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

T. Godoy¹ ; L. Saal¹

¹

Colloquium Mathematicum, Tome 86 (2000) no. 2, pp. 177-187.

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

DOI : 10.4064/cm-86-2-177-187