THE JACOBSON–MOROZOV MORPHISM FOR LANGLANDS PARAMETERS IN THE RELATIVE SETTING

. We construct a moduli space LP G of SL 2 -parameters over Q , and show that it has good geometric properties (e.g. explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism JM : LP G → WDP G (where WDP G is the moduli space of Weil–Deligne parameters considered by several other authors). We show that JM is an isomorphism over a dense open of WDP G , that it induces an isomorphism between the discrete loci LP disc G → WDP disc G , and that for any Q -algebra A it induces a bijection between Frobenius semi-simple equivalence classes in LP G ( A ) and Frobenius semi-simple equivalence classes in WDP G ( A ) with constant (up to conjugacy) monodromy operator.


Introduction
Motivation. A problem of fundamental importance in the study of harmonic analysis is the classification of irreducible complex admissible representations of G(F ) where F is a nonarchimedean local field, and G is a reductive group over F . The local Langlands correspondence, a guiding principle for many areas of number theory in the last 40 years, posits a parameterization of such admissible representations in terms of equivalence classes of parameters related to the Galois theory of F . These parameters come in several forms. Chief amongst these are the complex L-parameters which are homomorphisms ψ : W F × SL 2 (C) → L G(C) satisfying certain properties (cf. [SZ18,§3]), and complex Weil-Deligne parameters which are pairs (ϕ, N ) where ϕ : W F → L G(C) is a homomorphism and N is a nilpotent element of the Lie algebra of G(C), satisfying certain properties (cf. [GR10, §2.1]). The notion of equivalence in both cases is that of G(C)-conjugacy.

Some group theoretic preliminaries
In this section we establish some notation, definitions, and basic well-known results that we shall often use without comment in the sequel. We encourage the reader to skip this section on first reading, referring back only when necessary.
2.1. The nilpotent variety, unipotent variety, and exponential map. Let us fix k to be a field of characteristic 0 and H to be a reductive group over k. We denote by h the Lie algebra of H thought of both as a vector k-space and as a k-scheme.
Let A be a k-algebra and x an element of h A . Recall then that as in [DG70, II, §6, №3] one may associate an element exp(T x) in H(A T ) to x. We then say that x is nilpotent if it satisfies any of the following equivalent conditions. Proposition 2.1. The following are equivalent: (1) for all finite-dimensional representations ρ : H → GL(V ) the endomorphism dρ(x) of V A is nilpotent, (2) there exists a faithful finite-dimensional representation ρ : H → GL(V ) such that the endomorphism dρ(x) of V A is nilpotent, (3) exp(T x) belongs to H(A[T ]), (4) there exists a morphism of group A-schemes α : G a,A → H A such that x = dα(1), if A is in addition reduced, then (1)-(4) are equivalent to (5) x belongs to h der A and ad(x) is a nilpotent transformation of h der A . Proof. The equivalence of (1)-(4) is given by [DG70, II, §6, №3, Corollaire 3.5]. To see the equivalence of (1) and (5), in the case when A is reduced, we may assume that A is a field. Let σ : H/Z(H der ) → GL(W ) be the faithful representation given by taking a direct sum of Ad : H → GL(h der ) and the composition of H → H ab with a faithful representation of H ab . It is clear that applying (1) to σ shows that (5) holds. Conversely, suppose that (5) holds, so then dσ(x) is nilpotent. Let ρ be as in (1). We may assume that ρ is irreducible. We put n = |Z(H der )|. Then ρ ⊗n : H → GL(V ⊗n ) factors through H/Z(H der ). Hence by [Del82, Proposition 3.1 (a)] dρ ⊗n (x) is nilpotent. This implies that dρ(x) is nilpotent.
Let us consider the symmetric algebra on h * (resp. the graded ideal of positive degree tensors) S(h * ) = The nilpotent variety of H is the closed subshceme of h given by N := V S + (h * ) H (or N H when we want to emphasize H). This is not a misnomer as for any extension k ′ of k we have [Jan04, §6.1, Lemma]). In particular, N is the unique reduced subscheme of h whose k-points consist of the nilpotent elements of h k . The nilpotent variety N is an integral (cf. [Jan04, §6.2, Lemma]) finite type affine k-scheme of dimension dim(H)−r where r is the geometric rank of H (see [Jan04,§6.4]). In fact, as k is of characteristic 0, it is normal by the results of [Kos63]. Observe that the nilpotent variety is stable under the adjoint action of H. Also observe that if f : H → H ′ is a morphism of reductive groups over k it induces a morphism df : N H → N H ′ and satisfies df (Ad(h)(x)) = Ad(f (h))(df (x)).
Example 2.2. Let Mat n,k be the scheme of n-by-n matrices over k, and let I ⊆ O(Mat n,k ) be generated by those polynomials corresponding to (a ij ) n = 0. Then, N GL n,k = V ( √ I).
From this example, and the functoriality of the nilpotent variety, it's easy to see that if A is a k-algebra, then one has the containment N (A) ⊆ {x ∈ h A : x is nilpotent}, which is an equality if A is reduced, but can differ otherwise. That said, from this containment we see that for any element x of N (A) we may define an element exp(x) of H(A) as in [DG70, II, §6, №3, 3.7]. As this association is functorial we obtain an H-equivariant morphism of schemes N → H called the exponential morphism and denoted by exp (or exp H when we want to emphasize H) which is functorial in H. We would now like to describe the image of exp.
To this end, note that there exists a unique reduced closed subscheme U (or U H when we want to emphasize H) of H such that for all extensions k ′ of k (see [Spr69, Proposition 1.1]). We call U the unipotent variety associated to H. It is an integral finite type affine k-scheme of dimension dim(H) − r which is stable under the conjugation action of H (see loc. cit.). Moreover, as k is of characteristic 0, it is normal (see [Spr69,Proposition 1.3]). We observe that U is stable under the conjugation action of H.
Observe that exp factorizes through U , as both are reduced, and so this may be checked on the level of k-points. We have the following ombnibus result concerning the exponential morphism. 2.2. The L-group and C-group. Fix F to be a non-archimedean local field, and let G be a reductive group over F . In this subsection we define the C-group of G, which is a modification of the L-group of G that is better suited to the theory of parameters over a general Q-algebra.
To begin, let Ψ(G) denote the canonical based root datum of G F (see [Kot84,§1.1] and [Mil17, §21.42]) which comes equipped with an action of Γ F . We fix once and for all a Langlands dual group of G by which we mean a pinned reductive group ( G, B, T , {x α }) over Q (see [Mil17,§23.d]) together with an isomorphism between Ψ( G, B, T ) and Ψ(G) ∨ . We denote by g the Lie algebra of G, and by N the nilpotent variety of G.
Next, let W F denote the Weil group scheme over Q associated to F as in [Tat79,(4.1)]. For a Q-algebra A one may identify W F (A) with the set of continuous maps f : π 0 (Spec(A)) → W F where here π 0 (Spec(A)) is thought of as a profinite space (cf. [Sta21, Tag 0906]) and W F is given its usual topology. In particular, W F (A) = W F (A) when π 0 (Spec(A)) is discrete (e.g. if A is connected or Noetherian), but can differ otherwise. For w in W F we shall occasionally abuse notation and use w to also denote its image in W F (A).
Note that if d : W F → Z is the degree map sending a lift of arithmetic Frobenius to −1, then there is a morphism of Q-group schemes d : W F → Z which takes a map f to d • f . Observe that Z admits an embedding of group Q-schemes into G m,Q corresponding to 1 → q −1 and we denote the composition of d with this map by · : W F → G m,A . We define I F = ker( · ), which is an affine scheme equal to lim ← − I F /I K as K travels over all finite extensions of F . Note that if A is a Q-algebra and X an A-scheme locally of finite presentation then any morphism of A-schemes I F,A → X must factorize through I F /I K for some K (cf. [Sta21, Tag 01ZC]).
Remark 2.4. One reason to prefer W F over the constant group scheme W F is that the topological group π 0 (W F ) is equal to W F with its usual topology, and similarly for I F .
