Is there an analytic theory of automorphic functions for complex algebraic curves?

The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands raised the possibility of developing an analytic theory of automorphic forms on the moduli of G-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of certain Hecke operators, which he attempted to define. In these notes I show that Hecke operators are well-defined if G is abelian and give a complete description of their eigenfunctions and eigenvalues in this case. However, for non-abelian G, Hecke operators involve integration, which is problematic in the context of complex curves. Nonetheless, automorphic forms can be defined in a different way -- not as eigenfunctions of Hecke operators, but rather as eigenfunctions of a commutative algebra of global differential operators on the line bundle of half-densities on the moduli of G-bundles.

1. Introduction 1.1. The foundations of the Langlands Program were laid by Robert Langlands in the late 1960s [L1]. Originally, these ideas were applied in two realms: that of number fields, i.e. finite extensions of the field Q of rational numbers, and that of function fields, where by a function field one understands the field of rational functions on a smooth projective curve over a finite field F q . In both cases, the objects of interest are automorphic forms, which are, roughly speaking, functions on the quotient of the form G(F )\G(A F )/K, where G is a reductive algebraic group over F , the field in question (a number field or a function field), A F is the ring of adeles of F , and K is a compact subgroup of G(A F ). There is a family of mutually commuting Hecke operators acting on this space of functions, and one wishes to describe the common eigenfunctions of these operators as well as their eigenvalues. The idea is that those eigenvalues can be packaged as the "Langlands parameters" which can be described in terms of homomorphisms from a group closely related to the Galois group of F to the Langlands dual group L G associated to G, and perhaps some additional data.
To be more specific, let F be the field of rational functions on a curve X over F q and G = GL n . Let us further restrict ourselves to the unramified case, so that K is the maximal compact subgroup K = GL n (O F ), where O F ⊂ A F is the ring of integer adeles. In this case, a theorem of V. Drinfeld [Dr1,Dr2] for n = 2 and L. Lafforgue [Laf] for n > 2 states that (if we impose the so-called cuspidality condition and place a restriction on the action of the center of GL n ) the Hecke eigenfunctions on GL n (F )\GL n (A F )/GL n (O F ) are in one-toone correspondence with n-dimensional irreducible unramified representations of the Galois group of F (with a matching restriction on its determinant).
Notes for a talk at the 6th Abel Conference, University of Minnesota, November 14, 2018.
1.2. Number fields and function fields for curves over F q are two "languages" in André Weil's famous trilingual "Rosetta stone" [We], the third language being the theory of algebraic curves over the field C of complex numbers. Hence it is tempting to build an analogue of the Langlands correspondence in the setting of a complex curve X. Such a theory has indeed been developed starting from the mid-1980s, initially by V. Drinfeld [Dr2] and G. Laumon [La1] (and relying on ideas of an earlier work of P. Deligne), then by A. Beilinson and V. Drinfeld [BD], and subsequently by many others. See, for example, the surveys [Fr2,Gai3] for more details. However, this theory, dubbed "geometric Langlands Program," is quite different from the Langlands Program in its original formulation for number fields and function fields.
The most striking difference is that in the geometric theory the vector space of automorphic functions on the double quotient G(F )\G(A F )/K is replaced by a (derived) category of sheaves on an algebraic stack whose set of C-points is this quotient. For example, in the unramified case K = G(O F ), this is the moduli stack Bun G of principal G-bundles on our complex curve X. Instead of the Hecke operators of the classical theory, which act on functions, we then have Hecke functors acting on suitable categories of sheaves, and instead of Hecke eigenfunctions we have Hecke eigensheaves.
For example, in the unramified case a Hecke eigensheaf F is a sheaf on Bun G (more precisely, an object in the category of D-modules on Bun G , or the category of perverse sheaves on Bun G ) with the property that its images under the Hecke functors are isomorphic to F itself, tensored with a vector space (this is the categorical analogue of the statement that under the action of the Hecke operators eigenfunctions are multiplied by scalars). Furthermore, since the Hecke functors (just like the Hecke operators acting on functions) are parametrized by closed points of X, a Hecke eigensheaf actually yields a family of vector spaces parametrized by points of X. We then impose an additional requirement that these vector spaces be stalks of a local system on X for the Langlands dual group L G (taken in the representation of L G corresponding to the Hecke functor under consideration). This neat formulation enables us to directly link Hecke eigensheaves and (equivalence classes of) L G-local systems on X, which are the same as (equivalence classes of) homomorphisms from the fundamental group π 1 (X, p 0 ) of X to L G.
This makes sense from the point of view of Weil's Rosetta stone, because the fundamental group can be seen as a geometric analogue of the unramified quotient of the Galois group of a function field. We note that for G = GL n , in the unramified case, the Hecke eigensheaves have been been constructed in [Dr1] for n = 2 and in [FGV,Gai1] for n > 2. More precisely, the following theorem has been proved: for any irreducible rank n local system E on X, there exists a Hecke eigensheaf on Bun GLn whose "eigenvalues" correspond to E. 1 Many results of that nature have been obtained for other groups as well. For example, in [BD] Hecke eigensheaves on Bun G were constructed for all L G-local systems having the structure of an L G-oper (these local systems form a Lagrangian subspace in the moduli of all L G-local systems). Furthermore, a more satisfying categorical version of the geometric Langlands correspondence has been proposed by A. Beilinson and V. Drinfeld and developed further in the works of D. Arinkin and D. Gaitsgory [AG,Gai2] (see [Gai3] for a survey).
To summarize, the salient difference between the original formulation of the Langlands Program (for number fields and function fields of curves over F q ) and the geometric formulation is that the former is concerned with functions and the latter is concerned with sheaves.
What makes this geometric formulation appealing is that in the intermediate case -that of curves over F q -which serves as a kind of a bridge in the Rosetta stone between the number field case and the case of curves over C, both function-theoretic and sheaf-theoretic formulations make sense. Moreover, it is quite common that the same geometric construction works for curves over F q and C. For example, essentially the same construction produces Hecke eigensheaves on Bun GLn for an irreducible rank n local system on a curve over F q and over C [Dr1,FGV,Gai1]. 2 Furthermore, in the realm of curves over F q , the function-theoretic and sheaf-theoretic formulations are connected to each other by Alexander Grothendieck's "functions-sheaves dictionary." This dictionary assigns to a (ℓ-adic) sheaf F on a variety (or an algebraic stack) V over F q , a function on the set of closed points of V whose value at a given closed point v is the alternating sum of the traces of the Frobenius (a generator of the Galois group of the residue field of v) on the stalk cohomologies of F at v (see [La2], Sect. 1.2 or [Fr2], Sect. 3.3 for details). Thus, for curves over F q the geometric formulation of the Langlands Program may be viewed as a refinement of the original formulation: the goal is to produce, for each L G-local system on X, the corresponding Hecke eigensheaf on Bun G , but at the end of the day we can always go back to the more familiar Hecke eigenfunctions by taking the traces of the Frobenius on the stalks of the Hecke eigensheaf at the F q -points of Bun G . Thus, the function-theoretic and the sheaf-theoretic formulations go hand-in-hand for curves over F q .
1.3. In the case of curves over C there is no Frobenius, and hence no direct way to get functions out of Hecke eigensheaves on Bun G . However, since a Hecke eigensheaf is a Dmodule on Bun G , we could view its sections as analogues of automorphic functions of the analytic theory. The problem is that for non-abelian G, these D-modules -and hence their sections -are known to have complicated singularities. Outside of the singularity locus, a Hecke eigensheaf is a holomorphic vector bundle with a holomorphic flat connection, but its horizontal sections undergo non-trivial monodromies as we move around the multiple components of the singularity locus. So instead of functions we get multi-valued sections of a vector bundle. On top of that, in the non-abelian case the rank of this vector bundle grows exponentially as a function of the genus of X, and furthermore, the components of the singularity locus have a rather complicated structure. Thus, it is the D-modules themselves, rather than their sections, that are more meaningful objects of study, and that's why traditionally, in the geometric formulation of the Langlands Program for curves over C, we focus on these D-modules rather than their multi-valued sections. For this reason, the geometric theory in the case of complex curves becomes inherently sheaf-theoretic. It appears to be far away from the more familiar world of automorphic functions (though, as we will see later on, this appearance is somewhat misleading).
1.4. Against this backdrop, in a recent preprint [L2] Robert Langlands asked whether it is possible to develop a function-theoretic version of the Langlands correspondence for complex algebraic curves. He suggested considering the space of L 2 functions on Bun G (with respect to some unspecified integration measure) and defining Hecke operators acting on it. By analogy with the Satake isomorphism of the original formulation, he anticipated the eigenvalues of these Hecke operators to give rise to functions on the curve X with values in conjugacy classes of L G. He proposed that these functions could be expressed as holonomies of some (projectively) flat L G-connections on X. By taking the monodromy representation of these flat connections, he argued, one would obtain a link between Hecke eigenfunctions (in some space of L 2 functions on Bun G ) and homomorphisms from the fundamental group of X to L G, and furthermore, he argued that they should take values in a maximal compact subgroup of L G. Langlands presented some computations in the case that X is an elliptic curve and G = GL 1 or GL 2 .
In the present paper I discuss this proposal. Let's consider first the case of GL 1 . In this case, the Hecke operators do not involve integration and are well-defined on the space of L 2 functions on the Picard variety of a complex curve X (it is a good substitute for Bun GL 1 , see Section 2.1). The question of finding their eigenfunctions and eigenvalues is well-posed. I give a complete answer to this question in Section 2: first for elliptic curves in Sections 2.1 and 2.3 and then for curves of an arbitrary genus in Section 2.4. Furthermore, in Section 2.5 I generalize these results to the case of an arbitrary torus T instead of GL 1 . In particular, I show that Hecke eigenfunctions are labeled by H 1 (X, Λ * (T )), the first cohomology group of X with coefficients in the lattice of cocharacters of T , and give an explicit formula for the corresponding eigenvalues. The construction uses the Abel-Jacobi map.
