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.

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