${Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}}$${Z_{{M^{!}}^{\vee}}/Z_{{G^{!}[s]}^{\vee}}}$${Z_{\tilde{R}^{\vee}}^{\circ}/Z_{\tilde{L}^{\vee}}^{\circ}}$${Z_{{R^{!}}^{\vee}}/Z_{{L^{!}}^{\vee}}}$$\scriptstyle{\phi^{G}_{M}}$$\scriptstyle{p_{2}}$$\scriptstyle{\phi^{L}_{R}}$
(5.4.5)

with evident arrows. Denote the diagonal composition as $p_{1}(s)$. The
discussions above lead to

$X(s)^{-1}=\\#\operatorname{ker}\left(p_{1}(s)\right).$ (5.4.6)

On the other hand, Lemma 5.4.3 together with (5.4.5) imply that when
$d^{G}_{R}(M,L)\neq 0$, we have

$\frac{\\#\operatorname{ker}(p_{2})}{\\#\operatorname{ker}(p_{1}(s))}=\left(\\#\operatorname{ker}(\phi^{L}_{R})\right)^{-1}=i_{R^{!}}(\tilde{L},L^{!}).$

To conclude the proof, we return to (5.4.4). From $L^{!}=L^{!}[r^{L}]$,
(5.4.5) and (5.4.6) we see that $X(s)$ depends only on the image
$r^{L}\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})$ of $s$. In view of Lemma
5.4.3, we conclude that

$I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta[t]^{M^{!}},f\right)=\sum_{(L,s)}d^{G}_{R}(M,L)X(s)S^{L^{!}}_{R^{!}}\left(\delta[r^{L}],B^{\tilde{L}},(f_{\tilde{L}})^{L^{!}}\right)\\\
=\sum_{L\in\mathcal{L}^{G}(R)}d^{G}_{R}(M,L)\sum_{r^{L}\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})}\frac{\\#\operatorname{ker}(p_{2})}{\\#\operatorname{ker}(p_{1}(s))}S^{L^{!}}_{R^{!}}\left(\delta[r^{L}],B^{\tilde{L}},(f_{\tilde{L}})^{L^{!}}\right)\\\
=\sum_{L\in\mathcal{L}^{G}(R)}d^{G}_{R}(M,L)\sum_{r^{L}\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})}i_{R^{!}}(\tilde{L},L^{!})S^{L^{!}}_{R^{!}}\left(\delta[r^{L}],B^{\tilde{L}},(f_{\tilde{L}})^{L^{!}}\right)\\\
=\sum_{L\in\mathcal{L}^{G}(R)}d^{G}_{R}(M,L)I^{\tilde{L},\mathcal{E}}_{\tilde{R}}\left(\mathbf{R}^{!},\delta,f_{\tilde{L}}\right),$

as asserted. ∎

######  Corollary 5.4.5.

In the circumstance of Proposition 5.4.4, we have

$I^{\mathcal{E}}_{\tilde{M}}\left(\tilde{\gamma}^{\tilde{M}},f\right)=\sum_{L\in\mathcal{L}(R)}d^{G}_{R}(M,L)I^{\tilde{L},\mathcal{E}}_{\tilde{R}}\left(\tilde{\gamma},f_{\tilde{L}}\right).$

###### Proof.

Combine Proposition 5.4.4, Definition–Proposition 5.3.10 with Proposition
3.8.4. ∎

We will encounter many other endoscopic descent formulas of the same form.
Because of the purely combinatorial nature of the arguments presented above,
the proof will not be repeated anymore.

###  5.5 Reduction to the $\tilde{G}$-regular case

Let $F$ be a local field of characteristic zero, and let
$\mathbf{p}:\tilde{G}\twoheadrightarrow G(F)$ be of metaplectic type. We begin
by studying the local behavior of
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$, where $M\subset G$ is a Levi
subgroup.

######  Lemma 5.5.1.

Let $\mathcal{O}$ be a $G$-equisingular stable semisimple conjugacy class in
$M(F)$. For every
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
there exists
$g\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
such that

$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{M}}(\tilde{\gamma},g)$

holds if one of the following conditions hold:

  1. (i)

$\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, such that
$\operatorname{Supp}(\tilde{\gamma})$ is formed by $\tilde{G}$-regular
elements which are sufficiently close to $\mathbf{p}^{-1}(\mathcal{O})$;

  2. (ii)

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

###### Proof.

Our arguments are modeled on [60, V.1.9 Lemme]. As in the proof of Proposition
5.4.4, without loss of generality we may assume
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$.

Consider a Levi subgroup $R\subset M$ and
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$. Let
$\mathcal{O}_{R^{!}}$ be the finite union of stable semisimple classes in
$R^{!}(F)$ corresponding to $\mathcal{O}$ via
$\Sigma_{\mathrm{ss}}(R^{!})\to\Sigma_{\mathrm{ss}}(R)\to\Sigma_{\mathrm{ss}}(M)$.

For $L\in\mathcal{L}(R)$ with $d^{G}_{R}(M,L)\neq 0$ and
$s\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})$, we claim that
$\mathcal{O}_{R^{!}}[s]$ is $L^{!}[s]$-equisingular. Indeed, let
$\delta\in\mathcal{O}_{R^{!}}$ and $\gamma\in R(F)_{\mathrm{ss}}$ such that
$\delta\leftrightarrow\gamma$ via endoscopy, then $\gamma$ is equisingular
relative to $M\subset G$, hence Lemma 5.2.4 implies that $\gamma$ is
equisingular relative to $R\subset L$. In view of Definition 5.3.7 and Lemma
5.3.8, the $L^{!}[s]$-equisingularity of $\delta[s]$ follows.

By [60, V.1.4 (2), (3)] and its non-Archimedean analogues in [60, III], for
all quadruplets $(\tilde{R},\mathbf{R}^{!},\tilde{L},s)$ above, there exists
$g_{\tilde{R},\mathbf{R}^{!},\tilde{L},s}\in
S\mathcal{I}(R^{!})\otimes\operatorname{mes}(R^{!})$ such that

$S^{L^{!}[s]}_{R^{!}}\left(\delta[s],B^{\tilde{L}},\mathcal{T}_{\mathbf{L}^{!}[s],\tilde{L}}(f_{\tilde{L}})\right)=S^{R^{!}}\left(\delta,g_{\tilde{R},\mathbf{R}^{!},\tilde{L},s}\right)$
(5.5.1)

holds in each of the following circumstances:

  1. (i)′

$\delta\in SD_{\mathrm{orb}}(R^{!})\otimes\operatorname{mes}(R)^{\vee}$, such
that $\operatorname{Supp}(\delta)$ is formed by elements which are
sufficiently regular and close to $\mathcal{O}_{R^{!}}$;

  2. (ii)′

$\delta\in
SD_{\mathrm{geom}}(R^{!},\mathcal{O}_{R^{!}})\otimes\operatorname{mes}(R^{!})^{\vee}$.

Furthermore, when $R\subset M$ and $\mathbf{R}^{!}$ are kept fixed, $W^{M}(R)$
(as a subgroup of $W^{G}_{0}$) acts on $R^{!}$ and transports the data
$(\tilde{L},s)$. By transport of structure, the value of the left hand side of
(5.5.1) is unaltered under $W^{M}(R)$-action. Taking averages, we may assume
that each $g_{\tilde{R},\mathbf{R}^{!},\tilde{L},s}$ is $W^{M}(R)$-invariant.

Given $R\subset M$ and
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$, put

$g_{\tilde{R},\mathbf{R}^{!}}:=\sum_{L\in\mathcal{L}(R)}d^{G}_{R}(M,L)\sum_{s\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})}i_{R^{!}}(\tilde{L},L^{!}[s])g_{\tilde{R},\mathbf{R}^{!},\tilde{L},s}.$

We claim that, upon multiplying by a cut-off function (see below),
$\left(g_{\tilde{M},\mathbf{M}^{!}}\right)_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})}$
belongs to the space $\mathcal{I}^{\mathcal{E}}(\tilde{M})$ in Definition
3.8.5.

Consider $R\subset M$,
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$ and
$t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$; put
$\mathbf{M}^{!}:=\mathbf{M}^{!}[t]$. Take $\delta\in
SD_{\mathrm{geom}}(R^{!})\otimes\operatorname{mes}(R)^{\vee}$ subject to (i)′
or (ii)′. Since $d^{G}_{M}(M,L)\neq 0$ only when $L=G$, we have

$\displaystyle
S^{R^{!}}\left(\delta,(g_{\tilde{M},\mathbf{M}^{!}})^{R^{!}}[t]\right)$
$\displaystyle=S^{M^{!}}\left(\delta[t]^{M^{!}},g_{\tilde{M},\mathbf{M}^{!}}\right)$
(5.5.2)
$\displaystyle=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])S^{M^{!}}\left(\delta[t]^{M^{!}},g_{\tilde{M},\mathbf{M}^{!},\tilde{G},s}\right)$
$\displaystyle=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta[t]^{M^{!}},f\right)$
$\displaystyle=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}\left(\delta[t]^{M^{!}}\right),f\right)=I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{R}^{!},\tilde{R}}(\delta)^{\tilde{M}},f\right)$

where we invoked Definition–Proposition 5.3.10 and Proposition 3.8.4. The last
expression in (5.5.2) is hence independent of $t$ and $W^{M}(R)$-invariant.

So far we take $\delta$ with $\operatorname{Supp}(\delta)$ close to
$\mathcal{O}_{R^{!}}$, which implies
$\operatorname{Supp}\left(\delta[t]^{M^{!}}\right)$ is close to
$\mathcal{O}_{M^{!}}$. Consider the diagram of $F$-varieties

${M^{!}}$${M}$${T^{!}/\\!/W(M^{!},T^{!})}$${T/\\!/W(M,T)}$

where $T^{!}$, $T$ are split maximal $F$-tori, the vertical arrows are those
of Chevalley, and the horizontal arrow comes from endoscopy. Fix a
$C^{\infty}$ cut-off function on $(T/\\!/W(M,T))(F)$ centered at the point
classifying $\mathcal{O}$, of sufficiently small support, and pull it back to
$(T^{!}/\\!/W(M^{!},T^{!}))(F)$. As $g_{\tilde{M},\mathbf{M}^{!}}\in
S\mathcal{I}(M^{!})\otimes\operatorname{mes}(M)$ can be viewed as functions on
the strongly regular locus of $(T^{!}/\\!/W^{M^{!}}(T^{!}))(F)$, they may be
multiplied by the cut-off function to remove the constraint on supports.

Hence Theorem 3.8.10 affords
$g\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
with
$\mathcal{T}^{\mathcal{E}}(g)=\left(g_{\tilde{M},\mathbf{M}^{!}}\right)_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})}$.

Finally, let $\tilde{\gamma}$ be as in the assertions. By Theorem 3.10.6, we
may write

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

where $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\delta\in SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$
fulfills either (i)′ or (ii)′ with $R=M$. Hence (5.5.2) with $R=M$ implies

$I^{\tilde{M}}(\tilde{\gamma},g)=\sum_{(\mathbf{M}^{!},\delta)}S^{M^{!}}\left(\delta,g_{\tilde{M},\mathbf{M}^{!}}\right)=\sum_{(\mathbf{M}^{!},\delta)}I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,f\right).$

This yields $I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$ as desired. ∎

######  Proposition 5.5.2 (Cf. [60, V.1.11 Lemme]).

Let $F$ be a local field of characteristic zero. If the assertion in Theorem
5.3.11 holds whenever $\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ satisfies
$\operatorname{Supp}(\tilde{\gamma})\subset\tilde{G}_{\mathrm{reg}}$, then it
holds for all $\tilde{\gamma}\in
D_{\mathrm{geom},G\mathrm{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

###### Proof.

Let $\mathcal{O}\subset M(F)$ be a $G$-equisingular stable semisimple
conjugacy class. Fix
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
Definition 5.2.2 and Lemma 5.5.1 afford
$g_{1},g_{2}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
such that

$I_{\tilde{M}}\left(\tilde{\gamma},f\right)=I^{\tilde{M}}\left(\tilde{\gamma},g_{1}\right),\quad
I^{\mathcal{E}}_{\tilde{M}}\left(\tilde{\gamma},f\right)=I^{\tilde{M}}\left(\tilde{\gamma},g_{2}\right)$

in each of the following cases:

  1. (i)

$\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ whose
support is sufficiently regular and close to $\mathbf{p}^{-1}(\mathcal{O})$;

  2. (ii)

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

Case (i) and the assumptions imply $g_{1}=g_{2}$ in some neighborhood of
$\mathbf{p}^{-1}(\mathcal{O})$. By plugging this into case (ii) and varying
$\mathcal{O}$, we obtain
$I_{\tilde{M}}\left(\tilde{\gamma},f\right)=I^{\mathcal{E}}_{\tilde{M}}\left(\tilde{\gamma},f\right)$
for all $\tilde{\gamma}\in D_{\text{geom,
$G$-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. ∎

###  5.6 Statement of the weighted fundamental lemma

Consider a group of metaplectic type $\tilde{G}\xrightarrow{\mathbf{p}}G(F)$
in the unramified situation as in §3.11. In particular, we have a hyperspecial
subgroup $K=G(\mathfrak{o}_{F})$ of $G(F)$, over which $\mathbf{p}$ splits
canonically. It is harmless to assume in what follows that
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$, for the sake of simplicity.

Take the Haar measure on $G(F)$ with $\operatorname{mes}(K)=1$. Let $f_{K}$ be
the unit of $\mathcal{H}_{\dashrule[.7ex]{2222}{.4}}(K\backslash\tilde{G}/K)$.
If $M\subset G$ is a Levi subgroup in good position relative to $K$, we take
the Haar measure on $M(F)$ with $\operatorname{mes}(K\cap M(F))=1$. This
trivializes the lines $\operatorname{mes}(\cdots)$ in various formulas, and we
will omit them from the notation.

######  Definition 5.6.1.

Let $M\subset G$ be a Levi subgroup in good relative position to $K$. For
every $\tilde{\gamma}\in D_{\mathrm{geom},-}(\tilde{M})$, set

$r^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},K):=J^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f_{K}).$

The descent formula for weighted orbital integrals [45, Corollaire 5.4.3]
leads to

$r^{\tilde{G}}_{\tilde{L}}\left(\tilde{\gamma}^{L},K\right)=\sum_{L^{\dagger}\in\mathcal{L}^{G}(M)}d^{G}_{M}(L,L^{\dagger})r^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma},K\cap
L(F))$

where $L\in\mathcal{L}^{G}(M)$ and $\tilde{\gamma}^{L}$ is as in Proposition
5.1.5.

On the other hand, given a quasisplit group $G^{!}$ and its Levi subgroup
$M^{!}$, in [60, II.4.2] are defined the stable avatars

$s^{G^{!}}_{M^{!}}(\delta)=s^{G^{!}}_{M^{!}}(\delta,K^{!}),\quad\delta\in
SD_{\mathrm{geom}}(M^{!}).$

Here $K^{!}\subset G^{!}(F)$ is a hyperspecial subgroup and $M^{!}$ is in good
position relative to $K^{!}$. The Haar measures on $G^{!}(F)$ and $M^{!}(F)$
are determined by $K^{!}$ as before. The choice of $K^{!}$ is immaterial here
because they are conjugate under $G^{!}_{\mathrm{AD}}(F)$. On the other hand,
these quantities depend on the choice of an invariant positive-definite
quadratic form on $\mathfrak{a}^{G^{!}}_{M^{!}}$. Such a form is available
when we are in the situation (3.4.1), namely

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

######  Definition 5.6.2.

Given $\tilde{M}\subset\tilde{G}$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ as above, we
set

$r^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta):=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])s^{G^{!}[s]}_{M^{!}}(\delta[s])$

for all $\delta\in SD_{\mathrm{geom}}(M^{!})$.

