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

arXiv:2106.05243v1 [math.AG] 9 Jun 2021 NLTCLNLNSCREPNEC FOR CORRESPONDENCE LANGLANDS ANALYTIC .1 rcso oesof powers of Traces 3.11. ..Aypoiso ek prtr as operators Hecke of Asymptotics eigenvalue leading 3.8. The operators Hecke of 3.7. Compactness operators Hecke of 3.6. Boundedness 3.6 Theorem of 3.5. Proof operators 3.4. Hecke 3.3. .2 oml o h ek operator Hecke the for formula A 3.12. .0 h ulaigtr fteaypoisof asymptotics the of term subleading The theorem spectral 3.10. The 3.9. ..Agoercdsrpino aacdpairs balanced of description geometric A 4.9. ..Telaigegnau fteHceoeao 37 35 sys and local pairs balanced balanced and of case variety real The the in operator Hecke decomposition 4.8. the Spectral of eigenvalue case leading complex 4.7. The the in decomposition 4.6. eigenvalues Spectral for equation differential 4.5. The space Schwartz 4.4. operators The Hecke for equations 4.3. Differential system Gaudin 4.2. case The archimedian the zero, 4.1. Genus 4. ..TeHcecrepnec for correspondence Hecke The 3.2. ..Hcemdfiain n h ek orsodne7 5 of parametrizations zero Birational genus in zero operators bundles 3.1. genus Hecke stable in very divisor cone, wobbly 3. nilpotent The system, Hitchin correspondence fields, 2.5. Hecke Higgs the and modifications 2.4. bundles Hecke stable fields of local spaces 2.3. over Moduli manifolds analytic on 2.2. Measures 2.1. Preliminaries 2. Introduction 1. oto h ojcue f[F1 F2 nti case. this in EFK2] [EFK1, of conjectures the of most edwt aaoi tutrsa ntl aypit o h group the for points many finitely at structures parabolic with field aey esuyteHceoeaositoue n[F2 nteca the in [EFK2] in introduced a operators and Hecke Langlands the of suggestion study a we following Namely, EFK2] [EFK1, in initiated fields Abstract. AAOI TUTRSOE OA FIELDS LOCAL OVER STRUCTURES PARABOLIC A E TNO,EWR RNE,ADDVDKAZHDAN DAVID AND FRENKEL, EDWARD ETINGOF, PAVEL ecniu odvlpteaayi agad rga o uvso curves for program Langlands analytic the develop to continue We | H x | Bun X Contents = T G H x P sses41 -systems ( 1 x P → 1 1 for with t , t i 0 .,t ..., , x and m H ∈ x aaoi ons13 points parabolic 2 + S x m as n +1 ∞ → X ) x ( 0 F ∞ → and 25 ) Bun 22 . GL P eof se GL P oko Teschner. of work G es38 tems 2 ( eestablish We . P P 2 1 1 vralocal a over t , ON 0 e local ver .,t ..., , P 1 m WITH +1 ) 1 19 18 18 16 14 24 20 19 44 35 33 29 29 28 12 12 11 9 6 5 2 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. In Subsection 4.7 we show that the spectrum of Hecke operators in genus zero for F = R can be realized as a subset of . While in general we do not expect eigenfunctions or eigenvalues of Hecke operatorsB to be explicitly computable in terms of special functions of hypergeometric type,2 in some special cases this is possible. As an example, we compute the largest eigenvalue of the Hecke operator in the case when the configuration of parabolic points admits a cyclic symmetry, over F = R and C. In these cases solutions of the oper equation are expressible via the classical hypergeometric function, which gives a hypergeometric expression for this eigenvalue (Subsection 4.10). We then proceed to study in detail the simplest nontrivial special case m = 2, when we have 4 parabolic points, over a general local field F (Section 5). Then the varieties of stable 1 bundles of degree 0 and 1 are Bun0◦ = Bun1◦ = P 0, t, 1, , the Hitchin hamiltonian is the Laméoperator with parameter 1/2, and eigenfunctions\{ ∞} and eigenvalues of Hecke operators can be written in terms of− Laméfunctions with this parameter. In this case the Schwartz kernel K(x, y, z) of the suitably normalized Hecke operator Hx is an explicit locally L1-function, given by the formula from [K] (Subsection 5.2). We also reprove the compactness of Hecke operators by direct analysis of this kernel (Subsection 5.3).3 It turns out that the kernel K(x, y, z) is symmetric not just under the switch of y and z (which is equivalent to the fact that the operator Hx is symmetric) but has the full S3- symmetry. This is a special feature of the 4-point case which is related to the Okamoto symmetries of the PainlevéVI equation. As a result, eigenvalues of Hx as functions of x and eigenvectors of Hx are actually the same functions. In the case F = C they are single-valued solutions of the Laméequation real analytic outside the four singular points, as considered in [Be], and for F = R they are solutions satisfying appropriate gluing conditions at the real parabolic points. Moreover, in the case F = C we show that the spectrum of the Hecke operators coincides with the full set of Laméopers with real monodromy. We also describe

1More precisely, Conjecture 1.11 of [EFK2] concerns the Hecke operators on L2(Bun◦(F )) = 2 ◦ 2 ◦ 0 Hx L (Bun0(F )) L (Bun1(F )), which have the form . The eigenvalues of these operators thus ⊕ Hx 0 2 ◦ come in pairs (βk(x), βk(x)), where βk(x) are the eigenvalues of Hx acting on L (Bun0(F )) (using the −2 ◦ 2 ◦ identification Sm+1 : L (Bun0(F )) L (Bun1(F ))). So each real oper (which occurs in the spectrum) defines two eigenvalues of Hecke operators→ on L2(Bun◦(F )) differing by sign, exactly as stated in [EFK2], Conjecture 1.11. 2E.g., in the case of 4 points they express via Laméfunctions with parameter 1/2, which don’t have an explicit integral representation for generic eigenvalue (see [De, CC]). − 3 1 In the case F = C, X = P with 4 parabolic points and G = P GL2 the existence of the natural normal extension for the Hitchin hamiltonian (=Laméoperator) was shown in [EFK1], but Hecke operators, which are the main new objects in the present paper, provide a much simpler proof. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 5 the spectrum in the case F = R in terms of a suitable Sturm-Liouville problem for the Lamé operator (Subsections 5.5, 5.6). Finally, in Subsections 5.7, 5.8 we compute the subleading term of the asymptotics of the Hecke operator Hx and explain how it is connected to the work of Ruijsenaars [Ru]. In Section 6 we study the example of 4 points over a non-archimedian local field. Namely, in Subsection 6.1 we give a proof of the statement from [K] that the eigenvalues of Hecke operators are algebraic numbers. It is expected that this holds not just in this example but for a general group and general curve. In Subsection 6.2 we compute the “first batch” of eigenvalues of Hecke operators, and in Subection 6.3 we show that (except for 5 special eigenvalues) they are the same as eigenvalues of the usual Hecke operators over the finite (residue) field. This agrees with predictions in [K]. In Section 7 we study the behavior of eigenfunctions near their singularities in the archimedian case, using the quantum Gaudin system (we expect the same type of singularities over a non-archimedian field, even though there is no obvious analogue of the Gaudin system in that case). These singularities occur on the so-called wobbly divisor, which is the divisor of bundles that admit a nonzero nilpotent Higgs field. Namely, in Subsection 7.1 we compute the local behavior of solutions of the quantum Gaudin system near a generic point of the wobbly divisor. This allows us to describe the local behavior of eigenfunctions and monodromy (Subsection 2.5). Moreover, in the case of 5 points we prove that eigenfunctions of Hecke operators are continuous (but not differentiable) with square root singularities near the wobbly divisor (which has normal crossings), and give a geometric description of the Schwartz space (Subsection 7.3). This settles all the main conjectures from [EFK1],[EFK2] over the complex field in the case of 5 points.4 Finally, in the appendix we collect auxiliary results. Acknowledgements. We are grateful to M. Kontsevich for sharing his ideas on Hecke operators and to A. Braverman, V. Drinfeld, D. Gaiotto, D. Gaitsgory and E. Witten for useful discussions. We thank T. Pantev for his help with the proof of Prop. 2.11. P. E.’s work was partially supported by the NSF grant DMS - 1916120. The project has received funding from ERC under grant agreement No. 669655.

2. Preliminaries 2.1. Measures on analytic manifolds over local ﬁelds. Let F be a local ﬁeld with absolute value x x (i.e., the Haar measure on F multiplies by λ under rescaling by λ F ). For instance,7→ k fork F = R we have x = x , for F = C we havek k x = x 2, and for ∈ v(x) k k | | k k | | F = Qp we have x = p− , where v(x) is the p-adic valuation of x. Let dx denotek thek Haar measure on F normalized so that k k dx k k = log R 1 x

4For N > 5 points, it still remains to be proved that all single-valued eigenfunctions of Gaudin operators (as distributions) are in L2, and thus every real oper deﬁnes a point in the spectrum of Hecke operators. 6 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

F -manifold Y , we can deﬁne the corresponding measure ω , see [We2].5 This agrees with the notation dx for the Haar measure on F . k k k k

2.2. Moduli spaces of stable bundles. The Langlands correspondence over the field Fq(X) of rational functions on a curve X over Fq is formulated in terms of complex-valued functions on Fq-points of the moduli stack Bun = BunG(X), which includes isomorphism classes of all G-bundles on X (including unstable bundles with arbitrarily large automorphism groups); this is needed because the Hecke operators arising in this correspondence involve a summation over all Hecke modifications of a given bundle at a point x X(Fq), and each modification gives a nonzero contribution. Thus one cannot define the∈ action of Hecke operators on the space of functions on the set of stable bundles, since a Hecke modification of a stable bundle could be unstable. On the other hand, in our setting, when Fq is replaced by a local field F , summation is replaced by integration. Thus if the locus of stable bundles is open and dense (which happens under the conditions given in the next paragraph) then non-stable bundles constitute “a set of measure zero”. So we can restrict ourselves to the space of square integrable functions (or, more precisely, half-densities) defined only on stable bundles, which (at least in the case when G = P GLn) form a smooth quasiprojective variety. This is convenient for doing harmonic analysis. From now on let X be a smooth projective curve with distinct marked points t0, ..., tN 1 defined over a field F of characteristic =2, where N 1 for genus 1 and N 3 for genus− 6 ≥ ≥ 6 0, and let G = P GL2 (this guarantees that the locus of stable bundles is open and dense). Recall that a G-bundle on X is a GL2-bundle (i.e., a rank 2 vector bundle) up to tensoring with line bundles. Let BunG(X, t0, ..., tN 1) be the moduli stack of principal G-bundles on − X with parabolic structures at t0, ..., tN 1 (i.e., points yi in the fibers at ti of the associated P1-bundle). Such bundles are called quasiparabolic− .7

Deﬁnition 2.1. ([S, MS]) The parabolic slope of a rank 2 vector bundle E on X with parabolic structures y0, ..., yN 1 at t0, ..., tN 1 is the number − − 1 N µ(E) := 2 deg(E)+ 4 . If L E is a line subbundle of E then the parabolic slope of L is ⊂ NL µ(L) := deg(L)+ 2 ,

5Indeed, recall that the bundle of densities (or measures) on Y is the R-bundle whose sections transform under local coordinate changes x x′ according to the rule of change of variable in the integral: { i} 7→ { i}

′ ∂xi µ = det ′ µ. ∂xj ′ ∂xi On the other hand, a top differential form on M changes according to the rule ω = det( ′ )ω. Thus any ∂xj top form ω defines a measure ω . 6 k k N Our analysis can be generalized to the case when (t0, ..., tN−1) S X(F ) but ti are not necessarily defined over F . ∈ 7Note that this is slightly different from the notion of a parabolic bundle where all parabolic points are equipped with parabolic weights. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 7

where NL is the number of those i for which Lti = yi. The quasiparabolic bundle E is called stable, respectively semistable, if for any line subbundle L E, one has µ(L) < µ(E), respectively µ(L) µ(E). 8 ⊂ ≤ Let BunG◦ (X, t0, ..., tN 1) BunG(X, t0, ..., tN 1) be the open substack of stable quasiparabolic bundles ([S, MS]).− ⊂ Every stable quasiparabolic− bundle has a trivial automorphism group, so BunG◦ (X, t0, ..., tN 1) can be viewed as a scheme, and moreover it is known to be a smooth quasiprojective− variety of dimension 3(g 1) + N. We will denote this va- − 9 riety by BunG◦ (X, t0, ..., tN 1) or shortly by Bun◦ when no confusion is possible. In our − case, when G = P GL2, the variety BunG◦ (X, t0, ..., tN 1) is the union of two connected − components BunG◦ (X, t0, ..., tN 1)0 and BunG◦ (X, t0, ..., tN 1)1, the moduli spaces of bundles − − of even and odd degrees, respectively. Moreover, the varieties BunG◦ (X, t0, ..., tN 1)0 and − BunG◦ (X, t0, ..., tN 1)1 are naturally identiﬁed with the moduli spaces of stable rank 2 quasiparabolic vector bundles− on X of degree 0 and 1, respectively, modulo tensoring with line bundles of degree 0.

Remark 2.2. If N = 3,g = 0 then BunG◦ (X, t0, ..., tN 1)0 and BunG◦ (X, t0, ..., tN 1)1 each consist of one point. Therefore, if g = 0, we will usually− assume that N 4. − ≥ 2.3. Hecke modifications and the Hecke correspondence. Assume that X(F ) = . 6 ∅ Let HMx,s(E) be the Hecke modification of a rank 2 vector bundle E on X at the point x X(F ) along the line s Ex. Namely, regular sections of HMx,s(E) are rational sections of∈E with no poles outside∈ of x and a possible first order pole at x with residue belonging to s (more precisely, this residue is well defined only up to scaling, but it still makes sense to say that it belongs to s). So we have a short exact sequence of coherent sheaves 0 E HM (E) δ 0 → → x,s → x → where δx is the skyscraper sheaf at x, which gives rise to a natural short exact sequence of vector spaces 0 E /s HM (E) s 0. → x → x,s x → → Note that if E is stable then HMx,s(E) need not be stable (nor even semistable) in general. However, it is easy to see that if E is stable and x is fixed then for generic s the bundle HMx,s(E) is stable. Moreover, on this generic locus, the assignment (x,E,s) HMx,s(E) is a regular map. 7→ The notion of a Hecke modification is also defined on P GL2-bundles with parabolic structures. Namely, if x = ti, then there is no change at ti, and if x = ti then we define the fixed 6 10 line yi′ in HMx,s(E)x to be Ex/s (regardless of yi). In particular, if s = yi then yi′ = Ex/yi.

8If N = 0 (no parabolic points) then these notions coincide with the usual notions of slope and (semi)stability for vector bundles. Also, these notions coincide with the usual notions of (semi)stability 1 in GIT, for weights 0, 2 at parabolic points, see [Mu], Section 12, and also the original papers [S, MS]. 9 ◦ Namely, as explained in [S, MS], BunG(X,t0, ..., tN−1) is a smooth open subset of the normal projective variety BunG(X,t0, ..., tN−1) – the (coarse) moduli space of semistable bundles. This space, however, may be singular and not contained in the stack BunG(X,t0, ..., tN−1), since diﬀerent semistable bundles with the same associated graded under a Harder-Narasimhan ﬁltration may correspond to the same point of BunG(X,t0, ..., tN−1). Also a semistable bundle may have a nontrivial group of automorphisms. 10It is easy to show that HM is the limit of HM as x t , s y, as long as y = y . This is not so, ti,y x,s → i → 6 i however, if y = yi, in which case this limit does not exist. This follows, for instance, from Proposition 3.2(i) below. 8 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

This construction gives rise to the Hecke correspondence Z for stable bundles. Namely, let Z BunG◦ (X, t0, ..., tN 1) BunG◦ (X, t0, ..., tN 1) (X t0, ..., tN 1 ) ⊂ − × − × \{ − } be the set of triples (E,F,x) such that F is obtained from E by a Hecke modiﬁcation at x along some s Ex, and q1, q2 : Z BunG◦ (X, t0, ..., tN 1) and q3 : Z X t0, ..., tN 1 ∈ → − → \{ − } be the natural projections. We deﬁne the Hecke correspondence at x X t0, ..., tN 1 1 ∈ \{ 1 − } by Zx := q3− (x). Note that the maps q1, q2 : Zx BunG◦ (X, t0, ..., tN 1) are P -bundles → − restricted to a dense open subset of the total space, so Z,Zx are irreducible varieties of dimensions 3g + N 1 and 3g + N 2. The following lemma− is easy. − Lemma 2.3. There are unique isomorphisms HM HM (E) = E x,Ex/s ◦ x,s ∼ and HM HM ′ ′ (E) = HM ′ ′ HM (E) x,s ◦ x ,s ∼ x ,s ◦ x,s if x = x′, restricting to the identity outside x, x′. 6 It follows from Lemma 2.3 that Z is symmetric under the swap of the two copies of

BunG◦ (X, t0, ..., tN 1). − For any i = 1, ..., N we denote by Si : BunG◦ (X, t0, ..., tN 1) BunG◦ (X, t0, ..., tN 1) the − → − (a priori) rational morphism given by Si := HMti,yi .

Proposition 2.4. (i) The map Si extends to an involutive automorphism of the variety 11 BunG◦ (X, t0, ..., tN 1) that exchanges BunG◦ (X, t0, ..., tN 1)0 and BunG◦ (X, t0, ..., tN 1)1. − − − (ii) SiSj = SjSi. N Thus the automorphisms Si define an action of (Z/2) on BunG◦ (X, t0, ..., tN 1), which N N 1 − gives rise to an action of the subgroup V := (Z/2)0 = (Z/2) − (vectors with zero sum of ∼ 12 coordinates) on each of the two components of BunG◦ (X, t0, ..., tN 1). − Proof. Let us first show that the morphisms Si are regular (not just rational). For this it is sufficient to check that Si preserves stability. 1 Let E′ := Si(E). Then µ(E′)= µ(E)+ 2 , where µ denotes the parabolic slope. Let L be a line subbundle of E. If L = y then L defines a subbundle L′ of the same degree in E′, ti 6 i but now Lt′i = yi′. On the other hand, if Lti = yi then the corresponding subbundle L′ E′ 1 ⊂ has degree deg(L) + 1 but L′ = y′. So in both cases µ(L′) = µ(L)+ . So if µ(L) <µ(E) ti 6 i 2 then µ(L′) <µ(E′). 2 The equalities Si = Id and SiSj = SjSi follow from Lemma 2.3. Remark 2.5. If the number N of parabolic points is odd then every semistable rank 2 bundle is stable (since 2µ(E) is not an integer while 2µ(L) is an integer for every line subbundle L E). Thus, in this case the varieties BunG◦ (X, t0, ..., tN 1)0 and BunG◦ (X, t0, ..., tN 1)1 ⊂ − − are smooth projective varieties, isomorphic to each other via Si. Likewise, if N = 0 then 11 The involutions Si extend to BunG(X,t0, ..., tN−1). Thus, if N > 0 then BunG(X,t0, ..., tN−1)0 ∼= BunG(X,t0, ..., tN−1)1. However, if N = 0 (no parabolic points) then BunG(X)0 may be non-isomorphic to 3 BunG(X)1. For example, for genus 2, by a theorem of Narasimhan and Ramanan ([NR]), BunG(X)0 = P , 5 ∼ while BunG(X)1 is the intersection of two general quadrics in P ([C]), so these 3-folds (both smooth in this case) have different middle cohomology: b3(BunG(X)1)=4 = b3(BunG(X)0) = 0. 6 3 12This action is faithful for N 5, but for N = 4 there is a kernel generated by S . ≥ i=0 i Q 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 9 every rank 2 bundle of odd degree is stable, so BunG◦ (X, t0, ..., tN 1)1 is a smooth projective variety. − 2.4. Higgs fields, Hitchin system, nilpotent cone, very stable bundles. In this subsection we recall basics about Hitchin systems for quasiparabolic bundles for G = P GL2 (much of it is also recalled in [EFK1, EFK2], but we repeat it here for reader’s convenience). Let E be a quasiparabolic P GL2-bundle on a smooth irreducible projective curve X of genus g with parabolic points t0, ..., tN 1. − 0 N 1 Definition 2.6. A Higgs field for E is an element φ H (X, ad(E) K − O(t )) ∈ ⊗ X ⊗ i=0 i such that for all i = 0, ..., N 1, the residue of φ at ti is nilpotent and preserves the − N parabolic structure at ti (i.e., acts by zero on the corresponding line). A quasiparabolic Higgs bundle is a pair (E,φ) of a quasiparabolic bundle and a Higgs field for this bundle. Thus, if E is stable then a Higgs field for E is just a cotangent vector φ at E to the moduli space Bun◦ of stable quasiparabolic bundles on X with parabolic points t0, ..., tN 1, − and T ∗Bun◦ is the variety of quasiparabolic Higgs bundles (E,φ) such that E is stable. 0 2 N 1 Let := H (X,K⊗ − O(t )). This is a vector space of the same dimension d = B X ⊗ i=0 i 3g 3+N as Bun◦, and it is called the Hitchin base. The Hitchin map det : T ∗Bun◦ is− defined by the formulaN (E,φ) det φ (note that since the residue of φ is nilpotent→ at B 7→ ti, det(φ) has at most first order poles). It is well known that the Hitchin map is flat and generically a Lagrangian fibration, so it defines an algebraic integrable system called the Hitchin system ([Hi, BD1]). Namely, a choice of linear coordinates on defines a B collection of algebraically independent Poisson commuting regular functions H1, ..., Hd on T ∗Bun◦ (quadratic on fibers), which are called the Hitchin hamiltonians. The nilpotent cone T ∗Bun◦ is the zero-fiber of the Hitchin map, i.e., the subvariety of (E,φ) such that φ isN nilpotent ⊂ (that is, φ2 = 0). Since the Hitchin map is flat and generically a Lagrangian fibration, is a Lagrangian subvariety of T ∗Bun◦. N Remark 2.7. The variety T ∗Bun◦ is an open subset in the Hitchin moduli space H of Higgs pairs (E,φ) which are stable in the sense of geometric invariant theory. If ME is stable then so is (E,φ) for any φ, but (E,φ) may be stable for unstable E, so the inclusion T ∗Bun◦ ֒ H is strict. The Hitchin system and nilpotent cone are actually defined for the whole→ Hitchin M moduli space, but here we restrict ourselves only to the open subset T ∗Bun◦ . ⊂MH Beilinson and Drinfeld ([BD1]) quantized the Hitchin integrable system and defined the

quantum Hitchin system (H1, ..., Hd) where Hi are the quantum Hitchin hamiltonians – commuting twisted diﬀerential operators on Bun◦ whose symbols are Hi. The twisting here is by half-forms on Bun◦b. b b We may therefore consider the system of diﬀerential equations

Hiψ = µiψ with respect to a (multivalued) half-form ψ on Bun , where µ are scalars. It follows that b ◦ i this system deﬁnes a holonomic twisted D-module ∆µ on Bun◦ whose singular support is contained in the nilpotent cone . These D-modules were studied in [BD1] and play a key role in the geometric LanglandsN correspondence, as well as in our previous works [EFK1, EFK2] and this paper. 10 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Let D be the projection of to Bun◦. This is a divisor in Bun◦ which is called the wobbly divisor ([DP1]). TheN complement of D is thus the locus of bundles E such that every nilpotent Higgs field for E is zero. Such bundles are called very stable. So we will vs denote the locus of such bundles by Bun = Bun◦ D Bun◦. \ ⊂ Since the singular support of the quantum Hitchin D-module ∆µ is contained in , this D-module is -coherent (i.e., smooth) on Bunvs. Thus it defines a de Rham localN system O vs (vector bundle with a flat connection) µ on Bun . Moreover, the rank of this local system equals the degree of the Hitchin map restrictedV to the generic fiber of the cotangent bundle d 3g 3+N of Bun◦, i.e., the product of degrees of Hi restricted to this fiber, which is 2 =2 − . The same definitions and results apply verbatim to GL2-bundles (or, equivalently, rank 2 vector bundles). Lemma 2.8. Let E be a very stable quasiparabolic rank 2 vector bundle on a curve X of N g 1 genus g with N parabolic points, and L E a line subbundle. Then µ(E) µ(L) + − . ⊂ − ≥ 4 2 Proof. Suppose L contains k of the parabolic lines. Let M = E/L. Then N N 2(µ(E) µ(L)) = deg E + 2 deg L k = deg M deg L k + . − 2 − − − − 2 0 N 1 Let σ H (L M ∗ K − O(t )) be such that σ acts by zero on the parabolic line ∈ ⊗ ⊗ X ⊗ i=0 i si for all i [0, N 1]. The last condition is vacuous if si = Lti , so the dimension of the space V of∈ such elements− σ satisfiesN the inequality N 1 − 0 dim V H (L M ∗ K O(t )) N + k deg L deg M + k + g 1, ≥ ⊗ ⊗ X ⊗ i − ≥ − − i=0 O where in the last inequality we used the Riemann-Roch theorem. But every σ V defines a ∈ nilpotent Higgs field φσ, so for a very stable E we must have V = 0. Thus deg M deg L k g +1 0, − − − ≥ which yields N 2(µ(E) µ(L)) + g 1, − ≥ 2 − as claimed. N 2+2g Corollary 2.9. A very stable bundle E remains stable after < −2 Hecke modifications N 2+2g at non-parabolic points, and remains semistable under − such modifications. ≤ 2 1 Proof. A Hecke modification increases in µ(E) by 2 and either increases µ(L) by 1 or keeps 1 it unchanged. Thus µ(E) µ(L) either increases or decreases by 2 . Hence the statement follows from Lemma 2.8. −

2 Corollary 2.10. Let H be the Hecke operator on L (Bun◦(F )) at a point x X(F ) x ∈ deﬁned in [EFK2]. Let ψ be a smooth compactly supported half-density on Bun◦(F ) with vs N 2+2g support in Bun (F ). Then for any positive integer r < −2 and any non-parabolic points x , ..., x X(F ), the half-density H ...H ψ is smooth and compactly supported on 1 r ∈ x1 xr Bun◦(F ).

