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Split and almost split Kac-Moody groups. (Groupes de Kac-Moody déployés et presque déployés.) (French) Zbl 1001.22018
Astérisque. 277. Paris: Astérisque. vii, 348 p. (2002).
The purpose of this long text (340 pages) is to formulate a relative theory of Kac-Moody groups for Kac-Moody algebras over a field of non-zero characteristic. It generalizes former work of G. Rousseau in zero characteristic. A detailed introduction gives an overview of the motivations and of the main difficulties and choices.
The main purpose is to handle the following problem: Let \(\overline {\mathbb{K}}/\mathbb{K}\) be a field extension, where \(\overline{\mathbb{K}}\) is algebraically closed. Let \(\mathcal{D}\) be a Kac-Moody root datum, which defines a Tits functor \(\widetilde{\mathcal{G}_{\mathcal{D}}}\) and a group \(G=\widetilde{\mathcal{G}_{\mathcal{D}}}(\overline{\mathbb{K}}\)). (1) Give a definition for \(\mathbb{K}\)-forms of the split Kac-Moody group \(\widetilde {\mathcal{G}_{\mathcal{D}}}(\overline{\mathbb{K}}\)). (2) Define a class of \(\mathbb{K}\)-forms which is suitable for study using the theory of buildings. (3) Prove a Galois descent theorem for these \(\mathbb{K}\)-forms.
The work is divided into two parts.
The first part is devoted to the abstract study of a class of groups which contains isotropic reductive algebraic groups as well as split Kac-Moody groups. To each group a twin building is attached. The first chapters are devoted to twin buildings, geometrical realizations of buildings, negative curvature, convexity, Levi decomposition in twin root data.
The second part belongs to Kac-Moody theory. The first chapters of this part present algebras of Kac-Moody type, Tits functors and Kac-Moody groups, adjoint representations of Tits functors. The following chapters are devoted to split Kac-Moody groups for fields of any characteristic and to the study of \(\mathbb{K}\)-forms. Chapter 12 studies descent properties and the last chapter gives some examples.
Reviewer: Guy Roos (Paris)

22E67 Loop groups and related constructions, group-theoretic treatment
20G44 Kac-Moody groups
20E42 Groups with a \(BN\)-pair; buildings
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras