SD_{\mathrm{unip}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})^{\vee}\quad\text{that
maps to both}\\\ &\sigma^{G^{1}}_{J_{s}}(\delta_{\epsilon})\quad\text{and}\\\
&\left(Z_{\check{G}^{1}}^{\Gamma_{F}}:Z_{{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\right)e^{G^{!}_{J}[t]}_{M^{!}}(\epsilon[t])e^{G^{!}[s]}_{M^{!}}(\epsilon[s])^{-1}\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)^{-1}\sigma^{G^{2}}_{J_{t}}(\delta_{\epsilon}).\end{split}$
(6.4.5)

This is a variant of [60, V.5.6 (6)]. Also note that $J_{s}$ and $J_{t}$ are
the same lattice in $\mathfrak{a}_{M^{!}}^{*}$.

Before proving (6.4.5), let us prove Lemma 6.4.4 first.

###### Proof of Lemma 6.4.4.

Suppose that $J\in\mathcal{J}^{G}_{M}\smallsetminus\\{J_{\max}\\}$ and
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ maps to
$t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$. For each
$\alpha\in\Sigma(G^{!}[s]_{\epsilon[s]},T^{!})$, we write

$\beta:=B(\alpha)^{-1}\alpha\;\in
X^{*}(T^{!}_{\overline{F}})\otimes\mathbb{Q}.$

By our choice of $B$-functions in §5.3, $\alpha\mapsto\beta$ transforms a root
system of type $\mathrm{B}_{n}$ into type $\mathrm{C}_{n}$, and $\alpha$ is
short if and only if $\beta$ is long.

Hereafter, we assume $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ for the ease of
notations. In this case
$Z_{\tilde{G}^{\vee}}^{\circ}=Z_{\widetilde{G_{J}}^{\vee}}^{\circ}=\\{1\\}$.
Viewing $s$ and $t$ as elements in $\tilde{M}^{\vee}$, we infer from $s\mapsto
t$ that $s=t$.

By Remark 3.6.3, the chosen diagram $(\epsilon,B^{!},T^{!},B,T,\eta)$ induces
$X^{*}(T^{!}_{\overline{F}})\simeq X^{*}(T_{\overline{F}})\simeq
X_{*}(T^{\vee})$ where $T^{\vee}\subset\tilde{G}^{\vee}$ is part of the root
datum for $\tilde{G}^{\vee}$. In what follows, $\alpha\in
X^{*}(T^{!}_{\overline{F}})\simeq X_{*}(T^{\vee})$ denotes a priori a coroot
for $\tilde{G}^{\vee}\supset T^{\vee}$,. Denote by $\check{\alpha}\in
X^{*}(T^{\vee})$ the corresponding root. Then the image of $\beta$ by
$X^{*}(T_{\overline{F}})\simeq X^{*}(T^{!}_{\overline{F}})$ is in
$\Sigma(G,T)$. The sets $\Sigma(G^{i},T^{!})$ arise from these $\alpha$ as
follows ($i=1,2$).

  * •

The roots of $G^{1}\supset T^{!}$ are characterized by

    1. 1.

$\check{\alpha}(s)=1$ (so that $\alpha\in\Sigma(G^{!}[s],T^{!})$),

    2. 2.

$\alpha(\epsilon[s])=1$ (so that
$\alpha\in\Sigma(G^{!}[s]_{\epsilon[s]},T^{!})$),

    3. 3.

$\beta|_{\mathfrak{a}_{M^{!}}}\in R_{J_{s}}$ (so that
$\alpha\in\Sigma(G^{1},T^{!})$); this is tantamount to
$\xi^{*}(\beta|_{\mathfrak{a}_{M^{!}}})\in R_{J}$.

Moreover, let
$\Psi:X^{*}(T^{!}_{\overline{F}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X_{*}(T^{!}_{\overline{F}})$
be induced from the invariant quadratic form chosen in §3.1 and the diagram;
its effect is $e^{*}_{i}\mapsto e_{i,*}$ in the standard bases of
$X^{*}(T^{\vee})$ and $X_{*}(T^{\vee})$ in §3.1. The coroot of
$\alpha\in\Sigma(G^{1},T^{!})$ can be taken on the level of
$\Sigma(G^{!}[s],T^{!})$.

Via the descriptions of $B$ and $G^{!}[s]$ (a product of odd
$\operatorname{SO}$), the coroot of $\alpha$ is seen to be
$B(\alpha)^{-1}\Psi(\alpha)$; i.e. short roots become long.

  * •

The roots of $G^{2}\supset T^{!}$ are characterized by

    1. 1.

$\xi^{*}(\beta|_{\mathfrak{a}_{M^{!}}})\in R_{J}$ (so that
$\beta\in\Sigma(G_{J},T)$, or equivalently
$\alpha\in\check{\Sigma}(\widetilde{G_{J}}^{\vee},T^{\vee})$),

    2. 2.

$\check{\alpha}(t)=1$ (so that $\alpha\in\Sigma(G_{J}^{!}[t],T^{!})$),

    3. 3.

$\alpha(\epsilon[t])=1$ (so that $\alpha\in\Sigma(G^{2},T^{!})$).

The coroot of $\alpha\in\Sigma(G^{2},T^{!})$ can be taken in
$\Sigma(G_{J}^{!}[t],T^{!})$. Since $G^{!}_{J}[t]$ is a product of odd
$\operatorname{SO}$ and the effect of $B_{J}(\cdot)^{-1}$ is to make short
roots long, the coroot of $\alpha$ is seen to be
$B_{J}(\alpha)^{-1}\Psi(\alpha)$.

  * •

As $s=t$ and $\epsilon[s]=\epsilon[t]$, it follows from the above that
$\Sigma(G^{1},T^{!})=\Sigma(G^{2},T^{!})$.

We contend that the vertical arrows below of rescaling

${\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!}\right)}$${\Sigma(G^{1},T^{!})}$${\Sigma(G^{2},T^{!})}$${\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)}$${\Sigma(G^{2},T^{!},B_{J})}$$\scriptstyle{\cdot
B(\cdot)^{-1}}$$\scriptstyle{\eqref{eqn:J-G1}}$$\scriptstyle{\cdot
B_{J}(\cdot)^{-1}}$

coincide. Indeed, this follows from the description of $G_{J}$ (Propositions
6.1.3, 6.1.4) and the effects of $B(\cdot)^{-1}$ and $B_{J}(\cdot)^{-1}$ (see
the earlier discussion on coroots). Lemma 6.4.4 follows. ∎

Here is a by-product of the proof: we have actually shown that

$\Sigma(G^{1},T^{!})=\Sigma(G^{2},T^{!}),\quad\check{\Sigma}(G^{1},T^{!})=\check{\Sigma}(G^{2},T^{!}).$

###### Proof of Lemma 6.4.7.

In order to verify (6.4.5), it suffices to prove:

  1. (i)

there exists an isomorphism $G^{1}\simeq G^{2}$, compatible with the
embeddings of $M^{!}_{\epsilon[s]}=M^{!}_{\epsilon[t]}$ in both sides;

  2. (ii)

we have

$\left(Z_{\check{G}^{1}}^{\Gamma_{F}}:Z_{{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\right)e^{G_{J}^{!}[t]}_{M^{!}}(\epsilon[t])e^{G^{!}[s]}_{M^{!}}(\epsilon[s])^{-1}\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)^{-1}=1.$

To prove (i), observe that $G^{1}$ and $G^{2}$ share the maximal $F$-torus
$T^{!}$, and have the same roots and coroots; this is the by-product of the
proof of Lemma 6.4.4.

As for the compatibility of the embeddings of
$M^{!}_{\epsilon[s]}=M^{!}_{\epsilon[t]}$, note that the images of
$\Sigma(M^{!},T^{!})$ in $\Sigma(G^{i},T^{!})$ have the same characterizations
($i=1,2$), namely $\beta|_{A_{M^{!}}}=1$,
$\check{\alpha}(s)=\check{\alpha}(t)=1$, and
$\alpha(\epsilon[s])=\alpha(\epsilon[t])=1$ in the notation of the proof of
Lemma 6.4.4.

To prove (ii), we may assume $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$. Recall
that $Z_{G^{!}[s]^{\vee}}\subset Z_{G^{!}_{J}[t]^{\vee}}$ as subgroups of
$Z_{{M^{!}}^{\vee}}$ by the proof of Lemma 6.4.4. Consider the diagram

${Z_{{M^{!}}^{\vee}}\bigg{/}Z_{{G^{!}[s]}^{\vee}}}$${Z_{{M^{!}_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\bigg{/}Z_{{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}}$${Z_{{M^{!}_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\bigg{/}Z_{{G^{!}[s]_{\epsilon[s],J_{s}}}^{\vee}}^{\Gamma_{F}}}$${Z_{{M^{!}_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\bigg{/}Z_{\check{G}^{1}}^{\Gamma_{F}}}$${Z_{{M^{!}}^{\vee}}\bigg{/}\mathcal{A}}$${Z_{{M^{!}}^{\vee}}\bigg{/}Z_{{G_{J}^{!}[t]}^{\vee}}}$${Z_{{M^{!}_{\epsilon[t]}}^{\vee}}^{\Gamma_{F}}\bigg{/}Z_{{G_{J}^{!}[t]_{\epsilon[t],J_{t}}}^{\vee}}^{\Gamma_{F}}}$${Z_{{M^{!}_{\epsilon[t]}}^{\vee}}^{\Gamma_{F}}\bigg{/}Z_{\check{G}^{2}}^{\Gamma_{F}}}$$\scriptstyle{\varphi}$quot$\scriptstyle{\because\;\text{(i)}}$quotquot$\scriptstyle{\varphi}$

where $\mathcal{A}$ is as in (6.4.3), $\varphi$ stands for the homomorphisms
in Definition 4.6.3 and quot denotes the evident quotient homomorphism. All
arrows are surjective with finite kernels.

The diagram commutes. Computing the cardinality of the kernel in two ways, we
obtain

$\left(Z_{{G^{!}[s]}^{\vee}}:\mathcal{A}\right)e^{G^{!}[s]}_{M^{!}}(\epsilon[s])^{-1}\left(Z_{\check{G}^{1}}^{\Gamma_{F}}:Z_{{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\right)=\left(Z_{{G^{!}_{J}[t]}^{\vee}}:\mathcal{A}\right)e^{G_{J}^{!}[t]}_{M^{!}}(\epsilon[t])^{-1},$

hence

$e^{G^{!}[s]}_{M^{!}}(\epsilon[s])^{-1}\left(Z_{\check{G}^{1}}^{\Gamma_{F}}:Z_{{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\right)=\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)e^{G_{J}^{!}[t]}_{M^{!}}(\epsilon[t])^{-1}.$

This is exactly (ii). ∎

######  Remark 6.4.10.

In the counterpart of (ii) in [60, V.5.6], one has to treat a case of type
$\mathrm{A}_{2n}$ in which $G^{1}$ is related to $G^{2}$ by nonstandard
endoscopy. Here we are in the “typical case”, for which the nonstandard
endoscopic datum in question is tautological.

###  6.5 Summary for general local fields

As before, let $\mathbf{p}:\tilde{G}\to G(F)$ be of metaplectic type and let
$M\subset G$ be a Levi subgroup. Here $F$ can be any local field of
characteristic zero.

The key observation is that the methods of §6.3 also work for non-Archimedean
$F$. Given such a local field $F$, the method in §§6.3–6.4 can be simplified
in the following manner.

  1. 1.

One replaces all $D_{\text{tr-orb},-}$, $SD_{\text{tr-orb}}$ by
$D_{\mathrm{geom},-}$, $SD_{\mathrm{geom}}$, respectively. The objects (O1) —
(O4) are thus all unconditionally defined.

  2. 2.

Among the conditions (C1) — (C8), only (C2) (i.e. stabilization of
$I_{\tilde{M}}(\tilde{\gamma},\cdot)$) and (C3) (i.e. stabilization of
$\rho^{\tilde{G}}_{J}$) need proof. Again, we are reduced to proving that

$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)\mapsto\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$

for all $J\in\mathcal{J}^{G}_{M}$,
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, elliptic
$\mathcal{O}^{!}\subset M^{!}(F)$ and $\delta\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$.
In fact, the arguments simplify in the non-Archimedean case.

  3. 3.

To be more specific, the results on harmonic analysis are already available in
the non-Archimedean case. For example the reduction of local geometric Theorem
5.3.11 from $G$-equisingular to $G$-regular case is in Proposition 5.5.2;
alternatively, one can appeal to the upcoming Corollary 6.6.12 via Shalika
germs. In the proof of Lemma 6.4.1, the usage of [60, V.4.4 Lemme] can also be
replaced by the easier non-Archimedean version established in [60, I.5.10],
based on the property that $R^{!}_{\epsilon}=M^{!}_{\epsilon}$.

The real hardcore of the arguments in §6.4 is combinatorial, and the proofs
carry over verbatim.

Of course, the $G$-regular case of local geometric theorem (Hypothesis 6.2.14)
is still needed as input. The inductive assumptions also remain in force,
namely: all the results about $I^{\tilde{G}}_{\tilde{M}}$,
$\rho^{\tilde{G}}_{J}$, etc. are established in either of the following cases:

  * •

$\tilde{G}$ is replaced by a group of metaplectic type of smaller dimension;

  * •

$M$ is replaced by some strictly larger Levi subgroup of $G$ (the case $M=G$
is already known).

######  Theorem 6.5.1.

Let $F$ be a local field of characteristic zero. Make the inductive
assumptions above, and assume that for all $\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ such that
$\operatorname{Supp}(\tilde{\gamma})\subset\tilde{M}_{G\text{-reg}}$, we have

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

Then:

  1. 1.

if $F$ is non-Archimedean, the displayed equality extends to all
$\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$;

  2. 2.

if $F$ is Archimedean, the displayed equality extends to all

$\tilde{\gamma}\in\left(D_{\text{tr-orb},-}(\tilde{M})+D_{\text{geom,
$G$-equi},-}(\tilde{M})\right)\otimes\operatorname{mes}(M)^{\vee}.$

In particular, the statement of Theorem 5.3.11 holds true under these
assumptions.

###### Proof.

This is merely a summary of the discussions above. In the Archimedean case,
the fact that $I^{\tilde{G}}_{\tilde{M}}$ can be well-defined on $D_{\text{tr-
orb},-}(\tilde{M})+D_{\text{geom, $G$-equi},-}(\tilde{M})$ is the content of
(C8). ∎

###  6.6 Shalika germs

Fix a non-Archimedean local field $F$ of characteristic zero.

####  6.6.1 Basics

The Shalika germs for orbital integrals on coverings are discussed in [46,
§4]. Here we follow the paradigm of [60, II.2.1] to rephrase the theory.

Let $\mathbf{p}:\tilde{G}\twoheadrightarrow G(F)$ be a group of metaplectic
type, and let $\mathcal{O}$ be a semisimple conjugacy class in $G(F)$. The
Shalika germ in question is a germ of linear maps

$g_{\mathcal{O}}:D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}\to
D_{\mathrm{geom},-}(\tilde{G},\mathcal{O})\otimes\operatorname{mes}(G)^{\vee}$

characterized by

$I^{\tilde{G}}(\tilde{\gamma},f)=I^{\tilde{G}}(g_{\mathcal{O}}(\tilde{\gamma}),f)$

for all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$ such that
$\operatorname{Supp}(\tilde{\gamma})$ is sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$. Being a germ means: two such linear maps are
identified if they coincide when
$\operatorname{Supp}(\tilde{\gamma})\subset\mathbf{p}^{-1}(\mathcal{V})$,
where $\mathcal{V}$ is an invariant neighborhood of $\mathcal{O}$.

The _separation property_ of Shalika germs is rephrased as follows. For every
$\tau\in
D_{\mathrm{geom},-}(\tilde{G},\mathcal{O})\otimes\operatorname{mes}(G)^{\vee}$,
there exists $\tilde{\gamma}$, with
$\operatorname{Supp}(\tilde{\gamma})\subset\tilde{G}_{\mathrm{reg}}$ and close
to $\mathbf{p}^{-1}(\mathcal{O})$, such that
$g_{\mathcal{O}}(\tilde{\gamma})=\tau$. In fact, $\tilde{G}_{\mathrm{reg}}$
may be replaced by $\mathbf{p}^{-1}(\mathcal{W})$ where $\mathcal{W}$ is any
open dense invariant subset of $G_{\mathrm{reg}}(F)$.

By linearity, the formalism extends to the case that $\mathcal{O}$ is a finite
union of semisimple conjugacy classes.

The stable case is analogous. Let $G^{!}$ be a quasisplit $F$-group and let
$\mathcal{O}^{!}$ be a stable semisimple conjugacy class in $G^{!}(F)$. By
replacing $I^{\tilde{G}}$ by $S^{G^{!}}$, we have the germ of linear maps

$Sg_{\mathcal{O}}:SD_{\mathrm{geom}}(G^{!})\otimes\operatorname{mes}(G^{!})^{\vee}\to
SD_{\mathrm{geom}}(G^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(G^{!})^{\vee}$

with a similar characterization and separation property.

We also need the Shalika germs of invariant weighted orbital integrals. The
uncovered case without coverings is due to Arthur [22, §9]. Since the
combinatorial ingredients in the weighted orbital integrals on $\tilde{G}$ are
the same (see §5.1), whilst the remaining parts are based on harmonic
analysis, Arthur’s results on germs carry over to $\tilde{G}$.

