$\Sigma_{+}(\mu^{!})\simeq\Sigma^{H}_{+}$ that involves only some
“nonstandard” rescaling. The construction of $H$ implies

$\omega^{!}(\sigma)\in W^{H}\cdot\text{(the pinned $\Gamma_{F}$-action on
$H^{\vee}$)}\cdot\omega(\sigma)$

for all $\sigma\in\Gamma_{F}$. Since $\omega^{!}(\sigma)\sigma$ fixes
$\Sigma_{+}(\mu^{!})$, and when $w\in W^{H}$ we have
$w\Sigma_{+}(\mu^{!})=\Sigma_{+}(\mu^{!})$ if and only if $w=1$, the above
determines $\omega^{!}(\sigma)$.

######  Lemma 11.3.5.

The map (11.3.2) is bijective.

###### Proof.

Identify $G_{\eta,\overline{F}}$ with $\overline{G}$, and similarly for their
dual groups. Fix $\mathbf{H}\in E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$.

We begin by proving that the fiber over $\mathbf{H}$ has at most one element.
The determination of $s$ is clear from (11.3.1) since $T^{\vee}$ is shared by
$\tilde{G}^{\vee}$ and $\overline{G}^{\vee}$. The determination of
$(\mu^{!},\omega^{!})$ has been given in Remark 11.3.4.

It remains to show that the fiber over every
$\mathbf{H}=(H,\mathcal{H},\overline{s})$ is non-empty. For each direct factor
of $G_{\eta}$, say a symplectic or unitary group, $\overline{s}^{2}=1$ always
holds; cf. the explicit description in [70, §1.8] for the local case. As
$T^{\vee}$ is shared by $\tilde{G}^{\vee}$ and $\overline{G}^{\vee}$, we
define $s$ from (11.3.1) and obtain
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell},T^{\vee}}(\tilde{G})$.

By the general theory of endoscopy, using
$\mathcal{H}\hookrightarrow{}^{\mathrm{L}}G_{\eta}$ one can obtain a
continuous map $g:\mathrm{W}_{F}\to\overline{G}^{\vee}$ is given such that

  * •

(11.3.3) holds;

  * •

$\operatorname{Ad}(g(\cdot))$ stabilizes the Borel pair of $H^{\vee}$;

  * •

it induces a continuous $1$-cocycle $\omega_{H}:\Gamma_{F}\to
W^{\overline{G}}$, where the $\Gamma_{F}$-action on $W^{\overline{G}}$ is
induced by its action on $\overline{T}\subset\overline{G}$, involving a twist
by $\omega$.

Explicit choices of $g$ can be found in [70]. If $\mathbf{H}$ is unramified at
a place $v$, then $\omega_{H}|_{I_{F_{v}}}=1$.

Next, return to the description of
$\mathrm{Stab}(G^{!}(F))\to\mathrm{Stab}(G(F))$ in Definition 11.2.7. Define

$\mu^{!}:=\Psi^{-1}(\mu),\quad\underline{\omega^{!}}:=\omega_{H}\omega:\Gamma_{F}\to
W^{G}.$

We claim that

$(\mu^{!},\underline{\omega^{!}})\in\underline{\mathrm{Stab}}(G^{!}(F)).$

Once this is verified, its representative
$(\mu^{!},\omega^{!})\in\mathrm{Stab}(G^{!}(F))$ will map to $(\mu,\omega)$
under Definition 11.2.7, and
$(\mathbf{G}^{!},\mu^{!},\omega^{!})\mapsto\mathbf{H}$ by construction.
Moreover, the ellipticity of $(\mu^{!},\omega^{!})$ will follow easily from
that of $\mathbf{H}$ and $(\eta,\omega)$.

  1. 1.

To prove the claim, observe first that by (11.3.3), the $\Gamma_{F}$-action on
$T^{\vee}\subset H^{\vee}$ determined by $1\to
H^{\vee}\to\mathcal{H}\to\mathrm{W}_{F}\to 1$ comes from the one on
$T^{\vee}\subset\overline{G}^{\vee}$ (namely the homomorphism
$\omega:\Gamma_{F}\to W^{G}$) followed by $\omega_{H}$; this is just
$\underline{\omega}^{!}$. Hence $\underline{\omega}^{!}$ defines a
homomorphism $\Gamma_{F}\to W^{H^{\vee}}$.

As a consequence, we also deduce $\underline{\omega}^{!}(\sigma)\in
W^{\overline{G}^{\vee}}=W^{\overline{G}}$ for all $\sigma\in\Gamma_{F}$.

  2. 2.

Let us show $\underline{\omega}^{!}(\sigma)$ stabilizes $s\in T^{\vee}$ for
all $\sigma\in\Gamma_{F}$. As $Z_{\tilde{G}^{\vee}}(s)$ is connected, it will
then follow that $\underline{\omega^{!}}$ takes value in $W^{G^{!}}$, by [64,
E-35, 4.1].

Inspect the relation (11.3.1) between $s$ and $\overline{s}$: they only differ
by $-1$ on the factor $(G_{\eta}^{-})^{\vee}$ of $\overline{G}^{\vee}\simeq
G_{\eta}^{\vee}$. The meaning of $-1$ is explicated in the discussions after
(11.3.1); as such, it is evidently fixed by the Weyl group of
$(G^{-}_{\eta})^{\vee}$. Hence it suffices to show
$\underline{\omega}^{!}(\sigma)\overline{s}=\overline{s}$.

The explicit description in [70] decomposes $H=H_{+}\times H_{-}$ according to
the signs in $\overline{s}$, and one checks case-by-case that
$\overline{s}\in(H^{\vee})^{\Gamma_{F}}$. As noted in the first step, this
entails $\underline{\omega}^{!}(\sigma)\overline{s}=\overline{s}$.

  3. 3.

Also, $\underline{\omega^{!}}(\sigma)\sigma$ stabilizes $\mu\in
T(\overline{F})$ since $\omega_{H}(\sigma)\in W^{\overline{G}}$ and
$\omega_{H}(\sigma)\omega(\sigma)\sigma\mu=\omega_{H}(\sigma)\mu=\mu$. As
$\Psi$ is equivariant with respect to $\Gamma_{F}$ and
$W^{G^{!}}\hookrightarrow W^{G}$, it stabilizes $\mu^{!}$ as well. The claim
follows.

Finally, let us verify $S\supset S(\mathcal{O}^{!},K^{!})$ using Corollary
11.2.4. For $v\notin S$, we have $\mu\in T(\mathfrak{o}_{\overline{F_{v}}})$
and $\omega|_{I_{F_{v}}}=1$. Thus $\mu^{!}=\Psi^{-1}(\mu)\in
T^{!}(\mathfrak{o}_{\overline{F_{v}}})$; this verifies the condition (i).
Consider (ii): if $\beta$ is a long (resp. short) root of $G^{!}$,
corresponding to a short (resp. long) root $\alpha$ of $G$, then
$\beta(\mu^{!})-1=\alpha(\mu)-1$ (resp.
$(\beta(\mu^{!})-1)(\beta(\mu^{!})+1)=\alpha(\mu)-1$), so $\beta(\mu^{!})-1$
is either zero or invertible in $\mathfrak{o}_{F_{v}}$ since so is
$\alpha(\mu)-1$. The minus signs in $\Psi$ do not bother here.

Consider (iii). Since $\mathbf{H}_{v}$ is unramified,
$\omega_{H}|_{I_{F_{v}}}=1$, hence $\underline{\omega^{!}}|_{I_{F_{v}}}=1$.
The same property passes easily from $(\mu^{!},\underline{\omega^{!}})$ to its
representative $(\mu^{!},\omega^{!})\in\mathrm{Stab}(G^{!}(F))$. ∎

######  Definition 11.3.6.

For all $\mathbf{H}\in E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$, let

$\displaystyle\mathcal{J}(\mathbf{H})$ $\displaystyle:=\text{the fiber of
\eqref{eqn:JH-map} over}\;\mathbf{H},$
$\displaystyle\mathcal{J}(\mathbf{H}/\sim)$
$\displaystyle:=\bigsqcup_{\mathbf{H}^{\prime}\sim\mathbf{H}}\mathcal{J}(\mathbf{H}^{\prime}).$

It follows from Lemma 11.3.5 that $\mathcal{J}(\mathbf{H})$ is a singleton for
every $\mathbf{H}\in E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$.

Continue the discussions on (11.2.3). In order apply the method of descent,
recall that $SA^{G^{!}}(S,\mathcal{O}^{!})=0$ unless $\mathcal{O}^{!}$ is
elliptic and $S\supset S(\mathcal{O}^{!},K^{!})$. It is thus justified to
fiber the sum in (11.2.2) over $E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$
to obtain

$A^{\tilde{G},\mathcal{E}}(S,\mu,\omega)_{\operatorname{ell}}=\sum_{\mathbf{H}\in
E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)}A^{\tilde{G},\mathcal{E}}(S,\mathbf{H})_{\operatorname{ell}},$

where

$A^{\tilde{G},\mathcal{E}}(S,\mathbf{H})_{\operatorname{ell}}:=|W^{G}(\mu)|^{-1}\sum_{(\mathbf{G}^{!},\mu^{!},\omega^{!})\in\mathcal{J}(\mathbf{H})}\iota(\tilde{G},G^{!},\mathcal{O}^{!})\check{\mathcal{T}}_{\mathbf{G}^{!},\tilde{G}}\left(SA^{G^{!}}(S,\mathcal{O}^{!})_{\operatorname{ell}}\right)$
(11.3.4)

with $\mathcal{O}^{!}\subset G^{!}(F)$ satisfies
$(\mu^{!},\omega^{!})\in\mathrm{Fiber}(\mathcal{O}^{!})$.

Hence, for all $(\mu,\omega)\in\mathrm{Fiber}(\mathcal{O})$, we get

$\displaystyle A^{\tilde{G},\mathcal{E}}(S,\mu,\omega)_{\operatorname{ell}}$
$\displaystyle=\sum_{\mathbf{H}/\sim}A^{\tilde{G},\mathcal{E}}(S,\mathbf{H}/\sim)_{\operatorname{ell}},$
(11.3.5) $\displaystyle
A^{\tilde{G},\mathcal{E}}(S,\mathbf{H}/\sim)_{\operatorname{ell}}$
$\displaystyle:=\sum_{\begin{subarray}{c}\mathbf{H}^{\prime}\sim\mathbf{H}\end{subarray}}A^{\tilde{G},\mathcal{E}}(S,\mathbf{H}^{\prime})_{\operatorname{ell}}.$

###  11.4 Descent of coefficients

The scenario is the same as §11.3. In order to localize the problem, for every
place $v$ of $F$ we use Definition 2.9.4 to define the pointed set

$\mathcal{Y}_{v}:=\mathcal{Y}^{G_{F_{v}}}(\eta_{v})\;\subset
G(\overline{F_{v}}).$

It is acted by $G_{\eta}(\overline{F_{v}})\times G(F)$. Each
$G_{\eta}(\overline{F_{v}})$-orbit $G_{\eta}(\overline{F}_{v})y_{v}$ in
$\mathcal{Y}_{v}$ gives rise to the stable conjugate $\eta[y_{v}]\in
G(F_{v})$, whereas $\eta[y_{v}g_{v}]=g_{v}^{-1}\eta[y_{v}]g_{v}$ for all
$g_{v}\in G(F_{v})$.

Each $y_{v}\in\mathcal{Y}_{v}$ affords a pure inner twist

$\operatorname{Ad}(y_{v}):G_{\eta[y_{v}],\overline{F_{v}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G_{\eta,\overline{F_{v}}},$

making every $\mathbf{H}\in E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$ into
an elliptic endoscopic datum of $G_{\eta[y_{v}]}$. Those $y_{v}$ for which
$\mathbf{H}$ is relevant form a subset
$\mathcal{Y}_{v}^{\mathrm{rel}}(\mathbf{H})\subset\mathcal{Y}_{v}$. Define

$\mathcal{Y}_{S}:=\prod_{v\in
S}\mathcal{Y}_{v},\quad\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H}):=\prod_{v\in
S}\mathcal{Y}_{v}^{\mathrm{rel}}(\mathbf{H}).$

For all $y_{S}\in\mathcal{Y}_{S}$, define $\eta[y_{S}]:=\prod_{v\in
S}\eta[y_{v}]\in G(F_{S})$.

######  Lemma 11.4.1.

We have $\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})\neq\varnothing$ for all
$\mathbf{H}$; in fact, it contains the base-point $(1_{v})_{v\in S}$.

###### Proof.

The simple reason is that $G_{\eta}$ is assumed to be quasisplit. ∎

Choose a representative in $E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$ for
each class in $E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)/\sim$. Given such a
representative $\mathbf{H}=(H,\mathcal{H},\overline{s})$ and
$y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$, take the semi-local
transfer factors

$\displaystyle\Delta[y_{S}]$
$\displaystyle=\Delta_{\mathbf{H}}[y_{S}]:=\prod_{v\in S}\Delta_{v}[y_{v}],$
$\displaystyle\Delta_{v}[y_{v}]$
$\displaystyle=\Delta_{v}[y_{v}]:H_{\mathrm{reg}}(F_{v})\times
G_{\eta[y_{v}],\mathrm{reg}}(F_{v})\to\mathbb{C}$

determined by the quasisplit pure inner twists $\operatorname{Ad}(y_{v})$ and
the chosen $F$-pinning $\mathcal{E}$ of $G_{\eta}$. See §4.2 for a review.

######  Notation 11.4.2.

In what follows, we denote such “standard” representatives in a $\sim$ class
by $\mathbf{H}_{\mathrm{std}}$.

Based on the assumptions imposed on $\eta\in G(F)$ at the end of §11.1, we
define $\eta^{S}\in K^{S}$ and $\tilde{\eta}_{S}=\prod_{v\in
S}\tilde{\eta}_{v}$ via (10.4.6).

Let $f_{S}=\prod_{v\in
S}f_{v}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{S})\otimes\operatorname{mes}(G(F_{S}))$,
supported in a $\bm{\mu}_{8}$-invariant, small and stably invariant
neighborhood of $\tilde{\eta}_{S}$. For all $\mathbf{H}=(H,\mathcal{H},s)\in
E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$,
$y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$ and $v\in S$, take
$\hat{H}$ with $H\simeq\overline{\hat{H}}$ (Remark 11.3.2), so that we have
the maps

${\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{v})\otimes\operatorname{mes}(G(F_{v}))}$${f_{v}}$${\mathcal{I}(G_{\eta[y_{v}]}(F_{v}))\otimes\operatorname{mes}(G_{\eta[y_{v}]}(F_{v}))}$${{f_{v}[y_{v}]}}$${S\mathcal{I}(H(F_{v}))\otimes\operatorname{mes}(H(F_{v}))}$${{\overline{f}[y_{v}]}}$${S\mathcal{I}(\hat{H}(F_{v}))\otimes\operatorname{mes}(\hat{H}(F_{v}))}$${{\hat{f}[y_{v}]}}$$\scriptstyle{\operatorname{desc}^{\tilde{G}}_{\tilde{\eta}[y_{v}]}}$transfer$\scriptstyle{\sim}$nonstandard
transfer (11.4.1)

and taking products over $v$ yields $f_{S}\mapsto
f_{S}[y_{S}]\mapsto\overline{f}[y_{S}]\mapsto\hat{f}[y_{S}]$. Here,

  * •

the metaplectic stable conjugate $\tilde{\eta}[y_{v}]\in\tilde{G}_{v}$ of
$\tilde{\eta}_{v}$ is defined in [48, Définition 4.1.3], which depends only on
$G_{\eta}(\overline{F_{v}})y_{v}$ and satisfies
$\tilde{\eta}[y_{v}g_{v}]=g_{v}^{-1}\tilde{\eta}[y_{v}]g_{v}$ for $g_{v}\in
G(F_{v})$;

  * •

the transfer from $G_{\eta[y_{v}]}(F_{v})$ to $H(F_{v})$ uses the
representative $\mathbf{H}_{\mathrm{std}}$ in the $\sim$ class of $\mathbf{H}$
and the $\Delta_{\mathbf{H}_{\mathrm{std}}}[y_{S}]$ defined earlier, noting
that $\sim$ has no effect on the endoscopic groups and the correspondence of
orbits;

  * •

the nonstandard transfer between $H(F_{v})$ and $\hat{H}(F_{v})$ is described
in §4.3; it depends only on the identification $H\simeq\overline{\hat{H}}$.

######  Definition 11.4.3.

To $j=(\mathbf{G}^{!},\mu^{!},\omega^{!})\in\mathcal{J}(\mathbf{H}/\sim)$, we
have the transfer factor

$\Delta_{j}=\prod_{v\in S}\Delta_{\mathbf{G}^{!}_{v},\tilde{G}_{v}}$

for $\tilde{G}_{S}$. Let
$y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})=\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H}_{\mathrm{std}})$,
define

$d_{j}(y_{S}):=\prod_{v\in S}d_{j}(y_{v})\in\mathbb{C}^{\times}$

by (4.4.1). In other words, $d_{j}(y_{v})$ is the ratio of the descent at
$(\epsilon,\tilde{\eta}[y_{v}])$ of $\Delta_{j}$ over
$\Delta_{\mathbf{H}_{\mathrm{std}}}[y_{v}]$.

Given $\mathbf{H}$, choose a system
$\dot{\mathcal{Y}}_{S}^{\mathrm{rel}}(\mathbf{H})$ of representatives of
$G_{\eta}(\overline{F_{S}})\backslash\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})/G(F_{S})$
in $\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$. Note that
$\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$ and
$\dot{\mathcal{Y}}_{S}^{\mathrm{rel}}(\mathbf{H})$ depend only on the class
$\mathbf{H}/\sim$, since the notion of relevance is not affected by
automorphisms of endoscopic data. Thus we can write
$\dot{\mathcal{Y}}_{S}^{\mathrm{rel}}(\mathbf{H}/\sim)$, etc.

######  Lemma 11.4.4.

For all class $\mathbf{H}/\sim$ and $f_{S}$ as above, we have

$I^{\tilde{G}_{S}}\left(A^{\tilde{G},\mathcal{E}}(S,\mathbf{H}/\sim)_{\operatorname{ell}},f_{S}\right)\\\
=|W^{G}(\mu)|^{-1}|W^{H}|\cdot\frac{\tau(G)}{\tau(H)}\cdot\sum_{y_{S}\in\dot{\mathcal{Y}}_{S}^{\mathrm{rel}}(\mathbf{H}/\sim)}S^{H_{S}}\left(SA^{H}_{\mathrm{unip}}(S),\overline{f}[y_{S}]\right)\cdot\sum_{j\in\mathcal{J}(\mathbf{H}/\sim)}d_{j}(y_{S}).$

###### Proof.

For $j=(\mathbf{G}^{!},\mu^{!},\omega^{!})\in\mathcal{J}(\mathbf{H}/\sim)$,
write $f_{j}^{!}:=\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}(f_{S})$. Consider any
$\mathbf{H}$ in the $\sim$ class. Plug Lemma 11.2.6 into (11.3.4) to obtain

$I^{\tilde{G}_{S}}\left(A^{\tilde{G},\mathcal{E}}(S,\mathbf{H})_{\operatorname{ell}},f_{S}\right)\\\
=|W^{G}(\mu)|^{-1}\sum_{j\in\mathcal{J}(\mathbf{H})}|W^{G^{!}_{\epsilon}}|\cdot\frac{\tau(G)}{\tau(G^{!}_{\epsilon})}\cdot
S^{G^{!}_{\epsilon,S}}\left(SA^{G^{!}_{\epsilon}}_{\mathrm{unip}}(S),S\operatorname{desc}^{G^{!}}_{\epsilon}(f_{j}^{!})\right).$

Next, take $j\in\mathcal{J}(\mathbf{H})$. The descent of endoscopy in §4.4
furnishes an isomorphism
$\theta_{j}:\hat{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G^{!}_{\epsilon}$.
By the results cited there — specifically, the transposed form of Lemma 4.4.2
— we have

$\theta_{j}^{*}\left(S\operatorname{desc}^{G^{!}}_{\epsilon}\left(f_{j}^{!}\right)\right)=\sum_{y_{S}\in\dot{\mathcal{Y}}_{S}^{\mathrm{rel}}(\mathbf{H}/\sim)}d_{j}(y_{S})\hat{f}[y_{S}].$

In view of the canonicity of $SA_{\mathrm{unip}}$ [61, VI.2.2, 5.2], we may
pass to $\hat{H}$ and deduce

$S^{G^{!}_{\epsilon,S}}\left(SA^{G^{!}_{\epsilon}}_{\mathrm{unip}}(S),S\operatorname{desc}^{G^{!}}_{\epsilon}(f_{j}^{!})\right)=\sum_{y_{S}\in\dot{\mathcal{Y}}_{S}^{\mathrm{rel}}(\mathbf{H}/\sim)}d_{j}(y_{S})S^{\hat{H}_{S}}\left(SA^{\hat{H}}_{\mathrm{unip}}(S),\hat{f}[y_{S}]\right).$

By construction, $\overline{f}[y_{S}]$ and $\hat{f}[y_{S}]$ depend only on
$\mathbf{H}/\sim$ and $y_{S}$, whereas $\tau(G^{!}_{\epsilon})=\tau(\hat{H})$.
The only remnant of $j$ now resides in $d_{j}(y_{S})$.

