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

G.F. Helminck, A.G. Helminck, A.G. Helminck

Research output: Book/ReportReportProfessional

### Abstract

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 $k$.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 22 Published - 2002

### Publication series

Name Memorandum Faculty Mathematical Sciences University of Twente, Department of Applied Mathematics 1666 0169-2690

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

### Cite this

Helminck, G. F., Helminck, A. G., & Helminck, A. G. (2002). Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$. (Memorandum Faculty Mathematical Sciences; No. 1666). Enschede: University of Twente, Department of Applied Mathematics.
Helminck, G.F. ; Helminck, A.G. ; Helminck, A.G. / Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$. Enschede : University of Twente, Department of Applied Mathematics, 2002. 22 p. (Memorandum Faculty Mathematical Sciences; 1666).
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abstract = "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 $k$.",
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author = "G.F. Helminck and A.G. Helminck and A.G. Helminck",
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year = "2002",
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Helminck, GF, Helminck, AG & Helminck, AG 2002, Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$. Memorandum Faculty Mathematical Sciences, no. 1666, University of Twente, Department of Applied Mathematics, Enschede.

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

Enschede : University of Twente, Department of Applied Mathematics, 2002. 22 p. (Memorandum Faculty Mathematical Sciences; No. 1666).

Research output: Book/ReportReportProfessional

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 - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Helminck GF, Helminck AG, Helminck AG. Multiplicity one for representations corresponding to spherical distribution vectors of class $\rho$. Enschede: University of Twente, Department of Applied Mathematics, 2002. 22 p. (Memorandum Faculty Mathematical Sciences; 1666).