The Mirkovi?–Vilonen basis and Duistermaat–Heckman measures

Using the geometric Satake correspondence, Mirkovic-Vilonen cycles in affine Grasssmannian give bases for representations of a semisimple group G . We prove that these are perfect, i.e. compatible with action Chevelley generators positive half Lie algebra g. compute this terms intersection multiplicities Grassmannian. stitch together to basis C[N] regular functions on unipotent subgroup. multiplication MV using Beilinson-Drinfeld Grassmannian, thus proving conjecture Anderson. In third part paper, we define map from convolution measures dual Cartan subalgebra characterize universal centralizer space G. measure associated an element equals Duistermaat-Heckman corresponding cycle. This leads proof Muthiah. Finally, use compare and Lusztig's semicanonical basis. formulate conjectures relating algebraic invariants preprojective modules (which underlie basis) cycles. appendix, ideas do not coincide SL_6.