# Satake equivalence for Hodge modules on affine Grassmannians

Roman Fedorov  University of Pittsburgh, USA<EMAIL_ADDRESS>
###### Abstract.

For a reductive group $G$ we equip the category of
$G_{\mathcal{O}}$-equivariant polarizable pure Hodge modules on the affine
Grassmannian $\operatorname{Gr}_{G}$ with a structure of neutral Tannakian
category. We show that it is equivalent to a twisted tensor product of the
category of representations of the Langlands dual group and the category of
pure polarizable Hodge structures.

##  1\. Introduction and Main Results

We fix a complex reductive group $G$. Consider the affine Grassmannian
$\operatorname{Gr}_{G}$ defined as the étale sheafification of the functor
sending a $\mathbb{C}$-algebra $R$ to the quotient
$G\bigl{(}R((t))\bigr{)}/G(R[[t]])$, where $R((t)):=R[[t]](t^{-1})$. Let
$G_{\mathcal{O}}$ be the algebraic group of infinite type representing the
functor $R\mapsto G(R[[t]])$, that is, the group of jets. Clearly,
$G_{\mathcal{O}}$ acts on $\operatorname{Gr}_{G}$.

It is well-known that the affine Grassmannian is a strict ind-projective ind-
scheme, that is, we can write

(1) $\operatorname{Gr}_{G}=\lim_{l\to\infty}\operatorname{Gr}_{G}^{(l)},$

where $(\operatorname{Gr}_{G}^{(l)})$ is an inductive systems of projective
schemes with morphisms being closed embeddings. We may assume that the schemes
$\operatorname{Gr}_{G}^{(l)}$ are $G_{\mathcal{O}}$-equivariant.

###  1.1. Recollection on the Geometric Satake Equivalence

The first work in the direction of geometrizing the Satake isomorphism was
Lusztig’s paper [Lus]. Independently, Drinfeld understood that geometrizing
the Satake isomorphism is crucial for the geometric Langlands correspondence.
Following Drinfeld’s suggestion, Ginzburg in [Gin] proved the characteristic
zero case of the Geometric Satake equivalence. We follow [MV], where Mirkovic
and Vilonen generalized the above results.

In more details, Mirkovic and Vilonen equip the category
$\operatorname{Perv}_{\mathbb{Q}}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$
with a convolution product and a commutativity constraint for this product,
making it into a symmetric monoidal category. They also construct a fiber
functor making
$\operatorname{Perv}_{\mathbb{Q}}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$
into a neutral Tannakian category. Thus, according to Tannakian formalism (see
for example [DM, Thm. 2.11]), there is an affine group
$\check{G}_{\mathbb{Q}}$ over $\mathbb{Q}$ such that we have an equivalence of
categories

$\operatorname{Perv}_{\mathbb{Q}}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})=\operatorname{Rep}_{\mathbb{Q}}(\check{G}_{\mathbb{Q}}).$

Here $\operatorname{Rep}_{\mathbb{Q}}$ stands for the category of finite
dimensional representations on $\mathbb{Q}$-vector spaces. The main result of
[MV] shows that $\check{G}_{\mathbb{Q}}$ is split reductive and is Langlands
dual to $G$. The main goal of our paper is to “upgrade” this theorem from
perverse sheaves to pure polarizable Hodge modules.

Recently the geometric Satake equivalence was upgraded to the motivic setup in
[RS]. It seems plausible that some of our results can be obtained from Hodge
realization of the results in loc. cit. It is, however, important to have a
direct construction. Our result is also, in a sense, simpler, as it does not
use the theory of motives.

###  1.2. Geometric Satake Equivalence for pure polarizable Hodge modules

For a complex variety $Y$ (=reduced separated scheme of finite type over
$\mathbb{C}$) one has the category
$\operatorname{MH}(Y,n)=\operatorname{MH}(Y,\mathbb{Q},n)$ of pure polarizable
rational Hodge modules of weight $n$ (cf. [Sai1, Sai2]). In Section 2 we
extend the notion of Hodge modules to ind-schemes and to the equivariant
setting so that we have the category
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n)$ of
$G_{\mathcal{O}}$-equivariant pure polarizable Hodge modules of weight $n$ on
$\operatorname{Gr}_{G}$. This is a semisimple abelian $\mathbb{Q}$-linear
category. Consider the graded category

$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G}):=\bigoplus_{n\in\mathbb{Z}}\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n).$

We equip this category with a structure of neutral Tannakian category in
Section 3.

Let $\operatorname{MH}(pt,n)$ stand for the category of polarizable Hodge
structures of weight $n$ (which is the same as the polarizable Hodge modules
of weight $n$ on a point, thus the notation; see Section 2.1.3) and set

$\operatorname{MH}(pt):=\bigoplus_{n\in\mathbb{Z}}\operatorname{MH}(pt,n).$

Recall that we have the Tate twist, which is a degree $-2$ autoequivalence of
$\operatorname{MH}(pt)$.

Our main goal in this paper is to identify the category
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ as a “twisted”
tensor product. We will need two constructions.

####  1.2.1. Adding the square root of the Tate twist

Let $\mathcal{C}$ be a category with an auto-equivalence $T$. We define the
new category $\mathcal{C}[\sqrt{T}]:=\mathcal{C}^{+}\times\mathcal{C}^{-}$ to
be the product of two copies of $\mathcal{C}$. This decomposition is viewed as
a $\mathbb{Z}/2\mathbb{Z}$ grading. If $\mathcal{F}$ is an object of
$\mathcal{C}$, we write $\mathcal{F}^{\pm}$ for $\mathcal{F}$ viewed as an
object of $\mathcal{C}^{\pm}\subset\mathcal{C}[\sqrt{T}]$.

We define an autoequivalence $\sqrt{T}$ of $\mathcal{C}[\sqrt{T}]$ by

$\sqrt{T}(\mathcal{F}^{+})=\mathcal{F}^{-},\quad\sqrt{T}(\mathcal{F}^{-})=T(\mathcal{F})^{+}.$

Note that $T=(\sqrt{T})^{2}$. We will also write $\mathcal{F}(n/2)$ for
$\sqrt{T}^{n}(\mathcal{F})$.

Assume that $\mathcal{C}=\bigoplus_{i\in\mathbb{Z}}\mathcal{C}^{i}$ is a
graded additive category and that $T$ has degree $-2$. Then the category
$\mathcal{C}[\sqrt{T}]$ also has a $\mathbb{Z}$-grading defined by
$\mathcal{C}[\sqrt{T}]^{i}:=(\mathcal{C}^{+})^{i}\times(\mathcal{C}^{-})^{i+1}$.
Note that $\sqrt{T}$ has degree $-1$.

Assume now that $\mathcal{C}$ is a monoidal category and $T$ is compatible
with the monoidal structure in the sense that we have natural isomorphisms

$T(\mathcal{F}\otimes\mathcal{G})=T(\mathcal{F})\otimes\mathcal{G}=\mathcal{F}\otimes
T(\mathcal{G})$

for any objects $\mathcal{F}$ and $\mathcal{G}$. Then we define a monoidal
structure on $\mathcal{C}[\sqrt{T}]$ by

$\mathcal{F}^{+}\otimes\mathcal{G}^{+}:=(\mathcal{F}\otimes\mathcal{G})^{+},\quad\mathcal{F}^{+}\otimes\mathcal{G}^{-}:=(\mathcal{F}\otimes\mathcal{G})^{-},\quad\mathcal{F}^{-}\otimes\mathcal{G}^{+}:=(\mathcal{F}\otimes\mathcal{G})^{-},\quad\mathcal{F}^{-}\otimes\mathcal{G}^{-}:=(T(\mathcal{F}\otimes\mathcal{G}))^{+}$

The product is compatible with $\sqrt{T}$. If $\mathcal{C}$ is
$\mathbb{Z}$-graded, then this product is compatible with gradings.

####  1.2.2. Tensor product of semisimple categories

Recall that for abelian semisimple $\mathbb{Q}$-linear categories
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ one can define their tensor product
$\mathcal{C}_{1}\boxtimes\mathcal{C}_{2}$ (see, e.g. [BK, Def. 1.1.15]). The
objects of this category are formal expressions
$\oplus_{i=1}^{n}V_{1,i}\boxtimes V_{2,i}$, where $V_{j,i}$ are objects of
$\mathcal{C}_{j}$; the morphisms are given by

$\operatorname{Hom}(\oplus_{i=1}^{n}V_{1,i}\boxtimes
V_{2,i},\oplus_{j=1}^{m}V^{\prime}_{1,j}\boxtimes
V^{\prime}_{2,j}):=\oplus_{i,j}\operatorname{Hom}(V_{1,i},V^{\prime}_{1,j})\otimes_{\mathbb{Q}}\operatorname{Hom}(V_{2,i},V^{\prime}_{2,j}).$

The tensor product of neutral Tannakian categories is again a neutral
Tannakian category.

#### 1.2.3.

Recall that $\check{G}_{\mathbb{Q}}$ is the split reductive group over
$\mathbb{Q}$ Langlands dual to $G$. Fix a Borel subgroup
$\check{B}_{\mathbb{Q}}\subset\check{G}_{\mathbb{Q}}$ and a split maximal
torus $\check{T}_{\mathbb{Q}}\subset\check{B}_{\mathbb{Q}}$. The sum of all
positive coroots of $\check{G}_{\mathbb{Q}}$ with respect to
$\check{T}_{\mathbb{Q}}\subset\check{B}_{\mathbb{Q}}$ is a cocharacter
$2\rho\colon\mathbf{G_{m,\mathbb{Q}}}\hookrightarrow\check{G}_{\mathbb{Q}}$.

The simple objects of $\operatorname{Rep}_{\mathbb{Q}}\check{G}_{\mathbb{Q}}$
correspond to the dominant weights of $\check{G}_{\mathbb{Q}}$. If $\lambda$
is such a weight, we call the corresponding representation even or odd
depending on the parity of $2\rho(\lambda)$. Thus, we equip the category
$\operatorname{Rep}_{\mathbb{Q}}\check{G}_{\mathbb{Q}}$ with a
$\mathbb{Z}/2\mathbb{Z}$ grading. Now we can formulate our first main result.

######  Theorem 1.

The category $\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ has
a natural structure of a neutral Tannakian category. There is a canonical
equivalence of Tannakian categories compatible with the gradings and the fiber
functors

$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})=\left(\operatorname{Rep}_{\mathbb{Q}}\check{G}_{\mathbb{Q}}\boxtimes\operatorname{MH}(pt)[\sqrt{T}]\right)^{+},$

where $\operatorname{MH}(pt)[\sqrt{T}]$ is an instance of the construction
defined in Section 1.2.1, $T$ is the Tate twist; the superscript $+$ stands
for the even component of the $\mathbb{Z}/2\mathbb{Z}$-graded category.

We note that if $-1$ is in the kernel of
$2\rho\colon\mathbf{G_{m,\mathbb{Q}}}\hookrightarrow\check{G}_{\mathbb{Q}}$,
then the $\mathbb{Z}/2\mathbb{Z}$-grading on
$\operatorname{Rep}_{\mathbb{Q}}\check{G}_{\mathbb{Q}}$ is trivial, and the
previous theorem becomes

$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})=\operatorname{Rep}_{\mathbb{Q}}\check{G}_{\mathbb{Q}}\boxtimes\operatorname{MH}(pt).$

###  1.3. Geometric Satake Equivalence for Tate’s Hodge modules

Let $\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ be the
strictly full subcategory of
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ whose objects are
isomorphic to direct sums of Tate twists of IC Hodge modules on the
$G_{\mathcal{O}}$-orbits in $\operatorname{Gr}_{G}$. This is a semisimple
monoidal subcategory of
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$, see Proposition
4.3 below. We can give a more direct Tannakian description of this category.

We can form a twisted product

$\check{G}_{\mathbb{Q}}\times^{\mu_{2}}\mathbf{G_{m,\mathbb{Q}}}:=(\check{G}_{\mathbb{Q}}\times\mathbf{G_{m,\mathbb{Q}}})/((2\rho\times\iota)(\mu_{2})),$

where $\iota\colon\mu_{2}\to\mathbf{G_{m,\mathbb{Q}}}$ is the standard
embedding. This is Deligne’s modified Langlands dual group.

######  Theorem 2.

There is a canonical equivalence of Tannakian categories compatible with the
gradings and the fiber functors

$\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\simeq\operatorname{Rep}_{\mathbb{Q}}(\check{G}_{\mathbb{Q}}\times^{\mu_{2}}\mathbf{G_{m,\mathbb{Q}}}).$

This should be compared with [RS, Thm. C].

###  1.4. Geometric Satake equivalence for mixed Hodge modules

Similarly to the above, one can define the category
$\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ of
$G_{\mathcal{O}}$-equivariant mixed Hodge modules on $\operatorname{Gr}_{G}$.
In Section 5 we sketch the proof of the following theorem.

######  Theorem 3.

The category $\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ has
a natural structure of a neutral Tannakian category. There is a canonical
equivalence of Tannakian categories compatible with the fiber functors

$\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\simeq\left(\operatorname{Rep}_{\mathbb{Q}}\check{G}_{\mathbb{Q}}\boxtimes\operatorname{MHM}(pt)[\sqrt{T}]\right)^{+},$

where, as before, $T$ is the Tate twist and the superscript $+$ stands for the
even component of the $\mathbb{Z}/2\mathbb{Z}$-graded category.

We note that the category $\operatorname{MHM}(pt)$ is not semisimple. However,
the tensor product of two $\mathbb{Q}$-linear categories also makes sense if
only one of the categories is semisimple. Thus, the right hand side of the
above equivalence of categories makes sense.

###  1.5. Future directions

Consider the cotangent stack

$\operatorname{St}_{G}:=T^{*}(G_{\mathcal{O}}\backslash\operatorname{Gr}_{G}),$

which we call the _Steinberg Stack_. Here the stack
$G_{\mathcal{O}}\backslash\operatorname{Gr}_{G}$ is the quotient of
$\operatorname{Gr}_{G}$ by the action of $G_{\mathcal{O}}$. Note that
$\operatorname{St}_{G}$ is _not_ an ind-algebraic stack, since some points
have infinite-dimensional automorphism groups. However, one can show that it
is a quotient of an ind-scheme by $G_{\mathcal{O}}$. In a subsequent work we
will construct a functor

$\operatorname{Rep}_{\mathbb{Q}}(\check{G}_{\mathbb{Q}}\times^{\mu_{2}}\mathbf{G_{m,\mathbb{Q}}})\to\operatorname{Coh}(\operatorname{St}_{G}),$

where $\operatorname{Coh}(\bullet)$ is the abelian category of coherent
sheaves. We will equip $\operatorname{Coh}(\operatorname{St}_{G})$ with a
monoidal structure and show that the functor is uniquely defined by certain
axioms, provided that the quotient of $G$ by its radical is of adjoint type.
We call this functor the _semi-classical Satake functor_. We expect that this
functor will play a crucial role in the local Langlands duality for Hitchin
systems. Let us sketch the construction of this functor.

Assume first that $Y$ is a separated scheme smooth over $\mathbb{C}$. Then
$\operatorname{MH}(Y)$ is a strictly full subcategory of the category of
$\mathcal{D}$-modules on $Y$ with good filtrations (see Section 2.1.2 below).
Since the associated graded of a filtered $\mathcal{D}$-module is a coherent
sheaf on the cotangent bundle, we get a functor
$\operatorname{gr}\colon\operatorname{MH}(Y)\to\operatorname{Coh}(T^{*}Y)$.
This can be extended to ind-schemes and to the equivariant setting so that we
get a functor

$\operatorname{gr}\colon\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\to\operatorname{Coh}(\operatorname{St}_{G}).$

The semi-classical Satake functor is defined as the composition

$\operatorname{Rep}_{\mathbb{Q}}(\check{G}_{\mathbb{Q}}\times^{\mu_{2}}\mathbf{G_{m,\mathbb{Q}}})\xrightarrow{\simeq}\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\hookrightarrow\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\xrightarrow{\operatorname{gr}}\operatorname{Coh}(\operatorname{St}_{G}).$

The biggest difficulty in defining this functor is that
$\operatorname{Gr}_{G}$ is not a limit of smooth projective schemes.

###  1.6. Notation and conventions

We will be working with left $\mathcal{D}$-modules. We will freely work with
perverse sheaves on ind-schemes. The category of perverse sheaves of vector
spaces over $\mathbb{Q}$ on $Y$, where $\mathbb{Q}$ is a field, is denoted by
$\operatorname{Perv}_{\mathbb{Q}}(Y)$.

