Corollary X.I.3], there exists a $\overline{\mathbb{F}}_{\ell}$-linear action

$\mathrm{Perf}(\mathfrak{X}_{\hat{G}})\rightarrow\mathrm{End}(\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{\omega})$
$C\mapsto\\{A\mapsto C\star A\\}$

which, extending by colimits, gives

$\mathrm{Ind}(\mathrm{Perf}(\mathfrak{X}_{\hat{G}}))\rightarrow\mathrm{End}(\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell}))$

We recall the following basic properties of this action.

  1. (1)

For $V=\boxtimes_{i\in
I}V_{i}\in\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\phantom{}^{L}G^{I})$,
there is an attached vector bundle
$C_{V}\in\mathrm{Perf}(\mathfrak{X}_{\hat{G}})^{BW_{\mathbb{Q}_{p}}^{I}}$
whose evaluation at a $\overline{\mathbb{F}}_{\ell}$-point of
$\mathfrak{X}_{\hat{G}}$ corresponding to a (not necessarily semi-simple)
$L$-parameter
$\tilde{\phi}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{F}}_{\ell})$
is the vector space $V$ with $W_{\mathbb{Q}_{p}}^{I}$-action given by
$\boxtimes_{i\in I}r_{V_{i}}\circ\tilde{\phi}$. The endomorphism

$C_{V}\star(-):\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{\mathbb{Q}_{p}}^{I}}$

is the Hecke operator $T_{V}$ attached to $V$.

  2. (2)

The action is monoidal in the sense that, given
$C_{1},C_{2}\in\mathrm{Perf}(\mathfrak{X}_{\hat{G}})$, we have a natural
equivalence of endofunctors

$(C_{1}\otimes^{\mathbb{L}}C_{2})\star(-)\simeq C_{1}\star(C_{2}\star(-)).$

If
$\phi:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{F}}_{\ell})$
is a semi-simple $L$-parameter then this defines a closed
$\overline{\mathbb{F}}_{\ell}$-point $x$ in the moduli stack of Langlands
parameters, which maps to a closed point in the coarse moduli space. We let
$\mathfrak{m}_{\phi}\subset\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$
denote the corresponding maximal ideal. We recall that, for all
$f\in\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$ and
$A\in\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$, one obtains
an endomorphism

$A\simeq\mathcal{O}_{\mathfrak{X}_{\hat{G}}}\star
A\rightarrow\mathcal{O}_{\mathfrak{X}_{\hat{G}}}\star A\simeq A$

induced by multiplication by $f$. Under the description of
$\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$ in terms of the
excursion algebra, this encodes the action of the excursion algebra on
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ [Zou22, Theorem
5.2.1]. More precisely, we recall that, since $\ell$ is very good [FS21, Page
33], to any Schur-irreducible
$A\in\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ we can, by
[FS21, Proposition I.9.3], attach a conjugacy class of semi-simple
$L$-parameters

$\phi_{A}^{\mathrm{FS}}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{F}}_{\ell})$

called the Fargues-Scholze parameter of $A$. By [FS21, Theorem VIII.3.6], we
have an identification between the ring of global functions
$\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$ and excursion
operators. Since $A$ is Schur irreducible the endomorphisms corresponding to
$f\in\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$ determine a
non-zero scalar in $\overline{\mathbb{F}}_{\ell}$ which will be determined by
the excursion datum evaluated at the Fargues-Scholze parameter
$\phi_{A}^{\mathrm{FS}}$.

With this in hand, we can make our key definition.

######  Definition 4.1.

We define
$\iota_{\phi}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}\hookrightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
to be the full-subcategory of objects $A$ for which the endomorphisms
$A\rightarrow A$ induced by
$f\in\mathcal{O}_{\mathfrak{X}_{\hat{G}}}\setminus\mathfrak{m}_{\phi}$ are
isomorphisms.

It is easy to check that the subcategory
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}\subset\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
is preserved under colimits and limits, and therefore, by the
$\infty$-categorical adjoint functor theorem [Lur09, Corollary 5.5.2.9], there
exists a left adjoint to the inclusion $\iota_{\phi}$ denoted by
$\mathcal{L}_{\phi}$. We define
$(-)_{\phi}:=\iota_{\phi}\mathcal{L}_{\phi}(-)$. This, by the fully
faithfulness of $\iota_{\phi}$, will define an idempotent functor (See
Appendix A for details).

We now have the following key lemma.

######  Lemma 4.2.

The following is true.

  1. (1)

Any Schur irreducible object
$A\in\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$ has
Fargues-Scholze parameter equal to $\phi$ as conjugacy classes of parameters.

  2. (2)

Given $V\in\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\phantom{}^{L}G^{I})$,
the Hecke operator
$T_{V}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{\mathbb{Q}_{p}}^{I}}$
takes the subcategory
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$ to
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}^{BW_{\mathbb{Q}_{p}}^{I}}$,
and there is a natural isomorphism $T_{V}((-)_{\phi})\simeq(T_{V}(-))_{\phi}$.

  3. (3)

Given
$A\in\mathrm{D}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})\subset\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$,
we have an isomorphism

$R\Gamma(K_{p}^{\mathrm{hs}},A)_{\mathfrak{m}}\simeq
R\Gamma(K_{p}^{\mathrm{hs}},A_{\phi_{\mathfrak{m}}}),$

where the LHS is the usual localization under the smooth Hecke algebra.

  4. (4)

If
$A\in\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
is ULA then one has a direct sum decomposition

$A\simeq\bigoplus_{\phi}A_{\phi}$

ranging over all semi-simple $L$-parameters.

###### Proof.

Claims (2) and (4) follow from Proposition A.2 and Proposition A.5,
respectively, where for claim (2) we use the relationship between Hecke
operators and the spectral action described above.

For (1), this follows since the action of
$\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$ on $A$ will
factor through the maximal ideal $\mathfrak{m}_{A}$ defined by the semi-simple
$L$-parameter $\phi_{A}^{\mathrm{FS}}$ attached to $A$ by the above
discussion, and therefore
$A\in\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$ forces
an equality of maximal ideals: $\mathfrak{m}_{A}=\mathfrak{m}_{\phi}$.

For (3), we use the arguments in Koshikawa [Kos21, Page 6]. Consider the map

$\mathcal{O}_{\mathfrak{X}_{\hat{G}}}(\mathfrak{X}_{\hat{G}})\rightarrow\mathrm{End}_{G(\mathbb{Q}_{p})}(\mathrm{cInd}_{K_{p}^{\mathrm{hs}}}^{G(\mathbb{Q}_{p})}(\overline{\mathbb{F}}_{\ell}))\simeq
H_{K_{p}^{\mathrm{hs}}}^{\mathrm{op}}$

given by the spectral action, where
$\mathrm{cInd}_{K_{p}^{\mathrm{hs}}}^{G(\mathbb{Q}_{p})}(\overline{\mathbb{F}}_{\ell})$
is regarded as a right $H_{K_{p}^{\mathrm{hs}}}$-module. It is shown that
after through the usual action by the unramified Hecke algebra composed with
the involution $KhK\rightarrow Kh^{-1}K$ gives rise to a map which is
compatible with usual $L$-parameters for unramified irreducible
representations. In particular, the pullback of the maximal ideal
$\mathfrak{m}\subset H_{K_{p}^{\mathrm{hs}}}$ is given by the maximal ideal
$\mathfrak{m}_{\phi_{\mathfrak{m}}}\subset\mathcal{O}_{X_{\hat{G}}}(\mathfrak{X}_{\hat{G}})$.
Now, by arguing as in Proposition A.3, we have an identification:

$R\mathrm{Hom}(\mathrm{cInd}_{K_{p}^{\mathrm{hs}}}^{G(\mathbb{Q}_{p})}(\overline{\mathbb{F}}_{\ell}),A_{\phi_{\mathfrak{m}}})\simeq
R\mathrm{Hom}(\mathrm{cInd}_{K_{p}^{\mathrm{hs}}}^{G(\mathbb{Q}_{p})}(\overline{\mathbb{F}}_{\ell}),A)_{\mathfrak{m}_{\phi_{\mathfrak{m}}}}.$

Using Frobenius reciprocity, this gives an identification:

$R\Gamma(K_{p}^{\mathrm{hs}},A_{\phi_{\mathfrak{m}}})\simeq
R\Gamma(K_{p}^{\mathrm{hs}},A)_{\mathfrak{m}_{\phi_{\mathfrak{m}}}},$

but the RHS identifies with $R\Gamma(K_{p}^{\mathrm{hs}},A)_{\mathfrak{m}}$,
as explained above. ∎

We note that we get the following Corollary of this.

######  Corollary 4.3.

Let $A$ be a complex of smooth $G(\mathbb{Q}_{p})$-representations which is
admissible (i.e $A^{K}$ is a perfect complex for all compact open $K\subset
G(\mathbb{Q}_{p})$). We then have a decomposition

$A\simeq\bigoplus_{\phi}A_{\phi}$

running over semisimple $L$-parameters, where any irreducible constituent
$\pi$ of $A_{\phi}$ has Fargues-Scholze parameter equal to
$\phi_{\pi}^{\mathrm{FS}}$, as conjugacy classes of parameters.

###### Proof.

This follows by applying to Lemma 4.2 (1) and (4) to the full subcategory
$\mathrm{D}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})\subset\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
∎

Now our goal is to describe the subcategory
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$ more
explicitly, using the results of [Ham22] in the case that $\phi$ is induced
from a generic toral parameter $\phi_{T}$. To do this, we will need to have
some information about the Fargues-Scholze local Langlands correspondence.
First, let us introduce some notation.

We let $B(G)_{\mathrm{un}}:=\text{Im}{(B(T)\rightarrow B(G))}$. We recall that
these are precisely the elements $b\in B(G)$ such that the
$\sigma$-centralizer $J_{b}$ is quasi-split ([Ham22, Lemma 2.12]). In
particular, the fixed choice of Borel $B\subset G$ transfers to a Borel
subgroup $B_{b}$ for all $b\in B(G)_{\mathrm{un}}$, and $J_{b}\simeq M_{b}$
under the inner twisting, where $M_{b}\subset G$ is the Levi subgroup of $G$
determined by the centralizer of the slope homomorphism of $b$ in $G$. We let
$\delta_{P_{b}}$ denote the modulus character of the standard parabolic
$P_{b}$ of $G$ with Levi factor $M_{b}$ transferred to $J_{b}$ under the inner
twisting. We set $W_{b}:=W_{G}/W_{M_{b}}$ to be the quotient of the relative
Weyl group of $G$ by the relative Weyl group of $M_{b}$. We identify $W_{b}$
with a choice of representatives in $w\in W_{G}$ of minimal length. We set
$\rho_{b,w}:=i_{B_{b}}^{J_{b}}(\chi^{w})\otimes\delta_{P_{b}}^{1/2}$ to be the
normalized parabolic induction of $\chi^{w}$, where $\chi$ is the character of
$T(\mathbb{Q}_{p})$ attached to a toral parameter $\phi_{T}$ under local class
field theory and $\delta_{P_{b}}^{1/2}$ is the modulus character of
$M_{b}\simeq J_{b}$.

We will need to assume the following properties of the Fargues-Scholze local
Langlands correspondence, as in [Ham22, Assumption 7.5].

######  Assumption 4.4.

For a connected reductive group $H/\mathbb{Q}_{p}$, we have

  * •

the set $\Pi(H)$ of smooth irreducible
$\overline{\mathbb{Q}}_{\ell}$-representations of $H(\mathbb{Q}_{p})$,

  * •

the set $\Phi(H)$ of conjugacy classes of continuous maps

$\mathrm{WD}_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}H(\overline{\mathbb{Q}}_{\ell})$

where $\overline{\mathbb{Q}}_{\ell}$ has the discrete topology,
$\mathrm{SL}(2,\overline{\mathbb{Q}}_{\ell})$ acts via an algebraic
representation, and the map respects the action of $W_{\mathbb{Q}_{p}}$ on
$\phantom{}{}^{L}H(\overline{\mathbb{Q}}_{\ell})$, the $L$-group of $H$,

  * •

the set $\Phi^{\mathrm{ss}}(H)$ of continuous semi-simple homomorphisms

$W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}H(\overline{\mathbb{Q}}_{\ell}),$

  * •

and the semi-simplification map
$(-)^{\mathrm{ss}}:\Phi(H)\rightarrow\Phi^{\mathrm{ss}}(H)$ defined by
precomposition with

$W_{\mathbb{Q}_{p}}\rightarrow
W_{\mathbb{Q}_{p}}\times\mathrm{SL}(2,\overline{\mathbb{Q}}_{\ell})$
$g\mapsto(g,\begin{pmatrix}|g|^{1/2}&0\\\ 0&|g|^{-1/2}\end{pmatrix}).$

Then, we assume, for all $b\in B(G)$, that there exists a map

$\mathrm{LLC}_{b}:\Pi(J_{b})\rightarrow\Phi(J_{b})$ $\rho\mapsto\phi_{\rho}$

satisfying the following properties:

  1. (1)

The diagram

${\Pi(J_{b})}$${\Phi(J_{b})}$${\Phi^{\mathrm{ss}}(J_{b})}$$\scriptstyle{\mathrm{LLC}_{b}}$$\scriptstyle{\mathrm{LLC}^{\mathrm{FS}}_{b}}$$\scriptstyle{(-)^{\mathrm{ss}}}$

commutes, where $\mathrm{LLC}_{b}^{\mathrm{FS}}$ is the Fargues-Scholze local
Langlands correspondence for $J_{b}$.

  2. (2)

Consider $\phi_{\rho}$ as an element of $\Phi(G)$ given by composing with the
twisted embedding
$\phantom{}{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\simeq\phantom{}^{L}M_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$
(as defined in [FS21, Section IX.7.1]). Then $\phi_{\rho}$ factors through the
natural embedding $\phantom{}{}^{L}T\rightarrow\phantom{}^{L}G$ if and only if
$b\in B(G)_{\mathrm{un}}$.

  3. (3)

If $\rho$ is a representation such that
$W_{\mathbb{Q}_{p}}\times\mathrm{SL}(2,\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$
factors through $\phantom{}{}^{L}T$, where the last map is the twisted
embedding then, by (2), the element $b$ is unramified, and we require that
$\rho$ is isomorphic to an irreducible constituent of $\rho_{b,w}$ for $w\in
W_{b}$ and $\chi$ the character attached to the induced toral parameter
$\phi_{T}$.

The importance of this assumption is that it allows us to deduce the following
Proposition.

######  Proposition 4.5.

Assuming 4.4, we have that the following is true for a parameter $\phi$
induced from a generic parameter $\phi_{T}$. Given any Schur-irreducible
object
$A\in\mathrm{D}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})_{\phi}\subset\mathrm{D}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})\simeq\mathrm{D}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$
then $A$ is non-zero if and only if $b\in B(G)_{\mathrm{un}}$, and in this
case it must be an irreducible sub-quotient of $\rho_{b,w}$, for some $w\in
W_{b}$.

###### Proof.

This follows from combining the proof of [Ham22, Corollary 7.7] with Lemma 4.2
(1). ∎

Since we want some flexibility in the groups for which we have the above
results, we discuss how Assumption 4.4 behaves under central isogenies.

####  4.1.1. Assumption 4.4 under Central Isogenies

We consider an injective map $\psi:G^{\prime}\hookrightarrow G$ of connected
reductive groups which induces an isomorphism of adjoint and derived groups,
and the induced map $\psi_{\mathrm{Bun}}:B(G^{\prime})\rightarrow B(G)$ on the
associated Kottwitz sets. We now have the following lemma.

######  Lemma 4.6.

If $\psi:G^{\prime}\rightarrow G$ is an injective map which induces an
isomorphism on adjoint and derived groups then it follows that
$\psi_{\mathrm{Bun}}:B(G^{\prime})\rightarrow B(G)$ induces an injection
$J_{b^{\prime}}\rightarrow J_{b}$ which is an isomorphism of the derived group
and adjoint groups for all $b=\psi_{\mathrm{Bun}}(b^{\prime})$ and
$b^{\prime}\in B(G^{\prime})$.

