Theta Correspondence for Dummies (Correspondance Theta pour les - PowerPoint PPT Presentation

Theta Correspondence for Dummies (Correspondance Theta pour les nuls) Jeffrey Adams Dipendra Prasad Gordan Savin Conference in honor of Roger Howe Yale University June 1-5, 2015

Theta Correspondence Mp = � Sp ( 2 n , F ) : metaplectic group ( G , G ′ ) a reductive dual pair: G ′ = Cent Mp ( G ) G = Cent Mp ( G ′ ) , ψ character of F , → oscillator representation ω = ω ψ G , π ′ ∈ � → π ′ if Definition: π ∈ � G ′ , say π ← Hom G × G ′ ( ω, π ⊠ π ′ ) � = 0 Howe Duality Theorem (Howe, Waldspurger, Gan-Takeda) F local → π ′ is a bijection π ← (between subsets of � G and � G ′ ) Definition: π ′ = θ ( π ) , π = θ ( π ′ )

� � � � � Computing θ ( π ) Describe π → θ ( π ) (in terms of some kinds of parameters) Properties of the map: preserving tempered, unitary, relation on wave front sets, functoriality (Langlands/Arthur). . . Typically there are some easy cases, and some hard ones O ( 2 n + 4 ) θ m , 2 n + 4 ( π ) = non-tempered � � � � � � � � � � � � � � � O ( 2 n + 2 ) θ m , 2 n + 2 ( π ) = non-tempered � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � O ( 2 n ) θ m , 2 n ( π ) = discrete series π Sp ( 2 m , F ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � O ( 2 n − 2 ) θ m , 2 n − 2 ( π ) = discrete series � � � � � � � � � � � � � � � � O ( 2 n − 4 ) 0

Θ( π ) θ ( π ) irreducible, ω → π ⊠ θ ( π ) Defintion (Howe) ω ( π ) =the maximal π -isotypic quotient of ω Θ( π ) (“big-theta” of π ): ω ( π ) ≃ π ⊠ Θ( π ) Proof of the duality theorem: θ ( π ) is the unique irreducible quotient of Θ( π ) Generically, Θ( π ) is irreducible and θ ( π ) = Θ( π )

The structure of Θ( π ) Θ( π ) is important, interesting, complicated Θ( 1 ) (Kudla, Rallis, . . . ) Structure of reducible principal series (Howe. . . ) Lee/Zhu: Sp ( 2 n , R ) :

� � � � � � � � � � � � Example: see-saw pairs and reciprocity Howe G ′ H � � � � � � � � � � � � � � � � H ′ G Θ( σ ′ ) Θ( π ) ��������� � � � � � � � � � � π σ ′ Θ( σ ′ )[ π ] ⊠ σ ′ ≃ π ⊠ Θ( π )[ σ ′ ] Roughly: mult G ( π, Θ( σ ′ )) = mult H ′ ( σ ′ , Θ( π ))

� � � � Theta correspondence and induction GL ( n + r ) � Θ m , n + r � � � � � � � � � Θ m , n GL ( m ) GL ( n ) m = n : Θ n , n ( π ) = θ n , n ( π ) = π ∗ Kudla: P = MN , M = GL ( n ) × GL ( r ) Hom GL ( m ) ( ω m , n + r , π ⊠ Ind GL ( n + r ) ( θ m , n ( π ) ⊗ 1 )) � = 0 P Ind GL ( n + r ) ( θ m , n ( π ) ⊗ 1 ) P � Θ m , n + r � � � � � � � � � � � � � θ m , n � � π � θ n ( π )

Yale Freshman graduate student’s dream Θ m , n + r ( π ) ? = Ind GL ( n + r ) ( 1 ⊗ θ m , n ( π )) P θ m , n + r ( π ) is ( ? ) the unique irreducible quotient of Ind GL ( n + r ) ( 1 ⊗ θ m , n ( π )) P Neither is true in general ω = S ( M m , n ) filtration: ω k : functions supported on matrices of rank ≥ k : 0 = ω t ⊂ ω t − 1 ⊂ · · · ⊂ ω 0 = ω Serious issues with extensions here. . . also reducibility of induced representations

