# American Institute of Mathematical Sciences

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

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.
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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