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

Abstract

For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π 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.

37 pages, submitted to Advances in Mathematics

References in corpus (6)

• Quantum cluster algebras and quantum nilpotent algebras
• Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties
• Bott-Samelson varieties and Poisson Ore extensions
• On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups
• Deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes
• Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras

Cited by in corpus (4)

• Bott-Samelson varieties and Poisson Ore extensions
• Cluster Structures on Double Bott-Samelson Cells
• Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
• Double Bruhat cells and symplectic groupoids