On the normality of commuting scheme for general linear Lie algebra

TL;DR

This paper proves that the commuting scheme for the general linear Lie algebra is Cohen-Macaulay and normal, confirming a key conjecture and establishing a higher-dimensional Chevalley restriction theorem in positive characteristic.

Contribution

It demonstrates the normality and Cohen-Macaulay property of the commuting scheme for , providing new insights into its geometric structure and confirming a longstanding conjecture.

Findings

The commuting scheme  is Cohen-Macaulay.

The commuting scheme  is normal.

A 2-dimensional Chevalley restriction theorem is established in positive characteristic.

Abstract

The commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ 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 $\mathfrak{C}^{d}_{\mathfrak{g}}$ is reduced. Moreover, a higher dimensional analog of Chevalley restriction conjecture was conjectured by Chen-Ng\^{o}. We show that the commuting scheme of $\mathfrak{C}^{2}_{\mathfrak{g}l_{n}}$ is Cohen-Macaulay and normal. As a corollary, we prove a 2-dimensional Chevalley restriction theorem for general linear group in positive characteristic.