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Harish-Chandra characters and the local Langlands correspondence - - PowerPoint PPT Presentation

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Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha Local Langlands Correspondence Global Langlands correspondence Galois representations Tasho Kaletha Local

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Harish-Chandra characters and the local Langlands correspondence

Tasho Kaletha

University of Michigan

  • 16. November 2018

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence ϕ ↔ π

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence ϕ ↔ π π automorphic rep.:

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence ϕ ↔ π π automorphic rep.: L2(Σ \ GLn(R)) GLn(R)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence ϕ ↔ π π automorphic rep.: L2(GLn(Q) \ GLn(A)) GLn(A)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence ϕ ↔ π π automorphic rep.: L2(GLn(Q) \ GLn(A)) GLn(A) L(s, ϕ) = L(s, π)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

Galois representations Γ = Gal(¯ Q/Q) compact topological group encodes symmetries of solutions to rational polynomial equations ϕ : Γ → GLn(C) matrix representation L(s, ϕ) Artin L-function Global Langlands correspondence ϕ ↔ π π automorphic rep.: L2(GLn(Q) \ GLn(A)) GLn(A) L(s, ϕ) = L(s, π) Application Langlands 1980: Proves many cases of the 2-dimensional Artin conjecture

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C),

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C), . . .

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C), . . . ϕ ↔ π

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C), . . . ϕ ↔ π π automorphic representation:

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C), . . . ϕ ↔ π π automorphic representation: L2(G(Q) \ G(A)) G(A)

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C), . . . ϕ ↔ π π automorphic representation: L2(G(Q) \ G(A)) G(A) G ↔ G Langlands dual groups

Tasho Kaletha Local Langlands Correspondence

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Global Langlands correspondence

General reductive groups ϕ : Γ → G,

  • G = Sp2n(C),

SOn(C), . . . ϕ ↔ π π automorphic representation: L2(G(Q) \ G(A)) G(A) G ↔ G Langlands dual groups

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . .

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞)

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp πp irreducible (admissible) representation of G(Qp)

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp πp irreducible (admissible) representation of G(Qp) Results

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp πp irreducible (admissible) representation of G(Qp) Results GLN: Harris-Taylor, Henniart

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp πp irreducible (admissible) representation of G(Qp) Results GLN: Harris-Taylor, Henniart SpN, SON, UN, Arthur, Mok, KMSW

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp πp irreducible (admissible) representation of G(Qp) Results GLN: Harris-Taylor, Henniart SpN, SON, UN, Arthur, Mok, KMSW General G?

Tasho Kaletha Local Langlands Correspondence

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Local Langlands correspondence

Decomposition groups Γp = Gal(¯ Qp/Qp) ⊂ Γ , p = ∞, 2, 3, 5, 7, . . . Γ∞ = Z/2Z, generated by complex conjugation Γp infinite, compact, (p < ∞) Local correspondence ϕp ↔ πp πp irreducible (admissible) representation of G(Qp) Results GLN: Harris-Taylor, Henniart SpN, SON, UN, Arthur, Mok, KMSW General G? partial results in positive characteristic Lafforgue-Genestier

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods G(Qp): totally disconnected, little analysis

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods G(Qp): totally disconnected, little analysis

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods G(Qp): totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G(R) and G(Qp) ought to behave similarly

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods G(Qp): totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G(R) and G(Qp) ought to behave similarly Langlands classification π admissible ↔ (M, σ, ν), M ⊂ G Levi, σ ∈ Irr(M) tempered

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods G(Qp): totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G(R) and G(Qp) ought to behave similarly Langlands classification π admissible ↔ (M, σ, ν), M ⊂ G Levi, σ ∈ Irr(M) tempered R-groups π tempered ↔ (M, σ, τ), M ⊂ G Levi, σ ∈ Irr(M) discrete

