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Register a new userOn the $T$-leaves and the ranks of a Poisson structure on twisted conjugacy classes
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Jiang-Hua Lu
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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.
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