Regular Functions on the K-Nilpotent Cone

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}$-invariantsubvariety$$\mathcal{N}${\theta}$ of the (usual) nilpotent cone $\mathcal{N}\subset {\mathfrak{g}}^{\ast }$. In this article, we will derive a formula for the ring of regular functions $\mathbb{C}[{\mathcal{N}}_{\theta }]$ as a representation of $K \times $\mathbb{C}$^{\times}$. Some motivation comes from Hodge theory. In arXiv:1206.5547, Schmid and Vilonen use ideas from Saito’s theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If $G_{\mathbb{R}}$ is split, and $X$ is the spherical principal series representation of infinitesimal character $0$, then conjecturally $$\mathrm{g}\mathrm{r}$(X) \simeq $\mathbb{C}$[$\mathcal{N}${\theta}]$asrepresentationsof$K \times $\mathbb{C}$^{\times}$\mathrm{.}Soaformulafor$$\mathbb{C}$[$\mathcal{N}$_{\theta}]$ is an essential ingredient for computing Hodge filtrations.