An example of the Jantzen filtration of a D-module

We compute the Jantzen filtration of a D-module on the flag variety of ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$. At each step in the computation, we illustrate the ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$-module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the D-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson–Bernstein.