# Multiplicity formula and stable trace formula

and  Peng Zhifeng  Department of Mathematics, National University of Singapore
<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a connected reductive group over $\mathbb{Q}$. In this paper, we
give the stabilization of the local trace formula. In particular, we construct
the explicit form of the spectral side of the stable local trace formula in
the Archimedean case, when one component of the test function is cuspidal. We
also give the multiplicity formula for discrete series. At the same time, we
obtain the stable version of $L^{2}$-Lefschetz number formula.

##  1\. Introduction

Suppose that $G$ is a connected reductive group over $\mathbb{Q}$, and
$\Gamma$ is an arithmetic subgroup of $G(\mathbb{R})$ defined by congruence
conditions. Consider the regular representation $R$ with $G(\mathbb{R})$
acting on $L^{2}(\Gamma\backslash G(\mathbb{R}))$ through the right
translation. The fundamental problem is to decompose $R$ into a direct sum of
irreducible representations. In general, we decompose $R$ into two parts

$R=R_{\operatorname{disc}}\oplus R_{\operatorname{cont}},$

where $R_{\operatorname{disc}}$ is the sum of discrete series, and
$R_{\operatorname{cont}}$ is the continuous spectrum. The continuous spectrum
can be understood by Eisenstein series, which was studied by Langlands [23].
It suffices to study $R_{\operatorname{disc}}$. If $\pi_{\mathbb{R}}\in
R_{\operatorname{disc}}$ is an irreducible representation, we denote
$R_{\operatorname{disc}}(\pi_{\mathbb{R}})$ for the
$\pi_{\mathbb{R}}$-isotypical subspace of $R_{\operatorname{disc}}$. Then

$R_{\operatorname{disc}}(\pi_{\mathbb{R}})=\pi_{\mathbb{R}}^{\oplus
m_{\operatorname{disc}}(\pi_{\mathbb{R}})},$

where $m_{\operatorname{disc}}(\pi_{\mathbb{R}})$ is the multiplicity. A
classical problem is to find a finite summation formula for
$m_{\operatorname{disc}}(\pi_{\mathbb{R}})$.

If $\pi_{\mathbb{R}}$ belongs to the square integrable discrete series, and
$\Gamma\backslash G(\mathbb{R})$ is compact, then Langlands [21] gave a
formula for $m_{\operatorname{disc}}(\pi_{\mathbb{R}})$. If $\Gamma\backslash
G(\mathbb{R})$ is noncompact, the first result is for
$G(\mathbb{R})=SL_{2}(\mathbb{R})$, the formula for
$m_{\operatorname{disc}}(\pi_{\mathbb{R}})$ appeared in Selberg’s paper [26].
For $G$ having $\mathbb{R}$-rank one, there is a formula for it in [25]. In
general, for $G$ having any $\mathbb{R}$-rank, Arthur [3] studied the sum of
multiplicities

(1.1)
$\sum_{\pi_{\mathbb{R}}\in\Pi_{\operatorname{disc}}(\mu)}m_{\operatorname{disc}}(\pi_{\mathbb{R}},K_{0})$

by using the invariant trace formula, where the $L$-packet
$\Pi_{\operatorname{disc}}(\mu)$ is a finite set of discrete series
representations of $G(\mathbb{R})$ with the same infinitesimal character
$\mu$, and $K_{0}$ is an open compact subgroup of the finite adelic group
$G(\mathbb{A}_{\operatorname{fin}})$. More generally, we can consider Hecke
operators $h$ on $L^{2}(\Gamma\backslash G)$ that commute with $R$, and write
$R_{\operatorname{disc}}(\pi_{\mathbb{R}},h)$ for the restriction of $h$ to
$R_{\operatorname{disc}}(\pi_{\mathbb{R}})$. Arthur [3] obtained a formula for

(1.2)
$\sum_{\pi_{\mathbb{R}}\in\Pi_{\operatorname{disc}}(\mu)}\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h)),$

under a weak regularity condition on $\mu$. The spectral side of invariant
trace formula corresponds to (1.2), if the test function is taken to be a
stable cuspidal function $f_{\mu}$ associated to $\mu$. Therefore, the
explicit formula for (1.2) follows from the geometric side of invariant trace
formula. A key point is that the invariant distribution
$I_{M}(\gamma,f_{\mu})$ vanishes, if $\gamma$ is not semisimple.

We shall give a formula for the multiplicity of single representation
$m_{\operatorname{disc}}(\pi_{\mathbb{R}})$ and

$\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h)),$

which was conjectured by Spallone and Wakatsuki [31, Conjecture 1], who also
had checked two special cases. When one tries to use the invariant trace
formula to obtain the multiplicity formula, with the test function being the
pseudo-coefficient $f_{\pi_{\mathbb{R}}}$ for a single representation
$\pi_{\mathbb{R}}$, the invariant distribution
$I_{M}(\gamma,f_{\pi_{\mathbb{R}}})$ in general does not vanish for $\gamma$
having nontrivial unipotent part. Now we do not know how to obtain an explicit
formula for $I_{M}(\gamma,f_{\pi_{\mathbb{R}}})$ in this case. To capture a
single representation and obtain the stable cuspidal function, we need to use
the endoscopy theory and the stable trace formula.

Fortunately, when $G$ is a $K$-group, Arthur [9], [10], [11] has obtained the
stabilization of the general trace formula in 2003, assuming the Fundamental
Lemma and the the weighted Fundamental Lemma. In 2008, Ngo [20] proved the
Fundamental Lemma, and the weighted Fundamental Lemma was proved by
Chaudouard-Laumon. As a result, we have an unconditional stabilization of
global and local formal trace formula. From this we obtain the following
proposition by combining with the splitting formula.

######  Proposition 1.1.

For any $h\in\mathcal{H}(G(\mathbb{A}_{\operatorname{fin}})),$ and
$\pi_{\mathbb{R}}\in\Pi_{\operatorname{disc}}(\mu),$ with the infinitesimal
character $\mu$ being regular, we have

(1.3)
$\begin{split}\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h))&=I(f_{\pi_{\mathbb{R}}}h)\\\
&=\sum_{G^{{}^{\prime}}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{{}^{\prime}})\sum_{M^{{}^{\prime}}\in\mathcal{L}^{G^{{}^{\prime}}}}|W^{M^{{}^{\prime}}}_{0}||W^{G^{{}^{\prime}}}_{0}|^{-1}\sum_{\delta\in\Delta(M^{\prime},V,\zeta)}b^{M^{\prime}}(\delta)S^{G^{{}^{\prime}}}_{M^{{}^{\prime}}}(\delta,f_{\pi_{\mathbb{R}}})(h_{M})^{M^{{}^{\prime}}}(\delta)\end{split}$

See section 4 for more details. The stable coefficients
$b^{M^{\prime}}(\delta)$ and the stable distributions
$S^{G^{\prime}}_{M^{\prime}}(\delta,f_{\pi_{\mathbb{R}}})$ are not explicit.
However, the transfer function $f^{G^{\prime}}_{\pi_{\mathbb{R}}}$ is a stable
cuspidal function for endoscopic group $G^{\prime}$ of $G$, we obtain the
vanishing of $S^{G^{\prime}}_{M^{\prime}}(\delta,f_{\pi_{\mathbb{R}}})$, if
$\delta$ is not semisimple. It remains to study this distribution in the case
of semisimple elements of $G^{\prime}(\mathbb{R})$. We shall give the explicit
formula for the stable distribution
$S^{G^{\prime}}_{M^{\prime}}(\delta,f_{\pi_{\mathbb{R}}})$ through comparison
of the stable local trace formula with the stable Weyl integral formula. We
need to stabilize directly the spectral side of the local trace formula in the
Archimedean place.

In the $p$-adic case, when the test function is cuspidal at two places, Arthur
[7] gave a concrete formula about the geometric side of stable local trace
formula, which is just an inner product. We shall establish a formula in the
Archimedean case, with the test function being cuspidal at only one place, so
it is more complicated. It suffices to study the stable distribution on
quasisplit groups. Our work relies on the harmonic analysis on reductive
groups of Harish-Chandra [13], [14], [15], and on the Langlands [22]
classification of the irreducible representations of real algebraic groups. We
also need the work of Shelstad [29], [30], who classified the tempered
representations, directly constructed the spectral transfer factors, and gave
the inverse adjoint relations. For the stable local trace formula, we need to
build the transfer factors with respect to the virtual tempered
representations $\tau=(M,\pi,r)$ and the Langlands parameters. The main
obstruction is to show that the representation theoretic $R$-groups $R_{\pi}$
and the endoscopic $R$-groups are compatible. This is the essential part that
allows us to stabilize the local trace formula. We will also build the general
inverse adjoint relations, and obtain an explicit formula for the spectral
side of the stable local trace formula, c.f. Section $6$.

######  Theorem 1.2.

If $f=f_{1}\times\bar{f_{2}}$, $f_{1}\in
C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$, and $f_{2}\in
C(G(\mathbb{R}),\zeta)$, then

$\displaystyle I_{\operatorname{disc}}(f)=$
$\displaystyle\int_{T_{\operatorname{ell}}(G,\zeta)}i^{G}(\tau)f_{1,G}(\tau)\overline{f_{2,G}(\tau)}d\tau$
$\displaystyle=$
$\displaystyle\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\widehat{S}^{G^{\prime}}(f^{\prime}),$

and $\widehat{S}^{G^{\prime}}(f^{\prime})$ is a stable distribution on
$G^{\prime}$, where

$\displaystyle i^{G}(\tau)=$ $\displaystyle|d(\tau)|^{-1}|R_{\pi,r}|^{-1},$
$\displaystyle\widehat{S}^{G^{\prime}}(f^{\prime})=$
$\displaystyle\int_{\Phi_{2}(G^{\prime},\zeta)}S^{G^{\prime}}(\phi^{\prime})\widetilde{f_{1}}^{\prime}(\phi^{\prime})\overline{f_{2}^{\prime}(\phi^{\prime})}d\phi^{\prime},$
$\displaystyle S^{G^{\prime}}(\phi^{\prime})=$
$\displaystyle\frac{1}{|\mathcal{S}_{\phi^{\prime}}|},\qquad\phi=\xi^{\prime}\circ\phi^{\prime}.$

We shall obtain a stable local trace formula
$S_{\operatorname{geo}}=S_{\operatorname{spec}}$ in Theorem 7.1 and an
explicit formula for the stable distribution. We then have the following main
theorem.

######  Theorem 1.3.

If $h\in\mathcal{H}(G(\mathbb{A}_{\operatorname{fin}}))$, and the highest
weight of representation $\mu$ is regular, then we have

$\begin{split}&\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h))\\\
=&\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\sum_{M^{\prime}\in\mathcal{L}^{G^{\prime}}}(-1)^{\operatorname{dim}(A_{M^{\prime}}/A_{G^{\prime}})}|W^{M^{\prime}}_{0}||W^{G^{\prime}}_{0}|^{-1}\sum_{\delta\in\\{M^{\prime}(\mathbb{Q})\\}}P_{\mu}(M^{\prime})S\Phi_{M^{\prime}}(\phi^{\prime}_{\mu},\delta)(h_{M})^{M^{\prime}}(\delta),\end{split}$

and the multiplicity formula of the discrete series is

$m_{\operatorname{disc}}(\pi_{\mathbb{R}},K_{0})=\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},\mathit{1}_{K_{0}})).$

We will also give a stable $L^{2}$-Lefschetz formula, when the test function
is stable cuspidal. This case is also studied in an unpublished work of
Kottwitz [19].

######  Theorem 1.4.

For any $h\in\mathcal{H}(G(\mathbb{A}_{\operatorname{fin}}))$, we have

$\mathcal{L}_{\mu}(h)=\sum_{G^{{}^{\prime}}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{{}^{\prime}})\sum_{M^{{}^{\prime}}\in\mathcal{L}^{G^{{}^{\prime}}}}|W^{M^{{}^{\prime}}}_{0}||W^{G^{{}^{\prime}}}_{0}|^{-1}\sum_{\delta\in\\{M^{\prime}(\mathbb{Q})\\}}F_{\mu}(M^{\prime})S\Phi_{M^{\prime}}(\phi^{\prime},\delta)(h_{M})^{M^{\prime}}(\delta).$

The content of the paper is as follows. In section 2, we will introduce
$K$-groups, which are unions of connected reductive groups. If $F$ is a
$p$-adic field, then a $K$-group over $F$ is still a connected reductive
group. However, if $F$ is an Archimedean field, then a $K$-group over $F$ is
not connected. Any connected reductive group $G_{1}$ is a component of a
unique $K$-group $G$. The invariant and the stable distributions can be
extended to $K$-group as in [8].

In section 3, we will obtain the relation between multiplicity and invariant
trace formula. However when the test function is a pseudo-coefficient, we
cannot give an explicit invariant trace formula. This is because we cannot
cancel the contributions coming from the unipotent elements. For test
functions which are not stable, we need to stabilize the invariant trace
formula to overcome this obstruction. In general, the invariant trace formula
is the identity obtained from two different expansions of a certain linear
form $I(f)$. One side is the geometric expansion

$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{\gamma\in\Gamma(M,S)}a^{M}(S,\gamma)I_{M}(\gamma,f),$

which is a linear combination of distributions parameterized by conjugacy
classes $\gamma$ in Levi subgroups $M(F_{S})$. The other side is the spectral
expansion

$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Pi(M,S)}a^{M}(S,\pi)I_{M}(\pi,f)\,d\pi,$

which is a linear combination of distributions parameterized by
representations $\pi$ of Levi subgroup $M(F_{S})$. Here $f\in\mathcal{H}(G,S)$
in the Hecke algebra of $G(F_{S})$ (see [1], [2]), $S$ is the finite set of
place in $F$. Arthur [11] had stabilized the invariant trace formula. So we
can express the multiplicity by using the geometric side of the stable global
trace formula.

In section 4, We apply the splitting formula to reduce the local component of
the global trace formula for the Archimedean case. Compared with the local
trace formula, the local component of the global trace formula is more
complicated, because it contains contributions from nontrivial unipotent
elements. When the test function is stable cuspidal, those contributions
vanish, and the local components of the local trace formula coincide with the
local components of the global trace formula. Moreover, the pseudo-coefficient
of a representation can be transferred to a stable cuspidal function under the
Shelstad transfer mapping. So it is enough to study the stabilization of local
trace formula. In general, the geometric side of the local trace formula
concerns semisimple regular elements. The spectral side of the local trace
formula concerns the tempered representations and is thus much simpler in
comparison with the global trace formula.

The spectral side of the invariant local trace formula contains a natural
object, which is called the virtual character. A simple invariant local trace
formula can be given in terms of the virtual representation as in [5]. In
section 5, we introduce the virtual representation for the Archimedean case,
and define the transfer factors $\Delta(\tau,\phi)$ together with the inverse
factors $\Delta(\phi,\tau)$, building on Shelstad’s work [29]. Then we
stabilize the spectral side of the local trace formula. When one component of
the test function is cuspidal, we just need to consider elliptic
representations. In section 6, we obtain a formula for the spectral side of
the stable local trace formula.

In section 7, we study the main term $S^{G}_{M}(\delta,\phi)$. To do this, we
need to stabilize the Weyl integral formula, which connects between the
geometric and spectral sides of a local component of the local trace formula.
We can then compare with the stable local trace formula, and the main term of
the formula will appear.

In section 8, we establish the relation between $S^{G}_{M}(\delta,f)$ and the
invariant main term $\Phi_{M}(\gamma,f)$. This allows us to overcome the key
obstacle. We have $S^{G}_{M}(\delta,f_{\phi})=0$, if $\delta$ is not
semisimple, where $f_{\phi}$ is a stable cuspidal function which arises by a
transfer from a pseudo-coefficient of $\pi$. We shall collect various terms,
and they will be combined into our main formula in Theorem 8.3. In section 9,
we shall give a stable formula for $L^{2}$-Lefschetz number, when the test
function is stable cuspidal.

Acknowledgements. I would like to thank my advisor James Arthur’s help and
encouragement with this project. This paper is indebted to Chung Pang Mok for
much useful conversation. I would like to thank Bin Xu for helpful
conversation. We gratefully acknowledge generous support provided by the
National Natural Science Foundation of P.R. China Grant No.11601503.

##  2\. Preliminaries and distributions of K-groups

###  2.1. Notation

Let $G$ be a reductive group over $\mathbb{Q}$, $M$ be a Levi subgroup of $G$,
$P$ be a parabolic subgroup, and $A_{G}$ be the $\mathbb{Q}$-split component
of the center of $G$. If $\gamma$ is a semisimple element of $G$, we denote by
$G({\mathbb{Q},\gamma})$ the centralizer of $\gamma$ in $G(\mathbb{Q})$, and
write $G_{\gamma}$ for the identity component of $G(\mathbb{Q},\gamma)$. Write
$G_{\operatorname{der}}$ for the derived group of $G$, $G_{\operatorname{sc}}$
for the simply connected cover of $G_{\operatorname{der}}$. We say that $G$ is
cuspidal, if $G(\mathbb{R})$ contains a maximal $\mathbb{R}$-torus $T$ such
that $T/A_{G}$ is anisotropic over $\mathbb{R}$. In other words, the
$\mathbb{Q}$-split component coincide with $\mathbb{R}$-split component
$A_{G(\mathbb{R})}$ and the real group $G(\mathbb{R})$ contains an
$\mathbb{R}$-elliptic maximal torus. A torus $T$ in $G$ is elliptic if
$T/A_{G}$ is anisotropic. An element of $G(\mathbb{R})$ is elliptic if its
centralizer in $G(\mathbb{R})$ is an elliptic torus of $G$. Let
$X(G)_{\mathbb{R}}$ be the module of $\mathbb{R}$-rational character on $G$,
and $\mathfrak{a}_{G}=\operatorname{Hom}(X(G)_{\mathbb{R}},\mathbb{R})$.

We denote by $\mathbb{A}$ the ring of adéles of $\mathbb{Q}$, and denote by
$\mathbb{A}_{\operatorname{fin}}$ the finite part of adéles ring over
$\mathbb{Q}$, so that
$\mathbb{A}=\mathbb{R}\times\mathbb{A}_{\operatorname{fin}}$. We denote by
$G^{\prime}$ an endoscopy group of $G$. The center of $G$ is denoted as
$Z(G)$, and we will denote by $Z$ a central induced torus in $G$ over
$\mathbb{Q}$. Throughout the paper, $F$ will be a field of characteristic 0.

###  2.2. $K$-groups

$K$-groups are natural objects for studying the stabilization of general trace
formula, which contain several connected components. To work with several
groups simultaneously is suitable for studying the transfer properties of the
various objects in the trace formula. The use of several inner forms is due
originally to Vogan. When Kottwitz learned of Vogan’s idea, he applied it to
the Langlands-Shelstad transfer factors. We shall follow Arthur’s discussion
[8], where he extended the geometric transfer factors to $K$-groups.

######  Definition 2.1.

$G$ is called a $K$-group over a local field $F$, if

  1. (1)

$G$ is an algebraic variety whose connected components are reductive algebraic
groups over $F$, endowed with an equivalence class of objects
$\\{(\psi,u)\\}$. Here
$(\psi,u)=\\{(\psi_{\alpha\beta},u_{\alpha\beta}):\alpha,\beta\in\pi_{0}(G)\\}$,
$\psi_{\alpha\beta}:G_{\beta}\to G_{\alpha}$ is an isomorphism over
$\bar{\mathnormal{F}}$, and $u_{\alpha\beta}:\Gamma\to
G_{\alpha,\operatorname{sc}}$ is an 1-cocycle, where
$\Gamma=\operatorname{Gal}(\bar{\mathnormal{F}}/\mathnormal{F})$. We require
that $\\{(\psi,u)\\}$ satisfy the compatibility conditions,

(i)
$\psi_{\alpha\beta}\tau(\psi_{\alpha\beta})^{-1}=\operatorname{Int}(u_{\alpha\beta}(\tau));$

(ii) $\psi_{\alpha\gamma}=\psi_{\alpha\beta}\psi_{\beta\gamma};$

(iii)
$u_{\alpha\gamma}(\tau)=\psi_{\alpha\beta,\operatorname{sc}}(u_{\beta\gamma}(\tau))u_{\alpha\beta}(\tau),$
for any $\alpha,\beta,\gamma\in\pi_{0}(G)$ and $\tau\in\Gamma$,

  2. (2)

the corresponding sequence

$\\{1\\}\longrightarrow\\{u_{\alpha\beta}:\beta\in\pi_{0}(G)\\}\longrightarrow\mathnormal{H}^{1}(F,G_{\alpha})\xrightarrow{K_{G_{\alpha}}}\pi_{0}(Z(\widehat{G})^{\Gamma})^{\ast}.$

of pointed sets is exact. Here $\alpha\in\pi_{0}(G)$, the map $K_{G_{\alpha}}$
is defined in [17, §1].

The notation $\pi_{0}(G)$ is a set of indices for the components of $G$, and
we write $\pi_{0}(Z(\widehat{G})^{\Gamma})$ for the set of connected
components of $Z(\widehat{G})^{\Gamma}$ as usual.

