2.1 Notation and conventions

When k is a field, we denote by $\operatorname {\mathrm {\textbf {Vec}}}_k$ the category of finite-dimensional k-vector spaces. When R is a k-algebra,Footnote ¹ we denote by $\operatorname {\mathrm {\textbf {Mod}}}_R$ the category of R-modules, and by $\operatorname {\mathrm {\textbf {Alg}}}_R$ the category of R-algebras. When $V,W \in \operatorname {\mathrm {\textbf {Vec}}}_k$ , we write $V_R$ for $V\bigotimes _k R$ , and write $\alpha _R:= \alpha \otimes R$ , the R-linear extension of $\alpha $ , for all k-linear maps $\alpha \colon V\to W$ . When G is an affine algebraic k-group, we denote by $k[G]$ the Hopf algebra of G, by $G_R:= G \times _{\operatorname {\mathrm {Spec}} k} \operatorname {\mathrm {Spec}} R$ the base extension, by $H^1(k,G):= H^1 \big (\operatorname {\mathrm {Gal}}(k^{\operatorname {\mathrm {sep}}}/k),G(k^{\operatorname {\mathrm {sep}}}) \big )$ the first Galois cohomology set, and by $\operatorname {\mathrm {\textbf {Rep}}}_k(G)$ the category of representations of G on finite-dimensional k-vector spaces.

By a reductive k-group, we mean a (not necessarily connected) affine algebraic k-group G such that every smooth connected unipotent normal subgroup of $G_{\bar k}$ is trivial, where $\bar {k}$ is an algebraic closure of k.

For the rest of this paper, we work under the following hypothesis.

Hypothesis 2.1 Let p be a prime number and $q=p^f$ an integral power of p. Let F be a finite extension of $\mathbb {Q}_p$ with the ring of integers $\mathcal {O}_F$ , a fixed uniformizer $\varpi _F$ and the residue field $\mathbb {F}_q$ of q elements. Let $\kappa $ be a perfect field containing $\mathbb {F}_q$ . Let $\mathcal {O}_K=\mathcal {O}_F\bigotimes _{W(\mathbb {F}_q)} W(\kappa )$ , where $W(\mathbb {F}_q)$ (resp. $W(\kappa )$ ) denotes the ring of Witt vectors with coefficients in $\mathbb {F}_q$ (resp. in $\kappa $ ). Then, $K:= \operatorname {\mathrm {Frac}}(\mathcal {O}_K)\cong F\bigotimes _{W(\mathbb {F}_q)} W(\kappa )$ is a complete discretely valued field with ring of integers $\mathcal {O}_K$ , a uniformizer $\varpi := \varpi _F\otimes 1$ and residue field $\kappa $ . Let $\operatorname {\mathrm {Frob}}$ be the ring endomorphism of $W(\kappa )$ induced by the p-power map on $\kappa $ , and let

$$ \begin{align*}\varphi:= \operatorname{\mathrm{Id}}_F\otimes \operatorname{\mathrm{Frob}}^f \colon K\operatorname{\mathrm{\longrightarrow}} K\end{align*} $$

be the Frobenius automorphism on K relative to F. Then, $\varphi $ reduces to the q-power map on $\kappa $ , and the fixed field of $\varphi $ on K is $F\bigotimes _{W(\mathbb {F}_q)} W(\mathbb {F}_q) \cong F$ .

2.2 The Robba ring and its variants

For $\alpha \in (0,1)$ , we put

$$ \begin{align*}\mathcal{R}_\alpha:= \Big\{\operatorname*{\mathrm{\sum}}\limits_{i\in\mathbb{Z}} c_it^i~\Big|~\ c_i\in K, \lim\limits_{i\to\pm\infty} |c_i|\rho^i=0,~\forall\rho\in[\alpha,1) \Big\}.\end{align*} $$

For any $\rho \in [\alpha ,1)$ , we define the $\rho $ -Gauss norm on $\mathcal {R}_\alpha $ by setting $\big |\sum \limits _i c_it^i \big |_\rho :=\sup _i\{|c_i|\rho ^i\}$ . The Robba ring is defined to be the union $\mathcal {R} := \mathcal {R}(K,t):= \bigcup \limits _{\alpha \in (0,1)}\mathcal {R}_\alpha $ . For any $\sum \limits _i c_it^i\in \mathcal {R}$ , we define $\big |\sum \limits _i c_it^i\big |_1:=\sup _i\{|c_i|\}\in \mathbb {R}_{\geq 0}\cup \{\infty \}$ , the $1$ -Gauss norm.

The bounded Robba ring $\mathcal {E}^{\dagger }=\mathcal {E}^{\dagger }(K,t)$ is the subring of $\mathcal {R}$ consisting of bounded elements (i.e., elements with finite $1$ -Gauss norm), which is actually a Henselian discretely valued field w.r.t. the $1$ -Gauss norm with residue field $\kappa (\!( t)\!)$ .

Let $R\in \{\mathcal {R}, \mathcal {E}^{\dagger }\}$ . An absolute q-power Frobenius lift on R is a ring endomorphism $\varphi \colon R\to R$ given by $\sum \limits _{i\in \mathbb {Z}} c_it^i \longmapsto \sum \limits _{i\in \mathbb {Z}} \varphi (c_i)u^i$ for $u=\varphi (t)\in R$ such that $|u-t^q|_1<1$ .

For any $\alpha \in (0,1)$ , we define $\tilde {\mathcal {R}}_\alpha $ to be the ring of formal sums $\operatorname *{\mathrm {\sum }}\limits _{i\in \mathbb {Q}} c_it^i$ with $c_i\in K$ , subject to the following properties.

• • For any $c>0$ , the set $\{i\in \mathbb {Q}\mid |c_i|\geq c\}$ is well-ordered.

• • For any $\rho \in [\alpha ,1)$ , we have $\lim \limits _{i\to \pm \infty } |c_i|\rho ^i=0$ .

For any $\rho \in [\alpha ,1)$ , we define the $\rho $ -Gauss norm on $\tilde {\mathcal {R}}_\alpha $ by setting

$$ \begin{align*}\Big|\operatorname*{\mathrm{\sum}}\limits_i c_it^i \Big|_\rho=\sup\limits_{i\in\mathbb{Q}}\{|c_i|\rho^i\}.\end{align*} $$

We define $\tilde {\mathcal {R}} := \tilde {\mathcal {R}}(K,t):= \bigcup \limits _{\alpha \in (0,1)} \tilde {\mathcal {R}}_\alpha $ , the extended Robba ring. The absolute Frobenius lift on $\tilde {\mathcal {R}}$ is the ring automorphism on $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ given by $\sum \limits _{i\in \mathbb {Q}} c_it^i \mapsto \sum \limits _{i\in \mathbb {Q}} \varphi (c_i)t^{iq}$ . We denote by $\operatorname {\mathrm {\tilde {\mathcal {E}}}}^{\dagger }$ the subring of $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ consisting of bounded elements. By [Reference Kedlaya11, Proposition 2.2.6], we have a $\varphi $ -equivariant embedding $\psi :\mathcal {R}\to \tilde {\mathcal {R}}$ such that $|\psi (x)|_\rho =|x|_\rho $ for $\rho $ sufficiently close to $1$ .

2.3 The slope filtration theorem

We recall Kedlaya’s theory of slopes. Let $R\in \{\mathcal {E}^{\dagger },\mathcal {R},\operatorname {\mathrm {\tilde {\mathcal {E}}}}^{\dagger },\operatorname {\mathrm {\tilde {\mathcal {R}}}}\}$ equipped with a Frobenius lift $\varphi $ . For the notions of $\varphi $ -modules and $(\varphi ,\nabla )$ -modules over R, we refer to [Reference Kedlaya9, Section 2.5]. We denote by $\operatorname {\mathrm {\textbf {Mod}^{\varphi }_R}}$ (resp. $\operatorname {\mathrm {\textbf {Mod}^{\varphi ,\nabla }_R}}$ ) the category of $\varphi $ -modules (resp. $(\varphi ,\nabla )$ -modules) over R.

Let $(M,\varphi )\in \operatorname {\mathrm {\textbf {Mod}^{\varphi }_R}}$ , and let n be a positive integer. Then, $(M,\varphi ^n)$ is a $\varphi ^n$ -module over $(R,\varphi ^n)$ . The n-pushforward functor is given by

$$ \begin{align*}[n]_*\colon \operatorname{\mathrm{\textbf{Mod}^{\varphi}_R}} \operatorname{\mathrm{\longrightarrow}} \textbf{Mod}^{\varphi^n}_R, \;\;\;\;\; (M,\varphi) \longmapsto (M,\varphi^n).\end{align*} $$

For any $s\in \mathbb {Z}$ , we define the twist $M(s)$ of $(M,\varphi )$ by s to be the $\varphi $ -module $(M,\varpi ^s\Phi )$ .

Now, let M be a $\varphi $ -module over R of rank d.

1. (i) We say that M is a unit-root $\varphi $ -module if there exists a basis $\textbf {v}_1,\ldots ,\textbf {v}_d$ of M over R in which $\varphi $ acts via an invertible matrix in $\operatorname {\mathrm {GL}}_d(\mathcal {O}_{\mathcal {E}^{\dagger }})$ if $R\in \{\mathcal {E}^{\dagger },\mathcal {R}\}$ , or $\operatorname {\mathrm {GL}}_d(\mathcal {O}_{\operatorname {\mathrm {\tilde {\mathcal {E}}}}^{\dagger }})$ if $R\in \{\operatorname {\mathrm {\tilde {\mathcal {E}}}}^{\dagger },\operatorname {\mathrm {\tilde {\mathcal {R}}}}\}$ .

2. (ii) Let $\mu =s/r \in \mathbb {Q}$ with $r>0$ and $(s,r)=1$ . We say that M is pure of slope $\mu $ if $([r]_*M)(-s)$ is unit-root.

Let $M\in \operatorname {\mathrm {\textbf {Mod}^{\varphi }_R}}$ . We have a canonical filtration $0=M_0 \subseteq M_1 \subseteq \cdots \subseteq M_l=M$ of $\varphi $ -submodules over R such that each quotient $M_i/M_{i-1}$ is pure of some slope $\mu _i$ with $\mu _1< \cdots <\mu _l$ , by [Reference Kedlaya11, Theorem 1.7.1] if $R=\mathcal {R}$ or [Reference Kedlaya11, Proposition 1.4.15 and Theorem 2.1.8]) if $R=\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ . This is called the slope filtration of M. We call $\mu _1,\ldots ,\mu _l$ the jumps of the slope filtration. The (uniquely determined, not necessarily strictly) increasing sequence $(\mu _1,\ldots ,\mu _1,\ldots ,\mu _l,\ldots ,\mu _l)$ , with each $\mu _i$ appearing $\operatorname {\mathrm {rk}}_R(M_i/M_{i-1})$ times, is said to be the slope sequence for M. We call $\operatorname {\mathrm {rk}}_R (M_i/M_{i-1})$ the multiplicity of $\mu _i$ for all $1\leq i\leq l$ .

Moreover, if M is a $(\varphi ,\nabla )$ -module over $\mathcal {R}$ , then the slope filtration can be refined to a filtration of $(\varphi ,\nabla )$ -submodules. This is [Reference Kedlaya9, Theorem 6.12], and is referred to as the slope filtration theorem for $(\varphi ,\nabla )$ -modules over $\mathcal {R}$ .

To continue, we need to recall some notions introduced in [Reference Kedlaya12, Section 14]. A difference ring (resp. difference field) is a ring (resp. field) R equipped with an endomorphism $\phi $ . A difference module over R is an R-module M equipped with a $\phi $ -linear endomorphism $\Phi $ . A finite free difference module M over R is said to be dualizable (resp. trivial) if M admits a basis over R such that $\Phi $ acts via an invertible matrix (resp. the identity matrix). For example, a $\varphi $ -module over R is a dualizable difference module over R where R is any of the rings constructed in Section 2.2. A dualizable difference module M over R is said to be standard if it admits a basis $e_1,\ldots ,e_d$ such that $e_i=\Phi (e_{i-1})$ for $2\leq i \leq d$ and $\Phi (e_d)=\lambda e_1$ for some $\lambda \in R^{\times }$ . A difference field $(k,\phi _k)$ is called strongly difference-closed if $\phi _k$ is an automorphism and any dualizable difference module over k is trivial.

Let k be a complete nonarchimedean valued field and $(k,\phi _k)$ is a difference field in which $\phi _k$ is bijective. An admissible extension of $(k,\phi _k)$ is a difference field $(\ell ,\phi _\ell )$ , where $\ell $ is a field extension of k complete for the valuation extending the one on k with the same value group, and $\phi _\ell $ is an automorphism of $\ell $ extending $\phi _k$ . (See [Reference Kedlaya11, Definition 3.2.1].)

The following lemma is a recollection of some results which will be used in the sequel.

Assertion (i) is [Reference Liu15, Proposition 1.5.6]. Assertion (ii) is [Reference Liu15, Proposition 1.5.11], and assertion (iii) is Proposition 1.5.12 in loc. cit. Assertion (iv) is [Reference Kedlaya11, Proposition 1.4.18].

2.4 The Tannakian duality

In this subsection, k denotes a field unless otherwise specified. We follow the definitions and notations in [Reference Deligne and Milne7]. We denote by $\omega ^G$ the forgetful functor $\operatorname {\mathrm {\textbf {Rep}}}_k(G)\to \operatorname {\mathrm {\textbf {Vec}}}_k$ , which is called the fiber functor.

