Strong representation equivalence for compact symmetric spaces of real rank one

LAURET, EMILIO A.; MIATELLO, ROBERTO J.

PACIFIC JOURNAL OF MATHEMATICS

PACIFIC JOURNAL MATHEMATICS

Año: 2021 vol. 314 p. 333 - 373

0030-8730

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $au$ we compare the notions of $au$-representation equivalence with $au$-isospectrality. We exhibit infinitely many $K$-types $au$ 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 $au$-vector bundles over $Gammaackslash G/K$ and $Gamma´ackslash G/K$ agree for all but finitely many $lambda$, then $Gamma$ and $Gamma´$ are $au$-representation equivalent in $G$ (i.e. $dim operatorname{Hom}_G(V_pi, L^2(Gammaackslash G))=dim operatorname{Hom}_G(V_pi, L^2(Gamma´ackslash G))$ for all $piin widehat G$ satisfying $operatorname{Hom}_K(V_au,V_pi)eq0$). In particular $Gammaackslash G/K$ and $Gamma´ackslash G/K$ are $au$-isospectral (i.e. the multiplicities agree for all $lambda$). We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $au$ of the representation $au_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $igwedge^p T_{mathbb C}^*M$. We show that for such $au$, in most cases $au$-isospectrality implies $au$-representation equivalence. We construct an explicit counter-example for $G/K= operatorname{SO}(4n)/ operatorname{SO}(4n-1)simeq S^{4n-1}$.