Let $G$ be a connected complex semi-simple Lie group and ${mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, wh
ich contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.
Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${mathcal{A}}_{rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its
coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${mathcal{A}}_{rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, pi_{G/Q}, {mathcal{A}}_{rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T subset G$ is a maximal torus, $B subset G$ a Borel subgroup, and $N$ the uniradical of $B$.
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 unde
r 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.