However, the results presented in Section 2 do not agree with the proposal made by Langlands in [L2] that Hecke eigenfunctions in the case of GL 1 (and X an elliptic curve) should be in one-to-one correspondence with one-dimensional representations of the fundamental group π 1 (X, p 0 ) of X with finite image. He suggested to establish this correspondence by expressing Hecke eigenvalues as holonomies of a flat connection on a line bundle on X and then taking the monodromy representation of this connection. I show in Section 2 that the Hecke eigenvalues may indeed be written as holonomies of certain flat connections on the trivial line bundle on X (this is so not only for elliptic curves but for curves of an arbitrary genus). But each of these connections gives rise to the trivial monodromy representation of π 1 (X, p 0 ).
Indeed, the collection of the Hecke eigenvalues, labeled by x ∈ X, on a specific Hecke eigenfunction gives rise to a single-valued function on X, and this function is, by definition, a horizontal section of the connection. Therefore, one cannot possibly obtain non-trivial representations of π 1 (X, p 0 ) this way, see Remark 1 in Section 2.2 for more details (see also Remark 2 in which I discuss an alternative proposal made in [L2] to link Hecke eigenvalues and representations of π 1 (X, p 0 )).
What is true is that we have a one-to-one correspondence between the Hecke eigenvalues for GL 1 and certain flat connections on the trivial line bundle on X which give rise to the trivial monodromy representation of π 1 (X, p 0 ). These connections (being defined on the trivial line bundle) are the same as one-forms on X. I show that these are precisely the harmonic one-forms corresponding to the integral cohomology classes of X. The integrality of the cohomology class assures that the corresponding connection has trivial monodromy (note, however, that in Section 3.7 I will discuss a different interpretation of these oneforms). This one-to-one correspondence can be generalized to the case of an arbitrary torus T (and a curve of an arbitrary genus).
1.5. Now consider the case of GL 2 . A special feature of the Hecke operators in the case of GL 1 (which also holds for arbitrary tori), and one that is crucial for the Hecke operators being well-defined in this case, is that they are pull-backs of functions under a map on the moduli space of line bundles on X, which is the Picard variety of X (there is an equivalent reformulation, explained in Section 2, in which the Hecke operators act by pull-backs of functions on the neutral component of the Picard variety). Therefore no integration is needed, and no measure of integration needs to be defined. This is why the Hecke operators are well-defined and the question of finding their eigenfunctions and eigenvalues is well-posed in the abelian case (and furthermore, has a simple answer for a general curve and a general abelian group G, as I show in Section 2).
However, in the non-abelian case, in order to define Hecke operators, one cannot avoid integration, and this necessitates defining a measure of integration on the group G(C((t))). This group is not locally compact (unlike the group G(F q ((t))), for example), and therefore it does not have a Haar measure, which is defined for locally compact groups. The question of defining the measure of integration on G(C((t))) is not addressed in [L2]. Instead, in [L2] an attempt is made to define Hecke operators acting on a particular version of an L 2 space of Bun GL 2 of an elliptic curve, essentially by decree.
I discuss all this in detail in Section 3. First, I explain the obstacles to defining a measure of integration on G(G((t))) that would give rise to a meaningful spherical Hecke algebra having an analogue of the Satake isomorphism. I then discuss whether one could define Hecke operators directly, as operators acting on some space of functions associated to Bun G . Here again we face what appear to be insurmountable obstacles, which I illustrate with concrete examples in Sections 3.3 and 3.4. From this analysis, it appears that for non-abelian G (in fact, already for G = GL 2 and an elliptic curve) it is not possible to give a meaningful definition of Hecke operators from the first principles, and therefore the question of finding their eigenfunctions and eigenvalues is not well-posed.
1.6. There is, however, another possibility: rather than looking for the eigenfunctions of Hecke operators, let's look for the eigenfunctions of global differential operators on Bun G . These eigenfunctions and the corresponding eigenvalues have been recently studied for G = SL 2 in the framework of conformal field theory by Joerg Teschner [T]. In an ongoing joint work with David Kazhdan [FK], we are attempting to extend this analysis to other groups.
According to a theorem of Beilinson and Drinfeld [BD], there is a large commutative algebra of global holomorphic differential operators acting on sections of a square root K 1/2 of the canonical line bundle K on Bun G (this square root always exists, and is unique if G is simply-connected [BD]). The complex conjugates of these differential operators are antiholomorphic and act on sections of the complex conjugate line bundle K 1/2 on Bun G . The tensor product of these two algebras is therefore a commutative algebra that naturally acts on sections of the line bundle K 1/2 ⊗ K 1/2 which we refer to as the bundle of half-densities on Bun G . The space of global sections of the line bundle K 1/2 ⊗ K 1/2 on Bun G (or rather, on its open subspace of stable G-bundles) has a natural norm. Taking the completion of the space of sections with finite norm, we obtain a natural Hilbert space. Our differential operators act on this space (in fact, one can show that they are normal), and we can ask what are their eigenfunctions and eigenvalues. In Section 3.6, as a preview of [FK], we give some more details of this construction. We then look at the abelian case of G = GL 1 in Section 3.7. In the case of GL 1 , the global differential operators are polynomials in the shift vector fields, holomorphic and anti-holomorphic, on the neutral component Pic 0 of Picard variety. These operators commute with each other (and with the Hecke operators, which are available in the abelian case), and their joint eigenfunctions are the standard Fourier harmonics on Pic 0 . What about the eigenvalues? The spectrum of the commutative algebra of global holomorphic differential operators on Pic 0 can be identified with the space of holomorphic connections on the trivial line bundle on X. Hence every eigenvalue of this algebra can be encoded by a point in this space. It turns out that the points corresponding to the eigenvalues of this algebra on the space of L 2 functions on Pic 0 are precisely those holomorphic connections on the trivial line bundle on X that give rise to the homomorphisms π 1 (X, p 0 ) → C × with image in R × ⊂ C × . In other words, these are the connections with real monodromy. This dovetails nicely with the conjecture of Teschner [T] in the case of G = SL 2 . We expect an analogous statement to hold for a general reductive group G [FK].
Suppose for simplicity that G is simply-connected. Then, according to a theorem of Beilinson and Drinfeld [BD], the spectrum of the algebra of global holomorphic differential operators on Bun G is canonically identified with the space of L G-opers on X. In the case G = SL 2 , L G = P GL 2 and P GL 2 -opers are the same as projective connections. Teschner [T] proposed that in this case, the eigenvalues correspond to the projective connections with real monodromy (these projective connections have been described by Goldman [Gol]). For general G, we expect the joint eigenvalues of the global holomorphic differential operators on Bun G to correspond to those L G-opers that have monodromy in the split real form of L G (up to conjugation). If so, then the spectra of the global differential operators on Bun G can be described by analogues of the Langlands parameters of the classical theory: namely, certain homomorphisms from the fundamental group of X to the Langlands dual group L G. A somewhat surprising element is that the homomorphisms that appear here are the ones whose image is in the real form of L G (rather than the compact form). More details will appear in [FK]. 1.7. Thus, there is a rich and meaningful analytic theory of automorphic forms on Bun G , but the key role in this theory is played not by Hecke operators but by the global differential operators on Bun G . This raises the question: is there a connection between this analytic theory and the geometric theory?
Valuable insights into this question may be gleaned from two-dimensional conformal field theory (CFT). In CFT, one has two types of correlation functions. The first type is chiral correlation functions, also known as conformal blocks. They form a vector space for fixed values of the parameters of the CFT. Hence we obtain a vector bundle of conformal blocks on the space of parameters. In addition, the data of conformal field theory give rise to a projectively flat connection on this bundle. The conformal blocks are multi-valued horizontal sections of this bundle. The second type is the "true" correlation functions. They can be expressed as sesquilinear combinations of conformal blocks and their complex conjugates (anti-conformal blocks), chosen so that the combination is a single-valued function of the parameters (see, e.g., [Gaw], Lecture 4). 3 Now, the Hecke eigensheaves on Bun G constructed in [BD] may be viewed as sheaves of conformal blocks of a certain two-dimensional conformal field theory, see [Fr2]. Away from a singularity locus, these sheaves are vector bundles with a flat connection, and conformal blocks are their multi-valued horizontal sections (see Section 1.3 above). It turns out that in some cases there exist linear combinations of products of these conformal blocks and their complex conjugates which give rise to single-valued functions on Bun G . These functions are precisely the automorphic forms of the analytic theory. In other words, the objects of the analytic theory of automorphic forms on Bun G can be constructed from the objects of the geometric Langlands theory in roughly the same way as the correlation functions of CFT are constructed from conformal blocks (as predicted in [Fr4] and [T]). A crucial difference with the CFT is that whereas in CFT the monodromy of conformal blocks is typically unitary, here we expect the monodromy to be real.
Acknowledgments I am grateful Roberto Alvarenga, Julia Gordon, Ivan Fesenko, David Kazhdan, and Raven Waller for valuable discussions.
2. The abelian case 2.1. The case of an elliptic curve. Let's start with the case of an elliptic curve E τ with complex parameter τ . Let's choose, once and for all, a reference point p 0 on this curve. Then we can identify it with E τ ≃ C/(Z + Zτ ). (2.1) Next, consider the Picard variety Pic(E τ ) of E τ . This is the (fine) moduli space of line bundles on E τ (note that the corresponding moduli stack Bun GL 1 (E τ ) of line bundles on E τ is the quotient of Pic(E τ ) by the trivial action of the multiplicative group G m = GL 1 , which is the group of automorphisms of every line bundle on E τ ). It is a disjoint union of connected components Pic d (E τ ) corresponding to line bundles of degree d. Using the reference point p 0 , we can identity Pic d (E τ ) with Pic 0 (E τ ) by sending a line bundle L of degree d to L(−d · p 0 ). Furthermore, we can identify the degree 0 component Pic 0 (E τ ), which is the Jacobian variety of E τ , with E τ itself using the Abel-Jacobi map; namely, we map a point p ∈ E τ to the degree 0 line bundle O(p − p 0 ). Now we define the Hecke operators H p . They are labeled by points p of the curve E τ . The operator H p is the pull-back of functions with respect to the geometric map These operators commute with each other. Formula (2.2) implies that if f is a joint eigenfunction of the Hecke operators H p , p ∈ E τ , on Pic(E τ ), then its restriction f 0 to the connected component Pic 0 (E τ ) is an eigenfunction of the operators Conversely, given an eigenfunction f 0 of p 0 H p , p ∈ X, on Pic 0 (X) and µ p 0 ∈ C × , there is a unique extension of f 0 to an eigenfunction f of H p , p ∈ X, such that the eigenvalue of H p 0 on f is equal to µ p 0 . Namely, any line bundle L of degree d may be represented uniquely as L 0 (d · p 0 ), where L 0 is a line bundle of degree 0. We then set By construction, the eigenvalue µ p of H p on f is then equal to λ p · µ p 0 , where λ p is the eigenvalue of p 0 H p on f 0 (note that since p 0 H p 0 = Id, the eigenvalue λ p 0 is always equal to 1).
