论文标题
特殊nilpotent轨道封闭的镜像对称性
Mirror symmetry for special nilpotent orbit closures
论文作者
论文摘要
由几何兰兰兹（Legotric Langlands）启动，我们启动了一个程序，以研究半神经谎言代数的nilpotent轨道闭合与其兰格兰双重双重的镜像对称性。最有趣的情况是$ b_n $通过$ c_n $。从经典上讲，特殊轨道之间有著名的弹簧二元性。因此，自然而然地推测我们寻求的镜像对称性可能与特殊轨道背景下的Springer二元性一致。不幸的是，这样的天真陈述失败了。为了解决情况，我们提出了一个猜想，该猜想主张某些抛物线/诱发轨道覆盖物的镜像对称性。然后，我们证明了Richardson Orbits的猜想，并总体上获得了某些部分结果。在此过程中，我们揭示了这些有限覆盖物的一些非常有趣但又微妙的结构，这些结构与Lusztig的特殊Nilpotent轨道的规范商有关。例如，这些有限封面的足迹或程度范围内存在神秘的不对称性。最后，我们提供了两个示例，以表明镜像对称性在足迹之外失败。
Motivated by geometric Langlands, we initiate a program to study the mirror symmetry between nilpotent orbit closures of a semisimple Lie algebra and those of its Langlands dual. The most interesting case is $B_n$ via $C_n$. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. In the process, we reveal some very interesting and yet subtle structures of these finite covers, which are related to Lusztig's canonical quotients of special nilpotent orbits. For example, there is a mysterious asymmetry in the footprint or range of degrees of these finite covers. Finally, we provide two examples to show that the mirror symmetry fails outside the footprint.