###### Proof.

Since $\psi$ is an inclusion it easily follows that it induces an inclusion
$J_{b^{\prime}}\rightarrow J_{b}$ of $\sigma$-centralizers. To see that it
induces an isomorphism on derived/adjoint groups, recall that $J_{b}$ is an
inner form of a Levi subgroup $M_{b}$ of $G$ given by the centralizer of the
slope homomorphism of $b$. The preimage of $M_{b}$ under $\phi$ defines a Levi
subgroup $M_{b^{\prime}}$ of $G$ which will be the centralizer of the slope
homomorphism of $b^{\prime}$, since $\phi$ induces an isomorphism on adjoint
groups. Moreover, the inner twisting from $J_{b}$ to $M_{b}$ and
$J_{b^{\prime}}$ to $M_{b^{\prime}}$ are compatible with the inclusion in the
sense that the inclusion $J_{b^{\prime}}\rightarrow J_{b}$ is given by
applying the inner twist of $M_{b}$ to the inclusion
$M_{b^{\prime}}\rightarrow M_{b}$. Since the formation of derived/adjoint
groups respects inner twists, this reduces us to showing that the map
$M_{b^{\prime}}\rightarrow M_{b}$ on Levi subgroups induces an isomorphism on
the derived/adjoint groups, and this is clear. ∎

We now consider a map $\mathrm{LLC}_{\mathrm{Bun}_{G}}:\bigsqcup_{b\in
B(G)}\Pi(J_{b})\rightarrow\Phi(G)$ determined by components
$\mathrm{LLC}_{b}:\Pi(J_{b})\rightarrow\Phi(J_{b})$ and satisfying Assumption
4.4. We now wish to define
$\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}:\bigsqcup_{b^{\prime}\in
B(G^{\prime})}\Pi(J_{b})\rightarrow\Phi(G^{\prime})$ in terms of
$\mathrm{LLC}_{\mathrm{Bun}_{G}}$, and show that it also satisfies Assumption
4.4. To do this, we note that, for varying $b^{\prime}\in B(G^{\prime})$, we
define
$\mathrm{LLC}_{b^{\prime}}:\Pi(J_{b^{\prime}})\rightarrow\Phi(J_{b^{\prime}})$
to be the correspondence that makes the following diagram

${\Pi(J_{b})}$${\Phi(J_{b})}$${\Pi(J_{b^{\prime}})}$${\Phi(J_{b^{\prime}})}$$\scriptstyle{\mathrm{LLC}_{b}}$$\scriptstyle{\mathrm{LLC}_{b^{\prime}}}$

commute, where $b:=\psi_{\mathrm{Bun}}(b^{\prime})$. Here the right vertical
arrow is given by composing a parameter
$\phi:\mathrm{WD}_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})$
with the induced map
$\phantom{}{}^{L}J_{b}\rightarrow\phantom{}^{L}J_{b^{\prime}}$ on the dual
groups, and the left vertical arrow is not a map at all, it is a
correspondence defined by the subset of $\Pi(J_{b})\times\Pi(J_{b^{\prime}})$
consisting of pairs $(\pi_{b},\pi_{b^{\prime}})$ such that $\pi_{b^{\prime}}$
is a constituent of the restriction of $\pi_{b}$ to
$J_{b^{\prime}}(\mathbb{Q}_{p})$. We will now show that this gives rise to a
well-defined map under our assumptions on $\psi$. Given a representation
$\pi_{b^{\prime}}\in\Pi(J_{b^{\prime}})$, it follows by [GK82, Lemma 2.3] and
the previous Lemma that we can find a lift $\pi_{b}\in\Pi(J_{b})$ such that
$\pi_{b^{\prime}}$ is an irreducible constituent of
$\pi_{b}|_{J_{b^{\prime}}(\mathbb{Q}_{p})}$. It also follows from [GK82, Lemma
2.1] and [Tad92, Proposition 2.4,Corollary 2.5] that the set
$\Pi_{\pi_{b}}(J_{b^{\prime}})$ of representations of $J_{b^{\prime}}$
occurring in the restriction of $\pi_{b}$ is finite. Now, using the previous
Lemma, we have the following.

######  Lemma 4.7.

[GK82, Lemma 2.4] For the map $J_{b^{\prime}}\rightarrow J_{b}$ of
$\sigma$-centralizers induced by a map $\psi$ as above, and
$\pi_{b}^{1},\pi_{b}^{2}\in\Pi(J_{b})$ the following are equivalent.

  1. (1)

There exists a character
$\chi\in(J_{b}(\mathbb{Q}_{p})/J_{b^{\prime}}(\mathbb{Q}_{p}))^{\vee}$ such
that $\pi_{b}^{1}\simeq\pi_{b}^{2}\otimes\chi$, where $(-)^{\vee}$ denotes the
Pontryagin dual.

  2. (2)

$\Pi_{\pi_{b}^{1}}(J_{b^{\prime}})\cap\Pi_{\pi_{b}^{2}}(J_{b^{\prime}})\neq\emptyset$

  3. (3)

$\Pi_{\pi_{b}^{1}}(J_{b^{\prime}})=\Pi_{\pi_{b}^{2}}(J_{b^{\prime}})$.

Now we can use this to define
$\mathrm{LLC}_{b^{\prime}}:\Pi(J_{b^{\prime}})\rightarrow\Phi(J_{b^{\prime}})$
in terms of $\mathrm{LLC}_{b}:\Pi(J_{b})\rightarrow\Phi(J_{b})$ for
$\mathrm{LLC}_{\mathrm{Bun}_{G}}$ satisfying Assumption 4.4. Namely, for
$\pi_{b^{\prime}}\in\Pi(J_{b^{\prime}})$, we let $\pi_{b}\in\Pi(J_{b})$ be a
representation such that $\pi_{b^{\prime}}$ occurs as an irreducible
constituent of $\pi_{b}|_{J_{b^{\prime}}(\mathbb{Q}_{p})}$. We set
$\phi_{\pi_{b^{\prime}}}$ to be the parameter $\phi_{\pi_{b}}$ attached to
$\pi_{b}$ under $\mathrm{LLC}_{b}$ composed with the map
$\phantom{}{}^{L}J_{b}\rightarrow\phantom{}^{L}J_{b^{\prime}}$ on dual groups
induced by $\psi$. By the previous Lemma, any two choices of lifts
$\pi^{1}_{b}$ and $\pi^{2}_{b}$ of $\pi_{b^{\prime}}$ will differ by a
character twist of
$\chi\in(J_{b}(\mathbb{Q}_{p})/J_{b^{\prime}}(\mathbb{Q}_{p}))^{\vee}$. We
note that the Fargues-Scholze local Langlands correspondence is compatible
with character twists [FS21, Theorem I.9.6 (ii)]. Since $\mathrm{LLC}_{b}$ is
compatible with the Fargues-Scholze local Langlands after semi-simplification
by assumption, it follows that the same is true for $\mathrm{LLC}_{b}$.
Therefore, $\phi_{\pi^{1}_{b}}$ and $\phi_{\pi^{2}_{b}}$ differ by a character
twist that becomes trivial after composing with
$\phantom{}{}^{L}J_{b}\rightarrow\phantom{}^{L}J_{b^{\prime}}$, and so
$\phi_{\pi_{b}}$ does not depend on the choice of lift. We let
$\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}:\bigsqcup_{b^{\prime}\in
B(G^{\prime})}\Pi(J_{b^{\prime}})\rightarrow\Phi(G)$ be the local Langlands
correspondence defined by the $\mathrm{LLC}_{b^{\prime}}$ for $b^{\prime}$
varying. We now prove that our assumption is comaptible with central
isogenies.

######  Proposition 4.8.

Suppose we have an injective map $\psi:G^{\prime}\hookrightarrow G$ of quasi-
split connected reductive groups inducing an isomorphism on adjoint and
derived groups. Assume we have a local Langlands correspondence
$\mathrm{LLC}_{\mathrm{Bun}_{G}}:\bigsqcup_{b\in
B(G)}\Pi(J_{b})\rightarrow\Phi(G)$ such that Assumption 4.4 holds. If we let
$\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}:\bigsqcup_{b^{\prime}\in
B(G^{\prime})}\Pi(J_{b^{\prime}})\rightarrow\Phi(G^{\prime})$ be the local
Langlands correspondence induced by $\mathrm{LLC}_{\mathrm{Bun}_{G}}$ and
$\psi$ as above then $\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}$ satisfies
Assumption 4.4 as well.

###### Proof.

We note, since the Fargues-Scholze local Langlands correspondence is
compatible with maps $G^{\prime}\rightarrow G$ that induce an isomorphism of
adjoint groups [FS21, Theorem I.9.6 (v)], it follows by the above construction
that if Assumption 4.4 (1) holds true for $\mathrm{LLC}_{\mathrm{Bun}_{G}}$
then it also holds true for $\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}$.
Suppose we have $b^{\prime}\in B(G^{\prime})$ mapping to $b\in
B(G)_{\mathrm{un}}$. We let $B_{b}\subset J_{b}$ be the corresponding Borel
then, since the map $J_{b^{\prime}}\rightarrow J_{b}$ induces an isomorphism
on adjoint groups by Lemma 4.6, it follows that $B\cap
J_{b^{\prime}}=:B_{b^{\prime}}\subset J_{b^{\prime}}$ is a Borel of
$J_{b^{\prime}}$. In particular, $b^{\prime}$ must be an unramified element of
$B(G^{\prime})_{\mathrm{un}}$. Now, the map $J_{b^{\prime}}\rightarrow J_{b}$
induces an isomorphism

$J_{b^{\prime}}/B^{\prime}\simeq J_{b}/B.$

If we let $T$ be the maximal split torus of $J_{b}$ then the preimage
$T^{\prime}$ under $\phi$ is a maximal torus of $J_{b^{\prime}}$, and the
previous isomorphism of flag varieties implies that, given a character
$\chi:T(\mathbb{Q}_{p})\rightarrow\overline{\mathbb{Q}}_{\ell}^{*}$, we have
an isomorphism:

$i_{B_{b}}^{J_{b}}(\chi)|_{J_{b^{\prime}}(\mathbb{Q}_{p})}\simeq
i_{B_{b^{\prime}}}^{J_{b^{\prime}}}(\chi|_{T^{\prime}(\mathbb{Q}_{p})}).$

Given $\pi_{b^{\prime}}$ and a lift $\pi_{b}$ to $J_{b}$ then, by definition
of $\phi_{\pi_{b}}$, we have that it is equal to

$\mathrm{WD}_{\mathbb{Q}_{p}}\xrightarrow{\phi_{\pi_{b}}}\phantom{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}J_{b^{\prime}}(\overline{\mathbb{Q}}_{\ell})$

as a conjugacy class of parameters for $J_{b^{\prime}}$. Therefore,
$\phi_{\pi_{b^{\prime}}}$ factors through $\phantom{}{}^{L}T^{\prime}$ if and
only if $\phi_{\pi_{b}}$ factors through the preimage of
$\phantom{}{}^{L}T^{\prime}$ under the map
$\phantom{}{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}J_{b^{\prime}}(\overline{\mathbb{Q}}_{\ell})$
of $L$-groups, but this is precisely $\phantom{}{}^{L}T$, and so Assumption
4.4 (2) holds for $\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}$. Moreover, by
Assumption 4.4 (3) for $\mathrm{LLC}_{\mathrm{Bun}_{G}}$, we have that, in the
above situation, $\pi_{b}$ is an irreducible sub-quotient of
$i_{B_{b}}^{J_{b}}(\chi^{w})\otimes\delta_{P_{b}}^{1/2}$, but this implies
that $\pi_{b^{\prime}}$ is an irreducible constituent of the restriction
$i_{B_{b^{\prime}}}^{J_{b^{\prime}}}(\chi^{w}|_{T^{\prime}(\mathbb{Q}_{p})})\otimes\delta_{P_{b^{\prime}}}^{1/2}$.
From this, it follows that Assumption 4.4 (3) also holds for holds for
$\mathrm{LLC}_{\mathrm{Bun}_{G^{\prime}}}$. ∎

Now that we have shown this compatibility assumption is somewhat flexible, we
can state the groups we know to satisfy Assumption 4.4. This result is largely
contained in [Ham21, FS21, HKW22, BHN22], but we also want to consider an
additional group $\mathrm{GU}_{2}$, where we have the following construction
of $\mathrm{LLC}_{\mathrm{Bun}_{\mathrm{GU}_{2}}}$. Recall that
$\mathrm{GU}_{2}/L$ can be written as

$\mathrm{GU}_{2}:=(\mathrm{GL}_{2}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m})/\mathbb{G}_{m},$

where $\mathbb{G}_{m}$ is embedded in
$H:=\mathrm{GL}_{2}\times\mathrm{Res}_{L^{\prime}/L}(\mathbb{G}_{m})$ via
$a\mapsto(\mathrm{diag}(a,a),a^{-1})$, and $L^{\prime}/L$ is an unramified
quadratic extension. Let $\psi:B(H)\rightarrow B(\mathrm{GU}(2))$ and let
$\tilde{\psi}:B(H)\rightarrow B(\mathrm{GL}_{2})$ be the map of Kottwitz sets.
Given $b\in B(H)$, let $b^{\prime}=\psi(b)$, $\tilde{b}=\tilde{\psi}(b)$.

######  Lemma 4.9.

There is a bijection between $\Pi(J_{b^{\prime}})$ and the set of pairs
$(\tilde{\pi},\chi)$ such that $\tilde{\pi}\in\Pi(J_{\tilde{b}})$ and $\chi$
is a character of $(L^{\prime})^{\times}$ such that
$\chi|_{L^{\times}}=\omega_{\tilde{\pi}}|_{L^{\times}}$, where
$\omega_{\tilde{\pi}}$ is the central character of $\tilde{\pi}$.

###### Proof.

We will show that we have an isomorphism

$J_{b^{\prime}}\simeq(J_{\tilde{b}}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m})/\mathbb{G}_{m}$

of groups over $L$. In particular, we see that
$J_{b}=J_{\tilde{b}}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m}$, and the
quotient map $J_{b}\rightarrow J_{b^{\prime}}$ induces an isomorphism on
adjoint and derived subgroups of $J_{b^{\prime}}$. Moreover, by Hilbert’s
Theorem 90, we have $H^{1}(L,\mathbb{G}_{m})=0$, and thus we also have a
surjection on $L$-points, from which the lemma follows. To see this
isomorphism, recall that $J_{b}$ (resp. $J_{b^{\prime}},J_{\tilde{b}}$) is an
inner form of $M_{b}$ (resp. $M_{b^{\prime}},M_{\tilde{b}}$), the Levi
subgroup of $H$ (resp. $\mathrm{GU}_{2},\mathrm{GL}_{2}$) given by the
centralizer of the slope homomorphism of $b$ (resp. $b^{\prime},\tilde{b}$).
In particular, we see that we have an isomorphism

$M_{b^{\prime}}\simeq(M_{\tilde{b}}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m})/\mathbb{G}_{m}$

and thus we have a surjective map $M_{b}\rightarrow M_{b^{\prime}}$, since
$M_{b}=M_{\tilde{b}}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m}$.
Moreover, we see that under these maps, we have isomorphisms
$M_{b}^{\mathrm{ad}}\simeq M_{b^{\prime}}^{\mathrm{ad}}\simeq
M_{\tilde{b}}^{\mathrm{ad}}$, and the inner twist
$H^{1}(L,M_{b}^{\mathrm{ad}})$ corresponding to $J_{b}$ is, under this
identification, the inner twist inducing $J_{b^{\prime}}$ and $J_{\tilde{b}}$.
The identification of $J_{b^{\prime}}$ then follows. ∎

Moreover, we observe that we have an exact sequence of dual groups

$1\rightarrow\widehat{J}_{b^{\prime}}\rightarrow\widehat{J}_{b}\rightarrow\mathbb{G}_{m}\rightarrow
1,$

where the map $p:\widehat{J}_{b}\rightarrow\mathbb{G}_{m}$ is defined as
follows. We can write
$\widehat{J}_{b}=\widehat{J}_{\tilde{b}}\times\mathbb{G}_{m}^{2}$, and we have
maps $\hat{i}_{1}:\widehat{J}_{\tilde{b}}\rightarrow\mathbb{G}_{m}$,
$\hat{i}_{2}:\widehat{\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m}}=\mathbb{G}_{m}^{2}\rightarrow\mathbb{G}_{m}$
induced from the inclusion maps $i_{1}:\mathbb{G}_{m}\hookrightarrow
J_{\tilde{b}}$ and
$i_{2}:\mathbb{G}_{m}\hookrightarrow\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m}$,
and $p(g,h)=\hat{i}_{1}(g)\hat{i}_{2}(h)^{-1}$.

Now, we want to define $\mathrm{LLC}_{\mathrm{Bun}_{\mathrm{GU}_{2}}}$ in
terms of $\mathrm{LLC}_{\mathrm{Bun}_{H}}$. More precisely, for any
$b^{\prime}\in B(\mathrm{GU}_{2})$ we define
$\mathrm{LLC}_{b^{\prime}}:\Pi(J_{b^{\prime}})\rightarrow\Phi(J_{b^{\prime}})$
in terms of $\mathrm{LLC}_{b}:\Pi(J_{b})\rightarrow\Phi(J_{b})$ for
$\mathrm{LLC}_{\mathrm{Bun}_{H}}$. For
$\pi_{b^{\prime}}=(\tilde{\pi},\chi)\in\Pi(J_{b^{\prime}})$, we consider the
image $\phi=\mathrm{LLC}_{\mathrm{Bun}_{H}}((\tilde{\pi},\chi))$, and we want
to show that
$\phi:\mathrm{WD}_{L}\rightarrow\phantom{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})$
factors through
$\phantom{}{}^{L}J_{b^{\prime}}(\overline{\mathbb{Q}}_{\ell})$. To see this,
from the exact sequence above, it suffices to show that the composition of
$\phi$ with the map
$\phantom{}{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}\mathbb{G}_{m}(\overline{\mathbb{Q}}_{\ell})$
is trivial. However, we observe that the condition that
$\chi|_{L^{\times}}=\omega_{\tilde{\pi}}|_{L^{\times}}$ exactly implies that
this image is trivial, since the composition is the $L$-parameter associated
with the character $\omega_{\tilde{\pi}}\chi^{-1}|_{L^{\times}}$. Thus, we
have an $L$-parameter
$\phi^{\prime}:\mathrm{WD}_{L}\rightarrow\phantom{}^{L}J_{b^{\prime}}(\overline{\mathbb{Q}}_{\ell})$.
We thus define the map $\mathrm{LLC}_{b^{\prime}}$ to take $\pi_{b^{\prime}}$
to $\phi^{\prime}$.

We now have the following theorem about the groups we know to satisfy
Assumption 4.4.

######  Theorem 4.10.

[Ham21, FS21, HKW22, BHN22] Assumption 4.4 is true and $\ell$ is very good in
the following cases.

  1. (1)

The group $\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$ with $p>2$ and
$[L:\mathbb{Q}_{p}]\geq 2$ or $L=\mathbb{Q}_{p}$ for all $p$. In both cases,
we need to assume that $\ell\nmid 2[L:\mathbb{Q}_{p}]$.

  2. (2)

The groups $\mathrm{GU}_{n}$ or $\mathrm{U}_{n}$ for $n$ odd and defined with
respect to an unramified quadratic extension $E/\mathbb{Q}_{p}$, and $\ell\neq
2$.

  3. (3)

The group $\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GU}_{2})$ defined with
respect to an unramified quadratic extension $L^{\prime}/L$, and $\ell$ such
that $\ell\nmid[L:\mathbb{Q}_{p}]$.

  4. (4)

The group $\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GL}_{n})$ for all $p$ and
$\ell$ such that $\ell\nmid[L:\mathbb{Q}_{p}]$.

