# Connectedness of affine Deligne-Lusztig varieties for unramified groups

Sian Nie  Institute of Mathematics, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, 100190, Beijing, China
<EMAIL_ADDRESS>
###### Abstract.

For unramified reductive groups, we determine the connected components of
affine Deligne-Lusztig varieties in the partial affine flag varieties. Based
on the work of Hamacher-Kim and Zhou, this result allows us to verify, in the
unramified group case, the He-Rapoport axioms, the “almost product structure”
of Newton strata, and the precise description of mod $p$ isogeny classes
predicted by the Langlands-Rapoport conjecture, for the Kisin-Pappas integral
models of Shimura varieties of Hodge type with parahoric level structure.

## Introduction

### 0.1.

Let $F$ be a non-Archimedean local field with valuation ring $\mathcal{O}_{F}$
and residue field $\mathbb{F}_{q}$, where $q$ is a power of some prime $p$.
Denote by ${\breve{F}}$ the completion of a maximal unramified extension of
$F$. Let $G$ be a connected reductive group defined over $F$, and let $\sigma$
be the Frobenius automorphism of $G({\breve{F}})$. Fix an element $b\in
G({\breve{F}})$, a geometric cocharacter $\lambda$ of $G$, and a
$\sigma$-stable parahoric subgroup $K\subseteq G({\breve{F}})$. The attached
affine Deligne-Lusztig variety is defined by

$X(\lambda,b)_{K}=X^{G}(\lambda,b)_{K}=\\{g\in
G({\breve{F}})/K;g^{-1}b\sigma(g)\in K{\rm{Adm}}(\lambda)K\\},$

where ${\rm{Adm}}(\lambda)$ is the admissible set associated to the geometric
conjugacy class of $\lambda$. If $F$ is of equal characteristic,
$X(\lambda,b)_{K}$ is a locally closed and locally finite-type subvariety of
the partial affine flag variety $G({\breve{F}})/K$. If $F$ is of mixed
characteristic, $X(\lambda,b)_{K}$ is a perfect subscheme of the Witt vector
partial affine flag variety, in the sense of Bhatt-Scholze [1] and Zhu [37].

The variety $X(\lambda,b)_{K}$, first introduced by Rapoport [32], encodes
important arithmetic information of Shimura varieties. Let $(\mathbf{G},X)$ be
a Shimura datum with $G=\mathbf{G}_{\mathbb{Q}_{p}}$ and $\lambda$ the inverse
of the Hodge cocharacter. Suppose there is a good integral model for the
corresponding Shimura variety with parahoric level structure. Langlands [26],
and latter refined by Langlands-Rapoport [27] and Rapoport [32], conjectured a
precise description of $\overline{\mathbb{F}}_{p}$-points of the integral
model in terms of the varieties $X(\lambda,b)_{K}$. In the case of PEL Shimura
varieties, $X(\lambda,b)_{K}$ is also the set of
$\overline{\mathbb{F}}_{p}$-points of a moduli space of $p$-divisible groups
define by Rapoport-Zink [33].

### 0.2.

The main purpose of this paper is to study the set $\pi_{0}(X(\lambda,b)_{K})$
of connected components of $X(\lambda,b)_{K}$. Notice that $X(\lambda,b)_{K}$
only depends on $\lambda$ and the $\sigma$-conjugacy class $[b]$ of $b$.
Thanks to He [11], $X(\lambda,b)_{K}$ is non-empty if and only if $[b]$
belongs to the set of “neutral acceptable” $\sigma$-conjugacy classes of
$G({\breve{F}})$ with respect to $\lambda$.

Let $\pi_{1}(G)_{\Gamma_{0}}$ be the set of coinvariants of the fundamental
group $\pi_{1}(G)$ under the Galois group $\Gamma_{0}={\rm
Gal}(\overline{{\breve{F}}}/{\breve{F}})$. There is a natural map
$\eta_{G}:G({\breve{F}})/K\to\pi_{1}(G)_{\Gamma_{0}}$. To compute
$\pi_{0}(X(\lambda,b)_{K})$ we can assume that $G$ is adjoint and hence simple
by the following Cartesian diagram (see [18, Corollary 4.4])

$\textstyle{\pi_{0}(X^{G}(\lambda,b)_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{G}}$$\textstyle{\pi_{0}(X^{G_{{\rm
ad}}}(\lambda_{{\rm ad}},b_{{\rm ad}})_{K_{{\rm
ad}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{G_{{\rm
ad}}}}$$\textstyle{\pi_{1}(G)_{\Gamma_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(G_{{\rm
ad}})_{\Gamma_{0}}.}$

The map $\eta_{G}$ gives a natural obstruction to the connectedness of
$X(\lambda,b)_{K}$. Another more technical obstruction is given by the
following Hodge-Newton decomposition theorem.

######  Theorem 0.1 ([6, Theorem 4.17]).

Suppose $G$ is adjoint and simple. If the pair $(\lambda,b)$ is Hodge-Newton
decomposable (with respect to some proper Levi subgroup $M$) in the sense of
[6, §2.5.5], then $X(\lambda,b)_{K}$ is a disjoint union of open and closed
subsets, which are isomorphic to affine Deligne-Lusztig varieties attached to
$M$.

By Theorem 0.1 and induction on the dimension of $G$, it suffices to consider
the Hodge-Newton indecomposable case. This means that either $\lambda$ is a
central cocharacter or the pair $(\lambda,b)$ Hodge-Newton irreducible, see
[36, Lemma 5.3]. In the former case,

$X(\lambda,b)_{K}\cong\mathbb{J}_{b}/(K\cap\mathbb{J}_{b})$

is a discrete subset with $\mathbb{J}_{b}$ the $\sigma$-centralizer of $b$. In
the latter case, we have the following conjecture.

######  Conjecture 0.1 (see [36, Conjecture 5.4]).

Assume $G$ is adjoint and simple. If $(\lambda,b)$ is Hodge-Newton
irreducible, then the map $\eta_{G}$ induces a bijection

$\pi_{0}(X(\lambda,b)_{K})\cong\pi_{1}(G)_{\Gamma_{0}}^{\sigma},$

where $\pi_{1}(G)_{\Gamma_{0}}^{\sigma}$ is the set of $\sigma$-fixed point of
$\pi_{1}(G)_{\Gamma_{0}}$.

If $G$ is unramified (that is, $G$ extends to a reductive group over
$\mathcal{O}_{\breve{F}}$) and $K$ is hyperspecial, Conjecture 0.1 is
established by Viehmann [35], Chen-Kisin-Viehmann [2], and the author [29]. If
$b$ is basic, it is proved by He-Zhou [18]. If $G$ is split or $G={\rm
Res}_{E/F}{\rm GL}_{n}$ with $E/F$ a finite unramified field extension, it is
proved by L. Chen and the author in [3] and [4].

The main result of this paper is the following.

######  Theorem 0.2.

Conjecture 0.1 is true if $G$ is unramified.

In particular, Theorem 0.2 completes the computation of connected components
of affine Deligne-Lusztig varieties for unramified groups.

### 0.3.

We discuss some applications. Assume $p\neq 2$. Let $(\mathbf{G},X)$ be a
Shimura datum of Hodge type with parahoric level structure such that
$p\nmid|\pi_{1}(\mathbf{G}_{\rm der})|$, $\mathbf{G}_{\mathbb{Q}_{p}}$ is
tamely ramified, and the parahoric subgroup $K$ at $p$ is a connected
parahoric. Let $\mathscr{S}_{K}=\mathscr{S}_{K}(G,X)$ be the Kisin-Pappas
integral model of the corresponding Shimura variety constructed in [23]. Let
$F=\mathbb{Q}_{p}$, $G=\mathbf{G}_{\mathbb{Q}_{p}}$, and $\lambda$ be the
inverse of the Hodge cocharacter.

######  Remark 0.3.

In [31], Pappas and Rapoport obtained a new construction of integral models
for Hodge type Shimura varieties with parahoric level structure, without the
the tameness assumption on $G$. It would be desirable to extend the
applications discussed below to their integral models in the unramified group
case.

#### 0.3.1.

A major motivation to study $\pi_{0}(X(\lambda,b)_{K})$ comes from the
Langlands-Rapoport conjecture mentioned in §0.1. In the hyperspecial level
structure case, the conjecture is proved by Kottwitz [25] for PEL Shimura
varieties of types $A$ and $C$, and by Kisin [21] for his integral models [20]
of Shimura varieties of abelian type. Using the Kisin-Pappas integral models
[23] for Hodge type Shimura varieties with parahoric level structure $K$, Zhou
[36] proved that each mod $p$ isogeny class has the predicted form when $G$ is
residually split.

One of the key ingredients in the proofs of Kisin and Zhou is to construct
certain lifting map from $X(\lambda,b)_{K}$ to an isogeny class of
$\mathscr{S}_{K}(\overline{\mathbb{F}}_{p})$ (see also [14, Axiom A]), which
uses in a crucial way descriptions of $\pi_{0}(X(\lambda,b)_{K})$ in [2] and
[18] respectively. Combining [36, Proposition 6.5] with Theorem 0.2, we deduce
that such a lifting map always exists if $G$ is unramified.

######  Proposition 0.4.

If $G$ is unramified, then the Rapoport-Zink uniformisation map admits a
unique lift on $\overline{\mathbb{F}}_{p}$-points

$X(\lambda,b)_{K}\to\mathscr{S}_{K}(\overline{\mathbb{F}}_{p}),$

which respects canonical crystalline Tate tensors on both sides.

If $G$ is unramified and $K$ is hyperspecial, Proposition 0.4 is proved by
Kisin [21]. If $b$ is basic or $G$ is residually split, it is proved by Zhou
[36]. If $G$ is quasi-split and $K$ is absolutely special, it is proved by
Zhou in [12, Theorem A.4.3].

As an application, one can extend [36, Theorem 1.1] to the unramified group
case, by combining the methods in [36] and Proposition 0.4. This is pointed
out to us by Zhou.

######  Corollary 0.5.

If $G$ is unramified, then the isogeny classes in
$\mathscr{S}_{K}(\overline{\mathbb{F}}_{p})$ has the form predicted by the
Langlands-Rapoport conjecture. Moreover, each isogeny class contains a point
which lifts to a special point in the corresponding Shimura variety.

Corollary 0.5 was first proved by Pol van Hoften [12] using a different
approach.

#### 0.3.2.

In [17], He and Rapoport formulated five axioms on Shimura varieties with
parahoric level structure, which provide a group-theorectic way to study
certain characteristic subsets (such as Newton strata, Ekedahl-Oort strata,
Kottwitz-Rapoport strata, and so on) in the mod $p$ reductions of Shimura
varieties. Based on this axiomatic approach, Zhou [36] proved that all the
expected Newton strata are non-empty (see [22] using a different approach).
For more applications of these axioms, we refer to [17], [16], [6], [36] and
[34]. Combining [36, Theorem 8.1] with Proposition 0.4 we have

######  Corollary 0.6.

All the He-Rapoport axioms hold if $G$ is unramified.

These axioms are verified by He-Rapoport [17] in the Siegel case, and by He-
Zhou [18] for certain PEL Shimura varieties (unramified of types $A$ and $C$
and odd ramified unitary groups). In [36], Zhou verified all the axioms except
the surjectivity of [17, Axiom 4 (c)] in the general case, and all of them in
the case that $G$ is residually split.

#### 0.3.3.

In [28], Mantovan established a formula expressing the $l$-adic cohomology of
proper PEL Shimura varieties in terms of the $l$-adic cohomology with compact
supports of the Igusa varieties and of the Rapoport-Zink spaces for any prime
$l\neq p$. This formula encodes nicely the local-global compatibility of the
Langlands correspondence. A key part of its proof is to show that the products
of reduced fibers of Igusa varieties and Rapoport-Zink spaces form nice “pro-
étale covers up to perfection” for the Newton strata, of PEL Shimura varieties
with hyperspecial level structure. This is referred as the “almost product
structure” of Newton strata. In [14], Hamacher-Kim extended Mantovan’s results
to the Kisin-Pappas integral models under some mild assumptions. Combining
[14, Theorem 2] with Proposition 0.4 we have

######  Corollary 0.7.

The “almost product structure” of Newton strata holds if $G$ is unramified.

When $G$ is unramified and $K$ is hyperspecial, the “almost product structure”
of Newton strata is established by Mantovan [28] for PEL Shimura varieties.
The general case is proved by Hamacher-Kim provided the lifting property [14,
Axiom A] holds.

### 0.4.

We outline the strategy of the proof. First we show the $\sigma$-centralizer
$\mathbb{J}_{b}$ acts transitively on $\pi_{0}(X(\lambda,b)_{K})$. Then we
show the stabilizer of each connected component is the normal subgroup
$\mathbb{J}_{b}\cap\ker(\eta_{G})$. Combining these two results one deduces
that
$\pi_{0}(X(\lambda,b)_{K})\cong\mathbb{J}_{b}/(\mathbb{J}_{b}\cap\ker(\eta_{G}))\cong\pi_{1}(G)^{\sigma}$
as desired.

The stabilizers can be determined by adapting the computations in [30]. The
crucial part is to the show the transitivity of the $\mathbb{J}_{b}$ action.
Our starting point is the following natural surjection (see Theorem 2.1)

$\sqcup_{{\tilde{w}}\in\mathcal{S}_{\lambda,b}}\mathbb{J}_{b,{\tilde{w}}}\twoheadrightarrow\pi_{0}(X(\lambda,b)_{K}),$

where $\mathcal{S}_{\lambda,b}$ is the set of semi-standard elements (see
§1.4) contained in ${\rm{Adm}}(\lambda)$ and $[b]$, and
$\mathbb{J}_{b,{\tilde{w}}}=\\{g\in
G({\breve{F}});g^{-1}b\sigma(g)={\tilde{w}}\\}$ on which $\mathbb{J}_{b}$ acts
transitively. So it remains to connect all the subsets
$\mathbb{J}_{b,{\tilde{w}}}K/K$ in $X(\lambda,b)_{K}$. To this end we consider
the following decomposition

$\mathcal{S}_{\lambda,b}=\sqcup_{x\in\mathcal{S}_{\lambda,b}^{+}}\mathcal{S}_{\lambda,b,x},$

where $\mathcal{S}_{\lambda,b}^{+}$ consists of standard elements in
$\mathcal{S}_{\lambda,b}$, and $\mathcal{S}_{\lambda,b,x}$ consists of
elements in $\mathcal{S}_{\lambda,b}$ that are $\sigma$-conjugate to
$x\in\mathcal{S}_{\lambda,b}^{+}$ under the Weyl group of $G$. Note that
$\mathcal{S}_{\lambda,b}^{+}$ can be naturally identified with a subset of
cocharacters dominated by $\lambda$. So we can adapt the connecting algorithm
in [30] to connect $\mathbb{J}_{b,x}K/K$ for $x\in\mathcal{S}_{\lambda,b}^{+}$
with each other. In contrast, the structure of $\mathcal{S}_{\lambda,b,x}$ is
much more mysterious, which makes it an essential difficulty to connect
$\mathbb{J}_{b,{\tilde{w}}}K/K$ for ${\tilde{w}}\in\mathcal{S}_{\lambda,b,x}$
with each other. To overcome it, we show that each set
$\mathcal{S}_{\lambda,b,x}$ contains a unique (distinguished) element $x_{\rm
dist}$ which is of minimal length in its Weyl group coset, and then connect
$\mathbb{J}_{b,{\tilde{w}}}K/K$ with $\mathbb{J}_{b,x_{\rm dist}}K/K$ for all
${\tilde{w}}\in\mathcal{S}_{\lambda,b,x}$. This new connecting algorithm is
motivated from the partial conjugation method by He in [9] and [10].

### 0.5.

The paper is organized as follows. In §1 we recall some basic notions and
introduce the semi-standard elements. In §2 we outline the proof of the main
result. In §3 we introduce the set $\mathcal{P}_{\tilde{w}}$ which will play
an essential role in our new connecting algorithm. In §4, we introduce the new
connecting algorithm and use it to connect $\mathbb{J}_{b,{\tilde{w}}}K/K$ for
${\tilde{w}}\in\mathcal{S}_{\lambda,b,x}$ with each other. In §5 we connect
$\mathbb{J}_{b,x}K/K$ for $x\in\mathcal{S}_{\lambda,b}^{+}$ with each other.
In §6, §7 and §8 we compute the stabilizer in $\mathbb{J}_{b}$ of each
connected component of $X(\lambda,b)_{K}$.

### Acknowledgment

We are grateful to Xuhua He for his detailed comments which greatly improve
the exposition of the paper. We would like to thank Michael Rapoport for
bringing his joint work [31] with Georgios Pappas to our attention. We are
also grateful to Rong Zhou for pointing out the application (Corollary 0.5) to
the Langlans-Rapoport conjecture.

##  1\. Preliminaries

In the body of the paper we assume that $G$ is unramified, simple, and
adjoint. Without loss of generality, we assume further that
$F=\mathbb{F}_{q}((t))$. Then ${\breve{F}}=\mathbb{k}((t))$ with valuation
ring $\mathcal{O}_{\breve{F}}=\mathbb{k}[[t]]$ and residue field
$\mathbb{k}=\overline{\mathbb{F}}_{q}$.

### 1.1.

Let $T\subseteq B$ be a maximal torus and a Borel subgroup defined over
$\mathcal{O}_{F}$. Let $\mathcal{R}=(Y,\Phi^{\vee},X,\Phi,\SS_{0})$ be the
root datum associated to the triple $(T\subseteq B\subseteq F)$, where $X$ and
$Y$ are the character and cocharacter groups of $T$ respectively equipped with
a perfect pairing $\langle,\rangle:Y\times X\to\mathbb{Z}$;
$\Phi=\Phi_{G}\subseteq X$ (resp. $\Phi^{\vee}\subseteq Y$) is the set of
roots (resp. coroots); $\SS_{0}$ is the set of simple roots appearing in $B$.
For $\alpha\in\Phi$, we denote by $s_{\alpha}$ the reflection which sends
$\mu\in Y$ to $\mu-\langle\mu,\alpha\rangle\alpha^{\vee}$, where
$\alpha^{\vee}\in\Phi^{\vee}$ denotes the coroot of $\alpha$. Via the
bijection $\alpha\leftrightarrow s_{\alpha}$, we also denote by $\SS_{0}$ the
set of simple reflections.

Let $W_{0}=N_{T}({\breve{F}})/T({\breve{F}})$ be the Weyl group of $G$, where
$N_{T}$ is the normalizer of $T$ in $G$. The Iwahori-Weyl group of $G$ is
given by

$\tilde{W}_{G}=\tilde{W}=N_{T}({\breve{F}})/T(\mathcal{O}_{\breve{F}})=Y\rtimes
W_{0}=\\{t^{\mu}w;\mu\in Y,w\in W_{0}\\}.$

We can view $\tilde{W}$ as a subgroup of affine transformations of
$V:=Y\otimes_{\mathbb{Z}}\mathbb{R}$, where the action of
${\tilde{w}}=t^{\mu}w$ is given by $v\mapsto\mu+w(v)$. Let
$\Phi^{+}=\Phi\cap\mathbb{Z}_{\geqslant 0}\SS_{0}$ be the set of positive
roots and let $\mathbb{a}=\\{v\in
Y_{\mathbb{R}};0<\langle\alpha,v\rangle<1,\alpha\in\Phi^{+}\\}$ be the base
alcove.