On the other hand, by [61, VI.5.6 Théorème] (nonstandard transfer of
$SA_{\mathrm{unip}}$ for simply connected groups) and [61, VII.4.6
Proposition] (passage from $\operatorname{SO}(2m+1)$ to
$\operatorname{Spin}(2m+1)$ up to Tamagawa numbers), for all $y_{S}$ we have

$S^{\hat{H}_{S}}\left(SA^{\hat{H}}_{\mathrm{unip}}(S),\hat{f}[y_{S}]\right)=\frac{\tau(\hat{H})}{\tau(H)}\cdot
S^{H_{S}}\left(SA^{H}_{\mathrm{unip}}(S),\overline{f}[y_{S}]\right).$

We also have $W^{\hat{H}}\simeq W^{H}$. Finally, summing over $\mathbf{H}$ in
the $\sim$ class yields the desired equality. ∎

###  11.5 Cohomological arguments

Retain the conventions from §11.3. For all $v$, denote by
$[y_{v}]\in\operatorname{H}^{1}_{\mathrm{ab}}(F_{v},G_{\eta})$ the class
determined by $y_{v}\in\mathcal{Y}_{v}$ via
$G_{\eta}(\overline{F_{v}})\backslash\mathcal{Y}_{v}/G(F_{v})\to\operatorname{H}^{1}(F_{v},G_{\eta})$
composed with the abelianization map $\mathrm{ab}^{1}$.

For all $v\notin S$, define

$\mathcal{Y}_{v}^{\mathrm{nr}}:=\left\\{y_{v}\in\mathcal{Y}_{v}:\eta[y_{v}]\;\text{is
$G(F_{v})$-conjugate to}\;K_{v}\right\\}.$

######  Lemma 11.5.1.

The set $\mathcal{Y}_{v}^{\mathrm{nr}}$ equals the
$G_{\eta}(\overline{F_{v}})\times G(F_{v})$-orbit containing $1$. Moreover,
$\mathcal{Y}_{v}^{\mathrm{nr}}\subset\mathcal{Y}_{v}^{\mathrm{rel}}(\mathbf{H})$
for any $\mathbf{H}$.

###### Proof.

Since $S\supset S(\mathcal{O})$ and $\eta\in K_{v}$, the first assertion is
immediate from a result of Kottwitz [41, 7.1 Proposition]. That result also
implies $G_{\eta[y_{v}]}$ is unramified, thus quasisplit, and the second
assertion follows. ∎

######  Definition 11.5.2.

Let $\mathcal{Y}_{F}$ be the set of $y\in G(\overline{F})$ such that
$\eta[y]:=y\eta y^{-1}\in G(F)$, and
$\operatorname{Ad}(y):G_{\eta[y],\overline{F}}\to G_{\eta,\overline{F}}$ is a
pure inner twist. Denote the natural map $\mathcal{Y}_{F}\to\mathcal{Y}_{S}$
as $x\mapsto x_{S}$. Define

$\mathcal{Y}_{F}^{S\text{-nr}}:=\left\\{x\in\mathcal{Y}_{F}:v\notin S\implies
x_{v}\in\mathcal{Y}_{v}^{\mathrm{nr}}\right\\}.$

For all $y_{S}\in\mathcal{Y}_{S}$, define

$\mathcal{Y}_{F}[y_{S}]:=\left\\{x\in\mathcal{Y}_{F}^{S\text{-nr}}:x_{S}\in
G_{\eta}(\overline{F_{S}})y_{S}G(F_{S})\right\\}.$

As before, we fix systems of representatives
$\dot{\mathcal{Y}}_{F}^{S\text{-nr}}\subset\dot{\mathcal{Y}}_{F}$ of their
$G_{\eta}(\overline{F})\times G(F)$-orbits.

######  Lemma 11.5.3.

For all $y_{S}\in\mathcal{Y}_{S}$, the quotient set
$G_{\eta}(\overline{F})\backslash\mathcal{Y}_{F}[y_{S}]/G(F)$ is either empty
or a singleton.

###### Proof.

Identify $G_{\eta}(\overline{F})\backslash\mathcal{Y}_{F}/G(F)$ with
$\operatorname{H}^{1}(F,G_{\eta})$ as pointed sets. Lemma 11.5.1 identifies
$G_{\eta}(\overline{F})\backslash\mathcal{Y}_{F}[y_{S}]/G(F)$ as a fiber of

$\operatorname{H}^{1}(F,G_{\eta})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{H}_{\mathrm{ab}}^{1}(F,G_{\eta})\underset{\operatorname{H}_{\mathrm{ab}}^{1}(\mathbb{A}_{F},G_{\eta})}{\times}\operatorname{H}^{1}(\mathbb{A}_{F},G_{\eta})\xrightarrow{\text{pr}_{2}}\operatorname{H}^{1}(\mathbb{A}_{F},G_{\eta});$

the first arrow being an isomorphism by [42, Théorème 1.6.10]. Such a fiber,
if non-empty, is identifiable with a fiber of

$\operatorname{H}_{\mathrm{ab}}^{1}(F,G_{\eta})\to\operatorname{H}_{\mathrm{ab}}^{1}(\mathbb{A}_{F},G_{\eta}).$

Recall from §3.2 that $G_{\eta}$ is a product of symplectic and unitary
factors up to restriction of scalars. It suffices to show
$\operatorname{ker}_{\mathrm{ab}}^{1}(F,I)=0$ for any unitary factor $I$ in
$G_{\eta}$; the other cases are trivially true.

Without loss of generality, assume that $I$ is a unitary group relative to a
quadratic extension $E|F$ of fields. By [48, Exemple 3.1.2], the complex
$I_{\mathrm{ab}}$ computing $\operatorname{H}^{1}_{\mathrm{ab}}(\cdot,I)$ is
quasi-isomorphic to the norm-one torus relative to $E|F$. It remains to apply
the local-global principle in such a situation, see [62, Theorem 6.11]. ∎

Consider any $\mathbf{H}$ and take $j\in\mathcal{J}(\mathbf{H})$. In
Definition 11.4.3, the denominator of $d_{j}(y_{S})$ was
$\Delta_{\mathbf{H}_{\mathrm{std}}}[y_{S}]$, where $\mathbf{H}_{\mathrm{std}}$
is the standard representative in $\mathbf{H}/\sim$. We begin by inspecting
the variant $\hat{d}_{j}(y_{S})$ defined using the “direct descendant”
$\mathbf{H}$ of $j$ via Proposition 11.3.3. Results from §4.2.4 will be
needed.

######  Lemma 11.5.4.

For all $\mathbf{H}$, $j\in\mathcal{J}(\mathbf{H})$ and
$y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$, define
$\hat{d}_{j}(y_{S})=\prod_{v}\hat{d}_{j}(y_{v})\in\mathbb{C}^{\times}$, which
is the ratio of the descent of $\Delta_{j}$ over the transfer factor
$\Delta_{\mathbf{H}}[y_{S}]$. Then $\hat{d}_{j}(y_{S})=1$.

###### Proof.

For each place $v$ of $F$ and
$y_{v}\in\mathcal{Y}_{v}^{\mathrm{rel}}(\mathbf{H})$, the ratio
$\hat{d}_{j}(y_{v})$ of $\Delta_{j}$ over $\Delta_{\mathbf{H}}[y_{v}]$ is
determined in [52, Théorème 7.10]. Indeed, the endoscopic datum of
$G_{\eta[y_{v}]}$ used there is readily seen to be $\mathbf{H}$ localized at
$v$; the point is that in loc. cit. one picked out the $\overline{s}$ in
(11.3.1).

The ratio $\hat{d}_{j}(y_{v})$ is a _good constant_ in the parlance of loc.
cit. Let us be more precise.

  * •

Assume $y_{v}\in G_{\eta}(\overline{F_{v}})G(F_{v})$. Then
$\hat{d}_{j}(y_{v})=1$ in the unramified setting, and this is indeed the case
for all $v\notin S$ since the assumption on the $F$-pinning $\mathcal{E}$ of
$G_{\eta}$ implies that $\Delta_{\mathbf{H}}[y_{v}]$ is correctly normalized
in the unramified setting.

  * •

The constant $\hat{d}_{j}(y_{v})$ is independent of the pure inner twist
$\operatorname{Ad}(y_{v})$. To see this, note that in the arguments leading up
to [52, Théorème 7.10], the implied constants are made from

    * –

constants appearing in the parameter of the _stable_ classes of $\eta$ and
$\epsilon$ (see §3.2);

    * –

Weil’s constants $\gamma_{\psi_{v}}$, where $\psi_{v}$ is the $v$-component of
$\psi:\mathbb{A}_{F}/F\to\mathbb{S}^{1}$;

    * –

Hilbert symbols;

    * –

values at the “unitary part” of the data of the virtual character
$\omega_{\psi_{v}}^{+}-\omega_{\psi_{v}}^{-}$ where $\omega_{\psi_{v}}^{\pm}$
are Weil’s representations (see the first step in the proof of [52,
Proposition 7.20]), and such a value is stably invariant in the metaplectic
sense by [52, Définition 5.6]. 111 There is a typo in the proof of [52,
Proposition 7.21]: in the case $\bullet=u$, one should replace $N_{L/F}(u+1)$
by its original version $\det_{F}(u+1|W^{\prime}_{u})$. This has no
consequences elsewhere.

Consequently, $\hat{d}_{j}(y_{v})=\hat{d}_{j}(1_{v})$ for all $y_{v}$. We
claim that $\prod_{v}\hat{d}_{j}(1_{v})=1$. Indeed,

  * •

the $F$-pinning $\mathcal{E}$ of $G_{\eta}$ is assumed to be “unramified”
outside $S$,

  * •

Weil’s constants, Hilbert symbols, and the virtual character
$\omega_{\psi_{v}}^{+}-\omega_{\psi_{v}}^{-}$ (see [52, §4.7]) all have
product formula;

  * •

the other data appearing in $\hat{d}_{j}(1_{v})$ are all $F$-rational.

All in all, $\hat{d}_{j}(y_{S})=\prod_{v\in
S}\hat{d}_{j}(1_{v})=\prod_{v}\hat{d}_{j}(1_{v})=1$. ∎

######  Remark 11.5.5.

The phenomenon above is coherent with the cocycle property [52, Théorème
4.2.12] for $\Delta_{j}$, when $\eta$ and $\epsilon$ are in equisingular
correspondence. The reason is that [38, (5.1)] implies

$\Delta_{\mathbf{H}}[y_{v}]=\langle\overline{s},[y_{v}]\rangle_{\mathrm{Kott}}\Delta_{\mathbf{H}}[1_{v}]=\langle
s,[y_{v}]\rangle_{\mathrm{Kott}}\Delta_{\mathbf{H}}[1_{v}]$

although $\mathbf{H}$ is now a principal endoscopic datum; indeed, the
difference between $s$ and $\overline{s}$ in (11.3.1) does not affect
$\langle\cdot,\cdot\rangle_{\mathrm{Kott}}$. On the other hand,
$\Delta_{\mathbf{H}}[y_{v}]$ is replaced by the constant $1$ in loc. cit.,
thus independent of $y_{v}$. The product formula was also stated in loc. cit.
in the equisingular case.

In view of Lemma 11.3.5, for $j\in\mathcal{J}(\mathbf{H})$ and $z\in
Z_{\overline{G}^{\vee}}^{\Gamma_{F}}$, it makes sense to define

$zj:=\text{the unique element in}\;\mathcal{J}(z\mathbf{H}).$

This yields a free action of $Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$ on
$\mathcal{J}(\mathbf{H}/\sim)$.

######  Lemma 11.5.6.

Let $y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$.

  1. (i)

For all $j\in\mathcal{J}(\mathbf{H}/\sim)$ and $z\in
Z_{\overline{G}^{\vee}}^{\Gamma_{F}}$, we have

$d_{zj}(y_{S})=d_{j}(y_{S})\prod_{v\in
S}\langle[y_{v}],z\rangle_{\mathrm{Kott}}.$

  2. (ii)

If $\mathcal{Y}_{F}[y_{S}]\neq\varnothing$, then $d_{j}(y_{S})=1$.

###### Proof.

Consider (i). In view of Lemma 11.5.4, it suffices to show that

$\frac{d_{zj}(y_{S})}{\hat{d}_{zj}(y_{S})}=\frac{d_{j}(y_{S})}{\hat{d}_{j}(y_{S})}\prod_{v\in
S}\langle[y_{v}],z\rangle_{\mathrm{Kott}}.$

Denote by $d_{j}(y_{v})$ the component of $d_{j}(y_{S})$ at $v\in S$, and so
forth. We are reduced to showing

$\frac{d_{zj}(y_{v})}{\hat{d}_{zj}(y_{v})}=\frac{d_{j}(y_{v})}{\hat{d}_{j}(y_{v})}\cdot\langle[y_{v}],z\rangle_{\mathrm{Kott}},\quad
v\in S.$ (11.5.1)

Take $w\in Z_{\overline{G}^{\vee}}^{\Gamma_{F}}$ such that
$\mathbf{H}=w\mathbf{H}_{\mathrm{std}}$. Fix $v\in S$. Theorem 4.2.3 implies

$\frac{d_{j}(y_{v})}{\hat{d}_{j}(y_{v})}=\frac{\Delta_{\mathbf{H}}[y_{v}]}{\Delta_{\mathbf{H}_{\mathrm{std}}}[y_{v}]}=\langle[y_{v}],w\rangle_{\mathrm{Kott}}.$
(11.5.2)

Consider $zj$, which maps to $z\mathbf{H}=zw\mathbf{H}_{\mathrm{std}}$; the
same reasoning leads to

$\frac{d_{zj}(y_{v})}{\hat{d}_{zj}(y_{v})}=\frac{\Delta_{z\mathbf{H}}[y_{v}]}{\Delta_{\mathbf{H}_{\mathrm{std}}}[y_{v}]}=\langle[y_{v}],zw\rangle_{\mathrm{Kott}}.$

The identity (11.5.1) follows at once.

Consider (ii). Both
$[y_{S}]\in\operatorname{H}_{\mathrm{ab}}^{1}(\mathbb{A}_{F},G_{\eta})$ and
$w$ are of global origin, and the local pairings satisfy product formula.
Hence (11.5.2) entails

$\frac{d_{j}(y_{S})}{\hat{d}_{j}(y_{S})}=\prod_{v\in
S}\langle[y_{v}],w\rangle_{\mathrm{Kott}}=\prod_{v}\langle[y_{v}],w\rangle_{\mathrm{Kott}}=1.$

It remains to recall from Lemma 11.5.4 that $\hat{d}_{j}(y_{S})=1$. ∎

######  Corollary 11.5.7.

For all $\mathbf{H}/\sim$ and
$y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H}/\sim)$, we have

$\sum_{j\in\mathcal{J}(\mathbf{H}/\sim)}d_{j}(y_{S})=\begin{cases}0,&\text{if}\quad\mathcal{Y}_{F}[y_{S}]=\varnothing,\\\
\left|Z_{\overline{G}^{\vee}}^{\Gamma_{F}}\right|,&\text{if}\quad\mathcal{Y}_{F}[y_{S}]\neq\varnothing.\end{cases}$

###### Proof.

Take any $j_{0}\in\mathcal{J}(\mathbf{H}/\sim)$. The sum is indexed by $z\in
Z_{\overline{G}^{\vee}}^{\Gamma_{F}}$ where $z$ corresponds to
$d_{zj_{0}}(y_{S})$. By Lemma 11.5.6, this gives

$\sum_{j\in\mathcal{J}(\mathbf{H}/\sim)}d_{j}(y_{S})=d_{j_{0}}(y_{S})\sum_{z\in
Z_{\overline{G}^{\vee}}^{\Gamma_{F}}}\prod_{v}\langle[y_{v}],z\rangle_{\mathrm{Kott}}.$

Moreover, that Lemma asserts $d_{j}(y_{S})=1$ for all
$j\in\mathcal{J}(\mathbf{H}/\sim)$ when
$\mathcal{Y}_{F}[y_{S}]\neq\varnothing$. It remains to address the case
$\mathcal{Y}_{F}[y_{S}]=\varnothing$.

Let us adopt the formalism of abelianized Galois cohomologies from §2.9. In
our case, we will take $I=G_{\eta}$ and $H=G=\operatorname{Sp}(W)$, so that
the complex $G_{\mathrm{ab}}$ is trivial. Thus

$\displaystyle\mathfrak{D}(G_{\eta},G;\cdot)=\operatorname{H}^{1}(\cdot,G_{\eta}),\quad\mathfrak{E}(G_{\eta},G;\cdot)=\operatorname{H}_{\mathrm{ab}}^{1}(\cdot,G_{\eta}).$

On the other hand, to $y_{S}$ is attached a class in
$\mathfrak{D}(G_{\eta},G;\mathbb{A}_{F})$, whose components outside $S$ are
the base-points. The condition $\mathcal{Y}_{F}[y_{S}]\neq\varnothing$ is
equivalent to that the class of $y_{S}$ arises from
$\mathfrak{D}(G_{\eta},G;F)$.

Denote by $(\cdots)^{D}$ the Pontryagin dual. By [42, Proposition 1.7.3] we
also have

$\displaystyle\mathfrak{E}(G_{\eta},G;\mathbb{A}_{F}/F)$
$\displaystyle=\operatorname{H}_{\mathrm{ab}}^{1}(\mathbb{A}_{F}/F,G_{\eta})\simeq\pi_{0}\left(Z_{\overline{G}^{\vee}}^{\Gamma_{F}}\right)^{D},$
$\displaystyle\mathfrak{R}(G_{\eta},G;F)_{1}$
$\displaystyle=\mathfrak{R}(G_{\eta},G;F)=\operatorname{H}_{\mathrm{ab}}^{1}(\mathbb{A}_{F}/F,G_{\eta})^{D}$
$\displaystyle\simeq\pi_{0}\left(Z_{\overline{G}^{\vee}}^{\Gamma_{F}}\right).$

canonically. Via the map
$\mathfrak{E}(G_{\eta},G;\mathbb{A}_{F})\to\mathfrak{E}(G_{\eta},G;\mathbb{A}_{F}/F)$
in (2.9.2), the duality pairing between $\mathfrak{R}(G_{\eta},G;F)_{1}$ and
$\mathfrak{E}(G_{\eta},G;\mathbb{A}_{F}/F)$ is induced by the products of
$\langle\cdot,\cdot\rangle_{\mathrm{Kott}}$ over places $v$.

Now Corollary 2.9.3 entails that

$\mathcal{Y}_{F}[y_{S}]=\varnothing\implies\sum_{z\in
Z_{\overline{G}^{\vee}}^{\Gamma_{F}}}\prod_{v}\langle[y_{v}],z\rangle_{\mathrm{Kott}}=0$

since $\prod_{v}\langle[y_{v}],\cdot\rangle$ is then a non-trivial character.
This concludes the proof. ∎

The argument above should be compared with its local version in §6.7.

Note that $\left|Z_{\overline{G}^{\vee}}^{\Gamma_{F}}\right|=\tau(G_{\eta})$.
This is a consequence of Kottwitz’s formula for Tamagawa numbers [42,
Corollaire 1.7.4] and the fact
$\operatorname{ker}_{\mathrm{ab}}^{1}(F,G_{\eta})=0$, which has been observed
in the proof of Lemma 11.5.3. As a by-product,
$\tau(G_{\eta})=\tau(G_{\eta[y]})$ for all $y\in\mathcal{Y}_{F}$.

Next, when a class $\mathbf{H}$ is given, we impose relevance conditions on
$\mathcal{Y}_{F}^{S\text{-nr}}$ to define

$\mathcal{Y}_{F}^{S\text{-nr}}(\mathbf{H}):=\bigcup_{y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})}\mathcal{Y}_{F}[y_{S}]\subset\mathcal{Y}_{F}^{S\text{-nr}},$

and let
$\dot{\mathcal{Y}}_{F}^{S\text{-nr}}(\mathbf{H})\subset\dot{\mathcal{Y}}_{F}^{S\text{-nr}}$
be the system of representatives of $G_{\eta}(\overline{F})\times
G(F)$-orbits. Once again, they depend only on the class $\mathbf{H}/\sim$, and
we shall accordingly write
$\dot{\mathcal{Y}}_{F}^{S\text{-nr}}(\mathbf{H}/\sim)$, etc.

