Classification of upper motives of algebraic groups of inner type An

De Clercq, Charles

Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 433-436.

Soient A, A′ deux algèbres centrales simples sur un corps F et F un corps fini de caractéristique p. Nous prouvons que les facteurs directs indécomposables supérieurs des motifs de deux variétés anisotropes de drapeaux dʼidéaux à droite X(d1,…,dk;A) et X(d1′,…,dk′′;A′) à coefficients dans F sont isomorphes si et seulement si les valuations p-adiques de pgcd(d1,…,dk) et pgcd(d1′,…,dk′′) sont égales et les classes des composantes p-primaires Ap et Ap′ de A et A′ engendrent le même sous-groupe dans le groupe de Brauer de F. Ce résultat mène à une surprenante dichotomie entre les motifs supérieurs des groupes algébriques absolument simples, adjoints et intérieurs de type An.

Let A, A′ be two central simple algebras over a field F and F be a finite field of characteristic p. We prove that the upper indecomposable direct summands of the motives of two anisotropic varieties of flags of right ideals X(d1,…,dk;A) and X(d1′,…,dk′′;A′) with coefficients in F are isomorphic if and only if the p-adic valuations of gcd(d1,…,dk) and gcd(d1′,…,dk′′) are equal and the classes of the p-primary components Ap and Ap′ of A and A′ generate the same group in the Brauer group of F. This result leads to a surprising dichotomy between upper motives of absolutely simple adjoint algebraic groups of inner type An.

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