# Levi-Equivariant Restriction of Spherical Perverse Sheaves

Mark Macerato

###### Abstract.

We study the equivariant cohomology of spherical perverse sheaves on the
affine Grassmannian of a connected reductive group $G$ with support in the
affine Grassmannian of any Levi subgroup $L$ of $G$. In doing so, we extend
the work of Ginzburg and Riche on the $T$-equivariant cofibers of spherical
perverse sheaves. We obtain a description of this cohomology in terms of the
Langlands dual group $\check{G}$. More precisely, we identify the cohomology
of the regular sheaf on $\mathrm{Gr}_{G}$ with support along $\mathrm{Gr}_{L}$
with the algebra of functions on a hyperspherical Hamiltonian
$\check{G}$-variety $T^{*}(\check{G}/(\check{U},\psi_{L}))$, where the
Whittaker datum $\psi_{L}$ is an additive character (determined by $L$) of the
maximal unipotent subgroup $\check{U}$.

###### Contents

  1. 1 Introduction
    1. 1.1 Geometric Satake
    2. 1.2 Levi subgroups
    3. 1.3 Equivariant corestriction
    4. 1.4 The case of a torus and the work of Ginzburg-Riche
    5. 1.5 Our results
    6. 1.6 Notation and conventions
      1. 1.6.1 Group theoretic notation
      2. 1.6.2 Groups attached to the formal disc
      3. 1.6.3 Group actions
      4. 1.6.4 Sheaf-theoretic notation
    7. 1.7 Acknowledgements
  2. 2 Automorphic Side
    1. 2.1 Equivariant localization
    2. 2.2 Monoidal structure on corestriction
    3. 2.3 Fusion
    4. 2.4 Parabolic restriction
    5. 2.5 Action of equivariant homology
    6. 2.6 Summary
  3. 3 Spectral Side
    1. 3.1 Partial Kostant-Whittaker reduction
    2. 3.2 Action of the regular centralizer
    3. 3.3 Generic comparison of Hamiltonian and Kostant-Whittaker reductions
    4. 3.4 Anti-generic comparison of Hamiltonian and Kostant-Whittaker reductions
    5. 3.5 Affine closure
  4. 4 Comparison
    1. 4.1 Parabolic restriction
    2. 4.2 The Theorem of Ginzburg-Riche
    3. 4.3 The Theorem of Yun-Zhu
    4. 4.4 Proof of Theorem 1.5.2

##  1\. Introduction

###  1.1. Geometric Satake

Let $G$ denote a connected reductive group over $\mathbb{C}$. Following
Lusztig, we associate to $G$ its affine Grassmannian $\mathrm{Gr}_{G}$ [25,
§11], an ind-projective ind-variety111We will not distinguish here between
$\mathrm{Gr}_{G}$ and its reduction $\mathrm{Gr}_{G}^{\mathrm{red}}$, as our
considerations are purely topological. over $\mathbb{C}$ (we refer the reader
to [3, §1.2], [2, §9.1], [35, §1], and [5, §3] for its basic properties). The
topology of $\mathrm{Gr}_{G}$ gives rise to the Langlands dual group
$\check{G}$ through the geometric Satake equivalence of Ginzburg [18] and
Mirković-Vilonen [28]. More precisely, let $G(\mathcal{O})=G(\mathbb{C}[[t]])$
denote the arc-group222We abuse notation here and throughout the paper by
writing $G(\mathcal{O})$ in place of the arc-group scheme $L^{+}G$ [15,
Definition 1], whose group of $\mathbb{C}$-points is $G(\mathcal{O})$.  of
$G$. The pro-algebraic group $G(\mathcal{O})$ acts naturally on
$\mathrm{Gr}_{G}$, and we may consider the $G(\mathcal{O})$-equivariant
constructible derived category333See §1.6.4 for our notation and conventions
concerning categories of sheaves. $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ of
sheaves of complex vector spaces444Of course, one of the most profound aspects
of the approach of [28] is that the geometric Satake equivalence holds with
essentially arbitrary (in particular, integral) coefficients. We will not need
this generality here; our reliance on the equivariant localization theorem in
the body of the paper precludes our arguments from extending to the integral
setting. on $\mathrm{Gr}_{G}$. The triangulated category
$D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ carries a natural monoidal structure
$\star$ given by the convolution product of sheaves [28, §4]. Lusztig’s
“miraculous theorem” [6, §5.3.6] (reproven conceptually by Gaitsgory [16,
Proposition 6]) asserts that the convolution product $-\star-$ is $t$-exact,
and therefore restricts to an exact bifunctor on the heart of the perverse
$t$-structure $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\subseteq
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, the abelian category of
$G(\mathcal{O})$-equivariant perverse sheaves on $\mathrm{Gr}_{G}$.

The total cohomology functor

$H^{*}(\mathrm{Gr}_{G},-):\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow\mathrm{Vect}_{\mathbb{C}}$

admits a natural monoidal structure. Ginzburg’s approach to this fact in [18]
is to observe that the equivariant cohomology functor
$H_{G(\mathcal{O})}^{*}(\mathrm{Gr}_{G},-)$ has an obvious monoidal structure
(see, for example, [35, Proposition 5.2.3]), which induces a monoidal
structure on $H^{*}$ through the canonical isomorphism
$H^{*}(\mathrm{Gr}_{G},-)\simeq
H_{G(\mathcal{O})}^{*}(\mathrm{Gr}_{G},-)\otimes_{H_{G(\mathcal{O})}^{*}(\mathrm{pt},\mathbb{C})}H^{*}(\mathrm{pt},\mathbb{C})$
(which arises from the equivariant formality of all objects of
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$; compare to Proposition 2.1.1
below). See [28, §3] for the approach of Mirković-Vilonen, which works for
general coefficients. One moreover shows that $H^{*}(\mathrm{Gr}_{G},-)$ is
exact and conservative [28, Corollary 3.7].

Through a global reinterpretation of the convolution product [28, §5-6],
Mirković-Vilonen (following the ideas of Beilinson-Drinfeld [6]) equip the
monoidal abelian category $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ with
a symmetric braiding, the commutativity constraint. They show [28, Proposition
§6.3] that with respect to this symmetric monoidal structure on
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, the cohomology functor
$H^{*}(\mathrm{Gr}_{G},-)$ is symmetric monoidal.

######  Theorem 1.1.1 (Ginzburg [18], Mirković-Vilonen [28]).

There exists a canonically defined connected reductive group $\check{G}$ over
$\mathbb{C}$ equipped with an equivalence of symmetric monoidal abelian
categories

$\mathbb{S}_{G}:\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\simeq\operatorname{Rep}(\check{G})$

and a natural isomorphism of symmetric monoidal functors

$\mathrm{For}\circ\mathbb{S}_{G}\simeq
H^{*}(\mathrm{Gr}_{G},-):\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow\mathrm{Vect}_{\mathbb{C}},$

where
$\mathrm{For}:\operatorname{Rep}(\check{G})\rightarrow\mathrm{Vect}_{\mathbb{C}}$
is the forgetul functor from the category of finite dimensional
$\check{G}$-modules to the category of vector spaces. Moreover, $\check{G}$ is
equipped with a canonically defined Borel subgroup
$\check{B}\subseteq\check{G}$ and maximal torus $\check{T}\subseteq\check{B}$.
The root datum $(X^{*}(\check{T}),X_{*}(\check{T}),\Phi,\check{\Phi})$ of the
triple $\check{T}\subseteq\check{B}\subseteq\check{G}$ is dual to the abstract
root datum555See the discussion [13, §1.1] of Deligne-Lusztig for how one
defines the root datum of a connected reductive group $G$ without choosing
$T\subseteq B\subseteq G$. of $G$ (in the sense of Langlands).

The discussion above can be summarized by saying that the symmetric monoidal
abelian category $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ together with
the fiber functor $H^{*}(\mathrm{Gr}_{G},-)$ defines a neutralized Tannakian
category666Actually, there are a few technical conditions to check, such as
rigidity; these are easily verified for
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G}).$. The Tannakian reconstruction
theorem [5, Theorem 2.7] constructs the affine group scheme $\check{G}$ as the
automorphism group scheme of the fiber functor $H^{*}(\mathrm{Gr}_{G},-)$.
That $\check{G}$ is connected reductive is deduced from a Tannakian
characterization [5, Proposition 2.11] of this property. The most important
ingredient is the semisimplicity of the abelian category
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ [5, Theorem 4.2], which
follows from the parity vanishing property [5, Lemma 4.5] of the stalks of the
intersection complexes of the $G(\mathcal{O})$-orbit closures on
$\mathrm{Gr}_{G}$. This parity vanishing property is a geometric
reinterpretation of deep computations of Lusztig [25, §11] in the affine Hecke
algebra and can be deduced from the BBD Decomposition Theorem.

Observe that the fiber functor $H^{*}(\mathrm{Gr}_{G},-)$ is
$\mathbb{Z}$-graded by the cohomological index. By Tannakian formalism, this
$\mathbb{Z}$-grading on $H^{*}(\mathrm{Gr}_{G},-)$ defines a homomorphism
$2\check{\rho}_{G}:\mathbb{G}_{m}\rightarrow\check{G}$. It is well known that
the maximal torus $\check{T}\subseteq\check{G}$ (the Borel subgroup
$\check{B}\subseteq\check{G}$) arises as the centralizer (respectively, the
attracting scheme) of the cocharacter $2\check{\rho}_{G}$ in $\check{G}$.

###  1.2. Levi subgroups

We can now approach the problem considered in this paper. We fix a parabolic
subgroup $P\subseteq G$ with unipotent radical $V\subseteq P$ and Levi
quotient $L=P/V$. Note that $L$ is a connected reductive group in its own
right, so we also have the geometric Satake equivalence

$\mathbb{S}_{L}:\mathcal{P}_{L(\mathcal{O})}(\mathrm{Gr}_{L})\simeq\mathrm{Rep}(\check{L}).$

The categories $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ and
$\mathcal{P}_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ are directly related by a
functor, which we shall call the parabolic restriction, introduced by
Beilinson-Drinfeld in [6, §5.3.28] (generalizing the weight functors of
Mirković-Vilonen [28, §3] in the case that $L=T$ is a torus). We will have
much more to say about it in §§2.4, and also refer the reader to [5, §15.1]
for more details. For now, we note that it is an exact tensor functor

$\mathrm{Res}^{G,\natural}_{L}:\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow\mathcal{P}_{L(\mathcal{O})}(\mathrm{Gr}_{L}).$

Namely, we have a diagram of ind-schemes

${\mathrm{Gr}_{P}}$${\mathrm{Gr}_{L}}$${\mathrm{Gr}_{G}.}$$\scriptstyle{r}$$\scriptstyle{q}$

The functor $q_{*}r^{!}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ is not $t$-exact. However, let
$\mathcal{F}\in\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$. For each
connected component $\mathrm{Gr}_{L}^{\chi}\subseteq\mathrm{Gr}_{L}$ of
$\mathrm{Gr}_{L}$, the complex
$q_{*}r^{!}\mathcal{F}|_{\mathrm{Gr}_{L}^{\chi}}$ is concentrated in a unique
perverse degree. Thus, one defines the exact functor
$\mathrm{Res}^{G,\natural}_{L}$ by shifting the functor $r_{*}q^{!}$ by an
appropriate integer on each connected component $\mathrm{Gr}_{L}^{\chi}$ (see
Remark 2.4.3).

Tannakian formalism now defines a closed embedding
$\check{L}\hookrightarrow\check{G}$ and a canonical identification of the
tensor functor
$\mathbb{S}_{L}\circ\mathrm{Res}^{G,\natural}_{L}\circ\mathbb{S}_{G}^{-1}$
with the restriction functor
$\mathrm{Rep}(\check{G})\rightarrow\mathrm{Rep}(\check{L})$ [5, Proposition
15.3]. Moreover, the embedding $\check{L}\hookrightarrow\check{G}$ realizes
$\check{L}$ as a Levi subgroup of $\check{G}$ containing the maximal torus
$\check{T}$. The dual parabolic subgroup $\check{P}\subseteq\check{G}$ can now
be constructed as the attracting scheme in $\check{G}$ for the coweight
$2\check{\rho}_{G}-2\check{\rho}_{L}$ of $\check{T}$.

###  1.3. Equivariant corestriction

We will now fix a Levi factor $L\subseteq P$. We obtain a closed immersion

$i:\mathrm{Gr}_{L}\hookrightarrow\mathrm{Gr}_{G}$

identifying $\mathrm{Gr}_{L}$ with a closed subscheme of $\mathrm{Gr}_{G}$. In
fact, $\mathrm{Gr}_{L}\subseteq\mathrm{Gr}_{G}$ is exactly the fixed point
subscheme $\mathrm{Gr}_{G}^{Z^{\circ}(L)}\subseteq\mathrm{Gr}_{G}$ for the
action of the neutral component $Z(L)^{\circ}\subseteq Z(L)$ of the center of
$L$. We are interested in the (co)restriction functor

$i^{!}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$

given by the composition of the forgetful functor
$\mathrm{For}^{G(\mathcal{O})}_{L(\mathcal{O})}$ (forgetting
$G(\mathcal{O})$-equivariance down to $L(\mathcal{O})$-equivariance) followed
by the functor $i^{!}:D_{L(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ of $L(\mathcal{O})$-equivariant
$!$-restriction.777We consider $!$-restriction over $*$-restriction only
because the spectral “answer” to our question will be cleaner; of course, the
two functors differ only by Verdier duality. Unlike
$\mathrm{Res}^{G,\natural}_{L}$, the functor $i^{!}$ is not $t$-exact, and
there is no saving exactness with a mere grading shift. Nonetheless, it will
be convenient888Note that this grading shift will not be used in §2 to work
with the following “regraded” or “sheared” version of $i^{!}$:

$i^{!,\natural}:=\bigoplus_{\chi\in\pi_{0}(\mathrm{Gr}_{L})}i^{!}|_{\mathrm{Gr}_{L}^{\chi}}[-\langle
2\check{\rho}_{G}-2\check{\rho}_{L},\chi\rangle].$

Here, $\mathrm{Gr}_{L}^{\chi}\subseteq\mathrm{Gr}_{L}$ is the connected
component of $\mathrm{Gr}_{L}$ labelled by the element
$\chi\in\pi_{0}(\mathrm{Gr}_{L})\simeq\Lambda/Q_{L}$, where $\Lambda$ is the
weight lattice of $\check{T}$ and $Q_{L}\subseteq\Lambda$ is the root lattice
of $\check{L}$.

Unlike $\mathrm{Res}^{G,\natural}_{L}$, $i^{!,\natural}$ does not admit a
monoidal structure. However, in §§2.2, we will equip $i^{!,\natural}$ with a
lax monoidal structure. This is enough structure to ensure that
$i^{!,\natural}$ carries ring objects of $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$
to ring objects of $D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$. In fact, we will
carry out this construction for any connected reductive subgroup $L\subseteq
G$. We will give two constructions of this lax monoidal structure; the second
(given in §§2.3) uses Gaitsgory’s description [16, Proposition 6] of the
convolution product $-\star-$ in terms of a global nearby cycles construction.
Moreover, in §§2.4 (see Construction 2.4.5 and Remark 2.6.10), we will show
that the natural adjunction morphism

(1.3.1)
$\Xi^{\natural}:i^{!,\natural}\rightarrow\mathrm{Res}^{G,\natural}_{L}$

is compatible with the lax monoidal structures on these functors.

###  1.4. The case of a torus and the work of Ginzburg-Riche

We now specialize to the case in which $L=T$ is a maximal torus of $G$. The
affine Grassmannian $\mathrm{Gr}_{T}$ is the coweight lattice of $T$, which we
identify with the weight lattice $\Lambda$ of $\check{T}$. The closed
embedding $i:\Lambda=\mathrm{Gr}_{T}\hookrightarrow\mathrm{Gr}_{G}$ is the
familiar embedding of the coweight lattice into $\mathrm{Gr}_{G}$ taking the
coweight $\lambda\in\Lambda$ to the point $t^{\lambda}\in\mathrm{Gr}_{G}$ (the
image of the uniformizer $t\in\mathbb{C}((t))^{*}$ under the map
$\lambda(\mathbb{C}((t))):\mathbb{C}((t))^{*}\rightarrow
G(\mathbb{C}((t)))\twoheadrightarrow\mathrm{Gr}_{G}$). For each
$\mu\in\Lambda$, let $i_{\mu}:\\{t^{\mu}\\}\hookrightarrow\mathrm{Gr}_{G}$
denote the inclusion of the corresponding point of $\mathrm{Gr}_{T}$. Let
$\lambda\in\Lambda^{+}$ denote a dominant coweight, and let
$\mathrm{IC}_{\lambda}\in\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ denote
the intersection complex of
$\mathrm{Gr}_{G}^{\leq\lambda}=\overline{\mathrm{Gr}^{\lambda}_{G}}$. Note
that $L(\lambda):=\mathbb{S}(\mathrm{IC}_{\lambda})\in\mathrm{Rep}(\check{G})$
is the simple $\check{G}$-module of highest weight $\lambda\in\Lambda^{+}$.
Since the costalk $i_{\mu}^{!}\mathrm{IC}_{\lambda}$ is only non-trivial when
$\mu\leq\lambda$, we may assume that this is the case.

Lusztig’s description of the (non-equivariant) (co)stalks
$H^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})$ predates the geometric Satake
equivalence itself.

######  Theorem 1.4.1 (Lusztig [25, §11]).

Let $\widetilde{W}_{\mathrm{aff}}=\Lambda\rtimes W$ denote the extended affine
Weyl group and $\ell:\widetilde{W}_{\mathrm{aff}}\rightarrow\mathbb{Z}$ its
length function; let $p_{\mu},p_{\lambda}\in\widetilde{W}_{\mathrm{aff}}$
denote the coweights $\mu,\lambda\in\Lambda$ viewed as elements of
$\widetilde{W}_{\mathrm{aff}}$. Let $n_{\mu}\in\widetilde{W}_{\mathrm{aff}}$
(resp. $n_{\lambda}\in\widetilde{W}_{\mathrm{aff}}$) denote the element of
maximal length in the $W$-double coset
$Wp_{\mu}W\subseteq\widetilde{W}_{\mathrm{aff}}$ (resp.
$Wp_{\lambda}W\subseteq\widetilde{W}_{\mathrm{aff}}$). Then, we have the
vanishing $H^{i}(i_{\mu}^{!}\mathrm{IC}_{\lambda})\simeq 0$ for $i$ odd, as
well as the equality

$\sum_{i\geq
0}\dim{H^{2i}(i_{\mu}^{!}\mathrm{IC}_{\lambda})}q^{i}=P_{n_{\mu},n_{\lambda}}(q)=d_{\mu}(L(\lambda);q)$

in the ring $\mathbb{Z}[q]$. Here, for any elements
$y,w\in\widetilde{W}_{\mathrm{aff}}$ of the extended affine Weyl group,
$P_{y,w}\in\mathbb{Z}[q]$ denotes the corresponding Kazhdan-Lusztig polynomial
(see [25, §4]). Moreover, $d_{\mu}(L(\lambda);q)$ denotes Lusztig’s $q$-analog
of the weight multiplicity (see [25, §6]).

