  We study the Demazure-Lusztig operators induced by the left multiplication on
partial flag manifolds $G/P$. We prove that they generate the
Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant
cohomology), respectively their motivic Chern classes (in equivariant K
theory), in any partial flag manifold. Along the way we advertise many
properties of the left and right divided difference operators in cohomology and
K theory, and their actions on Schubert classes. We apply this to construct
left divided difference operators in equivariant quantum cohomology, and
equivariant quantum K theory, generating Schubert classes, and satisfying a
Leibniz rule compatible with the quantum product.
