Weight modules and gluing of sheaves on the flag variety
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911301114003456 |
|---|---|
| author | Alvarez, Pablo Boixeda Morton-Ferguson, Calder |
| author_facet | Alvarez, Pablo Boixeda Morton-Ferguson, Calder |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weight modules and gluing of sheaves on the flag variety Alvarez, Pablo Boixeda Morton-Ferguson, Calder Representation Theory Algebraic Geometry 17B10, 14F06 (Primary) 14M15, 17B55 (Secondary) 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. |
| title | Weight modules and gluing of sheaves on the flag variety |
| topic | Representation Theory Algebraic Geometry 17B10, 14F06 (Primary) 14M15, 17B55 (Secondary) |
| url | https://arxiv.org/abs/2509.25156 |