Cuspidality and Hecke algebras for Langlands ... - Cornell Math
Cuspidality and Hecke algebras for Langlands parameters Maarten Solleveld Universiteit Nijmegen joint with Anne-Marie Aubert and Ahmed Moussaoui
12 April 2016
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
1 / 25
Some aspects of the local Langlands program
Maarten Solleveld
p-adic side
Galois side
reductive p-adic group irreducible admissible reps supercuspidal reps Bernstein components affine Hecke algebras
Weil–Deligne group enhanced L-parameters ? ? ?
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
2 / 25
Some aspects of the local Langlands program p-adic side
Galois side
reductive p-adic group irreducible admissible reps supercuspidal reps Bernstein components affine Hecke algebras
Weil–Deligne group enhanced L-parameters ? ? ?
Goal of talk Define all this on the Galois side, so that it matches via the local Langlands correspondence
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
2 / 25
enhanced Langlands parameters Notations F : non-archimedean local field G = G(F ): connected reductive group over F G ∨ = G ∨ (C): complex dual group WF ⊂ Gal(F /F ): Weil group of F Assumption: G inner twist of F -split group G ∗
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
3 / 25
enhanced Langlands parameters Notations F : non-archimedean local field G = G(F ): connected reductive group over F G ∨ = G ∨ (C): complex dual group WF ⊂ Gal(F /F ): Weil group of F Assumption: G inner twist of F -split group G ∗
Definition A Langlands parameter for G ∨ is an ”admissible” homomorphism φ : WF × SL2 (C) → G ∨ ∨ : simply connected cover of G ∨ Gsc der Sφ = π0 (ZGsc∨ (φ)) An enhancement of φ is an irrep ρ of Sφ Φe (G ∨ ) = {enhanced L-parameters (φ, ρ)}/G ∨ -conjugation Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
3 / 25
Local Langlands Correspondence Definition (φ, ρ) ∈ Φe (G ∨ ) is relevant for G if ρ|Z (Gsc∨ ) is the Kottwitz parameter of G as an inner twist of a split group G ∗ Notation: Φe (G ) ⊂ Φe (G ∨ )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
4 / 25
Local Langlands Correspondence Definition (φ, ρ) ∈ Φe (G ∨ ) is relevant for G if ρ|Z (Gsc∨ ) is the Kottwitz parameter of G as an inner twist of a split group G ∗ Notation: Φe (G ) ⊂ Φe (G ∨ )
Conjecture (Langlands, Borel, Vogan...) There exists a bijection G
Irr(G ) ←→ Φe (G ∨ )
inner twists G of G ∗
which satisfies many nice properties, e.g. π ∈ Irr(G ) is essentially square-integrable (i.e. π|Gder is square-integrable) if and only if φπ is discrete (i.e. not a L-parameter for any proper Levi subgroup of G ) Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
4 / 25
Cuspidality for L-parameters Notations (φ, ρ) enhanced L-parameter for G ∨ Cφ = ZGsc∨ (φ(WF)): complex reductive group, possibly disconnected uφ = φ 1, ( 10 11 ) : unipotent element of Cφ◦
Lemma (Kazhdan–Lusztig) Natural isomorphism Sφ = π0 ZGsc∨ (φ(WF × SL2 (C))) → π0 (ZCφ (uφ ))
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
5 / 25
Cuspidality for L-parameters Notations (φ, ρ) enhanced L-parameter for G ∨ Cφ = ZGsc∨ (φ(WF)): complex reductive group, possibly disconnected uφ = φ 1, ( 10 11 ) : unipotent element of Cφ◦
Lemma (Kazhdan–Lusztig) Natural isomorphism Sφ = π0 ZGsc∨ (φ(WF × SL2 (C))) → π0 (ZCφ (uφ ))
Definition (φ, ρ) ∈ Φe (G ∨ ) is cuspidal if: φ is discrete; (uφ , ρ) is a cuspidal pairfor Cφ . i.e. ρ ∈ Irr π0 (ZCφ (uφ )) and (uφ , ρ) cannot be obtained from a proper Levi subgroup of Cφ via a certain induction procedure Notation: Φcusp (G ∨ ) ⊂ Φe (G ∨ ) Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
5 / 25
Cuspidal L-parameters for GLn (F ) By classification: (u, ρ) is a cuspidal pair for GLn (C) with ρ|Z (SLn (C)) = 1 ⇐⇒ n = 1, u = 1 and ρ = 1
Lemma (φ, ρ) ∈ Φe (GLn (F )) is cuspidal ⇐⇒ φ|SL2 (C) = 1, ρ = 1 and φ|WF is a n-dim irreducible rep
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
6 / 25
Cuspidal L-parameters for GLn (F ) By classification: (u, ρ) is a cuspidal pair for GLn (C) with ρ|Z (SLn (C)) = 1 ⇐⇒ n = 1, u = 1 and ρ = 1
Lemma (φ, ρ) ∈ Φe (GLn (F )) is cuspidal ⇐⇒ φ|SL2 (C) = 1, ρ = 1 and φ|WF is a n-dim irreducible rep
Proof ∨ ). ⇐ By the irreducibility, φ is discrete and ZGsc∨ (φ(WF )) = Z (Gsc ∨ The pair (u = 1, ρ = 1) is cuspidal for Z (Gsc ) = Z (SLn (C))
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
6 / 25
Cuspidal L-parameters for GLn (F ) By classification: (u, ρ) is a cuspidal pair for GLn (C) with ρ|Z (SLn (C)) = 1 ⇐⇒ n = 1, u = 1 and ρ = 1
Lemma (φ, ρ) ∈ Φe (GLn (F )) is cuspidal ⇐⇒ φ|SL2 (C) = 1, ρ = 1 and φ|WF is a n-dim irreducible rep
Proof ∨ ). ⇐ By the irreducibility, φ is discrete and ZGsc∨ (φ(WF )) = Z (Gsc ∨ The pair (u = 1, ρ = 1) is cuspidal for Z (Gsc ) = Z (SLn (C)) ⇒ ρ = 1 because G = GLn (F ) 1
Since φ is discrete, it is a n-dim irreducible rep of WF × SL2 (C)
2
Cn = V1 ⊗ V2 with V1 ∈ Irr(WF ) and V2 ∈ Irr(SL2 (C))
3
Cuspidality forces uφ = 1, hence V2 is the trivial SL2 (C)-rep
4
Cn = V1 ∈ Irr(WF )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
6 / 25
Cuspidal support of representations Theorem (Bernstein, 1984) Let π ∈ Irr(G ) = { irreducible smooth G -reps over C} There exist a parabolic subgroup P = L n U of G and a σ ∈ Irrcusp (L) such that π is a subquotient of the normalized parabolic induction IPG (σ) π determines the pair (L, σ) uniquely up to G -conjugation
