I hope my answer is not a reproduction of Jeffrey Adams answer. (I dont have the book at hand)

If I understand you right you want to know the indecomposable Harish-Chandra modules of $SL_2(\mathbb{R})$.

The category of Harish Chandra modules $HC$ can be decomposed by generalized infinitesimal characters and $K$-types (even or odd). Let $\chi$ be an infinitesimal and $k$ be a $K$-type. Let us write $HC_\chi^k$ for the category of Harish-Chandra modules with generalized infinitesimal character $\chi$ and $K$-type $k$.

Now the point is that every $HC_\chi^k$ is equivalent to some category of quiver representations where the quivers are quite simple. You can use this description to extract easily the indecomposeable modules.

**To give some example:**
If $\chi=-\rho$ ($\rho$ halfsum of positive roots) and $k$ is odd, then
$HC_\chi^k$ is equivalent to the category of finite dimensional representations of the quiver $\bullet \overset{\leftarrow}{\rightarrow}\bullet$ where the composition of arrows is nilpotent.

A full list is given in Chapter 3 of
Wolfgang Soergel, Langlands' Philosophy and Koszul Duality, even though you won't find any proofs there.

Alternativly you can check Andreas Glang's thesis:
http://www.freidok.uni-freiburg.de/volltexte/8876/pdf/arbeit.pdf The equivalences mentioned before are constructed explicitly.
However this thesis is written in german.