Paper detail

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

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 $θ$ of $G$, we introduce a holomorphic Poisson structure $π_θ$ 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 $π_θ$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $π_θ$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $π_θ$. We give the lowest rank of $π_θ$ in any given $θ$-twisted conjugacy class, and we relate the lowest possible rank locus of $π_θ$ in $G$ with spherical $θ$-twisted conjugacy classes of $G$. In particular, we show that $π_θ$ 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 $π_θ$ if and only if $C$ is spherical.