Let $\tilde{\Phi}=\tilde{\Phi}_{G}=\Phi\times\mathbb{Z}$ be the set of (real)
affine roots. Let ${\tilde{\alpha}}=\alpha+k\in\tilde{\Phi}$. Then
$\tilde{\alpha}$ is an affine function on $V$ such that
$\tilde{\alpha}(v)=-\langle\alpha,v\rangle+k$. The induced action of
$\tilde{W}$ on $\tilde{\Phi}$ is given by
${\tilde{w}}(\tilde{\alpha})(v)=\tilde{\alpha}({\tilde{w}}^{-1}(v))$. Let
$s_{\tilde{\alpha}}=t^{k\alpha^{\vee}}s_{\alpha}\in\tilde{W}$ be the
corresponding affine reflection. Then
$\\{s_{\tilde{\alpha}};\tilde{\alpha}\in\tilde{\Phi}\\}$ generates the affine
Weyl group

$W^{a}=W^{a}_{G}=\mathbb{Z}\Phi^{\vee}\rtimes
W_{0}=\\{t^{\mu}w;\mu\in\mathbb{Z}\Phi^{\vee},w\in W_{0}\\}.$

Moreover, we have $\tilde{W}=W^{a}\rtimes\Omega$, where
$\Omega=\Omega_{G}=\\{\omega\in\tilde{W};\omega(\mathbb{a})=\mathbb{a}\\}$.
Set
$\tilde{\Phi}^{+}=\tilde{\Phi}_{G}^{+}=\\{\tilde{\alpha}\in\tilde{\Phi};\tilde{\alpha}(\mathbb{a})>0\\}$
and $\tilde{\Phi}^{-}=-\tilde{\Phi}^{+}$. Then
$\tilde{\Phi}=\tilde{\Phi}^{+}\sqcup\tilde{\Phi}^{-}$. Note that
$\Phi^{\pm}\subseteq\tilde{\Phi}^{\mp}$. Let $\ell:\tilde{W}\to\mathbb{N}$ be
the length function given by
$\ell({\tilde{w}})=|\tilde{\Phi}^{-}\cap{\tilde{w}}(\tilde{\Phi}^{+})|$. Let
$\SS^{a}=\\{s_{\tilde{\alpha}};\tilde{\alpha}\in\tilde{\Phi},\ell(s_{\tilde{\alpha}})=1\\}$
be the set of simple affine reflections. Notice that $(W^{a},\SS^{a})$ is a
Coxeter system, and let $\leq$ be the associated Bruhat order on
$\tilde{W}=W^{a}\rtimes\Omega$.

For ${\tilde{\alpha}}=(\alpha,k)\in\tilde{\Phi}$, let
$U_{{\tilde{\alpha}}}:\mathbb{G}_{a}\to LG$ be the corresponding affine root
subgroup, where $LG$ denotes the loop group associated to $G$. More precisely,
$U_{{\tilde{\alpha}}}(z)=u_{\alpha}(zt^{k})$ for $z\in\mathbb{k}$, where
$u_{\alpha}:\mathbb{G}_{a}\to G$ is the root subgroup of $\alpha$. We set

$I=I_{G}=T(\mathcal{O}_{\breve{F}})\prod_{{\tilde{\alpha}}\in\tilde{\Phi}^{+}}U_{{\tilde{\alpha}}}(\mathbb{k})=T(\mathcal{O}_{\breve{F}})\prod_{\alpha\in\Phi^{+}}u_{\alpha}(t\mathcal{O}_{\breve{F}})\prod_{\beta\in\Phi^{+}}u_{-\beta}(\mathcal{O}_{\breve{F}}),$

which is called an Iwahori subgroup of $G({\breve{F}})$.

### 1.2.

Let $v\in V=Y\otimes\mathbb{R}$. We say $v$ is dominant if $\langle
v,\alpha\rangle\geqslant 0$ for each $\alpha\in\Phi^{+}$, and denote by
$\bar{v}$ the unique dominant $W_{0}$-conjugate of $v$. Let $Y^{+}$ and
$V^{+}$ be the set of dominant vectors in $Y$ and $V$ respectively. For
$v,v^{\prime}\in V$ we write $v^{\prime}\leqslant v$ if
$v-v^{\prime}\in\mathbb{R}_{\geqslant 0}(\Phi^{+})^{\vee}$.

Let $\sigma$ be the Frobenius automorphism of $G({\breve{F}})$. We also denote
by $\sigma$ the induced automorphism on the root datum $\mathcal{R}$. Then
$\sigma$ acts on $V$ as a linear transformation of finite order which
preserves $\mathbb{a}$. For ${\tilde{w}}\in\tilde{W}$ there exists a nonzero
integer $m$ such that $({\tilde{w}}\sigma)^{m}=t^{\xi}$ for some $\xi\in Y$.
Define $\nu_{\tilde{w}}=\xi/m\in V$, which does not depend on the choice of
$m$.

Let $b\in G(L)$. We denote by $[b]=[b]_{G}=\\{g^{-1}b\sigma(g);g\in G(L)\\}$
the $\sigma$-conjugate class of $b$. By [24], the $\sigma$-conjugacy class
$[b]$ is determined by two invariants: the Kottwitz point
$\kappa_{G}(b)\in\pi_{1}(G)_{\sigma}$ and the Newton point
$\nu_{G}(b)\in(V^{+})^{\sigma}$. Here
$\kappa_{G}:G({\breve{F}})\to\pi_{1}(G)_{\sigma}=\pi_{1}(G)/(\sigma-1)\pi_{1}(G)$
is the natural projection. To define $\nu_{G}(b)$, we note that there exists
${\tilde{w}}\in\tilde{W}$ such that ${\tilde{w}}\in[b]$. Then
$\nu_{G}(b)=\bar{\nu}_{{\tilde{w}}}$, which does not depend on the choice of
${\tilde{w}}$. For $b^{\prime}\in G({\breve{F}})$ we set

$\mathbb{J}_{b,b^{\prime}}=\mathbb{J}_{b,b^{\prime}}^{G}=\\{g\in
G({\breve{F}});g^{-1}b\sigma(g)=b^{\prime}\\},$

and put $\mathbb{J}_{b}=\mathbb{J}_{b,b^{\prime}}$ if $b=b^{\prime}$.

For $\lambda\in Y^{+}$ and $b\in G({\breve{F}})$ we define

$X(\lambda,b)=X^{G}(\lambda,b)_{I}=\\{g\in
G({\breve{F}})/I;g^{-1}b\sigma(g)\in I{\rm{Adm}}(\lambda)I\\},$

where ${\rm{Adm}}(\lambda)$ is the $\lambda$-admissible set defined by

${\rm{Adm}}(\lambda)=\\{x\in\tilde{W};x\leq t^{w(\lambda)}\text{ for some
$w\in W_{0}$}\\}.$

Note that $\mathbb{J}_{b}$ acts on $X(\lambda,b)$ by left multiplication. By
[11], $X(\lambda,b)\neq\emptyset$ if and only if
$\kappa_{G}(t^{\lambda})=\kappa_{G}(b)$ and
$\nu_{G}(b)\leqslant\lambda^{\diamond}$, where $t^{\lambda}:=\lambda(t)$ and
$\lambda^{\diamond}$ is the $\sigma$-average of $\lambda$. We say the pair
$(\lambda,b)$ is Hodge-Newton irreducible if $X(\lambda,b)\neq\emptyset$ and
$\lambda^{\diamond}-\nu_{G}(b)\in\mathbb{R}_{>0}(\Phi^{+})^{\vee}$.

### 1.3.

Let $M\supseteq T$ be a (semi-standard) Levi subgroup of $G$. Then $B\cap M$
is a Borel subgroup of $M$. By replacing the triple $(T,B,G)$ with $(T,B\cap
M,M)$, we can define, as in previous subsections, $\Phi_{M}^{+}$,
$\tilde{W}_{M}$, $\SS_{M}^{a}$, $\Omega_{M}$, $\tilde{\Phi}_{M}^{+}$, $I_{M}$,
$\kappa_{M}$ and so on.

For $v\in V$ we set $\Phi_{v}=\\{\alpha\in\Phi;\alpha(v)=0\\}$ and let
$M_{v}\subseteq G$ be the Levi subgroup generated by $T$ and the root
subgroups $u_{\alpha}$ for $\alpha\in\Phi_{v}$. We set
$\tilde{W}_{v}=\tilde{W}_{M_{v}}$, $\tilde{\Phi}_{v}=\tilde{\Phi}_{M_{v}}$,
and so on. If $v$ is dominant, let $J_{v}=\\{s\in\SS_{0};s(v)=v\\}$.

Let $J\subseteq\SS_{0}$. Then there exists some $v^{\prime}\in V^{+}$ such
that $J_{v^{\prime}}=J$, and we put $\Phi_{J}=\Phi_{M_{v^{\prime}}}$,
$\tilde{W}_{J}=\tilde{W}_{M_{v^{\prime}}}$,
$W_{J}^{a}=W_{M_{v^{\prime}}^{a}}$, $\Omega_{J}=\Omega_{M_{v^{\prime}}}$, and
so on. We say $\mu\in Y$ is $J$-dominant (resp. $J$-minuscule) if
$\langle\alpha,\mu\rangle\geqslant 0$ (resp.
$\langle\alpha,\mu\rangle\in\\{0,\pm 1\\}$) for $\alpha\in\Phi_{J}^{+}$.

Let $K\subseteq\SS^{a}$. Let $W_{K}\in W^{a}$ be the parabolic subgroup
generated by $K$. Set
${}^{K}\tilde{W}=\\{{\tilde{w}}\in\tilde{W};{\tilde{w}}<s{\tilde{w}}\text{ for
}s\in K\\}$ and $\tilde{W}^{K}=({}^{K}\tilde{W})^{-1}$. For
${\tilde{w}}\in{}^{K}\tilde{W}$ we define
$I(K,{\tilde{w}})=\max\\{K^{\prime}\subseteq
K;{\tilde{w}}\sigma(K^{\prime}){\tilde{w}}^{-1}=K^{\prime}\\}$.

Let ${\tilde{w}},{\tilde{w}}^{\prime}\in\tilde{W}$ and $s\in\SS^{a}$. Write
${\tilde{w}}\to_{s}{\tilde{w}}^{\prime}$ if
${\tilde{w}}^{\prime}=s{\tilde{w}}\sigma(s)$ and
$\ell({\tilde{w}}^{\prime})\leqslant\ell({\tilde{w}})$. For
$K\subseteq\SS^{a}$ we write ${\tilde{w}}\to_{K}{\tilde{w}}^{\prime}$ if there
is a sequence
${\tilde{w}}={\tilde{w}}_{0}\to_{s_{0}}{\tilde{w}}_{1}\to_{s_{1}}\cdots\to_{s_{n}}{\tilde{w}}_{n+1}={\tilde{w}}^{\prime}$
with $s_{i}\in K$ for $0\leqslant i\leqslant n$.

######  Lemma 1.1.

Let $K\subseteq\SS^{a}$ and ${\tilde{w}}\in{}^{K}\tilde{W}$. Then we have

(1) if ${\tilde{w}}<{\tilde{w}}s$ with $s\in\SS^{a}$, then either
${\tilde{w}}s\in{}^{K}\tilde{W}$ or ${\tilde{w}}s=s^{\prime}{\tilde{w}}$ for
some $s^{\prime}\in K$;

(2) ${\tilde{w}}$ is the unique element of its $W_{K}$-$\sigma$-conjugacy
class which lies in ${}^{K}\tilde{W}$.

######  Theorem 1.2.

[9, §3] Let $K\subseteq\SS^{a}$ and ${\tilde{w}}\in\tilde{W}$. Then there
exist $x\in{}^{K}\tilde{W}$ and $u\in I(x,K)$ such that
${\tilde{w}}\to_{K}ux$. Moreover, such $x$ is uniquely determined by the
$W_{K}$-$\sigma$-conjugacy class of ${\tilde{w}}$.

### 1.4.

We say ${\tilde{w}}\in\tilde{W}$ is semi-standard if
${\tilde{w}}\sigma(\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+})=\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+}$,
that is,
${}^{{\tilde{w}}\sigma}I_{M_{\nu_{\tilde{w}}}}=I_{M_{\nu_{\tilde{w}}}}$. We
say ${\tilde{w}}$ is standard if it is semi-standard and $\nu_{{\tilde{w}}}$
is dominant. Let $\mathcal{S}$ and $\mathcal{S}^{+}$ denote the set of semi-
standard elements and standard elements respectively.

By abuse of notation, we will freely identify an element of $\tilde{W}$ with
its lift in $N_{T}(\mathcal{O}_{\breve{F}})$, according to the context.

######  Lemma 1.3.

Let ${\tilde{w}}\in\mathcal{S}$. Then we have

(1) $z{\tilde{w}}\sigma(z)^{-1}\in\mathcal{S}$ if $z\in\tilde{W}$ such that
$z(\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+})\subseteq\tilde{\Phi}^{+}$;

(2) there exists a unique pair
$({\tilde{w}}^{\prime},z^{\prime})\in\mathcal{S}^{+}\times
W_{0}^{J_{\bar{\nu}_{{\tilde{w}}}}}$ such that
${\tilde{w}}=z^{\prime}{\tilde{w}}^{\prime}\sigma(z^{\prime})^{-1}$;

(3) $s{\tilde{w}}\sigma(s)^{-1}\in\mathcal{S}$ if $s\in\SS^{a}$ and either
$s{\tilde{w}}<{\tilde{w}}$ or ${\tilde{w}}\sigma(s)<{\tilde{w}}$;

(4) $\mathbb{J}_{{\tilde{w}}}$ is generated by $I\cap\mathbb{J}_{{\tilde{w}}}$
and $\tilde{W}\cap\mathbb{J}_{{\tilde{w}}}$.

###### Proof.

By assumption we have
$z(\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+})=\tilde{\Phi}_{\nu_{z{\tilde{w}}\sigma(z)^{-1}}}^{+}$.
So (1) follows by definition.

Let $z^{\prime}\in W_{0}^{J_{\bar{\nu}_{{\tilde{w}}}}}$ such that
$z^{\prime}(\bar{\nu}_{{\tilde{w}}})=\nu_{{\tilde{w}}}$. Let
${\tilde{w}}^{\prime}={z^{\prime}}^{-1}{\tilde{w}}\sigma(z^{\prime})$. Note
that
$z^{\prime}(\tilde{\Phi}_{\bar{\nu}_{{\tilde{w}}}}^{+})=\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+}$.
So ${\tilde{w}}^{\prime}\in\mathcal{S}^{+}$ by (1). Suppose there exists
another pair
$({\tilde{w}}^{\prime\prime},z^{\prime\prime})\in\mathcal{S}^{+}\times
W_{0}^{J_{\bar{\nu}_{{\tilde{w}}}}}$ satisfies (2). Then
$\nu_{{\tilde{w}}^{\prime}}=\nu_{{\tilde{w}}^{\prime\prime}}=\bar{\nu}_{{\tilde{w}}}$
and ${z^{\prime}}^{-1}z^{\prime\prime}\in W_{J_{\bar{\nu}_{{\tilde{w}}}}}$.
Thus $z^{\prime}=z^{\prime\prime}\in W_{0}^{J_{\bar{\nu}_{{\tilde{w}}}}}$ and
hence ${\tilde{w}}^{\prime}={\tilde{w}}^{\prime\prime}$.

By (1), to prove (3) it suffices to show
$s(\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+})\subseteq\tilde{\Phi}^{+}$. Otherwise,
the simple affine root of $s$ lies in $\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+}$.
Hence $s{\tilde{w}},{\tilde{w}}\sigma(s)>{\tilde{w}}$ (since
${\tilde{w}}\sigma(\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+})=\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{+}$),
contradicting our assumption.

Note that $\mathbb{J}_{\tilde{w}}\subseteq M_{\nu_{{\tilde{w}}}}$. Thus (4)
follows from that
${}^{{\tilde{w}}\sigma}I_{M_{\nu_{\tilde{w}}}}=I_{M_{\nu_{\tilde{w}}}}$,
${}^{{\tilde{w}}\sigma}\tilde{W}_{M_{\nu_{\tilde{w}}}}=\tilde{W}_{M_{\nu_{\tilde{w}}}}$,
and the Bruhat decomposition
$M_{\nu_{\tilde{w}}}({\breve{F}})=I_{M_{\nu_{\tilde{w}}}}\tilde{W}_{M_{\nu_{\tilde{w}}}}I_{M_{\nu_{\tilde{w}}}}$.
∎

##  2\. Proof of Theorem 0.2

We fix $\lambda\in Y^{+}$ and $b\in G({\breve{F}})$ such that
$X(\lambda,b)\neq\emptyset$. Let $J=J_{\nu_{G}(b)}\subseteq\SS_{0}$. Moreover,
we may and do assume that $b\in M_{J}({\breve{F}})$ and
$\nu_{M_{J}}(b)=\nu_{G}(b)$. In particular,
$\mathbb{J}_{b}=\mathbb{J}_{b}^{M_{J}}$.

For $x\in\pi_{1}(M_{J})\cong Y/\mathbb{Z}\Phi_{J}^{\vee}$ we set
${\tilde{w}}_{x}=t^{\mu_{x}}w_{x}\in\Omega_{M_{J}}$ with $\mu_{x}\in Y$ and
$w_{x}\in W_{J}$ such that $\eta_{M_{J}}({\tilde{w}}_{x})=x$. Here
$\eta_{M_{J}}:M_{J}({\breve{F}})\to\pi_{1}(M_{J})$ denotes the natural
projection. Define

$\displaystyle\mathcal{S}_{\lambda,b}^{+}$
$\displaystyle=\\{x\in\pi_{1}(M_{J});\kappa_{M_{J}}(x)=\kappa_{M_{J}}(b),\mu_{x}\preceq\lambda\\},$
$\displaystyle\mathcal{S}_{\lambda,b,x}$
$\displaystyle=\\{z{\tilde{w}}_{x}\sigma(z)^{-1}\in{\rm{Adm}}(\lambda);x\in\mathcal{S}_{\lambda,b}^{+},z\in
W_{0}^{J}\\}.$

Let $\mathcal{S}_{\lambda,b}$ be the set of semi-standard elements in
${\rm{Adm}}(\lambda)$ which are $\sigma$-conjugate to $b$. Then we have
$\mathcal{S}_{\lambda,b}=\sqcup_{x\in\mathcal{S}_{\lambda,b}^{+}}\mathcal{S}_{\lambda,b,x}$
by Lemma 1.3.

######  Theorem 2.1 ([18]).

Each connected component of $X(\lambda,b)$ intersects
$\mathbb{J}_{b,{\tilde{w}}}I/I$ for some
${\tilde{w}}\in\mathcal{S}_{\lambda,b}$.

