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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2018

OUTER AUTOMORPHISMS OF CLASSICAL ALGEBRAIC GROUPS

Résumé

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type 2 An with n even, or n = 5. More precisely, we prove a descent theorem for exponent 2 and degree 6 algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical types, namely 2 An with n odd, n ≥ 3 and 2 Dn, we provide explicit examples where the Tits class obstruction is satisfied, and yet the group does not have outer automorphism. As a crucial tool, we use "generic" sums of algebras with involution.
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Dates et versions

hal-03815212 , version 1 (14-10-2022)

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  • HAL Id : hal-03815212 , version 1

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Anne Quéguiner-Mathieu, Jean-Pierre Tignol. OUTER AUTOMORPHISMS OF CLASSICAL ALGEBRAIC GROUPS. Annales Scientifiques de l'École Normale Supérieure, 2018. ⟨hal-03815212⟩
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