Multiplicity Formulas for Perverse Coherent Sheaves on the Nilpotent Cone

Arinkin and Bezrukavnikov have given the construction of the category of equivariant perverse coherent sheaves on the nilpotent cone of a complex reductive algebraic group. Bezrukavnikov has shown that this category is in fact weakly quasi-hereditary with Andersen–Jantzen sheaves playing a role analogous to that of Verma modules in category O for a semi-simple Lie algebra. Our goal is to show that the category of perverse coherent sheaves possesses the added structure of a properly stratified category, and to use this structure to give an effective algorithm to compute multiplicities of simple objects in perverse coherent sheaves. The algorithm is developed by studying mixed constructible sheaves and Kazhdan–Lusztig polynomials on the affine Grassmannian, and their relation to certain complexes of coherent sheaves on the nilpotent cone.