American Institute of Mathematical Sciences

Advanced
  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,, \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

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,, \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

[1]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[2]

Ana-Maria Acu, Madalina Dancs, Voichiţa Adriana Radu. Representations for the inverses of certain operators. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4097-4109. doi: 10.3934/cpaa.2020182
[3]

Daniel Gerth, Andreas Hofinger, Ronny Ramlau. On the lifting of deterministic convergence rates for inverse problems with stochastic noise. Inverse Problems & Imaging, 2017, 11 (4) : 663-687. doi: 10.3934/ipi.2017031
[4]

Yves Frederix, Giovanni Samaey, Christophe Vandekerckhove, Ting Li, Erik Nies, Dirk Roose. Lifting in equation-free methods for molecular dynamics simulations of dense fluids. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 855-874. doi: 10.3934/dcdsb.2009.11.855
[5]

David Kazhdan and Yakov Varshavsky. Endoscopic decomposition of characters of certain cuspidal representations. Electronic Research Announcements, 2004, 10: 11-20.

[6]

Artem Dudko, Rostislav Grigorchuk. On spectra of Koopman, groupoid and quasi-regular representations. Journal of Modern Dynamics, 2017, 11: 99-123. doi: 10.3934/jmd.2017005
[7]

Peter Vandendriessche. LDPC codes associated with linear representations of geometries. Advances in Mathematics of Communications, 2010, 4 (3) : 405-417. doi: 10.3934/amc.2010.4.405
[8]

Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795
[9]

Uri Bader, Roman Muchnik. Boundary unitary representations-irreducibility and rigidity. Journal of Modern Dynamics, 2011, 5 (1) : 49-69. doi: 10.3934/jmd.2011.5.49
[10]

Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229
[11]

Diego Rapoport. Random representations of viscous fluids and the passive magnetic fields transported on them. Conference Publications, 2001, 2001 (Special) : 327-336. doi: 10.3934/proc.2001.2001.327
[12]

Constantin N. Beli. Representations of integral quadratic forms over dyadic local fields. Electronic Research Announcements, 2006, 12: 100-112.

[13]

Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345
[14]

Kanghui Guo, Demetrio Labate. Optimally sparse 3D approximations using shearlet representations. Electronic Research Announcements, 2010, 17: 125-137. doi: 10.3934/era.2010.17.125
[15]

Catarina Carvalho, Victor Nistor, Yu Qiao. Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras. Electronic Research Announcements, 2017, 24: 68-77. doi: 10.3934/era.2017.24.008
[16]

Uri Bader, Jan Dymara. Boundary unitary representations—right-angled hyperbolic buildings. Journal of Modern Dynamics, 2016, 10: 413-437. doi: 10.3934/jmd.2016.10.413
[17]

Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339
[18]

Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid. Networks & Heterogeneous Media, 2010, 5 (3) : 385-404. doi: 10.3934/nhm.2010.5.385
[19]

Lasse Kiviluoto, Patric R. J. Östergård, Vesa P. Vaskelainen. Sperner capacity of small digraphs. Advances in Mathematics of Communications, 2009, 3 (2) : 125-133. doi: 10.3934/amc.2009.3.125
[20]

Frank Fiedler. Small Golay sequences. Advances in Mathematics of Communications, 2013, 7 (4) : 379-407. doi: 10.3934/amc.2013.7.379

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences