Poisson structures on complex flag manifolds associated with real forms

Abstract

Abstract. For a complex semisimple Lie group G and a real form G 0 we define a Poisson structure on the variety of Borel subgroups of G with the property that all G 0 -orbits in X as well as all Bruhat cells (for a suitable choice of a Borel subgroup of G) are Poisson submanifolds. In particular, we show that every non-empty intersection of a G 0 -orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

New Realizations of the Maximal Satake Compactifications of Riemannian Symmetric Spaces of Noncompact Type

“…[2] on our study of certain 'moduli space' of Poisson homogeneous spaces, which was in turn motivated by the theory of quantum groups. One can show (see [3,Section 3]) that there is a natural Poisson structure p on X ¼ G=K which extends tolðXÞ. Corollary 1.3 will enable us to use the structure theory of X S max to study the boundary behavior of p on X.…”

“…Corollary 1.3 will enable us to use the structure theory of X S max to study the boundary behavior of p on X. We will carry out this study in a future paper, and we refer to [3] and [2] for the related background on Poisson geometry. As is explained in [4], there are characterizations of X S max from various points of view, such as that of Riemannian geometry, of the theory of random walks, and of harmonic analysis on X, each of which has its own advantage and sheds lights on the others.…”

“…It is easy to prove that (4) is a disjoint union, and cðA I;þ Þ Á d I ffi cðA I;þ Þ for each I. Moreover, a computation similar to the one that leads to (3) shows that a sequence expðk n Þ Á d I 1 2 cðA I 1 ;þ Þ Á d I 1 converges to expðkÞ Á d…”

The Poisson geometry of SU(1,1)

“…where the first term is the so-called Π v structure from [5], and the second is an SU(1, 1)-invariant Poisson structure on S 2 .…”

A Note on Iwasawa‐Type Decomposition

“…In Poisson geometry the groups SU(p, q) and AN (the upper-triangular subgroup of SL(n, C) with real positive diagonal entries) are naturally dual to each other [4]. Therefore, it is important to know the geometry of the orbits of the dressing action.…”