Now we can state the _weighted fundamental lemma_.

######  Theorem 5.6.3.

In the unramified situation, suppose moreover that the residual characteristic
$p$ of $F$ satisfies $p>5$. Given $\tilde{M}\subset\tilde{G}$ as above, we
have

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

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

When $\delta\in SD_{\mathrm{geom},\tilde{G}\text{-reg}}(M^{!})$, this reduces
to [49, Théorème 4.2.1]. The proof of the general case will be given in §6.8.

##  Chapter 6 Theory of germs

Consider a covering of metaplectic type $\tilde{G}$ over a local field $F$ of
characteristic zero. In the first part, we will follow [60, II, III] to
develop a theory of germs. Let $M\in\mathcal{L}(M_{0})$. The germs in question
are associated with certain subsets $J$ of $\Sigma(A_{M})$ with
$|J|=\dim\mathfrak{a}^{G}_{M}$, taken up to the equivalence that $J\sim
J^{\prime}$ if they generate the same $\mathbb{Z}$-module in
$\mathfrak{a}^{G,*}_{M}$. Fix a semisimple conjugacy class $\mathcal{O}$ in
$M(F)$. The germs are represented by linear maps

$\rho^{\tilde{G}}_{J}:D_{\mathrm{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\to
U_{J}\otimes\left(D_{\mathrm{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$

where

  * •

$U_{J}$ is a certain space of germs of functions on $A_{M}(F)$ near $a=1$,

  * •

$\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$ denotes the kernel of parabolic
induction to $\tilde{G}$,

  * •

we only consider the restriction of $\rho^{\tilde{G}}_{J}$ to distributions of
support sufficiently close to $\mathbf{p}^{-1}(\mathcal{O})$.

They are related to invariant weighted orbital integrals through the
equivalence

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

between germs of functions in $a\in A_{M}(F)$, $a\to 1$; see Definition 6.2.1
for the meaning of $\sim$. Note that $a\tilde{\gamma}$ is
$\tilde{G}$-equisingular for general $a$, which explains the usefulness of
germs towards the local geometric Theorem 5.3.11.

Note that $D_{\mathrm{geom},-}=D_{\mathrm{orb},-}$ for non-Archimedean $F$. As
for the Archimedean case, in §6.3 we will lay down a program to extend the
definitions of $\rho^{\tilde{G}}_{J}(\tilde{\gamma})$,
$I_{\tilde{M}}(\tilde{\gamma},f)$ and
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$ to a larger subspace
$D_{\text{tr-orb},-}(\tilde{M})$ of $D_{\mathrm{geom},-}(\tilde{M})$ that
accommodates the transfers of similar distributions from various
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. Furthermore,
we expect that the $\rho^{\tilde{G}}_{J}$ and $I_{\tilde{M}}$ so extended
match their endoscopic counterparts $\rho^{\tilde{G},\mathcal{E}}_{J}$ and
$I^{\mathcal{E}}_{\tilde{M}}$.

This program will be carried out in §6.4, under the Hypothesis 6.3.3 that the
local geometric Theorem 5.3.11 holds for $\tilde{G}$-regular $\tilde{\gamma}$.
In this process, subgroups $G_{J}\subset G$ and the corresponding coverings
$\widetilde{G_{J}}\subset\tilde{G}$ will appear naturally: in our case,
$G_{J}$ are simply products of smaller symplectic groups.

It turns out that the same argument can also handle the non-Archimedean case,
if one replaces $D_{\mathrm{tr-orb},-}$ by $D_{\mathrm{geom},-}$ everywhere.
In particular, assume that the local geometric Theorem 5.3.11 holds in the
$\tilde{G}$-regular case, then it holds in general, and
$\rho^{\tilde{G}}_{J}=\rho^{\tilde{G},\mathcal{E}}_{J}$. This is the content
of §6.5.

In the second part, we introduce the weighted Shalika germs on $\tilde{G}$ in
the non-Archimedean case, and define their endoscopic counterparts. The
matching Theorem 6.6.7 for Shalika germs can be proved directly via Harish-
Chandra descent: it is ultimately reduced to the unipotent matching settled in
[60, III], for both standard and nonstandard endoscopy. The matching of
Shalika germs can be employed in the following ways.

  * •

It furnishes another way to reduce the local geometric Theorem 5.3.11 to the
$\tilde{G}$-regular case (Corollary 6.6.12).

  * •

It will be used in the construction of the map $\epsilon_{\tilde{M}}$ in §7.5.

  * •

Certain arguments in the proof will be re-used in the Archimedean case in
§9.7.

Finally, we apply the technique of germs to prove the weighted fundamental
lemma (Theorem 5.6.3) in §6.8.

###  6.1 Algebraic constructions

This section is largely a review of [60, II.3.1]. We begin with a general
connected reductive $F$-group $G$ and its Levi subgroup $M$. Here $F$ can be
any field of characteristic zero.

####  6.1.1 Definition of $G_{J}$

View $\Sigma(A_{M})=\Sigma^{G}(A_{M})$ as a subset of $\mathfrak{a}^{G,*}_{M}$
and define

$\mathcal{J}^{G}_{M}:=\left\\{J\subset\Sigma(A_{M}):|J|=\dim\mathfrak{a}^{G}_{M},\;\text{linearly
independent}\right\\}\big{/}\sim$

where we write $J\sim J^{\prime}$ if they generate the same
$\mathbb{Z}$-submodule in $\mathfrak{a}^{G,*}_{M}$.

For each $J\in\mathcal{J}^{G}_{M}$, let $R_{J}$ be the $\mathbb{Z}$-submodule
of $\Sigma(A_{M})$ it generates. Make $\mathcal{J}^{G}_{M}$ into a partially
ordered set by setting $J\leq J^{\prime}$ if and only if $R_{J}\subset
R_{J^{\prime}}$. There is then a unique maximal element $J_{\max}$ of
$(\mathcal{J}^{G}_{M},\leq)$, characterized by $R_{J_{\max}}=\Sigma(A_{M})$.

Fix a Borel pair $(B_{G},T)$ for $G_{\overline{F}}$ such that $T$ is defined
over $F$, and $M_{\overline{F}}$ is a standard Levi subgroup relative to
$(B_{G},T)$; in particular $A_{M}\subset T$. Given $J\in\mathcal{J}^{G}_{M}$,
we have the root subsystem

$\left\\{\beta\in\Sigma(G,T):\beta|_{A_{M}}\in R_{J}\right\\}.$

It is the absolute root system (relative to $T$) of a connected reductive
$F$-group $G_{J}$ such that $M\subset G_{J}\subset G$. One readily checks that
$G_{J}$ is independent of $(B_{G},T)$. Moreover, $M$ is still a Levi subgroup
of $G_{J}$.

Note that $G_{J_{\max}}=G$. When $J\neq J_{\max}$, we have $\dim G_{J}<\dim
G$.

For all $J\in\mathcal{J}^{G}_{M}$, we have
$\Sigma^{G_{J}}(A_{M})\subset\Sigma^{G}(A_{M})$, and

$\mathcal{J}^{G_{J}}_{M}=\left\\{J^{\prime}\in\mathcal{J}^{G}_{M}:R_{J^{\prime}}\subset\Sigma^{G_{J}}(A_{M})\right\\}.$

Moreover, $J\in\mathcal{J}^{G_{J}}_{M}$ and it is the unique maximal element
of $\mathcal{J}^{G_{J}}_{M}$.

######  Proposition 6.1.1.

For every $J\in\mathcal{J}^{G}_{M}$, the groups $G_{\mathrm{der}}$ and
$G_{J,\mathrm{der}}$ have the same rank over $\overline{F}$.

###### Proof.

They share the Levi subgroup $M$ and
$\mathfrak{a}^{G}_{M}=|J|=\mathfrak{a}^{G_{J}}_{M}$. ∎

####  6.1.2 The case with $B$-functions

The following is a review of [60, II.3.3]. Let $L$ be a connected reductive
$F$-group with Levi subgroup $M$.

Fix a $B$-function $\Sigma(L,T)\to\mathbb{Q}_{>0}$ as in Definition 5.3.1. We
define the analogue $\mathcal{J}^{L}_{M}(B)$ of $\mathcal{J}^{L}_{M}$ by the
same recipe, replacing $\Sigma(A_{M})$, $\Sigma(L,T)$ by $\Sigma(A_{M};B)$,
$\Sigma(L,T;B)$ systematically.

Given $J\in\mathcal{J}^{L}_{M}(B)$, let

$\Sigma_{J}:=\left\\{\alpha\in\Sigma(L,T):B(\alpha)^{-1}\alpha|_{\mathfrak{a}_{M}}\in
R_{J}\right\\},\quad\check{\Sigma}_{J}:=\left\\{\check{\alpha}:\alpha\in\Sigma_{J}\right\\}.$

Then we construct a connected reductive $F$-group $L_{J}$ containing $T$,
whose roots and coroots are given by $\Sigma_{J}$ and $\check{\Sigma}_{J}$
respectively. It reverts to the prior construction when $B$ is trivial. It
still satisfies:

  * •

$M$ is canonically embedded as a Levi subgroup of $L_{J}$,

  * •

$Z_{L}\subset Z_{L_{J}}$,

  * •

$Z_{\check{L}}\hookrightarrow Z_{\check{L}_{J}}$, compatibly with
$\Gamma_{F}$-actions.

However, $L_{J}$ is not necessarily a subgroup of $L$ for non-constant $B$.

Now, suppose that $G^{!}$ is a quasisplit group endowed with a system of
$B$-functions (Definition 5.3.2), and $M^{!}\subset G^{!}$ is a Levi subgroup.
In the discussions after that definition, we defined a subset
$\Sigma(A_{M^{!}},B_{\mathcal{O}^{!}})$ of $\mathfrak{a}_{M^{!}}^{*}$ for
every semisimple conjugacy class $\mathcal{O}^{!}$ in $M^{!}(F)$. It consists
of nonzero restrictions to $A_{M^{!}}$ of elements from
$\Sigma(A_{M^{!}_{\epsilon}},B_{\epsilon})$ (taken inside $G^{!}_{\epsilon}$)
where $\epsilon\in\mathcal{O}^{!}$ is arbitrary. In particular, we can define
the analogue $\mathcal{J}^{G^{!}}_{M^{!}}(B_{\mathcal{O}^{!}})$ of
$\mathcal{J}^{G^{!}}_{M^{!}}$ by considering linearly independent subsets of
$\Sigma(A_{M^{!}},B_{\mathcal{O}^{!}})$ of size $\dim\mathfrak{a}^{G}_{M}$.

It follows immediately from Definition 5.3.2 that
$\mathcal{J}^{G^{!}}_{M^{!}}(B_{\mathcal{O}^{!}})$ depends only on the stable
semisimple conjugacy class containing $\mathcal{O}^{!}$. It can be empty.

In contrast with the previous case, in general $G^{!}_{J}$ does not make sense
for $J\in\mathcal{J}^{G^{!}}_{M^{!}}(B_{\mathcal{O}^{!}})$, except when
$\mathcal{O}^{!}$ is central.

Let us specialize this formalism to the situation

${G^{!}[s]}$${\tilde{G}}$${M^{!}}$${\tilde{M}}$ell.endo.ell.endo.LeviLevi

where $\tilde{G}$ is a group of metaplectic type,
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. Consider the family of
$B$-functions $B^{\tilde{G}}$ for $G^{!}$ from Definition 5.3.5. Let
$\mathcal{O}^{!}\subset M^{!}(F)$ be a stable semisimple conjugacy class and
put

$\mathcal{O}^{!}[s]:=\mathcal{O}^{!}\cdot z[s].$

Also recall that endoscopy affords an isomorphism
$\xi:A_{M}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{M^{!}}$ between
$F$-tori. In turn, it induces
$\xi^{*}:\mathfrak{a}_{M^{!}}^{*}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{a}_{M}$.
The following result is somewhat “built-in” to $B^{\tilde{G}}$.

######  Lemma 6.1.2.

In the situation above,
$\xi^{*}:\mathfrak{a}_{M^{!}}^{*}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{a}_{M}^{*}$
induces an injection

$\Sigma^{G^{!}[s]}\left(A_{M^{!}},B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\right)\hookrightarrow\Sigma^{G}(A_{M}),$

in turn, this induces an injection
$\mathcal{J}^{G^{!}[s]}_{M^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\right)\hookrightarrow\mathcal{J}^{G}_{M}$.

###### Proof.

Take $\epsilon\in\mathcal{O}^{!}$. Without loss of generality, we may assume
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ and $M^{!}_{\epsilon}$ is quasisplit.
Choose by Corollary 4.5.2 a diagram $(\epsilon,B^{!},T^{!},B,T,\eta)$ joining
$\epsilon$ and some $\eta\in M(F)_{\mathrm{ss}}$; the data include an
isomorphism $\xi_{T^{!},T}:T^{!}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T$
compatible with $\xi^{-1}$; see Definition 3.6.1. We have
$\epsilon\leftrightarrow\eta$ (resp. $\epsilon[s]\leftrightarrow\eta$) with
respect to $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$
(resp. $\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$).

The descent of endoscopic data in §4.4 shows that the roots of
$G^{!}[s]_{\epsilon[s]}\supset T^{!}$ injects into those of $G_{\eta}\supset
T$ in this way, up to explicit scaling. By definition, the scaling is exactly
prescribed by $B^{\tilde{G}}_{\epsilon[s]}$. Restricting the roots to $A_{M}$
and $A_{M^{!}}$ gives the first injection. The second follows immediately. ∎

####  6.1.3 Determination of $G_{J}$ in the symplectic case

In this subsection, we take $G=\operatorname{Sp}(W)$ and let $M=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W^{\flat})$ be a Levi
subgroup of $G$, where $W^{\flat}\subset W$ are symplectic $F$-vector spaces
and $I$ is a finite set.

######  Proposition 6.1.3.

Up to conjugation, the subgroups $G_{J}\subset G$ attached to various
$J\in\mathcal{J}^{G}_{M}$ are exactly of the form

$\prod_{i\in\mathbf{I}}\operatorname{Sp}(W^{i})\times\operatorname{Sp}(W^{\natural})$

where

  * •

$\mathbf{I}$ is a finite set (possibly empty) equipped with a map
$f:I\to\mathbf{I}\sqcup\\{\natural\\}$;

  * •

$W^{i}$ and $W^{\natural}$ are symplectic subspaces of $W$ such that
$W=\bigoplus_{i\in\mathbf{I}}W^{i}\oplus W^{\natural}$ (orthogonal sum) and
$W^{\flat}\subset W^{\natural}$;

  * •

$\dim W^{i}=2\sum_{f(k)=i}n_{k}$ for all $i\in\mathbf{I}$;

  * •

$\dim W^{\natural}=2\sum_{f(k)=\natural}n_{k}+\dim W^{\flat}$.

The embedding $M\hookrightarrow G_{J}$ is given by
$\operatorname{GL}(n_{k})\hookrightarrow\operatorname{Sp}(W^{f(k)})$ for $k\in
I$ and
$\operatorname{Sp}(W^{\flat})\hookrightarrow\operatorname{Sp}(W^{\natural})$.

