An analytic version of the Langlands correspondence for...

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An analytic version of the Langlands correspondence forcomplex curves

Edward Frenkel

University of California, Berkeley

January 2021

Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 1 / 35

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Introduction

In this talk I will present some results and conjectures from ajoint work with Pavel Etingof and David Kazhdan.

P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version ofthe Langlands correspondence for complex curves,arXiv:1908.09677, to appear in the Dubrovin Memorial Volume.

and two papers in preparation

Answering a question posed by R.P. Langlands, we propose ananalytic version of the Langlands correspondence for complex curves.

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Introduction

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Introduction

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Introduction

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The unramified Langlands correspondence for a curve X/Fq:on one side, joint spectrum of the commuting Hecke operators

acting on the space of L2 functions on the set of Fq-points of thestack BunG of G-bundles on X;

on the other side, Galois data associated to X and the Langlandsdual group LG.

If X is a curve over C, the Langlands correspondence has beentraditionally formulated in terms of sheaves rather than functions. It isusually referred to as geometric or categorical.

It turns out that there is a function-theoretic (or analytic) versionfor complex curves as well. The two versions complement each other.

Analogy: correlation functions in 2D conformal field theory aresingle-valued bilinear combinations of (multi-valued) conformal andanti-conformal blocks.

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Namely, it is possible to associate to BunG of X/C (and moregenerally X/F , where F is a local field) a natural Hilbert space HGand define analogues of the Hecke operators acting on a densesubspace of HG. We conjecture that they give rise to mutuallycommuting normal compact operators on HG.

In the case F = C, these Hecke operators commute with theglobal holomorphic differential operators on BunG introduced byBeilinson and Drinfeld, as well as their complex conjugates.

We conjecture that the joint spectrum of this commutativealgebra (properly understood) can be identified with the set ofLG-opers on X whose monodromy is in the split real form of LG, upto conjugation (these play the role of the Galois data).

This statement may be viewed as an anaytic Langlandsconjecture for complex curves.

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Basic definitions

X – smooth projective irreducible curve over CS ⊂ X – reduced divisorKX – canonical line bundle on X

G – connected simple algebraic group over CLG – the Langlands dual group

BunG = BunG(X,S) – algebraic stack of pairs (F , rS), where Fis a G-bundle on X and rS is a B-reduction of F|S

BunsG = BunsG(X,S) ⊂ BunG(X,S) – substack of those stable

pairs (F , rS) whose group of automorphisms is equal to the centerZ(G) of G

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Basic definitions

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Basic definitions

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Basic definitions

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Basic definitions

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Basic definitions

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Basic definitions

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Assumption:

BunsG(X,S) is open and dense in BunG(X,S), i.e. one of thefollowing cases:

1 the genus of X is greater than 1, and S is arbitrary;2 X is an elliptic curve and |S| ≥ 1;3 X = P1 and |S| ≥ 3.

The stack BunsG(X,S) is a Z(G)-gerbe over a smooth algebraicvariety BunsG(X,S).

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Hilbert space

KBun – the canonical line bundle on BunG

For simply-connected G, Beilinson and Drinfeld have constructed

a square root K1/2Bun of KBun. For a general G, their construction

sometimes requires a choice of a square root of the canonical linebundle KX on X. If so, we will make such a choice (however, the

bundle Ω1/2Bun below does not depend on this choice).

We’ll use the same notation for the restriction of this K1/2Bun to Bun

sG.

Given a holomorphic line bundle L on a variety Y , let|L| := L ⊗ L

Set Ω1/2Bun := |K

1/2Bun| – the line bundle of half-densities on BunsG.

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Hilbert space

sG.

Set Ω1/2Bun := |K

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Hilbert space

sG.

Set Ω1/2Bun := |K

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Hilbert space

sG.

Set Ω1/2Bun := |K

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Hilbert space

sG.

Set Ω1/2Bun := |K

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Hilbert space

Let VG – space of smooth compactly supported sections of Ω1/2Bun

over BunsG, and let

〈·, ·〉 – positive-definite Hermitian form on VG given by

〈v, w〉 :=∫BunsG

v · w, v, w ∈ VG

HG – the Hilbert space completion of VG

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Hilbert space

over BunsG, and let

v · w, v, w ∈ VG

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Hilbert space

over BunsG, and let

v · w, v, w ∈ VG

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What kind of operators could act on the Hilbert space HG?

1 holomorphic differential operators;2 anti-holomorphic differential operators;3 Hecke (integral) operators.

