# zbMATH — the first resource for mathematics

Smooth vectors for highest weight representations. (English) Zbl 1029.17007
Summary: Let $$(\pi_\lambda,{\mathcal H}_\lambda)$$ be a unitary highest weight representation of the connected Lie group $$G$$ and $${\mathfrak g}$$ its Lie algebra. Then $${\mathfrak g}$$ contains an invariant closed convex cone $$W_{\max}$$ such that, for each $$X\in W^0_{\max}$$, the selfadjoint operator $$i\cdot d\pi_\lambda (X)$$ is bounded from above. We show that for each such $$X$$, the space $${\mathcal H}_\lambda^\infty$$ of smooth vectors for the action of $$G$$ on $${\mathcal H}_\lambda$$ coincides with the set $${\mathcal D}^\infty (d\pi_\lambda (X))$$ of smooth vectors for the generally unbounded operator $$d\pi_\lambda (X)$$.

##### MSC:
 17B15 Representations of Lie algebras and Lie superalgebras, analytic theory 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E60 Lie algebras of Lie groups
Full Text: