Certain representations with unique models Yuanqing Cai Kyoto - PowerPoint PPT Presentation

Certain representations with unique models Yuanqing Cai Kyoto University November 2020 New York

◮ H : the quaternion algebra over R ◮ ν : H × → R > 0 ◮ tr : H → R ◮ ψ : R → C × nontrivial additive character � � 1 � � x ◮ N = u = ∈ GL 2 ( H ) . 1 ◮ ψ N ( u ) = ψ ( tr ( x ))

◮ π : an irreducible 5-dimensional representation of H × ◮ the normalized parabolic induction π × νπ has a unique irreducible subrepresentation θ ( π ). Question dim Hom N ( θ ( π ) , ψ N ) =? A 25 B 10 C 1 D 0

Uniqueness of Whittaker models, I ◮ F : non-Archimedean local field ◮ ψ : F → C × , a nontrivial additive character ◮ GL n (more generally, quasi-split groups) ◮ Write GL n for GL n ( F ) ◮ ν = | det | : GL n → C × ◮     1 u 12 ∗ · · · ∗           1 · · · ∗  u 23        1 · · · ∗ N n = u = ∈ GL n .       .   .      .      1 A generic character ψ n : N n → C × is of the form ψ n ( u ) = ψ ( u 12 + u 23 + · · · + u n − 1 , n ) .

Uniqueness of Whittaker models, II Theorem (Uniqueness of Whittaker models) For π ∈ Irr ( GL n ), Hom N n ( π, ψ n ) = Hom GL n ( π, ind GL n N n ψ n ) is of dimension ≤ 1. Equivalently, dim J N n ,ψ n ( π ) ≤ 1 . When the dimension is 1, we say that π is generic (or ψ N -generic) or π has a Whittaker model.

Uniqueness of Whittaker models, III Applications ◮ Such properties play important roles in the construction of many global integrals. (Use unique models to obtain Eulerian integrals.) ◮ Can be used to study the analytic properties of certain Langlands L -functions. ◮ For example, the Rankin-Selberg integrals and Langlands-Shahidi method.

Non-generic representations When π does not have any Whittaker model, we say that π is non-generic. Degenerate models Non-generic representations admit unique models of degenerate type.

Derivatives, I ◮ Mirabolic subgroup �� g � � v : g ∈ GL n − 1 , v ∈ F n − 1 P n = . 0 1 ◮ �� I n − 1 � � v : v ∈ F n − 1 U n = . 0 1 ◮ P n = GL n − 1 ⋉ U n ◮ the restriction of ψ n gives a character of U n

Derivatives, II Several functors ◮ Ψ − ( π ) = J U n ( π ) = π/ � π ( u ) v − v : u ∈ U n , v ∈ π � . This gives Ψ − : Rep ( P n ) → Rep ( GL n − 1 ) . ◮ Φ − ( π ) = J U n ,ψ n ( π ) = π/ � π ( u ) v − ψ n ( u ) v : u ∈ U n , v ∈ π � and this gives Φ − : Rep ( P n ) → Rep ( P n − 1 ) . ◮ k -th derivative π ( k ) = Ψ − ◦ (Φ − ) ( k − 1) ( π | P n ) . This gives a functor Rep ( GL n ) → Rep ( GL n − k ) .

Derivatives, III ◮ The n -th derivative is the functor J N n ,ψ n . ◮ Let k 0 be the maximal k such that π ( k ) � = 0. Then π ( k 0 ) is called the highest derivative of π . Notation: k 0 = ht ( π ). ◮ If π is generic, then the highest derivative of π is the n -th derivative.

Derivatives, IV Example (Speh representations) If τ ∈ Irr ( GL n ) is discrete series, then the normalized parabolic induction τ × τν × · · · × τν ℓ − 1 has a unique irreducible subrepresentation θ ( τ, ℓ ) ∈ Irr ( GL n ℓ ). In particular, if τ : GL 1 → C × is a character, then θ ( τ, ℓ ) = τ ◦ det . Generally If τ ∈ Irr ( GL n ) is generic and unitary, then τ = τ 1 × · · · × τ m for τ 1 , · · · , τ m essentially discrete series. Define θ ( τ, ℓ ) = θ ( τ 1 , ℓ ) × · · · × θ ( τ m , ℓ ) .

Derivatives, V ◮ the highest derivative of θ ( τ, ℓ ) is θ ( τ, ℓ ) ( n ) “ = ′′ θ ( τ, ℓ − 1). More generally, Theorem (Zelevinsky) If π is irreducible, then its highest derivative π ( k ) is also irreducible.

Derivatives, VI Given π ∈ Irr ( GL n ), we can take highest derivatives repeatedly: π 1 = π ( k 1 ) , k 1 = ht ( π ) , π 2 = π ( k 2 ) k 2 = ht ( π 1 ) , , 1 · · · π m = π ( k m ) k m = ht ( π m − 1 ) , m − 1 . This gives a partition ( k 1 k 2 · · · k m ) of n . ◮ π m is of the form J N n ,ψ ( k 1 ··· km ) ( π ) for some degenerate character ψ ( k 1 ··· k m ) . ◮ By the Frobenius reciprocity, this gives a degenerate model for π . ◮ By the theorem of Zelevinsky, π m is an irreducible representation of GL 0 , which must be one-dimensional.

Nilpotent orbits, I Summary: ◮ by computing derivatives, one can find a partition ( k 1 k 2 · · · k m ) and a unique model for π . ◮ ( k 1 k 2 · · · k m ) is the “maximal” partition (or nilpotent orbit) that support nonzero models for π .