###### Proof.

We first start with the conditions on $\ell$. If $G$ is of type $A_{n}$ then
all $\ell$ are very good. However, when $G$ is a unitary group with $n>2$, we
also need to impose the additional assumption that the action of
$W_{\mathbb{Q}_{p}}$ on $\hat{G}$ is of order prime to $\ell$, and this gives
us the condition that $\ell\neq 2$. Similarly, this gives rise to the
condition that $\ell\nmid[L:\mathbb{Q}_{p}]$ in all of the cases. If
$G=\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$ then it is of type $C$
and we also need to impose the additional condition that $\ell\neq 2$.

Now, we turn to Assumption 4.4 (1). For $\mathrm{GL}_{n}$, this follows from
[FS21, Theorem I.9.6] and [HKW22, Theorem 1.0.3], where $\mathrm{LLC}_{b}$ is
given by the Harris-Taylor correspondence. For
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$ and $L/\mathbb{Q}_{p}$ as
described above, this follows from [Ham21, Theorem 1.1], where
$\mathrm{LLC}_{b}$ is given by Harris-Taylor for the non-basic $b$ and Gan-
Takeda [GT11] and Gan-Tantono [GT14] for the basic elements666In the current
version of [Ham21], the assumption that $p>2$ is only used to invoke basic
uniformization of abelian type Shimura varieties, but when $L=\mathbb{Q}_{p}$
one can just use Rappoport-Zink uniformization, so this assumption is
unnecessary.. For $\mathrm{U}_{n}$ or $\mathrm{GU}_{n}$, this is [BHN22,
Theorem 1.1], where $\mathrm{LLC}_{b}$ for $b\in B(G)$ was constructed by Mok
[Mok15] and Kaletha-Minguez-Shin-White [Kal+14]. For $\mathrm{GU}_{2}$ this
follows from the compatibility for $\mathrm{GL}_{2}$, and the fact that the
Fargues-Scholze local Langlands correspondence is compatible with taking
products as well as maps $G^{\prime}\rightarrow G$ that induces an isomorphism
of adjoint groups [FS21, Theorem I.9.6 (v),(vi)].

Now we explain why Assumption 4.4 (2) is satisfied. We recall that if $J_{b}$
is a non quasi-split group then the fibers of the $\mathrm{LLC}_{b}$ over an
$L$-parameter
$\phi:\mathrm{WD}_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$
should be empty if $\phi$ factors through
$\phantom{}{}^{L}M(\overline{\mathbb{Q}}_{\ell})$ for a Levi subgroup
$M\subset G$ which does not transfer to a Levi subgroup of $J_{b}$. In
particular, such parameters are called irrelevant, and we expect the fiber to
be empty if and only if $\phi$ is irrelevant [Kal16, Conjecture A.2]. For the
Harris-Taylor correspondence, it is know that the fibers over irrelevant
parameters are empty by the standard properties of Jacquet-Langlands. For
$\mathrm{GU}_{n}$ or $\mathrm{U}_{n}$, odd unitary groups and their Levi
subgroups are always quasi-split, so it is reduced to the previous case of
$\mathrm{GL}_{n}$ using compatibility of the correspondence with parabolic
induction. For $\mathrm{GSp}_{4}$, one needs to show this for
$\mathrm{LLC}_{\mathrm{GU}_{2}(D)}$, where $\mathrm{GU}_{2}(D)$ is the unique
non-split inner form of $\mathrm{GSp}_{4}$. Here this follows from the
construction of Gan-Tantono (See the discussion before the main Theorem in
[GT14]). For $\mathrm{GU}_{2}$, we observe that since
$\mathrm{GU}_{2},H,\mathrm{GL}_{2}$ all have the same adjoint group,
$b^{\prime}\in B(\mathrm{GU}_{2})_{\mathrm{un}}$ is unramified exactly when
$b$ is unramified with notation as in Lemma 4.9. Now, let $\tilde{T}$ be a
maximal split torus of $\mathrm{GL}_{2}$, and observe that
$T^{\prime}=(\tilde{T}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m})/\mathbb{G}_{m}$
is a maximal torus of $\mathrm{GU}_{2}$ over $L$. Given
$\pi_{b^{\prime}}=(\tilde{\pi},\chi)$, and
$\phi_{\pi_{b^{\prime}}}:\mathrm{WD}_{L}\rightarrow\phantom{}^{L}J_{b^{\prime}}(\overline{\mathbb{Q}}_{\ell})$,
we see from the construction that this factors through
$\phantom{}{}^{L}T^{\prime}(\overline{\mathbb{Q}}_{\ell})$ exactly when the
associated $L$-parameter for $H$,
$\phi_{(\tilde{\pi},\chi)}:\mathrm{WD}_{L}\rightarrow\phantom{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})$,
factors through $\phantom{}{}^{L}T(\overline{\mathbb{Q}}_{\ell})$, where
$T=\tilde{T}\times\mathrm{Res}_{L^{\prime}/L}\mathbb{G}_{m}$. Since Assumption
4.4 (2) is clearly compatible with taking products, $H$ satisfies this
assumption, and thus so does $\mathrm{GU}_{2}$.

Now we explain why Assumption 4.4 (3) is satisfied. First, note that any
parameter
$\phi:\mathrm{WD}_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$
induced from a toral parameter $\phi_{T}$ has necessarily trivial monodromy,
since $\phantom{}{}^{L}T(\overline{\mathbb{Q}}_{\ell})$ consists only of semi-
simple elements. Moreover, since $J_{b}$ is an inner form of $M_{b}$, it
follows that the set of all distinct conjugacy classes of parameters
$\phi^{\prime}:\mathrm{WD}_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})$
which can give rise to $\phi$ under the twisted embedding
$\phantom{}{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$
are parameterized by a set of minimal length representatives of
$W_{b}=W_{G}/W_{M_{b}}$ via conjugating $\phi^{\prime}$. We expect (See
[Kal16, Conjecture A.5]) that the fiber of $\mathrm{LLC}_{b}$ over such a
$\phi^{\prime}$ inducing $\phi$ to be the irreducible constituents of the
normalized induction of the $L$-packet of $\phi_{T}^{w}$, which is just
$\chi^{w}$ by local class field theory for $w\in W_{b}$. This is indeed true
in all the cases we consider (See for example [BHN22, Section 2.3.3] for this
discussed in the case of unitary groups, and for $\mathrm{GSp}_{4}$ and its
unique non-split inner form $\mathrm{GU}_{2}(D)$ it follows directly from the
construction). We note that the twists by $\delta_{P_{b}}^{1/2}$ appear to
account for the half Tate twists appearing in the definition of the _twisted_
embedding
$\phantom{}{}^{L}J_{b}(\overline{\mathbb{Q}}_{\ell})\rightarrow\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$.
For $\mathrm{GU}_{2}$, we see that when $b^{\prime}$ is unramified, we have
isomorphisms of flag varieties

$J_{b^{\prime}}/B_{b^{\prime}}\simeq J_{\tilde{b}}/B_{\tilde{b}}\simeq
J_{b}/B_{b}.$

In the above situation where $\phi_{\pi_{b^{\prime}}}$ factors through
$\phantom{}{}^{L}T^{\prime}$, we see that since $H,\mathrm{GL}_{2}$ satisfy
Assumption 4.4 (3), the corresponding representation of $H(L)$ is of the form
$(\tilde{\pi},\chi)$, where $\tilde{\pi}$ is an irreducible constitutent of
$i_{B_{\tilde{b}}}^{J_{\tilde{b}}}(\chi_{1}^{w})\otimes\delta_{P_{\tilde{b}}}^{1/2}$,
for the associated character $\chi_{1}$ of $\tilde{T}$. In particular, we see
that $\pi_{b^{\prime}}$ is a constituent of
$i_{B_{b^{\prime}}}^{J_{b^{\prime}}}(\chi_{1}^{w}\otimes\chi)\otimes\delta_{P_{b^{\prime}}}^{1/2}$,
as desired. ∎

We now turn our attention to deriving our desired consequences.

###  4.2. Perverse $t$-exactness

We recall that $\mathrm{Bun}_{G}^{b}\simeq[\ast/\mathcal{J}_{b}]$, where
$\mathcal{J}_{b}:=\mathrm{Aut}(\mathcal{E}_{b})$ is the group diamond
parameterizing automorphisms of the bundle $\mathcal{E}_{b}$ attached to $b\in
B(G)$ on $X$. The diamond $\mathcal{J}_{b}$ has pure cohomological
$\ell$-dimension over the base (in the sense of [FS21, Definition IV.1.17])
equal to $\langle 2\rho_{G},\nu_{b}\rangle$, where $\nu_{b}$ is the slope
homomorphism of $b$. Moreover, we have that $\mathrm{Bun}_{G}$ is
cohomologically smooth of pure $\ell$-dimension equal to $0$ over the base.
This motivates the following definition.

######  Definition 4.11.

We define a perverse $t$-structure $(\phantom{}^{\mathrm{p}}\mathrm{D}^{\leq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell}),\phantom{}^{\mathrm{p}}\mathrm{D}^{\geq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell}))$ on
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ such that
$A\in\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ lies in
$\phantom{}{}^{\mathrm{p}}\mathrm{D}^{\leq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ (resp.
$\phantom{}{}^{\mathrm{p}}\mathrm{D}^{\geq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$) if and only if
$j_{b}^{*}(A)$ (resp. $j_{b}^{!}(A)$) sits in cohomological degrees
$\leq\langle 2\rho_{G},\nu_{b}\rangle$ ($\geq\langle
2\rho_{G},\nu_{b}\rangle$).

For $\phi$ induced from a generic toral parameter, we write
$(\phantom{}^{\mathrm{p}}\mathrm{D}^{\geq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi},\phantom{}^{\mathrm{p}}\mathrm{D}^{\leq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi})$ for the restriction
of this $t$-structure to
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$. Let
$\mathrm{Perv}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$ denote
the heart. We are almost ready to formulate our first big result. To do this,
we need the following definition.

######  Definition 4.12.

We say that $\phi_{T}$ is weakly normalized regular if it is generic and if
$\chi$ denotes the character attached to $\phi_{T}$ under local class field
theory, we have, for all $w\in W_{G}$ non-trivial, that

(15)
${}\chi\otimes\delta_{B}^{1/2}\not\simeq(\chi\otimes\delta_{B}^{-1/2})^{w}$

holds. Similarly, we say $\phi_{T}$ is regular if for all $w\in W_{G}$ non-
trivial we have that $\chi\not\simeq\chi^{w}$.

To motivate this, we recall that, since $\phi_{T}$ is weakly normalized
regular, we have by [Ham22, Theorem 10.10] an object
$\mathrm{nEis}(\mathcal{S}_{\phi_{T}})\in\mathrm{Perv}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$,
which is a perverse filtered Hecke eigensheaf on $\mathrm{Bun}_{G}$, assuming
4.4 holds. Moreover, it is supported on the set of unramified elements and,
for $b\in B(G)_{\mathrm{un}}$, its stalks are given by

$\mathrm{Red}_{b,\phi}^{\mathrm{tw}}:=\bigoplus_{w\in
W_{b}}\rho_{b,w}[-\langle 2\rho_{G},\nu_{b}\rangle],$

where we recall that
$\rho_{b,w}:=i_{B_{b}}^{J_{b}}(\chi^{w})\otimes\delta_{P_{b}}^{1/2}$. In
particular, by Proposition 4.5 it defines an object in the localized category
$\mathrm{Perv}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$. To show
the desired perverse $t$-exactness property, we would like to use the Hecke
eigensheaf property of $\mathrm{nEis}(\mathcal{S}_{\phi_{T}})$. Given a
geometric dominant cocharacter $\mu$, we consider the highest weight tilting
module $\mathcal{T}_{\mu}$ attached to $\mu$. We let

$T_{\mu}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}$

be the Hecke operator attached to the representation $\mathcal{T}_{\mu}$,
where $E_{\mu}$ denotes the reflex field of $\mu$. The sheaf
$T_{\mu}(\mathrm{nEis}(\mathcal{S}_{\phi_{T}}))$ carries a filtration which,
if it splits, guarantees an isomorphism
$\mathrm{nEis}(\mathcal{S}_{\phi_{T}})\boxtimes r_{\mu}\circ\phi\simeq
T_{\mu}(\mathrm{nEis}(\mathcal{S}_{\phi_{T}}))$, and we say that $\phi_{T}$ is
$\mu$-regular ([Ham22, Definition 10.11]) if such a splitting exists. Here
$r_{\mu}:\hat{G}\rightarrow\mathrm{GL}(\mathcal{T}_{\mu})$ is the map defined
by the tilting module $\mathcal{T}_{\mu}$. The condition of being
$\mu$-regular is guaranteed by the following stronger condition, using [Ham22,
Theorem 1.17].

######  Definition 4.13.

We write
$(-)^{\Gamma}:\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})\rightarrow\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})/\Gamma$
for the natural map from geometric cocharacters to their $\Gamma$-orbits. For
a toral parameter
$\phi_{T}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}T(\overline{\mathbb{F}}_{\ell})$
and a geometric dominant cocharacter $\mu$, we say $\phi_{T}$ is strongly
$\mu$-regular if the Galois cohomology complexes

$R\Gamma(W_{\mathbb{Q}_{p}},(\nu-\nu^{\prime})^{\Gamma}\circ\phi_{T})$

are trivial for $\nu$,$\nu^{\prime}$ defining distinct $\Gamma$-orbits of
weights in the highest weight tilting module $\mathcal{T}_{\mu}$.

######  Remark 4.14.

In particular, strong $\mu$-regularity implies $\mu$-regularity, and if we
know strong $\mu$-regularity then it implies $\mu^{\prime}$-regularity for any
$\mathcal{T}_{\mu^{\prime}}$ which occurs as a direct summand of the tensor
product $\mathcal{T}_{\mu}^{\otimes n}$, by [Ham21, Proposition 10.12]. Also,
as we will see, strong $\mu$-regularity is often implied by generic for some
suitably chosen $\mu$.

More importantly, we can use this to deduce the following.

######  Proposition 4.15.

For any $\phi$ induced from a generic $\phi_{T}$, assume, for all $b\in
B(G)_{\mathrm{un}}$ and $w\in W_{b}$, the representations $\rho_{b,w}$ are
semi-simple, and that Assumption 4.4 is true. Then we have a direct sum
decomposition

$\bigoplus_{b\in
B(G)_{\mathrm{un}}}\mathrm{D}^{\mathrm{adm}}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})_{\phi}\simeq\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi},$

where
$\mathrm{D}^{\mathrm{adm}}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})\subset\mathrm{D}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})\simeq\mathrm{D}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$
denotes the subcategory of admissible complexes.

Moreover, for any
$A\in\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})_{\phi}\simeq\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi}$,
we have that the $!$ and $*$ pushforwards agree with respect to the inclusion
$j_{b}:\mathrm{Bun}_{G}^{b}\rightarrow\mathrm{Bun}_{G}$.

###### Proof.

The first part of the Proposition follows from the second part. To see this,
we use the semi-orthogonal decomposition of
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ into
$\mathrm{D}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})\simeq\mathrm{D}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$
via the excision spectral sequence. Using that the $!$ and $*$-pushforwards
agree for all objects
$A\in\mathrm{D}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})_{\phi}$, we
see that the excision spectral sequence degenerates and the first part of the
claim follows. To see the second part, we now use Proposition 4.5 to see that
an object
$A\in\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$ can
only be supported on the HN-strata $\mathrm{Bun}_{G}^{b}$ for $b\in
B(G)_{\mathrm{un}}$, and that the restriction of $A$ to $\mathrm{Bun}_{G}^{b}$
has irreducible constituents valued in subquotients of the representations
$\rho_{b,w}$ for $w\in W_{b}$ varying. For the representations $\rho_{b,w}$,
we have the following.

######  Proposition 4.16.

[Ham22, Proposition 11.13] For all $b\in B(G)_{\mathrm{un}}$ and $w\in W_{b}$,
the natural map

$j_{b!}(\rho_{b,w})\rightarrow Rj_{b*}(\rho_{b,w})$

is an isomorphism assuming $\phi_{T}$ is generic.

So the $!$ and $*$ pushforwards agree on the $\rho_{b,w}$, and, since we are
assuming the representations $\rho_{b,w}$ are semisimple, the claim follows
for any constituent of $\rho_{b,w}$. This is enough to conclude the claim for
any
$A\in\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}^{b}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}\simeq\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi}$
using the following claim.

######  Lemma 4.17.

Assuming 4.4, for $\phi$ a generic parameter and any
$A\in\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi}$
the cohomology of $A$ has finite length.

###### Proof.

By Assumption 4.4, we know that any irreducible constituent of the cohomology
of $A$ is an irreducible constituent of $\rho_{b,w}$ for some $w\in W_{b}$. It
follows by [Vig96, p. II.5.13] that there are only finitely many possibilities
for the possible irreducible constituents. Therefore, by choosing $K\subset
G(\mathbb{Q}_{p})$ a sufficiently small open compact such that all these
representations have an invariant vector, we deduce, since $A^{K}$ is a
perfect complex by assumption, that $A$ must have finite length cohomology. ∎

∎

We note that the semi-simplicity of $\rho_{1,1}=i_{B}^{G}(\chi)$ is implied by
the conditions discussed above.

######  Lemma 4.18.

Let
$\phi_{T}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}T(\overline{\mathbb{F}}_{\ell})$
be a weakly normalized regular and regular. Suppose there exists a $\mu$ which
is not fixed under any $w\in W_{G}$ and $\phi_{T}$ is $\mu$-regular. Then,
$i_{B}^{G}(\chi)$ is irreducible.

###### Proof.

It follows, by [Ham22, Corollary 11.23] and the assumed $\mu$-regularity, that
we have an isomorphism $i_{B}^{G}(\chi)\simeq
i_{B}^{G}(\chi^{w})=i_{B^{w}}^{G}(\chi)$ for all $w\in W_{G}$. Here $B^{w}$ is
the conjugate of $B$ by $w$. We write $r_{B}^{G}$ for the normalized parabolic
restriction functor. We recall that we are working with $\ell$-modular
coefficients in possibly non-banal characteristic so $i_{B}^{G}(\chi)$ may
have cuspidal constituents. In particular, we will need the following lemma.

