  We give an interpretation of the map $\pi^c$ defined by Reading, which is a
map from the elements of a Coxeter group to the $c$-sortable elements, in terms
of the representation theory of preprojective algebras. Moreover, we study a
close relationship between $c$-sortable elements and torsion pairs, and give an
explicit description of the cofinite torsion classes in the context of the
Coxeter group. As a consequence, we give a proof of some conjectures proposed
by Oppermann, Reiten, and the second author.
