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

  1. Local Langlands correspondence Decomposition groups Γ p = Gal (¯ Q p / Q p ) ⊂ Γ , p = ∞ , 2 , 3 , 5 , 7 , . . . Γ ∞ = Z / 2 Z , generated by complex conjugation Γ p infinite, compact, ( p < ∞ ) Local correspondence ϕ p ↔ π p π p irreducible (admissible) representation of G ( Q p ) Results GL N : Harris-Taylor, Henniart Sp N , SO N , U N , Arthur, Mok, KMSW Tasho Kaletha Local Langlands Correspondence

  2. Local Langlands correspondence Decomposition groups Γ p = Gal (¯ Q p / Q p ) ⊂ Γ , p = ∞ , 2 , 3 , 5 , 7 , . . . Γ ∞ = Z / 2 Z , generated by complex conjugation Γ p infinite, compact, ( p < ∞ ) Local correspondence ϕ p ↔ π p π p irreducible (admissible) representation of G ( Q p ) Results GL N : Harris-Taylor, Henniart Sp N , SO N , U N , Arthur, Mok, KMSW General G ? Tasho Kaletha Local Langlands Correspondence

  3. Local Langlands correspondence Decomposition groups Γ p = Gal (¯ Q p / Q p ) ⊂ Γ , p = ∞ , 2 , 3 , 5 , 7 , . . . Γ ∞ = Z / 2 Z , generated by complex conjugation Γ p infinite, compact, ( p < ∞ ) Local correspondence ϕ p ↔ π p π p irreducible (admissible) representation of G ( Q p ) Results GL N : Harris-Taylor, Henniart Sp N , SO N , U N , Arthur, Mok, KMSW General G ? partial results in positive characteristic Lafforgue-Genestier Tasho Kaletha Local Langlands Correspondence

  4. Local representation theory The groups Tasho Kaletha Local Langlands Correspondence

  5. Local representation theory The groups G ( R ) : locally connected Tasho Kaletha Local Langlands Correspondence

  6. Local representation theory The groups G ( R ) : locally connected, analytic methods Tasho Kaletha Local Langlands Correspondence

  7. Local representation theory The groups G ( R ) : locally connected, analytic methods G ( Q p ) : totally disconnected, little analysis Tasho Kaletha Local Langlands Correspondence

  8. Local representation theory The groups G ( R ) : locally connected, analytic methods G ( Q p ) : totally disconnected, little analysis Tasho Kaletha Local Langlands Correspondence

  9. Local representation theory The groups G ( R ) : locally connected, analytic methods G ( Q p ) : totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G ( R ) and G ( Q p ) ought to behave similarly Tasho Kaletha Local Langlands Correspondence

  10. Local representation theory The groups G ( R ) : locally connected, analytic methods G ( Q p ) : totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G ( R ) and G ( Q p ) ought to behave similarly Langlands classification π admissible ↔ ( M , σ, ν ) , M ⊂ G Levi, σ ∈ Irr ( M ) tempered Tasho Kaletha Local Langlands Correspondence

  11. Local representation theory The groups G ( R ) : locally connected, analytic methods G ( Q p ) : totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G ( R ) and G ( Q p ) ought to behave similarly Langlands classification π admissible ↔ ( M , σ, ν ) , M ⊂ G Levi, σ ∈ Irr ( M ) tempered R -groups π tempered ↔ ( M , σ, τ ) , M ⊂ G Levi, σ ∈ Irr ( M ) discrete Tasho Kaletha Local Langlands Correspondence

  12. Local representation theory The groups G ( R ) : locally connected, analytic methods G ( Q p ) : totally disconnected, little analysis Harish-Chandra’s Lefschetz Principle G ( R ) and G ( Q p ) 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 I n : I G P ( σ ) → I G P ( σ ) , n ∈ W ( M , G )( F ) σ Tasho Kaletha Local Langlands Correspondence

  13. Discrete series dissonance Discrete series π discrete series: Tasho Kaletha Local Langlands Correspondence

  14. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: Tasho Kaletha Local Langlands Correspondence

  15. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Tasho Kaletha Local Langlands Correspondence

  16. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 Tasho Kaletha Local Langlands Correspondence

  17. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: 2 Tasho Kaletha Local Langlands Correspondence

  18. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 Tasho Kaletha Local Langlands Correspondence

  19. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 π does not appear in any parabolic induction 3 Tasho Kaletha Local Langlands Correspondence

  20. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 π does not appear in any parabolic induction 3 π induced from a compact open subgroup 4 Tasho Kaletha Local Langlands Correspondence

