)\rightleftarrows\mathsf{Rep}(G)\otimes\Upsilon(\mathfrak{n}_{P})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(M))$

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ after tensoring with
$\operatorname{id}_{\mathsf{Rep}(G)}$. Applying the construction (3.5.1)
yields an adjunction

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))\rightleftarrows\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))$ (5.4.2)

in $\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$,
with the left adjoint belonging to $\mathsf{FactCat}^{\operatorname{lax-
fact}}$. Note that by construction, the right adjoint makes the square

${\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))}$${\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))}$${\mathsf{Rep}(G\times M)}$${\mathsf{Rep}(G\times
P)}$$\scriptstyle{\operatorname{oblv}_{\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})}}$$\scriptstyle{\operatorname{oblv}_{\mathcal{O}_{G}}}$$\scriptstyle{\operatorname{id}\otimes\operatorname{res}^{P}_{M}}$

commute, and in particular is conservative over $X^{I}_{\operatorname{dR}}$
for any finite set $I$.

### 5.5.

We now define a t-structure on the category

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{X^{I}_{\operatorname{dR}}}$

for any finite set $I$. Recall that by Proposition 4.2.1, the category

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}$

admits a unique t-structure which makes the forgetful functor to
$\mathsf{Rep}(G\times P)_{X^{I}_{\operatorname{dR}}}$ t-exact.

######  Lemma 5.5.1.

For any finite set $I$, there is a unique t-structure on the category

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{X^{I}_{\operatorname{dR}}}$

which makes the left adjoint in (5.4.2) t-exact over
$X^{I}_{\operatorname{dR}}$, and this functor moreover restricts to an
equivalence on the hearts of the t-structures.

###### Proof.

Since the right adjoint in (5.4.2) is conservative, there is a unique
t-structure which makes the left adjoint right t-exact. To see that it is left
t-exact, it suffices to check that the corresponding monad on

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}$

is left t-exact. The forgetful functor

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(G\times
P)_{X^{I}_{\operatorname{dR}}}$

is t-exact and intertwines the monad in question with the monad

$\operatorname{id}_{\mathsf{Rep}(G)}\otimes(\operatorname{oblv}^{\operatorname{ch}\to*}\circ\operatorname{ind}^{*\to\operatorname{ch}}\circ\operatorname{res}^{N_{P}}_{\mathfrak{n}_{P}})$

on the target. Thus it suffices to show that the latter monad is left t-exact,
which follows from the PBW theorem for factorization modules as in the proof
of Proposition 7.11.1 in [Ras16].

The left adjoint in (5.4.2) is an equivalence on hearts by a similar
calculation using the PBW theorem (see the proof of Lemma 7.16.1 in _loc.
cit._ ; cf. also Proposition 4.4.4 above).

∎

By Proposition 4.4.4, the functor

$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}\cong\mathcal{O}_{G}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}\xrightarrow{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}$

restricts to an equivalence on the hearts of the t-structures. Combining this
with Lemma 5.5.1, we obtain an equivalence

$(\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}})^{\heartsuit}\tilde{\longrightarrow}(\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{X^{I}_{\operatorname{dR}}})^{\heartsuit}.$

######  Proposition 5.5.2.

For any finite set $I$, the t-structure on

$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))_{X^{I}_{\operatorname{dR}}}$

satisfies the hypotheses of Proposition 3.9.2. In particular, this category is
almost ULA generated over $X^{I}$.

###### Proof.

First, we show that the t-structure is almost ULA generated. Coherence follows
from Lemma 5.5.1 using the criterion of Lemma 2.4.7, since the left adjoint in
(5.4.2) preserves almost compact objects, the category

$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$

is compactly generated, and

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}$

is almost compactly generated by Corollary 4.6.1.1. Recall that

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))_{X^{I}_{\operatorname{dR}}}$

admits a set of almost ULA generators by Corollary 4.6.1.1, and since the
right adjoint in (5.4.2) is left t-exact, conservative, and
$\mathsf{D}(X^{I})$-linear, the claim follows.

Hypothesis (ii) of Proposition 3.9.1 follows from the corresponding property
of

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))$

via the right adjoint in (5.4.2), which is left t-exact and conservative over
$X^{I}_{\operatorname{dR}}$ for all $I$. As for hypothesis (iii), by right
completeness it suffices to check that the structure functors in question are
left t-exact on the heart of the t-structure. Since

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))$

has this property by Corollary 4.6.1.1, the claim follows from Lemma 5.5.1.

Similarly, the hypothesis in Proposition 3.9.2 is satisfied because of Lemma
5.5.1 and the corresponding property of

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P)).$

∎

###  5.6. Definition of the spectral Hecke categories

Applying Propositions 3.9.2 and 5.5.2, we obtain an object

$\mathsf{Sph}^{\operatorname{spec}}_{G,P}:=\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M))^{\operatorname{ren}}$

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

$\mathsf{Rep}(P)\cong\mathcal{O}_{G}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(G\times
P))\xrightarrow{\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}}\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
P))$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$ with the left adjoint in
(5.4.2), we obtain a morphism

$\mathsf{Rep}(P)\longrightarrow\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(G\times
M)).$ (5.6.1)

This functor is t-exact over $X^{I}_{\operatorname{dR}}$ for every finite set
$I$, and hence renormalizes to a morphism

$\mathsf{Rep}(P)\longrightarrow\mathsf{Sph}^{\operatorname{spec}}_{G,P}$
(5.6.2)

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$, where we used Proposition
4.1.2 to identify $\mathsf{Rep}(P)^{\operatorname{ren}}\cong\mathsf{Rep}(P)$.
By Lemma 5.5.1, this induces an equivalence

$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}^{\heartsuit}\tilde{\longrightarrow}(\mathsf{Sph}^{\operatorname{spec}}_{G,P})_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$

for any finite set $I$.

######  Proposition 5.6.1.

The morphism (5.6.2) admits a right adjoint in
$\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$, which
is conservative over $X^{I}_{\operatorname{dR}}$ for any finite set $I$. In
particular, the object $\mathsf{Sph}^{\operatorname{spec}}_{G,P}$ belongs to
$\mathsf{FactCat}$.

###### Proof.

Theorem 4.6.1, together with the existence of the adjunction (5.4.2), implies
that

$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}\longrightarrow(\mathsf{Sph}^{\operatorname{spec}}_{G,P})_{X^{I}_{\operatorname{dR}}}$

preserves ULA objects for any finite set $I$. Moreover, the essential image of
this functor generates its target, since it induces an equivalence on the
hearts of the t-structures. Since
$\mathsf{Rep}(P)_{X^{I}_{\operatorname{dR}}}$ is ULA generated by Proposition
4.1.2, we obtain the desired right adjoint. It follows that
$\mathsf{Sph}^{\operatorname{spec}}_{G,P}$ factorizes strictly, since
$\mathsf{Rep}(P)$ does and we have just shown that (5.6.2) has a monadic right
adjoint.