Recall (11.4.1). Given $\mathbf{H}/\sim$ in
$E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)/\sim$ and
$y_{S}\in\mathcal{Y}_{S}$, let us denote

$\overline{f}^{\mathbf{H}/\sim}[y_{S}]:=\begin{cases}\overline{f}[y_{S}],&\text{if}\quad
y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H}/\sim)\\\
0,&\text{otherwise}.\end{cases}$

Combining Lemmas 11.4.4, 11.5.3 and Corollary 11.5.7, we obtain

$I^{\tilde{G}_{S}}\left(A^{\tilde{G},\mathcal{E}}(S,\mathbf{H}/\sim)_{\operatorname{ell}},f_{S}\right)\\\
=|W^{G}(\mu)|^{-1}|W^{H}|\cdot\frac{\tau(G)}{\tau(H)}\cdot\tau(G_{\eta[y]})\sum_{y\in\dot{\mathcal{Y}}_{F}^{S\text{-nr}}}S^{H_{S}}\left(SA^{H}_{\mathrm{unip}}(S),\overline{f}^{\mathbf{H}/\sim}[y_{S}]\right);$
(11.5.3)

as usual, $\mathbf{H}=(H,\mathcal{H},\overline{s})$.

###  11.6 Global ascent

The starting point is (11.3.5) and (11.5.3). Summing the latter formula over
all classes $\mathbf{H}/\sim$ and noting $\tau(G)=1$, we obtain

$\begin{gathered}I^{\tilde{G}_{S}}\left(A^{\tilde{G},\mathcal{E}}(S,\mu,\omega)_{\operatorname{ell}},f_{S}\right)=|W^{G}(\mu)|^{-1}\sum_{y\in\dot{\mathcal{Y}}_{F}^{S\text{-nr}}}X[y],\\\
X[y]:=\sum_{\begin{subarray}{c}\mathbf{H}/\sim\\\ \in
E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)/\sim\end{subarray}}\frac{\tau(G_{\eta[y]})}{\tau(H)}\cdot|W^{H}|\cdot
S^{H_{S}}\left(SA^{H}_{\mathrm{unip}}(S),\overline{f}^{\mathbf{H}/\sim}[y_{S}]\right).\end{gathered}$
(11.6.1)

For each $y\in\dot{\mathcal{Y}}_{F}^{S\text{-nr}}$, denote by
$\mathcal{E}_{\operatorname{ell}}(G_{\eta[y]},S)$ the set of equivalence
classes of endoscopic data of $G_{\eta[y]}$ that are unramified outside $S$
and elliptic. There is an evident map

$\displaystyle E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$
$\displaystyle\to\mathcal{E}_{\operatorname{ell}}(G_{\eta[y]},S)$
$\displaystyle\mathbf{H}$ $\displaystyle\mapsto[\mathbf{H}]$

that factors through $\sim$ classes.

######  Lemma 11.6.1.

The map
$\left(E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)/\sim\right)\to\mathcal{E}_{\operatorname{ell}}(G_{\eta[y]},S)$
induced by $\mathbf{H}\mapsto[\mathbf{H}]$ is surjective, and each fiber has
cardinality equal to

$|W^{G_{\eta}}|\cdot|W^{H}|^{-1}|\mathrm{Out}(\mathbf{H})|^{-1}.$

###### Proof.

This assertion has nothing to do with coverings. An easy proof can be found in
[61, p.812]. ∎

On the other hand, the coefficients in the stabilization of the trace formula
for $G_{\eta[y]}$ read

$\iota(G_{\eta[y]},H)=\frac{\tau(G_{\eta[y]})}{\tau(H)}\cdot|\mathrm{Out}(\mathbf{H})|^{-1}$
(11.6.2)

for every $[\mathbf{H}]\in\mathcal{E}_{\operatorname{ell}}(G_{\eta[y]},S)$.

Before proceeding to the next step, note that for all $\mathbf{H}\in
E_{T^{\vee},\operatorname{ell}}(G_{\eta},S)$ and
$y\in\dot{\mathcal{Y}}_{F}^{S\text{-nr}}(\mathbf{H})$, say
$y\in\mathcal{Y}_{F}[y_{S}]$ for some
$y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$, the data defining
$\Delta_{\mathbf{H}}[y_{S}]$ are all global, and everything is unramified
outside $S$ relative to $K_{v}\cap G_{\eta}(F_{v})$. This implies that
$\Delta_{\mathbf{H}}[y_{S}]$ depends only on $[\mathbf{H}]$ and $y$; in fact,
it is the global transfer factor mentioned in §4.2.4. Therefore we can write
$\check{\mathcal{T}}_{\mathbf{H},G_{\eta[y]}}$ unambiguously, for each $y$ and
$[\mathbf{H}]$; it is the $\otimes$-product over $v\in S$ of local transfers.

On the other hand, the same $\otimes$-product over $v\in S$ yields
$\check{\mathcal{T}}_{\mathbf{H},G_{\eta[y]}}(\cdot)=0$ when
$y_{S}\in\mathcal{Y}_{S}\smallsetminus\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H})$.

###### Proof of Theorem 10.5.4.

Continue the foregoing discussions. Substituting Lemma 11.6.1 and (11.6.2)
into (11.6.1), we see

$\displaystyle X[y]$
$\displaystyle=|W^{G_{\eta}}|\sum_{[\mathbf{H}]\in\mathcal{E}_{\operatorname{ell}}(G_{\eta[y]},S)}\iota(G_{\eta[y]},H)I^{G_{\eta[y],S}}\left(\check{\mathcal{T}}_{\mathbf{H},G_{\eta[y]}}SA^{H}_{\mathrm{unip}}(S),f[y_{S}]\right)$
$\displaystyle=|W^{G}(\mu)|\cdot
I^{G_{\eta[y],S}}\left(\sum_{[\mathbf{H}]\in\mathcal{E}_{\operatorname{ell}}(G_{\eta[y]},S)}\iota(G_{\eta[y]},H)\check{\mathcal{T}}_{\mathbf{H},G_{\eta[y]}}SA^{H}_{\mathrm{unip}}(S),\;f[y_{S}]\right),$

where $f[y_{S}]:=\prod_{v\in S}f_{v}[y_{v}]$ as in (11.4.1), with
$f_{v}[y_{v}]:=\operatorname{desc}^{\tilde{G}}_{\tilde{\eta}[y_{v}]}(f_{v})$.
Also recall that

$\check{\mathcal{T}}_{\mathbf{H},G_{\eta[y]}}SA^{H}_{\mathrm{unip}}(S)\neq
0\implies y_{S}\in\mathcal{Y}_{S}^{\mathrm{rel}}(\mathbf{H}).$

Our normalization of transfer factors $\Delta_{\mathbf{H}}[y_{S}]$ is
compatible with [61, VI.3]. Now apply Arthur’s theorem on the matching of
coefficients in [4, p.239, Global Theorem 1] to arrive at

$X[y]=|W^{G}(\mu)|\cdot
I^{G_{\eta[y],S}}\left(A^{G_{\eta[y]}}_{\mathrm{unip}}(S),f[y_{S}]\right).$

We are going to ascend to $\tilde{G}$. Note that for
$y\in\dot{\mathcal{Y}}_{F}^{S\text{-nr}}$, according to [48, Lemme 4.1.7], the
product over local stable conjugates $\tilde{\eta}_{v}[y_{v}]\in\tilde{G}_{v}$
coincides with $\widetilde{\eta[y]}_{S}$, the component in the decomposition
$\eta[y]=\widetilde{\eta[y]}_{S}\eta[y]^{S}$ in (10.4.6). For discussions on
how $\eta[y]^{S}$, an ordinary conjugate of $\eta^{S}\in K^{S}$, is lifted to
$\tilde{G}^{S}$, see the last part of Remark 10.4.8.

For each $y\in\mathcal{Y}_{F}$, let $\mathcal{O}_{y}\subset G(F)$ denote the
semisimple $G(F)$-conjugacy class containing $\eta[y]$. Taking account of
Proposition 10.4.11, now (11.6.1) becomes

$I^{\tilde{G}_{S}}\left(A^{\tilde{G},\mathcal{E}}(S,\mu,\omega)_{\operatorname{ell}},f_{S}\right)=\sum_{y\in\dot{\mathcal{Y}}_{F}^{S\text{-nr}}}I^{\tilde{G}_{S}}\left(A^{\tilde{G}}(S,\mathcal{O}_{y})_{\operatorname{ell}},f_{S}\right).$

We claim that the sum above equals the same sum over the larger set
$\dot{\mathcal{Y}}_{F}$. Indeed, by §10.4.3,
$A^{\tilde{G}}(S,\mathcal{O}_{y})\neq 0$ only when $\mathcal{O}_{y}$ cuts
$K^{S}$, so

$y\in\dot{\mathcal{Y}}_{F}\smallsetminus\dot{\mathcal{Y}}_{F}^{S\text{-nr}}\implies
A^{\tilde{G}}(S,\mathcal{O}_{y})_{\operatorname{ell}}=0$

by the definition of $\mathcal{Y}_{v}^{\mathrm{nr}}$.

The enlarged sum is thus over all $G(F)$-conjugacy classes inside the stable
class $\mathcal{O}$. By varying $f_{S}$, we obtain
$A^{\tilde{G},\mathcal{E}}(S,\mu,\omega)_{\operatorname{ell}}=A^{\tilde{G}}(S,\mathcal{O})_{\operatorname{ell}}$.
This is exactly (11.2.3). ∎

##  Chapter 12 Stabilization of local trace formula

Let $F$ be a local field of characteristic zero. Consider a covering of
metaplectic type $\mathbf{p}:\tilde{G}\to G(F)$ where $G=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W)$, with chosen
$(W,\langle\cdot|\cdot\rangle)$, symplectic basis and non-trivial additive
character $\psi$ of $F$.

The invariant local trace formula for $\tilde{G}$ established in [46] is cast
into the form

$I_{\mathrm{geom}}(\overline{f_{1}},f_{2})=I_{\mathrm{disc}}(\overline{f_{1}},f_{2}),\quad
f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G).$

Philosophically, $\overline{f_{1}}$ should be viewed as an element of
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}^{\dagger})\otimes\operatorname{mes}(G)$,
where $\tilde{G}^{\dagger}$ is the _antipodal covering_ of $\tilde{G}$. This
means that $\tilde{G}^{\dagger}=\tilde{G}$ as topological groups, but the
embeddings of $\bm{\mu}_{8}$ differ by the automorphism $z\mapsto z^{-1}$.

In §12.1, we will show that when $G=\operatorname{Sp}(W)$, there is a
canonical isomorphism between $\tilde{G}^{\dagger}$ and the covering group
associated with $(W,-\langle\cdot|\cdot\rangle,\psi)$, as topological central
extensions of $G(F)$ by $\bm{\mu}_{8}$. Using this property, we can relate the
metaplectic transfer factors for $\tilde{G}$ and $\tilde{G}^{\dagger}$, then
show that

$I_{\tilde{M}^{\dagger}}(\tilde{\gamma},\overline{f})=\overline{I_{\tilde{M}}(\tilde{\gamma},f)},\quad
I^{\mathcal{E}}_{\tilde{M}^{\dagger}}(\tilde{\gamma},\overline{f})=\overline{I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)}$

for all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

The structure of the invariant local trace formula is reviewed in §12.2. In
§§12.3–12.4, we define endoscopic counterparts for the distributions in the
invariant local trace formula. The stabilization of local trace formula will
be a consequence of the $\tilde{G}$-regular case of the local geometric
Theorem 5.3.11.

On the other hand, given $M\in\mathcal{L}(M_{0})$, assume inductively that the
$\tilde{G}$-regular case of Theorem 5.3.11 is known when $M$ (resp. $G$) is
replaced by a larger (resp. proper) Levi subgroup, we will formulate a key
geometric Hypothesis 12.5.1, which produces a smooth function
$\epsilon(\mathbf{M}^{!},\cdot)$ on $M^{!}_{G\text{-reg}}(F)$ for all
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, such that
$\epsilon_{\tilde{M}}=0$ if and only if $\epsilon(\mathbf{M}^{!},\cdot)=0$ for
all $\mathbf{M}^{!}$. Then we will prove that

  * •

$\epsilon(\mathbf{M}^{!},\cdot)+\overline{\epsilon(\mathbf{M}^{!},\cdot)}=0$
by using the local trace formula (Lemma 12.5.4);

  * •

$\epsilon(\mathbf{M}^{!},\cdot)=\overline{\epsilon(\mathbf{M}^{!},\cdot)}$ by
showing that “the endoscopic transfer is isomorphic to its complex conjugate”,
via the MVW-involution on metaplectic groups (Lemma 12.5.5).

The combination of these facts implies $\epsilon_{\tilde{M}}=0$.

This reduces both the stabilization of local trace formula and the local
geometric Theorem 5.3.11 to the key geometric Hypothesis 12.5.1. That
hypothesis will ultimately be settled by a local–global argument in §13.8,
after proving a special case of global spectral stabilization; the notion of
$\tilde{M}$-cuspidality introduced in Definition 12.4.6 will continue to play
a key role there.

###  12.1 The antipodal covering

To begin with, consider a general covering

$1\to\bm{\mu}_{m}\to\tilde{G}\xrightarrow{\mathbf{p}}G(F)\to 1$

as in §2.2, where $G$ is any connected reductive $F$-group and
$m\in\mathbb{Z}_{\geq 1}$.

######  Definition 12.1.1.

The antipodal covering $\tilde{G}^{\dagger}$ of $\tilde{G}$ is defined as its
push-out via the map
$\bm{\mu}_{m}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bm{\mu}_{m}$ given
by $z\mapsto z^{-1}$. In other words,

$\tilde{G}^{\dagger}=\dfrac{\bm{\mu}_{m}\times\tilde{G}}{(zw,\tilde{x})\sim(z,w^{-1}\tilde{x})},\quad
z,w\in\bm{\mu}_{m},\;\tilde{x}\in\tilde{G},$

and $\bm{\mu}_{m}\hookrightarrow\tilde{G}^{\dagger}$ (resp.
$\tilde{G}^{\dagger}\to G(F)$) is given by $z\mapsto[z,1]$ (resp.
$[z,\tilde{x}]\mapsto\mathbf{p}(\tilde{x})$).

Alternatively, it is often convenient to identify $\tilde{G}^{\dagger}$ with
$\tilde{G}$, with the same $\mathbf{p}$, but the embedding
$\bm{\mu}_{m}\hookrightarrow\tilde{G}$ is modified by $z\mapsto z^{-1}$.

Recalling the definition in §2.3 of genuine functions (resp. anti-genuine
representations) over $\tilde{G}$, we have

$C^{\infty}_{-}(\tilde{G})=C^{\infty}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}^{\dagger}),\quad\Pi_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})=\Pi_{-}(\tilde{G}^{\dagger}).$

Also, by writing $f\mapsto\overline{f}$ for the complex conjugation,

$C^{\infty}_{-}(\tilde{G})=\left\\{\overline{f}:f\in
C^{\infty}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\right\\}.$

Ditto for $\mathcal{I}_{-}(\tilde{G})$ and so forth. The contragredient (resp.
complex conjugate) of a genuine representation $\pi$ of $\tilde{G}$ is a
genuine representation of $\tilde{G}^{\dagger}$.

These operations are obviously compatible with inclusion of Levi subgroups
$\tilde{M}\hookrightarrow\tilde{G}$. The maps
$C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})\twoheadrightarrow\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$
and
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
are readily seen to be “defined over $\mathbb{R}$”, hence commutes with
complex conjugation.

Hereafter, we return to the case where $\mathbf{p}:\tilde{G}\to G(F)$ is a
coverings of metaplectic type; in particular, we take $m=8$. What is special
about coverings of metaplectic type is the relation between antipodals and
endoscopy. Without loss of generality, let us assume
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$. Recall that the covering is determined
by $\psi\circ\langle\cdot|\cdot\rangle:W\times W\to\mathbb{S}^{1}$.

######  Definition–Proposition 12.1.2.

Let $\tilde{G}_{-}$ denote the covering of $\operatorname{Sp}(W)$ obtained
from the datum $(W,\langle\cdot|\cdot\rangle,\psi^{-1})$, or equivalently from
$(W,-\langle\cdot|\cdot\rangle,\psi)$. There is then a unique isomorphism
$\Xi:\tilde{G}^{\dagger}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\tilde{G}_{-}$
of topological groups, such that the following diagram commutes:

${1}$${\bm{\mu}_{8}}$${\tilde{G}^{\dagger}}$${G(F)}$${1}$${1}$${\bm{\mu}_{8}}$${\tilde{G}_{-}}$${G(F)}$${1.}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\Xi}$$\scriptstyle{\mathrm{id}}$

Equivalently, by identifying $\tilde{G}^{\dagger}$ with $\tilde{G}$ as groups,
the diagram can also be rendered as

${1}$${\bm{\mu}_{8}}$${\tilde{G}}$${G(F)}$${1}$${1}$${\bm{\mu}_{8}}$${\tilde{G}_{-}}$${G(F)}$${1.}$$\scriptstyle{z\mapsto
z^{-1}}$$\scriptstyle{\Xi}$$\scriptstyle{\mathrm{id}}$

###### Proof.

Uniqueness is straightforward: any two such isomorphisms of coverings differ
by a homomorphism $G(F)\to\bm{\mu}_{8}$, which must be trivial.

As for the existence, it suffices to deal with the second diagram. Fix a
polarization $W=\ell\oplus\ell^{\prime}$ where $\ell$ and $\ell^{\prime}$ are
Lagrangian subspaces. Recall the description of $\tilde{G}$ in terms of the
Maslov cocycle: according to [65] and [52, Théorème 2.6], it arises from the
section

$G(F)\to\tilde{G},\quad x\mapsto(x,M_{\ell}[x])$

in the notation of loc. cit., where $M_{\ell}[x]$ is some endomorphism of the
Schwartz space $\mathcal{S}(\ell^{\prime})$, and $\bm{\mu}_{8}$ acts by
dilation on the second component. Multiplication is described by

$(x,M_{\ell}[x])(y,M_{\ell}[y])=\gamma_{\psi}(\underbracket{\tau(\ell,y\ell,xy\ell)}_{\text{Maslov
index}})\left(xy,M_{\ell}[xy]\right).$

Recall from loc. cit. that the Maslov index here is understood as a class in
the Witt group of quadratic $F$-vector spaces. Since

$\gamma_{\psi^{-1}}(q)=\gamma_{\psi}(-q)=\gamma_{\psi}(q)^{-1},\quad(V,q):\text{any
quadratic $F$-vector space},$

we see that replacing $\psi$ (resp. $\langle\cdot|\cdot\rangle$) by
$\psi^{-1}$ (resp. $-\langle\cdot|\cdot\rangle$) has the effect of flipping
the Maslov cocycle, which yields $\tilde{G}^{\dagger}$. This concludes the
proof. ∎

Note that $\tilde{G}$ and $\tilde{G}_{-}$ have the same endoscopic data, and
the correspondence of semisimple classes is also the same. What differs is the
transfer factor.

######  Lemma 12.1.3.

For every
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\mathbf{G}^{!}$, we have $\overline{f}^{G^{!}}=\overline{f^{G^{!}}}$
where $f^{G^{!}}:=\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}(f)$ and
$\overline{f}^{G^{!}}:=\mathcal{T}_{\mathbf{G}^{!},\tilde{G}^{\dagger}}(\overline{f})$.

###### Proof.

The transfer is characterized by

$f^{G^{!}}(\delta)=\sum_{\gamma\in\Gamma_{\text{reg}}(G)}\Delta_{\mathbf{G}^{!},\tilde{G}}(\delta,\tilde{\gamma})f(\tilde{\gamma})$

for all $\delta\in\Sigma_{G\text{-reg}}(G^{!})$. Using
$\Xi:\tilde{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\tilde{G}_{-}$ and
taking complex conjugates, the assertion reduces to the claim that

$\Delta_{\mathbf{G}^{!},\tilde{G}_{-}}(\delta,\Xi(\tilde{\gamma}))=\overline{\Delta_{\mathbf{G}^{!},\tilde{G}}(\delta,\tilde{\gamma})}.$
(12.1.1)

Let us write $\tilde{\gamma}=z(\gamma,M_{\ell}[\gamma])$ as in the proof of
Definition–Proposition 12.1.2, where $\ell\subset W$ is Lagrangian and
$z\in\bm{\mu}_{8}$. By construction,
$\Xi(\tilde{\gamma})=z^{-1}(\gamma,M_{\ell}[\gamma])$, but everything is now
taken with respect to $\psi$ and $(W,-\langle\cdot|\cdot\rangle)$. To prove
(12.1.1), we may assume $z=1$.

