K-invariants in the universal enveloping algebra of the de Sitter group
Let G be a connected semisimple noncompact real Lie group, K its maximal compact subgroup, and \(G=KAN\) the corresponding Iwasawa decomposition of G. The universal enveloping algebras (over \({\mathbb{C}})\) of Lie algebras of the groups G, K, A are denoted by \({\mathfrak g}\), \({\mathfrak k}\), \({\mathfrak a}\), correspondingly. Let M be the centralizer of A in K, \({\mathfrak g}^ K\) the centralizer of K in \({\mathfrak g}\), and \({\mathfrak k}^ M\) the centralizer of M in \({\mathfrak k}\). There is an injective anti-homomorphism \(f: {\mathfrak g}^ K\to {\mathfrak k}^ M\otimes {\mathfrak a}\). The subalgebra f(\({\mathfrak g}^ k)\) is of great importance for the representation theory of G. The authors explicitly compute and study f(\({\mathfrak g}^ K)\) for the group \(G=SO_ 0(4,1)\), which is the simplest example of G when M is not abelian. It is shown that the algebra f(\({\mathfrak g}^ K)\) leads to the algebraic description of all irreducible representations of \(SO_ 0(4,1)\). It is said that the purpose of this paper is to test some ideas concerning the general case of G. For example, a detailed study of f(\({\mathfrak g}^ K)\) for \(SO_ 0(4,1)\) shows that Conjecture 10.4 of the paper by \textit{J. Lepowsky} [Trans. Am. Math. Soc. 176, 1-44 (1973; Zbl 0264.22012)], concerning the case of a general group G, is not correct.
scientific article; zbMATH DE number 4036029