Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916849219796992 |
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| author | Zeng, Yang Shu, Bin |
| author_facet | Zeng, Yang Shu, Bin |
| contents | Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_θ\in\mathfrak{g}_{\bar0}$ with $-θ$ being a minimal root of $\mathfrak{g}$. Set $U(\mathfrak{g},e)$ to be the minimal finite $W$-superalgebras associated with the pair $(\mathfrak{g},e)$. In this paper we study the highest weight theory for $U(\mathfrak{g},e)$, introduce the Verma modules and give a complete isomorphism classification of finite-dimensional irreducible modules, via the parameter set consisting of pairs of weights and levels. Those Verma modules can be further described via parabolic induction from Whittaker modules for $\mathfrak{osp}(1|2)$ or $\mathfrak{sl}(2)$ respectively, depending on the detecting parity of $\textsf{r}:=\dim\mathfrak{g}(-1)_{\bar1}$. We then introduce and investigate the BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highest weight theory, as a counterpart of the works for finite $W$-algebras by Brundan-Goodwin-Kleshchev and Losev, respectively.
In comparison with the non-super case, the significant difference here lies in the situation when $\textsf{r}$ is odd, which is a completely new phenomenon. The difficulty and complicated computation arise from there. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_00921 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories Zeng, Yang Shu, Bin Representation Theory Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_θ\in\mathfrak{g}_{\bar0}$ with $-θ$ being a minimal root of $\mathfrak{g}$. Set $U(\mathfrak{g},e)$ to be the minimal finite $W$-superalgebras associated with the pair $(\mathfrak{g},e)$. In this paper we study the highest weight theory for $U(\mathfrak{g},e)$, introduce the Verma modules and give a complete isomorphism classification of finite-dimensional irreducible modules, via the parameter set consisting of pairs of weights and levels. Those Verma modules can be further described via parabolic induction from Whittaker modules for $\mathfrak{osp}(1|2)$ or $\mathfrak{sl}(2)$ respectively, depending on the detecting parity of $\textsf{r}:=\dim\mathfrak{g}(-1)_{\bar1}$. We then introduce and investigate the BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highest weight theory, as a counterpart of the works for finite $W$-algebras by Brundan-Goodwin-Kleshchev and Losev, respectively. In comparison with the non-super case, the significant difference here lies in the situation when $\textsf{r}$ is odd, which is a completely new phenomenon. The difficulty and complicated computation arise from there. |
| title | Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2209.00921 |