A partition of the set of enhanced Langlands parameters of a reductive p-adic group joint work with Ahmed Moussaoui and Maarten Solleveld Anne-Marie Aubert Institut de Mathématiques de Jussieu - Paris Rive Gauche New Developments in Representation Theory National University of Singapore. Institute of Mathematical Sciences. 6-28 March 2016. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 1 / 31
Plan of the talk I 1 Motivation 2 The Harish-Chandra philosophy of cusp forms 3 Adaption of HC s philosophy to enhanced L-parameters 4 The generalized Springer correspondence for disconnected complex reductive groups 5 Plugging the GSC into the LLC Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 2 / 31
Motivation The L-group I The main actors They will be the enhanced Langlands parameters (in French : paramètres de Langlands enrichis) for p-adic reductive groups. Intuitively we will view them as labels (φ, ρ) with two arguments. Goal The goal is to extend the Harish-Chandra philosophy of cusp forms to enhanced Langlands parameters in a way that would be compatible with the local Langlands correspondence. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 3 / 31
Motivation The L-group II The Weil-Deligne group Let F be a local non-archimedean field, with absolute Weil group denoted by W F, a subgroup of the absolute Galois group Gal(F/F ). We have an exact sequence 1 I F Gal(F/F ) Gal(F q /F q ) 1 where I F is the inertia group and F q is the residual field of F. Then Gal(F q /F q ) Ẑ = p Z p and W F is the inverse image of Z under Gal(F/F ) Gal(F q /F q ). We define the Weil-Deligne group of F as W F := W F SL 2 (C). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 4 / 31
Motivation The L-group III The Langlands dual and the L-group of G Let G be the group of the F -rational points of a connected reductive algebraic group G defined over F. The Langlands dual of G (or G, all inner forms of G share the same dual group) is the group G = G is a complex Lie group with root system dual to that of G : G (X, R, X, R ) G (X, R, X, R). We have : GL n = GL n (C), SL n = PGL n (C), Sp 2n = SO 2n+1 (C), SO 2n+1 = Sp 2n (C), SO 2n = SO 2n (C), G = G(C) if G is of exceptional type. The L-group of G is L G := G W F. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 5 / 31
Motivation L-parameters and L-packets Langlands parameters (L-parameters, for short) are morphisms φ: W F L G subject to certain conditions. Let Φ(G, F ) denote the set of L-parameters. The group G is acting on Φ(G, F ) by conjugation. Let Irr(G) be the set of equivalence classes of irreducible smooth representations of G. The local Langlands correspondence predicts the existence of a partition of Irr(G) into finite subsets : Irr(G) = The Π φ are called L-packets. φ Φ(G,F ) Π φ (G). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 6 / 31
Motivation Enhanced L-parameters Definition/Question The π inside an L-packet are called L-indistinguishable. Q : How distinguish them? A : by putting a label on each. Creating the labels (Lusztig, Vogan, Arthur) The idea is to enhance φ by some additional datum : an irreducible representation ρ of a certain component group S φ in such a way that Π φ would be in bijection with a relevant subset Irr(S φ ) rel of the set of (equivalence classes of) irreducible representation of S φ. Let Φ(G, F ) e := {(φ, ρ) φ Φ(G, F ), ρ Irr(S φ ) rel } be the set of enhanced L-parameters for G. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 7 / 31
The Harish-Chandra philosophy of cusp forms The Harish-Chandra partition of Irr(G) I Harish-Chandra provided to us another way to partition Irr(G) (without passing through LLC). Parabolic Induction Let P be a parabolic subgroup of G, with unipotent radical U and Levi factor L (σ, V L ) be an irreducible smooth representation of L ( σ, V L ) its inflation to P I G L P (σ) := IndG P ( σ). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 8 / 31
The Harish-Chandra philosophy of cusp forms The Harish-Chandra partition of Irr(G) II Definition An irreducible smooth representation π of G that does not occur as a subquotient of any I G L P (σ) with P a proper parabolic subgroup of G is called supercuspidal. Theorem (Harish-Chandra) Every irreducible smooth representation π of G is contained in a parabolically induced representation I G L P (σ), with σ supercuspidal irreducible. The pair (L, σ) is unique up-to G-conjugation. Its G-conjugacy class is called the cuspidal support of π. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 9 / 31
