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Immeubles de type affine. (Buildings of affine type). (French) Zbl 0611.20026
Buildings and the geometry of diagrams, Lect. 3rd 1984 Sess. C.I.M.E., Como/Italy 1984, Lect. Notes Math. 1181, 159-190 (1986).
[For the entire collection see Zbl 0577.00009.]
This paper clarifies and extends the classification announced at the ICM 1974 in Vancouver (1975; Zbl 0336.57009). It classifies the irreducible buildings of affine type of rank \(\geq 3\). This is made in terms of a spherical building \({\mathcal J}^{\infty}\) associated with the building \({\mathcal J}\) and certain ”valuations” attached to \({\mathcal J}^{\infty}\) (but depending on \({\mathcal J})\). For rank \({\mathcal J}\geq 3\) \({\mathcal J}^{\infty}\) is determined by a family \((U_ a)\) of automorphisms of \({\mathcal J}\), called the root data. Replacing the ”valuations” of \({\mathcal J}^{\infty}\) by ”valuations of \((U_ a)\), which in turn are obtained from a single valuation of a ”base field” K corresponding to the root data, he arrives to the classification.
Reviewer: F.Levstein

20G15 Linear algebraic groups over arbitrary fields
20G25 Linear algebraic groups over local fields and their integers
14L35 Classical groups (algebro-geometric aspects)