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

TL;DR

This paper proves that if the symmetric algebra of a G-module has a good filtration, then the invariants under the unipotent radical of a parabolic subgroup are strongly F-regular, linking representation theory with singularity properties.

Contribution

It establishes a new connection between good filtrations in representation theory and strong F-regularity of invariant rings in positive characteristic.

Findings

Symmetric algebra with a good filtration implies strongly F-regular invariants.

Invariants under unipotent radicals exhibit strong F-regularity under certain conditions.

Bridges the gap between representation theory and singularity theory in algebraic geometry.

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.