Definition The cuspidal support map for G is G Sc : Irr(G ) → {L} × Irrcusp (L) G -conjugation Levi L
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
7 / 25
Cuspidal support of representations Theorem (Bernstein, 1984) Let π ∈ Irr(G ) = { irreducible smooth G -reps over C} There exist a parabolic subgroup P = L n U of G and a σ ∈ Irrcusp (L) such that π is a subquotient of the normalized parabolic induction IPG (σ) π determines the pair (L, σ) uniquely up to G -conjugation
Alternative presentation of the cuspidal support map Lev(G ): representatives for the conjugacy classes of Levi subgroups of G W (G , L) = NG (L)/L G Sc : Irr(G ) → {L} × Irrcusp (L)/W (G , L) L∈Lev(G )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
7 / 25
Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
8 / 25
Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
3
ψ = φ on the inertia group IF ⊂ Gal(F /F ) 1/2 1/2 qF 0 q ψ FrobF , = φ FrobF , F −1/2
4
0
qF
0
0
−1/2 qF
(3) and (4) say that Sc preserves infinitesimal characters
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
8 / 25
Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
3
ψ = φ on the inertia group IF ⊂ Gal(F /F ) 1/2 1/2 qF 0 q ψ FrobF , = φ FrobF , F −1/2
4
0
5
qF
0
0
−1/2 qF
L∨ sc , uψ , )
(Cφ ∩ is the cuspidal support of (uφ , ρ), for the group Cφ = ZGsc∨ (φ(WF )) (3) and (4) say that Sc preserves infinitesimal characters If (φ, ρ) ∈ Φcusp (G ), then by (5): Sc(φ, ρ) = (G ∨ , φ, ρ)
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
8 / 25
Examples of the cuspidal support map G = GL5m (F ) φ ∈ Φ(G ), ρ = 1 Suppose φ = φ1 ⊗ (R2 ⊕ R2 ⊕ R1 ) with φ1 ∈ Irr(WF ) and Ri = i-dim irrep of SL2 (C) L∨ = GLm (C)5
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
9 / 25
Examples of the cuspidal support map G = GL5m (F ) φ ∈ Φ(G ), ρ = 1 Suppose φ = φ1 ⊗ (R2 ⊕ R2 ⊕ R1 ) with φ1 ∈ Irr(WF ) and Ri = i-dim irrep of SL2 (C) L∨ = GLm (C)5 for w ∈ IF , x ∈ SL2 (C) : ψ(w , x) = φ1 (w ) ⊗ I5 = φ(w ) 1/2
−1/2
ψ(FrobF ) = φ(FrobF ) ⊗ (qF , qF Then Sc(φ, ρ) =
Maarten Solleveld
(L∨ , ψ,
1/2
−1/2
, qF , qF
, 1)
= 1)
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
9 / 25
Examples of the cuspidal support map G = GL5m (F ) φ ∈ Φ(G ), ρ = 1 Suppose φ = φ1 ⊗ (R2 ⊕ R2 ⊕ R1 ) with φ1 ∈ Irr(WF ) and Ri = i-dim irrep of SL2 (C) L∨ = GLm (C)5 for w ∈ IF , x ∈ SL2 (C) : ψ(w , x) = φ1 (w ) ⊗ I5 = φ(w ) 1/2
−1/2
ψ(FrobF ) = φ(FrobF ) ⊗ (qF , qF Then Sc(φ, ρ) =
(L∨ , ψ,
1/2
−1/2
, qF , qF
, 1)
= 1)
This works also for GLn (F ) It fits with the Zelevinsky classification of Irr(GLn (F ))
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
9 / 25
Examples of the cuspidal support map G = G2 (F ) φ|WF = 1, uφ = subregular unipotent ρ ∈ Irr π0 (ZG2 (C) (uφ )) ∼ = Irr(S3 ) if ρ = sign, then (φ, ρ) ∈ Φe (G ) is cuspidal
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
10 / 25
Examples of the cuspidal support map G = G2 (F ) φ|WF = 1, uφ = subregular unipotent ρ ∈ Irr π0 (ZG2 (C) (uφ )) ∼ = Irr(S3 ) if ρ = sign, then (φ, ρ) ∈ Φe (G ) is cuspidal if ρ 6= sign, then Sc(φ, ρ) = (T ∨ , ψ, 1), where T is a maximalsplit torus of G and n/2 q 0 ψ(FrobnF w , x) = φ 1, F −n/2 for w ∈ IF 0
qF
The cuspidal support really depends on the enhancements of L-parameters
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
10 / 25
Comparison of cuspidal support maps For L a Levi subgroup of G W (G , L) = NG (L)/L is isomorphic with NG ∨ (L∨ )/L∨ Provides an action of W (G , L) on Φcusp (L)
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
11 / 25
Comparison of cuspidal support maps For L a Levi subgroup of G W (G , L) = NG (L)/L is isomorphic with NG ∨ (L∨ )/L∨ Provides an action of W (G , L) on Φcusp (L)
Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φe (G ) o
F
Sc
L∈Lev(G ) Φcusp (L)/W (G , L)
Maarten Solleveld
/ Irr(G )
LLC
o
LLC
/
F
Sc
L∈Lev(G ) Irrcusp (L)/W (G , L)
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
11 / 25
Bernstein components for representations Xnr (G ): group of unramified characters G → C×
Definition Let σ ∈ Irrcusp (L) sL = (L, Xnr (L) ⊗ σ) ⊂ {L} × Irrcusp (L) s = [L, σ]G = G -conjugacy class of sL s is an inertial equivalence class for G Ω(G ): set of such classes Irr(G )s = Sc −1 (sL ), a Bernstein component of Irr(G )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
12 / 25
Bernstein components for representations Xnr (G ): group of unramified characters G → C×
Definition Let σ ∈ Irrcusp (L) sL = (L, Xnr (L) ⊗ σ) ⊂ {L} × Irrcusp (L) s = [L, σ]G = G -conjugacy class of sL s is an inertial equivalence class for G Ω(G ): set of such classes Irr(G )s = Sc −1 (sL ), a Bernstein component of Irr(G )
Theorem (Bernstein, 1984) F s Irr(G ) = Q s∈Ω(G ) Irr(G ) Rep(G ) = s∈Ω(G ) Rep(G )s
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
12 / 25
Bernstein components for L-parameters G inner twist, so Xnr (G ) ∼ = Z (G ∨ )◦
Definition Let (φ, ρ) ∈ Φcusp (L∨ ) ∨ ∨ ◦ ∨ ∨ s∨ L = (L , Z (L ) φ, ρ) ⊂ {L } × Φcusp (L ) s∨ = [L∨ , φ, ρ]G ∨ = G ∨ -conjugacy class of s∨ L s∨ is an inertial equivalence class for Φe (G ∨ )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
13 / 25
