engleski

Some Homological Properties of Category O, V

We compute projective dimension of translated simple modules in the regular block of the Bernstein–Gelfand–Gelfand 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 Kåhrström. 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 prove that Kostant’s problem is equivalent to a homological problem of decomposing translated simple modules in 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.

Bernstein–Gelfand–Gelfand category O ; Kazhdan–Lusztig combinatorics ; Kostant’s problem

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