The following Tannakian duality will be repeatedly used in this paper, whose proof can be found, e.g., in [Reference Milne18, Theorem 9.2].

Proof Notice that we have an action

$$ \begin{align*}\operatorname{\mathrm{\underline{Hom}}}^{\otimes} (\omega^G,\eta) \times \operatorname{\mathrm{\underline{Aut}}}^{\otimes} (\omega^G) \operatorname{\mathrm{\longrightarrow}} \operatorname{\mathrm{\underline{Hom}}}^{\otimes} (\omega^G,\eta)\end{align*} $$

by precomposition. By [Reference Deligne and Milne7, Theorem 3.2(i)], $\operatorname {\mathrm {\underline {Hom}}}^{\otimes } (\omega ^G,\eta )$ is an $\operatorname {\mathrm {\underline {Aut}}}^{\otimes } (\omega ^G)$ -torsor. In particular, it is a G-torsor over $\ell $ by Corollary 2.5.

Because G is a $\ell $ -group variety, G-torsors over $\ell $ are $\ell $ -varieties by [Reference Milne18, Proposition 2.69], whose isomorphism classes are classified by $H^1(\ell ,G)$ . It follows from the triviality of $H^1(\ell ,G)$ that $\operatorname {\mathrm {\underline {Hom}}}^{\otimes } (\omega ^G,\eta )(\ell )\cong G(\ell )$ ; hence, $\operatorname {\mathrm {\underline {Hom}}}^{\otimes } (\omega ^G,\eta )(\ell )\neq \emptyset $ . [Reference Deligne and Milne7, Proposition 1.13] then implies the second assertion.▪

To end this subsection, we give a Lie algebra version of Theorem 2.4. We start with recalling the notion of the Lie algebra of a k-group functor. (See [Reference Demazure and Gabriel8, Chapitre II, Section 4] for more details. Notice that k denotes a ring in loc. cit.)

For any $R\in \operatorname {\mathrm {\textbf {Alg}}}_k$ , we define the R-algebra of dual numbers $R[\varepsilon ]:= R[X]/(X^2)$ . Put $\varepsilon := X+(X^2)$ ; we then have the canonical projection $\pi _R\colon R[\varepsilon ]\to R,~\varepsilon \mapsto 0$ . Let G be a k-group functor. We define

$$ \begin{align*}\operatorname{\mathrm{Lie}}(G)(R) := \operatorname{\mathrm{Ker}} G(\pi_R).\end{align*} $$

Let $f\colon G\to H$ be a morphism of k-group functors. The commutative diagram

(1)

implies that $f(R[\epsilon ])\circ \iota _G(X)\in \operatorname {\mathrm {Lie}}(H)(R)$ for all $X\in \operatorname {\mathrm {Lie}}(G)(R)$ . We define $\operatorname {\mathrm {Lie}}(f):= f(R[\epsilon ])\circ \iota _G\colon \operatorname {\mathrm {Lie}}(G)(R) \to \operatorname {\mathrm {Lie}}(H)(R)$ . Hence, $\operatorname {\mathrm {Lie}}(\rule {2mm}{0.15mm})(R)$ is a functor from the category of k-group functors to that of abelian groups.

For an affine algebraic k-group G, we write I for the kernel of the counit $\epsilon _G\colon k[G]\to k$ . Because $k[G]$ is Noetherian, $I/I^2$ is a finite-dimensional vector space over $k\cong k[G]/I$ . We then have $\operatorname {\mathrm {Hom}}_k(I/I^2,R) \cong \operatorname {\mathrm {Hom}}_k(I/I^2,k) \bigotimes _k R$ . By [Reference Demazure and Gabriel8, Corollaire II.3.3], we have canonical group isomorphisms $\operatorname {\mathrm {Lie}}(G)(R)\cong \operatorname {\mathrm {Hom}}_k(I/I^2,R)$ and $\mathfrak {g}=\operatorname {\mathrm {Lie}}(G)(k) \cong \operatorname {\mathrm {Hom}}_k(I/I^2,k)$ , whence $\operatorname {\mathrm {Lie}}(G)(R) \cong \mathfrak {g}_R$ . The Lie structure on $\mathfrak {g}$ then canonically gives a Lie structure on $\mathfrak {g}_R$ and hence on $\operatorname {\mathrm {Lie}}(G)(R)$ . We call $\operatorname {\mathrm {Lie}}(G)(R)$ the Lie algebra of G over R, and will identify it with $\mathfrak {g}_R$ . Moreover, $\operatorname {\mathrm {Lie}}(\rule {2mm}{0.15mm})(R)$ is a functor from the category of affine algebraic k-groups to that of Lie algebras over R.

Applying the functor $\operatorname {\mathrm {Lie}}(\rule {2mm}{0.15mm})(R)$ on both sides of the isomorphism in Corollary 2.5 gives us an isomorphism $\mathfrak {g}_R\cong \operatorname {\mathrm {Lie}}(\operatorname {\mathrm {\underline {Aut}}}^{\otimes } (\omega ^G))(R)$ of Lie algebras over R. The following corollary indicates that elements in $\operatorname {\mathrm {Lie}}(\operatorname {\mathrm {\underline {Aut}}}^{\otimes } (\omega ^G))(R)$ are exactly the derivatives (in the sense of taking derivations of conditions (i–iii) in Theorem 2.4) of elements in $\operatorname {\mathrm {\underline {Aut}}}^{\otimes } (\omega ^G)(R)$ .

Proof For any $(V,\rho _V)\in \operatorname {\mathrm {\textbf {Rep}}}_k(G)$ and $\theta _V\colon V_R\to V_R$ , we define an $R[\varepsilon ]$ -linear map

$$ \begin{align*}\varepsilon \theta_V \colon V_{R[\varepsilon]} \operatorname{\mathrm{\longrightarrow}} V_{R[\varepsilon]}, ~~~~~ v\otimes(x+y\varepsilon) \longmapsto \theta_V(v\otimes x)\otimes \varepsilon.\end{align*} $$

We define an $R[\varepsilon ]$ -linear endomorphism

$$ \begin{align*}\tilde \theta_V := \operatorname{\mathrm{Id}}_{V_{R[\varepsilon]}}+\varepsilon \theta_V \colon V_{R[\varepsilon]} \operatorname{\mathrm{\longrightarrow}} V_{R[\varepsilon]}. \end{align*} $$

Then, $\tilde \theta _V\in \operatorname {\mathrm {Lie}}(\operatorname {\mathrm {GL}}_V)(R)\subseteq \operatorname {\mathrm {GL}}_V(R[\varepsilon ])$ , because $\pi _R(\tilde \theta _V)=\operatorname {\mathrm {Id}}_{V_R}$ .

We claim that the family

(2) $$ \begin{align} \big\{\tilde \theta_V\colon V_{R[\varepsilon]}\to V_{R[\varepsilon]} \mid (V,\rho_V)\in \operatorname{\mathrm{\textbf{Rep}}}_k (G) \big\} \end{align} $$

of $R[\varepsilon ]$ -linear endomorphisms satisfies conditions (i–iii) in Theorem 2.4. Granting this claim for a moment, we conclude that $\tilde \theta \in \operatorname {\mathrm {\underline {Aut}}}^{\otimes }(\omega ^G)(R[\varepsilon ])$ . In particular, there exists a unique element $X\in G(R[\varepsilon ])$ such that $\tilde \theta _V=\rho _V(X)$ for all $(V,\rho _V)\in \operatorname {\mathrm {\textbf {Rep}}}_k(G)$ . Because $\pi _R(\tilde \theta )=\operatorname {\mathrm {Id}}\in \operatorname {\mathrm {\underline {Aut}}}^{\otimes }(\omega ^G)(R)$ , we have $\tilde \theta \in \operatorname {\mathrm {Lie}}(\operatorname {\mathrm {\underline {Aut}}}^{\otimes }(\omega ^G))(R)$ . The isomorphism $\mathfrak {g}_R\cong \operatorname {\mathrm {Lie}}(\operatorname {\mathrm {\underline {Aut}}}^{\otimes } (\omega ^G))(R)$ then implies that $X\in \mathfrak {g}_R$ . Furthermore, it follows from the construction that $\theta _V=\operatorname {\mathrm {Lie}}(\rho _V)(X)$ for all $(V,\rho _V)\in \operatorname {\mathrm {\textbf {Rep}}}_k(G)$ , and the corollary follows.

It remains to prove the claim. Condition (ii) is clear from the construction. Given $(W,\rho _W)\in \operatorname {\mathrm {\textbf {Rep}}}_k(G)$ , we compute

$$ \begin{align*} \tilde \theta_{V\bigotimes W} &=\operatorname{\mathrm{Id}}_{(V\bigotimes W)_R}+\varepsilon \theta_{V\bigotimes W}\\ &=\operatorname{\mathrm{Id}}_{(V\otimes W)_R}+\varepsilon(\theta_V\otimes \operatorname{\mathrm{Id}}_{W_R} +\operatorname{\mathrm{Id}}_{V_R}\otimes \theta_W)\\ &=(\operatorname{\mathrm{Id}}_{V_R}+\varepsilon \theta_V)\otimes (\operatorname{\mathrm{Id}}_{W_R}+\varepsilon \theta_W)\\ &=\tilde \theta_V\otimes \tilde \theta_W. \end{align*} $$

Hence, (2) satisfies condition (i). It remains to show that Theorem 2.4 satisfies condition (iii). Let $\alpha \in \operatorname {\mathrm {Hom}}_G(V,W)$ . For any $v\otimes (x+y\varepsilon )\in V_{R[\varepsilon ]}$ , we compute

$$ \begin{align*} \alpha_{R[\varepsilon]}\circ \varepsilon \theta_V(v\otimes(x+y\varepsilon)) &=\alpha_{R[\varepsilon]} (\theta_V(v \otimes x)\otimes \varepsilon) =(\alpha_R\circ \theta_V)(v\otimes x )\otimes \varepsilon \\ &=(\theta_W\circ \alpha_R)(v\otimes x ) \otimes \varepsilon =\theta_W (\alpha(v)\otimes x)\otimes \varepsilon \\ &=\varepsilon\theta_W ( \alpha(v)\otimes (x+y\epsilon)) =\varepsilon\theta_W \circ \alpha_{R[\varepsilon]} (v\otimes (x+y\varepsilon)). \end{align*} $$

It follows that

$$ \begin{align*} \alpha_{R[\varepsilon]}\circ \tilde \theta_V&=\alpha_{R[\varepsilon]}\circ (\operatorname{\mathrm{Id}}_{V_{R[\varepsilon]}}+\varepsilon \theta_V) =\alpha_{R[\varepsilon]}+\alpha_{R[\varepsilon]}\circ \varepsilon \theta_V\\ &=\alpha_{R[\varepsilon]}+\varepsilon\theta_W \circ \alpha_{R[\varepsilon]} =(\operatorname{\mathrm{Id}}_{W_{R[\varepsilon]}}+\varepsilon \theta_W)\circ \alpha_{R[\varepsilon]}\\ &=\tilde \theta_W \circ \alpha_{R[\varepsilon]}, \end{align*} $$

as desired.▪

2.5 Filtered and graded fiber functors

We recall the notions of filtered and graded fiber functors on Tannakian categories following [Reference Ziegler25]. Let $\Gamma $ be a totally ordered abelian group (written additively), and let $R\in \operatorname {\mathrm {\textbf {Alg}}}_k$ . A $\Gamma $ -graded R-module is an R-module M together with a direct sum decomposition $M=\bigoplus \limits _{\gamma \in \Gamma } M_\gamma $ . A morphism between two $\Gamma $ -graded R modules M and N is an R-linear map $f\colon M\to N$ such that $f(M_\gamma )\subseteq N_\gamma $ for all $\gamma \in \Gamma $ . We denote by $\Gamma $ - $\operatorname {\mathrm {\textbf {Grad}}}_R$ the category of $\Gamma $ -graded modules over R. For $M,N\in \Gamma $ - $\operatorname {\mathrm {\textbf {Grad}}}_R$ , we define the tensor product $(M\bigotimes _R N)_\gamma = \bigoplus \limits _{\gamma '+\gamma ''=\gamma } \big ( M_{\gamma '}\bigotimes _R N_{\gamma ''}\big )$ .

Let M be an R-module. A $\Gamma $ -filtration on M is an increasing map

$$ \begin{align*}\mathcal{F}\colon \Gamma \operatorname{\mathrm{\longrightarrow}} \{R\text{-submodules of}~ M\},~~~\gamma \longmapsto \mathcal{F}^{\gamma} M ,\end{align*} $$

such that $\mathcal {F}^{\gamma } M=0$ for $\gamma \ll 0$ and $\mathcal {F}^{\gamma } M=M$ for $\gamma \gg 0$ , which is increasing in the sense that $\mathcal {F}^{\gamma } M \subseteq \mathcal {F}^{\gamma '} M$ whenever $\gamma \leq \gamma '$ . A $\Gamma $ -filtered R-module is an R-module M with a $\Gamma $ -filtration. To abbreviate notations, we sometimes denote $\mathcal {F}^{\gamma } M$ by $M^{\gamma }$ if no confusion shall arise. A morphism between two $\Gamma $ -filtered R-modules M and N is an R-linear map $f\colon M\to N$ such that $f(M^{\gamma })\subseteq N^{\gamma }$ for all $\gamma \in \Gamma $ . We denote by $\Gamma $ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ the category of $\Gamma $ -filtered modules over R.

