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Transactions of the American Mathematical Society

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Module structure of certain induced representations of compact Lie groups

Author: E. James Funderburk
Journal: Trans. Amer. Math. Soc. 228 (1977), 269-285
MSC: Primary 22E45
MathSciNet review: 0439992
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Abstract: Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight $ \lambda $, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight $ \lambda $. This paper studies the G-module structure of the space $ \Gamma $ of square integrable sections of the Borel-Weil line bundle. It is found that $ \Gamma = {\lim _{n \to \infty }}\Gamma (n)$, where $ \Gamma (n) \subset \Gamma (n + 1) \subset \Gamma $ and $ \Gamma (n)$ is isomorphic, as G-module, to

$\displaystyle V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast}),$

where $ V(\mu )$ denotes the irreducible representation of highest weight $ \mu $, '+' is the Cartan semigroup operation, and '$ ^\ast$' is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.

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Keywords: Compact connected Lie group, highest weight, lowest weight, opposition involution, Cartan semigroup, tensor product, cyclic representation, Borel-Weil realization
Article copyright: © Copyright 1977 American Mathematical Society