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Limit formulas for groups with one conjugacy class of Cartan subgroups. (English) Zbl 1153.22012
Let $$G$$ be a connected semisimple real Lie group with Lie algebra $${\mathfrak g}$$. Let $$f$$ be an element of the vector dual $${\mathfrak g}^{\star}$$ of $${\mathfrak g}$$. The orbit $${\mathcal O}_{f}$$ of $$f$$, under the action of $$G$$, is naturally equipped with a symplectic form $$\sigma_{f}$$ induced by the map $$(X,Y)\mapsto f([ X,Y])$$ on $${\mathfrak g}\times{\mathfrak g}$$. The Liouville measure on $${\mathcal O}_{f}$$ is then some multiple of the $$\frac{1}{2}\dim({\mathcal O}_{f})$$ of $$\sigma_{f}$$.
In the paper under review, the author proves a formula describing the Liouville measure on a nilpotent coadjoint orbit as a limit of (derivatives of) Liouville measures on orbits of regular elements in $$\mathfrak g$$. This is done when $$G$$ is linear with one conjugacy class of Cartan subgroups, combining results of Rossmann and of Schmid and Vilonen.

##### MSC:
 22E46 Semisimple Lie groups and their representations 22E30 Analysis on real and complex Lie groups 43A80 Analysis on other specific Lie groups
##### Keywords:
nilpotent orbit; Liouville measure; Weyl group; limit formula
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##### References:
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