Tasho Kaletha Local Langlands Correspondence

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Local representation theory

The groups G(R): locally connected, analytic methods G(Qp): totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G(R) and G(Qp) ought to behave similarly Langlands classification π admissible ↔ (M, σ, ν), M ⊂ G Levi, σ ∈ Irr(M) tempered R-groups π tempered ↔ (M, σ, τ), M ⊂ G Levi, σ ∈ Irr(M) discrete In : IG

P (σ) → IG P (σ),

n ∈ W(M, G)(F)σ

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series:

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z)

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal:

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

3

π does not appear in any parabolic induction

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

3

π does not appear in any parabolic induction

4

π induced from a compact open subgroup

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

3

π does not appear in any parabolic induction

4

π induced from a compact open subgroup real case

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

3

π does not appear in any parabolic induction

4

π induced from a compact open subgroup real case

1

There are no supercuspidal representations

Tasho Kaletha Local Langlands Correspondence

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SLIDE 60

Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

3

π does not appear in any parabolic induction

4

π induced from a compact open subgroup real case

1

There are no supercuspidal representations

2

Casselman: Every irreducible representation appears in parabolic induction

Tasho Kaletha Local Langlands Correspondence

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Discrete series dissonance

Discrete series π discrete series: aij(g) ∈ L2(G/Z) p-adic case

1

Many discrete series are supercuspidal

2

π supercuspidal: aij(g) compact modulo center

3

π does not appear in any parabolic induction

4

π induced from a compact open subgroup real case

1

There are no supercuspidal representations

2

Casselman: Every irreducible representation appears in parabolic induction

3

There are no compact open subgroups

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

Tasho Kaletha Local Langlands Correspondence

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Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Tasho Kaletha Local Langlands Correspondence

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SLIDE 70

Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Tasho Kaletha Local Langlands Correspondence

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SLIDE 71

Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Local Langlands correspondence

Tasho Kaletha Local Langlands Correspondence

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SLIDE 72

Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Local Langlands correspondence Langlands: ϕ

Tasho Kaletha Local Langlands Correspondence

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SLIDE 73

Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Local Langlands correspondence Langlands: ϕ → (S, B, θ)/G(C)

Tasho Kaletha Local Langlands Correspondence

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SLIDE 74

Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Local Langlands correspondence Langlands: ϕ → (S, B, θ)/G(C) → {π1, . . . , πk}

Tasho Kaletha Local Langlands Correspondence

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SLIDE 75

Real discrete series

Harish-Chandra parameterization

1

{π discrete} H−C ← → {(S, B, θ)}/G(R)

2

S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, θ : S(R) → C×, dθ is B-dominant

3

Θπ(s) = (−1)q(G)

  • w∈N(S,G)(R)/S(R)

θ(γw)

  • α>0(1−α(γw)−1)

Highest weight theory

1

π algebraic irrep of G(C) ↔ {(S, B, θ)}/G(C)

2

Θπ(s) =

  • w∈N(S,G)(C)/S(C)

θ(γw)

  • α>0(1−α(γw)−1)

Local Langlands correspondence Langlands: ϕ → (S, B, θ)/G(C) → {π1, . . . , πk} Θπ1 + · · · + Θπk conjugation invariant under G(C)

Tasho Kaletha Local Langlands Correspondence

slide-76
SLIDE 76

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Tasho Kaletha Local Langlands Correspondence

slide-77
SLIDE 77

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

Tasho Kaletha Local Langlands Correspondence

slide-78
SLIDE 78

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs,

Tasho Kaletha Local Langlands Correspondence

slide-79
SLIDE 79

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

Tasho Kaletha Local Langlands Correspondence

slide-80
SLIDE 80

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f Tasho Kaletha Local Langlands Correspondence

slide-81
SLIDE 81

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

Tasho Kaletha Local Langlands Correspondence

slide-82
SLIDE 82

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗,

Tasho Kaletha Local Langlands Correspondence

slide-83
SLIDE 83

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

Tasho Kaletha Local Langlands Correspondence

slide-84
SLIDE 84

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F),

Tasho Kaletha Local Langlands Correspondence

slide-85
SLIDE 85

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Tasho Kaletha Local Langlands Correspondence

slide-86
SLIDE 86

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

Tasho Kaletha Local Langlands Correspondence

slide-87
SLIDE 87

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G,

Tasho Kaletha Local Langlands Correspondence

slide-88
SLIDE 88

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk}

Tasho Kaletha Local Langlands Correspondence

slide-89
SLIDE 89

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk} ↔ Irr(Sϕ)

