We develop a group-theoretic method to generalize the Laplace-Beltrami operators
on the classical domains. In K. Okamoto, "Harmonic analysis on homogeneous
vector bundles," Lecture Notes in Mathematics, Springer-Verlag, 266
(1971), 255–271, inspired by Helgason's paper, "A duality for symmetric spaces
with applications to group representations," Advan. Math. 5 (1970),
1–154, we defined the "Poisson transforms" for homogeneous vector bundles over
symmetric spaces. In K. Okamoto, M. Tsukamoto and K. Yokota, "Generalized
Poisson and Cauchy kernel functions on classical domains," Japan. J.
Math. 26 No. 1 (2000), 51–103., we defined the generalized Poisson-Cauchy
transforms for homogeneous holomorphic line bundles over hermitian symmetric
spaces and computed explicitly the kernel functions for each type of the
classical domains. In E. Imamura, K. Okamoto, M. Tsukamoto and A. Yamamori,
"Generalized Laplacians for Generalized Poisson-Cauchy transforms on classical
domains," Proc. Japan Acad., 82, Ser. A (2006), 167–172., making use of
the Casimir operator, we defined the "generalized Laplacians" on homogeneous
holomorphic line bundles over hermitian symmetric spaces and showed that the
generalized Poisson-Cauchy transforms give rise to eigenfunctions of the
"generalized Laplacians". In this paper, using the canonical coordinates for
each type of the classical domains, we carry out the direct computation to
obtain the explicit formulas of (line bundle valued) invariant differential
operators which we call the generalized Laplacians and compute their eigenvalues
evaluated at the generalized Poisson-Cauchy kernel functions