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Kac-Moody groups split over a local field, microaffine buildings. (Groupes de Kac-Moody déployés sur un corps local, immeubles microaffines.) (French. English summary) Zbl 1094.22003
Summary: If \(G\) is a (split) Kac-Moody group over a field \(K\) endowed with a real valuation \(\omega\), we build an action of \(G\) on a geometric object \({\mathcal I}\). This object is called a building, as it is a union of apartments, with the classical properties of systems of apartments. However, these apartments are more exotic: those associated to a torus \(T\) may be seen as the gluing of all Satake compactifications of affine apartments of \(T\) with respect to spherical parabolic subgroups of \(G\) containing \(T\). Another geometric realization of these apartments makes them look more like the apartments of \(\Lambda\)-buildings; then the translations of the Weyl group act only on infinitely small elements of the apartment, so we call these buildings microaffine.

MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E67 Loop groups and related constructions, group-theoretic treatment
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20G25 Linear algebraic groups over local fields and their integers
51E24 Buildings and the geometry of diagrams
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