Limit formulas for groups with one conjugacy class of Cartan subgroups

Božičević, Mladen

Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1213-1232.

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.

Classification : 22E46, 22E30, 43A80
Keywords: nilpotent orbit, Liouville measure, Weyl group, limit formula
Mot clés : orbite nilpotente, mesure de Liouville, groupe de Weyl, formule limite

[1] Barbasch, D.; Vogan, D. Primitive ideals and orbital integrals in complex classical groups, Math. Ann., Volume 259 (1982), pp. 153-199 | DOI | MR | Zbl

[2] Barbasch, D.; Vogan, D. Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra, Volume 80 (1983), pp. 350-382 | DOI | MR | Zbl

[3] Bernstein, J.; Lunts, V. Equivariant Sheaves and Functors, Lecture Notes in Mathematics 1578, Springer-Verlag, 1994 | MR | Zbl

[4] Borel, A. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compactes, Ann. of Math., Volume 57 (1953), pp. 115-207 | DOI | MR | Zbl

[5] Borho, W.; MacPhearson, R. Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris, Volume 292 (1981), pp. 707-710 | Zbl

[6] Božičević, M. Characteristic cycles of standard sheaves associated with open orbits (preprint 2006, to appear in Proc. Amer. Math. Soc.)

[7] Božičević, M. Homology groups of conormal varieties (preprint 2006, to appear in Mediterranean Jour. Math.)

[8] Božičević, M. A limit formula for elliptic orbital integrals, Duke Math. J., Volume 113 (2002), pp. 331-353 | DOI | MR | Zbl

[9] Collingwood, D.; McGowern, W. Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993 | Zbl

[10] Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978 | MR | Zbl

[11] Hörmander, L. The Analysis of Linear Partial Differential Operators I, 256, Grundlehren Math. Wiss., Springer, Berlin Heidelberg, 1983 | MR | Zbl

[12] Hotta, R.; Kashiwara, M. The invariant holonomic system on a semisimple Lie algebra, Invent. Math., Volume 75 (1984), pp. 327-358 | DOI | MR | Zbl

[13] Kashiwara, M.; Schapira, P. Sheaves on Manifolds, 292, Grundlehren Math. Wiss., Springer, Berlin, 1990 | MR | Zbl

[14] Matsuki, T. The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, Volume 31 (1979), pp. 331-357 | DOI | MR | Zbl

[15] Rossmann, W. 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) (Progr. Math.), Volume 92, Birkhäuser, Boston, 1990, pp. 263-287 | MR | Zbl

[16] Rossmann, W. Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety, I, II, J. Funct. Anal., Volume 96 (1991), p. 130-154; 155-193 | DOI | Zbl

[17] Rossmann, W. Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra, Invent. Math., Volume 121 (1995), pp. 531-578 | DOI | MR | Zbl

[18] Schmid, W.; Vilonen, K. Characteristic cycles of constructible sheaves, Invent. Math., Volume 124 (1996), pp. 451-502 | DOI | MR | Zbl

[19] Schmid, W.; Vilonen, K. Two geometric character formulas for reductive Lie groups, J. Amer. Math. Soc., Volume 11 (1998), pp. 799-867 | DOI | MR | Zbl

[20] Schmid, W.; Vilonen, K. Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math., Volume 151 (2001) no. 2, pp. 1071-1118 | MR | Zbl

[21] Sekiguchi, J. Remarks on nilpotent orbits of a symmetric pair, J. Math. Soc. Japan, Volume 39 (1987), pp. 127-138 | DOI | MR | Zbl

[22] Tanisaki, T. Holonomic systems on a flag variety associated to Harish-Chandra modules and representations of a Weyl group, Algebraic groups and related topics (Adv. Studies in Pure Math.), Volume 6, North-Holland, 1985, pp. 139-154 | MR | Zbl

[23] Vogan, D. Irreducible characters of semisimple Lie groups IV. Character-multiplicity duality, Duke Math. J., Volume 49 (1982), pp. 943-1073 | MR | Zbl