Projection Theorems and Isometries of Hyperbolic Spaces

We prove a restricted projection theorem for an n-2 dimensional family of projections from ${\mathbb{R}}^n$ to $\mathbb{R}$. The family we consider arises naturally in the context of the adjoint representation of the maximal unipotent subgroup of $\mathrm{S}\mathrm{O}(n-1,1)$ on the Lie algebra of $\mathrm{S}\mathrm{O}(n,1)$.