Bernstein components for L-parameters G inner twist, so Xnr (G ) ∼ = Z (G ∨ )◦
Definition Let (φ, ρ) ∈ Φcusp (L∨ ) ∨ ∨ ◦ ∨ ∨ s∨ L = (L , Z (L ) φ, ρ) ⊂ {L } × Φcusp (L ) s∨ = [L∨ , φ, ρ]G ∨ = G ∨ -conjugacy class of s∨ L s∨ is an inertial equivalence class for Φe (G ∨ ) Ω(G ∨ ): set of such classes ∨ ∨ Φe (G ∨ )s = Sc −1 (s∨ L ), a Bernstein component of Φe (G )
Lemma Any Bernstein component of Φe (G ∨ ) is relevant for a unique inner twist of G∗ G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
13 / 25
Bernstein components for L-parameters
Example Suppose: L∨ = maximal torus of G ∨ φ1 (IF × SL2 (C)) = 1, φ1 (FrobF ) ∈ L∨ ∨ s∨ L = (L , φ1 , trivSφ1 )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Bernstein components for L-parameters
Example Suppose: L∨ = maximal torus of G ∨ φ1 (IF × SL2 (C)) = 1, φ1 (FrobF ) ∈ L∨ ∨ s∨ L = (L , φ1 , trivSφ1 ) Then: s∨ is relevant for the split form of G ∨ Φe (G ∨ )s = {(φ, ρ) ∈ Φe (G ∨ ) : φ(IF ) = 1, ρ appears in H∗ (B φ )} B φ = variety of Borel subgroups of G ∨ which contain the image of φ
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
14 / 25
Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φe (G ) o
F
/ Irr(G )
LLC
Sc
L∈Lev(G ) Φcusp (L)/W (G , L)
o
LLCcusp
/
F
Sc
L∈Lev(G ) Irrcusp (L)/W (G , L)
If this holds and LLCcusp is compatible with unramified twists, then LLC induces a bijection between Bernstein components
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
15 / 25
Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φe (G ) o
F
/ Irr(G )
LLC
Sc
L∈Lev(G ) Φcusp (L)/W (G , L)
o
LLCcusp
/
F
Sc
L∈Lev(G ) Irrcusp (L)/W (G , L)
If this holds and LLCcusp is compatible with unramified twists, then LLC induces a bijection between Bernstein components
Known for: inner twists of GLn (F ) and SLn (F ) (ABPS) Sp2n (F ) and SOn (F ) (Moussaoui) principal series representations of split groups (ABPS) unipotent representations of adjoint groups (Lusztig) Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Hecke algebras for Bernstein blocks of representations Example If G is F -split and I is an Iwahori subgroup, then H(G , I ) = Cc∞ (I \G /I ) is the affine Hecke algebra associated to the root datum of G ∨ and the parameter qF Mod(H(G , I )) is Morita equivalent to Rep(G )[T ,1]G
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Hecke algebras for Bernstein blocks of representations Example If G is F -split and I is an Iwahori subgroup, then H(G , I ) = Cc∞ (I \G /I ) is the affine Hecke algebra associated to the root datum of G ∨ and the parameter qF Mod(H(G , I )) is Morita equivalent to Rep(G )[T ,1]G
Conjecture For every inertial equivalence class s for G there exists a slight generalization Hs of an affine Hecke algebra, such that Rep(G )s ∼ = Mod(Hs ). There is a bijection Irr(G )s ←→ Irr(Hs )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Hecke algebras for Bernstein components of L-parameters ∨
Data from a Bernstein component Φe (G ∨ )s ∨ ∨ ◦ s∨ L = (L , Z (L ) φ, ρ)
torus Ts∨ := (Z (L∨ )◦ φ, ρ) ⊂ Φcusp (L) complex reductive group ZGsc∨ (φ(IF )) finite group Ws∨ = StabW (G ∨ ,L∨ ) (s∨ L)
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Hecke algebras for Bernstein components of L-parameters ∨
Data from a Bernstein component Φe (G ∨ )s ∨ ∨ ◦ s∨ L = (L , Z (L ) φ, ρ)
torus Ts∨ := (Z (L∨ )◦ φ, ρ) ⊂ Φcusp (L) complex reductive group ZGsc∨ (φ(IF )) finite group Ws∨ = StabW (G ∨ ,L∨ ) (s∨ L)
Example L∨ = G ∨ , s∨ = s∨ L is cuspidal algebra: O(Ts∨ )
Example L∨ = maximal torus, uφ = 1, Ws∨ = W (G ∨ , L∨ ) algebra: H(Ts∨ , Ws∨ , v), the affine Hecke algebra for the root datum of (G ∨ , L∨ ) and the single parameter qF = v2 Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Hecke algebras for Bernstein components of L-parameters Theorem Canonically associated to s∨ is an algebra H(Ts∨ , Ws∨ , v) such that: H(Ts∨ , Ws∨ , v) is an extension of an affine Hecke algebra by a finite dimensional algebra For every v ∈ R>0 there is a canonical bijection ∨ Φe (G ∨ )s ←→ Irr H(Ts∨ , Ws∨ , v)/(v − v )
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
18 / 25
Hecke algebras for Bernstein components of L-parameters Theorem Canonically associated to s∨ is an algebra H(Ts∨ , Ws∨ , v) such that: H(Ts∨ , Ws∨ , v) is an extension of an affine Hecke algebra by a finite dimensional algebra For every v ∈ R>0 there is a canonical bijection ∨ Φe (G ∨ )s ←→ Irr H(Ts∨ , Ws∨ , v)/(v − v ) ∨
Mod H(Ts∨ , Ws∨ , v) provides a categorification of Φe (G ∨ )s
The module category depends only on ZGsc∨ (φ(IF )) and some cuspidal data. This explains many equivalences between different Bernstein blocks The Galois side of the LLC can be phrased entirely in terms of cuspidal L-parameters and the groups Ws∨ Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Overview Rep(G )s
JLG
/ Rep(L)sL /Ws O
Sc
/ Irrcusp (L)sL /Ws O
O
Irr(G )s O
LLCcusp
LLC
∨
Φe (G )s
Sc
/ Φcusp (L)s∨L /Ws∨
Res O(Ts∨ ) / Mod O(Ts∨ ) /Ws∨ Mod H(Ts∨ , Ws∨ , v)
This helps to reduce the proof of the LLC to the cuspidal case. Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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1/2
Specialization at v = qF
JLG
Rep(G )s
/ Rep(L)sL O
O
? 1/2
Mod H(Ts∨ , Ws∨ , v)/(v − qF )
Maarten Solleveld
ResO(T
s∨ )
?
