is compact. Since $I_{x,\mathbb{Q}_\ell}$ is connected it follows from of [5, Propositions 8.4, Proposition 9.3] that it contains a maximal trigonalizable subgroup of $\mathbb{G}_{\mathbb{Q}_\ell}$. Since $\mathbb{G}_{\mathbb{Q}_\ell}$ is split, $I_{x,\mathbb{Q}_\ell}$ must contain a Borel subgroup of $\mathbb{G}_{\mathbb{Q}_\ell}$, and because it is reductive it follows that $I_{x,\mathbb{Q}_\ell} = \mathbb{G}_{\mathbb{Q}_\ell}$. It is well known that this only happens when $x$ is contained in the basic locus. $\square$

**Lemma 8.6.3.** Assume that $\mathrm{Sh}_{\mathbb{G},U}$ is proper. If a reduced closed subscheme $Z \subseteq \mathrm{Sh}_{\mathbb{G},U,\lfloor b \rfloor}$ is stable under the action of $\mathbb{G}(\mathbb{Q}_\ell)$ for some $\ell \neq p$ such that $\mathbb{G}^{\mathrm{sc}} \otimes \mathbb{Q}_\ell$ is split reductive, then $Z$ is stable under the action of $\mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)$.

*Proof.* The proof is almost exactly the same as the proof of [12, Proposition 4.6]. Nevertheless, we will give a complete proof for the benefit of the reader.

**Step 1.** A standard argument (see e.g. [84, The proof of Proposition 3.3.1]) using the quasi-affineness of the Ekedahl–Oort stratification ([36, Corollary I.2.6]) and the properness of $\mathrm{Sh}_{\mathbb{G},U}$ shows that the Zariski closure $\bar{Z}$ of $Z$ in $\mathrm{Sh}_{\mathbb{G},U}$ contains a point $x \in \mathrm{Sh}_{\mathbb{G},U}(\bar{\mathbb{F}}_p)$ with finite $\ell$-power Hecke orbit. In fact, this argument shows that for any irreducible component $V$ of $Z$ the closure $\bar{V}$ contains a point with a finite $\ell$-power orbit.

It follows from Lemma 8.6.2 that $x$ is contained in the basic locus of $\mathrm{Sh}_{\mathbb{G},U}$, and moreover that $I_x$ is an inner form of $\mathbb{G}$ (see [40, Corollary 5.2.11]). Thus the isomorphism $I_x \otimes \mathbb{A}_f^p \to \mathbb{G} \otimes \mathbb{A}_f^p$ induces an isomorphism $I_x^{\mathrm{sc}} \otimes \mathbb{A}_f^p \to \mathbb{G}^{\mathrm{sc}} \otimes \mathbb{A}_f^p$. Strong approximation away from $\infty$, $\ell$ (see [68, Theorem 7.8]), using the fact that $\mathbb{G}_{\mathbb{Q}_\ell}$ is split, tells us that the image of

$$I_x^{\mathrm{sc}}(\mathbb{Z}_{(p)}) \to \mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^{p,\ell})$$

is dense. Since the inclusion of the $\ell$-adic Hecke orbit of $x$ inside the prime-to-$p$ Hecke orbit of $x$ can be identified with

$$I_x(\mathbb{Z}_{(p)}) \backslash \mathbb{G}(\mathbb{Q}_\ell) U^p / U^p \subseteq I_x(\mathbb{Z}_{(p)}) \backslash \mathbb{G}(\mathbb{A}_f^p) / U^p,$$

we see that the $\ell$-adic Hecke orbit of $x$ is $\mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)$-stable. Moreover the local stabiliser principle, see Remark 8.2.2, tells us that

$$\bar{Z}^{/x} \subseteq C_{\mathbb{G}, \lfloor b \rfloor}^{/x}$$

is kept stable under the action of

$$I_x(\mathbb{Z}_{(p)}) \cap \mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p) \subseteq \mathrm{Aut}_{\mathbb{G}}(Y)(\bar{\mathbb{F}}_p)$$

and by continuity it is also kept stable by its closure.

