$J$-invariant of linear algebraic groups

[$J$-invariant des groupes algébriques linéaires]

Petrov, Viktor ; Semenov, Nikita ; Zainoulline, Kirill

Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008), p. 1023-1053 / Harvested from Numdam

Résumé

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 $G$.

Abstract

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 $G$.

Détail

Publié le : 2008-01-01

MR 2504112 | Zbl 1206.14017

DOI : https://doi.org/10.24033/asens.2088
Classification:  14C25,  20G15
Mots clés: motif, groupe algébrique, espace homogène

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