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Since $f(x)$ must be either constant or balanced, there are only 4 possibilities for what $f(x)$ could be. For each of these four cases, you can explicitly write $U_f$ as a matrix, which you can easily check for unitarity.


No there are none (apart from the trivial rep.). This follows not because of the metric but because the transformations on Euclidean space still form a non-compact group, like Poincaré. The best you can do apparently is to have indecomposable representations, and that’s a mess because that representation theory is “wild”. The case of $E(2)$ is somewhat ...

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