Positivity in equivariant Schubert calculus

We prove a conjecture of Dale Peterson on positivity in the multiplication in the T-equivariant cohomology of the flag variety. The theorem follows from a more general positivity result about the equivariant cohomology of varieties with actions of a solvable group with finitely many orbits. This more general result is an equivariant version of a theorem of Kumar and Nori.