One-parameter groups generated by operators in an enveloping algebra

AbstractLet π be a unitary representation of a connected Lie group G, and let ∂π be the associated representation of the complex enveloping algebra U(G)C of the Lie algebra G. Let h be a commutative subalgebra of G. The commutation relation eTXe−T = eadTX, (∗) with X ϵ ∂π(G) and T a skew-Hermitian element of ∂π(U(h)C), is established as an operator identity on the space of differentiable vectors for π, under the hypothesis that adG(h) is nilpotent. Relation (∗) is then used to prove that a spectral subspace corresponding to a compact, connected component of the spectrum of T is invariant under π(G)