One of the most fundamental problems in representation theory is to
compute the dimensions of the weight spaces of the irreducible
representations of groups. In the case of a reductive group (for
example GL_n(C)) there are many combinatorial models that answer this
question, for example by counting the number of semistrandard
tableaux. The dimensions of the weight spaces of irreducible representations of
reductive groups have a natural q-deformation, called the
Kostka-Foulkes polynomials. However, finding combinatorial statistics
which express these polynomials is a long-standing open problem in
algebraic combinatorics, which has only been solved in type A.
We develop a new approach based on the geometry of the affine
Grassmannian, where we construct the charge in terms of the associated
crystal graph, starting from a decomposition of the crystal into
atoms. This not only allows us to recover the results of Lascoux and
Schützenberger in type A geometrically, but also to define a charge
statistic in type C2 (joint with J. Torres).
