Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

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Let G be a connected linear semisimple Lie group with Lie algebra g, and let KC --> Aut(pC) be the complexified isotropy representation at the identity coset of the corresponding symmetric space G/K. Suppose that O is a nilpotent G-orbit in g and O* is the nilpotent KC-orbit in pC associated to O by the Kostant-Sekiguchi correspondence. We show that the complexity of O* as a KC variety measures the failure of the Poisson algebra of smooth K-invariant functions on O to be commutative.