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 ${\rm SO}(n-1,1)$ on the Lie algebra of ${\rm SO}(n,1)$.