Liftings of pseudo-reflection groups of toric quotients of Krull schemes

TL;DR

This paper investigates the conditions under which pseudo-reflections in the quotient of a Krull scheme by a torus action can be lifted to the original scheme, focusing on algebraic group actions and quotient fields.

Contribution

It establishes that pseudo-reflections of a group action on the invariant ring can be lifted to the original ring when the group is the centralizer of a torus, under certain quotient field conditions.

Findings

Pseudo-reflections can be lifted when G is the centralizer of T.

The quotient field condition ${Q}(R)^{T}= {Q}(R^{T})$ is crucial.

The result applies to affine Krull schemes with reductive group actions.

Abstract

Let $G$ be an affine algebraic group with a reductive identity component $G^{0}$ acting regularly on an affine Krull scheme $X = {Spec} (R)$ over an algebraically closed field. Let $T$ be an algebraic subtorus of $G$ and suppose that ${Q}(R)^{T}= {Q}(R^{T})$ of quotient fields. We will show: If $G$ is the centralizer of $T$ in $G$, then the pseudo-reflections of the action of $G$ on $R^{T}$ can be lifted to those on $R$.