We call an affine group scheme over $\mathbb{Q}$ a group for brevity.

We write $\simeq$ for an isomorphism and simply $=$ for a canonical
isomorphism.

###  1.7. Organization of the paper

In Section 2 we recall all the statements we need about Hodge modules. We
extend the definition of Hodge modules to the case of ind-schemes and to the
equivariant setup.

In Section 3 we equip the category
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ with a structure
of neutral Tannakian category. In Section 4 we prove Theorems 1 and 2.
Finally, in Section 5 we sketch a proof of Theorem 3.

###  1.8. Acknowledgements

A large part of this paper was written while the author visited Max Planck
Institute for Mathematics in Bonn (July 2018) and also Research Institute for
Mathematical Sciences in Kyoto (December 2018). The author would like to thank
Claude Sabbah for answering numerous questions about Hodge modules. The author
also benefited from talks with Roman Bezrukavnikov. The author is thankful to
Dennis Gaitsgory for the idea of the proof of Proposition 4.3 and to George
Lusztig for correcting a historical mistake. Of great importance was advice
from Dima Arinkin. The paper would not have been written without his help.

The author is partially supported by NSF grants DMS–1406532, DMS–2001516.

##  2\. Hodge modules on ind-schemes

The categories of Hodge modules on varieties have been defined and studied by
M. Saito in [Sai1, Sai2]. In this section we extend this to the case of ind-
separated ind-schemes with ‘nice’ actions of affine groups. We formulate all
the statements we will need later and prove most of them (leaving a few
simpler statements to the reader). It seems that some of the results about
pure polarizable Hodge modules (especially the results related to the inverse
images) can only be proved with the help of mixed Hodge modules, so we will
use mixed Hodge modules in the proofs as well.

###  2.1. Hodge modules on schemes

We briefly recall M. Saito’s theory of Hodge modules.

####  2.1.1. Filtered $\mathcal{D}$-modules with rational structures on smooth
schemes

Let $Y$ be a separated scheme smooth over $\mathbb{C}$. Let $\mathcal{D}_{Y}$
denote the sheaf of differential operators on $Y$ with the usual filtration
$\mathcal{D}^{\leq p}_{Y}$ by the degree. Let $M$ be a
$\mathcal{D}_{Y}$-module. Recall that an increasing filtration $F_{\bullet}$
of $M$ by $\mathcal{O}_{Y}$-coherent submodules is _good_ if it is separated
and exhaustive, it satisfies $\mathcal{D}^{\leq p}_{Y}\cdot F_{q}M\subset
F_{p+q}M$, and there is $q_{0}\in\mathbb{Z}$ such that for all $p\geq 0$ and
$q\geq q_{0}$ we have $\mathcal{D}^{\leq p}_{Y}\cdot F_{q}M=F_{p+q}M$.

Let $\operatorname{FM}_{rs}(\mathcal{D}_{Y})$ be the category of quadruples
$(M,F_{\bullet},\mathcal{M},\alpha)$, where $M$ is a holonomic
$\mathcal{D}_{Y}$-module with regular singularities, $F_{\bullet}$ is a good
filtration on $M$, $\mathcal{M}\in\operatorname{Perv}_{\mathbb{Q}}(Y)$, and
$\alpha$ is an isomorphism
$\alpha\colon\operatorname{DR}(M)\xrightarrow{\simeq}\mathcal{M}\otimes_{\mathbb{Q}}\mathbb{C}$
($\operatorname{DR}$ is the de Rham functor).

Since $\operatorname{DR}$ is an equivalence of categories, we see that
$\operatorname{FM}_{rs}(\mathcal{D}_{Y})$ can be defined equivalently as the
category of pairs $(\mathcal{M},F_{\bullet})$, where
$\mathcal{M}\in\operatorname{Perv}_{\mathbb{Q}}(Y)$, $F_{\bullet}$ is good
filtration of
$\operatorname{DR}^{-1}(\mathcal{M}\otimes_{\mathbb{Q}}\mathbb{C}$). It is
clear that $\operatorname{FM}_{rs}(\mathcal{D}_{Y})$ is an additive
$\mathbb{Q}$-linear category. We define the functor
$\mathrm{Rat}\colon\operatorname{FM}_{rs}(\mathcal{D}_{Y})\to\operatorname{Perv}_{\mathbb{Q}}(Y)$
by $\mathrm{Rat}(M,F_{\bullet},\mathcal{M},\alpha)=\mathcal{M}$. We see that
$\mathrm{Rat}$ is faithful.

If $j\colon U\to Y$ is an open embedding, we have an obvious restriction
functor

(2)
$j^{*}\colon\operatorname{FM}_{rs}(\mathcal{D}_{Y})\to\operatorname{FM}_{rs}(\mathcal{D}_{U}).$

On the other hand, let $i\colon Y^{\prime}\to Y$ be a closed embedding of
smooth varieties with connected $Y^{\prime}$. Following [Sai1, Intro.], we
define the direct image functor

(3)
$i_{*}\colon\operatorname{FM}_{rs}(\mathcal{D}_{Y^{\prime}})\to\operatorname{FM}_{rs}(\mathcal{D}_{Y})$

by

$i_{*}(M,F_{\bullet},\mathcal{M},\alpha):=(i_{*}M,{{}^{\prime}\\!F}^{\bullet},i_{*}\mathcal{M},i_{*}(\alpha)).$

We are using the fact that $\operatorname{DR}$ commutes with $i_{*}$. The
filtration ${}^{\prime}\\!F_{\bullet}$ is defined as follows. Recall that
$i_{*}M=i_{ab,*}(\mathcal{D}_{Y\leftarrow
Y^{\prime}}\otimes_{\mathcal{D}_{Y^{\prime}}}M)$, where $i_{ab,*}$ is the
direct image for sheaves of abelian groups. Then

${{}^{\prime}\\!F}^{p}i_{*}M:=i_{ab,*}\left(\sum_{q+\nu\leq
p-n}F_{\nu}\mathcal{D}_{Y\leftarrow Y^{\prime}}\otimes F_{q}M\right),$

where $n$ is the codimension of $Y^{\prime}$ in $Y$. It is clear that the
restriction functor for the composition of open embeddings is the composition
of restriction functors, and the direct image functor for the composition of
closed embeddings is the composition of direct image functors. It is also
clear that $i_{*}$ commutes with restriction to open subsets (this is a
particular case of the base change). We define the support of a filtered
$\mathcal{D}_{Y}$-module with a rational structure by

$\operatorname{supp}(M,F_{\bullet},\mathcal{M},\alpha):=\operatorname{supp}(M)=\operatorname{supp}(\mathcal{M}).$

Thus $\operatorname{supp}(M,F_{\bullet},\mathcal{M},\alpha)$ is a reduced
closed subscheme of $Y$.

####  2.1.2. Polarizable Hodge modules on separated schemes of finite type

In [Sai1] for every smooth over $\mathbb{C}$ separated scheme $Y$ Saito
defines the category $\operatorname{MH}(Y,\mathbb{Q},n)^{p}$ of polarizable
Hodge module of weight $n$. This is a strictly full subcategory of
$\operatorname{FM}_{rs}(\mathcal{D}_{Y})$. More precisely, Saito works with an
analogue of $\operatorname{FM}_{rs}(\mathcal{D}_{Y})$, where the
$\mathcal{D}_{Y}$-modules are only assumed to be holonomic. However, in
retrospect all $\mathcal{D}_{Y}$-modules underlying polarizable Hodge modules
have regular singularities [Sai1, p. 858 and Rem. 5.1.18].

_Convention_. We will only work with rational polarizable Hodge modules, so we
denote $\operatorname{MH}(Y,\mathbb{Q},n)^{p}$ by $\operatorname{MH}(Y,n)$ and
sometimes skip the adjective ‘polarizable’.

Let now $Y$ be a not necessarily smooth reduced separated scheme of finite
type over $\mathbb{C}$. In [Sai1, Sect. 5.3.12] (see also [Sai2, Sect. 2.1])
Saito extends the definition of $\operatorname{MH}(Y,n)$ to this case as
follows. Let $Y=\bigcup_{i}U_{i}$ be a Zariski cover and $U_{i}\to V_{i}$ be
closed embeddings into smooth schemes. Then a Hodge module on $Y$ is a
collection $\mathcal{F}_{i}$ of Hodge modules on $V_{i}$ such that
$\operatorname{supp}(\mathcal{F}_{i})\subset U_{i}$ together with isomorphisms
on intersections satisfying the cocycle condition. The category
$\operatorname{MH}(Y,n)$ does not depend on the cover and the embeddings
because Hodge modules on smooth separated schemes satisfy the Zariski descent
and Kashiwara’s equivalence.

######  Remark.

Polarizable Hodge modules satisfy Kashiwara’s equivalence according to [Sai1,
Lemme 5.1.9, Lemme 5.2.11]. It follows implicitly from [Sai1, Sect. 5.3.12]
that they satisfy Zariski descent. The reason is, however, not quite clear. In
fact, Zariski descent follows from definitions for Hodge modules, but it is
not obvious that a locally polarizable module is polarizable. This, however,
follows from [Sai2, Thm. 3.21] after reducing to the case of a module with
strict support (cf. also Proposition 2.1 below).

To summarize, for a reduced separated $\mathbb{C}$-scheme of finite type, we
have the category of Hodge modules; the Hodge modules satisfy the Zariski
descent and Kashiwara’s equivalence. In particular, the restriction functors
(2) to open subsets of smooth schemes preserve the subcategories of Hodge
modules and the definition of these functors easily extends to the case of
singular schemes. The same applies to the direct image functors (3).

We can easily extend the definition of Hodge modules to non-reduced schemes by
setting $\operatorname{MH}(Y,n):=\operatorname{MH}(Y_{red},n)$. Kashiwara’s
equivalence and the Zariski locality extend to this case. One also extends the
definition of the functor
$\mathrm{Rat}\colon\operatorname{MH}(Y,n)\to\operatorname{Perv}_{\mathbb{Q}}(Y)$
to not necessarily reduced schemes. Note that the support of a Hodge module is
always reduced.

######  Remark.

We can also extend the definition of Hodge modules to not necessarily
separated schemes locally of finite type over $\mathbb{C}$ using Zariski
locality. However, we will not need it. In fact, all our schemes will be
quasi-projective. Thus, we leave the generalization to the reader.

####  2.1.3. Hodge modules with strict support and polarizable variations of
Hodge structures

Let $Y$ be any separated scheme of finite type over $\mathbb{C}$. Let
$Z\subset Y$ be an integral closed subscheme. Recall from [Sai2, Sect. 2.1]
that $\mathcal{F}\in\operatorname{MH}(Y,n)$ _has strict support_ $Z$ if
$\operatorname{supp}\mathcal{F}=Z$ and $\mathrm{Rat}(\mathcal{F})$ is an IC-
sheaf (that is, $\mathrm{Rat}(\mathcal{F})$ has neither quotient objects nor
subobjects supported on a proper subscheme of $Z$, cf. also [Sai1, p. 3]).
Denote the full subcategory of Hodge modules of weight $n$ with strict support
$Z$ by $\operatorname{MH}_{Z}(Y,n)$. If $Z$ is an integral separated scheme of
finite type over $\mathbb{C}$ we set
$\operatorname{MH}_{Z}(n):=\operatorname{MH}_{Z}(Z,n)$.

Let $U$ be an integral smooth scheme. Recall that a rational variation of
Hodge structures (VHS for short) of weight $n$ on $U$ is a pair
$(\mathcal{S},F_{\bullet})$, where $\mathcal{S}$ is a locally constant sheaf
of $\mathbb{Q}$-vector spaces, $F_{\bullet}$ is a filtration of
$\mathcal{S}\otimes_{\mathbb{Q}}\mathcal{O}_{U}$ satisfying Griffiths
transversality condition and such that for all $u\in U$ the pair
$(\mathcal{S}_{u},(F_{\bullet})_{u})$ is a Hodge structure of weight $n$. We
note that in this case $\mathcal{S}[\dim U]$ is a perverse sheaf and
$\operatorname{DR}^{-1}(\mathcal{S}[\dim
U]\otimes_{\mathbb{Q}}\mathbb{C})=(\mathcal{S}\otimes_{\mathbb{Q}}\mathcal{O}_{U},\nabla)$,
where $\nabla$ is the connection whose sheaf of flat sections is
$\mathcal{S}\otimes_{\mathbb{Q}}\mathbb{C}$. Then $F_{\bullet}$ is a good
filtration of $(\mathcal{S}\otimes_{\mathbb{Q}}\mathcal{O}_{U},\nabla)$ and we
see that a VHS is a particular case of a filtered $\mathcal{D}$-module with
rational structure.

Denote by $\operatorname{VHS}(U,n)$ the category of polarizable variations of
Hodge structure of weight $n$ with coefficients in $\mathbb{Q}$ with quasi-
unipotent local monodromies. If $W\subset U$ is an open subscheme, then we
have a restriction functor

$\operatorname{VHS}(U,n)\to\operatorname{VHS}(W,n).$

For an integral separated scheme $Z$ of finite type over $\mathbb{C}$, let
$\operatorname{VHS}_{gen}(Z,n)$ denote the direct limit of categories
$\operatorname{VHS}(U,n)$, where $U$ ranges over smooth open non-empty
subschemes of $Z$.

######  Proposition 2.1.

Let $Z$ be an integral separated scheme of finite type over $\mathbb{C}$. Then
there is a canonical equivalence of categories

$\operatorname{VHS}_{gen}(Z,n-\dim
Z)\xrightarrow{\simeq}\operatorname{MH}_{Z}(n)$

satisfying the following compatibilities
(i) If $Z$ is smooth, then this equivalence associates to a VHS on $Z$ the
corresponding filtered $\mathcal{D}$-module with rational structure.
(ii) If $j\colon Z^{\prime}\subset Z$ is an open embedding of a non-empty
subscheme, then we have a canonical commutative diagram of functors

$\begin{CD}\operatorname{VHS}_{gen}(Z,n-\dim
Z)@>{\simeq}>{}>\operatorname{MH}_{Z}(n)\\\ \Big{\|}@V{}V{j^{*}}V\\\
\operatorname{VHS}_{gen}(Z^{\prime},n-\dim
Z^{\prime})@>{\simeq}>{}>\operatorname{MH}_{Z^{\prime}}(n)\end{CD}$

In particular, $j^{*}$ is an equivalence of categories.

###### Proof.

The construction of this equivalence of categories is [Sai2, Thm. 3.21]. The
compatibility with restriction to open subsets is clear from the proof. ∎

In particular, $\operatorname{MH}(pt,n)$ is the category of rational
polarizable Hodge structures of weight $n$. Note that the functor
$\mathrm{Rat}$ associates to a Hodge structure, viewed as an object of
$\operatorname{MH}(pt,n)$, the underlying vector space.

###  2.2. Hodge modules on ind-schemes

All the ind-schemes we consider are strict ind-separated ind-schemes of ind-
finite type over $\mathbb{C}$. That is, we work with étale sheaves
$Y\colon\mathrm{Aff}/\mathbb{C}\to\mathrm{Sets}^{op}$ representable as
$Y=\lim_{l\to\infty}Y^{(l)}$, where $Y^{(l)}$ are separated schemes of finite
type over $\mathbb{C}$, and for $l<l^{\prime}$ the morphism $Y^{(l)}\to
Y^{(l^{\prime})}$ is a closed embedding. Recall that any morphism $Z\to Y$,
where $Z$ is a scheme of finite type over $\mathbb{C}$, factors through some
$Y^{(l)}$.

We define

$\operatorname{MH}(Y,n):=\lim_{l\to\infty}\operatorname{MH}(Y^{(l)},n),$

where the direct limit is with respect to the system of direct image functors
corresponding to closed embeddings $Y^{(l)}\to Y^{(l^{\prime})}$,
$l<l^{\prime}$. By Kashiwara’s equivalence, this category does not depend on
the presentation of $Y$ as a limit of schemes. Explicitly, a Hodge module on
$Y$ is a pair $(l,\mathcal{F})$, where $\mathcal{F}$ is a Hodge module on
$Y^{(l)}$.

We note that the definitions of the functor of restriction to an open
subscheme and of the direct image under a closed embedding extend easily to
the case of ind-schemes. One also extends the definition of the functor
$\mathrm{Rat}$ to the case of ind-schemes.

