# Poincare Series and Miraculous Duality

Kevin Lin

###### Abstract.

In the setting of global geometric Langlands, we show that miraculous duality
on the stack of principal bundles on a curve intertwines the functor of
Poincare series with the dual functor to Whittaker coefficients. We construct,
for arbitrary parabolic subgroups, Jacquet functors controlling constant terms
of Poincare series and Whittaker coefficients of Eisenstein series in local
terms.

##  1\. Introduction

###  1.1. Statement of the theorem

#### 1.1.1.

We fix a smooth proper curve $X$ over an algebraically closed field of
characteristic zero, as well as a square root $\omega_{X}^{1/2}$ of the
canonical line bundle.

#### 1.1.2.

The geometric Langlands conjecture [AG15] predicts an equivalence of
categories

$\mathbf{L}_{G}:\mathrm{D}(\mathrm{Bun}_{G})\overset{\sim}{\rightarrow}\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}}).$

Let us review some of the compatibilities expected from this functor.

####  1.1.3. Duality

The category appearing on the spectral side of the Langlands equivalence has a
self-duality given by Serre duality

$\mathbf{D}:\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})\overset{\sim}{\rightarrow}\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})^{\vee}\qquad\mathscr{F}\mapsto\Gamma(X,\mathscr{F}\overset{!}{\otimes}-)$

The work [Gai17] constructs a self duality

$\mathrm{PsId}_{!}:\mathrm{D}(\mathrm{Bun}_{G})^{\vee}\overset{\sim}{\rightarrow}\mathrm{D}(\mathrm{Bun}_{G})$

which is expected to be the automorphic counterpart of Serre duality in the
sense that the following diagram commutes up to a Cartan involution and
cohomological shift.

(1.1.3.1)
${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})^{\vee}}$$\scriptstyle{\mathbf{L}_{G}}$${\tau\cdot{[?]}}$$\scriptstyle{\mathbf{D}}$$\scriptstyle{\mathrm{PsId}_{!}}$$\scriptstyle{\mathbf{L}_{G}^{\vee}}$

By definition, $\mathrm{PsId}_{!}$ is given by the object

$\Delta_{!}\cdot
k_{\mathrm{Bun}_{G}}\in\mathrm{D}(\mathrm{Bun}_{G}\times\mathrm{Bun}_{G})\overset{\sim}{\leftarrow}\mathrm{D}(\mathrm{Bun}_{G})\otimes\mathrm{D}(\mathrm{Bun}_{G}),$

where $\Delta:\mathrm{Bun}_{G}\to\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}$ is
the diagonal.

####  1.1.4. Whittaker coefficients

Let ${\check{\rho}}$ be the half-sum of the positive coroots. By using the
chosen square root of the canonical bundle, we obtain a point of
$\mathrm{Bun}_{T}$ denoted $\omega^{\check{\rho}}$ and define

$\mathrm{Bun}_{N}=\\{\omega^{\check{\rho}}\\}\underset{\mathrm{Bun}_{T}}{\times}\mathrm{Bun}_{B}.$

This stack is equipped with canonical maps

${\mathrm{Bun}_{N}}$${\mathbf{G}_{\mathrm{a}}}$${\mathrm{Bun}_{G}}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$

Now let $\mathrm{AS}$ denote the exponential D-module on
$\mathbf{G}_{\mathrm{a}}$, cohomologically shifted to be a character sheaf
with respect to $!$-convolution. We write

$\mathrm{Vect}\overset{\sim}{\rightarrow}\mathrm{Whit}(\mathrm{Bun}_{N})\overset{\mathrm{oblv}}{\longrightarrow}\mathrm{D}(\mathrm{Bun}_{N})$

for the full subcategory generated by the object
$\mathscr{W}_{\mathrm{vac}}=\mathrm{ev}^{!}\cdot\mathrm{AS}$. The inclusion
admits a right adjoint

$\mathrm{D}(\mathrm{Bun}_{N})\overset{\mathrm{Av}_{*}}{\longrightarrow}\mathrm{Whit}(\mathrm{Bun}_{N})\simeq\mathrm{Vect},$

and the functor of Whittaker coefficients is

$\mathrm{coeff}_{*}=\mathrm{Av}_{*}\cdot\pi^{!}.$

It is supposed to match the functor of pushforward along
$f:\mathrm{LS}_{\check{G}}\to\mathrm{pt}$.

(1.1.4.1)
${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$${\mathrm{Whit}(\mathrm{Bun}_{N})}$${\mathrm{QCoh}(\mathrm{pt})}$$\scriptstyle{\mathrm{coeff}_{*}}$$\scriptstyle{\mathbf{L}_{G}}$$\scriptstyle{f_{*}}$$\scriptstyle{\sim}$
${\mathrm{Whit}(\mathrm{Bun}_{N})}$${\mathrm{QCoh}(\mathrm{pt})}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{f^{*}}$$\scriptstyle{\mathbf{L}_{G}}$

Passing to left adjoints gives the diagram on the right, where

$\mathrm{Poinc}_{!}=\pi_{!}\cdot\mathrm{oblv}.$

#### 1.1.5.

It is a general fact that Serre duality exchanges $f_{*}$ with $f^{!}$.
However, a special feature of $\mathrm{LS}_{\check{G}}$ is that there is a
canonical isomorphism

$\omega_{\mathrm{LS}_{\check{G}}}\simeq\mathscr{O}_{\mathrm{LS}_{\check{G}}}\,[2\delta_{G}]$

where

$2\delta_{G}=(2g-2)\cdot\dim({\check{G}})=2\dim\mathrm{Bun}_{G}.$

Therefore the dual functor to $f_{*}$ identifies with a shift of $f^{*}$.

${\mathrm{QCoh}(\mathrm{pt})}$${\mathrm{QCoh}(\mathrm{pt})^{\vee}}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})^{\vee}}$$\scriptstyle{\sim}$$\scriptstyle{f^{*}{[2\delta_{G}]}}$$\scriptstyle{(f_{*})^{\vee}}$$\scriptstyle{\mathbf{D}}$

Let us translate this statement to the automophic side. We will use the
Verdier duality pairing

$\mathrm{Whit}_{-\psi}(\mathrm{Bun}_{N})\otimes\mathrm{Whit}_{\psi}(\mathrm{Bun}_{N})\to\mathrm{Vect}\qquad\langle\mathscr{W}_{1},\mathscr{W}_{2}\rangle=\Gamma(\mathrm{Bun}_{N},\mathscr{W}_{1}\overset{!}{\otimes}\mathscr{W}_{2}).$

The resulting statement is our main theorem:

###### 1.1.5 Theorem.

There is a canonical commutative diagram

${\mathrm{Whit}_{-\psi}(\mathrm{Bun}_{N})^{\vee}}$${\mathrm{Whit}_{\psi}(\mathrm{Bun}_{N})}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})}$$\scriptstyle{\mathrm{coeff}_{*}^{\vee}{[2\delta_{G}]}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{PsId}_{!}}$

#### 1.1.6.

As we have just explained, Theorem 1.1.5 is predicted by the geometric
Langlands conjecture. However, we expect that it will play an essential rôle
in the construction of the Langlands equivalence $\mathbf{L}_{G}$ and its
attendant compatibilities.

####  1.1.7. Cohomological shifts

Due the ambiguity in (1.1.3.1), the above discussion does not actually predict
the cohomological shift appearing in our theorem. (One can guess this shift by
considering the compatibility expressed by Proposition 1.4.4, which only
involves the automorphic side of the Langlands correspondence.) Let us use the
theorem statement to determine the shift that should appear in (1.1.3.1). The
Verdier duality on $\mathrm{Whit}(\mathrm{Bun}_{N})$ is related to Serre
duality as follows:

${\mathrm{Whit}_{\psi}(\mathrm{Bun}_{N})}$${\mathrm{QCoh}(\mathrm{pt})}$${\mathrm{Whit}_{-\psi}(\mathrm{Bun}_{N})^{\vee}}$${\mathrm{QCoh}(\mathrm{pt})^{\vee}}$$\scriptstyle{\sim}$$\scriptstyle{\mathbf{D}{[-2\delta_{N}]}}$$\scriptstyle{\mathbf{D}}$$\scriptstyle{\sim}$

The shift appearing here is actually to be expected: it accounts for the
difference between Verdier duality and the more natural local Whittaker
duality. In any case we obtain:

${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})^{\vee}}$$\scriptstyle{\mathbf{L}_{G}}$${\tau}$$\scriptstyle{\mathbf{D}{[2\delta_{G}]}}$$\scriptstyle{\mathrm{PsId}_{!}{[2\delta_{G}+2\delta_{N}]}}$$\scriptstyle{\mathbf{L}_{G}^{\vee}}$

###  1.2. Upgraded version

#### 1.2.1.

Our main theorem has a $\mathrm{Sph}_{G,\mathrm{Ran}}$-linear upgrade
involving the entire factorization category $\mathrm{Whit}(G)_{\mathrm{Ran}}$.
Let us recall some basic facts about Whittaker categories and Hecke actions
before giving the statement.

####  1.2.2. Global Whittaker model

Let $x$ be a finite set of points of $X$. We will use the global model for the
Whittaker category

$\mathrm{Whit}(G)_{x}\simeq\mathrm{Whit}\left(\overline{\mathrm{Bun}}_{N,x}^{\mathrm{pol}}\right)\hookrightarrow\mathrm{D}\left(\overline{\mathrm{Bun}}_{N,x}^{\mathrm{pol}}\right).$

It is naturally a module over the spherical Hecke category
$\mathrm{Sph}_{G,x}$, equipped with a pair of $\mathrm{Sph}_{G,x}$-linear
adjoint functors

$\mathrm{Poinc}_{!}:\mathrm{Whit}(G)_{x}\to\mathrm{D}(\mathrm{Bun}_{G})\qquad\mathrm{coeff}_{*}:\mathrm{D}(\mathrm{Bun}_{G})\to\mathrm{Whit}(G)_{x}.$

Here, as before, $\mathrm{Poinc}_{!}$ is simply $!$-pushforward of the
underlying sheaf to $\mathrm{Bun}_{G}$. The Verdier pairing

$\mathrm{Whit}_{-\psi}(G)_{x}\otimes\mathrm{Whit}_{\psi}(G)_{x}\to\mathrm{Vect}$

is a perfect pairing.

####  1.2.3. Left and right modules

To formulate the sense in which our dualities are $\mathrm{Sph}_{G}$-linear,
recall that every convolution stack comes with a canonical involution

$\sigma:X\underset{S}{\times}X\to X\underset{S}{\times}X$

given by swapping the two factors. For example, this involution on
$G\simeq\mathrm{pt}\underset{\mathbf{B}G}{\times}\mathrm{pt}$ is group
inversion. We obtain an involutive anti-automorphism

$\sigma:\mathrm{Sph}_{G,x}\overset{\sim}{\rightarrow}\mathrm{Sph}_{G,x}^{\mathrm{rev}}.$

Now let us view $\mathrm{D}(\mathrm{Bun}_{G})$ as a right
$\mathrm{Sph}_{G,x}$-module. Then $\mathrm{D}(\mathrm{Bun}_{G})^{\vee}$ is
naturally a left $\mathrm{Sph}_{G,x}$-module, but the involution $\sigma$
allows us to identify left and right modules for $\mathrm{Sph}_{G,x}$.
Therefore, it makes sense to say that the functor

$\mathrm{PsId}_{!}:\mathrm{D}(\mathrm{Bun}_{G})^{\vee}\to\mathrm{D}(\mathrm{Bun}_{G})$

is canonically $\mathrm{Sph}_{G,x}$-linear. This is easy to check using the
ULA property of Hecke kernels and the fact that $\sigma$ interchanges the
geometrically defined left and right $\mathrm{Sph}_{G,x}$ actions on
$\mathrm{D}(\mathrm{Bun}_{G})$. It is tautological that the duality

$\mathrm{Whit}_{-\psi}(G)_{x}^{\vee}\simeq\mathrm{Whit}_{\psi}(G)_{x}$

is $\mathrm{Sph}_{G,x}$-linear.

#### 1.2.4.

Allowing the set of points $x$ to vary in families and writing

$\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Ran}}=\mathrm{D}(\mathrm{Bun}_{G}\times\mathrm{Ran}),$

we obtain the following variant of the main theorem:

###### 1.2.4 Theorem.

There is a canonical $\mathrm{Sph}_{G,\mathrm{Ran}}$-linear commutative
diagram

${\mathrm{Whit}_{-\psi}(G)_{\mathrm{Ran}}^{\vee}}$${\mathrm{Whit}_{\psi}(G)_{\mathrm{Ran}}}$${\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Ran}}^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Ran}}}$$\scriptstyle{\mathrm{coeff}_{*}^{\vee}{[2\delta_{G}]}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{PsId}_{!}}$

#### 1.2.5.

Let us explain why this theorem is to be expected from the point of view of
Langlands duality. In our derivation of the main theorem for the Whittaker
vacuum, we actually ignored the Cartan involution without comment. This was
acceptable because the Cartan involution sends
$\mathscr{O}_{\mathrm{LS}_{\check{G}}}$ to itself. But in what follows we will
have to track this involution more closely.

#### 1.2.6.

Recall that geometric Satake defines a monoidal functor

$\mathrm{Sat}_{x}:\mathrm{Rep}({\check{G}})_{x}\to\mathrm{Sph}_{G,x}.$

The Langlands functor is supposed to intertwine the Hecke action of
$\mathrm{Sph}_{G,x}$ with the action of $\mathrm{Rep}({\check{G}})_{x}$ on
$\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})$ induced by the map

$\mathrm{ev}_{x}:\mathrm{LS}_{\check{G}}(X)\to\mathrm{LS}_{\check{G}}(\mathscr{D}_{x})\simeq(\mathbf{B}{\check{G}})_{x}$

and the action of $\mathrm{QCoh}$ on $\mathrm{IndCoh}$. Therefore, viewing
$\mathscr{W}_{\mathrm{vac}}$ as an object of $\mathrm{Whit}(G)_{x}$ and acting
on (1.1.4.1), we obtain:

${\mathrm{Whit}(G)_{x}}$${\mathrm{Rep}({\check{G}})_{x}}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{ev}_{x}^{*}}$$\scriptstyle{\mathrm{CS}}$$\scriptstyle{\mathbf{L}_{G}}$
${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$${\mathrm{Whit}(G)_{x}}$${\mathrm{Rep}({\check{G}})_{x}}$$\scriptstyle{\mathrm{coeff}_{*}}$$\scriptstyle{\mathbf{L}_{G}}$$\scriptstyle{\mathrm{ev}_{x,*}}$$\scriptstyle{\mathrm{CS}}$

####  1.2.7. Cartan involution

Let us recall the behavior of the involution $\sigma$ with respect to
geometric Satake. To reduce confusion, we will temporarily consider the
derived Satake isomorphism

$\mathrm{Sph}_{G,x}\simeq\mathrm{Sph}_{{\check{G}},x}^{\mathrm{spec}}\simeq\mathrm{IndCoh}_{\mathscr{N}}\left((\mathrm{pt}\underset{{\check{\mathfrak{g}}}}{\times}\mathrm{pt})/{\check{G}}\right).$

The naïve diagram

${\mathrm{Sph}_{G,x}}$${\mathrm{Sph}_{{\check{G}},x}^{\mathrm{spec}}}$${\mathrm{Sph}_{G,x}^{\mathrm{rev}}}$${\mathrm{Sph}_{{\check{G}},x}^{\mathrm{spec},\mathrm{rev}}}$$\scriptstyle{\mathrm{Sat}}$$\scriptstyle{\sigma}$${(\tau)}$$\scriptstyle{\sigma}$$\scriptstyle{\mathrm{Sat}^{\mathrm{rev}}}$

of algbera isomorphisms is not commutative: the commutativity is only restored
after inserting the Cartan involution. Note that the diagram

(1.2.7.1)
${\mathrm{Rep}({\check{G}})}$${\mathrm{Sph}_{{\check{G}},x}^{\mathrm{spec}}}$${\mathrm{Rep}({\check{G}})^{\mathrm{rev}}}$${\mathrm{Sph}_{{\check{G}},x}^{\mathrm{spec},\mathrm{rev}}}$$\scriptstyle{\mathbf{1}}$$\scriptstyle{\sigma}$

is canonically commutative because the horizonal map is given by pushforward
along the diagonal

$\mathbf{B}{\check{G}}\to\mathbf{B}{\check{G}}\underset{{\check{\mathfrak{g}}}/{\check{G}}}{\times}\mathbf{B}{\check{G}}.$

Since Serre duality on
$\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})$ is naturally
$\mathrm{Rep}({\check{G}})$-linear, the Cartan involution in (1.2.7.1)
explains the presence of the Cartan involution in (1.1.3.1). Similarly, the
Cartan involution also appears in the Whittaker duality:

${\mathrm{Whit}_{\psi}(G)_{x}}$${\mathrm{Rep}({\check{G}})_{x}}$${\mathrm{Whit}_{-\psi}(G)_{x}^{\vee}}$${\mathrm{Rep}({\check{G}})_{x}^{\vee}}$$\scriptstyle{\mathbf{D}{[-2\delta_{N}]}}$${\tau}$$\scriptstyle{\mathrm{CS}}$$\scriptstyle{\mathrm{CS}^{\vee}}$

#### 1.2.8.

Applying the expected compatibilities to every object appearing Theorem 1.2.4,
we obtain the diagram

${\mathrm{Rep}({\check{G}})_{x}^{\vee}}$${\mathrm{Rep}({\check{G}})_{x}}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})^{\vee}}$${\mathrm{IndCoh}_{\mathscr{N}}(\mathrm{LS}_{\check{G}})}$$\scriptstyle{\mathrm{ev}_{x,*}^{\vee}}$$\scriptstyle{\mathrm{ev}_{x}^{*}}$$\scriptstyle{\mathbf{D}{[2\delta_{G}]}}$

###  1.3. Outline of the proof

####  1.3.1. Notation

Let us write $(-)_{\mathrm{co}}$ for the Verdier duality $\mathbf{D}(-)$ and

$\mathrm{PsId}_{\mathrm{Whit}}:\mathrm{Whit}_{-\psi}(\mathrm{Bun}_{N})^{\vee}\to\mathrm{Whit}_{\psi}(\mathrm{Bun}_{N})$

for its inverse.

#### 1.3.2.

Our first task will be the construction of a natural transformation

$\eta_{G}:\mathrm{Poinc}_{!}\cdot\mathrm{PsId}_{\mathrm{Whit}}\to\mathrm{PsId}_{!}\cdot\mathrm{coeff}_{*}^{\vee}\,[2\delta_{G}].$

This problem is actually very concrete: the data of $\eta_{G}$ is equivalent
to a morphism

$\mathrm{Poinc}_{!}\cdot\mathscr{W}_{\mathrm{vac}}\to\mathrm{PsId}_{!}\cdot\mathrm{coeff}_{*}^{\vee}\cdot\mathscr{W}_{\mathrm{vac},\mathrm{co}}\,[2\delta_{G}]$

between two objects of $\mathrm{D}(\mathrm{Bun}_{G})$. We will perform this
construction in §2. Once this has been done, we will have to prove that
$\eta_{G}$ is an isomorphism.

#### 1.3.3.

Our proof that $\eta_{G}$ is an isomorphism is based on the decomposition of
$\mathrm{D}(\mathrm{Bun}_{G})$ into its cuspidal and Eisenstein parts. Recall
that the full subcategory
$\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{cusp}}\hookrightarrow\mathrm{D}(\mathrm{Bun}_{G})$
of cuspidal objects has the following equivalent characterizations:

  * –

It consists of objects $\mathscr{F}$ which are annihilated by
$\mathrm{CT}_{P,*}$ for every proper parabolic $P$.

  * –

It consists of objects $\mathscr{F}$ which are annihilated by
$\mathrm{CT}_{P,!}$ for every proper parabolic $P$.

  * –

It is the right orthogonal to $\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Eis}}$.

Here the full subcategory
$\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Eis}}\hookrightarrow\mathrm{D}(\mathrm{Bun}_{G})$
is characterized by the following equivalent characterizations:

  * –

It is the full subcategory generated by the images of $\mathrm{Eis}_{P,!}$ for
all proper parabolics $P$.

  * –

It is the left orthogonal to $\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{cusp}}$.

The category $\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Eis}}$ contains but is not
usually generated by the images of $\mathrm{Eis}_{P,*}$ for all proper
parabolics $P$. We will prove Theorem 1.1.5 by establishing the following:

###### 1.3.3 Lemma.

The cofiber of $\eta_{G}$ is Eisenstein series.

###### 1.3.3 Lemma.

The cofiber of $\eta_{G}$ is cuspidal.

