La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale

Li, Wen-Wei

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 5, pp. 787-859.

On établit des résultats de l'analyse harmonique locale nécessaires pour la formule des traces invariante d'Arthur pour les revêtements de groupes réductifs connexes. Plus précisément, on démontre pour les revêtements locaux (1) la formule de Plancherel et des préparatifs reliés, (2) la normalisation des opérateurs d'entrelacement soumise aux conditions d'Arthur, (3) le comportement local de caractères de représentations admissibles dans le cas non archimédien, et (4) la partie spécifique de la formule des traces locale invariante. Comme un sous-produit de la démonstration de la formule des traces locale invariante, on obtient aussi la densité de caractères tempérés pour les revêtements.

We establish some results in local harmonic analysis which are necessary for Arthur's invariant trace formula for coverings of connected reductive groups. More precisely, for local coverings we will study (1) the Plancherel formula and its preparations, (2) the normalization of intertwining operators subject to Arthur's conditions, (3) the local behavior of characters of admissible representations in the nonarchimedean case, and (4) the genuine part of the invariant local trace formula. As a byproduct of the invariant local trace formula, we deduce the density of tempered characters for coverings.

[1] J. Adams, D. Barbasch, A. Paul, P. Trapa & D. A. J. Vogan, Unitary Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007), 701-751. | MR | Zbl

[2] J. D. Adler & S. Debacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive $p$-adic group, Michigan Math. J. 50 (2002), 263-286. | MR | Zbl

[3] J. Arthur, A theorem on the Schwartz space of a reductive Lie group, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 4718-4719. | MR | Zbl

[4] J. Arthur, The trace formula in invariant form, Ann. of Math. 114 (1981), 1-74. | MR | Zbl

[5] J. Arthur, The invariant trace formula. I. Local theory, J. Amer. Math. Soc. 1 (1988), 323-383. | MR | Zbl

[6] J. Arthur, The invariant trace formula. II. Global theory, J. Amer. Math. Soc. 1 (1988), 501-554. | MR | Zbl

[7] J. Arthur, Harmonic analysis of tempered distributions on semisimple Lie groups of real rank one, in Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr. 31, Amer. Math. Soc., 1989, Dissertation, Yale University, New Haven, CT, 1970, 13-100. | MR | Zbl

[8] J. Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), 19-84. | MR | Zbl

[9] J. Arthur, A local trace formula, Publ. Math. I.H.É.S. 73 (1991), 5-96. | Numdam | MR | Zbl

[10] J. Arthur, On elliptic tempered characters, Acta Math. 171 (1993), 73-138. | MR | Zbl

[11] J. Arthur, On the Fourier transforms of weighted orbital integrals, J. reine angew. Math. 452 (1994), 163-217. | MR | Zbl

[12] J. Arthur, The trace Paley-Wiener theorem for Schwartz functions, in Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993), Contemp. Math. 177, Amer. Math. Soc., 1994, 171-180. | MR | Zbl

[13] J. Arthur, Canonical normalization of weighted characters and a transfer conjecture, C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), 33-52. | MR | Zbl

[14] J. Arthur, A stable trace formula. III. Proof of the main theorems, Ann. of Math. 158 (2003), 769-873. | MR | Zbl

[15] I. N. Bernšteĭn & A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), 5-70. | MR | Zbl

[16] I. N. Bernstein & A. V. Zelevinsky, Induced representations of reductive $𝔭$-adic groups. I, Ann. Sci. École Norm. Sup. 10 (1977), 441-472. | Numdam | MR | Zbl

[17] A. Bouaziz, Intégrales orbitales sur les algèbres de Lie réductives, Invent. Math. 115 (1994), 163-207. | MR | Zbl

[18] A. Bouaziz, Intégrales orbitales sur les groupes de Lie réductifs, Ann. Sci. École Norm. Sup. 27 (1994), 573-609. | Numdam | MR | Zbl

[19] J.-L. Brylinski & P. Deligne, Central extensions of reductive groups by ${𝐊}_2$, Publ. Math. I.H.É.S. 94 (2001), 5-85. | Numdam | MR | Zbl

[20] L. Clozel, Characters of nonconnected, reductive $p$-adic groups, Canad. J. Math. 39 (1987), 149-167. | MR | Zbl

[21] L. Clozel, J.-P. Labesse & R. Langlands, Morning seminar on the trace formula, Institute for Advanced Study, 1984, lecture notes.

[22] M. Hanzer & G. Muić, Parabolic induction and Jacquet functors for metaplectic groups, J. Algebra 323 (2010), 241-260. | MR | Zbl

[23] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1-111. | MR | Zbl

[24] Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. 104 (1976), 117-201. | MR | Zbl

[25] Harish-Chandra 1999. | MR | Zbl

[26] D. Kazhdan, Cuspidal geometry of $p$-adic groups, J. Analyse Math. 47 (1986), 1-36. | MR | Zbl

[27] D. A. Kazhdan & S. J. Patterson, Metaplectic forms, Publ. Math. I.H.É.S. 59 (1984), 35-142. | Numdam | MR | Zbl

[28] A. W. Knapp & D. A. J. Vogan, Cohomological induction and unitary representations, Princeton Mathematical Series 45, Princeton Univ. Press, 1995. | MR | Zbl

[29] A. W. Knapp & G. Zuckerman, Classification theorems for representations of semisimple Lie groups, in Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), 587, Springer, 1977, 138-159. | MR | Zbl

[30] T. Konno, A note on the Langlands classification and irreducibility of induced representations of $p$-adic groups, Kyushu J. Math. 57 (2003), 383-409. | MR | Zbl

[31] W.-W. Li, La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin, preprint arXiv:1004.4011, 2010.

[32] P. J. Mcnamara, Metaplectic Whittaker functions and crystal bases, Duke Math. J. 156 (2011), 1-31. | MR | Zbl

[33] P. Mezo, Comparisons of general linear groups and their metaplectic coverings. II, Represent. Theory 5 (2001), 524-580. | MR | Zbl

[34] A. J. Silberger, The Knapp-Stein dimension theorem for $p$-adic groups, Proc. Amer. Math. Soc. 68 (1978), 243-246. | MR | Zbl

[35] A. J. Silberger, CorrectionProc. Amer. Math. Soc. 76 (1979), 169-170. | MR | Zbl

[36] J.-L. Waldspurger, La formule de Plancherel pour les groupes $p$-adiques (d'après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), 235-333. | MR | Zbl

[37] N. R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press Inc., 1988. | MR | Zbl

[38] N. R. Wallach, Real reductive groups. II, Pure and Applied Mathematics 132-II, Academic Press Inc., 1992. | MR | Zbl

[39] M. H. Weissman, Metaplectic tori over local fields, Pacific J. Math. 241 (2009), 169-200. | MR | Zbl