Let G be the finite unitary group U_n(\hbox{\tenbx F}_q) over a finite field \hbox{\tenbx F}_q of characteristic p. Let U be a Sylow p-subgroup of G. We prove that, for any irreducible character χ of G that is contained in a certain class, there is a linear character λ of U such that (λG,χ)G=1. As an application, we shall determine the local Schur indices of an irreducible character of G which belongs to such class