Local Langlands correspondences in $\ell$-adic coefficients

Let $\ell$ be a prime number different from the residue characteristic of a non-archimedean local field $F$. We give formulations of $\ell$-adic local Langlands correspondences for connected reductive algebraic groups over $F$, which we conjecture to be independent of a choice of an isomorphism between the $\ell$-adic coefficient field and the complex number field.


Introduction
The local Langlands correspondence for a connected reductive algebraic group over a non-archimedean local field F is usually formulated with coefficients in C because of its relation with automorphic representations.On the other hand, when we discuss a realization of the local Langlands correspondence in ℓ-adic cohomology, we need a correspondence over Q ℓ , where ℓ is a prime number different from the residue characteristic of F .We can take an isomorphism ι : C ∼ → Q ℓ and use it to transfer the local Langlands correspondence over C to a local Langlands correspondence over Q ℓ .However, the obtained correspondence over Q ℓ depends on the choice of ι.In [BH06], Bushnell-Henniart formulated an ℓ-adic local Langlands correspondence for GL 2 , which is independent of a choice of an isomorphism C ∼ → Q ℓ by making some twists of L-parameters.The ℓ-adic local Langlands correspondence is suitable to describe the non-abelian Lubin-Tate theory in the sense that it is canonically defined over Q ℓ (cf.[IT22,5]).
In this paper, we discuss formulations of ℓ-adic local Langlands correspondences for general connected reductive groups.A natural idea is to make similar twists as GL 2 -case for L-parameters.However, we can not make such twists in general as explained in Example 2.1.A problem is that we do not have enough space inside the Langlands dual group to make an appropriate twist.To overcome this problem, we have two approaches.One is to introduce ℓ-adic C-parameters using C-groups, which make spaces for twists.Another is to introduce Tannakian ℓ-adic L-parameters that incorporate necessary twists in the realizations.Using these parameters, we formulate ℓ-adic local Langlands correspondences, which we conjecture to be independent of a choice of an isomorphism C ∼ → Q ℓ .We show that two conjectures formulated by using ℓ-adic C-parameters and Tannakian ℓ-adic L-parameters are equivalent.Further, we confirm that the conjectures are true for GL n and PGL n .The formulation of the ℓ-adic 2010 Mathematics Subject Classification.Primary: 11F70; Secondary: 11F80.local Langlands correspondence using Tannakian ℓ-adic L-parameters is motivated by the Kottwitz conjecture for local Shimura varieties in [RV14,Conjecture 7.4].In the number field case, a relation between C-groups and the Kottwitz conjecture for Shimura varieties is discussed in [Joh13].
In Section 1, we recall various versions of L-parameters and explain their relations.In Section 2, after explaining the problem, we give formulations of the ℓ-adic local Langlands correspondences introducing the ℓ-adic C-parameter and the Tannakian ℓ-adic L-parameter.
After we put a former version of this paper on arXiv, related papers [Ber20] and [Zhu20] appeared on arXiv.[Ber20] also explains a relation between C-groups and the local Langlands correspondence.In a similar philosophy as this paper, formulations of Satake isomorphisms using C-groups are explained in [Ber20] and [Zhu20].

Acknowledgments
The author would like to thank Teruhisa Koshikawa for his helpful comments.He is grateful to referees and an editor for their comments and fruitful suggestions.This work was supported by JSPS KAKENHI Grant Numbers JP18H01109, JP22H00093.

Notation
For a field F , let Γ F denote the absolute Galois group of F .For a non-archimedean local field F , let W F and I F denote the Weil group of F and its inertia subgroup.For a homomorphism φ : G → H of groups and g ∈ G, let φ g denote the homomorphism defined by φ g (g ′ ) = φ(gg ′ g −1 ) for g ′ ∈ G.For a group G, let Z(G) denote the center of G.For a group G, a subgroup H ⊂ G and a subset S ⊂ G, let Z H (S) denote the centralizer of S in H. Let Ad denote the adjoint action of a group, and ad denote the adjoint action of a Lie algebra.

