Smooth vectors for highest weight representations. (English) Zbl 1029.17007

Neeb, Karl-Hermann

Glasg. Math. J. 42, No. 3, 469-477 (2000).

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

Keywords:

unitary highest weight representation; connected Lie group; invariant closed convex cone; selfadjoint operator