On the Moy-Prasad filtration

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.


Introduction
The introduction of Moy-Prasad filtrations in the 1990s revolutionized the study of the representation theory of p-adic groups. As one example, their introduction enabled a construction of supercuspidal representations -the building blocks in the representation theory of p-adic groups -that is exhaustive for large primes p under certain tameness assumptions. However, while this and similar advances are remarkable, the restrictions on the prime p are unsatisfying. Given their critical role, we expect that a better understanding of the Moy-Prasad filtrations will be a key ingredient for future progress. To that end, we introduce a "global" model for the Moy-Prasad filtration quotients. This allows us to compare the Moy-Prasad filtrations for different primes p and to deduce results for all primes p that were previously only known for large primes. Our global model also enables us to express the Moy-Prasad filtration quotients in terms of more traditional, well studied concepts, e.g. as explicit Weyl modules or in terms of a generalized Vinberg-Levy theory. As an application, we exhibit new supercuspidal representations for non-split p-adic groups, including non-tame groups.
To explain the content and background of the paper in more detail, let us introduce some notation. Let k be a nonarchimedean local field with residual characteristic p > 0. Let K be a maximal unramified extension of k and identify its residue field with F p . Let G be a reductive group over K. In [BT72,BT84], Bruhat and Tits defined a building B(G, K) associated to G. For each point x in B(G, K), they constructed a compact subgroup G x of G(K), called a parahoric subgroup. In [MP94,MP96], Moy and Prasad defined a filtration of these parahoric subgroups by smaller subgroups G x = G x,0 G x,r 1 G x,r 2 . . . , where 0 < r 1 < r 2 < . . . are real numbers depending on x. For simplicity, we assume that r 1 , r 2 , . . . are rational numbers. The quotient G x,0 /G x,r 1 can be identified with the F p -points of a reductive group G x , and G x,r i /G x,r i+1 (i > 0) can be identified with an F p -vector space V x,r i on which G x acts.
Results about Moy-Prasad filtrations. We show for a large class of reductive groups G, which we call good groups (see Definition 3.1.1), that Moy-Prasad filtrations are in a certain sense (made precise below) independent of the residue-field characteristic p. The class of good groups contains reductive groups that split over a tamely ramified field extension (which is the class that many authors restrict to), as well as simply connected and adjoint semisimple groups, and products and restriction of scalars along finite separable (not necessarily tamely ramified) field extensions of any of these. The restriction to this (large) subclass of reductive groups is necessary as the main result (Theorem 3.4.1) fails in general. Given a good reductive group G over K, a rational point x of the Bruhat-Tits building B(G, K) and an arbitrary prime q coprime to a certain integer N that depends on the splitting field of G (for details see Definition 3.1.1), we construct a finite extension K q of Q ur q , a reductive group G q over K q and a point x q in B(G q , K q ). To these data, one can attach a Moy-Prasad filtration as above. The corresponding reductive quotient G xq is a reductive group over F q that acts on the quotients V xq,r i , which are identified with F q -vector spaces. For a given positive integer i, we show in Theorem 3.4.1 that there exists a split reductive group scheme H over Z[1/N ] acting on a free Z[1/N ]-module V such that the special fibers of this representation are the above constructed Moy-Prasad filtration representations of G xq on V xq,r i . This allows us to compare the Moy-Prasad filtration representations for different primes.
We also give a new description of the Moy-Prasad filtration representations, i.e. of G x acting on V x,r i , for reductive groups that split over a tamely ramified field extension of K. Let m be the order of x. We define an action of the group scheme µ m of m-th roots of unity on a reductive group G Fp over F p , and denote by G µ m ,0 Fp the identity component of the fixed-point group scheme. In addition, we define a related action of µ m on the Lie algebra Lie(G Fp ), which yields a decomposition . Then we prove that the action of G x on V x,r i corresponds to the action of G µ m ,0 Fp on one of the graded pieces Lie(G ) j (F p ) of the Lie algebra of G Fp . This was previously known by [RY14] for sufficiently large primes p, and representations of the latter kind have been studied by Vinberg [Vin76] in characteristic zero and generalized to positive characteristic coprime to m by Levy [Lev09]. To be precise, in this paper we even prove a global version of the above mentioned result. See Theorem 4.1.1 for details. We also show that the same statement holds true for all good reductive groups after base change of H and V to Q, see Corollary 4.2.1.
Moreover, the global version of the Moy-Prasad filtration representations given by Theorem 3.4.1 allows us to describe the representations occurring in the Moy-Prasad filtrations of good reductive groups explicitly in terms of Weyl modules if p = 2 or if the restricted root system of G contains no multiple root, see Section 6 for precise formulas.
An application to supercuspidal representations. Suppose G is defined over k. In 1998, Adler ([Adl98]) used the Moy-Prasad filtrations to construct supercuspidal representations of G(k), and Yu ([Yu01]) generalized his construction three years later. Kim ([Kim07]) showed that, for large primes p, Yu's construction yields all supercuspidal representations if we impose certain tameness assumptions on G. However, it is known that the construction does not give rise to all supercuspidal representations for small primes.
In 2014, Reeder and Yu ([RY14]) gave a new construction of supercuspidal representations of smallest positive depth, which they called epipelagic representations. A vector in the dualV x,r 1 = (G x,r 1 /G x,r 2 ) ∨ of the first Moy-Prasad filtration quotient is called stable (in the sense of geometric invariant theory) if its orbit under G x is closed and its stabilizer in G x is finite. The only input for the new construction of supercuspidal representations in [RY14] is such a stable vector. Assuming that G is a semisimple group that splits over a tamely ramified field extension, Reeder and Yu gave a necessary and sufficient criterion for the existence of stable vectors for sufficiently large primes p. In [FR17], Romano and the author removed the assumption on the prime p for absolutely simple split reductive groups G, which yielded new supercuspidal representations for split groups.
One application of our results on Moy-Prasad filtrations is a criterion for the existence of stable vectors for all primes p for a much larger class of semisimple groups, see Corollary 5.2.2. As a consequence we obtain new supercuspidal representations for a class of non-split p-adic reductive groups, including non-tame groups. Similarly, we prove in Theorem 5.1.1 that the existence of semistable vectors is independent of the residue-field characteristic. Semistable vectors play an important role when moving from epipelagic representations to representations of higher depth.
We expect that our new descriptions of the Moy-Prasad filtration will be useful for further applications beyond the ones explored in this paper.
Structure of the paper. In Section, 2 we first recall the Moy-Prasad filtration of G, and then in Section 2.5 we introduce a Chevalley system for the reductive quotient that will be used for the construction of the reductive group scheme H that appears in Theorem 3.4.1. In Section 2.6, we construct an inclusion of the Moy-Prasad filtration representation of G into that of G F for a sufficiently large field extension F of K that will allow us to define the action of H on V in Theorem 3.4.1. Afterwards, in Section 3, we move from a previously fixed residue-field characteristic p to other residue-field characteristics q. More precisely, we first introduce the notion of a good group and define K q /Q ur q , G q over K q , and x q ∈ B(G q , K q ). In Section 3.4, we prove our first main theorem, Theorem 3.4.1. Section 4 is devoted to giving a different description of the Moy-Prasad filtration representations and their global version as generalized Vinberg-Levy representations (Theorem 4.1.1). In Section 5, we use the results of the previous sections to show that the existence of (semi)stable vectors is independent of the residue characteristic. This leads to new supercuspidal representations. We conclude the paper by giving a description of the Moy-Prasad filtration representations in term of Weyl modules in Section 6.
Conventions and notation. If M is a free module over some ring A, and if there is no danger of confusion, then we denote the associated scheme whose functor of points is B → M ⊗ A B for any A-algebra B also by M . In addition, if G and T are schemes over a scheme S, then we may abbreviate the base change G × S T by G T ; and, if T = Spec A for some ring A, then we may also write G A instead of G T .
When we talk about the identity component of a smooth group scheme G of finite presentation, we mean the unique open subgroup scheme whose fibers are the connected components of the respective fibers of the original scheme that contains the identity. The identity component of G will be denoted by G 0 .
Throughout the paper, we require reductive groups to be connected.
For each prime number q, we fix an algebraic closure Q q of Q q and an algebraic closure F q ((t)) of F q ((t)). All field extensions of Q q and F q ((t)) are assumed to be contained in Q q and F q ((t)), respectively. We then denote by Q ur q the maximal unramified extension of Q q (inside Q q ), and by F q ((t)) ur the maximal unramified extension of F q ((t)). For any field extension F of Q q (or of F q ((t))), we denote by F tame its maximal tamely ramified field extension. Similarly, we fix an algebraic closure Q of Q, and we denote by Z the integral closure of Z in Q and by Z q the integral closure of Z q in Q q .
In addition, we will use the following notation throughout the paper: p denotes a fixed prime number, k is a nonarchimedean local field (of arbitrary characteristic) with residual characteristic p, and K is the maximal unramified extension of k contained in the fixed algebraic closure above. We write O for the ring of integers of K, v : K → Z ∪ {∞} for a valuation on K with image Z ∪ {∞}, and for a uniformizer. G is a reductive group over K, and E denotes a splitting field of G, i.e., E is a minimal field extension of K such that G E is split. Note that all reductive groups over K are quasi-split and hence E is unique up to conjugation. Let e be the degree of E over K, O E the ring of integers of E, and E a uniformizer of E. Without loss of generality, we assume that is chosen to equal e E modulo e+1 E O E . We denote the (absolute) root datum of G by R(G), and its root system by Φ = Φ(G). We fix a point x in the Bruhat-Tits building B(G, K) of G, denote by S a maximal split torus of G such that x is contained in the apartment A (S, K) associated to S, and let T be the centralizer of S, which is a maximal torus of G. Moreover, we fix a Borel subgroup B of G containing T , which yields a choice of simple roots ∆ in Φ. In addition, we denote by Φ K = Φ K (G) the restricted root system of G, i.e., the restrictions of the roots in Φ from T to S. Restriction yields a surjection from Φ to Φ K , and for a ∈ Φ K , we denote its preimage in Φ by Φ a . Moreover, to help the reader, we will adhere to the convention of labeling roots in Φ by Greek letters: α, β, . . ., and roots in Φ K by Latin letters: a, b, . . ..