By [52, §5.3], the only ingredient in $\Delta$ affected by flipping
$\langle\cdot|\cdot\rangle$ is the character of Weil representation
$\omega_{\psi}^{+}\oplus\omega_{\psi}^{-}$; or more precisely its value at
regular semisimple elements divided by $|\cdots|$. This value lies in
$\bm{\mu}_{8}$. In Maktouf’s character formula [52, Corollaire 4.4], this is
given through Weil’s constant $\gamma_{\psi}(\cdot)$ attached to a Maslov
index $\tau(\Gamma_{\gamma},\Gamma_{1},\ell\oplus\ell)$, which gets multiplied
by $-1$ if $\langle\cdot|\cdot\rangle$ is replaced by
$-\langle\cdot|\cdot\rangle$. As $\gamma_{\psi}(-q)=\gamma_{\psi}(q)^{-1}$ for
any quadratic $F$-vector space $(V,q)$, we obtain (12.1.1). ∎

######  Proposition 12.1.4.

For all $M\in\mathcal{L}(M_{0})$, $\tilde{\gamma}\in\tilde{M}$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
Fix Haar measures needed to define weighted orbital integrals, and identify
$\tilde{\gamma}$ with an element of $\tilde{M}^{\dagger}$. Then

$I_{\tilde{M}^{\dagger}}(\tilde{\gamma},\overline{f})=\overline{I_{\tilde{M}}(\tilde{\gamma},f)},\quad
I^{\mathcal{E}}_{\tilde{M}^{\dagger}}(\tilde{\gamma},\overline{f})=\overline{I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)}.$

In fact, the first equation holds for coverings in general.

###### Proof.

Consider the first equality. It is clear that
$J_{\tilde{M}^{\dagger}}(\tilde{\gamma},\overline{f})=\overline{J_{\tilde{M}}(\tilde{\gamma},f)}$,
as the weighted orbital integrals are defined over $\mathbb{R}$. To show the
equality, it suffices to show that
$\phi_{\tilde{L}^{\dagger}}(\overline{f})=\overline{\phi_{\tilde{L}}(f)}$ for
all Levi subgroup $L$, or equivalently,

$J_{\tilde{L}^{\dagger}}(\overline{\pi},\overline{f})=\overline{J_{\tilde{L}}(\pi,f)}$

for all $\pi\in\Pi_{\mathrm{temp},-}(\tilde{L})$, with complex conjugate
$\overline{\pi}\simeq\check{\pi}$.

Recall the definition of canonically normalized tempered weighted character in
[46, §5.7]: fix $P\in\mathcal{P}(L)$,

$\displaystyle J_{\tilde{L}}(\pi,f)$
$\displaystyle:=\operatorname{tr}\left(\mathcal{M}_{\tilde{L}}(\pi,\tilde{P})I_{\tilde{P}}(\pi,f)\right),$
$\displaystyle\mathcal{M}_{\tilde{L}}(\pi,\tilde{P})$
$\displaystyle:=\lim_{\Lambda\to
0}\sum_{Q\in\mathcal{P}(L)}\mathcal{M}_{\tilde{Q}}(\Lambda,\pi,\tilde{P})\theta_{Q}(\Lambda)^{-1},\quad\Lambda\in\mathfrak{a}^{*}_{L,\mathbb{C}},$
$\displaystyle\mathcal{M}_{\tilde{Q}}(\Lambda,\pi,\tilde{P})$
$\displaystyle:=\mu_{\tilde{Q}|\tilde{P}}(\pi)^{-1}\mu_{\tilde{Q}|\tilde{P}}(\pi_{\Lambda/2})J_{\tilde{Q}|\tilde{P}}(\pi)^{-1}J_{\tilde{Q}|\tilde{P}}(\pi_{\Lambda}).$

All these objects are well-defined upon perturbing $\pi$ by
$i\mathfrak{a}^{*}_{L,F}$.

  * •

The standard intertwining operators are defined over $\mathbb{R}$ in the sense
that
$J_{\tilde{Q}^{\dagger}|\tilde{P}^{\dagger}}(\overline{\pi_{\Lambda}})=\overline{J_{\tilde{Q}|\tilde{P}}(\pi_{\Lambda})}$
for all $\Pi_{-}(\tilde{L})$, which can be checked by looking at the integrals
in the range of convergence, as both sides are anti-holomorphic in $\Lambda$.

  * •

So are the Harish-Chandra $\mu$-functions: see [46, Proposition 2.4.3].

  * •

Finally,
$\theta_{Q}(\Lambda):=\mathrm{vol}(\mathfrak{a}^{G}_{L}/\mathbb{Z}\Delta_{Q}^{\vee})^{-1}\prod_{\alpha\in\Delta_{Q}}\langle\Lambda,\check{\alpha}\rangle$
is defined over $\mathbb{R}$.

The required identity for $\phi_{\tilde{L}^{\dagger}}$ follows at once. These
arguments apply to coverings in general.

There are also stable counterparts of the first equality for
$S^{G^{!}}_{M^{!}}(\delta,f^{!})$, where $G^{!}$ (resp. $M^{!}$) is an
elliptic endoscopic group of $\tilde{G}$ (resp. $\tilde{M}$). This is implicit
in [15, §9], and follows by a standard inductive argument.

To deduce the second equality, it suffices to combine the previous
observation, Lemma 12.1.3 and the first equality. ∎

The coverings $\tilde{G}$ and $\tilde{G}_{-}$ or $\tilde{G}^{\dagger}$ are
related by an explicitly constructed isomorphism. Denote by
$\operatorname{GSp}(W)$ the group of symplectic similitudes on $W$, and by
$\nu:\operatorname{GSp}(W)\to\mathbb{G}_{\mathrm{m}}$ the similitude
character.

######  Proposition 12.1.5 ([57, p.36]).

Take any $g\in\operatorname{GSp}(W)$ with $\nu(g)=-1$. Then there exists an
isomorphism
$\Theta_{g}:\tilde{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\tilde{G}_{-}$
of topological groups making the diagram below commutative:

${1}$${\bm{\mu}_{8}}$${\tilde{G}}$${G(F)}$${1}$${1}$${\bm{\mu}_{8}}$${\tilde{G}_{-}}$${G(F)}$${1.}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\Theta_{g}}$$\scriptstyle{\operatorname{Ad}(g)}$

###### Proof.

In the cited reference, this is stated for the coverings obtained by pushing-
out along $\bm{\mu}_{8}\hookrightarrow\mathbb{C}^{\times}$. Choose any
Lagrangian subspace $\ell$ of $W$ with respect to $\langle\cdot|\cdot\rangle$,
or equivalently for $-\langle\cdot|\cdot\rangle$. Using the construction of
$\tilde{G}$ (or $\tilde{G}_{-}$) via Schrödinger models (see the proof of
Definition–Proposition 12.1.2), the map in loc. cit. is simply
$(x,M_{\ell}[x])\mapsto(\operatorname{Ad}(g)x,M_{\ell}[x])$. Thus it restricts
to $\tilde{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\tilde{G}_{-}$. ∎

The isomorphism above depends on the choice of $g$, but only up to conjugacy:
$\Theta_{h^{\prime}gh}=\operatorname{Ad}(h^{\prime})\Theta_{g}\operatorname{Ad}(h)$
for all $h,h^{\prime}\in G(F)$. It is customary to fix $g$ and use the
shorthand $\Theta=\Theta_{g}$. The following result will also be needed.

######  Lemma 12.1.6.

Given $M\in\mathcal{L}(M_{0})$, written as $M=\prod_{j\in
J}\operatorname{GL}(m_{j})\times\operatorname{Sp}(W^{\flat})$, one can choose
$g\in\operatorname{GSp}(W)$ with $\nu(g)=-1$ such that
$\operatorname{Ad}(g)(M)=M$, and that $\operatorname{Ad}(g)$ acts as identity
(resp. as in Proposition 12.1.5) on each $\operatorname{GL}(m_{j})$ (resp. on
$\operatorname{Sp}(W^{\flat})$).

Such $\Theta_{g}$ is said to be _adapted to $\tilde{M}$_.

###### Proof.

Describe $M$ by a decomposition
$W=W^{\flat}\oplus\bigoplus_{j=1}^{r}(\ell_{j}\oplus\ell^{\prime}_{j})$ where
$W^{\flat}$ and $\ell_{j}\oplus\ell^{\prime}_{j}$ are symplectic vector
subspaces of $W$, and each $\ell_{j}$ is totally isotropic of dimension
$m_{j}$. Then $M\simeq\prod_{j\in
J}\operatorname{GL}(\ell_{j})\times\operatorname{Sp}(W^{\flat})$. Take
$g=\left((g_{j})_{j\in J},g^{\flat}\right)\in\operatorname{GL}(W)$ such that
$g^{\flat}\in\operatorname{GSp}(W^{\flat})$ satisfies $\nu(g^{\flat})=-1$, and

$g_{j}=\begin{pmatrix}\mathrm{id}_{\ell_{j}}&\\\
&-\mathrm{id}_{\ell_{j}}\end{pmatrix}\in\operatorname{GL}(\ell_{j}\oplus\ell^{\prime}_{j})$

for all $j\in J$. Then $\nu(g)=-1$ with respect to
$(W,\langle\cdot|\cdot\rangle)$, and $g$ has the desired behavior. ∎

By choosing $g$ in this way, $\Theta_{g}$ can be restricted to $\tilde{M}$.
The proof above actually implies that $g$ can be chosen so that
$\Theta_{g}^{2}=\mathrm{id}$.

######  Remark 12.1.7.

In view of the previous result, one can define the automorphisms $\Theta$ for
any group of metaplectic type, by requiring them to be inner on each
$\operatorname{GL}$-factor. Nonetheless, what matters in §12.5 is just the
$W^{G}(M)$-coset of $\Theta|_{\tilde{M}}$ modulo inner automorphisms. In view
of the more general scenario, in which $\operatorname{Ad}(g)$ is to be
replaced by a Chevalley involution of $G$, it is probably better to require
that $\Theta$ is the transpose-inverse on each $\operatorname{GL}$-factor. In
the case of $G=\operatorname{Sp}(W)$ and its Levi subgroup $M$, transpose-
inverses are realizable by $W^{G}(M)$, so this does not affect the upcoming
applications.

###  12.2 Local trace formulas

We are going to state the invariant local trace formula for $\tilde{G}$ in the
form of [46, §5.8]. Subsequently, we will review the stable version for the
endoscopic groups of $\tilde{G}$.

####  12.2.1 Invariant local trace formula

When talking about normalized intertwining operators and the resulting
weighted characters, etc., the normalizing factors for genuine representations
of $\tilde{G}$ and $\tilde{G}^{\dagger}$ are taken to be _complementary_ ; see
[12, p.85] and [46, p.849]. Specifically, we impose

$r_{\tilde{Q}|\tilde{P}}(\pi^{\vee})=r_{\tilde{P}|\tilde{Q}}(\pi),\quad\pi\in\Pi_{\mathrm{unit},-}(\tilde{M}),\quad
P,Q\in\mathcal{P}(M),$

where $\pi^{\vee}$ is viewed as an element of
$\Pi_{\mathrm{unit},-}(\tilde{M}^{\dagger})$. With this choice, the $R$-groups
and their central extensions in §2.6 satisfy

$\tilde{R}_{\pi^{\vee}}\simeq\tilde{R}_{\pi},\quad R_{\pi^{\vee}}\simeq
R_{\pi}$ (12.2.1)

canonically; see [12, p.92, p.95].

Such normalizing factors for $\tilde{G}^{\dagger}$ were called _weak
normalizing factors_ in §3.1 of loc. cit., as the condition (R7) is not
assumed.

For each $M\in\mathcal{L}(M_{0})$, we define the spaces $T_{-}(\tilde{M})$,
$T_{\operatorname{ell},-}(\tilde{M})$, etc. and the distributions
$\Theta_{\tau}$ using the normalizing factors; see §2.6. The anti-genuine
versions $T_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})$, etc. are defined using
the complementary factors, and will be identified with
$T_{-}(\tilde{M}^{\dagger})$, etc.

######  Definition 12.2.1.

The map

$\displaystyle\tilde{T}_{-}(\tilde{G})$
$\displaystyle\to\tilde{T}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})=\tilde{T}_{-}(\tilde{G}^{\dagger})$
$\displaystyle(M,\pi,r)$ $\displaystyle\mapsto(M,\pi^{\vee},r),$

is well-defined in view of (12.2.1). It factors as
$T_{-}(\tilde{G})\to\tilde{T}_{-}(\tilde{G}^{\dagger})$ as well as their
quotients by $\mathbb{S}^{1}$; the resulting maps will all be denoted as
$\tau\mapsto\tau^{\vee}$.

In this manner, one gets $\Theta_{\tau^{\vee}}\in
D_{\mathrm{spec},-}(\tilde{G}^{\dagger})\otimes\operatorname{mes}(G)^{\vee}=D_{\mathrm{spec},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$
from each $\tau\in T_{-}(\tilde{G})$.

Denote by $\mathscr{C}(\tilde{G})$ the Schwartz–Harish–Chandra space of
$\tilde{G}$, and denote by $\mathscr{C}_{-}(\tilde{G})$ (resp.
$\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$) its genuine (resp. anti-
genuine) part. Their quotient spaces $I\mathscr{C}_{-}$ and
$I\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}$ are defined by the familiar recipe.
The invariant local trace formula for $\tilde{G}$ is the following
identity.111In [46, p.851], one takes test functions from
$\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}\times\tilde{G})\subset\mathscr{C}(\tilde{G}\times\tilde{G})$,
and $f_{1}\otimes f_{2}$ belongs to this space. An obvious typo appeared
there: the subscript $\dashrule[.7ex]{2222}{.4}$ should mean the space of
functions that are anti-genuine under $z\mapsto(z^{-1},1)$ and
$z\mapsto(1,z)$.

######  Theorem 12.2.2 ([46, Théorème 5.8.6]).

For all $h_{1}\in I\mathscr{C}_{-}(\tilde{G})\otimes\operatorname{mes}(G)$ and
$h_{2}\in
I\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
we have

$I_{\mathrm{geom}}(h_{1},h_{2})=I_{\mathrm{disc}}(h_{1},h_{2}).$

Both sides are separately continuous in $(h_{1},h_{2})$ with respect to the
topology on $\mathscr{C}(\tilde{G})$; this is proved in [46, Proposition
5.7.6]. The equality will also be written as
$I^{\tilde{G}}_{\mathrm{geom}}=I^{\tilde{G}}_{\mathrm{disc}}$ in case of
possible confusions. The distributions are explicated below.

Geometric side

For each $M\in\mathcal{L}(M_{0})$, equip $A_{M}(F)$ with the Haar measure
determined by the invariant positive-definite quadratic form on
$\mathfrak{a}_{M}\subset\mathfrak{a}_{0}$. For each elliptic maximal torus
$T\subset M$, use the Haar measure such that
$\operatorname{mes}(T(F)/A_{M}(F))=1$. We set

$\displaystyle I_{\tilde{M},\mathrm{geom}}(h_{1},h_{2})$
$\displaystyle=I^{\tilde{G}}_{\tilde{M},\mathrm{geom}}(h_{1},h_{2})$
$\displaystyle:=\int_{\Gamma_{G\text{-reg,
ell}}(M)}I_{\tilde{M}}\left(\gamma,h_{1},h_{2}\right)\mathop{}\\!\mathrm{d}\gamma$

where $\Gamma_{G\text{-reg, ell}}(M)$ is equipped with the measure à the Weyl
integration formula, and
$I_{\tilde{M}}\left(\gamma,h_{1},h_{2}\right)=I_{\tilde{M}}\left(\gamma,h_{1}\otimes
h_{2}\right)$ is defined in [46, p.852] using the canonically normalized
invariant weighted characters. Here we follow the paradigm of the semi-local
distributions $I_{\tilde{M}_{V}}(\tilde{\gamma},\cdots)$ in §10.3, by imaging
that $V=\\{v_{1},v_{2}\\}$ with the same local fields $F$. The new feature
here is that it depends only on $\gamma$ and not on
$\tilde{\gamma}\in\mathbf{p}^{-1}(\gamma)$.

Now set

$I_{\mathrm{geom}}(h_{1},h_{2}):=\sum_{M\in\mathcal{L}(M_{0})}\frac{|W^{M}_{0}|}{|W^{G}_{0}|}(-1)^{\dim\mathfrak{a}^{G}_{M}}I_{\tilde{M},\mathrm{geom}}(h_{1},h_{2}).$

Compared with the general setting in p.823 of loc. cit., here we have
$A_{M}(F)^{\dagger}:=A_{M}(F)$ and $\iota(\gamma)=1$.

Spectral side

We have

$I_{\mathrm{disc}}(h_{1},h_{2}):=\int_{\mathbb{S}^{1}\backslash
T_{\mathrm{disc},-}(\tilde{G})}i(\tau)\Theta_{\tau^{\vee}}(h_{1})\Theta_{\tau}(h_{2})\mathop{}\\!\mathrm{d}\tau,$

where we choose representatives $\tau\in T_{\mathrm{disc},-}(\tilde{G})$ for
elements of $\mathbb{S}^{1}\backslash T_{\mathrm{disc},-}(\tilde{G})$. We
refer to §2.7 for the definition of $T_{\mathrm{disc},-}(\tilde{G})\subset
T_{-}(\tilde{G})$ and the Radon measure on $\mathbb{S}^{1}\backslash
T_{\mathrm{disc},-}(\tilde{G})$.

For $\tau=(L,\pi,r)$ and $P\in\mathcal{P}(L)$, the function $i(\tau)$
appearing above is given by

$i(\tau)=i^{\tilde{G}}(\tau):=\sum_{t\in W_{\pi}^{0}r\cap
W^{G}(L)_{\mathrm{reg}}}\epsilon_{\sigma}(t)\left|\det\left(1-t\middle|\mathfrak{a}^{G}_{L}\right)\right|^{-1},$

where $\epsilon_{\sigma}(t)=(-1)^{\ell(w_{0})}$ if $t=w_{0}r$ with $w_{0}\in
W_{\pi}^{0}$ and $r\in R_{\pi}$; see §2.7. The sign $\ell(w_{0})$ is defined
since by the theory of $R$-groups, $W_{\pi}^{0}$ is known to be the Weyl group
of the root system generated
$\left\\{\beta\in\Sigma_{P}^{\mathrm{red}}:\mu_{\beta}(\pi)=0\right\\}$.

It is convenient to set $i(\tau)=0$ if $\tau\notin\mathbb{S}^{1}\backslash
T_{\mathrm{disc},-}(\tilde{G})$. Note that the coefficients $i(\tau)$ are
invariant under $\mathbb{S}^{1}$ and $i\mathfrak{a}^{*}_{G,F}$-twists.

######  Remark 12.2.3.

This is the formula considered in [6]. It differs from the “symmetric” version
utilized in §9.9.

What we will use is a slightly different formulation. The distributions above
can also be written as

$I_{\mathrm{geom}}(\overline{f_{1}},f_{2}),\quad
I_{\mathrm{disc}}(\overline{f_{1}},f_{2}),\quad f_{1},f_{2}\in
I\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}).$ (12.2.2)

Let us relate them with the genuine distributions on $\tilde{G}$ already
defined. Consider the geometric side. Take any
$\tilde{\gamma}\in\mathbf{p}^{-1}(\tilde{\gamma})$ for a representative of
$\gamma\in\Gamma_{G\text{-reg, ell}}(M)$ in $M(F)$. There is a splitting
formula

$I_{\tilde{M}}(\gamma,h_{1},h_{2})=\sum_{L_{1},L_{2}\in\mathcal{L}(M)}d^{G}_{M}(L_{1},L_{2})I^{\tilde{L}_{1}}_{\tilde{M}}(\tilde{\gamma},h_{1,\tilde{L}_{1}})I^{\tilde{L}_{2}}_{\tilde{M}}(\tilde{\gamma},h_{2,\tilde{L}_{2}});$

see the discussions in §10.3 or the proof of [46, Corollaire 5.8.7]. Note that
the right hand side does not depend on the choice of $\tilde{\gamma}$. Since
$d^{G}_{M}(L_{1},L_{2})\in\mathbb{R}_{\geq 0}$, whilst
$\mathscr{C}(\tilde{G})\to I\mathscr{C}(\tilde{G})$ and $h_{i}\mapsto
h_{i,\tilde{L}_{i}}$ are both “defined over $\mathbb{R}$”, the first equality
of Proposition 12.1.4 (extended to Schwartz–Harish-Chandra spaces by
continuity) yields

$I_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})=\sum_{L_{1},L_{2}\in\mathcal{L}(M)}d^{G}_{M}(L_{1},L_{2})\overline{I^{\tilde{L}_{1}}_{\tilde{M}}(\tilde{\gamma},f_{1,\tilde{L}_{1}})}I^{\tilde{L}_{2}}_{\tilde{M}}(\tilde{\gamma},f_{2,\tilde{L}_{2}})$

for all $f_{1},f_{2}\in I\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$.

