  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.
