Remarks on a Paper of Hermann

Victor W. Guillemin, Shlomo Sternberg
1968 Transactions of the American Mathematical Society  
Let G be a Lie group acting differentiably on a manifold and let p be a point left fixed by G. If G is compact, a well-known result of H. Cartan asserts that the action of G in a neighborhood of p is equivalent to the linear action of G on the tangent space at p, cf. [4]. Palais and Smale have suggested extending this theorem to noncompact Lie groups. Hermann, in the preceding paper has shown that the corresponding formal problem can be reduced to a cohomology question which can always be
more » ... if the group is semisimple (cf. the discussion of this point in §1 below). In this paper we will concentrate on the semisimple case. If the group G is connected, then instead of trying to find a linear system of coordinates for G, one can try to find a linear system of coordinates for the vector fields corresponding to the one-parameter subgroups of G. More generally one can just consider a representation of a semisimple Lie algebra as an algebra of vector fields on a manifold, and try to determine the local behavior in the neighborhood of a common zero of all the vector fields. Hermann [2] proved that with the hypotheses above and C°° or analytic data one can always find a formal power series change of coordinates which formally linearizes all the vector fields at the critical point. However, it seems quite difficult to get a convergence proof by Hermann's method except in the relatively few cases where the power series change of coordinates is unique. In this paper we use an alternate approach based on complexification and the Weyl unitary trick to prove a linearization theorem if the data are analytic. We will also show that the analogous theorem is false in the C* case unless some restrictions are placed on the algebra. What restrictions is unclear at present, but it seems that the algebra sl(2, R) has to be singled out for special consideration.
doi:10.2307/1994774 fatcat:4zhb6cc6sbc77ivhlytdkte6i4