The parity vanishing in Lusztig’s theorem implies that
$H_{T}^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})$ is a free module over the ring
$R_{T}:=H^{*}_{T}(\mathrm{pt},\mathbb{C})$. Hence, it implies the (non-
canonical) isomorphism of $R_{T}$-modules
$H_{T}^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})\simeq
H^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})\otimes R_{T}$ and gives the graded
dimension of $H^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})$.

However, a more intrinsic description of the $R_{T}$-module
$H_{T}^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})$ has been obtained by Ginzburg
and Riche [20]. It is their work which we aim to extend to arbitrary Levi
subgroups of $G$. We point out that Ginzburg and Riche go further and describe
the equivariant costalks
$H_{T\rtimes\mathbb{G}_{m}}^{*}(i_{\mu}^{!}\mathrm{IC}_{\lambda})$, where
$\mathbb{G}_{m}$ denotes the loop rotation torus. In this paper, we will not
consider the loop rotation equivariance (because it is somewhat orthogonal to
the applications that we have in mind), although we anticipate that our
arguments will extend to this setting.

We can evaluate the natural transformation (1.3.1) on the object
$\mathrm{IC}_{\lambda}\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, restrict to
the component $t^{\mu}\in\mathrm{Gr}_{T}$ of $\mathrm{Gr}_{T}$, and take
cohomology to obtain an $R_{T}$-module map

(1.4.2) $H_{T}^{*}(i^{!,\natural}_{\mu}\mathrm{IC}_{\lambda})\rightarrow
H_{T}^{*}(\\{t^{\mu}\\},\mathrm{Res}^{G,\natural}_{T}(\mathrm{IC}_{\lambda}))=H^{*}(\\{t^{\mu}\\},\mathrm{Res}^{G,\natural}_{T}(\mathrm{IC}_{\lambda}))\otimes
R_{T}.$

To formulate their description of
$H_{T}^{*}(i^{!,\natural}_{\mu}\mathrm{IC}_{\lambda})$, we bring in the Lie
algebra $\check{\mathfrak{g}}=\mathrm{Lie}(\check{G})$, the Cartan subalgebra
$\check{\mathfrak{t}}=\mathrm{Lie}(\check{T})$, the maximal nilpotent
subalgebra $\check{\mathfrak{u}}=\mathrm{Lie}(\check{{U}})$, and the opposite
maximal nilpotent subalgebra
$\check{\mathfrak{u}}^{-}=\mathrm{Lie}(\check{U}^{-})$. We have the triangular
decomposition

$\check{\mathfrak{g}}=\check{\mathfrak{u}}\oplus\check{\mathfrak{t}}\oplus\check{\mathfrak{u}}^{-},$

which induces a projection
$\check{\mathfrak{g}}/\check{\mathfrak{u}}\twoheadrightarrow\check{\mathfrak{t}}$.
Let $\mathbb{C}(-\mu)$ denote the one dimensional $\check{B}$-module
corresponding to the weight $-\mu\in\Lambda$ of $\check{T}$. We obtain a
canonical map

(1.4.3)
$\displaystyle\left(L(\lambda)\otimes\operatorname{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}})\otimes\mathbb{C}(-\mu)\right)^{\check{B}}$
$\displaystyle\hookrightarrow\left(L(\lambda)\otimes\operatorname{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}})\otimes\mathbb{C}(-\mu)\right)^{\check{T}}$
$\displaystyle\twoheadrightarrow\left(L(\lambda)\otimes\operatorname{Sym}(\check{\mathfrak{t}})\otimes\mathbb{C}(-\mu)\right)^{\check{T}}$
$\displaystyle=\,\,L(\lambda)_{\mu}\otimes\operatorname{Sym}(\check{\mathfrak{t}}).$

Here, $L(\lambda)_{\mu}\subseteq L(\lambda)$ denotes the $\mu$ weight space.
Recall, moreover, that there is a canonical isomorphism
$R_{T}\simeq\operatorname{Sym}(\mathfrak{t}^{*})\simeq\operatorname{Sym}(\check{\mathfrak{t}})$
which places $\check{\mathfrak{t}}$ in graded degree $2$ (see, for example,
[2, Theorem 6.6.8]).

######  Theorem 1.4.4 (Ginzburg-Riche, [20, Theorem 2.3.1]).

There exists a unique graded
$R_{T}\simeq\operatorname{Sym}(\check{\mathfrak{t}})$-module isomorphism

(1.4.5)
$H_{T}^{*}(i^{!,\natural}_{\mu}\mathrm{IC}_{\lambda})\simeq\left(L(\lambda)\otimes\operatorname{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}})\otimes\mathbb{C}(-\mu)\right)^{\check{B}}$

such that the diagram

${H_{T}^{*}(i^{!,\natural}_{\mu}\mathrm{IC}_{\lambda})}$${H^{*}(\\{t^{\mu}\\},\mathrm{Res}^{G,\natural}_{T}(\mathrm{IC}_{\lambda}))\otimes
R_{T}}$${\left(L(\lambda)\otimes\operatorname{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}})\otimes\mathbb{C}(-\mu)\right)^{\check{B}}}$${L(\lambda)_{\mu}\otimes\operatorname{Sym}(\check{\mathfrak{t}})}$$\scriptstyle{\eqref{ginzburg
riche automorphic map}}$$\scriptstyle{\eqref{introduction ginzburg riche
isomorphism}}$$\scriptstyle{\sim}$$\scriptstyle{\eqref{introduction ginzburg
riche spectral map}}$

commutes. Here, $\check{\mathfrak{g}}/\check{\mathfrak{u}}$ is placed in
graded degree $2$ (so that (1.4.3) is a graded map).

We can formulate their theorem as follows. Let
$\mathcal{F}_{\mathrm{reg}}\in\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$
denote the regular sheaf999Strictly speaking, $\mathcal{F}_{\mathrm{reg}}$
does not belong to $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, but rather to a
suitable ind-completion (just as the regular representation
$\mathcal{O}(\check{G})$ does not belong to $\mathrm{Rep}(\check{G})$). See
§§1.6.4. We ignore this point for simplicity., the $\check{G}$-equivariant
ring object of $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ corresponding
to the left regular representation
$\mathcal{O}(\check{G})\in\mathrm{Rep}(\check{G})$ under the geometric Satake
equivalence. It admits the Peter-Weyl decomposition

$\mathcal{F}_{\mathrm{reg}}=\bigoplus_{\lambda\in\Lambda^{+}}\mathrm{IC}_{\lambda}\boxtimes
L(\lambda)^{*}.$

We identify $\mathrm{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}})$ with the
algebra of functions
$\mathcal{O}\left((\check{\mathfrak{g}}/\check{\mathfrak{u}})^{*}\right)$. By
tensoring $\eqref{introduction ginzburg riche isomorphism}$ with
$L(\lambda)^{*}$ and passing to the direct sum over $\lambda\in\Lambda^{+}$
and $\mu\in\Lambda$, we obtain a canonical isomorphism of $\mathbb{Z}$-graded
$\Lambda$-graded $\check{G}$-equivariant
$\mathrm{Sym}(\check{\mathfrak{t}})\simeq\mathcal{O}(\check{\mathfrak{t}}^{*})$-algebras

$\displaystyle H_{T}^{*}(i^{!,\natural}\mathcal{F}_{\mathrm{reg}})$
$\displaystyle\simeq\left(\left(\bigoplus_{\lambda\in\Lambda^{+}}L(\lambda)\boxtimes
L(\lambda)^{*}\right)\otimes\mathcal{O}\left((\check{\mathfrak{g}}/\check{\mathfrak{u}})^{*}\right)\right)^{\check{U}}$
$\displaystyle\simeq\left(\mathcal{O}(\check{G})\otimes\mathcal{O}((\check{\mathfrak{g}}/\check{\mathfrak{u}})^{*})\right)^{\check{U}}$
$\displaystyle\simeq\mathcal{O}\left(\check{G}\times(\check{\mathfrak{g}}/\check{\mathfrak{u}})^{*}\right)^{\check{U}}$
$\displaystyle\simeq\mathcal{O}(T^{*}(\check{G}/\check{U})).$

We can interpret the $\Lambda$-grading as arising from the $\check{T}$-action
on the variety $T^{*}(\check{G}/\check{U})$ (through the identification
$\check{T}\simeq\check{B}/\check{U}$ and the natural $\check{B}$-action by
right translation). The $\mathcal{O}(\check{\mathfrak{t}}^{*})$-algebra
structure arises from the $\check{T}$-equivariant moment map

$\mu:T^{*}(\check{G}/\check{U})\rightarrow\check{\mathfrak{t}}^{*}.$

The $\check{G}$-module structure arises from the natural action of $\check{G}$
on the variety $T^{*}(\check{G}/\check{U})$ by left translation. Finally, the
$\mathbb{Z}$-grading on $\mathcal{O}(T^{*}(\check{G}/\check{U}))$ is induced
by the $\mathbb{G}_{m}$-action on the fibers of the vector bundle
$T^{*}(\check{G}/\check{U})$ through which $t\in\mathbb{G}_{m}$ acts by
$t^{-2}$.

The isomorphism
$H^{*}_{T}(i^{!,\natural}\mathcal{F}_{\mathrm{reg}})\simeq\mathcal{O}(T^{*}(\check{G}/\check{U}))$
conversely yields Theorem 1.4.4 by passage to the $\check{G}$-isotypic and
$\check{T}$-isotypic components.

###  1.5. Our results

The geometric Satake equivalence induces a canonical pinning of the dual group
$\check{G}$. We have already explained that $\check{G}$ is equipped with a
canonical Borel subgroup $\check{B}\subseteq\check{G}$ (concretely, the
subgroup of automorphisms of the fiber functor $H^{*}$ preserving the
Mirković-Vilonen filtration) and a canonical maximal torus
$\check{T}\subseteq\check{B}$; the point is that one can moreover find
distinguished non-zero root vectors
$X_{\alpha}\in\mathfrak{\mathfrak{g}}_{\alpha}$. We refer the reader to [20,
§6.5] for a discussion of the precise pinning that we will use. Let $\Delta$
denote the set of simple roots of $\check{G}$ and $I\subseteq\Delta$ the
subset of simple roots of $\check{L}$.

We can use this pinning to define a distinguished additive character
$\psi_{I}$ of $\check{U}$ as the composition

${\check{U}}$${\check{U}/[\check{U},\check{U}]}$${\bigoplus_{\alpha\in\Delta}\mathbb{G}_{a}}$${\bigoplus_{\alpha\in
I}\mathbb{G}_{a}}$${\mathbb{G}_{a}.}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{pr}_{I}}$$\scriptstyle{+}$

Here, $\mathrm{pr}_{I}$ is given by projection onto the factors indexed by
$I$, and the last map is summation. We also denote by $\psi_{I}$ the linear
form $(d\psi_{I})_{1}\in\check{\mathfrak{u}}^{*}$, where $1\in\check{U}$ is
the identity. We write
$\check{\mathfrak{u}}^{\perp}\subseteq\check{\mathfrak{g}}^{*}$ for the linear
complement to $\check{\mathfrak{u}}\subseteq\check{\mathfrak{g}}$. We may
regard $\psi_{I}$ as an element of $\check{\mathfrak{g}}^{*}$ by extending it
trivially over $\check{\mathfrak{b}}^{-}$. Let $\psi:=\psi_{\Delta}$ (the non-
degenerate additive character).

The algebraic variety $T^{*}\check{G}$ is equipped with a pair of commuting
actions of $\check{G}$ by left and right translation. We have the
$\check{G}$-equivariant (for the left translation $\check{G}$-action) moment
map (for the right translation $\check{G}$-action)

$\mu:T^{*}\check{G}\simeq\check{G}\times\check{\mathfrak{g}}^{*}\xrightarrow{\mathrm{pr}_{2}}\check{\mathfrak{g}}^{*}$

given by projection to the second factor. The moment map for the induced
$\check{U}$-action is the composition
$T^{*}\check{G}\xrightarrow{\mathrm{pr}_{2}}\check{\mathfrak{g}}^{*}\twoheadrightarrow\check{\mathfrak{u}}^{*}$.
Since $\psi_{I}\in\check{\mathfrak{u}}^{*}$ is $\check{U}$-invariant, we can
form the Hamiltonian reduction of the Hamiltonian $\check{U}$-variety
$T^{*}\check{G}\rightarrow\check{\mathfrak{u}}^{*}$ at the level $\psi_{I}$ to
obtain the partial Kostant-Whittaker reduction

$T^{*}(\check{G}/(\check{U},\psi_{I})):=T^{*}\check{G}/\\!\\!/\\!\\!/(\check{U},\psi_{I}):=(T^{*}\check{G}\times_{\check{\mathfrak{u}}^{*}}\\{\psi_{I}\\})/\check{U}.$

Our entire §3 is dedicated to a discussion of this construction and its basic
properties.

The variety $T^{*}(\check{G}/(\check{U},\psi_{I}))$ carries a natural left
$\check{G}$-action. Moreover, we have a canonical $\check{G}$-equivariant
projection

$\pi_{I}:T^{*}(\check{G}/(\check{U},\psi_{I}))\hookrightarrow(T^{*}\check{G})/\check{U}\twoheadrightarrow\check{\mathfrak{g}}^{*}/\check{U}\twoheadrightarrow\check{\mathfrak{l}}^{*}/\check{U}\rightarrow\check{\mathfrak{c}}_{I}$

to the GIT quotient
$\check{\mathfrak{c}}_{I}:=\check{\mathfrak{l}}^{*}/\\!/\check{L}$. The
homomorphism $\pi_{I}^{*}$ equips
$\mathcal{O}(T^{*}(\check{G}/(\check{U},\psi_{I})))$ with the structure of a
$\check{G}$-equivariant $\mathcal{O}(\check{\mathfrak{c}}_{I})$-algebra.

The action of $\mathbb{G}_{m}$ on $\check{\mathfrak{l}}^{*}$ given by
$t\cdot\xi=t^{-2}\xi$ commutes with the coadjoint action of $\check{L}$, hence
descends to a $\mathbb{G}_{m}$-action on $\check{\mathfrak{c}}_{I}$ (which
places the generators of the polynomial algebra
$\mathcal{O}(\check{\mathfrak{c}}_{I})$ in degrees given by twice the
characteristic exponents of $\check{L}$). We let $\mathbb{G}_{m}$ act on
$T^{*}\check{G}\simeq\check{G}\times\check{\mathfrak{g}}^{*}$ by the
cocharacter $2\check{\rho}_{I}$ on the $\check{G}$ factor and by $t\mapsto
t^{-2}$ on $\check{\mathfrak{g}}^{*}$ (we will discuss gradings more carefully
in Construction 4.4.2). These actions of $\mathbb{G}_{m}$ make $\pi_{I}$ a
$\mathbb{G}_{m}$-equivariant map. Hence, $\pi_{I}^{*}$ equips
$\mathcal{O}(T^{*}(\check{G}/(\check{U},\psi_{I})))$ with the structure of a
graded $\check{G}$-equivariant
$\mathcal{O}(\check{\mathfrak{c}}_{I})$-algebra.

Let $\mathfrak{J}_{I}\rightarrow\check{\mathfrak{c}}_{I}$ denote Ngô’s regular
centralizer group scheme [30, Lemme 2.1.1] (see also Riche’s treatment [32])
associated to the group $\check{L}_{I}$. It is a commutative affine group
scheme over $\check{\mathfrak{c}}_{I}$. Moreover, when we discuss the regular
centralizer in more detail in §§3.2, we shall see that it acts canonically on
the variety $T^{*}(\check{G}/(\check{U},\psi_{I}))$ (or any partial Kostant-
Whittaker reduction, for that matter). When the derived subgroup of $G$ is
almost simple, Yun-Zhu [34] provide a canonical isomorphism of affine
$\check{\mathfrak{c}}_{I}$-group schemes

(1.5.1)
$\operatorname{Spec}{H^{L(\mathcal{O})}_{*}(\mathrm{Gr}_{L},\mathbb{C})}\simeq\mathfrak{J}_{I}.$

We will discuss their results in detail in both §§2.5 and §§4.3.

On the other hand, we will recall (again following Yun-Zhu [34]) in §§2.5 how
the group scheme
$\operatorname{Spec}{H^{L(\mathcal{O})}_{*}(\mathrm{Gr}_{L},\mathbb{C})}$ acts
canonically on
$H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!,\natural}\mathcal{F}_{\mathrm{reg}})$.

Finally, recall that there is a canonical algebra isomorphism
$R_{L}:=H_{L}^{*}(\mathrm{pt},\mathbb{C})\simeq\mathcal{O}(\check{\mathfrak{c}}_{I})$.
This isomorphism identifies our grading on
$\mathcal{O}(\check{\mathfrak{c}}_{I})$ with the cohomological grading on
$R_{L}$.

With this notational setup in place, we can state our main result in its basic
form.

######  Theorem 1.5.2.

There is a canonical $\check{G}$-equivariant isomorphism of graded
$R_{L}$-algebras

$\Upsilon_{I}:H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!,\natural}\mathcal{F}_{\mathrm{reg}})\simeq\mathcal{O}(T^{*}(\check{G}/(\check{U},\psi_{I}))).$

The left hand side is regarded as a ring via the lax monoidal structure on
$i^{!,\natural}$ of §§2.2. Assume that the derived subgroup of $G$ is almost
simple (so that the Yun-Zhu isomorphism (1.5.1) is available). Then,
$\Upsilon_{I}$ is moreover $\mathfrak{J}_{I}$-equivariant.

In fact, our result is stronger, and describes the canonical
$\check{G}$-equivariant graded $R_{L}$-algebra homomorphism of (1.3.1)

$\xi^{\natural}:=\Xi^{\natural}(\mathcal{F}_{\mathrm{reg}}):H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!,\natural}\mathcal{F}_{\mathrm{reg}})\rightarrow
H^{*}_{L(\mathcal{O})}(\mathrm{Gr}_{I},\operatorname{Res}^{G,\natural}_{L}(\mathcal{F}_{\mathrm{reg}}))$

in terms of the Langlands dual group. However, to avoid defining yet more
notation in this introduction, we will simply note that the version of Theorem
1.5.2 stated and proved in §§4.4 as Theorem 4.4.5 asserts and demonstrates
these additional compatibliities.

