Universal transitivity of simple and 2-simple prehomogeneous vector spaces
Annales de l'Institut Fourier, Volume 38 (1988) no. 2, pp. 11-41.

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.

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.

@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},
     pages = {11--41},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {2},
     year = {1988},
     doi = {10.5802/aif.1133},
     zbl = {0606.14037},
     mrnumber = {89e:20083},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1133/}
}
TY  - JOUR
AU  - Kimura, T.
AU  - Kasai, S.
AU  - Hosokawa, H.
TI  - Universal transitivity of simple and 2-simple prehomogeneous vector spaces
JO  - Annales de l'Institut Fourier
PY  - 1988
SP  - 11
EP  - 41
VL  - 38
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1133/
DO  - 10.5802/aif.1133
LA  - en
ID  - AIF_1988__38_2_11_0
ER  - 
%0 Journal Article
%A Kimura, T.
%A Kasai, S.
%A Hosokawa, H.
%T Universal transitivity of simple and 2-simple prehomogeneous vector spaces
%J Annales de l'Institut Fourier
%D 1988
%P 11-41
%V 38
%N 2
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1133/
%R 10.5802/aif.1133
%G en
%F 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, Volume 38 (1988) no. 2, pp. 11-41. doi : 10.5802/aif.1133. https://aif.centre-mersenne.org/articles/10.5802/aif.1133/

[1] J. Igusa, On functional equations of complex powers, Invent. Math., 85 (1986), 1-29. | MR | Zbl

[2] J. Igusa, On a certain class of prehomogeneous vector spaces, to appear in Journal of Algebra.

[3] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1-155. | MR | Zbl

[4] T. Kimura, A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications, Journal of Algebra, Vol. 83, N° 1, July (1983), 72-100. | MR | Zbl

[5] T. Kimura, S. Kasai, M. Inuzuka and O. Yasukura, A classification of 2-simple prehomogeneous vector spaces of type I, to appear in Journal of Algebra. | Zbl

[6] T. Kimura, S. Kasai, M. Taguchi and M. Inuzuka, Some P.V.-equivalences and a classification of 2-simple prehomogeneous vector spaces of type II, to appear in Transaction of A.M.S. | Zbl

[7] J. Serre, Cohomologie Galoisienne, Springer Lecture Note, 5 (1965). | MR | Zbl

[8] H. Rubentheler, Formes réelles des espaces préhomogènes irréductibles de type parabolique, Annales de l'Institut Fourier, Grenoble, 36-1 (1986), 1-38. | Numdam | MR | Zbl

Cited by Sources: