Admissible wavelets on the Siegel domain of type one

LetSp(n, R) be the sympletic group, and letK _n * be its maximal compact subgroup. ThenG=Sp(n,R)/K _n * can be realized as the Siegel domain of type one. The square-integrable representation ofG gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent state. With the Laguerre polynomials, Hermite polynomials and Jacobi polynomials, two kinds of orthogonal bases for AW are given, and they then give orthogonal decompositions ofL ²-space on the Siegel domain of type one ℒ(ℋ _n , |y| *dxdy). Project supported in part by the National Natural Science Foundation of China (Grant No. 19631080).