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Analytic Langlands Correspondence for PGL (2) on P^ 1 with Parabolic

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Analytic Langlands Correspondence for PGL (2) on P^ 1 with Parabolic 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P WITH PARABOLIC STRUCTURES OVER LOCAL FIELDS PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN Abstract. We continue to develop the analytic Langlands program for curves over local fields initiated in [EFK1, EFK2] following a suggestion of Langlands and a work of Teschner. Namely, we study the Hecke operators introduced in [EFK2] in the case of P1 over a local field with parabolic structures at finitely many points for the group P GL2. We establish most of the conjectures of [EFK1, EFK2] in this case. Contents 1. Introduction 2 2. Preliminaries 5 2.1. Measures on analytic manifolds over local fields 5 2.2. Moduli spaces of stable bundles 6 2.3. Hecke modifications and the Hecke correspondence 7 2.4. Higgs fields, Hitchin system, nilpotent cone, very stable bundles 9 2.5. The wobbly divisor in genus zero 11 3. Hecke operators in genus zero 12 1 1 3.1. Birational parametrizations of BunG(P , t0, ..., tm+1)0 and BunG(P , t0, ..., tm+1)1 12 3.2. The Hecke correspondence for X = P1 with m + 2 parabolic points 13 3.3. Hecke operators 14 3.4. Proof of Theorem 3.6 16 3.5. Boundedness of Hecke operators 18 3.6. Compactness of Hecke operators 18 3.7. The leading eigenvalue 19 3.8. Asymptotics of Hecke operators as x ti and x 19 3.9. The spectral theorem → →∞ 20 arXiv:2106.05243v1 [math.AG] 9 Jun 2021 3.10. The subleading term of the asymptotics of H as x . 22 x →∞ 3.11. Traces of powers of Hx 24 | | n 3.12. A formula for the Hecke operator Hx for x S X(F ) 25 4. Genus zero, the archimedian case ∈ 28 4.1. The Gaudin system 29 4.2. Differential equations for Hecke operators 29 4.3. The Schwartz space 33 4.4. The differential equation for eigenvalues 35 4.5. Spectral decomposition in the complex case 35 4.6. The leading eigenvalue of the Hecke operator 37 4.7. Spectral decomposition in the real case and balanced local systems 38 4.8. The variety of balanced pairs and T -systems 41 4.9. A geometric description of balanced pairs 44 1 2 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN 4.10. Hypergeometric opers 45 5. The case of X = P1 with four parabolic points 48 5.1. The moduli space of stable bundles 48 5.2. The Hecke correspondence and Hecke operators 49 5.3. Boundedness and compactness of Hx 51 5.4. The spectral decomposition 52 5.5. The archimedian case 54 5.6. The real case 55 5.7. The subleading term of asymptotics of Hx as x 56 5.8. ComparisonwiththeworkofS.Ruijsenaars→∞ 59 6. Hecke operators on P1 with four parabolic points over a non-archimedian local field 59 6.1. Mollified Hecke operators 59 6.2. Computation of eigenvalues of Hecke operators 64 6.3. Relation to Hecke operators over a finite field 69 7. Singularities of eigenfunctions 71 7.1. Singularities for N = m + 2 of parabolic points 71 7.2. The monodromy of the Gaudin system 73 7.3. The case of 5 points 74 8. Appendix: auxiliary results 75 8.1. Lemmas on integrals over local fields 75 8.2. Elliptic integrals 77 8.3. The Harish-Chandra theorem 78 8.4. The principal series representation W and its Harish-Chandra character 79 8.5. A lemma on algebraic groups 79 8.6. Irreducibility of the character variety 79 8.7. Interpolation of rational functions 80 8.8. The exponent of a sequence 81 8.9. Lemmas on compact operators 81 8.10. Self-adjoint second order differential operators on the circle 82 References 85 1. Introduction In our previous papers [EFK1, EFK2], motivated by a suggestion of Langlands ([L]) and a work of Teschner ([Te]), we proposed an “analytic Langlands program” for curves defined over local fields. In particular, in [EFK2] we constructed analogues of the Hecke operators for the moduli space of stable G-bundles on a smooth projective curve X over a local field F with parabolic structures at finitely many points. We conjectured that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F = C, we also conjectured that their joint spectrum is essentially in bijection with the set of LG-opers on X with real monodromy. Moreover, we conjectured an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in [EFK1]. