Smooth vectors forhighest weight representations

Abstract

Let(π_{λ}, ℋ_{λ}) be a unitary highest weight representation of the connected Lie group G and ℊ its Lie algebra. Thenℊ contains an invariant closed convex cone W_{\rm{max}} such that, for each X∈W_{\rm{max}}^0, the selfadjoint operatori·dπ_{λ}(X) is bounded from above. We show that for each suchX , the space ℋ_{λ}^{∞} of smooth vectors for the action of G on ℋ_{λ} coincides with the set𝒟^{∞}(dπ_{λ}(X)) of smooth vectors for the generally unbounded operator dπ_{λ}(X).