Admissible wavelets associated with the affine automorphism group of the Siegel upper half-plane
As pointed out by Grossmann and Morlet the wavelet theory can be interpreted in the scheme of square-integrable representations of locally compact Lie groups. In this paper the authors study a natural representation \(U\) of the minimal parabolic subgroup \(P\) of \(SU(n+1,1)\). Let \(G=NAK\) and \(P=NAM\) be the Iwasawa and Langlands decompositions of \(G\) and \(P\), respectively. Then \(P\) is the affine automorphism group of the Siegel upper half plane \(U^{n+1} \cong G/K\), and it also acts on the boundary \(\partial U^{n+1} \cong G/P\). We here note that \(N\) and \(\partial U^{n+1}\) can be identified with the Heisenberg group \(H^n\). Naturally, this action, as translation, dilation, and rotation on \(H^n\), induces a unitary representation \(U\) of \(P\) on \(L^2(H^n)\). The authors decompose \(L^2(H^n)\) into the direct sum of the irreducible invariant closed subspaces \(H^\sigma_k\) \((k\in \mathbb{Z}_+, \sigma= \pm)\), and find a vector \(\varphi\in H^\sigma_k\) for which the matrix coefficient \(\langle \varphi,U(p) \varphi \rangle_{L^2 (H^n)}\) \((p\in P)\) of \(U\) is square-integrable with respect to the left Haar measure \(d\mu\) on \(P\). Then, considering the Fourier transform of \(H^n\) defined by the Fock representation of \(H^n\), they obtain the so-called admissible condition of \(\varphi\). Finally, they define the wavelet transform \(W_\varphi f(p)= \langle f,U(p)\varphi\rangle_{L^2(H^n)}\) \((f\in H^\sigma_k, p\in P)\) and show that \(W_\varphi\) gives an isometry \(|W_\varphi f|_{L^2 (P,d\mu)} =C_\varphi |f|_{L^2 (H^n)}\) for \(f\in H^n_k\). The case in which the action of \(M\) is ignored is treated by \textit{H. Liu} and \textit{L. Peng} in [Research report of Peking University (1994)] and a similar approach using the Schrödinger representation of \(H^n\) has been used by the reviewer [Ann. Inst. Henri Poincaré, Phys. Théor. 65, 1-13 (1996; Zbl 0866.22013)].
scientific article; zbMATH DE number 1031594