is actually surjective. The proofs below hinge upon some results and
terminologies from §3.9 and §4, but there will be no worry of no circularity.

######  Theorem 3.8.10.

The linear maps $\mathcal{T}^{\mathcal{E}}$ and
$\mathcal{T}^{\mathcal{E}}_{\operatorname{cusp}}$ in Definition 3.8.6 are
isomorphisms; their inverses are given by the $\mathcal{T}_{\mathcal{E}}$ in
(3.8.4).

When $F$ is Archimedean, they are isomorphisms between topological vector
spaces and induce

$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{I}^{\mathcal{E}}(\tilde{G},\tilde{K}),\quad\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{I}^{\mathcal{E}}_{\operatorname{cusp}}(\tilde{G},\tilde{K}).$

###### Proof.

For non-Archimedean $F$, this is [50, Theorem 5.3.1 and Corollary 6.3.3]. It
remains to establish the Archimedean case. Note that the injectivity of
$\mathcal{T}^{\mathcal{E}}$ and the assertion about inverses follow from
Proposition 3.8.8.

Assume that $F$ is Archimedean. The first step is is to reduce the assertion
about $\mathcal{T}^{\mathcal{E}}$ to that about
$\mathcal{T}^{\tilde{M},\mathcal{E}}_{\operatorname{cusp}}$ for all Levi
subgroup $M\subset G$, using (3.9.4). It suffices to show the surjectivity of
$\mathcal{T}^{\tilde{M},\mathcal{E}}_{\operatorname{cusp}}$, since Proposition
3.8.9 will then imply that
$\mathcal{T}^{\tilde{M},\mathcal{E}}_{\operatorname{cusp}}$ is an isometry
between Hilbert spaces. By induction, we reduce further to the case $M=G$.

Secondly, we may assume $F=\mathbb{R}$ since
$\mathcal{I}_{\operatorname{cusp},\dashrule[.7ex]{2222}{.4}}(\tilde{G})=\\{0\\}=\mathcal{I}^{\mathcal{E}}_{\operatorname{cusp}}(\tilde{G})$
when $F=\mathbb{C}$.

Fix $\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$ and let
$\varphi^{!}\in
S\mathcal{I}_{\operatorname{cusp}}(G^{!})\otimes\operatorname{mes}(G^{!})$.
Define
$\varphi:=\mathcal{T}_{\mathcal{E}}(\varphi):\Gamma_{\mathrm{reg},-}(\tilde{G})\to\mathbb{C}$
(Definition 3.8.7 and (3.8.4)), namely

$\varphi(\tilde{\gamma}):=|\mathfrak{D}(G_{\gamma},G;F)|^{-1}\sum_{\delta\leftrightarrow\gamma}\Delta(\delta,\tilde{\gamma})^{-1}\varphi^{!}(\delta).$

Note that $\varphi(z\tilde{\gamma})=z^{-1}\varphi(\tilde{\gamma})$, and
$\varphi(\tilde{\gamma})=0$ for non-elliptic $\tilde{\gamma}$. In view of the
inversion formula [50, Lemma 5.2.2] for transfer and its adjoint, we are
reduced to show that
$\varphi\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
since it will then follow that
$\mathcal{T}^{\mathcal{E}}(\varphi)=\varphi^{!}$ (zero on the components
$\neq\mathbf{G}^{!}$).

The subsequent arguments follow [60, I.4.13] and we give only a brief sketch.
Suppose that a diagram $(\epsilon,B^{!},T^{!},B,T,\eta)$ (Definition 3.6.1) is
given, such that $T^{!}$ and $T$ are elliptic maximal tori in $G^{!}$ and $G$,
respectively. The data of diagram yield an isomorphism $T\simeq T^{!}$ of
$F$-tori. If $X\in\mathfrak{t}_{\mathrm{reg}}(F)$, we write
$Y\in\mathfrak{t}^{!}_{\mathrm{reg}}(F)$ for its image, and we have
$\exp(Y)\epsilon\leftrightarrow\exp(X)\eta$ under endoscopy.

For $X\in\mathfrak{t}_{\mathrm{reg}}(F)$, define

$\Delta^{G_{\eta}}_{T}(X):=\prod_{\begin{subarray}{c}\alpha\in\Sigma(G_{\eta},T)\\\
\alpha>0\end{subarray}}\operatorname{sgn}(i\alpha(X))\;\in\\{\pm 1\\}.$

The positive roots above are relative to $B$; they are all imaginary since $T$
is elliptic. Likewise, we define
$\Delta^{G^{!}_{\epsilon}}_{T^{!}}(Y)\in\\{\pm 1\\}$

Choose $\tilde{\gamma}\in\mathbf{p}^{-1}(\gamma)$. As in [60, p.104], using
the characterization [25] of
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{G})$, it
boils down to showing that

$X\mapsto\frac{\Delta^{G_{\eta}}_{T}(X)}{\Delta^{G^{!}_{\epsilon}}_{T^{!}}(Y)}\cdot\Delta(\exp(Y)\epsilon,\exp(X)\tilde{\eta})^{-1}$
(3.8.5)

extends to a $C^{\infty}$-function in a neighborhood of $0$ in
$\mathfrak{t}(F)$.

The next step involves the semisimple descent for $\Delta$ in [52, §7], to be
reviewed in §§4.4–4.5. The upshot is that one has a relevant endoscopic datum
$\overline{\mathbf{G}^{!}_{\epsilon}}$ of $G_{\eta}$ together with the
relations

${G^{!}_{\epsilon}}$${\overline{G^{!}_{\epsilon}}}$${G_{\eta}.}$nonstandardendoscopyendoscopy

We refer to §4.3 for the nonstandard endoscopic datum in question. As a
consequence of this construction, we may factor $T^{!}\simeq T$ into
$T^{!}\simeq\overline{T^{!}}\simeq T$ where $\overline{T^{!}}$ is an elliptic
maximal torus of $\overline{G^{!}_{\epsilon}}$ related to $T^{!}$ by
nonstandard endoscopy.

Denote by $\overline{Y}\in\overline{\mathfrak{t}^{!}}(F)$ the image of $Y$.
The descent of transfer factors (see (4.4.1)) asserts

$\Delta(\exp(Y)\epsilon,\exp(X)\tilde{\eta})=\Delta_{\overline{\mathbf{G}^{!}_{\epsilon}},G_{\eta}}(\overline{Y},X)$

where $\Delta_{\overline{\mathbf{G}^{!}_{\epsilon}},G_{\eta}}$ is a transfer
factor on Lie algebras for the endoscopic datum
$\overline{\mathbf{G}^{!}_{\epsilon}}$ of $G_{\eta}$.

Observe that
$\Delta^{G^{!}_{\epsilon}}_{T^{!}}(Y)=c\Delta^{\overline{G^{!}_{\epsilon}}}_{\overline{T^{!}}}(\overline{Y})$,
where $c$ is a nonzero constant depending solely on the choice of Borel
subgroups, since passing from $G^{!}_{\epsilon}\supset T^{!}$ to
$\overline{G^{!}_{\epsilon}}\supset\overline{T^{!}}$ only rescales the roots.
All in all, (3.8.5) reduces to the corresponding equality for the endoscopic
datum $\overline{\mathbf{G}^{!}_{\epsilon}}$ of $G_{\eta}$ established in [60,
p.104].

Finally, to show the isomorphisms for $\tilde{K}\times\tilde{K}$-finite
versions in the Archimedean case, it suffices to treat the case for
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{G})\otimes\operatorname{mes}(G)$.
In view of Remark 2.8.4, the latter case follows from the fact [50, Theorem
7.4.3] that $\Delta(\phi,\tau)$ (Definition 3.9.1) has finite support in each
variable. ∎

Of course, Theorem 3.8.10 extends to all groups of metaplectic type.

###  3.9 Spectral transfer

The conventions about $\tilde{G}$, etc. are the same as §3.8.

Following [50, §6.2] and the formalism from §2.6, we define the endoscopic
counterparts of $\mathbb{S}^{1}\backslash T_{\operatorname{ell},-}(\tilde{G})$
and $\mathbb{S}^{1}\backslash T_{-}(\tilde{G})$ as

$\displaystyle T^{\mathcal{E}}_{\operatorname{ell}}(\tilde{G})$
$\displaystyle:=\bigsqcup_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}\Phi_{2,\mathrm{bdd}}(G^{!}),$
$\displaystyle T^{\mathcal{E}}(\tilde{G})$
$\displaystyle:=\bigsqcup_{M\in\mathcal{L}(M_{0})/W^{G}_{0}}T^{\mathcal{E}}_{\operatorname{ell}}(\tilde{M})/W^{G}(M).$

Let $f^{\mathcal{E}}\in\mathcal{I}^{\mathcal{E}}(\tilde{G})$ and $\phi\in
T^{\mathcal{E}}(\tilde{G})$. Take $M\in\mathcal{L}(M_{0})$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ such that
$\phi$ comes from $\phi_{M}\in\Phi_{2,\mathrm{bdd}}(M^{!})$. We define

$f^{\mathcal{E}}(\phi):=f^{M^{!}}(\phi_{M})$

where $f^{\mathcal{E}}\mapsto f^{M^{!}}$ is as in (3.8.1).

######  Definition 3.9.1 (Spectral transfer factors [50, Definition 6.2.4]).

For every $(\phi,\tau)\in
T^{\mathcal{E}}_{\operatorname{ell}}(\tilde{G})\times
T_{\operatorname{ell},-}(\tilde{G})$, the spectral transfer factor
$\Delta(\phi,\tau)$ is characterized by

$\displaystyle\Delta(\phi,z\tau)$ $\displaystyle=z\Delta(\phi,\tau),\quad
z\in\mathbb{S}^{1},$ $\displaystyle\mathcal{T}^{\mathcal{E}}(f)(\phi)$
$\displaystyle=\sum_{\tau\in
T_{\operatorname{ell},-}(\tilde{G})/\mathbb{S}^{1}}\Delta(\phi,\tau)f_{\tilde{G}}(\tau),\quad\phi\in
T^{\mathcal{E}}_{\operatorname{ell}}(\tilde{G}),$

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

For general $(\phi,\tau)\in T^{\mathcal{E}}(\tilde{G})\times
T_{-}(\tilde{G})$, suppose that they come from $\phi_{M}\in
T^{\mathcal{E}}_{\operatorname{ell}}(\tilde{M})$ and $\tau_{M}\in
T_{\operatorname{ell},-}(\tilde{M})$ for the same
$M\in\mathcal{L}(M_{0})/W^{G}_{0}$. We put

$\Delta(\phi,\tau):=\sum_{\tau_{M}^{\dagger}\in
W^{G}(M)\cdot\tau_{M}}\Delta^{\tilde{M}}(\phi_{M},\tau_{M}^{\dagger})$

where $\Delta^{\tilde{M}}$ denotes the spectral transfer factor defined for
$\tilde{M}$ in the obvious way. If there is no such $M$, we put
$\Delta(\phi,\tau):=0$.

######  Theorem 3.9.2 (Local character relation in the tempered case).

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

$\mathcal{T}^{\mathcal{E}}(f)(\phi)=\sum_{\tau\in
T_{-}(\tilde{G})/\mathbb{S}^{1}}\Delta(\phi,\tau)f_{\tilde{G}}(\tau).$

###### Proof.

The non-Archimedean case is the main result of [50]. Consider the case
$F=\mathbb{R}$ next.

Write $f_{\tilde{G}}(\pi):=\Theta_{\pi}(f_{\tilde{G}})$ for each
$\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})$. The local character relation of [50,
Theorem 7.4.3] yields a function
$\Delta_{\mathrm{spec}}:T^{\mathcal{E}}(\tilde{G})\times\Pi_{\mathrm{temp},-}(\tilde{G})\to\\{\pm
1\\}$ satisfying

$\mathcal{T}^{\mathcal{E}}(f)(\phi)=\sum_{\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})}\Delta_{\mathrm{spec}}(\phi,\pi)f_{\tilde{G}}(\pi)$
(3.9.1)

for all $\phi$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
and $\Delta_{\mathrm{spec}}(\cdot,\pi)$ (resp.
$\Delta_{\mathrm{spec}}(\phi,\cdot)$) has finite support for each $\pi$ (resp.
for each $\phi$). These properties characterize $\Delta_{\mathrm{spec}}$.

Choose a representative in $T_{-}(\tilde{G})$ for every class in
$T_{-}(\tilde{G})/\mathbb{S}^{1}$. By the theory of $R$-groups,
$T_{-}(\tilde{G})/\mathbb{S}^{1}$ gives a basis of
$D_{\mathrm{temp},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$:
specifically, we may write

$\Theta_{\tau}=\sum_{\pi}\mathrm{mult}(\tau:\pi)\Theta_{\pi},\quad\Theta_{\pi}=\sum_{\tau}\mathrm{mult}(\pi:\tau)\Theta_{\tau}$

for all $\tau\in T_{-}(\tilde{G})/\mathbb{S}^{1}$ with its representative and
$\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})$, for uniquely determined coefficients
$\mathrm{mult}(\cdots)$. Switching between bases, (3.9.1) uniquely determines

$\Delta^{\circ}:T^{\mathcal{E}}(\tilde{G})\times
T_{-}(\tilde{G})\to\mathbb{S}^{1}$

such that

$\displaystyle\Delta^{\circ}(\phi,z\tau)$
$\displaystyle=z\Delta^{\circ}(\phi,\tau),\quad z\in\mathbb{S}^{1},$
$\displaystyle\mathcal{T}^{\mathcal{E}}(f)(\phi)$ $\displaystyle=\sum_{\tau\in
T_{-}(\tilde{G})/\mathbb{S}^{1}}\Delta^{\circ}(\phi,\tau)f_{\tilde{G}}(\tau)$

for all $f$. Specifically,
$\Delta^{\circ}(\phi,\tau)=\sum_{\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})}\Delta_{\mathrm{spec}}(\phi,\pi)\mathrm{mult}(\pi:\tau)$
for all $\tau\in T_{-}(\tilde{G})/\mathbb{S}^{1}$.

Our goal is thus to show

$\Delta^{\circ}(\phi,\tau)=\Delta(\phi,\tau),\quad(\phi,\tau)\in
T^{\mathcal{E}}(\tilde{G})\times T_{-}(\tilde{G}).$ (3.9.2)

The first step is to reduce to the elliptic setting. We say
$\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})$ is _elliptic_ if $\Theta_{\pi}$ is
not identically zero on $\Gamma_{\mathrm{reg,ell}}(\tilde{G})$. In [50,
Definition 7.4.1] one defined a subset $\Pi_{2\uparrow,-}(\tilde{G})$ of
$\Pi_{\mathrm{temp},-}(\tilde{G})$. All $\pi\in\Pi_{2\uparrow,-}(\tilde{G})$
are elliptic. Indeed, by [50, Remark 7.5.1] $\pi$ is a non-degenerate limit of
discrete series in the sense of Knapp–Zuckerman, and such representations are
known to be elliptic; see loc. cit. for the relevant references.

By [46, Proposition 5.4.4],
$T_{\operatorname{ell},-}(\tilde{G})/\mathbb{S}^{1}$ gives a basis for the
space spanned by the characters of all elliptic $\pi$.

Let $\phi\in T^{\mathcal{E}}(\tilde{G})$. Take $M\in\mathcal{L}(M_{0})$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ such that
$\phi$ comes from $\phi_{M^{!}}\in\Phi_{\mathrm{bdd},2}(M^{!})$ up to
$W^{G}(M)$. Denote the factors relative to $\tilde{M}$ as
$\Delta^{\tilde{M}}$, etc. By [50, Theorem 7.4.3],

$\Delta_{\mathrm{spec}}(\phi,\pi)=\sum_{\pi_{M}\in\Pi_{2\uparrow,-}(\tilde{M})}\Delta^{\tilde{M}}_{\mathrm{spec}}(\phi_{M^{!}},\pi_{M})\mathrm{mult}(I_{\tilde{P}}(\pi_{M}):\pi)$

for all $\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})$, where $P\in\mathcal{P}(M)$
and $\mathrm{mult}(I_{\tilde{P}}(\pi_{M}):\pi)$ denotes the multiplicity of
$\pi$ in $I_{\tilde{P}}(\pi_{M})$.

We claim that

$\Delta^{\circ}(\phi,\tau)=\sum_{\begin{subarray}{c}\tau_{M}\in
T_{\operatorname{ell},-}(\tilde{M})\\\
\tau_{M}\mapsto\tau\end{subarray}}\Delta^{\tilde{M},\circ}(\phi_{M^{!}},\tau_{M}).$
(3.9.3)

Note that the sum is actually over an orbit $W^{G}(M)\tau_{M}$ if such a
$\tau_{M}$ exists. We may assume $\tau$ is the representative of some element
from $T_{-}(\tilde{G})/\mathbb{S}^{1}$; we also choose representatives in
$T_{-}(\tilde{M})$ for $T_{-}(\tilde{M})/\mathbb{S}^{1}$ compatibly with
induction. We have

$\Delta^{\circ}(\phi,\tau)=\sum_{\pi\in\Pi_{\mathrm{temp},-}(\tilde{G})}\Delta_{\mathrm{spec}}(\phi,\pi)\mathrm{mult}(\pi:\tau)\\\
=\sum_{\pi}\sum_{\pi_{M}\in\Pi_{2\uparrow,-}(\tilde{M})}\sum_{\tau_{M}\in
T_{\operatorname{ell},-}(\tilde{M})/\mathbb{S}^{1}}\\\
\cdot\mathrm{mult}(\tau_{M}:\pi_{\tilde{M}})\mathrm{mult}(I_{\tilde{P}}(\pi_{\tilde{M}}):\pi)\mathrm{mult}(\pi:\tau)\Delta^{\tilde{M},\circ}(\phi_{M^{!}},\tau_{M}).$

Given $\tau_{M}$, the sum over $(\pi,\pi_{M})$ of the triple products of
$\mathrm{mult}(\cdots)$ is readily seen to be $1$ if $\tau_{M}\mapsto\tau$,
otherwise it is zero. This proves (3.9.3).

Observe that (3.9.3) take the same form as the induction formula in Definition
3.9.1. In order to prove (3.9.2), we may assume $\phi\in
T^{\mathcal{E}}_{\operatorname{ell}}(\tilde{G})$, in which case
$\Delta^{\circ}(\phi,\tau)=0$ unless $\tau\in
T_{\operatorname{ell},-}(\tilde{G})$, and ditto for $\Delta(\phi,\tau)$. It is
thus legitimate to take
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{G})\otimes\operatorname{mes}(G)$
in the characterization of $\Delta^{\circ}(\phi,\cdot)$. All in all,
$\Delta^{\circ}(\phi,\cdot)$ and $\Delta(\phi,\cdot)$ have the same
characterization, whence (3.9.2).

The case $F=\mathbb{C}$ is even simpler because it reduces to the case of
split maximal tori via parabolic induction: see [50, §7.6]. ∎

We remark that by the discussions in [50, §6.2], Theorem 3.9.2 is equivalent
to saying that $\mathcal{T}^{\mathcal{E}}$ equals the composition of

$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)\xrightarrow[\sim]{\eqref{eqn:orbI-
gr}}\bigoplus_{M\in\mathcal{L}(M_{0})/W^{G}_{0}}\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{M})^{W^{G}(M)}\otimes\operatorname{mes}(M)\\\
\xrightarrow{\bigoplus_{M}\mathcal{T}^{\tilde{M},\mathcal{E}}_{\operatorname{cusp}}}\bigoplus_{M\in\mathcal{L}(M_{0})/W^{G}_{0}}\mathcal{I}^{\mathcal{E}}_{\operatorname{cusp}}(\tilde{M})^{W^{G}(M)}\xrightarrow[\sim]{\eqref{eqn:orbIEndo-
gr}}\mathcal{I}^{\mathcal{E}}(\tilde{G})$ (3.9.4)

where $\mathcal{T}^{\tilde{M},\mathcal{E}}_{\operatorname{cusp}}$ stands for
the collective geometric transfer for $\tilde{M}$ restricted to the cuspidal
subspace.

