Niveau: Supérieur, Doctorat, Bac+8
EUCLIDEAN BUILDINGS By Guy Rousseau Buildings were introduced by Jacques Tits in the 1950s to give a systematic procedure for the geometric interpretation of the semi-simple Lie groups (in particular the exceptional groups) and for the construction and study of semi-simple groups over general fields. They were simplicial complexes and their apartments were euclidean spheres with a finite (Weyl) group of isometries. So these buildings were called of spherical type [Tits-74]. Later Franc¸ois Bruhat and Jacques Tits constructed buildings associated to semi-simple groups over fields endowed with a non archimedean valuation. When the valuation is discrete these Bruhat-Tits buildings are still simplicial (or polysimplicial) complexes, and their apartments are affine euclidean spaces tessellated by simplices (or polysimplices) with a group of affine isometries as Weyl group. So these buildings were called affine. But when the valuation is no longer discrete, the simplicial structure disappears ; so Bruhat and Tits construct (the geometric realization of) the building as a metric space, union of subspaces isometric to euclidean spaces, and they introduce facets as filters of subsets [Bruhat-Tits-72]. This is the point of view I wish to develop in these lectures, by giving a definition of euclidean buildings valid even in the non discrete case and independent of their construction. Actually such a definition has been already given by Tits [86a], but his definition emphasizes the role of sectors against that of facets.
- called
- group over
- group generated
- building
- all vectors orthogonal
- orthogonal direct product