LS galleries, the path model, and MV cycles

Abstract

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 desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The homology of ˆ Σ(λ) has a basis given by Bia lynicki–Birula cell’s, which are indexed by the T –fixed points in ˆ Σ(λ). Now the points of ˆ 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 non-empty intersection with G(C[[t]]).λ (identified with an open subset of ˆ Σ(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 ∨ .