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 3 The main goal of this paper is to prove the conjectures of [EFK1, EFK2] for G = P GL2 in genus 0 with parabolic points. In particular, we establish a spectral decomposition for 2 Hecke operators acting on the Hilbert space L (Bun◦(F )) of square integrable complex half- densities on the analytic manifold Bun◦(F ) of isomorphism classes of stable quasiparabolic 1 P GL2-bundles on P with m + 2 marked points defined over a local field F . We also study the corresponding eigenfunctions and eigenvalues. Thus this paper implements the analytic Langlands correspondence for G = P GL2 in genus 0. The content of the paper is as follows. After setting up preliminaries for a general curve X and G = P GL2 (Section 2), we focus on 1 the case X = P with parabolic points t0, ..., tm+1. We first define birational parametrizations of the moduli spaces Bun0◦ and Bun1◦ of bundles of degree 0 and 1 (Subsection 3.1), and m+2 observe that Bun◦ has a natural action of the group (Z/2) , whose generators Si, i = m+1 0, ..., m + 1 switch Bun0◦ with Bun1◦. This yields an action of V := (Z/2) on each component Bun◦, j =0, 1 by S S S . j i 7→ i m+1 This allows us to explicitly describe the Hecke correspondence (Subsection 3.2) and derive an explicit formula for the Hecke operator Hx in this case (Subsection 3.3). More precisely, 2 2 we identify L (Bun0◦(F )) with L (Bun1◦(F )) using the map Sm+1, which allows us to view 2 H as an (initially, densely defined) operator on the Hilbert space := L (Bun◦(F )), and x H 0 we write a formula for this operator in terms of the birational parametrization of Bun0◦. We then use this formula and basic representation theory of P GL2(F ) to prove the compactness of Hx (Subsections 3.5, 3.6). Then we compute the asymptotics of the Hecke operators when x approaches one of the parabolic points, and show that the leading terms of this asymptotics m+2 2 are given by the action of the generators Si of the group (Z/2) on L (Bun0◦(F )), with Sm+1 1 (Subsection 3.8). This implies that the common kernel of all Hx vanishes, hence we 7→ 2 have a spectral decomposition of L (Bun0◦(F )) into their finite dimensional joint eigenspaces (Subsection 3.9). Since the Hecke operators commute, their product Hx1 ...Hxn is symmetric in x1, ..., xn, but the formula for this product arising from the definition of Hx is not manifestly symmetric. Using the Cauchy-Jacobi interpolation formula for rational functions, we give a manifestly symmetric formula for this product in genus 0. This formula can then be extended to the n case when (x1, ..., xn) S X(F ) but individual coordinates xi are not necessarly defined over F (Subsection 3.12).∈ In the archimedian case (F = R, C) we reprove by an explicit computation (for X = P1) the result from [EFK2] showing that Hecke operators Hx commute with quantum Hitchin (i.e., Gaudin) hamiltonians (Subsection 4.3) and satisfy a second order ODE with respect to x – an operator version of the oper equation (Subsection 4.2). This implies that each eigenvalue of Hx is a solution of an SL2-oper with respect to x (Subsection 4.4). This gives rise to natural commuting normal extensions of Gaudin hamiltonians, yielding their joint spectral decomposition (namely, the same decomposition as for Hecke operators), which confirms Conjecture 1.5 of [EFK1] and its Corollary 1.6. From this decomposition we deduce that for F = C the spectrum Σ of Hecke operators is simple. Moreover, there is a natural injective map from Σ to a subset of the set of R SL2-opers with real monodromy, as conjectured in [EFK1], Conjecture 1.8,(1) (Subsection 4.5). Conjecturally, this map is bijective (i.e., Σ ∼= ), and we prove this for 4 and 5 points. Moreover, we express the eigenvalues of the Hecke operatorsR as bilinear combinations of the 4 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN solutions of the second order differential equation representing the corresponding oper, which proves Conjecture 1.11 of [EFK2] in this case.1 To describe the spectrum of Hecke operators for F = R (Subsection 4.7), we introduce the 1 notion of a balancing of an SL2 local system on CP t0, ..., tm+1 . An SL2 local system admits at most two balancings, and when it does, then\{ generically} only one. The space of local systems that admit a balancing is a middle-dimensional subvariety of the variety of all local systems, which we identify with the space of solutions of the T-system of type A1. Let be the set of balanced local systems that come from oper connections. It is equipped with aB natural, at most 2-to-1 map to the space of opers (1-to-1 for generic positions of parabolic points), whose image is expected to be discrete.
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