If $Y$ is a scheme viewed as an ind-scheme, then $\operatorname{MH}(Y,n)$
defined above is the same as $\operatorname{MH}(Y,n)$ discussed in Section
2.1.2.

We easily extend the notion of support to Hodge modules on ind-schemes. We
emphasize that the support of a Hodge module is a reduced subscheme of $Y$
(not an ind-subscheme). The notion of a Hodge module with strict support also
extends to the case of ind-schemes. Kashiwara’s equivalence gives a canonical
identification $\operatorname{MH}_{Z}(Y,n)=\operatorname{MH}_{Z}(n)$, whenever
$Z$ is an integral subscheme of $Y$. We have the following decomposition

(4) $\operatorname{MH}(Y,n)=\bigoplus_{Z\subset
Y}\operatorname{MH}_{Z}(Y,n)\simeq\bigoplus_{Z\subset
Y}\operatorname{MH}_{Z}(n),$

where the direct sum is over integral closed subschemes of $Y$. Indeed, if $Y$
is a smooth scheme, then this follows from [Sai1, p. 852–853]. If $Y$ can be
embedded into a smooth scheme, this follows from Kashiwara’s equivalence.
Thus, if $Y$ is any reduced scheme, then the decomposition exists locally. But
the decomposition of a Hodge module into the direct sum of modules with strict
support is unique because there are no morphisms between Hodge modules with
different strict supports. Thus the global decomposition follows from the
local one. Finally, the case of ind-schemes is reduced to the case of reduced
schemes using again Kashiwara’s equivalence.

######  Proposition 2.2.

Let $Y$ be a strict ind-scheme ind-separated and of ind-finite type over
$\mathbb{C}$. Then

(i) $\operatorname{MH}(Y,n)$ is a $\mathbb{Q}$-linear abelian semisimple
category.

(ii) The functor
$\mathrm{Rat}\colon\operatorname{MH}(Y,n)\to\operatorname{Perv}_{\mathbb{Q}}(Y)$
is exact, faithful, and conservative. It is compatible with the direct image
functor for closed embeddings and with restrictions to open sub-ind-schemes.

###### Proof.

(2.2) $\mathbb{Q}$-linearity is obvious from the definition. Let $Z$ be a
closed integral subscheme of $Y$. Then according to [Sai1, Lemme 5]
$\operatorname{MH}_{Z}(n)$ is abelian and semisimple. It remains to use (4).

(2.2) The compatibility with closed and open embeddings is clear. It is exact
because every additive functor from a semisimple category is exact. To show
that the functor is faithful, we first reduce the statement to the case when
$Y$ is a smooth scheme and then use the fact that $\operatorname{MH}(Y,n)$ is
a full subcategory of $\operatorname{FM}_{rs}(\mathcal{D}_{Y})$. Finally, any
faithful functor between abelian categories is conservative, since abelian
categories are balanced. ∎

###  2.3. Derived categories of polarizable Hodge modules

For a reduced separated scheme $Y$ of finite type over $\mathbb{C}$ Saito
defines the abelian category of mixed polarizable Hodge modules
$\operatorname{MHM}(Y)$ ([Sai2, (4.2.3)]) and its bounded derived category
$D^{b}\operatorname{MHM}(Y)$. By [Sai2, 2.17.5] mixed Hodge modules satisfy
Kashiwara’s equivalence. By [Sai2, 4.2.10] the derived category
$D^{b}\operatorname{MHM}(Y)$ satisfies Kashiwara’s equivalence as well. Thus
we can extend the definitions of the categories to the case, when $Y$ is an
ind-scheme. We still have a faithful exact functor
$\mathrm{Rat}\colon\operatorname{MHM}(Y)\to\operatorname{Perv}_{\mathbb{Q}}(Y)$.

Every mixed Hodge module has a weight filtration whose associated graded is a
pure Hodge module. The category $\operatorname{MH}(Y,n)$ is a full subcategory
of $\operatorname{MHM}(Y)$.

We define the derived category $D^{b}\operatorname{MH}(Y,n)$ of pure Hodge
modules of weight $n$ as the full subcategory of $D^{b}\operatorname{MHM}(Y)$
consisting of complexes $\mathcal{F}^{\bullet}$ such that for all
$i\in\mathbb{Z}$ we have
$H^{i}(\mathcal{F}^{\bullet})\in\operatorname{MH}(Y,n+i)$. We identify
$\operatorname{MH}(Y,n)$ with the full subcategory of
$D^{b}\operatorname{MH}(Y,n)$ consisting of complexes whose cohomology is
concentrated in degree 0.

######  Proposition 2.3.

(i) The functor

(5) $\bigoplus_{i\in\mathbb{Z}}\operatorname{MH}(Y,n+i)\to
D^{b}\operatorname{MH}(Y,n)\colon\bigoplus_{i\in\mathbb{Z}}\mathcal{F}_{i}\mapsto\bigoplus_{i\in\mathbb{Z}}\mathcal{F}_{i}[-i]$

is faithful and essentially surjective. A right inverse functor is given by

$D^{b}\operatorname{MH}(Y,n)\to\bigoplus_{i\in\mathbb{Z}}\operatorname{MH}(Y,n+i)\colon\mathcal{F}^{\bullet}\mapsto\bigoplus_{i\in\mathbb{Z}}H^{i}(\mathcal{F}^{\bullet}).$

(ii) Let $i\colon Y\to Y^{\prime}$ be a closed embedding. Then there is a
fully faithful canonical functor $i_{*}\colon D^{b}\operatorname{MH}(Y,n)\to
D^{b}\operatorname{MH}(Y^{\prime},n)$ compatible with functors from part (i)
whose essential image consists of complexes whose cohomology are supported on
$Y$.

###### Proof.

The essential surjectivity in part (i) follows easily from [Sai2, (4.5.4)] in
the case when $Y$ is a reduced scheme. It easily extends to the case of ind-
schemes. The rest of the proposition follows from the definitions. ∎

######  Remark.

Note that functor (5) is not full. Indeed, let a mixed Hodge module
$\mathcal{F}$ on $Y$ be an extension of a pure Hodge module $\mathcal{F}^{1}$
of weight 1 by a pure Hodge module $\mathcal{F}^{0}$ of weight $0$. Then we
have in $D^{b}\operatorname{MH}(Y,1)$:
$\operatorname{Hom}(\mathcal{F}^{1},\mathcal{F}^{0}[1])=\operatorname{Ext}^{1}(\mathcal{F}^{1},\mathcal{F}^{0})\neq
0$.

######  Proposition 2.4.

There is a canonical functor $\mathrm{Rat}\colon
D^{b}\operatorname{MH}(Y,n)\to D^{b}\operatorname{Perv}_{\mathbb{Q}}(Y)$. This
functor is exact and conservative. It is also compatible with the cohomology
functor and
$\mathrm{Rat}\colon\operatorname{MH}(Y,n)\to\operatorname{Perv}_{\mathbb{Q}}(Y)$.

###### Proof.

The proposition is easily reduced to the case, when $Y$ is a reduced scheme.
We define the functor by restricting the exact functor $\mathrm{Rat}\colon
D^{b}\operatorname{MHM}(Y)\to D^{b}\operatorname{Perv}_{\mathbb{Q}}(Y)$ from
[Sai2, Thm. 0.1].

Let $f$ be a morphism in $D^{b}\operatorname{MH}(Y)$ such that
$\mathrm{Rat}(f)$ is an isomorphism. Then
$\mathrm{Rat}(H^{i}(f))=H^{i}(\mathrm{Rat}(f))$ is an isomorphism for all $i$.
By Proposition 2.2(2.2) $H^{i}(f)$ is an isomorphism. Thus $f$ is an
isomorphism. We see that $\mathrm{Rat}$ is conservative. The compatibility
with cohomology follows from definitions. ∎

###  2.4. The intermediate extension functor

Let $j\colon Y^{\prime}\to Y$ be a locally closed embedding and let $Z\subset
Y^{\prime}$ be an integral closed subscheme. Then we have by Kashiwara’s
equivalence and Proposition 2.1(ii) an equivalence

$j_{!*}\colon\operatorname{MH}_{Z}(Y^{\prime},n)\xrightarrow{\simeq}\operatorname{MH}_{Z}(n)\xrightarrow{\simeq}\operatorname{MH}_{\overline{Z}}(n)\xrightarrow{\simeq}\operatorname{MH}_{\overline{Z}}(Y,n),$

where $\overline{Z}$ is the Zariski closure of $Z$ in $Y$. We extend $j_{!*}$
to a fully faithful functor called _intermediate extension_

$j_{!*}\colon\operatorname{MH}(Y^{\prime},n)\to\operatorname{MH}(Y,n)$

using (4).

The next lemma follows from definitions and Proposition 2.1.

######  Lemma 2.5.

(i) If $j\colon Y_{1}\to Y_{2}$ is an open embedding, then $j^{*}$ is left
inverse to $j_{!*}$.

(ii) If $j\colon Y_{1}\to Y_{2}$ is a closed embedding, then $j_{!*}=j_{*}$.

(iii) Let $j\colon Y_{1}\to Y_{2}$ and $j^{\prime}\colon Y_{2}\to Y_{3}$ be
locally closed embeddings. Then $(j^{\prime}\circ j)_{!*}=j^{\prime}_{!*}\circ
j_{!*}$.

We have the standard characterization of intermediate extensions.

######  Lemma 2.6.

Let $j\colon Y^{\prime}\to Y$ be an open embedding and let
$\mathcal{F}\in\operatorname{MH}(Y,n)$, then the following are equivalent
(i) $\mathrm{Rat}(\mathcal{F})$ has no subobjects supported on $Y-Y^{\prime}$.
(ii) $\mathrm{Rat}(\mathcal{F})$ has no quotient objects supported on
$Y-Y^{\prime}$.
(iii) The Hodge modules $j_{!*}j^{*}\mathcal{F}$ and $\mathcal{F}$ are
isomorphic.
In this case there is a unique isomorphism
$j_{!*}j^{*}\mathcal{F}\to\mathcal{F}$ whose restriction to $Y^{\prime}$ is
the identity.

###### Proof.

It is clear from the definition of $j_{!*}$ that for any
$\mathcal{F}^{\prime}\in\operatorname{MH}(Y^{\prime},n)$ the Hodge module
$j_{!*}\mathcal{F}^{\prime}$ has no sub- or quotient-objects supported on
$Y-Y^{\prime}$. Thus (iii) implies (i) and (ii). Let us show that (i) implies
(iii). By (4) we may assume that $\mathcal{F}$ has strict support $Z$ for a
closed subscheme $Z\subset Y$. Condition (i) implies that $Z\cap
Y^{\prime}\neq\varnothing$. Thus by the definition of intermediate extension
and Proposition 2.1 we can write $\mathcal{F}=j_{!*}\mathcal{F}^{\prime}$ for
$\mathcal{F}^{\prime}\in\operatorname{MH}(Y^{\prime},n)$. It remains to use
Lemma 2.5(i). We also see from Proposition 2.1 that there is a unique
isomorphism $j_{!*}j^{*}\mathcal{F}\to\mathcal{F}$ whose restriction to
$Y^{\prime}$ is the identity. Similarly we show that (ii) implies (iii). ∎

There is a related statement for mixed Hodge modules.

######  Lemma 2.7.

Let $j\colon Y^{\prime}\to Y$ be an open embedding and let
$\mathcal{F}\in\operatorname{MHM}(Y)$ be such that $j^{*}\mathcal{F}$ is pure
of weight $n$ and $\mathrm{Rat}(\mathcal{F})$ has no subobjects or quotient
objects supported on $Y-Y^{\prime}$. Then
$\mathcal{F}\in\operatorname{MH}(Y,n)$ and there is a unique isomorphism
$j_{!*}j^{*}\mathcal{F}\to\mathcal{F}$ whose restriction to $Y^{\prime}$ is
the identity.

###### Proof.

Considering the weight filtration on $\mathcal{F}$, we see that if
$\mathcal{F}$ is not pure of weight $n$, then there is either a submodule
$\mathcal{F}^{\prime}$ whose weights are less than $n$, or a quotient module
$\mathcal{F}^{\prime}$ whose weights are greater than $n$. Since the weight
filtration is Zariski local, we see that $j^{*}\mathcal{F}^{\prime}=0$. Thus
$\mathrm{Rat}(\mathcal{F}^{\prime})$ is supported on $Y-Y^{\prime}$, which is
a contradiction. We see that $\mathcal{F}$ is pure of weight $n$ and we can
apply Lemma 2.6. ∎

The next lemma follows from the definitions.

######  Lemma 2.8.

$j_{!*}$ is compatible with $\mathrm{Rat}$ and the intermediate extension for
perverse sheaves.

###  2.5. Direct image for ind-proper morphisms

We call a morphism $f\colon Y_{1}\to Y_{2}$ of ind-schemes _ind-proper_ if for
any closed subscheme $Z\subset Y_{2}$ the fibered product
$Y_{1}\times_{Y_{2}}Z$ is the inductive limit of proper $Z$-schemes (the
morphisms being closed embeddings).

######  Remark.

It is not clear whether this definition is Zariski local over the base.
However, the definition suffices for our purposes.

The direct image of complexes of mixed Hodge modules on schemes is defined in
[Sai2]. If the morphism is proper, a version of decomposition theorem ensures
that the direct image of a pure complex remains pure. We extend this to the
ind-proper morphisms of ind-schemes in the proposition below.

######  Proposition 2.9.

(i) Let $f\colon Y_{1}\to Y_{2}$ be an ind-proper morphism of ind-separated
ind-schemes of ind-finite type over $\mathbb{C}$. Then there is a canonical
exact functor

$f_{*}\colon D^{b}\operatorname{MH}(Y_{1},n)\to
D^{b}\operatorname{MH}(Y_{2},n)$

compatible with $\mathrm{Rat}$ and direct image for perverse sheaves in the
sense that for any $\mathcal{F}\in D^{b}\operatorname{MH}(Y_{1},n)$ we have a
canonical isomorphism

$f_{*}\mathrm{Rat}(\mathcal{F})=\mathrm{Rat}(f_{*}\mathcal{F}).$

(ii) Assume that $f$ is a closed embedding. Then $f_{*}$ is the closed
embedding functor from Proposition 2.3(ii).

(iii) Let $g\colon Y_{2}\to Y_{3}$ be another ind-proper morphism. Then we
have a canonical isomorphism $(g\circ f)_{*}=g_{*}\circ f_{*}$.

###### Proof.

Assume first that all our ind-schemes are, in fact, reduced schemes. By [Sai2,
Thm. 4.3] we have the functors $f_{*},f_{!}\colon
D^{b}\operatorname{MHM}(Y_{1})\to D^{b}\operatorname{MHM}(Y_{2})$. Since $f$
is proper, we have $f_{*}=f_{!}$ by [Sai2, (4.3.3)]. Since $f_{*}$ does not
decrease weights, and $f_{!}$ does not increase weights (see [PS, Sect.
14.1.1]), $f_{*}$ actually preserves weights. Thus $f_{*}$ takes
$D^{b}\operatorname{MH}(Y_{1},n)$ into $D^{b}\operatorname{MH}(Y_{2},n)$.

The other properties follow from the respective properties of the direct image
of complexes of mixed Hodge modules.

If $Y_{1}$ and $Y_{2}$ are not necessarily reduced schemes, consider
$f_{red}\colon Y_{1}^{red}\to Y_{2}^{red}$. We set $f_{*}:=(f_{red})_{*}$. The
required properties follow immediately.

Finally, let $Y_{i}=\lim Y_{i}^{(l)}$ be ind-schemes of finite type. Every
object of the category $D^{b}\operatorname{MH}(Y_{1},n)$ can be written as
$\iota^{(l)}_{1,*}\mathcal{F}$, where $\mathcal{F}$ is an object of
$D^{b}\operatorname{MH}(Y_{1}^{(l)},n)$ and $\iota_{1}^{(l)}\colon
Y_{1}^{(l)}\to Y_{1}$ is the embedding. Note that for large enough
$l^{\prime}$, the composition $f\circ\iota_{1}^{(l)}$ factors as
$\iota_{2}^{(l^{\prime})}\circ h$, where $\iota_{2}^{(l^{\prime})}\colon
Y_{2}^{(l^{\prime})}\to Y_{2}$ is the embedding and $h\colon Y_{1}^{(l)}\to
Y_{2}^{(l^{\prime})}$ is proper. We define
$f_{*}(\iota^{(l)}_{1,*}\mathcal{F}):=\iota_{2,*}^{(l^{\prime})}h_{*}\mathcal{F}$.
We define $f_{*}$ on morphisms similarly. We leave it to the reader to check
that the functor is well-defined and to derive (i)–(iii) from the case of
schemes. ∎

Note that $H^{i}(f_{*}\mathcal{F})=0$ for
$|i|>\dim\operatorname{supp}\mathcal{F}$.

