A REMARK ON THE N-INVARIANT GEOMETRY OF BOUNDED HOMOGENEOUS DOMAINS

Let $\mathbf {D}$ be a bounded homogeneous domain in ${\mathbb {C}}^n$ . In this note, we give a characterization of the Stein domains in $\mathbf {D}$ which are invariant under a maximal unipotent subgroup N of $Aut(\mathbf {D})$ . We also exhibit an N-invariant potential of the Bergman metric of $\mathbf {D}$ , expressed in a Lie theoretical fashion. These results extend the ones previously obtained by the authors in the symmetric case.