Introduction 0.1. Starting point 0.2. Summary 0.3. Some antecedents 0.4. Contents 0.5. Overview: the stack LocSys restr G (X) 0.6. Overview: spectral decomposition 0.7. Overview: Shv Nilp (BunG) 0.8. Overview: Langlands theory 0.9. Notations and conventions 0.10. Acknowledgements 1. The restricted version of the stack of local systems 1.1. Lisse sheaves 1.2. Another version of lisse sheaves 1.3. Definition of LocSys restr G (X) as a functor 1.4. Rigidification 1.5. Convergence 1.6. Deformation theory 1.7. A Tannakian intervention 1.8. Proof of ind-representability 2. Uniformization and the end of proof of Theorem 1.3.2 2.1. What is there left to prove? 2.2. Uniformization 2.3. Dominance of the uniformization morphism 2.4. Analysis of connected/irreducible components 3. Comparison with the Betti and de Rham versions of LocSys G (X) 3.1. Relation to the Rham version 3.2. A digression: ind-closed embeddings 3.3. Uniformization and the proof of Theorem 3.1.7 3.4. The Betti version of LocSys G (X) 3.5. Relationship of the restricted and Betti versions 3.6. The coarse moduli space of homomorphisms 4. The formal coarse moduli space 4.1. The "mock-properness" of red LocSys restr G 4.2. A digression: ind-algebraic stacks 4.3. Mock-affineness and coarse moduli spaces 4.4. Coarse moduli spaces for connected components of LocSys restr G (X)

No abstract

No abstract

We identify the category Shv Nilp (BunG) of automorphic sheaves with nilpotent singular support with its own dual, and relate this structure to the Serre functor on Shv Nilp (BunG) and miraculous duality.

We calculate the category of D-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures. G q. Example 1.2.15.2. The B-twisted theory YM G,B attaches to X the DG category QCohpLocSys G pXqq of quasi-coherent sheaves on LocSys G pXq, and to Dx attaches the DG 2-category ShvCat { LocSys G p Dxq .12 This may seem incomplete, given that 3d mirror symmetry is stated more symmetrically in terms of untwisted theories. However, as in Remark 1.2.3.1, the untwisted QFTs are on unsteady ground.13 This theory attaches Vect to X and DGCatcont to Dx. 20 These ideas were recorded in §1.1 of [DGGH] and in [BF] §7. 21 Because we are not aware of a publicly available account of this derivation, we direct the reader to [Cos2], which