A Lie-theoretic generalization of some Hilbert schemes
Oscar Kivinen
TL;DR
The authors generalize Hilbert schemes to arbitrary complex reductive Lie algebras by constructing three variants X_g (sgn, diag, symb) with isospectral counterparts, and they connect these spaces to rational Cherednik algebras and Coulomb branches. They establish a Gordon-Stafford-type localization linking H_c to a noncommutative/commutative degeneration framework via a Z-algebra, and show that X_g,symb serves as a natural, normal, symplectic object closely related to Calogero-Moser spaces, with type-A recovering the classical Hilbert scheme. They prove key structural results (normality, symplectic singularities) and provide detailed evidence in low-rank types (ABC, BC) including a quiver-variety realization in B/C and conjectural hyper-Kähler equivalences. The work also proposes deep conjectures linking fixed points to Weyl-group two-sided cells, endoscopy, and Shalika germs, suggesting broad unifications between representation theory, algebraic geometry, and geometric aspects of affine Springer fibers.
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.