######  Lemma 4.19.

Let $w_{0}\in W_{G}$ be the element of longest length for a character
$\chi:T(\mathbb{Q}_{p})\rightarrow\overline{\mathbb{F}}_{\ell}^{*}$. If we
have an isomorphism $i_{B}^{G}(\chi)\simeq i_{B}^{G}(\chi^{w_{0}})$ of
$G(\mathbb{Q}_{p})$-modules then any non-zero quotient $\sigma^{\prime}$ of
$i_{B}^{G}(\chi)$ satisfies $r_{B}^{G}(\sigma^{\prime})\neq 0$

###### Proof.

We apply second adjointness [Dat+22, Corollary 1.3] to the map

$i_{B^{w_{0}}}^{G}(\chi)\xrightarrow{\simeq}i_{B}^{G}(\chi)\rightarrow\sigma^{\prime}$

to conclude the existence of a non-zero map $\chi\rightarrow
r_{B}^{G}(\sigma^{\prime})$, which implies the claim. ∎

Now suppose for the sake of contradiction that $i_{B}^{G}(\chi)$ is not
irreducible. Then there exists an exact sequence

$0\rightarrow\sigma\rightarrow
i_{B}^{G}(\chi)\rightarrow\sigma^{\prime}\rightarrow 0.$

Since parabolic restriction is exact (for example by using second
adjointness), we get an exact sequence

$0\rightarrow r_{B}^{G}(\sigma)\rightarrow r_{B}^{G}i_{B}^{G}(\chi)\rightarrow
r_{B}^{G}(\sigma^{\prime})\rightarrow 0.$

This allows us to conclude an equality of lengths of representations:

$\ell(r_{B}^{G}(\sigma))+\ell(r_{B}^{G}(\sigma^{\prime}))=\ell(r_{B}^{G}(i_{B}^{G}(\chi)))\leq|W_{G}|,$

where the inequality follows from the geometric Lemma [Dat05, Section
2.8]777Note that this bound however fails without taking normalized
restriction because of the aforementioned cuspidal constituents of
$i_{B}^{G}(\chi)$ in non-banal characteristic (cf. [Dat05, Page 48]).. By the
previous lemma, we conclude that $\ell(r_{B}^{G}(\sigma))<W_{G}$. Now, since
we know that $\sigma\subset i_{B}^{G}(\chi)\simeq i_{B}^{G}(\chi^{w})$ for all
$w\in W_{G}$, Frobenius reciprocity implies that we have non-zero maps
$r_{B}^{G}(\sigma)\rightarrow\chi^{w}$ for all $w\in W_{G}$. This gives a
contradiction by the regularity of $\chi$. ∎

We now have the following key claim.

######  Theorem 4.20.

Let $\mu$ be a geometric dominant cocharacter. We write

$T_{\mu}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{\mathbb{Q}_{p}}}$

for the Hecke operator attached to the highest weight tilting module
$\mathcal{T}_{\mu}$ of highest weight $\mu$. Then the operator restricted to
$\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$
is perverse $t$-exact if $\phi_{T}$ is weakly normalized regular, Assumption
4.4 is true, the $\rho_{b,w}$ are semi-simple for all $b\in
B(G)_{\mathrm{un}}$ and $w\in W_{b}$, and $\phi_{T}$ is $\mu$-regular.

###### Proof.

Using Lemma 4.17, the commutation of Hecke operators with colimits,
Proposition 4.5, Proposition 4.15, and semi-simplicity of the representations
$\rho_{b,w}$, we can reduce to showing, for all $b\in B(G)_{\mathrm{un}}$,
that if we consider the complex

$\mathrm{Red}^{\mathrm{tw}}_{b,\phi}:=\bigoplus_{w\in
W_{b}}i_{B_{b}}^{J_{b}}(\chi^{w})\otimes\delta_{P_{b}}^{1/2}[-\langle
2\rho_{G},\nu_{b}\rangle]\in\mathrm{Perv}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$

then we have a containment

$T_{\mu}(j_{b!}(\mathrm{Red}^{\mathrm{tw}}_{b,\phi}))\in\mathrm{Perv}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}$

for the fixed $\mu$. However, $\mathrm{Red}^{\mathrm{tw}}_{b,\phi}$ are the
stalks of the perverse filtered Hecke eigensheaf
$\mathrm{nEis}(\mathcal{S}_{\phi_{T}})$ and, since $\phi_{T}$ is $\mu$-regular
by assumption, we have an isomorphism:

$T_{\mu}(\mathrm{nEis}(\mathcal{S}_{\phi_{T}}))\simeq\mathrm{nEis}(\mathcal{S}_{\phi_{T}})\boxtimes
r_{\mu}\circ\phi\in\mathrm{Perv}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi}^{BW_{E_{\mu}}}.$

This gives the desired claim. ∎

We are almost ready to deduce the result we need for torsion vanishing. To do
this, we will first need to discuss when the additional assumptions of weak
normalized regularity and $\mu$-regularity are superfluous, possibly under
certain assumptions on $\ell$.

###  4.3. Verification of additional assumptions

We first need the following lemma which will allow us to base change to
splitting fields.

######  Lemma 4.21.

Let $G$ be a quasi-split connected reductive group with splitting field $E$.
If $\phi_{T}$ is generic then
$R\Gamma(W_{E},\tilde{\alpha}\circ\phi_{T}|_{W_{E}})$ is trivial for all
absolute coroots
$\tilde{\alpha}\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})$.

###### Proof.

We recall that, given a $\Gamma$-orbit of positive absolute coroots
$\alpha\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}\backslash\Gamma$,
if $E_{\alpha}$ denotes the reflex field of $\alpha$ then the representation
of $\phantom{}{}^{L}T$ defined by $\alpha$ is given by choosing a
representative
$\tilde{\alpha}\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}$ of
$\alpha$, and inducing the representation of $\hat{T}\rtimes
W_{E_{\alpha}}/W_{E}$ defined by it to $W_{\mathbb{Q}_{p}}/W_{E}$. This
reduces the claim to Schapiro’s Lemma. ∎

Other than the groups listed in Theorem 4.10, there are two more groups of
interest to us. We will define them now.

Let $L/\mathbb{Q}_{p}$ be a finite extension. We have the similitude maps from
$\mathrm{GL}_{n}$ (resp. $\mathrm{GSp}_{4}$)

$\nu:\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GL}_{n}\rightarrow\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathbb{G}_{m}$

(resp.

$\nu:\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GSp}_{4}\rightarrow\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathbb{G}_{m}).$

We thus define

$G(\mathrm{SL}_{n,L}):=\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GL}_{n}\times_{\nu}\mathbb{G}_{m},$
$G(\mathrm{Sp}_{4,L}):=\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GL}_{n}\times_{\nu}\mathbb{G}_{m}.$

######  Lemma 4.22.

Let $L/\mathbb{Q}_{p}$ be a finite extension and $G$ be one of the following
groups:

  1. (1)

$\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{U}_{n}$,

  2. (2)

$\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GU}_{n}$,

  3. (3)

$\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GL}_{n}$,

  4. (4)

$G(\mathrm{SL}_{n,L})$.

If $\phi_{T}$ is a generic toral parameter for $G$ then $\phi_{T}$ is weakly
normalized regular and regular. Moreover, for $(1)-(3)$, $\phi_{T}$ will be
$\mu$-regular for all $\mu$, while, for $(4)$, $\phi_{T}$ will be
$\mu$-regular for $\mu$ which are of the form
$\prod_{\tau:L\hookrightarrow\overline{\mathbb{Q}}_{p}}\mu^{\prime}$ for
$\mu^{\prime}$ a cocharacter of $\mathrm{GL}_{n}$.

###### Proof.

We establish weak normalized regularity, and suppress giving the proof that
$\phi_{T}$ is regular as it is strictly easier. For cases (1), (2), and (3),
we may assume for simplicity that $L=\mathbb{Q}_{p}$ with the proof in general
essentially being the same. If $G=\mathrm{GL}_{n}$ then this is [Ham22, Lemma
3.10].

We now consider the case of $G=\mathrm{U}_{n}$ defined with respect to a
quadratic extension $E/\mathbb{Q}_{p}$. Suppose there exists a non-trivial
$w\in W_{G}$ such that we have an isomorphism:

$\chi\otimes\delta_{B}^{1/2}\simeq(\chi\otimes\delta_{B}^{-1/2})^{w}$

of characters on $T(\mathbb{Q}_{p})$. We recall that
$G_{E}\simeq\mathrm{GL}_{n,E}$ where $E/\mathbb{Q}_{p}$ denotes the quadratic
extension defining the unitary group. By the definition of the modulus
character in terms of the transformation character of Haar measures, we
observe that the precomposition of $\delta_{B}$ with the Norm map
$T(E)\rightarrow T(\mathbb{Q}_{p})$ gives the modulus character on the Borel
of $\mathrm{GL}_{n,E}$. Therefore, by precomposing the previous isomorphism
with this norm map, we obtain an analogous relationship of characters on the
torus $T(E)$, which is the maximal torus of $\mathrm{GL}_{n,E}$. Then Lemma
4.21 reduces us to the $\mathrm{GL}_{n}$ case.

The case of $\mathrm{GU}_{n}$ similarly reduces to the $\mathrm{U}_{n}$ case
by setting the coordinate on $T(\mathbb{Q}_{p})$ corresponding to the
similitude factor to be equal to $1$.

For case (4), let $d=[L:\mathbb{Q}_{p}]$. Observe that we have an isomorphism
$G(\mathrm{SL}_{n,L})_{L}\simeq H_{L}$, where

$H=\left\\{(g_{i})\in\prod_{L\hookrightarrow\overline{\mathbb{Q}}_{p}}\mathrm{GL}_{n}:\mathrm{det}(g_{i})=\mathrm{det}(g_{j})\leavevmode\nobreak\
\forall i,j\right\\}.$

Applying Lemma 4.21 again and arguing as for unitary groups, it suffices to
work with $H_{L}$. We assume $L=\mathbb{Q}_{p}$ for notation simplicity. The
maximal torus $T^{\prime}$ in $H$ can be identified with

$\mathbb{G}_{m}^{d}\times\mathbb{G}_{m},$

via the map $(t_{1},\dots,t_{d},t)\mapsto(\mathrm{diag}(t_{i},tt_{i}^{-1}))$.
Since $W_{H}=\prod W_{\mathrm{GL}_{2}}$, consider any element $w^{\prime}\in
W_{G}$, which we assume for notational simplicity is of the form

$(w,\dots,w,\mathrm{id},\dots,\mathrm{id}),$

where $w$ is the non-trivial element of the Weyl group of $\mathrm{GL}_{2}$,
and we have $w$ in the first $k$ entries, for some integer $0\leq k\leq d$.
The general case follows similarly. Observe that the isomorphism

$\chi\otimes\delta_{B}^{1/2}\simeq(\chi\otimes\delta_{B}^{-1/2})^{w^{\prime}}$

becomes

$\prod_{i=1}^{k}\chi_{i1}(t_{i}^{2}t^{-1})\chi_{i2}(tt_{i}^{-2})\simeq\prod_{i=k+1}^{d}|t_{i}^{-2}t|.$

If we substitute $t=x,t_{i}=x$ for $i=1,\dots,k$, while for $k+1\leq i\leq d$
we set $t_{i}=1$ if $i-k$ is odd and $t_{i}=x$ if $i-k$ is even, we see that
we get $\prod_{i=1}^{k}\chi_{i1}(x)\chi_{i2}^{-1}(x)$ is isomorphic to either
the trivial representation $\mathbf{1}$ or $|\cdot|$, which is a contradiction
to genericity.

We now show $\mu$-regularity. Again, for cases (1), (2), and (3), observe that
if $G$ is of the form $\mathrm{Res}_{L/\mathbb{Q}_{p}}G^{\prime}$ and
$T^{\prime}$ denotes the maximal torus of $G^{\prime}$ then we have an
isomorphism

$\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})\simeq\prod_{\phi\in\mathrm{Hom}_{\mathbb{Q}_{p}}(L,\overline{L})}\mathbb{X}_{*}(T^{\prime}_{\overline{L}})$

where $\overline{L}$ is an algebraic closure of $L$. Using this, we can
without loss of generality assume that $L=\mathbb{Q}_{p}$. In the case that
$G=\mathrm{GL}_{n},\mathrm{U}_{n}$, or $\mathrm{GU}_{n}$, this follows as in
the proof of [Ham22, Corollary 10.16]. We recall briefly how this goes.

One can consider the geometric dominant cocharacter $\mu=(1,0,\ldots,0,0)$ of
$\mathrm{GL}_{n}$. This defines the standard representation $V_{\mathrm{std}}$
of $\hat{G}\simeq\mathrm{GL}_{n}$. This cocharacter is in particular minuscule
so the weights form a closed Weyl group orbit with representative
$(1,0,\ldots,0,0)$. From here, it easily follows that the difference of the
weights appearing in $V_{\mathrm{std}}$ define coroots of $G$. In particular,
it follows that, if $\phi_{T}$ is generic then it is strongly $\mu$-regular
for $\mu=(1,0,\ldots,0)$ in the sense of Definition 4.13, and this implies the
filtration on $T_{\mu}(\mathrm{nEis}(\mathcal{S}_{\phi_{T}}))$ splits by
[Ham22, Theorem 10.10] for this $\mu$. Now, the tilting modules
$\mathcal{T}_{\omega_{i}}=\Lambda^{i}(V_{\mathrm{std}})$ attached to the other
fundamental coweights $\omega_{i}=(1^{i},0^{n-i})$ of $G$ can be realized as
direct summands of $V_{\mathrm{std}}^{\otimes i}$, and it follows that
$\phi_{T}$ is $\mu$-regular for $\mu=\omega_{i}$ by [Ham22, Proposition
10.12]. Since any dominant cocharacter can be written as a linear combination
of fundamental weights, the claim for any $\mu$ now follows from [Ham22,
Corollary 10.13]. The case of $\mathrm{GU}_{n}$ and $\mathrm{U}_{n}$ follows
in a very similar way, using Lemma 4.21.

For case (4), observe that as before, we can base change to $L$, and since $H$
is a subgroup of $\prod{\mathrm{GL}_{n}}$, all cocharacters $\mu$ of $H$
define products of cocharacters for $\mathrm{GL}_{n}$. Now, consider a
cocharacter of the form $\mu=\prod_{\tau}\mu_{\tau}$, where
$\mu_{\tau^{\prime}}=\mu_{\tau}$ for all $\tau,\tau^{\prime}$ and $\mu_{\tau}$
is a cocharacter of $\mathrm{GL}_{n}$. Note that every dominant minuscule
cocharacter of $H$ will be of this form. This is because a cocharacter
$\mu=\prod_{\tau}\mu_{\tau}$ of $\prod\mathrm{GL}_{n}$ factors through $H$
exactly when the composition with the determinant is equal for all $\tau$, and
we see that for $\mu$ to be minuscule, $\mu_{\tau}$ must be one of the
fundamental coweights $\omega_{i}$, which, after composing with the
determinant, give different characters for $i\neq j$. Now, we observe that the
same argument as above holds to show that the difference of Weyl conjugates
define a coroot of $H$ for the cocharacter $\mu_{1}=\prod(1,0,\dots,0)$, while
for all other cocharacters of the form $\mu=\prod\mu^{\prime}$, where
$\mu^{\prime}$ is a fundamental weight of $\mathrm{GL}_{n}$, they appear as
weights in some tensor power of the highest weight representation
corresponding to $\mu_{1}$. The claim for any $\mu=\prod\mu^{\prime}$, where
$\mu^{\prime}$ is a dominant cocharacter of $\mathrm{GL}_{n}$, follows by the
same argument as above, using [Ham22, Corollary 10.13]. ∎

######  Lemma 4.23.

Let $L/\mathbb{Q}_{p}$ be a finite extension, and $G$ be one of the following
groups:

  1. (1)

$\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GSp}_{4}$

  2. (2)

$G(\mathrm{Sp}_{4,L})$.

Suppose moreover that $\ell\neq 2$ and $\ell$ is banal with respect to $L$
(i.e. $(q^{4}-1,l)=1$, where $q$ is the size of the residue field of $L$). If
$\phi_{T}$ is a generic toral parameter for $G$ then $\phi_{T}$ is weakly
normalized regular and regular. Moreover, for $(1)$, $\phi_{T}$ will be
$\mu$-regular for all $\mu$, and, for $(2)$, $\phi_{T}$ will be $\mu$-regular
for $\mu$ which are of the form
$\prod_{\tau:L\hookrightarrow\overline{\mathbb{Q}}_{p}}\mu^{\prime}$ for
$\mu^{\prime}$ a cocharacter of $\mathrm{GSp}_{4}$.

###### Proof.

We will first establish weak normalized regularity, and again suppress giving
the proof that $\phi_{T}$ is regular as it is strictly easier. Again, for (1),
we assume that $L=\mathbb{Q}_{p}$ for this part with the proof in general
being more or less the same. We will show this by contradiction. Suppose on
the contrary that there exists some $w\in W_{G}$ such that we have an
isomorphism

(16) $\chi\otimes\delta_{B}^{1/2}\simeq(\chi\otimes\delta_{B}^{-1/2})^{w}.$

For case (1), consider the following parametrization of the maximal torus $T$

(17) ${}a:(\mathbb{Q}_{p}^{*})^{2}\times\mathbb{Q}_{p}^{*}\rightarrow
T(\mathbb{Q}_{p})$ $(t_{1},t_{2},t)\mapsto\begin{pmatrix}t_{1}&0&0&0\\\
0&t_{2}&0&0\\\ 0&0&tt_{2}^{-1}&0\\\ 0&0&0&tt_{1}^{-1}\end{pmatrix}$

as in [Tad94, Page 135]. This allows us to write the character
$\chi:T(\mathbb{Q}_{p})\rightarrow\overline{\mathbb{F}}_{\ell}^{*}$ as
$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)$, for characters
$\mathbb{Q}_{p}^{*}\rightarrow\overline{\mathbb{F}}_{\ell}^{*}$. Similarly, we
can express the modulus character as

$\delta_{B}(t_{1},t_{2},t)=|t_{1}|^{4}|t_{2}|^{2}|t|^{-3}$

where $|\cdot|$ is the norm character. We now check that (16) cannot hold for
all seven non-trivial elements of the Weyl group.

