On the T-leaves of some Poisson structures related to products of flag varieties

TL;DR

This paper develops a general theory of T-leaves and leaf stabilizers for Poisson structures influenced by Lie bialgebra actions, and applies it to describe the T-leaf decompositions of Poisson structures on products of flag varieties.

Contribution

It introduces a unified framework for T-leaves in Poisson structures from Lie bialgebra actions and applies it to complex semi-simple Lie groups' flag varieties.

Findings

Describes T-leaf decompositions via extended Richardson varieties and double Bruhat cells.

Computes leaf stabilizers and symplectic leaf dimensions within T-leaves.

Provides explicit descriptions for Poisson structures on products of flag varieties.

Abstract

For a connected abelian Lie group T acting on a Poisson manifold (Y,{\pi}) by Poisson isomorphisms, the T-leaves of {\pi} in Y are, by definition, the orbits of the symplectic leaves of {\pi} under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves in the T-leaf. In this paper, we first develop a general theory on T-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures on products of flag varieties and related spaces of a complex semi-simple Lie group G. We describe their T-leaf decompositions, where T is a maximal torus of G, in terms of (open) extended Richardson varieties and extended double Bruhat cells associated to conjugacy classes of G, and we compute their leaf stabilizers and the dimension of the symplectic leaves in each T -leaf.