On the Popov-Pommerening conjecture for linear algebraic groups

TL;DR

This paper proves the Popov-Pommerening conjecture for certain subgroups of classical groups, showing their invariant algebras are finitely generated, advancing understanding of invariants in algebraic group actions.

Contribution

The paper confirms the conjecture for subgroups of SL_n(k) under Borel actions and for some Borel regular subgroups of classical groups, providing partial results for general cases.

Findings

Proved the conjecture for subgroups of SL_n(k) with Borel action.

Established the conjecture for a class of Borel regular subgroups.

Provided partial results for the conjecture in the general case.

Abstract

Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H \subseteq G$ an observable subgroup normalized by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 70's that the invariant algebra $k[X]^H$ is finitely generated. We prove the conjecture for 1) subgroups of $\mathrm{SL}_n(k)$ closed under left (or right) Borel action and for 2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\mathrm{SL}_n(k)$.