  Let $k$ be an algebraically closed field of positive characteristic, $G$ a
reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $B$ be a
Borel subgroup of $G$, and $U$ its unipotent radical. We prove that if $S=\Sym
V$ has a good filtration, then $S^U$ is $F$-pure.