###  2.6. Duality

In [Sai2] Saito defines the duality $D\colon
D^{b}\operatorname{MHM}(Y)^{op}\to D^{b}\operatorname{MHM}(Y)$. The duality
interchanges the direct images $f_{*}$ and $f_{!}$ (see [Sai2, (4.3.5)]). In
particular, the duality commutes with $f_{*}=f_{!}$ if $f$ is proper. Applying
this in the case, when $f$ is a closed embedding, we see that the duality is
compatible with closed embeddings. This allows us to extend the duality to the
case of ind-schemes. Note that the duality functor is compatible with
$\mathrm{Rat}$ and the Verdier duality for perverse sheaves.

######  Proposition 2.10.

(i) The duality functor takes $D^{b}\operatorname{MH}(Y,n)^{op}$ to
$D^{b}\operatorname{MH}(Y,-n)$ and $\operatorname{MH}(Y,n)^{op}$ to
$\operatorname{MH}(Y,-n)$. Also, the duality is compatible with the duality
for variations of Hodge structures and Proposition 2.1.
(ii) The duality commutes with intermediate extensions and direct images for
ind-proper morphisms.

###### Proof.

It is enough to consider the case, when $Y$ is a reduced scheme. It follows,
from [PS, Prop 14.30(c)] and (4) that the duality takes
$\operatorname{MH}(Y,n)^{op}$ to $\operatorname{MH}(Y,-n)$. Thus it takes
$D^{b}\operatorname{MH}(Y,n)^{op}$ to $D^{b}\operatorname{MH}(Y,-n)$.

Let us show that the duality commutes with the intermediate extensions. Using
Kashiwara’s equivalence, we reduce to the case of an open embedding $j\colon
U\to Y$. Then for $\mathcal{F}\in\operatorname{MH}(Y,n)$ the intermediate
extension $j_{!*}\mathcal{F}$ has no quotient objects supported on $Y-U$ so
$Dj_{!*}\mathcal{F}$ has no subobjects supported on $Y-U$. Next, the duality
is compatible with restrictions to open subsets, so we have
$j^{*}Dj_{!*}\mathcal{F}=D\mathcal{F}$ (we used Lemma 2.5(i)) and the
statement follows from Lemma 2.6.

The compatibility with the duality for VHS follows from the definitions. The
compatibility with direct images has been already explained. ∎

###  2.7. Exterior product

In [Sai2], Saito defines the exterior product of mixed Hodge modules

$\boxtimes\colon\operatorname{MHM}(Y_{1})\times\operatorname{MHM}(Y_{2})\to\operatorname{MHM}(Y_{1}\times
Y_{2})$

satisfying the usual compatibilities. In particular, the exterior product is
compatible with Kashiwara’s equivalence, so we can extend it to the case of
ind-schemes.

######  Proposition 2.11.

(i) The exterior product is compatible with $Rat$.
(ii)The exterior product is compatible with the exterior product of VHS and
Proposition 2.1.
(iii) Let $j_{1}\colon Y_{1}^{\prime}\to Y_{1}$ and $j_{2}\colon
Y_{2}^{\prime}\to Y_{2}$ be locally closed embeddings and assume that
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are pure polarizable Hodge modules on
$Y^{\prime}_{1}$ and $Y^{\prime}_{2}$ respectively. Then

$(j_{1}\boxtimes
j_{2})_{!*}(\mathcal{F}_{1}\boxtimes\mathcal{F}_{2})=(j_{1})_{!*}\mathcal{F}_{1}\boxtimes(j_{2})_{!*}(\mathcal{F}_{2}).$

(iv) The exterior product restricts to a functor

$\boxtimes\colon\operatorname{MH}(Y_{1},n_{1})\times\operatorname{MH}(Y_{2},n_{2})\to\operatorname{MH}(Y_{1}\times
Y_{2},n_{1}+n_{2}).$

###### Proof.

The first two statements follow from the definitions. For (iii), using the
properties of perverse sheaves we show that
$\mathrm{Rat}((j_{1})_{!*}\mathcal{F}_{1}\boxtimes(j_{2})_{!*}\mathcal{F}_{2})$
has no subsheaves supported on

$Y_{1}\times Y_{2}-Y^{\prime}_{1}\times
Y^{\prime}_{2}=((Y_{1}-Y^{\prime}_{1})\times
Y_{2})\cup(Y_{1}\times(Y_{2}-Y^{\prime}_{2})).$

Now (iii) follows from Lemma 2.7.

It remains to prove (iv). It follows from (ii), (iii), and Proposition 2.1
that
$\operatorname{MH}_{Z_{1}}(n_{1})\boxtimes\operatorname{MH}_{Z_{2}}(n_{2})\subset\operatorname{MH}_{Z_{1}\times
Z_{2}}(n_{1}+n_{2})$, whenever $Z_{1}$ and $Z_{2}$ are integral schemes.
Finally, (iv) follows from (4). ∎

###  2.8. Tate twists

For the mixed Hodge modules on reduced separated schemes one has Tate twists
$\mathcal{F}\to\mathcal{F}(k)$, where $k\in\mathbb{Z}$. They are auto-
equivalences of $\operatorname{MHM}(X)$ sending $\operatorname{MH}(X,n)$ to
$\operatorname{MH}(X,n-2k)$ (see, e.g., [PS, XIV–17, Prop. 14.30(b)]). The
Tate twists are compatible with Kashiwara’s equivalence, so they can be
extended to ind-schemes. Note that the Tate twists are also compatible with
$\mathrm{Rat}$.

###  2.9. Inverse images

In [Sai2, Sect. 4.4] Saito defines the inverse images of mixed Hodge modules.
That is, if $f\colon Y_{1}\to Y_{2}$ is a morphism of reduced separated
schemes of finite type, then Saito defines functors $f^{!},f^{*}\colon
D^{b}\operatorname{MHM}(Y_{2})\to D^{b}\operatorname{MHM}(Y_{1})$ as right
(resp. left) adjoint to $f_{!}$ (resp. $f_{*}$).

These functors satisfy the usual properties, in particular, the base change
[Sai2, (4.4.3)] and compatibility with $\boxtimes$. This allows us to
generalize these functors to schematic morphisms of ind-schemes. The functors
$f^{!}$ and $f^{*}$ are also compatible with $\mathrm{Rat}$ and the
corresponding functors for perverse sheaves.

We will need a description of these inverse images in the case, when $f$ is a
smooth morphism of pure relative dimension. It will be convenient for us to
use the shifted inverse image $f^{\dagger}:=f^{!}[-d](-d)$, where $d$ is the
relative dimension of $f$ (this is also equal to $f^{*}[d]$ as follows from
the definition in [Sai2, Sect. 2.17]). Note that a similar functor for
perverse sheaves takes perverse sheaves to perverse sheaves, so, by
faithfulness of $\mathrm{Rat}$, $f^{\dagger}$ takes mixed Hodge modules to
mixed Hodge modules.

######  Proposition 2.12.

Let $f\colon Y_{1}\to Y_{2}$ be a smooth schematic morphism of relative
dimension $d$. Then
(i) If $f$ is an open embedding, then $f^{\dagger}$ is the restriction
functor.
(ii) If $Y_{2}$ is a smooth scheme, then $f^{\dagger}$ is compatible with
pullback of filtered $\mathcal{D}$-modules with rational structure and with
Proposition 2.1.
(iii) If $Y_{1}$ and $Y_{2}$ are integral schemes, then $f^{\dagger}$ takes
$\operatorname{MH}_{Y_{2}}(n)$ to $\operatorname{MH}_{Y_{1}}(n+d)$.
(iv) $f^{\dagger}$ takes $\operatorname{MH}(Y_{2},n)$ to
$\operatorname{MH}(Y_{1},n+d)$ and $D^{b}\operatorname{MH}(Y_{2},n)$ to
$D^{b}\operatorname{MH}(Y_{1},n+d)$.

###### Proof.

Parts (2.12) and (2.12) follow from the definition [Sai2, (2.17.6)] (see also
[Sai1, Sect. 3.5]).

Let us prove (2.12). If $Y_{2}$ is smooth, the statement follows from [Sai2,
Lemma 2.25] because being smooth $f$ is non-characteristic by [Sai1, Lemme
3.5.2]. In the general case, let $\mathcal{F}\in\operatorname{MH}_{Y_{2}}(n)$,
let $U_{2}$ be a non-empty smooth open subset of $Y_{2}$, and let
$\mathcal{F}^{\prime}:=\mathcal{F}|_{U_{2}}$. Let $g$ be the restriction of
$f$ to $U_{1}:=f^{-1}(U_{2})$. By the smooth case,
$g^{\dagger}\mathcal{F}^{\prime}\in\operatorname{MH}(U_{1},n+d)$. Let $j\colon
U_{1}\to Y_{1}$ be the open embedding. It is enough to show that
$f^{\dagger}\mathcal{F}=j_{!*}g^{\dagger}\mathcal{F}^{\prime}$, which follows
from Lemma 2.7 and the fact that smooth inverse images of perverse sheaves
commute with intermediate extensions. Finally, (2.12) follows from part
(2.12), (4), and Proposition 2.3. ∎

We need a version of base change for intermediate extensions.

######  Lemma 2.13.

Consider a Cartesian diagram of ind-schemes

$\begin{CD}Y_{1}@>{j^{\prime}}>{}>Y_{2}\\\ @V{f^{\prime}}V{}V@V{f}V{}V\\\
Z_{1}@>{j}>{}>Z_{2}\end{CD}$

where $j$ is a locally closed embedding and $f$ is smooth and schematic. Then
for all $\mathcal{F}\in\operatorname{MH}(Z_{1},n)$ we have functorially in
$\mathcal{F}$

$f^{\dagger}j_{!*}\mathcal{F}=j^{\prime}_{!*}(f^{\prime})^{\dagger}\mathcal{F}.$

###### Proof.

The proof is similar to that of Proposition 2.12(2.12) and is left to the
reader. ∎

######  Proposition 2.14.

Hodge modules satisfy descent for smooth schematic surjective morphisms.

###### Proof.

Let $f\colon Y_{1}\to Y_{2}$ be a smooth surjective schematic morphism.

We first prove the descent for morphisms. Let
$\mathcal{F},\mathcal{F}^{\prime}\in\operatorname{MH}(Y_{2},n)$ and let
$\varphi\colon f^{\dagger}\mathcal{F}\to f^{\dagger}\mathcal{F}^{\prime}$ be a
morphism. We need to show that if the two pullbacks of $\varphi$ to
$Y_{1}\times_{Y_{2}}Y_{1}$ coincide, then there is a unique morphism
$\psi\colon\mathcal{F}\to\mathcal{F}^{\prime}$ such that
$f^{\dagger}\psi=\varphi$.

First of all, we may assume that $Y_{2}$ is a quasi-projective reduced scheme.
Note that the natural map
$\operatorname{Hom}(\mathcal{F},\mathcal{F}^{\prime})\to\operatorname{Hom}(f^{\dagger}\mathcal{F},f^{\dagger}\mathcal{F}^{\prime})$
is injective (compose it with $\mathrm{Rat}$). It follows that the statement
is Zariski local over $Y_{2}$. Thus, we can assume that $Y_{2}$ is embeddable
into a smooth scheme, and, in turn, that $Y_{2}$ is a smooth scheme. It
remains to note that the category of Hodge modules on a smooth scheme is a
full subcategory of the category of filtered $\mathcal{D}$-modules with
rational structures and we have smooth descent for the latter category.

Now assume that $\mathcal{F}\in\operatorname{MH}(Y_{1},n)$ and suppose we are
given an isomorphism between the two pullbacks of $\mathcal{F}$ to
$Y_{1}\times_{Y_{2}}Y_{1}$ satisfying the cocycle condition. We need to show
that $\mathcal{F}\simeq f^{\dagger}(\mathcal{F}^{\prime})$ for some
$\mathcal{F}^{\prime}\in\operatorname{MH}(Y_{2},n-d)$, where $d$ is the
relative dimension of $f$. Since Hodge modules satisfy Zariski descent, we my
replace $Y_{1}$ with an affine cover, so we assume that $f$ is affine. We can
compactify $f$ to a projective morphism $\overline{Y}_{1}\to Y_{2}$. Using
resolution of singularities we may assume that $\overline{Y}_{1}-Y_{1}$ is a
locally principal divisor. If $j\colon Y_{1}\to\overline{Y}_{1}$ is the open
immersion, then $j_{!*}\mathcal{F}$ is an extension of $\mathcal{F}$ to
$\overline{Y}_{1}$. Next, $\mathrm{Rat}(\mathcal{F})$ descends to $Y_{2}$.
This follows from the faithfulness of $\mathrm{Rat}$ and the fact that
perverse sheaves satisfy smooth descent. Thus we can apply [Sai2, Lemma 2.27]
to conclude that there is a mixed Hodge module $\mathcal{F}^{\prime}$ such
that $\mathcal{F}\simeq f^{\dagger}(\mathcal{F}^{\prime})$. It remains to show
that $\mathcal{F}^{\prime}$ is pure of weight $n-d$. But this follows from
[Sai2, Lemma 2.25]. ∎

###  2.10. Equivariant Hodge modules on ind-schemes

Assume that $Y$ is a separated scheme of finite type over $\mathbb{C}$ acted
upon by a group $H$ of finite type over $\mathbb{C}$. Consider two smooth
morphisms: the projection $p\colon H\times Y\to Y$ and the action $a\colon
H\times Y\to Y$. An object of $\operatorname{MH}^{H}(Y,n)$ is a pair
$(\mathcal{F},s)$, where $\mathcal{F}\in\operatorname{MH}(Y,n)$, the
equivariance morphism $s\colon
p^{\dagger}\mathcal{F}\xrightarrow{\simeq}a^{\dagger}\mathcal{F}$ is an
isomorphism, and $s$ satisfies the cocycle condition (note that by Proposition
2.12(2.12) $p^{\dagger}\mathcal{F}$ and $a^{\dagger}\mathcal{F}$ are pure
Hodge modules). The morphisms in this category are morphisms of Hodge modules
compatible with the equivariance isomorphisms.

######  Lemma 2.15.

Assume that $H\to H^{\prime}$ is a surjective homomorphism of complex
algebraic groups with connected kernel. If $H$ acts on $Y$ through its
quotient $H^{\prime}$, then we have a canonical equivalence of the categories
$\operatorname{MH}^{H}(Y,n)=\operatorname{MH}^{H^{\prime}}(Y,n)$.

###### Proof.

Set $Z=H\times Y$, $Z^{\prime}=H^{\prime}\times Y$ and let $f\colon Z\to
Z^{\prime}$ be the projection. Let $\mathcal{F}\in\operatorname{MH}(Y,n)$ and
let $s$ be an equivariance isomorphism between the two pullbacks of
$\mathcal{F}$ to $Z$. We need to show that $s$ descends along $f$. Since a
similar statement is known for perverse sheaves, we see that $\mathrm{Rat}(s)$
descends along $f$, which means that the two pullbacks of $\mathrm{Rat}(s)$ to
$Z\times_{Z^{\prime}}Z$ are equal. Since $\mathrm{Rat}$ is faithful, this
implies that the two pullbacks of $s$ to $Z\times_{Z^{\prime}}Z$ are also
equal, and the statement follows from the descent (Proposition 2.14). ∎

Assume that $H=\lim\limits_{\infty\leftarrow m}H^{(m)}$ is a connected pro-
algebraic complex group acting on an ind-scheme $Y$. We will assume that the
morphisms $H^{(m)}\to H^{(m^{\prime})}$ are surjective with connected kernels.
Assume that $H$ acts _nicely_ , that is, every closed subscheme $Z\subset Y$
is contained in an $H$-invariant closed subscheme $Z^{\prime}$ on which $H$
acts via one of its quotients $H_{(m)}$ (see [Gai, A.3]). We keep these
assumptions on $H$ and the action through the end of Section 2.10. We are
going to define the category $\operatorname{MH}^{H}(Y,n)$ of pure polarizable
$H$-equivariant Hodge modules of weight $n$. We can write
$Y=\lim\limits_{l\to\infty}Y^{(l)}$, where $Y^{(l)}$ is $H$-invariant and $H$
acts through $H_{(m_{l})}$ for a certain $m_{l}$. We may assume that $m_{l}$
is a non-decreasing sequence.

