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 ${\scr O}_ S$: self-duality
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by Ronald S. Irving PDF

Trans. Amer. Math. Soc. 291 (1985), 701-732 Request permission

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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The

-homology of Harish-Chandra modules

generalizing a theorem of Kostant