Tasho Kaletha Local Langlands Correspondence

slide-90
SLIDE 90

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk} ↔ Irr(Sϕ)

2

SΘϕ =

π∈Πϕ(G) Θπ,

Tasho Kaletha Local Langlands Correspondence

slide-91
SLIDE 91

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk} ↔ Irr(Sϕ)

2

SΘϕ =

π∈Πϕ(G) Θπ,

invariant under G(¯ F)

Tasho Kaletha Local Langlands Correspondence

slide-92
SLIDE 92

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk} ↔ Irr(Sϕ)

2

SΘϕ =

π∈Πϕ(G) Θπ,

invariant under G(¯ F)

3

Fourier inversion: s ∈ Sϕ, Θs

ϕ = π∈Πϕ(G) ρπ(s)Θπ

Tasho Kaletha Local Langlands Correspondence

slide-93
SLIDE 93

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk} ↔ Irr(Sϕ)

2

SΘϕ =

π∈Πϕ(G) Θπ,

invariant under G(¯ F)

3

Fourier inversion: s ∈ Sϕ, Θs

ϕ = π∈Πϕ(G) ρπ(s)Θπ

4

  • Gs ⊂

G,

Tasho Kaletha Local Langlands Correspondence

slide-94
SLIDE 94

Endoscopy: G(¯ F)-conjugacy vs. G(F)-conjugacy

Geometric side

1

γ ∈ G(F)rs, Oγ(f) =

  • γG(F) f,

SOγ(f) =

  • γG(¯

F)∩G(F) f 2

γG(¯

F) ∩ G(F) = a∈H1(Γ,Tγ) aγG(F)

3

Fourier inversion: κ ∈ H1(Γ, Tγ)∗, Oκ

γ = a κ(a)Oaγ

4

  • Gκ ⊂

G, γκ ∈ Gκ(F), Oκ

γ,G(f) = ∆(γκ, γ)SOγκ,Gκ(fκ)

Spectral side

1

ϕ : Γp → G, Πϕ(G) = {π1, . . . , πk} ↔ Irr(Sϕ)

2

SΘϕ =

π∈Πϕ(G) Θπ,

invariant under G(¯ F)

3

Fourier inversion: s ∈ Sϕ, Θs

ϕ = π∈Πϕ(G) ρπ(s)Θπ

4

  • Gs ⊂

G, Θs

ϕ,G(f) = SΘϕ,Gs(fs)

Tasho Kaletha Local Langlands Correspondence

slide-95
SLIDE 95

Supercuspidal representations

Tasho Kaletha Local Langlands Correspondence

slide-96
SLIDE 96

Supercuspidal representations

Yu’s construction 2001    (G0 ⊂ G1 ⊂ · · · ⊂ Gd = G) π−1 (φ0, φ1, . . . , φd)   

J.K.Yu

− − − → {irred. s.c reps of G(F)}

Tasho Kaletha Local Langlands Correspondence

slide-97
SLIDE 97

Supercuspidal representations

Yu’s construction 2001    (G0 ⊂ G1 ⊂ · · · ⊂ Gd = G) π−1 (φ0, φ1, . . . , φd)   

J.K.Yu

− − − → {irred. s.c reps of G(F)} Properties

Tasho Kaletha Local Langlands Correspondence

slide-98
SLIDE 98

Supercuspidal representations

Yu’s construction 2001    (G0 ⊂ G1 ⊂ · · · ⊂ Gd = G) π−1 (φ0, φ1, . . . , φd)   

J.K.Yu

− − − → {irred. s.c reps of G(F)} Properties

1

Kim 2007: Surjective for p >> 0

Tasho Kaletha Local Langlands Correspondence

slide-99
SLIDE 99

Supercuspidal representations

Yu’s construction 2001    (G0 ⊂ G1 ⊂ · · · ⊂ Gd = G) π−1 (φ0, φ1, . . . , φd)   

J.K.Yu

− − − → {irred. s.c reps of G(F)} Properties

1

Kim 2007: Surjective for p >> 0

2

Hakim-Murnaghan 2008: Fibers as equivalence classes.

