where the functor ${\mathbb{L}}_{G,\operatorname{temp}}$ is the right adjoint
of the functor ${\mathbb{L}}^{L}_{G,\operatorname{temp}}$ of (1.8).

#### 5.1.5.

Set

$\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}:=\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})\cap\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\subset\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G}).$

It follows from (2.1) that

$\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}=\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\underset{{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})}{\otimes}{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})_{\operatorname{restr}},$

as subcategories of
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$.

It is easy to see that the functor ${\mathbf{u}}^{R}$ sends

$\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})\to\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}.$

Applying the functor

$-\underset{{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})}{\otimes}{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})_{\operatorname{restr}},$

from (5.1), we obtain a commutative diagram

(5.2)
$\begin{CD}\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})@>{{\mathbb{L}}^{\operatorname{restr}}_{G}}>{}>{\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})\\\
@V{{\mathbf{u}}^{R}}V{}V@V{}V{\Psi_{{\operatorname{Nilp}},\\{0\\}}}V\\\
\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}@>{{\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}}>{}>{\operatorname{QCoh}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}}).\end{CD}$

#### 5.1.6.

Suppose that the functor ${\mathbb{L}}^{\operatorname{restr}}_{G}$ is an
equivalence. We claim that this implies that
${\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}$ is also an
equivalence.

Indeed, the fact that ${\mathbb{L}}^{\operatorname{restr}}_{G}$ is an
equivalence implies that
${\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}$ is a Verdier
quotient. However, we also know that the functor
${\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}$ is conservative:
this is the main result of [FR].

######  Remark 5.1.7.

One can avoid appealing to [FR] for the proof of
“${\mathbb{L}}^{\operatorname{restr}}_{G}$ is an equivalence” $\Rightarrow$
“${\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}$ is an
equivalence”.

Indeed, it is not difficult to show that the functor ${\mathbb{L}}_{G}$
respects the action of the full Satake category (at a given point $x\in X$) on
the two sides121212This will be elaborated on in the subsequent paper in this
series.. Now,

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\simeq\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})\underset{\operatorname{Sat}_{G}}{\otimes}\operatorname{Sat}_{G,\operatorname{temp}}$

and

${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})\simeq{\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})\underset{\operatorname{Sat}_{G}}{\otimes}\operatorname{Sat}_{G,\operatorname{temp}},$

and also

$\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\simeq\operatorname{D-mod}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})\underset{\operatorname{Sat}_{G}}{\otimes}\operatorname{Sat}_{G,\operatorname{temp}}$

and

${\operatorname{QCoh}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})\simeq{\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})\underset{\operatorname{Sat}_{G}}{\otimes}\operatorname{Sat}_{G,\operatorname{temp}}.$

By a similar logic, if ${\mathbb{L}}_{G}$ is an equivalence, then it formally
follows that ${\mathbb{L}}_{G,\operatorname{temp}}$ is also an equivalence.

######  Remark 5.1.8.

In Sect. 5.2 we will prove an inverse implication (which is far less trivial):
the fact that ${\mathbb{L}}_{G,\operatorname{temp}}$ is an equivalence implies
that ${\mathbb{L}}_{G}$ is an equivalence.

The same argument will show that if
${\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}$ is an
equivalence then ${\mathbb{L}}^{\operatorname{restr}}_{G}$ is an equivalence.

#### 5.1.9.

We now claim that _if_
${\mathbb{L}}^{\operatorname{restr}}_{G,\operatorname{temp}}$ is an
equivalence (for all extensions of the ground field $k\subset k^{\prime}$),
_then_ ${\mathbb{L}}_{G,\operatorname{temp}}$ is an equivalence.

This follows by repeating verbatim the argument of [AGKRRV, Sect. 21.4].

######  Remark 5.1.10.

Both deductions:

$\text{Full tempered GLC}\,\Rightarrow\text{Restricted tempered GLC},$

carried out above, and

$\text{Full GLC}\,\Rightarrow\text{Restricted GLC},$

carried out in [AGKRRV, Sect. 21.4], follow the same logic, and both need that
the corresponding version of the Langlands functor admit an adjoint. Now, the
latter is immediate from the construction for
${\mathbb{L}}_{G,\operatorname{temp}}$, but requires more work for the
original Langlands functor ${\mathbb{L}}_{G}$.

#### 5.1.11.

Thus, we obtain that in order to prove the implication (i’) $\Rightarrow$ (i)
in Theorem 3.5.6, it suffices to show _if_
${\mathbb{L}}_{G,\operatorname{temp}}$ is an equivalence _then_
${\mathbb{L}}_{G}$ is an equivalence.

This will be done on the next section.

###  5.2. Full vs tempered Langlands

In this subsection we will assume that ${\mathbb{L}}_{G,\operatorname{temp}}$
is an equivalence and deduce that ${\mathbb{L}}_{G}$ is an equivalence.

The idea of the proof is that the behavior of
${\mathbb{L}}_{G,\operatorname{temp}}$ recovers the behavior of
${\mathbb{L}}_{G}$ on _compact objects_.

