2.1.1 Fargues–Fontaine curve.

We briefly recall the construction of the (relative) Fargues–Fontaine curve (cf. [Reference FarguesFar25, 1] or [Reference Fargues and ScholzeFS21, II.1]). Let $\pi_F$ be a uniformizer of F and let $\mathbb{F}_q$ be the residue field of F. For $S=\mathrm{Spa}(R, R^+)\in \mathbf{Perf}$ a perfectoid affinoid space over $\mathbb F_q$ , fix a pseudo-uniformizer $\omega\in R^+\subset R$ . Let

\[\mathcal{Y}_S:=\mathrm{Spa}(\mathbb{A}, \mathbb{A})\backslash V(\pi_F[\omega]),\]

where

\[\mathbb{A}=W_{\mathcal{O}_F}(R^+)=\bigg\{\sum_{n\geq0}[x_n]\pi_F^n\mid x_n\in R^+\bigg\}\]

is the ramified Witt ring with coefficients in $R^+$ equipped with the $(\pi_F,[\omega])$ -adic topology. It is known that $\mathcal Y_S$ is an analytic adic space over $\mathcal O_F$ [Reference Fargues and ScholzeFS21, Proposition II.1.1]. The qth Frobenius automorphism on $R^+$ induces an automorphism $\phi_S$ on $\mathcal Y_S$ over $\mathcal O_F$ , and this action is free and totally discontinuous [Reference Fargues and ScholzeFS21, Proposition II.1.16]. The relative adic Fargues–Fontaine curve $\mathcal{X}_S$ is by definition the following quotient

\[\mathcal{X}_S:=\mathcal{Y}_S/\phi_S^{\mathbb{Z}}.\]

We also have the relative schematic Fargues–Fontaine curve $X_S$ defined as

\[X_S:=\mathrm{Proj}\bigg( \bigoplus_{d=0}^{\infty} H^0(\mathcal{Y}_S,\mathcal{O}_{\mathcal{Y}_S})^{\phi_S=\pi_F^d}\bigg)\!.\]

When $S=\mathrm{Spa}(K,K^+)$ with K a perfectoid field, $X_S$ (sometimes also denoted by X or $X_K$ since it does not depend on the choice of $K^+$ ) is a noetherian scheme of dimension 1.

Let $S=\mathrm{Spa}(R, R^+)\in \mathbf{Perf}/\mathrm{Spd}(F)$ be a perfectoid affinoid space lying over the diamond $\mathrm{Spd}(F)$ , which corresponds to an untilt $S^\sharp=\mathrm{Spa}(R^{\sharp}, R^{\sharp +})$ over F of S. The untilt $S^{\sharp}$ gives rise to a natural closed immersion of adic spaces

\[S^{\sharp} \hookrightarrow \mathcal{Y}_S\]

that presents $S^{\sharp}$ as a closed Cartier divisor in $\mathcal Y_S$ [Reference Fargues and ScholzeFS21, Proposition II.3.1], with image $V(\xi)\subset \mathcal Y_S$ for some element $\xi\in \mathbb{A}$ . Furthermore, locally on S the closed Cartier divisor above can be also defined by an element in $H^0(\mathcal{X}_S,\mathcal{O}_{\mathcal{X}_S}(1))=H^0(\mathcal Y_S,\mathcal O_{\mathcal Y_S})^{\phi_S=\pi_F}$ [Reference Fargues and ScholzeFS21, Proposition II.2.3]. Consequently, it defines a Cartier divisor of the relative schematic Fargues–Fontaine curve

\[\mathrm{Spec}(R^{\sharp})\hookrightarrow X_{S}.\]

The Fontaine’s de Rham period ring $B_{\mathrm{dR}}^+(R^{\sharp})$ is defined to be the completion of $\mathcal{O}_{X_S}$ along this Cartier divisor (cf. [Reference FarguesFar25, 1.33]) which is also the $\xi$ -adic completion of $\mathbb{A}[{1}/{\pi_F}]$ . We write $B_{\mathrm{dR}}(R^{\sharp})= B_{\mathrm{dR}}^+(R^{\sharp})[{1}/{\xi}]$ .

2.1.2 The Kottwitz set.

Let B(G) be the set of $\sigma$ -conjugacy classes of $G(\breve F)$ , where $\sigma$ is the Frobenius with respective to $\breve F|F$ . For $b\in G(\breve F)$ , let $[b]\in B(G)$ be the $\sigma$ -conjugacy class of b. The Kottwitz set is characterized by two invariants. One is called the Newton map (cf. [Reference KottwitzKot85, 4]). For any $b\in G(\breve F)$ , we can define a slope morphism

\[\nu_b: \mathbb{D}_{\breve F}\longrightarrow G_{\breve F},\]

whose conjugacy class is defined over F, where $\mathbb{D}$ is the pro-torus with characters $X^*(\mathbb{D})=\mathbb{Q}$ . This defines the Newton map

\begin{align*}\nu: B(G)&\longrightarrow \mathcal{N}(G),\\{[b]}&\longmapsto [\nu_b],\end{align*}

where

\[\mathcal{N}(G):=[\mathrm{Hom}(\mathbb{D}_{\bar {F}}, G \bar F)/G(\bar F)\text{-conjugacy}]^{\Gamma}\]

with $\Gamma=\mathrm{Gal}(\bar F|F)$ . An element [b] in B(G) is said to be basic if $\nu_b$ is central. When G is quasi-split, there is a natural identification

\[\mathcal{N}(G)\simeq X_*(A)^+_{\mathbb{Q}}.\]

The other invariant is the Kottwitz map (cf. [Reference KottwitzKot97, 4.9, 7.5]):

\begin{align*}\kappa=\kappa_G: B(G)&\longrightarrow\pi_1(G)_{\Gamma},\\ {[b]}&\longmapsto \kappa([b]),\end{align*}

where $\pi_1(G)$ is the algebraic fundamental group of G and $\pi_1(G)_{\Gamma}$ is its Galois coinvariants.

By [Reference KottwitzKot97, 4.13], the induced map

(2.1) \begin{eqnarray} (\nu, \kappa): B(G)\longrightarrow \mathcal{N}(G)\times \pi_1(G)_{\Gamma}\end{eqnarray}

is injective.

Letting $[b]\in B(G)$ and $\mu\in X_*(T)^+$ , we define the Kottwitz set [Reference KottwitzKot97]

\[B(G, \mu):=\{[b]\in B(G)\mid [\nu_b]\leq \mu^{\diamond}, \kappa_G(b)=\mu^{\sharp}\},\]

where $\mu^{\diamond}\in \mathcal{N}(G)$ is the Galois average of $\mu$ , $\mu^{\sharp}\in \pi_1(G)_{\Gamma}$ is the image of $\mu$ via the natural quotient map $X_{*}(T)\rightarrow \pi_1(G)_{\Gamma}$ and the order $\leq$ on $\mathcal{N}(G)$ is the usual dominance order: $\nu_1\leq \nu_2$ if and only if $\nu_1$ lies in the convex hull of the Weyl orbit of $\nu_2$ .

We also need the following generalized Kottwitz set (cf. [Reference Chen, Fargues and ShenCFS21]). For $\epsilon\in \pi_1 (G)_\Gamma$ and $\delta\in \mathcal{N}(G)$ we set

\[B(G,\epsilon,\delta) = \{ [b]\in B(G)\ |\ \kappa_G (b)=\epsilon \text{ and } [\nu_b]\leq \delta\}.\]

We will need the following well-known fact, which can be deduced easily from a general result of Kottwitz [Reference KottwitzKot97, 3.6].

2.1.3 G-bundles.

Recall that a G-bundle on an F-scheme Z is a (right) G-torsor on Z locally trivial for the étale topology. It could also be viewed as an exact tensor functor

\[\mathrm{Rep} G \longrightarrow \mathrm{Bun}_{Z},\]

where $\mathrm{Rep } G$ is the category of rational algebraic representations of G and $\mathrm{Bun}_Z$ denotes the category of vector bundles (of finite rank) over Z.

