Cuspidality and Hecke algebras for Langlands parameters
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1 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 / 25
2 Some aspects of the local Langlands program p-adic side Galois side reductive p-adic group Weil Deligne group irreducible admissible reps enhanced L-parameters supercuspidal reps? Bernstein components? affine Hecke algebras? Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
3 Some aspects of the local Langlands program p-adic side Galois side reductive p-adic group Weil Deligne group irreducible admissible reps enhanced L-parameters supercuspidal reps? Bernstein components? affine Hecke algebras? 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 / 25
4 enhanced Langlands parameters Notations F : non-archimedean local field G = G(F ): connected reductive group over F G = G (C): complex dual group W F 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 / 25
5 enhanced Langlands parameters Notations F : non-archimedean local field G = G(F ): connected reductive group over F G = G (C): complex dual group W F 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 : W F SL 2 (C) G G sc: simply connected cover of G der S φ = π 0 (Z G sc (φ)) An enhancement of φ is an irrep ρ of S φ Φ e (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 / 25
6 Local Langlands Correspondence Definition (φ, ρ) Φ e (G ) is relevant for G if ρ Z(G sc ) is the Kottwitz parameter of G as an inner twist of a split group G Notation: Φ e (G) Φ e (G ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
7 Local Langlands Correspondence Definition (φ, ρ) Φ e (G ) is relevant for G if ρ Z(G sc ) is the Kottwitz parameter of G as an inner twist of a split group G Notation: Φ e (G) Φ e (G ) Conjecture (Langlands, Borel, Vogan...) There exists a bijection inner twists G of G Irr(G) Φ e (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 φ π 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 / 25
8 Cuspidality for L-parameters Notations (φ, ρ) enhanced L-parameter for G C φ = Z G sc (φ(w F )): complex reductive group, possibly disconnected u φ = φ ( 1, ( ) ) : unipotent element of C φ Lemma (Kazhdan Lusztig) Natural isomorphism S φ = π 0 ( ZG sc (φ(w F SL 2 (C))) ) π 0 (Z Cφ (u φ )) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
9 Cuspidality for L-parameters Notations (φ, ρ) enhanced L-parameter for G C φ = Z G sc (φ(w F )): complex reductive group, possibly disconnected u φ = φ ( 1, ( ) ) : unipotent element of C φ Lemma (Kazhdan Lusztig) Natural isomorphism S φ = π 0 ( ZG sc (φ(w F SL 2 (C))) ) π 0 (Z Cφ (u φ )) Definition (φ, ρ) Φ e (G ) is cuspidal if: φ is discrete; (u φ, ρ) is a cuspidal pair for C φ. i.e. ρ Irr ( π 0 (Z Cφ (u φ )) ) and (u φ, ρ) cannot be obtained from a proper Levi subgroup of C φ via a certain induction procedure Notation: Φ cusp (G ) Φ e (G ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
10 Cuspidal L-parameters for GL n (F ) By classification: (u, ρ) is a cuspidal pair for GL n (C) with ρ Z(SLn(C)) = 1 n = 1, u = 1 and ρ = 1 Lemma (φ, ρ) Φ e (GL n (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 / 25
11 Cuspidal L-parameters for GL n (F ) By classification: (u, ρ) is a cuspidal pair for GL n (C) with ρ Z(SLn(C)) = 1 n = 1, u = 1 and ρ = 1 Lemma (φ, ρ) Φ e (GL n (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 Z G sc (φ(w F )) = Z(Gsc). The pair (u = 1, ρ = 1) is cuspidal for Z(Gsc) = Z(SL n (C)) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
