Homomorphisms of generalized Verma modules, BGG parabolic category ${\fam2 O}^\gop$ and Juhl's conjecture
Abstrakt
Let $\km_\lambda(\gog,\gop),\km_\mu(\gog^\prime,\gop^\prime)$ be the generalized Verma modules for $\gog=so(p+1,q+1), \gog^\prime=so(p,q+1)$ induced from characters $\lambda ,\mu$ of the standard maximal parabolic (conformal) subalgebras $\gop ,\gop^\prime=\gog^\prime\cap\gop$. Motivated by questions about the existence of invariant differential operators in conformal geometry, we explain, reformulate and prove an extended version of Juhl's conjecture on the structure of ${\un(\gog^\prime)}$-homomorphisms of generalized Verma modules from $\km_\lambda(\gog^\prime,\gop^\prime)$ to $\km_\mu(\gog,\gop)$.
The answer has a natural formulation as a branching problem in the BGG parabolic category ${\sh}^{\gop^\prime}$ rather than the set of generalized Verma modules alone.