###### Proof.

By [18, Theorem 4.1], each connected component of $X(\lambda,b)$ intersects
$\mathbb{J}_{b,{\tilde{w}}}$ for some $\sigma$-straight element
${\tilde{w}}\in{\rm{Adm}}(\lambda)$ which is $\sigma$-conjugate to $b$. Then
the statement follows from the proof of [29, Theorm 1.3], which shows that
$\sigma$-straight elements are semi-standard. ∎

For $g,g^{\prime}\in G({\breve{F}})$ we write $gI\sim_{\lambda,b}g^{\prime}I$
if they are in the same connected component of $X(\lambda,b)$. For
${\tilde{w}},{\tilde{w}}^{\prime}\in\mathcal{S}_{\lambda,b}$, we write
$\mathbb{J}_{b,{\tilde{w}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}^{\prime}}$
if their natural images in $\pi_{0}(X(\lambda,b))$ coincide.

######  Proposition 2.2.

For $x\in\mathcal{S}_{\lambda,b}^{+}$ and
${\tilde{w}},{\tilde{w}}^{\prime}\in\mathcal{S}_{\lambda,b,x}$ we have
$\mathbb{J}_{b,{\tilde{w}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}^{\prime}}$.

In the following four propositions, we retain the assumptions in Theorem 0.2.
The proofs are given in the remaining sections.

######  Proposition 2.3.

For $x,x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ we have
$\mathbb{J}_{b,{\tilde{w}}_{x}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.
As a consequence, the natural projection
$\mathbb{J}_{b,{\tilde{w}}}\to\pi_{0}(X(\lambda,b))$ is surjective for
${\tilde{w}}\in\mathcal{S}_{\lambda,b}$.

######  Proposition 2.4.

The natural action of $\ker(\eta_{M_{J}})\cap\mathbb{J}_{b}$ on
$\pi_{0}(X(\lambda,b))$ is trivial.

######  Proposition 2.5.

The natural action of

$(\ker(\eta_{G})\cap\mathbb{J}_{b})/(\ker(\eta_{M_{J}})\cap\mathbb{J}_{b})\cong(\mathbb{Z}\Phi^{\vee}/\mathbb{Z}\Phi_{J}^{\vee})^{\sigma}$

on $\pi_{0}(X(\lambda,b))$ is trivial.

###### Proof of Theorem 0.2.

By [11, Theorem 1.1], it suffices to consider the Iwahori case $K=I$.
Combining Proposition 2.3, 2.4 and & 2.5 we have
$\pi_{0}(X(\lambda,b))\cong\mathbb{J}_{b}/(\mathbb{J}_{b}\cap\ker(\eta_{G}))$.
As $b$ is basic in $M_{J}$ and $\mathbb{J}_{b}=\mathbb{J}_{b}^{M_{J}}$, it
follows that $\mathbb{J}_{b}$ is generated by $I_{M_{J}}\cap\mathbb{J}_{b}$
and $\tilde{W}_{M_{J}}\cap\mathbb{J}_{b}$. So we have
$\mathbb{J}_{b}/(\ker(\eta_{G})\cap\mathbb{J}_{b})=(\Omega_{J}\cap\mathbb{J}_{b})/(\Omega_{J}\cap\ker{\eta_{G}}\cap\mathbb{J}_{b})=\Omega_{J}^{\sigma}/(\Omega_{J}^{\sigma}\cap\ker(\eta_{G}))\cong\pi_{1}(G)^{\sigma}$
as desired. ∎

##  3\. The set $\mathcal{P}_{\tilde{w}}$

In the rest of the paper, we assume that $G$ is adjoint, simple, and its root
system $\Phi$ has $d$ irreducible factors.

### 3.1.

Let ${\tilde{w}}\in{\rm{Adm}}(\lambda)$. For
$\alpha\in\Phi^{+}\setminus\Phi_{\nu_{\tilde{w}}}$ we set
$\alpha^{i}=({\tilde{w}}\sigma)^{i}(\alpha)\in\tilde{\Phi}$ for
$i\in\mathbb{Z}$, and

$m_{\alpha,{\tilde{w}}}=\min\\{i\in\mathbb{Z}_{\geqslant
1};\alpha^{-i}\in\tilde{\Phi}\setminus\Phi\\},$

which is well defined since $\langle\alpha,\nu_{\tilde{w}}\rangle\neq 0$.

We say $\alpha\in\Phi^{+}\setminus\Phi_{\nu_{\tilde{w}}}$ is
${\tilde{w}}$-permissible if
${\tilde{w}}\sigma(s_{\alpha})\in{\rm{Adm}}(\lambda)$ and
$\alpha^{-m_{\alpha,{\tilde{w}}}}\in\tilde{\Phi}^{+}$. Let
$\mathcal{P}_{{\tilde{w}}}$ denote the set of ${\tilde{w}}$-permissible roots.

Let $R\subseteq\SS_{0}$ be a $\sigma$-orbit of $\SS_{0}$. We say
${\tilde{w}}\in{\rm{Adm}}(\lambda)$ is left (resp. right) $R$-distinct if
$s{\tilde{w}}\notin{\rm{Adm}}(\lambda)$ (resp.
${\tilde{w}}s\notin{\rm{Adm}}(\lambda)$) for $s\in R$. Let $w_{R}$ denote the
longest root of $W_{R}$.

######  Lemma 3.1.

Let $R$ be a $\sigma$-orbit of $\SS_{0}$, and let
${\tilde{w}}\in{\rm{Adm}}(\lambda)$ be left $R$-distinct. Then
$w_{R}{\tilde{w}}w_{R}\in{\rm{Adm}}(\lambda)$ is right $R$-distinct. Moreover,
$\mathcal{P}_{w_{R}{\tilde{w}}w_{R}}\neq\emptyset$ if
$\mathcal{P}_{\tilde{w}}\neq\emptyset$.

###### Proof.

To show the first statement, we can assume $d=1$, and hence $R$ is either
commutative or is of type $A_{2}$. Then the first statement follows from Lemma
A.4.

Now we show the “Moreover” part. Let $\alpha\in\mathcal{P}_{\tilde{w}}$ and
let

$n_{\alpha}=\min\\{i\in\mathbb{Z}_{\geqslant
0};\alpha^{-i}\notin\Phi_{R}^{+}\\}\leqslant m_{\alpha,{\tilde{w}}}.$

We show $w_{R}(\alpha^{-n_{\alpha}})\in\mathcal{P}_{w_{R}{\tilde{w}}w_{R}}$,
and it suffices to check that $\alpha^{-n_{\alpha}}\in\Phi^{+}$ and

$\displaystyle
w_{R}{\tilde{w}}w_{R}\sigma(s_{w_{R}(\alpha^{-n_{\alpha}})})=w_{R}{\tilde{w}}\sigma(s_{\alpha^{-n_{\alpha}}})w_{R}\in{\rm{Adm}}(\lambda).$
(a)

If $n_{\alpha}=0$, then
$\alpha^{-n_{\alpha}}=\alpha\in\Phi^{+}\setminus\Phi_{R}$, and (a) follows
from Corollary A.6. Otherwise, $\alpha^{-n_{\alpha}+1}\in\Phi_{R}^{+}$, which
implies that
$\alpha^{-n_{\alpha}}=({\tilde{w}}\sigma)^{-1}(\alpha^{-n_{\alpha}+1})\in\tilde{\Phi}^{-}$
as ${\tilde{w}}$ is left $R$-distinct. Hence $\alpha^{-n_{\alpha}}\in\Phi^{+}$
since $n_{\alpha}\leqslant m_{\alpha,{\tilde{w}}}$ and
$\alpha^{-m_{\alpha,{\tilde{w}}}}\in\tilde{\Phi}^{+}$. Moreover,

$w_{R}{\tilde{w}}w_{R}\sigma(w_{R}(\alpha^{-n_{\alpha}}))=w_{R}(\alpha^{-n_{\alpha}+1})\in\Phi_{R}^{-},$

which means $w_{R}{\tilde{w}}w_{R}\sigma(s_{w_{R}(\alpha^{-n_{\alpha}})})\leq
w_{R}{\tilde{w}}w_{R}\in{\rm{Adm}}(\lambda)$, and (a) follows. ∎

### 3.2.

Fix $\eta\in Y$. Let $A=\max\\{|\langle\alpha,\eta\rangle|;\alpha\in\Phi\\}$.
Choose $M\in\mathbb{Z}_{\geqslant 2}$ such that
$M|\langle\alpha,\eta\rangle|>2A$ for any $\alpha\in\Phi$ with
$\langle\alpha,\eta\rangle\neq 0$. Let ${\tilde{w}}\in t^{\mu}W_{0}\subseteq
W_{0}t^{\eta}W_{0}$, where $\mu$ is some $W_{0}$-conjugate of $\eta$.
Motivated from the $\underline{\mathbf{a}}$-function in [10], we define

$\nu_{{\tilde{w}}}^{\flat}=\sum_{i=0}^{N-1}\frac{p({\tilde{w}}\sigma)^{i}(\mu)}{M^{i}},$

where $N$ is the order of $p({\tilde{w}}\sigma)$, and
$p:\tilde{W}\rtimes\langle\sigma\rangle\to W_{0}\rtimes\langle\sigma\rangle$
is the natural projection.

######  Lemma 3.2.

Let $\alpha\in\Phi$ and $0\leqslant n\leqslant N-1$ such that
$\langle\alpha,p({\tilde{w}}\sigma)^{n}(\mu)\rangle\neq 0$ and
$\langle\alpha,p({\tilde{w}}\sigma)^{i}(\mu)\rangle=0$ for $0\leqslant
i\leqslant n-1$. Then
$\langle\alpha,\nu_{{\tilde{w}}}^{\flat}\rangle\langle\alpha,p({\tilde{w}}\sigma)^{n}(\mu)\rangle>0$.

###### Proof.

Note that $\mu,\eta$ are conjugate by $W_{0}$. By the choice of $M\geqslant 2$
we have

$|\frac{\langle\alpha,p({\tilde{w}}\sigma)^{n}(\mu)\rangle}{M^{n}}|>\frac{2A}{M^{n+1}}>\frac{A}{M^{n+1}}\sum_{i=n+1}^{N-1}\frac{1}{M^{i-n-1}}\geqslant\sum_{i=n+1}^{N-1}|\frac{\langle\alpha,p({\tilde{w}}\sigma)^{i}(\mu)\rangle}{M^{i}}|.$

So the statement follows. ∎

######  Corollary 3.3.

We have the following:

(1) $\langle\alpha,\nu_{{\tilde{w}}}^{\flat}\rangle=0$ if and only if
$\langle\alpha,p({\tilde{w}}\sigma)^{i}(\mu)\rangle=0$ for $i\in\mathbb{Z}$;

(2) $\nu_{{\tilde{w}}}^{\flat}$ is dominant for $\Phi_{\nu_{{\tilde{w}}}}^{+}$
if ${\tilde{w}}\in\mathcal{S}$;

(3) $\nu_{z{\tilde{w}}\sigma(z)^{-1}}^{\flat}=z(\nu_{{\tilde{w}}}^{\flat})$
for $z\in W_{0}$;

(4)
${\tilde{w}}\sigma(\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}}^{\pm})=\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}}^{\pm}$
if ${\tilde{w}}\in\mathcal{S}$;

(5) if $\alpha\in\mathcal{P}_{\tilde{w}}$, then
$\langle\alpha^{i},\nu_{\tilde{w}}^{\flat}\rangle<0$ for
$1-m_{\alpha,{\tilde{w}}}\leqslant i\leqslant 0$, and the roots
$\alpha^{i}\in\Phi$ for $1-m_{\alpha,{\tilde{w}}}\leqslant i\leqslant 0$ for
are linearly independent. Here $\alpha^{i}=({\tilde{w}}\sigma)^{i}(\alpha)$.

###### Proof.

The statement (1) follows from Lemma 3.2 and the definition of
$\nu_{{\tilde{w}}}^{\flat}$.

Suppose there exists $\alpha\in\Phi_{\nu_{{\tilde{w}}}}^{+}$ such that
$\langle\alpha,\nu_{{\tilde{w}}}^{\flat}\rangle<0$. By Lemma 3.2, there exists
$n\in\mathbb{Z}_{\geqslant 0}$ such that
$\langle\alpha,p({\tilde{w}}\sigma)^{n}(\mu)\rangle<0$ and
$\langle\alpha,p({\tilde{w}}\sigma)^{i}(\mu)\rangle=0$ for $0\leqslant
i\leqslant n-1$. In particular, we have
$({\tilde{w}}\sigma)^{-i}(\alpha)=p({\tilde{w}}\sigma)^{-i}(\alpha)$ for
$1\leqslant i\leqslant n$ and
$({\tilde{w}}\sigma)^{-n-1}(\alpha)\in\tilde{\Phi}^{+}\setminus\Phi$,
contradicting that ${\tilde{w}}\in\mathcal{S}$. So (2) follows.

The statement (3) follows by definition.

By (1) we have
$\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}}={\tilde{w}}\sigma(\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}})\subseteq\tilde{\Phi}_{\nu_{{\tilde{w}}}}$.
As ${\tilde{w}}\in\mathcal{S}$, we have
${\tilde{w}}\sigma(\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{\pm})=\tilde{\Phi}_{\nu_{{\tilde{w}}}}^{\pm}$.
So (4) follows from that
$\tilde{\Phi}_{\nu_{\tilde{w}}^{\flat}}^{\pm}=\tilde{\Phi}_{\nu_{\tilde{w}}^{\flat}}\cap\tilde{\Phi}_{\nu_{\tilde{w}}}^{\pm}$.

Let $\alpha\in\mathcal{P}_{\tilde{w}}$ and let $m=m_{\alpha,{\tilde{w}}}$. By
definition, $\langle\alpha^{1-m},\mu\rangle<0$,
$\alpha^{-i}=p({\tilde{w}}\sigma)^{-i}(\alpha)$ and
$\langle\alpha^{1-i},\mu\rangle=\langle\alpha,p({\tilde{w}}\sigma)^{i-1}(\mu)\rangle=0$
for $1\leqslant i\leqslant m-1$. Thus it follows from Lemma 3.2 that
$\langle\alpha^{i},\nu_{\tilde{w}}^{\flat}\rangle<0$ for $1-m\leqslant
i\leqslant 0$. Suppose $\sum_{i=0}^{1-m}c_{i}\alpha^{i}=0$, where the
coefficients $c_{i}\in\mathbb{R}$ are not all zero. Let
$i_{0}=\min\\{1-m\leqslant i\leqslant 0;c_{i}\neq 0\\}$. Then

$0=\langle
p({\tilde{w}}\sigma)^{1-m-i_{0}}(\sum_{i=0}^{1-m}c_{i}\alpha^{i}),\mu\rangle=\sum_{i=0}^{i_{0}}c_{i}\langle\alpha^{1-m-i_{0}+i},\mu\rangle=c_{i_{0}}\langle\alpha^{1-m},\mu\rangle\neq
0,$

which is a contradiction. So (5) follows. ∎

######  Lemma 3.4.

Let ${\tilde{w}}\in\mathcal{S}$ and let $z_{0}\in W_{0}$ be the unique minimal
element such that $z_{0}(\nu_{{\tilde{w}}}^{\flat})$ is dominant. Then
$z_{0}{\tilde{w}}\sigma(z_{0})^{-1}\in{}^{\SS_{0}}\tilde{W}$. In particular,
${\tilde{w}}\in{}^{\SS_{0}}\tilde{W}$ if $\nu_{{\tilde{w}}}^{\flat}$ is
dominant.

###### Proof.

Let ${\tilde{w}}^{\prime}=z_{0}{\tilde{w}}\sigma(z_{0})^{-1}\in
t^{\mu^{\prime}}W_{0}$ with $\mu^{\prime}=z_{0}(\mu)$. By the choice of
$z_{0}$, we have
$\nu_{{\tilde{w}}^{\prime}}^{\flat}=z_{0}(\nu_{{\tilde{w}}}^{\flat})$ is
dominant and
$z_{0}(\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}}^{\pm})=\tilde{\Phi}_{\nu_{{\tilde{w}}^{\prime}}^{\flat}}^{\pm}$.
By Lemma 3.2, $\mu^{\prime}$ is dominant since
$\nu_{{\tilde{w}}^{\prime}}^{\flat}=z_{0}(\nu_{{\tilde{w}}}^{\flat})$ is
dominant. Let $\gamma\in\Phi^{+}$. We need to show that
${\tilde{w}}^{\prime}<s_{\gamma}{\tilde{w}}^{\prime}$. If
$\langle\gamma,\nu_{{\tilde{w}}^{\prime}}^{\flat}\rangle>0$, then either
$\langle\gamma,\mu^{\prime}\rangle>0$, or
$\langle\gamma,\mu^{\prime}\rangle=0$ and $\langle
p({\tilde{w}}^{\prime}\sigma)^{-1}(\gamma),\nu_{{\tilde{w}}^{\prime}}^{\flat}\rangle>0$
(hence $p({\tilde{w}}^{\prime}\sigma)^{-1}(\gamma)\in\Phi^{+}$) by Corollary
3.3, which means ${\tilde{w}}^{\prime}<s_{\gamma}{\tilde{w}}^{\prime}$ as
desired. Suppose $\langle\gamma,\nu_{{\tilde{w}}^{\prime}}^{\flat}\rangle=0$,
that is, $\gamma\in\Phi_{\nu_{{\tilde{w}}^{\prime}}^{\flat}}^{+}$. Then by
Corollary 3.3 (4) we have

$({\tilde{w}}^{\prime}\sigma)^{-1}(\gamma)\in
z_{0}({\tilde{w}}\sigma)^{-1}z_{0}^{-1}(\tilde{\Phi}_{\nu_{{\tilde{w}}^{\prime}}^{\flat}}^{-})=z_{0}({\tilde{w}}\sigma)^{-1}(\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}}^{-})=z_{0}(\tilde{\Phi}_{\nu_{{\tilde{w}}}^{\flat}}^{-})=\tilde{\Phi}_{\nu_{{\tilde{w}}^{\prime}}^{\flat}}^{-}.$

So we also have ${\tilde{w}}^{\prime}<s_{\gamma}{\tilde{w}}^{\prime}$ as
desired. ∎

######  Proposition 3.5.

Let $K\subseteq\SS_{0}$ and ${\tilde{w}}\in\mathcal{S}_{\lambda,b}$. Then
there exists a unique semi-standard element
${\tilde{w}}^{\prime}\in{}^{K}\tilde{W}$ which is $\sigma$-conjugate to
${\tilde{w}}$ by $W_{K}$. If, moreover, $K=\SS_{0}$ and $(\lambda,b)$ is
Hodge-Newton irreducible, then ${\tilde{w}}^{\prime}$ is not left $R$-distinct
for any $\sigma$-orbit $R$ of $\SS_{0}$.