Therefore, from now on we will consider the eigenproblem for the operators p 0 H p acting on the space L 2 (Pic 0 (E τ )). Using the above isomorphism between Pic 0 (E τ ) and E τ as above, we identify L 2 (Pic 0 (E τ )) with L 2 (E τ ) (with respect to the standard measure). The Hecke operator p 0 H p acting on L 2 (E τ ) is given by the formula In other words, it is simply the pull-back under the shift by p with respect to the (additive) abelian group structure on E τ , which can be described explicitly using the isomorphism (2.1). The subscript p 0 in p 0 H p serves as a reminder that this operator depends on the choice of the reference point p 0 . Now we would like to describe the joint eigenfunctions and eigenvalues of the operators To be even more concrete, let's start with the case τ = i, so E τ = E i which is identified with C/(Z + Zi) as above. Thus, we have a measure-preserving isomorphism between E i and the product of two circles (R/Z) × (R/Z) corresponding to the real and imaginary parts of z = x + iy. The space of L 2 functions on the curve E i is therefore the completed tensor product of two copies of L 2 (R/Z), and so it has the standard orthogonal Fourier basis: . The operator p 0 H p corresponds to the shift of z by p (with respect to the abelian group structure on E i ): It might be instructive to consider first the one-dimensional analogue of this picture, in which we have L 2 (S 1 ), where S 1 = C/Z with coordinate φ. Then the role of the family Then the Fourier harmonics f n (x) = e 2πinφ form an orthogonal eigenbasis of the operators H ′ α , α ∈ S 1 . The eigenvalue of H ′ α on f n is e 2πinα . Likewise, in the two-dimensional case of the elliptic curve E i , the Fourier harmonics f m,n form an orthogonal basis of eigenfunctions of the operators p 0 H p , p ∈ E i , in L 2 (E i ): (2.7) From this formula we see that the eigenvalue of p 0 H p on f m,n is e 2πi(mxp+nyp) . Thus, we have obtained a complete description of the Hecke eigenfunctions and eigenvalues for the curve X = E i and the group G = GL 1 . This will be generalized to the case of a general elliptic curve E τ in Section 2.3. However, the case of E i is representative enough so that we can already compare the above description of Hecke eigenfunctions and eigenvalues with what Langlands proposed in [L2].

2.2.
Is there a link with representations of π 1 (X, p 0 )? In [L2], Langlands conjectured that for a complex curve X, there is a one-to-one correspondence between the set H(X) of Hecke eigenfunctions (up to a scalar) for the group G = GL 1 and the set E(X) of equivalence classes of one-dimensional representations of the fundamental group π 1 (X, p 0 ) with finite image. He attempted to construct a map H(X) → E(X) in two different ways. The first is to express the Hecke eigenvalues of a given Hecke eigenfunction as holonomies of a flat connection on a line bundle on X and then take the monodromy representation of this connection. I show below that it is indeed possible to express the Hecke eigenvalues that we have found above for the curve X = E i as holonomies of a flat connection on the trivial line bundle on E i , and I will generalize this in Sections 2.3 and 2.4 to the case of an arbitrary curve X. But all of these connections have trivial monodromy representation. Thus, we obtain a trivial map H(X) → E(X). Herein lies an important difference between the analytic and geometric theories for curves over C, which I discuss in more detail in Remark 1 below.
Second, Langlands attempted to construct a map H(X) → E(X) explicitly in the case of an elliptic curve X = E τ . Unfortunately, this construction does not yield a bijective map, either, as I show in Remark 2 below.
In fact, there is no reason to expect a meaningful correspondence between the above sets H(X) and E(X). In the case of an elliptic curve X = E τ , elements of H(E τ ) are labeled by pairs of integers (as we have seen in the previous subsection for τ = i and will show in Theorem 1 for general τ ). Moreover, the harmonics f m,n given by formula (2.4) are all possible complex characters of E i viewed as an abelian group. Therefore the set H(E i ) is naturally a group, which is isomorphic to Z × Z (the same is true about H(E τ ) for a general τ ).
On the other hand, let E(E τ ) be the group of complex characters of π group of complex roots of unity (see Remark 2). Clearly, the groups Z × Z and µ × µ are not isomorphic as groups. Since each of them is countable, there exist bijections between them as sets. But it's hard to imagine that such a bijection would be pertinent to the questions at hand.
We now assign to the Hecke eigenfunction f m,n given by formula (2.4) a unitary flat connection ∇ (m,n) on the trivial line bundle over E i : (since the line bundle is trivial, a connection on it is the same as a one-form on the curve). In other words, the corresponding first order differential operators along x and y are given by the formulas (2.10) The horizontal sections of this connection are the solutions of the equations up to a scalar. The function Φ m,n is the unique solution of (2.11) normalized so that its value at the point 0 ∈ E i , corresponding to our reference point p 0 ∈ E i , is equal to 1. The value of this function Φ m,n at p = x p + iy p ∈ C/(Z + Zi) is indeed equal to the eigenvalue of the Hecke operator p 0 H p on the harmonic f m,n . In other words, this eigenvalue can be represented as the holonomy of the connection ∇ (m,n) over a path connecting our reference point p 0 ∈ E i , which corresponds to 0 ∈ C/(Z + Zi), with the point p ∈ E i . Since the connection is flat, it does not matter which path we choose.
However, and this is a crucial point, the connection ∇ (m,n) has trivial monodromy on E i . Indeed, Φ m,n (x + 1, y) = Φ m,n (x, y + 1) = Φ m,n (x, y) for all m, n ∈ Z.
Remark 1. Recall that in the classical unramified Langlands correspondence for a curve over F q , to each joint eigenfunction of the Hecke operators we assign a Langlands parameter. In the case of G = GL n , this is an equivalence class of ℓ-adic homomorphisms from theétale fundamental group of X to GL n (and more generally, one considers homomorphisms to the Langlands dual group L G of G). Given such a homomorphism σ, to each closed point x of X we can assign an ℓ-adic number, the trace of σ(Fr x ), where Fr x is the Frobenius conjugacy class, so we obtain a function from the set of closed points of X to the set of conjugacy classes in GL n (Q ℓ ).
In the geometric Langlands correspondence for curves over C, the picture is different. Now the role of theétale fundamental group is played by the topological fundamental group π 1 (X, p 0 ). Thus, the Langlands parameters are the equivalence classes of homomorphisms π 1 (X, p 0 ) → GL n (or, more generally, to L G). The question then is: how to interpret such a homomorphism as a Hecke "eigenvalue" on a Hecke eigensheaf?
The point is that for a Hecke eigensheaf, the "eigenvalue" of a Hecke operator (or rather, Hecke functor) is not a number but an n-dimensional vector space. As we move along a closed path on our curve (starting and ending at the point p 0 say), this vector space will in general undergo a non-trivial linear transformation, thus giving rise to a non-trivial homomorphism π 1 (X, p 0 ) → GL n .
Note that over C we have the Riemann-Hilbert correspondence, which sets up a bijection between the set of equivalence classes of homomorphisms π 1 (X, p 0 ) → GL n (or, more generally, π 1 (X, p 0 ) → L G) and the set of equivalence classes of pairs (P, ∇), where P is a rank n bundle on X (or, more generally, an L G-bundle) and ∇ is a flat connection on P. The map between the two data is defined by assigning to (P, ∇) the monodromy representation of ∇ (corresponding to a specific a trivialization of P at p 0 ). We may therefore take equivalence classes of the flat bundles (P, ∇) as our Langlands parameters instead of equivalence classes of homomorphisms π 1 (X, p 0 ) → GL n . As explained in the previous paragraph, these flat bundles (P, ∇) will in general have non-trivial monodromy.
However, in this section we consider (in the case of GL 1 and a curve X) the eigenfunctions of the Hecke operators p 0 H p , p ∈ X, on Pic 0 (X). Their eigenvalues are numbers, not vector spaces. Therefore they cannot undergo any transformations as we move along a closed path on our curve. In other words, these numbers give rise to a single-valued function from X to GL 1 (C) (it actually takes values in U 1 ⊂ GL 1 (C)). Because the function is single-valued, if we represent this function as the holonomy of a flat connection on a line bundle on X, then this connection necessarily has trivial monodromy. And indeed, we have seen above that each collection of joint eigenvalues of the Hecke operators p 0 H p , p ∈ E i , on functions on Pic 0 (E i ) can be represented as holonomies of a specific (unitary) connection ∇ m,n with trivial monodromy. The same is true for other curves, as we will see below.
Remark 2. On pp. 59-60 of [L2], another attempt is made to construct a map from the set H(E τ ) of Hecke eigenfunctions on Pic 0 (E τ ) to the set E(E τ ) of equivalence classes of homomorphisms π 1 (E τ , p 0 ) → GL 1 with finite image. As we have seen in the previous subsection for τ = i and will see in the next subsection for general τ (see Theorem 1), the set H(E τ ) can be identified with Z × Z. On the other hand, the set E(E τ ) can be identified with µ × µ, where µ is the group of complex roots of unity (we have an isomorphism Q/Z ≃ µ sending κ ∈ Q/Z to e 2πiκ ). Indeed, since π 1 (E τ , p 0 ) ≃ Z × Z, a homomorphism φ : π 1 (E τ , p 0 ) → GL 1 ≃ C × is uniquely determined by its values on the elements A = (1, 0) and B = (0, 1) of Z×Z. The homomorphism φ has finite image if and only if both φ(A), φ(B) belong to µ.
