  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.