Acknowledgment.
The author thanks her advisor, Benedict Gross, for his support. In addition, she thanks Mark Reeder for his lectures on epipelagic representations and inspiring discussions during the spring school and conference on representation theory and geometry of reductive groups. The author also thanks Jeffrey Adler, Stephen DeBacker, Brian Conrad, Wee Teck Gan, Thomas Haines, Tasho Kaletha, Gopal Prasad, Beth Romano, Cheng-Chiang Tsai and Zhiwei Yun for interesting discussions related to this paper. Moreover, the author is particularly grateful to Jeffrey Hakim for comments on an initial draft, to Loren Spice for carefully reading part of an earlier version of this paper, and to Stephen DeBacker for his feedback on several versions of the introduction.

Parahoric subgroups and Moy-Prasad filtration
In order to talk about the Moy-Prasad filtration, we will first recall the structure of the root groups following [BT84,Section 4]. For more details and proofs we refer to loc. cit.

Chevalley-Steinberg system
For α ∈ Φ, we denote by U E α the root subgroup of G E corresponding to α. Note that Γ = Gal(E/K) acts on Φ. We denote by E α the fixed subfield of E of the stabilizer Stab Γ (α) of α in Γ. In order to parameterize the root groups of G over K, we fix a Chevalley-Steinberg system {x E α : G a → U E α } α∈Φ of G with respect to T , i.e. a Chevalley system {x E α : G a → U E α } α∈Φ of G E (see Remark 2.1.1) satisfying the following additional properties for all roots α ∈ Φ: (iii) If the restriction a ∈ Φ K of α to S is divisible, then there exist β, β ∈ Φ restricting to a 2 , According to [BT84,4.1.3] such a Chevalley-Steinberg system does exist. It is a generalization of a Chevalley system for non-split groups and it will allow us to define a valuation of root groups in Section 2.2 even if the group G is non-split.
Remark 2.1.1. We follow the conventions resulting from [SGA 3 III new , XXIII Définition 6.1], so we do not add the requirement of Bruhat and Tits that for each root α, Definition 2.1.2. For α, β ∈ Φ, we define α,β ∈ {±1} by where s α denotes the reflection in the Weyl group W of Φ(G) corresponding to α. The integers α,β for α and β in Φ are called the signs of the Chevalley-Steinberg system {x E α } α∈Φ .

Parametrization and valuation of root groups
In this section, we associate a parametrization and a valuation to each root group of G.
Let a ∈ Φ K = Φ K (G), and let U a be the corresponding root subgroup of G, i.e., the connected unipotent (closed) subgroup of G normalized by S whose Lie algebra is the sum of the root spaces corresponding to the roots that are a positive integral multiple of a.
Let G a be the subgroup of G generated by U a and U −a , and let π : G a → G a be a simply connected cover. Note that π induces an isomorphism between a root group U + of G a and U a . We call U + the positive root group of G a . In order to describe the root group U a , we distinguish two cases.
Case 1: The root a ∈ Φ K is neither divisible nor multipliable, i.e. a 2 and 2a are both not in Φ K . Let α ∈ Φ a be a root that equals a when restricted to S. Then G a is isomorphic to the Weil restriction Res Eα/K SL 2 of SL 2 over E α to K, and U a Res which we call a parametrization of U a . Note that for u ∈ Res Eα/K G a (K) = E α , we have This allows us to define the valuation ϕ a : Case 2: The root a ∈ Φ K is divisible or multipliable, i.e. a 2 or 2a ∈ Φ K . We assume that a is multipliable and describe U a and U 2a . Let α, α ∈ Φ a be such that α + α is a root in Φ. Then G a is isomorphic to Res E α+ α /K SU 3 , where SU 3 is the special unitary group over E α+ α defined by the hermitian form (x, y, z) → σ(x)z + σ(y)y + σ(z)x on E 3 α with σ the nontrivial element in Gal(E α /E α+ α ). Hence, in order to parametrize U a , we first parametrize the positive root group U + of SU 3 . To simplify notation, write L = E α = E α and L 2 = E α+ α . Following [BT84], we define the subset H 0 (L, L 2 ) of L × L by Viewing L × L as a four dimensional vector space over L 2 , and considering the corresponding scheme over L 2 (as described in "Conventions and notation" in Section 1), we can view H 0 (L, L 2 ) as a closed subscheme of L × L over L 2 , which we will again denote by H 0 (L, L 2 ). Then there exists an L 2 -isomorphism µ : H 0 (L, L 2 ) → U + given by where σ is induced by the nontrivial element in Gal(L/L 2 ). Using this isomorphism, we can transfer the group structure of U + to H 0 (L, L 2 ) and thereby turn the latter into a group scheme over L 2 . Let us denote the restriction of scalars Res L 2 /K H 0 (L, L 2 ) of H 0 (L, L 2 ) from E α+ α = L 2 to K by H(L, L 2 ). Then, by identifying G a with Res E α+ α /K SU 3 , we obtain an isomorphism which we call the parametrization of U a . We can describe the isomorphism x a on K-points as follows.
Let [Φ a ] be a set of representatives in Φ a of the orbits of the action of Gal(E α /E α+ α ) = σ on Φ a . We will choose the sets of representatives for Φ a and Φ −a such that [Φ a ] and −[Φ −a ] are disjoint. For β ∈ [Φ a ], choose γ ∈ Gal(E/K) such that β = γ(α) and set β = γ( α) and u β = γ(u) for every u ∈ L. By replacing some x E where m β is defined as in Remark 2.1.1) 1 . Moreover, we choose the identification of G a with Res E α+ α /K SU 3 so that its restriction to the positive root group arises from the restriction of scalars of the identification that satisfies The root group U 2a corresponding to 2a is the subgroup of U a given by the image of x a (0, v). Hence U 2a (K) is identified with the group of elements in E α of trace zero with respect to the quadratic extension E α /E α+ α , which we denote by E 0 α . Using the parametrization x a , we define the valuation ϕ a of U a (K) and ϕ 2a of U 2a (K) by (ii) The valuation of the root groups U a can alternatively be defined for all roots a ∈ Φ K as follows. Let u ∈ U a (K), and write u = α∈Φa∪Φ 2a where ϕ E α (x α (v)) = v(v). The equivalence of the definitions is an easy exercise, see also [BT84,4 1 Note that our choice of x E β or x E β+ β for negative roots β, β deviates from Bruhat and Tits. It allows us a more uniform construction of the root group parameterizations that does not require us to distinguish between positive and negative roots, but that coincides with the ones defined by Bruhat and Tits in [BT84].

Affine roots
Recall that the apartment A = A (S, K) corresponding to the maximal split torus S of G is an affine space under the R-subspace of X * (S) ⊗ Z R spanned by the coroots of G, where X * (S) = Hom K (G m , S). The apartment A can be defined as corresponding to all valuations of (T (K), (U a (K)) a∈Φ K ) in the sense of [BT72, Section 6.2] that are equipolent to the one constructed in Section 2.2, i.e., families of maps ( ϕ a : U a (K) → R∪{∞}) a∈Φ K such that there exists v ∈ X * (S)⊗ Z R satisfying ϕ a (u) = ϕ a (u) + a(v) for all u ∈ U a (K), for all a ∈ Φ K . In particular, the valuation defined in Section 2.2 corresponds to a point in A that we denote by x 0 . Then the set of affine roots Ψ K on A consists of the affine functions on A given by It will turn out to be handy to introduce a more explicit description of Γ a . In order to do so, consider a multipliable root a and α ∈ Φ a , and define Then, by [BT84, 4.2.20, 4.2.21], the set (E α ) 1 max is nonempty, and, with λ any element of (E α ) 1 max and a still being multipliable, we have For a being neither multipliable nor divisible and α ∈ Φ a , we have Remark 2.3.1. Note that if the residue-field characteristic p is not 2, then 1 2 ∈ (E α ) 1 max for a a multipliable root and α ∈ Φ a , and hence Γ a = v(E α − {0}). If the residue-field characteristic is p = 2, then v(λ) < 0 for λ ∈ (E α ) 1 max .