For the next result, recall from [50, §7.3] that when $F$ is Archimedean,
there is a canonical map $\lambda^{!}\mapsto\lambda$ from the space of
infinitesimal characters for $G^{!}$ to that for $\tilde{G}$, where
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$. The map is
surjective with finite fibers. Hence we have a correspondence between
infinitesimal characters of members of $\Phi_{\mathrm{bdd}}(G^{!})$ and those
of $T_{-}(\tilde{G})/\mathbb{S}^{1}$.

######  Proposition 3.9.3.

When $F$ is Archimedean, we have $\Delta(\phi,\tau)\neq 0$ only when $\phi$
and $\tau$ have matching infinitesimal characters.

###### Proof.

Upon reviewing the proof of Theorem 3.9.2, this reduces immediately to the
case of the factors $\Delta(\phi,\pi)$. In turn, the latter case reduces to
$(\phi,\pi)\in\Phi_{\mathrm{bdd},2}(G^{!})\times\Pi_{2\uparrow,-}(\tilde{G})$
which is established in [50, §7.3]. ∎

###  3.10 Geometric distributions and their transfer

Consider a local field $F$ of characteristic zero and a group of metaplectic
type $\mathbf{p}:\tilde{M}\to M(F)$. The notion of genuine and anti-genuine
distributions has been defined in §3.3. In this section, we will focus on
genuine distributions of geometric origin, defined as follows.

For any $\tilde{\gamma}\in\tilde{M}$, we denote by
$\gamma=\gamma_{\text{ss}}\gamma_{\text{u}}$ the Jordan decomposition of
$\gamma:=\mathbf{p}(\tilde{\gamma})$ in $M(F)$, where $\gamma_{\text{ss}}$ and
$\gamma_{\text{u}}$ are the semisimple and unipotent parts, respectively.

It makes sense to define the support of $D\in D_{-}(\tilde{M})$ as a
$M(F)$-invariant closed subset $\operatorname{Supp}(D)\subset\tilde{M}$.
Similarly, for a quasisplit connected reductive $F$-group $M^{!}$ and $D\in
SD(M^{!})$, one can define $\operatorname{Supp}(D)\subset M^{!}(F)$.

######  Definition 3.10.1.

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

$\displaystyle D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})$
$\displaystyle:=\left\\{D\in
D_{-}(\tilde{M}):\tilde{\gamma}\in\operatorname{Supp}(D)\implies\gamma_{\text{ss}}\in\mathcal{O}\right\\},$
$\displaystyle D_{\mathrm{geom},-}(\tilde{M})$
$\displaystyle:=\bigoplus_{\begin{subarray}{c}\mathcal{O}\subset M(F)\\\
\text{ss.\ conj.\
class}\end{subarray}}D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\;\subset
D_{-}(\tilde{M}),$ $\displaystyle D_{\mathrm{unip},-}(\tilde{M})$
$\displaystyle:=D_{\mathrm{geom},-}(\tilde{M},\\{1\\}).$

The orbital integrals $I^{\tilde{M}}(\tilde{\gamma},\cdot)$ along conjugacy
classes whose closure contains $\mathbf{p}^{-1}(\mathcal{O})$ yield elements
of
$D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$.

######  Definition 3.10.2.

Denote by $D_{\mathrm{orb},-}(\tilde{M})$ (resp.
$D_{\mathrm{orb},-}(\tilde{M},\mathcal{O})$) the subspace of
$D_{\mathrm{geom},-}(\tilde{M})$ (resp.
$D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})$) spanned by the orbital integrals
above.

Note that when $F$ is non-Archimedean,
$D_{\mathrm{geom},-}=D_{\mathrm{orb},-}$, but the inclusion is proper for
Archimedean $F$.

######  Definition 3.10.3.

Let $\mathcal{O}$ be a finite union of semisimple conjugacy classes in $M(F)$.
Denote by $\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})_{\mathcal{O},0}$
the subspace of elements
$f_{\tilde{M}}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})$ such that
there exists an invariant neighborhood $\mathcal{V}$ of $\mathcal{O}$ in
$M(F)$ satisfying $f_{\tilde{M}}(\tilde{\gamma})=0$ for all
$\gamma\in\mathcal{V}\cap M_{\mathrm{reg}}(F)$. Define

$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})_{\mathcal{O},\mathrm{loc}}:=\begin{cases}\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\big{/}\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})_{\mathcal{O},0},&F:\text{non-
Archimedean},\\\
\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\big{/}\overline{\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})_{\mathcal{O},0}},&F:\text{Archimedean}.\end{cases}$

######  Lemma 3.10.4.

For every $\mathcal{O}$ as above, $D_{-}(\tilde{M},\mathcal{O})$ is identified
with the dual of
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})_{\mathcal{O},\mathrm{loc}}$
(continuous if $F$ is Archimedean).

###### Proof.

Same as [60, I.5.1, I.5.2], based on standard techniques of harmonic analysis.
∎

Suppose that $\tilde{M}$ is a Levi subgroup of $\tilde{G}$. If $\mathcal{O}$
is a finite union of semisimple conjugacy classes in $M(F)$, we denote by
$\mathcal{O}^{G}$ the finite union of the semisimple conjugacy classes in
$G(F)$ intersecting $\mathcal{O}$. Then

${D_{-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}}$${D_{-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}}$${D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}}$${D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}}$${D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}}$${D_{\mathrm{geom},-}(\tilde{G},\mathcal{O}^{G})\otimes\operatorname{mes}(G)^{\vee}.}$$\scriptstyle{\operatorname{Ind}^{\tilde{G}}_{\tilde{M}}}$${\subset}$${\subset}$${\subset}$${\subset}$
(3.10.1)

We recall the definition of the stable avatars below. See also [60, I.5.4 —
I.5.5].

######  Definition 3.10.5.

For a quasisplit connected reductive $F$-group $M^{!}$, we define the
subspaces $SD_{\mathrm{geom}}(M^{!},\mathcal{O})\subset
SD_{\mathrm{geom}}(M^{!})$ of $SD(M^{!})$ accordingly; here $\mathcal{O}$ is a
finite union of stable semisimple conjugacy classes in $M^{!}(F)$. We set
$SD_{\mathrm{unip}}(M^{!}):=SD_{\mathrm{geom}}(M^{!},\\{1\\})$.

We also define $SD_{\mathrm{orb}}(M^{!}):=SD_{\mathrm{geom}}(M^{!})\cap
D_{\mathrm{orb}}(M^{!})$.

As before, we define $S\mathcal{I}(M^{!})_{\mathcal{O},0}\subset
S\mathcal{I}(M^{!})$ to be the subspace of elements vanishing in an invariant
neighborhood of $\mathcal{O}$ in $M^{!}(F)$, and then define

$S\mathcal{I}(M^{!})_{\mathcal{O},\mathrm{loc}}:=\begin{cases}S\mathcal{I}(M^{!})\big{/}S\mathcal{I}(M^{!})_{\mathcal{O},0},&F:\text{non-
Archimedean},\\\
S\mathcal{I}(M^{!})\big{/}\overline{S\mathcal{I}(M^{!})_{\mathcal{O},0}},&F:\text{Archimedean}.\end{cases}$

The analogue of Lemma 3.10.4 holds here, namely
$SD_{\mathrm{geom}}(M^{!},\mathcal{O})$ is the dual of
$S\mathcal{I}(M^{!})_{\mathcal{O},\mathrm{loc}}$. See [60, I.5.4–I.5.5]. When
$M^{!}$ is a Levi subgroup of $G^{!}$, we also have the map
$\operatorname{Ind}^{G^{!}}_{M^{!}}:SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}\to
SD_{\mathrm{geom}}(G^{!})\otimes\operatorname{mes}(G^{!})^{\vee}$, also
denoted by $\delta\mapsto\delta^{G^{!}}$.

######  Theorem 3.10.6 (Cf. [60, I.5.7 Proposition]).

For any stable semisimple conjugacy class $\mathcal{O}$ in $G(F)$ and
$\mathbf{G}^{!}\in\mathcal{E}(\tilde{G})$, write

$\mathcal{O}_{G^{!}}:=\left\\{\delta\in
G^{!}(F)_{\mathrm{ss}}:\exists\gamma\in\mathcal{O},\;\delta\leftrightarrow\gamma\right\\}.$

Then $\mathcal{O}_{G^{!}}$ is a finite union of stable conjugacy classes in
$G^{!}(F)$. The map $\check{\mathcal{T}}^{\mathcal{E}}$ in Definition 3.8.6
restricts to a surjection

$\bigoplus_{\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}SD_{\mathrm{geom}}(G^{!},\mathcal{O}_{G^{!}})\otimes\operatorname{mes}(G^{!})^{\vee}\to
D_{\mathrm{geom},-}(\tilde{G},\mathcal{O})\otimes\operatorname{mes}(G)^{\vee},$

whose kernel is spanned by elements of the form

$\delta[s]^{G^{!}[s]}-\delta[t]^{G^{!}[t]},\quad\delta\in
SD_{\mathrm{geom}}(M^{!},\mathcal{O}_{M^{!}})\otimes\operatorname{mes}(M^{!})^{\vee},$

where $M\subset G$ is a Levi subgroup,
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$s,t\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$.

The same results holds if $\tilde{G}$ is replaced by any group of metaplectic
type.

###### Proof.

Since the morphism $\Sigma_{\mathrm{ss}}(G^{!})\to\Sigma_{\mathrm{ss}}(G)$ in
Definition 3.4.3 and the subsequent discussions has finite fibers,
$\mathcal{O}_{G^{!}}$ is a finite union of stable classes. The remaining
arguments are the same as in [60, I.5.8, I.5.9]. We shall omit the details and
give only the basic ingredients below.

  * •

Theorem 3.8.10, which describes the image of $\mathcal{T}^{\mathcal{E}}$.

  * •

The adjoint transfer discussed in §3.8, which expresses
$f_{\tilde{G}}(\tilde{\gamma})$ in terms of the values of
$\mathcal{T}^{\mathcal{E}}(f)$.

Our case is actually simpler than loc. cit., since the relevance of endoscopic
data and the endoscopic outer automorphisms on $G^{!}$ do not arise here (see
Remark 3.5.4). In contrast, the twists $\delta[s]$, $\delta[t]$ of $\delta$
are new: the same phenomenon of twists appeared in Proposition 3.4.9.

Finally, the extension to groups of metaplectic type is straightforward. ∎

###  3.11 Spherical fundamental lemma

Consider the local metaplectic covering $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$
with $(W,\langle\cdot|\cdot\rangle)$ and $\psi$ chosen. In this section, we
assume that $F$ is non-Archimedean with residual characteristic $p>2$, and fix
a self-dual $\mathfrak{o}_{F}$-lattice $L\subset W$ with respect to
$\psi\circ\langle\cdot|\cdot\rangle$.

Regard the resulting hyperspecial subgroup $K:=G(\mathfrak{o}_{F})$ as an open
compact subgroup of $\tilde{G}$ as in §3.3. Fix the Haar measure on $G(F)$
with $\operatorname{mes}(K)=1$, and take the corresponding measure on
$\tilde{G}$ by (2.2.2). Similarly, for each
$\mathbf{G}^{!}\in\mathcal{E}(\tilde{G})$, we use the Haar measure with
$\operatorname{mes}(K^{!})=1$ for some (equivalently, any) hyperspecial
subgroup $K^{!}\subset G^{!}(F)$. The lines $\operatorname{mes}(G)$ and
$\operatorname{mes}(G^{!})$ are thus trivialized.

We call this the _unramified situation_.

Consider the anti-genuine spherical Hecke algebra
$\mathcal{H}_{\dashrule[.7ex]{2222}{.4}}(K\backslash\tilde{G}/K)$ with unit
$f_{K}$. For each $\mathbf{G}^{!}\in\mathcal{E}(\tilde{G})$, we have the
spherical Hecke algebra $\mathcal{H}(K^{!}\backslash G^{!}(F)/K^{!})$ with
unit $\mathbf{1}_{K^{!}}$. The metaplectic Satake isomorphism (see eg. [56,
Theorem 3.1.3]) reads

$\mathcal{H}_{\dashrule[.7ex]{2222}{.4}}(K\backslash\tilde{G}/K)\simeq\mathbb{C}\left[\tilde{G}^{\vee}/\\!/\tilde{G}^{\vee}\right]$

between $\mathbb{C}$-algebras, where $\tilde{G}^{\vee}/\\!/\tilde{G}^{\vee}$
is the categorical quotient of $\tilde{G}^{\vee}$ under adjoint action, and
$\mathbb{C}[\cdots]$ means the algebra of regular functions. Similarly, the
classical Satake isomorphism reads

$\mathcal{H}(K^{!}\backslash
G^{!}(F)/K^{!})\simeq\mathbb{C}\left[\check{G}^{!}/\\!/\check{G}^{!}\right].$

The inclusion $\check{G}^{!}\hookrightarrow\tilde{G}^{\vee}$ being canonical
up to $\tilde{G}^{\vee}$-conjugacy, it induces a canonical homomorphism

$b_{\mathbf{G}^{!},\tilde{G}}:\mathcal{H}_{\dashrule[.7ex]{2222}{.4}}(K\backslash\tilde{G}/K)\to\mathcal{H}(K^{!}\backslash
G^{!}(F)/K^{!})$

between $\mathbb{C}$-algebras, In particular,
$b_{\mathbf{G}^{!},\tilde{G}}(f_{K})=\mathbf{1}_{K^{!}}$.

######  Theorem 3.11.1 (Spherical fundamental lemma).

In the unramified situation, the diagram

${\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})}$${S\mathcal{I}(G^{!})}$${\mathcal{H}_{\dashrule[.7ex]{2222}{.4}}(K\backslash\tilde{G}/K)}$${\mathcal{H}(K^{!}\backslash
G^{!}(F)/K^{!})}$$\scriptstyle{\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}}$$\scriptstyle{b_{\mathbf{G}^{!},\tilde{G}}}$

commutes for each $\mathbf{G}^{!}\in\mathcal{E}(\tilde{G})$.

###### Proof.

The case when $\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$
is the Main Theorem of [56], and the general case follows by parabolic descent
as usual. Note that in loc. cit., one assumes that $\psi$ has conductor
$\mathfrak{o}_{F}$ and $L$ is a self-dual lattice with respect to
$\langle\cdot|\cdot\rangle$. We can also reduce to that case by replacing
$\psi$ (resp. $\langle\cdot|\cdot\rangle$) by $\psi_{a}:x\mapsto ax$ (resp.
$a^{-1}\langle\cdot|\cdot\rangle$) for some $a\in F^{\times}$, so that
$\psi_{a}$ has conductor $\mathfrak{o}_{F}$. Then $L$ is self-dual with
respect to $a^{-1}\langle\cdot|\cdot\rangle$. This is legitimate since the
following depend only on $\psi\circ\langle\cdot|\cdot\rangle$:

  * •

the construction of $\tilde{G}$,

  * •

the transfer factors, and

  * •

the splitting of $\tilde{G}$ over $K$.

Details can be found in [52]; for instance, the last item can be checked using
the character formula for $\omega_{\psi}$ in [52, Lemme 4.20]. ∎

All the constructions and results above extend to groups of metaplectic type,
in the routine manner.

##  Chapter 4 Harish-Chandra descent

The Harish-Chandra descent, also known as semisimple descent, is one of the
most important tools in this work. We will review the descent of test
functions, endoscopic data and transfer factors around a semisimple element of
the metaplectic group, and set up the relevant notations. As a natural
consequence, the nonstandard endoscopy on Lie algebras enters into the
picture. These results are extracted from [52, 49].

These materials are supplemented with a discussion on diagrams in §4.5. The
setting with a Levi subgroup, addressed in §4.6, will play a vital role when
we study various “weighted” objects.

###  4.1 Harmonic analysis

Let $F$ be a local field of characteristic zero. The following theory works
for general coverings $\mathbf{p}:\tilde{G}\to G(F)$ with
$\operatorname{ker}(\mathbf{p})=\bm{\mu}_{m}$; see [46, §4].

Let $\eta\in G(F)_{\mathrm{ss}}$ and $\tilde{\eta}\in\mathbf{p}^{-1}(\eta)$.
Consider an open neighborhood $\mathcal{U}^{\flat}$ of
$0\in\mathfrak{g}_{\eta}(F)$ subject to the following conditions:

  * •

$X\in\mathfrak{g}_{\eta}(F)$ lies in $\mathcal{U}^{\flat}$ if and only if its
semisimple part does;

  * •

$\mathcal{U}^{\flat}$ is invariant under conjugation by $Z_{G}(\eta)(F)$;

  * •

the exponential map is defined on $\mathcal{U}^{\flat}$ and induces a
homeomorphism onto its image.

Specifically, there are two exponential maps fitting into a commutative
diagram

${\mathcal{U}^{\flat}}$${\widetilde{G_{\eta}}}$${G_{\eta}(F)}$$\scriptstyle{\exp}$$\scriptstyle{\exp}$$\scriptstyle{\mathbf{p}}$
(4.1.1)

and both will be denoted by $\exp$. Let $\tilde{\mathcal{U}}\subset\tilde{G}$
be the open subset consisting of elements which are conjugate to an element in
$\mathbf{p}^{-1}(\eta)\cdot\exp(\mathcal{U}^{\flat})$.

Define $\omega:Z_{G}(\eta)(F)\to\bm{\mu}_{m}$ by
$\omega(g)=\tilde{\eta}\tilde{g}\tilde{\eta}^{-1}\tilde{g}^{-1}$ by choosing
liftings. We have
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{\mathcal{U}})\subset\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$
and
$\mathcal{I}(\mathcal{U}^{\flat})_{\omega}\subset\mathcal{I}(\mathfrak{g}_{\eta})_{\omega}$,
where $(\cdots)_{\omega}$ is the space of $\omega$-coinvariants under
$Z_{G}(\eta)(F)$-action; see [46, §4.1].

Each $f^{\flat}\in\mathcal{I}(\mathfrak{g}_{\eta})_{\omega}$ can be evaluated
on $X\in\mathfrak{g}_{\eta,\mathrm{reg}}(F)$ once the Haar measures are
chosen, namely by taking orbital integrals twisted by the character
$\omega|_{G_{\eta}(F)}$. The value will be denoted by
$f^{\flat}_{G_{\eta},\omega}(X)$.

Following the terminology of [60, I.4.1], the _Harish-Chandra descent_ of test
functions is the canonical linear isomorphism

$\operatorname{desc}^{\tilde{G}}_{\tilde{\eta}}:\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{\mathcal{U}})\otimes\operatorname{mes}(G)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{I}(\mathcal{U}^{\flat})_{\omega}\otimes\operatorname{mes}(G_{\eta})$

characterized by

$f\mapsto f^{\flat}\iff\forall
X\in\mathcal{U}^{\flat}\cap\mathfrak{g}_{\eta,\mathrm{reg}}(F),\;f_{\tilde{G}}(\tilde{\eta}\exp
X)=f^{\flat}_{G_{\eta},\omega}(X).$

Moreover, if $\eta$ is elliptic in $G(F)$, it restricts to an isomorphism

$\mathcal{I}_{\dashrule[.7ex]{2222}{.4},\operatorname{cusp}}(\tilde{\mathcal{U}})\otimes\operatorname{mes}(G)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{I}_{\operatorname{cusp}}(\mathcal{U}^{\flat})_{\omega}\otimes\operatorname{mes}(G_{\eta})$

where $\mathcal{I}_{\operatorname{cusp}}(\cdots)_{\omega}$ is defined via
parabolic descent, as usual.

In the upcoming applications, $\mathbf{p}:\tilde{G}\to G(F)$ will be of
metaplectic type, therefore $\omega$ is trivial and $Z_{G}(\eta)=G_{\eta}$.

