On a spectral analog of the strong multiplicity one theorem

Abstract

We prove spectral analogs of the classical strong multiplicity one theorem for newforms. Let Γ₁ and Γ₂ be uniform lattices in a semisimple group G. Suppose all but finitely many irreducible unitary representations (resp. spherical) of G occur with equal multiplicities in L²(Γ₁\G) and L²(Γ₂\G). Then L²(Γ₁\G)≅L²(Γ₂\G) as G modules (resp. the spherical spectra of L²(Γ₁\G) and L²(Γ₂\G) are equal).