The situation on the spectral side is similar. Consider
$\tau=(M,\pi,r)\in\tilde{T}_{-}(\tilde{G})$ and fix $P\in\mathcal{P}(M)$. For
every $\tilde{x}\in\tilde{G}$, the endomorphism
$I_{\tilde{P}}(\pi^{\vee},\tilde{x})$ equals the transpose-inverse of
$I_{\tilde{P}}(\pi,\tilde{x})$. On the other hand, the formula (2.1) in [12,
p.86] says that the normalized intertwining operator
$R_{\tilde{P}}(r,\pi^{\vee})$ equals the transpose-inverse of
$R_{\tilde{P}}(r,\pi)$. All these operators are unitary. Hence the definition
of $\Theta_{\tau^{\vee}}$ and $\Theta_{\tau}$ entails that

$I_{\mathrm{disc}}(\overline{f_{1}},f_{2})=\int_{\mathbb{S}^{1}\backslash
T_{\mathrm{disc},-}(\tilde{G})}i(\tau)\overline{\Theta_{\tau}(f_{1})}\Theta_{\tau}(f_{2})\mathop{}\\!\mathrm{d}\tau.$

This is the form of the invariant local trace formula employed in [6, §6] and
[61, X.3.3].

####  12.2.2 Stable side

Consider the stable side $G^{!}\supset M^{!}$, in the usual notations. In
particular, $G^{!}$ is a direct product of factors of the form
$\operatorname{GL}(a)$ or $\operatorname{SO}(2b+1)$. Again, we fix an
invariant positive-definite quadratic form on $\mathfrak{a}_{0}^{!}$ in order
to normalize Haar measures. In the upcoming applications, they will come from
the prescribed invariant quadratic form on $\mathfrak{a}_{0}$ via endoscopy.

Denote by $S\mathscr{C}(G^{!})$ the stable analogue of $I\mathscr{C}(G^{!})$.
The stable version of the invariant local trace formula is established in [6,
Corollary 6.4], taking the form

$S^{G^{!}}_{\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2})=S^{G^{!}}_{\mathrm{disc}}(\overline{f^{!}_{1}},f^{!}_{2})$

for all $f^{!}_{1},f^{!}_{2}\in
S\mathscr{C}(G^{!})\otimes\operatorname{mes}(G^{!})$. See also [61, X.3.2].

Both sides are separately continuous in $(f^{!}_{1},f^{!}_{2})$ with respect
to the topologies on $\mathscr{C}(G^{!})$. Note that we are stating only the
analogue of (12.2.2), i.e. the version involving complex conjugates. The
precise construction is given inductively from the non-stable case settled in
[14, p.189]. Take the geometric side for example,

$S^{G^{!}}_{\mathrm{geom}}\left(\overline{f^{!}_{1}},f^{!}_{2}\right)=I^{G^{!}}_{\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2})-\sum_{\begin{subarray}{c}\mathbf{G}^{!!}\in\mathcal{E}_{\operatorname{ell}}(G^{!})\\\
G^{!!}\neq
G^{!}\end{subarray}}\iota(G^{!},G^{!!})S^{G^{!!}}_{\mathrm{geom}}\left(\overline{f_{1}^{!!}},f_{2}^{!!}\right)$

where $f_{i}^{!!}$ denotes the transfer of $f_{i}^{!}$ to the endoscopic group
$G^{!!}$, for $i=1,2$. We refer to [6, p.832] for details.

Geometric side

Following [6, (10.11)] or [61, p.1150], for each
$M^{!}\in\mathcal{L}(M^{!}_{0})$ we set

$\displaystyle S_{M^{!},\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2})$
$\displaystyle=S^{G^{!}}_{M^{!},\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2})$
$\displaystyle:=\int_{\Sigma_{G^{!}\text{-reg,
ell}}(M^{!})}\left|\mathfrak{E}(M^{!}_{\delta},M^{!};F)\right|^{-1}S^{G^{!}}_{M^{!}}(\delta,\overline{f^{!}_{1}},f^{!}_{2})\mathop{}\\!\mathrm{d}\delta,$

where $\mathfrak{E}(M^{!}_{\delta},M^{!};F)$ is as in §2.9, and
$\Sigma_{G^{!}\text{-reg, ell}}(M^{!})$ is endowed with the measure à la
stable Weyl integration formula. Again, $S^{G^{!}}_{M^{!}}(\delta,\cdots)$
follows the paradigm of semi-local versions in §10.3 with
$V=\\{v_{1},v_{2}\\}$, with the same local field $F$. In particular, there is
a stable splitting formula

$S^{G^{!}}_{M^{!}}(\delta,\overline{f^{!}_{1}},f^{!}_{2})=\sum_{L^{!}_{1},L^{!}_{2}\in\mathcal{L}(M^{!})}e^{G^{!}}_{M^{!}}(L^{!}_{1},L^{!}_{2})\overline{S^{L^{!}_{1}}_{M^{!}}(\delta,f_{1,L^{!}_{1}})}S^{L^{!}_{2}}_{M^{!}}(\delta,f_{2,L^{!}_{2}}).$

The geometric side is then expressed as

$S^{G^{!}}_{\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2})=\sum_{M^{!}\in\mathcal{L}(M^{!}_{0})}\frac{|W^{M^{!}}_{0}|}{|W^{G^{!}}_{0}|}(-1)^{\dim\mathfrak{a}^{G^{!}}_{M^{!}}}S^{G^{!}}_{M^{!},\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2}).$

Spectral side

By [6, Proposition 6.3 (b), Corollary 6.4 (b)], there are uniquely determined
coefficients $s^{G^{!}}(\phi)$ such that

$S^{G^{!}}_{\mathrm{disc}}(\overline{f^{!}_{1}},f^{!}_{2})=\int_{\Phi_{\mathrm{disc}}(G^{!})}s^{G^{!}}(\phi)\overline{S\Theta_{\phi}(f^{!}_{1})}S\Theta_{\phi}(f^{!}_{2})\mathop{}\\!\mathrm{d}\phi,$

where $\Phi_{\mathrm{disc}}(G^{!})$ is a subspace of
$\Phi_{\mathrm{bdd}}(G^{!})$, discrete modulo $i\mathfrak{a}^{*}_{G^{!},F}$,
that supports $s^{G^{!}}$. The measure on $\Phi_{\mathrm{disc}}(G^{!})$ is
defined by the same recipe as for
$T_{\mathrm{disc},-}(\tilde{G})/\mathbb{S}^{1}$; see (2.7.2).

The coefficients $s^{G^{!}}(\phi)$ are invariant under
$i\mathfrak{a}^{*}_{G^{!},F}$-twists. This follows inductively from the
corresponding property of $i^{G^{!}}$.

######  Remark 12.2.4.

When $f_{1}^{!}$ and $f_{2}^{!}$ are both cuspidal, the identity
$S^{G^{!}}_{\mathrm{geom}}(\overline{f^{!}_{1}},f^{!}_{2})=S^{G^{!}}_{\mathrm{disc}}(\overline{f^{!}_{1}},f^{!}_{2})$
simplifies into

$\int_{\Sigma_{\operatorname{ell}}(G^{!})}\left|\mathfrak{E}(M^{!}_{\delta},M^{!};F)\right|^{-1}\overline{f_{1}^{!}(\delta)}f_{2}^{!}(\delta)\mathop{}\\!\mathrm{d}\delta\\\
=\int_{\Phi_{\mathrm{bdd},2}(G^{!})}s^{G^{!}}(\phi)\overline{S\Theta_{\phi}(f^{!}_{1})}S\Theta_{\phi}(f^{!}_{2})\mathop{}\\!\mathrm{d}\phi.$

Recall that $f_{i}^{!}(\delta)=S^{G^{!}}(\delta,f_{i}^{!})$ by convention, and
the integral over $\Phi_{\mathrm{bdd},2}(G^{!})$ is discrete modulo
$i\mathfrak{a}^{*}_{G,F}$. The identity can be seen by using the stable
splitting formula for
$S^{G^{!}}_{M^{!}}(\delta,\overline{f^{!}_{1}},f^{!}_{2})$ to show that it
vanishes when $M^{!}\neq G^{!}$.

###  12.3 Endoscopic constructions: geometric side

We revert to the study of a covering of metaplectic type
$\mathbf{p}:\tilde{G}\to G(F)$.

Recall from Definition 3.7.1 that for every
$\mathbf{G}^{!}\in\mathcal{E}(\tilde{G})$, we have the constant

$\iota(\tilde{G},G^{!}):=\left(Z_{(G^{!})^{\vee}}:Z_{\tilde{G}^{\vee}}^{\circ}\right)^{-1}=\begin{cases}2^{-\\#(\operatorname{SO}\text{-factors
in}\;G^{!})},&\text{if}\;\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G}),\\\
0,&\text{otherwise}.\end{cases}$

The explicit formula for $\iota(\tilde{G},G^{!})$ is the same as the global
case (Definition 10.5.1).

For each $M\in\mathcal{L}(M_{0})$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, equip
$\mathfrak{a}_{M^{!}}$ with the invariant quadratic form induced from the one
on $\mathfrak{a}_{M}$, which is in turn restricted from $\mathfrak{a}_{0}$.
The isomorphism
$A_{M^{!}}(F)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{M}(F)$ preserves
the corresponding Haar measures. This determines the Haar measure on
$T^{!}(F)$ for each elliptic maximal torus $T^{!}\subset M^{!}$, such that
$\operatorname{mes}(T^{!}(F)/A_{M^{!}}(F))=1$.

If one starts with
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$ and take the
induced invariant quadratic form on $\mathfrak{a}^{!}_{0}$, then restriction
to $\mathfrak{a}_{M^{!}}$ for various $M^{!}\subset G^{!}$ gives the same Haar
measures.

Recall that $G=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W)$. The definition of
endoscopic datum is not affected by change of $\psi$ or dilation of
$(W,\langle\cdot|\cdot\rangle)$, hence
$\mathcal{E}_{\operatorname{ell}}(\tilde{G})=\mathcal{E}_{\operatorname{ell}}(\tilde{G}^{\dagger})$,
and so forth. On the other hand, the transfer factors are affected by such
changes.

######  Definition 12.3.1.

For all
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and all $\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$, let

  * •

$\overline{f_{1}}^{G^{!}}$ be the transfer of
$\overline{f_{1}}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}^{\dagger})\otimes\operatorname{mes}(G)$
via $\mathcal{T}_{\mathbf{G}^{!},\tilde{G}^{\dagger}}$, using the datum
$(W,\langle\cdot|\cdot\rangle,\psi^{-1})$ (or equivalently
$(W,-\langle\cdot|\cdot\rangle,\psi)$) via
$\Xi:\tilde{G}^{\dagger}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\tilde{G}^{-}$
(Definition–Proposition 12.1.2);

  * •

$f_{2}^{G^{!}}$ be the transfer of $f_{2}$ via
$\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}$.

We define

$\displaystyle I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})$
$\displaystyle=I^{\tilde{G},\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})$
$\displaystyle:=\sum_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}\iota(\tilde{G},G^{!})S^{G^{!}}_{\mathrm{geom}}\left(\overline{f_{1}}^{G^{!}},f_{2}^{G^{!}}\right).$

For each $M\in\mathcal{L}(M_{0})$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, we define

$\displaystyle
I^{\mathcal{E}}_{\tilde{M},\mathrm{geom}}(\mathbf{M}^{!},\overline{f_{1}},f_{2})$
$\displaystyle=I^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathrm{geom}}(\mathbf{M}^{!},\overline{f_{1}},f_{2})$
$\displaystyle:=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])S^{G^{!}[s]}_{M^{!},\mathrm{geom}}\left(\overline{f_{1}}^{G^{!}[s]},f_{2}^{G^{!}[s]}\right).$

Note that the $z[s]$-twists do not intervene here.

######  Definition 12.3.2.

Let

$\displaystyle
I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,\overline{f_{1}},f_{2}\right)$
$\displaystyle:=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}\iota_{M^{!}}(\tilde{G},G^{!}[s])S^{G^{!}[s]}_{M^{!}}\left(\delta[s],\overline{f_{1}}^{G^{!}[s]},f_{2}^{G^{!}[s]}\right)$
$\displaystyle
I^{\mathcal{E}}_{\tilde{M}}\left(\gamma,\overline{f_{1}},f_{2}\right)$
$\displaystyle:=\sum_{L_{1},L_{2}\in\mathcal{L}(M)}d^{G}_{M}(L_{1},L_{2})\overline{I^{\tilde{L}_{1},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f_{1,\tilde{L}_{1}})}I^{\tilde{L}_{2},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f_{2,\tilde{L}_{2}})$

where $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\delta\in\Sigma_{G\text{-reg, ell}}(M^{!})$ (in the first case), or
$\gamma\in\Gamma_{\text{reg}}(M)$ and
$\tilde{\gamma}\in\mathbf{p}^{-1}(\tilde{\gamma})$ is arbitrary (in the second
case). The paradigm follows the semi-local constructions in §10.6, say with
$V=\\{v_{1},v_{2}\\}$, and the details will not be repeated here.

In particular, the proof of Proposition 10.6.6 (i) adapts to this context, and
yields the splitting formula

$I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,\overline{f_{1}},f_{2}\right):=\sum_{L_{1},L_{2}\in\mathcal{L}(M)}d^{G}_{M}(L_{1},L_{2})\overline{I^{\tilde{L}_{1},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f_{1,\tilde{L}_{1}})}I^{\tilde{L}_{2},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f_{2,\tilde{L}_{2}}).$
(12.3.1)

We are ready to state the stabilization of the geometric side, as follows.

######  Theorem 12.3.3.

Let
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

  1. (i)

For all $M\in\mathcal{L}(M_{0})$, we have

$\sum_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})}\iota(\tilde{M},M^{!})I^{\mathcal{E}}_{\tilde{M},\mathrm{geom}}(\mathbf{M}^{!},\overline{f_{1}},f_{2})=I_{\tilde{M},\mathrm{geom}}(\overline{f_{1}},f_{2}).$

  2. (ii)

We have

$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})=I_{\mathrm{geom}}(\overline{f_{1}},f_{2}).$

A conditional proof based on the key geometric Hypothesis 12.5.1 will be given
in §12.5.

######  Remark 12.3.4.

Using the trace Paley–Wiener theorems for
$I\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$ and
$S\mathscr{C}(G^{!})$, together with the spectral transfer reviewed in §3.9,
one can show that $\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}$ and
$\mathcal{T}_{\mathbf{G}^{!},\tilde{G}^{\dagger}}$ extend by continuity to
Schwartz–Harish-Chandra spaces. The stabilization above extends automatically
to $f_{1},f_{2}\in I\mathscr{C}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$ by
continuity.

Also, by the continuity of these distributions, it suffices to prove the
matching in the case of $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$, since the
counterpart for the $\operatorname{GL}$-factors is tautological.

Before embarking on the proof of Theorem 12.3.3, observe that (i) $\implies$
(ii) in Theorem 12.3.3. To see this, apply Proposition 3.7.4 to obtain

$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})=\sum_{\mathbf{G}^{!}}\iota(\tilde{G},G^{!})\sum_{M^{!}}\frac{|W^{G^{!}}_{0}|}{|W^{M^{!}}_{0}|}\underbracket{(-1)^{\dim\mathfrak{a}^{G^{!}}_{M^{!}}}S^{G^{!}}_{M^{!},\mathrm{geom}}\left(\overline{f}_{1}^{G^{!}[s]},f_{2}^{G^{!}[s]}\right)}_{=:S(\mathbf{G}^{!},M^{!})}\\\
=\sum_{M}\frac{|W^{G}_{0}|}{|W^{M}_{0}|}(-1)^{\dim\mathfrak{a}^{G}_{M}}\sum_{\mathbf{M}^{!}}\iota(\tilde{M},M^{!})\sum_{s}i_{M^{!}}(\tilde{G},G^{!}[s])S^{G^{!}[s]}_{M^{!},\mathrm{geom}}\left(\overline{f}_{1}^{G^{!}[s]},f_{2}^{G^{!}[s]}\right)\\\
=\sum_{M}\frac{|W^{G}_{0}|}{|W^{M}_{0}|}(-1)^{\dim\mathfrak{a}^{G}_{M}}\sum_{\mathbf{M}^{!}}\iota(\tilde{M},M^{!})I^{\mathcal{E}}_{\tilde{M},\mathrm{geom}}(\mathbf{M}^{!},\overline{f_{1}},f_{2}).$
(12.3.2)

This equals $I_{\mathrm{geom}}(\overline{f_{1}},f_{2})$ if (i) holds.

######  Lemma 12.3.5 (Cf. [15, pp.276–277]).

For all $M\in\mathcal{L}(M_{0})$ and
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,

$\sum_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})}\iota(\tilde{M},M^{!})I^{\mathcal{E}}_{\tilde{M},\mathrm{geom}}(\mathbf{M}^{!},\overline{f_{1}},f_{2})=\int_{\Gamma_{G\text{-reg,
ell}}(M)}I^{\mathcal{E}}_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})\mathop{}\\!\mathrm{d}\gamma.$

###### Proof.

Put $n(\delta):=\left|\mathfrak{E}(M^{!}_{\delta},M^{!};F)\right|^{-1}$ for
each $(\mathbf{M}^{!},\delta)$. Since $\delta\mapsto\delta[s]$ does not change
the measure on $\Sigma_{\mathrm{reg}}(M^{!})$ and $n(\delta)=n(\delta[s])$,
our assertion is tantamount to

$\sum_{\mathbf{M}^{!}}\iota(\tilde{M},M^{!})\int_{\Sigma_{G\text{-reg,
ell}}(M^{!})}n(\delta)I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,\overline{f_{1}},f_{2}\right)\mathop{}\\!\mathrm{d}\delta\\\
=\int_{\Gamma_{G\text{-reg,
ell}}(M)}I^{\mathcal{E}}_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})\mathop{}\\!\mathrm{d}\gamma.$

Applying the definition of
$I^{\mathcal{E}}_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})$ and (12.3.1), we
are reduced to proving that

$\sum_{\mathbf{M}^{!}}\iota(\tilde{M},M^{!})\int_{\Sigma_{G\text{-reg,
ell}}(M^{!})}n(\delta)\overline{I^{\tilde{L}_{1},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f_{1,\tilde{L}_{1}})}I^{\tilde{L}_{2},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f_{2,\tilde{L}_{2}})\mathop{}\\!\mathrm{d}\delta\\\
=\int_{\Gamma_{G\text{-reg,
ell}}(M)}\overline{I^{\tilde{L}_{1},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f_{1,\tilde{L}_{1}})}I^{\tilde{L}_{2},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f_{2,\tilde{L}_{2}})\mathop{}\\!\mathrm{d}\gamma$

where $\tilde{\gamma}\in\mathbf{p}^{-1}(\gamma)$ is arbitrary, and
$L_{1},L_{2}\in\mathcal{L}(M_{0})$ satisfy $d^{G}_{M}(L_{1},L_{2})\neq 0$.

The above follows from the endoscopic inner product formula in [50, Corollary
5.2.5] for $\tilde{M}$. Indeed, it suffices to check that all the functions to
be paired are $L^{2}$. By ellipticity, it suffices to estimate them in the
direction of $A_{M}(F)$, or of $A_{M^{!}}(F)$ by expanding them in terms of
stable analogues. Ultimately, this reduces to the estimate for weighted
orbital integrals in [14, (5.7)]. For non-Archimedean case, the proof of that
estimate requires Howe’s conjecture for coverings, which is established in
[54]. ∎

######  Proposition 12.3.6 (Cf. [61, X.3.2 Proposition]).

Assume inductively that the local geometric Theorem 5.3.11 for $G$-regular
elements holds if $\tilde{G}$ is replaced by $\tilde{L}$ for every
$L\in\mathcal{L}(M)\smallsetminus\\{G\\}$. Let
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
For every $M\in\mathcal{L}(M_{0})$ and $\gamma\in\Gamma_{G\text{-reg,
ell}}(M)$, define $\mathfrak{d}_{M}(\gamma,\overline{f_{1}},f_{2})$ to be

$\left(\overline{I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f_{1})-I_{\tilde{M}}(\tilde{\gamma},f_{1})}\right)I^{\tilde{M}}(\tilde{\gamma},f_{2,\tilde{M}})+\overline{I^{\tilde{M}}(\tilde{\gamma},f_{1,\tilde{M}})}\left(I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f_{2})-I_{\tilde{M}}(\tilde{\gamma},f_{2})\right)$

where $\tilde{\gamma}\in\mathbf{p}^{-1}(\gamma)$ is arbitrary.

