We give an interpretation of the path model of a representation [18] of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS-galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure-Hansen-Bott-Samelson desingularizationΣ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The homology ofΣ(λ) has a basis given by Bia lynicki-Birula cell's, which are indexed by the T -fixed points in Σ(λ). Now the points ofΣ(λ) can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T -fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a non-empty intersection with G(C[[t]]).λ (identified with an open subset ofΣ(λ)), and we show that the closures of the strata associated to LS-galleries are exactly the MV-cycles [24], which form a basis of the representation V (λ) for the Langland's dual group G ∨ . for various helpful discussions. We are especially indebted to Guy Rousseau, who suggested to us the idea to use the retraction r −∞ . The first author would like to thank the CAALT for the support and the Aarhus Mathematics Department for the hospitality during the year 2002/2003. The second author is happy to thank the MSRI and the Department of Mathematics at UC Berkeley for the hospitality and support during the spring semester 2003, where part of this article has been worked out.We call S a (F ) the type of F , so S a (0) = S and S a (∆ f ) = ∅. For an arbitrary face F its type is defined as the type of F ′ , where F ′ is the unique face of ∆ f such that F = w(F ′ ) for some w ∈ W a .Remark 2. The alcoves are actually the chambers of the Coxeter complex A a associated to the Coxeter group (W a , S a ), but since we look at the same time also at the spherical complex, to not confuse the "affine" and the spherical chambers, we prefer the term alcove for the faces of maximal dimension.We view the (spherical) Weyl group W ⊂ W a as the subgroup generated by the reflections in S. By an open chamber we mean always an open spherical chamber, i.e., a connected component of A − β∈Φ + H β,0 , and a chamber is the closure of such a connected component. We have the dominant chamber C f corresponding to the choice of the Borel subgroup B and the anti-dominant chamber −C f . Definition 2. A sector s in A is a W a -translate of a chamber. Two sectors s, s ′ are called equivalent if there exists a third sector s ′′ in the intersection: s ′′ ⊂ s ∩ s ′ .

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Abstract. We define the Iwahori-Hecke algebra I H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer KI of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define I H as the algebra of some KI −bi-invariant functions on G with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G + of G. In the split case, we prove that the structure constants of I H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We give a presentation of this algebra I H, similar to the Bernstein-Lusztig presentation in the reductive case, and embed it in a greater algebra

We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer K of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define H as the algebra of K−bi-invariant functions on G with almost finite support. As two points in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G + of G. We prove that the structure constants of H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We also prove the Satake isomorphism between H and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, H is always commutative. Actually, our results apply to abstract "locally finite" hovels, so that we can define the spherical algebra with unequal parameters.