Consider now the stable version. Let $G^{!}$ be a quasisplit $F$-group. Let
$\epsilon\in G^{!}(F)_{\mathrm{ss}}$. Up to stable conjugacy, we may assume
$G^{!}_{\epsilon}$ is quasisplit by [40, Lemma 3.3]. Define the finite
$F$-group scheme

$\Xi_{\epsilon}=\Xi^{G^{!}}_{\epsilon}:=Z_{G^{!}}(\epsilon)/G^{!}_{\epsilon}.$

Consider an open neighborhood $\mathcal{U}^{\flat}$ of
$0\in\mathfrak{g}^{!}_{\epsilon}(F)$ subject to the conditions below:

  * •

$X\in\mathfrak{g}^{!}_{\epsilon}(F)$ lies in $\mathcal{U}^{\flat}$ if and only
if its semisimple part does;

  * •

$\mathcal{U}^{\flat}$ is invariant under stable conjugation and
$Z_{G^{!}}(\epsilon)(F)$-action;

  * •

the exponential map is defined on $\mathcal{U}^{\flat}$ and induces a
homeomorphism onto its image.

Define $\mathcal{U}\subset G^{!}(F)$ to be the open subset consisting of
elements which are conjugate to an element in
$\epsilon\cdot\exp(\mathcal{U}^{\flat})$. The stable Harish-Chandra descent is
a canonical linear isomorphism

$S\operatorname{desc}^{G^{!}}_{\epsilon}:S\mathcal{I}(\mathcal{U})\otimes\operatorname{mes}(G^{!})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}S\mathcal{I}(\mathcal{U}^{\flat})_{\Xi_{\epsilon}(F)}\otimes\operatorname{mes}(G^{!}_{\epsilon})$

characterized in terms of stable orbital integrals on Lie algebras, namely

$f\mapsto f^{\flat}\iff\forall
Y\in\mathcal{U}^{\flat}\cap\mathfrak{g}^{!}_{\epsilon,\mathrm{reg}}(F),\;f^{G^{!}}(\epsilon\exp(Y))=f^{\flat,G^{!}_{\epsilon}}(Y).$

Here $(\cdots)_{\Xi_{\epsilon}(F)}$ stands for the space of coinvariants under
$\Xi_{\epsilon}(F)$-action.

When $\epsilon$ is elliptic in $G^{!}(F)$, the map restricts to
$S\mathcal{I}_{\operatorname{cusp}}(\mathcal{U})\otimes\operatorname{mes}(G^{!})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}S\mathcal{I}_{\operatorname{cusp}}(\mathcal{U}^{\flat})_{\Xi_{\epsilon}(F)}\otimes\operatorname{mes}(G^{!}_{\epsilon})$.

These results are established in [60, I.4.8]. By taking transposes, we obtain

$\displaystyle\operatorname{desc}^{\tilde{G},*}_{\tilde{\eta}}:D(\mathcal{U}^{\flat})^{\omega}\otimes\operatorname{mes}(G_{\eta})^{\vee}\to
D_{-}(\tilde{\mathcal{U}})\otimes\operatorname{mes}(G)^{\vee},$ $\displaystyle
S\operatorname{desc}^{G^{!},*}_{\epsilon}:SD(\mathcal{U}^{\flat})^{\Xi_{\epsilon}(F)}\otimes\operatorname{mes}(G^{!}_{\epsilon})^{\vee}\to
D(\mathcal{U})\otimes\operatorname{mes}(G^{!})^{\vee},$

where $(\cdots)^{\omega}$ (resp. $(\cdots)^{\Xi_{\epsilon}(F)}$) stands for
the $\omega$-eigenspace under $Z_{G}(\eta)(F)$ (resp. the space of
$\Xi_{\epsilon}(F)$-invariants).

###  4.2 Endoscopy for linear groups

The formalism below will be applied to classical groups exclusively.

####  4.2.1 Endoscopic data

To begin with, let $F$ be a local or global field, and let $L$ be a connected
reductive $F$-group. Fix a quasisplit inner twist
$L_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}L^{*}_{\overline{F}}$
to define its $L$-group ${}^{\mathrm{L}}L={}^{\mathrm{L}}L^{*}$, which is
endowed with a $\Gamma_{F}$-stable pinning $\check{T}$, $\check{B}$ and so on.
We shall work with Weil forms of $L$-groups.

An _endoscopic datum_ of $L$ is a quadruplet
$\mathbf{H}=(H,\mathcal{H},s,\hat{\xi})$, or simply $(H,\mathcal{H},s)$ if no
confusion arises, such that

  * •

$H$ is a quasisplit $F$-group;

  * •

$\mathcal{H}$ sits in a split topological extension of groups

$1\to\check{H}\to\mathcal{H}\to\mathrm{W}_{F}\to 1$

such that the corresponding homomorphism
$\mathrm{W}_{F}\to\mathrm{Out}(\check{H})$ coincides with the one arising from
$H$;

  * •

$s\in\check{L}$ est semi-simple;

  * •

$\hat{\xi}:\mathcal{H}\to{}^{\mathrm{L}}L$ is an $L$-homomorphism such that

    * –

there exists a $1$-cocycle $a:\mathrm{W}_{F}\to Z_{\check{L}}$ satisfying
$\operatorname{Ad}(s)\circ\hat{\xi}(x)=a(w(x))\hat{\xi}(x)$ for all
$x\in\mathcal{H}$, where $w(x)$ denotes its projection in $\mathrm{W}_{F}$;

    * –

$\hat{\xi}$ induces an isomorphism
$\check{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}Z_{\check{L}}(s)^{\circ}$;

    * –

the cohomology class of $a$ is trivial (resp. locally trivial) if $F$ is local
(resp. global).

The group $H$ is called an endoscopic group of $L$. We say $\mathbf{H}$ is
elliptic if $\hat{\xi}\left(Z_{\check{H}}^{\Gamma_{F},\circ}\right)\subset
Z_{\check{L}}$.

An isomorphism between endoscopic data $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$
is an element $\hat{r}\in\check{L}$ such that

$\operatorname{Ad}(\hat{r})\hat{\xi}\left(\mathcal{H}_{1}\right)=\hat{\xi}^{\prime}\left(\mathcal{H}_{2}\right),\quad\operatorname{Ad}(\hat{r})(s_{1})\in
s_{2}Z_{\check{L}}.$

Isomorphic endoscopic data are often called equivalent. Denote the set of
equivalence classes endoscopic data of $L$ by $\mathcal{E}(L)$, and the subset
of elliptic ones as $\mathcal{E}_{\operatorname{ell}}(L)$.

Every auto-equivalence of $\mathbf{H}$ induces an outer automorphism of $H$.
We denote the resulting subgroup of $\mathrm{Out}(H)$ as
$\mathrm{Out}(\mathbf{H})$.

Given $\mathbf{H}\in\mathcal{E}(L)$, adjusting by isomorphisms, we may assume
that $s\in\check{T}$ and

$(\check{T}_{H},\check{B}_{H}):=(\hat{\xi}^{-1}(\check{T}),\hat{\xi}^{-1}(\check{B}))$

is the Borel pair coming with $\check{H}$. Hence $\hat{\xi}$ induces
$\check{T}_{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\check{T}$. Denote
by $W^{L}$ the absolute Weyl group of $L$. It turns out that there is a
$1$-cocycle $\omega_{H}:\Gamma_{F}\to W^{L}$ such that
$\hat{\xi}\circ\sigma(\hat{t})=\omega_{H}(\sigma)\sigma\hat{\xi}(\hat{t})$ for
all $\sigma\in\Gamma_{F}$ and $\hat{t}\in\check{T}_{H}$.

The obstruction $\omega_{H}$ for the $\Gamma_{F}$-equivariance of
$\hat{\xi}|_{\check{T}_{H}}$ and its dual
$\xi:T_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{H,\overline{F}}$
is hence $W^{L}$-valued. Here are some consequences:

  1. 1.

$\hat{\xi}$ restricts to a $\Gamma_{F}$-equivariant inclusion
$Z_{\check{L}}\hookrightarrow Z_{\check{H}}$;

  2. 2.

dually, $\xi$ induces an inclusion
$\mathfrak{a}_{L}\hookrightarrow\mathfrak{a}_{H}$.

In fact, $\mathbf{H}$ is elliptic if and only if
$Z_{\check{L}}^{\Gamma_{F},\circ}\simeq Z_{\check{H}}^{\Gamma_{F},\circ}$, or
equivalently
$\mathfrak{a}_{L}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{a}_{H}$.

When $F$ is a non-Archimedean local field, $\mathbf{H}$ is said to be
_unramified_ if

  * •

$L$ is unramified and the chosen inner twist
$L_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}L^{*}_{\overline{F}}$
is $\mathrm{id}$;

  * •

$H$ is unramified and $\hat{\xi}(\mathcal{H})\supset
Z_{\check{L}}(s)^{\circ}\rtimes I_{F}$.

####  4.2.2 Transfer

Given $\mathbf{H}$, we have a correspondence of stable semisimple conjugacy
classes between $L$ and $H$, or between stable semisimple classes in their Lie
algebras.

######  Definition 4.2.1.

An endoscopic datum $\mathbf{H}$ of $L$ is called _relevant_ if there exists
some $\delta\in\Sigma_{\mathrm{reg}}(H)$ that corresponds to some
$\gamma\in\Gamma_{\mathrm{reg}}(L)$.

When $\mathbf{H}$ is relevant, the $L$-regular locus $H_{L\text{-reg}}$ is
open and dense in $H$.

######  Remark 4.2.2.

In all the upcoming applications, $L$ and $L^{*}$ are classical groups and the
endoscopic datum $\mathbf{H}$ is endowed with an isomorphism
$\mathcal{H}\simeq{}^{\mathrm{L}}H$, which will not be altered. As a
consequence, we do not need $z$-extensions of $H$. See [70].

Assume $F$ to be local of characteristic in what follows. There is a notion of
_absolute transfer factors_

$\Delta=\Delta_{\mathbf{H},L}:\Sigma_{\mathrm{reg}}(H)\times\Gamma_{\mathrm{reg}}(L)\to\mathbb{C}$

such that $\Delta(\delta,\gamma)\neq 0$ only when
$\delta\leftrightarrow\gamma$. It suffices to consider transfer factors for
relevant $\mathbf{H}$.

In the original definition of Langlands–Shelstad [43], $\Delta_{\mathbf{H},L}$
is only canonical up to $\mathbb{C}^{\times}$. Let us take a relevant
$\mathbf{H}$. Once a choice of $\Delta_{\mathbf{H},L}$ is made, there is the
transfer of orbital integrals

$\mathcal{T}_{\mathbf{H},L}:\mathcal{I}(L)\otimes\operatorname{mes}(L)\to
S\mathcal{I}(H)\otimes\operatorname{mes}(H)$

characterized by matching orbital integrals

$\mathcal{T}_{\mathbf{H},L}(f_{L})(\delta)=\sum_{\gamma\leftrightarrow\delta}\Delta_{\mathbf{H},L}(\delta,\gamma)f_{L}(\gamma),\quad\delta\in\Sigma_{\mathrm{reg}}(H).$

The transfer is continuous for Archimedean $F$.

Dualization yields the transfer of distributions

$\check{\mathcal{T}}_{\mathbf{H},L}:SD(H)\otimes\operatorname{mes}(H)^{\vee}\to
D(L)\otimes\operatorname{mes}(L)^{\vee}.$

There are also Lie algebra versions, namely

$\displaystyle\mathcal{T}_{\mathbf{H},L}:\mathcal{I}(\mathfrak{l})\otimes\operatorname{mes}(L)$
$\displaystyle\to S\mathcal{I}(\mathfrak{h})\otimes\operatorname{mes}(H),$
$\displaystyle\check{\mathcal{T}}_{\mathbf{H},L}:SD(\mathfrak{h})\otimes\operatorname{mes}(H)^{\vee}$
$\displaystyle\to D(\mathfrak{l})\otimes\operatorname{mes}(L)^{\vee}.$

####  4.2.3 Levi subgroups

Suppose that $R$ is a Levi subgroup of $L$. The following construction is due
to Arthur [8]. Consider an endoscopic datum
$\mathbf{R}^{!}=(R^{!},\mathcal{R}^{!},s^{\flat},\hat{\xi})$ of $R$; our
notational scheme here is changed in order to conform to §3.4. Let $s\in
s^{\flat}Z_{\check{R}}^{\Gamma_{F}}/Z_{\check{L}}^{\Gamma_{F}}$. Put

$\displaystyle L^{!}[s]^{\vee}$ $\displaystyle:=Z_{\check{L}}(s)^{\circ},$
$\displaystyle\mathcal{L}^{!}[s]$
$\displaystyle:=L^{!}[s]^{\vee}\cdot\hat{\xi}(\mathcal{R}^{!}),$
$\displaystyle\hat{\xi}[s]$
$\displaystyle:\mathcal{L}^{!}[s]\hookrightarrow{}^{\mathrm{L}}L\quad\text{is
the inclusion}.$

This gives an endoscopic datum
$\mathbf{L}^{!}[s]=(L^{!}[s],\mathcal{L}^{!}[s],s,\hat{\xi}[s])$ for $L$.
Moreover, it is unramified whenever $\mathbf{R}^{!}$ is. Define

$\mathcal{E}_{\mathbf{R}^{!}}(L):=\\{s\in
s^{\flat}Z_{\check{R}}^{\Gamma_{F}}/Z_{\check{L}}^{\Gamma_{F}}:\mathbf{L}^{!}[s]\;\text{is
elliptic}\\}.$

This is a finite set by [8, §4].

Given $s\in\mathcal{E}_{\mathbf{R}^{!}}(L)$, one can view $R^{!}$ as a Levi
subgroup of $L^{!}[s]$. There is a canonical homomorphism
$W^{L^{!}[s]}(R^{!})\hookrightarrow W^{L}(R)$, for which
$\mathfrak{a}_{R}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{a}_{R^{!}}$
is equivariant. In this case, one can also define the constant

$i_{R^{!}}(L,L^{!}[s]):=\begin{cases}\frac{\left(Z_{\check{R}^{!}}^{\Gamma_{F}}:Z_{\check{R}}^{\Gamma_{F}}\right)}{\left(Z_{L^{!}[s]^{\vee}}^{\Gamma_{F}}:Z_{\check{L}}^{\Gamma_{F}}\right)},&s\in\mathcal{E}_{\mathbf{R}^{!}}(L),\\\
0,&\text{otherwise},\end{cases}$ (4.2.1)

for all $s\in s^{\flat}Z_{\check{R}}^{\Gamma_{F}}/Z_{\check{L}}^{\Gamma_{F}}$.

When $\mathbf{R}^{!}$ is relevant, one has the open dense subvariety
$R^{!}_{L-\text{reg}}$ of $L$-regular elements in $R^{!}$, and similarly on
Lie algebras. Every choice of $\Delta_{\mathbf{R}^{!},R}$ determines one for
$\Delta_{\mathbf{L}^{!}[s],L}$, characterized by

$\Delta_{\mathbf{L}^{!}[s],L}(\delta,\gamma)=\Delta_{\mathbf{R}^{!},R}(\delta,\gamma)$

if $\delta\in R^{!}_{L\text{-reg}}(F)$, $\gamma\in R_{\text{reg}}(F)$ and
$\delta\leftrightarrow\gamma$.

####  4.2.4 Complements on absolute transfer factors

Let $F$ be a local or global field. Fix the following data:

  * •

an $F$-pinning $\mathcal{E}$ of $L^{*}$,

  * •

a pure inner twist
$\varphi:L_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}L^{*}_{\overline{F}}$,

  * •

the $1$-cocycle $u:\Gamma_{K}\to L^{*}(\overline{F})$ associated with
$\varphi$.

Consider an endoscopic datum $\mathbf{H}$ of $L$. In view of future
applications, we still assume $\mathcal{H}\simeq{}^{\mathrm{L}}H$ and avoid
the use of $z$-extensions; see Remark 4.2.2.

Assume that $F$ is local hereafter. It has been observed by Kottwitz that
$\Delta_{\mathbf{H},L}$ can be canonically defined using the data
$(\mathcal{E},\varphi,u)$ fixed above. In the case where $L$ is classical,
explicit descriptions of the elliptic endoscopic data can be found in [70,
§1.5]; the data above can be specified by linear-algebraic data, and there are
also explicit formulas for those $\Delta_{\mathbf{H},L}$. The non-elliptic
case follows by passing to Levi subgroups.

Kottwitz’s idea is vastly amplified by Kaletha in [38, Section 5.3], but we
will only use the simplest case of pure inner twists. Note that Kaletha works
with Whittaker data of $L^{*}$ instead of pinnings.

Given a relevant endoscopic datum $\mathbf{H}=(H,\mathcal{H},s)$ of $L$, one
considers its translate $z\mathbf{H}:=(H,\mathcal{H},zs)$ where $z\in
Z_{L^{\vee}}^{\Gamma_{F}}$. Although $z\mathbf{H}$ is equivalent to
$\mathbf{H}$, Arthur [2] and Hiraga–Saito [36] observed that unless
$\varphi=\mathrm{id}_{L}$, the transfer factors may differ. In order to state
the version for pure inner twists we need, denote by

$\Delta_{\mathbf{H}}=\Delta_{\mathbf{H},L^{*},u}:H_{\mathrm{reg}}(F)\times
L_{\mathrm{reg}}(F)\to\mathbb{C}$

the absolute transfer factor determined by $(\mathcal{E},\varphi,u)$ and
$\mathbf{H}$. Denote by $[u]\in\operatorname{H}_{\mathrm{ab}}^{1}(F,L^{*})$
the abelianized cohomology class of $u$. We have the perfect pairing

$\langle\cdot,\cdot\rangle_{\mathrm{Kott}}:\operatorname{H}_{\mathrm{ab}}^{1}(F,L^{*})\times\pi_{0}\left(Z_{L^{\vee}}^{\Gamma_{F}}\right)\to\mathbb{C}^{\times,\mathrm{tors}}$

due to Kottwitz, see [42, Proposition 1.7.3]. It is normalized to make the
following hold.

######  Theorem 4.2.3.

For every $z\in Z_{L^{\vee}}^{\Gamma_{F}}$ and $\mathbf{H}$, we have

$\Delta_{z\mathbf{H}}=\langle[u],z\rangle_{\mathrm{Kott}}\Delta_{\mathbf{H}}.$

###### Proof.

For the general case, see [38, Section 5.3], especially (5.1). For elliptic
endoscopic data for classical groups over non-Archimedean $F$, this is also
documented in [70, §1.11 (1)]. ∎

On the other hand, once $\mathcal{E}$ and $L^{*}$ are chosen,
$\Delta_{\mathbf{H},L^{*},u}$ depends only on $[u]$; see the last assertion in
[38, Proposition 5.6].

######  Remark 4.2.4.

Even for principal endoscopic data $\mathbf{H}$, namely those with $s\in
Z_{L^{\vee}}^{\Gamma_{F}}$, translation by $Z_{L^{\vee}}^{\Gamma_{F}}$ or
change of pure inner twists can still alter $\Delta_{\mathbf{H}}$; see loc.
cit. Thus we cannot naively assume $\Delta=1$ for all principal endoscopic
data of $L$.

Finally, let $F$ be a number field. There is a well-defined absolute adélic or
_global transfer factor_ $\Delta_{\mathbf{H},\mathbb{A}_{F}}$ for every
endoscopic data $\mathbf{H}$ of $L$ that is _locally relevant_ ; see [43, §6]
and [4, §4]. By fixing a pinned quasisplit $F$-group $L^{*}$, a pure inner
twist
$\varphi:L_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}L^{*}_{\overline{F}}$
and the resulting $1$-cocycle $u$ globally (i.e. over $F$),
$\Delta_{\mathbf{H},\mathbb{A}_{F}}$ becomes the product over the local
absolute transfer factors. See [38, §5.7].

