On a spectral analogue of the strong multiplicity one theorem

We prove spectral analogues 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)