Local Langlands correspondence and examples of ABPS conjecture

Local Langlands correspondence and examples of ABPS conjecture Ahmed Moussaoui UPMC Paris VI - IMJ 23/08/203

Notation F non-archimedean local field : finite extension of Q p or F p ((t)) O F = {x F, v(x) 0}, ring of integers of F p F = {x F, v(x) > 0}, unique maximal ideal of O F k F = O F /p F F q, residual field of F (finite field) ϖ F O F uniformizer of F F separable algebraic closure of F

K/F finite extension ϖ F O K = ϖ e K O K,e ramification index of K/F k K /k F finite extension, f = [k K : k F ] residual degree of K/F K/F finite Galois extension, k K /k F is also Galois extension σ Gal(K/F), σ(o K ) = O K, σ(p K ) = p K I K/F Gal(K/F) Gal(k K /k F )

Definition An extension K/F is unramified if the ramification index e(k/f) = and k K /k F is separable. Theorem n,! F F n F unramified extension of degree n F n /F is Galois, k Fn F q n and Gal(F n /F) Gal(k Fn /k F ) φ n Gal(F n /F) (x x q ) Gal(k Fn /k F ) Φ n = φn, Φ n, Gal(F n /F) Z/nZ F ur = composite of all F n F ur /F is the unique maximal unramified extension of F Gal(F ur /F) lim Z/nZ = Ẑ n Φ F Gal(F ur /F), Φ F Fn = Φ n

Definition The absolute Galois group of F is Γ F = Gal(F/F) Γ F lim Gal(E/F) where E ranges over finite Galois extension such that F E F Topology on Γ F : Open neighborhood basis at the identity are Gal(F/K) with K/F finite extension Γ F is a compact Haussdorf profinite group

I F Gal(F/F) Gal(F ur /F)Ẑ 0 I F W F < Φ > ZZ 0 Definition The Weil group of F is the topological group, with underlying abstract group the inverse image in Γ F of < Φ >, such that : I F is an open subgroup of W F the topology on I F, as subspace of W F, coincides with its natural topology as subspace of Γ F

Let WF der = [W F, W F ] and WF ab = W F/WF der. Artin Reciprocity Map There is a canonical continuous group morphism with the following properties. a F : W F F a F induces a topological isomorphism W ab F F x W F is a geometric Frobenius, if and only if a F (x) is an uniformizer of F a F (I F ) = O F

The Artin Reciprocity Map induces a bijection between { irreducible representations of F } { admissible morphisms GL (F) GL (C) W F GL (C) } If T is a split torus over F, then { irreducible representations of T T GL (C) } { admissible morphisms W F T } (+ equality of L-functions) This is the local Langlands correspondence for GL() and for split tori.

For ψ : W F GL n (C) we can define a L-function by L(s, ψ) = det( ψ(φ) V I F q s ) Godement and Jacquet defined L-functions for the representations of GL n (F). For n = 2, for supercuspidal representations are for Steinberg representation (and their twist by character) involve one factor with q s for other involve two factors with q s

Definition The Weil-Deligne group is WD F = C W F where W F acts on C by wxw = w x for w W F, x C. A Langlands parameter for GL(n) is a continous morphism ψ : WD F GL n (C) such that the restriction to C is algebraic, the image of C consists of unipotent elements and the image of W F consists of semisimple elements. We can identify one Langlands parameter ψ to a pair (ρ, N) where ρ : W F GL n (C) is a continous morphism such that his image consists of semisimple elements and N gl n (C) is such that ρ(w)nρ(w) = w N for all w W F. ψ(z, w) = exp(zn)ρ(w), (z, w) WD F

( w /2 For w W F, let h w = w /2 ) SL 2 (C). There is an isomorphism ( between ) C and the subgroup G a of z SL 2 (C) of the matrices, z C. 0 The conjugation by h w in G a induce the same action as W F on C, and extend to SL 2 (C). We can consider the semidirect product SL 2 (C) W F with the product (x, w )(x 2, w 2 ) = (x h w x 2 h w, w w 2 ) This group is isomorphic to SL 2 (C) W F via the map (x, w) (xh w, w).

We denote W F = W F SL 2 (C). If a Langlands parameter is identified to (ρ, N), then it correspond ( ( )) to a morphism ψ : W F GL n(c) with ψ, = exp(n) and 0 ψ(w, h w ) = ρ(w).

Local Langlands correspondence (conjecture) Let G a split connected reductive group over F and Ĝ is Langlands dual. Let Π(G) the set of isomorphism classes of irreducible representations of G and Φ(G) the set of equivalence classes of admissible morphisms for G. There is finite-to-one map rec : Π(G) Φ(G) with these properties π Π(G) is essentially square-integrable, if and only if, the image of W F by φ π : W F Ĝ is not contained in any proper Levi subgroup of Ĝ. π Π(G) is tempered, if and only if, the image of W F by φ π : W F Ĝ is bounded π Π(G), χ a character of F, then rec(π χ) = rec(π)χ. Equality of L-functions and ε-factors.

