Preprojective algebras and c-sortable words

Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in \cite{Bua2} a finite dimensional algebra $\Lambda_w=\Lambda/I_w$. In this paper we look at filtrations of $\Lambda_w$ associated to any reduced expression $\ww$ of $w$. We are specially interested in the case where the word $\ww$ is $c$-sortable where $c$ is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting $kQ$-modules with finite torsionfree class. This nice description allows us to construct a triangle equivalence between the 2-Calabi-Yau triangulated category $\underline{\Sub}\Lambda_w$ and the generalized cluster category associated with an Auslander algebra.