∎

###  5.7. Associative algebra and bimodule structures

In the case $G=P$, we will write

$\mathsf{Sph}^{\operatorname{spec}}_{G}:=\mathsf{Sph}^{\operatorname{spec}}_{G,G}$

to simplify the notation. Here the factorization algebra
$\Upsilon(\mathfrak{n}_{P},\mathcal{O}_{G})=\mathcal{O}_{G}$ is the monoidal
unit in

$\mathsf{Rep}(G\times
G)\cong\underline{\operatorname{End}}_{\mathsf{FactCat}_{\operatorname{lax-
untl}}}(\mathsf{Rep}(G)).$

The object

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G))$

is obtained by applying the construction (3.7.2) to $\mathsf{Rep}(G\times G)$,
hence inherits an associative algebra structure in
$\mathsf{FactCat}^{\operatorname{lax-fact}}$. Similarly, the object

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

is naturally a bimodule for the pair

$(\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G)),\mathcal{O}_{M}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M\times
M))).$

######  Proposition 5.7.1.

The object $\mathsf{Sph}^{\operatorname{spec}}_{G}$ admits a unique structure
of associative algebra in $\mathsf{FactCat}$ compatible with the morphism

$\mathsf{Sph}^{\operatorname{spec}}_{G}\longrightarrow\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G)).$

Similarly, the object $\mathsf{Sph}^{\operatorname{spec}}_{G,P}$ admits a
unique structure of
$(\mathsf{Sph}^{\operatorname{spec}}_{G},\mathsf{Sph}^{\operatorname{spec}}_{M})$-bimodule
in $\mathsf{FactCat}$ compatible with the morphism

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

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

###### Proof.

For the first claim, by Proposition 3.9.3 it suffices to show that the unit
and multiplication morphisms for the associative algebra

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G))$

are unary and binary morphisms, respectively, in the pseudo-tensor category
$(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{aULA}}$. This means that for any finite set
$I$, the monoidal unit object in

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G))_{X^{I}_{\operatorname{dR}}}$

is eventually coconnective, and that eventually coconnective objects are
stable under the monoidal operation. By Lemmas 4.3.1 and 4.4.2, the unit
object $\mathcal{O}_{G,X^{I}}$ is concentrated in cohomological degree
$-\\#I$, and in particular is eventually coconnective. As for the
multiplication, the t-exact monoidal functor

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:\mathsf{Rep}(G)_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G))_{X^{I}_{\operatorname{dR}}}$

induces an equivalence on the hearts by Proposition 4.4.4, and monoidal
structure on $\mathsf{Rep}(G)_{X^{I}_{\operatorname{dR}}}$ is left t-exact in
each variable separately. The claim now follows by right completeness of the
t-structure on

$\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G))_{X^{I}_{\operatorname{dR}}}.$

Recall that the morphism (5.6.1) in $\mathsf{FactCat}^{\operatorname{lax-
fact}}$ is t-exact and induces an equivalence on the hearts over each
$X^{I}_{\operatorname{dR}}$. By right completeness of the t-structures, it
follows that the action of the pair

$(\mathcal{O}_{G}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(G\times
G)),\mathcal{O}_{M}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(M\times
M)))$

preserves the full subcategory

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

and that the action of the pair on this subcategory takes place in
$(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{aULA}}$. Then we are done by Proposition
3.9.3.

∎

##  6\. Construction of the derived Satake transform

In this section, we construct the factorization categories that appear on the
geometric (a.k.a. automorphic) side. We then construct functors to the
spectral side, and formulate our main results as Theorems 6.6.1 and 6.12.3.

###  6.1. Canonical twists

Let $\rho$ denote the half-sum of the simple coroots of $G$. Choose a square
root $(\Omega^{1}_{X})^{\otimes\frac{1}{2}}$ of the canonical line bundle
$\Omega^{1}_{X}$ on $X$, and put

$\rho(\Omega^{1}_{X}):=2\rho((\Omega^{1}_{X})^{\otimes\frac{1}{2}}),$

which is a well-defined $T$-bundle on $X$ because $2\rho\in\Lambda$ is an
integral coweight.

In what follows, we replace $G$ by its _canonical twist_ by
$\rho(\Omega^{1}_{X})$. By definition, this is the group scheme of
automorphisms of the $G$-bundle on $X$ defined by

$G\overset{T}{\times}\rho(\Omega^{1}_{X}).$

Similarly, we replace $P$ and $P^{-}$ by their canonical twists, defined in
the same way (the corresponding twist of $T$ itself is trivial because $T$ is
commutative). These are subgroup schemes of the canonical twist of $G$. We
remark that by construction, all of these twists are pure inner forms (cf.
§2.15 of [Ras16] of the corresponding constant group schemes $G$, $P$, and
$P^{-}$. In particular, replacing each of these groups with its canonical
twist does not affect the corresponding space of principal bundles.

The canonical twist of $N$ is defined as

$N\overset{T}{\times}\rho(\Omega^{1}_{X})$

using the adjoint action of $T$ on $N$, and similarly for $N^{-}$. These are
normal subgroup schemes of the canonical twists of $B$ and $B^{-}$
respectively, with the quotient in either case being the constant group scheme
$T$. The canonical twists of $N$ and $N^{-}$ are _not_ pure inner forms of the
constant group schemes, and their spaces of principal bundles generally
differ. One can similarly define the canonical twist of any subgroup of $N$
stable under the adjoint action of $T$, such as $N_{P}$ or $N_{M}:=N\cap M$.

These twists will be suppressed below in order to simplify the notation. Their
presence can be safely ignored in most situations, with the notable exception
of defining the Whittaker condition.

###  6.2. The spherical Hecke stack

Write
$\mathcal{H}_{G}:=\mathfrak{L}^{+}G\backslash\mathfrak{L}G/\mathfrak{L}^{+}G$
for the spherical Hecke stack, a groupoid in corr-unital factorization spaces
(i.e. factorization spaces with a unit correspondence rather than a single
morphism, cf. [CF21b] §10). The groupoid structure on $\mathcal{H}_{G}$
induces a monoidal structure on

$\mathsf{D}(\mathcal{H}_{G,X^{I}}):=\mathsf{D}((\mathfrak{L}G)_{X^{I}})^{\mathfrak{L}^{+}G\times\mathfrak{L}^{+}G}$

for each finite set $I$, and moreover these assemble into an object
$\mathsf{D}(\mathcal{H}_{G})$ of $\mathsf{AssocAlg}(\mathsf{FactCat})$
satisfying

$\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}=\mathsf{D}(\mathcal{H}_{G,X^{I}}).$

For any finite set $I$, the Beilinson-Drinfeld affine Grassmannian

$\operatorname{Gr}_{G,X^{I}}=(\mathfrak{L}G/\mathfrak{L}^{+}G)_{X^{I}}$

is an ind-scheme locally of finite type, and in particular
$\mathsf{D}(\operatorname{Gr}_{G,X^{I}})$ is equipped with a natural
t-structure. We give $\mathsf{D}(\mathcal{H}_{G,X^{I}})$ the unique
t-structure such that the forgetful functor

$\mathsf{D}(\mathcal{H}_{G,X^{I}})=\mathsf{D}(\operatorname{Gr}_{G,X^{I}})^{\mathfrak{L}^{+}G}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G,X^{I}})$

is t-exact.

###  6.3. Construction of the spherical Hecke category

The naïve Satake transform, as constructed in §6 of [Ras21a], is a morphism

$\operatorname{Sat}^{\operatorname{naive}}_{G}:\mathsf{Rep}(\check{G})\longrightarrow\mathsf{D}(\mathcal{H}_{G})$

in $\mathsf{AssocAlg}(\mathsf{FactCat})$. We recall from _loc. cit._ that for
any finite set $I$, the corresponding functor

$\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{D}(\mathcal{H}_{G,X^{I}})$

is t-exact and induces an equivalence

$\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}^{\heartsuit}\tilde{\longrightarrow}\mathsf{D}(\mathcal{H}_{G,X^{I}})^{\heartsuit}.$

######  Lemma 6.3.1.

For any finite set $I$, the functor

$\operatorname{Sat}^{\operatorname{naive}}_{G}:\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{D}(\mathcal{H}_{G,X^{I}})$

preserves almost ULA objects.