The Harish-Chandra philosophy of cusp forms The Harish-Chandra partition of Irr(G) III Notation Let P(G) be the set of h = (L, σ) G with L Levi subgroup of G and σ supercuspidal irreducible representation of L, and let Sc: Irr(G) P(G) be the map sending π to its cuspidal support. Pictorial definition We call P(G) the Harish-Chandra color palette. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 10 / 31
The Harish-Chandra philosophy of cusp forms The Harish-Chandra partition of Irr(G) IV Theorem (Harish-Chandra) We have a partition Irr(G) = h P(G) where Irr(G) h := {π Irr(G) Sc(π) = h}. Irr h (G), An intuitive version of Harish-Chandra Theorem via painting : Think of the h as different colors : all the elements in Irr h (G) are painted in the color h. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 11 / 31
The Harish-Chandra philosophy of cusp forms The Harish-Chandra partition of Irr(G) V Remarks If G = GL n (F ), each L-packet contains a unique element. Also : Elements of the same color may belong to distinct L-packets (for instance, for G = GL n (F ), already for n = 2, take irreducible subquotients of a non-irreducible principal series). A given L-packet may contain elements of different colors (for instance, already for G = Sp 4 (F ) or G of type G 2, one have L-packets containing non-generic supercuspidal representations and non-supercuspidal representations). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 12 / 31
Adaption of HC s philosophy to enhanced L-parameters Results and Conjecture I [A-Moussaoui-Solleveld] (1) Find a intrinsic (that is, without using the LLC) way of coloring the elements of Φ(G, F ) e /G. (2) Color Matching Conjecture : Assume that the LLC holds for G. For every π Irr(G), let denote by (φ π, ρ π ) its image under the LLC. Then the colors in (1) match with Harish-Chandra s way of coloring, that is, π Irr(G) and (φ π, ρ π ) are of the same color, for all π Irr(G). The conjecture is true for split classical groups if char(f ) = 0 (Moussaoui, Ph. D thesis UPMC, June 2015). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 13 / 31
Adaption of HC s philosophy to enhanced L-parameters Results and Conjecture II Remark An obvious necessary condition for the Color Matching Conjecture to be true is that we take the same color palettes in both the group side (for coloring the elements of Irr(G)) and the Galois side (for coloring the elements of Φ(G, F )/G ) : that is, one needs to attach to Φ(G, F )/G a set of cuspidal data which would be in bijection with the Harish-Chandra palette P(G). Main tool : The generalized Springer correspondence for arbitrary (that is, possibly disconnected) reductive algebraic groups over C. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 14 / 31
The generalized Springer correspondence for disconnected complex reductive groups The GSC for connected complex Lie groups (Lusztig, 1984) I Main references : G. Lusztig, Intersection cohomology complexes on a reductive group, Inventiones Math. 75 (1984), 205-272. P. Achar, D. Juteau, A. Henderson, S. Riche, Modular generalized Springer correspondence I : The general linear group, J. Eur. Math. Soc. to appear. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 15 / 31
The generalized Springer correspondence for disconnected complex reductive groups The GSC for connected complex Lie groups (Lusztig, 1984) II Let H be a complex algebraic group, let Unip(H) be the unipotent variety of H, let DH b (Unip(H)) denote the constructible H-equivariant derived category defined by Bernstein and Lunts, and let Perv H (Unip(H)) be its subcategory of H-equivariant perverse sheaves on Unip(H). Let G be a complex connected reductive algebraic group. Recall that G has finitely many orbits in Unip(G) and every simple object in Perv G (Unip(G)) is of the form IC(O, E), where O Unip(G) is a a G-orbit and E is an irreducible G-equivariant local system on O. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 16 / 31