For $l<l^{\prime}$ we have a functor

$F_{l,l^{\prime}}\colon\operatorname{MH}^{H_{(m_{l})}}(Y^{(l)},n)=\operatorname{MH}^{H_{(m_{l^{\prime}})}}(Y^{(l)},n)\to\operatorname{MH}^{H_{(m_{l^{\prime}})}}(Y^{(l^{\prime})},n),$

where the equality follows from Lemma 2.15 and the direct image functor is
defined with a help of base change. It is easy to check that we have a
coherent system of isomorphisms of functors
$F_{l^{\prime},l^{\prime\prime}}\circ F_{l,l^{\prime}}\simeq
F_{l,l^{\prime\prime}}$ defined whenever $l<l^{\prime}<l^{\prime\prime}$. Set

$\operatorname{MH}^{H}(Y,n):=\lim_{l\to\infty}\operatorname{MH}^{H_{(m_{l})}}(Y^{(l)},n).$

One checks that this category does not depend on the choices up to a canonical
equivalence.

######  Proposition 2.16.

(i) The forgetful functor
$\mathrm{For}\colon\operatorname{MH}^{H}(Y,n)\to\operatorname{MH}(Y,n)$ is a
fully faithful embedding.
(ii) $\operatorname{MH}^{H}(Y,n)$ is a semisimple abelian $\mathbb{Q}$-linear
category.

###### Proof.

(2.16) We may assume that $Y$ is a reduced scheme. It is clear that
$\mathrm{For}$ is faithful. Let us show that $\mathrm{For}$ is full. We take
$\mathcal{F},\mathcal{F}^{\prime}\in\operatorname{MH}^{H}(Y,n)$ and let
$\varphi\colon\mathrm{For}(\mathcal{F})\to\mathrm{For}(\mathcal{F}^{\prime})$
be a morphism. We need to show that it respects the $H$-equivariance
structure. A similar statement is well known for perverse sheaves. Thus
$\mathrm{Rat}(\varphi)$ respects the equivariant structures. It follows from
the faithfulness of $\mathrm{Rat}$ that $\varphi$ respects the equivariant
structure as well.

(2.16) It is easy to see that $\operatorname{MH}^{H}(Y,n)$ is a
$\mathbb{Q}$-linear abelian category. Now it follows from part (2.16) and
Proposition 2.2(2.2) that it is semisimple: indeed, a full abelian subcategory
of a semisimple abelian category is necessarily semisimple. ∎

We use this proposition to identify the category $\operatorname{MH}^{H}(Y,n)$
with a full subcategory of $\operatorname{MH}(Y,n)$. Thus, we may view
equivariance as a property rather than a structure. It is clear that
$\mathrm{Rat}$ takes $\operatorname{MH}^{H}(Y,n)$ to
$\operatorname{Perv}_{\mathbb{Q}}^{H}(Y,n)$. The proof of the following easy
lemma is left to the reader. We let $D^{b}\operatorname{MH}^{H}(Y,n)$ be the
full subcategory of $D^{b}\operatorname{MH}(Y,n)$ whose objects are complexes
with equivariant cohomology groups.

######  Lemma 2.17.

Let $H$ act nicely on ind-schemes $Y_{1}$ and $Y_{2}$ and let $f\colon
Y_{1}\to Y_{2}$ be an equivariant morphism, then:
(i) If $f$ is ind-proper, then
$f_{*}(D^{b}\operatorname{MH}^{H}(Y_{1},n))\subset
D^{b}\operatorname{MH}^{H}(Y_{2},n)$.
(ii) If $f$ is smooth of relative dimension $d$, then
$f^{\dagger}(\operatorname{MH}^{H}(Y_{2},n))\subset\operatorname{MH}^{H}(Y_{1},n+d)$.
(iii) If $f$ is a locally closed embedding, then
$f_{!*}(\operatorname{MH}^{H}(Y_{1},n))\subset\operatorname{MH}^{H}(Y_{2},n)$.

We have the following consequence of smooth descent (Proposition 2.14).

######  Proposition 2.18.

Let $f\colon Y_{1}\to Y_{2}$ be an $H$-torsor, where $H$ is an algebraic group
acting nicely on $Y_{1}$. Then the functor $f^{\dagger}$ induces an
equivalence of the categories
$\operatorname{MH}(Y_{2},n)\xrightarrow{\simeq}\operatorname{MH}^{H}(Y_{1},n+d)$,
where $d=\dim H$. In particular, if $H$ is connected, then $f^{\dagger}$
identifies $\operatorname{MH}(Y_{2},n)$ with a full subcategory of
$\operatorname{MH}(Y_{1},n+d)$.

Since $H$ is assumed to act nicely on $Y$, every orbit $\mathcal{O}\to Y$ is a
locally closed subscheme. Let $j_{\mathcal{O}}\colon\mathcal{O}\to Y$ be the
locally closed embedding and let $a_{\mathcal{O}}\colon\mathcal{O}\to pt$ be
the projection to the point. Let $\mathcal{H}$ be a pure polarizable Hodge
structure. Set
$\mathrm{IC}_{\mathcal{O}}(\mathcal{H}):=(j_{\mathcal{O}})_{!*}a_{\mathcal{O}}^{\dagger}\mathcal{H}$.

We will say that $H$ acts on $Y$ with _ind-finitely many orbits_ if we can
write $Y=\lim_{l\to\infty}Y^{(l)}$, where $Y^{(l)}$ is a closed $H$-invariant
subscheme on which $H$ acts with finitely many orbits. The following
proposition give full description of the abelian semisimple category
$\operatorname{MH}^{H}(Y,n)$ in the case when $H$ acts with ind-finitely many
orbits.

######  Proposition 2.19.

Let $H$ act nicely and with ind-finitely many orbits on an ind-scheme $Y$.

(i) The forgetful functor
$\operatorname{MH}^{H}(Y,n)\to\operatorname{MH}(Y,n)$ induces an equivalence
between $\operatorname{MH}^{H}(Y,n)$ and the strictly full subcategory in
$\operatorname{MH}(Y,n)$ whose simple objects are isomorphic to
$\mathrm{IC}_{\mathcal{O}}(\mathcal{H})$, where $\mathcal{O}$ ranges over the
set of orbits and $\mathcal{H}$ ranges over irreducible polarizable Hodge
structures of weight $n-\dim\mathcal{O}$.

(ii) For Hodge structures $\mathcal{H}$ and $\mathcal{H}^{\prime}$ and an
orbit $\mathcal{O}$ we have a canonical isomorphism

$\operatorname{Hom}(\mathrm{IC}_{\mathcal{O}}(\mathcal{H}),\mathrm{IC}_{\mathcal{O}}(\mathcal{H}^{\prime})=\operatorname{Hom}(\mathcal{H},\mathcal{H}^{\prime}).$

###### Proof.

(i) Let $\mathcal{H}$ be an irreducible polarizable Hodge structure, then it
follows from Lemma 2.17 that $\mathrm{IC}_{\mathcal{O}}(\mathcal{H})$ has an
equivariant structure. Assume that it can be written as the direct sum
$\mathcal{F}_{1}\oplus\mathcal{F}_{2}$. Then $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$ necessarily have strict support $\overline{\mathcal{O}}$.
Thus, over some open subset of $\mathcal{O}$ both $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$ are variations of Hodge structures. This, however,
contradicts irreducibility of $\mathcal{H}$. Thus
$\mathrm{IC}_{\mathcal{O}}(\mathcal{H})$ is a simple object.

Conversely, assume that $\mathcal{F}$ is a simple object of
$\operatorname{MH}^{H}(Y,n)$. Then it is also a simple object of
$\operatorname{MH}(Y,n)$ (as otherwise we would have a non-trivial idempotent
endomorphism). By (4) $Z=\operatorname{supp}(\mathcal{F})$ is an integral
subscheme and we can write $\mathcal{F}=i_{*}\mathcal{F}^{\prime}$, where
$i\colon Z\to Y$ is the closed embedding. Since $\mathrm{Rat}(\mathcal{F})$ is
$H$-equivariant, we see that $Z$ is $H$-invariant and that
$\mathcal{F}^{\prime}\in\operatorname{MH}^{H}(Z,n)$. Since the action has ind-
finitely many orbits, some orbit $\mathcal{O}$ is open in $Z$. By Proposition
2.1, generically on $Z$, $\mathcal{F}^{\prime}$ is a polarizable variation of
Hodge structures. Let $W$ be the largest open subset of $Z$ such that
$\mathcal{F}^{\prime}|_{W}$ is a variation of Hodge structures. It is easy to
see that $W$ is $H$-invariant. We see that $\mathcal{O}\subset W$. Thus,
$\mathcal{V}:=\mathcal{F}^{\prime}|_{\mathcal{O}}$ is an $H$-equivariant
variation of Hodge structures. Since $H$ acts transitively on $\mathcal{O}$,
it is easy to see that $\mathcal{V}$ is isomorphic to
$a_{\mathcal{O}}^{\dagger}\mathcal{H}$ for a Hodge structure $\mathcal{H}$. It
is clear that $\mathcal{H}$ is polarizable and has weight $n-\dim\mathcal{O}$.
Now, by Proposition 2.1 we have

$\mathcal{F}\simeq(j_{\mathcal{O}})_{!*}\mathcal{V}\simeq\mathrm{IC}_{\mathcal{O}}(\mathcal{H}).$

Clearly, $\mathcal{H}$ is irreducible.

(ii) It follows from the definitions that
$\mathrm{IC}_{\mathcal{O}}(\mathcal{H})$ and
$\mathrm{IC}_{\mathcal{O}}(\mathcal{H}^{\prime})$ have strict support equal to
the closure of $\mathcal{O}$. Now, using Kashiwara’s equivalence and
Proposition 2.1(ii) we get an isomorphism

$\operatorname{Hom}(\mathrm{IC}_{\mathcal{O}}(\mathcal{H}),\mathrm{IC}_{\mathcal{O}}(\mathcal{H}^{\prime})=\operatorname{Hom}(a_{\mathcal{O}}^{\dagger}\mathcal{H},a_{\mathcal{O}}^{\dagger}\mathcal{H}^{\prime})=\operatorname{Hom}(\mathcal{H},\mathcal{H}^{\prime}),$

where the last isomorphism follows, for example, from Proposition 2.12(2.12)
and a similar statement for constant VHS. ∎

##  3\. Tannakian structure on
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$

In this section we equip
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ with a structure
of neutral Tannakian category.

###  3.1. $G_{\mathcal{O}}$-equivariant Hodge modules on
$\operatorname{Gr}_{G}$

Put $\mathcal{K}:=\mathbb{C}((t))=\mathbb{C}[[t]][t^{-1}]$. Let
$G_{\mathcal{K}}$ denote the loop group representing the functor $R\mapsto
G\bigl{(}R((t))\bigr{)}$ from the category of $\mathbb{C}$-algebras to the
category of sets. Then the affine Grassmannian is canonically isomorphic to
the quotient $G_{\mathcal{K}}/G_{\mathcal{O}}$.

Fix a Borel subgroup $B\subset G$ and a split maximal torus $T\subset B$. Let
$X_{*}:=\operatorname{Hom}(\mathbf{G_{m,\mathbb{C}}},T)$ be the co-character
lattice and let $X_{+}\subset X_{*}$ be the semigroup of $B$-dominant co-
characters. Every $\lambda\in X_{*}$ gives a point $t^{\lambda}\in
G_{\mathcal{K}}(\mathbb{C})$ defined by post-composing $\lambda$ with
$T\hookrightarrow G$ and pre-composing with
$\operatorname{Spec}\mathcal{K}\to\mathbf{G_{m,\mathbb{C}}}$. Consider the
$G_{\mathcal{O}}$-orbit
$\operatorname{Gr}^{\lambda}:=G_{\mathcal{O}}\cdot(t^{\lambda}\cdot
G_{\mathcal{O}})\subset\operatorname{Gr}_{G}$. It is well-known that every
$G_{\mathcal{O}}$-orbit in $\operatorname{Gr}_{G}$ is of the form
$\operatorname{Gr}^{\lambda}$ and
$\operatorname{Gr}^{\lambda}=\operatorname{Gr}^{\lambda^{\prime}}$ if and only
if $\lambda$ and $\lambda^{\prime}$ are in the same orbit under the action of
the Weyl group of $G$ on $X_{*}$. Thus we have a stratification

$\operatorname{Gr}_{G}=\bigcup_{\lambda\in X_{+}}\operatorname{Gr}^{\lambda}.$

For the proofs we refer the reader to [BT] and [AHLH].

Recall that we have a presentation
$\operatorname{Gr}_{G}=\lim_{l\to\infty}\operatorname{Gr}_{G}^{(l)}$, where
$\operatorname{Gr}_{G}^{(l)}$ are $G_{\mathcal{O}}$-invariant projective
schemes (cf. (1)). Note that for $m\gg 0$ the group $G_{\mathcal{O}}$ acts on
$\operatorname{Gr}_{G}^{(l)}$ through a finite dimensional quotient $G_{(m)}$,
where $G_{(m)}$ represents the functor $R\mapsto G(R[t]/t^{m})$. In
particular, the group $G_{\mathcal{O}}$ acts nicely on $\operatorname{Gr}_{G}$
and we can consider the category
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n)$. Recall that
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})=\bigoplus_{n\in\mathbb{Z}}\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n)$.

Take $\lambda\in X_{+}$ and let $\mathcal{H}$ be a pure polarizable Hodge
structure of weight $n-\dim\operatorname{Gr}^{\lambda}$. Set
$\mathrm{IC}_{\lambda}(\mathcal{H}):=\mathrm{IC}_{\operatorname{Gr}^{\lambda}}(\mathcal{H})$
so that
$\mathrm{IC}_{\lambda}(\mathcal{H})\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n)$.

Combining Proposition 2.16 and Proposition 2.19 we get the description of
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ up to
equivalence:

######  Proposition 3.1.

(i) The forgetful functor
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\to\operatorname{MH}(\operatorname{Gr}_{G})$
induces an equivalence between
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ and a full
abelian semisimple subcategory in $\operatorname{MH}(\operatorname{Gr}_{G})$
whose simple objects are isomorphic to $\mathrm{IC}_{\lambda}(\mathcal{H})$,
where $\lambda$ ranges over $X_{+}$ and $\mathcal{H}$ ranges over irreducible
Hodge structures.

(ii) For Hodge structures $\mathcal{H}$ and $\mathcal{H}^{\prime}$ we have a
canonical isomorphism

$\operatorname{Hom}(\mathrm{IC}_{\lambda}(\mathcal{H}),\mathrm{IC}_{\lambda}(\mathcal{H}^{\prime}))=\operatorname{Hom}(\mathcal{H},\mathcal{H}^{\prime}).$

We often use this proposition tacitly to identify the two categories.

###  3.2. Convolution of Hodge modules

We want to define a convolution on the category
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$. Consider the
convolution Grassmannian

$\operatorname{Gr}_{conv}:=G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\operatorname{Gr}_{G}.$

This is a fibration over $\operatorname{Gr}_{G}$ with the fiber isomorphic to
$\operatorname{Gr}_{G}$. In particular, $\operatorname{Gr}_{conv}$ is an ind-
proper ind-scheme. Let
$\mathcal{F}_{1},\mathcal{F}_{2}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$.
We first need to define the twisted exterior product
$\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}$ on
$\operatorname{Gr}_{conv}$. Morally, the definition is as follows. Let
$\tilde{\mathcal{F}}_{1}$ be the pullback of $\mathcal{F}_{1}$ to
$G_{\mathcal{K}}$. Then the $G_{\mathcal{O}}$-equivariance shows that
$\tilde{\mathcal{F}}_{1}\boxtimes\mathcal{F}_{2}$ descends from
$G_{\mathcal{K}}\times\operatorname{Gr}_{G}$ to the Hodge module
$\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}$ on
$\operatorname{Gr}_{conv}$. We prefer to avoid using Hodge modules on
$G_{\mathcal{K}}$ because $G_{\mathcal{K}}$ is not of ind-finite type.

