Les formules limites qui relient la mesure canonique sur une orbite coadjointe nilpotente aux mesures canoniques sur les orbites semi-simples régulières jouent un rôle important dans les études des distributions invariantes sur les groupes de Lie réels réductifs. Le but de cet article est d’étudier un type particulier de la formule limite proposée par Rossmann. En utilisant les résultats de Schmid et Vilonen concernant les faisceaux équivariants sur la variété de drapeaux d’une algèbre de Lie réductifs, nous calculons les mesures invariantes associées aux orbites nilpotentes pour les groupes de Lie semi-simples ayant l’unique classe de conjugaison de sous-groupes de Cartan.
Limit formulas for the computation of the canonical measure on a nilpotent coadjoint orbit in terms of the canonical measures on regular semisimple coadjoint orbits arise naturally in the study of invariant eigendistributions on a reductive Lie algebra. In the present paper we consider a particular type of the limit formula for canonical measures which was proposed by Rossmann. The main technical tool in our analysis are the results of Schmid and Vilonen on the equivariant sheaves on the flag variety and their characteristic cycles. We combine the theory of Schmid and Vilonen, and the work of Rossmann to compute canonical measures on nilpotent orbits for the real semisimple Lie groups with one conjugacy class of Cartan subgroups.
@article{AIF_2008__58_4_1213_0,
author = {Bo\v zi\v cevi\'c, Mladen},
title = {Limit formulas for groups with one conjugacy class of Cartan subgroups},
journal = {Annales de l'Institut Fourier},
volume = {58},
year = {2008},
pages = {1213-1232},
doi = {10.5802/aif.2383},
zbl = {1153.22012},
mrnumber = {2427959},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2008__58_4_1213_0}
}
Božičević, Mladen. Limit formulas for groups with one conjugacy class of Cartan subgroups. Annales de l'Institut Fourier, Tome 58 (2008) pp. 1213-1232. doi : 10.5802/aif.2383. http://gdmltest.u-ga.fr/item/AIF_2008__58_4_1213_0/
[1] Primitive ideals and orbital integrals in complex classical groups, Math. Ann., Tome 259 (1982), pp. 153-199 | Article | MR 656661 | Zbl 0489.22010
[2] Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra, Tome 80 (1983), pp. 350-382 | Article | MR 691809 | Zbl 0513.22009
[3] Equivariant Sheaves and Functors, Springer-Verlag, Lecture Notes in Mathematics 1578 (1994) | MR 1299527 | Zbl 0808.14038
[4] Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compactes, Ann. of Math., Tome 57 (1953), pp. 115-207 | Article | MR 51508 | Zbl 0052.40001
[5] Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris, Tome 292 (1981), pp. 707-710 | Zbl 0467.20036
[6] Characteristic cycles of standard sheaves associated with open orbits (preprint 2006, to appear in Proc. Amer. Math. Soc.)
[7] Homology groups of conormal varieties (preprint 2006, to appear in Mediterranean Jour. Math.)
[8] A limit formula for elliptic orbital integrals, Duke Math. J., Tome 113 (2002), pp. 331-353 | Article | MR 1909221 | Zbl 1010.22022
[9] Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York (1993) | Zbl 0972.17008
[10] Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York (1978) | MR 514561 | Zbl 0451.53038
[11] The Analysis of Linear Partial Differential Operators I, Grundlehren Math. Wiss., Springer, Berlin Heidelberg Tome 256 (1983) | MR 717035 | Zbl 0521.35001
[12] The invariant holonomic system on a semisimple Lie algebra, Invent. Math., Tome 75 (1984), pp. 327-358 | Article | MR 732550 | Zbl 0538.22013
[13] Sheaves on Manifolds, Grundlehren Math. Wiss., Springer, Berlin Tome 292 (1990) | MR 1074006 | Zbl 0709.18001
[14] The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, Tome 31 (1979), pp. 331-357 | Article | MR 527548 | Zbl 0396.53025
[15] Nilpotent orbital integrals in a real semisimple Lie algebra and representations of the Weyl groups, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Birkhäuser, Boston (Progr. Math.) Tome 92 (1990), pp. 263-287 | MR 1103593 | Zbl 0744.22012
[16] Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety, I, II, J. Funct. Anal., Tome 96 (1991), p. 130-154; 155-193 | Article | Zbl 0755.22004
[17] Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra, Invent. Math., Tome 121 (1995), pp. 531-578 | Article | MR 1353308 | Zbl 0861.22008
[18] Characteristic cycles of constructible sheaves, Invent. Math., Tome 124 (1996), pp. 451-502 | Article | MR 1369425 | Zbl 0851.32011
[19] Two geometric character formulas for reductive Lie groups, J. Amer. Math. Soc., Tome 11 (1998), pp. 799-867 | Article | MR 1612634 | Zbl 0976.22010
[20] Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math., Tome 151 (2001) no. 2, pp. 1071-1118 | MR 1779564 | Zbl 0960.22009
[21] Remarks on nilpotent orbits of a symmetric pair, J. Math. Soc. Japan, Tome 39 (1987), pp. 127-138 | Article | MR 867991 | Zbl 0627.22008
[22] Holonomic systems on a flag variety associated to Harish-Chandra modules and representations of a Weyl group, Algebraic groups and related topics, North-Holland (Adv. Studies in Pure Math.) Tome 6 (1985), pp. 139-154 | MR 803333 | Zbl 0582.22011
[23] Irreducible characters of semisimple Lie groups IV. Character-multiplicity duality, Duke Math. J., Tome 49 (1982), pp. 943-1073 | MR 683010 | Zbl 0536.22022