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

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