Moy-Prasad filtration
Bruhat and Tits ( [BT72,BT84]) associated to each point x in the Bruhat-Tits building B(G, K) a parahoric group scheme over O, which we denote by P x , whose generic fiber is isomorphic to G. We will quickly recall the filtration of G x := P x (O) introduced by Moy and Prasad in [MP94,MP96] and thereby specify our convention for the involved parameter.
Then T 0 is a subgroup of finite index in the maximal bounded subgroup {t ∈ T (K) | v(χ(t)) = 0 ∀ χ ∈ X * (T ) = Hom K (T, G m )} of T (K). Note that this index equals one if G is split.
For every positive real number r, we define For every affine root ψ ∈ Ψ K , we denote byψ its gradient and define the subgroup U ψ of Uψ(K) by Then the Moy-Prasad filtration subgroups of G x are given by The quotient G x /G x,0+ can be identified with the F p -points of the reductive quotient of the special fiber P x × O F p of the parahoric group scheme P x , which we denote by G x . From [BT84, Corollaire 4.6.12] we deduce the following lemma.
We can define a filtration of the Lie algebra g = Lie(G)(K) similar to the filtration of G x . In order to do so, we denote the O-lattice Lie(P x ) of g by p. Define p a = p ∩ g a for a ∈ Φ K and t = Lie(T )(K).
We define the Moy-Prasad filtration of the Lie algebra t for r ∈ R to be For every root a ∈ Φ K , we define the Moy-Prasad filtration of g a as follows. Let ψ a be the smallest affine root with gradient a such that ψ a (x) ≥ 0. For every ψ ∈ Ψ K with gradient a, we let n ψ = e α (ψ − ψ a ), where e α = [E α : K] for some root α ∈ Φ a that restricts to a. Note that n ψ is an integer. Choosing a uniformizer α ∈ E α and viewing p a inside Lie(G)(E α ) we set 2 Then the Moy-Prasad filtration of the Lie algebra g is given by In general, the quotient G x,r /G x,r+ is not isomorphic to g x,r /g x,r+ for r > 0. However, it turns out that we can identify them (as F p -vector spaces) under the following assumption.
Assumption 2.4.2. The maximal split torus T of G becomes an induced torus over a tamely ramified extension.
Recall that the torus T is called induced if it is a product of separable Weil restrictions of G m , i.e.
Res K i /K G m for some integer N and finite separable field extensions For the rest of Section 2, we impose Assumption 2.4.2.
Remark 2.4.3. Assumption 2.4.2 holds, for example, if G is either adjoint or simply connected semisimple, or if G splits over a tamely ramified extension.
For r ∈ R, we denote the quotient g x,r /g x,r+ ( G x,r /G x,r+ for r > 0) by V x,r . The adjoint action of G x,0 on g x,r (or, equivalently, the conjugation action of G x,0 on G x,r for r > 0) induces an action of the algebraic group G x on the quotients V x,r .

Chevalley system for the reductive quotient
In this section we construct a Chevalley system for the reductive quotient G x by reduction of the root group parameterizations given in Section 2.2. Let U a denote the root group of G x corresponding to the root a ∈ Φ(G x ) ⊂ Φ K (G). We denote by O Q ur p the ring of integers in Q ur p . If K is an extension of Q ur p , we let χ : F p → O Q ur p be the Teichmüller lift, i.e. the unique multiplicative section of the surjection O Q ur ] be the usual inclusion.
; e.g., take λ 0 = λ = 1 2 if p = 2. Consider the map This implies (assuming 3 = 1, otherwise it's an easy exercise to add in the required constants) that for m a := x a (1)x −a ( a,a )x a (1) with a,a = α,α we have We obtain a similar result even if Φ a and Φ b are not singletons by the requirement that {x E α } α∈Φ is a Chevalley-Steinberg system, i.e. compatible with the Galois action as described in Section 2. Similarly, we can extend the result that Ad(m a )(Lie( Suppose now that a ∈ Φ(G x ) ⊂ Φ K is multipliable, and let α ∈ Φ a and α = σ(α) ∈ Φ a as above.
Then Bruhat and Tits Loc. cit. show that m a (U, V ) is in the normalizer of the maximal torus T and m a (U, V ) = m a,1 a(V ) and where m a,1 = π •Res E α+ α/K Note that we have and denote its image in G x,0 /G x,0+ by m a . Using that v(λ) = 0 if p = 2, and σ( E ) ≡ ± E ≡ E mod 2 E if p = 2 as well as the compatibility with Galois action properties of a Chevalley-Steinberg system, we obtain Moreover, using Equation (6) and (7), an easy calculation shows that as desired. We obtain analogous results for m −a being defined as above by substituting "a" by "−a". Moreover, m a = m −a , and hence Ad(m −a )(Lie(x a )(1)) = a,a Lie(x −a )(1).
In order to show that {x a } a∈Φ(Gx) forms a Chevalley system, it is left to check that holds for a, b ∈ Φ(G x ) with a = ±b and either a or b multipliable. Note that if x a and x −a commute with x b , then the statement is trivial. Note also that if b is multipliable and β ∈ Φ b , then β lies in the span of the roots of a connected component of the Dynkin digram Dyn(G) of Φ(G) of type A 2n for some positive integer n. Hence, for some α ∈ Φ a , α and β lie in the span of the roots of such a connected component. Moreover, by the compatibility of the Chevalley-Steinberg system {x E α } α∈Φ with the Galois action, it suffices to restrict to the case where Dyn(G) is of type A 2n with simple roots labeled by α n , α n−1 , . . . , β 1 , α 1 , β 1 , β 2 , . . . , β n as in Figure 1, and the K-structure of G arises from the unique outer automorphism of A 2n of order two that sends α i to β i . If a root in Φ K (G) is multipliable, then it is the image of ±(α 1 + . . . + α s ) in Φ K for some 1 ≤ s ≤ n. In particular, the positive multipliable roots are orthogonal to each other, by which we mean that ǎ, b = 0 for two distinct positive multipliable roots a and b. Equation (8) can now be verified by simple matrix calculations in SL 2n+1 .

Moy-Prasad filtration and field extensions
Let F be a field extension of K of degree d = [F : K], and denote by v : Using the definitions introduced in Section 2.4, but for notational convenience still with the valuation v (instead of replacing it by the normalized valuation d · v), we can define a Moy-Prasad filtration of G(F ) and g F at x, which we denote by G F x,r (r ≥ 0) and g F x,r (r ∈ R), as well as its quotients V F x,r (r ∈ R) and the reductive quotient G F x . Suppose now that G F is split, and that Γ a ⊂ v(F ) for all restricted roots a ∈ Φ K (G). This holds, for example, if F is an even-degree extension of the splitting field E. Then, using Remark 2.2.1(i) and the definition of the Moy-Prasad filtration, the inclusion G(K) → G(F ) maps G x,r into G F x,r . Furthermore, recalling that for split tori T the subgroup T 0 is the maximal bounded subgroup of the (rational points of) T and using the assumption that Γ a ⊂ v(F ) for all restricted roots a ∈ Φ K (G), we observe that this map induces an injection which yields a map of algebraic groups G x → G F x , also denoted by ι K,F . If p = 2 or d is odd, then ι K,F is a closed immersion.
Lemma 2.6.1. For every r ∈ R, there exists an injection such that we obtain a commutative diagram for the action described in Section 2.4 Proof. For p = 2, let ι K,F,r be induced by the inclusion g → g F = g ⊗ K F . This map is well defined, and it is easy to see that it is injective on (t ∩ g x,r )/g x,r+ and on (g a ∩ g x,r )/g x,r+ for . This follows from the trace of 1 2 being one, hence v − 1 2 σ(u)u is traceless and therefore has valuation outside 2Γ Moreover, Diagram (10) commutes.
max by reasoning analogous to that above. However, recall from Remark 2.3.1 that v(λ α ) < 0 for p = 2. This allows us to define ι K,F,r as follows. We define the linear morphism i K,F,r : g → g F to be the usual inclusion K are the non-multipliable roots in Φ K , and to be the linear map from where Φ mul K denotes the set of multipliable roots in Φ K , a ∈ Φ mul K and α ∈ Φ a . By restricting i K,F,r to g x,r and passing to the quotient, we obtain an injection ι K,F,r of V x,r into V F x,r . In order to show that ι is compatible with the action of G x for p = 2 as in Diagram (10), it suffices to show that ι K,F (G x ) stabilizes the subspace where the overline denotes the image in V x,r . First suppose that the Dynkin diagram Dyn(G) of Φ(G) is of type A 2n with simple roots labeled by α n , α n−1 , . . . , α 2 , α 1 , β 1 , β 2 , . . . , β n as in Figure  1 on page 1, and that the K-structure of G arises from the unique outer automorphism of A 2n of order two that sends (Note that the last two summands can be deleted unless s = t.) Thus V is preserved under the action of the image of x b in G F x . Similarly (but more easily) one can check that the action of the image of x b in G F x for all other b ∈ Φ K preserves V , and the same is true for the image of T ∩ G x,0 in G F x . Hence ι K,F (G x ) stabilizes V . The case of a general group G follows using the observation that, if a ∈ Φ K is multipliable, then each α ∈ Φ a is spanned by the roots of a connected component of the Dynkin diagram Dyn(G) of Φ(G) that is of type A 2n , together with the observation that the above explanation also works for Dyn(G) being a union of Dynkin diagrams of A 2n that are permuted transitively by the action of the absolute Galois group of K. Thus V is preserved under the action of ι K,F (G x ), and hence the Diagram (10) commutes.
In the sequel we might abuse notation and identify V x,r with its image in V F x,r under ι K,F .

