THE STACK OF LOCAL SYSTEMS WITH RESTRICTED VARIATION AND GEOMETRIC LANGLANDS THEORY WITH NILPOTENT SINGULAR SUPPORT D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY Preliminary version Contents Introduction 4 0.1. Starting point 4 0.2. Summary 4 0.3. Some antecedents 6 0.4. Contents 7 restr 0.5. Overview: the stack LocSysG (X) 7 0.6. Overview: spectral decomposition 11 0.7. Overview: ShvNilp(BunG) 13 0.8. Overview: Langlands theory 17 0.9. Notations and conventions 20 0.10. Acknowledgements 22 1. The restricted version of the stack of local systems 23 1.1. Lisse sheaves 23 1.2. Another version of lisse sheaves 25 restr 1.3. Definition of LocSysG (X) as a functor 26 1.4. Rigidification 28 1.5. Convergence 28 1.6. Deformation theory 30 1.7. A Tannakian intervention 32 1.8. Proof of ind-representability 34 2. Uniformization and the end of proof of Theorem 1.3.2 36 2.1. What is there left to prove? 37 2.2. Uniformization 37 2.3. Dominance of the uniformization morphism 40 2.4. Analysis of connected/irreducible components 42 3. Comparison with the Betti and de Rham versions of LocSysG(X) 44 3.1. Relation to the Rham version 44 3.2. A digression: ind-closed embeddings 46 3.3. Uniformization and the proof of Theorem 3.1.7 47 3.4. The Betti version of LocSysG(X) 50 3.5. Relationship of the restricted and Betti versions 51 3.6. The coarse moduli space of homomorphisms 53 4. The formal coarse moduli space 55 red restr 4.1. The \mock-properness" of LocSysG 56 4.2. A digression: ind-algebraic stacks 58 4.3. Mock-affineness and coarse moduli spaces 59 restr 4.4. Coarse moduli spaces for connected components of LocSysG (X) 60 Date: October 4, 2020. 1 2 D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY 4.5. Coarse moduli space in the Betti setting 61 4.6. Maps of groups 62 4.7. Proof of Theorem 4.6.6 63 4.8. Proof of Theorem 4.4.2 66 4.9. Proof of Theorem 4.8.6 68 4.10. Proof of Theorem 4.8.7 69 5. Properties of the category of quasi-coherent sheaves 70 5.1. Quasi-coherent sheaves on a formal affine scheme 70 5.2. Some 2-categorical properties 72 5.3. Independence of presentation 73 5.4. Self-duality of QCoh of a formal affine scheme 74 5.5. The functor of !-global sections 75 restr;rigidx 5.6. Consequences for LocSysG (X) 76 restr 5.7. Consequences for LocSysG (X) 78 restr 5.8. Compact generation of QCoh(LocSysG (X)) 80 6. The spectral decomposition theorem 82 6.1. Definition of a lisse action 83 6.2. Abstract setting for Theorem 6.1.4 83 6.3. The implication Theorem 6.2.11 ) Theorem 6.1.4 86 6.4. A rigidified version 87 7. Some particular cases 88 7.1. Proof in the Betti context 88 7.2. The implication Theorem 7.1.2 ) Theorem 6.1.4 89 7.3. Complements: Rham and Betti spectral actions 92 7.4. The case of Lie algebras 93 7.5. Consequences and interpretations of Theorem 7.4.5 95 7.6. Proof of Theorem 7.4.5 96 7.7. Back to the Betti case 98 7.8. A (failed?) attempt to prove that (6.6) is an equivalence 99 8. Proof of Theorem 6.2.11 for H = Rep(H) for an algebraic group H 101 8.1. The case when H is semi-simple 101 8.2. The case when H is a torus 103 8.3. The case of a general reductive group 105 8.4. The case of a general algebraic group (via Harish-Chandra pairs) 106 9. Proof of Theorem 6.2.11 108 9.1. The diagonal of coMaps(Rep(G); H) 109 9.2. Proof of fully-faithfulness 112 9.3. End of proof of Theorem 6.2.11 113 10. Automorphic sheaves with nilpotent singular support 119 10.1. Definition and basic properties 119 10.2. Hecke action on the category with nilpotent singular support 120 10.3. A converse to Theorem 10.2.3 122 10.4. The Betti context 124 10.5. Spectral decomposition of the category with nilpotent singular support 126 10.6. Proof of Theorem 10.5.2 127 11. Applications of Beilinson's spectral projector 129 11.1. Construction of the projector 129 11.2. Compact generation of ShvNilp(BunG) 131 11.3. The right adjoint 132 11.4. Proof of Theorem 11.3.4 133 11.5. The tensor product property 135 11.6. The de Rham context 138 11.7. The Betti context 139 GEOMETRIC LANGLANDS WITH NILPOTENT SINGULAR SUPPORT 3 12. Preservation of nilpotence of singular support 140 12.1. Statement of the result 141 12.2. Set-up for the proof 141 12.3. What do we need to show? 142 12.4. Singular support in a contractive situation 144 13. Proof of Theorem 10.3.3 145 13.1. Estimating singular support from below 146 13.2. The case of G = GL2 146 13.3. The case of G = GLn 149 13.4. The case of an arbitrary