###### Proof.

It suffices to show that the $\mathsf{IndCoh}(X^{I})$-linear functor

$\mathsf{Rep}(\check{G})_{X^{I}}\longrightarrow\mathsf{D}(\mathcal{H}_{G})_{X^{I}}$

preserves almost compact objects. By Proposition 4.1.2, almost compact objects
in $\mathsf{Rep}(\check{G})_{X^{I}}$ are compact. Moreover, the symmetric
monoidal structure is rigid, which means that compactness in
$\mathsf{Rep}(\check{G})_{X^{I}}$ is equivalent to dualizability. Since
$\operatorname{Sat}^{\operatorname{naive}}_{G}$ is monoidal, it preserves
dualizable objects, so it remains only to show that the unit object
$\delta_{\mathfrak{L}^{+}G,X^{I}}$ is almost compact in
$\mathsf{D}(\mathcal{H}_{G})_{X^{I}}$. For this, recall that the forgetful
functor

$\operatorname{oblv}_{\mathfrak{L}^{+}G}:\mathsf{D}(\mathcal{H}_{G})_{X^{I}}\longrightarrow\mathsf{D}(\operatorname{Gr}_{G})_{X^{I}}$

is t-exact, conservative, and admits an $\mathsf{IndCoh}(X^{I})$-linear right
adjoint, namely $*$-averaging with respect to $\mathfrak{L}^{+}G$. The image
of $\delta_{\mathfrak{L}^{+}G,X^{I}}$ under this functor is the direct image
of $\omega_{X^{I}}$ along the unit section

$X^{I}\longrightarrow
X^{I}\times_{X^{I}_{\operatorname{dR}}}(\operatorname{Gr}_{G,X^{I}})_{\operatorname{dR}},$

which is ind-proper. It follows that
$\operatorname{oblv}_{\mathfrak{L}^{+}G}(\delta_{\mathfrak{L}^{+}G,X^{I}})$ is
compact in $\mathsf{D}(\operatorname{Gr}_{G})_{X^{I}}$, and hence
$\delta_{\mathfrak{L}^{+}G,X^{I}}$ is almost compact in
$\mathsf{D}(\mathcal{H}_{G})_{X^{I}}$ by Lemma 2.4.1.

∎

######  Proposition 6.3.2.

The t-structure on $\mathsf{D}(\mathcal{H}_{G,X^{I}})$ satisfies the
hypotheses of Proposition 3.9.2. Moreover, the object

$\mathsf{Sph}_{G}:=\mathsf{D}(\mathcal{H}_{G})^{\operatorname{ren}}$

belongs to $\mathsf{FactCat}$ and admits a unique associative algebra
structure compatible with

$\mathsf{Sph}_{G}\longrightarrow\mathsf{D}(\mathcal{H}_{G}).$

###### Proof.

We first show that that the t-structure on $\mathsf{D}(\mathcal{H}_{G,X^{I}})$
is coherent, or equivalently that any object compact in
$\mathsf{D}(\mathcal{H}_{G,X^{I}})^{\geq 0}$ is almost compact in
$\mathsf{D}(\mathcal{H}_{G,X^{I}})$. The t-structure on
$\mathsf{D}(\operatorname{Gr}_{G,X^{I}})$ is coherent because
$\operatorname{Gr}_{G,X^{I}}$ is an ind-scheme locally of finite type. Thus
the claim follows from Lemma 2.4.1 applied to the functor

$\operatorname{oblv}_{\mathfrak{L}^{+}G}:\mathsf{D}(\mathcal{H}_{G,X^{I}})\longrightarrow\mathsf{D}(\operatorname{Gr}_{G,X^{I}}),$

which is t-exact, conservative, and admits a continuous right adjoint.

The functor

$\operatorname{Sat}^{\operatorname{naive}}_{G}:\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{D}(\mathcal{H}_{G,X^{I}})$

induces an equivalence on the hearts and preserves almost ULA objects by Lemma
6.3.1, and $\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}$ is ULA
generated by Proposition 4.1.2. It follows readily that
$\mathsf{D}(\mathcal{H}_{G,X^{I}})$ is almost ULA generated. Conditions (ii-
iii) of Proposition 3.9.1, as well as the hypothesis of Proposition 3.9.2, can
be deduced from the corresponding properties of
$\mathsf{D}(\operatorname{Gr}_{G,X^{I}})$.

By Proposition 4.1.2, renormalizing
$\operatorname{Sat}^{\operatorname{naive}}_{G}$ yields a morphism

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

in $\mathsf{AssocAlg}(\mathsf{FactCat}^{\operatorname{lax-fact}})$. Moreover,
by Lemma 6.3.1, this morphism admits a right adjoint in
$\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$. Since
$\operatorname{Sat}^{\operatorname{naive}}_{G}$ is an equivalence on the
hearts, this right adjoint is conservative and hence monadic over each
$X^{I}_{\operatorname{dR}}$. It follows that $\mathsf{Sph}_{G}$ belongs to
$\mathsf{FactCat}$, i.e. factorizes strictly, since $\mathsf{Rep}(\check{G})$
has this property.

For the last claim, by Proposition 3.9.3 it suffices to show that
$\mathsf{D}(\mathcal{H}_{G})$ is an associative algebra in the pseudo-tensor
category $(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{aULA}}$. This follows from right completeness
and the corresponding property of $\mathsf{Rep}(\check{G})$, once more using
the fact that the monoidal functor

$\operatorname{Sat}^{\operatorname{naive}}_{G}:\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Sph}_{G,X^{I}_{\operatorname{dR}}}$

is t-exact and induces an equivalence on the hearts for any finite set $I$.

∎

######  Remark 6.3.3.

The above result may be compared with [Noc23] Corollary 1.2 in the Betti
setting.

###  6.4. Whittaker (co)invariants

We have an isomorphism

$N^{-}/[N^{-},N^{-}]\cong\prod_{i\in\mathcal{I}_{G}}\mathbb{G}_{a}^{\Omega^{1}_{X}}$

of group schemes over $X$, where

$\mathbb{G}_{a}^{\Omega^{1}_{X}}:=\mathbb{G}_{a}\overset{\mathbb{G}_{m}}{\times}\Omega^{1}_{X}$

is the canonical twist of $\mathbb{G}_{a}$. Summing over $\mathcal{I}_{G}$, we
obtain a homomorphism $N^{-}\to\mathbb{G}_{a}^{\Omega^{1}_{X}}$ and hence a
homomorphism of factorizable group ind-schemes

$\mathfrak{L}N^{-}\longrightarrow\mathfrak{L}(\mathbb{G}_{a}^{\Omega^{1}_{X}}).$

Finally, we compose with the canonical residue homomorphism

$\mathfrak{L}(\mathbb{G}_{a}^{\Omega^{1}_{X}})\longrightarrow\mathbb{G}_{a}$

to obtain

$\psi:\mathfrak{L}N^{-}\longrightarrow\mathbb{G}_{a}.$

In particular, we obtain a multiplicative D-module (a.k.a. character D-module
of rank 1) $\psi^{!}\exp[1]$ on $\mathfrak{L}N^{-}$. For any category
$\mathcal{C}$ acted on by $\mathsf{D}((\mathfrak{L}G)_{X^{I}})$, we can
consider the associated Whittaker invariants

$\mathcal{C}^{\mathfrak{L}N^{-},\psi}:=\mathcal{C}^{\mathfrak{L}N^{-},\psi^{!}\exp[1]},$

and similarly for coinvariants (we suppress $X^{I}$ from the notation here for
simplicity). Theorem 2.2.1 of [Ras21b] asserts that there is a canonical
equivalence

$\mathcal{C}_{\mathfrak{L}N^{-},\psi}\tilde{\longrightarrow}\mathcal{C}^{\mathfrak{L}N^{-},\psi}.$

###  6.5. The spherical Whittaker category

As shown in [Ras16] Corollary 2.31.2, the $\mathsf{D}(X^{I})$-module
categories

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

assemble into an object

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

of $\mathsf{FactCat}$, with unit given by the so-called Whittaker vacuum
object. Convolution on the right defines an action of
$\mathsf{D}(\mathcal{H}_{G})$ on this category, and in particular, we obtain a
morphism

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

in $\mathsf{FactCat}$ by acting on the unit.

