  We study support $\tau$-tilting modules over preprojective algebras of Dynkin
type. We classify basic support $\tau$-tilting modules by giving a bijection
with elements in the corresponding Weyl groups. Moreover we show that they are
in bijection with the set of torsion classes, the set of torsion-free classes
and other important objects in representation theory. We also study
$g$-matrices of support $\tau$-tilting modules, which show terms of minimal
projective presentations of indecomposable direct summands. We give an explicit
description of $g$-matrices and prove that cones given by $g$-matrices coincide
with chambers of the associated root systems.