Let M be a $\Gamma $ -filtered module over R. For any $\gamma \in \Gamma $ , we put $\mathcal {F}^{\gamma -} M:= \operatorname *{\mathrm {\sum }}\limits _{\gamma '<\gamma } \mathcal {F}^{\gamma '} M$ . We define

$$ \begin{align*}\operatorname{\mathrm{gr}}_{\mathcal{F}}^{\gamma} M:= \mathcal{F}^{\gamma} M/\mathcal{F}^{\gamma-}M.\end{align*} $$

Then, $\operatorname {\mathrm {gr}}_{\mathcal {F}} M:= \bigoplus \limits _{\gamma \in \Gamma } \operatorname {\mathrm {gr}}_{\mathcal {F}}^{\gamma } M$ is a $\Gamma $ -graded R module, and is called the $\Gamma $ -graded R-module associated to $\mathcal {F}$ . We thus have a functor

$$ \begin{align*}\operatorname{\mathrm{gr}}\colon \Gamma\text{-}\operatorname{\mathrm{\textbf{Fil}}}_R \operatorname{\mathrm{\longrightarrow}} \Gamma\text{-}\operatorname{\mathrm{\textbf{Grad}}}_R.\end{align*} $$

Elements $\gamma \in \Gamma $ such that $\operatorname {\mathrm {gr}}_{\mathcal {F}}^{\gamma } M\neq 0$ are said to be the $\Gamma $ -jumps (or simply jumps) of $\mathcal {F}$ .

The tensor product structure in $\Gamma $ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ is defined by

$$ \begin{align*}\mathcal{F}^{\gamma} (M\bigotimes_R N)=\operatorname*{\mathrm{\sum}}\limits_{\gamma'+\gamma''=\gamma} \mathcal{F}^{\gamma'} M\bigotimes_R \mathcal{F}^{\gamma''}N,\end{align*} $$

for all $\Gamma $ -filtered modules M and N over R.

A morphism $f\colon M\to N$ in $\Gamma $ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ is said to be admissible (or strict) if

$$ \begin{align*}f(M^{\gamma})=f(M)\cap N^{\gamma},~~~~~\forall \gamma\in \Gamma.\end{align*} $$

Following [Reference Ziegler25, Section 4.1], we say that a short sequence in $\Gamma $ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ is exact if both of $f'$ and $f''$ are admissible, and the underlying short sequence in $\operatorname {\mathrm {\textbf {Mod}}}_R$ is exact.

Let $\mathcal T$ be a Tannakian category over k, and let R be a k-algebra.

1. (i) A $\Gamma $ -graded fiber functor on $\mathcal T$ over R is an exact faithful k-linear tensor functor $\tau \colon \mathcal T \to \Gamma $ - $\operatorname {\mathrm {\textbf {Grad}}}_R$ .

2. (ii) A $\Gamma $ -filtered fiber functor on $\mathcal T$ over R is an exact faithful k-linear tensor functor $\eta \colon \mathcal T \to \Gamma $ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ .

3. (iii) Given an object $M=\bigoplus \limits _{\gamma \in \Gamma } M_\gamma $ in $\Gamma $ - $\operatorname {\mathrm {\textbf {Grad}}}_R$ , we put $\mathcal {F}^{\gamma }(M):=\bigoplus \limits _{\gamma '\leq \gamma } M_{\gamma '}$ . This gives rise to a functor $\operatorname {\mathrm {fil}}\colon \Gamma $ - $\operatorname {\mathrm {\textbf {Grad}}}_R \to \Gamma $ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ .

4. (iv) A $\Gamma $ -filtered fiber functor $\eta $ is called splittable if there exists a $\Gamma $ -graded fiber functor $\tau $ such that $\eta =\operatorname {\mathrm {fil}}\circ \tau $ , and $\tau $ is called a splitting of $\eta $ .

Construction 2.11 Let $(M,\mathcal {F})\in \mathbb {Z}$ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ be a $\mathbb {Z}$ -filtered module with $\mathbb {Z}$ -jumps $\jmath _1< \cdots <\jmath _n$ . For any $\gamma \in \mathbb {Q}_{>0}$ , we define a $\mathbb {Q}$ -filtered module $(M,[\gamma ]_*\mathcal {F})$ by

$$ \begin{align*}([\gamma]_*\mathcal{F})^{x} M:= \left\{ \begin{array}{ll} 0 &\text{for}~x< \jmath_1\gamma, \\ M^{\jmath_i} ~~~ &\text{for}~ \jmath_i\gamma \leq x <\jmath_{i+1}\gamma,~ 1\leq i\leq n-1,\\ M &\text{for}~x\geq \jmath_n\gamma. \end{array} \right.\end{align*} $$

We then have a fully faithful embedding $[\gamma ]_*\colon \mathbb {Z}$ - $\operatorname {\mathrm {\textbf {Fil}}}_R \to \mathbb {Q}$ - $\operatorname {\mathrm {\textbf {Fil}}}_R$ . Similarly, we have a fully faithful embedding $[\gamma ]_*\colon \mathbb {Z}$ - $\operatorname {\mathrm {\textbf {Grad}}}_R \to \mathbb {Q}$ - $\operatorname {\mathrm {\textbf {Grad}}}_R$ by defining $[\gamma ]_* := \operatorname {\mathrm {gr}}\circ [\gamma ]_*\circ \operatorname {\mathrm {fil}}$ .

To end this subsection, we exhibit the following theorem for latter use. (Be aware that in [Reference Ziegler25], the author only considers $\Gamma $ -gradings and $\Gamma $ -filtrations for $\Gamma = \mathbb{Z}$ .)

3.1 Definition

Let $R\in \{\mathcal {E}^{\dagger },\mathcal {R},\operatorname {\mathrm {\tilde {\mathcal {E}}}}^{\dagger },\operatorname {\mathrm {\tilde {\mathcal {R}}}}\}$ equipped with an absolute Frobenius lift $\varphi $ . The following definition is motivated by that of G-isocrystals introduced in [Reference Dat, Orlik and Rapoport6, Section IX.1].

Definition 3.1 A G- $\varphi $ -module over R is an exact faithful F-linear tensor functor

$$ \begin{align*}\operatorname{\mathrm{I}}\colon \operatorname{\mathrm{\textbf{Rep}}}_F(G) \operatorname{\mathrm{\longrightarrow}} \operatorname{\mathrm{\textbf{Mod}^{\varphi}_R}},\end{align*} $$

which satisfies $\operatorname {\mathrm {forg}} \circ \operatorname {\mathrm {I}} = \omega ^G \otimes R$ , where $\operatorname {\mathrm {forg}} \colon \operatorname {\mathrm {\textbf {Mod}^{\varphi }_R}} \to \operatorname {\mathrm {\textbf {Mod}}}_R$ is the forgetful functor. The category of G- $\varphi $ -modules over R is denoted by $\operatorname {\mathrm {G-\textbf {Mod}^{\varphi }_R}}$ , whose morphisms are morphisms of tensor functors.

Let $(V,\rho )\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , and let $g\in G(R)$ . We define $\operatorname {\mathrm {I}}(g)(V):= (V_R,g\varphi )$ , where

$$ \begin{align*} g\varphi\colon V_R \operatorname{\mathrm{\longrightarrow}} V_R, \;\;\;\;\; v\otimes f \longmapsto \rho(g)(v\otimes 1)\varphi(f). \end{align*} $$

Let $V,W\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ . We have a canonical isomorphism $(V\otimes W)_{\mathcal {R}} \cong V_{\mathcal {R}} \otimes _{\mathcal {R}} W_{\mathcal {R}}$ , and we will henceforth identify them. Given any $\alpha \in \operatorname {\mathrm {Hom}}_G(V,W)$ , we define $\operatorname {\mathrm {I}}(g)(\alpha ):= \alpha _R$ . We thus have the following G- $\varphi $ -module over R (associated to g).

$$ \begin{align*}\operatorname{\mathrm{I}}(g)\colon \operatorname{\mathrm{\textbf{Rep}}}_F(G) \operatorname{\mathrm{\longrightarrow}} \operatorname{\mathrm{\textbf{Mod}^{\varphi}_R}},\;\;\;\;\; V\longmapsto (V_R,g\varphi).\end{align*} $$

We call $\operatorname {\mathrm {I}}(g)(V)=(V_R,g\varphi )$ the G- $\varphi $ -module over R associated to g.

For any $g\in G(R)$ , we sometimes write $\Phi _g= \Phi _{g,V}$ for the $\varphi $ -linear action $g\varphi $ on $V_R$ . Both notations have their own advantages in practice.

Remark 3.2 For any $g\in G(R)$ , we define $\Phi (g):= G(\varphi )(g)$ . For any $(V,\rho )\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , we have a commutative diagram

Hence, $\rho (\varphi (g))=\varphi (\rho (g))$ . For any $h\in G(R)$ and $n,m\geq 0$ , we have the following formula in $G(R)\rtimes \operatorname {\mathrm {\langle }}\varphi \operatorname {\mathrm {\rangle }}$ :

$$ \begin{align*}(h\varphi^n)\circ (g\varphi^m)=\big(h\varphi^n(g)\big)\varphi^{n+m}. \end{align*} $$

3.2 The $\mathbb {Q}$ -filtered fiber functor $\operatorname {\mathrm {{HN}}}_g$

We fix an element $g\in G(\mathcal {R})$ .

Construction 3.3 For any $V\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , we have a $\varphi $ -module $(V_{\mathcal {R}},g\varphi )$ over $\mathcal {R}$ . Kedlaya’s slope filtration theorem [Reference Kedlaya9, Theorem 6.10] then provides a filtration

$$ \begin{align*}0\subseteq V_{\mathcal{R}}^{\mu_1}\subseteq \cdots\subseteq V_{\mathcal{R}}^{\mu_l}=V_{\mathcal{R}},\end{align*} $$

satisfying

• • $V_{\mathcal {R}}^{\mu _1}$ is pure of some slope $\mu _1\in \mathbb {Q}$ and each $V_{\mathcal {R}}^{\mu _i}/V_{\mathcal {R}}^{\mu _{i-1}}$ is pure of some slope $\mu _i\in \mathbb {Q}$ for $2\leq i\leq l$ ;

• • $\mu _1<\cdots <\mu _l$ .

We thus have an increasing map

$$ \begin{align*} \mathcal{HN}_g \colon \mathbb{Q} &\operatorname{\mathrm{\longrightarrow}} \{\mathcal{R}\text{-modules of}~V_{\mathcal{R}}\}\\ x &\longmapsto \mathcal{HN}_g^x(V_{\mathcal{R}}), \end{align*} $$

where

$$ \begin{align*}\mathcal{HN}_g^x(V_{\mathcal{R}})=\left\{ \begin{array}{ll} 0 &\text{for}~x<\mu_1,\\ V_{\mathcal{R}}^{\mu_i} ~~~ &\text{for}~ \mu_i\leq x <\mu_{i+1}, 1\leq i\leq l-1,\\ V_{\mathcal{R}} &\text{for}~x\geq \mu_l. \end{array} \right.\end{align*} $$

Then, $(V_{\mathcal {R}},\mathcal {HN}_g)$ is a $\mathbb {Q}$ -filtered module over $\mathcal {R}$ with $\mathbb {Q}$ -jumps $\mu _1<\cdots <\mu _l$ . We will denote $\mathcal {HN}_g^x(V_{\mathcal {R}})$ by $V_{\mathcal {R}}^x$ when $\mathcal {HN}_g$ is clear in the context.

Theorem 3.4 The assignments

$$ \begin{align*}V \longmapsto (V_{\mathcal{R}},\mathcal{HN}_g) ~~~\text{and}~~~\alpha \longmapsto \alpha_{\mathcal{R}},\end{align*} $$

for all $\alpha \in \operatorname {\mathrm {Hom}}_G(V,W)$ , define a $\mathbb {Q}$ -filtered fiber functor

$$ \begin{align*}\operatorname{\mathrm{{HN}}}_g \colon \operatorname{\mathrm{\textbf{Rep}}}_F(G) \operatorname{\mathrm{\longrightarrow}} \mathbb{Q}\text{-}\operatorname{\mathrm{\textbf{Fil}}}_{\mathcal{R}}.\end{align*} $$

For any admissible extension E of K, we first remark that the $\varphi $ -equivariant embedding $\psi \colon \mathcal {R}\to \operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)$ is faithfully flat (see [Reference Kedlaya11, Remark 3.5.3]). We also remark that, if $M_1$ and $M_2$ are pure $\varphi $ -modules over $\mathcal {R}$ of slopes $\mu _1$ and $\mu _2$ , respectively, then $M_1 \bigotimes _{\mathcal {R}} M_2$ is pure of slope $\mu _1+\mu _2$ (cf. [Reference Kedlaya11, Corollary 1.6.4]). These facts will be repeatedly used in the sequel.

Proof Let be the trivial G-representation. Then, is of rank $1$ with slope $0$ , proving that $\operatorname {\mathrm {{HN}}}_g$ preserves identity objects.

