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- W4224252758 abstract "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 $mathcal{N}_{theta}$ of the (usual) nilpotent cone $mathcal{N} subset mathfrak{g}^*$. In this article, we will derive a formula for the ring of regular functions $mathbb{C}[mathcal{N}_{theta}]$ 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 $mathrm{gr}(X) simeq mathbb{C}[mathcal{N}_{theta}]$ as representations of $K times mathbb{C}^{times}$. So a formula for $mathbb{C}[mathcal{N}_{theta}]$ is an essential ingredient for computing Hodge filtrations." @default.
- W4224252758 created "2022-04-26" @default.
- W4224252758 creator A5080814993 @default.
- W4224252758 date "2022-04-21" @default.
- W4224252758 modified "2023-09-25" @default.
- W4224252758 title "Regular Functions on the K-Nilpotent Cone" @default.
- W4224252758 doi "https://doi.org/10.48550/arxiv.2204.10118" @default.
- W4224252758 hasPublicationYear "2022" @default.
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