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On certain Rankin-Selberg Integrals on GE6

Published online by Cambridge University Press:  11 January 2016

David Ginzburg  and
Joseph Hundley
David Ginzburg
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Israel, ginzburg@post.tau.ac.il
Joseph Hundley
Affiliation:
Mathematics Department, Mailcode 4408, Southern Illinois University Carbondale, 1245 Lincoln Drive, Carbondale, IL 62901, U.S.A., jhundley@math.siu.edu
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Abstract

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In this paper we begin the study of two Rankin-Selberg integrals defined on the exceptional group of type GE6. We show that each factorizes and that the contribution from the unramified places is, in one case, the degree 54 Euler product LS(π × τ, E6 × GL2,s) and in the other case the degree 30 Euler product LS(π × τ, ∧2 × GL2,s).

Keywords

11F6611F70
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 191 , 2008 , pp. 21 - 78
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[B] Brion, M., Invariants d’un sous-groupe unipotent maximal d’un groupe semi-simple, Ann. Inst. Fourier (Grenoble), 33 (1983), no. 1, 1–27.CrossRefGoogle Scholar
[B-F-G] Bump, D., Friedberg, S., and Ginzburg, D., Rankin-Selberg Integrals in two complex variables, Math. Ann., 313 (1999), 731–761.CrossRefGoogle Scholar
[C-S] Casselman, W. and Shalika, J., The unramified principal series of p-adic groups II: The Whittaker function, Compositio Math., 41 (1980), 207–231.Google Scholar
[F-H] Fulton, W. and Harris, J., Representation Theory: A first course, Springer GTM 129, New York, 1991.Google Scholar
[Ge-J] Gelbart, S. and Jacquet, H., A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. (4), 11 (1978), no. 4, 471–542.CrossRefGoogle Scholar
[Gk-Se] Gilkey, P. and Seitz, G., Some representations of exceptional Lie algebras, Geom. Dedicata, 25 (1988), 407–416.CrossRefGoogle Scholar
[G] Ginzburg, D., On standard L-functions for E6 and E7 , J. Reine Angew. Math., 465 (1995), 101–131.Google Scholar
[G-J] Ginzburg, D. and Jiang, D., Periods and Liftings: From G2 to C3 , Israel Journal of Math., 123 (2001), 29–59.CrossRefGoogle Scholar
[G-H] Ginzburg, D. and Hundley, J., Multi-variable Rankin-Selberg Integrals for Orthogonal Groups, Int. Math. Res. Not. 2004, no. 58, 3097–3119.CrossRefGoogle Scholar
[G-H2] Ginzburg, D. and Hundley, J., On Spin L-Functions for GSO10 , J. Reine Angew. Math., 603 (2007), 183–213.Google Scholar
[G-H3] Ginzburg, D. and Hundley, J., A New Tower of Rankin-Selberg Integrals, Electronic Research Announcements of the AMS, 12 (2006), 56–62.CrossRefGoogle Scholar
[G-R-S] Ginzburg, D., Rallis, S., and Soudry, D., On the automorphic theta representation for simply laced groups, Israel Journal of Mathematics, 100 (1997), 61–116.CrossRefGoogle Scholar
[G-R-S1] Ginzburg, D., Rallis, S., and Soudry, D., On Fourier Coefficients of Automorphic Forms of Symplectic Groups, Manuscripta Mathematica, 111 (2003), 1–16.CrossRefGoogle Scholar
[G-R-S2] Ginzburg, D., Rallis, S., and Soudry, D., Construction of CAP Representations for Symplectic Groups Using the Descent Method, Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005, pp. 193–224.Google Scholar
[Gu] Gurevich, N., A theta lift for Spin7 , Compositio Math., 136 (2003), no. 1, 25–59.CrossRefGoogle Scholar
[J-S] Jacquet, H. and Shalika, J., Exterior square L-functions, Automorphic forms, Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math., 11, Academic Press, Boston, MA, 1990, pp. 143–226.Google Scholar
[K] Kac, V. G., Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190–213.CrossRefGoogle Scholar
[L] Littelmann, P., On spherical double cones, J. Algebra, 166 (1994), no. 1, 142–157.CrossRefGoogle Scholar
[P1] Panyushev, D. I., A restriction theorem and the Poincaré series for U-invariants, Math. Ann., 301 (1995), no. 4, 655–675.CrossRefGoogle Scholar
[P2] Panyushev, D. I., Complexity and rank of double cones and tensor product decompositions, Comment. Math. Helv., 68 (1993), no. 3, 455–468.CrossRefGoogle Scholar
[P3] Panyushev, D. I., Orbits of highest dimension of solvable subgroups of reductive linear groups and reduction for U-invariants, (Russian) Mat. Sb. (N.S.), 132 (174) (1987), no. 3, 371–382, 445; translation in Math. USSR-Sb., 60 (1988), no. 2, 365–375.Google Scholar
[S] Shahidi, F., On certain L-functions, Amer. J. Math., 103 (1981), no. 2, 297–355.Google Scholar