We say that two such families $(\psi,u)$ and
$(\psi^{{}^{\prime}},u^{{}^{\prime}})$ are equivalent, if there are elements
$g_{\alpha\beta}\in G_{\alpha,\operatorname{sc}}$ such that
$\psi^{{}^{\prime}}_{\alpha\beta}=\operatorname{Int}(g_{\alpha\beta})\psi_{\alpha\beta}$
and
$u^{{}^{\prime}}_{\alpha\beta}(\tau)=g_{\alpha\beta}u_{\alpha\beta}(\tau)\tau(g_{\alpha\beta})^{-1}$,
for any $\alpha,\beta\in\pi_{0}(G)$ and $\tau\in\Gamma$. We call a
representative $(\psi,u)$ from the equivalence class as a frame for $G$. If
$F$ is $p$-adic, $K_{G_{\alpha}}$ is a bijection [17, Theorem 1.2], so a
$K$-group is just a connected reductive group. If $F$ is Archimedean, the
kernel of $K_{G_{\alpha}}$ is the image of
$\mathnormal{H}^{1}(F,G_{\alpha,\operatorname{sc}})$ in
$\mathnormal{H}^{1}(F,G_{\alpha})$ [17, Theorem 1.2], and the number of
components of a $K$-group over $\mathbb{R}$ therefore is equal to the number
of classes in this image.

Suppose that $G$ is a $K$-group, then we can write
$G=\coprod_{\alpha\in\pi_{0}(G)}G_{\alpha}$. A homomorphism between $K$-groups
$G$ and $\bar{G}$ over $F$ is a morphism

$\theta=\coprod_{\alpha}(\theta_{\alpha}:G_{\alpha}\to\bar{G}_{\bar{\alpha}})$

from $G$ to $\bar{G}$ (as varieties over $F$) that preserves all the
structures. In other words, it satisfies the following two properties:

  1. (1)

For any $\alpha\in\pi_{0}(G)$, and $\bar{\alpha}=\theta(\alpha)$ the image of
$\alpha$ in $\pi_{0}(G)$, the restriction
$\theta_{\alpha}:G_{\alpha}\rightarrow\bar{G}_{\bar{\alpha}}$ is a
homomorphism of connected algebraic groups.

  2. (2)

There are frames $(\psi,u)$ and $(\bar{\psi},\bar{u})$ for $G$ and $\bar{G}$,
such that
$\theta_{\alpha}\circ\psi_{\alpha\beta}=\bar{\psi}_{\bar{\alpha}\bar{\beta}}\circ\theta_{\beta}$,
and
$\bar{u}_{\bar{\alpha}\bar{\beta}}=\theta_{\alpha,\operatorname{sc}}(u_{\alpha\beta})$,
for each $\alpha,\beta\in\pi_{0}(G)$.

An isomorphism of $K$-groups is an invertible homomorphism. Arthur introduces
the notion of weak isomorphism in [9, §4]. It satisfies all the requirements
of an isomorphism except for the condition relating
$u_{\bar{\alpha}\bar{\beta}}$ with $u_{\alpha\beta}$, so that one can identify
$K$-groups that differ only by the choice of functions
$\\{u_{\alpha\beta}\\}$. If we are given a connected reductive group $G_{1}$
over $F$, we can find a $K$-group $G$ over $F$, such that
$G_{\alpha_{1}}=G_{1}$ for some $\alpha_{1}\in\pi_{0}(G)$. There could be
several such $G$, but the weak isomorphism class of $G$ is uniquely determined
by $G_{1}$. In particular, any connected quasisplit group $G^{\ast}$ has a
quasisplit inner $K$-form $G$, which is unique up to weak isomorphism. We say
that $K$-group $G$ is quasisplit if it has a connected component that is
quasisplit over $F$.

The Levi subgroups $M$ of a $K$-group $G$ was defined [8, §1]. For any such
$M$, we construct the associated objects $W(M)$, $\mathcal{P}(M)$,
$\mathcal{L}(M)$ and $\mathcal{F}(M)$ as in [8, §1], which represent the Weyl
group, the set of the parabolic subgroups for which the component of Levi
subgroup equals $M$, the set of Levi subgroups which contain the Levi subgroup
$M$ and the set of parabolic subgroups for which the component of the Levi
subgroups contain the Levi subgroup $M$ respectively. They play the same role
as in the connected case. We can also form a dual group $\widehat{G}$ for $G$,
and a dual Levi subgroup $\widehat{M}\subset\widehat{G}$ for $M$. For any such
group $\widehat{M}$, we also have the analogue object
$\mathcal{P}(\widehat{M})$, $\mathcal{L}(\widehat{M})$ and
$\mathcal{F}(\widehat{M})$, with the understanding that the sets contain only
$\Gamma$-stable elements. It comes with a bijection $L\to\widehat{L}$ from
$\mathcal{L}(M)$ to $\mathcal{L}(\widehat{M})$, and a bijection
$P\to\widehat{P}$ from $P(M)$ to $P(\widehat{M})$.

Invariant harmonic analysis for connected real groups extends in a natural way
to $K$-groups. For example, we have the Harish-Chandra’s Schwartz space,

$C(G)=\bigoplus_{\alpha\in\pi_{0}(G)}C(G_{\alpha})$

on $G(\mathbb{R})$, and its invariant analogue

$I(G)=\bigoplus_{\alpha\in\pi_{0}(G_{\alpha})}I(G_{\alpha}).$

Elements in $C(G)$ are the functions on $G(\mathbb{R})$, and elements in
$I(G)$ can be regarded as the functions on the disjoint union

$\Pi_{\operatorname{temp}}(G)=\coprod_{\alpha\in\pi_{0}(G)}\Pi_{\operatorname{temp}}(G_{\alpha})$

of sets of irreducible tempered representations on the groups
$G_{\alpha}(\mathbb{R})$, or as functions on the disjoint union

$\Gamma_{\operatorname{reg}}(G)=\coprod_{\alpha\in\pi_{0}(G)}\Gamma_{\operatorname{reg}}(G_{\alpha})$

of the sets of strongly regular conjugacy classes in the groups
$G_{\alpha}(\mathbb{R})$.

For purpose of induction argument, it is convenient to fix a central character
datum $(Z,\zeta)$ for $G$, where $Z$ is an induced torus over $\mathbb{R}$,
with the central embedding $Z\to Z_{\alpha}\subset{G_{\alpha}}$ that is
compatible with isomorphisms $\psi_{\alpha\beta}$. The second component
$\zeta$ is a character on $Z(\mathbb{R})$, which corresponds to a character
$\zeta_{\alpha}$ on $Z_{\alpha}(\mathbb{R})$ for each $\alpha$.

We can then form the space

$C(G,\zeta)=\bigoplus_{\alpha\in\pi_{0}(G)}C(G_{\alpha},\zeta_{\alpha})$

of $\zeta^{-1}$ equivariant Schwartz functions on $G(\mathbb{R})$, and its
invariant analogue

$I(G,\zeta)=\bigoplus_{\alpha\in\pi_{0}(G)}I(G_{\alpha},\zeta_{\alpha}).$

Elements in $I(G,\zeta)$ may be regarded either as $\zeta^{-1}$-equivariant
functions on $\Pi_{\operatorname{temp}}(G,\zeta)$ or on
$\Gamma_{\operatorname{reg}}(G/Z)$.

If $\gamma$ lies in $\Gamma(G_{\alpha})$, we write $G_{\gamma}$ for the
centralizer in $G_{\alpha}$ of (some representative of) $\gamma$. Two classes
$\gamma_{1}$ and $\gamma_{2}$ in $\Gamma(G)$ with
$\gamma_{i}\in\Gamma(G_{\alpha_{i}})$ for $i=1,2$ are stably conjugate, if
$\psi_{\alpha_{1}\alpha_{2}}(\gamma_{2})$ is conjugate in
$G_{\alpha_{1}}(\bar{F})$ to $\gamma_{1}$, for any frame $(\psi,u)$. We can
then write $\Delta_{\operatorname{reg}}(G(F))$ for the set of strongly regular
stable conjugacy classes in $G(F)$. There is a canonical injection
$\delta\to\delta^{\ast}$ from $\Delta_{\operatorname{reg}}(G)$ to the set
$\Delta_{\operatorname{reg}}(G^{\ast})=\Delta_{\operatorname{reg}}(G^{\ast}(F))$
of strongly regular stable conjugacy classes in the quasisplit inner twist
$G^{\ast}(F)$.

An endoscopic datum for $G$ is defined entirely in terms of the dual group
$\widehat{G}$, and is therefore no different from the connected case.
$\mathcal{E}(G)$ will stand for the set of isomorphism classes of endoscopic
data for $G$ that are relevant to $G$. An element in $\mathcal{E}(G)$ is
therefore the image of some elliptic endoscopic datum
$M^{{}^{\prime}}=(M^{{}^{\prime}},\mathcal{M}^{{}^{\prime}},s^{{}^{\prime}},\xi^{{}^{\prime}})$
in $\mathcal{E}_{\operatorname{ell}}(M)$, for a Levi subgroup $M$ of $G$ and a
dual Levi subgroup $\widehat{M}$ of $\widehat{G}$. Here the elliptic datum
means that the image of $\mathcal{M}^{{}^{\prime}}$ in ${}^{L}M$ is contained
in no proper parabolic subgroup ${}^{L}M$, or equivalently that
$(Z(\mathcal{M^{\prime}})^{\Gamma})^{0}=(Z(\widehat{M})^{\Gamma})^{0}$. The
set $\mathcal{\mathcal{E}}(G)$ embeds into the larger set
$\mathcal{\mathcal{E}}(G^{\ast})$, which we identify as the collection of all
isomorphism classes of endoscopic data for $G$. For each
$G^{\prime}=(G^{\prime},\mathcal{G}^{\prime},s,\xi^{\prime})\in\mathcal{\mathcal{E}}(G^{\ast})$,
we fix a central extension as in [7, §2].

$1\longrightarrow\widetilde{Z}^{\prime}\longrightarrow\widetilde{G}^{\prime}\longrightarrow
G^{\prime}\longrightarrow 1$

of $G^{\prime}$ by a central induced torus of $\widetilde{Z}^{\prime}$. Then
there exists an $L$-morphism
$\widetilde{\xi^{\prime}}:\mathcal{G}^{\prime}\to{}^{L}\widetilde{G}^{\prime}$.

A $K$-group is a natural domain for the transfer factors of [24]. If $F$ is
Archimedean, Arthur extends the transfer factors to the $K$-groups. It is
known [30] that the set of conjugacy classes in the stable conjugacy classes
of $K$-group can be parametrized by the set
$\mathcal{E}(T)=\operatorname{Im}(\mathnormal{H}^{1}(\Gamma,T_{\operatorname{sc}})\to\mathnormal{H}^{1}(\Gamma,T))$,
where $T$ is the maximal torus of $G$. When $G$ is a connected group, then
it’s parametrized by a subset $\mathscr{D}(T)$ of $\mathcal{\mathcal{E}}(T)$.
Here
$\mathscr{D}(T)=\operatorname{Ker}(\mathnormal{H}^{1}(\Gamma,T)\to\mathnormal{H}^{1}(\Gamma,G))$,
where $T$ is a compact maximal torus. Similarly, an $L$-packet of discrete
series representations is parametrized by $\mathcal{E}(T)$.

For example, If we consider the connected group
$G^{{}^{\prime}}=G^{{}^{\prime}}_{\operatorname{sc}}=SU(2,1)$ over
$\mathbb{R}$, its $K$-group is $G=G^{{}^{\prime}}\coprod G_{1}$, where
$G_{1}=SU(3)$. Then a stable conjugacy class of regular elliptic elements of
$G{{}^{\prime}}$ consists of three conjugacy classes parametrized by three of
the four elements of $\mathnormal{H}^{1}(\Gamma,T)$. Similarly, an L-packet of
discrete series representations of $G^{\prime}$ is parametrized by three
elements of the same group, and we can obtain the other conjugacy class and
the other representation from $G_{1}$.

We also extend Langlands-Shelstad transfer mapping to $K$-groups.

$\varphi:\mathcal{H}(G,\zeta)\to
SI(\widetilde{G}^{{}^{\prime}},\widetilde{\zeta}^{{}^{\prime}})$
$\varphi(f)=f^{{}^{\prime}}(\delta^{{}^{\prime}})=\sum_{\gamma\in\Gamma_{\operatorname{reg}}(G,\zeta)}\Delta(\delta^{{}^{\prime}},\gamma)f_{G}(\gamma).$

Here $\mathcal{H}(G,\zeta)$ is the Hecke algebra.
$SI(\widetilde{G}^{{}^{\prime}},\widetilde{\zeta^{{}^{\prime}}})$ is the space
of stable orbital integrals of functions. If $F$ is non-archimedean, this is
the main result of Waldspurger [33] and Ngo [20]. If $F$ is Archimedean, the
result was proved in Shelstad’s paper [27].

If $F$ is a global field, then there is a notion of $K$-group over $F$. Such a
$G$ satisfies the global analogue of the property as above, together with a
local product structure. A local product structure on $G$ is a family of local
$K$-groups $(G_{v},F_{v})$, indexed by the valuations of $F$, and a family of
homomorphisms $G\to G_{v}$ over $F_{v}$ whose restricted direct product
$G(\mathbb{A})\to\prod_{v}G_{v}(F_{v})$ is an isomorphism over $\mathbb{A}$.
Such a structure determines a surjective map

$\alpha\mapsto\alpha_{V}=\prod_{v\in
V}\alpha_{v},\qquad\alpha\in\pi_{0}(G),\alpha_{v}\in\pi_{0}(G_{v}),$

of components. We also have a group theoretic injection of $G_{\alpha}(F)$ to
$G_{\alpha_{v}}(F_{v})$ for each $\alpha\in\pi_{0}(G)$. We shall write

$G_{V}(F_{V})=\prod_{v\in V}G_{v}(F_{v})=\prod_{v\in
V}\coprod_{\alpha_{v}}G_{v,\alpha_{v}}(F_{v})=\coprod_{\alpha_{V}}G_{V,\alpha_{V}}(F_{V}).$

Suppose that $G$ is a $K$-group over $F$, and $G^{\ast}$ is a quasisplit inner
twist of $G$. Then $G^{\ast}$ is a connected quasisplit group over $F$,
together with a inner class of inner twists $\psi_{\alpha}:G_{\alpha}\to
G^{*}$ and a corresponding family of functions $u_{\alpha}:\Gamma\to
G^{\ast}_{\operatorname{sc}}$, for $\alpha\in\pi_{0}(G)$. Then $G^{\ast}$
determines a quasisplit inner twist $G^{\ast}_{v}$ of each local $K$-group
$G_{v}$. We shall refer to $G$ as an inner $K$-form of $G^{\ast}$.

If $\gamma_{V}$ is an element in the set $\Gamma_{G_{V}}(M_{V})$, let
$\alpha_{V}\in\pi_{0}(M_{V})$ be the index such that $\gamma_{V}$ belongs to
$\Gamma_{G_{V}}(M_{\alpha_{V}})$. We define the weighted orbital integral of
$f$ at $\gamma_{V}$ by

$J_{M_{V}}(\gamma_{V},f_{V})=J_{M_{V}}(\gamma_{V},f_{\alpha_{V}})\qquad
f_{V}\in\mathcal{H}(G_{V},\zeta_{V}),$

where $J_{M_{V}}(\gamma_{V},f_{\alpha_{V}})$ is the weighted orbital integral
on $G_{\alpha_{V}}(F_{V})$. Similarly, we set

$I_{M_{V}}(\gamma_{V},f_{V})=I_{M_{V}}(\gamma_{V},f_{\alpha_{V}})\qquad
f_{V}\in\mathcal{H}(G_{V},\zeta_{V}),$

where $I_{M_{V}}(\gamma_{V},f_{\alpha_{V}})$ is the invariant distribution on
$G_{\alpha_{V}}(F_{V})$.

If $f_{V}=\oplus_{\alpha_{V}}f_{\alpha_{V}}$ in
$\mathcal{H}(G_{V},\zeta_{V})$, $f_{G_{V}}(\gamma_{V})$ denotes the invariant
orbital integral $I_{G_{V}}(\gamma_{V},f_{\alpha_{V}})$,

$f^{{}^{\prime}}_{V}(\delta^{{}^{\prime}}_{V})=f^{\widetilde{G}^{{}^{\prime}}_{V}}_{V}(\delta^{{}^{\prime}}_{V})=\sum_{\gamma_{V}\in\Gamma(G_{V})}\Delta_{G_{V}}(\delta^{{}^{\prime}}_{V},\gamma_{V})f_{G_{V}}(\gamma_{V}),\qquad\delta^{{}^{\prime}}_{V}\in\bigtriangleup_{G_{V}}(\widetilde{G}^{{}^{\prime}}_{V}),$

then $f^{{}^{\prime}}_{V}=\oplus_{\alpha_{V}}f^{{}^{\prime}}_{\alpha_{V}}$.
The Langlands-Shelstad transfer Theorem, applied to each of the groups
$G_{\alpha_{V}}$, asserts that $f_{V}^{{}^{\prime}}$ belongs to the space
$SI\mathcal{H}(\widetilde{G}^{{}^{\prime}}_{V},\widetilde{\zeta}^{{}^{\prime}}_{V})$
of stable orbital integrals of functions in
$\mathcal{H}(\widetilde{G}^{{}^{\prime}}_{V},\widetilde{\zeta}^{{}^{\prime}}_{V})$,
where $\widetilde{G}^{{}^{\prime}}_{V}$ comes with a central data
$(\widetilde{Z}^{{}^{\prime}}_{V},\widetilde{\zeta}_{V}^{{}^{\prime}})$.

Similarly, we can define the objects on a $K$-group on the spectral side. We
will stabilize the spectral side of the local trace formula for the $K$-group
over $\mathbb{R}$ in section 5, section 6, and section 7.

##  3\. Multiplicity of discrete series and invariant trace formula

Suppose that $F$ is a number field, and $G$ is a connected reductive $K$-group
over $F$. We can form the adélic ring
$\mathbb{A}_{F}=\prod_{v}^{\prime}F_{v}$, and the group of adélic points of
$G$ is $G(\mathbb{A}_{F})=\prod_{v}^{\prime}G(F_{v})$.

Automorphic representations of $G$ over $F$ are irreducible constituents of
the right regular representation of $R$, defined by

$R(x)\varphi(y)=\varphi(yx),\qquad\varphi\in L^{2}(G(F)\backslash
G(\mathbb{A}_{F})).$

The fundamental problem in the automorphic representation theory is to
decompose $R$. It is well known that $R$ can be decomposed into a discrete
spectrum and a continuous spectrum,

$R=R_{\operatorname{disc}}\oplus R_{\operatorname{cont}}.$

Langlands [23] has studied the continuous spectrum by Eisenstein series. So it
remains to study the discrete part

(3.1)
$R_{\operatorname{disc}}=\oplus_{\pi\in\Pi_{\operatorname{disc}}(G(\mathbb{A}))}m_{\operatorname{disc}}(\pi)\pi$

where $\Pi_{\operatorname{disc}}(G(\mathbb{A}))$ stands for the set of
equivalence classes of irreducible representations of $G(\mathbb{A})$ on
$L^{2}_{\operatorname{disc}}(G(F)\backslash G(\mathbb{A}_{F}))$.

Suppose $\pi$ is an irreducible automorphic representation of
$G(\mathbb{A}_{F})$. Then we have a decomposition

$\pi=\otimes_{v}^{\prime}\pi_{v},$

such that

  1. (1)

$\pi_{v}$ is an irreducible admissible representation of $G(F_{v})$;

  2. (2)

$\pi_{v}$ is unramified for almost all $v$.

