Abstract
Let G be a non-compact connected semisimple Lie group with finite center and let GK denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over ℂ of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G K→K M⊗A, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over ℂ corresponding to K anda, respectively, andK M is the centralizer of M inK. The subalgebra P(G K) ofK M⊗A has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G K) when G=S0o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).
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References
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Partially supported by CONICET (Argentina) grants.
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Brega, A., Tirao, J. K-invariants in the universal enveloping algebra of the desitter group. Manuscripta Math 58, 1–36 (1987). https://doi.org/10.1007/BF01169080
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DOI: https://doi.org/10.1007/BF01169080