###### Proof.

By Theorem 1.2, there exist unique ${\tilde{w}}^{\prime}\in{}^{K}\tilde{W}$
and some $u\in I(K,{\tilde{w}}^{\prime})$ such that
${\tilde{w}}\to_{K}u{\tilde{w}}^{\prime}$. So
$\Phi_{I(K,{\tilde{w}}^{\prime})}\subseteq\Phi_{\nu_{{\tilde{w}}^{\prime}}}$
and $\ell(u{\tilde{w}}^{\prime})=\ell(u)+\ell({\tilde{w}}^{\prime})$. As
${\tilde{w}}\in\mathcal{S}$, we have $u{\tilde{w}}^{\prime}\in\mathcal{S}$ by
Lemma 1.3 (2). So $u{\tilde{w}}^{\prime}\leq
u^{-1}u{\tilde{w}}^{\prime}={\tilde{w}}^{\prime}$, which means $u=1$, and the
first statement follows. The second one is proved in [3, Lemma 6.11]. ∎

### 3.3.

Let ${\tilde{w}},{\tilde{w}}^{\prime}\in\tilde{W}$ and $s\in\SS^{a}$. Write
${\tilde{w}}\rightharpoonup_{s}{\tilde{w}}^{\prime}$ if
${\tilde{w}}^{\prime}=s{\tilde{w}}\sigma(s)$ and $s{\tilde{w}}<{\tilde{w}}$
(which implies that $\ell({\tilde{w}}^{\prime})\leqslant\ell({\tilde{w}})$).
For $K\subseteq\SS^{a}$ we write
${\tilde{w}}\rightharpoonup_{K}{\tilde{w}}^{\prime}$ if there is a sequence
${\tilde{w}}={\tilde{w}}_{0}\rightharpoonup_{s_{0}}{\tilde{w}}_{1}\rightharpoonup_{s_{1}}\cdots\rightharpoonup_{s_{n}}{\tilde{w}}_{n+1}={\tilde{w}}^{\prime}$
with $s_{i}\in K$ for $0\leqslant i\leqslant n$.

######  Lemma 3.6.

Let $K\subseteq\SS_{0}$ and ${\tilde{w}}\in\mathcal{S}$. There is no infinite
sequence

${\tilde{w}}={\tilde{w}}_{0}\rightharpoonup_{s_{0}}{\tilde{w}}_{1}\rightharpoonup_{s_{1}}\cdots,$

where $s_{i}\in K$ for $i\in\mathbb{Z}_{\geqslant 0}$. As a consequence, we
have ${\tilde{w}}\rightharpoonup_{K}{\tilde{w}}^{\prime}$ for some
${\tilde{w}}^{\prime}\in{}^{K}\tilde{W}$.

###### Proof.

We argue by induction on $|K|$. If $K=\emptyset$, the statement is trivial.
Assume $|K|\geqslant 1$. Suppose there exists such an infinite sequence. By
Lemma 1.3 we have ${\tilde{w}}_{i}\in\mathcal{S}$ for
$i\in\mathbb{Z}_{\geqslant 0}$. Moreover, by Lemma 1.1 (1) we can assume that
(a) $\ell({\tilde{w}}_{0})=\ell({\tilde{w}}_{1})=\cdots$, and (b) there exists
$y\in{}^{K}\tilde{W}$ such that ${\tilde{w}}_{i}\in W_{K}y$ and
$y\sigma(s_{i})y^{-1}\in K$ for $i\in\mathbb{Z}_{\geqslant 0}$. If each
element of $K$ appears infinitely many times in the sequence
$s_{0},s_{1},\dots$, then by (b) we have $K=I(K,y)\subseteq
W_{\nu_{\tilde{w}}}$, see §1.3. So ${\tilde{w}}_{i}=y\in{}^{K}\tilde{W}$ since
${\tilde{w}}_{i}\in\mathcal{S}$, which is impossible. Otherwise, there exists
a proper subset $K^{\prime}\subsetneq K$ such that $s_{i}\in K^{\prime}$ for
$i\gg 0$, which contradicts the induction hypothesis. The proof is finished. ∎

Let $R$ be a $\sigma$-orbit of $\SS_{0}$. For
${\tilde{w}},{\tilde{w}}^{\prime}\in\tilde{W}$ we write
${\tilde{w}}\Rightarrow_{R}{\tilde{w}}^{\prime}$ if
${\tilde{w}},{\tilde{w}}^{\prime}\in\mathcal{S}$ are
$W_{R}$-$\sigma$-conjugate and ${\tilde{w}}^{\prime}\in{}^{R}\tilde{W}$. Write
${\tilde{w}}\Rightarrow{\tilde{w}}^{\prime}$ if there is a sequence
${\tilde{w}}={\tilde{w}}_{0}\Rightarrow_{R_{0}}{\tilde{w}}_{1}\Rightarrow_{R_{1}}\cdots\Rightarrow_{R_{n}}{\tilde{w}}_{n+1}={\tilde{w}}^{\prime}$.

######  Proposition 3.7 ([3, Proposition 6.16]).

Let ${\tilde{w}}\in\mathcal{S}$. Then
${\tilde{w}}\Rightarrow{\tilde{w}}^{\prime}$, where
${\tilde{w}}^{\prime}\in{}^{\SS_{0}}\tilde{W}$ is the unique element in the
$W_{0}$-$\sigma$-conjugacy class of ${\tilde{w}}$.

###### Proof.

Assume otherwise. Then by Lemma 3.6 there is an infinite sequence

${\tilde{w}}={\tilde{w}}_{0}\rightharpoonup_{R_{0}}{\tilde{w}}_{1}\rightharpoonup_{R_{1}}\cdots,$

where ${\tilde{w}}_{i+1}\in{}^{R_{i}}\tilde{W}$ and $R_{i}$ is some
$\sigma$-orbit of $\SS_{0}$ for $i\in\mathbb{Z}_{\geqslant 0}$. This
contradicts Lemma 3.6. So the statement follows. ∎

######  Lemma 3.8.

Let $R$ be a $\sigma$-orbit of $\SS_{0}$. Let
${\tilde{w}}\in{\rm{Adm}}(\lambda)\cap\mathcal{S}$. If
${\tilde{w}}\notin{}^{R}\tilde{W}$ and ${\tilde{w}}$ is not right
$R$-distinct. Then $\mathcal{P}_{{\tilde{w}}}\neq\emptyset$.

###### Proof.

By assumption, there exists $s\in R$ and $0\leqslant k\leqslant|R|-1$ such
that $\sigma^{-k}(s){\tilde{w}}<{\tilde{w}}$,
${\tilde{w}}\sigma(s)\in{\rm{Adm}}(\lambda)$, and

$k=\min\\{0\leqslant
i\leqslant|R|-1;\sigma^{-i}(s^{\prime}){\tilde{w}}<{\tilde{w}},{\tilde{w}}\sigma(s^{\prime})\in{\rm{Adm}}(\lambda)\text{
for some }s^{\prime}\in R\\}.$

Let $\alpha\in\Phi^{+}$ be the simple root of $s$. We claim that

$\displaystyle\alpha^{-i}=\sigma^{-i}(\alpha)\text{ for }0\leqslant i\leqslant
k,\text{ and hence }m_{\alpha,{\tilde{w}}}\geqslant k+1.$ (a)

Let $0\leqslant i\leqslant k-1$. By the choice of $k$ we have
${\tilde{w}}<\sigma^{-i}(s){\tilde{w}}$ and
${\tilde{w}}\sigma^{-i}(s)\notin{\rm{Adm}}(\lambda)$, which means
$\sigma^{-i}(\alpha)={\tilde{w}}\sigma^{-i}(\alpha)$ by Lemma A.2. So (a) is
proved.

By (a) we have $\alpha^{-k}\in\Phi^{+}$. So
$\alpha^{-k-1}=({\tilde{w}}\sigma)^{-1}(\alpha^{-k})\in\tilde{\Phi}^{+}$ since
$\sigma^{-k}(s){\tilde{w}}<{\tilde{w}}$. As ${\tilde{w}}\in\mathcal{S}$, it
follows that $\alpha^{-k}\notin\Phi_{\nu_{\tilde{w}}}$ and hence
$\alpha^{i}\notin\Phi_{\nu_{\tilde{w}}}$ for $i\in\mathbb{Z}$. If
$\alpha\notin\mathcal{P}_{\tilde{w}}$, we have
$\alpha^{-m_{\alpha,{\tilde{w}}}}\in\tilde{\Phi}^{-}\setminus\Phi$ by
definition, which means $\alpha^{-k-1}\in\tilde{\Phi}^{+}\cap\Phi=\Phi^{-}$.
Let $\beta=-\alpha^{-k-1}\in\Phi^{+}\setminus\Phi_{\nu_{\tilde{w}}}$. Then
$\beta^{-m_{\beta,{\tilde{w}}}}=-\alpha^{-m_{\alpha,{\tilde{w}}}}\in\tilde{\Phi}^{+}\setminus\Phi$,
and ${\tilde{w}}\sigma(s_{\beta})<{\tilde{w}}\in{\rm{Adm}}(\lambda)$ since
${\tilde{w}}\sigma(\beta)=-\alpha^{-k}\in\Phi^{-}$. So
$\beta\in\mathcal{P}_{\tilde{w}}$ as desired. ∎

######  Corollary 3.9.

Assume $(\lambda,b)$ is Hodge-Newton irreducible. For
${\tilde{w}}\in\mathcal{S}_{\lambda,b}$ we have either
${\tilde{w}}\in{}^{\SS_{0}}\tilde{W}$ or
$\mathcal{P}_{\tilde{w}}\neq\emptyset$.

###### Proof.

By Proposition 3.7, there exists a sequence

${\tilde{w}}={\tilde{w}}_{0}\Rightarrow_{R_{0}}{\tilde{w}}_{1}\Rightarrow_{R_{1}}\cdots\Rightarrow_{R_{n-1}}{\tilde{w}}_{n}={\tilde{w}}^{\prime},$

where ${\tilde{w}}_{0},\dots,{\tilde{w}}_{n+1}$ are distinct semi-standard
elements, $R_{0},\dots,R_{n}$ are $\sigma$-orbits of $\SS_{0}$, and
${\tilde{w}}^{\prime}\in{}^{\SS_{0}}\tilde{W}$. We argue by induction on $n$.
If $n=0$, then ${\tilde{w}}\in{}^{\SS_{0}}\tilde{W}$ as desired. Assume
$n\geqslant 1$. If ${\tilde{w}}={\tilde{w}}_{0}$ is not right
$R_{0}$-distinct, then $\mathcal{P}_{{\tilde{w}}}\neq\emptyset$ by Lemma 3.8.
Otherwise, by Lemma A.4, $w_{R_{0}}{\tilde{w}}w_{R_{0}}\in{\rm{Adm}}(\lambda)$
is left $R_{0}$-distinct. So
$w_{R_{0}}{\tilde{w}}w_{R_{0}}={\tilde{w}}_{1}\in{}^{R_{0}}\tilde{W}$ by Lemma
1.1 (2). Moreover, ${\tilde{w}}_{1}\notin{}^{\SS_{0}}\tilde{W}$ by Proposition
3.5. By induction hypothesis, $\mathcal{P}_{{\tilde{w}}_{1}}\neq\emptyset$,
which implies $\mathcal{P}_{\tilde{w}}\neq\emptyset$ by Lemma 3.1. ∎

##  4\. Proof of Proposition 2.2

Assume $(\lambda,b)$ is Hodge-Newton irreducible. Recall that $d$ is the
number of connected components of $\SS_{0}$. For $g\in G({\breve{F}})$,
$\tilde{\gamma}\in\tilde{\Phi}$, ${\tilde{w}}\in\tilde{W}$, and
$m\in\mathbb{Z}_{\geqslant 0}$, we define

$\textsl{g}_{g,\tilde{\gamma},{\tilde{w}},m}:\mathbb{P}^{1}\to
G({\breve{F}})/I,\ z\mapsto
g{}^{({\tilde{w}}\sigma)^{1-m}}U_{\tilde{\gamma}}(z)\cdots{}^{({\tilde{w}}\sigma)^{-1}}U_{\tilde{\gamma}}(z)U_{\tilde{\gamma}}(z)I.$

######  Hypothesis 4.1.

Recall that $\mathbb{F}_{q}$ is the residue field of $F$. Assume that
$q^{d}>2$ (resp. $q^{d}>3$) if some/any connected component of $\SS_{0}$ is
non-simply-laced except of type $G_{2}$ (resp. is of type $G_{2}$).

Note that if Hypothesis 4.1 is not true, then $d=1$ and $\SS_{0}$ is non-
simply-laced, which implies that $G$ is residually split, and hence split
(since $G$ is unramified).

######  Lemma 4.1.

Suppose Hypothesis 4.1 is true. Let ${\tilde{w}}\in\tilde{W}$,
$\gamma\in\Phi$, and $m\in\mathbb{Z}_{\geqslant 0}$ such that the roots
$\gamma^{i}:=({\tilde{w}}\sigma)^{i}(\gamma)\in\Phi$ for $1-m\leqslant
i\leqslant 0$ are linearly independent. Let
$\textsl{g}=\textsl{g}_{1,\gamma,{\tilde{w}},m}$. Then there exist integers
$1-m\leqslant i_{r}<\cdots<i_{0}\leqslant 0$ such that

$\textsl{g}(\infty)=s_{\gamma^{i_{r}}}\cdots s_{\gamma^{i_{0}}}I,\text{ and
}s_{\gamma^{i_{0}}}\cdots
s_{\gamma^{i_{k-1}}}(\gamma^{i_{k}})\in\Phi^{+}\text{ for }0\leqslant
k\leqslant r.$

Moreover, if there exists $v\in V$ such that $\langle\gamma^{i},v\rangle<0$
for $1-m\leqslant i\leqslant 0$, then $v\leqslant(s_{\gamma^{i_{r}}}\cdots
s_{\gamma^{i_{0}}})^{-1}(v)$, where the equality holds if and only if $r<0$,
that is, $\gamma^{i}\in\Phi^{-}$ for $1-m\leqslant i\leqslant 0$.

###### Proof.

First notice that

$\displaystyle{}^{({\tilde{w}}\sigma)^{i}}U_{\gamma}(z)=U_{\gamma^{i}}(c_{i}z^{q^{i}})\text{
with }c_{i}\in\mathcal{O}_{\breve{F}}^{\times}\text{ for }1-m\leqslant
i\leqslant 0.$ (a)

We argue by induction on $m$. If $m=0$, the statement is trivial. Assume
$m\geqslant 1$. If $\gamma\in\Phi^{-}$, then
$\textsl{g}(\infty)=\textsl{g}_{1,\gamma^{-1},{\tilde{w}},m}(\infty)$, and it
follows by induction hypothesis. Otherwise, we have

$\textsl{g}(z)={}^{({\tilde{w}}\sigma)^{1-m}}U_{\gamma}(z)\cdots{}^{({\tilde{w}}\sigma)^{-1}}U_{\gamma}(z)U_{-\gamma}(z^{-1})s_{\gamma}I\text{
for }z\neq 0.$

As the roots $\gamma^{i}$ for $1-m\leqslant i\leqslant 0$ are linearly
independent, it follows by (a) and induction on $m$ that

${}^{{}^{({\tilde{w}}\sigma)^{1-m}}U_{\gamma}(z)\cdots{}^{({\tilde{w}}\sigma)^{-1}}U_{\gamma}(z)}U_{-\gamma}(z^{-1})=\prod_{(\beta,a_{\bullet})}U_{\beta}(c_{a_{\bullet}}z^{n_{a_{\bullet}}}),$

where $a_{\bullet}=(a_{i})_{0\leqslant i\leqslant
m-1}\in(\mathbb{Z}_{\geqslant 0})^{m}$ such that $a_{0}\geqslant 1$ and
$a_{i}=0$ unless $i\in d\mathbb{Z}$,
$\beta=-a_{0}\gamma+\sum_{i=1}^{m-1}a_{i}\gamma^{-i}\in\Phi$,
$c_{a_{\bullet}}\in\mathcal{O}_{\breve{F}}$ and
$n_{a_{\bullet}}=-a_{0}+\sum_{i=1}^{m-1}a_{i}q^{-i}$. Moreover, we have
$a_{jd}/a_{0}\leqslant 1$ (resp. $a_{jd}/a_{0}\leqslant 2$, resp.
$a_{jd}/a_{0}\leqslant 3$) for $j\geqslant 1$ if some/any connected component
of $\SS_{0}$ is simply-laced (resp. is non-simply-laced except of type
$G_{2}$, resp. is of type $G_{2}$). Thus by Hypothesis 4.1 we have
$a_{jd}/a_{0}\leqslant q^{d}-1$ for $j\geqslant 1$, which implies that
$n_{a_{\bullet}}<0$ and

$\lim_{z\to\infty}{}^{{}^{({\tilde{w}}\sigma)^{1-m}}U_{\gamma}(z)\cdots{}^{({\tilde{w}}\sigma)^{-1}}U_{\gamma}(z)}U_{-\gamma}(z^{-1})=1.$

Then
$\textsl{g}(\infty)=s_{\gamma}\textsl{g}_{1,s_{\gamma}(\gamma^{-1}),s_{\gamma}{\tilde{w}}\sigma(s_{\gamma}),m-1}(\infty)$,
and the first statement follows by induction hypothesis.

Set $\beta_{k}=s_{\gamma^{i_{0}}}\cdots
s_{\gamma^{i_{k-1}}}(\gamma^{i_{k}})\in\Phi^{+}$ and
$v_{k}=s_{\gamma^{i_{0}}}\cdots s_{\gamma^{i_{k}}}(v)$ for $0\leqslant
k\leqslant r$. As $\langle\gamma^{i_{k}},v\rangle<0$ we have

$v_{k}=s_{\beta_{k}}(v_{k-1})=v_{k-1}-\langle\beta_{k},v_{k-1}\rangle\beta_{k}^{\vee}=v_{k-1}-\langle\gamma^{i_{k}},v\rangle\beta_{k}^{\vee}>v_{k-1}.$

So the “Moreover” part follows. ∎

Let $x\in\mathcal{S}_{\lambda,b}^{+}$. Let $J_{x,0}$ be union of connected
components of $J$ on which $\sigma^{i}(\mu_{x})$ is central for
$i\in\mathbb{Z}$. Let $J_{x,1}=J\setminus J_{x,0}$. Let $H_{x}\subseteq
M_{J}({\breve{F}})$ be the subgroup generated by $I_{M_{J}}$, $W_{J_{x,0}}$,
and $W_{J_{x,1}}^{a}$, see §1.3. Note that $J_{x,1}$ commutes with $J_{x,0}$,
and ${\tilde{w}}_{x}\in\tilde{W}_{J_{x,1}}$.

Note that $\tilde{W}=\sqcup_{z\in W_{0}^{J}}z\tilde{W}_{J}=\sqcup_{z\in
W_{0}^{J}}\sqcup_{\omega\in\Omega_{J}}z\omega^{-1}W_{J}^{a}$.

