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

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.

References in corpus (1)

• Indecomposable manipulations with simple modules in category O

Cited by in corpus (1)

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