% Basic control strings and macros for tib bibliography formatting, % and default definitions. % % tib and the accompaning TeX macros assume the \catcode of plain TeX % e.g. { demarks the beginning of a group, % demarks a comment, etc. % See D.E. Knuth, The TeXbook, p. 37. % So far as possible, definitions have been reduced to primitive TeX % strings, to make tib as compatible with any TeX as possible. % Two possible exceptions: the fonts \rm, \sl, \sevenrm are assumed % to have been loaded by TeX. If not, this file should be modified below. % Also macros for footnotes are tailored to the particular TeX, since % the footnote macros of different TeXs differ. Footnote macro files % ending in .p are for plain TeX; those ending in .l are for LaTeX. % % \Resetstrings and \Refformat are called within a local group--- % the string names will not conflict with uses elsewhere in the document. % Other control strings are defined globally. The following is % a complete list of such globally defined control strings (except those % the user might define in a private tib formatting file). % % \Resetstrings \Refformat \Rpunct \Lpunct % \Lspace \Lperiod \Lcomma \Lquest % \Lcolon \Lscolon \Lbang \Lquote % \Lqquote \Lrquote \Rspace \Rperiod % \Rcomma \Rquest \Rcolon \Rscolon % \Rbang \Rquote \Rqquote \Rrquote % \Refstd \Refstda \Smallcapsaand \Smallcapseand % \Acomma \Aand \Aandd \Ecomma % \Eand \Eandd \acomma \aand % \aandd \ecomma \eand \eandd % \Namecomma \Nameand \Nameandd \Revcomma % \Initper \Initgap \Citefont \ACitefont % \Authfont \Titlefont \Tomefont \Volfont % \Flagfont \Reffont \Smallcapsfont \Underlinemark % \Citebrackets \Citeparen \Citesuper \Citenamedate % \Lcitemark \Rcitemark \LAcitemark \RAcitemark % \LIcitemark \RIcitemark \Citehyphen \Citecomma % \Citebreak \Resetstrings \ztest \zstr % \Ztest \Zstr % \journalarticleformat \conferencereportformat \bookformat % \technicalreportformat \bookarticleformat \otherformat % % This file is \input first in a document processed by tib. % Secondly any tib format files are processed. Then the input % document file is read and processed. Thus the default definitions % of this file can be overridden either in the tib format files % or the input document. % % \Refformat calls macros for creating the actual bibliography listings. % Such macros are generally kept in tibtex files and \input by a statement % in the tib format file. \def\Resetstrings{%Clears all strings before processing reference listing. % The strings (\Astr, etc.) are fields taken from the database. % If the string is present, the appropriate test (\Atest, etc) is set % equal to \present---thus allowing the macros to test for the presence % or absence of a field. All reference processing is done in a local % group--the control string names will not conflict with uses % elsewhere in the document. \def\present{ }\let\bgroup={\let\egroup=}%primitive TeX \def\Astr{}\def\astr{}\def\Atest{}\def\atest{}% \def\Bstr{}\def\bstr{}\def\Btest{}\def\btest{}% \def\Cstr{}\def\cstr{}\def\Ctest{}\def\ctest{}% \def\Dstr{}\def\dstr{}\def\Dtest{}\def\dtest{}% \def\Estr{}\def\estr{}\def\Etest{}\def\etest{}% \def\Fstr{}\def\fstr{}\def\Ftest{}\def\ftest{}% \def\Gstr{}\def\gstr{}\def\Gtest{}\def\gtest{}% \def\Hstr{}\def\hstr{}\def\Htest{}\def\htest{}% \def\Istr{}\def\istr{}\def\Itest{}\def\itest{}% \def\Jstr{}\def\jstr{}\def\Jtest{}\def\jtest{}% \def\Kstr{}\def\kstr{}\def\Ktest{}\def\ktest{}% \def\Lstr{}\def\lstr{}\def\Ltest{}\def\ltest{}% \def\Mstr{}\def\mstr{}\def\Mtest{}\def\mtest{}% \def\Nstr{}\def\nstr{}\def\Ntest{}\def\ntest{}% \def\Ostr{}\def\ostr{}\def\Otest{}\def\otest{}% \def\Pstr{}\def\pstr{}\def\Ptest{}\def\ptest{}% \def\Qstr{}\def\qstr{}\def\Qtest{}\def\qtest{}% \def\Rstr{}\def\rstr{}\def\Rtest{}\def\rtest{}% \def\Sstr{}\def\sstr{}\def\Stest{}\def\stest{}% \def\Tstr{}\def\tstr{}\def\Ttest{}\def\ttest{}% \def\Ustr{}\def\ustr{}\def\Utest{}\def\utest{}% \def\Vstr{}\def\vstr{}\def\Vtest{}\def\vtest{}% \def\Wstr{}\def\wstr{}\def\Wtest{}\def\wtest{}% \def\Xstr{}\def\xstr{}\def\Xtest{}\def\xtest{}% \def\Ystr{}\def\ystr{}\def\Ytest{}\def\ytest{}% } \Resetstrings\def\Ztest{}\def\ztest{} \def\Refformat{%Determines the kind of reference by the presence or % absence of certain fields in the database listing, and calls the % appropriate macro. \if\Jtest\present {\if\Vtest\present\journalarticleformat \else\conferencereportformat\fi} \else\if\Btest\present\bookarticleformat \else\if\Rtest\present\technicalreportformat \else\if\Itest\present\bookformat \else\otherformat\fi\fi\fi\fi} \def\Rpunct{%Default punctuation control strings if the punctuation % is to appear after the citation. (tib looks for punctuation to % precede the incomplete citation in the input document; the TeX % output puts it to the left or right depending on the style of citation.) \def\Lspace{ }% \def\Lperiod{ }% . \def\Lcomma{ }% , \def\Lquest{ }% ? \def\Lcolon{ }% : \def\Lscolon{ }% ; \def\Lbang{ }% ! \def\Lquote{ }% ' \def\Lqquote{ }% " \def\Lrquote{ }% ` \def\Rspace{}% \def\Rperiod{.}% . \def\Rcomma{,}% , \def\Rquest{?}% ? \def\Rcolon{:}% : \def\Rscolon{;}% ; \def\Rbang{!}% ! \def\Rquote{'}% ' \def\Rqquote{"}% " \def\Rrquote{`}% ` } \def\Lpunct{%Default punctuation control strings if the punctuation % is to appear before the citation. (tib looks for punctuation to % precede the incomplete citation in the input document; the TeX % output puts it to the left or right depending on the style of citation.) \def\Lspace{}% \def\Lperiod{\unskip.}% . \def\Lcomma{\unskip,}% , \def\Lquest{\unskip?}% ? \def\Lcolon{\unskip:}% : \def\Lscolon{\unskip;}% ; \def\Lbang{\unskip!}% ! \def\Lquote{\unskip'}% ' \def\Lqquote{\unskip"}% " \def\Lrquote{\unskip`}% ` \def\Rspace{\spacefactor=1000}% \def\Rperiod{\spacefactor=3000}% . \def\Rcomma{\spacefactor=1250}% , \def\Rquest{\spacefactor=3000}% ? \def\Rcolon{\spacefactor=2000}% : \def\Rscolon{\spacefactor=1250}% ; \def\Rbang{\spacefactor=3000}% ! \def\Rquote{\spacefactor=1000}% ' \def\Rqquote{\spacefactor=1000}% " \def\Rrquote{\spacefactor=1000}% ` } \def\Refstd{%Standard control strings for formatting bibliography listings. \def\Acomma{\unskip, }%between multiple author names \def\Aand{\unskip\ and }%between two author names \def\Aandd{\unskip\ and }%between last two of multiple author names \def\Ecomma{\unskip, }%between multiple editor names \def\Eand{\unskip\ and }%between two editor names \def\Eandd{\unskip\ and }%between last two of multiple author names \def\acomma{\unskip, }%same for authors of reviewed material \def\aand{\unskip\ and }%same for authors of reviewed material \def\aandd{\unskip\ and }%same for authors of reviewed material \def\ecomma{\unskip, }%same for translators \def\eand{\unskip\ and }%same for translators \def\eandd{\unskip\ and }%same for translators \def\Namecomma{\unskip, }%same for citations using authors' names \def\Nameand{\unskip\ and }%same for citations using authors' names \def\Nameandd{\unskip\ and }%same for citations using authors' names \def\Revcomma{\unskip, }%between last and first name of reversed name \def\Initper{.\ }%punctuation after initial \def\Initgap{\dimen0=\spaceskip\divide\dimen0 by 2\hskip-\dimen0}% %gap between initials of abbreviated first name } \def\Smallcapsaand{%Smallcaps redefinition of \Aand and \Aandd for \Refstd \def\Aand{\unskip\bgroup{\Smallcapsfont\ AND }\egroup}% \def\Aandd{\unskip\bgroup{\Smallcapsfont\ AND }\egroup}% \def\eand{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\eandd{\unskip\bgroup\Smallcapsfont\ AND \egroup}% } \def\Smallcapseand{%Smallcaps redefinition of \Eand, \Eeand, etc for Refstd \def\Eand{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\Eandd{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\aand{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\aandd{\unskip\bgroup\Smallcapsfont\ AND \egroup}% } \def\Refstda{%Standard control strings for formatting bibliography listings. % \Refstda sets an Ampersand instead of the word "and". \chardef\Ampersand=`\&%primitive TeX \def\Acomma{\unskip, }%between multiple author names \def\Aand{\unskip\ \Ampersand\ }%between two author names \def\Aandd{\unskip\ \Ampersand\ }%between last two of multiple author names \def\Ecomma{\unskip, }%between multiple editor names \def\Eand{\unskip\ \Ampersand\ }%between two editor names \def\Eandd{\unskip\ \Ampersand\ }%between last two of multiple author names \def\acomma{\unskip, }%same for authors of reviewed material \def\aand{\unskip\ \Ampersand\ }%same for authors of reviewed material \def\aandd{\unskip\ \Ampersand\ }%same for authors of reviewed material \def\ecomma{\unskip, }%same for translators \def\eand{\unskip\ \Ampersand\ }%same for translators \def\eandd{\unskip\ \Ampersand\ }%same for translators \def\Namecomma{\unskip, }%same for citations using authors' names \def\Nameand{\unskip\ \Ampersand\ }%same for citations using authors' names \def\Nameandd{\unskip\ \Ampersand\ }%same for citations using authors' names \def\Revcomma{\unskip, }%between last and first name of reversed name \def\Initper{.