/ Mod O(Ts∨ )
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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1/2
Specialization at v = qF
JLG
Rep(G )s
/ Rep(L)sL O
O
? 1/2
Mod H(Ts∨ , Ws∨ , v)/(v − qF )
ResO(T
s∨ )
?
/ Mod O(Ts∨ )
Specialization at v = 1 Irr(G )s
Sc
O
LLC
∨
Φe (G )s
Sc
O
Irr O(Ts∨ ) o C[Ws∨ , \s∨ ] Maarten Solleveld
/ Irrcusp (L)sL /Ws O
LLCcusp
/ Φcusp (L)s∨L /Ws∨ O / Ts∨ /Ws∨
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Generalizations of the Springer correspondence Unipe (H) = (u, ρ) : u ∈ H unipotent, ρ ∈ Irr π0 (ZH (u)) /H-conjugacy Unipcusp (H) = cuspidal part of Unipe (H)
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Generalizations of the Springer correspondence Unipe (H) = (u, ρ) : u ∈ H unipotent, ρ ∈ Irr π0 (ZH (u)) /H-conjugacy Unipcusp (H) = cuspidal part of Unipe (H)
Theorem (Lusztig, 1984) 1
There exists a unique cuspidal support map G {L} × Unipcusp (L)/H◦ -conjugacy ScH◦ : Unipe (H◦ ) → Levi L
such that (u, ρ) appears in some induction of ScH◦ (u, ρ).
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Generalizations of the Springer correspondence Unipe (H) = (u, ρ) : u ∈ H unipotent, ρ ∈ Irr π0 (ZH (u)) /H-conjugacy Unipcusp (H) = cuspidal part of Unipe (H)
Theorem (Lusztig, 1984) 1
There exists a unique cuspidal support map G {L} × Unipcusp (L)/H◦ -conjugacy ScH◦ : Unipe (H◦ ) → Levi L
such that (u, ρ) appears in some induction of ScH◦ (u, ρ). 2
Let t = (L, v , ) be a cuspidal support for H◦ . There exists a canonical bijection −1 ◦ Σt : ScH ◦ (t) → Irr(W (H , L)),
realized in the cohomology of a certain sheaf. Maarten Solleveld
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Generalizations of the Springer correspondence Desired: Lusztig–Springer correspondence for disconnected complex reductive groups H
Modifications 1
Cuspidal support (L, v , ) is replaced by a cuspidal quasi-support qt = (qL, v , q), where: I I I
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qL ⊂ H is quasi-Levi: qL = ZH (Z (L)◦ ) for a Levi subgroup L ⊂ H◦ v ∈ qL◦ is unipotent q ∈ Irr π0 (ZH (v )) such that q|π0 (ZH◦ (v )) is a sum of cuspidal reps
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Generalizations of the Springer correspondence Desired: Lusztig–Springer correspondence for disconnected complex reductive groups H
Modifications 1
Cuspidal support (L, v , ) is replaced by a cuspidal quasi-support qt = (qL, v , q), where: I I I
2 3
qL ⊂ H is quasi-Levi: qL = ZH (Z (L)◦ ) for a Levi subgroup L ⊂ H◦ v ∈ qL◦ is unipotent q ∈ Irr π0 (ZH (v )) such that q|π0 (ZH◦ (v )) is a sum of cuspidal reps
W (H◦ , L) is extended to W (H, qL, q) = NH (qL, q)/qL Irr(W (H, L)) is replaced by Irr(C[W (H, qL, q), \q ]), where \q : (W (H, qL, q)/W (H◦ , L))2 → C× is a 2-cocycle
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Theorem (Generalization of the Lusztig–Springer correspondence) Let H be a complex reductive group, possibly disconnected. 1
There exists a canonical cuspidal support map G ScH : Unipe (H) → {L} × Unipcusp (L)/H-conjugacy quasi-Levi L
ScCφ is used in the cuspidal support map for enhanced L-parameters
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Theorem (Generalization of the Lusztig–Springer correspondence) Let H be a complex reductive group, possibly disconnected. 1
There exists a canonical cuspidal support map G ScH : Unipe (H) → {L} × Unipcusp (L)/H-conjugacy quasi-Levi L
2
Let qt = (qL, v , q) be a cuspidal quasi-support for H. There exists a (almost canonical) bijection −1 Σqt : ScH (qt) → Irr(C[W (H, qL, q), \q ]),
realized in the cohomology of a certain sheaf. ScCφ is used in the cuspidal support map for enhanced L-parameters Sometimes the 2-cocycle \q is nontrivial. Used in H(Ts∨ , Ws∨ , v) Maarten Solleveld
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Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
3
ψ = φ on the inertia group IF ⊂ Gal(F /F ) 1/2 1/2 qF 0 q ψ FrobF , = φ FrobF , F −1/2
4
0
5
qF
0
0
−1/2
qF
(Cφ ∩ L∨ sc , uψ , ) is the cuspidal support of (uφ , ρ), for the group Cφ = ZGsc∨ (φ(WF ))
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Summary ∨
A Bernstein component Φe (G ∨ )s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
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Summary ∨
A Bernstein component Φe (G ∨ )s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
To every s∨ we can attach H(Ts∨ , Ws∨ , v), which is almost an affine Hecke algebra. For every v ∈ R>0 G G Φe (G ∨ ) = Irr H(Ts∨ , Ws∨ , v)/(v−v ) inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
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Summary ∨
A Bernstein component Φe (G ∨ )s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
To every s∨ we can attach H(Ts∨ , Ws∨ , v), which is almost an affine Hecke algebra. For every v ∈ R>0 G G Φe (G ∨ ) = Irr H(Ts∨ , Ws∨ , v)/(v−v ) inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
Can the local Langlands correspondence be categorified to G Rep(G ) ∼ Irr H(Ts∨ , Ws∨ , v) ? G -relevant s∨ ∈Ω(G ∨ )
Is there some sheaf with endomorphism algebra H(Ts∨ , Ws∨ , v) ? Maarten Solleveld
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12 April 2016
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Some aspects of the local Langlands program
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p-adic side
Galois side
reductive p-adic group irreducible admissible reps supercuspidal reps Bernstein components affine Hecke algebras
Weil–Deligne group enhanced L-parameters ? ? ?