Proof. Corollary 2.9 implies that the integral deﬁning Hx1 ...Hxr ψ is over a compact set and has no singularities, which implies the statement. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 11

2.5. The wobbly divisor in genus zero. In the case of X = P1 with N parabolic points the wobbly divisor can be described explicitly. Namely, for generic E Bun0◦, a line subbundle O(1 r) E contains at most 2r 1 parabolic points, and it∈ turns out that D is exactly the locus− ⊂ on which this condition is− violated for some r. In more detail, for a subset S [0, N 1] of even cardinality 2r, let DS Bun0◦ be the locus of bundles E which contain⊂ a subbundle− O(1 r) containing the parabolic⊂ lines y at − i ti, i S. Also∈ let S be the symmetric group and recall that V denotes its N 1-dimensional N − reﬂection representation over F2, i.e, the space of functions f : [0, N 1] F2 with sum of all values zero. Let us identify V with the set of subsets of [0, N 1]− of even→ cardinality by mapping f V to its support. Consider the Weyl group W :=−W (D ) := S ⋉V. This ∈ N N group acts naturally on the set W (DN )/W (AN 1)= SN ⋉V/SN = V, where V acts on itself − by translations and SN acts on V by permutations. The group W also acts on the set of components of D, with SN acting as the geometric Galois group permuting the parabolic points and V acting by the maps Si, and it is easy to see that for all g W we have g(D )= D . ∈ S g(S) Proposition 2.11. ([DP2]) (i) For each S [0, N 1], DS is an irreducible divisor in Bun1, and these are distinct for N 5. ⊂ − 0 ≥ (ii) The components of the wobbly divisor D are exactly the DS. Thus for N 5, D has N 1 ≥ 2 − irreducible components permuted transitively by the group W. Proof. (i) If S = i, j then D is the locus of bundles isomorphic to O O which contain { } S ⊕ a trivial line subbundle O containing yi and yj. It follows that DS is just the closure of the locus where y = y and there are no other equalities between y . Thus for S = 2, D is an i j k | | S irreducible divisor in Bun0◦. Since DS are transitively permuted by V, this holds for any S. Moreover, it follows from the formulas for S (see Proposition 3.3 below) that for N 5 the i ≥ stabilizer of DS in V is trivial, i.e., all DS are distinct. (ii) Let E DS, S = 2r. So we have a line subbundle O(1 r) E containing the parabolic lines∈y at t| ,| i S. Thus we have a short exact sequence− ⊂ i i ∈ 0 O(1 r) E O(r 1) 0. → − → → − →

Recall that K = O( 2). So the line bundle O(r 1)∗ O(1 r) K i S O(ti) has degree zero, hence is− trivial. Thus there exists a unique− up⊗ to scaling− ⊗nonzero⊗ ∈ N 0 1 0 1 φ H (P ,O(r 1)∗ O(1 r) K O(t )) H (P , ad(E) K O(t )). ∈ − ⊗ − ⊗ ⊗ i ⊂ ⊗ ⊗ i i S i S O∈ O∈ Moreover, since O(1 r) contains yi, φ is a nilpotent Higgs ﬁeld for E. Thus DS D. Conversely, let E −be not very stable and φ be a nonzero nilpotent Higgs ﬁeld⊂ for E. We have E = O(k) O( k) for some k 0, so the adjoint sl2-bundle has the triangular decomposition ad(E⊕) = O−(k) O O( ≥k). Thus if φ is regular then k 1, so E D . So assume that φ is not regular⊕ and⊕ has− minimal possible number of poles≥ occurring at∈ th∅e points ti, i S for some = S [0, N 1]. Then φ is nonvanishing (otherwise the number of poles of ∈φ can be reduced∅ 6 by⊂ renomalizing− it by a rational function). Thus L := Kerφ is a line subbundle of E containing the parabolic lines yi at ti, i S. So we have a short exact sequence ∈ 0 L E E/L 0 → → → → 12 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

and 0 1 φ H (P , (E/L)∗ L K O(t )). ∈ ⊗ ⊗ ⊗ i i S O∈ Thus the degree of the bundle (E/L)∗ L K i S O(ti), which equals S 2 deg L 2, S ⊗ ⊗ ⊗ ∈ | | − − is nonnegative, so deg(L) | | 1. Moreover, if this inequality is strict then we can reduce 2 N the number of poles of φ by≤ renormalizing− it by a section of O(1), a contradiction. Thus S is an even number 2r and L = O(1 r), i.e., E D , as claimed. | | ∼ − ∈ S Let us give an explicit description of the components D for S = . In this case the bundle S 6 ∅ E DS is generically trivial: E = O O. Note that an inclusion ι : O(1 r) ֒ O O is defined∈ by a rational function f(z)⊕ = p(z)/q(z) of degree r (i.e., p, q are− polynomials→ ⊕ of degree r without common roots, and at least one of them has degree exactly r). So the ≤ condition cutting out DS is that there exists such f with f(t )= y , i S, i i ∈ i.e., p(t ) y q(t )=0, i S. i − i i ∈ As explained in Subsection 8.7, this condition can be written as the vanishing condition of the determinant of a 2r by 2r matrix: det(tj,y tj, i S, 0 j r 1)=0. i i i ∈ ≤ ≤ − 3. Hecke operators in genus zero In this section we assume that X = P1 with N = m + 2 parabolic points. Also until Subsection 3.12 we assume that ti X(F ). In such a case we may assume that t0 = 0 and t = . ∈ m+1 ∞ 1 1 3.1. Birational parametrizations of BunG(P , t0, ..., tm+1)0 and BunG(P , t0, ..., tm+1)1. 1 We start with a construction of birational parametrizations of BunG(P , t0, ..., tm+1)0 and 1 BunG(P , t0, ..., tm+1)1. 1 Since a generic quasiparabolic bundle E BunG(P , t0, ..., tm+1)0 is isomorphic to O O as an ordinary vector bundle, it is determined∈ by an m+2-tuple of one-dimensional subspaces⊕ (lines) in A2. Moreover, since Aut(O O)= GL , we may assume that these lines are defined ⊕ 2 by vectors (1, 0), (1,y1),...,(1,ym), (0, 1), where yi are uniquely determined up to simultaneous scaling. Let Ey,0 be the bundle corresponding to y =(y1, ..., ym). The assignment y Ey,0 1 m 1 7→ gives rise to a rational parametrization of BunG(P , t0, ..., tm+1)0 by P − . A generic quasiparabolic bundle of degree 1 is isomorphic to O O(1) as an ordinary 1 ⊕ 1 bundle. We realize O(1) by gluing the charts U0 = P and U = P 0 using the gluing map g(w)= w. Thus a rational section of such bundle\∞ is given by∞ a pair of\ rational functions f0 on U0 and f on U such that f0(w) = wf (w). So the vector bundle O O(1) can realized similarly∞ using∞ the equation ∞ ⊕ 1 0 (3.1) f0(w)= f (w), 0 w ∞ where f0, f are now pairs of rational functions (written as column vectors). Namely, a ∞ 2 rational section of this bundle is given by a pair of A -valued rational functions f0 on U0 and f on U satisfying (3.1). We will realize lines in fibers of this bundle in the chart U0 except∞ at ∞where we will use the chart U . ∞ ∞ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 13

1 To define a point on BunG(P , t0, ..., tm+1)1, we have to fix m + 2 lines in the fiber of O O(1) at 0, t1, ..., tm, . Recall that the group Aut(O O(1)) consists of maps of the form⊕ (u, v) λ(u, (aw +∞b)u + cv). Thus at 0, we can fix⊕ standard lines spanned by the → ∞ vectors (1, 0), (1, 0) (the latter in the chart U ), and the remaining lines spanned by (1, zi), ∞ i = 1, ..., m, where zi are again uniquely determined up to simultaneous scaling. Let Ez,1 be the bundle corresponding to z = (z1, ..., zm). The assignment z Ez,1 gives rise to a 1 m 1 7→ rational parametrization of BunG(P , t0, ..., tm+1)1 by P − . We thus obtain m 1 Lemma 3.1. The assignments y Ey,0, z Ez,1 define birational isomorphisms P − 1 m 1 7→ 7→1 → Bun (P , t , ..., t ) and P − Bun (P , t , ..., t ) . G 0 m+1 0 → G 0 m+1 1 3.2. The Hecke correspondence for X = P1 with m +2 parabolic points. In this subsection we provide an explicit description of the Hecke correspondence on bundles over X = P1 with m + 2 parabolic points in terms of the rational parametrization of Lemma 3.1. By definition, the Hecke correspondence for stable bundles at x X is the variety Zx of 1 ∈ pairs (E,s) where E BunG(P , t0, ..., tm+1),s PEx. This variety is equipped with maps q : Z Bun (P1, t ∈, ..., t ), i =1, 2, where q∈ (E,s)= E and q (E,s)= HM (E). i x → G 0 m+1 1 2 x,s Proposition 3.2. (i) HMx,s(Ey,0) ∼= Ez,1, where t s xy (3.2) z (t, x, y,s)= i − i . i s y − i In particular, Sm+1(Ey,0) = Ey,1. ∼13 (ii) HMx,s(Ey,1) ∼= Ez,0. 1 Proof. (i) Since Ey,0 is trivial as an ordinary bundle, we have an identification PEx ∼= P for any point x X = P1. Let x be a point distinct from any of the parabolic points. Assume that the line∈s P1 is spanned by the vector (1,s).14 By definition, regular sections of ∈ HMx,s(Ey,0) (over some open set) are then pairs of functions (g, h) regular except possible first order poles at x, such that h sg is regular at x. This bundle is isomorphic to O O(1) as an ordinary bundle, but to compute− z we need to identify it with the standard realization⊕ of this bundle and see what happens at the parabolic points. This is achieved by using the change of variable (g, h) (h sg, (w x)g). 7→ − − So, consider what happens to the lines in the fibers at 0, t1, ..., tm, under this change of variable. At 0 we had the vector (1, 0), so after the change we get (∞s, x). At t we had − − i (1,yi), so after the change we will get (yi s, ti x). At we had (0, 1), so w x drops out and we get (1, 0). − − ∞ − Now we need to bring this m + 2-tuple of lines to the standard form. As noted above, automorphisms of O O(1) have the form (u, v) λ(u, (aw + b)u + cv), so in terms of ⊕ → 1 ζ := v/u we have ζ aw + b + cζ. Thus we have a = 0 and b + cxs− = 0, which gives b = x, c = s up to7→ scaling. So, we obtain − t x t s xy z (t, x, y,s)= x + s i − = i − i , i s y s y − i − i 13 ′ Note however that HMx′,s′ (HMx,s(Ey,1)) ≇ HMx′,s′ (Ez,0), since HMx,s does not preserve s , but maps ′ ′ x s−xs ′ ′ ′ ′ it to s−s′ . So we have HMx ,s (HMx,s(Ey,1)) = HM ′ x s−xs (Ez,0). ∼ x , s−s′ 14Here and below we abuse notation by using the same letter s to denote a point of A1 P1 and the corresponding line in the fiber of our vector bundle at x. ⊂ 14 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

as claimed. This proves the first statement of (i). The second statement then follows by taking the limit x and then s (note that the order of limits is important here!). →∞ →∞ (ii) follows from (i) and the fact that HMx,s commutes with Sm+1. It will be convenient to have two more variants of the formula for the Hecke modification. First, we can quotient out the dilation symmetry, i.e., impose the condition tm = ym = zm = 1. Then the formula for the Hecke modification looks like (t s xy )(s 1) (3.3) z = i − i − , 1 i m 1. i (s x)(s y ) ≤ ≤ − − − i On the other hand, instead of breaking the dilation symmetry, we may keep it and moreover restore the translation symmetry, no longer requiring that t0 = y0 = z0 = 0. Then the formula for the Hecke modification takes the form ti x (3.4) zi = − , 0 i m. s yi ≤ ≤ 1 − 1 From now on we identify BunG(P , t0, ..., tm+1)0 with BunG(P , t0, ..., tm+1)1 using the map Sm+1. By Proposition 3.2, in the coordinates yi, zi this will just be the identity map. 1 Now we express the maps Si in terms of the parametrizations of BunG(P , t0, ..., tm+1)0, 1 and BunG(P , t0, ..., tm+1)1 given by Lemma 3.1 (assuming that t0 = y0 = z0 = 0). Proposition 3.3. We have t t S (y , ..., y )= 1 , ..., m , 0 1 m y y 1 m and for 1 i m ≤ ≤ Si(y1, ..., ym)=(z1, ..., zm), where y t y t z = j i − i j , j = i, z = t . j y y 6 i i j − i Proof. The proposition follows by taking a limit in formula (3.2), first x ti and then s y . → → i 3.3. Hecke operators. We now pass from algebraic geometry to analysis. Consider the 1 1 set BunG◦ (P , t0, ..., tm+1)i(F ) of F -points of the variety BunG◦ (P , t0, ..., tm+1)i, i = 0, 1, an analytic F -manifold. To simplify notation, we will denote this analytic manifold by Buni◦(F ). 0 2 1 2 Consider the Hilbert spaces := L (Bun0◦(F )), = L (Bun1◦(F )), the spaces of square H H 0 integrable half-densities. The birational isomorphism Sm+1 provides an identification ∼= 1, so for brevity we will denote this space by . H H We have a natural regular map H 1 m 1 π : Bun◦ (P , t , ..., t ) P − G 0 m+1 i → (see e.g. [C]) which is a birational isomorphism (namely, the inverse is given by the parametriza- 2 m 1 tion of Lemma 3.1). Thus π defines an identification = L (P − (F )) of with the space m 1 H ∼ H of square integrable half-densities on P − (F ). As well known, the bundle of half-densities 1 m 1 m 1 on P − (F ) is K 2 where K = O( m) is the canonical bundle of P − . Thus we may k k − (and will) realize half-densities on Buni◦(F ) as homogeneous complex-valued functions ψ of m m (y1, ..., ym) F 0 of homogeneity degree 2 , i.e., functions ψ such that ∈ \ −m ψ(λy)= λ − 2 ψ(y), λ F. k k ∈ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 15

m Definition 3.4. Define U F 0 to be the open set of such points y = (y1, ..., ym) that ⊂ \ m for any y F the equality yi = y holds for fewer than 2 + 1 values of i for y = 0 and fewer m ∈ 6 than 2 values of i for y = 0. Definition 3.5. Let V be the space of continuous complex-valued functions ψ on ⊂ H F m 0 of homogeneity degree m (a dense subspace in ). Also let V V be the space of \ − 2 H ⊃ functions ψ of homogeneity degree m defined and continuous on U.15 − 2 e Recall that in [EFK2], Subsection 1.2 we defined the Hecke operator Hx depending on a point x X(F ), which is given by the convolution with the “δ-function” of the Hecke ∈ 1 correspondence Zx. Recall also that Hx is not a function of x but rather a 2 -density, i.e., 1 − a section of KX − 2 . For the purposes of computation, however, we will treat Hx as a k k 1 function, so that the actual Hecke operator is Hx dx − 2 . Then an explicit formula for the Hecke operators in the case of genus 0 is given byk thek following theorem (in which we set 16 t0 = 0). Theorem 3.6. The Hecke operator H is the operator V V given by the formula x → 1 m 2 e Hx = (ti x) Hx, − i=0 Y where m−2 t s xy t s xy s 2 ds (3.5) (H ψ)(y , ..., y ) := ψ 1 − 1 , ..., m − m k k k k . x 1 m s y s y m s y ZF − 1 − m i=1 k − ik We will call Hx the modified Hecke operator. Q Before proving Theorem 3.6, which is done in the next subsection, let us record its two equivalent formulations corresponding to breaking the dilation symmetry and to restoring the translation symmetry. For the first variant we assume that t0 = y0 =0, tm = ym = 1 (this can be achieved by shift 2 m 1 2 and rescaling). We now realize as the space L (F − ) of L -functions in m 1 variables H − w1, ..., wm 1 (without a homogeneity condition) with the norm given by the formula − 2 2 φ = φ(w1, ..., wm 1) dw1...dwm 1 , k k m−1 | − | k − k ZF via φ(w1, ..., wm 1)= ψ(0,w1, ..., wm 1, 1). − − Proposition 3.7. In terms of φ, the modified Hecke operator takes the form

(3.6) (Hxφ)(u1, ..., um 1)= − m 1 (t1s u1x)(s 1) (tm 1s um 1x)(s 1) s(s 1) 2 − ds − − φ − − , ..., − − k −m km 1 k k . F (s u1)(s x) (s um 1)(s x) s x 2 − s u Z − − − − − k − k i=1 k − ik Proof. This is equivalent to Theorem 3.6 using formula (3.3) and the homogeneQ ity of ψ.

15Note that V is not contained in . 16 H Our space V (the initial domain for Hx) here is slightly different than that in [EFK2]. This is a very minor modificatione that has no effect on the final result. 16 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

2 m 1 Deﬁne the operator U on L (F − ) for x = t by s,x 6 i (Us,xφ)(u1, ...., um 1) := − 1 m 1 2 − s(s 1)(ti x) (t1s u1x)(s 1) (tm 1s um 1x)(s 1) − − − − 2 φ − − , ..., − − . (s x)(s ui) (s u1)(s x) (s um 1)(s x) i=1 − − − − − − − Y m 1 It is easy to check that Us,x is a unitary operator on . Namely, let G := P GL2 − , and W H 1 be the unitary representation of P GL2(F ) of principal series, on half-densities on P (F ) (cf. m 1 Subsection 8.4). Then Us,x is the operator deﬁned in W ⊗ − by the element

(tis ux)(s 1) (3.7) gs,x := (gs,x,1, ..., gs,x,m 1) G(F ), gs,x,i(u) := − − . − ∈ (s u)(s x) − − We thus obtain the following equivalent form of Proposition 3.7 (and hence Theorem 3.6). Proposition 3.8. x(x 1) (3.8) H = U − ds . x s,x s(s 1)(s x) k k ZF s − − For the second variant, we will think of ψ as a function of y0, ..., ym which is semi-invariant m under the group y ay + b of degree 2 , no longer assuming that t0 = y0 = 0. Then we have 7→ − 1 Proposition 3.9. For the curve X = P with m +2 marked points t0, ..., tm, the modiﬁed Hecke operator is given by the formula ∞ t x t x ds (H ψ)(y , ..., y )= ψ 0 − , ..., m − k k . x 0 m s y s y m s y ZF − 0 − m i=0 k − ik Proof. This is equivalent to Theorem 3.6 using formula (3.4) andQ the translation invariance of ψ.

3.4. Proof of Theorem 3.6. We start with showing that Hx is precisely the Hecke operator deﬁned in [EFK2], Subsection 1.2. We will use the formulation from Proposition 3.8. By Proposition 3.3, the Hecke operator deﬁned in [EFK2] has the form

(3.9) Hx = Us,xdµ(s), ZF where dµ(s) is a certain measure on P1(F ), and our job is to compute dµ(s). To any point (y, z,s) Zx we can attach two 1-dimensional spaces T1,s, T2,s, where T1,s is ∈ 1 1 the tangent space at (y, z,s) to q1− (y) and T2,s is the tangent space at (y, z,s) to q2− (z). 1 These spaces define line bundles T1, T2 on the projective line P with coordinate s when 1 y is fixed. The bundle T1 is just the anticanonical bundle KP−1 , while the bundle T2 is 2 isomorphic to KP1 , and moreover there is a canonical isomorphism η : T1 T2 KP−1 ; 2 → ⊗ namely, η = (a∗)⊗ where a is the isomorphism of [EFK2], Theorem 1.1 (for fixed x). This 2 isomorphism has the form η(v) = η (v)γ(s)(ds)− , where v T1,s, η : T1 T2 is inverse ∗ ∈ 2 ∗ → to the map induced by the projection (y, z,s) s, and γ(s)(ds)− is a regular section of 2 7→ KP−1 serving to cancel the poles of η , to make sure that η is a well defined isomorphism. 1 ∗ Moreover, we have dµ(s)= γ(s) − 2 ds . Thus the explicit form of the map η completely determines γ(s) and hence dµk (s),k at leastk k up to a scalar (depending on x). ∗ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 17

To compute η , note that the ﬁber of q2 near (y, z,s) is the parametrized curve (y(u), z,u) with y(s) = y,∗ where the corresponding Hecke modiﬁcation z(u) = z remains constant. Thus for v T , ∈ 1,s η (v)=(y′(s), 0, 1)ds(v). ∗ Setting ti = t, yi = y, zi = z for brevity, we have dz(s) = 0, which yields

∂sz y′(s)= . −∂yz By formula (3.3) we have (ts xy)(s 1) z = − − . (s y)(s x) − − Thus 1 (t x)s z− ∂ z = − , y (s y)(ts xy) − − 1 1 1 1 1 z− ∂ z = + . s s xyt 1 s 1 − s x − s y − − − − − So (s y)(ts xy) 1 1 1 1 dy = − − + ds. − (t x)s s xyt 1 s 1 − s x − u y − − − − − − This 1-form has ﬁrst order poles at s =0, 1, x, and no other singularities. Thus the same 2 ∞ holds for η . It follows that γ(s)(ds)− has simple zeros at 0, 1, x, , i.e., ∗ ∞ 2 1 2 (ds) γ(s)− (ds) = C(x) . s(s 1)(s x) − − It remains to show that C(x) = x(x 1). This can be shown by a slightly more careful analysis, taking into account the variation− of x. One can also see that C(x) is proportional to x(x 1) by looking at the asymptotics x 0, 1, t, . Thus, we get − → ∞ x(x 1) dµ(s)= − ds , s(s 1)(s x) k k s − − hence the operator of Theorem 3.6 coincides with the Hecke operator deﬁned in [EFK2],

Subsection 1.2. Now we show that (3.5) defines a linear operator V V . Suppose ψ is continuous, and m → that less than 2 + 1 points yi coincide. Let us show that the integral defining Hx converges 2 (uniformly on compact sets in U). At s = the density ine (3.5) behaves as s− ds , so we ∞ k k only need to check convergence near s = yi, say, for i = 1. Using the homogeneity of ψ, we can rewrite (3.5) as follows: (Hxψ)(y1, ..., ym)= m m−2 (s y1)(t2s xy2) (s y1)(tms xym) s y1 2 s 2 ds = ψ t1s xy1, − − , ..., − − k − mk k k k k. F − s y2 s ym i=1 s yi Z − − m k − k k m 2 − By the definition of U, this density behaves near s = y1 as s y1 dsQ , where k < 2 +1 m−2 k m k − k k k 2 − if y1 = 0 and as s y1 ds , where k < 2 if y1 = 0. This density is integrable, so the integral6 in (3.5)k − converges.k k Finally,k observe that H ψ is homogeneous of degree m . x − 2 This implies that H : V V . x → e 18 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

3.5. Boundedness of Hecke operators. We now show that Hecke operators extend to bounded operators on . H Proposition 3.10. We have H (V ) , and H extends to a bounded operator on which x ⊂ H x H depends continuously on x when x = ti, . Moreover, 1 6 ∞ 1 H = O x t 2 log 1 , x t ; H = O( x 2 log x ), x . x i x ti i x k k k − k k − k → k k k k k k →∞ Proof. By Proposition 3.8, x(x 1) (3.10) H − ds . k xk ≤ s(s 1)(s x) k k ZF s − − This integral is convergent, so, using Lemma 8.2 and the symmetry between ti and 0, 1, , we get the result. ∞ We will see in Proposition 3.15 below that the bound of Proposition 3.10 is in fact sharp. We also have

Proposition 3.11. The operators Hx are self-adjoint and pairwise commuting.

Proof. By Proposition 3.10, the operators Hx are bounded. Also, it is easy to see that x(s 1) (3.11) U † = U , σ (s) := − s,x σx(s),x x s x − and the measure of integration in (3.8) is invariant under the involution s σx(s). This implies the ﬁrst statement. The second statement follows from Lemma 2.3. 7→

Example 3.12. Let m = 1. In this case the sets Bun0◦(F ) and Bun1◦(F ) consist of one point, so the space is 1-dimensional. Thus the operator U is the identity on this 1-dimensional H s,x space. So we see that the operator Hx is just a scalar function of x given by the formula

Hx = E(x), where E(x) is the elliptic integral deﬁned in Subsection 8.2. 3.6. Compactness of Hecke operators.

Proposition 3.13. The Hecke operator Hx is compact and strongly continuous (i.e., continuous in the operator norm) in x when x = t , . 6 i ∞ Proof. We ﬁrst show that Hx is compact. Let n = 3m 3. We will approximate the n − operator Hx in the operator norm by trace class (hence compact) operators. Since the space n of compact operators is closed with respect to the operator norm, this will imply that Hx is compact, hence Hx is also compact (as it is self-adjoint). We have n Hx = Us1,x, ..., Usn,xdνx(s1)...dνx(sn), n ZF where νx is the measure of integration in (3.8), i.e., x(x 1) (3.12) dν (s)= − ds . x s(s 1)(s x) k k s − − m 1 n Recall that G = P GL − . Let ξ : A G be the rational map given by 2 n → ξn(s1, ..., sn) := gs1,x...gsn,x, 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 19

where gs,x G is deﬁned by (3.7) (except that here we work with algebraic varieties and ∈ ⊠n ⊠n not yet with their points over F ). Let λ := ξn (νx ) be the direct image of the measure νx ∗ under ξn. Since n = dim G, ξn is dominant by Lemma 8.9. Hence

dλ = fx(g)dg, 1 where dg is the Haar measure on G(F ) and fx is an L -function on G(F ). Indeed, this follows from the fact that the preimage of a set of measure zero under ξn has measure zero. Recall that W denotes the principal series representation of P GL2(F ) on half-densities 1 m 1 on P (F ), and W ⊗ − the corresponding unitary representation of G(F ). Let ρ the corresponding representation map. We have

n Hx = ρ(g)fx(g)dg. ZG(F ) By Harish-Chandra’s Theorem (Theorem 8.6), for any smooth compactly supported function φ on G(F ), the operator

Aφ := ρ(g)φ(g)dg ZG(F ) 1 is trace class, therefore compact. But since the function fx is L , it can be approximated in L1-norm by a smooth compactly supported function φ with any precision ε > 0. This n implies that the operator Hx is compact, hence so is Hx. It remains to show that Hx is strongly continuous in x. To this end, ﬁx x and ε > 0. It 1 is easy to see that fy depends continuously on y in the L metric. Therefore, there is δ > 0 such that if y x < δ then f f <ε. Then H H <ε, as claimed. | − | k y − xkL1 k y − xk 3.7. The leading eigenvalue. By the spectral theorem for compact self-adjoint operators, the commuting operators Hx, being compact, have a common orthogonal eigenbasis. More- over, we have the following proposition.

Proposition 3.14. The largest eigenvalue β0(x) of Hx is positive and has multiplicity 1, with a unique positive normalized eigenfunction ψ (independent on x), and H = β (x). 0 k xk 0 Proof. This follows by the Krein-Rutman theorem (an inﬁnite-dimensional analog of the n Frobenius-Perron theorem, see [KR]), since the Schwartz kernel of Hx is a strictly positive function for large enough n.

3.8. Asymptotics of Hecke operators as x ti and x . When x ti and x , Hecke operators have singularities. So we would→ like to compute→∞ the leading→ coeﬃcient→∞ of the asymptotics. Proposition 3.15. (i) In the sense of weak convergence, we have17 1 1 H x t 2 log( x t − )S , x t ; x ∼k − ik k − ik i → i in other words, for any ψ we have ∈ H H ψ x S ψ 1 1 i x t 2 log( x t − ) → k − ik k − ik 1 − 1 17 − 2 dz 2 −1 This implies that the “true” Hecke operator Hx dx has the weak asymptotics z log z Si near the parabolic point t , where z is the local coordinatek k near the parabolic point. Note that this statement i is independent on the choice of the local coordinate z. 20 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

as x ti. Similarly, in the sense of weak convergence → 1 H x 2 log x , x . x ∼k k k k →∞ (ii) We have 1 1 1 H x t 2 log( x t − ), x t ; H x 2 log x , x . k xk∼k − ik k − ik → i k xk∼k k k k →∞ Proof. (i) We establish the asymptotics at x ; the case x ti then follows by symmetry. Hx →∞ → By Proposition 3.10, k1 k is bounded when x . Therefore, it suﬃces to show that x 2 log x k k k k →∞ Hxψ (3.13) lim 1 = ψ x →∞ x 2 log x k k k k for ψ belonging to a dense subspace of . In particular, it is enough to prove this in the case when ψ is continuous. H We have (Us, φ)(u1, ...., um 1)= ∞ − 1 m 1 2 − s(s 1) u1(s 1) um 1(s 1) − − 2 φ − , ..., − . (s ui) s u1 s um 1 i=1 − − − − Y Also by Lemma 8.2 the density 1 2 1 x(x 1) 1 Ex 1 − ds = 1 1 2 s(s 1)(s x) k k − x 2 x log x s x − log x k k k k − − k k k k converges as x to δ . This implies (3.13) for continuous ψ by formula (3.8), since →∞ ∞ u(s 1) lim − = u. s s u →∞ − (ii) By Proposition 3.14, H =(H ψ , ψ ), so the statement follows from (i). k xk x 0 0 Remark 3.16. It follows from Proposition 3.13 that the convergence in Proposition 3.15 is only weak, since a sequence of compact operators cannot converge strongly to an invertible operator. 3.9. The spectral theorem. Here is one of our main results. Theorem 3.17. (i) There is an orthogonal decomposition

= ∞ , H ⊕k=0Hk where are the eigenspaces of H : Hk x H ψ = β (x)ψ, ψ , x k ∈ Hk where βk are continuous real functions of x deﬁned for x = t0, ..., tm, . (ii) 6 ∞ 1 β (x) x 2 log x , x . k ∼k k k k →∞ In particular, the function βk is not identically zero for any k. Thus the spaces k are ﬁnite dimensional. H (iii) There is a leading positive eigenvalue β0(x) of Hx such that β (x) < β (x) | i | 0 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 21

for all i > 0, and the corresponding eigenspace 0 is 1-dimensional, spanned by a positive eigenfunction ψ (y). Moreover, H = β (x). H 0 k xk 0 Proof. (i) This follows from Propositions 3.11 and 3.13 and the spectral theorem for compact self-adjoint operators. (ii) This follows from (i) and Proposition 3.15. (iii) Follows from Proposition 3.14.

Theorem 3.17 implies Conjecture 1.2 of [EFK2] for G = P GL2 and curves of genus zero.

Corollary 3.18. On every compact subset C F t0, ..., tm , the sequence βk(x) is equicon- tinuous and converges uniformly to 0 as k ⊂ . \{ } →∞ Proof. The function x Hx is continuous on C by Proposition 3.13, therefore by Cantor’s theorem it is uniformly7→ continuous. So for any ε> 0 there is δ > 0 such that if y x < δ then H H <ε. Then β (y) β (x) <ε for all i, which proves equicontinuity.| − | k y − xk | k − k | Suppose βk does not go uniformly to 0 on C. Then there is a sequence kj such that sup β (x) ε for some ε > 0. The sequence β is uniformly bounded (since so are C | kj | ≥ k the operators Hx for x C), so by the Ascoli-Arzela theorem, the subsequence βkj has a uniformly convergent subsequence,∈ to some nonzero function β(x). This is a contradiction since by Proposition 3.13, for any x we have limk βk(x) = 0. →∞ Now recall that we have an action of the group V =(Z/2)m+1 on by the operators S . H i Since by Proposition 3.15, Si are the leading coeﬃcients of Hx as x approaches ti, it follows that Si act by scalars ( 1) on each eigenspace k. Let us denote this character of V by χk; i.e., S v = χ (S )v for v± . Note that χ =H 1. We thus obtain i k i ∈ Hk 0 Corollary 3.19. One has 1 1 β (x) x t 2 log( x t − )χ (S ), x t . k ∼k − ik k − ik k i → i The following corollary may be interpreted as the statement that the operator Hx is smooth in x.