We claim that $\operatorname {\mathrm {{HN}}}_g$ is functorial. Let $\alpha \in \operatorname {\mathrm {Hom}}_G(V,W)$ be a morphism of finite-dimensional G-modules. We need to show that $\alpha _{\mathcal {R}}(V_{\mathcal {R}}^x)\subseteq W_{\mathcal {R}}^x$ for all ${x\in \mathbb {Q}}$ . Choose by Lemma 2.2 an admissible extension E of K such that $\kappa _E$ is strongly difference-closed. For any fixed $x\in \mathbb {Q}$ , we set $V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x := V_{\mathcal {R}}^x\bigotimes _{\mathcal {R}} {\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}$ , and $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x := W_{\mathcal {R}}^x\bigotimes _{\mathcal {R}} {\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}$ . By Lemma 2.3(iii), we have a decomposition $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}=W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x \operatorname *{\mathrm {\bigoplus }} W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}'$ of $\varphi $ -modules over $\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)$ , where $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x$ (resp. $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}'$ ) has slopes less or equal to x (resp. greater than x). By Lemma 2.3(iv), the induced morphism $V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x \to W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}'$ of $\varphi $ -modules is zero. We thus have $\alpha _{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)} \big (V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x \big )\subseteq W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x$ . Given any $\textbf {v}\in V_{\mathcal {R}}^x$ , we may write $\alpha _{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}(\textbf {v}\otimes 1)=\alpha _{\mathcal {R}}(\textbf {v})\otimes 1=\operatorname *{\mathrm {\sum }}\limits _{i\in I} \textbf {w}_i\otimes s_i$ for some finite set I, with $\textbf {w}_i\in W_{\mathcal {R}}^x$ and $s_i\in {\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}$ for all $i\in I$ . Let M be the $\mathcal {R}$ -submodule of $W_{\mathcal {R}}$ generated by $\alpha _{\mathcal {R}}(\textbf {v})$ and the $\textbf {w}_i$ , and let N be the $\mathcal {R}$ -submodule of $W_{\mathcal {R}}^x$ generated by the $\textbf {w}_i$ . We then have $(M/N)\bigotimes _{\mathcal {R}} \operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t) \cong (M\bigotimes _{\mathcal {R}} \operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t))/ (N\bigotimes _{\mathcal {R}} \operatorname {\mathrm {\tilde {\mathcal {R}}}}{(E,t))=0}$ . It follows that $M/N=0$ as $\mathcal {R}\to \operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)$ is faithfully flat. We thus have $\alpha _{\mathcal {R}}(\textbf {v})\in N\subseteq W_{\mathcal {R}}^x$ , as desired.

It remains to show that $\operatorname {\mathrm {{HN}}}_g$ preserves tensor products (in the sense of Remark 2.10(ii)). Let V and W be two finite-dimensional G-modules, and suppose that the slope filtration of $(V_{\mathcal {R}},g\varphi )$ (resp. $(W_{\mathcal {R}},g\varphi )$ ) has jumps $\mu _1<\cdots <\mu _{l_V}$ (resp. ${\nu _1<\cdots <\nu _{l_W}}$ ). By [Reference Kedlaya12, Lemma 16.4.3], $\big ((V\bigotimes _F W)_{\mathcal {R}},g\varphi \big )$ has jumps $\{\mu _i+\nu _j \mid 1\leq i\leq {l_V}, 1\leq j\leq {l_W}\}$ . Fix any $1\leq l\leq {l_V}$ and $1\leq s\leq {l_W}$ ; we need to show

(3) $$ \begin{align} (V \bigotimes_F W)_{\mathcal{R}}^{\mu_{l}+\nu_{s}}= \operatorname*{\mathrm{\sum}}\limits_{\scriptstyle \begin{array}{c} x,y\in\mathbb{Q} \\ x+y=\mu_{l}+\nu_{s}\end{array}} V_{\mathcal{R}}^x \bigotimes_{\mathcal{R}} W_{\mathcal{R}}^y, \end{align} $$

and we will do so in the remainder of the proof.

We claim that

$$ \begin{align*}\operatorname*{\mathrm{\sum}}\limits_{\scriptstyle \begin{array}{c} x,y\in\mathbb{Q} \\ x+y=\mu_{l}+\nu_{s}\end{array}} V_{\mathcal{R}}^x \bigotimes_{\mathcal{R}} W_{\mathcal{R}}^y= \operatorname*{\mathrm{\sum}}\limits_{\scriptstyle \begin{array}{c} \mu_i+\nu_j\leq \mu_l+\nu_s \\ 1\leq i\leq {l_V}, 1\leq j\leq {l_W}\end{array}}V_{\mathcal{R}}^{\mu_i} \bigotimes_{\mathcal{R}} W_{\mathcal{R}}^{\nu_j}.\end{align*} $$

It is clear that the RHS is contained in the LHS; we now show the reverse inclusion. Let $x,y\in \mathbb {Q}$ such that $x+y=\mu _{l}+\nu _{s}$ . If $x<\mu _1$ or $y<\nu _1$ , then $V_{\mathcal {R}}^x\bigotimes _{\mathcal {R}} W_{\mathcal {R}}^y=0$ which is contained in the RHS. Otherwise, there exists the largest integer $1\leq i\leq {l_V}$ (resp. $1\leq j\leq {l_W}$ ) with the property that $\mu _i\leq x$ (resp. $\nu _j\leq y$ ). We then have $V_{\mathcal {R}}^x\bigotimes _{\mathcal {R}} W_{\mathcal {R}}^y=V_{\mathcal {R}}^{\mu _i}\bigotimes _{\mathcal {R}} W_{\mathcal {R}}^{\nu _j}$ and $\mu _i+\nu _j\leq \mu _{l}+\nu _{s}$ . The claim is thus proved.

From Lemma 2.3(iii), we see that

$$ \begin{align*} \big(V \bigotimes_F W \big)_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}^{\mu_{l}+\nu_{s}}= \Big( \operatorname*{\mathrm{\sum}}\limits_{\scriptstyle \begin{array}{c} \mu_i+\nu_j\leq \mu_{l}+\nu_{s} \\ 1\leq i\leq {l_V}, 1\leq j\leq {l_W}\end{array}} V_{\mathcal{R}}^{\mu_i}\bigotimes_{\mathcal{R}} W_{\mathcal{R}}^{\nu_j} \Big)\bigotimes_{\mathcal{R}} {\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}.\end{align*} $$

Therefore, we have

$$ \begin{align*} \big(V \bigotimes W \big)_{\mathcal{R}}^{\mu_{l}+\nu_{s}}=\operatorname*{\mathrm{\sum}}\limits_{\scriptstyle \begin{array}{c}\mu_i+\nu_j\leq \mu_{l}+\nu_{s} \\ 1\leq i\leq {l_V}, 1\leq j\leq {l_W}\end{array}} V_{\mathcal{R}}^{\mu_i}\bigotimes_{\mathcal{R}} W_{\mathcal{R}}^{\nu_j}\end{align*} $$

by Lemma 2.3(i) and the fact that $\mathcal {R} \to {\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}$ is faithfully flat. The desired equality (1) then follows from the preceding claim.▪

Let $(M,\varphi )$ be a $\varphi $ -module over $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ of rank d. Then, $\Phi $ is invertible, because the Frobenius lift on $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ is bijective, and $(M,\varphi ^{-1})$ is a $\varphi ^{-1}$ -module over $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ . More explicitly, let $A\in \operatorname {\mathrm {GL}}_d(\operatorname {\mathrm {\tilde {\mathcal {R}}}})$ be the matrix of action of $\varphi $ in some basis for M over $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ . Then, in the same basis, the matrix of action of $\varphi ^{-1}$ is $\varphi ^{-1}(A^{-1})$ . For example, if $M=V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}}$ for some $V\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , and $\Phi =\psi (g)\varphi $ where $\psi $ denotes (by abuse of notation) the group morphism $G(\mathcal {R}) \to G(\operatorname {\mathrm {\tilde {\mathcal {R}}}})$ induced by the embedding $\psi \colon \mathcal {R} \to \operatorname {\mathrm {\tilde {\mathcal {R}}}}$ recalled above Proposition 3.5, then

$$ \begin{align*} \big(\psi(g)\varphi \big) \cdot \big(\varphi^{-1}(\psi(g^{-1}))\varphi^{-1}\big)=1\end{align*} $$

in $G(\operatorname {\mathrm {\tilde {\mathcal {R}}}})\rtimes \operatorname {\mathrm {\langle }}\varphi \operatorname {\mathrm {\rangle }}$ (cf. Remark 3.2), which implies that $\varphi ^{-1}=\varphi ^{-1}(\psi (g^{-1}))\varphi ^{-1}$ .

Let M be a standard $\varphi $ -module over $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ of slope $\mu =s/r$ with $r>0$ and $(s,r)=1$ . Namely, we have a standard basis $e_1,\ldots ,e_r$ in which $\varphi $ acts via

$$ \begin{align*}A= \left(\begin{smallmatrix} 0 & & &\varpi^s\\ 1 &\ddots & & \\ &\;\ddots &\;\;\ddots &\\ && 1 &0 \end{smallmatrix}\right). \end{align*} $$

Then,

$$ \begin{align*}\varphi^{-1}(A^{-1})= \left(\begin{smallmatrix} 0 &1 & &\\ &\ddots &\ddots & \\ & & \;\;\ddots & {{\scriptstyle 1}} \\ \varpi^{-s} & & &0 \end{smallmatrix}\right), \end{align*} $$

which implies that $(M,\varphi ^{-1})$ is a standard $\varphi ^{-1}$ -module pure of slope $-\mu $ .

Proof Let $\alpha \in \operatorname {\mathrm {Hom}}_G(V,W)$ be a morphism of finite-dimensional G-modules. We need to show that $\alpha _{\mathcal {R}}(V_{\mathcal {R}}^x)=\alpha _{\mathcal {R}}(V_{\mathcal {R}})\cap W_{\mathcal {R}}^x$ for all $x\in \mathbb {Q}$ . For any fixed $x\in \mathbb {Q}$ , the functoriality in Proposition 3.5 already implies that $\alpha _{\mathcal {R}}(V_{\mathcal {R}}^x)\subseteq \alpha _{\mathcal {R}}(V_{\mathcal {R}})\cap W_{\mathcal {R}}^x$ . Thus, it suffices to show that for any nonzero element $\textbf {v}\in V_{\mathcal {R}}$ such that $\alpha _{\mathcal {R}}(\textbf {v})\in W_{\mathcal {R}}^x$ , there exists $\textbf {v}'\in V_{\mathcal {R}}^x$ with $\alpha _{\mathcal {R}}(\textbf {v})=\alpha _{\mathcal {R}}(\textbf {v}')$ .

By Lemma 2.3(iii), we have decompositions

(4) $$ \begin{align} V_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}=V_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}^x \operatorname*{\mathrm{\bigoplus}} V_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}'~~~\text{and}~~~W_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}=W_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}^x \operatorname*{\mathrm{\bigoplus}} W_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}' \end{align} $$

of $\varphi $ -modules over $\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)$ , in which $V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x$ and $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x$ have slopes less or equal to x, while $V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}'$ and $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}'$ have slopes greater than x. Notice that the composition

is a morphism of $\varphi $ -modules. We claim that $\xi =0$ . We write $\varphi =\psi (g)\varphi $ , then $\varphi ^{-1}=\varphi ^{-1}(\psi (g^{-1}))\varphi ^{-1}$ . Because $\alpha $ is G-equivariant and $\varphi ^{-1}(\psi (g^{-1}))\in G(\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t))$ , we have that $\alpha _{\operatorname {\mathrm {\tilde {\mathcal {R}}}}}\colon (V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)},\varphi ^{-1}) \to (W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)},\varphi ^{-1})$ is a morphism of $\varphi ^{-1}$ -modules. On the other hand, we also have decompositions of $\varphi ^{-1}$ -modules as in (2), together with the induced morphism $\xi \colon V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}' \to W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x$ of $\varphi ^{-1}$ -modules. But in this case, $V_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}'$ has slopes less than x, while $W_{\operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)}^x$ has slopes greater or equal to x. It then follows from Lemma 2.3(iv) that $\xi =0$ , as claimed.

