Groupes de Kac–Moody déployés sur un corps local, immeubles microaffines

Abstract

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 an union of apartments, with the classical properties of systems of apartments. However, these apartments are more exotic: that 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.