reference help indecomposable representations of SL(2,R) Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible $(\mathfrak{g},K)$-modules is well known.
Now I'm interested in indecomposable $(\mathfrak{g},K)$-modules. After searching the literature, I didn't find anythings.
So I'm here wondering if anyone would suggest any reference about indecomposable $(\mathfrak{g},K)$-modules for $SL_2(\mathbb{R})$? Or if this has been done before? Much appreciated for any help.
 A: There is a complete discussion of this in the book Non-Abelian Harmonic Analysis by Roger Howe and Eng Chye Tan, Chapter II
A: 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.