We proceed as follows. Let $l$ be large enough, then we may view
$\mathcal{F}_{2}$ as an object of
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G}^{(l)})$. Let
$G_{\mathcal{O}}$ act on $\operatorname{Gr}_{G}^{(l)}$ through its quotient
$G_{(m)}$. Thus, we may view $\mathcal{F}_{2}$ as an object of
$\operatorname{MH}^{G_{(m)}}(\operatorname{Gr}_{G}^{(l)})$ (by Lemma 2.15).

Denote the kernel of $G_{\mathcal{O}}\to G_{(m)}$ by $G^{(m)}$ (this is the
$m$-th congruence subgroup) and consider the ind-scheme of ind-finite type
$G_{\mathcal{K}}/G^{(m)}$; the morphism $p\colon
G_{\mathcal{K}}/G^{(m)}\to\operatorname{Gr}_{G}$ is a $G_{(m)}$-torsor.
Consider the closed embedding.

(6)
$i_{m,l}\colon(G_{\mathcal{K}}/G^{(m)})\times^{G_{(m)}}\operatorname{Gr}_{G}^{(l)}=G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\operatorname{Gr}_{G}^{(l)}\to\operatorname{Gr}_{conv}.$

The Hodge module
$p^{\dagger}\mathcal{F}_{1}\boxtimes\mathcal{F}_{2}\in\operatorname{MH}((G_{\mathcal{K}}/G^{(m)})\times\operatorname{Gr}_{G}^{(l)})$
is easily seen to be $G_{(m)}$-equivariant, so it descends to a Hodge module
$\tilde{\mathcal{F}}$ on
$(G_{\mathcal{K}}/G^{(m)})\times^{G_{(m)}}\operatorname{Gr}_{G}^{(l)}$ by
Proposition 2.18. We set

$\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}:=(i_{m,l})_{*}\tilde{\mathcal{F}}.$

Let $m\colon\operatorname{Gr}_{conv}\to\operatorname{Gr}_{G}$ be the morphism,
induced by the action of $G_{\mathcal{K}}$ on $\operatorname{Gr}_{G}$. It is
ind-proper because $\operatorname{Gr}_{conv}$ is ind-proper.

######  Definition 1.

$\mathcal{F}_{1}\star\mathcal{F}_{2}:=H^{0}(m_{*}(\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2})).$

It is easy to see that the convolution is functorial.

######  Lemma 3.2.

(i) We have
$H^{j}(m_{*}(\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}))=0$
for all $j\neq 0$.

(ii) $\mathrm{Rat}$ is a monoidal functor.

###### Proof.

Using the fact that $\mathrm{Rat}$ commutes with $m_{*}$ (Proposition 2.9),
with $\boxtimes$ (Proposition 2.11(i)), and with smooth pullbacks (Proposition
2.12), it is easy to see that

$\mathrm{Rat}(m_{*}(\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}))=\mathrm{Rat}(\mathcal{F}_{1})\star\mathrm{Rat}(\mathcal{F}_{2}),$

where the convolution on perverse sheaves is defined in [MV, (4.2)]. To prove
(i), since $\mathrm{Rat}$ is conservative (Proposition 2.4), we only need to
show that
$\mathrm{Rat}(m_{*}(\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}))$
is concentrated in cohomological degree zero. This follows from the above
equality and [MV, Prop 4.2]. Now (ii) is also clear. ∎

######  Lemma 3.3.

$\mathcal{F}_{1}\star\mathcal{F}_{2}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G}).$

###### Proof.

Let $G_{\mathcal{O}}$ act on $\operatorname{Gr}_{conv}$ via left
multiplication on $G_{\mathcal{K}}$. Let $G_{\mathcal{O}}$ act on
$G_{\mathcal{K}}/G^{(m)}$ similarly. It is clear that (6) is
$G_{\mathcal{O}}$-equivariant. Lemma 2.17(ii,iii) shows that
$\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}$ is
$G_{\mathcal{O}}$-equivariant. Since $m$ is $G_{\mathcal{O}}$-equivariant, the
statement follows from Lemma 2.17(i). ∎

We note that the convolution is compatible with weights, that is,

$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n_{1})\star\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n_{2})\subset\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n_{1}+n_{2}).$

One can define triple convolution along the same lines and use it to show that
the convolution is associative up to a canonical isomorphism, that is, the
pentagon identities are satisfied (one can also derive the pentagon identities
from the faithfulness of $\mathrm{Rat}$).

###  3.3. Fusion product and the commutativity constraint

Recall from [Gai, Sect. 3.1.1] the following version of Beilinson–Drinfeld
Grassmannian $\widetilde{\operatorname{Gr}}$. This is the ind-scheme
classifying the data $(x,\mathcal{E},s)$, where $x\in\mathbb{A}^{1}$,
$\mathcal{E}$ is a $G$-torsor on $\mathbb{A}^{1}$, and $s$ is a trivialization
of $\mathcal{E}$ over $\mathbb{A}^{1}-x-0$. (For simplicity, we work with the
curve $\mathbb{A}^{1}$, as it has a global coordinate but we could work with
any curve). The ind-scheme $\widetilde{\operatorname{Gr}}$ is ind-separated
and of ind-finite type over $\mathbb{C}$ so we can talk about Hodge modules on
$\widetilde{\operatorname{Gr}}$. We have a morphism
$\widetilde{\operatorname{Gr}}\to\mathbb{A}^{1}$ given by
$(x,\mathcal{E},s)\mapsto x$. We use this morphism to view
$\widetilde{\operatorname{Gr}}$ as a family over $\mathbb{A}^{1}$.

Let $X$ be a connected curve over $\mathbb{C}$ and let $x\in X$ be a smooth
closed point of $X$. Recall that the modular interpretation of
$\operatorname{Gr}_{G}$ identifies it with the moduli space of pairs
$(\mathcal{E},s)$ where $\mathcal{E}$ is a $G$-torsor over $X$, $s$ is a
trivialization of $\mathcal{E}$ on $X-x$. This identification depends only on
a choice of a formal coordinate at $x$. Applying this to $X=\mathbb{A}^{1}$,
$x=0$, we see that the 0-fiber of the family
$\widetilde{\operatorname{Gr}}\to\mathbb{A}^{1}$ is identified with
$\operatorname{Gr}_{G}$. The well-known factorization property gives a
canonical isomorphism (cf. [Gai, Prop. 5]).

$\widetilde{\operatorname{Gr}}|_{\mathbb{A}^{1}-0}\xrightarrow{=}\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}\colon(x,\mathcal{E},s)\mapsto((\mathcal{E}|_{\mathbb{A}^{1}-0},s),x,(\mathcal{E}|_{\mathbb{A}^{1}-x},s)).$

Thus we have an open embedding

$j\colon\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}=\widetilde{\operatorname{Gr}}|_{\mathbb{A}^{1}-0}\to\widetilde{\operatorname{Gr}}$

and a closed embedding

$i\colon\operatorname{Gr}_{G}=\widetilde{\operatorname{Gr}}|_{0}\to\widetilde{\operatorname{Gr}}.$

We make a convention. If $Y$ is an ind-scheme over $\mathbb{A}^{1}$, let $i$
be the closed embedding of the zero fiber, $j$ be the open embedding of the
complement. Then for $\mathcal{F}\in\operatorname{MH}(Y)$ we write

$\mathcal{F}|_{0}:=i^{*}\mathcal{F}[-1],\quad\mathcal{F}|_{\mathbb{A}^{1}-0}:=j^{*}\mathcal{F}.$

Thus $\mathcal{F}|_{\mathbb{A}^{1}-0}$ is again a direct sum of pure Hodge
modules, while $\mathcal{F}|_{0}$ may, in general, be a complex of mixed Hodge
modules. We note that

(7)
$\mathbb{Q}_{\mathbb{A}_{1}}^{\mathrm{Hdg}}|_{0}=i^{*}a^{*}\mathbb{Q}^{\mathrm{Hdg}}=\mathbb{Q}^{\mathrm{Hdg}},$

where $a\colon\mathbb{A}^{1}\to pt$ is the projection.

Let $\mathcal{H}_{0}$ be the unique Hodge structure on
$\mathbb{Q}=\mathbb{Q}^{1}$ of weight zero. For a smooth scheme $Z$ we will
denote the Hodge module $a^{\dagger}\mathcal{H}_{0}$ by
$\mathbb{Q}^{\mathrm{Hdg}}_{Z}$, where $a\colon Z\to pt$ is the projection.
The following is essentially the Hodge module version of [Gai, Prop. 6(a)].

######  Proposition 3.4.

Assume that $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are in
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$. Then we have a
canonical isomorphism

$\mathcal{F}_{1}\star\mathcal{F}_{2}=\bigl{(}j_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})\bigr{)}|_{0}.$

###### Proof.

We introduce the convolution Grassmannian
$\widetilde{\operatorname{Gr}}_{conv}$ classifying the data
$(x,\mathcal{E}_{1},\mathcal{E}_{2},s_{2},s_{21})$, where
$x\in\mathbb{A}^{1}$, $\mathcal{E}_{i}$ are $G$-torsors over $\mathbb{A}^{1}$,
$s_{2}$ is a trivialization of $\mathcal{E}_{2}$ over $\mathbb{A}^{1}-x$
(viewed as an isomorphism from the trivial torsor to $\mathcal{E}_{2}$),
$s_{21}$ is an isomorphism from $\mathcal{E}_{2}$ to $\mathcal{E}_{1}$ over
$\mathbb{A}^{1}-x$. We view $\widetilde{\operatorname{Gr}}_{conv}$ as a family
over $\mathbb{A}^{1}$. A modular description of $\operatorname{Gr}_{conv}$
identifies $\widetilde{\operatorname{Gr}}_{conv}|_{0}$ with
$\operatorname{Gr}_{conv}$. More precisely, let $(g,(\mathcal{E},s))\in
G_{\mathcal{K}}\times\operatorname{Gr}_{G}$ be a representative of a point of
$G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\operatorname{Gr}_{G}=\operatorname{Gr}_{conv}$.
Let $\mathcal{E}^{\prime}$ be obtained by gluing the trivial $G$-torsor over
$\mathbb{A}^{1}-0$ with the trivial $G$-torsor on the formal disc centered at
0 via $g$. Let $s^{\prime}$ be the canonical trivialization of
$\mathcal{E}^{\prime}$ on $\mathbb{A}^{1}-0$. Then the embedding
$\tilde{\imath}\colon\operatorname{Gr}_{conv}\to\widetilde{\operatorname{Gr}}_{conv}$
sends $(g,(\mathcal{E},s))$ to
$(0,\mathcal{E},\mathcal{E}^{\prime},s^{\prime},s)$.

We also have a factorization
$\widetilde{\operatorname{Gr}}_{conv}|_{\mathbb{A}^{1}-0}=\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}$
given by

$(x,\mathcal{E}_{1},\mathcal{E}_{2},s_{2},s_{21})\mapsto((\mathcal{E}_{1}|_{\mathbb{A}^{1}-0},s_{21}|_{\mathbb{A}^{1}-x-0}\circ
s_{2}|_{\mathbb{A}^{1}-x-0}),x,(\mathcal{E}_{2},s_{2})).$

We have a morphism
$\tilde{m}\colon\widetilde{\operatorname{Gr}}_{conv}\to\widetilde{\operatorname{Gr}}$
sending $(x,\mathcal{E}_{1},\mathcal{E}_{2},s_{2},s_{21})$ to
$(x,\mathcal{E}_{1},s_{21}|_{\mathbb{A}^{1}-x-0}\circ
s_{2}|_{\mathbb{A}^{1}-x-0})$. It is not difficult to see that
$\widetilde{\operatorname{Gr}}_{conv}$ is ind-proper over $\mathbb{A}^{1}$, so
$\tilde{m}$ is ind-proper. We have a commutative diagram with Cartesian
squares

(8)
$\begin{CD}\operatorname{Gr}_{conv}@>{\tilde{\imath}}>{}>\widetilde{\operatorname{Gr}}_{conv}@<{\tilde{\jmath}}<{}<\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}\\\
@V{m}V{}V@V{\tilde{m}}V{}V\Big{\|}\\\
\operatorname{Gr}_{G}@>{i}>{}>\widetilde{\operatorname{Gr}}@<{j}<{}<\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}\\\
@V{}V{}V@V{\pi}V{}V@V{}V{}V\\\
0<EMAIL_ADDRESS>
Let
$\mathcal{F}^{\prime}_{1}\in\operatorname{MH}(\operatorname{Gr}_{G}\times\mathbb{A}^{1})$
and $\mathcal{F}_{2}\in\operatorname{MH}(\operatorname{Gr}_{G})$. We define
the _outer convolution_ by

$\mathcal{F}^{\prime}_{1}\star_{o}\mathcal{F}_{2}:=\tilde{m}_{*}\tilde{\jmath}_{!*}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2})\in
D^{b}\operatorname{MH}(\widetilde{\operatorname{Gr}})$

(cf. [GR, Def. 5]).

######  Lemma 3.5.

Let
$\mathcal{F}^{\prime}_{1}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G}\times\mathbb{A}^{1})$
and
$\mathcal{F}_{2}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$.
Then we have

$\mathcal{F}^{\prime}_{1}\star_{o}\mathcal{F}_{2}=j_{!*}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2}).$

In particular, the outer convolution is concentrated in cohomological
dimension zero.

###### Proof.

We claim that

$\tilde{m}_{*}\tilde{\jmath}_{!*}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2})=j_{!*}j^{*}\tilde{m}_{*}\tilde{\jmath}_{!*}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2}).$

By Lemma 2.6, it is enough to prove a similar statement for the underlying
perverse sheaves, where it is equivalent to the statement that

$\tilde{m}_{*}\tilde{\jmath}_{!*}\mathrm{Rat}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2})$

has no vanishing cycles and its nearby cycles have trivial monodromy. This
follows from the fact that vanishing and nearby cycles commute with proper
direct images (cf. [GR, p. 7]).

Thus by base change and Lemma 2.5(i) we have

$\mathcal{F}^{\prime}_{1}\star_{o}\mathcal{F}_{2}=j_{!*}j^{*}\tilde{m}_{*}\tilde{\jmath}_{!*}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2})=j_{!*}((\mathcal{F}^{\prime}_{1}|_{\mathbb{A}^{1}-0})\boxtimes\mathcal{F}_{2}).$

∎

Next, we have the following lemma.

