Kac-Moody groups, hovels and Littelmann paths
Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2605-2657.

We give the definition of a kind of building for a symmetrizable Kac-Moody group over a field K endowed with a discrete valuation and with a residue field containing . Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if K=((t)), the geodesic segments in , ending in a special vertex and retracting onto a given path π, are parametrized by a Zariski open subset P of N . This dimension N is maximal when π is a LS path and then P is closely related to some Mirković-Vilonen cycle.

Nous définissons une sorte d’immeuble associé à un groupe de Kac-Moody symétrisable sur un corps K muni d’une valuation discrète avec un corps résiduel contenant . Nous l’appelons masure (hovel) à cause de l’absence d’une propriété importante des immeubles. Cependant, de bonnes propriétés restent, par exemple l’existence de retractions de centre un germe de quartier. Cela nous permet de généraliser plusieurs résultats prouvés par S. Gaussent et P. Littelmann dans le cas semi-simple. En particulier, si K=((t)), les segments géodésiques dans , d’extrémité un sommet spécial et se rétractant sur un chemin donné π, sont paramétrés par un ouvert de Zariski P de N . Cette dimension N est maximale quand π est un chemin LS et alors P est fortement associé à un cycle de Mirković-Vilonen.

Received:
Accepted:
DOI: 10.5802/aif.2423
Classification: 22E46,  20G05,  17B67,  22E65,  20E42,  51E24
Keywords: Kac-Moody group, valuated field, building, path
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Gaussent, Stéphane; Rousseau, Guy. Kac-Moody groups, hovels and Littelmann paths. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2605-2657. doi : 10.5802/aif.2423. https://aif.centre-mersenne.org/articles/10.5802/aif.2423/

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