Representation equivalence and p-Spectrum of constant curvature space forms
We study the $p$-spectrum of a locally symmetric space of constant curvature $Γ\backslash X$, in connection with the right regular representation of the full isometry group $G$ of $X$ on $L^2(Γ\backslash G)_{τ_p}$, where $τ_p$ 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 $d_λ(p,Γ)$ of the eigenvalues of the $p$-Hodge-Laplace operator in terms of multiplicities $n_Γ(π)$ 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 $Γ\backslash X$, and we compare $p$-isospectrality with $τ_p$-equivalence for $0\leq p\leq n$. For spherical space forms, we show that $τ$-isospectrality implies $τ$-equivalence for a class of $τ$'s that includes the case $τ=τ_p$. 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 $τ_p$-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 $τ_q$-equivalence for every $0\leq q\leq p$. As a byproduct of the methods we obtain several results relating $p$-isospectrality with $τ_p$-equivalence.