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On the Popov–Pommerening conjecture for linear algebraic groups

Published online by Cambridge University Press:  09 October 2017

Gergely Bérczi*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford OX2 6GG, UK email berczi@maths.ox.ac.uk

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 $\operatorname{SL}_{n}(k)$ .

Type
Research Article
Copyright
© The Author 2017 

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On the Popov–Pommerening conjecture for linear algebraic groups
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