Thu, 27 Nov 2003
17:00
L1

LS-galleries and MV-cycles

Peter Littlemann
(Wuppertal)
Abstract

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 $G^\vee$.

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