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An example of the Jantzen filtration of a D-module

Simon Bohun, Anna Romanov

TL;DR

This paper uses the $\mathfrak{sl}_2$ example to illuminate Beilinson--Bernstein’s geometric approach to Jantzen filtrations, linking algebraic and geometric perspectives via localization on the flag/base affine space. It constructs and analyzes deformed Verma and dual Verma modules through the maximal extension functor $\Xi_f$ and its monodromy filtration, then shows that the resulting geometric Jantzen filtration on certain $\mathcal{D}$-modules corresponds to the algebraic Jantzen filtration on their global sections. The main result is an explicit, calculational bridge: for $\mathfrak{sl}_2$ the geometric filtration coincides with the algebraic one, with the global sections revealing Verma/dual Verma decompositions and big-projective structures in category $\mathcal{O}$. The work provides concrete algebraic snapshots and visualizations that demystify the Beilinson--Bernstein framework and demonstrate how geometric filtrations recover classical representation-theoretic information. It thus offers a practical, detailed guide to the interplay between deformation, monodromy, weight filtrations, and Jantzen-type phenomena in a foundational SL$_2$ setting.

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.

An example of the Jantzen filtration of a D-module

TL;DR

This paper uses the example to illuminate Beilinson--Bernstein’s geometric approach to Jantzen filtrations, linking algebraic and geometric perspectives via localization on the flag/base affine space. It constructs and analyzes deformed Verma and dual Verma modules through the maximal extension functor and its monodromy filtration, then shows that the resulting geometric Jantzen filtration on certain -modules corresponds to the algebraic Jantzen filtration on their global sections. The main result is an explicit, calculational bridge: for the geometric filtration coincides with the algebraic one, with the global sections revealing Verma/dual Verma decompositions and big-projective structures in category . The work provides concrete algebraic snapshots and visualizations that demystify the Beilinson--Bernstein framework and demonstrate how geometric filtrations recover classical representation-theoretic information. It thus offers a practical, detailed guide to the interplay between deformation, monodromy, weight filtrations, and Jantzen-type phenomena in a foundational SL setting.

Abstract

We compute the Jantzen filtration of a D-module on the flag variety of . At each step in the computation, we illustrate the -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.
Paper Structure (25 sections, 2 theorems, 122 equations, 8 figures)

This paper contains 25 sections, 2 theorems, 122 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. The algebraic Jantzen filtration
  4. The geometric Jantzen filtration
  5. Monodromy filtrations
  6. Geometric Jantzen filtrations
  7. Geometric Jantzen filtrations on Harish-Chandra sheaves
  8. Relationship between monodromy and weight filtrations
  9. Relationship between algebraic and geometric Jantzen filtrations
  10. Structure of the paper
  11. Acknowledgements
  12. Example
  13. The map $\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{Z}(\mathfrak{g})} \mathcal{U}(\mathfrak{h}) \rightarrow \Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}})$
  14. Monodromic $\mathcal{D}_{X}$-modules
  15. Verma modules and dual Verma modules
  16. ...and 10 more sections

Key Result

Lemma 2.1

The homomorphism eq: the map U(g) tensor U(h) --> D factors through the quotient where $\mathcal{Z}(\mathfrak{g})$ acts on $\mathcal{U}(\mathfrak{h})$ via the Harish-Chandra projection $\gamma_\mathrm{HC}$eq: HC projection.

Figures (8)

  • Figure 1: Dual Verma modules arise as global sections of $j_+\mathcal{O}_U$.
  • Figure 2: Verma modules arise as global sections of $j_!\mathcal{O}_U$.
  • Figure 3: Deformed dual Verma modules arise as global sections of $j_+f^s\mathcal{O}_U^{(n)}$.
  • Figure 4: Deformed Verma modules arise as global sections of $j_!f^s\mathcal{O}_U^{(n)}$.
  • Figure 5: Caricature of the maximal extension $\Xi_\rho \mathcal{O}_U$.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 7 more