######  Remark 4.2.5.

Note that $\Delta_{\mathbf{H},\mathbb{A}_{F}}$ is invariant under
$Z_{L^{\vee}}^{\Gamma_{F}}$-twists. It is also independent of the choice of
global pure inner twist.

###  4.3 An instance of nonstandard endoscopy

To begin with, consider any field $F$ of characteristic $\neq 2$.

######  Notation 4.3.1.

Given a quasisplit $F$-group $L$ endowed with a decomposition

$L=\operatorname{SO}(2a+1)\times\operatorname{SO}(2b+1)\times L_{0}$

where $a,b\in\mathbb{Z}_{\geq 0}$ and $L_{0}$ is another quasisplit $F$-group.
We will write

$\overline{L}:=\operatorname{Sp}(2a)\times\operatorname{Sp}(2b)\times L_{0}.$

In order to compare $L$ and $\overline{L}$, one is reduced to the comparison
between $\operatorname{SO}(2n+1)$ and $\operatorname{Sp}(2n)$, for various
$n\in\mathbb{Z}_{\geq 1}$.

Now recall the _nonstandard endoscopy_ between simply connected groups of type
$\mathrm{B}_{n}$ and $\mathrm{C}_{n}$, which is discussed in [60, III.6.1],
[52, §8.2] or [49, §5.3]. The situation is

${\operatorname{Spin}(2n+1)}$${\operatorname{Sp}(2n)}$${\operatorname{SO}(2n+1)}$isogenynonstd.
endo.

where the $\operatorname{SO}$ and $\operatorname{Spin}$ groups are always
assumed split.

Put $G_{1}:=\operatorname{Spin}(2n+1)$ and $G_{2}:=\operatorname{Sp}(2n)$.
There is a bijection between $\Sigma_{\mathrm{reg}}(\mathfrak{g}_{1})$ and
$\Sigma_{\mathrm{reg}}(\mathfrak{g}_{2})$, written as $X_{1}\leftrightarrow
X_{2}$, namely by matching their eigenvalues. Matching elements have
isomorphic centralizers.

We now assume $F$ is a local field of characteristic zero. Consider $f_{i}\in
S\mathcal{I}(\mathfrak{g}_{i})\otimes\operatorname{mes}(G_{i})$ for $i=1,2$.
We say $f_{1}$ and $f_{2}$ have matching stable orbital integrals if

$\forall X_{1}\leftrightarrow X_{2},\quad
f_{1}^{\operatorname{Spin}(2n+1)}(X_{1})=f_{2}^{\operatorname{Sp}(2n)}(X_{2}).$

Therefore, one can say that the transfer factor for nonstandard endoscopy is
the constant $1$.

######  Theorem 4.3.2 (Nonstandard transfer).

Let $F$ be a local field of characteristic zero. The matching of stable
orbital integrals determines an isomorphism

$\mathcal{T}_{G_{2},G_{1}}:S\mathcal{I}(\mathfrak{g}_{1})\otimes\operatorname{mes}(G_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}S\mathcal{I}(\mathfrak{g}_{2})\otimes\operatorname{mes}(G_{2}).$

It is also a homeomorphism when $F$ is Archimedean.

###### Proof.

For non-Archimedean $F$, the isomorphism $\mathcal{T}_{\overline{L},L}$ is
established in [69, Proposition 1.8 (ii)]. The case $F=\mathbb{R}$ is included
in [60, V.5.1], and the case $F=\mathbb{C}$ follows by restriction of scalars;
the proof is based on Shelstad’s description of
$S\mathcal{I}(\mathfrak{g}_{i})$ in terms of jump conditions. ∎

In studying $SD_{\mathrm{unip}}$ at the level of groups, one can restrict to
arbitrarily small and stably invariant neighborhoods of $1$, on which the
exponential map is defined. Hence we obtain the dual of the transfer map

$\check{\mathcal{T}}_{G_{1},G_{2}}:SD_{\mathrm{unip}}(G_{1})\otimes\operatorname{mes}(G_{1})^{\vee}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}SD_{\mathrm{unip}}(G_{2})\otimes\operatorname{mes}(G_{2})^{\vee}.$

The isogeny $\operatorname{Spin}(2n+1,F)\to\operatorname{SO}(2n+1,F)$ being a
local homeomorphism, the respective spaces $SD_{\mathrm{unip}}(\cdots)$ and
lines $\operatorname{mes}(\cdots)$ can be identified. Hence we obtain

$SD_{\mathrm{unip}}(\operatorname{SO}(2n+1))\otimes\operatorname{mes}(\operatorname{SO}(2n+1))^{\vee}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}SD_{\mathrm{unip}}(\operatorname{Sp}(2n))\otimes\operatorname{mes}(\operatorname{Sp}(2n))^{\vee}.$

All in all, in the situation of Notation 4.3.1, we arrive at canonical
isomorphisms

$\displaystyle\mathcal{T}_{\overline{L},L}:S\mathcal{I}(\mathfrak{l})\otimes\operatorname{mes}(L)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}S\mathcal{I}(\overline{\mathfrak{l}})\otimes\operatorname{mes}(\overline{L}),$
$\displaystyle\mathcal{T}_{L,\overline{L}}:=\mathcal{T}_{\overline{L},L}^{-1},$
$\displaystyle\check{\mathcal{T}}_{L,\overline{L}}:SD_{\mathrm{unip}}(L)\otimes\operatorname{mes}(L)^{\vee}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}SD_{\mathrm{unip}}(\overline{L})\otimes\operatorname{mes}(\overline{L})^{\vee},$
$\displaystyle\check{\mathcal{T}}_{\overline{L},L}^{-1}:=\check{\mathcal{T}}_{L,\overline{L}}^{-1}.$

They will still be called the nonstandard transfer of orbital integrals and
distributions, respectively. In this setting, we still have a bijection
between $\Sigma_{\mathrm{reg}}(\overline{\mathfrak{l}})$ and
$\Sigma_{\mathrm{reg}}(\mathfrak{l})$, and $\mathcal{T}_{\overline{L},L}$ is
characterized by matching orbital integrals as before. Everything boils down
to the matching between $\mathfrak{so}(2n+1)$ and $\mathfrak{sp}(2n)$.

###  4.4 Descent of endoscopy

Fix a local (resp. global) field $F$ of characteristic zero. Let
$\mathbf{p}:\tilde{G}\to G(F)$ (resp. $\mathbf{p}:\tilde{G}\to
G(\mathbb{A}_{F})$) be a group of metaplectic type, and
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$. Write
$G=\prod_{i\in I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W)$. Suppose
that $\epsilon\in G^{!}(F)_{\mathrm{ss}}$ corresponds to $\eta\in
G(F)_{\mathrm{ss}}$ under endoscopy. Upon replacing $\epsilon$ by a stable
conjugate, we may and do assume that $G^{!}_{\epsilon}$ is quasisplit.

The group $G^{!}_{\epsilon}$ is a product of classical groups by Remark 3.2.6:
it comes endowed with the data as in Notation 4.3.1, hence
$\overline{G^{!}_{\epsilon}}$ makes sense. Let us write

$\displaystyle G$ $\displaystyle=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W),$ $\displaystyle G^{!}$
$\displaystyle=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{SO}(2n^{\prime}+1)\times\operatorname{SO}(2n^{\prime\prime}+1).$

Accordingly,
$\epsilon=(\epsilon_{\operatorname{GL}},\epsilon^{\prime},\epsilon^{\prime\prime})$.
The odd orthogonal factors in $G^{!}_{\epsilon}$ correspond to the
$(+1)$-eigenvalues of $\epsilon^{\prime}$ and $\epsilon^{\prime\prime}$.

One of the main results in [52, §7] asserts that there is an endoscopic datum
$\mathbf{G}_{\eta}^{!}$ for $G_{\eta}$, canonical up to equivalences, such
that $G^{!}_{\eta}=\overline{G_{\epsilon}}$ and

${G^{!}_{\epsilon}}$${\overline{G^{!}_{\epsilon}}}$${G_{\eta}.}$nonstd.
endo.endo.

Henceforth, $F$ will be a local field.

Consider elements that are close to zero:

$Y\in\mathfrak{g}^{!}_{\epsilon,\mathrm{reg}}(F),\quad
X\in\mathfrak{g}_{\eta,\mathrm{reg}}(F)$

Denote by
$\overline{Y}\in(\overline{\mathfrak{g}^{!}_{\epsilon}})_{\mathrm{reg}}(F)$
any element corresponding to $Y$ via nonstandard endoscopy. Then
$\exp(Y)\epsilon$ corresponds to $\exp(X)\eta$ via
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$ if and only if
$\overline{Y}$ corresponds to $X$ via
$\mathbf{G}^{!}_{\eta}\in\mathcal{E}(G_{\eta})$.

We may also replace $\eta$ by its stable conjugate $\eta[y]$, where
$y\in\mathcal{Y}(\eta)$. Given $\tilde{\eta}$ and $y$, one defines a canonical
element $\tilde{\eta}[y]\in\mathbf{p}^{-1}(\eta[y])$ according to [48,
Définition 4.1.3]; $\tilde{\eta}[y]$ and $\eta[y]$ depend only on the class of
$y$ in $\dot{\mathcal{Y}}(\eta)$. For each $y\in\dot{\mathcal{Y}}[y]$, we
choose a transfer factor $\Delta(y)$ for the endoscopic datum
$\mathbf{G}^{!}_{\eta[y]}$ of $G_{\eta[y]}$, if it is relevant (Definition
4.2.1).

In the relevant case, we may choose
$X[y]\in\mathfrak{g}_{\eta[y],\mathrm{reg}}(F)$ and
$\overline{Y}\in\overline{\mathfrak{g}^{!}_{\epsilon}}_{,\mathrm{reg}}(F)$
which are sufficiently close to zero and $\overline{Y}\leftrightarrow X[y]$.
We also take $Y\in\mathfrak{g}^{!}_{\epsilon,\mathrm{reg}}(F)$ such that
$\overline{Y}\leftrightarrow Y$; in particular $Y$ is also close to zero.
Define

$d(y):=\begin{cases}\dfrac{\Delta_{\mathbf{G}^{!},\tilde{G}}(\exp(Y)\epsilon,\exp(X[y])\tilde{\eta}[y])}{\Delta(y)(\exp\overline{Y},\exp
X[y])},&\mathbf{G}^{!}_{\eta[y]}\;\text{is relevant},\\\
0,&\text{otherwise}.\end{cases}$ (4.4.1)

The descent of transfer factors [52, Théorème 7.23] asserts that $d(y)$ is
constant for $\overline{Y}$ and $X$ sufficiently close to zero, when
$\mathbf{G}^{!}_{\eta[y]}$ is relevant. This holds in both the non-Archimedean
and Archimedean cases. We repeat it as follows.

######  Theorem 4.4.1.

The value of $d(y)$ is independent of $\overline{Y}$ and $X$. It depends only
on the choice of $\Delta(y)$.

######  Lemma 4.4.2 (Cf. [60, III.5.2]).

Assume $F$ to be non-Archimedean. In the situation above, suppose that
$\delta\in
SD_{\mathrm{geom}}(\mathfrak{g}^{!}_{\epsilon})\otimes\operatorname{mes}(G^{!}_{\epsilon})$
is given and $\operatorname{Supp}(\delta)$ is close to zero. For each
$y\in\dot{\mathcal{Y}}(\eta)$, define

$\displaystyle\delta^{G^{!}}$
$\displaystyle:=S\operatorname{desc}^{G^{!},*}_{\epsilon}(\delta)\;\in
SD_{\mathrm{geom}}(G^{!})\otimes\operatorname{mes}(G^{!})^{\vee},$
$\displaystyle\delta[y]$
$\displaystyle:=\check{\mathcal{T}}_{\mathbf{G}^{!}_{\eta[y]},G_{\eta[y]}}\check{\mathcal{T}}_{G^{!}_{\epsilon},\overline{G^{!}_{\epsilon}}}(\delta)\;\in
D_{\mathrm{geom}}(\mathfrak{g}_{\eta[y]})\otimes\operatorname{mes}(G_{\eta[y]})^{\vee},$
$\displaystyle\delta[y]^{\tilde{G}}$
$\displaystyle:=\operatorname{desc}^{\tilde{G},*}_{\tilde{\eta}[y]}\left(\delta[y]\right)\;\in
D_{\mathrm{geom},-}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}.$

When $\mathbf{G}^{!}_{\eta[y]}$ is relevant,
$\check{\mathcal{T}}_{\mathbf{G}^{!}_{\eta[y]},G_{\eta[y]}}$ (on the level of
Lie algebras) is defined relative to the $\Delta(y)$. Then we have

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

###### Proof.

To begin with, let us assume $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$. Identify
$\delta$ with some $Y\in\Sigma_{\mathrm{reg}}(\mathfrak{g}^{!}_{\epsilon})$
via exponential map by fixing Haar measures. For all
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
we obtain

$\displaystyle\langle\check{\mathcal{T}}_{\mathbf{G}^{!},\tilde{G}}(\delta^{G^{!}}),f\rangle$
$\displaystyle=\langle\exp(Y)\epsilon,\mathcal{T}_{\mathbf{G}^{!},\tilde{G}}(f)\rangle$
$\displaystyle=\sum_{y\in\dot{\mathcal{Y}}(\eta)}d(y)\langle\delta[y],\operatorname{desc}^{\tilde{G}}_{\tilde{\eta}[y]}f\rangle$
$\displaystyle=\sum_{y\in\dot{\mathcal{Y}}(\eta)}d(y)\langle\delta[y]^{\tilde{G}},f\rangle$

where the second equality follows from the proof of [52, Lemme 8.5]. It is
routine to deduce the general case when $\tilde{G}$ is only of metaplectic
type. ∎

###  4.5 Complements on diagrams

We resume the discussion on “diagrams” in §3.6. In particular, $F$ is allowed
to be a field of characteristic zero in the first part of this section.

We begin with the regular semisimple case.

######  Lemma 4.5.1.

Let $\eta\in G(F)_{\mathrm{reg}}$ and $\epsilon\in G^{!}(F)_{\mathrm{reg}}$.
If $\epsilon$ corresponds to $\eta$ via endoscopy, then there exists a diagram
joining $\epsilon$ and $\eta$.

###### Proof.

Let $T^{!}:=G^{!}_{\epsilon}$. Choose $h\in G^{!}(\overline{F})$ such that
$\operatorname{Ad}(h)$ induces
$T^{!}_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{G^{!},\overline{F}}$.
We use the following terminology from Galois descent: for any $F$-varieties
$A$, $B$ and a morphism $f:A_{\overline{F}}\to B_{\overline{F}}$, we write
${}^{\sigma}f=\sigma f\sigma^{-1}:A_{\overline{F}}\to B_{\overline{F}}$, for
every $\sigma\in\Gamma_{F}$. Since $T$ and $T_{G^{!}}$ are $F$-tori,

${}^{\sigma}\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}=w_{\sigma}\circ\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}},\quad
w_{\sigma}\in W^{G^{!}}$

and $\sigma\mapsto w_{\sigma}$ is a $1$-cocycle. We also have
${}^{\sigma}\Phi=\Phi$. Since $\Phi$ is equivariant with respect to
$W^{G^{!}}\to W^{G}$, as embeddings $T^{!}_{\overline{F}}\to G_{\overline{F}}$
we have

${}^{\sigma}\left(\Phi\circ\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}\right)=w_{\sigma}\circ\Phi\circ\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}},\quad\sigma\in\Gamma_{F}.$

Using Kottwitz’s result on embeddings of tori [40, Corollary 2.2], we conclude
that there exists $g\in G(\overline{F})$ such that
$\operatorname{Ad}(g)^{-1}\Phi\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}$ is
defined over $F$, written as
$\xi_{T^{!},T}:T^{!}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T$ for some
maximal $F$-torus $T\subset G$.

Note that
$\operatorname{Ad}(g):T_{\overline{F}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{G,\overline{F}}$
also satisfies
${}^{\sigma}\operatorname{Ad}(g)|_{T_{\overline{F}}}=v_{\sigma}\circ\operatorname{Ad}(g)|_{T_{\overline{F}}}$
for some $v_{\sigma}\in W^{G}$. By comparing with the cocycle $w_{\sigma}$, we
see that $v_{\sigma}=w_{\sigma}$ since
$\operatorname{Ad}(g)^{-1}\Phi\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}$ is
defined over $F$.

Next, note that $\Psi=\tau\circ\Phi$ where $\tau$ is the translation by the
$W^{G^{!}}$-invariant element $(1,1,-1)\in T(F)$. Hence

$\displaystyle{}^{\sigma}\left(\operatorname{Ad}(g)^{-1}\Psi\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}\right)$
$\displaystyle=\operatorname{Ad}(g)^{-1}v_{\sigma}^{-1}\tau\Phi
w_{\sigma}\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}$
$\displaystyle=\operatorname{Ad}(g)^{-1}v_{\sigma}^{-1}w_{\sigma}\tau\Phi\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}$
$\displaystyle=\operatorname{Ad}(g)^{-1}\Psi\operatorname{Ad}(h)|_{T^{!}_{\overline{F}}}.$

This yields a morphism
$\tilde{\xi}_{T^{!},T}:T^{!}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T$
defined over $F$. By construction,
$\eta^{\prime}:=\tilde{\xi}_{T^{!},T}(\epsilon)$ corresponds to $\epsilon$
under endoscopy. Hence there exists $g_{1}\in G(\overline{F})$ with

$\operatorname{Ad}(g_{1})(\eta)=\eta^{\prime},\quad\forall\sigma\in\Gamma_{F},\;{}^{\sigma}g_{1}\in
T(\overline{F})g_{1}.$

Therefore
$\operatorname{Ad}(g_{1}):T_{1}:=G_{\eta}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T$
is a stable conjugacy between maximal $F$-tori in $G$, sending $\eta$ to
$\eta^{\prime}$ and defined over $F$.

Set $B^{!}:=\operatorname{Ad}(h)^{-1}B_{G^{!}}$ and
$B:=\operatorname{Ad}(gg_{1})^{-1}B_{G}$. We conclude that
$(\epsilon,B^{!},T^{!},B,T_{1},\eta)$ is a diagram; the corresponding
$\xi_{T^{!},T_{1}}$ (resp. $\tilde{\xi}_{T^{!},T}$) being
$\operatorname{Ad}(g_{1})^{-1}\xi_{T^{!},T}$ (resp.
$\operatorname{Ad}(g_{1})^{-1}\tilde{\xi}_{T^{!},T}$). ∎

######  Corollary 4.5.2.

For any $\epsilon\in G^{!}(F)_{\mathrm{ss}}$ and a maximal $F$-torus
$T^{!}\subset G^{!}$ containing $\epsilon$, there exists a diagram joining
$\epsilon$ to some $\eta\in G(F)_{\mathrm{ss}}$, with $T^{!}$ being part of
the data.

###### Proof.

Pick $\delta\in T^{!}_{G\text{-reg}}(F)$. It corresponds to some $\gamma\in
G_{\mathrm{reg}}(F)$. Lemma 4.5.1 furnishes a diagram
$(\delta,B^{!},T^{!},B,T,\gamma)$. Now take
$\eta:=\tilde{\xi}_{T^{!},T}(\epsilon)$. ∎

We give a supplementary result on the transfer of ellipticity.

######  Lemma 4.5.3.

Suppose that $\epsilon\in G^{!}(F)_{\mathrm{ss}}$ corresponds to $\eta\in
G(F)_{\mathrm{ss}}$ under endoscopy. If $\epsilon$ is elliptic in $G^{!}$,
then $\eta$ is elliptic in $G$.