Specifically, let $M\subset G$ be a Levi subgroup, $\mathcal{O}$ a semisimple
conjugacy class in $M(F)$, and denote by $\mathcal{O}^{G}$ the semisimple
conjugacy class in $G(F)$ containing $\mathcal{O}$. We follow [60, II.2.3] to
define the Shalika germ as a germ of linear maps

$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}:D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}\to
D_{\mathrm{geom},-}(\tilde{G},\mathcal{O}^{G})\otimes\operatorname{mes}(G)^{\vee}$

characterized recursively by

$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{G}}_{\tilde{L}}\left(g^{\tilde{L}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)$

for all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\tilde{\gamma}$ with $\operatorname{Supp}(\tilde{\gamma})$ close to
$\mathbf{p}^{-1}(\tilde{O})$. When $M=G$ we recover the previous version.

Again, the definition extends to the case when $\mathcal{O}$ is a finite union
of semisimple conjugacy classes.

As in Arthur’s setting [22, p.270, Remark], we have the following crucial
property:

$\mathcal{O}\;\text{is $G$-equisingular and $M\neq G$}\implies
g^{\tilde{G}}_{\tilde{M},\mathcal{O}}=0.$

The weighted germs also satisfy the following descent formula.

######  Proposition 6.6.1.

Let $L\in\mathcal{L}^{G}(M)$. Write
$\tilde{\gamma}^{\tilde{L}}:=\operatorname{Ind}^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma})$
(see (2.4.1)) and define $\mathcal{O}^{L}$ to be the semisimple conjugacy
class in $L(F)$ containing $\mathcal{O}$. Then

$g^{\tilde{G}}_{\tilde{L},\mathcal{O}^{L}}\left(\tilde{\gamma}^{\tilde{L}}\right)=\sum_{L^{\dagger}\in\mathcal{L}(M)}d^{G}_{M}(L,L^{\dagger})g^{\tilde{L}^{\dagger}}_{\tilde{M},\mathcal{O}}\left(\tilde{\gamma}\right)^{\tilde{G}}.$

whenever $\operatorname{Supp}(\tilde{\gamma})$ is sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$.

###### Proof.

Same as [60, II.2.11]: this is based on the descent formula of Proposition
5.1.5. ∎

######  Remark 6.6.2.

Shalika germs are related to the germs $\rho^{\tilde{G}}_{J}$ in §6.2 by the
equivalence

$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}(a\tilde{\gamma})\sim\sum_{J\in\mathcal{J}^{G}_{M}}\rho^{\tilde{G}}_{J}(\tilde{\gamma},a)^{\tilde{G}},\quad\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}.$

between germs of functions in $a\in A_{M}(F)$ in general position, $a\to 1$.
This is immediate from Definition–Proposition 6.2.6 (ii).

Consider now the stable setting, namely $G^{!}$ is a quasisplit $F$-group with
Levi subgroup $M^{!}$. We follow [60, II.2.4] to define

$SD_{\mathrm{geom},G^{!}\text{-equi}}(M^{!}):=SD_{\mathrm{geom}}(M^{!})\cap
D_{\mathrm{geom},G^{!}\text{-equi}}(M^{!}).$

Suppose that a system of $B$-functions have been prescribed on $G^{!}$. Let
$\mathcal{O}^{!}$ be a stable semisimple conjugacy class in $M^{!}(F)$, and
let $\mathcal{O}^{G^{!}}$ be the stable semisimple conjugacy class in
$G^{!}(F)$ containing $\mathcal{O}^{!}$. In view of [60, II.2.4, Théorème;
II.2.5 Proposition], we have the germ of linear maps

$Sg^{G^{!}}_{M^{!},\mathcal{O}^{!}}(B):SD_{\mathrm{geom},G^{!}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}\to
SD_{\mathrm{geom}}(G^{!},\mathcal{O}^{G^{!}})\otimes\operatorname{mes}(G^{!})^{\vee}$

satisfying

$S^{G^{!}}_{M^{!}}(\delta,B,f)=\sum_{L^{!}\in\mathcal{L}(M^{!})}I^{G^{!}}_{L^{!}}\left(Sg^{L^{!}}_{M^{!},\mathcal{O}^{!}}(\delta,B|_{L^{!}}),B,f\right)$

for all $f$ and $\delta$ with $\operatorname{Supp}(\delta)$ close to
$\mathcal{O}^{!}$.

Again, when $M^{!}=G^{!}$ we recover $Sg_{\mathcal{O}^{!}}$. Also,

$\mathcal{O}^{!}\;\text{is $G^{!}$-equisingular and $M^{!}\neq G^{!}$}\implies
Sg^{G^{!}}_{M^{!},\mathcal{O}^{!}}=0.$

####  6.6.2 Statement of the matching

Let $\tilde{M}\subset\tilde{G}$ as before. Fix a stable semisimple conjugacy
class $\mathcal{O}$ in $M(F)$. Denote by $\mathcal{O}^{G}$ the stable
semisimple conjugacy class in $G(F)$ containing $\mathcal{O}$.

######  Definition 6.6.3.

Let $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and let
$\mathcal{O}^{!}$ be the finite union of stable semisimple conjugacy classes
in $M^{!}(F)$ that map to $\mathcal{O}$. Define the germ of linear maps

$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!}):SD_{\mathrm{geom},\tilde{G}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}\to
D_{\mathrm{geom},-}(\tilde{G},\mathcal{O}^{G})\otimes\operatorname{mes}(G)^{\vee}$

by the formula

$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta):=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])\check{\mathcal{T}}_{\mathbf{G}^{!}[s],\tilde{G}}\left(Sg^{G^{!}[s]}_{M^{!},\mathcal{O}^{!}[s]}(\delta[s],B^{\tilde{G}})\right)$

where $\operatorname{Supp}(\delta)$ is assumed to be close to
$\mathcal{O}^{!}$.

######  Lemma 6.6.4.

In the circumstance above, we have

$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(g^{\tilde{L},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta),f\right)$

for all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and all $\delta$ with $\operatorname{Supp}(\delta)$ close to
$\mathcal{O}^{!}$, where the left hand side is as Definition 5.3.9.

###### Proof.

By the properties of stable Shalika germs,
$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)$ equals

$\sum_{s,L^{!}}i_{M^{!}}(\tilde{G},G^{!}[s])S^{G^{!}[s]}_{L^{!}}\left(Sg^{L^{!}}_{M^{!},\mathcal{O}^{!}[s]}(\delta[s],B^{\tilde{G}}|_{L^{!}}),B^{\tilde{G}},f^{G^{!}[s]}\right)$

where $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$,
$L^{!}\in\mathcal{L}^{G^{!}[s]}(M^{!})$ and
$f^{G^{!}[s]}=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)$. Apply Lemma 5.4.2
with $R=M$ to transform this into a sum over $(L,s)$ where
$L\in\mathcal{L}^{G}(M)$ and $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$
factors uniquely through a pair $(s^{L},s_{L})$ with

$s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L}),\quad
s_{L}\in\mathcal{E}_{\mathbf{L}^{!}[s^{L}]}(\tilde{G}),$

so that $L^{!}$ (resp. $\mathbf{G}^{!}[s]$) corresponds to $L^{!}[s^{L}]$
(resp. $\mathbf{G}^{!}[s_{L}]$).

Lemma 3.5.7 asserts
$i_{M^{!}}(\tilde{G},G^{!}[s])=i_{M^{!}}(\tilde{L},L^{!})i_{L^{!}}(\tilde{G},G^{!}[s_{L}])$.
Lemma 5.3.6 implies $B^{\tilde{G}}|_{L^{!}}=B^{\tilde{L}}$. We also have
$\delta[s]=\delta[s^{L}][s_{L}]$; as $z[s_{L}]$ is central in $L^{!}(F)$,
translation by $z[s_{L}]$ commutes with
$Sg^{L^{!}}_{M^{!},\mathcal{O}^{!}[\star]}(B^{\tilde{L}})$. All in all,
$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)$ becomes

$\sum_{\begin{subarray}{c}L\in\mathcal{L}^{G}(M)\\\
s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})\\\
L^{!}:=L^{!}[s^{L}]\end{subarray}}i_{M^{!}}(\tilde{L},L^{!})\sum_{\begin{subarray}{c}s_{L}\in\mathcal{E}_{\mathbf{L}^{!}}(\tilde{G})\\\
G^{!}:=G^{!}[s_{L}]\end{subarray}}i_{L^{!}}(\tilde{G},G^{!})S^{G^{!}}_{L^{!}}\left(Sg^{L^{!}}_{M^{!},\mathcal{O}^{!}[s^{L}]}(\delta[s^{L}],B^{\tilde{L}})[s_{L}],B^{\tilde{G}},f^{G^{!}}\right)\\\
=\sum_{L\in\mathcal{L}^{G}(M)}\sum_{\begin{subarray}{c}s^{L}\in\mathcal{E}_{\mathbf{M^{!}}(\tilde{L})}\\\
L^{!}:=L^{!}[s^{L}]\end{subarray}}i_{M^{!}}(\tilde{L},L^{!})I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}\left(Sg^{L^{!}}_{M^{!},\mathcal{O}^{!}[s^{L}]}(\delta[s^{L}],B^{\tilde{L}})\right),f\right)\\\
=\sum_{L\in\mathcal{L}^{G}(M)}I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(g^{\tilde{L},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta),f\right).$

This is what we seek. ∎

######  Definition–Proposition 6.6.5.

Express $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$
with $\operatorname{Supp}(\tilde{\gamma})$ sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$ as

$\tilde{\gamma}=\sum_{\mathbf{M}^{!}}\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta_{\mathbf{M}^{!}})$

where $(\mathbf{M}^{!},\delta_{\mathbf{M}^{!}})$ is as Definition 6.6.3 for
each $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. Put

$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}):=\sum_{\mathbf{M}^{!}}g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta_{\mathbf{M}^{!}}).$

Then $g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})$
depends only on $\tilde{\gamma}$, and it satisfies

$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(g^{\tilde{L},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right),\quad
f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G).$

###### Proof.

By Lemma 6.6.4, we have

$\displaystyle I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$
$\displaystyle=\sum_{\mathbf{M}^{!}}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta_{\mathbf{M}^{!}},f)$
$\displaystyle=\sum_{L\in\mathcal{L}(M)}I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\sum_{\mathbf{M}^{!}}g^{\tilde{L},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta_{\mathbf{M}^{!}}),f\right),$

the first equality being Definition–Proposition 5.3.10. The first term depends
only on $\tilde{\gamma}$, and so do those
$\sum_{\mathbf{M}^{!}}g^{\tilde{L},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta_{\mathbf{M}^{!}})$
with $L\neq G$ by induction. Hence the required case $L=G$ follows by varying
$f$. ∎

######  Remark 6.6.6.

In view of Proposition 6.2.12, we deduce inductively an obvious endoscopic
analogue of Remark 6.6.2, namely the equivalence

$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\xi(a)\delta)\sim\sum_{J\in\mathcal{J}^{G}_{M}}\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta,a)^{\tilde{G}},\quad\delta\in
SD_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}.$

between germs of functions in $a\in A_{M}(F)$ in general position, $a\to 1$;
here $\xi:A_{M}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{M^{!}}$ is the
natural identification from endoscopy.

We can now state the matching of weighted Shalika germs.

######  Theorem 6.6.7 (Cf. [60, III.8.5 Proposition]).

We have the equality
$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}=g^{\tilde{G}}_{\tilde{M},\mathcal{O}}$
between germs of linear maps from
$D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$
to
$D_{\mathrm{geom},-}(\tilde{G},\mathcal{O}^{G})\otimes\operatorname{mes}(G)^{\vee}$.

The simplest case $M=G$ of Theorem 6.6.7 will be established in Corollary
6.6.11. The proof of the general case will be completed in §6.7. As the first
step, let us reduce it to the case when $\mathcal{O}$ is elliptic, using
Proposition 6.6.1 and its endoscopic counterpart below.

######  Proposition 6.6.8.

Let $L\in\mathcal{L}^{G}(M)$ and let $\mathcal{O}^{L}$ be the semisimple
conjugacy class in $L(F)$ containing $\mathcal{O}$. Then

$g^{\tilde{G},\mathcal{E}}_{\tilde{L},\mathcal{O}^{L}}\left(\tilde{\gamma}^{\tilde{L}}\right)=\sum_{L^{\dagger}\in\mathcal{L}(M)}d^{G}_{M}(L,L^{\dagger})g^{\tilde{L}^{\dagger},\mathcal{E}}_{\tilde{M},\mathcal{O}}\left(\tilde{\gamma}\right)^{\tilde{G}}$

whenever $\operatorname{Supp}(\tilde{\gamma})$ is sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$.

###### Proof.

Same as [60, II.2.12]. Compared to the proof of Proposition 6.6.1, the inputs
here are the descent formula of Corollary 5.4.5 and Definition–Proposition
6.6.5. ∎

Similarly, the following property of
$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}$ mirrors that of
$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}$.

######  Lemma 6.6.9.

Suppose that $\mathcal{O}$ is $G$-equisingular and $M\neq G$. Then
$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}=0$.

###### Proof.

For every $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, let
$\mathcal{O}^{!}$ be a stable semisimple conjugacy class in $M^{!}(F)$ with
$\mathcal{O}^{!}\mapsto\mathcal{O}$. Then $\mathcal{O}^{!}$ is
$\tilde{G}$-equisingular. Let $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$.
By [60, II.2.4 (2)], we have

$Sg^{G^{!}[s]}_{M^{!},\mathcal{O}^{!}[s]}(B^{\tilde{G}})=0$

since $\mathcal{O}^{!}[s]$ is $G^{!}[s]$-equisingular by Lemma 5.3.8. Hence
$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}=0$. ∎

####  6.6.3 Applications to the local geometric theorem

Conserve the notations and assumptions from §6.6. Fix a semisimple conjugacy
class $\mathcal{O}$ in $M(F)$.

The local geometric Theorem 5.3.11 asserts that
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$
for all $\tilde{\gamma}$ and $f$. We assume inductively that

  * •

the statement of Theorem 5.3.11 holds true when $G$ (resp. $M$) is replaced by
a strictly smaller (resp. larger) Levi subgroup;

  * •

ditto for Theorem 6.6.7.

######  Lemma 6.6.10.

Let $\mathcal{D}$ be a subset of
$D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$
with the following separation property: for all $\tau\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$,
there exists $\tilde{\gamma}\in\mathcal{D}$ with
$\operatorname{Supp}(\tilde{\gamma})$ arbitrarily close to
$\mathbf{p}^{-1}(\mathcal{O})$, such that
$g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})=\tau$.

Suppose that
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
for all $\tilde{\gamma}\in\mathcal{D}$. Then the following are equivalent:

  1. (i)

$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
for all $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$;

  2. (ii)

$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})=g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})$
for all $\tilde{\gamma}\in\mathcal{D}$ with
$\operatorname{Supp}(\tilde{\gamma})$ sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$.

###### Proof.

In view of various inductive assumptions, for $\tilde{\gamma}\in\mathcal{D}$
with $\operatorname{Supp}(\tilde{\gamma})$ sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$ and all $f$, the germ expansions for
$I^{\tilde{G}}_{\tilde{M}}$ and $I^{\tilde{G},\mathcal{E}}_{\tilde{M}}$ imply

$\displaystyle 0$
$\displaystyle=I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)-I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$
$\displaystyle=I^{\tilde{G}}_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)-I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)$
$\displaystyle\quad+I^{\tilde{G}}_{\tilde{G}}\left(g^{\tilde{G}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})-g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right).$

Assuming (i), we deduce that
$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})-g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})$
has vanishing orbital integrals for these $\tilde{\gamma}$, hence (ii) holds.

Assuming (ii), we deduce that
$I^{\tilde{G}}_{\tilde{M}}(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),\cdot)$
for these $\tilde{\gamma}$. By the separation property, this implies (i). ∎

######  Corollary 6.6.11.

We have
$g^{\tilde{G}}_{\tilde{G},\mathcal{O}}=g^{\tilde{G},\mathcal{E}}_{\tilde{G},\mathcal{O}}$
for all stable semisimple conjugacy class $\mathcal{O}$ in $G(F)$.

###### Proof.

Take $M=G$ and
$\mathcal{D}=D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$
in Lemma 6.6.10. The identity
$I^{\tilde{G}}_{\tilde{G}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{G}}(\tilde{\gamma},\cdot)$
is trivially true for all $\tilde{\gamma}\in\mathcal{D}$, and the matching of
Shalika germs follows. ∎

We may also take $\mathcal{D}$ spanned by those $\tilde{\gamma}$ with supports
in $\tilde{M}_{G\text{-reg}}$ and sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$. This yields the following consequence.

######  Corollary 6.6.12.

Suppose that $\mathcal{O}$ is $G$-equisingular and fix an invariant
neighborhood $\mathcal{V}$ of $\mathcal{O}$ in $M(F)$. If
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
for all $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ such that
$\operatorname{Supp}(\tilde{\gamma})\subset\mathbf{p}^{-1}(\mathcal{V}\cap
M_{G\text{-reg}}(F))$, then
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
for all $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$.

