Let $\Gamma$ be a discrete subgroup acting without fixed points on a
simply connected constant curvature space $X=G/K$, where $G$ is the full
isometry group of $X$ and $K=O(n)$. We will relate the eigenspaces of the
Hodge-Laplace operator on $p$-forms on $X$ with the irreducible
constituents of the right regular representation of $G$ on $L^2(\Gamma\backslash G)$. As a consequence, we will relate the notions of p-isospectrality with
$\tau_p$-representation equivalence, extending results of Pesce in this
context.