arXiv:2611.12892v1 Announce Type: new Abstract: In various special cases, we provide an alternative argument for equivalence between Kostant positivity of some involutions in type $A$ and the conjunction of the categorical K{\aa}hrstr{\"o}m's condition for these involutions with indecomposability of the images of the corresponding simple modules in category $\mathcal{O}$ under projective functors.
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