2.
We consider the conormal bundle of a Schubert variety
\(S_I\) in the cotangent bundle
\(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian
\({{\mathrm{\mathrm {Gr}}}}\) of
\(k\) -planes in
\({{\mathrm{\mathbb {C}}}}^n\) . This conormal bundle has a fundamental class
\({\kappa _I}\) in the equivariant cohomology
\(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . Here
\({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\) . The torus
\(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on
\(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor
\({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum
\(Y_I\) of the Yangian
\(Y(\mathfrak {gl}_2)\) weight functions
\((W_J)_J\) . We describe a relation of
\(Y_I\) with the double Schur polynomial
\([S_I]\) . A modified version of the
\(\kappa _I\) classes, named
\(\kappa '_I\) , satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold
\(T^*\!{{\mathrm{\mathrm {Gr}}}}\) . This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on
\({{\mathrm{\mathrm {Gr}}}}\) . The classes
\((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology
\(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . This basis depends on the choice of the coordinate flag in
\({{\mathrm{\mathbb {C}}}}^n\) . We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.
相似文献