# Is a smooth action of a semi-simple Lie group linearizable near a staionary point?

Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any stationary point of the action? This was conjectured by Steve Smale and myself in 1965, and was proved for the case that $M$ and the action were analytic by Bob Herman and Guillemin and Sternberg in two papers from long ago:

Hermann, R.: The formal linearization of a semi-simple Lie algebra of vector fields about a singular point. Trans. Am. Math. Soc. 130, 105-109 (1968)

Guillemin, V., Sternberg, S.: Remarks on a paper of Hermann. Trans. Am. Math. Soc. 130,110-116 (1968)

I have not heard whether any progress has been made since then and I would be interested to hear from anyone who has heard of a proof or a counter-example. The reason is not just idle curiousity; this is the missing step in a proof that what I call The Principle of Symmetric Criticality is valid for smooth finite dimensional actions of a semi-simple group: see (particularly page 29 of) the paper downloadable here:

There are smooth counter-examples by Cairns and Ghys [Ens. Math. 43, 1997], for instance a smooth non-linearizable action of $SL(2,\mathbb{R})$ on $\mathbb{R}^3$ (fixing the origin) or of $SL(3,\mathbb{R})$ on $\mathbb{R}^8$. By contrast, they show that any $C^k$ action of $SL(n,\mathbb{R})$ on $\mathbb{R}^n$ (same $n$, fixing the origin) is $C^k$-linearizable. Here is a link to their paper.