%Here is the file sample.tex % %- - - - - - - - - - - - cut here - - - - - - - - - - - - - - % % \input amssym.def \input amssym.tex \hsize=4.5truein \vsize=7.5truein \hoffset=6.25pc \voffset=0.5pc \font\titlefont=cmb10 scaled\magstep3 \font\authorfont=cmb10 scaled\magstep1 %\mathchardef\rtimes="786F \countdef\startno=0 \global\def\makeheadline{\vbox to 0pt{\vskip-38.5pt \line{\vbox to 8.5pt{}% \the\headline}\vss} \nointerlineskip} \headline={\ifnum\startno=10\hfill\else \ifodd\pageno\oddheadline\else\evenheadline\fi \fi} \footline={\ifnum\startno=10\tenrm\hfil\folio\hfil \else\hfil\fi} \global\def\evenheadline{\tenrm\folio\hfil John Doe and Jane Doe\hfil} \global\def\oddheadline{\tenrm \hfil Descent for Transfer Factors\hfil\folio} \voffset=2\baselineskip \def\Cent{\rm Cent} \def\dbR{{\Bbb R}} \def\Gal{\rm Gal} \def\st{\rm st} \def\calA{{\cal A}} \def\calN{{\cal N}} \def\calB{{\cal B}} \def\calO{{\cal O}} \def\calC{{\cal C}} \def\calP{{\cal P}} \def\calD{{\cal D}} \def\calQ{{\cal Q}} \def\calE{{\cal E}} \def\calR{{\cal R}} \def\calF{{\cal F}} \def\calS{{\cal S}} \def\calG{{\cal G}} \def\calT{{\cal T}} \def\calH{{\cal H}} \def\calU{{\cal U}} \def\calI{{\cal I}} \def\calV{{\cal V}} \def\calJ{{\cal J}} \def\calW{{\cal W}} \def\calK{{\cal K}} \def\calX{{\cal X}} \def\calL{{\cal L}} \def\calY{{\cal Y}} \def\calM{{\cal M}} \def\calZ{{\cal Z}} \def\nspace{\lineskip=1pt\baselineskip=12pt\lineskiplimit=0pt} \def\references#1{\bigskip\par\centerline{\bf References}\medskip \parindent=#1pt\nspace} \def\Ref[#1]{\par\smallskip\hang\indent\llap{\hbox to\parindent {[#1]\hfil\enspace}}\ignorespaces} \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