松弛最高权模：来自 Kashiwara 旗簇上的 $\mathcal{D}$-模
Relaxed Highest Weight Modules from $\mathcal{D}$-Modules on the Kashiwara Flag Scheme
AI总结：
本文将松弛最高权表示推广到任意仿射 Kac-Moody 代数，并在 Kashiwara 旗簇上通过扭曲 D-模的整体截面实现此类模，给出上同调完整描述，推广了 Kashiwara-Tanisaki 的结果。
摘要：
Feigin 等人引入的松弛最高权表示是仿射 Kac-Moody 代数 $\hat{\mathfrak{sl}_2}$ 的一类表示，它们没有最高（或最低）权。我们将这一概念推广到任意仿射 Kac-Moody 代数 $\mathfrak{g}$。随后，我们将此类诱导 $\mathfrak{g}$-模及其对偶实现为与 $\mathfrak{g}$ 相关联的 Kashiwara 旗簇 $X$ 上扭曲 $\mathcal{D}$-模的整体截面。我们构造中出现的 $\mathcal{D}$-模是从 $X$ 的子概形的直接像，这些子概形是有限维 Schubert 胞腔与其经简单反射平移的交。除了扭曲 $λ$ 之外，它们还依赖于一个复数，该复数描述我们在这些交上构造的局部系统的单值性。我们将 $*$-直接像的整体截面描述为 $\mathfrak{g}$ 的 Cartan 子代数上的模，并证明高上同调消失。在以下两种情形中，我们获得了直接像的上同调群作为 $\mathfrak{g}$-模的完整描述。首先，我们处理交同构于 $\mathbb{G}_{\text{m}}$ 的情形。其次，我们处理扭曲正则反支配且单值性平凡时，从任意交的 $*$-直接像的情形。为了证明这一情形，我们引入了由简单反射的提升定义的 $X$ 的自同构所诱导的 $\mathcal{D}$-模范畴 $\text{Hol}(λ)$ 的自等价。这些结果首次描述了显式的非最高权 $\mathfrak{g}$-模作为 Kashiwara 旗簇上的整体截面，并将 Kashiwara-Tanisaki 的若干结果推广到松弛最高权表示的情形。
英文摘要：
The relaxed highest weight representations introduced by Feigin et al. are a class of representations of the affine Kac-Moody algebra $\hat{\mathfrak{sl}_2}$, which do not have a highest (or lowest) weight. We formulate a generalization of this notion for an arbitrary affine Kac-Moody algebra $\mathfrak{g}$. We then realize induced $\mathfrak{g}$-modules of this type and their duals as global sections of twisted $\mathcal{D}$-modules on the Kashiwara flag scheme $X$ associated to $\mathfrak{g}$. The $\mathcal{D}$-modules that appear in our construction are direct images from subschemes of $X$ that are intersections of finite dimensional Schubert cells with their translate by a simple reflection. Besides the twist $λ$, they depend on a complex number describing the monodromy of the local systems we construct on these intersections. We describe the global sections of the $*$-direct images as a module over the Cartan subalgebra of $\mathfrak{g}$ and show that the higher cohomology vanishes. We obtain a complete description of the cohomology groups of the direct images as $\mathfrak{g}$-modules in the following two cases. First, we address the case when the intersection is isomorphic to $\mathbb{G}_{\text{m}}$. Second, we address the case of the $*$-direct image from an arbitrary intersection when the twist is regular antidominant and the monodromy is trivial. For the proof of this case we introduce an auto-equivalence of the category of $\mathcal{D}$-modules $\text{Hol}(λ)$ induced by the automorphism of $X$ defined by a lift of a simple reflection. These results describe for the first time explicit non-highest weight $\mathfrak{g}$-modules as global sections on the Kashiwara flag scheme and extend several results of Kashiwara-Tanisaki to the case of relaxed highest weight representations.
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