AN EXAMPLE OF THE JANTZEN FILTRATION OF A D-MODULE

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