Tasho Kaletha Local Langlands Correspondence

slide-100
SLIDE 100

Supercuspidal representations

Yu’s construction 2001    (G0 ⊂ G1 ⊂ · · · ⊂ Gd = G) π−1 (φ0, φ1, . . . , φd)   

J.K.Yu

− − − → {irred. s.c reps of G(F)} Properties

1

Kim 2007: Surjective for p >> 0

2

Hakim-Murnaghan 2008: Fibers as equivalence classes.

3

Adler-DeBacker-Spice 2008++: Character formula.

Tasho Kaletha Local Langlands Correspondence

slide-101
SLIDE 101

Supercuspidal representations

Yu’s construction 2001    (G0 ⊂ G1 ⊂ · · · ⊂ Gd = G) π−1 (φ0, φ1, . . . , φd)   

J.K.Yu

− − − → {irred. s.c reps of G(F)} Properties

1

Kim 2007: Surjective for p >> 0

2

Hakim-Murnaghan 2008: Fibers as equivalence classes.

3

Adler-DeBacker-Spice 2008++: Character formula.

4

Fintzen 2018: Surjective for p ∤ |W| and in positive characteristic.

Tasho Kaletha Local Langlands Correspondence

slide-102
SLIDE 102

Discrete series harmony (K. 2015)

Tasho Kaletha Local Langlands Correspondence

slide-103
SLIDE 103

Discrete series harmony (K. 2015)

Regular real discrete series {π d.s. of G(R)} ↔ {(S, B, θ)}/G(R)

Tasho Kaletha Local Langlands Correspondence

slide-104
SLIDE 104

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R)

Tasho Kaletha Local Langlands Correspondence

slide-105
SLIDE 105

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R) Regular supercuspidal representations {π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp)

Tasho Kaletha Local Langlands Correspondence

slide-106
SLIDE 106

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R) Regular supercuspidal representations {π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp) Character formula

Tasho Kaletha Local Langlands Correspondence

slide-107
SLIDE 107

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R) Regular supercuspidal representations {π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp) Character formula Θπ(γ) = e(G)ǫ(X ∗(S)C − X ∗(T)C, Λ)

  • w∈N(S,G)(F)/S(F)

∆abs

II (γw)θ(γw)

Tasho Kaletha Local Langlands Correspondence

slide-108
SLIDE 108

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R) Regular supercuspidal representations {π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp) Character formula Θπ(γ) = e(G)ǫ(X ∗(S)C − X ∗(T)C, Λ)

  • w∈N(S,G)(F)/S(F)

∆abs

II (γw)θ(γw)

F = Qp, γ ∈ S(F) shallow,

Tasho Kaletha Local Langlands Correspondence

slide-109
SLIDE 109

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R) Regular supercuspidal representations {π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp) Character formula Θπ(γ) = e(G)ǫ(X ∗(S)C − X ∗(T)C, Λ)

  • w∈N(S,G)(F)/S(F)

∆abs

II (γw)θ(γw)

F = Qp, γ ∈ S(F) shallow,

  • r F = R, γ ∈ S(F),

Tasho Kaletha Local Langlands Correspondence

slide-110
SLIDE 110

Discrete series harmony (K. 2015)

Regular real discrete series {π reg. d.s. of G(R)} ↔ {(S, θ)}/G(R) Regular supercuspidal representations {π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp) Character formula Θπ(γ) = e(G)ǫ(X ∗(S)C − X ∗(T)C, Λ)

  • w∈N(S,G)(F)/S(F)

∆abs

II (γw)θ(γw)

F = Qp, γ ∈ S(F) shallow,

  • r F = R, γ ∈ S(F), recovers H-C formula!