Consider the Weyl group element corresponding to the translation:

$w_{1}:a(t_{1},t_{2},t)\mapsto a(t_{2},t_{1},t)$

If we consider equation (16) with respect to this element and evaluate on
$(x,1,x)=(t_{1},t_{2},t)$ then we obtain the equation

$\chi_{1}(x)|x|^{2}|x|^{-3/2}\simeq\chi_{2}(x)|x|^{-1}|x|^{3/2}$

which gives an isomorphism $\chi_{1}\chi_{2}^{-1}(x)\simeq\mathbf{1}$
contradicting genericity.

Similarly, if we consider the simple Weyl group element

$w_{2}:a(t_{1},t_{2},t)\mapsto a(t_{1},t_{2}^{-1}t,t)$

then evaluating equation (16) for this relationship reduces to

$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t_{2}||t|^{-3/2}\simeq\chi_{1}(t_{1})\chi_{2}(t_{2}^{-1}t)\nu(t)|t_{1}|^{-2}|t_{2}||t|^{-1}|t|^{3/2}$

cancelling terms we obtain that

$\chi_{2}(t)^{-1}\chi_{2}(t_{2})^{2}\simeq|t_{1}|^{-4}|t|^{2}$

so if we evaluate at $(t_{1},t_{2},t)=(x^{3},x^{2},x^{4})$ then we obtain

$\mathbf{1}\simeq|x|^{-4}$

which contradicts the assumption that $(p^{4}-1,\ell)=1$.

Consider now the Weyl group element

$w_{3}:a(t_{1},t_{2},t)\mapsto a(t_{2}^{-1}t,t_{1},t)$

if we evaluate equation (16) then we obtain

$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t_{2}||t|^{-3/2}\simeq\chi_{1}(t_{2})^{-1}\chi_{1}(t)\chi_{2}(t_{1})\nu(t)|t_{2}|^{2}|t|^{-2}|t_{1}|^{-1}|t|^{3/2}$

rearranging and cancelling terms we obtain

$\chi_{1}\chi_{2}^{-1}(t_{1})\chi_{2}\chi_{1}(t_{2})\chi_{1}(t)^{-1}\simeq|t_{1}|^{-3}|t_{2}||t|$

so if we evaluate at $(t_{1},t_{2},t)=(1,1,x)$ we obtain that

$\chi_{1}^{-1}(x)\simeq|x|$

which contradicts genericity (See [Tad94, Page 167] for the enumeration of
$1$-parameter subgroups attached to the coroots in the parametrization (17).
Note that we could also have substituted $(t_{1},t_{2},t)=(x,x,x)$ to obtain

$\chi_{1}(x)\simeq|x|^{-1}.$

Consider the reflection

$w_{4}:a(t_{1},t_{2},t)\mapsto a(t_{1}^{-1}t,t_{2},t)$

then equation (16) becomes

$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t_{2}||t|^{-3/2}\simeq\chi_{1}(t_{1}^{-1}t)\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t|^{-2}|t_{2}|^{-1}|t|^{3/2}$

which gives

$\chi_{1}(t_{1}^{2})\chi_{1}(t)^{-1}\simeq|t_{2}|^{-2}|t|$

so if we evaluate at $(t_{1},t_{2},t)=(1,1,x)$, this becomes

$\chi_{1}(x)^{-1}\simeq|x|$

which contradicts genericity. Note that we could also have substituted
$(t_{1},t_{2},t)=(1,x,x)$ to obtain

$\chi_{1}(x)^{-1}\simeq|x|^{-1}.$

Now consider the Weyl group element

$w_{5}:a(t_{1},t_{2},t)\mapsto a(t_{2},t_{1}^{-1}t,t)$

then equation (16)

$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t_{2}||t|^{-3/2}\simeq\chi_{1}(t_{2})\chi_{2}(t_{1}^{-1}t)\nu(t)|t_{2}|^{-2}|t_{1}||t|^{-1}|t|^{3/2}$

which simplifies to

$\chi_{2}\chi_{1}^{-1}(t_{2})\chi_{1}\chi_{2}(t_{1})\chi_{2}(t)^{-1}\simeq|t_{1}|^{-1}|t_{2}|^{-3}|t|^{2}$

so if we evaluate at $(t_{1},t_{2},t)=(x,1,x)$ then this gives

$\chi_{1}(x)\simeq|x|$

which contradicts genericity. Note that we could also have substituted
$(t_{1},t_{2},t)=(1,x,x)$ to obtain

$\chi_{1}^{-1}(x)\simeq|x|^{-1}.$

Now consider the Weyl group element

$w_{6}:a(t_{1},t_{2},t)\mapsto a(t_{1}^{-1}t,t_{2}^{-1}t,t)$

then equation (16) becomes

$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t_{2}||t|^{-3/2}\simeq\chi_{1}(t_{1}^{-1}t)\chi_{2}(t_{2}^{-1}t)\nu(t)|t_{1}|^{2}|t|^{-2}|t_{2}||t|^{-1}|t|^{3/2}$

which simplifies to

$\chi_{1}^{2}(t_{1})\chi_{2}^{2}(t_{2})\chi_{1}\chi_{2}(t)^{-1}\simeq\mathbf{1}$

so if we evaluate at $(t_{1},t_{2},t)=(1,1,x)$ then this becomes

$\chi_{1}\chi_{2}(x)\simeq\mathbf{1}$

which contradicts genericity.

Now finally we consider

$w_{7}:a(t_{1},t_{2},t)\mapsto a(t_{2}^{-1}t,t_{1}^{-1}t,t)$

then equation (16) becomes

$\chi_{1}(t_{1})\chi_{2}(t_{2})\nu(t)|t_{1}|^{2}|t_{2}||t|^{-3/2}\simeq\chi_{1}(t_{2}^{-1}t)\chi_{2}(t_{1}^{-1}t)\nu(t)|t_{2}|^{2}|t|^{-2}|t_{1}||t|^{-1}|t|^{3/2}$

which simplifies to

$\chi_{1}\chi_{2}(t_{1})\chi_{1}\chi_{2}(t_{2})\chi_{1}\chi_{2}(t^{-1})\simeq|t_{1}|^{-1}|t_{2}|$

evaluated at $(t_{1},t_{2},t)=(1,1,x)$ simplifies to

$\chi_{1}\chi_{2}(x)\simeq\mathbf{1}$

which contradicts genericity.

This concludes our discussion of weakly normalized regularity for
$\mathrm{Res}_{L/\mathbb{Q}_{p}}\mathrm{GSp}_{4}$.

We now turn to the case of $G(\mathrm{Sp}_{4,L})$. As in the proof of the
previous lemma, observe that if we let

(18)
$H=\\{(g_{i})\in\prod_{L\hookrightarrow\overline{\mathbb{Q}}_{p}}\mathrm{GSp}_{4}\text{
such that }\nu(g_{i})=\nu(g_{j}),\forall i,j\\},$

then we have $H_{L}\simeq G(\mathrm{Sp}_{4,L})_{L}$. Thus, we may reduce to
the case of $H$. Since $H\subset\prod\mathrm{GSp}_{4}$, we may also use the
parametrization in [Tad94, 135] to see that the maximal torus $T^{\prime}$ is
given by a parametrization

$(\mathbb{Q}_{p}^{*})^{2d}\times\mathbb{Q}_{p}^{*}\rightarrow
T^{\prime}(\mathbb{Q}_{p}),$ $((t_{\tau 1},t_{\tau
2})_{\tau:L\hookrightarrow\overline{\mathbb{Q}}_{p}},t)\mapsto\begin{pmatrix}t_{\tau
1}&0&0&0\\\ 0&t_{\tau 2}&0&0\\\ 0&0&tt_{\tau 2}^{-1}&0\\\ 0&0&0&tt_{\tau
1}^{-1}\end{pmatrix}_{\tau:L\hookrightarrow\overline{\mathbb{Q}}_{p}}$

where we note that the common similitude factor is the last coordinate $t$.

Since $\delta_{B}$ is just the restriction of the character for the Borel of
$\prod\mathrm{GSp}_{4}$ from the torus $\prod T$ to $T^{\prime}$, we see that
the modulus character is

$\delta_{B}((t_{\tau 1},t_{\tau 2})_{\tau},t)=|t|^{-3d}\prod_{\tau}|t_{\tau
1}|^{4}|t_{\tau 2}|^{2}.$

Since $W_{G}=\prod_{\tau}W_{\mathrm{GSp}_{4}}$, consider any element
$w=(w_{\tau})\in W_{G}$, where $w_{\tau}\in W_{\mathrm{GSp}_{4}}$. Observe
that the expression obtained from the isomorphism (16) for $w=(w_{\tau})$ is
simply the product of the isomorphisms for $\mathrm{GSp}_{4}$ each $w_{\tau}$.
Thus, if we wanted to argue by contradiction, using the notation of the proof
above, when $w_{\tau}=w_{i}$ for $i=1,\dots,7$, we should substitute for
$t_{\tau 1},t_{\tau 2},t$ the values we considered above, subject to the
additional constraint that we must have $t$, the similitude factor, being
equal for all $\tau$.

We thus have two possibilities: either some $w_{\tau}$ is the Weyl group
element $w_{2}$ (i.e. corresponding to the reflection

$w_{2}:a(t_{1},t_{2},t)\mapsto a(t_{1},t_{2}^{-1}t,t)$

or none of the $w_{\tau}$ are this element.

In the first situation, suppose that for some $\tau$, $w_{\tau}$ is the Weyl
reflection $w_{2}$. If we consider the equation (16), evaluated on the element
$t_{\tau 1}=x,t_{\tau 2}=x^{2}$,and
$t_{\tau^{\prime}1}=t_{\tau^{\prime}2}=x^{2},t=x^{4}$ for all
$\tau^{\prime}\neq\tau$, then equation (16) simplifies to

(19) $\mathbf{1}\simeq|x|^{-4},$

since one can check that substituting $t_{1}=t_{2}=x^{2},t=x^{4}$ into the
isomorphism (16) for $\mathrm{GSp}_{4}$ for all the Weyl elements not equal to
$w_{2}$ above simply gives the isomorphism $\mathbf{1}\simeq\mathbf{1}$ after
simplification, and thus does not matter when taking products). This
contradicts the banality assumption that $(p^{4}-1,\ell)=1$.

Now, we suppose we are in the second situation, i.e. no $w_{\tau}=w_{2}$. For
any $w=(w_{\tau})$, let
$J=\\{\tau:L\hookrightarrow\overline{\mathbb{Q}}_{p}:\text{$w_{\tau}=\mathrm{id},w_{3},w_{4},w_{5}$}\\}$,
and
$J^{\prime}=\\{\tau:L\hookrightarrow\overline{\mathbb{Q}}_{p}:\text{$w_{\tau}\neq\mathrm{id},w_{3},w_{4},w_{5}$}\\}$.
For some choice of $((t_{\tau 1},t_{\tau 2})_{\tau},t)$ we see the equation
(16) becomes

(20) $\prod_{\tau\in J^{\prime}}\chi_{\tau 1}\chi_{\tau 2}\prod_{\tau\in
J}\chi_{\tau 1}^{-1}\simeq\mathbf{1}$

or

(21) $\prod_{\tau\in J^{\prime}}\chi_{\tau 1}\chi_{\tau 2}\prod_{\tau\in
J}\chi_{\tau 1}^{-1}\simeq|\cdot|$

which contradicts genericity. Indeed, if we chose some $((t_{\tau 1},t_{\tau
2})_{\tau},t)$ for each $w_{\tau}$ such that we derived a contradiction in the
case of $\mathrm{GSp}_{4}$ as above, then we see that the right-hand side of
the isomorphism (16) simplifies to $|\cdot|^{n}$, for some $n$. If $n\neq
0,1$, then we see that by changing the values of $(t_{\tau 1},t_{\tau 2})$ for
some $\tau\in J$, to the other choice of substitution, the right-hand side
evaluates to $|\cdot|^{n-2}$, and continuing this process as necessary we get
either equation (20) or (21).

We now show $\mu$-regularity. As in the proof of the previous lemma, for case
(1) it suffices to check the claim when $G=\mathrm{GSp}_{4}$, the Langlands
dual group is given by $\mathrm{GSpin}_{5}$, which is isomorphic to
$\mathrm{GSp}_{4}$. The spin representation

$\mathrm{spin}:\mathrm{GSpin}_{5}\rightarrow\mathrm{GL}_{4}(V_{\mathrm{spin}})$

defines a minuscule highest weight representation, which, under the
isomorphism $\mathrm{GSpin}_{5}\simeq\mathrm{GSp}_{4}$, identifies with the
defining representation of $\mathrm{GSp}_{4}$. From here it is easy to see
that the differences of the weights are roots of $\mathrm{GSp}_{4}$ (= coroots
of $\mathrm{GSpin}_{5}$). For example, by using the parametrization of the
maximal torus, as in (17), and the description of the roots in this
parametrization provided on [Tad94, Page 167]. Therefore, genericity
guarantees strong $\mu$-regularity for this representation which implies
$\mu$-regularity as before. The other fundamental tilting module of
$\mathrm{GSpin}_{5}$ is given by the defining representation
$\mathrm{GSpin}_{5}\rightarrow\mathrm{SO}_{5}\rightarrow\mathrm{GL}_{5}$
assuming $\ell\neq 2$ (See [Ham22, Appendix B]). Moreover, this occurs as a
$5$-dimensional summand of $V_{\mathrm{spin}}\otimes V_{\mathrm{spin}}$, and
it follows by [Ham22, Proposition 10.12] that we know $\mu$-regularity for
this representation as well. Therefore, since we know $\mu$-regularity for the
fundamental coweights, we are now done by [Ham22, Corollary 10.13] as before.

Now, for case (2), note that, as in the previous lemma, all cocharacters of
$H$ are of the form $\mu=\prod_{\tau}\mu_{\tau}$ for some cocharacters
$\mu_{\tau}$ of $\mathrm{GSp}_{4}$. The argument given above for
$\mathrm{GSp}_{4}$ shows that if we let $\mu^{\prime}$ be one of the
fundamental coweights of $\mathrm{GSp}_{4}$, then if we take
$\mu=\prod\mu^{\prime}$, (i.e. $\mu_{\tau}=\mu^{\prime}$ for all $\tau$) then
we are $\mu$-regular for such $\mu$. Applying [Ham22, Corollary 10.13] again
shows that if $\mu=\prod_{\tau}\mu^{\prime}$, where $\mu^{\prime}$ is a
dominant cocharacter of $\mathrm{GSp}_{4}$, then $\phi_{T}$ will be
$\mu$-regular for such $\mu$. Note that, as in the case of
$G(\mathrm{SL}_{n})$, every dominant minuscule cocharacter of $H$ is of this
form. ∎

Now let us package our final result in a nice form. We first consider the
following table, summarizing the groups and primes for which our results
apply. We have left the entry blank if no constraint is imposed, and just
mentioned the groups that appear as local constituents of global groups that
admit a Shimura datum and for which $G$ is unramified.

(22) $G$ | Constraint on $G$ | $\ell$ | $p$
---|---|---|---
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GL}_{n})$ |  $L/\mathbb{Q}_{p}$ unramified | $(\ell,[L:\mathbb{Q}_{p}])=1$ |
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$ | $L=\mathbb{Q}_{p}$ | $(\ell,2(p^{4}-1))=1$ |
|  $L/\mathbb{Q}_{p}$ unramified | $(\ell,2[L:\mathbb{Q}_{p}](p^{4[L:\mathbb{Q}_{p}]}-1))=1$ | $p\neq 2$
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GU}_{2})$ |  $L/\mathbb{Q}_{p}$ unramified | $(\ell,[L:\mathbb{Q}_{p}])=1$ |
$G=\mathrm{U}_{n}(L/\mathbb{Q}_{p})$ |  $n$ odd $L$ unramified | $\ell\neq 2$ |
$G=\mathrm{GU}_{n}(L/\mathbb{Q}_{p})$ |  $n$ odd $L$ unramified | $\ell\neq 2$ |
$G(\mathrm{SL}_{2,L})$ |  $L/\mathbb{Q}_{p}$ unramified | $(\ell,[L:\mathbb{Q}_{p}])=1$ |
$G(\mathrm{Sp}_{4,L})$ |  $L/\mathbb{Q}_{p}$ unramified, $L\neq\mathbb{Q}_{p}$ | $(\ell,2[L:\mathbb{Q}_{p}](p^{4[L:\mathbb{Q}_{p}]}-1))=1$ | $p\neq 2$

We now apply Theorem 4.20.

######  Corollary 4.24.

Assume $G$ is a product of the groups appearing in Table (22) with $p$ and
$\ell$ satisfying the corresponding conditions. Then, we have that the natural
map

$j_{1}^{*}T_{\mu}(-):\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\rightarrow\mathrm{D}^{\mathrm{adm}}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$

is exact with respect to the perverse $t$-structure on the source and the
natural $t$-structure (= perverse $t$-structure) on the target.

###### Proof.

First note that, using the decomposition $\mathrm{Bun}_{G_{1}\times
G_{2}}:=\mathrm{Bun}_{G_{1}}\times\mathrm{Bun}_{G_{2}}$, we can assume that
$G$ is isomorphic to one of the groups appearing in Table (22).

