What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup?
The explicit description in terms of the Plücker embedding I know is that the torus acts on the Stiefel manifold by acting on the column vectors; this action descends to one on the image of the Plücker embedding - i.e. of the Grassmanian.
Is there a highest weight description of the action?