**Step 2.** Take a $\lambda$-adic Hecke operator $g_\lambda \in \mathbb{G}^{\mathrm{sc}}(\mathbb{Q}_\lambda)$ for $\lambda \neq p, \ell$ and let $W$ be the image of $\bar{Z}$ under $g_\lambda$. Since every irreducible component of $\bar{Z}$ contains a point $x$ with finite $\ell$-power Hecke orbit, it follows that every irreducible component of $W$ contains an element in the $g_\lambda$-orbit of such an $x$. Since $\bar{Z}$ contains the $\mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)$ orbit of $x$, it follows that every irreducible component $W_i$ of $W$ intersects $\bar{Z}$ in a point $y_i$ with finite $\ell$-power Hecke orbit.

The reduced prime-to-$p$ Hecke orbit of $y_i$ has the form

$$I_{y_i}^{\mathrm{sc}}(\mathbb{Z}_{(p)}) \backslash \mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p) U^p / U^p.$$

By strong approximation ([68, Theorem 7.8]) away from $\ell$ for $I_{y_i}^{\mathrm{sc}}$, using the fact that $\mathbb{G}_{\mathbb{Q}_\ell}$ is split, we can choose $\delta \in I_{y_i}^{\mathrm{sc}}(\mathbb{Q})$ which lands in $U^{p,\ell,\lambda}$ and such that there is an element $g_\ell \in \mathbb{G}^{\mathrm{sc}}(\mathbb{Q}_\ell)$ such that $\delta \cdot g_\ell = g_\lambda^{-1}$ in $g_\lambda$ in $I_{y_i}(\mathbb{Z}_{(p)}) \backslash \mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p) U^p / U^p$.
We see that $g_λ \circ g_ℓ$ fixes $y_i$, and since $\bar{Z}$ is $g_ℓ$-stable, the image of $\bar{Z}$ under $g_λ \circ g_ℓ$ is equal to $W$. Now we consider the closed subschemes

$$W_i^{/y_i}, \bar{Z}^{/y_i} \subseteq \mathrm{Sh}_{G,U}^{/y_i}.$$

The subscheme $W_i^{/y_i}$ is the image of $\bar{Z}^{/y_i}$ under the Hecke operator $g_λ \circ g_ℓ$. Since $\bar{Z}^{/y_i}$ is stable under the action of the closure of $I_x(Z(p))$ and since $g_λ \circ g_ℓ$ is contained in that closure by construction, it follows that

$$W_i^{/y_i} \subseteq \bar{Z}^{/y_i}.$$

From this we deduce that $W_i \subseteq \bar{Z}$. Thus every irreducible component $W_i$ of $W$ is contained in $\bar{Z}$, and we conclude that $\bar{Z}$ is stable under the action of $g_λ$. Since $\lambda$ and $g_λ$ were arbitrary, it follows that $\bar{Z}$ is stable under the action of $\mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)$.

We know that $Z$ is the intersection of $\bar{Z}$ with $\mathrm{Sh}_{G,U,[b]}$ and as the intersection of two $\mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)$-stable subschemes it must itself be $\mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)$-stable. $\square$

**Proof of Theorem 8.6.1.** If we let $\mathcal{M}$ be the isocrystal attached to the universal abelian variety up to prime-to-$p$ isogeny over $Z$ and if we let $x \in Z$ be a smooth point, then arguing as in the proof of Theorem 8.3.2, we can combine [43, Corollary 3.3.5] with Theorem II to deduce that the monodromy of the isocrystal $\mathcal{M}$ over $\mathrm{Spec} \hat{\mathcal{O}}_{Z,x}$ is isomorphic to $U_{\nu_b}$.

