Jet schemes and invariant theory

Abstract

Let \(G\) be a complex reductive group and \(V\) a \(G\)-module. Then the \(m\)th jet scheme \(G_m\) acts on the \(m\)th jet scheme \(V_m\) for all \(m\geq 0\). We are interested in the invariant ring \(\mathcal{O}(V_m)^{G_m}\) and whether the map \(p_m^*\colon\mathcal{O}((V//G)_m) \rightarrow \mathcal{O}(V_m)^{G_m}\) induced by the categorical quotient map \(p\colon V\rightarrow V//G\) is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for \(p_m^*\) to be an isomorphism for all \(m\), and we prove this when \(G=SL_n\), \(GL_n\), \(SO_n\), or \(Sp_{2n}\) and \(V\) is a sum of copies of the standard representation and its dual, such that \(V//G\) is smooth or a complete intersection. We classify all representations of \(\mathbb{C}^*\) for which \(p^*_{\infty}\) is surjective or an isomorphism. Finally, we give examples where \(p^*_m\) is surjective for \(m=\infty\) but not for finite \(m\), and where it is surjective but not injective.