Journal of Lie Theory 21 (2011), No. 4, 771--785
Copyright Heldermann Verlag 2011
On Differentiability of Vectors in Lie Group Representations
Ingrid Beltita
Institute of Mathematics "Simion Stoilow", Romanian Academy of Sciences, P. O. Box 1-764, Bucharest, Romania
Ingrid.Beltita@imar.ro
Daniel Beltita
Institute of Mathematics "Simion Stoilow", Romanian Academy of Sciences, P. O. Box 1-764, Bucharest, Romania
Daniel.Beltita@imar.ro
[Abstract-pdf]
\def\g{{\frak g}} We address a linearity problem for differentiable vectors in representations of infinite-dimensional Lie groups on locally convex spaces, which is similar to the linearity problem for the directional derivatives of functions. In particular, we find conditions ensuring that if $\pi\colon G\to{\rm End}({\cal Y})$ is such a representation, and $y\in{\cal Y}$ is a vector such that ${\rm d}\pi(x)y$ makes sense for every $x$ in the Lie algebra $\g$ of $G$, then the mapping ${\rm d}\pi(\cdot)y\colon\g\to{\cal Y}$ is linear and continuous.
Keywords: Lie group, topological group, unitary representation, smooth vector.
MSC: 22E65; 22E66, 22A10, 22A25
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