Tasho Kaletha Local Langlands Correspondence

slide-111
SLIDE 111

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|.

Tasho Kaletha Local Langlands Correspondence

slide-112
SLIDE 112

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular

Tasho Kaletha Local Langlands Correspondence

slide-113
SLIDE 113

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

Tasho Kaletha Local Langlands Correspondence

slide-114
SLIDE 114

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

Tasho Kaletha Local Langlands Correspondence

slide-115
SLIDE 115

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian

Tasho Kaletha Local Langlands Correspondence

slide-116
SLIDE 116

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian

Tasho Kaletha Local Langlands Correspondence

slide-117
SLIDE 117

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian The general case, K. in progress

Tasho Kaletha Local Langlands Correspondence

slide-118
SLIDE 118

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian The general case, K. in progress

1

ϕ : WF → G discrete

Tasho Kaletha Local Langlands Correspondence

slide-119
SLIDE 119

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian The general case, K. in progress

1

ϕ : WF → G discrete ↔ Πϕ(G)

Tasho Kaletha Local Langlands Correspondence

slide-120
SLIDE 120

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian The general case, K. in progress

1

ϕ : WF → G discrete ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

Tasho Kaletha Local Langlands Correspondence

slide-121
SLIDE 121

Supercuspidal Local Langlands Correspondence

Assume p ∤ |W|. The regular case, K. 2015

1

ϕ : WF → G discrete,regular ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ abelian The general case, K. in progress

1

ϕ : WF → G discrete ↔ Πϕ(G)

2

Πϕ(G) ↔ Irr(Sϕ)

3

Sϕ no longer abelian, structure of Πϕ(G) more complex

Tasho Kaletha Local Langlands Correspondence

slide-122
SLIDE 122

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

Tasho Kaletha Local Langlands Correspondence

slide-123
SLIDE 123

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

Tasho Kaletha Local Langlands Correspondence

slide-124
SLIDE 124

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

Tasho Kaletha Local Langlands Correspondence

slide-125
SLIDE 125

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function Θ(γ) = “ǫ(X ∗(S)C − X ∗(T)C, Λ)∆abs

II (γw)θ(γw)′′

Tasho Kaletha Local Langlands Correspondence

slide-126
SLIDE 126

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function Θ(γ) = “ǫ(X ∗(S)C − X ∗(T)C, Λ)∆abs

II (γw)θ(γw)′′

Each j : S → G provides unique irrep πj of G(F) by Θπj = e(G)

  • w∈N(jS,G)(F)/jS(F)

Θ(γw).

Tasho Kaletha Local Langlands Correspondence

slide-127
SLIDE 127

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function Θ(γ) = “ǫ(X ∗(S)C − X ∗(T)C, Λ)∆abs

II (γw)θ(γw)′′

Each j : S → G provides unique irrep πj of G(F) by Θπj = e(G)

  • w∈N(jS,G)(F)/jS(F)

Θ(γw). The L-packet, together with internal structure:

Tasho Kaletha Local Langlands Correspondence

slide-128
SLIDE 128

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function Θ(γ) = “ǫ(X ∗(S)C − X ∗(T)C, Λ)∆abs

II (γw)θ(γw)′′

Each j : S → G provides unique irrep πj of G(F) by Θπj = e(G)

  • w∈N(jS,G)(F)/jS(F)

Θ(γw). The L-packet, together with internal structure:

1

Πϕ(G) = {πj|j : S → G}

Tasho Kaletha Local Langlands Correspondence

slide-129
SLIDE 129

Construction of LLC: Regular case

From ϕ : WF → LG discrete and regular we obtain canonically

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function Θ(γ) = “ǫ(X ∗(S)C − X ∗(T)C, Λ)∆abs

II (γw)θ(γw)′′

Each j : S → G provides unique irrep πj of G(F) by Θπj = e(G)

  • w∈N(jS,G)(F)/jS(F)

Θ(γw). The L-packet, together with internal structure:

1

Πϕ(G) = {πj|j : S → G}

2

Sϕ = SΓ ↔ H1(Γ, S) {j : S → G}

Tasho Kaletha Local Langlands Correspondence

slide-130
SLIDE 130

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.

