Let $G$ denote a noncompact semi-simple Lie group of real rank $1$ and let $K$ be a maximal compact subgroup. The Casimir (Laplace-Beltrami) operator acts by scalar on every irreducible representation of $G$. Thus every irreducible representation of $G$, which occures in the decomposition of the regular representation on $L^2(G/K)$ gives an eigenvalue of the Casimir (Laplace) operator on $L^2(G/K)$.
My questions:
1.) Is it right, that only representations $(\pi,V)$ does occur in the decomposition for which the space $V_{\pi}^K=\{v\in V : \pi(k)v=v\ \forall k\in K\}$ is not zero? (I know it can only have dimension one or zero).
2.) Is it right, that the discrete series representations do not gives eigenvalues? If yes, how to show that the space $V_{\pi}^K$ is zero if $\pi$ is out of the discrete series?
Many thanks in advance.
