1 results
On the Popov–Pommerening conjecture for linear algebraic groups
- Part of
-
- Journal:
- Compositio Mathematica / Volume 154 / Issue 1 / January 2018
- Published online by Cambridge University Press:
- 09 October 2017, pp. 36-79
- Print publication:
- January 2018
-
- Article
-
- You have access
- HTML
- Export citation
-
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 
$\operatorname{SL}_{n}(k)$.
