Papers › Some homological properties of category 𝒪, V
Some homological properties of category 𝒪, V
Hankyung Ko, Volodymyr Mazorchuk, Rafael Mrđen
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We compute projective dimension of translated simple modules in the regular block of the BGG category 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 K{\aa}hrstr\"om. We also derive some inequalities, in terms of Lusztig's 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 K{\aa}hrstr\"om.
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