Therefore, we find $\textbf {v}_1,\ldots ,\textbf {v}_n \in V_{\mathcal {R}}^x$ and $s_1,\ldots ,s_n\in \operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)$ such that

$$ \begin{align*} \alpha_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t)}(\textbf{v}\otimes 1)=\alpha_{\mathcal{R}}(\textbf{v})\otimes 1 =\operatorname*{\mathrm{\sum}}\limits_{i=1}^n \alpha_{\mathcal{R}}(\textbf{v}_i)\otimes s_i.\end{align*} $$

Let M be the submodule of $W_{\mathcal {R}}$ generated by $\alpha _{\mathcal {R}}(\textbf {v})$ and the $\alpha _{\mathcal {R}}(\textbf {v}_i)$ , and let N be the submodule generated by the $\alpha _{\mathcal {R}}(\textbf {v}_i)$ . We then have

$$ \begin{align*}(M/N)\bigotimes_{\mathcal{R}} \operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t) \cong (M\bigotimes_{\mathcal{R}} \operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t))/ (N\bigotimes_{\mathcal{R}} \operatorname{\mathrm{\tilde{\mathcal{R}}}}(E,t))=0.\end{align*} $$

It follows that $M/N=0$ as $\mathcal {R}\to \operatorname {\mathrm {\tilde {\mathcal {R}}}}(E,t)$ is faithfully flat, and hence, $\alpha _{\mathcal {R}}(\textbf {v})=\sum \limits _{i=1}^n r_i\alpha _{\mathcal {R}}(\textbf {v}_i)\in W_{\mathcal {R}}^x$ for some $r_i\in \mathcal {R}$ . Put $\textbf {v}':= \sum \limits _{i=1}^n r_i\textbf {v}_i \in V_{\mathcal {R}}^x$ , we then have $\alpha _{\mathcal {R}}(\textbf {v}')=\alpha _{\mathcal {R}}(\textbf {v})$ , as desired.▪

3.3 Splittings of $\operatorname {\mathrm {{HN}}}_g$

As before, we fix an element $g\in G(\mathcal {R})$ . In Section 3.2, we have constructed a $\mathbb {Q}$ -filtered fiber functor $\operatorname {\mathrm {{HN}}}_g\colon \operatorname {\mathrm {\textbf {Rep}}}_F(G) \to \mathbb {Q}$ - $\operatorname {\mathrm {\textbf {Fil}}}_{\mathcal {R}}$ . In this subsection, we show that $\operatorname {\mathrm {{HN}}}_g$ is splittable whenever G is smooth. Our strategy goes as follows. We first use Lemma 3.10 reducing $\operatorname {\mathrm {{HN}}}_g$ to a $\mathbb {Z}$ -filtered fiber functor $\operatorname {\mathrm {{HN}}}_g^{\mathbb {Z}}$ to which Theorem 2.12 is applicable. This $\operatorname {\mathrm {{HN}}}_g^{\mathbb {Z}}$ then admits a $\mathbb {Z}$ -splitting. Finally, in Theorem 3.12, we lift such a $\mathbb {Z}$ -splitting to a $\mathbb {Q}$ -splitting of $\operatorname {\mathrm {{HN}}}_g$ .

Definition 3.7 We define the support of $\operatorname {\mathrm {{HN}}}_g$ by

$$ \begin{align*}\operatorname{\mathrm{Supp}}(\operatorname{\mathrm{{HN}}}_g) := \{x\in \mathbb{Q} \mid \operatorname{\mathrm{gr}}^x_{\operatorname{\mathrm{{HN}}}_g}(V)\neq 0~\text{for some}~ V\in \operatorname{\mathrm{\textbf{Rep}}}_F(G)\}.\end{align*} $$

Notice that $\operatorname {\mathrm {Supp}}(\operatorname {\mathrm {{HN}}}_g)$ is the set of jumps of the slope filtrations of $(V_{\mathcal {R}},g\varphi )$ for all $V\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ .

The general idea of the following construction was addressed in [Reference Anschütz2], after Definition 2.5 in loc. cit.; we will make it more explicit in our case.

Construction 3.8 Let $W\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ be a faithful representation. Because G is algebraic, W is a tensor generator for $\operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , i.e., any representation V of G is a subquotient of a direct sum of representations $\bigotimes ^m (W \bigoplus W^{\vee })$ for various $m\in \mathbb {N}$ . (See [Reference Milne18, Theorem 4.14].) Therefore, $\operatorname {\mathrm {Supp}}(\operatorname {\mathrm {{HN}}}_g)$ is the additive subgroup of $\mathbb {Q}$ finitely generated by the $\mathbb {Q}$ -jumps $\nu _1,\ldots ,\nu _n$ of $(W_{\mathcal {R}},g\varphi )$ . We write $\nu _i=s_i/d_i$ with $d_i>0$ and $(s_i,d_i)=1$ for $1\leq i\leq n$ . Let $d_g\in \mathbb {N}$ be the least common multiple of the $d_i$ . We then have $d_g\nu _i\in \mathbb {Z}$ for $1\leq i\leq n$ . In particular, we have

$$ \begin{align*}d_g=\min \{m\in\mathbb{N} \mid mx\in \mathbb{Z}, \forall x\in \operatorname{\mathrm{Supp}}(\operatorname{\mathrm{{HN}}}_g)\}.\end{align*} $$

Therefore, $d_g$ is uniquely determined by g. We call $d_g$ the least common denominator of g.

Lemma 3.10 $\operatorname {\mathrm {{HN}}}_g$ factors through a $\mathbb {Z}$ -filtered fiber functor $\operatorname {\mathrm {{HN}}}^{\mathbb {Z}}_g\colon \operatorname {\mathrm {\textbf {Rep}}}_F(G) \to \mathbb {Z}$ - $\operatorname {\mathrm {\textbf {Fil}}}_{\mathcal {R}}$ which makes the diagram

commute.

We remark that the functor ${[d_g^{-1}]}_*$ (see Construction 2.11) is nothing but relabeling the jumps by multiplying all jumps with $d_g^{-1}$ . In particular, this lemma implies that $\operatorname {\mathrm {gr}}^x_{\operatorname {\mathrm {{HN}}}_g}(V)=\operatorname {\mathrm {gr}}^{d_g^{-1}x}_{\operatorname {\mathrm {{HN}}}^{\mathbb {Z}}_g}(V)$ for all $x\in \mathbb {Q}$ and $V\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ .

Proof of Lemma 3.10 Let $V\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , and let $\mu _1,\ldots ,\mu _l$ be the $\mathbb {Q}$ -jumps of $(V_{\mathcal {R}},g\varphi )$ . We then have $d_g\mu _i\in \mathbb {Z}$ for all i. We have an increasing map

$$ \begin{align*} \mathcal{F}_g \colon \mathbb{Z} &\operatorname{\mathrm{\longrightarrow}} \{\mathcal{R}\text{-submodules of}~V_{\mathcal{R}}\},\\ x &\longmapsto \mathcal{F}_g^x(V_{\mathcal{R}}), \end{align*} $$

where

$$ \begin{align*}\mathcal{F}_g^x(V_{\mathcal{R}}):= \left\{ \begin{array}{ll} 0 &\text{for}~x<d_g\mu_1,\\ \mathcal{HN}_g^{\mu_i}(V_{\mathcal{R}}) ~~~ &\text{for}~ d_g\mu_i\leq x <d_g\mu_{i+1}, 1\leq i\leq l-1,\\ V_{\mathcal{R}} &\text{for}~x\geq d_g\mu_l. \end{array} \right.\end{align*} $$

Then, $(V_{\mathcal {R}},\mathcal {F}_g)$ is a $\mathbb {Z}$ -filtered module over $\mathcal {R}$ with $\mathbb {Z}$ -jumps $d_g\mu _1<\cdots <d_g\mu _l$ . We thus have a $\mathbb {Z}$ -filtered fiber functor

$$ \begin{align*} \operatorname{\mathrm{{HN}}}^{\mathbb{Z}}_g \colon \operatorname{\mathrm{\textbf{Rep}}}_F(G) &\operatorname{\mathrm{\longrightarrow}} \mathbb{Z}\text{-}\operatorname{\mathrm{\textbf{Fil}}}_{\mathcal{R}},\\ V &\longmapsto (V_{\mathcal{R}},\mathcal{F}_g), \end{align*} $$

satisfying $\operatorname {\mathrm {{HN}}}_g=[d_g^{-1}]_*\circ \operatorname {\mathrm {{HN}}}_g^{\mathbb {Z}}$ .▪

By the definition of $\operatorname {\mathrm {\underline {Aut}}}^{\otimes }$ and Corollary 2.5, we have $\operatorname {\mathrm {\underline {Aut}}}^{\otimes }(\omega ^G)(R)=\operatorname {\mathrm {Aut}}^{\otimes } (\omega ^G_R)\cong G(R)$ for all $R\in \operatorname {\mathrm {\textbf {Alg}}}_k$ . For any R-algebra S, we then have

$$ \begin{align*}\operatorname{\mathrm{\underline{Aut}}}^{\otimes}(\omega_R^G)(S)=\operatorname{\mathrm{Aut}}^{\otimes} (\omega^G_R\otimes S)=\operatorname{\mathrm{Aut}}^{\otimes} (\omega^G_S)\cong G_R(S).\end{align*} $$

Proof Because $\operatorname {\mathrm {forg}}\circ \operatorname {\mathrm {{HN}}}^{\mathbb {Z}}_g=\omega ^G\otimes \mathcal {R}$ , we have

$$ \begin{align*} \operatorname{\mathrm{\underline{Aut}}}^{\otimes}_{\mathcal{R}}(\operatorname{\mathrm{forg}}\circ \operatorname{\mathrm{{HN}}}^{\mathbb{Z}}_g)= \operatorname{\mathrm{\underline{Aut}}}^{\otimes}_{\mathcal{R}}(\omega^G_{\mathcal{R}})\cong G_{\mathcal{R}}.\end{align*} $$

Notice that $G_{\mathcal {R}}$ is smooth over $\mathcal {R}$ ; the proposition then follows from Theorem 2.12.▪

Proof Choose a splitting $\tau _g\colon \operatorname {\mathrm {\textbf {Rep}}}_F(G)\to \mathbb {Z}$ - $\operatorname {\mathrm {\textbf {Grad}}}_{\mathcal {R}}$ of $\operatorname {\mathrm {{HN}}}^{\mathbb {Z}}_g$ by Proposition 3.11, we then have a $\mathbb {Q}$ -graded fiber functor ${[d_g^{-1}]}_*\circ \tau _g\colon \operatorname {\mathrm {\textbf {Rep}}}_F(G)\to \mathbb {Q}$ - $\operatorname {\mathrm {\textbf {Grad}}}_{\mathcal {R}}$ . On the other hand, we have the diagram

(5)

with the upper-left, the upper-right, and the bottom triangles commutative. Here, the commutativity of the upper-left (resp. the upper-right) triangle follows from Proposition 3.11 (resp. Lemma 3.10); for the bottom one, we note that ${[d_g^{-1}]}_*\circ \operatorname {\mathrm {fil}}=\operatorname {\mathrm {fil}}\circ {[d_g^{-1}]}_*$ . Hence, the outer diagram also commutes, which implies that $\operatorname {\mathrm {{HN}}}_g$ factors through the $\mathbb {Q}$ -graded fiber functor ${[d_g^{-1}]}_*\circ \tau _g$ , as desired.▪

3.4 The slope morphism

Let R be a commutative ring with $1$ , and let $\Gamma $ be an abelian group (not necessarily finitely generated). We first continue the discussions in Section 2.5 to see how $\Gamma $ -gradings over R are related to $D_R(\Gamma )$ -modules, for some affine group scheme $D_R(\Gamma )$ which will be defined as follows.

The group algebra $R[\Gamma ]:= \bigoplus \limits _{\gamma \in \Gamma } Re_\gamma $ carries a Hopf algebra structure, where the comultiplication is given by $\Delta (e_\gamma )=e_\gamma \otimes e_\gamma $ , the counit is given by $\epsilon (e_\gamma )=1$ , and the antipode is given by $S(e_\gamma )=e_{-\gamma }$ , for all $\gamma \in \Gamma $ . We denote by $D_R(\Gamma )$ the affine R-group scheme represented by $R[\Gamma ]$ . For any $\gamma \in \Gamma $ , the Hopf algebra morphism $R[\mathbb {Z}] \to R[\Gamma ], e_1 \mapsto e_\gamma $ gives rise to a character $\chi _\gamma \colon D_R(\Gamma ) \to \mathbb {G}_{m,R}$ of $D_R(\Gamma )$ . For the remainder of this paper, we denote by $\mathbb {D}_R$ the R-group scheme $D_R(\mathbb {Q})$ .

Let $M=\bigoplus _{\gamma \in \Gamma } M_\gamma $ be a $\Gamma $ -graded module over R. Then, M becomes a $D_R(\Gamma )$ -module where $D_R(\Gamma )$ acts on each $M_\gamma $ via $\chi _\gamma $ . The following lemma shows that this assignment gives an equivalence of categories.

Proof Let $M\in \mathbb {Z}\text {-}\operatorname {\mathrm {\textbf {Grad}}}_R$ . By Lemma 3.13, we may write $M=\bigoplus \limits _{n\in \mathbb {Z}} M_n$ as a direct sum of eigenmodules. By construction, we have $[\gamma ]_*(M)=\bigoplus \limits _{n\in \mathbb {Z}} ([\gamma ]_*(M))_{\gamma n}$ with $([\gamma ]_*(M))_{\gamma n}=M_n$ for all n, which is also a decomposition into eigenmodules. Therefore, giving $[\gamma ]_*$ is equivalent to giving the commutative diagram

of R-modules for all $n\in \mathbb {Z}$ such that $M_n\neq 0$ . Here, the left (resp. the right) vertical arrow is given by $m\mapsto m\otimes e_n$ (resp. $m\mapsto m\otimes e_{\gamma n}$ ). The diagram then corresponds to $R[\mathbb {Z}]\to R[\mathbb {Q}],~e_1\mapsto e_\gamma $ , as desired.▪

We now apply the preceding discussions to the functors constructed in Section 3.3, following [Reference Kottwitz14, Section 4].