###### Proof.

The $G$-equisingularity implies
$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}=0=g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}$
when $M\neq G$, by Lemma 6.6.9. On the other hand, we have seen that
$g^{\tilde{G}}_{\tilde{G},\mathcal{O}}=g^{\tilde{G},\mathcal{E}}_{\tilde{G},\mathcal{O}}$.
Lemma 6.6.10 can thus be applied. ∎

This gives another proof of the non-Archimedean case of Proposition 5.5.2.

###  6.7 Proof of the matching of Shalika germs

Fix $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, a stable
semisimple conjugacy class $\mathcal{O}^{!}\subset M^{!}(F)$ and take
$\epsilon\in\mathcal{O}^{!}$ such that $M^{!}_{\epsilon}$ is quasisplit.
Denote by $\mathcal{O}$ the image of $\mathcal{O}^{!}$, which is a stable
semisimple conjugacy class in $M(F)$. For all
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, we define
$\overline{M^{!}_{\epsilon}}\subset\overline{G^{!}[s]_{\epsilon[s]}}$ by the
convention in §4.3.

The first and easiest step is to reduce Theorem 6.6.7 to the elliptic case.

###### Proof of Theorem 6.6.7 in the non-elliptic case.

We may assume
$\tilde{\gamma}=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)$ for
some

$\delta\in
SD_{\mathrm{geom},\tilde{G}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$

with $\operatorname{Supp}(\delta)$ close to $\mathcal{O}^{!}$. Put

$R^{!}:=Z_{M^{!}}\left(A_{M^{!}_{\epsilon}}\right),\quad\mathcal{O}_{R^{!}}:=\mathcal{O}^{!}\cap
R^{!}(F).$

By construction, $R^{!}\hookrightarrow M^{!}$ is a Levi subgroup containing
$\epsilon$, and $\mathcal{O}^{!}=(\mathcal{O}_{R^{!}})^{M^{!}}$. By
Proposition 3.7.2, these data extend uniquely (up to conjugacy) to

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Observe that $R^{!}_{\epsilon}=M^{!}_{\epsilon}$. Indeed, every element of
$M^{!}_{\epsilon}$ centralizes $A_{M^{!}_{\epsilon}}$, hence lies in
$Z_{R^{!}}(\epsilon)$ and we conclude that $M^{!}_{\epsilon}\subset
R^{!}_{\epsilon}$. Also note that
$\delta\in\operatorname{im}\left(S\operatorname{desc}^{M^{!},*}_{\epsilon}\right)$.
These observations entail $\delta=\nu^{M^{!}}$ for some $\nu\in
SD_{\mathrm{geom}}(R^{!})\otimes\operatorname{mes}(R^{!})^{\vee}$ with
$\operatorname{Supp}(\nu)$ close to $\mathcal{O}_{R^{!}}$. Cf. the first step
of the proof in [60, III.7.4 Proposition].

Put $\nu[t^{-1}]:=\nu\cdot z[t]^{-1}$ as usual, and suppose that
$\mathcal{O}_{R^{!}}[t^{-1}]\mapsto\mathcal{O}_{R}$, where $\mathcal{O}_{R}$
is a stable semisimple conjugacy class in $R(F)$. Then
$\mathcal{O}=(\mathcal{O}_{R})^{M}$ and

$\tilde{\gamma}=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}\left(\nu^{M^{!}}\right)=\left(\check{\mathcal{T}}_{\mathbf{R}^{!},\tilde{R}}(\nu[t^{-1}])\right)^{\tilde{M}}.$

Suppose that $\mathcal{O}^{!}$ is not elliptic in $M^{!}$, then $R^{!}$ is a
proper Levi subgroup of $M^{!}$. The descent formulas from Propositions 6.6.1,
6.6.8 and inductive assumptions then imply
$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})=g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma})$.
∎

In what follows, we assume $\epsilon$ is elliptic in $M^{!}$ and take
$\eta\in\mathcal{O}$ such that $M_{\eta}$ is quasisplit. Note that $\eta$ is
elliptic in $M$ by Lemma 4.5.3. Define

$\mathcal{Z}:=\operatorname{ker}\left[\frac{Z_{\tilde{M}^{\vee}}^{\circ}}{Z_{\tilde{G}^{\vee}}^{\circ}}\xrightarrow{\text{descent}}\frac{Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}}{Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}}\right],$
(6.7.1)

where “descent” is the map in Remark 4.6.4. It is finite when $\eta$ is
elliptic in $G$.

######  Lemma 6.7.1.

Let $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. Suppose that $\eta$ is
elliptic in $G^{!}[s]$, then $\overline{G^{!}[s]_{\epsilon[s]}}$ is an
elliptic endoscopic group of $G_{\eta}$ if and only if $\epsilon[s]$ is
elliptic in $G^{!}[s]$.

###### Proof.

The following inequalities hold

$\dim A_{\overline{G^{!}_{\epsilon[s]}}}=\dim
A_{G^{!}[s]_{\epsilon[s]}}\geq\dim A_{G^{!}[s]}=\dim A_{G}=\dim A_{G_{\eta}},$

and the assertion follows at once. ∎

######  Lemma 6.7.2.

Let $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. Suppose that $\epsilon[s]$
is elliptic in $G^{!}[s]$. Then $\eta$ is elliptic in $G$ and
$\overline{G^{!}[s]_{\epsilon[s]}}$ is an elliptic endoscopic group of
$G_{\eta}$. Let

$c(s):=c^{\overline{G^{!}[s]_{\epsilon[s]}},G^{!}[s]_{\epsilon[s]}}_{\overline{M^{!}_{\epsilon}},M^{!}_{\epsilon}}\in
2^{\mathbb{Z}_{\leq 0}}$

be the factor in [49, Proposition 5.3.3]. Then

$e^{G^{!}[s]}_{M^{!}}(\epsilon[s])i_{M^{!}}\left(\tilde{G},G^{!}[s]\right)=|\mathcal{Z}|^{-1}c(s)i_{\overline{M^{!}_{\epsilon}}}\left(G_{\eta},\overline{G^{!}[s]_{\epsilon[s]}}\right)$

where $e^{G^{!}[s]}_{M^{!}}(\epsilon[s])$ is as in Definition 4.6.3.

###### Proof.

Since $\epsilon[s]\leftrightarrow\eta$ with respect to $\mathbf{G}^{!}[s]$,
Lemma 4.5.3 implies $\eta$ is elliptic in $G$, and Lemma 6.7.1 implies that
$\overline{G^{!}[s]_{\epsilon[s]}}$ is an elliptic endoscopic group of
$G_{\eta}$. Consider the diagram:

${Z_{(M^{!})^{\vee}}\big{/}Z_{(G^{!}[s])^{\vee}}}$${Z_{(M^{!}_{\epsilon})^{\vee}}^{\Gamma_{F}}\big{/}Z_{(G^{!}[s]_{\epsilon[s]})^{\vee}}^{\Gamma_{F}}}$${Z_{\tilde{M}^{\vee}}^{\circ}\big{/}Z_{\tilde{G}^{\vee}}^{\circ}}$${Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\big{/}Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}}$${Z_{\overline{M^{!}_{\epsilon}}^{\vee}}^{\Gamma_{F}}\big{/}Z_{\overline{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}}$descentdescentstd.
endo.

Every quotient group is connected by [15, Lemma 1.1], and every arrow is
surjective by comparing dimensions using ellipticity. The rightmost arrow
comes from switching factors of type $\mathrm{B}$ and $\mathrm{C}$, namely

$\displaystyle
L=\operatorname{SO}(2n+1),\quad\overline{L}=\operatorname{Sp}(2n),$
$\displaystyle Z_{\overline{L}^{\vee}}^{\Gamma_{F}}=\\{1\\}\to\\{\pm
1\\}=Z_{L^{\vee}}^{\Gamma_{F}}.$

Count the cardinality of fibers in two ways. The composite along  yields

$i_{M^{!}}\left(\tilde{G},G^{!}[s]\right)^{-1}e^{G^{!}[s]}_{M^{!}}(\epsilon[s])^{-1},$

whilst  yields $|\mathcal{Z}|\cdot
i_{\overline{M^{!}_{\epsilon}}}\left(G_{\eta},\overline{G^{!}[s]_{\epsilon[s]}}\right)^{-1}c(s)^{-1}$,
the $c(s)^{-1}$ coming from the rightmost arrow. This proves the desired
equality. ∎

It remains to establish the following

######  Proposition 6.7.3 (Cf. [60, III.8.1 Proposition]).

Assume that $\mathcal{O}^{!}$ is elliptic in $M^{!}$. For all $\delta\in
SD_{\mathrm{geom},\tilde{G}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$,
with $\operatorname{Supp}(\delta)$ sufficiently close to $\mathcal{O}^{!}$, we
have

$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)=g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)\right).$

###### Proof.

Choose $\epsilon$ and $\eta$ as before. Recall that

$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])\check{\mathcal{T}}_{\mathbf{G}^{!}[s],\tilde{G}}Sg^{G^{!}[s]}_{M^{!},\mathcal{O}[s]}\left(\delta[s],B^{\tilde{G}}\right).$

Take $\delta_{\epsilon}\in
SD_{\mathrm{geom}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})^{\vee}$
with $\operatorname{Supp}(\delta_{\epsilon})$ close to $1$ and

$\delta=S\operatorname{desc}^{M^{!},*}_{\epsilon}(\delta_{\epsilon}),\quad\delta[s]=S\operatorname{desc}^{M^{!},*}_{\epsilon[s]}(\delta_{\epsilon}).$

For every $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, we define

$\tau(s):=\begin{cases}Sg^{G^{!}[s]_{\epsilon[s]}}_{M^{!}_{\epsilon},\mathrm{unip}}\left(\delta_{\epsilon},B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\right),&\text{if}\;\epsilon[s]\;\text{is
elliptic in}\;G^{!}[s]\\\ 0,&\text{otherwise.}\end{cases}$

It is of unipotent support, so we can set
$\tau(s)^{G^{!}[s]}:=S\operatorname{desc}^{G^{!}[s],*}_{\epsilon[s]}\left(\tau(s)\right)$.
By [60, III.4.4 Proposition],

$Sg^{G^{!}[s]}_{M^{!},\mathcal{O}^{!}[s]}\left(\delta[s],B^{\tilde{G}}\right)=e^{G^{!}[s]}_{M^{!}}(\epsilon[s])\tau(s)^{G^{!}[s]}$

Hereafter, we adopt the conventions from §4.6 on descent of endoscopic data
and transfer factors. For every $y\in\mathcal{Y}$, use the chosen transfer
factors to define

$\displaystyle\tau[s,y]$
$\displaystyle:=\underbracket{\check{\mathcal{T}}_{\mathbf{G}^{!}_{\eta[y]},G_{\eta[y]}}}_{\text{std.}}\underbracket{\check{\mathcal{T}}_{G^{!}[s]_{\epsilon[s]},\overline{G^{!}[s]_{\epsilon[s]}}}}_{\text{non-
std.}}(\tau(s))\;\in
D_{\mathrm{unip}}(G_{\eta[y]})\otimes\operatorname{mes}(G_{\eta[y]})^{\vee},$
$\displaystyle\tau[s,y]^{\tilde{G}}$
$\displaystyle:=\operatorname{desc}^{\tilde{G},*}_{\eta[y]}(\tau[s,y])\;\in
D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}.$

Applying Lemma 4.4.2 relative to
$\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$, with the
$d(s,y)$ from (4.6.1) replacing $d(y)$, we see

$\check{\mathcal{T}}_{\mathbf{G}^{!}[s],\tilde{G}}Sg^{G^{!}[s]}_{M^{!},\mathcal{O}^{!}[s]}\left(\delta[s],B^{\tilde{G}}\right)=e^{G^{!}[s]}_{M^{!}}(\epsilon[s])\displaystyle\sum_{y\in\dot{\mathcal{Y}}}d(s,y)\tau[s,y]^{\tilde{G}}.$

Note that $\tau[s,y]$ is nonzero only when $\epsilon[s]$ is elliptic in
$G^{!}[s]$.

Next, suppose that
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ arises from an
elliptic semisimple element $s^{\flat}\in\tilde{M}^{\vee}$. Choose
$\overline{s^{\flat}}\in M_{\eta}^{\vee}$ that is part of the endoscopic datum
$\mathbf{M}^{!}_{\eta}$ of $M_{\eta}$ obtained by descent at
$(\epsilon,\eta)$. By Remark 4.6.4, the
$\mathbf{G}_{\eta}^{!}\in\mathcal{E}(G_{\eta})$ obtained by descent at
($\epsilon[s],\eta$) is equivalent to the one associated with
$\overline{s}\in\mathcal{E}_{\mathbf{M}^{!}_{\eta}}(G_{\eta})$, where
$\overline{s}$ is determined by $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$
as in (4.6.2). Also recall that the endoscopic group underlying
$\mathbf{G}^{!}_{\eta}$ is identified with
$\overline{G^{!}[s]_{\epsilon[s]}}$.

Now let $s$ vary. Put

$\displaystyle\mathcal{S}$
$\displaystyle:=\left\\{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G}):\epsilon[s]\;\text{is
elliptic in}\;G^{!}[s]\right\\},$ $\displaystyle\overline{\delta_{\epsilon}}$
$\displaystyle:=\check{\mathcal{T}}_{M^{!}_{\epsilon},\overline{M^{!}_{\epsilon}}}(\delta_{\epsilon}).$

To each $s\in\mathcal{S}$ we attach the constant $c(s)$ of Lemma 6.7.2. By
[60, III.6.6 Proposition],

$\check{\mathcal{T}}_{G^{!}[s]_{\epsilon[s]},\overline{G^{!}[s]_{\epsilon[s]}}}\left(\tau(s)\right)=\begin{cases}c(s)^{-1}Sg^{\overline{G^{!}[s]_{\epsilon[s]}}}_{\overline{M^{!}_{\epsilon}},\mathrm{unip}}(\overline{\delta_{\epsilon}}),&s\in\mathcal{S}\\\
0,&\text{otherwise,}\end{cases}$

Indeed, the cited result is applicable by our choice of $B$-functions in
Definition 5.3.5. Therefore

$\tau[s,y]=\begin{cases}c(s)^{-1}\check{\mathcal{T}}_{\mathbf{G}^{!}_{\eta[y]},G_{\eta[y]}}Sg^{\overline{G^{!}[s]_{\epsilon[s]}}}_{\overline{M^{!}_{\epsilon}},\mathrm{unip}}\left(\overline{\delta_{\epsilon}}\right),&s\in\mathcal{S}\\\
0,&s\notin\mathcal{S}.\end{cases}$

Fix representatives in $\mathcal{Y}$ for $\dot{\mathcal{Y}}$. We infer that
$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)$
equals

$\sum_{y\in\dot{\mathcal{Y}}}\operatorname{desc}^{\tilde{G},*}_{\eta[y]}\sum_{s\in\mathcal{S}}i_{M^{!}}(\tilde{G},G^{!}[s])e^{G^{!}[s]}_{M^{!}}(\epsilon[s])d(s,y)c(s)^{-1}\check{\mathcal{T}}_{\mathbf{G}^{!}_{\eta[y]},G_{\eta[y]}}\left(Sg^{\overline{G^{!}[s]_{\epsilon[s]}}}_{\overline{M^{!}_{\epsilon}},\mathrm{unip}}(\overline{\delta_{\epsilon}})\right).$
(6.7.2)

Hereafter, suppose $\eta$ is elliptic in $G$. Interpret
$\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ (thus $\mathcal{S}$) as a subset of
$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}\big{/}Z_{\tilde{G}^{\vee}}^{\circ}$.
Consider the map $s\mapsto\overline{s}$ from
$\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ to
$\overline{s^{\flat}}Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\big{/}Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$
in (4.6.2). Lemma 6.7.1 implies

$\mathcal{S}=\left\\{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G}):\overline{s}\;\text{determines
an elliptic endoscopic datum of}\;G_{\eta}\right\\}.$ (6.7.3)

Combining Lemma 6.7.2 and (6.7.2), we arrive at

$\displaystyle
g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)=\sum_{y\in\dot{\mathcal{Y}}}\operatorname{desc}^{\tilde{G},*}_{\eta[y]}\left(\sum_{\overline{s}\in\overline{s^{\flat}}Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}/Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}}x(\overline{s},y)\check{\mathcal{T}}_{\mathbf{G}^{!}(\overline{s}),G_{\eta[y]}}Sg^{G^{!}(\overline{s})}_{\overline{M^{!}_{\epsilon}},\mathrm{unip}}(\overline{\delta_{\epsilon}})\right),$

where

  * •

$\mathbf{G}^{!}(\overline{s})$ denotes the endoscopic datum of $G_{\eta}$
(equivalently, for $G_{\eta[y]}$) determined by $\overline{s}$ and the
endoscopic datum of $M_{\eta}$ obtained by descent, taken up to equivalence;

  * •

for each $y$ and $\overline{s}$, we set

$x(\overline{s},y):=|\mathcal{Z}|^{-1}\displaystyle\sum_{\begin{subarray}{c}s\in\mathcal{S}\\\
s\mapsto\overline{s}\end{subarray}}d(s,y)i_{\overline{M^{!}_{\epsilon}}}(G_{\eta},G^{!}(\overline{s})).$
(6.7.4)

By fiat, $i_{\overline{M^{!}_{\epsilon}}}(G_{\eta},G^{!}(\overline{s}))=0$
when $\mathbf{G}^{!}(\overline{s})$ is non-elliptic. We will prove in Lemma
6.7.4 that

$x(\overline{s},y)=\begin{cases}i_{\overline{M^{!}_{\epsilon}}}(G_{\eta},G^{!}(\overline{s}))d^{M}(y),&y\in\dot{\mathcal{Y}}^{M}\\\
0,&y\in\dot{\mathcal{Y}}\smallsetminus\dot{\mathcal{Y}}^{M}.\end{cases}$

Still assuming $\eta$ (hence each $\eta[y]$) is elliptic in $G$, we set

$\delta[y]:=\check{\mathcal{T}}_{\overline{M^{!}_{\epsilon}},M_{\eta[y]}}(\overline{\delta_{\epsilon}})$

and deduce

$\displaystyle
g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)=\sum_{y\in\dot{\mathcal{Y}}^{M}}d^{M}(y)\operatorname{desc}^{\tilde{G},*}_{\eta[y]}\left(g^{G_{\eta[y]},\mathcal{E}}_{M_{\eta[y]},\mathrm{unip}}(\delta[y])\right).$

By Arthur’s result [60, III.1.5 Corollaire] in standard endoscopy, we obtain

$\displaystyle
g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)$
$\displaystyle=\sum_{y\in\dot{\mathcal{Y}}^{M}}d^{M}(y)\operatorname{desc}^{\tilde{G},*}_{\eta[y]}\left(g^{G_{\eta[y]}}_{M_{\eta[y]},\mathrm{unip}}(\delta[y])\right)$
$\displaystyle=\sum_{y\in\dot{\mathcal{Y}}^{M}}d^{M}(y)g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\left(\delta[y]^{\tilde{M}}\right)=g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\left(\sum_{y\in\dot{\mathcal{Y}}^{M}}d^{M}(y)\delta[y]^{\tilde{M}}\right),$
$\displaystyle\text{with}\;\delta[y]^{\tilde{M}}$
$\displaystyle:=\operatorname{desc}^{\tilde{M},*}_{\eta[y]}(\delta[y]),$

the second equality being the descent of ordinary germs [60, III.4.2
Proposition] at elliptic elements. The last expression equals
$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)\right)$
by Lemma 4.4.2. This settles the case when $\eta$ is elliptic in $G$.

