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Journal of Lie Theory 22 (2012), No. 2, 541--555
Copyright Heldermann Verlag 2012

Homomorphisms of Generalized Verma Modules, BGG Parabolic Category Op and Juhl's Conjecture

Petr Somberg
Faculty of Mathematics and Physics, Sokolovska 83, Praha 8 - Karlin, Czech Republic


\def\g{{\frak g}} \def\p{{\frak p}} Let ${\cal M}_\lambda(\g,\p)$, ${\cal M}_\mu(\g^\prime, \p^\prime)$ be the generalized Verma modules for $\g={\rm so}(p+1,q+1), \g^\prime={\rm so}(p,q+1)$ induced from characters $\lambda$ ,$\mu$ of the standard maximal parabolic (conformal) subalgebras $\p$, $\p^\prime=\g^\prime\cap\p$. 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 ${\cal U}(\g^\prime)$-homomorphisms of generalized Verma modules from ${\cal M}_\lambda(\g^\prime,\p^\prime)$ to ${\cal M}_\mu(\g,\p)$. The answer has a natural formulation as a branching problem in the BGG parabolic category ${\cal O}^{\p^\prime}$ rather than the set of generalized Verma modules alone.

Keywords: Branching rules, generalized Verma modules, BGG parabolic category Op, Juhl's conjectures.

MSC: 22E47, 17B10, 13C10

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