Divided difference operators in the
equivariant cohomology of Grassmannians

Divided difference operators were introduced by Bernstein, Gelfand, and Gelfand to find explicit maps between algebraic constructions of the cohomology of G/P and geometric constructions. A third, more combinatorial construction of the (equivariant) cohomology of G/P was discovered independently by Kostant and Kumar and, in more generality, by Goresky, Kottwitz, and MacPherson (GKM). This method builds the equivariant cohomology of G/P by applying an algebraic algorithm to a combinatorial graph based on the geometry of G/P. We show how to construct divided difference operators using certain automorphisms of this combinatorial graph. One application is a generalization of Billey's formula for the localizations of equivariant Schubert classes to arbitrary G/P; other applications will be discussed as time permits.