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.

24 pages, Indagationes Mathematicae, Volume 25, Issue 5, October 2014

References in corpus (2)

• Symplectic structures associated to Lie-Poisson groups
• On spherical twisted conjugacy classes