###### Proof.

Without loss of generality, we may assume $G=\operatorname{Sp}(W)$. If
$\epsilon$ is elliptic, its parameter introduced in §3.2 has no hyperbolic
indices by Remark 3.2.7. The same holds for the parameter of $\eta$, hence
$\eta$ is elliptic by Corollary 3.2.3. ∎

The converse of Lemma 4.5.3 is not true. For example, take
$G=\operatorname{Sp}(2)$ and $G^{!}=\operatorname{SO}(3)$, arising from
$(1,0)\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$. The elliptic element
$\eta=-1\in G(F)$ corresponds to some $\epsilon\in G^{!}(F)$ with
$G^{!}_{\epsilon}$ isomorphic to the split
$\operatorname{SO}(2)\simeq\mathbb{G}_{\mathrm{m}}$, so $\epsilon$ is not
elliptic.

######  Proposition 4.5.4.

Assume that $F$ is a local field. Let $\eta\in G(F)_{\mathrm{ss}}$ and
$\epsilon\in G^{!}(F)_{\mathrm{ss}}$. Suppose that $\epsilon$ corresponds to
$\eta$ via endoscopy, $G^{!}_{\epsilon}$ is quasisplit, and the endoscopic
datum $\mathbf{G}^{!}_{\eta}$ of $G_{\eta}$ is relevant (see §4.4), then there
exists a diagram $(\epsilon,B^{!},T^{!},B,T,\eta)$ joining $\epsilon$ and
$\eta$ for $\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$,
compatible with descent in the sense that

  * •

$T^{!}\subset G^{!}_{\epsilon}$ and $T\subset G_{\eta}$,

  * •

$T^{!}$ transfer to a maximal torus
$\overline{T^{!}}\subset\overline{G^{!}_{\epsilon}}$ under nonstandard
endoscopy,

  * •

$\overline{T^{!}}$ transfers to $T$ under
$\mathbf{G}^{!}_{\eta}\in\mathcal{E}(G_{\eta})$.

In fact, we can take any $T^{!}$, $\overline{T^{!}}$ and $T$ subject to the
requirements above.

###### Proof.

By assumption and the descent of endoscopy, there exist
$Y\in\mathfrak{g}^{!}_{\epsilon,\mathrm{reg}}(F)$,
$\overline{Y}\in\overline{\mathfrak{g}^{!}_{\epsilon}}_{\mathrm{reg}}(F)$ and
$X\in\mathfrak{g}_{\eta,\mathrm{reg}}(F)$, arbitrarily close to $0$, such that

  * •

$Y\leftrightarrow\overline{Y}\leftrightarrow X$, the first (resp. second)
correspondence being relative to nonstandard endoscopy (resp.
$\mathbf{G}^{!}_{\eta}\in\mathcal{E}(G_{\eta})$),

  * •

$\exp(Y)\epsilon\leftrightarrow\exp(X)\eta$ relative to
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$.

Let $T^{!}\subset G^{!}_{\epsilon}$,
$\overline{T^{!}}\subset\overline{G^{!}_{\epsilon}}$ and $T\subset G_{\eta}$
be the centralizers of $Y$, $\overline{Y}$ and $X$ respectively. It follows
that $T^{!}=G^{!}_{\exp(Y)\epsilon}$ and $T=G_{\exp(X)\eta}$.

By Lemma 4.5.1, we obtain a diagram with Borel subgroups $B^{!}\supset T^{!}$
and $B\supset T$, for each $(Y,\overline{Y},X)$, such that the
$\tilde{\xi}_{T^{!},T}$ maps $\exp(Y)\epsilon$ to $\exp(X)\eta$. By the
finiteness of Borel subgroups containing a given maximal torus, we may now
take sequences $Y_{n},\overline{Y}_{n},X_{n}\to 0$, such that there is a
single choice of $(B^{!},B)$ giving rise to the
$\tilde{\xi}_{T^{!},T}:\exp(Y_{n})\epsilon\mapsto\exp(X_{n})\eta$, for all
$n$.

Pass to the limit to conclude $\tilde{\xi}_{T^{!},T}(\epsilon)=\eta$. This is
the required $(\epsilon,B^{!},T^{!},B,T,\eta)$. ∎

The relevance of $\mathbf{G}^{!}_{\eta}\in\mathcal{E}(G_{\eta})$ is automatic
if we assume $G_{\eta}$ is quasisplit. This is a direct consequence of [40,
§2]. We also make the following observation.

######  Proposition 4.5.5.

Let $F$ be a local field, $\eta\in G(F)_{\mathrm{ss}}$ and $\epsilon\in
G^{!}(F)_{\mathrm{ss}}$. Suppose that $\epsilon$ corresponds to $\eta$ via
endoscopy, and $\epsilon$ in elliptic in $G^{!}$. Then the endoscopic datum
$\mathbf{G}^{!}_{\eta}$ obtained by descent is relevant.

###### Proof.

In this case, $\eta$ is elliptic by Lemma 4.5.3, so $\mathbf{G}^{!}_{\eta}$ is
elliptic. Without loss of generality, we assume $G=\operatorname{Sp}(W)$ in
what follows.

There exists $y\in\mathcal{Y}(\eta)$ such that $G_{\eta[y]}$ is quasisplit.
Since the parameter of $\epsilon$ (see §3.2) contains no hyperbolic indices,
the concrete description of $G^{!}_{\epsilon}$ implies that there is an
elliptic maximal torus $T^{!}\subset G^{!}_{\epsilon}$; its nonstandard
transfer $\overline{T^{!}}\subset\overline{G^{!}_{\epsilon}}$ is still
elliptic. In turn, $\overline{T^{!}}$ transfers to $T^{\prime}\subset
G_{\eta[y]}$ via standard endoscopy; $T^{\prime}$ is still elliptic in
$G_{\eta[y]}$.

By [41, 10.2 Lemma], $T^{\prime}\subset G_{\eta[y]}$ transfers to some
$T\subset G_{\eta}$ under a pure inner twist which lies in the same class as
$\operatorname{Ad}(y)$ in $\operatorname{H}^{1}(F,G_{\eta})$. This shows that
$\overline{T^{!}}\subset\overline{G^{!}_{\epsilon}}$ transfers to $T\subset
G_{\eta}$, as desired. ∎

###  4.6 Descent with Levi subgroups

Return to the setting in §4.4 and consider
$\mathbf{p}:\tilde{G}\twoheadrightarrow G(F)$ together with a Levi subgroup
$M\subset G$. Let
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. We also fix
$\epsilon\in M^{!}(F)_{\mathrm{ss}}$ and $\eta\in M(F)_{\mathrm{ss}}$ in
endoscopic correspondence. Replacing $\epsilon$ by a stable conjugate if
necessary, we may assume $M^{!}_{\epsilon}$ is quasisplit.

Using $\eta$, we define the pointed sets $\mathcal{Y}^{M}$, $\mathcal{Y}$ and
$\dot{\mathcal{Y}}^{M}$, $\dot{\mathcal{Y}}$.

For each $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, we have the
corresponding
$\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$. Thus
$\epsilon[s]$ corresponds to $\eta$ with respect to
$\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$.

######  Lemma 4.6.1.

For each $s$, the group $G^{!}[s]_{\epsilon[s]}$ is quasisplit.

###### Proof.

It contains $M^{!}_{\epsilon[s]}=M^{!}_{\epsilon}$ as a Levi subgroup. The
quasisplit-ness of $M^{!}_{\epsilon}$ amounts to the existence of a maximal
$F$-torus $T^{!}$ which is a simultaneously a minimal Levi. Then $T^{!}$ has
the same properties relative to $G^{!}_{\epsilon[s]}$, hence
$G^{!}_{\epsilon[s]}$ is quasisplit as well. ∎

For all $y\in\mathcal{Y}$ and $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$,
we obtain an endoscopic datum $\mathbf{G}^{!}_{\eta[y]}$ of $G_{\eta[y]}$
obtained from $\mathbf{G}^{!}[s]$ by descent at $\epsilon[s]$ and $\eta[y]$,
canonical up to equivalence, as reviewed in §4.4. Given $y\in\mathcal{Y}$, let
us pick a representative in each equivalence class of relevant endoscopic data
of $G_{\eta[y]}$ arising from some
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, and fix a transfer factor for
it, denoted hereafter by $\Delta(\overline{s},y)$.

One defines the following constants of proportionality.

  1. 1.

For each $y\in\mathcal{Y}^{M}$, we apply (4.4.1) to
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, $\epsilon\in
M^{!}(F)_{\mathrm{ss}}$ and $\eta[y]\in M(F)_{\mathrm{ss}}$ to define
$d^{M}(y)$. This depends on the choice of transfer factor $\Delta(y)$ for the
$\mathbf{M}^{!}_{\eta[y]}\in\mathcal{E}(M_{\eta[y]})$ obtained by descent,
whenever it is relevant.

  2. 2.

For each $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ and $y\in\mathcal{Y}$,
we apply (4.4.1) to
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ to define

$d(s,y):=\begin{cases}\dfrac{\Delta_{\mathbf{G}^{!}[s],\tilde{G}}(\exp(Y)\epsilon[s],\exp(X[y])\tilde{\eta}[y])}{\Delta(\overline{s},y)(\exp(\overline{Y}),\exp(X[y]))},&\mathbf{G}^{!}_{\eta[y]}\;\text{is
relevant},\\\ 0,&\text{otherwise},\end{cases}$ (4.6.1)

where $Y\leftrightarrow\overline{Y}\leftrightarrow X[y]$ are all close to
zero.

This is similar to [60, III.5.4 (2)]; our case is simpler since there are no
$z$-extensions.

These factors depend on various choices. The factor $d(s,y)$ is more delicate
since different choices of $s$ (which determines the numerator in (4.6.1)) may
give rise to the same equivalence class of endoscopic data of $G_{\eta[y]}$
(which determines the denominator). Ergo, there is no straightforward
normalization for $d(s,y)$.

Nevertheless, the problem simplifies when $y\in\mathcal{Y}^{M}$. In this case,
$d^{M}(y)\neq 0$ if the endoscopic datum $\mathbf{M}^{!}_{\eta[y]}$ of
$M_{\eta[y]}$ obtained by descent is relevant. We explicate as follows.

######  Proposition 4.6.2.

Let $y\in\mathcal{Y}^{M}$. The transfer factors $\Delta(\overline{s},y)$ may
be chosen so that $d(s,y)=d^{M}(y)$ when $\mathbf{M}^{!}_{\eta[y]}$ is
relevant, for all $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$.

###### Proof.

Suppose $\mathbf{M}^{!}_{\eta[y]}$ is relevant. We may choose
$Y\in\mathfrak{m}^{!}_{\epsilon,\mathrm{reg}}(F)$ corresponding to
$\overline{Y}\in\overline{\mathfrak{m}^{!}_{\epsilon}}_{,\mathrm{reg}}(F)$,
such that $\overline{Y}$ transfers to
$X[y]\in\mathfrak{m}_{\eta[y],\mathrm{reg}}(F)$. By [49, Proposition 3.3.6],

$\Delta_{\mathbf{G}^{!},\tilde{G}}\left(\exp(Y)\epsilon[s],\exp(X[y])\tilde{\eta}[y]\right)=\Delta_{\mathbf{M}^{!},\tilde{M}}\left(\exp(Y)\epsilon,\exp(X[y])\tilde{\eta}[y]\right).$

Equation (4.6.1) simplifies into

$d(s,y)=\frac{\Delta_{\mathbf{M}^{!},\tilde{M}}\left(\exp(Y)\epsilon,\exp(X[y])\tilde{\eta}[y]\right)}{\Delta(\overline{s},y)(\exp(\overline{Y}),\exp(X[y]))}.$

Consider the denominator: we are in the situation

${\overline{G^{!}[s]_{\epsilon[s]}}}$${G_{\eta[y]}}$${\overline{M^{!}_{\epsilon}}=\overline{M^{!}_{\epsilon[s]}}}$${M_{\eta[y]};}$endo.endo.LeviLevi

see Remark 4.6.4 below for a more precise description.

By the parabolic descent of transfer factors in standard endoscopy, we may
normalize $\Delta(\overline{s},y)$, for all $s$, so that

$\Delta(\overline{s},y)(\exp(\overline{Y}),\exp(X[y]))=\Delta(y)\left(\exp(\overline{Y}),\exp(X[y])\right)$

where $\Delta(y)$ is the chosen transfer factor for
$\mathbf{M}^{!}_{\eta[y]}$. A comparison with (4.4.1) yields
$d(s,y)=d^{M}(y)$. ∎

For future reference, we record another definition pertaining to the method of
descent on the dual side.

######  Definition 4.6.3.

Recall from [60, p.362] that for all quasisplit $F$-group $L$, a Levi subgroup
$R\subset L$ and an element $\mu\in R(F)_{\mathrm{ss}}$, we have a canonical
homomorphism

$\mathrm{descent}:Z_{\check{R}}^{\Gamma_{F}}/Z_{\check{L}}^{\Gamma_{F}}\to
Z_{R_{\mu}^{\vee}}^{\Gamma_{F}}/Z_{L_{\mu}^{\vee}}^{\Gamma_{F}}.$

When $\mu$ is elliptic in $L$ (hence in $R$), one defines

$e^{L}_{R}(\mu):=\left|\operatorname{ker}\left(Z_{\check{R}}^{\Gamma_{F}}/Z_{\check{L}}^{\Gamma_{F}}\xrightarrow{\text{descent}}Z_{R_{\mu}^{\vee}}^{\Gamma_{F}}/Z_{L_{\mu}^{\vee}}^{\Gamma_{F}}\right)\right|^{-1}.$

By loc. cit., the homomorphism is surjective with finite kernel, thus
$e^{L}_{R}(\mu)$ is well-defined.

######  Remark 4.6.4.

We will have to consider the descent of metaplectic endoscopic data from the
dual side. Given
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, $\epsilon\in
M^{!}(F)_{\mathrm{ss}}$ and $\eta\in M(F)_{\mathrm{ss}}$ as before, such that
$\epsilon\leftrightarrow\eta$ and $M^{!}_{\epsilon}$ is quasisplit, we take

  * •

an elliptic element $s^{\flat}\in\tilde{M}^{\vee}$ that determines
$\mathbf{M}^{!}$, trivial on the $\operatorname{GL}$-factors;

  * •

$\overline{s^{\flat}}\in M_{\eta}^{\vee}$: part of the
$\mathbf{M}^{!}_{\eta}\in\mathcal{E}(M_{\eta})$ obtained by descent at
$\epsilon$ and $\eta$ (see [49, p.524] for a more explicit description).

The metaplectic analogue of Definition 4.6.3 is the composite

$\mathrm{descent}:Z_{\tilde{M}^{\vee}}^{\circ}\big{/}Z_{\tilde{G}^{\vee}}^{\circ}\simeq
Z_{\check{M}}^{\circ}\big{/}Z_{\check{G}}^{\circ}\to
Z_{\check{M}}\big{/}Z_{\check{G}}\to
Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\big{/}Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}$

that was denoted by $\tau$ and explained in detail in [49, pp.525–526]. Now
interpret $\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ as a subset of
$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}$ by
Proposition 3.5.5. For any $s=s^{\flat}tZ_{\tilde{G}^{\vee}}^{\circ}$ in that
subset, let

$\overline{s}:=\overline{s^{\flat}}\cdot\mathrm{descent}(t)\;\in\overline{s^{\flat}}\cdot
Z_{M_{\eta}^{\vee}}^{\Gamma_{F}}\big{/}Z_{G_{\eta}^{\vee}}^{\Gamma_{F}}.$
(4.6.2)

It is shown in [49, Lemme 6.3.2] that the
$\mathbf{G}_{\eta}\in\mathcal{E}(G_{\eta})$ obtained by descent at
$\epsilon[s]$ and $\eta$ is equivalent to the one associated with
$\overline{s}\in\mathcal{E}_{\mathbf{M}^{!}_{\eta}}(G_{\eta})$ by Arthur’s
recipe reviewed in §4.2.

##  Chapter 5 The local geometric theorem

Fix a local field $F$ of characteristic zero, a non-trivial additive character
$\psi$ of $F$, and a symplectic $F$-vector space
$(W,\langle\cdot|\cdot\rangle)$ to form the covering

$1\to\bm{\mu}_{8}\to\tilde{G}\xrightarrow{\mathbf{p}}G(F)\to 1,\quad
G:=\operatorname{Sp}(W).$

As in §3.1, we also fix a symplectic basis for $W$ to obtain the corresponding
minimal Levi subgroup $M_{0}$. This gives rise to an invariant positive-
definite quadratic form on $\mathfrak{a}_{0}$, thus on various
$\mathfrak{a}^{G}_{M}$ (Definition 3.1.2).

In fact, we will work with covering groups $\tilde{G}$ of metaplectic type in
order to perform inductive arguments.

Given $M\in\mathcal{L}(M_{0})$, we will introduce the invariant weighted
orbital integrals $I_{\tilde{M}}(\tilde{\gamma},f)$, where
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

  * •

When $F$ is non-Archimedean, $\tilde{\gamma}$ can be any element in
$D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$.

  * •

The case of Archimedean $F$ is more subtle. Although
$I_{\tilde{M}}(\tilde{\gamma},f)$ can be defined for “orbital” distributions
$\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$, that
subspace is not large enough to accommodate all endoscopic transfers. On the
other hand, general $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ might
involve derivatives in the normal direction of orbits. Nonetheless, it is
still possible to define $I_{\tilde{M}}(\tilde{\gamma},f)$ when
$\tilde{\gamma}$ is $\tilde{G}$-equisingular in the sense of Definition 2.2.3.

Given $\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$, we set
(Definition 5.3.9)

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

where $f^{G^{!}[s]}:=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)$ is the
transfer, $\delta\in
SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$ is assumed
to be $\tilde{G}$-equisingular in the Archimedean case, and
$\delta[s]:=\delta\cdot z[s]$ is the central twist. Here appears a new
ingredient $B^{\tilde{G}}$ in the stable weighted orbital integral
$S^{G^{!}[s]}_{M^{!}}(\cdots)$, to be defined in §5.3. In brief,
$B^{\tilde{G}}$ prescribes the rescaling of roots in $G^{!}[s]_{\delta[s]}$;
it first appeared in the study of twisted endoscopy in [60, II].

From $I^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)$, one builds the
endoscopic counterpart $I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$ of
$I_{\tilde{M}}(\tilde{\gamma},f)$ by expressing $\tilde{\gamma}$ via transfers
from various $(\mathbf{M}^{!},\delta)$ (Definition–Proposition 5.3.10). The
local geometric Theorem 5.3.11 asserts that

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

for all $f$ and $\tilde{\gamma}$, the latter assumed to be
$\tilde{G}$-equisingular in the Archimedean case.

The proof of this theorem will only be achieved at the end of this work. At
this stage, we prove in §5.4 that
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$ satisfies the same descent
formula as $I_{\tilde{M}}(\tilde{\gamma},f)$; then we reduce the
$\tilde{G}$-equisingular case (for both Archimedean and non-Archimedean $F$)
to the $\tilde{G}$-regular case in §5.5. The combinatorial technique for
proving the endoscopic descent formula is a recurring theme in this work. It
will be invoked repeatedly later on.

In the unramified situation, the weighted fundamental lemma for general
$\tilde{\gamma}$ is stated as Theorem 5.6.3. Its proof as well as further
reductions for Theorem 5.3.11 demand the theory of germs.

The results in §§5.1–5.2 will only be stated for coverings of metaplectic
type, although they can also be adapted to general coverings; see Remark
5.2.6.

###  5.1 Weighted orbital integrals: non-Archimedean case

Assume $F$ to be non-Archimedean.

####  5.1.1 Non-invariant version

Choose a maximal compact subgroup $K\subset G(F)$ in good position relative to
$M_{0}$. Let $M\in\mathcal{L}(M_{0})$ and $\tilde{\gamma}\in\tilde{M}$. We
write the Jordan decomposition of $\tilde{\gamma}$ reviewed in §2.2 as

$\tilde{\gamma}=\tilde{\eta}u=u\tilde{\eta},\quad\tilde{\eta}:=\tilde{\gamma}_{\mathrm{ss}}.$

Let

$v_{M}:M(F)\backslash G(F)/K\to\mathbb{R}_{\geq 0}$

be Arthur’s weight factor attached to the $(G,M)$-family
$v_{P}(\lambda,x):=e^{-\langle\lambda,H_{P}(x)\rangle}$, where
$P\in\mathcal{P}(M)$ and $\lambda\in i\mathfrak{a}^{*}_{M}$.

