Poisson structures on affine spaces and flag varieties. II. General case

TL;DR

This paper studies the geometry of Poisson structures on flag varieties, revealing the structure of symplectic leaf orbits, their closures, and special Poisson subvarieties, with new proofs and explicit computations.

Contribution

It provides new proofs and explicit descriptions of symplectic leaf orbits and their closures in flag varieties with Poisson structures, especially for Hermitian symmetric spaces.

Findings

Orbits of symplectic leaves are smooth, irreducible, and isomorphic to intersections of dual Schubert cells.

Zariski closures of these orbits are explicitly computed.

Poisson structures vanish at special base points related to Richardson, R"ohrle, and Steinberg.

Abstract

The standard Poisson structures on the flag varieties G/P of a complex reductive algebraic group G are investigated. It is shown that the orbits of symplectic leaves in G/P under a fixed maximal torus of G are smooth irreducible locally closed subvarieties of G/P, isomorphic to intersections of dual Schubert cells in the full flag variety G/B of G, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson--Dirac submanifolds. For a parabolic subgroup P with abelian unipotent radical (in which case G/P is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor L of P on G/P are complete Poisson subvarieties which are quotients of L, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on G/P vanishes at all special base points for the L-orbits on G/P constructed by Richardson, R\"ohrle, and Steinberg.