LS-Galleries, the path model and MV-cycles

We give an interpretation of the path model of a representation \cite{Lit1} 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 $\widehat{\mathrm{\Sigma }}(\text{lam})$ of the closure of an orbit $G(\text{bc}[[t]])\mathrm{.}\text{lam}$ in the affine Grassmannian. The homology of $\widehat{\mathrm{\Sigma }}(\text{lam})$ has a basis given by Bia{\l}ynicki–Birula cell’s, which are indexed by the $T$–fixed points in $\widehat{\mathrm{\Sigma }}(\text{lam})$. Now the points of $\widehat{\mathrm{\Sigma }}(\text{lam})$ 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 $G(\text{bc}[[t]])\mathrm{.}\text{lam}$ (identified with an open subset of $\widehat{\mathrm{\Sigma }}(\text{lam})$), and we show that the closures of the strata associated to LS-galleries are exactly the MV–cycles \cite{MV}, which form a basis of the representation $V(\text{lam})$ for the Langland’s dual group $G^{\vee }$.