Is every (finite-dimensional, complex) representation of $SO(3)$ the restriction of a representation of $U(3)$?
I actually 'know' all the representations of $SO(3)$ and $U(3)$ using the theory of weights on $U(n)$ (for $SO(3)$, note that it is covered by $SU(2)$), but the truth is I only know their characters, and the formulas for $SO(3)$ look ugly. So I am failing to see if the answer to my question is true or not. Any help is appreciated, even a yes/no answer would be nice.
The answer to the title question (v1) is No. The real subgroup $SO(3)\subsetneq SU(3)\subsetneq U(3)$ has e.g. a 5-dimensional irrep, while the Lie groups $SU(3)$ and $U(3)$ don't have a 5-dimensional (complex) irrep.