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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.25185 |
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| _version_ | 1866914513238884352 |
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| author | Zheng, Xiaoyao Zhao, Yufang Liu, Genqiang |
| author_facet | Zheng, Xiaoyao Zhao, Yufang Liu, Genqiang |
| contents | Let $\bar{S}_2$ be the Lie algebra of polynomial vector fields on $A_2=\mathbb{C}[t_1,t_2]$ with constant divergence.In this paper, we first show that each block $Ω^{\widetilde{S}_2}_{\mathbf{a}}$ of the category of $(A_2, \bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional module category over the parabolic subalgebra $\bar{S}_2^{\geq 0}$. Then we classify all simple Whittaker $\bar{S}_2$-modules with finite-dimensional Whittaker vector spaces using $\mathfrak{gl}_2$-modules. Finally, we establish an equivalence between $Ω^{\bar{S}_2}_{\mathbf{1}}$ and the category $H_{\mathbf{1}}$-fmod of finite-dimensional modules over an associative algebra $H_{\mathbf{1}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25185 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The category of Whittaker modules over the Cartan Type Lie algebra $\bar{S}_2$ Zheng, Xiaoyao Zhao, Yufang Liu, Genqiang Representation Theory Let $\bar{S}_2$ be the Lie algebra of polynomial vector fields on $A_2=\mathbb{C}[t_1,t_2]$ with constant divergence.In this paper, we first show that each block $Ω^{\widetilde{S}_2}_{\mathbf{a}}$ of the category of $(A_2, \bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional module category over the parabolic subalgebra $\bar{S}_2^{\geq 0}$. Then we classify all simple Whittaker $\bar{S}_2$-modules with finite-dimensional Whittaker vector spaces using $\mathfrak{gl}_2$-modules. Finally, we establish an equivalence between $Ω^{\bar{S}_2}_{\mathbf{1}}$ and the category $H_{\mathbf{1}}$-fmod of finite-dimensional modules over an associative algebra $H_{\mathbf{1}}$. |
| title | The category of Whittaker modules over the Cartan Type Lie algebra $\bar{S}_2$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2604.25185 |