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Dichotomy for generic supercuspidal representations of G2

Part of: Lie groups Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  15 February 2011

Gordan Savin  and
Martin H. Weissman
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA (email: savin@math.utah.edu)
Martin H. Weissman
Affiliation:
Department of Mathematics, University of California, Santa Cruz, CA 95064, USA (email: weissman@ucsc.edu)
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Abstract

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The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 or PGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This correspondence arises from theta correspondences in E6 and E7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of G2 to a single conjecture about the parameterization for PGSp 6.

Keywords

G2dichotomyLanglandslocal fieldp-adicShalikasupercuspidal

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11R39: Langlands-Weil conjectures, nonabelian class field theory 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 735 - 783
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Allison, B. N., A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), 133–156.CrossRefGoogle Scholar
[2]Allison, B. N., Models of isotropic simple Lie algebras, Comm. Algebra 7 (1979), 1835–1875.CrossRefGoogle Scholar
[3]Allison, B. N., Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras, Trans. Amer. Math. Soc. 306 (1988), 667–695.CrossRefGoogle Scholar
[4]Arthur, J., Unipotent automorphic representations: conjectures, Astérisque 171–172 (1989), 13–71, Orbites unipotentes et représentations, II.Google Scholar
[5]Aschbacher, M., Chevalley groups of type G 2 as the group of a trilinear form, J. Algebra 109 (1987), 193–259.CrossRefGoogle Scholar
[6]Aubert, A.-M., Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique, Trans. Amer. Math. Soc. 347 (1995), 2179–2189.Google Scholar
[7]Ban, D. and Zhang, Y., Arthur R-groups, classical R-groups, and Aubert involutions for SO(2n+1), Compositio Math. 141 (2005), 323–343.CrossRefGoogle Scholar
[8]Borel, A. and De Siebenthal, J., Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221.CrossRefGoogle Scholar
[9]Bump, D. and Ginzburg, D., Spin L-functions on symplectic groups, Int. Math. Res. Not. 1992 (1992), 153–160.CrossRefGoogle Scholar
[10]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
[11]Cogdell, J. W., Kim, H. H., Piatetski-Shapiro, I. I. and Shahidi, F., Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163–233.CrossRefGoogle Scholar
[12]Gan, W. T. and Savin, G., Real and global lifts from PGL3 to G 2, Int. Math. Res. Not. 2003 (2003), 2699–2724.CrossRefGoogle Scholar
[13]Gan, W. T. and Savin, G., Endoscopic lifts from PGL3 to G 2, Compositio Math. 140 (2004), 793–808.CrossRefGoogle Scholar
[14]Gan, W. T. and Savin, G., On minimal representations: definitions and properties, Represent. Theory 9 (2005), 46–93 (electronic).CrossRefGoogle Scholar
[15]Gelbart, S. and Jacquet, H., Forms of GL(2) from the analytic point of view, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State University, Corvallis, OR, 1977), Part 1, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 213–251.Google Scholar
[16]Gelbart, S. and Shahidi, F., Analytic properties of automorphic L-functions, Perspectives in Mathematics, vol. 6 (Academic Press, Boston, MA, 1988).Google Scholar
[17]Gel’fand, I. M. and Kajdan, D. A., Representations of the group GL(n,K) where K is a local field, in Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) (Halsted, New York, 1975), 95–118.Google Scholar
[18]Ginzburg, D. and Jiang, D., Periods and liftings: from G 2 to C 3, Israel J. Math. 123 (2001), 29–59.CrossRefGoogle Scholar
[19]Ginzburg, D., Rallis, S. and Soudry, D., A tower of theta correspondences for G 2, Duke Math. J. 88 (1997), 537–624.CrossRefGoogle Scholar