We write $\pi=\pi_{\mathbb{R}}\otimes\pi_{\operatorname{fin}}$, where
$\pi_{\mathbb{R}}$ and $\pi_{\operatorname{fin}}$ are irreducible
representations of $G(\mathbb{R})$ and $G(\mathbb{A}_{\operatorname{fin}})$
respectively. In this paper we study the multiplicity formula of
$m_{\operatorname{disc}}(\pi_{\mathbb{R}})$ for connected $K$-group over
$\mathbb{Q}$. The original problem comes from [3, p,284], where Arthur applied
the invariant trace formula to compute the $L^{2}$-Lefschetz numbers of Hecke
operators. He obtained the sum of multiplicity formula for
$\sum_{\pi\in\Pi_{\operatorname{disc}}(\mu)}m_{\operatorname{disc}}(\pi_{\mathbb{R}},K_{0})$,
under a weak regularity assumption on the representations in
$\Pi_{\operatorname{disc}}(\mu)$, where $G$ is a connected reductive group,
and $K_{0}$ is an open compact subgroup of the finite adélic group
$G(\mathbb{A}_{\operatorname{fin}})$. The packet
$\Pi_{\operatorname{disc}}(\mu)$ consists of the set of discrete series
representations with the same infinitesimal character $\mu$. We will give a
formula for single multiplicity
$m_{\operatorname{disc}}(\pi_{\mathbb{R}},K_{0})$ again under regularity
assumption on infinitesimal character on $\pi_{\mathbb{R}}$. Since the double
coset space

$G(\mathbb{Q})\backslash G(\mathbb{A})/G(\mathbb{R})K_{0}$

is finite, we denote by ${x_{1}=1,x_{2}\cdot\cdot\cdot,x_{n}}$ the set of
representatives in $G(\mathbb{A}_{\operatorname{fin}})$. The groups

$\Gamma_{i}=(G(\mathbb{Q})\cdot x_{i}K_{0}x_{i}^{-1})\cap G(\mathbb{R}),1\leq
i\leq n$

are arithmetic subgroups of $G(\mathbb{R})$, and $G(\mathbb{Q})\backslash
G(\mathbb{A})/K_{0}$ is the disjoint union of space $\Gamma_{i}\backslash
G(\mathbb{R})$. The question of the multiplicity is quite natural from the
point of view of spectral theory. More generally, one can consider Hecke
operators $h$ on $L^{2}(G(F)\backslash{G(\mathbb{A})})$, where $h$ is a
$K_{0}$-bi-invariant function in
$\mathcal{H}(\mathbb{A}_{\operatorname{fin}})$. Any such operator commutes
with the action of $G(\mathbb{R})$. Its restriction to the subspace
$R_{\operatorname{disc}}(\pi_{\mathbb{R}})$ is denoted as
$R_{\operatorname{disc}}(\pi_{\mathbb{R}},h)$.

If $G$ is a connected reductive group over $\mathbb{Q}$, the multiplicities of
discrete series have a homological interpretation. The global multiplicity
$m_{\operatorname{disc}}(\pi)$ occurs in the well known isomorphism

$\mathnormal{H}^{q}_{(2)}(h,\mathcal{F}_{\mu})\cong\bigoplus_{\pi\in\Pi(G(\mathbb{A}),\zeta)}(m_{\operatorname{disc}}(\pi)\operatorname{dim}\mathnormal{H}^{q}(\mathfrak{g}(\mathbb{R}),K_{\mathbb{R}};\pi_{\mathbb{R}}\otimes\mu))\pi_{\operatorname{fin}}(h).$

(see [3, §2], the coefficient $m_{\operatorname{disc}}(\pi)$ stands for the
multiplicity of $\pi$ in $R_{\operatorname{disc}}$), where
$\Pi(G(\mathbb{A}),\zeta)$ is the set of equivalence classes of irreducible
representations of $G(\mathbb{A})$, whose the central character coincides with
a given quasi-character $\zeta$ on $Z$. $\mathfrak{g}$ is the Lie algebra of
$G$; Both of $\mathfrak{g}(\mathbb{R})$ and $K_{\mathbb{R}}$ act on the space
of $K_{\mathbb{R}}$-finite vectors of the representation
$\pi_{\mathbb{R}}\otimes\mu$ of $G(\mathbb{R})$. The relative Lie algebra
cohomology groups
$\mathnormal{H}^{q}(\mathfrak{g}(\mathbb{R}),K_{\mathbb{R}};\pi_{\mathbb{R}}\otimes\mu)$
give the contribution of $\pi_{\mathbb{R}}$ to the cohomology.
$V(\pi^{K_{0}}_{\operatorname{fin}})$ denotes the subspace of vectors in the
underlying space of $\pi_{\operatorname{fin}}$ which are fixed by $K_{0}$,
this is a finite dimensional subspace, which gives the contribution of
$\pi_{\operatorname{fin}}$ to the cohomology. If the multiplicity of
$\pi_{\mathbb{R}}$ occurs discretely in the representation of $G(\mathbb{R})$
on $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{0},\zeta)$, then

(3.2)
$m_{\operatorname{disc}}(\pi_{\mathbb{R}})=\sum_{\begin{subarray}{c}\pi=\pi_{\mathbb{R}}\otimes\pi_{\operatorname{fin}}\\\
\pi\in\Pi(G(\mathbb{A}),\zeta)\end{subarray}}m_{\operatorname{disc}}(\pi)\operatorname{dim}(V(\pi^{K_{0}}_{\operatorname{fin}})).$

A representation of $K$-group $G$ is determined by the representations of
connected components of $G$. If
$\pi_{\alpha}\in\Pi(G_{\alpha},\zeta),f_{\alpha}\in C(G_{\alpha},\zeta),f\in
C(G,\zeta)$, then we define

$f_{G}(\pi_{\alpha})=f_{G_{\alpha}}(\pi_{\alpha}),\quad
f=\oplus_{\alpha}f_{\alpha},\qquad\alpha\in\pi_{0}(G).$

So we can naturally extend the homological interpretation for the
representation on the connected components of $K$-group to the $K$-group. Then
we can extend (3.2) to $K$-group.

We write $\Pi_{2}(G(\mathbb{R}),\zeta)$,
$\Pi_{\operatorname{temp}}(G(\mathbb{R}),\zeta)$ and
$\Pi(G(\mathbb{R}),\zeta)$ for the set of equivalence classes of discrete
series representations, the set of equivalence classes of tempered
representations and the set of equivalence classes of all irreducible
admissible representations of $G(\mathbb{R})$, whose central character
coincides with a given character $\zeta$ on $Z$. $C(G(\mathbb{R}),\zeta)$ is
the Schwartz space of $G(\mathbb{R})$, whose central character is given by a
character $\zeta^{-1}$ on $Z$.

If $f$ is any function in $C(G(\mathbb{R}),\zeta)$ and $\pi_{\mathbb{R}}$
belongs to $\Pi(G(\mathbb{R}),\zeta)$, we can set

$\pi_{\mathbb{R}}(f)=\int_{G(\mathbb{R})/Z}f(x)\pi_{\mathbb{R}}(x)\,dx$

######  Lemma 3.1.

Let $\pi_{\mathbb{R}}\in\Pi_{2}(G(\mathbb{R}),\zeta)$, there is a function
$f_{\pi_{\mathbb{R}}}\in C(G(\mathbb{R}),\zeta)$, such that for any
$\sigma\in\Pi_{\operatorname{temp}}(G(\mathbb{R}),\zeta)$,

$\operatorname{tr}(\sigma(f_{\pi_{\mathbb{R}}}))=\begin{cases}1&\text{if
}\sigma=\pi_{\mathbb{R}},\\\ 0&\text{ otherwise}.\end{cases}$

The above lemma is an immediate consequence of the trace Paley-Wiener theorem
of Arthur [6]. Such $f_{\pi_{\mathbb{R}}}$ is called for a pseudo-coefficient
of $\pi_{\mathbb{R}}$. We say that a function $f\in C(G(\mathbb{R}),\zeta)$ is
cuspidal, if $\operatorname{tr}\pi(f)$ is viewed as a function on
$\Pi_{\operatorname{temp}}(G(\mathbb{R}),\zeta)$, is supported on
$\Pi_{2}(G(\mathbb{R}),\zeta)$. So $f_{\pi_{\mathbb{R}}}$ is cuspidal.

Now, we fix a function $h\in\mathcal{H}(G(\mathbb{A}_{\operatorname{fin}}))$,
and set

$(f_{\pi_{\mathbb{R}}}h)(x)=f_{\pi_{\mathbb{R}}}(x_{\mathbb{R}})h(x_{\operatorname{fin}}),\qquad
x=x_{\mathbb{R}}x_{\operatorname{fin}}\in G(\mathbb{A}),x_{\mathbb{R}}\in
G(\mathbb{R}),x_{\operatorname{fin}}\in G(\mathbb{A}_{\operatorname{fin}}).$

If $\pi=\pi_{\mathbb{R}}\otimes\pi_{\operatorname{fin}}$ is any representation
in $\Pi(G(\mathbb{A}),\zeta)$, we have

$\displaystyle\operatorname{tr}\pi(f_{\pi_{\mathbb{R}}}h)$
$\displaystyle=\operatorname{tr}(\int_{G(\mathbb{A})/Z}(f_{\pi_{\mathbb{R}}}h)(x)\pi(x)\,dx)$
$\displaystyle=\operatorname{tr}\pi_{\mathbb{R}}(f_{\pi_{\mathbb{R}}})\operatorname{tr}\pi_{\operatorname{fin}}(h).$

Since $f_{\pi_{\mathbb{R}}}$ is cuspidal, which will cancel the contributions
from Levi subgroups, so the invariant trace formula simplifies. The spectral
side of the invariant trace formula is

$\displaystyle I(f_{\pi_{\mathbb{R}}}h)$
$\displaystyle=\sum_{\pi\in\Pi(G(\mathbb{A}),\zeta)}m_{\operatorname{disc}}(\pi)\operatorname{tr}\pi(f_{\pi_{\mathbb{R}}}h)$
$\displaystyle=\sum_{\pi\in\Pi(G(\mathbb{A}),\zeta)}m_{\operatorname{disc}}(\pi)\operatorname{tr}\pi_{\mathbb{R}}(f_{\pi_{\mathbb{R}}})\operatorname{tr}\pi_{\operatorname{fin}}(h)$
$\displaystyle=\sum_{\pi=\pi_{\mathbb{R}}\otimes\pi_{\operatorname{fin}}}m_{\operatorname{disc}}(\pi)\operatorname{tr}\pi_{\operatorname{fin}}(h).$

If we take $h$ for the characteristic function $I_{K_{0}}$ in
$\mathcal{H}(\mathbb{A}_{\operatorname{fin}})$, then
$\operatorname{tr}\pi_{\operatorname{fin}}(h)=\operatorname{dim}(V(\pi^{K_{0}}_{\operatorname{fin}}))$.
So

(3.3)
$I(f_{\pi_{\mathbb{R}}}I_{k_{0}})=m_{\operatorname{disc}}(\pi_{\mathbb{R}},K_{0}),$

and

(3.4)
$I(f_{\pi_{\mathbb{R}}}h)=\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h)).$

We can expand $m_{\operatorname{disc}}(\pi_{\mathbb{R}},K_{0})$ and
$\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h))$ by using the
geometric side of invariant trace formula, but we cannot obtain an explicit
formula directly. This is because $f_{\pi_{\mathbb{R}}}$ is not a stable
function, so we can not cancel the contribution of the non-trivial unipotent
elements in the invariant trace formula.

If the test function $f$ is stable cuspidal. We can obtain an explicit formula
through the geometric side of the invariant trace formula [3], and the results
are easily extended to $K$-group. In the rest of this section, we review the
setup in [3]. Firstly, if $G$ is a connected $K$-group over $\mathbb{R}$,
$f\in C(G(\mathbb{R}),\zeta)$ is stable cuspidal and $\gamma\in
M(\mathbb{R})$, where $M(\mathbb{R})$ is a Levi subgroup of $G(\mathbb{R})$,
then following the section $4$ and section $5$ of [3], when $\gamma$ is
$G$-regular, one defines

$\Phi_{M}(\gamma,f)=|D^{M}(\gamma)|^{-1/2}I_{M}(\gamma,f).$

Then one has the formula:

(3.5)
$\Phi_{M}(\gamma,f)=(-1)^{\operatorname{dim}(A_{M}/A_{G})}\upsilon{(M_{\gamma})^{-1}}\sum_{\uptau\in\Pi(G^{\ast}(\mathbb{R}),\zeta)}\Phi_{M}(\gamma,\uptau)\operatorname{tr}\widetilde{\uptau}(f),$

and $\Phi_{M}(\gamma,f)$ is equal to zero if $\gamma$ is not semisimple (see
[3, Theorem 5.1]). Here we denote by

$D^{M}(\gamma)=\operatorname{det}((1-Ad(\sigma))_{\mathfrak{m}/\mathfrak{m}_{\sigma}}),$

$\gamma$ is an element in $M(\mathbb{R})$ with the Jordan decomposition
$\gamma=\sigma u$, $G^{\ast}$ is isomorphic to $G$ over $\mathbb{C}$, and
$G^{\ast}/A_{G}(\mathbb{R})$ is compact. $\widetilde{\uptau}$ is the
contragredient of $\uptau$. The other terms are explained as follow:

$\upsilon(G)=(-1)^{q(G)}\operatorname{vol}(G^{\ast}/A_{G}(\mathbb{R})^{\circ})|\mathscr{D}(G,B)|^{-1},$
$\Phi_{M}(\gamma,\uptau)=(-1)^{q(G)}|D^{G}_{M}(\gamma)|^{1/2}\sum_{\pi\in{\Pi}_{\operatorname{disc}}(\uptau)}\Theta_{\pi}(\gamma),$

where $\Pi_{\operatorname{disc}}(\uptau)$ is an $L$-packet of discrete series
of $G(\mathbb{R})$ parameterized by $\uptau$,

$q(G)=\frac{1}{2}\operatorname{dim}(G(\mathbb{R})/K_{\mathbb{R}}A_{G}(\mathbb{R})),$

$B$ is a maximal torus of $G(\mathbb{R})$ that is contained in
$K_{\mathbb{R}}$, and $K_{\mathbb{R}}$ is a maximal compact subgroup of
$G(\mathbb{R})$. The invariant distribution $\Phi_{M}(\gamma,\uptau)$ is given
in terms of Harish-Chandra’s formula for stable character of discrete series.
Let us review the explicit formula for the averaged discrete series
characters. We can naturally extend it to the $K$-group $G(\mathbb{R})$. Let
$Z(B)$ be the centralizer of the connected component $G(\mathbb{R})^{\circ}$
in $K_{\mathbb{R}}.$ $B(\mathbb{R})$ (refer to [13, Lemma 3.4]) equals the
product of its connected component $B(\mathbb{R})^{\circ}$ with $Z(B)$. We set
$\rho_{B}$ as usual to be half of the sum of positive roots of $(G,B)$. Let
$\Lambda(\zeta)$ denote the set of pairs

$(\zeta,\lambda),\quad\zeta\in
Z(B)^{\ast},\lambda\in\mathfrak{b}(\mathbb{C})^{\ast},$

such that $z\operatorname{exp}H\to\zeta(z)e^{(\lambda-\rho_{B})(H)},z\in
Z(B),H\in\mathfrak{b}(\mathbb{R})$ is a well defined quasi-character on
$B(\mathbb{R})$ whose restriction to $A_{G}(\mathbb{R})^{\circ}$ equals
$\zeta$, and $\lambda$ is regular. $\Lambda(\zeta)$ equipped with an action of
Weyl group $W(G,B)$. The discrete series are parameterized by the
$W(G(\mathbb{R}),B(\mathbb{R}))$-orbits in $\Lambda(\zeta)$. We denote by
$\phi$ a discrete Langlands parameter, and then we can find that an $L$-packet
$\Pi_{\phi}$ corresponds to the partition of a given $W(G,B)$-orbit into
$W(G(\mathbb{R}),B(\mathbb{R}))$-orbits [27]. We set
$\mathscr{D}(G,B)=W(G,B)/W(G(\mathbb{R}),B(\mathbb{R}))$, then

$|\mathscr{D}(T)|=|\mathscr{D}(G,B)|,$

where $T$ is a maximal torus which is $\mathbb{R}$-anisotropic modulo
$A(\mathbb{R})^{\circ}$, and $T$ is conjugate with $B$.

Let $\eta:G^{\ast}\rightarrow G$ be an isomorphism over $\mathbb{C}$ such that
the automorphism $\eta^{\sigma}\eta^{-1}$ is inner for
$\sigma\in\operatorname{Gal}(\mathbb{C}/R)$. We use $\eta$ to identity $A_{G}$
with the $\mathbb{R}$-split component of the center of $G^{\ast}$, and we
assume that $G^{\ast}(\mathbb{R})/A_{G}(\mathbb{R})$ is compact. Then the
representations in $\Pi(G^{\ast},\zeta)$ are all finite dimensional, According
to Langlands classification [22], the set
$\Pi_{\operatorname{disc}}(G(\mathbb{R}),\zeta)$ is a disjoint union of finite
subsets $\Pi_{\operatorname{disc}}(\uptau)$, which are parametrized by the
irreducible representation $\uptau$ in $\Pi(G^{\ast}(\mathbb{R}),\zeta)$. If
$\uptau$ and $\phi$ are parametrized by the same infinitesimal character, then
the $L$-packet $\Pi_{\phi}$ equals the finite set
$\Pi_{\operatorname{disc}}(\uptau)$. We can set
$\Phi_{M}(\gamma,\phi)=\Phi_{M}(\gamma,\uptau)$.

For given $\uptau\in\Pi(G^{\ast}(\mathbb{R}),\zeta)$, let
$(\zeta,\lambda)\in\Lambda(\zeta)$ be the point in the corresponding orbit,
such that $\lambda$ is positive on all the positive co-roots of $(G,B)$. Then
if

$\gamma=z\operatorname{exp}H,z\in Z(B),H\in\mathfrak{b}(\mathbb{R})$

is a regular point in $B(\mathbb{R})$, we have

$\Phi_{G}(\gamma,\uptau)=\operatorname{tr}\uptau(\gamma)=\Delta^{G}_{B}(H)^{-1}\zeta(z)\sum_{s\in
W(G,B)}\varepsilon(s)e^{(s\lambda)(H)}.$

Here
$\Delta^{G}_{B}(H)=\Pi_{\alpha>0}(e^{\frac{1}{2}\alpha(H)}-e^{-\frac{1}{2}\alpha(H)}).$

For the general averaged discrete series character $\Phi_{M}(\gamma,\uptau)$,
let $T$ be a maximal torus in $M$, which is $\mathbb{R}$-anisotropic modulo
$A_{M}(\mathbb{R})^{\circ}$. We take $R$ to be the set of real roots of
$(G,T)$. The existence of torus $B$ means that $W(R)$ contains an element that
acts as $-1$. We can take $T$ from its $M(\mathbb{R})$-conjugates so that
$T=(T\cap B)A_{M}$. Then there is an element $y\in G(\mathbb{C})$, such that
$Ad(y)(\mathfrak{b}(\mathbb{C}))=\mathfrak{t}(\mathbb{C})$, where
$\mathfrak{t}$ is the Lie algebra of $T$.

Suppose that $\uptau\in\Pi(G(\mathbb{R}),\xi)$. $(\xi,\lambda)\in\Lambda(\xi)$
is a point in the corresponding $W(G,B)$-orbit such that $y\lambda$ is
positive on all positive co-roots of $(G,T)$. Then $\Phi_{M}(\gamma,\uptau)$
vanishes for any regular point $\gamma\in T(\mathbb{R})$ unless $\gamma$ is of
the form

$\gamma=z\operatorname{exp}(H),\quad z\in Z(B),H\in\mathfrak{t}(\mathbb{R}),$

in which case

(3.6)
$\Phi_{M}(\gamma,\uptau)=\Delta^{M}_{T}(H)^{-1}\varepsilon_{R}(H)\xi(z)\sum_{s\in
W(G,B)}\varepsilon(s)\bar{C}(Q^{+}_{ys\lambda},R^{+}_{H})e^{(ys\lambda)(H)}.$

Here $\varepsilon_{R}(H)=(-1)^{|R^{+}_{H}\cap(-R^{+})|}$. $H$ is a regular
point in $\mathfrak{t}(\mathbb{R})$, which is the Lie algebra of
$T(\mathbb{R})$, and $R^{+}_{H}$ for the set of roots which are positive on
$H$. $\varepsilon(s)$ is a sign function on $W(G,B)$, and
$\bar{C}(Q^{+},R^{+})$ is an integer valued function, which is defined for
root systems $R$ whose Weyl group $W(R)$ contains $-1$. The function
$\bar{C}(Q^{+},R^{+})$ is uniquely determined by the following four
properties.

  1. (1)

$\bar{C}(sQ^{+},sR^{+})=\bar{C}(Q^{+},R^{+}),s\in W(R)$.

  2. (2)

The number $\bar{C}(Q^{+},R^{+})$ vanishes unless $\nu(X)$ negative for every
$X\in\mathfrak{a}_{R^{+}}$, and $\nu\in\mathfrak{a}_{Q^{+}}$.

  3. (3)

$\bar{C}(Q^{+},R^{+})+\bar{C}(s_{\alpha}Q^{+},R^{+})=2\bar{C}(Q^{+}\cap
Q_{\alpha},R^{+}\cap R_{\alpha})$, for any reflection $s_{\alpha}\in W(R)$
corresponding to a root $\alpha\in R$.

  4. (4)

If $R$ is the empty root system, then $\bar{C}(Q^{+},R^{+})=1$.

Here $R^{+}$ is a system of positive roots for $R$, and $Q^{+}$ is a positive
system for the set $Q=R^{\vee}$ of co-roots, $Q^{+}_{\lambda}$ is the set of
co-roots $\alpha^{\vee}$ of $Q=R^{\vee}$ for which $\lambda(\alpha^{\vee})$
are all positive.

