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Transactions of the American Mathematical Society

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



On the Harish-Chandra homomorphism

Author: J. Lepowsky
Journal: Trans. Amer. Math. Soc. 208 (1975), 193-218
MSC: Primary 17B35; Secondary 22E45
MathSciNet review: 0376792
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Abstract: Using the Iwasawa decomposition $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ of a real semisimple Lie algebra $ \mathfrak{g}$, Harish-Chandra has defined a now-classical homomorphism from the centralizer of $ \mathfrak{k}$ in the universal enveloping algebra of $ \mathfrak{g}$ into the enveloping algebra $ \mathcal{A}$ of $ \mathfrak{a}$. He proved, using analysis, that its image is the space of Weyl group invariants in $ \mathcal{A}$. Here the weaker fact that the image is contained in this space of invariants is proved ``purely algebraically". In fact, this proof is carried out in the general setting of semisimple symmetric Lie algebras over arbitrary fields of characteristic zero, so that Harish-Chandra's result is generalized. Related results are also obtained.

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Keywords: Harish-Chandra homomorphism, real semisimple Lie algebra, Iwasawa decomposition, universal enveloping algebra, restricted Weyl group, semisimple symmetric Lie algebra, symmetric decomposition, splitting Cartan subspace, restricted root system
Article copyright: © Copyright 1975 American Mathematical Society