Langlands parameters 1.Over fields of characteristic zero
Let F be a non-archimedean local field of residue characteristic p.Let q be the number of elements of the residue field of be the Artin reciprocity isomorphism normalized so that a uniformizer is sent to a lift of the geometric Frobenius element.For w ∈ W F , we put w) , where w denotes the image of w in W ab F .Let G be a connected reductive algebraic group over F .Let C be a field of characteristic 0. Let G be the dual group of G over C.
Definition 1.1.Let H be a group with action of W F .Let φ : H ⋊ W F → L G(C) be a homomorphism of groups over W F .
(1) We say that φ is semisimple if all the elements of φ(W F ) are semisimple.
(2) We say that φ is relevant if any parabolic subgroup P of L G containing Im φ is relevant in the sense of [Bor79,3.3].
We say that two homomorphisms H ⋊ W F → L G(C) of groups over W F are equivalent if they are conjugate by an element of G(C).
We say that a map from W F to a set is smooth if it is locally constant.
Lemma 1.2.Let φ : W F → L G(C) be a homomorphism of groups over W F such that p G • φ is smooth.Let σ q ∈ W F be a lift of the q-th power Frobenius element.Assume that φ(σ q ) is semisimple.Then φ is semisimple.
Proof.We write w ∈ W F as σ m q σ for m ∈ Z and σ ∈ I F .Since (p G • φ)(I F ) is finite, there is a positive integer d such that φ((σ m q σ) d ) = φ(σ dm q ).Then the claim follows, because φ(w) is semisimple if and only if φ(w) d is semisimple.
Let WD F = G a ⋊ W F be the Weil-Deligne group scheme over Q for F defined in [Del73,8.3.6].(2) A Weil-Deligne L-parameter for G over C is a Weil-Deligne L-homomorphism (τ, N) for G over C such that τ is semisimple and any parabolic subgroup P of L G containing τ (W F ) and satisfying N ∈ Lie(P ∩ G)(C) is relevant.
Lemma 1.6.Let (τ, N) be a Weil-Deligne L-homomorphism for G over C. Then N is an nilpotent element of Lie( G der (C)).
Proof.We take a finite separable extension F ′ of F that splits G.By the condition Ad(τ (w))N = |w|N for w ∈ W F ′ , we have N ∈ Lie( G der (C)).Further, the adjoint endomorphism ad(N) ∈ End(Lie( G der (C))) satisfies for w ∈ W F .Hence N is an nilpotent element.

Over C
Assume that C = C in this subsection.
Let Φ(G) denote the set of equivalence classes of L-parameters for G.
be the restriction of φ to SU 2 (R).The continuous homomorphism φ SU 2 extends to an algebraic morphism φ SL 2 : SL 2 (C) → H(C) uniquely by [OV90, 5.2.5 Theorem 11], since any continuous homomorphism between real Lie groups are differentiable.Let Lemma 1.12.Let G be a reductive group over C. Let s be a semisimple element of G(C) and u be a unipotent element of G(C) such that sus −1 = u q .Then there is an algebraic homomorphism Further, such θ is unique up to conjugation by Z G(C) ({s, u}).
Proof.The first claim is proved in the same way as [Hei06, Proposition 3.5].We recall the argument briefly.We take an algebraic homomorphism θ : SL 2 (C) → G(C) such that Then we can see that Hence, by replacing θ by its conjugate under S G ′ (C) (u), we may assume that (s, q 2 ) ∈ S G ′ (C) (θ).Then θ satisfies the conditions in the first claim.
The second claim is proved in the same way as [KL87, 2.4 (h)].
The following proposition is proved in [GR10, Proposition 2.2].We give a different proof here.
Proof.Let ξ be a Weil-Deligne L-parameter for G over C. Let H be the centralizer of ξ(I F ) in G. Then H is reductive by Lemma 1.11.Take a lift σ q ∈ W F of the q-th power Frobenius element.Then H is stable under conjugation by ξ(σ q ), since I F is a normal subgroup of W F .Take a positive integer m 0 such that Then G is a reductive group, because the identity component of G is equal to the identity component of H.We view H as an algebraic subgroup of G. Let s be the image of ξ(0, σ q ) in G(C).Since ξ is semisimple, the element s is semisimple.Let u ∈ G(C) be the image of 1 ∈ G a (C) under ξ.Then u belongs to H(C).Hence we can view u as an element of G(C).By Lemma 1.12, there is a morphism (1.1) Since θ factors through the identity component of G(C), it factors through H(C).
Remark 1.14.The bijectivity in Proposition 1.13 does not hold in general if we drop the Frobenius semisimplicity conditions from the both sides (cf.[BMIY24, Example 3.5]).