Finally, assume $\eta$ (hence each $\eta[y]$) is non-elliptic in $G$, then
$\mathcal{S}=\varnothing$ by Lemma 6.7.2, hence (6.7.2) gives
$g^{\tilde{G},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\mathbf{M}^{!},\delta)=0$.
On the other hand,
$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\circ\operatorname{desc}^{\tilde{M},*}_{\eta[y]}=0$
in this case by loc. cit., therefore

$g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)\right)=g^{\tilde{G}}_{\tilde{M},\mathcal{O}}\left(\sum_{y\in\dot{\mathcal{Y}}^{M}}\delta[y]^{\tilde{M}}\right)=0.$

This settles the remaining cases. ∎

######  Lemma 6.7.4 (Cf. [60, III.8.3]).

Keep the assumptions of Proposition 6.7.3. In addition, assume that $\eta$ is
elliptic in $G$. For each
$\overline{s}\in\overline{s^{\flat}}Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\big{/}Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$
we have

$x(\overline{s},y)=\begin{cases}i_{\overline{M^{!}_{\epsilon}}}(G_{\eta},G^{!}(\overline{s}))d^{M}(y),&y\in\dot{\mathcal{Y}}^{M}\\\
0,&y\in\dot{\mathcal{Y}}\smallsetminus\dot{\mathcal{Y}}^{M}.\end{cases}$

###### Proof.

We still interpret $\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ as a subset of
$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}\big{/}Z_{\tilde{G}^{\vee}}^{\circ}$. By
Lemma 6.7.1, $\mathbf{G}^{!}(\overline{s})$ is elliptic if $\overline{s}$
comes from $\mathcal{S}$. Conversely, suppose that
$\overline{s}\in\overline{s^{\flat}}Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\big{/}Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$
and $\mathbf{G}^{!}(\overline{s})$ is an elliptic endoscopic datum of
$G_{\eta}$, then we have

$\left\\{s\in\mathcal{S}:s\mapsto\overline{s}\right\\}=\left\\{s^{\flat}t\in
s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}\big{/}Z_{\tilde{G}^{\vee}}^{\circ}:\overline{s^{\flat}}\cdot\mathrm{descent}(t)=\overline{s}\right\\}.$
(6.7.5)

Indeed, $\subset$ is trivial. As for $\supset$, recall that one constructs
$\mathbf{G}^{!}[s]\in\mathcal{E}(\tilde{G})$ from $\mathbf{M}^{!}$ and any
$s\in
s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}\big{/}Z_{\tilde{G}^{\vee}}^{\circ}$. If
$\overline{s^{\flat}}\cdot\mathrm{descent}(s)=\overline{s}$, then
$\mathbf{G}^{!}[s]$ must be elliptic, otherwise $\mathbf{G}^{!}(\overline{s})$
would factor through an endoscopic datum of $L_{\eta}$ for some proper
$L\in\mathcal{L}^{G}(M)$, which is absurd for the following reason:
$\mathbf{G}^{!}(\overline{s})$ is an elliptic endoscopic datum of $G_{\eta}$,
thus

$\dim A_{L_{\eta}}\leq\dim A_{G^{!}(\overline{s})}=\dim A_{G_{\eta}}.$

As $\eta\in G(F)$ is elliptic, $\dim A_{G_{\eta}}=\dim A_{G}$, we get $\dim
A_{L_{\eta}}\leq\dim A_{G}$, which contradicts $\dim A_{G}<\dim A_{L}\leq\dim
A_{L_{\eta}}$. The conclusion is that
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, and this shows (6.7.5).

As observed in the proof of Lemma 6.7.2, the map “descent” in Remark 4.6.4 is
surjective since $\eta$ is elliptic in $M$ and $G$. Hence both sides of
(6.7.5) are nonempty.

By the prior discussion, the definition (6.7.4) can be written as

$x(\overline{s},y)=|\mathcal{Z}|^{-1}\underbracket{i_{\overline{M^{!}_{\epsilon}}}(G_{\eta},G^{!}(\overline{s}))}_{=0\;\text{unless}\;\mathbf{G}^{!}(\overline{s})\;\text{is
ell.}}\displaystyle\sum_{\begin{subarray}{c}s\in\mathcal{S}\\\
s\mapsto\overline{s}\end{subarray}}d(s,y),$

and the sum above is taken over a $\mathcal{Z}$-coset. The case
$y\in\dot{\mathcal{Y}}^{M}$ becomes straightforward since $d(s,y)=d^{M}(y)$ by
Proposition 4.6.2.

Assume from now onward that $y\notin\dot{\mathcal{Y}}^{M}$. Our goal is to
show that when $\mathbf{G}^{!}(\overline{s})$ is elliptic, we have

$\sum_{z\in\mathcal{Z}}d(sz,y)=0$

for any $s\in\mathcal{S}$ with $s\mapsto\overline{s}$. In this case
$\mathbf{G}^{!}(\overline{s})$ is relevant for $G_{\eta[y]}$ by Proposition
4.5.5 (true for general $y$). Also note that $d(sz,1)=d^{M}(1)\neq 0$ since
all endoscopic data of the quasisplit group $M_{\eta}$ are relevant. We are
thus reduced to show that

$s\in\mathcal{S},\;y\notin\dot{\mathcal{Y}}^{M}\implies\sum_{z\in\mathcal{Z}}\frac{d(sz,y)}{d(sz,1)}=0.$

By definition,

$\frac{d(sz,y)}{d(sz,1)}=\frac{\Delta_{\mathbf{G}^{!}[sz],\tilde{G}}(\exp(\overline{Y})\epsilon[sz],\exp(X[y])\tilde{\eta}[y])}{\Delta_{\mathbf{G}^{!}[sz],\tilde{G}}(\exp(\overline{Y})\epsilon[sz],\exp(X)\tilde{\eta})}\cdot\frac{\Delta(\overline{s},1)(\exp(\overline{Y}),\exp(X))}{\Delta(\overline{s},y)(\exp(\overline{Y}),\exp(X[y]))}.$

On the right hand side, the second quotient is nonzero and independent of $z$;
denote by $\chi(z)$ the first quotient, as a function in $z\in\mathcal{Z}$. It
suffices to show that

$y\notin\dot{\mathcal{Y}}^{M}\implies\chi(z)=\text{constant}\times\;\text{a
nontrivial character of}\;\mathcal{Z}.$ (6.7.6)

Since $\mathbf{G}^{!}(\overline{s})$ is relevant, Proposition 4.5.4 says that
there exists a diagram $(\epsilon[s],B^{!},T^{!},B,T,\eta)$ for
$\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$, compatibly
with descent. In fact, since $\epsilon[s]$ (resp. $\eta$) is elliptic in
$G^{!}[s]$ (resp. $G$), we may take $T^{!}$ (resp. $T$) elliptic in
$G^{!}[s]_{\epsilon[s]}$ (resp. $G_{\eta}$).

Furthermore, [41, 10.2 Lemma] guarantees that there is a representative of $y$
such that the pure inner twist
$\operatorname{Ad}(y):G_{\eta,\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G_{\eta[y],\overline{F}}$
restricts to an isomorphism
$T\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T[y]$ between $F$-tori, which
determines $\mathrm{inv}(y)\in\operatorname{H}^{1}(F,T)$. By construction, the
image $\mathrm{inv}^{G}(y)$ in $\operatorname{H}^{1}(F,G_{\eta})$ of
$\mathrm{inv}(y)$ is the class corresponding to $y\in\dot{\mathcal{Y}}$.

The $Y$, $\overline{Y}$, $X$ and $X[y]$ above may be taken in the Lie algebras
of these tori; one can suppose that they match under isomorphisms. The
relative position $\mathrm{inv}(\cdot,\cdot)$ satisfies

$\mathrm{inv}\left(\exp(X)\eta,\exp(X[y])\eta[y]\right)=\mathrm{inv}(y)\in\operatorname{H}^{1}(F,T).$

Recall that $\exp(X[y])\tilde{\eta}[y]$ is stably conjugate in the metaplectic
sense to $\exp(X)\tilde{\eta}$ in $\tilde{G}$. The cocycle property [52,
Proposition 5.13] for metaplectic transfer factors extends immediately to
groups of metaplectic type. It implies

$\chi(z):=\frac{\Delta_{\mathbf{G}^{!}[sz],\tilde{G}}(\exp(\overline{Y})\epsilon[sz],\exp(X[y])\tilde{\eta}[y])}{\Delta_{\mathbf{G}^{!}[sz],\tilde{G}}(\exp(\overline{Y})\epsilon[sz],\exp(X)\tilde{\eta})}=\langle\mathrm{inv}(y),sz\rangle;$

the pairing is formed by

  * •

regarding $sz$ as an element of $\check{T}^{\Gamma_{F}}$ (with the help of
diagrams, see §3.6),

  * •

the canonical perfect pairing
$\operatorname{H}^{1}(F,T)\times\pi_{0}(\check{T}^{\Gamma_{F}})\to\mathbb{C}^{\times}$
due to Kottwitz [41, 1.2 Theorem].

In particular, $\chi$ is a nonzero constant times a character of
$\mathcal{Z}$.

Now we use the functoriality of the perfect pairings

${\langle\cdot,\cdot\rangle:\operatorname{H}^{1}(F,T)}$${\pi_{0}\left(\check{T}^{\Gamma_{F}}\right)}$${\langle\cdot,\cdot\rangle_{G}:\operatorname{H}^{1}(F,G_{\eta})}$${\pi_{0}\left(Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}\right)}$${\mathbb{C}^{\times}.}$${\langle\cdot,\cdot\rangle_{M}:\operatorname{H}^{1}(F,M_{\eta})}$${\pi_{0}\left(Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\right)}$${\times}$${\times}$${\times}$

As $\eta$ is elliptic in $G$, we have
$\pi_{0}\left(Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}\right)=Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$.
We contend that

$\operatorname{ker}\left[\pi_{0}\left(Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}\right)\twoheadrightarrow\pi_{0}\left(Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\right)\right]\subset\operatorname{im}\left[\mathcal{Z}\to
Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}\right].$ (6.7.7)

To see this, use the commutative diagram

${Z_{\tilde{M}^{\vee}}^{\circ}}$${Z_{M^{\vee}}^{\circ}}$${Z_{M_{\eta}^{\vee}}^{\Gamma_{F},\circ}}$${Z_{M^{\vee}}}$${Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}}$${Z_{G_{\eta}^{\vee}}^{\Gamma_{F}},}$$\scriptstyle{\sim}$

the surjectivity being due to the ellipticity of $\eta$ in $M$. Suppose $x\in
Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$ maps to
$1\in\pi_{0}(Z_{M_{\eta}^{\vee}}^{\Gamma_{F}})$, then $x\in
Z_{M_{\eta}^{\vee}}^{\Gamma_{F},\circ}$, hence there exists $z\in
Z_{\tilde{M}^{\vee}}^{\circ}$ with $z\mapsto x$. This easily leads to (6.7.7).

Let us prove (6.7.6) now. If $\chi$ is constant on $\mathcal{Z}$, the
functoriality of pairings would imply

$\langle\cdot,\mathrm{inv}^{G}(y)\rangle_{G}\;\text{is trivial
on}\;\operatorname{ker}\left[\pi_{0}\left(Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}\right)\twoheadrightarrow\pi_{0}\left(Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\right)\right].$

Hence $\mathrm{inv}^{G}(y)$ is in the image of
$\operatorname{H}^{1}(F,M_{\eta})$. By (2.9.3), this contradicts
$y\notin\dot{\mathcal{Y}}^{M}$. ∎

######  Remark 6.7.5.

The last part of the proof concerning $\langle\cdot,\cdot\rangle$ and
$\mathcal{Z}$ also works for Archimedean $F$, upon replacing
$\operatorname{H}^{1}(F,\cdot)$ by
$\operatorname{H}^{1}_{\mathrm{ab}}(F,\cdot)$; of course,
$\operatorname{H}^{1}(F,T)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{H}^{1}_{\mathrm{ab}}(F,T)$.
There is one caveat: for a connected reductive $F$-group $L$, Kottwitz’s
natural homomorphism

$\displaystyle\operatorname{H}^{1}_{\mathrm{ab}}(F,L)\to\pi_{0}\left(Z_{L^{\vee}}^{\Gamma_{F}}\right)^{D},$
$\displaystyle(\cdots)^{D}:=\text{the Pontryagin dual}$

is bijective for non-Archimedean $F$, but only injective when $F=\mathbb{R}$;
see also [42, Proposition 1.7.3]. The arguments before will carry over if the
surjectivity holds for both $L=G_{\eta}$ and $L=M_{\eta}$. Such an adaptation
will be needed in §9.7.

###  6.8 Proof of the weighted fundamental lemma

Let $F$, $\tilde{M}$, $\tilde{G}$, $K$ be as in §5.6. In particular, we are in
the unramified situation defined in §3.11. Our goal is to prove Theorem 5.6.3
by reducing it to the $\tilde{G}$-regular case solved in [49].

Fix $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. The lines
$\operatorname{mes}(G)$, $\operatorname{mes}(M)$, $\operatorname{mes}(M^{!})$,
etc. are trivialized by the Haar measures determined by hyperspecial
subgroups. The protagonists here are
$r^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},K)$ and
$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta)$, where
$\tilde{\gamma}\in D_{\mathrm{geom},-}(\tilde{M})$ and $\delta\in
SD_{\mathrm{geom}}(M^{!})$.

Fix a stable semisimple conjugacy class $\mathcal{O}^{!}$ (resp.
$\mathcal{O}$) in $M^{!}(F)$ (resp. $M(F)$), and assume
$\mathcal{O}^{!}\mapsto\mathcal{O}$. We have the following expansions into
Shalika germs.

  * •

If $\operatorname{Supp}(\tilde{\gamma})$ is sufficiently close to
$\mathbf{p}^{-1}(\mathcal{O})$, we have

$r^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},K)=\sum_{L\in\mathcal{L}(M)}r^{\tilde{G}}_{\tilde{L}}\left(g^{\tilde{L}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),K\right).$

This is the metaplectic analogue of [60, II.4.5 (1)]. It is based on the fact
that non-invariant weighted orbital integrals have the same expansion into
weighted Shalika germs as the invariant ones, see [22, §9]. The proof is based
on harmonic analysis and has nothing to do with metaplectic covers, so it
carries over verbatim.

  * •

When $\mathcal{O}$ is $G$-equisingular, the equality above reduces to
$r^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},K)=r^{\tilde{G}}_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),K\right)$.

  * •

Let $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. If $\mathcal{O}^{!}$ is
$\tilde{G}$-equisingular and $\operatorname{Supp}(\delta)$ is sufficiently
close to $\mathcal{O}^{!}$, we have

$s^{G^{!}[s]}_{M^{!}}(\delta[s])=s^{G^{!}[s]}_{M^{!}}\left(g^{M^{!}}_{M^{!},\mathcal{O}^{!}[s]}(\delta[s])\right);$

here $g^{M^{!}}_{M^{!},\mathcal{O}^{!}[s]}(\delta[s])\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!}[s])$ by [60, II.2.2 Lemme], so we may
take $s^{G^{!}[s]}_{M^{!}}$. Indeed, $\mathcal{O}^{!}[s]$ is
$G^{!}[s]$-equisingular by Lemma 5.3.8, so this reduces to [60, II.4.5 (2)].

