Lie-Theoretic Generalization of Some Hilbert Schemes

Haiman’s construction of the Hilbert scheme of points on the plane and its isospectral variant has several different generalizations to other reductive Lie algebras. We explore these constructions and single out a particularly interesting candidate among these. This yields a class of varieties with conical symplectic singularities. In types ABC, and conjecturally in general, the varieties we propose are hyper-Kähler rotations of (possibly singular) Calogero–Moser spaces and their fixed points correspond to two-sided cells in the Weyl group. Time permitting, I will explain how the geometry of these varieties encodes Hochschild homology of Soergel bimodules as well as topological properties of affine Springer fibers.