Transverse Poisson structures to adjoint orbits in semisimple Lie algebras

Pantelis A. Damianou, Hervé Sabourin and Pol Vanhaecke

Vol. 232 (2007), No. 1, 111–138

DOI: 10.2140/pjm.2007.232.111

Abstract

We study the transverse Poisson structure to adjoint orbits in a complex semisimple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasihomogeneous coordinates, the quasidegree of the transverse Poisson structure is −2. For subregular nilpotent orbits, we show that the structure may be computed using a simple determinantal formula that involves 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, which is intimately related to the simple rational singularity that corresponds to the subregular orbit.

Keywords

transverse Poisson structure, nilpotent orbits, Kleinian singularities

Mathematical Subject Classification 2000

Primary: 53D17

Secondary: 17B10, 14J17

Milestones

Received: 27 May 2006

Revised: 27 September 2006

Accepted: 28 September 2006

Published: 1 September 2007