Let A be a symmetric positive definite Cartan matrix. As in [4], we denote by U the quantum group arising from A and $U_lambda$ be the corresponding quantum group at a root of unity $lambda$. In [4], Lusztig constructed irreducible highest weight $U_lambda$-modules $L_lambda(z)$ for $z in Z^n$ and showed that $L_lambda(z)$ is of finite dimension over C if and only if $z in (Z^+)^n$.