Positivity in equivariant Schubert calculus

We prove a positivity property for the cup product in the T-equivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of S. Billey. The result for the flag variety follows from a more general result about algebraic varieties with an action of a solvable linear algebraic group such that the unipotent radical acts with finitely many orbits. The methods are those used by S. Kumar and M. Nori.