Global solvability of the laplacians on pseudo-Riemannian symmetric spaces

Let G be a noncompact semisimple Lie group with finite center and H an open subgroup of the fixed point group of an involution of G. $\text{G}\text{H}$ becomes a pseudo-Riemannian manifold. We prove that the Laplacian P on $\text{G}\text{H}$ is globally solvable in the sense that $\text{PC}{}^{\mathrm{\infty }}\text{(}\text{G}\text{H}\text{) = C}{}^{\mathrm{\infty }}\text{(}\text{G}\text{H}\text{)}$. This generalizes the global solvability of the Casimir operators on non-compact semisimple Lie groups with finite center due to J. Rauch and D. Wigner.