###  1.4. Comparison with the naïve pseudo-identity

#### 1.4.1.

Recall that there is a naïve variant of the pseudo-identity functor, denoted

$\mathrm{PsId}_{*}:\mathrm{D}(\mathrm{Bun}_{G})^{\vee}\to\mathrm{D}(\mathrm{Bun}_{G}),$

corresponding to the object

$\Delta_{*}\cdot\omega_{\mathrm{Bun}_{G}}\in\mathrm{D}(\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}).$

It is related to $\mathrm{PsId}_{!}$ by a canonical map

$\mathrm{PsId}_{!}\,[2\delta_{G}+z_{G}]\to\mathrm{PsId}_{*}\qquad z_{G}=\dim
Z_{G}$

whose cofiber is Eisenstein series.

#### 1.4.2.

The functor $\mathrm{Poinc}_{!}$ has a variant

$\mathrm{Poinc}_{*}=\pi_{*}\cdot\mathrm{oblv},$

related to $\mathrm{Poinc}_{!}$ by a canonical map

(1.4.2.1) $\mathrm{Poinc}_{!}\,[z_{G}]\to\mathrm{Poinc}_{*}.$

The version of our main theorem with $\mathrm{PsId}_{*}$ and
$\mathrm{Poinc}_{*}$ is actually trivial:

###### 1.4.2 Proposition.

There is a canonical commutative diagram

${\mathrm{Whit}_{-\psi}(\mathrm{Bun}_{N})^{\vee}}$${\mathrm{Whit}_{\psi}(\mathrm{Bun}_{N})}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})}$$\scriptstyle{\mathrm{coeff}_{*}^{\vee}}$$\scriptstyle{\mathrm{PsId}}$$\scriptstyle{\mathrm{Poinc}_{*}}$$\scriptstyle{\mathrm{PsId}_{*}}$

####  1.4.3. Proof

Performing base change on the diagram

${\mathrm{Bun}_{N}}$${\mathrm{Bun}_{G}}$${\mathrm{Bun}_{N}\times\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}}$$\scriptstyle{\Delta}$$\scriptstyle{\pi\times\mathbf{1}}$$\scriptstyle{\pi_{2}}$

gives

$\mathrm{PsId}_{*}\cdot\mathrm{coeff}_{*}^{\vee}\cdot\mathscr{W}_{\mathrm{co}}\simeq\pi_{2,*}\cdot\left\\{\mathscr{W}\overset{!}{\otimes}(\pi\times\mathbf{1})^{!}\cdot\Delta_{*}\cdot\omega_{\mathrm{Bun}_{G}}\right\\}\simeq\mathrm{Poinc}_{*}\cdot\mathscr{W}.$

#### 1.4.4.

Combining $\eta_{G}$ with the morphism relating $\mathrm{PsId}_{!}$ to
$\mathrm{PsId}_{*}$, we obtain a map

(1.4.4.1)
$\begin{split}\mathrm{Poinc}_{!}\,[z_{G}]&\to\mathrm{PsId}_{!}\cdot\mathrm{coeff}^{\vee}\cdot\mathrm{PsId}_{\mathrm{Whit}}^{-1}\,[2\delta_{G}+z_{G}]\\\
&\qquad\qquad\to\mathrm{PsId}_{*}\cdot\mathrm{coeff}^{\vee}\cdot\mathrm{PsId}_{\mathrm{Whit}}^{-1}\simeq\mathrm{Poinc}_{*}.\end{split}$

Then Lemma 1.3.3 is equivalent to the assertion that the cofiber of this map
is Eisenstein series. We will prove this in §3 by verifying the following two
assertions:

###### 1.4.4 Proposition.

The map (1.4.4.1) canonically identifies with (1.4.2.1).

###### 1.4.4 Proposition.

The cofiber of (1.4.2.1) is Eisenstein series.

####  1.4.5. Remark

Combining Theorem 1.1.5 with Proposition 1.4.2 gives

$\mathrm{Poinc}_{*}\simeq\mathrm{DL}\cdot\mathrm{Poinc}_{!}$

where

$\mathrm{DL}=\mathrm{PsId}_{*}\cdot\mathrm{PsId}_{!}^{-1}\,[-2\delta_{G}]$

is the Deligne–Lusztig functor. This means that
$\mathrm{Poinc}_{*}\cdot\mathscr{W}_{\mathrm{vac}}$ should correspond under
Langlands duality to the Steinberg object constructed by D. Beraldo [Ber21].

###  1.5. Jacquet functors

#### 1.5.1.

In order to show that the cofiber of $\eta_{G}$ is cuspidal, we need to prove
that the morphism

$\mathrm{CT}_{P,!}^{-}\cdot\mathrm{Poinc}_{!}\cdot\mathscr{W}_{\mathrm{vac}}\to\mathrm{CT}_{P,!}^{-}\cdot\mathrm{PsId}_{!}\cdot\mathrm{coeff}^{\vee}\cdot\mathscr{W}_{\mathrm{vac},\mathrm{co}}\,[2\delta_{G}]$

is an equivalence for every proper parabolic $P$. We will do this by writing
both sides in terms of Poincare series for $M$ and applying induction. This
will force us to consider the Ran version of the Whittaker category.

####  1.5.2. The LHS

We will construct a Jacquet functor

$\mathrm{Jac}:\mathrm{Whit}(\mathrm{Bun}_{N})\to\mathrm{Whit}(M)_{\mathrm{Ran}}$

fitting into a commutative square:

${\mathrm{Whit}(G)}$${\mathrm{Whit}(M)_{\mathrm{Ran}}}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{D}(\mathrm{Bun}_{M})}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{Jac}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{CT}_{P}^{-}}$

The right vertical functor appearing in this diagram is the composite

$\mathrm{Whit}(M)_{\mathrm{Ran}}\to\mathrm{D}(\mathrm{Bun}_{M})_{\mathrm{Ran}}\to\mathrm{D}(\mathrm{Bun}_{M}),$

where the first arrow is the family version of Poincare series appearing in
Theorem 1.2.4 for $M$, and the second arrow is integration over the Ran space.
The proof that this square is commutative will require geometric input in the
form of a ULA property established in §5.

####  1.5.3. The RHS

First recall that

$\mathrm{CT}^{-}_{P,!}\cdot\mathrm{PsId}_{G,!}\simeq\mathrm{PsId}_{M,!}\cdot(\mathrm{Eis}_{P,!}^{-})^{\vee}$

because, as objects of $\mathrm{D}(\mathrm{Bun}_{G}\times\mathrm{Bun}_{M})$,
both identify by base-change with

$(p\times q)_{!}\cdot k_{\mathrm{Bun}_{P^{-}}}.$

Then, similarly to the above, we will construct a functor

$\mathrm{Jac}_{\mathrm{co}}:\mathrm{Whit}(M)_{\mathrm{Ran}}\to\mathrm{Whit}(\mathrm{Bun}_{N})$

equipped with a commutative square:

${\mathrm{D}(\mathrm{Bun}_{M})}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{Whit}(M)_{\mathrm{Ran}}}$${\mathrm{Whit}(G)}$$\scriptstyle{\mathrm{Eis}_{!}^{-}}$$\scriptstyle{\mathrm{coeff}_{*}}$$\scriptstyle{\mathrm{coeff}_{*}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{co}}}$

The functors $\mathrm{Jac}$ and $\mathrm{Jac}_{\mathrm{co}}$ will be dual to
each other in the sense that there a commutative diagram:

${\mathrm{Whit}(G)}$${\mathrm{Whit}(M)_{\mathrm{Ran}}}$${\mathrm{Whit}(G)^{\vee}}$${\mathrm{Whit}(M)_{\mathrm{Ran}}^{\vee}}$$\scriptstyle{\mathrm{Jac}}$$\scriptstyle{\mathbf{D}{[2\delta_{G}]}}$$\scriptstyle{\mathbf{D}{[2\delta_{M}]}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{co}}^{\vee}}$

#### 1.5.4.

We organize this discussion by pasting everything into a large diagram.

(1.5.4.1)
${\mathrm{Whit}(M)_{\mathrm{Ran}}^{\vee}}$${\mathrm{Whit}(M)_{\mathrm{Ran}}}$${\mathrm{Whit}(G)^{\vee}}$${\mathrm{Whit}(G)}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{D}(\mathrm{Bun}_{M})^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{M})}$$\scriptstyle{\mathrm{coeff}_{*}^{\vee}}$$\scriptstyle{\mathbf{D}{[2\delta_{M}]}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{coeff}_{*}^{\vee}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{co}}^{\vee}}$$\scriptstyle{\mathbf{D}{[2\delta_{G}]}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{Jac}}$$\scriptstyle{\mathrm{Eis}_{P,!}^{-,\vee}}$$\scriptstyle{\mathrm{PsId}_{!}}$$\scriptstyle{\mathrm{CT}_{P,!}^{-}}$$\scriptstyle{\mathrm{PsId}_{M,!}}$

In §8, we shall extend the construction of $\eta_{G}$ to a
$\mathrm{Sph}_{G,\mathrm{Ran}}$-linear morphism

$\eta_{G,\mathrm{Ran}}:\mathrm{Poinc}_{\mathrm{Ran},!}\cdot\mathrm{PsId}_{\mathrm{Ran},!}^{\mathrm{Whit}}\to\mathrm{PsId}_{G,!}\cdot\mathrm{coeff}_{\mathrm{Ran},*}^{\vee}\,[2\delta_{G}].$

It is compatible in the sense that its evaluation on
$\mathscr{W}_{\mathrm{vac}}$ agrees with $\eta_{G}$. Therefore, Theorem 1.1.5
for a given group implies Theorem 1.2.4 for that same group.

###### 1.5.4 Lemma.

Complete (1.5.4.1) into a hexahedron of functors and natural transformations
by pasting $\eta_{M,\mathrm{Ran}}$ along the outer edges. Then there exists a
3-morphism rendering the cube commutative.

#### 1.5.5.

We were not able to prove Lemma 1.5.4 directly. Instead, the proof of the
theorem goes through a slightly different statement from which this lemma can
be deduced. This is a rather unsatisfactory situation because the resulting
3-morphism is non-canonical.

#### 1.5.6.

Let us explain how Theorem 1.1.5 would follow by induction on $G$ if we knew
Lemma 1.5.4. Suppose that $\eta_{M}$ is an isomorphism for all proper Levi
subgroups of $G$. By $\mathrm{Sph}_{M,\mathrm{Ran}}$-linearity,
$\eta_{M,\mathrm{Ran}}$ is a an isomorphism for all proper Levis $M$. Then
Lemma 1.5.4 implies that $\mathrm{CT}_{P,!}^{-}\cdot\eta_{G}$ is an
isomorphism for all proper Levis $M$. This proves Lemma 1.3.3 for $G$.
Combining this with Lemma 1.3.3 proves that $\eta_{G}$ is an isomorphism.

###  1.6. Acknowledgements

#### 1.6.1.

This problem was suggested by our advisor D. Gaitsgory, and this work would
not have existed without his steady guidance and support. We are grateful to
S. Raskin for conversations which lead to the argument of §5. We are also
grateful to L. Chen and Y. Fu for many patient explanations and for their
encouragement.

#### 1.6.2.

We thank L. Chen and D. Gaitsgory for their comments on an earlier draft of
this work.

##  2\. Construction of the map

#### 2.0.1.

In this section we will construct the map

$\eta_{G}:\mathrm{Poinc}_{!}\cdot\mathrm{PsId}_{\mathrm{Whit}}^{-1}\to\mathrm{PsId}_{!}\cdot\mathrm{coeff}_{*}^{\vee}[2\delta_{G}].$

###  2.1. Reminders on Vinberg’s semigroup

#### 2.1.1.

We write

$T_{\mathrm{ad}}=T/Z_{G}\simeq(\mathbf{G}_{m})^{\Delta}$

for the adjoint torus. We write

$T_{\mathrm{ad}}^{+}=(\mathbf{A}^{1})^{\Delta},$

viewed as a semigroup. The points in $T_{\mathrm{ad}}^{+}$ with zero-one
coördinates are in natural bijection with subsets of $\Delta$ and hence
standard parabolics. We write $c_{P}$ for the point corresponding to the
parabolic $P$. The $T$ orbit of $c_{P}$ identifies with
$(T_{M})_{\mathrm{ad}}\simeq T/Z_{M}$, and these orbits form a stratification
of $T_{\mathrm{ad}}^{+}$.

####  2.1.2. Vinberg’s semigroup

Consider the group homomorphism

$G\overset{Z_{G}}{\times}T\to T_{\mathrm{ad}}.$

This map is naturally $G\times T\times G$ equivariant, where $G\times G$ acts
on the left and right on the source and trivially on the target. Vinberg’s
semigroup $\mathrm{Vin}_{G}$ is an algebraic semigroup equipped with a
$G\times G\times T$-equivariant semigroup morphism

$\mathrm{Vin}_{G}\to T_{\mathrm{ad}}^{+}$

extending the above group homomorphism. This morphism has a canonical “unit
section”

$s:T^{+}_{\mathrm{ad}}\to\mathrm{Vin}_{G}$

whose orbit under $G\times G$ is an open subscheme denoted
$\mathrm{Vin}_{G}^{\circ}$.

####  2.1.3. Stratification

The fiber of

$T_{\mathrm{ad}}^{+}\to\mathrm{Vin}_{G}^{\circ}\hookrightarrow\mathrm{Vin}_{G}$

over $c_{P}$ is canonically identified with

(2.1.3.1) $\mathrm{pt}\simeq P^{-}/U_{P}^{-}\overset{M}{\times}U_{P}\backslash
P\to G/U_{P}^{-}\overset{M}{\times}U_{P}\backslash
G\hookrightarrow\overline{G/U_{P}^{-}\overset{M}{\times}U_{P}\backslash G}$

with the natural $G\times G$ action. The residual action of $Z_{M}$ is
descended from the right and left actions on $G/U_{P}^{-}$ and
$U_{P}\backslash G$, respectively.

####  2.1.4. Generic maps

Let us recall the notion of generic maps from [J. Barlev]. For any scheme $S$,
an open subset $U\hookrightarrow X\times S$ is called a domain if it is open
dense after any base change $S^{\prime}\to S$. For any target $\mathscr{Y}$,
we define the prestack

$\mathrm{Maps}^{\mathrm{gen}}(X,\mathscr{Y}):S\mapsto\lim_{U}\mathrm{Maps}(U,\mathscr{Y})$

where the limit is taken over all domains $U\hookrightarrow X\times S$. Given
a map $\mathscr{Y}_{1}\to\mathscr{Y}_{2}$, we define
$\mathrm{Maps}^{\mathrm{gen}}(X,\mathscr{Y}_{1}\to\mathscr{Y}_{2})$ as the
fiber product:

${\mathrm{Maps}^{\mathrm{gen}}(X,\mathscr{Y}_{1}\to\mathscr{Y}_{2})}$${\mathrm{Maps}(X,\mathscr{Y}_{2})}$${\mathrm{Maps}^{\mathrm{gen}}(X,\mathscr{Y}_{1})}$${\mathrm{Maps}^{\mathrm{gen}}(X,\mathscr{Y}_{1})}$

#### 2.1.5.

We define

$\mathrm{Vin}\mathrm{Bun}_{G}=\mathrm{Maps}^{\mathrm{gen}}\left(X,G\backslash\mathrm{Vin}^{\circ}_{G}/G\hookrightarrow
G\backslash\mathrm{Vin}_{G}/G\right).$

This a stack over $\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}\times
T_{\mathrm{ad}}^{+}$, and we may take the quotient by $T$ to obtain

$\overline{\mathrm{Bun}}_{G}=\mathrm{VinBun}_{G}/T,$

which is a stack over $\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}\times
T_{\mathrm{ad}}^{+}/T$. The stratification of $T_{\mathrm{ad}}^{+}/T$ by the
poset of standard parabolics induces a stratification of
$\overline{\mathrm{Bun}}_{G}$, whose pieces we shall denote
$\overline{\mathrm{Bun}}_{G,P}$. Note that the open stratum identifies with

$\overline{\mathrm{Bun}}_{G,G}\simeq\mathrm{Bun}_{G}\times\mathbf{B}Z_{G}.$

####  2.1.6. Reminders on $\mathrm{PsId}_{*}$ and $\mathrm{PsId}_{!}$

There is a sequence of maps

$\mathrm{Bun}_{G}\overset{r}{\longrightarrow}\mathrm{Bun}_{G}\times\mathbf{B}Z_{G}\overset{j}{\hookrightarrow}\overline{\mathrm{Bun}}_{G}\overset{\overline{\Delta}}{\longrightarrow}\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}$

with the following features:

  * –

We have $r_{!}\simeq r_{*}[z_{G}]$.

  * –

The map $\overline{\Delta}$ is proper.

Therefore, we obtain a map

$\Delta_{!}\cdot k_{\mathrm{Bun}_{G}}\to\Delta_{*}\cdot
k_{\mathrm{Bun}_{G}}[z_{G}]\simeq\Delta_{*}\cdot\omega_{\mathrm{Bun}_{G}}[z_{G}-2\delta_{G}].$

This gives the map

$\mathrm{PsId}_{!}\to\mathrm{PsId}_{*}[z_{G}-2\delta_{G}].$

###  2.2. Definition of $\eta_{G}$

####  2.2.1. Notation

We will write

$r_{G,!}=j_{!}\cdot r_{!}\cdot
k_{\mathrm{Bun}_{G}}\in\mathrm{D}\left(\overline{\mathrm{Bun}}_{G}\right).$

#### 2.2.2.

The object $\mathscr{W}_{\mathrm{vac}}$ extends cleanly along
$\mathrm{Bun}_{N}\hookrightarrow\overline{\mathrm{Bun}}_{N}$. We use the same
symbol for this extension, and we make the identification

$\mathrm{Whit}\left(\overline{\mathrm{Bun}}_{N}\right)\simeq\mathrm{Whit}(\mathrm{Bun}_{N}).$

We shall write the definition of $\eta_{G}$ using
$\overline{\mathrm{Bun}}_{N}$ even though it is slightly more complicated
because the $\overline{\mathrm{Bun}}_{N}$ version will be needed in the
construction of the Jacquet functor.

#### 2.2.3.

Consider the Cartesian diagram below.

${\overline{\mathrm{Bun}}_{N}}$${\mathrm{Bun}_{G}}$${\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}}$${\overline{\mathrm{Bun}}_{G}}$${\overline{\mathrm{Bun}}_{N}\times\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}}$$\scriptstyle{{{}^{\prime}}\overline{\Delta}}$$\scriptstyle{\mathrm{pr}_{2}}$

We write

$\pi_{2}=\mathrm{pr}_{2}\cdot{{}^{\prime}}\overline{\Delta}.$

Note that we have the identifications

$\mathrm{Poinc}_{!}\cdot\mathscr{W}_{\mathrm{vac}}\simeq\pi_{2,!}\left(\mathscr{W}_{\mathrm{vac}}\overset{*}{\boxtimes}r_{G,!}\right)\qquad\mathrm{PsId}_{!}\cdot\mathrm{coeff}_{*}^{\vee}\cdot\mathscr{W}_{\mathrm{vac},\mathrm{co}}\simeq\pi_{2,*}\left(\mathscr{W}_{\mathrm{vac}}\overset{!}{\boxtimes}r_{G,!}\right)$

by base change. Therefore, we expect that the purity morphism and forgetting
supports combine to give a map

$\pi_{2,!}\left(\mathscr{W}_{\mathrm{vac}}\overset{!}{\boxtimes}r_{G,!}\right)\to\pi_{2,*}\left(\mathscr{W}_{\mathrm{vac}}\overset{*}{\boxtimes}r_{G,!}\right)[2\delta_{G}].$

Unfortunately, $\pi_{2}$ is not representable, so there is no forgetting
supports morphism $\pi_{2,!}\to\pi_{2,*}$. To circumvent this issue, we will
construct an open substack

(2.2.3.1)
$j:\left(\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{tr}}\hookrightarrow\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\overline{\mathrm{Bun}}_{G}$

containing $\overline{\mathrm{Bun}}_{N}\times\mathbf{B}Z_{G}$ with the
following features:

###### 2.2.3 Proposition.

Let

$i:\left(\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{cis}}\hookrightarrow\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\overline{\mathrm{Bun}}_{G}$

denote the closed complement to $j$. Then

$\pi_{2,*}\cdot i_{*}\cdot
i^{!}\left(\mathscr{W}\overset{!}{\boxtimes}r_{G,!}\right)\simeq 0.$

###### 2.2.3 Proposition.

The projection $\pi_{2}\cdot j$ onto $\mathrm{Bun}_{G}$ is representable.

