Good filtrations and strong $F$-regularity of the ring of $U_P$-invariants

Abstract: Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $P$ be a parabolic subgroup of $G$, and $U_P$ its unipotent radical. We prove that if $S$=\textyen $Sym V$ has a good filtration, then the ring of invariants $S^{U_P}$ is strongly $F$-regular.