\ }%punctuation after initial \def\Initgap{\dimen0=\spaceskip\divide\dimen0 by 2\hskip-\dimen0}% %gap between initials of abbreviated first name } %default fonts \def\Citefont{}%citations \def\ACitefont{}%alternate citations \def\Authfont{}%authors \def\Titlefont{}%titles \def\Tomefont{\sl}%journals or books \def\Volfont{}%volume number of journal \def\Flagfont{}%citation flag \def\Reffont{\rm}%set at beginning of reference listing \def\Smallcapsfont{\sevenrm}%small caps \def\Flagstyle#1{\hangindent\parindent\indent\hbox to0pt%flag style {\hss[{\Flagfont#1}]\kern.5em}\ignorespaces}% for references \def\Underlinemark{\vrule height .7pt depth 0pt width 3pc}%for replacing % successive listings of identical author(s) by underline (U option % in tib format file or -u flag on call). \def\Citebrackets{\Rpunct%defaults for putting citations in brackets []. \def\Lcitemark{\def\Cfont{\Citefont}[\bgroup\Cfont}%mark at left of citation \def\Rcitemark{\egroup]}%mark at right of citation \def\LAcitemark{\def\Cfont{\ACitefont}\bgroup\ACitefont}% %mark at left of alternate citation \def\RAcitemark{\egroup}%mark at right of alternate citation \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Cfont}%mark at right of insertion in citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\Citecomma{\egroup,\hskip0pt\bgroup\Cfont}% %separater for multiple citations \def\Citebreak{}%mark between parts of citation (e.g. author\Citebreak date) } \def\Citeparen{\Rpunct%defaults for putting citations in parenthesis (). \def\Lcitemark{\def\Cfont{\Citefont}(\bgroup\Cfont}%mark at left of citation \def\Rcitemark{\egroup)}%mark at right of citation \def\LAcitemark{\def\Cfont{\ACitefont}\bgroup\ACitefont}% %mark at left of alternate citation \def\RAcitemark{\egroup}%mark at right of alternate citation \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Cfont}%mark at right of insertion in citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\Citecomma{\egroup,\hskip0pt\bgroup\Cfont}% %separater for multiple citations \def\Citebreak{}%mark between parts of citation (e.g. author\Citebreak date) } \def\Citesuper{\Lpunct%defaults for making superscript citations \def\Lcitemark{\def\Cfont{\Citefont}\raise1ex\hbox\bgroup\bgroup\Cfont}% %mark at left of citation \def\Rcitemark{\egroup\egroup}%mark at right of citation \def\LAcitemark{\def\Cfont{\ACitefont}\bgroup\ACitefont}% %mark at left of alternate citation \def\RAcitemark{\egroup}%mark at right of alternate citation \def\LIcitemark{\egroup\egroup}%mark at left of insertion in citation \def\RIcitemark{\raise1ex\hbox\bgroup\bgroup\Cfont}% %mark at right of insertion in citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\Citecomma{\egroup,\hskip0pt\bgroup% \Cfont}%separater for multiple citations \def\Citebreak{}%mark between parts of citation (e.g. author\Citebreak date) } \def\Citenamedate{\Rpunct%defaults for making name-date citations \def\Lcitemark{%mark at left of citation--also sets internal punctuation \def\Citebreak{\egroup\ [\bgroup\Citefont}%separater in citation \def\Citecomma{\egroup]; %between multiple citations \bgroup\let\uchyph=1\Citefont}(\bgroup\let\uchyph=1\Citefont}% \def\Rcitemark{\egroup])}%mark at right of citation \def\LAcitemark{%mark at left of alternate citation \def\Citebreak{\egroup\ [\bgroup\Citefont}\def\Citecomma{\egroup], % \bgroup\ACitefont }\bgroup\let\uchyph=1\ACitefont}% \def\RAcitemark{\egroup]}%mark at right of alternate citation \def\Citehyphen{\egroup--\bgroup\Citefont}%separater for string of citations \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Citefont}%mark at right of insertion in citation } %ams numeric or alpha flag style %flag, author, title, etc., volume (date) pages, gov't no., other \Refstd\Citebrackets%set general formats for reference list and citations \def\Citefont{\bf}\def\Titlefont{\sl}\def\Volfont{\bf}\def\Tomefont{\Reffont}%redefine some fonts \def\journalarticleformat{\Reffont\let\uchyph=1\parindent=1.25pc\def\Comma{}% \sfcode`\.=1000\sfcode`\?=1000\sfcode`\!=1000\sfcode`\:=1000\sfcode`\;=1000\sfcode`\,=1000%\frenchspacing \par\vfil\penalty-200\vfilneg%\filbreak \if\Ftest\present\Flagstyle\Fstr\fi% \if\Atest\present\bgroup\Authfont\Astr\egroup\def\Comma{\unskip, }\fi% \if\Ttest\present\Comma\bgroup\Titlefont\Tstr\egroup\def\Comma{, }\fi% \if\etest\present\hskip.2em(\bgroup\estr\egroup)\def\Comma{\unskip, }\fi% \if\Jtest\present\Comma\bgroup\Tomefont\Jstr\/\egroup\def\Comma{, }\fi% \if\Vtest\present\if\Jtest\present\hskip.2em\else\Comma\fi\bgroup\Volfont\Vstr\egroup\def\Comma{, }\fi% \if\Dtest\present\hskip.2em(\bgroup\Dstr\egroup)\def\Comma{, }\fi% \if\Ptest\present\bgroup, \Pstr\egroup\def\Comma{, }\fi% \if\ttest\present\Comma\bgroup\Titlefont\tstr\egroup\def\Comma{, }\fi% \if\jtest\present\Comma\bgroup\Tomefont\jstr\/\egroup\def\Comma{, }\fi% \if\vtest,\present\if\jtest\present\hskip.2em\else\Comma\fi\bgroup\Volfont\vstr\egroup\def\Comma{, }\fi% \if\dtest\present\hskip.2em(\bgroup\dstr\egroup)\def\Comma{, }\fi% \if\ptest\present\bgroup, \pstr\egroup\def\Comma{, }\fi% \if\Gtest\present{\Comma Gov't ordering no. }\bgroup\Gstr\egroup\def\Comma{, }\fi% \if\Mtest\present\Comma MR \#\bgroup\Mstr\egroup\def\Comma{, }\fi% \if\Otest\present{\Comma\bgroup\Ostr\egroup.}\else{.}\fi% \vskip3ptplus1ptminus1pt}%\smallskip \def\conferencereportformat{\Reffont\let\uchyph=1\parindent=1.25pc\def\Comma{}% \sfcode`\.=1000\sfcode`\?=1000\sfcode`\!=1000\sfcode`\:=1000\sfcode`\;=1000\sfcode`\,=1000%\frenchspacing \par\vfil\penalty-200\vfilneg%\filbreak \if\Ftest\present\Flagstyle\Fstr\fi% \if\Atest\present\bgroup\Authfont\Astr\egroup\def\Comma{\unskip, }\fi% \if\Ttest\present\Comma\bgroup\Titlefont\Tstr\egroup\def\Comma{, }\fi% \if\Jtest\present\Comma\bgroup\Tomefont\Jstr\/\egroup\def\Comma{, }\fi% \if\Ctest\present\Comma\bgroup\Cstr\egroup\def\Comma{, }\fi% \if\Dtest\present\hskip.2em(\bgroup\Dstr\egroup)\def\Comma{, }\fi% \if\Mtest\present\Comma MR \#\bgroup\Mstr\egroup\def\Comma{, }\fi% \if\Otest\present{\Comma\bgroup\Ostr\egroup.}\else{.}\fi% \vskip3ptplus1ptminus1pt}%\smallskip \def\bookarticleformat{\Reffont\let\uchyph=1\parindent=1.25pc\def\Comma{}% \sfcode`\.=1000\sfcode`\?=1000\sfcode`\!=1000\sfcode`\:=1000\sfcode`\;=1000\sfcode`\,=1000%\frenchspacing \par\vfil\penalty-200\vfilneg%\filbreak \if\Ftest\present\Flagstyle\Fstr\fi% \if\Atest\present\bgroup\Authfont\Astr\egroup\def\Comma{\unskip, }\fi% \if\Ttest\present\Comma\bgroup\Titlefont\Tstr\egroup\def\Comma{, }\fi% \if\etest\present\hskip.2em(\bgroup\estr\egroup)\def\Comma{\unskip, }\fi% \if\Btest\present\Comma in \bgroup\Tomefont\Bstr\/\egroup\def\Comma{\unskip, }\fi% \if\otest\present\ \bgroup\ostr\egroup\def\Comma{, }\fi% \if\Etest\present\Comma\bgroup\Estr\egroup\unskip, \ifnum\Ecnt>1eds.\else ed.\fi\def\Comma{, }\fi% \if\Stest\present\Comma\bgroup\Sstr\egroup\def\Comma{, }\fi% \if\Vtest\present\Comma vol. \bgroup\Vstr\egroup\def\Comma{, }\fi% \if\Ntest\present\Comma no. \bgroup\Nstr\egroup\def\Comma{, }\fi% \if\Itest\present\Comma\bgroup\Istr\egroup\def\Comma{, }\fi% \if\Ctest\present\Comma\bgroup\Cstr\egroup\def\Comma{, }\fi% \if\Dtest\present\Comma\bgroup\Dstr\egroup\def\Comma{, }\fi% \if\Ptest\present\Comma\Pstr\def\Comma{, }\fi% \if\ttest\present\Comma\bgroup\Titlefont\Tstr\egroup\def\Comma{, }\fi% \if\btest\present\Comma in \bgroup\Tomefont\bstr\egroup\def\Comma{, }\fi% \if\atest\present\Comma\bgroup\astr\egroup\unskip, \if\acnt\present eds.\else ed.\fi\def\Comma{, }\fi% \if\stest\present\Comma\bgroup\sstr\egroup\def\Comma{, }\fi% \if\vtest\present\Comma vol. \bgroup\vstr\egroup\def\Comma{, }\fi% \if\ntest\present\Comma no. \bgroup\nstr\egroup\def\Comma{, }\fi% \if\itest\present\Comma\bgroup\istr\egroup\def\Comma{, }\fi% \if\ctest\present\Comma\bgroup\cstr\egroup\def\Comma{, }\fi% \if\dtest\present\Comma\bgroup\dstr\egroup\def\Comma{, }\fi% \if\ptest\present\Comma\pstr\def\Comma{, }\fi% \if\Gtest\present{\Comma Gov't ordering no. }\bgroup\Gstr\egroup\def\Comma{, }\fi% \if\Mtest\present\Comma MR \#\bgroup\Mstr\egroup\def\Comma{, }\fi% \if\Otest\present{\Comma\bgroup\Ostr\egroup.}\else{.}\fi% \vskip3ptplus1ptminus1pt}%\smallskip \def\bookformat{\Reffont\let\uchyph=1\parindent=1.25pc\def\Comma{}% \sfcode`\.=1000\sfcode`\?=1000\sfcode`\!=1000\sfcode`\:=1000\sfcode`\;=1000\sfcode`\,=1000%\frenchspacing \par\vfil\penalty-200\vfilneg%\filbreak \if\Ftest\present\Flagstyle\Fstr\fi% \if\Atest\present\bgroup\Authfont\Astr\egroup\def\Comma{\unskip, }% \else\if\Etest\present\bgroup\def\Eand{\Aand}\def\Eandd{\Aandd}\Authfont\Estr\egroup\unskip, \ifnum\Ecnt>1eds.\else ed.\fi\def\Comma{, }% \else\if\Itest\present\bgroup\Authfont\Istr\egroup\def\Comma{, }\fi\fi\fi% \if\Ttest\present\Comma\bgroup\Titlefont\Tstr\/\egroup\def\Comma{\unskip, }% \else\if\Btest\present\Comma\bgroup\Titlefont\Bstr\/\egroup\def\Comma{\unskip, }\fi\fi% \if\otest\present\ \bgroup\ostr\egroup\def\Comma{, }\fi% \if\etest\present\hskip.2em(\bgroup\estr\egroup)\def\Comma{\unskip, }\fi% \if\Stest\present\Comma\bgroup\Sstr\egroup\def\Comma{, }\fi% \if\Vtest\present\Comma vol. \bgroup\Vstr\egroup\def\Comma{, }\fi% \if\Ntest\present\Comma no. \bgroup\Nstr\egroup\def\Comma{, }\fi% \if\Atest\present\if\Itest\present \Comma\bgroup\Istr\egroup\def\Comma{\unskip, }\fi% \else\if\Etest\present\if\Itest\present \Comma\bgroup\Istr\egroup\def\Comma{\unskip, }\fi\fi\fi% \if\Ctest\present\Comma\bgroup\Cstr\egroup\def\Comma{, }\fi% \if\Dtest\present\Comma\bgroup\Dstr\egroup\def\Comma{, }\fi% \if\ttest\present\Comma\bgroup\Titlefont\tstr\egroup\def\Comma{, }% \else\if\btest\present\Comma\bgroup\Titlefont\bstr\egroup\def\Comma{, }\fi\fi% \if\stest\present\Comma\bgroup\sstr\egroup\def\Comma{, }\fi% \if\vtest\present\Comma vol. \bgroup\vstr\egroup\def\Comma{, }\fi% \if\ntest\present\Comma no. \bgroup\nstr\egroup\def\Comma{, }\fi% \if\itest\present\Comma\bgroup\istr\egroup\def\Comma{, }\fi% \if\ctest\present\Comma\bgroup\cstr\egroup\def\Comma{, }\fi% \if\dtest\present\Comma\bgroup\dstr\egroup\def\Comma{, }\fi% \if\Gtest\present{\Comma Gov't ordering no. }\bgroup\Gstr\egroup\def\Comma{, }\fi% \if\Mtest\present\Comma MR \#\bgroup\Mstr\egroup\def\Comma{, }\fi% \if\Otest\present{\Comma\bgroup\Ostr\egroup.}\else{.}\fi% \vskip3ptplus1ptminus1pt}%\smallskip \def\technicalreportformat{\Reffont\let\uchyph=1\parindent=1.25pc\def\Comma{}% \sfcode`\.=1000\sfcode`\?=1000\sfcode`\!=1000\sfcode`\:=1000\sfcode`\;=1000\sfcode`\,=1000%\frenchspacing \par\vfil\penalty-200\vfilneg%\filbreak \if\Ftest\present\Flagstyle\Fstr\fi% \if\Atest\present\bgroup\Authfont\Astr\egroup\def\Comma{\unskip, }% \else\if\Etest\present\bgroup\def\Eand{\Aand}\def\Eandd{\Aandd}\Authfont\Estr\egroup\unskip, \ifnum\Ecnt>1eds.\else ed.