Let S be a perfectoid affinoid space over the residue field $\bar{\mathbb F}_q$ of $\breve F$ . For each $b\in G(\breve F)$ , we can associate a G-bundle

\[\mathcal{E}_b\]

on the relative Fargues–Fontaine curve $X_S$ as follows. Suppose $S=\mathrm{Spa}(R,R^+)$ . In particular, $X_S=\mathrm{Proj}(\oplus_d B^{\phi_S=\pi_F^d})$ , with $B=H^0(\mathcal Y_S,\mathcal O_{\mathcal Y_S})$ . As an exact tensor functor the G-bundle $\mathcal E_b$ is given by

\[\mathcal E_b:\mathrm{Rep}G\longrightarrow \mathrm{Bun}_{X_S}, \quad (V,\rho)\mapsto \mathcal E_b(V,\rho),\]

with $\mathcal E_b(V,\rho)$ the vector bundle on $X_S$ corresponding to the graded module

\[\bigoplus_{d\in \mathbb Z} \left(V\otimes_F B\right)^{\rho(b)\phi_S=\pi_F^d}\]

over the graded ring $\oplus_d B^ {\phi_S=\pi_F^d}$ . Here we use the same notation $\rho(b)$ to denote the B-linear automorphism on $V\otimes_FB=(V\otimes_F\breve F)\otimes_{\breve F}B$ induced by $\rho(b)\in \mathrm{GL}(V\otimes_F \breve F)$ . The isomorphism class of $\mathcal E_b$ only depends on the $\sigma$ -conjugacy class of b.

If moreover $S=\mathrm{Spa}(C,C^+)$ with C an algebraically closed perfectoid field containing $\bar{\mathbb F}_q$ and we write $X=X_S$ , this construction gives a bijection of pointed sets by a fundamental theorem of Fargues [Reference FarguesFar20]:

\begin{align*}B(G)&\stackrel{\sim}{\longrightarrow} H^1_{\acute{e}t}(X, G),\\{[b]}&\longmapsto \mathcal{E}_b.\end{align*}

2.1.4 Harder–Narasimhan reduction in the quasi-split case.

In this subsection, suppose G is quasi-split.

Recall the following definition of semi-stable G-bundles on X. See for example [Reference Chen, Fargues and ShenCFS21, Definition 1.5].

For any G-bundle $\mathcal{E}$ on X, by [Reference Chen, Fargues and ShenCFS21, Theorem 1.7], there exist a unique standard parabolic subgroup P of G and a unique reduction $\mathcal{E}_P$ of $\mathcal{E}$ to P, such that the associated M-bundle $\mathcal{E}_P\times^P M$ is semi-stable and P is maximal among all the standard parabolic with the previous condition. The reduction $\mathcal{E}_P$ is called the Harder–Narasimhan reduction or canonical reduction of $\mathcal{E}$ . This defines the Harder–Narasimhan polygon of $\mathcal{E}$ :

\[\nu_{\mathcal{E}}\in X_*(A)^+_{\mathbb{Q}},\]

such that for any $\chi\in X^*(P)$ , we have $\langle \chi, \nu_{\mathcal{E}}\rangle= \operatorname{deg}\chi_*(\mathcal{E}_P)$ . By [Reference FarguesFar20, Proposition 6.6], we have $\nu_{\mathcal{E}_{b}}=-w_0\nu_b$ .

The Harder–Narasimhan reduction has the following characterization.

Proposition 2.5 [Reference SchiederSch15, Theorem 4.5.1] For $\mathcal{E}$ a G-bundle on X equipped with a reduction $\mathcal E_Q$ to the standard parabolic subgroup Q of G, consider the vector

\[\begin{split}v: X^*(Q)&\longrightarrow\mathbb{Z}, \\\chi&\longmapsto \operatorname{deg}\chi_*\mathcal{E}_Q, \end{split}\]

seen as an element of $X_*(A)_{\mathbb{Q}}$ . Then:

1. (i) $v\leq \nu_\mathcal{E}$ ; and, if this inequality is an equality,

2. (ii) $Q\subset P$ and the canonical reduction $\mathcal{E}_P\simeq \mathcal{E}_Q\times^Q P$ .

2.2.1 Schubert varieties.

Let $\bar F$ be an algebraic closure of F. We choose a maximal torus $T\subset G_{\bar F}$ and a Borel subgroup containing T. Write $\mathrm{Gr}_{G,\bar F}$ the restriction of $\mathrm{Gr}_G$ to $\mathbf{Perf}/\mathrm{Spd}(\bar F)$ . Let $S=\mathrm{Spa}(C,C^+)\rightarrow \mathrm{Spd}(\bar F)$ be a geometric point of $\mathrm{Spd}(\bar F)$ , given by an untilt $\mathrm{Spa}(C^{\sharp},C^{\sharp+})$ of S over $\bar F$ .

By the Cartan decomposition,

\[G(B_{\mathrm{dR}}(C^{\sharp}))=\coprod_{\mu\in X_*\left(T\right)^+}G(B_{\mathrm{dR}}^+(C^{\sharp}))\mu\left(\xi\right)^{-1}G(B_{\mathrm{dR}}^+(C^{\sharp})).\]

In particular,

\begin{align*}\mathrm{Gr}_G(S)&=G(B_{\mathrm{dR}}(C^{\sharp}))/G(B_{\mathrm{dR}}^+(C^{\sharp}))\\ &=\coprod_{\mu\in X_*(T)^+} G(B_{\mathrm{dR}}^+(C^{\sharp}))\mu(\xi)^{-1}G(B_{\mathrm{dR}}^+(C^{\sharp}))/G(B_{\mathrm{dR}}^+(C^{\sharp})).\end{align*}

Here we have the first equality since C is algebraically closed and thus every G-bundle over $\mathrm{Spec}(B_{\mathrm{dR}}^+(C^{\sharp}))$ is trivial. By abuse of notation, for any geometric conjugacy class $\{\mu\}\in X_*(G)/G(\bar F)$ , we denote by

\[\mathrm{Gr}_{G,\mu}\subset \mathrm{Gr}_{G,\leq \mu}\subset \mathrm{Gr}_{G,\bar F}\]

the subfunctors of $\mathrm{Gr}_{G,\bar F}$ defined by the condition that a map

\[S'\longrightarrow \mathrm{Gr}_{G,\bar F}\]

with $S'\in \mathbf{Perf}/\mathrm{Spd}(\bar F)$ factors through $\mathrm{Gr}_{G,\mu}$ , or respectively $\mathrm{Gr}_{G,\leq \mu}$ , if and only if, for all geometric points $S=\mathrm{Spa}(C,{C}^{+})\rightarrow S'$ , the corresponding S-valued point of $\mathrm{Gr}_{G,\bar F}$ lies in

\[G(B_{\mathrm{dR}}^+(C^{\sharp}))\mu(\xi)^{-1}G(B_{\mathrm{dR}}^+(C^{\sharp}))/G(B_{\mathrm{dR}}^+(C^{\sharp})),\]

or respectively lies in

\[\coprod_{\mu'\leq \mu}G(B_{\mathrm{dR}}^+(C^{\sharp}))\mu'(\xi)^{-1}G(B_{\mathrm{dR}}^+(C^{\sharp}))/G(B_{\mathrm{dR}}^+(C^{\sharp})).\]

According to [Reference Scholze and WeinsteinSW20, Proposition 19.2.3], $\mathrm{Gr}_{G,\leq \mu}\subset \mathrm{Gr}_{G,\bar F}$ is a closed subfunctor proper over $\mathrm{Spd}(\bar F)$ , and $\mathrm{Gr}_{G,\mu}\subset \mathrm{Gr}_{G,\leq \mu}$ is an open subfunctor. Furthermore, the small v-sheaf $\mathrm{Gr}_{G,\leq \mu}$ is a spatial diamond [Reference Scholze and WeinsteinSW20, Theorem 19.2.4].

