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Spectra for Gelfand pairs associated with the Heisenberg group

Chal Benson; Joe Jenkins; Gail Ratcliff; Tefera Worku — 1996

Colloquium Mathematicae

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...

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Moment sets and the unitary dual of a nilpotent Lie group.

Invariant theory for the orthogonal group via star products.

Spectra for Gelfand pairs associated with the Heisenberg group