Moy-Prasad filtration for different residual characteristics
In this section we compare the Moy-Prasad filtration quotients for groups over nonarchimedean local fields of different residue-field characteristics. In order to do so, we first introduce in Definition 3.1.1 the class of reductive groups that we are going to work with. We then show in Proposition 3.1.4 that this class contains reductive groups that split over a tamely ramified extension, i.e. those groups considered in [RY14], but also general simply connected and adjoint semisimple groups, among others. The restriction to this (large) class of reductive groups is necessary as the main result (Theorem 3.4.1) about the comparison of Moy-Prasad filtrations for different residue-field characteristics does not hold true for some reductive groups that are not good groups.

Definition and properties of good groups
Definition 3.1.1. We say that a reductive group G over K, split over E, is good if there exist · an action of a finite cyclic group Γ = γ on the root datum R(G) = (X, Φ,X,Φ) preserving the simple roots ∆, · an element u generating the cyclic group Gal(E∩K tame /K) and whose order Gal(E ∩ K tame /K) is divisible by the prime-to-p part of the order of Γ such that the following two conditions are satisfied.
(i) The orbits of Gal(E/K) and Γ on Φ coincide, and, for every root α ∈ Φ, there exists (ii) There exists a basis B of X stabilized by Gal(E/E ∩K tame ) and γ N on which the Gal(E/E ∩ K tame )-orbits and γ N -orbits agree, and such that for any B ∈ B, there exists an element v 1,B ∈ Gal(E/K) satisfying In the sequel, we will write |Γ | = p s · N for some integers s and N with (N, p) = 1.
Remark 3.1.2. Note that condition (i) of Definition 3.1.1 is equivalent to the condition (i') The orbits of Gal(E/K) on Φ coincide with the orbits of Γ on Φ, and there exist representatives C 1 , . . . , C n of the orbits of Γ on the connected components of the Dynkin diagram of Φ(G) satisfying the following. Denote by Φ i the roots in Φ that are a linear combination of roots corresponding to C i (1 ≤ i ≤ n). Then for every root α ∈ Φ 1 ∪ . . . ∪ Φ n and Condition (ii) of Definition 3.1.1 is equivalent to the condition (ii') There exists a basis B of X stabilized by Gal(E/E ∩ K tame ) and by γ N on which the Gal(E/E ∩ K tame )-orbits and γ N -orbits agree, and such that there exist representatives {B 1 , . . . , B n } for these orbits on B, and elements v t 1 ,i ∈ Gal(E/K) for all 1 ≤ t 1 ≤ p s N and 1 ≤ i ≤ n satisfying Before showing in Proposition 3.1.4 that a large class of reductive groups is good, we prove a lemma that shows some more properties of good groups.
Lemma 3.1.3. We assume that G is a good group, use the notation introduced in Definition 3.1.1 and Remark 3.1.2, and denote by E t the tamely ramified Galois extension of K of degree N contained in E. Then the following statements hold.
(a) The basis B of X given in Property (ii) is stabilized by Gal(E/E t ) and the Gal(E/E t )-orbits and γ N -orbits on B agree.
which coincide with the γ N -orbits. This proves part (a).
Part (b) follows from part (a) by the definition of an induced torus.
In order to show part (c), note that X Gal(E/Et) is spanned (over Z) by The Z-span of the latter equals X γ N , which implies X γ N = X Gal(E/Et) . Using Definition 3.1.1(ii) and the observation that u mod Gal(E ∩K tame /E t ) is a generator of Gal(E t /K), we conclude that the action of u on X Gal(E/Et) agrees with the action of γ on X γ N = X Gal(E/Et) and that (b) Part (b) can be deduced from (c) and (d) (whose proofs do not depend on (b)) as follows.
If G is a simply connected or adjoint group then G is the direct product of restrictions of scalars of simply connected or adjoint absolutely simple groups. Hence by (c) and (d) it suffices to show that, if G is a simply connected or adjoint absolutely simple group, then G is good. Recall that these groups are classified by choosing the attribute "simply connected" or "adjoint" and giving a connected finite Dynkin diagram together with an action of the absolute Galois group Gal(Q p /K) on it. We distinguish the two possible cases. Case 1: G splits over a cyclic field extension E of K. Then take Γ = Gal(E/K) and u = γ or u = 1 according as the field extension is tamely ramified or wildly ramified, and choose B to be the set of simple roots of G, if G is adjoint, and the set of fundamental weights dual to the simple co-roots of G (i.e. those weights pairing with one simple co-root to 1, and with all others to 0), if G is simply connected. Case 2: G does not split over a cyclic field extension. Then G has to be of type D 4 and split over a field extension E of K of degree six with Gal(E/K) S 3 , where S 3 is the symmetric group on three letters. In this case we observe (using that G is simply connected or adjoint) that the orbits of the action of Gal(E/K) on X are the same as the orbits of a subgroup Z/3Z ⊂ Gal(E/K) S 3 . Moreover, as S 3 does not contain a normal subgroup of order two, i.e. there does not exist a tamely ramified Galois extension of K of degree three, this case can only occur if p = 3, and we can choose Γ = Z/3Z, u the nontrivial element in Gal(E ∩ K tame /K) Z/2Z, and B as in Case 1 to see that G is good.
(c) In order to show part (c), suppose that G 1 , . . . , G k are good groups with splitting fields E 1 , . . . , E k and corresponding cyclic groups Γ 1 = γ 1 , . . . , Γ k = γ k and generators u i ∈ Gal( Then G splits over the composition field E of E 1 , . . . , E k , and Gal(E ∩ K tame /K) is the smallest common multiple of Gal( , which we assume to be coprime to p by adding Gal(E i ∩ K tame /K) if necessary. Hence (γ i ) r i is a generator of Γ i , and we define γ = (γ 1 ) r 1 × . . . × (γ k ) r k and Γ = γ . Note that the order |Γ | = p s N of Γ is the smallest common multiple of |Γ i | , 1 ≤ i ≤ k, and hence N divides Gal Let u 1,α be a preimage of u 1,α in Gal(E/K). Using that we obtain by considering the factors prime to p that Gal( where the map arises from reduction mod Gal(E ∩E tame i /E ∩K tame ), has oder a power of p, hence is trivial; so we deduce that the map is an isomorphism. Thus we can choose an element , by multiplying u 1,α with powers of u 0 ∈ Gal(E/E i ) if necessary we can ensure that u 1,α ≡ u mod Gal(E/E ∩ K tame ). As Gal(E/E i ) fixes α, we also have γ (α) = u 1,α (α), and we conclude that G satisfies Property (i) of Definition 3.1.1 for all α ∈ Φ(G) = k i=1 Φ(G i ). Choosing B to be the union of the bases B i corresponding to the good groups G i (by viewing X i embedded into X := X 1 × . . . × X k ), we conclude similarly that G satisfies Property (ii). This proves that G is a good group and finishes part (c).
We suppose without loss of generality that the fixed field of Stab Γ (X 1 ) is F , i.e. Stab Γ (X 1 ) = Gal(E/F ), and we write d = d p · d p , where d p is a power of p and d p is coprime to p. As G is good, there exist a cyclic group Γ = γ acting on (X 1 , Φ 1 , ∆ 1 ) and a generator u of Gal(E ∩ F tame /F ) satisfying the conditions in Definition 3.1.1. Fix a splitting Gal(E ∩ F tame /F ) → Gal(E/F ), and let u 0 be the image of u under the composition Hence we can choose u 0 ∈ Gal(E/K) such that Then the cyclic group Γ = γ preserves ∆, and we claim that Γ and u satisfy the conditions for G in Definition 3.1.1.
Property (i) of Definition 3.1.1 is satisfied by the construction of γ .
In order to check Property (ii), let B be a basis of Note that γ N (X 1 ) = 0≤i<dp X 1+id p , and hence, setting n = n · d p , the set forms a basis of X (because γ N has order d p ). We will show that B satisfies Property (ii') of Remark 3.1.2 with set of orbit representatives {B i } 1≤i≤n (and hence satisfies Property (ii) of Definition 3.1.1).
and γ k ≡ u mod Gal(E ∩ K tame /K) for all 1 ≤ k < d by definition, we obtain v t,i ≡ u t mod Gal(E/E ∩ K tame ) for all 1 ≤ t ≤ p s N and 1 ≤ i ≤ n .
This shows that the action of (γ ) t 1 on B i for 1 ≤ t 1 ≤ p s N and 1 ≤ i ≤ n is as required by Condition (ii') of Remark 3.1.2. It remains to show that B is Gal(E/E ∩ K tame )-stable and that the Gal(E/E ∩ K tame )-orbits coincide with the γ N -orbits.
In order to do so, note that Equation (11) implies in particular that for where E t is the tamely ramified degree N field extension of K inside E. Let us denote by E t the tamely ramified Galois extension of F of degree N/d p contained in E. Note that E t is the maximal tamely ramified subextension of E t over K, and [ E t : E t ] = d p . As G is good, we obtain from Property (ii) of Definition 3.1.1 and Lemma 3.1.3 (a) that Using γ N (X 1 ) = 0≤i<dp X 1+id p and the inclusion (12), we deduce that In particular, we obtain that and hence the Gal(E/E ∩ K tame )-orbits on B agree with the γ N -orbits on B. This finishes the proof that Property (ii') of Remark 3.1.2 and hence Property (ii) of Definition 3.1.1 is satisfied for our choice of Γ and u, and hence G is good.
From now on we assume that our group G is good.