#### 2.2.4.

We think of this open substack a locus of transversality, and we will write
$(-)^{\mathrm{tr}}$ for restriction along $j$. Then $\eta_{G}$ is defined via
the diagram:

${\pi_{2,!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}r_{!}\right)^{\mathrm{tr}}}$${\pi_{2,*}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}r_{!}\right)[2\delta_{G}]}$${\pi_{2,!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}r_{!}\right)}$${\pi_{2,*}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}r_{!}\right)^{\mathrm{tr}}[2\delta_{G}]}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

The left diagonal map is an isomorphism because
$\mathscr{W}\overset{*}{\boxtimes}r_{!}$ is $!$-extended from
$\mathrm{Bun}_{N}\times\mathbf{B}Z_{G}$. The right diagonal map is an
isomorphism by virtue of Proposition 2.2.3. The middle diagonal map is a
combination of the purity morphism and the forgetting supports morphism for
$\pi_{2}\cdot j$, which makes sense by Proposition 2.2.3.

#### 2.2.5.

We now give the definition of (2.2.3.1). The proofs of Propositions 2.2.3 and
2.2.3 will appear in the subsequent subsections.

###### 2.2.5 Proposition.

The $N\times G$ orbit of $s$ is an open subscheme
$\mathrm{Vin}_{G}^{\circ\circ}\hookrightarrow\mathrm{Vin}_{G}^{\circ}$. It is
equal to the $B\times G$ orbit of $s$.

####  2.2.6. Proof of Proposition 2.2.5

Since smooth morphisms are open, it suffices to show that the morphism

$N\times T_{\mathrm{ad}}^{+}\times G\to N\times\mathrm{Vin}_{G}^{\circ}\times
G\to\mathrm{Vin}_{G}^{\circ}$

given by acting on the unit section is smooth. This is a map of schemes smooth
over $T_{\mathrm{ad}}^{+}$, so we can check the smoothness fiberwise using the
explicit description (2.1.3.1) of the fibers.

#### 2.2.7.

Now

$\overline{\mathrm{Bun}}_{B}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}\simeq\mathrm{Maps}^{\mathrm{gen}}\left(X,B\backslash\mathrm{Vin}_{G}^{\circ}/G\hookrightarrow
T\backslash\overline{N\backslash
G}\overset{G}{\times}\mathrm{Vin}_{G}/G\right)$

and we define

$\left(\overline{\mathrm{Bun}}_{B}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{tr}}=\mathrm{Maps}^{\mathrm{gen}}\left(X,B\backslash\mathrm{Vin}_{G}^{\circ\circ}/G\hookrightarrow
T\backslash\overline{N\backslash
G}\overset{G}{\times}\mathrm{Vin}_{G}/G\right)$

We take

$\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{tr}}=\overline{\mathrm{Bun}}_{N}\underset{\overline{\mathrm{Bun}}_{B}}{\times}\left(\overline{\mathrm{Bun}}_{B}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{tr}}$

The map

$\overline{\mathrm{Bun}}_{N}\times\mathbf{B}Z_{G}\to\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}$

factors through the transverse locus because $\mathrm{Vin}_{G}^{\circ\circ}$
contains
$\mathrm{Vin}_{G}^{\circ}\underset{T^{+}_{\mathrm{ad}}}{\times}T_{\mathrm{ad}}$.

####  2.2.8. Sanity check

Let us study the extent to which this device depends on the choice of open
substack. Suppose that we have a diagram

${\mathscr{U}}$${\mathscr{Y}}$${S}$$\scriptstyle{j}$$\scriptstyle{\pi}$

in which $j$ is an open embedding and $\pi$ is representable. Then for any map
$\mathscr{F}\to\mathscr{G}$ of sheaves on $\mathscr{Y}$, we have an obvious
commutative diagram:

${j_{!}\cdot\mathscr{F}_{\mathscr{U}}}$${\mathscr{F}}$${j_{*}\cdot\mathscr{F}_{\mathscr{U}}}$${j_{!}\cdot\mathscr{G}_{\mathscr{U}}}$${\mathscr{G}}$${j_{*}\cdot\mathscr{G}_{\mathscr{U}}}$

This induces

${\pi_{!}\cdot
j_{!}\cdot\mathscr{F}_{\mathscr{U}}}$${\pi_{!}\cdot\mathscr{F}}$${\pi_{*}\cdot
j_{*}\cdot\mathscr{F}_{\mathscr{U}}}$${\pi_{!}\cdot
j_{!}\cdot\mathscr{G}_{\mathscr{U}}}$${\pi_{*}\cdot\mathscr{G}}$${\pi_{*}\cdot
j_{*}\cdot\mathscr{G}_{\mathscr{U}}}$

This diagram exhibits the sense in which our morphism does not depend on the
choice of $\mathscr{U}$.

#### 2.2.9.

This sanity check shows that the definition of $\eta_{G}$ would be unchanged
if we replaced $\overline{\mathrm{Bun}}_{N}$ with $\mathrm{Bun}_{N}$
everywhere.

###  2.3. Example: what happens for $\mathrm{SL}_{2}$

#### 2.3.1.

The rest of this paper has no logical dependencies on this subsection.

#### 2.3.2.

We fix $G=\mathrm{SL}_{2}$ for this subsection. In this case the objects at
hand admit the following descriptions:

  * –

The stack $\mathrm{Vin}\mathrm{Bun}_{G}$ classifies
$(\mathscr{E}_{1},\mathscr{E}_{2},\alpha)$ where the $\mathscr{E}_{i}$ are a
pair of $\mathrm{SL}_{2}$-bundles and $\alpha$ is a nonzero map of coherent
sheaves

$\alpha:\mathscr{E}_{1}\to\mathscr{E}_{2}.$

The torus $T\simeq\mathbf{G}_{m}$ acts by scaling $\alpha$.

  * –

The quotient $\overline{\mathrm{Bun}}_{G}$ classifies
$(\mathscr{E}_{1},\mathscr{E}_{2},L,\alpha)$ where the $\mathscr{E}_{i}$ are a
pair of $\mathrm{SL}_{2}$-bundles, $L$ is a line, and $\alpha$ is a nonzero
map of coherent sheaves

$\alpha:\mathscr{E}_{1}\to\mathscr{E}_{2}\otimes L.$

The open stratum $\overline{\mathrm{Bun}}_{G,G}$ is the locus where $\alpha$
is an isomorphism, and the closed stratum $\overline{\mathrm{Bun}}_{G,B}$ is
the locus where $\alpha$ has rank one.

  * –

The fiber product
$\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}$
classifies the data of

$\omega^{1/2}\hookrightarrow\mathscr{E}_{1}\to\mathscr{E}_{2}\otimes L.$

The transverse locus is the open substack where the composed map
$\omega^{1/2}\to\mathscr{E}_{2}\otimes L$ is nonzero.

#### 2.3.3.

Let’s verify by hand that the transverse locus is representable over
$\mathrm{Bun}_{G}$. The $G$-stratum of the transverse locus is simply

$\overline{\mathrm{Bun}}_{N}\times\mathbf{B}Z_{G}\to\mathrm{Bun}_{G}$

so we may restrict attention to the $B$-stratum. This stratum has a further
stratification by the degree of $\mathrm{im}(\alpha)$. Now the stratum where
$\mathrm{im}(\alpha)$ has degree $d$ can be perceived as classifying a point
$\mathscr{E}_{2}$ of $\mathrm{Bun}_{G}$ along with:

  * (i)

a choice of degree $d$ divisor $D$,

  * (ii)

a line $L$ along with an injective map of coherent sheaves

$\omega^{1/2}(D)\hookrightarrow\mathscr{E}_{2}\otimes L,$

and

  * (iii)

an $\mathrm{SL}_{2}$ bundle $\mathscr{E}_{1}$ fitting into a triangle

${\omega^{1/2}}$${\mathscr{E}_{1}}$${\omega^{1/2}(D)}$

We claim that the stacks parameterizing each additional data are schematic
over the previous ones. So for (i) we are looking at the projection

$\mathrm{Sym}^{d}(X)\times\mathrm{Bun}_{G}\to\mathrm{Bun}_{G},$

which is schematic because $\mathrm{Sym}^{d}(X)$ is a scheme. Then the choices
in (ii) are parameterized by the generalized projective bundle over
$\mathrm{Sym}^{d}(X)\times\mathrm{Bun}_{G}$ whose fibers are

$\mathbf{P}H^{0}\left(\mathrm{Maps}(\omega^{1/2}(D),\mathscr{E}_{1})\right).$

Finally, the data in (iii) are parametrized by the vector bundle over this
projective bundle whose fibers are

$\displaystyle\mathrm{fib}\left\\{\mathrm{Maps}\left(\omega^{1/2}(D),\omega^{-1/2}(-D)[1]\right)\to\mathrm{Maps}\left(\omega^{1/2},\omega^{-1/2}(-D)[1]\right)\right\\}$
$\displaystyle\qquad\qquad\simeq\mathrm{Maps}\left(\omega^{1/2}_{D},\omega^{-1/2}(-D)[1]\right)\simeq\mathrm{Maps}\left(\mathscr{O},(\omega_{D})^{\otimes
2}\right)^{\vee}.$

###  2.4. Proof of Proposition 2.2.3

#### 2.4.1.

We prove Proposition 2.2.3 by a series of reductions. Let us write

$\mathscr{F}=\mathscr{W}\overset{!}{\boxtimes}r_{G,!}.$

####  2.4.2. First reduction

We use the stratification of $\overline{\mathrm{Bun}}_{G}$ into the strata
$\overline{\mathrm{Bun}}_{G,P}$. It suffices to show that the !-restrictions
of $\mathscr{F}$ to each

$\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G,P}\right)^{\mathrm{cis}}$

are killed by $*$-pushforward to $\mathrm{Bun}_{G}$.

####  2.4.3. A generality

Let $T$ be any algebraic torus, and consider a principal $T$-bundle

${P}$${\mathrm{pt}}$${S}$${\mathbf{B}T}$$\scriptstyle{f}$$\scriptstyle{\alpha}$

For any sheaf $\mathscr{F}$ on $S$ we have

$\Gamma(S,\mathscr{F})\simeq\mathrm{Maps}(k_{\mathbf{B}T},\alpha_{*}\cdot\mathscr{F})\simeq\mathrm{Maps}_{\mathrm{H}_{*}(T)}\left(r^{!}\cdot
k_{\mathbf{B}T},\Gamma(P,\mathscr{F})\right).$

####  2.4.4. Second reduction

Let $\mathrm{Vin}\mathrm{Bun}_{G,P}$ denote the fiber of
$\mathrm{Vin}\mathrm{Bun}_{G}$ over the point $c_{P}\in T_{\mathrm{ad}}^{+}$.
We have

$\mathrm{Bun}_{G,P}\simeq\mathrm{Vin}\mathrm{Bun}_{G,P}/Z_{M}.$

By the above, it suffices to show that the !-pullbacks of $\mathscr{F}$ to

$\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{VinBun}_{G,P}\right)^{\mathrm{cis}}$

are killed by $*$-pushforward to $\mathrm{Bun}_{G}$.

####  2.4.5. Third reduction

Recall that $\mathrm{VinBun}_{G,P}$ has a stratification of the form

(2.4.5.1)
$\mathrm{Bun}_{P^{-}}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+}\underset{\mathrm{Bun}_{M}}{\times}\mathrm{Bun}_{P}\to\mathrm{VinBun}_{G,P}.$

Therefore, it suffices to show that the !-pullback of $\mathscr{F}$ to this
stratification is killed by $*$-pushforward along

$\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+}\underset{\mathrm{Bun}_{M}}{\times}\mathrm{Bun}_{P}\right)^{\mathrm{cis}}\to\mathrm{Bun}_{P}\to\mathrm{Bun}_{G}.$

####  2.4.6. Fourth reduction

Finally, we recall that

$\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}$

has a relative position stratification by

$W/W_{P}.$

It suffices to prove the analogous statement for

$\pi_{2}^{\mathrm{gr}}:\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}\right)^{w}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+}\underset{\mathrm{Bun}_{M}}{\times}\mathrm{Bun}_{P}\to\mathscr{H}_{M}^{+}\underset{\mathrm{Bun}_{M}}{\times}\mathrm{Bun}_{P}$

where $w\neq 1$.

####  2.4.7. End of the proof

By [Che20, Lemma 4.2.1], the !-restriction of $r_{!}$ to the above stack is
!-pulled back along $\pi_{2}^{\mathrm{gr}}$. Therefore, it suffices to observe
that $\mathscr{W}$ is annihilated by $*$-pushforward along

$\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}\right)^{w}\to\mathrm{Bun}_{M}.$

This is a standard fact whose proof we review in §4.4.

###  2.5. Proof of Proposition 2.2.3

#### 2.5.1.

To simplify notation, we will prove the analogue of Proposition 2.2.3 for the
_untwisted_ version of $\mathrm{Bun}_{N}$. The twisted version can be proved
in the same way by using the appropriate non-constant group scheme twist of
$N$ and replacing the notion of generic maps with generic sections.

####  2.5.2. Proof of Proposition 2.2.3

Recall that an Artin stack is an algebraic space if the stabilizer group of
every geometric point is trivial. Therefore, we only need to check that our
map induces injections on stabilizer groups when evaluated at geometric points
(note that this property is stable under base change). We shall do this after
passage to a stratification

####  2.5.3. Stratification

We will again use the stratification of $\overline{\mathrm{Bun}}_{G}$ into
$\overline{\mathrm{Bun}}_{G,P}$. We further stratify
$\overline{\mathrm{Bun}}_{G,P}$ using the quotient of (2.4.5.1) by $Z_{M}$.
Therefore, it suffices to show that the map

$\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}\right)^{\mathrm{tr}}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+}/Z_{M}\underset{\mathrm{Bun}_{M}}{\times}\mathrm{Bun}_{P}\to\mathrm{Bun}_{G}$

is representable. Since the morphism $\mathrm{Bun}_{P}\to\mathrm{Bun}_{G}$ is
schematic, we only need to check that the map

$\alpha:\left(\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}\right)^{\mathrm{tr}}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+}/Z_{M}\to\mathrm{Bun}_{M}$

is representable.

#### 2.5.4.

Now recall that the source of the stratification (2.4.5.1) is given by

$\mathrm{Maps}^{\mathrm{gen}}\left(X,P^{-}\backslash M/P\hookrightarrow
P^{-}\backslash M^{+}/P\right)$

for a certain monoid $M^{+}$ containing $M$ as its unit group. Therefore, in
terms of generic maps, the morphism $\alpha$ can be written as the composite:

$\displaystyle\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathbf{B}N_{M}\hookrightarrow
N\backslash G\overset{P^{-}}{\times}M^{+}/M\right)/Z_{M}$
$\displaystyle\qquad\hookrightarrow\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathbf{B}(N_{M}\cdot
Z_{M})\hookrightarrow N\backslash G\overset{P^{-}}{\times}M^{+}/M\cdot
Z_{M}\right)$
$\displaystyle\qquad\qquad\overset{\beta}{\longrightarrow}\mathrm{Maps}(X,\mathbf{B}M)$

The first arrow is a closed immersion because it is a base change of
$\mathbf{B}M\to\mathrm{Bun}_{M}$. Therefore, it suffices that to show that
$\beta$ induces injections on stabilizer groups when evaluated at geometric
points.

#### 2.5.5.

Let us write

$\mathscr{Y}=N\backslash G\overset{P^{-}}{\times}M^{+}/Z_{M}\cdot M\qquad
V=N_{M}\cdot Z_{M}$

for convenience. Then $\beta$ fits into the following Cartesian diagram.

${\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathbf{B}V\hookrightarrow\mathscr{Y}\right)}$${\mathrm{Maps}(X,\mathscr{Y})}$${\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathbf{B}V\to\mathbf{B}M\right)}$${\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathscr{Y}\to\mathbf{B}M\right)}$${\mathrm{Maps}(X,\mathbf{B}M)}$${\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathbf{B}V\right)}$${\mathrm{Maps}^{\mathrm{gen}}\left(X,\mathscr{Y}\right)}$${\mathrm{Maps}^{\mathrm{gen}}(X,\mathbf{B}M)}$$\scriptstyle{\star}$

The composite of the lower row induces injections on stabilizers of geometric
points, and hence so does the composite of the middle row. Therefore, it
suffices to show that the map ($\star$) has the same property. But in fact
this property holds for the composite

$\mathrm{Maps}(X,\mathscr{Y})\to\mathrm{Maps}^{\mathrm{gen}}(X,\mathscr{Y})$

because $\mathscr{Y}$ is the quotient of a scheme by an algebraic group.

##  3\. The cofiber of $\eta_{G}$ is Eisenstein series

#### 3.0.1.

The purpose of this section is to prove Lemma 1.3.3. We will use
$\mathrm{Whit}(\mathrm{Bun}_{N})$ and not
$\mathrm{Whit}\left(\overline{\mathrm{Bun}}_{N}\right)$.

###  3.1. A compatibility

#### 3.1.1.

The purpose of this subsection is to prove Proposition 1.4.4. Let us define
the map (1.4.2.1) first. Consider the factorization

$\mathrm{Bun}_{N}\overset{r}{\longrightarrow}\mathrm{Bun}_{N}/Z_{G}\overset{\pi}{\longrightarrow}\mathrm{Bun}_{G}.$

The action of $Z_{G}$ on $\mathrm{Bun}_{N}$ is trivial, so we have

$r_{!}\,[z_{G}]\simeq r_{*}.$

Combining this with the forgetting supports map for $\pi$ gives the desired
morphism

$\mathrm{Poinc}_{!}\,[z_{G}]\to\mathrm{Poinc}_{*}.$

####  3.1.2. Notation

We write

$r_{G,*}=j_{*}\cdot r_{*}\cdot k_{\mathrm{Bun}_{G}}\simeq j_{*}\cdot
r_{*}\cdot\omega_{\mathrm{Bun}_{G}}[-2\delta_{G}]\in\mathrm{D}\left(\overline{\mathrm{Bun}}_{G}\right).$

There is a canonical map

$r_{G,!}\,[z_{G}]\to r_{G,*}.$

####  3.1.3. Proof of Proposition 1.4.4

Recall that (1.4.4.1) is defined by applying the forgetting supports morphisms
$\pi_{2,!}\to\pi_{2,*}$ to the composite

(3.1.3.1)
$\left(\mathscr{W}\overset{*}{\boxtimes}r_{G,!}\right)^{\mathrm{tr}}[z_{G}]\to\left(\mathscr{W}\overset{!}{\boxtimes}r_{G,!}\right)^{\mathrm{tr}}[2\delta_{G}+z_{G}]\to\left(\mathscr{W}\overset{!}{\boxtimes}r_{G,*}\right)^{\mathrm{tr}}[2\delta_{G}].$

The source of this morphism is $!$-extended from the open substack

$j:\mathrm{Bun}_{N}\times\mathbf{B}Z_{G}\hookrightarrow\left(\mathrm{Bun}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{tr}}$

and the target is $*$-extended from the same locus. Therefore, (1.4.4.1)
identifies with the forgetting supports morphism for

$\mathrm{Bun}_{N}\times\mathbf{B}Z_{G}\to\mathrm{Bun}_{G}$

applied to the restriction of (3.1.3.1) along $j$. This restriction identifies
with the canonical map

$r_{!}\cdot\mathscr{W}\,[z_{G}]\to r_{*}\cdot\mathscr{W}.$

###  3.2. The cofiber is Eisenstein series

#### 3.2.1.

Let us form the diagram below.

${\mathrm{Bun}_{N}}$${\mathrm{Bun}_{N}/Z_{G}}$${\mathrm{Bun}_{N}/T}$${\mathrm{Bun}_{B}}$${\overline{\mathrm{Bun}}_{N}}$${\overline{\mathrm{Bun}}_{N}/Z_{G}}$${\overline{\mathrm{Bun}}_{N}/T}$${\overline{\mathrm{Bun}}_{B}}$$\scriptstyle{r}$$\scriptstyle{\rho}$$\scriptstyle{j}$$\scriptstyle{i}$

Observe that $i$ is a closed immersion because it is a base change of
$\mathbf{B}T\hookrightarrow\mathrm{Bun}_{T}$. So the morphism

$\mathrm{Poinc}_{!}[z_{G}]\to\mathrm{Poinc}_{*}$

is induced by the proper pushforward of

$j_{!}\cdot\rho_{!}\cdot
r_{!}\overset{\alpha}{\longrightarrow}j_{*}\cdot\rho_{!}\cdot
r_{!}\overset{\beta}{\longrightarrow}j_{*}\cdot\rho_{*}\cdot r_{!}\simeq
j_{*}\cdot\rho_{*}\cdot r_{*}\,[-z_{G}].$

along

$\pi:\overline{\mathrm{Bun}}_{N}/T\to\mathrm{Bun}_{G}.$

Therefore, Proposition 1.4.4 is a consequence of the following two assertions:

  * –

The cofiber of $\pi_{*}\cdot\beta$ belongs to the image of
$\mathrm{Eis}_{B,*}$. This will follow from Proposition 3.2.1.

