Cohomology of line bundles on the cotangent bundle of a Grassmannian

We show that certain line bundles on the cotangent bundle of a Grassmannian arising from an anti-dominant character $\lambda$ have cohomology groups isomorphic to those of a line bundle on the cotangent bundle of the dual Grassmannian arising from the dominant character $w_0(\lambda)$, where $w_0$ is the longest element of the Weyl group of $SL_{l+1}(k)$.