The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck-Springer resolution

TL;DR

This paper studies the geometric structure of a Hamiltonian reduction related to the affine quotient of the cotangent bundle of the Grothendieck-Springer resolution, showing it is smooth, affine, and isomorphic to a dense open subset of complex affine space.

Contribution

It proves the smoothness and affine nature of the Hamiltonian reduction of the regular semisimple locus and describes the invariant functions explicitly.

Findings

The Hamiltonian reduction is smooth, affine, and reduced.

It is scheme-theoretically isomorphic to a dense open subset of ^{2n}.

Invariant functions are given by traces of matrix products.

Abstract

Let $G= GL_n(\mathbb{C})$, the general linear group over the complex numbers, and let $B$ be the set of invertible upper triangular matrices in $G$. Let $\mathfrak{b}=\text{Lie}(B)$. For $\mu:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, where $\mathfrak{b}^*\cong \mathfrak{g}/\mathfrak{u}$ and $\mathfrak{u}$ being strictly upper triangular matrices in $\mathfrak{g}=\text{Lie}(G)$, we prove that the Hamiltonian reduction $\mu^{-1}(0)^{rss}/\!\!/B$ of the extended regular semisimple locus $\mathfrak{b}^{rss}$ of the Borel subalgebra is smooth, affine, reduced, and scheme-theoretically isomorphic to a dense open locus of $\mathbb{C}^{2n}$. We also show that the $B$-invariant functions on the regular semisimple locus of the Hamiltonian reduction of $\mathfrak{b}\times \mathbb{C}^n$ arise as the trace of a certain product of matrices.