The geometric Satake isomorphism is an equivalence of categories between the representations of a reductive group over any commutative ring, and the perverse sheaves on the complex affine Grassmannian of the Langlands dual group, with coefficients in that ring. In particular, the decomposition numbers for a reductive group can be seen as decomposition numbers for perverse sheaves.
We use this bridge in both directions, in the case of the minimal degeneration singularities of the affine Grassmannian which were classified by Malkin, Ostrik and Vybornov. They correspond to pairs of adjacent weights, which were classified by Stembridge.
From geometry to representation theory, we use our previous work on simple and minimal singularities to recover known decomposition numbers for reductive groups. From representation theory to geoemtry, we prove non-smoothness results and non-equivalences of singularities, using decomposition numbers for reductive groups.
(Organized by D.Juteau, Z.Lin)
|