######  Theorem 6.5.1.

The composite functor

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

is an isomorphism in $\mathsf{FactCat}$.

###### Proof.

This is Theorem 6.36.1 of [Ras21a].

∎

In particular, the equivalence in the theorem induces an action of
$\mathsf{D}(\mathcal{H}_{G})$ on

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

in $\mathsf{FactCat}$. By Proposition 3.4.1, this action corresponds to a
morphism

$\mathsf{D}(\mathcal{H}_{G})\longrightarrow\underline{\operatorname{End}}_{\mathsf{FactCat}_{\operatorname{lax-
untl}}}(\mathsf{Rep}(\check{G}))\cong\mathsf{Rep}(\check{G}\times\check{G})$
(6.5.1)

in $\mathsf{AssocAlg}(\mathsf{FactCat}_{\operatorname{lax-untl}})$.

We will need the following observation.

######  Lemma 6.5.2.

For any finite set $I$, the functor

$\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}\xrightarrow{(\ref{sphtobimod})}\mathsf{Rep}(\check{G}\times\check{G})_{X^{I}_{\operatorname{dR}}}$

is t-exact.

###### Proof.

First, we claim that the composite functor

$\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}\xrightarrow{\operatorname{Sat}^{\operatorname{naive}}_{G}}\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(\check{G}\times\check{G})_{X^{I}_{\operatorname{dR}}}$

is t-exact. By the construction of (6.5.1), this composite agrees with the
induction functor
$\operatorname{coind}_{\check{G}}^{\check{G}\times\check{G}}$, i.e. the right
adjoint to restriction along the diagonal homomorphism
$\check{G}\to\check{G}\times\check{G}$. The t-exactness of
$\operatorname{coind}_{\check{G}}^{\check{G}\times\check{G}}$ be proved in the
same way as (or deduced from) that of $\operatorname{coind}_{1}^{\check{G}}$,
which is Proposition 6.24.2 of [Ras21a].

This immediately implies that the functor in the proposition is right t-exact,
since $\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}^{\leq 0}$ is
generated under colimits by

$\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}^{\heartsuit}\cong\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}^{\heartsuit}.$

Right completeness of the t-structure implies that
$\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}^{\geq 0}$ is
generated by
$\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$ under
filtered colimits, shifts, and extensions, whence the functor in question is
left t-exact.

∎

###  6.6. Derived Satake transform

Note that the functor (6.5.1) is not strictly unital with respect to the
factorization structure. Namely, it preserves monoidal units, but the monoidal
and factorization units in $\mathsf{Rep}(\check{G}\times\check{G})$ do not
coincide.

Applying the functor (3.7.2), we obtain a morphism

$\mathsf{D}(\mathcal{H}_{G})\longrightarrow\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{G}))$
(6.6.1)

in $\mathsf{AssocAlg}(\mathsf{FactCat}^{\operatorname{lax-fact}})$

Observe that the monoidal functor

$\mathsf{D}(\mathcal{H}_{G})_{X^{I}_{\operatorname{dR}}}\xrightarrow{(\ref{presatake})}\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{G}))_{X^{I}_{\operatorname{dR}}}$

is t-exact for any finite set $I$, since the forgetful functor

$\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{G}))_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(\check{G}\times\check{G})_{X^{I}_{\operatorname{dR}}}$

is t-exact and conservative, and its composition with (6.6.1) is t-exact by
Proposition 6.5.2.

By Propositions 3.9.3 and 6.3.2, together with Corollary 4.6.1.1, we can
renormalize (6.6.1) to obtain a morphism

$\operatorname{Sat}_{G}:\mathsf{Sph}_{G}\longrightarrow\mathsf{Sph}_{\check{G}}^{\operatorname{spec}}$

in $\mathsf{AssocAlg}(\mathsf{FactCat})$. Now we can state our first main
theorem precisely.

######  Theorem 6.6.1.

The functor $\operatorname{Sat}_{G}$ is an equivalence.

This theorem will be proved in §9.

###  6.7. The semi-infinite Whittaker category

Recall from [Ras16] §2.34 that the categories

$((\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi})_{X^{I}_{\operatorname{dR}}}$

assemble into an object of $\mathsf{FactCat}$, denoted there by
$\mathsf{Whit}^{\frac{\infty}{2}}$. Following _loc. cit._ in the case $P=B$,
we will now construct a morphism

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}\longrightarrow\mathsf{Rep}(\check{M})$
(6.7.1)

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$.

First, we restrict the Whittaker condition to the subgroup
$N^{-}_{M}:=N^{-}\cap M$, i.e. apply the forgetful functor

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}\longrightarrow(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N_{M}^{-},\psi}.$

At the next stage, restrict along the inclusion
$\mathfrak{L}P\to\mathfrak{L}G$ to obtain

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N_{M}^{-},\psi}\longrightarrow(\mathsf{D}(\mathfrak{L}P)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N_{M}^{-},\psi}\cong\mathsf{D}(\operatorname{Gr}_{M})^{\mathfrak{L}N_{M}^{-},\psi}.$

Recall that for any finite set $I$ there is a degree map

$\deg_{M}:\operatorname{Gr}_{M,X^{I}}\longrightarrow\pi_{1}(M),$

compatible with factorization, where $\pi_{1}(M)$ is viewed as discrete
abelian group. We can identify

$\pi_{1}(M)=\Lambda/(\mathbb{Z}\cdot R_{M}),$

the lattice of coweights modulo the coroot lattice of $M$. Note that the
homomorphism

$\displaystyle\ell_{P}:\pi_{1}(M)$ $\displaystyle\longrightarrow\mathbb{Z}$
$\displaystyle\lambda$ $\displaystyle\mapsto\langle
2\check{\rho}_{G}-2\check{\rho}_{M},\lambda\rangle$

is well-defined, and hence we obtain an automorphism

$\displaystyle\mathsf{D}(\operatorname{Gr}_{M})$
$\displaystyle\longrightarrow\mathsf{D}(\operatorname{Gr}_{M})$
$\displaystyle\mathcal{M}$
$\displaystyle\mapsto\mathcal{M}[-\ell_{P}(\deg_{M})]$

in $\mathsf{FactCat}$ given by a cohomological shift which depends on the
connected component.

Since $\mathfrak{L}N^{-}_{M}$ is connected, this automorpism of
$\mathsf{D}(\operatorname{Gr}_{M})$ preserves the Whittaker condition. Compose
this automorphism with the previously constructed morphism

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

to obtain another such morphism. Finally, applying Theorem 6.5.1 to $M$, we
have

$\mathsf{D}(\operatorname{Gr}_{M})^{\mathfrak{L}N_{M}^{-},\psi}\tilde{\longrightarrow}\mathsf{Rep}(\check{M}).$

This completes the construction of the morphism (6.7.1).

######  Theorem 6.7.1.

The morphism (6.7.1) sends the unit object to the factorization algebra
$\Upsilon(\check{\mathfrak{n}}_{\check{P}})$ in $\mathsf{Rep}(\check{M})$.

###### Proof.

In the case $P=G$, the morphism (6.7.1) is inverse to the equivalence of
Theorem 6.5.1, and $\Upsilon(\check{\mathfrak{n}}_{\check{P}})$ is the unit
object in $\mathsf{Rep}(\check{M})=\mathsf{Rep}(\check{G})$, so there is
nothing to show.

