Summary: 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 \(\Gamma\setminus G/K\) and \(\Gamma'\setminus G/K\) agree for all but finitely many \(\lambda \), then \(\Gamma\) and \(\Gamma'\) are \(\tau \)-representation equivalent in \(G\) (i.e., \( \dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma\setminus G))=\dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma'\setminus G))\) for all \(\pi\in \widehat G\) satisfying \(\operatorname{Hom}_K(V_\tau,V_\pi)\neq0)\). In particular, \( \Gamma\setminus G/K\) and \(\Gamma'\setminus G/K\) 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 counterexample for \(G/K= \operatorname{SO}(4n)/ \operatorname{SO}(4n-1)\simeq S^{4n-1} \).