Motivation: Consider $\mathfrak{sl}_2$. Given an integral weight $\lambda\in\mathbf Z$, then the only indecomposable modules contained in $\mathcal O_\lambda$ are the Verma modules and the simple modules with highest weight linked to $\lambda$ as well es their projective covers and injective hulls, cf. for example prop. 3.12 in Humphreys’ category 𝓞.

Question: Do similar statements hold more generally? At least for $\mathfrak{sl}_3$ or $\mathfrak{sl}_n$? I doubt, but cannot find any clear statements.

  • $\begingroup$ Does the paper below answer your question? $\endgroup$ – Hanno Apr 18 '17 at 5:54

The representation types of the blocks of ${\mathscr O}$ have been completely determined here.

  • $\begingroup$ I'm not sure how to correctly understand the paper. According to table 2, $\mathcal O_\lambda$ for 𝔰𝔩₃ has finite representation type (does this mean there are finitely many indecomposable modules?) iff 𝛷₀ = A₁, which is the case if 𝜆 lies on a reflection plane. For regular 𝜆, no such list exists. Is this the correct meaning of that paper? $\endgroup$ – Bubaya Apr 19 '17 at 17:55

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