2.2.2 Generalized semi-infinite orbits.

We keep the notation in § 2.2.1. Let $P\subset G_{\bar F}$ be a standard parabolic subgroup. Let M be the Levi quotient of P, $M^{ab}$ the maximal torus quotient of M and $N\subset P$ the unipotent radical. We have a natural identification of v-sheaves on $\mathbf{Perf}/\mathrm{Spd}(\bar F)$ ,

\[\mathrm{Gr}_{M^{ab}}\stackrel{\sim}{\longrightarrow} X_*(M^{ab}), \]

which sends $v(\xi)^{-1}M^{ab}(B_{\mathrm{dR}}^+)\in \mathrm{Gr}_{M^{ab}}(C)$ to v for every cocharacter $v\in X_*(M^{ab})$ .

On the other hand, consider the $B_{\mathrm{dR}}^+$ -affine Grassmannian $\mathrm{Gr}_P$ associated with the parabolic subgroup P. For $v\in X_*(M^{ab})$ , write

\[\mathrm{Gr}_{P}^v\subset \mathrm{Gr}_{P,\bar F},\]

the open and closed subfunctor of $\mathrm{Gr}_{P,\bar F}$ (over $\mathbf{Perf}/\mathrm{Spd}(\bar F)$ ) obtained as the preimage of $v\in X_*(M^{ab})$ under the following composed map:

\[\mathrm{Gr}_{P,\bar F}\longrightarrow \mathrm{Gr}_{M,\bar F}\longrightarrow \mathrm{Gr}_{M^{ab},\bar F}\stackrel{\sim}{\longrightarrow} X_*(M^{ab}).\]

Proposition 2.8 [Reference Fargues and ScholzeFS21, Proposition VI 3.1]) The map

\[\mathrm{Gr}_{P}=\coprod_{v\in X_*\left(M^{ab}\right)}\mathrm{Gr}_{P}^v\longrightarrow \mathrm{Gr}_{G,\bar F}\]

is bijective on geometric points, and it is a locally closed immersion on each $\mathrm{Gr}_{P}^{v}$ . Moreover, the image of the union

\[\bigcup_{v\leq v'}\mathrm{Gr}_P^{v'}\]

is closed in $\mathrm{Gr}_{G,\bar F}$ . Here $X_*(M^{ab})\subset X_*(T)_{\mathbb Q}$ is equipped with the dominance order, where $v\leq v' $ if $v'-v\in X_*(T)_{\mathbb Q}$ is a sum of positive coroots with $\mathbb Z_{\geq 0}$ -coefficients.

We also have a finer version of $\mathrm{Gr}_P^{v}$ obtained as follows. Let $\lambda\in X_*(T)$ be an M-dominant cocharacter. We set

\[\mathrm{Gr}_{P,\lambda}\subset \mathrm{Gr}_{P,\bar F},\]

the locally closed subfunctor given by the preimage of $\mathrm{Gr}_{M,\lambda}\subset \mathrm{Gr}_M$ through the natural projection

\[\mathrm{pr}_M:\mathrm{Gr}_P\longrightarrow \mathrm{Gr}_M.\]

Clearly,

\[\mathrm{Gr}_{P,\lambda}\subset \mathrm{Gr}_P^{\bar \lambda},\]

where $\bar \lambda$ denotes the image of $\lambda\in X_*(M)$ in $X_{*}(M^{ab})$ . So $\mathrm{Gr}_{P,\lambda}\subset \mathrm{Gr}_{G,\bar F}$ is locally closed. Moreover, for a geometric point $\mathrm{Spa}(C,C^+)\rightarrow \mathrm{Spd}(\bar F)$ of $\mathrm{Spd}(\bar F)$ corresponding to an untilt $(C^{\sharp},C^{\sharp+})$ over $\bar F$ , we have

\[\mathrm{Gr}_{P,\lambda}(C,C^+)=N(B_{\mathrm{dR}}(C^{\sharp}))M(B_{\mathrm{dR}}^+(C^{\sharp}))\lambda(\xi)^{-1}G(B_{\mathrm{dR}}^+(C^{\sharp}))/G(B_{\mathrm{dR}}^+(C^{\sharp})).\]

Motivated by [Reference Görtz, Haines, Kottwitz and ReumanGHKR06], for a given G-dominant cocharacter $\mu$ we write $S_{M}(\mu;C,C^+)$ for the set of M-dominant cocharacters $\lambda$ for which the intersection

\[N(B_{\mathrm{dR}}(C^{\sharp}))\lambda^{-1}(\xi)\cap G(B_{\mathrm{dR}}^+(C^{\sharp}))\mu^{-1}(\xi) G(B_{\mathrm{dR}}^+(C^{\sharp}))\neq\emptyset.\]

We shall see in the next lemma that the set $S_M(\mu;C,C^+)$ does not depend on the choice of the geometric point $\mathrm{Spa}(C,C^+)\rightarrow \mathrm{Spd}(\bar F)$ and hence there will be no confusion to write $S_M(\mu):=S_M(\mu; C, C^+)$ .

Proof. It is enough to prove (i). Recall that non-canonically $B_{\mathrm{dR}}^+(C^{\sharp})\simeq C^{\sharp}[\![\xi]\!]$ . Write $\mathscr{G}_G$ for the classical affine Grassmannian, that is, the étale sheafification of the functor on the category $\mathbf{Aff}_{\bar F}$ of affine $\bar F$ -schemes sending $\mathrm{Spec}(R)\in \mathbf{Aff}_{\bar F}$ to the coset

\[G(R(\!(\xi)\!))/G(R[\![\xi]\!]).\]

Let $\mathscr{G}_{G,\mu}\subset \mathscr G_G$ be the corresponding Schubert variety attached to $\mu$ (with our convention on the minus sign), and let $\mathscr{S}_{\lambda}\subset \mathscr G_G$ be the orbit of the $\bar F$ -point

\[\lambda(\xi)^{-1}G(\bar F[\![\xi]\!])\in \mathscr G_G(\bar F),\]

under the action of the ind-group scheme

\[\mathscr{N}:\mathbf{Aff}_{\bar F}\longrightarrow \mathbf{Grps}, \quad \mathrm{Spec}(R)\longmapsto N(R(\!(\xi)\!)),\]

by left multiplication on $\mathscr G_G$ . Using the (non-canonical) identification $B_{\mathrm{dR}}^+\simeq C^{\sharp}[\![\xi]\!]$ , we have $\lambda\in S_{M}(\mu;C,C^+)$ if and only if

\[N(C^{\sharp}[\![\xi]\!])\lambda^{-1}(\xi)\cap G(C^{\sharp}[\![\xi]\!])\mu^{-1}(\xi) G(C^{\sharp}[\![\xi]\!])\neq\emptyset,\]

or equivalently $(\mathscr G_{G,\mu}\cap \mathscr S_{\lambda})(C^{\sharp})\neq \emptyset$ . But $\mathscr G_{G,\mu}\cap \mathscr S_{\lambda}$ is an $\bar F$ -scheme of finite type, so it has a point with values in the algebraically closed field $C^{\sharp}$ if and only if $\mathscr{G}_{G,\mu}\cap \mathscr S_{\lambda}\neq \emptyset$ . This shows that the set $S_{M}(\mu;C,C^+)$ is actually independent on the choice of the geometric point $\mathrm{Spa}(C,C^+)\rightarrow \mathrm{Spd}(\bar F)$ .

For the description of $S_M(\mu)$ , we have an analogous result as [Reference Görtz, Haines, Kottwitz and ReumanGHKR06, Lemma 5.4.1].