Corollary 3.20. The function βk(x) is smooth in x for x = ti, (meaning locally constant in the non-archimedian case). 6 ∞

Proof. This follows from the fact that βk(x)=(Hxψ, ψ) for an (x-independent) normalized eigenfunction ψ . The integral deﬁning (H ψ, ψ) is a smooth function of x. ∈ Hk x Let us also derive a formula for the Schwartz kernel K(x1, ..., xr, y, z) of the product

Hx1 ...Hxr of Hecke operators (which may be a singular distribution) in terms of eigenfunctions, i.e. as a “reproducing kernel”. We may choose an orthonormal basis ψ in each , k,j Hk so that all ψk,j are real-valued; this is possible since the Schwartz kernel of a power of the Hecke operator is real-valued. Then we have

(3.14) K(x1, ..., xr, y, z)= βk(x1)...βk(xr)ψk,j(y)ψk,j(z). Xk,j In particular, for n 1 we have ≥

(3.15) Tr(Hx1 ...Hxn )= dkβk(x1)...βk(xN ), Xk where d := dim , provided that this series is absolutely convergent. k Hk 22 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

3.10. The subleading term of the asymptotics of H as x . x →∞ 1 Proposition 3.21. The operator Hx := x − 2 Hx log x has a weak limit M = H as x , which is an unbounded self-adjointk k operator− on k,k acting diagonally in the basis∞ of →∞ H eigenfunctions ψn(y): e e (k) Mψk = µ ψk. That is, for any ψ in the domain of M, we have

lim Hxψ = Mψ. x →∞ Here e 1 (k) 2 µ = lim ( x − βk(x) log x ). x →∞ k k − k k Namely, the operator M is given by the formula

(Mψ)(y1, ..., ym) :=

y1 ym ψ −1 , ..., −1 1 y1s 1 yms ds ψ(y t s, ..., y t s)+ − − ψ(y , ..., y ) k k.  1 − 1 m − m m 1 y s 1 − 1 m  s ZF i=1 k − i − k k k   Note that integral (3.21) is convergentQ (say, for ψ smooth) since the third summand cancels the logarithmic divergences at s = 0 and s = generated by the first and the second summand, respectively. ∞ Proof. We first prove that M is given by the claimed formula up to adding the operator of multiplication by a function h(y). For this it suffices to check that for a fixed generic y =(y1, ..., ym), if ψ is smooth with ψ(y) = 0 then

y1 ym ψ −1 , ..., −1 ds 1 y1s 1 yms ds (3.16) (Mψ)(y)= ψ(y t s, ..., y t s)k k + − − k k. 1 − 1 m − m s m 1 y s 1 s ZF k k ZF i=1 k − i − k k k m 1 To prove this formula, consider the closure Zx,y in P − of theQ preimage of y in Zx. This is m 1 a parametrized rational curve of degree m in P − given by t s xy z (s)= i − i . i s y − i Since (Hxψ)(y) is defined by integration over Zx,y(F ), the function (Mψ)(y) is defined by integration over Z ,y(F ), where Z ,y is the degeneration of the curve Zx,y as x . By ∞ ∞ 1/n → ∞ definition, z Z ,y if there exist s,λ C((x− )) for some n such that ∈ ∞ ∈ tis(x) xyi lim λ(x) − = zi x s(x) y →∞ − i (the factor λ(x) is needed since zi are defined only up to simultaneous scaling). If s has a finite limit s0 at x = , we have ∞ yi zi = 1 1 y s− − i 0 up to scaling (namely, if s0 = yi for some i, we get zi = 1 and zj = 0 for j = i, and if s0 =0 then z = 1 for all i). On the other hand, if the order of s(x) at is 6 r for a rational i ∞ − 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 23

number r > 0 then if 0 1 we get zi = ti for all i, so the only interesting case is r = 1, i.e., s(x)= s x + o(x), s = 0. In this case we have 0 0 6 z = y t s i i − i 0 1 up to scaling. This shows that Z ,y is the union of two components: Z ,y of degree m 1 deﬁned by the parametric equations∞ ∞ − y z (s)= i , i 1 y s 1 − i − 2 and Z ,y of degree 1 (a line) deﬁned by the parametric equations ∞ z (s)= y t s i i − i m 1 ti (which are permuted by the birational involution S of P − transforming y into ). Thus 0 i yi 1 2 (Mψ)(y) is the sum of two integrals, over Zx,y(F ) and Zx,y(F ), which yields the desired formula. It remains to show that h = 0. To this end, let us apply the operator M to the function m 1 ψ(y , ..., y )= y − 2 . We have 1 m i=1 k ik m Q 1 (Mψ)(y , ..., y )= I(y , ..., y ) y − 2 , 1 m 1 m k ik i=1 Y where

1 ds I(y1, ..., ym) := lim 1 k k log x . x  m 1 1 1 2 s − k k →∞ F (1 s y )(1 st y− x ) Z i=1 − − i − i i − k k But direct asymptotic analysis shows that  Q 1 1 ds I(y , ..., y )= + 1 k k, 1 m m 1 1  1 2 m 1 2 −  s F 1 s− yi 1 st y− Z i=1 k − k i=1 − i i k k   which implies that h = 0. Q Q

We note that the two summands in formula (3.16) are permuted by S0, while Si for 1 i m preserves each summand (in fact, this holds even before integration if we change ≤ ≤1 s to s− in one of the summands). Thus we have

S0M = MS0 = S0Q + QS0, where Q is a self-adjoint operator such that ds (Qψ)(y)= ψ(y t s, ..., y t s)k k 1 − 1 m − m s ZF k k for ψ smooth with ψ(y) = 0 for a given point y. Moreover, [Q, Si]=0for1 i m. For m = 2 (4 parabolic points), the operator M is studied in more detail≤ in≤ Subsection 5.7. In the case, it is easy to check that the two summands in formula (3.16) coincide with m each other. This agrees with the fact that in this special case i=0 Si = S0S1S2 = 1 (as we have mentioned, this is not so for m> 2). Q 24 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

3.11. Traces of powers of Hx . Let m 2. Note that the series (3.15) (say, with all xi equal) cannot be absolutely convergent| | if n≥ 2(m 2). Indeed, the Schwartz kernel of the m 2 ≤ − operator H − is supported on the m 2-th convolution power of the Hecke correspondence x − Zx, which is a proper subvariety (since codimZx = m 2), hence a set of measure zero. So −2(m 2) this operator cannot be Hilbert-Schmidt and thus Tr(Hx − )= . On the other hand, if n is suﬃciently large, the series (3.15) does∞ have to converge absolutely. To show this, let n(ε)= n(Hx,ε) be the number of eigenvalues of Hx (counted with multiplicities) of magnitude ε, and deﬁne ≥ log n(ε) bm := limsupε 0 1 . → log(ε− )

Lemma 3.22. For any a > bm, Tr H a = d β (x) a < , | x| k| k | ∞ k 0 X≥ and for any a < bm Tr H a = d β (x) a = , | x| k| k | ∞ k 0 X≥ where d = dim . k Hk Proof. This follows from Lemma 8.13 applied to the sequence β (x) (repeated d times). | i | i Proposition 3.23. (i) We have 2(m 1) bm < . (ii) For sufficiently large n the series− (3.15)≤ converges∞ absolutely. Proof. (i) We first show that b < . Let I be the union of the ε-neighborhood of the points m ∞ ε 0, x F and its image under the map σx defined by (3.11). Consider the cutoff measure dν ∈(s) on F which equals dν (s) (defined by (3.12)) when s / I and zero otherwise. Set x,ε x ∈ ε

Hx,ε = Us,xdνx,ε(s). ZF This operator is self-adjoint since νx,ε is invariant under σx. Moreover, it is compact (the 1 proof is the same as Proposition 3.13). Finally, ν ν Cε 2 for some C > 0. Thus k x,ε − xkL1 ≤ 1 H H Cε 2 . k x,ε − xk ≤ By Lemma 8.15(ii), this implies that 1 1 (3.17) n(H , 2Cε 2 ) n(H ,Cε 2 ). x ≤ x,ε ℓ c Lemma 3.24. There exist an even integer ℓ> 0 and a real c> 0 such that TrHx,ε = O(ε− ) as ε 0. → Proof. Similarly to the proof of Proposition 3.13, one can show that

3(m 1) Hx,ε − = ρ(g)fx,ε(g)dg, ZG(F ) where f (g) f (g) is an L1-function with compact support. Since f (g)dg is the direct x,ε ≤ x x image of the norm of a rational function under a ﬁnite covering map ξ3(m 1), it follows that r − − fx L r 1 for suﬃciently large even integer r. Then using Young’s inequality for convolutions r ∈ k r−k ([We1], p.54-55), it follows by induction that the convolution power fx∗ belongs to L when 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 25

r k r; in particular, f ∗ belongs to L∞. We then denote by M the L∞-norm of this function, ≤ x r and by Sε its (compact) support. Clearly, r f ∗ ∞ M . x,ε L ≤ r Let ℓ := 3(m 1)r. Then − ℓ m 1 r TrHx,ε = χ⊗ − (g)fx,ε∗ (g)dg, ZG(F ) 1 where χ = χW is the character of the principal series representation W (a locally L function given by Proposition 8.8). It follows from Proposition 8.8 that χ(g) 0 and for some c> 0, ≥ m 1 c χ⊗ − (g)dg = O(ε− ). ZSε Thus

m 1 r m 1 r m 1 c χ⊗ − (g)f ∗ (g)dg = χ⊗ − (g)f ∗ (g)dg M χ⊗ − (g)dg = O(ε− ). x,ε x,ε ≤ r ZG(F ) ZSε ZSε This implies the statement. Now, using Lemma 3.24 and (3.17), we get

1 1 1 ℓ ℓ c ℓ/2 n(H , 2Cε 2 ) n(H ,Cε 2 ) (Cε 2 )− TrH = O(ε− − ). x ≤ x,ε ≤ x,ε This implies that 2c ℓ n(Hx,ε)= O(ε− − ), i.e., b 2c + ℓ< . m ≤ ∞ On the other hand, by Lemma 8.16 bm 2(m 1), which completes the proof (i). (ii) follows from (i), Lemma 3.22 and the≥ arithmetic− and geometric mean inequality. 1 Thus, the exponent of the sequence βi(x) repeated di times (Subsection 8.8) is . | | bm Remark 3.25. It would be interesting to determine this exponent precisely. As we have 1 1 1 shown, it is 2(m 1) . Moreover, one can show that for m = 2 the exponent equals 2(m 1) = 2 . ≤ − − n 3.12. A formula for the Hecke operator Hx for x S X(F ). Let x , ..., x F and ∈ 1 n ∈ x =(x1, ..., xn). Consider the operator

Hx := Hx1 ...Hxn .

Since the factors commute, this product is invariant under permutations of xi. However, if we write an explicit formula for Hx by iterating the deﬁnition of Hx, the resulting expression won’t be manifestly Sn-invariant. The goal of this subsection is to write another, manifestly symmetric formula for Hx. More precisely, we will write a symmetric formula for

Hx := Hx1 ...Hxn . This formula will then also make sense for x =(x , ..., x ) SnX(F ), where the individual 1 n ∈ coordinates xi are not required to be deﬁned over F . Let y = (y0, ..., ym) and Ey,0, Ey,1 be the vector bundle O O, respectively O O(1), with parabolic structures at t given by the vectors (1,y ), 0 ⊕i m and the vector⊕ (0, 1), i i ≤ ≤ respectively (1, 0) at (see Subsection 3.1). Let E be the vector bundle obtained from Ey ∞ ,0 by simultaneous Hecke modiﬁcation at points x1, ..., xn along the lines s1, ..., sn. The sections 26 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

of E are pairs of rational functions (g, h) with at most simple poles at x , ..., x and h s g 1 n − i regular at xi for i =1, ..., n. We are now ready to write a formula for Hx. We will use Lemma 8.12. First consider the case of even n =2r. In this case generically E ∼= O(r) O(r). Explicitly, this isomorphism is given by the map ⊕ (g, h) (q h p g, q h p g), 7→ 1 − 1 2 − 2 where p1, q1,p2, q2 are polynomials of degrees r, r 1,r 1,r with p1, q2 monic such that the rational functions − − pk(w) fk(x, s,w)= fk(w)= qk(w) satisfy the conditions fk(xi)= si, i =1, ..., n.

(this uniquely determines the coeﬃcients of p1, q1,p2, q2 from a system of linear equations). This means that p1 1 p2 1 f1 = = ιx−, (s, ), f2 = = ιx−, (s, 0). q1 ∞ ∞ q2 ∞ where ι is the map from Lemma 8.12. Thus we get E = Ez O(r), where ,0 ⊗ p (x, s, t ) q (x, s, t )y (3.18) z = 2 i − 2 i i . i p (x, s, t ) q (x, s, t )y 1 i − 1 i i Now consider the case of odd n =2r+1. Then generically E = O(r) O(r+1). Explicitly ∼ ⊕ this isomorphism is given as in the even case, except with p1, q1,p2, q2 being polynomials of degrees r, r, r +1,r 1 with q ,p monic, and − 1 2 p1 1 p2 1 2 f1 = = ιx− (s), f2 = = ιx−, (s, ), q1 q2 ∞ ∞ where 1 2 1 ιx−, (s, ) := lim ιx−,b, (s, , ) ∞ b ∞ ∞ →∞ ∞ ∞ is the rational function of degree r + 1 taking values s at x for 1 i 2r + 1 and growing i i ≤ ≤ quadratically at . So we get E = Ez,1 O(r), with zi given by (3.18). Alternatively, formula∞ (3.18) for n 2⊗ can be written as ≥ n j 1 sj yi ( 1) − R(x , s ) − j=1 j j ti xj − − zi = n sj yi , ( 1)j 1T (x , s ) − Pj=1 − j j ti xj − b b − where xi, si are obtained from x, Ps by omitting the i-th coordinate, and R, T are the cofactors arising from Lemma 8.12 when expandingb theb determinants in the numerator and denominatorb b in the 0-th column. To write explicit formulas for these cofactors, deﬁne the matrix Mk,l(x, s) with k + l = n by the formula j 1 j k 1 M (x, s) = s x − , 1 j k; M (x, s) = x − − , k +1 j n. k,l ij i i ≤ ≤ k,l ij i ≤ ≤ Then for n =2r 1 we have − T (x, s) = det Mr 1,r(x, s), R(x, s) = det Mr,r 1(x, s), − − while for n =2r we have

T (x, s) = det Mr,r(x, s), R(x, s) = det Mr+1,r 1(x, s). − Altogether we obtain the following proposition. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 27

Proposition 3.26. We have

(Hxψ)(y0, ..., ym)= −2 p2(x, s,t0) − q2(x, s,t0)y0 p2(x, s,tm) − q2(x, s,tm)ym ∆(x)T (x, s) ds ψ , ..., m , Z n p1(x, s,t0) − q1(x, s,t1)y0 p1(x, s,tm) − q1(x, s,tm)ym (p1(x, s,ti) − q1(x, s,ti)yi) F i=0

where ∆(x) := (x x ) is the Vandermonde determinant. Q i

(Hx1,x2 ψ)(y0, ..., ym)=

s1 y0 s2 y0 s1 ym s2 ym − s − s − s − s 2 t0 x1 2 t0 x2 1 tm x1 2 tm x2 1 (x1 x2)(s1 s2)− ds1ds2 (3.19) ψ − − − , ..., − − − . s1 y0 s2 y0 s1 ym s2 ym m − s1x2 −s2x1 x1 x2 2 − − − − (ti − − yi) F t0 x1 t0 x2 tm x1 tm x2 ! i=1 s1 s2 s1 s2 Z − − − − − − − − − −

Remark 3.28. Formula (3.27) (as well as its analogs for Qn > 2) may be checked by direct composition of 1-point Hecke operators. Namely, in the case n = 2, if s is the integration variable associated to x2 and s′ to x1 then the composition formula for Hx1 Hx2 turns into x2 x1 the symmetric formula (3.27) by the change of variable s = s ,s′ = − . Thus the change 2 s1 s2 − of variable in the composition formula needed to exhibit the commutativity of Hx1 and Hx2 is

x2 x1 ((s′, x ), (s, x )) ((s + −′ , x ), (s′, x )). 1 2 7→ s 1 2 Proposition 3.26 is useful for generalizing the theory of Hecke operators to the case when the point x F is not defined over F . In this case, we cannot define the Hecke operator ∈ Hx attached to x but can define the Hecke operator attached to the Galois orbit x := Γx = x , ..., x , where Γ := Gal(F /F ); it is an analog of H ...H but the individual { 1 n} x1 xn factors in this product are now not defined. Let Γi Γ be the stabilizers of xi, and let Γi ⊂ n Ki := F = F (xi). Let K be the set of s =(s1, ..., sn) F such that si Ki and gsi = sj whenever gx = x , g Γ. Note that K is a local field,∈ and we have a natural∈ identification i j ∈ K ∼= Ki sending s to si. Now we can define the modified Hecke operator Hx : V V attached to x by the same formula as in Proposition 3.26 but with integration over K:→ e (Hxψ)(y0, ..., ym)=

−2 p2(x, s,t0) − q2(x, s,t0)y0 p2(x, s,tm) − q2(x, s,tm)ym ∆(x)T (x, s) ds ψ , ..., m , ZK p1(x, s,t0) − q1(x, s,t0)y0 p1(x, s,tm) − q1(x, s,tm)ym (p1(x, s,ti) − q1(x, s,ti)yi) i=0

Q 18 This case n = 1 is somewhat degenerate since q2 = 0,so f2 is not well deﬁned. Nevertheless, computation of the coeﬃcients of pk, qk using the corresponding system of linear equations produces the given answer. 28 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

and deﬁne the usual Hecke operator by the formula

1 2

Hx = (ti sj) Hx − i,j Y

(note that pk(x, s, ti), qk(x, s, ti) F because of symmetry). In fact, using this formula, we can define the Hecke operator corresponding∈ to any effective divisor x on P1 defined over F and not containing the parabolic points; namely, we write x as the sum of Galois orbits and then take the product of the corresponding operators Hx.

Theorem 3.29. The operators Hx extend to commuting self-adjoint compact operators on . H Proof. The proof is analogous to the proof of Theorem 3.13, using the representation theory of P GL2(K). Example 3.30. Let F = R and x = (x, x) where x C. Let s = (s, s). Then the formula of Example 3.27(2) yields ∈

(Hx,xψ)(y0, ..., ym)=

m 1 1 Im((s y )(t x)s) Im((s y )(t x)s) Im(x)Im(s) − dsds ψ − 0 0 − , ..., − m m − | | . 8π Im((s y )(t x)) Im((s y )(t x)) m Im((s y )(t x)) ZC − 0 0 − − m m − i=0 | − j j − | where dsds is the ordinary Lebesgue measure on C. Thus, introducingQ real coordinates by s := u + iv, x := a + ib, we get

(Hx,xψ)(y0, ..., ym)=

2 2 2 2 m−1 1 (v(t0 − a) − ub)y0 +(u + v )b (v(tm − a) − ub)ym +(u + v )b |bv |dudv ψ , ..., m . 2 8π ZR by0 + v(t0 − a) − ub bym + v(tm − a) − ub i=0 |byi + v(ti − a) − ub| Q

4. Genus zero, the archimedian case In this section we will consider the archimedian case, i.e., we assume that F = R or 19 F = C. The modiﬁed Hecke operators Hx are given by the formula 1 t x t x ds (H ψ)(y , ..., y )= ψ 0 − , ..., m − x 0 m 2 s y s y m s y ZR − 0 − m i=0 | − i| for F = R and Q 1 t x t x dsds (H ψ)(y , ..., y )= ψ 0 − , ..., m − x 0 m π s y s y m s y 2 ZC − 0 − m i=0 | − i| for F = C. Q

19 For simplicity for F = R we restrict to the case ti R, although the results can be generalized to the case when there are complex conjugate pairs of parabolic∈ points. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 29

4.1. The Gaudin system. Definition 4.1. The Gaudin operators are second order differential operators in the variables y0, ..., ym defined by the formula

1 2 1 Gi := (yi yj) ∂i∂j +(yi yj)(∂i ∂j)+ . ti tj − − − − 2 0 j m,j=i ≤ X≤ 6 − It is easy to see that

Gi =0, Gi∗ = Gi i X (i.e., Gi are algebraically symmetric), and on translation invariant functions of homogeneity degree m one has − 2 m t G = . i i 4 i X It is well known and easy to check that [Gi,Gj] = 0. Therefore the operators Gi form a quantum integrable system. Namely, it is a the quantum Hitchin system for G = P GL2 1 20 m+1 for X = P with parabolic points. Hence for every µ = (µ0, ..., µm) C such that m ∈ i µi = 0 and i tiµi = 4 , we have the holonomic system of differential equations G ψ = µ ψ, i =1, ..., m 1 P P i i − on a translation-invariant function ψ(y0, ..., ym) (under simultaneous translation of all variables) of homogeneity degree m , which is called the Gaudin system. As explained in − 2 Subsection 2.4, the Gaudin system defines an -coherent twisted D-module21 M(µ) on the vs O m 1 open subset Bun Bun◦ of very stable bundles, of rank 2 − . 0 ⊂ 0 Proposition 4.2. The D-module M(µ) is irreducible. Proof. The lemma follows from the explicit identification of M(µ) with the D-module obtained by Drinfeld’s first construction [Dr1] from the symmetric power of the D-module on P1 attached to the corresponding oper. This identification (called the separation of variables transform) is explained in [Fr1], Subsections 6.5 and 6.6, using the results of [Sk]. The irreducibility of the oper D-module on P1 then implies the irreducibility of Drinfeld’s D-module, and hence the irreducibility of the D-module M(µ) corresponding to the Gaudin system.22 4.2. Differential equations for Hecke operators. In this section we show that the Hecke operators Hx satisfy a second order differential equation with respect to x which can be used to describe their spectrum more explicitly. Let 1 Gi := Gi . − 2(ti tj) j=i X6 − 20For more details on the Gaudin operatorsb see [EFK1], Section 7 and references therein. 1 21The twisting here is by the line bundle of half-forms K 2 . We will often regard the Gaudin system as a usual D-module by tensoring it with the dual bundle. 22Note that using the results of Gaitsgory [Ga] on the uniqueness of Hecke eigensheaves one can identify

the quantum Hitchin D-module on BunPGL2 constructed by Beilinson and Drinfeld [BD1], which generalizes the Gaudin D-module, with the D-module obtained by Drinfeld’s ﬁrst construction [Dr1] for an arbitrary curve. 30 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Proposition 4.3. (i) We have 1 G ∂2 + ∂ H H i =0. x x t x x − x x t j 0 j ! i 0 i X≥ − X≥ −b More precisely, let U F m 0 be the open set defined in Subsection 3.3 and ψ be a smooth ⊂ \ m function on U homogeneous of degree 2 whose support modulo dilations is compact. Then the function x H ψ is smooth− for x = t , and we have 7→ x ∈ H 6 i ∞ 1 G ψ ∂2 + ∂ (H ψ) H i =0 x x t x x − x x t j 0 j ! i 0 i X≥ − X≥ b− in . (ii)H In the same sense, we have 1 G ∂2 + H H i =0. x 4(x t )2 x − x x t i 0 i ! i 0 i X≥ − X≥ − Remark 4.4. This is essentially a special case of [EFK2], Theorem 1.15, but here we will give a more elementary proof by direct computation. Proof. Let u = u (s) := y s and ψ , ψ be the first and second derivatives of ψ evaluated i i i − i ij ti x ds at the point z with coordinates z := − . Also let dµ(s) := m . Let η be another i ui Q (s ym) i=0 − smooth function on U homogeneous of degree m with compact support modulo dilations. − 2 Note that j 0 ψj = 0 (as ψ is translation invariant) and j 0 ∂jψ0 = ∂sψ0. Thus we have ≥ ≥ − P P ψ ∂2(η, H ψ)= η, ij dµ(s) , x x u u F i,j 0 i j ! Z X≥ t x tj x t x tj x i − 2 i − H G ψ ( u− u ) ψij +( u− u )(ψi ψj) η, x i = η, − i − j i − j − dµ(s) . x ti (x ti)(ti tj) i 0 ! F i=j ! X≥ −b Z X6 − − Subtracting, we get G ψ ∂2(η, H ψ) η, H i = x x − x x t i 0 i ! X≥ b− 2 2 (ti x) (tj x) t x tj x − − i − ( 2 + 2 )ψij ( u− u )(ψi ψj) ψii − ui uj − i − j − η, 2 dµ(s) =   u − (x ti)(ti tj)   F i i i=j Z X X6 − −   ti x tj x   t x ψ ( − − )(ψi ψj) ( i ) ij ui − uj − η, − 2 dµ(s) . (tj x)u − (x ti)(ti tj) F i,j 0 i i=j ! ! Z X≥ − X6 − − Now, using integration by parts (justified since η, ψ have compact support modulo dilations which is contained in U), we have (t x)ψ 1 η, i − ij dµ(s) = η, ∂ ψ dµ(s) = (t x)u2 x t i j F i,j 0 j i ! F j 0 j i 0 ! Z X≥ − Z X≥ − X≥ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 31

1 1 η, ∂ ψ dµ(s) = η, ψ dµ(s) . t x s j (t x)u j F j 0 j ! F i,j 0 j i ! Z X≥ − Z X≥ − Thus we get G ψ ∂2(η, H ψ) η, H i = x x − x x t i 0 i ! X≥ b− ti x tj x ( − − )(ψ ψ ) 1 1 ui uj i j η, ψj + − − dµ(s) = (tj x)ui 2 (ti x)(tj x) F i,j 0 i=j ! ! Z X≥ − X6 − − 1 1 η, ψi + ψi dµ(s) = (tj x)ui (tj x)ui F i 0 i=j ! ! Z X≥ − X6 − 1 ψ 1 η, i dµ(s) = ∂ (η, H ψ) . t x u x t x x j 0 j F i 0 i ! j 0 j X≥ − Z X≥ X≥ − Thus 1 G ψ ∂2 ∂ (η, H ψ)= η, H i . x − x t x x x x t j 0 j ! i 0 i ! X≥ − X≥ b− which implies that G 1 (4.1) ∂2(η,H ψ)= η,H i ψ . x x x x t − 4(x t )2 i 0 i i ! X≥ − − At the algebraic level, we are now done, as this is the claimed equation for the matrix coefficient (η,Hxψ). Analytically, however, getting rid of η is not automatic, as we are working in an infinite dimensional Hilbert space. To start with, we need to show that Hxψ is twice differentiable. So in order to dispose of η and complete the proof, we will perform a double integration to turn differential equation (4.1) into an integral equation.23 Namely, picking a point x = t for any i, we get (for fixed ψ): 0 6 i x y G 1 (η,H ψ)= η,H i ψ dtdy + c (η)+ c (η)x = x x x t − 4(x t )2 0 1 x0 x0 i 0 i i ! Z Z X≥ − − x y G 1 η, H i ψdtdy + c (η)+ c (η)x. x x t − 4(x t )2 0 1 x0 x0 i 0 i i ! Z Z X≥ − − By Proposition 3.13, the left hand side of this equation and the first summand on the right hand side are continuous in η in the metric of . Therefore, the second and third summands on the right hand side are also continuous. So,H since the possible η are dense in , we have H x y G 1 H ψ = H i ψdtdy + c + c x. x x x t − 4(x t )2 0 1 x0 x0 i 0 i i Z Z X≥ − − 23This helps because integrals are better behaved analytically than derivatives, and is similar to the beginning of the standard proof of Picard’s theorem on the existence of solutions of ODE. 32 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

for some c0,c1 . This implies that Hxψ is twice diﬀerentiable in x. So diﬀerentiating twice, we get ∈ H 1 G ψ ∂2 + (H ψ) H i =0 x 4(x t )2 x − x x t i 0 i ! i 0 i X≥ − X≥ − and 1 G ψ ∂2 + ∂ (H ψ) H i =0, x x t x x − x x t j 0 j ! i 0 i X≥ − X≥ b− as claimed.

Example 4.5. Let m = 1, t0 =0, t1 = 1. In this case, as explained in Example 3.12, is 1 H 2 1-dimensional and Hx = x(x 1) E(x). Thus Hx = E(x) is given by the elliptic integral. k − k 1 1 1 1 Also Gi, i =0, 1 act by the numbers µi such that µ0 = 4 ,µ1 = 4 , so G0 = 4 , G1 = 4 . So the equation of Proposition 4.3 takes the form − − b b 1 1 1 ∂2 + + ∂ + E(x)=0. x x x 1 x 4x(x 1) − − This is the classical Picard-Fuchs equation for the elliptic integral.

