Representation theory of $GL(2,\mathbb{C})$ 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. 
 A: The group $GL_2(\mathbb C)$ is the product of its centre $\mathbb C^{\times}$ and its subgroup $SL_2(\mathbb C)$.  This product is not direct; the two subgroups intersect in $\{\pm 1\}$ (the centre of $SL_2(\mathbb C)$).  
Suppose $\pi$ is an irred. unitary rep. of $SL_2(\mathbb C)$, and let $\varepsilon$ be the character through which $\{\pm 1\}$ acts on $\pi$.  Choose an extension of $\epsilon$ to a unitary character of $\mathbb C^{\times}$.  Then there is a unique extension of $\pi$ to a unitary rep. of $GL_2(\mathbb C)$,
on which $\mathbb C^{\times}$ acts via the character $\chi$.
Conversely, any irred. unitary rep. of $GL_2(\mathbb C)$ arises from one of $SL_2(\mathbb C)$ by such a construction.
[The case of $GL_2(\mathbb R)$ is a little more subtle, and this is what I was thinking of when I wrote my comment above.  The reason is that in this case $GL_2(\mathbb R)$ is not the product of its centre and $SL_2(\mathbb R)$,
because the squaring map $\mathbb R^{\times} \to \mathbb R^{\times}$ --- which is what arises from restricting the determinant map to the centre of $GL_2$ --- is not surjective, unlike in the case of $GL_2(\mathbb C)$.]
