American Institute of Mathematical Sciences

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

Constructing automorphic representations in split classical groups

David Ginzburg 1,

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.
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, 99 (2004), 163-233.  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), 1123-1140. doi: 10.1155/S1073792803212228.  Google Scholar

[5]

D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations, Israel Journal of Mathematics, 151 (2006), 323-356. 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), 447-503. 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), 1499-1518.  Google Scholar

[8]

D. Ginzburg and D. Jiang, Periods and liftings: From $G_2$ to $C_3$, Israel Journal of Math., 123 (2001), 29-59. 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), 179-211. 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), 319-358. 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 "Automorphic Representations, $L$ Functions and Applications: Progress and Prospects," de-Gruyter, (2005), 193-224.  Google Scholar

[15]

H. Jacquet, On the residual spectrum of $GL(n)$, in "Lie Group Representations," II (College Park, Md., 1982/1983), Lecture Notes in Math., 1041, Springer, Berlin, (1984), 185-208.  Google Scholar

[16]

I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math., 71 (1983), 309-338. doi: 10.1007/BF01389101.  Google Scholar

show all references

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, 99 (2004), 163-233.  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), 1123-1140. doi: 10.1155/S1073792803212228.  Google Scholar

[5]

D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations, Israel Journal of Mathematics, 151 (2006), 323-356. 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), 447-503. 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), 1499-1518.  Google Scholar

[8]

D. Ginzburg and D. Jiang, Periods and liftings: From $G_2$ to $C_3$, Israel Journal of Math., 123 (2001), 29-59. 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), 179-211. 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), 319-358. 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 "Automorphic Representations, $L$ Functions and Applications: Progress and Prospects," de-Gruyter, (2005), 193-224.  Google Scholar

[15]

H. Jacquet, On the residual spectrum of $GL(n)$, in "Lie Group Representations," II (College Park, Md., 1982/1983), Lecture Notes in Math., 1041, Springer, Berlin, (1984), 185-208.  Google Scholar

[16]

I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math., 71 (1983), 309-338. doi: 10.1007/BF01389101.  Google Scholar

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