######  Lemma 3.6.

For
$\mathcal{F}_{1},\mathcal{F}_{2}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$
we have a canonical isomorphism

$\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2}=(\tilde{\jmath}_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})|_{0}.$

###### Proof.

Let $l$ and $m$ be as in the definition of $\mathrel{\tilde{\boxtimes}}$
(Section 3.2). Consider the closed sub-ind-scheme
$\widetilde{\operatorname{Gr}}_{conv}^{(l)}$ of
$\widetilde{\operatorname{Gr}}_{conv}$ consisting of
$(x,\mathcal{E}_{1},\mathcal{E}_{2},s_{2},s_{21})$ such that
$(\mathcal{E}_{2},s_{2})\in\operatorname{Gr}_{G}^{(l)}$. Next, consider the
ind-scheme $\widetilde{\operatorname{Gr}}_{conv}^{(l,m)}$ classifying
$(x,\mathcal{E}_{1},\mathcal{E}_{2},s_{2},s_{21},t_{2})$, where
$(x,\mathcal{E}_{1},\mathcal{E}_{2},s_{2},s_{21})\in\widetilde{\operatorname{Gr}}_{conv}^{(l)}$,
$t_{2}$ is a trivialization of $\mathcal{E}_{2}$ on the $m$-th infinitesimal
neighborhood of 0. Clearly, this is a $G_{(m)}$-torsor over
$\widetilde{\operatorname{Gr}}_{conv}^{(l)}$. One checks that
$\widetilde{\operatorname{Gr}}_{conv}^{(l,m)}$ factors as
$(G_{\mathcal{K}}/G^{(m)})\times\mathbb{A}^{1}\times\operatorname{Gr}_{G}^{(l)}$
and we have a diagram with cartesian squares

$\begin{CD}G_{\mathcal{K}}/G^{(m)}\times\operatorname{Gr}_{G}^{(l)}@>{}>{}>G_{\mathcal{K}}/G^{(m)}\times\mathbb{A}^{1}\times\operatorname{Gr}_{G}^{(l)}@<{}<{}<G_{\mathcal{K}}/G^{(m)}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}^{(l)}\\\
@V{q}V{}V@V{\tilde{p}}V{}V@V{}V{}V\\\
G_{\mathcal{K}}/G^{(m)}\times^{G_{(m)}}\operatorname{Gr}_{G}^{(l)}@>{}>{}>\widetilde{\operatorname{Gr}}_{conv}^{(l)}@<{}<{}<\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}^{(l)}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
\operatorname{Gr}_{conv}@>{\tilde{\imath}}>{}>\widetilde{\operatorname{Gr}}_{conv}@<{\tilde{\jmath}}<{}<\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
0<EMAIL_ADDRESS>
We use Kashiwara’s equivalence to identify Hodge modules on the ind-schemes in
the second row with Hodge modules on the corresponding ind-schemes on the
third row. Then the base change Lemma 2.13, the compatibility of exterior
products with inverse images and intermediate extensions (Proposition
2.11(iii)), and Lemma 2.6 give

$\tilde{p}^{\dagger}(\tilde{\jmath}_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2}))=\tilde{\mathcal{F}}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}}\boxtimes\mathcal{F}_{2},$

where $\tilde{\mathcal{F}}_{1}$ is the pullback of $\mathcal{F}_{1}$ to
$G_{\mathcal{K}}/G^{(m)}$. Using the compatibility of $\boxtimes$ with inverse
images and (7), we get

$q^{\dagger}\left(\tilde{\jmath}_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})|_{0}\right)=\tilde{\mathcal{F}}_{1}\boxtimes\mathcal{F}_{2}.$

Since this Hodge module is $G_{(m)}$-equivariant, the statement follows from
the descent (Proposition 2.18). ∎

Now we can prove Proposition 3.4. Using Definition 1, Lemma 3.6, base change,
the definition of the outer product, Lemma 3.5, and compatibility of
$\boxtimes$ with restrictions, we get

$\mathcal{F}_{1}\star\mathcal{F}_{2}=m_{*}(\mathcal{F}_{1}\mathrel{\tilde{\boxtimes}}\mathcal{F}_{2})=m_{*}((\tilde{\jmath}_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})|_{0})=\\\
(\tilde{m}_{*}\tilde{\jmath}_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2}))|_{0}=((\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}})\star_{o}\mathcal{F}_{2})|_{0}=(j_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2}))|_{0}.$

∎

Now we can define the commutativity constraint. Let $sw$ be the automorphism
of $\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}$
switching two copies of $\operatorname{Gr}_{G}$ and taking
$x\in\mathbb{A}^{1}$ to $-x$. Let
$\widetilde{sw}\colon\widetilde{\operatorname{Gr}}\to\widetilde{\operatorname{Gr}}$
be the automorphism given by
$(x,\mathcal{E},s)\mapsto(-x,(sh_{-x})^{*}\mathcal{E},(sh_{-x})^{*}s)$, where
$sh_{x}\colon\mathbb{A}^{1}\to\mathbb{A}^{1}$ is the shift by $-x$. It is easy
to see that under the factorization isomorphism we have

$sw=\widetilde{sw}|_{\mathbb{A}^{1}-0}.$

Thus by Proposition 3.4

$\mathcal{F}_{2}\star\mathcal{F}_{1}=\bigl{(}j_{!*}(\mathcal{F}_{2}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{1})\bigr{)}\bigl{|}_{0}=\bigl{(}j_{!*}sw^{*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})\bigr{)}\bigl{|}_{0}=\\\
\bigl{(}\widetilde{sw}^{*}j_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})\bigr{)}\bigl{|}_{0}=\bigl{(}j_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})\bigr{)}\bigl{|}_{0}=\mathcal{F}_{1}\star\mathcal{F}_{2},$

We used Lemma 2.13 and the equation $\widetilde{sw}\circ i=i$. It is easy to
see that we indeed get a commutativity constraint, that is, the isomorphism is
involutive. This commutativity constraint is also easily seen to be compatible
with $\mathrm{Rat}$ and the commutativity constraint from [MV]. Thus the
hexagon identities follow from faithfulness of $\mathrm{Rat}$.

###  3.4. Identity object

######  Lemma 3.7.

For $\lambda,\mu\in X_{+}$ and for polarizable pure Hodge structures
$\mathcal{H}$ and $\mathcal{H}^{\prime}$ let $l$ be such that
$\operatorname{Gr}^{\mu}$ factors through $\operatorname{Gr}_{G}^{(l)}$. Let
$\operatorname{Gr}^{\lambda,(m)}$ be the preimage of
$\operatorname{Gr}^{\lambda}$ in $G_{\mathcal{K}}/G^{(m)}$. Consider the
locally closed embedding defined as the composition

(9)
$j_{\lambda,\mu}:\operatorname{Gr}^{\lambda,(m)}\times^{G_{\mathcal{O}}^{(m)}}\operatorname{Gr}^{\mu}\to(G_{\mathcal{K}}/K^{(m)})\times^{G_{\mathcal{O}}^{(m)}}\operatorname{Gr}_{G}^{(l)}=G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\operatorname{Gr}_{G}^{(l)}\to\operatorname{Gr}_{conv}.$

Define the projection
$a_{\lambda,\mu}:\operatorname{Gr}^{\lambda,(m)}\times^{G_{\mathcal{O}}^{(m)}}\operatorname{Gr}^{\mu}\to
pt$. Then we have a canonical isomorphism

(10)
$\mathrm{IC}_{\lambda}(\mathcal{H})\mathrel{\tilde{\boxtimes}}\mathrm{IC}_{\mu}(\mathcal{H}^{\prime}):=(j_{\lambda,m})_{!*}a_{\lambda,m}^{\dagger}(\mathcal{H}\otimes\mathcal{H}^{\prime}).$

###### Proof.

This follows from base change (Lemma 2.13), the properties of $\boxtimes$
(Proposition 2.11), and the descent for torsors (Proposition 2.18). ∎

######  Lemma 3.8.

For $\mu\in X_{+}$ and for polarizable pure Hodge structures $\mathcal{H}$ and
$\mathcal{H}^{\prime}$ we have a canonical isomorphism

(11)
$\mathrm{IC}_{\mu}(\mathcal{H}\otimes\mathcal{H}^{\prime})=\mathrm{IC}_{0}(\mathcal{H})\star\mathrm{IC}_{\mu}(\mathcal{H}^{\prime}).$

###### Proof.

Consider the closed embedding

$i_{0}\colon\operatorname{Gr}_{G}=G_{\mathcal{O}}\times^{G_{\mathcal{O}}}\operatorname{Gr}_{G}\to
G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\operatorname{Gr}_{G}=\operatorname{Gr}_{conv}.$

It is easy to see that when $\lambda=0$ the morphism (9) factors as
$j_{0,\mu}=i_{0}\circ j_{\mu}$, where
$j_{\mu}\colon\operatorname{Gr}^{\mu}\to\operatorname{Gr}_{G}$ is the
embedding (note that
$\operatorname{Gr}^{0,(m)}\times^{G_{(m)}}\operatorname{Gr}^{\mu}=\operatorname{Gr}^{\mu}$
and $a_{0,\mu}=a_{\mu}$). Thus, according to the previous lemma,

$\mathrm{IC}_{0}(\mathcal{H})\star\mathrm{IC}_{\mu}(\mathcal{H}^{\prime})=m_{*}(j_{0,\mu})_{!*}a_{\mu}^{\dagger}(\mathcal{H}\otimes\mathcal{H}^{\prime})=m_{*}(i_{0})_{*}(j_{\mu})_{!*}a_{\mu}^{\dagger}(\mathcal{H}\otimes\mathcal{H}^{\prime})=\mathrm{IC}_{\mu}(\mathcal{H}\otimes\mathcal{H}^{\prime}),$

where we used that $m\circ i_{0}=\operatorname{Id}_{\operatorname{Gr}_{G}}$. ∎

It follows from Lemma 3.8 and Proposition 3.1 that the Hodge module
$\delta_{1}:=\mathrm{IC}_{0}(\mathcal{H}_{0})$ is the identity for the
convolution. Thus, we have equipped
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ with a structure
of monoidal category in the sense of [DM, Def. 1.1].

###  3.5. Rigidity

Let $\mathcal{F}$ be an object of
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$. Recall from [BK,
Def. 2.1.1] that an object $\mathcal{F}^{\vee}$ is the left dual of
$\mathcal{F}$ if there are morphisms
$e_{\mathcal{F}}\colon\mathcal{F}^{\vee}\star\mathcal{F}\to\delta_{1}$ (called
the counit) and
$i_{\mathcal{F}}\colon\delta_{1}\to\mathcal{F}\star\mathcal{F}^{\vee}$ (called
the unit) such that the composition

$\mathcal{F}\xrightarrow{i_{\mathcal{F}}\otimes\operatorname{Id}_{\mathcal{F}}}\mathcal{F}\star\mathcal{F}^{\vee}\star\mathcal{F}\xrightarrow{\operatorname{Id}_{\mathcal{F}}\otimes
e_{\mathcal{F}}}\mathcal{F}$

is equal to $\operatorname{Id}_{\mathcal{F}}$ and the composition

$\mathcal{F}^{\vee}\xrightarrow{\operatorname{Id}_{\mathcal{F}^{\vee}}\otimes
i_{\mathcal{F}}}\mathcal{F}^{\vee}\star\mathcal{F}\star\mathcal{F}^{\vee}\xrightarrow{\operatorname{Id}_{\mathcal{F}^{\vee}}\otimes
e_{\mathcal{F}}}\mathcal{F}^{\vee}$

is equal to $\operatorname{Id}_{\mathcal{F}^{\vee}}$. Right dual objects are
defined similarly. Our goal is to show that
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ is a rigid
monoidal category, that is, every object has a left and a right dual.

Our first observation is that a monoidal functor obviously preserves duals.
The functor
$\operatorname{MH}(pt)\to\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$
sending $\mathcal{H}$ to $\mathrm{IC}_{0}(\mathcal{H})$ is easily seen to be
monoidal, and the category of pure polarizable Hodge structures is rigid. Thus
every object of the form $\mathrm{IC}_{0}(\mathcal{H})$ has duals.

Secondly, for $\lambda\in X_{+}$ set $\lambda^{\prime}:=w_{0}(-\lambda)\in
X_{+}$, where $w_{0}$ is the longest element of the Weyl group of $G$. Set
$\mathcal{F}:=\mathrm{IC}_{\lambda}(\mathcal{H}_{0})$ and
$\mathcal{F}^{\prime}:=\mathrm{IC}_{\lambda^{\prime}}(\mathcal{H}_{0})$. We
claim that $\mathcal{F}^{\prime}(i)$, where
$i=\dim\operatorname{Gr}^{\lambda}$, is the dual of $\mathcal{F}$.

Indeed, first of all, we have $G_{\mathcal{O}}\cdot(t^{-\lambda}\cdot
G_{\mathcal{O}})=\operatorname{Gr}^{\lambda^{\prime}}$. Now it follows from
[MV, (11.10)] that we have
$\mathrm{Rat}(\mathcal{F}^{\prime}(i))\simeq(\mathrm{Rat}(\mathcal{F}))^{\vee}$
(note that for perverse sheaves we can trivialize the Tate twists). Next, we
claim that the injective map

(12)
$\operatorname{Hom}(\mathcal{F}^{\prime}(i)\star\mathcal{F},\delta_{1})\hookrightarrow\operatorname{Hom}(\mathrm{Rat}(\mathcal{F}^{\prime}(i)\star\mathrm{Rat}(\mathcal{F}),\mathrm{Rat}(\delta_{1}))$

is an isomorphism. Indeed, since the RHS is equal to
$\operatorname{Hom}(\mathrm{Rat}(\mathcal{F}),\mathrm{Rat}(\mathcal{F}))$, it
is a 1-dimensional vector space over $\mathbb{Q}$, so it is enough to show
that the LHS is non-zero. However, we know that
$\mathcal{F}^{\vee}\star\mathcal{F}$ is the direct sum of Hodge modules
isomorphic to $\mathrm{IC}_{\lambda}(\mathcal{H})$, where $\mathcal{H}$ are
pure polarizable Hodge structures. Applying $\mathrm{Rat}$, and noting once
again that the RHS of (12) is a 1-dimensional vector space, we see that there
is exactly one summand with $\lambda=0$ in this sum, and the rank of the
corresponding Hodge structure $\mathcal{H}$ is one. Thus
$\mathcal{H}=\mathcal{H}_{0}(j)$ is Tate’s Hodge structure.

It remains to compare weights: we have

$\dim\operatorname{Gr}^{\lambda^{\prime}}=(2\rho,\lambda^{\prime})=(w_{0}^{-1}(2\rho),-\lambda)=(-2\rho,-\lambda)=\dim\operatorname{Gr}^{\lambda}.$

Thus, $\mathcal{F}^{\prime}(i)\star\mathcal{F}$ has weight
$\dim\operatorname{Gr}^{\lambda^{\prime}}-2\dim\operatorname{Gr}^{\lambda}+\dim\operatorname{Gr}^{\lambda}=0$,
so $j=0$. Thus the LHS of (12) is non-zero, so (12) is an isomorphism. We
define the counit $e_{\mathcal{F}}$ as the preimage of the counit for
$\mathrm{Rat}(\mathcal{F})$ under this isomorphism.

We define the unit $i_{\mathcal{F}}$ similarly. It follows from the fact that
$\mathrm{Rat}$ is faithful and monoidal that $e_{\mathcal{F}}$ and
$i_{\mathcal{F}}$ satisfy the above conditions for the unit and the counit.
The existence of a right dual is proved similarly.

One checks that the dual of $\mathcal{F}_{1}\star\mathcal{F}_{2}$ is
$\mathcal{F}_{2}^{\vee}\star\mathcal{F}_{1}^{\vee}$. Thus, according to (11),
for every $\lambda\in X_{+}$ and for every pure polarizable Hodge structure
$\mathcal{H}$, the Hodge module $\mathrm{IC}_{\lambda}(\mathcal{H})$ has a
dual. Finally, every object is the direct sum of the objects isomorphic to
$\mathrm{IC}_{\lambda}(\mathcal{H})$, so every object has a dual and the
category is proved to be rigid.

###  3.6. Fiber functor

We define the fiber functor
$\mathrm{Fib}\colon\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\to\operatorname{Vect}_{\mathbb{Q}}$
as the composition of $\mathrm{Rat}$ and the global cohomology functor from
[MV, Sect. 3]. According to Proposition 2.2(ii), Lemma 3.2(ii)), and [MV, Cor.
3.7, Prop 6.2]$\mathrm{Fib}$ is a faithful exact monoidal functor. We have
equipped $\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ with a
structure of a neutral Tannakian category.

##  4\. Geometric Satake equivalence for Hodge modules

###  4.1. Fiber functor to Hodge structures

First of all, we would like to upgrade the fiber functor to a functor to Hodge
structures. Let $a$ denote the projection $\operatorname{Gr}_{G}\to pt$.
Consider the functor

$a_{+}\colon\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})\to\operatorname{MH}(pt)\colon\mathcal{F}\mapsto\bigoplus_{i\in\mathbb{Z}}H^{2i}(a_{*}(\mathcal{F}))(i).$

We note that $a_{+}$ takes
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G},n)$ to
$\operatorname{MH}(pt,n)$. Here $\operatorname{MH}(pt)$ is the category of
pure Hodge structures, $(i)$ stands for the $i$-th Tate twist of a pure Hodge
structure. Note that by [MV, Thm. 3.6] the odd direct images vanish, so the
composition of $a_{+}$ with $\mathrm{Rat}$ is the fiber functor defined in
Section 3.6.

######  Proposition 4.1.

The functor $a_{+}$ is monoidal.

