Generalization of Harish-chandra’s Basic Theorem for Riemannian Symmetric Spaces of Non-compact Type

Let g be a real semisimple Lie algebra and θ a fixed Cartan involution of g. In this paper the subscript C is used for indicating the complexification of a real object. Denote the universal enveloping algebra of the complex Lie algebra gC by U(gC), the center of U(gC) by Z(gC), and the symmetric algebra of gC by S (gC). Similar notation is used for other complex Lie algebras or vector spaces. Let Gad be the adjoint group of g, Gθ the subgroup of the adjoint group of gC consisting of all the elements that leave g stable, and G an arbitrary group such that Gad ⊂ G ⊂ Gθ. The adjoint action of Gad or G or Gθ (resp. gC) is denoted by Ad (resp. ad). Let g = k ⊕ p be the Cartan decomposition. Take a maximal Abelian subspace a of p and fix a basis Π of the restricted root system Σ for (g, a). Π defines the system Σ of positive roots. Put K = G (the subgroup of G which commutes with θ), M = ZK(a) (the centralizer of a in K), and m = Lie(M). Using NK(a) (the normalizer of a in K) define the Weyl group W = NK(a)/M. For each α ∈ Σ, let Hα be the corresponding element of a via the Killing form B(·, ·) of g and put |α| = B(Hα, Hα) 1 2 , α∨ = 2Hα |α|2 (the coroot of α). Denote the restricted root space for α ∈ Σ by gα and put n = ∑ α∈Σ+ gα, ρ = 1 2 ∑ α∈Σ+ (dim gα)α. Let us now define the map γ of U(gC) into S (aC) by the projection U(gC) = U(aC)⊕ ( nCU(gC) + U(gC)kC ) → U(aC) ≃ S (aC)