$S{L}_2$-action on Hilbert schemes and Calogero-Moser spaces

We study the natural $G{L}_2$-action on the Hilbert scheme of points in the plane, resp. $S{L}_2$-action on the Calogero-Moser space. We describe the closure of the $G{L}_2$-orbit, resp. $S{L}_2$-orbit, of each point fixed by the corresponding diagonal torus. We also find the character of the representation of the group $G{L}_2$ in the fiber of the Procesi bundle, and its Calogero-Moser analogue, over the $S{L}_2$-fixed point.