###### Proof.

Since $\mathrm{char}(F)=0$ and the descriptions of $R_{J}$, $G_{J}$,
$\mathcal{J}^{G}_{M}$ are insensitive to field extensions, it suffices to
describe the corresponding Lie subalgebras $\mathfrak{g}_{J}$ over
$\overline{F}$. Following Dynkin, a subalgebra
$\mathfrak{f}\subset\mathfrak{g}$ is called _regular_ if
$[\mathfrak{h},\mathfrak{f}]\subset\mathfrak{f}$ for some Cartan subalgebra
$\mathfrak{h}\subset\mathfrak{g}$. We claim that $\mathfrak{g}_{J}$ must be a
semisimple regular subalgebra of $\mathfrak{g}$. Indeed, $\mathfrak{g}_{J}$ is
regular since it contains $\mathfrak{m}$, and semisimplicity follows from
Proposition 6.1.1.

Now we can appeal to Dynkin’s classification of semisimple regular subalgebras
of $\mathfrak{g}$; see [29, p.383] or [68, Chapter 6]: $\mathfrak{g}_{J}$ must
be a product of simple Lie algebras of type $\mathrm{C}_{m}$, for various $m$,
and they arise from an orthogonal decomposition of $W$ as indicated. The other
conditions are imposed in order to make $M$ a Levi subgroup of $G_{J}$.

To show that all subgroups in the assertion arise as some $G_{J}$, assume that
the data $\mathbf{I}$, $f$, $W^{i}$ and $W^{\natural}$ are given. Take the
standard $\mathbb{Z}$-basis $\\{\epsilon_{i}:i\in I\\}$ for $X^{*}(A_{M})$,
put a total order on $I$, and let

$\tilde{J}:=\left\\{\epsilon_{i}-\epsilon_{j}:i<j,\;f(i)=f(j)\right\\}\sqcup\left\\{2\epsilon_{i}:i\in
I\right\\}.$

This set is linearly independent in $\mathfrak{a}^{G,*}_{M}$, and the lattice
$R_{\tilde{J}}$ generated by $\tilde{J}$ in $X^{*}(A_{M})$ equals

$\left\\{\sum_{k\in I}x_{k}\epsilon_{k}:\forall
i\in\mathbf{I}\sqcup\\{\natural\\},\;\sum_{f(k)=i}x_{k}\in
2\mathbb{Z}\right\\}.$

Take $J\in\mathcal{J}^{G}_{M}$ to be the equivalence class generated by
$\tilde{J}$. One verifies that
$G_{J}\simeq\prod_{i\in\mathbf{I}}\operatorname{Sp}(W^{i})\times\operatorname{Sp}(W^{\natural})$,
compatibly with the given embedding of $M$. ∎

Next, consider $G^{!}:=\operatorname{SO}(2n+1)$ with Levi subgroup
$M^{!}:=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{SO}(2n^{\flat}+1)$, recalling
that they are all split. Equip $G^{!}$ with the $B$-function specified in
Definition 5.3.5 (with $n^{\prime}=n$, $n^{\prime\prime}=0$ and $\epsilon=1$).

######  Proposition 6.1.4.

Up to isomorphisms, the groups $G^{!}_{J^{!}}$ attached to various
$J^{!}\in\mathcal{J}^{G^{!}}_{M^{!}}(B)$ are exactly of the form

$\prod_{i\in\mathbf{I}}\operatorname{SO}(2n^{i}+1)\times\operatorname{SO}(2n^{\natural}+1)$

where

  * •

$\mathbf{I}$ is a finite set (possibly empty) equipped with a map
$f:I\to\mathbf{I}\sqcup\\{\natural\\}$;

  * •

the $n^{i}$ and $n^{\natural}$ are in $\mathbb{Z}_{\geq 0}$ such that
$n=\sum_{i\in\mathbf{I}}n^{i}+n^{\natural}$;

  * •

$n^{i}=2\sum_{f(k)=i}n_{k}$ for all $i\in\mathbf{I}$;

  * •

$n^{\natural}=2\sum_{f(k)=\natural}n_{k}+n^{\flat}$.

The embedding $M^{!}\hookrightarrow G^{!}_{J^{!}}$ is given by
$\operatorname{GL}(n_{i})\hookrightarrow\operatorname{SO}\left(2n^{f(i)}+1\right)$
for $i\in I$ and
$\operatorname{SO}(2n^{\flat}+1)\hookrightarrow\operatorname{SO}(2n^{\natural}+1)$.

###### Proof.

All these groups are split. In view of the remarks in [60, p.283], the problem
can be translated to the dual side

${M^{!}}^{\vee}=\prod_{i\in
I}\operatorname{GL}(n_{i},\mathbb{C})\times\operatorname{Sp}(2n^{\flat},\mathbb{C})\hookrightarrow{G^{!}}^{\vee}=\operatorname{Sp}(2n,\mathbb{C})$

endowed with the trivial $B$-function. Proposition 6.1.3 can thus be applied
over $\mathbb{C}$ to give a complete description of $(G^{!}_{J^{!}})^{\vee}$.
By duality, the case for $G^{!}_{J^{!}}$ follows at once. ∎

The evident duality between $G_{J}$ (type $\mathrm{C}$) and $G^{!}_{J^{!}}$
(type $\mathrm{B}$) is compatible with the identification between
$\mathcal{J}^{G}_{M}$ and $\mathcal{J}^{G^{!}}_{M^{!}}(B)$ which intervenes in
nonstandard endoscopy. See §4.3.

###  6.2 The maps $\rho$, $\sigma$ and $\rho^{\mathcal{E}}$

Let $F$ be a local field of characteristic zero.

####  6.2.1 The spaces $U_{J}$

In what follows, $a$ will denote points of $A_{M}(F)$ which are close to $1$.
Such elements can be canonically viewed as elements of $\widetilde{A_{M}}$, by
using the two exponential maps in (4.1.1).

We will consider germs at $a=1$ of $\mathbb{C}$-valued functions defined
almost everywhere on $A_{M}(F)$ (or on $\widetilde{A_{M}}$ as explained
above), i.e. off the $F$-points of some closed subvariety of $A_{M}$.
Hereafter they will be abbreviated as “germs”.

For each class $J\in\mathcal{J}^{G}_{M}$, define the $\mathbb{C}$-vector space
of germs

$\displaystyle U_{J}$ $\displaystyle=U_{J}^{G\supset M}$ (6.2.1)
$\displaystyle:=\text{the span
of}\;\left\\{a\mapsto\prod_{i}\left|\alpha_{i}(a)-\alpha_{i}(a)^{-1}\right|:\left\\{\alpha_{1},\alpha_{2},\ldots\right\\}\in
J\right\\}.$

When $M=G$ we have $\mathcal{J}^{G}_{M}=\\{\varnothing\\}$ and
$U_{\varnothing}=\\{0\\}$.

######  Definition 6.2.1.

Fix a function $a\mapsto d(a)$ for $a\in A_{M}(F)$ close to $1$ as in the
discussions after Definition 5.1.1. Let $u$, $u^{\prime}$ be germs. We say
that $u\sim u^{\prime}$ if there exists $r>0$ such that over all domains in
$A_{M}(F)$ of the form

$\forall\alpha\in\Sigma(A_{M}),\;|\alpha(a)-1|>c\cdot d(a),$

there exists $C>0$ such that for almost all $a$ in that domain which are close
to $1$,

$\left|u^{\prime}(a)-u(a)\right|<C\cdot d(a)^{r}.$

The following result is established in [60, II.3.1 and V.2.3].

######  Proposition 6.2.2.

Let $u$ be a germ.

  1. (i)

If $u\sim 0$ and $\alpha_{1},\ldots,\alpha_{n}$ are elements in
$\Sigma(A_{M})$, then
$u\cdot\prod_{i=1}^{n}\log\left|\alpha_{i}-\alpha_{i}^{-1}\right|\sim 0$.

  2. (ii)

Suppose that $F$ is non-Archimedean. If
$u\in\sum_{L\in\mathcal{L}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}U_{J}$, then
$u\sim 0$ if and only if $u=0$.

  3. (iii)

Suppose that $M\neq G$, $J\in\mathcal{J}^{G}_{M}$ and $u\in U_{J}$. If $u\sim
c$ where $c\in\mathbb{C}$, then $u=0$ and $c=0$. Consequently, if
$u_{1}+c_{1}\sim u_{2}+c_{2}$, where $c_{i}\in\mathbb{C}$ and $u_{i}\in U_{J}$
for $i=1,2$, then $u_{1}=u_{2}$ and $c_{1}=c_{2}$.

Define the spaces of germs on $A_{M}(F)$:

$\displaystyle U^{G}_{M}$
$\displaystyle:=\sum_{L\in\mathcal{L}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}U_{J}^{L\supset
M},$ $\displaystyle U^{G,+}_{M}$
$\displaystyle:=\sum_{\begin{subarray}{c}L\in\mathcal{L}(M)\\\ L\neq
M\end{subarray}}\sum_{J\in\mathcal{J}^{L}_{M}}U_{J}^{L\supset M}.$

Note the the summand with $L=M$ is $U_{\varnothing}=\mathbb{C}$ (constant
functions).

######  Lemma 6.2.3 ([60, II.4.6 Lemme]).

Assume that $F$ is non-Archimedean with residual characteristic $p>5$. Then
the sum $U^{G}_{M}=\mathbb{C}+U^{G,+}_{M}$ is direct.

This lemma allows us to define the constant term of any element of $U^{G}_{M}$
in the non-Archimedean case.

Now let $G^{!}$ be a quasisplit $F$-group endowed with a system of
$B$-functions, and fix a Levi subgroup $M^{!}\subset G^{!}$. Let
$\mathcal{O}^{!}$ be a stable semisimple conjugacy class in $M^{!}(F)$. For
every $J\in\mathcal{J}^{G^{!}}_{M^{!}}(B_{\mathcal{O}^{!}})$, we iterate same
construction as (6.2.1) to define the linear subspace $U_{J}$ of the space of
germs. Similarly, we define $U^{G^{!}}_{M^{!}}$ in this setting. It depends on
the $B$-function and $\mathcal{O}^{!}$.

####  6.2.2 The coverings $\widetilde{G_{J}}$ and annihilators

Consider a covering $\mathbf{p}:\tilde{G}\twoheadrightarrow G(F)$ in general.
Let $M\subset G$ be a Levi subgroup.

######  Definition 6.2.4.

Given $J\in\mathcal{J}^{G}_{M}$, let

$\displaystyle\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}$
$\displaystyle:=\operatorname{ker}\left[D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}\xrightarrow{\operatorname{Ind}^{\tilde{G}}_{\tilde{M}}}D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}\right],$
$\displaystyle\widetilde{G_{J}}$
$\displaystyle:=\mathbf{p}^{-1}\left(G_{J}(F)\right).$

######  Lemma 6.2.5.

We have
$\mathrm{Ann}^{\widetilde{G_{J}}}_{\tilde{M}}\subset\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}$.

###### Proof.

The arguments in [60, p.279] carry over. It boils down to the easy fact that
$W^{G_{J}}(M)\subset W^{G}(M)$ operates on the trace Paley–Wiener spaces of
$\tilde{M}$, $\tilde{G}$ and $\widetilde{G_{J}}$. Note that in the Archimedean
case, we shall work with the $C^{\infty}_{c}$-version
$\mathrm{PW}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})^{\infty}$ (Remark 2.8.4).
∎

Let $\mathcal{O}$ be a finite union of semisimple conjugacy classes in $M(F)$.
Define

$\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}:=\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}\cap\left(D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right).$

The uncovered case $\tilde{G}=G(F)$ has been discussed in loc. cit. In that
case, we also write

$\mathrm{Ann}^{G}_{\mathrm{unip}}:=\mathrm{Ann}^{G}_{\\{1\\}}.$

Similarly, for a quasisplit group $G^{!}$ with Levi subgroup $M^{!}$, we
define the stable avatars

$S\mathrm{Ann}^{G^{!}}_{M^{!}},\quad
S\mathrm{Ann}^{G^{!}}_{\mathcal{O}^{!}},\quad
S\mathrm{Ann}^{G^{!}}_{\mathrm{unip}}:=S\mathrm{Ann}^{G^{!}}_{\\{1\\}},$

where $\mathcal{O}^{!}$ is any finite union of stable semisimple conjugacy
classes in $M^{!}(F)$. It suffices to replace $D_{\mathrm{geom},-}(\tilde{M})$
by $SD_{\mathrm{geom}}(M^{!})$, and so forth. Clearly,
$S\mathrm{Ann}^{G^{!}}_{\mathcal{O}^{!}}\subset\mathrm{Ann}^{G^{!}}_{\mathcal{O}^{!}}$.

####  6.2.3 Non-Archimedean case

Assume furthermore that $F$ is non-Archimedean. Let
$\mathbf{p}:\tilde{G}\twoheadrightarrow G(F)$ be a covering, $M\subset G$ a
Levi subgroup. Fix a semisimple conjugacy class $\mathcal{O}$ in $M(F)$.

In fact, the coverings we need will either be of metaplectic type, or of the
form $\widetilde{G_{J}}$ (Definition 6.2.4) where $\tilde{G}$ is of
metaplectic type, hence the situation is not too different from the essential
case $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$.

######  Definition–Proposition 6.2.6.

For all $J\in\mathcal{J}^{G}_{M}$, there exists a unique linear map

$\rho^{\tilde{G}}_{J}:D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\to
U_{J}\otimes\left(D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$

such that

  1. (i)

it factorizes into

$D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\xrightarrow{\rho^{\widetilde{G_{J}}}_{J}}U_{J}\otimes(\cdots)\big{/}\mathrm{Ann}^{\widetilde{G_{J}}}_{\mathcal{O}}\twoheadrightarrow
U_{J}\otimes(\cdots)\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$

where we used Lemma 6.2.5;

  2. (ii)

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

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

as germs in $a\in A_{M}(F)$.

Here, $a\tilde{\gamma}$ makes sense since $a$ is naturally lifted to
$\mathbf{p}^{-1}\left(A_{M}(F)\right)$ for $a\to 1$, and
$(\cdots)^{\tilde{L}}$ stands for the map

$\mathrm{id}\otimes\operatorname{Ind}^{\tilde{L}}_{\tilde{M}}:U_{J}\otimes\left(D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{L}}_{\tilde{M}}\to
U_{J}\otimes
D_{\mathrm{geom},-}(\tilde{L})\otimes\operatorname{mes}(L)^{\vee}$

where $U_{J}=U_{J}^{L\supset M}$.

###### Proof.

Identical to [60, II.3.2 Proposition]. It boils down to the following
ingredients:

  * •

generalities on the $(G,M)$-family defining weighted orbital integrals,

  * •

standard properties of $I_{\tilde{M}}(\tilde{\gamma},\cdots)$, see Proposition
5.1.5;

  * •

standard properties of germs, see Proposition 6.2.2.

In particular, the coverings do not really intervene in the arguments. For a
more explicit formula for $\rho^{\tilde{G}}_{J}$, see [60, II.3.2 (5)]. ∎

Note that our conventions imply that
$\rho^{\tilde{G}}_{\varnothing}=\mathrm{id}$ when $M=G$.

######  Lemma 6.2.7.

Let $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$
where $\mathcal{O}$ is $G$-equisingular. Then
$\rho^{\tilde{G}}_{J}(\tilde{\gamma})=0$ unless $M=G$.

