Equivariant Chern-Schwartz-MacPherson classes in partial flag varieties: interpolation and formulae

Consider the natural torus action on a partial flag manifold $Fl$. Let ${\mathrm{\Omega }}_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}({\mathrm{\Omega }}_I)$, as well as a formula, of `localization type’, for $c^{sm}({\mathrm{\Omega }}_I)$. In fact, we proved similar results for a class ${\kappa }_I\in H_T^{\ast }(Fl)$ — in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}({\mathrm{\Omega }}_I)={\kappa }_I$.