12 Cuspidal L-parameters for GL n (F ) By classification: (u, ρ) is a cuspidal pair for GL n (C) with ρ Z(SLn(C)) = 1 n = 1, u = 1 and ρ = 1 Lemma (φ, ρ) Φ e (GL n (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 Z G sc (φ(w F )) = Z(Gsc). The pair (u = 1, ρ = 1) is cuspidal for Z(Gsc) = Z(SL n (C)) ρ = 1 because G = GL n (F ) 1 Since φ is discrete, it is a n-dim irreducible rep of W F SL 2 (C) 2 C n = V 1 V 2 with V 1 Irr(W F ) and V 2 Irr(SL 2 (C)) 3 Cuspidality forces u φ = 1, hence V 2 is the trivial SL 2 (C)-rep 4 C n = V 1 Irr(W F ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
13 Cuspidal support of representations Theorem (Bernstein, 1984) Let π Irr(G) = { irreducible smooth G-reps over C} There exist a parabolic subgroup P = L U of G and a σ Irr cusp (L) such that π is a subquotient of the normalized parabolic induction I G P (σ) π determines the pair (L, σ) uniquely up to G-conjugation Definition The cuspidal support map for G is Sc : Irr(G) {L} Irr cusp (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 / 25
14 Cuspidal support of representations Theorem (Bernstein, 1984) Let π Irr(G) = { irreducible smooth G-reps over C} There exist a parabolic subgroup P = L U of G and a σ Irr cusp (L) such that π is a subquotient of the normalized parabolic induction I G P (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) = N G (L)/L Sc : Irr(G) L Lev(G) {L} ( Irr cusp (L)/W (G, L) ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
15 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, ɛ), where: 1 L is a Levi subgroup of G 2 (ψ, ɛ) 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 / 25
16 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, ɛ), where: 1 L is a Levi subgroup of G 2 (ψ, ɛ) is a cuspidal L-parameter for L 3 ψ = φ on the inertia group I F Gal(F /F ) ) ( ) ( q = φ FrobF, 1/2 F 0 4 ψ ( Frob F, ( q 1/2 F 0 0 q 1/2 F 0 q 1/2 F ) ) (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 / 25
17 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, ɛ), where: 1 L is a Levi subgroup of G 2 (ψ, ɛ) is a cuspidal L-parameter for L 3 ψ = φ on the inertia group I F Gal(F /F ) ) ( ) ( q = φ FrobF, 1/2 F 0 4 ψ ( Frob F, ( q 1/2 F 0 0 q 1/2 F 0 q 1/2 F 5 (C φ L sc, u ψ, ɛ) is the cuspidal support of (u φ, ρ), for the group C φ = Z G sc (φ(w F )) ) ) (3) and (4) say that Sc preserves infinitesimal characters If (φ, ρ) Φ cusp (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 / 25
18 Examples of the cuspidal support map G = GL 5m (F ) φ Φ(G), ρ = 1 Suppose φ = φ 1 (R 2 R 2 R 1 ) with φ 1 Irr(W F ) and R i = i-dim irrep of SL 2 (C) L = GL m (C) 5 Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
19 Examples of the cuspidal support map G = GL 5m (F ) φ Φ(G), ρ = 1 Suppose φ = φ 1 (R 2 R 2 R 1 ) with φ 1 Irr(W F ) and R i = i-dim irrep of SL 2 (C) L = GL m (C) 5 for w I F, x SL 2 (C) : ψ(w, x) = φ 1 (w) I 5 = φ(w) ψ(frob F ) = φ(frob F ) (q 1/2 F, q 1/2 F, q 1/2 F, q 1/2 F, 1) Then Sc(φ, ρ) = (L, ψ, ɛ = 1) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
20 Examples of the cuspidal support map G = GL 5m (F ) φ Φ(G), ρ = 1 Suppose φ = φ 1 (R 2 R 2 R 1 ) with φ 1 Irr(W F ) and R i = i-dim irrep of SL 2 (C) L = GL m (C) 5 for w I F, x SL 2 (C) : ψ(w, x) = φ 1 (w) I 5 = φ(w) ψ(frob F ) = φ(frob F ) (q 1/2 F, q 1/2 F, q 1/2 F, q 1/2 F, 1) Then Sc(φ, ρ) = (L, ψ, ɛ = 1) This works also for GL n (F ) It fits with the Zelevinsky classification of Irr(GL n (F )) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui 12 April / 25
21 Examples of the cuspidal support map G = G 2 (F ) φ WF = 1, u φ = subregular unipotent ρ Irr ( π 0 (Z G2 (C)(u φ )) ) = Irr(S3 ) if ρ = sign, then (φ, ρ) Φ e (G) is cuspidal Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
22 Examples of the cuspidal support map G = G 2 (F ) φ WF = 1, u φ = subregular unipotent ρ Irr ( π 0 (Z G2 (C)(u φ )) ) = Irr(S3 ) if ρ = sign, then (φ, ρ) Φ e (G) is cuspidal if ρ sign, then Sc(φ, ρ) = (T, ψ, 1), where T is a maximal split torus ) of G and ) ψ(frob n F w, x) = φ( 1, for w IF ( q n/2 F 0 0 q n/2 F 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 Moussaoui12 April / 25
