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

22 May 2018
•
Im Mee Seong

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.(read more)