###### Proof.

Let
$\mathcal{F}_{1},\mathcal{F}_{2}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$,
let $\pi$ and $j$ be as in diagram (8). Set

$\mathcal{G}:=\pi_{*}j_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})\in
D^{b}\operatorname{MH}(\mathbb{A}^{1}).$

######  Lemma 4.2.

Let $a\colon\operatorname{Gr}_{G}\to pt$ be the projection. We have

(13) $\mathcal{G}|_{\mathbb{A}^{1}-0}=a_{*}(\mathcal{F}_{1})\otimes
a_{*}(\mathcal{F}_{2})\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}.$

###### Proof.

Using base change and Lemma 2.5(i), we get

$\mathcal{G}|_{\mathbb{A}^{1}-0}=p_{2,*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2}),$

where
$p_{2}\colon\operatorname{Gr}_{G}\times(\mathbb{A}^{1}-0)\times\operatorname{Gr}_{G}\to\mathbb{A}^{1}-0$
is the projection. Since
$p_{2}=a\times\operatorname{Id}_{\mathbb{A}^{1}-0}\times a$, it remains to use
the compatibility of $\boxtimes$ with direct images. ∎

Using the base change and Proposition 3.4, we get

$\mathcal{G}|_{0}=a_{*}(j_{!*}(\mathcal{F}_{1}\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}-0}\boxtimes\mathcal{F}_{2})|_{0})=a_{*}(\mathcal{F}_{1}\star\mathcal{F}_{2}).$

It follows from (13), (4), and Proposition 2.1 that
$\mathcal{G}=a_{*}(\mathcal{F}_{1})\otimes
a_{*}(\mathcal{F}_{2})\boxtimes\mathbb{Q}^{\mathrm{Hdg}}_{\mathbb{A}^{1}}\oplus\mathcal{G}^{\prime}$,
where $\mathcal{G}^{\prime}\in D^{b}\operatorname{MH}(\mathbb{A}^{1})$ is
supported at zero. Thus we get a split injective homomorphism in
$D^{b}\operatorname{MH}(pt)$:

$a_{*}(\mathcal{F}_{1})\otimes
a_{*}(\mathcal{F}_{2})\hookrightarrow\mathcal{G}|_{0}=a_{*}(\mathcal{F}_{1}\star\mathcal{F}_{2}).$

Applying the cohomology functor and using the Künneth decomposition (which
follows from Proposition 2.3(i)) we get an injective graded homomorphism

$a_{+}(\mathcal{F}_{1})\otimes a_{+}(\mathcal{F}_{2})\hookrightarrow
a_{+}(\mathcal{F}_{1}\star\mathcal{F}_{2}).$

Since the global cohomology functor from [MV] is monoidal, the graded
dimensions are equal, so this homomorphism is an isomorphism. ∎

###  4.2. The category of Tate’s Hodge modules

Recall that $\mathcal{H}_{0}$ stands for the trivial 1-dimensional Hodge
structure. The following proposition is crucial for the proofs of Theorems 1,
2, and 3.

######  Proposition 4.3.

Let $\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ be the
strictly full subcategory of
$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ whose objects are
isomorphic to the direct sums of $\mathrm{IC}_{\lambda}(\mathcal{H}_{0}(i))$,
where $\lambda$ ranges over $X_{+}$ and $i$ ranges over $\mathbb{Z}$.

(i) For
$\mathcal{F}\in\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ the
Hodge structure $a_{+}(\mathcal{F})$ is the direct sum of Tate twists of the
trivial Hodge structure if and only if
$\mathcal{F}\in\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$.

(ii) The category
$\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ is closed under
the convolution.

###### Proof.

(i) Using (11) and Lemma 4.1, we see that

$a_{+}(\mathrm{IC}_{\lambda}(\mathcal{H}))=a_{+}(\mathrm{IC}_{0}(\mathcal{H})\star\mathrm{IC}_{\lambda}(\mathcal{H}_{0}))=\mathcal{H}\otimes
a_{+}(\mathrm{IC}_{\lambda}(\mathcal{H}_{0})).$

Now the ‘only if’ direction follows easily from Proposition 3.1 and
faithfulness of $a_{+}$.

For the ‘if’ direction, let
$j_{\lambda}\colon\operatorname{Gr}^{\lambda}\to\operatorname{Gr}$ be the
locally closed embedding and $a_{\lambda}\colon\operatorname{Gr}^{\lambda}\to
pt$ be the constant morphism. Consider the mixed Hodge module
$\mathcal{F}_{\lambda}:=H^{0}((j_{\lambda})_{!}a_{\lambda}^{\dagger}\mathbb{Q}^{\mathrm{Hdg}})\in\operatorname{MHM}(\operatorname{Gr}_{G})$.
Recall that $a\colon\operatorname{Gr}_{G}\to pt$ is the constant morphism. We
need a lemma.

######  Lemma 4.4.

For $i\in\mathbb{Z}$ we have $H^{2i+1}(a_{*}\mathcal{F}_{\lambda})=0$ and
$H^{2i}(a_{*}\mathcal{F}_{\lambda})$ is a Tate twist of a trivial Hodge
structure.

###### Proof.

Let $N\subset G$ be the unipotent radical of the Borel subgroup $B$. Let
$\nu\in X_{*}$ be a co-character and let
$S_{\nu}:=N_{\mathcal{K}}\cdot(t^{\nu}G_{\mathcal{O}})$ be a semi-infinite
Schubert cell (here $t^{\nu}$ is as in Section 3.1). Let $\rho$ be the half
sum of positive roots of $G$ with respect to $B$. Recall from [MV, Thm. 3.2]
that the intersection $S_{\nu}\cap\operatorname{Gr}^{\lambda}$ is of pure
dimension $\rho(\nu+\lambda)$ as long as it is non-empty.

Let $a_{\nu,\lambda}\colon S_{\nu}\cap\operatorname{Gr}^{\lambda}\to pt$ be
the projection to the point. Consider the top degree cohomology with compact
support

$H_{c}^{2\rho(\nu+\lambda)}(S_{\nu}\cap\operatorname{Gr}^{\lambda},\mathbb{Q}^{\mathrm{Hdg}}):=H^{2\rho(\nu+\lambda)}((a_{\nu,\lambda})_{!}a_{\nu,\lambda}^{\dagger}\mathbb{Q}^{\mathrm{Hdg}})\in\operatorname{MHM}(pt).$

Repeating the arguments of [MV, Sect. 3] (see especially Prop. 3.10 in loc.
cit.), we identify $H^{i}(a_{*}\mathcal{F}_{\lambda})$ with

$\bigoplus_{\nu\in
X_{*},2\rho(\nu)=i}H_{c}^{2\rho(\nu+\lambda)}(S_{\nu}\cap\operatorname{Gr}^{\lambda},\mathbb{Q}).$

In particular, we see that $H^{i}(a_{*}\mathcal{F}_{\lambda})=0$ for odd $i$.

It remains to show that
$H_{c}^{2\rho(\nu+\lambda)}(S_{\nu}\cap\operatorname{Gr}^{\lambda},\mathbb{Q})$
is a Tate twist of a trivial Hodge structure. In fact, the top cohomology with
compact support is always Tate. One way to show this is to reduce to the case
when the scheme is irreducible and to note that the top cohomology is
1-dimensional in this case. ∎

We return to the proof of Proposition 4.3. Since $a_{*}=a_{!}$ preserves
weights, we have
$\operatorname{gr}^{0}(a_{*}\mathcal{F}_{\lambda})=a_{*}(\operatorname{gr}^{0}(\mathcal{F}_{\lambda}))$.
Now the previous lemma implies that
$a_{*}(\operatorname{gr}^{0}(\mathcal{F}_{\lambda}))$ is the direct sum of
Tate twists of trivial Hodge structures.

Note that
$\operatorname{gr}^{0}\mathcal{F}_{\lambda}\in\operatorname{MH}^{0}(\operatorname{Gr}_{G})$
and it follows from semisimplicity that
$\mathrm{IC}_{\lambda}(\mathcal{H}_{0})$ is its direct summand. Thus
$a_{*}\mathrm{IC}_{\lambda}(\mathcal{H}_{0})$ is a direct summand of
$a_{*}(\operatorname{gr}^{0}(\mathcal{F}_{\lambda}))$ and the claim follows.

(ii) Since $a_{+}$ is monoidal (Proposition 4.1), part (ii) follows from part
(i). ∎

###  4.3. Proof of Theorem 1

We will construct an equivalence

$\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]=\operatorname{Perv}_{\mathbb{Q}}(\operatorname{Gr}_{G})\boxtimes\operatorname{MH}(pt)[\sqrt{T}].$

The theorem will follow by applying the usual geometric Satake equivalence
$\operatorname{Perv}_{\mathbb{Q}}(\operatorname{Gr}_{G})=\operatorname{Rep}\mathbb{Q}(\check{G}_{\mathbb{Q}})$
and taking the even components of the both sides.

Consider the functor
$F\colon\operatorname{Perv}_{\mathbb{Q}}(\operatorname{Gr}_{G})\to\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]$
sending $\mathrm{IC}_{\lambda}$ to
$\mathrm{IC}_{\lambda}(\mathcal{H}_{0})^{+}(\frac{1}{2}\dim\operatorname{Gr}^{\lambda})$.
We claim that this is a monoidal functor. Indeed, take $\lambda,\mu\in X_{*}$.
By Proposition 4.3,
$\mathrm{IC}_{\lambda}(\mathcal{H}_{0})(\frac{1}{2}\dim\operatorname{Gr}^{\lambda})\star\mathrm{IC}_{\mu}(\mathcal{H}_{0})(\frac{1}{2}\dim\operatorname{Gr}^{\mu})$
is the direct sum of objects of the form
$\mathrm{IC}_{\nu}(\mathcal{H}_{0})(i)$, where $\nu\in X_{*}$ and
$i\in\frac{1}{2}\mathbb{Z}$. However, this object has weight zero, and the
statement follows.

Next, we have an obvious embedding
$G\colon\operatorname{MH}(pt)[\sqrt{T}]\to\operatorname{MH}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]$.
We claim that $F\boxtimes G$ is an equivalence of categories. According to
(11) and the commutativity constraints, we have

$\mathrm{IC}_{\lambda}(\mathcal{H})^{+}=\mathrm{IC}_{\lambda}(\mathcal{H}_{0})^{+}\left(\frac{1}{2}\dim\operatorname{Gr}^{\lambda}\right)\star\mathrm{IC}_{0}(\mathcal{H})^{+}\left(-\frac{1}{2}\dim\operatorname{Gr}^{\lambda}\right).$

Similarly,

$\mathrm{IC}_{\lambda}(\mathcal{H})^{-}=\mathrm{IC}_{\lambda}(\mathcal{H}_{0})^{+}\left(\frac{1}{2}\dim\operatorname{Gr}^{\lambda}\right)\star\mathrm{IC}_{0}(\mathcal{H})^{-}\left(-\frac{1}{2}\dim\operatorname{Gr}^{\lambda}\right).$

We see that by Proposition 3.1(i), the functor is essentially surjective. The
fact that the functor is fully faithful follows from the definitions and
Proposition 3.1(ii). ∎

###  4.4. Proof of Theorem 2

Consider the functor
$F\colon\operatorname{Perv}_{\mathbb{Q}}(\operatorname{Gr}_{G})\to\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]$
sending $\mathrm{IC}_{\lambda}$ to
$\mathrm{IC}_{\lambda}(\mathcal{H}_{0})^{+}(\frac{1}{2}\dim\operatorname{Gr}^{\lambda})$.
As in the proof of Theorem 1, we check that this functor is monoidal.

Consider the category $\operatorname{Rep}\mathbf{G_{m,\mathbb{Q}}}$. Its
irreducible objects are in a natural bijection with $\mathbb{Z}$, and we equip
the category with the $\mathbb{Z}/2\mathbb{Z}$ parity grading. Consider the
functor

$G\colon\operatorname{Rep}\mathbf{G_{m,\mathbb{Q}}}\to\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]$

sending the representation with weight $i\in\mathbb{Z}$ to
$\mathrm{IC}(\mathcal{H}_{0})^{+}(i/2)$. As in the proof of Theorem 1, we
check that $F\boxtimes G$ is an equivalence of categories. Thus, using the
usual geometric Satake, we get

$\operatorname{Tate}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]=\operatorname{Perv}_{\mathbb{Q}}(\operatorname{Gr}_{G})\boxtimes\operatorname{Rep}_{\mathbb{Q}}\mathbf{G_{m,\mathbb{Q}}}=\operatorname{Rep}_{\mathbb{Q}}(\check{G}_{\mathbb{Q}}\times\mathbf{G_{m,\mathbb{Q}}}).$

We note that the $\mathbb{Z}/2\mathbb{Z}$-grading on this category corresponds
to a central embedding
$\mu_{2}\to\check{G}_{\mathbb{Q}}\times\mathbf{G_{m,\mathbb{Q}}}$. It is easy
to see that this embedding is exactly $2\rho\times\iota$. The theorem follows.
∎

##  5\. Mixed Hodge modules

In this Section we sketch a proof of Theorem 3. In Section 2.3 we defined the
category $\operatorname{MHM}(Y)$ of mixed Hodge modules on an ind-scheme $Y$
with an exact faithful functor
$\operatorname{MHM}(Y)\to\operatorname{Perv}_{\mathbb{Q}}(Y)$. We note that we
have smooth descents for mixed Hodge modules, proof being the same as that of
Proposition 2.14.

###  5.1. Equivariant mixed Hodge modules

Let, as in Section 2.10, $H$ be a connected pro-algebraic group acting nicely
on an ind-scheme $Y$. In this case, using smooth descent for morphisms of
mixed Hodge modules, we can define the category $\operatorname{MHM}^{H}(Y)$ of
equivariant mixed Hodge modules. Similarly to Proposition 2.16, we show that
the forgetful functor $\operatorname{MHM}^{H}(Y)\to\operatorname{MHM}(Y)$ is
fully faithful and we identify $\operatorname{MHM}^{H}(Y)$ with a full
subcategory of $\operatorname{MHM}(Y)$. Finally, we have an analogue of Lemma
2.17.

Recall that an object of $\operatorname{MHM}(Y)$ has a weight filtration. It
follows from [Sai2, Lemma 2.25] that for an object in
$\operatorname{MHM}^{H}(Y)$ this is a filtration by equivariant submodules so
that the associated graded is in $\operatorname{MH}^{H}(Y)$.

###  5.2. $G_{\mathcal{O}}$-equivariant mixed Hodge modules on
$\operatorname{Gr}_{G}$

By the above, we can consider the category
$\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$. We want to
describe the objects. Using the weight filtration and Proposition 3.1, we see
that the simple objects are pure Hodge modules of the form
$\mathrm{IC}_{\lambda}(\mathcal{H})$, where $\lambda\in X_{+}$ and
$\mathcal{H}$ is an irreducible Hodge structure.

The definition of $\mathrm{IC}_{\lambda}(\mathcal{H})$, given in Section 3.1,
generalizes to the mixed Hodge structures $\mathcal{H}$ as follows:
$\mathrm{IC}_{\lambda}(\mathcal{H}):=\mathcal{H}\otimes\mathrm{IC}_{\lambda}(\mathcal{H}_{0})$.
(The original definition cannot be used because we do not define intermediate
extensions for mixed Hodge modules.)

######  Proposition 5.1.

Every object of $\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$
can be uniquely written as the direct sum of subobjects isomorphic to
$\mathrm{IC}_{\lambda}(\mathcal{H})$ for some distinct co-characters
$\lambda\in X_{+}$ and some mixed Hodge structures $\mathcal{H}$.

###### Proof.

Let $\lambda,\mu\in X_{+}$, $\lambda\neq\mu$, let $\mathcal{H}$ and
$\mathcal{H}^{\prime}$ be mixed Hodge structures. It is enough to show that
$\operatorname{Ext}^{1}(\mathrm{IC}_{\lambda}(\mathcal{H}),\mathrm{IC}_{\mu}(\mathcal{H}^{\prime}))=0$.
Without loss of generality we may assume that
$\operatorname{Gr}^{\mu}\subset\overline{\operatorname{Gr}}^{\lambda}$. Let
$j\colon\operatorname{Gr}^{\mu}\to\overline{\operatorname{Gr}}^{\lambda}$ be
the locally closed embedding. It is enough to show that
$j^{*}\mathrm{IC}_{\lambda}(\mathcal{H})$ lives in cohomological degrees
$\leq-2$. Since $\mathrm{Rat}$ is exact and faithful, it is enough to show
that $j^{*}\mathrm{IC}_{\lambda}$ lives in perverse cohomological degrees
$\leq-2$. This is explained in [Gai, Sect. 2.1.3]. ∎

We define the convolution of mixed Hodge modules similarly to the convolution
of pure Hodge modules.

The constructions of rigidity, commutativity constraint, unit object, and the
fiber functor are the same as in the pure case with minor modifications, thus
$\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ becomes a
neutral Tannakian category.

Finally, as in the proof of Theorem 1 we construct an equivalence of
categories.

$\operatorname{MHM}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})[\sqrt{T}]=\operatorname{Perv}_{\mathbb{Q}}(\operatorname{Gr}_{G})\boxtimes\operatorname{MHM}(pt)[\sqrt{T}].$

and the theorem follows. ∎

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