Cohomology of symplectic reductions of generic coadjoint orbits

Let $\mathcal{O}_\lambda$ be a generic coadjoint orbit of a compact semi-simple Lie group $K$. Weight varieties are the symplectic reductions of $\mathcal{O}_\lambda$ by the maximal torus $T$ in $K$. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of $\mathcal{O}_\lambda$, and it makes explicit the dependence on $\lambda$ and a parameter in $Lie(T)^*=:\mathfrak{t}^*$.