  21. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 π does not appear in any parabolic induction 3 π induced from a compact open subgroup 4 real case Tasho Kaletha Local Langlands Correspondence

  22. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 π does not appear in any parabolic induction 3 π induced from a compact open subgroup 4 real case There are no supercuspidal representations 1 Tasho Kaletha Local Langlands Correspondence

  23. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 π does not appear in any parabolic induction 3 π induced from a compact open subgroup 4 real case There are no supercuspidal representations 1 Casselman: Every irreducible representation appears in 2 parabolic induction Tasho Kaletha Local Langlands Correspondence

  24. Discrete series dissonance Discrete series a ij ( g ) ∈ L 2 ( G / Z ) π discrete series: p -adic case Many discrete series are supercuspidal 1 π supercuspidal: a ij ( g ) compact modulo center 2 π does not appear in any parabolic induction 3 π induced from a compact open subgroup 4 real case There are no supercuspidal representations 1 Casselman: Every irreducible representation appears in 2 parabolic induction There are no compact open subgroups 3 Tasho Kaletha Local Langlands Correspondence

  25. Real discrete series Harish-Chandra parameterization Tasho Kaletha Local Langlands Correspondence

  26. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 Tasho Kaletha Local Langlands Correspondence

  27. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant Tasho Kaletha Local Langlands Correspondence

  28. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Tasho Kaletha Local Langlands Correspondence

  29. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Tasho Kaletha Local Langlands Correspondence

  30. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory Tasho Kaletha Local Langlands Correspondence

  31. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 Tasho Kaletha Local Langlands Correspondence

  32. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Tasho Kaletha Local Langlands Correspondence

  33. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Tasho Kaletha Local Langlands Correspondence

  34. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Local Langlands correspondence Tasho Kaletha Local Langlands Correspondence

  35. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Local Langlands correspondence Langlands: ϕ Tasho Kaletha Local Langlands Correspondence

  36. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Local Langlands correspondence Langlands: ϕ → ( S , B , θ ) / G ( C ) Tasho Kaletha Local Langlands Correspondence

  37. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Local Langlands correspondence Langlands: ϕ → ( S , B , θ ) / G ( C ) → { π 1 , . . . , π k } Tasho Kaletha Local Langlands Correspondence

  38. Real discrete series Harish-Chandra parameterization { π discrete } H − C ← → { ( S , B , θ ) } / G ( R ) 1 S ⊂ G elliptic max torus, S ⊂ B complex Borel subgroup, 2 θ : S ( R ) → C × , d θ is B -dominant � θ ( γ w ) Θ π ( s ) = ( − 1 ) q ( G ) 3 � α> 0 ( 1 − α ( γ w ) − 1 ) w ∈ N ( S , G )( R ) / S ( R ) Highest weight theory π algebraic irrep of G ( C ) ↔ { ( S , B , θ ) } / G ( C ) 1 � θ ( γ w ) Θ π ( s ) = 2 α> 0 ( 1 − α ( γ w ) − 1 ) � w ∈ N ( S , G )( C ) / S ( C ) Local Langlands correspondence Langlands: ϕ → ( S , B , θ ) / G ( C ) → { π 1 , . . . , π k } Θ π 1 + · · · + Θ π k conjugation invariant under G ( C ) Tasho Kaletha Local Langlands Correspondence

  39. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Tasho Kaletha Local Langlands Correspondence

  40. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side Tasho Kaletha Local Langlands Correspondence

  41. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side γ ∈ G ( F ) rs , 1 Tasho Kaletha Local Langlands Correspondence

  42. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , 1 Tasho Kaletha Local Langlands Correspondence

  43. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ Tasho Kaletha Local Langlands Correspondence

  44. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 Tasho Kaletha Local Langlands Correspondence

  45. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , 3 Tasho Kaletha Local Langlands Correspondence

  46. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 Tasho Kaletha Local Langlands Correspondence

  47. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � G , γ κ ∈ G κ ( F ) , 4 Tasho Kaletha Local Langlands Correspondence

  48. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Tasho Kaletha Local Langlands Correspondence

  49. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side Tasho Kaletha Local Langlands Correspondence

  50. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , 1 Tasho Kaletha Local Langlands Correspondence

  51. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } 1 Tasho Kaletha Local Langlands Correspondence

  52. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } ↔ Irr ( S ϕ ) 1 Tasho Kaletha Local Langlands Correspondence

  53. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } ↔ Irr ( S ϕ ) 1 S Θ ϕ = � π ∈ Π ϕ ( G ) Θ π , 2 Tasho Kaletha Local Langlands Correspondence

