A Lie-theoretic generalization of some Hilbert schemes

TL;DR

This paper introduces Lie-theoretic generalizations of Hilbert schemes, exploring their geometric properties, localization theorems, and connections to Cherednik algebras and affine Springer fibers, with conjectures on fixed points and Weyl group cells.

Contribution

It defines new varieties associated with Lie algebras, proves a localization theorem, and relates these to existing structures like Cherednik algebras and Springer fibers, extending the theory of Hilbert schemes.

Findings

Proved Gordon-Stafford localization theorem for these varieties.

Established a conjectural bijection between fixed points and Weyl group cells.

Connected the varieties to Calogero-Moser spaces.

Abstract

We define several versions of a class of varieties $X_{\mathfrak{g}}$ attached to a complex reductive Lie algebra $\mathfrak{g}$, generalizing the Hilbert scheme of points on the plane. These include trigonometric and elliptic versions attached to the corresponding groups. We also define the corresponding isospectral varieties $Y_{\mathfrak{g}}$. We prove a Gordon-Stafford localization theorem for $X_{\mathfrak{g}}$ and the corresponding equal-parameter rational Cherednik algebras, relate these varieties to the affine Springer fiber-sheaf correspondence of arXiv:2204.00303, and discuss examples. We conjecture that the torus-fixed points of our varieties are in bijection with two-sided cells in the finite Weyl group and prove this in types $ABC$. We relate these results to known results about Calogero-Moser spaces.