Let J =... and J = For M GL 4 (F), τ M = J t MJ. Let G = Sp 4 (F) = {g SL 4 (F) t gj g = J } and B = minimal parabolic subgroup of Sp 4 (F). There are three parabolic subgroups of Sp 4 (F) : B, P, Q with respective Lévi T, M, L..

t t T = 2 t 2, t i F, T (F ) 2 t {( ) } g M = τ g, g GL 2 (F), M GL 2 (F) t L = g, t F, g SL 2 (F), L GL (F) SL 2 (F) t

The Weyl group of G is W = N G (T)/T S 2 (Z/2Z) 2, generated by a =, b =

Action of W on T and on their irreducible representations w w (t, t 2 ) w (χ χ 2 ) e (t, t 2 ) χ χ 2 a (t 2, t ) χ 2 χ b (t, t2 ) χ2 ab (t2 ) χ 2 χ ba (t 2, t ) 2 χ aba (t 2) χ χ 2 bab (t2 ) χ 2 χ abab (t 2 ) χ χ2

Let χ a character of F such that χ 2 O F and χ(ϖ F ) =, ζ a character of F such that ζ 2 O F =, ζ OF and ζ (ϖ F ) =. We consider two inertial classes s = [T, χ ] and t = [T, χ ζ ]. The isotropy subgroup of s is the subgroup of elements w in W such that there exists an unramified character φ ψ(t), w (χ ) = (χ ) φ. For s and t we have W s = W t = {, b} and T s = T t = ψ(t) (C ) 2.

The extended quotient is T s /W s = T s /W s T b s /Z W s (b) Let (t, t 2 ) T s. We have b (t, t 2 ) = (t, t 2 ) (t, t2 ) = (t, t 2 ) t 2 = ±. Ts b /W s = { (b, (z, )), z C } { (b, (z, )), z C } Remark : It is the same extended quotient for s and t

Theorem (Sally, Tadic) Let ψ, ψ 2 two characters of F. Then ψ ψ 2 is reducible, if and only if ψ ± = ν ± ψ 2 ψ i = ν ± ψ i is order two ψ ζ = ψ T ζ + ψ Tζ 2, Tζ i tempered representation of SL 2 (F) (ζ character of F of order two). ψ ν has two subquotients : ψ SL2 and ψ St SL2

s = [T, χ ] point of T s /W s cocharacter constituant representation (e, (z, z 2 )) (, ) ψ χ ψ 2 ψ χ ψ 2 (e, (z, q )) (, ) ψχ SL2 ψχ ν (b, (z, )) (, τ 2 ) ψχ St SL2 (e, (z, )) (, ) ψχ T ε ψχ ε (b, (z, )) (, ) ψχ T ε 2 T s /W s = (T s /W s ) u0 (T s /W s ) ue (T s /W s ) u0 = T s /W s Ts b /W s, cocharacter h u0 (τ) = (, ) (T s /W s ) ue = Ts b+ /W s, cocharacter h ue (τ) = (, τ 2 ) Remark : (e, (z, )) and (b, (z, )) are in the same L-packet

t = [T, χ ζ ] point of T t /W t cocharacter constituant representation (e, (z, z 2 )) (, ) ψ χ ψ 2 ψ χ ψ 2 (e, (z, )) (, ) ψχ T ζ ψχ ζ (b, (z, )) (, ) ψχ T ζ 2 (e, (z, )) (, ) ψχ T εζ (b, (z, )) (, ) ψχ T εζ 2 ψχ εζ T t /W t = (T t /W t ) u0 (T t /W t ) ue (T t /W t ) u0 = T s /W s T b s /W s, cocharacter h u0 (τ) = (, ) (T t /W t ) ue = Remark : (e, (z, )) and (b, (z, )) are in the same L-packet, for (e, (z, )) and (b, (z, )) is the same

The Langlands dual group of Sp 4 (F) is SO 5 (C), with SO 5 (C) = {g SL 5 (C), t gjg = J} and the maximal torus z z 2 T = z 2 z, z i C (C ) 2 The Lie algebra of SO 5 (C) is y x 2 x 3 x 4 0 x 2 y 2 x 23 0 x 4 so 5 (C) = x 3 x 32 0 x 23 x 3, x ij, y i C x 4 0 x 32 y 2 x 2 0 x 4 x 3 x 2 y

The Langlands parameters for χ χ 2 is ρ(χ, χ 2 ) : W F T The χ (w) χ 2 (w) w χ 2 (w) χ (w) Langlands parameters for the irreducible subquotients of ψ χ ψ 2 are (ρ(ψ χ, ψ 2 ), 0), (ρ(ψ χ, ν), 0) and (ρ(ψ χ, ν), N) with 0 0 N = 0 0 0

The Langlands parameters are also the morphisms from W F to SO 5 (C). For (ρ(ψ χ, ν), N) the corresponding morphism is φ : W F T (ψ χ)(w) (w, s) S 3 (s) (ψ χ)(w) where S 3 is the irreducible representation of dimension 3 of SL 2 (C). ( ( )) τ τ 2 We have φ, = τ τ 2

For the irreducible subquotient of ψ χ ψ 2 ζ there are (ρ(ψ χ, ψ 2 ζ ), 0). We find in this way the cocharacters and the decomposition.

s = [T, χ ] point of T s /Ws cocharacter Langands parameter constituant representation (e, (z, z 2 )) (, ) (ψ χ, ψ 2 ) T ψ χ ψ 2 ψ χ ψ 2 (e, (z, q )) (, ) (ψχ, ψχ ν) T SL2 ψχ ν (ψ χ)(w) (b, (z, )) (, τ 2 ) S 3 (s) ψχ St (ψ χ)(w) SL2 (e, (z, )) (, ) (ψχ, ψχ T ε) T ε ψχ ε (b, (z, )) (, ) (ψχ, ψχ T ε) T 2 ε

t = [T, χ ζ ] point of T t /W t cocharacter Langands parameter constituant representation (e, (z, z 2 )) (, ) (ψ χ, ψ 2 ) T ψ χ ψ 2 ψ χ ψ 2 (e, (z, )) (, ) (ψχ, ζ ) T ψχ T ζ ψχ ζ (b, (z, )) (, ) (ψχ, ζ ) T (e, (z, )) (, ) (ψχ, εζ ) T (b, (z, )) (, ) (ψχ, εζ ) T ψχ T ζ 2 ψχ T εζ ψχ T εζ 2 ψχ εζ

Thank you for your attention.