The case $P=B$ is Theorem 4.4.1 of [Ras16].

The general case will be proved in forthcoming work by Faergeman and Hayash.

∎

In particular, applying the construction (3.5.1) to (6.7.1), we obtain a
morphism

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}\longrightarrow\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))$
(6.7.2)

in $\mathsf{FactCat}$.

###  6.8. The accessible Whittaker category

The morphism

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}\xrightarrow{\operatorname{oblv}_{\mathfrak{L}N^{-},\psi}}\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\xrightarrow{\operatorname{Av}^{\mathfrak{L}^{+}G}_{*}}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ admits a left adjoint (see
[Ras16] Proposition 5.4.1), which we denote by
$\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}$. Moreover, by the
construction of unital structures in _loc. cit._ , the morphism
$\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}$ is strictly unital.

Define the factorization subcategory

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}\subset(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}$

to be generated under colimits by the essential image of

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

As explained in _loc. cit._ , this subcategory admits a unique unital
factorization structure which makes the inclusion a morphism in
$\mathsf{FactCat}$.

######  Theorem 6.8.1.

The morphism (6.7.2) restricts to an equivalence

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

###### Proof.

In the case $P=G$, the accessibility condition is vacuous, and this is inverse
to the equivalence of Theorem 6.5.1.

In the case $P=B$, Theorem 5.7.1 of [Ras16] says that this functor is fully
faithful. To prove essential surjectivity, we claim that for arbitrary $P$,
the image of the functor generates

$\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))$

under colimits. The functor in question is $\mathsf{Rep}(\check{M})$-linear,
where $\mathsf{Rep}(\check{M})$ acts on

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}$

via $\operatorname{Sat}_{M}^{\operatorname{naive}}$. We have a commutative
triangle

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

in $\mathsf{FactCat}$, where both diagonal morphisms are given by acting on
the unit for the factorization structure. Similarly, the triangle

${\mathsf{Rep}(\check{M})}$${\mathsf{Rep}(\check{P})}$${\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))}$$\scriptstyle{\operatorname{triv}_{\check{N}_{\check{P}}}}$$\scriptstyle{(\ref{chiralind2})}$

commutes because both circuits are $\mathsf{Rep}(\check{M})$-linear morphisms
in $\mathsf{FactCat}$. Since the images of
$\operatorname{triv}_{\check{N}_{\check{P}}}$ and (5.3.1) generate their
targets under colimits, the claim follows.

For the general case, by the argument above it suffices to show fully
faithfulness. This can be deduced from Theorem 6.7.1 by adapting the argument
of [Ras16] to the parabolic case. One also needs a parabolic version of the
Arkhipov-Bezrukavnikov equivalence [AB09], which can be deduced from the case
$P=B$, as explained in §6 of [BL18] (cf. also [ACR18]).

∎

###  6.9. The t-structure on the semi-infinite spherical category

We now prepare to define the spherical Hecke category $\mathsf{Sph}_{G,P}$
attached to $P$. Appealing again to [Ras16] §2, we recall that the categories

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}$

assemble into an object of $\mathsf{FactCat}$. We will denote the unit object
by $\Delta^{0}$.

Consider the correspondence of corr-unital factorization spaces

${\mathfrak{L}^{+}P\backslash\mathfrak{L}P}$${\mathfrak{L}^{+}G\backslash\mathfrak{L}G}$${\mathfrak{L}^{+}M\backslash\mathfrak{L}M.}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\mathfrak{q}}$

The composite morphism

$\mathsf{D}(\mathcal{H}_{M})\xrightarrow{\mathfrak{q}^{!}}\mathsf{D}(\mathfrak{L}^{+}P\backslash\mathfrak{L}P/\mathfrak{L}^{+}M)\longrightarrow\mathsf{D}(\mathfrak{L}^{+}P\backslash\mathfrak{L}P)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$

is an isomorphism in $\mathsf{FactCat}$. We will abuse notation and denote it
simply by $\mathfrak{q}^{!}$, and its inverse by $\mathfrak{q}_{!}$.

The morphism

$\mathfrak{p}^{!}:\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\longrightarrow\mathsf{D}(\mathfrak{L}^{+}P\backslash\mathfrak{L}P)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$

admits a left adjoint $\mathfrak{p}_{!}$. Explicitly, the composite

$\mathsf{D}(\mathcal{H}_{M})\xrightarrow{\mathfrak{q}^{!}}\mathsf{D}(\mathfrak{L}^{+}P\backslash\mathfrak{L}P)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\xrightarrow{\mathfrak{p}_{!}}\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$

is given by acting on the unit object.

Now consider the adjunction

$\mathfrak{p}_{!}\mathfrak{q}^{!}[-\ell_{P}(\deg_{M})]:\mathsf{D}(\mathcal{H}_{M})\rightleftarrows\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}:\mathfrak{q}_{!}\mathfrak{p}^{!}[\ell_{P}(\deg_{M})],$
(6.9.1)

which takes place in $\mathsf{FactCat}_{\operatorname{lax-untl}}$, with the
left adjoint belonging to $\mathsf{FactCat}$. We also remark that the right
adjoint is conservative, which follows from the fact that $\mathfrak{p}$ is an
ind-locally closed stratification.

We equip
$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}$
with the t-structure characterized by the requirement that

$\mathfrak{q}_{!}\mathfrak{p}^{!}[\ell_{P}(\deg_{M})]:(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{D}(\mathcal{H}_{M})_{X^{I}_{\operatorname{dR}}}$

be left t-exact. Equivalently, the connective objects are generated by the
image of $\mathsf{D}(\mathcal{H}_{M,X^{I}})^{\leq 0}$ under
$\mathfrak{p}_{!}\mathfrak{q}^{!}[-\ell_{P}(\deg_{M})]$.

We remark that this does _not_ agree with the t-structure on this category
defined (in the case $P=B$) by Gaitsgory in [Gai22]. Namely, unlike the
t-structure introduced in _loc. cit._ , our t-structure is local on the curve
$X$ (since the t-structure on $\mathsf{D}(\mathcal{H}_{M,X^{I}})$ has this
feature). However, the two t-structures do agree when restricted to objects
supported at a single point $x\in X(k)$.

######  Lemma 6.9.1.

For any finite set $I$, the functor

$\mathfrak{p}_{!}\mathfrak{q}^{!}[-\ell_{P}(\deg_{M})]:\mathsf{D}(\mathcal{H}_{M})_{X^{I}_{\operatorname{dR}}}\longrightarrow(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}$

is t-exact.