Construction of G q
In this section we define reductive groups G q over nonarchimedean local fields with arbitrary positive residue-field characteristic q whose Moy-Prasad filtration quotients are in a certain way (made precise in Theorem 3.4.1) the "same" as those of the given good group G over K.
For the rest of the paper, assume x ∈ B(G, K) is a rational point of order m. Here rational means that ψ(x) is in Q for all affine roots ψ ∈ Ψ K , and the order m of x is defined to be the smallest positive integer such that ψ(x) ∈ 1 m Z for all affine roots ψ ∈ Ψ K . Fix a prime number q, and let Γ be the finite cyclic group acting on R(G) as in Definition 3.1.1. Let F be a Galois extension of K containing the splitting field of (x 2 − 2) over E, such that This implies that the image of x in B(G F , F ) is hyperspecial, and by the last condition the set of valuations Γ a (defined in Section 2.3) is contained in v(F ) for all a ∈ Φ K . In particular, F satisfies all assumptions made in Section 2.6 in order to define ι K,F and ι K,F,r . For later use, denote by F a uniformizer of F such that , and let O F be the ring of integers of F . Let K q be the splitting field of x M − 1 over Q ur q , with ring of integers O q and uniformizer q . Let F q = K q [x]/(x M − q ) with uniformizer Fq satisfying M Fq = q and ring of integers O Fq . Recall that every reductive group over K q is quasi-split, and that there is a one to one correspondence between (quasi-split) reductive groups over K q with root datum R(G) and elements of Hom(Gal(Q q /K q ), Aut(R(G), ∆))/Conjugation by Aut(R(G), ∆), where Aut(R(G), ∆) denotes the group of automorphisms of the root datum R(G) that stabilize ∆. Thus we can define a reductive group G q over K q by requiring that G q has root datum R(G) and that the action of Gal(Q q /K q ) on R(G) defining the K q -structure factors through Gal(F q /K q ) and is given by where the last map is the action of Γ on R(G) as in Definition 3.1.1. This means that G q is already split over E q := K q [x]/(x p s N − q ). Note that by construction, Definition 3.1.1 and Lemma 3.1.3, the restricted root data of G q and G agree: All objects introduced in Section 2 can also be constructed for G q , and we will denote them by the same letter(s), but with a G q in parentheses to specify the group; e.g., we write Γ a (G q ).

Construction of x q
In order to compare the Moy-Prasad filtration quotients of G q with those of G at x, we need to specify a point x q in the Bruhat-Tits building B(G q , K q ) of G q . To do so, choose a maximal split torus S q in G q with centralizer denoted by T q , and fix a Chevalley-Steinberg system {x Fq α } α∈Φ for G q with respect to T q . For later use, we choose the Chevalley-Steinberg system to have signs α,β as in Definition 2.1.2, i.e.
where N Gq (T q ) denotes the normalizer of T q in G q , and Ad(m Using the valuation constructed in Section 2.2 attached to this Chevalley-Steinberg system, we obtain a point x 0,q in the apartment A q of B(G q , K q ) corresponding to S q . Fixing an isomorphism f S,q : X * (S) → X * (S q ) that identifies R K (G) with R Kq (G q ), we define an isomorphism of affine spaces f A ,q : A → A q by where Φ +,mul K are the positive multipliable roots in Φ K , λ a ∈ (E α ) 1 max (G) for some α ∈ Φ a , andǎ is the coroot of a, so we haveǎ(a) = 2. We define x q := f A ,q (x).
Moreover, we have for all a ∈ Φ K and r ∈ R that r−a(x−x 0 ) ∈ Γ a (G) if and only if r−a(x q −x 0,q ) ∈ Γ a (G q ).
Proof. As the set of affine roots for G on A (and analogously for G q on A q ) is Let us fix a ∈ Φ K , and α ∈ Φ a ⊂ Φ = Φ(G) = Φ(G q ). Recall that E α (G) is the fixed subfield of E under the action of Stab Gal(E/K) (α). Using Equation (13) on page 22, we obtain Note that the Dynkin diagram Dyn(G) of Φ(G) is a disjoint union of irreducible Dynkin diagrams, and if a is a multipliable root, then α is contained in the span of the simple roots of a Dynkin diagram of type A 2n . Thus by Equation (16) and the description of Γ a as in Equation (4) on page 9, the Equality (15) holds for α in the span of simple roots of an irreducible Dynkin diagram of any type other than A 2n , n ∈ Z >0 , or in the span of an irreducible Dynkin diagram of type A 2n whose 2n simple roots lie in 2n distinct Galois orbits. We are therefore left to prove the lemma in the case of Dyn(G) being a disjoint union of finitely many A 2n whose simple roots form n orbits under the action of Gal(E/K). An easy calculation (see the proof of Lemma 2.6.1 for details) shows that, in this case, the positive multipliable roots of Φ K form an orthogonal basis for the subspace of X * (S) ⊗ R generated by Φ K , where by "orthogonal" we mean thatb(a) = 0 if a and b are distinct positive multipliable roots, and that, if b ∈ Φ K and b = a∈Φ +,mul K κ a a is not multipliable, then a∈Φ +,mul K κ a ∈ 2 · Z. Moreover, by the definition of K q and F q , it is easy to check that for . Thus using the description of Γ a as in Equation (2) on page 9 and Equation (3) on page 9, we see that the desired Equation (15) holds.
The second claim of the lemma follows from combining Equation (15) and the definition of x q using the map in Equation (14) on page 22. Note that Lemma 3.3.1 implies in particular that x q is also a rational point of order m. Let us denote the reductive quotient of G q at x q by G xq ; the corresponding Moy-Prasad filtration groups by G xq,r , r ≥ 0; the Lie algebra filtration by g xq,r , r ∈ R; and the filtration quotients of the Lie algebra by V xq,r , r ∈ R. Then using Lemma 2.4.1, we obtain the following corollary to Lemma 3.3.1.
Corollary 3.3.2. The root data R(G x ) and R(G xq ) are isomorphic.

Global Moy-Prasad filtration representation
Since R(G x ) = R(G xq ) (Corollary 3.3.2), we can define a split reductive group scheme H over Z by requiring that R(H ) = R(G x ), and then H Fp G x and H Fq G xq ; i.e., we can define the reductive quotient "globally". In this section we show that we can define not only the reductive quotient globally, but also the action of the reductive quotient on the Moy-Prasad filtration quotients. More precisely, we will prove the following theorem.
Theorem 3.4.1. Let r be a real number, and keep the notation from Section 3.2 and 3.3, so G is a good reductive group over K and x a rational point of B(G, K). Then there exists a split reductive group scheme H over Z[1/N ] acting on a free Z[1/N ]-module V satisfying the following. For every prime q coprime to N , there exist isomorphisms H Fq G xq and V Fq V xq,r such that the induced representation of H Fq on V Fq corresponds to the usual adjoint representation of G xq on V xq,r . Moreover, there are isomorphisms H Fp G x and V Fp V x,r such that the induced representation of H Fp on V Fp is the usual adjoint representation of G x on V x,r . In other words, we have commutative diagrams Remark 3.4.2. The theorem fails for some reductive groups that are not good groups.
We prove the theorem in two steps. In Section 3.4.1 we construct a morphism from H to an auxiliary split reductive group scheme G , and in Section 3.4.2 we construct V inside the Lie algebra of G and use the adjoint action of G on its Lie algebra to define the action of H on V .

