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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.

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    $\begingroup$ The representations $Ind= Ind_P^G (\chi)$, where $\Chi$ is a one dimensional representation of the group $P$ of the upper triangular matrices in $G=SL_2({\mathbb R})$ are indecomposable; unless $InD$ has finite dimensional sub or quotient, it is irreducible; if $Ind$ does have finite dimensional sub or quotient, then it is indecomposable (the Jordan holder series contains two discrete series and the f.d representation. I expect all indecomposables are of this form. $\endgroup$ – Venkataramana Aug 25 '14 at 11:15
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There is a complete discussion of this in the book Non-Abelian Harmonic Analysis by Roger Howe and Eng Chye Tan, Chapter II

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  • $\begingroup$ Maybe "complete" is an overstatement, but in any case what Howe and Tan do is quite explicit. $\endgroup$ – Jim Humphreys Dec 16 '14 at 20:13
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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.

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