Some Homological Properties of Category 𝒪, V

Abstract

Abstract We compute projective dimension of translated simple modules in the regular block of the Bernstein–Gelfand–Gelfand 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 $\textbf{a}$-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we prove that Kostant’s problem is equivalent to a homological problem of decomposing translated simple modules in $\mathcal O$. This gives a conjectural answer to Kostant’s problem in terms of the Kazhdan–Lusztig basis and addresses yet another conjecture by Johan Kåhrström.