Exotic algebraic group actions. (Actions exotiques des groupes algébriques). (English) Zbl 0688.14040

Schwarz, Gerald W.

C. R. Acad. Sci., Paris, Sér. I 309, No. 2, 89-94 (1989).

Let G be a reductive complex algebraic group and let V and W be G- modules. Let E be an element of the class \(Vec_ G(V,W)\) of all G-vector bundles over V with typical fiber W. By a theorem of Bass and Haboush the G-action on \(E\simeq {\mathbb{A}}^ n\) is algebraically linearizable only if the Whitney sum \(E\oplus (V\times V)\in Vec_ G(V,W\oplus V)\) is G- isomorphic to the trivial bundle. Using this theorem the author gives examples for non-linearizable G-actions on \({\mathbb{A}}^ n.\)
The paper contains the following results: Let \(I_ G(V,W)\) denote the set of G-isomorphism classes in \(Vec_ G(V,W)\). If the categorical quotient V//G is one-dimensional, then \(I_ G(V,W)\) is the abelian group \({\mathbb{A}}^ r\) for some \(r\in {\mathbb{N}}\). In particular for \(G:=O(2,{\mathbb{C}})={\mathbb{Z}}_ 2\ltimes {\mathbb{C}}^*\) and \(V_ m\) the two- dimensional irreducible G-module with weight m and \(-m\) as \({\mathbb{C}}^*\)- module, \(I_ G(V_ 2,V_ m)\simeq {\mathbb{A}}^{(m-1)/2}\) for m odd. The operator \(E\to E\oplus (V_ 2\times V_ 2)\) induces an isomorphism \(I_ G(V_ 2,V_ m){\tilde \to}I_ G(V_ 2,V_ m\oplus V_ 2)\) and therefore ensures the existence of an algebraically non-linearizable O(2,\({\mathbb{C}})\)-action on \({\mathbb{A}}^ 4\). An explicit example is given for \(m=3\). With \(G:=SL(2,{\mathbb{C}})\) and \(R_ m\) the G-module of binary forms of degree \( m\) the operator \(E\to E\oplus (R_ m\times R_ m)\) induces an isomorphism \(I_ G(R_ 2,R_ m){\tilde \to}I_ G(R_ 2,R_ m\oplus R_ 2)\simeq {\mathbb{A}}^{k^ 2}\) for \(m=2k+1\), \(k\geq 1\), and has a non-trivial image for m even, \(m\geq 3\). This ensures the existence of algebraically non-linearizable \(SL(2,{\mathbb{C}})\)-actions on \({\mathbb{A}}^ 7\) and \(SO(3,{\mathbb{C}})\)-actions on \({\mathbb{A}}^{10}\).

Reviewer: E.Oeljeklaus

MSC:

14L30	Group actions on varieties or schemes (quotients)
20G20	Linear algebraic groups over the reals, the complexes, the quaternions
20G05	Representation theory for linear algebraic groups

Keywords:

rench; reductive complex algebraic group; examples for non-linearizable G-actions