  For a dominant integral weight $lambda $, we introduce a family of $U_q ^+
(mathfrak{g})$-submodules $V_w (lambda)$ of the irreducible highest weight $U_q
(mathfrak{g})$-module $V(lambda)$ of highest weight $lambda $ for a generalized
Kac--Moody algebra $mathfrak{g}$. We prove that the module $V_w (lambda)$ is
spanned by its global basis, and then give a character formula for $V_w
(lambda)$, which generalizes the Demazure character formula for ordinary
Kac--Moody algebras.
