I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a Lie group $G$ on some Hilbert space $H$ (we can choose other types of topological vector spaces, but I'm interested in particular in Hilbert spaces). We say that a vector $\xi \in H$ is smooth, if the function $x \mapsto \pi(x) \xi$, is a smooth $H$-valued function on $G$. The space of smooth vectors, $H^{\infty}$, is dense in $H$ (a result of Garding). Furthermore, one can endow this space with a Frechet space topology, that in general is strictly finer then the topology induced from $H$.

Harish-Chandra proved that for every semi-simple $G$, the space $H^{\infty}$ is nuclear (I'm not familiar with the details of the proof). Another result is that for a nilpotent $G$, this space is nuclear (it follows from the fact that there is a topological isomorphism between this space and a space of Schwartz functions, know to be nuclear). I couldn't find an example of a group and a representation, such that this space is not nuclear. Maybe someone is familiar with such an example or can point me to some source that shows some general conditions, under which this space is nuclear.

I've posted this question on StackExchange, but unfortunately got no answers so far.

Thank you in advance.


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