Quotients by actions of the derived group of a maximal unipotent subgroup

TL;DR

This paper investigates the properties of quotient morphisms arising from the action of the derived subgroup of a maximal unipotent subgroup on affine varieties, including fiber structure and null-cone criteria.

Contribution

It establishes finiteness of invariants, analyzes fiber dimensions, and extends the Hilbert-Mumford criterion to $U'$-invariants in affine $G$-varieties.

Findings

All fibers of the quotient are equidimensional.

The algebra of $U'$-invariants is finitely generated.

An analogue of the Hilbert-Mumford criterion for null-cones is proven.

Abstract

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. If $X$ is an affine $G$-variety, then the algebra of $U'$-invariants, $k[X]^U'$, is finitely generated and the quotient morphism $\pi: X \to X//U'$ is well-defined. In this article, we study properties of such quotient morphisms, e.g. the property that all the fibres of $\pi$ are equidimensional. We also establish an analogue of the Hilbert-Mumford criterion for the null-cones with respect to $U'$-invariants.