Challenges: Differential operators are unbounded. It is a highlynon-trivial task to define their self-adjoint (or normal) extensions,which is necessary to be able to make sense of the notion of theirjoint spectra on HG (and there could be different choices).

Hecke operators are also initially defined on a dense subspace ofHG. But we conjecture that they extend by continuity to normalcompact operators on the entire HG. If one proves this, one gets agood spectral problem for both Hecke & differential operators sinceone can show that they commute (in the sense we’ll discuss later).

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Holomorphic differential operators

Consider the case of simply-connected G and |S| = ∅.Let DG be the sheaf of algebraic (hence holomorphic) differential

operators acting on the line bundle K1/2Bun on BunG.

DG := Γ(BunG,DG)

Theorem 1 (Beilinson & Drinfeld)

DG ' Fun OpLG(X), where OpLG(X) – space of LG-opers on X.

Definition. An LG-oper on a curve X is a holomorphicLG-bundle with a holomorphic connection ∇ and a reduction to aBorel subgroup LB which is in a special relative position with ∇.

Example (to be discussed later). A PGL2-oper on X is aprojective connection, i.e. a second-order holomorphic differential

operator of the form ∂2z − v(z): K−1/2X → K

3/2X .

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DG := Γ(BunG,DG)

3/2X .

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DG := Γ(BunG,DG)

3/2X .

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DG := Γ(BunG,DG)

3/2X .

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DG := Γ(BunG,DG)

3/2X .

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DG := Γ(BunG,DG)

3/2X .

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DG := Γ(BunG,DG)

3/2X .

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Anti-holomorphic differential operators

Complex conjugates of elements of DG are global

anti-holomorphic differential operators acting on K1/2

Bun.

They generate a commutative algebra DG.

DG ' Fun OpLG(X)

AG := DG ⊗DG is a commutative algebra acting on C∞

sections of the line bundle Ω1/2Bun = K

1/2Bun ⊗K

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Bun on BunsG.

Let ṼG be the space of smooth sections of Ω1/2Bun on

BunvsG ⊂ BunsG, the moduli space of very stable G-bundles (thoseF which do not admit non-zero φ ∈ Γ(X, gF ⊗KX) taking nilpotentvalues everywhere).

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Bun.

DG ' Fun OpLG(X)

1/2Bun ⊗K

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Bun on BunsG.

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Bun.

DG ' Fun OpLG(X)

1/2Bun ⊗K

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Bun on BunsG.

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Bun.

DG ' Fun OpLG(X)

1/2Bun ⊗K

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Bun on BunsG.

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“Doubling” of the quantum Hitchin system

Given a homomorphism Λ : AG → C, denote by ṼG,Λ thecorresponding eigenspace of AG in ṼG.

Λ = (χ, µ), where χ ∈ OpLG(X), µ ∈ OpLG(X).

If f is a non-zero element of ṼG,(χ,µ), then it satisfies twosystems of differential equations:

(1) P · f = χ(P )f, P ∈ DG(2) Q · f = µ(Q)f, Q ∈ DGSystem (1) is known as the quantum Hitchin system.

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The corresponding left DG-module∆χ := DG ⊗

DGCχ

was introduced and studied by Beilinson and Drinfeld, who haveproved that ∆χ is a Hecke eigensheaf corresponding to the

LG-oper χunder the geometric/categorical Langlands correspondence.

Moreover, they have shown that the restriction of ∆χ to BunvsG

is a vector bundle with a projectively flat connection (of a rank thatgrows exponentially with the genus of X).

Local sections of ∆χ over BunvsG are local holomorphic solutions

of system (1). They are multi-valued and the monodromy is rathercomplicated, which is why it’s impossible to attach to a given χ aspecific holomorphic half-form. (Even if there were single-valuedsolutions, it wouldn’t be clear which one to choose.) Instead, weattach a whole DG-module on BunG to χ.

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DGCχ

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DGCχ

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DGCχ

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Likewise, to µ ∈ OpLG(X) we attach an anti-holomorphicD-module ∆µ whose local sections on Bun

vsG are local

anti-holomorphic solutions of system (2), also multi-valued.

However, if we look for smooth solutions of systems (1) and (2)simultaneously, it is possible that for some χ and µ there will be asingle-valued solution, which can be written locally in bilinear form

f =∑

i,j aij φi(z)ψj(z)

{φi} – local sections of ∆χ{ψj} – local sections of ∆µ.

