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

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

### 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 Department of Applied Mathematics, University of Twente 22 Published - 2002

### Publication series

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

### Fingerprint

Multiplicity
Decompose
Algebraic groups
Uniqueness
Closed
Spherical distribution
Gelfand pairs
Second countable
Unitary representation
Reductive group
Locally compact group
Commutativity
Locally compact
Homogeneous space
Vector bundle
Involution
Convolution
Hilbert space
Fixed point
Fiber

• 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: Department of Applied Mathematics, University of Twente.

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

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

Research output: ProfessionalReport

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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, 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.

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

Research output: ProfessionalReport

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 -

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