Araki Cohomology of the de Sitter Scalar Discrete Series

Building on the recently established de Sitter (dS)-covariant indecomposable Krein realization of the scalar discrete-series unitary irreducible representations ${\mathrm{\Pi }}_{p,0}$, $p=1,2,\cdots$, of ${\mathrm{S}\mathrm{O}}_0(1,4)$, naturally organized into a Gupta-Bleuler triplet, we uncover its underlying Araki-cohomological structure. We show that the Krein-Gupta-Bleuler construction provides a concrete realization of Araki’s standard Gupta-Bleuler triplet, with its non-split indecomposable structure encoded by non-trivial first cohomology classes related by adjunction. We further show that the reducible intermediate sector naturally resolves into its two irreducible components, inducing a corresponding decomposition of the extension classes and yielding a four-layer refinement of Araki’s standard triplet. The resulting Gupta-Bleuler quadruplet provides a finer cohomological resolution, distinguishing the physical representation from its complex-conjugate sector.