# Spectra for Gelfand pairs associated with the Heisenberg group

Chal Benson; Joe Jenkins; Gail Ratcliff; Tefera Worku

Colloquium Mathematicae (1996)

- Volume: 71, Issue: 2, page 305-328
- ISSN: 0010-1354

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topBenson, Chal, et al. "Spectra for Gelfand pairs associated with the Heisenberg group." Colloquium Mathematicae 71.2 (1996): 305-328. <http://eudml.org/doc/210444>.

@article{Benson1996,

abstract = {Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L^1_K(H_n)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L^1_K(H_n)$ can be identified with the set $Δ(K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $Δ(K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $Δ(K,H_n)$ is a complete metric space and that the ’type 1’ K-spherical functions are dense in $Δ(K,H_n)$. Our main result shows that one can embed $Δ(K,H_n)$ quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of ($K ⋉ H_n$)-invariant differential operators on $H_n$. This viewpoint on the spectrum for $Δ(K,H_n)$ was previously known for K=U(n) and is referred to as ’the Heisenberg fan’.},

author = {Benson, Chal, Jenkins, Joe, Ratcliff, Gail, Worku, Tefera},

journal = {Colloquium Mathematicae},

keywords = {Gelfand pairs; spherical function; Heisenberg group; left invariant differential operators},

language = {eng},

number = {2},

pages = {305-328},

title = {Spectra for Gelfand pairs associated with the Heisenberg group},

url = {http://eudml.org/doc/210444},

volume = {71},

year = {1996},

}

TY - JOUR

AU - Benson, Chal

AU - Jenkins, Joe

AU - Ratcliff, Gail

AU - Worku, Tefera

TI - Spectra for Gelfand pairs associated with the Heisenberg group

JO - Colloquium Mathematicae

PY - 1996

VL - 71

IS - 2

SP - 305

EP - 328

AB - Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L^1_K(H_n)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L^1_K(H_n)$ can be identified with the set $Δ(K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $Δ(K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $Δ(K,H_n)$ is a complete metric space and that the ’type 1’ K-spherical functions are dense in $Δ(K,H_n)$. Our main result shows that one can embed $Δ(K,H_n)$ quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of ($K ⋉ H_n$)-invariant differential operators on $H_n$. This viewpoint on the spectrum for $Δ(K,H_n)$ was previously known for K=U(n) and is referred to as ’the Heisenberg fan’.

LA - eng

KW - Gelfand pairs; spherical function; Heisenberg group; left invariant differential operators

UR - http://eudml.org/doc/210444

ER -

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