Now recall that we have a spectral decomposition = with respect to the action H ⊕kHk of the operators Hx. Recall also that is naturally a subspace of the space of distributions on U. H

Proposition 4.6. Let η k. Then for all i the distribution Giη on U belongs to and equals µ η for some scalar∈ Hµ C. H i,k i,k ∈ Proof. Let η . By Proposition 4.3, ∈ Hk 1 G ψ ∂2 + (H ψ, η) H i , η =0. x 4(x t )2 x − x x t i 0 i ! i 0 i ! X≥ − X≥ −

But η , so H η = β (x)η. Thus, using that H is self-adjoint and G∗ = G , we get ∈ Hk x k x i i 1 1 ∂2 + β (x)(ψ, η) β (x)(ψ, G η)=0. x 4(x t )2 k − x t k i i 0 i ! i 0 i X≥ − X≥ − This holds for all test functions ψ, so we get

G ∂2β (x) 1 i η = x k + η. x t β (x) 4(x t )2 i 0 i k i 0 i ! X≥ − X≥ − Thus, for any x the operator Gi acts on by a scalar. Applying this statement for i 0 x ti k ≥ − 1 H distinct x0, ..., xm and using that det( ) = 0 (the Cauchy determinant), we deduce that P xj ti 6 for all i the operator G acts on the space− by some scalar µ , as claimed. i Hk i,k 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 33

4.3. The Schwartz space. For ψ let ψ be the projection of ψ to . ∈ H k Hk Definition 4.7. Define the Schwartz space to be the space of vectors ψ such S ⊂ H ∈ H that for any i1, ..., ir µ ...µ 2 ψ 2 < . | i1,k ir,k| k kk ∞ Xk Let be the commutative algebra of differential operators on Bun◦(F ) (regarded as a A 0 real analytic manifold) generated by Gi for F = R and by Gi, Gi for F = C. This algebra has a conjugation map given by G† = G (where G = G for F = R). † i i i i Let µk : C be the conjugation-equivariant character defined by µk(Gi)= µi,k. Thus is the spaceA→ of ψ such that for every A one has S ∈A µ (A) 2 ψ 2 < . | k | k kk ∞ Xk Recall that U F m 0 denotes the open set defined in Subsection 3.3. ⊂ \ Definition 4.8. Define = C0∞ to be the space of smooth functions on U of homogeneity degree m with compactV support modulo dilations. − 2 Note that acts on . A V Proposition 4.9. One has . Moreover, for φ , A we have V ⊂ S ∈V ∈A Aφ = µk(A)φk. Xk Proof. Let φ . Then for any A , we have Aφ . Thus we get ∈V ∈A ∈V⊂H Aφ 2 = (Aφ) 2 < . k k k kk ∞ Xk But by Proposition 4.6 for v we have ∈ Hk (v, (Aφ)k)=(v,Aφ)=(Av,φ)= µk(A)(v,φ)= µk(A)(v,φk)=(v, µk(A)φk). Thus (Aφ)k = µk(A)φk. This implies both statements. We can now define a representation of the algebra on by the formula A S Aψ := µk(A)ψk. Xk By Proposition 4.9, this extends to the usual action of on . So from now on let us regard as an algebra of endomorphismsS of . Note that forA eachV v,u , we have A S ∈ S (Av,u)= ((Av)k,uk)= µk(A)(vk,uk)= (vk, (A†u)k)=(v, A†u). Xk Xk Xk Thus the subalgebra R of elements A such that A† = A acts on by symmetric operators. A ∈ A S Recall ([EFK1], Definition 1.17) that S( ) denotes the space of u such that the linear functional v (Av,u) on is continuousA in the metric of for all∈A H . 7→ S H ∈A Proposition 4.10. One has S( )= . A S 34 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Proof. If u then (Av,u)=(v, A†u), so it is continuous in v, hence u S( ). Conversely, suppose u ∈S S( ). Then (Av,u)=(v,w) for some w . So we have ∈ A ∈ A ∈ H

(vk,wk)=(v,w)=(Av,u)= ((Av)k,uk)= µk(A)(vk,uk). Xk Xk Xk Taking v = v , we have w = µ (A)u for all k. Thus k ∈ Hk ⊂ S k k k µ (A) 2 u 2 = w 2 < , | k | k kk k kk ∞ Xk Xk i.e., u . ∈ S

Recall ([EFK1], Deﬁnition 11.8) that the algebra R is said to be essentially self-adjoint A on if every A R is essentially self-adjoint on S( ). S ∈A A

Proposition 4.11. The algebra R is essentially self-adjoint on . In particular, Gi, Gi are unbounded normal operators on A (self-adjoint for F = R) whichS (strongly) commute with H each other and with the Hecke operators Hx (see [EFK1], Subsection 11.2). Proof. This follows immediately from the fact that S( )= contains for all k. A S Hk Proposition 4.11 immediately yields

Corollary 4.12. For every φ , x = (x , ..., x ) (with x = t , ) and 0 i m, the ∈ V 1 n i 6 j ∞ ≤ ≤ distribution G Hxφ belongs to and we have G Hxφ = HxG φ. i H i i We also obtain

Corollary 4.13. For ψ , the map x Hxψ is twice diﬀerentiable in x as a function with values in distributions∈ H on U for x = t7→, , and we have 6 i ∞ 1 G ∂2 + (H ψ)= i H ψ x 4(x t )2 x x t x i 0 i ! i 0 i X≥ − X≥ − as distributions on U.

Proof. For a test function φ , we have (Hxψ,φ)=(ψ,Hxφ), which is twice diﬀerentiable in x by Proposition 4.3. Thus∈V again using Proposition 4.3,

1 1 ∂2 + (H ψ),φ = ∂2 + (H ψ,φ)= x 4(x t )2 x x 4(x t )2 x i 0 i ! ! i 0 i ! X≥ − X≥ −

1 G † G ∂2 + (ψ,H φ)= ψ, H i φ = i H ψ,φ , x 4(x t )2 x  x x t  x t x i 0 i ! i i ! i i ! X≥ − X − X −   as claimed. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 35

4.4. The diﬀerential equation for eigenvalues. Proposition 4.3 implies

Corollary 4.14. The function βk(x) satisﬁes the diﬀerential equation

(4.2) L(µk)βk(x)=0, where 1 µ L(µ) := ∂2 + i,k x 4(x t )2 − x t i 0 i i 0 i ≥ − ≥ − X m X is an SL2-oper (where i µi =0, i tiµi = 4 ). 1 1 Note that equation (4.2)P is FuchsianP at the points ti with characteristic exponents 2 , 2 and, since m µ =0, t µ = , i,k i i,k 4 i i X X 1 1 it is also Fuchsian at with characteristic exponents 2 , 2 . In other words, basic solutions 1 ∞ 1 − − 1 1 24 behave as (x ti) 2 and (x ti) 2 log(x ti) near ti and as x 2 , x 2 log x at . Thus the − − − 1 1 ∞ monodromy operators of (4.2) at ti and are conjugate to − . ∞ 0 1 − m 1 4.5. Spectral decomposition in the complex case. Let F = C and C − be the set of points µ such that the oper L(µ), when viewed as a rank 2 local system,R ⊂ has a real monodromy representation (i.e., landing in SL2(R) up to conjugation). It is known that is discrete (see [Fa]). R Theorem 4.15. (i) The Hecke operators Hx have a simple joint spectrum Σ on . (ii) For any µ there is a unique up to scaling single-valued real analytic hHalf-density ∈ R ψµ, deﬁned outside the wobbly divisor D in Bun0◦(C) (see Subsectioin 2.4), such that

(4.3) Giψµ = µiψµ, Giψµ = µiψµ. . ֒ iii) There is a natural inclusion Σ) → R (iv) Let ψ be such that Giψ = µiψ as distributions on U for some µi C. Then H ψ = β(x)ψ∈for H some function β(x). Thus ψ for some k and β(x)= β (∈x). x ∈ Hk k (v) µ belongs to Σ (i.e., ψµ is an eigenfunction) if and only if ψµ belongs to the Schwartz∈ space R . S Remark 4.16. 1. We expect that the condition in (v) always holds, so Σ = (this is Conjecture 1.8(2) in [EFK2]). We will see below that this is true in the case of fourR and five points. 2. See Footnote 1 on the difference in normalization of eigenvalues of the Hecke operators used in the present paper and in Conjecture 1.11 of [EFK2]. Proof. (i),(ii),(iii) Consider a joint eigenspace of H . By Proposition 4.6, the operators Hk x Gi act on this space by some scalars µi,k, and by Corollary 4.14 the eigenvalue βk(x) of Hx on satisfies the differential equations Hk (4.4) L(µk)βk =0, L(µk)βk =0.

1 3 24We remind the reader that since the oper L(µ) is a map from K− 2 to K 2 , solutions of the equation 1 1 L(µ)β = 0 are sections of K− 2 , but we view them as functions by multiplying by (dx)− 2 . 36 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Since these equations have a non-zero single-valued solution βk(x), the oper L(µk) has a real monodromy representation, i.e., µk (see e.g. [EFK1], Subsection 3.4, [EFK2], Subsection 5.1). Furthermore, since an oper∈ Rconnection is always irreducible ([BD1, BD3]), this single-valued solution is unique up to scaling. Hence by Corollary 3.19 it is uniquely determined by µk (namely, its asymptotics at ﬁxes the scaling). This gives an inclusion ∞ . ֒ Σ → R Also every element ψ k is a single-valued real analytic solution of the holonomic system (4.3). By Proposition 4.2,∈ H the holomorphic part of this system is an irreducible D-module. This implies that ψ is unique up to scaling, i.e., ψ = ψµ , and dim k = 1. k H (iv) For N = 4 this is shown in the next section, so assume that N 5. In this case let ≥vs 0 be the subspace of functions φ supported on the open set Bun (F ) U of very stableV ⊂ V bundles (i.e., ones that have no nonzero nilpotent Higgs ﬁeld, see Subsection⊂ 2.4).Then by Corollary 2.10, Hxφ . By Corollary 4.12, GiHxφ = HxGiφ as distributions, hence as functions on U (as they∈V are both elements of ). Thus for φ we have V ∈V0 (GiHxψ, φ)=(Hxψ, Giφ)=(ψ, HxGiφ)=(ψ, GiHxφ), Moreover, since H φ , we have x ∈V (ψ, GiHxφ)=(Giψ,Hxφ)= µi(ψ,Hxφ)=(µiHxψ,φ). Thus GiHxψ = µiHxψ vs as distributions on Bun . Since the monodromy representation of the system Giψ = µiψ is irreducible, there is at most one single-valued solution up to scaling. Therefore generically on U we have Hxψ = β(x)ψ, as claimed. (v) It is clear that ψµ . Conversely, if ψµ for some µ then by (iv) we k ∈S⊂H ∈ H ∈ R have ψµ k for some k. ∈ H Theorem 4.15 implies the validity of the main conjectures of [EFK1] and [EFK2] (namely Conjectures 1.4, 1.9, 1.10 and 1.11 of [EFK1] and Conjectures 1.5, 1.11 of [EFK2]) for 25 G = P GL2 and curves of genus zero. Remark 4.17. Let Y be the (2-dimensional) space of solutions the oper equation

L(µk)f =0 1 near some point x0 CP , x0 = t0, ..., tm+1. Then Y carries a monodromy-invariant sym- plectic form sending∈ (f,g) to the6 Wronskian W (f,g) (which is a constant function since L(µk) has no ﬁrst derivative term). It also carries a monodromy-invariant pseudo-Hermitian inner product (f,g) B(f,g), since the monodromy of the corresponding local system R 7→ is in SL2( ) ∼= SU(1, 1). Let f0, f1 be a basis of Y with B(f0, f0) = B(f1, f1) = 0 and B(f0, f1) = 1. Assume that B is normalized so that W (f0, f1) = 1 (there are two such normalizations diﬀering by sign). | | Then by Corollary 4.14 the corresponding eigenvalue βk has the form β (x, x)= f (x)f (x)+ f (x)f (x). ± k 0 1 1 0 Similarly, if B(f , f ) = 1, B(f , f )= 1 and B(f , f ) = 0, then we have 0 0 1 1 − 0 1 β (x, x)= f (x) 2 f (x) 2. ± k | 0 | −| 1 | 25 Note that the normalization of the eigenvalues βk(x) is uniquely determined by Theorem 3.17(ii). This way of normalization is possible due to presence of parabolic points. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 37

These are special cases of the formula in [EFK2], Conjecture 1.11. Note that we can ﬁx the sign by imposing the condition that β (x, x) is positive near . k ∞ 4.6. The leading eigenvalue of the Hecke operator. The positive (leading) eigenvalue β0(x) of Hx has a special meaning.

Proposition 4.18. β0(x) is a single-valued solution of the system 4.4 corresponding to the 1 analytic uniformization of the punctured Riemann surface X◦ := CP t0, ..., tm+1 (see [Fa, Go, Ta]). \{ } Proof. As explained, e.g., in [Ta], the only oper with real monodromy and positive single- valued solution β(x) of the system (4.4) is the uniformization oper, and in this case the complete hyperbolic conformal metric of constant negative curvature on X◦ is given by the formula 2 2 ds = β(x)− (which is well deﬁned as a conformal metric independently of the choice of the coordinate 1 x since β(x) is naturally a -density on X◦). Since the leading eigenvalue β (x) of H is − 2 0 x positive, it must correspond to the uniformization oper, i.e., β0(x)= β(x). 1 In more detail, let J : C+ CP t0, ..., tm+1 be an analytic unformization map of the Riemann surface CP1 t→, ..., t \{ by the upper} half-plane (recall that it is unique \{ 0 m+1} up to the action of P SL2(R) on C+). Then we have the multivalued holomorphic function 1 1 1 K(x) := J − (x) on CP t , ..., t . Deﬁne the multivalued holomorphic -densities \{ 0 m+1} − 2 πi πi e 4 1 e− 4 K(x) 1 f0(x) := (dx)− 2 , f1(x) := (dx)− 2 . K′(x) K′(x)

One can show that they formp a basis of solutions of the operp equation L(µ0)β = 0 (near some point x CP1 t , ..., t ) with Wronskian 1 in which the monodromy of this equation 0 ∈ \{ 0 m+1} is in SU(1, 1), the group of symmetries of the pseudo-Hermitian form Re(z1z2) (namely, it is the corresponding Fuchsian group Γ SL2(R) ∼= SU(1, 1)), and the action of SL2(R) on J is simply transitive on bases of solutions⊂ with this property.26 Thus the real analytic 1 -density − 2 2ImK(x) 1 (4.5) β(x, x) := f (x)f (x)+ f (x)f (x)= (dxdx)− 2 0 1 1 0 K (x) | ′ | satisﬁes the system of the oper and anti-oper equations (4.4):

L(µ0)β =0, L(µ0)β =0, and is the only nonzero single-valued solution of these equations up to scaling (it is single- 2ImK(x) ′ R valued because the expression K (x) is invariant under the action of SL(2, ) given by aK+b | | 2 K cK+d ). It is manifestly positive, and the hyperbolic metric is expressed as β− , as explained7→ above.

Thus we obtain an explicit formula (4.5) for the leading eigenvalues β0(x) = β(x) of the Hecke operators Hx in terms of the uniformization map. Other eigenvalues βk(x) can also be written in the form (4.5) but with K(x) now taking values in the complex plane (rather than the upper half-plane). For this reason, βk(x) vanishes on the union of ﬁnitely many analytic

26 Here the group PSL2(R) gets replaced by its double cover SL2(R) due to the ambiguity of the square root in the deﬁnition of fi(x). 38 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

1 2 contours in CP t0, ..., tm+1 . Thus the corresponding metric β− has constant negative curvature away from\{ these contours} and the singularities at the contours which locally look dx2+dy2 like the Poincar´emetric y2 near the boundary y = 0 of the upper half-plane (see [Go] and Section 4 of [Ta]).

4.7. Spectral decomposition in the real case and balanced local systems. Let F = R and t0 < t1 < ... < tm. In this case the oper equation L(µ)β = 0 for the eigenvalue β(x) of the Hecke operator Hx derived in Subsection 4.4 is a second order linear differential equation 1 1 on the circle RP = S with regular singularities at t0, ..., tm, tm+1 = . To characterize the spectrum of Hecke operators in the real case,∞ we need a replacement for the reality condition for opers used in the complex case. To this end, even though we are now working over R, we will need to consider oper connections in the complex domain. This agrees with the general principle that on the spectral side of the Langlands correspondence one should always consider a complex Lie group, regardless of the field F . Let Locm be the variety of irreducible rank 2 local systems (i.e., locally constant sheaves) 1 1 1 on CP t0, ..., tm+1 with monodromies around ti conjugate to − . By Lemma \{ } 0 1 − 8.10, this is an irreducible smooth variety of dimension 2(m 1) (and for odd m any local system with such monodromies is automatically irreducible).− + Let Loc (C). Let V be the space of sections of on (t , t + ε), and V − be the ∇ ∈ m j ∇ j j j space of sections of on (tj, tj ε) for small ε > 0 (where we use the addition law of the ∇ − + circle). We have an isomorphism ξj : Vj ∼= Vj− which assigns to a section f the section i i ξj(f) := (f f+) when j = m + 1 and ξj(f) := (f f+) for j = m + 1, where 2 − − 6 − 2 − − f+, f are the continuations of f above and below the real axis, respectively. Using these − + 27 isomorphisms, we identify Vj− with Vj and denote the resulting space just by Vj. For j Z/(m + 2) we denote by ∈ + Bj : Vj = Vj− Vj 1 = Vj 1 ∼ → − ∼ − the operator of continuation of sections along the real axis, and define

B := B0...Bm+1.

Definition 4.19. Let us say that is nondegenerate if there exists nonzero f Vm+1 which is not an eigenvector of the monodromy∇ of around t = but Bf = λf for∈ some ∇ m+1 ∞ λ C×. ∈ It is clear that for a generic the operator B is regular semisimple, so is nondegenerate ∇ ∇ and there are two choices of f corresponding to two different eigenvalues λ1,λ2 of B, up to scaling. However, if B is a Jordan block (for nondegenerate ) then there is only one choice of f up to scaling, while for scalar B such choices form an affine∇ line. Now let be nondegenerate. Choose an eigenvector f as in Definition 4.19 and let ∇ f := B ...B f, g := f + if , j = m +1, g := f + if . j j+1 m+1 j j − j 6 m+1 − m+1 − m+1 Definition 4.20. Let us say that the pair ( , f) is nondegenerate if for all j [0, m + 1], ∇ ∈ the vector fj is not an eigenvector of the monodromy around tj.

27The diﬀerent sign for j = m + 1 is chosen because for opers we identify half-densities with functions 1/2 using the half-density dx which has a singularity at t +1 = . | | m ∞ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 39

In this case fj,gj form a basis of Vj. Again, it is clear that for generic any pair ( , f) is nondegenerate. ∇ ∇ Let ( , f) be nondegenerate. Then in the bases fj,gj, the half-monodromy around tj in ∇ 1 the negative direction above and below the real axis is given by the matrices J, J − , where i 0 1 J := , except j = m + 1 when we get J, J − . 1 i − − Moreover, we have

1 bj λ b0 (4.6) Bj = , j = 0; B0 = . 0 aj 6 0 a0 − − It is clear that these matrices don’t change under rescaling of f. Finally, we have two matrix equations m+1 m+1 1 (4.7) B J = 1, B J − = 1 j − j − j=0 j=0 Y Y (the monodromy around R iε is trivial). Taking the determinant, this gives ± m+1 1 (4.8) λ = aj− , j=0 Y Conversely, a collection of matrices B = (Bi) satisfying (4.6), (4.7), (4.8) defines a local 1 system on CP1 t , ..., t as well as a vector f = . ∇B \{ 0 m+1} B 0 Now let Loc be the variety of nondegenerate pairs ( , f) (up to scaling of f) where m ∇ Locm. We have seen that the map π : Locm Locm given by ( , f) has degree 2, ∇with ∈ at mostd 1-dimensional fibers which occur in→ codimension 2,∇ for scalar7→ ∇B. Since Loc ≥ m is irreducible, it follows that so is Locm. d Proposition 4.21. Let Y be the variety of m +2-tuples of matrices (B , ..., B ) of the m d 0 m+2 form (4.6) which satisfy the equations (4.7),(4.8) such that the local system is irreducible. ∇B Then the assignment ( , f) B = (B0, ..., Bm+2) is an isomorphism η : Locm Ym. In particular, Y is an irreducible∇ 7→ variety of dimension 2(m 1). → m − 1 d Proof. The inverse to η is given by η− (B)=( , f ). ∇B B Definition 4.22. Let us say that ( , f) Locm is a balanced pair if aj = 1 for all j (hence λ = 1). In this case f is said to∇ be a∈balancing of . ∇ d In other words, ( , f) is a balanced pair if ∇ 1 b B = j . j 0 1 − Lemma 4.23. If ( , f) is a balanced pair then the two equations in (4.7) are equivalent to each other. ∇ Proof. We have

1 bj i ibj (4.9) B J ± = j ±1 ± i − ∓ 40 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

1 2i Thus, conjugating B J by S := , we get j 0 1 1 bj i ibj 1 SB JS− = = B J − , j −1 −i j − which implies the statement.

Remark 4.24. Note that if aj = 1 and bj = 0 for all j then equations (4.7) are not satisfied. Thus every collection B = (B0, ..., Bm+1) satisfying the equations for a balanced pair automatically defines an irreducible local system . ∇B A description of the spectrum of the Hecke operators in the real case results from the following proposition. Proposition 4.25. The local system of solutions of the oper equation L(µ )β =0 belongs ∇ k to Locm(C), admits a balancing by f = βk (the eigenvalue of the Hecke operators Hx) and has b R. j ∈ 1 1 Proof. First, since is irreducible and has monordromies around ti conjugate to − , ∇ 0 1 − it belongs to Locm(C). Now fix k and let fj± be the restrictions of the eigenvalue βk to the positive and negative + part of a neighborhood of tj. By Propositions 3.15 and 3.21, we have ξj(fj ) = fj−, so fj± give rise to a well defined vector fj Vj. We also have Bjfj = fj 1 for all j, so λ = 1. Also by Proposition 3.15, ∈ − 1 1 f (x) x t 2 log x t , x t , j = m +1, f (x) x 2 log x , x , j ∼ ∓| − j| | − j| → j 6 m+1 ∼| | | | →∞ so setting g := f + if , j = m +1, g = f + if j j − j 6 m+1 − m+1 − m+1 as above, we get 1 1 g (x) π x t 2 , x t , j = m + 1; g (x) π x 2 , x . j ∼ ± | − j| → j 6 m+1 ∼ | | →∞ This implies that the Wronskian W (f (x),g (x)) equals π on the right of t and π on the j j − j left of tj, which implies that det Bj = 1. Thus aj = 1. Finally, since the functions f ,g are− real-valued, we have b R. j j i ∈ Now let be the set of balanced opers, i.e., balanced pairs ( , f) such that = L is the localB system of solutions of an oper L with real coefficients. We∇ have a surjective∇ map∇ m 1 R − sending ( (µ), f) to µ. B → B∗ ⊂ ∇ Proposition 4.26. Every point of has at most two preimages in . B∗ B Proof. If ( , f) is a preimage of L and β(x) the corresponding function on the ∇ ∈ B ∈ B∗ circle then the leading coefficients of the asymptotics of β(x) as x ti are 1, while the leading coefficient at is 1. So if ( i, f i) , i = 1, 2, 3 are three→ distinct± preimages of L and βi are the corresponding∞ functions∇ on∈ the B circle (thus also distinct) then the leading coefficients for β1 β2, β1 β3 are 0 or 2. But since L has order 2, these functions must be proportional (as− they both− have leading± coefficient 0 at ). So, since β2 = β3, we must have ∞ 6 β1 β2 = β2 β3 − − 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 41

as well as all transformations of this equation under permutations of indices. This contradicts the fact that βi span a 2-dimensional space. Remark 4.27. The map is bijective if m is odd. Indeed, in this case det(B)= 1, so B has eigenvalues 1, 1B and → Bf∗ is uniquely determined by if exists. However, for even− m in a non-generic situation− (when B = 1), a ﬁber of the map∇ can consist of two points, see e.g. Subsection 4.10. B → B∗ m 1 Remark 4.28. We expect that is a discrete subset of R − . B∗ Theorem 4.29. The joint spectrum of the Hecke operators H on is a subset of . x H B Proof. This follows from Proposition 4.25. 4.8. The variety of balanced pairs and T -systems. Theorem 4.29 shows that it is m+2 interesting to consider the aﬃne scheme Xm which is cut out inside A with coordinates b0, ..., bm+1 by either of the two equations (4.7) (i.e., we assume that aj = 1 for all j). To describe this scheme, introduce the polynomial Pr(b1, ..., br) which is the lower left corner entry of the matrix B0JB1J...BrJ (it is easy to see that it does not depend on b0). For instance, − P =1, P (b )= b , P (b , b )= b b 1, P (b , b , b )= b b b b b , 0 1 1 1 2 1 2 1 2 − 3 1 2 3 1 2 3 − 1 − 3 and so on. It is easy to see that these polynomials are determined from the recursion

(4.10) Pr(b1, ..., br)= Pr 1(b1, ..., br 1)br Pr 2(b1, ..., br 2). − − − − − This shows, in particular, that Pr have real (in fact, integer) coeﬃcients, even though the matrix J is not real. m Proposition 4.30. (i) The scheme Xm is the irreducible rational hypersurface in A deﬁned by the equation

(4.11) Pm(b1, ..., bm)=1.

The values of b0 and bm+1 are recovered by the formulas

(4.12) b0 = Pm 1(b2, ..., bm), bm+1 = Pm 1(b1, ..., bm 1). − − − (ii) Xm is smooth. (iii) Xm has a stratification 1 2 Xm = Um 1 Um 3 A Um 5 A ... − ⊔ − × ⊔ − × where U is the open subset of Ar defined by the condition P (b , ..., b ) =0. r r 1 r 6 (iv) Xm(R) is a smooth connected real manifold of dimension m 1 with stratification as in (iii). − Proof. (i) Given (b , ..., b ) X , equation (4.11) is obtained from the identity 0 m+1 ∈ m 1 B JB J...B J =(B J)− − 0 1 m m+1 by comparing left lower corner entries. Equations (4.12) are obtained similarly from the identities 1 B J...B J =(B JB J)− − 1 m m+1 0 and 1 B0J...Bm 1J =(BmJBm+1J)− . − − 42 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Equation (4.10) implies that the hypersurface Pm = 1 is reduced and irreducible. Also it is rational since we can solve the equation Pm = 1 for bm. This implies (i), since it is clear that dim Xm = m 1. (ii) The proof− is by induction in m. The base is trivial. To make the induction step, note that a singular point of Xm would be a solution of the equations Pm(b1, ..., bm) = 1, dPm(b1, ..., bm) = 0. So using (4.10) we have

∂Pm 0= (b1, ..., bm)= Pm 1(b1, ..., bm 1), ∂bm − −

hence Pm 2(b1, ..., bm 2) = 1. Also − − ∂Pm(b1, ..., bm) ∂Pm 1(b1, ..., bm 1) 0= = − − bm = Pm 2(b1, ..., bm 2)bm = bm. ∂bm 1 ∂bm 1 − − − − Thus b =0. So for i m 2 we have m ≤ − ∂Pm(b1, ..., bm) ∂Pm 2(b1, ..., bm 2) = − − =0. ∂bi ∂bi

So (b1, ..., bm 2) is a singular point of Xm 2. But by the induction assumption there are no such points.− This is a contradiction which− completes the induction step. 1 (iii) Equation (4.10) implies that we have a decomposition Xm ∼= Um 1 Xm 2 A , and the result follows by iteration. − ⊔ − × (iv) Follows from (i),(ii),(iii).

Remark 4.31. Suppose all the bj are equal: bj = b. Then equation (4.10) takes the form

(4.13) Pr(b)= bPr 1(b) Pr 2(b), − − − with P0 = 1 and P1 = b. Hence Pr is the Chebyshev polynomial of the second kind encoding sin(r+1)x SL2-characters, i.e., Pr(2 cos x)= sin x . Thus equations (4.11),(4.12) look like

Pm 1(b)= b, Pm(b)=1. − πk m+2 So Pm+1(b) = 0, i.e. b = 2 cos m+2 for 1 k m + 1, k odd (so we have [ 2 ] solutions). ≤ ≤ π Note that the solution for k = 1, i.e., b = 2 cos m+2 , arises from a balanced oper for the leading eigenvalue of Hecke operator for the Z/(m + 2)-invariant conﬁguration of points, see Subsection 4.10.

