$SL_2$-action on Hilbert schemes and Calogero-Moser spaces

TL;DR

This paper explores the actions of $GL_2$ and $SL_2$ on Hilbert schemes and Calogero-Moser spaces, describing orbit closures and group representations at fixed points, advancing understanding of geometric and algebraic structures in these spaces.

Contribution

It provides a detailed description of orbit closures and character formulas for group actions on Hilbert schemes and Calogero-Moser spaces, including fixed point analysis.

Findings

Orbit closures of fixed points are explicitly described.

Characters of $GL_2$ representations at fixed points are computed.

Connections between group actions and geometric structures are established.

Abstract

We study the natural $GL_2$-action on the Hilbert scheme of points in the plane, resp. $SL_2$-action on the Calogero-Moser space. We describe the closure of the $GL_2$-orbit, resp. $SL_2$-orbit, of each point fixed by the corresponding diagonal torus. We also find the character of the representation of the group $GL_2$ in the fiber of the Procesi bundle, and its Calogero-Moser analogue, over the $SL_2$-fixed point.