Kac-Moody groups, hovels and Littelmann's paths

We give the definition of a kind of building I for a symmetrizable Kac-Moody group over a field K endowed with a dicrete valuation and with a residue field containing C. Due to some bad properties, we call this I a hovel. Nevertheless I has some good properties, for example the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semi-simple case by S. Gaussent and P. Littelmann [Duke Math. J; 127 (2005), 35-88]. In particular, if K= C((t)), the geodesic segments in I, with a given special vertex as end point and a good image under some retraction, are parametrized by a Zariski open subset P of C^N. This dimension N is maximum when this image is a LS path and then P is closely related to some Mirkovic-Vilonen cycle.