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

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

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

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

Local representation theory The groups Tasho Kaletha Local Langlands Correspondence

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Supercuspidal representations Tasho Kaletha Local Langlands Correspondence

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

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

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

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

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