Strong representation equivalence for compact symmetric spaces of real rank one

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

arXiv admin note: text overlap with arXiv:1804.08288

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