Let $M=G/K$ with $G$ compact and let $(\tau,W_\tau)$ be an irreducible representation of $K$, let $E_\tau$ be the associated $G$-homogeneous vector bundle of $M$. Then, the space of $L^2$-sections of $E_\tau$ decomposes as $$ L^2(E_\tau) = \sum_{\pi\in\widehat G} V_\pi\otimes \operatorname{Hom}_K(V_\pi,W_\tau). $$
The proof follows by Peter-Weyl: $\Gamma^\infty(E_\tau)\simeq C^\infty(G;\tau):=\{f:G\to W_\tau:f(gk)=\tau(k)^{-1}f(g)\}$. For each $\pi\in\widehat G$, we have $V_\pi\otimes \operatorname{Hom}_K(V_\pi,W_\tau) \to C^\infty(G;\tau)$ given by $v\otimes L \mapsto (x\mapsto L(\pi(x^{-1}).v))$. This map is $G$-equivariant. By Peter-Weyl, we have that the formula above.
When $M$ is symmetric (e.g. $M=S^3$) and $G$ semisimple, the Laplace operator acts as the Casimir element of $\mathfrak g$ (the Lie algebra of $G$). Furthermore, if $S^3=\operatorname{Spin}(4)/\operatorname{Spin}(3)$, the square of the Dirac operator $D^2$ coincides with the Casimir element plus $1/8 R$ ($R$ is the curvature curvature of $M$).