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.