Kac-Moody groups over ultrametric fields

The Kac-Moody groups studied here are the minimal (=algebraic) and split ones, as introduced by J. Tits in [8]. When they are defined over an ultrametric field, it seems natural to associate to them some analogues of the Bruhat-Tits buildings. Actually I came to this problem when I was trying to build new buildings of nondiscrete type. If G is a Kac-Moody group over an ultrametric field K, I was able to build a microaffine building I on which G(K) acts [5]. This building is an union of apartments in one to one correspondence with the maximal split tori and the usual axioms of buildings are satisfied, among them the fundamental axiom: any two points are in a same apartment. It is closely related to the Satake (or polyhedral) compactification of the Bruhat-Tits building of a semi-simple group over K. One knows that this compactification is the disjoint union of the Bruhat-Tits buildings of the semi-simple quotients of all parabolic subgroups of this semi-simple group. For a Kac-Moody group the same definition gives the microaffine building, but now the parabolic subgroups give something in I only when they are of finite type, so G itself gives nothing. We just have to define the apartments and prove the usual axioms of buildings, see [5]. Unfortunately I seems to give only a few informations about the structure of G(K). Moreover P. Littelmann asked me whether it could be used to generalize his results with S. Gaussent in the semi-simple case [2]: they proved in particular that a LS-path may be seen (in an apartment of a Bruhat-Tits building over the field of Laurent series C((t)) ) as an image of a segment of the building under some fixed retraction (with center a sector-germ), satisfying also some numerical condition. It was soon clear that, in the Kac-Moody case, I is not suitable. One has to mimic more closely the Bruhat-Tits construction. The normalizer of the standard maximal split torus in G(K) acts on the corresponding apartment A by a group of affine transformations, generated by reflections on walls. But there is a lot of walls (infinitely many directions), moreover in the loop group situation, H. Garland in [1] had proved that there is no Cartan decomposition, so the expected building would not satisfy the fundamental axiom of buildings: it seemed at first too ugly. Nevertheless it is possible to build this close analogue to Bruhat-Tits buildings for some split Kac-Moody groups (joint work with Stéphane Gaussent):