One can interpret Xm as the space of solutions of the T -system of type A1 of level m (also known as Hirota-Miwa equations). Namely, recall ([KNS]) that the T -system is the following system of equations for a function Ti(k) of two integer variables i, k with even i+k:

Ti(k 1)Ti(k +1) = Ti 1(k)Ti+1(k)+1. − − A solution of the T -system of level m is a solution T (k) deﬁned for 0 i m such that i ≤ ≤ T0(k)=1, Tm(k) = 1 for all k. Proposition 4.32. (i) If (b , ..., b ) X (C) then the assignment 0 m+1 ∈ m Ti(k) := Pi(b k−i , ..., b k+i−2 ) 2 2

deﬁnes a solution of the T -system of level m. Moreover, for generic (bj) this solution is nonvanishing. (ii) Any nonvanishing solution of the T -system of level m is of this form for a unique (b , ..., b ) X (C). 0 m+1 ∈ m 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 43

Proof. (i) We prove that Ti(k) satisﬁes the T -system by induction in i. The base case i =1 is easy, so let us perform the induction step from i 1 to i, with i 2. Using (4.10) and the induction assumption, we have − ≥

Pi+1(b0, ..., bi)Pi 1(b1, ..., bi 1)= − − Pi(b0, ..., bi 1)biPi 1(b1, ..., bi 1) Pi 1(b0, ..., bi 2)Pi 1(b1, ..., bi 1)= − − − − − − − − Pi(b0, ..., bi 1)biPi 1(b1, ..., bi 1) Pi(b0, ..., bi 1)Pi 2(b1, ..., bi 2) 1= − − − − − − − − Pi(b0, ..., bi 1)Pi(b1, ..., bi) 1, − − which completes the induction step. The fact that T (k) = 0 for generic (b ) is obvious. i 6 j (ii) Let Ti(k) be a nonvanishing solution of the T -system of level m. The proof of (i) implies by induction in i that

Ti(k) := Pi(b k−i , ..., b k+i−2 ) 2 2

where bi := T1(2i + 1). In particular, this means that

Pm(br, ..., br+m 1)=1 − and Pm+1(br, ..., br+m)=0 for all r. Thus by (4.10), equations (4.12) hold. So (b , ..., b ) X (C), as claimed. 0 m+1 ∈ m Let us say that a solution Ti(k) of the T -system of level m is half-periodic with period m + 2 if Tm i(k + m +2)= Ti(k) − for all i, k (clearly, such a solution is periodic with period 2(m + 2)). Since 1 B0J...Bi 1J =(BiJ...Bm+1J)− , − − the solution Ti(k) of the T -system obtained from a point of Xm is half-periodic with period m + 2. In particular, we see that any nonvanishing solution is half-periodic with period m + 2, which is the well known “Zamolodchikov conjecture” (now a theorem, see [FS, GT]). We also obtain

Corollary 4.33. Xm is the closure of the variety of nonvanishing solutions of the T -system of level m. Example 4.34. 1. If m = 1 then there is only one solution of the T -system of level m, T (k) = 1. This corresponds to the fact that X A1 is a point, defined by the equation i 1 ⊂ b1 = 1. So we have bj = 1 for all j. This reproduces the monodromy representation of the Picard-Fuchs equation, which agrees with Example 4.5.28 2. If m = 2 then the general solution is defined by the formula T2(2j)= b if j is even and T2(2j)=2/b if j is odd, for some nonzero number b. So we have b2r = b and b2r+1 = 2/b. 2 This corresponds to the fact that X2 A is the hyperbola defined by the equation b1b2 = 2. 3. If m = 3 then we have T (2j)⊂ = b for some numbers b , j Z, and T (2j +1) = 2 j j ∈ 3 bjbj+1 1. So half-periodic solutions correspond to collections of numbers bj, j Z/5 such that − ∈ bj+2 = bj 1bj 1. − − The space of such solutions is the surface X A3 defined by the equation b b b b b = 1. 3 ⊂ 1 2 3 − 1 − 3 28Note that this representation is real since it can be conjugated into an index 12 congruence subgroup of SL(2, Z). However, for m> 1 the corresponding representations are not real, in general. 44 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

So if b = b, b = c and bc = 1 then we have 1 2 6 1+ b 1+ c b = b, b = c, b = , b = bc 1, b = . 1 2 3 bc 1 4 − 0 bc 1 − − If bc = 1 then we must have b = c = 1 and we have − b = 1, b = 1, b = d, b =0, b = d 1 1 − 2 − 3 4 0 − − for some number d. In other words, bi form a 5-cycle occurring in the 5-term relation for the dilogarithm, see [Z], Subsection II.2.− 4.9. A geometric description of balanced pairs. Let

Q := SL2(C)/SL2(R)= P SL2(C)/PSL2(R). The following lemma is well known.

Lemma 4.35. Q can be naturally identified with the set H of A SL2(C) such that AA =1, 1 ∈ by T Q T T − . This identification is SL2(C)-equivariant, where SL2(C) acts on H by ∈ 7→ 1 g A := gAg− . ◦ a ib Proof. Let A = SL (C). Then the equation AA = 1 reduces to the equations ic d ∈ 2 d = a, b = b, c = c, ad + bc =1. This yields 2 2 a1 + a2 + bc =1 (where a1 + ia2 = a), which defines a one-sheeted hyperboloid H of signature (3, 1) in 4 the space R with coordinates a1, a2,b,c. The Lorentz group SO(3, 1) = P SL2(C) thus acts transitively on H, and the stabilizer of the point Id = (1, 0, 0, 0) H is SO(2, 1) = P SL2(R). Thus we get an isomorphism ξ : Q = SO(3, 1)/SO(2, 1) H, and∈ it is easy to see that it is 1 → given precisely by ξ(T )= T T − and is equivariant. N 1 N N Now let N 3 and (A0, ..., AN 1) Q . Let Jj := Aj− Aj 1, j Z/N. Let Qirr Q ≥ − ∈ − ∈ ⊂ be the set of points such that the collection of operators Jj is irreducible. Let N := N { } M Qirr/PSL2(C), where P SL2(C) acts diagonally by 1 1 g (A0, ..., AN 1) := (gA0g− , ..., gAN 1g− ). ◦ − − This action is free by Schur’s lemma and the irreducibility of Jj , so N is the real locus of a real smooth algebraic variety of dimension 3(N 2). { } M 0 − Let N N be the real algebraic subset of tuples (A0, ..., AN 1) N such that for M ⊂M − ∈M 1 1 1 each j the matrix Aj− Aj 1 is conjugate to − . It follows from Lemma 8.10 that the − 0 1 0 − closure of N is an irreducible complete intersection inside N cut out by N equations 1 M M Tr(Aj− Aj 1)= 2, so − − dim 0 = 3(N 2) N = 2(N 3). MN − − − 1 Let tj RP , j = 0, ..., N 1 be distinct points occurring in the given order. Let ∈1 − X◦ := CP t0, ..., tN 1 . \{ − } Definition 4.36. A local system on X◦ equivariant under complex conjugation is a local system E on X◦ equipped with an isomorphism g : E E such that g g = 1. → ◦ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 45

Lemma 4.37. The real algebraic set 0 is isomorphic to the moduli space of irreducible MN SL2 local systems on X◦ equivariant under complex conjugation and having monodromy around t conjugate to a Jordan block with eigenvalue 1. j − Proof. To prove the lemma, we will realize local systems (i.e., locally constant sheaves) on X◦ as representations of its fundamental groupoid. Namely, pick a base point p on the upper half-plane, and let γi be the path from p to 0 p passing between the points tj 1 and tj. Then given A = (A0, ..., AN 1) N , we can − − ∈ M define a local system ρ = ρA on X◦ by the formula ρ(γj) = Aj, giving a representation of the fundamental groupoid π1(X◦, p, p). This local system is equivariant under complex 1 conjugation since AjAj = 1. Then for the closed paths δi := γj− γj 1 beginning and ending 1 − at p we have ρ(δi)= Aj− Aj 1, defining a representation of the fundamental group π1(X◦,p). − Conversely, the same formula defines a point A = A 0 from an equivariant local system ρ ∈MN ρ on X◦. 0 1 1 Now given A = (A0, ..., AN 1) N , let Tj := ξ− (Aj), Cj := Tj− 1Tj, and Bj := − ∈ M − ImCj be the imaginary part of Cj. Then the tuple (B0, ..., BN 1) gives a well defined real N 1 − representation V (A)= j=0− Vj(A) of the cyclic quiver with N vertices with dimension vector ⊕ 1 − (2, ..., 2) and Bj : Vj(A) Vj 1(A). Also since the eigenvalues of Cj Cj are 1, we have 1 → − − − 2 Cj Cj = 1+ iEj, where the map Ej : Vj(A) Vj(A) is real and Ej = 0. − N 1 N 1 → Let B = − B , E = − E be endomorphisms of V (A). ⊕j=0 j ⊕j=0 j Proposition 4.38. Balancings of the local system defined by the representation ρ for A A ∈ N correspond to subrepresentations L V (A) with dimension vector (1, ..., 1) which are M ⊂ 0 invariant under the operator BE + EB. Thus elements of N which admit a balancing bal M0 form a connected irreducible real algebraic subset in N N of dimension N 3 (and every local system admits at most two balancings). M ⊂M − Proof. The first statement is proved by a direct calculation. The rest follows from Proposition 4.30. bal Remark 4.39. We see that the natural map XN 2(R) N is a normalization map (which is an isomorphism for odd N but not for even− N).→ For M example, as we have seen, the variety X is isomorphic to A1 0 via b (b, 2/b). One can show that the local system 2 \ 7→ b attached to b determines b uniquely except when b = √2, in which case b and b give ∇ bal 2 − rise to the same local system, and that 4 is the curve in R defined by the equation 2 2 M bal v u(u 2) , a punctured affine nodal cubic. The natural map X2(R) 4 is then the normalization− − map given by b (b2, b(b2 2)). This example will be revisited→M in Subsection 5.6. 7→ − 4.10. Hypergeometric opers. In general, the solutions of the oper equation appearing in the above formula for the eigenvalues of Hecke operators are not expected to be explicitly computable. However, if the configuration of points ti is very symmetric then the leading eigenvalue β0(x) of the Hecke operators Hx may be expressed via the hypergeometric function. Let us describe this situation in more detail. 2πij 1 First consider F = C and tj = e N , where N = m + 2. Then the curve X = P with marked points tj has a symmetry group Γ = Z/N, and the quotient X(C)/Γ is the orbifold CP1 with two orbifold points 0, with stabilizer Z/N and one marked point 1. Thus the ∞ oper corresponding to the leading eigenvalue β0(x) of Hx (i.e., the uniformization oper) 46 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

is invariant under Γ and reduces to a hypergeometric connection on X(C)/Γ = CP1 with singularities at 0, 1, . Explicitly, it is easy to compute that the corresponding oper equation ∞ for β0 has the form N 2 xN 2 ∂2 + − β =0. 4 (xN 1)2 0 − This gives β (x)= x γ(xN ), where γ(y) satisﬁes the equation 0 | | 1 y (4.14) (y∂)2 + γ =0, − 4N 2 4(y 1)2 − which reduces to a hypergeometric equation. Namely, we have N N 2 2 N 2 (4.15) β0(x)= C 1 x F (x ) λ F+(x ) , | − | | − | − | | where 1 1 1 1 1 1 1 1 1 F (y) := F ( , , 1 ; y), F+(y) := y N F ( , + , 1+ ; y) − 2 2 − N − N 2 2 N N are the basic solutions of the Euler hypergeometric equation

y(1 y)F ′′ +(c (a + b + 1)y)F ′ abF =0 − − − with parameters a = 1 , b = 1 1 , c =1 1 . 2 2 − N − N Namely, F is the hypergeometric function (a) (b) F (x)= F (x)= n n xn. 2 1 (c) n! n 0 n X≥ Note that the function (4.15) is real analytic at x = 0 (for any C,λ). It remains to determine the constants C and λ. The constant λ is determined from the condition that the function β0 is single-valued. Namely, using the transformation formula for F from 0 to , a direct calculation yields ∞ Γ 1 + 1 λ = 2 N . Γ 1 1 2 − N Finally, to determine C, consider the asymptotics of β0(x) near x = 1. By deﬁnition, we should have 2 β (x) x 1 log( x 1 − ), x . 0 ∼| − | | − | →∞ Using again the transformation formulas for F (this time from 0 to 1), from this we obtain after a calculation: Γ 1 1 Γ 1+ 1 C = 2 − N N . Γ 1 + 1 Γ 1 1 2 N − N Thus, we obtain Proposition 4.40. For F = C we have

β0(x)=

N 1 1 2 1 1 1 1 N 2 1 1 2 1 1 1 1 N 2 2 1 x Γ 2 N F ( 2 , 2 N , 1 N ; x ) Γ 2 + N xF ( 2 , 2 + N , 1+ N ; x ) | − | − | − − | − | | . 1 1 2π Γ( N )Γ(1 + N ) sinN 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 47

2πij Remark 4.41. Similar analysis can be carried out for the conﬁgurations t0 = 0, tj = e m+1 , 2πij 1 j m + 1 with Z/(m + 1)-symmetry and t0 = 0, tj = e m , 1 j m, tm+1 = with Z/m≤ -symmetry,≤ giving a hypergeometric formula for the leading eigenva≤ ≤lue. ∞

1 1 Now consider the case F = R. Fix the real structure on P given by x∗ = x− . Then the real locus is the unit circle x = 1 (with upper half plane x < 1) and the same conﬁguration | | | | of the points tj has a symmetry group Γ = Z/N. It is clear that the function β0 is invariant π π under this group, so we can regard it as a function of the angle θ ( 2 , 2 ) such that 2i (θ+ π ) ∈ − x = e N 2 . So changing variables (remembering that the eigenvalue is a 1/2-form) we − ﬁnd that β0(θ) is an even solution of the equation 1 1 ∂2 + + β =0, θ N 2 4 cos2 θ 0 which is equation (4.14) with y = e2iθ. So we get − iθ( 1 1 ) 1 1 1 1 2iθ β (θ)= C√2 cos θ Re e 2 − N F ( , , 1 ; e ) . 0 · 2 2 − N − N − The constant C can be found by looking at the asymptotics at θ = π/2. This yields ± π Γ( 1 1 ) C = 2 2 − N . Γ(1 1 ) cos π ( 1 1 ) −pN 2 2 − N Thus we get

Proposition 4.42. The leading eigenvalue β0 of the Hecke operators for F = R in the Z/N-symmetric case is given by the formula 1 1 √πΓ( 2 N ) iθ( 1 1 ) 1 1 1 1 2iθ β (θ)= − √cos θ Re e 2 − N F ( , , 1 ; e ) 0 Γ(1 1 ) cos π ( 1 1 ) · 2 2 − N − N − − N 2 2 − N extended periodically. Moreover, if N is even then the same oper admits another balancing 1 1 √πΓ( 2 N ) iθ( 1 1 ) 1 1 1 1 2iθ β (θ)= − √cos θ Im e 2 − N F ( , , 1 ; e ) 1 Γ(1 1 ) cos π ( 1 + 1 ) · 2 2 − N − N − − N 2 2 N extended antiperiodically, so we have two eigenvalues of Hecke operators corresponding to the same oper.

Now consider the difference h := β0 β1, which is a solution of the oper equation on π π − ( 2 , 2 ) regular at θ = π/2 (defined for both even and odd N). Then analytic continuation from− π/2 to π/2 gives± Bβ = β , Bβ = β , so Bh = h +2β . Moreover, the half- − 0 0 1 − 1 − 0 monodromy around π/2 is given by Jh = ih, Jβj = iβj + λh for some constant λ. To find λ, let c = cos π ( 1 1 ), and consider the function ± 2 2 ± N 1 1 1 1 √πΓ( 2 N ) iθ( ) 1 1 1 1 2iθ β = c β0 + ic+β1 = − √cos θ e 2 − N F , , 1 ; e . − Γ(1 1 ) · 2 2 − N − N − − N 2 πi Then JBβ = ζ β, where ζ := e 2N , so 2 J(c β0 ic+β1)= ζ (c β0 + ic+β1). − − − Thus 2 i(c β0 ic+β1)+ λ(c ic+)(β0 β1)= ζ (c β0 + ic+β1). − − − − − − 48 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

πi Note that c ic+ = e− 4 ζ. So we get − − πi 2 πi 2 ic + λe− 4 ζ = ζ c , c+ λe− 4 ζ = iζ c+. − − − It is easy to see that these equations are equivalent to each other, and yield

1 2 2 π λ = (ζ + ζ− ) = cos . 2 N B π Thus setting g = λh, f = β0, we get Jf = if + g and g = g +2λf = g + 2 cos N f. − − · π This shows that the numbers bj attached to this balanced oper are all equal to 2 cos N . 5. The case of X = P1 with four parabolic points The goal of this section is to consider in more detail the special case m = 2, i.e., X = P1 with four parabolic points, which we may assume to be t0 = 0, t1 = t, t2 = 1, t3 = . In particular, we will provide more explicit versions and alternative proofs of some of the∞ results of the previous section in this case. 5.1. The moduli space of stable bundles. 1 Proposition 5.1. The variety Bun◦ is isomorphic to P 0, t, 1, . 0 \{ ∞} Proof. The proof is well known but we give it for reader’s convenience. Any bundle E Bun0◦ 1 ∈ has the form E = O(r) O( r) for some r 0, and its parabolic degree is 4 2 = 2, so the parabolic slope is 1.⊕ This− implies that for≥ a stable bundle we must have r ·= 0, since otherwise O(r) would have parabolic degree (hence slope) 1 which is forbidden for stable bundles. ≥ Thus E = O O. So the parabolic structure on E is defined by a choice of four lines 1⊕ y0,y1,y2,y3 P at the four marked points. If a subbundle O O O contains k of these four lines then∈ its parabolic degree (=slope) is k/2, so for stable⊂ bundles⊕ we must have k 1. ≤ This means that all yi are distinct. Such quadruples modulo Möbius transformations are 1 (y0 y1)(y2 y3) parametrized by P 0, 1, , using the cross ratio u = − − . \{ ∞} (y0 y2)(y1 y3) However, the condition that y = y for i = j does not guarantee− − the stability. There are i 6 j 6 many embeddings L ∼= O( 1) ֒ O O as a subbundle and if L contains k of the four lines, its parabolic degree will− be→ 1+⊕k/2. Thus we must have k 3. So we have a single forbidden case k = 4, i.e. the case− when the (distinct!) parabolic≤ lines at the four points are the fibers L. This removes one more point (namely t) from the moduli space and we see 1 that the stable locus is Bun◦ = P 0, t, 1, . 0 \{ ∞} Remark 5.2. By Proposition 2.4 and Remark 2.5(3), the Hecke modification at along ∞ the parabolic line defines a natural isomorphism between Bun0◦ and Bun1◦. It is nevertheless instructive to see directly in this example that the sets of stable bundles of degrees 0 and 1 have exactly the same structure.29 We may realize Bun1◦ as the space of stable rank 2 vector bundles of degree 1. Such a bundle has the form E = O( r) O(r + 1), where r 0. The parabolic degree of such a 1 − ⊕ ≥ bundle is 1 + 4 2 = 3, so the parabolic slope is 3/2. Thus in the stable case we must have r = 0, i.e., E =·O O(1) as a bundle. We realize O(1)⊕ as we did in Subsection 3.1. The conditions of stability are that the (unique) subbundle of E isomorphic to O(1) contains none of the fixed lines (as its slope is

29 P1 It is easy to show that the moduli spaces of semistable bundles Bun0 ∼= Bun1 are isomorphic to , but we will not use these spaces. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 49

1), while any subbundle isomorphic to O contains at most two. The first condition means that the generating vectors of the four fixed lines over the marked points have a nonzero first coordinate, which we may assume to be 1, and encode the vectors by the second coordinate. Now let us see what constraints are imposed by the second condition. It is easy to see that = ((all embeddings O ֒ O O(1) as a subbundle form a single orbit under PAut(O O(1 2 → ⊕ ⊕ Gm ⋉ Ga. Also, it is clear that for any E, some copy of O in O O(1) contains any given two of the four fixed lines. We may therefore assume that the stan⊕dard copy O O O(1) contains the lines y at 0 and y at . Thus y ,y are both generated by the vector⊂ ⊕ (1, 0). 1 4 ∞ 1 4 Now, the line y2 at the point 1 cannot be the same, so after rescaling it is spanned by a vector (1, 1). Finally, the line y3 at the point t cannot lie in the copy of O passing through y ,y , so it is spanned by a vector (1, z) with z = 0. Also it can’t lie in the copy of O through 1 4 6 y1,y2, which yields z = t. Finally, it can’t lie in the copy of O through y2,y4, which yields 6 1 z = 1. Thus, Bun◦ = P 0, t, 1, , i.e., is isomorphic to Bun◦. 6 1 \{ ∞} 0 If we identify Bun0◦ and Bun1◦ using the involution S3 then the involutions Si take the form t t(y 1) y t S (y)= , S (y)= − , S (y)= − , S (y)= y. 0 y 1 y t 2 y 1 3 − − Thus they define an action of the Klein 4-group (Z/2)2, which acts transitively on the singular points 0, t, 1, . Note that this is a special feature of the case of 4 points: for N 5 points ∞ N ≥ we have a faithful action of the group V = (Z/2)0 on Bun0◦, while for N = 4 the action of 4 3 2 V =(Z/2)0 =(Z/2) factors through (Z/2) , as S0S1S2 = 1.

5.2. The Hecke correspondence and Hecke operators. Let Ey,0 denote the bundle of 1 degree 0 corresponding to y P 0, t, 1, and Ez,1 the bundle of degree 1 corresponding to z P1 0, t, 1, . Specializing∈ \{ Proposition∞} 3.2 to the case m = 2, we obtain ∈ \{ ∞} Proposition 5.3. We have HMx,s(Ey,0)= Ez,1, where (1 s)(xy st) z(t,x,y,s)= − − . (x s)(y s) − − This implies Corollary 5.4. The modiﬁed Hecke operator has the form given by the specialization of (3.6): (1 s)(xy st) ds (5.1) (H ψ)(y)= ψ − − . x (x s)(y s) (x s)(y s) ZF − − − − (s x)(t x) It turns out that a more convenient coordinate than s is r := − − . In this coordinate, (s y)(t y) the equation of the Hecke correspondence looks like − − ((x 1)(x t) (y 1)(y t)r)(yr x) z = − − − − − − . r(y x)2 − (see [udB], Theorem 7.3). This gives rise to a quadratic equation for r in terms of t, x, y, z: y(y 1)(y t)r2 (y(x 1)(x t)+ x(y 1)(y t) z(x y)2)r + x(x 1)(x t)=0. − − − − − − − − − − − 2 2 The discriminant of this equation is D := ((x y) rz′(r)) , and − 1 x(x 1)(x t)r− y(y 1)(y t)r rz′(r)= − − − − − . (x y)2 − 50 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Thus 1 2 (y(y 1)(y t)r x(x 1)(x t)r− ) = − − − − − 1 2 (y(y 1)(y t)r + x(x 1)(x t)r− ) 4x(x 1)(x t)y(y 1)(y t)= − − − − − − − − − ((x y)2z y(x 1)(x t) x(y 1)(y t))2 4x(x 1)(x t)y(y 1)(y t)=(x y)2f (x, y, z), − − − − − − − − − − − − − t where f (x, y, z) := (t+xy zx zy)2 4(z t)(z 1)xy =(xy+xz+yz t)2 +4(1+t x y z)xyz t − − − − − − − − − is the polynomial considered by Kontsevich in [K], p.3. We also see that the degree 2 map r z encoding the projection 7→ 1 p : p− (y) Bun◦ 1y 0 ⊂ Hx → 1 x(x 1)(x t) − − branches at the points r = r := y(y 1)(y t) (zeros of D). This means that ± ± − − q 1 1 p1∗y(K) = K O(r+)− O(r )− , ∼ ⊗ ⊗ − 1 1 where K is the canonical bundle on P . But we have an isomorphism O(r+) O(r ) ∼= K− , 2 1/2 ⊗ − so we get p∗ (K)= K . Thus p∗ ( K )= K , the bundle of densities. This means that 1y 1y k k k k integration of p1∗yψ will be well defined as soon as we choose an identification 1 O(r+) O(r ) = K− ⊗ − ∼ up to a phase factor. Such an identification is determined by a 1-form on P1 (with coordinate r) which has simple poles at r = r and no other singularities. Such a form, up to scaling depending on x, y, is dr , and± the correct scaling turns out to be (r r+)(r r−) − − 1 dr dr ω = = 2 . y(y 1)(y t) (r r+)(r r ) y(y 1)(y t)r x(x 1)(x t) − − − − − − − − − − This implies that the modified Hecke operator is given by the formula dr (5.2) (H ψ)(y)=2 ψ(z)θ(f (x, y, z)) , x t r(x y) ZF − where θ(a) = 1 if a F is a square and θ(a) = 0 otherwise, and we view functions on P1(F ) ∈ 1 as half-densities by using the map ψ(z) ψ(z) dz 2 . (Here the factor 2 appears because the map r z has degree 2). 7→ k k Hence we7→ have dz (H ψ)(y)=2 ψ(z)θ(f (x, y, z)) = x t rz (r)(x y) ZF ′ − (x y)dz 2 ψ(z)θ(f (x, y, z)) − . t y(y 1)(y t)r x(x 1)(x t)r 1 ZF − − − − − − Thus we obtain

Proposition 5.5. The modiﬁed Hecke operator Hx is given by the formula

θ(ft(x, y, z)) dz (Hxψ)(y)=2 ψ(z) k k. F f (x, y, z) Z k t k p 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 51

For example, for F = C we have θ = 1 so we just get 2 dzdz (H ψ)(y)= ψ(z) . x π f (x, y, z) ZC | t | Note also that ft(x, y, z) is symmetric in x, z, so the operator Hx is manifestly symmetric.

5.3. Boundedness and compactness of Hx. We already know that the operator Hx is bounded and moreover compact on (Proposition 3.10, Proposition 3.13). In this subsection we provide alternative proofs of theseH facts in the case of four points. We start with boundedness. Let φ be a positive half-density on P1(F ) 0, t, 1, with \{ ∞} logarithmic singularities at 0, 1, t, , i.e. φ = φ(w) dw 1/2 in a local coordinate near each ∞ 1 k k of these points, with φ(w) const log w− as w 0 (note that such a half-density is 2 ∼ · k k → automatically in L ). Note that for a ﬁxed y the function ft(x, y, z) is quadratic in z with simple zeros, which collide into a double zero when y = 0, t, 1, .30 Therefore, Proposition ∞ 5.5 and Lemma 8.1 imply that there exists C > 0 such that (Hxφ)(y) Cφ(y) for all y P1(F ) 0, t, 1, . Thus by Schur’s test ([HS], Theorem 5.2), H is≤ bounded with ∈ \{ ∞} x Hx C. k Thus,k ≤ H is a bounded self-adjoint operator on . x H Now let us establish the compactness of Hx. The compactness would follow from Hx being 1 trace class. At ﬁrst sight this appears possible since the Schwartz kernel of Hx is locally L (this is a special feature of the case of 4 points). However, it turns out that Hx is not trace class, nor even Hilbert-Schmidt, since

2 dydz Tr(Hx)=4 θ(ft(x, y, z)) = ; 2 f (x, y, z) ∞ ZF t

namely, the integral logarithmically diverges at the divisor of zeros of the polynomial ft(x, y, z) (for ﬁxed t, x).31 Nevertheless, we have the following result.

2 Proposition 5.6. The operator Hx is Hilbert-Schmidt. Hence the operator Hx is compact. Proof. We have

2 dydz (Hxψ)(u)=4 ψ(z)θ(ft(x, y, z))θ(ft(x,y,u)) k k . F 2 f (x, y, z)f (x,y,u) Z k t t k 2 Thus the Schwartz kernel of Hx is p dy K(u, z) := 4 θ(ft(x, y, z))θ(ft(x,y,u)) k k . F f (x, y, z)f (x,y,u) Z k t t k 1 It follows from Lemma 8.1 that K(u, z) K0 log u pz near the diagonal u = z for some K > 0. This implies that ≤ − 0

4 2 Tr(Hx)= K(u, z) dudz < , 2 k k ∞ ZF hence the proposition.