For (i), let us start with the case where $M=T$ is the maximal torus (and thus $N=U$ is the unipotent radical of the Borel subgroup $B\subset G$ ), $\Sigma(\mu)_{M\text{-}\text{max}}=S_M(\mu)=\Sigma(\mu)_{M\text{-}\text{dom}}$ . In this case, our lemma says that, for $\lambda\in X_*(T)$ ,

\[U(B_{\mathrm{dR}}(C^{\sharp}))\lambda^{-1}(\xi)\cap G(B_{\mathrm{dR}}^+(C^{\sharp}))\mu^{-1}(\xi)G(B_{\mathrm{dR}}^+(C^{\sharp}))\neq \emptyset\]

if and only if $(\lambda)_{\mathrm{dom}}\leq \mu$ . Using a non-canonical identification $B_{\mathrm{dR}}^+(C^{\sharp})\simeq C^{\sharp}[\![\xi]\!]$ , the desired assertion follows from the geometry of semi-infinite orbits of the usual affine Grassmannian for the group G (cf. [Reference Mirković and VilonenMV07, 3.2 Theorem] or [Reference ZhuZhu17, Lemma 5.3.7 and Theorem 5.3.9]). The proof for the general standard Levi $M\subset G$ is then the same as that of [Reference Görtz, Haines, Kottwitz and ReumanGHKR06, Lemma 5.4.1].

Remark 2.12. Fixing $\mu$ , there are only finitely many $v\in X_*(M^{ab})$ such that $\mathrm{Gr}_P^v\cap \mathrm{Gr}_{G, \mu}\neq \emptyset$ . Indeed,

\[\coprod_{\lambda\in X_*(T)\atop \lambda=v\text{ in }\pi_1(M^{ab})}\mathrm{Gr}_{P, \lambda}\longrightarrow \mathrm{Gr}_P^v\]

induces a bijection on geometric points. By Lemma 2.2, $\mathrm{Gr}_P^v\cap \mathrm{Gr}_{G, \mu}\neq \emptyset$ if and only if v is contained in the image of $S_M(\mu)$ via $\theta: X_*(T)\rightarrow X_*(M^{ab})$ , if and only if $v\in \theta(W\mu)$ by the previous lemma.

2.2.3 $B_{\mathrm{dR}}^+$ -affine Grassmannian and modifications of G-bundles.

Let S be a perfectoid affinoid space over $\mathbb F_q$ lying above $\mathrm{Spd}(\breve F)$ , given by an untilt $S^{\sharp}$ of S over $\breve F$ . Let $b\in G(\breve F)$ . Consider the associated G-bundle $\mathcal E_b$ on the relative Fargues–Fontaine curve $X_S$ (whose construction is recalled in § 2.1.3). An element $x\in \mathrm{Gr}_G(S)$ gives rise to a modification

\[\mathcal E_{b,x}\]

of $\mathcal E_b$ à la Beauville and Laszlo whose construction is recalled briefly as follows. Write $S=\mathrm{Spa}(R,R^+)$ , and $S^{\sharp}=\mathrm{Spa}(R^{\sharp}, R^{\sharp +})$ . Let $x\in \mathrm{Gr}_{G}(S)$ correspond to the G-bundle $\mathcal F$ on $B_{\mathrm{dR}}^+(R^{\sharp})$ together with a trivialization of G-bundles

\[\iota: G|_{\mathrm{Spec}(B_{\mathrm{dR}}(R^{\sharp}))}\stackrel{\sim}{\longrightarrow}\mathcal F|_{\mathrm{Spec}(B_{\mathrm{dR}}(R^{\sharp}))}.\]

Recall that the completion of $X_{S}$ along the Cartier divisor $\mathrm{Spec}(R^{\sharp})\hookrightarrow X_{S}$ is given by the de Rham period ring $B_{\mathrm{dR}}^+(R^{\sharp})$ (cf. [Reference FarguesFar25, 1.33]). Moreover, the G-bundle $\mathcal E_{b}$ is canonically trivialized on $\mathrm{Spec}(B_{\mathrm{dR}}^+(R^{\sharp}))$ through the natural map $\mathrm{Spec}(B_{\mathrm{dR}}^+(R^{\sharp}))\rightarrow X_S$ (cf. [Reference Chen and TongCT22, Remark 1.7]). So by a theorem of Beauville and Laszlo (cf. [Reference Beauville and LaszloBL95]), we can glue $\mathcal E_{b}$ and the G-bundle $\mathcal F$ along the trivialization $\iota$ . In this way we obtain the G-bundle $\mathcal E_{b,x}$ on $X_{S}$ .

Recall also the following useful fact.

Proposition 2.14 [Reference Chen, Fargues and ShenCFS21, Lemma 2.6, Reference ViehmannVie24, Lemma 3.10, Reference ShenShe23, Lemma 6.6]. Assume G is quasi-split. Let $M\subset G$ be a standard Levi subgroup. Suppose that $(b_M,g)$ is a reduction of b to M, so $b_M\in M(\breve F)$ and $g\in G(\breve F)$ such that $b=gb_M\sigma(g)^{-1}$ . For a geometric point $x\in \mathrm{Gr}_G(C,C^+)$ , also viewed as an element of $\mathrm{Gr}_P(C,C^+)$ via the bijection

\[\mathrm{Gr}_P(C,C^+)\stackrel{\sim}{\longrightarrow}\mathrm{Gr}_G(C,C^+)\]

given by the Iwasawa decomposition, we have a natural isomorphism

\[(\mathcal{E}_{b, g\cdot x})_P\times^P M\simeq (\mathcal{E}_{b_M,x})_P\times^PM \simeq \mathcal{E}_{b_M, \mathrm{pr}_M(x)},\]

where $\mathrm{pr}_M: \mathrm{Gr}_P\rightarrow \mathrm{Gr}_M$ is the natural projection.

Proof. Indeed, [Reference Chen, Fargues and ShenCFS21, Lemma 2.6] and [Reference ViehmannVie24, Lemma 3.10] deals with the case b basic, while [Reference ShenShe23, Lemma 6.6] deals with the general case. For the convenience of the readers, we still give the proof here. Since $b=gb_M\sigma(g)^{-1}$ , the element g gives an isomorphism of G-bundles

\[\mathcal E_{b_M,x}\stackrel{\sim}{\longrightarrow}\mathcal E_{b,g\cdot x}.\]

On the other hand, as $b_M\in M(\breve F)$ the G-bundle $\mathcal E_{b_M,x}$ has a natural reduction to the P-bundle $(\mathcal E_{b_M,x})_P=\mathcal E_{b_M,x}^P$ where we view $x\in \mathrm{Gr}_{G}(C,C^+)$ as an element of $\mathrm{Gr}_{P}(C,C^+)$ using the Iwasawa decomposition, yielding a reduction $(\mathcal E_{b,g\cdot x})_P$ to P of $\mathcal E_{b,g\cdot x}$ . By functoriality, $\mathcal E_{b_M,x}^P\times^PM\simeq \mathcal E_{b_M,\mathrm{pr}_M(x)}$ , so we get the required isomorphisms

\begin{align*}(\mathcal{E}_{b, g\cdot x})_P\times^P M\simeq(\mathcal{E}_{b_M,x})_P\times^PM \simeq \mathcal{E}_{b_M,\mathrm{pr}_M(x)}.\\[-36pt]\end{align*}

3.3.1 The vectorial Tits building for $\mathbf{GL}_n$ .

Let L be a field and V a finite-dimensional L-vector space. Let $\mathbf F(V)$ be the set of separated exhaustive decreasing $\mathbb{R}$ -filtrations $f=(V_f^a)_{a\in \mathbb R}$ of V by sub-spaces so that

\[V_f^{a}=V_f^{a,-}:=\bigcap_{a'<a}V_{f}^{a'}\quad \textrm{for all } a\in \mathbb R.\]

For such a filtration f of V we denote as usual by

\[\mathrm{gr}_{f}V:=\bigoplus_{a\in \mathbb R}\mathrm{gr}_f^aV\]

the associated graded L-vector space, where

\[\mathrm{gr}_f^aV:=V_{f}^a/V_{f}^{a,+},\]

with $V_f^{a,+}:=\bigcup_{a'>a}V_f^{a'}\subset V_f^a$ .

