Kac-moody Groups, Hovels and Littelmann's Paths
نویسنده
چکیده
We give the definition of a kind of building I for a symmetrizable KacMoody 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 ρπ under some retraction ρ, are parametrized by a Zariski open subset P of C . This dimension N is maximum when ρπ is a LS path and then P is closely related to some Mirković-Vilonen cycle. Introduction In the representation theory of a complex symmetrizable Kac-Moody algebra ǧ , P. Littelmann [94; 95] introduced the path model. It gives a method to compute the multiplicities of a weight μ in an irreducible representation of highest weight λ (a dominant weight) by counting some “Lakshmibai-Seshadri” (or LS) paths of shape λ starting from 0 and ending in μ. When ǧ is semi-simple and G is an algebraic group of Lie algebra the Langlands dual g of ǧ , I. Mirković and K. Vilonen [00] gave a new interpretation of this multiplicity: it is the number of irreducible components (the MV cycles) in some subvariety X λ of the affine grassmannian G = G(C((t)) )/G(C[[t]] ). S. Gaussent and P. Littelmann [05] gave a link between these two theories (when G is semi-simple). Actually the LS paths are drawn in a vector space V which is a Bruhat-Tits apartment A of G (over any non archimedean ”local” field K, in particular K = C((t)) ). So they replaced the LS paths of shape λ from 0 to μ by ”LS galleries” of type λ from 0 to μ; this gives a new “gallery model” for the representations of 2000 Mathematics Subject classification: 22E46 (primary), 20G05, 17B67, 22E65, 20E42, 51E24.
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