The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space

Let $\mathcal{F}_1(n,m)$ be the space of ordered m-tuples of pairwise distinct points in $\partial \mathbf{H}_{\mathbb{H}}^n$ up to its isometry group $PSp(n,1)$. It is a real $2m^2-6m+5-\sum^{m-n-1}_{i=1}{m-2 \choose n-1+i}$ dimensional algebraic variety when $m>n+1$. In this paper, we construct and describe the moduli space of $\mathcal{F}_1(n,m)$, in terms of the Cartan's angle and cross-ratio invariants, by applying the Moore's determinant.