Abstract: 
Motivated by geometric Langlands, we initiate a program to study the mirror symmetry for the moduli space of parabolic GHiggs bundles. This talk will focus on G=Sp_{2n} and its Langlands dual SO_{2n+1}. Our goal is to prove the SYZ mirror symmetry and topological mirror symmetry (TMS). The parabolic structure of the parabolic Higgs bundle is related to the nilpotent orbit closure. So we need to first figure out the mirror pair for nilpotent orbits. 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, together with Prof. Ruan and Prof. Fu (arXiv:2207.10533), 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. After understanding the mirror parabolic structures, together with W. He, X. Su, B. Wang, X. Wen, we are working to prove the SYZ and TMS for the moduli space of parabolic Sp_{2n}/SO_{2n+1}Higgs bundles.
