  Let $\mathfrak{g}$ be a complex semisimple Lie algebra. The
Beilinson-Bernstein localization theorem establishes an equivalence of the
category of $\mathfrak{g}$-modules of a fixed infinitesimal character and a
category of modules over a twisted sheaf of differential operators on the flag
variety of $\mathfrak{g}$. In this expository paper, we give four detailed
examples of this theorem when $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$.
Specifically, we describe the $\mathcal{D}$-modules associated to
finite-dimensional irreducible $\mathfrak{g}$-modules, Verma modules, Whittaker
modules, discrete series representations of $SL(2,\mathbb{R})$, and principal
series representations of $SL(2,\mathbb{R})$.
