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Bernstein centre for enhanced Langlands parameters Ahmed Moussaoui University of Calgary December 5, 2015 Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 1 / 37

Bernstein decomposition Let G be a connected reductive group over a p -adic field F . The set of (equivalences classes of) irreducibles representations of G is decomposed as � Irr ( G ) = Irr ( G ) s , s ∈B ( G ) where s = [ M , σ ] with M a Levi subgroup of G and σ ∈ Irr ( M ) cuspidal. There is a map Sc : Irr ( G ) − → Ω( G ) which associate to an irreducible representation its cuspidal support. Question How to define the Bernstein decomposition for Langlands parameters ? What is the notion of cuspidal Langlands parameter ? of cuspidal support ? Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 2 / 37

Generalized Springer correspondence Let H be a complex connected reductive group For all x ∈ H , we denote A H ( x ) = Z H ( x ) / Z H ( x ) ◦ . � � � � N + ( C H H = u , η ) � u ∈ H unipotent , η ∈ Irr ( A H ( u )) We denote by S H the set of ( H -conjugacy classes of) triples ( L , C L v , ε ) with L a Levi subgroup of H ; C L v an unipotent L -orbit ; ε ∈ Irr ( A L ( v )) cuspidal. For all H , the triple ( T , { 1 } , 1 ) ∈ S H , and N H ( T ) / T is the Weyl group of H ; A H ( u ) ε H condition unipotent orbi n = 1 O ( 1 ) { 1 } 1 GL n ( Z / 2 Z ) d ε ( z 2 i ) = ( − 1 ) i 2 n = d ( d + 1 ) O ( 2 d , 2 d − 2 ,..., 4 , 2 ) Sp 2 n n = d 2 ( Z / 2 Z ) d − 1 SO n O ( 2 d − 1 , 2 d − 3 ,..., 3 , 1 ) ε ( z 2 i − 1 z 2 i + 1 ) = − 1 Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 3 / 37

Let u ∈ G be a unipotent element and ε ∈ Irr ( A H ( u )) . Let P = LU be a parabolic subgroup of H and v ∈ L be a unipotent element. We define � � gZ L ( v ) ◦ U | g ∈ H , g − 1 ug ∈ vU Y u , v = and d u , v = 1 2 ( dim Z H ( u ) − dim Z L ( v )) . Then dim Y u , v � d u , v and Z H ( u ) acts on Y u , v by left translation. We denote by S u , v the permutation representation on the irreducibles components of Y u , v of dimension d u , v . If P = B = TU , then � � � B ′ ∈ B | u ∈ B ′ � gB ∈ H / B | g ∈ H , g − 1 ug ∈ U Y u , 1 = = = B u . Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 4 / 37

Definition We say that ε is cuspidal, if and only if, for all proper parabolic subgroup P = LU , for all unipotent v ∈ L , we have Hom A G ( u ) ( ε, S u , v ) = 0. Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 5 / 37

Generalized Springer correspondence � � � � N + ( C H � u ∈ H unipotent , η ∈ Irr ( A H ( u )) H = u , η ) S H the set of ( H -conjugacy classes of) triples ( L , C L v , ε ) with L a Levi subgroup of H ; C L v an unipotent L -orbit ; ε ∈ Irr ( A L ( v )) cuspidal. We denote by W H L = N H ( L ) / L . Theorem (Lusztig,1984) � N + Irr ( W H H ≃ L ) ( L , C L v ,ε ) ∈S H ( C H → ( L , C L u , η ) ← v , ε ; ρ ) Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 6 / 37

Generalized Springer correspondence in a disconnected case We suppose now that H is a reductive not necessarily connected H acts by conjugation on N + H ◦ and S H ◦ . Proposition (M.) The generalized Springer correspondence for H ◦ is H -équivariante, i.e. h · ( C H ◦ → h · ( L ◦ , C L ◦ u , η ) ← v , ε ; ρ ) . Définition We call quasi-Levi subgroup of H a subgroup of the form L = Z H ( A ) , where A is a torus contained in H ◦ . The neutral component of a quasi-Levi subgroup of H is a Levi subgroup of H ◦ . L ◦ = N H ◦ ( L ◦ ) / L ◦ as normal subgroup. W H L = N H ( A ) / Z H ( A ) admits W H ◦ Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 7 / 37

Generalized Springer correspondence in a disconnected case Let u ∈ H ◦ be unipotent et ε ∈ Irr ( A H ( u )) . We say that ε is cuspidal if all irreducibles subrepresentations of A H ◦ ( u ) which appear in the restrcition to A H ◦ ( u ) are cuspidals. We denote by � � u , η ) , u ∈ H ◦ unipotent , η ∈ Irr ( A H ( u )) N + ( C H H = S H the set of ( H -conjugacy classes of) triples ( L , C L v , ε ) avec L quasi-Levi subgroup of H ; C L v a unipotent L -orbit ; ε ∈ Irr ( A L ( v )) cuspidal. Theorem (M.) For H = O n , � N + Irr ( W H H ≃ L ) ( L , C L v ,ε ) ∈S H ( C H → ( L , C L u , η ) ← v , ε, ρ ) Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 8 / 37

