SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

• Published in: Transformation groups Vol. 22; no. 2; pp. 475 - 485
• Main Authors: KUTZSCHEBAUCH, FRANK, LÁRUSSON, FINNUR, SCHWARZ, GERALD W.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2017; Springer Nature B.V
• Subjects: Algebra; Isomorphism; Lie Groups; Linearization; Mathematical analysis; Mathematics; Mathematics and Statistics; Representations; Stratification; Stratigraphy; Subgroups; Topological Groups; Closed Orbit; Categorical Quotient; Stein Manifold; Principal Orbit; Reductive Subgroup
• ISSN: 1083-4362; 1531-586X
• DOI: 10.1007/s00031-016-9376-7

Let G be a reductive complex Lie group acting holomorphically on X = ℂ n . The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂ n such that the G -action becomes linear. Equivalently, is there a G -equivariant biholomorphism Φ: X → V where V is a G -module? There is an intrinsic stratification of the categorical quotient Q X , called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G . Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: Q X → Q V which is stratified, i.e., the stratum of Q X with a given label is sent isomorphically to the stratum of Q V with the same label. The counterexamples to the Linearisation Problem construct an action of G such that Q X is not stratified biholomorphic to any Q V .Our main theorem shows that, for most X , a stratified biholomorphism of Q X to some Q V is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to ℂ n , only that X is a Stein manifold.