Nilpotent orbits, II More generally, given a reductive group G , to every coadjoint nilpotent orbit O ⊂ g ∗ and every π ∈ Rep ( G ), we associate a certain generalized Whittaker quotient π O . ◮ Let WO ( π ) denote the set of all nipotent orbit O with π O � = 0 ◮ WS ( π ) denote the set of maximal orbits in WO ( π ) with respect to the closure ordering. Example ◮ Nilpotent orbits of GL n are classified by the partitions of n via the Jordan canonical decomposition. ◮ WS ( θ ( τ, ℓ )) = { ( n ℓ ) } .

Nilpotent orbits, III Character expansion One can define the character χ π of π as a distribution and we have a charater expansion � χ π = c O ˆ µ O . O where the sum is over the set of nilpotent orbits. Theorem (Mœglin-Waldspurger, Varma) The set WS ( π ) is the same as the maximal elements such that c O � = 0. Moreover, for O ∈ WS ( π ), dim π O = c O .

Nilpotent orbits, IV Example For θ ( τ, ℓ ), c ( n ℓ ) = 1 and χ θ ( τ,ℓ ) = ˆ µ ( n ℓ ) + other terms. Archimedean case ◮ There are irreducible representations without unique models. ◮ The Archimedean version of Mœglin-Waldspurger’s theorem has not been proven.

Division algebras, I ◮ D : central division algebra over F of dimension d 2 ◮ Consider GL n , D ◮ Nilpotent orbits of GL n , D are classified by partitions of n . Notation: ( n 1 · · · n m ) D . Unique models? Unfortunately, uniqueness of models fails in general.

Division algebras, II Question Find representations of GL n , D with unique models. Example (Case n = 1) There is no non-trivial nilpotent elements in D × but there are irreducible finite-dimensional representations of D × of dimension greater than 1. Only one-dimensional representations have unique models.

Jacquet-Langlands correspondence, I How to construct representations of GL n , D ? ◮ For g ′ ∈ GL n , D , one can define characteristic polynomial ◮ g ∈ GL nd , g ′ ∈ GL n , D ◮ Define: g ↔ g ′ if and only if g and g ′ are both regular semi-simple and have the same characteristic polynomials. nd corresponds to O ′ = ( n 1 · · · n m ) D in m ) in gl ∗ ◮ O = ( n d 1 · · · n d gl ∗ n , D ◮ D n : discrete series of GL n ◮ D ′ n : discrete series of GL n , D

Jacquet-Langlands correspondence, II Theorem (Deligne-Kazhdan-Vign´ eras) There is a unique bijection C : D nd → D ′ n such that for all π ∈ D nd we have χ π ( g ) = ( − 1) nd − n χ C( π ) ( g ′ ) for all g ∈ GL nd and g ′ ∈ GL n , D such that g ↔ g ′ . Theorem (Badulescu, Badulescu-Renard) If π is a ‘ d -compatible’ irreducible unitary representation of GL nd , then there exists a unique irreducible unitary representation π ′ of GL n , D and a unique sign ε π ∈ {− 1 , 1 } such that χ π ( g ) = ε π χ π ′ ( g ′ ) for all g ′ ↔ g . Notation: π ′ = LJ( π ) .

Jacquet-Langlands correspondence, III We will take the later version as it is compatible with a global correspondence. Non-Archimedean Strategy ◮ (Prasad’s result) character relation implies identities nd corresponds to O ′ ⊂ gl ∗ c O = ε π c O ′ , where O ⊂ gl ∗ n , D . ◮ Idea: find representations of GL nd with suitable size such that O ∈ WS ( π ) corresponds to O ′ ∈ WS ( LJ ( π )). ◮ (Important!) find representations such that ε π = 1

Jacquet-Langlands correspondence, IV Definition For a positive integer ℓ and an irreducible generic unitary τ , define θ D ( τ, ℓ ) = LJ( θ ( τ, d ℓ )) . ◮ θ ( τ, d ℓ ) is d -compatible. ◮ ε θ ( τ, d ℓ ) = 1 ◮ WS ( θ ( τ, d ℓ )) = ( n d ℓ ) ◮ one can check that WS ( θ D ( τ, ℓ )) = ( n ℓ ) D with unique models. ◮ If τ is one-dimensional, then θ ( τ, d ℓ ) = τ ◦ det and θ D ( τ, ℓ ) = τ ◦ Nm .

Jacquet-Langlands correspondence, V ◮ D : unique quaternion algebra over F ◮ π : Steinberg representation of GL 2 . ◮ 1 GL 2 , 1 D × : trivial representations Then ◮ C( π ) = 1 D × , but χ π ( g ) = − χ 1 D × ( g ′ ) for all g ↔ g ′ . ◮ LJ(1 GL 2 ) = 1 D × and χ 1 GL 2 ( g ) = χ 1 D × ( g ′ ) for all g ↔ g ′ .

Jacquet-Langlands correspondence, VI Archimedean case The definition of θ H ( τ, ℓ ) works. Similar results are expected but a different approach is required. Global definition Given a cuspidal representation τ = ⊗ ′ v τ v of GL n ( A ), one can define θ D ( τ, ℓ ) = ⊗ ′ v θ D v ( τ v , ℓ ) , and this is a discrete series of GL n ℓ, D ( A ). One can ask similar questions for global representations (in terms of degenerate Whittaker coefficients). Note: for central simple algebra D v = M r v ( A v ), θ D v ( τ v , ℓ ) = θ A v ( τ v , r v ℓ ) .