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Some aspects of the local Langlands program p-adic side
Galois side
reductive p-adic group irreducible admissible reps supercuspidal reps Bernstein components affine Hecke algebras
Weil–Deligne group enhanced L-parameters ? ? ?
Goal of talk Define all this on the Galois side, so that it matches via the local Langlands correspondence
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enhanced Langlands parameters Notations F : non-archimedean local field G = G(F ): connected reductive group over F G ∨ = G ∨ (C): complex dual group WF ⊂ Gal(F /F ): Weil group of F Assumption: G inner twist of F -split group G ∗
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enhanced Langlands parameters Notations F : non-archimedean local field G = G(F ): connected reductive group over F G ∨ = G ∨ (C): complex dual group WF ⊂ Gal(F /F ): Weil group of F Assumption: G inner twist of F -split group G ∗
Definition A Langlands parameter for G ∨ is an ”admissible” homomorphism φ : WF × SL2 (C) → G ∨ ∨ : simply connected cover of G ∨ Gsc der Sφ = π0 (ZGsc∨ (φ)) An enhancement of φ is an irrep ρ of Sφ Φe (G ∨ ) = {enhanced L-parameters (φ, ρ)}/G ∨ -conjugation Maarten Solleveld
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Local Langlands Correspondence Definition (φ, ρ) ∈ Φe (G ∨ ) is relevant for G if ρ|Z (Gsc∨ ) is the Kottwitz parameter of G as an inner twist of a split group G ∗ Notation: Φe (G ) ⊂ Φe (G ∨ )
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Local Langlands Correspondence Definition (φ, ρ) ∈ Φe (G ∨ ) is relevant for G if ρ|Z (Gsc∨ ) is the Kottwitz parameter of G as an inner twist of a split group G ∗ Notation: Φe (G ) ⊂ Φe (G ∨ )
Conjecture (Langlands, Borel, Vogan...) There exists a bijection G
Irr(G ) ←→ Φe (G ∨ )
inner twists G of G ∗
which satisfies many nice properties, e.g. π ∈ Irr(G ) is essentially square-integrable (i.e. π|Gder is square-integrable) if and only if φπ is discrete (i.e. not a L-parameter for any proper Levi subgroup of G ) Maarten Solleveld
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Cuspidality for L-parameters Notations (φ, ρ) enhanced L-parameter for G ∨ Cφ = ZGsc∨ (φ(WF)): complex reductive group, possibly disconnected uφ = φ 1, ( 10 11 ) : unipotent element of Cφ◦
Lemma (Kazhdan–Lusztig) Natural isomorphism Sφ = π0 ZGsc∨ (φ(WF × SL2 (C))) → π0 (ZCφ (uφ ))
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Cuspidality for L-parameters Notations (φ, ρ) enhanced L-parameter for G ∨ Cφ = ZGsc∨ (φ(WF)): complex reductive group, possibly disconnected uφ = φ 1, ( 10 11 ) : unipotent element of Cφ◦
Lemma (Kazhdan–Lusztig) Natural isomorphism Sφ = π0 ZGsc∨ (φ(WF × SL2 (C))) → π0 (ZCφ (uφ ))
Definition (φ, ρ) ∈ Φe (G ∨ ) is cuspidal if: φ is discrete; (uφ , ρ) is a cuspidal pairfor Cφ . i.e. ρ ∈ Irr π0 (ZCφ (uφ )) and (uφ , ρ) cannot be obtained from a proper Levi subgroup of Cφ via a certain induction procedure Notation: Φcusp (G ∨ ) ⊂ Φe (G ∨ ) Maarten Solleveld
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Cuspidal L-parameters for GLn (F ) By classification: (u, ρ) is a cuspidal pair for GLn (C) with ρ|Z (SLn (C)) = 1 ⇐⇒ n = 1, u = 1 and ρ = 1
Lemma (φ, ρ) ∈ Φe (GLn (F )) is cuspidal ⇐⇒ φ|SL2 (C) = 1, ρ = 1 and φ|WF is a n-dim irreducible rep
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Cuspidal L-parameters for GLn (F ) By classification: (u, ρ) is a cuspidal pair for GLn (C) with ρ|Z (SLn (C)) = 1 ⇐⇒ n = 1, u = 1 and ρ = 1
Lemma (φ, ρ) ∈ Φe (GLn (F )) is cuspidal ⇐⇒ φ|SL2 (C) = 1, ρ = 1 and φ|WF is a n-dim irreducible rep
Proof ∨ ). ⇐ By the irreducibility, φ is discrete and ZGsc∨ (φ(WF )) = Z (Gsc ∨ The pair (u = 1, ρ = 1) is cuspidal for Z (Gsc ) = Z (SLn (C))
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Cuspidal L-parameters for GLn (F ) By classification: (u, ρ) is a cuspidal pair for GLn (C) with ρ|Z (SLn (C)) = 1 ⇐⇒ n = 1, u = 1 and ρ = 1
Lemma (φ, ρ) ∈ Φe (GLn (F )) is cuspidal ⇐⇒ φ|SL2 (C) = 1, ρ = 1 and φ|WF is a n-dim irreducible rep
Proof ∨ ). ⇐ By the irreducibility, φ is discrete and ZGsc∨ (φ(WF )) = Z (Gsc ∨ The pair (u = 1, ρ = 1) is cuspidal for Z (Gsc ) = Z (SLn (C)) ⇒ ρ = 1 because G = GLn (F ) 1
Since φ is discrete, it is a n-dim irreducible rep of WF × SL2 (C)
2
Cn = V1 ⊗ V2 with V1 ∈ Irr(WF ) and V2 ∈ Irr(SL2 (C))
3
Cuspidality forces uφ = 1, hence V2 is the trivial SL2 (C)-rep
4
Cn = V1 ∈ Irr(WF )
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Cuspidal support of representations Theorem (Bernstein, 1984) Let π ∈ Irr(G ) = { irreducible smooth G -reps over C} There exist a parabolic subgroup P = L n U of G and a σ ∈ Irrcusp (L) such that π is a subquotient of the normalized parabolic induction IPG (σ) π determines the pair (L, σ) uniquely up to G -conjugation
Definition The cuspidal support map for G is G Sc : Irr(G ) → {L} × Irrcusp (L) G -conjugation Levi L
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Cuspidal support of representations Theorem (Bernstein, 1984) Let π ∈ Irr(G ) = { irreducible smooth G -reps over C} There exist a parabolic subgroup P = L n U of G and a σ ∈ Irrcusp (L) such that π is a subquotient of the normalized parabolic induction IPG (σ) π determines the pair (L, σ) uniquely up to G -conjugation