Langlands attempts to construct a map (Z × Z) → (µ × µ) as follows (see pp. 59-60 of [L2]): he sets (0, 0) → (1, 1). (2.12) Next, given a non-zero element (k, l) ∈ Z×Z, there exists a matrix g k,l = α β γ δ ∈ SL 2 (Z) such that Two comments on (2.13): first, as noted in [L2], the matrix g k,l is not uniquely determined by formula (2.13). Indeed, this formula will still be satisfied if we multiply g k,l on the left by any lower triangular matrix in SL 2 (Z). Second, formula (2.13) implies that where, for a pair of integers (k, l) = (0, 0), we define gcd(k, l) as l if k = 0, as k if l = 0, and gcd(|k|, |l|) times the product of the signs of k and l if they are both non-zero. Therefore Using a particular choice of the matrix g k,l , Langlands defines a new set of generators {A ′ , B ′ } of the group π 1 (E τ , p 0 ): He then defines a homomorphism φ k,l : π 1 (E τ , p 0 ) → GL 1 corresponding to (k, l) by the formulas and so we find the values of φ k,l on the original generators A and B: Langlands writes in [L2], "This has a peculiar property that part of the numerator becomes the denominator, which baffles me and may well baffle the reader." He goes on to say, "To be honest, this worries me." In fact, this construction does not give us a well-defined map (Z × Z) → (µ × µ). Indeed, g k,l is only defined up to left multiplication by a lower triangular matrix: under which we have the following transformation: (2.20) But then the homomorphism (2.18) gets transformed to the homomorphism sending The homomorphisms (2.18) and (2.21) can only coincide for all x ∈ Z if both α and β are divisible by k ′ . But if k ′ = 1, this contradicts the condition, established in formula (2.14), that α and β are relatively prime. Hence (2.18) and (2.21) will in general differ from each other, and so we don't get a well-defined map (Z × Z) → (µ × µ).
We could try to fix this problem by replacing the relation (2.16) with Then the homomorphism φ k,l would send This way, we get a well-defined map (Z × Z) → (µ × µ), but it's not a bijection.
2.3. General elliptic curve. Here we generalize the description of Hecke eigenfunctions and eigenvalues obtained in Section 2.1 to the case of an arbitrary elliptic curve E τ ≃ C/(Z + Zτ ), where Im τ > 0. Let us again identify every component of Pic(E τ ) with E τ using the reference point p 0 . Then we obtain the Hecke operators p 0 H p labeled by p ∈ E τ corresponding to the shift by p naturally acting on E τ . The eigenfunctions and eigenvalues of these operators are then given by the following theorem.

Theorem 1. The joint eigenfunctions of the Hecke operators
The eigenvalues are given by the right hand side of the following formula: In the next subsection we will give an alternative formula for these eigenfunctions (for an arbitrary smooth projective curve instead of E τ ).
Finally, let us express the Hecke eigenvalues as holonomies of a flat connection on the trivial line bundle on E τ , generalizing the results of Section 2.2.
Introduce a flat unitary connection τ ∇ (m,n) on the trivial line bundle on E τ (which becomes the connection ∇ (m,n) from Section 2.1 when we specialize τ = i): The first order operators corresponding to z and z are (2.28) Just as in the case τ = i (see Section 2.2), for every p ∈ E τ , the holonomy of the connection τ ∇ (m,n) over a path connecting p 0 ∈ E τ and p ∈ E τ is equal to the eigenvalue of p 0 H p on f τ m,n given by the right hand side of formula (2.25). As in the case of τ = i, all connections τ ∇ (m,n) yield the trivial monodromy representation π 1 (E τ , p 0 ) → GL 1 (see Remark 1 above).
2.4. Higher genus curves. Let X be a smooth projective connected curve over C. Denote by Pic(X) the Picard variety of X, i.e. the moduli space of line bundles on X (as before, the moduli stack Bun GL 1 (X) of line bundles on X is the quotient of Pic(X) by the trivial action of G m = GL 1 ). We have a decomposition of Pic(X) into a disjoint union of connected components Pic d (X) corresponding to line bundles of degree d. The Hecke operator H p , p ∈ X, is the pull-back of functions with respect to the map (see formula (2.2) for X = E τ ): The Hecke operators H p with different p ∈ X commute with each other, and it is natural to consider the problem of finding joint eigenfunctions and eigenvalues of these operators on functions on Pic(X). In the same way as in Section 2.1, we find that this problem is equivalent to the problem of finding joint eigenfunctions and eigenvalues of the operators where p 0 is a reference point on X that we choose once and for all. The operator p 0 H p is the pull-back of functions with respect to the map p 0 T p : Pic 0 (X) → Pic 0 (X) sending a line bundle L to L(p − p 0 ). Now, Pic 0 (X) is the Jacobian of X, which is a 2g-dimensional torus (see, e.g., [GH]) where H 1 (X, Z) is embedded into the space of linear functionals on the space H 0 (X, Ω 1,0 ) of holomorphic one-forms on X by sending β ∈ H 1 (X, Z) to the linear functional (2.30) Motivated by Theorem 1, it is natural to guess that the standard Fourier harmonics in L 2 (Pic 0 (X)) form an orthogonal eigenbasis of the Hecke operators. This is indeed the case.
To see that, we give an explicit formula for these harmonics. They can be written in the form e 2πiϕ , where ϕ : H 0 (X, Ω 1,0 ) * → R is an R-linear functional such that ϕ(β) ∈ Z for all β ∈ H 1 (X, Z). To write them down explicitly, we use the Hodge decomposition H 1 (X, C) = H 0 (X, Ω 1,0 ) ⊕ H 0 (X, Ω 0,1 ) = H 0 (X, Ω 1,0 ) ⊕ H 0 (X, Ω 1,0 ) to identify H 0 (X, Ω 1,0 ), viewed as an R-vector space, with H 1 (X, R) by the formula ω ∈ H 0 (X, Ω 1,0 ) → ω + ω. (2.31) In particular, for any class c ∈ H 1 (X, R), there is a unique holomorphic one-form ω c such that c is represented by the real-valued harmonic one-form ω c + ω c , Viewed as a real manifold, where H 1 (X, Z) is embedded into H 1 (X, R) * by sending β ∈ H 1 (X, Z) to the linear functional on H 1 (X, R) given by the formula (compare with formulas (2.30) and (2.31)) (2.33) Now, to each γ ∈ H 1 (X, Z) we attach the corresponding element of the vector space H 1 (X, R), which can be viewed as a linear functional ϕ γ on the dual vector space H 1 (X, R) * , It has the desired property: ϕ γ (β) ∈ Z for all β ∈ H 1 (X, Z). The corresponding functions are the Fourier harmonics that form an orthogonal basis of the Hilbert space L 2 (Pic 0 (X)).
We claim that each of these functions is an eigenfunction of the Hecke operators p 0 H p , p ∈ X, so that together they give us a sought-after orthogonal eigenbasis of the Hecke operators. To see that, we use the Abel-Jacobi map.
For d > 0, let X (d) be the dth symmetric power of X, and p d : (2.35) We can lift the map T p to a map so that we have a commutative diagram Denote by H p the corresponding pull-back operator on functions. Now let f 0 be a non-zero function on Pic 0 (X). Identifying Pic d (X) with Pic 0 (X) using the reference point p 0 : where λ p = 0 for all p and λ p 0 = 1. This is equivalent to the following factorization formula for f d : The surjectivity of p d with d ≥ g and the commutativity of the diagram (2.37) then implies that (2.41) But then it follows from the definition of f d that f 0 is an eigenfunction of the operators ). This observation gives us an effective way to demonstrate that a given function f 0 on Pic 0 (X) is a Hecke eigenfunction.
Let us use it in the case of the function f 0 = e 2πiϕγ , γ ∈ H 1 (X, Z), on Pic 0 (X) given by formula (2.34). For that, denote by d e 2πiϕγ , γ ∈ H 1 (X, Z), (2.42) the corresponding functions f d on Pic d (X) obtained via the identification (2.38). We claim that for any γ ∈ H 1 (X, Z), the pull-backs of d e 2πiϕγ to X (d) , d > 0, via the Abel-Jacobi maps have the form (2.40), and hence e 2πiϕγ is a Hecke eigenfunction on Pic 0 (X).
To see that, we recall an explicit formula for the composition where the second map is given by formula (2.38) (see, e.g., [GH]). Namely, the composition Composing the map (2.43) with the isomorphism H 0 (X, Ω 1,0 ) ≃ H 1 (X, R) defined above, we obtain a map (2.45) (see formula (2.32) for the definition of ω c ). If c ∈ H 1 (X, R) is the image of an integral cohomology class γ ∈ H 1 (X, Z), we will write the corresponding holomorphic one-form ω c as ω γ .
Let p 0 f d,γ be the pull-back of the function d e 2πiϕγ (see formula (2.42)) to X (d) . Equivalently, p 0 f d,γ is the pull-back of the function e 2πiϕγ under the map p 0 Φ d . It follows from the definition Thus, we obtain that p 0 f d,γ is given by the formula We conclude that the functions p 0 f d,γ satisfy the factorization property (2.40). Therefore the function e 2πiϕγ on Pic 0 (X) is indeed an eigenfunction of p 0 H p , with the eigenvalue λ γ p given by formula (2.47), which is what we wanted to prove. 4 Thus, we have proved the following theorem.
Theorem 2. The joint eigenfunctions of the Hecke operators p 0 H p , p ∈ X, on L 2 (Pic 0 (X)) are the functions e 2πiϕγ , γ ∈ H 1 (X, Z). The eigenvalues of p 0 H p are given by formula (2.47), so that we have on the trivial line bundle on X, taken along (no matter which) path from p 0 to p. As in the case of elliptic curves, the monodromy representation of each of these connections is trivial, ensuring that the Hecke eigenvalues λ γ p , viewed as functions of p ∈ X, are single-valued (see Remark 1).
2.5. General torus. Let now T be a connected torus over C, and Bun T (X) the moduli space of T -bundles on X (note that the moduli stack Bun T (X) is the quotient of Bun T (X) by the trivial action of T ). In Section 2.4 we find the joint eigenfunctions and eigenvalues of the Hecke operators in the case of Bun T (X) where T = G m ; in this case Bun Gm (X) = Pic(X).