Observe that all the groups in Table (22) satisfy Assumption 4.4, where the
first five rows follows from Theorem 4.10, and the last two from Proposition
4.8. We now apply Theorem 4.20. To do this, we also need to check that if
$\phi_{T}$ is a generic toral parameter then is also weakly normalized
regular, and $\mu$-regular for all minuscule $\mu$. This follows from Lemma
4.22 and Lemma 4.23.

Lastly, we need to check that the representations
$\rho_{b,w}:=i_{B_{b}}^{J_{b}}(\chi^{w})\otimes\delta_{P_{b}}^{1/2}$ are semi-
simple for all $b\in B(G)_{\mathrm{un}}$ and $w\in W_{b}$. We claim that they
are in fact irreducible. Recall that $J_{b}\simeq M_{b}\subset G$, where
$M_{b}$ is a Levi of $G$. Moreover, we note that any such Levi $M_{b}$ is a
product of groups also appearing in (22). Therefore, the desired
irreducibility follows from the $\mu$-regularity, weak normalized regularity,
and regularity of $\phi_{T}$ combined with Lemma 4.18. Note that in the case
of $G(\mathrm{SL}_{2,L})$ and $G(\mathrm{Sp}_{4,L})$, we can always find a
cocharacter $\mu$ of the form $\prod_{\tau}\mu^{\prime}$ which is not fixed by
the Weyl group, since we can simply look at any cocharacter of $\mu^{\prime}$
of $\mathrm{GL}_{2}$ (resp. $\mathrm{GSp}_{4}$) which is not fixed by the Weyl
group, and take $\mu$ to be the product of these $\mu^{\prime}$. ∎

We also have the following.

######  Corollary 4.25.

Assume $G$ is a product of the groups appearing in Table (22) with $p$ and
$\ell$ satisfying the corresponding conditions. Then, for $\mathfrak{m}\subset
H_{K_{p}}^{\mathrm{hs}}$ a generic maximal ideal, we have that

$\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\simeq\bigoplus_{b\in
B(G)_{\mathrm{un}}}\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}.$

Moreover, the $!$ and $*$ pushforwards agree for any
$A\in\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\simeq\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$

###### Proof.

This follows from Proposition 4.15, where the semisimplicity of the
$\rho_{b,w}$ follows as in the proof of the Previous Corollary. ∎

##  5\. The Proof of Theorems 1.15 and 1.17

###  5.1. Proof of Theorems 1.14 and 1.15

Throughout this section, we assume $(\mathbf{G},X)$ is a PEL datum of type $A$
or $C$ such that $\mathbf{G}_{\mathbb{Q}_{p}}$ is a product of simple groups
as in Table (1) with $p$ and $\ell$ satisfying the corresponding conditions,
and that Assumption 1.11 holds.

###### Proof.

(Theorem 1.15) By Corollary 3.17, the complex
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})$
has a $G(\mathbb{Q}_{p})\times W_{E}$-equivariant filtration with graded
pieces isomorphic to $j_{1}^{*}T_{\mu}j_{b!}(V_{b})[-d](-\frac{d}{2})$. The
cohomology of Igusa varieties $V_{b}$ and the global Shimura variety is
admissible [Zha23, Proposition 8.21], so we can apply the results of the
previous section to them. We let $\mathfrak{m}$ be a generic maximal ideal of
the spherical Hecke algebra, and consider the localization

$(j_{1}^{*}T_{\mu}j_{b!}(V_{b}))_{\phi_{\mathfrak{m}}}[-d](-\frac{d}{2}).$

This defines a filtration on
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$.
The filtration on
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$
considered above comes from applying $(-)_{\phi_{\mathfrak{m}}}$ to
$R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}],i_{b!}i_{b}^{*}(R\pi_{\mathrm{HT!}}(\overline{\mathbb{F}}_{\ell})))$
viewed as a $G(\mathbb{Q}_{p})$-representation. Using Corollary 4.25, we see
that these graded pieces are also isomorphic to

$(j_{1}^{*}T_{\mu}j_{b*}(V_{b}))_{\phi_{\mathfrak{m}}}[-d](-\frac{d}{2})$

via the natural transformation $j_{b!}\rightarrow j_{b*}$ and are trivial for
$b\in B(G,\mu)_{\mathrm{un}}$. However, using Lemma 3.14, this implies that
the natural map

$R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}],i_{b!}i_{b}^{*}(R\pi_{\mathrm{HT!}}(\overline{\mathbb{F}}_{\ell})))_{\phi_{\mathfrak{m}}}\rightarrow
R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}],i_{b*}i_{b}^{*}(R\pi_{\mathrm{HT!}}(\overline{\mathbb{F}}_{\ell})))_{\phi_{\mathfrak{m}}}$

is an isomorphism, using Remark 3.15. Therefore, we see that the edge maps in
the excision spectral sequence actually degenerate after applying
$(-)_{\phi_{\mathfrak{m}}}$, giving us a direct sum decomposition

$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\simeq\bigoplus_{b\in
B(G,\mu)_{\mathrm{un}}}j_{1}^{*}T_{\mu}j_{b!}(V_{b})[-d](-\frac{d}{2})_{\phi_{\mathfrak{m}}}.$

By applying $R\Gamma(K_{p}^{\mathrm{hs}},-)$ and invoking Lemma 4.2 (3), we
obtain that

$R\Gamma(K_{p}^{\mathrm{hs}},R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}})\simeq
R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$

has a filtration with graded pieces isomorphic to

$R\Gamma(K_{p}^{\mathrm{hs}},j_{1}^{*}T_{\mu}j_{b!}(V_{b}))_{\mathfrak{m}}[-d](-\frac{d}{2}).$

Just as in the proof of Corollary 3.17, we can rewrite this as

$(R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell}(d_{b}))_{\mathfrak{m}}\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}V_{b})[2d_{b}],$

as desired. ∎

###### Proof.

(Theorem 1.17) We recall, by Proposition 3.7, that $V_{b}$ is a complex of
smooth $J_{b}(\mathbb{Q}_{p})$-representations concentrated in degree $\leq
d_{b}$. It follows that we have an inclusion $\bigoplus_{b\in
B(G,\mu)}j_{b!}(V_{b})_{\phi_{\mathfrak{m}}}\in\phantom{}^{\mathrm{p}}\mathrm{D}^{\leq
0,\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$,
using Proposition A.5. Corollary 4.24 implies that

$j_{1}^{*}T_{\mu}j_{b!}(V_{b})[-d]_{\phi_{\mathfrak{m}}}\in\mathrm{D}^{\leq
d}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$

after forgetting the Weil group action. Therefore, we conclude that

$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$

is concentrated in degrees $0\leq i\leq d$. By applying Poincaré duality at
finite level and Corollary A.7, this allows us to conclude that the non-
compactly supported cohomology

$R\Gamma(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}^{\vee}}$

localized at $\phi_{\mathfrak{m}}^{\vee}$ is concentrated in degrees $d\leq
i\leq 2d$, where we define this to be the colimit over the non-compactly
supported cohomology of finite levels (cf. Remark 3.1). It therefore follows
that

$R\Gamma(K_{p}^{\mathrm{hs}},R\Gamma(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}^{\vee}})$

is also concentrated in degrees $\geq d$, but this isomorphic to

$R\Gamma(\mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}^{\mathrm{hs}},C},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}^{\vee}}$

by Lemma 4.2 (3). This establishes Theorem 1.17, by applying Poincaré duality
to the Shimura variety at finite level again. ∎

###  5.2. Proof of Corollary 1.19

We would like to obtain the main theorem for Shimura varities of non-PEL type,
especially Hilbert-Siegel modular varieties (attached to
$\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_{2}$ or
$\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GSp}_{4}$). We will show this in a more
general setup, as follows.

Let $(\mathbf{G},X)$, $(\mathbf{G}_{2},X_{2})$ be a pair of abelian type
Shimura data such that $\mathbf{G},\mathbf{G}_{2}$ are centrally isogenous,
and we have an isomorphism of derived subgroups

$\mathbf{G}^{\mathrm{der}}\xrightarrow{\sim}\mathbf{G}_{2}^{\mathrm{der}},$

as well as adjoint quotients. Consider the associated Shimura varieties
$\mathrm{Sh}(\mathbf{G},X)_{K}$ and
$\mathrm{Sh}(\mathbf{G}_{2},X_{2})_{K_{2}}$, where we choose the level
$K,K_{2}$ such that the level at $p$, satisfies that we have an equality
$K_{p}\cap G^{\mathrm{der}}(\mathbb{Q}_{p})=K_{2,p}\cap
G_{2}^{\mathrm{der}}(\mathbb{Q}_{p})$.

We now assume $K_{p}$ and $K_{2,p}$ are both hyperspecial. Observe that this
implies that $K^{\prime}_{p}=K_{p}\cap G^{\mathrm{der}}(\mathbb{Q}_{p})$ is
also hyperspecial. By the Satake isomorphism, we have an isomorphism of
$\overline{\mathbb{F}}_{\ell}$-algebras

$H_{K_{p}}\simeq\overline{\mathbb{F}}_{\ell}[X_{*}(T)]^{W_{G}},$

and, since $G,G^{\mathrm{der}}$ have isomorphic adjoint groups, the inclusion
of cocharacters $X_{*}(T^{\prime})\subset X_{*}(T)$ induces an inclusion of
Hecke algebras $H^{\prime}_{K^{\prime}_{p}}\subset H_{K_{p}}$, where
$T^{\prime}$ denotes the torus $T\cap G^{\mathrm{der}}$, and
$H^{\prime}_{K^{\prime}_{p}}$ denotes the spherical Hecke algebra for
$G^{\mathrm{der}}$. Moreover, given a maximal ideal $\mathfrak{m}\subset
H_{K_{p}}$, then $\mathfrak{m}^{\prime}=\mathfrak{m}\cap
H^{\prime}_{K^{\prime}_{p}}$ is a maximal ideal of
$H^{\prime}_{K^{\prime}_{p}}$.

Fix a connected component $X^{+}\subset X$. This also fixes a
$X^{+}_{2}\subset X_{2}$, and an isomorphism $X^{+}\simeq X^{+}_{2}$, since
$\mathbf{G},\mathbf{G}_{2}$ have isomorphic adjoint quotients. For any compact
open subgroup $K\subset\mathbf{G}(\mathbb{A}_{f})$, we let
$\mathrm{Sh}^{+}(\mathbf{G},X)_{K}$ be the geometrically connected component
which is the image of $X^{+}\times 1$. Moreover, we will let

$\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}=\mathop{\rm
lim}\nolimits\limits_{{\begin{subarray}{c}\leftarrow\\\
K^{p}\end{subarray}}}\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}K^{p}}.$

Note that since all the transition morphisms are finite étale and hence
affine, $\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}$ is also a qcqs scheme by
[Sta23, Lemma 01YX].

Since $\mathbf{G},\mathbf{G}_{2}$ have isomorphic derived subgroups, this
implies that we have an isomorphism of geometric connected components

$\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}\simeq\mathrm{Sh}^{+}(\mathbf{G}_{2},X^{\prime})_{K_{2,p}}.$

Moreover, we see that the action of $H^{\prime}_{K^{\prime}_{p}}$ on
$\mathrm{Sh}(\mathbf{G},X)_{K_{p}}$ preserves the geometric connected
component, since we see that $\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}$ is simply
the connected Shimura variety associated to
$(\mathbf{G}^{\mathrm{der}},X^{+})$. Indeed, we see that the set of
$\mathbb{C}$-points of $\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}$ is given by

$\mathbf{G}^{\mathrm{der}}(\mathbb{Q})^{(p),\mathrm{cl}}_{+}\backslash
X^{+}\times(\mathbf{G}^{\mathrm{der}}(\mathbb{A}_{f}^{p})),$

where $\mathbf{G}^{\mathrm{der}}(\mathbb{Q})^{(p),\mathrm{cl}}_{+}$ denotes
the closure in $\mathbf{G}^{\mathrm{der}}(\mathbb{A}_{f}^{p})$ of
$\mathbf{G}^{\mathrm{der}}(\mathbb{R})_{+}\cap\mathbf{G}^{\mathrm{der}}(\mathbb{Q})\cap
K_{p}$, and $\mathbf{G}^{\mathrm{der}}(\mathbb{R})_{+}$ denotes the preimage
of the neutral connected component $\mathbf{G}^{\mathrm{ad}}(\mathbb{R})^{+}$
in $\mathbf{G}^{\mathrm{der}}(\mathbb{R})$ under the quotient map.

Following the description of the connected components of Shimura varities from
[Del79, §2], and using the notation of [Kis10, §3.3] we see that there exist
groups
$\mathscr{A}(\mathbf{G})^{\circ},\mathscr{A}(\mathbf{G}_{2}),\mathscr{A}(\mathbf{G})$
such that we have $\mathbf{G}(\mathbb{A}_{f})$ (resp.
$\mathbf{G}_{2}(\mathbb{A}_{f})$) equivariant isomorphisms

$\mathrm{Sh}(\mathbf{G},X)_{K_{p}}\simeq[\mathscr{A}(\mathbf{G})\times\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}/\mathscr{A}(\mathbf{G})^{\circ}]$

and

$\mathrm{Sh}(\mathbf{G}_{2},X_{2})_{K_{2,p}}\simeq[\mathscr{A}(\mathbf{G}_{2})\times\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}/\mathscr{A}(\mathbf{G})^{\circ}].$

Observe that, since
$\mathscr{A}(\mathbf{G})^{\circ}=\mathscr{A}(\mathbf{G}^{\mathrm{der}})^{\circ}$
is a subgroup of $\mathscr{A}(\mathbf{G}_{2})$, the Shimura variety
$\mathrm{Sh}(\mathbf{G}_{2},X_{2})_{K_{2,p}}$ is simply an (infinite) union of
copies of $\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}$. Moreover, we see that the
action of $H^{\prime}_{K^{\prime}_{p}}$ on the right-hand side of the above
isomorphism is given by the action on $\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}}$.
In particular, we observe that

$H^{i}(\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m^{\prime}}}$

vanishes if and only if

$H^{i}(\mathrm{Sh}(\mathbf{G}_{2},X)_{K_{2,p}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m^{\prime}}}$

does as well. We thus have the following proposition.

######  Proposition 5.1.

Suppose that $(\mathbf{G},X)$ is of PEL type A or C satisfying the conditions
in Theorem 1.17. Then, Conjecture 1.2 also holds for $(\mathbf{G}_{2},X_{2})$.

###### Proof.

Since Conjecture 1.2 is true for $(\mathbf{G},X)$, we will first show that a
maximal ideal $\mathfrak{m}$ of $H_{K_{p}}$ is generic if and only if the
maximal ideal $\mathfrak{m}^{\prime}$ of $H^{\prime}_{K^{\prime}_{p}}$ is
generic. To see this, we will reformulate this in terms of $L$-parameters.
This is equivalent to showing that an $L$-parameter

$\phi:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}T(\overline{\mathbb{F}}_{\ell})$

is generic if and only if the composition $\phi^{\prime}$ with the map
$g:\phantom{}^{L}T(\overline{\mathbb{F}}_{\ell})\rightarrow\phantom{}^{L}T^{\prime}(\overline{\mathbb{F}}_{\ell})$
induced by the inclusion of tori $T^{\prime}\hookrightarrow T$ is generic.
(Here, $T^{\prime}=G^{\mathrm{der}}\cap T$). This follows from the observation
that any coroot $\alpha$ factors through $G^{\mathrm{der}}$, and hence the
composition $\alpha\circ\phi$ is equal to $\alpha\circ\phi^{\prime}$.

Now, we consider the limit

$\mathrm{Sh}(\mathbf{G},X):=\mathop{\rm
lim}\nolimits\limits_{{\begin{subarray}{c}\leftarrow\\\
K^{p}\end{subarray}}}\mathrm{Sh}(\mathbf{G},X)_{K^{p}}.$

Note that since all schemes appearing in the limit are qcqs, by [Sta23,
Theorem 09YQ] we have an isomorphism of cohomology groups

$H^{i}(\mathrm{Sh}(\mathbf{G},X),\overline{\mathbb{F}}_{\ell})\simeq\mathop{\rm
lim}\nolimits\limits_{{\begin{subarray}{c}\rightarrow\\\
K^{p}\end{subarray}}}H^{i}(\mathrm{Sh}(\mathbf{G},X)_{K^{p}},\overline{\mathbb{F}}_{\ell}).$

We now have the following lemma.

######  Lemma 5.2.

Let $G^{\prime}\rightarrow G$ be a map inducing an isomorphism on adjoint
groups with $g:\phantom{}^{L}G\rightarrow\phantom{}^{L}G^{\prime}$, the
induced map on dual groups. For
$\phi:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{F}}_{\ell})$
a L-parameter and $A$ an admissible complex of $G(\mathbb{Q}_{p})$-modules,
there is a natural isomorphism of $G^{\prime}(\mathbb{Q}_{p})$-modules

$(A|_{G^{\prime}(\mathbb{Q}_{p})})_{\phi^{\prime}}\simeq\bigoplus_{\begin{subarray}{c}\phi\\\
\phi^{\prime}=g\circ\phi\end{subarray}}A_{\phi}|_{G^{\prime}(\mathbb{Q}_{p})},$

with notation as in Corollary 4.3.

