Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space

Abstract

The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M_{m,n}(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GL_{m+n}(C). Three different presentations of the T-orbits of symplectic leaves in M_{m,n}(C) are obtained - (a) as pullbacks of Bruhat cells in GL_{m+n}(C) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GL_m(C) and GL_n(C). In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group S_{m+n}, such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed row-echelon form. Finally, decompositions of generalized double Bruhat cells in M_{m,n}(C) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.