Existence of Hilbert quotients for arbitrary closed Lorentz subgroups

Determine whether a Hilbert quotient exists for the action of an arbitrary closed subgroup of \(\textup{SO}_0(1,d)\) in the setting of the extended future tube construction.

Background

The paper studies domains of the form $Z=G^{\mathbb C}\cdot T^M$, where $T^M$ is a product of future tubes and $G$ is a subgroup of $\textup{SO}_0(1,d)$. Its main results establish that the relevant orbit space and domain are Stein when $G$ is a connected unipotent subgroup. The general quotient-theoretic strategy would benefit from an analytic Hilbert quotient for the ambient action, but the existence of such a quotient is not established for arbitrary closed subgroups of the Lorentz group. The paper instead develops conditions for constructing a geometric quotient directly in the unipotent setting.