Spectra for Gelfand pairs associated with the Heisenberg group

Benson, Chal; Jenkins, Joe; Ratcliff, Gail; Worku, Tefera

Benson, Chal ; Jenkins, Joe ; Ratcliff, Gail ; Worku, Tefera

Colloquium Mathematicae, Tome 70 (1996), p. 305-328 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $L_{K}^{1} (H_{n})$ of integrable K-invariant functions on $H_{n}$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L_{K}^{1} (H_{n})$ 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’.

Publié le : 1996-01-01

Zbl 0876.22011

EUDML-ID : urn:eudml:doc:210444

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     author = {Chal Benson and Joe Jenkins and Gail Ratcliff and Tefera Worku},
     title = {Spectra for Gelfand pairs associated with the Heisenberg group},
     journal = {Colloquium Mathematicae},
     volume = {70},
     year = {1996},
     pages = {305-328},
     zbl = {0876.22011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv71i2p305bwm}
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Benson, Chal; Jenkins, Joe; Ratcliff, Gail; Worku, Tefera. Spectra for Gelfand pairs associated with the Heisenberg group. Colloquium Mathematicae, Tome 70 (1996) pp. 305-328. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv71i2p305bwm/

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