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

Abstract

We consider the conormal bundle of a Schubert variety $S_I$ in the cotangent bundle $T^{\ast }Gr$ of the Grassmannian $Gr$ of $k$-planes in $C^n$. This conormal bundle has a fundamental class $\hat{I}{º}_I$ in the equivariant cohomology $H_T^{\ast }(T^{\ast }Gr)$. Here $T=(C^{\ast }{)}^n\times C^{\ast }$. The torus $(C^{\ast }{)}^n$ acts on $T^{\ast }Gr$ in the standard way and the last factor $C^{\ast }$ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum $Y_I$ of the Yangian $Y(g{l}_2)$ weight functions $(W_J{)}_J$. We describe a relation of $Y_I$ with the double Schur polynomial $[S_I]$. A modified version of the $\hat{I}{º}_I$ classes, named $\hat{I}{º}_I^{\prime }$, satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold $T^{\ast }Gr$. This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on $Gr$. The classes $(\hat{I}{º}_I^{\prime }{)}_I$ form a basis in the suitably localized equivariant cohomology $H_T^{\ast }(T^{\ast }Gr)$. This basis depends on the choice of the coordinate flag in $C^n$. We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.

New sections and references are added, introduction is extended. Typo corrected on page 5