30 An exception is y = x, where ft(x,y,z) is linear in z. 31 It is easy to show that the curve ft(x,y,z) = 0 always has points over F . Indeed, the discriminant of f (x,y,z) as a polynomial of z is 16x(x 1)(x t)y(y 1)(y t), and this is a square if y is close to x. t − − − − 52 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

2 Remark 5.7. In fact, since the Schwartz kernel of Hx has a logarithmic singularity on the diagonal, we have Tr H 2+ε < for any ε> 0. | x| ∞ We also recover Proposition 3.15 on the asymptotics of Hecke operators near the parabolic points: 1 1 2 2 1 Hx x log x , x ; Hx x log x S0, x 0; ∼k k1 k k →∞ ∼k k 1 → 2 1 2 1 Hx x t log x t S1, x t; Hx x 1 log x 1 S2, x 1. ∼k − k − → ∼k − k − → Indeed, this follows from Proposition 5.5 and the formulas

f (0,y,z)=(t yz)2, f (1,y,z)=(y + z t yz)2, f (t, y, z)=(ty + tz t yz)2. t − t − − t − − 2 Namely, these formulas show that for i = 0, 1, 2 we have ft(ti,y,z) = hi(y, z) , where the graph of Si is defined by the equation hi(y, z) = 0. 5.4. The spectral decomposition. By the spectral theorem for compact self-adjoint operators, the commuting operators Hx have a common eigenbasis ψn(y), n 0 of , with ψ = 1 and ≥ H k nk (5.3) Hxψn = βn(x)ψn 1/2 for real-valued functions β (x)= x(x 1)(x t) − β (x), none of them identically zero. n k − −e k n Thus all joint eigenspaces of Hx are finite dimensional. Equation (5.3) implies that ψn are Hxψn smooth outside 0, t, 1, (choosinge x such that βn(x) = 0 and writing ψn = ). Moreover, ∞ 6 βn(x) ψn can be chosen real-valued since Hx has a real-valued Schwartz kernel. Finally, we can pick the sign of ψ so that it is positive near , which fixes ψ uniquely. n ∞ n Corollary 5.8. We have βn = cnψn for some cn > 0. Thus

2θ(ft(x, y, z)) e = cnψn(x)ψn(y)ψn(z). ft(x, y, z) n k k X Proof. It follows from (5.3)p that

2θ(ft(x, y, z)) = βn(x)ψn(y)ψn(z). ft(x, y, z) n 0 k k X≥ e Since ft is symmetric, we havep βn(x)= cnψn(x), as claimed. Corollary 5.9. We have e 2θ(ft(x, y, z)) HxHy = Hz dz . F f (x, y, z) k k Z k t k Proof. Both sides act by the same eigenvaluesp on the basis ψn. 1 Corollary 5.10. We have cn = ψn,− , where ∞ ψn(x) ψn, := lim 1 . ∞ x x − log x →∞ k k k k In other words, we have 2θ(f (x, y, z)) ψ (x)ψ (y)ψ (z) (5.4) t = n n n . ψ ft(x, y, z) n n, k k X ∞ p 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 53

Proof. This follows from Proposition 3.15. Corollary 5.11. Let 2 Q := Hx dx . F k k Z 2 Then Q is a positive Hilbert-Schmidt operator such that Qψn = cnψn. Moreover, s 2s Tr(Q )= cn . n X In particular, c4 < . n ∞ n X Corollary 5.12. In the case of 4 points, the joint eigenspaces of the Hecke operators Hx are 1-dimensional. Proof. This follows from formula (5.4). This recovers Theorem 2 of [K] as well as the package of properties from [K], p.3. Remark 5.13. It is interesting to consider the trace of the modiﬁed Hecke operator

Tr(Hx)= βn(x). n X Unfortunately, as we have seen, this trace is note well deﬁned since Hx is not trace class, i.e., Tr Hx = n βn(x) = (so the series n βn(x) is not absolutely convergent and the sum depends| | on the| order| of∞ summation). Thus we can only talk about this trace in the regularized sense,P ande have to choose a regularizationP e procedure. Let us choose a “geometric” 2θ(ft(x,y,z)) regularization procedure, i.e. write the Schwartz kernel Kt(x, y, z) = of Hx as √ ft(x,y,z) k k a limit limN Kt,N (x, y, z) (uniform on compact sets not containing the singularities of →∞ Kt(x, y, z)) of a pointwise increasing sequence of positive continuous kernels (for example, we can set Kt,N := min(Kt, N)). Then we can deﬁne the regularized trace of Hx by the formula

T (t, x) = lim Kt,N (x, z, z) dz . N k k →∞ ZF It is easy to see that T (t, x)= K (x, z, z) dz . t k k ZF We have f (x, z, z)=(t z2)2 4z(z 1)(z t)x. t − − − − So we get 2θ((t z2)2 4z(z 1)(z t)x) T (t, x)= − − − − dz . 2 2 F (t z ) 4z(z 1)(z t)x k k Z k − − − − k The discriminant of the polynomial P (z) := (t z2)2 4z(z 1)(z t)x in z is p t,x − − − − D =212(x(x 1)t(t 1)(x t))2. − − − So the trace integral converges for x =0, 1, t, where it diverges logarithmically. 6 54 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

It can be shown by a direct computation that the cross-ratio of the roots of the polynomial 2 2 t(1 x) − (t z ) 4z(z 1)(z t)x (whose Galois group is Z/2 Z/2) is x(1 t) . Thus T (t, x) can − − − − ×t(1 x) − − be expressed in terms of the modified elliptic integral E+( x(1 t) ) (see Subsection 8.2). − 5.5. The archimedian case. As we showed in the previous section, in the archimedian case the operators Hx commute with Gaudin (=quantum Hitchin) Hamiltonians (Proposition 4.11) and also satisfy a second order ODE (an oper equation) with respect to x (Proposition 4.3). In the special case of 4 points, because of the symmetry of ft(x, y, z) with respect to permutations of x, y, z, these two (in general, completely different) types of equations turn out to be equivalent. Namely, both boil down to the following result. Let L = ∂ z(z 1)(z t)∂ + z z z − − z be the Laméoperator (=the quantum Hitchin Hamiltonian for 4 points). Proposition 5.14. For any t, x we have 1 (Ly Lz) =0. − ft(x, y, z) This follows from Proposition 4.11 orp Proposition 4.3 or by a (rather tedious) direct computation. In fact, an even stronger statement holds (and can be checked similarly): we have 1 (Ly Lz) =0 − f (x, y, z) k t k in the sense of distributions. So in the realp case we have 1 (Ly Lz) =0 − f (x, y, z) | t | and in the complex case p 1 1 (L L ) =(L L ) =0. y − z f (x, y, z) y − z f (x, y, z) | t | | t | Moreover, in the real case we also have

2θ(ft(x, y, z)) (Ly Lz) =0 − ft(x, y, z) as distributions. These statements are equivalentp to the statement that the operator Hx commutes with L (and in the complex case also with L) and satisﬁes the oper equation. Thus in the archimedian case we see that

Lψn =Λnψn for certain eigenvalues Λ . So for F = C the operators L Λ for various n correspond n − n exactly to the real opers, and thus the eigenvalues Λn are distinct. So we have 2 ψ (x)ψ (y)ψ (z) (5.5) = n n n , π f (x, y, z) ψ t n n, | | X ∞ where ψn runs over single-valued eigenfunctions of L normalized to be positive near and have norm 1. ∞ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 55

5.6. The real case. On the other hand, for F = R the situation is more subtle. To explain what is going on, fix an oper L(µ) that admits an eigenfunction, i.e., a solution ψ of the equation L(µ)ψ = 0 which near looks like ∞ ψ(x)= x 1/2(C log x + h (x)) | | 3 | | 3 for a continuous function h3, and at each tj, j =0, 1, 2 looks like ψ(x)= x t 1/2(C log x t + h (x)) | − j| j | − j| j for continuous hj. One of the constants Cj must be nonzero, so without loss of generality we may assume that C = 0, and set C = 1. Moreover if C = C = C = 0 then the function 3 6 3 0 1 2 ψ(x) ψ(x) := x(x 1)(x t) − − is entire and vanishes at , whichb is a contradiction,p so Cj = 0 for some j 0, 1, 2 . It is therefore easy to see∞ that any possible configuration is6 equivalent to∈{ one of following} four. (1) all Cj are nonzero; (2) C = 0, C ,C = 0; 0 1 2 6 (3) C0,C1 = 0, C2 = 0; (4) C ,C = 0, C =6 0. 0 2 1 6 We define the functions fj,gj near tj as follows. First, f3 is the restriction of ψ and + g3 = f3 if3. Next, for 0 j 2, if Cj = 0 then we set fj to be the restriction of ψ and −+ − ≤ ≤ 6 gj = fj ifj. Finally, if Cj = 0, we choose fj to be any solution with leading asymptotics −1 + x t 2 log x t near t and g := f if (thus in this case we have a freedom of ±| − j| | − j| j j j − j replacing fj by fj + λgj). Then (fj,gj) is a basis of solutions for each j. Now consider± cases 1-4 one by one. Case 1: all Cj are nonzero. In this case we have

Bjfj = fj 1, Bjgj = ajgj 1 + bjfj 1, − − − − where a0a1a2a3 = 1 (here aj, bj R). The equation j BjJ = 1 then yields by a direct calculation: ∈ − 1 Q 1 b0 = b2 = b, b1 = b3 =2b− , a0 = a2 = a, a1 = a3 = a− , 2 2 and either a = 1 and b =2(case 1a) or b =2a (case 1b). In case 1a, Cj = 1 for all j, so β = ψ defines a balancing6 of the local system (µ) corresponding to the oper±L(µ), and we get that µ . Moreover, this balancing is unique,∇ so the fiber in over µ consists of one point. ∈ B∗ B On the other hand, in case 1b, we have BjBj+1 = 1 for all j, so besides ψ we have another eigenfunction η which is regular at t1 and t3. Thus there are two values of c such that β = ψ + cη is a balancing for (µ). So µ and the fiber over µ in consists of two points. ∇ ∈ B∗ B Case 2: C = 0, C ,C = 0. Thus B , B are as in Case 1, but 0 1 2 6 2 3 1 B1f1 = a0g0, B1g1 = a1f0, B0f0 = b0f3 + a0g3, B0g0 = a0− f3

(choosing f0 to be a multiple of B1g1 and using that det B0 = 1). Then the lower left entry 1 − of the equation 4.7 written as B JB J = (B JB J)− yields b = 0, hence the upper right 0 1 − 2 3 3 entry yields a1 = 0, which is a contradiction. So this case is impossible. 56 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Case 3: In this case B3 is as in Case 1 but 1 B f = a a g , B g = a f , B f = a f + b g , B g = a− g , 2 2 0 1 1 2 2 2 1 1 1 − 1 0 1 0 1 1 1 0 1 B0f0 = a0g3, B0g0 = a0− f3 1 (choosing f0 to be a multiple of B0− g3, f1 to be a multiple of B2g2 and using that det B0 = det B = 1). Then the upper right entry of the same equation as in Case 2 gives a a a a = 1 − 0 1 2 3 1, while the determinant of this equation gives a0a1a2a3 = 1, again a contradiction. So −this case is impossible as well. Case 4: C ,C = 0, C = 0. In this case we have 0 2 1 6 1 B3f3 = a3− g2, B3g3 = a3f2, B2f2 = b2f1 + a2g1, B2g2 = cf1, 1 1 B1f1 = c− a0a3g0, B1g1 = a1f0 + b1g0, B0f0 = a0g3, B0g0 = a0− f3 1 (choosing f2 to be a multiple of B3g3, f0 to be a multiple of B0− g3, and using that det B3 = det B0 = 1). Then a direct calculation shows that for a suitable choice of signs of f0, f2 − 1 the equation B JB J = (B JB J)− is equivalent to the equations 0 1 − 2 3 1 1 1 b0 = b1 = b2 = b3 =0, a0 = , a1 = a, a2 = a− , a3 = √2, c =2a− . √2 a2 a 1 Since det(B B ) = , we have f x t 2 log x t near x = t . Thus deﬁning 0 1 2 1 √2 1 1 1 1 ∼ ± | − | | − | β = f3 g3, we see that β gives rise to a balancing for the local system (µ), so again ± ± √2 ± ∇ µ . In other words, Case 4 is equivalent to Case 1b by changing the choice of a periodic ∈ B∗ π √ eigenfunction ψ. Note that this case arises for 4 points in Subsection 4.10, as 2 cos 4 = 2. Now let βn, n 0, be all possible balancings for local systems attached to opers; so they run over the set ≥. By the S -symmetry of the Schwartz kernel of Hecke operators, we have B 3 Hxβn = βn(x)βn, Let ψ = β / β . Thus we obtain n n k nk θ(f (x, y, z)) ψ (x)ψ (y)ψ (z) (5.6) t = n n n . ψ ft(x, y, z) n n, X ∞ Thus we see that the Heckep operators Hx ﬁx a particular self-adjoint extension of the operator L in the real case and a particular normal extension of L in the complex case. In the complex case, this is exactly the extension described in [EFK1], Part II, while in the real case it is the extension corresponding to the space V1 of Subsection 8.10.

5.7. The subleading term of asymptotics of Hx as x . We have shown in Propo- →∞1 sition 3.15 that the operator Hx has the asymptotics Hx x − log x as x . So one may ask for the next (subleading) term of the asymptotics.∼k k k k →∞ Proposition 5.15. There exists a weak limit (on the Schwartz space ) S M := lim ( x Hx log x ), x →∞ k k − k k which extends to an unbounded self-adjoint operator on , essentially self-adjoint on . This operator is deﬁned by the formula H S (k) Mψk = µ ψk, where (k) 1 µ := lim ( x ψ− ψk(x) log x ) R. x k, →∞ k k ∞ − k k ∈ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 57

This proposition will follow from the explicit computation of M. We already gave a formula for M for any number of parabolic points in Proposition 3.21, but here we would like to do the same computation in a slightly different way. Namely, we will compute the Schwartz kernel KM (y, z) of M. Note that ft(x, y, z) is a quadratic polynomial in x with leading coefficient (z y)2. This implies that outside of the diagonal we have − 2 K (y, z)= , y = z. M z y 6 k − k However, we are not yet done since KM turns out to have a singular part concentrated on 2 the diagonal, and in any case the kernel z y does not give rise to a well defined operator since it is not locally L1. One possible regularizationk − k is given by 2f(z) (1 sign(log z y ))f(y) (M f)(y) := − − k − k dz . 0 z y k k ZF k − k It remains to compute the operator M M . We have − 0 (M M )ψ = hψ, − 0 where 2 x θ(f (x, y, z)) 1 + sign(log z y ) h(y) := lim k k t k − k dz log x . x →∞ F ft(x, y, z) − z y ! k k − k k! Z k k k − k 2(1+t)xy (x+y)(t+xy) 1 p− By translating z by (x y)2 = y + O(x− ), x to complete the square and 1 − − →∞ neglecting O(x− ), we obtain

2θ(z2 D(x,y) ) 1 + sign(log z ) h(y) := lim   − 4 k k  dz log x  , x − z k k − k k →∞ ZF z2 D(x,y) k k   4     −    r   where D(x, y) is the discriminant of the quadratic polynomial f (x, y, z)/(x y)2 in z. By t − Lemma 8.3, the integral under the limit equals log D(x,y) . But − 16

D(x, y) x(x 1)(x t)y(y 1)(y t) y(y 1)(y t) = − − − − − − , x . 16 (x y)4 ∼ x →∞ − Therefore Lemma 8.3 implies that h(y)= log y(y 1)(y t) . − k − − k We thus obtain the following proposition. Proposition 5.16. We have 2f(z) (1 sign(log z y ))f(y) (Mf)(y)= − − k − k dz f(y) log y(y 1)(y t) . z y k k − k − − k ZF k − k In other words, we have 2 K (y, z)= δ(z y) log y(y 1)(y t) , M z y − − k − − k k − k 2 where z y stands for the Schwartz kernel of the operator M0. k − k 58 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

1 Note that the distribution y regularized as above, i.e. by setting k k 2f(y) (1 sign(log y )f(0) ( 2 , f) := − − k k dy , y y k k k k ZF k k has Fourier transform 2 = 2 log p . F y − k k k k Thus the operator M can be written as 1 M = 2 log y − log y(y 1)(y t) . −F ◦ k k◦F − k − − k Thus for F = R we have M = 2 log ∂ log y(y 1)(y t) − | | − | − − | and for F = C we have M = 2 log ∂ 2 log y(y 1)(y t) 2. − | | − | − − | In these two cases, we can easily reprove directly that M indeed commutes with the Lam´e operator L = ∂z(z 1)(z t)∂ + z. Namely, one just needs to establish the formal algebraic identity − − (5.7) [2 log ∂ + log P,∂P∂ + z]=0 for any monic cubic polynomial P = P (z), and then apply it to P (z) = z(z 1)(z t).32 But (5.7) easily follows by a direct calculation in the fraction ﬁeld of the Weyl− algebra.−

Remark 5.17. Let F = R or C. Then we have Lψn =Λnψn. Let fn,gn be the solutions of 1 1 this equation near such that fn(z) z− (log z + o(1)), gn(z) z− (1 + o(1)), z + . Then for F = C ∞ ∼ ∼ → ∞ 1 (n) 2 ψn,− ψn(z)= fn(z)gn(z)+ gn(z)fn(z)+ µ gn(z) , ∞ | | and µ(n) is the unique constant for which this expression is monodromy invariant. On the other hand, if F = R then 1 (n) ψn,− ψn(z)= fn(z)+ µ gn(z). ∞ Remark 5.18. In the archimedian case, this analysis suggests that the eigenvalues of the operator M go to + logarithmically. Indeed, we expect that M obeys the Weyl law: the number of eigenvalues∞ N can be obtained by semiclassical analysis from the asymptotics of the volume of the region≤ 2 log p + log x(x 1)(x t) K k k k − − k ≤ 1 K/2 in T ∗P (F ) as K . But this volume equals E(t)e , where E(t) is the elliptic integral deﬁned in Subsection→∞ 8.2. This would imply that the number of eigenvalues of M which are N/2 2 1 N is asymptotic to CF E(t)e , where CR = π and CC = 4 , i.e., that the eigenvalues grow logarithmically.≤

32This identity makes sense, as both [2 log ∂, ∂P ∂ + z] and [log P, ∂P ∂ + z] are well deﬁned elements in the Weyl algebra C[z, ∂]. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 59

5.8. Comparison with the work of S. Ruijsenaars. In this subsection we would like to explain the connection of our results in the case of four points and F = R with the work of Ruijsenaars ([Ru]). 1 The points 0, t, 1, divide Bun0(R) = RP into four intervals I0 = [0, t], I1 = [t, 1], I = [1, ], I = [ ∞, 0]. Thus 2 ∞ 3 −∞ (5.8) = , H H0 ⊕ H1 ⊕ H2 ⊕ H3 where is the subspace of half-densities supported on I . Consider the self-adjoint extension Hj j L0 of the Laméoperator L corresponding to the space V0 of Subsection 8.10 (note that it does not coincide with the extension L1 of L defined by the Hecke operators, which corresponds to the space V1 of Subsection 8.10) It is clear that the operator L0 (unlike L1) preserves the subspaces , i.e., it is block-diagonal with respect to decomposition (5.8), since elements Hj of V0 are not required to satisfy any gluing conditions at xi (instead, they are just required to be bounded). Also, the subleading term M of Hx as x computed in Subsection 5.7 is given by a 4-by-4 block matrix with entries M : →. ∞ It follows that [M , L ] = 0 for all i, j. Note ij Hj → Hi ij 0 also that while M is not bounded, the operators Mj,j+2 with j Z/4 are Hilbert-Schmidt, 2 ∈ since they have a continuous Schwartz kernel y z (it is continuous since the intervals Ij,Ij+2 are disjoint). Thus, the operator L : | −commutes| with the Hilbert-Schmidt operator 0 Hj → Hj M † M = M M on , which can thus be used to fix a self-adjoint extension of L on ij ij ji ij Hj Hj (which, of course, coincides with L0). This method was proposed by S. Ruijsenaars in [Ru], and the kernel 2 coincides (up to a change of variable) with the kernel (u, v) used in y z S [Ru]. | − | This shows that the spectrum of L0 on j (i.e., the spectrum of L with bounded boundary conditions on I ) is the same as its spectrumH on . This is also easy to show directly since j Hj+2 the equivalence class of the operator L0 on [tj 1, tj] only depends on the cross-ratio of the − points (tj 1, tj, tj+1, tj+2), and this cross-ratio is unchanged under the map j j + 2. This spectrum− is the spectrum of the Sturm-Liouville problem (1) from [EFK1], Subsection7→ 10.5. On the other hand, the spectrum of L0 on 1 and 3 is also the same and coincides with the spectrum of the Sturm-Liouville problem (2)H fromH [EFK1], Subsection 10.5. These singular Sturm-Liouville problems were introduced in [Ta] following the work of Klein, Hilbert and V. I. Smirnov.

6. Hecke operators on P1 with four parabolic points over a non-archimedian local field In this subsection we study Hecke operators over non-archimedian local ﬁelds for G = 1 P GL2 in the simplest case of X = P with four parabolic points. We give a proof of the statement in [K] that eigenvalues of Hecke operators are algebraic numbers, and relate these eigenvalues to eigenvalues of the usual Hecke operators over the residue ﬁeld.

6.1. Mollified Hecke operators. Let F be a non-archimedian local field with residue field Fq. Let p be the characteristic of Fq, and assume that p > 2. Let x0 F , m Z, and consider the mollified Hecke operator ∈ ∈

Hx0,m := Hx dx . −m x x0 q k k Zk − k≤ 60 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

The Schwartz kernel of this operator has the form

θ(ft(x, y, z)) Kx0,m(y, z) := 2 dx . −m x x0 q ft(x, y, z) k k Zk − k≤ k k The main result of this subsection is the followingp theorem. Theorem 6.1. If x =0, 1, t, and m 0 then the operator H has ﬁnite rank. 0 6 ≫ x0,m The proof of Theorem 6.1 is given below.

Corollary 6.2. ([K]) The eigenvalues of the Hecke operators Hx are algebraic numbers. Proof. Since Hecke operators commute, they preserve the ﬁnite dimensional vector spaces

Im(Hx0,m). Also it is clear that the restrictions of Hx to Im(Hx0,m) are expressed in a

suitable basis by matrices with algebraic entries. Thus the eigenvalues of Hx on Im(Hx0,m)

are algebraic numbers. On the other hand, it is clear that the sum of the spaces Im(Hx0,m) over various x , m is dense in . This implies the statement. 0 H Remark 6.3. Note that Corollary 6.2 is a very special property, since eigenvalues of “generic” p-adic integral operators are usually transcendental (see [EK]). According to M. Kontsevich, this has to do with the fact that Hecke operators comprise an “integrable system over a local ﬁeld” in the sense of [K], Subsection 2.4. The rest of the subsection is devoted to the proof of Theorem 6.1. For this purpose it is 1 1 enough to show that the kernel of Kx0,m is of ﬁnite rank near each point of P P . This is achieved in Proposition 6.8 at the end of this subsection. ×

The operator Hx0,m is invariant under the group Z2 Z2 acting simply transitively on the points 0, 1, t, . Thus it suffices to consider only the finite× region A1 A1 P1 P1. Recall that∞ × ⊂ × f (x, y, z)=(y z)2x2 + 2(2(1 + t)yz (y + z)(t + yz))x +(t yz)2. t − − − Thus f (x, 0, 0) = t2, f (x, 1, 1)=(t 1)2, f (x, t, t)= t2(t 1)2, t t − t − f (x, 0, 1)=(x t)2, f (x, 0, t)= t2(x 1)2, f (x, 1, t)=(t 1)2x2. x − t − t − This implies that for m 0 the kernel Kx0,m(y, z) is locally constant (hence finite rank) near (y, z) where y, z 0≫, 1, t . Let us now study neighborhoods∈{ } of other points. When y = z, we have 6 f (x, y, z)=(y z)2(x2 + bx + c), t − where 2(1 + t)yz (y + z)(t + yz) (t yz)2 b := 2 − , c := − . (y z)2 (y z)2 − − Lemma 6.4. For m Z, c F let ∈ ∈ θ(x2 + c) Jm(c) := . 2 x 1 q−m x + c Zk − k≤ k k b Then for y = z and x0 + =0 we have p 6 2 6 2 c b − 4 2Jm k b 2 − (x0+ 2 ) K (y, z)= . x0,m y z k − k 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 61

where k := log x + b . − q 0 2 Proof. We have 2J (x ,b,c) K (y, z)= m 0 , x0,m y z k − k where for b, c F , ∈ θ(x2 + bx + c) Jm(x0,b,c) := dx . −m 2 x x0 q x + bx + c k k Zk − k≤ k k b x+ 2 Making the change of variable u = b , we seep that x0+ 2 b2 c 4 Jm(x0,b,c)= Jm k − . − b 2 (x0 + 2 ) ! This implies the statement. Let D = D (y, z) := 16y(y 1)(y t)z(z 1)(z t)=(b2 4c)(y z)4 t − − − − − − and P = P (x ,y,z) := (x + b )(y z)2 = x (y z)2 +2(1+ t)yz (y + z)(t + yz). t 0 0 2 − 0 − − Then 2 c b D − 4 = . b 2 2 (x0 + 2 ) −4P Thus Lemma 6.4 implies Corollary 6.5. If y = z and x + b =0 then 6 0 2 6 D 2Jm k( 2 ) K (y, z)= − − 4P . x0,m y z k − k Let us now proceed with computation of Jm(c). m Case 1: m> 0. In this case, if x 1 q− then x = 1. Note also that k − k ≤ k k x2 + c =(x + 1)(x 1)+(1+ c) − and x +1 = 1. Thus we have the following cases. k k r m Case 1a. 1+ c = q− > q− . Then we have k k m θ(1 + c) Jm(c)= q− . 1+ c k k In particular, if 1+ c > 1 (or, equivalently, c > 1) then k k kpk m θ(c) Jm(c)= q− . c k k r m Case 1b. 1+ c = q− q− . In this case we’ll need the following lemma. k k ≤ p r Lemma 6.6. Let a = q− < 1. Then k k ℓ θ(u + a) ∞ 1+( 1) ℓ r 1 1 2 2 1 = 2 (1 q− ) − q− q− − θ(a). u =q−r u + a − 2 − Zk k k k Xℓ=r p 62 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Proof. We have ℓ θ(u + a) θ(u) ∞ 1+( 1) ℓ 1 1 2 = = 2 (1 q− ) − q− . u q−r u + a u =q−r u − 2 Zk k≤ k k Zk k k k Xℓ=r On the other hand, p p θ(u + a) r 1 = q− 2 − θ(a). u q−r−1 u + a Zk k≤ k k Subtracting, we obtain the desired statement.p Now, we have ∞ θ(x2 + c) Jm(c)= dx . 2 x 1 =q−ℓ x + c k k ℓX=m Zk − k k k Splitting the sum into three parts ℓrp , we get r 1 − ℓ 2 1 1 1+( 1) ℓ θ(x + c) r 1 Jm(c)= (1 q− ) − q− 2 + dx + q− 2 − θ(1 + c). 2 2 − 2 x 1 =q−r x + c k k Xℓ=m Zk − k k k To compute the integral, we use the change of variablepx2 1= u and Lemma 6.6. Then we get − r 1 − ℓ ∞ ℓ 1 1 1+( 1) ℓ 1 1 1+( 1) ℓ r 1 r 1 J (c)= (1 q− ) − q− 2 + (1 q− ) − q− 2 q− 2 − θ(1+c)+q− 2 − θ(1+c)= m 2 − 2 2 − 2 − ℓX=m Xℓ=r ∞ ℓ 1 1 1+( 1) ℓ (1 q− ) − q− 2 . 2 − 2 ℓX=m Thus m 1 2 Jm(c)= 2 q− if m is even and m+1 1 2 Jm(c)= 2 q− if m is odd. Case 2: m 0. In this case we have ≤ θ(x2 + c) Jm(c)= dx . 2 x q−m x + c k k Zk k≤ k k Setting x = πmy, where π F is a uniformizer, we get ∈ p 2 2m θ(y + cπ− ) 2m Jm(c)= dy = J0(cπ− ). 2 2m y 1 y + cπ− k k Zk k≤ k k So we need to compute J0(c). p Case 2a. If c > 1 then we have k k θ(c) J0(c)= . c k k 2r+1 Case 2b. If c = q− < 1 then p k k 1 1 1 J (c)=(1 q− )r = (1 q− )(log c 1). 0 − − 2 − q k k − 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 63

2r On the other hand, if c = q− 1 then we have k k ≤ 2 1 1 θ(x + c) 1 J0(c)= (1 q− ) log c + dx + q− θ(c). 2 q 2 − − k k x =q−r x + c k k Zk k k k We also have p θ(x2 + c) θ(y2 + cπ 2r) dx = − dy . − 2 2 2r x =q r x + c k k y =1 y + cπ− k k Zk k k k Zk k k k We will now use the followingp lemma. p Lemma 6.7. Let a =1. Then k k 2 θ(y a) 1 1 1 − dy = (1 q− ) q− θ( a). 2 2 y =1 y a k k − − − Zk k k − k Proof. Denote the integralp in question by I. Assume first that a is a non-square. Then 2 1 y a = 1 for all y with y =1. So I = q− N, where N is the number of y F× such k − k k k ∈ q that y2 a = x2 for some x. Then the number of solutions of the equation y2 a = x2 in F such− that y = 0 is 2N. But this equation can be written as (y x)(y + x)= a−, and y x q 6 − − can be chosen any nonzero element in Fq, which completely determines y + x, hence y and x. However, we need to exclude the case y = 0, which gives two solutions iff a is a square. Thus 2N = q 1 2θ( a), which implies the statement. − Now assume− that− a is− a square. Then we have θ(y2 a) I = + + − dy . 2 y =1,y= √a mod π y=√a mod π y= √a mod π y a k k Zk k 6 ± Z Z − k − k 1 1 2 2 The first integral equals 2 q− times the number of solutions of thep equation y a = x over 1 1 − F excluding (0, √a) with y = 0, which is q− (q 3 2θ( a)). On the other hand, q ± 6 2 − − − 2 θ(y a) θ(2√a(y √a)) θ(x) 1 1 − dy = − dy = dx = q− . 2 2 y=√a mod π y a k k y=√a mod π y √a k k x q−1 x k k Z k − k Z k − k Zk k≤ k k This impliesp the statement. p p Lemma 6.7 implies that 2 θ(x + c) 1 1 1 dx = (1 q− ) q− θ(c). 2 2 x =q−r x + c k k − − Zk k k k Thus p 1 1 J (c)= (1 q− )(log c 1). 0 − 2 − q k k − We see that this formula in fact holds for both odd and even powers of q.

