Extensions of Verma modules

Carlin, Kevin J.

doi:10.1090/S0002-9947-1986-0819933-4

Extensions of Verma modules

Author: Kevin J. Carlin
Journal: Trans. Amer. Math. Soc. 294 (1986), 29-43
MSC: Primary 17B10; Secondary 17B20, 22E47
DOI: https://doi.org/10.1090/S0002-9947-1986-0819933-4
MathSciNet review: 819933
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Abstract: A spectral sequence is introduced which computes extensions in category $\mathcal{O}$ in terms of derived functors associated to coherent translation functors.

This is applied to the problem of computing extensions of one Verma module by another when the highest weights are integral and regular. Some results are obtained which are consistent with the Gabber-Joseph conjecture. The main result is that the highest-degree nonzero extension is one-dimensional.

The spectral sequence is also applied to the Kazhdan-Lusztig conjecture and related to the work of Vogan in this area.

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