Is a smooth action of a semi-simple Lie group linearizable near a staionary point? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T05:34:23Z http://mathoverflow.net/feeds/question/48213 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48213/is-a-smooth-action-of-a-semi-simple-lie-group-linearizable-near-a-staionary-point Is a smooth action of a semi-simple Lie group linearizable near a staionary point? Dick Palais 2010-12-03T19:45:13Z 2010-12-10T14:15:27Z <p>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:</p> <p>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) </p> <p>Guillemin, V., Sternberg, S.: Remarks on a paper of Hermann. Trans. Am. Math. Soc. 130,110-116 (1968)</p> <p>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:</p> <p><a href="http://www.springerlink.com/content/wur75t1t65371812/" rel="nofollow">http://www.springerlink.com/content/wur75t1t65371812/</a></p> <p>for more details on this principle and why it is important, particularly in mathematical physics.</p> http://mathoverflow.net/questions/48213/is-a-smooth-action-of-a-semi-simple-lie-group-linearizable-near-a-staionary-point/48925#48925 Answer by BS for Is a smooth action of a semi-simple Lie group linearizable near a staionary point? BS 2010-12-10T14:15:27Z 2010-12-10T14:15:27Z <p>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 <a href="http://retro.seals.ch/cntmng?type=pdf&amp;rid=ensmat-001%3A1997%3A43%3A%3A80&amp;subp=hires" rel="nofollow">link</a> to their paper.</p>