23 Comparison of cuspidal support maps For L a Levi subgroup of G W (G, L) = N G (L)/L is isomorphic with N G (L )/L Provides an action of W (G, L) on Φ cusp (L) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
24 Comparison of cuspidal support maps For L a Levi subgroup of G W (G, L) = N G (L)/L is isomorphic with N G (L )/L Provides an action of W (G, L) on Φ cusp (L) Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φ e (G) Sc L Lev(G) Φ cusp(l)/w (G, L) LLC LLC Irr(G) Sc L Lev(G) Irr cusp(l)/w (G, L) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
25 Bernstein components for representations X nr (G): group of unramified characters G C Definition Let σ Irr cusp (L) s L = (L, X nr (L) σ) {L} Irr cusp (L) s = [L, σ] G = G-conjugacy class of s L s is an inertial equivalence class for G Ω(G): set of such classes Irr(G) s = Sc 1 (s L ), a Bernstein component of Irr(G) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
26 Bernstein components for representations X nr (G): group of unramified characters G C Definition Let σ Irr cusp (L) s L = (L, X nr (L) σ) {L} Irr cusp (L) s = [L, σ] G = G-conjugacy class of s L s is an inertial equivalence class for G Ω(G): set of such classes Irr(G) s = Sc 1 (s L ), a Bernstein component of Irr(G) Theorem (Bernstein, 1984) Irr(G) = s Ω(G) Irr(G)s 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 Moussaoui12 April / 25
27 Bernstein components for L-parameters G inner twist, so X nr (G) = Z(G ) Definition Let (φ, ρ) Φ cusp (L ) s L = (L, Z(L ) φ, ρ) {L } Φ cusp (L ) s = [L, φ, ρ] G = G -conjugacy class of s L s is an inertial equivalence class for Φ e (G ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
28 Bernstein components for L-parameters G inner twist, so X nr (G) = Z(G ) Definition Let (φ, ρ) Φ cusp (L ) s L = (L, Z(L ) φ, ρ) {L } Φ cusp (L ) s = [L, φ, ρ] G = G -conjugacy class of s L s is an inertial equivalence class for Φ e (G ) Ω(G ): set of such classes Φ e (G ) s = Sc 1 (s L ), a Bernstein component of Φ e(g ) Lemma Any Bernstein component of Φ e (G ) is relevant for a unique inner twist of G Φ e (G ) = Φ e (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 Moussaoui12 April / 25
29 Bernstein components for L-parameters Example Suppose: L = maximal torus of G φ 1 (I F SL 2 (C)) = 1, φ 1 (Frob F ) L s L = (L, φ 1, triv Sφ1 ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
30 Bernstein components for L-parameters Example Suppose: L = maximal torus of G φ 1 (I F SL 2 (C)) = 1, φ 1 (Frob F ) L s L = (L, φ 1, triv Sφ1 ) Then: s is relevant for the split form of G Φ e (G ) s = {(φ, ρ) Φ e (G ) : φ(i F ) = 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 Moussaoui12 April / 25
31 Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φ e (G) Sc L Lev(G) Φ cusp(l)/w (G, L) LLC LLC cusp Irr(G) Sc L Lev(G) Irr cusp(l)/w (G, L) If this holds and LLC cusp 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 Moussaoui12 April / 25
32 Conjecture Let G be a connected reductive p-adic group. The local Langlands correspondence makes following diagram commute Φ e (G) Sc L Lev(G) Φ cusp(l)/w (G, L) LLC LLC cusp Irr(G) Sc L Lev(G) Irr cusp(l)/w (G, L) If this holds and LLC cusp is compatible with unramified twists, then LLC induces a bijection between Bernstein components Known for: inner twists of GL n (F ) and SL n (F ) (ABPS) Sp 2n (F ) and SO n (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 Moussaoui12 April / 25
33 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 q F 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 Moussaoui12 April / 25