Arthur [3] extended $\Phi(\gamma,\uptau)$ to a continuous, $W(M,T)$-invariant
function on $T(\mathbb{R})$, and then extended to a function on
$M(\mathbb{R})$, which is a connected reductive group, and $\Phi(.,\uptau)$ is
a $M(\mathbb{R})$-invariant function on $M(\mathbb{R})$ which is supported on
the $M(\mathbb{R})$-elliptic conjugacy classes. If $M\in\mathcal{L}$ is not
cuspidal, then we set $\Phi_{M}(\gamma,\uptau)$ to be identically zero.

Finally we come to the geometric side of the invariant trace formula. This
will be reviewed in more details in section $4$ below. If the representative
of the conjugacy class $\gamma\in\Gamma(M,S)$ is semisimple, it is independent
of $S$. Moreover, for any semisimple element $\gamma\in M(\mathbb{Q})$,

$\mathnormal{a}^{M}(S,\gamma)=|\iota^{M}(\gamma)|^{-1}\operatorname{vol}(M_{\gamma}(\mathbb{Q})\backslash{M_{\gamma}(\mathbb{A})^{1}}).$

If $\gamma$ is not $\mathbb{Q}$-elliptic in $M$, then
$\mathnormal{a}^{M}(S,\gamma)$ vanishes. Here
$|\iota^{M}(\gamma)|=|M_{\gamma}(\mathbb{Q})\backslash{M(\mathbb{Q},\gamma)}|$,
$M_{\gamma}(\mathbb{A})^{1}=A_{M}(\mathbb{R})^{0}\backslash{M_{\gamma}(\mathbb{A})}$.
So if $f$ is stable cuspidal, we have an explicit geometric expansion of
general global trace formula,

(3.7)
$I(fh)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{\gamma\in\Gamma(M,S)}a^{M}(S,\gamma)I_{M}(\gamma,fh),$

and
$I_{M}(\gamma,fh)=I^{G}_{M}(\gamma,f)I^{M}_{M}(\gamma,h)=|D^{M}(\gamma)|\Phi_{M}(\gamma,f)h_{M}(\gamma)$.

##  4\. Global stable trace formula

Suppose now $G$ is a $K$-group over a number field $F$. The invariant trace
formula can be extended to $K$-group, which is stabilized by Arthur in [9],
[10], [11]. We need to recall the basic information about the stable trace
formula before applying it to obtain the multiplicity formula. Let $S$ be a
finite set of valuations of $F$ that contains the set of archimedean places
and the set of places at which $G$ ramifies. The general invariant trace
formula is the identity obtained from two different expansions of a certain
linear form $I(f)$ in [2] for $f\in\mathcal{H}(G,S)$, the Hecke algebra of
$G(F_{S})$. The geometric expansion

(4.1)
$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{\gamma\in\Gamma(M,S)}a^{M}(S,\gamma)I_{M}(\gamma,f)$

is a linear combination of distributions parametrized by conjugacy classes
$\gamma$ in Levi subgroups $M(F_{S})$. The spectral expansion

(4.2) $I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{t\geq
0}\int_{\Pi_{t}(M,S)}a^{M}(S,\pi)I_{M}(\pi,f)\,d\pi$

is a linear combination of distributions parameterized by representations
$\pi$ of Levi subgroup $M(F_{S})$.

Arthur studied the more general coefficients $a^{M}(\gamma)$ and $a^{M}(\pi)$,
which are the global objects. The terms $I_{M}(\gamma,f)$ and $I_{M}(\pi,f)$
are the local objects. We can apply splitting formula and descent formula to
reduce the distribution to Levi subgroup of $G$. The multiplicity formula
results are described by the geometric side of the global trace formula.

We need to stabilize the coefficient $a^{M}(\gamma)$ and the invariant
distribution $I_{M}(\gamma,f)$. Those are the corresponding global theorem and
local theorem [11]. Firstly, we recall the relation between the coefficient
$a^{M}(\gamma)$ and $a^{M}(S,\gamma)$. We will freely use the notation in [9].
It suffices to consider $M=G$. The coefficient $a^{G}(S,\gamma)$ is defined on
$\Gamma(G,S)$, where $\Gamma(G,S)$ is the set of the $(G,S)$-equivalence
classes in $G(F)$. The two elements $\gamma$ and $\gamma_{1}$ in $G(F)$, with
standard Jordan decompositions $\gamma=c\alpha$ and
$\gamma_{1}=c_{1}\alpha_{1}$, are defined to be $(G,S)$-equivalent if there is
an element $\delta\in G(F)$ such that $\delta^{-1}c_{1}\delta=c$, and such
that $\delta^{-1}\alpha_{1}\delta$ is conjugate to $\alpha$ in $G_{c}(F_{S})$.
It is the usual conjugacy class if $\gamma$ is semisimple.

For a general element $\gamma=c\alpha$, $c$ being the semisimple part,
$\alpha$ being the unipotent part, the coefficient is defined by a descent
formula,

$a^{G}(S,\gamma)=i^{G}(S,c)|\operatorname{stab}(c,\alpha)|^{-1}a^{G_{c}}(S,\alpha),$

where $\operatorname{stab}(c,\alpha)$ stands for the stabilizer of $\alpha$ in
the finite group $(G_{c,+}(F)/G_{c}(F))$, which acts on the set of unipotent
conjugacy classes in $G_{c}(F_{S})$. $i^{G}(S,c)$ is equal to 1, if $c$ is
$F$-elliptic in $G$, and the $G(\mathbb{A}^{S})$ conjugacy class of $c$ meets
$K^{S}$; otherwise equal to 0. The descent formula reduces the study of
$a^{G}(S,\gamma)$ to the case of unipotent elements.

We set

$a^{G}_{\operatorname{ell}}(\gamma)=\sum_{\\{\gamma\\}}|Z(F,\gamma)|^{-1}a^{G}(S,\gamma)(\gamma_{S}/\gamma)^{-1},$

where $\\{\gamma\\}$ is summed over those $Z_{S,O}=Z(F)\cap Z_{S}Z(O)^{S}$
orbit in $(G(F))_{G,S}$ that map to $\gamma_{S}$, and such that the
$G(\mathbb{A}^{S})$ conjugacy class of $\gamma$ in $G(\mathbb{A}^{S})$ meets
$K^{S}$, $Z(F,\gamma)=\\{z\in Z(F):z\gamma=\gamma\\}=\\{z\in
Z_{S,O}:z\gamma=\gamma\\}$, and $\gamma_{S}/\gamma$ is the ratio of the
invariant measure on $\gamma_{S}$ and the signed measure on $\gamma_{S}$ that
comes with $\gamma$. The coefficient $a^{G}_{\operatorname{ell}}(\gamma)$ is
supported on the set of admissible elements in the discrete subset
$\Gamma_{\operatorname{ell}}(G,S,\zeta)$ (see [9, (2.6)]) of
$\Gamma(G^{Z}_{S},\zeta_{V})$, where admissible elements is defined in [9,
§1]. We denote by

$G^{Z}=\\{x\in G:H_{G}(x)\in\operatorname{Image}(a_{Z}\mapsto a_{G})\\}.$

If $M$ is a Levi subgroup of $G$, and $\mu$ belongs to
$\Gamma(M^{Z}_{S},\zeta_{S})$, the induced distribution $\mu^{G}$ is a finite
linear combination of elements in $\Gamma(G^{Z}_{S},\zeta_{S})$. We write
$\Gamma(G,S,\zeta)$ for the set of elements so obtained, as $M$ ranges over
$\mathcal{L}$ and $\mu$ runs over the element
$\Gamma_{\operatorname{ell}}(M,S,\zeta)$, these objects are compatible with
the spectral side.

If $\gamma$ belongs to $\Gamma(G^{Z}_{V},\zeta_{V})$,
$V_{\operatorname{ram}}\subset V\subset S$, we denote [9, (2.8)]

$a^{G}(\gamma)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{k\in\mathcal{K}^{V}_{\operatorname{ell}}(M/Z,S)}a^{M}_{\operatorname{ell}}(\gamma_{M}\times
k)r^{G}_{M}(k),$

where
$r^{G}_{M}(k)=\mathcal{J}_{M}(r^{V}_{S}(k),u^{V}_{S}),k\in\mathcal{K}((M/Z)^{V}_{S})$
is the unramified weighted orbital integrals,
$\mathcal{K}^{V}_{\operatorname{ell}}(G/Z,S)$ for the set of $k$ in
$\mathcal{K}((G/Z)^{V}_{S})$ such that $\gamma\times k$ belongs to
$\Gamma_{\operatorname{ell}}(G,S,\zeta)$ for some $\gamma$.

If $f\in\mathcal{H}(G,V,\zeta)$, then the linear form $I(f)$ has a geometric
expansion [9, Proposition 2.2]

(4.3)
$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{\gamma\in\Gamma(M,V,\zeta)}a^{M}(\gamma)I_{M}(\gamma,f).$

The general trace formula was stabilized by Arthur, who inductively defined a
general stable distribution on the quasisplit group $G^{\prime}$ which is
independent of the $G$, and then built up the endoscopy trace formula.

If $G$ is quasisplit, Arthur [11] proved that

$S^{G}(f)=I(f)-\sum_{G^{{}^{\prime}}\in\mathcal{\mathcal{E}}^{0}_{\operatorname{ell}}(G)}\iota(G,G^{{}^{\prime}})\widehat{S}^{\widetilde{G}^{{}^{\prime}}}(f^{{}^{\prime}})$

is stable. For $G$ general, he [11] proved

$I(f)=I^{\mathcal{E}}(f)=\sum_{G^{{}^{\prime}}\in\mathcal{\mathcal{E}}_{\operatorname{ell}}(G)}\iota(G,G^{{}^{\prime}})\widehat{S}^{G^{{}^{\prime}}}(f^{{}^{\prime}}).$

The endoscopic trace formula $I^{\mathcal{E}}(f)$ and the stable trace formula
$S^{G}(f)$ both have a geometric expansion

(4.4)
$I^{\mathcal{E}}(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{\gamma\in\Gamma^{\mathcal{E}}(M,V,\zeta)}a^{M,\mathcal{E}}(\gamma)I^{\mathcal{E}}_{M}(\gamma,f),$

and

(4.5)
$S^{G}(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{\delta\in\Delta(M,V,\zeta)}b^{M}(\delta)S^{G}_{M}(\delta,f).$

Here the relation between coefficients is given by

$a^{G}(\gamma)=a^{G,\mathcal{E}}(\gamma)=\sum_{G^{{}^{\prime}}}\sum_{\delta^{{}^{\prime}}}\iota(G,G^{{}^{\prime}})b^{\widetilde{G^{{}^{\prime}}}}(\delta^{{}^{\prime}})\Delta_{G}(\delta^{{}^{\prime}},\gamma)+\varepsilon(G)\sum_{\delta}b^{G}(\delta)\Delta_{G}(\delta,\gamma),$

with $\gamma\in\Gamma(G^{Z}_{V},\zeta_{V})$,
$G^{{}^{\prime}},\delta^{{}^{\prime}}$ and $\delta$ summed over
$\mathcal{E}^{0}_{\operatorname{ell}}(G)=\mathcal{E}_{\operatorname{ell}}\backslash\\{G^{\ast}\\}$,
$\Delta((\widetilde{G^{{}^{\prime}}}_{V})^{Z^{{}^{\prime}}},\widetilde{\zeta}^{{}^{\prime}}_{V})$
and $\Delta^{\mathcal{E}}(G^{Z}_{V},\zeta_{V})$ respectively, where $G^{\ast}$
is the quasisplit inner form of $G$. The stable coefficient $b^{G}(\delta)$ is
stable, meaning that $b^{G}(\delta)$ is supported on $\Delta(G,V,\zeta)$, and
the coefficients $a^{G}(\gamma)$ and $b^{G}(\delta)$ are independent of $S$.

To stabilize the general trace formula, it essentially amounts to comparing
the two expansions (4.4) and (4.5). Assuming the Fundamental Lemma and the
weighted Fundamental Lemma, Arthur [11] has proved this, and in 2008 Ngo [20]
proved the Fundamental Lemma. The weighted version was proved by Chaudouard-
Laumon. So now the stable trace formula is available.

We have the formula for the stable coefficient

$b^{G}(\delta)=b^{G}_{\operatorname{ell}}(\delta,S)+\sum_{M\in\mathcal{L}^{0}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{k\in\mathcal{K}^{V}_{\operatorname{ell}}(M/Z,S)}b^{M}_{\operatorname{ell}}(\delta_{M}\times
k)S^{G}_{M}(k),$

where the stable term $S^{G}_{M}(k)$ comes from the unramified weighted
orbital integrals $r^{G}_{M}(k)$ (the weighted Fundamental Lemma was applied
to this part). Here $b^{G}_{\operatorname{ell}}(\delta)$ also satisfies a
descent formula [10], if $\delta_{S}$ is an admissible element in
$\Delta_{\operatorname{ell}}(G,S,\zeta)$ with Jordan decomposition
$\delta_{S}=d_{S}\beta_{S}$, then

$b^{G}_{\operatorname{ell}}(\delta_{S})=\sum_{d}\sum_{\beta}j^{G^{\ast}}(S,d)b^{G^{\ast}_{d}}_{\operatorname{ell}}(\beta),$

where $d$ is summed over the set of elements in $\Delta_{ss}(G^{\ast})$ whose
image in $\Delta_{\operatorname{ss}}(G^{\ast}_{S})$ equals $d_{S}$, where
$G^{\ast}$ is quasisplit inner form of $G$, and $\beta$ is summed over the
orbit of $(\overline{G^{\ast}}/\overline{G^{\ast}}_{d,S})(F)$ in
$\Delta_{\operatorname{unip}}(G^{\ast}_{d,S},\zeta)$. Moreover,
$b^{G}_{\operatorname{ell}}(\delta)$ is equal to zero if $\delta$ in the
complement of $\Delta_{\operatorname{ell}}(G,S,\zeta)$ in the set of
admissible elements in $\Delta^{\mathcal{E}}_{\operatorname{ell}}(G,S,\zeta)$
(see [10, Theorem 1.1]), and
$j^{G^{\prime}}(S,d^{\prime})=i^{G^{\prime}}(S,d^{\prime})\tau(G^{\prime})\tau(G^{\prime}_{d^{\prime}})^{-1}$
(see [10, (1.7)]).

If $\delta^{\prime}$ is semisimple, elliptic in
$\Delta_{\operatorname{ell}}(G^{\prime},V,\zeta)$, then by [17, Theorem
8.3.1], or [11, page 105],

$\displaystyle b^{G^{\prime}}(\delta^{\prime})$
$\displaystyle=b^{G^{\prime}}_{\operatorname{ell}}(\delta^{\prime})=j^{G^{\prime}}(S,\delta^{\prime})b^{G^{\prime}_{\delta^{\prime}}}_{\operatorname{ell}}(1)$
$\displaystyle=\tau(G^{\prime})\tau(T)^{-1}\tau(T)=\tau(G^{\prime}),$

where $T=G^{\prime}_{\delta^{\prime}}$ and
$\tau(G^{\prime})=|\pi_{0}(Z(\widehat{G^{\prime}})^{\Gamma})||\operatorname{Ker}^{1}(F,Z(\widehat{G^{\prime}}))|^{-1}$
is the Tamagawa number of $G^{\prime}$ [16, (5.1.1)], or [18].

We obtain a stable form of the invariant trace formula

$I(f)=\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\sum_{M^{\prime}\in\mathcal{L}^{G^{\prime}}}|W^{M^{\prime}}_{0}||W^{G{\prime}}_{0}|^{-1}\sum_{\delta\in\Delta(M^{\prime},V,\zeta)}b^{M^{\prime}}(\delta)S^{G^{\prime}}_{M^{\prime}}(\delta,f).$

Where $\Delta(M^{\prime},V,\zeta)$ is the basis of $S\mathcal{D}(M^{\prime
Z}_{V},\zeta)$, which is the subspace of stable distributions in
$\mathcal{D}(M^{\prime Z}_{V},\zeta)$. Here $\mathcal{D}(M^{\prime
Z}_{V},\zeta)$ is the vector space of distributions $D$ on $M^{\prime Z}_{V}$
that satisfy the following three conditions:

  1. (1)

$D$ is invariant under the conjugation by $M^{\prime Z}_{V}$,

  2. (2)

$D$ is $\zeta$-equivariant under translation by $Z_{V}$,

  3. (3)

$D$ is supported on the preimage in $M^{\prime Z}_{V}$ of a finite union of
conjugacy classes in $\overline{M^{\prime Z}_{V}}=M^{\prime Z}_{V}/Z_{V}$.

The Langlands’s global coefficients

$\iota(G,G^{\prime})=\iota(G_{\alpha},G^{\prime})=\tau(G)\tau(G^{\prime})^{-1}|\operatorname{Out}_{G}(G^{\prime})|^{-1},$

where
$\alpha\in\pi_{0}(G),G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G),\operatorname{Out}_{G}(G^{\prime})=\operatorname{Aut}_{G}(G^{\prime})/\widehat{G^{\prime}}$.

If $f=f_{\infty}h$, $f_{\infty}$ is cuspidal in $C(G(\mathbb{R}),\zeta)$ and
$h\in\mathcal{H}(G(\mathbb{A}_{\operatorname{fin}})),$ then the term

$S^{G}_{M}(\delta,f)=S^{G}_{M}(\delta,f_{\infty})(f^{\infty}_{V})^{M}(\delta),\qquad\delta\in\Delta(M,V,\zeta).$

This is a consequence of the stable splitting formula [8, Theorem 6.1]. Indeed
we set $V=V_{1}\coprod V_{2}$, where $V_{1}$ is the set of archimedean places,
$f=f_{V_{1}}f_{V_{2}}$, then

$S^{G}_{M}(\delta,f)=\sum_{L_{1},L_{2}\in\mathcal{L}(M)}e^{G}_{M}(L_{1},L_{2})\widehat{S}^{L_{1}}_{M}(\delta,f_{V_{1},L_{1}})\widehat{S}^{L_{2}}_{M}(\delta,f_{V_{2},L_{2}}),$

where $e^{G}_{M}(L_{1},L_{2})$ was defined [8, Theorem 6.1]. Since $f_{V_{1}}$
is cuspidal, thus $(f_{V_{1}})_{L_{1}}=0$ for $L_{1}\neq G$. On the other
hand, $e^{G}_{M}(G,L_{2})\neq 0$ only when $L_{2}=M$ in which case it is equal
to 1. Then

$S^{G}_{M}(\delta,f)=\widehat{S}^{G}_{M}(\delta,f_{V_{1},G})\widehat{S}^{M}_{M}(\delta,f_{V_{2},M}).$

However $V$ is finite, so we continue this process to obtain

$S^{G}_{M}(\delta,f_{\infty}h)=S^{G}_{M}(\delta,f_{\infty})(h^{\infty}_{V})^{M}(\delta).$

We now get the following proposition by combining (3.4).

######  Proposition 4.1.

For any $h\in\mathcal{H}(G(\mathbb{A}_{\operatorname{fin}}))$, and
$f_{\pi_{\mathbb{R}}}\in C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$ is
pseudo-coefficient of $\pi_{\mathbb{R}}\in\Pi_{2}(G(\mathbb{R}),\zeta)$. We
have

(4.6)
$\begin{split}\operatorname{tr}(R_{\operatorname{disc}}(\pi_{\mathbb{R}},h))&=I(f_{\pi_{\mathbb{R}}}h)\\\
&=\sum_{G^{{}^{\prime}}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{{}^{\prime}})\sum_{M^{{}^{\prime}}\in\mathcal{L}^{G^{{}^{\prime}}}}|W^{M^{{}^{\prime}}}_{0}||W^{G^{{}^{\prime}}}_{0}|^{-1}\sum_{\delta\in\Delta(M^{{}^{\prime}},V,\zeta)}b^{M^{{}^{\prime}}}(\delta)S^{G^{{}^{\prime}}}_{M^{{}^{\prime}}}(\delta,f_{\pi_{\mathbb{R}}})(h_{M})^{M^{{}^{\prime}}}(\delta).\end{split}$

We now have a stable trace formula. In order to make this more explicit, we
need to examine the individual terms more closely. The stable orbital integral
$(h_{M})^{M^{{}^{\prime}}}(\delta)$ is treated by the transfer theorem. The
term $S^{G}_{M}(\delta,f_{\pi_{\mathbb{R}}})$ is a stable distribution
attached to a invariant distribution, which is more complicated. We need to
give an explicit formula for $S^{G}_{M}(\delta,f_{\pi_{\mathbb{R}}})$. We know
that the local component of stable global trace formula is more complicated
than the local component of stable local trace formula. But if the test
function is stable cuspidal, they are the same, because the component of
invariant local trace formula and the local component of invariant trace
formula are the same, when $G$ is a connected reductive group [3]. Moreover we
have the following proposition.

######  Proposition 4.2.

If $G(\mathbb{R})$ is a $K$-group, $G^{\prime}$ is the endoscopy group of $G$,
$f$ is a cuspidal function in
$\mathcal{H}_{\operatorname{ac}}(G(\mathbb{R}),\zeta)$, $\delta\in
M^{\prime}(\mathbb{R})$ is any element with Jordan decomposition $\delta=cu$,
and $u$ is a non-trivial unipotent element, then
$S^{G^{\prime}}_{M^{\prime}}(\delta,f)=0$.

