K-structure of U(g) for su(n,1) and so(n,1)

Original · EN

Let G be the adjoint group of a real simple Lie algebra g₀ equal either su(n,1) or so(n,1), K its maximal compact subgroup, U(g) the universal enveloping algebra of the complexification g of g₀ and U(g)ᵏ its subalgebra of K-invariant elements. By a result of F. Knopp [3] U(g) is free as a U(g)ᵏ-module, so there exists a K-submodule E of U(g) such that the multiplication defines an isomorphism of K-modules U(g)ᵏ⊗ E U(g). We prove that E is equivalent to the regular representation of K, i.e. that the multiplicity of every δ∈K in E equals its dimension. As a consequence we get that for any finitedimensional complex K-module V the space (U(g)⊗ V)ᵏ of K-invariants is free U(g)ᵏ-module of rank V.