The Isotropic Subgroup Of Normal Siegel Domain

In 1961, Piatetski-Shapiro[1] defined Siegel domains and proved that any Siegel domain is holomorphically isomorphic to a bounded domain. Successively, in 1963, Vinberg, Gindikin and Piatetski-Shapiro[2] proved that any homogeneous bounded domain is holomorphically isomorphic to a homogeneous Siegel domain. In 1976 and 1977, Yichao Xu[3] constructed a class of special homogeneous Siegel domain, i.e. normal Siegel domain D(Vh,F), and proved that any homogeneous Siegel domain is affine equivalent to a normal Siegel domain in [4]. Also in 1976, Xu determined the Lie algebra aut(Z)(VN,F)) of the holomorphic automorphism group Aut (D(VN,F)) and the generating elements of the affine automorphism group Aff (D(VN, F)). At the same time, Dorfmeister gave an algebraic realization of a homogeneous bounded domain and the holomorphic automorphism group, but the existence condition of some holomorphic automorphisms in his work is not clear.From Cauchy-Szego kernel S(z,ξ), Hua[6] constructed the formal Poisson kernel P(z,z, ξ,ξ) = |S(z,ξ)|2/S(z,z) on a classical domain, and proved that the formal Poisson kernel is the Poisson kernel function. In 1965, Koranyi proved that the formal Poisson kernel is the Poisson kernel on the symmetric Siegel domains using the Lie group theory. When D is an indecomposable normal Siegel domain, Xu proved that a formal Poisson kernel is Poisson kernel if and only if D is a symmetric siegel domain. Note that the Silov boundary S(D) of a symmetric domain is a transitive boundary acted by the isotropic subgroup, but in the case of non-symmetric domain, the Silov boundary S(D) is not a transitive boundary acted by the isotropic subgroup. So we can pose the following problem: when D is a non-symmetric homogeneous Siegel domain, what is the integral representation from the continuous function class on Silov boundary S(D) to the Laplace-Beltrami harmonic function class with respect to the Bergman metric on D?In order to consider the Poisson integral, we need to give the explicit expressions of the following two sets: one is the generating elements of the maximal connected holomorphic automorphism group Aut (D(VN, F)) acting on the normal Siegel domains, the other is the generating elements of the isotropic subgroup Iso (D(VN, F)) of the fixed point ((?)-1v0, 0). Particularly, we give the explicit expression of all orbits in the Silov boundary under the action by the isotropic subgroup Iso (D(VN, F)). In this thesis, we will give the generating element set of the isotropic subgroup Iso (D(VN, F)) of the fixed point ((?)-1v0, 0).The thesis consists of three chapters. Chapter one gives a brief introduction to the relevant background and poses the problem which we will solve in this thesis. Chapter two introduces some necessary definitions and theorems needed for further arguments. In chapter three, based on professor Yichao Xu's work, we obtain the main results of this thesis: give some one parameter subgroups by solving some ordinary differential equations and get the generating element set of isotropic subgroup of normal Siegel domain.