######  Remark 1.5.3.

We want to draw attention to what we find to be one of the more interesting
aspects of the proof of Theorem 1.5.2. Namely, our choice to treat the regular
object $\mathcal{F}_{\mathrm{reg}}$ rather than the complexes
$\mathrm{IC}_{\lambda}$ individually is essential to our argument. In [20,
Proposition 3.2.3], Ginzburg-Riche establish that for any dominant
$\lambda\in\Lambda^{+}$, the $\mathcal{O}(\check{\mathfrak{t}}^{*})$-module

$H^{0}(\widetilde{\check{\mathfrak{g}}},\mathcal{O}_{\widetilde{\check{\mathfrak{g}}}}(\lambda))=\mathrm{Ind}_{\check{B}}^{\check{G}}(\mathrm{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}})\otimes\mathbb{C}(-\lambda))^{\check{B}}$

is free. Here,
$\widetilde{\check{\mathfrak{g}}}\rightarrow{\check{\mathfrak{g}}}$ is the
Grothendieck-Springer alteration and
$\mathcal{O}_{\widetilde{\check{\mathfrak{g}}}}(\lambda)$ is the pullback of
the Borel-Weil line bundle $\mathcal{O}_{\check{G}/\check{B}}(\lambda)$ on the
flag variety $\check{G}/\check{B}$ along the natural projection
$q:\widetilde{\check{\mathfrak{g}}}\rightarrow\check{G}/\check{B}$. Their
proof appeals to cohomological computations of Broer [23, Proposition 2.5]. By
working instead with the entire coordinate ring
$\mathcal{O}(T^{*}(\check{G}/\check{U}))$, (a suitable adaptation of) our
application of Hartog’s principle (Lemma 3.5.1) can be used in place of this
freeness. We then obtain the freeness of
$H^{0}(\widetilde{\check{\mathfrak{g}}},\mathcal{O}_{\widetilde{\check{\mathfrak{g}}}}(\lambda))$
as a corollary to the Ginzburg-Riche theorem.

We can pass to $\check{G}$-isotypic components in Theorem 1.5.2 and undo the
grading twist in the definition of $i^{!,\natural}$ to obtain the following
more concrete result.

######  Corollary 1.5.4.

Let $\lambda\in\Lambda^{+}$ denote a dominant weight of $\check{T}$ and
$L(\lambda)=\mathbb{S}_{G}(\mathrm{IC}_{\lambda})$ the corresponding simple
$\check{G}$-module. Then, we have a canonical graded
$R_{L}\simeq\mathcal{O}(\check{\mathfrak{c}}_{I})$-module isomorphism

$\Upsilon_{I,\lambda}:H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!}\mathrm{IC}_{\lambda})\simeq(L(\lambda)\otimes\mathcal{O}(\check{\mathfrak{u}}^{\perp}+\psi_{I}))^{\check{U}}.$

The grading of
$H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!}\mathrm{IC}_{\lambda})$ by the
component group $\pi_{0}(\mathrm{Gr}_{L})\simeq\Lambda/Q_{L}\simeq
X^{*}(Z(\check{L}))$ corresponds to the grading of
$(L(\lambda)\otimes\mathcal{O}(\check{\mathfrak{u}}^{\perp}+\psi_{I}))^{\check{U}}$
by the eigenvalues of the center $Z(\check{L})$. Moreover, the cohomological
grading on
$H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!}\mathrm{IC}_{\lambda})$
corresponds under $\Upsilon_{I,\lambda}$ to the _Brylinski-Kostant_ grading on
$(L(\lambda)\otimes\mathcal{O}(\check{\mathfrak{u}}^{\perp}+\psi_{I}))^{\check{U}}$.

Assume moreover that $G$ has almost simple derived group. Then,
$\Upsilon_{I,\lambda}$ is an isomorphism of $\mathfrak{J}_{I}$-modules.

To reiterate Remark 1.5.3, we don’t know how to prove Corollary 1.5.4 without
first proving Theorem 1.5.2, since our approach to the latter leverages the
algebraic geometry of the moment map
$T^{*}(\check{G}/\check{V})\rightarrow\check{\mathfrak{l}}^{*}$.

###  1.6. Notation and conventions

We will now review some of our notation pertaining to root data and dual
groups.

####  1.6.1. Group theoretic notation

We fix a connected reductive group $G$ over $\mathbb{C}$. At certain points in
§4, we will take the additional hypothesis that the derived subgroup
$G^{\mathrm{der}}\subseteq G$ is almost simple; we will make this assumption
explicit in theorem statements whenever it is needed. We will fix a Borel
subgroup $B\subseteq G$ and a maximal torus $T\subseteq B$ with weight lattice
$\check{\Lambda}=X^{*}(T)$ and coweight lattice $\Lambda=X_{*}(T)$. Let
$U\subseteq B$ denote the unipotent radical. We also have the opposite Borel
subgroup $B^{-}\subseteq G$ and its unipotent radical $U^{-}\subseteq B^{-}$.

Let $\check{\Phi}\subseteq X^{*}(T)$ denote the set of roots,
$\check{\Phi}^{+}\subseteq\check{\Phi}$ the subset of positive roots (relative
to $B$), and $\check{\Delta}\subseteq\check{\Phi}^{+}$ the subset of simple
roots. Let $\Phi\subseteq\Lambda=X_{*}(T)$ denote the set of coroots,
${\Phi}^{+}\subseteq{\Phi}$ the positive coroots, and
${\Delta}\subseteq{\Phi}^{+}$ the simple coroots. We typically denote coroots
$\alpha\in\Phi$ without a check; the corresponding root is denoted
$\check{\alpha}\in\check{\Phi}$. Let $\Lambda^{+}\subseteq\Lambda$ denote the
subset of dominant coweights.

Given a subset $I\subseteq\check{\Delta}$, we let $P_{I}\subseteq G$ denote
the corresponding standard parabolic subgroup. That is, $P_{I}$ is the unique
parabolic subgroup of $G$ containing $B$ such that the negative simple root
spaces $\mathfrak{g}_{-\check{\alpha}}\subseteq\mathfrak{g}$ contained in
$\mathfrak{p}_{I}:=\mathrm{Lie}(P_{I})$ are exactly those labelled by the
negative simple roots $\check{\alpha}\in I$. We let $V_{I}\subseteq P_{I}$
denote the unipotent radical of $P^{-}_{I}$, $P^{-}_{I}\subseteq G$ the
opposite parabolic subgroup, and $V_{I}^{-}\subseteq P^{-}_{I}$ its unipotent
radical. Let $L_{I}=P_{I}\cap P_{I}^{-}$ denote the unique Levi subgroup of
$P_{I}$ containing $T$. Let $B_{I}=B/V_{I}\subseteq L_{I}$ and
$U_{I}=U/V_{I}\subseteq L_{I}$; $B_{I}$ is a Borel subgroup of $L_{I}$ and
$U_{I}\subseteq B_{I}$ is the unipotent radical of $B_{I}$.

Let $\check{\Phi}_{I}\subseteq\check{\Phi}$ denote the set of roots of $L_{I}$
and $\check{\Phi}^{+}_{I}\subseteq\check{\Phi}_{I}$ the subset of positive
roots (relative to $\check{B}_{I}$). Let
$2\check{\rho}_{I}=\sum_{\check{\alpha}\in\check{\Phi}_{I}^{+}}\check{\alpha}\in
X^{*}(T)$ and $2\rho_{I}=\sum_{\alpha\in\Phi_{I}^{+}}\alpha$ denote the sum of
the positive roots of $L_{I}$ and positive coroots of $L_{I}$, repsectively.
Note that $L_{I}=Z_{G}(2\rho_{I})$ is the centralizer of the homomorphism
$2\rho_{I}:\mathbb{G}_{m}\rightarrow G$ in $G$. When $I=\Delta$, we simply
write $2\rho$ and $2\check{\rho}$ in place of $2\rho_{\Delta}$ and
$2\check{\rho}_{\Delta}$, respectively.

Let $\check{G}$ denote the Langlands dual group. Recall that the geometric
Satake equivalence induces a canonical pinning of $\check{G}$ (independently
of our choice of $T\subseteq B\subseteq G$, which is made only for notational
convenience). We adopt the conventions of [20, Section 6.5] and refer the
reader there for a discussion of the pinning that we use (although note that
their notation differs from ours in that what we call $G$, they call
$\check{G}$). In particular, let
$\check{T}\subseteq\check{B}\subseteq\check{G}$ denote the maximal torus and
Borel subgroup, $\check{B}^{-}\subseteq\check{G}$ the opposite Borel subgroup,
$\check{U}\subseteq\check{B}$ and $\check{U}^{-}\subseteq\check{B}^{-}$ the
unipotent radical and opposite unipotent radical, respectively. We have
canonical identifications $X^{*}(T)=X_{*}(\check{T})$ and
$X_{*}(T)=X^{*}(\check{T})$ under which $\check{\Phi}\subseteq X^{*}(T)$ and
$\Phi\subseteq X_{*}(T)$ correspond to the coroots and roots of $\check{G}$,
respectively. We abuse notation and let $I\subseteq\Delta$ denote the set of
simple coroots of $G$ (roots of $\check{G}$) corresponding a given subset
$I\subseteq\check{\Delta}$ of simple roots of $G$ (coroots of $\check{G}$).
Given a subset $I\subseteq\check{\Delta}$, we let
$\check{P}_{I}\subseteq\check{G}$, $\check{P}_{I}^{-}\subseteq\check{G}$,
$\check{V}_{I}\subseteq\check{P}_{I}$,
$\check{V}_{I}^{-}\subseteq\check{P}_{I}^{-}$,
$\check{L}_{I}=\check{P}_{I}\cap\check{P}^{-}_{I}$ denote the corresponding
parabolic subgroup, opposite parabolic subgroup, unipotent radical, opposite
unipotent radical, and Levi subgroup, respectively.

If the subset $I\subseteq\Delta$ is understood, we will sometimes drop the
subscript $I$ if it is unlikely to cause confusion; for example, we may write
$\check{L}$ in place of $\check{L}_{I}$.

Given a subset $I\subseteq\Delta$, we let
$i_{I}:\mathrm{Gr}_{I}:=\mathrm{Gr}_{L_{I}}\hookrightarrow\mathrm{Gr}_{G}$
denote the induced closed immersion of affine Grassmannians. When
$I=\emptyset$, we often write $\mathrm{Gr}_{T}$ in place of
$\mathrm{Gr}_{\emptyset}$ and $i_{T}$ in place of $i_{\emptyset}$.

####  1.6.2. Groups attached to the formal disc

As in the introduction, we write $G(\mathcal{O})$ for the arc group associated
to $G$ in place of the more technically correct notation $L^{+}G$ (where
$L^{+}G(R):=G(R[[t]])$), except when we are intentionally being more precise.
Similarly, we write $G(\mathcal{K})$ for the loop group $LG$ (where
$LG(R)=G(R((t)))$).

####  1.6.3. Group actions

All group actions are left actions. Given an action of an algebraic group $G$
on an affine variety $X$, the induced action of $G$ on the coordinate ring
$\mathcal{O}(X)$ is characterized by the formula $(g\cdot f)(x)=f(g^{-1}x)$
(where $g\in G$, $f\in\mathcal{O}(X)$ and $x\in X$). These conventions are
relevant for determining the gradings produced by $\mathbb{G}_{m}$-actions.
For example, suppose that $\mathbb{G}_{m}$ acts on the vector space
$\mathfrak{\check{t}}^{*}$ by the formula $t\cdot\xi=t^{-2}\xi$ (for
$t\in\mathbb{G}_{m}$, $\xi\in\mathfrak{\check{t}}^{*}$). Then,
$t\in\mathbb{G}_{m}$ acts on a linear form $x\in\mathfrak{\mathfrak{t}}$ by
$t\cdot x=t^{2}x$, since $(t\cdot
x)(\xi)=x(t^{-1}\cdot\xi)=x(t^{2}\xi)=t^{2}x(\xi)$ for any
$\xi\in\mathfrak{\check{t}}^{*}$. Therefore, this $\mathbb{G}_{m}$-action on
$\check{\mathfrak{t}}$ equips the polynomial algebra
$\mathrm{Sym}({\mathfrak{t}})\simeq\mathcal{O}(\check{\mathfrak{t}}^{*})$ with
the grading in which the generators have degree $+2$, i.e. the cohomological
grading on $H_{T}^{*}(\mathrm{pt},\mathbb{C})$.

####  1.6.4. Sheaf-theoretic notation

Let $H$ denote a complex algebraic group acting on a complex algebraic variety
$X$. We write $D_{H}(X):=D_{H}(X,\mathbb{C})$ for the triangulated
$H$-equivariant constructible derived category of sheaves of complex vector
spaces on the underlying topological space $X^{\mathrm{an}}(\mathbb{C})$, as
defined in [2, Definition 6.2.11]. We omit the superscript “$b$” on $D_{H}(X)$
as we will not have occasion to consider the non-constructible derived
category. Let $\mathcal{P}_{H}(X)\subseteq D_{H}(X)$ denote the heart of the
perverse $t$-structure, the abelian category of $H$-equivariant perverse
sheaves on $X$. Of course, we will also need to consider the situation in
which $X$ is a ind-scheme over $\mathbb{C}$ and $H$ is a pro-algebraic
$\mathbb{C}$-group. In this situation, we follow the conventions of [29, §2.2]
(see also [2, §9.1]) in defining $D_{H}(X)$. In particular, our convention is
that every object $F\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ of the spherical
Hecke category is supported on a closed stratum
$\mathrm{Gr}_{G}^{\leq\lambda}$ (for some $\lambda\in\Lambda^{+}$).

We have found it convenient to organize the constructions of this paper around
the “regular object” $\mathcal{F}_{\mathrm{reg}}$ (see Remark 2.2.1), the
“perverse sheaf” on $\mathrm{Gr}_{G}$ corresponding under the geometric Satake
equivalence
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\simeq\mathrm{Rep}(\check{G})$
to the (left) regular representation $\mathcal{O}(\check{G})$ of $\check{G}$.
Of course, under the conventions above, there is no such object in the
category $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ (just as the regular
representation $\mathcal{O}(\check{G})$ does not belong to the category
$\mathrm{Rep}(\check{G})$ of finite dimensional $\check{G}$-modules). It is
equally obvious that this observation does not pose a genuine issue. The least
technical solution is to simply define $\mathcal{F}_{\mathrm{reg}}$ as a
$\Lambda^{+}$-graded object of the abelian category
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ (in the sense of [29, §2.1.3])
through the formula

$\mathcal{F}_{\mathrm{reg}}:=\bigoplus_{\lambda\in\Lambda^{+}}\mathrm{IC}_{\lambda}\boxtimes
L(\lambda)^{*}.$

Here, $L(\lambda)\in\mathrm{Rep}(\check{G})$ is the simple $\check{G}$-module
of highest weight $\lambda\in\Lambda^{+}$ and
$\mathrm{IC}_{\lambda}\in\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$
denotes the corresponding irreducible perverse sheaf supported on
$\mathrm{Gr}_{G}^{\leq\lambda}:=\overline{\mathrm{Gr}^{\lambda}_{G}}$, where
$\mathrm{Gr}^{\lambda}_{G}=G(\mathcal{O})t^{\lambda}\subseteq\mathrm{Gr}_{G}$
is the corresponding spherical orbit. The corestriction
$i_{I}^{!}\mathcal{F}_{\mathrm{reg}}$ should then be understood as the
$\Lambda^{+}$-graded object

$i_{I}^{!}\mathcal{F}_{\mathrm{reg}}:=\bigoplus_{\lambda\in\Lambda^{+}}i_{I}^{!}\mathrm{IC}_{\lambda}\boxtimes
L(\lambda)^{*}$

of $D_{L_{I}(\mathcal{O})}(\mathrm{Gr}_{I})$. However, we have chosen to
completely suppress this ultimately notational issue in the body of the paper,
and will instead refer to $\mathcal{F}_{\mathrm{reg}}$ as an object of
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$; the concerned reader should
have no issue in making this abuse of notation precise. Similarly, we will
refer to the regular representation $\mathcal{O}(\check{G})$ as an object of
$\mathrm{Rep}(\check{G})$.

###  1.7. Acknowledgements

We thank David Nadler, Tsao-Hsien Chen, John O’Brien, Jeremy Taylor, Yixuan
Li, Peter Haine, Guglielmo Nocera, and Connor Halleck-Dube for many useful
comments and discussions. We are especially grateful for the extensive
mathematical and academic support that we have received from David Nadler over
the past four years.

Of course, this paper was directly motivated by the works of Ginzburg-Riche
[20] and Ginzburg-Kazhdan [19] and borrows heavily from their ideas. Moreover,
the foundational works of Achar [2] and Achar-Riche [3] were invaluable in
preparing this work.

The author was supported by an NSF Graduate Research Fellowship during the
writing of this paper.

##  2\. Automorphic Side

###  2.1. Equivariant localization

We begin on the “geometric side” of Theorem 1.5.2. Let
$I\subseteq\check{\Delta}$ denote a set of simple roots of $G$. Let
$\lambda\in\Lambda^{+}$ denote a dominant coweight and let
$\mathrm{IC}_{\lambda}\in\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ denote
the corresponding irreducible perverse sheaf supported on
$\mathrm{Gr}_{G}^{\leq\lambda}:=\overline{\mathrm{Gr}^{\lambda}_{G}}$, where
$\mathrm{Gr}^{\lambda}_{G}=G(\mathcal{O})t^{\lambda}\subseteq\mathrm{Gr}_{G}$
is the corresponding spherical orbit. Let
$\mathrm{Gr}_{I}=\mathrm{Gr}_{L_{I}}$ denote the affine Grassmannian of the
Levi subgroup $L_{I}\subseteq G$ and
$i_{I}:\mathrm{Gr}_{I}\hookrightarrow\mathrm{Gr}_{G}$ the inclusion. We will
study the cohomology
$H^{*}_{L_{I}(\mathcal{O})}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})$
through the technique of equivariant localization. We abuse notation and write
$i_{I}^{!}\mathrm{IC}_{\lambda}\in D_{T}(\mathrm{Gr}_{L})$ for the image of
$i_{I}^{!}\mathrm{IC}_{\lambda}\in D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ under
the functor forgetting the $L(\mathcal{O})$-equivariance down to
$T$-equivariance. Recall from [21] (especially the discussion surrounding [21,
Theorem 1.6.2]) that a complex $F\in D_{T}(\mathrm{Gr}_{L})$ is equivariantly
formal if the spectral sequence

$E_{2}^{p,q}=H_{T}^{p}(\mathrm{pt},H^{q}(\mathrm{Gr}_{I},F))\implies
H^{p+q}_{T}(\mathrm{Gr}_{I},F)$

degenerates at $E_{2}$.

######  Proposition 2.1.1.

The complex $i_{I}^{!}\mathrm{IC}_{\lambda}\in D_{T}(\mathrm{Gr}_{L})$ is
equivariantly formal.

###### Proof.

For each point $x\in\mathrm{Gr}_{I}$, we have that
$H^{j}(i_{x}^{!}i_{I}^{!}\mathrm{IC}_{\lambda})\simeq
H^{j}(i_{x}^{!}\mathrm{IC}_{\lambda})$ vanishes for $j\not\equiv\langle
2\check{\rho},\lambda\rangle$ (mod 2), by the well-known parity vanishing
property of $\mathrm{IC}_{\lambda}$ [5, Lemma 4.5]. Since
$i^{!}\mathrm{IC}_{\lambda}$ is constructible with respect to an affine paving
of $\mathrm{Gr}_{I}$ (the Schubert stratification by the orbits of an Iwahori
subgroup $\mathcal{I}\subseteq L(\mathcal{O})$), a standard argument with the
Cousin spectral sequence implies that
$H^{j}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})$ vanishes for
$j\not\equiv\langle 2\check{\rho},\lambda\rangle$.

