论文标题
类别$ \ mathcal {o} $的某些同源属性，v
Some homological properties of category $\mathcal{O}$, V
论文作者
论文摘要
我们根据Kazhdan-Lusztig Combinatorics计算BGG类别$ \ MATHCAL {O} $的常规块中翻译的简单模块的投影维度。这使我们能够确定在最小的投影分辨率的最后一步，用于翻译的简单模块，确认了JohanKåhrström的猜想。我们还从Lusztig的$ \ Mathbf {a} $函数方面得出了一些不等式，对于可能的程度，在其中翻译的简单模块可以使用的顶部（或SOCLE）。最后，我们将Kostant的问题与翻译简单模块的可分解性和同构性联系在一起，并解决了JohanKåhrström的另一个猜想。
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.
