Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local...

Harish-Chandra characters and the localLanglands correspondence

Tasho Kaletha

University of Michigan

16. November 2018

Tasho Kaletha Local Langlands Correspondence

Global Langlands correspondence

Galois representations

Γ = Gal(Q/Q) compact topological group encodessymmetries of solutions to rational polynomial equationsϕ : Γ→ GLn(C) matrix representationL(s, ϕ) Artin L-function

Global Langlands correspondenceϕ↔ π

π automorphic rep.:L(s, ϕ) = L(s, π)

Application

Langlands 1980: Proves many cases of the 2-dimensionalArtin conjecture




Γ = Gal(Q/Q)

compact topological group encodessymmetries of solutions to rational polynomial equationsϕ : Γ→ GLn(C) matrix representationL(s, ϕ) Artin L-function



Application





Γ = Gal(Q/Q) compact topological group

encodessymmetries of solutions to rational polynomial equationsϕ : Γ→ GLn(C) matrix representationL(s, ϕ) Artin L-function



Application





Γ = Gal(Q/Q) compact topological group encodessymmetries of solutions to rational polynomial equations

ϕ : Γ→ GLn(C) matrix representationL(s, ϕ) Artin L-function



Application





Γ = Gal(Q/Q) compact topological group encodessymmetries of solutions to rational polynomial equationsϕ : Γ→ GLn(C)

matrix representationL(s, ϕ) Artin L-function



Application





Γ = Gal(Q/Q) compact topological group encodessymmetries of solutions to rational polynomial equationsϕ : Γ→ GLn(C) matrix representation

L(s, ϕ) Artin L-function



Application





Γ = Gal(Q/Q) compact topological group encodessymmetries of solutions to rational polynomial equationsϕ : Γ→ GLn(C) matrix representationL(s, ϕ)

Artin L-function



Application








Application







ϕ↔ π


Application








Application







π automorphic rep.:

L(s, ϕ) = L(s, π)

Application







π automorphic rep.: L2(Σ \GLn(R)) x GLn(R)

L(s, ϕ) = L(s, π)

Application







π automorphic rep.: L2(GLn(Q) \GLn(A)) x GLn(A)

L(s, ϕ) = L(s, π)

Application







π automorphic rep.: L2(GLn(Q) \GLn(A)) x GLn(A)

L(s, ϕ) = L(s, π)

ApplicationLanglands 1980: Proves many cases of the 2-dimensionalArtin conjecture



General reductive groups

ϕ : Γ→ G, G = Sp2n(C), SOn(C), . . .

ϕ↔ π

π automorphic representation: L2(G(Q) \G(A)) x G(A)

G↔ G Langlands dual groups




ϕ : Γ→ G

, G = Sp2n(C), SOn(C), . . .

ϕ↔ π






ϕ : Γ→ G, G = Sp2n(C)

, SOn(C), . . .

ϕ↔ π






ϕ : Γ→ G, G = Sp2n(C), SOn(C),

. . .

ϕ↔ π






ϕ : Γ→ G, G = Sp2n(C), SOn(C), . . .

ϕ↔ π






ϕ : Γ→ G, G = Sp2n(C), SOn(C), . . .

ϕ↔ π

π automorphic representation:

L2(G(Q) \G(A)) x G(A)





ϕ : Γ→ G, G = Sp2n(C), SOn(C), . . .

ϕ↔ π




Local Langlands correspondence

Decomposition groups

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

Local correspondenceϕp ↔ πp

πp irreducible (admissible) representation of G(Qp)

ResultsGLN : Harris-Taylor, HenniartSpN , SON ,UN , Arthur, Mok, KMSWGeneral G? partial results in positive characteristicLafforgue-Genestier




Γp = Gal(Qp/Qp) ⊂ Γ

, p =∞,2,3,5,7, . . .Γ∞ = Z/2Z, generated by complex conjugationΓp infinite, compact, (p <∞)







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

Γ∞ = Z/2Z, generated by complex conjugationΓp infinite, compact, (p <∞)







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

, generated by complex conjugationΓp infinite, compact, (p <∞)







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









Results

GLN : Harris-Taylor, HenniartSpN , SON ,UN , Arthur, Mok, KMSWGeneral G? partial results in positive characteristicLafforgue-Genestier







ResultsGLN : Harris-Taylor, Henniart

SpN , SON ,UN , Arthur, Mok, KMSWGeneral G? partial results in positive characteristicLafforgue-Genestier







ResultsGLN : Harris-Taylor, HenniartSpN , SON ,UN , Arthur, Mok, KMSW

General G? partial results in positive characteristicLafforgue-Genestier







ResultsGLN : Harris-Taylor, HenniartSpN , SON ,UN , Arthur, Mok, KMSWGeneral G?

partial results in positive characteristicLafforgue-Genestier


Local representation theory

The groups

G(R): locally connected, analytic methodsG(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 : IGP (σ)→ IG

