论文标题
所有最高权重模块的权重成型和高阶抛物线类别$ \ mathcal {o} $
A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$
论文作者
论文摘要
令$ \ mathfrak {g} $成为一个复杂的kac-moody代数，带有cartan subalgebra $ \ mathfrak {h} $。还要修复\ mathfrak {h}^*$中的重量$λ\。对于$ m（λ）\ tweatheadrightArrow v $一个任意的最高权重$ \ mathfrak {g} $ - 模块，我们为$ v $的权重提供了无取消的非恢复公式。这甚至在有限类型中也是新颖的，并且是从$λ$和一个与$ v $相关的$ \ mathfrak {g} $的dynkin图中的独立集的$ \ mathcal {h} = \ mathcal {h} = \ mathcal {h} _v $获得的。 我们的证明使用并揭示了“高阶Verma模块”的有限家族（对于每个$λ$）$ \ MATHBB {M}（λ，\ Mathcal {h}）$ - 这些都是重量调节的通用模块。他们（i）概括并集中抛物线抛物线模块$ m（λ，j）$和（ii）具有成对不同的重量集，这耗尽了所有模块的重量集$ m（λ）\ tixheadHeadReftRightArrow v $。作为一个应用程序，我们解释了Lepowsky的模块$ m（λ）$和$ m（λ，j_v）$的意义，分别是每个$ v $的零件和一阶上阶较高的上限，并将继续更高阶，并继续更高阶，并将上阶的上订单上级上限$ \ mathbb {m} _k（mathc $λ，\ mathcal（and}）较低的附属物）。我们确定$ v $的每$ k $ th订单集成性。 然后，我们介绍类别$ \ MATHCAL {O}^\ MATHCAL {H} \ subset \ Mathcal {O} $，这是一个更高阶抛物线词，包含更高阶的Verma模块$ \ MATHBB {M MATHBB {M}（λ，λ，\ Mathcal {H}）$。我们表明，$ \ MATHCAL {O}^\ MATHCAL {H} $具有足够的预测，并且还通过证明所有$ \ Mathcal {O}^\ Mathcal {h} $在$ \ Mathfrak {g} = \ Mathfrak} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ {最后，在某些情况下，包括所有级别3 $ \ mathfrak {g} $，我们为我们的通用模块$ \ mathbb {m}（λ，\ mathcal {h}）$提供了BGG分辨率;这产生了其Weyl型特征公式，并具有抛物线Weyl Semigroups的作用。
Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $λ\in\mathfrak{h}^*$. For $M(λ)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-module, we provide a cancellation-free, non-recursive formula for the weights of $V$. This is novel even in finite type, and is obtained from $λ$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$. Our proofs use and reveal a finite family (for each $λ$) of "higher order Verma modules" $\mathbb{M}(λ,\mathcal{H})$ - these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules $M(λ,J)$, and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules $M(λ)\twoheadrightarrow V$. As an application, we explain the sense in which the modules $M(λ)$ of Verma and $M(λ,J_V)$ of Lepowsky are respectively the zeroth and first order upper-approximations of every $V$, and continue to higher order upper-approximations $\mathbb{M}_k(λ,\mathcal{H}_V)$ (and to lower-approximations). We determine every $k$th order integrability of $V$. We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$, which is a higher order parabolic analogue that contains the higher order Verma modules $\mathbb{M}(λ,\mathcal{H})$. We show that $\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over $\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution for our universal modules $\mathbb{M}(λ,\mathcal{H})$ in certain cases including all rank-3 $\mathfrak{g}$; this yields their Weyl-type character formulas, with the actions of parabolic Weyl semigroups.