Tasho Kaletha Local Langlands Correspondence

slide-131
SLIDE 131

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

Tasho Kaletha Local Langlands Correspondence

slide-132
SLIDE 132

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

Tasho Kaletha Local Langlands Correspondence

slide-133
SLIDE 133

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

Tasho Kaletha Local Langlands Correspondence

slide-134
SLIDE 134

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

3

πj for each j : S → G

Tasho Kaletha Local Langlands Correspondence

slide-135
SLIDE 135

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

3

πj for each j : S → G First big difference: πj is reducible!

Tasho Kaletha Local Langlands Correspondence

slide-136
SLIDE 136

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

3

πj for each j : S → G First big difference: πj is reducible! Πϕ(G) = {irr. const. πj|j : S → G}

Tasho Kaletha Local Langlands Correspondence

slide-137
SLIDE 137

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

3

πj for each j : S → G First big difference: πj is reducible! Πϕ(G) = {irr. const. πj|j : S → G} Second big difference: Sϕ no longer abelian!

Tasho Kaletha Local Langlands Correspondence

slide-138
SLIDE 138

Construction of LLC: Singular case

Let ϕ : WF → LG be discrete, not assumed regular.As before:

1

S algebraic torus, with a family of embeddings j : S → G

2

Θ : S(F) → C function

3

πj for each j : S → G First big difference: πj is reducible! Πϕ(G) = {irr. const. πj|j : S → G} Second big difference: Sϕ no longer abelian! Main Challenge: Construct Irr(Sϕ) ↔ Πϕ(G)

Tasho Kaletha Local Langlands Correspondence

slide-139
SLIDE 139

Geometric intertwining operators

Reductions

Tasho Kaletha Local Langlands Correspondence

slide-140
SLIDE 140

Geometric intertwining operators

Reductions Understanding the structure of πj

Tasho Kaletha Local Langlands Correspondence

slide-141
SLIDE 141

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero

Tasho Kaletha Local Langlands Correspondence

slide-142
SLIDE 142

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

Tasho Kaletha Local Langlands Correspondence

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SLIDE 143

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],

Tasho Kaletha Local Langlands Correspondence

slide-144
SLIDE 144

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

Tasho Kaletha Local Langlands Correspondence

slide-145
SLIDE 145

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ W(S, G)(k)θ

Tasho Kaletha Local Langlands Correspondence

slide-146
SLIDE 146

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ W(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • Tasho Kaletha

Local Langlands Correspondence

slide-147
SLIDE 147

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ W(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • 3

As in classical case: In don’t compose well

Tasho Kaletha Local Langlands Correspondence

slide-148
SLIDE 148

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ W(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • 3

As in classical case: In don’t compose well p-adic field

Tasho Kaletha Local Langlands Correspondence

slide-149
SLIDE 149

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ Aut(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • 3

As in classical case: In don’t compose well p-adic field

Tasho Kaletha Local Langlands Correspondence

slide-150
SLIDE 150

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ Aut(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • 3

As in classical case: In don’t compose well p-adic field

1

The In can be normalized to compose well

Tasho Kaletha Local Langlands Correspondence

slide-151
SLIDE 151

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ Aut(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • 3

As in classical case: In don’t compose well p-adic field

1

The In can be normalized to compose well

2

[πj] ↔ Irr(N(S, G)(F)θ, θ)