34 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 q F 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 H s of an affine Hecke algebra, such that Rep(G) s = Mod(H s ). There is a bijection Irr(G) s Irr(H s ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
35 Hecke algebras for Bernstein components of L-parameters Data from a Bernstein component Φ e (G ) s s L = (L, Z(L ) φ, ρ) torus T s := (Z(L ) φ, ρ) Φ cusp (L) complex reductive group Z G sc (φ(i F )) finite group W s = Stab W (G,L )(s L ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
36 Hecke algebras for Bernstein components of L-parameters Data from a Bernstein component Φ e (G ) s s L = (L, Z(L ) φ, ρ) torus T s := (Z(L ) φ, ρ) Φ cusp (L) complex reductive group Z G sc (φ(i F )) finite group W s = Stab W (G,L )(s L ) Example L = G, s = s L algebra: O(T s ) is cuspidal Example L = maximal torus, u φ = 1, W s = W (G, L ) algebra: H(T s, W s, v), the affine Hecke algebra for the root datum of (G, L ) and the single parameter q F = v 2 Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
37 Hecke algebras for Bernstein components of L-parameters Theorem Canonically associated to s is an algebra H(T s, W s, v) such that: H(T s, W s, v) is an extension of an affine Hecke algebra by a finite dimensional algebra For every v R >0 there is a canonical bijection Φ e (G ) s Irr ( H(T s, W s, v)/(v v) ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
38 Hecke algebras for Bernstein components of L-parameters Theorem Canonically associated to s is an algebra H(T s, W s, v) such that: H(T s, W s, v) is an extension of an affine Hecke algebra by a finite dimensional algebra For every v R >0 there is a canonical bijection Φ e (G ) s Irr ( H(T s, W s, v)/(v v) ) Mod H(T s, W s, v) provides a categorification of Φ e (G ) s The module category depends only on Z G sc (φ(i F )) 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 W s Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
39 Overview Rep(G) s J G L Rep(L) s L/W s Irr(G) s Sc Irr cusp (L) s L/W s LLC LLC cusp Φ e (G) s Sc Φ cusp (L) s L /W s Mod ( Res H(T s, W s, v) ) O(Ts ) Mod ( O(T s ) ) /W s 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 Moussaoui12 April / 25
40 Specialization at v = q 1/2 F Rep(G) s J G L? Mod ( H(T s, W s, v)/(v q 1/2 F )) Res O(T s ) Rep(L) s L? Mod ( O(T s ) ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
41 Specialization at v = q 1/2 F Rep(G) s J G L? Mod ( H(T s, W s, v)/(v q 1/2 F )) Res O(T s ) Specialization at v = 1 Irr(G) s Sc LLC Φ e (G) s Sc Rep(L) s L? Mod ( O(T s ) ) Irr cusp (L) s L/W s LLC cusp Φ cusp (L) s L /W s Irr ( O(T s ) C[W s, s ] ) T s /W s Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
42 Generalizations of the Springer correspondence Unip e (H) = { (u, ρ) :u H unipotent, ρ Irr ( π 0 (Z H (u)) )} /H-conjugacy Unip cusp (H) = cuspidal part of Unip e (H) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
43 Generalizations of the Springer correspondence Unip e (H) = { (u, ρ) :u H unipotent, ρ Irr ( π 0 (Z H (u)) )} /H-conjugacy Unip cusp (H) = cuspidal part of Unip e (H) Theorem (Lusztig, 1984) 1 There exists a unique cuspidal support map Sc H : Unip e (H ) Levi L {L} Unip cusp (L)/H -conjugacy such that (u, ρ) appears in some induction of Sc H (u, ρ). Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
44 Generalizations of the Springer correspondence Unip e (H) = { (u, ρ) :u H unipotent, ρ Irr ( π 0 (Z H (u)) )} /H-conjugacy Unip cusp (H) = cuspidal part of Unip e (H) Theorem (Lusztig, 1984) 1 There exists a unique cuspidal support map Sc H : Unip e (H ) Levi L {L} Unip cusp (L)/H -conjugacy such that (u, ρ) appears in some induction of Sc H (u, ρ). 2 Let t = (L, v, ɛ) be a cuspidal support for H. There exists a canonical bijection Σ t : Sc 1 H (t) Irr(W (H, L)), realized in the cohomology of a certain sheaf. Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
