# zbMATH — the first resource for mathematics

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
Full Text: