STRONG REPRESENTATION EQUIVALENCE FOR COMPACT SYMMETRIC SPACES OF REAL RANK ONE

Let G/K be a simply connected compact irreducible symmetric space of real rank one. For each K -type tau we compare the notions of tau-representation equivalence with tau-isospectrality. We exhibit infinitely many K-types tau so that, for arbitrary discrete subgroups Gamma and Gamma' of G, if the multiplicities of lambda in the spectra of the Laplace operators acting on sections of the induced tau-vector bundles over Gamma\G/K and Gamma'\G/K agree for all but finitely many lambda, then Gamma and Gamma' are tau-representation equivalent in G (i.e., dim Hom(G)(V-pi, L-2(Gamma \G)) = dim Hom(G) (V-pi, L-2 (Gamma'\G)) for all pi is an element of (G) over cap satisfying Hom(K) (V tau, V-pi) not equal 0). In particular, Gamma\G/K and Gamma'\G/K are tau-isospectral (i.e., the multiplicities agree for all lambda). We specially study the case of p-form representations, i.e., the irreducible subrepresentations tau of the representation tau(p) of K on the p-exterior power of the complexified cotangent bundle Lambda(p) T-C*M. We show that for such tau, in most cases tau-isospectrality implies tau-representation equivalence. We construct an explicit counterexample for G/K = SO(4n)/ SO(4n -1) similar or equal to S4n-1.

Automated summary

This paper investigates the relationship between different K-types in a symmetric space, comparing equivalence and isospectrality between different types. It particularly focuses on p-form representations and shows that in most cases isospectrality implies representation equivalence. An explicit counterexample is also constructed for a specific scenario.