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

Abstract LetP = NAMbe the minimal parabolic subgroup ofSU(n + 1, 1), which can be regarded as the affine automorphism group of the Siegel upper half-planeUn + 1,Palso acts on the Heisenberg groupHn, the boundary ofUn + 1. ThereforePhas a natural representationUonL2(Hn). We decomposeL2(Hn) into the direct sum of the irreducible invariant closed subspaces underU. The restrictions ofUon 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 ofL2(Hn) toL2(P).