On Quantization of a Nilpotent Orbit Closure in $G_2$

Let $G$ be the complex exceptional Lie group of type $G_2$. Among the five nilpotent orbits in its Lie algebra $\mathfrak{g}$, only the 8-dimensional orbit ${\mathcal{O}}_8$ has non-normal orbit closure $\overline{{\mathcal{O}}_8}$. In this short note, we will give a quantization model of $\overline{{\mathcal{O}}_8}$, verifying a conjecture of Vogan in 1984.