#### 5.2.1.

Recall that the category
${\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})$
is compactly generated (see [AG, Sect. 11.1.6]); its subcategory of compact
objects is
${\operatorname{Coh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})$,
the category of _coherent_ sheaves with nilpotent singular support.

Note that the functor

$\Psi:{\operatorname{IndCoh}}(\operatorname{LS}_{\check{G}})\to{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})$

is fully faithful, when restricted to the subcategory of compact objects.
Indeed, this restriction is the natural embedding

${\operatorname{Coh}}(\operatorname{LS}_{\check{G}})\hookrightarrow{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})$

.

In particular, we obtain that the functor
$\Psi_{{\operatorname{Nilp}},\\{0\\}}$ is also fully faithful when restricted
to

${\operatorname{Coh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})={\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{restr}}_{\check{G}})^{c}.$

#### 5.2.2.

The proof of the desired implication is a formal consequence of the
combination of the following two assertions:

######  Proposition 5.2.3.

The functor

${\mathbf{u}}^{R}|_{\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{c}}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{c}\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$

is fully faithful.

######  Proposition 5.2.4.

Assume that ${\mathbb{L}}_{G,\operatorname{temp}}$ is an equivalence (for the
group $G$ and all its Levi subgroups). Then the essential image of
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{c}$ in
${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})$ of
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{c}$ under

${\mathbb{L}}_{G,\operatorname{temp}}\circ{\mathbf{u}}^{R}\simeq{\mathbb{L}}_{G,\operatorname{coarse}}$

equals
${\operatorname{Coh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})$.

###  5.3. Proof of Proposition 5.2.3

The proof will be obtained by combining the following two ingredients. One is
the _miraculous functor_ on $\operatorname{Bun}_{G}$. The other is a result of
[Be1], which says that objects
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$ that are
*-extended from quasi-compact open substacks are tempered.

#### 5.3.1.

For an object
${\mathcal{F}}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$,
let

${\mathcal{F}}_{\operatorname{temp}}\to{\mathcal{F}}\to{\mathcal{F}}_{\operatorname{anti-
temp}}$

be the fiber sequence associated with

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\hookrightarrow\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G}).$

Explicitly,

${\mathcal{F}}\simeq{\mathbf{u}}\circ{\mathbf{u}}^{R}({\mathcal{F}}).$

We have to show that for a pair of compact object
${\mathcal{F}}_{1},{\mathcal{F}}_{2}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$,
the map

(5.3)
$\operatorname{Hom}_{\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})}({\mathcal{F}}_{1},{\mathcal{F}}_{2})\to\operatorname{Hom}_{\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})}({\mathcal{F}}_{1,\operatorname{temp}},{\mathcal{F}}_{2,\operatorname{temp}})$

is an isomorphism. As we shall see, just the assumption that
${\mathcal{F}}_{2}$ be compact will suffice.

The map (5.3) is tautologically an isomorphism if ${\mathcal{F}}_{1}$ is
tempered (for any ${\mathcal{F}}_{2}$). Hence, it suffices to show that if
${\mathcal{F}}_{2}$ is compact and ${\mathcal{F}}_{1}$ is _anti-tempered_ ,
i.e., if ${\mathcal{F}}_{1,\operatorname{temp}}=0$ or, equivalently, if

${\mathcal{F}}_{1}\to{\mathcal{F}}_{1,\operatorname{anti-temp}}$

is an isomorphism, then

$\operatorname{Hom}_{\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})}({\mathcal{F}}_{1},{\mathcal{F}}_{2})=0.$

#### 5.3.2.

We will now use the _miraculous functor_

$\operatorname{Mir}_{\operatorname{Bun}_{G}}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{co}}\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G}),$

see [Ga1, Sect. 3.1.1].

According to [Ga1, Theorem 3.1.5]), the functor
$\operatorname{Mir}_{\operatorname{Bun}_{G}}$ is an equivalence. Hence, it is
enough to show that for ${\mathcal{F}}_{2}$ compact and ${\mathcal{F}}_{1}$
anti-tempered,

(5.4)
$\operatorname{Hom}_{\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{co}}}(\operatorname{Mir}^{-1}_{\operatorname{Bun}_{G}}({\mathcal{F}}_{1}),\operatorname{Mir}^{-1}_{\operatorname{Bun}_{G}}({\mathcal{F}}_{2}))=0.$

#### 5.3.3.

Recall that compact objects in
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$ are of the form

$(j_{U})_{!}({\mathcal{F}}_{U}),$

where

$U\overset{j_{U}}{\hookrightarrow}\operatorname{Bun}_{G}$

is the embedding of a quasi-compact open, and
${\mathcal{F}}_{U}\in\operatorname{D-mod}(U)^{c}$.

