Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

Abstract

We consider the conormal bundle of a Schubert variety SI in the cotangent bundle T∗Gr of the Grassmannian Gr of k-planes in Cn. This conormal bundle has a fundamental class κI in the equivariant cohomology H∗T(T∗Gr). Here T=(C∗)n×C∗. The torus (C∗)n acts on T∗Gr in the standard way and the last factor C∗ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum YI of the Yangian Y(gl2) weight functions (WJ)J. We describe a relation of YI with the double Schur polynomial [SI]. A modified version of the κI classes, named κ′I, satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold T∗Gr. This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on Gr. The classes (κ′I)I form a basis in the suitably localized equivariant cohomology H∗T(T∗Gr). This basis depends on the choice of the coordinate flag in Cn. We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.