  1. (i)

For all $M\in\mathcal{L}(M_{0})$, we have

$\sum_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})}\iota(\tilde{M},M^{!})I^{\mathcal{E}}_{\tilde{M},\mathrm{geom}}(\mathbf{M}^{!},\overline{f_{1}},f_{2})-I_{\tilde{M},\mathrm{geom}}(\overline{f_{1}},f_{2})\\\
=\int_{\Gamma_{G\text{-reg,
ell}}(M)}\mathfrak{d}_{M}(\gamma,\overline{f_{1}},f_{2})\mathop{}\\!\mathrm{d}\gamma.$

  2. (ii)

We have

$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})-I_{\mathrm{geom}}(\overline{f_{1}},f_{2})\\\
=\sum_{M\in\mathcal{L}(M_{0})}\frac{|W^{M}_{0}|}{|W^{G}_{0}|}(-1)^{\dim\mathfrak{a}^{G}_{M}}\int_{\Gamma_{G\text{-reg,
ell}}(M)}\mathfrak{d}_{M}(\gamma,\overline{f_{1}},f_{2})\mathop{}\\!\mathrm{d}\gamma.$

  3. (iii)

Choose $P_{0}\in\mathcal{P}(M_{0})$ to define the notion of standard Levi
subgroups, then

$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})-I_{\mathrm{geom}}(\overline{f_{1}},f_{2})\\\
=\sum_{M:\text{standard
Levi}}|W^{G}(M)|^{-1}(-1)^{\dim\mathfrak{a}^{G}_{M}}\int_{\Gamma_{G\text{-reg,
ell}}(M)}\mathfrak{d}_{M}(\gamma,\overline{f_{1}},f_{2})\mathop{}\\!\mathrm{d}\gamma.$

###### Proof.

We only have to show (i); one can then deduce (ii) from (12.3.2), and (iii)
will follow from invariance and the familiar combinatorics. By Lemma 12.3.5,
the left hand side of (i) equals

$\int_{\Gamma_{G\text{-reg,
ell}}(M)}\left(I^{\mathcal{E}}_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})-I_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})\right)\mathop{}\\!\mathrm{d}\gamma.$

The integrand may be expanded by splitting
$I_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})$ and
$I^{\mathcal{E}}_{\tilde{M}}(\gamma,\overline{f_{1}},f_{2})$: both are
expressed as a sum over $L_{1},L_{2}\in\mathcal{L}(M)$ weighted by
$d^{G}_{M}(L_{1},L_{2})\neq 0$. The summands take the form
$\overline{I^{\tilde{L}_{1},\mathcal{E}}_{\tilde{M}}}I^{\tilde{L}_{2},\mathcal{E}}_{\tilde{M}}-\overline{I^{\tilde{L}_{1}}_{\tilde{M}}}I^{\tilde{L}_{2}}_{\tilde{M}}$.
By assumption, only the terms with either $L_{1}=G$ or $L_{2}=G$ can survive.
The only possibilities are $(L_{1},L_{2})=(G,M)$ or $(M,G)$. This yields
$\mathfrak{d}_{M}(\gamma,\overline{f_{1}},f_{2})$. ∎

As a consequence, the $G$-regular case of the local geometric Theorem 5.3.11
will imply Theorem 12.3.3 (i), which in turn implies (ii) as already seen. We
also obtain the following consequence.

######  Corollary 12.3.7.

Retain the assumptions in Proposition 12.3.6. If
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{G})\otimes\operatorname{mes}(G)$,
then
$I_{\mathrm{geom}}(\overline{f_{1}},f_{2})=I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})$.

###### Proof.

In this case, $\mathfrak{d}_{M}=0$ unless $M=G$, but
$I_{\tilde{G}}=I^{\mathcal{E}}_{\tilde{G}}$ is already known. ∎

###  12.4 Endoscopic constructions: spectral side

Now comes the endoscopic construction on the spectral side. Consider

$I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2}):=\sum_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}\iota(\tilde{G},G^{!})S^{G^{!}}_{\mathrm{disc}}(\overline{f_{1}}^{G^{!}},f_{2}^{G^{!}})$

where
$\overline{f_{1}}^{G^{!}}:=\mathcal{T}_{\mathbf{G}^{!},\tilde{G}^{\dagger}}(\overline{f_{1}})$
and $f_{2}^{G^{!}}:=\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}(f_{2})$.

Recall that $S^{G^{!}}_{\mathrm{disc}}$ is an integral over
$\Phi_{\mathrm{disc}}(G^{!})$, a space discrete modulo
$i\mathfrak{a}^{*}_{G^{!},F}$. Therefore,
$I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})$ is an integral taken
over the space

$T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G}):=\bigsqcup_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}\Phi_{\mathrm{disc}}(G^{!}),$

which is equipped with the Radon measure such that

$\int_{T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})}\beta=\sum_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}\int_{\Phi_{\mathrm{disc}}(G^{!})}\beta(\phi)\mathop{}\\!\mathrm{d}\phi$

for all $\beta\in C_{c}(T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G}))$.

On the other hand, the spectral transfer factor $\Delta(\phi,\tau)$ makes
sense for each $\phi=(\mathbf{G}^{!},\phi)\in
T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})$ and $\tau\in T_{-}(\tilde{G})$.
Define the $\mathbb{S}^{1}$-stable subset

$T_{\mathcal{E}\text{-disc},-}(\tilde{G}):=\left\\{\begin{array}[]{r|l}\tau\in
T_{-}(\tilde{G})&\exists\;(\mathbf{G}^{!},\phi)\in\Phi^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})\\\
&\Delta(\phi,\tau)\neq 0\end{array}\right\\}\cup
T_{\mathrm{disc},-}(\tilde{G})$

of $T_{-}(\tilde{G})$, make it into a space that is discrete modulo
$\mathbb{S}^{1}\times i\mathfrak{a}^{*}_{G,F}$. As
$i\mathfrak{a}^{*}_{G^{!},F}\simeq i\mathfrak{a}^{*}_{G,F}$, this space can be
equipped with the Radon measure222For metaplectic groups
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$, there are no outer automorphisms of
endoscopic origin; see Remark 3.5.4. such that each
$i\mathfrak{a}^{*}_{G,F}$-orbit containing $(L,\pi,r)$ carries the quotient
measure divided by $|Z_{R_{\pi}}(r)|$.

######  Lemma 12.4.1.

With the measures defined above, the formula of “change of variables”

$\ \int_{\begin{subarray}{c}\tau\in\mathbb{S}^{1}\backslash
T_{\mathcal{E}\text{-disc},-}(\tilde{G})\\\
\tau=(L,\pi,r)\end{subarray}}\sum_{\phi\in
T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})}\beta(\phi)|\Delta(\phi,\tau)|^{2}\alpha(\tau)|Z_{R_{\pi}}(r)|\mathop{}\\!\mathrm{d}\tau\\\
=\int_{T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})}\sum_{\tau\in\mathbb{S}^{1}\backslash
T_{\mathcal{E}\text{-disc},-}(\tilde{G})}\beta(\phi)|\Delta(\phi,\tau)|^{2}\alpha(\tau)\mathop{}\\!\mathrm{d}\phi$
(12.4.1)

holds for all $\alpha\in C_{c}(\mathbb{S}^{1}\backslash
T_{\mathcal{E}\text{-disc},-}(\tilde{G}))$ and $\beta\in
C_{c}(T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G}))$.

###### Proof.

This is formally similar to the elliptic version [13, Lemma 5.3], with the
same proof. Compared with (2.7.2), the measure $\mathop{}\\!\mathrm{d}\tau$ in
loc. cit. does not involve $|Z_{R_{\pi}}(r)|^{-1}$, whence the factor
$|Z_{R_{\pi}}(r)|\mathop{}\\!\mathrm{d}\tau$ in our version. 333Besides,
$|Z_{R_{\pi}}(r)|$ is also missing in the elliptic version stated in [50,
Lemma 6.2.7]. That result is only used in the proof of [50, Proposition
8.3.5], and does not affect the proof since $Z_{R_{\pi}}(r)$ is not altered
under $i\mathfrak{a}^{*}_{G,F}$-twists. ∎

######  Definition 12.4.2.

For all $\tau=(L,\pi,r)\in\tilde{T}_{-}(\tilde{G})$, set

$i^{\mathcal{E}}(\tau):=i^{\tilde{G},\mathcal{E}}(\tau):=\sum_{\phi\in\Phi^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})}\iota(\tilde{G},G^{!})|\Delta(\phi,\tau)|^{2}s^{G^{!}}(\phi)|Z_{R_{\pi}}(r)|$

where $\Delta(\phi,\tau)$ is the spectral transfer factor in Definition
§3.9.1. This depends only on the $\mathbb{S}^{1}$-orbit of $\tau$, and
vanishes outside $T_{\mathcal{E}\text{-disc},-}(\tilde{G})$.

The following is analogous to [6, Proposition 6.3 (a)].

######  Lemma 12.4.3.

For all
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
we have

$I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})=\int_{\mathbb{S}^{1}\backslash
T_{\mathcal{E}\text{-disc},-}(\tilde{G})}i^{\mathcal{E}}(\tau)\overline{\Theta_{\tau}(f_{1})}\Theta_{\tau}(f_{2})\mathop{}\\!\mathrm{d}\tau.$

###### Proof.

Lemma 12.1.3 implies $\overline{f_{1}}^{G^{!}}=\overline{f_{1}^{G^{!}}}$. The
left hand side is hence the integral over $(\mathbf{G}^{!},\phi)\in
T^{\mathcal{E}}_{\mathrm{disc}}(\tilde{G})$ of

$\iota(\tilde{G},G^{!})s^{G^{!}}(\phi)\overline{S\Theta_{\phi}(f_{1}^{!})}S\Theta_{\phi}(f_{2}^{!}).$

Via spectral transfer, the expression above equals

$\iota(\tilde{G},G^{!})s^{G^{!}}(\phi)\sum_{\tau_{1},\tau_{2}\in
T_{-}(\tilde{G})/\mathbb{S}^{1}}\overline{\Delta(\phi,\tau_{1})}\Delta(\phi,\tau_{2})\overline{\Theta_{\tau_{1}}(f_{1})}\Theta_{\tau_{2}}(f_{2}).$

In the integration over $\phi$, only the terms with $\tau_{1}=\tau_{2}\in
T_{\mathcal{E}\text{-disc},-}(\tilde{G})$ contribute. This is due to the
orthogonality relation in [50, Definition 6.2.4, Lemma 6.2.6]; for the
Archimedean case, see also §7.5 of loc. cit. It remains to apply Lemma 12.4.1.
∎

The assertion of Theorem 12.3.3 (ii) is equivalent to that
$I_{\mathrm{disc}}(\overline{f_{1}},f_{2})=I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})$
for all $f_{1},f_{2}$. It also entails the stabilization of the spectral
coefficients $i(\tau)$. We record this formally as follows.

######  Theorem 12.4.4 (Cf. [6, Corollary 6.4]).

For all $f_{1}$, $f_{2}$, we have

$I_{\mathrm{disc}}(\overline{f_{1}},f_{2})=I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2}),$

and $i^{\mathcal{E}}(\tau)=i(\tau)$ for all $\tau$. In particular,
$i^{\mathcal{E}}(\tau)$ is nonzero only when $\tau\in
T_{\mathrm{disc},-}(\tilde{G})$.

###### Proof.

The validity of Theorem 12.3.3 (ii) is assumed throughout the proof. The first
equality is already seen. Now compare
$I_{\mathrm{disc}}(\overline{f_{1}},f_{2})$ with the integral in Lemma 12.4.3,
noting that $\mathbb{S}^{1}\backslash T_{\mathrm{disc},-}(\tilde{G})$ is a
union of connected components in $\mathbb{S}^{1}\backslash
T_{\mathcal{E}\text{-disc},-}(\tilde{G})$, in a way compatible with the Radon
measures. Since $f_{1}$ and $f_{2}$ are arbitrary, it follows that
$i^{\mathcal{E}}(\tau)=i(\tau)$. ∎

######  Proposition 12.4.5.

Under the assumptions of Proposition 12.3.6, we have

$I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})-I_{\mathrm{disc}}(\overline{f_{1}},f_{2})=\int_{i\mathfrak{a}^{*}_{G,F}}\sum_{M,\chi_{M},\tau}a(\tilde{M},\chi_{M},\tau)\overline{\Theta_{\tau_{\lambda}}(f_{1,\tilde{M}})}\Theta_{\tau_{\lambda}}(f_{2,\tilde{M}})\mathop{}\\!\mathrm{d}\lambda$

for all $f_{1},f_{2}$, where

  * •

$M$ ranges over proper standard Levi subgroups of $G$,

  * •

$\chi_{M}$ ranges over $i\mathfrak{a}^{*}_{G,F}$-orbits of genuine characters
$\widetilde{A_{M}}\to\mathbb{S}^{1}$,

  * •

$\tau$ ranges over $\mathbb{S}^{1}\backslash
T_{\operatorname{ell},-}(\tilde{M})/i\mathfrak{a}^{*}_{G,F}$, for which we
choose representatives in $T_{\operatorname{ell},-}(\tilde{M})$, and we
require that $\tau$ has central character equal to $\chi_{M}$ on
$\widetilde{A_{M}}$, modulo $i\mathfrak{a}^{*}_{G,F}$;

  * •

$a(\tilde{M},\chi_{M},\cdot)$ are identically zero for all but finitely many
pairs $(M,\chi_{M})$.

For all proper standard Levi subgroup $M$, we may choose
$(\chi_{M},\tau)\mapsto a(\tilde{M},\chi_{M},\tau)$ to be
$W^{G}(M)$-invariant.

###### Proof.

Such an expression follows by sorting out the integrals in $I_{\mathrm{disc}}$
and $I^{\mathcal{E}}_{\mathrm{disc}}$, using Lemma 12.4.3. To verify the
vanishing of $a(\tilde{M},\chi_{M},\cdot)$ for all but finitely many pairs
$(M,\chi_{M})$, we use the stable counterparts in [61, p.1160] for the
contribution from $I^{\mathcal{E}}_{\mathrm{disc}}$. For the contribution from
$I_{\mathrm{disc}}$, simply repeat the argument from [61, X.3.3 Lemme,
p.1157]; the only needed input is the first part of Lemma 2.7.2.

The $W^{G}(M)$-invariance of $a(\tilde{M},\chi_{M},\tau)$ can be ensured by
averaging, since $\Theta_{\tau_{\lambda}}(f_{i,\tilde{M}})$ is
$W^{G}(M)$-invariant for $i=1,2$.

Finally, to show that only the terms with $M\neq G$ contribute, we use the
vanishing of the left hand side when $f_{1}$, $f_{2}$ are both cuspidal
(Corollary 12.3.7), together with the trace Paley–Wiener theorem which implies
that $\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{G})$
separates points in $\mathbb{S}^{1}\backslash
T_{\operatorname{ell},-}(\tilde{G})$. ∎

######  Definition 12.4.6.

Let $M\in\mathcal{L}(M_{0})$. We say that
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
is $\tilde{M}$-cuspidal if we have $f_{\tilde{L}}=0$ for all Levi subgroups
$L$ of $G$ that does not contain any conjugate of $M$.

For example, cuspidality is equivalent to $\tilde{G}$-cuspidality, and every
function is $M_{0}$-cuspidal. The next two results are taken from [61, X.3.4].

######  Corollary 12.4.7.

Fix $M\in\mathcal{L}(M_{0})$ and an $\tilde{M}$-cuspidal
$f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
If
$I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})=I_{\mathrm{disc}}(\overline{f_{1}},f_{2})$
holds for all
$f_{1}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
which is $\tilde{M}$-cuspidal, then it holds for all
$f_{1}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

###### Proof.

Apply the expansion furnished by Proposition 12.4.5 and the trace Paley–Wiener
theorem. ∎

######  Corollary 12.4.8.

Suppose that
$I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},\cdot)=I_{\tilde{L}}(\tilde{\gamma},\cdot)$
holds for all $L\supset M$ and all $\tilde{\gamma}\in\Gamma_{G\text{-reg,
ell}}(\tilde{L})$. Then
$I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},f)=I_{\tilde{L}}(\tilde{\gamma},f)$
holds for all $L\in\mathcal{L}(M_{0})$, $\tilde{\gamma}\in\Gamma_{G\text{-reg,
ell}}(\tilde{L})$ and $\tilde{M}$-cuspidal $f$.

###### Proof.

Suppose $f_{1},f_{2}$ are both $\tilde{M}$-cuspidal. The expansion of
Proposition 12.3.6 involves only terms indexed by $L\in\mathcal{L}(M_{0})$
that contains a conjugate of $M$. In particular,
$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})=I_{\mathrm{geom}}(\overline{f_{1}},f_{2})$
holds. Therefore, Corollary 12.4.7 implies that
$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})=I_{\mathrm{geom}}(\overline{f_{1}},f_{2})$
if only $f_{2}$ is assumed to be $\tilde{M}$-cuspidal.

Looking back at the expansion in Proposition 12.3.6, with $f_{2}$ being
$\tilde{M}$-cuspidal, it now becomes

$\sum_{L\in\mathcal{L}(M_{0})}\frac{|W^{L}_{0}|}{|W^{G}_{0}|}(-1)^{\dim\mathfrak{a}^{G}_{L}}\int_{\Gamma_{G\text{-reg,
ell}}(L)}\overline{I^{\tilde{L}}(\tilde{\gamma},f_{1,\tilde{L}})}\left(I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},f_{2})-I_{\tilde{L}}(\tilde{\gamma},f_{2})\right)\mathop{}\\!\mathrm{d}\gamma.$

Observe that $I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},f_{2})$ and
$I_{\tilde{L}}(\tilde{\gamma},f_{2})$ are both $W^{G}(L)$-invariant in
$\tilde{\gamma}$. As $f_{1}$ is arbitrary, we deduce
$I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},f_{2})=I_{\tilde{L}}(\tilde{\gamma},f_{2})$
for all $L$ and $\tilde{\gamma}\in\Gamma_{G\text{-reg, ell}}(\tilde{L})$. ∎

###  12.5 Key geometric hypothesis

Fix $M\in\mathcal{L}(M_{0})$. We assume inductively that

$I^{\tilde{S},\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},\cdot)=I^{\tilde{S}}_{\tilde{L}}(\tilde{\gamma},\cdot)$
(12.5.1)

for all $\tilde{\gamma}\in\Gamma_{S\text{-reg}}(\tilde{L})$, where $L\subset
S$ are Levi subgroups of $G$ containing $M$ such that

$\text{either}\quad S\subsetneq G\quad\text{or}\quad M\subsetneq L.$

Let
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
Recall the
$\epsilon_{\tilde{M}}(f)\in\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{M})\otimes\operatorname{mes}(M)$
constructed in Definition–Propositions 7.5.1 and 9.8.5. For all
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, one has the
function

$\delta\mapsto\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}\left(\epsilon_{\tilde{M}}(f)\right)(\delta),\quad\delta\in\Sigma_{\mathrm{reg}}(M^{!})$

given by stable orbital integrals; see also (3.8.2). On the other hand, it
also makes sense to consider

$\delta\mapsto\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}\left(f_{\tilde{M}}\right)(\delta).$

We now state the _key geometric hypothesis_.

######  Hypothesis 12.5.1.

For all $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, there
exists a smooth function

$\epsilon(\mathbf{M}^{!},\cdot):M^{!}_{G\text{-reg}}(F)\to\mathbb{C}$

characterized uniquely by

$\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}\left(\epsilon_{\tilde{M}}(f)\right)(\delta)=\epsilon(\mathbf{M}^{!},\delta)\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}\left(f_{\tilde{M}}\right)(\delta)$

for all $\delta\in\Sigma_{G\text{-reg}}(M^{!})$ and all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

######  Remark 12.5.2.

The group $W^{G}(M)$ acts on the $\operatorname{GL}$-factors of $M^{!}$, and
$\epsilon(\mathbf{M}^{!},\cdot)$ is $W^{G}(M)$-invariant. To see this, we need
the $W^{G}(M)$-invariance of $f_{\tilde{M}}$ and $\epsilon_{\tilde{M}}(f)$;
for the latter, we use the invariance of $I_{\tilde{M}}$ under automorphisms
[8, Lemma 3.3], as well as its endoscopic version.

######  Remark 12.5.3.

Since $\epsilon_{\tilde{M}}(f)$ is cuspidal for all $f$, we have
$\epsilon(\mathbf{M}^{!},\delta)=0$ unless $\delta$ is elliptic.

Our aim is to show that
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)=I_{\tilde{M}}(\tilde{\gamma},f)$
for all $\tilde{\gamma}$ and $f$, or equivalently that
$\epsilon_{\tilde{M}}(f)=0$. By adjoint geometric transfer (Proposition
3.8.8), this amounts to showing
$\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(\epsilon_{\tilde{M}}(f))=0$ for all
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. This will be
achieved, under Hypothesis 12.5.1, by showing that
$\epsilon(\mathbf{M}^{!},\delta)=0$ for all $(\mathbf{M}^{!},\delta)$. We
shall proceed in steps.

######  Lemma 12.5.4 (Cf. [6, Lemma 6.5]).

Under the inductive assumptions (12.5.1) and the key geometric Hypothesis
12.5.1, we have
$\epsilon(\mathbf{M}^{!},\delta)+\overline{\epsilon(\mathbf{M}^{!},\delta)}=0$
for all $(\mathbf{M}^{!},\delta)$.

