Some spectral results on $L^{2}(H_{n})$ related to the action of U(p,q)

Godoy, T. ; Saal, L.

Colloquium Mathematicae, Tome 84/85 (2000), p. 177-187 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $H_{n}$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_{n}$. So $L^{2}(H_{n})$ has a natural structure of G-module. We obtain a decomposition of $L^{2}(H_{n})$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^{2}(H_{n})$ where $L=\sum _{j=1}^{p} (X_{j}^{2}+Y_{j}^{2}) - \sum _{j=p+1}^{n} (X_{j}^{2}+Y_{j}^{2})$, and where ${X_{1},Y_{1},...,X_{n},Y_{n},T}$ denotes the standard basis of the Lie algebra of $H_{n}$. Finally, we obtain a spectral characterization of the bounded operators on $L^{2}(H_{n})$ that commute with the action of G.

Détail

Publié le : 2000-01-01

Zbl 0961.43008

EUDML-ID : urn:eudml:doc:210848

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