######  Lemma 4.2.

Let $x\in\mathcal{S}_{\lambda,b}^{+}$. Let
${\tilde{w}}\in\mathcal{S}_{\lambda,b,x}$ and $z\in W_{0}^{J}$ such that
${\tilde{w}}=z{\tilde{w}}_{x}\sigma(z)^{-1}$. Let $y\in\tilde{W}$ (resp. $y\in
W_{0}$) such that $y{\tilde{w}}\sigma(y)^{-1}\in{\rm{Adm}}(\lambda)$. Let
$z^{\prime}\in W_{0}^{J}$ and $\omega\in\Omega_{J}$ such that $yz\in
z^{\prime}\omega^{-1}W_{J}^{a}$. Then
${\tilde{w}}^{\prime}:=z^{\prime}\omega^{-1}{\tilde{w}}_{x}\sigma(\omega)\sigma(z^{\prime})^{-1}\in\mathcal{S}_{\lambda,b}$.
Moreover, there exists
$h\in\ker(\eta_{M_{J}})\cap\mathbb{J}_{{\tilde{w}}_{x}}$ (resp. $h\in
H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$) such that
$gy^{-1}I\sim_{\lambda,b}gzh\omega{z^{\prime}}^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}}$.

###### Proof.

Write $yz=z^{\prime}\omega^{-1}u$ for some $u\in W_{J}^{a}$. Let
$x^{\prime}\in\pi_{1}(M_{J})$ such that
${\tilde{w}}_{x^{\prime}}=\omega^{-1}{\tilde{w}}_{x}\sigma(\omega)\sigma(z^{\prime})^{-1}$.
By [3, Lemma 1.3] we have

${\tilde{w}}^{\prime}=z^{\prime}{\tilde{w}}_{x^{\prime}}\sigma(z^{\prime})^{-1}\leq
z^{\prime}\omega^{-1}\delta{\tilde{w}}_{x}\sigma(\omega)\sigma(z^{\prime})^{-1}=y{\tilde{w}}\sigma(y)^{-1}\in{\rm{Adm}}(\lambda)$

where $\delta=u{\tilde{w}}_{x}\sigma(u)^{-1}{\tilde{w}}_{x}^{-1}\in
W_{J}^{a}$. So ${\tilde{w}}^{\prime}\in{\rm{Adm}}(\lambda)$ is semi-standard
by Lemma 1.3. In particular,
${\tilde{w}}_{x^{\prime}}=\omega^{-1}{\tilde{w}}_{x}\sigma(\omega)\sigma(z^{\prime})^{-1}\in{\rm{Adm}}(\lambda)$
and hence $x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$.

To show the“Moreover” part we set

$Z=\\{m\in
M_{J}({\breve{F}})/I_{M_{J}};m^{-1}{\tilde{w}}_{x}\sigma(m)\in\cup_{\delta^{\prime}\leq_{J}\delta}I_{M_{J}}\delta^{\prime}{\tilde{w}}_{x}I_{M_{J}}\\}.$

Note that $u^{-1}I_{M_{J}}\in Z$. As ${\tilde{w}}_{x}\in\Omega_{J}$, by [18,
Theorem 4.1] (resp. [4, Lemma 6.13]), there exists
$h\in\ker(\eta_{M_{J}})\cap\mathbb{J}_{{\tilde{w}}_{x}}$ (resp. $h\in
H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ if $y\in W_{0}$) such that
$u^{-1}I_{M_{J}},hI_{M_{J}}$ are connected in $Z$. For
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ there is an embedding

$Z\hookrightarrow X(\lambda,b),\quad mI_{M_{J}}\mapsto
gzm\omega{z^{\prime}}^{-1}I,$

from which we have
$gy^{-1}I=gzu^{-1}\omega{z^{\prime}}^{-1}I\sim_{\lambda,b}gzh\omega{z^{\prime}}^{-1}$
as desired. ∎

######  Lemma 4.3.

Assume $G$ is not split. Let $x\in\mathcal{S}_{\lambda,b}^{+}$ and
${\tilde{w}}\in\mathcal{S}_{\lambda,b,x}$. If
${\tilde{w}}\notin{}^{\SS_{0}}\tilde{W}$, then there exist $h\in
H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ and
${\tilde{w}}^{\prime}\in\mathcal{S}_{\lambda,b,x}$ such that
$\nu_{{\tilde{w}}}^{\flat}<\nu_{{\tilde{w}}^{\prime}}^{\flat}$ and
$gI\sim_{\lambda,b}gzh{z^{\prime}}^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}}$. Here $z,z^{\prime}\in W_{0}^{J}$ such that
${\tilde{w}}=z{\tilde{w}}_{x}\sigma(z)^{-1}$ and
${\tilde{w}}^{\prime}=z^{\prime}{\tilde{w}}_{x}\sigma(z^{\prime})^{-1}$.

###### Proof.

By Corollary 3.9, there exists $\alpha\in\mathcal{P}_{{\tilde{w}}}$. Set
$\alpha^{i}=({\tilde{w}}\sigma)^{i}(\alpha)$ for $i\in\mathbb{Z}$. Let
$\textsl{g}=\textsl{g}_{g,\alpha,{\tilde{w}},m_{\alpha,{\tilde{w}}}}$ for
$g\in\mathbb{J}_{b,{\tilde{w}}}$. Since
$\alpha^{-m_{\alpha,{\tilde{w}}}}\in\tilde{\Phi}^{+}\setminus\Phi$ and
$\alpha^{i}\in\Phi$ for $1-m_{\alpha,{\tilde{w}}}\leqslant i\leqslant 0$, we
have

$\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq{\tilde{w}}U_{\sigma(\alpha)}I\subseteq
I\\{{\tilde{w}}\sigma,{\tilde{w}}\sigma(s_{\alpha})\\}I\subseteq
I{\rm{Adm}}(\lambda)I.$

As $G$ is not split, then Hypothesis 4.1 is true. Moreover, by Lemma 3.3 (5),
the conditions in Lemma 4.1 are satisfied (for
$(\gamma,m,v)=(\alpha,m_{{\tilde{w}},\alpha},\nu_{\tilde{w}}^{\flat})$). Thus,
by Lemma 4.1 we have
$gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=gy^{-1}I$ for some $y\in
W_{0}$ such that $y(v_{\tilde{w}}^{\flat})>v_{\tilde{w}}^{\flat}$. Then
${\tilde{w}}^{\prime\prime}:=y{\tilde{w}}\sigma(y)^{-1}\in{\rm{Adm}}(\lambda)$
and
$\nu_{{\tilde{w}}}^{\flat}<y(\nu_{{\tilde{w}}}^{\flat})=\nu_{{\tilde{w}}^{\prime\prime}}^{\flat}$.
Let $h\in H_{x}$, ${\tilde{w}}^{\prime}\in\mathcal{S}_{\lambda,b,x}$, and
$z^{\prime}\in W_{0}^{J}$ be as in Lemma 4.2 such that
$gI\sim_{\lambda,b}gy^{-1}I\sim_{\lambda,b}gzh{z^{\prime}}^{-1}I$. By
construction, ${\tilde{w}}^{\prime}$ and ${\tilde{w}}^{\prime\prime}$ are
$\sigma$-conjugate by
$W_{\nu_{{\tilde{w}}^{\prime}}}=z^{\prime}W_{J}{z^{\prime}}^{-1}$, and hence
$\nu_{{\tilde{w}}^{\prime}}^{\flat}$ and
$\nu_{{\tilde{w}}^{\prime\prime}}^{\flat}$ are conjugate by
$W_{\nu_{{\tilde{w}}^{\prime}}}$. By Corollary 3.3 (2),
$\nu_{{\tilde{w}}^{\prime}}^{\flat}$ is dominant for
$\Phi_{\nu_{{\tilde{w}}^{\prime}}}^{+}$, which means
$\nu_{{\tilde{w}}}^{\flat}<\nu_{{\tilde{w}}^{\prime\prime}}^{\flat}\leqslant\nu_{{\tilde{w}}^{\prime}}^{\flat}$
as desired. ∎

######  Corollary 4.4.

Let $x\in\mathcal{S}_{\lambda,b}^{+}$ and
${\tilde{w}},{\tilde{w}}^{\prime}\in\mathcal{S}_{\lambda,b,x}$ with
${\tilde{w}}^{\prime}$ the unique element in ${}^{\SS_{0}}\tilde{W}$. Then
there exists $h\in H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ such that
$gI\sim_{\lambda,b}gzh{z^{\prime}}^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}}$, where $z,z^{\prime}\in W_{0}^{J}$ such that
${\tilde{w}}=z{\tilde{w}}_{x}\sigma(z)^{-1}$ and
${\tilde{w}}^{\prime}=z^{\prime}{\tilde{w}}_{x}\sigma(z^{\prime})^{-1}$.

###### Proof.

Note that the statement follows from Theorem 0.2, which is proved in [3] when
$G$ is split. So we assume that $G$ is not split. If
${\tilde{w}}\notin{}^{\SS_{0}}\tilde{W}$, by Lemma 4.3, there exist $h\in
H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ and
${\tilde{w}}^{\prime}\in\mathcal{S}_{\lambda,b,x}$ such that
$\nu_{\tilde{w}}^{\flat}<\nu_{{\tilde{w}}^{\prime}}^{\flat}$ and
$gI\sim_{\lambda,b}gzh{z^{\prime}}^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}}$, where $z^{\prime}\in W_{0}^{J}$ such that
${\tilde{w}}^{\prime}=z^{\prime}{\tilde{w}}_{x}\sigma(z^{\prime})^{-1}$.
Repeating this process, we may assume either
${\tilde{w}}^{\prime}\in{}^{\SS_{0}}\tilde{W}$ or
$\nu_{{\tilde{w}}^{\prime}}^{\flat}$ is dominant. In either case, we have
${\tilde{w}}^{\prime}\in{}^{\SS_{0}}\tilde{W}$ by Lemma 3.4. So the statement
follows. ∎

Proposition 2.2 is a consequence of the following result.

######  Proposition 4.5.

Let $x\in\mathcal{S}_{\lambda,b}^{+}$ and
${\tilde{w}}\in\mathcal{S}_{\lambda,b,x}$. Then there exists $h\in
H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ such that
$gI\sim_{\lambda,b}ghz^{-1}I$, or equivalently,
$gh^{-1}I\sim_{\lambda,b}gz^{-1}I$ for $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$,
where $z\in W_{0}^{J}$ such that ${\tilde{w}}=z{\tilde{w}}_{x}\sigma(z)^{-1}$.
In particular,
$\mathbb{J}_{b,{\tilde{w}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x}}$.

###### Proof.

Let $z^{\prime}\in W_{0}^{J}$ such that
$z^{\prime}{\tilde{w}}_{x}\sigma(z^{\prime})^{-1}\in{}^{\SS_{0}}\tilde{W}$
(see Lemma 1.3 and Proposition 3.5). By Corollary 4.4, there exist
$h_{1},h_{2}\in H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ such that
$gI\sim_{\lambda,b}gh_{1}{z^{\prime}}^{-1}I$ and
$gz^{-1}I\sim_{\lambda,b}gh_{2}{z^{\prime}}^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$. Then we have

$ghz^{-1}I=jgz^{-1}I\sim_{\lambda,b}jgh_{2}{z^{\prime}}^{-1}I=gh_{1}{z^{\prime}}^{-1}I\sim_{\lambda,b}gI,$

where $h=h_{1}h_{2}^{-1}\in H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ and
$j=gh_{1}h_{2}^{-1}g^{-1}\in\mathbb{J}_{b}$. ∎

######  Corollary 4.6.

Let $x\in\mathcal{S}_{\lambda,b}^{+}$ and $y\in\tilde{W}$ such that
$y{\tilde{w}}_{x}\sigma(y)^{-1}\in{\rm{Adm}}(\lambda)$. Then there exists
$h\in\ker(\eta_{M_{J}})\cap\mathbb{J}_{{\tilde{w}}_{x}}$ such that
$gy^{-1}I\sim_{\lambda,b}gh\omega I$ for $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$,
where $\omega\in\Omega_{J}$ such that $y\in W_{0}^{J}\omega^{-1}W_{J}^{a}$.

###### Proof.

It follows from Lemma 4.2 and Proposition 4.5. ∎

##  5\. Proof of Proposition 2.3

### 5.1.

Let $K\subseteq\SS_{0}$. Let ${\tilde{w}}=t^{\mu}w\in\Omega_{K}$ with $\mu\in
Y$ and $w\in W_{K}$. Let $\gamma\in\Phi^{+}\setminus\Phi_{K}$ such that
$\gamma^{\vee}$ is $K$-dominant and $K$-minuscule. Set
$\tilde{\gamma}=\gamma+1\in\tilde{\Phi}^{+}$. Suppose

$\mu,\mu-\gamma^{\vee},\mu+w\sigma^{r}(\gamma^{\vee}),\mu-\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})\preceq\lambda\text{
for some }r\in\mathbb{Z}_{\geqslant 0}.$

######  Lemma 5.1.

Let $K$, ${\tilde{w}}=t^{\mu}w$, $\gamma$, $\tilde{\gamma}$, and $r$ be as in
§5.1. Then we have

(1)
$\mu-\gamma^{\vee},\mu+w(\sigma^{r}(\gamma^{\vee})),\mu-\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})$
are $K$-minuscule;

(2)
${\tilde{w}},s_{\tilde{\gamma}}{\tilde{w}},{\tilde{w}}s_{\sigma^{r}(\tilde{\gamma})},s_{\tilde{\gamma}}{\tilde{w}}s_{\tilde{\gamma}}\in{\rm{Adm}}(\lambda)$;

(3)
$s_{\tilde{\gamma}}{\tilde{w}}s_{\sigma^{r}(\tilde{\gamma})}\in{\rm{Adm}}(\lambda)$
if $\gamma\neq\sigma^{r}(\gamma)$ and $-\langle
w\sigma^{r}(\gamma),\mu\rangle,\langle\gamma,\mu\rangle\leqslant 1$;

###### Proof.

Note that (1) and (2) are proved in [2, Lemma 4.4.6] and [4, Lemma 1.5]
respectively. To show (3) we claim that

there is a $W_{K}$-conjugate $\eta$ of $\mu$ such that
$\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})$ is $K$-minuscule. (a)

Indeed, let $\eta$ be a $W_{K}$-conjugate of $\mu$ such that
$\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})$ is minimal under the partial
order $\preceq$. If $\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})$ is not
$K$-minuscule, then there exists $\alpha\in\Phi_{K}$ such that
$\langle\alpha,\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})\rangle\geqslant
2$. As $\eta$ is $K$-minuscule, and $\gamma^{\vee},\sigma^{r}(\gamma^{\vee})$
are $K$-dominant and $K$-minuscule, we deduce that
$\langle\alpha,\eta\rangle=1$. Let
$\eta^{\prime}=s_{\alpha}(\eta)=\eta-\alpha^{\vee}$. Then we have

$\eta^{\prime}-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})=\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})-\alpha^{\vee}\prec\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee}),$

which contradicts the choice of $\eta$. So (a) is proved.

By (1) and (a), $\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})$,
$\mu-\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})$ are conjugate by $W_{K}$. In
particular, $\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})\preceq\lambda$. Then
(3) follows from that

$s_{\tilde{\gamma}}{\tilde{w}}s_{\sigma^{r}(\tilde{\gamma})}\leq
s_{\tilde{\gamma}}t^{\eta}s_{\sigma^{r}(\tilde{\gamma})}=s_{\gamma}t^{\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})}s_{\sigma^{r}(\gamma)}\leq
t^{\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})}\in{\rm{Adm}}(\lambda),$

where the first $\leq$ follows from [3, Lemma 1.3], and the second $\leq$
follows from that

$\displaystyle\langle\gamma,\sigma^{r}(\gamma^{\vee})\rangle$
$\displaystyle\leqslant 0\text{ since }\gamma\neq\sigma^{r}(\gamma);$
$\displaystyle\langle\gamma,\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})\rangle$
$\displaystyle\leqslant\langle\gamma,\mu\rangle-2\leqslant-1;$
$\displaystyle\langle\sigma^{r}(\gamma),\eta-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})\rangle$
$\displaystyle\geqslant\langle w\sigma^{r}(\gamma),\mu\rangle+2\geqslant 1.$

The proof is finished. ∎

For $K\subseteq\SS_{0}$ we say $\gamma^{\vee}$ with
$\gamma\in\Phi^{+}\setminus\Phi_{K}$ is strongly $K$-minuscule if
$\gamma^{\vee}$ is $K$-minuscule, and moreover, $\gamma$ is a long root if (1)
some/any connected component of $\SS_{0}$ is of type $G_{2}$, and (2) $K$ is
the set of short simple roots.

######  Lemma 5.2.

Let $K$, ${\tilde{w}}=t^{\mu}w$, $\gamma$, $\tilde{\gamma}$, and $r$ be as in
§5.1. Assume furthermore that $\gamma^{\vee}$ is strongly $K$-minuscule. Then
$U_{-\tilde{\gamma}}{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}\subseteq
I{\rm{Adm}}(\lambda)I$ unless

$\displaystyle\langle\gamma,\mu\rangle=-\langle
w\sigma^{r}(\gamma),\mu\rangle=1\text{ and
}\langle\gamma,w\sigma^{r}(\gamma^{\vee})\rangle=-1,$ (*)

in which case we have

${\tilde{w}}\neq{\tilde{w}}^{\prime},\
U_{-\sigma^{r}(\tilde{\gamma})}{\tilde{w}}^{\prime}U_{-\tilde{\gamma}}\subseteq
I{\rm{Adm}}(\lambda)I,\text{ and
}\mu\pm(\gamma+w\sigma^{r}(\gamma))^{\vee}\preceq\lambda.$

Here
${\tilde{w}}^{\prime}=\mu-\gamma^{\vee}+\sigma^{r}(\gamma^{\vee})\in\pi_{1}(M_{K})\cong\Omega_{K}$.

###### Proof.

First we claim that

$\Psi:=\Phi\cap(\mathbb{Z}\gamma+\mathbb{Z}w\sigma^{r}(\gamma))$ is of type
$A_{2}$, or $A_{1}\times A_{1}$, or $A_{1}$. (a)

Otherwise, then $\Psi$ is of type $B_{2}$ or $G_{2}$. In particular,
$\gamma=\sigma^{r}(\gamma)$ (since $\sigma^{d}=\textrm{id}$), $\gamma\neq
w\sigma^{r}(\gamma)=w(\gamma)$, and hence $K\neq\emptyset$. If $\Psi$ is of
$B_{2}$, then $\gamma\pm w\sigma^{r}(\gamma)\in\Phi$ and
$\langle\gamma,w\sigma^{r}(\gamma^{\vee})\rangle=0$ since
$\gamma,w\sigma^{r}(\gamma)$ are of the same length. Thus
$\gamma-w\sigma^{r}(\gamma)\in\Phi_{K}$ and
$\langle\gamma-w\sigma^{r}(\gamma),\gamma^{\vee}\rangle=2$, contradicting that
$\gamma^{\vee}$ is $K$-minuscule. So $\Psi$ is of type $G_{2}$. As
$\gamma^{\vee}$ is strongly $K$-minuscule, $\gamma\neq w\sigma^{r}(\gamma)$
are short roots and $K$ consists of long simple roots, which contradicts that
$\gamma^{\vee}$ is $K$-minuscule. So (a) is proved.