Let $\underline{e}=(e_1,\ldots, e_n)$ be an L-basis of V. We say that an $\mathbb R$ -filtration $f\in \mathbf{F}(V)$ is split by $\underline e$ if each subspace $V_f^a$ is spanned by a subset of $\{e_1,\ldots, e_n\}$ . We have a natural injective map

(3.2) \begin{equation} \mathbb R^n\longrightarrow \mathbf F(V)\end{equation}

sending $\alpha=(a_1,\ldots,a_n)\in \mathbb R^n$ to the $\mathbb R$ -filtration $f_{\alpha}$ of V given by setting

\[V_{f_{\alpha}}^a:=\mathrm{Span}\{e_i| a_i\geq a\}\subset V.\]

The image of (3.2) is an apartment of $\mathbf{F}(V)$ , denoted by $\mathbf{F}(\underline{e})$ . Hence, $\mathbf{F}(V)$ is the union of all its apartments. It is known that any two $\mathbb R$ -filtrations of V can be split by the same L-basis, and thus they are contained in a common apartment.

Example 3.12. Let $f,g\in \mathbf{F}(V)$ and $\lambda\in \mathbb R_{>0}$ . We can define two new elements $f+g$ and $\lambda \cdot f$ of $\mathbf F(V)$ by

\[V_{f+g}^a:=\sum_{b+c=a}V_f^b\bigcap V_g^c \quad \textrm{and}\quad V_{\lambda\cdot f}^a:=V_{f}^{\lambda^{-1}a}.\]

Note that the formula defining $f+g$ makes sense since the sets

\[\{V_f^b \ |\ b\in \mathbb R\}\quad \textrm{and} \quad \{V_g^c\ |\ c\in \mathbb R\}\]

consist of only finitely many subspaces of V. Moreover, if f,g are contained in the apartment $\mathbf{F}(\underline{e})$ for a certain L-basis, and we write $f=f_{\alpha}$ and $g=f_{\beta}$ for some $\alpha,\beta\in \mathbb R^n$ , then $f+g=f_{\alpha+\beta}$ and $\lambda \cdot f=f_{\lambda \alpha}$ .

The set $\mathbf{F}(V)$ is naturally a metric space [Reference CornutCor18, §2.3]: let $f,g\in \mathbf F(V)$ , and write

\begin{align*}\langle f,g\rangle&: = \sum_{s,t\in \mathbb R}st\dim \mathrm{gr}_{f}^s(\mathrm{gr}^t_{g}V), \\ |f|& := \sqrt{\langle f,f\rangle},\\d(f,g)& := \sqrt{|f|^2+|g|^2-2\langle f,g\rangle }.\end{align*}

Here $\mathrm{gr}_f^s(\mathrm{gr}_g^tV)$ is the subquotient of the L-vector space $\mathrm{gr}_g^tV$ associated to the filtration induced from f.

Proof. This can be viewed as the special case of some general results of Cornut. Indeed, as V is a finite-dimensional L-vector space, the set $\mathscr X$ of all L-subspaces, equipped with the partial order $\leq $ given by the inclusion relation of subspaces, is a bounded modular lattice of finite length $\dim_KV$ , with

\[W_1\vee W_2:=W_1+W_2 \quad \textrm{and}\quad W_1\wedge W_2:=W_1\cap W_2.\]

Moreover, the set $\mathbf{F}(V)$ is naturally identified with the set $\mathbf{F}(\mathscr X)$ of all $\mathbb R$ -filtrations on $\mathscr X$ considered in [Reference CornutCor18, §2.2], and the map

\[\mathscr X\longrightarrow \mathbb R, \quad W\longmapsto \dim_{L}W\]

gives a rank function (in the sense of [Reference CornutCor18, § 2.1.3]) on $\mathscr X$ . So our proposition follows from the corresponding general statements for bounded modular lattices of finite length established in [Reference CornutCor18, Propositions 2.6 and 2.7].

Remark 3.15. Let $\underline{e}$ be an L-basis of V. Restricting to the apartment $\mathbf{F}(\underline e)$ , identified with $\mathbb R^n$ via (3.2), the pairing above is just the standard scalar product $\langle -,-\rangle$ in $\mathbb R^n$ , i.e.,

\[\langle f_{\alpha},f_{\beta}\rangle =\langle \alpha,\beta\rangle, \quad \forall \alpha,\beta\in \mathbb R^n.\]

In particular, for $f,g,h\in \mathbf{F}(\underline e)$ , we have

\[\langle f,g+h\rangle=\langle f,g\rangle +\langle f,h\rangle.\]

In general, the pairing $\langle -,-\rangle $ is only concave [Reference CornutCor18, Lemma 2.5]: we have

(3.3) \begin{equation} \langle f,g+h\rangle \geq \langle f,g\rangle +\langle f,h\rangle, \quad \forall f,g,h\in \mathbf F(V).\end{equation}

Let $C\subset \mathbf{F}(V)$ be a closed subset. Assume moreover that it is convex, i.e., for any $f,g\in C$ ,

\[\lambda \cdot f+(1-\lambda )\cdot g\in C, \quad \forall \lambda\in [0,1].\]

Then, for every $f\in \mathbf{F}(V)$ , there exists a unique $p_C(f)\in C$ , called the convex projection of f to C, such that

\[d(f,p_C(f))=\min\{d(f,g)\mid g\in C\}.\]

Lemma 3.16. Let $f\in \mathbf{F}(V)$ , with convex projection $p_C(f)\in C$ . We have

\[|p_C(f)|\leq |f|.\]

Moreover, if $g\in \mathbf F(V)$ so that $p_C(f)+t\cdot g\in C$ for sufficiently small positive real t, we have

\[\langle p_C(f),g\rangle\geq \langle f,g\rangle.\]

Proof. The inequality $|p_C(f)|\leq |f|$ follows from the fact that the convex projection $p_C$ is non-expanding [Reference Bridson and HaefligerBH99, II.2.4]. The second assertion is also well known and let us recall the proof for the sake of completeness. Note that, for $t\in \mathbb R_{>0}$ with $p_C(f)+tg\in C$ ,

\begin{align*}d(f,p_C(f))^2 & \leq d(f,p_C(f)+tg)^2 \\ & = |f|^2+|p_C(f)+tg|^2-2\langle f,p_C(f)+tg\rangle \\ & = |f|^2+|p_C(f)|^2+|g|^2t^2+2\langle p_C(f),g\rangle t -2\langle f,p_C(f)+tg\rangle \\ & \leq |f|^2+|p_C(f)|^2+|g|^2t^2+2\langle p_C(f),g\rangle t -2\langle f,p_C(f)\rangle -2\langle f,tg\rangle \\& = |f|^2+|p_C(f)|^2+|g|^2t^2+2\langle p_C(f),g\rangle t -2\langle f,p_C(f)\rangle -2t\langle f,g\rangle \\ & = |g|^2t^2+2\left(\langle p_C(f),g\rangle-\langle f,g\rangle\right) t + |f|^2+|p_C(f)|^2-2\langle f,p_C(f)\rangle \\ & = |g|^2t^2+2\left(\langle p_C(f),g\rangle-\langle f,g\rangle\right) t + d(f,p_C(f))^2.\end{align*}

Here the second inequality follows from the fact that the pairing $\langle -,-\rangle $ is concave (cf. (3.3)). Since these inequalities hold for any sufficiently small positive real number t, it follows that the polynomial

\[|g|^2t^2+2\left(\langle p_C(f),g\rangle-\langle f,g\rangle\right) t,\]

attains its minimum at some $t_0\leq 0$ . As a result, we find $\langle p_C(f),g\rangle \geq \langle f,g\rangle$ , as required.