###### Proof.

One uses the explicit construction [60, p.282, (5)] of
$\rho^{\tilde{G}}_{J}(\tilde{\gamma})$, which applies to coverings as well.
Suppose $M\neq G$. The terms $m(\underline{\alpha},\tilde{\gamma})$ therein
are all zero because so are the terms $\rho(\alpha,\tilde{\gamma})$, which is
in turn a consequence of the definition in [60, p.194] of
$\rho(\alpha,\tilde{\gamma})$ since
$G_{\gamma_{\mathrm{ss}}}=M_{\gamma_{\mathrm{ss}}}$. ∎

Before stating the next descent formula, we collect some facts from [60,
p.297]. Let $L,L^{\dagger}\in\mathcal{L}^{G}(M)$ be such that
$d^{G}_{M}(L,L^{\dagger})\neq 0$. Then:

  * •

the inclusion $A_{L}\subset A_{M}$ induces an injection
$\Sigma^{L^{\dagger}}(A_{M})\hookrightarrow\Sigma^{G}(A_{L})$;

  * •

in turn, this induces an injection
$\mathcal{J}^{L^{\dagger}}_{M}\hookrightarrow\mathcal{J}^{G}_{L}$;

  * •

for each $J\in\mathcal{J}^{L^{\dagger}}_{M}$, also viewed as an element in
$\mathcal{J}^{G}_{L}$, we have

$U_{J}^{G\supset L}=U_{J}^{L^{\dagger}\supset M}\big{|}_{A_{L}(F)}$

as spaces of germs over $A_{L}(F)$.

On the other hand, the parabolic induction
$\operatorname{Ind}^{\tilde{L}}_{\tilde{M}}$ of (2.4.1) always maps
$\mathrm{Ann}^{\tilde{L}^{\dagger}}_{\tilde{M}}$ to
$\mathrm{Ann}^{\tilde{G}}_{\tilde{L}}$. Therefore, when
$d^{G}_{M}(L,L^{\dagger})\neq 0$, we obtain the linear map

$U_{J}^{L^{\dagger}\supset
M}\otimes\left(D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{L}^{\dagger}}_{\tilde{M}}\to
U_{J}^{G\supset
L}\otimes\left(D_{\mathrm{geom},-}(\tilde{L})\otimes\operatorname{mes}(L)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{G}}_{\tilde{L}},$

which will again be denoted by $\rho\mapsto\rho^{\tilde{L}}$.

######  Proposition 6.2.8.

Let $L\in\mathcal{L}(M)$, $J\in\mathcal{J}^{G}_{L}$ and $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. Then

$\rho^{\tilde{G}}_{J}\left(\tilde{\gamma}^{\tilde{L}}\right)=\sum_{\begin{subarray}{c}L^{\dagger}\in\mathcal{L}^{G}(M)\\\
\text{s.t.}\;J\in\mathcal{J}^{L^{\dagger}}_{M}\end{subarray}}d^{G}_{M}(L,L^{\dagger})\rho^{\tilde{L}^{\dagger}}_{J}\left(\tilde{\gamma}\right)^{\tilde{L}}$

as germs on $A_{M}(F)$.

###### Proof.

This is the analogue of [60, II.3.10 Lemme] for coverings, and the proof in
loc. cit. carries over, cf. the explanations for Definition–Proposition 6.2.6.
∎

Consider now a quasisplit $F$-group $G^{!}$ endowed with a system of
$B$-functions and a Levi subgroup $M^{!}\subset G^{!}$. Let $\mathcal{O}^{!}$
be a stable semisimple conjugacy class in $M^{!}(F)$. The stable avatar of
Definition–Proposition 6.2.6 is given in [60, II.3.5 Proposition]: for every
$J\in\mathcal{J}^{G^{!}}_{M^{!}}(B_{\mathcal{O}^{!}})$ we have a canonical
linear map

$\sigma^{G^{!}}_{J}:SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}\to
U_{J}\otimes\left(SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}S\mathrm{Ann}^{G^{!}}_{\mathcal{O}^{!}}.$

Note that it involves the system of $B$-functions.

We are now ready to define the endoscopic avatar of $\rho^{\tilde{G}}_{J}$.
Assume henceforth that $\tilde{G}$ is a group of metaplectic type, $M\subset
G$ is a Levi subgroup, and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. Let
$\mathcal{O}^{!}$ be a stable semisimple conjugacy class in $M^{!}(F)$. Denote
the identification
$A_{M}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{M^{!}}$ as $\xi$.
Finally, let $\mathcal{O}\subset M(F)$ be the union of semisimple conjugacy
classes corresponding to $\mathcal{O}^{!}$.

######  Definition 6.2.9.

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

$\displaystyle\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}^{s}:\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
$\displaystyle\longrightarrow
S\mathcal{I}(M^{!})\otimes\operatorname{mes}(M^{!})$ $\displaystyle f$
$\displaystyle\longmapsto\left(\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}(f)\right)[s]^{-1}$

where $g^{M^{!}}\mapsto g^{M^{!}}[s]^{-1}$ is the inverse of the automorphism
$g^{M^{!}}\mapsto g^{M^{!}}[s]$ in Definition 3.8.3. Denote its transpose by

$\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}:SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}\longrightarrow
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}.$

Set $\mathcal{O}^{!}[s]:=\mathcal{O}^{!}z[s]$. It is routine to verify that
$\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}$ restricts to

$SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!}[s])\otimes\operatorname{mes}(M)^{\vee}\to
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}.$

######  Lemma 6.2.10.

For every $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, the map
$\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}$ maps
$S\mathrm{Ann}^{G^{!}[s]}_{M^{!}}$ (resp.
$S\mathrm{Ann}^{G^{!}[s]}_{\mathcal{O}^{!}[s]}$) to
$\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}$ (resp.
$\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$).

###### Proof.

Proposition 3.8.4 asserts the commutativity of

${\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)}$${S\mathcal{I}(M^{!})\otimes\operatorname{mes}(M^{!})}$${\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)}$${S\mathcal{I}(G^{!}[s])\otimes\operatorname{mes}(G^{!}[s])}$$\scriptstyle{\mathcal{T}_{\mathbf{M}^{!},\tilde{M}}^{s}}$$\scriptstyle{\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}}$

the vertical arrows being parabolic descent. Taking transpose yields
$\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}\left(S\mathrm{Ann}^{G^{!}[s]}_{M^{!}}\right)\subset\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}$.
Restriction to
$SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!}[s])\otimes\operatorname{mes}(M^{!})$
yields the case with $\mathcal{O}^{!}[s]$ and $\mathcal{O}$. ∎

Observe that if $J^{!}\mapsto J$ as in Lemma 6.1.2, then $\xi^{*}:u\mapsto
u\xi$ induces an injective linear map $U_{J^{!}}\to U_{J}$.

Hence for each
$J^{!}\in\mathcal{J}^{G^{!}[s]}_{M^{!}}(B^{\tilde{G}}_{\mathcal{O}^{!}[s]})$
and $\delta\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$,
we can form

$\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}\left(\sigma^{G^{!}[s]}_{J^{!}}(\delta[s])\right)\in
U_{J}\otimes\left(D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}.$

Unraveling definitions, it involves two twists inside
$SD_{\mathrm{geom}}(M^{!})$: one by $z[s]$ (for $\delta\mapsto\delta[s]$) and
one by $z[s]^{-1}$ (for $\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}$).
Due to the presence of $S\mathrm{Ann}^{G^{!}[s]}_{M^{!}}$ and
$\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}$, they do not cancel.

Observe that the constructions above also work for Archimedean $F$.

######  Definition 6.2.11.

For each $\delta\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$
and $J\in\mathcal{J}^{G}_{M}$, set

$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta,a):=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{G^{!}[s]}_{M^{!}}(B^{\tilde{G}}_{\mathcal{O}^{!}[s]})\\\
J^{!}\mapsto
J\end{subarray}}\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}\left(\sigma^{G^{!}[s]}_{J^{!}}\left(\delta[s],\xi(a)\right)\right)$

where $a\in A_{M}(F)$ is in general position and close to $1$. Viewing $a$ as
a variable, we obtain an element

$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)\in
U_{J}\otimes\left(D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}.$

######  Proposition 6.2.12.

For all $(\mathbf{M}^{!},\delta)$ above, we have

$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\xi(a)\delta,f\right)\sim\sum_{L\in\mathcal{L}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\rho^{\tilde{L},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta,a)^{\tilde{L}},f\right)$

as germs in $a\in A_{M}(F)$, $a\to 1$.

###### Proof.

Using [60, II.3.7 Proposition], we have

$\displaystyle
I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\xi(a)\delta,f\right)$
$\displaystyle=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])S^{G^{!}[s]}_{M^{!}}\left(\xi(a)\delta[s],B^{\tilde{G}},f^{G^{!}[s]}\right)$
$\displaystyle\sim\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])\sum_{L^{!}\in\mathcal{L}^{G^{!}[s]}(M^{!})}\sum_{J^{!}}S^{G^{!}[s]}_{L^{!}}\left(\sigma^{L^{!}}_{J^{!}}(\delta[s],\xi
a)^{L^{!}},B^{\tilde{G}},f^{G^{!}[s]}\right).$

Apply Lemma 5.4.2 with $R=M$ to transform $\sum_{s,L^{!}}$ into $\sum_{L,s}$
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
$i_{M^{!}}(\tilde{G},G^{!}[s])=i_{M^{!}}(\tilde{L},L^{!})i_{L^{!}}(\tilde{G},G^{!}[s_{L}])$.
We also have $\delta[s]=\delta[s^{L}][s_{L}]$ and $z[s_{L}]$ is central in
$L^{!}(F)$. The last expression becomes

$\sum_{\begin{subarray}{c}L\in\mathcal{L}^{G}(M)\\\
s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})\\\
L^{!}:=L^{!}[s^{L}]\end{subarray}}i_{M^{!}}(\tilde{L},L^{!})\sum_{\begin{subarray}{c}s_{L}\in\mathcal{E}_{\mathbf{L}^{!}}(\tilde{G})\\\
G^{!}:=G^{!}[s_{L}]\end{subarray}}i_{L^{!}}(\tilde{G},G^{!})\sum_{J^{!}}S^{G^{!}}_{L^{!}}\left(\sigma^{L^{!}}_{J^{!}}(\delta[s^{L}],\xi
a)^{L^{!}}[s_{L}],B^{\tilde{G}},f^{G^{!}}\right)\\\
=\sum_{L\in\mathcal{L}^{G}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}\sum_{\begin{subarray}{c}s^{L}\in\mathcal{E}_{\mathbf{M^{!}}(\tilde{L})}\\\
L^{!}:=L^{!}[s^{L}]\end{subarray}}i_{M^{!}}(\tilde{L},L^{!})I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\mathbf{L}^{!},\sum_{J^{!}\mapsto
J}\sigma^{L^{!}}_{J^{!}}(\delta[s^{L}],\xi a)^{L^{!}},f\right).$

It remains to recall that
$\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}\left(\sigma^{L^{!}}_{J^{!}}(\delta[s^{L}],\xi
a)^{L^{!}}\right)$ equals
$\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s^{L}}\left(\sigma^{L^{!}}_{J^{!}}(\delta[s^{L}],\xi
a)\right)^{\tilde{L}}$. ∎

######  Theorem 6.2.13.

Recall that $F$ is non-Archimedean here. For every $\delta\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})$ and $J\in\mathcal{J}^{G}_{M}$, we
have

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

as germs on $A_{M}(F)$.

A conditional proof of Theorem 6.2.13 will be given in §6.5, under the
following

######  Hypothesis 6.2.14.

For all $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ with
$\operatorname{Supp}(\tilde{\gamma})\subset\tilde{M}_{G\text{-reg}}$, and all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
we have

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

The hypothesis will be resolved in §13.9.

####  6.2.4 Archimedean case

Assume that $F$ is an Archimedean local field. In this case
$D_{\mathrm{orb},-}(\tilde{M})$ and $D_{\mathrm{geom},-}(\tilde{M})$ are no
longer equal. In §5.2 we have defined
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)$ for all $\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. Let
$J\in\mathcal{J}^{G}_{M}$. The same construction as in the non-Archimedean
case yields a linear map

$\rho^{\tilde{G}}_{J}:D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}\to
U_{J}\otimes\left(D_{\mathrm{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$

with the same properties as in Definition–Proposition 6.2.6. There is also a
counterpart of Lemma 6.2.7.

######  Lemma 6.2.15.

Let $\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$
where $\mathcal{O}$ is $G$-equisingular. Then
$\rho^{\tilde{G}}_{J}(\tilde{\gamma})=0$ unless $M=G$.

Nevertheless, the constructions in the stable and endoscopic contexts are more
tricky. The reason is that we have to allow invariant distributions which are
not necessarily orbital. We will extend these constructions to a broader class
of $\tilde{\gamma}$ and formulate the Archimedean analogue of Theorem 6.2.13;
besides, we need to postulate an Archimedean analogue of Hypothesis 6.2.14.
This is the topic of §6.3.

###  6.3 Extension of definitions in the Archimedean case

In this section, $F$ is an Archimedean local field.

####  6.3.1 The space $D_{\text{tr-orb}}$

Let $\mathbf{p}:\tilde{G}\to G(F)$ be of metaplectic type. Following [60,
V.2], we are going to define a subspace $D_{\text{tr-orb},-}(\tilde{G})$ of
$D_{\text{geom},-}(\tilde{G})$ in the Archimedean setting.

Let us fix Haar measures temporarily to trivialize the lines
$\operatorname{mes}(\cdots)$; they will not affect the following definitions.

First off, recall from [60, V.2.1] that for every quasisplit $F$-group $L$,
one defines inductively a vector subspace

$D_{\text{tr-orb}}(L)\subset D_{\mathrm{geom}}(L)$

generated by $D_{\mathrm{orb}}(L)$ together with the images of

$SD_{\text{tr-orb}}(L^{!}):=D_{\text{tr-orb}}(L^{!})\cap
SD_{\mathrm{geom}}(L^{!})$

under endoscopic transfer, where $\mathbf{L}^{!}$ ranges over
$\mathcal{E}_{\operatorname{ell}}(L)$ with $\dim L^{!}<\dim L$, modulo some
technical details from $z$-extensions (see loc. cit.). If $\mathcal{O}^{!}$ is
a finite union of stable semisimple conjugacy classes in $L(F)$, we set

$SD_{\text{tr-orb}}(L,\mathcal{O}^{!}):=SD_{\text{tr-orb}}(L)\cap
SD_{\text{geom}}(L,\mathcal{O}^{!}).$

By taking $\mathcal{O}^{!}=\\{1\\}$, we obtain the space $SD_{\text{tr-
unip}}(L)$.

######  Definition 6.3.1.

Let

$D_{\text{tr-
orb},-}(\tilde{G}):=D_{\mathrm{orb},-}(\tilde{G})+\sum_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}\check{\mathcal{T}}_{\mathbf{G}^{!},\tilde{G}}\left(SD_{\text{tr-
orb}}(G^{!})\right).$

It is a vector subspace of $D_{\mathrm{geom},-}(\tilde{G})$.

If $\mathcal{O}$ is a finite union of semisimple conjugacy classes in $G(F)$,
we write

$\displaystyle D_{\text{tr-orb},-}(\tilde{G},\mathcal{O})$
$\displaystyle:=D_{\text{tr-orb},-}(\tilde{G})\cap
D_{\text{geom},-}(\tilde{G},\mathcal{O}),$ $\displaystyle D_{\text{tr-
unip},-}(\tilde{G})$ $\displaystyle:=D_{\text{tr-orb},-}(\tilde{G},\\{1\\}).$

Hereafter, we relax the choice of Haar measures.

