Strong representation equivalence for compact symmetric spaces of real rank one

Abstract

Let G∕K be a simply connected compact irreducible symmetric space of real rank one. For each K-type τ we compare the notions of τ-representation equivalence with τ-isospectrality. We exhibit infinitely many K-types τ so that, for arbitrary discrete subgroups Γ and Γ´ of G, if the multiplicities of λ in the spectra of the Laplace operators acting on sections of the induced τ-vector bundles over Γ∖G∕K and Γ´∖G/K agree for all but finitely many λ, then Γ and Γ´ are τ-representation equivalent in G (i.e., dimHomG(Vπ,L2(Γ∖G))=dimHomG(Vπ,L2(Γ´∖G)) for all π∈?G satisfying HomK(Vτ,Vπ)≠0). In particular, Γ∖G∕K and Γ´∖G/K are τ-isospectral (i.e., the multiplicities agree for all λ).We specially study the case of p-form representations, i.e., the irreducible subrepresentations τ of the representation τp of K on the p-exterior power of the complexified cotangent bundle ∧pT∗CM. We show that for such τ, in most cases τ-isospectrality implies τ-representation equivalence. We construct an explicit counterexample for G/K=SO(4n)/SO(4n−1)≃S4n−1.