It is now clear from parity considerations that the spectral sequence

$E_{2}^{p,q}=H^{p}_{T}(\mathrm{pt},H^{q}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda}))\implies
H^{p+q}_{T}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})$

degenerates at $E_{2}$. ∎

######  Remark 2.1.2.

Proposition 2.1.1 is a special case of the following very simple but quite
general observation. Let $X$ denote a $T$-variety equipped with a $T$-stable
affine paving. Let $F\in D_{T}(X)$ denote a $T$-equivariant complex whose
underlying non-equivariant complex $\mathrm{For}^{T}(F)\in D(X)$ is $!$-parity
(in the language of [22], and with respect to the dimension pariversity
$\diamond$). Then, $F$ is equivariantly formal. Moreover, the property of
being $!$-parity is preserved by $!$-restriction to any $T$-stable closed
subvariety $i:Y\hookrightarrow X$. Thus, if $Y$ also carries a $T$-stable
affine paving, then $i^{!}F$ is equivariantly formal.

######  Remark 2.1.3.

Let $\mathrm{IC}_{\lambda}$ be equipped with its natural structure of
$T$-equivariant mixed Hodge module (the theory of equivariant mixed Hodge
modules is not very well documented, but see [1] for a treatment) on
$\mathrm{Gr}_{G}$, for which it is pure of weight 0. Note that
$i_{I}^{!}\mathrm{IC}_{\lambda}$ is not necessarily pure. Indeed,
$i^{!}_{I}\mathrm{IC}_{\lambda}$ is usually not a semisimple complex (which
implies that it is not pure, by the BBD decomposition theorem). Therefore, the
equivariant formality of Proposition 2.1.1 really requires the costalk parity
considerations used in the proof and does not follow from the proof of [21,
Theorem 14.1(7)]. On the other hand, the same argument works without change in
settings which lack a theory of weights but in which strong parity vanishing
results hold; for example, on the quaternionic affine Grassmannian of [12].

Recall (see [28, Section 3]) that $\mathrm{Gr}_{G}$ admits the Iwasawa
decomposition

$\mathrm{Gr}_{G}=\coprod_{\nu\in\Lambda}S_{\nu},$

where $S_{\nu}=U(\mathcal{K})\cdot t^{\nu}$ is the orbit of the group ind-
scheme $U(\mathcal{K})$ through the point $t^{\nu}\in\mathrm{Gr}_{G}$.
Following [15], we write $L^{-}B$ for the ind-affine group ind-scheme
representing the functor $R\mapsto L^{-}B(R)=B(R[t^{-1}])$ and
$L^{--}B\subseteq L^{-}B$ for the kernel of the reduction morphism
$L^{-}B\twoheadrightarrow B$ given by $t^{-1}\mapsto 0$. Then, by the
discussion in [3] surrounding Equation 1.2.6, there is a $T$-equivariant
isomorphism

(2.1.4) $L^{--}B\simeq S_{\nu}$

given on $\mathbb{C}$-points by $g\mapsto
t^{\nu}gG(\mathcal{O})\in\mathrm{Gr}_{G}$. Here, $T$ acts on $L^{--}B$ acts by
conjugation. In the proof of Proposition 2.1.5, we write $V=V_{I}$ for the
unipotent radical of $P=P_{I}$ to ease notation.

######  Proposition 2.1.5.

For $x\in\mathrm{Gr}_{G}$, let $T_{x}\subseteq T$ denote the stabilizer of $x$
in $T$. Assume that $\alpha|_{T_{x}}:T_{x}\rightarrow\mathbb{G}_{m}$ is a
dominant morphism for every $\alpha\in\Phi^{+}_{G}\setminus\Phi_{L}^{+}$.
Then, $x\in\mathrm{Gr}_{L}$.

###### Proof.

The point $x$ belongs to the semi-infinite orbit $S_{\nu}$ for a unique
$\nu\in\Lambda$. We will use the $T$-equivariant isomorphism $L^{--}B\simeq
S_{\nu}$ of Equation 2.1.4. Let $B^{\prime}=B\cap L$. We also have a
$T$-equivariant isomorphism of schemes $V\times B^{\prime}\simeq B$
($(v,b^{\prime})\mapsto vb^{\prime}$), which induces a $T$-equivariant
isomorphism $L^{--}V\times L^{--}B^{\prime}\simeq L^{--}B$. Since $B^{\prime}$
is a Borel subgroup of $L$, we also obtain a $T$-equivariant isomorphism
$L^{--}B^{\prime}\simeq S_{\nu}^{\prime}$, where $S_{\nu}^{\prime}$ is the
semi-infinite orbit in $\mathrm{Gr}_{L}$ through $t^{\nu}$. Hence, there is a
$T$-equivariant isomorphism $S_{\nu}\simeq L^{--}V\times S_{\nu}^{\prime}$.
Let $\pi_{1}:S_{\nu}\rightarrow L^{--}V$ denote the first projection. To show
that $x\in\mathrm{Gr}_{L}$, it suffices to show that $\pi_{1}(x)$ is the
identity element $e\in L^{--}V$. Since $\pi_{1}$ is $T$-equivariant, the
subgroup $T_{x}\subseteq T$ also fixes $\pi_{1}(x)$. Furthermore, we have a
$T$-equivariant isomorphism of ind-affine ind-schemes

$L^{--}V\simeq\prod_{\alpha\in\Phi^{+}_{G}\setminus\Phi^{+}_{L}}L^{--}U_{\alpha},$

where $U_{\alpha}\subseteq U$ is the root subgroup corresponding to the root
$\alpha$. Let $\pi_{\alpha}$ denote the projection onto the factor indexed by
$\alpha$. We have an isomorphism $L^{--}U_{\alpha}\simeq L^{--}\mathbb{A}^{1}$
under which the adjoint action of $T$ on $L^{--}U_{\alpha}$ is induced by the
homomorphism $\alpha:T\rightarrow\mathbb{G}_{m}$ and the natural scaling
action of $\mathbb{G}_{m}$ on $L^{--}\mathbb{A}^{1}$. Since $\alpha|_{T_{x}}$
dominates $\mathbb{G}_{m}$, we deduce that
$(L^{--}\mathbb{A}^{1})^{T_{x}}=(L^{--}\mathbb{A}^{1})^{\mathbb{G}_{m}}=\\{0\\}$.
Therefore, $\pi_{\alpha}(\pi_{1}(x))=\pi_{\alpha}(e)\in L^{--}U_{\alpha}$ for
each $\alpha\in\Phi^{+}_{G}\setminus\Phi^{+}_{L}$, which implies that
$\pi_{1}(x)=e$, as claimed. ∎

We will now apply the localization theorem in $T$-equivariant cohomology. We
again refer the reader to [21] for a general treatment as well as [35, Theorem
A.1.13] for the specific statement that we use. Recall that we have a
canonical graded ring isomorphism

$H^{*}_{T}(\mathrm{pt},\mathbb{C})\simeq\operatorname{Sym}({\mathfrak{t}}^{*})$

obtained (for example) by identifying $H_{T}^{*}(\mathrm{pt},\mathbb{C})$ with
the cohomology $H^{*}(BT,\mathbb{C})$ of the (topological) classifying space
of $T$. Here, the elements of $\mathfrak{t}^{*}$ are placed in graded degree
2.

Let $i_{T}:\mathrm{Gr}_{T}\hookrightarrow\mathrm{Gr}_{G}$ denote the inclusion
of the Grassmannian of $T$. We use the same symbol to denote the inclusion
$\mathrm{Gr}_{T}\hookrightarrow\mathrm{Gr}_{I}$.

######  Proposition 2.1.6.

(a) Let

$f_{I}=\prod_{\alpha\in\Phi^{+}_{G}\setminus\Phi^{+}_{L}}\alpha\in\operatorname{Sym}{\mathfrak{h}^{*}}=H_{T}^{*}(\mathrm{pt}).$

The natural map

(2.1.7) $H^{*}_{T}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})\rightarrow
H^{*}_{T}(\mathrm{Gr}_{G},\mathrm{IC}_{\lambda})$

is an injective map of $H_{T}^{*}(\mathrm{pt},\mathbb{C})$-modules, and
becomes an isomorphism after inverting $f_{I}$.

(b) Let

$g_{I}=\prod_{\alpha\in\Phi_{L}^{+}}\alpha\in\operatorname{Sym}{\mathfrak{h}^{*}}=H_{T}^{*}(\mathrm{pt}).$

The natural map

$H_{T}^{*}(\mathrm{Gr}_{T},i_{T}^{!}\mathrm{IC}_{\lambda})\simeq
H_{T}^{*}(\mathrm{Gr}_{T},i_{T}^{!}i_{I}^{!}\mathrm{IC}_{\lambda})\rightarrow
H_{T}^{*}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})$

is an injective map of $H_{T}^{*}(\mathrm{pt},\mathbb{C})$-modules, and
becomes an isomorphism after inverting $g_{I}$.

###### Proof.

(a) Let $Z=\mathrm{Gr}^{\leq\lambda}_{G}$, $Z^{\prime}=Z\cap\mathrm{Gr}_{L}$,
and $U=Z\setminus Z^{\prime}$. By Proposition 2.1.5, $f_{I}$ vanishes on
$\mathrm{Lie}(T_{x})\subseteq\mathfrak{t}$ for any point $x\in
U\subseteq\mathrm{Gr}_{G}\setminus\mathrm{Gr}_{L}$ (at least one of the roots
$\alpha\in\Phi^{+}_{G}\setminus\Phi_{L}^{+}$ is trivial on $T_{x}$ since
$x\not\in\mathrm{Gr}_{I}$). Hence, by the equivariant localization theorem
(applied to the projective $T$-scheme $Z$, the $T$-stable closed subscheme
$Z^{\prime}$, and the complex $\mathrm{IC}_{\lambda}\in D_{T}(Z)$), say in the
form of [35, Theorem A.1.13], the morphism 2.1.7 becomes an isomorphism after
inverting $f_{I}$. By Proposition 2.1.1,
$H_{T}^{*}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})$ is a free
$H_{T}^{*}(\mathrm{pt},\mathbb{C})$-module, so the injectivity of the map
2.1.7 follows. The proof of (b) is similar, with the pair $(G,L)$ replaced by
$(L,T)$. ∎

######  Remark 2.1.8.

Suppose that the Levi subgroup $L=L_{I}$ is replaced by an arbitrary connected
reductive subgroup $K\subseteq G$. Let
$i:\mathrm{Gr}_{K}\hookrightarrow\mathrm{Gr}_{G}$ denote the induced closed
immersion. If $K$ contains a maximal torus $T$ of $G$, then the arguments of
this section carry over to show that the natural map

$H_{T}^{*}(\mathrm{Gr}_{K},i^{!}\mathrm{IC}_{\lambda})\rightarrow
H_{T}^{*}(\mathrm{Gr}_{G},\mathrm{IC}_{\lambda})$

is injective and becomes an isomorphism after inverting a suitable element of
$H_{T}^{*}(\mathrm{pt},\mathbb{C})$. However, without an analysis of the
$T$-stabilizers of points $x\in\mathrm{Gr}_{G}\setminus\mathrm{Gr}_{K}$
analogous to Proposition 2.1.5, it is not clear which element of
$H_{T}^{*}(\mathrm{pt},\mathbb{C})$ must be inverted.

######  Remark 2.1.9.

Proposition 2.1.6 can be understood in geometric terms as follows. The
morphisms

$H^{*}_{T}(\mathrm{Gr}_{T},i_{T}^{!}\mathrm{IC}_{\lambda})\rightarrow
H_{T}^{*}(\mathrm{Gr}_{I},i_{I}^{!}\mathrm{IC}_{\lambda})\rightarrow
H^{*}_{T}(\mathrm{Gr}_{G},\mathrm{IC}_{\lambda})$

of free
$H_{T}^{*}(\mathrm{pt},\mathbb{C})\simeq\operatorname{Sym}(\mathfrak{t}^{*})$-modules
can be viewed as morphisms of (trivial) vector bundles over the affine space
$\operatorname{Spec}{\operatorname{Sym}(\mathfrak{t}^{*})}=\mathfrak{t}$.
Then, the second morphism restricts to an isomorphism away from the walls of
the root system of $\check{G}$ which are not walls of the root system of
$\check{L}_{I}$. On the other hand, the first morphism restricts to an
isomorphism away from the walls of the root system of $\check{L}_{I}$.

###  2.2. Monoidal structure on corestriction

We will write $L=L_{I}$ and $i=i_{I}$ for now. We emphasize that the
isomorphism of Theorem 1.5.2 is an isomorphism of algebras over the ring
$R_{L}:=H_{L}^{*}(\mathrm{pt},\mathbb{C})$. However, our proof of Theorem
1.5.2 constructs this isomorphism of $R_{L}$-modules without any appeal to the
existence of an algebra structure on the left hand side. The existence of a
ring structure on
$H^{*}_{L(\mathcal{O})}(\mathrm{Gr}_{L},i^{!}\mathcal{F}_{\mathrm{reg}})$
appeared to us as an unexpected consequence of Theorem 1.5.2, since the
spectral side $\mathcal{O}(T^{*}(\check{G}/(\check{U},\psi_{I})))$ is
naturally a ring under the pointwise multiplication of regular functions.

The purpose of this section is therefore to equip
$H^{*}_{L(\mathcal{O})}(\mathrm{Gr}_{L},i^{!}\mathcal{F}_{\mathrm{reg}})$ with
an $R_{L}$-algebra structure defined entirely on the geometric side. In the
proof of Theorem 1.5.2, we will show that our identification
$H_{L(\mathcal{O})}^{*}(\mathrm{Gr}_{L},i^{!}\mathcal{F}_{\mathrm{reg}})\simeq\mathcal{O}(T^{*}(\check{G}/(\check{U},\psi_{I})))$
is in fact an isomorphism of $R_{L}$-algebras.

We note that the existence of an algebra structure on
$H^{*}_{L(\mathcal{O})}(\mathrm{Gr}_{L},i^{!}\mathcal{F}_{\mathrm{reg}})$ is
asserted in [11, Section 5(xi)]. However, we are unaware of a complete
construction, so we will provide two in this section (and we will use both).
The reader may wish to skim §§2.6 to see how the results of this section fit
together before reading further.

######  Remark 2.2.1.

Consider the case $L=G$. The $R_{G}:=H_{G}^{*}(\mathrm{pt})$-module
$H^{*}_{G(\mathcal{O})}(\mathcal{F}_{\mathrm{reg}})$ carries an evident
$R_{G}$-algebra structure which was extensively exploited in [4]. It also
plays a prominent role in the construction of Coulomb branches in [11]. Under
the geometric Satake equivalence
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\simeq\operatorname{Rep}(\check{G})$,
$\mathcal{F}_{\mathrm{reg}}$ corresponds to the left regular representation
$\mathcal{O}(\check{G})$. The pointwise multiplication of regular functions
equips $\mathcal{O}(\check{G})$ with the structure of a ring object in the
monoidal category $\operatorname{Rep}(\check{G})$. Passing back along
geometric Satake to perverse sheaves, we find that
$\mathcal{F}_{\mathrm{reg}}$ carries a natural ring structure in the monoidal
category $\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$.

Recall that the fiber functor
$H^{*}:\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow\mathrm{Mod}(\mathbb{C})$
carries a natural monoidal structure [28, Lemma 6.1]. This construction of
Mirković-Vilonen can be adapted (as in [34, Section 2.3] and [34, Remark 2.5])
to produce a monoidal structure on the equivariant cohomology functor
$H^{*}_{G(\mathcal{O})}:\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow\operatorname{Mod}(R_{G})$.
Since a monoidal functor takes ring objects to ring objects, we find a ring
structure on
$H_{G(\mathcal{O})}^{*}(\mathrm{Gr}_{G},\mathcal{F}_{\mathrm{reg}})\in\operatorname{Mod}(R_{G})$.

######  Remark 2.2.2.

We will generalize Remark 2.2.1 by equipping the object
$i^{!}\mathcal{F}_{\mathrm{reg}}\in D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ with
a ring structure. However, we emphasize that the functor
$i^{!}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ is not monoidal. For example, consider
the case $L=T$. The functor

$i_{0}^{!}:D_{T(\mathcal{O})}(\mathrm{Gr}_{T})\rightarrow\mathrm{Mod}(R_{T})$

of costalk at the basepoint $t^{0}\in\mathrm{Gr}_{T}$ is in fact monoidal.
However, the composition $i_{0}^{!}\circ
i^{!}\simeq\mathrm{Hom}_{T(\mathcal{O})}(\mathrm{IC}_{0},-)$ does not admit a
monoidal structure: its restriction to
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})\simeq\mathrm{Rep}(\check{G})$
is naturally isomorphic to the functor
$V\mapsto(V\otimes\mathrm{Sym}(\check{\mathfrak{g}}/\check{\mathfrak{u}}))^{\check{B}}$
(by the main result of [20], recalled above as Theorem 1.4.4). In particular,
$i^{!}$ does not admit a monoidal structure.

Therefore, we will only aim to establish the weaker (but sufficient) result
that $i^{!}$ admits a natural lax monoidal structure. This will suffice: lax
monoidal functors carry algebras to algebras, so
$i^{!}\mathcal{F}_{\mathrm{reg}}$ will acquire a ring structure in
$D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$.

######  Remark 2.2.3.

It is surely possible to give a more elegant construction of the lax monoidal
structure on $i^{!}$ (even at the “cochain level”) by appealing to a suitable
theory of sheaves and correspondences, such as that of [26, Section A.5].
However, as our Construction 2.2.14 makes essential use of non-invertible
natural transformations (specifically, base change morphisms arising from non-
Cartesian commutative squares), the kind of formalism described in [26] is not
sufficient. It would be necessary to use a theory like that of [17]. However,
as we do not wish to make our relatively pedestrian computations dependent on
the (only partially documented) machinery of $(\infty,2)$-category theory, we
will proceed directly.

######  Remark 2.2.4.

In the remainder of this section, we will allow the Levi subgroup $L\subseteq
G$ to be replaced by an arbitrary connected reductive subgroup of $G$.
However, our only use case in this paper is when $L$ is a Levi subgroup.

######  Remark 2.2.5.

Verdier duality defines an equivalence of categories

$\mathbb{D}_{G}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\simeq
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})^{\mathrm{op}}.$

For objects $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, there is moreover a
natural isomorphism

$\mathbb{D}_{G}(A\star B)\simeq\mathbb{D}_{G}(A)\star\mathbb{D}_{G}(B).$

See [2, Lemma 7.2.8] for the analogous fact about the convolution of
$B$-equivariant sheaves on the flag variety $G/B$; the proof in our setting is
identical. We also have a natural isomorphism $\mathbb{D}_{L}\circ i^{!}\simeq
i^{*}\circ\mathbb{D}_{G}$. Therefore, the problem of equipping $i^{!}$ with a
lax monoidal structure is formally equivalent to that of equipping $i^{*}$
with a colax monoidal structure (and that is what we will do).

######  Remark 2.2.6.

In order to fix notation, we recall the definition of the convolution product
on $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ (from [28, Section 4]), together with
its associativity constraint (described briefly in [28, Proposition 4.6]). Let
$\mathrm{Gr}_{G}^{(2)}=G(\mathcal{K})\times^{G(\mathcal{O})}\mathrm{Gr}_{G}=:\mathrm{Gr}_{G}\operatorname{\tilde{\times}}\mathrm{Gr}_{G}$
denote the convolution Grassmannian. Let
$m:\mathrm{Gr}_{G}^{(2)}\rightarrow\mathrm{Gr}_{G}$ denote the multiplication
map. We denote by
$-\operatorname{\widetilde{\boxtimes}}-:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(2)})$ the twisted external product.

