Let A: be a global field and denote by A its ring of adeles. Let 0 be a unitary character of A x trivial on k. For the group = GLn, let C^ = L§(GA/Gfc,0) be the space of cusp forms on G with central character . The multiplicity one theorem, first proved by Jacquet-Langlands in [2] for n — 2, and by Shalika in [11] for all n, states that each irreducible unitary representation of G occurs with multiplicity at most one in C$. It is also well known that each irreducible unitary representation of G occurring in C^ has a factorization 7r = ® v 7rv, where each TTV is a representation of the local group Gv = GLn(kv). As a complement to the multiplicity one theorem, one has the following rigidity theorem (see [9, p. 209; 4, p. 552]; for n = 2, [1, p. 307; 7, p. 187]).