Smooth linearization of sl(2,R) and SL(2,R) actions ?

This is a particular case of a larger problem of linearization of Lie group and Lie algebra actions. The case of sl(2,R) and SL(2,R) is already non-trivial.

Guillemin and Sternberg gave an example of a smooth sl(2,R) action on R3 which is not linearizable.

Cairns and Ghys gave an example of a smooth SL(2,R) action on R3 which is not linearizable. The example of Cairns & Ghys is a perturbation of the coadjoint action, which admits a 3-dimensional orbit, so its nonlinearizable for obvious topologivcal reasons.

Conjecture: Assume that we have the following conditions

– the dimension of the orbits = the dimension of the orbits of the linear action.

– a closedness (or adherence) condition for the orbits

then the action linearizable.

In other words, if the action is non-lineariable, then it’s due to topological reasons which are easy to detect.

If there is no “ovbious” topological obstruction then the action is linearizable.

Remark: Since sl(2) is simple, the action is always formally linearizable near 0.

Remark: Cairns & Ghys also constructed perturbations of the coadjoint representation of SL(3,R) with open orbits.

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