Returning to G, note that the action of Γ F on Ψ(G) gives rise to an action of Γ F on ( G, B, T , {x α }) and, in particular, on G as a group Q-scheme. We define a finite Galois extension F * of F characterized by the equality Γ F * = ker(W F → Aut( G)). Equivalently, F * is the minimal field splitting G * , the quasi-split inner form of G. We write Γ * for Γ F * /F . As Γ * acts on G and W F admits Γ * as a quotient, we obtain an action of W F on G. Define the L-group scheme of G to be the group Q-scheme L G = G ⋊ W F . Observe that there is a natural inclusion G ֒→ L G which identifies G as a normal subgroup scheme of L G. In particular, there is a natural conjugation action of L G on G, which in turn induces an adjoint action of L G on g.
As the action of W F on G factorizes through a finite quotient, we see by Lemma 2.5 below that the group presheaf associating a Q-algebra Lemma 2.5. Let A be a Q-algebra, H a reductive group over A, and Σ a finite group acting on H by group A-scheme automorphisms. Then, the group functor is represented by a subgroup scheme of H smooth over A, with H • reductive over A, and such that for all A-algebras B one has the equality Lie(H Σ )(B) = Lie(H)(B) Σ .
Proof. Write H = Spec(R), then one easily verifies that Spec(R Σ ), where R Σ is the ring of coinvariants, represents H Σ . As A is a Q-algebra, it is evident that R Σ is a direct summand of R and thus H Σ is flat over A, and thus smooth. By [SGA3-1, Exposé VIB, Corollaire 4.4] we know that H • is representable and smooth over A, and it is then reductive by [PY02, Theorem 2.1]). The claim about Lie algebras is clear as the functor of Σ-invariants preserves kernels.
Let X * denote the cocharacter component of Ψ(G) and R + the positive root component, and define δ to be the element of X * given by the sum over the elements of R + . By our identification between Ψ( G, B, T ) and Ψ(G) ∨ we see that δ corresponds to an element of X * ( T ) which we also denote by δ. Let us set z G := δ(−1) ∈ T (Q) [2]. By the proof of [BG14, Proposition 5.39], z G lies in Z 0 ( G)(Q). Thus, the action of W F on G × G m,Q (with trivial action on the second component) fixes the pair (z G , −1). Therefore, W F acts onǦ := (G × G m,Q )/ (z G , −1) . We then define the C-group scheme of G to be C G =Ǧ ⋊ W F . Note that by [BG14,Proposition 5.39] there exists a central extension G of G such that C G is naturally isomorphic to L G.
The group G admits a natural embedding intoǦ, with normal image, via the first factor, and therefore we obtain a conjugation action of C G on G, and thus an adjoint action of C G on g. Also, the morphism annihilates (z G , −1) , and thus induces a morphism Finally, we observe that if k is an extension of Q, and c is any element of k such that c 2 = q, then there is a morphism i c : L G k → C G k obtained as the composition 3. Scheme of homomorphisms and cross-section homomorphisms. We establish here some terminology and basic results concering the scheme of homomorphisms as well as the scheme of cross-section homomorphisms (in the sense of [DHKM20, Appendix A]). Throughout the following we fix k to be field of characteristic 0.
for any k-algebra A the natural map Proof. Statements (1) and (2)  Proposition 2.7. Suppose that X and Y are finite type integral k-schemes with X unirational. Then for any k-algebra A, the natural map is injective.
Proof. One quickly reduces to the case when X = D(w) ⊆ A n k for w in k[x 1 , . . . , x n ], Y = A 1 k , f lies in A[x 1 , . . . , x n ] and g is the zero map. As X(F ) → X(A) is injective, we will be done if we can show that f does not vanish on D(w)(k). If {a i } i∈I is a basis of A as a k-vector space then we may write f = i∈I a i f i where f i ∈ k[x 1 , . . . , x n ]. As f is non-zero there exists some i such that f i is non-zero. As D(w)(k) is Zariski dense in A n k as k is infinite, there then exists some x in D(w)(k) such that f i (x) = 0. Then, by setup, f (x) = 0.
In the future, we call a homomorphism of groups H(A) → H ′ (A) algebraic if it is the map on A-points of a morphism (necessarily unique) of group A-schemes H A → H ′ A . Schemes of cross-section homomorphisms. Fix an abstract group Σ and a reductive group H over k. We then consider the presheaf This presheaf clearly carries an H-conjugation action. If, in addition, Σ acts on H by group k-scheme morphisms then for a k-algebra A we say a homomorphism f : 2.4. Transporter and centralizer schemes. Let R be a ring, H a group-valued functor on Alg R , and X a set-valued functor on Alg R . Then, for an R-algebra S and two elements α and β of X(S) we define the transporter set to be We then define the transporter presheaf to be the presheaf We abbreviate Transp H (β, β) to Z H (β) and call it the centralizer presheaf, which is clearly a group presheaf. We then have the following obvious proposition.
Proposition 2.9. Suppose that H is a group R-scheme and that X is a separated R-scheme of finite presentation. Then, for any R-algebra S and any elements α and β of X(S), the presheaves Transp H (α, β) and Z H (β) are representable by closed finitely presented subschemes of H S . Moreover, for any S-algebra T one has the natural equalities

The classical setting
In this section we recall the Jacobson-Morozov theorem and the Jacobson-Morozov theorem for parameters in their classical settings. This will not only serve to emphasize the results we wish to geometrize, but will play an important role in the proof of these more general results.
3.1. The Jacobson-Morozov theorem. Let k be a field of characteristic 0 and H an algebraic group over k such that H • is reductive. It will be useful to explicitly name the matrices which form a k-basis of the Lie algebra sl 2,k . We then have the Jacobson-Morozov Theorem as follows.  We end this subsection by explaining the relationship between the centralizers of θ and N = JM(θ). Namely, let us set u N = im(ad(N )) ∩ ker(ad(N )), Then, we have the following Levi decomposition statement.
Proposition 3.3. The equality Z H (N ) = U N ⋊ Z H (θ) holds. Further we have where for an integer i we set Proof. The first claim is proved in the same way as [ 3.2. The Jacobson-Morozov theorem for parameters. We now recall the analogue of the Jacobson-Morozov theorem for parameters. We use the notation from §2.2.
Definition 3.4. Topologize L G(C) by giving G(C) the classical topology. (

such that
• ψ| W F : W F → L G(C) is a continuous cross-section homomorphism, • ψ| SL 2 (C) : SL 2 (C) → L G(C) takes values in G(C) and is algebraic.
For τ ∈ {L, WD} let us denote by Φ τ, G the set of complex τ -parameters for G. Recall that a Weil-Deligne parameter (ϕ, N ) (resp. an L-parameter ψ) is called Frobenius semi-simple if for one (equiv. for any) lift w 0 of arithmetic Frobenius the element ϕ(w 0 ) (resp. ψ(w 0 )) is semisimple (in the sense of [Bor79, §8.2]). We denote by Φ τ,ss, G the subset of Frobenius semi-simple τ -parameters. For each τ there is a natural action of G(C) on Φ τ, G which stabilizes the subset Φ τ,ss, G . We then define Φ τ G := Φ τ, G / G(C) and Φ τ,ss G := Φ τ,ss, G / G(C). For an element ψ of Φ L, G we denote by θ (or θ ψ when we want to emphasize ψ) the morphism ψ| SL 2 (C) : SL 2 (C) → G(C). To upgrade Theorem 3.1 to the parameter setting, we need to associate a Weil-Deligne parameter to any L-parameter. To this end, let us define a morphism of groups We then define the Jacobson-Morozov map to be the G(C)-equivariant map It is easy to check that JM −1 (Φ WD,ss,  Example 3.5. Set G = GL 4 and as G is split we may replace L G(C) with G(C) = GL 4 (C). Consider the Weil-Deligne parameter (ϕ, N ) given as follows Suppose that (ϕ, N ) = JM(ψ). Then, ψ is of the form ρ ⊠ Std, where Std is the standard representation of SL 2 (C). Indeed, from the Jacobson-Morozov theorem one sees that as an SL 2 (C) representation C 4 is isomorphic to Std 2 . One may then check that the morphism is an isomorphism of W F × SL 2 (C)-representations. That said, note that the twist of ρ by the unramified character w → w −1/2 must be isomorphic to the representation on KerN induced by ϕ. In particular ρ is semi-simple. Hence the Weil-Deligne parameter attached to ψ must be Frobenius semi-simple, but the original (ϕ, N ) is not Frobenius semi-simple.