  * •

Recall that
$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta)=\sum_{s}i_{M^{!}}(\tilde{G},G^{!}[s])s^{G^{!}[s]}_{M^{!}}(\delta[s])$.
Under the assumptions of the previous item,

$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta)=r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},g^{M^{!}}_{M^{!},\mathcal{O}^{!}}(\delta)\right).$

This is because $z[s]\in Z_{M^{!}}(F)$ implies
$g^{M^{!}}_{M^{!},\mathcal{O}^{!}}(\delta)[s]=g^{M^{!}}_{M^{!},\mathcal{O}^{!}[s]}(\delta[s])$.

We shall prove the equisingular case of Theorem 5.6.3 first.

######  Lemma 6.8.1.

The statements of Theorem 5.6.3 hold when $\delta\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})$ and $\mathcal{O}^{!}$ is
$\tilde{G}$-equisingular.

###### Proof.

By [60, II.2.2 Lemme], there exists $\delta_{\mathrm{reg}}\in
SD_{\mathrm{geom},\tilde{G}\text{-reg}}(M^{!})$ with
$\operatorname{Supp}(\delta_{\mathrm{reg}})$ arbitrarily close to
$\mathcal{O}^{!}$, such that
$g^{M^{!}}_{M^{!},\mathcal{O}^{!}}(\delta_{\mathrm{reg}})=\delta$. The germs
expansions above imply

$\displaystyle
r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta\right)=r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},g^{M^{!}}_{M^{!},\mathcal{O}^{!}}(\delta_{\mathrm{reg}})\right)=r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta_{\mathrm{reg}}\right)$
$\displaystyle
r^{\tilde{G}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta_{\mathrm{reg}}),K\right)=r^{\tilde{G}}_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta_{\mathrm{reg}})\right),K\right).$

Corollary 6.6.11 asserts

$g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta_{\mathrm{reg}}))=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}\left(g^{M^{!}}_{M^{!},\mathcal{O}^{!}}(\delta_{\mathrm{reg}})\right).$

By the $\tilde{G}$-regular case [49, Théorème 4.2.1] of Theorem 5.6.3,

$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta_{\mathrm{reg}}\right)=r^{\tilde{G}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta_{\mathrm{reg}}),K\right).$

This implies that
$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta\right)=r^{\tilde{G}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),K\right)$,
as required. ∎

In order to access to the general case, we need the following germ expansions
from §6.2. Recall the spaces of germs $U^{G}_{M}\supset U^{G,+}_{M}$,
$U^{G^{!}[s]}_{M^{!}}\supset U^{G^{!}[s],+}_{M^{!}}$ defined before Lemma
6.2.3.

  * •

Let $\tilde{\gamma}\in D_{\mathrm{geom},-}(\tilde{M})$, we have the
equivalence

$r^{\tilde{G}}_{\tilde{M}}(a\tilde{\gamma},K)\sim\sum_{L\in\mathcal{L}(M)}\sum_{J\in\mathcal{J}^{G}_{M}}r^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}(\tilde{\gamma},a)^{\tilde{L}},K\right)$

between germs of functions of $a\in A_{M}(F)$ in general position, $a\to 1$.
Again, this is a fact from harmonic analysis having nothing to do with
metaplectic covers; see [60, p.307] for explanations.

The terms $a\mapsto
r^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}(\tilde{\gamma},a)^{\tilde{L}},K\right)$
belongs to $U_{J}^{L\supset M}$. The right hand side is thus in

$r^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},K)+U^{G,+}_{M}\;\subset U^{G}_{M}.$

  * •

The stable side is similar: for each
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, we use the germ expansion of
$s^{G^{!}[s]}_{M^{!}}$ from [60, II.4.7] to obtain

$s^{G^{!}[s]}_{M^{!}}(a^{!}\delta[s])\sim
s^{G^{!}[s]}_{M^{!}}(\delta[s])+U^{G^{!}[s],+}_{M^{!}}\;\subset
U^{G^{!}[s]}_{M^{!}},$

where $a^{!}\in A_{M^{!}}(F)$ is in general position, $a^{!}\to 1$.

  * •

Consider the endoscopic case. Recall that we have
$\xi:A_{M}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{M^{!}}$. Given the
previous case, the upcoming Lemma 6.8.2 will guarantee that

$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\xi(a)\delta)\sim
r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta)+U^{G,+}_{M}\;\subset
U^{G}_{M}.$

######  Lemma 6.8.2.

Let $s\in\mathcal{E}_{\mathbf{M^{!}}}(\tilde{G})$ and consider the spaces
$U^{G}_{M}$, $U^{G^{!}[s]}_{M^{!}}$.

  1. (i)

For all $u^{!}\in U^{G^{!}[s]}_{M^{!}}$, put $u:=u^{!}\circ\xi$. Then $u\in
U^{G}_{M}$ and it has the same constant term as $u^{\prime}$.

  2. (ii)

If $u^{!}\sim v^{!}$ in $U^{G^{!}[s]}_{M^{!}}$, then $u\sim v$ in $U^{G}_{M}$.

Consequently, $u^{!}\mapsto u$ induces a linear map from
$U^{G^{!}[s]}_{M^{!}}/\sim$ to $U^{G}_{M}/\sim$.

###### Proof.

First, we show $u^{!}\in U_{J^{!}}\implies u\in U_{J}$ for all
$J^{!}\in\mathcal{J}^{G^{!}[s]}_{M^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\right)$
and $J\in\mathcal{J}^{G}_{M}$. Let
$\gamma_{1},\ldots,\gamma_{m}\in\Sigma^{G^{!}[s]}\left(A_{M^{!}},B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\right)$
be linearly independent with $m=\dim\mathfrak{a}^{G^{!}[s]}_{M^{!}}$. They
determine $J^{!}$ with image $J$. Set
$u^{!}:=\prod_{i=1}^{m}\log\left|\gamma_{i}-\gamma_{i}^{-1}\right|\in
U_{J^{!}}$. To see $u\in U_{J}$, it suffices to recall Lemma 6.1.2.

As each $L^{!}\in\mathcal{L}^{G^{!}[s]}(M^{!})$ comes from some
$L\in\mathcal{L}^{G}(M)$ (see Lemma 5.4.2 with $M=R$), (i) follows by
replacing $G^{!}[s]$ by various $L^{!}$.

The proof of (ii) is essentially the same as [60, II.4.7 (6)]. Let us be more
precise. The relation $\sim$ between germs is defined in terms of

  * •

a function $d$ with $d(\exp H)=\|H\|$, which works simultaneously for $A_{M}$
and $A_{M^{!}}$ via $\xi$;

  * •

estimates of functions on domains of the form $|\alpha(a)-1|>c\cdot d(a)$
where $\alpha\in\Sigma^{G}(A_{M})$ or $\Sigma^{G^{!}[s]}(A_{M^{!}})$.

The sets $\Sigma^{G}(A_{M})$ and $\Sigma^{G^{!}[s]}(A_{M^{!}})$ are not in
bijection via $\xi$. However, they match under dilation by $B$-functions.
Since $B^{\tilde{G}}$ takes value in $\\{1,\frac{1}{2}\\}$, $p>2$ and $a\to
1$, one readily sees that the dilation is harmless. ∎

We are ready to prove the weighted fundamental lemma in general.

###### Proof of Theorem 5.6.3.

Fix $\delta\in SD_{\mathrm{geom}}(M^{!})$. For $a\in A_{M}(F)$ in general
position, $a\to 1$, the $\tilde{G}$-equisingular matching (Lemma 6.8.1)
implies

$\displaystyle
r^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\xi(a)\delta\right)$
$\displaystyle=r^{\tilde{G}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\xi(a)\delta),K\right)$
$\displaystyle=r^{\tilde{G}}_{\tilde{M}}\left(a\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),K\right).$

By Lemma 6.8.2, both sides are germs in $U^{G}_{M}$, and we deduce the
equivalence of germs

$r^{\tilde{G}}_{\tilde{M}}(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),K)-r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta)+(\text{element
of}\;U^{G,+}_{M})\sim 0\quad\text{in}\;U^{G}_{M}.$

Hence Proposition 6.2.2 (ii) and Lemma 6.2.3 yield
$r^{\tilde{G}}_{\tilde{M}}(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),K)=r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta)$.
∎

##  Chapter 7 Weighted characters: non-Archimedean case, and construction of
$\epsilon_{\tilde{M}}$

Let $F$ be a non-Archimedean local field of characteristic zero. We will work
with a covering of metaplectic type $\mathbf{p}:\tilde{G}\to G(F)$ where
$G=\prod_{i\in I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W)$; here we
fix the symplectic $F$-vector space $(W,\langle\cdot|\cdot\rangle)$ and the
additive character $\psi$ of $F$ to construct the eightfold covering
$\tilde{G}$. We also fix a minimal Levi subgroup $M_{0}$ of $G$.

What follows is closely modeled on Arthur’s strategy in [20, 6]. First off, we
will review the unitary weighted characters $J_{\tilde{M}}(\pi,f)$ and the
associated invariant distributions $I_{\tilde{M}}(\pi,\cdot)$ on $\tilde{G}$,
where $\pi\in\Pi_{\mathrm{unit},-}(\tilde{M})$. They were already defined in
[44, 47] via the _canonical normalization_ , i.e. in terms of Harish-Chandra’s
$\mu$-functions instead of chosen normalizing factors.

Next, we define the compactly supported weighted characters
${}^{c}J_{\tilde{M}}(\pi,f)$, the corresponding variant
${}^{c}\phi_{\tilde{M}}$ of $\phi_{\tilde{M}}$, and the resulting invariant
distributions ${}^{c}I_{\tilde{M}}(\pi,f)$ in §§7.1–7.3. There is a family of
linear maps ${}^{c}\theta_{\tilde{M}}$ linking $\phi_{\tilde{M}}$ and
${}^{c}\phi_{\tilde{M}}$, where $M$ ranges over $\mathcal{L}(M_{0})$.

In §7.4, we define the endoscopic counterparts
${}^{c}I^{\mathcal{E}}_{\tilde{M}}$ and
${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}$, and formulate the corresponding
matching Theorem 7.4.8 for them.

These definitions are geared to establish the existence of a linear map

$\epsilon_{\tilde{M}}:\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{M})\otimes\operatorname{mes}(M)$

for a given $M\in\mathcal{L}(M_{0})$ under inductive assumptions. It is
characterized by

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}\left(\tilde{\gamma},\epsilon_{\tilde{M}}(f)\right)$

for all $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. In other
words, $\epsilon_{\tilde{M}}$ is the obstruction to the local geometric
Theorem 5.3.11. The point is to consider
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)$
as a function in $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$,
where $f$ is kept fixed, and then the construction divides into two steps.

  * •

Locally, the matching of weighted Shalika germs establishes the existence of
such a $\epsilon_{\tilde{M}}(f)$ around any given stable semisimple class
$\mathcal{O}\subset M(F)$.

  * •

The global problem is more delicate, since neither
$I_{\tilde{M}}(\tilde{\gamma},f)$ nor ${}^{c}I_{\tilde{M}}(\tilde{\gamma},f)$
are compactly supported in $\tilde{\gamma}$ modulo conjugation. This is where
${}^{c}I_{\tilde{M}}$, ${}^{c}\theta_{\tilde{M}}$ and their endoscopic avatars
enter.

By induction, it is legitimate to assume the matching theorems for
$I_{\tilde{M}}(\tilde{\gamma},f)$, ${}^{c}I_{\tilde{M}}(\tilde{\gamma},f)$ and
${}^{c}\theta_{\tilde{M}}$ hold when $M$ (resp. $G$) is replaced a larger
(resp. proper) Levi subgroup. Then the ingredients above and an argument via
partition of unity furnish the required $\epsilon_{\tilde{M}}(f)$. This is the
content of Proposition 7.5.1.

###  7.1 Review of unitary weighted characters

For unitary genuine representations, their _weighted characters_ on coverings
are defined in [46, §5.7] and [47, §4.1]. Specifically, let
$M\in\mathcal{L}(M_{0})$ and let $\pi\in\Pi_{\mathrm{unit},-}(\tilde{M})$. One
defines

$J_{\tilde{M}}(\pi,f)=J^{\tilde{G}}_{\tilde{M}}(\pi,f)=\operatorname{tr}\left(\mathcal{M}_{M}(\pi,\tilde{P})I_{\tilde{P}}(\pi,f)\right)$

where

  * •

$f\in C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})$,

  * •

$P\in\mathcal{P}(M)$ is arbitrary,

  * •

$\mathcal{M}_{M}(\pi,\tilde{P})$ is the endomorphism of $I_{\tilde{P}}(\pi)$
arising from the $(G,M)$-family

$\begin{gathered}\mu_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})^{-1}\mu_{\tilde{Q}|\tilde{P}}\left(\pi_{\lambda+\frac{\Lambda}{2}}\right)J_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})^{-1}J_{\tilde{Q}|\tilde{P}}(\pi_{\lambda+\Lambda}),\\\
Q\in\mathcal{P}(M),\;\Lambda\in\mathfrak{a}_{M,\mathbb{C}}^{*},\end{gathered}$
(7.1.1)

where
$\mu_{\tilde{Q}|\tilde{P}}(\pi_{\lambda}):=\prod_{\alpha\in\Sigma_{P}^{\mathrm{red}}\cap\Sigma_{Q^{-}}^{\mathrm{red}}}\mu_{\alpha}(\pi_{\lambda})$
is made from Harish-Chandra’s $\mu$-functions; this is called the _canonical
normalization_ of weighted characters in [8].

The result turns out to be independent of $P$. It also differs from Arthur’s
earlier recipe in [10], which is based on the choice of normalizing factors.
When $M=G$, we revert to the usual characters.

Define

$\displaystyle\mathfrak{a}_{M,F}$ $\displaystyle:=H_{M}(M(F)),$
$\displaystyle\mathfrak{a}_{M,F}^{\vee}$
$\displaystyle:=\left\\{H\in\mathfrak{a}_{M}^{*}:\forall
X\in\mathfrak{a}_{M,F},\;\langle H,X\rangle\in 2\pi\mathbb{Z}\right\\}.$

We have thus the compact torus
$i\mathfrak{a}^{*}_{M,F}:=i\mathfrak{a}_{M}^{*}/i\mathfrak{a}_{M,F}^{\vee}$
and the complex torus
$\mathfrak{a}_{M,\mathbb{C}}^{*}/i\mathfrak{a}_{M,F}^{\vee}$.

The function $\lambda\mapsto J_{\tilde{M}}(\pi_{\lambda},f)$ on
$i\mathfrak{a}^{*}_{M}$ is analytic and periodic relative to the lattice
$i\mathfrak{a}_{M,F}^{\vee}$. One defines

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

Then
$J_{\tilde{M}}(\pi_{\lambda},X,f)=J_{\tilde{M}}(\pi,X,f)e^{\langle\lambda,X\rangle}$,
and the resulting distribution on $\tilde{G}$ is concentrated at $H_{G}=X$.
Therefore $J_{\tilde{M}}(\pi,X,\cdot)$ extends to
$C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})$. See [47,
Corollaire 4.6].

Using the trace Paley–Wiener theorem, the map

$\phi_{\tilde{M}}:C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$

is defined by mapping $f$ to the function $(\pi,X)\mapsto
J_{\tilde{M}}(\pi,X,f)$. See [47, Théorème 4.13].

The non-unitary weighted characters will be reviewed in §13.1.

###  7.2 The maps ${}^{c}\phi_{\tilde{M}}$ and ${}^{c}\theta_{\tilde{M}}$

Let $M\in\mathcal{L}(M_{0})$. It makes sense to say that a rational function
on $\mathfrak{a}_{M,\mathbb{C}}^{*}/i\mathfrak{a}_{M,F}^{\vee}$ has
singularities along “hyperplanes” of the form
$e^{\langle\cdot,\check{\alpha}\rangle}=\text{const}$, where
$\alpha\in\Sigma(A_{M})$. See [61, VIII.1.2].

The basic example is

$\mathfrak{a}_{R,\mathbb{C}}^{*}\ni\lambda\mapsto
J^{\tilde{G}}_{\tilde{M}}\left(\pi_{\lambda}^{\tilde{M}},f\right)$

where

  * •

$R\subset\mathcal{L}^{M}(M_{0})$,

  * •

$\pi$ is a genuine admissible representation of $\tilde{R}$,

  * •

$\pi_{\lambda}^{\tilde{M}}=(\pi_{\lambda})^{\tilde{M}}$ is the normalized
parabolic induction,

  * •

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

It is rational with a finite number of polar hyperplanes of the form
$e^{\langle\cdot,\check{\alpha}\rangle}=\text{const}$, where
$\alpha\in\Sigma(A_{R})$.

