Abstract

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 ${\Gamma }^{\prime }$ 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 ${\Gamma }^{\prime }\setminus G∕K$ agree for all but finitely many $\lambda$, then $\Gamma$ and ${\Gamma }^{\prime }$ are $\tau$-representation equivalent in $G$ (i.e., $dim{Hom}_G\left(V_{\pi },L^2\left(\Gamma \setminus G\right)\right)=dim{Hom}_G\left(V_{\pi },L^2\left({\Gamma }^{\prime }\setminus G\right)\right)$ for all $\pi \in \hat{G}$ satisfying ${Hom}_K\left(V_{\tau },V_{\pi }\right)\ne 0$). In particular, $\Gamma \setminus G∕K$ and ${\Gamma }^{\prime }\setminus G∕K$ 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 ${\wedge }^p{T}_{ℂ}^{\ast }M$. We show that for such $\tau$, in most cases $\tau$-isospectrality implies $\tau$-representation equivalence. We construct an explicit counterexample for $G∕K=SO\left(4n\right)∕SO\left(4n-1\right)\simeq S^{4n-1}$.

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