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.