Over Q ℓ
Let ℓ be a prime number different from p. Assume that C = Q ℓ in this subsection.
Definition 1.15.(1) An ℓ-adic L-homomorphism for G is a continuous homomorphism ϕ : of groups over W F .We say that an ℓ-adic L-homomorphism ϕ for G is Frobeniussemisimple if ϕ(σ q ) is semisimple for any lift σ q ∈ W F of the q-th power Frobenius element.
Let L ℓ (G) denote the set of the equivalence classes of ℓ-adic L-homomorphisms.Let Φ ℓ (G) denote the set of the equivalence classes of ℓ-adic L-parameters of G.
Let t ℓ : I F → Z ℓ (1) be the ℓ-adic tame character.We take an isomorphism Z ℓ (1) ≃ Z ℓ and let t ′ ℓ : Let ξ be an L-homomorphism of Weil-Deligne type for G over Q ℓ .We take a lift σ q ∈ W F of the q-th power Frobenius element and define ϕ ξ : for m ∈ Z and σ ∈ I F .
Proof.Let σ ′ q be another choice of a lift of the q-th power Frobenius element.Then σ ′ q = σ q σ ′ for some σ ′ ∈ I F .We define ϕ ′ ξ similarly as ϕ ξ using σ ′ q instead of σ q .We put be a homomorphism obtained from another choice of an isomorphism Z ℓ (1) ≃ Z ℓ .
Then we have t ′′ ℓ = ut ′ ℓ for some u ∈ Z × ℓ .Take a positive integer m 0 such that σ m 0 q commutes with G in L G and for any σ ∈ I F .We put h 0 = (p G • ξ)(0, σ m 0 q ).If ξ| Ga(Q ℓ ) is trivial, there is nothing to prove.Hence we assume that ξ| Ga(Q ℓ ) is non-trivial.Let U ξ be the algebraic subgroup of G defined by ξ(G a (Q ℓ )).Let H be the intersection of the normalizer of U ξ in G and the centralizer of ξ(W F ) in G.We have Ad(h 0 )(ξ(a, 1)) = Ad(ξ(σ m 0 q ))(ξ(a, 1)) = ξ(q m 0 a, 1) for a ∈ G a (Q ℓ ) and for σ ∈ I F using (1.2).Hence we have h 0 ∈ H(Q ℓ ).The morphism induced by the adjoint action is surjective, because f (h 0 ) = q m 0 is not of finite order.Hence we can take h ∈ H(Q ℓ ) such that f (h) = u.Then we have We define Θ : Proposition 1.17.The map Θ is a bijection.Further it induces a bijection Proof.Let ϕ be an ℓ-adic L-homomorphism for G. Take a finite Galois extension F ′ of F such that G splits over F ′ .Take a representation which factors through a faithful algebraic representation for m ∈ Z and σ ∈ I F .Take a finite separable extension Corollary 1.18.Let σ q ∈ W F be a lift of the q-th power Frobenius element.Then an ℓ-adic L-homomorphism ϕ for G is Frobenius-semisimple if ϕ(σ q ) is semisimple.
We define a bijection Λ : Then LL GL 2 ,ℓ is independent of the choice of ι.We can not make a similar twist for an L-parameter of a general connected reductive group G as the following example suggests.
Example 2.1.We have a commutative diagram by functoriality.On the other hand, there does not exist a map which makes the commutative diagram