reductive group G 152 13.5. Proof of point (c) and affine Springer fibers 156 13.6. Proof of Proposition 13.5.4 157 13.7. Proof of Theorem 10.3.3 for non-holonomic D-modules 160 14. Geometric Langlands Conjecture with nilpotent singular support 163 14.1. Digression: coherent singular support 163 14.2. Geometric Langlands Conjecture for ShvNilp(BunG) 164 14.3. Comparison to other forms of the Geometric Langlands Conjecture 166 14.4. An inverse implication 167 15. The trace conjecture 169 15.1. The categorical trace of Frobenius 169 15.2. The true local term 173 15.3. From geometric to classical: the Trace Conjecture 175 15.4. The case of G = Gm 176 15.5. A generalization: cohomologies of shtukas 178 16. Localization of the space of automorphic functions 180 restr 16.1. The arithmetic LocSysG 180 16.2. The excursion algebra 181 16.3. Enhanced trace and the universal shtuka 181 16.4. Relation to shtukas 183 16.5. Arithmetic Arthur parameters 185 16.6. Towards an explicit spectral description of the space of automorphic functions 186 17. Proofs of Theorems 16.1.4 and 16.1.6 188 17.1. Proof of Theorem 16.1.4 188 17.2. Proof of Proposition 17.1.7: the pure case 190 17.3. Reduction to the pure case 191 17.4. Proof of Theorem 16.1.6 194 Appendix A. Formal affine schemes vs ind-schemes 196 A.1. Creating the ring 196 A.2. Analysis of the classical truncation 196 A.3. Derived structure 198 Appendix B. Ind-constructible sheaves on schemes 200 B.1. The left completeness theorem 200 B.2. Categorical K(π; 1)'s 201 B.3. The dual of QLisse(Y ) 203 B.4. Specifying singular support 204 B.5. The tensor product theorems 204 Appendix C. Constructible sheaves on an algebraic stack 206 C.1. Generalities 206 C.2. Constructible vs compact 207 C.3. Duality-adapted stacks 209 C.4. The renormalized category of sheaves 212 C.5. Singular support on stacks 214 C.6. Product theorems for stacks 216 4 D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY C.7. Compactness in the Betti situation 218 Appendix D. Proof of Theorem 13.1.2 220 D.1. Strategy of proof 220 D.2. Construction of (C; Φ2; Φ1) 221 D.3. Proof of Proposition D.2.2 223 References 225 Introduction 0.1. Starting point. Classically, Langlands conjectured a bijection between irreducible automorphic representations for a reductive group G and spectral data involving the dual group Gˇ. P. Deligne (for GL1), V. Drinfeld (for GL2) and G. Laumon (for GLn) realized Langlands-style phenomena in algebraic geometry. In their setting, the fundamental objects of interest are Hecke eigensheaves. This theory works over an arbitrary ground field k, and takes as an additional input a sheaf theory for varieties over that field. Specializing to k = Fq and ´etale sheaves, one recovers special cases of Langlands's conjectures by taking the trace of Frobenius. Inspired by these works, Beilinson and Drinfeld proposed the categorical Geometric Langlands Con- jecture (0.1) D-mod(BunG) ' IndCohNilp(LocSysGˇ (X)) for X a smooth projective curve over a field k of characteristic zero. Here the left-hand side D-mod(BunG) is a sheaf-theoretic analogue of the space of unramified automorphic functions, and the right hand side is defined in [AG]. There are (related) discrepancies between this categorical conjecture and more classical conjectures. (i) Hecke eigensheaves make sense in any sheaf theory, while the Beilinson-Drinfeld conjecture applies only in the setting of D-modules. (ii) Hecke eigensheaves categorify the arithmetic Langlands correspondence through the trace of Frobe- nius construction, while the Beilinson-Drinfeld conjecture bears no direct relation to automorphic functions. (iii) Langlands's conjecture parametrizes irreducible automorphic representations, while the Beilinson- Drinfeld conjecture provides a spectral decomposition of (a sheaf-theoretic analogue of) the whole space of (unramified) automorphic functions. These differences provoke natural questions: {Is there a categorical geometric Langlands conjecture that applies in any sheaf-theoretic context, in particular, in the ´etalesetting over finite fields? {The trace construction attaches automorphic functions to particular ´etalesheaves on BunG; is there a direct relationship between the category of ´etalesheaves on BunG and the space of automorphic functions? {Is it possible to give a spectral description of the space of classical automorphic functions, not merely its irreducible constituents? 0.2.