Mirkovic-Vilonen cycles, preprojective algebra modules and Duistermaat-Heckman measures

Via the geometric Satake correspondence, the Mirkovic-Vilonen cycles give bases for representations of semisimple Lie algebras.  Similarly, by work of Lusztig, generic preprojective algebra modules give bases for these representations as well.  It is a long-standing open problem to compare these bases.  We will explain a new geometric way to make this comparison.  As an application, we will show that these bases do not agree in $SL_6$.