Structure of the degenerate principal series on symmetric R-spaces and small representations

TL;DR

This paper analyzes the structure and unitarizability of degenerate principal series representations on symmetric R-spaces, identifying small representations with minimal nilpotent orbit associated varieties.

Contribution

It provides a detailed study of reducibility, composition series, and unitarizable constituents of these representations, highlighting small representations with minimal orbit varieties.

Findings

Identified points of reducibility for the representations.

Determined composition series and unitarizable constituents.

Found small representations with minimal nilpotent orbit associated varieties.

Abstract

Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\infty(X)$ in the case where $P$ is not conjugate to its opposite parabolic. We find the points of reducibility, the composition series and all unitarizable constituents. Among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent $K_{\mathbb{C}}$-orbit in $\mathfrak{p}_{\mathbb{C}}^*$, where $K_{\mathbb{C}}$ is the complexification of a maximal compact subgroup $K\subseteq G$ and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ the corresponding Cartan decomposition.