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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Jantzen filtration of a certain class of Verma modules


Author: Jong Min Ku
Journal: Proc. Amer. Math. Soc. 99 (1987), 35-40
MSC: Primary 17B67; Secondary 17B10, 22E47
MathSciNet review: 866425
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Abstract: Let $ G = {N_ + } \oplus H \oplus {N_ - }$ be a Kac-Moody Lie algebra. For each $ M$ in the category $ \mathcal{O}$ of $ G$-modules, there is a filtration $ {({M_i})_{i \geq 0}}$ by $ G$-submodules of $ M$ naturally associated with the set $ \left\{ {\upsilon \in M\left\vert {{N_ + }\upsilon } \right. = 0} \right\}$. If $ G$ is symmetrizable and $ M$ is a Verma module, $ {M_i} = {M^i}$ for all $ i$ if and only if $ \left[ {M:L(\mu )} \right] = \dim \operatorname{Hom}_G(M(\mu ),M)$ for all $ \mu \in {H^ * }$ where $ {({M^i})_{i \geq 0}}$ is the Jantzen filtration of $ M$. The main tools used are the nondegenerate form on each $ {M^i}/{M^{i + 1}}$ together with the $ \Gamma $-operator of $ G$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0866425-9
PII: S 0002-9939(1987)0866425-9
Article copyright: © Copyright 1987 American Mathematical Society