Quotients of Commuting Schemes Associated with Symmetric Pairs

Let G be a classical group, i.e., either isomorphic to \(\textrm{GL}_n\) ${\text{GL}}_n$ or the group of isometries of a non-degenerate symmetric or anti-symmetric bilinear form on a vector space over an algebraically closed field k of characteristic zero. Let \(\theta \) $\theta$ be an involution on G and let \(G_0\) $G_0$ be the fixed point subgroup \(G^\theta \) $G^{\theta }$ . Let \(\mathfrak {g}\) $\mathfrak{g}$ and \(\mathfrak {g}_0\) ${\mathfrak{g}}_0$ be the Lie algebras of G and \(G_0\) $G_0$ respectively. Let \(\tau \) $\tau$ be the automorphism on \(\mathfrak {g}\) $\mathfrak{g}$ induced by \(\theta \) $\theta$ . Let \(\mathfrak {g}_1\) ${\mathfrak{g}}_1$ be the \(-1\) $-1$ eigenspace of \(\tau \) $\tau$ . For \(d \ge 2\) $d\ge 2$ , let \(\mathfrak {C}^d(\mathfrak {g}_1)\) ${\mathfrak{C}}^d({\mathfrak{g}}_1)$ be the d-th commuting scheme associated with the symmetric pair \((\mathfrak g, \mathfrak g_0)\) $(\mathfrak{g},{\mathfrak{g}}_0)$ . In this article, we study the reducedness of the quotient scheme \(\mathfrak {C}^d(\mathfrak {g}_1)//{G_0}\) ${\mathfrak{C}}^d({\mathfrak{g}}_1)//G_0$ via the Chevalley restriction map. As a part of the proof, we describe a generating set for the algebra \(k[\mathfrak {g}_1^d]^{G_0}\) $k{[{\mathfrak{g}}_1^d]}^{G_0}$ , which is of independent interest.