Extended Future Tube Conjecture for Unipotent Subgroups

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