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

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