3.3.2 Vectorial Tits building and Harder–Narasimhan filtration.

We use the notation fixed at the beginning of § 3.1. Let $(D,\alpha)$ be a normed isocrystal over $K/F$ , with V the underlying $\breve F$ -vector space. Let $\mathbf{F}(D)\subset \mathbf{F}(V)$ be the subset of $\mathbb R$ -filtrations of D by sub-isocrystals. As in the proof of Proposition 3.14, the set of all sub-isocrystals of D, equipped with the partial order $\leq$ given by the inclusion relation, is naturally a bounded modular lattice of finite length $\leq \dim_{\breve F}D$ . So, by [Reference CornutCor18, Proposition 2.7], the subset $\mathbf{F}(D)$ , equipped with the distance function induced from that on $\mathbf{F}(V)$ is complete, and thus is closed in $\mathbf{F}(D)$ . The norm $\alpha$ on $D\otimes_{\breve F}K$ defines a map

\[\langle \alpha,-\rangle : \mathbf{F}(D)\longrightarrow \mathbb R,\quad f\longmapsto \langle \alpha,f\rangle =\sum_{s\in \mathbb R}s\cdot \operatorname{deg}(\mathrm{gr}_f^sD,\alpha).\]

Using the vectorial Tits building, the Harder–Narasimhan filtration for the normed isocrystal $(D,\alpha)$ has the following characterization.

Proposition 3.17 [Reference CornutCor18, Proposition 2.12]. The Harder–Narasimhan filtration for $(D,\alpha)$ is the unique filtration $\mathcal F\in \mathbf{F}(D)$ such that

\[|\mathcal F|^2-2\langle \alpha,\mathcal F\rangle\leq |f|^2-2\langle \alpha,f\rangle\]

for any $f\in\mathbf{F}(D)$ .

On can slightly strengthen the above inequality.

Corollary 3.18. Let $(D,\alpha)$ be a normed isocrystal, with $\mathcal F=\mathcal F_{\text{HN}}(D,\alpha)$ its Harder–Narasimhan filtration. Then, for any $f\in \mathbf{F}(D)$ ,

\[|\mathcal F|^2-2\langle \alpha,\mathcal F\rangle\leq |f|^2-2\langle \alpha,f\rangle-\frac{(\operatorname{deg}(f)-\operatorname{deg}(D,\alpha))^2}{\dim_{\breve F}D}.\]

Here $\operatorname{deg}(f)$ denotes the degree of the filtration f, i.e.,

\[\operatorname{deg}(f):=\sum_{a\in \mathbb R}a\cdot \dim_{\breve F} \mathrm{gr}_f^aD.\]

Proof. For $r\in \mathbb R$ , let $f_r$ be the $\mathbb R$ -filtration on D with

\[D_{f_r}^a:=D_f^{a-r} \quad \textrm{for all }a\in \mathbb R.\]

Then

\begin{align*}|\mathcal F|^2-2\langle \alpha,\mathcal F\rangle & \leq|f_r|^2-2\langle \alpha, f_r\rangle \\ & = \sum_{a\in \mathbb R}(a+r)^2\dim_{\breve{F}}\mathrm{gr}_f^aD-2\sum_{a\in \mathbb R}(a+r)\operatorname{deg}(\mathrm{gr}_f^aD,\alpha). \\ & = r^2\cdot \dim_{\breve F}D+2r\cdot (\operatorname{deg}(f)-\operatorname{deg}(D,\alpha))+|f|^2-2\langle \alpha, f\rangle. \end{align*}

Since this holds for any r, taking $r=-({\operatorname{deg}(f)-\operatorname{deg}(D,\alpha)})/{\dim_{\breve F}D}$ , we obtain the desired inequality.

3.3.3 Compatibility of Harder–Narasimhan filtration for normed isocrystals with tensor products.

Let $(D,\alpha)$ and $(D',\alpha')$ be two normed isocrystals on $K/F$ , with underlying $\breve F$ -vector spaces V and V’, respectively. Consider the isocrystal

\[E=D\oplus D'\oplus (D\otimes D')\]

equipped with the norm $\beta$ on $E_K$ induced from $\alpha$ and $\alpha'$ . We have the following map:

\[\mathbf F(D)\times \mathbf{F}(D')\longrightarrow \mathbf{F}(E), \quad (f,f' )\mapsto f\oplus f' \oplus (f\otimes f').\]

Proof. The injectivity of the above map is clear. So we can view $\mathbf{F}(D)\times \mathbf{F}(D')$ as a subset of $\mathbf{F}(E)$ . To check that $\mathbf F(D)\times \mathbf F(D')$ is convex in $\mathbf F(E)$ , we claim that, for (f,f’) and (g,g’) in $\mathbf F(D)\times \mathbf F(D')$ ,

\[(f+g)\otimes (f'+g')=f\otimes f'+g\otimes g'\in \mathbf F(D\otimes D').\]

For this, let $\underline{e}=(e_1,\ldots,e_n)$ (respectively $\underline e'=(e_{1}',\ldots, e_m')$ ) be an $\breve F$ -basis that split f and g (respectively f’ and g’) at the same time, with the corresponding n-tuples (respectively m-tuples) $\alpha=(a_1,\ldots, a_n)$ and $\beta=(b_1,\ldots,b_n)$ (respectively $\alpha'=(a_1',\ldots,a_m')$ and $\beta'=(b_1',\ldots,b_m')$ ). Then the $\breve F$ -basis $(e_i\otimes e_{j}')_{i,j}$ splits both the filtrations $(f+g)\otimes(f'+g')$ and $f\otimes f'+g\otimes g'$ , and the corresponding mn-tuples are the same, given by

\[((a_i+b_i)+(a_j'+b_j' ))_{i,j}=( (a_i+a_j')+(b_i+b_{j} '))_{i,j},\]

giving our claim.

It remains to check that $\mathbf{F}(D)\times \mathbf{F}(D')\subset \mathbf{F}(E)$ is closed. Let (f,f’) and (g,g’) be two elements in $\mathbf{F}(D)\times \mathbf F(D')$ . Viewed as two elements in $\mathbf F(E)$ , their distance is

\[\sqrt{d(f,g)^2+d(f',g')^2+d(f\otimes f',g\otimes g')^2}.\]

On the other hand, consider the $\breve F$ -bases $\underline e$ and $\underline e '$ , and the tuples $\alpha, \alpha',\beta$ and $\beta'$ as above, so we then have

\begin{align*}d(f\otimes f',g\otimes g')^2 & = \sum_{i=1}^n\sum_{j=1}^m (a_i+a_j'-b_i-b_j')^2 \\ & \leq \sum_{i=1}^n\sum_{j=1}^m 2\left((a_i-b_i)^2+(a_j'-b_j')^2\right) \\ & = 2m\cdot d(f,g)^2+2n\cdot d(f',g'),\end{align*}

with $m=\dim_{\breve F}D'$ and $n=\dim_{\breve F}D$ . Consequently,

\begin{align*}\sqrt{d(f,g)^2+d(f',g')^2} & \leq \sqrt{d(f,g)^2+d(f',g')^2+d(f\otimes f',g\otimes g')^2} \\ & \leq \sqrt{(2m+1)d(f,g)^2+(2n+1)d(f',g')^2}.\end{align*}

In particular, the distance function on $\mathbf{F}(D)\times \mathbf F(D')$ induced from the distance function on $\mathbf{F}(E)$ is equivalent to the product of the distance functions on $\mathbf{F}(D)$ and on $\mathbf{F}(D')$ . Since $(\mathbf{F}(D),d)$ and $(\mathbf{F}(D'),d)$ are complete, it follows that, as a subset of $\mathbf {F}(E)$ , $\mathbf{F}(D)\times \mathbf{F}(D')$ is also complete, and thus is closed, as required.

