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AN EXAMPLE OF THE JANTZEN FILTRATION OF A D-MODULE

Part of: (Co)homology theory Lie algebras and Lie superalgebras Lie groups

Published online by Cambridge University Press:  11 November 2024

SIMON BOHUN  and
ANNA ROMANOV Open the ORCID record for ANNA ROMANOV [Opens in a new window]
SIMON BOHUN
Affiliation:
University of Utah, 201 Presidents’ Cir, Salt Lake City, UT 84112, USA e-mail: u1473792@utah.edu
ANNA ROMANOV*
Affiliation:
The University of New South Wales, Sydney NSW 2033, Australia
*
e-mail: a.romanov@unsw.edu.au
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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.

Keywords

representations of complex Lie algebrasWhittaker modulesJantzen filtrationBeilinson–Bernstein localisationmonodromic 𝒟-modules

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Secondary: 17B10: Representations, algebraic theory (weights) 22E47: Representations of Lie and real algebraic groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 118 , Issue 3 , June 2025 , pp. 265 - 296
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Oded Yacobi

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