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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.

Transverse Poisson structures to adjoint orbits in semisimple Lie algebras