45 Generalizations of the Springer correspondence Desired: Lusztig Springer correspondence for disconnected complex reductive groups H Modifications 1 Cuspidal support (L, v, ɛ) is replaced by a cuspidal quasi-support qt = (ql, v, qɛ), where: ql H is quasi-levi: ql = ZH (Z(L) ) for a Levi subgroup L H v ql is unipotent qɛ Irr ( π 0 (Z H (v)) ) such that qɛ π0(z H (v)) is a sum of cuspidal reps Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
46 Generalizations of the Springer correspondence Desired: Lusztig Springer correspondence for disconnected complex reductive groups H Modifications 1 Cuspidal support (L, v, ɛ) is replaced by a cuspidal quasi-support qt = (ql, v, qɛ), where: ql H is quasi-levi: ql = ZH (Z(L) ) for a Levi subgroup L H v ql is unipotent qɛ Irr ( π 0 (Z H (v)) ) such that qɛ π0(z H (v)) is a sum of cuspidal reps 2 W (H, L) is extended to W (H, ql, qɛ) = N H (ql, qɛ)/ql 3 Irr(W (H, L)) is replaced by Irr(C[W (H, ql, qɛ), qɛ ]), where qɛ : (W (H, ql, qɛ)/w (H, L)) 2 C is a 2-cocycle Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
47 Theorem (Generalization of the Lusztig Springer correspondence) Let H be a complex reductive group, possibly disconnected. 1 There exists a canonical cuspidal support map Sc H : Unip e (H) {L} Unip cusp (L)/H-conjugacy quasi-levi L Sc Cφ is used in the cuspidal support map for enhanced L-parameters Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
48 Theorem (Generalization of the Lusztig Springer correspondence) Let H be a complex reductive group, possibly disconnected. 1 There exists a canonical cuspidal support map Sc H : Unip e (H) {L} Unip cusp (L)/H-conjugacy quasi-levi L 2 Let qt = (ql, v, qɛ) be a cuspidal quasi-support for H. There exists a (almost canonical) bijection Σ qt : Sc 1 H (qt) Irr(C[W (H, ql, qɛ), qɛ]), realized in the cohomology of a certain sheaf. Sc Cφ is used in the cuspidal support map for enhanced L-parameters Sometimes the 2-cocycle qɛ is nontrivial. Used in H(T s, W s, v) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
49 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, ɛ), where: 1 L is a Levi subgroup of G 2 (ψ, ɛ) is a cuspidal L-parameter for L 3 ψ = φ on the inertia group I F Gal(F /F ) ) ( ) ( q = φ FrobF, 1/2 F 0 4 ψ ( Frob F, ( q 1/2 F 0 0 q 1/2 F 0 q 1/2 F 5 (C φ L sc, u ψ, ɛ) is the cuspidal support of (u φ, ρ), for the group C φ = Z G sc (φ(w F )) ) ) Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
50 Summary A Bernstein component Φ e (G ) s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists Φ e (G ) = Φ e (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 Moussaoui12 April / 25
51 Summary A Bernstein component Φ e (G ) s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists Φ e (G ) = Φ e (G ) s inner twists G of G G-relevant s Ω(G ) To every s we can attach H(T s, W s, v), which is almost an affine Hecke algebra. For every v R >0 Φ e (G ) = Irr ( H(T s, W s, v)/(v v) ) 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 Moussaoui12 April / 25
52 Summary A Bernstein component Φ e (G ) s consists of the enhanced L-parameters with the same cuspidal support, up to unramified twists Φ e (G ) = Φ e (G ) s inner twists G of G G-relevant s Ω(G ) To every s we can attach H(T s, W s, v), which is almost an affine Hecke algebra. For every v R >0 Φ e (G ) = Irr ( H(T s, W s, v)/(v v) ) inner twists G of G G-relevant s Ω(G ) Can the local Langlands correspondence be categorified to Rep(G) Irr ( H(T s, W s, v) )? G-relevant s Ω(G ) Is there some sheaf with endomorphism algebra H(T s, W s, v)? Maarten Solleveld Universiteit Nijmegen Cuspidality [3mm] joint andwith Hecke Anne-Marie algebras Aubert for Langlands and Ahmed parameters Moussaoui12 April / 25
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