SL₂-action on Hilbert schemes and Calogero-Moser spaces

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We study the natural GL₂-action on the Hilbert scheme of points in the plane, resp. SL₂-action on the Calogero-Moser space. We describe the closure of the GL₂-orbit, resp. SL₂-orbit, of each point fixed by the corresponding diagonal torus. We also find the character of the representation of the group GL₂ in the fiber of the Procesi bundle, and its Calogero-Moser analogue, over the SL₂-fixed point.