Regularity Conjecture for antiunitary representations

Establish that for every Euler element h in the Lie algebra of a connected Lie group G, every antiunitary representation (U,H) of the semidirect product G_{τ_h} = G ⋊ {1, τ_h} is h-regular, meaning that there exists an identity neighborhood N ⊂ G such that the intersection of translates ∩_{g ∈ N} U(g) V is cyclic for the canonical standard subspace V = (h,U) determined by Δ_V = exp(2π i · ∂U(h)) and J_V = U(τ_h).

Background

The paper studies nets of real subspaces associated with antiunitary representations of Lie groups and their Bisognano–Wichmann-type properties. A central technical notion is h-regularity: for an Euler element h, an antiunitary representation (U,H) of G_{τh} is called h-regular if there exists an identity neighborhood N ⊂ G such that the intersection of translates ∩{g ∈ N} U(g) V is cyclic, where V = (h,U) is the standard subspace specified by Δ_V = exp(2π i * ∂U(h)) and J_V = U(τ_h).

The authors show that h-regularity is equivalent to the existence of nets on G satisfying isotony, covariance, Reeh–Schlieder, and Bisognano–Wichmann properties (Theorem 5.9), and they prove the conjecture for connected reductive groups (Corollary 5.12). Extending h-regularity to all antiunitary representations, regardless of the group’s structure, would provide a unified existence theory for such nets across broad classes of Lie groups.