Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups

TL;DR

This paper analyzes principal series representations of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups, identifying their structure, reducibility, and unitarity properties.

Contribution

It provides a detailed analysis of principal series representations for specific exceptional Lie groups, including their $K$-types, Lie algebra actions, and unitarity criteria, which was previously not fully understood.

Findings

Identified $K$-types via circle bundle structures.

Computed Lie algebra actions on the representation space.

Determined complementary series, reducible points, and unitary subrepresentations.

Abstract

Let $G/K$ be an irreducible quaternionic symmetric space of rank $4$. We study the principal series representation $\pi_\nu=\text{Ind}_P^G(1\otimes e^\nu\otimes 1)$ of $G$ induced from the Heisenberg parabolic subgroup $P=MAN$ realized on $L^2(K/L)$, $L=K\cap M$. We find the $K$-types in the induced representation via a double cover $K/L_0\to K/L$ and a circle bundle $K/L_0\to K/L_1$ over a compact Hermitian symmetric space $K/L_1$. We compute the Lie algebra $\mathfrak g$-action of $G$ on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations.