An example of the Jantzen filtration of a D-module

TL;DR

This paper explicitly computes the Jantzen filtration of a D-module on the flag variety of SL_2(C), illustrating the module structure and confirming a key theorem linking geometric and algebraic filtrations.

Contribution

It provides a detailed algebraic and geometric analysis of the Jantzen filtration for a specific D-module, illustrating the module structure and confirming the Beilinson--Bernstein theorem.

Findings

Jantzen filtration computed explicitly for the D-module

Module structure on global sections illustrated

Confirmation of the algebraic and geometric filtration correspondence

Abstract

We compute the Jantzen filtration of a D-module on the flag variety of $\mathrm{SL}_2(\mathbb{C})$. 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 D-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson--Bernstein.