###### Proof.

By applying Corollary 4.3, we obtain a decomposition

$A\simeq\bigoplus_{\phi}A_{\phi}$

of $A$ as a $G(\mathbb{Q}_{p})$-module. We restrict to
$G^{\prime}(\mathbb{Q}_{p})$ and apply the localization map
$(-)_{\phi^{\prime}}$. This gives an isomorphism

$(A|_{G^{\prime}(\mathbb{Q}_{p})})_{\phi^{\prime}}\simeq\bigoplus_{\phi}(A_{\phi}|_{G^{\prime}(\mathbb{Q}_{p})})_{\phi^{\prime}},$

where we have used that localization commutes with direct sums since it is a
left adjoint by definition. Now, using the compatibility of the Fargues-
Scholze correspondence with central isogenies [FS21, Theorem IX.6.1], either
$\phi^{\prime}=g\circ\phi$ and
$A_{\phi}|_{G^{\prime}(\mathbb{Q}_{p})}\in\mathrm{D}(G^{\prime}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi^{\prime}}$
and, by the idempotence of the localization map, we have that
$(A_{\phi}|_{G^{\prime}(\mathbb{Q}_{p})})_{\phi^{\prime}}=A_{\phi}|_{G^{\prime}(\mathbb{Q}_{p})}$
or $(A_{\phi}|_{G^{\prime}(\mathbb{Q}_{p})})_{\phi^{\prime}}$ is $0$. The
claim follows. ∎

By the previous lemma applied to $G^{\mathrm{der}}=G^{\prime}\subset G$, we
have a natural decomposition

$H^{i}(\mathrm{Sh}(\mathbf{G},X)_{K^{p}},\overline{\mathbb{F}}_{\ell})_{\phi^{\prime}}\simeq\bigoplus_{\begin{subarray}{c}\phi\\\
\phi^{\prime}=g\circ\phi\end{subarray}}H^{i}(\mathrm{Sh}(\mathbf{G},X)_{K^{p}},\overline{\mathbb{F}}_{\ell})_{\phi}.$

Taking limits over $K^{p}$, we obtain

$H^{i}(\mathrm{Sh}(\mathbf{G},X),\overline{\mathbb{F}}_{\ell})_{\phi^{\prime}}\simeq\bigoplus_{\begin{subarray}{c}\phi\\\
\phi^{\prime}=g\circ\phi\end{subarray}}H^{i}(\mathrm{Sh}(\mathbf{G},X),\overline{\mathbb{F}}_{\ell})_{\phi}.$

Hence, we see that
$H^{i}(\mathrm{Sh}(\mathbf{G},X),\overline{\mathbb{F}}_{\ell})_{\phi^{\prime}}$
vanishes for $i<d$, since all the $\phi$ appearing on the right-hand side are
generic, and hence we can apply Theorem 1.17 and take limits to see that all
the direct summands vanish. Applying $R\Gamma(K_{p},-)$ and Lemma 4.2 (3), we
see that

$H^{i}(\mathrm{Sh}(\mathbf{G},X)_{K_{p}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}^{\prime}}$

vanishes for $i<d$. Thus, we see that
$H^{i}(\mathrm{Sh}^{+}(\mathbf{G},X)_{K_{p}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m^{\prime}}}$
vanishes for $i<d$, and therefore the same is true for
$H^{i}(\mathrm{Sh}(\mathbf{G}_{2},X_{2})_{K_{2,p}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}^{\prime}}$
from the discussion above. By the Hochschild-Serre spectral sequence, we see
that for all sufficiently small $K_{2}^{p}$,
$H^{i}_{c}(\mathrm{Sh}(\mathbf{G}_{2},X_{2})_{K_{2,p}K_{2}^{p}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}^{\prime}}$
vanishes for $i<d$.

Now, consider a generic maximal ideal $\mathfrak{m}_{2}$ for the spherical
Hecke algebra $H_{K_{2,p}}$ of $G_{2}$. This corresponds to a generic maximal
ideal $\mathfrak{m}^{\prime}$ of $H^{\prime}_{K^{\prime}_{p}}$. It remains to
observe that for any finitely-generated $H_{K_{2,p}}$-module $A$, if the
localization $A_{\mathfrak{m}^{\prime}}=0$, then there is some element in
$r\in H^{\prime}_{K^{\prime}_{p}}\backslash\mathfrak{m}^{\prime}$ such that
$rA=0$. Thus we must have $A_{\mathfrak{m}_{2}}=0$ as well since
$H^{\prime}_{K^{\prime}_{p}}\backslash\mathfrak{m}^{\prime}\subset
H_{K_{2,p}}\backslash\mathfrak{m}_{2}$. ∎

As a corollary, we can strengthen previous results of Caraiani-Tamiozzo [CT21,
Theorem B], who previously showed torsion vanishing for Hilbert modular
varieties under the additional assumption that $p$ was split in the totally
real field $F$ (though we also note that they showed torsion vanishing under a
hypothesis on $\mathfrak{m}$ which is weaker than the genericity considered
here, see Remark 1.7).

######  Corollary 5.3.

Conjecture 1.2 is true for Hilbert-Siegel Shimura varieties (attached to
$\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_{2}$,
$\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GSp}_{4}$) and quaternionic Shimura
varieties.

###### Proof.

Observe that for Hilbert-Siegel Shimura varieties (attached to
$\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_{2}$,
$\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GSp}_{4}$), there is a cover by a PEL-type
Shimura variety with local group $G$ of the form $G(\mathrm{SL}_{2})$ and
$G(\mathrm{Sp}_{4})$ respectively. For the case of quaternionic Shimura
varieties, we can relate their geometric connected components to unitary PEL-
type Shimura varieties with local group $\mathrm{GU}_{2}$, as described in
[TX16, Corollary 3.11]. Therefore, the result follows from Theorem 1.17. ∎

##  6\. Conjectures and Concluding Remarks

###  6.1. Relationship to Xiao-Zhu

Assume that the basic element $b\in B(G,\mu)_{\mathrm{un}}$ is unramified (See
[XZ17, Remark 4.2.11] for a classification). Let us look at the middle degree
cohomology
$H^{d}(R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}})$.
By Theorem 1.15, it has a summand isomorphic to

$H^{d}(R\Gamma_{c}(G,b,\mu)\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})).$

To describe this, let $\mathbf{G}^{\prime}$ be the unique $\mathbb{Q}$-inner
form of $\mathbf{G}$ such that
$\mathbf{G}(\mathbb{A}^{p\infty})\simeq\mathbf{G}^{\prime}(\mathbb{A}^{p\infty})$,
$\mathbf{G}^{\prime}(\mathbb{R})$ is compact modulo center, and
$\mathbf{G}_{\mathbb{Q}_{p}}\simeq J_{b}$ (See [Han20, Proposition 3.1] for
the existence). We write
$C(\mathbf{G}^{\prime}(\mathbb{Q})\backslash\mathbf{G}^{\prime}(\mathbb{A}_{f})/K^{p},\overline{\mathbb{F}}_{\ell})$
for the set of all continuous functions on the profinite set
$\mathbf{G}^{\prime}(\mathbb{Q})\backslash\mathbf{G}^{\prime}(\mathbb{A}_{f})/K^{p}$.
It is easy to show that one has an isomorphism

$C(K^{p}\backslash\mathbf{G}^{\prime}(\mathbb{A}_{f})/\mathbf{G}^{\prime}(\mathbb{Q}),\overline{\mathbb{F}}_{\ell})\simeq
R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$

for example by combining [Han20, Theorem 3.4] and Corollary 3.6. We let
$V_{\mu}\in\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\hat{G})$ be the usual
highest weight module of highest weight $\mu$, which in particular agrees with
the highest weight tilting module, since $\mu$ is minuscule. We let $b_{T}$
denote the unique (since $b$ is basic) reduction of $b\in B(G)$ to $B(T)$, and
regard it as an element in $B(T)\simeq\mathbb{X}^{*}(\hat{T}^{\Gamma})$ in
what follows. It should be the case that, under possible additional
constraints on $\mathfrak{m}$ depending on $\mu$ (See for example [Ham22,
Conjecture 1.25] and [XZ17, Definition 1.4.2]), we have an isomorphism

(23)
${}C(K^{p}\backslash\mathbf{G}^{\prime}(\mathbb{A}_{f})/\mathbf{G}^{\prime}(\mathbb{Q}),\overline{\mathbb{F}}_{\ell})\otimes^{\mathbb{L}}R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}\simeq
C(K^{p}\backslash\mathbf{G}^{\prime}(\mathbb{A}_{f})/\mathbf{G}^{\prime}(\mathbb{Q}),\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}\otimes
V_{\mu}|_{\hat{G}^{\Gamma}}(b_{T})[-d](-\frac{d}{2})$

of $G(\mathbb{Q}_{p})$-representations888One should also be able describe the
Weil group action, as in [Ham22, Conjecture 1.25]., where we note that
$J_{b}\simeq G$ if $b\in B(G,\mu)_{\mathrm{un}}$ since $b$ is basic, and
$J_{b}$ must be quasi-split since $b$ is unramified. In particular, by arguing
as in Koshikawa [Kos21, Page 6], we know that
$R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$
will have irreducible constituents given by the representations of
$J_{b}(\mathbb{Q}_{p})$ with Fargues-Scholze parameter equal
$\phi_{\mathfrak{m}}$ as conjugacy classes of parameters. Moreover, using that
Assumption 4.4 holds for the groups appearing in Table (1), we know by
Propsoition 4.5 that they have to be constituents of $i_{B}^{G}(\chi)$, which
will also be irreducible under the generic assumption and the constraints
appearing in Table (1) (See the proof of Corollary 4.24). Then [Ham22,
Conjecture 1.25] would imply that $R\Gamma_{c}(G,b,\mu)[i_{B}^{G}(\chi)]\simeq
i_{B}^{G}(\chi)\otimes V_{\mu}|_{\hat{G}^{\Gamma}}(b_{T})[-d](\frac{-d}{2})$
as $G(\mathbb{Q}_{p})$-modules. Assume $\ell$ is banal (i.e coprime to the
pro-order of $K_{p}^{\mathrm{hs}}$) then passing to
$K_{p}^{\mathrm{hs}}$-invariants, recalling that it is exact under the banal
hypothesis, gives us the isomorphism (23).999For this comparison, it would
have been more natural to consider an analogue of Theorem 1.15 with
$\overline{\mathbb{Q}}_{\ell}$-coefficients. This is indeed doable assuming
that $\phi_{\mathfrak{m}}$ admits a $\overline{\mathbb{Z}}_{\ell}$-lattice as
in [Ham22, Theorem 1.17]. This integrality condition is however an artifiact
of the theory of solid $\overline{\mathbb{Q}}_{\ell}$-sheaves not being
properly understood (e.g excision fails) and should be removable with more
technology.

######  Remark 6.1.

If $B(G,\mu)_{\mathrm{un}}$ consists of only the basic element and the
$\mu$-ordinary element and $\phi_{T}$ is strongly $\mu$-regular (Definition
4.13) then [Ham22, Conjecture 1.25] is true. In particular, it follows from
[Ham22, Theorem 1.27] that the isomorphism (23) can be made unconditional.

We note that this description of the middle degree cohomology on the generic
fiber of the Shimura variety at hyperspecial level parallels Theorem [XZ17,
Theorem 1.14 (1)], describing the middle degree cohomology on the special
fiber of the natural integral model.

###  6.2. A General Torsion Vanishing Conjecture

Consider now a general Shimura datum $(\mathbf{G},X)$. Let
$\Lambda\in\\{\overline{\mathbb{Q}}_{\ell},\overline{\mathbb{F}}_{\ell}\\}$.
If $\Lambda=\overline{\mathbb{F}}_{\ell}$ assume that $\ell$ is very good with
respect to $G:=\mathbf{G}_{\mathbb{Q}_{p}}$, as in [FS21, Page 33]. We can
then look at the $G(\mathbb{Q}_{p})\times
W_{E_{\mathfrak{p}}}$-representation.

$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)$

defined by the cohomology at infinite level. By applying Corollary 4.3, we
obtain a $G(\mathbb{Q}_{p})\times W_{E_{\mathfrak{p}}}$-equivariant
decomposition of this

$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)=\bigoplus_{\phi}R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}$

running over semi-simple $L$-parameters
$\phi:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\Lambda)$. For such a
$\phi$, we let $(\phi_{M},M)$ denote a cuspidal support. I.e $M$ is a Levi of
$G$ and $\phi_{M}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}M(\Lambda)$ is a
supercuspidal $L$-parameter such that $\phi$ is induced by composing with the
natural embedding
$\phantom{}{}^{L}M(\Lambda)\rightarrow\phantom{}^{L}G(\Lambda)$. We want to
describe the degrees of cohomology that
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}$ sits in for
suitably nice $\phi$. The case where $\phi$ factors through $M=T$ is covered
by Conjecture 1.2. To go beyond this, we give the following definition.

######  Definition 6.2.

For a semi-simple $L$-parameter $\phi$ with a cuspidal support $(M,\phi_{M})$,
we let $P$ be a parabolic with Levi factor $M$ and unipotent radical $N$. We
consider the representation $r$ given by looking at the action of
$\phantom{}{}^{L}M$ on the Lie algebra of $\phantom{}{}^{L}N$ via the adjoint
action. We say $\phi$ is of Langlands-Shahidi type if the Galois cohomology
groups

$R\Gamma(W_{\mathbb{Q}_{p}},r\circ\phi_{M})$

and

$R\Gamma(W_{\mathbb{Q}_{p}},(r\circ\phi_{M})^{\vee})$

are trivial. Similarly, we say $\phi$ is of weakly Langlands-Shahidi type if

$H^{2}(R\Gamma(W_{\mathbb{Q}_{p}},r\circ\phi_{M}))$

and

$H^{2}(R\Gamma(W_{\mathbb{Q}_{p}},(r\circ\phi_{M})^{\vee}))$

are trivial.

######  Remark 6.3.

We note that since we enforced this condition on both $r\circ\phi_{M}$ and its
dual that this is independent of the choice of parabolic $P$ and the choice of
cuspidal support. Moreover, it is easy to check that, if $M=T$, this precisely
recovers Definition 1.1.

The terminology of "Langlands-Shahidi type" comes from the fact that the
representation $r\circ\phi_{M}$ is precisely the representation which appears
in the description of the constant term of the usual Eisenstein series via the
Langlands-Shahidi method. The motivation for this definition comes from
considering the behavior of geometric Eisenstein series over the Fargues-
Fontaine curve for general parabolics, by making analogies with the classical
theory over function fields, as developed in [BG02, Lau90]. In particular,
this should be the correct definition that guarantees that the eigensheaves
$\mathcal{S}_{\phi}$ on $\mathrm{Bun}_{G}$ with eigenvalue $\phi$ are as
simple as possible, and the analysis carried out in [Ham22] generalizes to the
non-principal case. This is discussed in more detail in [Ham23, Chapter 3]. In
addition, we expect that the consequences derived from the analysis in [Ham22]
in the principal case should also generalize. More precisely, we conjecture
the following generalization of Proposition 4.15 and Corollary 4.24

######  Conjecture 6.4.

Let $B(G)_{M}:=\mathrm{Im}(B(M)_{\mathrm{basic}}\rightarrow B(G))$ be the set
of $M$-reducible elements, and let $\phi$ be a semi-simple $L$-parameter of
Langlands-Shahidi type with cuspidal support $(M,\phi_{M})$. The category
$\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)_{\phi}$ of $\phi$-local
lisse-étale $\Lambda$-sheaves (as defined in Appendix A) breaks up as direct
sum

$\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)_{\phi}\simeq\bigoplus_{b\in
B(G)_{M}}\mathrm{D}(\mathrm{Bun}_{G}^{b},\Lambda)_{\phi}$

via excision, and the $!$ and $*$ pushhforwards agree for any smooth
irreducible representation $\rho$ of $J_{b}(\mathbb{Q}_{p})$ lying in
$\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G}^{b},\Lambda)_{\phi}$ for $b\in
B(G)_{M}$.

Given a tilting module $V\in\mathrm{Tilt}_{\Lambda}(\phantom{}^{L}G^{I})$, if
$\phi$ is of weakly Langlands-Shahidi type then the map induced by associated
the Hecke operator

$T_{V}:\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)_{\phi}\rightarrow\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{BW_{\mathbb{Q}_{p}}^{I}}_{\phi}$

is perverse $t$-exact, where the fact the Hecke operator preserves this
subcategory is proven as in Lemma 4.2 (2).

######  Remark 6.5.

During the preparation of this manuscript, Hansen formulated similar
conjectures with rational coefficients [Han23]. He refers to Langlands-Shahidi
parameters as generous parameters [Han23, Definition 2.5] and to weakly
Langlands-Shahidi parameters as generic semi-simple parameters [Han23, Section
2.3]. One can show that these two definitions are equivalent. Indeed, note
that the Galois cohomology $H^{1}(R\Gamma(W_{\mathbb{Q}_{p}},r\circ\phi_{M}))$
controls the lifts of a semi-simple parameter
$\phi_{M}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}M(\Lambda)$ to a
$\phantom{}{}^{L}P(\Lambda)$-valued parameter and that such lifts correspond
to finding parameters whose semi-simplification is equal to $\phi$. Moreover,
insisting that $H^{1}(R\Gamma(W_{\mathbb{Q}_{p}},r\circ\phi_{M}))$ is trivial
is equivalent to insisting that $R\Gamma(W_{\mathbb{Q}_{p}},r\circ\phi_{M})$
is trivial using local Tate-duality and that the Euler-Poincaré characteristic
of this complex is $0$. This shows the equivalence of the generous condition
with the Langlands-Shahidi type condition, using that the stack of Langlands
parameters with rational coefficients is reduced. Lastly, the set of such
lifts coming from classes in
$H^{0}(R\Gamma(W_{\mathbb{Q}_{p}},r\circ\phi_{M}))$ will give rise to non
Frobenius semi-simple L-parameters allowing one to see that weakly Langlands-
Shahidi is equivalent to generic semi-simple.

In particular, by combining this with a generalization of Theorem 1.13 to
arbitrary Shimura varieties and the analysis carried out in §5, we could
deduce the following as a consequence.

######  Conjecture 6.6.

Let $\phi$ be a semi-simple $L$-parameter of weakly Langlands-Shahidi type
with cuspidal support $(M,\phi_{M})$. Then the complex
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}$ (resp.
$R\Gamma(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}$) is concentrated
in degrees $0\leq i\leq d$ (resp. $d\leq i\leq 2d$).

