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

Abstract

Let $G=G{L}_n(\mathbb{C})$, 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 $\mu :T^{\ast }(\mathfrak{b}\times {\mathbb{C}}^n)\to {\mathfrak{b}}^{\ast }$, where ${\mathfrak{b}}^{\ast }\cong \mathfrak{g}/\mathfrak{u}$ 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.

arXiv comment17 pages, to appear in Journal of Geometry and Physics