Hermitian Lie Algebras and Metaplectic Representations. I

Abstract

A notion of “hermitian Lie algebra” is introduced which relates ordinary and graded Lie algebra structures. In the case of real-symplectic and arbitrary-signature-unitary Lie algebras, it leads to an analysis of the minimal dimensional coadjoint orbits, and then to the metaplectic representations and their restrictions to unitary groups of arbitrary signature and parabolic subgroups of these unitary groups.