On the existence of metric polarizations (Q1070049)

Let G be a Lie group. In a previous paper the author proved that to any co-adjoint orbit in \({\mathfrak g}^*\) that admits a metric polarization with certain additional properties one can associate a harmonically induced representation of G which is irreducible and whose class is independent of the polarization [J. Reine Angew. Math. 344, 120-148 (1983; Zbl 0538.22008)]. The aim of the paper under review is to find conditions under which such (or almost such) polarizations exist. The first main result asserts that in the case of a real algebraic group there always exist essentially harmonic polarizations for admissible, well-polarizable functionals in \({\mathfrak g}^*\). Such a polarization is not metric (it is not invariant). The second main result asserts that for an orbit whose reductive data (a canonically associated reductive subgroup of G) is Harish-Chandra class and has abelian Cartan subgroups, there exists a harmonic polarization (in a particular metric).