Similarly, we let

$\mathrm{Gr}_{G}^{(3)}=G(\mathcal{K})\times^{G(\mathcal{O})}G(\mathcal{K})\times^{G(\mathcal{O})}\mathrm{Gr}_{G}=:\mathrm{Gr}_{G}\operatorname{\tilde{\times}}\mathrm{Gr}_{G}\operatorname{\tilde{\times}}\mathrm{Gr}_{G}$

denote the threefold convolution Grassmannian. Let
$m^{23}:\mathrm{Gr}_{G}^{(3)}\rightarrow\mathrm{Gr}_{G}^{(2)}$ denote the
action map (given by
$m^{23}(g_{1},g_{2},g_{3}G(\mathcal{O}))=(g_{1},g_{2}g_{3}G(\mathcal{O}))$),
and let $m^{12}:\mathrm{Gr}_{G}^{(3)}\rightarrow\mathrm{Gr}_{G}^{(2)}$ denote
the multiplication of the first two factors (given by
$m^{12}(g_{1},g_{2},g_{3}G(\mathcal{O}))=(g_{1}g_{2},g_{3}G(\mathcal{O}))$).
We also have the twisted external product functors:

$\displaystyle-\operatorname{\widetilde{\boxtimes}}-$
$\displaystyle:D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(2)})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(3)})$
$\displaystyle-\operatorname{\widetilde{\boxtimes}}-$
$\displaystyle:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(2)})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(3)})$

Finally, there is a triple multiplication map

$m^{123}:\mathrm{Gr}_{G}^{(3)}\rightarrow\mathrm{Gr}_{G}$

given by
$m^{123}(g_{1},g_{2},g_{3}G(\mathcal{O}))=g_{1}g_{2}g_{3}G(\mathcal{O})$, and
a triple external product

$-\operatorname{\widetilde{\boxtimes}}-\operatorname{\widetilde{\boxtimes}}-:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(3)}).$

The convolution product of $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ is
defined by

$A\star B:=m_{*}(A\operatorname{\widetilde{\boxtimes}}B).$

The triple convolution product of $A,B,C\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ is similarly defined by

$A\star B\star
C=m^{123}_{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C).$

The associativity constaint $\alpha_{A,B,C}$ is a trifunctorial isomorphism

(2.2.7) $\alpha_{A,B,C}:(A\star B)\star C\simeq A\star(B\star C)$

defined as a composition
$\alpha_{A,B,C}=\gamma_{A,B,C}^{-1}\circ\beta_{A,B,C}$, where $\beta_{A,B,C}$
is a natural isomorphism

(2.2.8) $\beta_{A,B,C}:(A\star B)\star C\simeq A\star B\star C$

and $\gamma_{A,B,C}$ is a natural isomorphism

$\gamma_{A,B,C}:A\star(B\star C)\simeq A\star B\star C.$

We spell out the definition of $\beta_{A,B,C}$ (that of $\gamma_{A,B,C}$ is
analogous). We have $m^{123}=m\circ m^{12}$. Hence, we have a compositional
isomorphism $m^{123}_{*}\simeq m_{*}\circ m_{*}^{12}$. It defines an
isomorphism

$\mathrm{comp}_{A,B,C}^{12}:m_{*}m_{*}^{12}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)\simeq
m^{123}_{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)=A\star
B\star C.$

The next ingredient is the associativity constraint on the twisted external
product

$\delta_{A,B,C}:(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C\simeq
A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C.$

Applying $m^{12}_{*}$ yields an isomorphism

$m^{12}_{*}(\delta_{A,B,C}):m^{12}_{*}((A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)\simeq
m_{*}^{12}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C).$

For any objects $E\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(2)})$, $F\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, we have an isomorphism

(2.2.9) $\epsilon_{E,F}:m_{*}E\operatorname{\widetilde{\boxtimes}}F\simeq
m^{12}_{*}(E\operatorname{\widetilde{\boxtimes}}F).$

By definition, $\epsilon_{E,F}$ is obtained by adjunction from the composition
of the evident natural maps

$m^{12,*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)\xrightarrow{\xi^{12}_{E,F}}m^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}F\xrightarrow{\eta_{E}\operatorname{\widetilde{\boxtimes}}1}E\operatorname{\widetilde{\boxtimes}}F.$

We may now take $E=A\operatorname{\widetilde{\boxtimes}}B$ and $F=C$ to obtain
the isomorphism

$\epsilon_{A,B,C}:=\epsilon_{A\operatorname{\widetilde{\boxtimes}}B,C}:m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C\simeq
m_{*}^{12}((A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C).$

Let

(2.2.10)
$\sigma_{A,B,C}=m_{*}(m_{*}^{12}(\delta_{A,B,C})\circ\epsilon_{A,B,C}).$

The isomorphism $\beta_{A,B,C}$ is now defined to be the composition

$\beta_{A,B,C}=\mathrm{comp}^{12}_{A,B,C}\circ\sigma_{A,B,C}.$

When dealing with both groups $G$ and $L$, we will generally use the same
notation for the analogous maps and isomorphisms.

######  Remark 2.2.11.

In addition to the morphism $i:\mathrm{Gr}_{L}\rightarrow\mathrm{Gr}_{G}$, we
obtain morphisms $i_{2}:\mathrm{Gr}^{(2)}_{L}\rightarrow\mathrm{Gr}^{(2)}_{G}$
and $i_{3}:\mathrm{Gr}^{(3)}_{L}\rightarrow\mathrm{Gr}^{(3)}_{G}$. We have the
identities $m\circ i_{2}=i\circ m$, $m^{12}\circ i_{3}=i_{2}\circ m^{12}$,
$m^{23}\circ i_{3}=i_{2}\circ m^{23}$, and $m^{123}\circ i_{3}=i\circ
m^{123}$. We also have evident natural isomorphisms

(2.2.12) $\displaystyle\omega_{A,B}$
$\displaystyle:i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\simeq
i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B$ (2.2.13)
$\displaystyle\omega_{A,B,C}$
$\displaystyle:i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)\simeq
i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B\operatorname{\widetilde{\boxtimes}}i^{*}C.$

######  Construction 2.2.14.

We can now define the (non-unital part of the) colax monoidal structure on
$i^{*}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$. For objects $A,B\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, we must define a bifunctorial map

$\theta_{A,B}:i^{*}(A\star B)\rightarrow i^{*}A\star i^{*}B.$

By definition, $A\star B=m_{*}(A\operatorname{\widetilde{\boxtimes}}B)$. Since
$m\circ i_{2}=i\circ m$, we deduce the existence of a base change morphism
$\rho:i^{*}m_{*}\rightarrow m_{*}i_{2}^{*}$. Hence, we have a natural map

(2.2.15)
$\rho_{A,B,C}:=\rho_{A\operatorname{\widetilde{\boxtimes}}B}:i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\rightarrow
m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B).$

We also have the natural isomorphism
$\omega_{A,B}:i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\simeq
i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B$. Thus, we can define

(2.2.16) $\theta_{A,B}=m_{*}(\omega_{A,B})\circ\rho_{A,B,C}$

The proof of the following proposition is the price that we pay for Remark
2.2.3, i.e. our choice to construct $\theta_{A,B}$ directly without the use of
a higher categorical machinery that would presumably encode the commutativity
of the diagrams below implicitly. The reader is encouraged to skip it on a
first reading.

######  Proposition 2.2.17.

The natural transformation $\theta:i^{*}(-\star-)\rightarrow i^{*}(-)\star
i^{*}(-)$ of functors $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{G})$ of Construction 2.2.14 is compatible with
the associativity constraints of Remark 2.2.6 underlying the monoidal
categories $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ and
$D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$. That is, for objects $A,B,C\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, the following diagram in
$D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ commutes:

${i^{*}((A\star B)\star C)}$${i^{*}(A\star(B\star C))}$${i^{*}(A\star B)\star
i^{*}C}$${i^{*}A\star i^{*}(B\star C)}$${(i^{*}A\star i^{*}B)\star
i^{*}C}$${i^{*}A\star(i^{*}B\star
i^{*}C)}$$\scriptstyle{i^{*}\alpha_{A,B,C}}$$\scriptstyle{\theta_{A\star
B,C}}$$\scriptstyle{\theta_{A,B\star
C}}$$\scriptstyle{\operatorname{\theta}_{A,B}\star\operatorname{id}}$$\scriptstyle{\operatorname{id}\star\operatorname{\theta}_{B,C}}$$\scriptstyle{\alpha_{i^{*}A,i^{*}B,i^{*}C}}$

###### Proof.

We expand the diagram horizontally, using the definition (2.2.7) of the
associativity isomorphism $\alpha$:

${i^{*}((A\star B)\star C)}$${i^{*}(A\star B\star C)}$${i^{*}(A\star(B\star
C))}$${i^{*}(A\star B)\star i^{*}C}$${i^{*}A\star i^{*}(B\star
C)}$${(i^{*}A\star i^{*}B)\star i^{*}C}$${i^{*}A\star i^{*}B\star
i^{*}C}$${i^{*}A\star(i^{*}B\star
i^{*}C)}$$\scriptstyle{i^{*}\beta_{A,B,C}}$$\scriptstyle{\theta_{A\star
B,C}}$$\scriptstyle{i^{*}\gamma_{A,B,C}^{-1}}$$\scriptstyle{\theta_{A,B,C}}$$\scriptstyle{\theta_{A,B\star
C}}$$\scriptstyle{\operatorname{\theta}_{A,B}\star\operatorname{id}}$$\scriptstyle{\operatorname{id}\star\operatorname{\theta}_{B,C}}$$\scriptstyle{\beta_{i^{*}A,i^{*}B,i^{*}C}}$$\scriptstyle{\gamma_{i^{*}A,i^{*}B,i^{*}C}^{-1}}$

It suffices to show that both of the small rectangles in this diagram commute.
We will show that the one on the left commutes; the proof that the one on the
right commutes is completely analogous. We now expand the diagram on the left,
using the definitions (2.2.16) and (2.2.8) of $\theta$ and $\beta$,
respectively:

${i^{*}m_{*}(m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${i^{*}m_{*}m_{*}^{12}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${i^{*}m^{123}_{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}i_{2}^{*}(m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}i_{2}^{*}m_{*}^{12}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}(i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C)}$${m_{*}m_{*}^{12}i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}^{123}i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}(m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C)}$${m_{*}(m_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)\operatorname{\widetilde{\boxtimes}}i^{*}C)}$${m_{*}m_{*}^{12}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B\operatorname{\widetilde{\boxtimes}}i^{*}C)}$${m^{123}_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B\operatorname{\widetilde{\boxtimes}}i^{*}C)}$$\scriptstyle{i^{*}m_{*}\sigma_{A,B,C}}$$\scriptstyle{\rho_{m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C}}$$\scriptstyle{\mathrm{comp}^{12}_{A,B,C}}$$\scriptstyle{\rho_{m^{12}_{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}}$$\scriptstyle{\rho^{123}_{A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C}}$$\scriptstyle{m_{*}\omega_{m_{*}(A\operatorname{\widetilde{\boxtimes}}B),C}}$$\scriptstyle{m_{*}i_{2}^{*}\sigma_{A,B,C}}$$\scriptstyle{m_{*}\rho^{12,3}_{A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C}}$$\scriptstyle{m_{*}(\rho_{A,B}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{\mathrm{comp}^{12}_{i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}}$$\scriptstyle{m_{*}m_{*}^{12}\omega_{A,B,C}}$$\scriptstyle{m^{123}_{*}\omega_{A,B,C}}$$\scriptstyle{m_{*}(m_{*}\omega_{A,B}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{m_{*}\sigma_{i^{*}A,i^{*}B,i^{*}C}}$$\scriptstyle{\mathrm{comp}_{i^{*}A,i^{*}B,i^{*}C}^{12}}$

Here, we have introduced the base change maps
$\rho^{12,3}:i_{2}^{*}m_{*}^{12}\rightarrow m_{*}^{12}i_{3}^{*}$ and
$\rho^{123}:i^{*}m_{*}^{123}\rightarrow m_{*}^{123}i_{3}^{*}$ deduced from the
equalities $i_{2}m^{12}=m^{12}i_{3}$ and $im^{123}=m^{123}i_{3}$,
respectively. The upper left rectangle commutes by the naturality of the base
change transformation $\rho:i^{*}m_{*}\rightarrow m_{*}i_{2}^{*}$ of (2.2.15)
applied to the isomorphism $\sigma_{A,B,C}$. The upper right rectangle
commutes by the compatiblity of compositional isomorphisms with base change
(see Proposition 2.11.7 of [2] and note that the Cartesian hypothesis is
superfluous). The lower right rectangle commutes by the naturality of the
compositional isomorphism $\mathrm{comp}^{12}:m_{*}m_{*}^{12}\simeq
m_{*}^{123}$ applied to the isomorphism $\omega_{A,B,C}$.

It therefore suffices to show that the lower left rectangle commutes. It is
obtained by applying $m_{*}$ to the the following diagram:

${i_{2}^{*}(m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${i_{2}^{*}m_{*}^{12}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${m_{*}^{12}i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${m_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${m_{*}^{12}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B\operatorname{\widetilde{\boxtimes}}i^{*}C)}$$\scriptstyle{\omega_{m_{*}(A\operatorname{\widetilde{\boxtimes}}B),C}}$$\scriptstyle{i_{2}^{*}\sigma_{A,B,C}}$$\scriptstyle{\rho^{12,3}_{A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C}}$$\scriptstyle{\rho_{A,B}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{m_{*}^{12}\omega_{A,B,C}}$$\scriptstyle{m_{*}\omega_{A,B}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{\sigma_{i^{*}A,i^{*}B,i^{*}C}}$

To show that the above diagram commutes, we expand it horizontally using the
definition (2.2.10) of $\sigma$ to obtain:

${i_{2}^{*}(m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${i_{2}^{*}m_{*}^{12}((A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${i_{2}^{*}m_{*}^{12}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${(\star)}$${m_{*}^{12}i_{3}^{*}((A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}^{12}i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${m_{*}^{12}(i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C)}$${m_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${m_{*}^{12}((i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)\operatorname{\widetilde{\boxtimes}}i^{*}C)}$${m_{*}^{12}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B\operatorname{\widetilde{\boxtimes}}i^{*}C)}$$\scriptstyle{\omega_{m_{*}(A\operatorname{\widetilde{\boxtimes}}B),C}}$$\scriptstyle{i_{2}^{*}\epsilon_{A,B,C}}$$\scriptstyle{i_{2}^{*}m_{*}^{12}\delta_{A,B,C}}$$\scriptstyle{\rho^{12,3}_{(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C}}$$\scriptstyle{\rho^{12,3}_{A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C}}$$\scriptstyle{\rho_{A,B}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{m^{12}_{*}i_{3}^{*}\delta_{A,B,C}}$$\scriptstyle{m_{*}^{12}\omega_{A\operatorname{\widetilde{\boxtimes}}B,C}}$$\scriptstyle{m_{*}^{12}\omega_{A,B,C}}$$\scriptstyle{m_{*}\omega_{A,B}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{\epsilon_{i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B),i^{*}C}}$$\scriptstyle{m_{*}^{12}(\omega_{A,B}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{\epsilon_{i^{*}A,i^{*}B,i^{*}C}}$$\scriptstyle{m_{*}^{12}\delta_{i^{*}A,i^{*}B,i^{*}C}}$

The upper right rectangle commutes by the naturality of the base change map
$\rho^{12,3}:i_{2}^{*}m_{*}^{12}\rightarrow m^{12}_{*}i^{*}_{3}$ applied to
the isomorphism $\delta_{A,B,C}$. The lower left rectangle commutes by the
naturality of the isomorphism
$\epsilon_{-,i^{*}C}:m_{*}(-)\operatorname{\widetilde{\boxtimes}}i^{*}C\simeq
m_{*}^{12}(-\operatorname{\widetilde{\boxtimes}}i^{*}C)$ of (2.2.9) applied to
the isomorphism
$\omega_{A,B}:i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\simeq
i_{2}^{*}A\operatorname{\widetilde{\boxtimes}}i_{2}^{*}B$ of (2.2.12). The
lower right rectangle commutes because it is obtained by applying the functor
$m_{*}^{12}$ to the commutative diagram:

${i_{3}^{*}((A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}C)}$${i_{3}^{*}(A\operatorname{\widetilde{\boxtimes}}B\operatorname{\widetilde{\boxtimes}}C)}$${i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)\operatorname{\widetilde{\boxtimes}}i^{*}C}$${i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B\operatorname{\widetilde{\boxtimes}}i^{*}C}$$\scriptstyle{i_{3}^{*}\delta_{A,B,C}}$$\scriptstyle{\omega_{A\operatorname{\widetilde{\boxtimes}}B,C}}$$\scriptstyle{\omega_{A,B,C}}$$\scriptstyle{\omega_{A,B}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{\delta_{i^{*}A,i^{*}B,i^{*}C}}$

It remains to show that the upper left rectangle $(\star)$ commutes, which
follows from the more general assertion that for $E\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}^{(2)})$, $F\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, the following diagram commutes:

${i_{2}^{*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${i_{2}^{*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)}$${i^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${m_{*}^{12}i_{3}^{*}(E\operatorname{\widetilde{\boxtimes}}F)}$${m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${m_{*}^{12}(i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$$\scriptstyle{\omega_{m_{*}E,F}}$$\scriptstyle{i_{2}^{*}\epsilon_{E,F}}$$\scriptstyle{\rho^{12,3}_{E\operatorname{\widetilde{\boxtimes}}F}}$$\scriptstyle{\rho_{E}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{m^{12}_{*}\omega_{E,F}}$$\scriptstyle{\epsilon_{i_{2}^{*}E,i^{*}F}}$

By adjunction (and the definition (2.2.9) of $\epsilon$, which is used below
to expand the bottom row horizontally), it suffices to show that the outer
rectangle in the following diagram commutes (note that the middle vertical
arrow $\mathrm{comp}$ refers to the compositional isomorphism
$m^{12,*}i_{2}^{*}\simeq i_{3}^{*}m^{12,*}$ evaluated on
$i_{2}^{*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)$):

${m^{12,*}i_{2}^{*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}i_{2}^{*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}i_{2}^{*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)}$${i_{3}^{*}m^{12,*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}m^{12}_{*}i_{3}^{*}(E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}(i^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$${i_{3}^{*}(E\operatorname{\widetilde{\boxtimes}}F)}$${i_{3}^{*}(E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}(m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$${m^{*}m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$$\scriptstyle{m^{12,*}\omega_{m_{*}E,F}}$$\scriptstyle{m^{12,*}i_{2}^{*}\epsilon_{E,F}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{m^{12,*}\rho^{12,3}_{E\operatorname{\widetilde{\boxtimes}}F}}$$\scriptstyle{i_{3}^{*}\eta^{12}_{E\operatorname{\widetilde{\boxtimes}}F}}$$\scriptstyle{\eta^{12}_{i_{3}^{*}(E\operatorname{\widetilde{\boxtimes}}F)}}$$\scriptstyle{m^{12,*}(\rho^{12}_{E}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{\omega_{E,F}}$$\scriptstyle{\xi^{12}_{m_{*}i_{2}^{*}E,i^{*}F}}$$\scriptstyle{\eta_{i_{2}^{*}E}\operatorname{\widetilde{\boxtimes}}1}$

The upper right rectangle commutes by the very definition of the base change
morphism $\rho^{12,3}_{E\operatorname{\widetilde{\boxtimes}}F}$. Thus, it
suffices to show that the outer rectangle in the following diagram commutes.

