Quotients of commuting schemes associated to Symmetric Pairs

TL;DR

This paper investigates the structure of commuting schemes associated with symmetric pairs of classical Lie algebras, proving their normality and reducedness, and describes generators for related invariant algebras.

Contribution

It provides a detailed analysis of the categorical quotient of commuting schemes for symmetric pairs, establishing their geometric properties and identifying generators of invariant algebras.

Findings

The quotient scheme $\,rak C^d(rak g_1)//G_0$ is normal.

The quotient scheme $\,rak C^d(rak g_1)//G_0$ is reduced.

A generating set for the algebra of invariants $k[rak g_1^d]^{G_0}$ is described.

Abstract

Let $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ be a $\mathbb Z_2$-grading of a classical Lie algebra such that $(\mathfrak{g}, \mathfrak{g}_0)$ is a classical symmetric pair. Let $G$ be a classical group with Lie algebra $\mathfrak{g}$ and let $G_0$ be the connected subgroup of $G$ with ${\rm Lie} (G_0)=\mathfrak g_0$. For $d \geq 2$, let $\mathfrak{C}^d(\mathfrak{g}_1)$ be the $d$-th commuting scheme associated with the symmetric pair $(\mathfrak g, \mathfrak g_0)$. In this article, we study the categorical quotient $\mathfrak{C}^d(\mathfrak{g}_1)//{G_0}$ via the Chevalley restriction map. As a consequence we show that the categorical quotient scheme $\mathfrak C^d(\mathfrak g_1)//G_0$ is normal and reduced. As a part of the proof, we describe a generating set for the algebra $k[\mathfrak{g}_1^d]^{G_0}$, which are of independent interest.