Complete representation theory of $\mathrm{SL}(2,\mathbb R)$ $\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense that one can explicitly describe all (nonunitary, projective) irreducible representations, the decomposition of their tensor products, and the structure of all indecomposable reducible representations?
Where would I find the current state of the art?
 A: Yes, this is all completely settled and for the finite dimensional case it is the basic underpinning of much of Lie theory. The irreducible finite-dimensional representations of $\operatorname{SL}(2,\mathbb{R})$ are indexed by the positive integers and can be simply described as $V_m = \operatorname{Sym}^m\mathbb{R}^2$ (no need for projective representations all of these are linear). $\operatorname{SL}(2,\mathbb{C})$ is pretty much identical. None of the finite dimensional representations are unitary. You should find this all in any basic textbook on Lie theory.
The general linear versions are a little more subtle (see here for example) but mostly amounts to taking a $\operatorname{SL}(2,\mathbb{C})$-rep and deciding how the centre acts.
Decomposition of tensor products is again well understood. You can find information under the name Schur-Weyl duality (the 2 dimensional case is sometimes called Clebsch-Gordan theory I believe).
I'm less familiar with what happens with infinite dimensional representations but it is a little more complicated see here
A: For unitary infinite dimensional representations of $SL(2, \mathbb{R})$ and $SL(2, \mathbb{C})$, I would look at

*

*A. Knapp, Representation theory of semisimple groups, 1986


*R. Takahashi, $SL(2, \mathbb{R})$, École d'Été "Analyse harmonique", Université de Nancy I, 1980


*S. Lang, $SL(2, \mathbb{R})$, 1985.
