Lifted infinitesimal holomorphic representation for the n-dimensional complex hyperbolic ball and for Cartan domains of type I

The Laplacian and Ornstein-Uhlenbeck operators on the finite dimensional complex ball are obtained from the infinitesimal holomorphic representation of the group U(n,1). We compare the invariant measures for these operators with the unitarizing measures of the discrete series representation. Then with Hua differential calculus, we show how to extend the results to domains with matrix elements.