STRONG MULTIPLICITY ONE THEOREMS FOR LOCALLY HOMOGENEOUS SPACES OF COMPACT-TYPE

Let G be a compact connected semisimple Lie group, let K be a closed subgroup of G, let Gamma be a finite subgroup of G, and let tau be a finite-dimensional representation of K. For pi in the unitary dual (G) over cap of G, denote by n(Gamma)(pi) its multiplicity in L-2 (Gamma\G). We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the n(Gamma)(pi) for pi in the set (G) over cap (tau) of irreducible tau-spherical representations of G. More precisely, for Gamma and Gamma' finite subgroups of G, we prove that if n(Gamma)(pi) = n(Gamma')(pi) for all but finitely many pi is an element of (G) over cap (tau), then Gamma and Gamma' are tau-representation equivalent, that is, n(Gamma)(pi) = n(Gamma')(pi) for all pi is an element of (G) over cap (tau). Moreover, when (G) over cap (tau) can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset (F) over cap (tau) of (G) over cap (tau) verifying some mild conditions, the values of the n(Gamma)(pi) for pi is an element of (F) over cap (tau. )determine the n(Gamma)(pi)'s for all pi is an element of (G) over cap tau. In particular, for two finite subgroups Gamma and Gamma' of G, if n(Gamma)(pi) = n(Gamma')(pi) for all pi is an element of (F) over cap tau, then the equality holds for every pi is an element of (G) over cap tau. We use algebraic methods involving generating functions and some facts from the representation theory of G.