论文标题
BGG类别的Dirac Somology $ \ Mathcal {O} $
Dirac cohomology for the BGG category $\mathcal{O}$
论文作者
论文摘要
我们研究狄拉克的同胞$ h_d^{\ mathfrak {g}，\ mathfrak {h}}}（m）$属于类别的模块$ \ mathcal {o} $的模块。我们证明了Vogan的猜想，这是$ h_d^{\ mathfrak {g}，\ Mathfrak {h}}}（m）$的一个不变结果，而我们证明在Hermitian Symmetric Pair $（\ Mathfrak {g}的情况下， $ m \ in \ Mathcal {o} $，Dirac共同体与nilpotent Lie代数同胞相吻合，与$ m $中的系数相吻合。在最后一部分中，我们表明，Pandžić和Somberg引入的较高的Dirac共同体和索引满足了\ Mathcal {O} $中的$ M \的良好同源性能。
We study Dirac cohomology $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ for modules belonging to category $\mathcal{O}$ of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ while we show that in the case of a Hermitian symmetric pair $(\mathfrak{g},\mathfrak{k})$ and an irreducible unitary module $M\in\mathcal{O}$, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in $M$. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for $M\in\mathcal{O}$.
