  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 $\hat\Sigma(\lam)$ of the closure of an orbit
$G(\bc[[t]]).\lam$ in the affine Grassmannian. The homology of
$\hat\Sigma(\lam)$ has a basis given by Bia{\l}ynicki--Birula cell's, which are
indexed by the $T$--fixed points in $\hat\Sigma(\lam)$. Now the points of
$\hat\Sigma(\lam)$ 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 $G(\bc[[t]]).\lam$ (identified with an open subset
of $\hat\Sigma(\lam)$), 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 $V(\lam)$ for the Langland's dual group $G^\vee$.
