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
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
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
Galois representations
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
Global Langlands correspondenceϕ↔ π
π automorphic rep.:L(s, ϕ) = L(s, π)
Application
Langlands 1980: Proves many cases of the 2-dimensionalArtin conjecture
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
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 representation
L(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
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
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
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 correspondence
ϕ↔ π
π automorphic rep.:L(s, ϕ) = L(s, π)
Application
Langlands 1980: Proves many cases of the 2-dimensionalArtin conjecture
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
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
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.: L2(Σ \GLn(R)) x GLn(R)
L(s, ϕ) = L(s, π)
Application
Langlands 1980: Proves many cases of the 2-dimensionalArtin conjecture
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.: L2(GLn(Q) \GLn(A)) x GLn(A)
L(s, ϕ) = L(s, π)
Application
Langlands 1980: Proves many cases of the 2-dimensionalArtin conjecture
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.: L2(GLn(Q) \GLn(A)) x GLn(A)
L(s, ϕ) = L(s, π)
Application
Langlands 1980: Proves many cases of the 2-dimensionalArtin conjecture
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.: L2(GLn(Q) \GLn(A)) x GLn(A)
L(s, ϕ) = L(s, π)
ApplicationLanglands 1980: Proves many cases of the 2-dimensionalArtin conjecture
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
Global Langlands correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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 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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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 correspondence
ϕp ↔ πp
πp irreducible (admissible) representation of G(Qp)
ResultsGLN : Harris-Taylor, HenniartSpN , SON ,UN , Arthur, Mok, KMSWGeneral G? partial results in positive characteristicLafforgue-Genestier
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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)
Results
GLN : Harris-Taylor, HenniartSpN , SON ,UN , Arthur, Mok, KMSWGeneral G? partial results in positive characteristicLafforgue-Genestier
Tasho Kaletha Local Langlands Correspondence
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, Henniart
SpN , SON ,UN , Arthur, Mok, KMSWGeneral G? partial results in positive characteristicLafforgue-Genestier
Tasho Kaletha Local Langlands Correspondence
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, KMSW
General G? partial results in positive characteristicLafforgue-Genestier
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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 : IGP (σ)→ IG
P (σ), n ∈W (M,G)(F )σ
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
Discrete series dissonance
Discrete series
π discrete series: aij(g) ∈ L2(G/Z )
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
real case1 There are no supercuspidal representations2 Casselman: Every irreducible representation appears in
parabolic induction3 There are no compact open subgroups
Tasho Kaletha Local Langlands Correspondence
Discrete series dissonance
Discrete series
π discrete series: aij(g) ∈ L2(G/Z )
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
real case1 There are no supercuspidal representations2 Casselman: Every irreducible representation appears in
parabolic induction3 There are no compact open subgroups
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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 center
3 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
Tasho Kaletha Local Langlands Correspondence
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 induction
4 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
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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 case
1 There are no supercuspidal representations2 Casselman: Every irreducible representation appears in
parabolic induction3 There are no compact open subgroups
Tasho Kaletha Local Langlands Correspondence
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 representations
2 Casselman: Every irreducible representation appears inparabolic induction
3 There are no compact open subgroups
Tasho Kaletha Local Langlands Correspondence
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 induction
3 There are no compact open subgroups
Tasho Kaletha Local Langlands Correspondence
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
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ
→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)
→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}
Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Local Langlands correspondence
Langlands: ϕ→ (S,B, θ)/G(C)→ {π1, . . . , πk}Θπ1 + · · ·+ Θπk conjugation invariant under G(C)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)∗, 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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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)
Tasho Kaletha Local Langlands Correspondence
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 )}
Properties1 Kim 2007: Surjective for p >> 02 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
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 )}
Properties1 Kim 2007: Surjective for p >> 02 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
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 >> 02 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
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 )}
Properties1 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
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 )}
Properties1 Kim 2007: Surjective for p >> 02 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
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 )}
Properties1 Kim 2007: Surjective for p >> 02 Hakim-Murnaghan 2008: Fibers as equivalence classes.3 Adler-DeBacker-Spice 2008++: Character formula.
4 Fintzen 2018: Surjective for p - |W | and in positivecharacteristic.
Tasho Kaletha Local Langlands Correspondence
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 )}
Properties1 Kim 2007: Surjective for p >> 02 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
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
Discrete series harmony (K. 2015)
Regular real discrete series
{π d.s. of G(R)} ↔ {(S,B, θ)}/G(R)
Regular supercuspidal representations
{π reg. s.c. of G(Qp)} ↔ {(S, θ)}/G(Qp)
Character formula
Θπ(γ) = e(G)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,
or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ),
recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(S)C − X ∗(T )C,Λ)∑
w∈N(S,G)(F )/S(F )
∆absII (γw )θ(γw )
F = Qp, γ ∈ S(F ) shallow,or F = R, γ ∈ S(F ), recovers H-C formula!