With no restriction of generality, we can assume that $U$ is _co-truncative_
(see [DG1, Sect. 3.1] for what this means). In this case, the functor

$\operatorname{Mir}_{\operatorname{Bun}_{U}}:\operatorname{D-mod}_{\frac{1}{2}}(U)\to\operatorname{D-mod}_{\frac{1}{2}}(U)$

is an equivalence (see [DG1, Lemma 4.5.7]) and we have

$\operatorname{Mir}^{-1}_{\operatorname{Bun}_{G}}\circ(j_{U})_{!}\simeq(j_{U})_{*,\operatorname{co}}\circ\operatorname{Mir}^{-1}_{U},$

where

$(j_{U})_{*,\operatorname{co}}:\operatorname{D-mod}(U)\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{co}}$

is the tautological functor, see [DG1, Lemma 4.4.12].

Hence, in order the prove (5.4), it suffices to show that if
${\mathcal{F}}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$
is anti-tempered, then

$(j_{U})^{*}_{\operatorname{co}}\circ\operatorname{Mir}^{-1}_{\operatorname{Bun}_{G}}({\mathcal{F}})=0,$

where $(j_{U})^{*}_{\operatorname{co}}$ is the left adjoint of
$(j_{U})_{*,\operatorname{co}}$.

#### 5.3.4.

Let

$\operatorname{Id}^{\operatorname{nv}}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{co}}\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$

be the naive functor (see [Ga1, Sect. 2.1]). Recall that

$(j_{U})^{*}_{\operatorname{co}}\simeq(j_{U})^{*}\circ\operatorname{Id}^{\operatorname{nv}}$

(see [Ga1, Corollary 2.1.5]).

Hence, it suffices to show that the composite functor

$\operatorname{Id}^{\operatorname{nv}}\circ\operatorname{Mir}^{-1}_{\operatorname{Bun}_{G}}$

annihilates the anti-tempered subcategory.

It is easy to see that both functors $\operatorname{Id}^{\operatorname{nv}}$
and $\operatorname{Mir}_{\operatorname{Bun}_{G}}$ commute with the Hecke
action. Hence, it suffices to show that for any
${\mathcal{F}}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$,
we have

$(\operatorname{Id}^{\operatorname{nv}}\circ\operatorname{Mir}^{-1}_{\operatorname{Bun}_{G}}({\mathcal{F}}))_{\operatorname{anti-
temp}}=0.$

However, this follows follows from the fact the essential image of the functor
$\operatorname{Id}^{\operatorname{nv}}$ is contained in
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$,
see [Be1, Theorem B].

∎[Proposition 5.2.3]

###  5.4. Proof of Proposition 5.2.4

The idea of the proof is that compact generators of
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$ can be assembled
from tempered compact objects, and _Eisenstein series_.

The behavior of the former is controlled by the assumption that
${\mathbb{L}}_{G,\operatorname{temp}}$ is an equivalence. The behavior of the
latter is an expression of a basic compatibility between Eisenstein series and
the Langlands functor.

#### 5.4.1.

Let

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Eis}}\subset\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$

be the full subcategory, generated by the Eisenstein functors

$\operatorname{Eis}_{!}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{M})\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$

for _proper_ parabolic subgroups of $G$.

#### 5.4.2.

We claim:

######  Lemma 5.4.3.

The objects in

$\left(\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\right)^{c}\text{
and
}\left(\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Eis}}\right)^{c}$

generate $\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$.

###### Proof.

Let

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{cusp}}:=\left(\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Eis}}\right)^{\perp}.$

Denote by ${\mathbf{e}}$ the embedding

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{cusp}}\hookrightarrow\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G}),$

and let ${\mathbf{e}}^{L}$ denote its right adjoint.

Since the category
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$
is compactly generated131313On the one hand, since we have assumed that
${\mathbb{L}}_{G,\operatorname{coarse}}$ is an equivalence, the compact
generation of
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$
follows from that of ${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})$.
On the other hand, this compact generation can be proved unconditionally: in
[FR] it is shown that the objects obtained by acting by
${\mathop{\operatorname{\rm Rep}}}({\check{G}})_{\operatorname{Ran}}$ on
$\operatorname{Poinc}^{\operatorname{Vac,glob}}_{!}$ generate
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$.,
it suffices to show that the essential image of the composite functor

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\overset{{\mathbf{u}}}{\hookrightarrow}\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})\overset{{\mathbf{e}}^{L}}{\twoheadrightarrow}\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{cusp}}$

generates the target category. We will show that this functor is in fact a
localization (i.e., its right adjoint is fully faithful).

Indeed, the right adjoint to the above functor is the composition

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\overset{{\mathbf{u}}^{R}}{\hookleftarrow}\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})\overset{{\mathbf{e}}}{\twoheadleftarrow}\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{cusp}}.$

Now, according to [Be1, Theorem A], the essential image of the functor
${\mathbf{e}}$ is contained in
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$,
i.e., ${\mathbf{e}}$ factors as

${\mathbf{u}}\circ{\mathbf{e}}_{\operatorname{temp}}$

for a uniquely defined (fully faithful) functor

${\mathbf{e}}_{\operatorname{temp}}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{cusp}}\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}.$

Hence, the above right adjoint identifies with

${\mathbf{u}}^{R}\circ{\mathbf{u}}\circ{\mathbf{e}}_{\operatorname{temp}}\simeq{\mathbf{e}}_{\operatorname{temp}},$

as required.

