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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Projective modules in the category ${\scr O}_ S$: 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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Keywords: Generalized Verma modules, category <!– MATH $\mathcal {O}$ –> <IMG WIDTH="22" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathcal {O}$">
Article copyright: © Copyright 1985 American Mathematical Society