###### Proof.

This functor is right t-exact by definition, so it suffices to show left
t-exactness. Since its right adjoint is conservative, it is enough to prove
that the resulting monad on $\mathsf{D}(\mathcal{H}_{M,X^{I}})$ is left
t-exact. This monad is $\mathsf{D}(\mathcal{H}_{M,X^{I}})$-linear, so its
value on an object $\mathcal{M}$ in $\mathsf{D}(\mathcal{H}_{M,X^{I}})$ is

$(\mathfrak{q}_{!}\mathfrak{p}^{!}\Delta^{0}_{X^{I}})[\ell_{P}(\deg_{M})]\star\mathcal{M}.$

Using $\mathsf{D}(X^{I})$-linearity, we can reduce to the case that $I$ has
one element. So we must show that if $\mathcal{M}$ belongs to
$\mathsf{D}(\mathcal{H}_{M,X})^{\geq 0}$, then

$(\mathfrak{q}_{!}\mathfrak{p}^{!}\Delta^{0}_{X})[\ell_{P}(\deg_{M})]\star\mathcal{M}$

is coconnective. Since $\mathsf{D}(\mathcal{H}_{M,X})$ is right complete, we
can even assume that $\mathcal{M}$ belongs to

$\mathsf{D}(\mathcal{H}_{M,X})^{\heartsuit}\cong\mathsf{Rep}(\check{M})_{X_{\operatorname{dR}}}^{\heartsuit}\cong(\mathsf{Rep}(\check{M})\otimes\mathsf{D}(X))^{\heartsuit}.$

Choose an étale coordinate on $X$, so that
$\Delta^{0}_{X}\cong\Delta^{0}_{x}\boxtimes\omega_{X}$ for a fixed point $x\in
X(k)$. Thus it suffices to show that for any
$V\in\mathsf{Rep}(\check{M})^{\heartsuit}$, the object

$(\mathfrak{q}_{!}\mathfrak{p}^{!}\Delta^{0}_{x})[\ell_{P}(\deg_{M})]\star\operatorname{Sat}_{M,x}^{\operatorname{naive}}(V)$

belongs to $\mathsf{D}(\mathcal{H}_{M,x})^{\geq 0}$. Theorem 1.5.5 in [Gai19]
says that

$(\mathfrak{q}_{!}\mathfrak{p}^{!}\Delta^{0}_{x})[\ell_{P}(\deg_{M})]$

is coconnective when $P=B$, and the general case can be proved similarly. Now
we are done by t-exactness of the convolution product on
$\mathsf{D}(\mathcal{H}_{M,x})$.

∎

###  6.10. The t-structure on the semi-infinite Whittaker category

Consider the morphism

$\mathsf{Rep}(\check{M})\longrightarrow(\mathsf{D}(\mathfrak{L}N^{-},\psi\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{\operatorname{acc}}$
(6.10.1)

in $\mathsf{FactCat}$ given by acting on the unit of the factorization
structure via $\operatorname{Sat}_{M}^{\operatorname{naive}}$. For any finite
set $I$, we equip the category

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}_{X^{I}_{\operatorname{dR}}}$

with the t-structure whose connective objects are generated under colimits by
the image of $\mathsf{Rep}(\check{M})^{\leq 0}$ under (6.10.1).

######  Proposition 6.10.1.

The equivalence of Theorem 6.8.1 is exact over $X^{I}_{\operatorname{dR}}$ for
every finite set $I$.

###### Proof.

By definition, the connective objects in

$\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))_{X^{I}_{\operatorname{dR}}}$

are generated by the image of
$\mathsf{Rep}(\check{P})_{X^{I}_{\operatorname{dR}}}^{\leq 0}$ under the left
adjoint in (5.3.1). But the latter is generated by the image of
$\mathsf{Rep}(\check{M})_{X^{I}_{\operatorname{dR}}}^{\leq 0}$ under

$\operatorname{triv}_{\check{N}}:\mathsf{Rep}(\check{M})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(\check{P})_{X^{I}_{\operatorname{dR}}},$

so it suffices to show that the triangle

${\mathsf{Rep}(\check{M})}$${(\mathsf{D}(\mathfrak{L}N^{-},\psi\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{\operatorname{acc}}}$${\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))}$$\scriptstyle{(\ref{whitgen})}$$\scriptstyle{\operatorname{ind}^{*\to\operatorname{ch}}\circ\operatorname{triv}_{\check{N}}}$$\scriptstyle{\sim}$

in $\mathsf{FactCat}$ commutes. This is immediate from the fact that all three
functors are $\mathsf{Rep}(\check{M})$-linear and unital.

∎

######  Proposition 6.10.2.

The morphism (6.10.1) is t-exact over $X^{I}_{\operatorname{dR}}$ for any
finite set $I$ and preserves ULA objects. The resulting right adjoint

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

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ is conservative over
$X^{I}_{\operatorname{dR}}$ for any finite set $I$.

###### Proof.

These are consequences of Theorem 6.8.1. Alternatively, one can prove the
t-exactness more directly along the same lines as Lemma 6.9.1, and the
existence of a factorizable and conservative right adjoint follows from
Theorem 5.10.1 in [Ras16].

∎

###  6.11. Definition of the spherical Hecke category for a parabolic

In order to renormalize the spherical semi-infinite category, we need a couple
of technical lemmas.

######  Lemma 6.11.1.

For any finite set $I$, the functor

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

is t-exact. Moreover, this functor is conservative on eventually coconnective
objects.

###### Proof.

To see that the functor is right t-exact, consider the commutative triangle

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

in $\mathsf{FactCat}$, where both diagonal morphisms are given by acting on
the unit objects. Over $X^{I}_{\operatorname{dR}}$, the diagonal functors are
t-exact and generate their targets under colimits, which implies that the
horizontal functor is right t-exact.

For left t-exactness, the Cousin filtration and factorization immediately
reduce the claim to the case where $I$ has one element. After étale
localization on $X$, we have a $\mathsf{D}(X)$-linear equivalence

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X_{\operatorname{dR}}}\cong(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}\otimes\mathsf{D}(X)$

for a fixed $x\in X(k)$, and similarly for

$(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}_{X_{\operatorname{dR}}}.$

Thus, by Lemma 2.2.1 we can even reduce to proving the claim at a single point
$x$. Since the t-structure on
$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}$
is right complete, it is enough to prove that the functor is left t-exact when
restricted to the heart. By (parabolic variants of) Proposition 1.5.7 and
Theorem 4.2.3 in [Gai19], there is a functor

$\mathsf{Rep}(\check{P})\longrightarrow(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}$
(6.11.1)

which induces an equivalence

$\mathsf{Rep}(\check{P})^{\heartsuit}\tilde{\longrightarrow}(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\heartsuit}.$

Since $\mathsf{Rep}(\check{P})^{\heartsuit}$ is generated under extensions and
direct sums by the image of

$\operatorname{triv}_{\check{N}_{\check{P}}}:\mathsf{Rep}(\check{M})^{\heartsuit}\longrightarrow\mathsf{Rep}(\check{P})^{\heartsuit},$

the claim follows from Proposition 6.10.2.

Finally, we prove conservativity on eventually coconnective objects. Again,
using factorization we can reduce to the case that $I$ has a single element.
The functor $\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}$ is left adjoint
to $\operatorname{Av}^{\mathfrak{L}^{+}G}_{*}$, and by the t-exactness
statements proved above, the resulting monad on

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X_{\operatorname{dR}}}$

is left t-exact. Thus it suffices to show that this monad is conservative on
eventually coconnective objects. Observe that the monad is given by the action
of the object

$\operatorname{Av}^{\mathfrak{L}^{+}G}_{*}\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}\delta_{1,X}$

in $\mathsf{D}(\mathcal{H}_{G,X})$. After étale localization on $X$ we have
$(\mathfrak{L}G)_{X}\cong(\mathfrak{L}G)_{x}\times X$ for a fixed $x\in X(k)$,
from which it follows that the object

$\operatorname{Av}^{\mathfrak{L}^{+}G}_{*}\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}\delta_{1,X}$

has the form $\mathcal{M}\boxtimes\omega_{X}$ with respect to the
decomposition

$\mathsf{D}(\mathcal{H}_{G,X})\cong\mathsf{D}(\mathcal{H}_{G,x})\otimes\mathsf{D}(X).$

The naïve Satake equivalence implies that for any $i\in\mathbb{Z}$, we have
$H^{i}\mathcal{M}\cong\operatorname{Sat}_{G,x}^{\operatorname{naive}}V$ for
some $V$ in $\mathsf{Rep}(\check{G})^{\heartsuit}$. Since the monad in
question is left t-exact, we have $H^{i}\mathcal{M}=0$ for $i<0$, and Theorem
6.5.1 implies that $H^{0}\mathcal{M}=\delta_{1,x}$. Thus it suffices show that
the action of the symmetric monoidal category $\mathsf{Rep}(\check{G})$ on
$(D(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}$
is left t-exact. This follows from the fact that the functor (6.11.1) is
$\mathsf{Rep}(\check{G})$-linear and induces an equivalence on the hearts.

