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# Jet schemes and invariant theory

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### 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.

### Author and article information

###### Journal
2011-12-29
2015-06-08
###### Article
1112.6230