######  Proposition 6.3.2.

Let $M\subset G$ be a Levi subgroup. Then
$\operatorname{Ind}^{\tilde{G}}_{\tilde{M}}$ maps $D_{\text{tr-
orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ to $D_{\text{tr-
orb},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$.

###### Proof.

We may assume $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$. The spaces
$SD_{\text{tr-orb}}(L)\otimes\operatorname{mes}(L)^{\vee}$ are known to be
stable under parabolic induction. By considering the situation (3.4.1) for
various $\mathbf{M}^{!}$ and $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$,
then applying Proposition 3.8.4, it suffices to show that
$\delta\mapsto\delta\cdot z[s]$ induces an automorphism of $SD_{\text{tr-
orb}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$. This is [60, V.2.1 (7)].
∎

The shorthand
$\tilde{\gamma}^{\tilde{G}}:=\operatorname{Ind}^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma})$
will be used.

####  6.3.2 A program

This subsection is a variant of the situation (A) in [60, V.2.4]. Consider a
group of metaplectic type $\mathbf{p}:\tilde{G}\to G(F)$ and a Levi subgroup
$M\subset G$. Our aim is to define the following objects.

  1. (O1)

For each $J\in\mathcal{J}^{G}_{M}$ and a semisimple conjugacy class
$\mathcal{O}\subset M(F)$, a linear map

$\rho^{\tilde{G}}_{J}:D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\to
U_{J}\otimes\left(D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)/\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}.$

It extends by linearity to the case when $\mathcal{O}$ is a finite union of
semisimple conjugacy classes.

  2. (O2)

For each $\tilde{\gamma}\in D_{\text{tr-
orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, a continuous linear
functional

$f\mapsto I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)$

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

  3. (O3)

Given

     * •

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

     * •

a stable semisimple conjugacy class $\mathcal{O}^{!}\subset M^{!}(F)$,

     * •

the image $\mathcal{O}$ of $\mathcal{O}^{!}$, which is a stable semisimple
conjugacy class in $M(F)$,

     * •

$\delta\in SD_{\text{tr-
orb}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$,

we associate the germ
$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$ in $a\in A_{M}(F)$,
$a\to 1$ in general position:

$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta,a):=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{G^{!}[s]}_{M^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\right)\\\
J^{!}\mapsto
J\end{subarray}}\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{s}\left(\sigma^{G^{!}[s]}_{J^{!}}\left(\delta[s],\xi(a)\right)\right).$

It is an element of
$U_{J}\otimes\left(D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\right)/\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$.
Note that $\sigma^{G^{!}[s]}_{J^{!}}\left(\cdots\right)$ is defined in the
situation (B) of [60, V.2.4], as $\delta[s]\in SD_{\text{tr-
orb}}(M^{!},\mathcal{O}^{!}[s])\otimes\operatorname{mes}(M^{!})^{\vee}$ (see
the proof of Proposition 6.3.2).

  4. (O4)

For $\mathbf{M}^{!}$, $\delta$ as above,

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

where $f^{G^{!}[s]}:=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)$. Again,
$S^{G^{!}[s]}_{M^{!}}(\delta[s],B^{\tilde{G}},\cdot)$ is defined in the
situation (B) of [60, V.2.4].

Observe that the definitions in (O3) and (O4) are unconditional.

The objects postulated above must be subject to the following conditions.

  1. (C1)

When $\tilde{\gamma}\in
D_{\text{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$,
the $\rho^{\tilde{G}}_{J}(\tilde{\gamma})$ in (O1) (resp.
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)$ in (O2)) coincides with the one
defined in §6.2.4 (resp. in §5.2).

  2. (C2)

For all $(\mathbf{M}^{!},\delta)$ as in (O4) and all $f$, we have

$\underbracket{I^{\tilde{G}}_{\tilde{M}}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),f\right)}_{\text{in
(O2)}}=\underbracket{I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)}_{\text{in
(O4)}}.$

  3. (C3)

For all $(\mathbf{M}^{!},\delta)$ as in (O3), we have the equality of germs

$\underbracket{\rho^{\tilde{G}}_{J}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)\right)}_{\text{in
(O1)}}=\underbracket{\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)}_{\text{in
(O3)}}.$

  4. (C4)

For each $\tilde{\gamma}\in D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$,
$J\in\mathcal{J}^{G}_{M}$ and $a\in A_{M}(F)$ in general position, $a\to 1$,
we have

$\rho^{\tilde{G}}_{J}(\tilde{\gamma},a)^{\tilde{G}}\in D_{\text{tr-
orb},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}.$

  5. (C5)

Let $\tilde{\gamma}$ be as above. The germ at $1$ of $a\mapsto
I^{\tilde{G}}_{\tilde{M}}(a\tilde{\gamma},f)$ is equivalent in the sense of
§6.2 to

$a\mapsto\sum_{L\in\mathcal{L}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}(\tilde{\gamma},a)^{\tilde{L}},f\right),$

which is well-defined by (C4). Observe that for $a$ in general position,
$a\tilde{\gamma}$ is $\tilde{G}$-equisingular and
$I^{\tilde{G}}_{\tilde{M}}(a\tilde{\gamma},f)$ should be understood by
Definition 5.2.2; see (C8) below.

  6. (C6)

For each $\tilde{\gamma}\in D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$ and
$L\in\mathcal{L}^{G}(M)$, we have

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

Cf. Proposition 5.2.5.

  7. (C7)

For each $\tilde{\gamma}\in D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$,
$L\in\mathcal{L}^{G}(M)$ and $J\in\mathcal{J}^{L}_{M}$, we have

$\rho^{\tilde{G}}_{J}\left(\tilde{\gamma}^{\tilde{L}}\right)=\sum_{\begin{subarray}{c}L^{\dagger}\in\mathcal{L}^{G}(M)\\\
\text{s.t.}\;J\in\mathcal{J}^{L^{\dagger}}_{M}\end{subarray}}d^{G}_{M}(L,L^{\dagger})\rho^{\tilde{L}^{\dagger}}_{J}\left(\tilde{\gamma}\right)^{\tilde{L}}.$

Cf. Proposition 6.2.8.

  8. (C8)

When $\tilde{\gamma}\in D_{\text{tr-
orb}}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$ and
$\mathcal{O}$ is $G$-equisingular, then the
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)$ in (O2) coincides with the
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)$ of Definition 5.2.2.

The condition (C8) is not explicitly stated in [60]. Nonetheless, the
arguments below for (C8) will also apply to the scenario in loc. cit.. In
particular, the stable analogues for (C8) hold on the endoscopic side.

Write $\tilde{G}=\prod_{i\in
I}\operatorname{GL}(n_{i},F)\times\widetilde{\mathrm{Sp}}(W)$. We shall
establish these objectives by assuming

  * •

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

  * •

the case when $M$ is replaced by some strictly larger Levi subgroup of $G$,
which is legitimate since the case $M=G$ is already known;

  * •

the following hypothesis.

######  Hypothesis 6.3.3.

For all $\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ with
$\operatorname{Supp}(\tilde{\gamma})\subset\tilde{M}_{G\text{-reg}}$, and all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
we have

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

Note that the equality above extends automatically to $\tilde{\gamma}\in
D_{\text{geom, $G$-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ by
Proposition 5.5.2.

####  6.3.3 Preliminary reductions

This part is modeled on [60, V.2.5]. Let $\tilde{G}\supset\tilde{M}$ be as in
§6.3.2. First off, recall that
$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)$, etc. (resp.
$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$, etc.) are well-
defined in (O4) (resp. (O3)).

######  Lemma 6.3.4.

Let $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$,
$\mathcal{O}^{!}\subset M^{!}(F)$ be a stable semisimple conjugacy class, and
$\delta\in SD_{\text{tr-
orb}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$. For all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
the germ near $a=1$ of

$a\mapsto
I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\xi(a)\delta,f\right)$

is equivalent to that of

$a\mapsto
I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,f\right)+\sum_{J\in\mathcal{J}^{G}_{M}}I^{\tilde{G}}_{\tilde{G}}\left(\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta,a)^{\tilde{G}},f\right)\\\
+\sum_{L\in\mathcal{L}^{G}(M)\smallsetminus\\{M,G\\}}\sum_{J\in\mathcal{J}^{L}_{M}}I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),a\right)^{\tilde{L}},f\right),$

where $a\in A_{M}(F)$ is in general position so that $\xi(a)\delta$ is
$\tilde{G}$-equisingular.

###### Proof.

Arguments from [60, II.3.9] show that the germ of
$I^{\tilde{G},\mathcal{E}}_{\mathbf{M}^{!}}\left(\mathbf{M}^{!},\xi(a)\delta,f\right)$
is equivalent to

$\sum_{L\in\mathcal{L}^{G}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})}i_{M^{!}}(\tilde{L},L^{!}[s])\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{L^{!}[s]}_{M^{!}}(B^{\tilde{G}}_{\mathcal{O}^{!}[s]})\\\
J^{!}\mapsto
J\end{subarray}}I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\mathbf{L}^{!}[s],\sigma^{L^{!}[s]}_{J^{!}}(\delta[s],\xi(a))^{L^{!}[s]},f\right).$

This step is purely combinatorial, the only inputs being Lemma 3.5.7 and Lemma
5.4.2.

Note that
$B^{\tilde{G}}_{\mathcal{O}^{!}[s]}\big{|}_{L^{!}[s]}=B^{\tilde{L}}_{\mathcal{O}^{!}[s]}$
by Lemma 5.3.6. We shall show that the asserted decomposition corresponds to
the contributions of $L=M$, $L=G$ and $L\notin\\{M,G\\}$, respectively. This
step requires some care because of the $z[s]$-twists.

For all $(L,s)$ with $L\neq M$, Proposition 3.8.4 and the induction hypotheses
applied to (C2) yield

$\displaystyle
I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\mathbf{L}^{!}[s],\sigma^{L^{!}[s]}_{J^{!}}(\delta[s],\xi(a))^{L^{!}[s]},f\right)$
$\displaystyle=I^{\tilde{G}}_{\tilde{L}}\left(\check{\mathcal{T}}_{\mathbf{L}^{!}[s],\tilde{L}}\left(\left(\sigma^{L^{!}[s]}_{J^{!}}(\delta[s],\xi(a))\right)^{L^{!}[s]}\right),f\right)$
$\displaystyle=I^{\tilde{G}}_{\tilde{L}}\left(\check{\mathcal{T}}^{s}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{L^{!}[s]}_{J^{!}}(\delta[s],\xi(a))\right)^{\tilde{L}},f\right).$

For fixed $L\neq M$ and $J$, the corresponding contribution is thus equal to

$I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta,a)^{\tilde{L}},f\right).$

  * •

When $L\neq M,G$, the induction hypotheses applied to (C3) show that the
contribution is
$I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}\left(\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta),a\right)^{\tilde{L}},f\right)$.

  * •

When $L=G$, the contribution is
$I^{\tilde{G}}_{\tilde{G}}\left(\rho^{\tilde{G},\mathcal{E}}(\mathbf{M}^{!},\delta,a)\right)$.

Finally, when $L=M$ we have: $s$ is trivial,
$\mathcal{J}^{L}_{M}=\\{\varnothing\\}=\mathcal{J}^{L^{!}[s]}_{M^{!}}\left(B^{\tilde{L}}_{\mathcal{O}^{!}[s]}\right)$
and $i_{M^{!}}(\tilde{L},L^{!}[s])=1$. The contribution is thus
$I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,f\right)$. ∎

We need the endoscopic version below of the conditions (C6) and (C7).

######  Lemma 6.3.5.

Let $(\mathbf{M}^{!},\delta)$ and $\mathcal{O}^{!}$ be as before. Suppose that
we are in the situation

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Then for all $\delta\in SD_{\text{tr-
orb}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M)^{!}$,
$J\in\mathcal{J}^{G}_{L}$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
we have

$\displaystyle
I^{\tilde{G},\mathcal{E}}_{\tilde{L}}\left(\mathbf{L}^{!},\delta[t]^{L^{!}},f\right)$
$\displaystyle=\sum_{L^{\dagger}\in\mathcal{L}^{G}(M)}d^{G}_{M}(L,L^{\dagger})I^{\tilde{L}^{\dagger},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta,f_{\tilde{L}^{\dagger}}\right),$
$\displaystyle\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{L}^{!},\delta[t]^{L^{!}}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}L^{\dagger}\in\mathcal{L}^{G}(M)\\\
\text{s.t.}\;J\in\mathcal{J}^{L^{\dagger}}_{M}\end{subarray}}d^{G}_{M}(L,L^{\dagger})\rho^{\tilde{L}^{\dagger},\mathcal{E}}_{J}\left(\mathbf{M}^{!},\delta\right)^{\tilde{L}}.$

###### Proof.

The arguments are combinatorial. The first formula follows from exactly the
same proof of Proposition 5.4.4. The second formula requires extra care, as
explained below.

Let us assume $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ without loss of
generality. We claim that

$\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{L}^{!},\delta[t]^{L^{!}}\right)=\sum_{s\in\mathcal{E}_{\mathbf{L}^{!}}(\tilde{G})}i_{L^{!}}(\tilde{G},G^{!}[s])\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{G^{!}[s]}_{L^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[t][s]}\right)\\\
J^{!}\mapsto
J\end{subarray}}\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}^{s}\left(\sigma^{G^{!}[s]}_{J^{!}}(\delta[t]^{L^{!}}[s])\right)\\\
=\sum_{s\in\mathcal{E}_{\mathbf{L}^{!}}(\tilde{G})}i_{L^{!}}(\tilde{G},G^{!}[s])\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{G^{!}[r]}_{L^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[r]}\right)\\\
J^{!}\mapsto
J\end{subarray}}\sum_{\begin{subarray}{c}L^{\dagger!}\in\mathcal{L}^{G^{!}[r]}(M^{!})\\\
\text{s.t.}\;J^{!}\in\mathcal{J}^{L^{\dagger!}}_{M^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[r]}\big{|}_{L^{\dagger!}}\right)\end{subarray}}e^{G^{!}[r]}_{M^{!}}(L^{!},L^{\dagger!})\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}^{s}\left(\sigma^{L^{\dagger!}}_{J^{!}}(\delta[r])^{L^{!}}\right),$

where $\mathcal{O}^{!}[t][s]$ also stands for the stable class it generates in
$L^{!}(F)$, by abusing notation slightly. Indeed, the first equality follows
from the definition. Next, as in the proof of Proposition 5.4.4,

  * •

we write $r=r(s,t)\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$;

  * •

to each $(s,L^{\dagger!})$ we attach a canonical
$L^{\dagger}\in\mathcal{L}^{G}(M)$ and decompose $r$ into
$r^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L}^{\dagger})$,
$r_{L}\in\mathcal{E}_{\mathbf{L}^{\dagger!}}(\tilde{G})$ so that
$L^{\dagger!}=L^{\dagger!}[r^{L}]$ is an endoscopic group of
$\tilde{L}^{\dagger}$;

  * •

the situation can be summarized as

${G^{!}[s]}$${L^{!}}$${L^{\dagger!}}$${M^{!};}$$\scriptstyle{s}$$\scriptstyle{r_{L}}$$\scriptstyle{t}$$\scriptstyle{r^{L}}$$\scriptstyle{r}$

hence the second equality follows from
$\delta[t]^{L^{!}}[s]=\delta[r]^{L^{!}}$ and the descent formula [60, II.3.11
Lemme] for $\sigma^{G^{!}[s]}_{J^{!}}$.

