论文标题
k-nilpotent锥的定期功能
Regular Functions on the K-Nilpotent Cone
论文作者
论文摘要
让$ g $成为一个复杂的还原代数群，带有lie代数$ \ mathfrak {g} $，让$ g _ {\ mathbb {r}} $是$ g $的真实形式，具有最大的紧凑型亚组$ k _ {\ mathbb {r}} $。与$ g _ {\ mathbb {r}} $相关联是a $ k \ times \ times \ mathbb {c}^{\ times} $ - 不变的subvarietiant subVarieT $ \ mathcal {n}_θ$（nilpotent）nilpotent cone $ \ natcal $ \ n n} n}在本文中，我们将得出常规函数环$ \ mathbb {c} [\ mathcal {n}_θ] $的公式，为$ k \ times \ times \ times \ athbb {c}^{\ times} $的表示。 一些动机来自霍奇理论。在Arxiv：1206.5547中，Schmid和Vilonen使用了Saito的混合霍奇模块理论中的想法来定义许多Harish-Chandra模块的规范良好过滤（包括所有标准和不可修复的Harish-Chandra模块）。使用这些过滤，它们制定了对统一双重的猜想描述。如果$ g _ {\ mathbb {r}} $被拆分，而$ x $是无穷小字符$ 0 $的球形主序列表示，则可以猜测$ \ mathrm {gr}（x）\ simeq \ simeq \ simeq \ simeq \ mathbb {c} \ mathbb {c}^{\ times} $。因此，$ \ Mathbb {C} [\ Mathcal {n}_θ] $的公式是计算Hodge过滤的必要成分。
Let $G$ be a complex reductive algebraic group with Lie algebra $\mathfrak{g}$ and let $G_{\mathbb{R}}$ be a real form of $G$ with maximal compact subgroup $K_{\mathbb{R}}$. Associated to $G_{\mathbb{R}}$ is a $K \times \mathbb{C}^{\times}$-invariant subvariety $\mathcal{N}_θ$ of the (usual) nilpotent cone $\mathcal{N} \subset \mathfrak{g}^*$. In this article, we will derive a formula for the ring of regular functions $\mathbb{C}[\mathcal{N}_θ]$ as a representation of $K \times \mathbb{C}^{\times}$. Some motivation comes from Hodge theory. In arXiv:1206.5547, Schmid and Vilonen use ideas from Saito's theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If $G_{\mathbb{R}}$ is split, and $X$ is the spherical principal series representation of infinitesimal character $0$, then conjecturally $\mathrm{gr}(X) \simeq \mathbb{C}[\mathcal{N}_θ]$ as representations of $K \times \mathbb{C}^{\times}$. So a formula for $\mathbb{C}[\mathcal{N}_θ]$ is an essential ingredient for computing Hodge filtrations.
