Abstract: Let H_n be the n-direct proded of Heisenbery group H, P the affine group of _n. Then P ho a natural unitary representation U on L²(H_n). In this paper,the direct sum decomposition of irreducible invariant closed subspaces under unitary representation U for L²(H_n) is given. The restrictins of U on these subspaces are square-integrable. The charactedsation of admissible condition is obtained in ierms of the Fourier transform. By seleting appropriately an orthogonal wavelet basis and the wavelet transform,the authors obtain the orthogonal direct chin decomposinon of function space L²(P,dμ₁).

Key words: Admissible condition, Wavelet transform, Heisenbery group, Orthogonal direct sum decomposition, Generalized upper half-plance