\fi\def\Comma{, }% \else\if\Itest\present\bgroup\Authfont\Istr\egroup\def\Comma{, }\fi\fi\fi% \if\Ttest\present\Comma\bgroup\Titlefont\Tstr\egroup\def\Comma{, }\fi% \if\Atest\present\if\Itest\present \Comma\bgroup\Istr\egroup\def\Comma{, }\fi% \else\if\Etest\present\if\Itest\present \Comma\bgroup\Istr\egroup\def\Comma{, }\fi\fi\fi% \if\Rtest\present\Comma\bgroup\Rstr\egroup\def\Comma{, }\fi% \if\Ctest\present\Comma\bgroup\Cstr\egroup\def\Comma{, }\fi% \if\Dtest\present\Comma\bgroup\Dstr\egroup\def\Comma{, }\fi% \if\ttest\present\Comma\bgroup\Titlefont\tstr\egroup\def\Comma{, }\fi% \if\itest\present\Comma\bgroup\istr\egroup\def\Comma{, }\fi% \if\rtest\present\Comma\bgroup\rstr\egroup\def\Comma{, }\fi% \if\ctest\present\Comma\bgroup\cstr\egroup\def\Comma{, }\fi% \if\dtest\present\Comma\bgroup\dstr\egroup\def\Comma{, }\fi% \if\Gtest\present{\Comma Gov't ordering no. }\bgroup\Gstr\egroup\def\Comma{, }\fi% \if\Mtest\present\Comma MR \#\bgroup\Mstr\egroup\def\Comma{, }\fi% \if\Otest\present{\Comma\bgroup\Ostr\egroup.}\else{.}\fi% \vskip3ptplus1ptminus1pt}%\smallskip \def\otherformat{\Reffont\let\uchyph=1\parindent=1.25pc\def\Comma{}% 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\magnification =\magstep1 \hcorrection{.25in} \NoBlackBoxes \def\D{\Delta} \predefine\Sec{\S} \redefine\L{\Lambda} \def\O{\Omega} \def\P{\Phi} \redefine\S{\Sigma} \def\a{\alpha} \def\b{\beta} \def\d{\delta} \def\e{\varepsilon} \def\g{\gamma} \def\l{\lambda} \def\o{\omega} \def\ph{\varphi} \def\p{\psi} \def\s{\sigma} \def\t{\chi} \def\T{\theta} \def\C{\Bbb C} \def\R{\Bbb R} \def\Z{\Bbb Z} \def\A{\text{\bf A}} \def\B{\text{\bf B}} \def\G{\text{\bf G}} \def\M{\text{\bf M}} \def\N{\bold N} \def\PP{\text{\bf P}} \def\TT{\text{\bf T}} \def\U{\text{\bf U}} \def\aa{\text{\bf a}} \def\x{\text{\bf x}} \def\xx{\overset *\to x} \def\diag{\operatorname{diag}} \def\Hom{\operatorname{Hom}} \def\Ind{\operatorname{Ind}} \def\ind{\operatorname{ind}} \def\mod{\operatorname{mod}} \def\Res{\operatorname{Res}} \def\supp{\operatorname{supp}} \def\cc{{C_c^\infty}} \def\ot{{\omega_\chi}} \def\tot{{\tilde{\omega}_\chi}} \def\DD{{\Cal D}} \def\PPP{{\Cal P}} \loadeurm \topmatter \title On local coefficients for non-generic representations of some classical groups \endtitle \author Solomon Friedberg and David Goldberg\endauthor %\affil University of California, Santa Cruz \\ %Purdue University \endaffil \address Solomon Friedberg, Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064 \endaddress \email friedbe\@cats.ucsc.edu \endemail \address David Goldberg, Department of Mathematics, Purdue University, West Lafayette, IN 47907 \endaddress \email goldberg\@math.purdue.edu \endemail \rightheadtext{On local coefficients for non-generic representations} \abstract This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Last, local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. \endabstract \subjclass Primary 22E50, Secondary 22E35\endsubjclass %\leftheadtext{Solomon Friedberg and David Goldberg} \thanks{Friedberg was partially supported by National Security Administration Grant MDA904-95-H-1053. Goldberg was partially supported by National Science Foundation Fellowship DMS-9206246 and National Science Foundation Career Grant DMS-9501868. The research of both authors at MSRI was supported in part by National Science Foundation Grant DMS-9022140.} \endthanks \endtopmatter \document \subheading{Introduction} $L$-functions are a central object of study in representation theory and number theory. Over a global field, one has the Langlands Conjectures, which assert in particular the meromorphic continuation and functional equation of a class of Euler products. Over a local field one has additional conjectures due to Langlands, expressing the Plancherel measure arithmetically as the ratio of certain local $L$-functions and root numbers. In many cases these conjectures have been established by Shahidi\Lspace \Lcitemark Shab\Citecomma Shac\Citecomma Shad\Rcitemark \Rspace{}, following a path laid out by Langlands\Lspace \Lcitemark Lana\Rcitemark \Rspace{}. The framework for Shahidi's work is the study of Eisenstein series or their local analogues, induced representations. One knows the continuation of these Eisenstein series due to Langlands \Lcitemark Lanb\Rcitemark \Rspace{}. Langlands also showed that the constant coefficients of the Eisenstein series may be expressed in terms of local intertwining operators which are almost everywhere quotients of certain $L$-functions. It remains to study these intertwining operators for the finite set of `bad' places. If the inducing data is generic, that is, admits a Whittaker model, then Shahidi has succeeded in relating them to local $L$-functions. Thus the careful study of the Eisenstein series, both local and global, affords a proof of certain of the Langlands conjectures for these $L$-functions. The aim of this work is to suggest that the Langlands-Shahidi method may be extended beyond the generic spectrum by the use of other models. The Whittaker model is unique (an irreducible admissible representation admits at most one such model up to scalars). In this paper we study the properties of local representations of split orthogonal groups and quasi-split unitary groups which are not generic, but which support a unique model of a different kind, the generalized Bessel model. These models involve a character of a proper subgroup of the unipotent radical of a Borel subgroup, but transform under a reductive group of some, in general non-zero, rank. The uniqueness of the models has been proved by S. Rallis \Lcitemark Ral\Rcitemark \Rspace{} in the orthogonal case, but as the argument has not yet been written out in full detail in the unitary case we make it a hypothesis throughout the paper. We first study the properties of Bessel models under induction, and prove an analogue of Rodier's Theorem\Lspace \Lcitemark Rodb\Rcitemark \Rspace{} concerning the induction of Whittaker models. Our analogue, Theorem 2.1, states that if one parabolically induces a representation with a Bessel model of minimal rank, or more generally one which is minimal in the sense of Definition 1.5 below, then the induced representation has a unique Bessel model of the same rank and compatible type. In the case of rank 0, we recover Rodier's theorem. To carry out the proof we use Bruhat's extension\Lspace \Lcitemark Bru\Rcitemark \Rspace{} of Mackey theory and investigate precisely which double cosets of the appropriate type may support a functional with the desired equivariance property. We show that there is a unique such double coset by an extensive combinatorial argument. Next, we establish the holomorphicity of the Bessel functional which arises from one which is minimal by parabolic induction of the underlying representation. Our approach is based on Bernstein's theorem\Lspace \Lcitemark Ber\Rcitemark \Rspace{}, which uses uniqueness to conclude meromorphicity under some regularity hypotheses, and Banks's extension\Lspace \Lcitemark Ban\Rcitemark \Rspace{}, which allows one to prove holomorphicity as well. We show in Theorem 3.6 that there is a non-zero Bessel functional $\Lambda(\nu,\pi)$, attached to an irreducible admissible representation $\pi$ of the Levi subgroup $M$ and a parameter $\nu$ in the complexified dual of the Lie algebra of the split component of $M$, which is holomorphic in $\nu$. If $\pi$ is supercuspidal and has a Bessel model, or more generally if $\pi$ is irreducible and carries a Bessel model corresponding to a minimal Bessel model of the supercuspidal from which it is induced, these results allow us to establish the existence of a local coefficient. In the generic case, such a local coefficient was crucial for Shahidi's study of the intertwining operators and of the relation between Plancherel measures and $L$--functions; see Shahidi\Lspace \Lcitemark Shad\Rcitemark \Rspace{}. Let $A(\nu,\pi,w)$ denote the standard intertwining operator attached to inducing data $\nu,\pi$ and Weyl group element representative $w$ (see (3.4) below). We shall prove (cf. Theorem 3.8): \proclaim{Theorem} Let $\pi$ be an irreducible representation of $M$ which is a component of the representation parabolically induced from an irreducible admissible supercuspidal representation $\rho$ of a parabolic subgroup of $M.$ Suppose that $\pi$ carries a Bessel model corresponding (in the sense of Theorem 2.1) to a minimal Bessel model of $\rho$. For each $\tilde w$ in the Weyl group, choose a representative $w$ for $\tilde w.$ Then there is a complex number $C(\nu,\pi,w)$ so that $$\L(\nu,\pi)=C(\nu,\pi,w)\L(\tilde w\nu,\tilde w\pi)A(\nu,\pi,w).$$ Moreover, the function $\nu\mapsto C(\nu,\pi,w)$ is meromorphic and depends only on the class of $\pi$ and the choice of the representative $w.$ \endproclaim We call $C(\nu,\pi,w)$ the {\it local coefficient} attached to $\pi$, $\nu$, and $w$. We then show in Corollary 3.9 that the local coefficients behave as expected with respect to the Langlands decomposition of the intertwining operators. This generalizes a property of the local coefficients introduced by Shahidi\Lspace \Lcitemark Shaa\Rcitemark \Rspace{} in the generic case. The authors wish to thank D.\ Ramakrishnan, who suggested we begin our study of using the Bessel functional to extend the Langlands-Shahidi method by seeking to generalize Rodier's theorem, and F.\ Shahidi for his interest in this project. The authors also express their appreciation to W.\ Banks, D.\ Bump, M.\ Furusawa, D.\ Ginzburg, S.\ Rallis, F.\ Shahidi, and D.\ Soudry for helpful discussions of their work. Finally, the authors wish to thank the Mathematical Sciences Research Institute and the organizers of the 1994-95 Special Year in Automorphic Forms. The authors met and began this work while both were in residence there; without their support this project would not have started. \subheading{\Sec 1 Preliminaries on Bessel Models} In this section we recall the notion of a Bessel model following \Lspace \Lcitemark Ral\Rcitemark \Rspace{} and \Lcitemark GPR\Rcitemark \Rspace{}, and review some properties of such models. Let $F$ be a nonarchimedean local field of characteristic zero. Let $\G$ be one of the classical groups $SO_{2r+1}, U_{2r+1}, U_{2r},$ or $SO_{2r},$ defined over $F.