I want to classify all unitary representations of $GL(2,\mathbb{C})$ from the representation theory of $SL(2, \mathbb{C})$. Is this possible? Knapp claims that one obtains all irreducible representation by pasting a character on $\mathbb{C}^\times$, which coincides on $-1$ onto $SL(2, \mathbb{C})$ and that one exhausts in this fashion all irreducible representation, but I worry about the well definedness of the square root $g \mapsto \sqrt{\det g}$. How does this work?
Alternatively, a reference which list all irreducible representations is also enough for my current purpose.