Equivalence motivique des groupes algebriques semisimples

TL;DR

This paper introduces combinatorial invariants called Tits p-indices that characterize motivic equivalence of semisimple algebraic groups, generalizing Vishik's criterion for quadrics and linking motives with rational geometry.

Contribution

It constructs the Tits p-index invariants as motivic analogues of Tits indices, providing algebraic criteria for motivic equivalence of semisimple algebraic groups.

Findings

Defines Tits p-indexes in terms of classical invariants.

Provides algebraic criteria for motivic equivalence.

Clarifies the relation between motives and rational geometry.

Abstract

Two semisimple algebraic groups are said to be motivic equivalent if the motives of the associated twisted flag varieties are isomorphic modulo any prime p. The purpose of this note is to construct the combinatorial invariants which characterize motivic equivalence and which are the motivic analogues of the Tits indices which appear in the classification of semisimple algebraic groups. The expression of these invariants -the Tits p-indexes- in terms of the classical invariants associated to the natural underlying structures of semisimple algebraic groups allow to produce algebraic criteria of motivic equivalence, generalizing Vishik's criterion of motivic equivalence for the motives of quadrics. It also clarifies the relation between the motives and the rational geometry of twisted flag varieties. Deux groupes semisimples sont dits motiviquement equivalents si les motifs des varietes de drapeaux generalisees associees sont isomorphes modulo tout nombre premier p. L'objet de cette note est de construire les invariants combinatoires qui caracterisent l'equivalence motivique et sont les analogues motiviques des indices de Tits apparaissant dans la classification des groupes algebriques semisimples. L'expression de ces invariants -les p-indices de Tits superieurs- en fonction des indices classiques associes aux structures naturelles sous-jacentes aux groupes semisimples permet de produire des criteres algebriques d'equivalence motivique, generalisant le critere de Vishik d'equivalence motivique des quadriques. Elle permet en outre de clarifier le lien qu'entretiennent les motifs et la geometrie birationnelle des varietes de drapeaux.