This actually implies that dim ṼG,(χ,µ)

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vsG are local

f =∑

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vsG are local

f =∑

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vsG are local

f =∑

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vsG are local

f =∑

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vsG are local

f =∑

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Conjecture 1

1 All ṼG,(χ,µ) ⊂ HG2 There is an orthogonal decomposition

HG =⊕̂

(χ,µ) ṼG,(χ,µ)

3 If ṼG,(χ,µ) 6= 0, then µ = τ(χ), where τ is the Chevalley involutionon LG and χ ∈ OpLG(X)R.

Definition. OpLG(X)R is the set ofLG-opers on X such that

the monodromy representation ρχ : π1(X, p0)→ LG(C) is isomorphicto its complex conjugate, i.e. ρχ ' ρχ.

We expect that OpLG(X)R is a discrete subset of OpLG(X).This is known for LG = PGL2 (G. Faltings).

For G = PGL2, Conjecture 1 implements ideas of J.Teschner.

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Conjecture 1

HG =⊕̂

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Conjecture 1

HG =⊕̂

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Conjecture 1

HG =⊕̂

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We expect that OpLG(X)R coincides with the set of allLG-opers

on X with real monodromy, i.e. such that the image in LG(C) of themonodromy representation

ρχ : π1(X, p0)→ LG

associated to χ is contained, up to conjugation, in the split real formLG(R) of LG(C).

This is known for G = PGL2 and we can prove it for general Gin the case when there is at least one point with Borel reduction (i.e.|S| 6= ∅).

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We expect that OpLG(X)R coincides with the set of allLG-opers

on X with real monodromy, i.e. such that the image in LG(C) of themonodromy representation

ρχ : π1(X, p0)→ LG

associated to χ is contained, up to conjugation, in the split real formLG(R) of LG(C).

This is known for G = PGL2 and we can prove it for general Gin the case when there is at least one point with Borel reduction (i.e.|S| 6= ∅).

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Proving Conjecture 1 directly is a daunting task. This is wherethe third set of operators on HG – integral Hecke operators – comesin handy.

Though they are also initially defined on a dense subspace of HG(like diff. operators), we conjecture that, unlike the differentialoperators, they extend to (mutually commuting) continuous operatorson the entire HG, which are moreover normal and compact with trivialcommon kernel.

If so, then by a general result of functional analysis, HGdecomposes into a (completed) direct sum of mutually orthogonalfinite-dimensional eigenspaces of the Hecke operators. Moreover, wecan show that they commute with the differential operators, and sothe Compactness Conjecture can be used to prove Conjecture 1.

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Hecke operators

In fact, Hecke operators can be defined for curves over any local field.

For non-archimedian local fields, these operators were essentiallydefined by A. Braverman and D. Kazhdan inSome examples of Hecke algebras for two-dimensional local fields,Nagoya Math. J. Volume 184 (2006), 57-84.

For G = PGL2, X = P1, Hecke operators were studied by M.Kontsevich in his paper Notes on motives in finite characteristic(2007). In his letters to us (2019) he conjectured compactness ofaverages of the Hecke operators over sufficiently many points.

The idea that Hecke operators over C could be used to constructan analogue of the Langlands correspondence was suggested in 2018by R.P. Langlands, who attempted to construct them in the casewhen G = GL2, X is an elliptic curve, and S = ∅ (however, for anelliptic curve X we can only define Hecke operators if |S| 6= ∅).

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Hecke operators

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Hecke operators

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Hecke operators

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Hecke operators

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For a dominant coweight λ of G, denote by

q : Z(λ)→ BunG × BunG ×Xthe Hecke correspondence attached to λ. Let

p1,2 : BunG×BunG×X → BunG, p3 : BunG×BunG×X → Xbe the projections, and set qi := pi ◦ q.The following is due to Beilinson–Drinfeld and Braverman–Kazhdan.

Theorem 2

There exists an isomorphism

a : q∗1(K1/2Bun) ' q

∗2(K

1/2Bun)⊗ ω2 ⊗ q

∗3(K

−〈λ,ρ〉X )

where ω2 is the relative canonical bundle along the fibers of q2 × q3and ρ is the half sum of positive roots.