The generalized Springer correspondence for disconnected complex reductive groups The GSC for connected complex Lie groups (Lusztig, 1984) III The parabolic induction functor Let P = LU be a parabolic subgroup of G. Let m : Unip(P) Unip(G) and p : Unip(P) Unip(L) denote inclusion and projection, respectively. Then i G L P := γg P m! p : Perv L (Unip(L)) Perv G (Unip(G)), where γp G : Db P (Unip(G)) Db G (Unip(G)) denotes the integration functor (left adjoint of the forgetful functor), is called the parabolic induction functor. It commutes with Verdier duality. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 17 / 31
The generalized Springer correspondence for disconnected complex reductive groups The GSC for connected complex Lie groups (Lusztig, 1984) IV Remark 1 The functor i G L P is left adjoint to rg L P := p! m and right adjoint to r G L P := p m!. These functors are exchanged by Verdier duality. Remark 2 : If F L is a simple object in Perv L (Unip(L), then i G L P (F L) is semisimple. Definition (Achar, Henderson, Juteau, Riche, 2014) A simple object F in Perv G (Unip(G)) is cuspidal if for any simple object F L in Perv L (Unip(L)), F does not occur in i G L P (F L) (equivalently, if r G L P (F) = 0, resp, r G L P (F) = 0) for any proper parabolic subgroup P of G, with Levi factor L. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 18 / 31
The generalized Springer correspondence for disconnected complex reductive groups The GSC for connected complex Lie groups (Lusztig, 1984) V Remark 1 Cuspidality is preserved by Verdier duality. Remark 2 The above definition of cuspidality has been chosen here for the purpose of the talk, due to its similarity with Harish-Chandra s notion of cuspidality for irreducible representations of p-adic groups. It is equivalent to the definition of cuspidality given by Lusztig in 1984 in [loc. cit.]. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 19 / 31
The generalized Springer correspondence for disconnected complex reductive groups The enhancement of the unipotent variety I Notation For u a given unipotent element in G, let A G (u) denote the component group of its centralizer Z G (u). Let O = (u) G be the G-conjugacy class of u. We set A G (O) := A G (u). We will denote by ρ E ρ the bijection between Irr(A G (u)) and the irreducible G-equivariant local systems E on O = (u) G. Definitions (A., Moussaoui, Solleveld, 2015) We set Unip(G) e := {(O, ρ) : with O unipotent, ρ Irr(A G (O))}. We call Unip(G) e the enhancement of the unipotent variety of G, and call a pair (O, ρ) an enhanced unipotent class O. We say that an enhanced unipotent class (O, ρ) is cuspidal if the perverse sheaf IC(O, E ρ ) is cuspidal. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 20 / 31
The generalized Springer correspondence for disconnected complex reductive groups The enhancement of the unipotent variety II Proposition (Lusztig, 1984) If (O, ρ) is cuspidal, with O = (u) G, then u is a distinguished unipotent element in G (i.e., u does not meet Unip(L) for any proper Levi subgroup L). The Lusztig cuspidal support map for Unip(G) e Let P(Unip(G) e ) be the set of G-conjugacy classes of triples τ := (L, O L, ɛ), where L is a Levi subgroup of G, and (O L, ɛ) is a cuspidal enhanced unipotent class in L. Lusztig has defined a map Sc: Unip(G) e P(Unip(G) e ). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 21 / 31
The generalized Springer correspondence for disconnected complex reductive groups The enhancement of the unipotent variety III Theorem (Lusztig, 1984) For each τ P(Unip(G) e ), let Unip(G) τ e denotes the fiber of τ under Sc. Let W L := N G (L)/L. It is a Weyl group and Unip(G) e = τ P(Unip(G) e) where Unip(G) τ e is in bijection with Irr(W L ). Unip(G) τ e, Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 22 / 31
The generalized Springer correspondence for disconnected complex reductive groups The enhancement of the unipotent variety IV A generalized Springer correspondence for disconnected groups Remove the assumption of the connectedness of G. Let G be the identity component of G. Let u Unip(G ) and set O := (u) G. We have A G (u) A G (u). We extend the previous notation and terminology to the disconnected case. We say that an enhanced unipotent class (O, ρ) with O = (u) G and ρ Irr(A G (O)) is cuspidal if the restriction of ρ to A G (u) is a direct sum of cuspidal irreducible representations of A G (u). The color palette It is the set P(Unip(G) e ) of G-conjugacy classes of triples τ := (M, O M, ɛ), where M is a quasi-levi subgroup of G, (i.e., a subgroup of the form Z G ((Z(L) ), where L is a Levi subgroup of G ), and (O M, ɛ) is a cuspidal enhanced unipotent class in M. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 23 / 31
