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On a Generalization of a Result of Waldspurger

Published online by Cambridge University Press:  20 November 2018

Jiandong Guo
Jiandong Guo*
Affiliation:
Department of Mathematics Stanford University Stanford, California 94305 U.S.A., e-mail: [email protected]
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Abstract

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We consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

Keywords

11F7011F6722E50
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 1 , 01 February 1996 , pp. 105 - 142
Copyright
Copyright © Canadian Mathematical Society 1997

References

[AC] Arthur, J. and Clozel, L., Simple algebras, base change, and advanced theory of the trace formula. Ann. Math. Stud. 120, Princeton Univ. Press, 1989.Google Scholar
[BF] Bump, D. and Friedberg, S., The exterior square L-fiinctions on GL(n). In: Festschrift in honor of 1.1. Piatetski-Shapiro, Part II, Israel Math. Conf. Proc. 3, 1990. 47–65.Google Scholar
[C] Carrier, P., Representations ofp-adic groups. In: Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, Rhode Island, 1979. 111–155.Google Scholar
[CI] Clozel, L., The fundamental lemma for stable base change, Duke Math. J. 61(1990), 255–302.Google Scholar
[DKV] Deligne, P., Kazhdan, D. and Vigneras, M.-F., Representations des algebres centrales simples p-adiques.. In: Representations des Groupes Reductifs sur un Corps Local, Herman, Paris, 1984. 33–117.Google Scholar
[FJ] Friedberg, S. and Jacquet, H., Linear periods, J. Reine Angew. Math. 443(1993), 91–139.Google Scholar
[G1] Guo, J., On thepositivity of the central critical values ofautomorphic L-functions for GL(2), Duke Math. J., to appear.Google Scholar
[G2] Guo, J., Uniqueness of generalized Waldspurger model for GL(2/i), (1995), preprint.Google Scholar
[J1] Jacquet, H., Sur une resultat de Waldspurger, Ann. Sci. Ecole Norm. Sup. (4) 19 (1986), 185–229.Google Scholar
[J2] Jacquet, H., On the nonvanishing of some L-functions, Proc. Indian Acad. Sci. Math. Sci. 97(1987), 117—155.Google Scholar
[JR] Jacquet, H. and Rallis, S., Uniqueness of linear periods, Compositio Math., to appear.Google Scholar
[JS] Jacquet, H. and Shalika, J., On the exterior square L-function. In: Automorphic Forms, Shimura Varieties, and L-functions, (eds. Clozel, L. and Milne, J.S.), Perspectives in Math. 11, Academic Press, 143–225.Google Scholar
[K] Kottwitz, R., Orbital integrals on GL(3), Amer. J. Math. 102(1980), 327–384.Google Scholar
[L] Labesse, J.P., Fonctions elementaires et lemme fondamental pour le changement de base stable, Duke Math. J. 61(1990), 519–530.Google Scholar
[R] Richardson, R., Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66(1982), 287–312.Google Scholar
[W1] Waldspurger, J.-L., Sur les valeurs de certains fonctions L automorphes en leur centre de symetrie, Composite Math. 54(1985), 173–242.Google Scholar
[W2] Waldspurger, J.-L., Quelques proprietes arithmetiques des forms modulaires depoids demi-entier, Compositio Math. 54(1985), 121–171.Google Scholar