Representation equivalence and p-spectrum of constant curvature space forms

E.A.LAURET,; R.J. MIATELLO; J.P. ROSSETTI

THE JOURNAL OF GEOMETRIC ANALYSIS

SPRINGER

Lugar: Berlin; Año: 2015 vol. 25 p. 564 - 591

1050-6926

We study the $p$-spectrum of a locally symmetric space of constant curvature $Ga X$, in connection with the right regular representation of the full isometry group $G$ of $X$ on $L^2(Ga G)_{ au_p}$, where $ au_p$ is the complexified $p$-exterior representation of $Ot(n)$ on $igwedge^p(R^n)_C$. We give an expression of the multiplicity $d_lambda(p,G)$ of the eigenvalues of the $p$-Hodge-Laplace operator in terms of multiplicities $n_G(pi)$ 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 $Ga X$, and we compare $p$-isospectrality with $ au_p$-equivalence for $0le ple n$. For spherical space forms, we show that $ au$-isospectrality implies $ au$-equivalence for a class of $ au$´s that includes the case $ au= au_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 $ au_p$-equivalence, but a variant of Pesce´s result remains true. Namely, for each fixed $p$, $q$-isospectrality for every $0le qle p$ implies $ au_q$-equivalence for every $0le qle p$. As a byproduct of the methods we obtain several results relating $p$-isospectrality with $ au_p$-equivalence.