Consider $\tilde{\gamma}\in\tilde{M}$ with Jordan decomposition
$\tilde{\gamma}=\tilde{\eta}u=u\tilde{\eta}$. Suppose that $\tilde{\gamma}$ is
$\tilde{G}$-equisingular (Definition 2.2.3), we define

$J_{\tilde{M}}(\tilde{\gamma},f)=J^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f):=|D^{G}(\gamma)|^{1/2}\int_{M_{\gamma}(F)\backslash
G(F)}f(g^{-1}\tilde{\gamma}g)v_{M}(g)\mathop{}\\!\mathrm{d}g$

where $f\in C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})$ and the Haar
measures on $M_{\gamma}(F)$, $G(F)$ are chosen. The weighted orbital integrals
for general coverings have been defined in [45, §6.3]; the case for
$\tilde{G}$ simplifies since all elements are “good”.

Define $A_{M,\mathrm{reg}}\subset A_{M}$ be the open subset of elements such
that

$\prod_{\beta\in\Sigma_{P}^{\mathrm{red}}}\left(\beta(a)-\beta(a)^{-1}\right),\quad
P\in\mathcal{P}(M)$

are all invertible. For general $\tilde{\gamma}$, in [45] the weighted orbital
integral was defined as

$J_{\tilde{M}}^{\mathrm{Art}}(\tilde{\gamma},f):=\lim_{\begin{subarray}{c}a\in
A_{M,\mathrm{reg}}(F)\\\ a\to
1\end{subarray}}\sum_{L\in\mathcal{L}(M)}r^{L,\mathrm{Art}}_{M}(\gamma,a)J_{\tilde{L}}(a\tilde{\gamma},f),\quad
a\gamma:G\text{-equisingular}$

by noting that $M_{a\gamma}=M_{\gamma}$, and that
$\mathbf{p}:\widetilde{A_{M}}\to A_{M}(F)$ splits canonically in some
neighborhood of $1$, namely by comparing two exponential maps (see (4.1.1)).

Assume $a\to 1$ is in general position as above. Here
$r^{\mathrm{Art}}_{M}(\gamma,a)=r^{\mathrm{Art}}_{M}(\gamma,a;0)$ is attached
to the $(G,M)$-family

$\displaystyle r^{\mathrm{Art}}_{P}(\gamma,a;\lambda)$
$\displaystyle=\prod_{\alpha\in\Sigma_{P}}r_{\alpha}^{\mathrm{Art}}\left(\gamma,a;\frac{\lambda}{2}\right),$
$\displaystyle r_{\alpha}^{\mathrm{Art}}\left(\gamma,a;\lambda\right)$
$\displaystyle=\prod_{\begin{subarray}{c}\beta\in\Sigma^{G_{\eta}}_{\mathrm{red}}(A_{M_{\eta}})\\\
\beta_{M}=\alpha\end{subarray}}\left|\alpha(a)-\alpha(a)^{-1}\right|^{\langle\lambda,\rho^{\mathrm{Art}}(\beta,u)\check{\beta}_{M}\rangle}$

where $\check{\beta}\in\mathfrak{a}_{M_{\eta}}$ is the “coroot” of $\beta$
defined by Arthur [22, p.229], $\check{\beta}_{M}\in\mathfrak{a}_{M}$ is its
natural projection, and $\beta_{M}:=\beta|_{A_{M}}$. The crucial factor
$\rho^{\mathrm{Art}}(\beta,u)\in\mathbb{R}$ is also defined by Arthur via the
Grothendieck–Springer resolution. It depends only on $G_{\eta}\supset
M_{\eta}$, $\beta$, $u$ and has nothing to do with the covering.

Our definition below is based on the corrected version
$\rho(\beta,u)\in\mathfrak{a}_{M_{\eta}}$ proposed in [60, II.1.4], for all
$u\in M_{\eta,\mathrm{unip}}(F)$ and
$\beta\in\Sigma^{G_{\eta}}(A_{M_{\eta}})$. It is designed to make [22,
Corollary 6.3] and its variant [45, Proposition 5.4.2] true; it satisfies

$\rho^{\mathrm{Art}}(\beta,u)\check{\beta}=\sum_{n\geq 1}\rho(n\beta,u)$

for all $\beta\in\Sigma^{G_{\eta}}_{\mathrm{red}}(A_{M_{\eta}})$, where
$\rho(n\beta,u):=0$ if $n\beta\notin\Sigma^{G_{\eta}}(A_{M_{\eta}})$. Next,
for $\alpha\in\Sigma(A_{M})$,

$\displaystyle\rho(\alpha,\gamma)$
$\displaystyle:=\sum_{\begin{subarray}{c}\beta\in\Sigma^{G_{\eta}}(A_{M_{\eta}})\\\
\beta_{M}=\alpha\end{subarray}}\underbracket{\rho(\beta,u)}_{\in\mathfrak{a}_{M_{\eta}}}\quad\text{projected
to }\mathfrak{a}_{M},$ $\displaystyle r_{\alpha}\left(\gamma,a;\lambda\right)$
$\displaystyle:=\left|\alpha(a)-\alpha(a)^{-1}\right|^{\langle\lambda,\rho(\alpha,\gamma)\rangle}.$

Then we define $r_{P}(\gamma,a;\lambda)$, $r_{M}(\gamma,a)$ in the same way.
The weighted orbital integral considered in this work is defined as follows.

######  Definition 5.1.1.

For all $\tilde{\gamma}\in\tilde{M}$ and $f$, set

$J_{\tilde{M}}(\tilde{\gamma},f):=\lim_{\begin{subarray}{c}a\in
A_{M,\mathrm{reg}}(F)\\\ a\to
1\end{subarray}}\sum_{L\in\mathcal{L}(M)}r^{L}_{M}(\gamma,a)J_{\tilde{L}}(a\tilde{\gamma},f),\quad
a\gamma:G\text{-equisingular}.$

Here are some basic properties of $J_{\tilde{M}}(\tilde{\gamma},f)$.

  * •

The limit above exists.

  * •

It satisfies the usual properties used in the coarse trace formula, such as
the descent formulas and splitting formulas.

  * •

Moreover, the estimate in [60, II.1.5 (3)] also holds in our context: there
exists $r>0$ such that

$\left|J_{\tilde{M}}(\tilde{\gamma},f)-\sum_{L\in\mathcal{L}(M)}r^{L}_{M}(\gamma,a)J_{\tilde{L}}(a\tilde{\gamma},f)\right|\ll
d(a)^{r},\quad\text{as}\;a\in A_{M,\mathrm{reg}}(F),\;a\to 1;$ (5.1.1)

Here $d(\exp H)=\|H\|$ for a chosen norm $\|\cdot\|$ on the Lie algebra of
$A_{M}$, when $H$ is close to zero.

######  Remark 5.1.2.

To prescribe $\tilde{\gamma}\in\tilde{M}$ and a Haar measure on
$M_{\gamma}(F)$ is the same as to prescribe an element of
$D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. In this
way, we shall view $J_{\tilde{M}}(\tilde{\gamma},f)$ as a function of

$\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},\quad f\in
C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$

since $D_{\mathrm{geom},-}=D_{\mathrm{orb},-}$ for non-Archimedean $F$.

####  5.1.2 Invariant version

Let $M$ be a Levi subgroup of $G$. Since
$J^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)$ is clearly concentrated at
$H_{\tilde{G}}(\tilde{\gamma})\in\mathfrak{a}_{G}$ when
$\operatorname{Supp}(\tilde{\gamma})$ is a single conjugacy class, it extends
to
$C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.

Now let $L$ be a Levi subgroup of $G$. We have the linear map from [47,
Théorème 4.13]

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

defined through tempered weighted characters. For a summary of these weighted
characters, we refer to §7.1.

######  Definition 5.1.3.

The invariant version of weighted orbital integrals is defined recursively by

$I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f):=J_{\tilde{M}}(\tilde{\gamma},f)-\sum_{L\in\mathcal{L}(M)}I^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma},\phi_{\tilde{L}}(f)).$

Following Remark 5.1.2, here we take

  * •

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

  * •

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

to get rid of any choice of Haar measures. By induction,
$I_{\tilde{M}}(\tilde{\gamma},\cdot)$ is concentrated at
$H_{\tilde{G}}(\tilde{\gamma})\in\mathfrak{a}_{G}$ when
$\operatorname{Supp}(\tilde{\gamma})$ is a single conjugacy class.

As in [60, II.1.7],

$\displaystyle I_{\tilde{M}}(\tilde{\gamma},f)=\lim_{\begin{subarray}{c}a\in
A_{M,\mathrm{reg}}(F)\\\ a\to
1\end{subarray}}\sum_{L\in\mathcal{L}(M)}r^{L}_{M}(\gamma,a)I_{\tilde{L}}(a\tilde{\gamma},f),\quad
a\gamma:G\text{-equisingular},$
$\displaystyle\left|I_{\tilde{M}}(\tilde{\gamma},f)-\sum_{L\in\mathcal{L}(M)}r^{L}_{M}(\gamma,a)I_{\tilde{L}}(a\tilde{\gamma},f)\right|\ll
d(a)^{r},\quad\text{as}\;a\in A_{M,\mathrm{reg}}(F),\;a\to 1.$

The proofs are formal: see loc. cit.

######  Remark 5.1.4.

In fact, $I_{\tilde{M}}(\tilde{\gamma},f)$ is independent on the choice of
$K$. To see this, it suffices to copy the arguments from [8, Lemma 3.4].

Now we can state the descent formula for invariant weighted orbital integrals
for $\tilde{G}$.

######  Proposition 5.1.5 (Cf. [60, II.1.7 Lemme]).

Let $L\in\mathcal{L}(M)$,
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. Write
$\tilde{\gamma}^{\tilde{L}}:=\operatorname{Ind}^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma})$;
see (3.10.1). Then

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

###### Proof.

Identical as the proof in loc. cit., which is based on harmonic analysis for
$\tilde{G}$ and general properties of the constants
$d^{G}_{M}(L,L^{\dagger})$. ∎

###  5.2 Weighted orbital integrals: Archimedean case

Consider an Archimedean local field $F$. The same recipe yields

$J_{\tilde{M}}(\tilde{\gamma},f)\quad\text{for all}\quad\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},\quad f\in
C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G).$

Moreover, it is continuous in $f$. The construction works when $\tilde{G}$ is
replaced by any group of metaplectic type. As before, for these
$\tilde{\gamma}$ we can define

$I_{\tilde{M}}(\tilde{\gamma},f)=I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f):=J_{\tilde{M}}(\tilde{\gamma},f)-\sum_{L\in\mathcal{L}(M)}I^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma},\phi_{\tilde{L}}(f)).$

The $\phi_{\tilde{L}}$ here is a continuous linear map defined through
tempered weighted characters (see §8.2). One can proceed in one of the
following ways.

  1. 1.

One can choose a maximal compact subgroup $K\subset G(F)$ in good position
relative to $M_{0}$, take
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})\otimes\operatorname{mes}(G)$
(see Definition 2.3.8), and use

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

for all $L\in\mathcal{L}(M_{0})$ with $K^{L}:=K\cap L(F)$. This is the map
used in [47, §4.3], based on the $\tilde{K}\times\tilde{K}$-finite trace
Paley–Wiener theorem (Remark 2.8.4) for groups of metaplectic type.

Since
$\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G},\tilde{K})$,
etc. are independent of $\tilde{K}$, so is $I_{\tilde{M}}(\tilde{\gamma},f)$.

  2. 2.

One can also consider the broader setting of $f\in
C^{\infty}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and the corresponding

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

This is based on the $C^{\infty}_{c}$ trace Paley–Wiener theorem [25], valid
for general coverings; see [60, IV.1].

Note that $I_{\tilde{M}}(\tilde{\gamma},\cdot)$ is concentrated at
$H_{\tilde{G}}(\tilde{\gamma})\in\mathfrak{a}_{G}$ when
$\operatorname{Supp}(\tilde{\gamma})$ is a single conjugacy class.

So far we defined $I_{\tilde{M}}(\tilde{\gamma},\cdot)$ only for
$\tilde{\gamma}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. Following
the recipe in [60, V.1.3], we propose a partial extension to some other
$\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$. It is
based on the notion of equisingularity (Definition 2.2.3).

######  Lemma 5.2.1.

Fix a $G$-equisingular semisimple class $\mathcal{O}\subset M(F)$,
$\tilde{\gamma}\in\mathbf{p}^{-1}(\mathcal{O})$ and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$.
Then there exist
$f_{1}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
and an invariant neighborhood $\mathcal{U}$ of $\mathcal{O}$ such that
$I^{\tilde{M}}(\tilde{\gamma}^{\prime},f_{1})=I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma}^{\prime},f)$
for all $\tilde{\gamma}^{\prime}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ supported in
$\mathbf{p}^{-1}(\mathcal{U})$.

###### Proof.

Assume inductively that the property holds when $G$ is replaced by any
$L\in\mathcal{L}(M)$ such that $L\neq G$; note that the case $L=M$ is trivial.

Choose Haar measures to trivialize the lines $\operatorname{mes}(\cdots)$. By
[45, Proposition 6.4.1], there exist $f_{0}\in
C^{\infty}_{c,\dashrule[.7ex]{2222}{.4}}(\tilde{G})$ and an invariant
neighborhood $\mathcal{V}$ of $\mathcal{O}$ such that
$I^{\tilde{M}}(\tilde{\gamma}^{\prime},f_{0})=J^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma}^{\prime},f)$
for all $\tilde{\gamma}^{\prime}\in\Gamma_{\mathrm{orb},-}(\tilde{M})$
supported in $\mathbf{p}^{-1}(\mathcal{V})$. The assertion then follows from
the definition of $I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma}^{\prime},f)$ and
induction hypotheses. ∎

######  Definition 5.2.2 (Extension to $G$-equisingular classes).

Let $\mathcal{O}\subset M(F)$ be a $G$-equisingular semisimple class. For
every $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$
and
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$,
set

$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f):=I^{\tilde{M}}(\tilde{\gamma},f_{1})$

where
$f_{1}\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{M})\otimes\operatorname{mes}(M)$
is given as in Lemma 5.2.1.

Let us show that this is well-defined. For simplicity, assume
$\operatorname{Supp}(\tilde{\gamma})$ is a single conjugacy class. Definition
5.2.2 is independent of the choice of $f_{1}$ since
$I^{\tilde{M}}(\tilde{\gamma},f_{1})$ can be expressed as a linear combination
of derivatives at $X=0$ of

$X\mapsto
I^{\tilde{M}}(\exp(X)\tilde{\gamma}_{\mathrm{orb}},f_{1})=I^{\tilde{G}}_{\tilde{M}}(\exp(X)\tilde{\gamma}_{\mathrm{orb}},f),\quad
X\in\mathfrak{t}(F)$

where

  * •

$\tilde{\gamma}_{\mathrm{orb}}\in
D_{\mathrm{orb},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee}$ is the
“orbital part” of $\tilde{\gamma}$;

  * •

$T$ is any maximal $F$-torus in $M$ with $\gamma\in T(F)$;

  * •

we assume that
$\exp(X)\tilde{\gamma}_{\mathrm{orb}}\in\tilde{G}_{\mathrm{reg}}$.

It is clear that $I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},f)$ gives back the
original definition when $\tilde{\gamma}$ is already “orbital”.

The description above implies that
$I^{\tilde{G}}_{\tilde{M}}(\tilde{\gamma},\cdot)$ is concentrated at
$H_{\tilde{G}}(\tilde{\gamma})$. Therefore $I_{\tilde{M}}(\tilde{\gamma},f)$
is defined for all
$f\in\mathcal{I}_{\mathrm{ac},\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$.

######  Definition 5.2.3.

Let
$D_{\mathrm{geom},G\text{-equi},-}(\tilde{M}):=\bigoplus_{\mathcal{O}}D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})$
where $\mathcal{O}$ ranges over the $G$-equisingular semisimple classes in
$M(F)$ (Definition 2.2.3).

Varying $\mathcal{O}$, we arrive at the definition of

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

In the group-theoretic result below, $F$ can be any field.

######  Lemma 5.2.4.

Let $L\in\mathcal{L}(M)$ and $\mathcal{O}$ be a $G$-equisingular semisimple
conjugacy class in $M(F)$. Denote by $\mathcal{O}^{L}$ the conjugacy class in
$L(F)$ containing $\mathcal{O}$. Assume that $\mathcal{O}^{L}$ is
$G$-equisingular. For all $L^{\dagger}\in\mathcal{L}(M)$, we have

$d^{G}_{M}(L,L^{\dagger})\neq 0\implies\mathcal{O}\;\text{is
$L^{\dagger}$-equisingular}.$

###### Proof.

This is part (i) of [60, V.1.3 Lemme]. ∎

Now comes the Archimedean counterpart of Proposition 5.1.5.

######  Proposition 5.2.5 (Cf. part (ii) of [60, V.1.3 Lemme]).

Let $L\in\mathcal{L}(M)$,
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and $\tilde{\gamma}\in
D_{\mathrm{geom},-}(\tilde{M},\mathcal{O})\otimes\operatorname{mes}(M)^{\vee}$,
where $\mathcal{O}$ is a semisimple conjugacy class in $M(F)$ such that
$\mathcal{O}^{L}$ is $G$-equisingular. Write

$\tilde{\gamma}^{\tilde{L}}:=\operatorname{Ind}^{\tilde{L}}_{\tilde{M}}(\tilde{\gamma})\in
D_{\mathrm{geom},-}(\tilde{L},\mathcal{O}^{L})\otimes\operatorname{mes}(L)^{\vee}$

(see (2.4.1)). Then

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

###### Proof.

Note that
$d^{G}_{M}(L,L^{\dagger})I^{\tilde{L}^{\dagger}}_{\tilde{M}}\left(\tilde{\gamma},f_{\tilde{L}}\right)$
makes sense by Lemma 5.2.4. The remaining arguments are identical to loc.
cit.. ∎

######  Remark 5.2.6.

The constructions in the non-Archimedean case §5.1 work for all covering group
$\tilde{G}$, its Levi subgroup $\tilde{M}$ and all good conjugacy classes
$\tilde{\gamma}\in\Gamma(\tilde{M})$. See for example [45, §6.3].

The same also holds in the Archimedean case. One can avoid assuming the
$\tilde{K}\times\tilde{K}$-finite trace Paley–Wiener theorem by using the
$C^{\infty}_{c}$-version, which works for general coverings.

###  5.3 Systems of $B$-functions and the local geometric theorem

To begin with, consider a connected reductive $F$-group $L$. Its
$\overline{F}$-pinnings $(B_{L},T,(E_{\beta})_{\beta})$ form a torsor under
$L_{\mathrm{AD}}(\overline{F})$-action. One defines universal versions of the
root data and the Weyl group as compatible systems indexed by pinnings, with
transition maps given by $L_{\mathrm{AD}}$. Given
$(B_{L},T,(E_{\beta})_{\beta})$, we obtain a Galois action on the universal
$\Sigma(L,T)$ and $\check{\Sigma}(L,T)$, denoted hereafter as
$\sigma\mapsto\sigma_{L}^{*}$ where $\sigma\in\Gamma_{F}$; it factors through
$\operatorname{Gal}(E|F)$ where $E|F$ is any Galois extension splitting $G$.
This is also the viewpoint adopted in [60, I.1.2].

