On a Zelevinsky Theorem and the Schur Indices of the Finite Unitary Groups

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 $(λ^G, χ)_G=1$. As an application, we shall determine the local Schur indices of an irreducible character of $G$ which belongs to such class