# Irreducible finite dimensional complex representation of $GL_2(\Bbb C)$

I know the basic theory of representation theory of compact Lie groups and I want to understand a non-compact example:

How to find all irreducible finite dimensional complex representations of $GL_2(\Bbb C)$? Are its finite dimensional complex representations always completely irreducible?

I think there cannot exist an invariant inner product on it's representations, is there some way to find it's representations using something like it's lie algebra?

Or, for example, instead of $G=GL_2$, trying $G=SL_2$ avoids the counter-example.
No. Consider the action $\rho$ of $GL_2(\mathbb{C})$ on $\mathbb{C}^2$ such that, for each $g\in GL_2(\mathbb{C})$, the matrix of $\rho(g)$ with respect to the standard basis is$$\begin{pmatrix}1&\log|\det g|\\0&1\end{pmatrix}.$$ It is not completely reducible, because $\{(z,0)\,|\,z\in\mathbb{C}\}$ is invariant, but it is the only one-dimensional invariant subspace.