Construction 3.15 Let $g\in G(\mathcal {R})$ ; we fix a splitting $\tau _g$ of $\operatorname {\mathrm {{HN}}}_g^{\mathbb {Z}}$ given by Proposition 3.11. For any $(V,\rho )\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , $\tau _g$ gives a decomposition of $V_{\mathcal {R}}$ , which induces a morphism $\lambda _{\rho ,g}\colon \mathbb {G}_{m,\mathcal {R}} \to \operatorname {\mathrm {GL}}_{V,\mathcal {R}}$ by Lemma 3.13. Let S be an $\mathcal {R}$ -algebra, and let $a\in \mathbb {G}_{m,\mathcal {R}}(S)$ . We then have a family

$$ \begin{align*} \big\{ \lambda_{\rho,g}(a)\colon V_S \to V_S \mid (V,\rho)\in\operatorname{\mathrm{\textbf{Rep}}}_F(G) \big\} \end{align*} $$

of S-linear maps. Because $\tau _g$ is a tensor functor, this family satisfies conditions (i–iii) in Theorem 2.4. We thus find a unique element $b\in G_{\mathcal {R}}(S)$ such that $\lambda _{\rho ,g}(a)=\rho (b)$ for all $(V,\rho )\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ . The assignment $a\mapsto b$ is functorial in S, because both $\lambda _{\rho ,g}$ and $\rho $ are functorial. We then have a morphism of $\mathcal {R}$ -groups

$$ \begin{align*}\lambda_g\colon \mathbb{G}_{m,\mathcal{R}} \operatorname{\mathrm{\longrightarrow}} G_{\mathcal{R}},\end{align*} $$

which is said to be the $\mathbb {Z}$ -slope morphism of g.

By Corollary 3.14, ${[d_g^{-1}]}_*$ gives a unique morphism $\chi _{d_g^{-1}}\colon \mathbb {D}_{\mathcal {R}} \to \mathbb {G}_{m,\mathcal {R}}$ . We define

$$ \begin{align*}\nu_g:= \lambda_g\circ \chi_{d_g^{-1}} \colon \mathbb{D}_{\mathcal{R}} \operatorname{\mathrm{\longrightarrow}} G_{\mathcal{R}},\end{align*} $$

which is said to be the $\mathbb {Q}$ -slope morphism of g.

The following example demonstrates explicitly how $\lambda _g$ and $\nu _g$ are related to the splittings constructed in Section 3.3 (see Diagram 3).

Example 3.16 Let $(V,\rho )\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ and suppose that the slope filtration of $(V_{\mathcal {R}},g\varphi )$ is

$$ \begin{align*}0\subseteq V_{\mathcal{R}}^{\mu_1} \subseteq \cdots \subseteq V_{\mathcal{R}}^{\mu_l}=V_{\mathcal{R}}\end{align*} $$

with jumps $\mu _1<\cdots <\mu _l$ . By Theorem 3.12, the functor $[d_g^{-1}]_* \circ \tau _g \colon \operatorname {\mathrm {\textbf {Rep}}}_F(G) \to \mathbb {Q}\text {-}\operatorname {\mathrm {\textbf {Grad}}}_{\mathcal {R}}$ gives a splitting

(6) $$ \begin{align} \textstyle V_{\mathcal{R}}=V_{\mathcal{R},\mu_1}\operatorname*{\mathrm{\bigoplus}}\cdots \operatorname*{\mathrm{\bigoplus}} V_{\mathcal{R},\mu_l} \end{align} $$

of $\operatorname {\mathrm {{HN}}}_g(V)$ , i.e., we have $\bigoplus \limits _{i=1}^j V_{\mathcal {R},\mu _i}=V_{\mathcal {R}}^{\mu _j}$ for all $1\leq j\leq l$ .

First, we fix $1\leq i\leq l$ . Let $S\in \operatorname {\mathrm {\textbf {Alg}}}_{\mathcal {R}}$ and $a\in \mathbb {D}_{\mathcal {R}}(S)$ , then $\rho \circ \nu _g(a)$ acts on $V_{\mathcal {R},\mu _i}\bigotimes _{\mathcal {R}} S$ via multiplication by $\chi _{\mu _i}(a)$ . By Lemma 3.10, $\rho \circ \lambda _g(b)$ acts on $V_{\mathcal {R},\mu _i}$ via multiplication by $b^{d_g\mu _i}$ , for all $b\in \mathbb {G}_{m,\mathcal {R}}(S)$ . Notice that for any $\frac {m}{n}\in \mathbb {Q}$ , we have $e_{\frac {m}{n}}=(e_{\frac {1}{n}})^m \in \mathcal {R}[\mathbb {Q}]$ , and hence, $\chi _{\frac {m}{n}}=(\chi _{\frac {1}{n}})^m$ . In particular, we have $\chi _{\mu _i}=\chi _{\frac {d_g\mu _i}{d_g}}=(\chi _{d_g^{-1}})^{d_g\mu _i}$ . Then, on $V_{\mathcal {R},\mu _i}\bigotimes _{\mathcal {R}} S$ , we have

$$ \begin{align*}\rho\circ \nu_g(a)=\chi_{\mu_i}(a)=(\chi_{d_g^{-1}}(a))^{d_g\mu_i}=\rho\circ \lambda_g\big( \chi_{d_g^{-1}}(a)\big)=\rho\circ \lambda_g\circ \chi_{d_g^{-1}}(a).\end{align*} $$

We next apply this result to all $1\leq i\leq l$ . Because $V_{\mathcal {R}}=\bigoplus \limits _{i=1}^l V_{\mathcal {R},\mu _i}$ , we conclude that $\rho \circ \nu _g=\rho \circ \lambda _g\circ \chi _{d_g^{-1}}$ . It follows that $\nu _g=\lambda _g\circ \chi _{d_g^{-1}}$ once we choose a faithful representation, as is expected from the definition of $\nu _g$ .

If $G=\operatorname {\mathrm {GL}}_V$ for some $V\in \operatorname {\mathrm {\textbf {Vec}}}_F$ , we consider the standard representation $(V,\rho )$ of G where $\rho $ is the identity. The discussion in the above example then indicates that the image of $\lambda _g$ is contained in a split maximal torus in $G_{\mathcal {R}}$ ; we conjecture that this property holds true for an arbitrary split reductive F-group G, and we shall give one more evidence as follows.

Example 3.17 Fix a d-dimensional F-vector space V. For any $R\in \operatorname {\mathrm {\textbf {Alg}}}_F$ , we define

$$ \begin{align*}\operatorname{\mathrm{SL}}_V(R) := \{ g\in \operatorname{\mathrm{GL}}_V(R) \mid \det(g)=1\} .\end{align*} $$

The affine algebraic F-group $\operatorname {\mathrm {SL}}_V$ is called the special linear group (associated to V).

Fix an arbitrary $g\in \operatorname {\mathrm {SL}}_V(\mathcal {R})$ . With the notation as in Construction 4.14, we suppose the jumps of the slope filtration of $(V_{\mathcal {R}},\Phi _g)$ are $\mu _1,\ldots ,\mu _l$ and $\xi _g(V)=\bigoplus \limits _{i=1}^l V_{\mathcal {R},\mu _i}$ . For each i, we write $r_i=\operatorname {\mathrm {rk}}_{\mathcal {R}} (V_{\mathcal {R},\mu _i})$ , then the $r_i$ -th exterior product $\Lambda ^{r_i}(V_{\mathcal {R},\mu _i})$ is of rank  $1$ . We choose a generator $m_i$ , then $\Lambda ^{r_i}(\Phi _{g,\mu _i})(m_i)=f_i m_i$ for some $f_i\in \mathcal {R}^{\times }=(\mathcal {E}^{\dagger })^{\times }$ . Let $\nu $ be the valuation of the $1$ -Gauss norm on $\mathcal {E}^{\dagger }$ . We then have $\mu _i=\frac {\nu (f_i)}{r_i}$ by [Reference Kedlaya11, Definition 1.4.4].

Let $e_1,\ldots ,e_d$ be a basis for V over F, and let $A\in \operatorname {\mathrm {SL}}_d(\mathcal {R})$ be the matrix of action of $\Phi _g$ in $e_1\otimes 1,\ldots , e_d\otimes 1$ . Let $B\in \operatorname {\mathrm {GL}}_d(\mathcal {R})$ represent a change-of-basis over $\mathcal {R}$ . Then, in the new basis, the matrix of action of $\Phi _g$ is $B^{-1}A \varphi (B)$ . Notice that $\det (B)\in (\mathcal {E}^{\dagger })^{\times }$ and $\varphi $ preserves $\nu $ , we then have

$$ \begin{align*}\nu\big( \det(B^{-1}A \varphi(B) )\big)= \nu \big(\det(B^{-1}) \det(A) \varphi(\det(B))\big)=\nu(\det(A)), \end{align*} $$

which implies that the valuation of the determinant of the matrix of action of $\Phi _g$ is invariant under change-of-basis. We denote by $\nu (\det (\Phi _g))$ this invariant. In particular, we have $\nu (\det (\Phi _g))=0$ , because $\det (A)=1$ by assumption. Put $\Phi ^{\prime }_g := \bigoplus \limits _{i=1}^l \Phi _{g,\mu _i}$ , where each $\Phi _{g,\mu _i}$ is the projection of $\Phi _g$ to the $\mu _i$ -th graded piece of $\xi _g(V)$ (cf. Construction 4.14 below). We thus have

$$ \begin{align*}0=\nu(\det (\Phi_g))=\nu(\det (\Phi^{\prime}_g))=\nu(f_1)+\cdots +\nu(f_l)=r_1\mu_1+\cdots +r_l\mu_l.\end{align*} $$

Let $S\in \operatorname {\mathrm {\textbf {Alg}}}_{\mathcal {R}}$ and $t\in \mathbb {G}_{m,\mathcal {R}}(S)$ . Because $\lambda _g(t)$ acts on each $V_{\mathcal {R},\mu _i}\bigotimes _{\mathcal {R}} S$ via multiplication by $t^{d_g\mu _i}$ where $d_g$ is the least common denominator of g, we then have

$$ \begin{align*}\det(\lambda_g(t))= t^{d_g(r_1\mu_1+\cdots +r_l \mu_l)}=1.\end{align*} $$

Therefore, the image of $\lambda _g$ is contained in a split maximal torus in $\operatorname {\mathrm {SL}}_{V,\mathcal {R}}$ .

4.1 Definition and an identification

Let $R\in \{\mathcal {E}^{\dagger },\mathcal {R}\}$ equipped with an absolute Frobenius lift $\varphi $ and the usual derivation $\partial =\partial _t=d/dt$ on R.

Definition 4.1 A G- $(\varphi ,\nabla )$ -module over R is an exact faithful F-linear tensor functor

$$ \begin{align*}\operatorname{\mathrm{I}}\colon \operatorname{\mathrm{\textbf{Rep}}}_F(G) \operatorname{\mathrm{\longrightarrow}} \textbf{Mod}^{\varphi,\nabla}_R,\end{align*} $$

which satisfies $\operatorname {\mathrm {forg}} \circ \operatorname {\mathrm {I}} = \omega ^G \otimes R$ , where $\operatorname {\mathrm {forg}} \colon \operatorname {\mathrm {\textbf {Mod}^{\varphi }_R}} \to \operatorname {\mathrm {\textbf {Mod}}}_R$ is the forgetful functor. The category of G- $(\varphi ,\nabla )$ -modules over R is denoted by $\operatorname {\mathrm {G-\textbf {Mod}^{\varphi ,\nabla }_R}}$ , whose morphisms are morphisms of tensor functors. A G- $(\varphi ,\nabla )$ -module $\operatorname {\mathrm {I}}$ over R is called unit-root if $\operatorname {\mathrm {I}}(V,\rho )$ is a unit-root $(\varphi ,\nabla )$ -module over R for all $(V,\rho ) \in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ .

We put

$$ \begin{align*}\operatorname{\mathrm{\boldsymbol\mu}} := \operatorname{\mathrm{\boldsymbol\mu}} (\varphi,t):= \partial(\varphi(t)).\end{align*} $$

Let $\Omega _R^1 := \Omega _{R/K}^1$ be the free R-module generated by the symbol $dt$ , with the K-linear derivation $d\colon R \to \Omega ^1_R, f\mapsto \partial (f) dt$ . We also define a $\varphi $ -linear endomorphism

$$ \begin{align*} d\varphi\colon \Omega^1_R \operatorname{\mathrm{\longrightarrow}} \Omega^1_R,\;\;\;\;\; fdt \longmapsto \operatorname{\mathrm{\boldsymbol\mu}}\varphi(f)dt. \end{align*} $$

Given a finite-dimensional representation $\rho \colon G\to \operatorname {\mathrm {GL}}_V$ , we have a morphism $\operatorname {\mathrm {Lie}}(\rho ) \colon \mathfrak {g}\to \operatorname {\mathrm {\mathfrak {gl}}}_V$ of Lie algebras, and hence a morphism $ \mathfrak {g}_R \to \operatorname {\mathrm {\mathfrak {gl}}}_V\bigotimes R\cong \operatorname {\mathrm {End}}_R(V_R)$ of Lie algebras over R (which is injective if $\rho $ is a closed embedding). For any $X\in \mathfrak {g}_R$ , we denote by X the action of $\operatorname {\mathrm {Lie}}(\rho )(X)$ on $V_R$ (see Remark 2.8). We define the connection $\nabla _X$ of $V_R$ associated to X by

$$ \begin{align*} \nabla_X := \nabla_{X,V} \colon V_R &\operatorname{\mathrm{\longrightarrow}} V_R \bigotimes_R \Omega_R^1,\\ v\otimes f &\longmapsto (v\otimes 1)\otimes d(f) + X(v\otimes f)\otimes dt. \end{align*} $$

Because $fdt\mapsto f$ gives an isomorphism $\Omega _R^1 \cong R$ , we have an isomorphism $\iota \colon V_R\bigotimes _R \Omega ^1_R\to V_R$ . Let $\Theta _X := \Theta _{X,V}$ be the differential operator associated to $\nabla _X$ given by the following composition:

We have that $\Theta _X(v\otimes f)=v\otimes \partial (f)+X(v\otimes f)$ for all $v\otimes f\in V_R$ .