However, we have the following Jacobson-Morozov theorem for parameters. 3.3. Bijection over reductive centralizer locus and applications. The Jacobson-Morozov theorem for parameters is stated at the level of G(C)-orbits. While this is a non-issue for now, when we attempt to geometrize this result it becomes more problematic due to the subtle nature of quotients in algebraic geometry. So, we wish to upgrade the Jacobson-Morozov theorem for parameters to a bijectivity statement before quotienting by G(C).
To begin, we give an analogue of Proposition 3.3 for parameters. To state it, let (ϕ, N ) be an element of Φ WD,  Proof. Given Proposition 3.3 it suffices to show that if ua belongs to Z G(C) (ϕ, N ), where u is in U N (C) and a is in Z G(C) (θ), then in fact u belongs to U N (ϕ) and a belongs to Z G(C) (ψ). To prove this, we note that conjugation by an element in the image of ϕ stabilizes both U N (C) and Z G(C) (θ). Indeed, since Ad(ϕ(w))(N ) = w N , we have that conjugation by ϕ(w) stabilizes Z G(C) (N ) and hence its unipotent radical U N . On the other hand, as ϕ(w) equals ψ(w, 1)θ(i 2 (w)), and ψ(w, 1) commutes with θ, one may easily check the claim that ϕ(w) normalizes Z G(C) (θ). Now for each w ∈ W F , ua equals Int(ϕ(w))(u) Int(ϕ(w))(a). Therefore, Int(ϕ(w))(a)a −1 equals Int(ϕ(w))(u) −1 u. By what we have proven, the former is an element of Z G(C) (θ) and the latter is an element of U N (C). Since U N (C) and Z G(C) (θ) have trivial intersection, we have that both sides are trivial and so a and u commute with ϕ(w) as desired.
We may use this decomposition to exhibit an example of a semi-simple L-parameter ψ whose associated Weil-Deligne parameter has strictly larger centralizer.
Example 3.8. Let G = GL 3 and consider the element ψ in Φ L,ss, G given by the following and set (ϕ, N ) = JM(ψ). In this case, we have but it does not belong to Z G(C) (ψ) by Proposition 3.7.
Remark 3.9. We remark that although Z G(C) (ψ) need not equal Z G(C) (JM(ψ)), these groups are the same for the purposes of parametrizing L-packets as in [Kal16] as they have the same component groups by Proposition 3.7. More generally, one can consider the group S ♮ ψ (resp. S ♮ JM(ψ) ) that is related to [Kal16, Conjecture F] and is defined by These groups are equal by Proposition 3.7 as This decomposition also allows us to give an algebraic condition for when a Weil-Deligne parameter is the image under the Jacobson-Morozov map of a semi-simple L-parameter with the same centralizer. In the rest of this section, we use Proposition 5.11, but the proof of the proposition does not depend on the rest of this section.
Proof. Suppose first that Z G(C) (ϕ, N ) • is reductive. We shall show in Proposition 5.11 that this implies that (ϕ, N ) is Frobenius semi-simple. Let ψ be any element of Φ L,ss, G such that JM(ψ) = (ϕ, N ). By Proposition 3.7 the reductivity of Z G(C) (ϕ, N ) • implies that U N (ϕ) is trivial, and Corollary 3.11. The map JM : Proof. This follows from Theorem 3.6, Proposition 3.10 and that ψ is Frobenius semi-simple if and only if JM(ψ) is for ψ ∈ Φ L, G .
3.4. Essentially tempered parameters. To make Corollary 3.11 useful, we now show that JM −1 (Φ WD,rc, G ) contains a large class of important L-parameters. To this end, let us call an element ψ of Φ L, G essentially tempered if the projection of ψ(W F ) to G(C)/Z 0 ( G)(C) is relatively compact. Let Φ L,est, G be the set consisting of essentially tempered L-parameters. We will soon show that every essentially tempered L-parameter maps into the reductive centralizer locus, but first we must establish some results concerning Frobenius semi-simple parameters.
Proposition 3.12. Any element ψ of Φ L,est, G is Frobenius semi-simple.
Proof. The map ψ ′ obtained by composing ψ| W F * with the projection to G(C)/Z 0 ( G)(C) is a homomorphism. By Lemma 3.13 below it suffices to show that if w 0 is an arithmetic Frobenius lift and m is divisible by [F * : F ], then ψ ′ (w m 0 ) is semi-simple. But, by essentially temperedness we know that the image of Up to conjugation, we may then assume that  Proof. Fix any representation r : The following shows that the naming of essentially tempered L-parameters is reasonable.
Proposition 3.14. For ψ ∈ Φ L, G , the following conditions are equivalent: We now relate Φ L,est, G to the reductive centralizer locus of Φ WD, Proposition 3.15. The containment Φ L,est, Proof. Let ψ be an element of Φ L,est, G and set (ϕ, N ) = JM(ψ). Then ψ is Frobenius semi-simple by Proposition 3.12. We claim that Z G(C) (ψ) = Z G(C) (ϕ, N ), from where we will be done by Proposition 3.10. By Proposition 3.7, it suffices to show that U N (ϕ) is trivial. We assume that U N (ϕ) is non-trivial and take a non-trivial weight vector v of Lie(U N (ϕ)) with respect to the adjoint action of θ| T 2 , where T 2 is the standard maximal torus of SL 2,C . We put u = exp(v). For each w ∈ W F we have that ϕ(w) = ψ(w, 1)θ(i 2 (w)). Since ϕ(w) commutes with u, we see that Int(ψ(w, 1) −1 )(u) is equal to Int(θ(i 2 (w)))(u), and therefore But, observe that if w 0 is a lift of arithmetic Frobenius in W F then i 2 (w 2n 0 ) = q n 0 0 q −n . By Proposition 3.3, we deduce that Ad(θ(i 2 (w 2n 0 )))(v) = q jn v for some j 1. Letting n tend towards infinity, and using the fact that u is non-trivial, we deduce that the adjoint orbit of W F on v is non-compact, which is a contradiction.
We now state a corollary to Proposition 3.15. Before doing so, we recall an even smaller subset of Φ L,est, G that will feature prominently below. Namely, recall that (ϕ, N ) in Φ WD, ) the set of discrete parameters and Φ WD,disc and [SZ18, Lemma 5.2]), and thus ψ is discrete if and only if JM(ψ) discrete as they have the same centralizers by Proposition 3.15 and its proof.
Corollary 3.16. The map Note that implicit in the above is the following result of independent interest.
Proposition 3.17. Any element of Φ WD,disc, Proof. The first claim is a special case of Proposition 5.11. The second claim follows from Φ L,disc, and Proposition 3.12.
We end this subsection by showing that one may apply Corollary 3.16 to show that the assocation of ψ • i to ψ is injective when restricted to the set of discrete L-parameters. This result plays an important technical role in [BMY20].
Proposition 3.18. The maps are injective.
Proof. By Corollary 3.16 it suffices to show that the former map is injective. Fix λ in the set Hom(W F , L G(C)). By Proposition 3.17 it then suffices to show that (if non-empty) the set intersects at most one G(C)-orbit of discrete parameters. As in [Vog93,§4], set G(C) λ to be Z G(C) (λ), and where w 0 is any lift of arithmetic Frobenius. Both P (G, λ) and g , and that the latter space has only finitely many orbits. Therefore, P (G, λ) carries the structure of a vector space on which G(C) λ acts algebraically and with only finitely many orbits.
Suppose then that (λ, N ) is a discrete element of P (G, λ) and Proposition 5.23 and Proposition 7.12]). But, note that H = Z G(C) (λ, N ) and so contains Z 0 ( G)(C) as a finite index subgroup. We deduce that dim(O) is equal to dim( G(C) λ ) − dim(Z 0 ( G)(C)). But, as G(C) λ acts through G(C) λ /Z 0 ( G)(C), and has finitely many (locally closed) orbits, we see that dim P (G, λ) is at most dim( G(C) λ ) − dim(Z 0 ( G)(C)). Thus, we deduce that dim(O) = dim(P (G, λ)). As O is locally closed in P (G, λ) we deduce 13 that O is open. As P (G, λ) is a vector space it is irreducible, so open orbits are unique, and the conclusion follows.

