  Divided difference operators are degree-reducing operators on the cohomology
of flag varieties that are used to compute algebraic invariants of the ring
(for instance, structure constants). We identify divided difference operators
on the equivariant cohomology of G/P for arbitrary partial flag varieties of
arbitrary Lie type, and show how to use them in the ordinary cohomology of G/P.
We provide three applications. The first shows that all Schubert classes of
partial flag varieties can be generated from a sequence of divided difference
operators on the highest-degree Schubert class. The second is a generalization
of Billey's formula for the localizations of equivariant Schubert classes of
flag varieties to arbitrary partial flag varieties. The third gives a choice of
Schubert polynomials for partial flag varieties as well as an explicit formula
for each. We focus on the example of maximal Grassmannians, including
Grassmannians of k-planes in a complex n-dimensional vector space.