Alternative presentation of the cuspidal support map Lev(G ): representatives for the conjugacy classes of Levi subgroups of G W (G , L) = NG (L)/L G Sc : Irr(G ) → {L} × Irrcusp (L)/W (G , L) L∈Lev(G )
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Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
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Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
3
ψ = φ on the inertia group IF ⊂ Gal(F /F ) 1/2 1/2 qF 0 q ψ FrobF , = φ FrobF , F −1/2
4
0
qF
0
0
−1/2 qF
(3) and (4) say that Sc preserves infinitesimal characters
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Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
3
ψ = φ on the inertia group IF ⊂ Gal(F /F ) 1/2 1/2 qF 0 q ψ FrobF , = φ FrobF , F −1/2
4
0
5
qF
0
0
−1/2 qF
L∨ sc , uψ , )
(Cφ ∩ is the cuspidal support of (uφ , ρ), for the group Cφ = ZGsc∨ (φ(WF )) (3) and (4) say that Sc preserves infinitesimal characters If (φ, ρ) ∈ Φcusp (G ), then by (5): Sc(φ, ρ) = (G ∨ , φ, ρ)
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Examples of the cuspidal support map G = GL5m (F ) φ ∈ Φ(G ), ρ = 1 Suppose φ = φ1 ⊗ (R2 ⊕ R2 ⊕ R1 ) with φ1 ∈ Irr(WF ) and Ri = i-dim irrep of SL2 (C) L∨ = GLm (C)5
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Examples of the cuspidal support map G = GL5m (F ) φ ∈ Φ(G ), ρ = 1 Suppose φ = φ1 ⊗ (R2 ⊕ R2 ⊕ R1 ) with φ1 ∈ Irr(WF ) and Ri = i-dim irrep of SL2 (C) L∨ = GLm (C)5 for w ∈ IF , x ∈ SL2 (C) : ψ(w , x) = φ1 (w ) ⊗ I5 = φ(w ) 1/2
−1/2
ψ(FrobF ) = φ(FrobF ) ⊗ (qF , qF Then Sc(φ, ρ) =
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(L∨ , ψ,
1/2
−1/2
, qF , qF
, 1)
= 1)
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Examples of the cuspidal support map G = GL5m (F ) φ ∈ Φ(G ), ρ = 1 Suppose φ = φ1 ⊗ (R2 ⊕ R2 ⊕ R1 ) with φ1 ∈ Irr(WF ) and Ri = i-dim irrep of SL2 (C) L∨ = GLm (C)5 for w ∈ IF , x ∈ SL2 (C) : ψ(w , x) = φ1 (w ) ⊗ I5 = φ(w ) 1/2
−1/2
ψ(FrobF ) = φ(FrobF ) ⊗ (qF , qF Then Sc(φ, ρ) =
(L∨ , ψ,
1/2
−1/2
, qF , qF
, 1)
= 1)
This works also for GLn (F ) It fits with the Zelevinsky classification of Irr(GLn (F ))
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Examples of the cuspidal support map G = G2 (F ) φ|WF = 1, uφ = subregular unipotent ρ ∈ Irr π0 (ZG2 (C) (uφ )) ∼ = Irr(S3 ) if ρ = sign, then (φ, ρ) ∈ Φe (G ) is cuspidal
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Examples of the cuspidal support map G = G2 (F ) φ|WF = 1, uφ = subregular unipotent ρ ∈ Irr π0 (ZG2 (C) (uφ )) ∼ = Irr(S3 ) if ρ = sign, then (φ, ρ) ∈ Φe (G ) is cuspidal if ρ 6= sign, then Sc(φ, ρ) = (T ∨ , ψ, 1), where T is a maximalsplit torus of G and n/2 q 0 ψ(FrobnF w , x) = φ 1, F −n/2 for w ∈ IF 0
qF
The cuspidal support really depends on the enhancements of L-parameters
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Comparison of cuspidal support maps For L a Levi subgroup of G W (G , L) = NG (L)/L is isomorphic with NG ∨ (L∨ )/L∨ Provides an action of W (G , L) on Φcusp (L)
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Comparison of cuspidal support maps For L a Levi subgroup of G W (G , L) = NG (L)/L is isomorphic with NG ∨ (L∨ )/L∨ Provides an action of W (G , L) on Φcusp (L)
Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φe (G ) o
F
Sc
L∈Lev(G ) Φcusp (L)/W (G , L)
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/ Irr(G )
LLC
o
LLC
/
F
Sc
L∈Lev(G ) Irrcusp (L)/W (G , L)
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Bernstein components for representations Xnr (G ): group of unramified characters G → C×
Definition Let σ ∈ Irrcusp (L) sL = (L, Xnr (L) ⊗ σ) ⊂ {L} × Irrcusp (L) s = [L, σ]G = G -conjugacy class of sL s is an inertial equivalence class for G Ω(G ): set of such classes Irr(G )s = Sc −1 (sL ), a Bernstein component of Irr(G )
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Bernstein components for representations Xnr (G ): group of unramified characters G → C×
Definition Let σ ∈ Irrcusp (L) sL = (L, Xnr (L) ⊗ σ) ⊂ {L} × Irrcusp (L) s = [L, σ]G = G -conjugacy class of sL s is an inertial equivalence class for G Ω(G ): set of such classes Irr(G )s = Sc −1 (sL ), a Bernstein component of Irr(G )
Theorem (Bernstein, 1984) F s Irr(G ) = Q s∈Ω(G ) Irr(G ) Rep(G ) = s∈Ω(G ) Rep(G )s
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Bernstein components for L-parameters G inner twist, so Xnr (G ) ∼ = Z (G ∨ )◦
Definition Let (φ, ρ) ∈ Φcusp (L∨ ) ∨ ∨ ◦ ∨ ∨ s∨ L = (L , Z (L ) φ, ρ) ⊂ {L } × Φcusp (L ) s∨ = [L∨ , φ, ρ]G ∨ = G ∨ -conjugacy class of s∨ L s∨ is an inertial equivalence class for Φe (G ∨ )
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Bernstein components for L-parameters G inner twist, so Xnr (G ) ∼ = Z (G ∨ )◦
Definition Let (φ, ρ) ∈ Φcusp (L∨ ) ∨ ∨ ◦ ∨ ∨ s∨ L = (L , Z (L ) φ, ρ) ⊂ {L } × Φcusp (L ) s∨ = [L∨ , φ, ρ]G ∨ = G ∨ -conjugacy class of s∨ L s∨ is an inertial equivalence class for Φe (G ∨ ) Ω(G ∨ ): set of such classes ∨ ∨ Φe (G ∨ )s = Sc −1 (s∨ L ), a Bernstein component of Φe (G )