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Theorem 2

a : q∗1(K1/2Bun) ' q

∗2(K

1/2Bun)⊗ ω2 ⊗ q

∗3(K

−〈λ,ρ〉X )

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Theorem 2

a : q∗1(K1/2Bun) ' q

∗2(K

1/2Bun)⊗ ω2 ⊗ q

∗3(K

−〈λ,ρ〉X )

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Theorem 2

a : q∗1(K1/2Bun) ' q

∗2(K

1/2Bun)⊗ ω2 ⊗ q

∗3(K

−〈λ,ρ〉X )

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The isomorphism a gives rise to an isomorphism

|a| : q∗1(Ω1/2Bun) ' q∗2(Ω

1/2Bun)⊗ Ω2 ⊗ q∗3(|KX |−〈λ,ρ〉)

where Ω2 := |ω2| is the relative line bundle of densities along thefibers of q2 × q3. Let

UG(λ) := {F ∈ BunsG|(q2(q−11 (F)) ⊂ BunsG}This is an open subset of BunsG, which is dense if

dim BunG = dimG · (g − 1) + dimG/B · |S| (g > 1)is sufficiently large. (For example, for G = PGL2, λ = ω1, this is soif dim BunG > 1.)

Assume that UG(λ) ⊂ BunsG is dense and let VG(λ) ⊂ VG bethe subspace of half-densities f such that supp(f) ⊂ UG(λ).

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|a| : q∗1(Ω1/2Bun) ' q∗2(Ω

1/2Bun)⊗ Ω2 ⊗ q∗3(|KX |−〈λ,ρ〉)

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|a| : q∗1(Ω1/2Bun) ' q∗2(Ω

1/2Bun)⊗ Ω2 ⊗ q∗3(|KX |−〈λ,ρ〉)

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|a| : q∗1(Ω1/2Bun) ' q∗2(Ω

1/2Bun)⊗ Ω2 ⊗ q∗3(|KX |−〈λ,ρ〉)

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|a| : q∗1(Ω1/2Bun) ' q∗2(Ω

1/2Bun)⊗ Ω2 ⊗ q∗3(|KX |−〈λ,ρ〉)

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ZG,x := (q2 × q3)−1(G × x), G ∈ BunG(C), x ∈ X(C)It is compact and isomorphic to the closure Grλ of the

G[[z]]-orbit Grλ in the affine Grassmannian of G.

The results of Braverman–Kazhdan imply that for any f ∈ VG(λ)and x ∈ X(C), the restriction of the pull-back q∗1(f) to ZG,x is awell-defined measure with values in the line |ΩBun|1/2G ⊗ |KX |

−〈λ,ρ〉x .

Hence for any f ∈ VG(λ), the integral

(Ĥλ(x) · f)(G) :=∫ZxG(F )

q∗1(f),

is absolutely convergent for all G ∈ BunsG(C) and belongs to thespace V̂G of smooth functions on Bun

sG(C).

Therefore this integral defines a Hecke operator

Ĥλ(x) : VG(λ)→ V̂G

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−〈λ,ρ〉x .

(Ĥλ(x) · f)(G) :=∫ZxG(F )

q∗1(f),

sG(C).

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−〈λ,ρ〉x .

(Ĥλ(x) · f)(G) :=∫ZxG(F )

q∗1(f),

sG(C).

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−〈λ,ρ〉x .

(Ĥλ(x) · f)(G) :=∫ZxG(F )

q∗1(f),

sG(C).

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−〈λ,ρ〉x .

(Ĥλ(x) · f)(G) :=∫ZxG(F )

q∗1(f),

sG(C).

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Conjecture 2 (Compactness Conjecture)

1 For any f ∈ VG(λ) and x ∈ X(C), the section Ĥλ(x) · f issquare-integrable (i.e. belongs to HG) and hence we obtain anoperator

Hλ(x) : VG(λ)→ HG ⊗ |KX |−〈λ,ρ〉x .2 For any identification (K

1/2X )x

∼= C, the corresponding operatorsVG(λ)→ HG extend to a family of commuting compact normaloperators on HG, which we also denote by Hλ(x).

3 Hλ(x)† = H−w0(λ)(x).

4

⋂λ,x KerHλ(x) = {0}.

Remark. We expect that integrals defining Hecke operatorsHλ(x) are absolutely convergent for all f ∈ VG.From now on we assume that Compactness Conjecture holds.

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1/2X )x

3 Hλ(x)† = H−w0(λ)(x).

4

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1/2X )x

3 Hλ(x)† = H−w0(λ)(x).

4

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Let HG be the commutative algebra generated by operatorsHλ(x), λ ∈ P̌+, x ∈ X. Denote by Spec(HG) its spectrum.

Corollary 3

There is an orthogonal decomposition

HG =⊕̂

s∈Spec(HG)HG(s)

where HG(s), s ∈ Spec(HG), are the finite-dimensional jointeigenspaces of HG in HG.

Conjecture 3

Every HG(s) is an eigenspace of AG.

Corollary 4

If (χ, µ) ∈ SpecAG, then µ = τ(χ) and χ ∈ OpγLG(X)R.

