# Department of Mathematics

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

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### DOI

• Jan Möllers
• Benjamin Schwarz, Universität Paderborn, Germany
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 language English Journal of Functional Analysis 266 6 3508–3542 35 0022-1236 https://doi.org/10.1016/j.jfa.2014.01.006 Published - 2014

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