When $G=\operatorname {\mathrm {GL}}_V$ for some $V\in \operatorname {\mathrm {\textbf {Vec}}}_F$ , we may canonically associate to any G- $(\varphi ,\nabla )$ -module $\operatorname {\mathrm {I}}$ over R a $(\varphi ,\nabla )$ -module $(V_R,\Phi ,\nabla )$ over R, where $(V_R,\Phi ,\nabla ) := \operatorname {\mathrm {I}}(V,\rho )$ and $\rho \colon G\to G$ is the identity. Choose a basis $e_1,\ldots ,e_d$ of V, we define elements $g\in G(R)$ and $X\in \mathfrak {g}_R$ by setting $g(e_i\otimes 1) := \Phi (e_i\otimes 1)$ and $X(e_i\otimes 1) := \iota \circ \nabla (e_i \otimes 1)$ , respectively. We then have $\Phi =g\varphi $ and $\nabla =\nabla _X$ .

Lemma 4.3 Let $V,W \in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , and let $\alpha \in \operatorname {\mathrm {Hom}}_G(V,W)$ . We then have

$$ \begin{align*}\alpha_R\circ\Theta_{X,V}=\Theta_{X,W}\circ\alpha_R,\;\;\;\;\text{and}\;\;\;\;\Theta_{X,V\otimes W}=\Theta_{X,V}\otimes \operatorname{\mathrm{Id}}_{W_{\mathcal{R}}}+ \operatorname{\mathrm{Id}}_{V_{\mathcal{R}}}\otimes \Theta_{X,W}.\end{align*} $$

Construction 4.4 We consider the R-algebra morphism

$$ \begin{align*}\hat\partial\colon R\operatorname{\mathrm{\longrightarrow}} R[\varepsilon],~~~ r\longmapsto r+\partial(r)\varepsilon,\end{align*} $$

which induces a morphism $G(\hat \partial )\colon G(R)\to G(R[\varepsilon ])$ . Notice that $\pi _R\circ \hat \partial =\operatorname {\mathrm {Id}}_R$ ; we then have $G(\pi _R)\circ G(\hat \partial )= \operatorname {\mathrm {Id}}_{G(R)}$ , in particular, $G(\pi _R)\big (G(\hat \partial ) (g)\big )=g$ . Identifying g with its image in $G(R[\varepsilon ])$ induced by the inclusion $R\to R[\varepsilon ],r\mapsto r$ , we then have

$$ \begin{align*}G(\hat\partial)(g)g^{-1}\in \operatorname{\mathrm{Ker}} G(\pi_R)=\mathfrak{g}_R.\end{align*} $$

For $g\in G(R)$ , we define $\partial (g):= G(\hat \partial ) (g)\in G(R[\varepsilon ])$ , and put

$$ \begin{align*}\operatorname{\mathrm{dlog}}(g):= \partial(g)g^{-1}\in \mathfrak{g}_R.\end{align*} $$

Proof Choose a basis $e_1,\ldots ,e_d$ for V over F where $d=\dim _F V$ . Let $A=(a_{ij})_{i,j} \in \operatorname {\mathrm {GL}}_d(R)$ (resp. $N=(n_{ij})_{i,j} \in \operatorname {\mathrm {Mat}}_{n,n}(R)$ ) be the representing matrix of $\rho (g)$ (resp. X). For any $\textbf {v}=\sum \limits _{i=1}^d e_i\otimes f_i \in V_{\mathcal {R}}$ , we compute

$$ \begin{align*} g\varphi(\Theta_X(\textbf{v})) &= g\varphi \Big(\operatorname*{\mathrm{\sum}}\limits_{i=1}^d e_i \otimes \partial(f_i) + \operatorname*{\mathrm{\sum}}\limits_{j=1}^d e_j \otimes \operatorname*{\mathrm{\sum}}\limits_{i=1}^d n_{ji}f_i \Big)\\ &=\operatorname*{\mathrm{\sum}} \limits_{j=1}^d e_j \otimes \operatorname*{\mathrm{\sum}} \limits_{i=1}^d a_{ji}\varphi(\partial(f_i)) + \operatorname*{\mathrm{\sum}}\limits_{k=1}^d e_k \otimes \operatorname*{\mathrm{\sum}} \limits_{i=1}^d \operatorname*{\mathrm{\sum}} \limits_{j=1}^d a_{kj}\varphi(n_{ji} f_i), \end{align*} $$

and

$$ \begin{align*} \Theta_X (g\varphi(\textbf{v})) &=\Theta_X \Big( \operatorname*{\mathrm{\sum}}\limits_{j=1}^d e_j \otimes \operatorname*{\mathrm{\sum}}\limits_{i=1}^d a_{ji} \varphi(f_i) \Big) \\ &=\operatorname*{\mathrm{\sum}}\limits_{j=1}^d e_j \otimes \operatorname*{\mathrm{\sum}}\limits_{i=1}^d \partial(a_{ji}) \varphi(f_i) +\operatorname*{\mathrm{\sum}}\limits_{j=1}^d e_j \otimes \operatorname*{\mathrm{\sum}}\limits_{i=1}^d a_{ji}\partial( \varphi(f_i))\\ &\quad{}+ \operatorname*{\mathrm{\sum}}\limits_{k=1}^d e_k \otimes \operatorname*{\mathrm{\sum}} \limits_{i=1}^d \operatorname*{\mathrm{\sum}} \limits_{j=1}^d n_{kj} a_{ji} \varphi(f_i). \end{align*} $$

Because $\operatorname {\mathrm {\boldsymbol \mu }}\cdot \sum \limits _{j=1}^d e_j \otimes \sum \limits _{i=1}^d a_{ji}\varphi (\partial (f_i))= \sum \limits _{j=1}^d e_j \otimes \sum \limits _{i=1}^d a_{ji}\partial ( \varphi (f_i))$ , we have that $\operatorname {\mathrm {\boldsymbol \mu }}\cdot g\varphi \circ \Theta _X= \Theta _X \circ g\varphi $ if and only if $\mu A \varphi (N)=\partial (A)+NA$ , i.e., $N=\operatorname {\mathrm {\boldsymbol \mu }} A\varphi (N)A^{-1}-\partial (A)A^{-1}$ . The last equality holds because of the assumption $X=\Gamma _g\big (\operatorname {\mathrm {\boldsymbol \mu }}\varphi (X)\big )$ , which completes the proof.▪

As a consequence, we may define a functor

(7) $$ \begin{align} \textbf{B}^{\varphi,\nabla}(G,R) \operatorname{\mathrm{\longrightarrow}} \operatorname{\mathrm{G-\textbf{Mod}^{\varphi,\nabla}_R}}, \;\;\;\;\; (g,X)\longmapsto \operatorname{\mathrm{I}}(g,X), \end{align} $$

where $\operatorname {\mathrm {I}}(g,X)(V):= (V_R,g\varphi ,\nabla _X)$ . We next show that this functor is an isomorphism. To do this, we need the following elementary descent result.

Proof We have a diagram

with the outer triangle commutative in which $\rho ^*$ is surjective. We prove the lemma by constructing a unique k-algebra morphism $\alpha \colon A\to S$ such that $\tilde z=\iota \circ \alpha $ , as follows. For any $a\in A$ , the surjectivity of $\rho ^*$ gives us some $b\in B$ such that $\rho ^*(b)=a$ . We define $\alpha (a):= \beta (b)$ . Because $\iota $ is injective, we have $\operatorname {\mathrm {Ker}} \rho ^* \subseteq \operatorname {\mathrm {Ker}} \beta $ , which implies that $\alpha $ is well-defined. We then have $\tilde z\circ \rho ^*=\iota \circ \beta =\iota \circ \alpha \circ \rho ^*$ , which implies that $\tilde z=\iota \circ \alpha $ , because $\rho ^*$ is surjective. Moreover, $\alpha $ is a k-algebra morphism, because $\iota $ is injective and both $\iota $ and $\tilde z=\iota \circ \alpha $ are k-algebra morphisms. Finally, we see that $\alpha $ is unique, again because $\iota $ is injective.▪

Proof The proof is similar to that of [Reference Dat, Orlik and Rapoport6, Lemma 9.1.4]. We first show that the functor is fully faithful. Let $(g,X), (g',X')\in \textbf {B}^{\varphi ,\nabla }(G,R)$ . Then, any morphism $\eta \colon \operatorname {\mathrm {I}}(g,x) \to \operatorname {\mathrm {I}}(g',X')$ is an isomorphism according to [Reference Deligne and Milne7, Proposition 1.13] (see also Remark 4.2). By composing $\eta $ with the forgetful functor, we then have an automorphism of the fiber functor $\omega ^G\otimes R$ . By Corollary 2.5, this automorphism is given by a unique element $x\in G(R)$ , which then gives an isomorphism between $(g,X)$ and $(g',X')$ , as desired.

It remains to show that, for any $\operatorname {\mathrm {I}}\in \operatorname {\mathrm {G-\textbf {Mod}^{\varphi ,\nabla }_R}}$ , there exists a unique $(g,X)\in \textbf {B}^{\varphi ,\nabla }(G,R)$ such that $\operatorname {\mathrm {I}}=\operatorname {\mathrm {I}}(g,X)$ . For any $(V,\rho )\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , we write $\operatorname {\mathrm {I}}(V,\rho _V)=(V_R,\Phi _V,\nabla _V)$ for a $\varphi $ -linear map $\Phi _V$ and a connection $\nabla _V$ on $V_R$ . The proof consists of two steps.

Step 1: There exists a unique $X\in \mathfrak {g}_R$ such that $\nabla _V=\nabla _X$ . We write $\Theta _V$ for the composition of

where $\iota $ is induced by $fdt\mapsto f$ , and put $\theta _V:= \Theta _V-\operatorname {\mathrm {Id}}_V\otimes \partial $ . It is clear that

, where

denotes the trivial representation. Lemma 4.3 then implies that the family

$$ \begin{align*} \big\{\theta_V\colon V_R \to V_R \mid (V,\rho_V)\in \operatorname{\mathrm{\textbf{Rep}}}_F(G) \big\} \end{align*} $$

of R-linear endomorphisms satisfies conditions (i–iii) in Corollary 2.9. We thus find a unique $X\in \mathfrak {g}_R$ such that $\theta _V=\operatorname {\mathrm {Lie}}(\rho _V)(X)$ for all $(V,\rho _V)\in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ , which implies that $\nabla _V=\nabla _X$ .

Step 2: There exists a unique $g\in G(R)$ such that $\Phi _V=g\varphi $ . We put $\tilde \Phi _V := \Phi _V\otimes \varphi $ , where $\varphi $ is the Frobenius lift on $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ (in particular, $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ is viewed as an $\mathcal {R}$ -module via the $\varphi $ -equivariant embedding $\psi $ described in Section 2.3). The family

$$ \begin{align*}\big\{\lambda_V:= \tilde\Phi_V \circ (\operatorname{\mathrm{Id}}_V\otimes \varphi^{-1})\colon V_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}}\to V_{\operatorname{\mathrm{\tilde{\mathcal{R}}}}} \mid V\in \operatorname{\mathrm{\textbf{Rep}}}_F(G) \big \}\end{align*} $$

of $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ -linear endomorphisms satisfies conditions (i–) in Theorem 2.4, which provides a unique element $\tilde g\in G(\operatorname {\mathrm {\tilde {\mathcal {R}}}})$ such that $\lambda _V=\rho _V(\tilde g)$ for all $(V,\rho _V) \in \operatorname {\mathrm {\textbf {Rep}}}_F(G)$ . We next reduce $\tilde g$ to an element in $G(\mathcal {R})$ . We compute

$$ \begin{align*}\tilde\Phi_V\circ (\operatorname{\mathrm{Id}}_V\otimes \varphi^{-1})(v\otimes f) =\tilde\Phi_V(v\otimes \varphi^{-1}(f))=\rho_V(\tilde g)(v\otimes f),\end{align*} $$

which implies that $\tilde \Phi _V(v\otimes f)=\rho _V(\tilde g)(v\otimes \varphi (f))$ , and hence, $\tilde \Phi _V=\tilde g \varphi $ . We now fix a d-dimensional faithful representation $(V,\rho _V)$ , and an F-basis $e_1,\ldots ,e_d$ for V. Suppose that $\Phi _V(e_i)=\sum \limits _{j=1}^d a_{ji}e_j$ , where $a_{ij}\in R$ for all $1\leq i,j \leq d$ . Put $A=(a_{ij})_{i,j} \in \operatorname {\mathrm {GL}}_d(R)$ . Then, $\psi (A)=(\psi (a_{ij}))_{i,j}\in \operatorname {\mathrm {GL}}_d(\operatorname {\mathrm {\tilde {\mathcal {R}}}})$ describes the $\varphi $ -linear action of $\tilde \Phi _V$ as well as the $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ -linear action $\rho (\tilde g)$ in the basis $e_1\otimes 1,\ldots , e_d \otimes 1$ . By replacing X with G, Y with $\operatorname {\mathrm {GL}}_d$ , S with R, $\tilde S$ with $\operatorname {\mathrm {\tilde {\mathcal {R}}}}$ , and $\iota $ with $\psi $ in Lemma 4.8, we find a unique element $g\in G(R)$ such that $\psi (g)=\tilde g$ . It follows that $\Phi _V=g\varphi $ , as desired.▪

Example 4.10 Let $d\in \mathbb {N}$ . The affine algebraic F-group $\operatorname {\mathrm {SL}}_d$ is defined by

$$ \begin{align*}\operatorname{\mathrm{SL}}_d(S)=\{ A\in \operatorname{\mathrm{GL}}_d(S) \mid \det(A)=1\}\end{align*} $$

for all $S\in \operatorname {\mathrm {\textbf {Alg}}}_F$ , whose Lie algebra $\operatorname {\mathrm {\mathfrak {sl}}}_d$ consists of $d\times d$ matrices with entries in F and with trace zero.

