On the Zassenhaus varieties of finite $W$-algebras in prime characteristic
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911495321812992 |
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| author | Shu, Bin Zeng, Yang |
| author_facet | Shu, Bin Zeng, Yang |
| contents | Let $Z(\mathcal{W})$ be the center of the finite $W$-algebra $\mathcal{W}({\mathfrak{g}},e)$ associated with $\mathfrak{g}=\text{Lie}(G)$ and a nilpotent element $e\in\mathfrak{g}$ for a connected reductive algebraic group $G$ over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$ under the standard hypotheses (H1)-(H3) in [Jantzen]. In this paper, we first demonstrate that our previous results in [Shu-Zeng] on the structure and geometric properties of $Z(\mathcal{W})$ for $p>>0$ are still true under the present weakened restriction on $p$. Then we study the Zassenhaus variety $\mathscr{Z}$ of $\mathcal{W}(\mathfrak{g},e)$, which is by definition the maximal spectrum $\text{Specm}(Z(\mathcal{W}))$ of $Z({\mathcal{W}})$. On basis of the structure properties of $Z({\mathcal{W}})$, we describe $\mathscr{Z}$ via a good transverse slice $\mathcal{S}$ and show that $\mathscr{Z}$ is birationally equivalent to $\mathcal{S}$, thereby a rational affine scheme. In the special case when $e=0$, we reobtain one of the main results of [Tange] on the rationality of the Zassenhaus varieites for reductive Lie algebras in prime characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06940 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Zassenhaus varieties of finite $W$-algebras in prime characteristic Shu, Bin Zeng, Yang Representation Theory Primary 17B35, Secondary 17B45 and 17B50 Let $Z(\mathcal{W})$ be the center of the finite $W$-algebra $\mathcal{W}({\mathfrak{g}},e)$ associated with $\mathfrak{g}=\text{Lie}(G)$ and a nilpotent element $e\in\mathfrak{g}$ for a connected reductive algebraic group $G$ over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$ under the standard hypotheses (H1)-(H3) in [Jantzen]. In this paper, we first demonstrate that our previous results in [Shu-Zeng] on the structure and geometric properties of $Z(\mathcal{W})$ for $p>>0$ are still true under the present weakened restriction on $p$. Then we study the Zassenhaus variety $\mathscr{Z}$ of $\mathcal{W}(\mathfrak{g},e)$, which is by definition the maximal spectrum $\text{Specm}(Z(\mathcal{W}))$ of $Z({\mathcal{W}})$. On basis of the structure properties of $Z({\mathcal{W}})$, we describe $\mathscr{Z}$ via a good transverse slice $\mathcal{S}$ and show that $\mathscr{Z}$ is birationally equivalent to $\mathcal{S}$, thereby a rational affine scheme. In the special case when $e=0$, we reobtain one of the main results of [Tange] on the rationality of the Zassenhaus varieites for reductive Lie algebras in prime characteristic. |
| title | On the Zassenhaus varieties of finite $W$-algebras in prime characteristic |
| topic | Representation Theory Primary 17B35, Secondary 17B45 and 17B50 |
| url | https://arxiv.org/abs/2603.06940 |