The generalized Springer correspondence for disconnected complex reductive groups The enhancement of the unipotent variety V Theorem (A, Moussaoui, Solleveld, 2015) The Lusztig cuspidal support map Sc extends to the disconnected case. Let τ P(Unip(G) e ). Then there exists a (non always trivial) 2-cocycle κ τ : W L /W L W L /W L C and a twisted algebra C[W L, κ τ ] such that Unip(G) e = τ L(G) Unip(G) τ e, where the fiber Unip(G) τ e of τ under Sc is in bijection with Irr(C[W L, κ τ ]). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 24 / 31
Plugging the GSC into the LLC Plugging the GSC into the LLC I Recall that G is a connected reductive group over F and L G = G W F its L-group. Let Gad be the adjoint group of G, and let Gsc be the simply connected cover of the derived group of Gad. For φ Φ(G, F ) e, let Z G ad (φ) be the centralizer of φ(w F ) in G ad. Arthur (2006) defined the group S φ as the component group of the inverse image in Gsc of Z G ad (φ). The adaptator : Let G φ be inverse image in Gsc of Z G ad (φ W F ). Set u φ = φ(1, ( 1 0 1 1 )) and O φ := (u φ ) G. Then we have S φ A Gφ (u φ ). Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 25 / 31
Plugging the GSC into the LLC Plugging the GSC into the LLC II The enhanced L-parameters for G We set Φ(G, F ) e := {(φ, ρ) φ Φ(G, F ), ρ Irr(S φ ) rel }, where Irr(S φ ) rel is the subset Irr(S φ ) of the relevant ρ s (i.e., that admit a certain character when restricted to the image of Z(G sc ) in S φ ). Definition (A., Moussaoui, Solleveld, 2015) An enhanced L-parameter (φ, ρ) Φ(G, F ) e is called cuspidal if φ(w F ) is not contained in any proper Levi subgroup of G, and (O φ, ρ) is a cuspidal enhanced unipotent class in G φ. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 26 / 31
Plugging the GSC into the LLC Plugging the GSC into the LLC III Conjecture (A-Moussaoui-Solleveld, 2015) The cuspidal enhanced Langlands parameters correspond to the irreducible supercuspidal representations of G by the LLC. The conjecture above is known to be true in the case of general linear groups and split classical groups (any representation) : Moussaoui, Centre de Bernstein enrichi pour les groupes classiques, arxiv :1511.02521 ; inner forms of linear groups and of special linear groups, quasi-split unitary groups (any representation), and also for unipotent representations of simple p-adic groups of adjoint type, and for Deligne-Lusztig depth-zero representations : A., Moussaoui, Solleveld, Generalizations of the Springer correspondence and cuspidal Langlands parameters, arxiv :1511.05335. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 27 / 31
Plugging the GSC into the LLC Plugging the GSC into the LLC IV The color palette on the Galois side : It is the set P(Φ(G, F ) e ) of G -conjugacy classes of quadruples l = (ZL G (Z(M φ ) 0 ), φ WF, O Mφ, ρ) such that M φ is quasi-levi subgroup of G φ, and (O Mφ, ρ) is a cuspidal enhanced unipotent class in G φ. Theorem (A-Moussaoui-Solleveld, 2015) One can define a cuspidal support map Sc: Φ(G, F ) e P(Φ(G, F ) e ), such that Φ(G, F ) e = Φ(G, F ) l e, l P(Φ(G,F ) e) where Φ(G, F ) l e is the fiber of l under Sc. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 28 / 31
Plugging the GSC into the LLC Plugging the GSC into the LLC V Corollary By taking a more restrictive palette of colors on both sides (meaning up-to twisting by unramified characters), one obtains a decomposition à la Bernstein of the set Φ(G, F ) e. Theorem (A, Moussaoui, Solleveld) One may define a Galois analog of the ABPS conjecture, and this analog holds true. For a more precise statement, see Theorem 9.3 in [A., Moussaoui, Solleveld, Generalizations of the Springer correspondence and cuspidal Langlands parameters, arxiv :1511.05335]. Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 29 / 31
Plugging the GSC into the LLC An example of almost perfect color matching... Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 30 / 31
Plugging the GSC into the LLC Thank you for your attention! Anne-Marie Aubert A partition of the set of enhanced Langlands parameters of a reductive p-adic group 31 / 31