Universal transitivity of simple and 2-simple prehomogeneous vector spaces

Kimura, T.; Kasai, S.; Hosokawa, H.

doi:10.5802/aif.1133

Kimura, T. ; Kasai, S. ; Hosokawa, H.

Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 11-41.

Résumé
Abstract

Nous notons $k$ un champ de caractéristique 0 satisfaisant $H^{1} (k, Aut (S L_{2})) \neq 0$ . 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 $k Y (k)$ 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 $H^{1} (k, Aut (S L_{2})) \neq 0$ . 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.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1133

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     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},
     pages = {11--41},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {2},
     year = {1988},
     doi = {10.5802/aif.1133},
     mrnumber = {89e:20083},
     zbl = {0606.14037},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1133/}
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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. https://www.numdam.org/articles/10.5802/aif.1133/

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Ryu, Inhyun Universal transitivity of reductive prehomogeneous vector spaces with a finite number of orbits, Journal of Algebra, Volume 370 (2012), p. 361 | DOI:10.1016/j.jalgebra.2012.07.044
WAKATSUKI, Satoshi The Igusa local zeta function of the simple prehomogeneous vector space $(G L (1)^{4} \times S L (2 n + 1), Λ_{2} \oplus Λ_{1} \oplus Λ_{1} \oplus Λ_{1})$ , Journal of the Mathematical Society of Japan, Volume 57 (2005) no. 1 | DOI:10.2969/jmsj/1160745816
Kimura, Tatsuo Arithmetic calculus of Fourier transforms by Igusa local zeta functions, Transactions of the American Mathematical Society, Volume 346 (1994) no. 1, p. 297 | DOI:10.1090/s0002-9947-1994-1267223-5

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