与刚性零势轨道相关的无多重性原始理想数

Pub Date : 2024-03-26 DOI:10.4310/pamq.2024.v20.n1.a12
Alexander Premet, David I. Stewart
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引用次数: 0

摘要

让 $G$ 是定义在 $\mathbb{C}$ 上的一个简单代数群,让 $e$ 是 $g = \operatorname{Lie} (G)$ 中的一个刚性无势元素。在本文中,我们将证明有限 $W$-algebra $U(\mathfrak{g}, e)$ 允许一个或两个 $1$维表示。由于前面得到的结果,这可以归结为证明了与 E_8$ 型李代数中维度为 202 的刚性零potent 轨道相关的有限 $W$-gebras 恰好包含两个一维表示。作为推论,我们完成了与 $\mathfrak{g}$ 的刚性零potent $G$-orbits 相关的 $U(\mathfrak{g})$ 的无多重性基元理想的描述。在本文的最后,我们应用我们的结果列举了相关的还原包络代数的小不可还原表示。
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The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits
Let $G$ be a simple algebraic group defined over $\mathbb{C}$ and let $e$ be a rigid nilpotent element in $g = \operatorname{Lie} (G)$. In this paper we prove that the finite $W$-algebra $U(\mathfrak{g}, e)$ admits either one or two $1$-dimensional representations. Thanks to the results obtained earlier this boils down to showing that the finite $W$-algebras associated with the rigid nilpotent orbits of dimension 202 in the Lie algebras of type $E_8$ admit exactly two 1‑dimensional representations. As a corollary, we complete the description of the multiplicity-free primitive ideals of $U(\mathfrak{g})$ associated with the rigid nilpotent $G$-orbits of $\mathfrak{g}$. At the end of the paper, we apply our results to enumerate the small irreducible representations of the related reduced enveloping algebras.
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