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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 \(d\pi_\lambda (X)\).

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
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