Lemma Any Bernstein component of Φe (G ∨ ) is relevant for a unique inner twist of G∗ G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
Maarten Solleveld
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Bernstein components for L-parameters
Example Suppose: L∨ = maximal torus of G ∨ φ1 (IF × SL2 (C)) = 1, φ1 (FrobF ) ∈ L∨ ∨ s∨ L = (L , φ1 , trivSφ1 )
Maarten Solleveld
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Bernstein components for L-parameters
Example Suppose: L∨ = maximal torus of G ∨ φ1 (IF × SL2 (C)) = 1, φ1 (FrobF ) ∈ L∨ ∨ s∨ L = (L , φ1 , trivSφ1 ) Then: s∨ is relevant for the split form of G ∨ Φe (G ∨ )s = {(φ, ρ) ∈ Φe (G ∨ ) : φ(IF ) = 1, ρ appears in H∗ (B φ )} B φ = variety of Borel subgroups of G ∨ which contain the image of φ
Maarten Solleveld
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Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φe (G ) o
F
/ Irr(G )
LLC
Sc
L∈Lev(G ) Φcusp (L)/W (G , L)
o
LLCcusp
/
F
Sc
L∈Lev(G ) Irrcusp (L)/W (G , L)
If this holds and LLCcusp is compatible with unramified twists, then LLC induces a bijection between Bernstein components
Maarten Solleveld
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Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φe (G ) o
F
/ Irr(G )
LLC
Sc
L∈Lev(G ) Φcusp (L)/W (G , L)
o
LLCcusp
/
F
Sc
L∈Lev(G ) Irrcusp (L)/W (G , L)
If this holds and LLCcusp is compatible with unramified twists, then LLC induces a bijection between Bernstein components
Known for: inner twists of GLn (F ) and SLn (F ) (ABPS) Sp2n (F ) and SOn (F ) (Moussaoui) principal series representations of split groups (ABPS) unipotent representations of adjoint groups (Lusztig) Maarten Solleveld
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Hecke algebras for Bernstein blocks of representations Example If G is F -split and I is an Iwahori subgroup, then H(G , I ) = Cc∞ (I \G /I ) is the affine Hecke algebra associated to the root datum of G ∨ and the parameter qF Mod(H(G , I )) is Morita equivalent to Rep(G )[T ,1]G
Maarten Solleveld
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Hecke algebras for Bernstein blocks of representations Example If G is F -split and I is an Iwahori subgroup, then H(G , I ) = Cc∞ (I \G /I ) is the affine Hecke algebra associated to the root datum of G ∨ and the parameter qF Mod(H(G , I )) is Morita equivalent to Rep(G )[T ,1]G
Conjecture For every inertial equivalence class s for G there exists a slight generalization Hs of an affine Hecke algebra, such that Rep(G )s ∼ = Mod(Hs ). There is a bijection Irr(G )s ←→ Irr(Hs )
Maarten Solleveld
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Hecke algebras for Bernstein components of L-parameters ∨
Data from a Bernstein component Φe (G ∨ )s ∨ ∨ ◦ s∨ L = (L , Z (L ) φ, ρ)
torus Ts∨ := (Z (L∨ )◦ φ, ρ) ⊂ Φcusp (L) complex reductive group ZGsc∨ (φ(IF )) finite group Ws∨ = StabW (G ∨ ,L∨ ) (s∨ L)
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Hecke algebras for Bernstein components of L-parameters ∨
Data from a Bernstein component Φe (G ∨ )s ∨ ∨ ◦ s∨ L = (L , Z (L ) φ, ρ)
torus Ts∨ := (Z (L∨ )◦ φ, ρ) ⊂ Φcusp (L) complex reductive group ZGsc∨ (φ(IF )) finite group Ws∨ = StabW (G ∨ ,L∨ ) (s∨ L)
Example L∨ = G ∨ , s∨ = s∨ L is cuspidal algebra: O(Ts∨ )
Example L∨ = maximal torus, uφ = 1, Ws∨ = W (G ∨ , L∨ ) algebra: H(Ts∨ , Ws∨ , v), the affine Hecke algebra for the root datum of (G ∨ , L∨ ) and the single parameter qF = v2 Maarten Solleveld
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Hecke algebras for Bernstein components of L-parameters Theorem Canonically associated to s∨ is an algebra H(Ts∨ , Ws∨ , v) such that: H(Ts∨ , Ws∨ , v) is an extension of an affine Hecke algebra by a finite dimensional algebra For every v ∈ R>0 there is a canonical bijection ∨ Φe (G ∨ )s ←→ Irr H(Ts∨ , Ws∨ , v)/(v − v )
Maarten Solleveld
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Hecke algebras for Bernstein components of L-parameters Theorem Canonically associated to s∨ is an algebra H(Ts∨ , Ws∨ , v) such that: H(Ts∨ , Ws∨ , v) is an extension of an affine Hecke algebra by a finite dimensional algebra For every v ∈ R>0 there is a canonical bijection ∨ Φe (G ∨ )s ←→ Irr H(Ts∨ , Ws∨ , v)/(v − v ) ∨
Mod H(Ts∨ , Ws∨ , v) provides a categorification of Φe (G ∨ )s
The module category depends only on ZGsc∨ (φ(IF )) and some cuspidal data. This explains many equivalences between different Bernstein blocks The Galois side of the LLC can be phrased entirely in terms of cuspidal L-parameters and the groups Ws∨ Maarten Solleveld
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Overview Rep(G )s
JLG
/ Rep(L)sL /Ws O
Sc
/ Irrcusp (L)sL /Ws O
O
Irr(G )s O
LLCcusp
LLC
∨
Φe (G )s
Sc
/ Φcusp (L)s∨L /Ws∨
Res O(Ts∨ ) / Mod O(Ts∨ ) /Ws∨ Mod H(Ts∨ , Ws∨ , v)
This helps to reduce the proof of the LLC to the cuspidal case. Maarten Solleveld
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1/2
Specialization at v = qF
JLG
Rep(G )s
/ Rep(L)sL O
O
? 1/2
Mod H(Ts∨ , Ws∨ , v)/(v − qF )
Maarten Solleveld
ResO(T
s∨ )
?
