Extended Future Tube Conjecture for Unipotent Subgroups

Let $\mathrm{\Omega }$ be the Lorentz future cone in ${\mathbb{R}}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T={\mathbb{R}}^{d+1}+i\mathrm{\Omega }$. The Lorentz group ${\mathrm{S}\mathrm{O}}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{S}\mathrm{O}(1,d{)}^{\mathbb{C}}$ acts on ${\mathbb{C}}^{(d+1)\times M}$. We prove that the domain $G^{\mathbb{C}}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of ${\mathrm{S}\mathrm{O}}_0(1,d)$.