As a result, we can consider the convex projection of $\mathbf{F}(E)$ to the closed convex subset $\mathbf{F}(D)\times \mathbf {F}(D')$ , denoted by

\[p: \mathbf{F}(E)\longrightarrow \mathbf{F}(D)\times \mathbf{F}(D')\]

in what follows. Recall that the normed isocrystals have underlying $\breve{F}$ -vector spaces V and V’, respectively. Let

\[W=V\oplus V'\oplus (V\otimes V')\]

be the underlying $\breve{F}$ -vector space of the isocrystal E. A similar argument shows that the natural map

(3.4) \begin{equation} \mathbf{F}(V)\times \mathbf{F}(V')\longrightarrow F(W), \quad (f,f' )\longmapsto f\oplus f'\oplus (f\otimes f')\end{equation}

identifies $\mathbf{F}(V)\times \mathbf{F}(V' )$ as a closed convex subset of $\mathbf{F}(W)$ . Let us denote the corresponding convex projection by q.

Lemma 3.20. The following square is commutative.

Proof. By the uniqueness of convex projection, it suffices to check that, for $f\in \mathbf{F}(E)$ viewed as an element in $\mathbf{F}(W)$ , its convex projection $q(f)\in \mathbf{F}(V)\times \mathbf F(W)$ is an $\mathbb R$ -filtration by sub-isocrystals. To show this, by abuse of notation, we denote by $\phi$ the Frobenius on the related isocrystals in the proof. So for $\mathcal D\in \{D, D',E\} $ , with underlying $\breve F$ -vector space $\mathcal V$ , we have an isomorphism of $\breve F$ -spaces

\[\phi\otimes 1: \mathcal V\otimes_{\breve F,\sigma}\breve F\longrightarrow \mathcal V.\]

In particular, for $g\in \mathbf{F}(\mathcal V)$ , we obtain a filtration $\phi(g)$ of $\mathcal V$ by setting

\[\mathcal V_{\phi(g)}^a=(\phi\otimes 1)(\mathcal V_{g}^a)\subset \mathcal V.\]

As the Frobenius map $\phi$ is bijective, we obtain the bijection

\[\phi: \mathbf{F}(\mathcal V)\rightarrow \mathbf{F}(\mathcal V)\]

and $\mathbf{F}(\mathcal D)\subset \mathbf{F}(\mathcal V)$ is precisely the subset of elements fixed by $\phi$ . For g’ a second filtration of $\mathbf{F}(\mathcal V)$ , we have

\[\langle g,g'\rangle=\langle \phi(g),\phi(g')\rangle,\]

and thus $d(\phi(g),\phi(g' ))=d(g,g' )$ . Moreover, the map (3.4) is equivariant with respect to the bijection $\phi$ on both sides. It follows that, for $f\in \mathbf{F}(E)\subset \mathbf F(W)$ , q(f) and $\phi(q(f))$ are two elements in $\mathbf F(V)\times \mathbf{F}(V' )$ minimizing the function

\[\mathbf{F}(V)\times \mathbf{F}(V')\ni g\longmapsto d( f,g).\]

As a result, $\phi(q(f))=q(f)\in \mathbf{F}(V)\times \mathbf{F}(V')$ and thus $q(f)\in \mathbf{F}(D)\times \mathbf{F}(D' )$ , as required.

We are now ready to prove the main result of this section.

Proof of Theorem 3.10. Recall that $(D,\alpha)$ and $(D',\alpha')$ are two normed isocrystals on $K/F$ , and

\[(E,\beta):=(D,\alpha)\oplus (D',\alpha')\oplus ((D,\alpha)\otimes (D',\alpha')).\]

Let

\[\mathcal F=(\mathcal F_{\text{HN}}(D,\alpha),\mathcal F_{\text{HN}}(D' ,\alpha' ))\in \mathbf{F}(D)\times \mathbf{F}(D'),\]

and view it as the element

\[\mathcal F_{\text{HN}}(D,\alpha)\oplus \mathcal F_{\text{HN}}(D',\alpha')\oplus (\mathcal F_{\text{HN}}(D,\alpha)\otimes \mathcal F_{\text{HN}}(D',\alpha'))\]

in $\mathbf {F}(E)$ . We want to show that this coincides with the Harder–Narasimhan filtration $\mathcal F_{\text{HN}}(E,\beta)$ for $(E, \beta)$ . For this, it suffices to check that $\mathcal F$ satisfies the inequality characterizing the Harder–Narasimhan filtration $\mathcal F_{\text{HN}}(E,\beta)$ :

(3.5) \begin{equation} |\mathcal F|^2-2\langle \beta,\mathcal F\rangle \leq |f|^2-2\langle \beta,f\rangle, \quad \forall f\in \mathbf{F}(E).\end{equation}

For the moment, let us check the inequality (3.5) for $f=(g,g' )\in \mathbf{F}(D)\times \mathbf F(D')\subset \mathbf F(E)$ . Firstly,

\begin{align*}|f|^2& = \sum_{a\in \mathbb R} a^2\dim \mathrm{gr}_f^ a(D\oplus D'\oplus D\otimes D' ) \\ & = \sum_{a\in \mathbb R} a^2(\dim\mathrm{gr}_g^ a(D)\oplus \mathrm{gr}_{g'}^a(D') \oplus\mathrm{gr}_{g\otimes g' }^ a(D\otimes D') ))\\ & =|g|^2+|g'|^2+\sum_{a}a^2\dim\bigg(\bigoplus_{r+s=a}\mathrm{gr}_g^rD\otimes \mathrm{gr}_{g'}^s D' \bigg) \\ & = |g|^2+|g' |^2+\sum_{r,s\in \mathbb R}(r+s)^2\dim(\mathrm{gr}_g^r D)\cdot \dim(\mathrm{gr}_{g'}^s D') \\ & = (\dim D'+1)|g|^2+(\dim D+1)|g'|^2+2\operatorname{deg}(g)\operatorname{deg}(g').\end{align*}

Secondly,

\begin{align*}\langle \alpha\otimes \alpha',g\otimes g' \rangle & = \sum_{a\in \mathbb R}a \cdot \operatorname{deg}(\mathrm{gr}_{g\otimes g' }^a,\alpha\otimes \alpha' ) \\ & = \sum_{r,s\in \mathbb R}(r+s)\operatorname{deg}(\mathrm{gr}_g^rD\otimes \mathrm{gr}_{g' }^s D',\alpha\otimes \alpha' ) \\ & = \sum_{r,s\in \mathbb R}(r+s)(\operatorname{deg}(\mathrm{gr}_g^rD,\alpha)\cdot \dim (\mathrm{gr}_{g' }^sD') + \operatorname{deg}(\mathrm{gr}_{g' }^sD',\alpha')\cdot \dim(\mathrm{gr}_{g }^r D)) \\ & = \langle \alpha,g\rangle \dim D'+\langle \alpha',g' \rangle \dim D+\operatorname{deg}(D,\alpha)\operatorname{deg}(g')+\operatorname{deg}(D',\alpha' )\operatorname{deg}(g),\end{align*}

yielding

\begin{align*}\langle \beta, f\rangle &= \langle \alpha,g\rangle+\langle \alpha',g' \rangle+\langle \alpha\otimes \alpha',g\otimes g' \rangle \\ & = \langle \alpha,g\rangle (\dim D'+1)+\langle \alpha',g' \rangle (\dim D+1)+\operatorname{deg}(D,\alpha)\operatorname{deg}(g' )+\operatorname{deg}(D',\alpha' )\operatorname{deg}(g).\end{align*}

So

\begin{align*}|f|^2-2\langle \beta,f\rangle & = (\dim D'+1)(|g|^2-2\langle\alpha,g\rangle) +(\dim D+1)(|g' |^2-2\langle \alpha' ,g'\rangle) \\&\quad +2\operatorname{deg}(g)\operatorname{deg}(g')-2\operatorname{deg}(D,\alpha)\operatorname{deg}(g')-2\operatorname{deg}(D',\alpha' )\operatorname{deg}(g).\end{align*}