ℓ-adic C-parameter
A C-group is constructed in [BG14, Definition 5.38] for a connected reductive group over a number field.We recall the construction here in our setting.Let G ad be the adjoint quotient of G, and G sc be the simply-connected cover of G ad .Let γ : Z(G sc ) → G m be the restriction to Z(G sc ) of the half sum of the positive roots of G sc , where we take a maximal torus T sc and a Borel subgroup B sc of G sc F such that T sc ⊂ B sc to define the positive roots, but γ is independent of the choice.By pushing forward the exact sequence By taking the pullback of this extension along the natural morphism G → G ad , we obtain an extension The character (2.1) We take a Borel subgroup B ⊂ G F and a maximal torus T ⊂ B over F .Let ρ G denote the half sum of positive roots of G with respect to T and B. Then 2ρ G defines a cocharacter δ G : G m → T .We put Then z G is central in G and independent of choices of B and T as in [BG14, Proposition 5.39].By the independent of choices, we see that Then the morphism (2.1) induces the isomorphism as in [BG14, Proposition 5.39].We sometimes express a point of C G as [(g, z, w)] using the above isomorphism.We define Let Φ C ℓ (G) denote the equivalence classes of ℓ-adic C-parameters for G.We take c ∈ Q ℓ such that c 2 = q.We define For an ℓ-adic L-parameter ϕ for G, we put Lemma 2.3.We have a bijection between the set of the ℓ-adic L-parameters for G and the set of the ℓ-adic C-parameters for G given by sending ϕ to ϕ c .Further, this induces a bijection Φ ℓ (G) ≃ Φ C ℓ (G).Proof.The first claim follows from the definitions.If two ℓ-adic L-parameters for G are conjugate by an element of G(Q ℓ ), then they are conjugate by an element of Hence the second claim follows from the first one.
We define a map

Tannakian ℓ-adic L-parameter
Assume that C = Q ℓ .For a topological group H, let Rep Q ℓ (H) be the category of continuous finite dimensional representations of H over Q ℓ .For an algebraic group H over a field, a character χ of H and a cocharacter µ of H, we define χ, µ H ∈ Z by For a cocharacter µ ∈ X * (T ) of a torus T over F , let µ ∈ X * ( T ) denote the corresponding character of the dual torus T .Let M G be the conjugacy classes of cocharacters where we take a Borel subgroup B ⊂ G F , a maximal torus T ⊂ B defined over F and a dominant representative µ ∈ X * (T ).Let E We take c ∈ Q ℓ such that c 2 = q.For an integer m, let denote the twist by the character Then we have a decomposition as representations of G(Q ℓ ) where V [µ] is the r G,[µ] -typic part of V .For an ℓ-adic L-parameter ϕ, we define (r which means that we twist r • ϕ : For an ℓ-adic L-parameter ϕ for G, we define a tensor functor Definition 2.5.A Tannakian ℓ-adic L-parameter for G is a functor which is equal to F ϕ,c for an ℓ-adic L-parameter ϕ for G.We say that two Tannakian ℓ-adic L-parameters F and F ′ are equivalent if there is g ∈ G(Q ℓ ) such that, for all r ∈ Rep alg Lemma 2.6.The set of the Tannakian ℓ-adic L-parameters for G is independent of a choice of c ∈ Q ℓ such that c 2 = q.
Proof.We have be the character sending w to z . By (2.3), we have for an ℓ-adic L-parameter ϕ and r ∈ Rep alg Since ω z G ϕ is also an ℓ-adic Lparameter, the claim follows.

