J-invariant of linear algebraic groups
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 6, p. 1023-1053

Let G be a semisimple linear algebraic group of inner type over a field F, and let X be a projective homogeneous G-variety such that G splits over the function field of X. We introduce the J-invariant of G which characterizes the motivic behavior of X, and generalizes the J-invariant defined by A. Vishik in the context of quadratic forms. We use this J-invariant to provide motivic decompositions of all generically split projective homogeneous G-varieties, e.g. Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, varieties of Borel subgroups of G. We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G.

Soit G un groupe algébrique linéaire semi-simple de type intérieur sur un corps F et soit X un G-espace homogène projectif tel que le groupe G soit déployé sur le point générique de X. Nous introduisons le J-invariant de G qui caractérise le comportement motivique de X et généralise le J-invariant défini par A. Vishik dans le cadre des formes quadratiques. Nous utilisons cet invariant pour obtenir les décompositions motiviques de tous les G-espaces homogènes projectifs qui sont génériquement déployés, par exemple les variétés de Severi-Brauer, les quadriques de Pfister, la grassmannienne des sous-espaces totalement isotropes maximaux d’une forme quadratique, la variété des sous-groupes de Borel de G. Nous discutons également les relations avec les indices de torsion, la dimension canonique et les invariants cohomologiques du groupe G.

DOI : https://doi.org/10.24033/asens.2088
Classification:  14C25,  20G15
Keywords: motive, algebraic group, homogeneous variety
@article{ASENS_2008_4_41_6_1023_0,
     author = {Petrov, Viktor and Semenov, Nikita and Zainoulline, Kirill},
     title = {$J$-invariant of linear algebraic groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {6},
     year = {2008},
     pages = {1023-1053},
     doi = {10.24033/asens.2088},
     zbl = {1206.14017},
     mrnumber = {2504112},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2008_4_41_6_1023_0}
}
Petrov, Viktor; Semenov, Nikita; Zainoulline, Kirill. $J$-invariant of linear algebraic groups. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 6, pp. 1023-1053. doi : 10.24033/asens.2088. http://www.numdam.org/item/ASENS_2008_4_41_6_1023_0/

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