$ We assume that the orthogonal groups are split, and that the unitary groups are quasi-split, and split over a quadratic extension $E/F.$ Let $r_0=2r$ if $\G=U_{2r}$ or $SO_{2r},$ and $r_0=2r+1$ otherwise. Denote by $\B=\TT\U$ the Borel subgroup of $\G,$ where $\TT$ contains the maximal split subtorus of diagonal elements, and $\U$ is the subgroup of upper triangular unipotent matrices in $\G.$ We use $G$ to denote the $F$\snug-rational points of $\G,$ and use this notational convention for other algebraic groups defined over $F.$ Denote by $\P(\G,\TT)$ the root system of $\G$ with respect to $\TT.$ We choose the ordering on the roots corresponding to our choice of Borel subgroup. Let $W=W(\G,\TT_d)$ be the Weyl group of $\G$ with respect to the maximal split subtorus $\TT_d$ of $\TT.$ Thus, $W=N_G(\TT_d)/\TT.$ Then, $$W\simeq\cases S_r\ltimes\Z_2^r&\text{if }\G\neq SO_{2r}\\ S_r\ltimes \Z_2^{r-1}&\text{if }\G=SO_{2r}.\endcases$$ (See\Lspace \Lcitemark Gola\Citecomma Golb\Rcitemark \Rspace{} for a more explicit description of $\TT$ and $W.$) Here we will denote all elements of $W$ as permutations on $r_0$ letters. Thus, the permutation $(ij)\in S_r$ corresponds to the permutation $(ij)(r_0+1-j\, r_0+1-i)$ in $S_{r_0}.$ Similarly, the sign change $c_i$ which generates that $i$\snug-th copy of $\Z_2$ corresponds to the permutation $(i\,r_0+1-i)$ in $S_{r_0}.$ Fix an $\elln,$ then $ a_{i+1}\leq n.$ This argument also shows that we may assume that $t$ is even. Now we see that $$\align w&\equiv cs\cdot ( a_1\,a_{t-1}\, a_{t-3}\dots a_3)(a_1'\,a_{t-1}'\,a_{t-3}'\dots a_3')\\&=c( a_1\, a_t)( a_3\, a_2)\dots ( a_{t-1}\, a_{t-2})( a_1'\, a_t')( a_3'\, a_2')\dots ( a_{t-1}'\, a_{t-2}'). \endalign$$ Now write $c=(b_1\,b_1')(b_2\,b_2')\dots (b_s\, b_s'),$ with $b_i\neq b_j,$ for $i\neq j.$ If, for a fixed even $i\geq 2,$ $\{a_i,a_{i+1}\}\subset \{b_j\}_{j=1}^s,$ then the product $$(a_i\, a_i')(a_{i+1}\,a_{i+1}')(a_{i+1}\,a_{i})(a_{i+1}'\,a_{i}')=(a_i\,a_{i+1}')(a_{i+1}\,a_{i}')$$ appears in the reduced product for $w.$ The same is true if $\{a_1,a_t\}\subset\{b_j\}_{j=1}^s,$ i.e., $(a_1\,a_t')(a_t\,a_1')$ appears in $w.$ If $i\geq 2$ is even and $\{a_i,a_{i+1}\}\cap\{b_j\}_{j=1}^s=\emptyset,$ then $c$ commutes with $(a_i\,a_{i+1})(a_i'\,a_{i+1}'),$ and so this product of transpositions appears in $w.$ Similarly, if $\{a_1,\,a_t\}\cap\{b_j\}_{j=1}^s=\emptyset,$ then $(a_1\,a_t)(a_1'\,a_t')$ appears in $w.$ Suppose $i\geq 2$ is even and that exactly one element of $\{a_i,a_{i+1}\}$ belongs to $\{b_j\}_{j=1}^s.$ Then, if $\G\neq SO_{2r},$ we can replace $w$ by $w(a_i\, a_i'),$ and we see that either $(a_{i+1}\,a_i)(a_i'\,a_{i+1}')$ or $(a_{i+1}\,a_i')(a_i\, a_{i+1}')$ appears in $w(a_i\, a_i'),$ depending on whether $a_{i+1}$ or $a_i$ is in $\{b_j\}_{j=1}^s.$ If $\G=SO_{2r}$ and $w$ either fixes some $d_0>n,$ or interchanges some $d_0$ and $d_0',$ then we can instead multiply $w$ by $(a_i\, a_i')(d_0\, d_0'),$ which shows that one of $(a_{i+1}\,a_i)(a_i'\,a_{i+1}')$ or $(a_{i+1}\,a_i')(a_i\, a_{i+1}')$ appears. We see that the above considerations apply equally well to the pair $\{a_1,a_t\}.$ By fixing the element $d_0$ before starting the above process, we can guarantee that, when we have concluded, $w_2=1$ or $w_2=(d_0\, d_0').$ Finally suppose that no such $d_0$ exists. Thus, $w(d)\neq d,d'$ for all $n+1\leq d\leq r.$ So we may now assume that $d\in\{a_i\},$ for each $d,$ $n+1\leq d\leq r.$ Suppose that the number of $d$ for which $d=a_i\in \{b_j\}$ with $a_{i+1}\not\in \{b_j\}$ is even. (Here we are including $\{a_1,\, a_t\}$ as one possible pair.) Then we see that $$w\equiv w\prod (d\, d'),$$ where the product is over precisely those $d=a_i$ for which $a_{i+1}\not\in \{b_j\},$ is of the form $w_1$ as claimed. Finally if $\big|\{n+1\leq d\leq r|d\in \{b_j\}\}\big|$ is odd, then we fix some such $d_0.$ Without loss of generality, assume that $d_0=a_t.$ Multiplying on the right by the elements $(d\, d'),$ for the other such $d,$ we see that we have a factor of $(a_t\, a_t')(a_1\, a_t)(a_1'\, a_t')$ remaining to be dealt with. But this product is indeed $w_2=(a_1\, a_t'\, a_1'\, a_t),$ as claimed. \qed \enddemo If $\G=SO_{2r},$ and $w_2$ is of this final form, then there is some flexibility as to the indices appearing in $w_2.$ That is, we may choose, for $d_0,$ any of the $a_i>n$ for which $(a_i\, a_i')$ appears in $c,$ but $(a_{i+1}\, a_{i+1}')$ does not. We will need this below. Recall that $\ell_0=r_0-2\ell.$ Let $s=\ell_0-2.$ Suppose $\x=(x_1,\dots,x_{s})\in F^{s}$ if $\G$ is orthogonal and $\x\in E^s$ if $\G$ is unitary. Let $$n(\x)=\pmatrix I_{\ell}\\ &1&x_1&\dots&x_s&*\\ &&1&0&\dots&-\bar x_s\\ &&&\ddots&0&\vdots\\ &&&&1&-\bar x_1\\ &&&&&1\\ &&&&&&I_{\ell}\endpmatrix,$$ where $\bar x$ is the Galois conjugate of $x$ if $\G$ is unitary, and is $x$ if $\G$ is orthogonal. The next result is a straightforward consequence of Witt's Theorem and the Bruhat decomposition. Here and for the rest of this section, we pass between a Weyl group element and its coset representative without changing the notation. \proclaim{Proposition 2.3} \roster \item"(a)" Let $g\in G.$ Then for some $w\in W$ and some $\x\in F^s,$ we have $R_\t gP=R_\t n(\x)wP.$ Clearly we can choose $w$ up to $W_{\M},$ i.e., we may assume $w=w_1$ is of the form given in Lemma 2.2. \item"(b)" Denote by $||\x||$ the standard length of $\x\in F^s$ or $E^s$ accordingly. If $||\x||=||\x_1||,$ then $R_\t n(\x)wP=R_\t n(\x_1)wP,$ for all $w.$ \endroster \endproclaim If $H\subset G,$ we will use $\ind_H^G(\pi)$ to denote the representation of $G$ compactly induced from $\pi$\Lspace \Lcitemark BeZa\Citecomma Cas\Rcitemark \Rspace{}. Recall that $\Ind_P^G(\pi)=\ind_P^G(\pi),$ by the Iwasawa decomposition. If $V$ is a complex vector space, let $C^\infty(G,V)$ denote the space of locally constant $V$--valued functions on $G,$ and let $C^\infty_c(G,V)$ denote the subspace of elements of $C^\infty(G,V)$ with compact support. Let $\DD(G,V)=C^{\infty}_c(G,V)^*$ be the space of $V$--distributions on $G.$ Let $V_\s$ be the space of $\s,$ $V_\tau$ be the space of $\tau,$ and $V_{\o}$ the space of $\o$ (and hence the space of $\o_\t$). We let $V_{\pi}=V_\s\otimes V_\tau.$ Denote by $V$ the vector space $\tilde V_{\o}\otimes V_{\pi},$ where $\tilde V_\o$ is the space of the smooth contragredient $\tilde \o$ of $\o.$ We wish to analyze the space $\Hom_G(\Ind_P^G(\pi),\Ind_{R_\t}^G(\o_\t)).$ Dualizing, and using Theorem 2.4.2 of\Lspace \Lcitemark Cas\Rcitemark \Rspace{}, this is isomorphic to the space $$\Hom_G(\ind_{R_\t}^G(\tilde\o_\t),\Ind_P^G(\tilde\s\otimes\tilde\tau)).$$ This space, in turn, is isomorphic to the space of intertwining forms on $$\ind_{R_\t}^G(\tilde\o_\t) \bigotimes\Ind_P^G(\pi)$$ \Lcitemark Har\LIcitemark{}, Lemma 4\RIcitemark \Rcitemark \Rspace{}. Now by Bruhat's thesis (see\Lspace \Lcitemark Rodb\LIcitemark{}, Theorem 4\RIcitemark \Rcitemark \Rspace{}) this is isomorphic to the space of $V$--distributions $T$ on $G$ satisfying $$\e(r)*T*\e(p^{-1})=\d_{P}^{1/2}(p)T\circ[\tilde\o_\t(r)\otimes\pi(p)],\tag 2.4$$ for all $r\in R_\t$ and $p\in P.$ The analysis of this space of distributions will make use of the following proposition. Its proof requires a combinatorial argument, and will be given in several steps later in this section. \proclaim{Proposition 2.4} If there is a non-zero $V$\snug-distribution $T$ satisfying {\rm (2.4)} for all $r\in R_\t$ and $p\in P$ which is supported on $R_\t n(\x) w P,$ then $R_\t n(\x)wP=R_{\t} w_0 P$ and $\s$ is generic. \endproclaim \proclaim {Lemma 2.5} Suppose that $T$ satisfies {\rm (2.4)}. Then $T$ is completely determined by its restriction to $R_\t w_0P.$ \endproclaim \demo{Proof} First note that a straightforward matrix computation shows that $n(\x)w_0=w_0n(\x),$ for any $\x.$ Thus $$R_\t w_0P=\bigcup_{\x}R_{\t}n(\x)w_0 P=Pw_0P,$$ is open. Therefore $C=G\setminus R_\t {w_0}P$ is closed. Therefore, we have the exact sequence\Lspace \Lcitemark BeZa\LIcitemark{}, \Sec 1.7\RIcitemark \Rcitemark \Rspace{} $$0\longrightarrow C_c^\infty(R_\t{w_0}P)\longrightarrow C_c^\infty(G)\longrightarrow C_c^\infty(C)\longrightarrow 0.$$ Then, by tensoring with $V,$ the above exact sequence yields the exact sequence $$0\longrightarrow C_c^\infty(R_\t {w_0P},V)\longrightarrow C_c^\infty(G,V)\longrightarrow C_c^\infty(C,V)\longrightarrow 0.$$ Dualizing, we get the exact sequence $$0\longrightarrow\DD(C,V)\longrightarrow\DD(G,V)\longrightarrow\DD(R_\t w_0 P,V)\longrightarrow 0.$$ Let $\DD_{R_\t,P}$ be the subspace of distributions satisfying (2.4). Then Proposition 2.4 implies that if $T\in\DD(G,V)_{R_\t,P}$ and $f\in C_c^\infty(C,V),$ then $T(f)=0.$ Thus, the above sequence tells us that $\DD(G,V)_{R_\t,P}\simeq\DD(R_\t w_0P,V)_{R_\t,P},$ which completes the proof of the Lemma.\qed \enddemo Let $R_\t^{w_0}=w_0^{-1}R_\t w_0,$ and denote by $\o_\t^{w_0}$ the representation of $R_\t^{w_0}$ defined by $\o_\t^{w_0}(r)=\o_\t(w_0rw_0^{-1}).$ Recall that $\bold P=\M\N$ is the Levi decomposition of $\bold P.$ \proclaim{Lemma 2.6} There exists an isomorphism between the vector space $\DD(R_\t w_0P,V)_{R_\t,P}$ and the vector space of distributions in $\DD(U_\ell)\otimes\DD(P,V)$ of the form $$\t(u)\, du\otimes\d_{P}^{-1/2}(m)\, dQ(m)\, dn,$$ where $Q\in\DD(M,V)$ satisfies $$\e(r)*Q*\e(m^{-1})=Q\circ[\tilde\o^{w_0}_\t(r)\otimes\pi(m)],\tag 2.5$$ for all $r\in R_\t^{w_0}\cap M,$ $m\in M.$ \endproclaim \demo{Proof} Define a projection $$\PPP:\cc(U_\ell)\otimes \cc(P,V)\to\cc(U_\ell w_0P,V)$$ by specifying that for all $f_1\in\cc(U_\ell)$ and $f_2\in\cc(P,V)$, one has $$\PPP(f_1\otimes f_2)(uw_0p)= \int\limits_{U_\ell\cap w_0Pw_0^{-1}}f_1(uu_1)\,f_2(w_0^{-1}u_1^{-1}w_0p) \,du_1.$$ Then it follows from\Lspace \Lcitemark Sil\LIcitemark{}, Lemma 1.2.1\RIcitemark \Rcitemark \Rspace{} that $\PPP$ is onto. Let $T\in \DD(R_\t w_0P,V)_{R_\t ,P}$. For $f_1, f_2$ as above, define $T'\in\DD(U_\ell)\otimes\DD(P,V)$ by $$T'(f_1\otimes f_2)=T(\PPP(f_1\otimes f_2)).