Let Φ T
ℓ (G) be the set of equivalence classes of Tannakian ℓ-adic L-parameters for G.This set is independent of a choice of c by Lemma 2.6.Lemma 2.7.We have a bijection between the set of the ℓ-adic L-parameters for G and the set of the Tannakian ℓ-adic L-parameters for G given by sending ϕ to F ϕ,c .Further, this induces a bijection Φ ℓ (G) ≃ Φ T ℓ (G).
Proof.We show the first claim.The map is surjective by Definition 2.5.Assume that ϕ and ϕ ′ are different ℓ-adic L-parameters for G and F ϕ,c = F ϕ ′ ,c .We take w ∈ W F such that ϕ(w) = ϕ ′ (w).Further, we take a finite Galois extension F ′ of F such that G splits over F ′ and the images of ϕ(w) and ϕ . Hence the map is injective.
The second claim follows from the first one.
For a Tannakian ℓ-adic L-parameter F for G, we take an ℓ-adic L-parameter ϕ for G such that F = F ϕ,c .Then the centralizer S ϕ = Z G(Q ℓ ) (Im ϕ) is independent of a choice of c by (2.4).We write S F for S ϕ .Then F naturally factors through For a finite separable extension F ′ of F , we define the restriction of F to F ′ by the usual restriction to W F ′ of an ℓ-adic L-parameter for G and bijections given by Lemma 2.7.Let and the natural functor induced by the restriction with respect to S F ⊂ S F | F ′ .
Remark 2.13.We can also show Corollary 2.12 using the geometric realization of the local Langlands correspondence for GL n in the ℓ-adic etale cohomology of the Lubin-Tate spaces after the reduction to the supercuspidal case in the same spirit as Remark 2.9 (cf.[HT01, Lemma VII.1.6]).Here we gave a proof by the characterization without appealing to such a geometric realization.
Definition 1.3 (cf.[Bor79, 8.2]).(1) An L-homomorphism of Weil-Deligne type for G over C is a homomorphism ξ : WD F (C) → L G(C) of groups over W F such that ξ| Ga(C) is algebraic and (p G • ξ)| W F is smooth.(2) An L-parameter of Weil-Deligne type for G over C is a semisimple relevant L-homomorphism of Weil-Deligne type for G over C. Let L WD C (G) denote the set of equivalence classes of L-homomorphisms of Weil-Deligne type for G over C. Let Φ WD C (G) denote the set of equivalence classes of L-parameters of Weil-Deligne type for G over C. Definition 1.4.(1) A Weil-Deligne L-homomorphism for G over C is a pair (τ, N) of a homomorphism τ : W F → L G(C) of groups over W F and N ∈ Lie( G)(C) such that p G • τ is smooth and Ad(τ (w))N = |w|N for w ∈ W F .The second component N of a Weil-Deligne L-homomorphism (τ, N) is called a monodromy operator.We say that two Weil-Deligne L-homomorphisms for G over C are equivalent if they are conjugate by an element of G(C).
denote the set of equivalence classes of Weil-Deligne L-homomorphisms for G over C. Let Φ M C (G) denote the set of equivalence classes of Weil-Deligne Lparameters for G over C. Remark 1.5.Let ι : C ∼ → C ′ be an isomorphism of fields of characteristic 0. Then ι induces bijections L WD C 10.1.2Lemma].Hence we obtain the claim.The following is a slight generalization of [Hei06, Proposition 3.5] and [KL87, 2.4].

2. 1
Problem Let Irr(G(F )) denote the set of isomorphism classes of irreducible smooth representations of G(F ) over C. The conjectured local Langlands correspondence is a surjective map LL G : Irr(G(F )) → Φ(G) with finite fibers satisfying various properties (cf.[Bor79, 10], [Kal16, Conjecture G]).We assume the existence of the local Langlands correspondence in the sequel.Let Irr ℓ (G(F )) denote the set of isomorphism classes of irreducible smooth representations of G(F ) over Q ℓ .If we fix an isomorphism ι : C ℓ of an element of Irr ℓ (GL 2 (F )) coming from Irr ℓ (PGL 2 (F )) by the construction of LL GL 2 ,ℓ and [Bor79, 10.1].
1 by the construction of γ.It further induces a character G → G m by taking composition with the natural morphism G → G 1 .Hence we obtain a morphism G → G × G m .
[µ] be the field of definition of [µ].Let r G,[µ] be the irreducible representation of G(Q ℓ ) of highest weight µ viewed as a dominant character of a maximal torus of G.