As in [61, VIII.1.1] or [20, §4], we fix a family of functions
$\omega_{S}:\mathfrak{a}_{R}\to[0,1]$ for every Levi subgroup $R$ of $G$ and
$S\in\mathcal{P}(R)$, such that

  * •

$\operatorname{Supp}(\omega_{S})$ lies in some translate of the chamber
$\mathfrak{a}_{S}^{+}$;

  * •

$\sum_{S\in\mathcal{P}(R)}\omega_{S}=1$;

  * •

the families $(R,(\omega_{S})_{S\in\mathcal{P}(R)})$ are invariant under
$G(F)$-conjugation.

Moreover, we can replace $G$ by any of its Levi subgroup and impose the
following compatibility: for every $L\in\mathcal{L}(R)$ and
$S\in\mathcal{P}^{L}(R)$, we have

$\omega_{S}=\sum_{\begin{subarray}{c}S^{\prime}\in\mathcal{P}(R)\\\
S^{\prime}\cap L=S\end{subarray}}\omega_{S^{\prime}}.$ (7.2.1)

######  Definition 7.2.1.

Let $\pi\in
D_{\operatorname{ell},-}(\tilde{R})\otimes\operatorname{mes}(R)^{\vee}$. Set

${}^{c}J_{\tilde{M}}\left(\pi^{\tilde{M}},f\right)={}^{c}J^{\tilde{G}}_{\tilde{M}}\left(\pi^{\tilde{M}},f\right):=\sum_{X\in\mathfrak{a}_{R,F}}\sum_{S\in\mathcal{P}^{G}(R)}\omega_{S}(X)\\\
\int_{\nu_{S}+i\mathfrak{a}^{*}_{R,F}}J_{\tilde{M}}\left(\pi_{\lambda}^{\tilde{M}},f\right)e^{-\langle\lambda,X\rangle}\mathop{}\\!\mathrm{d}\lambda$

where

  * •

$i\mathfrak{a}^{*}_{R,F}$ carries the Haar measure dual to the counting
measure on $\mathfrak{a}_{R,F}$,

  * •

$\nu_{S}\in\mathfrak{a}_{R}^{*}$ satisfies
$\langle\nu_{S},\check{\alpha}\rangle\gg 0$ for all
$\alpha\in\Sigma^{S}(A_{R})$.

This is independent of the choice of $(\nu_{S})_{S}$. In view of (2.6.2), we
can thus define ${}^{c}J_{\tilde{M}}(\pi,f)$ for all $\pi\in
D_{\mathrm{temp},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

######  Definition–Proposition 7.2.2.

There is a linear map

${}^{c}\phi_{\tilde{M}}={}^{c}\phi^{\tilde{G}}_{\tilde{M}}:C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M(F))$

characterized by
$I^{\tilde{M}}\left(\pi,{}^{c}\phi_{\tilde{M}}(f)\right)={}^{c}J_{\tilde{M}}\left(\pi,f\right)$
for all $\pi\in
D_{\mathrm{temp},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

###### Proof.

This is the analogue of [61, VIII.1.3 Proposition], which carries over to
coverings because of its analytic nature. ∎

Similarly, one has

  1. 1.

the descent formula for ${}^{c}\phi_{\tilde{M}}$ as in [61, VIII.1.4 (1)];

  2. 2.

the explicit behavior of ${}^{c}\phi_{\tilde{M}}$ under conjugation, cf. [61,
VIII.1.4 (2)];

  3. 3.

the extension to
$C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\mathrm{ac}}(\tilde{M})\otimes\operatorname{mes}(M)$
as in [61, VIII.1.4 (6)].

All these proofs carry over to coverings without change.

######  Definition 7.2.3 (Cf. [61, VIII.1.5]).

We define the linear maps

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

characterized inductively by

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

These linear maps also satisfy a descent formula relative to parabolic
descent. See [61, VIII.1.6 Lemme], whose proof carries over to coverings as
usual.

######  Definition 7.2.4.

We say that
$f\in\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
is a _Schwartz function_ if $X\mapsto I^{\tilde{G}}(\pi,X,f)$ is rapidly
decreasing on $\mathfrak{a}_{G,F}$ for each $\pi\in
D_{\mathrm{temp},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$.

The definition is the same as [61, VIII.1.7]. If $f$ is Schwartz, one can
define

$I^{\tilde{G}}(\pi,\lambda,f):=\sum_{X\in\mathfrak{a}_{G,F}}e^{\langle\lambda,X\rangle}I^{\tilde{G}}(\pi,X,f),\quad\lambda\in
i\mathfrak{a}_{G}^{*}/i\mathfrak{a}_{G,F}^{\vee}.$

We record the following basic properties of $I^{\tilde{G}}(\pi,\lambda,f)$
from loc. cit.

  * •

It is $C^{\infty}$ in $\lambda$.

  * •

Every element of
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
is Schwartz. Fourier inversion implies that

$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\implies
I^{\tilde{G}}(\pi,\lambda,f)=I^{\tilde{G}}(\pi_{\lambda},f).$

  * •

Suppose that $I^{\tilde{G}}(\pi,\lambda,f)$ extends to a rational function in
$\lambda$ on $\mathfrak{a}_{G,\mathbb{C}}^{*}/i\mathfrak{a}_{G,F}^{\vee}$. By
setting

$I^{\tilde{G}}(\pi,\nu,X,f):=\int_{\underbracket{\nu+i\mathfrak{a}_{G,F}^{*}}_{\text{compact}}}I^{\tilde{G}}(\pi,\lambda,f)e^{-\langle\lambda,X\rangle}\mathop{}\\!\mathrm{d}\lambda,\quad
X\in\mathfrak{a}_{G,F},$

provided that $\nu+i\mathfrak{a}_{G,F}$ contains no poles, we have

$I^{\tilde{G}}(\pi,\lambda,f)=\sum_{X\in\mathfrak{a}_{G,F}}e^{\langle\lambda,X\rangle}I^{\tilde{G}}(\pi,\nu,X,f)\quad\text{when}\quad\operatorname{Re}(\lambda)=\nu.$

In the special case
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
we have $I^{\tilde{G}}(\pi,\lambda,f)=I^{\tilde{G}}(\pi_{\lambda},f)$ for all
$\lambda\in\mathfrak{a}^{*}_{G,\mathbb{C}}$, thus $I^{\tilde{G}}(\pi,\nu,X,f)$
is independent of $\nu$ by shifting contours; in fact, it equals
$I^{\tilde{G}}(\pi,0,X,f)=I^{\tilde{G}}(\pi,X,f)$.

######  Proposition 7.2.5 (Cf. [61, VIII.1.8]).

Let
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $M\in\mathcal{L}(M_{0})$. Then

  * •

${}^{c}\theta^{\tilde{G}}_{\tilde{M}}(f)$ is Schwartz on $\tilde{M}$;

  * •

for all $\pi\in
D_{\mathrm{temp},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, the
function

$\lambda\mapsto
I^{\tilde{M}}(\pi,\lambda,{}^{c}\theta^{\tilde{G}}_{\tilde{M}}(f))$

extends to a rational function on
$\mathfrak{a}^{*}_{M,\mathbb{C}}/i\mathfrak{a}_{M,F}^{\vee}$ with only a
finite number of polar hyperplanes, all of which are of the form
$e^{\langle\cdot,\check{\alpha}\rangle}=\mathrm{const}$ with
$\alpha\in\Sigma^{G}(A_{M})$;

  * •

if $M\neq G$ and $\pi\in
D_{\mathrm{ell},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, we have

$\sum_{S\in\mathcal{P}(M)}\omega_{S}(X)I^{\tilde{M}}\left(\pi,\nu_{S},X,{}^{c}\theta^{\tilde{G}}_{\tilde{M}}(f)\right)=0,\quad
X\in\mathfrak{a}_{M,F}$

provided that $\nu_{S}$ is sufficiently deep relative to
$\mathfrak{a}_{S}^{+}$, for each $S\in\mathcal{P}(M)$.

Because of the analytic nature of the arguments, we refer to loc. cit. for the
proofs.

To conclude this section, we consider the stable analogues

${}^{c}S\theta^{G^{!}}_{M^{!}}:S\mathcal{I}(G^{!})\otimes\operatorname{mes}(G^{!})\to
S\mathcal{I}_{\mathrm{ac}}(M^{!})^{W^{G^{!}}(M^{!})}\otimes\operatorname{mes}(M^{!})$

in [61, VIII.2], where $G^{!}$ is quasisplit and $M^{!}$ is a Levi subgroup.
Suppose that $M^{!}$ arises from
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\mathbf{G}^{!}=\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$
as in (3.4.1). We assume that the data $(\omega_{P})_{P\in\mathcal{P}(M)}$ and
$(\omega_{P^{!}})_{P^{!}\in\mathcal{P}(M^{!})}$ are chosen compatibly, namely

$\omega_{P^{!}}=\sum_{\begin{subarray}{c}P\in\mathcal{P}(M)\\\ P\mapsto
P^{!}\end{subarray}}\omega_{P}$ (7.2.2)

where we used the isomorphism $\mathfrak{a}_{M}\simeq\mathfrak{a}_{M^{!}}$
from endoscopy, and $P\mapsto P^{!}$ signifies that the corresponding chambers
satisfy
$\overline{\mathfrak{a}_{P^{!}}^{+}}\supset\overline{\mathfrak{a}_{P}^{+}}$.
In fact, $\overline{\mathfrak{a}_{P^{!}}^{+}}$ is the union of various
$\overline{\mathfrak{a}_{P}^{+}}$, as there are “more walls” in $G$ than in
$G^{!}$. See [61, VIII.2.1].

There are also stable analogues of the notion of Schwartz functions $f^{!}$.
It suffices to consider $S^{G^{!}}(\pi,\lambda,f^{!})$ where $\pi\in
SD_{\mathrm{temp}}(G^{!})\otimes\operatorname{mes}(G^{!})^{\vee}$. Also recall
that the tempered local Langlands correspondence for $G^{!}$ is available; see
§2.6.

###  7.3 The distributions
${}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)$

Let $M\in\mathcal{L}(M_{0})$, $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. Define
inductively the distribution
${}^{c}I_{\tilde{M}}(\tilde{\gamma},\cdot)={}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
by

$\sum_{L\in\mathcal{L}(M)}{}^{c}I^{\tilde{L}}_{\tilde{M}}\left(\tilde{\gamma},{}^{c}\phi^{\tilde{G}}_{\tilde{L}}(f)\right)=J^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f),\quad
f\in
C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G).$

The following facts follow in exactly the same way as in [61, VIII.1.9].

  * •

They factor through
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

  * •

For all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
there exists a compact subset $\Omega\subset M(F)$ such that
${}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)=0$ whenever
$\operatorname{Supp}(\tilde{\gamma})\cap\mathbf{p}^{-1}(\Omega)=\varnothing$.

  * •

${}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)=\displaystyle\sum_{L\in\mathcal{L}^{G}(M)}I^{\tilde{L}}_{\tilde{M}}\left(\tilde{\gamma},{}^{c}\theta^{\tilde{G}}_{\tilde{L}}(f)\right)$.

On the stable side, we also have the stable distributions
${}^{c}S^{G^{!}}_{M^{!}}\left(\delta,B^{\tilde{G}},\cdot\right)$ for
$\delta\in SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M)^{\vee}$. This
is the content of [61, VIII.2]. Here the auxiliary functions $\omega_{P^{!}}$
are chosen compatibly with the $\omega_{P}$ on the metaplectic side, as
explained in (7.2.2).

######  Remark 7.3.1.

The system $B^{\tilde{G}}$ of $B$-functions in ${}^{c}S^{G^{!}}_{M^{!}}$ will
be dropped, since we will ultimately focus on the case of $\delta\in
SD_{\mathrm{geom},\tilde{G}\text{-reg}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$.

###  7.4 The distributions
${}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$ and
statement of matching theorems

Let $M\in\mathcal{L}(M_{0})$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. For every
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, we have the automorphism
$g\mapsto g[s]:=g(\cdot z[s])$ of
$S\mathcal{I}(M^{!})\otimes\operatorname{mes}(M)$.

######  Definition 7.4.1.

Let
${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\cdot)={}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\cdot)$
be the linear map

$\displaystyle\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
$\displaystyle\to
S\mathcal{I}_{\mathrm{ac}}(M^{!})\otimes\operatorname{mes}(M^{!})$
$\displaystyle f$
$\displaystyle\mapsto\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s]){}^{c}S\theta^{G^{!}[s]}_{M^{!}}\left(f^{G^{!}[s]}\right)[s]$

where $f^{G^{!}[s]}:=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)$ as usual.

Again, we have the following standard properties.

  * •

The image of ${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\cdot)$ is
$W(M)$-invariant. Cf. [61, VIII.3.2].

  * •

For all $f$, the function
${}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},f)$ is
Schwartz. For every $\pi\in
SD_{\mathrm{temp}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$, the
function

$\lambda\mapsto
S^{M^{!}}\left(\pi,\lambda,{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},f)\right)$

extends to a rational function on
$i\mathfrak{a}_{M^{!}}^{*}/i\mathfrak{a}_{M^{!},F}^{\vee}$, with only finitely
many polar hyperplanes of the form
$e^{\langle\cdot,\check{\alpha}\rangle}=\mathrm{const}$ where
$\alpha\in\bigcup_{s}\Sigma^{G^{!}[s]}(A_{M^{!}})$. Cf. [61, VIII.3.1 (4)].

Due to the presence of $z[s]$-twists, the descent formula requires more care.

######  Proposition 7.4.2 (Cf. [61, VIII.3.3 Proposition (i)]).

Suppose that $R\in\mathcal{L}^{M}(M_{0})$,
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$ and
$t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$ so that
$\mathbf{M}^{!}=\mathbf{M}^{!}[t]$. In other words, we are in the situation

${M^{!}}$${\tilde{M}}$${R^{!}}$${\tilde{R}.}$endo.ell.endo.ell.LeviLevi

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

${}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},f)_{R^{!}}[t]=\sum_{L\in\mathcal{L}^{G}(R)}d^{G}_{R}(M,L){}^{c}\theta^{\tilde{L},\mathcal{E}}_{\tilde{R}}\left(\mathbf{R}^{!},f_{\tilde{L}}\right).$

Here $f\mapsto f_{\tilde{L}}$ denotes the parabolic descent and
$(\cdots)_{R^{!}}$ is its stable avatar.

###### Proof.

Follow the paradigm of the proof of Proposition 5.4.4. Apart from the
combinatorics from the cited result, the only input is the descent formula
[61, VIII.2.3 Lemme] for ${}^{c}S\theta^{G^{!}[s]}_{M^{!}}$. ∎

######  Proposition 7.4.3 (Cf. [61, VIII.3.4]).

Given $M\in\mathcal{L}(M_{0})$, there exists a unique linear map

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

characterized by

$\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}\left({}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(f)\right)={}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},f)$

for all $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$.

###### Proof.

It amounts to a straightforward gluing of various
${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\cdot)$. See the
arguments for Definition–Proposition 5.3.10. ∎

######  Proposition 7.4.4.

Given $M\in\mathcal{L}(M_{0})$ and $R\in\mathcal{L}^{M}(M_{0})$, we have the
descent formula

${}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(f)_{\tilde{R}}=\sum_{L\in\mathcal{L}^{G}(R)}d^{G}_{R}(M,L){}^{c}\theta^{\tilde{L},\mathcal{E}}_{\tilde{R}}(f_{\tilde{L}}).$

###### Proof.

It suffices to show that for each
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$, both sides in
the assertion have the same transfer to $R^{!}$. Take
$t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$ and set
$\mathbf{M}^{!}:=\mathbf{M}^{!}[t]$. The left hand side becomes
${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},f)_{R^{!}}[t]$ whilst
the right hand side becomes
$\sum_{L\in\mathcal{L}^{G}(R)}d^{G}_{R}(M,L){}^{c}\theta^{\tilde{L},\mathcal{E}}_{\tilde{R}}(\mathbf{R}^{!},f_{\tilde{L}})$.
The required equality follows at once from Proposition 7.4.2. ∎

Moreover, the following properties hold for the endoscopic counterparts.

  * •

The function ${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(f)$ is Schwartz, and
$\lambda\mapsto
I^{\tilde{G}}(\pi,\lambda,{}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(f))$ extends
to a rational function with only finitely many polar hyperplanes of the form
$e^{\langle\cdot,\check{\alpha}\rangle}=\mathrm{const}$, where
$\alpha\in\Sigma^{G}(A_{M})$.

Indeed, this is the same as (and easier than) [61, VIII.3.6], the main
ingredients being the Proposition 7.4.4 and the case for ${}^{c}S\theta$. Note
that the coroots from $\Sigma^{G^{!}}(A_{M^{!}})$ and $\Sigma^{G}(A_{M})$ may
differ by a scaling by $2$, but this does not affect the required properties.

  * •

If $M\neq G$ and $\pi\in
D_{\mathrm{ell},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, then

$\sum_{S\in\mathcal{P}(M)}\omega_{S}(X)I^{\tilde{M}}\left(\pi,\nu_{S},X,{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(f)\right)=0$

for all $X\in\mathfrak{a}_{M,F}$, provided that $\nu_{S}$ is sufficiently deep
for each $S\in\mathcal{P}(M)$. Again, this reduces to the stable counterparts
for various $G^{!}\supset M^{!}$: see [61, VIII.3.7 Proposition]. Our case is
even easier since $\mathfrak{a}_{M,F}=\mathfrak{a}_{M^{!},F}$ for all
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$.

