  We construct twisted $\mathcal{D}$-modules on the projective line
$\mathbb{P}^1$ that are equivariant for the action of the diagonal torus
subgroup of $SL_2$. In the most interesting case these arise as extensions from
local systems on $\mathbb{C}^{\times}$. We discuss their subquotient structure.
Their sheaf cohomology groups are weight modules for the Lie algebra
$\mathfrak{sl}_2$. We also discuss their subquotient structure and in case
these modules are not the familiar highest or lowest weight modules, we give an
explicit presentation for them. Our computations illustrate some basic
$\mathcal{D}$-module concepts and the Beilinson-Bernstein equivalence. They are
the first step in a program that aims to describe categories of modules over
semisimple and affine Kac-Moody Lie algebras that are next to highest (or
lowest) weight via $\mathcal{D}$-modules on the flag variety.
