Analytic invariants of bounded symmetric domains

Abstract

Introduction. In order to characterize equivalence classes of bounded domains in Cn under holomorphic homeomorphisms S. Bergman introduced various invariants with the help of his kernel function and invariant metric. For general domains one does not have, at least at the moment, a convenient complete system of invariants; for certain very restricted classes of domains, however, it is possible to find such complete systems. K. H. Look [10] has shown that within the class of irreducible bounded symmetric domains of classical type each holomorphic equivalence class can be characterized by three invariants. In the present paper it will be pointed out that three constants are sufficient also for the class of all irreducible bounded symmetric domains. Beside this slight extension of Look's result we shall also compute certain invariants connected with the Bergman metric in terms of some fundamental invariants for our class. For the special case of the classical domains the invariants of Proposition 3 were also computed in [10], using explicit realizations of the domains. The results of [10] concerning the Schwarz constant were extended to the case of arbitrary symmetric domains in [9].