Jet schemes and invariant theory

[Schémas des jets et théorie des invariants]

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

Annales de l'Institut Fourier, Tome 65 (2015), p. 2571-2599 / Harvested from Numdam

Résumé

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 ${\mathcal{O}}(V_m)^{G_m}$ et au morphisme $p_m^*\colon {\mathcal{O}}((V/\!\!/G)_m) \rightarrow {\mathcal{O}}(V_m)^{G_m}$ induit par le morphisme du quotient catégorique $p\colon V\rightarrow V/\!\!/G$  : 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 $G=\operatorname{SL}_n$, $\operatorname{GL}_n$, $\operatorname{SO}_n$, ou $\operatorname{Sp}_{2n}$ 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 $\mathbb{C}^*$ 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.

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 $p_m^*\colon {\mathcal{O}}((V/\!\!/G)_m) \rightarrow {\mathcal{O}}(V_m)^{G_m}$ 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 $G=\operatorname{SL}_n$, $\operatorname{GL}_n$, $\operatorname{SO}_n$, or $\operatorname{Sp}_{2n}$ 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 $\mathbb{C}^*$ 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.

Détail

Publié le : 2015-01-01

Zbl 1342.13009

DOI : https://doi.org/10.5802/aif.2996
Classification:  13A50,  14L24,  14L30
Mots clés: schémas des jets, théorie classique des invariants

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