Kac-Moody groups, hovels and Littelmann's paths

Stéphane Gaussent (1) , Guy Rousseau (1)

1 IECN - Institut Élie Cartan de Nancy

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.

Mots clés: Kac-Moody group; local field; building; path

Domaines: Théorie des groupes [math.GR]; Théorie des représentations [math.RT]

Stéphane Gaussent, Guy Rousseau. Kac-Moody groups, hovels and Littelmann's paths. Annales de l'Institut Fourier, 2008, 58 (7), pp.2605-2657. ⟨10.5802/aif.2423⟩. ⟨hal-00137735v2⟩