∎

######  Lemma 6.11.2.

For any finite set $I$, the t-structure on
$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}$
satisfies the conditions of Proposition 3.9.2.

###### Proof.

Most of the conditions can be easily deduced from the corresponding properties
of the t-structure on
$\mathsf{D}(\mathcal{H}_{M})_{X^{I}_{\operatorname{dR}}}$, with the exception
of coherence.

For coherence, we consider the adjunction of $\mathsf{D}(X^{I})$-module
categories

$\operatorname{Av}^{\mathfrak{L}N^{-},\psi}_{!}:(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}\rightleftarrows(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi,\operatorname{acc}}_{X^{I}_{\operatorname{dR}}}:\operatorname{Av}^{\mathfrak{L}^{+}G}_{*}.$

Using Theorem 6.8.1, we can replace the right side by its spectral
counterpart:

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}\rightleftarrows\Upsilon(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}))_{X^{I}_{\operatorname{dR}}}.$

Proposition 6.11.1 says that the left adjoint is t-exact, and also
conservative on eventually coconnective objects. By Lemma 2.4.1, an object in
$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}$
is almost compact if and only if it is so after applying the left adjoint. Now
coherence of the left side follows from coherence of the right side, which was
proved in Proposition 5.3.1.

∎

Thus we can define

$\mathsf{Sph}_{G,P}:=(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\operatorname{ren}},$

which is _a priori_ an object of $\mathsf{FactCat}^{\operatorname{lax-fact}}$.
Note that the adjunction (6.9.1) renormalizes to a monadic adjunction

$\mathsf{Sph}_{M}\rightleftarrows\mathsf{Sph}_{G,P},$

and in particular $\mathsf{Sph}_{G,P}$ belongs to $\mathsf{FactCat}$ because
$\mathsf{Sph}_{M}$ does.

######  Proposition 6.11.3.

The object $\mathsf{Sph}_{G,P}$ admits a unique structure of
$(\mathsf{Sph}_{G},\mathsf{Sph}_{M})$-bimodule in $\mathsf{FactCat}$
compatible with the morphism

$\mathsf{Sph}_{G,P}\longrightarrow\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}.$

###### Proof.

By Proposition 3.9.3, it suffices to show that the
$(\mathsf{D}(\mathcal{H}_{G}),\mathsf{D}(\mathcal{H}_{M}))$-bimodule structure
on
$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$
takes place in the pseudo-tensor category
$\mathsf{FactCat}^{\otimes}_{\operatorname{aULA}}$. For
$\mathsf{D}(\mathcal{H}_{M})$, this follows from the existence of the right
adjoint in (6.9.1), which is $\mathsf{D}(\mathcal{H}_{M})$-linear as well as
conservative and left t-exact over $X^{I}_{\operatorname{dR}}$ for every
finite set $I$. As for $\mathsf{D}(\mathcal{H}_{G})$, by right completeness of
the t-structure and the naïve Satake equivalence, it is enough to show that
any object of
$\mathsf{Rep}(\check{G})_{X^{I}_{\operatorname{dR}}}^{\heartsuit}$ acts by a
left t-exact functor. Using factorization, this immediately reduces to the
corresponding claim at a point $x\in X(k)$, which was established in the proof
of Lemma 6.11.1.

∎

###  6.12. Derived Satake transform for a parabolic

The first step is to construct a morphism

$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\longrightarrow\mathsf{Rep}(\check{G}\times\check{M})$
(6.12.1)

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$. By Proposition 3.4.1, this is
the same datum as a morphism

$\mathsf{Rep}(\check{G})\otimes\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\longrightarrow\mathsf{Rep}(\check{M}).$

By Theorem 6.5.1, we can identify

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

so we will construct a morphism

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

First, convolve in the middle to obtain a morphism

$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\otimes\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\longrightarrow(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}.$

Then compose with (6.7.1) to obtain the desired morphism (6.12.1).

######  Theorem 6.12.1.

The image of the unit under the morphism (6.12.1) is canonically isomorphic to
$\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})$ as a
factorization algebra.

###### Proof.

We have already seen this in the case $P=G$, where the functor (6.12.1) is
monoidal and the
$\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})$ is the
monoidal unit $\mathcal{O}_{\check{G}}$ in
$\mathsf{Rep}(\check{G}\times\check{G})$.

For the general case, a parabolic variant of Theorem 7.9.1 in [Ras21a] says
that the triangle

${\mathsf{Rep}(\check{G})}$${(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}}$${\mathsf{Rep}(\check{M})}$$\scriptstyle{\Upsilon(\check{\mathfrak{n}}_{\check{P}},-)}$$\scriptstyle{(\ref{semiinfres})}$

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$ commutes. Here the left
diagonal morphism is the composite

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

where the first functor is given by acting on the unit via
$\operatorname{Sat}^{\operatorname{naive}}_{G}$. Since
$\Upsilon(\check{\mathfrak{n}},-)$ corresponds to
$\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})$ under the
equivalence

$\underline{\operatorname{Hom}}_{\mathsf{FactCat}_{\operatorname{lax-
untl}}}(\mathsf{Rep}(\check{G}),\mathsf{Rep}(\check{M}))\cong\mathsf{Rep}(\check{G}\times\check{M}),$

the claim follows.

∎

Applying (3.5.1), we therefore obtain a morphism

$\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\longrightarrow\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{M})).$
(6.12.2)

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$. By construction, it is a
morphism of
$(\mathsf{D}(\mathcal{H}_{G}),\mathsf{D}(\mathcal{H}_{M}))$-bimodules, where
the bimodule structure on the right side comes from the action of

$(\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{G})),\mathcal{O}_{\check{M}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{M}\times\check{M})))$

via (6.6.1).

######  Proposition 6.12.2.

The image of the morphism (6.12.2) is contained in

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

For any finite set $I$, the resulting functor

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{X^{I}_{\operatorname{dR}}}\longrightarrow\Upsilon(\check{\mathfrak{n}}_{\check{P}},\mathcal{O}_{\check{G}})\text{\textendash}\mathsf{mod}_{0}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{M}))_{X^{I}_{\operatorname{dR}}}$

is t-exact.

###### Proof.

The first claim follows from the fact that the image of the left adjoint in
(6.9.1) generates the target under colimits, and hence the image of (6.12.2)
is contained in the $\mathsf{D}(\mathcal{H}_{M})$-submodule category generated
by the unit.

Right t-exactness of the functor in question follows from the commutative
triangle

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

where the diagonal functors are given by acting on the unit objects via
$\operatorname{Sat}_{M}^{\operatorname{naive}}$. Namely, over each
$X^{I}_{\operatorname{dR}}$, the diagonal functors are t-exact and generate
their targets under colimits.

Using factorization, left t-exactness reduces to the corresponding claim at a
point $x\in X$. By right completeness of the t-structure, it suffices to show
that (6.12.2) sends any object in

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\heartsuit}$

to a coconnective object. As in the proof of Proposition 6.11.1, we have an
equivalence

$(\mathsf{D}(\mathfrak{L}^{+}G\backslash\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})_{x}^{\heartsuit}\cong\mathsf{Rep}(\check{P})^{\heartsuit},$

and $\mathsf{Rep}(\check{P})^{\heartsuit}$ is generated under extensions and
filtered colimits by the image of

$\operatorname{triv}_{\check{N}_{\check{P}}}:\mathsf{Rep}(\check{M})^{\heartsuit}\longrightarrow\mathsf{Rep}(\check{P})^{\heartsuit}.$

In view of the commutative triangle above, the claim follows from Lemma 5.5.1.

