harish-chandra characters and the local langlands ...harish-chandra characters and the local...
TRANSCRIPT
![Page 1: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/1.jpg)
Harish-Chandra characters and the localLanglands correspondence
Tasho Kaletha
University of Michigan
16. November 2018
Tasho Kaletha Local Langlands Correspondence
![Page 2: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/2.jpg)
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
![Page 3: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/3.jpg)
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
![Page 4: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/4.jpg)
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
![Page 5: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/5.jpg)
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
![Page 6: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/6.jpg)
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
![Page 7: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/7.jpg)
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
![Page 8: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/8.jpg)
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
![Page 9: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/9.jpg)
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
![Page 10: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/10.jpg)
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
![Page 11: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/11.jpg)
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
![Page 12: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/12.jpg)
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
![Page 13: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/13.jpg)
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
![Page 14: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/14.jpg)
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
![Page 15: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/15.jpg)
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
![Page 16: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/16.jpg)
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
![Page 17: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/17.jpg)
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
![Page 18: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/18.jpg)
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
![Page 19: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/19.jpg)
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
![Page 20: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/20.jpg)
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
![Page 21: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/21.jpg)
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
![Page 22: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/22.jpg)
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
![Page 23: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/23.jpg)
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
![Page 24: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/24.jpg)
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
![Page 25: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/25.jpg)
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
![Page 26: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/26.jpg)
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
![Page 27: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/27.jpg)
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
![Page 28: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/28.jpg)
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
![Page 29: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/29.jpg)
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
![Page 30: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/30.jpg)
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
![Page 31: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/31.jpg)
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
![Page 32: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/32.jpg)
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
![Page 33: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/33.jpg)
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
![Page 34: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/34.jpg)
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
![Page 35: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/35.jpg)
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
![Page 36: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/36.jpg)
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
![Page 37: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/37.jpg)
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
![Page 38: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/38.jpg)
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
![Page 39: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/39.jpg)
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
![Page 40: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/40.jpg)
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
![Page 41: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/41.jpg)
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
![Page 42: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/42.jpg)
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
![Page 43: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/43.jpg)
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
![Page 44: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/44.jpg)
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
![Page 45: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/45.jpg)
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
![Page 46: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/46.jpg)
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
![Page 47: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/47.jpg)
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
![Page 48: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/48.jpg)
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
![Page 49: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/49.jpg)
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
![Page 50: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/50.jpg)
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
![Page 51: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/51.jpg)
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
![Page 52: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/52.jpg)
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
![Page 53: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/53.jpg)
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
![Page 54: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/54.jpg)
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
![Page 55: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/55.jpg)
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
![Page 56: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/56.jpg)
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
![Page 57: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/57.jpg)
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
![Page 58: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/58.jpg)
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
![Page 59: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/59.jpg)
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
![Page 60: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/60.jpg)
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
![Page 61: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/61.jpg)
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
![Page 62: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/62.jpg)
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
![Page 63: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/63.jpg)
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
![Page 64: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/64.jpg)
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
![Page 65: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/65.jpg)
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
![Page 66: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/66.jpg)
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
![Page 67: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/67.jpg)
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
![Page 68: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/68.jpg)
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
![Page 69: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/69.jpg)
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
![Page 70: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/70.jpg)
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
![Page 71: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/71.jpg)
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
![Page 72: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/72.jpg)
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
![Page 73: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/73.jpg)
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
![Page 74: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/74.jpg)
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
![Page 75: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/75.jpg)
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
![Page 76: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/76.jpg)
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
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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
![Page 78: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/78.jpg)
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
![Page 79: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/79.jpg)
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
![Page 80: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/80.jpg)
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
![Page 81: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/81.jpg)
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
![Page 82: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/82.jpg)
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
![Page 83: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/83.jpg)
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
![Page 84: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/84.jpg)
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
![Page 85: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/85.jpg)
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
![Page 86: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/86.jpg)
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
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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
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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
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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
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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
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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
![Page 92: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/92.jpg)
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
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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
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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
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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
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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
![Page 97: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/97.jpg)
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
![Page 98: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/98.jpg)
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
![Page 99: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/99.jpg)
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
![Page 100: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/100.jpg)
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
![Page 101: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/101.jpg)
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
![Page 102: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/102.jpg)
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
![Page 103: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/103.jpg)
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
![Page 104: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/104.jpg)
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
![Page 105: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/105.jpg)
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
![Page 106: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/106.jpg)
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
![Page 107: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/107.jpg)
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
![Page 108: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/108.jpg)
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
![Page 109: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/109.jpg)
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
![Page 110: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/110.jpg)
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
![Page 111: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/111.jpg)
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
![Page 112: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/112.jpg)
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
![Page 113: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/113.jpg)
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
![Page 114: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/114.jpg)
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
![Page 115: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/115.jpg)
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
![Page 116: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/116.jpg)
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
![Page 117: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/117.jpg)
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
![Page 118: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/118.jpg)
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
![Page 119: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/119.jpg)
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
![Page 120: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/120.jpg)
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
![Page 121: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/121.jpg)
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
![Page 122: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/122.jpg)
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
![Page 123: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/123.jpg)
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
![Page 124: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/124.jpg)
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
![Page 125: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/125.jpg)
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
![Page 126: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/126.jpg)
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
![Page 127: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/127.jpg)
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
![Page 128: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/128.jpg)
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
![Page 129: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/129.jpg)
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
![Page 130: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/130.jpg)
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
![Page 131: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/131.jpg)
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
![Page 132: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/132.jpg)
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
![Page 133: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/133.jpg)
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
![Page 134: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/134.jpg)
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
![Page 135: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/135.jpg)
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
![Page 136: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/136.jpg)
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
![Page 137: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/137.jpg)
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
![Page 138: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/138.jpg)
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
![Page 139: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/139.jpg)
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
![Page 140: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/140.jpg)
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
![Page 141: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/141.jpg)
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
![Page 142: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/142.jpg)
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
![Page 143: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/143.jpg)
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
![Page 144: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/144.jpg)
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
![Page 145: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/145.jpg)
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
![Page 146: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/146.jpg)
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
![Page 147: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/147.jpg)
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
![Page 148: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/148.jpg)
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
![Page 149: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/149.jpg)
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
![Page 150: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/150.jpg)
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
![Page 151: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/151.jpg)
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
![Page 152: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/152.jpg)
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
![Page 153: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/153.jpg)
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
![Page 154: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/154.jpg)
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
![Page 155: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/155.jpg)
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
![Page 156: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/156.jpg)
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
![Page 157: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/157.jpg)
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
![Page 158: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/158.jpg)
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
![Page 159: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/159.jpg)
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
![Page 160: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/160.jpg)
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
![Page 161: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/161.jpg)
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
![Page 162: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/162.jpg)
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
![Page 163: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/163.jpg)
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
![Page 164: Harish-Chandra characters and the local Langlands ...Harish-Chandra characters and the local Langlands correspondence Tasho Kaletha University of Michigan 16. November 2018 Tasho Kaletha](https://reader033.vdocuments.site/reader033/viewer/2022060319/5f0cc1767e708231d436f95a/html5/thumbnails/164.jpg)
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