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arXiv · 2007.00342

Some homological properties of category $\mathcal{O}$, V

Hankyung Ko·Volodymyr Mazorchuk·Rafael Mrđen
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Abstract

We compute projective dimension of translated simple modules in the regular block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan Kåhrström. We also derive some inequalities, in terms of Lusztig's $\mathbf{a}$-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we relate Kostant's problem with decomposability and isomorphism of translated simple modules, addressing yet another conjecture by Johan Kåhrström.

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Hankyung Ko, Volodymyr Mazorchuk, Rafael Mrđen. 2020-10-23. Some homological properties of category $\mathcal{O}$, V. https://doi.org/10.1093/imrn%2Frnab330

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