Recall that
$\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}^{s}(\cdots)=\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}(\cdots[s]^{-1})$
and factorize $[s]^{-1}$ into $[r_{L}]^{-1}[r^{L}]^{-1}[t]$. We have

$\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r]\right)^{L^{!}}[s]^{-1}=\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}][r_{L}]\right)[s]^{-1}\right)^{L^{!}}=\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}]\right)[r^{L}]^{-1}[t]\right)^{L^{!}}$

Since $\mathbf{L}^{!}=\mathbf{L}^{!}[t]$, Proposition 3.8.4 thus yields

$\displaystyle\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}^{s}\left(\sigma^{L^{\dagger!}}_{J^{!}}(\delta[r])^{L^{!}}\right)$
$\displaystyle=\check{\mathcal{T}}_{\mathbf{L}^{!},\tilde{L}}\left(\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}]\right)[r^{L}]^{-1}[t]\right)^{L^{!}}\right)$
$\displaystyle=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}]\right)[r^{L}]^{-1}\right)^{\tilde{L}}$
$\displaystyle=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{r^{L}}\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}]\right)\right)^{\tilde{L}}.$

Next, we have
$\mathcal{J}^{L^{\dagger!}}_{M^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[r]}\big{|}_{L^{\dagger!}}\right)=\mathcal{J}^{L^{\dagger!}}_{M^{!}}\left(B^{\tilde{L}}_{\mathcal{O}^{!}[r]}\right)=\mathcal{J}^{L^{\dagger!}}_{M^{!}}\left(B^{\tilde{L}}_{\mathcal{O}^{!}[r^{L}]}\right)$
as the central twist by $z[r_{L}]$ has no effect here. Furthermore, when
$e^{G^{!}[r]}_{M^{!}}(L^{!},L^{\dagger!})\neq 0$ (thus
$d^{G}_{M}(L,L^{\dagger})\neq 0$), Lemma 6.1.2 and the discussions preceding
Proposition 6.2.8 lead to the commutative diagram

${\mathcal{J}^{L^{\dagger!}}_{M^{!}}\left(B^{\tilde{L}}_{\mathcal{O}^{!}[r^{L}]}\right)}$${\mathcal{J}^{G^{!}[r]}_{L^{!}}\left(B^{\tilde{G}}_{\mathcal{O}^{!}[r]}\right)}$${\mathcal{J}^{L^{\dagger}}_{M}}$${\mathcal{J}^{G}_{L}.}$

With the notations above, we arrive at

$\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{L}^{!},\delta[t]^{L^{!}}\right)=\sum_{(L^{\dagger!},s)}i_{L^{!}}(\tilde{G},G^{!}[s])e^{G^{!}[r]}_{M^{!}}\left(L^{!},L^{\dagger!}\right)\\\
\xi_{J}(L^{\dagger})\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{\tilde{L}^{\dagger}}_{M^{!}}\left(B^{\tilde{L}}_{\mathcal{O}^{!}[r^{L}]}\right)\\\
J^{!}\mapsto
J\end{subarray}}\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{r^{L}}\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}]\right)\right)^{\tilde{L}}$

where $\xi_{J}(L^{\dagger})=1$ when $d^{G}_{M}(L,L^{\dagger})\neq 0$ and
$J\in\mathcal{J}^{L^{\dagger}}_{M}$, otherwise $\xi_{J}(L^{\dagger})=0$.

The combinatorial proof of Proposition 5.4.4 can now be applied to transform
this expression into

$\sum_{L^{\dagger}\in\mathcal{L}^{G}(M)}d^{G}_{M}(L^{\dagger},L)\xi_{J}(L^{\dagger})\sum_{r^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L}^{\dagger})}i_{\mathbf{M}^{!}}(\tilde{L}^{\dagger},L^{\dagger!})\\\
\sum_{\begin{subarray}{c}J^{!}\in\mathcal{J}^{\tilde{L}^{\dagger}}_{M^{!}}\left(B^{\tilde{L}}_{\mathcal{O}^{!}[r^{L}]}\right)\\\
J^{!}\mapsto
J\end{subarray}}\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}^{r^{L}}\left(\sigma^{L^{\dagger!}}_{J^{!}}\left(\delta[r^{L}]\right)\right)^{\tilde{L}}.$

It remains to recall the definition of
$\rho^{\tilde{L}^{\dagger},\mathcal{E}}_{J}\left(\mathbf{M}^{!},\delta\right)$.
∎

######  Proposition 6.3.6 (Cf. [60, V.2.5]).

Assume that Hypothesis 6.3.3 holds. Suppose that for all
$J\in\mathcal{J}^{G}_{M}\smallsetminus\\{J_{\max}\\}$ and semisimple conjugacy
class $\mathcal{O}\subset M(F)$, we have $\rho^{\tilde{G}}_{J}$ as in (O1)
meeting the conditions (C1), (C3). Then the program in §6.3.2 can be realized.

###### Proof.

We may assume $M\neq G$. Let $J\in\mathcal{J}^{G}_{M}$. Our conditions force
that $\rho^{\tilde{G}}_{J}$ and $I^{\tilde{G}}_{\tilde{M}}$ must be defined as
follows. Express $\tilde{\gamma}\in D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$ as

$\tilde{\gamma}=\tilde{\gamma}_{\text{orb}}+\sum_{i=1}^{n}\check{\mathcal{T}}_{\mathbf{M}^{!}_{i},\tilde{M}}\left(\delta_{i}\right)$
(6.3.1)

with $\tilde{\gamma}\in
D_{\text{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ and
$\delta_{i}\in SD_{\text{tr-
orb}}(M_{i}^{!},\mathcal{O}^{!}_{i})\otimes\operatorname{mes}(M_{i}^{!})^{\vee}$,
where $\mathbf{M}_{i}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\mathcal{O}^{!}_{i}\mapsto\mathcal{O}$ for all $i$. Then

$\displaystyle\rho^{\tilde{G}}_{J}\left(\tilde{\gamma}\right)$
$\displaystyle=\rho^{\tilde{G}}_{J}\left(\tilde{\gamma}_{\text{orb}}\right)+\sum_{i}\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{M}_{i}^{!},\delta_{i}\right),$
$\displaystyle I^{\tilde{G}}_{\tilde{M}}\left(\tilde{\gamma},f\right)$
$\displaystyle=I^{\tilde{G}}_{\tilde{M}}\left(\tilde{\gamma}_{\text{orb}},f\right)+\sum_{i}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}_{i}^{!},\delta_{i},f\right).$

What remains is to show that they are independent of (6.3.1) and satisfy the
required conditions.

Consider the germ at $a=1$ of
$I^{\tilde{G}}_{\tilde{M}}(a\tilde{\gamma}_{\mathrm{orb}},f)+\sum_{i}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!}_{i},\xi_{i}(a)\delta_{i},f)$,
where $\xi_{i}:A_{M}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{M^{!}_{i}}$
comes from endoscopy. The germ of the first term is equivalent to

$\sum_{L\in\mathcal{L}(M)}\sum_{J\in\mathcal{J}^{L}_{M}}I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}(\tilde{\gamma}_{\mathrm{orb}},a)^{\tilde{L}},f\right),$
(6.3.2)

whilst by Lemma 6.3.4, the second term is equivalent to

$\sum_{i}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!}_{i},\delta_{i},f\right)+\sum_{J\in\mathcal{J}^{G}_{M}}I^{\tilde{G}}_{\tilde{G}}\left(\sum_{i}\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!}_{i},\delta_{i},a)^{\tilde{G}},f\right)\\\
+\sum_{\begin{subarray}{c}L\in\mathcal{L}^{G}(M)\\\ L\neq
M,G\end{subarray}}\sum_{J\in\mathcal{J}^{L}_{M}}I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}\left(\sum_{i}\check{\mathcal{T}}_{\mathbf{M}^{!}_{i},\tilde{M}}(\delta_{i}),a\right)^{\tilde{L}},f\right).$
(6.3.3)

To simplify the expression, put

$\underline{\rho}_{J}(\tilde{\gamma}):=\rho^{\tilde{G}}_{J}(\tilde{\gamma}_{\mathrm{orb}})+\sum_{i=1}^{n}\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!}_{i},\delta_{i}).$

When $J\in\mathcal{J}^{G}_{M}\smallsetminus\\{J_{\max}\\}$, we have
$\underline{\rho}_{J}(\tilde{\gamma})=\rho^{\tilde{G}}_{J}(\tilde{\gamma})$.
The sum of (6.3.2) and (6.3.3) becomes

$\begin{gathered}\underbracket{I^{\tilde{G}}_{\tilde{M}}\left(\tilde{\gamma}_{\text{orb}},f\right)}_{\text{$L=M$
in
\eqref{eqn:Jmax-1}}}+\underbracket{\sum_{i}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}_{i}^{!},\delta_{i},f\right)}_{\text{the
first term in
\eqref{eqn:Jmax-2}}}+\underbracket{I^{\tilde{G}}_{\tilde{G}}\left(\underline{\rho}_{J_{\max}}(\tilde{\gamma},a)^{\tilde{G}}\right)}_{\begin{subarray}{c}L=G,\;J=J_{\max}\;\text{in
\eqref{eqn:Jmax-1}}\\\ J=J_{\max}\in\mathcal{J}^{G}_{M}\;\text{in
\eqref{eqn:Jmax-2}}\end{subarray}}\\\
+\underbracket{\sum_{\begin{subarray}{c}J\in\mathcal{J}^{G}_{M}\\\ J\neq
J_{\max}\end{subarray}}I^{\tilde{G}}_{\tilde{G}}\left(\rho^{\tilde{G}}_{J}(\tilde{\gamma},a)^{\tilde{G}},f\right)}_{\begin{subarray}{c}L=G,J\neq
J_{\max}\;\text{in \eqref{eqn:Jmax-1}}\\\
J\in\mathcal{J}^{G}_{M}\smallsetminus\\{J_{\max}\\}\;\text{in
\eqref{eqn:Jmax-2}}\end{subarray}}+\underbracket{\sum_{\begin{subarray}{c}L\in\mathcal{L}^{G}(M)\\\
L\neq
M,G\end{subarray}}\sum_{J\in\mathcal{J}^{L}_{M}}I^{\tilde{G}}_{\tilde{L}}\left(\rho^{\tilde{L}}_{J}(\tilde{\gamma},a)^{\tilde{L}},f\right)}_{\begin{subarray}{c}L\neq
M,G\;\text{in \eqref{eqn:Jmax-1}}\\\ \text{the final sum in
\eqref{eqn:Jmax-2}}\end{subarray}}.\end{gathered}$ (6.3.4)

Up to equivalence of germs, (6.3.4) is determined by
$I^{\tilde{G}}_{\tilde{M}}(a\tilde{\gamma},f)$ for $a\to 1$ in general
position so that $a\tilde{\gamma}$ is $\tilde{G}$-equisingular, by Hypothesis
6.3.3. Hence

$\left(I^{\tilde{G}}_{\tilde{M}}\left(\tilde{\gamma}_{\text{orb}},f\right)+\sum_{i}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}_{i}^{!},\delta_{i},f\right)\right)+I^{\tilde{G}}_{\tilde{G}}\left(\underline{\rho}_{J_{\max}}(\tilde{\gamma})^{\tilde{G}},f\right)$

is independent of (6.3.1) up to equivalence of germs.

Consider the formula displayed above. Since the first term in parentheses is
constant in $a$ whereas the second term is in $U_{J_{\max}}$, Proposition
6.2.2 (iii) implies that both are independent of (6.3.1). Hence

$I^{\tilde{G}}_{\tilde{M}}\left(\tilde{\gamma},f\right):=I^{\tilde{G}}_{\tilde{M}}\left(\tilde{\gamma}_{\text{orb}},f\right)+\sum_{i}I^{\tilde{G},\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}_{i}^{!},\delta_{i},f\right)$

is well-defined. By varying $f$, we see that
$\underline{\rho}_{J_{\max}}(\tilde{\gamma})$ is also independent of (6.3.1)
(since it is a coset under $\mathrm{Ann}^{\tilde{G}}_{\tilde{M}}$). One
obtains the well-defined germ

$\rho^{\tilde{G}}_{J_{\max}}(\tilde{\gamma}):=\underline{\rho}_{J_{\max}}(\tilde{\gamma}).$

The condition (C5) is verified by (6.3.4), and (C1), (C2), (C3) all follow by
construction. Now consider the condition (C4). The case $\tilde{\gamma}\in
D_{\text{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$ is
known. In view of (C3), we may assume
$\tilde{\gamma}=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)$ for
some $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\delta$. Then by Proposition 3.8.4,

$\displaystyle\rho^{\tilde{G}}_{J}\left(\tilde{\gamma},a\right)^{\tilde{G}}$
$\displaystyle=\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{M}^{!},\delta,a\right)^{\tilde{G}}$
$\displaystyle=\sum_{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}i_{M^{!}}(\tilde{G},G^{!}[s])\sum_{J^{!}\mapsto
J}\left(\check{\mathcal{T}}^{s}_{\mathbf{M^{!}},\tilde{M}}\left(\sigma^{G^{!}[s]}_{J^{!}}(\delta[s],\xi(a))\right)\right)^{\tilde{G}}$
$\displaystyle=\sum_{s}i_{M^{!}}(\tilde{G},G^{!}[s])\sum_{J^{!}\mapsto
J}\check{\mathcal{T}}_{\mathbf{G}^{!}[s],\tilde{G}}\left(\sigma^{G^{!}[s]}_{J^{!}}(\delta[s],\xi(a))^{G^{!}[s]}\right)$
$\displaystyle\in D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}.$

The descent formulas (C6) and (C7) are known when $\tilde{\gamma}\in
D_{\text{orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$.
The case of
$\tilde{\gamma}=\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}(\delta)$ for
some $(\mathbf{M}^{!},\delta)$ is covered by Lemma 6.3.5.

Finally, let $\mathcal{O}$ and $\tilde{\gamma}$ be as in (C8). Fix $f$ and
denote by $I_{1}(\tilde{\gamma})$ (resp. $I_{2}(\tilde{\gamma})$) the
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)$ of Definition 5.2.2 (resp. of
(O2)). By (C5) and Lemma 6.2.15, the germ of $a\mapsto I_{1}(a\tilde{\gamma})$
is equivalent to $I_{2}(\tilde{\gamma})$.

However, it is clear from Definition 5.2.2 and basic properties of
$I^{\tilde{M}}_{\tilde{M}}(a\tilde{\gamma},\cdot)$ that $a\mapsto
I_{1}(a\tilde{\gamma})$ is $C^{\infty}$ near $a=1$. We conclude by Proposition
6.2.2 (iii) that $I_{1}(\tilde{\gamma})=I_{2}(\tilde{\gamma})$, which is (C8).
∎

Next, let $J\in\mathcal{J}^{G}_{M}$. By Proposition 6.1.3, we have

$\displaystyle G_{J}$ $\displaystyle=\prod_{k\in
K}\operatorname{GL}(n_{k})\times\operatorname{Sp}(W_{1})\times\cdots\times\operatorname{Sp}(W_{n}),$
$\displaystyle\widetilde{G_{J}}$ $\displaystyle=\prod_{k\in
K}\operatorname{GL}(n_{k},F)\times\left(\widetilde{\mathrm{Sp}}(W_{1})\overset{\bm{\mu}_{8}}{\times}\cdots\overset{\bm{\mu}_{8}}{\times}\widetilde{\mathrm{Sp}}(W_{n})\right),$

where

  * •

$K$ is a finite set and $n_{k}\in\mathbb{Z}_{\geq 1}$ for all $k\in K$;

  * •

$W_{1},\ldots,W_{n}$ are symplectic $F$-vector spaces, $n\in\mathbb{Z}_{\geq
0}$;

  * •

$\widetilde{\mathrm{Sp}}(W_{1})\overset{\bm{\mu}_{8}}{\times}\cdots\overset{\bm{\mu}_{8}}{\times}\widetilde{\mathrm{Sp}}(W_{n})$
is the quotient of $\prod_{i=1}^{n}\widetilde{\mathrm{Sp}}(W_{i})$ by
$\left\\{(z_{i})_{i=1}^{n}\in\bm{\mu}_{8}^{n}:z_{1}\cdots z_{n}=1\right\\}$.

