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Une formule intégrale reliée à la conjecture locale de Gross–Prasad

Part of: Lie groups

Published online by Cambridge University Press:  04 May 2010

J.-L. Waldspurger
J.-L. Waldspurger*
Affiliation:
Institut de mathématiques de Jussieu - CNRS, 175 rue du Chevaleret, 75013 Paris, France (email: [email protected])
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Abstract

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Let V be a vector space over a p-adic field F, of finite dimension, let q be a non-degenerate quadratic form over V and let D be a non-isotropic line in V. Denote by W the hyperplane orthogonal to D, and by G and H the special orthogonal groups of V and W. Let π, respectively σ, be an irreducible admissible representation of G(F) , respectively H(F) . The representation σ appears as a quotient of the restriction of π to H(F)with a certain multiplicity m(π,σ) . We know that m(π,σ)≤1 . We assume that π is supercuspidal. Then we prove a formula that computes m(π,σ)as an integral of functions deduced from the characters of π and σ. Let Π, respectively Σ, be an L-packet of tempered irreducible representations of G(F) , respectively H(F) . Here we use the sophisticated notion of L-packet due to Vogan and we assume some usual conjectural properties of those packets. A weak form of the local Gross–Prasad conjecture says that there exists a unique pair (π,σ)∈Π×Σ such that m(π,σ)=1 . Assuming that the elements of Π are supercuspidal, we prove this assertion.

Keywords

représentations des groupes spéciaux orthogonauxconjecture locale de Gross–Prasad

MSC classification

Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 5 , September 2010 , pp. 1180 - 1290
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Aizenbud, A., Gourevitch, D., Rallis, S. and Schiffmann, G., Multiplicity one theorems, Preprint (2007), arxiv RT 0709.4215.Google Scholar
[2]Arthur, J., The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74.Google Scholar
[3]Arthur, J., The characters of supercuspidal representations as weighted orbital integrals, Proc. Indian Acad. Sci. 97 (1987), 3–19.CrossRefGoogle Scholar
[4]Arthur, J., The invariant trace formula I. Local theory, J. Amer. Math. Soc. 1 (1988), 323–383.CrossRefGoogle Scholar
[5]Arthur, J., The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223–293.CrossRefGoogle Scholar
[6]Arthur, J., A local trace formula, Publ. Math. Inst. Hautes Études Sci. 73 (1991), 5–96.Google Scholar
[7]Arthur, J., On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99 (1999), 209–283.Google Scholar
[8]Gan, W. T., Gross, B. and Prasad, D., Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups, Preprint (2008).Google Scholar
[9]Gross, B. and Prasad, D., On irreducible representations of SO 2n+1×SO 2m, Canad. J. Math. 46 (1994), 930–950.CrossRefGoogle Scholar
[10]Harish-Chandra, , Harmonic analysis on reductive p-adic groups, notes par G. van Dijk, Lecture Notes in Mathematics, vol. 162 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[11]Harish-Chandra, , Admissible invariant distributions on reductive p-adic groups, in notes par S. DeBacker et P. Sally, University Lecture Series, vol. 16 (American Mathematical Society, Providence, RI, 1999).Google Scholar
[12]Konno, T., Twisted endoscopy implies the generic packet conjecture, Israël J. Math. 129 (2002), 253–289.CrossRefGoogle Scholar
[13]Kottwitz, R., Transfer factors for Lie algebras, Represent. Theory 3 (1999), 127–138.CrossRefGoogle Scholar
[14]Ngo, B. C., Le lemme fondamental pour les algèbres de Lie, Preprint (2008), arxiv:0801.0446.Google Scholar
[15]Rodier, F., Modèle de Whittaker et caractères de représentations, in Non commutative harmonic analysis, Lecture Notes in Mathematics, vol. 466, eds Carmona, J., Dixmier, J. and Vergne, M. (Springer, Berlin, 1981), 151–171.Google Scholar
[16]Shelstad, D., A formula for regular unipotent germs, in Orbites unipotentes et représentations II. Groupes p-adiques et réels, Astérisque 171–172 (1989), 275–277.Google Scholar
[17]Waldspurger, J.-L., Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, p⁄=2, in Festschrift in honor of I.I. Piatetski–Shapiro, eds Gelbart, S., Howe, R. and Sarnak, P. (The Weizmann Science Press, Israël, 1990).Google Scholar
[18]Waldspurger, J.-L., Une formule des traces locale pour les algèbres de Lie p-adiques, J. Reine Angew. Math. 465 (1995), 41–99.Google Scholar
[19]Waldspurger, J.-L., Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), 153–236.CrossRefGoogle Scholar
[20]Waldspurger, J.-L., Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque 269 (2001).Google Scholar
[21]Waldspurger, J.-L., Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann, Preprint (2009), arxiv:0911.1618.Google Scholar