Homomorphisms of generalized Verma modules, BGG parabolic category \(\mathcal O^{\mathfrak p}\) and Juhl's conjecture (Q2892868)

In [\textit{A. Juhl}, Families of conformally covariant differential operators, Q-curvature and holography. Progress in Mathematics 275. Basel: Birkhäuser (2009; Zbl 1177.53001)], when \(\mathfrak{g} = \mathfrak{so}(n+1,1)\) and \(\mathfrak{g}' = \mathfrak{so}(n,1)\), for given \(\lambda \in \mathbb{C}\) and each natural number \(N\), Juhl constructed a \(U(\mathfrak{g}')\)-homomorphism \(D_N(\lambda): M(\mathfrak{g}', \mathfrak{p}', \mathbb{C}_{\lambda-N}) \to M(\mathfrak{g}, \mathfrak{p},\mathbb{C}_{\lambda})\) between generalized Verma modules \(M(\mathfrak{g}', \mathfrak{p}', \mathbb{C}_{\lambda-N})\) and \(M(\mathfrak{g}, \mathfrak{p},\mathbb{C}_{\lambda})\), where \(\mathfrak{p}\) is the maximal parabolic subalgebra of \(\mathfrak{g}\) determined by the first simple root and \(\mathfrak{p}' = \mathfrak{p} \cap \mathfrak{g}'\). He conjectured that the family \(\{D_N(\lambda): N \in \mathbb{N}\}\) generates all the \(U(\mathfrak{g}')\)-homomorphisms between the two generalized Verma modules. In this article, the author first constructs the \(U(\mathfrak{g}')\)-homomorphism \(D_N(\lambda)\) for \(\mathfrak{g} = \mathfrak{so}(p+1, q+1)\) and \(\mathfrak{g}' = \mathfrak{so}(p, q+1)\) with \(p, q\) arbitrary. He then proves Juhl's conjecture in generality.NEWLINENEWLINETo prove the conjecture, he also shows the explicit decomposition of \(M(\mathfrak{g}, \mathfrak{p}, \mathbb{C}_{\lambda})\) in the parabolic category \(\mathcal{O}^{\mathfrak{p}'}\) in terms of the generalized Verma modules and projective modules.