Then we claim that

$U_{-\tilde{\gamma}}{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}\subseteq
I{\rm{Adm}}(\lambda)I$ if one of the following holds: (b) either
$\langle\gamma,\mu\rangle\geqslant 2$ or $\langle\gamma,\mu\rangle=1$ and
$\langle\gamma,w\sigma(\gamma^{\vee})\rangle\geqslant 0$; (b1) either $\langle
w\sigma^{r}(\gamma),\mu\rangle\leqslant-2$ or $\langle
w\sigma^{r}(\gamma),\mu\rangle=-1$ and
$\langle\gamma,w\sigma^{r}(\gamma^{\vee})\rangle\geqslant 0$. (b2)

By symmetry we assume (b1) occurs. Then
$U_{-{\tilde{w}}^{-1}(\tilde{\gamma})},[U_{-{\tilde{w}}^{-1}(\tilde{\gamma})},U_{-\sigma^{r}(\tilde{\gamma})}]\subseteq
I$ by (a). Thus

$U_{-\tilde{\gamma}}{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}\subseteq{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}I\subseteq
I\\{{\tilde{w}},{\tilde{w}}s_{\sigma^{r}(\tilde{\gamma})}\\}\subseteq
I{\rm{Adm}}(\lambda)I,$

where the last inclusion follows from Lemma 5.1 (2). So (b) is proved.

Suppose
$U_{-\tilde{\gamma}}{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}\nsubseteq
I{\rm{Adm}}(\lambda)I$. Then $-\langle
w\sigma^{r}(\gamma),\mu\rangle,\langle\gamma,\mu\rangle\leqslant 1$ by (b),
and ${\tilde{w}}^{-1}(\gamma)\neq\sigma^{r}(\gamma)$. Assume
$\langle\gamma,\mu\rangle\leqslant 0$. Then
$U_{{\tilde{w}}^{-1}(\tilde{\gamma})},[U_{{\tilde{w}}^{-1}(\tilde{\gamma})},U_{-\sigma^{r}(\tilde{\gamma})}]\subseteq
I$ by (a) and that $\gamma,\sigma^{r}(\gamma)$ are $K$-dominant. Thus by Lemma
5.1 we have

$U_{-\tilde{\gamma}}{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}\subseteq
Is_{\tilde{\gamma}}{\tilde{w}}U_{-\sigma^{r}(\tilde{\gamma})}I\subseteq
I\\{s_{\tilde{\gamma}}{\tilde{w}},s_{\tilde{\gamma}}{\tilde{w}}s_{\sigma^{r}(\tilde{\gamma})}\\}I\subseteq
I{\rm{Adm}}(\lambda)I,$

which contradicts our assumption. So $\langle\gamma,\mu\rangle=1$, and
$\langle w\sigma^{r}(\gamma),\mu\rangle=-1$ by symmetry. Moreover, we have
$\langle\gamma,w\sigma^{r}(\gamma^{\vee})\rangle=-1$ by (b) and (a).

Write ${\tilde{w}}^{\prime}=t^{\mu^{\prime}}w^{\prime}\in\Omega_{K}$ with
$\mu^{\prime}\in Y$ and $w^{\prime}\in W_{K}$. Then
$\mu^{\prime},\mu-\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})$ (resp.
$\mu^{\prime}-\sigma^{r}(\gamma^{\vee}),\mu-\gamma^{\vee}$, resp.
$\mu^{\prime}+w^{\prime}(\gamma^{\vee}),\mu+w\sigma^{r}(\gamma^{\vee})$) are
conjugate by $W_{K}$ by Lemma 5.1 (1). Since
$\langle\gamma,\mu\rangle=-\langle
w\sigma^{r}(\gamma),\mu\rangle=-\langle\gamma,w\sigma^{r}(\gamma^{\vee})\rangle=1$,
it follows that $\mu-\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})$ and
$\mu\pm(\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee}))$ are conjugate by $W_{0}$.
Hence
$\mu\pm(\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})),\mu^{\prime}\preceq\lambda$.
As $w_{K}(\gamma)$ (with $w_{K}$ the longest element of $W_{K}$) is $K$-anti-
dominant, we have

$\langle w^{\prime}(\gamma),\mu^{\prime}\rangle=\langle
w_{K}(\gamma),\mu^{\prime}\rangle\leqslant\langle\gamma,\mu-\gamma^{\vee}+w\sigma^{r}(\gamma^{\vee})\rangle=-2.$

Hence $\sigma^{r}(\gamma)\neq\gamma$, that is,
${\tilde{w}}\neq{\tilde{w}}^{\prime}$, and
$U_{-\sigma^{r}(\tilde{\gamma})}{\tilde{w}}^{\prime}U_{-\tilde{\gamma}}\subseteq
I{\rm{Adm}}(\lambda)I$ by (b2). ∎

### 5.2.

Let $x,x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}\subseteq\pi_{1}(M_{J})$. Write
$x\overset{(\gamma,r)}{\to}x^{\prime}$ for some
$\gamma\in\Phi\setminus\Phi_{J}$ and $r\in\mathbb{Z}_{\geqslant 1}$ if
$x^{\prime}-x=\sigma^{r}(\gamma^{\vee})-\gamma^{\vee}$ and
$\mu_{x-\gamma^{\vee}},\mu_{x+\sigma^{r}(\gamma^{\vee})}\preceq\lambda$, see
§2. Moreover, write $x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime}$ if
$x\overset{(\gamma,r)}{\to}x^{\prime}$, and for each $1\leqslant i\leqslant
r-1$ we have

$\displaystyle\text{neither }\quad\
x\overset{(\gamma,i)}{\to}x-\gamma^{\vee}+\sigma^{i}(\gamma^{\vee})\overset{(\sigma^{i}(\gamma),r-i)}{\to}x^{\prime},$
$\displaystyle\text{ nor }\quad\
x\overset{(\sigma^{i}(\gamma),r-i)}{\to}x-\sigma^{i}(\gamma^{\vee})+\sigma^{r}(\gamma^{\vee})\overset{(\gamma,i)}{\to}x^{\prime}.$

Notice that $x\overset{(\gamma,r)}{\to}x^{\prime}$ is equivalent to
$x^{\prime}\overset{(-\gamma,r)}{\to}x$.

######  Lemma 5.3 ([2, Remark 4.5.2]).

Let $x\neq x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such that
$x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime}$ for some
$\gamma\in\Phi\setminus\Phi_{J}$ and $r\in\mathbb{Z}_{\geqslant 1}$. Then
${\tilde{w}}_{x}\sigma^{i}(\delta)=\sigma^{i}(\delta)$ for any
$W_{0}$-conjugate $\delta$ of $\gamma$ and $1\leqslant i\leqslant r-1$ with
$i,i-r\notin d\mathbb{Z}$.

For $\gamma\in\Phi$ we denote by $\mathcal{O}_{\gamma}$ the $\sigma$-orbit of
$\gamma$.

######  Proposition 5.4 ([30, Lemma 6.7]).

Let $x\neq x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$. Then there exist
distinct elements
$x=x_{0},x_{1},\dots,x_{m}=x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such that
for each $1\leqslant i\leqslant m$ we have

(1) $x_{i-1}\overset{(\gamma_{i},r_{1})}{\rightarrowtail}x_{i}$ with
$\gamma_{i}\in\Phi\setminus\Phi_{J}$ such that $\gamma_{i}^{\vee}$
$J$-dominant and $J$-minuscule;

(2) $1\leqslant r_{i}\leqslant d-1$ if $|\mathcal{O}_{\gamma_{i}}|=d$;
$1\leqslant r_{i}\leqslant d$ if $|\mathcal{O}_{\gamma_{i}}|=2d$; $1\leqslant
r_{i}\leqslant 2d-1$ if $|\mathcal{O}_{\gamma_{i}}|\leqslant 3d$.

###### Proof of Proposition 2.3.

The case that $\sigma$ has order $3d$ is handled in §8.2. We consider the case
that $\sigma$ has order $\leqslant 2d$. Without loss of generality, we can
assume that $|\mathcal{O}_{\gamma}|=2d$. By Proposition 5.4 and symmetry, we
may assume $x\overset{(\gamma,r)}{\to}x^{\prime}$ for some $1\leqslant
r\leqslant d$ and $\gamma\in\Phi^{+}\setminus\Phi_{J}$ with
$\gamma_{i}^{\vee}$ $J$-dominant and $J$-minuscule. Moreover, we can assume

$\displaystyle
U_{-\tilde{\gamma}}{\tilde{w}}_{x}U_{-\sigma^{r}(\tilde{\gamma})}\subseteq
I{\rm{Adm}}(\lambda)I.$ (a)

Indeed, if $1\leqslant r\leqslant d-1$, (a) follows from Lemma 5.1 (2). If
$r=d$, by Lemma 5.2 we can switch the pairs $(x,\gamma)$ and
$(x^{\prime},\sigma^{d}(\gamma))$ if necessary so that (a) still holds.

Now we can assume further that
$x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime}$. Let
$\tilde{\gamma}=\gamma+1\in\tilde{\Phi}^{+}$, and let
$\textsl{g}=\textsl{g}_{g,-\sigma^{r-1}(\tilde{\gamma}),{\tilde{w}}_{x},r}$
for $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ (see §4). By Lemma 5.3,
$({\tilde{w}}\sigma)^{i}(\gamma)=\sigma^{i}(\gamma)$ for $1\leqslant
i\leqslant r-1$. Then by (a) we have
$\textsl{g}^{-1}{\tilde{w}}\sigma(\textsl{g})\subseteq
U_{-\tilde{\gamma}}{\tilde{w}}_{x}U_{-\sigma^{r}(\tilde{\gamma})}\subseteq
I{\rm{Adm}}(\lambda)I$, which means that
$gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=gsI$, where
$s=s_{\tilde{\gamma}}\cdots s_{\sigma^{r-1}(\tilde{\gamma})}$. By [4, Lemma
1.3] we can write $s=\omega z^{-1}$, where $z\in W_{0}^{J}$ and
$\omega=\gamma^{\vee}+\cdots+\sigma^{r-1}(\gamma^{\vee})\in\Omega_{J}\cong\pi_{1}(M_{J})$.
By Proposition 4.5, there is
$h^{\prime}\in\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$ such that
$gI\sim_{\lambda,b}g\omega z^{-1}I\sim_{\lambda,b}g\omega h^{\prime}I$. So we
have
$\mathbb{J}_{b,{\tilde{w}}_{x}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$
as desired. ∎

##  6\. Proof of Proposition 2.4

Retain the assumptions and notations in previous sections.

For $K\subseteq\SS_{0}$ we denote by ${\rm
pr}_{K}:\mathbb{R}\Phi^{\vee}\to(\mathbb{R}\Phi_{K}^{\vee})^{\perp}$ the
orthogonal projection.

######  Lemma 6.1.

Let $x\in\mathcal{S}_{\lambda,b}^{+}$ and let $\mathcal{O}$ be a
$\sigma$-orbit of $J$-anti-dominant roots in $\Phi^{+}\setminus\Phi_{J}$. Then
we have (1) $\sum_{\alpha\in\mathcal{O}}\langle\alpha,{\rm
pr}_{J}(\mu_{x})\rangle>0$, and (2) $\langle
w_{J}(\beta),\mu_{x}\rangle\geqslant 1$ for some $\beta\in\mathcal{O}$. Here
$w_{J}$ denotes the longest element of $W_{J}$.

###### Proof.

Let $\gamma\in\mathcal{O}$. By definition,
$\langle\gamma,\nu_{G}(b)\rangle=\langle\gamma,{\rm
pr}_{J}(\mu_{x})^{\diamond}\rangle>0$. So (1) follows as

$\sum_{\alpha\in\mathcal{O}}\langle\alpha,{\rm
pr}_{J}(\mu_{x})\rangle=\sum_{\alpha\in\mathcal{O}}\langle\alpha,{\rm
pr}_{J}(\mu_{x})^{\diamond}\rangle=|\mathcal{O}|\langle\gamma,\nu_{G}(b)\rangle>0.$

By (1), there exists $\beta\in\mathcal{O}$ such that $\langle\beta,{\rm
pr}_{J}(\mu_{x})\rangle>0$. As $w_{J}(\beta)$ is $J$-dominant and
$\mu_{x}-{\rm pr}_{J}(\mu_{x})\in\mathbb{R}_{\geqslant
0}(\Phi_{J}^{+})^{\vee}$, we have

$\langle w_{J}(\beta),\mu_{x}\rangle\geqslant\langle w_{J}(\beta),{\rm
pr}_{J}(\mu_{x})\rangle=\langle\beta,{\rm pr}_{J}(\mu_{x})\rangle>0.$

So (2) follows. ∎

######  Lemma 6.2 ([4, Lemma 1.6]).

Let $K\subseteq\SS_{0}$ and ${\tilde{w}}=t^{\mu}w\in\Omega_{K}$ with $\mu\in
Y$ and $w\in W_{K}$. Let $\alpha\in\Phi^{+}$ be $K$-anti-dominant. Then (1)
${\tilde{w}}s_{\alpha}\in{\rm{Adm}}(\lambda)$ if
$\mu+\alpha^{\vee}\preceq\lambda$; (2)
$s_{\alpha}{\tilde{w}}\in{\rm{Adm}}(\lambda)$ if
$\mu-w(\alpha)^{\vee}\preceq\lambda$; (3)
$z{\tilde{w}}z^{-1}\in{\rm{Adm}}(\lambda)$ for $z\in\tilde{W}^{K}$.

### 6.1.

Define $J_{1}=\cup_{x\in\mathcal{S}_{\lambda,b}^{+}}J_{x,1}$ and
$J_{0}=J\setminus J_{1}$. Define
$H_{J^{\prime}}=M_{J^{\prime}}({\breve{F}})\cap\ker(\eta_{M_{J^{\prime}}})$
for $J^{\prime}\subseteq J$.

######  Theorem 6.3 ([18, Theorem 6.3]).

Let $x\in\mathcal{S}_{\lambda,b}^{+}$. Then
$H_{J_{x,1}}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ fixes each connected component
of $X^{M_{J_{x,1}}}(\mu_{x},{\tilde{w}}_{x})$.

######  Lemma 6.4.

We have that $H_{J_{1}}\cap\mathbb{J}_{b}$ fixes each connected component of
$X(\lambda,b)$.

###### Proof.

Let $C$ be a connected component of $X(\lambda,b)$. Let
$x\in\mathcal{S}_{\lambda,b}^{+}$. By Proposition 2.3, there exists
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}\subseteq M_{J}({\breve{F}})$ such that
$gI\in C$. Moreover, $gI$ also lies in the image of the embedding

$X^{M_{J_{x,1}}}(\mu_{x},{\tilde{w}}_{x})\hookrightarrow X(\lambda,b),\
hI_{M_{J_{x,1}}}\mapsto ghI.$

Thus
$g(H_{J_{x,1}}\cap\mathbb{J}_{{\tilde{w}}_{x}})g^{-1}=H_{J_{x,1}}\cap\mathbb{J}_{b}$
fixes $C$ by Theorem 6.3. So the statement follows by noticing that
$H_{J_{1}}\cap\mathbb{J}_{b}$ is generated by $H_{J_{x,1}}\cap\mathbb{J}_{b}$
for $x\in\mathcal{S}_{\lambda,b}^{+}$. ∎

### 6.2.

Let $K\subseteq J_{0}$ be the union of some $\sigma$-orbit of connected
components of $J_{0}$.

######  Lemma 6.5.

If $\mu_{x}+\alpha^{\vee}\preceq\lambda$ for some
$x\in\mathcal{S}_{\lambda,b}^{+}$ and $\alpha\in K$, then
$H_{K}\cap\mathbb{J}_{b}$ fixes each connected component of $X(\lambda,b)$.

###### Proof.

As $\mu_{x}$ is central on $\Phi_{K}$, we can assume $\alpha$ is $K$-dominant
and hence $\sigma^{d}(\alpha)=\alpha$. Let $C$ be a connected component of
$X(\lambda,b)$. Then $gI\in C$ for some $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$.
So the stabilizer of $C$ in $\mathbb{J}_{b}$ equals $gQg^{-1}$, where
$Q\subseteq\mathbb{J}_{{\tilde{w}}_{x}}$ a standard parahoric subgroup
containing $I_{M_{J}}\cap\mathbb{J}_{{\tilde{w}}_{x}}$. By Lemma 1.3 (4) it
remains to show $W_{K}^{a}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq Q$. Let
$\textsl{g}=\textsl{g}_{g,\alpha,{\tilde{w}}_{x},d}$ and
$\textsl{g}^{\prime}=\textsl{g}_{g,-\alpha-1,{\tilde{w}}_{x},d}$ (see §4). By
Lemma 6.2 and Lemma 5.1,

$g^{-1}b\sigma(g)\subseteq U_{\alpha}{\tilde{w}}_{x}\subseteq
I{\rm{Adm}}(\lambda)I,\text{ and
}{g^{\prime}}^{-1}b\sigma(g^{\prime})\subseteq{\tilde{w}}_{x}U_{-\alpha-1}\subseteq
I{\rm{Adm}}(\lambda)I,$

which means

$gsI=\textsl{g}(\infty)\sim_{\lambda,b}\textsl{g}(0)=\textsl{g}^{\prime}(0)\sim_{\lambda,b}\textsl{g}^{\prime}(\infty)=gs^{\prime}I,$

where $s=s_{\alpha}\cdots
s_{\sigma^{d-1}(\alpha)},s^{\prime}=s_{\alpha+1}\cdots
s_{\sigma^{d-1}(\alpha)+1}\in\mathbb{J}_{{\tilde{w}}_{x}}$. So we have
$s,s^{\prime}\in Q$, which means
$W_{K}^{a}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq Q$ since
$\alpha\in\Phi_{K}^{+}$ is $K$-dominant. ∎

The following technical lemma is proved in §6.3

######  Lemma 6.6.

If $\mu_{x^{\prime\prime}}+\delta^{\vee}\npreceq\lambda$ for any
$x^{\prime\prime}\in\mathcal{S}_{\lambda,b}^{+}$ and $\delta\in K$, then there
exist $x\in\mathcal{S}_{\lambda,b}^{+}$ and
$\beta\in\Phi^{+}\setminus\Phi_{J}$ with $\beta^{\vee}$ $J$-anti-dominant and
$J$-minuscule such that

(1) $\mu_{x}+\beta^{\vee}\preceq\lambda$, and $\beta^{\vee}$ is non-central on
$K$;

(2) ${\tilde{w}}_{x}\sigma^{i}(\beta)=\sigma^{i}(\beta)$ for
$i\in\mathbb{Z}\setminus n\mathbb{Z}$;

(3) $\langle w_{x}\sigma^{n}(\beta),\mu_{x}\rangle\geqslant 1$;

(4) if $\sigma^{n}$ does not act trivially on $\Psi_{\beta}\cap J_{0}$, then
$\Psi=\Phi$, $\Psi_{\beta}$ is of type $E_{6}$, $\Psi_{\beta}\cap
J_{0}=\\{\alpha_{1},\alpha_{6}\\}$, $\Psi_{\beta}\cap
J_{1}=\\{\alpha_{2},\alpha_{4}\\}$, $\beta=\alpha_{3}$,
$\mu_{x}|_{\Psi_{\beta}}=\omega_{4}^{\vee}-\omega_{3}^{\vee}$, and
$\mu_{x}|_{\Psi\setminus\Psi_{\beta}}=0$.

