Towards a Global Springer Theory III: Endoscopy and Langlands duality

Abstract

We prove three new results about the global Springer action defined in \cite{GSI}. The first one determines the support of the perverse cohomology sheaves of the parabolic Hitchin complex, which serves as a technical tool for the next results. The second one (the Endoscopic Decomposition Theorem) links certain direct summands of the parabolic Hitchin complex of $G$ to the endoscopic groups of $G$. This result generalizes Ngô's geometric stabilization of the trace formula in \cite{NgoFL}. The third result links the stable parts of the parabolic Hitchin complexes for Langlands dual groups, and establishes a relation between the global Springer action on one hand and certain Chern class action on the other. This result is inspired by the mirror symmetry between dual Hitchin fibrations. Finally, we present the first nontrivial example in the global Springer theory.

48 pages

References in corpus (3)

• Le lemme fondamental pour les algebres de Lie
• Towards a Global Springer Theory II: the double affine action
• Towards a Global Springer Theory I: The affine Weyl group action

Cited by in corpus (6)

• Global topology of the Hitchin system
• Towards a Global Springer Theory II: the double affine action
• Towards a Global Springer Theory I: The affine Weyl group action
• Endoscopic decompositions and the Hausel-Thaddeus conjecture
• On the intersection cohomology of the moduli of ${\mathrm{SL}}_n$-Higgs bundles on a curve
• The fundamental lemma and the Hitchin fibration [after Ngo Bao Chau]