Abstract. The moduli of Higgs bundles on curve was firstly studied by Hitchin in 1980’s and has been vastly studied during the last over 30 years. One of the most remarkable observation about Higgs moduli was proposed by Hausel and Thaddeus in 2003 that there is a mirror relation between the G-Higgs moduli and G^L-Higgs moduli, here G^L is the Langlands dual group of G. They conjectured that there is a topological mirror symmetry between SL_n/PGL_n Higgs moduli and this conjecture was proved by Groechenig, Wyss and Ziegler also by Maulik and Shen using different methods. 

The mirror phenomenon of Higgs moduli was also noticed by Gokov, Kapustin and Witten by the viewpoint of physics. Especially, Gokov and Witten proposed in physics that there should be a mirror relation of parabolic Higgs bundles for Langlands dual groups and nilpotent orbits inserted at the marked points. 

If one considers Higgs bundles with nilpotent orbits inserted, the most interesting case is type B/C. In this talk, we will show that how to relate nilpotent orbits in type B/C using Kazhdan-Lusztig map and loop Lie algebra, and then use these local computations to prove a topological mirror symmetry statement for parabolic Higgs moduli of type B/C using p-adic integration. This program was suggested by Prof. Ruan, and it is a joint work with Weiqiang He, Xiaoyu Su, Bin Wang and Yaoxiong Wen.