On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains

We present a canonical construction of an injective normal symplectic representation of a given normal j-algebra. On the other hand, the image of the holomorphic map corresponding to any normal symplectic representation is described as a subset of the Siegel upper half plane by using vector spaces of matrices satisfying certain axioms. Combining these results, we naturally obtain Xu's realization of homogeneous Siegel domains [Y.C. Xu, Automorphism groups of homogeneous bounded domains, Acta Math. Sinica 19 (1976) 161-191; Y.C. Xu, On the isomorphism of homogeneous bounded domains, Acta Math. Sinica 20 (1977) 248-266].