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