1. (i) We claim that any pair $(A,N) \in \operatorname {\mathrm {SL}}_d(\mathcal {R}) \times \operatorname {\mathrm {Mat}}_{d,d}(\mathcal {E}^{\dagger })$ satisfying $N=\operatorname {\mathrm {\boldsymbol \mu }} A \varphi (N) A^{-1}-\partial (A)A^{-1}$ is already an object in $\textbf {B}^{\varphi ,\nabla }(\operatorname {\mathrm {SL}}_d,\mathcal {R})$ . It is equivalent to showing that the trace $\operatorname {\mathrm {Tr}}(N)$ of N is zero. Recall that the Frobenius lift $\varphi $ on $\mathcal {E}^{\dagger }$ is given by $\varphi \big (\sum \limits _{i\in \mathbb {Z}} c_it^i \big )=\sum \limits _{i\in \mathbb {Z}} \varphi (c_i) u^i$ , where $u=\varphi (t)$ satisfies $|u-t^q|_1<1$ . If we write $u=\sum \limits _{i\in \mathbb {Z}} u_it^i, u_i\in K$ , we then have $|u_j|<1$ for all $j \neq q$ and $|u_q|=1$ . It follows that $|\operatorname {\mathrm {\boldsymbol \mu }}|_1=|\partial (u)|_1=|\sum \limits _{i\in \mathbb {Z}} i u_i t^{i-1}|_1 <1$ . On the other hand, we have $\operatorname {\mathrm {Tr}}\big (\partial (A)A^{-1} \big )=0$ , because $\partial (A)A^{-1} \in \operatorname {\mathrm {\mathfrak {sl}}}_{d,\mathcal {R}}$ (see Construction 4.4). Assume, to the contrary, that $\operatorname {\mathrm {Tr}}(N)\neq 0$ , we have

   $$ \begin{align*} |\operatorname{\mathrm{Tr}}(N)|_1= |\operatorname{\mathrm{\boldsymbol\mu}} \operatorname{\mathrm{Tr}}(\varphi(N))|_1 = |\operatorname{\mathrm{\boldsymbol\mu}} \varphi(\operatorname{\mathrm{Tr}}(N))|_1 < |\varphi(\operatorname{\mathrm{Tr}}(N))|_1= |\operatorname{\mathrm{Tr}}(N)|_1, \end{align*} $$
   a contradiction (we have the last equality, because $\varphi $ preserves the $1$ -Gauss norm on $\mathcal {E}^{\dagger }$ ).

2. (ii) We use the Bessel isocrystal as described in [Reference Kedlaya12, Example 20.2.1] (see also [Reference Kedlaya9, Section 1.5] and [Reference Tsuzuki24, Example 6.2.6]) to construct an object in $\textbf {B}^{\varphi ,\nabla }(\operatorname {\mathrm {SL}}_2,\mathcal {R})$ . We first briefly recall the Bessel isocrystal. In Hypothesis 2.1, we let $q=p$ be an odd prime, $\kappa =\mathbb {F}_p$ , and $F=\mathbb {Q}_p(\pi )$ , where $\pi $ is a $(p-1)$ st root of $-p$ in $\bar {\mathbb {Q}}_p$ . Then, the (p-power) Frobenius on $K=F$ is the identity. Let $\varphi $ be the Frobenius lift on $\mathcal {R}$ given by $\varphi (t)=t^p$ . Then, [Reference Kedlaya12, Example 20.2.1] gives rise to a pair $(A_0,N_0)\in \operatorname {\mathrm {GL}}_2(\mathcal {R}) \times \operatorname {\mathrm {Mat}}_{2,2}(\mathcal {E}^{\dagger })$ with $\det (A_0)=p$ satisfying the gauge compatibility condition, in which $N_0= \left ( \begin {smallmatrix} 0 & t^{-1}\\ \pi ^2 t^{-2} & 0 \end {smallmatrix}\right ) \in \operatorname {\mathrm {\mathfrak {sl}}}_{2,\mathcal {E}^{\dagger }}.$ We now assume that $p \equiv 1 (\operatorname {\mathrm {mod}} 4)$ , and $\textbf {i}$ is a square root of $-1$ in $\mathbb {Q}_p$ . Because $p-1$ is even, we may set $\alpha := \frac {\textbf {i}}{\pi ^{(p-1)/2}}\in F^{\times }$ . We then have $\alpha ^2=p^{-1}=\det (A_0)^{-1}$ . Put $D_0 = \left ( \begin {smallmatrix} 0 & 1\\ \alpha & 0 \end {smallmatrix}\right ) \in \operatorname {\mathrm {GL}}_2(F).$ Then, $D_0A_0D_0\in \operatorname {\mathrm {SL}}_2(\mathcal {R})$ . Moreover, we see that $D_0N_0D_0^{-1}=D_0^{-1}N_0 D_0 \in \operatorname {\mathrm {\mathfrak {sl}}}_{2,\mathcal {E}^{\dagger }}$ . Put $A := D_0A_0 D_0$ and $N := D_0 N_0 D_0^{-1}$ . Then, a straightforward verification shows $N=\operatorname {\mathrm {\boldsymbol \mu }} A \varphi (N)A^{-1}-\partial (A)A^{-1}$ (noting that $\varphi (D_0)=D_0$ and $\partial (D_0)=0$ ). We thus have $(A,N)\in \textbf {B}^{\varphi ,\nabla }(\operatorname {\mathrm {SL}}_2,\mathcal {R})$ , as desired.

3. (iii) Let $(A,N) \in \textbf {B}^{\varphi ,\nabla }(\operatorname {\mathrm {SL}}_d,\mathcal {R})$ . We show that $(A,N)$ is“ $\operatorname {\mathrm {SL}}_d$ -quasi-unipotent” (as described in the introduction) by modifying the classical monodromy as follows. By the classical pLMT, we find a finite separable extension L of $\kappa (\!( t )\!)$ and $B\in \operatorname {\mathrm {GL}}_d(\mathcal {R}_L)$ such that $BNB^{-1}-\partial (B)B^{-1}$ has trace zero being an upper-triangular block matrix with zero blocks in the diagonal. We wish to replace B with an element in $\operatorname {\mathrm {SL}}_d(\mathcal {R}_L)$ . To this end, we deduce first that $\operatorname {\mathrm {Tr}}(\partial (B)B^{-1})= \operatorname {\mathrm {Tr}}(BNB^{-1})=\operatorname {\mathrm {Tr}}(N)=0$ . It then follows from Jacobi’s formula that $\partial (\det (B))= \det (B) \cdot \operatorname {\mathrm {Tr}}(B^{-1} \partial (B))=0$ . Put $D := \operatorname {\mathrm {Diag}}( \det (B)^{-1},1,\ldots ,1)$ . Then, $DB\in \operatorname {\mathrm {SL}}_d(\mathcal {R}_L)$ and $\partial (D)=0$ . We then have

   $$ \begin{align*} (DB) N (DB)^{-1}- \partial(DB)(DB)^{-1} = D \big( B N B^{-1} -\partial(B)B^{-1} \big) D^{-1}, \end{align*} $$
   which is an upper-triangular block matrix with zero blocks, and the sizes of the blocks are the same as those in $B N B^{-1} -\partial (B)B^{-1}$ (the said properties are preserved under conjugation by a diagonal matrix). Hence, $DB$ is a desired replacement of B and we are done.

Example 4.11 For any matrix X, we denote by $X^{\operatorname {\mathrm {T}}}$ its transpose, and by $X^{-\operatorname {\mathrm {T}}}$ the inverse of transpose if X is invertible.

We fix the skew-symmetric matrix $J=\left (\begin {smallmatrix} & & & 1 \\ & & -1 & \\ & 1 & &\\ -1 & & & \end {smallmatrix}\right )$ . The affine algebraic F-group $\operatorname {\mathrm {Sp}}_4$ is defined by

$$ \begin{align*} \operatorname{\mathrm{Sp}}_4(S) := \{ A\in \operatorname{\mathrm{GL}}_4(S) \mid A^{-1}=J^{-1} A^{\operatorname{\mathrm{T}}} J\}, \end{align*} $$

for all $S\in \operatorname {\mathrm {\textbf {Alg}}}_F$ . We denote by $\operatorname {\mathrm {\mathfrak {sp}}}_4$ the Lie algebra of $\operatorname {\mathrm {Sp}}_4$ . For any $S\in \operatorname {\mathrm {\textbf {Alg}}}_F$ , we then have $\operatorname {\mathrm {\mathfrak {sp}}}_{4,S}=\{X\in \operatorname {\mathrm {Mat}}_{4,4}(S) \mid X=J X^{\operatorname {\mathrm {T}}} J\}$ . We remark that the specific choice of J preserves Borel subgroups under conjugation, which will be useful in the monodromy considered below.

Given any $(\varphi ,\nabla )$ -module over $\mathcal {R}$ of rank $2$ , e.g., the Bessel isocrystal described above, we obtain a pair $(A_0,N_0)\in \operatorname {\mathrm {GL}}_2(\mathcal {R}) \times \operatorname {\mathrm {Mat}}_{2,2}(\mathcal {R})$ satisfying $N_0=\operatorname {\mathrm {\boldsymbol \mu }} A_0\varphi (N_0)A_0^{-1}-\partial (A_0)A_0^{-1}$ . Put

$$ \begin{align*} A := \begin{pmatrix} A_0 & 0 \\ 0 & \left(\begin{smallmatrix} & 1\\ -1 & \end{smallmatrix}\right)^{-1} A_0^{-\operatorname{\mathrm{T}}} \left(\begin{smallmatrix} & 1\\ -1 & \end{smallmatrix}\right) \end{pmatrix} ~~~\text{and}~~~ N := \begin{pmatrix} N_0 &0 \\ 0 & \left(\begin{smallmatrix} & 1\\ -1 & \end{smallmatrix}\right) N_0^{\operatorname{\mathrm{T}}} \left(\begin{smallmatrix} & 1\\ -1 & \end{smallmatrix}\right) \end{pmatrix}. \end{align*} $$

A straightforward verification shows that $A\in \operatorname {\mathrm {Sp}}_4(\mathcal {R}),N\in \operatorname {\mathrm {\mathfrak {sp}}}_{4,\mathcal {R}}$ , and, moreover, $N=\operatorname {\mathrm {\boldsymbol \mu }} A \varphi (N)A^{-1}-\partial (A)A^{-1}$ (noting that $\left (\begin {smallmatrix} 0 & 1\\ -1 & 0 \end {smallmatrix}\right )^{-1} =-\left (\begin {smallmatrix} 0 & 1\\ -1 & 0\end {smallmatrix} \right )$ ), which implies that $(A,N) \in \textbf {B}^{\varphi ,\nabla }(\operatorname {\mathrm {Sp}}_4,\mathcal {R})$ .

We next show that $(A,N)$ is “ $\operatorname {\mathrm {Sp}}_4$ -quasi-unipotent.” By the classical pLMT, we find a finite separable extension L of $\kappa (\!( t )\!)$ and $B_0 \in \operatorname {\mathrm {GL}}_2(\mathcal {R}_L)$ such that

$$ \begin{align*} B_0 N_0 B_0^{-1}-\partial(B_0)B_0^{-1}= \left( \begin{smallmatrix} 0 & n\\ 0 &0 \end{smallmatrix}\right), \end{align*} $$

for some $n\in \mathcal {R}_L$ (n could be $0$ ). Put

$$ \begin{align*} B := \begin{pmatrix} B_0 & 0 \\ 0 & \left(\begin{smallmatrix} & 1\\ -1 & \end{smallmatrix}\right)^{-1} B_0^{-\operatorname{\mathrm{T}}} \left(\begin{smallmatrix} & 1\\ -1 & \end{smallmatrix}\right) \end{pmatrix}. \end{align*} $$

We then have $B\in \operatorname {\mathrm {Sp}}_4(\mathcal {R}_L)$ , and

$$ \begin{align*} BNB^{-1}-\partial(B)B^{-1}= \left(\begin{smallmatrix} 0 & n & 0 & 0\\ 0 &0 & 0 & 0\\ 0 &0 & 0 & n\\ 0 &0 & 0 & 0 \end{smallmatrix}\right), \end{align*} $$

again by straightforward computations.