OpγLG(X)R – subset of realLG-opers in a component of OpLG(X).

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Corollary 3

HG =⊕̂

s∈Spec(HG)HG(s)

Conjecture 3

Corollary 4

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Corollary 3

HG =⊕̂

s∈Spec(HG)HG(s)

Conjecture 3

Corollary 4

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Corollary 3

HG =⊕̂

s∈Spec(HG)HG(s)

Conjecture 3

Corollary 4

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Corollary 3

HG =⊕̂

s∈Spec(HG)HG(s)

Conjecture 3

Corollary 4

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Remark. Recall that first we defined a Hecke operatorĤλ(x) : VG(λ)→ V̂G.

The algebra AG naturally acts on both VG(λ) and V̂G. Hence thecommutators [P, Ĥλ(x)], P ∈ AG, make sense.

We have [P, Ĥλ(x)] = 0, ∀P ∈ AG.To see this, realize BunG as G(X\x)\G(Fx)/G(Ox).

Then Ĥλ(x) acts from the right, whereas AG can be obtainedfrom the action of the center of Ũ(ĝ)crit from the left.

However, to prove Conjecture 3 we need a stronger form ofcommutativity, and a crucial element in proving it is the system ofdifferential equations satisfied by Ĥλ(x) which we discuss below.

Pavel will explain this in more detail in his talk.

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With a choice of K1/2X , we can identify OpLGad(X) with a

specific component OpγLG(X) in OpLG(X).

An oper χ ∈ OpγLG(X) is a triple (FγLG,FγLB,∇, ), where F

γLG

is aspecific LG-bundle on X equipped with a reduction FγLB to a Borelsubgroup LB ⊂ LG, and ∇χ is a holomorphic connection on FγLG,satisfying a transversality condition with respect to FγLB.

Consider the case G = SL2 (following Beilinson and Drinfeld).

OpSL2(X) =⊔

γ∈θ(X)

OpγSL2(X)

where θ(X) is the set of isomorphism classes of square roots of KX .

Consider a component OpγSL2(X) of OpSL2(X).

K1/2X – a square root of KX in the isomorphism class γ.Vω1 – the rank 2 vector bundle associated to F

γSL2

. Then

0→ K1/2X → Vω1 −→ K−1/2X → 0

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γLG

OpSL2(X) =⊔

γ∈θ(X)

OpγSL2(X)

γSL2

. Then

0→ K1/2X → Vω1 −→ K−1/2X → 0

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γLG

OpSL2(X) =⊔

γ∈θ(X)

OpγSL2(X)

γSL2

. Then

0→ K1/2X → Vω1 −→ K−1/2X → 0

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γLG

OpSL2(X) =⊔

γ∈θ(X)

OpγSL2(X)

γSL2

. Then

0→ K1/2X → Vω1 −→ K−1/2X → 0

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Each component OpγSL2(X) is isomorphic to OpPGL2(X). Here’san altenative description of this component.

A projective connection associated to K1/2X is a second-order

differential operator P : K−1/2X → K

3/2X such that

1 symb(P ) = 1 ∈ OX , and2 P is algebraically self-adjoint.

They form an affine space Projγ(X). Locally, P = ∂2z − v(z).

Lemma 5

There is a bijection OpγSL2(X) ' Projγ(X)

χ ∈ OpγSL2(X) 7→ Pχ ∈ Projγ(X)such that the section sω1 ∈ Γ(X,K

−1/2X ⊗ Vω1) corresponding to the

embedding K1/2X ↪→ Vω1 satisfies Pχ · sω1 = 0

(here we use the DX-module structure on Vω1 corresponding to ∇χ).

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We will say that χ ∈ OpγLG(X)R if the monodromyrepresentation ρχ : π1(X, p0)→ LG(C) is isomorphic to its complexconjugate, i.e. ρχ ' ρχ.

According to Corollary 4, we expect that there is a map

Φ : OpγLG(X)R → Spec(HG)(possibly multivalued) and we wish to describe it explicitly. Thiswould give a description of the eigenvalues of the Hecke operators.

As x varies along X, the Hecke operators Hλ(x) combine into asection of the C∞ line bundle |KX |−〈λ,ρ〉 on X with values inoperators HG → HG. We denote it by Hλ.

Thus, each eigenvalue of Hλ defines a section of the C∞ line

bundle |KX |−〈λ,ρ〉 on X.We will now write an explicit formula for the eigenvalue Φλ(χ)

corresponding to χ ∈ OpγLG(X)R.

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