${m^{12,*}i_{2}^{*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}i_{2}^{*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}i_{2}^{*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)}$${i_{3}^{*}m^{12,*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${i_{3}^{*}m^{12,*}m_{*}^{12}(E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}(i^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$${i_{3}^{*}(m^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${i_{3}^{*}(E\operatorname{\widetilde{\boxtimes}}F)}$${i_{2}^{*}m^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}F}$${i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${m^{12,*}(m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$${m^{*}m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$$\scriptstyle{m^{12,*}\omega_{m_{*}E,F}}$$\scriptstyle{m^{12,*}i_{2}^{*}\epsilon_{E,F}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{i_{3}^{*}\xi^{12}_{m_{*}E,F}}$$\scriptstyle{i_{3}^{*}m^{12,*}\epsilon_{E,F}}$$\scriptstyle{i_{3}^{*}\eta^{12}_{E\operatorname{\widetilde{\boxtimes}}F}}$$\scriptstyle{m^{12,*}(\rho^{12}_{E}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{i_{3}^{*}(\eta_{E}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{\omega_{m^{*}m_{*}E,F}}$$\scriptstyle{\omega_{E,F}}$$\scriptstyle{i_{2}^{*}\eta_{E}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{\xi^{12}_{m_{*}i_{2}^{*}E,i^{*}F}}$$\scriptstyle{\eta_{i_{2}^{*}E}\operatorname{\widetilde{\boxtimes}}1}$

The upper right rectangle commutes by the naturality of the isomorphism
$\mathrm{comp}:m^{12,*}i_{2}^{*}\simeq i_{3}^{*}m^{12,*}$. The rectangle below
it commutes by ($i_{3}^{*}$ applied to) the definition (2.2.9) of
$\epsilon_{E,F}$. Hence, it suffices to show that the outer rectangle in the
following diagram commutes:

${m^{12,*}i_{2}^{*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${i_{3}^{*}m^{12,*}(m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${(\dagger)}$${i_{3}^{*}(m^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}F)}$${m^{12,*}(i_{2}^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$${m^{*}i_{2}^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${i_{2}^{*}m^{*}m_{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${m^{12,*}(m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F)}$${m^{*}m_{*}i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$${i_{2}^{*}E\operatorname{\widetilde{\boxtimes}}i^{*}F}$$\scriptstyle{m^{12,*}\omega_{m_{*}E,F}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{i_{3}^{*}\xi^{12}_{m_{*}E,F}}$$\scriptstyle{\omega_{m^{*}m_{*}E,F}}$$\scriptstyle{m^{12,*}(\rho^{12}_{E}\operatorname{\widetilde{\boxtimes}}1)}$$\scriptstyle{\xi^{12}_{i_{2}^{*}m_{*}E,i^{*}F}}$$\scriptstyle{\mathrm{comp}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{m^{*}\rho_{E}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{i_{2}^{*}\eta_{E}\operatorname{\widetilde{\boxtimes}}1}$$\scriptstyle{\xi^{12}_{m_{*}i_{2}^{*}E,i^{*}F}}$$\scriptstyle{\eta_{i_{2}^{*}E}\operatorname{\widetilde{\boxtimes}}1}$

The lower left rectangle commutes by the naturality of the isomorphism
$\xi^{12}:m^{12,*}(-\operatorname{\widetilde{\boxtimes}}-)\simeq
m^{*}(-)\operatorname{\widetilde{\boxtimes}}-$ applied to the map
$\rho_{E}^{12}\operatorname{\widetilde{\boxtimes}}1$. The lower right
rectangle is obtained by applying the functor
$-\operatorname{\widetilde{\boxtimes}}i^{*}F$ to the diagram

${m^{*}i_{2}^{*}m_{*}E}$${i_{2}^{*}m^{*}m_{*}E}$${m^{*}m_{*}i_{2}^{*}E}$${i_{2}^{*}E.}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{m^{*}\rho^{12}_{E}}$$\scriptstyle{i_{2}^{*}\eta_{E}}$$\scriptstyle{\eta_{i_{2}^{*}E}}$

This square commutes by the very definition of the base change map
$\rho^{12}_{E}:i_{2}^{*}m_{*}E\rightarrow m_{*}i_{2}^{*}E$. We must verify the
commutativity of the upper rectangle $(\dagger)$. The claim is that for
$X,Y\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, the following diagram commutes.

${m^{12,*}i_{2}^{*}(X\operatorname{\widetilde{\boxtimes}}Y)}$${i_{3}^{*}m^{12,*}(X\operatorname{\widetilde{\boxtimes}}Y)}$${i_{3}^{*}(m^{*}X\operatorname{\widetilde{\boxtimes}}Y)}$${m^{12,*}(i_{2}^{*}X\operatorname{\widetilde{\boxtimes}}Y)}$${m^{*}i_{2}^{*}X\operatorname{\widetilde{\boxtimes}}i^{*}Y}$${i_{2}^{*}m^{*}X\operatorname{\widetilde{\boxtimes}}i^{*}Y}$$\scriptstyle{m^{12,*}\omega_{X,Y}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{i_{3}^{*}\xi^{12}_{X,Y}}$$\scriptstyle{\omega_{m^{*}X,Y}}$$\scriptstyle{\xi^{12}_{i_{2}^{*}X,i^{*}Y}}$$\scriptstyle{\mathrm{comp}\operatorname{\widetilde{\boxtimes}}1}$

Checking the commutativity of this diagram is left to the reader (reduce to
verifying the commutativity of the corresponding diagram with untwisted
external products, which follows from a straightforward compatiblity between
the isomorphism $f^{*}(-\otimes-)\simeq f^{*}(-)\otimes f^{*}(-)$ and the
compositional isomorphisms $(gf)^{*}\simeq f^{*}g^{*}$). ∎

Next, we turn to the following problem. Suppose that $M\subseteq L$ is a
connected reductive subgroup of $L$ (the reader should keep in the mind the
case in which $M\subseteq L$ is an inclusion of Levi subgroups of $G$; for
instance, when $M=T$ is a maximal torus). Let
$j:\mathrm{Gr}_{M}\hookrightarrow\mathrm{Gr}_{L}$ denote the induced map on
affine Grassmannians. We also have the composition $k=i\circ
j:\mathrm{Gr}_{M}\hookrightarrow\mathrm{Gr}_{G}$. In Construction 2.2.14, we
equipped the functors $i^{*}$, $j^{*}$, and $k^{*}$ with (non-unital, for now)
colax monoidal structures. On the other hand, we have a canonical isomorphism
$k^{*}\simeq j^{*}\circ i^{*}$. We would like to show that this natural
isomorphism is an isomorphism of (non-unital) colax monoidal functors.

######  Proposition 2.2.18.

Let $j:\mathrm{Gr}_{M}\hookrightarrow\mathrm{Gr}_{L}$,
$i:\mathrm{Gr}_{L}\hookrightarrow\mathrm{Gr}_{G}$, and $k=i\circ
j:\mathrm{Gr}_{M}\hookrightarrow\mathrm{Gr}_{G}$ be as above. Equip the
functors $i^{*}$, $j^{*}$, and $k^{*}$ with the (non-unital) colax monoidal
structures of Construction 2.2.14. Then, the compositional isomorphism
$k^{*}\simeq j^{*}\circ i^{*}$ is an isomorphism of (non-unital) colax
monoidal functors. More precisely, let $A,B\in
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$. Then, the following diagram in
$D_{M(\mathcal{O})}(\mathrm{Gr}_{M})$ commutes:

(2.2.19) ${j^{*}i^{*}(A\star B)}$${j^{*}(i^{*}A\star
i^{*}B)}$${j^{*}i^{*}A\star j^{*}i^{*}B}$${k^{*}(A\star B)}$${k^{*}A\star
k^{*}B}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{j^{*}\theta_{A,B}}$$\scriptstyle{\theta_{i^{*}A,i^{*}B}}$$\scriptstyle{\operatorname{comp}\star\operatorname{comp}}$$\scriptstyle{\theta_{A,B}}$

###### Proof.

We will notationally suppress all composition isomorphisms in the following
argument (there are several that appear, but there is never ambiguity about
which is meant). We can more explicitly express the above diagram (2.2.19) as
follows.

(2.2.20)
${j^{*}i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${j^{*}m_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)}$${m_{*}(j^{*}i^{*}A\operatorname{\widetilde{\boxtimes}}j^{*}i^{*}B)}$${k^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}(k^{*}A\operatorname{\widetilde{\boxtimes}}k^{*}B)}$$\scriptstyle{j^{*}\theta_{A,B}}$$\scriptstyle{\theta_{i^{*}A,i^{*}B}}$$\scriptstyle{\theta_{A,B}}$

The following diagram commutes by the compatibility of composition with base
change (Proposition 2.11.7 of [2]).

(2.2.21)
${j^{*}i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${j^{*}m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}j_{2}^{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${k^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}k_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$$\scriptstyle{j^{*}\rho_{A\operatorname{\widetilde{\boxtimes}}B}}$$\scriptstyle{\rho_{i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}}$$\scriptstyle{\rho_{A\operatorname{\widetilde{\boxtimes}}B}}$

The following diagram commutes because it is obtained by applying the functor
$m_{*}$ to the diagram which witnesses the compatibility of $\omega_{A,B}$
with composition (see the last step in the proof of Proposition 2.2.17).

(2.2.22)
${m_{*}j_{2}^{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}j_{2}^{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)}$${m_{*}(j^{*}i^{*}A\operatorname{\widetilde{\boxtimes}}j^{*}i^{*}B)}$${m_{*}k_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}(k^{*}A\operatorname{\widetilde{\boxtimes}}k^{*}B).}$$\scriptstyle{m_{*}\omega_{A,B}}$$\scriptstyle{m_{*}\omega_{i^{*}A,i^{*}B}}$$\scriptstyle{m_{*}\omega_{A,B}}$

Pasting diagrams (2.2.21) and (2.2.22) together horizontally yields a
commutative rectangle in which the left vertical, right vertical, and bottom
horizontal arrows are equal to those of the original diagram (2.2.20). It
therefore suffices to show that the top horizontal arrows coincide. Since the
first and last morphisms in these four-fold compositions are equal, the claim
amounts to verifying the commutativity of the following diagram.

${j^{*}m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${j^{*}m_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)}$${m_{*}j_{2}^{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}j_{2}^{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)}$

This diagram commutes by the naturality of the base change map
$j^{*}m_{*}\rightarrow m_{*}j_{2}^{*}$ applied to the isomorphism
$i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)\simeq
i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B$. ∎

######  Remark 2.2.23.

Recall (for example, from [24, Tag 00CC]) that to give a monoidal structure on
a category $\mathcal{C}$, it suffices to give an associative product $\star$
on $\mathcal{C}$ together with a unit object of $\mathcal{C}$ (with respect to
$\star$), that is, an object $1_{\mathcal{C}}\in\mathcal{C}$ equipped with an
isomorphism

$1_{\mathcal{C}}\star 1_{\mathcal{C}}\simeq 1_{\mathcal{C}}.$

We use this simplification below when discussing the unit constraint on the
derived Satake category.

######  Remark 2.2.24.

We recall the definition of the unit constraint on
$(D_{G(\mathcal{O})}(\mathrm{Gr}_{G}),\star)$ (see [2, Lemma 9.2.2]). Let
$\mathrm{IC}_{0}=\underline{\mathbb{C}}_{\mathrm{Gr}_{G}^{0}}$ denote the
denote ${G}(\mathcal{O})$-equivariant skyscraper at the basepoint
$t^{0}\in\mathrm{Gr}_{G}$ and let
$j:\mathrm{pt}\hookrightarrow\mathrm{Gr}_{G}$ denote the closed inclusion of
$t^{0}$. Let
$j_{2}:\mathrm{pt}\hookrightarrow\mathrm{Gr}_{G}\operatorname{\tilde{\times}}\mathrm{Gr}_{G}$
denote the inclusion of the point
$\mathrm{Gr}_{G}^{0}\operatorname{\tilde{\times}}\mathrm{Gr}_{G}^{0}$.
Firstly, there is an isomorphism

$\xi:\mathrm{IC}_{0}\operatorname{\widetilde{\boxtimes}}\mathrm{IC}_{0}=j_{*}\underline{\mathbb{C}}_{\mathrm{pt}}\operatorname{\widetilde{\boxtimes}}j_{*}\underline{\mathbb{C}}_{\mathrm{pt}}\simeq(j_{2})_{*}(\underline{\mathbb{C}}_{\mathrm{pt}}\boxtimes\underline{\mathbb{C}}_{\mathrm{pt}})\simeq(j_{2})_{*}\underline{\mathbb{C}}_{\mathrm{pt}}.$

Since $m\circ j_{2}=j$, we have an isomorphism $(mj_{2})_{*}\simeq
m_{*}(j_{2})_{*}$. Applying $m_{*}$ to $\xi$ therefore yields an isomorphism

$\eta:\mathrm{IC}_{0}\star\mathrm{IC}_{0}\simeq
m_{*}(j_{2})_{*}\underline{\mathbb{C}}_{\mathrm{pt}}\simeq
j_{*}\underline{\mathbb{C}}_{\mathrm{pt}}=\mathrm{IC}_{0}.$

The isomorphism $\eta$ is the unit contraint.

######  Construction 2.2.25.

Let $\mathrm{IC}_{0}^{\prime}\in D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ denote
the monoidal unit and $j^{\prime}:\mathrm{pt}\hookrightarrow\mathrm{Gr}_{L}$
the inclusion of the basepoint. We have an isomorphism
$\chi:i^{*}\mathrm{IC}_{0}\xrightarrow{\sim}\mathrm{IC}_{0}^{\prime}$, which
can be precisely defined as the composition

$\chi:i^{*}\mathrm{IC}_{0}=i^{*}j_{*}\underline{\mathbb{C}}_{\mathrm{pt}}\simeq
i^{*}i_{*}j^{\prime}_{*}\underline{\mathbb{C}}_{\mathrm{pt}}\xrightarrow{\sim}j^{\prime}_{*}\underline{\mathbb{C}}_{\mathrm{pt}}=\mathrm{IC}_{0}^{\prime}.$

######  Proposition 2.2.26.

The natural transformation $\theta:i^{*}(-\star-)\rightarrow i^{*}(-)\star
i^{*}(-)$ of functors $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{G})$ defined in Construction 2.2.14 is
compatible with the unit constraints underlying the monoidal categories
$D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ and
$D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$. More precisely, the following diagram
in $D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ commutes:

(2.2.27)
${i^{*}(\mathrm{IC}_{0}\star\mathrm{IC}_{0})}$${i^{*}\mathrm{IC}_{0}}$${i^{*}\mathrm{IC}_{0}\star
i^{*}\mathrm{IC}_{0}}$${\mathrm{IC}^{\prime}_{0}\star\mathrm{IC}^{\prime}_{0}}$${\mathrm{IC}_{0}^{\prime}}$$\scriptstyle{i^{*}\eta}$$\scriptstyle{\theta_{\mathrm{IC}_{0},\mathrm{IC}_{0}}}$$\scriptstyle{\chi}$$\scriptstyle{\chi\star\chi}$$\scriptstyle{\eta}$

###### Proof.

It is at least obvious that all of the terms in (2.2.27) are isomorphic and
supported at $t^{0}\in\mathrm{Gr}_{L}$. Hence, it suffices to prove the
commutativity of the diagram after applying the functor $j^{*}$, where
$j:\mathrm{pt}\rightarrow\mathrm{Gr}_{L}$ denotes the inclusion of the
basepoint. Let $k=ji:\mathrm{pt}\rightarrow\mathrm{Gr}_{G}$ denote the
composition. The diagram $j^{*}$(2.2.27) can be embedded into the following
diagram:

${k^{*}(\mathrm{IC}_{0}\star\mathrm{IC}_{0})}$${k^{*}(\mathrm{IC}_{0})}$${j^{*}i^{*}(\mathrm{IC}_{0}\star\mathrm{IC}_{0})}$${j^{*}i^{*}\mathrm{IC}_{0}}$${j^{*}(i^{*}\mathrm{IC}_{0}\star
i^{*}\mathrm{IC}_{0})}$${j^{*}(\mathrm{IC}^{\prime}_{0}\star\mathrm{IC}^{\prime}_{0})}$${j^{*}\mathrm{IC}_{0}^{\prime}}$${j^{*}i^{*}\mathrm{IC}_{0}\star
j^{*}i^{*}\mathrm{IC}_{0}}$${j^{*}\mathrm{IC}_{0}^{\prime}\star
j^{*}\mathrm{IC}_{0}^{\prime}}$${\underline{\mathbb{C}}_{\mathrm{pt}}}$${k^{*}\mathrm{IC}_{0}\star
k^{*}\mathrm{IC}_{0}}$${\underline{\mathbb{C}}_{\mathrm{pt}}\star\underline{\mathbb{C}}_{\mathrm{pt}}}$${\underline{\mathbb{C}}_{\mathrm{pt}}}$$\scriptstyle{k^{*}\eta}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{j^{*}i^{*}\eta}$$\scriptstyle{j^{*}\theta_{\mathrm{IC}_{0},\mathrm{IC}_{0}}}$$\scriptstyle{j^{*}\chi}$$\scriptstyle{j^{*}(\chi\,\star\,\chi)}$$\scriptstyle{\theta_{i^{*}\mathrm{IC}_{0},i^{*}\mathrm{IC}_{0}}}$$\scriptstyle{\theta_{\mathrm{IC}^{\prime}_{0},\mathrm{IC}^{\prime}_{0}}}$$\scriptstyle{j^{*}\eta}$$\scriptstyle{\chi^{\prime}}$$\scriptstyle{(j^{*}\chi)\,\star\,(j^{*}\chi)}$$\scriptstyle{\operatorname{comp}\star\operatorname{comp}}$$\scriptstyle{\chi^{\prime}\star\chi^{\prime}}$$\scriptstyle{\chi^{\prime\prime}\,\star\,\chi^{\prime\prime}}$$\scriptstyle{1\otimes
1\mapsto 1}$

Here, the isomorphism
$\chi^{\prime}:j^{*}\mathrm{IC}_{0}^{\prime}\rightarrow\underline{\mathbb{C}}_{\mathrm{pt}}$
(respectively,
$\chi^{\prime\prime}:k^{*}\mathrm{IC}_{0}\underline{\mathbb{C}}_{\mathrm{pt}}$)
is obtained from Construction 2.2.25 after replacing the pair $L\subseteq G$
by the pair $\mathrm{pt}\subseteq L$ (respectively, $\mathrm{pt}\subseteq G$).
Moreover, the unlabelled map $j^{*}\mathrm{IC}^{\prime}_{0}\star
j^{*}\mathrm{IC}_{0}^{\prime}\rightarrow\underline{\mathbb{C}}_{\mathrm{pt}}$
is defined as the composition $j^{*}\mathrm{IC}_{0}^{\prime}\star
j^{*}\mathrm{IC}_{0}^{\prime}\rightarrow\underline{\mathbb{C}}_{\mathrm{pt}}\star\underline{\mathbb{C}}_{\mathrm{pt}}\simeq\underline{\mathbb{C}}_{\mathrm{pt}}$
(and is included only for formatting purposes).