Here we generalize these results to the case of an arbitrary T .
Let Λ * (T ) and Λ * (T ) be the lattices of characters and cocharacters of T , respectively. Any P ∈ Bun T (X) is uniquely determined by the G m -bundles (equivalently, line bundles) P × Gm χ associated to the characters χ : T → G m in Λ * (T ). This yields a canonical isomorphism

The neutral component
Bun 0 is non-canonically isomorphic to Pic 0 (X) r , where r is the rank of the lattice Λ * (T ). The Hecke operators Hμ p are now labeled by p ∈ X andμ ∈ Λ * (T ). The operator Hμ p corresponds to the pull-back under the map where P(μ · p) is defined by the formula 4 Note that Abel's theorem implies that each function p 0 f d,γ , γ ∈ H 1 (X, Z), is constant along the fibers of the Abel-Jacobi map X (d) → Pic d and therefore descends to Pic d . This suggests another proof of Theorem 2: we start from the functions p 0 f d,γ on X (d) , d > 0. Formula (2.46) shows that they combine into an eigenfunction of the operators Hp. Hence the function on Pic d (X), d ≥ g, to which p 0 f d,γ descends, viewed as a function on Pic 0 (X) under the identification (2.38), is a Hecke eigenfunction. One can then show that this function is equal to e 2πiϕγ .
As in the case of T = G m , we choose, once and for all, a reference point p 0 ∈ X. As in the case of T = G m , finding eigenfunctions and eigenvalues of the commuting operators Hμ p on functions on Bun T (X) is equivalent to finding eigenfunctions and eigenvalues of the operators p 0 Hμ p = (Hμ p 0 ) −1 • Hμ p on functions on Bun 0 T (X). As in the case of T = G m , we represent Bun 0 T (X) as is the split real form of the complex Lie algebra t of T .
On the other hand, the image of γ in H 1 (X, t * R ) gives rise to a linear functional ϕ γ : H 1 (X, t * R ) * → R satisfying ϕ γ (β) ∈ Z for all β ∈ H 1 (X, Λ * (T )). Therefore, according to formula (2.50), e 2πiϕγ is a well-defined function on Bun 0 T (X). These are the Fourier harmonics on Bun 0 T (X). In the same way as in Section 2.4, we prove the following result.
(2.53) Let us explain the notation we used on the right hand side of formula (2.53): denote by L T the Langlands dual torus to T . We have Λ * ( L T ) = Λ * (T ) and Λ * ( L T ) = Λ * (T ). The eigenvalue of the Hecke operator p 0 Hμ p , p ∈ X,μ ∈ Λ * ( L T ), on the function e 2πiϕγ is equal to the value of the characterμ of L T on the L T -valued function F γ on X F γ (p) = e on the trivial L T u -bundle on X over (no matter which) path from p 0 to p. As in the case of T = G m , each of these connections has trivial monodromy.

Non-abelian case
In this section we try to generalize to the case of a non-abelian group G the results obtained in the previous section for abelian G.
3.1. Spherical Hecke algebra for groups over F q ((t)). In the case of the function field of a curve X over a finite field, the Hecke operators attached to a closed point x of X generate the spherical Hecke algebra H(G(F q ((t))), G(F q [[t]])). As a vector space, it is the space of C-valued functions on the group G(F q ((t))) that are bi-invariant with respect to the subgroup G(F q [[t]]) (here F q is the residue field of x). This vector space is endowed with the convolution product defined by the formula where dh stands for the Haar measure on G(F q ((t))) normalized so that the volume of the sub- is the unit element of the convolution algebra). The Haar measure can be defined because G(F q ((t))) is a locally compact group. The resulting convolution algebra H(G(F q ((t))), G(F q [[t]])) is commutative and we have the Satake isomorphism between this algebra and the complexified representation ring Rep L G of the Langlands dual group L G.

3.2.
Is there a spherical Hecke algebra for groups over C((t))? In contrast to the group G(F q ((t))), the group G(C((t))) is not locally compact. Therefore it does not carry a Haar measure. Indeed, the field C((t)) is an example of a two-dimensional local field, in the terminology of [Fe1], more akin to F q ((z))((t)) or Q p ((t)) than to F q ((t)) or Q p . Ivan Fesenko has developed integration theory for the two-dimensional local fields [Fe1,Fe2], and his students have extended it to algebraic groups over such fields [Mo1,Mo2,Wa1], but this theory is quite different from the familiar case of G(F q ((t))).
First, integrals over C((t)) and G(C((t))) take values not in real numbers, but in formal Laurent power series R((X)), where X is a formal variable. Under certain restrictions, the value of the integral is a polynomial in X; if so, then one could set X to be equal to a real number. But this way one might lose some important properties of the integration that we normally take for granted.
Second, if S is a Lebesque measurable subset of C, then according to [Fe1,Fe2], the measure of a subset of C((t)) of the form is equal to µ(S)X i , where µ(S) is the usual Lebesque measure of S. In particular, this means that the measure of the subset C[[t]] of C((t)) is equal to 0, as is the measure of the subset t n C[[t]] for any n ∈ Z. Contrast this with the fact that under a suitably normalized Haar measure on F q ((t)), the measure of t n F q [[t]] is equal to q −n . Thus, if we take as G the additive group, it's not even clear how to define a unit element in the would-be spherical Hecke algebra (which would be the characteristic function of the subset F q [[t]] in the case of the field F q ((t))). The situation is similar in the case of a general group G. For this reason, according to Waller [Wa2], from the point of view of the two-dimensional integration theory it would make more sense to consider distributions on G((t)) that are bi-invariant not with respect to G(C[[t]]), but its subgroup K consisting of those elements g(t) ∈ G(C[[t]]) for which g(0) belongs to a compact subgroup K of G(C). This would be similar to the construction used in representation theory of real Lie groups, where one considers, for example, the space of distributions on the group G(C) supported on a compact subgroup K with a natural convolution product [KV]. 5 For instance, if K = {1}, the resulting algebra is U (g), the universal enveloping algebra of the Lie algebra g of G(C). (Thus, we see that the objects that are meaningful over C are closer to U (g) rather than the spherical Hecke algebra H(G(F q ((t))), G(F q [[t]])), a point we will also discuss in Remark 3 below.) Perhaps, a convolution product on some space of distributions of this kind can be defined for G(C((t))), but from the structure of the double cosets of K in G(C((t))) it is clear that the resulting algebra cannot be isomorphic to Rep L G; nor can its elements act on functions on Bun G (which is what we need).
Another option is to consider motivic integration theory. A motivic version of the Haar measure has in fact been defined by Julia Gordon [Gor] for the group G(C((t))) (it may also be obtained in the framework of the general theories of Cluckers-Loeser [CL] or Hrushovski-Kazhdan [HK1]). Also, in a recent paper [CCH] it was shown that the spherical Hecke algebra H(G(F q ((t))), G(F q [[t]])) can be obtained by a certain specialization from its version in which the ordinary integration with respect to the Haar measure on G(F q ((t))) is replaced by the motivic integration with respect to the motivic Haar measure. Presumably, one could carry some of the results of [CCH] over to the case of C((t)).
However, this does not seem to give us much help, for the following reason: the motivic integrals over a ground field k take values in a certain algebra M k , which is roughly speaking a localization of the Grothendieck ring of algebraic varieties over k. In the case of the ground field F q , the algebra M Fq is rich, and ordinary integrals may be recovered from the motivic ones by taking a homomorphism from M Fq to R sending the class of the affine line over F q to q. But in the case of the ground field C, the structure of the algebra M C appears to be very different (for example, it has divisors of zero), and this construction does not work. In fact, it seems that there are very few (if any) known homomorphisms from M C to positive real numbers, besides the Euler characteristic. 6 Perhaps, taking the Euler characteristic, one can obtain a non-trivial convolution algebra structure on the space of G(C[[t]]) bi-invariant functions on G(((t))) (one may wonder whether it could be interpreted as a kind of q → 1 limit of H(G(F q ((t))), G(F q [[t]]))), but it is doubtful that this algebra could be useful in any way for defining an analytic theory of automorphic forms on Bun G for complex algebraic curves.
3.3. An attempt to define Hecke operators. What we are interested in here, however, is not the spherical Hecke algebra itself, but rather the action of the corresponding Hecke operators on automorphic functions. After all, in Section 2 we were able to define Hecke operators without any reference to a convolution algebra on the group C((t)) × . As we will see below though, the abelian case was an exception in that the action of the Hecke operators did not require integration. In the non-abelian case (in fact, already for G = GL 2 ), it appears that integration is necessary, and this creates serious -perhaps, insurmountableproblems. I illustrate this below with some concrete examples in the case of GL 2 .
Recall that for curves over F q , the unramified automorphic functions are functions on the double quotient where F = F q (X), and X is a curve over F q . The action of Hecke operators on functions on this double quotient can be defined by means of certain correspondences, and we can try to imitate this definition for complex curves.
To this end, we take the same double quotient (3.3) with F = C(X), where X is a curve over C. As in the case of F q , this is the set of equivalence classes of principle G-bundles on X. The Hecke correspondences can be conveniently defined in these terms.
For instance, consider the case of GL 2 and the first Hecke operator (for a survey of the general case, see, e.g., [Fr2], Sect. 3.7). Then we have the Hecke correspondence Hecke 1,x , where x is a closed point of X: Here Hecke 1,x is the moduli stack classifying the quadruples where M and M ′ are points of Bun GL 2 , which means that they are rank two vector bundles on X, and β is an embedding of their sheaves of (holomorphic) sections β : M ′ ֒→ M such that M/M ′ is supported at x and is isomorphic to the skyscraper sheaf Thus, the fiber of Hecke 1,x over M is isomorphic to the projectivization of the twodimensional vector space M * x , i.e. to CP 1 . We conclude that Hecke 1,x is a CP 1 -fibration over over the "left" Bun GL 2 in the diagram (3.4) . Likewise, we obtain that Hecke 1,x is a CP 1 -fibration over the "right" Bun GL 2 in (3.4).