∎

#### 5.4.4.

Let

${\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})_{\operatorname{Eis}}\subset{\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})$

be the full subcategory generated by the essential images of the _spectral
Eisenstein functors_

$\operatorname{Eis}^{\operatorname{spec}}:{\operatorname{QCoh}}(\operatorname{LS}_{\check{M}})\to{\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})$

for proper parabolic subgroups.

According to [AG, Theorem 13.3.6], the category
${\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})$
is compactly generated by

${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})^{c}\text{ and
}\left({\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})_{\operatorname{Eis}}\right)^{c},$

where we view ${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})$ as a
subcategory of
${\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})$
via the functor $\Xi_{\\{0\\},{\operatorname{Nilp}}}$, the left adjoint of
$\Psi_{{\operatorname{Nilp}},\\{0\\}}$.

Hence, given Lemma 5.4.3, in order to prove Proposition 5.2.4, it suffices to
show that:

  1. (1)

The essential image of
$\left(\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\right)^{c}$
in ${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})$ under the functor
${\mathbb{L}}_{G,\operatorname{coarse}}$ equals
${\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})^{c}$;

  2. (2)

The essential image of
$\left(\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Eis}}\right)^{c}$
under ${\mathbb{L}}_{G,\operatorname{coarse}}$ equals
$\left({\operatorname{IndCoh}}_{\operatorname{Nilp}}(\operatorname{LS}_{\check{G}})_{\operatorname{Eis}}\right)^{c}$.

#### 5.4.5.

Point (1) is immediate from the fact that
${\mathbb{L}}_{G,\operatorname{temp}}$ is an equivalence.

Point (2) follows by induction on the semi-simple rank from the fact that
${\mathbb{L}}_{G,\operatorname{temp}}$ is fully faithful from the next
assertion (which is well-known, and will be discussed in detail in a
subsequent paper in the series):

######  Lemma 5.4.6.

For a given parabolic subgroup $P$ with Levi quotient $M$, the diagram

$\begin{CD}\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{M})@>{{\mathbb{L}}_{M,\operatorname{coarse}}}>{}>{\operatorname{QCoh}}(\operatorname{LS}_{\check{M}})\\\
@V{\operatorname{Eis}_{!}}V{}V@V{}V{\operatorname{Eis}^{\operatorname{spec}}_{\operatorname{coarse}}}V\\\
\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})@>{{\mathbb{L}}_{G,\operatorname{coarse}}}>{}>{\operatorname{QCoh}}(\operatorname{LS}_{\check{G}})\end{CD}$

commutes up to cohomological shifts and tensoring by line bundles, where

$\operatorname{Eis}^{\operatorname{spec}}_{\operatorname{coarse}}:=\Psi_{{\operatorname{Nilp}},\\{0\\}}\circ\operatorname{Eis}^{\operatorname{spec}}.$

∎[Proposition 5.2.4]

###  5.5. The Betti context

The goal of this subsection is to prove the implication (ii’) $\Rightarrow$
(ii) in Theorem 3.5.6. The argument will be parallel to the proof of the
implication (i’) $\Rightarrow$ (i) given above.

#### 5.5.1.

Define the full subcategory

$\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\subset\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})$

as in [AG, Sect. 12.8.4].

Then the material in Sects. 5.1.4-5.1.11 applies as is. I.e., it suffices to
show that if the functor

${\mathbb{L}}^{\operatorname{Betti}}_{G,\operatorname{temp}}:\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}\to{\operatorname{QCoh}}(\operatorname{LS}^{\operatorname{Betti}}_{\check{G}})$

is an equivalence, then so is ${\mathbb{L}}^{\operatorname{Betti}}_{G}$.

The latter in turn reduces to the combination of the following two
propositions:

######  Proposition 5.5.2.

The functor

${\mathbf{u}}^{R}|_{\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})^{c}}:\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})^{c}\to\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$

is fully faithful.

######  Proposition 5.5.3.

Assume that ${\mathbb{L}}^{\operatorname{Betti}}_{G,\operatorname{temp}}$ is
an equivalence (for the group $G$ and all its Levi subgroups). Then the
essential image of
$\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})^{c}$
in
${\operatorname{QCoh}}(\operatorname{LS}^{\operatorname{Betti}}_{\check{G}})$
under

${\mathbb{L}}^{\operatorname{Betti}}_{G,\operatorname{temp}}\circ{\mathbf{u}}^{R}\simeq{\mathbb{L}}^{\operatorname{Betti}}_{G,\operatorname{coarse}}$

equals
${\operatorname{Coh}}_{\operatorname{Nilp}}(\operatorname{LS}^{\operatorname{Betti}}_{\check{G}})$.

