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Weight modules and gluing of sheaves on the flag variety

Pablo Boixeda Alvarez, Calder Morton-Ferguson

TL;DR

The paper develops two natural enlargements of Category O for semisimple Lie algebras and proves a derived Koszul duality between them: the algebraic category of finite weight modules A_f (via O_0^w extensions) and the Kazhdan–Laumon glued category KL_O (via gluing data). Using Beilinson–Bernstein localization, the authors provide a geometric realization through unipotent monodromy and singular support conditions on D-modules or perverse sheaves on G/B, and they recast the derived categories as algebras over a natural monad. They establish a unified framework—monads, Barr–Beck–Lurie, and t-structures—that yields equivalences D^{alg} ≅ KL_O^∨ ≅ D^{geom}, with KL_O^∨ being the Koszul dual of KL_O. The work connects to microlocal sheaves on affine Springer fibers and coherent families in the sense of Fernando and Mathieu, and it conjectures deeper links to the geometry of the small quantum group and semi-infinite flag varieties. Overall, the paper provides a geometric reframing of weight-module theory and lays groundwork for further Koszul-duality connections in representation theory and quantum geometry.

Abstract

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric realization of this category as unipotently monodromic sheaves on the flag variety satisfying a singular support condition. We then explain that a derived version $\mathcal{D}$ of this category is Koszul dual to the Kazhdan-Laumon Category O, a different enlargement of the BGG Category O obtained from a gluing construction for sheaves on the flag variety. This characterizes $\mathcal{D}$ as the category of algebras over a natural monad on a direct sum of copies of the derived Category O. These results give new interpretations of classical algebraic constructions of weight modules over semisimple Lie algebras due to Fernando and Mathieu. We also conjecture that our results fit naturally into a proposed Koszul duality relating the small quantum group and the semi-infinite flag variety to the geometry of affine Springer fibers.

Weight modules and gluing of sheaves on the flag variety

TL;DR

The paper develops two natural enlargements of Category O for semisimple Lie algebras and proves a derived Koszul duality between them: the algebraic category of finite weight modules A_f (via O_0^w extensions) and the Kazhdan–Laumon glued category KL_O (via gluing data). Using Beilinson–Bernstein localization, the authors provide a geometric realization through unipotent monodromy and singular support conditions on D-modules or perverse sheaves on G/B, and they recast the derived categories as algebras over a natural monad. They establish a unified framework—monads, Barr–Beck–Lurie, and t-structures—that yields equivalences D^{alg} ≅ KL_O^∨ ≅ D^{geom}, with KL_O^∨ being the Koszul dual of KL_O. The work connects to microlocal sheaves on affine Springer fibers and coherent families in the sense of Fernando and Mathieu, and it conjectures deeper links to the geometry of the small quantum group and semi-infinite flag varieties. Overall, the paper provides a geometric reframing of weight-module theory and lays groundwork for further Koszul-duality connections in representation theory and quantum geometry.

Abstract

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric realization of this category as unipotently monodromic sheaves on the flag variety satisfying a singular support condition. We then explain that a derived version of this category is Koszul dual to the Kazhdan-Laumon Category O, a different enlargement of the BGG Category O obtained from a gluing construction for sheaves on the flag variety. This characterizes as the category of algebras over a natural monad on a direct sum of copies of the derived Category O. These results give new interpretations of classical algebraic constructions of weight modules over semisimple Lie algebras due to Fernando and Mathieu. We also conjecture that our results fit naturally into a proposed Koszul duality relating the small quantum group and the semi-infinite flag variety to the geometry of affine Springer fibers.

Paper Structure

This paper contains 56 sections, 49 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Two categories
  3. Connection to microlocal sheaves on affine Springer fibers
  4. Overview of the paper
  5. Acknowledgments
  6. The category of finite weight modules
  7. Preliminaries
  8. The BGG Category O and a generalization
  9. The BGG Category O
  10. Finite weight modules
  11. The categories $\mathcal{O}_{0}^w$
  12. Simple objects in $\mathcal{A}_f$
  13. Singular support, algebraically
  14. The associated variety of a $\mathfrak{g}$-module
  15. Functors and derived categories
  16. ...and 41 more sections

Key Result

Proposition 1.2

The category $\mathcal{A}_f$ is the full subcategory of modules $M \in \mathfrak{g}\mathrm{-mod}$ satisfying the following conditions.

Figures (4)

  • Figure 1: A schematic picture of the module $M_0$.
  • Figure 2: A schematic picture of the module $M_{-2}$.
  • Figure 3: A schematic picture of the module $M_2$.
  • Figure 4: A schematic picture of the module $\mathcal{M}[0]$.

Theorems & Definitions (122)

  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Claim 1.8
  • Conjecture 1.9
  • Definition 2.1
  • Definition 2.2
  • ...and 112 more