/ Mod O(Ts∨ )
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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1/2
Specialization at v = qF
JLG
Rep(G )s
/ Rep(L)sL O
O
? 1/2
Mod H(Ts∨ , Ws∨ , v)/(v − qF )
ResO(T
s∨ )
?
/ Mod O(Ts∨ )
Specialization at v = 1 Irr(G )s
Sc
O
LLC
∨
Φe (G )s
Sc
O
Irr O(Ts∨ ) o C[Ws∨ , \s∨ ] Maarten Solleveld
/ Irrcusp (L)sL /Ws O
LLCcusp
/ Φcusp (L)s∨L /Ws∨ O / Ts∨ /Ws∨
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Generalizations of the Springer correspondence Unipe (H) = (u, ρ) : u ∈ H unipotent, ρ ∈ Irr π0 (ZH (u)) /H-conjugacy Unipcusp (H) = cuspidal part of Unipe (H)
Maarten Solleveld
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Generalizations of the Springer correspondence Unipe (H) = (u, ρ) : u ∈ H unipotent, ρ ∈ Irr π0 (ZH (u)) /H-conjugacy Unipcusp (H) = cuspidal part of Unipe (H)
Theorem (Lusztig, 1984) 1
There exists a unique cuspidal support map G {L} × Unipcusp (L)/H◦ -conjugacy ScH◦ : Unipe (H◦ ) → Levi L
such that (u, ρ) appears in some induction of ScH◦ (u, ρ).
Maarten Solleveld
Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April 2016
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Generalizations of the Springer correspondence Unipe (H) = (u, ρ) : u ∈ H unipotent, ρ ∈ Irr π0 (ZH (u)) /H-conjugacy Unipcusp (H) = cuspidal part of Unipe (H)
Theorem (Lusztig, 1984) 1
There exists a unique cuspidal support map G {L} × Unipcusp (L)/H◦ -conjugacy ScH◦ : Unipe (H◦ ) → Levi L
such that (u, ρ) appears in some induction of ScH◦ (u, ρ). 2
Let t = (L, v , ) be a cuspidal support for H◦ . There exists a canonical bijection −1 ◦ Σt : ScH ◦ (t) → Irr(W (H , L)),
realized in the cohomology of a certain sheaf. Maarten Solleveld
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Generalizations of the Springer correspondence Desired: Lusztig–Springer correspondence for disconnected complex reductive groups H
Modifications 1
Cuspidal support (L, v , ) is replaced by a cuspidal quasi-support qt = (qL, v , q), where: I I I
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qL ⊂ H is quasi-Levi: qL = ZH (Z (L)◦ ) for a Levi subgroup L ⊂ H◦ v ∈ qL◦ is unipotent q ∈ Irr π0 (ZH (v )) such that q|π0 (ZH◦ (v )) is a sum of cuspidal reps
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Generalizations of the Springer correspondence Desired: Lusztig–Springer correspondence for disconnected complex reductive groups H
Modifications 1
Cuspidal support (L, v , ) is replaced by a cuspidal quasi-support qt = (qL, v , q), where: I I I
2 3
qL ⊂ H is quasi-Levi: qL = ZH (Z (L)◦ ) for a Levi subgroup L ⊂ H◦ v ∈ qL◦ is unipotent q ∈ Irr π0 (ZH (v )) such that q|π0 (ZH◦ (v )) is a sum of cuspidal reps
W (H◦ , L) is extended to W (H, qL, q) = NH (qL, q)/qL Irr(W (H, L)) is replaced by Irr(C[W (H, qL, q), \q ]), where \q : (W (H, qL, q)/W (H◦ , L))2 → C× is a 2-cocycle
Maarten Solleveld
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Theorem (Generalization of the Lusztig–Springer correspondence) Let H be a complex reductive group, possibly disconnected. 1
There exists a canonical cuspidal support map G ScH : Unipe (H) → {L} × Unipcusp (L)/H-conjugacy quasi-Levi L
ScCφ is used in the cuspidal support map for enhanced L-parameters
Maarten Solleveld
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Theorem (Generalization of the Lusztig–Springer correspondence) Let H be a complex reductive group, possibly disconnected. 1
There exists a canonical cuspidal support map G ScH : Unipe (H) → {L} × Unipcusp (L)/H-conjugacy quasi-Levi L
2
Let qt = (qL, v , q) be a cuspidal quasi-support for H. There exists a (almost canonical) bijection −1 Σqt : ScH (qt) → Irr(C[W (H, qL, q), \q ]),
realized in the cohomology of a certain sheaf. ScCφ is used in the cuspidal support map for enhanced L-parameters Sometimes the 2-cocycle \q is nontrivial. Used in H(Ts∨ , Ws∨ , v) Maarten Solleveld
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Cuspidal support of enhanced L-parameters (φ, ρ) enhanced L-parameter for G
Definition The cuspidal support Sc(φ, ρ) is the G ∨ -conjugacy class of (L∨ , ψ, ), where: 1
L is a Levi subgroup of G
2
(ψ, ) is a cuspidal L-parameter for L
3
ψ = φ on the inertia group IF ⊂ Gal(F /F ) 1/2 1/2 qF 0 q ψ FrobF , = φ FrobF , F −1/2
4
0
5
qF
0
0
−1/2
qF
(Cφ ∩ L∨ sc , uψ , ) is the cuspidal support of (uφ , ρ), for the group Cφ = ZGsc∨ (φ(WF ))
Maarten Solleveld
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Summary ∨
A Bernstein component Φe (G ∨ )s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
Maarten Solleveld
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Summary ∨
A Bernstein component Φe (G ∨ )s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
To every s∨ we can attach H(Ts∨ , Ws∨ , v), which is almost an affine Hecke algebra. For every v ∈ R>0 G G Φe (G ∨ ) = Irr H(Ts∨ , Ws∨ , v)/(v−v ) inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
Maarten Solleveld
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Summary ∨
A Bernstein component Φe (G ∨ )s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists G G ∨ Φe (G ∨ ) = Φe (G ∨ )s inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
To every s∨ we can attach H(Ts∨ , Ws∨ , v), which is almost an affine Hecke algebra. For every v ∈ R>0 G G Φe (G ∨ ) = Irr H(Ts∨ , Ws∨ , v)/(v−v ) inner twists G of G ∗ G -relevant s∨ ∈Ω(G ∨ )
Can the local Langlands correspondence be categorified to G Rep(G ) ∼ Irr H(Ts∨ , Ws∨ , v) ? G -relevant s∨ ∈Ω(G ∨ )
Is there some sheaf with endomorphism algebra H(Ts∨ , Ws∨ , v) ? Maarten Solleveld
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