Langlands duality and global Springer theory (Q2894206)

This paper is a continuation of the author's paper [\textit{Z. Yun}, Adv. Math. 228, No. 1, 266--328 (2011; Zbl 1230.14048)], in which he constructs an action of the graded double affine Hecke algebra (DAHA) on the cohomologies of parabolic Hitchin fibers. In the paper under review, the author compares this action for Langlands dual groups \(G\) and \(G^\vee\). The main result states that the stable parts of the parabolic Hitchin complexes for Langlands dual groups are naturally isomorphic up to a shift under the Verdier duality after passing to the associated graded of the perverse filtration, and that the global Springer action and Chern class cup-product action of the graded DAHA get interchanged under the Verdier duality. These results should be considered as shadows of the conjectured mirror symmetry between Hitchin moduli spaces for Langlands dual groups. To prove the results, the author extends the support theorem of Ngô to the parabolic setting, and calculates explicitly the Springer action and the Chern class cup-product action on the good locus.