Category $\mathcal{O}$ and asymptotic characters
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911069638754304 |
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| author | Leroux-Lapierre, Alexis |
| author_facet | Leroux-Lapierre, Alexis |
| contents | This paper defines an asymptotic character map which is a morphism from the Grothendieck group of category $\mathcal{O}$ of an integral filtered quantization to rational functions on the Lie algebra of a torus. We show that the asymptotic character of a module computes the equivariant multiplicity of its characteristic cycle. We then apply this construction to truncated shifted Yangians coming from simple, simply-laced Lie algebras and draw connections with characters of modules over KLR algebras using an equivalence of categories of arXiv:1806.07519. Our main theorem shows how this new formalism gives formulas relating equivariant multiplicities of Mirković-Vilonen cycles and characters of modules over cyclotomic KLR algebras. We explain how this result provides evidence that the change-of-basis between Lusztig's dual canonical basis and the Mirković-Vilonen basis of $\mathbb{C}[N]$ is computed by a characteristic cycle map whose domain is category $\mathcal{O}$ for truncated shifted Yangians, implying that the coefficients are non-negative integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Category $\mathcal{O}$ and asymptotic characters Leroux-Lapierre, Alexis Representation Theory This paper defines an asymptotic character map which is a morphism from the Grothendieck group of category $\mathcal{O}$ of an integral filtered quantization to rational functions on the Lie algebra of a torus. We show that the asymptotic character of a module computes the equivariant multiplicity of its characteristic cycle. We then apply this construction to truncated shifted Yangians coming from simple, simply-laced Lie algebras and draw connections with characters of modules over KLR algebras using an equivalence of categories of arXiv:1806.07519. Our main theorem shows how this new formalism gives formulas relating equivariant multiplicities of Mirković-Vilonen cycles and characters of modules over cyclotomic KLR algebras. We explain how this result provides evidence that the change-of-basis between Lusztig's dual canonical basis and the Mirković-Vilonen basis of $\mathbb{C}[N]$ is computed by a characteristic cycle map whose domain is category $\mathcal{O}$ for truncated shifted Yangians, implying that the coefficients are non-negative integers. |
| title | Category $\mathcal{O}$ and asymptotic characters |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2507.16215 |