Langlands correspondence Let be F a p -adic field and G a split reductive connected group over F . We denote by � G the Langlands dual group of G , W F the Weil group of F and W ′ F = W F × SL 2 ( C ) the Weil-Deligne group. Définition A Langlands parameter of G is a continous morphism → � φ : W ′ F − G , such that φ SL 2 ( C ) is algebraic ; φ ( W F ) consist of semisimple elements. We denote by Φ( G ) the set of � G -conjugation classes of Langlands parameters of G . Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 9 / 37

Langlands correspondence We denote by Irr ( G ) the set of (smooth) irreducible representations of G . Conjecture There exists a finite to one map rec G : Irr ( G ) − → Φ( G ) . Hence, � Irr ( G ) = Π φ ( G ) . φ ∈ Φ( G ) There exists a bijection Π φ ( G ) ≃ Irr ( S G φ ) , avec S G G ( φ ) ◦ Z � φ = Z � G ( φ ) / Z � G . + other proprieties. Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 10 / 37

Langlands correspondence The Langlands correspondence is proved for GL n by Harris et Taylor ; Henniart et Scholze, for SO n et Sp 2 n by Arthur. We denote by � � � � Φ( G ) + = � φ ∈ Φ( G ) , η ∈ Irr ( S G ( φ, η ) φ ) . Then rec + G : Irr ( G ) ≃ Φ( G ) + . Properties of the Langlands correspondence : for all φ ∈ Φ( G ) , the following are equivalent one element in Π φ ( G ) is in the discrete serie ; all elements in Π φ ( G ) are in the discrete serie ; φ ∈ Φ( G ) 2 (is discrete). Supercuspidal ? Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 11 / 37

Jordan bloc of discrete series Let G be one of the split groups Sp 2 n ( F ) or SO n ( F ) . For all unitary irreducible supercuspidal representation π de GL d π ( F ) and for all integer a � 1, the induced representation a − 1 a − 3 1 − a 2 × π | | 2 × . . . × π | | 2 , π | | admit a unique irreducible subrepresentation of GL ad π ( F ) : St ( π, a ) . Let τ be an irreducible discrete serie of G . We denote by Jord ( τ ) = { ( π, a ) } with π an unitary irreducible supercuspidal representation of a GL d π ( F ) and a � 1 such that there exists an integer a ′ which verify :  a ≡ a ′ mod 2  St ( π, a ) ⋊ τ irréductible  St ( π, a ′ ) ⋊ τ réductible Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 12 / 37

Jordan bloc of discrete series Let ϕ ∈ Φ( G ) a discrete parameter. The decomposition of Std ◦ φ , where Std : � → GL N � G ֒ G ( C ) is : � � Std ◦ φ = π ⊠ S a . π ∈ I ϕ a ∈ J π We call Jordan bloc of ϕ and we denote by Jord ( ϕ ) = { ( π, a ) | π ∈ I ϕ , a ∈ J π } . Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 13 / 37

Jordan bloc of discrete series Jord ( ϕ ) without hole (or jump) ⇐ ⇒ (( π, a ) ∈ Jord ( ϕ ) et a � 3 = ⇒ ( π, a − 2 ) ∈ Jord ( ϕ )) . A � G ( ϕ ) is generated by � z π, a pour ( π, a ) ∈ Jord ( ϕ ) et a pair pour ( π, a ) , ( π, a ′ ) ∈ Jord ( ϕ ) without parity condition on a , a ′ z π, a z π, a ′ ( π, a ) , ( π, a ′ ) ∈ Jord ( ϕ ) , with a ′ < a , consecutive ⇔ for all b ∈ � a ′ + 1 , a − 1 � , ( π, b ) �∈ Jord ( ϕ ) . a π, min the smallest integer a � 1 such that ( π, a ) ∈ Jord ( ϕ ) . Définition G ( ϕ ) is alternate if for all ( π, a ) , ( π, a ′ ) ∈ Jord ( ϕ ) A character ε of A � consecutive, ε ( z π, a z π, a ′ ) = − 1 and if for all ( π, a π, min ) ∈ Jord ( ϕ ) with a π, min evenn, ε ( z π, a π, min ) = − 1. Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 14 / 37

Jordan bloc of discrete series Theorem (Mœglin) The Langlands classification of discrete series of G by Arthur induce a bijection between the set of irreducible supercuspidal representation of G and the set of pairs ( ϕ, ε ) such that Jord ( ϕ ) is without holes and ε is alternate ; the bijection τ �→ ( ϕ, ε ) is defined by Jord ( ϕ ) = Jord ( τ ) et ε = ε τ . Ahmed Moussaoui (University of Calgary) Bernstein centre for enhanced Langlands parameters December 5, 2015 15 / 37