Mirror symmetry for special nilpotent orbit closures In Memory of Professor Gang Xiao

作     者：Baohua Fu Yongbin Ruan Yaoxiong Wen 

作者机构：AMSS HLM and MCM Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Institute for Advanced Study in Mathematics Zhejiang University Korea Institute for Advanced Study 

出 版 物：《Science China Mathematics》 (中国科学:数学(英文版))

年 卷 期：2024年

核心收录：

学科分类：06[历史学] 060207[历史学-专门史] 07[理学] 070104[理学-应用数学] 0602[历史学-中国史] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant No.12288201) supported by a KIAS Individual Grant (Grant No.MG083902) at Korea Institute for Advanced Study a POSCO Science fellowship 

摘      要：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 Bn via Cn. 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 ***, 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.