$$ Then one sees easily that (2.4) implies the equality $$\e(u)*T'*\e(p^{-1}) =\tot(u)T \circ[\pi(p)]\tag 2.6$$ for all $u\in U_\ell$, $p\in P$ (where $\pi$ acts on the second factor of $V$). As in\Lspace \Lcitemark Sil\LIcitemark{}, Section 1.8\RIcitemark \Rcitemark \Rspace{}, this implies that $T'$ is in fact a pure tensor of the form $$\chi(u)\,du\otimes\delta_P(m)^{-1/2}\,dQ(m)\,dn.\tag2.7$$ where $Q\in \DD(M,V)$. (Here we are using that $\pi(mn)=\pi(m)$.) It is a formal consequence of the definitions that (2.6) implies that $$Q*\e(m^{-1})=Q\circ[\pi(m)]$$ for all $m\in M$. We claim that, more strongly, equation (2.5) holds. To see this, write $$dQ(p)=\delta_{P}(m)^{-1/2}\,dQ(m)\,dn.$$ Let $f_1\in \cc(U_\ell)$, $f_2\in \cc(P,V)$ and $r\in R_\t\cap w_0Mw_0^{-1}.$ Then by (2.4) we have $$\align \int\limits_{U_\ell}f_1(u)\t(u)\,du\int\limits_P\tilde\o_\t(r)f_2(p)\,dQ(p)&= \int\limits_{U_\ell\times P}f_1(u)\, \tilde\o_\t(r)f_2(p)\,dT'(u,p)\\ &= T\left(\tilde\o_\t(r)\PPP(f_1\otimes f_2)\right)\\ &= \int\limits_{U_\ell w_0P}\PPP(f_1\otimes f_2)(ruw_0p)\,dT(uw_0p). \endalign$$ But $ruw_0p=(rur^{-1})w_0(w_0^{-1}rw_0p)$, so this expression is equal to $$\align \int\limits_{U_\ell\times P}f_1(rur^{-1})f_2(w_0^{-1}rw_0p)&\,dT'(u,p)\\ &=\int\limits_{U_\ell}f_1(rur^{-1})\t(u)\,du\,\int\limits_Pf_2(w_0^{-1}rw_0p)\,dQ(p)\\ &=\int\limits_{U_\ell}f_1(u)\t(u)\,du\int\limits_P f_2(p)\,d(\e(w_0^{-1}rw_0)*Q)(p), \endalign$$ where in this last equality the defining properties of $R_\t=M_\t U_\ell$ have been used to simplify the $U_\ell$ integral. Since this holds for all $f_1\in\cc(U_\ell)$ one concludes that $$\e(w_0^{-1}rw_0)*Q=Q\circ\tilde\o_\t(r)$$ for all $r\in R_\t\cap w_0Mw_0^{-1}$, as desired. Conversely, given a distribution $Q$ satisfying equation (2.5), one reverses the above steps to arrive at a distribution $T'\in\DD(U_\ell)\otimes\DD(P,V)$ satisfying (2.6). Since the map $\PPP$ is onto, one may define a distribution $T\in\DD(U_\ell w_0P,V)$ by the formula $$T(\PPP(f_1\otimes f_2))=T'(f_1\otimes f_2)$$ provided one shows that if $\PPP(\sum_if_{1,i}\otimes f_{2,i})=0$, then $T'(\sum_if_{1,i}\otimes f_{2,i})=0$. This follows as in\Lspace \Lcitemark HeR\LIcitemark{}, Theorem 15.24\RIcitemark \Rcitemark \Rspace{}. Since $M_\t\subseteq w_0^{-1}Mw_0$ and $R_\t=U_\ell M_\t$, it follows from (2.5) and (2.7) that the $T$ so-obtained satisfies (2.4). The maps $T\mapsto Q$, $Q\mapsto T$ described above are clearly inverses. This completes the proof of the Lemma.\qed\enddemo We now complete the proof of Theorem 2.1, modulo the proof of Proposition 2.4. Let $Q$ be as in the proof of Lemma 2.6. Then by Bruhat's thesis once again, $Q$ corresponds to an element of $$\Hom_{M}(\ind_{R_\t^{w_0}\cap M}^M(\tilde \o_\t^{w_0}),\tilde\s\otimes\tilde\tau),$$ which, by duality gives an element of $\Hom_{M}(\pi,\Ind_{R_\t^{w_0}\cap M}^M(\o_\t^{w_0})).$ Since $M=GL_n(\eurm k)\times G(m),$ where $G_1$ is either $GL_n(F)$ or $GL_n(E),$ depending on whether $\G$ is orthogonal or unitary, we see that this last space is exactly the space of Whittaker models for $\s$ tensored with the space of $\o_\t^{w_0}$--Bessel models for $\tau.$ %The minimality of the $\omega_\chi^{w_0}$-Bessel %model for $\Ind_P^G(\pi)$ follows from this argument %and the proof %of Proposition 2.4 below. \qed \demo{Proof of Proposition 2.4} The remainder of the section will consist of a proof of Proposition 2.4. This is carried out in several steps. We begin by showing that, on many double cosets, the compatibility condition $\pi(p)=\o_\t(wpw^{-1})$ can not be satisfied for some $p\in P$ with $r=wpw^{-1}\in R_\t.$ By \Lcitemark Sil\LIcitemark{}, Theorem 1.9.5\RIcitemark \Rcitemark \Rspace{}, this is sufficient to imply the Proposition. Let $\S_{\PP}^+$ denote the set of positive roots in $\N.$ Let $\D$ denote the simple roots of $\TT$ in $\G$ which give rise to our choice of Borel subgroup. If $\a\in \P(\G,\TT),$ then we let $X_\a$ be the corresponding element of a Chevalley basis for the Lie algebra of $\U$ or $\bar{\U}.$ Let $\a_i$ denote the root $e_i-e_{i+1},$ and $\b=e_{\ell}+e_{\ell+1}.$ Let $X=\{\a_1,\a_2,\dots,\a_\ell,\b\}.$ Then $X$ is the set of roots where the character $\t$ is non-trivial. For $\a\in X$ we have $\t(I+tX_\a)=\p_{\a}(t).$ Also, note that $X\cap\S_{\PP}^+=\{\a_n\}.$ We list the elements of $\S_{\PP}^+,$ for future reference. If $\G=SO_{2r+1},$ then $$\align \S_{\PP}^+=&\left\{e_i\pm e_j\,|\,1\leq i\leq nn;$ \item"(b)" For all $i$ with $1\leq i\leq n,$ we have $w(i)\neq i.$ \endroster \endproclaim \demo{Proof} (a) First suppose that $w(\ell)\leq n.$ If $w(\ell +1)\leq n,$ then $w(\b)\in\S_{\PP}^+,$ contradicting our choice of $w.$ If $w(\ell+1)=\ell+1,$ then again $w(\b)\in\S_{\PP}^+.$ Finally, if $w(\ell+1)\geq n',$ then $w(\a_\ell)\in\S_{\PP}^+.$ So we must have $w(\ell)>n.$ Now suppose that for some $k,$ $n+1\leq k\leq \ell -1,$ we have $w(k)\leq n.$ If $w(k+1)=k+1,$ or $w(k+1)\geq n',$ then $w(\a_k)\in\S_{\PP}^+,$ which is a contradiction. Therefore, $w(k+1)\leq n.$ However, this implies, by induction, that $w(\ell)\leq n,$ which we have already seen is impossible. Therefore, $w(k)>n.$ (b) Suppose that $w(i)=i$ for some $i,$ $1\leq i\leq n-1.$ If $w(i+1)\neq i+1,$ then $w(i+1)>n,$ and so $w(\a_i)\in\S_{\PP}^+.$ Since this contradicts our choice of $w,$ we have $w(i+1)=i+1.$ We may thus suppose that $w$ fixes $n.$ Now by part (a), we have $w(n+1)\geq n+1,$ and therefore, $w\a_n\in\S^+_{\PP}.$ This again is a contradiction, so $w$ cannot fix $n.$ Therefore, $w$ fixes none of the integers $1,2,\dots,n.$ \qed\enddemo \proclaim{Lemma 2.9} Suppose that $w$ is as in Lemma 2.8 and assume that $w(n)\neq n'.$ Then for $n+1\leq k\leq\ell,$ we have $w(k)=k.$ \endproclaim \demo{Proof} By Lemma 2.8(a) it is enough to show that it is impossible that $w(k)\geq n'$ for any such $k.$ Suppose to the contrary that there is some $k,$ with $n+1\leq k\leq \ell,$ for which $w(k)\geq n'.$ Then there some $i\leq n,$ for which $k=w(i').$ If $w(k-1)=k-1,$ then $\a_{k-1}=w(e_{i}+e_{k-1})\in w\S_{\PP}^+\cap X.$ Since this contradicts our choice of $w,$ we must have $w(k-1)\geq n'.$ Therefore, by (downwards) induction, $w(n+1)\geq n'.$ Set $w(n+1)=i'.$ Since $w(n)\neq n,$ and, by assumption, $w(n)\neq n',$ either $w(n)=k$ or $w(n)=k'$ for some $k,$ with $n+1\leq k\leq n'-1.$ Therefore, $\a_n=w(e_{i}+e_{k})$ or $\a_n=w(e_i-e_k).$ Either one of these possibilities contradicts our assumption on $w.$ Thus $w(n+1)n.$ \endproclaim \demo{Proof} Suppose that $w^{-1}(k)=j\leq n.$ If $w(k+1)=k+1,$ then $w^{-1}\a_k=e_j-e_{k+1}\in\S_{\PP}^+.$ If $w(k+1)=i'$ for some $i\leq n,$ then $w^{-1}\a_k=e_j+e_i.$ Either case contradicts our hypotheses. Therefore $w^{-1}(k+1)\leq n.$ Now by induction, $w^{-1}(\ell-1)\leq n.$ On the other hand, by Lemma 2.11, $w(\ell)=\ell.$ Therefore $w^{-1}\a_{\ell-1}\in\S_{\PP}^+,$ contradicting our choice of $w.$ Consequently, $w^{-1}(k)>n.$\qed \enddemo \proclaim{Lemma 2.13} Suppose that $w$ is as in Lemma 2.11. Then $w(n)\neq n.$ \endproclaim \demo{Proof} Suppose that $w(n)=n.$ If $w(n+1)=n+1,$ then $w$ fixes $\a_n,$ which is in the intersection of $X$ and $\S_{\PP}^+.$ If $w^{-1}(n+1)=j'$ for some $j\leq n,$ then $w^{-1}\a_n=e_n+e_j.$ Both of these possibilities contradict our choice of $w.$ By Lemma 2.12, $w^{-1}(n+1)>n,$ and so these are the only two choices for $w(n+1).$ Since each leads to a contradiction, $w(n)\neq n.$\qed \enddemo \proclaim{Lemma 2.14} Suppose that $w$ is as in Lemma 2.11. \roster \item"(a)" For all $i\leq n$ we have $w(i)\neq i.$ \item"(b)" If $w(i_0)=i_0',$ for some $i_0\leq n$ then $w(i)=i'$ for all $i\leq i_0.$ \endroster \endproclaim \demo{Proof} (a) Suppose that $w(i)=i$ for some $i\leq n.$ Choose the maximal such $i.$ By Lemma 2.13, $i0,$ then by Lemma 2.14(b) $w=(1\, r_0)(2\, r_0-1)\dots (n_1\, n_1')w_2w',$ where $w'(i)=i$ for all $i\leq n_1,$ and $w_2$ and $w'$ are disjoint. Now $w'(i)\neq i$ and $w'(i)\neq i'$ for $n_1+1\leq i\leq n,$ and therefore $n+1\leq w'(i)\leq n'-1.$ However, by Lemma 2.16, $\ell+1\leq w'(i)\leq (\ell+1)'.$ Thus, $$w'=(n_1+1\,k_1)(k_1'\, n'_1-1)\dots (n\, k_{n_2})(k_{n_2}'\,n'),$$ as claimed. Finally, Lemma 2.11 implies $w(\ell+1)\neq\ell+1,$ and so we must have $(w')^{-1}(\ell+1)=w^{-1}(\ell+1)=j',$ for some $j\leq n.$ \qed \enddemo We now finish the proof of Proposition 2.4. If $w=w_0,$ then $n(\x)w_0=w_0n(\x),$ and since $n(\x)\in P,$ we have $R_\t w_0P=R_\t n(\x)w_0P.$ If $w\S_{\PP}^+\cap X=\{\a_{\ell}\},$ or $w\S_{\PP}^+\cap X=\emptyset,$ then Lemma 2.16 and equation (2.8) show that $w(e_n+e_\ell)=e_\ell\pm e_k,$ for some $k,$ with $\ell+2\leq k\leq r.$ Let $\a=w(e_n+e_\ell),$ and denote $I+tX_{e_n+e_\ell}$ by $p.$ As before, choose $\x_0$ so that $R_\t n(\x_0)wP=R_\t n(\x)wP,$ and such that $\x_0$ has a non-zero entry $y$ with $\o_\t(n(\x)wpw^{-1}n(\x)^{-1})=\p_\a(yt).$ Note that $\pi(p)=1.$ Choosing $t$ for which $\p_\a(yt)\neq 1,$ we see that $R_\t n(\x)wP$ cannot support a $V$\snug-distribution of the desired form. \qed \enddemo From the argument above, it is apparent that $\Ind_P^G(\pi)$ cannot have a Bessel model of rank less than $\Cal B(\tau)$. Hence we obtain the following Corollary. \proclaim{Corollary 2.18} Let the notation be as in Theorem 2.1. Suppose that the $\o_\chi$\snug-Bessel model for $\tau$ is of rank $\Cal B(\tau)$. Then the $\omega_\chi^{w_0}$\snug-Bessel model for $\Ind_P^G(\pi)$ is also minimal, and of rank $\Cal B(\tau)$. \endproclaim The proof of Theorem 2.1 also gives the following result. \proclaim{Corollary 2.19} For any $\s,\, \tau,\,\ell,\, \t,\,$ and $\o,$ the support of the twisted Jacquet functor $\pi_{U_\ell,\t}$ is a finite number of double cosets. \qed \endproclaim \subheading{\Sec 3 Holomorphicity and local coefficients} In this section we prove the holomorphicity of the Bessel functional and the existence of a local coefficient. To do so, we first adapt the argument used by Banks\Lspace \Lcitemark Ban\Rcitemark \Rspace{} to prove the holomorphicity of Whittaker functions for metaplectic covers of $GL_n.