The geometric and relative Jacobson-Morozov theorems
Before we can geometrize the Jacobson-Morozov theorem for parameters, we now first geometrize the Jacobson-Morozov theorem. After doing so, we derive a version of the Jacobson-Morozov on the level of A-points. We fix for the remainder of this section a field k of characteristic 0 and H a reductive group over k.
Remark 4.1. In this section we often assume that H is split. This will be sufficient for us as G is a split group. That said, most of these statements admit obvious generalizations to arbitrary reductive H, with similar proofs. The exception is Theorem 4.15, but we suspect that the statement is still true and that one can employ a similar strategy to prove it.
4.1. The orbit separation space. Pivotal to our formulation of a geometric version of the Jacobson-Morozov theorem is a certain construction which, in a precise sense, replaces a variety with group action with the disjoint union of its orbits. Throughout this subsection we fix a reduced quasi-projective scheme X over k equipped with an action of H. We also assume that the map is surjective (although one may deal with the general case by Galois descent).
For each element x of X(k) let us denote by O x the orbit scheme given as the fppf sheafification of the presheaf It will be useful to have a more explicit description of the A-points of O x for a k-algebra A. and Proof. The first claim follows from the fact that the orbit map µ When A is a reduced k algebra, one may give a simpler description. Say an element x of X(A) is everywhere geometrically conjugate (egc) to x if for all geometric points Spec(k ′ ) → Spec(A) one has that x and x have images in X(k ′ ) belonging to the same H(k ′ )-orbit. Proof. Evidently any element of O x (A) is egc to x. If x is egc to x then the morphism x : Spec(A) → X has the property that x(| Spec(A)|) ⊆ |O x |. As Spec(A) is reduced this implies that x factorizes through O x as desired.
We then assemble the spaces O x into one as follows.
Definition 4.5. We define the orbit separation of X, denoted by X ⊔ , to be the space We have a tautological map X ⊔ → X, and we have the following omnibus result concerning its properties in the case when X(k)/H(k) is finite, which is the case of most interest to us. Below, and in the sequel, we call a morphism of schemes f : Proposition 4.6. Suppose that X(k)/H(k) is finite. Then, the map X ⊔ → X is a weakly birational surjective monomorphism and it is an isomorphism if and only if the action map µ : H × X → X is smooth.

As the last condition is equivalent to the claim that
Lemma 3.5] and [Sta21, Tag 05VJ]), this is a special case of Lemma 4.7 below.
Proof. As all of these claims may be checked over k we may assume without loss of generality that k is algebraically closed. The final claim is clear, thus we focus on the first two claims. For the first claim, To see the second claim, it suffices to show the if direction. For each irreducible component Z of X note that {Y i ∩ Z} is a finite set of locally closed subsets with dense union. This implies that there exists some i 0 such that Y i 0 ∩Z is open. Let C be the union of irreducible components of X which intersect Z at a proper non-empty subset. Set Finally, observe that the orbit separation space is a functorial construction. Namely, if Y is another quasi-projective scheme over k equipped with an action of H with the same properties, then for any H-equivariant morphism X → Y , the composition 4.2. The geometric Jacobson-Morozov theorem. We now move to the geometrization of the Jacobson-Morozov theorem. Let us now assume that H is split. To begin, observe that one has a Jacobson-Morozov morphism JM : Hom(SL 2,k , H) → N , θ → dθ(e 0 ).
We would like to apply the orbit separation construction from the last subsection to this map, but before we do so, we should first observe that the actions of H on Hom(SL 2,k , H) and N satisfy the properties used in the last section. Proof. By Theorem 3.1 it suffices to show the first map is a surjection. Let N be an element of N (k). Bala-Carter theory (see [Jan04,§4]) says that there exists a Levi subgroup L of H k and a parabolic subgroup P of L such that N is conjugate to an element contained in the unique open orbit of P acting on Lie(R u (P )). Now, as H is split, we may assume up to conjugacy, that L = L k for a Levi subgroup L of H (see [Sol20]). As L is also split we may also assume, up to conjugacy, that P = P k for a parabolic subgroup P of L.
As the unique open orbit of P acting on Lie(R u (P )) has a k-point, being a Zariski open of a vector k-space, we are done. Before we show that our two spaces with H-action have finitely many H(k)-orbits, we observe the following.  Proof. By Theorem 3.1 these two sets are in bijection, so it suffices to prove the finiteness of either. The finiteness of the latter set is a classical result (e.g. see [Jan04, §2.8, Theorem 1]). Alternatively, one may prove the finiteness of the former set by observing that by Proposition 4.10 the sets Hom(SL 2,k , H k )/H(k) and π 0 (Hom(SL 2,k , H k )) are equipotent. But, by Proposition 2.6 the scheme Hom(SL 2,k , H k ) is finite type over k and thus π 0 (Hom(SL 2,k , H k )) is finite.
By the functoriality of the orbit separation construction the Jacobson-Morozov morphism factors uniquely through N ⊔ and we also denote the resulting map Hom(SL 2,k , H) → N ⊔ by JM. But, unlike Hom(SL 2,k , H), the orbit separation space N ⊔ is essentially never equal to N . Proof. If H is abelian then N is a single point. If N ⊔ → N is an isomorphism then by Proposition 4.6 the orbit of 0 is open, but as it is also closed and N is connected we deduce that it is equal to N . As dim(N ) is equal to dim(H) − r(H), we see that H is a torus as desired.      Thus, a fortiori, we see that p (ad(h)| a e ) = 0.
Proof. For each i = 0, . . . , m + 1 let us set Observe that h e A = d 0 ⊇ · · · ⊇ d m+1 = 0. We claim then that (ad(h)−i)(d i ) ⊆ d i+1 . Note that d i is generated as an A-algebra by elements of the form ad(e) i (z) for z in h. The exact same algebra as in [ as desired.

Moduli spaces of Weil-Deligne parameters
To give a geometrization of the results of §3.2 it is useful to first develop a space intermediary between the moduli space of L-parameters (see §6) and the moduli space of Weil-Deligne parameters. We give such a space in this section which, in short, parameterizes Weil-Deligne parameters whose monodromy operator lies in N ⊔ . 5.1. The moduli space of Weil-Deligne parameters. We first recall the moduli space of Weil-Deligne parameters roughly following the presentation as in [Zhu20].
Initial definitions. We begin by defining the relative analogue of a Weil-Deligne parameter.
Definition 5.1. For a Q-algebra A, we define a Weil-Deligne parameter over A to be a pair (ϕ, N ) where (WDP1) ϕ : W F,A → C G A is a morphism of group A-schemes such that p C • ϕ = ( · , id), (WDP2) N is an element of N (A) such that Ad(ϕ(w))(N ) = w N for all w ∈ W F (A).
We denote the set of Weil-Deligne parameters over A by WDP G (A) which clearly constitutes a presheaf on Q-algebras. The presheaf WDP G has a natural action by G given by g(ϕ, N )g −1 := (Int(g) • ϕ, Ad(g)(N )).
So, for a Weil-Deligne parameter (ϕ, N ) we may consider the centralizer group presheaf Z G (ϕ, N ).
We define the morphismφ : W F,A →Ǧ A of schemes as the composition of ϕ with the projection toǦ A . We denote by ϕ the homomorphism W F,A → ( G ⋊ Γ * ) A obtained by composing ϕ with the quotient map C G A → ( G ⋊ Γ * ) A . Observe that whileφ may not be a homomorphism, it becomes so after restriction to W F * ,A . In particular, for any w ∈ W F (A) the restriction ofφ to w m is a homomorphism whenever [F * : F ] divides m.
Let K be a finite extension of F * Galois over F , and let us define for a Q-algebra A the set We observe that WDP K G forms a G-stable subfunctor of WDP G . In fact, one sees that there is an equality of functors WDP G = lim − → WDP K G as K travels over all such extensions.
We finally observe that WDP G has a more familiar form over an extension k of Q containing an element c such that c 2 = q. More precisely, for a k-algebra A, we equip G(A) with the discrete topology and put WDP ′ G,k (A) := (ϕ, N ) : (1) ϕ : W F → G(A) ⋊ W F is a a continuous cross-section homomorphism, (2) N ∈ N (A) is such that Ad(ϕ(w))(N ) = w N for all w ∈ W F .
