Good filtrations and $F$-purity of invariant subrings

TL;DR

This paper proves that if the symmetric algebra of a G-module has a good filtration, then its invariants under the unipotent radical are F-pure, linking representation theory with Frobenius properties in positive characteristic.

Contribution

It establishes a new connection between good filtrations of symmetric algebras and F-purity of invariant subrings in positive characteristic.

Findings

Symmetric algebra with good filtration implies F-purity of U-invariants.

Links representation-theoretic properties to Frobenius splitting.

Provides conditions for F-purity in invariant theory.

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 $B$ 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.