$ Banks's result is an extension of Bernstein's Theorem, which establishes the meromorphicity under uniqueness and regularity hypotheses. We show that the desired regularity holds in the case of Bessel functionals. We then use an argument similar to Harish-Chandra's and to Shahidi's in the generic case to establish the existence of the local coefficient under certain conditions (Theorem 3.8). Corollary 3.9 shows that the local coefficient factors in a manner analogous to the generic case. Let $\G$ be as in Section 1. We use the conventions found in\Lspace \Lcitemark Cas\LIcitemark{}, \Sec 1\RIcitemark \Citecomma Shaa\Rcitemark \Rspace{} for subsets of simple roots, Weyl groups, and arbitrary parabolic subgroups. Suppose that $\Delta$ is the collection of simple roots corresponding to our choice of Borel subgroup. Let $\theta\subset \Delta$ be a collection of simple roots and set $\PP=\PP_{\theta}$. Then $\PP$ has Levi decomposition $\PP=\M_{\T}\N_{\T},$ with $$\M=\M_\T\simeq GL_{n_1}\times\dots\times GL_{n_k}\times\G(m),$$ for some $n_i,\, m$ such that $r=n_1+\dots+n_k+m.$ We abbreviate this by writing $\M\simeq \G_1\times\G(m).$ We also write $\N=\N_\T.$ Let $\A=\A_\T$ be the split component of $\M.$ Denote by $\frak a_{\C}^*=\left(\frak a_{\T}\right)^*_\C$ the complexified dual of the real Lie algebra of $\A,$ $q_F$ the residual characteristic of $F,$ and denote by $H_P$ the Harish-Chandra homomorphism \Lcitemark Har\Citecomma Shaa\Rcitemark \Rspace{}. Suppose that $\s\in\Cal E(G_1)$ and $\tau\in\Cal E(G(m))$, and let $\pi=\sigma\otimes\tau$. For $\nu\in\frak a_{\C}^*,$ let $I(\nu,\pi,\theta)$ denote the induced representation $$\Ind_{P}^G(\pi\otimes q_{F}^{<\nu, H_P()>})$$ and let $V(\nu,\pi,\theta)$ denote the space of associated functions. We also use $\Pi_\nu$ to denote the representation $I(\nu,\pi,\theta).$ Assume that $\s$ is generic and that $\tau$ has an $\o_{\t'}$--Bessel model which is minimal and of rank $\ell_0.$ Let $\t$ be the character of $U_\ell$ whose restriction to $U_\ell\cap G(m)$ is $\t'$ and whose restriction to $G_1(F)\cap U_\ell$ is a $\psi$--generic character $\t_1.$ We will construct a non-zero functional $\L_\t(\nu,\pi,\theta)$ on $X_\nu=I(\nu,\pi,\theta)\otimes\tilde V_{\o}$ so that, for a certain character $\d$ of $M_\t,$ $$\L_\t(\nu,\pi,\theta)(\Pi_\nu(mu)(f_\nu\otimes\tilde v))= \d(m)\t(u)^{-1}\L_\t(\nu,\pi,\theta)(f_\nu\otimes\tilde \o(m^{-1})\tilde v),$$ for all choices of $f_\nu\otimes\tilde v\in X_\nu$ and $mu\in R_{\t}.$ Then we will show in Theorem 3.6 that the function $\nu\mapsto\L(\nu,\pi,\T)(x_\nu)$ is holomorphic, for a holomorphic section $\nu\mapsto x_\nu.$ Let $K=G(\Cal O_F),$ where $\Cal O_F$ is the ring of integers in $F.$ Then $K$ is a good maximal compact subgroup of $G$\Lspace \Lcitemark Cas\Rcitemark \Rspace{}. Let $K_m$ be the corresponding $m$\snug-th principal congruence subgroup. Then each $K_m$ is normal in $K.$ Let $\Gamma_m$ be a complete set of coset representatives for $P\cap K\backslash K/K_m.$ Note that $\Gamma_m$ is of finite cardinality. Let $$Y=\left\{f\in C^\infty(K,V_{\pi})\,\big|\, f(pk)=\left(\pi\right)(p)f(k),\,\forall p\in P\cap K,\, k\in K\right\}.$$ Then $F\mapsto F|_K$ is a $K$\snug-isomorphism from $V(\nu,\pi,\theta)$ to $Y,$ by the Iwasawa decomposition of $G.$ We will define a certain functional on $Y,$ and use this realization to define an associated functional on $X_\nu.$ Let $$Y_m=\left\{f\in Y|f(kk_1)=f(k),\forall k\in K,k_1\in K_m\right\}.$$ Thus, $Y_m$ is the set of $K_m$--fixed vectors of $Y$ under the action of $K.$ Furthermore, the Iwasawa decomposition allows us to realize $I(\nu,\pi,\theta)$ on $Y$ for each $\nu.$ Denote by $V_{\pi,m}$ the subspace of $V_{\pi}$ consisting of $P\cap K_m$--fixed vectors. Since $\pi$ is admissible, $V_{\pi,m}$ is finite dimensional. The next three results are standard. We include the proof of the first two for completeness. The third is a straightforward consequence of the Iwasawa decomposition. \proclaim{Lemma 3.1} $Y_m$ has a basis $\{f_j\}$ which satisfies the following properties: \roster \item If $\g\in\Gamma_m,$ then the non-zero vectors among $\{f_j(\g)\}$ are a basis for $V_{\pi,m}.$ \item If $f_j$ is fixed, then $f_j(\g)\neq 0$ for some $\g\in\Gamma_m.$ \endroster \endproclaim \demo{Proof} Suppose that $f\in Y_m.$ Then $f(pkk_1)=\pi(p)f(k),$ for all $p\in P\cap K,$ $k\in K,$ and $k_1\in K_m.$ Thus, $f$ is completely determined by its values on $\Gamma_m.$ Fix $\g\in\Gamma_m,$ and let $p\in P\cap K_m.$ Since $\g^{-1} K_m\g= K_m,$ we have $\g^{-1}p\g\in K_m.$ Therefore, $$f(\g)=f(\g\g^{-1}p\g)=f(p\g)=\pi(p)f(\g).$$ This says that $f(\g)$ is an element of $V_{\pi,m}.$ Fix a basis $\{v_{m,i}\}$ of $V_{\pi,m}.$ Let $f_{\g,i}:K\longrightarrow V_{\pi,m}$ be given by $$f_{\g,i}(k)=\cases \pi(p)v_{m,i}&\text{ if }k=p\g k_1,\text{ for some }p\in P\cap K,k_1\in K_m,\\0&\text{otherwise.}\endcases$$ Then it is immediate that $f_{\g,i}$ is a well-defined element of $Y_m.$ We claim that $\{f_{\g,i}\}$ is a basis for $Y_m.$ Suppose $f\in Y_m.$ If $\g'\in\Gamma_m,\, p\in P\cap K,$ and $k\in K_m,$ then $$f(p\g' k)=\pi(p)f(\g').$$ Since $f(\g')\in V_{\pi,m},$ $$f(\g')=\sum_{i}c_{\g',i}v_{m,i}.$$ This implies that $$f(p\g' k)=\sum_i c_{\g',i}\pi(p)v_{m,i}=\sum_ic_{\g',i}f_{\g',i}(p\g' k).$$ Now, taking the collection $\{c_{\g,i}\}$ for all $\g\in\Gamma_m,$ and noting that $f_{\g,i}(p\g'k)=0$ for $\g\neq\g',$ $$f(p\g' k)=\sum_{\g,i} c_{\g,i}f_{\g,i}(p\g' k),$$ which says that $\{f_{\g,i}\}$ spans $Y_m.$ On the other hand, suppose that $\sum\limits_{\g,i}c_{\g,i}f_{\g,i}=0.$ Then, for any $\g'\in \Gamma_m,$ we have $$\gather \sum_{\g,i} c_{\g,i}f_{\g,i}(\g')=0,\\ \intertext{which implies that} \sum_i c_{\g',i} f_{\g',i}(\g')=\sum_i c_{\g',i}v_{m,i}=0. \endgather$$ But, since the $v_{m,i}$ are linearly independent, $c_{\g',i}=0,$ for each $\g'$ and $i.$ Thus, $\{f_{\g,i}\}$ are also linearly independent. The collection $f_{\g,i}$ clearly has properties (1) and (2). \qed\enddemo Denote by $X$ the space $Y\otimes\tilde V_\o.$ For each $\nu\in\frak a^*_\C$ let $X_\nu=V(\nu,\pi,\theta)\otimes\tilde V_\o.$ For $f\in Y$ denote by $f_{\nu}$ the unique element of $V(\nu,\pi,\T)$ satisfying $f_{\nu}|_K=f.$ Then $\{f_{\nu}\otimes\tilde v|f\in Y,\tilde v\in \tilde V_\o\}$ spans $X_\nu.$ Recall that $\Pi_{\nu}$ can be realized on $Y$ via $\Pi_{\nu}(g)f=\left[\Pi_{\nu}(g)f_{\nu}\right]|_K.$ This gives the context in which we discuss the holomorphicity of the map $\nu\mapsto\Pi_{\nu}(g)f_{\nu}$ for a fixed choice of $g$ and $f.$ \proclaim{Lemma 3.2} Fix $g\in G,\, f\in Y$ and $\tilde v\in \tilde V_\o.$ Then the function $\nu\mapsto\Pi_{\nu}(g)f\otimes\tilde v$ is a regular function from $\frak a_\C^*$ to $X.$ \endproclaim \demo{Proof} Choose $m_0$ so that $f\in Y_{m_0},$ and choose $m>m_0$ satisfying $g^{-1}K_mg\subset K_{m_0}.$ Then $f\in Y_m$ and, for all $\nu\in\frak a^*_{\C}$ and $k\in K_m,$ $$\Pi_{\nu}(k)(\Pi_{\nu}(g)f_{\nu})(x)=f_{\nu}(xgg^{-1}kg)=f_{\nu}(xg)=\Pi_{\nu}(g)f_{\nu}(x),$$ which says that $\Pi_{\nu}(g)f\in Y_m$ for all $\nu.$ Now, by Lemma 3.1, $$\Pi_{\nu}(g)f=\sum\limits_{\g,i}c_{\g,i}(\nu) f_{\g,i}$$ for a unique choice of $c_{\g,i}(\nu)\in \C.$ It suffices to show that $c_{\g,i}:\frak a_{\C}^*\longrightarrow \C$ is holomorphic. Fix $\g'\in\Gamma_m.$ Then $\g' g=p\g'' k,$ for some $p\in P,$ $\g''\in\Gamma_m,$ and $k\in K_m.$ Then $$\align \Pi_\nu(g)f(\g')&=q_F^{<\nu,H_P(p)>}\d_P^{1/2}(p)\pi(p)f(\g'' k)\\ &=q_F^{<\nu,H_P(p)>}\d_P^{1/2}(p)\pi(p)\sum\limits_{\g,i}c_{\g,i}(\nu)f_{\g,i}(\g'')\\ &=q_F^{<\nu,H_P(p)>}\d_P^{1/2}(p)\pi(p)\sum\limits_i c_{\g'',i}(\nu)v_i. \endalign$$ Set $c'_{\g'',i}(\nu)=q_F^{<-\nu,H_p(p)>}\d_P^{-1/2}(p)c_{\g'',i}(\nu).$ Then $$\pi(p)f(\g'')=\sum\limits_i c_{\g'',i}(\nu) v_i,$$ for all $\nu.$ Since the left hand side in the equation above is independent of $\nu$ and the $v_i$ are linearly independent, $c'_{\g'',i}(\nu)$ is constant for each $i.$ This implies that $c_{\g'',i}(\nu)$ is holomorphic. \qed\enddemo From now on we need to distinguish between a Weyl group element $\tilde w\in W(\G,\A),$ for some torus $\A,$ and a representative $w\in N_{G}(A)$ for $\tilde w.$ Let $\tilde w_{\T}=\tilde w_{l,\D} \tilde w_{l,\T},$ where $\tilde w_{l,\D}$ is the longest element of the Weyl group $W(\G,\TT),$ and $\tilde w_{l,\T}$ is the longest element of $W(\G,\A_\T).$ Fix a representative $w_{\T}$ for $\tilde w_{\T}$ with $w_{\T}\in K.$ Note that $\tilde w_\T(\T)\subset \D.$ Now let $\M'=\M_{\tilde w_\T(\T)}=w_\T \M_\T w_{\T}^{-1}.$ Then $\M'$ is a standard Levi subgroup of $\G.$ Let $\N'$ be the standard unipotent subgroup of $\U$ so that $\PP'=\M'\N'$ is a standard parabolic subgroup of $\G.$ Since $\M'\simeq\M,$ we have $\U_\ell\supset\N'.$ \proclaim{Lemma 3.3} For each $m>0,$ we have $w_\T^{-1}N'\cap Pw_\T^{-1}K_m$ is compact. \endproclaim For $m\in M_\t,$ let $\d(m)=(\d_P^{-1/2}\d_{P'})(m).$ Let $X_{R_\t,\o,\nu,\theta}$ be the subspace spanned by functions of the form $$\Pi_{\nu}(mu)f\otimes \tilde v-\d(m)\t(u)f\otimes\tilde\o(m^{-1})\tilde v,$$ for $m\in M_\t,$ $u\in U_\ell,$ $f\in Y,$ and $\tilde v\in V_{\tilde\o}.$ Then a non-zero functional $\L$ on $X$ is a $(\d\o_\t)$\snug-Bessel functional for $\Pi_{\nu}$ if and only if $\L|_{X_{R_\t,\o,\nu,\theta}}\equiv 0.$ By Theorem 2.1 the space of such functionals is one-dimensional. Thus $X/X_{R_\t,\o,\nu,\theta}$ is one-dimensional. The construction of this functional will be obtained by taking a direct limit of functionals given by integrating over compact subsets of $N'.$ We show that such a limit exists and is not identically zero. Moreover, we show that there is a function in $X$ which is a complement to $X_{R_\chi,\omega,\nu,\theta}$ for all $\nu$. This will give the regularity condition necessary to apply Bernstein's Theorem and to obtain the holomorphicity of the functional. Now let us fix a Whittaker functional for $\s$ and a Bessel functional for $\o.