It is clear that WDP ′ G is a functor on the category of k-algebras and comes equipped with a natural action of G k . Let us also observe that if i c is the map from §2.2 then there is a morphism i WD c : WDP ′ G,k → WDP G,k which on A-points is given by sending (ϕ ′ , N ) to the unique element of WDP G (A) of the form (ϕ, N ) which is equal to (i c • ϕ ′ , N ) on A-points.
Proposition 5.2. The morphism of functors i WD c : WDP ′ G,k → WDP G,k is an isomorphism. Proof. This follows from the cartesian diagram and that any morphism W F,A →Ǧ A of schemes over A factors through (W F /I K ) A for a finite extension K of F .
Representability. We now establish the representability of the functor WDP G . To this end, let us fix K a finite extension of F * Galois over F . Note that for a Q-algebra A and an element (ϕ, N ) of WDP K G (A) we may define an element φ of Z 1 (I F /I K , G)(A) as follows. First observe that condition (WDP1) implies that ϕ| I given by We then have the following explicit description of WDP K G . Proposition 5.3. The morphism j w 0 identifies WDP K G with the equalizer Eq(D WD ). Thus, WDP K G is representable by a finite type affine Q-scheme and j w 0 is a closed embedding. Observe that for an extension K ⊆ K ′ of Galois extensions of F containing F * there is a restriction morphism Z 1 (I F /I K ′ , G) → Z 1 (I K /I K ′ , G). By Proposition 2.8 and Proposition 4.6 the subspace consisting of only the trivial homomorphism is a clopen subset of the target, and thus so is its preimage in Z 1 (I F /I K ′ , G), but this is precisely Z 1 (I F /I K , G). We deduce that the natural inclusion of functors WDP K G → WDP K ′ G is a clopen embedding. From the identification WDP G = lim − →K WDP K G we deduce from Proposition 5.3 that WDP G is representable by a scheme locally of finite type over Q, all of whose connected components are affine.
The following non-trivial result will play an important technical role below. are reduced for all K, and thus, a fortiori, WDP G is reduced.

5.2.
Semi-simplicity of parameters. As in the Theorem 3.6 one requires Frobenius semisimplicity conditions to get a Jacobson-Morozov result in the relative setting. Therefore, we now develop a sufficient notion of Frobenius semi-simplicity for a Weil-Deligne and L-parameter over a Q-algebra A.
Definition 5.5. Let R be a Q-algebra and H is a smooth group R-scheme such that H • is reductive. We then say that an element h of H(R) is semi-simple if there exists some m 1, anétale cover Spec(S) → Spec(R), and a torus T of H • S such that h m is in T (S).
Proposition 5.6. Let R be a Q-algebra and H is a smooth group R-scheme such that H • is reductive, and let h be an element of H(R). Then, the following statements are true.
(1) If h is semi-simple, there exists anétale cover Spec(S) → Spec(R), an integer m 1, and a split maximal torus T of H • S such that h m is in T (S). (2) S is a Q(ζ r )-algebra, where r := [ h : h m ] and ζ r is a primitive r th -root of unity, (3) and S contains an r th -root of all λ such that M (h r , λ) = 0.
Proof. Take anétale cover Spec(S) → Spec(R) and m 1 such that h m is contained in a split torus T of H • S and h m commutes with I. Then h r ∈ h m is contained in T and commutes with I. By [CGP15, Lemma A.8.8] one may decompose M S into character spaces M S (χ). One then observes that M S (h r , λ) is precisely the direct sum of those character spaces M S (χ) such that χ(h r ) = λ. So, M S admits a direct sum decomposition with respect to the spaces M S (h r , λ).
As M S is finitely generated, we know that M S (h r , λ) is trivial for all but finitely many λ 1 , . . . , λ e . In particular, we may further pass to theétale extension S ′ := S[λ 1 /r 1 , . . . , λ 1 /r e , ζ r ]. We extend the action of I on each nontrivial M S ′ (h r , λ) by ρ to the action of the finite where ν travels over the characters We then see that for each τ ∈ (S ′ ) × such that τ r = λ the space M I S ′ (h, τ ) admits a direct decomposition into the spaces M S ′ (h r , λ)[ν] as ν ranges over those characters with ν(h) = λ − 1 /r τ .
One may then check that the module τ M I S ′ (h, τ ) as τ ranges over those elements of (S ′ ) × − R × is stabilized under theétale descent data associated to M I S ′ , and therefore (see [Sta21, Tag 023N]) descends to a submodule M ′ of M I . One sees that M ′ is a complement of λ M I (h, λ) as λ travels over the elements of R × , as this may be checked over the faithfully flat extension S ′ . One may then check that M ′ is independent of all choices, and satisfies the desired conditions.
The following proposition will be helpful to define Frobenius semi-simple in a way that does not require the choice of an explicit arithmetic Frobenius lift.
To define the notion of Frobenius semi-simple parameters, it is useful to have the following analogue of Lemma 3.13.
Proposition 5.9. Let (ϕ, N ) be an element of WDP G (A). Then, the following are equivalent: (1) for any (equiv. one) lift w 0 ∈ W F of arithmetic Frobenius, ϕ(w 0 ) is semi-simple, (2) for some m as in Proposition 5.8, the morphismφ mé tale locally factorizes through a torus ofǦ A .
Proof. By definition, (1) holds if and only if ϕ(w 0 ) has the property that ϕ(w 0 ) mé tale locally lies in a torus of (Ǧ ⋊ Γ * ) • A =Ǧ A for some m as in Proposition 5.8. But, as an element ofǦ A , one easily sees that ϕ(w 0 ) m is preciselyφ m (1). As it is clear that (2) is equivalent to claim that etale locally on A there exists a torus containingφ m (1) the claim follows.
Definition 5.10. For a Q-algebra A, we call an element (ϕ, N ) of WDP G (A) Frobenius semi-simple if it satisfies any of the equivalent conditions of Proposition 5.9.
For each Q-algebra A, let us denote by WDP ss G (A) (resp. WDP K,ss G (A)) the subset of WDP G (A) (resp. WDP K G (A)) consisting of Frobenius semi-simple parameters. It is clear that this forms a G-stable subpresheaf 2 of WDP G (resp. WDP K G ). Note also that by Proposition 5.6, under the bijection of WDP G (C) with Φ WD, G the set WDP ss G (C) corresponds to Φ WD,ss, G . The following technical result will play an important role later in the paper. then (ϕ, N ) is Frobenius semisimple.
Proof. Define S(N ) to be the closed subgroup scheme ofǦ A cut out by the closed condition gN g −1 = p Gm (g)N . We have the equality Z G (ϕ, N ) = ker(p Gm | Z S(N) (ϕ) ). Note that for all x in Spec(A) one has a short exact sequence Definition 5.12. We denote by WDP K,⊔ Now, let us fix a finite extension K of F * Galois over F and a lift w 0 of arithmetic Frobenius. Then, by Proposition 5.3 we have an identification (2) p Gm (γ) = q, N ).
As this definition is clearly functorial, we observe that we may define a subpresheaf U (γ, φ, N ) of WDP K,⊔ G,Q whose A-points are given by is locally movable to (γ, φ, N ) . We then have the following.
Before we prove this proposition, we observe its major consequence. To this end, let us define an equivalence relation on WDP K G (Q) by declaring that (γ, φ, N ) is equivalent to (γ ′ , φ ′ , N ′ ) if there exists some (g, h) ∈ ( G × Z φ,N )(Q) such that (γ ′ , φ ′ , N ′ ) is equal to g(hγ, φ, N )g −1 . Let us denote an equivalence class under this relation by [(γ, φ, N )]. Observe that as we do not require that h to actually lie in Z • φ,N (Q) that [(γ, φ, N )] differs from U (γ, φ, N )(Q). For each such equivalence class, let us choose an element (γ, φ, N ). We consider π 0 (Z φ,N ) as a finite abstract group, and we define an equivalence relation on it by declaring that c is equivalent to c 1 cγc −1 1 γ −1 for any c 1 in π 0 (Z φ,N ). We denote by [c] an equivalence class for this relation.
