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     \else\hfil\fi}
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     John Doe and Jane Doe\hfil}
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     \hfil Descent for Transfer Factors\hfil\folio}
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\def\Cent{\rm Cent}      \def\dbR{{\Bbb R}}
\def\Gal{\rm Gal}
\def\st{\rm st}

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\vglue 1.00true in
\startno=10
\centerline{\titlefont Descent for Transfer Factors}

\medskip
\centerline{\authorfont John Doe and Jane Doe%
     \footnote*{\hbox{\rm Supported in part by the National
Science Foundation.}}}

\smallskip
\centerline{Department of Mathematics}
\centerline{University of Anywhere}
\centerline{Anytown, USA}

\vglue2pc
\centerline{\bf Abstract}
\smallskip

If there was an abstract in this paper, this is where it would
go, and the heading would be centered in 10pt bold,
2pc down from the last line in the affiliation.

\vglue2pc
\centerline{\bf Introduction}
\smallskip

In [I] we introduced the notion of transfer from a group over a
local field to an associated endoscopic group, but did not  prove its
existence, nor do we do so in the present paper.
Nonetheless we carry out what is probably an unavoidable step in any
proof of existence: reduction to a local statement at the identity in
the centralizer of a semisimple element, a favorite procedure of
Harish~Chandra that he referred to as descent.

The principal difficulty is to show that the transfer factors of
[I] for the original group $G$ are compatible with those on the
connected centralizer $G_\epsilon$ of the semisimple element $\epsilon$.
After some preliminary explanations in Section~1, the compatibility is
stated as Theorem~1.6.A. In Section~2 we show  that this
compatibility indeed reduces the problem of existence to a local
problem at the identity on the groups $G_\epsilon$, 
and in passing we note some other applications
$\,\,\,\ldots$

\vglue2pc
\centerline{\bf Acknowledgements}

\smallskip
If there were acknowledgements, they would normally go here.
\vfill\eject

\centerline{\bf \S1. Descent Principles}
\smallskip\noindent
{\bf 1.1. Notation.}\enspace
We follow closely the notation of [I].
In particular, $G$ is a connected reductive group over a field
$F$ of characteristic zero, now assumed local.

As in [I, Sect. 1.2.] $G^*$, $\psi$ are quasisplit data and
${}^LG=\hat G\rtimes W_F$ is the $L$-group.
To conserve notation we fix an $F$-splitting $(\calB,\calT,
\{X_{\alpha^\vee}\})$ of $\hat G$ and given a class of endoscopic
data choose a representation $(H,\calH,s,\xi)$ with
$\xi\colon\calH\hookrightarrow {}^LG$ as inclusion and $s$ an
element of $\calT$.
It is also convenient $\,\,\,\ldots$

\medskip
\noindent
{\bf 1.2. Images of semisimple elements.}\enspace
For $\epsilon$ in $G$ the {\it identity component} of
$\Cent(\epsilon,G)$ will be denoted $G_\epsilon$.
If $\epsilon$ in $G(F)$ is semisimple then, following [K1], the
stable conjugacy class of $\epsilon$ is
$$
\{g^{-1}\epsilon g\colon g\sigma(g)^{-1}\in
G_\epsilon,\sigma\in\Gamma\},\leqno(1)
$$
where $\Gamma=\Gal(\bar F/F)$.
If $\Cent(\epsilon,G)$ is connected then
this coincides with the set of $F$-rational points in the
conjugacy class of $\epsilon$ in $G(\bar F)$.
In general, $\,\,\,\ldots$

\medskip\noindent
{\bf Theorem.}\enspace 
{\it There is a constant $c$ such that}
$$
\Delta(\gamma_{_H},\gamma_{_G})=c\Delta^{\dbR}(\gamma_{_H},\gamma
_{_G})\leqno(2)
$$
{\it for all $G$-regular $\gamma_{_H}$ in $H(\dbR)$}

\medskip\noindent
{\bf Proof.}\enspace
By continuity we can assume $\,\,\,\ldots$

\vglue2pc
\centerline{\bf \S2. Consequences}
\smallskip\noindent
{\bf 2.1. Local transfer.}\enspace
We say that $(G,H)$ {\it admits $\Delta$-transfer} if for each
$f\in C_c^\infty(G(F))$ there exists $f^{\tilde H}\in
C_c^\infty(\hat H(F),\hat\lambda)$ (notation of 1.3) such that
$f$ and $f^{\tilde H}$ have $\delta$-matching orbital integrals,
that is,
$$
\Phi^{\st}(\tilde\gamma_{_H},f^{\tilde H})=
\sum\limits_{\gamma_{_G}}\Delta(\gamma_{_H},\gamma_{_G})
\Phi(\gamma_{_G},f)\leqno(3)
$$
for all strongly $G$-regular$\,\,\,\ldots$

\medskip\noindent
{\bf Lemma 6.5.}\enspace
{\it $\lambda'$ is a coboundary.}

\medskip\noindent
{\bf Proof.}\enspace
Because $\sigma_j^{-1}A_j/A_j=\sigma_j^{-1}B_j/B_j$ we may write
$\lambda'$ as
$$
\prod\limits_{j}{\sigma_j^{-1}A_j\over B_j}
\sigma_j^{-1}C_j^{-1}\left({\xi\over \rho\xi}\right)
^{\delta(\alpha_{j'}(\sigma))}.
$$
>From the definition of $A_j$ we see that
$$
\eqalign{
{\sigma_j^{-1}A_j\over B_j} &=\left({\sigma_j^{-1}(i)\over i}
\right)^{\delta(\alpha_j(\rho))\delta(\alpha_{j'}(\sigma))}\cr
&=\eta_j^{2\delta(\alpha_j(\rho))\delta(\alpha_{j'}(\sigma))}\cr}
\leqno(3)
$$
\vglue2pc

\references{40}
{\nspace{
\Ref[I]
R. Langlands and D. Shelstad,
\sl On the definition of transfer factors,
\rm Math. Ann., 278, 219-271 (1987).
\smallskip
\Ref[B]
N. Bourbaki,
\sl Groupes et Alg\`ebres de Lie,
\rm Chs. 4, 5, 6, Hermann (1968).
\smallskip
\Ref[C-D]
L. Clozel and P. Delorme,
\sl Le th\'eor\`eme de Paley-Wiener invariant pour les groupes
de Lie r\'eductifs,
\rm Inv. Math. 77, 427-453 (1984).

}}

\bye