∎

Thus (6.12.2) renormalizes to a morphism

$\operatorname{Sat}_{G,P}:\mathsf{Sph}_{G,P}\longrightarrow\mathsf{Sph}_{\check{G},\check{P}}^{\operatorname{spec}}$

in $\mathsf{FactCat}$. By construction, it is a morphism of
$(\mathsf{Sph}_{G},\mathsf{Sph}_{M})$-bimodules, where the bimodule structure
on the target comes from $\operatorname{Sat}_{G}$ and
$\operatorname{Sat}_{M}$.

The following is our second main theorem.

######  Theorem 6.12.3.

The morphism $\operatorname{Sat}_{G,P}$ is an isomorphism.

The proof of this theorem is contained in §10.

###  6.13. Reduction to a point

Before proceeding, we show that in order to prove Theorem 6.12.3 (and hence
its special case Theorem 6.6.1), it is enough to show that

$\operatorname{Sat}_{G,P,x}:\mathsf{Sph}_{G,P,x}\longrightarrow\mathsf{Sph}_{\check{G},\check{P},x}^{\operatorname{spec}}$

is an equivalence for any $x\in X(k)$.

Consider the commutative triangle

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

in $\mathsf{FactCat}$, where the diagonal morphisms are uniquely characterized
by $\mathsf{Rep}(\check{M})$-linearity. For any finite set $I$, the diagonal
functors preserve ULA objects over $X^{I}$ and generate their targets, which
implies that the functor $\operatorname{Sat}_{G,P}$ preserves ULA objects.
Thus, by Proposition B.8.1 in [Ras21a], it suffices to show that this functor
is an equivalence over each stratum in $X^{I}$. Using factorization, we can
therefore assume that $I$ is a singleton, i.e. we must show that
$\operatorname{Sat}_{G,P,X_{\operatorname{dR}}}$ is an equivalence.

Choosing an étale coordinate on $X$, we can moreover assume that we have
$\mathsf{D}(X)$-linear equivalences

$\mathsf{Sph}_{G,P,X_{\operatorname{dR}}}\cong\mathsf{Sph}_{G,P,x}\otimes\mathsf{D}(X)$

and

$\mathsf{Sph}_{\check{G},\check{P},X_{\operatorname{dR}}}^{\operatorname{spec}}\cong\mathsf{Sph}_{\check{G},\check{P},x}^{\operatorname{spec}}\otimes\mathsf{D}(X)$

for a fixed $x\in X(k)$, and that the $\mathsf{D}(X)$-linear functor
$\operatorname{Sat}_{G,P,X_{\operatorname{dR}}}$ is induced from
$\operatorname{Sat}_{G,P,x}$. Thus it suffices to show that

$\operatorname{Sat}_{G,P,x}:\mathsf{Sph}_{G,P,x}\longrightarrow\mathsf{Sph}_{\check{G},\check{P},x}^{\operatorname{spec}}$

is an equivalence.

##  7\. The case of a torus

The goal of this section is to formulate and prove Theorem 7.3.1, a
factorizable version of local geometric class field theory in the de Rham
setting, then deduce Theorem 6.6.1 in the case $G=T$.

###  7.1. Gauge forms

Let $H$ denote an arbitrary algebraic group. For any finite set $I$, the space
$(\mathfrak{h}\otimes\Omega^{1}_{D})_{X^{I}_{\operatorname{dR}}}$ is defined
as follows: a point
$S\to(\mathfrak{h}\otimes\Omega^{1}_{D})_{X^{I}_{\operatorname{dR}}}$ consists
of $x_{I}:S\to X^{I}_{\operatorname{dR}}$ together with a section of the
vector bundle $(\mathfrak{h}\otimes\Omega^{1}_{X})|_{D_{x_{I}}}$. The space
$(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}$
is defined similarly, but the section is allowed to have a pole along
$\\{x\\}\times S$.

These assemble into factorization spaces $\mathfrak{h}\otimes\Omega^{1}_{D}$
and $(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}$ which are co-
unital and corr-unital, respectively. The inclusion

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

is a morphism of corr-unital factorization spaces.

Fix a point $x\in X(k)$. We let
$(\mathfrak{L}^{+}H)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}$ be
the group classifying maps $D_{x_{I}}\to H$ which are possibly meromorphic
along $\\{x\\}\times S$. It also forms a corr-unital factorization space, and
receives a homomorphism

$\mathfrak{L}^{+}H\longrightarrow(\mathfrak{L}^{+}H)^{x\operatorname{-ram}}$

compatible with this structure.

The group $(\mathfrak{L}^{+}H)^{x\operatorname{-ram}}$ acts on
$(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}$, compatibly with
the factorization structure, via the gauge action (cf. §1.12 of [Ras15c]). The
subgroup $\mathfrak{L}^{+}H$ preserves the subspace
$\mathfrak{h}\otimes\Omega^{1}_{D}$. Define

$\operatorname{LS}_{H}(D):=(\mathfrak{h}\otimes\Omega^{1}_{D})/\mathfrak{L}^{+}H$

and

$\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}:=(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}/(\mathfrak{L}^{+}H)^{x\operatorname{-ram}},$

a co-unital and corr-unital factorization space respectively. We have a
canonical morphism of corr-unital factorization spaces

$\iota:\operatorname{LS}_{H}(D)\longrightarrow\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}.$

The co-unital factorization structure on $\operatorname{LS}_{H}(D)$ makes
$\mathsf{QCoh}(\operatorname{LS}_{H}(D))$ into an object of
$\mathsf{FactCat}^{\operatorname{lax-fact}}$. We recall from [Ras16], Lemma
9.8.1 that we have a canonical equivalence

$\mathsf{QCoh}(\operatorname{LS}_{H}(D))\tilde{\longrightarrow}\mathsf{Rep}(H)$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$, and in particular
$\mathsf{QCoh}(\operatorname{LS}_{H}(D))$ actually belongs to
$\mathsf{FactCat}$.

###  7.2. Unital structures

We would like to upgrade
$\mathsf{QCoh}(\operatorname{LS}_{H}(D)^{x\operatorname{-ram}})$ to a unital
factorization category. Given an injection of finite sets $J\to I$, the corr-
unital structure on $\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}$
determines a correspondence

${\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{J\to
I}}$${\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X^{J}_{\operatorname{dR}}}}$${\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}.}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

The morphism $\beta$ is ind-schematic, but generally not schematic unless $H$
is unipotent. In particular, the usual direct image of quasicoherent sheaves
along this map is poorly behaved.

######  Proposition 7.2.1.

If $H$ is reductive, then for any injection of finite sets $J\to I$, the
functor

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

admits a left adjoint.

###### Proof.

Writing
$\operatorname{LS}_{H}(D)^{x\operatorname{-ram},\log}:=(\mathfrak{h}\otimes\Omega^{1}_{D})^{x\operatorname{-ram}}/\mathfrak{L}^{+}H$,
we can factorize $\beta$ as

$\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{J\to
I}\xrightarrow{\beta_{1}}\operatorname{LS}_{H}(D)^{x\operatorname{-ram},\log}_{X^{I}_{\operatorname{dR}}}\xrightarrow{\beta_{2}}\operatorname{LS}_{H}(D)^{x\operatorname{-ram}}_{X^{I}_{\operatorname{dR}}}.$

To see that $\beta_{1}^{*}$ admits a left adjoint, it suffices to show the
same for inverse image along the inclusion

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