  * –

The cofiber of $\pi_{*}\cdot\alpha$ lies in the full subcategory generated by
all $\mathrm{Eis}_{P,*}$. This will follow from Proposition 3.2.3.

###### 3.2.1 Proposition.

The object

$\mathrm{cofib}\left(\rho_{!}\cdot r_{!}\cdot\mathscr{W}\to\rho_{*}\cdot
r_{!}\cdot\mathscr{W}\right)$

lies in the image of

$q^{!}:\mathrm{D}(\mathrm{Bun}_{T})\to\mathrm{D}(\mathrm{Bun}_{B}).$

#### 3.2.2.

The mechanism responsible for this result is the following computation of G.
Laumon [Lau03]. Consider the diagram:

${\mathbf{G}_{\mathrm{a}}}$${\mathrm{pt}}$${\mathbf{G}_{\mathrm{a}}/\mathbf{G}_{m}}$${\mathbf{B}\mathbf{G}_{m}}$$\scriptstyle{\varrho}$$\scriptstyle{j}$$\scriptstyle{i}$

By the contraction principle,

$i^{*}\cdot\varrho_{*}\cdot\mathrm{AS}\simeq\Gamma(\mathbf{G}_{\mathrm{a}}/\mathbf{G}_{m},\varrho_{*}\cdot\mathrm{AS})\simeq
0.$

Then one obtains

$\rho_{*}\cdot\mathrm{AS}\simeq j_{!}\cdot k[1]$

by using base change to compute $j^{!}\cdot\varrho_{*}\cdot\mathrm{AS}$.
Therefore, we have fiber sequences

$k\,[1]\to\varrho_{!}\cdot\mathrm{AS}\to i_{*}\cdot
k_{\mathbf{B}\mathbf{G}_{m}}\qquad i_{*}\cdot
k_{\mathbf{B}\mathbf{G}_{m}}\to\varrho_{*}\cdot\mathrm{AS}\to k\,[1]$

One computes that the forgetting supports map for $\mathrm{AS}$ identifies
with

$\varrho_{!}\cdot\mathrm{AS}\to i_{*}\cdot
k_{\mathbf{B}\mathbf{G}_{m}}\to\varrho_{*}\cdot\mathrm{AS},$

so its cofiber is an extension of $k[1]$ by itself.

####  3.2.3. Proof

Recall that the Whittaker character is defined via a composite

$\mathrm{Bun}_{N}\overset{\mathrm{ev}}{\longrightarrow}N^{\mathrm{ab}}\simeq
T_{\mathrm{ad}}^{+}\overset{\mu}{\longrightarrow}\mathbf{G}_{\mathrm{a}}.$

Since the map $\mathrm{ev}$ is $T$-equivariant, we may form the Cartesian
diagram

${\mathrm{Bun}_{N}}$${\mathrm{Bun}_{N}\times\mathbf{B}Z_{G}}$${\mathrm{Bun}_{N}/T}$${T_{\mathrm{ad}}^{+}}$${T_{\mathrm{ad}}^{+}\times\mathbf{B}Z_{G}}$${T_{\mathrm{ad}}^{+}/T_{\mathrm{ad}}\times\mathbf{B}Z_{G}.}$$\scriptstyle{r}$$\scriptstyle{\mathrm{ev}}$$\scriptstyle{\rho}$$\scriptstyle{\mathrm{ev}/T}$$\scriptstyle{\varrho\times\mathbf{1}}$

Since $\mathrm{ev}$ is smooth, we have

$\displaystyle(\rho_{!}\to\rho_{*})\cdot r_{!}\cdot\mathscr{W}$
$\displaystyle\qquad\simeq(\rho_{!}\to\rho_{*})\cdot(\mathrm{ev}\times\mathbf{1})^{!}\cdot(\mu^{!}\cdot\mathrm{AS}\boxtimes
r_{!}\cdot k)$
$\displaystyle\qquad\qquad\simeq(\mathrm{ev}/T)^{!}\cdot(\varrho_{!}\to\varrho_{*})\cdot(\mu^{!}\cdot\mathrm{AS}\boxtimes
r_{!}\cdot k)$

by base change. Therefore, it suffices to prove that the object

$\mathrm{cofib}\left(\varrho_{!}\cdot\mu^{!}\cdot\mathrm{AS}\to\varrho_{*}\cdot\mu^{!}\cdot\mathrm{AS}\right)$

lies in the full subcategory of
$\mathrm{D}(T_{\mathrm{ad}}^{+}/T_{\mathrm{ad}})$ generated by the constant
sheaf. But $\varrho$ identifies with

$\mathbf{G}_{\mathrm{a}}\times\cdots\times\mathbf{G}_{\mathrm{a}}\to(\mathbf{G}_{\mathrm{a}}/\mathbf{G}_{m})\times\cdots\times(\mathbf{G}_{\mathrm{a}}/\mathbf{G}_{m})$

and

$\mu^{!}\cdot\mathrm{AS}\simeq\mathrm{AS}\boxtimes\cdots\boxtimes\mathrm{AS}$

because $\mathrm{AS}$ is a character sheaf. Therefore, the aforementioned
calculation (§3.2.2) induces a filtration on the cofiber in question by the
power poset of $\Delta$, each of whose graded pieces is

$k[1]\boxtimes\cdots\boxtimes k[1].$

###### 3.2.3 Proposition.

The direct image along the proper morphism
$\overline{\mathrm{Bun}}_{N}/T\to\mathrm{Bun}_{G}$ of any object supported on
the complement of

$\mathrm{Bun}_{N}/T\hookrightarrow\overline{\mathrm{Bun}}_{N}/T$

is Eisenstein series.

#### 3.2.4.

In what follows, we will prove this proposition in several steps under the
assumption that $X$ does not have genus zero. In the genus zero case, the
entire category $\mathrm{D}(\mathrm{Bun}_{G})$ is Eisenstein series if $G$ is
not a torus. The proposition is vacuous when $G$ is a torus.

####  3.2.5. First step

To analyse this complement, recall that $\overline{\mathrm{Bun}}_{B}$ has a
defect stratification indexed by dominant cocharacters

$\left(X^{\lambda}\times\mathrm{Bun}_{T}\right)\underset{\mathrm{Bun}_{T}}{\times}\mathrm{Bun}_{B}\simeq\overline{\mathrm{Bun}}_{B}^{\lambda}\hookrightarrow\overline{\mathrm{Bun}}_{B}.$

Here we are using the map

$X^{\lambda}\times\mathrm{Bun}_{T}\to\mathrm{Bun}_{T}\qquad(D,\mathscr{F}_{T})\mapsto\mathscr{F}_{T}(D).$

We will use the notation $\overline{\mathrm{Bun}}_{N}^{\lambda}/T$ for the
induced stratification of $\overline{\mathrm{Bun}}_{N}/T$. It suffices to
prove that for every nonzero $\lambda$, the image of $*$-pushforward along the
map

$\overline{\mathrm{Bun}}_{N}^{\lambda}/T\to\mathrm{Bun}_{G}$

is Eisenstein series.

####  3.2.6. Second step

Let $P$ be an arbitrary standard parabolic. Then we have the following
Cartesian diagram:

${\overline{\mathrm{Bun}}_{B}^{\lambda}}$${\mathrm{Bun}_{B}}$${\mathrm{Bun}_{P}}$${\overline{\mathrm{Bun}}_{B_{M}}^{\lambda}}$${\mathrm{Bun}_{B_{M}}}$${\mathrm{Bun}_{M}}$${X^{\lambda}\times\mathrm{Bun}_{T}}$${\mathrm{Bun}_{T}}$

This gives the Cartesian left face of the following diagram:

${\overline{\mathrm{Bun}}_{N}^{\lambda}/T}$${\overline{\mathrm{Bun}}_{N}/T}$${\overline{\mathrm{Bun}}_{N_{M}}^{\lambda}/T}$${\mathrm{Bun}_{P}}$${\mathrm{Bun}_{M}}$${\mathrm{Bun}_{G}}$$\scriptstyle{{}^{\prime}q}$

By base change, it suffices to show that for every nonzero $\lambda$, there
exists a proper parabolic for which the image of the functor

${{}^{\prime}q}^{!}:\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M}}^{\lambda}/T\right)\to\mathrm{D}\left(\overline{\mathrm{Bun}}_{N}^{\lambda}/T\right)$

generates the target.

####  3.2.7. Third step

For each $\lambda$, let $P_{\lambda}$ be the parabolic corresponding to the
set of positive roots $\alpha$ for which $\lambda(\alpha)\neq 0$. We claim
that if the genus of $X$ is nonzero, then after taking $P=P_{\lambda}$ in the
above notation, ${}^{\prime}q$ is tower of unipotent gerbes. This will prove
the proposition because $P_{\lambda}$ is a proper parabolic whenever
$\lambda\neq 0$.

####  3.2.8. Facts

Let us recall some basic facts before proving the claim.

  * (i)

Let $H\to G$ be a map of groups. Then the fiber of
$\mathrm{Bun}_{H}\to\mathrm{Bun}_{G}$ over a point $\mathscr{F}_{G}$
classifies sections of

$\mathscr{F}_{G}\overset{G}{\times}G/H.$

  * (ii)

Let $\mathbf{G}_{m}$ act on $\mathbf{B}\mathbf{G}_{\mathrm{a}}$ through the
dilation action on $\mathbf{G}_{\mathrm{a}}$. For any line bundle
$\mathscr{L}$ on $X$, sections of
$\mathscr{L}\overset{\mathbf{G}_{m}}{\times}\mathbf{B}\mathbf{G}_{\mathrm{a}}$
are classified by $\Gamma(X,\mathscr{L}[1])$.

Note that (ii) is obtained from (i) by taking
$H\hookrightarrow\mathrm{GL}_{2}$ to be the automorphism group of
$\mathbf{A}^{1}$.

####  3.2.9. Notation

Choose a sequence of quotients

$B=B_{0}\twoheadrightarrow B_{1}\twoheadrightarrow\cdots\twoheadrightarrow
B_{n}=B/U_{P}$

with the property that

$\ker(B_{i}\twoheadrightarrow B_{i+1})\simeq\mathbf{G}_{\mathrm{a}}$

is central in the unipotent radical of $B_{i}$. Then the action of $B_{i+1}$
on $B_{i+1}/B_{i}\simeq\mathbf{B}\mathbf{G}_{\mathrm{a}}$ factors through
$B_{i+1}\twoheadrightarrow T$. The weight of the $T$-action is some positive
root $\alpha_{i}$ of $M$.

####  3.2.10. Proof of the claim

It suffices to show that the map ${{}^{\prime}}q_{i}$ in the Cartesian diagram
below is a unipotent gerbe.

${\mathscr{Y}_{i}}$${\mathscr{Y}_{i+1}}$${X^{\lambda}}$${\mathrm{Bun}_{B_{i}}}$${\mathrm{Bun}_{B_{i+1}}}$${\mathrm{Bun}_{T}}$$\scriptstyle{{{}^{\prime}}q_{i}}$$\scriptstyle{\mathrm{AJ}_{\omega}}$

So fix a point of $\mathscr{Y}_{i+1}$, and say that its image in $X^{\lambda}$
is $D$. Then the fiber of ${{}^{\prime}}q_{i}$ over our chosen point is

(3.2.10.1)
$\Gamma\left(X,\omega^{\alpha_{i}({\check{\rho}})}(\alpha_{i}(D))[1]\right)\simeq\Gamma\left(X,\omega^{1-\alpha_{i}({\check{\rho}})}(-\alpha_{i}(D))\right)^{\vee}$

We claim that this complex is concentrated in negative degrees. Indeed, the
line bundle appearing in the RHS of this expression has degree

$(1-\alpha_{i}({\check{\rho}}))\cdot\deg(\omega_{X})-\alpha_{i}(D).$

This integer is negative because $\alpha_{i}({\check{\rho}})$ is a positive
integer, $\alpha_{i}(D)$ is a positive divisor of degree
$\alpha_{i}(\lambda)>0$, and $\deg(\omega_{X})\geq 0$ by our assumption on the
genus.

##  4\. The Jacquet functor

#### 4.0.1.

In this section we will define a certain category
$\mathrm{Whit}(M)_{\mathrm{Conf}}$ along with functors

$\mathrm{Jac}_{\mathrm{Conf}}:\mathrm{Whit}(G)\to\mathrm{Whit}(M)_{\mathrm{Conf}}\qquad\mathrm{Poinc}_{!}:\mathrm{Whit}(M)_{\mathrm{Conf}}\to\mathrm{D}(\mathrm{Bun}_{M}).$

Furthermore, we will construct a commutative square

(4.0.1.1)
${\mathrm{Whit}(G)}$${\mathrm{Whit}(M)_{\mathrm{Conf}}}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{D}(\mathrm{Bun}_{M})}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{Conf}}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{CT}_{P,!}^{-}}$

modulo a certain ULA result whose proof we defer to the next section.

###  4.1. Definition of the Jacquet functor

####  4.1.1. Zastava spaces

Write

${\widetilde{\mathscr{Z}}}=\mathscr{Z}_{\overline{N},\widetilde{P}^{-}}\hookrightarrow\overline{\mathrm{Bun}}_{N}\underset{\mathrm{Bun}_{G}}{\times}\widetilde{\mathrm{Bun}}_{P^{-}}$

for the open substack where the generic $N$ and $P^{-}$ reductions are
transverse. We will use the notation $(-)_{\widetilde{\mathscr{Z}}}$ to denote
restriction to ${\widetilde{\mathscr{Z}}}$. We will also use notation such as
$\mathscr{Z}_{\overline{B},\widetilde{P}^{-}}$,
$\mathscr{Z}_{\overline{N},P^{-}}$, and $\mathscr{Z}_{N,P^{-}}$ to denote the
similarly defined stacks with other subgroups and degeneracy conditions.

####  4.1.2. Configuration space

Recall that we have a map

$\Lambda_{G}\to\Lambda_{G,P}$

from the cocharacter lattice of $T$ to the cocharacter lattice of
$M^{\mathrm{ab}}$. We write $\Lambda_{G,P}^{\mathrm{pos}}$ for the submonoid
generated by the images of the simple roots not lying in the Dynkin subdiagram
corresponding to $M$. We write

$\mathrm{Conf}_{G,P}=\sqcup_{\theta\in\Lambda_{G,P}^{\mathrm{pos}}}X^{\theta}$

for the space of divisors colored by $\alpha_{i}$, $i\notin\Delta_{M}$.

#### 4.1.3.

Let us regard $X$ as a constant curve over $\mathrm{Conf}_{G,P}$ and write
$\overset{\circ}{X}\hookrightarrow X$ for the complement of the tautological
divisor.

  * –

We define

$\displaystyle\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}$
$\displaystyle=\mathrm{Maps}^{\mathrm{gen}}\left(\overset{\circ}{X},\mathbf{B}B_{M}\hookrightarrow
M\backslash\left(\overline{M/N_{M}}\right)/T\right)$
$\displaystyle\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{pol}}$
$\displaystyle=\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}\underset{\mathrm{Maps}(\overset{\circ}{X},\mathbf{B}M)}{\times}\left(\mathrm{Bun}_{M}\times\mathrm{Conf}\right).$

  * –

Consider the map

$\mathrm{Conf}\to\mathrm{Maps}\left(\overset{\circ}{X},\mathbf{B}T\right)$

given by restricting $\omega^{\check{\rho}}$ to $\overset{\circ}{X}$. We
define

$\displaystyle\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}$
$\displaystyle=\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}\underset{\mathrm{Maps}(\overset{\circ}{X},\mathbf{B}T)}{\times}\mathrm{Conf}$
$\displaystyle\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}$
$\displaystyle=\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{pol}}\underset{\mathrm{Maps}(\overset{\circ}{X},\mathbf{B}T)}{\times}\mathrm{Conf}.$

So $\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}$
classifies a point of $\mathrm{Conf}$ along with a degenerate $N_{M}$-bundle
on $\overset{\circ}{X}$. The stack
$\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}$
classifies this data along with an extension of the induced $M$-bundle to all
of $X$. The reason for introducing
$\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}$ is that it
receives a natural map from ${\widetilde{\mathscr{Z}}}$. The following will
appear in §4.3.

###### 4.1.3 Construction.

The map ${\widetilde{\mathscr{Z}}}\to\mathrm{Bun}_{M}$ admits a lift

$u:{\widetilde{\mathscr{Z}}}\to\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}.$

Furthermore, this morphism is proper.

#### 4.1.4.

The definition of the full subcategory

$\mathrm{Whit}(M)_{\mathrm{Conf}}=\mathrm{Whit}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)\hookrightarrow\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right).$

follows a standard pattern. Write

$\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}\right)^{\mathrm{good}}\hookrightarrow\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}$

for the open subset cut out by the condition that the point of $\mathrm{Ran}$
is disjoint from both the support of the divisor and the degeneracy locus of
the $N_{M}$-reduction. Let

(4.1.4.1)
${\mathscr{H}_{N_{M},\mathrm{Conf}}}$${\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\mathbf{G}_{\mathrm{a}}}$$\scriptstyle{\psi}$

denote the Hecke groupoid which acts by modifying the $N_{M}$-bundle, equipped
with its natural character. We will say that $\mathscr{W}$ belongs to
$\mathrm{Whit}(M)_{\mathrm{Conf}}$ if

$(\mathscr{W}\boxtimes\omega_{\mathrm{Ran}})^{\mathrm{good}}$

is equivariant against $\psi^{!}\cdot\mathrm{AS}$.

#### 4.1.5.

The functor

$\mathrm{oblv}:\mathrm{Whit}(M)_{\mathrm{Conf}}\hookrightarrow\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)$

admits a continuous right adjoint

$\mathrm{Av}_{*}:\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)\to\mathrm{Whit}(M)_{\mathrm{Conf}}$

and we define

$\mathrm{Poinc}_{\mathrm{Conf},!}=\pi_{\mathrm{Conf},!}\cdot\mathrm{oblv}\qquad\mathrm{coeff}_{\mathrm{Conf},*}=\mathrm{Av}_{*}\cdot\pi_{\mathrm{Conf}}^{!},$

where $\pi_{\mathrm{Conf}}$ is the projection to $\mathrm{Bun}_{M}$ displayed
in the diagram (4.1.6.1) below.

#### 4.1.6.

Now we are ready to describe the construction of
$\mathrm{Jac}_{\mathrm{Conf}}$. Consider the diagram:

(4.1.6.1)
${{\widetilde{\mathscr{Z}}}}$${\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\widetilde{\mathrm{Bun}}_{P^{-}}}$${\overline{\mathrm{Bun}}_{N}}$${\widetilde{\mathrm{Bun}}_{P^{-}}}$${\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}}$${\mathrm{Bun}_{G}}$${\mathrm{Bun}_{M}}$$\scriptstyle{u}$$\scriptstyle{{{}^{\prime}}\widetilde{p}}$$\scriptstyle{\pi}$$\scriptstyle{\widetilde{p}}$$\scriptstyle{\widetilde{q}}$$\scriptstyle{\pi_{\mathrm{Conf}}}$

Write $j_{P,!}$ for the $!$-extension of the constant sheaf from
$\mathrm{Bun}_{P^{-}}$ to $\widetilde{\mathrm{Bun}}_{P^{-}}$. The following
assertion is standard, but we will recall its proof at the end of the section.

###### 4.1.6 Proposition.

Let $i$ denote the closed complement to
${\widetilde{\mathscr{Z}}}\hookrightarrow\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\widetilde{\mathrm{Bun}}_{P^{-}}$.
Then the $!$-pushforward of

$i^{*}\left(\mathscr{W}\overset{*}{\boxtimes}j_{P,!}^{-}\right)$

to $\mathrm{Bun}_{M}$ is zero.

#### 4.1.7.

This vanishing gives an isomorphism

$\mathrm{CT}_{P}^{-}\cdot\mathrm{Poinc}_{!}\cdot\mathscr{W}\simeq(\widetilde{q}\cdot\pi)_{!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}.$

Since $u$ is proper, the map

$(\widetilde{q}\cdot\pi)_{!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\simeq\pi_{\mathrm{Conf},!}\cdot
u_{!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\to\pi_{\mathrm{Conf},!}\cdot
u_{*}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\,[2\delta_{G}].$

is an isomorphism by the following result, whose proof will occupy §5.

