### Abstract

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | Department of Applied Mathematics, University of Twente |

Number of pages | 22 |

State | Published - 2002 |

### Publication series

Name | Memorandum Faculty Mathematical Sciences |
---|---|

Publisher | University of Twente, Department of Applied Mathematics |

No. | 1666 |

ISSN (Print) | 0169-2690 |

### Fingerprint

### Keywords

- EWI-3486
- IR-65852
- MSC-22E46
- MSC-22E15
- METIS-208649
- MSC-20G20
- MSC-20G15

### Cite this

*Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$*. (Memorandum Faculty Mathematical Sciences; No. 1666). Enschede: Department of Applied Mathematics, University of Twente.

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*Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$*. Memorandum Faculty Mathematical Sciences, no. 1666, Department of Applied Mathematics, University of Twente, Enschede.

**Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$.** / Helminck, G.F.; Helminck, A.G.; Helminck, A.G.

Research output: Professional › Report

TY - BOOK

T1 - Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$

AU - Helminck,G.F.

AU - Helminck,A.G.

AU - Helminck,A.G.

N1 - Imported from MEMORANDA

PY - 2002

Y1 - 2002

N2 - In this paper one considers a unimodular second countable locally compact group $G$ and the homogeneous space $X:=H /G$, where $H$ is a closed unimodular subgroup of $G$. Over $X$ complex vector bundles are considered such that $H$ acts on the fibers by a unitary representation $\rho$ with closed image. The natural action of $G$ on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class $\rho$. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that $H$ is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for $X$ a symmetric $k$-variety $\mathcal{H}_k / \mathcal{G}_k$, with $k$ a locally compact field of characteristic not equal to two. Here $\mathcal{G}$ is a reductive algebraic group defined over $k$ and $\mathcal{H}$ is the fixed point group of an involution $\sigma$ of $\mathcal{G}$ defined over $k$. It is shown then that the natural representation $\mathcal{L}$ of $G_k$ on the Hilbert space $L^2(\mathcal{H}_k /\mathcal{G}_k)$ is multiplicity free if $\mathcal{H}$ is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the nonarchimedean case, a general procedure is given that might lead to showing that a pair $(\mathcal{G}_k,\mathcal{H}_k)$ is a generalized Gelfand pair. Here $\mathcal{G}$ and $\mathcal{H}$ are suitable algebraic groups defined over $k$.

AB - In this paper one considers a unimodular second countable locally compact group $G$ and the homogeneous space $X:=H /G$, where $H$ is a closed unimodular subgroup of $G$. Over $X$ complex vector bundles are considered such that $H$ acts on the fibers by a unitary representation $\rho$ with closed image. The natural action of $G$ on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class $\rho$. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that $H$ is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for $X$ a symmetric $k$-variety $\mathcal{H}_k / \mathcal{G}_k$, with $k$ a locally compact field of characteristic not equal to two. Here $\mathcal{G}$ is a reductive algebraic group defined over $k$ and $\mathcal{H}$ is the fixed point group of an involution $\sigma$ of $\mathcal{G}$ defined over $k$. It is shown then that the natural representation $\mathcal{L}$ of $G_k$ on the Hilbert space $L^2(\mathcal{H}_k /\mathcal{G}_k)$ is multiplicity free if $\mathcal{H}$ is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the nonarchimedean case, a general procedure is given that might lead to showing that a pair $(\mathcal{G}_k,\mathcal{H}_k)$ is a generalized Gelfand pair. Here $\mathcal{G}$ and $\mathcal{H}$ are suitable algebraic groups defined over $k$.

KW - EWI-3486

KW - IR-65852

KW - MSC-22E46

KW - MSC-22E15

KW - METIS-208649

KW - MSC-20G20

KW - MSC-20G15

M3 - Report

T3 - Memorandum Faculty Mathematical Sciences

BT - Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$

PB - Department of Applied Mathematics, University of Twente

ER -