Poisson geometry of the principal series and nonlinearizable structures
If \({\mathfrak g}\) is a Lie algebra (over \({\mathbb{R}})\), then the dual space \({\mathfrak g}^*\) carries a linear Poisson structure \(\pi_ 0\) called the Lie-Poisson structure. The linearization question is whether a Poisson structure \(\pi\) defined near 0 on \({\mathfrak g}^*\) and having the same 1- jet as \(\pi_ 0\) at 0 is equivalent to \(\pi_ 0\) on a neighborhood of 0. The answer to the linearization question is yes if \({\mathfrak g}\) is semisimple and \(\pi\) is analytic [\textit{J. F. Conn}, Ann. Math., II. Ser. 119, 577-601 (1984; Zbl 0553.58004)] or if \({\mathfrak g}\) is semisimple of compact type and \(\pi\) is \(C^{\infty}\) [\textit{J. F. Conn}, ibid. 121, 565-593 (1985; see the following review)], but it can be no if \({\mathfrak g}={\mathfrak sl}(2,{\mathbb{R}})\) and \(\pi\) is \(C^{\infty}\) [the author, J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011)]. In this paper, we show that the answer can be no for a \(C^{\infty}\) \(\pi\) if \({\mathfrak g}\) is any semisimple algebra of noncompact type with real rank at least two. The cases of real rank one other than \({\mathfrak sl}(2,{\mathbb{R}})\) are still open. The construction of nonlinearizable examples involves an analysis of the Poisson geometry of the subset of \({\mathfrak g}^*\) corresponding to the principal series representations. This analysis, in turn, relies on showing the functorial nature of the phase space for a classical particle in a Yang-Mills field [\textit{S. Sternberg}, Proc. Natl. Acad. Sci., USA 74, 5253-5254 (1977); the author, Lett. Math. Phys. 2, 417-420 (1978; Zbl 0388.58010)]. A by-product of our results is an analog in Poisson geometry to the correspondence between certain representations of \({\mathfrak g}\) and of the associated Cartan motion algebra [see, for instance, \textit{G. W. Mackey}, in Lie groups and their representations, 1971, 339-363 (1975; Zbl 0324.22006)].
scientific article; zbMATH DE number 3951634