On the T-leaves and the ranks of a Poisson structure on twisted conjugacy classes

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Let G be a connected complex semisimple Lie group with a fixed maximal torus T and a Borel subgroup B ⊃ T. For an arbitrary automorphism θ of G, we introduce a holomorphic Poisson structure πθ on G which is invariant under the θ-twisted conjugation by T and has the property that every θ-twisted conjugacy class of G is a Poisson subvariety with respect to πθ. We describe the T-orbits of symplectic leaves, called T-leaves, of πθ and compute the dimensions of the symplectic leaves (i.e, the ranks) of πθ. We give the lowest rank of πθ in any given θ-twisted conjugacy class, and we relate the lowest possible rank locus of πθ in G with spherical θ-twisted conjugacy classes of G. In particular, we show that πθ vanishes somewhere on G if and only if θ induces an involution on the Dynkin diagram of G, and that in such a case a θ-twisted conjugacy class C contains a vanishing point of πθ if and only if C is spherical.