Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups

TL;DR

This paper investigates principal series representations of Hermitian Lie groups induced from Heisenberg parabolic subgroups, identifying their unitarity, reducibility, and complementary series regions.

Contribution

It provides a detailed analysis of the unitarity and reducibility of principal series representations induced from Heisenberg parabolics in Hermitian Lie groups, including new classifications.

Findings

Identified complementary series regions.

Determined reducibility points.

Classified unitary subrepresentations.

Abstract

Let $G$ be an irreducible Hermitian Lie group and $D=G/K$ its bounded symmetric domain in $\mathbb C^d$ of rank $r$. Each $\gamma$ of the Harish-Chandra strongly orthogonal roots $\{\gamma_1, \cdots, \gamma_r\}$ defines a Heisenberg parabolic subgroup $P=MAN$ of $G$. We study the principal series representations $\Ind_P^G(1\otimes e^\nu\otimes 1)$ of $G$ induced from $P$. We find the complementary series, reduction points, and unitary subrepresentations in this family of representations.