On the normality of commuting scheme for general linear Lie algebra
Artan Sheshmani, Xiaopeng Xia, Beihui Yuan
TL;DR
The paper proves that the commuting scheme for $\mathfrak{gl}_n$ with two commuting elements is Cohen–Macaulay and normal, establishing integrality and reducedness in positive characteristic for the corresponding quotient. The authors achieve this via an intricate induction on $n$, introducing a cascade of rings $R(n)$, $\tilde{R}(n)$, $R_1(n)$, $R'(n)$, and $R_2(n)$, and showing CM and normal properties are inherited through carefully controlled flat deformations and dimension counts. Key technical contributions include proving CM-ness for $\tilde{R}(n-1)/(w_{n-1},t)$ and establishing precise equidimensionality and regularity of various quotient schemes, culminating in the CM and normal structure of $R(n)$ and hence of $\mathfrak{C}^2_{\mathfrak{gl}_n}$. A corollary yields a 2-dimensional Chevalley restriction isomorphism in positive characteristic, linking to Hitchin-type spectral data and broadening understanding of commuting schemes and their quotients.
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-Ngô. 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.