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

Jan Möllers, Benjamin Schwarz

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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.
Original languageEnglish
JournalJournal of Functional Analysis
Volume266
Issue6
Pages (from-to)3508–3542
Number of pages35
ISSN0022-1236
DOIs
Publication statusPublished - 2014

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