Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups

Abstract

Let G be an irreducible Hermitian Lie group and D=G/K its bounded symmetric domain in Cd of rank r. Each γ of the Harish-Chandra strongly orthogonal roots {γ1,⋯,γr} defines a Heisenberg parabolic subgroup P=MAN of G. We study the principal series representations IndPG(1⊗eν⊗1) of G induced from P. These representations can be realized as the L2-space on the minimal K-orbit S=Ke=K/L of a root vector e of γ in Cd, and S is a circle bundle over a compact Hermitian symmetric space K/L0 of K of rank one or two. We find the complementary series, reduction points, and unitary sub-quotients in this family of representations.