Here, $n\in\\{d,2d,3d\\}$ denotes the minimal integer such that
$\beta,\sigma^{n}(\beta)$ are in the same connected component $\Psi_{\beta}$
of $\Psi:=\Phi\cap\mathbb{Z}(J\cup\mathcal{O}_{\beta})$, whose simple roots
$\alpha_{i}$ and fundamental coweights $\omega_{i}^{\vee}$ for $1\leqslant
i\leqslant 6$ are labeled as in [19].

######  Lemma 6.7.

Retain the situation of Lemma 6.6. Let $\alpha\in\Phi_{K}^{+}$ such that
$\langle\alpha,\beta^{\vee}\rangle=-1$. If $\alpha=\sigma^{n}(\alpha)$, then
$U_{\beta}{\tilde{w}}_{x}U_{\sigma^{n}(\beta)},U_{\alpha}s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}U_{\alpha}\subseteq
I{\rm{Adm}}(\lambda)I$.

###### Proof.

Note that
$\mu_{x}+s_{\alpha}(\beta)^{\vee}=\mu_{x}+\beta^{\vee}+\alpha^{\vee}\preceq\lambda$,
$s_{\alpha},s_{\beta},s_{s_{\alpha}(\beta)}\in W^{J_{1}}$, and
${\tilde{w}}_{x}\in\Omega_{J_{1}}$. By Lemma 6.6 (2) and Lemma 6.2,
$s_{\beta}{\tilde{w}}_{x},s_{s_{\alpha}(\beta)}{\tilde{w}}_{x}\in{\rm{Adm}}(\lambda)$.
As ${\tilde{w}}_{x}\sigma^{n}(\beta)\in\tilde{\Phi}^{+}\setminus\Phi$, we have

$U_{\beta}{\tilde{w}}_{x}U_{\sigma^{n}(\beta)}\subseteq
IU_{\beta}{\tilde{w}}_{x},\
U_{\alpha}s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}U_{\alpha}\subseteq
IU_{\alpha}s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)},$

and it remains to show
$s_{\alpha}s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)},s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}\in{\rm{Adm}}(\lambda)$.
As ${\tilde{w}}_{x}\sigma^{n}(\beta)\in\tilde{\Phi}^{+}\setminus\Phi$,
${\tilde{w}}_{x}(\alpha)=\alpha$ and $s_{\alpha}s_{\beta}(\alpha)\in\Phi^{+}$,
we have $s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}\leq
s_{\beta}{\tilde{w}}_{x}\in{\rm{Adm}}(\lambda)$ and

$s_{\alpha}s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}\leq
s_{\alpha}s_{\beta}{\tilde{w}}_{x}\leq
s_{\alpha}s_{\beta}s_{\alpha}{\tilde{w}}_{x}=s_{s_{\alpha}(\beta)}{\tilde{w}}_{x}\in{\rm{Adm}}(\lambda).$

The proof is finished. ∎

###### Proof of Proposition 2.4.

Let $K$ be the union of some $\sigma$-orbit of connected components of
$J_{0}$. By Lemma 6.4 and Lemma 6.5, it remains to show
$H_{K}\cap\mathbb{J}_{b}$ acts trivially on $\pi_{0}(X(\lambda,b))$. Let $x$,
$\beta$ and $n$ be as in Lemma 6.6. Let $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$
and $I_{M_{J}}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq
Q\subseteq\mathbb{J}_{{\tilde{w}}_{x}}$ be as in the proof of Lemma 6.5. It
suffices to show $W_{K}^{a}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq Q$.

Case(1): $\sigma^{n}$ acts trivially on $\Psi_{\beta}\cap J_{0}$. Let
$\alpha\in\Phi_{K}^{+}$ be a highest root such that
$\langle\alpha,\beta^{\vee}\rangle=-1$. Then it suffices to show
$s,s^{\prime}\in Q$, where $s=s_{\alpha}\cdots
s_{\sigma^{n-1}(\alpha)},s^{\prime}=s_{\alpha+1}\cdots
s_{\sigma^{n-1}(\alpha)+1}\in\mathbb{J}_{{\tilde{w}}_{x}}$.

Let $r=s_{\beta}\cdots\sigma^{n-1}(s_{\beta})$. We claim that

$\displaystyle
gI\sim_{\lambda,b}grI\sim_{\lambda,b}grsI\sim_{\lambda,b}gsI,\text{ and hence
}s\in Q$ (a)

To show the first relation $\sim_{\lambda,b}$ in (a) we define
$\textsl{g}=\textsl{g}_{g,\sigma^{n-1}(\beta),{\tilde{w}}_{x},n}$. By Lemma
6.6 (2) and Lemma 6.7 we have

$\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq
U_{\beta}{\tilde{w}}_{x}U_{\sigma^{n}(\beta)}\subseteq I{\rm{Adm}}(\lambda)I,$

which means $gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=grI$ as
desired. The last relation $\sim_{\lambda,b}$ in (a) follows the same way by
replacing $g$, $\beta$ with $gs$, $s_{\alpha}(\beta)$ respectively.

To show the second relation $\sim_{\lambda,b}$ in (b) we define
$\textsl{g}^{\prime}=\textsl{g}_{gr,\sigma^{n-1}(\alpha),{\tilde{w}}_{x},n}$.
Notice that
$r^{-1}{\tilde{w}}_{x}\sigma(r)=s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}$.
Then by Lemma 6.7 we have

${\textsl{g}^{\prime}}^{-1}b\sigma(\textsl{g}^{\prime})\subseteq
U_{\alpha}s_{\beta}{\tilde{w}}_{x}s_{\sigma^{n}(\beta)}U_{\alpha}\subseteq
I{\rm{Adm}}(\lambda)I,$

which means
$grI=\textsl{g}^{\prime}(0)\sim_{\lambda,b}\textsl{g}^{\prime}(\infty)=grsI$.
So (a) is proved.

By Lemma 6.4, Lemma 6.5, and (a) we have
$(W_{J_{1}}^{a}W_{J_{0}})\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq Q$, and
hence

$\displaystyle H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq Q.$ (b)

Let $x^{\prime}=x+\beta^{\vee}-\sigma^{n}(\beta)^{\vee}\in\pi_{1}(M_{J})$. If
$\beta\neq\sigma^{n}(\beta)$, then $\beta,\sigma^{n}(\beta)$ are neighbors of
$\Psi_{\beta}\cap K$ on which $\sigma^{n}$ acts trivially, which means they
are in distinct connected components of $\Psi_{\beta}\setminus K$. Thus

$\displaystyle\langle w(\beta),\sigma^{n}(\beta)^{\vee}\rangle=0\text{ for any
}w\in W_{J_{1}}\text{ if }\beta\neq\sigma^{n}(\beta^{\vee}).$ (c)

By Lemma 6.6 (1) & (3) and (c) we have
$x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$. Moreover,
$\mu_{x}+\beta^{\vee}-w_{x}(\beta^{\vee}),\mu_{x^{\prime}}$ are conjugate by
$W_{J_{1}}$ as they are conjugate by $W_{J}$ and $\mu_{x^{\prime}}$ is central
on $J_{0}$. Let $\gamma_{1}=w_{J_{1}}(\beta)$ and
$\gamma_{2}=w_{J_{1}}(s_{\alpha}(\beta))$ which are $J_{1}$-dominant. By Lemma
5.1 (1) and that $\sigma^{n}$ acts trivially on $\Psi_{\beta}\cap J_{0}$,

$\mu_{x},\mu_{x}-\sigma^{n}(\gamma_{i}^{\vee}),\mu_{x}+w_{x}(\gamma_{i}^{\vee}),\mu_{x}-\sigma^{n}(\gamma_{i}^{\vee})+w_{x}(\gamma_{i}^{\vee})\preceq\lambda$

are conjugate to

$\mu_{x^{\prime}}-\gamma_{i}^{\vee}+w_{x^{\prime}}\sigma^{n}(\gamma_{i}^{\vee}),\mu_{x^{\prime}}-\gamma_{i}^{\vee},\mu_{x^{\prime}}+w_{x^{\prime}}\sigma^{n}(\gamma_{i}^{\vee}),\mu_{x^{\prime}}\preceq\lambda$

under $W_{J_{1}}$ respectively.

Let
$\tau=\beta^{\vee}+\cdots\sigma^{n-1}(\beta)^{\vee}\in\pi_{1}(M_{J_{1}})\cong\Omega_{J_{1}}$.
Then ${\tilde{w}}_{x}=\tau^{-1}{\tilde{w}}_{x^{\prime}}\sigma(\tau)$ and hence
$g\tau^{-1}\in\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$. Define
$\textsl{g}_{i}=\textsl{g}_{g\tau^{-1},-\sigma^{n-1}(\gamma_{i})-1,{\tilde{w}}_{x^{\prime}},n}$.
As $J_{0}\neq\emptyset$, $\gamma_{i}^{\vee}$ is strongly $J_{1}$-minuscule.
Then it follows from Lemma 6.6 (2), Lemma 5.2 and (c) that

${\textsl{g}_{i}}^{-1}b\sigma(\textsl{g}_{i})\subseteq
U_{-\gamma_{i}-1}{\tilde{w}}_{x^{\prime}}U_{-\sigma^{n}(\gamma_{i})-1}\subseteq
I{\rm{Adm}}(\lambda)I,$

which means
$g\tau^{-1}I=\textsl{g}_{i}(0)\sim_{\lambda,b}\textsl{g}_{i}(\infty)=g\tau^{-1}s_{i}I$,
where $s_{i}=s_{\gamma_{i}+1}\cdots s_{\sigma^{n-1}(\gamma_{i})+1}$. As
$\gamma_{i}^{\vee}$ is $J_{1}$-minuscule and $J_{1}$-dominant, we have
$s_{i}=\tau_{i}y_{i}^{-1}$, where $\tau_{i}\in\Omega_{J_{1}}$ and $y_{i}\in
W_{0}$. Notice that $g\tau^{-1}\tau_{i}\in\mathbb{J}_{b,{\tilde{w}}_{x}}$,
$\tau=\tau_{1}$, and
$\tau^{-1}\tau_{2}=s^{\prime}s\in\mathbb{J}_{{\tilde{w}}_{x}}$. By Lemma 4.2
and Proposition 4.5, there exist $h_{i}\in
H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ such that

$g\tau^{-1}s_{i}I=g\tau^{-1}\tau_{i}y_{i}^{-1}I\sim_{\lambda,b}g\tau^{-1}\tau_{i}h_{i}I.$

In particular, by (b) we have
$gI\sim_{\lambda,b}gh_{1}I\sim_{\lambda,b}g\tau^{-1}I\sim_{\lambda,b}g\tau^{-1}\tau_{2}h_{2}I$,
that is, $\tau^{-1}\tau_{2}h_{2}\in Q$. It follows from (b) and (a) that
$\tau^{-1}\tau_{2}=s^{\prime}s\in Q$ and $s^{\prime}\in Q$ as desired.

Case(2): $\sigma^{n}$ acts nontrivially on $\Psi_{\beta}\cap J_{0}$. By Lemma
6.6 (4), $\Psi=\Phi$ and $\mu_{x}|_{\Psi\setminus\Psi_{\beta}}=0$. So we can
assume that $n=d=1$, $\sigma$ is of order $2$, and $\Phi$ is of type $E_{6}$.
Then $w_{x}=s_{\alpha_{4}}s_{\alpha_{2}}$, and it suffices to show
$s,s^{\prime}\in Q$, where $s=s_{\alpha_{1}}s_{\alpha_{6}}$ and
$s^{\prime}=s_{\alpha_{1}+1}s_{\alpha_{6}+1}$ are all the simple affine
reflections of $W_{J}^{a}\cap\mathbb{J}_{{\tilde{w}}_{x}}$.

Let $\theta_{0}=\alpha_{2}+\alpha_{4}+\alpha_{5}+\alpha_{6}$,
$\theta_{1}=\alpha_{2}+\alpha_{4}+\alpha_{5}$,
$\eta_{i}=(w_{x}\sigma)^{-1}(\theta_{i})$ and
$\vartheta_{i}=\eta_{i}+\theta_{i}$. Define
$\textsl{g}_{i}=\textsl{g}_{g,-\theta_{i}-1,{\tilde{w}}_{x},2}$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$. As
$\mu+\alpha_{3}^{\vee},\mu+\alpha_{3}^{\vee}+\alpha_{1}^{\vee}\preceq\lambda$,
we have ${\tilde{w}}_{x}s_{\sigma(\theta_{i})+1}\in{\rm{Adm}}(\lambda)$ by
Lemma 5.1. Then

$\textsl{g}_{i}^{-1}b\sigma(\textsl{g}_{i})\subseteq
IU_{-\vartheta_{i}-1}{\tilde{w}}_{x}U_{-\sigma(\theta_{i})-1}\subseteq
I{\tilde{w}}_{x}U_{-\sigma(\theta_{i})-1}I\subseteq I{\rm{Adm}}(\lambda)I,$

which means

$gs_{\vartheta_{0}+1}s_{\eta_{0}}I=\textsl{g}_{0}(\infty)\sim_{\lambda,b}\textsl{g}_{0}(0)=gI=\textsl{g}_{1}(0)\sim_{\lambda,b}\textsl{g}_{1}(\infty)=gs_{\vartheta_{1}+1}s_{\eta_{1}}I.$

As $\vartheta_{0}^{\vee}$ is $J$-dominant and $J$-minuscule,
$s_{\vartheta_{0}+1}s_{\eta_{0}}=\omega y_{0}^{-1}$, where
$\omega=\vartheta_{0}^{\vee}\in\Omega_{J}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ and
$y_{0}\in W_{0}$. Then
$s_{\tilde{\vartheta}_{1}}s_{\eta_{1}}=ss_{\tilde{\vartheta}_{0}}s_{\eta_{0}}s=s\omega
y_{1}^{-1}$ for some $y_{1}\in W_{0}$. By Proposition 4.2 & 4.5, there exist
$h_{0},h_{1}\in H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}$ such that $g\omega
h_{0}I\sim_{\lambda,b}gI\sim_{\lambda,b}gs\omega h_{1}I$, that is, $\omega
h_{0},s\omega h_{1}\in Q$, and hence

$\displaystyle s\omega h_{1}h_{0}^{-1}\omega^{-1}\in Q.$ (d)

As $h_{0}h_{1}^{-1}\in H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq
I((W_{J_{0}}W_{J_{1}}^{a})\cap\mathbb{J}_{{\tilde{w}}_{x}})I=I\\{1,s\\}I$ and
$\omega s\omega^{-1}=s^{\prime}$, by (d) we have $s\omega
h_{0}h_{1}^{-1}\omega^{-1}\in Q\cap(I\\{s,ss^{\prime}\\}I)$, which means $s\in
Q$. Hence $H_{x}\cap\mathbb{J}_{{\tilde{w}}_{x}}\subseteq Q$, $\omega\in Q$
and $s^{\prime}=\omega s\omega^{-1}\in Q$ as desired. ∎

######  Corollary 6.8.

Let $x\in\mathcal{S}_{\lambda,b}^{+}$, $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$,
and $y\in\tilde{W}$ such that $gI\sim_{\lambda,b}gy^{-1}I$. Then we have
$gI\sim_{\lambda,b}g\omega{z^{\prime}}^{-1}\sim_{\lambda,b}g\omega I$, where
$z^{\prime}\in W_{0}^{J}$ and $\omega\in\Omega_{J}$ such that $y\in
z^{\prime}\omega^{-1}W_{J}^{a}$.

###### Proof.

It follows from Corollary 4.6 and Proposition 2.4. ∎

### 6.3.

To prove Lemma 6.6, we start with a general lemma on root systems.

######  Lemma 6.9.

Let $\mu\in Y$, $\lambda\in Y^{+}$ and $\alpha\in\Phi^{+}$ such that
$\mu\preceq\lambda$, $\mu+\alpha^{\vee}\leqslant\lambda$, and
$\mu+\alpha^{\vee}\npreceq\lambda$. Then there exists $\beta\in\Phi^{+}$ such
that $\langle\beta,\mu+\alpha^{\vee}\rangle\leqslant-2$, and either
$\mu+\beta^{\vee}\preceq\lambda$ or
$\mu+\alpha^{\vee}+\beta^{\vee}\leqslant\lambda$.

