GEOMETRIC ENDOSCOPY AND MIRROR SYMMETRY EDWARD FRENKEL1 AND EDWARD WITTEN2 Abstract. The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T –duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of A-branes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.-C. Ngˆ o, a fact which could be significant for understanding the trace formula.

Contents 1. Introduction 1.1. T –duality Of Singular Fibers 1.2. A-branes And D-modules 1.3. From Curves Over C To Curves Over Fq 1.4. Classical Endoscopy 1.5. Geometric Endoscopy 1.6. Connection With The Work Of B.-C. Ngˆo 1.7. Quantum Field Theory 1.8. Plan Of The Paper 1.9. Acknowledgments 2. Duality, Branes, and Endoscopy 2.1. Geometric Langlands Duality And Mirror Symmetry 2.2. Branes And Their Duals 2.3. From A-Branes To D-Modules Date: October 2007. 1 Supported in part by DARPA and AFOSR through the grant FA9550-07-1-0543. 2 Supported in part by NSF Grant PHY-0503584. 1

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EDWARD FRENKEL AND EDWARD WITTEN

3. Explicit Example In Genus One 3.1. Higgs Bundles In Genus One 3.2. Ramification 3.3. The Moduli Space 3.4. The Hitchin Fibration 3.5. Symmetry Group 3.6. Langlands Dual Group 3.7. O2 -Bundles 3.8. Second Component 3.9. Relation To The Cotangent Bundle 3.10. Mirror Symmetry Of Orbifolds 3.11. Relation To Seiberg-Witten Theory 4. A-Branes And D-Modules 4.1. Spectral Curves 4.2. Relation To A-Branes 4.3. Map From A-Branes To Twisted D-Modules 4.4. Poles 4.5. Application To Our Example 4.6. The Central Twist 4.7. The B-Field 4.8. Tame And Irregular Singularities 4.9. The Multi-Dimensional Case 5. Spectral Covers, Hecke Operators, and Higher Genus 5.1. Genus One Revisited 5.2. Extension To Higher Genus 5.3. ’t Hooft/Hecke Operators 6. Categories Of Eigensheaves 6.1. Generalities On Categories 6.2. Examples 6.3. Hecke Eigensheaves 6.4. Category Of Hecke Eigensheaves In The Endoscopic Example 6.5. Fractional Hecke Eigensheaves 6.6. Other Examples 7. The Classical Story 7.1. Local And Global Langlands Correspondence 7.2. L-packets 7.3. Spaces Of Invariant Vectors 7.4. The Improper Hecke Operators 8. From Hecke Eigensheaves To Hecke Eigenfunctions 8.1. Hecke Eigensheaves In Positive Characteristic 8.2. Equivariance And Commutativity Conditions For Hecke Eigensheaves 8.3. Back To SL2 8.4. From Curves Over C To Curves Over Fq 8.5. Fractional Hecke Property 8.6. Fractional Hecke Eigenfunctions

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8.7. The Improper Hecke Functors 8.8. L-packets Associated To σ And σ ′ 8.9. Abelian Case 8.10. The Iwahori Case 9. Other groups 9.1. Overview 9.2. Categories Of Branes Corresponding To The Endoscopic Local Systems 9.3. Fractional Eigenbranes And Eigensheaves 9.4. Computations With Hecke Eigenfunctions 10. Gerbes 10.1. A Subtle