#### Vol. 314, No. 2, 2021

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Strong representation equivalence for compact symmetric spaces of real rank one

### Emilio A. Lauret and Roberto J. Miatello

Vol. 314 (2021), No. 2, 333–373
##### 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 $\pi \in \stackrel{̂}{G}$ 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 $G∕K=SO\left(4n\right)∕SO\left(4n-1\right)\simeq {S}^{4n-1}$.

##### Keywords
representation equivalent, isospectral, $\tau$-spectrum
##### Mathematical Subject Classification
Primary: 58J53
Secondary: 22C05, 22E46, 58J50