$ (Actually, for notational convenience, we twist $\o$ by $\d_{R\t}^{-1/2}.$) That is, suppose that $\l_{\t}:V_{\pi}\otimes \tilde V_\o\longrightarrow\C$ satisfies $$\l_{\t}((\s(u_1)\otimes\tau(mu_2))(v_1\otimes v_2)\otimes\tilde v)= \t_1(u_1)\t'(u_2)\l_{\t}(v_1\otimes v_2\otimes\tilde\o(m^{-1})\tilde v)$$ for all $u_1\in U_{\ell}\cap G_1,$ $u_2\in U_{\ell}\cap G(m),$ and $m\in M_\t.$ Let $\O$ be a compact subgroup of $N'.$ Define a functional on $X$ by $$\l_{\pi,\nu,\T}^{\O}(f\otimes\tilde v)=\int\limits_{\O}\l_\t(\Pi_{\nu}(w_\T^{-1}u)f_\nu(e)\otimes\tilde v)\t(u)^{-1}\,du.\tag 3.1$$ This functional depends on the choice of the representative $w_{\T}$ for $\tilde w_{\T}.$ Since $N'$ is exhausted by compact subgroups, the compact subgroups of $N'$ form a directed set. The following Lemma was suggested to the authors by Prof.\ Steve Rallis. \proclaim{Lemma 3.4} For every $f\otimes\tilde v\in X,$ the limit $\lim\limits_{\O}\l_{\pi,\nu,\T}^{\O}(f\otimes\tilde v)$ exists, where the limit is the direct limit taken over all compact subgroups of $N'.$ \endproclaim \demo{Proof} Fix $f\otimes\tilde v\in X$. Let $\Omega_r$ be the subset of $N'$ of elements with all entries of absolute value at most $q_F^r$. It suffices to show that there is an $r$ sufficiently large such that if $\Omega\supset \Omega_r$ then $\l_{\pi,\nu,\T}^{\O}(f\otimes\tilde v)= \l_{\pi,\nu,\T}^{\O_r}(f\otimes\tilde v).$ This follows since, for $r$ sufficiently large, if $\gamma\in N'\setminus\O_r$ then $$\int\limits_{\O_r\gamma}\l_\t(\Pi_{\nu}(w_\T^{-1}u) f_\nu(e)\otimes\tilde v)\t(u)^{-1}\,du=0.$$ Indeed, there is a subgroup $K\subset\O_r$ such that $f_\nu(w_\T^{-1}ku)=f_\nu(w_\T^{-1}u)$ for $k\in K$, $u\in \O_r\gamma$, but such that $\t$ is not identically 1 on $K$. For one may choose $K$ such that the relevant minors of $w_\T^{-1}u$ and $w_\T^{-1}ku$ are highly congruent, and use the Iwasawa decomposition. \qed \enddemo Define a functional on $X$ by $$\L_\t(\nu,\pi,\T)(f\otimes\tilde v)=\lim\limits_\O \l_{\pi,\nu,\T}^{\O}(f\otimes\tilde v).\tag 3.2$$ Again, this functional depends on the choice of $w_{\T}.$ \proclaim{Proposition 3.5} Let $\L_\t(\nu,\pi,\T)$ be defined as in {\rm (3.2)}, and extend $\L_\t$ to $X_\nu$ by the section $f\otimes \tilde v\mapsto f_\nu\otimes \tilde v$. Then $\L_\t(\nu,\pi,\T)$ defines a non-zero $\d\o_\chi$--Bessel functional for $\Pi_{\nu}.$\qed \endproclaim \demo{Proof} Suppose that $u_1\in U_\ell.$ Since $U_\ell\subset P',$ we can write $u_1=m_1n_1,$ with $m_1\in M'\cap U_\ell,$ and $n_1\in N'.$ Suppose first that $u_1=n_1\in N'.$ Since $N'$ is exhausted by compact subgroups, we can choose $\O_0$ compact with $n_1\in\O_0.$ If $\O_0\subset \O,$ then $$\gather \l_{\pi,\nu,\T}^{\O}(\Pi_\nu(n_1)f\otimes\tilde v)=\int\limits_{\O} \l_\t(f_\nu(w_\T^{-1}un_1)\otimes\tilde v)\t^{-1}(u)\, du\\ =\int\limits_{\O}\l_{\t}(f_\nu(w_\T^{-1}u)\otimes\tilde v)\t^{-1}(un_1^{-1})\,du\\ =\t(n_1)\l_{\pi,\nu,\T}^{\O}(f\otimes\tilde v). \endgather$$ Therefore, $$\L_\t(\nu,\pi,\T)(\Pi_\nu(n_1)f\otimes\tilde v)=\t(n_1)\L_\t(\nu,\pi,\T)(f\otimes\tilde v).$$ If $u=m_1\in U_\ell\cap M',$ then since $\t|_{G_1\cap U_\ell}$ is $\psi$--generic, $\t^{w_\T}(m_1)=\t(m_1).$ Thus, $$\gather \l_{\pi,\nu,\T}^{\O}(\Pi_\nu(m_1)f\otimes\tilde v)=\int\limits_{\O}\l_\t(f_\nu(w_\T^{-1}um_1)\otimes\tilde v)\t^{-1}(u)\, du\\ =\int\limits_{\O}\l_\t(f_\nu(w_\T^{-1}m_1w_\T w_\T^{-1}m_1^{-1}um_1)\otimes\tilde v)\t^{-1}(u)\, du\\ =\int\limits_{\O}\l_\t(\pi(w_\T^{-1}m_1 w_\T)f_\nu(w_\T^{-1}m_1^{-1}um_1)\otimes\tilde v)\t^{-1}(u)\, du\\ =\t(m_1)\int\limits_{m_1^{-1}\O m_1}\l_{\t}(f_\nu(w_\T^{-1}u)\otimes\tilde v)\t^{-1}(u)\, du\\ =\t(m_1)\l_{\pi,\nu,\T}^{m_1^{-1}\O m_1}(f\otimes\tilde v). \endgather$$ Therefore, $$\L_\t(\nu,\pi,\T)(\Pi_\nu(m_1)f\otimes\tilde v)=\t(m_1)\L_\t(\nu,\pi,\T)(f\otimes\tilde v).$$ Similarly, if $m\in M_\t\subset M',$ then $$\gather \l_{\pi,\nu,\T}^{\O}(\Pi_\nu(m)f\otimes\tilde v)=\int\limits_{\O}\l_{\t}(f_\nu(w_\T^{-1} u m)\otimes\tilde v)\t^{-1}(u)\, du\\ =\int\limits_{\O}\l_\t(\pi(w_\T^{-1}mw_{\T})\d_{P}^{1/2}(w_\T^{-1}mw_\T) f_\nu(w_\T^{-1}m^{-1}um)\otimes\tilde v)\t^{-1}(u)\, du\\ =\d_P^{1/2}(w_\T^{-1}mw_\T)\int\limits_{m^{-1}\O m} \l_\t(f_\nu(w_\T^{-1}m^{-1}um)\otimes\tilde\o(m^{-1})\tilde v)\t^{-1}(u)\, du\\ =\d_P^{-1/2}\d_{P'}(m)\int\limits_{m^{-1}\O m}\l_{\t}(f_\nu(w_\T^{-1} u)\otimes\tilde\o(m^{-1})\tilde v)\t^{-1}(mum^{-1})\, du\\ =\d(m)\l_{\pi,\nu,\T}^{m^{-1}\O m}(f\otimes\tilde\o(m^{-1})\tilde v). \endgather$$ Taking the limit on $\O$ on the right and left sides of the above equation completes the proof that $\L_\t(\nu,\pi,\T)$ is a Bessel functional for $\Pi_\nu$ with respect to the representation $\d(m)\o_\t.$ It remains to show that $\L_\t(\nu,\pi,\T)$ is not identically zero. Let $\bar{\PP'}$ be the parabolic opposite to $\PP'$. Then $\bar{\PP'}=w_\T\PP w_\T^{-1}.$ By Lemma 3.4, $\bar P' K_m$ is compact, and if $pw_\T^{-1}k\in Pw_\T^{-1}K_m\cap N',$ then in fact $p\in P\cap K_m.$ Choose a $v\in V_{\pi}$ and $\tilde v\in \tilde V_\o$ such that $\l_\t(v\otimes\tilde v)\neq 0.$ Choose $m>>0$ such that $v\in V_{\pi,m}$ and such that $\t|{N'\cap\bar P' K_m}\equiv 1.$ Consider the function in $Y$ defined by $$f_0(k)=\cases \pi(p)v&\text{if }k=pw_\T^{-1}k_1, p\in P\cap K,k_1\in K_m\\ 0&\text{otherwise.}\endcases\tag 3.3 $$ Then $$\gather \L_\t(\nu,\pi,\T)(f_0\otimes\tilde v)=\int\limits_{N'\cap \bar P' K_m}\l_\t(f_\nu(w_\T^{-1}u)\otimes\tilde v)\, du\\ =\l_\t(v\otimes\tilde v)\, |N'\cap\bar P'K_m|\neq 0. \endgather$$ Thus, $\L_\t(\nu,\pi,\T)$ is non-zero, and $f_0$ is a complement to $X_{R_\t,\o,\nu,\T}$ for all $\nu.\qed$ \enddemo Suppose $r=mu\in R_{\t},$ $f\in Y,$ and $\tilde v\in \tilde V_{\o}.$ Define an $X$--valued function on $\frak a_{\C}^*$ by $$x_{r,f,\tilde v,\theta}(\nu)=\Pi_{\nu}(r)(f)\otimes\tilde v-\d(m)\t(u)(f\otimes\tilde\o(m^{-1})\tilde v).$$ \proclaim{Theorem 3.6} The function $\nu\mapsto \L_\t(\nu,\pi,\T)(x)$ is holomorphic for each $x\in X.$ \endproclaim \demo{Proof} We will apply Banks's extension of Bernstein's Theorem. Let $$\Cal R=\{(r,f\otimes\tilde v)|r\in R_{\t}, f\in Y,\tilde v\in V_{\tilde\o}\}\cup\{*\}.$$ For $\a=(r,f\otimes\tilde v)\in\Cal R,$ we let $x_\a(\nu)=x_{r,f,\tilde v,\T}(\nu)$ in $X$ and let $c_\a(\nu)=0.$ Fix $m,$ $\tilde v,$ $v,$ and $f_0$ as in (3.3). For $\a=*,$ we set $x_*(\nu)=f_0\otimes\tilde v$ and $c_*(\nu)=|N'\cap\bar P' K_m\,|\,\l_\t(v\otimes \tilde v).$ Now for every $\nu\in\frak a_{\C}^*,$ we consider the systems of equations in $X\times\C$ given by: $$\Xi(\nu)=\left\{(x_\a(\nu),c_\a(\nu))|\,\a\in \Cal R\right\}.$$ By Lemma 3.2, the function $\nu\mapsto x_\a(\nu)$ is holomorphic for each $\a$ of the form $(r,f\otimes\tilde v).$ For $\a=*,$ the function $x_\a(\nu)=f_0\otimes\tilde v$ is constant on $\frak a^*_\C.$ Note that each $c_\a$ is constant, hence holomorphic as well. Now, for each $\nu$ the functional $\L_\t(\nu,\pi,\T)$ is a solution to the system $\Xi(\nu).$ Moreover, such a solution is unique by the results of Section 2. Thus, Banks's extension of Bernstein's theorem\Lspace \Lcitemark Ban\Rcitemark \Rspace{} implies that $\nu\mapsto\L(\nu,\pi,\T)(f\otimes\tilde v)$ is holomorphic for all choices of $f$ and $\tilde v.$\qed\enddemo We turn to the question of local coefficients. Let $\tilde w\in W,$ and fix a representative $w$ for $\tilde w$ with $w\in K.$ We recall that the intertwining operator $$A(\nu,\pi,w):V(\nu,\pi,\T)\rightarrow V(\tilde w(\nu),\tilde w\pi,\tilde w(\T))$$ is defined for $\nu >>0$ by $$A(\nu,\pi,w)f(g)=\int\limits_{N_{\tilde w}} f(w^{-1}ng)\, dn,\tag 3.4$$ where $\N_{\tilde w}=\U\cap w\bar\N w^{-1},$ and $\bar{\N}$ is the unipotent radical opposite to $\N.$ Then $A(\nu,\pi,w)$ is defined on all of $\frak a_{\C}^*$ by analytic continuation. Note that the intertwining operator depends on the choice of $w$ representing $\tilde w.$ We also recall the Langlands decomposition of the intertwining operator, described in Lemma 2.1.2 of\Lspace \Lcitemark Shaa\Rcitemark \Rspace{}. For the convenience of the reader, let us restate this here. For two associate subsets $\theta$ and $\theta'$ of $\Delta,$ we let $$W(\theta,\theta')=\left\{\tilde w\in W\, |\, \tilde w\theta=\theta'\right\}.$$ \proclaim{Lemma 3.7 (Langlands (see\Lspace \Lcitemark Shaa\LIcitemark{}, Lemma 2.1.2\RIcitemark \Rcitemark \Rspace{}))} Suppose that $\T,\T'\subset \D$ are associate. Let $\tilde w\in W(\T,\T').$ Then there exists a family $\T_1,\T_2,\dots,\T_n\subset\D$ so that \roster \item $\T_1=\T$ and $\T_n=\T';$ \item For each $1\leq i\leq n$ there is a root $\a_i\in\D\setminus\T_i$ so that $\T_{i+1}$ is the conjugate of $\T_i$ in $\D_i=\T_i\cup\{\a_i\};$ \item For each $1\leq i\leq n-1,$ we let $\tilde w_i=\tilde w_{\ell,\D_i}\tilde w_{\ell,\T_i}$ in $W(\T_i,\T_{i+1}).$ Then $\tilde w=\tilde w_{n-1}\dots \tilde w_1;$ \item Set $\tilde w_1'=\tilde w,$ and $\tilde w'_{i+1}= \tilde w_i'\tilde w_i^{-1}$ for $1\leq i\leq n-1.$ Then $\tilde w_n'=1$ and $$\frak n_{\tilde w_i'}=\frak n_{\tilde w_i}\oplus\text{{\rm Ad}}(w_i^{-1})\frak n_{\tilde w'_{i+1}}.$$ Here $\frak n$ is the Lie algebra of $\N.$ \endroster \endproclaim Let $\T_{*}\subset\T$ and let $\rho$ be an irreducible supercuspidal representation of $M_{\T_*}$. If $\rho$ is generic, then Rodier's Theorem implies that there is a unique constituent $\pi$ of $\Ind_{P_{\T_*}}^{M}(\rho)$ which is generic with compatible character. For this constituent, Shahidi proved that there is a complex number $C_\t(\nu,\pi,\T,w)$ which satisfies $$\L_\t(\nu,\pi,\T)=C_\t(\nu,\pi,\T,w)\L_\t(\tilde w\nu,\tilde w\pi,\tilde w\T)A(\nu,\pi,w),$$ where $\L_\t$ is the Whittaker functional. Moreover, the function $\nu\mapsto C_\t(\nu,\pi,\T,w)$ is a meromorphic function on $(\frak a_\T)^*_\C.$ The value of the local coefficient depends on the choice of representative $w$ for $\tilde w.