###### Proof.

$G$ is a $K$-group over $\mathbb{R}$, we can write
$G=\amalg_{\alpha\in\pi_{0}(G)}G_{\alpha}$, where $G_{\alpha}$ is a connected
reductive group. If $\delta\in M^{\prime}_{\alpha}$ and $\delta=cu$, $u$ is
not trivial, then we have

$S^{G^{\prime}}_{M^{\prime}}(\delta,f)=S^{G^{\prime}_{\alpha}}_{M^{\prime}_{\alpha}}(\delta,f_{\alpha})=\widehat{S}^{G^{\prime}_{\alpha}}_{M^{\prime}_{\alpha}}(\delta,(f)^{G^{\prime}_{\alpha}})=0.$

The third equality is from Arthur’s result [3, Theorem 5.1]. Indeed we know
that when $G$ is a connected reductive group, and when the test function is
stable cuspidal, then the corresponding invariant distribution vanishes on the
non-semisimple element. However $f^{G^{\prime}_{\alpha}}(\delta)$ is stable
cuspidal function of $G^{\prime}_{\alpha}$ by Shelstad’s transfer theorem, so
we obtain $S^{G^{\prime}}_{M^{\prime}}(\delta,f)=0$. ∎

From the above proposition, we see that the component of stable local trace
formula and the component of stable global formula are compatible. So we have
reduced the study of the stable trace formula to the stable local trace
formula in the archimedean case. There are two ways to study the stable
distribution $S^{G}_{M}(\delta,f_{\infty})$. One way is to explicitly
stabilize the invariant trace formula in this special case. The other is to
stabilize the local trace formula, and then we compare the stable Weyl
integral formula with the stable local trace formula. In this paper we follow
the second way.

##  5\.  The transfer factors of spectral side and characters

Suppose that $G$ is a reductive $K$-group over $\mathbb{R}$, $Z$ stands for a
central induced torus in $G$ over $\mathbb{R}$, $\zeta$ is a character on
$Z(\mathbb{R})$. Before we discuss the stabilization, we need to recall the
invariant local trace formula[4] and virtual characters in the Archimedean
case [5]. We set $V=\\{\infty_{1},\infty_{2}\\}$ for two Archimedean places.
Then $G_{V}=G(\mathbb{R})\times G(\mathbb{R})$ and
$\zeta_{V}=\zeta\times\zeta^{-1}$, while $f=f_{1}\times\bar{f}_{2}$, where
$f_{1},f_{2}\in C(G(\mathbb{R}),\zeta)$. Then $f$ is a function in the
Schwartz space $C(G_{V},\zeta_{V})$.

The geometric side of the local trace formula is the linear form

(5.1)
$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Gamma_{\operatorname{ell}}(M,V,\zeta)}I_{M}(\gamma,f)\,{d}\gamma,$

where
$\Gamma_{\operatorname{ell}}(M,V,\zeta)=\\{(\gamma,\gamma):\gamma\in\Gamma_{\operatorname{ell}}(M,\zeta)\\}$.
(The set $\Gamma_{\operatorname{ell}}(M,V,\zeta)$ is in bijection with the
family $\Gamma_{\operatorname{ell}}(\bar{M})$ of elliptic conjugacy classes in
$\bar{M}(\mathbb{R})=M(\mathbb{R})/Z(\mathbb{R}).)$

The spectral side is the linear form

(5.2)
$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{T_{\operatorname{disc}}(M,V,\zeta)}i^{M}(\tau)I_{M}(\tau,f_{1}\times\bar{f_{2}})\,{d}\tau,$

where
$I_{M}(\tau,f_{1}\times\bar{f_{2}})=r_{M}(\tau,P)\theta(\tau,f_{1,P})\overline{\theta(\tau,f_{2,P})}$
as in [5], $T_{\operatorname{disc}}(M,V,\zeta)$ stands for the diagonal image
$\\{(\tau,\tau^{\vee}):\tau\in T_{\operatorname{disc}}(M,\zeta)\\}$ in
$T_{\operatorname{temp}}(M_{V},\zeta_{V})$, defined as in [5, §3].

We denote the leading term ( i.e. $M=G$ in (5.2)) by
$I_{\operatorname{disc}}(f)$. Then

(5.3)
$I_{\operatorname{disc}}(f)=\int_{T_{\operatorname{disc}}(G,V,\zeta)}i^{G}(\tau)f_{G}(\tau)\,{d}\tau,$

where
$f_{G}(\tau)=(f_{1})_{G}(\tau)(\bar{f}_{2})_{G}(\tau^{\vee})=f_{1,G}(\tau)\overline{f_{2,G}(\tau)}$,
and

$i^{G}(\tau)=|W^{0}_{\pi}|^{-1}|R_{\pi,r}|^{-1}\sum_{w\in
W_{\pi}(r)_{\operatorname{reg}}}\varepsilon_{\pi}(w)|\operatorname{det}(1-w)_{\mathfrak{a}^{G}_{M}}|^{-1},$

where $\tau=(M,\pi,r).$

$W_{\pi}=\\{w\in W(\mathfrak{a}_{M}):w\pi\cong\pi\\}.$

$W^{0}_{\pi}$ is the subgroup of elements $w\in W_{\pi}$ such that the
operator $R(w,\pi)$(see [5, §2]) is a scalar. $R_{\pi}=W_{\pi}/W^{0}_{\pi}$,
$R_{\pi,r}$ is the centralizer of $r$ in the group $R_{\pi}$.
$W_{\pi}(r)_{\operatorname{reg}}$ is the intersection of the
$W^{0}_{\pi}$-coset $W_{\pi}(r)=W^{0}_{\pi}r$ in $W_{\pi}$ with the set

$W_{\pi,\operatorname{reg}}=\\{w\in
W_{\pi}:\mathfrak{a}^{w}_{M}=\mathfrak{a}_{G}\\}$

of regular elements. $\varepsilon_{\pi}(w)$ stands for the sign of projection
of $w$ onto the Weyl group $W^{0}_{\pi},$ taken relative to the decomposition
$W_{\pi}=W^{0}_{\pi}\rtimes R_{\pi}$.

If $f=f_{1}\times\bar{f}_{2}$ and $f_{1}\in
C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta),f_{2}\in C(G(\mathbb{R}),\zeta)$,
then

$I(f)=I_{\operatorname{disc}}(f).$

Here

$I(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Gamma_{G-\operatorname{reg},\operatorname{ell}}(M,V,\zeta)}I_{M}(\gamma,f)\,{d}\gamma,$

where $\Gamma_{G-\operatorname{reg},\operatorname{ell}}(M,V,\zeta)$ is the
subset of strongly $G$ regular, elliptic elements in the basis
$\Gamma(M,\zeta),$ and $i^{G}(\tau)=|d(\tau)|^{-1}|R_{\pi,r}|^{-1},$
$d(\tau)=d(r)=\operatorname{det}(1-r)_{\mathfrak{a}_{M}/\mathfrak{a}_{G}}$.

We write $\widehat{G}$ for the complex dual of $G$, and ${}^{L}G$ for the
$L$-group $\widehat{G}\rtimes W_{\mathbb{R}}$, which acts through
$W_{\mathbb{R}}\to\Gamma$, Galois group $\Gamma=\\{1,\sigma\\}$.

An endoscopic data for $G$ is a tuple
$(G^{\prime},\mathcal{G^{\prime}},s^{\prime},\xi)$, where

  1. (1)

$G^{\prime}$ is a $K$-group and quasi-split over $\mathbb{R}$, and so has dual
Galois automorphism $\sigma_{\widehat{G^{\prime}}}$.

  2. (2)

$\mathcal{G}^{\prime}$ is a split extension of $W_{\mathbb{R}}$ by
$\widehat{G^{\prime}}$, where $W_{\mathbb{R}}$ acts through
$W_{\mathbb{R}}\to\Gamma$, and $\sigma$ act as $\sigma_{\widehat{G^{\prime}}}$
up to an inner automorphism of $\widehat{G^{\prime}}$.

  3. (3)

$s^{\prime}$ is a semisimple element of $\widehat{G}$,

  4. (4)

$\xi^{\prime}:\mathcal{G}^{\prime}\to{}^{L}G$ is an embedding of extensions
under which the image of $\widehat{G^{\prime}}$ is the identity component of
$\operatorname{Cent}(s^{\prime},\widehat{G})$, and the full image lies in
$\operatorname{Cent}(s^{{}^{\prime\prime}},{}^{L}G)$, for some
$s^{{}^{\prime\prime}}$ congruent to $s^{\prime}$ modulo the center
$Z(\widehat{G})$ of $\widehat{G}$.

We denote $\mathcal{E}(G)$ for the set of equivalence of endoscopic data for
$G$. Then
$\mathcal{E}(G)=\coprod_{\\{M\\}}(\mathcal{E}_{\operatorname{ell}}(M)/W(M))$
or
$\mathcal{E}(G)=(\coprod_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}}\mathcal{L}^{G^{\prime}})/\sim$,
where the equivalence relation is defined by $\widehat{G}$ conjugacy.

We first recall the geometric side of stable local trace formula, which is
given by Arthur [11, §6]. We shall identify $G^{\prime}$ with the diagonal
endoscopic datum $G^{\prime}_{V}=G^{\prime}\times\bar{G^{\prime}}$ for
$G_{V}=G\times G$, where $G^{\prime}$ represents the datum
$(G^{\prime},\mathcal{G}^{\prime},s^{\prime},\xi^{\prime})$, and
$\bar{G}^{\prime}$ represents the adjoint datum
$(G^{\prime},\mathcal{G}^{\prime},(s^{\prime})^{-1},\xi^{\prime})$. The
Langlands-Shelstad transfer factors attached to $(G,G^{\prime})$ depend on an
auxiliary data $\widetilde{G^{\prime}}$ for $G^{\prime}$, such that
$\widetilde{\xi^{\prime}}:\mathcal{G}^{\prime}\to{}^{L}\widetilde{G^{\prime}}$
for $L$-morphism. We can choose a compatible auxiliary data for
$\bar{G^{\prime}}$, so that the relative transfer factor for
$(G,\bar{G^{\prime}})$ is the inverse of the relative transfer factor for
$(G,G^{\prime})$, then we have transfer property,
$\bar{f_{2}}^{\bar{G^{\prime}}}=\overline{f_{2}^{G^{\prime}}}$. The transfer
mappings [8] were used to construct supplementary linear forms
$I^{\mathcal{E}}(f)$ and $S^{G}(f)$ from $I(f)$, where

$I^{\mathcal{E}}(f)=\sum_{G^{\prime}\in\mathcal{E}^{0}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\widehat{S^{\prime}}(f^{\prime})+\varepsilon(G)S^{G}(f),$

the linear forms $\widehat{S^{\prime}}=\widehat{S}^{\widetilde{G^{\prime}}}$
on $SIC(\widetilde{G^{\prime}_{V}},\widetilde{\zeta^{\prime}_{V}})$ are
determined inductively by the further requirement such that
$I^{\mathcal{E}}(f)=I(f)$ if $G$ is a quasi-split group. Here

$\iota(G,G^{\prime})=|\operatorname{Out}_{G}(G^{\prime})|^{-1}|Z(\widehat{G^{\prime}})^{\Gamma}/Z(\widehat{G})^{\Gamma}|^{-1},$

$\mathcal{E}^{0}_{\operatorname{ell}}(G)=\mathcal{E}_{\operatorname{ell}}\backslash\\{G\\},$
and

$\varepsilon(G)=\begin{cases}1&\text{if $G$ is quasi-split},\\\
0&\text{otherwise}.\end{cases}$

For $G$ general, we have $I(f)=I^{\mathcal{E}}(f)$ [11, §6]. If $G$ is quasi-
split, we have a geometric expansion [11, (10.11)]

$S^{G}(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Delta_{G-\operatorname{reg},\operatorname{ell}}(\widetilde{M^{\prime}},V,\widetilde{\zeta^{\prime}})}n(\delta)^{-1}S^{G}_{M}(\delta,f)\,d\delta.$

If $f=f_{1}\times\overline{f_{2}}$, $f_{1}\in
C_{\operatorname{cusp}}(G,\zeta)$, $f_{2}\in C(G,\zeta)$, then

$S^{G}_{M}(\delta,f_{1}\times\bar{f_{2}})=S^{G}_{M}(\delta,f_{1})\times\overline{f^{G}_{2}(\delta)}.$

The spectral side of the local trace formula was stabilized in [8], when the
test function $f\in C_{\operatorname{cusp}}(G,\zeta)\times C(G,\zeta^{-1})$.
But it is just a formal formula that matches the geometric side. The point of
the present work is to directly construct the spectral side of the stable
local trace formula. We need to review a few facts about the spectral side of
invariant local trace formula, before constructing a stabilization.

The irreducible tempered representations could well be regarded as the objects
dual to semisimple conjugacy classes in $G(\mathbb{R})$. It is better to take
the family of virtual characters [5], which is parametrized by a set
$T(G,\zeta)$.

######  Definition 5.1.

$T(G,\zeta)$ is the set of $W_{0}$ orbits of the triplet
$\tau=(M,\pi,r),M\in\mathcal{L},\pi\in\Pi_{2}(M,\zeta),r\in\widetilde{R_{\pi}}$,
where $\Pi_{2}(M,\zeta)$ stands for the equivalence classes of irreducible
unitary representations of $M(\mathbb{R})$ which are square integrable modulo
the center, whose central character is $\zeta$, and $\widetilde{R_{\pi}}$ is a
fixed central extension

$1\longrightarrow\widetilde{Z_{\pi}}\longrightarrow\widetilde{R_{\pi}}\longrightarrow
R_{\pi}\longrightarrow 1$

of the $R$-group of $\pi$.

The purpose of the extension is to ensure that the normalized intertwining
operators,
$r\mapsto\widetilde{R_{P}}(r,\pi),r\in\widetilde{R_{\pi}},P\in\mathcal{P}(M)$
for the induced representation $I_{P}(\pi)$, give a representation of
$\widetilde{R_{\pi}}$ instead of just a projective representation of
$R_{\pi}$. However, in the Archimedean case, $R_{\pi}$ is a product of groups
$\mathbb{Z}/2\mathbb{Z}$, the cocycle which defines $\widetilde{R_{\pi}}$
splits, so we take $\widetilde{R_{\pi}}=R_{\pi}$.

There is a bijection $\rho\mapsto\pi_{\rho}$ from $\Pi(R_{\pi})$ the set of
irreducible representations of $R_{\pi}$ onto the set of irreducible
constituents of $I_{P}(\pi)$, with the properties that

(5.4)
$\Theta(\tau,f)=\operatorname{tr}(R_{P}(r,\pi)I_{P}(\pi,f))=\sum_{\rho\in\Pi(R_{\pi})}\operatorname{tr}(\rho^{\vee}(r))\operatorname{tr}(\pi_{\rho}(f)),$

and

(5.5) $\operatorname{tr}(\pi_{\rho}(f))=|R_{\pi}|^{-1}\sum_{r\in
R_{\pi}}\operatorname{tr}(\rho(r))\operatorname{tr}(R_{P}(r,\pi)I_{P}(\pi,f)).$

We can write

$T(G,\zeta)=\coprod_{\\{M\\}}(T_{\operatorname{ell}}(M,\zeta)/W(M)),$

where $T_{\operatorname{ell}}(M,\zeta)=\\{\tau=(M^{\prime},\pi^{\prime},r)\in
T(M,\zeta):\mathfrak{a}^{r}_{M^{\prime}}=\mathfrak{a}_{M},r\in R_{\pi}\\}$.
$\\{M\\}$ as usual runs over the orbits in $\mathcal{L}/W^{G}_{0}$. If
$\tau=(M,\pi,r)$ is any triplet, the isotropy subspace $\mathfrak{a}^{r}_{M}$
of $\mathfrak{a}_{M}$ equals $\mathfrak{a}_{L}$ for some $L$. There is an
action

$\tau\mapsto\tau_{\lambda}=(M,\pi_{\lambda},r),\tau\in
T_{ell}(L,\zeta),\lambda\in i\mathfrak{a}^{\ast}_{L,Z}$

of $i\mathfrak{a}^{\ast}_{L,Z}$ on $T_{\operatorname{ell}}(L,\zeta)$, where
$\pi_{\lambda}(x)=\pi(x)e^{\lambda(H_{M}(x))}$ for any $x\in M(\mathbb{R})$.
This gives the structure for $T(G,\zeta)$ a disjoint union of finite quotients
of compact tori. We will write $T(G,\zeta)_{\mathbb{C}}$ to be the disjoint
union over $\\{M\\}$ of the spaces of $W(M)$ orbits in
$T_{\operatorname{ell}}(M,\zeta)_{\mathbb{C}}=\\{\tau_{\lambda}:\tau\in
T_{\operatorname{ell}}(M,\zeta),\lambda\in\mathfrak{a}^{\ast}_{M,Z,\mathbb{C}}\\}$,
where $\mathfrak{a}^{\ast}_{M,Z}$ is the subspace of linear forms on
$\mathfrak{a}_{M}$ which are trivial on the image of $\mathfrak{a}_{Z}$ in
$\mathfrak{a}_{M}$, and
$\mathfrak{a}^{\ast}_{M,\mathbb{C}}=X(M)_{\mathbb{R}}\otimes\mathbb{C}$, where
$X(M)$ is the set of rational characters on $M$.

We denote $T_{\operatorname{disc}}(G,\zeta)$ for a set of orbits $(M,\pi,r)$
in $T(G,\zeta)$ such that $W_{\pi}(r)_{\operatorname{reg}}$ is not empty. Then

$T_{\operatorname{ell}}(G,\zeta)\subset
T_{\operatorname{disc}}(G,\zeta)\subset T(G,\zeta).$

Let $I(G(\mathbb{R}),\zeta)$ be the space of functions

$\alpha:T(G,\zeta)\to\mathbb{C},$

which satisfy the following three conditions;

  1. (1)

$\alpha$ is supported on finitely many components of $T(G,\zeta)$,

  2. (2)

$\alpha$ is symmetric under $W^{G}_{0}$,

  3. (3)

$\alpha\in S(T(G,\zeta))$.

Here $S(T(G,\zeta))$ is the space of smooth functions $\alpha$ on
$T(G,\zeta)$, such that for each $M\in\mathcal{L}$, each integer $n$ and each
invariant differential operator $D=D_{\lambda}$ on
$i\mathfrak{a}^{\ast}_{M,Z}$ transferred in the obvious way
$D_{\tau}\alpha(\tau)=lim_{\lambda\to
0}D_{\lambda}\alpha(\tau_{\lambda}),\tau\in T_{\operatorname{ell}}(M,\zeta)$
to $T_{\operatorname{ell}}(M,\zeta)$, and such that the semi-norm

$\|\alpha\|_{M,D,n}=Sup_{\tau\in
T_{\operatorname{ell}}(M,\zeta)}(|D_{\tau}\alpha(\tau)|(1+\|\mu_{\tau}\|)^{n})$

is finite, where $\mu_{\tau}=\mu_{\pi}$ for $\tau=(M,\pi,r)$. $\mu_{\pi}$ is
the linear form determined by the infinitesimal character of $\pi$. Then there
is a natural topology which makes $I(G(\mathbb{R}),\zeta)$ into a complete
topological vector space. By means of the inversion formula (5.5), we can
identify $I(G(\mathbb{R}),\zeta)$ with the topological vector space of
functions on $\Pi_{\operatorname{temp}}(G(\mathbb{R}),\zeta)$, and also
denoted by $I(G(\mathbb{R}),\zeta)$.

The trace Paley-Wiener theorem [6] is equivalent to the assertion that the map
which sends $f\in C(G(\mathbb{R}),\zeta)$ to the function
$f_{G}(\tau)=\Theta(\tau,f)$ is an open, continuous and surjective linear
transformation from $C(G(\mathbb{R}),\zeta)$ onto $I(G(\mathbb{R}),\zeta)$.
Observe that if $\tau\in T_{\operatorname{ell}}(G,\zeta)$, there is a function
$f\in C(G(\mathbb{R}),\zeta)$ with $f_{G}(\tau)=1$, and such that $f_{G}$
vanishes away from the $i\mathfrak{a}^{\ast}_{G,Z}$ orbit of $\tau\in
T(G,\zeta)$. We call such a function $f$ for a pseudo-coefficient of $\tau$.

Stabilization of the spectral side of local trace formula depends on
Shelstad’s works. She [28], [29] directly constructs the spectral transfer
factors and obtains the transfer theorem. The adjoint relation on $K$-groups
and the structure of tempered $L$-packets are given in [30].

We denote by $\Pi_{\operatorname{temp}}(G,\zeta)$ the set of tempered
representations, with central character for $\zeta$. We have
$\Pi_{\operatorname{temp}}(G,\zeta)=\coprod_{\\{M\\}}\Pi_{2}(M,\zeta)/W(M)$,
here $\\{M\\}$ for the set of $W^{G}_{0}$-orbits of Levi subgroups of $G$.