Characteristic Basic Principle � ( − 1 ) i Ext i Hom → Ext → EP = i (+ vanishing. . . ) Problem: Study Ext i G × G ′ ( ω, π ⊠ π ′ ) , EP G × G ′ ( ω, π ⊠ π ′ ) alternatively: Ext i G ( ω, π ) , EP G ( ω, π ) as (virtual) representations of G ′ Idea: EP G ( ω, π ) is like Hom G ( ω, π ) with everything made completely reducible. . . all “boundary terms” vanish

Example: GL ( 1 ) , or Tate’s Thesis ( G , G ′ ) = ( GL ( 1 ) , GL ( 1 )) ⊂ SL ( 2 , F ) ω : S ( F ) ( S = C ∞ c , the Schwarz space) ω ( g , h )( f )( x ) = f ( g − 1 xh ) (up to | det | ± 1 2 ) χ character of GL ( 1 ) Question: Hom GL ( 1 ) ( S ( F ) , χ ) =? 0 → S ( F × ) → S ( F ) → C → 0 Hom ( , χ ) = Hom GL ( 1 ) ( , χ ) 0 → Hom ( C , χ ) → Hom ( S ( F ) , χ ) → Hom ( S ( F × ) , χ ) → Ext ( C , χ )

Example: GL ( 1 ) , or Tate’s Thesis 0 → Hom ( C , χ ) → Hom ( S ( F ) , χ ) → Hom ( S ( F × ) , χ ) → Ext ( C , χ ) χ � = 1: 0 → 0 → Hom ( S ( F ) , χ ) → Hom ( S ( F × ) , χ ) → 0 Hom GL ( 1 ) ( S ( F ) , χ ) = Hom GL ( 1 ) ( S ( F ∗ ) , χ ) = C χ = 1: 0 → C → Hom ( S ( F ) , χ ) → Hom ( S ( F × ) , χ ) → C → Ext 1 ( S ( F ) , C ) = 0 Hom GL ( 1 ) ( S ( F ) , χ ) = 1 in all cases Remark: Tate’s thesis: this is true provided | χ ( x ) | = | x | s with s > 1. General case: analytic continuation in χ of Tate L-functions.

Theta and the Euler Poincare Characteristic Punch line: Theorem (Adams/Prasad/Savin) Fix m , and consider the dual pairs ( G = GL ( m ) , GL ( n )) n ≥ 0. π ∈ � G � 0 n < m EP G ( ω m , n , π ) ∞ ≃ Ind GL ( n ) ( 1 ⊗ π ) n ≥ m P where M = GL ( n − m ) × GL ( m ) More details. . .

Euler-Poincare Characteristic Reference: D. Prasad, Ext Analogues of Branching Laws F : p -adic field, G : reductive group/ F C = C G : category of smooth representations S ( G ) = C ∞ c ( G ) , smooth compactly supported functions, smooth representation of G × G Lemma: C has enough projectives and injectives Ext i G ( X , Y ) : derived functors of Hom G ( _ , Y ) or Hom G ( X , _ ) .

Euler-Poincare Characteristic P = MN ⊂ G , Ind G P normalized induction r G P normalized Jacquet functor X , Y smooth 1. Ext i G ( X , Y ) = 0 for i > split rank of G 2. S ( G ) is projective (as a left G -module) 3. Hom G ( S ( G ) , X ) G −∞ ≃ X 4. EP GL ( m ) ( X , Y ) = 0 ( X , Y finite length) G ( X , Ind G 5. Ext i P ( Y )) ≃ Ext i M ( r G P ( X ) , Y ) 6. Ext i G ( Ind G P ( X ) , Y ) ≃ Ext i M ( X , r P G ( Y )) 7. Kunneth Formula ( X 1 admissible): � Ext j Ext i G 1 ( X 1 , Y 1 ) ⊗ Ext k G 1 × G 2 ( X 1 ⊠ X 2 , Y 1 ⊠ Y 2 ) ≃ G 2 ( X 2 , Y 2 ) j + k = i