#### 5.5.4.

Proposition 5.5.2 is proved along the same lines as Proposition 5.2.3 using
the following ingredients:

  * •

The category
$\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{co}}$
is well-defined (see [AGKRRV, Theorem 14.1.5]), and the functor

$\operatorname{Mir}_{\operatorname{Bun}_{G}}:\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{co}}\to\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})$

is an equivalence;

  * •

The essential image of the functor

$\operatorname{Id}^{\operatorname{nv}}:\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{co}}\to\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})$

lies in
$\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{temp}}$.

Both these assertions are proved in the same way as in the de Rham setting.

#### 5.5.5.

Proposition 5.5.3 is proved along the same lines as Proposition 5.2.4 using
the following ingredients:

  * •

The subcategory

$\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})_{\operatorname{cusp}}\subset\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})$

is contained in the essential image of the functor
$\operatorname{Id}^{\operatorname{nv}}$.

  * •

The diagram

$\begin{CD}\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{M})@>{{\mathbb{L}}^{\operatorname{Betti}}_{M,\operatorname{coarse}}}>{}>{\operatorname{QCoh}}(\operatorname{LS}^{\operatorname{Betti}}_{\check{M}})\\\
@V{\operatorname{Eis}_{!}}V{}V@V{}V{\operatorname{Eis}^{\operatorname{spec}}_{\operatorname{coarse}}}V\\\
\operatorname{Shv}^{\operatorname{Betti}}_{\frac{1}{2},{\operatorname{Nilp}}}(\operatorname{Bun}_{G})@>{{\mathbb{L}}^{\operatorname{Betti}}_{G,\operatorname{coarse}}}>{}>{\operatorname{QCoh}}(\operatorname{LS}^{\operatorname{Betti}}_{\check{G}})\end{CD}$

commutes up to cohomological shifts and tensoring by line bundles.

Both these assertions are proved in the same way as in the de Rham setting.

##  6\. The structure of Hecke eigensheaves

In this section, we discuss the structure of Hecke eigensheaves for
irreducible spectral parameters, building on recent results from [AGKRRV] and
[FR].

Below, we will assume GLC in the de Rham setting, and hence in the Betti
setting.

###  6.1. Statement of the main result

####  6.1.1. The Hecke eigensheaf

Let $\sigma\in\operatorname{LS}_{{\check{G}}}$ be a $k$-point of
$\operatorname{LS}_{{\check{G}}}$.

We define the category

$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Hecke},\sigma}:=\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})\underset{{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}}})}{\otimes}\operatorname{Vect}$

of _Hecke eigensheaves_ for $\sigma$. Here
${\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}}})$ acts on
$\operatorname{Vect}$ by pullback along
$\sigma:\operatorname{Spec}(k)\to\operatorname{LS}_{{\check{G}}}$.

Using the pushforward functor
$\sigma_{*}:\operatorname{Vect}\to{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}}})$,
we obtain a functor functor

${\mathbf{oblv}}_{\operatorname{Hecke},\sigma}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Hecke},\sigma}\to\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G}).$

Now assume that $\sigma$ is an _irreducible_ local system, i.e., $\sigma$ does
not admit a reduction to any proper parabolic
${\check{P}}\subsetneq{\check{G}}$. By [AG, Prop. 13.3.3] and our running
assumption that GLC holds, the functor ${\mathbb{L}}_{G}$ induces an
equivalence:

${\mathbb{L}}_{G,\sigma}:\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Hecke},\sigma}\simeq\operatorname{Vect}.$

Throughout this section, we let

$\widetilde{{\mathcal{F}}}_{\sigma}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})_{\operatorname{Hecke},\sigma}$

denote the unique object with

${\mathbb{L}}_{G,\sigma}(\widetilde{{\mathcal{F}}}_{\sigma})=k[-\dim(\operatorname{Bun}_{G})+\dim(\operatorname{Bun}_{N,\rho(\omega_{X})})].$

The shift here is motivated by Theorem 2.3.8.

We then let

${\mathcal{F}}_{\sigma}:={\mathbf{oblv}}_{\operatorname{Hecke},\sigma}(\widetilde{{\mathcal{F}}}_{\sigma})\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$

denote the underlying D-module of the canonical Hecke eigensheaf.

#### 6.1.2.

We will prove the following result. Most parts of it are not original to this
work; we will discuss attributions after stating the result.

######  Theorem 6.1.3.

As above, let $\sigma$ be an irreducible local system and assume the geometric
Langlands conjecture.

  1. (1)

${\mathcal{F}}_{\sigma}$ is holonomic and has regular singularities.

  2. (2)

The singular support of ${\mathcal{F}}_{\sigma}$ lies in the nilpotent cone
${\operatorname{Nilp}}\subset T^{*}(\operatorname{Bun}_{G})$.

  3. (3)

${\mathcal{F}}_{\sigma}$ lies in the heart
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{\heartsuit}$ of
$\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})$.

