On the normality of commuting scheme for general linear Lie algebra

Abstract

The commuting scheme ${\mathfrak{C}}_{\mathfrak{g}}^d$ for reductive Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathbb{K}$ is the subscheme of ${\mathfrak{g}}^d$ defined by quadratic equations, whose $\mathbb{K}$-valued points are $d$-tuples of commuting elements in $\mathfrak{g}$ over $\mathbb{K}$. There is a long-standing conjecture that the commuting scheme ${\mathfrak{C}}_{\mathfrak{g}}^d$ is reduced. Moreover, a higher dimensional analog of Chevalley restriction conjecture was conjectured by Chen-Ngô. We show that the commuting scheme of ${\mathfrak{C}}_{\mathfrak{g}{l}_n}^2$ is Cohen-Macaulay and normal. As a corollary, we prove a 2-dimensional Chevalley restriction theorem for general linear group in positive characteristic.

arXiv comment30 pages, comments are welcome