Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen

The paper studies reductive groups acting (algebraically) on an affine space. Gerald Schwarz found the first examples which are not linearizable, i.e., where the action is not conjugate to a linear action.

The main result of the paper is that every connected, non-abelian, reductive group admits a non-linearizable action on some affine space. In particular, every semisimple group has such an action.

The method is reduction to SL(2,k) and then using Schwarz's examples for that group.

Appeared in: Inventiones Mathematicae 105 (1991) 217-220

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Remarks: 1. Using similar methods as in the paper one can construct non-linearizable actions for any reductive group whose connected component of unity is not a torus (unpublished).

2. Examples for some finite and other groups have been constructed by Masuda, Moser-Jauslin, Petrie.

3. No non-linearizable action is known (Oct. 1994) for an abelian reductive group.

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Last updated: August 25, 2006