Representation equivalence and p-Spectrum of constant curvature space forms

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 $0\leq p\leq n$. 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 $0\leq q\leq p$ implies $\tau_q$-equivalence for every $0\leq q\leq p$. As a byproduct of the methods we obtain several results relating $p$-isospectrality with $\tau_p$-equivalence.