LS galleries, the path model, and MV cycles

We give an interpretation of the path model of a representation (see [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 Lakshmibai-Seshadri (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 desingularization Σ(λ) of the closure of an orbit G(C[[t]])·λ in the affine Grassmannian. The homology of Σ(λ) has a basis given by Białynicki-Birula (BB) cells, which are indexed by the T-fixed points in Σ(λ). Now the points of Σ(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 nonempty intersection with G(C[[t]])·λ (identified with an open subset of Σ(λ)), and we show that the closures of the strata associated to LS galleries are exactly the Mirković-Vilonen (MV) cycles (see [25]) which form a basis of the representation V(λ) for the Langlands dual group G^.