$ Now suppose that $\rho$ is any irreducible supercuspidal which has a minimal Bessel model of a particular type. Then we prove a similar result for the constituent $\pi$ of $\Ind_{P_{\T_*}}^{M}(\rho)$ which has a Bessel model of compatible type; such a constituent exists and is unique by Theorem 2.1. %Below we prove a similar result with one additional assumption, %which we now describe. %Write $\rho=\rho_1\otimes\rho_2,$ where $\rho_1$ %is an irreducible supercuspidal representation of %$$GL_{a_1}(\eurm k)\times\dots\times GL_{a_s}(\eurm k),$$ %and $\rho_2$ is an irreducible supercuspidal representation of $G(m')$ %for some $m'\leq m.$ Then $\rho_1$ is generic and $\Cal B(\rho_2)\leq \Cal B(\tau).$ %Suppose that $\Cal B(\rho_2)=\Cal B(\tau).$ %Note that $I(\nu,\pi,\T)\hookrightarrow I(\tilde\nu,\rho,\T_*),$ %when $\tilde\nu$ restricts to $\nu.$ Furthermore, Theorem 2.1 %guarantees that both of these representations have a unique %Bessel model of a given type of rank $\Cal B(\rho_2).$ Thus, the %constituent of $I(\nu,\pi,\T)$ which carries a minimal Bessel model, %is also the unique such constituent of $I(\tilde\nu,\rho,\T_*).$ %It is in this case that we can attach a local coefficient to %$\pi.$ \proclaim{Theorem 3.8} Let $\T$ and $\T'$ be associate subsets of $\D.$ Let $\T_{*}\subset\T$ and let $\rho$ be an irreducible supercuspidal representation of $M_{\T_*}$. Suppose that $\rho$ has an $\omega_\chi$--Bessel model which is minimal. Let $\pi$ be the constituent of $\Ind_{P_{\T_*}}^{M}(\rho)$ such that $\pi$ has an $\omega_\chi^{w_0}$-Bessel model, as in Theorem 2.1. For each $\tilde w\in W(\T,\T')$ fix a representative $w$ for $\tilde w.$ Then there is a complex number $C_\t(\nu,\pi,\T,w)$ so that $$\L_\t(\nu,\pi,\T)=C_\t(\nu,\pi,\T,w)\L_\t(\tilde w\nu,\tilde w\pi,\tilde w\T)A(\nu,\pi,w).\tag 3.5$$ Moreover, the function $\nu\mapsto C_\t(\nu,\pi,\T,w)$ is meromorphic on $\frak a_\C^*,$ and depends only on the class of $\pi$ and the choice of $w.$ \endproclaim \demo{Proof} We first show how to define $C_\t(\tilde\nu,\pi,\T_*,w)$ for $\tilde\nu\in (\frak a_{\T_*})^*_{\C}.$ By\Lspace \Lcitemark Sil\LIcitemark{}, Theorem 5.4.3.7\RIcitemark \Rcitemark \Rspace{} the representation $I(\tilde\nu,\rho,\T_*)$ is irreducible unless the Plancherel measure $\mu(\tilde\nu,\rho)=0$ and $(\tilde\nu,\rho)$ is fixed by a nontrivial element of the Weyl group $W_{\T_*}$ (i.e., is singular). Thus, on an open dense subset of $(\frak a_{\T_*})^*_\C$ the representation $I(\tilde\nu,\rho,\T_*)$ is irreducible, and so $\L_\t(\tilde w\tilde\nu,\tilde w\rho,\tilde w\T_*)A(\tilde\nu,\rho,w)$ defines a non-zero Bessel functional on $V(\tilde\nu,\rho,\T_*)\otimes\tilde V_\o.$ By the uniqueness of such a functional (Theorem/Conjecture 1.4), we get the existence of $C(\tilde\nu,\rho,\T_*,w)$ satisfying $$\L_\t(\tilde\nu,\rho,\T_*)=C_\t(\tilde\nu,\rho,\T_*,w)\L_\t(\tilde w\tilde\nu,\tilde w\rho,\tilde w\T_*)A(\tilde\nu,\rho,w)$$ on the open dense subset. Moreover, it is holomorphic there since both $$\L_\t(\tilde w\tilde\nu,\tilde w\rho,\tilde w\T_*)$$ and $A(\tilde\nu,\rho,w)$ are holomorphic there. Thus, $C(\tilde\nu,\rho,\T_*,w)$ extends to a meromorphic function on $(\frak a_{\T_*})^*_\C.$ Now, write $\tilde w=\tilde w_{n-1}\dots \tilde w_1$ as in Lemma 3.7. Since $C_\t(\tilde\nu,\rho,\T_*,w)$ is now defined, it admits a factorization compatible with the decomposition of the intertwining operators given in Lemma 3.7. (See corollary 3.9.) This implies that on an open dense subset of $\nu\in(\frak a_\T)^*_\C,$ the local coefficient $C_\t(\nu,\rho,\T_*,w)$ may be defined by the equation $C_\t(\nu,\rho,\T_*,w)=C_\t(\tilde\nu,\rho,\T_*,w),$ where $\tilde\nu$ is the restriction of $\nu$ to $(\frak a_{\T_*})_\C.$ Suppose that, for some $\nu$ in this open dense subset, $\L_\t(\tilde w\nu,\tilde w\pi,\tilde w\T)A(\nu,\pi,w)$ was the zero functional. Then, by inducing in stages and using the discussion preceding this Theorem, we would conclude that $\L_\t(\tilde w\tilde\nu,\tilde w\rho,\tilde w\T_*)A(\tilde\nu,\rho,w)$ is also zero. However, since $C_\t(\tilde\nu,\rho,\T_*,w)$ is defined there, this would be a contradiction. Thus we may define $C(\nu,\pi,\T,w)$ by the relation (3.5) on this open dense subset, and we have $C(\nu,\pi,\T,w)=C(\tilde\nu,\rho,\T_*,w).$ Since $A(\nu,\pi,w)$ has a meromorphic continuation to $(\frak a_\T)^*_\C,$ and $\L_\t(\tilde w\nu,\tilde w\pi,\tilde w\T)$ is holomorphic on $(\frak a_\T)^*_\C,$ the function $\nu\mapsto C_\t(\nu,\pi,\T,w)$ must have a meromorphic continuation. \qed \enddemo \proclaim{Corollary 3.9} Let the notation be as in Lemma 3.7 and Theorem 3.8. Let $\pi_1=\pi,$ and $\nu_1=\nu.$ For each $i,$ $2\leq i\leq n-1,$ set $\pi_i=\tilde w_i\pi_{i-1},\nu_i=\tilde w\nu_{i-1}.$ Then the local coefficient factors as $$C_\t(\nu,\pi,\T,w)=\prod_{i=1}^{n-1} C_\t(\pi_i,\T_i,w_i).$$ \endproclaim \demo{Proof} Let $f_1=f\in V(\nu,\pi,\T)$ and for $2\leq i\leq n-1,$ let $f_i=A(\nu_{i-1},\pi_{i-1},w_{i-1})f_{i-1}.$ Then $$\align \L_\t(\nu_i,\pi_i,\T_i)f_i=&C_\t(\nu_i,\pi_i,\T_i,w_i)\L_\t(\nu_{i+1},\pi_{i+1},\T_{i+1})\\ &\cdot A(\nu_i,\pi_i,w_i)f_i, \endalign$$ for each $1\leq i\leq n-1.$ The corollary now follows immediately from Lemma 3.7 and iteration of the above equality.\qed \enddemo \vfil\eject \Refs \message{REFERENCE LIST} \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Ban}% \def\Atest{ }\def\Astr{W. B\bgroup\Smallcapsfont ANKS\egroup{}}% \def\Ttest{ }\def\Tstr{Exceptional representations on the metaplectic group}% \def\Otest{ }\def\Ostr{preprint}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{BeZa}% \def\Atest{ }\def\Astr{I.N. B\bgroup\Smallcapsfont ERNSTEIN\egroup{}% \Aand A.V. Z\bgroup\Smallcapsfont ELEVINSKY\egroup{}}% \def\Ttest{ }\def\Tstr{Representations of the group $GL(n,F)$ where $F$ is a local non-archimedean field}% \def\Jtest{ }\def\Jstr{Russian Math. Surveys}% \def\Vtest{ }\def\Vstr{33}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1--68}% \def\Dtest{ }\def\Dstr{1976}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{BeZb}% \def\Atest{ }\def\Astr{I.N. B\bgroup\Smallcapsfont ERNSTEIN\egroup{}% \Aand A.V. Z\bgroup\Smallcapsfont ELEVINSKY\egroup{}}% \def\Ttest{ }\def\Tstr{Induced representations of reductive $p$--adic groups. I}% \def\Jtest{ }\def\Jstr{Ann. Sci. \'Ecole Norm. Sup. (4)}% \def\Vtest{ }\def\Vstr{10}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{441--472}% \def\Dtest{ }\def\Dstr{1977}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Ber}% \def\Atest{ }\def\Astr{J. B\bgroup\Smallcapsfont ERNSTEIN\egroup{}}% \def\Ttest{ }\def\Tstr{Letter to I.I. Piatetski-Shapiro, Fall 1985}% \def\Otest{ }\def\Ostr{to appear in a book by J. Cogdell and I.I. Piatetski-Shapiro}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Bru}% \def\Atest{ }\def\Astr{F. B\bgroup\Smallcapsfont RUHAT\egroup{}}% \def\Ttest{ }\def\Tstr{Sur les repr\'esentations induites des groupes de Lie}% \def\Jtest{ }\def\Jstr{Bull. Soc. Math. France}% \def\Dtest{ }\def\Dstr{1956}% \def\Vtest{ }\def\Vstr{84}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{97--205}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Cas}% \def\Atest{ }\def\Astr{W. C\bgroup\Smallcapsfont ASSELMAN\egroup{}}% \def\Ttest{ }\def\Tstr{Introduction to the theory of admissible representations of $p$--adic reductive groups}% \def\Otest{ }\def\Ostr{preprint}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{3}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{GPR}% \def\Atest{ }\def\Astr{D. G\bgroup\Smallcapsfont INZBURG\egroup{}% \Acomma I. P\bgroup\Smallcapsfont IATETSKII\egroup{}-S\bgroup\Smallcapsfont HAPIRO\egroup{}% \Aandd S. 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R\bgroup\Smallcapsfont ALLIS\egroup{}}% \def\Ttest{ }\def\Tstr{On certain Gelfand Graev models which are Gelfand pairs}% \def\Otest{ }\def\Ostr{preprint}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Roda}% \def\Atest{ }\def\Astr{F. 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Pure Math.}% \def\Itest{ }\def\Istr{AMS}% \def\Ctest{ }\def\Cstr{Providence, RI}% \def\Vtest{ }\def\Vstr{26}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{425--430}% \def\Dtest{ }\def\Dstr{1973}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Shaa}% \def\Atest{ }\def\Astr{F. S\bgroup\Smallcapsfont HAHIDI\egroup{}}% \def\Ttest{ }\def\Tstr{On certain $L$--functions}% \def\Jtest{ }\def\Jstr{Amer. J. 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J.}% \def\Vtest{ }\def\Vstr{52}% \def\Dtest{ }\def\Dstr{1985}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{973--1007}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Shac}% \def\Atest{ }\def\Astr{F. S\bgroup\Smallcapsfont HAHIDI\egroup{}}% \def\Ttest{ }\def\Tstr{On the Ramanujan conjecture and finiteness of poles for certain $L$--functions}% \def\Jtest{ }\def\Jstr{Ann. of Math. (2)}% \def\Vtest{ }\def\Vstr{127}% \def\Dtest{ }\def\Dstr{1988}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{547--584}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Shad}% \def\Atest{ }\def\Astr{F. S\bgroup\Smallcapsfont HAHIDI\egroup{}}% \def\Ttest{ }\def\Tstr{A proof of Langlands conjecture for Plancherel measures; complementary series for $p$--adic groups}% \def\Jtest{ }\def\Jstr{Ann. of Math. (2)}% \def\Dtest{ }\def\Dstr{1990}% \def\Vtest{ }\def\Vstr{132}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{273--330}% \def\Astr{\Underlinemark}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{}\def\Capssmallcapstest{ }\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{ }% \def\NoArev{0}\def\NoErev{0}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Fstr{Sil}% \def\Atest{ }\def\Astr{A.J. S\bgroup\Smallcapsfont ILBERGER\egroup{}}% \def\Ttest{ }\def\Tstr{Introduction to Harmonic Analysis on Reductive $p$--adic Groups}% \def\Itest{ }\def\Istr{Princeton University Press}% \def\Ctest{ }\def\Cstr{Princeton, NJ}% \def\Stest{ }\def\Sstr{Mathematical Notes}% \def\Ntest{ }\def\Nstr{23}% \def\Dtest{ }\def\Dstr{1979}% \Refformat\egroup% \endRefs \enddocument \end