###### 4.1.7 Lemma.

The purity morphism

$\left(\mathscr{W}\overset{*}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\to\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\,[2\delta_{G}]$

is an isomorphism.

#### 4.1.8.

Therefore, we define

$\widetilde{\mathrm{Jac}}_{\mathrm{Conf}}=u_{*}\cdot\left(-\overset{*}{\boxtimes}j_{P,!}^{-}\right)_{\mathscr{Z}}\,[2\delta_{G}].$

and consider the diagram

${\mathrm{Whit}(G)}$${\mathrm{Whit}(M)_{\mathrm{Conf}}}$${\mathrm{D}\left(\overline{\mathrm{Bun}}_{N}\right)}$${\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)}$$\scriptstyle{\mathrm{Jac}}$$\scriptstyle{\widetilde{\mathrm{Jac}}}$

We define $\mathrm{Jac}_{\mathrm{Conf}}$ to be the upper dashed arrow, which
makes by virtue of the proposition below. It follows that

$\mathrm{CT}_{P,!}^{-}\cdot\mathrm{Poinc}_{!}\simeq\mathrm{Poinc}_{!}\cdot\mathrm{Jac}_{\mathrm{Conf}}.$

###### 4.1.8 Proposition.

The functor $\widetilde{\mathrm{Jac}}_{\mathrm{Conf}}$ takes Whittaker sheaves
to Whittaker sheaves.

###  4.2. Some invariant theory

#### 4.2.1.

Construction 4.1.3 is based on some elementary facts about the invariant
theory of $G$ and its subgroups. Everything we will need is summarized in
Proposition 4.2.3, whose proof can be obtained by a close reading of [Wan17,
§3].

####  4.2.2. Notation

For a classical affine scheme $X=\mathrm{Spec}(A)$ with an action of a group
$G$, we write

$X/\\!\\!/G=\mathrm{Spec}(A^{G})$

for the naive GIT quotient. Here we are taking invariants in the underived
sense. So, for example, we have $\overline{G/N}=G/\\!\\!/N$ in this notation.

#### 4.2.3.

Recall that the closure of

$M\simeq P/U_{P}\hookrightarrow G/\\!\\!/U_{P}$

is a reductive monoid containing $M$ as its group of units. We will denote it
by $M^{+}$. Similarly, the closure of

$M^{\mathrm{ab}}\simeq P/[P,P]\hookrightarrow G/\\!\\!/[P,P]$

is a reductive monoid containing $M^{\mathrm{ab}}$ as its group of units. We
denote it $M^{\mathrm{ab},+}$.

###### 4.2.3 Proposition.

Consider the commutative diagram

(4.2.3.1) ${U_{P}^{-}\backslash\\!\\!\backslash
P^{-}\overset{M}{\times}P/\\!\\!/N}$${U_{P}^{-}\backslash\\!\\!\backslash
G\overset{G}{\times}G/\\!\\!/N}$${U_{P}^{-}\backslash\\!\\!\backslash
P^{-}\overset{M}{\times}P/\\!\\!/[P,P]}$${U_{P}^{-}\backslash\\!\\!\backslash
G/\\!\\!/[P,P]}$

of affine schemes with $M\times T$ action.

  * (i)

This diagram is Cartesian.

  * (ii)

The lower row identifies with $M^{\mathrm{ab}}\hookrightarrow
M^{\mathrm{ab},+}$.

####  4.2.4. Order of operations

Our convention is that GIT quotient takes precedence over balanced product.

####  4.2.5. Proof of Proposition 4.2.3.(ii)

Actually (ii) is a consequence of a more precise assertion, which we state
separately:

###### 4.2.5 Proposition.

The composite

$M^{\mathrm{ab},+}\to G/\\!\\!/[P,P]\to U_{P}^{-}\backslash\\!\\!\backslash
G/\\!\\!/[P,P]$

is an isomorphism.

####  4.2.6. Proof

Consider the diagram below obtained by functoriality.

${M^{+}}$${G/\\!\\!/U_{P}}$${U_{P}^{-}\backslash\\!\\!\backslash
G/\\!\\!/U_{P}}$${M^{+}/\\!\\!/[M,M]}$${G/\\!\\!/[P,P]}$${U_{P}\backslash\\!\\!\backslash
G/\\!\\!/[P,P]}$

According to [Wan17, Theorem 3.2.8], the upper row is a retract. Therefore,
the lower row is also a retract. In fact, it is a retract of affine schemes,
so the lower left arrow is a closed immersion. Therefore, $M^{+}/\\!\\!/[M,M]$
contains $M^{\mathrm{ab},+}$ as a closed subscheme. But $M$ maps dominantly
onto both, so they are equal.

#### 4.2.7.

In order to prove Proposition 4.2.3.(i), it will be useful to have some
generalities on GIT quotients. Let $\mathscr{U}\hookrightarrow X$ be integral
affine schemes with $G$-action. We will say that the pair
$(\mathscr{U}\hookrightarrow X,G)$ is _good_ if the following square is
Cartesian:

${\mathscr{U}}$${X}$${\mathscr{U}/\\!\\!/G}$${X/\\!\\!/G}$

We will further say that the pair is _quite good_ if the lower arrow is a
distinguished open affine.

####  4.2.8. Example

Let $\mathscr{U}\hookrightarrow\mathbf{A}^{2}$ be the complement of a line
through the origin, and let $\mathbf{G}_{m}$ act by dilation. Then the pair
$(\mathscr{U}\hookrightarrow X,\mathbf{G}_{m})$ is not good.

###### 4.2.8 Proposition.

Let $X$ be an integral affine scheme with $G$-action.

  * (i)

For any distinguished open affine $V\hookrightarrow X/\\!\\!/G$, the pair

$\left(V\\!\underset{X/\\!\\!/G}{\times}\\!X\hookrightarrow X,G\right)$

is quite good.

  * (ii)

Suppose that $(\mathscr{U}\hookrightarrow X,G)$ is quite good. For any
subgroup $H\hookrightarrow G$, the pair $(\mathscr{U}\hookrightarrow X,H)$ is
quite good and the square

${\mathscr{U}/\\!\\!/H}$${X/\\!\\!/H}$${\mathscr{U}/\\!\\!/G}$${X/\\!\\!/G}$

is Cartesian.

  * (iii)

Let $H\hookrightarrow G$ be a normal subgroup. If $(\mathscr{U}\hookrightarrow
X,H)$ is good and $(\mathscr{U}/\\!\\!/H\hookrightarrow X/\\!\\!/H,G/H)$ is
quite good, then $(\mathscr{U}\hookrightarrow X,G)$ is quite good.

####  4.2.9. Proof of (i)

Let $\mathscr{U}$ denote the fiber product indicated by the diagram.

${\mathscr{U}}$${X}$${\mathscr{U}/\\!\\!/G}$${V}$${X/\\!\\!/G}$

We need to check that the dashed arrow is an isomorphism. Write
$X=\mathrm{Spec}(A)$ and let $f\in A^{G}$ be a function cutting out the
complement of $V$. Then the dashed arrow corresponds to the inclusion of rings

$A^{G}[\tfrac{1}{f}]\hookrightarrow A[\tfrac{1}{f}]^{G}.$

This is an isomorphism.

####  4.2.10. Proof of (ii)

Write $V$ for the fiber product appearing in the following Cartesian diagram:

${\mathscr{U}}$${X}$${\mathscr{U}/\\!\\!/H}$${V}$${X/\\!\\!/H}$${\mathscr{U}/\\!\\!/G}$${X/\\!\\!/G}$

Note that $V\hookrightarrow X/\\!\\!/H$ is a distinguished open affine because
$\mathscr{U}/\\!\\!/G\hookrightarrow X/\\!\\!/G$ is. Therefore we may apply
(i).

####  4.2.11. Proof of (iii)

This is a formal consequence of the pullback lemma.

####  4.2.12. Proof of Proposition 4.2.3.(i)

We give a sequence of reductions.

####  4.2.13. First reduction

It suffices to show that (4.2.3.1) is Cartesian after pullback along the
$G$-torsor

$U_{P}^{-}\backslash\\!\\!\backslash G\times G/\\!\\!/N\to
U_{P}^{-}\backslash\\!\\!\backslash G\overset{G}{\times}G/\\!\\!/N.$

So we need to show that the diagram

${U_{P}^{-}\backslash\\!\\!\backslash P^{-}\overset{M}{\times}P\times
G/\\!\\!/N}$${U_{P}^{-}\backslash\\!\\!\backslash G\times
G/\\!\\!/N}$${U_{P}^{-}\backslash\\!\\!\backslash
P^{-}\overset{M}{\times}P/\\!\\!/[P,P]}$${U_{P}^{-}\backslash\\!\\!\backslash
G/\\!\\!/[P,P]}$

is Cartesian. Let us spell out the precise definition of the upper arrow.
Consider the pair

(4.2.13.1) $\left(P^{-}\overset{M}{\times}P\times G\hookrightarrow G\times
G,U_{P}^{-}\times G\times[P,P]\right)$

where the map is

$(x,h)\mapsto(xh^{-1},h).$

and the group actions are

$(u,g,p)\cdot(x,h)=(u\cdot x\cdot p,g\cdot h\cdot
p)\qquad(u,g,p)\cdot(h,h^{\prime})=(u\cdot h\cdot g^{-1},g\cdot
h^{\prime}\cdot p).$

Taking the GIT quotient by the subgroup $U_{P}^{-}\times N\hookrightarrow
U_{P}^{-}\times G\times[P,P]$ gives the upper arrow.

####  4.2.14. Second reduction

Now Proposition 4.2.8.(ii) implies that it suffices to show that (4.2.13.1) is
quite good. Note that

$\left(P^{-}\overset{M}{\times}P\times G\hookrightarrow G\times G,G\right)$

is good. Therefore, by Proposition 4.2.8(iii), it is enough to prove that

$\left(P^{-}\overset{M}{\times}P\hookrightarrow G,U_{P}^{-}\times[P,P]\right)$

is quite good.

####  4.2.15. Third reduction

By Proposition 4.2.5, the diagram

${U_{P}^{-}\backslash\\!\\!\backslash\left(P^{-}\overset{M}{\times}P\right)/\\!\\!/U_{P}}$${U_{P}^{-}\backslash\\!\\!\backslash
G/\\!\\!/P}$${U_{P}^{-}\backslash\\!\\!\backslash\left(P^{-}\overset{M}{\times}P\right)/\\!\\!/[P,P]}$${U_{P}^{-}\backslash\\!\\!\backslash
G/\\!\\!/[P,P]}$

identifies with

${M}$${M^{+}}$${M^{\mathrm{ab}}}$${M^{\mathrm{ab},+}}$

This square is Cartesian by [Che20, Appendix C.4.3]. Therefore, by Proposition
4.2.8.(iii), it suffices to show that

$\left(P^{-}\overset{M}{\times}P\hookrightarrow G,U_{P}^{-}\times
U_{P}\right)$

is good.

####  4.2.16. End of the proof

It remains to show that the diagram

(4.2.16.1) ${P^{-}\overset{M}{\times}P}$${G}$${M}$${M^{+}}$

is Cartesian. For $P=B$ one can verify this by direct calculation [Wan17,
_Proof_ of Lemma 3.2.10]. The corresponding diagram is displayed on the left.

${B^{-}\overset{T}{\times}B}$${G}$${T}$${N^{-}\backslash\\!\\!\backslash
G/\\!\\!/N}$
${B^{-}\overset{T}{\times}B}$${G}$${B_{M}^{-}\overset{T}{\times}B_{M}}$${M^{+}}$

Proposition 4.2.8.(ii) now implies that the diagram on the right is also
Cartesian. But this diagram will remain Cartesian if we replace the left
vertical arrow with its $M$-orbit. The result of this operation is (4.2.16.1).

###  4.3. Zastava spaces

#### 4.3.1.

Now we will obtain a precise description of ${\widetilde{\mathscr{Z}}}$ in
local terms. In order to state the result, we will need to introduce some more
stacks over $\mathrm{Conf}_{G,P}$.

#### 4.3.2.

As before, let us view $X$ as a constant curve over $\mathrm{Conf}$. Write

$\overset{\circ}{\mathscr{D}}\hookrightarrow\mathscr{D}\hookrightarrow X$

for the punctured formal disc inside the formal disc around the support of the
tautological divisor. For a map $\mathscr{Y}_{1}\to\mathscr{Y}_{2}$, we use
the notation

$\mathrm{Maps}^{\mathrm{gen}}(\mathscr{D},\mathscr{Y}_{1}\to\mathscr{Y}_{2})=\mathrm{Maps}(\mathscr{D},\mathscr{Y}_{2})\underset{\mathrm{Maps}(\overset{\circ}{\mathscr{D}},\mathscr{Y}_{2})}{\times}\mathrm{Maps}(\overset{\circ}{\mathscr{D}},\mathscr{Y}_{1}).$

####  4.3.3. Notation

  * –

We define

$\displaystyle\mathrm{Gr}_{G,\mathrm{Conf}}$
$\displaystyle=\mathrm{Maps}^{\mathrm{gen}}\left(\mathscr{D},\mathrm{pt}\to\mathbf{B}G\right)$
$\displaystyle\overline{\mathrm{Gr}}_{B,\mathrm{Conf}}$
$\displaystyle=\mathrm{Maps}^{\mathrm{gen}}\left(\mathscr{D},\mathrm{pt}\to
T\backslash(N\backslash\\!\\!\backslash G)/G\right)$
$\displaystyle\widetilde{\mathrm{Gr}}_{P^{-},\mathrm{Conf}}$
$\displaystyle=\mathrm{Maps}^{\mathrm{gen}}\left(\mathscr{D},\mathrm{pt}\to
G\backslash(G/\\!\\!/U_{P}^{-})/M\right)$

  * –

Note that there is a tautological section

(4.3.3.1)
$\mathrm{Conf}\to\mathrm{Maps}^{\mathrm{gen}}(\mathscr{D},M^{\mathrm{ab}}\backslash
M^{\mathrm{ab},+}).$

The map

$N\backslash\\!\\!\backslash G\overset{G}{\times}G/\\!\\!/U_{P}^{-}\to
M^{\mathrm{ab},+}$

induces a morphism

$\overline{\mathrm{Gr}}_{B,\mathrm{Conf}}\underset{\mathrm{Gr}_{G,\mathrm{Conf}}}{\times}\widetilde{\mathrm{Gr}}_{P^{-},\mathrm{Conf}}\to\mathrm{Maps}^{\mathrm{gen}}(\mathscr{D},M^{\mathrm{ab}}\backslash
M^{\mathrm{ab},+})$

and we define $\mathscr{Z}_{\overline{B},\widetilde{P},\mathrm{Conf}}$ to be
its base change along (4.3.3.1).

####  4.3.4. Variants

  * –

Consider the map

$\mathrm{Conf}\to\mathrm{Maps}\left(\overset{\circ}{X},\mathbf{B}T\right)$

given by restricting $\omega^{\check{\rho}}$ to $\overset{\circ}{X}$. We
define

$\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}=\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}\underset{\mathrm{Maps}(\overset{\circ}{X},\mathbf{B}T)}{\times}\mathrm{Conf}.$

  * –

Consider the map

$\mathrm{Conf}\to\mathrm{Maps}(\mathscr{D},\mathbf{B}T)$

given by restricting $\omega^{\check{\rho}}$ to $\mathbf{D}$. We define

$\displaystyle\overline{\mathrm{Gr}}_{N,\mathrm{Conf}}$
$\displaystyle=\overline{\mathrm{Gr}}_{B,\mathrm{Conf}}\underset{\mathrm{Maps}(\mathscr{D},\mathbf{B}T)}{\times}\\{\omega_{\mathscr{D}}^{\check{\rho}}\\}$
$\displaystyle\mathscr{Z}_{\overline{N},\widetilde{P}^{-},\mathrm{Conf}}$
$\displaystyle=\mathscr{Z}_{\overline{B},\widetilde{P}^{-},\mathrm{Conf}}\underset{\mathrm{Maps}(\mathscr{D},\mathbf{B}T)}{\times}\\{\omega_{\mathscr{D}}^{\check{\rho}}\\}$

###### 4.3.4 Proposition.

Define $\mathscr{Z}_{\overline{B},\widetilde{P}^{-}}$ by replacing $N$ with
$B$ in the definition of
${\widetilde{\mathscr{Z}}}=\mathscr{Z}_{\overline{N},\widetilde{P}^{-}}$.
There are canonical isomorphisms

$\displaystyle\mathscr{Z}_{\overline{B},\widetilde{P}^{-}}$
$\displaystyle\simeq\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\widetilde{\times}\mathscr{Z}_{\overline{B},\widetilde{P}^{-},\mathrm{Conf}}$
$\displaystyle\mathscr{Z}_{\overline{N},\widetilde{P}^{-}}$
$\displaystyle\simeq\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\widetilde{\times}\mathscr{Z}_{\overline{N},\widetilde{P}^{-},\mathrm{Conf}}.$

#### 4.3.5.

Let us explain the meaning of these twisted products, which _prima facie_
involve fiber products over $\mathbf{B}\mathscr{L}_{\mathrm{Conf}}B_{M}$ and
$\mathbf{B}\mathscr{L}_{\mathrm{Conf}}N_{M}^{\omega}$. To avoid the problem of
taking quotients by ind-infinite dimensional group schemes, we use

$\mathrm{Maps}(\overset{\circ}{\mathscr{D}},\mathbf{B}B_{M})$

as a substitute for $\mathbf{B}\mathscr{L}_{\mathrm{Conf}}B_{M}$. Then the
twisted product involving $B$ is defined to be the pullback of

(4.3.5.1)
$\mathrm{Maps}^{\mathrm{gen}}\left(\mathscr{D},\mathbf{B}B_{M}\hookrightarrow
T\backslash\left(N\backslash\\!\\!\backslash
G\overset{G}{\times}G/\\!\\!/U_{P}^{-}\right)/M\right)$

along the natural map

$\alpha:\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}\to\mathrm{Maps}(\overset{\circ}{\mathscr{D}},\mathbf{B}B_{M}).$

The second twisted product involving $N$ is defined similarly.

####  4.3.6. Sanity check

The expression in (4.3.5.1) is a substitute for

$\mathscr{L}_{\mathrm{Conf}}B_{M}\backslash\mathscr{Z}_{\overline{B},\widetilde{P}^{-},\mathrm{Conf}}$

in the sense that its base change along

$\beta:\mathrm{Conf}\to\mathrm{Maps}(\overset{\circ}{\mathscr{D}},\mathbf{B}B_{M})$

identifies with $\mathscr{Z}_{\overline{B},\widetilde{P}^{-},\mathrm{Conf}}$.
Since $\alpha$ Zariski locally factors through $\beta$, the twisted product is
Zariski locally on
$\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}$ a
product

$\overline{\mathrm{Bun}}_{B_{M},\mathrm{Conf}}^{\mathrm{v.}\mathrm{pol}}\underset{\mathrm{Conf}}{\times}\mathscr{Z}_{\overline{B},\widetilde{P}^{-},\mathrm{Conf}}$

over the configuration space.

####  4.3.7. Completion of Construction 4.1.3

Using the identification of Proposition 4.3.4, we define the morphism $u$ to
be

$\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\widetilde{\times}\mathscr{Z}_{\overline{N},\widetilde{P}^{-},\mathrm{Conf}}\to\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\widetilde{\times}\mathrm{Gr}_{M,\mathrm{Conf}}\simeq\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}.$

It is proper because
$\mathscr{Z}_{\overline{N},\widetilde{P}^{-},\mathrm{Conf}}\to\mathrm{Conf}$
is proper.

####  4.3.8. Proof of Proposition 4.3.4

Let us rewrite (4.2.3.1) as

${M/\\!\\!/N_{M}}$${U_{P}^{-}\backslash\\!\\!\backslash
G\overset{G}{\times}G/\\!\\!/N}$${M^{\mathrm{ab}}}$${M^{\mathrm{ab},+}}$

The quotient of the lower row by $M\times T$ identifies with

$\mathbf{B}[M,M]\times\mathbf{B}T\hookrightarrow\mathbf{B}[M,M]\times
M^{\mathrm{ab}}\backslash M^{\mathrm{ab},+}\times\mathbf{B}T.$

Therefore, we obtain a Cartesian diagram

${M\backslash\left(M/\\!\\!/N_{M}\right)/T}$${M\backslash\left(U_{P}^{-}\backslash\\!\\!\backslash
G\overset{G}{\times}G/\\!\\!/N\right)/T}$${\mathrm{pt}}$${M^{\mathrm{ab}}\backslash
M^{\mathrm{ab},+}}$

Now Proposition 4.3.4 is a formal consequence of Beauville–Laszlo gluing.

