Projective modules in the category 𝒪_{𝒮}: self-duality

Irving, Ronald S.

doi:10.1090/S0002-9947-1985-0800259-9

Projective modules in the category ${\scr O}\sb S$ : self-duality

Author: Ronald S. Irving
Journal: Trans. Amer. Math. Soc. 291 (1985), 701-732
MSC: Primary 17B10; Secondary 22E47
DOI: https://doi.org/10.1090/S0002-9947-1985-0800259-9
MathSciNet review: 800259
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Abstract: Given a parabolic subalgebra ${\mathfrak{p}_S}$ of a complex, semisimple Lie algebra $\mathfrak{g}$ , there is a naturally defined category ${\mathcal{O}_S}$ of $\mathfrak{g}$ -modules which includes all the $\mathfrak{g}$ -modules induced from finite-dimensional ${\mathfrak{p}_S}$ -modules. This paper treats the question of whether certain projective modules in ${\mathcal{O}_S}$ are isomorphic to their dual modules. The projectives in question are the projective covers of those simple modules occurring in the socles of generalized Verma modules. Their self-duality is proved in a number of cases, and additional information is obtained on the generalized Verma modules.

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