Mirkovic and Vilonen have shown that a category of perverse sheaves on the Affine Grassmannian for a reductive algebraic group is equivalent to the category of representations of the Langlands dual of this group. This equivalence gives a geometric analogue of the Satake isomorphism and opens a new avenue of attack on problems in the representation theory of algebraic group schemes. This paper gives a derivation of the weight multiplicities of the irreducible representations of the algebraic group SL 2 over a field of positive characteristic. The derivation will depend only upon the geometry of the Affine Grassmannian and will not rely on any representation-theoretic constructions.
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