Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)$. Also, what is the relation of the diffeomorphism group with the automorphism group of $U(n)$ that we know is a semidirect product of $U(n)$ with a finite group. Any literature about the topic, specifically of $U(n)$ as a manifold, would be highly appreciated.
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asked Dec 20, 2023 at 15:14
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4$\begingroup$ Do you have any reason to think that there should be an interesting relationship between the automorphism group of $\operatorname U(n)$ (relatively small, as you point out) and its diffeomorphism group (very large), on the level of the groups themselves or of their representations? In particular, do you have any criterion for what would be a satisfactory answer to your first question? $\endgroup$– LSpiceCommented Dec 20, 2023 at 15:18
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$\begingroup$ It is more an intuition from the simple idea that from a specific representation of $U(n)$ I should be able to describe any diffeomorphism explicitly as an operation on matrices. I was wondering if the linear ones are precisely the ones described by the automorphism group or there is something else. $\endgroup$– Nicolas Medina SanchezCommented Dec 20, 2023 at 18:39
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