A Schwarz lemma for bounded symmetric domains

Abstract

The purpose of this note is to generalize Pick's invariant formulation of the classical Schwarz lemma. For the four classical types of bounded symmetric domains such a generalization was given by K. H. Look [3]; the present treatment will be independent of classification theory and will also include the exceptional Cartan domains. The results are also independent of the particular realization of the domain in Cn, they depend only on its structure as a hermitian manifold. The results will therefore be formulated for hermitian symmetric spaces of noncompact type; these are known to be in one-to-one correspondence with the holomorphic equivalence classes of bounded symmetric domains. We shall make use of Harish-Chandra's canonical realization of the hermitian symmetric spaces as bounded domains; this could perhaps be avoided, but it makes the proofs considerably simpler. In the following M=G/K will be a hermitian symmetric space of noncompact type; the identity coset will be denoted by po, 9 and 1 will denote the Lie algebras of G and K, respectively, and Ha, Ea, ■ ■ • will be a Weyl basis of q with respect to a Cartan subalgebra of Q contained in f. By a result of Harish-Chandra there exists a set A of strongly orthogonal roots of Q such that ct= ^„eA P(P«4-P-a) is a Cartan subalgebra of the symmetric pair (g, f). So every point pEM can be represented in the form p = kexp(^2aeAta(Ea-\-E-a)) -p0 with kEK, ta^0. For any p, qEM we denote by d(p, q) the distance of p and q in the metric induced by the hermitian structure of M. In any realization of M as a complex domain this is the Bergman metric. We denote by d*(p, q) the Caratheodory distance, which is defined by