Tasho Kaletha Local Langlands Correspondence
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
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
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
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
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
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
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
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
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
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
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
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 → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G
2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
Construction of LLC: Regular case
From ϕ : WF → LG discrete and regular we obtain canonically1 S algebraic torus, with a family of embeddings j : S → G2 Θ : S(F )→ C function
Θ(γ) = “ε(X ∗(S)C − X ∗(T )C,Λ)∆absII (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) y {j : S → G}
Tasho Kaletha Local Langlands Correspondence
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 → G2 Θ : S(F )→ C function3 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
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 → G2 Θ : S(F )→ C function3 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
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 function3 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
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 → G2 Θ : S(F )→ C function
3 πj for each j : S → GFirst 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
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 → G2 Θ : S(F )→ C function3 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
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 → G2 Θ : S(F )→ C function3 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
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 → G2 Θ : S(F )→ C function3 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
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 → G2 Θ : S(F )→ C function3 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
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 → G2 Θ : S(F )→ C function3 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
Geometric intertwining operators
Reductions
Understanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 Construct In : H∗(YB)θ → H∗(YB)θ for n ∈
Y (2)Bn,B
}} ""
BDR 2017 Y (2)B,FBn
{{ ""YB YBn,B // YB,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj
of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 Construct In : H∗(YB)θ → H∗(YB)θ for n ∈
Y (2)Bn,B
}} ""
BDR 2017 Y (2)B,FBn
{{ ""YB YBn,B // YB,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 Construct In : H∗(YB)θ → H∗(YB)θ for n ∈
Y (2)Bn,B
}} ""
BDR 2017 Y (2)B,FBn
{{ ""YB YBn,B // YB,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field
1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 Construct In : H∗(YB)θ → H∗(YB)θ for n ∈
Y (2)Bn,B
}} ""
BDR 2017 Y (2)B,FBn
{{ ""YB YBn,B // YB,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],
reminiscent of R-group2 Construct In : H∗(YB)θ → H∗(YB)θ for n ∈
Y (2)Bn,B
}} ""
BDR 2017 Y (2)B,FBn
{{ ""YB YBn,B // YB,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group
2 Construct In : H∗(YB)θ → H∗(YB)θ for n ∈Y (2)
Bn,B
}} ""
BDR 2017 Y (2)B,FBn
{{ ""YB YBn,B // YB,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field
1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field
1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 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
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Geometric intertwining operators
ReductionsUnderstanding the structure of πj of depth zero
Finite field1 Lusztig:W (S,G)(k)∗θy[H∗(YB, Ql)θ],reminiscent of R-group2 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,FBnoo YBn
nmm
3 As in classical case: In don’t compose well
p-adic field1 The In can be normalized to compose well2 [πj ]↔ Irr(N(S,G)(F )θ, θ)
3 Sϕ ↔⋃
j,G′ N(jS,G′)(F )θ
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 Hochs, Wang 2017: Character computed via Atiyah-Singer
fixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series
1 Langlands,Schmid 1960-1970: π d.s. of G(R) can befound in L2-cohomology of the flag manifold
2 Hochs, Wang 2017: Character computed via Atiyah-Singerfixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 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-Singerfixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 Hochs, Wang 2017: Character computed via Atiyah-Singer
fixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 Hochs, Wang 2017: Character computed via Atiyah-Singer
fixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 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-adiccohomology of local Shtuka spaces
2 Can we compute the character using a generalizedLefschetz fixed-point formula?
3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 Hochs, Wang 2017: Character computed via Atiyah-Singer
fixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces
2 Can we compute the character using a generalizedLefschetz fixed-point formula?
3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 Hochs, Wang 2017: Character computed via Atiyah-Singer
fixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?
3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
Speculation: Character formula via fixed-point formula
Real discrete series1 Langlands,Schmid 1960-1970: π d.s. of G(R) can be
found in L2-cohomology of the flag manifold2 Hochs, Wang 2017: Character computed via Atiyah-Singer
fixed point formula for non-compact domains
Supercuspidal representations1 Expect to find supercuspidal representations in l-adic
cohomology of local Shtuka spaces2 Can we compute the character using a generalized
Lefschetz fixed-point formula?3 K.-Weinstein 2017: Partial results.
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(T )C − X ∗(T ′)C,Λ)∑
w∈W (G′,G)(F )
∆G/G′II (γw )SΘϕ,G′(γ
w ).
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(T )C − X ∗(T ′)C,Λ)∑
w∈W (G′,G)(F )
∆G/G′II (γw )SΘϕ,G′(γ
w ).
Tasho Kaletha Local Langlands Correspondence
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)ε(X ∗(T )C − X ∗(T ′)C,Λ)∑
w∈W (G′,G)(F )
∆G/G′II (γw )SΘϕ,G′(γ
w ).
Tasho Kaletha Local Langlands Correspondence
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