Almost commuting scheme of symplectic matrices and quantum Hamiltonian reduction

TL;DR

This paper explores the geometric structure of almost commuting symplectic matrices, constructs a special Lagrangian subscheme, and links quantum Hamiltonian reduction to rational Cherednik algebras, advancing understanding of symplectic and algebraic geometry.

Contribution

It constructs a Lagrangian subscheme of almost commuting symplectic matrices and establishes an isomorphism between quantum Hamiltonian reduction and a rational Cherednik algebra of Type C.

Findings

Constructed a Lagrangian subscheme $X^{nil}$ with specific dimension and irreducible components.

Proved the quantum Hamiltonian reduction is isomorphic to the spherical subalgebra of a Type C rational Cherednik algebra.

Analyzed the algebro-geometric properties of the scheme of almost commuting symplectic matrices.

Abstract

Losev introduced the scheme $X$ of almost commuting elements (i.e., elements commuting upto a rank one element) of $\mathfrak{g}=\mathfrak{sp}(V)$ for a symplectic vector space $V$ and discussed its algebro-geometric properties. We construct a Lagrangian subscheme $X^{nil}$ of $X$ and show that it is a complete intersection of dimension $\text{dim}(\mathfrak{g})+\frac{1}{2}\text{dim}(V)$ and compute its irreducible components. We also study the quantum Hamiltonian reduction of the algebra $\mathcal{D}(\mathfrak{g})$ of differential operators on the Lie algebra $\mathfrak{g}$ tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type $C$.