Whence the evident central extension

$1\to\bm{\mu}_{8}\to\widetilde{G_{J}}\xrightarrow{\mathbf{p}_{J}}G_{J}(F)\to
1.$

Although it is not necessarily a group of metaplectic type, there is still a
theory of elliptic endoscopy for $\widetilde{G_{J}}$. Obviously, it is the
“product” of the theories for $\operatorname{GL}(n_{k})$ and those for
$\widetilde{\mathrm{Sp}}(W_{i})$, for all $k,i$. In particular, it makes sense
to talk about the transfers
$\mathcal{T}_{\mathbf{G_{J}}^{!},\widetilde{G_{J}}}$ and
$\check{\mathcal{T}}_{\mathbf{G_{J}}^{!},\widetilde{G_{J}}}$ for each
$\mathbf{G}_{J}^{!}\in\mathcal{E}_{\operatorname{ell}}(\widetilde{G_{J}})$.
The dual group of $\widetilde{G_{J}}$, with trivial Galois action, is simply

$\widetilde{G_{J}}^{\vee}=\prod_{k\in
K}\operatorname{GL}(n_{k},\mathbb{C})\times\prod_{i=1}^{n}\operatorname{Sp}(2m_{i},\mathbb{C}),\quad
2m_{i}:=\dim_{F}W_{i}.$

In the same vein, one defines
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$ as in (O3).
It satisfies the analogues of Lemmas 6.3.4 and 6.3.5 for $\widetilde{G_{J}}$.
If the program in §6.3.2 can be achieved for each
$\widetilde{\mathrm{Sp}}(W_{i})$, the same can be said for
$\widetilde{G_{J}}$.

######  Proposition 6.3.7.

Assume that Hypothesis 6.3.3 holds. Suppose that for all
$J\in\mathcal{J}^{G}_{M}$,
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and $\delta\in
SD_{\text{tr-
orb}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$, where
$\mathcal{O}^{!}\subset M^{!}(F)$ and $\mathcal{O}\subset M(F)$ are stable
semisimple conjugacy classes with $\mathcal{O}^{!}\mapsto\mathcal{O}$, the map
induced by Lemma 6.2.5

$D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\big{/}\mathrm{Ann}^{\widetilde{G_{J}}}_{\mathcal{O}}\twoheadrightarrow
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}\big{/}\mathrm{Ann}^{\tilde{G}}_{\mathcal{O}}$

sends $\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$ to
$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$. Then the premises
of Proposition 6.3.6 hold.

###### Proof.

Given $\tilde{\gamma}\in D_{\text{tr-
orb},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$, choose an
expression

$\tilde{\gamma}=\tilde{\gamma}_{\text{orb}}+\sum_{i=1}^{n}\check{\mathcal{T}}_{\mathbf{M}^{!}_{i},\tilde{M}}\left(\delta_{i}\right).$

In order to verify the premises of Proposition 6.3.6, it suffices to show that
for all $J\in\mathcal{J}^{G}_{M}\smallsetminus\\{J_{\max}\\}$, the germ

$\rho^{\tilde{G}}_{J}(\tilde{\gamma}):=\rho^{\tilde{G}}_{J}(\tilde{\gamma}_{\mathrm{orb}})+\sum_{i}\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!}_{i},\delta_{i})$

is independent of the expression above of $\tilde{\gamma}$.

The Archimedean version of Definition–Proposition 6.2.6 implies that
$\rho^{\widetilde{G_{J}}}_{J}(\tilde{\gamma}_{\mathrm{orb}})\mapsto\rho^{\tilde{G}}_{J}(\tilde{\gamma}_{\mathrm{orb}})$
under the natural surjection. By fiat,
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}_{i}^{!},\delta_{i})\mapsto\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!}_{i},\delta_{i})$
for all $i$. Since $\dim G_{J}<\dim G$, by induction we see that

$\rho^{\widetilde{G_{J}}}_{J}(\tilde{\gamma}):=\rho^{\widetilde{G_{J}}}_{J}(\tilde{\gamma}_{\mathrm{orb}})+\sum_{i}\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}_{i}^{!},\delta_{i})$

is independent of the expression of $\tilde{\gamma}$. Hence so is its image
$\rho^{\tilde{G}}_{J}(\tilde{\gamma})$. ∎

Note that
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)\mapsto\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$
is tautologically true when $J=J_{\max}$, since $G_{J_{\max}}=G$.

###  6.4 Conditional proof of the local theorems: Archimedean case

Let $F$ be an Archimedean local field. Consider a group of metaplectic type
$\mathbf{p}:\tilde{G}\to G(F)$ and a Levi subgroup $M\subset G$. The goal here
is to realize the program in §6.3.2 under the Hypothesis 6.3.3. We will also
take the induction hypotheses for granted. See the discussions preceding
Hypothesis 6.3.3.

In view of Propositions 6.3.6 and 6.3.7, it suffices to show that
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)\mapsto\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$
under the natural map, for each $J\in\mathcal{J}^{G}_{M}$. Here
$(\mathbf{M}^{!},\delta)$ and $\mathcal{O}^{!}$ are as in Proposition 6.3.7.

####  6.4.1 Reduction to the elliptic case

Fix $\epsilon\in\mathcal{O}^{!}$ and $\eta\in\mathcal{O}$. By [40, Lemma 3.3],
we may and do assume that $M^{!}_{\epsilon}$ is quasisplit.

######  Lemma 6.4.1.

Consider the statement that
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)\mapsto\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$
under the natural map, for various $\tilde{M}\subset\tilde{G}$,
$J\in\mathcal{J}^{G}_{M}$, $\mathcal{O}^{!}\subset M^{!}(F)_{\mathrm{ss}}$ and
$(\mathbf{M}^{!},\delta)$. If it holds whenever $\mathcal{O}^{!}$ is elliptic
in $M^{!}$, then it holds in general.

###### Proof.

Define the Levi subgroup

$R^{!}:=Z_{M^{!}}(A_{M^{!}_{\epsilon}})\subset M^{!}$

and let $\mathcal{O}_{R^{!}}$ be the stable semisimple class in $R^{!}(F)$
containing $\epsilon$. By Proposition 3.7.2, to
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$R^{!}\hookrightarrow M^{!}$ is associated a Levi subgroup $R\subset M$,
unique up to conjugacy, together with a unique
$t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$ such that

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By [60, V.4.4 Lemme], there exists $\delta_{R^{!}}\in SD_{\text{tr-
orb}}(R^{!},\mathcal{O}_{R^{!}}[t]^{-1})\otimes\operatorname{mes}(R^{!})^{\vee}$
such that $\delta=\left(\delta_{R^{!}}[t]\right)^{M^{!}}$, where
$\mathcal{O}_{R^{!}}[t^{-1}]:=\mathcal{O}_{R^{!}}\cdot z[t]^{-1}$.

Lemma 6.3.5 implies

$\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{M}^{!},\delta\right)=\sum_{\begin{subarray}{c}L\in\mathcal{L}^{G}(R)\\\
\text{s.t.}\;J\in\mathcal{J}^{L}_{R}\end{subarray}}d^{G}_{R}(M,L)\rho^{\tilde{L},\mathcal{E}}_{J}\left(\mathbf{R}^{!},\delta_{R^{!}}\right)^{\tilde{M}}.$

The analogue of Lemma 6.3.5 for $\widetilde{G_{J}}$ also yields

$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}\left(\mathbf{M}^{!},\delta\right)=\sum_{\begin{subarray}{c}L^{\dagger}\in\mathcal{L}^{G_{J}}(R)\\\
\text{s.t.}\;J\in\mathcal{J}^{L^{\dagger}}_{R}\end{subarray}}d^{G_{J}}_{R}(M,L^{\dagger})\rho^{\widetilde{L^{\dagger}},\mathcal{E}}_{J}\left(\mathbf{R}^{!},\delta_{R^{!}}\right)^{\tilde{M}}.$

We contend that $L\mapsto L_{J}$ gives a bijection

${\left\\{L\in\mathcal{L}^{G}(R):J\in\mathcal{J}^{L}_{R}\right\\}}$${\left\\{L^{\dagger}\in\mathcal{L}^{G_{J}}(R):J\in\mathcal{J}^{L^{\dagger}}_{R}\right\\}}$$\scriptstyle{1:1}$

and we have $d^{G}_{R}(M,L)=d^{G_{J}}_{R}(M,L_{J})$. Indeed, the desired
bijection is in [60, p.546], and so is the equality for $d_{R}(\cdot,\cdot)$
which is based on $\mathfrak{a}_{G_{J}}=\mathfrak{a}_{G}$ as subspaces of
$\mathfrak{a}_{M}$.

We may assume that $\epsilon$ is not elliptic in $M^{!}$. Then $R^{!}\neq
M^{!}$, and those $L$ with $d^{G}_{R}(M,L)\neq 0$ must satisfy $L\neq G$. By
induction,
$\rho^{\widetilde{L_{J}},\mathcal{E}}_{J}\left(\mathbf{R}^{!},\delta_{R^{!}}\right)$
maps
$\rho^{\tilde{L},\mathcal{E}}_{J}\left(\mathbf{R}^{!},\delta_{R^{!}}\right)$.
It follows that
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$ maps to
$\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$. ∎

####  6.4.2 The elliptic case

In what follows, we assume $\mathcal{O}^{!}$ is elliptic. Then $\mathcal{O}$
is also elliptic by Lemma 4.5.3.

The endoscopic datum of $M_{\eta}$ obtained by descent is relevant
(Proposition 4.5.5), hence we can take a diagram
$(\epsilon,B^{!},T^{!},B,T,\eta)$ for the endoscopic datum $\mathbf{M}^{!}$ of
$\tilde{M}$, compatibly with descent in the sense of Proposition 4.5.4. It
affords an isomorphisms between tori

${T^{!}}$${T}$${A_{M^{!}}}$${A_{M}.}$$\scriptstyle{\xi_{T^{!},T}}$${\subset}$$\scriptstyle{\xi^{-1}}$${\subset}$

In turn, we obtain
$\xi^{*}:\mathfrak{a}_{M^{!}}^{*}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{a}_{M}^{*}$;
see Remark 3.6.2.

######  Notation 6.4.2.

Fix $J\in\mathcal{J}^{G}_{M}\smallsetminus\\{J_{\max}\\}$ and $\delta\in
SD_{\text{tr-orb}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$. For the
sake of simplicity, $B$ (resp. $B_{J}$) will denote the systems of
$B$-functions $B_{\mathcal{O}^{!}}$ (resp. $B^{J}_{\mathcal{O}^{!}}$, defined
relative to $\tilde{M}\subset\widetilde{G_{J}}$).

Let $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. Definition 5.3.1 furnish
the following finite subsets of
$X^{*}(T^{!}_{\overline{F}})\otimes\mathbb{Q}$:

$\displaystyle\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)$
$\displaystyle:=\left\\{\beta^{\prime}=B(\beta)^{-1}\beta:\beta\in\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!}\right)\right\\},$
$\displaystyle\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)$
$\displaystyle:=\left\\{\beta^{\prime}\in\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right):\xi^{*}(\beta^{\prime}|_{\mathfrak{a}_{M^{!}}})\in
R_{J}\right\\},$
$\displaystyle\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!}\right)$
$\displaystyle:=\left\\{\beta\in\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!}\right):B(\beta)^{-1}\beta\in\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)\right\\}.$

Note that $\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)$ (resp.
$\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)$) is obtained from
$\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!}\right)$ (resp.
$\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!}\right)$) by multiplying by
$B(\cdot)^{-1}$.

Similarly, for $t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$ we define
the subsets of $X^{*}(T^{!}_{\overline{F}})\otimes\mathbb{Q}$:

$\Sigma\left(G_{J}^{!}[t]_{\epsilon[t]},T^{!}\right),\quad\Sigma\left(G_{J}^{!}[t]_{\epsilon[t]},T^{!},B_{J}\right),$

which are related by dilation by $B_{J}(\cdot)^{-1}$.

By Lemma 6.1.2, there is at most one
$J^{!}\in\mathcal{J}^{G^{!}[s]}_{M^{!}}(B)$ with $J^{!}\mapsto J$. If it
exists, denote it as $J_{s}$. Similarly, for each
$t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$, denote by
$J_{t}\in\mathcal{J}^{G_{J}^{!}[t]}_{M^{!}}(B_{J})$ the unique element mapping
to $J$, if it exists.

Given $s$ and $J_{s}$, we construct the connected reductive group
$G^{!}[s]_{\epsilon[s],J_{s}}$ using the recipe of §6.1.2 and the
$A_{M^{!}_{\epsilon[s]}}=A_{M^{!}}$. It contains $T^{!}$ as a maximal
$F$-torus, and satisfies

$\displaystyle\Sigma\left(G^{!}[s]_{\epsilon[s],J_{s}},T^{!}\right)$
$\displaystyle=\left\\{\beta\in\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!}\right):B(\beta)^{-1}\beta|_{\mathfrak{a}_{M^{!}}}\in
R_{J_{s}}\right\\}$ (6.4.1)
$\displaystyle=\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!}\right),$

the last equality being a consequence of the (rather roundabout) definitions
above.

There is a natural surjection

$\displaystyle\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$
$\displaystyle\twoheadrightarrow\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$
$\displaystyle s$ $\displaystyle\mapsto t$

which can be interpreted in two ways.

  1. 1.

Let $s^{\flat}\in\tilde{M}^{\vee}$ be an elliptic semisimple element
determining $\mathbf{M}^{!}$. The map above is induced by

$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}\to
s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\widetilde{G_{J}}^{\vee}}^{\circ}$

using $Z_{\widetilde{G_{J}}^{\vee}}\supset Z_{\tilde{G}^{\vee}}$.

  2. 2.

Alternatively, let us assume for simplicity that
$\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ and $\tilde{M}=\prod_{i\in
I}\operatorname{GL}(n_{i},F)\times\widetilde{\mathrm{Sp}}(W^{\flat})$. With
the notation of Proposition 6.1.3, write

$G_{J}=\prod_{i\in\mathbf{I}}\operatorname{Sp}(W^{i})\times\operatorname{Sp}(W^{\natural})$

and assume that $M\hookrightarrow G_{J}$ is given by a map
$f:I\to\mathbf{I}\sqcup\\{\natural\\}$, so that

     * •

$\operatorname{Sp}(W^{\flat})\hookrightarrow\operatorname{Sp}(W^{\natural})$,

     * •

$\operatorname{GL}(n_{k})\hookrightarrow\operatorname{Sp}(W^{f(k)})$ for all
$k\in I$.

