Strong multiplicity one theorems for locally homogeneous spaces of compact type

Abstract

Let G be a compact connected semisimple Lie group, let K be a closed subgroup of G, let Γ be a finite subgroup of G, and let τ be a finitedimensional representation of K. For π in the unitary dual G of G, denote by nΓ(π) its multiplicity in L2(Γ\G). We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the nΓ(π) for π in the set Gτ of irreducible τ-spherical representations of G. More precisely, for Γ and Γ finite subgroups of G, we prove that if nΓ(π) = nΓ (π) for all but finitely many π ∈ Gτ , then Γ and Γ are τ-representation equivalent, that is, nΓ(π) = nΓ (π) for all π ∈ Gτ . Moreover, when Gτ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset Fτ of Gτ verifying some mild conditions, the values of the nΓ(π) for π ∈ Fτ determine the nΓ(π)’s for all π ∈ Gτ . In particular, for two finite subgroups Γ and Γ of G, if nΓ(π) = nΓ (π) for all π ∈ Fτ , then the equality holds for every π ∈ Gτ . We use algebraic methods involving generating functions and some facts from the representation theory of G.