P (σ), n ∈W (M,G)(F )σ



The groups

G(R): locally connected

, analytic methodsG(Qp): totally disconnected, little analysis





R-groups


In : IGP (σ)→ IG

P (σ), n ∈W (M,G)(F )σ



The groups

G(R): locally connected, analytic methods

G(Qp): totally disconnected, little analysis





R-groups


In : IGP (σ)→ IG

P (σ), n ∈W (M,G)(F )σ



The groups

G(R): locally connected, analytic methodsG(Qp): totally disconnected, little analysis





R-groups


In : IGP (σ)→ IG

P (σ), n ∈W (M,G)(F )σ


Discrete series dissonance

Discrete series

π discrete series:

aij(g) ∈ L2(G/Z )

p-adic case1 Many discrete series are supercuspidal2 π supercuspidal: aij(g) compact modulo center3 π does not appear in any parabolic induction4 π induced from a compact open subgroup

real case1 There are no supercuspidal representations2 Casselman: Every irreducible representation appears in

parabolic induction3 There are no compact open subgroups



Discrete series

π discrete series: aij(g) ∈ L2(G/Z )






Discrete series


p-adic case

1 Many discrete series are supercuspidal2 π supercuspidal: aij(g) compact modulo center3 π does not appear in any parabolic induction4 π induced from a compact open subgroup





Discrete series


p-adic case1 Many discrete series are supercuspidal

2 π supercuspidal: aij(g) compact modulo center3 π does not appear in any parabolic induction4 π induced from a compact open subgroup





Discrete series


p-adic case1 Many discrete series are supercuspidal2 π supercuspidal:

aij(g) compact modulo center3 π does not appear in any parabolic induction4 π induced from a compact open subgroup





Discrete series


p-adic case1 Many discrete series are supercuspidal2 π supercuspidal: aij(g) compact modulo center

3 π does not appear in any parabolic induction4 π induced from a compact open subgroup





Discrete series


p-adic case1 Many discrete series are supercuspidal2 π supercuspidal: aij(g) compact modulo center3 π does not appear in any parabolic induction

4 π induced from a compact open subgroup





Discrete series







Discrete series



real case

1 There are no supercuspidal representations2 Casselman: Every irreducible representation appears in




Discrete series



real case1 There are no supercuspidal representations

2 Casselman: Every irreducible representation appears inparabolic induction

3 There are no compact open subgroups



Discrete series




parabolic induction

3 There are no compact open subgroups



Discrete series






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 theory1 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)


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






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)








3 Θπ(s) = (−1)q(G)∑

w∈N(S,G)(R)/S(R)

θ(γw )∏α>0(1−α(γw )−1)


2 Θπ(s) =∑

w∈N(S,G)(C)/S(C)

θ(γw )∏α>0(1−α(γw )−1)








3 Θπ(s) = (−1)q(G)∑

w∈N(S,G)(R)/S(R)

θ(γw )∏α>0(1−α(γw )−1)


2 Θπ(s) =∑

w∈N(S,G)(C)/S(C)

θ(γw )∏α>0(1−α(γw )−1)


Langlands: ϕ

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






3 Θπ(s) = (−1)q(G)∑

w∈N(S,G)(R)/S(R)

θ(γw )∏α>0(1−α(γw )−1)


2 Θπ(s) =∑

w∈N(S,G)(C)/S(C)

θ(γw )∏α>0(1−α(γw )−1)


Langlands: ϕ→ (S,B, θ)/G(C)

→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)






3 Θπ(s) = (−1)q(G)∑

w∈N(S,G)(R)/S(R)

θ(γw )∏α>0(1−α(γw )−1)


2 Θπ(s) =∑

w∈N(S,G)(C)/S(C)

θ(γw )∏α>0(1−α(γw )−1)


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

Θπ1 + · · ·+ Θπk conjugation invariant under G(C)






3 Θπ(s) = (−1)q(G)∑

w∈N(S,G)(R)/S(R)

θ(γw )∏α>0(1−α(γw )−1)


2 Θπ(s) =∑

w∈N(S,G)(C)/S(C)

θ(γw )∏α>0(1−α(γw )−1)




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

Geometric side1 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)∗, Oκγ =

∑a κ(a)Oaγ

4 Gκ ⊂ G, γκ ∈ Gκ(F ), Oκγ,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, Θsϕ =

∑π∈Πϕ(G) ρπ(s)Θπ

4 Gs ⊂ G, Θsϕ,G(f ) = SΘϕ,Gs (fs)



Geometric side

1 γ ∈ G(F )rs, Oγ(f ) =∫γG(F ) f , SOγ(f ) =


2 γG(F ) ∩G(F ) =⋃



∑a κ(a)Oaγ


Spectral side


2 SΘϕ =∑







Geometric side1 γ ∈ G(F )rs,

Oγ(f ) =∫γG(F ) f , SOγ(f ) =


2 γG(F ) ∩G(F ) =⋃



∑a κ(a)Oaγ


Spectral side


2 SΘϕ =∑








∫γG(F ) f ,

SOγ(f ) =∫γG(F )∩G(F ) f

2 γG(F ) ∩G(F ) =⋃



∑a κ(a)Oaγ


Spectral side


2 SΘϕ =∑