The Langlands parameter $\phi:W_{\mathbb{R}}\to{}^{L}G$ is an
$L$-homomorphism, which maps from $W_{\mathbb{R}}$ into the $L$-group
${}^{L}G$. We denote $\Phi(G)$ for the set of $\widehat{G}$-orbits of
parameters which are tempered, which means that the image of $W_{\mathbb{R}}$
in $\widehat{G}$ is bounded. We denote $\Phi_{2}(G)$ for the subset of
parameters in $\Phi(G)$ which are cuspidal. The cuspidal condition means that
the image of $W_{\mathbb{R}}$ is contained in no proper parabolic subgroup.
There is a canonical decomposition

$\Phi(G)=\coprod_{\\{M\\}}(\Phi_{2}(M)/W(M)).$

For any $\phi$, we denote $S_{\phi}$ as the centralizer of the image of $\phi$
in $\widehat{G}$, and $\mathcal{S}_{\phi}$ stands for the group of connected
components in $\bar{S}_{\phi}=S_{\phi}/Z(\widehat{G})^{\Gamma}$. We say that
$\phi$ is elliptic, if $\bar{S}_{\phi,s}$ is finite for some semisimple
element $s\in\bar{S}_{\phi}$. For any parameter $\phi\in\Phi(G)$, we denote
the central character $\zeta$ for $\phi$, whose Langlands parameter is just
the composition

$W_{F}\xrightarrow{\phi}{}^{L}G\rightarrow{}^{L}Z.$

The entire set $\Phi(G)$ decomposes into a disjoint union of the subsets
$\Phi(G,\zeta)$. The set $\Phi_{2}(G,\zeta)$ also comes with an action
$\phi\mapsto\phi_{\lambda}=\phi\circ\rho_{\lambda}$ of
$i\mathfrak{a}^{\ast}_{G,Z}$, where $\rho_{\lambda}\in
H^{1}(W_{\mathbb{R}},Z(\widehat{G})^{\Gamma})$, which corresponds to the
character $\pi_{\rho_{\lambda}}(x)=\mathnormal{e}^{\lambda(H_{G}(x))}$.

Given $s\in\bar{S}_{\phi}$, we attach an endoscopic data
$G^{\prime}=G^{s}=(G^{s},\mathcal{G}^{s},s,\xi^{s})$. Where $\mathcal{G}^{s}$
is the subgroup of ${}^{L}G$ generated by
$\operatorname{Cent}(s,\widehat{G})^{\circ}$ and the image of $\phi$.
$\xi^{s}$ is the inclusion $\mathcal{G}^{s}\hookrightarrow{}^{L}G$ and $G^{s}$
is a quasi-split group. Usually, $\mathcal{G}^{s}$ need not be an $L$-group,
namely that there might not be an $L$-isomorphism from $\mathcal{G}^{s}$ to
${}^{L}G^{s}$ which is the identity component of $\widehat{G^{\prime}}$. To
deal with this problem, we need to make a $z$-extension
$\widetilde{G^{\prime}}$ of $G^{\prime}$. For simplicity, we assume
$\widetilde{G^{\prime}}=G^{\prime}$, for any $G^{\prime}$.

Shelstad established the spectral transfer mapping, which is given by a linear
combination

(5.6)
$f^{\prime}(\phi^{\prime})=\sum_{\pi\in\Pi_{\operatorname{temp}}(G(\mathbb{R}))}\Delta(\phi^{\prime},\pi)f_{G}(\pi)$

of irreducible tempered characters
$f_{G}(\pi)=\operatorname{tr}(\pi(f)),\pi\in\Pi_{\operatorname{temp}}(G,\zeta),$
on $G(\mathbb{R})$. The coefficients are spectral transfer factors
$\Delta(\phi^{\prime},\pi)$. They are established explicitly by Shelstad in
[29], which are compatible with the geometric transfer factors. We assume
implicitly that the Langlands parameter $\phi$ is relevant to $G$, in the
sense that if its image is contained in a parabolic subgroup
${}^{L}P\subset{}^{L}{G}$, then ${}^{L}P$ is dual to a $\mathbb{Q}$-rational
parabolic subgroup $P\subset G$. It then gives rise to the $L$-packet
$\Pi_{\phi}$ that was an integral part of Langlands’s classification of
representations of real groups [22]. $\Pi_{\phi}$ is a finite subset of
representations in $\Pi_{\operatorname{temp}}(G,\zeta)$ whose constituents
have the same local $L$-functions and $\varepsilon$-factors, and
$\Pi_{\operatorname{temp}}(G,\zeta)$ is a disjoint union over
$\phi\in\Phi(G,\zeta)$ of the subsets $\Pi_{\phi}$. Shelstad established that
for any $\phi$, the distribution

$f^{G}(\phi)=\sum_{\pi\in\Pi_{\phi}}f_{G}(\pi)$

is stable, in the sense that it depends only on the image $f^{G}$ of $f$ in
$S(G,\zeta)$. Here $S(G,\zeta)=\\{f^{G}:f\in C(G,\zeta)\\}$, and $f^{G}$ is
the stable orbital integral

$f^{G}(\delta)=|D(\delta)|^{1/2}\int_{G_{\delta}(\mathbb{R})\backslash
G(\mathbb{R})}f(x^{-1}\delta x)\,dx=\sum_{\gamma\to\delta}f_{G}(\gamma).$

When applied to $G^{\prime}$ instead of $G$, this gives the left hand side of
(5.6). We also assume that the given pair $(G^{\prime},\phi^{\prime})$ is
relevant to $G$, in the sense that the composite Langlands parameter
$\phi=\xi^{\prime}\circ\phi^{\prime}:W_{\mathbb{R}}\to{}^{L}G$ is relevant to
$G$. Then we have a bijection mapping

(5.7) $(G^{\prime},\phi^{\prime})\to(\phi,s),$

where $s\in\bar{S}_{\phi}$.

Shelstad also established the inversion of the transfer mapping, which is
given by a linear combination

$f_{G}(\pi)=\sum_{s_{\operatorname{sc}}}\Delta(\pi,\phi^{s})f^{\prime}(\phi^{s})$

of stable characters on endoscopic groups, where
$s_{\operatorname{sc}}\in\widetilde{\mathcal{S}_{\phi}}$ comes from an
extension

$1\rightarrow\widehat{Z}_{\operatorname{sc}}\rightarrow\widetilde{\mathcal{S}_{\phi}}=\pi_{0}(S_{\phi,\operatorname{sc}})\rightarrow\mathcal{S_{\phi}}\rightarrow
1,$

where $S_{\phi,\operatorname{sc}}$ is the preimage of $\bar{S}_{\phi}$ in
$\widehat{G}_{\operatorname{sc}}$, the simply connected cover the derived
group of $\widehat{G}$ and
$\widehat{Z}_{\operatorname{sc}}=Z(\widehat{G}_{\operatorname{sc}})$,
$\phi^{s}$ stands for the parameter $\phi^{\prime}$ that corresponds to the
pair $(\phi,s)$ under the map (5.7). The inversion of transfer mapping rests
on explicit adjoint relations for spectral transfer factors
$\Delta_{\operatorname{spec}}$ defined initially as a product
$\Delta_{I}\Delta_{II}\Delta_{III}$ in the $G$-regular case in [29], and we
have adjoint relations

$\sum_{s_{\operatorname{sc}}}\Delta(\pi,\phi^{s})\Delta(\phi^{s},\pi^{\prime})=\delta(\pi,\pi^{\prime}),$

where the sum is over semisimple representative $s_{\operatorname{sc}}$ for

$\widetilde{\mathcal{S}_{\mathcal{\phi}}}/\operatorname{Ker}(\widetilde{\mathcal{S}_{\mathcal{\phi}}}\to\mathcal{S}_{\phi})\simeq\mathcal{S}_{\phi},$

$\delta(.,.)$ denotes the Kronecker delta function. If $\pi$ and $\phi^{s}$
correspond to the same $L$-parameter $\phi$, then

$\Delta(\pi,\phi^{s})=\frac{1}{n(\pi)}\Delta(\phi^{s},\pi)^{-1},$

and

$\Delta(\phi^{s},\pi)^{-1}=\overline{\Delta(\phi^{s},\pi)}/\|\Delta\|^{2},$

$n(\pi)=|\mathcal{S_{\phi}}|$ is the cardinality of the $L$-packet of $\pi$,
$\|\Delta\|=|\Delta(\phi^{s},\pi)|$ is a constant (independent s), which is
compatible with geometric transfer factor. However, the geometric transfer
factor is unitary, so we take $\|\Delta\|=1$.

We also obtain the other adjoint relations

$\sum_{\pi\in\Pi_{\phi}}\Delta(\phi^{s},\pi)\Delta(\pi,\phi^{s^{\prime}})=\delta(s,s^{\prime}),$

where $\phi=\xi^{s}\circ\phi^{s}$, $s,s^{\prime}$ in $\mathcal{S}_{\phi}$.

So we can define an inversion adjoint transfer factor

$\Delta(\tau,\phi^{s})=\sum_{\chi\in\widehat{R}_{\pi}}\overline{\chi(r)}\Delta(\pi^{\chi},\phi^{s}),$

where $\overline{\chi(r)}=\operatorname{tr}(\rho^{\vee}(r))$. (Recall that
$R_{\pi}$ is finite abelian group.)

We obtain

(5.8)
$\Theta(\tau,f)=\sum_{s\in\mathcal{S}_{\phi}}\Delta(\tau,\phi^{s})f^{\prime}(\phi^{s}).$

However, when we define the transfer factor $\Delta(\phi^{s},\tau)$, we need
to assume that the Langlands parameter $\phi$ to be elliptic. This means that
$\Pi_{\phi}$ contains elliptic representations. $\phi$ factors through a
discrete parameter for a cuspidal Levi subgroup ${}^{L}M$, and so through
${}^{L}T_{M}$ [22], where $T_{M}$ is the maximal torus which is compact modulo
the center of $M$. We consider the associated short exact sequence [29]

$1\rightarrow\mathcal{E}(T_{M})\rightarrow\mathcal{S}_{\phi}\rightarrow
R_{\phi}\rightarrow 1,$

where $R_{\phi}$ is the Langlands $R$-group, and the group
$\mathcal{E}(T_{M})$ is isomorphic to $\mathcal{S}_{\phi_{M}}$, where $M$ is a
Levi subgroup of $G$, and $\phi_{M}:W_{\mathbb{R}}\to{}^{L}M$ is a Langlands
parameter for $M$ whose image in ${}^{L}G$ equals $\phi$, and whose L-packet
$\Pi_{\phi_{M}}$ consists of representations in the discrete series of
$M(\mathbb{R})$.

Suppose $\pi_{M}\in\Pi_{\phi_{M}}$ corresponds to the character $\chi$ on the
group $\mathcal{S}_{\phi_{M}}$. Since $\mathcal{S}_{\phi}$ is an abelian
group, $R_{\chi}$ equals the full group $R_{\phi}$, where $R_{\chi}$ is the
subgroup of elements in $R_{\phi}$ that stabilize $\chi$, and $\chi$ extends
to a character $\theta$ on $\mathcal{S}_{\phi}$. The set of such extension
$\theta$ is a torsor under the action of the characters in $R_{\phi}$. It
corresponds to subset $\Pi_{\phi,\pi_{M}}$ of $\Pi_{\phi}$, composed of the
irreducible constituents of the induced representation $I^{G}_{P}(\pi_{M})$,
where $P$ belongs to the set $\mathcal{P}(M)$ of parabolic subgroups of $G$
with Levi component $M$. So we have $|R_{\pi_{M}}|=|R_{\phi}|$.

On the other hand, we identity the stabilizer $W_{\phi}$ of $\phi_{M}$ with a
subgroup of $W(M)$, where $W_{\phi}=W(S_{\phi},A_{\widehat{M}})$, which is to
say, the group of automorphisms of $A_{\widehat{M}}$ induced from
$\bar{S}_{\phi}$, and $A_{\widehat{M}}=(Z(\widehat{M})^{\Gamma})^{\circ}$.
This Weyl group contains the stabilizer $W_{\pi_{M}}$ of $\pi_{M}$. It is a
consequence of the disjointness of tempered $L$-packets for $M$ that
$W_{\phi}$ contains $W_{\pi_{M}}$. From the above discussion, we know that any
element in $W_{\phi}$ stabilizes $\pi_{M}$. Therefore $W_{\phi}$ equals
$W_{\pi_{M}}$. Moreover, we know that elements in the subgroup $W^{0}_{\phi}$
of $W_{\phi}$ give scalar intertwining operators for the induced
representation $I^{G}_{P}(\pi_{M})$, where
$W^{\circ}_{\phi}=W(S^{\circ}_{\phi},A_{\widehat{M}})$ is to the normal
subgroup of automorphisms in $W_{\phi}$ that are induced from the connected
component $\overline{S}^{\circ}_{\phi}$. It follows that $W^{0}_{\phi}$ is
contained in the subgroup $W^{0}_{\pi_{M}}$ of $W_{\pi_{M}}$. We have a
surjective mapping

$R_{\phi}\cong W_{\phi}/W^{0}_{\phi}\mapsto W_{\pi_{M}}/W^{0}_{\pi_{M}}\cong
R_{\pi_{M}},\quad\pi_{M}\in\Pi_{\phi_{M}},$

for any elliptic parameter $\phi\in\Phi_{\operatorname{ell}}(G,\xi)$. So we
obtain $R_{\pi_{M}}=R_{\phi},W^{\circ}_{\pi_{M}}=W^{\circ}_{\phi},$ and
$W_{\pi_{M}}=W_{\phi}$. We define the subset
$T_{\phi}=\\{(M,\pi,r):M=M_{\phi},\pi\in\Pi_{\phi_{M}},r\in R_{\phi}\\}$ of
$T(G)$, then
$|T_{\phi}|=|\Pi_{\phi_{M}}||R_{\phi}|=|\mathcal{S}_{\phi}|=|\Pi_{\phi}|$. And
we get a bijection from $T_{\phi}$ to $\mathcal{S}_{\phi}$.

We can define the adjoint transfer factor

$\Delta(\phi^{s},\tau)=\sum_{\chi\in\widehat{R}_{\pi}}\frac{1}{|R_{\pi}|}\chi(r)\Delta(\phi^{s},\pi^{\chi}),$

where $\tau=(M,\pi,r)$, $R_{\pi}$ is a $2$-group, and $\pi^{\chi}$ is the
irreducible component of the induced representation of $\pi$, which
corresponds to $\chi$ by Arthur’s classification Theorem in [5, §2]. So we
obtain

(5.9) $f^{\prime}(\phi^{s})=\sum_{\tau\in
T_{\phi}}\Delta(\phi^{s},\tau)\Theta(\tau,f).$

The transfer factors have the following properties.

######  Proposition 5.2.

If $\phi$ is elliptic, then

  1. (1)

$\Delta(\tau,\phi^{s})=\frac{|R_{\phi}|}{|\mathcal{S}_{\phi}|}\overline{\Delta(\phi^{s},\tau)},$

  2. (2)

we have the adjoint relations,

(5.10) $\displaystyle\sum_{\tau\in
T_{\phi}}\Delta(\phi^{s_{1}},\tau)\Delta(\tau,\phi^{s_{2}})$
$\displaystyle=\delta(\phi^{s_{1}},\phi^{s_{2}}),$ (5.11)
$\displaystyle\sum_{s\in\mathcal{S}_{\phi}}\Delta(\tau,\phi^{s})\Delta(\phi^{s},\tau_{1})$
$\displaystyle=\delta(\tau,\tau_{1}).$

###### Proof.

We first check (1).

$\Delta(\tau,\phi^{s})=\sum_{\chi\in\widehat{R}_{\pi}}\overline{\chi(r)}\Delta(\pi^{\chi},\phi^{s}),$

and

$\overline{\Delta(\phi^{s},\tau)}=\sum_{\chi\in\widehat{R}_{\pi}}\frac{1}{|R_{\pi}|}\overline{\chi(r)}\overline{\Delta(\phi^{s},\pi^{\chi})}=\sum_{\chi\in\widehat{R}_{\pi}}\frac{n(\pi)}{|R_{\pi}|}\overline{\chi(r)}\Delta(\pi^{\chi},\phi^{s}),$

where $|R_{\pi}|=|R_{\phi}|$, and $n(\pi)=|\mathcal{S}_{\phi}|$. We obtain the
equation (1).

We now prove (2). We only check (5.10), as (5.11) is similar. Observe that
$\phi$ is relevant to $G$, and

$f^{\prime}(\phi^{s_{1}})=\sum_{\tau\in
T_{\phi}}\Delta(\phi^{s_{1}},\tau)\Theta(\tau,f)=\sum_{s\in\mathcal{S}_{\phi}}\sum_{\tau\in
T_{\phi}}\Delta(\phi^{s_{1}},\tau)\Delta(\tau,\phi^{s})f^{\prime}(\phi^{s}).$

Using the fact that the characters of representations are linear independent,
then we obtain the identity (5.10). ∎

We return to analyze the transfer factors

$\Delta(\phi^{s},\tau)=\sum_{\chi\in\widehat{R}_{\pi}}\frac{1}{|R_{\pi}|}\chi(r)\Delta(\phi^{s},\pi^{\chi}),$

where $\Delta(\phi^{s},\pi^{\chi})$ is defined directly by Shelstad in [29],
and extended to a $K$-group [30]. She also checked Arthur’s conjecture for the
transfer factors in [12], and obtained the formula for
$\Delta(\phi^{s},\pi^{\chi})$ in [30]:

$\Delta(\phi^{s},\pi^{\chi})=\rho(\Delta,s_{\operatorname{sc}})\langle{s_{\operatorname{sc}}},{\pi^{\chi}}\rangle,$

where $\rho(\Delta,s_{\operatorname{sc}})$ satisfies
$\rho(t\Delta,z_{\operatorname{sc}}s_{\operatorname{sc}})=t\rho(\Delta,s_{\operatorname{sc}})\zeta_{G}(z_{\operatorname{sc}})^{-1}$
for $t\in\mathbb{C}^{\times}$, $z_{\operatorname{sc}}\in
Z(\widehat{G}_{\operatorname{sc}}$), and $s_{\operatorname{sc}}$ is the
preimage $s$ of the mapping
$\widetilde{\mathcal{S}}_{\phi}\to\mathcal{S}_{\phi}$. $\zeta_{G}$ comes from
Arthur’s paper [8]. And

$\rho(\Delta,s_{\operatorname{sc}})=\zeta_{G}(s_{\operatorname{sc}})^{-1}\delta(\pi^{s},\pi^{base}),\quad\zeta_{G}(s_{\operatorname{sc}})=\langle{s_{\operatorname{sc}}},\pi^{base}\rangle.$

If $G$ is quasi-split, $\rho(\Delta,s_{\operatorname{sc}})=1$, so the formula
for the transfer factor simplifies:

$\Delta(\phi^{s},\pi^{\chi})=\langle{s_{\operatorname{sc}}},{\pi^{\chi}}\rangle.$

In section 6, we will stabilize the spectral side of the invariant local trace
formula when one of the component of test function is cuspidal, which is
enough to give the multiplicity formula.

##  6\. Stabilization of the elliptic terms

We now consider the test function $f=f_{1}\times\bar{f_{2}}$, $f_{1}\in
C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$, $f_{2}\in
C(G(\mathbb{R}),\zeta)$, $G$ is a $K$-group over $\mathbb{R}$. Recall that
$C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$ stands for the space of
functions $f_{1}$ in $C(G(\mathbb{R}),\zeta)$ that are cuspidal, in the sense
that the orbital integral

$\gamma\mapsto f_{1,G}(\gamma)=J_{G}(\gamma,f_{1}),\quad\gamma\in\Gamma(G)$

is supported on the subset $\Gamma_{\operatorname{ell}}(G)$ of elliptic
classes in $\Gamma(G)$.