In the geometric theory, we use the correspondence (3.4) to define a Hecke functor H 1,x on the (derived) category of D-modules on Bun GL 2 : (3.6) and in addition have similar isomorphisms for the second set of Hecke functors H 2,x , x ∈ X. These are defined similarly to the Hecke operators for GL 1 , as the pull-backs with respect to the morphisms sending a rank two bundle M to M(x) (they change the degree of M by 2). (As explained in [FGV], Sect. 1.1, the second Hecke eigensheaf property follows from the first together with a certain S 2 -equivariance condition.) Thus, if K is a Hecke eigensheaf, we obtain a family of isomorphisms (3.6) for all x ∈ X, and similarly for the second set of Hecke functors. We then impose a stronger requirement that the two-dimensional vector spaces appearing on the right hand side of (3.6) as "eigenvalues" fit together as stalks of a single rank two local system E on X (and similarly for the second set of Hecke functors, where the eigenvalues should be the stalks of the rank one local system ∧ 2 E on X; this is, however, automatic if we impose the S 2 -equivariance condition from [FGV], Sect. 1.1). If that's the case, we say that K is a Hecke eigensheaf with the eigenvalue E. This is explained in more detail, e.g., in [Fr2], Sect. 3.8.
The first task of the geometric theory (in the case of G = GL 2 ) is to show that such a Hecke eigensheaf on Bun GL 2 exists for every irreducible rank two local system E on X. This was accomplished by Drinfeld in [Dr1], a groundbreaking work that was the starting point of the geometric theory. We now know that the same is true for G = GL n [FGV,Gai1] and in many other cases. Now let's try to adapt the diagram (3.4) to functions. Thus, given a function f on the set of C-points, we wish to define the action of the first Hecke operator H 1,x on it by the formula Thus, we see that the result must be an integral over the complex projective line h −1 ℓ,x (M). The key question is: what is the measure dM ′ ?
Herein lies a crucial difference with the abelian case considered in Section 2: in the abelian case every Hecke operator acted by pull-back of a function, so no integration was needed. But in the non-abelian case, already for the first Hecke operators H 1,x in the case of G = GL 2 , we must integrate functions over the projective lines h −1 ℓ,x (M), where M ∈ Bun GL 2 (C). Note that if our curve were over a finite field, this integration is in fact a summation over a finite set of q + 1 elements, the number of points of P 1 over F q , where F q is the residue field of the closed point x at which we take the Hecke operator. The terms of this summation correspond to points of the fibers h −1 ℓ,x (M). Being finite sums, these integrals are always well-defined if our curve is over F q . For curves over C, this is not so, and this creates major problems, as we will see below.
3.4. The case of an elliptic curve. Let's look at the case of an elliptic curve X. If it is defined over a finite field, the fibers h −1 ℓ,x (M) appearing in the Hecke operators have been described explicitly in [Lo, Al], using the classification of rank two bundles on elliptic curves due to Atiyah [At].
For a complex elliptic curve X, the fibers h −1 ℓ,x (M) have been described explicitly in [Bo]. In [L2], Langlands attempts to describe them in the language of adeles, which is more unwieldy than the vector bundle language used in [Bo] and hence more prone to errors. As the result of his computations, Langlands states on p.18 of [L2]: if M is the rank two bundle corresponding the adele g ∈ GL 2 (A F )] is always equal to 0... Hence the domain of integration in [adelic version of our formula (3.7) above] is a finite set." This statement is incorrect. First of all, as we show below, there are rank two bundles M on an elliptic curve for which there are infinitely many non-isomorphic bundles in the fiber h −1 ℓ,x (M) (in fact, we have a continuous family of non-isomorphic bundles parametrized by the points of h −1 ℓ,x (M)). Second, even if there are finitely many isomorphism classes among those M ′ which appear in the fiber h −1 ℓ,x (M) for a fixed M, this does not mean that we are integrating over a finite set.
In fact, according to formula (3.7) (whose adelic version is formula (10) of [L2]), for every M, the fiber h −1 ℓ,x (M) of the Hecke correspondence over which we are supposed to integrate is always isomorphic to CP 1 if we take into account the automorphism groups of the bundles involved. In the adelic language, the automorphism group Aut(M) of a bundle M may be described, up to an isomorphism, as follows: M corresponds to a point in the double quotient (3.3); we lift this point to G(F )\G(A F ) and take its stabilizer subgroup in G(O F ). The necessity to take into account these automorphism groups is clear from the fact that over F q , the measure on the double quotient (3.3) induced by the Tamagawa measure assigns (up to an overall factor) to a point not 1 but 1/| Aut(M)| (this measure is well-defined over F q because then the group Aut(M) is finite for any M; however, this is not so over C). With respect to this correctly defined measure, the fiber h −1 ℓ,x (M) for any M can be identified with the the set of F q -points of the projective line, with each point having measure 1.
As a concrete illustration, consider the case that M = L 1 ⊕ L 2 , where L 1 and L 2 are two line bundles of degrees d 1 and d 2 such that d 1 > d 2 + 1. Then the vector bundles M ′ that appear in the fiber h −1 ℓ,x (M) are isomorphic to either M ′ 1 = L 1 (−x)⊕L 2 or M ′ 2 = L 1 ⊕L 2 (−x). However, the groups of automorphisms of these bundles are different: each of them is a semidirect product of the group (C × ) 2 of rescalings of the two line bundles appearing in a direct sum decomposition and an additive group, which is Hom(L 2 , L 1 ) for M; Hom(L 2 , L 1 (−x)) for M ′ 1 ; and Hom(L 2 (−x), L 1 ) for M ′ 2 . Under our assumption that d 1 > d 2 + 1, we find that the latter groups are isomorphic to C d 1 −d 2 −1 , C d 1 −d 2 −2 , and C d 1 −d 2 , respectively. Thus, the automorphism group of M ′ 1 is "smaller" by one copy of the additive group G a than that of M, whereas the automorphism group of M ′ 2 is larger than that of M by the same amount. This implies that the fiber h −1 ℓ,x (M) is the union of a complex affine line worth of points corresponding to M ′ 1 and a single point corresponding to M ′ 2 . If we worked over F q , we would find that , with the factor of q representing the number of points of the affine line (see [Lo, Al]). Over C, we formally obtain the sum of two terms: (1) an integral of the constant function taking value f (M ′ 1 ) on an open dense subset of CP 1 isomorphic to the affine line, and (2) a single term f (M ′ 2 ) corresponding to the remaining point. Is there an integration measure of CP 1 that would render this sum meaningful?
If we use a standard integration measure on CP 1 , then the answer would be f (M ′ 1 ) multiplied by the measure of the affine line. The second term would drop out, as it would correspond to a subset (namely, a point) of measure zero. If we want to include the second bundle (which we certainly do for Hecke operators to be meaningful), then the measure of this point has to be non-zero. But we also expect our measure on CP 1 to be invariant (indeed, we cannot a priori distinguish a special point on each of these projective lines). Therefore the measure of every point of CP 1 would have to be given by the same non-zero number. But then our integral would diverge. It seems highly unlikely that one could somehow regularize these divergent integrals in a uniform and meaningful way.
Next, we give an example in which the fiber h −1 ℓ,x (M) is a continuous family of nonisomorphic vector bundles (thus directly contradicting the above statement from [L2]). This is the case if M is the indecomposable rank two bundle degree 1 vector bundle F 2 (x) (in the notation of [L2]) which is a unique, up to an isomorphism, non-trivial extension In this case, as shown in [Bo], Sect. 4.3, the fiber h −1 ℓ,x (F 2 (x)) may be described in terms of a canonical two-sheeted covering π : Pic 0 (X) → CP 1 = h −1 ℓ,x (F 2 (x)) ramified at 4 points such that (1) if a ∈ h −1 ℓ,x (F 2 (x)) is outside of the ramification locus, then π −1 (a) = {L a , L −1 a }, where L a is a line bundle on X; and (2) the fibers over the 4 ramification points are the four square roots L i , i = 1, . . . , 4, of the trivial line bundle on X. Namely, the vector bundle M ′ (a) corresponding to a point a ∈ h −1 ℓ,x (F 2 (x)) is described in terms of π as follows (note that in [Bo] the bundle F 2 (x) is denoted by G 2 (x)): where F 2 is the unique, up to an isomorphism, non-trivial extension of O X by itself.
According to the Atiyah's classification, the bundles M ′ (a) and M ′ (b) corresponding to different points a = b in h −1 ℓ,x (F 2 (x)) are non-isomorphic. Thus, there is an infinite continuous family of non-isomorphic vector bundles appearing in the fiber h −1 ℓ,x (F 2 (x)) in this case. One gets a similar answer for M = F 2 (x) ⊗ L, where L is an arbitrary line bundle on X (note that unlike the vector bundles discussed in the previous example, all of the bundles F 2 (x) ⊗ L are stable). This means that the value of H 1,x · f at bundles M of this form depends in a crucial way on the choice of a measure of integration on h −1 ℓ,x (M). It seems highly unlikely that one could define these measures for different M and different x ∈ X in a consistent and meaningful way, so that they would not only yield well-defined integrals but that the corresponding operators H 1,x , x ∈ X, would commute with each other and with the second set of Hecke operators H 2,x , x ∈ X (whose definition, discussed above, does not involve integration).
In [L2] (pp. 20-21), Langlands postulates, essentially by decree (as he writes on p.21, "указом!"), that Hecke operators should act diagonally on the direct sum of the spaces of L 2 functions on two subsets of Bun GL 2 , denoted by D and U. The former parametrizes rank two vector bundles on X decomposable as direct sums of line bundles; the latter parametrizes indecomposable rank two vector bundles (both can be explicitly described using Atiyah's classification results [At]).
However, the idea of treating the moduli of rank two bundles on X as a disjoint union of the subsets D and U appears to be highly problematic. Indeed, in the moduli stack Bun GL 2 they are "glued" together in a non-trivial way. 7 If we tear them apart, we are at the same time tearing apart the projective lines h −1 ℓ,x (M) appearing as the fibers of the Hecke correspondences.
