The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck-Springer resolution (1607.02208v2)
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.