Volume of Bounded Symmetric Domains and Compactification of Jordan Triple Systems

Abstract

For an irreducible bounded complex circled homogeneous domain, there is a natural normalization of the Euclidean volume, such that this volume is an integer of some projective realization of its compact dual. We give an explanation of this phenomenon in the language of Jordan triple systems. First, we give a simplified version of the projective imbedding of a compactification introduced by O. Loos. We then compute the pullback of the invariant projective volume element by this imbedding; this pullback turns out to be ‘dual’ to the Bergman kernel of the domain. Finally, we prove the equality mentioned above (and more general identities) using some special (real analytic) isomorphism, defined via the Jordan structure, between the bounded domain and its ambient vector space.