######  Definition 7.4.5.

Let $M\in\mathcal{L}(M_{0})$,
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and $\delta\in
SD_{\mathrm{geom},\tilde{G}\text{-reg}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$,
set

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

for all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G^{!})$,
where $f^{G^{!}[s]}:=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)$. See also
Remark 7.3.1.

As in Definition–Proposition 5.3.10, it is now routine to obtain the following

######  Definition–Proposition 7.4.6.

Let $M\in\mathcal{L}(M_{0})$. There are invariant distributions

${}^{c}I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)={}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot):\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathbb{C},$

where $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-reg},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$,
which are characterized by

${}^{c}I^{\mathcal{E}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),f\right)={}^{c}I^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)$

for all $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\delta\in
SD_{\mathrm{geom},\tilde{G}\text{-reg}}(M^{!})\otimes\operatorname{mes}(M^{!})\vee$.

######  Proposition 7.4.7.

For $f$ fixed, there exists a compact subset $\Omega\subset M(F)$ such that
${}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)=0$ whenever
$\operatorname{Supp}(\tilde{\gamma})\cap\mathbf{p}^{-1}(\Omega)=\varnothing$.

###### Proof.

This reduces to the similar property enjoyed by ${}^{c}S^{G^{!}}_{M^{!}}$ for
various $G^{!}\supset M^{!}$, which is in turn deduced from the same property
for ${}^{c}I^{G^{!}}_{M^{!}}$. ∎

We now state the matching between the distributions
${}^{c}\theta^{\tilde{G}}_{\tilde{M}}$, ${}^{c}I^{\tilde{G}}_{\tilde{M}}$ and
their endoscopic counterparts.

######  Theorem 7.4.8.

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

  1. (i)

${}^{c}\theta^{\tilde{G}}_{\tilde{M}}={}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}$.

  2. (ii)

${}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)={}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$,
for all $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-reg},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

The case $M=G$ of Theorem 7.4.8 is already available. In Corollary 7.4.10, the
general case will be reduced to the $G$-regular case of the local geometric
Theorem 5.3.11, which will be established in §13.9.

We proceed to study the comparison between compact and non-compact endoscopic
distributions. Fix $M\in\mathcal{L}(M_{0})$. The following is the endoscopic
counterpart of an identity in §7.3.

######  Proposition 7.4.9 (Cf. [61, VIII.4.1]).

For all $\tilde{\gamma}\in
D_{\mathrm{geom},\tilde{G}\text{-reg},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$,
we have

${}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{L},\mathcal{E}}_{\tilde{M}}\left(\tilde{\gamma},{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}(\cdot)\right).$

###### Proof.

By linearity in $\tilde{\gamma}$, it suffices to fix
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, $\delta\in
SD_{\mathrm{geom},\tilde{G}\text{-reg}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$
such that
$\tilde{\gamma}=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)$ and
show that

$\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s]){}^{c}S^{G^{!}[s]}_{M^{!}}\left(\delta[s],f^{G^{!}[s]}\right)=\sum_{L\in\mathcal{L}(M)}I^{\tilde{L},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}(f)\right).$
(7.4.1)

The stable counterpart [61, VIII.4.1] implies that the left-hand side of
(7.4.1) is

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

We apply Lemma 5.4.2 (with $R=M$) and Lemma 3.5.7 to transform the above into

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

Since $z[s_{L}]$ is central in $L^{!}[s^{L}]$, this equals

$\sum_{(L,s^{L})}i_{M^{!}}(\tilde{L},L^{!}[s^{L}])S^{L^{!}[s^{L}]}_{M^{!}}\left(\delta[s^{L}],A(L,s^{L})\right)$

where

$\displaystyle A(L,s^{L})$
$\displaystyle:=\sum_{s_{L}}i_{L^{!}[s^{L}]}(\tilde{G},G^{!}[s_{L}]){}^{c}S\theta^{G^{!}[s_{L}]}_{L^{!}[s^{L}]}\left(f^{G^{!}[s_{L}]}\right)[s_{L}]$
$\displaystyle={}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\mathbf{L}^{!}[s^{L}],f\right)=\mathcal{T}_{\mathbf{L}^{!}[s^{L}],\tilde{L}}\left({}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}(f)\right).$

The left-hand side of (7.4.1) thus equals

$\sum_{L}\sum_{s^{L}}i_{M^{!}}(\tilde{L},L^{!}[s^{L}])S^{L^{!}[s^{L}]}_{M^{!}}\left(\delta[s^{L}],\mathcal{T}_{\mathbf{L}^{!}[s^{L}],\tilde{L}}\left({}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}(f)\right)\right).$

By definition, this equals the right-hand side of (7.4.1). ∎

######  Corollary 7.4.10 (Cf. [61, VIII.4.3]).

Suppose that

  * •

$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot)$
for all $M\in\mathcal{L}(M_{0})$ and all $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-reg},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

  * •

the assertions in Theorem 7.4.8 are known when $M$ is replaced by any
$L\in\mathcal{L}(M)$ with $L\neq M$, or when $G$ is replaced by any
$L\in\mathcal{L}(M)$ with $L\neq G$.

Then the assertions (i) and (ii) in Theorem 7.4.8 hold for $M$.

###### Proof.

The cited proof is reproduced below for the reader’s convenience. Expanding
${}^{c}I^{\tilde{G}}_{\tilde{M}}$ and
${}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}$ in terms of non-compact
analogues (by Proposition 7.4.9), the assumptions imply

${}^{c}I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)-{}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}\left(\tilde{\gamma},{}^{c}\theta^{\tilde{G}}_{\tilde{M}}(f)-{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(f)\right).$

Denote the difference inside $I^{\tilde{M}}(\tilde{\gamma},\cdot)$ as
$\varphi$. The equality combined with the compact-support property of
${}^{c}I^{\tilde{G}}_{\tilde{M}}$ and
${}^{c}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}$ implies that
$\varphi\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$.
On the other hand, a comparison of the descent formulas for
${}^{c}\theta^{\tilde{G}}_{\tilde{M}}$ (see [61, VIII.1.6 Lemme]) and
${}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}$ (Proposition 7.4.4) shows
that $\varphi$ is cuspidal.

If $M=G$, it is clear that $\varphi=0$. If $M\neq G$, we apply Proposition
7.2.5 and its endoscopic analogue (which follows from the stable case in [61,
VIII.3.7 Proposition]) to see

$\sum_{S\in\mathcal{P}(M)}\omega_{S}(X)I^{\tilde{M}}\left(\pi,\nu_{S},X,\varphi\right)=0$

for all $X\in\mathfrak{a}_{M,F}$ and $\pi\in
D_{\operatorname{ell},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$,
provided that $\nu_{S}$ is sufficiently deep for each $S\in\mathcal{P}(M)$.
Since
$\varphi\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$,
we can shift $\nu_{S}$ to $0$ in the displayed equality to obtain
$I^{\tilde{M}}(\pi,X,\varphi)=I^{\tilde{M}}(\pi,X,0,\varphi)=0$.

By varying $X$ and $\pi$, we deduce $\varphi=0$ in the case $M\neq G$, since
$\varphi\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{M})\otimes\operatorname{mes}(M)$.
The assertion follows. ∎

###  7.5 Construction of $\epsilon_{\tilde{M}}$

This section is parallel to [61, VIII.4.4]. Our situation is simpler since the
matching of Shalika germs (Theorem 6.6.7) is already available to us.

As before, fix $M\in\mathcal{L}(M_{0})$ and assume inductively that whenever
$\tilde{\gamma}$ is $\tilde{G}$-regular,

$\displaystyle
I^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma},\cdot)=I^{\tilde{L},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},\cdot),\quad
L\in\mathcal{L}(M),\;L\neq G,$ $\displaystyle
I^{\tilde{G}}_{\tilde{L}}(\tilde{\gamma},\cdot)=I^{\tilde{G},\mathcal{E}}_{\tilde{L}}(\tilde{\gamma},\cdot),\quad
L\in\mathcal{L}(M),\;L\neq M.$

Similarly, we assume that the assertions of Theorem 7.4.8 hold when $G$ (resp.
$M$) becomes smaller (resp. larger).

######  Proposition 7.5.1.

There exists a linear map

$\epsilon_{\tilde{M}}:\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\to\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{M})\otimes\operatorname{mes}(M)$

characterized uniquely by

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}\left(\tilde{\gamma},\epsilon_{\tilde{M}}(f)\right)$

for all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

###### Proof.

We may assume $M\neq G$. The uniqueness of $f\mapsto\epsilon_{\tilde{M}}(f)$
is clear. Let us fix $f$ and prove that for each stable semisimple conjugacy
class $\mathcal{O}\subset M(F)$, there exists
$\varphi_{1}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
such that

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}(\tilde{\gamma},\varphi_{1})$
(7.5.1)

for all $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$
such that $\operatorname{Supp}(\tilde{\gamma})$ is close to
$\mathbf{p}^{-1}(\mathcal{O})$.

The induction hypotheses and the matching of weighted Shalika germs (Theorem
6.6.7) imply that

$\displaystyle
I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)$
$\displaystyle=\sum_{L\in\mathcal{L}(M)}\left(I^{\mathcal{E}}_{\tilde{L}}\left(g^{\tilde{L},\mathcal{E}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)-I_{\tilde{L}}\left(g^{\tilde{L}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)\right)$
$\displaystyle=I^{\mathcal{E}}_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)-I_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right).$

Since $D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})$ is finite-dimensional, we
may choose $\varphi_{1}$ such that

$I^{\tilde{M}}(\tilde{\eta},\varphi_{1})=I^{\mathcal{E}}_{\tilde{M}}(\tilde{\eta},f)-I_{\tilde{M}}(\tilde{\eta},f)$

for all $\tilde{\eta}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$.
Hence

$I^{\mathcal{E}}_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)-I_{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),f\right)=I^{\tilde{M}}\left(g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}),\varphi_{1}\right)=I^{\tilde{M}}(\tilde{\gamma},\varphi_{1})$

by the definition of Shalika germs. This proves (7.5.1).

Next, we claim that there exist a compact subset $\Omega\subset M(F)$ and
$\varphi_{2}\in\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
such that

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}(\tilde{\gamma},\varphi_{2})$
(7.5.2)

for all $f$ and $\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-reg},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$
such that
$\operatorname{Supp}(\tilde{\gamma})\cap\mathbf{p}^{-1}(\Omega)=\varnothing$.

To establish (7.5.2), apply Proposition 7.4.9 and its counterpart for
${}^{c}I_{\tilde{M}}$ to express the left-hand side as

${}^{c}I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-{}^{c}I_{\tilde{M}}(\tilde{\gamma},f)-\sum_{\begin{subarray}{c}L\in\mathcal{L}(M)\\\
L\neq
G\end{subarray}}\left(I^{\tilde{L},\mathcal{E}}_{\tilde{M}}\left(\tilde{\gamma},{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}(f)\right)-I^{\tilde{L}}_{\tilde{M}}\left(\tilde{\gamma},{}^{c}\theta^{\tilde{G}}_{\tilde{L}}(f)\right)\right).$

By induction hypotheses, $L\neq G$ (resp. $L\neq M$) implies
$I^{\tilde{L},\mathcal{E}}_{\tilde{M}}=I^{\tilde{L}}_{\tilde{M}}$ (resp.
${}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{L}}={}^{c}\theta^{\tilde{G}}_{\tilde{L}}$).
Hence the previous expression becomes

${}^{c}I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-{}^{c}I_{\tilde{M}}(\tilde{\gamma},f)-I^{\tilde{M}}\left(\tilde{\gamma},{}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(f)-{}^{c}\theta^{\tilde{G}}_{\tilde{M}}(f)\right).$

Combining §7.3 and Proposition 7.4.7, the term
${}^{c}I^{\mathcal{E}}_{\tilde{M}}(\cdot,f)-{}^{c}I_{\tilde{M}}(\cdot,f)$ is
seen to vanish off some compact $\Omega\subset M(F)$. One takes
$\varphi_{2}:={}^{c}\theta^{\tilde{G},\mathcal{E}}_{\tilde{M}}(f)-{}^{c}\theta^{\tilde{G}}_{\tilde{M}}(f)$
to obtain (7.5.2).

Now we can use an argument of partition of unity to glue $\varphi_{1}$ (for
various $\mathcal{O}$) and $\varphi_{2}$ from (7.5.1) and (7.5.2), and obtain
$\varphi\in\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
such that

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}(\tilde{\gamma},\varphi),\quad\tilde{\gamma}\in
D_{\mathrm{geom},G\text{-reg},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}.$
(7.5.3)

We have to extend (7.5.3) to general $\tilde{\gamma}$. It suffices to fix
$\mathcal{O}\subset M(F)$ and assume that
$\operatorname{Supp}(\tilde{\gamma})=\mathbf{p}^{-1}(\mathcal{O})$. The
separation property of Shalika germs (see §6.6) furnishes
$\tilde{\gamma}^{\prime}$ with $G$-regular support close to
$\mathbf{p}^{-1}(\mathcal{O})$, such that
$g^{\tilde{M}}_{\tilde{M},\mathcal{O}}(\tilde{\gamma}^{\prime})=\tilde{\gamma}$.
Then

$\displaystyle
I^{\tilde{M}}(\tilde{\gamma},\varphi)=I^{\tilde{M}}(\tilde{\gamma}^{\prime},\varphi)$
$\displaystyle=I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma}^{\prime},f)-I_{\tilde{M}}(\tilde{\gamma}^{\prime},f)$
$\displaystyle=\sum_{L\in\mathcal{L}(M)}\left(I^{\mathcal{E}}_{\tilde{L}}\left(g^{\tilde{L},\mathcal{E}}_{\tilde{M}}(\tilde{\gamma}^{\prime}),f\right)-I_{\tilde{L}}\left(g^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma}^{\prime}),f\right)\right).$

In the last expression, all the differences with $L\neq M$ vanish by the
induction assumption $I^{\mathcal{E}}_{\tilde{L}}=I_{\tilde{L}}$ and the
matching of Shalika germs. What remains is
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)-I_{\tilde{M}}(\tilde{\gamma},f)$.
This extends (7.5.3) to all $\tilde{\gamma}$.

All in all, we take $\epsilon_{\tilde{M}}(f):=\varphi$, which is uniquely
characterized and linear in $f$. It remains to show
$\epsilon_{\tilde{M}}(f)\in\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{M})\otimes\operatorname{mes}(M)$.
Indeed, for all $R\in\mathcal{L}^{M}(M_{0})$, $\tilde{\eta}\in
D_{\mathrm{geom},-}(\tilde{R})\otimes\operatorname{mes}(R)^{\vee}$ with $R\neq
M$, a comparison between descent formulas (Proposition 5.1.5, Corollary 5.4.5)
using induction hypotheses entails

$I^{\tilde{R}}\left(\tilde{\eta},\epsilon_{\tilde{M}}(f)_{\tilde{R}}\right)=I^{\mathcal{E}}_{\tilde{M}}\left(\tilde{\eta}^{\tilde{M}},f\right)-I_{\tilde{M}}\left(\tilde{\eta}^{\tilde{M}},f\right)=0.$

This completes the proof. ∎

######  Remark 7.5.2.

As a by-product of the proof, we see that $\epsilon_{\tilde{M}}(f)$ can be
written as

${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(f)-{}^{c}\theta_{\tilde{M}}(f)+\;\text{an
element
of}\;\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M).$

######  Corollary 7.5.3.

Let
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
Then $\epsilon_{\tilde{M}}(f)$ is Schwartz, and for every $\pi\in
D_{\mathrm{temp},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, the
function

$\lambda\mapsto
I^{\tilde{M}}\left(\pi,\lambda,\epsilon_{\tilde{M}}(f)\right),\quad\lambda\in
i\mathfrak{a}_{M}^{*}/i\mathfrak{a}_{M,F}^{\vee}$

extends to a rational function on
$\mathfrak{a}_{M,\mathbb{C}}^{*}/i\mathfrak{a}_{M,F}^{\vee}$ with only
finitely many polar hyperplanes of the form
$e^{\langle\cdot,\check{\alpha}\rangle}=\mathrm{const}$ where
$\alpha\in\Sigma^{G}(A_{M})$.

###### Proof.

In view of Remark 7.5.2, this reduces to the corresponding properties of
${}^{c}\theta_{\tilde{M}}(f)$ (Proposition 7.2.5),
${}^{c}\theta^{\mathcal{E}}_{\tilde{M}}(f)$ (see §7.4), and of
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
(see §7.2). ∎

##  Chapter 8 Weighted characters: Archimedean case

Let $F$ be a an Archimedean local field. Consider a covering of metaplectic
type $\mathbf{p}:\tilde{G}\to G(F)$ where $G=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W)$; here we fix the
auxiliary data $(W,\langle\cdot|\cdot\rangle)$, $\psi$ as in §7. We also fix a
maximal compact subgroup $K\subset G(F)$ arising from a Cartan involution
$\theta$, in good position relative to $M_{0}$.