The commutativity of the top rectangle follows from the naturality of the
isomorphism $\mathrm{comp}:j^{*}i^{*}\simeq k^{*}$ applied to the morphism
$\eta$. The commutativity of the upper left small rectangle follows from the
naturality of the morphism $\theta:j^{*}(-\star-)\rightarrow j^{*}(-)\star
j^{*}(-)$ applied to $\chi\star\chi$. The commutativity of the upper right
small rectangle will follow from the case $L=\mathrm{pt}$. The commutativity
of the lower right small rectangle is trivial. The lower left small rectangle
commutes by the compatiblity of $\chi$ with composition. The commutativity of
the large outer rectangle will follow from the case $L=\mathrm{pt}$. These
observations imply the commutativity of the second rectangle from the top
(that is, the diagram $j^{*}$(2.2.27)).

We may therefore assume that $L=\mathrm{pt}$ is the trivial subgroup. In this
case, the claim is obvious. ∎

######  Corollary 2.2.28.

  1. (1)

The constructions of 2.2.14 and 2.2.25 equip the functor
$i^{*}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ with a colax monoidal structure.

  2. (2)

The constructions of 2.2.5, 2.2.14, and 2.2.25 equip the functor
$i^{!}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ with a lax monoidal structure.

###  2.3. Fusion

In this section, we will give an alternative description of our lax monoidal
structure on $i^{!}$. We recall some definitions and constructions from [16,
Section 2.1.2], to which we refer the reader for more details. Let
$\mathrm{Aut}$ denote the automorphism group scheme of the
$\mathbb{C}$-algebra $\mathcal{O}$. It is a pro-algebraic group and acts
naturally on the affine Grassmannian $\mathrm{Gr}_{G}$ by loop rotation. Let
$X=\mathbb{A}^{1}$ and let $\tilde{X}\rightarrow X$ denote the canonical
$\mathrm{Aut}$-torsor over $X$ (whose fiber over $x\in X$ is the
$\mathrm{Aut}$-torsor parameterizing uniformizers of the completed local ring
$\widehat{\mathcal{O}}_{X,x}$). Recall that the Beilinson-Drinfeld
Grassmannian is the ind-scheme over $X$ given by the twisted product

$\mathrm{Gr}_{G,X}=\tilde{X}\times^{\mathrm{Aut}}\mathrm{Gr}_{G}\rightarrow
X.$

For any affine $X$-scheme $x:S\rightarrow X$, the set $\mathrm{Gr}_{G,X}(S)$
can be identified canonically with the set of isomorphism classes of pairs
$(\mathcal{P},\sigma)$, where $\mathcal{P}$ is a $G$-torsor over
$X_{S}=X\times S$ and $\sigma$ is a trivialization of $\mathcal{P}$ over the
subscheme $X_{S}\setminus\Gamma_{x}$, the complement of the graph
$\Gamma_{x}:S\rightarrow X_{S}$ (Lemma 3 of [16, Section 2.1.2]). The group
scheme $\mathrm{Aut}$ acts through group automorphisms on the arc group
$G(\mathcal{O})$. Therefore, we can form the group scheme $\mathcal{G}_{X}$
over $X$ given by

$\mathcal{G}_{X}:=\tilde{X}\times^{\mathrm{Aut}}G(\mathcal{O})\rightarrow X.$

Since the action
$G(\mathcal{O})\times\mathrm{Gr}_{G}\rightarrow\mathrm{Gr}_{G}$ is
$\mathrm{Aut}$-equivariant, we obtain an action of the group $X$-scheme
$\mathcal{G}_{X}$ on the $X$-ind-scheme $\mathrm{Gr}_{G,X}$. The twisted
external product defines a functor

$p_{2}^{\dagger}:=\underline{\mathbb{C}}_{\tilde{X}}[1]\operatorname{\widetilde{\boxtimes}}-:D_{G(\mathcal{O})\rtimes\mathrm{Aut}}(\mathrm{Gr}_{G})\rightarrow
D_{\mathcal{G}_{X}}(\mathrm{Gr}_{G,X}).$

See [3, Section 2.4.1] for a brief discussion of the definition of the above
$\mathcal{G}_{X}$-equivariant derived category, and [3, Chapter 10] for a
thorough treatment. On the other hand, we have Gaitsgory’s degeneration
$\pi:\mathrm{Gr}_{G,X}^{\prime}\rightarrow X$ from [16, Section 3.1.1]. It is
the ind-scheme over $X$ whose points over an affine $X$-scheme $x:S\rightarrow
X$ are given by isomorphism classes of pairs $(\mathcal{P},\sigma)$, where
$\mathcal{P}$ is a $G$-torsor over $X_{S}$ and $\sigma$ is a trivialization of
$\mathcal{P}$ over the subscheme $X_{S}\setminus(\Gamma_{x}\cup\Gamma_{0})$,
where $\Gamma_{0}$ denotes the graph of the map
$S\rightarrow\operatorname{Spec}{\mathbb{C}}\xrightarrow{0}\mathbb{A}^{1}=X$.
Note that $\mathcal{G}_{X}$ acts on $\mathrm{Gr}_{G,X}^{\prime}$ through
modification of the trivialization $\sigma$. There are canonical isomorphisms

$\displaystyle\mathrm{Gr}_{G,0}^{\prime}:=\mathrm{Gr}_{G,X}^{\prime}|_{\\{0\\}}\simeq\mathrm{Gr}_{G}$
$\displaystyle\mathrm{Gr}^{\prime}_{G,X-0}:=\mathrm{Gr}_{G,X}^{\prime}|_{X\setminus\\{0\\}}\simeq\mathrm{Gr}_{G}\times\mathrm{Gr}_{G,X}|_{X\setminus\\{0\\}}=\mathrm{Gr}_{G}\times\mathrm{Gr}_{G,X-0}.$

Hence, we obtain a functor

$-\boxtimes
p_{2}^{\dagger}(-)|_{\mathrm{Gr}_{G,X-0}}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})\rtimes\mathrm{Aut}}(\mathrm{Gr}_{G})\rightarrow
D_{\mathcal{G}_{X-0}}(\mathrm{Gr}_{G,X-0}^{\prime}).$

Here, we write $\mathcal{G}_{X-0}$ for the restriction of $\mathcal{G}_{X}$ to
the open subscheme $X\setminus\\{0\\}$. Finally, let
$\Psi:D_{\mathcal{G}_{X-0}}(\mathrm{Gr}_{G,X}^{\prime})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ denote the functor of nearby cycles in
the family $\pi$ (we are implicitly using the canonical identification of
group schemes $\mathcal{G}_{X}|_{\\{0\\}}\simeq G(\mathcal{O})$ provided by
the uniformizer $t\in\widehat{\mathcal{O}}_{X,0}=\mathbb{C}[[t]]$ here).
Following Gaitsgory, we define

(2.3.1) $C(A,B):=\Psi(A\boxtimes p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}}).$

Because the forgetful functor
$D_{G(\mathcal{O})\rtimes\mathrm{Aut}}(\mathrm{Gr}_{G})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ is an equivalence of categories (it is
fully faithful because $\mathrm{Aut}$ is pro-unipotent, and essentially
surjective because its image contains
$\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ by [29, Proposition 3.2.2]
which generates $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ as a triangulated
category), we may regard $C(-,-)$ as a bifunctor

$C(-,-):D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{G(\mathcal{O})}(\mathrm{Gr}_{G}).$

As usual, we will use the same notation for the corresponding objects defined
using the subgroup $L$. We have closed immersions

$i_{X}:\mathrm{Gr}_{L,X}=\tilde{X}\times^{\mathrm{Aut}}\mathrm{Gr}_{L}\xrightarrow{\mathrm{id}_{X}\operatorname{\widetilde{\boxtimes}}i}\tilde{X}\times^{\mathrm{Aut}}\mathrm{Gr}_{G}=\mathrm{Gr}_{G,X}$
$i^{\prime}_{X}:\mathrm{Gr}^{\prime}_{L,X}\rightarrow\mathrm{Gr}_{G,X}^{\prime}$

globalizing the $\mathrm{Aut}$-equivariant closed immersion
$i:\mathrm{Gr}_{L}\hookrightarrow\mathrm{Gr}_{G}$. The morphism
$i_{X}^{\prime}$ takes a point $(\mathcal{P},\sigma)$ of
$\mathrm{Gr}_{L,X}^{\prime}$ over an affine $X$-scheme $x:S\rightarrow X$ to
the point
$(\mathcal{P}^{\prime},\sigma^{\prime})\in\mathrm{Gr}_{G,X}^{\prime}(S)$ given
by the induced $G$-torsor $\mathcal{P^{\prime}}=\mathcal{P}\times^{L}G$ and
the trivialization $\sigma^{\prime}$ of $\mathcal{P}^{\prime}$ over
$X_{S}\setminus(\Gamma_{x}\cup\Gamma_{0})$ induced by $\sigma$. Let $i_{X-0}$,
$i^{\prime}_{X-0}$ denote their restrictions to $X\setminus\\{0\\}$.

######  Construction 2.3.2.

Let $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$. We will define a natural map

$\Theta_{A,B}:i^{*}C(A,B)\rightarrow C(i^{*}A,i^{*}B)$

in $D_{L(\mathcal{O})}(\mathrm{Gr}_{G})$. We have a natural isomorphism
(witnessing the monoidality of $*$-pullback with respect to the tensor
product)

(2.3.3) $\Omega_{A,B}:(i_{X-0}^{\prime})^{*}(A\boxtimes
p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}})\xrightarrow{\Omega_{1}}i^{*}A\boxtimes
i^{*}p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}}\xrightarrow{1\,\boxtimes\,\Omega_{2}}i^{*}A\boxtimes
p_{2}^{\dagger}(i^{*}B)|_{\mathrm{Gr}_{G,X-0}}.$

We have also used the compositional isomorphism $i_{2}^{*}p_{2}^{*}\simeq
p_{2}^{*}i^{*}$ to commute $i^{*}$ with $p_{2}^{\dagger}$. There is also a
natural transformation [2, Lemma 4.4.8]

(2.3.4) $\zeta:i^{*}\circ\Psi\rightarrow\Psi\circ(i_{X-0}^{\prime})^{*}.$

Applying $\Psi$ to the isomorphism $\Omega_{A,B}$ and then precomposing with
$\zeta$ defines the natural transformation $\Theta_{A,B}$. That is,

$\Theta_{A,B}=\Psi(\Omega_{A,B})\circ\zeta_{A\,\boxtimes\,p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}}}.$

The following fundamental result of Gaitsgory relates the bifunctor $C(-,-)$
to the convolution product.

######  Theorem 2.3.5 (Gaitsgory [16, Proposition 6]).

Let $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$. There is a natural
isomorphism in $D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$

$\varpi_{A,B}:C(A,B)\simeq A\star B.$

######  Remark 2.3.6.

Gaitsgory’s result only requires $G(\mathcal{O})$-equivariance on one of the
factors; the other need only belong to the constructible derived category
$D(\mathrm{Gr}_{G})$ (with the obvious caveat that $\varpi_{A,B}$ is in that
case only an isomorphism of non-equivariant complexes). However, we will only
need the version stated above.

######  Construction 2.3.7.

We need to unpack the construction of the isomorphism $\varpi_{A,B}$ from
Gaitsgory’s proof of [16, Proposition 6]. Let
$\widetilde{\mathrm{Gr}}^{\prime}_{G,X}\rightarrow X$ denote the ind-scheme
over $X$ whose set of points $\widetilde{\mathrm{Gr}}^{\prime}_{G,X}(S)$ over
an affine $X$-scheme $x:S\rightarrow X$ is the set of isomorphism classes of
tuples $(\mathcal{P}_{0},\mathcal{P}_{1},\eta_{0},\eta_{1})$, where
$\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ are $G$-torsors over $X_{S}$,
$\eta_{0}$ is an isomorphism of $G$-torsors
$\eta_{0}:\mathcal{P}_{0}|_{X_{S}\setminus\Gamma_{x}}\simeq\mathcal{P}_{1}|_{X_{S}\setminus\Gamma_{x}}$,
and $\eta_{1}$ is a trivialization of the $G$-torsor $\mathcal{P}_{1}$ over
$X_{S}\setminus\Gamma_{0}$. We have a map
$p:\widetilde{\mathrm{Gr}}_{G,X}^{\prime}\rightarrow\mathrm{Gr}_{G,X}$ of
$X$-ind-schemes taking an $S$-point
$(\mathcal{P}_{0},\mathcal{P}_{1},\eta_{0},\eta_{1})\in\widetilde{\mathrm{Gr}}^{\prime}_{G,X}(S)$
to the $S$-point $(\mathcal{P}_{1},\eta_{1})\in\mathrm{Gr}_{G,X}(S)$ and a map
$\mu:\widetilde{\mathrm{Gr}}_{G,X}^{\prime}\rightarrow\mathrm{Gr}_{G,X}^{\prime}$
taking an $S$-point
$(\mathcal{P}_{0},\mathcal{P}_{1},\eta_{0},\eta_{1})\in\widetilde{\mathrm{Gr}}^{\prime}_{G,X}(S)$
to the $S$-point
$(\mathcal{P}_{0},\eta_{0}\circ\eta_{1}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{0})})\in{\mathrm{Gr}}^{\prime}_{G,X}(S)$.

On the other hand, consider the $G(\mathcal{O})$-torsor
$\widetilde{\mathrm{Gr}}_{G,X}\rightarrow\mathrm{Gr}_{G,X}$, defined by the
fact that for any affine $X$-scheme $x:S\rightarrow X$,
$\widetilde{\mathrm{Gr}}_{G,X}(S)$ is the set of isomorphism classes of tuples
$(\mathcal{P},\eta,\gamma)$, where $(\mathcal{P},\eta)$ is an $S$-point of
$\mathrm{Gr}_{G,X}$ and $\gamma$ is a trivialization of $\mathcal{P}$ over the
formal neighborhood $\widehat{\Gamma}_{0}$ of $\Gamma_{0}\subseteq X_{S}$.
Then, there is an isomorphism

$\widetilde{\mathrm{Gr}}_{G,X}^{\prime}\simeq\mathrm{Gr}_{G}\times^{G(\mathcal{O})}\widetilde{\mathrm{Gr}}_{G,X}=:\mathrm{Gr}_{G}\operatorname{\tilde{\times}}\mathrm{Gr}_{G,X}$

of ind-schemes over $X$, under which $p$ corresponds to the natural projection
to the second factor. This description allows us to consider the twisted
external product

$-\operatorname{\widetilde{\boxtimes}}-:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\times
D_{\mathcal{G}_{X}}(\mathrm{Gr}_{G,X})\rightarrow
D_{\mathcal{G}_{X}}(\widetilde{\mathrm{Gr}}_{G,X}^{\prime}).$

Let $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$. Form the complex
$A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B\in
D_{\mathcal{G}_{X}}(\widetilde{\mathrm{Gr}}^{\prime}_{G,X})$. Then, there is a
natural isomorphism [16, Lemma 5(b)]

(2.3.8)
$\mu_{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)|_{\mathrm{Gr}_{G,X-0}}\simeq
A\boxtimes(p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}})$

induced by the projection formula. Let $\mu_{0}$ denote the fiber of $\mu$
over $0\in X$. There is a natural transformation [2, Lemma 4.8.8]

(2.3.9) $\Psi\mu_{*}\rightarrow\mu_{0*}\Psi,$

which is an isomorphism by the ind-properness of $\mu$ (note that the nearby
cycles functor $\Psi$ appearing in the domain of this natural transformation
is that corresponding to the family $\mathrm{Gr}_{G,X}^{\prime}\rightarrow X$
whereas that in the domain is nearby cycles in the family
$\widetilde{\mathrm{Gr}}_{G,X}^{\prime}\rightarrow X$). Combining it with
(2.3.8) yields an isomorphism

$\mu_{0*}\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)\simeq
C(A,B).$

Since $\widetilde{\mathrm{Gr}}_{G,X}^{\prime}\rightarrow X$ can be trivialized
by the choice of a global coordinate $t$ on $\mathbb{A}^{1}$, we also have a
natural isomorphism

(2.3.10)
$\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)\simeq(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)|_{\mathrm{Gr}_{G}^{(2)}}\simeq
A\operatorname{\widetilde{\boxtimes}}B,$

where $\mathrm{Gr}_{G}^{(2)}$ is identified with with the fiber of
$\widetilde{\mathrm{Gr}}_{G,X}^{\prime}\rightarrow X$ over $0\in X$. Since
$\mu_{0}=m$ is the convolution map, we obtain the desired isomorphism

$\varpi_{A,B}:A\star
B=m_{*}(A\operatorname{\widetilde{\boxtimes}}B)\simeq\mu_{0*}\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)\simeq
C(A,B).$

######  Construction 2.3.11.

We can define a morphism

$\tilde{i}^{\prime}_{X}:\widetilde{\mathrm{Gr}}_{L,X}^{\prime}\rightarrow\widetilde{\mathrm{Gr}}^{\prime}_{G,X}$

by sending an $S$-point (for $x:S\rightarrow X$ an affine $X$-scheme) of
$\widetilde{\mathrm{Gr}}_{L,X}^{\prime}$ given by a tuple
$(\mathcal{P}_{0},\mathcal{P}_{1},\eta_{0},\eta_{1})$ to the $S$-point of
$\widetilde{\mathrm{Gr}}_{G,X}^{\prime}$ given by
$(\mathcal{P}^{\prime}_{0},\mathcal{P}^{\prime}_{1},\eta^{\prime}_{0},\eta^{\prime}_{1})$,
where $\mathcal{P}^{\prime}_{i}=\mathcal{P}_{i}\times^{L}G$ is the $G$-torsor
associated to the $L$-torsor $\mathcal{P}_{i}$, $\eta_{0}^{\prime}$ is the
isomorphism
$\mathcal{P}^{\prime}_{0}|_{X_{S}\setminus\Gamma_{x}}\simeq\mathcal{P}^{\prime}_{1}|_{X_{S}\setminus\Gamma_{x}}$
induced from $\eta_{0}^{\prime}$, and $\eta_{1}^{\prime}$ is the
trivialization of $\mathcal{P}^{\prime}_{1}$ away from $\Gamma_{0}$ induced
from the trivialization $\eta_{1}$ of $\mathcal{P}_{1}$ away from
$\Gamma_{0}$. Given $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$, we have a
natural isomorphism

(2.3.12)
$\widetilde{\Omega}_{A,B}:(\tilde{i}^{\prime}_{X})^{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)\simeq
i^{*}A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}i^{*}B.$

Note that the fiber of $\tilde{i}_{X}^{\prime}$ over $0\in X$ identifies with
the map $i_{2}:\mathrm{Gr}_{L}^{(2)}\rightarrow\mathrm{Gr}_{G}^{(2)}$. Hence,
the construction of (2.3.4) defines a natural transformation

$\zeta_{2}:i_{2}^{*}\Psi\rightarrow\Psi(\tilde{i}_{X}^{\prime})^{*}.$

We can now formulate the global interpretation of our lax monoidal structure
on $i^{!}$ (equivalently, of the colax monoidal structure on $i^{*}$).

