Tilting theory of preprojective algebras and c-sortable elements

For a finite acyclic quiver Q and the corresponding preprojective algebra Π, we study the factor algebra Πw associated with an element w in the Coxeter group of Q introduced by Buan-Iyama-Reiten-Scott. The algebra Πw has a natural Z-grading. We prove that Sub_ZΠw has a tilting object M if w is c-sortable. Moreover, we show that the endomorphism algebra of M is isomorphic to the stable Auslander algebra of a certain torsion free class of modkQ.