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The group SU(1,d) acts naturally on the Hilbert space , where B is the unit ball of and the weighted measure . It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...
It is well known that (U(p,q),Hₙ) is a generalized Gelfand pair. Applying the associated spectral analysis, we prove a theorem of Wiener Tauberian type for the reduced Heisenberg group, which generalizes a known result for the case p = n, q = 0.
We define partial spectral integrals on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by , where is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that converges a.e. to f(x) as R → ∞.
Let be a right-invariant sub-Laplacian on a connected Lie group and let denote the associated “spherical partial sums,” where is the spectral resolution of We prove that converges a.e. to as under the assumption
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