  54. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } ↔ Irr ( S ϕ ) 1 S Θ ϕ = � invariant under G (¯ π ∈ Π ϕ ( G ) Θ π , F ) 2 Tasho Kaletha Local Langlands Correspondence

  55. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } ↔ Irr ( S ϕ ) 1 S Θ ϕ = � invariant under G (¯ π ∈ Π ϕ ( G ) Θ π , F ) 2 ϕ = � Fourier inversion: s ∈ S ϕ , Θ s π ∈ Π ϕ ( G ) ρ π ( s )Θ π 3 Tasho Kaletha Local Langlands Correspondence

  56. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } ↔ Irr ( S ϕ ) 1 S Θ ϕ = � invariant under G (¯ π ∈ Π ϕ ( G ) Θ π , F ) 2 ϕ = � Fourier inversion: s ∈ S ϕ , Θ s π ∈ Π ϕ ( G ) ρ π ( s )Θ π 3 G s ⊂ � � G , 4 Tasho Kaletha Local Langlands Correspondence

  57. Endoscopy: G (¯ F ) -conjugacy vs. G ( F ) -conjugacy Geometric side � � γ ∈ G ( F ) rs , O γ ( f ) = γ G ( F ) f , SO γ ( f ) = F ) ∩ G ( F ) f 1 γ G (¯ F ) ∩ G ( F ) = � γ G (¯ a ∈ H 1 (Γ , T γ ) a γ G ( F ) 2 γ = � Fourier inversion: κ ∈ H 1 (Γ , T γ ) ∗ , O κ a κ ( a ) O a γ 3 G κ ⊂ � � O κ G , γ κ ∈ G κ ( F ) , γ, G ( f ) = ∆( γ κ , γ ) SO γ κ , G κ ( f κ ) 4 Spectral side ϕ : Γ p → � G , Π ϕ ( G ) = { π 1 , . . . , π k } ↔ Irr ( S ϕ ) 1 S Θ ϕ = � invariant under G (¯ π ∈ Π ϕ ( G ) Θ π , F ) 2 ϕ = � Fourier inversion: s ∈ S ϕ , Θ s π ∈ Π ϕ ( G ) ρ π ( s )Θ π 3 G s ⊂ � � Θ s G , ϕ, G ( f ) = S Θ ϕ, G s ( f s ) 4 Tasho Kaletha Local Langlands Correspondence

  58. Supercuspidal representations Tasho Kaletha Local Langlands Correspondence

  59. Supercuspidal representations Yu’s construction 2001   ( G 0 ⊂ G 1 ⊂ · · · ⊂ G d = G )   J.K.Yu π − 1 − − − → { irred. s.c reps of G ( F ) }   ( φ 0 , φ 1 , . . . , φ d ) Tasho Kaletha Local Langlands Correspondence

  60. Supercuspidal representations Yu’s construction 2001   ( G 0 ⊂ G 1 ⊂ · · · ⊂ G d = G )   J.K.Yu π − 1 − − − → { irred. s.c reps of G ( F ) }   ( φ 0 , φ 1 , . . . , φ d ) Properties Tasho Kaletha Local Langlands Correspondence

  61. Supercuspidal representations Yu’s construction 2001   ( G 0 ⊂ G 1 ⊂ · · · ⊂ G d = G )   J.K.Yu π − 1 − − − → { irred. s.c reps of G ( F ) }   ( φ 0 , φ 1 , . . . , φ d ) Properties Kim 2007: Surjective for p >> 0 1 Tasho Kaletha Local Langlands Correspondence

  62. Supercuspidal representations Yu’s construction 2001   ( G 0 ⊂ G 1 ⊂ · · · ⊂ G d = G )   J.K.Yu π − 1 − − − → { irred. s.c reps of G ( F ) }   ( φ 0 , φ 1 , . . . , φ d ) Properties Kim 2007: Surjective for p >> 0 1 Hakim-Murnaghan 2008: Fibers as equivalence classes. 2 Tasho Kaletha Local Langlands Correspondence

  63. Supercuspidal representations Yu’s construction 2001   ( G 0 ⊂ G 1 ⊂ · · · ⊂ G d = G )   J.K.Yu π − 1 − − − → { irred. s.c reps of G ( F ) }   ( φ 0 , φ 1 , . . . , φ d ) Properties Kim 2007: Surjective for p >> 0 1 Hakim-Murnaghan 2008: Fibers as equivalence classes. 2 Adler-DeBacker-Spice 2008++: Character formula. 3 Tasho Kaletha Local Langlands Correspondence

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