We then have the following decomposition of WDP K,⊔ G,Q into explicit connected components. U (cγ, φ, N ).
Proof. From Proposition 5.14 we know that each U (cγ, φ, N ) is an open subset of WDP K,⊔ G,Q . As WDP K,⊔ G,Q is a finite type Q-scheme, it thus suffices to prove this claim at the level of Q-points. But, note that by Proposition 5.3, if (γ, φ, N ) satisfies the conditions to be in WDP K G (Q) then (γ ′ , φ, N ) does if and only if γ ′ = hγ for h in Z φ,N (Q). Thus, we have a decomposition Next observe that an element (hγ, φ, N ) may be written in the form g(h ′ γ, φ, N )g −1 if and only if g is in Z φ,N (Q) and hγ = gh ′ γg −1 which implies that h = gh ′ γg −1 γ −1 . With this, it is easy to see that from where the desired equality follows.
From this we deduce the following non-trivial result. Let us denote the set of equivalence classes for WDP K G (Q) (resp. π 0 (Z φ,N )) by [WDP K G (Q)] (resp. [π 0 (Z φ,N )]). Corollary 5.17. The Q-scheme WDP K,⊔ G is smooth, and there is a non-canonical Γ Q -equivariant bijection where the Γ Q action on the target is inherited from WDP K,⊔ G and G.

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The proof of Proposition 5.14. Define the morphism π K : WDP K,⊔ G → Z 1 (I F /I K , G) × N ⊔ by π K (ϕ, N ) = (φ, N ). This morphism is G-equivariant when the target is endowed with the diagonal G-action. Now, by Proposition 2.8 there is a decomposition where J is the set of G(Q) 2 orbits of (Z 1 (I F /I K , G)× N ⊔ )(Q). Observe though that if (ϕ, N ) is in WDP K,⊔ G (Q) with π K (ϕ, N ) = (φ, N ) then φ centralizes N . So, if we set J ′ to be the subset of J consisting of those [(φ 0 , N 0 )] with φ 0 centralizing N 0 then we may produce a factorization As As these schemes are Noetherian, to finish it suffices to show that for all i and all Noetherian Q-algebras A every morphism Spec(A) → U i factorizes through O L,N i . As O N = O N i we may assume without loss of generality that i = 1, and so N i = N . Let N be the element of l A coresponding to Spec(A) → U i . We must then show thatétale locally on A, N is conjugate to N . Let I denotes the nilradical of A, and write A 0 = A/I. As A is Noetherian, I m = (0) for some m, and thus by inducting we may assume that I 2 = (0). Now, as A 0 is reduced the map Spec(A 0 ) → U i factorizes through O L,N and thus N A 0 isétale locally conjugate to N . As thé etale covers of A and A 0 are equivalent (see [Sta21, Tag 04DY]), and we are free to workétale locally on A, we may assume without loss of generality that Ad(l 0 )(N A 0 ) = N for some l 0 in L(A 0 ). As L is smooth, we may apply the infinitesimal lifting criterion to find a lift l in L(A) of l 0 . Replacing N by Ad(l)(N) we may assume without loss of generality that N A 0 = N . Now, as Transp G (N, N ) → Spec(A) is a Z G (N )-torsor, and thus smooth, we know by the infinitesimal lifting criterion that there exists some g in Transp G (N, N )(A) lifting the identity. Using the notation of [DG70, II, §4, №3, 3.7], we may write g = e x for x in I g A . Then, by [DG70, II, §4, №4, 4.2] we have N = Ad(g)(N) = N + ad(x)(N).
As N and N lie in l A , they are invariant for the action of the finite group φ(I F /I K ), and so if y denotes the average of x over the action of φ(I F /I K ) then N = N + ad(y)(N).
But, by loc. cit. this right-hand side is equal to Ad(e y )(N). By Lemma 2.5 we see that e y lies in L(A), from where the claim follows.
Let us now denote by (γ univ , φ univ , N univ ) the universal object over X(φ, N ). Consider the transporter scheme Transp G (φ univ , φ) → Z 1 (I F /I K , G) and set T to be the pullback to X(φ, N ). Set b : T → X(φ, N ) to be the tautological map, which is smooth as T is visibly an L-torsor. Note that we have a morphism a : T → O N ∩ N L given by a(g) = Ad(g)(N univ ) and observe then that we have a scheme-theoretic decomposition T = i a −1 (O L,N i ). But, for each i we also have a map κ i : a −1 (O L,N i ) → π 0 (Z G (φ, N )) given by sending g to the component containing Int(g)(γ univ )γ −1 , and we define for each i and each c ∈ π 0 (Z G (φ, N )) the open subscheme Finally, to show that U (γ, φ, N ) is smooth and irreducible consider the natural morphism G×Z • φ,N → U (γ, φ, N ). To simplify notation let us write S = G×Z • φ,N . Note that, by definition, S → U (γ, φ, N ) is surjective asétale sheaves and thus a fortiori surjective as schemes, and thus U (γ, φ, N ) is irreducible. To see that U (γ, φ, N ) is smooth, note that as S → U (γ, φ, N ) is surjective asétale sheaves there exists anétale cover V → U (γ, φ, N ) such that p : S V → V admits a section. Note though that as S V → S isétale and the target is reduced, so is the source (see [Sta21,Tag 025O]). But, as p has a section, this implies that V is reduced as the morphism of sheaves of rings O V → p * O S has a section and thus is injective. This implies that U (γ, φ, N ) is reduced by [Sta21,Tag 033F]. But, as we're in characteristic 0, this implies that U (γ, φ, N ) is generically smooth over Q (see [Sta21,Tag 056V]). But, as S(Q) acts U (γ, φ, N ) by scheme automorphisms acting transitively on U (γ, φ, N )(Q) we deduce that every point of U (γ, φ, N )(Q) has regular local ring, and thus U (γ, φ, N ) is smooth over Q as desired (see [Sta21,Tag 0B8X]). This completes the proof of Proposition 5.14.

The moduli space of L-parameters and the Jacobson-Morozov morphism
In this section we define the moduli space LP K G of L-parameters for G, show it has favorable geometric properties, construct the Jacobson-Morozov morphism LP K G → WDP K,⊔ G , and show that an analogue of Theorem 3.6 holds for any Q-algebra A.
6.1. The moduli space of L-parameters. We begin with a slight modification of the Langlands group scheme W F × SL 2,Q better suited to arithmetic discussions over Q.
Definition 6.1. We call the Q-scheme representing the functor the twisted Langlands group scheme and denote it L tw F .

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To justify the naming of L tw F , note that if k is any extension of Q and c is any element of k such that c 2 = q, then the morphism , is an isomorphism. For future reference, we observe that we have a morphism Let us also observe that there is a natural embedding of group schemes SL 2,Q → L tw F given by sending g to (1, g), as well as an embedding With these embeddings, we shall implicitly think of SL 2,Q and I F as subfunctors of L tw F . Finally, we observe that the embedding of W K into W F for any finite extension K of F gives rise to an embedding of L tw K → L tw F which we implicitly use to think of L tw K as a subgroup scheme of L tw F .
Definition 6.2. For a Q-algebra A we define an L-parameter over A to be a homomorphism of group A-schemes ψ : Denote by LP G (A) the set of L-parameters over A, which is functorial in A. Note that LP G has a natural conjugation action by G and so one has the centralizer group presheaf Z G (ψ).
For an L-parameter ψ over A we define the morphismψ : L tw F,A →Ǧ A as the composition of ψ with the projection C G A →Ǧ A . We denote by ψ the homomorphism of group A-schemes L tw F,A → ( G⋊Γ * ) A obtained by composing ψ with the quotient homomorphismǦ A → ( G⋊Γ * ) A . Let us observe that whileψ may not be a homomorphism, it becomes so after restriction to L tw F * ,A . Finally, by our assumptions on ψ the restriction to SL 2,A takes values in G A and we denote this resulting morphism SL 2,A → G A by θ (or θ ψ when we want to emphasize ψ).
