Post on 18-Feb-2021
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1/35
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
2/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 2 / 35
2/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 2 / 35
2/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 2 / 35
2/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 2 / 35
3/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 3 / 35
3/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 3 / 35
3/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 3 / 35
3/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 3 / 35
3/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 3 / 35
3/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 3 / 35
4/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 4 / 35
4/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 4 / 35
4/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 4 / 35
4/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 4 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
5/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 5 / 35
6/35
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 6 / 35
7/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 7 / 35
7/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 7 / 35
7/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 7 / 35
7/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 7 / 35
7/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 7 / 35
8/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 8 / 35
8/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 8 / 35
8/35
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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 8 / 35
9/35
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 9 / 35
9/35
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 9 / 35
9/35
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 9 / 35
9/35
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 9 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
10/35
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 .
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 10 / 35
11/35
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
1/2
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 11 / 35
11/35
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
1/2
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 11 / 35
11/35
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
1/2
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 11 / 35
11/35
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
1/2
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 11 / 35
12/35
“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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 12 / 35
12/35
“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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 12 / 35
12/35
“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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 12 / 35
12/35
“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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 12 / 35
12/35
“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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 12 / 35
13/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 13 / 35
13/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 13 / 35
13/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 13 / 35
13/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 13 / 35
14/35
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)
14/35
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)
14/35
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)
14/35
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)
14/35
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)
14/35
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)
15/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 15 / 35
15/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 15 / 35
15/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 15 / 35
15/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 15 / 35
16/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 16 / 35
16/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 16 / 35
17/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 17 / 35
17/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 17 / 35
17/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 17 / 35
18/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 18 / 35
18/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 18 / 35
18/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 18 / 35
18/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 18 / 35
18/35
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= ∅).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 18 / 35
19/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 19 / 35
19/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 19 / 35
19/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 19 / 35
19/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 19 / 35
20/35
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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 20 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 20 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 20 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 20 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 20 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 21 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 21 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 21 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 21 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 21 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 22 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 22 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 22 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 23 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 23 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 23 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 23 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 23 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 24 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 24 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 24 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 24 / 35
24/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 24 / 35
24/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 24 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 25 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 25 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 25 / 35
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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
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 25 / 35
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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).
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 26 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 27 / 35
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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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 27 / 35
27/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 27 / 35
27/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 27 / 35
27/35
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.
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 27 / 35
28/35
Consider first the case G = SL2, λ = ω1.
Let χ ∈ OpγSL2(X)R. The corresponding eigenvalue of Hω1 is asection Φω1(χ) of |KX |−1/2.
Recall 0→ K1/2X → Vω1 −→ K1/2X → 0
and sω1 ∈ Γ(X,K−1/2X ⊗ Vω1) corresponding to K
−1/2X ↪→ Vω1 .
By definition of OpγSL2(X)R,
(Vω1 ,∇χ,ω1) ' (Vω1 ,∇χ,ω1)
as C∞ flat bundles. Since Vω1 ' V∗ω1 , we get an Hermitian form
hχ,ω1(·, ·) : (Vω1 ,∇χ,ω1)⊗ (Vω1 ,∇χ,ω1)→ (C∞X , d)
Conjecture 4
Φω1(χ) = ±hχ,ω1(sω1 , sω1)
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 28 / 35
28/35
Consider first the case G = SL2, λ = ω1.
Let χ ∈ OpγSL2(X)R. The corresponding eigenvalue of Hω1 is asection Φω1(χ) of |KX |−1/2.
Recall 0→ K1/2X → Vω1 −→ K1/2X → 0
and sω1 ∈ Γ(X,K−1/2X ⊗ Vω1) corresponding to K
−1/2X ↪→ Vω1 .
By definition of OpγSL2(X)R,
(Vω1 ,∇χ,ω1) ' (Vω1 ,∇χ,ω1)
as C∞ flat bundles. Since Vω1 ' V∗ω1 , we get an Hermitian form
hχ,ω1(·, ·) : (Vω1 ,∇χ,ω1)⊗ (Vω1 ,∇χ,ω1)→ (C∞X , d)
Conjecture 4
Φω1(χ) = ±hχ,ω1(sω1 , sω1)
Edward Frenkel (UC Berkeley) Analytic version of the Langlands correspondence January 2021 28 / 35
28/35
Consider first the case G = SL2, λ = ω1.
Let χ ∈ OpγSL2(X)R. The corresponding eigenvalue of Hω1 is asection Φω1(χ) of |KX |−1/2.
Recall 0→ K1/2X → Vω1 −→ K1/2X → 0
and sω1 ∈ Γ(X,K−1/2X ⊗ Vω1) corresponding to K
−1/2X ↪→ Vω1 .
By definition of OpγSL2(X)R,
(Vω1 ,∇