2 2 D ec 1 99 9 Linearization Problems for Lie Algebroids and Lie Groupoids In memory of Moshé Flato , 1938 - 1999

Why is it so hard to prove the linearizability of Poisson structures with semisimple linear part? Conn published proofs about 15 years ago in a pair of papers [5][6] full of elaborate estimates. Except for the somewhat more conceptual reformulation by Desolneux-Moulis [9] in the smooth case, no simplification of Conn’s proofs has appeared. This is a mystery to me, because analogous theorems about the linearizability of actions of semisimple groups near their fixed points were proven in the compact (smooth or analytic) case by Bochner [3] using a simple averaging method and in the noncompact analytic case by Guillemin–Sternberg [13] and Kushnirenko [18], who used analytic continuation from the compact case–a nonlinear version of “Weyl’s unitary trick”. Hermann [16] established formal linearization for actions of general semisimple algebras, using cohomological methods similar to those which will appear several times in the present report.