We assume that $f_{1}$ is cuspidal. We set

$I_{\operatorname{disc}}(f)=\int_{T_{\operatorname{ell}}(G,\zeta)}i^{G}(\tau)f_{1,G}(\tau)\overline{f_{2,G}(\tau)}d\tau,$

where
$i^{G}(\tau)=|R_{\pi,r}|^{-1}|\operatorname{det}(1-r)_{\mathfrak{a}_{M}/\mathfrak{a}_{G}}|^{-1}$,
$\tau=(M,\pi,r)$,
$T_{\operatorname{ell}}(G,\zeta)=\prod_{\alpha\in\pi_{0}(G)}T_{\operatorname{ell}}(G_{\alpha},\zeta_{\alpha})$.
For the given $f$, $I_{\operatorname{disc}}(f)$ equals the spectral side of
the local trace formula. We have

$I(f)=I_{\operatorname{disc}}(f).$

Now we can regard $I_{\operatorname{disc}}$ as a linear form on the subspace

$C_{1-\operatorname{cusp}}(G_{V},\zeta_{V})=C_{\operatorname{cusp}}(G,\zeta)\otimes
C(G,\zeta^{-1})$

of $C(G_{V},\zeta_{V})$.
If $f_{1}\in C_{\operatorname{cusp}}(G,\zeta)$, then $f_{1}$ is supported on
$T_{\operatorname{ell}}(G,\zeta)$. For stabilization, we need to consider
Langlands parameters $\phi$ which are elliptic. We can define a corresponding
set

$\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)=\\{(G^{\prime},\phi^{\prime}):G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G),\phi^{\prime}\in\Phi_{2}(G^{\prime},G,\zeta)\\},$

where
$\Phi_{2}(G^{\prime},G,\zeta)=\Phi_{2}(G,\zeta)/\operatorname{Out}_{G}(G^{\prime})$,
$\operatorname{Out}_{G}(G^{\prime})=\operatorname{Aut}_{G}(G^{\prime})/\xi^{\prime}(\widehat{G^{\prime}})$,
and

$\operatorname{Aut}_{G}(G^{\prime})=\\{g\in\widehat{G}:gs^{\prime}g^{-1}\in
s^{\prime}Z(\widehat{G}),g\mathcal{G}^{\prime}g^{-1}=\mathcal{G}^{\prime}\\}.$

We denote $SI_{\operatorname{cusp}}(G^{\prime},\zeta)$ for the set of linear
forms $f^{\prime}(\phi^{\prime})$ on $\Phi_{2}(G^{\prime},\zeta)$ obtained
from the transfer map. $f^{\prime}(\phi^{\prime})$ depends only on the image
of $\phi^{\prime}$ in $\Phi_{2}(G^{\prime},G,\zeta)$, which is the set of
$\operatorname{Out}_{G}(G^{\prime})$-orbit in $\Phi_{2}(G^{\prime},\zeta)$
[30]. If $f^{\prime}(\phi^{\prime})\in
SI_{\operatorname{cusp}}(G^{\prime},\zeta)$, then $f^{\prime}(\phi^{\prime})$
is supported on $\Phi_{2}(G^{\prime},G,\zeta)$.

We have

$\displaystyle\theta(\tau,f)$
$\displaystyle=\sum_{\phi^{\prime}\in\Phi_{\operatorname{ell}}^{\mathcal{E}}(G)}\Delta(\tau,\phi^{\prime})f^{\mathcal{E}}(\phi^{\prime})$
$\displaystyle=\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\sum_{\phi^{\prime}\in\Phi_{2}(G^{\prime},G,\zeta)}\Delta(\tau,\phi^{\prime})f^{G^{\prime}}(\phi^{\prime}),$

and

$f^{G^{\prime}}(\phi^{\prime})=\sum_{\tau\in
T_{\operatorname{ell}}(G,\zeta)}\Delta(\phi^{\prime},\tau)\theta(\tau,f).$

######  Lemma 6.1.

The transfer factors $\Delta(\tau,\phi)$ and $\Delta(\phi,\tau)$ have finite
support in $\phi$ for fixed $\tau$, and finite support in $\tau$ for fixed
$\phi$. Moreover,

$\displaystyle\sum_{\phi^{\prime}\in\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}\Delta(\tau,\phi^{\prime})\Delta(\phi^{\prime},\tau_{1})=\delta(\tau,\tau_{1}),$
$\displaystyle\tau,\tau_{1}\in T_{\operatorname{ell}}(G,\zeta),$
$\displaystyle and$ $\displaystyle\sum_{\tau\in
T_{\operatorname{ell}}(G,\zeta)}\Delta(\phi^{\prime},\tau)\Delta(\tau,\phi_{1}^{\prime})=\delta(\phi^{\prime},\phi_{1}^{\prime}),$
$\displaystyle\phi^{\prime},\phi_{1}^{\prime}\in\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta),$

where $\delta(\tau,\tau_{1})$ and $\delta(\phi^{\prime},\phi_{1}^{\prime})$
are Kronecker delta functions.

###### Proof.

That $\Delta(\tau,\phi)$ and $\Delta(\phi,\tau)$ have finite support, which is
equivalent to saying that Shelstad’s transfer factor $\Delta(\pi,\phi)$ and
$\Delta(\phi,\pi)$ have finite support, following from [29, §7]. The proof of
second part is similar to Proposition 5.2 (2). ∎

We define a measure on $\Phi_{2}(G,\zeta)$ by setting

$\int_{\Phi_{2}(G,\zeta)}\beta(\phi)d\phi=\sum_{\phi\in\Phi_{2}(G)/i\mathfrak{a}_{G,Z}^{\ast}}\int_{i\mathfrak{a}_{G,Z}^{\ast}}\beta(\phi_{\lambda})d\lambda$

for any $\beta\in C(\Phi_{2}(G,\zeta))$. So
$\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)$ has the measure obtained
from the quotient measures on the set $\Phi_{2}(G^{\prime},G,\zeta)$ and the
transfer factors govern the change of variables of integration.

######  Lemma 6.2.

Suppose that $\alpha\in C(T_{\operatorname{ell}}(G),\zeta)$, and that
$\beta\in
C_{\operatorname{cusp}}(\Phi_{\operatorname{ell}}^{\mathcal{E}}(G),\zeta)$.
Then

$\displaystyle\int_{T_{\operatorname{ell}}(G,\zeta)}\sum_{\phi\in\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}\beta(\phi)\Delta(\phi,\tau)\alpha(\tau)d\tau$
$\displaystyle=$
$\displaystyle\int_{\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}\sum_{\tau\in
T_{\operatorname{ell}}(G,\zeta)}\beta(\phi)\Delta(\phi,\tau)\alpha(\tau)d\phi.$

###### Proof.

According to the definition of the measure $d\tau$, we can decompose the left
hand side of the required identity into an expression

$\sum_{\tau}\int_{i\mathfrak{a}_{G,Z}^{\ast}}\sum_{\phi}\sum_{\mu}\beta(\phi_{\mu})\Delta(\phi_{\mu},\tau_{\lambda})\alpha(\tau_{\lambda})d\lambda.$

Here $\tau\in T_{\operatorname{ell}}(G,\zeta)/i\mathfrak{a}^{\ast}_{G,Z}$,
$\phi\in\Phi^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)/i\mathfrak{a}^{\ast}_{G,Z},$
$\mu\in i\mathfrak{a}^{\ast}_{G,Z}.$ We recall that the transfer factor
$\Delta(\phi,\tau)$ vanishes unless $\tau\in T_{\phi}$. We observe from the
definition that $\phi_{\mu}=\phi\circ\rho_{\mu}$. Here $\rho_{\mu}\in
H^{1}(W_{\mathbb{R}},Z(\widehat{G})^{\Gamma})$ and it corresponds to the
$L$-packet which is
$\Pi_{\phi_{\mu}}=\\{\pi_{\rho_{\mu}}\otimes\pi|\pi\in\Pi_{\phi}\\}$, where
$\pi_{\rho_{\mu}}(x)=e^{\mu(H_{G}(x))}$. However,
$\tau_{\lambda}=(M,\pi_{\lambda},r)$, where
$\pi_{\lambda}(x)=\pi(x)e^{\lambda(H_{G}(x))}$. If
$\Delta(\phi_{\mu},\tau_{\lambda})$ doesn’t vanish, then $\phi_{\mu}$ and
$\tau_{\lambda}$ correspond to the same parameter $\phi_{\lambda}$. So,
$\mu=\lambda$. We see that the sum over $\mu$ reduces to the one element
$\mu=\lambda$. The expression becomes

$\sum_{(\tau,\phi)}\int_{i\mathfrak{a}_{G,Z}^{\ast}}\beta(\phi_{\lambda})\Delta(\phi_{\lambda},\tau_{\lambda})\alpha(\tau_{\lambda})d\lambda,$

where $(\tau,\phi)$ is summed over pairs in
$(T_{\operatorname{ell}}(G,\zeta)\times\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta))/i\mathfrak{a}_{G,Z}^{\ast}$.
From its obvious symmetry, we conclude that the expression must also be equal
to the right hand side of the required identity. The identity is therefore
valid. ∎

In the following, we will stabilize the elliptic terms of the spectral side of
invariant local trace formula and obtain the explicit formula for the
coefficients .

Assume that $f_{1}$ is a cuspidal function, we denote
$|d(\tau)|=|d(r)|=|\operatorname{det}(1-r)_{\mathfrak{a}_{M}/\mathfrak{a}_{G}}|^{-1}$,
$i^{G}(\tau)=|R_{\pi,r}|^{-1}|d(\tau)|^{-1}$. We need to stabilize
$d(\tau)f_{1,G}(\tau)$.

######  Lemma 6.3.

If $f_{1}\in C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$, which is supported
on $T_{\operatorname{ell}}(G,\zeta)$, there exists a cuspidal function
$\widetilde{f}_{1}\in C(G(R),\zeta)$, satisfying

$\widetilde{f}_{1,G}(\tau)=d(\tau)f_{1,G}(\tau).$

###### Proof.

Notice that $\tau\to d(\tau)f_{1,G}(\tau)$ is also a function in
$I(G(\mathbb{R}),\zeta)$, which is supported on
$T_{\operatorname{ell}}(G,\zeta)$. Applying the trace Paley-Wiener theorem,
there exists a cuspidal function $\widetilde{f_{1}}\in C(G(\mathbb{R}),\zeta)$
such that

$\widetilde{f}_{1,G}(\tau)=\theta(\tau,\widetilde{f_{1}})=d(\tau)f_{1,G}(\tau).$

∎

For computing the coefficient of the spectral side of the stable local trace
formula, we need to consider two cases. One is that $\pi$ is elliptic as well
as regular, the other is that $\pi$ is elliptic and not regular. Consider the
first case, then $\pi$ is a discrete series representation. Thus there exists
$\phi$ which is cuspidal, and satisfies $\pi\in\Pi_{\phi}$. The image of
$\phi$ is contained in no proper parabolic subgroup of ${}^{L}G$. For each
$s_{\operatorname{sc}}$ in $\mathcal{S}^{\operatorname{sc}}_{\phi}$, the
parameter $\phi$ is discrete, so that $\pi^{s}$ is also a discrete series
representations of $G^{\prime}=(G^{\prime},s,\mathcal{G}^{\prime},\xi)$.

We need to recall some details of Langlands parameters $\phi$. We fixed a
splitting
$spl_{\widehat{G}}=(\mathcal{B},\mathcal{T},\\{X_{\alpha^{\vee}}\\})$ of
$\widehat{G}$, where $\mathcal{B}$ is a Borel subgroup in $\widehat{G}$,
$\mathcal{T}$ is a maximal torus in $\widehat{G}$ included in $\mathcal{B}$,
$X_{\alpha^{\vee}}$ is a eigenvector of $\alpha^{\vee}$. Let $\iota$ be half
of the sum of the positive roots of $\mathcal{T}$ in $\mathcal{B}$. Then
according to [22] or [27, §7], one considers pairs
$(\mu,\lambda)\in(X_{\ast}(\mathcal{T})\otimes\mathbb{C})^{2}$, which
parameterize Langlands parameters $\phi$, with $\phi=\phi(\mu,\lambda)$ being
defined by

$\phi(z\times 1)=z^{\mu}\bar{z}^{\sigma_{T}(\mu)}$

for $z\in\mathbb{C}^{\ast}$, and

$\phi(1\times\sigma)=e^{2\pi i\lambda}n(\sigma_{T})\times(1\times\sigma),$

where $T$ is a maximal torus of $G(\mathbb{R})$, there is a canonical
$\widehat{G}$-conjugacy class of admissible embedding
$\xi:{}^{L}T\longrightarrow{}^{L}G$, then $\xi$ maps $\widehat{T}$ to
$\mathcal{T}$ by the isomorphism attached to the pair
$(\mathcal{B},\mathcal{T})$ and the choice of a Borel subgroup $B$ in $G$
containing $T$. There exists $h\in\widehat{G}$ such that
$(h^{-1}\widehat{B}h,h^{-1}\widehat{T}h)=(\mathcal{B},\mathcal{T})$. We define
$\omega_{T}(\sigma):=\operatorname{Int}(h^{-1}\sigma(h))$, which is an element
in the Weyl group $W(\widehat{G},\mathcal{T})$, then
$\sigma_{T}=\omega_{T}(\sigma)\times\sigma$. We have
$n(\sigma_{T})=n(\omega_{T}(\sigma))$, and $n(\omega_{T}(\sigma))$ was defined
by $\\{X_{\alpha^{\vee}}\\}$, as in [24, section 2.1]. $\mu$ and $\lambda$
satisfy the property

$\frac{1}{2}(\mu-\sigma_{T}\mu)-\iota+(\lambda+\sigma_{T}\lambda)\in
X_{\ast}(\mathcal{T}).$

Here $\mu$ is determined uniquely, while $\lambda$ is determined modulo

$X_{\ast}(\mathcal{T})+\\{\nu-\sigma_{T}\nu:\nu\in
X_{\ast}(\mathcal{T})\otimes\mathbb{C}\\},$

and $\phi$ is determined uniquely up to $\mathcal{T}$-conjugacy.

For the endoscopic datum
$G^{\prime}=(G^{\prime},s,\mathcal{G^{\prime}},\xi^{\prime})$, where
$\xi^{\prime}:\mathcal{G^{\prime}}\to{}^{L}G$ is an embedding, and
$\xi^{\prime}$ can be parameterized by $(\mu^{\ast},\lambda^{\ast})$. We
assume that ${}^{L}G^{\prime}$ equals $\mathcal{G^{\prime}}$,
$\phi^{\prime}:W_{\mathbb{R}}\to{}^{L}G^{\prime}$ is a Langlands parameter,
and $\phi^{\prime}=\phi^{\prime}(\mu^{\prime},\lambda^{\prime})$,
$\phi=\xi^{\prime}\circ\phi^{\prime}$. We then have a relation:
$\mu=\mu^{\prime}+\mu^{\ast}$, and $\lambda=\lambda^{\prime}+\lambda^{\ast}$.
If $\mu$ is dominant and regular, then $\pi$ is a discrete series, which
belongs to $\Pi_{\phi}$, and $\pi^{\prime}\in\Pi_{\phi^{\prime}}$ is also a
discrete series. In this case, we know the cardinal number of the $L$-packet
$\Pi_{\phi}$, which equals $|\mathcal{S}_{\phi}|$ in [30, Corollary 7.6], and

$|\mathcal{S}_{\phi}|=|\mathcal{E}(T)|.$

In this case the $R$-group $R_{\phi}$ is trivial. Then we obtain

$\frac{|\mathcal{S}_{\phi}|}{|\mathcal{S}_{\phi^{\prime}}|}=\frac{|\mathcal{E}(T)|}{|\mathcal{E}(T^{\prime})|}=\frac{|\mathcal{K}_{T}|}{|\mathcal{K}_{T^{\prime}}|}=\frac{|Z(\widehat{G^{\prime}})^{\Gamma}|}{|Z(\widehat{G})^{\Gamma}|}.$

Here
$\mathcal{K}_{T}=\pi_{0}((\widehat{T})^{\Gamma}/Z(\widehat{G})^{\Gamma})$.
Since the tori $T$ and $T^{\prime}$ are isomorphic, the third equality holds,
and the second equality comes from the Tate-Nakayama duality.

In the following we deal with the singular elliptic case. In other words, if
$\pi\in\Pi_{\phi_{1}}$ is elliptic representation, then the Langlands
parameter $\phi_{1}$ factors through a discrete parameter for a cuspidal Levi
subgroup ${}^{L}M$ and so through ${}^{L}T_{M}$, where the maximal torus
$T_{M}$ is compact modulo the center of $M$ [22]. Following the argument of
[27], we can construct a new Langlands parameter $\phi$ in the conjugacy class
of $\phi_{1}$ and factoring through ${}^{L}T$, where $T$ is compact modulo the
center of $G$. This new Langlands parameter $\phi$ will be of the form
$\phi(\mu,\lambda)$ as in the first case, but now the regularity requirement
on $\mu$ is not necessary, we call such a $\phi$ for a limit of discrete
parameters. We obtain, by transfer of a discrete parameter $\phi^{\prime}$ for
an elliptic endoscopy group, a limit of discrete parameters $\phi$ [30,
section 9], so we obtain

$\frac{|\mathcal{S}_{\phi}|}{|\mathcal{S}_{\phi^{\prime}}|}=\frac{|Z(\widehat{G^{\prime}})^{\Gamma}||\operatorname{Out}_{G}(G^{\prime},\phi^{\prime})|}{|Z(\widehat{G})^{\Gamma}|}=\frac{|Z(\widehat{G^{\prime}})^{\Gamma}|}{|Z(\widehat{G})^{\Gamma}|}.$

The second equality comes from the discrete case, where
$\operatorname{Out}_{G}(G^{\prime},\phi^{\prime})$ is the stabilizer of
$\phi^{\prime}$ (as a $\widehat{G^{\prime}}$-orbit) in the finite group
$\operatorname{Out}_{G}(G^{\prime})$. Thus we have the following lemma.

######  Lemma 6.4.

If $\phi$ is elliptic, then we have the coefficient relation

$\frac{|S_{\phi}|}{|S_{\phi^{\prime}}|}=|Z(\widehat{G^{\prime}})^{\Gamma}/Z(\widehat{G})^{\Gamma}|.$

######  Theorem 6.5.

If $f=f_{1}\times\bar{f_{2}}$, $f_{1}\in
C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$, $f_{2}\in
C(G(\mathbb{R}),\zeta)$, then

$\displaystyle I_{\operatorname{disc}}(f)=$
$\displaystyle\int_{T_{\operatorname{ell}}(G,\zeta)}i^{G}(\tau)f_{1,G}(\tau)\overline{f_{2,G}(\tau)}d\tau$
$\displaystyle=$
$\displaystyle\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\widehat{S}^{G^{\prime}}(f^{\prime}),$

and $\widehat{S}^{G^{\prime}}(f^{\prime})$ is a stable distribution on
$G^{\prime}$, where

$\displaystyle i^{G}(\tau)=$ $\displaystyle|d(\tau)|^{-1}|R_{\pi,r}|^{-1},$
$\displaystyle\widehat{S}^{G^{\prime}}(f^{\prime})=$
$\displaystyle\int_{\Phi_{2}(G^{\prime},\zeta)}S^{G^{\prime}}(\phi^{\prime})\widetilde{f_{1}}^{\prime}(\phi^{\prime})\overline{f_{2}^{\prime}(\phi^{\prime})}d\phi^{\prime},$
$\displaystyle S^{G^{\prime}}(\phi^{\prime})=$
$\displaystyle\frac{1}{|\mathcal{S}_{\phi^{\prime}}|},\qquad\phi=\xi^{\prime}\circ\phi^{\prime}.$

###### Proof.

By (3.7) we have

(6.1)
$I_{\operatorname{disc}}(f)=\int_{T_{\operatorname{ell}}(G,\zeta)}|R_{\pi,r}|^{-1}|d(\tau)|^{-1}f_{1,G}(\tau)\overline{f_{2,G}(\tau)}d\tau.$

Applying Lemma 6.3, then (6.1) equals

$\displaystyle\int_{T_{\operatorname{ell}}(G,\zeta)}|R_{\pi,r}|^{-1}\widetilde{f}_{1,G}(\tau)\overline{f_{2,G}(\tau)}d\tau$
$\displaystyle=$
$\displaystyle\int_{T_{\operatorname{ell}}(G,\zeta)}|R_{\phi}|^{-1}\sum_{\phi\in\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}\Delta(\tau,\phi)\widetilde{f_{1}}^{\mathcal{E}}(\phi)\overline{f_{2,G}(\tau)}d\tau$
$\displaystyle=$
$\displaystyle\int_{T_{\operatorname{ell}}(G,\zeta)}|R_{\phi}|^{-1}\sum_{\phi\in\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}\frac{|R_{\phi}|}{|\mathcal{S}_{\phi}|}\widetilde{f_{1}}^{\mathcal{E}}(\phi)\overline{\Delta(\phi,\tau)f_{2,G}(\tau)}d\tau.$

Applying Lemma 6.2, We see that this last expression can be written as

$\int_{\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}|\mathcal{S}_{\phi}|^{-1}\widetilde{f_{1}}^{\mathcal{E}}(\phi)\sum_{\tau\in
T_{\operatorname{ell}}(G,\zeta)}\overline{\Delta(\phi,\tau)f_{2,G}(\tau)}d\phi,$

which is just

$\displaystyle\int_{\Phi_{\operatorname{ell}}^{\mathcal{E}}(G,\zeta)}|\mathcal{S}_{\phi}|^{-1}\widetilde{f_{1}}^{\mathcal{E}}(\phi)\overline{f_{2}^{\mathcal{E}}(\phi)}d\phi$
$\displaystyle=$
$\displaystyle\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\int_{\Phi_{2}(G^{\prime},G,\zeta)}|\mathcal{S}_{\phi}|^{-1}\widetilde{f_{1}}^{\prime}(\phi^{\prime})\overline{f_{2}^{\prime}(\phi^{\prime})}d\phi^{\prime}.$

We obtain

$I_{\operatorname{disc}}(f)=\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\int_{\Phi_{2}(G^{\prime},\zeta)}|\mathcal{S}_{\phi}|^{-1}|Z(\widehat{G}^{\prime})^{\Gamma}/Z(\widehat{G})^{\Gamma}|\widetilde{f_{1}}^{\prime}(\phi^{\prime})\overline{f_{2}^{\prime}(\phi^{\prime})}d\phi^{\prime}.$

We denote the coefficient as
$S^{G^{\prime}}(\phi^{\prime})=|\mathcal{S}_{\phi}|^{-1}|Z(\widehat{G}^{\prime})^{\Gamma}/Z(\widehat{G})^{\Gamma}|=|\mathcal{S}_{\phi^{\prime}}|^{-1}$,
whence the last equality by the Lemma 6.4.
Then we obtain the required formula

$I_{\operatorname{disc}}(f)=\sum_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\widehat{S}^{G^{\prime}}(f^{\prime}).$

Since $\widetilde{f_{1}}^{\prime}(\phi^{\prime})$, and
$f_{2}^{\prime}(\phi^{\prime})$ are stable as distributions on $G^{\prime}$,
$\widehat{S}^{G^{\prime}}(f^{\prime})$ is stable as distributions on
$G^{\prime}$. We have thus proved the theorem. ∎

##  7\. Characters and Stable orbital integral

Assume that the test function $f=f_{1}\times\bar{f_{2}}$, $f_{1}\in
C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta),f_{2}\in C(G(\mathbb{R}),\zeta)$,
$G$ is a reductive $K$-group. The key point in the stable trace formula is
that the stable distribution $S^{G^{\prime}}_{M^{\prime}}(\delta,f)$ only
depends on the quasisplit group $G^{\prime}$. We have the following theorem.

