2006/05/25 by Pantelis A. Damianou, Damianou, Pantelis A., Herve Sabourin +3
Mathematics · #14J17 #17B10 #53D17 #Differential Geometry (math.DG) #FOS: Mathematics #Representation Theory (math.RT) #math.DG #math.RT #msc:14J17 #msc:17B10 #msc:53D17
paper · pdf · doi:10.48550/arxiv.math/0605660
22 Pages, 1 Figure
arxiv created 2006/05/25 · arxiv updated 2009/12/01
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of \emph subregular nilpotent orbits we show that the structure may be computed by means of a simple determinantal formula, involving the restriction of the Chevalley invariants on the slice. In addition, using results of Brieskorn and Slodowy, the Poisson structure is reduced to a three dimensional Poisson bracket, intimately related to the simple rational singularity that corresponds to the subregular orbit.