Euler-Poincare Characteristic X : G × G ′ -modules (for example: ω ) Y : G -module Ext i G ( X , Y ) is an G ′ -module (not necessarily smooth) Definition: G ( X , Y ) ∞ = Ext i G ( X , Y ) G ′ −∞ Ext i (a smooth G ′ -module) Dangerous bend: Ext i G ( X , Y ) is (probably) not the derived functors of Y → Hom G ( X , Y ) G ′ −∞ Definition: Assume Ext i G ( X , Y ) has finite length for all i EP G ( X , Y ) = � i ( − 1 ) i Ext G ( X , Y ) ∞ is a well-defined element of the Grothendieck group of smooth representations of G ′

Back to Θ( π ) ( G , G ′ ) dual pair, ω , π irreducible representation of G EP G ( ω, π ) ∞ ω → π ⊠ Θ( π ) Proposition: Hom G ( ω, π ) ∞ = Θ( π ) ∨ ∨ : smooth dual proof: 0 → ω [ π ] → ω → π ⊠ Θ( π ) → 0 Hom ( , π ) is left exact: φ 0 → Hom G ( π ⊠ Θ( π ) , π ) → Hom G ( ω, π ) → Hom G ( ω [ π ] , π ) φ = 0 ⇒ Hom ( ω, π ) ≃ Θ( π ) ∗ , take the smooth vectors

Computing EP Recall: ω k = S ( matrices of rank ≥ k ) 0 = ω t ⊂ ω t − 1 ⊂ · · · ⊂ ω 0 = ω ω k /ω k + 1 = S (Ω k ) (Ω k = matrices of rank k ) S (Ω k ) ≃ Ind GL ( m ) × GL ( n ) GL ( k ) × GL ( m − k ) × GL ( k ) × GL ( n − k ) ( S ( GL ( k )) ⊠ 1 ) . Compute Ext i GL ( m ) ( S (Ω k ) , π ) By Frobenius reciprocity, Kunneth formula, other basic properties. . .

ℓ � GL ( m ) ( S (Ω k ) , π ) ∞ ≃ Ind GL ( n ) Ext i GL ( k ) × GL ( n − k ) ( σ j ⊠ 1 ) ⊗ Ext i GL ( m − k ) ( 1 , τ j ) j = 1 r P ( π ) = � σ j ⊠ τ j implies Lemma Ext i GL ( m ) ( S (Ω k ) , π ) is a finite length GL ( n ) -module EP GL ( m ) ( S (Ω k ) , π ) is well defined EP GL ( m ) ( S (Ω k ) , π ) = 0 unless k = m .

Main Theorem: Type II Theorem Fix m , and consider the dual pairs ( G = GL ( m ) , GL ( n )) n ≥ 0. π ∈ � G � 0 n < m EP G ( ω m , n , π ) ∞ ≃ Ind GL ( n ) ( 1 ⊗ π ) n ≥ m P where M = GL ( n − m ) × GL ( m )

Main Theorem: Type I Similar idea, using Kudla (and MVW) calculation of the Jacquet module of the oscillator representation For simplicity: state it for ( Sp ( 2 m ) , O ( 2 n )) (split orthogonal groups) ω = ω m , n oscillator representation for ( G , G ′ ) = ( Sp ( 2 m ) , O ( 2 n )) t < m → M ( t ) = GL ( t ) × Sp ( 2 m − 2 t ) ⊂ Sp ( 2 m ) P ( t ) = M ( t ) N ( t ) , Ind G P ( t ) () t < n → M ′ ( t ) = GL ( t ) × O ( 2 n − 2 t ) ⊂ O ( 2 m ) P ′ ( t ) = M ′ ( t ) N ′ ( t ) , Ind G ′ P ′ ( t ) () ω M ( t ) , M ′ ( t ) oscillator representation for dual pair ( M ( t ) , M ′ ( t ))

Main Theorem: Type I Theorem Fix an irreducible representation π of M ( t ) . Then � 0 t > n P ( t ) ( π )) ∞ ≃ EP G ( ω G , G ′ , Ind G Ind G ′ P ′ ( t ) ( EP M ( t ) ( ω M ( t ) , M ′ ( t ) , π ) ∞ ) t ≤ n . EP( ω ,_) ∞ commutes with induction