[20]Ginzburg, D., Rallis, S. and Soudry, D., On a correspondence between cuspidal representations of GL2n and , J. Amer. Math. Soc. 12 (1999), 849–907.CrossRefGoogle Scholar
[21]Ginzburg, D., Rallis, S. and Soudry, D., On explicit lifts of cusp forms from GLm to classical groups, Ann. of Math. (2) 150 (1999), 807–866.CrossRefGoogle Scholar
[22]Gross, B. H., Some applications of Gel’fand pairs to number theory, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 277–301.CrossRefGoogle Scholar
[23]Gross, B. H. and Savin, G., Motives with Galois group of type G 2: an exceptional theta-correspondence, Compositio Math. 114 (1998), 153–217.CrossRefGoogle Scholar
[24]Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton, NJ, 2001), With an appendix by Vladimir G. Berkovich.Google Scholar
[25]Henniart, G., La conjecture de Langlands locale pour GL(3), Mém. Soc. Math. France (N.S.) 11–12 (1984), 186.Google Scholar
[26]Henniart, G., La conjecture de Langlands locale pour GL(p), C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 73–76.Google Scholar
[27]Henniart, G., Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), 439–455.CrossRefGoogle Scholar
[28]Hiraga, K., On functoriality of Zelevinski involutions, Compositio Math. 140 (2004), 1625–1656.CrossRefGoogle Scholar
[29]Huang, J.-S., Magaard, K. and Savin, G., Unipotent representations of G 2 arising from the minimal representation of D E4, J. Reine Angew. Math. 500 (1998), 65–81.Google Scholar
[30]Jacobson, N., Derivation algebras and multiplication algebras of semi-simple Jordan algebras, Ann. of Math. (2) 50 (1949), 866–874.CrossRefGoogle Scholar
[31]Jacquet, H. and Rallis, S., Uniqueness of linear periods, Compositio Math. 102 (1996), 65–123.Google Scholar
[32]Jiang, D., Nien, C. and Qin, Y., Local Shalika models and functoriality, Manuscripta Math. 127 (2008), 187–217.CrossRefGoogle Scholar
[33]Kantor, I. L., Certain generalizations of Jordan algebras, Tr. Semin. Vektor. Tenzor. Anal. 16 (1972), 407–499.Google Scholar
[34]Koecher, M., Imbedding of Jordan algebras into Lie algebras. I, Amer. J. Math. 89 (1967), 787–816.CrossRefGoogle Scholar
[35]Krutelevich, S., Jordan algebras, exceptional groups, and Bhargava composition, J. Algebra 314 (2007), 924–977.CrossRefGoogle Scholar
[36]Kudla, S. S., On the local theta-correspondence, Invent. Math. 83 (1986), 229–255.CrossRefGoogle Scholar
[37]Kutzko, P. and Moy, A., On the local Langlands conjecture in prime dimension, Ann. of Math. (2) 121 (1985), 495–517.CrossRefGoogle Scholar
[38]Loke, H. Y. and Savin, G., On local lifts from to and , Israel J. Math. 159 (2007), 349–371.CrossRefGoogle Scholar
[39]Magaard, K. and Savin, G., Exceptional Θ-correspondences. I, Compositio Math. 107 (1997), 89–123.CrossRefGoogle Scholar
[40]Muić, G. and Savin, G., Symplectic-orthogonal theta lifts of generic discrete series, Duke Math. J. 101 (2000), 317–333.CrossRefGoogle Scholar
[41]Savin, G., Dual pair G 𝒥×PGL2 where G 𝒥 is the automorphism group of the Jordan algebra 𝒥, Invent. Math. 118 (1994), 141–160.CrossRefGoogle Scholar
[42]Savin, G., A class of supercuspidal representations of G 2(k), Canad. Math. Bull. 42 (1999), 393–400.CrossRefGoogle Scholar
[43]Savin, G. and Gan, W. T., The dual pair G 2×PU3(D) (p-adic case), Canad. J. Math. 51 (1999), 130–146.CrossRefGoogle Scholar
[44]Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), 273–330.CrossRefGoogle Scholar
[45]Vo, S. C., The spin L-function on the symplectic group GSp(6), Israel J. Math. 101 (1997), 1–71.CrossRefGoogle Scholar
[46]Vogan, D. A. Jr., The local Langlands conjecture, in Representation theory of groups and algebras, Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), 305–379.CrossRefGoogle Scholar
[47]Zhang, Y., L-packets and reducibilities, J. Reine Angew. Math. 510 (1999), 83–102.CrossRefGoogle Scholar