These “universal” objects are insensitive to inner twists. They can be
manipulated concretely by choosing an $\overline{F}$-pinning
$(B_{L},T,(E_{\beta})_{\beta})$. If $(B_{L},T,(E_{\beta})_{\beta})$ is an
$F$-pinning, then $\sigma\mapsto\sigma_{L^{*}}$ is induced by the
$F$-structure of $T$.

Let us fix a positive definite quadratic form on the
$X_{*}(T)\otimes\mathbb{R}$ that is invariant under $\Gamma_{F}$ and the Weyl
group. Here $X_{*}(T)$, the Weyl group and $\Gamma_{F}$-actions are understood
in the universal sense above. By duality, we deduce a similar form on
$X^{*}(T)\otimes\mathbb{R}$, denoted as $(\cdot|\cdot)$. The definitions below
are from [60, II.1.8, II.1.9].

######  Definition 5.3.1.

Consider the universal set of absolute roots $\Sigma(L,T)$ in $L$ defined
above.

  1. 1.

A $B$-function on $\Sigma(L,T)$ is a map $\Sigma(L,T)\to\mathbb{Q}_{>0}$
satisfying

     * •

$B$ is invariant under $\Gamma_{F}$ and the Weyl group,

     * •

$B(-\beta)=B(\beta)$ for all $\beta\in\Sigma(L,T)$,

     * •

over any irreducible factor of the root system associated attached to
$\Sigma(L,T)$, either $B$ or $\beta\mapsto(\beta|\beta)^{-1}B(\beta)$ is
constant.

  2. 2.

Given a $B$-function as above, set

$\displaystyle\Sigma(L,T;B)$
$\displaystyle:=\left\\{B(\beta)^{-1}\beta:\beta\in\Sigma(L,T)\right\\}\subset
X^{*}(T)\otimes\mathbb{Q},$ $\displaystyle\check{\Sigma}(L,T;B)$
$\displaystyle:=\left\\{B(\beta)\check{\beta}:\beta\in\Sigma(L,T)\right\\}\subset
X_{*}(T)\otimes\mathbb{Q}.$

According to [60, pp.204–205], $\Sigma(L,T;B)$ form a root system such that
$\check{\Sigma}(L,T;B)$ is the set of coroots, and the Weyl group is the same
as the original one.

For any Levi subgroup $M\subset L$, denote by $\Sigma(A_{M};B)$ the
restrictions to $\mathfrak{a}_{M}$ of the elements of $\Sigma(L,T;B)$.

Now let $G^{!}$ be a quasisplit connected reductive $F$-group, and take any
$F$-pinning

$(B^{!},T^{!},(E_{\alpha})_{\alpha})\quad\text{of}\;G^{!}.$

Fix a positive definite quadratic form $(\cdot|\cdot)$ on
$X_{*}(T^{!})\otimes\mathbb{R}$ with the same invariance properties as before.
For every $\epsilon\in G^{!}(F)_{\mathrm{ss}}$, there exists $g\in
G^{!}_{\mathrm{AD}}(\overline{F})$ such that $\epsilon\in
gT^{!}(\overline{F})g^{-1}$; in this manner, we obtain a similar quadratic
form for $L:=G^{!}_{\epsilon}$ by forming the $\overline{F}$-pinning

$\left(gB^{!}g^{-1}\cap
L,\;gT^{!}g^{-1},\;(gE_{\alpha}g^{-1})_{\alpha:\alpha(g^{-1}\epsilon
g)=1}\right)$

of $L$. One readily checks that the form is independent of all choices.

######  Definition 5.3.2.

Let $G^{!}$ be a quasisplit connected reductive $F$-group, with a chosen
quadratic form $(\cdot|\cdot)$ as above. A _system of $B$-functions_ is an
assignment $B:\epsilon\mapsto B_{\epsilon}$ for each $\epsilon\in
G^{!}(F)_{\mathrm{ss}}$ such that

  * •

$B_{\epsilon}$ is a $B$-function for $G^{!}_{\epsilon}$;

  * •

if $\epsilon,\epsilon^{\prime}\in G^{!}(F)_{\mathrm{ss}}$ and $g\in
G^{!}(\overline{F})$ satisfy $g\epsilon g^{-1}=\epsilon^{\prime}$, then the
inner twist
$\operatorname{Ad}_{g}:(G^{!}_{\epsilon})_{\overline{F}}\to(G^{!}_{\epsilon^{\prime}})_{\overline{F}}$
identifies $B_{\epsilon}$ with $B_{\epsilon^{\prime}}$.

The assignment $B_{\epsilon}=1$ for all $\epsilon\in G^{!}(F)_{\mathrm{ss}}$
is called the _trivial system of $B$-functions_.

The system of $B$-functions prescribe a coherent way to rescale the roots.
Given a system of $B$-functions for $G^{!}$, a Levi subgroup $M^{!}$ and
$\gamma=\epsilon u\in M^{!}(F)$ (Jordan decomposition), there are variants

$\rho(\beta,u;B),\quad\rho(\alpha,\gamma;B),\quad r_{P}(u,a;B,\lambda)$

of the objects in §5.1, where
$\beta\in\Sigma(A_{M^{!}_{\epsilon}};B_{\epsilon})$ and
$\alpha\in\Sigma(A_{M^{!}};B_{\epsilon})$. Here
$\Sigma(A_{M^{!}};B_{\epsilon})$ is the set of non-trivial restrictions of
elements of $\Sigma(A_{M^{!}_{\epsilon}};B_{\epsilon})$ to
$\mathfrak{a}_{M^{!}}$. See [60, pp.206–209] for details.

Since $\Sigma(A_{M^{!}};B_{\epsilon})$ depends only on the semisimple stable
conjugacy class $\mathcal{O}^{!}$ in $M(F)$ generated by $\epsilon$, it may be
denoted as $\Sigma(A_{M^{!}},B_{\mathcal{O}^{!}})$.

######  Remark 5.3.3.

Given a system of $B$-functions for $G^{!}$ and a Levi subgroup $L^{!}\subset
G^{!}$, we obtain a system $B|_{L^{!}}$ since $L^{!}_{\epsilon}\subset
G^{!}_{\epsilon}$ is also a Levi subgroup for every $\epsilon\in
L^{!}(F)_{\mathrm{ss}}$, and one can restrict $B_{\epsilon}$ to the roots
which are trivial on $Z_{L^{!}}^{\circ}$.

Using the factors $r^{L^{!}}_{M^{!}}(u,a;B)$ for various
$L^{!}\in\mathcal{L}(M^{!})$, we are then able to define

$J^{G^{!}}_{M^{!}}(\gamma,B,f^{!}),\quad I^{G^{!}}_{M^{!}}(\gamma,B,f^{!})$

as before, with similar properties.

With the given system of $B$-functions, the _stable weighted orbital
integrals_ are defined in [60, II.1.10, V.1.4]:

$S^{G^{!}}_{M^{!}}(\delta,B,f^{!}),\quad f^{!}\in
S\mathcal{I}(G^{!})\otimes\operatorname{mes}(G^{!}),$

and $\delta$ ranges over
$SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$ for non-
Archimedean $F$ (resp.
$SD_{\mathrm{geom},G^{!}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}$
for Archimedean $F$, as in §5.2). The distribution is concentrated at
$H_{G^{!}}(\delta)$ when $\operatorname{Supp}(\delta)$ is a single stable
class. It turns out that $S^{G^{!}}_{M^{!}}(\delta,B,f^{!})$ is continuous in
$f^{!}$ when $F$ is Archimedean. The stability of these distributions is the
content of [60, Theorem II.1.10].

When $B$ is trivial, all these objects reduce to the usual versions without
reference $B$.

######  Remark 5.3.4.

When $\gamma$ is $G$-equisingular, we have
$S^{G^{!}}_{M^{!}}(\delta,B,f^{!})=S^{G^{!}}_{M^{!}}(\delta,f^{!})$ by [60,
II.1.10 (23)]. This is the case when $\gamma\in M^{!}_{G^{!}\text{-reg}}(F)$,
for example.

Return to the case of a group of metaplectic type $\tilde{G}=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\widetilde{\mathrm{Sp}}(W)$. The hardcore is
of course the case $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$. Let
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$. We are going
to define a system of $B$-functions for $G^{!}$, denoted as $B^{\tilde{G}}$.
Recall that invariant quadratic forms for $G^{!}$ have been chosen in
Definition 3.1.2. By Definition 5.3.2 and [40, Lemma 3.3], it suffices to
prescribe $B^{\tilde{G}}_{\epsilon}$ for each $\epsilon\in
G^{!}(F)_{\mathrm{ss}}$ such that $G^{!}_{\epsilon}$ is quasisplit.

######  Definition 5.3.5.

Suppose $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ and let
$\mathbf{G}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$ be associated
with the pair $(n^{\prime},n^{\prime\prime})\in\mathbb{Z}_{\geq 0}^{2}$. The
system $B^{\tilde{G}}$ for
$G^{!}=\operatorname{SO}(2n^{\prime}+1)\times\operatorname{SO}(2n^{\prime\prime}+1)$
is defined as follows. Let
$\epsilon=(\epsilon^{\prime},\epsilon^{\prime\prime})\in
G^{!}(F)_{\mathrm{ss}}$. According to §3.2, there are canonical decompositions

$\operatorname{SO}(2n^{\prime}+1)_{\epsilon^{\prime}}=U^{\prime}\times\operatorname{SO}(V^{\prime}_{+})\times\operatorname{SO}(V^{\prime}_{-}),\quad\operatorname{SO}(2n^{\prime\prime}+1)_{\epsilon^{\prime\prime}}=U^{\prime\prime}\times\operatorname{SO}(V^{\prime\prime}_{+})\times\operatorname{SO}(V^{\prime\prime}_{-}),$

where $U^{\prime}$ is a direct product of unitary groups and general linear
groups up to restriction of scalars, $V^{\prime}_{+}$ (resp. $V^{\prime}_{-}$)
is a quadratic $F$-vector space of odd (resp. even) dimension; ditto for
$U^{\prime\prime}$, $V^{\prime\prime}_{+}$ and $V^{\prime\prime}_{-}$. Since
$G^{!}_{\epsilon}$ is quasisplit, $\operatorname{SO}(V^{\prime}_{+})$ and
$\operatorname{SO}(V^{\prime\prime}_{+})$ are split.

Let $\beta$ be an (absolute) root of $G^{!}_{\epsilon}$. If $\beta$ comes from
the factor $\operatorname{SO}(V^{\prime}_{+})$ or
$\operatorname{SO}(V^{\prime\prime}_{+})$, set

$B^{\tilde{G}}_{\epsilon}(\beta):=\begin{cases}1,&\beta:\;\text{long},\\\
\frac{1}{2},&\beta:\;\text{short},\end{cases}$

and in the other cases we set $B^{\tilde{G}}(\beta)=1$; note that when
$\operatorname{SO}(V^{\prime}_{+})$ or
$\operatorname{SO}(V^{\prime\prime}_{+})$ has rank $1$, every root is short by
convention.

The recipe extends to groups of metaplectic type and their elliptic endoscopic
data: simply set $B^{\tilde{G}}$ to be trivial on the
$\operatorname{GL}$-factors. Observe that
$B^{\tilde{G}}_{z\epsilon}=B^{\tilde{G}}_{\epsilon}$ for all $z\in
Z_{G^{!}}(F)$.

It is straightforward to verify the conditions in Definition 5.3.2 for
$B^{\tilde{G}}$.

######  Lemma 5.3.6.

Suppose $M$ is a Levi subgroup of $G$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. For every
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, consider the system of
$B$-functions $B^{\tilde{G}}$ on $G^{!}[s]$. Then
$B^{\tilde{G}}|_{M^{!}}=B^{\tilde{M}}$ (see Remark 5.3.3).

###### Proof.

It is clear that $B^{\tilde{G}}_{\epsilon}$ restricted to the roots in
$M^{!}_{\epsilon}(F)=M^{!}_{\epsilon[s]}(F)$ equals
$B^{\tilde{M}}_{\epsilon}=B^{\tilde{M}}_{\epsilon[s]}$: all are described in
terms of root-lengths in $M^{!}_{\epsilon}$. ∎

######  Definition 5.3.7.

Suppose $M$ is a Levi subgroup of $G$ and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. Let
$\mathcal{O}^{!}$ be a stable semisimple conjugacy class in $M^{!}(F)$.

  * •

We say that $\mathcal{O}^{!}$ is $\tilde{G}$-equisingular if $\gamma$ is
$G$-equisingular (Definition 2.2.3) for all element $\gamma\in
M(F)_{\mathrm{ss}}$ corresponding to $\mathcal{O}^{!}$ under
$\Sigma_{\mathrm{ss}}(M^{!})\xrightarrow{\text{endoscopy}}\Sigma_{\mathrm{ss}}(M)$.

  * •

Denote by $SD_{\mathrm{geom},\tilde{G}\text{-equi}}(M^{!})$ the subspace
$\bigoplus_{\mathcal{O}^{!}}SD_{\mathrm{geom}}(M^{!},\mathcal{O}^{!})$ of
$SD_{\mathrm{geom}}(M^{!})$, where $\mathcal{O}^{!}$ ranges over the
$\tilde{G}$-equisingular classes.

The next result involves the Definitions 3.4.6 and 3.4.7.

######  Lemma 5.3.8 (Cf. [60, p.483]).

Let $M\subset G$,
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ and
$\mathcal{O}^{!}\subset M^{!}(F)_{\mathrm{ss}}$ be as above. For all
$\delta\in\mathcal{O}^{!}$ and $s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$,
if $\delta$ is $\tilde{G}$-equisingular then $\delta[s]$ is
$G^{!}[s]$-equisingular.

###### Proof.

There exist $\gamma\in M(F)_{\mathrm{ss}}$ and a diagram
$(\delta,B^{!},T^{!},B,T,\gamma)$, by Corollary 4.5.2. Since $A_{M}\subset
M_{\gamma}$, we have $\gamma$ is $G$-equisingular if and only if $A_{M}$ is
central in $G_{\gamma}$, or equivalently that all
$\alpha\in\Sigma(G_{\gamma},T)$ are trivial on $A_{M}$. There is a similar
criterion for the $G^{!}[s]$-equisingularity of $\delta[s]$.

By Proposition 3.4.9, $\delta[s]$ corresponds to $\gamma$ via
$\mathbf{G}^{!}[s]\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})$. The descent
of endoscopy in §4.4 implies that up to rescaling by a factor of two,
$\Sigma\left(G^{!}[s]_{\delta[s]},T^{!}\right)$ embeds into
$\Sigma\left(G_{\gamma},T\right)$, compatibly with the isomorphisms
$T^{!}\simeq T$ and $A_{M^{!}}\simeq A_{M}$ from the chosen diagram. As
$A_{M}$ is connected, the scaling is immaterial, so all
$\beta\in\Sigma\left(G^{!}[s]_{\delta[s]},T^{!}\right)$ are trivial on
$A_{M^{!}}$. ∎

######  Definition 5.3.9.

Let $M\subset G$ be a Levi subgroup and
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$. Given
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)$
and

$\delta\in\begin{cases}SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!}),&\text{when
$F$ is non-Archimedean},\\\
SD_{\mathrm{geom},\tilde{G}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!}),&\text{when
$F$ is Archimedean.}\end{cases}$

We define

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

where $f^{G^{!}[s]}:=\mathcal{T}_{\mathbf{G}^{!}[s],\tilde{G}}(f)\in
S\mathcal{I}(G^{!}[s])\otimes\operatorname{mes}(G^{!}[s])$.

Note that
$S^{G^{!}[s]}_{M^{!}}\left(\delta[s],B^{\tilde{G}},f^{G^{!}[s]}\right)$ is
well-defined in the Archimedean case by Lemma 5.3.8.

######  Definition–Proposition 5.3.10.

Let $M$ be a Levi subgroup of $G$. Suppose that

$\delta_{\mathbf{M}^{!}}\in
SD_{\mathrm{geom}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee},\quad\text{(resp.\
}SD_{\mathrm{geom},\tilde{G}\text{-equi}}(M^{!})\otimes\operatorname{mes}(M^{!})^{\vee}\text{
)}$

is given for each
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ when $F$ is
non-Archimedean (resp. Archimedean), and set

$\tilde{\gamma}:=\sum_{\mathbf{M}^{!}}\check{\mathcal{T}}_{\mathbf{M}^{!},\tilde{M}}\left(\delta_{\mathbf{M}^{!}}\right)\;\in\begin{cases}D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},&F:\;\text{non-
Archimedean}\\\
D_{\mathrm{geom},G\text{-equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},&F:\;\text{Archimedean}.\end{cases}$

Then
$I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f):=\sum_{\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{G})}I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta_{\mathbf{M}^{!}},f\right)$
depends only on $f$ and $\tilde{\gamma}$. This defines

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

for all

$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G),\quad\tilde{\gamma}\in\begin{cases}D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},&F:\;\text{non-
Archimedean}\\\
D_{\mathrm{geom},\text{equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},&F:\;\text{Archimedean}.\end{cases}$

Moreover, it is continuous in $f$ for each $\tilde{\gamma}$ when $F$ is
Archimedean.

###### Proof.

In view of Theorem 3.10.6 and the notation thereof, it suffice to show that
given a Levi subgroup $R\subset M$ and
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$, we have

$I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!}[s],\delta[s]^{M^{!}[s]},f\right)=I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!}[t],\delta[t]^{M^{!}[t]},f\right)$

for all $\delta\in
SD_{\mathrm{geom}}(R^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(R)^{!}$
where $\mathcal{O}^{!}$ is a stable semisimple class in $R^{!}(F)$ which is
$\tilde{G}$-equisingular when $F$ is Archimedean,
$s,t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$ and all $f$. Indeed, this is
immediate from the endoscopic descent formula (Proposition 5.4.4) to be proved
later on.

When $F$ is Archimedean, the continuity in $f$ results from that of
$I^{\mathcal{E}}_{\tilde{M}}(\mathbf{M}^{!},\delta,f)$ for various $\delta$.
In turn, this reduces to the continuity of
$S^{G^{!}[s]}_{M^{!}}(\delta[s],B^{\tilde{G}},\cdot)$ for all
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$. ∎

######  Theorem 5.3.11 (Local geometric theorem).

For all Levi subgroup $M\subset G$, we have
$I_{\tilde{M}}(\tilde{\gamma},f)=I^{\mathcal{E}}_{\tilde{M}}(\tilde{\gamma},f)$
for all

$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G),\quad\tilde{\gamma}\in\begin{cases}D_{\mathrm{geom},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},&F:\;\text{non-
Archimedean}\\\
D_{\mathrm{geom},G-\mathrm{equi},-}(\tilde{M})\otimes\operatorname{mes}(M)^{\vee},&F:\;\text{Archimedean}.\end{cases}$

The proof of Theorem 5.3.11 will be completed at the end of this work; see
§13.9. As the first step, we will reduce it to the $\tilde{G}$-regular case in
Proposition 5.5.2.

###  5.4 An endoscopic descent formula

Let $\tilde{M}\subset\tilde{G}$ be as in §5.3. We begin with some
combinatorics.

