Regularity of (anti-)unitary representations of Euler-involutive Z2-graded Lie groups

Determine whether every (anti-)unitary representation U of the Z2-graded Lie group G_{τ_h}, obtained from a connected Lie group G by adjoining the Euler involution τ_h associated with an Euler element h ∈ E(𝔤), is regular with respect to h; specifically, ascertain whether there exists an identity neighborhood N ⊂ G such that the intersection ∩_{g∈N} U(g) H_h is cyclic, where H_h denotes the Brunetti–Guido–Longo wedge standard subspace corresponding to the Euler wedge (h, τ_h).

Background

In the paper, the authors define a regularity property for (anti-)unitary representations U of Lie groups equipped with an Euler element h. Given the Brunetti–Guido–Longo (BGL) construction of the wedge standard subspace associated with the Euler wedge (h, τ_h), U is said to be h-regular if there exists an identity neighborhood N in G such that the intersection of the images of this standard subspace under U(g), for g in N, is cyclic.

The authors establish regularity in several settings, including positive-energy representations for certain simple and semidirect product Lie groups, and via localizability results on causal symmetric spaces. However, a general proof of regularity for all (anti-)unitary representations of groups of the form G_{τ_h} remains unresolved. They suggest that the case of solvable groups is particularly critical for addressing this question.