  4. (4)

Let $S_{\sigma}$ denote the (non-derived) group of automorphisms of $\sigma$
as a local system. Then ${\mathcal{F}}_{\sigma}$ has a decomposition:

${\mathcal{F}}_{\sigma}\simeq\oplus_{\rho\in\operatorname{Irrep}(S_{\sigma})}{\mathcal{F}}_{\sigma,\rho}^{\dim\rho}$

where each object

${\mathcal{F}}_{\sigma,\rho}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{\heartsuit}$

is a _simple_ (holonomic) $D$-module. Moreover, these simple objects are
distinct:
${\mathcal{F}}_{\sigma,\rho_{1}}\simeq{\mathcal{F}}_{\sigma,\rho_{2}}$ if and
only if $\rho_{1}=\rho_{2}$.

In particular,
${\mathcal{F}}_{\sigma}\in\operatorname{D-mod}_{\frac{1}{2}}(\operatorname{Bun}_{G})^{\heartsuit}$
is semi-simple.

  5. (5)

Assume the genus of $X$ is at least two and $G$ has connected center.

Let $[{\operatorname{Nilp}}]$ denote the cycle on
$T^{*}(\operatorname{Bun}_{G})$ defined by the nilpotent cone. In other words,
the multiplicities of $[{\operatorname{Nilp}}]$ are the multiplicities of
irreducible components of the closed substack141414We recall that
${\operatorname{Nilp}}$ is defined scheme-theoretically as the preimage of $0$
under the Hitchin map. ${\operatorname{Nilp}}\subset
T^{*}(\operatorname{Bun}_{G})$. Then the characteristic cycle
$\operatorname{CC}({\mathcal{F}}_{\sigma})$ equals $[{\operatorname{Nilp}}]$.

In the above, (1) and (2) are due to [AGKRRV] and (3) is due to [FR]. Also,
[FR] observed that its results imply (4) assuming GLC, and proved a special
case of this result without GLC. The last result (5) is new.

The above results have old folklore status as conjectures. Most appear in
[Laum, Conj. 6.3.2]. We expect (5) to hold without the restriction on $G$, and
our methods below prove something in this direction in general (e.g., we prove
that (5) holds for a non-trivial class of local systems $\sigma$).

######  Remark 6.1.4.

We note that one can check directly that the multiplicity of the zero section
in $[{\operatorname{Nilp}}]$ is $\prod_{d_{i}\geq
2}d_{i}^{\dim\Gamma(\Omega_{X}^{1,\otimes d_{i}})}=\prod
d_{i}^{(2d_{i}-1)(g-1)}$ for $d_{i}$ running over the exponents of $G$; it
follows that this is the rank of ${\mathcal{F}}_{\sigma}$ at a generic point
of $\operatorname{Bun}_{G}$.

###### Proof of Theorem 6.1.3, (1)-(4).

First, (2) is [AGKRRV, Cor. 14.4.10]. As ${\operatorname{Nilp}}$ is
Lagrangian, we deduce that Hecke eigensheaves are holonomic. They have regular
singularities by [AGKRRV, Cor. 16.5.7]. Then (3) is [FR, Thm. 11.2.1.2].

We now turn to (4). Recall that $S_{\sigma}$ is an extension of a finite group
by the center $Z_{{\check{G}}}$ of ${\check{G}}$; in particular, $S_{\sigma}$
is a (disconnected) reductive group.

Let
$\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}}\subset\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}}$
denote the connected component containing $\sigma$. Define:

$\operatorname{D-mod}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})_{\sigma}:=\operatorname{D-mod}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})\underset{{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}}}^{\operatorname{restr}})}{\otimes}{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}}).$

As in [AGKRRV, Cor. 14.3.5], this category is a direct summand of
$\operatorname{D-mod}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})$.

By Lemma 5.4.6, every object of
$\operatorname{D-mod}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})_{\sigma}$
is cuspidal. Therefore, by [Be1], every object of
$\operatorname{D-mod}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})_{\sigma}$
is tempered. Therefore, ${\mathbb{L}}_{G}$ induces an equivalence

${\mathbb{L}}_{G,\widehat{\sigma}}:\operatorname{D-mod}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})_{\sigma}\simeq{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}}).$

We then have a diagram

${\operatorname{Vect}}$${\operatorname{Shv}_{{\operatorname{Nilp}},\frac{1}{2}}(\operatorname{Bun}_{G})_{\sigma}}$${{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}})}$$\scriptstyle{{\mathcal{F}}_{\sigma}}$$\scriptstyle{\sigma_{*}}$$\scriptstyle{{\mathbb{L}}_{G,\widehat{\sigma}}{[C]}}$

that commutes by definition of ${\mathcal{F}}_{\sigma}$. Here
$C:=\dim(\operatorname{Bun}_{G})-\dim(\operatorname{Bun}_{N,\rho(\omega_{X})})$
and $\sigma_{*}$ denotes pushforward along
$\sigma:\operatorname{Spec}(k)\to\operatorname{LS}_{{\check{G}}}^{\operatorname{restr}}$.

The bottom arrow in the above diagram is a t-exact equivalence by GLC and
Theorem 2.3.8. Therefore, Jordan-Hölder questions for ${\mathcal{F}}_{\sigma}$
are the same as for $\sigma_{*}(k)$.

