Structure of the degenerate principal series on symmetric R-spaces and small representations - Danish National Research Database-Den Danske Forskningsdatabase

^{1} Department of Mathematics, Science and Technology, Aarhus University^{2} Universität Paderborn^{3} Department of Mathematics, Science and Technology, Aarhus University

DOI:

10.1016/j.jfa.2014.01.006

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.

Type:

Journal article

Language:

English

Published in:

Journal of Functional Analysis, 2014, Vol 266, Issue 6