K-invariants in the universal enveloping algebra of the desitter group

Abstract

Let G be a non-compact connected semisimple Lie group with finite center and let G^K 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=S0_o(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).