We now remind ([AGKRRV, Prop. 4.3.5]) that $\sigma$ factors as
$\operatorname{Spec}(k)\to\operatorname{Spec}(k)/S_{\sigma}\to\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}}$
with the latter map being a closed embedding. Therefore, there is a fully
faithful embedding ${\mathop{\operatorname{\rm
Rep}}}(S_{\sigma})^{\heartsuit}\subset{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}},\sigma}^{\operatorname{restr}})^{\heartsuit}$
with $\sigma_{*}(k)$ corresponding to the regular representation of
$S_{\sigma}$.

Now the result follows from the fact that $S_{\sigma}$ is reductive, so its
(left) regular representation has the form
$R_{S_{\sigma}}\simeq\oplus_{\rho\in\operatorname{Irrep}(S_{\sigma})}\rho^{\dim\rho}$.

∎

###  6.2. Characteristic cycles for eigensheaves

We now turn to the proof of Theorem 6.1.3 (5).

####  6.2.1. Idea of the proof

Our main new result is the following. We always assume genus $\geq 2$ in what
follows, but $G$ is allowed to be arbitrary.

######  Theorem 6.2.2.

For every pair of irreducible local systems
$\sigma_{1},\sigma_{2}\in\operatorname{LS}_{{\check{G}}}$ lying in the same
irreducible component of $\operatorname{LS}_{{\check{G}}}$,
$\operatorname{CC}({\mathcal{F}}_{\sigma_{1}})=\operatorname{CC}({\mathcal{F}}_{\sigma_{2}})$.

The following is [BD, Prop. 5.1.2]:

######  Theorem 6.2.3 (Beilinson-Drinfeld).

Let
$\operatorname{LS}_{{\check{G}},\operatorname{neut}}\subset\operatorname{LS}_{{\check{G}}}$
be the irreducible component of $\operatorname{LS}_{{\check{G}}}$ containing
the trivial local system. Then there exists an irreducible local system
$\sigma\in\operatorname{LS}_{{\check{G}},\operatorname{neut}}$ such that
$\operatorname{CC}({\mathcal{F}}_{\sigma})=[{\operatorname{Nilp}}]$.

We now recall that $\operatorname{LS}_{{\check{G}}}$ is irreducible when
${\check{G}}$ has simply-connected derived group (and the genus is $\geq 2$),
see [BD, Prop. 2.11.4]. Therefore, the above two results imply the claim.

######  Remark 6.2.4.

Theorem 6.2.3 is proved by Rham methods, but Theorem 6.2.2 will be proved by
Betti methods.

###  6.3. Proof of Theorem 6.2.2

We will prove this result using some theory of Betti constructible sheaves.

By the Lefschetz principle, note that Theorem 6.2.2 reduces to its Betti
version. Therefore, we put ourselves in this setting in what follows.

####  6.3.1. Review of microstalks

Let $k={\mathbb{C}}$, let ${\mathcal{Y}}$ be a smooth algebraic stack of
finite type over ${\mathbb{C}}$, and recall that
$\operatorname{Shv}^{\operatorname{Betti}}({\mathcal{Y}})$ is the category of
all Betti sheaves on ${\mathcal{Y}}$. Let $\Lambda\subset
T^{*}({\mathcal{Y}})$ be a conical (algebraic) Lagrangian and consider
$\operatorname{Shv}_{\Lambda}^{\operatorname{Betti}}({\mathcal{Y}})\subset\operatorname{Shv}^{\operatorname{Betti}}({\mathcal{Y}})$,
which we recall is the subcategory of sheaves with singular support in
$\Lambda$ and is defined in [AGKRRV, Sect. F.6] for stacks.

Let $\xi\in\Lambda$ be a point lying in the smooth locus of $\Lambda$. The
following result summarizes the microstalk theory for our needs.

######  Theorem 6.3.2.

There is a functor
$\mu_{\xi}:\operatorname{Shv}_{\Lambda}({\mathcal{Y}})\to\operatorname{Vect}$
with the following properties:

  1. (1)

$\mu_{\xi}$ is t-exact.

  2. (2)

$\mu_{\xi}$ admits a left adjoint, i.e., $\mu_{\xi}$ is corepresented by an
object $\mu\delta_{\xi}\in\operatorname{Shv}_{\Lambda}({\mathcal{Y}})$.

  3. (3)

For a constructible sheaf
${\mathcal{F}}\in\operatorname{Shv}_{\Lambda}^{\operatorname{Betti}}({\mathcal{Y}})$,
$\dim(\mu_{\xi}({\mathcal{F}}))$ equals the order of the characteristic cycle
of ${\mathcal{F}}$ at $\xi$.

###### Proof.

First, assume ${\mathcal{Y}}$ is a smooth scheme, see e.g. [GPS, Sect. 4.4]
for a review of the construction and the corepresentability. Exactness (up to
a normalizing shift) follows from [KS, Cor. 10.3.13]. The relation to the
characteristic cycle is [KS, Examples 9.5.7].