Proposition 6.8. The kernel Kx0,m(y, z) is locally of ﬁnite rank. That is, for any y0, z0 1 ∈ P (F ) there exists ε> 0 such that when y y0 < ε, y y0 <ε, we can write Kx0,m(y, z) in the form k − k k − k N

Kx0,m(y, z)= ai(y)bi(z). i=1 X 64 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Proof. We ﬁrst consider the case y = z, so y z > 0. b m 6 k − k Case A. x0 + 2 > q− , i.e., k < m. This is equivalent to m 2 P > q− y z k k k − k

and means we are in Case 1. Then Kx0,m is locally constant, as desired. b m Case B. x + q− , i.e., k m. This is equivalent to the condition that 0 2 ≤ ≥ m 2 P q− y z . k k ≤ k − k This means we are in Case 2. Then the condition for Case 2a is 2m 4 D > q− y z k k k − k

which means that Kx0,m is locally constant. So it remains to consider the case 2m 4 D q− y z , k k ≤ k − k

which means we are in Case 2b. Then it is easy to check that Kx0,m is locally constant unless D = 0, in which case it is not locally constant but is the product of a function of y and a function of z (as so is D), so still of ﬁnite rank. Finally, consider the case y = z. Case C. y = z. In this case P = 2y(y 1)(y t). So if P = 0 then we get y(y 1)(y t)= 0 so y = z = 0, 1, t, a case that− has already− − been considered. Thus we may− restrict− to

the case when P > 0. In this case Kx0,m is locally constant unless ft(x0,y,y) = 0. k k s On the other hand, if ft(x0,y,y) = 0, assume that P = q− . Then near this point 2 2 k k k−s b s k s 2 x + 2 y z = q− , i.e., q− y z = q− , which yields that y z = q . In particular,k this− k implies that k s kis even,− k so the parity of m k is thek same− k as the parity − D − of m s (a ﬁxed number). Also in this case 1 2 = 0, so near this point we are in Case − − P 1b. Thus in this case Kx0,m is also locally constant. This completes the proof of Theorem 6.1.

Remark 6.9. This proof shows that Kx0,m fails to be locally constant near (y, z) if and only m 2 if D = 0 and P q− x y . As m , this set shrinks to the following 8 points: k k ≤ k − k →∞ t x0 t t(x0 1) t x0 t t(x0 1) (0, ), (1, − ), (t, − ), ( , x ), ( , 0), ( − , 1), ( − , t), (x , ). x0 x0 1 x0 t 0 x0 x0 1 x0 t 0 − − ∞ − − ∞ 6.2. Computation of eigenvalues of Hecke operators. We would now like to compute the first “batch” of eigenvalues of the Hecke operators, namely the eigenvalues on the finite dimensional space generated under the Hecke operators by the characteristic functions of balls 1 of radius q− . We will show that this space has dimension q + 5 and relate the eigenvalues of the Hecke operators on this space to eigenvalues of the usual Hecke operators over the finite field Fq.

1 6.2.1. Computation of Kx,1(y, z). Let 1x be the indicator function of the ball of radius q− around x F , and let us compute Hz1x, where z = 1, z =0, 1, t mod π (where, as before, π F is the∈ uniformizer). So we may assume thatk k x 1.6 We also recall that ∈ k k ≥ (y z)2f (x, y, z)=4P 2 D. − t − We have

(Hz1x)(y)= Kx,1(y, z). 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 65

So, as shown in the previous subsection, generically we get D 2J1 k( 2 ) (H 1 )(y)= − − 4P . z x y z k − k b where k = logq x + 2 . Thus we have the following cases, corresponding to the cases with the same− numbers considered in the previous subsection. 2 Case 1. Pt(x, y, z) y z . Case 1a.k f (x, y, zk≥k) P−(x,k y, z) . Then we have k t k≥k t k 1 2q− θ(ft(x, y, z)) (Hz1x)(y)= . f (x, y, z) k t k 1 Case 1b. ft(x, y, z) q− Pt(x, y, z) .p Then we have k k ≤ k k s q− y z (Hz1x)(y)= k − k , P (x, y, z) k t k 1 where s = 1 if P is an even power of q andps = 2 if P is an odd power of q. k k 1 2 k k Case 2. Pt(x, y, z) q− y z . In this case, we have the following cases. k k ≤ k1 − k 4 1 2 Case 2a. If Dt(y, z) q− y z (equivalently, ft(x, y, z) q− y z ) then we have k k ≥ k − k k k ≥ k − k 1 2q− θ(ft(x, y, z)) (Hz1x)(y)= . f (x, y, z) k t k 2 4 Case 2b. If Dt(y, z) q− y z thenp k k ≤ k − k 1 Dt(y,z) (1 q− )(logq (y z)4 + 1) (H 1 )(y)= − − − . z x y z k − k 6.2.2. Diagonalization of Hecke operators. Let = be the ring of integers of F . For O OF x denote by x0 its reduction to the residue ﬁeld Fq. Note that the function 1x depends only∈ O on x . Let t be such that t = 0, 1. Let z , z = 0, 1, t , and x . Assume 0 ∈ O 0 6 ∈ O 0 6 0 ∈ O ﬁrst that y 1. We have y z 1 while Pt(x, y, z) 1, ft(x, y, z) 1. So if P k(xk,y ≤ , z ) = 0 andk f−(xk,y ≤ , z ) = 0k then we arek ≤ in Casek 1a andk ≤ t0 0 0 0 6 t0 0 0 0 6 1 (Hz1x)(y)=2q− θ(ft0 (x0,y0, z0)). However, if P (x ,y , z ) = 0 but f (x ,y , z ) =0 then we are in Case 1b and t0 0 0 0 6 t0 0 0 0 1 (Hz1x)(y)= q− .

On the other hand, if Pt0 (x0,y0, z0) = 0 (which necessarily implies y0 = z0) then we have D (y, z) 1. So if D (y, z) = 1 (i.e., y =0, 1, t ) then we are in Case6 2a and we have k t k ≤ k t k 0 6 0 1 1 (H 1 )(y)=2q− θ(f (x ,y , z ))=2q− θ( D (y , z )). z x t0 0 0 0 − t0 0 0 1 1 If Dt(y, z) q− (i.e., y0 =0, 1, t0) then we are in Case 2a ( Dt(y, z) = q− ) or Case 2b k k ≤ 2 k k ( D (y, z) q− ) but in both cases we have k t k ≤ 1 (H 1 )(y)= (1 q− )(log y r + 1). z x − − q k − k where r =0, 1, t respectively. 66 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

2 3 Now assume y q. Then y z = y , Pt(x, y, z) y , Dt(x, y, z) = y , so k k ≥2 k − k k k k k≤k k k k k k if P (x, y, z) = y (i.e., x = z ) we are in Case 1a and k t k k k 0 6 0 2q 1 (H 1 )(y)= − , z x y k k 1 2 while if Pt(x, y, z) q− y (i.e., x0 = z0) then if y = q we are in Case 2a and if y q2kthen we arek ≤ in Casek 2b,k but in both cases we havek k k k ≥ 1 1 (1 q− )(log y− + 1) (H 1 )(y)= − − q k k . z x y k k For r =0, 1, t let us introduce the functions 1 1 φ := (1 q− )(log y r + 1), y r q− r − − q k − k k − k ≤ and vanishing otherwise. We also have a similar function φ which diﬀers by dividing by y (to account for the fact that we have half-forms): ∞ k k 1 1 (1 q− )(log y− + 1) φ := − − q k k , y q ∞ y k k ≥ k k and vanishing otherwise. Also we introduce the function 1 1 (y)= if y q ∞ y k k ≥ k k and zero otherwise. Then we have

Hz1x = ax0j1y + bx0rφr,

j P1(Fq ) r=0,1,t, ∈X X ∞ where the coeﬃcients aij, bir are as follows. Case A. i =0, 1, t0, . Case A1. 6If j =0, 1∞, t , then if f (i, j, z ) = 0, we have 6 0 ∞ t0 0 6 1 aij =2q− θ(ft0 (i, j, z0)) while if ft0 (i, j, z0) = 0 then 1 aij = q− . Case A2. If j = 0 then if i = t /z , we have 6 0 0 1 aij =2q−

while if i = t0/z0, we have aij =0. t0 z0 The same holds for j =1, t , , where the corresponding equations have the form i = − , 0 1 z0 ∞ − t0(z0 1) i = − , i = z respectively. z0 t0 0 Also− if i = t /z , we have 6 0 0 bi0 =0 while if i = t0/z0 then bi0 = q. t0 z0 The same holds for r =1, t, , where the corresponding equations have the form i = − , 1 z0 ∞ − t0(z0 1) i = − , i = z respectively. z0 t0 0 Case− B. i =0, 1, t, . Consider ﬁrst the case i = 0. ∞ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 67

Case B1. If j =0, 1, t, then if f (0, j, z )=(t jz )2 = 0 (i.e., j = t /z ) then 6 ∞ t0 0 0 − 0 6 6 0 0 1 a0j =2q− ,

while if j = t0/z0 then 1 a0j = q− . 2 Case B2. j = 0. We have ft0 (0, 0, z0)= t0 so 1 a0j =2q− . Case B3. j = 1. We have f (0, 1, z )=(z t )2 so t0 0 0 − 0 1 a0j =2q− . Case B4. j = t . We have f (0, t , z )= t2(z 1)2 so 0 t0 0 0 0 0 − 1 a0j =2q− . Case B5. j = . Since z = 0, we have ∞ 0 6 1 a0j =2q− . In the cases i =1, t, the answer is the same. Also it is easy to see∞ that bir =0 for all i, r =0, 1, t, . ∞ 2 1 Now note that the functions 1 are orthogonal with 1 = q− . We would like to correct x k xk the functions φr so that they become orthogonal to each other and to 1x. So let us introduce ψ := φ β1 r r − r and choose β so this is orthogonal to 1r. Then

β = q(ψr, 1r). So for r =0, 1, t we have

1 1 β = q(1 q− ) (logq y r + 1) dy = q− . − − y q−1 k − k k k Zk k≤ Thus for r =0, 1, t 1 1 ψ (y)= (1 q− ) log y r 1, y r q− r − − q k − k − k − k ≤ and zero otherwise, and 1 1 (1 q− ) log y− 1 ψ (y)= − − q k k − , y q ∞ y k k ≥ k k and zero otherwise. Let us now normalize these vectors. We have

2 2 2 ψ0 = ψ0(y) dy = q− . k k y q−1 | | ·k k Zk k≤ So the normalized basis vectors are 1/2 1x := q 1x, ψr := qψr.

In this basis the matrix of the operator Hz is symmetric, and has the following form. b b 68 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Proposition 6.10. 1. Let x ,y =0, 1, t . If f (x ,y , z ) is not a square then 0 0 6 0 t0 0 0 0 (1x,Hz1y)=0.

If ft0 (x0,y0, z0)=0 then b b 1 (1x,Hz1y)= q− . 2. We have b b 1 (1x,Hz1y)= q− at the following 8 positions: b b t0 z0 t0 t0(z0 1) t0 z0 t0 t0(z0 1) (6.1) (0, ), (1, − ), (t, − ), ( , z ), ( , 0), ( − , 1), ( − , t ), (z , ). z0 z0 1 z0 t0 0 z0 z0 1 z0 t0 0 0 − − ∞ − − ∞ 3. In all other cases 1 (1x,Hz1y)=2q− . 4. We have b b 1 (1x,Hzψr)=(ψr,Hz1x)= q− 2 at the 8 positions of (6.1). Otherwise b b b b (1x,Hzψr)=(ψr,Hz1x)=0. 5. We have b b b b (ψs,Hzψr)=0. Proof. (1)-(4) follow from the formulas for a , b and the self-adjointness of H . (5) is b bij ij z checked by an easy direct computation, using that the function ft(x, y, z) is a square and 1 has norm 1 when the distances from x to r and from y to s are q− . ≤ In particular, we see that in the orthogonal (but not orthonormal) basis 1x, ψr the matrix qHz (of size q +5) has integer entries 0, 1, 2, q (even though it is non-symmetric in this basis).

Proposition 6.11. (i) The Perron-Frobenius eigenvalue of Hz equals

32q 1+ 1+ 2 1 (q+1) 1 2 λ =(1+ q− ) =1+9q− + O(q− ). + q 2 The corresponding eigenvector is

v+ = λ+ 1i + (1r + ψr).

i P1(Fq ) r=0,1,t, ∈X X ∞ The operator Hz also has the eigenvalue

32q 1 1+ 2 1 − (q+1) 1 2 λ =(1+ q− ) = 8q− + O(q− ). − q 2 − with eigenvector

v = λ 1i + (1r + ψr). − − i P1(Fq ) r=0,1,t, ∈X X ∞ These eigenvalues are the roots of the equation qλ2 (q + 1)λ 8=0. − − 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 69

(ii) Hz has eigenvalue 0 with multiplicity at least 3. Namely, the vectors

ar(1r + ψr) r=0,1,t, X ∞ with a1 + a2 + a3 + a4 =0 are null vectors for Hz. 6.3. Relation to Hecke operators over a ﬁnite ﬁeld. The eigenvectors of Proposition 1 1 1 6.11 span the 5-dimensional space V0 with basis := i P1(Fq ) i and er := r + ψr, r = ∈ 0, 1, t, . Note that the matrix of the Hecke operator Hz on V0 in this basis does not depend on t. ∞ P Let us now consider the orthogonal complement V = V1 of this space, which has dimension q. Namely, V is the space of vectors a 1 q a ψ i i − r r i P1(Fq) r ∈X X 1 1 where i P1(Fq) ai = 0. Its basis is formed by the vectors vk := k + qψ if k =0, 1, t ∈ − ∞ ∞ 6 and vk := 1k 1 qψk + qψ if k =0, 1, t. P − ∞ − ∞ Thus the matrix of the operator qHz on V in this basis is given by the formula

t z−t − [qHz V ]x,y = N(t,z,x,y) 2+(q + 1)δx,z qδ0,yδx, qδ1,yδx, qδt,yδx, t(z 1) . | − − z − z−1 − z−t 2 where N(t,z,x,y) is the number of solutions of the equation w = ft(x, y, z) in Fq (which 1 is 2, 1 or 0). This means that H = q− T , where T is the cuspidal component of the z|V − z z Hecke operator over Fq (see [K], p.4). Namely, the formula for the matrix Tz (also valid if z =0, 1, t) is

t z−t − (Tz)x,y =2 N(t,z,x,y) czδx,z + (1 δz(z 1)(z t),0)(qδ0,yδx, + qδ1,yδx, + qδt,yδx, t(z 1) ), − − − − − z z−1 z−t where c = 1 if z =0, 1, t and c = q + 1 if z =0, 1, t.33 z z 6 34 Example 6.12. Let q = 5, t = 4, z = 2. Then the 10-by-10 matrix of qHz in the basis 1x, ψr is 221222 222122 12 211 1 21 12 11    222122    221222     5     5   5     5    (where empty positions stand for zeros, for better viewing). The eigenvalues of this matrix are 0 (with multiplicity 5), 10, 4, 2, 2√3, 2√3. The Perron-Frobenius eigenvector (with eigenvalue 10) has coordinates (3−, 3, 2, 2, 3, 3−, 1, 1, 1, 1).

33 Here we correct a misprint in the formula on the top of p.4 in [K], where it is written that cz = q + 1 if z =0, 1,t and c = 1 if z =0, 1,t. So these should be switched. z 6 34We thank M. Vlasenko for sharing her computations in this case. 70 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

The matrix qH = T has the form − z|V z 00 1 0 0 00 0 1 0 0 6 4 4 6  − − − − 06 2 2 6    00 0 1 0      and has eigenvalues 2, 2√3, 2√3, 0, 0. − − Note that this describes all the cases for q = 5. Indeed, we have the group S3 acting on the values of t (generated by t 1 t and t 1/t) which acts transitively on the possible values t = 2, 3, 4, and the stabilizer7→ − of t = 47→ is generated by t 1/t which exchanges the possible values z =2, 3. 7→

Example 6.13. Let q = 7, t = 6, z = 2. Then the 12-by-12 matrix of qHz in the basis 1x, ψr is 22212222 22212222 22 2 11 11 112 2211    22 22    22 22    22122222    22122222     7   7     7     7    The eigenvalues of this matrix are 0 (multiplicity 6), 4, 2, 1 √17, 4 6√2. The Perron- − − ± ± Frobenius eigenvalue is 4 + 6√2. Similarly, if instead z = 3 (with the same q, t) then the eigenvalues are 0 (multiplicity 6), 4, 2, 1 √17, 4 6√2. The Perron-Frobenius eigenvalue is still 4 + 6√2. − ± ± This covers all the cases with q = 7, t =2, 4, 6 due to the S3-symmetry.

Example 6.14. Let q = 7, t = 3, z = 2. Then the 12-by-12 matrix of qHz in the basis 1x, ψr is 22222122 22222211 222 12 1 22221222    22 1 1 2 1    1221 2 1    2112 21    22122222     7     7   7     7      1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 71

1 √33 1 √33 The eigenvalues of this matrix are 0 (multiplicity 3), − ±2 (each with multiplicity 2), ±2 , 1, 4 6√2. The Perron-Frobenius eigenvalue is again 4 + 6√2. ± Via the S3-symmetry this covers all cases with t = 3 except z = 6. In this case the 1 √33 eigenvalues are 0 (multiplicity 3), ± (each with multiplicity 3), 1, 4 6√2. The Perron- 2 ± Frobenius eigenvalue is again 4 + 6√2. This also covers the case t = 5 via the S3-symmetry. Thus we have described all the cases with q = 7.

Remark 6.15. By a theorem of Drinfeld ([Dr2]), the eigenvalues of Hz on V = V1 have 1 1 absolute value 2q− 2 and have the form 2q− 2 Reλ, where λ = 1 and λ are asymptotically uniformly distributed≤ on the circle for large q. | |

7. Singularities of eigenfunctions In this section we study the singularities of eigenfunctions of Hecke operators for G = P GL2 in genus zero (with parabolic points). We will use the Gaudin operators, so this analysis only applies to the archimedian case. However, we expect that the behavior of eigenfunctions near singularities is essentially the same over all local ﬁelds.

7.1. Singularities for N = m +2 of parabolic points. Consider the (modiﬁed) Gaudin operators

1 2 Gi := (yi yj) ∂i∂j +(yi yj)(∂i ∂j) . ti tj − − − − 0 j m,j=i − ≤ X≤ 6 To study the behavior of eigenfunctions, we will quotient out the symmetry under the 1 group Gm ⋉ Ga of aﬃne transformations of A and write these operators in the coordinates y1, ..., ym 1 on Bun0◦. − First we quotient out the translation group Ga. Set t0 = 0 and consider the action of Gi on shift-invariant functions, i.e., such that i ∂if = 0. Thus ∂ f = P ∂ f. 0 − i 1 i m ≤X≤ Substituting this and setting y = 0, we have, for 1 i m: 0 ≤ ≤

1 2 Gi = yi ∂i( ∂j)+ yi(2∂i + ∂j) + ti 1 j m j=i,1 j m ! ≤X≤ 6 X≤ ≤

1 2 (yi yj) ∂i∂j +(yi yj)(∂i ∂j) . ti tj − − − − j=i,1 j m − 6 X≤ ≤ Now we quotient out the dilation group Gm. To this end consider the action of the Gaudin operators on functions of homogeneity degree m/2. On such functions − m 1 − 1 m ∂ f = y− ( + y ∂ )f. m − m 2 i i i=1 X Substituting this and setting ym = 1, tm = 1 we get the following proposition 72 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Proposition 7.1. For 1 i m 1 we have ≤ ≤ −

1 2 m m Gi = yi ∂i (1 yj)∂j 2 + yi (2 yi)∂i + (1 yj)∂j 2 + ti − − − − − 1 j m 1 ! j=i,1 j m 1 !! ≤X≤ − 6 X≤ ≤ − 1 2 (yi yj) ∂i∂j +(yi yj)(∂i ∂j) + ti tj − − − − j=i,1 j m 1 − 6 X≤ ≤ −

1 2 m m (yi 1) ∂i( yj∂j + 2 )+(yi 1)((1 + yi)∂i + yj∂j + 2 ) . ti 1 − − 1 j m 1 1 j m 1,j=i ! − ≤X≤ − ≤ ≤X− 6 Consider now the quantum Gaudin system

Giψ = µiψ. N 3 As explained in Subsection 2.4, generically on Bun0◦, this is a holonomic system of rank 2 − whose singularities are located on the wobbly divisor. Let us consider its solutions near a 0 generic point y of the divisor ym 1 = 0, which is a component of the wobbly divisor. So we 0 0 − will set ym 1 = 0, yi = ai for i = 1, ..., m 2 with ai generic, and z = ym 1. We want to ﬁnd a solution− in the form − − λ+n ψ(y1, ..., ym 2, z)= an(y1, ..., ym 2)z . − − n 0 X≥ 0 To this end, we have to compute the leading term Gi of Gi with respect to z. For i < m 1 we get: − 0 tm 1 Gi = − yi(yi∂i + 1)∂z, tm 1 ti − − and for i = m 1 we get −

0 1 1 m 2 Gm 1 = yj(yj∂j + 1)∂z + ∂z(z∂z + yj∂j + 2− ) , − t t t 1 1 j m 2 j m 1 m 1 j m 2 ! ≤X≤ − − − − − ≤X− both of degree 1. Thus we have − 0 λ Gi (a0(y1, ..., ym 2)z )=0, 1 i m 1. − ≤ ≤ − This yields

(7.1) λ(yi∂i + 1)a0 =0, and 2 m 2 (λ − λ)a =0. − 2 0 m 2 m 2 We thus obtain λ = 0 or λ = 2− , and for m > 2 the space of solutions with λ = 2− is 1-dimensional. Also replacing λ with ε where ε2 = 0 (over the base ring C[ε]/ε2), we see that if m > 2 then there are no solutions of the form a0(y1, ..., ym 2) log(z)+ O(1) with a0 = 0. Thus, applying permutations of indices, we arrive at the following− proposition. 6 Proposition 7.2. For N 5 points all solutions of the Gaudin system are bounded near a ≥ generic point y of the divisor yi = yj. Hence single-valued eigenfunctions of Gaudin operators and in particular eigenfunctions of the Hecke operators are continuous (albeit not C∞) at y. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 73

Remark 7.3. 1. More precisely, our analysis shows that eigenfunctions belong to the local − m 2 m 2 35 Hölder space H 2 for F = R and to H − for F = C near y. 2. On the contrary, we have seen that for 4 points the eigenfunctions grow logarithmically at the singularities. 7.2. The monodromy of the Gaudin system. Proposition 7.4. For N 5 points all solutions of the Gaudin system are bounded near a generic point y of the wobbly≥ divisor D. Hence single-valued eigenfunctions of the Gaudin operators and in particular eigenfunctions of the Hecke operators are continuous (although not C∞) at y. Proof. This follows from Proposition 2.11 and Proposition 7.2. We also obtain the following results on the monodromy of Gaudin systems for F = C. Proposition 7.5. The monodromy operator γ of the Gaudin system around a component of D is a reflection for odd number of points N (diagonalizable, one eigenvalue 1, other eigenvalues 1) and a transvection for even N (unipotent, γ 1 has rank 1). − − Proof. It follows from our description of the local behavior of solutions that for even N all eigenvalues of γ are 1, and for odd N all of them but one are 1 and one eigenvalue is 1, and that γ 1 has rank 1. But if γ = 1 then the corresponding D-module on an open− N 3− ≤ set of P − is monodromy-free, hence a multiple of . Thus it cannot be irreducible, a contradiction with Proposition 4.2. O Corollary 7.6. The eigenfunctions of Gaudin (or Hecke) operators near a generic point y of the wobbly divisor D have the form m 2 f = f + f z − 0 1| | for odd m and m 2 f = f + f z − log z 0 1| | | | for even m, where f0 and f1 are real analytic functions near y and z is a complex coordinate on Bun0◦ such that D is locally near y defined by the equation z =0. Proof. Let y D. The above analysis implies that for odd m there is a basis of solutions of the Gaudin system∈ in which all but one element are holomorphic at y and one element is of m−2 the form hz 2 where h is holomorphic at y, and for even m there is a basis with all but m−2 two elements holomorphic at y, and the remaining two elements have the form hz 2 and m−2 g + hz 2 log z, where g, h are holomorphic at y. This implies the statement. Corollary 7.7. The monodromy group of the quantum Gaudin system with any eigenvalues µi is generated by reflections for odd N and by transvections for even N. Corollary 7.8. The monodromy groups of the quantum Gaudin systems corresponding to real opers (i.e., satisfied by eigenfunctions of Hecke operators) are real, i.e., contained in N 3 GL(2 − , R) up to conjugation.

35Here for a nonnegative integer k and 0 < α 1, we write Hk+α for the local H¨older space Ck,α. In particular, for a positive integer k, we have Hk+δ ≤ Ck for any δ > 0, but Ck is a proper subspace of Hk. ⊂ 74 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

N 3 Proof. Consider the corresponding monodromy group in GL(2 − , C). By the argument of [EFK2], Remark 1.8, this monodromy group is contained in some inner real form of N 3 GL(2 − , C). There are only two such forms – the split one with the group of real points N 3 N 4 GL(2 − , R) and the quaternionic one with group of real points GL(2 − , H). But the quaternionic form does not contain images of transvections or reﬂections, since the space of invariants of such an element has odd complex dimension, hence cannot be a quaternionic vector space. This implies the statement. 7.3. The case of 5 points. Consider now the case of N = 5 points, i.e., m = 3. We let t1 = s, t2 = t, y1 = y, y2 = z.

2 7.3.1. Behavior of eigenfunctions. Recall (see e.g. [C],[DP1]) that Bun0◦ is P blown up at points (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1), (s, t, 1). According to [DP1], Subsection 5.4 or Proposition 2.11, the components of the divisor D are as follows: 1) exceptional fibers over these 5 points; 2) 10 lines y =0, y =1, y = s, z =0, z =1, z = s,y = z, ty = sz, (t 1)(y 1)=(s 1)(z 1), line at ; − − − − ∞ 3) quadric st(y z)+(t s)yz + sz ty = 0. − − − So there are 16 components, permuted transitively by the Weyl group W (D5) = S5 ⋉V, where V is the 4-dimensional reflection representation over F2. Namely, the set of components is W (D5)/W (A4)= S5 ⋉V/S5 = V, where V acts by translations and S5 by reflections. The origin in V corresponds to the component over (s, t, 1) which is the component of bundles isomorphic to O(1) O( 1). The components of⊕ the− divisor D can be subdivided according to the invariant r = S /2. Namely, | | 5 r = 0 corresponds to the exceptional fiber over (t, s, 1) (bundles O( 1) O(1); 0 =1 component);• − ⊕ r = 1 corresponds to lines y =0,y =1, z =0, z =1, z = y, the line at infinity, and the • 5 remaining 4 exceptional fibers at (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) ( 2 = 10 components); r = 2 corresponds to lines y = s, z = t, ty = sz, (t 1)(y 1)=(s 1)(z 1), and the • 5 − − − − quadric ( 4 = 5 components). The divisor has normal crossings, and the crossings define the Clebsch graph, which is the regular 16-vertex, 40-edge graph obtained by the 5-dimensional hypercube graph by quotienting by the central symmetry. Thus using the standard theory of holonomic systems with regular singularities on a normal crossing divisor, we get the following theorem for F = C.

Theorem 7.9. (i) Near a generic point of D there is a basis f1, f2, f3, f4 of solutions of the Gaudin system such that f1, f2, f3 are holomorphic and f4 = h√z where h is holomorphic, where D is locally deﬁned by the equation z =0. (ii) Near the intersection point of two components of D there is a basis f1, f2, f3, f4 of solutions of the Gaudin system such that f1, f2 are holomorphic, f3 = h1√w and f4 = h2√z where h1, h2 are holomorphic and D is locally deﬁned by the equation zw =0. (iii) Single-valued eigenfunctions of the Gaudin operators are continuous, near a generic point of the divisor D are of the form ψ + ψ z , where ψ , ψ are real analytic, and near the 0 1| | 0 1 intersection of two components of D are of the form ψ0 + ψ1 z + ψ2 w where ψ0, ψ1, ψ2 are real analytic. Thus all of them are in L2 and satisfy the Gaudin| | equations| | as distributions, 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 75 i.e., coincide with eigenfunctions of the Hecke operators. Thus eigenvalues of Hecke operators .(are in bijection with real opers (i.e., the inclusion Σ ֒ is a bijection → R (iv) The monodromy of the Gaudin system can be conjugated into GL4(R). Moreover, the monodromy around components of D is conjugate to diag(1, 1, 1, 1) and near intersection of two components to (diag(1, 1, 1, 1), diag(1, 1, 1, 1)). − − − 1 Thus in the case of G = P GL2 and X = P with 5 parabolic points we have proved all the conjectures from [EFK2] (for F = C).

7.3.2. The Schwartz space. We can now geometrically describe the Schwartz space in the case of 5 points for F = C.36 (The geometric description of the Schwartz space forS 4 points was given in [EFK1]). Namely, consider an eigenfunction ψ near a generic point y of one of the components of D, 2 say D , the exceptional line over (s, t, 1) P . By Theorem 7.9(iii), we have ψ = ψ0 + ψ1 z ∅ ∈ | | for smooth ψ0, ψ1 near y. A more careful analysis shows that ψ1 is not only smooth but has an additional property. To fomulate this property, recall that ψ is not a function but rather a half-density. It turns out that the condition on the half-density ψ1 z is that it has to be the pullback of a smooth half-density from a neighborhood of (s, t, 1)| | P2 under the map 2 ∈ π : Bun◦ P . Note that in suitable local coordinates π is given by the formula 0 → π(z,w)=(z,zw), so a pullback of a smooth half-density f(z,u) dz du looks like | ∧ | f(z,zw) dz d(zw) = f(z,zw) z dz dw . | ∧ | | |·| ∧ | (note that it is not smooth!). In other words, we ﬁnd that in such coordinates

ψ1(z,w)= f(z,zw) for a smooth function f. This can be deduced from the Gaudin system for ψ. For example, the fact that ψ1(0,w) = f(0, 0) is constant follows from equation (7.1) after passing to the coordinates z,w. This leads to the following description of the Schwartz space deﬁned in Subsection 4.3. S ⊂ H

Theorem 7.10. For N =5 the Schwartz space consists of continuous half-forms on Bun0◦ which are S (1) smooth outside D; (2) Near a generic point of a component D D have the form ψ + ψ z , where ψ is S ⊂ 0 1| | 0 smooth and ψ z is the pullback of a smooth half-density from the blow-down of Bun◦ along 1| | 0 DS;

(3) Near the intersection of two components DS1 ,DS2 D have the form ψ0+ψ1 z +ψ2 w , where ψ is smooth and ψ z , ψ w are pullbacks of smooth⊂ half-densities from| the| blow-| | 0 1| | 2| | downs of Bun0◦ along DS1 and DS2 , respectively.