Assuming temporarily that $\tilde{G}$ is a metaplectic group
$\widetilde{\mathrm{Sp}}(W)$, we may write

$M=\prod_{i\in I}\operatorname{GL}(n_{i})\times\operatorname{Sp}(W^{\flat})$

where $I$ is a finite set, $(n_{i})_{i}\in\mathbb{Z}_{\geq 1}^{I}$ and
$W^{\flat}$ is a symplectic subspace of $W$. To prescribe $L\in\mathcal{L}(M)$
is the same as to prescribe

  * •

a finite set $K$ and $(l_{k})_{k\in K}\in\mathbb{Z}_{\geq 1}^{K}$,

  * •

a decomposition $I=I_{0}\sqcup I_{1}$ together with a surjection
$\eta:I_{0}\twoheadrightarrow K$,

  * •

a symplectic subspace $W^{\Diamond}$ of $W$ such that $W^{\Diamond}\supset
W^{\flat}$,

satisfying $\dim W=\dim W^{\Diamond}+2\sum_{k\in K}l_{k}$; specifically, we
have then

$M\subset L=\prod_{k\in
K}\operatorname{GL}(l_{k})\times\operatorname{Sp}(W^{\Diamond})\subset G,$

where $M\hookrightarrow L$ is given by

$\operatorname{Sp}(W^{\flat})\hookrightarrow\operatorname{Sp}(W^{\Diamond}),\quad\operatorname{GL}(n_{i})\hookrightarrow\begin{cases}\operatorname{GL}(l_{\eta(i)}),&i\in
I_{0}\\\ \operatorname{Sp}(W^{\Diamond}),&i\in I_{1}.\end{cases}$

Suppose $M\subset L\subset G$ are parameterized as above, and fix
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$ with

$M^{!}=\prod_{i\in
I}\operatorname{GL}(n_{i})\times\operatorname{SO}(2a^{\prime}+1)\times\operatorname{SO}(2a^{\prime\prime}+1),\quad
a^{\prime}+a^{\prime\prime}=\frac{1}{2}\dim W^{\flat}.$

Given $s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})$ and
$s_{L}\in\mathcal{E}_{\mathbf{L}^{!}[s^{L}]}(\tilde{G})$, they assemble into
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ with
$\mathbf{G}^{!}[s]=\mathbf{G}^{!}[s_{L}]$.

Consider now the reciprocal question: given $M\subset L\subset G$ and
$\mathbf{M}^{!}$, when does a given
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$ arise from a pair
$(s^{L},s_{L})$? Recall that $s$ corresponds to a decomposition
$I=I^{\prime}\sqcup I^{\prime\prime}$. It turns out that following condition
on $s$ is necessary and sufficient:

The fibers of $\eta:I_{0}\twoheadrightarrow K$ are either contained in
$I^{\prime}$ or in $I^{\prime\prime}$. (5.4.1)

When (5.4.1) holds for $(L,s)$, we may define
$s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})$ corresponding to

$I_{1}=I^{\prime}_{1}\sqcup I^{\prime\prime}_{1},\quad
I^{\prime}_{1}:=I_{1}\cap I^{\prime},\quad I^{\prime\prime}_{1}:=I_{1}\cap
I^{\prime\prime},$ (5.4.2)

as well as $s_{L}\in\mathcal{E}_{\mathbf{L}^{!}[s^{L}]}(\tilde{G})$
corresponding to

$K=K^{\prime}\sqcup K^{\prime\prime},\quad
K^{\prime}:=\left\\{k:\eta^{-1}(k)\subset I^{\prime}\right\\},\quad
K^{\prime}:=\left\\{k:\eta^{-1}(k)\subset I^{\prime\prime}\right\\}.$ (5.4.3)

Evidently, $\mathbf{G}^{!}[s]$ (relative to
$\mathbf{M}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{M})$) equals
$\mathbf{G}^{!}[s_{L}]$ (relative to
$\mathbf{L}^{!}[s^{L}]\in\mathcal{E}_{\operatorname{ell}}(\tilde{L})$). The
choice of $(s^{L},s_{L})$ is unique.

######  Remark 5.4.1.

It is often more flexible to work inside dual groups as in (3.5.2). Represent
$\mathbf{M}^{!}$ by an elliptic semisimple element
$s^{\flat}\in\tilde{M}^{\vee}$. Consider an element $s$ of
$s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}$. For
every $L\in\mathcal{L}(M)$, these data can always be decomposed into $s^{L}\in
s^{\flat}Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{L}^{\vee}}^{\circ}$ and
$s_{L}\in s^{L}Z_{\tilde{L}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}$.
However, even if the endoscopic datum
$\mathbf{G}^{!}[s]=\mathbf{G}^{!}[s_{L}]$ of $\tilde{G}$ is elliptic (i.e. if
$s$ lands in $\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$), it is not necessarily
the case for $\mathbf{L}^{!}[s^{L}]$ for $\tilde{L}$. This is why we impose
conditions like (5.4.1).

######  Lemma 5.4.2.

Suppose that we are in the situation (cf. (3.4.1)):

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

for some given $\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$
and $\mathbf{M}^{!}=\mathbf{M}^{!}[t]$ with
$t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$. Make the identifications
$\mathfrak{a}_{R^{!}}=\mathfrak{a}_{R}$ and
$\mathfrak{a}_{M^{!}}=\mathfrak{a}_{M}$ accordingly. There is a bijection

$\displaystyle\left\\{\begin{array}[]{r|l}(s,L^{!})&s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})\\\
&L^{!}\in\mathcal{L}^{G^{!}[s]}(R^{!})\end{array}\right\\}$
$\displaystyle\xrightarrow{1:1}\left\\{\begin{array}[]{r|l}(L,s)&L\in\mathcal{L}^{G}(R),\;s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G}),\\\
&r=r(s,t)\text{ satisfies \eqref{eqn:descent-ellipticity}}\\\
&\text{w.r.t.}\;R\subset L\subset G\end{array}\right\\}$

where $r(s,t)\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{G})$ is assembled from $t$
and $s$ as above. Specifically,

  * •

it is characterized by $\mathfrak{a}_{L}=\mathfrak{a}_{L^{!}}$ (as subspaces
of $\mathfrak{a}_{R}=\mathfrak{a}_{R^{!}}$);

  * •

if $(s,L^{!})$ is mapped to $(L,s)$, then $L^{!}=L^{!}[r^{L}]$.

###### Proof.

This is a metaplectic version of a similar assertion in [60, p.236] proved by
working on the dual side. One can either adapt the cited proof, or use the
direct arguments below.

First, taking $r:=r(s,t)$ yields embeddings

$\displaystyle\left\\{\begin{array}[]{r|l}(s,L^{!})&s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})\\\
&L^{!}\in\mathcal{L}^{G^{!}[s]}(R^{!})\end{array}\right\\}\hookrightarrow\left\\{\begin{array}[]{r|l}(s,L^{!})&r\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{G})\\\
&L^{!}\in\mathcal{L}^{G^{!}[r]}(R^{!})\end{array}\right\\},$
$\displaystyle\left\\{\begin{array}[]{r|l}(L,s)&L\in\mathcal{L}^{G}(R),\;s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G}),\\\
&r=r(s,t)\text{ satisfies \eqref{eqn:descent-ellipticity}}\\\
&\text{w.r.t.}\;R\subset L\subset
G\end{array}\right\\}\hookrightarrow\left\\{\begin{array}[]{r|l}(L,r)&L\in\mathcal{L}^{G}(R),\;r\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{G}),\\\
&r=r(s,t)\text{ satisfies \eqref{eqn:descent-ellipticity}}\\\
&\text{w.r.t.}\;R\subset L\subset G\end{array}\right\\},$

whose images are characterized by the condition (5.4.1) with respect to
$R\subset M\subset G$, and the same reason gives injectivity. We are thus
reduced to the case $M=R$, and $r=s$ in this case.

Given $(s,L^{!})$, write
$G^{!}[s]=\operatorname{SO}(2n^{\prime}+1)\times\operatorname{SO}(2n^{\prime\prime}+1)$
and

$L^{!}=\left(\prod_{k\in
K^{\prime}}\operatorname{GL}(l_{k})\times\operatorname{SO}(2b^{\prime}+1)\right)\times\left(\prod_{k\in
K^{\prime\prime}}\operatorname{GL}(l_{k})\times\operatorname{SO}(2b^{\prime\prime}+1)\right)\subset
G^{!}[s].$

The inclusions $M^{!}\hookrightarrow L^{!}\hookrightarrow G^{!}[s]$ of Levi
subgroups yield the data

$I^{\prime}=I^{\prime}_{0}\sqcup
I^{\prime}_{1},\;\eta^{\prime}:I^{\prime}_{0}\twoheadrightarrow
K^{\prime},\quad I^{\prime\prime}=I^{\prime\prime}_{0}\sqcup
I^{\prime\prime}_{1},\;\eta^{\prime\prime}:I^{\prime\prime}_{0}\twoheadrightarrow
K^{\prime\prime}.$

Take $I_{0}:=I^{\prime}_{0}\sqcup I^{\prime\prime}_{0}$,
$I_{1}:=I^{\prime}_{1}\sqcup I^{\prime\prime}_{1}$ (thus $I=I_{0}\sqcup
I_{1}$), $K:=K^{\prime}\sqcup K^{\prime\prime}$ and

$\eta:=\eta^{\prime}\sqcup\eta^{\prime\prime}:I_{0}\twoheadrightarrow K.$

Moreover, take a symplectic subspace $W^{\Diamond}$ of $W$ containing
$W^{\flat}$ with $\frac{1}{2}\dim W^{\Diamond}=b^{\prime}+b^{\prime\prime}$.
Then $(K,(l_{k})_{k},I_{0},I_{1},\eta)$ determines $L\in\mathcal{L}(M)$. By
construction, $\mathfrak{a}_{L}=\mathfrak{a}_{L^{!}}$ and (5.4.1) is satisfied
by $(L,s)$. It is also clear that $L^{!}=L^{!}[s^{L}]$ because (5.4.2) holds
automatically.

Reciprocally, given $(L,s)$ satisfying (5.4.1) with respect to $M\subset
L\subset G$, we construct $s^{L}\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{L})$
and $L^{!}:=L^{!}[s^{L}]$ as performed in (5.4.2) and (5.4.3), so that
$\mathfrak{a}_{L^{!}}=\mathfrak{a}_{L}$. Since $L$ (resp. $L^{!}$) is uniquely
determined by $\mathfrak{a}_{L}\subset\mathfrak{a}_{M}$ (resp.
$\mathfrak{a}_{L^{!}}\subset\mathfrak{a}_{M^{!}}$), these two maps are indeed
mutually inverse. ∎

######  Lemma 5.4.3.

In the setting of Lemma 5.4.2, fix $L\in\mathcal{L}^{G}(R)$ and assume
$d^{G}_{R}(M,L)\neq 0$. Then the map

$\displaystyle\left\\{\begin{array}[]{r|l}s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})&r=r(s,t)\text{
satisfies \eqref{eqn:descent-ellipticity}}\\\ &\text{w.r.t.}\;R\subset
L\subset G\end{array}\right\\}$
$\displaystyle\to\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})$ $\displaystyle s$
$\displaystyle\mapsto r^{L}$

is surjective, each fiber having cardinality equal to
$\\#\operatorname{ker}(p_{2})$ where
$p_{2}:Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}\to
Z_{\tilde{R}^{\vee}}^{\circ}/Z_{\tilde{L}^{\vee}}^{\circ}$ is the natural
homomorphism. Moreover, $p_{2}$ is surjective with finite kernel.

###### Proof.

In what follows, we interpret endoscopic data on dual groups; see §3.5. Take
dual Levi subgroups
$\tilde{M}^{\vee}\supset\tilde{R}^{\vee}\subset\tilde{L}^{\vee}$ inside
$\tilde{G}^{\vee}$. Fix a maximal torus $T^{\vee}\subset\tilde{R}^{\vee}$.
Suppose that $s^{\flat}\in T^{\vee}\subset\tilde{R}^{\vee}$ determines
$\mathbf{R}^{!}$; we may identify $t$ (resp. $s$) as an element of
$Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{R}^{\vee}}^{\circ}$ (resp.
$Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}$) such that
$s^{\flat}t\in\tilde{M}^{\vee}$ (resp. $s^{\flat}ts$) determines the elliptic
endoscopic datum $\mathbf{M}^{!}[t]$ (resp. $\mathbf{G}^{!}[s]$). Ditto for
$R\subset L$ and $r^{L}$, etc.

We contend that $s\mapsto r^{L}$ can be interpreted in terms of $p_{2}$; in
particular, $\\{s:s\mapsto r^{L}\\}$ has the same size as $p_{2}^{-1}(r^{L})$
if we view $r^{L}$ as an element of
$Z_{\tilde{R}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}$ giving rise to an
elliptic endoscopic datum of $\tilde{L}$.

Indeed, if one removes the exponent $\circ$, i.e. working with endoscopy of
$\operatorname{SO}(2n+1)$ instead of $\widetilde{\mathrm{Sp}}(2n)$, this is
exactly the arguments in [60, pp.236–237]. Adding $(\cdots)^{\circ}$ passes to
the metaplectic setting. The assertions about kernel and surjectivity of
$p_{2}$ also reduce to the version in loc. cit. ∎

######  Proposition 5.4.4.

Let $R$ be a Levi subgroup of $M$ and
$\mathbf{R}^{!}\in\mathcal{E}_{\operatorname{ell}}(\tilde{R})$. Fix a stable
semisimple class $\mathcal{O}^{!}$ in $R^{!}(F)$ which is
$\tilde{G}$-equisingular when $F$ is Archimedean. Then

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

for all $t\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{M})$,
$f\in\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})\otimes\operatorname{mes}(G)^{\vee}$
and $\delta\in
SD_{\mathrm{geom}}(R^{!},\mathcal{O}^{!})\otimes\operatorname{mes}(R)^{!}$.

###### Proof.

We may and do assume that $\tilde{G}=\widetilde{\mathrm{Sp}}(W)$ without loss
of generality. To see this, note that if
$\tilde{G}=\prod_{i}\operatorname{GL}(n_{i})\times\widetilde{\mathrm{Sp}}(W)$,
we have
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})=\bigotimes_{i}\mathcal{I}(\operatorname{GL}(n_{i}))\otimes\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\widetilde{\mathrm{Sp}}(W))$
when $F$ is non-Archimedean; in this case, $I_{\tilde{M}}^{\mathcal{E}}$,
$\mathcal{L}(R)$, $d^{G}_{R}(M,L)$ and $I^{\tilde{L},\mathcal{E}}_{\tilde{R}}$
all decompose accordingly so that we are reduced to the case of
$\operatorname{GL}(n_{i})$ (which is trivial) and the case of
$\widetilde{\mathrm{Sp}}(W)$. When $F$ is Archimedean,
$\mathcal{I}_{\dashrule[.7ex]{2222}{.4}}(\tilde{G})$ decomposes into a
completed $\otimes$-product of nuclear spaces; again, we are reduced to the
cases of $\operatorname{GL}(n_{i})$ and $\widetilde{\mathrm{Sp}}(W)$ by
continuity.

For the ease of notations, write $\mathbf{M}^{!}=\mathbf{M}^{!}[t]$. For every
$s\in\mathcal{E}_{\mathbf{M}^{!}}(\tilde{G})$, define
$r=r(s,t)\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{G})$ and
$r^{L}\in\mathcal{E}_{\mathbf{R}^{!}}(\tilde{L})$ as in Lemma 5.4.2. By
construction, $\mathbf{G}^{!}[s]=\mathbf{G}^{!}[r]$ and we have

$\delta[t]^{M^{!}}[s]=(\delta\cdot z[t])^{M^{!}}\cdot\underbracket{z[s]}_{\in
Z_{M^{!}}(F)}=(\delta\cdot z[s]\cdot z[t])^{M^{!}}=\delta[r]^{M^{!}}.$

By the stable descent formula [60, II.1.14 Proposition (ii)] and Remark 5.3.3,
we have

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

Next, use the bijection $(s,L^{!})\leftrightarrow(L,s)$ in Lemma 5.4.2 to
rewrite the above as the sum over $(L,s)$ of

$\displaystyle
i_{M^{!}}(\tilde{G},G^{!}[s])e^{G^{!}[r]}_{R^{!}}(M^{!},L^{!})S^{L^{!}}_{R^{!}}\left(\delta[r^{L}],B^{\tilde{L}},(f_{\tilde{L}})^{L^{!}}\right).$

This is based on the following observations:

  * •

we have $L^{!}=L^{!}[r^{L}]$ and the situation can be summarized as

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

  * •

$B^{\tilde{G}}|_{L^{!}}=B^{\tilde{L}}$ by Lemma 5.3.6;

  * •

$\delta[r]=\delta[r^{L}]\cdot z[r_{L}]$ and Proposition 3.8.4 entails

$S^{L^{!}}_{R^{!}}\left(\delta[r],B^{\tilde{L}},(f^{G^{!}[r]})_{L^{!}}\right)=S^{L^{!}}_{R^{!}}\left(\delta[r]\cdot
z[r_{L}]^{-1},B^{\tilde{L}},(f_{\tilde{L}})^{L^{!}}\right).$

Therefore

$I^{\mathcal{E}}_{\tilde{M}}\left(\mathbf{M}^{!},\delta[t]^{M^{!}},f\right)=\sum_{(L,s)}d^{G}_{R}(M,L)X(s)S^{L^{!}}_{R^{!}}\left(\delta[r^{L}],B^{\tilde{L}},(f_{\tilde{L}})^{L^{!}}\right)$
(5.4.4)

where $(s,L^{!})\leftrightarrow(L,s)$ as before and

$\displaystyle X(s)$
$\displaystyle:=i_{M^{!}}(\tilde{G},G^{!}[s])\left(Z_{{M^{!}}^{\vee}}\cap
Z_{{L^{!}}^{\vee}}:Z_{{G^{!}[s]}^{\vee}}\right)^{-1}$
$\displaystyle=i_{M^{!}}(\tilde{G},G^{!}[s])\cdot\\#\operatorname{ker}\left(\frac{Z_{{M^{!}}^{\vee}}}{Z_{{G^{!}[s]}^{\vee}}}\to\frac{Z_{{R^{!}}^{\vee}}}{Z_{{L^{!}}^{\vee}}}\right)^{-1}.$

We claim that

$i_{M^{!}}(\tilde{G},G^{!}[s])=\\#\operatorname{ker}\left(\phi^{G}_{M}:\frac{Z_{\tilde{M}^{\vee}}^{\circ}}{Z_{\tilde{G}^{\vee}}^{\circ}}\twoheadrightarrow\frac{Z_{{R^{!}}^{\vee}}}{Z_{{L^{!}}^{\vee}}}\right)^{-1}$

where $\phi^{G}_{M}$ is the evident map. To see this, apply Snake Lemma to

${1}$${Z_{\tilde{G}^{\vee}}^{\circ}}$${Z_{\tilde{M}^{\vee}}^{\circ}}$${Z_{\tilde{M}^{\vee}}^{\circ}/Z_{\tilde{G}^{\vee}}^{\circ}}$${1}$${1}$${Z_{{G^{!}[s]}^{\vee}}}$${Z_{{M^{!}}^{\vee}}}$${Z_{{M^{!}}^{\vee}}/Z_{{G^{!}[s]}^{\vee}}}$${0}$$\scriptstyle{\phi^{G}_{M}}$

and observe that $\phi^{G}_{M}$ is surjective since [15, Lemma 1.1] implies

$Z_{{M^{!}}^{\vee}}=Z_{{M^{!}}^{\vee}}^{\circ}Z_{{G^{!}[s]}^{\vee}}=Z_{\tilde{M}^{\vee}}^{\circ}Z_{{G^{!}[s]}^{\vee}}.$

For the same reason, when
$\mathfrak{a}^{M^{!}}_{R^{!}}\oplus\mathfrak{a}^{L^{!}}_{R^{!}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{a}^{G^{!}[s]}_{R^{!}}$
(equivalently, $d^{G}_{R}(M,L)\neq 0$) we have

$Z_{{R^{!}}^{\vee}}=Z_{{M^{!}}^{\vee}}Z_{{R^{!}}^{\vee}}^{\circ}=Z_{{M^{!}}^{\vee}}Z_{{M^{!}}^{\vee}}^{\circ}Z_{{L^{!}}^{\vee}}^{\circ}=Z_{{M^{!}}^{\vee}}Z_{{L^{!}}^{\vee}}.$

Therefore $Z_{{M^{!}}^{\vee}}/Z_{{G^{!}[s]}^{\vee}}\twoheadrightarrow
Z_{{R^{!}}^{\vee}}/Z_{{L^{!}}^{\vee}}$.

Assuming $d^{G}_{R}(M,L)\neq 0$, we arrive at the commutative diagram of
abelian groups