Let me illustrate this with a concrete example: let M = O X ⊕ O X (x). Then, as explained in [Bo], there are two points in the fiber h −1 ℓ,x (M), corresponding to M ′ 1 = O X ⊕O X and M ′ 2 = O X (−x)⊕O X (x), and each point in the complement (which is isomorphic to C × ) corresponds to the indecomposable bundle F 2 . Thus, in particular, we see that there exist continuous families of rank two vector bundles on X over an affine line A 1 which are isomorphic to F 2 away from 0 ∈ A 1 and to as belonging to a different connected component of Bun GL 2 than F 2 , then what to make of the integral (3.7)? It would seemingly break into the sum of two points and an integral over their complement. 7 If we were to consider instead the moduli space of semi-stable bundles, then, depending on the stability condition we choose, some of the decomposable bundles in D would have to be removed, or identified with the indecomposable ones in U. Considering the moduli space of semi-stable bundles is, however, problematic for a different reason: it is not preserved by the Hecke correspondences, and therefore it is not clear how to define the Hecke operators.
That would be fine in the case of a curve over F q : we would simply obtain the formula , with the factor of (q − 1) being the number of points of P 1 without two points (see [Lo, Al]). But over complex numbers we have to integrate over C × . We would therefore have to somehow combine summation over two points and integration over their complement. As in another example of this nature that we considered above, it seems unlikely that there exists an integration measure that could achieve this in a consistent and meaningful fashion.
Similarly, if M = L 1 ⊕ L 2 , where L 1 and L 2 are non-isomorphic line bundles of degree 0, then the fiber of the Hecke correspondence h −1 ℓ,x (M) over M has two points corresponding , and every point in the complement of these two points (which is isomorphic to C × ) corresponds to the indecomposable vector bundle F 2 (x)⊗L(−x), where L ⊗2 ≃ L 1 ⊗L 2 . Again, it is not clear how one could possibly integrate over h −1 ℓ,x (M) in this case. 8 The purported Hecke operators (acting diagonally on L 2 (D) ⊕ L 2 (U)) are constructed in [L2] without addressing any of these problems, without specifying the pertinent integration measures. Our examples above show that attempts to properly define these integration measures (and hence the Hecke operators) in a systematic way run into what appear to be insurmountable obstacles. The same applies to other non-abelian groups.
3.5. A toy model. At this point, it might be instructive to consider a toy model of the questions we have been discussing. Over F q , there is a well-understood finite-dimensional analogue of the spherical Hecke algebra of G(F q ((t))); namely, the Hecke algebra H q (G) of B(F q ) bi-invariant C-valued functions on the group G(F q ), where B is a Borel subgroup of a simple algebraic group G.
As a vector space, this algebra has a basis labeled by the characteristic functions c w of the Bruhat-Schubert cells B(F q )wB(F q ), where w runs over the Weyl group of G. The convolution product on H q (G) is defined using the constant measure µ q on the finite group G(F q ) normalized so that the measure of B(F q ) is equal to 1. Then the function c 1 is a unit element of H q (G).
It is convenient to describe the convolution product on H q (G) as follows: identify the B bi-invariant functions on G with B-invariant functions on G/B and then with G-invariant functions on (G/B) × (G/B) (with respect to the diagonal action). Given two G-invariant functions f 1 and f 2 on (G/B) × (G/B), we define their convolution product by the formula (3.9) Under this convolution product, the algebra H q (G) is generated by the functions c s i , where the s i are the simple reflections in W . They satisfy the well-known relations.
Observe also that the algebra H q (G) naturally acts on the space C[G(F q )/B(F q )] of C-valued functions on G(F q )/B(F q ). It acts on the right and commutes with the natural left action of G(F q ). Unlike the spherical Hecke algebra, H q (G) is non-commutative.
8 Incidentally, this example shows that for any x ∈ X and any pair of degree zero line bundles L, L1 on X, there is a family of rank two vector bundles on X over an affine line A 1 that are isomorphic to F2(x) ⊗ L away from 0 ∈ A 1 and to L1(x) ⊕ (L ⊗2 ⊗ L −1 1 ) at the point 0 ∈ A 1 . This and the previous example illustrate the intricate topology of BunGL 2 of an elliptic curve; in particular, the fact that the subsets D and U are glued together in a sophisticated way.
Nevertheless, we can use the decomposition of the space C[G(F q )/B(F q )] into irreducible representations of H q (G) to describe it as direct sum of irreducible representations of G(F q ).
Now suppose that we wish to generalize this construction to the complex case. Thus, we consider the group G(C), its Borel subgroup B(C), and the quotient G(C)/B(C), which is the set of C-points of the flag variety G/B over C. A naive analogue of H q (G) would be the space H C (G) of B(C) bi-invariant functions on G(C). Therefore we have the following analogues of the questions that we discussed above in the case of the spherical Hecke algebra: Is it possible to define a measure of integration on G(C) that gives rise to a meaningful convolution product on H C (G)? Is it possible to use the resulting algebra to decompose the space of L 2 functions on G(C)/B(C)?
For example, consider the case of G = SL 2 . Then G/B = P 1 . The Hecke algebra H q (SL 2 ) has a basis consisting of two elements, c 1 and c s , which (in its realization as G-invariant functions on (G/B) × (G/B) explained above) correspond to the characteristic functions of the two SL 2 -orbits in P 1 × P 1 : the diagonal and its complement, respectively. Applying formula (3.9), we obtain that The two formulas in (3.10) follow from the fact that for each x and y, in formula (3.9) there is either a unique value of z for which the integrand is non-zero, or no such values. The coefficients in formula (3.11) have the following meaning: q = µ q (A 1 ), q − 1 = µ q (A 1 \0). Now, if we try to adopt this to the case of P 1 over C, we quickly run into trouble. Indeed, if we want c 1 to be the unit element, we want to keep the two formulas in (3.10). But in order to reproduce the second formula in (3.10), we need a measure dz on CP 1 that would give us χ u dz = 1 for every point u ∈ CP 1 , where χ u is the characteristic function of u. However, then the integral of this measure over the affine line inside CP 1 would diverge, rendering the convolution product c s ⋆ c s meaningless.
Likewise, we run into trouble if we attempt to define an action of H C (G) on the space of functions on G(C)/B(C). Thus, we see that the questions we asked above do not have satisfactory answers, and the reasons for that are similar to those we discussed in the previous section, concerning the spherical Hecke algebra and the possibility of defining an action of Hecke operators on functions on Bun G .
However, there are two natural variations of these questions that do have satisfactory answers. The first possibility is to consider a categorical version of the Hecke algebra, i.e., instead of the space of B-invariant functions, the category D(G/B) B -mod of B-equivariant D-modules on G/B. According to a theorem of Beilinson and Bernstein [BB], we have an exact functor of global sections (as O-modules) from this category to the category of modules over the Lie algebra g of G, which is an equivalence with the category of those g-module which have a fixed character of the center of U (g) (the character of the trivial representation of g). This is the category that appears in the Kazhdan-Lusztig theory, which gives rise, among other things, to character formulas for irreducible g-modules from the category O. Furthermore, instead of the convolution product on functions, we now have convolution functors on a derived version of D(G/B) B -mod. This is the categorical Hecke algebra which has many applications. For example, Beilinson and Bernstein have defined a categorical action of this category on the derived category of the category O (which may be viewed as the category of (g, B) Harish-Chandra modules). This is a special case of a rich theory.
Note that a closely related category of perverse sheaves may also be defined over F q . Taking the traces of the Frobenius on the stalks of those sheaves, we obtain the elements of the original Hecke algebra H q (G). This operation transforms convolution product of sheaves into convolution product of functions. Thus, we see many parallels with the geometric Langlands Program (for more on this, see [Fr3], Sect. 1.3.3). In particular, the spherical Hecke algebra has a categorical analogue, for which a categorical version of the Satake isomorphism has been proved [Gin, Lu, MV]. In other words, the path of categorification of the Hecke algebra H q (G) is parallel to the path taken in the geometric Langlands theory.
But there is also a second option: We can consider the space of L 2 functions on G(C)/B(C) with respect to the natural measure of integration coming from a symplectic structure. Or, alternatively, we can define L 2 (G(C)/B(C)) as the completion of the space of half-densities on G(C)/B(C) with respect to the natural Hermitian inner product.
Both are meaningful objects, but we no longer have an action of a Hecke algebra on it. However, and this is a key point, there is a meaningful substitute for it: differential operators on G/B.
The Lie algebra g acts on L 2 (G(C)/B(C)) by holomorphic vector fields, and we have a commuting action of another copy of g by anti-holomorphic vector fields. Therefore, the tensor product of two copies of the center of U (g) acts by mutually commuting differential operators. As we mentioned above, both holomorphic and anti-holomorphic ones act according to the central character of the trivial representation. However, the center of U (g c ), where g c is a compact form of the Lie algebra g, also acts on L 2 (G(C)/B(C)) by commuting differential operators, and this action is non-trivial. It includes the Laplace operator, which corresponds to the Casimir element of U (g c ).
We then ask what are the eigenfunctions and eigenvalues of these commuting differential operators. This question has a meaningful answer. Indeed, using the isomorphism G/B ≃ G c /T c , where T c is a maximal torus of the compact form of G, and the Peter-Weyl theorem, we obtain that L 2 (G(C)/B(C)) can be decomposed as a direct sum of irreducible finitedimensional representations of g c which can be exponentiated to the group G c of adjoint type, each representation appearing with multiplicity one. Therefore the combined action of the center of U (g c ) and the Cartan subalgebra t c of T c (acting by vector fields from the right) has as eigenspaces, various weight components of the irreducible finite-dimensional representations of g c . All of these eigenspaces are therefore finite-dimensional.
For instance, for G = SL 2 every eigenspace is one-dimensional, and so we find that these differential operators have simple spectrum. In fact, suitably normalized joint eigenfunctions of the center of U (g c ) and t c are in this case the standard spherical harmonics (note that in this case G(C)/B(C) ≃ S 2 ).
This discussion suggests we may be able to build a meaningful analytic theory of automorphic forms on Bun G if, rather than looking for the eigenfunctions of Hecke operators (whose existence is questionable in the non-abelian case, as we have seen), we look for the eigenfunctions of a commutative algebra of global differential operators on Bun G . It turns out that we are in luck: there exists a large commutative algebra of differential operators acting on the line bundle of half-densities on Bun G .