######  Remark 6.7.

For $(\mathbf{G},X)$ of PEL type $A$ or $C$, assuming 1.11 and that $\phi$ of
Langlands-Shahidi type, we should also obtain a $W_{E_{\mathfrak{p}}}\times
G(\mathbb{Q}_{p})$-equivariant direct sum decomposition

$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}\simeq\bigoplus_{b\in
B(G,\mu)_{M}}(R\Gamma_{c}(G,b,\mu)_{\phi}\otimes^{\mathbb{L}}V_{b})[2d_{b}],$

where $R\Gamma_{c}(G,b,\mu):=\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}},\Lambda(d_{b}))$
and $R\Gamma_{c}(G,b,\mu)_{\phi}$ is the projection applied to the complex
viewed as a $G(\mathbb{Q}_{p})$-representation. This should also generalize
once one has appropriate general definitions of $\mathrm{Ig}^{b}$ and
$\mathrm{Ig}^{b,*}$ so that one can actually define
$V_{b}:=R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\Lambda)$. Under possible
additional constraints on $\phi$, one should also be able to describe the
contribution of $R\Gamma_{c}(G,b,\mu)_{\phi}$ in terms of the decomposition
$V_{\mu}|_{Z(\hat{M}^{\Gamma})}=\mathcal{T}_{\mu}|_{Z(\hat{M}^{\Gamma})}$ for
$b\in B(G)_{M}$ (along the lines of [Ham22, Conjecture 1.25]), as is explained
in the toral case in §6.1. It would be interesting to formulate an optimal
conjecture.

######  Remark 6.8.

We believe that this conjecture should be true under just the weakly
Langlands-Shahidi condition. However, we strongly suspect that the splitting
of the semi-orthogonal decomposition and in turn the splitting of Mantovan’s
filtration discussed in the previous Remark should not hold unless the set
$B(G,\mu)_{M}$ is a singleton. In particular, in [Han23, Section 2.2] Hansen
conjectures the existence of perverse sheaves lying
$\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)_{\phi}$, for which the
semi-orthogonal decomposition does not split. Nonetheless, one still expects
perverse $t$-exactness of Hecke operators to hold in these cases [Han23,
Conjecture 2.32].

##  Appendix A Spectral Decomposition of Sheaves on $\mathrm{Bun}_{G}$, by
David Hansen

Let $G/\mathbb{Q}_{p}$ be a connected reductive group,
$\Lambda/\mathbf{Z}_{\ell}$ an algebraically closed field. If
$\mathrm{char}(\Lambda)\neq 0$ we assume $\ell$ is very good for $G$.

Set
$\mathrm{D}(\mathrm{Bun}_{G})=\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$
to be the derived category of lisse-étale $\Lambda$-sheaves, regarded as a
stable $\infty$-category whenever convenient. Let
$\mathfrak{X}_{\hat{G}}=Z^{1}(W_{E},\hat{G})_{\Lambda}/\hat{G}$ be the stack
of $L$-parameters over $\Lambda$, and let $X_{\hat{G}}$ be its coarse moduli
space, $q:\mathfrak{X}_{\hat{G}}\to X_{\hat{G}}$ the natural map. We will
regard $\mathfrak{X}_{\hat{G}}$ as a disjoint union of finite type algebraic
stacks over $\Lambda$, and $X_{\hat{G}}$ as a disjoint union of finite type
affine $\Lambda$-schemes. As in [FS21], we have the spectral action of
$\mathrm{Perf}(\mathfrak{X}_{\hat{G}})$ on $\mathrm{D}(\mathrm{Bun}_{G})$, and
there is a natural map
$\Psi_{G}:\mathcal{O}(\mathfrak{X}_{\hat{G}})=\mathcal{O}(X_{\hat{G}})\to\mathfrak{Z}(\mathrm{D}(\mathrm{Bun}_{G})):=\pi_{0}(\mathrm{id}_{\mathrm{D}(\mathrm{Bun}_{G})})$,
where we recall that $Z^{1}(W_{E},\hat{G})_{\Lambda}$ is a disjoint union of
affine schemes by [FS21, Theorem VIII.1.3]. These two structures are
compatible (as proven by Zou [Zou22, Theorem 5.2.1]).

By [FS21, Prop. VIII.3.8], the set of closed points $X_{\hat{G}}(\Lambda)$ is
naturally in bijection with the set of isomorphism classes of semisimple
$L$-parameters $\phi:W_{E}\to\phantom{}^{L}\\!G(\Lambda)$. Let
$\mathfrak{m}_{\phi}\subset\mathcal{O}(X_{\hat{G}})$ be the maximal ideal
associated with a given $\phi$.

######  Definition A.1.

Given any $\phi$ as above,
$\mathrm{D}(\mathrm{Bun}_{G})_{\phi}\subset\mathrm{D}(\mathrm{Bun}_{G})$ is
the full subcategory of sheaves $A\in\mathrm{D}(\mathrm{Bun}_{G})$ such that
for every $f\in\mathcal{O}(X_{\hat{G}})\smallsetminus\mathfrak{m}_{\phi}$,
$A\overset{\cdot f}{\to}A$ is an isomorphism. Here $\cdot f$ is the
endomorphism of $A$ induced by $\Psi_{G}$.

We will call objects of $\mathrm{D}(\mathrm{Bun}_{G})_{\phi}$ $\phi$-_local_
sheaves.

By construction, $\mathrm{D}(\mathrm{Bun}_{G})_{\phi}$ is a full subcategory
of $\mathrm{D}(\mathrm{Bun}_{G})$ stable under arbitrary limits and colimits,
and the tautological inclusion functor
$\iota_{\phi}:\mathrm{D}(\mathrm{Bun}_{G})_{\phi}\hookrightarrow\mathrm{D}(\mathrm{Bun}_{G})$
commutes with limits and colimits. By the $\infty$-categorical adjoint functor
theorem [Lur09, Cor. 5.5.2.9.(2)], it therefore admits a left adjoint
$\mathcal{L}_{\phi}:\mathrm{D}(\mathrm{Bun}_{G})\to\mathrm{D}(\mathrm{Bun}_{G})_{\phi}$.101010To
see that $\iota_{\phi}$ is accessible, use [Lur09, Prop. 5.4.7.7] together
with the fact that $\iota_{\phi}$ admits a right adjoint, which follows from
[Lur09, Cor. 5.5.2.9.(1)]. The unit of the adjunction gives a map
$A\to\iota_{\phi}\mathcal{L}_{\phi}A=:A_{\phi}$ functorially in $A$. Since
$\iota_{\phi}$ is fully faithful,
$\mathcal{L}_{\phi}\iota_{\phi}=\mathrm{id}$, so $(A_{\phi})_{\phi}=A_{\phi}$,
i.e. the endofunctor $A\rightsquigarrow A_{\phi}$ is idempotent. We remark
that $\mathrm{D}(\mathrm{Bun}_{G})_{\phi}$ is a Bousfield localization of
$\mathrm{D}(\mathrm{Bun}_{G})$, and the map $A\to A_{\phi}$ is the initial map
from $A$ to a $\phi$-local sheaf.

######  Proposition A.2.

The full subcategory $\mathrm{D}(\mathrm{Bun}_{G})_{\phi}$ is preserved by the
spectral action, and $A\rightsquigarrow A_{\phi}$ commutes with the spectral
action. Moreover, $\mathrm{supp}(A_{\phi})\subseteq\mathrm{supp}(A)$.

###### Proof.

The first claim is clear, since the spectral action commutes with the action
of $\mathcal{O}(X_{\hat{G}})$. For the remaining claims (and some later
arguments), it is useful to give an explicit formula for $A_{\phi}$. Let
$\mathcal{I}_{\phi}$ be the diagram category whose objects are elements of
$\mathcal{O}(X_{\hat{G}})\smallsetminus\mathfrak{m}_{\phi}$ and where a
morphism $f\to g$ is an element
$h\in\mathcal{O}(X_{\hat{G}})\smallsetminus\mathfrak{m}_{\phi}$ such that
$g=fh$. This is clearly cofiltered. Let
$F\in\mathrm{Fun}(\mathcal{I}_{\phi},\mathrm{D}(\mathrm{Bun}_{G}))$ be the
functor sending $f$ to $A$ and sending a morphism $h\in\mathrm{Mor}(f,g)$ to
$\cdot h\in\mathrm{End}(A)$. Then
$A_{\phi}=\mathrm{colim}_{i\in\mathcal{I}_{\phi}}F(i)$. The remaining claims
are now immediate. ∎

To make sense of the next proposition, note that for any
$A,B\in\mathrm{D}(\mathrm{Bun}_{G})$, $\mathrm{Hom}(B,A)$ is naturally a
$\mathfrak{Z}(\mathrm{D}(\mathrm{Bun}_{G}))$-module, whence a
$\mathcal{O}(X_{\hat{G}})$-module.

######  Proposition A.3.

If $C\in\mathrm{D}(\mathrm{Bun}_{G})$ is compact, then
$\mathrm{Hom}(C,A_{\phi})\cong\mathrm{Hom}(C,A)_{\mathfrak{m}_{\phi}}$
functorially in $A$ and $C$, where the RHS is the usual localization as an
$\mathcal{O}(X_{\hat{G}})$-module.

###### Proof.

Notation as in the previous proof, we have

$\displaystyle\mathrm{Hom}(C,A_{\phi})$
$\displaystyle\cong\mathrm{Hom}(C,\mathrm{colim}_{i\in\mathcal{I}_{\phi}}F(i))$
$\displaystyle\cong\mathrm{colim}_{i\in\mathcal{I}_{\phi}}\mathrm{Hom}(C,F(i))$
$\displaystyle\cong\mathrm{Hom}(C,A)_{\mathfrak{m}_{\phi}}$

where the second isomorphism follows from the compactness of $C$ and the third
isomorphism is immediate from the definition of $(-)_{\mathfrak{m}_{\phi}}$. ∎

######  Proposition A.4.

If $A$ is ULA, then also $A_{\phi}$ is ULA.

###### Proof.

Recall from [FS21, Prop. VII.7.9] that $B\in\mathrm{D}(\mathrm{Bun}_{G})$ is
ULA iff $R\mathrm{Hom}(C,B)\in\mathrm{Perf}(\Lambda)$ is a perfect complex for
all compact objects $C\in\mathrm{D}(\mathrm{Bun}_{G})$. Now, if $C$ is
compact, $R\mathrm{Hom}(C,-)$ commutes with filtered colimits, so

$\displaystyle R\mathrm{Hom}(C,A_{\phi})$ $\displaystyle\simeq
R\mathrm{Hom}(C,\mathrm{colim}_{i\in\mathcal{I}_{\phi}}F(i))$
$\displaystyle\simeq\mathrm{colim}_{i\in\mathcal{I}_{\phi}}R\mathrm{Hom}(C,F(i))$

with notation as in the proof of Proposition A.2. Since $F(i)\simeq A$ for all
$i$, $\mathrm{colim}_{i\in\mathcal{I}_{\phi}}R\mathrm{Hom}(C,F(i))$ is a
filtered colimit of perfect complexes $P_{i}$ which vanish outside a finite
interval independent of $n$, and with $\mathrm{dim}_{\Lambda}(H^{j}(P_{i}))$
bounded independently of $i$. It then easily follows that
$\mathrm{colim}_{i\in\mathcal{I}_{\phi}}R\mathrm{Hom}(C,F(i))$ is perfect,
whence the claim. ∎

######  Proposition A.5.

If $A$ is ULA, the natural maps
$A\to\prod_{\phi}A_{\phi}\leftarrow\oplus_{\phi}A_{\phi}$ are isomorphisms,
where the direct sum and direct product are taken over all semi-simple
$L$-parameters. In particular, $A_{\phi}$ is functorially a direct summand of
$A$ for ULA sheaves $A$, and the functor $(-)_{\phi}$ on ULA sheaves is
perverse t-exact.

######  Remark A.6.

The isomorphism $\oplus_{\phi}A_{\phi}\overset{\sim}{\to}\prod_{\phi}A_{\phi}$
may be surprising at first glance. To put this in context, we remind the
reader that if $(\pi_{i})_{i\in I}$ is a collection of admissible smooth
$\Lambda[G(\mathbb{Q}_{p})]$-modules whose product $\prod_{i}\pi_{i}$ is
admissible, then $\oplus_{i}\pi_{i}\overset{\sim}{\to}\prod_{i}\pi_{i}$
automatically, because admissibility of $\prod_{i}\pi_{i}$ implies that for
any given compact open subgroup $K\subset G(\mathbb{Q}_{p})$ we have
$\pi_{i}^{K}=0$ for all but finitely many $i$. A similar argument occurs in
the following proof, which actually shows that if $(A_{i})_{i\in I}$ is any
collection of ULA sheaves on $\mathrm{Bun}_{G}$ whose product $\prod_{i}A_{i}$
is ULA, then $\oplus_{i}A_{i}\overset{\sim}{\to}\prod_{i}A_{i}$ automatically.

###### Proof.

We first show that $A\to\prod_{\phi}A_{\phi}$ is an isomorphism. Let $C$ be
any compact object. It suffices to prove that the natural map

$\mathrm{Hom}(C,A)\to\prod_{\phi}\mathrm{Hom}(C,A_{\phi})\cong\mathrm{Hom}(C,\prod_{\phi}A_{\phi})$

is an isomorphism, since $\mathrm{D}(\mathrm{Bun}_{G})$ is compactly generated
[FS21, Theorem I.5.1 (iii)]. As in the previous proof, $R\mathrm{Hom}(C,A)$ is
a perfect complex, so $\mathrm{Hom}(C,A)$ is a finite $\Lambda$-vector space.
In particular, it is a finite length $\mathcal{O}(X_{\hat{G}})$-module
supported at a finite set of closed points $S\subset X_{\hat{G}}(\Lambda)$, so
if $\phi\notin S$ then
$\mathrm{Hom}(C,A_{\phi})=\mathrm{Hom}(C,A)_{\mathfrak{m}_{\phi}}=0$ using
Proposition A.3. We then conclude that

$\displaystyle\mathrm{Hom}(C,A)$ $\displaystyle=\oplus_{\phi\in
S}\mathrm{Hom}(C,A)_{\mathfrak{m}_{\phi}}$ $\displaystyle=\oplus_{\phi\in
S}\mathrm{Hom}(C,A_{\phi})$
$\displaystyle=\prod_{\phi}\mathrm{Hom}(C,A_{\phi})$

where the first equality follows from general nonsense about finite length
modules over commutative rings, the second equality follows from Proposition
A.3, and the third equality follows from the vanishing of
$\mathrm{Hom}(C,A_{\phi})$ for all but finitely many $\phi$. This also shows
that
$\mathrm{Hom}(C,\oplus_{\phi}A_{\phi})\cong\oplus_{\phi}\mathrm{Hom}(C,A_{\phi})\to\prod_{\phi}\mathrm{Hom}(C,A_{\phi})$
is an isomorphism (here again the first isomorphism follows from compactness
of $C$), which implies that
$\oplus_{\phi}A_{\phi}\overset{\sim}{\to}\prod_{\phi}A_{\phi}$ is an
isomorphism. ∎

Next, recall the Verdier duality functor $\mathbb{D}_{\mathrm{Bun}_{G}}$ on
$\mathrm{D}(\mathrm{Bun}_{G})$, which induces an involutive anti-equivalence
on the subcategory of ULA sheaves. Recall also that, for any $A$, the diagram

$\textstyle{\mathcal{O}(X_{\hat{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{G}}$$\scriptstyle{f\mapsto
f^{\vee}}$$\textstyle{\mathrm{End}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(X_{\hat{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{G}}$$\textstyle{\mathrm{End}(\mathbb{D}_{\mathrm{Bun}_{G}}(A))}$

commutes, where $f\mapsto f^{\vee}$ is the involution of
$\mathcal{O}(X_{\hat{G}})$ induced by composition with the Chevalley
involution at the level of $L$-parameters. Since $f\in\mathfrak{m}_{\phi}$ iff
$f^{\vee}\in\mathfrak{m}_{\phi^{\vee}},$ we deduce that if $A$ is $\phi$-local
then $\mathbb{D}_{\mathrm{Bun}_{G}}(A)$ is $\phi^{\vee}$-local. Using
biduality, we also get that if $A$ is ULA then $A$ is $\phi$-local if and only
if $\mathbb{D}_{\mathrm{Bun}_{G}}(A)$ is $\phi^{\vee}$-local.

######  Corollary A.7.

If $A$ is ULA, then
$\mathbb{D}_{\mathrm{Bun}_{G}}(A_{\phi})\cong(\mathbb{D}_{\mathrm{Bun}_{G}}(A))_{\phi^{\vee}}$.

###### Proof.

By Proposition A.5 and the remarks preceding its proof, the decomposition
$A=\oplus_{\psi}A_{\psi}$ dualizes to a decomposition

$\mathbb{D}_{\mathrm{Bun}_{G}}(A)=\prod_{\psi}\mathbb{D}_{\mathrm{Bun}_{G}}(A_{\psi})\cong\oplus_{\psi}\mathbb{D}_{\mathrm{Bun}_{G}}(A_{\psi})$

where the second isomorphism follows from the discussion in Remark A.6. On the
other hand, applying Proposition A.5 directly to
$\mathbb{D}_{\mathrm{Bun}_{G}}(A)$ gives a decomposition

$\mathbb{D}_{\mathrm{Bun}_{G}}(A)\cong\oplus_{\psi^{\prime}}(\mathbb{D}_{\mathrm{Bun}_{G}}(A))_{\psi^{\prime}},$

so comparing these we get a natural isomorphism

$\oplus_{\psi}\mathbb{D}_{\mathrm{Bun}_{G}}(A_{\psi})\cong\oplus_{\psi^{\prime}}(\mathbb{D}_{\mathrm{Bun}_{G}}(A))_{\psi^{\prime}}.$

Applying $(-)_{\phi^{\vee}}$ to both sides, we get
$\mathbb{D}_{\mathrm{Bun}_{G}}(A_{\phi})$ on the left side (using that
$\mathbb{D}_{\mathrm{Bun}_{G}}(A_{\phi})$ is $\phi^{\vee}$-local), and
$(\mathbb{D}_{\mathrm{Bun}_{G}}(A))_{\phi^{\vee}}$ on the right side. This
gives the claim. ∎

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