Jet schemes and invariant theory
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599.

Let G be a complex reductive group and V a G-module. Then the mth jet scheme G m acts on the mth jet scheme V m for all m0. We are interested in the invariant ring 𝒪(V m ) G m and whether the map p m * :𝒪((V//G) m )𝒪(V m ) G m induced by the categorical quotient map p:VV//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 * for which p * is surjective or an isomorphism. Finally, we give examples where p m * is surjective for m= but not for finite m, and where it is surjective but not injective.

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 m0. Nous nous intéressons à l’anneau des invariants 𝒪(V m ) G m et au morphisme p m * :𝒪((V//G) m )𝒪(V m ) G m induit par le morphisme du quotient catégorique p:VV//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 * telles que p * soit surjectif ou un isomorphisme. Enfin, nous donnons des exemples où p m * est surjectif pour m= mais non surjectif pour m fini, et d’autres exemples où p m * est surjectif mais non injectif.

DOI: 10.5802/aif.2996
Classification: 13A50,  14L24,  14L30
Keywords: jet schemes, classical invariant theory
Linshaw, Andrew R. 1; Schwarz, Gerald W. 2; Song, Bailin 3

1 Department of Mathematics University of Denver 2280 S. Vine St. Denver, CO 80208 (USA)
2 Department of Mathematics Brandeis University 415 South Street Waltham, MA 02453 (USA)
3 School of Mathematical Sciences University of Science and Technology of China No. 96 Jinzhai Road Hefei, Anhui Province, 230026 (P. R. China)
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Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin. Jet schemes and invariant theory. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599. doi : 10.5802/aif.2996. https://aif.centre-mersenne.org/articles/10.5802/aif.2996/

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