8. Appendix: auxiliary results 8.1. Lemmas on integrals over local ﬁelds. Let F be a local ﬁeld.

36We are grateful to D. Gaiotto and E. Witten for helping us formulate this description. 76 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Lemma 8.1. We have

du 1 k k log ε− , ε 0. u F : u 1 u(u + ε) ∼ → Z ∈ k k≤ k k Proof. This is easily obtained fromp the change of variable u = εv.

Now for x F , x =0, 1 consider the distribution on P1(F ) given by ∈ 6 ds Ex(φ) := φ(s) k k , F s(s 1)(s x) Z k − − k where φ is a smooth function on P1(F ). p

Lemma 8.2. We have 1 Ex x − 2 log x δ , x , ∼k k k k ∞ →∞ where δ is the delta distribution at . In other words, for any test function φ, ∞ ∞ Ex(φ) lim 1 = φ( ). x →∞ x − 2 log x ∞ k k k k 1 Proof. This follows easily from Lemma 8.1 by making a change of variable s s− . 7→ Let θ : F C be the function deﬁned by the formula θ(a) = 1 if a is a square and θ(a)=0 otherwise. →

Lemma 8.3. For b F × one has ∈ 2θ(x2 4b) 1 + sign(log x ) − k k dx = log b . 2 F x 4b − x ! k k − k k Z k − k k k Proof. We have p 2θ(x2 4b) dx dx − k k = k k. 2 x R x 4b x R,x2 y2=4b y Zk k≤ k − k Zk k≤ − k k p x y x+y 1 1 1 Making the change of variables u = −2 , v = 2 , we get v = bu− , x = u+bu− , y = u bu− . dx du − Thus we get y = u , hence

dx du k k = . x R,x2 y2=4b y u+bu−1 R u Zk k≤ − Zk k≤ k k

For large R, the latter integral is close to du du 1 + sign(log x ) = = k k dx log b , u , bu−1 R u R−1 b u R u x R x k k − k k Zk k k k≤ Z k k≤k k≤ Zk k≤ k k in the sense that the diﬀerence goes to zero as R . This implies the statement. →∞ 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 77

8.2. Elliptic integrals. In particular, the asymptotics of Lemma 8.2 applies to the function ds E(x) := Ex(1) = k k . F s(s 1)(s x) Z k − − k If F = R, this function is a classical elliptic integral,p and it is also expressible in terms of elliptic integrals for F = C. Thus we will call E(x) the elliptic integral of F . Example 8.4. Let us compute the elliptic integral E(x) over a non-archimedian local ﬁeld F with residue ﬁeld Fq. Proposition 8.5. For x F with 0 < x 1, 1 x =1, we have ∈ k k ≤ k − k 1 1 1+4q 2 + q E(x)= − − log q log x , 1 q 1 − k k − − For other x =0, 1, the value of E(x) is computed from this formula using the equalities 6 1 1 E(x)= E(1 x), E(x− )= x 2 E(x). − k k 1 k Proof. Let π F be a uniformizer, then π = q− . Since x 1, we have x = q− for k 0. ∈ k k k k ≤ k k ≥ 1 q−1 We will compute E (x) := − E(x). We have ∗ log q

E (x)= En(x), ∗ n Z X∈ where n ds En(x) := q− 2 k k , s: s =qn (s 1)(s x) Z k k k − − k 1 with the usual normalization of ds . Note thatp the change of variable s′ = xs− yields the identity k k

En(x)= E k n(x). − − n n For s F with s = q lets ¯ F× be the image of π s. Thus ∈ k k ∈ q

En(x)= En,a(x), × a Fq X∈ where n ds En,a(x) := q− 2 k k . s: s =qn,s¯=a (s 1)(s x) Z k k k − − k Thus if n> 0, we get p n 1 n 1 E (x) := q− 2 − , E (x)=(q 1)q− 2 − . n,a n − So 3 q 1 1 2 2 1 En(x)= En(x)= q− − 1 = q− + q− . 2 n>0 n< k 1 q− X X− − Assume now that k = 0 (which can happen only if q 3). Then it remains to compute E (x). Here we will use that 1 x = 1. For a =1, x¯ we≥ have 0 k − k 6 1 E0,a(x)= q− , 78 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

and 1 du 1 ∞ n 1 2 2 1 2 2 1 E0,1(x)= E0,x¯(x)= q− k k1 = q− (1 q− )q− = q− + q− . 2 − u 1 u n=0 Zk k≤ k k X So altogether we get

1 1 1 1 1 E (x)=4(q− 2 + q− )+(q 3)q− =1+4q− 2 + q− . ∗ − Now assume that k> 0. Then similarly for 0

1 1 1 1 1 1 1 E (x)=2(q− 2 + q− )+(k + 1)(q 1)q− +2q− 2 =1+4q− 2 + q− + k(1 q− ). ∗ − − Thus 1 1 1 1 1+4q− 2 + q 1+4q− 2 + q E(x)= − + k log q = − log q log x , 1 q 1 1 q 1 − k k − − ! − − as claimed. It is also useful to consider (over any local ﬁeld F ) the modiﬁed elliptic integral θ(s(s 1)(s x)) ds E+(x) := 2 − − k k, F s(s 1)(s x) Z k − − k where we recall that θ(a) = 1 if a is a squarep and θ(a) = 0 otherwise. This function computes the volume of the elliptic curve Cx given by the equation v2 = u(u 1)(u x) − − du over F with respect to the Haar measure v .

8.3. The Harish-Chandra theorem. Let G be a reductive algebraic group deﬁned over a local ﬁeld F , and ρ : G(F ) Aut(V ) be an irreducible unitary representation of G(F ). Let dg be a Haar measure on G→(F ). Theorem 8.6. (see [HC] for the archimedian case and [J] for the non-archimedian case) For any smooth compactly supported function φ on G(F ),37 the operator

Aφ := ρ(g)φ(g)dg ZG(F ) on V is trace class.

Remark 8.7. In the non-archimedian case, the operator Aφ is actually of ﬁnite rank. This allows us to deﬁne the Harish-Chandra character of ρ to be the distribution χ = χρ on G such that TrAφ =(χ, φ).

37As usual, over a non-archimedian ﬁeld smooth means locally constant. 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 79

8.4. The principal series representation W and its Harish-Chandra character. Let F be a local ﬁeld. Then the group G = P GL2(F ) acts by unitary operators on the space of square-integrable half-densities W := L2(P1(F )). Explicitly, this space can be realized as the space L2(F ) with the norm deﬁned by the Haar measure and right action of G given by the formula 1 a b ad bc 2 az + b ρ f (z)= k − k f . c d cz + d cz + d k k This is the principal series representation with parameter ν = 0. The following proposition is classical.

Proposition 8.8. [GGP] (i) The Harish-Chandra character χW of the representation W is a locally integrable function on G; (ii) χW is given by the formula 2 χW (g)= x y y − x

if g is hyperbolic with a lift to GL2(F ) having eigenvalues x, y F , and χW (g)=0 otherwise. ∈ 8.5. A lemma on algebraic groups. Let k be a ﬁeld, C an irreducible algebraic curve over k. Let G be a connected algebraic group over k of dimension d and ξ : C G a regular map. Suppose that G is generated by the image of ξ. Let ξ : Cn G be→ the map given n → by ξn(s1, ..., sn)= ξ(s1)...ξ(sn), and Zn be the closure of the image of ξn.

Lemma 8.9. dim Zn = n for n d. In particular, Zn = G and ξn is dominant for any n d. ≤ ≥ Proof. We have Zn Zn+1 and Zn are irreducible closed subvarieties of G. So if Zn = Zn+1 then dim Z > dim⊂Z . On the other hand, if Z = Z then ξ(s)Z Z for all s6, hence n+1 n n n+1 n ⊂ n Zn = G. This implies the statement.

8.6. Irreducibility of the character variety. Let c0, ..., cm+1 C× and Locm(c0, ..., cm+1) 1 ∈ be the variety of irreducible rank 2 local systems on CP t0, ..., tm+1 with local mon- 1 \{ } odromies at ti regular with eigenvalues ci,ci− . It is easy to show that this is a smooth variety of pure dimension 2(m 1). The following lemma is well− known, but we provide a proof for reader’s convenience.

Lemma 8.10. For m 1, the variety Locm(c0, ..., cm+1) is irreducible (equivalently, connected), and for m 2 ≥it is nonempty. ≥ Example 8.11. A simple computation shows that the variety Loc1(c0,c1,c2) consists of one 1 1 1 point or is empty, and the latter happens iﬀ c0± c1± c2± = 1 for some choice of signs.

Proof. Let Locm(c0, ..., cm+1) be the variety of irreducible m + 2-tuples of non-scalar 2-by-2 m+1 1 matrices (C0, ..., Cm+1) such that i=0 Ci = 1 and the eigenvalues of Ci are ci,ci− . Then P GL actsg freely on Loc (c , ..., c ) and Loc (c , ..., c ) = Loc (c , ..., c )/PGL . 2 m 0 Qm+1 m 0 m+1 m 0 m+1 2 Thus Locm(c0, ..., cm+1) is a smooth variety of pure dimension 2m + 1, and it suﬃces to show that it is irreducible. g g red Firstg consider the variety Locm(c0, ..., cm+1) of reducible tuples satisfying the same conditions (which is non-empty only for special eigenvalues). Such a tuple preserves a line in g 80 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

the 2-dimensional space, and once such line is chosen, is determined by an extension between two 1-dimensional representations of the free group Fm+1 in which the generators map to 1 ci± . Thus (8.1) dim Loc (c , ..., c )red m +2. m 0 m+1 ≤ We will prove the lemma by induction in m. The base case m = 1 is discussed in Example g 8.11. So let m 2 and suppose the statement is known for smaller values. Let C = CmCm+1. It is easy to see≥ that the matrix C can be any non-scalar matrix. So we have a dominant map π : Loc (c , ..., c ) SL m 0 m+1 → 2 which sends (C0, ..., Cm+1) to CmCm+1. 1 1 Let us compute the fiber π− (C). If C is not a scalar and has eigenvalues c,c− then 1 π− (C) consists of four parts: 1 (i) (C0, ..., Cm 1,C) and (C− ,Cm,Cm+1) are both irreducible. By the induction assumption the first one− runs over a variety of dimension 2m 3 and the second over a variety of dimension 1 (as C is fixed), so this piece has dimension− (2m 3)+1=2m 2. Moreover, it is non-empty for generic C. So the union over all C is non-empty− of dimension− 2m + 1, and it is irreducible by the induction assumption. 1 (ii) (C0, ..., Cm,C) is irreducible, but (C− ,Cm+1,Cm) is reducible. This can happen only for special eigenvalues of C and gives dimension (2m 3)+1=2m 2 as well. So the union over all C is of dimension 2m. − − ≤ 1 (iii) (C0, ..., Cm 1,C) is reducible, but (C− ,Cm+1,Cm) is irreducible. Then by (8.1) the first tuple runs over− a variety of dimension m 1 (as C is fixed), while the second one runs over a variety of dimension 1, so this piece≤ has− dimension (m 1)+1= m. As this occurs only for special eigenvalues of C, the union over all C has≤ dimension− m + 2. It remains to consider the case when C = 1 is a scalar. These two cases≤ are equivalent so we only consider C = 1. Thus we have two± options: (i) (C0, ..., Cm 1) is irreducible. Then we get a piece of dimension (2m 3)+2=2m 1 − − − (the summand 2 accounts for the choice of Cm in its conjugacy class). (ii) (C0, ..., Cm 1) is reducible. By (8.1) this gives a piece of dimension m + 2. − ≤ Since m +2 2m and the variety Locm(c0, ..., cm+1) has pure dimension, we obtain that this variety is irreducible,≤ as claimed. g 8.7. Interpolation of rational functions. Let x0, ..., x2n be a fixed collection of distinct 1 points on P over a field k. Let n be the variety of rational functions f(x) of degree n, i.e., 1 U1 1 2n+1 defining a degree n map P P . We have a regular map ιx : (P ) given by → Un → ιx(f) := (f(x0), ..., f(x2n)).

Lemma 8.12. (i) ιx is an open embedding. 1 det A(x,s,w) (ii) If x k and s =(s , ..., s ) then ι− (s)= , where A =(a ), B =(b ) are i ∈ 0 2n x det B(x,s,w) ij ij the following square matrices of size 2n +1: s 1 j 1 j n 1 a = i , b = ; a = b = s x − , 1 j n; a = b = x − − , n+1 j 2n. i0 w xi i0 w xi ij ij i i ij ij i − − ≤ ≤ ≤ ≤ Proof. The lemma is classical but we give a proof for reader’s convenience. If ιx(f1)= ιx(f2) and fk = pk/qk, k = 1, 2, where pk, qk are polynomials of degree n, then p1q2 q1p2 is a polynomial of degree 2n which vanishes at 2n + 1 points x , x , ..., x , so it− vanishes ≤ 0 1 2n identically and f = f . Thus ιx is injective. Since dim = 2n + 1, this implies that ιx 1 2 Un 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 81

is dominant. Thus ιx is birational, and moreover Zariski’s main theorem implies that it is actually an open embedding, proving (i). Part (ii) (due to Cauchy and Jacobi) follows from Cramer’s rule (see e.g. [GM]). 8.8. The exponent of a sequence. Let a := a , i 1 be a sequence of nonnegative { i ≥ } numbers such that ai 0. Let na(ε) be the number of i such that ai ε. Then we have the following well known→ lemma from calculus. | | ≥ b Lemma 8.13. (i) Suppose that na(ε) = O(ε− ) as ε 0, for some b 0. Then for any p → ≥ p > b we have i 1 ai < . p ≥ ∞ p (ii) If i 1 ai < for some p> 0 then na(ε)= o(ε− ) as ε 0. ≥ P ∞ → Proof. ByP rescaling we may assume that ai < 1 for all i. Then

p 1 1 1 1 a na C i ≤ mp − (m + 1)p m +1 ≤ mp+1 b i 1 m 1 m 1 − X≥ X≥ X≥ for some C > 0, which implies the statement. p (ii) Assume the contrary, then there is a sequence εm 0 such that na(εm) Kεm− for some K > 0. We may choose this sequence in such a way→ that ε εm . Then≥ m+1 ≤ 2 p p p (ε ε )na(ε ) a < . m − m+1 m ≤ i ∞ m 1 i 1 X≥ X≥ Thus p p p (ε ε )ε− < , m − m+1 m ∞ m 1 ≥ p X p p p which is a contradiction since (ε ε )ε− > 1 2− . m − m+1 m − 1 Thus if na(ε) has at most polynomial growth in ε− then the number b = b(a) := p log na(ε) inf p : a < may also be described as limsup − . { i i ∞} log(ε 1) DeﬁnitionP 8.14. We will call the reciprocal 1/b of this number the exponent of a. 1 For example, the exponent of the sequence am = mp is p. 8.9. Lemmas on compact operators. Let A be a compact self-adjoint operator on a separable Hilbert space . Denote by n(A, ε) the number of eigenvalues of A of magnitude ε counted with multiplicities.H ≥ Lemma 8.15. (i) Let A = B + E, where B has rank r and E has norm < ε. Then n(A, ε) r. (ii) Let≤ D be another compact self-adjoint operator such A D δ. Then for any ε> 0, k − k ≤ n(A, δ + ε) n(D,ε). ≤ Proof. (i) Let V be the span of all eigenvectors of A with eigenvalues of magnitude ε. Then for any nonzero vector v V , we have (Av, v) ε(v, v). On the other hand,≥ (Av, v)=(Bv,v)+(Ev,v) and (∈Ev,v) <ε(v, v).| Thus |Bv ≥= 0. Hence dim V r. (ii) Let Π be the orthogonal projection| | to the sum of eigenspaces6 of D with≤ eigenvalues of magnitude ε. Then D = ΠD + (1 Π)D, so A = B + E, where B = ΠD and E = (1 Π)D≥+(A D). Since (1 Π)D− < ε, we have E < δ + ε. Also, the rank of ΠD is n−(D,ε). Thus− the result followsk − fromk (i). k k 82 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

Lemma 8.16. Let T : L2(M) L2(M) be a bounded integral operator on an n-dimensional analytic F -manifold M whose→ Schwartz kernel is supported on a submanifold of M M of dimension n + k, where k < n. If Tr T 2p < then p n .38 × | | ∞ ≥ k Proof. First suppose that k +1 < n and p> 1 is such that Tr T 2p < . Pick an integral operator A on L2(M) with Schwartz kernel supported on an n+1-dimensional| | ∞ submanifold such 2n n k 1 that Tr A < and A − − T † = 0. Such A exists; namely, this holds if the Schwartz kernel of A| is| a smooth∞ compactly supported6 half-density concentrated on an n + 1-dimensional analytic submanifold of M M, both chosen in a sufficiently generic way. Then the opera- n k 1 × tor A − − T † has Schwartz kernel supported in codimension 1, so it is not Hilbert-Schmidt. Hence n k 1 n k 1 2 n k 1 2 Tr(T (A − − )†A − − T †) = Tr( T A − − )= , | | | | ∞ Thus by the Hölder inequality for Schatten norms p p− 2p 1 n k 1 2 1 (Tr T ) p (Tr A − − p−1 ) p = . | | | | ∞ 2p 2(n−k−1)p n k 1 − − Hence Tr A − − p 1 = . Using the Hölder inequality again, this yields Tr A p 1 = . This implies| that| ∞ | | ∞ (n k 1)p n> − − , p 1 − i.e. n (8.2) p> . k +1 N N Now consider the operator T ⊗ . It is an integral operator on the manifold M of dimension Nn with Schwartz kernel supported on a submanifold of dimension Nn + Nk. Also N 2p 2p N N Tr T ⊗ = (Tr T ) < . So applying the bound (8.2) to T ⊗ , we obtain | | | | ∞ Nn p> . Nk +1 Now sending N , we get p n , as claimed. →∞ ≥ k 8.10. Self-adjoint second order differential operators on the circle. Let L := ∂b∂+U be a second order differential operator on R/Z, where b, U are smooth real functions such that b has simple zeros (note that the number of zeros of b is necessarily even). We’d like to study self-adjoint extensions of L initially defined on the domain C∞(R/Z), on which L is symmetric. Our main example will be the Laméoperator L = ∂z(z 1)(z t)∂ + z, t R, − 1 − ∈ t = 0, 1, acting on half-densities on R, which in the coordinate x = π− arccot(z) takes the form6 L = ∂b∂ + U for 2 b(x)= π− cos πx sin πx(sin πx cos πx)(t sin πx cos πx), − − 1 1 1 which has four simple zeros on R/Z, namely 0, 4 , 2 , π− arccot(t). 2 Consider the closure of L on C∞(R/Z), and let V L (R/Z) be the domain of the ⊂ adjoint operator L†. Then L† is not symmetric on V if b has zeros. So we may define a skew-hermitian form on V given by ω(v,u)=(L†v,u) (v, L†u). − Proposition 8.17. ω has rank 2r, where r is the number of zeros of b, and signature (r, r). Moreover, the domain of the closure of L is V ⊥ V , so V/V ⊥ has dimension 2r. ⊂ 38Here T := √T †T . | | 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 83

2 Proof. Let xi, i = 1, ..., r, be the zeros of b. Note that V is the space of L functions f on 2 1 R/Z such that (bf ′)′ L (thus f is in C outside xi). Hence bf ′ belongs to the Sobolev 1,2 ∈ space W , so it is continuous. Let bf ′(xi)= ci. Let h be a smooth function on R/Z except at x such that h = c log x x near x . Then h V and bh′(x ) = c . Thus we have a i i | − i| i ∈ i i subspace V0 V of codimension r which is the space of f V such that ci = 0. ⊂ x ∈ 2 Let f V0. Then b(x)f ′(x)= φ(t)dt, for each i, where φ L (R/Z). Thus ∈ xi ∈ x R φ(t)dt f(x)= xi dx + a θ(x x )+ C b(x) i − i R i Z X near x = xi, where θ is the Heaviside function. By the Cauchy-Schwarz inequality we have x 2 x φ(t)dt x x φ(t) 2dt, ≤| − i| | | Zxi Zxi which means that x 1 φ(t)dt = o( x x 2 ), x x . | − i| → i Zxi Thus f is continuous on each side near xi (but may jump at xi). Now, we have 1 1 ω(v, u)= (bv′)′udx v(bu′)′dx. − Z0 Z0 Integrating this by parts, we get

ω(v, u) = lim (b(v′u u′v)(xi + ε) b(v′u u′v)(xi ε)). ε 0 − − − − → i X Assume that v,u V0. Then, since bu′(xi)= bv′(xi), and u and v are continuous near xi on each side, we get ∈ ω(v, u)=0. Thus V is an isotropic subspace of V with respect to ω, which shows that rank(ω) 2r and 0 ≤ dim V/V ⊥ 2r. It remains≤ to show that this dimension is exactly 2r. For this purpose for each i let v = log x x and u = θ(x x ) near x = x (and smooth at other x ). Then | − i| − i i j 1 ω(vi, ui) = lim b(x xi)(x xi)− = b′(xi) =0. x xi → − − 6 On the other hand, ω(vi, vj)= ω(ui, uj)= ω(vi, uj)=0 for i = j, and ω(u , u )= ω(v , v ) = 0 for all i. This implies the statement. 6 i i i i From the proof we also obtain the following corollary.

Corollary 8.18. (i) V ⊥ = V C(R/Z), the space of all continuous functions in V . (ii) Near each point x every∩ element f V can be uniquely written as i ∈ f(x)= c log x x + a θ(x x )+ f (x), i | − i| i − i i where fi is continuous. Moreover, we have r ω(v,u)= (c (v)a (u) a (v)c (u)). i i − i i i=1 X 84 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

(iii) The subspaces W V containing V ⊥ on which L† is closed symmetric (respectively, self-adjoint) are in natural⊂ bijection with isotropic (respectively, Lagrangian) subspaces in 2r V/V ⊥ = C with respect to the skew-hermitian form (c a′ a c′ ), via W W/V ⊥. i i i − i i 7→ (iii) Let C∞ be the space of functions in W which are smooth outside the points xi and of W P the form (ii) with f smooth at each x . Then for any V ⊥ W V , the operator L maps i i ⊂ ⊂ CW∞ to itself, and W is the domain of the closure of L initially deﬁned on CW∞. Moreover, L is symmetric (respectively, essentially self-adjoint) on CW∞ if and only if W is isotropic (respectively, Lagrangian).

We have seen that one example of a space W with Lagrangian W/V ⊥ is W = V0, which may be characterized as the space of bounded functions in V . Another example is the space W = V1 of functions which near xi look like f(x)= c log x x + f (x), i | − i| i where fi is continuous. Corollary 8.19. The operator L is essentially self-adjoint on the spaces CV∞0 and CV∞1 . Remark 8.20. 1. When L is essentially self-adjoint, it has a discrete spectrum, determined from the corresponding singular Sturm-Liouville problem. 2. If we want to have an invariant domain that contains the eigenfunctions of L, we should

slightly enlarge the spaces CV∞0 , CV∞1 .

Namely, they can be replaced with the space CV∞0 of functions in V which are smooth outside xi and near xi are of the form f(x)= g (x)θ(x xe)+ f (x), i − i i

where fi,gi are smooth, and the space CV∞1 of functions in V which are smooth outside xi and near xi are of the form f(x)= g (xe) log x x + f (x), i | − i| i where fi,gi are smooth.

Remark 8.21. 1. The space V is a module over C∞(R/Z). Indeed, if φ C∞(R/Z) then ∈ 2 L(φf) = fLφ + φLf +2φ′bf ′. It is clear that the ﬁrst two summands are in L , and so is 1,2 2 the last summand, since bf ′ W , hence in L . ∈ 2. V is invariant under changes of variable x which do not move the points xi. Indeed, 1 1 let x = g(u) be a change of variable. Then L transforms to L′ = g′(u)− ∂ub(g(u))g′(u)− ∂u. 1 2 b(g(u)) Thus we need to check that for f V we have ∂ub(g(u))g′(u)− ∂uf L . Let h(u)= b(u)g′(u) . ∈ 2 ∈ So we need to check that (bhf ′)′ L . We have ∈ (bhf ′)′ = hLf + h′bf ′, 2 1,2 2 which is in L since bf ′ W , hence in L . ∈ 3. Let c C∞(R/Z), and consider the operator ∈ L(b, c, U)= ∂b∂ + i(bc∂ + ∂bc)+ U (so that the previous setting is recovered by putting c = 0). This operator is symmetric on smooth functions if b, c, U are real-valued. Then the space V = V (b, c, U) of functions f L2(R/Z) and L(b, c, U)f L2(R/Z) is independent on b, c, U (as long as the points x ∈ ∈ i are ﬁxed). Indeed, by the C∞-version of [EFK1], Lemma 13.2, locally near xi by changes of 1 ANALYTIC LANGLANDS CORRESPONDENCE FOR P GL2 ON P 85

variable and left and right multiplication by invertible functions, L(b, c, U) can be uniquely brought to the form ∂x∂, and we have seen that such transformations preserve the spaces V . 4. Our arguments show that the space V ⊥ (for any b, c, U) has the following explicit description. It is the space of functions of the form f = gdx + C, where g W 1,2 is such b ∈ that g(x ) = 0 and gdx = 0. Namely, g = bf . Note that the integral is (absolutely) i R/Z b ′ R convergent since any function in W 1,2 is 1/2-Hölder continuous (Morrey inequality). For the R same reason (continuity of g) this description is independent on the choice of b. Thus, we get the following proposition. Proposition 8.22. The space V (b, c, U) is independent on b, c, U and can be described as the space of functions g(x) f(x)= c θ(x x )+ dx + C, c C, i − i b(x) i ∈ i X Z where g W 1,2 is such that g(x) dx = c (where the integral is understood in the ∈ R/Z b(x) − i i sense of principal value). In particular, the continuous part f(x) ciθ(x xi) of f at xi is 1 R P 1 − − -Hölder continuous at x (so is const + O( x x 2 ) as x x ). 2 i | − i| → i References [Be] F. Beukers, Unitary Monodromy of LaméDifferential Operators, Regular and Chaotic Dynamics 12, No. 6 (2007) 630–641. [BD1] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint http://www.math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf [BD3] A. Beilinson and V. Drinfeld, Opers, arXiv:math/0501398. [C] C. Casagrande, Rank 2 quasiparabolic vector bundles on P1 and the variety of linear subspaces contained in two odd-dimensional quadrics, Math. Z. 280 (2015) 981–988. [CC] D.V. Chudnovsky and G. V. Chudnovsky, Computational problems in arithmetic of linear differential equations. Some Diophantine applications, in Number theory, eds. D.V. Chudnovsky, e.a. pp. 12–49, Lecture Notes in Math. 1383, Springer, Berlin, 1989. [De] P. Deligne, Un théoréme de finitude pour la monodromie, in Discrete groups in geometry and analysis, ed. R. Howe, pp. 1–19, Prog. in Math. 67, Birkhäuser, 1987. [Dr1] V. G. Drinfeld, Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2), Amer. J. Math. 105 (1983) 85–114. [Dr2] V. Drinfeld, Langlands conjecture for GL(2) over functional fields, Proc. of Int. Congress of Math- ematicians (Helsinki, 1978), pp. 565—574, Acad. Sci. Fennica, Helsinki, 1980. [DP1] R. Donagi and T. Pantev, Parabolic Hecke eigensheaves, arXiv:1910.02357. [DP2] R. Donagi and T. Pantev, private communication. [EFK1] P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, dedicated to Boris Dubrovin, Vol. II, eds. S. Novikov, e.a., pp. 137–202, Proc. Symp. Pure Math. 103.2, AMS, 2021 (arXiv:1908.09677). [EFK2] P. Etingof, E. Frenkel, and D. Kazhdan, Hecke operators and analytic Langlands correspondence for curves over local fields, arXiv:2103.01509. [EK] P. Etingof and D. Kazhdan, Characteristic functions of p-adic integral operators, arXiv:2101.05185. [Fa] G. Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (1983) 223–269. [Fr1] E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in Proc. of Int. Congress of Math. Phys. (Paris, 1994), ed. D. Iagolnitzer, pp. 606–642, International Press, 1995 (arXiv:qalg/9506003). [FS] E. Frenkel and A. Szenes, Thermodynamic Bethe ansatz and dilogarithm identities. I, Math.Res. Lett. 2 (1995) 677–693. 86 PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN

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Department of Mathematics, MIT, Cambridge, MA 02139, USA

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel