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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.

22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
Full Text: DOI Numdam
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