###  4.4. The Whittaker condition

####  4.4.1. Proof of Proposition 4.1.6

We follow the usual pattern of passing to strata and applying character
orthogonality.

####  4.4.2. First reduction

The object $\mathscr{W}\overset{*}{\boxtimes}j_{P,!}$ in question is $!$
extended from the locus

$\mathrm{Bun}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}.$

Therefore, it suffices to show that for each locally closed substack

$i_{w}:\mathscr{Z}_{N,P^{-}}^{w}=\left(\mathrm{Bun}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}\right)^{w}\hookrightarrow\mathrm{Bun}_{N}\underset{\mathrm{Bun}_{G}}{\times}\mathrm{Bun}_{P^{-}}$

with $w\neq 1$, the object

$i_{w}^{*}\left(\mathscr{W}\overset{*}{\boxtimes}k_{\mathrm{Bun}_{P^{-}}}\right)\simeq{{}^{\prime}}p^{*}\cdot\mathscr{W}$

is annihilated by $!$-pushforward to $\mathrm{Bun}_{M}$.

####  4.4.3. Second reduction

For any finite set $x\hookrightarrow X$ we may consider the open substack

$\mathscr{Z}_{N,P^{-}}^{w,\mathrm{good}\,x}\hookrightarrow\mathscr{Z}_{N,P^{-}}^{w}$

where the $N$ and $P^{-}$ reductions are in relative position $w$ at $x$.
Since $\mathscr{Z}_{N,P^{-}}^{w}$ is a colimit of open substacks of this form,
it suffices to show that the $*$-pullback of $\mathscr{W}$ to each
$\mathscr{Z}_{N,P^{-}}^{w,\mathrm{good}\,x}$ is annihilated by $!$-pushforward
to $\mathrm{Bun}_{M}$.

####  4.4.4. End of the proof

Now consider the groupoid acting on
$\mathscr{Z}_{N,P^{-}}^{w,\mathrm{good}\,x}$ by $N^{w}\cap P^{-}$-Hecke
modifications at $x$. The unipotent subgroupoid corresponding to

$N^{w}\cap U_{P}^{-}\simeq\ker\left(N^{w}\cap P^{-}\to M\right)$

acts along the fibers of the projection to $\mathrm{Bun}_{M}$. Now the
$*$-pullback of $\mathscr{W}$ is equivariant for the character associated to
the pullback of $\psi$ along

$N^{w}\cap U_{P}^{-}\hookrightarrow N^{w}\simeq N\to\mathbf{G}_{\mathrm{a}},$

which is nontrivial because $w\neq 1$.

####  4.4.5. Proof of Proposition 4.1.8

This claim is verified by doing some bookkeeping with Hecke groupoids.

####  4.4.6. Step one

We form an extension of the diagram (4.1.4.1) appearing in the definition of
Whittaker condition for $\mathrm{Whit}(M)_{\mathrm{Conf}}$.

${\mathscr{H}_{\widetilde{\mathscr{Z}}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\mathscr{H}_{N_{M},\mathrm{Conf}}}$${\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\mathbf{G}_{\mathrm{a}}}$$\scriptstyle{\psi}$

Here are the definitions:

  * –

We write $({\widetilde{\mathscr{Z}}}\times\mathrm{Ran})^{\mathrm{good}}$ for
the preimage of

$\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}\right)^{\mathrm{good}}\hookrightarrow\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}$

along the projection

$u\times\mathbf{1}:{\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\to\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\times\mathrm{Ran}.$

  * –

The fiber products of the two faces are naturally identified. We write
$\mathscr{H}_{\widetilde{\mathscr{Z}}}$ for this common value; it is naturally
a groupoid acting on
$({\widetilde{\mathscr{Z}}}\times\mathrm{Ran})^{\mathrm{good}}$.

Therefore, it remains to show that

$\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}$

is $(\mathscr{H}_{\widetilde{\mathscr{Z}}},\psi)$-equivariant, since $u_{*}$
sends such objects into $\mathrm{Whit}(M)_{\mathrm{Conf}}$.

####  4.4.7. Step two

The $!$-pullback of $\mathscr{W}$ along
${\widetilde{\mathscr{Z}}}\to\overline{\mathrm{Bun}}_{N}$ is
$(\mathscr{H}_{\widetilde{\mathscr{Z}}},\psi)$-equivariant. Indeed, we have a
commutative diagram

${\mathscr{H}_{\widetilde{\mathscr{Z}}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\mathscr{H}_{N}}$${\left(\overline{\mathrm{Bun}}_{N}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left(\overline{\mathrm{Bun}}_{N}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\mathbf{G}_{\mathrm{a}}}$$\scriptstyle{\psi}$

in which the two faces are not Cartesian, and the pullback of the Whittaker
character from $\mathscr{H}_{N}$ agrees with the pullback of the Whittaker
character from $\mathscr{H}_{N_{M},\mathrm{Conf}}$.

####  4.4.8. Step three

The $!$-pullback of $j_{P,!}$ along
${\widetilde{\mathscr{Z}}}\to\widetilde{\mathrm{Bun}}_{P^{-}}$ is
$\mathscr{H}_{\widetilde{\mathscr{Z}}}$-equivariant. Indeed, we have a
commutative diagram

${\mathscr{H}_{\widetilde{\mathscr{Z}}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\mathscr{H}_{P^{-}}}$${\left(\widetilde{\mathrm{Bun}}_{P^{-}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$${\left(\widetilde{\mathrm{Bun}}_{P^{-}}\times\mathrm{Ran}\right)^{\mathrm{good}}}$

in which the two faces are not Cartesian, and $j_{P,!}^{-}$ is equivariant for
$\mathscr{H}_{P^{-}}$.

##  5\. ULA proofs

#### 5.0.1.

The purpose of this section is to prove Lemma 4.1.7. In the case $P=B$, this
follows from the work of J. Campbell [Cam19]. The main input to that work was
the Kontsevich compactification, whose parabolic version
$\overline{\mathrm{Bun}}_{P}^{K}\to\overline{\mathrm{Bun}}_{P}$ does not seem
to lift along the map
$\widetilde{\mathrm{Bun}}_{P}\to\overline{\mathrm{Bun}}_{P}$.

###  5.1. Outline of the proof

#### 5.1.1.

Let us write $\mathscr{Z}\hookrightarrow{\widetilde{\mathscr{Z}}}$ for
$\mathscr{Z}_{N,P^{-}}\hookrightarrow\mathscr{Z}_{\overline{N},\widetilde{P}^{-}}$.
We first reformulate the assertion of the lemma in terms that are smooth-local
with respect to the data of

$j:\mathscr{Z}\hookrightarrow{\widetilde{\mathscr{Z}}}\qquad\qquad\mathscr{F}=\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}.$

So let us examine the purity morphism

$\left(\mathscr{W}\overset{*}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\to\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\,[2\delta_{G}]$

in question. Note that the LHS is $!$-extended along $j$. Furthermore, the
restriction of the purity map to $\mathscr{Z}$ is an isomorphism because
$\mathscr{W}\boxtimes j_{P,!}$ is a local system on
$\mathrm{Bun}_{N}\times\mathrm{Bun}_{P^{-}}$. Therefore, our purity morphism
identifies with the canonical morphism

(5.1.1.1) $j_{!}\cdot j^{!}\cdot\mathscr{F}\to\mathscr{F}$

shifted by $[2\delta_{G}]$. So Lemma 4.1.7 is equivalent to the following
assertion, which is smooth-local in the desired sense:

###### 5.1.1 Lemma.

The sheaf $\mathscr{F}$ is $!$-extended from
$\mathscr{Z}\hookrightarrow{\widetilde{\mathscr{Z}}}$.

####  5.1.2. Proof of Lemma 5.1.1, modulo the rest of the section

Write $\mathscr{U}\hookrightarrow{\widetilde{\mathscr{Z}}}$ for the largest
open subset over which (5.1.1.1) is an isomorphism. In Construction 5.2.0, we
will form a correspondence

${\widetilde{\mathscr{H}}}$${{\widetilde{\mathscr{Z}}}}$${{\widetilde{\mathscr{Z}}}}$$\scriptstyle{\mu}$$\scriptstyle{\pi}$

whose action preserves $\mathscr{U}$ in the sense that

$\mu^{-1}(\mathscr{U})=\pi^{-1}(\mathscr{U}).$

This last assertion is the content of Proposition 5.2.0.

####  5.1.3. Informal description of $\widetilde{\mathscr{H}}$

This paragraph has no mathematical meaning. According to Proposition 4.3.4, a
point of ${\widetilde{\mathscr{Z}}}$ over $D\in\mathrm{Conf}_{G,P}$ may be
viewed as a degenerate $N_{M}$-bundle away from $D$ glued to some Zastava data
on the formal neighborhood of $|D|$. Actually there is a constraint on the
nature of this local Zastava data in terms of the value of $D$, but we will
ignore it. In these terms, the action of $\widetilde{\mathscr{H}}$ from right
to left admits the following description:

  * –

Choose a divisor $D_{\mathrm{good}}$ whose support is disjoint from that of
$D$, and impose the open condition that the $N_{M}$-bundle is nondegenerate at
$D_{\mathrm{good}}$.

  * –

Cut out the $N_{M}$-bundle at $D_{\mathrm{good}}$ and glue in some Zastava
data coming from

$\mathrm{Gr}_{N}\cap\mathrm{Gr}_{P^{-}}\to\mathrm{Gr}_{N}\to\mathrm{Gr}_{N_{M}}.$

The result is a new point of ${\widetilde{\mathscr{Z}}}$ lying over the
divisor $D+D_{\mathrm{good}}$.

#### 5.1.4.

We will then prove Proposition 5.3.3, which gives an upper bound on the
singular support of $j_{P,!}^{-}$. This will allow us to exhibit an open
substack

${\widetilde{\mathscr{Z}}}_{\mathrm{SS}}\hookrightarrow\mathscr{U}$

with a fairly simple membership criterion. Finally, we prove Proposition
5.4.3, which says that that every point of ${\widetilde{\mathscr{Z}}}$ can be
related by the correspondence $\widetilde{\mathscr{H}}$ to a point in
${\widetilde{\mathscr{Z}}}_{\mathrm{SS}}$.

###  5.2. Construction of the correspondence

###### 5.2.0 Construction.

There is a correspondence

${\mathscr{H}}$${\mathscr{Z}}$${\mathscr{Z}}$${\widetilde{\mathscr{H}}}$${{\widetilde{\mathscr{Z}}}}$${{\widetilde{\mathscr{Z}}}}$$\scriptstyle{\mu}$$\scriptstyle{\pi}$

with the feature that both squares are Cartesian.

###### 5.2.0 Proposition.

The following hold:

  * (i)

The map $\mu$ is etale.

  * (ii)

The map $\pi$ is smooth-locally a product.

  * (ii)

The two pullbacks

$\mu^{!}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}\qquad\pi^{!}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}j_{P,!}^{-}\right)_{\widetilde{\mathscr{Z}}}$

are smooth-locally isomorphic up to tensoring with the shift of a rank one
local system.

####  5.2.1. Notation

Our construction of $\widetilde{\mathscr{H}}$ will be given in terms of the
local description of ${\widetilde{\mathscr{Z}}}$ recorded as Proposition
4.3.4. We write

$\mathscr{Z}_{\mathrm{Conf}}=\mathscr{Z}_{N,P^{-},\mathrm{Conf}}\qquad{\widetilde{\mathscr{Z}}}_{\mathrm{Conf}}=\mathscr{Z}_{\overline{N},\widetilde{P}^{-},\mathrm{Conf}}.$

The definition of $\mathscr{Z}_{N,P^{-},\mathrm{Conf}}$ repeats the definition
of $\mathscr{Z}_{\overline{N},\widetilde{P}^{-},\mathrm{Conf}}$ given in §4.3
but with $\mathrm{Gr}_{N}$ and $\mathrm{Gr}_{P^{-}}$ replacing
$\overline{\mathrm{Gr}}_{N}$ and $\widetilde{\mathrm{Gr}}_{P^{-}}$. There is a
natural map

$\mathscr{Z}_{\mathrm{Conf}}\to\mathrm{Gr}_{N,\mathrm{Conf}}\to\mathrm{Gr}_{N_{M},\mathrm{Conf}}.$

#### 5.2.2.

We take

$\widetilde{\mathscr{H}}=\left\\{\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}\times\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\right\\}_{\mathrm{disj}}\underset{[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}}{\widetilde{\times}}\left\\{{\widetilde{\mathscr{Z}}}_{\mathrm{Conf}}\times\mathscr{Z}_{\mathrm{Conf}}\right\\}_{\mathrm{disj}}$

Then the leftward map is

$\displaystyle\mu:\widetilde{\mathscr{H}}\hookrightarrow$
$\displaystyle\left\\{\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}\times\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\right\\}_{\mathrm{disj}}\underset{[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}}{\widetilde{\times}}\left\\{{\widetilde{\mathscr{Z}}}_{\mathrm{Conf}}\times{\widetilde{\mathscr{Z}}}_{\mathrm{Conf}}\right\\}_{\mathrm{disj}}$
$\displaystyle\quad\simeq{\widetilde{\mathscr{Z}}}\underset{\mathrm{Conf}}{\times}[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}$
$\displaystyle\qquad\to{\widetilde{\mathscr{Z}}},$

where we are using the sum map

$m:[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}\to\mathrm{Conf}$

The rightward map is

$\displaystyle\pi:\widetilde{\mathscr{H}}\to$
$\displaystyle\left\\{\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}\times\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\right\\}_{\mathrm{disj}}\underset{[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}}{\widetilde{\times}}\left\\{{\widetilde{\mathscr{Z}}}_{\mathrm{Conf}}\times\mathrm{Gr}_{N_{M},\mathrm{Conf}}\right\\}_{\mathrm{disj}}$
$\displaystyle\quad\simeq\left({\widetilde{\mathscr{Z}}}\underset{\mathrm{Conf}}{\times}[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}\right)^{\text{good
at $D_{2}$}}$
$\displaystyle\qquad\hookrightarrow{\widetilde{\mathscr{Z}}}\underset{\mathrm{Conf}}{\times}[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}$
$\displaystyle\qquad\qquad\to{\widetilde{\mathscr{Z}}}$

where we use the map

$\mathrm{pr}_{1}:[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}\to\mathrm{Conf}.$

The base change of $\mathscr{Z}\hookrightarrow{\widetilde{\mathscr{Z}}}$ along
either map identifies with

$\mathscr{H}=\left\\{\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}\times\mathrm{Conf}}^{{\mathrm{v.}\mathrm{pol}}}\right\\}_{\mathrm{disj}}\underset{[\mathrm{Conf}\times\mathrm{Conf}]_{\mathrm{disj}}}{\widetilde{\times}}\left\\{\mathscr{Z}_{\mathrm{Conf}}\times\mathscr{Z}_{\mathrm{Conf}}\right\\}_{\mathrm{disj}}.$

#### 5.2.3.

This completes Construction 5.2.0. We now turn to the proof of Proposition
5.2.0.

####  5.2.4. Proofs of (i) & (ii)

These assertions are easily verified by examining the definitions. Indeed,
$\mu$ is an open immersion followed by a base change of $m$, which is etale.
This proves (i). To prove (ii), we note that twisted products are smooth-
locally products and that $\mathrm{pr}_{1}$ is an open subset of a product.

####  5.2.5. Proof of (iii)

This argument is very similar to the proof of Proposition 4.1.8 appearing in
§4.4. As before, we define
$({\widetilde{\mathscr{Z}}}\times\mathrm{Conf})^{\mathrm{good}}$ so that the
lower square of the diagram

${\widetilde{\mathscr{H}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Conf}\right)^{\mathrm{good}}}$${{\widetilde{\mathscr{Z}}}}$${\left(\overline{\mathrm{Bun}}_{N}\times\mathrm{Conf}\right)^{\mathrm{good}}}$${\overline{\mathrm{Bun}}_{N}}$$\scriptstyle\widehat{\mu}$
or $\scriptstyle\widehat{\pi}$$\scriptstyle\mu$ or $\scriptstyle\pi$

is Cartesian. As the dashed arrow indicates, both $\mu$ and $\pi$ have natural
lifts fitting into a commutative diagram

${\widetilde{\mathscr{H}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Conf}\right)^{\mathrm{good}}}$${\left({\widetilde{\mathscr{Z}}}\times\mathrm{Conf}\right)^{\mathrm{good}}}$${\mathscr{H}_{N}}$${\left(\overline{\mathrm{Bun}}_{N}\times\mathrm{Conf}\right)^{\mathrm{good}}}$${\left(\overline{\mathrm{Bun}}_{N}\times\mathrm{Conf}\right)^{\mathrm{good}}}$${\mathbf{G}_{\mathrm{a}}}$$\scriptstyle{\widehat{\mu}}$$\scriptstyle{\widehat{\pi}}$$\scriptstyle{\psi}$

Therefore, the two $!$-pullbacks of $\mathscr{W}$ to
${\widetilde{\mathscr{Z}}}$ differ by the $!$-pullback of the Whittaker
character along $\widetilde{\mathscr{H}}\to\mathscr{H}_{N}$. A similar
argument shows that the two $!$-pullbacks of $j_{P,!}^{-}$ to
${\widetilde{\mathscr{Z}}}$ are isomorphic.

###  5.3. A singular support estimate

#### 5.3.1.

The goal of this subsection is to obtain an upper bound on the singular
support of $j_{P,!}^{-}$. In order to state the result, we will need to
describe a stratification of $\widetilde{\mathrm{Bun}}_{P^{-}}$ into smooth
stacks

${}_{\vartheta^{\prime}\leq\vartheta}\widetilde{\mathrm{Bun}}_{P^{-}}\hookrightarrow\widetilde{\mathrm{Bun}}_{P^{-}}$

indexed by pairs of partitions.

#### 5.3.2.

Recall that $\overline{\mathrm{Bun}}_{P^{-}}$ has a smooth stratification by

$X^{\theta}\times\mathrm{Bun}_{P^{-}}\simeq{}_{\theta}\overline{\mathrm{Bun}}_{P^{-}}\hookrightarrow\overline{\mathrm{Bun}}_{P^{-}}.$

Under this identification. the map
${}_{\theta}\overline{\mathrm{Bun}}_{P^{-}}\to\mathrm{Bun}_{G}$ is given by
projection on $\mathrm{Bun}_{P^{-}}$, followed by the usual map
$\mathrm{Bun}_{P^{-}}\to\mathrm{Bun}_{G}$.

#### 5.3.3.

Taking the pullback along
$\widetilde{\mathrm{Bun}}_{P^{-}}\to\overline{\mathrm{Bun}}_{P^{-}}$, we
obtain a stratification by

$\mathrm{Bun}_{P^{-}}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+,\theta}\simeq{}_{\theta}\widetilde{\mathrm{Bun}}_{P^{-}}\hookrightarrow\widetilde{\mathrm{Bun}}_{P^{-}}.$

These strata are not smooth because $\mathscr{H}_{M}^{+,\theta}$ is not smooth
over $\mathrm{Bun}_{M}$, so let us examine $\mathscr{H}_{M}^{+,\theta}$ more
closely. Consider first the map

$\mathscr{H}_{M}^{+,\theta}\to X^{\theta}.$

We have a stratification $X^{\vartheta}\hookrightarrow X^{\theta}$ indexed by
partitions $\vartheta\vdash\theta$. The restriction
$\mathscr{H}_{M}^{+,\vartheta}$ of $\mathscr{H}_{M}^{+,\theta}$ to each
$X^{\vartheta}$ is a closed substack of $\mathscr{H}_{M,X^{\vartheta}}$.
Therefore, $\mathscr{H}_{M}^{+,\vartheta}$ inherits a relative position
stratification, indexed by $\vartheta^{\prime}\leq\vartheta$. These strata
$\mathscr{H}_{M}^{+,\vartheta^{\prime}\leq\vartheta}$ are smooth over
$\mathrm{Bun}_{M}\times X^{\vartheta}$, and we define

${}_{\vartheta^{\prime}\leq\vartheta}\widetilde{\mathrm{Bun}}_{P^{-}}=\mathrm{Bun}_{P^{-}}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+,\vartheta\leq\vartheta}.$

###### 5.3.3 Proposition.

The singular support of $j_{P,!}^{-}$ is contained in the conormals of the
strata

${}_{\vartheta^{\prime}\leq\vartheta}\widetilde{\mathrm{Bun}}_{P^{-}}\hookrightarrow\widetilde{\mathrm{Bun}}_{P^{-}}.$

#### 5.3.4.

The proof of Proposition 5.3.3 will use Zastava spaces as a local model for
$\widetilde{\mathrm{Bun}}_{P^{-}}$, following [Bra+02]. Let us recall this.
For the sake of notational consistency, we write

$\mathrm{Mod}_{M}^{+}=\mathrm{pt}\underset{\mathrm{Bun}_{M}}{\times}\mathscr{H}_{M}^{+}.$

There are natural maps

$\mathscr{Z}_{U_{P},\widetilde{P}^{-}}\to\mathrm{Mod}_{M}^{+}\to\mathrm{Conf}_{G,P}.$

and the first arrow has a canonical section. We write $(-)^{\theta}$ to denote
base change to the component
$\mathrm{Conf}^{\theta}\hookrightarrow\mathrm{Conf}$. Now according to
[Bra+04, (1)], the diagram of Zastava spaces on the left

${\mathrm{Mod}_{M}^{+,\theta}}$${\mathscr{Z}_{U_{P},\widetilde{P}^{-}}^{\theta}}$${\mathscr{Z}_{U_{P},P^{-}}^{\theta}}$${\mathrm{Mod}_{M}^{+,\theta}}$$\scriptstyle{j_{\mathscr{Z}}}$
${{}_{\theta}\widetilde{\mathrm{Bun}}_{P^{-}}}$${\widetilde{\mathrm{Bun}}_{P^{-}}}$${\mathrm{Bun}_{P^{-}}}$${\mathscr{H}_{M}^{+,\theta}}$

is a smooth-local model for the diagram of stacks on the right.

