Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-PlaneU

Let PsNAM be the minimal parabolic subgroup of SU nq1, 1., which can be regarded as the affine automorphism group of the Siegel upper half-plane Unq1, P also acts on the Heisenberg group Hn, the boundary of Unq1. Therefore P has a natural representation U on L2 Hn.. We decompose L2 Hn. into the direct sum of the irreducible invariant closed subspaces under U. The restrictions of U on these subspaces are square-integrable. We give the characterization of the admissible condition in terms of the Fourier transform and define the wavelet transform with respect to admissible wavelets. The wavelet transform gives isometric operators from the irreducible invariant closed subspaces of L2 Hn. to L2 P..