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

Abstract

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.

Mixed Product Poisson Structures Associated to Poisson Lie Groups and Lie Bialgebras

“…, Q n are closed subgroups of G, let (1.2) Z = G × Q1 · · · × Qn−1 G/Q n T -actions on the first factors of G n and (G × G) n , preserve the Poisson structures π Z . In [30], a sequel to the present paper, we describe the T -leaves and compute the ranks of all the symplectic leaves in (Z, π Z ) for Z ∈ {F n , F n , F n , F n } by first developing a general theory on torus orbits of symplectic leaves for Poisson structures defined by quasitriangular r-matrices. Our descriptions of the T -leaves of these Poisson manifolds are in terms of extended Bruhat cells, extended Richardson varieties, and extended Double Bruhat cells associated to conjugacy classes (see [30] for detail).…”

On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

“…Let T be a complex torus. A T-Poisson manifold, defined in [16,Section 1.1], is a complex Poisson manifold (X, π X ) with a T-action preserving the Poisson structure π X . A T-Poisson manifold gives rise to a decomposition of X into the union of T-leaves of π X , a term defined in [16,Section 2.2].…”

“…A T-Poisson manifold, defined in [16,Section 1.1], is a complex Poisson manifold (X, π X ) with a T-action preserving the Poisson structure π X . A T-Poisson manifold gives rise to a decomposition of X into the union of T-leaves of π X , a term defined in [16,Section 2.2]. Every single T-leaf in (X, π X ) admits a non-zero anti-canonical section on X called the Poisson T-Pfaffian, a term suggested by A. Knutson and M. Yakimov (see [16,Section 1.1]).…”

“…A T-Poisson manifold gives rise to a decomposition of X into the union of T-leaves of π X , a term defined in [16,Section 2.2]. Every single T-leaf in (X, π X ) admits a non-zero anti-canonical section on X called the Poisson T-Pfaffian, a term suggested by A. Knutson and M. Yakimov (see [16,Section 1.1]). When k = C, G becomes a complex semi-simple Lie group.…”

“…Denote by k an algebraically closed field with char(k) = p > 2. The purpose of this section is to explain a few aspects of Poisson algebras over k, and in particular the notion of Poisson T-Pfaffians, T being an algebraic torus, which in the case of characteristic 0 is introduced in[16]. Note that a special feature is that, given a Poisson algebra A and any element a ∈ A, the ideal a p is a Poisson ideal since {a p b, c} = a p {b, c} + pa p−1 b{a, c} = a p {b, c} ∈ a p , b, c ∈ A.…”