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

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