Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Unitarity and subrepresentations

TL;DR

This paper investigates a class of induced representations of Hermitian Lie groups using Fourier analysis on Heisenberg groups, providing explicit formulas for intertwining operators and new insights into unitarity and subrepresentations.

Contribution

It introduces a novel approach to analyze induced representations via Fourier transform on Heisenberg groups, leading to explicit intertwining operators and new unitarity results.

Findings

Explicit formula for Knapp-Stein intertwining operators.

Construction of new complementary series representations.

Identification of unitarizable subrepresentations at reducibility points.

Abstract

For a Hermitian Lie group $G$, we study the family of representations induced from a character of the maximal parabolic subgroup $P=MAN$ whose unipotent radical $N$ is a Heisenberg group. Realizing these representations in the non-compact picture on a space $I(\nu)$ of functions on the opposite unipotent radical $\bar{N}$, we apply the Heisenberg group Fourier transform mapping functions on $\bar N$ to operators on Fock spaces. The main result is an explicit expression for the Knapp-Stein intertwining operators $I(\nu)\to I(-\nu)$ on the Fourier transformed side. This gives a new construction of the complementary series and of certain unitarizable subrepresentations at points of reducibility. Further auxiliary results are a Bernstein-Sato identity for the Knapp-Stein kernel on $\bar{N}$ and the decomposition of the metaplectic representation under the non-compact group $M$.