Heisenberg Parabolically Induced Representations of Hermitian Lie Groups, Part II: Next-to-Minimal Representations and Branching Rules

Book chapter, 2025

Every simple Hermitian Lie group has a unique family of spherical representations induced from a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. For most Hermitian groups, this family contains a complementary series, and at its endpoint sits a proper unitarizable subrepresentation. We show that this subrepresentation is next-to-minimal in the sense that its associated variety is a next-to-minimal nilpotent coadjoint orbit. Moreover, for the Hermitian groups SO0(2,n) and E6(−14) we study some branching problems of these next-to-minimal representations.