To relate this to more familiar objects, fix k to be an extension of Q containing an element c such that c 2 = q. For a k-algebra A, we endow G(A) with the discrete topology and set (1) ψ is a homomorphism over W F , There is a morphism i L c : LP ′ G,k → LP G,k which on A-points is given by sending ψ ′ to the element ψ of LP K G (A) that is equal to i c • ψ ′ • η −1 c on A-points, where ψ is uniquely determined by Proposition 2.6. We can show the following proposition in the same way as Proposition 5.2.
which clearly forms a subpresheaf of LP G . We have the equality of presheaves LP G = lim − →K LP K G . As in the case of Weil-Deligne parameters, may associate to an L-parameter ψ in LP K G (A) an element φ of Z 1 (I F /I K , G)(A) and thus obtain a morphism of presheaves LP K G → Z 1 (I F /I K , G). given by the two maps We then have the following explicit description of LP K G . Proposition 6.4. The morphism j w 0 gives an identification of LP K G with Eq(D L ). In particular, LP K G is representable by a finite type affine Q-scheme and j w 0 is a closed embedding. As already observed, for an extension K ⊆ K ′ of finite extensions of F * Galois over F , there is a restriction morphism Z 1 (I F /I K ′ , G) → Z 1 (I K /I K ′ , G) which is a clopen embedding, and thus LP K G → LP K ′ G is also a clopen embedding. As we have the identification of presheaves LP G = lim − →K LP K G we deduce from Proposition 5.3 that LP G is representable by a scheme locally of finite type over Q, all of whose connected components are affine.
6.2. Decomposition into connected components. We now establish the analogue of Theorem 5.16 for LP G . Let us fix K a finite extension of F * Galois over F , and a lift w 0 of arithmetic Frobenius. Then, by Proposition 6.4 we have an identification . This is a linear algebraic group over Q whose identity component is reductive. Let us then say that an element (γ ′ , φ ′ , θ ′ ) in LP K G (A), for a Q-algebra A, is locally movable to (γ, φ, θ) if there exists anétale cover Spec(A ′ ) → Spec(A) and (g, h) ∈ ( G × Z • φ,θ )(A ′ ) such that (γ ′ , φ ′ , θ ′ ) = g(hγ, φ, θ)g −1 . As this definition is clearly functorial, we obtain a subpresheaf of LP K G,Q as follows: is locally movable to (γ, φ, θ) . We then have the following, whose proof is identical to Proposition 5.14 except the analogue of Lemma 5.18 is simpler considering Proposition 4.10.
We then have the following decomposition of LP K G,Q into explicit connected components, whose proof is exactly the same as that of Theorem 5.16.
We derive from this two corollaries neither of which is a priori obvious.
Corollary 6.8. The affine Q-scheme LP K G is smooth, and there is a non-canonical Γ Q -equivariant bijection where the Γ Q action on the target is inherited from LP K G and G. 6.3. The Jacobson-Morozov morphism. We now come to the definition of the Jacobson-Morozov map in the geometric setting.
It is clear that JM is G-equivariant. By Theorem 4.14 it is also clear that JM factorizes uniquely through WDP ⊔ G . Moreover, for any finite extension K of F * Galois over F , one sees that JM −1 (WDP K G ) is precisely LP K G and so we get factorizations LP K G → WDP K G and LP K G → WDP K,⊔ G . We denote all these factorizations also by JM. Observe that over Q we may give a simpler description of the Jacobson-Morozov morphism on each connected component. Namely, let us fix (γ, φ, θ) in LP K G (Q) as in the notation of §6.2. Then, first observe that JM(γ, φ, θ) is equal to (γ, φ, N ) where N = JM(θ). We may then observe that JM restricted to U (γ, φ, θ) maps into U (γ, φ, N ) and is theétale sheafification of the map which on A-points is the map given by sending g(hγ, φ, θ)g −1 to g(hγ, φ, N )g −1 .
We also observe that if k is an extension of Q and c is an element of k such that c 2 = q then under the isomorphisms described in Proposition 5.2 and Proposition 6.3 that the Jacobson-Morozov corresponds to the morphism LP ′ G,k → WDP ′ G,k sending ψ to the map on A-points of (ψ • ι ′ , dθ ψ (e 0 )) where So, on the level of C-points we see that our Jacobson-Morozov map agrees with that from §3.2. We now move towards stating the analogue of Theorem 3.6 at the level of A-points. To begin, we must define the notion of semi-simplicity for L-parameters in the relative setting.
Proposition 6.10. Let ψ be an L-parameter over a Q-algebra A. Then there is a positive integer m divisible by [F * : F ] such that the morphism admits a factorization Proof. This is proved in the same way as Proposition 5.8.
Definition 6.11. For A a Q-algebra, we call an element ψ of LP G (A) Frobenius semi-simple if there exists an integer m as in Proposition 6.10 such thatψ m factors through a subtorus ofǦ A etale locally on A.
Let us denote by LP ss G (A) (resp. LP K,ss G (A)) the subset of Frobenius semi-simple elements of LP G (A) (resp. LP K G (A)). This evidently forms a G-stable subfunctor of LP G (resp. LP K G ).
Remark 6.12. To understand the reasoning for this definition, observe that under the isomorphism in Proposition 6.3, this condition corresponds to an element ψ ′ of LP ′ G,k (A) satisfying the property that the projection of ψ ′ (w 2m 0 , 1) to G(A) is semi-simple for some m as in Proposition 6.10. In particular, this notion of semi-simple agrees with that from §3.2 for C-points by Lemma 3.13.
We now prove the following surprisingly subtle semi-simplicity preservation property for the Jacobson-Morozov morphism.
Proposition 6.13. Let A be a Q-algebra and ψ an element of LP G (A). Then, ψ is Frobenius semi-simple if and only if JM(ψ) is.
Proof. Suppose that ψ is Frobenius semi-simple. As the conclusion is insensitive to passing to anétale extension and conjugating, we do so freely. Take m as in Proposition 6.10 and a split maximal torus T ofǦ A such thatψ m factors through T . Note that the eigenspaceǧ A (1) with respect toψ m (1) is the Lie algebra of a Levi subgroup L ofǦ A such thatψ m factors through Z(L). Indeed, we may assume that T = (T 0 ) A for a maximal torus T 0 ofǦ. Let L ′ be the Levi subgroup ofǦ generated by T 0 and the root groups for the roots α which annihilateψ m (1). Then, we may take L = L ′ A , whereψ m factors through Z(L) by [Con14, Corollary 3.3.6]. Note that θ factorizes through L as by Proposition 2.6 it suffices to check this on the level of Lie algebras, from where it is clear. Let T 2 denote the standard diagonal subtorus of SL 2,A . Since θ factorizes through L, by [Con14, Lemma 5.3.6] we may assume that the map θ| T 2 factorizes through a maximal torus T ′ of L. But, as Z(L) ⊆ T ′ both θ| T 2 andψ m factorize through T ′ . Hence, if we write JM(ψ) = (ϕ, N ) then the morphism W F,A →Ǧ A given by w → ϕ(w m ) factors through T ′ . This implies that JM(ψ) is Frobenius semi-simple.
Conversely, suppose that JM(ψ) = (ϕ, N ) is Frobenius semi-simple. Let m be any integer as Proposition 5.8. As above, we may build a reductive subgroup L m ofǦ A such that Lie(L m ) is identified withǧ A (1) with respect toφ m (1). We claim that the group L km stabilizes for k sufficiently large. Indeed, the roots of α ofǦ relative to T 0 that annihilateφ km (1) =φ m (1) k stabilize for k sufficiently large, from where the claim follows by the construction. Denote by L the group L km for k sufficiently large, say for k k 0 . Let us write Z for the torus Z(L) • (see [Con14,Theorem 3.3.4]). Observe that asφ km , for k k 0 , centralizes Lie(L) thatφ km factors through Z(L). So then, for some k 1 k 0 we have thatφ k 1 m factors through Z. We put m 1 = k 1 m. We will be done if we can show that θ| T 2 factorizes through the reductive group Ascheme ZǦ(Z) (see [ [Con14,Lemma 5.3.6], we know that after passing to anétale extension, θ| T 2 factorizes through a maximal torus T ′ of ZǦ(Z). Then θ| T 2 andφ m 1 factor through T ′ . Hence factors through T ′ . This implies that ψ is Frobenius semi-simple. Working etale locally, and by passing to aǦ(A)-conjugate, we may assume that Z is equal to Z ′ A for a split subtorus Z ′ ofǦ. Let R 0 be the set of nontrivial characters of Z ′ appearing in the adjoint action of Z ′ onǧ A . Note that these characters are already defined over Q. Consider the functor on Alg Q with Y (B) := z ∈ Z ′ (B) : (1) χ(z) = 1 for all χ ∈ R 0 , (2) χ(z) = q m 1 for all χ ∈ R 0 such that χ(φ m 1 (1)) = q m 1 .