In general, choose $\pi:Y\to{\mathcal{Y}}$ a smooth cover by a smooth scheme,
and we assume Let $y\in{\mathcal{Y}}$ be the point so $\xi\in
T_{y}^{*}({\mathcal{Y}})$; let $\widetilde{y}\in\pi^{-1}(y)$ be a lift of $y$
and let $\widetilde{\xi}:=\pi^{*}(\xi)\in T_{\widetilde{y}}^{*}(Y)$. Define:

$\mu_{\xi}({\mathcal{F}}):=\mu_{\widetilde{\xi}}\circ\pi^{!}({\mathcal{F}})[-\dim
Y+\dim{\mathcal{Y}}]=\mu_{\widetilde{\xi}}\circ\pi^{*}({\mathcal{F}})[\dim
Y-\dim{\mathcal{Y}}].$

Finally, recall from [AGKRRV, Cor. G.7.6] that the embedding
$\iota:\operatorname{Shv}_{\Lambda}^{\operatorname{Betti}}({\mathcal{Y}})\to\operatorname{Shv}^{\operatorname{Betti}}({\mathcal{Y}})$
admits a left adjoint $\iota^{L}$. Then:

$\mu\delta_{\xi}:=\iota^{L}(\pi_{!}(\mu\delta_{\widetilde{\xi}}))[\dim
Y-\dim{\mathcal{Y}}].$

evidently corepresents the microstalk at $\xi$.

∎

####  6.3.3. Microstalks and the Langlands equivalence

Let ${\operatorname{Nilp}}^{\alpha}\subset{\operatorname{Nilp}}$ be an
irreducible component and let $\xi\in{\operatorname{Nilp}}^{\alpha}$ be a
generic point lying in the smooth locus. We have a microstalk functor:

$\operatorname{Shv}_{{\operatorname{Nilp}},\frac{1}{2}}^{\operatorname{Betti}}(\operatorname{Bun}_{G})\to\operatorname{Vect}$

and a corepresenting object $\mu\delta_{\xi}$.

Define
${\mathcal{E}}_{\xi}\in{\operatorname{IndCoh}}_{{\operatorname{Nilp}}}(\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti}})$
as:

${\mathcal{E}}_{\xi}:={\mathbb{L}}_{G}^{\operatorname{Betti}}(\mu\delta_{\xi})[-\dim(\operatorname{Bun}_{G})+\dim(\operatorname{Bun}_{N,\rho(\omega_{X})})].$

We let
${\mathcal{E}}_{\xi}^{\operatorname{irred}}\in{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti},\operatorname{irred}})$
be the restriction of ${\mathcal{E}}_{\xi}$ to the irreducible locus.

#### 6.3.4.

Observe that ${\mathcal{E}}_{\xi}$ is compact, i.e., it lies in
${\operatorname{Coh}}_{{\operatorname{Nilp}}}(\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti}})$:
this follows as it corresponds to the compact object $\mu\delta_{\xi}$ under
the Betti Langlands equivalence.

In particular, ${\mathcal{E}}_{\xi}^{\operatorname{irred}}$ is perfect, so the
Euler characteristic of its fibers form a locally constant function on
$\operatorname{LS}_{\check{G}}^{\operatorname{Betti},\operatorname{irred}}$.

We compute the dual:

$\operatorname{Hom}_{\operatorname{Vect}}(\sigma^{*}({\mathcal{E}}_{\xi}),{\mathbb{C}})$

to this fiber at
$\sigma\in\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti},\operatorname{irred}}({\mathbb{C}})$
as:

$\operatorname{Hom}_{{\operatorname{QCoh}}(\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti}}}({\mathcal{E}}_{\xi},\sigma_{*}({\mathbb{C}}))\simeq\operatorname{Hom}_{\operatorname{Shv}_{{\operatorname{Nilp}},\frac{1}{2}}^{\operatorname{Betti}}}(\mu\delta_{\xi},{\mathcal{F}}_{\sigma})=\mu_{\xi}({\mathcal{F}}_{\sigma})$

for ${\mathcal{F}}_{\sigma}$ the (Betti) eigensheaf corresponding to $\sigma$,
including the shift as in Sect. 6.1.1.

Applying Theorem 6.3.2, we see (_i_) $\sigma^{*}({\mathcal{E}}_{\xi})$ is
actually concentrated in one degree, so
${\mathcal{E}}_{\xi}^{\operatorname{irred}}$ is actually a vector bundle, and
(_ii_) the fiber of ${\mathcal{E}}_{\xi}^{\operatorname{irred}}$ at $\sigma$
has the same dimension as the order of the characteristic cycle of
${\mathcal{F}}_{\sigma}$ at $\xi$.

Therefore, the function
$\sigma\mapsto\operatorname{ord}_{\xi}(\operatorname{CC}({\mathcal{F}}_{\sigma}))$
is a locally constant function on
$\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti},\operatorname{irred}}$.
It now suffices to observe that
$\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti},\operatorname{irred}}$
is smooth and dense in
$\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti}}$, so connected
components of
$\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti},\operatorname{irred}}$
are in bijection with irreducible components of
$\operatorname{LS}_{{\check{G}}}^{\operatorname{Betti}}$.

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