  We consider the space X = Sym^3(C^2) of binary cubic forms, equipped with the
natural action of the group GL_2 of invertible linear transformations of C^2.
We describe explicitly the category of GL_2-equivariant coherent D_X-modules as
the category of representations of a quiver with relations. We show moreover
that this quiver is of tame representation type and we classify its
indecomposable representations. We also give a construction of the simple
equivariant D_X-modules (of which there are 14), and give formulas for the
characters of their underlying GL_2-representations. We conclude the article
with an explicit calculation of (iterated) local cohomology groups with
supports given by orbit closures.
