On the Popov–Pommerening conjecture for linear algebraic groups. (9th October 2017)

• Record Type: Journal Article
• Authors: Bérczi, Gergely
• Is Part Of: Compositio mathematica. Volume 154:Number 1(2018)
• Page Start: 36
• Page End: 79
• Publication Date: 2017-10-09
• Subjects: 14L24, -- 13A50 (primary); invariants, -- non-reductive groups; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007473
• Languages: English
• ISSNs: 0010-437X

Abstract

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 normalised by a maximal torus of$G$ and$X$ an affine$k$ -variety acted on by $G$ . Popov and Pommerening conjectured in the late 1970s that the invariant algebra$k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of$\operatorname{SL}_{n}(k)$ 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$\operatorname{SL}_{n}(k)$ .