Past Representation Theory Seminar

27 November 2003
Peter Littlemann
Let $G$ be a complex semisimple algebraic group. We give an interpretation of the path model of a representation in terms of the geometry of the affine Grassmannian for $G$. In this setting, the paths are replaced by LS--galleries in the affine Coxeter complex associated to the Weyl group of $G$. The connection with geometry is obtained as follows: consider a Bott--Samelson desingularization of the closure of an orbit $G(\bc[[t]]).\lam$ in the affine Grassmannian. The points of this variety can be viewed as galleries of a fixed type in the affine Tits building associated to $G$. The retraction of the Tits building onto the affine Coxeter complex induces in this way, a stratification of the $G(\bc[[t]])$--orbit, indexed by certain folded galleries in the Coxeter complex. The connection with representation theory is given by the fact that the closures of the strata associated to LS-galleries are the Mirkovic-Vilonen--cycles, which form a basis of the representation $V(\lam)$ for the Langland's dual group $G^\vee$.
  • Representation Theory Seminar