Quotients of commuting schemes associated to Symmetric Pairs

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 $\mathrm{Lie}(G_0)={\mathfrak{g}}_0$. For $d\ge 2$, 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.

Preliminary version. Comments are welcome. 16 pages