Global reductive quotient
Let G be a split reductive group scheme over Z whose root datum is the root datum of G. In this section we construct a morphism ι : H → G that lifts all the morphisms ι K,F : G x,0 /G x,0+ → G F x,0 /G F x,0+ and ι K,Fq : G xq,0 /G xq,0+ → G Fq xq,0 /G Fq xq,0+ defined in Section 2.6. In order to do so, let us first describe the image of ι K,F more explicitly. In analogy to the root group parametrization x a defined in Section 2.2, and using the notation from that section, we define for a ∈ Φ K (G) multipliable the more general map X a : F × F → G(F ) by Note that X a | H 0 (Eα,E α+ α ) (α ∈ Φ a ) agrees with x a . We then have the following lemma.
if a is multipliable and p = 2 X a (0, otherwise.
Then the composition y a of y a with the quotient map G F x,0 Proof. If p = 2 or if a is not multipliable, the lemma follows immediately from Lemma 2.5.1.
In the case p = 2, note that (using the notation from Lemma 2. F O F , and hence y a (u) = ι K,F (x a (u)) by Lemma 2.5.1. Remark 3.4.4. An analogous statement holds for G xq . In the sequel we denote the root group parameterizations constructed for G xq analogously to Lemma 2.5.1 by x q a : G a → U q a , a ∈ Φ(G xq ).
Recall that x is hyperspecial in B(G F , F ), and hence the reductive quotient G F x of G F at x is a split reductive group over F p with root datum R(G F x ) = R(G). The analogous statement holds for x q . Thus G Fp is isomorphic to G F x , and G Fq is isomorphic to G Fq xq . In order to construct explicit isomorphisms, let us fix a split maximal torus T of G and a Chevalley system {x α : G a − → U α ⊂ G } α∈Φ(G )=Φ for (G , T ) with signs equal to α,β as in Definition 2.1.2; i.e., the Chevalley system {x α } α∈Φ for (G , T ) and the Chevalley-Steinberg system {x α } α∈Φ for (G, T ) have the same signs.
Moreover, the split maximal torus T F ⊂ G F and the Chevalley system {x F α } α∈Φ yield a split maximal torus T F x of G F x and a Chevalley system Moreover, we define constants c α,q ∈ O Fq and c α ∈ O F for α ∈ Φ as follows. We choose γ ∈ Gal(F/K) such that Let C 1 , . . . , C n be the representatives for the action of Γ = γ on the connected components of Dyn(G) as given in Remark 3.1.2(i'), and recall that Φ i denotes the roots that are a linear combination of simple roots corresponding to C i . For α ∈ Φ there exists a unique triple (i, α i , e q (α)) with i ∈ [1, n], α i ∈ Φ i and e q (α) minimal in Z ≥0 such that γ eq(α) q (α i ) = α. Note that e q (α) is independent of the choice of prime number q. We also write e(α) = e q (α). We define   The definitions of ζ G , ζ Gq and e(α) are chosen so that the following lemma holds.
Lemma 3.4.6. We keep the notation from above and let r ∈ R.
which shows part (i).
In order to prove part (ii), let γ ∈ Gal(F/K) with γ(α i ) = α, and write γ = γ e w for some integer e and w ∈ Gal(F/E∩K tame ). By Property (i) of Definition 3.1.1 and the definition of e(α) there exists w ∈ Gal(F/E ∩ K tame ) such that γ e(α) w(α i ) = α, and hence w −1 γ e(α)− e w(α i ) = α i , and therefore (γ ) e(α)− e (α i ) ∈ Gal(F/E ∩ K tame )(α i ). On the other hand, as the Γ -orbits on Φ agree with the Gal(F/K)-orbits on Φ and X γ N = X Gal(F/E∩K tame ) (by Property (ii) of Definition 3.1.1 and Lemma 3.1.3), the Gal(F/E ∩ K tame )-orbits on Gal(F/K)(α i ) coincide with the γ N -orbits, which are the same as the γ N i orbits, where N i is coprime to p such that |Gal(F/K)(α i )| = p s i N i for some F as all p-power roots of unity in F p are trivial. Hence which proves part (ii).
Now let f T : T F x → T Fp be an isomorphism that identifies the root data R(G x ) and R(G ). Then we can extend f T as follows.
Lemma 3.4.7. There exists an isomorphism f : G F x → G Fp extending f T such that for α ∈ Φ and u ∈ G a (F p ) we have f (x F α (u)) = x α (c α · u).
Similarly, for each prime q, let f T,q : T Fq xq → T Fq be an isomorphism that identifies the root data R(G xq ) and R(G ). Then we have the analogous statement.
Lemma 3.4.8. There exists an isomorphism f q : G F xq → G Fq extending f T,q such that for α ∈ Φ and u ∈ G a (F q ) we have This allows us to define a map ι from H to G as follows.
Let S be a split maximal torus of H . Then we have where the first identification arises from R(H ) = R(G x ), the second from Lemma 2.4.1 and the fourth from R(G ) = R(G). This yields a closed immersion f S : S → T . Note that f S also corresponds to the injection X * (S ) = X * (T xq ) = X * (S q ) = X * (T q ) Gal(Fq/Kq) → X * (T q ) = X * (T ), and we have commutative diagrams To define ι on root groups, let {x H a } a∈Φ(H )=Φ K =Φ(Gx) be a Chevalley system for (H , S ) such that there exists an isomorphism f H,q : H Fq → G xq mapping S Fq to T xq and identifying (x H a ) Fq with x q a , or equivalently having the same signs as the Chevalley system {x q a } a∈Φ K , for some q = 2.
Moreover, note that for a ∈ Φ K = Φ(H ), there exists a unique integer in [1, n], denoted by n(a), such that Φ a ∩Φ n(a) = ∅ (see Remark 3.1.2 for the definition of Φ i , i ∈ [1, n]). We label the elements in Φ a ∩ Φ n(a) by {α i } 1≤i≤|Φa∩Φ n(a) | so that they satisfy the following two properties: • If a is a multipliable root, we assume that α 1 ∈ [Φ a ], where [Φ a ] is as defined in Section 2.2.
Definition / Proposition 3.4.9. There exists a unique group scheme homomorphism ι : H Z → G Z extending f S such that for all Z-algebras A, a ∈ Φ(H ) = Φ K and u ∈ G a (A) we have if a is multipliable, where ζ i is a primitive i-th root of unity, i = 1, 2 or 3, and Γ n(a) = Stab Γ (Φ n(a) ).  In order to provide a different construction of H in Section 4, we use the following Lemma.

Moreover, we have commutative diagrams
Lemma 3.4.11. Let ι be as in Definition / Proposition 3.4.9. Then ι Q : H Q → G Q is a closed immersion.

Proof.
In order to show that ι Q is a closed immersion, it suffices to show that its kernel is trivial ([Con14, Proposition 1.1.1]). As Q is of characteristic zero, the kernel of ι Q (a group scheme of finite type) is smooth. Hence we only need to show that ι Q is injective on Q-points. Let g ∈ H (Q). LetẆ be a set of representatives of the Weyl group of H in the normalizer of S . Without loss of generality, we assume that the elements ofẆ are products of x H a (1)x H −a ( a,a )x H a (1). Let U be the unipotent radical of the Borel subgroup corresponding to ∆(H ), U − the one of the opposite Borel, and U w = U (Q) ∩ wU − (Q)w −1 . By the Bruhat decomposition, we can write g uniquely as u 1 wtu 2 with w ∈Ẇ , t ∈ S (Q), u 1 ∈ U w and u 2 ∈ U (Q). By the uniqueness 1 = ι(g) = ι(u 1 )ι(w)ι(t)ι(w 2 ) if and only if 1 = ι(u 1 ) = ι(w) = ι(t) = ι(u 2 ). Note that ι(w) = 1 implies w = 1 by our choice ofẆ , and ι(t) = implies t = 1. Choosing an order of Φ + K , there is a unique way to write with u a ∈ Q for all a ∈ Φ + K . By choosing a compatible ordering of the roots in Φ + and the uniqueness of writing ι(u 2 ) = α∈Φ + x α (u α ) with u α ∈ Q together with the explicit description of ι on root groups given in Definition / Proposition 3.4.9, we conclude that u a = 0 for all a ∈ Φ + K , and hence u 2 = 1. Similarly, u 1 = 1, which shows that the map ι is injective as desired.

Global Moy-Prasad filtration quotients
In this section we will also lift the injections ι K,F,r : V x,r → V F x,r and ι Kq,Fq,r : V xq,r → V Fq xq,r in such a way that we get a lift of the commutative Diagram (10). Using these injections we view V x,r as a subspace of V F x,r and V xq,r as a subspace of V Next consider a ∈ Φ K . We recall that Γ n(a) denotes the stabilizer of the component C n(a) of the
Also note that the G F x representation V F x,r is isomorphic to the adjoint representation of G F x on Lie(G F x ) and, similarly, the G  Moreover, for q coprime to N , we have df (V x,r ) = V Fp , df q (V xq,r ) = V Fq and the following diagrams commute Proof. We first show that df q (V xq,r ) = V Fq for q coprime to N and df (V For α ∈ Φ, denote by Γ α the stabilizer of α in Γ , and let X α = Lie(x Fq α )(1), n a = Φ a ∩ Φ n(a) ∈ {1, 2, 3} and ζ γ := ζ Gq e(γ (α 1 )) = ζ Gq e(γ (α i )) , 1 ≤ i ≤ n a .
Here the action of Γ on V xq,r is the one induced from the action on g Fq xq,r . Thus by definition of the Moy-Prasad filtration and the inclusion ι Fq,Kq,r constructed in the proof of Lemma 2.6.1 we obtain the equality In order to show the analogous statement for V x,r , we claim that ζ . This is obviously true for p = 2 as a(x − x 0 ) = a(x q − x 0,q ) in this case. If p = 2, then ζ 2 = −1 = 1 in F p and we only have the to consider the case n a = Φ a ∩ Φ n(a) = 3. However, n a = 3 implies that the corresponding component C n(a) of the Dynkin diagram Dyn(G) is of type D 4 , and henceb(a) = 0 for all multipliable roots b ∈ Φ +,mul follows. Let ζ γ = ζ G e(γ (α 1 )) , X α = Lie(x F α )(1), and use otherwise the same notation as above. Then there exists a set of representatives Gal(F/K)/ Stab Gal(F/K) (α) where the last equality follows from Lemma 3.4.6. Thus we obtain . Let us consider V T . From the definition of the Moy-Prasad filtration t Et x,r of the Lie algebra t Et of the torus T Et together with Lemma 3.1.3 and the observation that all p-power roots of unity in F p are trivial, we deduce (by sending ζ M ⊗ 1 to ζ G under the isomorphism Z[1/N ] ⊗ Z[1/N ] F p F p , as above) that df ι Et,F,r t Et x,r /t Et Moreover, by combining Proposition 4.6.2 and Proposition 4.6.1 from [Yu15, Section 4.6], we have t x,r = (t Et x,r ) Gal(Et/K) as E t is tamely ramified over K, and we obtain (using tameness of E t /K) that For q coprime to N , we denote by E t,q the tamely ramified extension of degree N of K q . Then we obtain by the same reasoning (substituting E t by E t,q ) Combing Equations (24) and (25), and (23) and (26), we obtain for q coprime to N that In order to show that the adjoint action of G Z[1/N ] on Lie(G )(Z[1/N ]) restricts to an action of H Z[1/N ] on V , we observe that the following diagram commutes Since ι Kq,Fq (G xq ) preserves V xq,r (Equation (10)