Then we obtain

$I=\bigsqcup_{i\in\mathbf{I}\sqcup\\{\natural\\}}I_{i},\quad
I_{i}:=f^{-1}(i).$

Given $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, corresponding to
$I=I^{\prime}\sqcup I^{\prime\prime}$, define $I^{\prime}_{i}:=I_{i}\cap
I^{\prime}$ and $I^{\prime\prime}_{i}:=I_{i}\cap I^{\prime\prime}$ for all
$i\in\mathbf{I}\sqcup\\{\natural\\}$. Then the map
$\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})\twoheadrightarrow\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$
maps $s$ to the element $t$ corresponding to the family of partitions
$\left(I_{i}=I^{\prime}_{i}\sqcup
I^{\prime\prime}_{i}\right)_{i\in\mathbf{I}\sqcup\\{\natural\\}}$.

The second interpretation leads immediately to the surjectivity of the map, as
well as

$z[s]=z[t]\;\in Z_{M^{!}}(F).$ (6.4.2)

######  Definition 6.4.3.

For $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ (resp.
$t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$), write $\exists J_{s}$
(resp. $\exists J_{t}$) to signify that there exists $J_{s}$ (resp. $J_{t}$)
mapping to $J$ via endoscopy. Let

${\mathcal{Z}}$${\left\\{s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G}):\exists
J_{s}\right\\}}$${\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})}$${\mathcal{Z}_{J}}$${\left\\{t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}}):\exists
J_{t}\right\\}}$${\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}}).}$${:=}$${\subset}$${:=}$${\subset}$

Moreover, we fix an elliptic semisimple element $s^{\flat}\in\tilde{M}^{\vee}$
to identify $\mathcal{Z}$ (resp. $\mathcal{Z}_{J}$) as a subset of
$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}$ (resp.
$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\widetilde{G_{J}}^{\vee}}^{\circ}$).

######  Lemma 6.4.4.

If $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ maps to
$t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$, then

$\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)=\Sigma\left(G_{J}^{!}[t]_{\epsilon[t]},T^{!},B_{J}\right).$

The proof is deferred to §6.4.3.

######  Lemma 6.4.5.

In the circumstance of Definition 6.4.3, $\mathcal{Z}$ equals the preimage of
$\mathcal{Z}_{J}$.

###### Proof.

Let $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. The condition
$s\in\mathcal{Z}$ is equivalent to that there exist linearly independent
$\beta_{1},\ldots,\beta_{r}\in\Sigma\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)$
such that $r=\dim\mathfrak{a}^{G}_{M}$ and
$\left\\{\xi^{*}(\beta_{i}|_{\mathfrak{a}_{M^{!}}})\right\\}_{i=1}^{r}$
generate $R_{J}$. These conditions force
$\beta_{i}\in\Sigma^{J}\left(G^{!}[s]_{\epsilon[s]},T^{!},B\right)$ for all
$i$.

Likewise, for $t\in\mathcal{E}_{\mathbf{M}^{!}}(\widetilde{G_{J}})$, the
condition $t\in\mathcal{Z}_{J}$ is equivalent to that there exist linearly
independent
$\beta_{1},\ldots,\beta_{r}\in\Sigma\left(G^{!}_{J}[t]_{\epsilon[t]},T^{!},B_{J}\right)$
such that $r=\dim\mathfrak{a}^{G_{J}}_{M}$ and
$\left\\{\xi^{*}(\beta_{i}|_{\mathfrak{a}_{M^{!}}})\right\\}_{i=1}^{r}$
generate $R_{J}$.

Since $\dim\mathfrak{a}^{G_{J}}_{M}=\dim\mathfrak{a}^{G}_{M}$, we conclude by
Lemma 6.4.4. ∎

Suppose that $s\in\mathcal{Z}$ maps to $t\in\mathcal{Z}_{J}$. View both
$Z_{{G^{!}_{J}[t]}^{\vee}}$ and $Z_{{G^{!}[s]}^{\vee}}$ as subgroups of
$Z_{{M^{!}}^{\vee}}$. Put $\mathcal{A}:=Z_{\tilde{G}^{\vee}}^{\circ}$, which
lies in their intersection, and define

$\displaystyle\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)$
$\displaystyle:=\frac{\left(Z_{{G^{!}_{J}[t]}^{\vee}}:\mathcal{A}\right)}{\left(Z_{{G^{!}[s]}^{\vee}}:\mathcal{A}\right)}\;\in\mathbb{Q}_{>0},$
(6.4.3)
$\displaystyle\left(Z_{{G^{!}[s]}^{\vee}}:Z_{{G^{!}_{J}[t]}^{\vee}}\right)$
$\displaystyle:=\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)^{-1}.$

######  Lemma 6.4.6 (Cf. [60, p.552 (7)]).

For $s\in\mathcal{Z}$ mapping to $t\in\mathcal{Z}_{J}$, we have

$i_{M^{!}}\left(\widetilde{G_{J}}:G^{!}_{J}[t]\right)\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)=\underbracket{\left(Z_{\widetilde{G_{J}}^{\vee}}^{\circ}:Z_{\tilde{G}^{\vee}}^{\circ}\right)}_{=1\;\text{in
our case}}i_{M^{!}}\left(\tilde{G},G^{!}[s]\right).$

###### Proof.

In the commutative diagram below of diagonalizable $\mathbb{C}$-groups, all
arrows are surjective with finite kernels:

${Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}}$${Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\widetilde{G_{J}}^{\vee}}^{\circ}}$${Z_{{M^{!}}^{\vee}}/\mathcal{A}}$${Z_{{M^{!}}^{\vee}}/Z_{{G^{!}_{J}[t]}^{\vee}}}$${Z_{{M^{!}}^{\vee}}/Z_{{G^{!}[s]^{\vee}}}.}$$\scriptstyle{\mathbf{A}}$$\scriptstyle{\mathbf{B}}$$\scriptstyle{\mathbf{C}}$$\scriptstyle{\mathbf{D}}$$\scriptstyle{\mathbf{E}}$

Hence

$|\operatorname{ker}(\mathbf{A})|\cdot|\operatorname{ker}(\mathbf{C})|=|\operatorname{ker}(\mathbf{B})|\cdot|\operatorname{ker}(\mathbf{D})|=\frac{|\operatorname{ker}(\mathbf{E}\mathbf{B})|}{|\operatorname{ker}(\mathbf{E})|}\cdot|\operatorname{ker}(\mathbf{D})|.$

On the other hand, one readily checks that

$\displaystyle
i_{M^{!}}\left(\widetilde{G_{J}}:G^{!}_{J}[t]\right)=|\operatorname{ker}(\mathbf{C})|^{-1},\quad\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)=\frac{|\operatorname{ker}(\mathbf{D})|}{|\operatorname{ker}(\mathbf{E})|},$
$\displaystyle\left(Z_{\widetilde{G_{J}}^{\vee}}^{\circ}:Z_{\tilde{G}^{\vee}}^{\circ}\right)=\underbracket{|\operatorname{ker}(\mathbf{A})|}_{=1\;\text{in
our case}},\quad
i_{M^{!}}\left(\tilde{G},G^{!}[s]\right)=|\operatorname{ker}(\mathbf{E}\mathbf{B})|^{-1}.$

The assertion follows at once. ∎

We keep the constants which equal to $1$ in order to maintain the analogy with
[60].

######  Lemma 6.4.7.

For $s\in\mathcal{Z}$ mapping to $t\in\mathcal{Z}_{J}$, we have the equality

$\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)\cdot\check{\mathcal{T}}^{s}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{G^{!}[s]}_{J_{s}}(\delta[s])\right)=\check{\mathcal{T}}^{t}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{G^{!}_{J}[t]}_{J_{t}}(\delta[t])\right)$

of germs, where $\delta\in SD_{\text{tr-
orb}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$.

The proof is deferred to §6.4.3. Let us show how it completes the program in
§6.3.2.

######  Theorem 6.4.8.

Under the Hypothesis 6.3.3, the program in §6.3.2 can be realized.
Specifically,
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)\mapsto\rho^{\tilde{G},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$
under the natural map for all $\tilde{M}\subset\tilde{G}$,
$(\mathbf{M}^{!},\delta)$, $\mathcal{O}^{!}$ and $J\in\mathcal{J}^{G}_{M}$.

###### Proof.

We may assume $J\neq J_{\max}$. Lemma 6.4.1 reduces the assertion to the case
of elliptic $\mathcal{O}^{!}$. By (O3) and Lemma 6.4.6, we have the equality
of germs

$\rho^{\tilde{G},\mathcal{E}}_{J}\left(\mathbf{M}^{!},\delta\right)=\sum_{s\in\mathcal{Z}}i_{M^{!}}(\tilde{G},G^{!}[s])\check{\mathcal{T}}^{s}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{G^{!}[s]}_{J_{s}}(\delta[s])\right)\\\
=\sum_{s\in\mathcal{Z}}\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)i_{M^{!}}\left(\widetilde{G_{J}}:G^{!}[t]\right)\check{\mathcal{T}}^{s}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{G^{!}[s]}_{J_{s}}(\delta[s])\right)$

where $s\mapsto t\in\mathcal{Z}_{J}$. Lemma 6.4.7 transforms the last
expression into

$\sum_{s\in\mathcal{Z}}i_{M^{!}}\left(\widetilde{G_{J}}:G^{!}[t]\right)\check{\mathcal{T}}^{t}_{\mathbf{M}^{!},\tilde{M}}\left(\sigma^{G_{J}^{!}[t]}_{J_{t}}(\delta[t])\right).$

In view of Lemma 6.4.5,
$\left(Z_{\widetilde{G_{J}}^{\vee}}^{\circ}:Z_{\tilde{G}^{\vee}}^{\circ}\right)=1$
is the common cardinality of fibers of
$\mathcal{Z}\twoheadrightarrow\mathcal{Z}_{J}$. All in all, we obtain the germ
$\rho^{\widetilde{G_{J}},\mathcal{E}}_{J}(\mathbf{M}^{!},\delta)$. ∎

####  6.4.3 Proof of crucial lemmas

Retain the notations of $J$, $s\mapsto t$, $J_{s}$, $J_{t}$, $\delta$ etc., as
in Lemma 6.4.7. By (6.4.2), we have

$\epsilon[s]=\epsilon[t],\quad\delta[s]=\delta[t],\quad
M^{!}_{\epsilon[s]}=M^{!}_{\epsilon}=M^{!}_{\epsilon[t]}.$

Define $\Xi_{\xi}$ to be the finite $F$-group scheme
$Z_{M^{!}}(\epsilon)/M^{!}_{\epsilon}$. Then $\Xi_{\epsilon}(F)$ acts on
$SD_{\mathrm{unip}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})^{\vee}$,
leaving $SD_{\text{tr-
unip}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})^{\vee}$
invariant. By the general theory of descent, the dual of stable descent maps

${SD_{\mathrm{unip}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})}$${SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!}[s])\otimes\operatorname{mes}(M^{!})^{\vee}}$${\delta[s]}$${SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(M^{!})^{\vee}}$${\delta}$$\scriptstyle{S\operatorname{desc}^{M^{!},*}_{\epsilon[s]}}$$\scriptstyle{S\operatorname{desc}^{M^{!},*}_{\epsilon}}$${\ni}$$\scriptstyle{\cdot
z[s]}$$\scriptstyle{\simeq}$${\ni}$

become injective after restriction to $\Xi_{\epsilon}(F)$-invariants. Note
that the diagram commutes.

This injectivity together with the “localization” of tr-orb spaces in [60,
V.4.3 Lemme (ii)] furnish a unique $\delta_{\epsilon}\in SD_{\text{tr-
unip}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})^{\vee}$
such that $\operatorname{Supp}(\delta_{\epsilon})$ is close to $1$ and

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

We assumed $A_{M^{!}_{\epsilon}}=A_{M^{!}}$. In proving Lemma 6.4.7, we may
also assume $A_{G^{!}[s]_{\epsilon[s]}}=A_{G^{!}[s]}$, since otherwise
$\mathcal{J}^{G^{!}[s]}_{M^{!}}(B)=\varnothing$, contradicting that
$s\in\mathcal{Z}$.

Under these assumptions, we have the bijection
$\mathcal{J}^{G^{!}[s]_{\epsilon[s]}}_{M^{!}_{\epsilon}}(B)\xrightarrow{1:1}\mathcal{J}^{G^{!}[s]}_{M^{!}}(B)$
such that $J_{s}\mapsto J$, and [60, V.4.6 (9)] entails

$\sigma^{G^{!}[s]}_{J_{s}}\left(\delta[s]\right)=S\operatorname{desc}^{M^{!},*}_{\epsilon[s]}\left(e^{G^{!}[s]}_{M^{!}}(\epsilon[s])\sigma^{G^{!}[s]_{\epsilon[s]}}_{J_{s}}(\delta_{\epsilon})\right),$

where $e^{G^{!}[s]}_{M^{!}}(\epsilon[s])$ is as in Definition 4.6.3. Analogous
constructions for $G_{J}$ lead to

$\sigma^{G_{J}^{!}[t]}_{J_{t}}\left(\delta[t]\right)=S\operatorname{desc}^{M^{!},*}_{\epsilon[t]}\left(e^{G_{J}^{!}[t]}_{M^{!}}(\epsilon[t])\sigma^{G_{J}^{!}[t]_{\epsilon[t]}}_{J_{t}}(\delta_{\epsilon})\right)$

where $J_{t}$ is identified with an element of
$\mathcal{J}^{G_{J}^{!}[t]_{\epsilon[t]}}_{M^{!}_{\epsilon}}(B_{J})$.

Note that the situation here is simpler than [60, V.5.6] since there are
neither simply connected covers nor $z$-extensions. The transition factors
$\lambda$ and the constants $c$ in [60, p.556] are thus trivial.

Define $\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)$ as in
(6.4.3). All in all, Lemma 6.4.7 reduces to the assertion:

$\begin{split}\text{There exists}\quad&\tau\in
SD_{\mathrm{unip}}(M^{!}_{\epsilon})\otimes\operatorname{mes}(M^{!}_{\epsilon})^{\vee}\quad\text{that
maps to both}\\\
&\sigma^{G^{!}[s]_{\epsilon[s]}}_{J_{s}}(\delta_{\epsilon})\quad\text{and}\\\
&e^{G^{!}_{J}[t]}_{M^{!}}(\epsilon[t])e^{G^{!}[s]}_{M^{!}}(\epsilon[s])^{-1}\left(Z_{{G^{!}_{J}[t]}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)^{-1}\sigma^{G^{!}_{J}[t]_{\epsilon[t]}}_{J_{t}}(\delta_{\epsilon})\end{split}$
(6.4.4)

under the obvious quotient maps. This is a variant of [60, V.5.6 (5)].

######  Definition 6.4.9.

Put

$\displaystyle G^{1}$ $\displaystyle:=G^{!}[s]_{\epsilon[s],J_{s}},$
$\displaystyle G^{2}$ $\displaystyle:=G^{!}_{J}[t]_{\epsilon[t]}.$

It has been shown in [60, II.3.3] that
$S\mathrm{Ann}^{G^{!}[s]_{\epsilon[s],J_{s}}}_{\mathrm{unip}}\subset
S\mathrm{Ann}^{G^{!}[s]_{\epsilon[s]}}_{\mathrm{unip}}$. By [60, V.3.2 (3) and
p.315] we obtain

$\left(Z_{\check{G}^{1}}^{\Gamma_{F}}:Z_{{G^{!}[s]_{\epsilon[s]}}^{\vee}}^{\Gamma_{F}}\right)^{-1}\sigma^{G^{1}}_{J_{s}}(\delta_{\epsilon})\mapsto\sigma^{G^{!}[s]_{\epsilon[s]}}_{J_{s}}(\delta_{\epsilon}).$

In view of (6.4.4), Lemma 6.4.7 is reduced to proving:

$\begin{split}\text{There exists}\quad&\tau\in