Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909982879907840 |
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| author | Liu, Genqiang Li, Mingjie |
| author_facet | Liu, Genqiang Li, Mingjie |
| contents | A weight $\mathfrak{sl}_{n+1}$-module with finite-dimensional weight spaces is called a cuspidal module, if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively on it. In \cite{LL}, it has been shown that any block with a generalized central character of the cuspidal $\mathfrak{sl}_{n+1}$-module category is equivalent to a block of the category of finite-dimensional modules over the minimal nilpotent finite $W$-algebra $W(e)$ for $\mathfrak{sl}_{n+1}$. In this paper, using a centralizer realization of $W(e)$ and an explicit embedding $W(e)\rightarrow U(\mathfrak{gl}_n)$, we show that every finite-dimensional irreducible $W(e)$-module is isomorphic to an irreducible $W(e)$-quotient module of some finite-dimensional irreducible $\mathfrak{gl}_n$-module. As an application, we can give very explicit realizations of all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules using finite-dimensional irreducible $\mathfrak{gl}_n$-modules, avoiding using the twisted localization method and the coherent family introduced in \cite{M}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra Liu, Genqiang Li, Mingjie Representation Theory Rings and Algebras A weight $\mathfrak{sl}_{n+1}$-module with finite-dimensional weight spaces is called a cuspidal module, if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively on it. In \cite{LL}, it has been shown that any block with a generalized central character of the cuspidal $\mathfrak{sl}_{n+1}$-module category is equivalent to a block of the category of finite-dimensional modules over the minimal nilpotent finite $W$-algebra $W(e)$ for $\mathfrak{sl}_{n+1}$. In this paper, using a centralizer realization of $W(e)$ and an explicit embedding $W(e)\rightarrow U(\mathfrak{gl}_n)$, we show that every finite-dimensional irreducible $W(e)$-module is isomorphic to an irreducible $W(e)$-quotient module of some finite-dimensional irreducible $\mathfrak{gl}_n$-module. As an application, we can give very explicit realizations of all irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules using finite-dimensional irreducible $\mathfrak{gl}_n$-modules, avoiding using the twisted localization method and the coherent family introduced in \cite{M}. |
| title | Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2505.19417 |