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