Nilpotent centralizers and good filtrations

TL;DR

This paper proves that modules with good filtrations for a connected reductive group also retain this property when restricted to the centralizer of a nilpotent element, under certain characteristic conditions.

Contribution

It establishes that good filtrations are preserved when restricting modules to the reductive part of a nilpotent centralizer in characteristic restrictions.

Findings

Modules with good filtrations remain good when restricted to nilpotent centralizers.

The result holds under mild restrictions on the characteristic of the base field.

Provides a link between module filtrations and nilpotent orbit structure.

Abstract

Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the reductive part of the centralizer of a nilpotent element $x$ in its Lie algebra.