## 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.

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 | Journal of Functional Analysis |

Volume | 266 |

Issue | 6 |

Pages (from-to) | 3508–3542 |

Number of pages | 35 |

ISSN | 0022-1236 |

DOIs | |

Publication status | Published - 2014 |