Homomorphisms of Generalized Verma Modules, BGG Parabolic Category O^p and Juhl's Conjecture

Petr Somberg¹

¹Faculty of Mathematics and Physics, Sokolovska 83, Praha 8 - Karlin, Czech Republic

Journal of Lie Theory, Tome 22 (2012) no. 2, pp. 541-555

Résumé

\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.

Détail

Classification : 22E47, 17B10, 13C10
Mots-clés : Branching rules, generalized Verma modules, BGG parabolic category O^p, Juhl's conjectures