Constructing automorphic representations in split classical groups

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January 2012, 19: 18-32. doi: 10.3934/era.2012.19.18

Constructing automorphic representations in split classical groups

1.	School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, 69978, Israel

Received June 2011 Revised December 2011 Published February 2012

Abstract
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In this paper we introduce a general construction for a correspondence between certain Automorphic representations in classical groups. This construction is based on the method of small representations, which we use to construct examples of CAP representations.

Keywords: small representations., Functorial lifting.

Mathematics Subject Classification: Primary: 11F7.

Citation: David Ginzburg. Constructing automorphic representations in split classical groups. Electronic Research Announcements, 2012, 19: 18-32. doi: 10.3934/era.2012.19.18

References:

[1]	R. Carter, "Finite Groups of Lie Type,", J. Wiley & Sons, (1985). Google Scholar
[2]	J. Cogdell, H. Kim, I. Piatetski-Shapiro and F. Shahidi, Functoriality for the classical groups,, \textbf{99} (2004), 99 (2004), 163. Google Scholar
[3]	D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras,", Van Nostrand Reinhold, (1991). Google Scholar
[4]	D. Ginzburg, "A construction of CAP representations for classical groups,", International Math. Research Notices, 20 (2003), 1123. doi: 10.1155/S1073792803212228. Google Scholar
[5]	D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations,, Israel Journal of Mathematics, 151 (2006), 323. doi: 10.1007/BF02777366. Google Scholar
[6]	D. Ginzburg, Endoscopic lifting in classical groups and poles of tensor $L$ functions,, Duke Math. Journal, 141 (2008), 447. doi: 10.1215/00127094-2007-002. Google Scholar
[7]	D. Ginzburg, On the lifting from $PGL_2\times PGL_2$ to $G_2$,, International Math. Research Notices, 25 (2005), 1499. Google Scholar
[8]	D. Ginzburg and D. Jiang, Periods and liftings: From $G_2$ to $C_3$,, Israel Journal of Math., 123 (2001), 29. doi: 10.1007/BF02784119. Google Scholar
[9]	D. Ginzburg and D. Jiang, Some conjectures on endoscopic representations in odd orthogonal groups,, Nagoya Mathematical Journal, (). Google Scholar
[10]	D. Ginzburg, D. Jiang and D. Soudry, On CAP representations for even orthogonal groups I: A correspondence of unramified representations,, preprint., (). Google Scholar
[11]	D. Ginzburg, D. Jiang and S. Rallis, On CAP automorphic representations of a split group of type $D_4$,, J. Reine Angew. Math., 552 (2002), 179. doi: 10.1515/crll.2002.090. Google Scholar
[12]	D. Ginzburg, D. Jiang and S. Rallis, Periods of residual representations of $SO(2l)$,, Manuscripta Mathematica, 113 (2004), 319. doi: 10.1007/s00229-003-0417-x. Google Scholar
[13]	D. Ginzburg, S. Rallis and D. Soudry, "The Descent Map from Automorphic Representations of $GL(n)$ to Classical Groups,", World Scientific, (2011). doi: 10.1142/9789814304993. Google Scholar
[14]	D. Ginzburg, S. Rallis and D. Soudry, Construction of CAP representations for symplectic groups using the descent method,, in, (2005), 193. Google Scholar
[15]	H. Jacquet, On the residual spectrum of $GL(n)$,, in, 1041 (1984), 185. Google Scholar
[16]	I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting,, Invent. Math., 71 (1983), 309. doi: 10.1007/BF01389101. Google Scholar

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References:

[1]	R. Carter, "Finite Groups of Lie Type,", J. Wiley & Sons, (1985). Google Scholar
[2]	J. Cogdell, H. Kim, I. Piatetski-Shapiro and F. Shahidi, Functoriality for the classical groups,, \textbf{99} (2004), 99 (2004), 163. Google Scholar
[3]	D. Collingwood and W. McGovern, "Nilpotent Orbits in Semisimple Lie Algebras,", Van Nostrand Reinhold, (1991). Google Scholar
[4]	D. Ginzburg, "A construction of CAP representations for classical groups,", International Math. Research Notices, 20 (2003), 1123. doi: 10.1155/S1073792803212228. Google Scholar
[5]	D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations,, Israel Journal of Mathematics, 151 (2006), 323. doi: 10.1007/BF02777366. Google Scholar
[6]	D. Ginzburg, Endoscopic lifting in classical groups and poles of tensor $L$ functions,, Duke Math. Journal, 141 (2008), 447. doi: 10.1215/00127094-2007-002. Google Scholar
[7]	D. Ginzburg, On the lifting from $PGL_2\times PGL_2$ to $G_2$,, International Math. Research Notices, 25 (2005), 1499. Google Scholar
[8]	D. Ginzburg and D. Jiang, Periods and liftings: From $G_2$ to $C_3$,, Israel Journal of Math., 123 (2001), 29. doi: 10.1007/BF02784119. Google Scholar
[9]	D. Ginzburg and D. Jiang, Some conjectures on endoscopic representations in odd orthogonal groups,, Nagoya Mathematical Journal, (). Google Scholar
[10]	D. Ginzburg, D. Jiang and D. Soudry, On CAP representations for even orthogonal groups I: A correspondence of unramified representations,, preprint., (). Google Scholar
[11]	D. Ginzburg, D. Jiang and S. Rallis, On CAP automorphic representations of a split group of type $D_4$,, J. Reine Angew. Math., 552 (2002), 179. doi: 10.1515/crll.2002.090. Google Scholar
[12]	D. Ginzburg, D. Jiang and S. Rallis, Periods of residual representations of $SO(2l)$,, Manuscripta Mathematica, 113 (2004), 319. doi: 10.1007/s00229-003-0417-x. Google Scholar
[13]	D. Ginzburg, S. Rallis and D. Soudry, "The Descent Map from Automorphic Representations of $GL(n)$ to Classical Groups,", World Scientific, (2011). doi: 10.1142/9789814304993. Google Scholar
[14]	D. Ginzburg, S. Rallis and D. Soudry, Construction of CAP representations for symplectic groups using the descent method,, in, (2005), 193. Google Scholar
[15]	H. Jacquet, On the residual spectrum of $GL(n)$,, in, 1041 (1984), 185. Google Scholar
[16]	I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting,, Invent. Math., 71 (1983), 309. doi: 10.1007/BF01389101. Google Scholar

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