Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces

On a symmetric coset space G/H the eigenvalues of the Laplacian and the Lichnerowicz operator acting on arbitrary tensor harmonics are given in terms of the eigenvalues of the quadratic Casimir operators of G and H. Explicit examples for Sn, CPn, and real (complex) Grassmann manifolds are analyzed.