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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

##### MSC:
 20G15 Linear algebraic groups over arbitrary fields 20G25 Linear algebraic groups over local fields and their integers 14L35 Classical groups (algebro-geometric aspects)
##### Keywords:
buildings of affine type; spherical building; root data