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Transactions of the American Mathematical Society

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



The restriction of admissible modules to parabolic subalgebras

Authors: J. T. Stafford and N. R. Wallach
Journal: Trans. Amer. Math. Soc. 272 (1982), 333-350
MSC: Primary 17B10; Secondary 17B20
MathSciNet review: 656493
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Abstract: This paper studies algebraic versions of Casselman's subrepresentation theorem. Let $ \mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field $ F$ of characteristic zero and $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ be an Iwasawa decomposition for $ \mathfrak{g}$. Then $ (\mathfrak{g},\mathfrak{k})$ is said to satisfy property $ (\mathfrak{n})$ if $ M \ne M$ for every admissible $ (\mathfrak{g},\mathfrak{k})$-module $ M$. We prove that, if $ (\mathfrak{g},\mathfrak{k})$ satisfies property $ (\mathfrak{n})$, then $ \mathfrak{n}N \ne N$ whenever $ N$ is a $ (\mathfrak{g},\mathfrak{k})$-module with $ \dim N < \operatorname{card} F$. This is then used to show (purely algebraically) that $ (\mathfrak{s}l(n,F),\mathfrak{s}o(n,F))$ satisfies property $ (\mathfrak{n})$. The subrepresentation theorem for $ \mathfrak{s}l(n)$ is an easy consequence of this.

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Keywords: Semisimple Lie algebra, parabolic subalgebra, admissible (Harish-Chandra)module, subrepresentation theorem, Artin-Rees property
Article copyright: © Copyright 1982 American Mathematical Society