Indecomposability preservation from isotropy to transvection holonomy

Determine whether, for a reductive Lorentzian homogeneous space M = G/H with isotropy H ⊆ O(1,m−1), the holonomy algebra \tilde{H} of the canonical Ambrose–Singer connection in the transvection presentation M = \tilde{G}/\tilde{H} necessarily remains indecomposable and non-irreducible; equivalently, ascertain whether the assumptions in the plane-wave characterization conjecture imply the assumptions in the main theorem.

Background

To connect the conjecture with the main theorem, the authors discuss the canonical Ambrose–Singer connection associated to a reductive homogeneous space and its transvection presentation M = \tilde{G}/\tilde{H}. While \tilde{H} is a subgroup of the original isotropy H, indecomposability can be lost when passing to \tilde{H}.

This possibility prevents a direct implication from the conjecture’s assumptions to those of the main theorem, leaving open whether indecomposability of isotropy at the level of G/H guarantees indecomposability of the canonical connection’s holonomy in the transvection representation.