Invariant differential operators on Hermitian symmetric spaces and their eigenvalues

Let (D) over bar be the invariant Cauchy Riemann operator and M-m = D-m (D) over bar (m) the corresponding invariant Laplacians on a bounded symmetric domain. We calculate the eigenvalues of M-m on spherical functions. In particular we prove that for a symmetric domain off rank two the operators M-1, M-3 generate all invariant differential operators. We also find the eigenvalues of the generators introduced by Shimura.