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](https://doi.org/10.1090/S0002-9947-1968-0217225-7). Trans. Am. Math. Soc. 130, 105-109 (1968) 


Guillemin, V., Sternberg, S.: [Remarks on a paper of Hermann](https://doi.org/10.1090/S0002-9947-1968-0217226-9). 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:


<http://www.springerlink.com/content/wur75t1t65371812/>


for more details on this principle and why it is important, particularly in mathematical physics.



Is a smooth action of a semi-simple Lie group linearizable near a stationary point?
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](https://www.e-periodica.ch/digbib/view?pid=ens-001:1997:43::341#341) to their paper.