Moy-Prasad filtration representations and global Vinberg-Levy theory
In this section we will give a different description of the reductive group scheme H and its action on V from Theorem 3.4.1 as a fixed-point group scheme of a larger split reductive scheme G acting on a graded piece of Lie G (see Theorem 4.1.1). This means we are in the setting of a global version of Vinberg-Levy theory and the special fibers correspond to (generalized) Vinberg-Levy representations for all primes q. In order to give such a description integrally (i.e. over Z[1/N ]), we will specialize to reductive groups G that become split over a tamely-ramified field extension in Section 4.1. Afterwards, in Section 4.2, we will then show that such a description holds over Q for all good groups. This will also allow us to study the existence of (semi)stable vectors in Section 5.

The case of G splitting over a tamely ramified extension
Let S be a scheme, then we denote by µ M,S the group scheme of M -th roots of unity over S. We will often omit S if it can be deduced from the context. Given an S-group scheme G , we denote by Aut G /S its automorphisms functor, i.e. the functor that sends an S-scheme S to the group of automorphisms of G S in the category of S -group schemes, and by Aut G /S its representing group scheme if it exists. We will often omit S if it can be deduced from the context. Given, in addition, a morphism θ : µ M,S → Aut G , we denote by G θ the scheme theoretic fixed locus of G under the action of µ M,S via θ, if it exists, i.e. G θ represents the functor that sends an S-scheme S to the elements of G (S ) on which µ M,S acts trivially. If G θ is a smooth group scheme over S of finite presentation, we denote by G θ,0 its identity component. Similarly, if F is a quasi-coherent O S -module, we denote by Aut F/O S its automorphism functor, and by Aut F/O S (or simply Aut F ) the group scheme representing Aut F/O S if it exists. In particular, this implies that for q coprime to e we have commutative diagrams Let e , f be integers such that e | e , M = e f , gcd(e , f ) = 1 and e is minimal satisfying these properties. Let E e be the splitting field of (x e − 1) over E, and let O e be the ring of integers in E e .
We let G be a split reductive group scheme over O e [ 1 e ] ⊂ Z[ 1 e ] whose root datum R(G ) coincides with the root datum R(G) of G, i.e. G is as defined in Section 3.4.1 base changed to O e [ 1 e ], and T denotes a split maximal torus of G . Let G ad be the adjoint group of G. We have an isogeny G → G ad , and we denote the image of T under this isogeny by T ad . The isogeny induces an injection X * (T ) → X * (T ad ) that yields an isomorphism X * (T ) ⊗ Z R − → X * (T ad ) ⊗ Z R, which we use to identify the two real vector spaces. This allows us to choose λ ∈ X * (T ad ) ⊂ X * (T ) ⊗ R such that x = x 0 + 1 M λ. Note that then, using the identification of X * (T ) with X * (T ad ), we have x q = x 0,q + 1 M λ. We also denote by λ the corresponding element in X * (T ad ) ⊂ X * (T ) ⊗ R under the identification of X * (T ) with X * (T ). Consider the action θ λ of µ m on G given by composition of the closed immersion µ m → G m with λ and the adjoint action of G ad on G , i.e. ]. Let us denote G θ,0 by H . We claim that T ϑ,0 is a maximal torus of H . In order to prove the claim for geometric fibers, we use a similar argument to one used in [Hai15,Section 4]. Let q be an arbitrary prime number coprime to e, B the Borel of G corresponding to the positive roots, and U its unipotent radical. As H Fq is a closed subgroup of  Let q be a large enough prime, to be more precise odd, not torsion for G and not dividing M . Then we deduce from the arguments used in [RY14, Section 4] that we have commutative diagrams: Moreover, the diagram on the right hand side is compatible with the action by the groups of the diagram on the left hand side.
Recall that we constructed in Section 3.4 a map ι : H → G Z[1/e] and V as a free Z[ 1 e ]-submodule of Lie(G )(Z[ 1 e ]) such that we have the following commutative diagrams for all primes q coprime to where the diagram on the right hand side is compatible with the action of the groups on the left hand side by Proposition 3.4.12. Note that ι Kq,Fq is a closed immersion as either q is odd or e is odd (see Section 2.6).
Thus we conclude that h −1 (ι(H Fq )) = H Fq for large enough q.
Let q now be any prime coprime to e, and let g ∈ H (F q ). As H (Z[ 1 e ]) surjects onto H (F q ) (because this holds for the root groups and the torus), we can choose g ∈ H (Z[ 1 e ]) whose image in H (F q ) is g. By combining the Diagrams (28) and (29), we see that the image of h −1 ι(g) in G (F q ) is actually contained in H (F q ) for all sufficiently large primes q . Hence for all primes q coprime to e. The same arguments show that Moreover, we claim that h −1 • ι(H Q ) = H Q . In order to prove the claim, note that (µ M ) Q Z/M Z Q , and hence the action of the group scheme µ M on G Q corresponds to the action of the finite group Z/M Z generated by ϑ·Inn(λ(ζ M )). Therefore, by the construction of ι : H Z[1/e] → G Z[1/e] (see Proposition 3.4.9) and the definition of h : G Z[1/e] → G Z[1/e] , we see that h −1 • ι(H (Q)) ⊂ G θ (Q).
As ι Q : H Q → G Q is a closed immersion by Lemma 3.4.11, h −1 • ι(H Q ) H Q G θ,0 Q = H Q , and we conclude that

Vinberg-Levy theory for all good groups
Even though the Moy-Prasad filtration representation of groups that do not split over a tamely ramified extension might not be described as in Vinberg-Levy theory, its lift to characteristic zero can be described using Vinberg theory, i.e. as the fixed-point subgroup of a finite order automorphism on a larger group acting on some eigenspace in the Lie algebra of the larger group.
To be more precise, we have the following corollary of Theorem 4.1.1 combined with Theorem 3.4.1.
Corollary 4.2.1. Let G be a good group over K, r = d M for some nonnegative integer d < M , and let the representation H acting on V be as in Theorem 3.4.1. Then there exist a reductive group scheme G Q over Q and morphisms , and the action of H Q on V Q corresponds via these isomorphisms to the restriction of the adjoint action of G Q on Lie(G Q )(Q).

Proof.
Let q be a prime larger than p s N . Then, by construction, the representation over Z[ 1 p s N ] associated to G q via the proof of Theorem 3.4.1 agrees with the representation of H Z[1/(p s N )] on V Z[1/(p s N )] . As G q splits over a tamely ramified extension, Theorem 4.1.1 allows us to deduce the corollary.

Semistable and stable vectors
In this section we apply our results of Section 3 and Section 4 to prove that the existence of stable and semistable vectors in the Moy-Prasad filtration representations is independent of the characteristic of the residue field. Recall that a vector v in a vector space V over an algebraically closed field is stable under the action of a reductive group G V on V if the orbit G V v is closed and the stabilizer Stab G V (v) of v in G V is finite. A vector v ∈ V is called semistable if the closure of the orbit G V v does not contain zero.

Semistable vectors
The global version of the Moy-Prasad filtration representation as provided by Theorem 3.4.1 allows us to show that the existence of semistable vectors is prime independent as follows.
Theorem 5.1.1. We keep the notation used in Theorem 3.4.1, in particular G is a good reductive group over K and x ∈ B(G, K). Then the following are equivalent (i) V x,r has semistable vectors under the action of G x .
(ii) V xq,r has semistable vectors under the action of G xq for some prime q coprime to N .
(iii) V xq,r has semistable vectors under the action of G xq for all primes q coprime to N .

