Universal transitivity of simple and 2-simple prehomogeneous vector spaces
Annales de l'Institut Fourier, Tome 38 (1988) no. 2, p. 11-41
Nous notons k un champ de caractéristique 0 satisfaisant H 1 (k, Aut (SL 2 ))0. Soit G un groupe algébrique linéaire connexe et déployé sur k agissant sur X= Aff n rationnellement par ρ avec une G-orbite Zariski dense. Un espace vectoriel préhomogène (G,ρ,X) est appelé “universellement transitif” si l’ensemble des points rationnels de kY(k) est une unique orbite ρ(G)(k) pour tout k. De tels espaces vectoriels préhomogènes sont classés par J. Igusa lorsque ρ est irréductible. Nous les classons lorsque G est réductible et son sous-groupe des commutateurs [G,G] est soit un groupe algébrique simple, soit un produit de deux groupes algébriques simples.
We denote by k a field of characteristic zero satisfying H 1 (k, Aut (SL 2 ))0. Let G be a connected k-split linear algebraic group acting on X= Aff n rationally by ρ with a Zariski-dense G-orbit Y. A prehomogeneous vector space (G,ρ,X) is called “universally transitive” if the set of k-rational points Y(k) is a single ρ (G)(k)-orbit for all such k. Such prehomogeneous vector spaces are classified by J. Igusa when ρ is irreducible. We classify them when G is reductive and its commutator subgroup [G,G] is either a simple algebraic group or a product of two simple algebraic groups.
@article{AIF_1988__38_2_11_0,
     author = {Kimura, T. and Kasai, S. and Hosokawa, H.},
     title = {Universal transitivity of simple and 2-simple prehomogeneous vector spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {38},
     number = {2},
     year = {1988},
     pages = {11-41},
     doi = {10.5802/aif.1133},
     zbl = {0606.14037},
     mrnumber = {89e:20083},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1988__38_2_11_0}
}
Kimura, T.; Kasai, S.; Hosokawa, H. Universal transitivity of simple and 2-simple prehomogeneous vector spaces. Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 11-41. doi : 10.5802/aif.1133. http://www.numdam.org/item/AIF_1988__38_2_11_0/

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