Representation equivalence and p-Spectrum of constant curvature space forms

Abstract

We study the $p$-spectrum of a locally symmetric space of constant curvature $\Gamma \backslash X$, in connection with the right regular representation of the full isometry group $G$ of $X$ on $L^2(\Gamma \backslash G{)}_{{\tau }_p}$, where ${\tau }_p$ is the complexified $p$-exterior representation of $\mathrm{O}(n)$ on ${\bigwedge }^p({\mathbb{R}}^n{)}_{\mathbb{C}}$. We give an expression of the multiplicity $d_{\lambda }(p,\Gamma )$ of the eigenvalues of the $p$-Hodge-Laplace operator in terms of multiplicities $n_{\Gamma }(\pi )$ of specific irreducible unitary representations of $G$. As a consequence, we extend results of Pesce for the spectrum on functions to the $p$-spectrum of the Hodge-Laplace operator on $p$-forms of $\Gamma \backslash X$, and we compare $p$-isospectrality with ${\tau }_p$-equivalence for $0\le p\le n$. For spherical space forms, we show that $\tau$-isospectrality implies $\tau$-equivalence for a class of $\tau$'s that includes the case $\tau ={\tau }_p$. Furthermore we prove that $p-1$ and $p+1$-isospectral implies $p$-isospectral. For nonpositive curvature space forms, we give examples showing that $p$-isospectrality is far from implying ${\tau }_p$-equivalence, but a variant of Pesce's result remains true. Namely, for each fixed $p$, $q$-isospectrality for every $0\le q\le p$ implies ${\tau }_q$-equivalence for every $0\le q\le p$. As a byproduct of the methods we obtain several results relating $p$-isospectrality with ${\tau }_p$-equivalence.

24 pages

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