######  Proposition 2.3.13.

Let $A,B\in D_{G(\mathcal{O})}(\mathrm{Gr}_{G})$. Then, the following diagram
in $D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$ commutes:

${i^{*}C(A,B)}$${C(i^{*}A,i^{*}B)}$${i^{*}(A\star B)}$${i^{*}A\star
i^{*}B}$$\scriptstyle{\varpi_{A,B}}$$\scriptstyle{\Theta_{A,B}}$$\scriptstyle{\varpi_{A,B}}$$\scriptstyle{\theta_{A,B}}$

###### Proof.

We expand the diagram using the definition (2.3.1) of $C(A,B)$ as well as the
definitions 2.3.7, 2.3.2, and 2.2.16 of $\varpi_{A,B}$, $\Theta_{A,B}$, and
$\theta_{A,B}$, respectively. We will suppress many subscripts on our natural
transformations (indicating the objects that they are evaluated on) to
preserve readability.

${i^{*}\Psi(A\boxtimes
p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}})}$${\Psi(i^{\prime}_{X-0})^{*}(A\boxtimes
p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}})}$${\Psi(i^{*}A\boxtimes
p_{2}^{\dagger}i^{*}B)}$${i^{*}\Psi\mu_{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\Psi(i^{\prime}_{X-0})^{*}\mu_{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\Psi\mu_{*}(\tilde{i}_{X}^{\prime})^{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\Psi\mu_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}i^{*}B)}$${i^{*}m_{*}\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${m_{*}i_{2}^{*}\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${m_{*}\Psi(\tilde{i}_{X}^{\prime})^{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${m_{*}\Psi(i^{*}A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}i^{*}B)}$${i^{*}m_{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${m_{*}(i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B)}$$\scriptstyle{\zeta}$$\scriptstyle{i^{*}\Psi{\eqref{gaitsgorylemma5b}}}$$\scriptstyle{\Psi(i_{X-0}^{\prime})^{*}{\eqref{gaitsgorylemma5b}}}$$\scriptstyle{\Psi(\Omega_{A,B})}$$\scriptstyle{\Psi{\eqref{gaitsgorylemma5b}}}$$\scriptstyle{i^{*}{\eqref{nearbycyclesandpushforward}}}$$\scriptstyle{\zeta}$$\scriptstyle{\Psi(\rho^{\prime})}$$\scriptstyle{\Psi(\mu_{*}\widetilde{\Omega})}$$\scriptstyle{\eqref{nearbycyclesandpushforward}}$$\scriptstyle{\eqref{nearbycyclesandpushforward}}$$\scriptstyle{i^{*}m_{*}{\eqref{nearbycyclesinatrivialfamily}}}$$\scriptstyle{\rho}$$\scriptstyle{m_{*}\zeta_{2}}$$\scriptstyle{m_{*}i_{2}^{*}{\eqref{nearbycyclesinatrivialfamily}}}$$\scriptstyle{m_{*}\Psi(\widetilde{\Omega})}$$\scriptstyle{m_{*}{\eqref{nearbycyclesinatrivialfamily}}}$$\scriptstyle{\rho}$$\scriptstyle{m_{*}\omega_{A,B}}$

The upper left rectangle commutes by the naturality of the map
$\zeta:i^{*}\Psi\rightarrow\Psi(i_{X-0}^{\prime})^{*}$ of (2.3.4) applied to
the morphism (2.3.8). The middle right rectangle commutes by the naturality of
the map $\Psi\mu_{*}\rightarrow m_{*}\Psi$ of (2.3.9) applied to the
isomorphism $\Omega_{A,B}$ of (2.3.3). The lower left rectangle commutes by
the naturality of the base change map $\rho:i^{*}m_{*}\rightarrow
m_{*}i_{2}^{*}$ of (2.2.15) applied to the isomorphism
$\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)\simeq
A\operatorname{\widetilde{\boxtimes}}B$ of (2.3.8). The lower right rectangle
is obtained by applying the functor $m_{*}$ to the diagram

${i_{2}^{*}\Psi(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\Psi(\tilde{i}_{X}^{\prime})^{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\Psi(i^{*}A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}i^{*}B)}$${i_{2}^{*}(A\operatorname{\widetilde{\boxtimes}}B)}$${(\tilde{i}_{X}^{\prime})^{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)|_{\mathrm{Gr}_{G}^{(2)}}}$${i^{*}A\operatorname{\widetilde{\boxtimes}}i^{*}B.}$$\scriptstyle{\zeta_{2}}$$\scriptstyle{i_{2}^{*}{\eqref{nearbycyclesinatrivialfamily}}}$$\scriptstyle{\Psi(\widetilde{\Omega}_{A,B})}$$\scriptstyle{\eqref{nearbycyclesinatrivialfamily}}$$\scriptstyle{\eqref{nearbycyclesinatrivialfamily}}$$\scriptstyle{\omega_{A,B}}$$\scriptstyle{\widetilde{\Omega}_{A,B}|_{\mathrm{Gr}^{(2)}_{G}}}$

The right rectangle commutes by the naturality of the isomorphism
$\Psi\simeq(-)|_{\mathrm{Gr}_{G}^{(2)}}$ of (2.3.10) applied to the
isomorphism $\widetilde{\Omega}_{A,B}$ of (2.3.12). Note that the isomorphism
$\Psi\simeq(-)|_{\mathrm{Gr}_{G}^{(2)}}$ of (2.3.10) is defined by applying
the construction of (2.3.4) to the closed inclusion
$\mathrm{Gr}_{G}^{(2)}\hookrightarrow\widetilde{\mathrm{Gr}}_{G,X}^{\prime}$
of the zero fiber. Hence, the commutativity of the left rectangle follows from
the straightforward compatiblity of the natural transformation
$g^{*}\Psi\rightarrow\Psi g^{*}$ of [2, Lemma 4.8.8] with the compositional
isomorphisms for $*$-pullback.

We return to the original diagram and consider its upper right rectangle,
which is obtained by applying $\Psi$ to the diagram

${(i^{\prime}_{X-0})^{*}(A\boxtimes
p_{2}^{\dagger}B|_{\mathrm{Gr}_{G,X-0}})}$${i^{*}A\boxtimes
p_{2}^{\dagger}i^{*}B}$${(i^{\prime}_{X-0})^{*}\mu_{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\mu_{*}(\tilde{i}_{X}^{\prime})^{*}(A\operatorname{\widetilde{\boxtimes}}p_{2}^{\dagger}B)}$${\mu_{*}(i^{*}A\boxtimes
p_{2}^{\dagger}i^{*}B).}$$\scriptstyle{\Omega_{A,B}}$$\scriptstyle{(i^{\prime}_{X-0})^{*}{\eqref{gaitsgorylemma5b}}}$$\scriptstyle{\eqref{gaitsgorylemma5b}}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\mu_{*}\widetilde{\Omega}}$

The proof that this diagram commutes is sufficiently similar to the arguments
in the proof of Proposition 2.2.17 that we omit it and leave it to the reader
(use the adjunction $\mu^{*}\dashv\mu_{*}$ and then unwind the definitions of
(2.3.8) and the base change map $\rho^{\prime}$ to reduce the commutativity of
this diagram to an assertion about the compatibility of $\Omega$ and
$\widetilde{\Omega}$ with the compositional isomorphisms for $*$-pullback).

The commutativity of the middle left rectangle of the original diagram follows
from the more general assertion that for any object $E\in
D_{\mathcal{G}_{X}}(\widetilde{\mathrm{Gr}}_{G,X}^{\prime})$, the following
diagram commutes in $D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$:

${i^{*}\Psi\mu_{*}E}$${\Psi(i^{\prime}_{X})^{*}\mu_{*}E}$${\Psi\mu_{*}(\tilde{i}_{X}^{\prime})^{*}E}$${i^{*}m_{*}\Psi(E)}$${m_{*}i_{2}^{*}\Psi(E)}$${m_{*}\Psi(\tilde{i}_{X}^{\prime})^{*}E}$$\scriptstyle{\zeta_{\mu_{*}E}}$$\scriptstyle{i^{*}\eqref{nearbycyclesandpushforward}}$$\scriptstyle{\Psi(\rho^{\prime})}$$\scriptstyle{\eqref{nearbycyclesandpushforward}}$$\scriptstyle{\rho}$$\scriptstyle{m_{*}\zeta_{2}}$

The commutativity of this diagram is equivalent, under the adjunction
$m^{*}\dashv m_{*}$, to the commutativity of the outer rectangle in the
following diagram:

${m^{*}i^{*}\Psi\mu_{*}E}$${m^{*}\Psi(i_{X}^{\prime})^{*}\mu_{*}E}$${m^{*}\Psi\mu_{*}(\tilde{i}_{X}^{\prime})^{*}E}$${\Psi\mu^{*}(i_{X}^{\prime})^{*}\mu_{*}E}$${\Psi\mu^{*}\mu_{*}(\tilde{i}^{\prime}_{X})^{*}E}$${\Psi(\tilde{i}^{\prime}_{X})^{*}\mu^{*}\mu_{*}E}$${\Psi(\tilde{i}^{\prime}_{X})^{*}E}$${i_{2}^{*}m^{*}m_{*}\Psi(E)}$${i_{2}^{*}\Psi\mu^{*}\mu_{*}E}$${m^{*}i^{*}m_{*}\Psi(E)}$${m^{*}m_{*}i_{2}^{*}\Psi(E)}$${i_{2}^{*}\Psi(E)}$${\Psi(\tilde{i}^{\prime}_{X})^{*}E}$$\scriptstyle{m^{*}\zeta_{\mu_{*}E}}$$\scriptstyle{m^{*}i^{*}\eqref{nearbycyclesandpushforward}}$$\scriptstyle{m^{*}\Psi(\rho^{\prime})}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{\Psi\mu^{*}(\rho^{\prime})}$$\scriptstyle{\Psi(\mathrm{counit})}$$\scriptstyle{\Psi
i_{2}^{*}(\mathrm{counit})}$$\scriptstyle{i_{2}^{*}\mathrm{counit}}$$\scriptstyle{i_{2}^{*}\mathrm{counit}}$$\scriptstyle{\zeta_{2}}$$\scriptstyle{m^{*}\rho_{\Psi(E)}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{\mathrm{counit}}$

The upper right rectangle commutes by the naturality of the map
$\gamma:m^{*}\Psi\rightarrow\Psi\mu^{*}$ of [2, Lemma 4.8.8]. The rectangle
below it commutes by the definition of the base change map
$\rho^{\prime}:(i_{X}^{\prime})^{*}\mu_{*}\rightarrow\mu_{*}(\tilde{i}_{X}^{\prime})^{*}$.
The rectangle below that commutes by the naturality of the map
$\zeta_{2}:i_{2}^{*}\Psi\rightarrow\Psi(\tilde{i}_{X}^{\prime})$ applied to
the counit map $\mu^{*}\mu_{*}E\rightarrow E$. The commutativity of the bottom
left triangle follows from the definition of the base change map
$\rho:i^{*}m_{*}\rightarrow m_{*}i_{2}^{*}$. Thus, we are reduced to showing
that the following diagram commutes.

${m^{*}i^{*}\Psi\mu_{*}E}$${m^{*}\Psi(i_{X}^{\prime})^{*}\mu_{*}E}$${i_{2}^{*}m^{*}\Psi\mu_{*}E}$${\Psi\mu^{*}(i^{\prime}_{X})^{*}\mu_{*}E}$${\Psi(\tilde{i}_{X}^{\prime})^{*}\mu^{*}\mu_{*}E}$${i_{2}^{*}\Psi\mu^{*}\mu_{*}E}$${m^{*}i^{*}m_{*}\Psi(E)}$${i_{2}^{*}m^{*}m_{*}\Psi(E)}$${i_{2}^{*}\Psi(E)}$$\scriptstyle{m^{*}\zeta_{\mu_{*}E}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{m^{*}i^{*}\eqref{nearbycyclesandpushforward}}$$\scriptstyle{\gamma}$$\scriptstyle{i_{2}^{*}\gamma_{\mu_{*}E}}$$\scriptstyle{i_{2}^{*}m^{*}\eqref{nearbycyclesandpushforward}}$${(\dagger)}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{i_{2}^{*}(\mathrm{counit})}$$\scriptstyle{\zeta_{2}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{i_{2}^{*}(\mathrm{counit})}$

The left quadrilateral commutes by the naturality of the compositional
isomorphism $\mathrm{comp}:m^{*}i^{*}\simeq i_{2}^{*}m^{*}$ applied to the map
$\Psi\mu_{*}\rightarrow m_{*}\Psi$ of (2.3.9). The lower right quadrilateral
is obtained by applying $i_{2}^{*}$ to the diagram

${m^{*}\Psi\mu_{*}E}$${\Psi\mu^{*}\mu_{*}E}$${m^{*}m_{*}\Psi(E)}$${\Psi(E).}$$\scriptstyle{m^{*}\eqref{nearbycyclesandpushforward}}$$\scriptstyle{\gamma_{\mu_{*}E}}$$\scriptstyle{\Psi(\mathrm{counit})}$$\scriptstyle{\mathrm{counit}}$

Both composite maps $m^{*}\Psi\mu_{*}E\rightarrow\Psi(E)$ are left adjunct to
the morphism $\Psi\mu_{*}E\rightarrow m_{*}\Psi(E)$ of (2.3.9), hence are
equal. Finally, the upper figure $(\dagger)$ commutes by the more general
assertion that for any $F\in D_{\mathcal{G}_{X}}(\mathrm{Gr}_{G,X}^{\prime})$,
the following diagram commutes:

${m^{*}i^{*}\Psi(F)}$${m^{*}\Psi(i_{X}^{\prime})^{*}(F)}$${\Psi\mu^{*}(i_{X}^{\prime})^{*}(F)}$${i_{2}^{*}m^{*}\Psi(F)}$${i_{2}^{*}\Psi\mu^{*}(F)}$${\Psi(\tilde{i}_{X}^{\prime})^{*}\mu^{*}(F)}$$\scriptstyle{m^{*}\zeta_{2}}$$\scriptstyle{\mathrm{comp}}$$\scriptstyle{\gamma_{(i^{\prime}_{X})^{*}F}}$$\scriptstyle{\Psi(\mathrm{comp})}$$\scriptstyle{i_{2}^{*}\gamma_{F}}$$\scriptstyle{\zeta_{2}}$

This diagram expresses a straightforward compatibility between the natural
transformation of [2, Lemma 4.8.8] with the compositional isomorphisms for
$*$-pullback, whose proof we leave to the reader (since it is easy but
requires unravelling the definition of the nearby cycles functor $\Psi$, which
we will avoid doing here). ∎

###  2.4. Parabolic restriction

We now return to the situation in which $L\subseteq G$ is a Levi subgroup of
$G$. In [6, Section 5.3.28], Beilinson and Drinfeld define a natural operation
taking complexes on $\mathrm{Gr}_{G}$ to complexes on $\mathrm{Gr}_{L}$ which
we will refer to as parabolic restriction (other possible names are the
constant term functor or the Jacquet functor). We also refer the reader to [5,
Section 15] for a complete treatment. It is a functor

$\mathrm{Res}^{G}_{L}:D(\mathrm{Gr}_{G})\rightarrow D(\mathrm{Gr}_{L})$

defined through the following procedure. We identify $L$ with the quotient
$P/V$ of $P$ by its unipotent radical. We obtain a correspondence of ind-
schemes (the induction diagram):

${\mathrm{Gr}_{P}}$${\mathrm{Gr}_{L}}$${\mathrm{Gr}_{G}}$$\scriptstyle{r}$$\scriptstyle{q}$

The functor of (unnormalized) parabolic restriction is defined as the
hyperbolic restriction

$\mathrm{Res}^{G}_{L}=q_{*}\circ r^{!}:D(\mathrm{Gr}_{G})\rightarrow
D(\mathrm{Gr}_{L}).$

We will also need an equivariant version of this functor. We define it as the
composition

(2.4.1) $\mathrm{Res}^{G}_{L}=q_{*}\circ
r^{!}\circ\mathrm{For}^{G(\mathcal{O})}_{L(\mathcal{O})}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L}),$

where by $r^{!}$ we mean the $L(\mathcal{O})$-equivariant $!$-pullback functor
and by $q_{*}$ we mean the $L(\mathcal{O})$-equivariant $*$-pushforward
functor.

######  Remark 2.4.2.

In keeping with the notational convention of not distinguishing between the
equivariant and non-equivariant versions of a sheaf functor, we continue to
use the notation $\mathrm{Res}^{G}_{L}$ for the equivariant (unnormalized)
parabolic restriction.

######  Remark 2.4.3.

Let $\mathrm{Gr}^{\chi}_{L}\subseteq\mathrm{Gr}_{L}$ denote a connected
component of $\mathrm{Gr}_{L}$ indexed by a character $\chi\in\Lambda$ of
$Z(\check{L})$. According to [6, Proposition 5.3.29(1)] (see also [5, Lemma
15.1]), if $F\in\mathcal{P}_{G(\mathcal{O})}(\mathrm{Gr}_{G})$ is perverse,
then the complex $\mathrm{Res}^{G}_{L}(F)|_{\mathrm{Gr}_{L}^{\chi}}$ is
concentrated in perverse degree $\langle 2\rho_{G}-2\rho_{L},\chi\rangle$. It
is therefore useful to introduce the normalized parabolic restriction functor
$\mathrm{Res}^{G,\natural}_{L}$ defined by

$\mathrm{Res}^{G,\natural}_{L}:=\bigoplus_{\chi\in\pi_{0}(\mathrm{Gr}_{L})}\mathrm{Res}^{G}_{L}|_{\mathrm{Gr}_{L}^{\chi}}[-\langle
2\rho_{G}-2\rho_{L},\chi\rangle].$

The functor $\mathrm{Res}^{G,\natural}_{L}$ is now $t$-exact. Of course, these
considerations apply verbatim to the equivariant version (2.4.1) of
$\mathrm{Res}_{L}^{G}$, and we let $\mathrm{Res}^{G,\natural}_{L}$ also denote
the normalized equivariant parabolic restriction functor.

######  Remark 2.4.4.

Throughout this work, we have used the notation $i^{!}$ to refer to the
functor

$i^{!}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{G})$

given by composing the forgetful functor
$\mathrm{For}^{G(\mathcal{O})}_{L(\mathcal{O})}:D_{G(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{G})$ with the $L(\mathcal{O})$-equivariant
$!$-pullback $i^{!}:D_{L(\mathcal{O})}(\mathrm{Gr}_{G})\rightarrow
D_{L(\mathcal{O})}(\mathrm{Gr}_{L})$. However, at some points in this section,