####  5.3.5. Proof of Proposition 5.3.3

We make a preliminary construction and the perform a sequence of reductions.

####  5.3.6. A group action

We claim that there is an action of $\mathscr{L}_{\mathrm{Conf}}^{+}M$ on
$\mathscr{Z}_{U_{P},\widetilde{P}^{-}}$ whose restriction to

$\mathrm{Mod}_{M}^{+,\vartheta}\hookrightarrow\mathrm{Gr}_{M,X^{\vartheta}}$

matches the usual action of $\mathscr{L}_{X^{\vartheta}}^{+}M$. This is
because $\mathscr{Z}_{U_{P},\widetilde{P}^{-}}$ admits a local description
similar to that of $\mathscr{Z}_{N,\widetilde{P}^{-}}$. Namely, it is the base
change of

$\mathrm{Maps}^{\mathrm{gen}}\left(\mathscr{D},\mathrm{pt}\hookrightarrow(U_{P}^{-}\backslash
G/\\!\\!/U_{P})/M\right)$

along the tautological section (4.3.3.1). The left action of $M$ on
$U_{P}^{-}\backslash G/\\!\\!/U_{P}$ induces the action we seek.

####  5.3.7. First reduction

This model matches the strata

$\mathrm{Mod}_{M}^{+,\vartheta^{\prime}\leq\vartheta}\hookrightarrow\mathrm{Mod}_{M}^{+,\theta}\qquad\mathscr{H}_{M}^{+,\vartheta^{\prime}\leq\vartheta}\hookrightarrow\mathscr{H}_{M}^{+,\theta}.$

Therefore, it suffices to show the singular support of $j_{\mathscr{Z},!}\cdot
k$ at every point of $\mathrm{Mod}_{M}^{+,\theta}$ is contained in the
conormals to the strata
$\mathrm{Mod}_{M}^{+,\vartheta^{\prime}\leq\vartheta}$.

####  5.3.8. Second reduction

By the factorization of the Zastava diagram, we only need to consider points
lying over the main diagonal $X\hookrightarrow X^{\theta}$. These are the
points belong to strata indexed by $\vartheta^{\prime}\leq(\theta)$ where
$(\theta)\vdash\theta$ is the trivial partition.

####  5.3.9. Third reduction

Since the Zastava diagram is etale local with respect to the curve, we may
take $X=\mathbf{A}^{1}$. Then the Zastava diagram admits a translation action
by $\mathbf{G}_{\mathrm{a}}$ commuting with the
$\mathscr{L}_{\mathrm{Conf}}^{+}M$ action constructed above. The result
follows because each stratum
$\mathrm{Mod}_{M}^{+,\vartheta^{\prime}\leq(\theta)}$ is an orbit for the
combined action of
$\mathscr{L}_{\mathrm{Conf}}^{+}M\times\mathbf{G}_{\mathrm{a}}$.

###  5.4. End of the proof

#### 5.4.1.

Let $\mathscr{U}_{\mathrm{SS}}\hookrightarrow\widetilde{\mathrm{Bun}}_{P^{-}}$
be the open subset where the singular support of $j_{P,!}^{-}$ is transverse
to the codifferential of
$\widetilde{\mathrm{Bun}}_{P^{-}}\to\mathrm{Bun}_{G}$. The singular support
bound implies:

###### 5.4.1 Proposition.

Let $x$ be a geometric point of ${}_{\theta}\widetilde{\mathrm{Bun}}_{P^{-}}$.
If the image of $x$ under the projection
${}_{\theta}\widetilde{\mathrm{Bun}}_{P^{-}}\to\mathrm{Bun}_{P^{-}}$ belongs
to the smooth locus of $\mathrm{Bun}_{P^{-}}\to\mathrm{Bun}_{G}$, then $x$
belongs to $\mathscr{U}_{\mathrm{SS}}$.

####  5.4.2. Proof

We have the following Cartesian diagram:

${\mathscr{H}_{M}^{+,\vartheta^{\prime}\leq\vartheta}}$${\mathscr{H}_{M}^{+,\theta}}$${{}_{\theta}\widetilde{\mathrm{Bun}}_{P^{-}}}$${\mathrm{Bun}_{M}}$${\mathrm{Bun}_{P^{-}}}$${\mathrm{Bun}_{G}}$

Therefore it suffices to observe that diagonal dashed arrow is smooth.

#### 5.4.3.

Define

${\widetilde{\mathscr{Z}}}_{\mathrm{SS}}=\mathscr{U}_{\mathrm{SS}}\underset{\widetilde{\mathrm{Bun}}_{P^{-}}}{\times}{\widetilde{\mathscr{Z}}}.$

###### 5.4.3 Proposition.

Every geometric point of ${\widetilde{\mathscr{Z}}}$ can be taken to a point
of ${\widetilde{\mathscr{Z}}}_{\mathrm{SS}}$ by the action of
$\widetilde{\mathscr{H}}$.

####  5.4.4. Proof of Proposition 5.4.3

The proof comes in two parts.

####  5.4.5. Part one

Let $z$ be a geometric point of ${\widetilde{\mathscr{Z}}}$, and let
$\mathscr{F}_{P^{-}}$ be its image in $\mathrm{Bun}_{P^{-}}$. Then
$\mathscr{F}_{P^{-}}$ belongs to the smooth locus of
$\mathrm{Bun}_{P^{-}}\to\mathrm{Bun}_{G}$ iff

(5.4.5.1)
$H^{0}\left(X,\mathscr{F}_{P^{-}}\overset{P^{-}}{\times}\mathfrak{g}/\mathfrak{p}^{-}\right)\simeq
0$

To control this cohomology group, we choose an arbitrary reduction
$\mathscr{F}_{B^{-}}$ of $\mathscr{F}_{P^{-}}$ to $B^{-}$ that is generically
transverse to the image of $z$ in $\overline{\mathrm{Bun}}_{N}$. Then the
vector bundle appearing in (5.4.5.1) acquires a complete flag whose graded
pieces are the line bundles $\mathscr{F}_{T}\overset{T}{\times}\alpha$ for
$\alpha$ ranging over the roots appearing in $\mathfrak{u}_{P}$.

####  5.4.6. Part two

From all of this data, we may extract the following defect divisors:

  * –

The image of $z$ in $\mathrm{Conf}_{G,P}$ is a divisor $D_{1}$, away from
which $\mathscr{F}_{M}$ has a degenerate $N_{M}$-reduction.

  * –

The degeneracy of this $N_{M}$-reduction defines a $\Lambda_{M}$-valued
divisor $D_{2}$ on $X\setminus|D_{1}|$.

  * –

From the choice of $\mathscr{F}_{B^{-}}$ we obtain a point of
$\mathscr{Z}_{\overline{N},B^{-}}$ and hence a divisor $D_{3}$ in
$\mathrm{Conf}_{G,B}$.

Now consider a point $E$ of $\mathrm{Conf}_{G,P}$ whose support is disjoint
from all of three of the above divisors, and let $E^{\prime}$ be a lift of $E$
to $\mathrm{Conf}_{G,B}$. Perform a Hecke modification coming from

$\left(\mathscr{Z}_{N,B^{-},\mathrm{Conf}}\right)_{E^{\prime}}\to\left(\mathscr{Z}_{N,P^{-},\mathrm{Conf}}\right)_{E}$

to obtain a new point of ${\widetilde{\mathscr{Z}}}$, say $z^{\prime}$. The
image of $z^{\prime}$ in $\mathrm{Bun}_{P^{-}}$ comes with a reduction
$\mathscr{F}_{B^{-}}^{\prime}$ to $B^{-}$, and we have

$\mathscr{F}_{T}^{\prime}\overset{T}{\times}\alpha\simeq\left(\mathscr{F}_{T}\overset{T}{\times}\alpha\right)(\alpha(E^{\prime}))$

But we may choose $E$ so that the divisors $\alpha(E^{\prime})=\alpha(E)$ for
$\alpha$ appearing in $\mathfrak{u}_{P}$ are arbitrarily negative, thereby
forcing the vanishing (5.4.5.1) for $\mathscr{F}_{P^{-}}^{\prime}$.

##  6\. The co-Jacquet functor

#### 6.0.1.

In this section we shall construct the following objects:

  * –

A Jacquet functor

$\mathrm{Jac}_{\mathrm{co},\mathrm{Conf}}:\mathrm{Whit}(M)_{\mathrm{Conf}}\to\mathrm{Whit}(G)$

along with a commutative diagram

(6.0.1.1)
${\mathrm{D}(\mathrm{Bun}_{M})}$${\mathrm{D}(\mathrm{Bun}_{G})}$${\mathrm{Whit}(M)_{\mathrm{Conf}}}$${\mathrm{Whit}(G)}$$\scriptstyle{\mathrm{Eis}_{!}^{-}}$$\scriptstyle{\mathrm{coeff}_{*}}$$\scriptstyle{\mathrm{coeff}_{*}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{co}}}$

  * –

A duality

$\mathrm{Whit}(M)_{\mathrm{Conf}}^{\vee}\simeq\mathrm{Whit}(M)_{\mathrm{Conf}}$

along with a commutative diagram

(6.0.1.2)
${\mathrm{Whit}(G)}$${\mathrm{Whit}(M)_{\mathrm{Conf}}}$${\mathrm{Whit}(G)^{\vee}}$${\mathrm{Whit}(M)_{\mathrm{Conf}}^{\vee}}$$\scriptstyle{\mathrm{Jac}}$$\scriptstyle{\mathbf{D}{[2\delta_{G}]}}$$\scriptstyle{\mathbf{D}{[2\delta_{M}]}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{co}}^{\vee}}$

###  6.1. Definition of the functor

#### 6.1.1.

Let us return to the diagram (4.1.6.1). We have, in particular, the following
morphisms:

${{\widetilde{\mathscr{Z}}}_{N,P^{-}}}$${\overline{\mathrm{Bun}}_{N}}$${\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}}$${\widetilde{\mathrm{Bun}}_{P^{-}}}$${\mathrm{Bun}_{M}}$$\scriptstyle{u}$$\scriptstyle{w}$$\scriptstyle{v}$$\scriptstyle{\pi_{\mathrm{Conf}}}$

Recall that

$\mathrm{Eis}_{!}^{-}=\widetilde{q}_{!}\cdot\left(\widetilde{p}^{*}(-)\overset{*}{\otimes}j_{P,!}^{-}\right)\simeq\widetilde{q}_{*}\cdot\left(\widetilde{p}^{!}(-)\overset{!}{\otimes}j_{P,!}^{-}\right)[2\delta_{M}].$

Therefore, we obtain a morphism

$\displaystyle\mathrm{coeff}_{*}\cdot\mathrm{Eis}_{!}^{-}$
$\displaystyle\simeq\mathrm{Av}_{*}^{\mathrm{Whit}}\cdot{{}^{\prime}}\widetilde{p}_{*}\cdot\left({{}^{\prime}}\pi^{!}\cdot\widetilde{q}^{!}(-)\overset{!}{\otimes}{{}^{\prime}}\pi^{!}\cdot
j_{P,!}^{-}\right)\,[2\delta_{M}]$
$\displaystyle\qquad\to\mathrm{Av}_{*}^{\mathrm{Whit}}\cdot
v_{*}\cdot\left(u^{!}\cdot\pi_{\mathrm{Conf}}^{!}(-)\overset{!}{\otimes}w^{!}\cdot
j_{P,!}^{-}\right)\,[2\delta_{M}]$

The same analysis as Proposition 4.1.6 shows that this map is an isomorphism.
Therefore, we consider the diagram

(6.1.1.1)
${\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)}$${\mathrm{D}(\overline{\mathrm{Bun}}_{N})}$${\mathrm{Whit}(M)_{\mathrm{Conf}}}$${\mathrm{Whit}(G)}$$\scriptstyle{\widetilde{\mathrm{Jac}}_{\mathrm{co}}}$$\scriptstyle{\mathrm{Av}_{*}}$$\scriptstyle{\mathrm{Av}_{*}}$$\scriptstyle{\mathrm{Jac}_{\mathrm{co}}}$

where

$\widetilde{\mathrm{Jac}}_{\mathrm{co}}=v_{*}\cdot\left(u^{!}(-)\overset{!}{\otimes}w^{!}\cdot
j_{P,!}^{-}\right)\,[2\delta_{M}].$

###### 6.1.1 Proposition.

The factorization $\mathrm{Jac}_{\mathrm{co}}$ indicated in the diagram
(6.1.1.1) exists.

#### 6.1.2.

Note that the functor $\mathrm{Jac}_{\mathrm{co}}$ is uniquely determined by
the commutativity of (6.1.1.1). This completes the construction of
$\mathrm{Jac}_{\mathrm{co}}$ and the commutative diagram (6.0.1.1) modulo the
proof of Proposition 6.1.1, which will appear in the next subsection.

###  6.2. Duality

#### 6.2.1.

For $\mathscr{W}\in\mathrm{Whit}(M)_{\mathrm{Conf}}$ the formula

$\langle\mathscr{W},-\rangle_{M}=\Gamma\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}},\mathscr{W}\overset{!}{\otimes}-\right)$

defines a continuous functor

$\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)\to\mathrm{Vect}.$

Furthermore, the map

$\left\langle\mathscr{W},\mathrm{Av}_{*}^{-\psi}\cdot\mathscr{F}\right\rangle\to\langle\mathscr{W},\mathscr{F}\rangle_{M}$

is an isomorphism, and we obtain a perfect pairing

$\mathrm{Whit}_{\psi}(M)_{\mathrm{Conf}}\otimes\mathrm{Whit}_{-\psi}(M)_{\mathrm{Conf}}\to\mathrm{Vect}.$

###### 6.2.1 Proposition.

For any $\mathscr{W}_{G}\in\mathrm{Whit}(G)$ and
$\mathscr{F}_{M}\in\mathrm{D}\left(\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}\right)$
we have

$\left\langle\mathrm{Jac}\cdot\mathscr{W}_{G},\mathscr{F}_{M}\right\rangle_{M}\simeq\left\langle\mathscr{W}_{G},\widetilde{\mathrm{Jac}}_{\mathrm{co},\mathrm{Conf}}\cdot\mathscr{F}_{M}\right\rangle_{G}[2\delta_{G}-2\delta_{M}].$

####  6.2.2. Proof

This is clear because

$\displaystyle\mathrm{LHS}$
$\displaystyle\simeq\Gamma\left(\overline{\mathrm{Bun}}_{N(M),\mathrm{Conf}}^{\mathrm{pol}},u_{*}\left\\{v^{!}\cdot\mathscr{W}_{G}\overset{!}{\otimes}w^{!}\cdot
j_{P,!}^{-}\right\\}\overset{!}{\otimes}\mathscr{F}_{M}\right)[2\delta_{G}]$
$\displaystyle\qquad\simeq\Gamma\left({\widetilde{\mathscr{Z}}}_{N,P^{-}},v^{!}\cdot\mathscr{W}_{G}\overset{!}{\otimes}w^{!}\cdot
j_{P,!}^{-}\overset{!}{\otimes}u^{!}\cdot\mathscr{F}_{M}\right)[2\delta_{G}]$
$\displaystyle\qquad\quad\simeq\Gamma\left(\overline{\mathrm{Bun}}_{N},\mathscr{W}_{G}\overset{!}{\otimes}v_{*}\left\\{w^{!}\cdot
j_{P,!}^{-}\overset{!}{\otimes}u^{!}\cdot\mathscr{F}_{M}\right\\}\right)[2\delta_{G}]\simeq\mathrm{RHS}.$

####  6.2.3. Proof of Proposition 6.1.1

Since the duality pairing factors through $\mathrm{Av}_{*}^{-\psi}$, the above
proposition gives

$\left\langle\mathscr{W}_{G},\mathrm{Av}_{*}^{-\psi}\cdot\widetilde{\mathrm{Jac}}_{\mathrm{co}}\cdot\mathscr{F}_{M}\right\rangle_{G}\simeq\left\langle\mathrm{Jac}\cdot\mathscr{W}_{G},\mathrm{Av}_{*}^{-\psi}\cdot\mathscr{F}_{M}\right\rangle_{M}[2\delta_{M}-2\delta_{G}].$

The result follows from the fact that $\langle-,-\rangle_{G}$ is a perfect
pairing.

####  6.2.4. Construction of (6.0.1.2)

We have

$\langle\mathrm{Jac}\cdot\mathscr{W}_{G},\mathscr{W}_{M}\rangle_{M}\,[2\delta_{M}]\simeq\langle\mathscr{W}_{G},\mathrm{Jac}_{\mathrm{co}}\cdot\mathscr{W}_{M}\rangle_{G}\,[2\delta_{G}]$

for every $\mathscr{W}_{G}\in\mathrm{Whit}(G)$ and
$\mathscr{W}_{M}\in\mathrm{Whit}(M)_{\mathrm{Conf}}$.

##  7\. The factorizable Whittaker category

#### 7.0.1.

In the section we will review the definitions of the Ran version
$\mathrm{Whit}(G)_{\mathrm{Ran}}$ of the Whittaker category from [Gai20] and
relate it to the configuration space version
$\mathrm{Whit}(G)_{\mathrm{Conf}}$. Although $\mathrm{Whit}(G)_{\mathrm{Ran}}$
is a naturally a unital factorization category, we will not make explicit use
of this additional structure.

###  7.1. Review of Whittaker categories

####  7.1.1. Local model

For a finite set $I$, we write

$\mathrm{Whit}(G)_{I}=\mathrm{D}(\mathrm{Gr}_{G,I})^{\mathscr{L}N,\psi}\qquad\mathrm{Whit}(G)_{I,\mathrm{co}}=\mathrm{D}(\mathrm{Gr}_{G,I})_{\mathscr{L}N,\psi}$

By [Che20, Appendix B], these assignments define factorization categories
which we denote

$\mathrm{Whit}(G)\qquad\mathrm{Whit}(G)_{\mathrm{co}}.$

Furthermore, we have a canonical $\mathrm{D}(X^{I})$-linear equivalence

$\mathrm{Whit}_{\psi}(G)_{I}^{\vee}\simeq\mathrm{Whit}_{-\psi}(G)_{I,\mathrm{co}}$

inducing an equivalence

$\mathrm{Whit}_{\psi}(G)^{\vee}\simeq\mathrm{Whit}_{-\psi}(G)_{\mathrm{co}}$

of factorization categories.

####  7.1.2. Local duality

The functor of renormalized $*$-averaging

$\mathrm{Av}_{*,\mathrm{ren}}:\mathrm{D}(\mathrm{Gr}_{G,I})\to\mathrm{Whit}(\mathrm{Gr}_{G,I})$

induces an equivalence

$\mathrm{PsId}_{\mathrm{Whit}}:\mathrm{Whit}(G)_{I,\mathrm{co}}\overset{\sim}{\rightarrow}\mathrm{Whit}(G)_{I}.$

We obtain an equivalence of factorization categories

$\mathrm{Whit}_{-\psi}(G)^{\vee}\overset{\sim}{\rightarrow}\mathrm{Whit}_{\psi}(G).$

####  7.1.3. Global model

Write $\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}$ for the stack classifying
a point $x\in X^{I}$, and a principal $G$-bundle on $X$ equipped with a
degenerate $N^{\omega}$-reduction away from $x$. We set

$\mathrm{Whit}(G)_{I}^{\mathrm{glob}}=\mathrm{Whit}\left(\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}\right).$

This defines a sheaf of categories over the Ran space, which we denote

$\mathrm{Whit}(G)^{\mathrm{glob}}.$

Note that $\mathrm{Whit}(G)^{\mathrm{glob}}$ acquires a weak factorization
structure from the weak factorization structure

$\overline{\mathrm{Bun}}_{N,\mathrm{Ran}}^{\mathrm{pol}}\underset{\mathrm{Ran}}{\times}\left[\mathrm{Ran}\times\mathrm{Ran}\right]_{\mathrm{disj}}\to\left[\overline{\mathrm{Bun}}_{N,\mathrm{Ran}}^{\mathrm{pol}}\times\overline{\mathrm{Bun}}_{N,\mathrm{Ran}}^{\mathrm{pol}}\right]_{\mathrm{disj}}.$

The comparison with the local category will show that
$\mathrm{Whit}(G)^{\mathrm{glob}}$ is a genuine factorization category.