######  Theorem 7.1.

If $G$ is a quasisplit $K$-group, we have

$S^{G}(f)=S^{G}_{\operatorname{disc}}(f).$

Here

(7.1)
$S^{G}(f)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Delta_{G-\operatorname{reg},\operatorname{ell}}(M,\zeta)}n(\delta)^{-1}S^{G}_{M}(\delta,f_{1})\overline{f^{G}_{2}(\delta)}\,d\delta,$

and

(7.2)
$S^{G}_{\operatorname{disc}}(f)=\int_{\Phi_{2}(G,\zeta)}S^{G}(\phi)\widetilde{f}^{G}_{1}(\phi)\overline{f^{G}_{2}(\phi)}\,d\phi.$

###### Proof.

Because we have

$S^{G}(f)=I(f)-\sum_{G^{\prime}\in\mathcal{E}^{\circ}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\widehat{S}^{G^{\prime}}(f^{\prime}),$

and

$S^{G}_{\operatorname{disc}}(f)=I_{\operatorname{disc}}(f)-\sum_{G^{\prime}\in\mathcal{E}^{\circ}_{\operatorname{ell}}(G)}\iota(G,G^{\prime})\widehat{S}^{G^{\prime}}_{\operatorname{disc}}(f^{\prime}),$

and $f_{1}$ is cuspidal, then $I(f)=I_{\operatorname{disc}}(f)$. We can prove
the theorem inductively. Since $G$ is quasisplit, we have
$\operatorname{dim}(G^{\prime})<\operatorname{dim}(G)$ for all
$G^{\prime}\in\mathcal{E}^{\circ}_{\operatorname{ell}}(G)=\mathcal{E}_{\operatorname{ell}}(G)\backslash\\{G\\}$.
So for all $G^{\prime}\in\mathcal{E}^{\circ}_{\operatorname{ell}}(G)$, we have
$\widehat{S}^{G^{\prime}}(f^{\prime})=\widehat{S}^{G^{\prime}}_{\operatorname{disc}}(f^{\prime})$
by the induction, thus $S^{G}(f)=S^{G}_{\operatorname{disc}}(f)$. ∎

We need to connect the distributions of the geometric side and the
distributions of the spectral side in the stable local trace formula. To do
this, we need the stable Weyl integral formula. We recall the Weyl integral
formula, which is an expansion

(7.3)
$\Theta(\pi,f_{2})=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}\Phi_{M}(\pi,\gamma)I_{G}(\gamma,f_{2})\,d\gamma,$

where $\Phi_{M}(\pi,\gamma)=|D(\gamma)|^{1/2}\Theta(\pi,\gamma)$, and
$\Theta(\pi,\gamma)=\operatorname{tr}\pi(\gamma).$

We know that

$f^{G}_{2}(\phi)=\sum_{\pi\in\Pi_{\phi}}\Theta(\pi,f_{2}).$

Substitute the Weyl integral formula 7.3 into this formula, we obtain an
expansion,

$f^{G}_{2}(\phi)=\sum_{\pi\in\Pi_{\phi}}\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}\Phi_{M}(\pi,\gamma)I_{G}(\gamma,f_{2})\,d\gamma.$

We need to recall the basic objects of geometric side before stabilizing the
Weyl integral formula. We shall make free use of the language and notation
[24] in this part, often without comments. To define the general transfer
factor $\Delta(\sigma^{\prime},\gamma),$ it is necessary to fix elements
$\bar{\sigma^{\prime}}$ and $\bar{\gamma}$ such that $\bar{\sigma^{\prime}}$
is an image of $\bar{\gamma}$, and to specify
$\Delta(\bar{\sigma^{\prime}},\bar{\gamma})$. We will take it to be any
complex number of absolute value 1, then $\Delta(\sigma^{\prime},\gamma)$ is
defined to be the product of $\Delta(\bar{\sigma^{\prime}},\bar{\gamma})$ with
the factor

$\Delta(\sigma^{\prime},\gamma;\bar{\sigma^{\prime}},\bar{\gamma})=\frac{\Delta_{I}(\sigma^{\prime},\gamma)}{\Delta_{I}(\bar{\sigma^{\prime}},\bar{\gamma})}\frac{\Delta_{II}(\sigma^{\prime},\gamma)}{\Delta_{II}(\bar{\sigma^{\prime}},\bar{\gamma})}\frac{\Delta_{2}(\sigma^{\prime},\gamma)}{\Delta_{2}(\bar{\sigma^{\prime}},\bar{\gamma})}\Delta_{1}(\sigma^{\prime},\gamma;\bar{\sigma^{\prime}},\bar{\gamma}).$

There is an additional factor
$\Delta_{IV}(\sigma^{\prime},\gamma)=|D^{G}(\gamma)||D^{G^{\prime}}(\gamma^{\prime})|^{-1}$
included in the definition of [24], but since we have already put these
normalizing factors into our orbital integrals, the term does not appear in
this equation. The remaining factors are all constructed from the special
values of unitary abelian characters, and therefore have absolute value 1.

There is a natural measure on $\Gamma_{\operatorname{ell}}(G,\zeta)$ given by

$\int_{\Gamma_{\operatorname{ell}}(G,\zeta)}\alpha(\gamma)\,d\gamma=\sum_{\\{T\\}}|W(G(\mathbb{R}),T(\mathbb{R}))|^{-1}\int_{T(\mathbb{R})/Z(\mathbb{R})}\alpha(t)\,dt$

for any $\alpha\in C(\Gamma_{\operatorname{ell}}(G,\zeta))$, where $\\{T\\}$
is a set of representatives of $G(\mathbb{R})$ conjugacy classes of elliptic
torus in $G$ over $\mathbb{R}$, $W(G(\mathbb{R}),T(\mathbb{R}))$ is the Weyl
group of $(G(\mathbb{R}),T(\mathbb{R}))$, and $dt$ is the Haar measure on
$T(\mathbb{R})$. $\Gamma_{\operatorname{ell}}(G,\zeta)$ is the set of
conjugacy classes $\gamma$ in $G(\mathbb{R})$ such that $G_{\gamma}$ is an
elliptic maximal torus in $G$ and as a distribution with a central character
$\zeta$ on $Z(\mathbb{R})$ as in [9]. Let
$\Delta_{\operatorname{ell}}(G,\zeta)$ be the set of stable conjugacy classes
in $\Gamma_{\operatorname{ell}}(G,\zeta)$. We define a measure on
$\Delta_{\operatorname{ell}}(G,\zeta)$ by setting

$\int_{\Delta_{\operatorname{ell}}(G,\zeta)}\beta(\delta)\,d\delta=\sum_{\\{T\\}_{\operatorname{stab}}}|W_{\mathbb{R}}(G,T)|^{-1}\int_{T(\mathbb{R})/Z(\mathbb{R})}\beta(t)\,dt$

for any $\beta\in C(\Delta_{\operatorname{ell}}(G,\zeta))$, where
$\\{T\\}_{\operatorname{stab}}$ is a set of representatives of stable
conjugacy classes of elliptic maximal tori in $G$ over $\mathbb{R}$. And
$W_{\mathbb{R}}(G,T)$ is the subgroup of elements in the absolute Weyl group
of $(G,T)$ defined over $\mathbb{R}$. The measure on
$\Gamma_{\operatorname{ell}}(G,\zeta)$ and
$\Delta_{\operatorname{ell}}(G,\zeta)$ are related by a formula

(7.4)
$\int_{\Delta_{\operatorname{ell}}(G,\zeta)}(\sum_{\gamma\to\delta}\alpha(\gamma))\,d\delta=\int_{\Gamma_{\operatorname{ell}}(G)}\alpha(\gamma)\,d\gamma.$

Let $\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)$ be the set of
isomorphism classes of pair $(G^{\prime},\sigma^{\prime})$, where $G^{\prime}$
is an elliptic endoscopic datum for $G$ and $\sigma^{\prime}$ in an element in
$\Delta_{G,\operatorname{ell}}(G^{\prime},\zeta)$. By an isomorphism from
$(G^{\prime},\sigma^{\prime})$ to a second pair
$(G^{\prime}_{1},\sigma^{\prime}_{1})$, we mean that an isomorphism from the
datum $G^{\prime}$ to $G^{\prime}_{1}$, which takes $\sigma^{\prime}$ to
$\sigma^{\prime}_{1}$. So we have a decomposition

$\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)=\coprod_{G^{\prime}\in\mathcal{E}_{\operatorname{ell}}(G)}\Delta_{\operatorname{ell}}(G^{\prime},G,\zeta),$

where
$\Delta_{\operatorname{ell}}(G^{\prime},G,\zeta)=\Delta_{G,\operatorname{ell}}(G^{\prime},\zeta)/\operatorname{Out}_{G}(G^{\prime})$.
We also have an analogue of Lemma 6.2 for the geometric transfer factor.

######  Lemma 7.2.

Suppose that $\alpha\in C(\Gamma_{\operatorname{ell}}(G,\zeta))$, and that
$\beta\in C(\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta))$. Then

$\int_{\Gamma_{\operatorname{ell}}(G,\zeta)}\sum_{\delta\in\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)}\beta(\delta)\Delta(\gamma,\delta)\alpha(\gamma)\,d\gamma=\int_{\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)}\sum_{\gamma\in\Gamma_{\operatorname{ell}}(G,\zeta)}\beta(\delta)\Delta(\gamma,\delta)\alpha(\gamma)\,d\delta.,$

###### Proof.

Let $\psi:G\to G^{\ast}$ be the underlying quasisplit inner twist of $G$.
According to (7.4) the integral over $\Gamma_{\operatorname{ell}}(G,\zeta)$
can be decomposed into an integral over
$\delta^{\ast}\in\Delta_{\operatorname{ell}}(G^{\ast},\zeta)$ and a sum over
the elements $\gamma\in\Gamma_{\operatorname{ell}}(G,\zeta)$ which is mapped
to $\delta^{\ast}$. Similarly, the integral over
$\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)$ can be decomposed into an
integral over $\delta^{\ast}\in\Delta_{\operatorname{ell}}(G^{\ast},\zeta)$,
and a summation over the element
$\delta^{\prime}\in\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)$ which
is mapped to $\delta^{\ast}$. This depends on the fact that the map
$\delta^{\prime}\to S_{T^{\ast}}(\delta^{\prime})$,
$T^{\ast}=G_{\delta^{\ast}}$ is a bijection from the preimage of
$\delta^{\ast}$ in $\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)$ onto
$\mathcal{K}(T^{\ast})=\pi_{0}((\widehat{T^{\ast}})^{\Gamma}/Z(\widehat{G})^{\Gamma})$,
and this bijection depends on $G$, being a $K$-group.

With the two decomposition, we can represent each side of the required
identity as an integral over $\Delta_{\operatorname{ell}}(G^{\ast},\zeta),$
and a double sum over $\delta$ and ${\gamma}$. The transfer factor
$\Delta(\delta,\gamma)$ vanishes unless $\delta$ and $\gamma$ have the same
image in the $\Delta_{\operatorname{ell}}(G^{\ast},\zeta)$. The double sum in
each case can therefore be taken over the preimages of $\delta^{\ast}$ in
$\Gamma^{\mathcal{E}}_{\operatorname{ell}}(G,\zeta)\times\Gamma_{\operatorname{ell}}(G,\zeta)$,
then the identity follows. ∎

We can now stabilize the Weyl integral formula.

######  Lemma 7.3.

For $f\in C(G(\mathbb{R}),\zeta).$ Then we have the stable Weyl integral
formula

$f^{G}_{2}(\phi)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}n(\delta)^{-1}\sum_{\pi\in\Pi_{\phi}}S\Phi_{M}(\pi,\delta)f^{\mathcal{E}}_{2,M}(\delta)\,d\delta,$

where
$S\Phi_{M}(\pi,\delta)=\sum_{\gamma\in\Gamma_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}\overline{\Delta(\delta,\gamma)}\Phi_{M}(\pi,\gamma)$.

###### Proof.

We observe that $I_{G}(\gamma,f_{2})=I^{M}_{M}(\gamma,f_{2})=f_{2,M}(\gamma)$
where $\gamma\in\Gamma_{\operatorname{ell}}(M(\mathbb{R}),\zeta)$, and

(7.5)
$\Theta(\pi,f_{2})=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}n(\delta)^{-1}\Phi_{M}(\pi,\gamma)\sum_{\delta\in\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}\overline{\Delta(\delta,\gamma)}f^{\mathcal{E}}_{2,M}(\delta)\,d\gamma.$

Applying Lemma 7.2, we have

$\Theta(\pi,f_{2})=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}n(\delta)^{-1}S\Phi_{M}(\pi,\delta)f^{\mathcal{E}}_{2,M}(\delta)\,d\delta,$

where
$S\Phi_{M}(\pi,\delta)=\sum_{\gamma\in\Gamma_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}\overline{\Delta(\delta,\gamma)}\Phi_{M}(\pi,\gamma)$.
So we obtain

$f^{G}_{2}(\phi)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}n(\delta)^{-1}\sum_{\pi\in\Pi_{\phi}}S\Phi_{M}(\pi,\delta)f^{\mathcal{E}}_{2,M}(\delta)\,d\delta.$

∎

We now obtain the formula for the stable distribution
$S^{G}_{M}(\delta,f_{1})$ by comparing the stable Weyl integral formula with
the stable local trace formula.

######  Theorem 7.4.

Assume that $G$ is a quasisplit $K$-group, and $f_{1}\in
C_{\operatorname{cusp}}(G(\mathbb{R}),\zeta)$. Then we have

(7.6)
$S^{G}_{M}(\delta,f_{1})=(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Phi_{2}(G,\zeta)}S^{G}(\phi)S\Phi_{M}(\phi,\delta)\widetilde{f}_{1}^{G}(\phi)\,d\phi,$

where

$S\Phi_{M}(\phi,\delta)=\sum_{\pi\in\Pi_{\phi}}\sum_{\gamma\in\Gamma_{\operatorname{ell}}(M,\zeta)}\overline{\Phi_{M}(\pi,\gamma)}\Delta(\delta,\gamma),$

and

$\Phi_{M}(\pi,\gamma)=\begin{cases}|D(\gamma)|^{1/2}\Theta(\pi,\gamma)&\text{if
$\gamma\in M(\mathbb{R})_{\operatorname{ell}}$ },\\\ 0&\text{
otherwise},\end{cases}$ $\widetilde{f}_{1}^{G}(\phi)=\sum_{\tau\in
T_{\operatorname{ell}}(G,\zeta)}\Delta(\phi,\tau)|d(\tau)|^{-1}\Theta(\tau,f_{1}),$

for Levi subgroup $M\in\mathcal{L}$ and $\delta$ is the stable strongly
$G$-regular conjugacy class in $M(\mathbb{R})$.

###### Proof.

Suppose that $\delta$ does not lie in
$\Delta_{\operatorname{ell}}(M(\mathbb{R}),\zeta)$. Then by descent formula
[8, §6] and the cuspidality of $f_{1}$, $S^{G}_{M}(\delta,f_{1})$ vanishes.
The right hand side of (7.6) vanishes by definition. So the formula holds in
the case. It is therefore enough to establish (7.6) when $\delta$ lies in
$\Delta_{\operatorname{ell}}(M(\mathbb{R}),\zeta)$.

To deal with the elliptic point in $M(\mathbb{R})$, we apply the simple
version of the stable local trace formula. Consider the two expression (7.1)
and (7.2) in Theorem 7.1, with $f_{1}$ for the given cuspidal function and
$f_{2}$ for a variable function in $C(G(\mathbb{R}),\zeta)$. The expressions
depend on $f_{2}$ through different distributions $f^{G}_{2}$ and
$f^{\mathcal{E}}_{2,M}$. However, the relation is given by a stable Weyl
integral formula, which has an expansion

$f^{G}_{2}(\phi)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}n(\delta)^{-1}\sum_{\pi\in\Pi_{\phi}}S\Phi_{M}(\pi,\delta)f^{\mathcal{E}}_{2,M}(\delta)\,d\delta.$

We obtain

$f^{G}_{2}(\phi)=\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\int_{\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)}n(\delta)^{-1}S\Phi_{M}(\phi,\delta)f^{\mathcal{E}}_{2,M}(\delta)\,d\delta.$

Substituting this into (7.2), we collect the coefficient of
$\overline{f^{M}_{2}(\delta)}$ in the resulting identity of (7.1) with (7.2).
We see that if we set:

$P_{M}(\delta,f_{1})=S^{G}_{M}(\delta,f_{1})-(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Phi_{2}(G,\zeta)}S^{G}(\phi)S\Phi_{M}(\phi,\delta)\widetilde{f}_{1}^{G}(\phi)\,d\phi,$

then the sum of

(7.7)
$\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Delta_{G,\operatorname{ell}}(M,\zeta)}n(\delta)^{-1}P_{M}(\delta,f_{1})\overline{f^{M}_{2}(\delta)}\,d\delta,$

and

(7.8)
$-\sum_{M\in\mathcal{L}}|W^{M}_{0}||W^{G}_{0}|^{-1}\sum_{M^{\prime}\in\mathcal{E}^{0}_{\operatorname{ell}}(M)}\int_{\Delta_{\operatorname{ell}}(M^{\prime},M,\zeta)}\int_{\Phi_{2}(G,\zeta)}n(\delta)^{-1}S^{G}(\phi)\widetilde{f}^{G}_{1}(\phi)S\Phi_{M}(\phi,\delta)\overline{f^{M^{\prime}}_{2}(\delta)}\,d\delta
d\phi$

vanishes.

We have written
$f^{\mathcal{E}}_{2,M}(\delta^{\prime})=f^{M^{\prime}}_{2}(\delta^{\prime})$,
where $\delta^{\prime}\in\Delta_{G,\operatorname{ell}}(M^{\prime},\zeta)$. As
in [8, §10], we choose $f_{2}$ so that $f^{\mathcal{E}}_{2,G}$ has compact
support modulo $Z(\mathbb{R})$ on
$\Gamma^{\mathcal{E}}(G)=\coprod_{\\{M\\}}\Gamma^{\mathcal{E}}_{\operatorname{ell}}(M(\mathbb{R}),\zeta)$
, and so that $f^{\mathcal{E}}_{2,G}$ approaches the $\zeta^{-1}$-equivariant
Dirac measure at the image of $\delta^{\prime}$ in $\Gamma^{\mathcal{E}}(G)$.
The expression then approaches a nonzero multiple of
$P_{M}(\delta^{\prime},f_{1})$ when
$\delta^{\prime}\in\Delta_{G,\operatorname{ell}}(M,\zeta)$, and approaches a
zero. We conclude that $P_{M}(\delta,f_{1})=0$. So we have obtained the
required formula

$S^{G}_{M}(\delta,f_{1})=(-1)^{\operatorname{dim}(A_{M}/A_{G})}\int_{\Phi_{2}(G,\zeta)}S^{G}(\phi)S\Phi_{M}(\phi,\delta)\widetilde{f_{1}^{G}}(\phi)\,d\phi.$

∎

######  Corollary 7.5.

If we set
$S^{G}_{M}(M^{\prime},\delta,f)=n(\delta)^{-1}\int_{\Phi_{2}(G,\zeta)}S^{G}(\phi)S\Phi_{M}(\phi,\delta)\widetilde{f}^{G}(\phi)\mathrm{d}\phi$,
and $f$ is a cuspidal function, where