Let $M\in\mathcal{L}(M_{0})$. Our goal is to define the compactly supported
variant ${}^{c}I_{\tilde{M}}(\tilde{\gamma},f)$ of
$I_{\tilde{M}}(\tilde{\gamma},f)$ for $\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G)$,
and formulate the matching theorems. In contrast with the non-Archimedean
case, a new problem here is that the canonically normalized weighted character
$J_{\tilde{M}}(\pi_{\lambda},f)$, as a meromorphic family in $\lambda$, may
have infinitely many polar hyperplanes. For this reason, we revert to Arthur’s
earlier recipe of using normalizing factors. This is inspired by [61, IX.5],
where a more complicated “rational normalization” is used in the twisted
setting.

Without loss of generality, we may assume
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ with $\dim W=2n$. Based on the results
of [46] (resp. [10]), the normalizing factors for $\widetilde{\mathrm{Sp}}(W)$
(resp. $\operatorname{SO}(2n+1)$) are explicitly and canonically given.
Furthermore, by matching the infinitesimal characters of discrete series
between $\widetilde{\mathrm{Sp}}(2n)$ and $\operatorname{SO}(2n+1)$, their
normalizing factors differ by a precise power of $\sqrt{2}$. This is the
content of §8.1, which is based on explicit computations. In this way, we
obtain the normalized weighted characters $J^{r}_{\tilde{M}}(\pi,f)$ for
$\pi\in\Pi_{\mathrm{unit},-}(\tilde{M})$, as well as the variant
${}^{r}\phi_{\tilde{M}}$ of $\phi_{\tilde{M}}$.

In §8.3, we proceed to define the compactly supported weighted characters
${}^{c,r}J_{\tilde{M}}(\pi,f)$ out from $J^{r}_{\tilde{M}}(\pi,f)$. We also
obtain the variant ${}^{c}\phi_{\tilde{M}}$ of $\phi_{\tilde{M}}$, from which
${}^{c}I_{\tilde{M}}(\tilde{\gamma},f)$ is built. The same construction also
applies to the stable side, although this was not documented before.

The triad of maps ${}^{c}\phi_{\tilde{M}}$, $\phi_{\tilde{M}}$ and
${}^{r}\phi_{\tilde{M}}$ are related by families of maps
${}^{c}\theta_{\tilde{M}}$, ${}^{c,r}\theta_{\tilde{M}}$ and
${}^{r}\theta_{\tilde{M}}$, as $M\in\mathcal{L}(M_{0})$ varies. Together with
the distributions ${}^{c}I_{\tilde{M}}(\tilde{\gamma},\cdot)$, each of them
comes with an endoscopic counterpart; see §8.9. The formulation of the
respective matching theorems is the content of §8.10.

Among these statements, the case of ${}^{r}\theta_{\tilde{M}}$ (Theorem
8.10.1) is immediately within reach. Roughly speaking, it reduces to the
stabilization of some factors $r^{\tilde{G}}_{\tilde{M}}(\pi)$ made from
normalizing factors; we seek to express them via similar factors
$s^{G^{!}[s]}_{M^{!}}(\phi^{!})$ on the stable side, where
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. This is achieved in §8.8 by
reducing to the case of $\operatorname{SO}(2n+1)$ settled in [9].

###  8.1 Normalizing factors

####  8.1.1 The setting

Consider a general covering $\tilde{G}\twoheadrightarrow G(F)$ over a local
field $F$ of characteristic zero and a Levi subgroup $M$ of $G$. For
$\pi\in\Pi_{-}(\tilde{M})$, the normalized intertwining operators take the
form

$R_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})=r_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})^{-1}J_{\tilde{Q}|\tilde{P}}(\pi_{\lambda}):I_{\tilde{P}}(\pi_{\lambda})\to
I_{\tilde{Q}}(\pi_{\lambda}),\quad P,Q\in\mathcal{P}(M),$

where $J_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})$ is the standard intertwining
operator, and $r_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})$ is a meromorphic
function in $\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{*}$. The _normalizing
factors_ $r_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})$ are subject to the
conditions in [46, §3.1], which are in turn based on [10, §2]. Their
definition is eventually reduced to the case when $M$ is maximal proper and
$\pi\in\Pi_{2,-}(\tilde{M})$ (i.e. essentially square-integrable, or discrete
series).

Specifically, fix a maximal $F$-torus $T\subset G$ and denote the elements of
$\Sigma(A_{M})$ as $[\alpha]$, viewed as equivalence classes of
$\alpha\in\Sigma(G,T)$ restricting to $[\alpha]$, we have the property

$r_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})=\prod_{[\alpha]\in\Sigma_{Q}^{\mathrm{red}}\cap\Sigma_{P}^{\mathrm{red}}}r_{[\alpha]}(\pi_{\lambda}),$
(8.1.1)

where $r_{[\alpha]}$ is the normalizing factor relative to the Levi subgroup
$\widetilde{M_{[\alpha]}}\subset\tilde{G}$ containing $\tilde{M}$ minimally
with $\pm\alpha$ in $\Sigma(M_{[\alpha]},T)$.

Now revert to the case where $\tilde{G}$ is of metaplectic type.

  * •

For non-Archimedean $F$, the existence of normalizing factors for general
coverings is settled in [46, §3.1], although they are not canonically given
except in the unramified case (see §13.3). This suffices for the purposes of
this work.

  * •

For Archimedean $F$, we will have to compare them with their counterparts for
$\operatorname{SO}(2n+1)$, and this requires explicit formulas.

We remark that in the uncovered case, Langlands proposes an explicit choice of
normalizing factors in terms of local factors. Although the metaplectic
counterpart is accessible through $\Theta$-correspondence, we do not need this
level of precision in this work.

Henceforth, $F\in\\{\mathbb{R},\mathbb{C}\\}$ and $G=\operatorname{Sp}(W)$,
$\mathbf{p}:\tilde{G}\to G(F)$. Our strategy is to work out the formula [46,
§3.1, (10)]. In view of the reduction procedure for normalizing factors, we
fix the following data

  * •

a maximal proper Levi subgroup $M$ of $G$ and $P,Q\in\mathcal{P}(M)$ which are
opposite to each other;

  * •

$\pi\in\Pi_{2,-}(\tilde{M})$.

Then there is only one $[\alpha]$, we have
$r_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})=r_{[\alpha]}(\pi_{\lambda})$, and
$M_{[\alpha]}=G$. The case when $\tilde{G}$ is of metaplectic type will then
follow immediately.

######  Notation 8.1.1.

We write $\Gamma_{\mathbb{R}}(z):=\pi^{-z/2}\Gamma\left(\frac{z}{2}\right)$
and $\Gamma_{\mathbb{C}}(z):=2(2\pi)^{-z}\Gamma(z)$, for $z\in\mathbb{C}$.

####  8.1.2 The real case

Let $F=\mathbb{R}$. Take a maximal $\mathbb{R}$-torus $T\subset M$ such that
$T/A_{M}$ is anisotropic. Define $\rho^{M}$ to be the half-sum of positive
roots of $(\mathfrak{m}_{\mathbb{C}},\mathfrak{t}_{\mathbb{C}})$ relative to
some choice of Borel subgroup over $M_{\mathbb{C}}$. Denote by $\sigma$ the
complex conjugation in $\operatorname{Gal}(\mathbb{C}|\mathbb{R})$.

Fix a Cartan involution $\theta$ of $\mathfrak{g}$ so that the associated
maximal compact subgroup is in good position relative to $M$. Fix an invariant
bilinear form $\mathsf{B}$ on $\mathfrak{g}$ such that
$X\mapsto-\mathsf{B}(X,\theta X)$ is positive definite.

######  Notation 8.1.2.

Unless otherwise stated, we will use the _twofold_ version of the metaplectic
covering $\mathbf{p}:\tilde{G}\to G(\mathbb{R})$ in what follows.

Take $\mathop{}\\!\mathrm{d}\chi\in\mathfrak{t}^{*}_{\mathbb{C}}$ such that
$\mathop{}\\!\mathrm{d}\chi+\rho^{M}$ represents the infinitesimal character
of $\pi$. By Harish-Chandra’s theory of discrete series, $\chi$ lifts to a
genuine character of $\tilde{T}$. Take
$\lambda_{0}\in\mathfrak{t}_{\mathbb{C}}^{*}$ and $\mu\in i\mathfrak{t}^{*}$
such that

$\langle\mathop{}\\!\mathrm{d}\chi,H\rangle=\langle\lambda_{0},H-\sigma
H\rangle+\frac{1}{2}\langle\mu-\rho^{M},H+\sigma H\rangle,\quad
H\in\mathfrak{t}_{\mathbb{C}}.$

If $\pi$ is replaced by $\pi_{\lambda}$ with $\lambda\in
i\mathfrak{a}^{*}_{M}$, then $(\lambda_{0},\mu)$ is replaced by
$(\lambda_{0}+\lambda,\mu+\lambda)$.

Let $\Sigma_{P}(G,T)\subset\Sigma(G,T)$ be the subset of roots restricting to
$\Sigma_{P}$. An element of $\Sigma(G,T)$ is said to be _real_ if it is
$\sigma$-invariant. Since $M$ is maximal and $T/A_{M}$ is anisotropic, there
is at most one real root in $\Sigma_{P}(G,T)$. Let $\alpha$ be such a real
root, and let $H_{\alpha}\in\mathfrak{t}$ be characterized by
$\mathsf{B}(H,H_{\alpha})=\langle\alpha,H\rangle$ for all $H\in\mathfrak{t}$.
Take a triplet $(H^{\prime},X^{\prime},Y^{\prime})$ such that
$H^{\prime}=2\langle\alpha,H_{\alpha}\rangle^{-1}H_{\alpha}$,
$X^{\prime}\in\mathfrak{g}_{\alpha}$, $X^{\prime}\in\mathfrak{g}_{-\alpha}$,
and

$[H^{\prime},X^{\prime}]=2X^{\prime},\quad[H^{\prime},Y^{\prime}]=-2Y^{\prime},\quad\quad[X^{\prime},Y^{\prime}]=H^{\prime}.$

From the triplet we obtain a homomorphism $\operatorname{SL}(2)\to G$. Denote
the pullback along $\mathbf{p}$ of this homomorphism as
$\Phi_{\alpha}:\widetilde{\operatorname{SL}}(2)\to\tilde{G}$. Calculating with

$H^{\prime}=\begin{pmatrix}1&\\\ &-1\end{pmatrix},\quad
X^{\prime}=\begin{pmatrix}&1\\\ &\end{pmatrix},\quad
Y^{\prime}=\begin{pmatrix}&\\\ 1&\end{pmatrix}$

in $\mathfrak{sl}_{2}$, where the blank entries are zero, we see that
$\pi(X^{\prime}-Y^{\prime})=\bigl{(}\begin{smallmatrix}0&\pi\\\
-\pi&0\end{smallmatrix}\bigr{)}$. Put

$\gamma:=\exp\left(\mathop{}\\!\mathrm{d}\Phi_{\alpha}(\pi(X^{\prime}-Y^{\prime}))\right)\;\in\tilde{T}.$

Choose a Borel subgroup $B$ of $G_{\mathbb{C}}$ such that
$T_{\mathbb{C}}\subset B\subset P_{\mathbb{C}}$, and that $B\cap
M_{\mathbb{C}}$ is the Borel subgroup of $M_{\mathbb{C}}$ chosen before.
Define $\rho_{B}$ in this way. Since we work with twofold coverings here
($m=2$), the discussions above imply $\gamma^{4}=1$ and there exists
$k\in\\{0,1,2,3\\}$ such that

$(-1)^{\langle\rho_{P},\check{\alpha}\rangle}\chi(\gamma)=\exp\left(\frac{2\pi
k}{4i}\right).$

When $\frac{k}{4}=0$ (resp. $\frac{k}{4}=\frac{1}{2}$), put $N_{0}:=1$ (resp.
$N_{0}:=0$). Furthermore, we follow [46, Remarques 3.2.2, 3.2.3] to define

$\mathsf{G}(z):=\begin{cases}\displaystyle\frac{\Gamma_{\mathbb{R}}(z+N_{0})}{\Gamma_{\mathbb{R}}(z+N_{0}+1)},&\frac{k}{4}\in\\{0,\frac{1}{2}\\}\\\
\sqrt{2}\cdot\displaystyle\frac{\Gamma_{\mathbb{R}}(2z)}{\Gamma_{\mathbb{R}}(2z+1)},&\frac{k}{4}\in\\{\frac{1}{4},\frac{3}{4}\\}.\end{cases}$
(8.1.2)

Recall that $\rho_{P}$ is defined as the half-sum of roots appearing in
$U_{P}$, and $\langle\rho_{P},\check{\alpha}\rangle$ is interpreted
accordingly. This quantity appeared in Harish-Chandra’s work [35, p.178,
p.189] on $\mu$-functions, and he denotes
$\langle\rho_{P},\check{\alpha}\rangle$ by $\rho_{\alpha}$. One can work with
$\rho_{B}$ instead, by virtue of the following result.

######  Lemma 8.1.3.

In the circumstance above, we have
$\langle\rho_{B},\check{\alpha}\rangle=\langle\rho_{P},\check{\alpha}\rangle$.

###### Proof.

By definition, we have $\rho_{B}=\rho_{P}+\rho^{M}_{B\cap M_{\mathbb{C}}}$,
where $\rho^{M}_{B\cap M_{\mathbb{C}}}$ is the half-sum of
$B\cap{M_{\mathbb{C}}}$-positive roots in $M_{\mathbb{C}}$. Since $T/A_{M}$ is
anisotropic, $\sigma\rho^{M}_{B\cap M_{\mathbb{C}}}=-\rho^{M}_{B\cap
M_{\mathbb{C}}}$. On the other hand, $\sigma\check{\alpha}=\check{\alpha}$
since $\alpha$ is real. It follows that $\langle\rho^{M}_{B\cap
M_{\mathbb{C}}},\check{\alpha}\rangle=0$ since $\langle\cdot,\cdot\rangle$ is
$\sigma$-invariant. ∎

######  Notation 8.1.4.

Let us say that we are in “case I” if there exists a real root in
$\Sigma_{P}(G,T)$, denoted as $\alpha_{0}$; otherwise we are in “case II”. In
case I, we pick for each $\operatorname{Gal}(\mathbb{C}|\mathbb{R})$-orbit in
$\Sigma_{P}(G,T)\smallsetminus\\{\alpha_{0}\\}$ a representative $\alpha$ such
that

$\langle\sigma\mu-\mu,\check{\alpha}\rangle\in\mathbb{Z}_{\leq 0}.$ (8.1.3)

######  Definition–Proposition 8.1.5.

In the circumstance above, the normalizing factors can be taken as

$r_{\tilde{Q}|\tilde{P}}(\pi_{\lambda})=\begin{cases}\displaystyle\prod_{\begin{subarray}{c}\alpha\in\Sigma_{P}(G,T)\\\
\bmod\;\operatorname{Gal}(\mathbb{C}|\mathbb{R})\\\
\alpha\neq\alpha_{0}\end{subarray}}\frac{\Gamma_{\mathbb{C}}(\langle\mu+\lambda,\check{\alpha}\rangle)}{\Gamma_{\mathbb{C}}(\langle\mu+\lambda,\;\check{\alpha}\rangle+1)}\cdot\mathsf{G}(\langle\mu+\lambda,\check{\alpha}_{0}\rangle),&\text{case
I},\\\ \displaystyle\prod_{\begin{subarray}{c}\alpha\in\Sigma_{P}(G,T)\\\
\bmod\;\operatorname{Gal}(\mathbb{C}|\mathbb{R})\end{subarray}}\frac{\Gamma_{\mathbb{C}}(\langle\mu+\lambda,\;\check{\alpha}\rangle)}{\Gamma_{\mathbb{C}}(\langle\mu+\lambda,\;\check{\alpha}\rangle+1)},&\text{case
II},\\\ \end{cases}$

for all $\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{*}$. The representatives
$\alpha$ in each orbit are chosen as in (8.1.3).

###### Proof.

This is the formula [46, (10)] which intervenes in the Théorème 3.2.1 thereof.
∎

We have to be more precise. Since $M$ is maximal proper in $G$ and admits a
maximal torus $T$ with $T/A_{M}$ anisotropic, there exist $a\in\\{1,2\\}$ and
a symplectic subspace $W^{\flat}\subset W$ of codimension $2a$ such that

$M=\operatorname{GL}(a)\times\operatorname{Sp}(W^{\flat})\hookrightarrow\operatorname{Sp}(W)=G.$

Decompose $T$ into $T_{\operatorname{GL}}\times T^{\flat}$ accordingly,
$T^{\flat}\subset\operatorname{Sp}(W^{\flat})$ being an anisotropic torus. Let
$n:=\frac{1}{2}\dim W$.

  * •

As before, choose a Borel subgroup $B$ of $G_{\mathbb{C}}$ appropriately.

  * •

Take a symplectic basis for $W^{\flat}_{\mathbb{C}}$, enlarge it to a
symplectic basis of $W_{\mathbb{C}}$, so that $X^{*}(T_{\mathbb{C}})$ has the
standard basis $e^{*}_{1},\ldots,e^{*}_{n}$ satisfying

$X^{*}(T_{\operatorname{GL},\mathbb{C}})=\bigoplus_{k=1}^{a}\mathbb{Z}e^{*}_{k},\quad
X^{*}(T^{\flat}_{\mathbb{C}})=\bigoplus_{k=a+1}^{n}\mathbb{Z}e^{*}_{k},$