####  7.1.4. Global duality

The perfect pairing

$\mathrm{Whit}_{-\psi}\left(\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}\right)\otimes\mathrm{Whit}_{\psi}\left(\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}\right)\to\mathrm{Vect}\qquad\langle\mathscr{W}_{1},\mathscr{W}_{2}\rangle=\Gamma\left(\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}},\mathscr{W}_{1}\overset{!}{\otimes}\mathscr{W}_{2}\right)$

defines a factorizable equivalence

$\mathbf{D}:\mathrm{Whit}_{\psi}(G)^{\mathrm{glob}}\overset{\sim}{\rightarrow}\mathrm{Whit}_{-\psi}(G)^{\mathrm{glob},\vee}.$

####  7.1.5. Local v global

The map

$\pi:\mathrm{Gr}_{G,I}\to\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}$

induces equivalences

$\pi^{!}:\mathrm{Whit}(G)_{I}^{\mathrm{glob}}\to\mathrm{Whit}(G)_{I}\qquad\pi_{*}:\mathrm{Whit}(G)_{\mathrm{co},I}\to\mathrm{Whit}(G)_{\mathrm{co},I}^{\mathrm{glob}}.$

We use the same notation for the Ran versions. The interaction of this
comparison with duality is summarized by the diagram below.

(7.1.5.1)
${\mathrm{Whit}_{-\psi}(G)^{\mathrm{glob},\vee}}$${\mathrm{Whit}_{\psi}(G)^{\mathrm{glob}}}$${\mathrm{Whit}_{-\psi}(G)^{\vee}}$${\mathrm{Whit}_{\psi}(G)_{\mathrm{co}}}$${\mathrm{Whit}_{\psi}(G)}$$\scriptstyle{\mathbf{D}}$$\scriptstyle{\pi^{!}}$$\scriptstyle{\sim}$$\scriptstyle{(\pi^{!})^{\vee}}$$\scriptstyle{\mathrm{PsId}{[-2\delta_{N}]}}$$\scriptstyle{\pi_{*}}$

####  7.1.6. Hecke action

  * –

We write $\mathrm{Bun}_{G,I}$ for $\mathrm{Bun}_{G}\times X^{I}$ and
$\mathrm{D}(\mathrm{Bun}_{G})_{I}$ for $\mathrm{D}(\mathrm{Bun}_{G,I})$,
viewed as a $\mathrm{D}(X^{I})$-module.

  * –

Recall the Hecke stack

${\mathscr{H}_{G,I}}$${\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}}$${X^{I}}$$\scriptstyle{{\mathop{h}\limits^{\vbox
to-1.07639pt{\kern-1.61458pt\hbox{$\scriptstyle\leftharpoonup$}\vss}}}}$$\scriptstyle{{\mathop{h}\limits^{\vbox
to-1.07639pt{\kern-1.61458pt\hbox{$\scriptstyle\rightharpoonup$}\vss}}}}$$\scriptstyle{s}$

Note that $\mathrm{D}(\mathscr{H}_{G,I})$ is naturally a
$\mathrm{D}(X^{I})$-algebra acting on $\mathrm{D}(\mathrm{Bun}_{G})_{I}$.

  * –

We write $\mathrm{Sph}_{G,I}$ for the spherical Hecke category over $X^{I}$.
It is a $\mathrm{D}(X^{I})$-algebra acting on
$\mathrm{D}(\mathrm{Bun}_{G})_{I}$ through the canonical monoidal functor

$\mathrm{Sph}_{G,I}\to\mathrm{D}(\mathscr{H}_{G,I})\qquad\SS\mapsto\mathscr{H}(\SS).$

  * –

We write $\mathrm{Sph}_{G,\mathrm{Ran}}$ for the Ran version of
$\mathrm{Sph}_{G,I}$. It is a monoidal factorization category.

Now $\mathrm{Sph}_{G,I}$ acts on all of the previously defined Whittaker
categories over $X^{I}$. Let us discuss the compatibility of (7.1.5.1) with
the $\mathrm{Sph}_{G}$ action. The right triangle of (7.1.5.1) is a diagram of
right $\mathrm{Sph}_{G}$-modules, and the left arrow $(\pi^{!})^{\vee}$ is a
morphism of left $\mathrm{Sph}_{G}$-modules. The global Verdier duality is
$\mathrm{Sph}_{G}$-linear with respect to the canonical anti-involution
$\tau:\mathrm{Sph}_{G}\to\mathrm{Sph}_{G}^{\mathrm{rev}}$, as is the
identification
$\mathrm{Whit}_{-\psi}(G)^{\vee}\overset{\sim}{\rightarrow}\mathrm{Whit}_{\psi}(G)_{\mathrm{co}}$.

####  7.1.7. Whittaker coefficients and Poincare series

The functor of Whittaker coefficients

$\mathrm{coeff}_{I,*}:\mathrm{D}(\mathrm{Bun}_{G})_{I}\overset{\pi^{!}}{\longrightarrow}\mathrm{D}\left(\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}\right)\overset{\mathrm{Av}_{*}}{\longrightarrow}\mathrm{Whit}(G)_{I}^{\mathrm{glob}}$

is $\mathrm{Sph}_{G,I}$-linear. Therefore, its left adjoint

$\mathrm{Poinc}_{I,!}:\mathrm{Whit}(G)_{I}^{\mathrm{glob}}\to\mathrm{D}(\mathrm{Bun}_{G,I})$

is automatically co-lax $\mathrm{Sph}_{G,I}$-linear. We write
$\mathrm{coeff}_{\mathrm{Ran},*}$ and $\mathrm{Poinc}_{\mathrm{Ran},!}$ for
the Ran variants of these functors.

###### 7.1.7 Proposition.

The functor $\mathrm{Poinc}_{I,!}$ is actually $\mathrm{Sph}_{G,I}$-linear.

####  7.1.8. A triviality

We will use the following observation in both the proof of Proposition 7.1.7
and in later sections without comment. Suppose that $f:M_{1}\to M_{2}$ is a
colax morphism of modules over a monoidal category $A$. Let us say that an
object $m$ of $M_{1}$ is _strict_ if the colax constraint

$f(a\otimes m)\to a\otimes f(m)$

is an isomorphism for all $a\in A$. Then the full subcategory of strict
objects is a monoidal ideal.

####  7.1.9. Proof of Proposition 7.1.7

Since $\mathrm{Whit}(G)_{I}$ is generated as a $\mathrm{Sph}_{G,I}$-module by
the vacuum object

$\mathscr{W}_{\mathrm{vac},I}=\mathscr{W}_{\mathrm{vac}}\boxtimes\omega_{X^{I}}\in\mathrm{Whit}\left(\overline{\mathrm{Bun}}_{N}\times
X^{I}\right)\hookrightarrow\mathrm{Whit}(G)_{I},$

it suffices to show that the colax constraint

$\mathrm{Poinc}_{I,!}\cdot(\SS\star\mathscr{W}_{\mathrm{vac},I})\to\SS\star\left(\mathrm{Poinc}_{I,!}\cdot\mathscr{W}_{\mathrm{vac},I}\right)$

is an isomorphism. Now consider the following diagram.

${\overline{\mathrm{Bun}}_{N}\\!\underset{\mathrm{Bun}_{G}}{\times}\\!\mathscr{H}_{G,I}}$${\overline{\mathrm{Bun}}_{N}}$${\mathscr{H}_{G,I}}$${\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}}$${\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G,I}}$$\scriptstyle{{\mathop{h}\limits^{\vbox
to-1.07639pt{\kern-1.61458pt\hbox{$\scriptstyle\leftharpoonup$}\vss}}}}$$\scriptstyle{{\mathop{h}\limits^{\vbox
to-1.07639pt{\kern-1.61458pt\hbox{$\scriptstyle\rightharpoonup$}\vss}}}_{\\!N}}$$\scriptstyle{\pi}$$\scriptstyle{{\mathop{h}\limits^{\vbox
to-1.07639pt{\kern-1.61458pt\hbox{$\scriptstyle\leftharpoonup$}\vss}}}}$$\scriptstyle{{\mathop{h}\limits^{\vbox
to-1.07639pt{\kern-1.61458pt\hbox{$\scriptstyle\rightharpoonup$}\vss}}}\times
s}$$\scriptstyle{\pi_{I}}$

The colax constraint identifies with

$\pi_{I,!}\cdot{\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\rightharpoonup$}\vss}}}_{\\!N,!}\cdot\left(\mathscr{W}_{\mathrm{vac}}\overset{!}{\boxtimes}\SS_{\mathscr{H}}\right)\to({\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\rightharpoonup$}\vss}}}\times
s)_{*}\cdot\left({\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\leftharpoonup$}\vss}}}^{!}\cdot\pi_{!}\cdot\mathscr{W}_{\mathrm{vac}}\overset{!}{\otimes}\SS_{\mathscr{H}}\right),$

which is an isomorphism because ${\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\rightharpoonup$}\vss}}}\times s$ and
${\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\rightharpoonup$}\vss}}}_{\\!N}$ are proper
and $\SS_{\mathscr{H}}$ is ULA over $\mathrm{Bun}_{G}$.

#### 7.1.10.

We write

$\Gamma(\mathrm{Ran},-):\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Ran}}\to\mathrm{D}(\mathrm{Bun}_{G})$

for pushforward along the projection

$\mathrm{pr}:\mathrm{Bun}_{G}\times\mathrm{Ran}\to\mathrm{Bun}_{G}.$

It is left adjoint to $\mathrm{pr}^{!}$, and we have a commutative diagram

${\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Ran}}^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})_{\mathrm{Ran}}}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})}$$\scriptstyle{\mathrm{PsId}_{!}}$$\scriptstyle{(\mathrm{pr}^{!})^{\vee}}$$\scriptstyle{\Gamma(\mathrm{Ran},-)}$$\scriptstyle{\mathrm{PsId}_{!}}$

###  7.2. Conf v Ran

#### 7.2.1.

The purpose of this subsection is to construct a functor

$\mathrm{sprd}_{*}:\mathrm{Whit}(M)_{\mathrm{Conf}}\to\mathrm{Whit}(M)_{\mathrm{Ran}}$

along with the following compatibilities:

  * –

We have

(7.2.1.1)
$\Gamma(\mathrm{Ran},\mathrm{Poinc}_{\mathrm{Ran},!}\cdot\mathrm{sprd}_{*})\simeq\mathrm{Poinc}_{\mathrm{Conf},!}.$

  * –

Let $\mathrm{sprd}^{!}$ denote the right adjoint to $\mathrm{sprd}_{*}$. Then
there is a commutative diagram:

(7.2.1.2)
${\mathrm{Whit}(M)_{\mathrm{Conf}}^{\vee}}$${\mathrm{Whit}(M)_{\mathrm{Conf}}}$${\mathrm{Whit}(M)_{\mathrm{Ran}}^{\vee}}$${\mathrm{Whit}(M)_{\mathrm{Ran}}}$$\scriptstyle{\sim}$$\scriptstyle{(\mathrm{sprd}^{!})^{\vee}}$$\scriptstyle{\mathrm{sprd}_{*}}$$\scriptstyle{\sim}$

#### 7.2.2.

The device relating $\mathrm{Conf}$ to $\mathrm{Ran}$ is a Cartesian diagram

(7.2.2.1)
${\overline{\mathrm{Bun}}_{N_{M},\mathrm{Ran}}^{\mathrm{pol},\mathrm{neg}}}$${\overline{\mathrm{Bun}}_{N_{M},\mathrm{Conf}}^{\mathrm{pol}}}$${\overline{\mathrm{Bun}}_{N_{M},\mathrm{Ran}}^{\mathrm{pol}}}$${\mathrm{Gr}_{M^{\mathrm{ab}},\mathrm{Ran}}^{\omega,\mathrm{neg}}}$${\mathrm{Conf}_{G,P}}$${\mathrm{Gr}^{\omega}_{M^{\mathrm{ab}},\mathrm{Ran}}}$$\scriptstyle{{{}^{\prime}}\pi}$$\scriptstyle{{{}^{\prime}}i^{\mathrm{neg}}}$$\scriptstyle{\pi}$$\scriptstyle{i^{\mathrm{neg}}}$

whose terms are defined as follows:

  * –

We write $\mathrm{Gr}_{M^{\mathrm{ab}},\mathrm{Ran}}^{\omega}$ for the
$\omega^{\check{\rho}}$-twisted form of the affine Grassmannian. It classifies
a point $x$ of $\mathrm{Ran}$, a principal $M^{\mathrm{ab}}$-bundle
$\mathscr{F}_{M^{\mathrm{ab}}}$, and a system of compatible maps

$\overset{\circ}{\mathscr{F}}{}_{M^{\mathrm{ab}}}\overset{M^{\mathrm{ab}}}{\times}\theta\to\omega^{\theta({\check{\rho}})}\qquad\theta\in\Lambda_{G,P}.$

We write $\mathrm{Gr}_{M^{\mathrm{ab}}}^{\omega,\mathrm{neg}}$ for the closed
sub-prestack where these maps extend to regular maps for all
$\theta\in\Lambda_{G,P}^{\mathrm{pos}}$. When this happens we obtain a divisor
$D\in\mathrm{Conf}_{G,P}$ supported on $x$. This defines the leftward map
denoted $\pi$.

  * –

The prestack $\overline{\mathrm{Bun}}_{N_{M},\mathrm{Ran}}^{\mathrm{pol}}$ is
the Ran version of $\overline{\mathrm{Bun}}_{N_{M},I}^{\mathrm{pol}}$. It
classifies a point $x$ of the Ran space, a principal $M$-bundle
$\mathscr{F}_{M}$ on $X$, and an $N_{M}^{\omega}$-Plücker data away from $x$.
The right vertical map is defined by observing that such a Plücker data
provides an identification between the bundles

$\mathscr{F}_{M}\overset{M}{\times}M^{\mathrm{ab}}\qquad\text{and}\qquad\omega^{\check{\rho}}\overset{T}{\times}M^{\mathrm{ab}}$

away from $x$.

  * –

We may define
$\overline{\mathrm{Bun}}_{N_{M},\mathrm{Ran}}^{\mathrm{pol},\mathrm{neg}}$ as
the fiber product of either face; the two are canonically identified.

Then we define

$\mathrm{sprd}_{*}=({{}^{\prime}}i^{\mathrm{neg}})_{*}\cdot{{}^{\prime}}\pi^{!}.$

#### 7.2.3.

In fact, the map

$\mathrm{Gr}_{M^{\mathrm{ab}}}^{\omega,\mathrm{neg}}\hookrightarrow\mathrm{Conf}\times\mathrm{Ran}$

identifies $\mathrm{Gr}_{M^{\mathrm{ab}}}^{\omega,\mathrm{neg}}$ with the
locus of pairs $(D,x)$ cut out by the closed condition $|D|\hookrightarrow x$.
Let us use this to obtain the desired compatibilities.

  * –

Since $\pi$ is pseudo-ind-proper and $i^{\mathrm{neg}}$ is a closed immersion,
$\mathrm{sprd}_{*}$ admits a continuous right adjoint

$\mathrm{sprd}^{!}\simeq{{}^{\prime}}\pi_{*}\cdot({{}^{\prime}}i^{\mathrm{neg}})^{!}.$

This gives the compatibility (7.2.1.2) with duality.

  * –

The compatibility (7.2.1.1) with Poincare series follows from the universal
homological contractibility of $\pi$.

##  8\. Ran upgrades

#### 8.0.1.

In this section we will upgrade our previous constructions to the Ran setting.

###  8.1. The Ran version of $\eta_{G}$

#### 8.1.1.

For convenience, we will write

$\mathrm{Poinc}_{?}=\mathrm{PsId}_{!}\cdot\mathrm{coeff}_{*}^{\vee}\cdot\mathrm{PsId}_{\mathrm{Whit}}^{-1}.$

We write $\mathrm{Poinc}_{\mathrm{Ran},?}$ and $\mathrm{Poinc}_{I,?}$ the
variants of this functor over $\mathrm{Ran}$ and $X^{I}$.

#### 8.1.2.

Consider the diagram

${\mathrm{Whit}(G)_{\mathrm{Ran}}^{\vee}}$${\mathrm{Whit}(G)_{\mathrm{Ran}}}$${\mathrm{D}(\mathrm{Bun}_{G})^{\vee}}$${\mathrm{D}(\mathrm{Bun}_{G})}$$\scriptstyle{\mathrm{coeff}_{*}^{\vee}}$$\scriptstyle{\mathrm{Poinc}_{!}}$$\scriptstyle{\mathrm{PsId}_{!}}$

of $\mathrm{Sph}_{G,\mathrm{Ran}}$-linear categories. The goal of this section
is to extend $\eta_{G}$ to a map

$\eta_{G,\mathrm{Ran}}:\mathrm{Poinc}_{\mathrm{Ran},!}\to\mathrm{Poinc}_{\mathrm{Ran},?}[2\delta_{G}]$

of $\mathrm{Sph}_{G,\mathrm{Ran}}$-linear functors. We will write the
construction over each power of the curve.

#### 8.1.3.

In fact we can simply repeat the definition of $\eta_{G}$ in order to define a
map

$\eta_{G,I}:\mathrm{Poinc}_{I,!}\to\mathrm{Poinc}_{I,?}[2\delta_{G}]$

of plain functors. That is, we consider the diagram

${\left(\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}\right)^{\mathrm{tr}}}$${\overline{\mathrm{Bun}}_{N,I}^{\mathrm{pol}}\underset{\mathrm{Bun}_{G}}{\times}\overline{\mathrm{Bun}}_{G}}$${\mathrm{Bun}_{G}}$$\scriptstyle{\pi_{2}}$

and define $\eta_{G,I}$ to be

${\pi_{2,!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}r_{G,!}\right)^{\mathrm{tr}}}$${\pi_{2,*}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}r_{G,!}\right)}$${\pi_{2,!}\cdot\left(\mathscr{W}\overset{*}{\boxtimes}r_{G,!}\right)[2\delta_{G}]}$${\pi_{2,*}\cdot\left(\mathscr{W}\overset{!}{\boxtimes}r_{G,!}\right)^{\mathrm{tr}}[2\delta_{G}]}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

The left arrow is obviously an isomorphism, and the right arrow is an
isomorphism by the same argument as Proposition 2.2.3.

###### 8.1.3 Proposition.

The map $\eta_{G,I}$ is canonically $\mathrm{Sph}_{G,I}$-linear.

####  8.1.4. Remarks

  * (i)

This result is not tautological. It requires the functorial construction of
commutative diagrams:

${\SS\star(\mathrm{Poinc}_{!}\cdot\mathscr{W})}$${\SS\star(\mathrm{Poinc}_{?}\cdot\mathscr{W})}$${\mathrm{Poinc}_{!}\cdot(\SS\star\mathscr{W})}$${\mathrm{Poinc}_{?}\cdot(\SS\star\mathscr{W})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

The right vertical arrow uses the fact that $\mathrm{PsId}_{G,I,!}$ is linear
for the action of $\mathrm{Sph}_{G,I}$. It is easy to prove this by observing
that

$\mathrm{PsId}_{G,I,!}\star\SS\simeq\SS\star\mathrm{PsId}_{G,I,!}\in\mathrm{D}(\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}\times
X^{I})$

is given by the $!$-pushforward of $\SS$ along the map

${\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\leftharpoonup$}\vss}}}\times{\mathop{h}\limits^{\vbox
to-1.50694pt{\kern-2.2604pt\hbox{$\rightharpoonup$}\vss}}}\times
s:\mathscr{H}_{G,I}\to\mathrm{Bun}_{G}\times\mathrm{Bun}_{G}\times X^{I}.$

But this identification is difficult to use because has the wrong (i.e.
$*$-pull and $!$-push) nature. Therefore, we should remedy this by
constructing a compactification of ${\mathop{h}\limits^{\vbox