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Kac-Moody groups, hovels and Littelmann paths. (English) Zbl 1161.22007
The authors give the definition of a kind of building $$\mathcal{I}$$ for a symmetrizable Kac-Moody group over a field $$K$$ endowed with a discrete valuation and with a residue field containing $$\mathbb{C}$$, which they call a hovel, due to its pathological behaviour. This concept allows them to generalize some previous results obtained by themselves, separately, and in a joint paper of St. Gaussent and P. Littelmann [Duke Math. J. 127, 35–88 (2005; Zbl 1078.22007)], by introducing a new set $${\mathcal I}$$ associated with the Kac-Moody group $$G$$ over $$K = {\mathbb C}((t))$$, or, even more generally, over any discretely valuated field $$K$$ with residue field containing $${\mathbb C}$$. By using that concept they get good generalizations of Gaussent-Littelmann’s results and obtain a new result which is an analogue of some previous ones in the semisimple case. In the two last sections they give the definitions of LS paths, Hecke paths, dual dimension and codimension, and they prove some characterizations of them.

##### MSC:
 22E46 Semisimple Lie groups and their representations 20G05 Representation theory for linear algebraic groups 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 20E42 Groups with a $$BN$$-pair; buildings 51E24 Buildings and the geometry of diagrams
##### Keywords:
Kac-Moody group; valuated field; building; path
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