Vector Bundle-Valued Poisson and Cauchy Kernel Functions on Classical Domains

In this paper we investigate the vector bundle-valued Cauchy and Poisson kernel functions. We compute explicitly matrix-valued eigenfunctions of an invariant differential operator on the classical domain of Type I. Furthermore, a special choice of a vector bundle gives us a matrix-valued Cauchy and Poisson kernel function which satisfies the matrix-valued Laplacian operator on the classical domain of Type I.