SL2-action on Hilbert Schemes and Calogero-Moser spaces

We study the natural GL2GL2-action on the Hilbert scheme of points in the plane, resp. SL2SL2-action on the Calogero–Moser space. We describe the closure of the GL2GL2-orbit, resp. SL2SL2-orbit, of each point fixed by the corresponding diagonal torus. We also find the character of the representation of the group GL2GL2 in the fiber of the Procesi bundle and its Calogero–Moser analogue over the SL2SL2-fixed point