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We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classesof Schubert cells in a generalized flag manifold G/B. In analogy to how theordinary divided difference operators act on Schubert classes, each CSM classof a Schubert class is obtained by applying certain Demazure-Lusztig typeoperators to the CSM class of a cell of dimension one less. These operatorsdefine a representation of the Weyl group on the homology of G/B. Byfunctoriality, we deduce algorithmic expressions for CSM classes of Schubertcells in any flag manifold G/P. We conjecture that the CSM classes of Schubertcells are an effective combination of (homology) Schubert classes, and provethat this is the case in several classes of examples. We also extend ourresults and conjectures to the torus equivariant setting.

Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds