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Forthcoming events in this series
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14:00
14:30
14:30
14:30
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14:30
Smash products of linear categories and the Cartan-Leray spectral sequence
16:00
Representations of real semi-simple Lie groups and the geometry of flag manifolds
(Clown Meeting)
12:00
Symplectic leaves of nxn matrices and representation theory of quantum groups
(Clown Meeting)
14:30
17:00
LS-galleries and MV-cycles
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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