Mirror symmetry for special nilpotent orbit closures

TL;DR

This paper explores a new form of mirror symmetry between special nilpotent orbit closures in Lie algebras and their Langlands duals, proposing conjectures and proving results for specific cases, revealing intricate structures and limitations.

Contribution

It introduces a conjecture on mirror symmetry for covers of special orbits, proves it for Richardson orbits, and uncovers subtle structures related to Lusztig's quotients.

Findings

Proved mirror symmetry conjecture for Richardson orbits

Discovered asymmetry in finite covers related to Lusztig's quotients

Identified limitations of mirror symmetry outside certain ranges

Abstract

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.