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

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 ${\mathfrak{p}}_{\mathbb{C}}^{\ast }$, 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.

arXiv comment29 pages, added Lemma 2.2 which fills a gap in the proof of Corollary 2.3

References in corpus (6)

• Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}
• Explicit Hilbert spaces for certain unipotent representations II
• Classification of symmetric pairs with discretely decomposable restrictions of (g,K)-modules
• Minimal representations via Bessel operators
• Smallest complex nilpotent orbits with real points
• Intertwining operators for the generalized principal series on symmetric R-spaces

Cited by in corpus (5)

• Norm computation and analytic continuation of vector valued holomorphic discrete series representations
• Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Unitarity and subrepresentations
• Symmetry breaking operators for line bundles over real projective spaces
• The Berezin form on symmetric $R$-spaces and reflection positivity
• Branching laws for small unitary representations of GL(n,C)