Vol. 314, No. 2, 2021

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Strong representation equivalence for compact symmetric spaces of real rank one

Emilio A. Lauret and Roberto J. Miatello

Vol. 314 (2021), No. 2, 333–373

Let GK 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 ΓGK and ΓGK agree for all but finitely many λ, then Γ and Γ are τ-representation equivalent in G (i.e., dimHomG(V π,L2(ΓG)) = dimHomG(V π,L2(ΓG)) for all π Ĝ satisfying HomK(V τ,V π)0). In particular, ΓGK and ΓGK 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 pTM. We show that for such τ, in most cases τ-isospectrality implies τ-representation equivalence. We construct an explicit counterexample for GK = SO(4n)SO(4n 1) S4n1.

representation equivalent, isospectral, $\tau$-spectrum
Mathematical Subject Classification
Primary: 58J53
Secondary: 22C05, 22E46, 58J50
Received: 9 May 2020
Revised: 17 May 2021
Accepted: 31 May 2021
Published: 10 November 2021
Emilio A. Lauret
Departamento de Matemática
Instituto de Matemática (INMABB)
Universidad Nacional del Sur (UNS)-CONICET
B8000CPB Bahía Blanca
Roberto J. Miatello
Universidad Nacional de Córdoba
5000 Córdoba