Quotients of commuting schemes associated to Symmetric Pairs

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.