Proof.
We first show that (ii) implies (i). Suppose that (ii) holds, i.e. that V xq,r contains semistable vectors under G xq for some prime q coprime to N . This implies by [MP94,Proposition 4.3] that V Q q has semistable vectors under the action of H Q q , where H and V are as in Theorem 3.4.1. By [Mum77, p. 41] (based on [Mum65, Definition 1.7 and Proposition 2.2]) this means that there exists a H Q q -invariant non-constant homogeneous element P q in SymV Q q . Moreover, there exists X ∈ V Q ⊂ V Q q such that P q (X) = 0, i.e. X is semistable in V Q q under the action of H Q q . Hence X = 0 is also semistable in V Q under the action of H Q , which implies (SymV Q ) H (Q) = Q. Thus, there does also exist a H (Z)-invariant non-constant homogeneous element P in SymV Z . As P is non-constant and homogeneous, we can assume without loss of generality that the image P of P in SymV Z ⊗ F p SymV Fp is non-constant. Note that H (Z) surjects onto H (F p ), which follows from the surjections on all root groups and the split maximal torus.
The same arguments show that if G x,r has semistable vectors, then G xq,r has semistable vectors for all primes q coprime to N , i.e. (i) implies (iii). As (iii) implies (ii), we conclude that all three statements are equivalent.
Note that the same holds for the linear dualsV x,r andV xq,r of V x,r and V xq,r usingV instead of V in the proof above: Corollary 5.1.2. We use the same notation as above. ThenV x,r has semistable vectors under the action of G x if and only ifV xq,r has semistable vectors under the action of G xq for some prime q coprime to N if and only ifV xq,r has semistable vectors under the action of G xq for all primes q coprime to N .
Remark 5.1.3. For semisimple groups G that split over a tamely ramified extension and sufficiently large residue-field characteristic p, Reeder and Yu classified in [RY14,Theorem 8.3] those x for whichV x,r contains semistable vectors in terms of conditions that are independent of the prime p. Corollary 5.1.2 allows us to conclude that these prime independent conditions also classify points x such that V x,r contains semistable vectors for all good semisimple groups G (without any restriction on the residue-field characteristic). Note that the removal of the restriction on the residue-field characteristic for absolutely simple split reductive groups G is also contained in a joint paper of the author with Romano, see [FR17]. For this result, it suffices to construct H acting on V over Z p .

Stable vectors
In this section we show an analogous result to the one in Section 5.1 for stable vectors. This allows us to generalize the criterion in [RY14] for the existence of stable vectors in the dual of the first Moy-Prasad filtration quotient to arbitrary residual characteristics p and all good semisimple groups, which in turn produces new supercuspidal representations.
Theorem 5.2.1. We keep the notation used above, in particular G is a good reductive group over K and x ∈ B(G, K). Then the following are equivalent (i) V x,r has stable vectors under the action of G x .
(ii) V xq,r has stable vectors under the action of G xq for some prime q coprime to N .
(iii) V xq,r has stable vectors under the action of G xq for all primes q coprime to N .
Before we prove the theorem, we would like to mention that part of the following proof appears as well in [FR17] in order to proof Corollary 5.2.3 below in the case of G being absolutely simple and split.
Proof. We suppose without loss of generality that r = d M for some nonnegative integer d < M . Assume that (ii) is satisfied, i.e. there exists a prime q coprime to N such that V xq,r contains stable vectors under the action of G xq . Hence there exists a nonzero polynomial x n ] such that the Q p -points of the closed reduced subvariety V (P ) of V Q p defined by the vanishing locus of P contain V Q p − (V Q p ) s 0. We can assume without loss of generality that the coefficients of P are in Z p , i.e. P ∈ O(V Zp ) ⊂ O(V Q p ), and that at least one coefficient of P has p-adic valuation zero. Let P be the image of P under the reduction map O(V Zp ) Then P is not constant, because P (0) = 0, and there exists X ∈ V Fp V x,r such that P (X) = 0.
We claim that X is a stable vector under the action of G x . We will prove the claim using the Hilbert-Mumford Criterion that states that a vector is stable if and only if it has positive and negative weights for every non-trivial one-parameter subgroup, see be a lift of X, and write X = X − + X 0 + X + with X − ∈ V − , X 0 ∈ V 0 , X + ∈ V + . Note that the weight decomposition of V Fp under the action of G m via the composition of λ with the action of H Fp on V Fp is the image of the decomposition Suppose that v(X − ) > 0. Then P (X) ≡ P (X 0 + X + ) modulo the maximal ideal of Z p . However, X 0 + X + is not a stable vector, because it has no negative weights with respect to the non-trivial one parameter subgroup λ × Zp Q p , which implies P (X 0 + X + ) = 0. Hence P (X) = 0 contradicting the choice of X. The same contradiction arises if we assume that v(X − ) > 0. Thus, X has positive and negative weights for every non trivial one parameter subgroup, i.e. X is stable by the Hilbert-Mumford criterion. Hence, statement (i) of the theorem holds.
The same arguments show that if G x,r has stable vectors, then G xq,r has stable vectors for all q coprime to N , i.e. (i) implies (iii). As (iii) implies (ii), the three statements are equivalent.
As in the semistable case, the same proof works for the linear duals of the Moy-Prasad filtration quotients: Corollary 5.2.2. We use the same notation as above. ThenV x,r has stable vectors under the action of G x if and only ifV xq,r has stable vectors under the action of G xq for some prime q coprime to N if and only ifV xq,r has stable vectors under the action of G xq for all primes q coprime to N .
Denote by r(x) the smallest positive real number such that V x,r(x) = {0}, and letρ = 1 2 α∈Φ +α , where Φ + are the positive roots of Φ = Φ(G) (with respect to the fixed Borel B). Then Corollary 5.2.2 allows us to classify the existence of stable vectors inV x,r(x) for arbitrary primes p and good semisimple groups below. This generalizes the result of [RY14, Corollary 5.1] for large primes p and semisimple groups that split over a tamely ramified extensions. Proof. Note that by Lemma 3.3.1, the order of x q is m, and by Theorem 3.4.1, we have r(x q ) = r(x). Let q be sufficiently large, i.e. coprime to M , not torsion and odd. Then G q is a semisimple group that splits over a tamely ramified extension, and we deduce from the proof of [RY14, Lemma 3.1] thatV xq,r(xq) can only admit stable vectors under G xq if x q is a barycenter of some facet of A q = A (S q , K q ), and hence r(x q ) = 1/m. Therefore, as q is chosen sufficiently large, we obtain by [RY14, Corollary 5.1] thatV xq,r(xq) has stable vectors if and only if x q is conjugate under the affine Weyl group W aff q of the restricted root system of G q to x 0,q +ρ/m, r(x) = 1/m and there exists an elliptic Z-regular element wγ of order m in W γ , because W is isomorphic to the absolute Weil group of G q . Note that x q ∼ W aff q x 0,q +ρ/m if and only if x ∼ W aff x 0 + 1 4 a∈Φ +,mul K v(λ a ) ·ǎ +ρ m , and x 0 + 1 4 a∈Φ +,mul K v(λ a ) ·ǎ +ρ/m is conjugate to x 0 +ρ/m under the extended affine Weyl group of the restricted root system of G. However, by checking the tables for all possible points x q whose first Moy-Prasad filtration quotientV xq,r(xq) admits stable vectors in [RLYG12] and [RY14], we observe that the latter conjugacy can be replaced by conjugacy under the (unextended) affine Weyl group. Hence using Corollary 5.2.2, we conclude thatV x,r(x) contains stable vectors under the action of G x if and only if x ∼ W aff x 0 +ρ/m, r(x) = 1/m, and there exists an elliptic Z-regular element of order m in W γ .
Recall that k is a nonarchimedean local field with maximal unramified extension K.
Corollary 5.2.4. Let G be a good semisimple group, and suppose that G is defined over k. Assume that W γ contains an elliptic Z-regular element. Then using the construction of [RY14, Section 2.5] we obtain supercuspidal (epipelagic) representations of G(k ) for some finite unramified field extension k of k.
Proof. Let m be the order of an elliptic Z-regular element of W γ , and x = x 0 +ρ/m ∈ A (S, K). By Corollary 5.2.3,V x,r(x) contains stable vectors under the action of G x . Since x is fixed under the action of the Galois group Gal(K/k), the vector spaceV x,r(x) is defined over the residue field f of k. Hence there exists a finite unramified field extension k of k with residue field f sucȟ V x,r(x) contains a stable vector defined over f . Applying [RY14, Proposition 2.4] yields the desired result.

Moy-Prasad filtration representations as Weyl modules
In this section we describe the Moy-Prasad filtration representations in terms of Weyl modules.
Recall that for λ ∈ X * (S ) a dominant weight, the Weyl module V (λ) (over Z[1/N ]) is given by where B H is the Borel subgroup of H corresponding to ∆(H ), B − H is the opposite Borel subgroup, w 0 is the longest element of the Weyl group of Φ(H ), and (.) ∨ denotes the dual ([Jan03, II.8.9]). We define Φ x,r = a ∈ Φ K | r − a(x − x 0 ) ∈ Γ a (G) Φ max x,r = a ∈ Φ x,r | a + b ∈ Φ x,r for all b ∈ Φ + (H ) ⊂ Φ K .
To simplify notation, we will prove the claim for the case that b is the image of α 1 and a + N a,b b is the image of −α 2 . All other cases work analogously. Combining Equation Recall that for α ∈ Φ and H α := Lie(α)(1), we have ([Con14, Corollary 5.1.12]) x −α (u)(X α ) = X α + α,α uH −α − α,α u 2 X −α x −α (u)(H) = H + Lie(α)(H)uX −α for all u ∈ G a (Z[1/N ]) and all H ∈ Lie(T )(Z[1/N ]). Using these identities, we obtain Corollary 6.2.2. Let G be a good reductive group, r / ∈ 1 p s N Z a real number, and x a rational point of B(G, K). Suppose that either p is odd or that Φ K does not contain any multipliable root. Then Proof. If r / ∈ 1 p s N Z, then V T = {0}, and the claim follows by combining Theorem 6.2.1 and Theorem 3.4.1.