Let NAK be the Iwasawa
decomposition of group SU(n + 1,1). The Iwasawa subgroup P =NA can be
identified with the generalized upper half–plane Un+1 and has a natural
representation U on the L2–space of the Heisenberg group 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 characterize the
admissible condition in terms of the Fourier transform and define the wavelet
transform with respect to admissible wavelets. The wavelet transform leads
to isometric operators from the irreducible invariant closed subspaces of
L2(Hn) to L2,ν(Un+1), the weighted L2–spaces on Un+1. By selecting a set of
mutual orthogonal admissible wavelets, we get the direct sum decomposition of
L2,ν(Un+1) with the first component Aν(Un+1), the (weighted) Bergman
space.