###### Proof.

We may and do assume that $M$ is a proper standard Levi subgroup, and $\delta$
is elliptic in $M^{!}$. Take
$f_{1},f_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and expand
$I^{\mathcal{E}}_{\mathrm{geom}}(\overline{f_{1}},f_{2})-I_{\mathrm{geom}}(\overline{f_{1}},f_{2})=I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})-I_{\mathrm{disc}}(\overline{f_{1}},f_{2})$
in two ways, namely via Propositions 12.3.6 (iii) and 12.4.5, to obtain

$\sum_{L:\text{standard
Levi}}|W^{G}(L)|^{-1}(-1)^{\dim\mathfrak{a}^{G}_{L}}\int_{\Gamma_{G\text{-reg,
ell}}(L)}\mathop{}\\!\mathrm{d}\gamma\cdot\\\
\overline{\left(I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},f_{1})-I_{\tilde{L}}(\tilde{\gamma},f_{1})\right)}I^{\tilde{M}}(\tilde{\gamma},f_{2,\tilde{L}})+\overline{I^{\tilde{L}}(\tilde{\gamma},f_{1,\tilde{L}})}\left(I^{\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},f_{2})-I_{\tilde{L}}(\tilde{\gamma},f_{2})\right)\\\
=\int_{i\mathfrak{a}^{*}_{G,F}}\sum_{\begin{subarray}{c}L,\chi_{L},\tau\\\
L:\text{standard Levi}\\\ L\neq
G\end{subarray}}a(\tilde{L},\chi_{L},\tau)\overline{\Theta_{\tau_{\lambda}}(f_{1,\tilde{L}})}\Theta_{\tau_{\lambda}}(f_{2,\tilde{L}})\mathop{}\\!\mathrm{d}\lambda.$

Consider any standard Levi subgroup $L^{\prime}\supsetneq M$. We contend that
$a(\tilde{L}^{\prime},\chi_{L^{\prime}},\tau)=0$ for all $\chi_{L^{\prime}}$
and $\tau$. Take $f_{1},f_{2}$ to be $\tilde{L}^{\prime}$-cuspidal. The
geometric (i.e. left hand) side of the displayed equation vanishes by
inductive assumptions. By choosing $f_{1}=f_{2}$ suitably and using the trace
Paley–Wiener theorem plus the properties stated in Proposition 12.4.5, it
follows that $a(\tilde{L}^{\prime},\chi_{L^{\prime}},\tau)=0$ as desired.

Next, take $f_{1},f_{2}$ to be $\tilde{M}$-cuspidal. On the spectral side,
only the terms with $L=M$ survive. On the geometric side, only the terms with
$L\supset M$ survive; furthermore, inductive assumptions kill the terms with
$L\neq M$. Hence the geometric side can be simplified into a sum of two inner
products taken over $\Gamma_{G\text{-reg, ell}}(\tilde{M})$, and we also have

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)-I_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{M}}(\tilde{\gamma},\epsilon_{\tilde{M}}(\cdot)).$

In turn, by using

  * •

the endoscopic inner product formula for $\tilde{M}$ in [50, Corollary 5.2.5],

  * •

the key geometric Hypothesis 12.5.1,

the geometric side is seen to equal

$(\text{nonzero
constant})\cdot\sum_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})}\iota(\tilde{M},M^{!})\\\
\int_{\Sigma_{\tilde{M}\text{-reg,
ell}}(M^{!})}n(\delta)\left(\epsilon(\mathbf{M}^{!},\delta)+\overline{\epsilon(\mathbf{M}^{!},\delta)}\right)\overline{\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f_{1,\tilde{M}})(\delta)}\cdot\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f_{2,\tilde{M}})(\delta)\mathop{}\\!\mathrm{d}\delta.$
(12.5.2)

Fix an $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. In what
follows, we take $f_{1}$ such that

  * •

the $W^{G}(M)$-invariant function
$\delta^{\prime}\mapsto\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f_{1,\tilde{M}})(\delta^{\prime})$
on $\Sigma_{\mathrm{reg}}(M^{!})$ has compact supported in
$\Sigma_{\operatorname{ell},G\text{-reg}}(M^{!})$;

  * •

for any $\mathbf{M}^{!!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ with
$\mathbf{M}^{!!}\neq\mathbf{M}^{!}$, we have
$\mathcal{T}_{\mathbf{M}^{!!},\tilde{M}}(f_{1,\tilde{M}})=0$.

In fact, we will take a sequence of $f_{1}$ such that
$\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f_{1,\tilde{M}})$ approximates the
Dirac measure attached to the $W^{G}(M)$-orbit of any given
$\delta\in\Sigma_{\operatorname{ell},G\text{-reg}}(M^{!})$. The existence such
a sequence follows from Theorem 3.8.10 and the characterization of orbital
integrals in [24, 67] (cf. the proof of Proposition 9.8.3).

With these assumptions, we infer that

$c_{1}:=\left(\overline{\epsilon(\mathbf{M}^{!},\delta)}+\epsilon(\mathbf{M}^{!},\delta)\right)\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f_{1,\tilde{M}})(\delta)\in
S\mathcal{I}_{\operatorname{cusp}}(M^{!})\otimes\operatorname{mes}(M^{!}),$

whilst by $\tilde{M}$-cuspidality of $f_{2}$, we also have

$c_{2}:=\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f_{2,\tilde{M}})\in
S\mathcal{I}_{\operatorname{cusp}}(M^{!})\otimes\operatorname{mes}(M^{!}).$

Hence the integral in (12.5.2) can be expressed using
$S^{M^{!}}_{\mathrm{geom}}(\overline{c_{1}},c_{2})=S^{M^{!}}_{\mathrm{disc}}(\overline{c_{1}},c_{2})$,
simplified according to Remark 12.2.4, as an inner product of stable
characters indexed by $\Phi_{\mathrm{bdd},2}(M^{!})$, defined using linear
combinations of integrals over $i\mathfrak{a}^{*}_{M^{!},F}\simeq
i\mathfrak{a}^{*}_{M,F}$.

On the other hand, the expression

$I^{\mathcal{E}}_{\mathrm{disc}}(\overline{f_{1}},f_{2})-I_{\mathrm{disc}}(\overline{f_{1}},f_{2})=\int_{i\mathfrak{a}^{*}_{G,F}}\sum_{\chi_{M},\tau}a(\tilde{M},\chi_{M},\tau)\overline{\Theta_{\tau_{\lambda}}(f_{1,\tilde{M}})}\Theta_{\tau_{\lambda}}(f_{2,\tilde{M}})$

can be translated to $\mathbf{M}^{!}$ on the endoscopic side via spectral
transfer. Keep $f_{1}$ fixed and vary the $\tilde{M}$-cuspidal test function
$f_{2}$. Since $M\neq G$, a familiar comparison with the previous integrals
over $i\mathfrak{a}^{*}_{M,F}$ shows that both expansions are zero. In
particular, the integral over
$\Sigma_{G\text{-reg},\operatorname{ell}}(M^{!})$ in (12.5.2) vanishes for
such $(f_{1},f_{2})$.

Notice that the three terms and $n(\delta)$ in the integrand of (12.5.2) are
all $W^{G}(M)$-invariant by Remark 12.5.2. Given $(\mathbf{M}^{!},\delta)$,
let $f_{1,\tilde{M}}$ approach the Dirac measure attached to $W^{G}(M)\delta$
and vary $f_{2}$. This implies
$\epsilon(\mathbf{M}^{!},\delta)+\overline{\epsilon(\mathbf{M}^{!},\delta)}=0$,
as asserted. ∎

######  Lemma 12.5.5.

Under the inductive assumptions (12.5.1) and the key geometric Hypothesis
12.5.1, we have
$\epsilon(\mathbf{M}^{!},\delta)=\overline{\epsilon(\mathbf{M}^{!},\delta)}$
for all $(\mathbf{M}^{!},\delta)$.

###### Proof.

By taking complex conjugates in the characterization of
$\epsilon(\mathbf{M}^{!},\cdot)$ and using Lemma 12.1.3, we obtain

$\mathcal{T}_{\mathbf{M}^{!},\tilde{M}^{\dagger}}(\overline{\epsilon_{\tilde{M}}(f)})(\delta)=\overline{\epsilon(\mathbf{M}^{!},\delta)}\mathcal{T}_{\mathbf{M}^{!},\tilde{M}^{\dagger}}(\overline{f}_{\tilde{M}^{\dagger}})(\delta).$

Proposition 12.1.4 implies
$\overline{\epsilon_{\tilde{M}}(f)}=\epsilon_{\tilde{M}^{\dagger}}(\overline{f})$
for all $f$. Therefore, we see that Hypothesis 12.5.1 also holds for
$\tilde{M}^{\dagger}\subset\tilde{G}^{\dagger}$. Moreover, by denoting the
antipodal version of $\epsilon(\mathbf{M}^{!},\cdot)$ as
$\epsilon(\mathbf{M}^{!},\cdot)^{\dagger}$, we have
$\epsilon(\mathbf{M}^{!},\cdot)^{\dagger}=\overline{\epsilon(\mathbf{M}^{!},\cdot)}$.

Next, describe $M$ by a decomposition $W=\bigoplus_{j\in
J}(\ell_{j}\oplus\ell^{\prime}_{j})\oplus W^{\flat}$ into symplectic vector
subspaces. Consider the isomorphism $\Theta_{g}$ in Proposition 12.1.5 that is
adapted to $\tilde{M}$ (Lemma 12.1.6). Note that $g$ induces isomorphisms of
symplectic $F$-vector spaces

$(W,\langle\cdot|\cdot\rangle)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(W,-\langle\cdot|\cdot\rangle),\quad(W^{\flat},\langle\cdot|\cdot\rangle)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(W^{\flat},-\langle\cdot|\cdot\rangle)$

and is inner on each $\operatorname{GL}$-factor of $\tilde{M}$. The endoscopy
for metaplectic groups is defined in terms of linear algebraic data, when
$\psi$ is kept fixed; consequently, $\Theta_{g}$ transports the whole
formalism of endoscopy for $\tilde{M}\subset\tilde{G}$ to
$\tilde{M}_{-}\subset\tilde{G}_{-}$.

Identifying $\tilde{M}^{\dagger}\subset\tilde{G}^{\dagger}$ with
$\tilde{M}_{-}\subset\tilde{G}_{-}$ by Proposition 12.1.5, we conclude that
$\epsilon(\mathbf{M}^{!},\cdot)=\epsilon(\mathbf{M}^{!},\cdot)^{\dagger}$. The
assertion follows immediately. ∎

######  Corollary 12.5.6.

Under the inductive assumptions (12.5.1) and the key geometric Hypothesis
12.5.1, we have
$I_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
for all $\tilde{\gamma}\in\Gamma_{G\text{-reg}}(\tilde{M})$; equivalently,
$\epsilon_{\tilde{M}}(f)=0$.

In particular, the assertions in Theorem 12.3.3 hold under these assumptions.

###### Proof.

Combining Lemmas 12.5.4 and 12.5.4, we obtain
$\epsilon(\mathbf{M}^{!},\cdot)=0$ for all
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. This implies
$\epsilon_{\tilde{M}}(f)=0$, and the first part follows. As for the second
part, see the discussions after Proposition 12.3.6. ∎

In the local setting, we have reduced everything to the Hypothesis 12.5.1. Its
verification is the topic of §13.8.

######  Remark 12.5.7.

The proof of $\epsilon_{\tilde{M}}(f)=0$ given above can be compared with the
simple case treated in [61, p.1169] for twisted trace formulas. This should
not be over-generalized: the case of general coverings will require more
sophisticated techniques, provided that a theory of endoscopy is available.

##  Chapter 13 Global trace formula: the spectral side

This part marks the end of stabilization. Let $F$ be a number field and
consider a covering $\mathbf{p}:\tilde{G}\to G(\mathbb{A}_{F})$ of metaplectic
type, together with auxiliary data chosen as in §10. Take a finite set $V$ of
places of $F$ containing all ramified places.

Let us disrupt the ordering of sections and begin with §13.5. The spectral
side of the invariant trace formula [47] is phrased as

$I_{\mathrm{spec}}(f)=\sum_{t\geq 0}I_{t}(f),\quad
f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V})),$

where $t$ specifies the height of the imaginary part of infinitesimal
characters in the spectral expansion. Each $I_{t}$ can be decomposed into
$\sum_{\begin{subarray}{c}\nu\in\mathfrak{h}^{*}_{\mathbb{C}}/W_{\infty}\\\
\|\operatorname{Im}(\nu)\|=t\end{subarray}}I_{\nu}$, according to
infinitesimal characters.

Similarly to the geometric side $I_{\mathrm{geom}}$, these distributions are
“compressed” and it is also necessary to deal with the pre-compressed form,
built upon the discrete parts

$I_{\mathrm{disc},t}(\dot{f})=\sum_{\pi\in\Pi_{t,-}(\tilde{G}^{1})}a^{\tilde{G}}_{\mathrm{disc}}(\pi)\operatorname{tr}\pi(\dot{f}|_{\tilde{G}^{1}})$

where
$\dot{f}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G(\mathbb{A}_{F}))$,
as well as their counterparts $I^{\tilde{M}}_{\mathrm{disc},t}$ for various
$M\in\mathcal{L}(M_{0})$. We shall view $I_{\mathrm{disc},t}$ as a formal
linear combination of genuine irreducible characters of $\tilde{G}$. Its
evaluation at $\dot{f}$ makes sense; it is convenient to justify such
operations by employing the notion of _semi-finiteness_ (Definition 13.5.3)
from [61, X.5.3], which we adopt here. Let $\rho$ be such a formal linear
combination, then $\rho$ can be decomposed according to

  * •

the infinitesimal characters $\nu$, written as $\rho=\sum_{\nu}\rho_{\nu}$;

  * •

the Satake parameters $c^{V}$ off $V$, if we only look at the part
$\rho^{V\text{-nr}}$ unramified outside $V$, so that
$\rho^{V\text{-nr}}=\sum_{\nu,c^{V}}\rho_{\nu,c^{V}}$.

Applied to $\rho:=I_{\mathrm{disc},t}$, this yields

$I_{\mathrm{disc},\nu}(\dot{f}),\quad I_{\mathrm{disc},\nu,c^{V}}(\dot{f}).$

We also obtain distributions $I_{\mathrm{disc},\nu,c^{V}}$, etc. on
$\tilde{G}_{V}$, by evaluating the earlier distributions at
$\dot{f}:=f\prod_{v\notin V}f_{K_{v}}$ for
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$.

Given $\nu$, we can express $I_{\nu}(f)$ as

$I_{\nu}(f)=\sum_{M\in\mathcal{L}(M_{0})}\frac{|W^{M}_{0}|}{|W^{G}_{0}|}I_{\tilde{M}}^{\mathrm{glob}}\left(I^{\tilde{M}}_{\mathrm{disc},\nu},f\right),$

where $I_{\tilde{M}}^{\mathrm{glob}}$ is the (invariant) _global weighted
character_ to be defined in §13.3. The global weighted character involves the
following two ingredients introduced in §13.1:

  * •

the semi-local weighted characters $I_{\tilde{M}_{V}}(\pi_{V},\lambda,X,f)$
for not necessarily unitary representations
$\pi_{V}\in\Pi_{-}(\tilde{M}_{V})$;

  * •

the factors $r^{\tilde{G}}_{\tilde{M}}(c^{V})$ related to Satake parameters
$c^{V}$ of $\tilde{M}$, assuming $c^{V}$ is automorphic or merely quasi-
automorphic.

All these objects admit endoscopic counterparts. This step requires the
transfer of centers of universal enveloping algebras, obtained in [50], as
well as the spherical fundamental lemma [56]. The stabilization of the
spectral side of the trace formula is stated in §13.6. It boils down to the
stabilization of $I_{\tilde{M}}^{\mathrm{glob}}$ and
$I_{\mathrm{disc},\nu,c^{V}}$.

The case of $I_{\tilde{M}}^{\mathrm{glob}}$ is relatively easy. One ingredient
is a special case of the stabilization of local weighted characters, settled
in §13.2; we remark that the general case will be deduced from the local
geometric Theorem 5.3.11. The other ingredient is the stabilization of
$r^{\tilde{G}}_{\tilde{M}}(c^{V})$: this is some sort of spectral weighted
fundamental lemma, and can be proved directly by the method in §8.8.

The case of $I_{\mathrm{disc},\nu,c^{V}}$ is the hardcore of this work. In
Lemma 13.7.5, we will use the available stabilization for
$A^{\tilde{G}}(V,\mathcal{O})$ to reduce it to the stabilization of
$I_{\mathrm{geom}}$. Recall that the stabilization of $I_{\mathrm{geom}}$ is a
consequence of the key geometric Hypothesis 12.5.1, as already seen in §12.5.

In §13.7, we pick $M\in\mathcal{L}(M_{0})$ and impose local and global
inductive assumptions on various matching theorems. We then prove the
stabilization of $I_{\mathrm{disc},\nu}$ in Proposition 13.7.7 for
$f=\prod_{v}f_{v}$ which is $\tilde{M}$-cuspidal (Definition 12.4.6) at two
distinct places, and has vanishing singular orbital integrals at some non-
Archimedean place.

In §13.8 a local-global argument reduces the key geometric Hypothesis 12.5.1
to the aforementioned special case of spectral stabilization. This will enable
us to prove all the remaining matching theorems in this work, thereby
completing the long induction process. The remaining steps are summarized in
§13.9.

The strategy sketched above roughly follows [6], but also incorporates some
viewpoints from [61]. However, the result of Finis–Lapid–Müller [30] on the
convergence of $I_{\mathrm{spec}}(f)$ has not been written down for coverings.
Therefore, instead of following the paradigm of [61], we revert to Arthur’s
original approach via the multiplier convergence estimates, which suffices for
our purposes here.

###  13.1 Invariant weighted characters

We will extend the definition of weighted characters in §7.1, §8.2 to non-
unitary case. Next, we will introduce the invariant weighted characters and
their semi-local counterparts, for not necessarily unitary representations.
The semi-local version will intervene in the spectral side of the invariant
trace formula.

####  13.1.1 Local version

To begin with, consider a local field $F$ of characteristic zero. We work with
a covering of metaplectic type

$\mathbf{p}:\tilde{G}\to G(F),\quad G=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W).$

We also fix a symplectic basis for $(W,\langle\cdot|\cdot\rangle)$, the
resulting minimal Levi subgroup $M_{0}$ and positive-definite invariant
quadratic form on $\mathfrak{a}_{0}$, and a maximal compact subgroups
$K\subset G(F)$ in good position relative to $M_{0}$.

Below is a review of [21] or [61, X.4.2] in the metaplectic setting. Let
$M\in\mathcal{L}(M_{0})$.

Non-Archimedean case

Define $\mathfrak{a}_{M,F}$ and its Pontryagin dual
$i\mathfrak{a}^{*}_{M,F}=i\mathfrak{a}_{M}^{*}/i\mathfrak{a}_{M,F}^{\vee}$ as
in §7.1. Set $\mathfrak{a}^{*}_{M,F,\mathbb{C}}$ to be the $\mathbb{C}$-torus
$\mathfrak{a}_{M,\mathbb{C}}^{*}/i\mathfrak{a}_{M,F}^{\vee}$.

Archimedean case

Define $\mathfrak{a}_{M,F}:=\mathfrak{a}_{M}$ and its Pontryagin dual
$i\mathfrak{a}^{*}_{M,F}:=i\mathfrak{a}^{*}_{M}$. Set
$\mathfrak{a}^{*}_{M,F,\mathbb{C}}$ to be the $\mathbb{C}$-vector space
$\mathfrak{a}^{*}_{M,\mathbb{C}}$.

In each case, we have $\mathfrak{a}_{M,F}=H_{M}(M(F))$. Equip
$\mathfrak{a}_{M,F}$ and $i\mathfrak{a}^{*}_{M,F}$ with dual Haar measures; in
the non-Archimedean case, we use the counting measure on $\mathfrak{a}_{M,F}$

Let $\pi\in\Pi_{-}(\tilde{M})$. The associated canonically normalized weighted
character is

$J_{\tilde{M}}(\pi_{\lambda},f)=J_{\tilde{M}}^{\tilde{G}}(\pi_{\lambda},f),\quad
f\in
C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G(F)),$

viewed as a meromorphic (resp. rational) family as $\lambda$ varies over
$\mathfrak{a}^{*}_{M,F,\mathbb{C}}$. For generic $\lambda$ such that
$\lambda+i\mathfrak{a}^{*}_{M,F}$ is disjoint from the singularities, one
defines

$\displaystyle J_{\tilde{M}}(\pi,\lambda,X,f)$
$\displaystyle=J_{\tilde{M}}^{\tilde{G}}(\pi,\lambda,X,f)$
$\displaystyle:=\int_{\lambda+i\mathfrak{a}^{*}_{M,F}}J_{\tilde{M}}(\pi_{\mu},f)e^{-\langle\mu,X\rangle}\mathop{}\\!\mathrm{d}\mu$

where $X\in\mathfrak{a}_{M}$. By shifting contours, $\lambda\mapsto
J_{\tilde{M}}(\pi,\lambda,X,f)$ is seen to be locally constant around each
point where it is defined. Also, $J_{\tilde{M}}(\pi,\lambda,X,\cdot)$ is
concentrated at $H_{\tilde{G}}(\cdot)=X_{G}$, where $X_{G}$ is the projection
of $X$ along the natural map $\mathfrak{a}_{M,F}\to\mathfrak{a}_{G,F}$. Thus
$J_{\tilde{M}}(\pi,\lambda,X,\cdot)$ extends to
$C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G(F))$.