Remark 3. The above discussion dovetails nicely with the intuition that comes from the theory of automorphic functions for a reductive group G over a number field F . Such a field has non-archimedian as well as archimedian places. The representation theories of the corresponding groups, such as G(Q p ) and G(C), are known to follow different paths: for the former we have, in the unramified case, the spherical Hecke algebra and the Satake isomorphism. For the latter, we never consider a naive analogue of the spherical Hecke algebra. Rather, its role is played by the center of U (g) (or, more generally, the convolution algebra of distributions on G(C) supported on its compact subgroup K, see [KV]). Now let's try to generalize this to the two-dimensional field F ((t)), where F is a number field. Then instead of G(Q p ) we have G(Q p ((t))), and instead of G(C) we have G(C((t))). In the former case, there are meaningful analogues of the spherical Hecke algebra and the Satake isomorphism which have been explored, in particular, in [K,KL,BK1,BK2,HK2]. But in the case of G(C((t))), searching for a spherical Hecke algebra is incongruent with what we know about the one-dimensional case (that of a number field F ). It seems a lot more prudent to consider the center of U (g((t))). As we show in the rest of this section, this approach leads to a rich and meaningful theory. Indeed, if we take the so-called critical central extension of g((t)), then the corresponding completed enveloping algebra does contain a large center, as shown in [FF] (see also [Fr1,Fr3]). This center gives rise to a large algebra of global commuting differential operators on Bun G .
3.6. Global differential operators on Bun G . In this subsection and the next one we report on a joint work in progress with David Kazhdan [FK]. Let us assume for simplicity that G is a connected, simply-connected, simple algebraic group over C. In [BD], Beilinson and Drinfeld have described the algebra D G of global holomorphic differential operators on Bun G acting on the square root K 1/2 of a canonical line bundle (which exists for any reductive G and is unique under our assumptions). They have proved that D G is commutative and is isomorphic to the algebra of functions on the space OpL G (X) of L G-opers on X. For a survey of this construction and the definition of OpL G (X), see, e.g., [Fr2], Sects. 8 and 9. Under the above assumptions on G, the space OpL G (X) may be identified with the space of all holomorphic connections on a particular holomorphic G-bundle F 0 on X. In particular, it is an affine space of dimension equal to dim Bun G .
The construction of these global differential operators is similar to the construction outlined in Section 3.5 above. Namely, they are obtained in [BD] from the central elements of the completed enveloping algebra of the affine Kac-Moody algebra g at the critical level, using the realization of Bun G as a double quotient of the formal loop group G(C((t))) and the Beilinson-Bernstein type localization functor. The critical level of g corresponds to the square root of the canonical line bundle on Bun G . A theorem of Feigin and myself [FF] (see also [Fr1,Fr3]) identifies the center of this enveloping algebra with the algebra of functions on the space of L G-opers on the formal punctured disc. This is a local statement that Beilinson and Drinfeld use in the proof of their theorem. Now, we can use the same method to construct the algebra D G of global anti-holomorphic differential operators on Bun G acting on the square root K 1/2 of the anti-canonical line bundle. The theorem of Beilinson and Drinfeld implies that D G is isomorphic to the algebra of functions on the complex conjugate space to the space of opers, which we denote by OpL G (X). Under the above assumptions on G, it can be identified with the space of all antiholomorphic connections on the G-bundle F 0 that is the complex conjugate of the G-bundle F 0 . While F 0 carries a holomorphic structure (i.e. a (0, 1)-connection), F 0 carries a (1, 0)connection (which one could call an "anti-holomorphic structure" on F 0 ). Just as a (1, 0), i.e. holomorphic, connection on F 0 completes its holomorphic structure to a flat connection, so does a (0, 1), i.e. anti-holomorphic, connection on F 0 completes its (1, 0)-connection to a flat connection.
Both OpL G (X) and OpL G (X) may be viewed as Lagrangian subspaces of the moduli stack of flat L G-bundles on X, and it turns out that it is their intersection that is relevant to the eigenfunctions of the global differential operators.
Indeed, we have a large commutative algebra D G ⊗ D G of global differential operators on the line bundle K 1/2 ⊗ K 1/2 of half-densities on Bun G . This algebra is isomorphic to the algebra of functions on OpL G (X) × OpL G (X).
On the other hand, there is a natural L 2 norm on the space of sections of K 1/2 ⊗ K 1/2 on Bun G . We can therefore define the space L 2 (Bun G ) as the completion of the subspace of sections with well-defined norm. These sections are uniquely determined by their restriction to an open dense subset Bun st G of stable G-bundles 9 (note that the fact that this subset is not preserved by Hecke correspondence is no longer an issue). Thus, we can define L 2 (Bun G ) as the completion of the space of sections of K 1/2 ⊗ K 1/2 with the finite norm on Bun st G . It is a Hilbert space with respect to the standard Hermitian inner product.
The elements of the algebra D G ⊗ D G are well-defined linear operators on L 2 (Bun G ). Furthermore, we expect that these operators are normal. Thus, we get a nice set-up for the problem of finding eigenfunctions and eigenvalues of these operators. It is natural to call these eigenfunctions the automorphic forms on Bun G (or Bun st G ) for a complex algebraic curve. The construction generalizes to arbitrary connected reductive complex groups G.
The joint eigenvalues of D G ⊗ D G on L 2 (Bun G ) correspond to points in OpL G (X) × OpL G (X), i.e. pairs (χ, ρ), where χ ∈ OpL G (X) and ρ ∈ OpL G (X). The question then is to describe the set of those pairs that occur as eigenvalues.
As far as I know, this spectral problem was first considered by Teschner [T], in the case of G = SL 2 . In this case, L G-opers are P GL 2 -opers, or equivalently, projective connections on X. Teschner conjectured in [T] that the eigenvalues of the algebra D SL 2 ⊗ D SL 2 on L 2 (Bun SL 2 ) are in one-to-one correspondence with those projective connections that have real monodromy. Given such a projective connection χ ∈ Op P GL 2 (X), the corresponding point in Op P GL 2 (X) × Op P GL 2 (X) is (χ, χ), where χ is determined by χ (it also has real monodromy).
Projective connections with real monodromy have been described by Goldman [Gol]. If the genus of X is greater than 1, then among them there is a special one, corresponding to the uniformization of X. But there are many others ones as well, and they have been the subject of interest for many years. It is fascinating that they now show up in the context of the Langlands correspondence for complex curves.
We expect that a similar description holds for other groups as well, i.e. the eigenvalues of the algebra D G ⊗ D G on L 2 (Bun G ) are in one-to-one correspondence with the L G-opers on X whose monodromy takes values in the split real form of L G. The details will be discussed in [FK].
In the next subsection I will illustrate how these opers appear in the abelian case.
3.7. The spectra of global differential operators for G = GL 1 . For simplicity, consider the elliptic curve X = E i = C/(Z + Zi) discussed in Section 2.1. We identify the neutral component Pic 0 (X) with X using a reference point p 0 , as in Section 2.1. Then the algebra D GL 1 (resp. D GL 1 ) coincides with the algebra of constant holomorphic (resp. anti-holomorphic) differential operators on X: The eigenfunctions of these operators are precisely the Fourier harmonics f m,n given by formula (2.4): f m,n = e 2πimx · e 2πiny , m, n ∈ Z. If we rewrite it in terms of z and z: f m,n = e πz(n+im) · e −πz(n−im) , then we find that the eigenvalues of ∂ z and ∂ z on f m,n are π(n + im) and −π(n − im) respectively. Let us recast these eigenvalues in terms of the corresponding GL 1 -opers.
By definition, a GL 1 -oper is a holomorphic connection on the trivial line bundle on X (see [Fr2], Sect. 4.5). The space of such connections is canonically isomorphic to the space of holomorphic one-forms on X which may be written as −λdz, where λ ∈ C. An element of the space of GL 1 -opers may therefore be represented as a holomorphic connection on the trivial line bundle, which together with its (0, 1) part ∂ z yields the flat connection ∇ = d − λ dz, λ ∈ C. (3.12) Under the isomorphism Spec D GL 1 ≃ Op GL 1 (X), the oper (3.12) corresponds to the eigenvalue λ of ∂ z (this is why we included the sign in (3.12)). Likewise, an element of the complex conjugate space Op GL 1 (X) is an anti-holomorphic connection on the trivial line bundle, which together with its (1, 0) part ∂ z yields the flat connection ∇ = d − µ dz, µ ∈ C.
We have found above that the eigenvalues of ∂ z and ∂ z on L 2 (Bun GL 1 ) are π(n + im) and −π(n − im), respectively, where m, n ∈ Z. The following lemma, which is proved by a direct computation, links them to GL 1 -opers with real monodromy.
This Lemma generalizes in a straightforward fashion to arbitrary curves and arbitrary abelian groups. Namely, the harmonics e 2πiϕγ , γ ∈ H 1 (X, Λ * (T )), introduced in Section 2.5 are the eigenfunctions of the global differential operators on Bun 0 T (X). The L T -oper on X encoding the eigenvalues of the holomorphic differential operators is the holomorphic connection on the trivial L T -bundle on X ∇ hol γ = d − iω γ (3.14) (compare with formula (2.55)). One can show that its monodromy representation takes values in the split real form of L T , and conversely, these are all the L T -opers on X that have real monodromy. Thus, the conjectural description of the spectra of global differential operators on Bun G in terms of opers with real monodromy (see the end of Section 3.6) holds in the abelian case.
Recall that in the abelian case we also have well-defined Hecke operators. It is interesting to note that they commute with the global differential operators and share the same eigenfunctions. Furthermore, the eigenvalues of the Hecke operators may be expressed in terms of the eigenvalues of the global differential operators. For non-abelian G, the definition of Hecke operators is problematic, as we have argued in this section. But the global differential operators are well-defined. We expect that their eigenvalues are given by the L G-opers satisfying a special condition: namely, their monodromy representation π 1 (X, p 0 ) → L G takes values in the split real form of L G. It is natural to view these homomorphisms as the Langlands parameters of the automorphic forms for curves over C.
The details will appear in [FK].