I understand that unitary representations arise naturally in quantum mechanics when groups act on the Hilbert space in a way that preserves probability.
I don't understand what details make unitary representations different from other representations. It seems as though physicists talk explicitly about "unitary representations" all the time.
Are there some theorems or examples that show why working with a "unitary" representation ought to be notable?
I've heard that compact Lie groups have only finite dimensional unitary representations, but seeing as so many important Lie groups in physics are non-compact this doesn't seem like the most vital reason.