**Proposition 8.2.1** (see Remark 8.2.2) tells us that $Z^x$ is stable under the action of

$$(I_x(Z(p)) \cap \mathbb{G}^{\mathrm{sc}}(\mathbb{A}_f^p)) \subseteq \mathrm{Aut}_G(Y)(\bar{\mathbb{F}}_p).$$

By continuity, it is also stable under the closure $\Gamma \subseteq \mathrm{Aut}_G(Y)(\bar{\mathbb{F}}_p)$. As in the proof of [43, Corollary 6.1.6], it follows that $\Gamma$ acts strongly non-trivially on $C_{G,[b]}^x = Z(\mathfrak{a}^+)$. The same argument as in the proof of Theorem 8.3.2 allows us to conclude that $Z$ is a union of connected components of $C$. $\square$

**8.7. Further questions of Chai-Oort.** Let $Z \subseteq C_{(Y,\lambda)}$ be an irreducible smooth closed subvariety and let $x \in Z(\bar{\mathbb{F}}_p)$. We call $Z$ strongly Tate-linear at $x$ if $Z^x \subseteq C_{(Y,\lambda)}^x$ is a strongly Tate-linear subvariety.

**Question 8.7.1.** Suppose that $Z$ is strongly Tate-linear at some closed point $z_0 \in Z(\bar{\mathbb{F}}_p)$. Is $Z$ then strongly Tate-linear at all closed points $z \in Z(\bar{\mathbb{F}}_p)$?

It follows from Theorem II that the monodromy group of $\mathcal{M}$ over $\mathrm{Spec} \hat{\mathcal{O}}_{Z,x}$ does not depend on $x$. Now the validity of Conjecture 5.5.7 would imply that the question above has an affirmative answer.

**Question 8.7.2.** Suppose that $Z$ is strongly Tate-linear at some closed point $z_0 \in Z(\bar{\mathbb{F}}_p)$. Is $Z$ an irreducible component of a central leaf in the mod $p$ reduction of a Shimura variety of Hodge type?

The stronger assertion that $Z$ must itself be an irreducible component of a Shimura variety of Hodge type is false in general, because only finitely many central leaves in a given Newton stratum contain the mod $p$ reductions of special points by [53, Theorem 1.3]. If $C_{(Y,\lambda)}$ is the ordinary locus and $Z$ is proper, then results towards this stronger assertion are proved in work of Moonen [62].

**8.8. Results at ramified primes and parahoric level.** The statements of our theorems make sense for central leaves in special fibres of the Kisin-Pappas [50] integral models of Shimura varieties of parahoric level at tamely ramified primes $p > 2$.
8.8.1. If $G$ is unramified over $\mathbb{Q}_p$ and the level is parahoric, then the Hecke orbit conjecture for central leaves at parahoric level follows immediately from Theorem 8.3.2. The main observation is that the forgetful map

$$C' \to C,$$

where $C'$ is a central leaf at Iwahori level and $C$ is a central leaf at hyperspecial level, is equivariant for the prime-to-$p$ Hecke operators and induces a bijection on $\pi_0$. This last statement can be proven using the surjectivity<fnref n="15" /> of $C' \to C$, and the explicit description of connected components of Igusa varieties in [44, 55].

Rapoport-Zink uniformisation of isogeny classes at parahoric level<fnref n="15" /> implies as before that isogeny classes are dense in the Newton strata containing them.

8.8.2. If $G$ is a ramified group over $\mathbb{Q}_p$, then it is not always true that $G(\mathbb{A}_f^p)$ acts transitively on $\pi_0(\mathrm{Sh}_{G,U})$ (see [64] for explicit counterexamples), and therefore it is not necessarily true that $G(\mathbb{A}_f^p)$ acts transitively on $\pi_0(C)$ either because $\pi_0(C) \to \pi_0(\mathrm{Sh}_{G,U})$ is surjective. Nevertheless, we expect that the continuous part of the Hecke orbit conjecture is true for ramified groups. In fact, we suspect that the strategy adopted in this paper can be made to work for ramified groups.

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INSTITUT DE RECHERCHE MATHÉMATIQUE AVANCÉE (IRMA), UNIVERSITÉ DE STRASBOURG, 7 RUE RENÉ-DESCARTES, 67000 STRASBOURG, FRANCE

Email address: daddezio@unistra.fr

STANFORD MATHEMATICS DEPARTMENT, 450 JANE STANFORD, WAY BUILDING 380X, STANFORD, CA 94305, USA

Email address: p.van.hoften@vu.nl