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Transfert d’intégrales orbitales pour le groupe métaplectique

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 September 2010

Wen-Wei Li
Wen-Wei Li*
Affiliation:
Institut de Mathématiques de Jussieu – Université Paris Diderot 7, 175 rue du Chevaleret, 75013 Paris, France (email: wenweili@math.jussieu.fr)
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Abstract

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We set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

Keywords

endoscopymetaplectic grouporbital integralLanglands–Shelstad conjecturefundamental lemma

MSC classification

Secondary: 22E35: Analysis on $p$-adic Lie groups 11F72: Spectral theory; Selberg trace formula 22E30: Analysis on real and complex Lie groups 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 524 - 590
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Adams, J., Lifting of characters on orthogonal and metaplectic groups, Duke Math. J. 92 (1998), 129–178.CrossRefGoogle Scholar
[2]Adams, J., Barbasch, D., Paul, A., Trapa, P. and Vogan, D. A. Jr., Unitary Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007), 701–751.CrossRefGoogle Scholar
[3]Hales, T. C., A simple definition of transfer factors for unramified groups, in Representation theory of groups and algebras, Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), 109–134.CrossRefGoogle Scholar
[4]Howard, T., Lifting of characters on p-adic orthogonal and metaplectic groups, PhD thesis, University of Maryland (2007).Google Scholar
[5]Kottwitz, R. E., Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365–399.CrossRefGoogle Scholar
[6]Labesse, J.-P., Cohomologie, stabilisation et changement de base, Astérisque 257 (1999), with Appendix A by L. Clozel and J.-P. Labesse and Appendix B by L. Breen.Google Scholar
[7]Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 2005).Google Scholar
[8]Langlands, R. P. and Shelstad, D., On the definition of transfer factors, Math. Ann. 278 (1987), 219–271.CrossRefGoogle Scholar
[9]Langlands, R. and Shelstad, D., Descent for transfer factors, in The Grothendieck Festschrift, Volume II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, MA, 1990), 485–563.Google Scholar
[10]Lion, G. and Perrin, P., Extension des représentations de groupes unipotents p-adiques. Calculs d’obstructions, in Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Mathematics, vol. 880 (Springer, Berlin, 1981), 337–356.CrossRefGoogle Scholar
[11]Lion, G. and Vergne, M., The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6 (Birkhäuser, Boston, MA, 1980).CrossRefGoogle Scholar
[12]Maktouf, K., Le caractère de la représentation métaplectique et la formule du caractère pour certaines représentations d’un groupe de Lie presque algébrique sur un corps p-adique, J. Funct. Anal. 164 (1999), 249–339.CrossRefGoogle Scholar
[13]Mœglin, C., Vignéras, M.-F. and Waldspurger, J.-L., Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 1291 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[14]Mœglin, C. and Waldspurger, J.-L., Décomposition spectrale et séries d’Eisenstein : une paraphrase de l’écriture, Progress in Mathematics, vol. 113 (Birkhäuser, Basel, 1994).Google Scholar
[15]Ngô, B.-C., Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169.CrossRefGoogle Scholar
[16]Perrin, P., Représentations de Schrödinger, indice de Maslov et groupe metaplectique, in Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Mathematics, vol. 880 (Springer, Berlin, 1981), 370–407.CrossRefGoogle Scholar
[17]Renard, D., Transfert d’intégrales orbitales entre Mp(2n,R) et SO(n+1,n), Duke Math. J. 95 (1998), 425–450.CrossRefGoogle Scholar
[18]Renard, D., Endoscopy for Mp(2n,R), Amer. J. Math. 121 (1999), 1215–1243.CrossRefGoogle Scholar
[19]Savin, G., Local Shimura correspondence, Math. Ann. 280 (1988), 185–190.CrossRefGoogle Scholar
[20]Schultz, J., Lifting of characters of  and SO1,2(F) for F a nonarchimedean local field, PhD thesis, University of Maryland (1998).Google Scholar
[21]Shelstad, D., Tempered endoscopy for real groups. I. Geometric transfer with canonical factors, in Representation theory of real reductive Lie groups, Contemporary Mathematics, vol. 472 (American Mathematical Society, Providence, RI, 2008), 215–246.CrossRefGoogle Scholar
[22]Thomas, T., The Maslov index as a quadratic space, Math. Res. Lett. 13 (2006), 985–999.CrossRefGoogle Scholar
[23]Thomas, T., The character of the Weil representation, J. Lond. Math. Soc. (2) 77 (2008), 221–239.CrossRefGoogle Scholar
[24]Thomas, T., The Weil representation and Cayley transform, Preprint (2008),http://www.maths.ed.ac.uk/∼jthomas7/texts/Weil2.pdf.Google Scholar
[25]Waldspurger, J.-L., Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque 269 (2001).Google Scholar
[26]Waldspurger, J.-L., L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194 (2008).Google Scholar
[27]Weil, A., Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211.CrossRefGoogle Scholar