Jet schemes and invariant theory

Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2571-2599.

Soient $G$ un groupe réductif complexe et $V$ un $G$-module. Alors $G_m$, le schéma des jets d’ordre $m$ de $G$, opère dans $V_m$, le schéma des jets d’ordre $m$ de $V$, pour tout $m\ge 0$. Nous nous intéressons à l’anneau des invariants $𝒪{\left(V_m\right)}^{G_m}$ et au morphisme $p_m^{*}:𝒪\left({(V//G)}_m\right)\to 𝒪{\left(V_m\right)}^{G_m}$ induit par le morphisme du quotient catégorique $p:V\to V//G$  : ce morphisme est-il un isomorphisme, surjectif, ou non ? En utilisant le théorème du slice de Luna, nous obtenons des critères pour que $p_m^{*}$ soit un isomorphisme pour tout $m$. Nous montrons que c’est bien le cas lorsque $G={SL}_n$, ${GL}_n$, ${SO}_n$, ou ${Sp}_{2n}$ et $V$ est un somme directe de copies du module standard et de son dual, pourvu que $V//G$ soit lisse ou une intersection complète. Nous classifions toutes les représentations de ${ℂ}^{*}$ telles que $p_{\infty }^{*}$ soit surjectif ou un isomorphisme. Enfin, nous donnons des exemples où $p_m^{*}$ est surjectif pour $m=\infty$ mais non surjectif pour $m$ fini, et d’autres exemples où $p_m^{*}$ est surjectif mais non injectif.

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 $𝒪{\left(V_m\right)}^{G_m}$ and whether the map $p_m^{*}:𝒪\left({(V//G)}_m\right)\to 𝒪{\left(V_m\right)}^{G_m}$ 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 $p_m^{*}$ 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 module and its dual, such that $V//G$ is smooth or a complete intersection. We classify all representations of ${ℂ}^{*}$ 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.

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