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 $m\ge 0$. We are interested in the invariant ring $\mathcal{O}(V_m{)}^{G_m}$ and whether the map $p_m^{\ast }:\mathcal{O}((V//G{)}_m)\to \mathcal{O}(V_m{)}^{G_m}$ induced by the categorical quotient map $p:V\to V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^{\ast }$ to be an isomorphism for all $m$, and we prove this when $G=S{L}_n$, $G{L}_n$, $S{O}_n$, or $S{p}_{2n}$ 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 ${\mathbb{C}}^{\ast }$ for which $p_{\infty }^{\ast }$ is surjective or an isomorphism. Finally, we give examples where $p_m^{\ast }$ is surjective for $m=\infty$ but not for finite $m$, and where it is surjective but not injective.

Final version, to appear in Annales de l'Institut Fourier

References in corpus (2)

• Arc spaces and the vertex algebra commutant problem
• The global sections of the chiral de Rham complex on a Kummer surface

Cited by in corpus (11)

• Arc spaces and the vertex algebra commutant problem
• Cosets of free field algebras via arc spaces
• Standard monomials and invariant theory for arc spaces I: general linear group
• Vector bundles induced from jet schemes
• The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds II
• Standard monomials and invariant theory for arc spaces III: special linear group
• The global sections of the chiral de Rham complex on a Kummer surface
• The global sections of chiral de Rham complexes on compact Ricci-flat Kähler manifolds
• Standard monomials and invariant theory of arc spaces II: Symplectic group
• Functorial constructions related to double Poisson vertex algebras
• Vertex Algebras and Commutative Algebras