Tasho Kaletha Local Langlands Correspondence

slide-152
SLIDE 152

Geometric intertwining operators

Reductions Understanding the structure of πj of depth zero Finite field

1

Lusztig: W(S, G)(k)∗

θ[

H∗(YB, ¯ Ql)θ],reminiscent of R-group

2

Construct In : H∗(YB)θ → H∗(YB)θ for n ∈ Aut(S, G)(k)θ

Y (2)

Bn,B

  • BDR 2017

Y (2)

B,FBn

  • YB

YBn,B

YB,FBn

  • YBn

n

  • 3

As in classical case: In don’t compose well p-adic field

1

The In can be normalized to compose well

2

[πj] ↔ Irr(N(S, G)(F)θ, θ)

3

Sϕ ↔

j,G′ N(jS, G′)(F)θ

Tasho Kaletha Local Langlands Correspondence

slide-153
SLIDE 153

Speculation: Character formula via fixed-point formula

Tasho Kaletha Local Langlands Correspondence

slide-154
SLIDE 154

Speculation: Character formula via fixed-point formula

Real discrete series

Tasho Kaletha Local Langlands Correspondence

slide-155
SLIDE 155

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

Tasho Kaletha Local Langlands Correspondence

slide-156
SLIDE 156

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

2

Hochs, Wang 2017: Character computed via Atiyah-Singer fixed point formula for non-compact domains

Tasho Kaletha Local Langlands Correspondence

slide-157
SLIDE 157

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

2

Hochs, Wang 2017: Character computed via Atiyah-Singer fixed point formula for non-compact domains

Tasho Kaletha Local Langlands Correspondence

slide-158
SLIDE 158

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

2

Hochs, Wang 2017: Character computed via Atiyah-Singer fixed point formula for non-compact domains Supercuspidal representations

Tasho Kaletha Local Langlands Correspondence

slide-159
SLIDE 159

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

2

Hochs, Wang 2017: Character computed via Atiyah-Singer fixed point formula for non-compact domains Supercuspidal representations

1

Expect to find supercuspidal representations in l-adic cohomology of local Shtuka spaces

Tasho Kaletha Local Langlands Correspondence

slide-160
SLIDE 160

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

2

Hochs, Wang 2017: Character computed via Atiyah-Singer fixed point formula for non-compact domains Supercuspidal representations

1

Expect to find supercuspidal representations in l-adic cohomology of local Shtuka spaces

2

Can we compute the character using a generalized Lefschetz fixed-point formula?

Tasho Kaletha Local Langlands Correspondence

slide-161
SLIDE 161

Speculation: Character formula via fixed-point formula

Real discrete series

1

Langlands,Schmid 1960-1970: π d.s. of G(R) can be found in L2-cohomology of the flag manifold

2

Hochs, Wang 2017: Character computed via Atiyah-Singer fixed point formula for non-compact domains Supercuspidal representations

1

Expect to find supercuspidal representations in l-adic cohomology of local Shtuka spaces

2

Can we compute the character using a generalized Lefschetz fixed-point formula?

3

K.-Weinstein 2017: Partial results.

Tasho Kaletha Local Langlands Correspondence

slide-162
SLIDE 162

Speculation: Beyond endoscopy and twisted Levis

Tasho Kaletha Local Langlands Correspondence

slide-163
SLIDE 163

Speculation: Beyond endoscopy and twisted Levis

Let G′ ⊂ G be an elliptic twisted Levi subgroup, ϕ : WF → LG′.

Tasho Kaletha Local Langlands Correspondence

slide-164
SLIDE 164

Speculation: Beyond endoscopy and twisted Levis

Let G′ ⊂ G be an elliptic twisted Levi subgroup, ϕ : WF → LG′. Expect for γ ∈ G′(F) elliptic that SΘϕ,G(γ) equals e(G′)e(G)ǫ(X ∗(T)C − X ∗(T ′)C, Λ)

  • w∈W(G′,G)(F)

∆G/G′

II

(γw)SΘϕ,G′(γw).

Tasho Kaletha Local Langlands Correspondence