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The Rankin–Selberg integral with a non-unique model for the standard $\mathcal{L}$-function of $G_2$

Part of: Lie groups

Published online by Cambridge University Press:  08 May 2014

Nadya Gurevich
Affiliation:
School of Mathematics, Ben Gurion University of the Negev, POB 653, Be’er Sheva 84105, Israel([email protected])([email protected])
Avner Segal
Affiliation:
School of Mathematics, Ben Gurion University of the Negev, POB 653, Be’er Sheva 84105, Israel([email protected])([email protected])

Abstract

Let $\mathcal{L}^{S}\left (s,\pi ,{\mathfrak{st}}\right )$ be a partial $\mathcal{L}$-function of degree $7$ of a cuspidal automorphic representation $\pi $ of the exceptional group $G_2$. In this paper we construct a Rankin–Selberg integral for representations having a certain Fourier coefficient.

Type
Research Article
Copyright
© Cambridge University Press 2014 

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References

Andrianov, A. N., Multiplicative arithmetic of Siegel’s modular forms, Uspekhi Mat. Nauk 34(1(205)) (1979), 67135.Google Scholar
Bump, D., The Rankin–Selberg method: an introduction and survey, in Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., Volume 11, pp. 4173 (de Gruyter, Berlin, 2005).Google Scholar
Bump, D., Furusawa, M. and Ginzburg, D., Non-unique models in the Rankin–Selberg method, J. Reine Angew. Math. 468 (1995), 77111.Google Scholar
Casselman, W., The unramified principal series of p-adic groups. I. The spherical function, Compos. Math. 40(3) (1980), 387406.Google Scholar
Dixmier, J. and Malliavin, P., Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2) 102(4) (1978), 307330.Google Scholar
Gan, W. T., Multiplicity formula for cubic unipotent Arthur packets, Duke Math. J. 130(2) (2005), 297320.CrossRefGoogle Scholar
Gan, W. T., Gross, B. and Savin, G., Fourier coefficients of modular forms on G 2 , Duke Math. J. 115(1) (2002), 105169.Google Scholar
Gan, W. T., Gurevich, N. and Jiang, D., Cubic unipotent Arthur parameters and multiplicities of square integrable automorphic forms, Invent. Math. 149(2) (2002), 225265.Google Scholar
Ginzburg, D., On the standard L-function for G 2 , Duke Math. J. 69(2) (1993), 315333.Google Scholar
Ginzburg, D. and Hundley, J., A doubling integral for $G_2$ , preprint.Google Scholar
Gross, B. H., On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., Volume 254, pp. 223237 (Cambridge Univ. Press, Cambridge, 1998).Google Scholar
Huang, J.-S., Magaard, K. and Savin, G., Unipotent representations of G 2arising from the minimal representation of D 4 E , J. Reine Angew. Math. 500 (1998), 6581.Google Scholar
Jiang, D., G 2-periods and residual representations, J. Reine Angew. Math. 497 (1998), 1746.Google Scholar
Piatetski-Shapiro, I. and Rallis, S., A new way to get Euler products, J. Reine Angew. Math. 392 (1988), 110124.Google Scholar
Rallis, S. and Schiffmann, G., Theta correspondence associated to G 2 , Amer. J. Math. 111(5) (1989), 801849.Google Scholar