Clearly Y defines a locally closed subscheme of Z ′ which is non-empty asφ m 1 (1) is an element of Y (A). Take y ∈ Y (F ) for a finite extension F of Q. By passing to anétale extension, we may assume that A contains F . We claim that inclusion ZǦ(Z) ⊆ ZǦ(y) • A is an equality. As ZǦ(Z) is flat over Spec(A), we know from the fibral criterion for isomorphism (see [EGA4-4, Corollaire 17.9.5]), that it suffices to check this after base change to every point of Spec(A). But, as A is Q-algebra, and ZǦ(Z) and ZǦ(y) • A are both connected, it then suffices to check they have the same Lie algebra (e.g. see [Mil17,Corollary 10.16]), but this is true by construction.
In the following, we use the notationǧ A (λ) for λ ∈ A × with respect toφ m 1 (1). By construction, we know that Int(y) acts onǧ A (q ±m 1 ) by multiplication by q ±m 1 . Moreover, the SL 2 -triple (N, f, h) associated to θ by Theorem 4.17 satisfies N ∈ǧ A (q m 1 ), f ∈ǧ A (q −m 1 ) and h ∈ǧ A (1). Therefore, the sl 2 -triple attached to Int(y) • θ is (q m 1 N, q −m 1 f, h). Thus, the sl 2 -triple attached to Int(y) • θ • µ is (N, f, h) where By Theorem 4.17 Int(y) • θ • µ = θ, so θ| T 2 factorizes through ZǦ(y) • A = ZǦ(Z) as desired. We end this section by proving a relative version of Proposition 3.7. Fix a Q-algebra A and let N be an element of N ⊔ (A). Let us denote by u N the A-submodule im(ad(N )) ∩ Ker(ad(N )) of g A , which we also treat as a subfunctor of g A in the obvious way. Note that u N is in fact a closed subscheme of N A and for all A-algebras B there is an equality u N (B) = im(ad(N ⊗ 1)) ∩ Ker(ad(N ⊗ 1)).
As these claims areétale local, we may assume that N = gN 0 g −1 for some N 0 in N (Q) and g in G(A). Observe then that u N is equal to g(u N 0 ) A g −1 where u N 0 ⊆ g is defined in the same way as u N . As N A is G(A)-equivariant it suffices to show that u N 0 factorizes through N which may be checked on Q-points which is then clear. One similarly proves the claimed equality.
As u N is a closed subscheme of N A , we obtain a closed subscheme U N := exp(u N ) of G A . We claim that U N is a closed subgroup scheme of G A flat over A. As this may be checked etale locally we are again reduced to checking that exp(u N 0 ) is a closed subgroup Q-scheme of G (automatically flat over Q), but this is true by Proposition 2.3. For an element (ϕ, N ) of WDP ⊔ G (A) we set U N (ϕ) := U N × G A Z G (ϕ).
Concretely this means that for every A-algebra B one has an identification of U N (ϕ)(B) with U N (B) ∩ Z G (ϕ)(B) where this intersection is taken in G(B). Let us first establish the following relative version of Proposition 3.3, which follows easily (using the same reduction arguments as already used above) from Proposition 3.3 Lemma 6.14. Let θ be an element of Hom(SL 2,Q , G)(A) and define N = JM(θ). Then, Z G (N ) = U N ⋊ Z G (θ).
Proof. Let B be an A-algebra. Given Lemma 6.14 it clearly suffices to show that conjugation by an element in the image of ϕ stabilizes U N , as the rest of the argument for Proposition 3.7 then goes through verbatim. Let u = exp(n) be an element of U N (B) and observe that Int(ϕ(w))(u) is equal to exp(Ad(ϕ(w))(n)), and so we are done as clearly Ad(ϕ(w))(n) ∈ u N (B). 6.4. The relative Jacobson-Morozov theorem for parameters. We now arrive at the relative analogue of Theorem 3.6. Let us set WDP ⊔,ss G to be the presheaf whose A-points consist of Frobenius semi-simple Weil-Deligne parameters (ϕ, N ) such that N lies in N ⊔ (A).
Theorem 6.16 (Relative Jacobson-Morozov theorem for parameters). The Jacobson-Morozov morphism JM : LP ss G → WDP ⊔,ss G is surjective, and induces an isomorphism of quotient presheaves JM : LP ss G / G ∼ − → WDP ⊔,ss G / G. Let us fix a Q-algebra A, an element (ϕ, N ) of WDP ⊔,ss G (A), and an arithmetic Frobenius lift w 0 ∈ W F,A . In the notation from Proposition 5.7, with ρ : (Ǧ ⋊ Γ * ) A → GL( g A ) the adjoint action, h = ϕ(w 0 ), and I = φ(I F /I K ), let h and h(λ) be g I A and g I A (λ) respectively.
Step 4: Replacing G with G der we may assume that Z 0 ( G) is finite. Proposition 5.11 together with Theorem 6.16 imply that (ϕ η , N η ) (resp. (ϕ s , N s )) comes from an L-parameter ψ 1 (resp. ψ 2 ). Write µ i for the restriction of θ ψ i to the diagonal maximal torus. Fix w 0 to be an arithmetic Frobenius lift. By Frobenius semi-simplicity and the fact that A is strictly Henselian, there is, up to conjugacy, a positive integer m 0 divisible by [F * : F ] such thatφ(w m 0 0 ) is contained in the A-points of a maximal torus T ofǦ Q . By the relationship between ψ i and ϕ i and the argument of [GR10, Lemma 3.1], we see that up to replacing m 0 by a power, we may further assume thatφ η (w 2m 0 0 ) = µ 1 (q m 0 ) andφ s (w 2m 0 0 ) = µ 2 (q m 0 ). From this first equality it is simple to see that µ 1 factorizes through T η , and thus there exists a unique lift µ A of µ 1 to T A where µ is a cocharacter of T . We note as N s = 0, that µ 2 is characterized by the property that the image of µ 2 containsφ s (w 2m 0 0 ) and Ad(µ 2 (q m 0 ))(N s ) = q 2m 0 N s . AsǦ A and g A are separated over A, we have that the image of µ contains ϕ(w 2m 0 0 ) and Ad(µ(q m 0 ))(N ) = q 2m 0 N . Hence, µ s satisfies the above characterization of µ 2 , so µ s = µ 2 . Let P (µ) be the parabolic subgroup of G Q associated to µ. Define g η (j) (resp. g s (i)) using µ η (resp. µ s ) as in [Car85,§5.7]. Then by [Car85,Proposition 5.7.3] N η (resp. N s ) is in the unique open P (µ) η -orbit (resp. P (µ) s ) of i≥2 g η (i) (resp. i≥2 g s (i)). But, by the uniqueness of this open orbit, we then see that N η and N s are both conjugate to any Q-point of the unique open orbit of P (µ) on i≥2 g(i), from where the conclusion follows. We are then done by Proposition 4.8.
We next show the pleasant property that WDP K,rc G actually has dense image in WDP K,⊔ G . Lemma 7.6. Let k be a field, X an irreducible finite type k-scheme equipped with an action of an algebraic k-group H, and Y an irreducible locally closed subscheme of X. Assume that the action morphism µ : H × Y → X is dominant. Then there is a dense open subset U of Y such that dim Z H (y) dim(H) + dim(Y ) − dim(X) for all y ∈ U .
Proof. By [GW20, Corollary 14.116] there exists a dense open subset V of X with the property that dim µ −1 (y) = dim H + dim Y − dim X for all y ∈ V . As µ is H-equivariant when H is made to act on the first component of H × Y , we may assume that V is H-stable. We put U = V ∩ Y , which is non-empty as µ is dominant and V is H-stable. As Z H (y) × {y} ⊆ µ −1 (y) for y ∈ U , we obtain the claim.