Cohomology of twisted $\mathcal{D}$-modules on $\mathbb{P}^1$ obtained as extensions from $\mathbb{C}^{\times}$

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.