Strong representation equivalence for compact symmetric spaces of real rank one

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $\tau$ we compare the notions $\tau$-representation equivalence with $\tau$-isospectrality. We exhibit infinitely many $K$-types so that, for arbitrary discrete subgroups $\Gamma$ and $\Gamma'$ $G$, if multiplicities $\lambda$ in spectra Laplace operators acting on sections induced $\tau$-vector bundles over $\Gamma\backslash G/K$ $\Gamma'\backslash agree all but finitely $\lambda$, then are equivalent $G$ (i.e.\ $\dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma\backslash G))=\dim L^2(\Gamma'\backslash G))$ $\pi\in \widehat G$ satisfying $\operatorname{Hom}_K(V_\tau,V_\pi)\neq0$). In particular $\tau$-isospectral $\lambda$). specially study case $p$-form representations, i.e. subrepresentations representation $\tau_p$ $K$ $p$-exterior power complexified cotangent bundle $\bigwedge^p T_{\mathbb C}^*M$. show that such $\tau$, most cases $\tau$-isospectrality implies equivalence. construct an explicit counter-example $G/K= \operatorname{SO}(4n)/ \operatorname{SO}(4n-1)\simeq S^{4n-1}$.