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.
"Positivity in equivariant Schubert calculus." Duke Math. J. 109 (3) 599 - 614, 15 September 2001. https://doi.org/10.1215/S0012-7094-01-10935-6