The invariant version
$I_{\tilde{M}}(\pi,\lambda,X,f)=I^{\tilde{G}}_{\tilde{M}}(\pi,\lambda,X,f)$ is
constructed inductively by

$\sum_{L\in\mathcal{L}(M_{0})}I^{\tilde{L}}_{\tilde{M}}\left(\pi,\lambda,X,\phi_{\tilde{L}}(f)\right)=J_{\tilde{M}}(\pi,\lambda,X,f)$

for $\lambda$ off the singular hyperplanes (infinitely many for Archimedean
$F$). They are concentrated at $H_{\tilde{G}}(\cdot)=X_{G}$, thus defined for
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\mathrm{ac}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G(F))$.
As usual, $\phi_{\tilde{L}}$ is defined in terms of canonically normalized
tempered weighted characters.

In the case $\pi\in\Pi_{\mathrm{unit},-}(\tilde{M})$, the weighted character
$J_{\tilde{M}}(\pi_{\lambda},f)$ is analytic for $\lambda\in
i\mathfrak{a}^{*}_{M,F}$, and the distributions

$J_{\tilde{M}}(\pi,X,f):=J_{\tilde{M}}(\pi,0,X,f),\quad
I_{\tilde{M}}(\pi,X,f):=I_{\tilde{M}}(\pi,0,X,f)$ (13.1.1)

have been introduced in §7.1 and §8.2. These definitions also extend by
linearity to the case when $\pi$ is merely a virtual genuine representation.
Their behavior under parabolic induction is described as follows.

######  Proposition 13.1.1.

Suppose that $R\in\mathcal{L}^{M}(M_{0})$ and $\sigma$ is a virtual genuine
representation of $\tilde{R}$. Then its normalized parabolic induction
$\sigma^{\tilde{M}}$ to $\tilde{M}$ satisfies

$I_{\tilde{M}}\left(\sigma^{\tilde{M}},\lambda,X,f\right)=\sum_{L\in\mathcal{L}(R)}d^{G}_{R}(L,M)\int_{\begin{subarray}{c}Y\in\mathfrak{a}_{R,F}\\\
Y_{M}=X\end{subarray}}I^{\tilde{L}}_{\tilde{R}}(\sigma,\lambda,Y,f_{\tilde{L}})\mathop{}\\!\mathrm{d}Y$

for all $\lambda$ off the singularities, $X$ and $f$.

###### Proof.

Same as [61, p.1184]. It is of an analytic-combinatorial nature. ∎

####  13.1.2 Semi-local version

Now let $F$ be a number field and consider

$\mathbf{p}:\tilde{G}\to G(\mathbb{A}_{F}),\quad G=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W)$

as in §10.2. Denote by $\tilde{G}_{\infty}$ the preimage of
$\prod_{v\mid\infty}G(F_{v})$ under $\mathbf{p}$. We also fix a symplectic
basis for $(W,\langle\cdot|\cdot\rangle)$, the resulting $M_{0}$, etc. as well
as maximal compact subgroups $K_{v}\subset G(F_{v})$ in good position relative
to $M_{0}$ for each place $v$, such that $K_{v}$ is hyperspecial whenever
$v\notin V_{\mathrm{ram}}$.

Take a finite set $V$ of places of $F$ with the _closure property_ [47,
Définition 3.1], namely

  * •

either $V$ contains an Archimedean place,

  * •

or that all $v\in V$ are non-Archimedean with the same residual
characteristic.

In view of the closure property, one can still define $\mathfrak{a}_{M,V}$,
its Pontryagin dual $i\mathfrak{a}^{*}_{M,V}$, and
$\mathfrak{a}^{*}_{M,V,\mathbb{C}}$ as in the local case, by setting
$\mathfrak{a}_{M,V}:=H_{M}(M(F_{V}))$. In any case, for $v\in V$ we have
natural maps

$\mathfrak{a}_{M,F_{v}}\hookrightarrow\mathfrak{a}_{M,V}\hookrightarrow\mathfrak{a}_{M},\quad\mathfrak{a}^{*}_{M,\mathbb{C}}\twoheadrightarrow\mathfrak{a}^{*}_{M,V,\mathbb{C}}\twoheadrightarrow\mathfrak{a}^{*}_{M,F_{v},\mathbb{C}}.$

Let $M\in\mathcal{L}(M_{0})$. Consider

  * •

$\pi_{V}\in\Pi_{-}(\tilde{M}_{V})$,

  * •

$X\in\mathfrak{a}_{M,V}$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$,

  * •

$\lambda\in\mathfrak{a}^{*}_{M,V,\mathbb{C}}$ in general position, that is,
lying off the singular hyperplanes.

Iterating the constructions in the local case, we get the semi-local weighted
characters

$J_{\tilde{M}_{V}}(\pi_{V,\lambda},f)=J_{\tilde{M}_{V}}^{\tilde{G}_{V}}(\pi_{V,\lambda},f),\quad
f\in
C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V})),$

meromorphic in $\lambda$ in general position, from which we construct

$\displaystyle
J_{\tilde{M}_{V}}\left(\pi_{V},\lambda,X,f\right)=J^{\tilde{G}_{V}}_{\tilde{M}_{V}}\left(\pi_{V},\lambda,X,f\right),$
$\displaystyle
I_{\tilde{M}_{V}}\left(\pi_{V},\lambda,X,f\right)=I^{\tilde{G}_{V}}_{\tilde{M}_{V}}\left(\pi_{V},\lambda,X,f\right).$

As before, they are locally constant in $\lambda$, concentrated at
$H_{\tilde{G}}(\cdot)=X_{G}\in\mathfrak{a}_{G,V}$, and extend to $f$ in
$C^{\infty}_{\dashrule[.7ex]{2222}{.4},\mathrm{ac}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$
and
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\mathrm{ac}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$,
respectively.

The following splitting formula is the spectral analogue of Proposition
10.3.1. The only difference comes from the parameter $X$.

######  Proposition 13.1.2.

Let $\pi_{V}=\bigotimes_{v}\pi_{v}\in\Pi_{-}(\tilde{M}_{V})$ and
$f=\prod_{v}f_{v}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$.
For all $X\in\mathfrak{a}_{M,V}$ and
$\lambda\in\mathfrak{a}^{*}_{M,V,\mathbb{C}}$ off singularities, let

$\mathcal{A}_{X}:=\left\\{(X_{v})_{v}\in\prod_{v\in
V}\mathfrak{a}_{M,F_{v}}:\sum_{v\in V}X_{v,M}=X\right\\}.$

Define $M^{V}$ so that $M^{v}=M$ for all $v\in V$. Then we have

$I_{\tilde{M}_{V}}(\pi_{V},\lambda,X,f)=\sum_{L^{V}\in\mathcal{L}(M^{V})}d^{G}_{M^{V}}\left(M,L^{V}\right)\int_{\mathcal{A}_{X}}\prod_{v\in
V}I^{\tilde{L}^{v}}_{\tilde{M}^{v}}(\pi_{v},\lambda,X_{v},f_{v,\tilde{L}^{v}})\mathop{}\\!\mathrm{d}(X_{v})_{v}$

for an appropriate measure on $\mathcal{A}_{X}$, where we used the maps
$\mathfrak{a}^{*}_{M,V,\mathbb{C}}\to\mathfrak{a}^{*}_{M,F_{v},\mathbb{C}}$.

###### Proof.

Same as [61, p.1198]. ∎

For coverings, the following result can be found in [47, §4.1].

######  Definition–Proposition 13.1.3.

Let $\pi_{V}\in\Pi_{\mathrm{unit},-}(\tilde{M}_{V})$. Then

$J_{\tilde{M}_{V}}(\pi_{V},\lambda,\cdot,\cdot),\quad
I_{\tilde{M}_{V}}(\pi_{V},\lambda,\cdot,\cdot)$

are both defined for $\lambda$ in some neighborhood of
$i\mathfrak{a}^{*}_{M,V}$. In this case, we define

$J_{\tilde{M}_{V}}(\pi_{V},X,f):=J_{\tilde{M}_{V}}(\pi_{V},0,X,f),\quad
I_{\tilde{M}_{V}}(\pi_{V},X,f):=I_{\tilde{M}_{V}}(\pi_{V},0,X,f).$ (13.1.2)

The above is just the semi-local version of (13.1.1). Also note that for
unitary $\pi_{V}$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V},\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$,
one can define

$I_{\tilde{M}_{V}}(\pi_{V},f):=\int_{\mathfrak{a}_{M,V}}I_{\tilde{M}_{V}}(\pi_{V},X,f)\mathop{}\\!\mathrm{d}X.$

Finally, for a quasisplit group $G^{!}$, its Levi subgroup $M^{!}$, and a
stable virtual character $\pi_{V}^{!}$ of $M^{!}(F_{V})$, we have the stable
counterpart

$S^{G^{!}_{V}}_{M^{!}_{V}}(\pi^{!}_{V},\lambda,X,\cdot),$

We refer to [61, X.4.3] for the precise construction and properties.

######  Remark 13.1.4.

The invariant distributions $I_{\tilde{M}_{V}}(\pi_{V},\lambda,X,f)$ also
appeared in [47, Hypothèse 4.17], albeit with a minor difference. In that
context, one also fixes a connected Lie subgroup $A\subset A_{G,\infty}$,
isomorphic to $\mathfrak{a}\subset A_{G,\infty}$ under $H_{G}$. Endow
$\mathfrak{a}$ (thus $A$) with the Haar measure determined by the chosen
positive-definite invariant quadratic form. In defining
$J_{\tilde{M}_{V}}(\pi_{V},\lambda,X,\cdot)$ and
$I_{\tilde{M}_{V}}(\pi_{V},\lambda,X,\cdot)$, the assumptions become:

  * •

$V$ contains all Archimedean places (so that
$\mathfrak{a}_{M,V}=\mathfrak{a}_{M}$);

  * •

$\pi_{V}\in\Pi_{-}(\tilde{M}_{V}/A)$;

  * •

$X\in\mathfrak{a}_{M}/\mathfrak{a}$ and $f^{A}\in
C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G}_{V}/A,\tilde{K}_{V})\otimes\operatorname{mes}(G(F_{V}))$
or with $C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}$ replaced by
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}$;

  * •

$\lambda\in\mathfrak{a}^{*}_{M,\mathbb{C}}$ is orthogonal to $\mathfrak{a}$
and lies in general position.

For example, in this setting we put

$\displaystyle J_{\tilde{M}_{V}}(\pi_{V},\lambda,X,f^{A})$
$\displaystyle=J^{\tilde{G}_{V}}_{\tilde{M}_{V}}(\pi,\lambda,X,f^{A})$
$\displaystyle:=\int_{\lambda+i\mathfrak{a}^{*}_{M}\cap\mathfrak{a}^{\perp}}J_{\tilde{M}_{V}}(\pi_{\mu},f^{A})e^{-\langle\mu,X\rangle}\mathop{}\\!\mathrm{d}\mu.$

In particular, for $M=G$ and $A=A_{G,\infty}$, the integral degenerates to
$I^{\tilde{G}_{V}}(\pi,f^{A})$.

When $\pi_{V}$ is unitary, $J_{\tilde{M}_{V}}(\pi_{V},\lambda,X,f^{A})$ and
$I_{\tilde{M}}(\pi_{V},\lambda,X,f^{A})$ are defined for $\lambda$ in some
neighborhood of $i\mathfrak{a}^{*}_{M}$. The same constructions applies in the
stable context, for all $M^{!}\subset G^{!}$ and $A^{!}\subset
A_{G^{!},\infty}$.

The study of these distributions can be reduced to the case $A=\\{1\\}$, and
then to the local case. We will return to this issue in §13.5.

######  Remark 13.1.5.

The foregoing constructions of $J^{\tilde{G}}_{\tilde{M}}$,
$I^{\tilde{G}}_{\tilde{M}}$, etc. apply to all adélic coverings subject to the
requirements in [47] (in particular, the Hypothèse 3.12 therein).

###  13.2 Local spectral stabilization

We continue to work in the local setting of §13.1. We will also use the maps
${}^{c}\phi_{\tilde{M}}$, ${}^{c}\theta_{\tilde{M}}$ defined in §7.2 and §8.3;
they intervened in studying the compactly supported version of invariant
weighted orbital integrals ${}^{c}I_{\tilde{M}}(\tilde{\gamma},\cdot)$. The
focus here is on the spectral aspect.

Let $\pi$ be a genuine irreducible representation of $\tilde{M}$, or more
generally a genuine virtual character. The genuine invariant distributions
${}^{c}I_{\tilde{M}}(\pi,\lambda,\cdot)={}^{c}I^{\tilde{G}}_{\tilde{M}}(\pi,\lambda,\cdot)$
are characterized by

$\sum_{L\in\mathcal{L}(M)}{}^{c}I^{\tilde{L}}_{\tilde{M}}(\pi,\lambda,{}^{c}\phi_{\tilde{L}}(f))=J_{\tilde{M}}(\pi_{\lambda},f);$

they are meromorphic in $\lambda$. We also have

$\displaystyle{}^{c}I_{\tilde{M}}(\pi,\lambda,X,f)$
$\displaystyle={}^{c}I^{\tilde{G}}_{\tilde{M}}(\pi,\lambda,X,f)$
$\displaystyle=\int_{\lambda+i\mathfrak{a}^{*}_{M,F}}e^{-\langle\mu,X\rangle}\cdot{}^{c}I_{\tilde{M}}(\pi,\mu,f)\mathop{}\\!\mathrm{d}\mu,$

which is locally constant in $\lambda$ off the singularities.

Recall that
${}^{c}\theta_{\tilde{M}}={}^{c}\theta^{\tilde{G}}_{\tilde{M}}:\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G(F))\to\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\mathrm{ac}}(\tilde{M})\otimes\operatorname{mes}(M(F))$
satisfies

$\phi_{\tilde{M}}=\sum_{L\in\mathcal{L}(M)}{}^{c}\theta^{\tilde{L}}_{\tilde{M}}\circ{}^{c}\phi_{\tilde{L}}.$

In Propositions 7.2.5 and 8.5.4, it is also shown that when $\pi$ is tempered,
$X\mapsto I^{\tilde{M}}(\pi,X,{}^{c}\theta_{\tilde{M}}(f))$ is the Fourier
transform of a function on $i\mathfrak{a}^{*}_{M,F}$, denoted reasonably by
$\lambda\mapsto
I^{\tilde{M}}\left(\pi_{\lambda},{}^{c}\theta_{\tilde{M}}(f)\right)$
hereafter, which extends meromorphically to
$\mathfrak{a}^{*}_{M,F,\mathbb{C}}$.

######  Lemma 13.2.1.

For all $\pi$, $X$ and $f$, we have

${}^{c}I_{\tilde{M}}(\pi,\lambda,X,f)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{L}}_{\tilde{M}}\left(\pi,\lambda,X,{}^{c}\theta_{\tilde{L}}(f)\right)$

for $\lambda$ in general position. If $\pi$ is tempered, then we have

${}^{c}I_{\tilde{M}}(\pi,\lambda,f)=I^{\tilde{M}}\left(\pi_{\lambda},{}^{c}\theta_{\tilde{M}}(f)\right).$

###### Proof.

Same as [61, pp.1186–1187]. It is based on standard properties of the maps
${}^{c}\theta_{\tilde{M}}$, etc., having nothing to do with special properties
of the covering. ∎

Consider now a quasisplit group $G^{!}$ with Levi subgroup $M^{!}$. In [61,
X.4.3], one defined the stable counterparts

${}^{c}S^{G^{!}}_{M^{!}}(\pi^{!},\lambda,X,\cdot)$

of the aforementioned distributions, where $\pi^{!}$ is now a stable virtual
representation of $M^{!}(F)$, $X\in\mathfrak{a}_{M,F}$ and
$\lambda\in\mathfrak{a}^{*}_{M,F,\mathbb{C}}$ (off the singularities). Their
stability is established in loc. cit. In the next definition, we take $M^{!}$
(resp. $G^{!}=G^{!}[s]$) arising from
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ (resp.
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$).

######  Definition 13.2.2.

Let $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\pi^{!},\lambda,X$ be as above. Identify $\mathfrak{a}_{M,F}$ and
$\mathfrak{a}_{M^{!},F}$ (resp. $\mathfrak{a}^{*}_{M,F,\mathbb{C}}$ and
$\mathfrak{a}^{*}_{M^{!},F,\mathbb{C}}$) by endoscopy. Consider a stable
virtual character $\pi^{!}$ of $M^{!}(F)$. For all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G(F))$,
put $f^{G^{!}[s]}:=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)$ and define

$\displaystyle I^{\mathcal{E}}_{\tilde{M}}(\pi^{!},\lambda,X,f)$
$\displaystyle=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\pi^{!},\lambda,X,f)$
$\displaystyle:=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])S^{G^{!}[s]}_{M^{!}}(\pi^{!}[s],\lambda,X,f^{G^{!}[s]}),$
$\displaystyle{}^{c}I^{\mathcal{E}}_{\tilde{M}}(\pi^{!},\lambda,X,f)$
$\displaystyle={}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\pi^{!},\lambda,X,f)$
$\displaystyle:=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s]){}^{c}S^{G^{!}[s]}_{M^{!}}(\pi^{!}[s],\lambda,X,f^{G^{!}[s]}).$

A more consistent notation would be
$I^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\pi^{!},\lambda,X,f)$ and so on.
The only reason for not doing so is typographic.

Recall Definition 7.4.1 for ${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\cdot)$
and ${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\cdot)$. The
equality ${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}={}^{c}\theta_{\tilde{M}}$ is
the content of Theorems 7.4.8 and 8.10.2, to be proved in §13.9.

######  Lemma 13.2.3.

Assume that the local geometric Theorem 5.3.11 holds when $M$ (resp. $G$) is
replaced by any $L\in\mathcal{L}(M)$ with $L\neq M$ (resp. $L\neq G$). Then

${}^{c}I^{\mathcal{E}}_{\tilde{M}}\left(\pi^{!},\lambda,X,f\right)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{L},\mathcal{E}}_{\tilde{M}}\left(\pi^{!},\lambda,X,{}^{c}\theta_{\tilde{L}}(f)\right)\\\
+I^{M^{!}}\left(\pi^{!},\lambda,X,{}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},f)-{}^{c}\theta_{\tilde{M}}(f)^{M^{!}}\right),$

where
${}^{c}\theta_{\tilde{M}}(f)^{M^{!}}:=\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}\left({}^{c}\theta_{\tilde{M}}(f)\right)$.

###### Proof.

By the same combinatorial argument as in the first step in proving Proposition
7.4.9, the left hand side gets transformed into

$\sum_{L\in\mathcal{L}^{G}(M)}\sum_{s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})}\sum_{s_{L}\in\mathcal{E}_{\mathbf{L}^{!}[s^{L}]}(\tilde{G})}i_{M^{!}}(\tilde{L},L^{!}[s^{L}])i_{L^{!}[s^{L}]}(\tilde{G},G^{!}[s_{L}])\\\
S^{L^{!}[s^{L}]}_{M^{!}}\left(\pi^{!}[s^{L}],\lambda,X,{}^{c}S\theta^{G^{!}[s_{L}]}_{L^{!}[s^{L}]}\left(f^{G^{!}[s_{L}]}\right)[s_{L}]\right)$

as $z[s_{L}]$ is central in $L^{!}[s^{L}]$. This equals

$\sum_{L\in\mathcal{L}^{G}(M)}\sum_{s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})}i_{M^{!}}(\tilde{L},L^{!}[s^{L}])S^{L^{!}[s^{L}]}_{M^{!}}\left(\pi^{!}[s^{L}],\lambda,X,{}^{c}\theta^{\mathcal{E}}_{\tilde{L}}(\mathbf{L}^{!}[s^{L}],f)\right)\\\
=\sum_{L\in\mathcal{L}^{G}(M)}I^{\tilde{L},\mathcal{E}}_{\tilde{M}}\left(\pi^{!},\lambda,X,{}^{c}\theta^{\mathcal{E}}_{\tilde{L}}(f)\right).$