Nilpotent orbital integrals in a real semisimple Lie algebra and representations of Weyl groups.

*(English)*Zbl 0744.22012
Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 263-287 (1990).

[For the entire collection see Zbl 0719.00018.]

Let \({\mathfrak g}\) be a semisimple real Lie algebra with adjoint group \(G\). Associated to each semisimple or nilpotent element \(X\) of \({\mathfrak g}^*\), there is a canonical invariant measure \(m_ X\) on the \(G\)-orbit of \(X\) in \({\mathfrak g}^*\). A well-known theorem of Harish-Chandra says that \(m_ 0\) can be obtained as a limit of derivatives of the measures associated to certain regular semisimple orbits. This paper deals with the problem of finding an analogous formula for arbitrary nilpotent orbits. The problem is solved for nilpotent orbits which satisfy a certain technical hypothesis. In this case the differential operator needed comes from a correspondence between real nilpotent orbits and certain representations of the Weyl group. It is not known in what generality this hypothesis is satisfied. It is satisfied for all nilpotent orbits when \({\mathfrak g}\) is complex, in which case the result was already known.

Let \({\mathfrak g}\) be a semisimple real Lie algebra with adjoint group \(G\). Associated to each semisimple or nilpotent element \(X\) of \({\mathfrak g}^*\), there is a canonical invariant measure \(m_ X\) on the \(G\)-orbit of \(X\) in \({\mathfrak g}^*\). A well-known theorem of Harish-Chandra says that \(m_ 0\) can be obtained as a limit of derivatives of the measures associated to certain regular semisimple orbits. This paper deals with the problem of finding an analogous formula for arbitrary nilpotent orbits. The problem is solved for nilpotent orbits which satisfy a certain technical hypothesis. In this case the differential operator needed comes from a correspondence between real nilpotent orbits and certain representations of the Weyl group. It is not known in what generality this hypothesis is satisfied. It is satisfied for all nilpotent orbits when \({\mathfrak g}\) is complex, in which case the result was already known.

##### MSC:

22E30 | Analysis on real and complex Lie groups |

22E46 | Semisimple Lie groups and their representations |

22E60 | Lie algebras of Lie groups |