Applying this equality to $f=\mathcal F$ , and noticing that

\[\operatorname{deg}(\mathcal F_{\text{HN}}(D,\alpha))=\operatorname{deg}(D,\alpha) \quad \textrm{and}\quad \operatorname{deg}(\mathcal F_{\text{HN}}(D' ,\alpha' ))=\operatorname{deg}(D',\alpha'),\]

we find that

\begin{align*}|\mathcal F|^2-2\langle \beta,\mathcal F\rangle & = (\dim D'+1)(|\mathcal F_{\text{HN}}(D,\alpha)|^2-2\langle \alpha,\mathcal F_{\text{HN}}(D,\alpha)\rangle)\\ &\quad +(\dim D+1)(|\mathcal F_{\text{HN}}(D',\alpha' )|^2-2\langle \alpha' ,\mathcal F_{\text{HN}}(D',\alpha' )\rangle) \\ &\quad -2\operatorname{deg}(D,\alpha)\operatorname{deg}(D',\alpha' ).\end{align*}

Using Corollary 3.18, we deduce

\begin{align*}& |\mathcal F|^2-2\langle \beta,\mathcal F\rangle -(|f|^2-2\langle\beta,f\rangle ) \\ &\quad \leq -\left( \frac{\dim D '+1}{\dim D}(\operatorname{deg}(g)-\operatorname{deg}(D,\alpha))^2+\frac{\dim D +1}{\dim D'}(\operatorname{deg}(g')-\operatorname{deg}(D',\alpha'))^2\right) \\ &\qquad-2(\operatorname{deg}(g)-\operatorname{deg}(D,\alpha))(\operatorname{deg}(g')-\operatorname{deg}(D',\alpha'))\\ &\quad\leq -\left( \frac{\dim D '}{\dim D}(\operatorname{deg}(g)-\operatorname{deg}(D,\alpha))^2+\frac{\dim D }{\dim D'}(\operatorname{deg}(g')-\operatorname{deg}(D',\alpha'))^2\right) \\ &\qquad-2(\operatorname{deg}(g)-\operatorname{deg}(D,\alpha))(\operatorname{deg}(g')-\operatorname{deg}(D',\alpha')) \\ &\quad \leq 0.\end{align*}

Here, for the last inequality, we use the classical inequality

\[a^2+b^2\geq 2ab, \quad \forall a,b\in \mathbb R.\]

As a result, we see that the inequality (3.5) holds once $f\in \mathbf{F}(D)\times \mathbf{F}(D')\subset \mathbf{F}(E)$ .

For general $f\in \mathbf{F}(E)$ , consider its convex projection p(f) to $\mathbf{F}(D)\times \mathbf{F}(D')$ . According to Lemma 3.16, we have

\[|p(f)|^2\leq |f|^2.\]

On the other hand, viewing f as an $\mathbb R$ -filtration of the underlying $\breve{F}$ -vector space W of E, we have by Remark 3.7 the following equality

\[\operatorname{deg}(\mathrm{gr}_f^ aE,\beta)=\operatorname{deg}(\mathrm{gr}_f^aW, \beta)-\operatorname{deg}(\mathrm{gr}_f^aE),\]

where the first term on the right-hand side is the degree of the normed vector space $\mathrm{gr}_f^a W$ , equipped with the norm induced from $\beta$ , and the second term $\operatorname{deg}(\mathrm{gr}_f^aE)$ is the degree of the isocrystsal $\mathrm{gr}_f^aW$ . Hence

\begin{align*}\langle \beta,f\rangle& = \sum_{a\in \mathbb R}a\cdot\operatorname{deg}(\mathrm{gr}_f^{a}E,\beta) \\& = \sum_{a\in \mathbb R}a\cdot\operatorname{deg}(\mathrm{gr}_f^{a}W,\beta)-\sum_a a\cdot \operatorname{deg}(\mathrm{gr}_f^ aE).\end{align*}

Now the first sum

\[\langle\beta,f\rangle ' :=\sum_a a\cdot \operatorname{deg}(\mathrm{gr}_f^aW,\beta)\]

has been already considered by Cornut in [Reference CornutCor18, § 5.2]. As p(f) is also the convex projection of $f\in \mathbf{F}(E)\subset \mathbf F(W)$ to $\mathbf{F}(V)\times \mathbf{F}(V')\subset \mathbf F(W)$ according to Lemma 3.20, by [Reference CornutCor18, Proposition 5.6 and §5.2.13], we have

(3.6) \begin{equation} \langle \beta,f\rangle ' \leq \langle \beta,p(f)\rangle '.\end{equation}

Furthermore, if we write $\mathcal F_{\textrm{N}}\in \mathbf{F}(E)$ as the Newton filtration on the isocrystal E and $\mathcal F_{\textrm{N}}^*$ its opposite (defined from the slope decomposition), it follows that

\[\sum_a a\cdot \operatorname{deg}(\mathrm{gr}_f^ aE)=\langle\mathcal F_{\textrm{N}},f\rangle=-\langle \mathcal F_{\textrm{N}}^*, f\rangle.\]

Clearly, $\mathcal F_{\textrm{N}},\mathcal F_{\textrm{N}}^*\in \mathbf{F}(D)\times \mathbf{F}(D' )\subset \mathbf F(E)$ , and both $ t\cdot \mathcal F_N+p(f)$ and $t\cdot \mathcal F_{\textrm{N}}^*+p(f)$ are contained in $\mathbf{F}(D)\times \mathbf{F}(D')$ for any $t\in \mathbb R_{>0}$ sufficiently close to 0. By Lemma 3.16 we have

\[\langle \mathcal F_{\textrm{N}},f\rangle \leq \langle \mathcal F_{\textrm{N}},p(f)\rangle \quad \textrm{and}\quad \langle \mathcal F_{\textrm{N}}^*,f\rangle \leq \langle \mathcal F_{\textrm{N}}^*,p(f)\rangle,\]

and thus an equality

\[\langle \mathcal F_{\textrm{N}},f\rangle =\langle \mathcal F_{\textrm{N}},p(f)\rangle.\]

Combining with the inequality (3.6), we find

\[\langle \beta,f\rangle =\langle \beta,f\rangle'-\langle \mathcal F_{\textrm{N}},f\rangle \leq \langle \beta, p(f)\rangle'-\langle \mathcal F_{\textrm{N}},p(f)\rangle=\langle \beta, p(f)\rangle,\]

and thus

\begin{align*}|\mathcal F|^2-2\langle \beta,\mathcal F\rangle &\leq|p(f)|^2-2\langle \beta,p(f)\rangle \\ & \leq |f|^2-2\langle \beta,f\rangle\end{align*}

for any $f\in \mathbf{F}(E)$ , where the first inequality follows from the previous step. Therefore, we find

\[\mathcal F_{\text{HN}}(D,\alpha)\oplus \mathcal F_{\text{HN}}(D',\alpha')\oplus (\mathcal F_{\text{HN}}(D,\alpha)\otimes \mathcal F_{\text{HN}}(D',\alpha'))=\mathcal F_{\text{HN}}(E,\beta),\]

while

\[\mathcal{F}_{\text{HN}}(E,\beta)=\mathcal F_{\text{HN}}(D,\alpha)\oplus \mathcal F_{\text{HN}}(D',\alpha')\oplus \mathcal F_{\text{HN}}(D\otimes D',\alpha\otimes \alpha')\]

by Proposition 3.9. In other words,

\[\mathcal F_{\text{HN}}(D,\alpha)\otimes \mathcal F_{\text{HN}}(D' ,\alpha' )=\mathcal F_{\text{HN}}(D\otimes D',\alpha\otimes \alpha' ),\]

and the functor (3.1) is compatible with tensor products.