Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials
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_ | 1866915634664701952 |
|---|---|
| author | Riche, Simon Situ, Quan |
| author_facet | Riche, Simon Situ, Quan |
| contents | Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in the principal block of the category of strongly $B$-equivariant $\mathfrak{g}$-modules (also called modular category $\mathcal{O}$), from a simple object to a costandard object, under the assumption that Lusztig's conjecture holds (which is known in large characteristic). The answer is given by a coefficient of a periodic Kazhdan--Lusztig polynomial associated with the corresponding affine Weyl group. Among other things, the proof uses a torus-equivariant version of the Koszul duality for $\mathfrak{g}$-modules constructed by the first author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials Riche, Simon Situ, Quan Representation Theory Algebraic Geometry Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in the principal block of the category of strongly $B$-equivariant $\mathfrak{g}$-modules (also called modular category $\mathcal{O}$), from a simple object to a costandard object, under the assumption that Lusztig's conjecture holds (which is known in large characteristic). The answer is given by a coefficient of a periodic Kazhdan--Lusztig polynomial associated with the corresponding affine Weyl group. Among other things, the proof uses a torus-equivariant version of the Koszul duality for $\mathfrak{g}$-modules constructed by the first author. |
| title | Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2511.18518 |