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Transactions of the American Mathematical Society

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

 

 

On the structure of certain subalgebras of a universal enveloping algebra


Authors: Bertram Kostant and Juan Tirao
Journal: Trans. Amer. Math. Soc. 218 (1976), 133-154
MSC: Primary 17B35
DOI: https://doi.org/10.1090/S0002-9947-1976-0404367-0
MathSciNet review: 0404367
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Abstract: The representation theory of a semisimple group G, from an algebraic point of view, reduces to determining the finite dimensional representation of the centralizer $ {U^\mathfrak{k}}$ of the maximal compact subgroup K of G in the universal enveloping algebra U of the Lie algebra $ \mathfrak{g}$ of G. The theory of spherical representations has been determined in this way since by a result of Harish-Chandra $ {U^\mathfrak{k}}$ modulo a suitable ideal I is isomorphic to the ring of Weyl group W invariants $ U{(\mathfrak{a})^W}$ in a suitable polynomial ring $ U(\mathfrak{a})$. To deal with the general case one must determine the image of $ {U^\mathfrak{k}}$ in $ U(\mathfrak{k}) \otimes U(\mathfrak{a})$, where $ \mathfrak{k}$ is the Lie algebra of K. We prove that if W is replaced by the Kunze-Stein intertwining operators $ \tilde W$ then $ {U^\mathfrak{k}}$ suitably localized and completed is indeed isomorphic to $ U(\mathfrak{k}) \otimes U{(\mathfrak{a})^{\tilde W}}$ suitably localized and completed.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0404367-0
Keywords: Universal enveloping algebra, intertwining operators, Weyl group, centralizer of the maximal compact subgroup, valuation on enveloping algebra
Article copyright: © Copyright 1976 American Mathematical Society