Representation Equivalence and p-Spectrum of Constant Curvature Space Forms

Abstract

We study the p-spectrum of a locally symmetric space of constant curvature Γ∖X, in connection with the right regular representation of the full isometry group G of X on $L^{2}(\varGamma \backslash G)_{\tau_{p}}$, where τ _p is the complexified p-exterior representation of $\operatorname{O}(n)$ 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∖X, and we compare p-isospectrality with τ _p -equivalence for 0≤p≤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≤q≤p implies τ _q -equivalence for every 0≤q≤p. As a byproduct of the methods we obtain several results relating p-isospectrality with τ _p -equivalence.