###### Proof.

We argue by induction on $\mu+\alpha^{\vee}$ via the partial order
$\leqslant$. If $\mu+\alpha^{\vee}\in Y^{+}$, then
$\mu+\alpha^{\vee}\preceq\lambda$, contradicting our assumption. So there
exists $\beta\in\SS_{0}$ such that
$\langle\beta,\mu+\alpha^{\vee}\rangle\leqslant-1$ and hence
$\mu+\alpha^{\vee}+\beta^{\vee}\leqslant\lambda$ (by [5, Proposition 2.2]). If
$\langle\beta,\mu+\alpha^{\vee}\rangle\leqslant-2$, the statement follows.
Assume $\langle\beta,\mu+\alpha^{\vee}\rangle=-1$. Then
$\mu+\alpha^{\vee}<s_{\beta}(\mu+\alpha^{\vee})\npreceq\lambda$. If
$\beta=\alpha$, then $\langle\alpha,\mu\rangle=-3$ and
$\mu+\alpha^{\vee}\preceq\mu\preceq\lambda$, a contradiction. So
$\beta\neq\alpha$ and $s_{\beta}(\alpha)\in\Phi^{+}$. By induction hypothesis,
for the pair $(s_{\beta}(\mu),s_{\beta}(\alpha))$ there exists
$\gamma\in\Phi^{+}$ such that

$\langle\gamma,s_{\beta}(\mu+\alpha^{\vee})\rangle=\langle
s_{\beta}(\gamma),\mu+\alpha^{\vee}\rangle\leqslant-2,$

(which means $\beta\neq\gamma$ and $s_{\beta}(\gamma)\in\Phi^{+}$), and either
$s_{\beta}(\mu)+\gamma^{\vee}\preceq\lambda$ or
$s_{\beta}(\mu+\alpha^{\vee})+\gamma^{\vee}\leqslant\lambda$. If the former
case occurs, we have $\mu+s_{\beta}(\gamma^{\vee})\preceq\lambda$, and the
statement follows. Otherwise, $\langle s_{\beta}(\gamma),\mu\rangle\geqslant
0$ and the latter case occurs. In particular, $\langle
s_{\beta}(\gamma),\alpha^{\vee}\rangle\leqslant-2$, and hence means $\gamma$
is a long root. So we have

$\mu+\alpha^{\vee}+s_{\beta}(\gamma^{\vee})\leqslant\mu+\alpha^{\vee}+\gamma^{\vee}+\beta^{\vee}=s_{\beta}(\mu+\alpha^{\vee})+\gamma^{\vee}\leqslant\lambda,$

and the statement also follows. ∎

###### Proof of Lemma 6.6.

By [30, Lemma 3.3], there exists $x\in\mathcal{S}_{\lambda,b}^{+}$ such that
$\mu_{x}$ is weakly dominant, that is,
$\langle\delta,\mu_{x}\rangle\geqslant-1$ for $\delta\in\Phi^{+}$. As
$(\lambda,b)$ is Hodge-Newton irreducible, there exists $\alpha\in K$ such
that $\mu_{x}+\alpha^{\vee}\leqslant\lambda$. We show that

$\displaystyle\text{(a1) there exists }\xi\in\Phi^{+}\setminus\Phi_{J}\text{
such that
}\langle\alpha,\xi^{\vee}\rangle\leqslant-1,\mu+\xi^{\vee}\preceq\lambda;$ (a)
$\displaystyle\text{(a2) if, moreover, $\Phi$ is simply-laced, then
}\langle\xi,\mu_{x}\rangle=-1\text{ and }\beta\in\Phi^{+}\setminus\Phi_{J}.$

By assumption, $\mu_{x}+\alpha^{\vee}\npreceq\lambda$. By Lemma 6.9, there
exists $\zeta\in\Phi^{+}$ such that
$\langle\zeta,\mu_{x}+\alpha^{\vee}\rangle\leqslant-2$, and either
$\mu_{x}+\zeta^{\vee}\preceq\lambda$ or
$\mu_{x}+\alpha^{\vee}+\zeta^{\vee}\leqslant\lambda$. As $\mu_{x}$ is weakly
dominant, we have either (i)
$\langle\zeta,\alpha^{\vee}\rangle\leqslant\langle\zeta,\mu_{x}\rangle=-1$ or
(ii) $\langle\zeta,\alpha^{\vee}\rangle\leqslant-2$ and
$\langle\zeta,\mu_{x}\rangle=0$ or (iii)
$\langle\zeta,\alpha^{\vee}\rangle=-3$ and $\langle\zeta,\mu_{x}\rangle=1$.
Take $\xi=\zeta$ if (i) occurs. Assume (ii) or (iii) occurs. Then $\Phi$ is
non-simply-laced and $\langle\alpha,\zeta^{\vee}\rangle=-1$. If
$\mu_{x}+\zeta^{\vee}\preceq\lambda$, take $\xi=\zeta$. Otherwise,
$\mu_{x}+\zeta^{\vee}\leqslant\lambda$ is not weakly dominant (by [5,
Proposition 2.2]). So there exists $\gamma\in\Phi^{+}$ such that
$\langle\gamma,\mu_{x}+\zeta^{\vee}\rangle\leqslant-2$, which means
$\langle\gamma,\zeta^{\vee}\rangle=\langle\gamma,\mu_{x}\rangle=-1$ since
$\mu_{x}$ is weakly dominant and $\zeta$ is a long root. Then
$\gamma\in\Phi^{+}\setminus\Phi_{J}$ and
$\mu_{x}+\gamma^{\vee}\preceq\lambda$. Note that $\alpha$ is a short root and
$\langle\alpha,\mu_{x}\rangle=0$. If $\langle\alpha,\gamma^{\vee}\rangle=-1$,
we take $\xi=\gamma$. If $\langle\alpha,\gamma^{\vee}\rangle=0$, then (ii)
occurs (since if (iii) occurs, then $\gamma=-3\alpha-2\zeta$, contradicting
that $\langle\gamma,\mu_{x}\rangle=-1$), which means
$\mu_{x}+\gamma^{\vee}+\zeta^{\vee}\preceq\lambda$. So we take
$\xi=s_{\gamma}(\zeta)$. If $\langle\alpha,\gamma^{\vee}\rangle=1$, we take
$\xi=s_{\alpha}(\gamma)$. It remains to show
$\xi\in\Phi^{+}\setminus\Phi_{J}^{+}$. Otherwise, $\xi\in\Phi_{K}$ since
$\langle\alpha,\xi^{\vee}\rangle\neq 0$, contradicting our assumption that
$\mu+\xi^{\vee}\npreceq\lambda$. So (a) is proved.

Let $\beta$ be the $J$-anti-dominant conjugate of $\xi$ under $W_{J}$. By (a)
we have

(b) $\langle\beta,\mu_{x}\rangle=-1$ if $\Phi$ is simply-laced; (c)
$\mu_{x}+\beta^{\vee}\preceq\lambda$; (d) $\beta^{\vee}$ is non-central on
$K_{0}$,

where $K_{0}\subseteq\Psi_{\beta}$ is the connected component of $K$
containing $\alpha$. We show that

$\beta^{\vee}$ is $K$-minuscule. (e)

Otherwise, $\langle\theta,\beta^{\vee}\rangle\leqslant-2$ for some
$\theta\in\Phi_{K}^{+}$. Then
$\mu_{x}+\beta^{\vee}+\theta^{\vee}\preceq\lambda$. If
$\langle\beta,\mu_{x}\rangle\geqslant 0$, then
$\langle\beta,\mu_{x}+\beta^{\vee}+\theta^{\vee}\rangle\geqslant 1$ and
$\mu_{x}+\theta^{\vee}\preceq\lambda$, contradicting our assumption.
Otherwise, $\langle\beta,\mu_{x}\rangle=-1$ and $\langle
s_{\beta}(\theta),\mu_{x}\rangle=-\langle\theta,\beta^{\vee}\rangle\langle\beta,\mu_{x}\rangle\leqslant-2$,
contradicting that $\mu_{x}$ is weakly dominant. So (e) follows.

Applying [30, Lemma 6.6] we can assume furthermore that $\beta^{\vee}$ is
$J$-anti-dominant and $J$-minuscule. Hence (1) is proved.

If $\langle w_{x}\sigma^{i}(\beta),\mu_{x}\rangle\geqslant 1$ for some
$i\in\mathbb{Z}\setminus n\mathbb{Z}$, then
$\mu_{1}:=\mu_{x}+\beta^{\vee}-w_{x}\sigma^{i}(\beta)^{\vee}\preceq\lambda$,
which means
$x_{1}:=x+\beta^{\vee}-\sigma^{i}(\beta)^{\vee}\in\mathcal{S}_{\lambda,b}^{+}$.
By (e), $\mu_{1}$ is non-central on $K_{0}$. As $\mu_{x_{1}},\mu_{1}$ are
conjugate by $W_{J}$ (see Lemma 5.1), $\mu_{x_{1}}$ is also non-central on
$K_{0}$, contradicting that $K_{0}\subseteq J_{0}$. So $\langle
w_{x}\sigma^{i}(\beta),\mu_{x}\rangle\leqslant 0$ for $i\in\mathbb{Z}\setminus
n\mathbb{Z}$. If $\langle\sigma^{i}(\beta),\mu_{x}\rangle\leqslant-1$ for some
$i\in\mathbb{Z}\setminus n\mathbb{Z}$, by Lemma 6.1 there exists $j\in
n\mathbb{Z}$ such that $\langle w_{x}\sigma^{j}(\beta),\mu_{x}\rangle\geqslant
1$. Then
$\mu_{2}:=\mu_{x}-w_{x}\sigma^{j}(\beta)^{\vee}+\sigma^{i}(\beta)^{\vee}\preceq\lambda$
and hence
$x_{2}:=x-\sigma^{j}(\beta)^{\vee}+\sigma^{i}(\beta)^{\vee}\in\mathcal{S}_{\lambda,b}^{+}$,
which is also impossible since $\mu_{2}$ is non-central on
$\sigma^{j}(K_{0})$. So $\langle\sigma^{i}(\beta),\mu_{x}\rangle=\langle
w_{x}\sigma^{i}(\beta),\mu_{x}\rangle=0$ for $i\in\mathbb{Z}\setminus
n\mathbb{Z}$ and (2) is proved.

If $\sigma^{2n}(\beta)\neq\beta$, then $\Phi=\Psi$ and $\Psi_{\beta}$ is of
type $D_{4}$, whose simple roots are
$\beta,\sigma^{n}(\beta),\sigma^{2n}(\beta),\alpha$ with
$\sigma^{n}(\alpha)=\alpha$. Moreover, $J=J_{0}=\mathcal{O}_{\alpha}$. By (2),
we have $\mu_{x}|_{\Psi\setminus\Psi_{\beta}}=0$. Hence
$\sum_{i=0}^{n}\langle\sigma^{i}(\beta),\mu_{x}\rangle\geqslant 1$ by Lemma
6.1. If $\langle\sigma^{n}(\beta),\mu_{x}\rangle\geqslant 1$, then (3)
follows. If $\langle\sigma^{n}(\beta),\mu_{x}\rangle\leqslant-1$, it follows
by replacing $\beta$ with $\sigma^{n}(\beta)$. If
$\langle\sigma^{n}(\beta),\mu_{x}\rangle=0$, it follows by replacing $x$ with
$x-\sigma^{2n}(\beta)^{\vee}+\sigma^{n}(\beta)^{\vee}\in\mathcal{S}_{\lambda,b}^{+}$.

Now we assume $\sigma^{2n}(\beta)=\beta$. By (2) and Lemma 6.1,

$\displaystyle\langle\beta+\sigma^{n}(\beta),{\rm
pr}_{J}(\mu_{x})\rangle=\langle\beta+\sigma^{n}(\beta),{\rm
pr}_{J_{1}}(\mu_{x})\rangle>0.$ (f)

So (3) follows if $\beta=\sigma^{n}(\beta)$. Assume
$\beta\neq\sigma^{n}(\beta)$. Then $\Phi$ is simply-laced, and hence
$\langle\beta,\mu_{x}\rangle=-1$ by (b). Moreover, $\mathcal{O}_{\beta}\cup J$
is a set of simple roots of $\Psi$ by [2, Proposition 4.2.11]. As $\beta$ is a
neighbor of $K_{0}$ in $\Psi_{\beta}$ and $\langle\beta,\mu_{x}\rangle=-1$,
one checks (on the type of $\Psi_{\beta}$) that $\langle\beta,{\rm
pr}_{J_{1}}(\mu_{x})\rangle<0$. By (f) we have $\langle
w_{x}\sigma^{n}(\beta),\mu_{x}\rangle\geqslant 1$ and (3) follows.

Assume $\sigma^{n}$ does not act trivially on $\Psi_{\beta}\cap J_{0}$. Then
$\Phi$ is simply-laced and $\langle\beta,\mu_{x}\rangle=-1$. We may assume
$\sigma^{n}$ does not fix each point of $K_{0}$. Let $\alpha\in K_{0}$ such
that $\langle\beta,\alpha^{\vee}\rangle=-1$. If $\sigma^{n}(\beta)=\beta$,
then one checks directly (on the type of $\Psi_{\beta}$ and using the
assumption on $K_{0}$) that $\langle\beta,{\rm pr}_{J}(\mu_{x})\rangle<0$,
which contradicts (f). So $\beta\neq\sigma^{n}(\beta)\in\Psi_{\beta}$. Let
$x_{3}=x+\beta^{\vee}-\sigma^{n}(\beta)^{\vee}\in\pi_{1}(M_{J})$. If
$\beta,\sigma^{n}(\beta)$ are in distinct connected components of
$\mathcal{O}_{\beta}\cup
J\setminus\\{\alpha,\sigma^{n}(\alpha)\\}\supseteq\mathcal{O}_{\beta}\cup
J_{1}$, then $x_{3}\in\mathcal{S}_{\lambda,b}^{+}$ by (2) that $\langle
w_{x}\sigma^{n}(\beta),\mu_{x}\rangle\geqslant 1$. As
$\langle\alpha,\mu_{x_{3}}\rangle=0$, we deduce that
$\alpha=\sigma^{n}(\alpha)$ is the common neighbor of
$\beta,\sigma^{n}(\beta)$ in $\Psi_{\beta}$, which implies that $\sigma^{n}$
fixes each point of $K_{0}$, contradicting our assumption. So
$\beta,\sigma^{n}(\beta)$ are connected in $\mathcal{O}_{\beta}\cup
J\setminus\\{\alpha,\sigma^{n}(\alpha)\\}$. Then
$\alpha\neq\sigma^{n}(\alpha)$, and it follows from (f) that either $\langle
w_{x}\sigma^{n}(\beta),\mu_{x}\rangle\geqslant 2$ or the case in (4) occurs.
The former case does not occur since $x_{3}\in\mathcal{S}_{\lambda,b}^{+}$ but
$\mu_{x_{3}}$ is non-central on $K_{0}$. So (4) follows. ∎

##  7\. Proof of Proposition 2.5

In this section, we assume that $(\lambda,b)$ is Hodge-Newton irreducible. Let
$\mathcal{O}$ be a $\sigma$-orbit of $\Phi^{+}$. We set

$\omega_{\mathcal{O}}=\sum_{\alpha\in\mathcal{O}}\alpha^{\vee}\in\pi_{1}(M_{J})^{\sigma}\cong\Omega_{J}^{\sigma}\subseteq\Omega_{J}\cap\mathbb{J}_{b}.$

Let $\Psi=\Phi\cap\mathbb{Z}(\mathcal{O}\cup J)$. We say $\mathcal{O}$ is of
type I (resp. type II, resp. type III) if $|\mathcal{O}|$ equals $n$ (resp.
$2n$, resp. $3n$). Here $n\in\\{d,2d,3d\\}$ is the minimal positive integer
such that $\alpha,\sigma^{n}(\alpha)$ are in the same connected component of
$\Psi$ for some/any $\alpha\in\mathcal{O}$. If $\mathcal{O}$ is of type II or
III, then $n=d$, $\Phi$ is simply-laced, and $\mathcal{O}\cap J$ is a set of
simple roots for $\Psi$. In this case, for $\alpha\in\mathcal{O}$ we denote by
$\vartheta_{\alpha}\in\Phi^{+}$ the sum of simple roots in the (unique)
minimal $\sigma^{n}$-stable connected subset of $\mathcal{O}\cup J$ which
contains $\alpha$, see [2, §4.7].

For $x\in\mathcal{S}_{\lambda,b}^{+}$ we define

$C_{\lambda,b,x}=\\{\alpha\in\Phi^{+}\setminus\Phi_{J};\mu_{x}+\alpha^{\vee}\preceq\lambda,\alpha^{\vee}\text{
is $J$-anti-dominant and strongly $J$-minuscule}\\}.$

### 7.1.

Fix a $\sigma$-orbit $\mathcal{O}$ of roots in $\Phi^{+}\setminus\Phi_{J}$
which are $J$-anti-dominant and $J$-minuscule.

######  Lemma 7.1.

Assume $x\overset{(\gamma,r)}{\to}x^{\prime}$ with
$x^{\prime}=x-\gamma^{\vee}+\sigma^{r}(\gamma)^{\vee}\in\mathcal{S}_{\lambda,b}^{+}$
for some $\gamma\in\mathcal{O}$ and $1\leqslant r\leqslant n$. Let
$\omega=\gamma^{\vee}+\cdots+\sigma^{r-1}(\gamma)^{\vee}\in\pi_{1}(M_{J})\cong\Omega_{J}$.
Then for $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ we have

$gI\sim_{\lambda,b}gy^{-1}I\sim_{\lambda,b}g\omega I\text{ for some }y\in
W_{0}^{J}\omega^{-1}W_{J}^{a}\text{ if
}U_{-w_{J}(\gamma)-1}{\tilde{w}}_{x}U_{-w_{J}\sigma^{r}(\gamma)-1}\subseteq
I{\rm{Adm}}(\lambda)I.$

In particular, if $\mathcal{O}=\mathcal{O}_{\alpha}$ for some $\alpha\in
C_{\lambda,b,x}$, the above inclusion holds if (1) $1\leqslant r\leqslant
n-1$; or (2) $x=x^{\prime}$; or (3)
$\mu_{x}+\vartheta_{\gamma}^{\vee}\npreceq\lambda$ when $\mathcal{O}$ is of
type II and $r=n$.

###### Proof.

Let $\tilde{\theta}=w_{J}\sigma^{r-1}(\gamma)+1\in\tilde{\Phi}^{+}$. Suppose

$\displaystyle
U_{-\sigma^{1-r}(\tilde{\theta})}{\tilde{w}}_{x}U_{-\sigma(\tilde{\theta})}=U_{-w_{J}(\gamma)-1}{\tilde{w}}_{x}U_{-w_{J}\sigma^{r}(\gamma)-1}\subseteq
I{\rm{Adm}}(\lambda)I.$ (a)

By [30, Lemma 6.5] we can assume that

$\displaystyle x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime},\text{ and
hence
}({\tilde{w}}_{x}\sigma)^{i}(\tilde{\theta})=\sigma^{i}(\tilde{\theta})\text{
for }1-r\leqslant i\leqslant 0.$ (b)

Define $\textsl{g}=\textsl{g}_{g,-\tilde{\theta},{\tilde{w}}_{x},r}$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$. By (a) and (b) we have

$\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq
U_{-\sigma^{1-r}(\tilde{\theta})}{\tilde{w}}_{x}U_{-\sigma(\tilde{\theta})}\subseteq
I{\rm{Adm}}(\lambda)I,$

which means

$gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=gs_{\tilde{\theta}}\cdots
s_{\sigma^{r-1}(\tilde{\theta})}I=g\omega u^{-1}I$

for some $u\in W_{0}$ as desired. The relation $gI\sim_{\lambda,b}g\omega I$
follows from Corollary 6.8.

If $\mathcal{O}$ is of type II and $r=n$, then $\vartheta_{\gamma}^{\vee}$ is
$J$-anti-dominant and $J$-minuscule, which means
$\mu_{x}+\vartheta_{\gamma}^{\vee}$ is $J$-minuscule and hence
$\mu_{x}+\vartheta_{\gamma}^{\vee}\preceq\mu_{x}+(w_{J}(\gamma)+w_{x}w_{J}\sigma^{r}(\gamma))^{\vee}$.
Thus the “ Moreover” part follows from Lemma 5.2 (*) by noticing that $\langle
w_{J}(\gamma),w_{x}w_{J}\sigma^{r}(\gamma^{\vee})\rangle=0$ if $1\leqslant
r\leqslant d-1$. ∎

Let $\mathcal{A}_{\lambda,b}$ be the group of elements
$\omega\in\pi_{1}(M_{J})^{\sigma}\cong\Omega_{J}^{\sigma}$ which fix some/any
connected component of $X(\lambda,b)$.

######  Lemma 7.2.

Suppose $\mathcal{O}=\mathcal{O}_{\xi}$ for some $\xi\in C_{\lambda,b,x}$. If
$\mathcal{O}$ is of type I, then there exist $\gamma\in\mathcal{O}$,
$1\leqslant r\leqslant n$, and $x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such
that $x\overset{(\gamma,r)}{\to}x^{\prime}$. Moreover,
$\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$.

###### Proof.

Note that $\mu+\alpha^{\vee}\preceq\lambda$. If $\langle
w_{J}\sigma^{r}(\alpha),\mu_{x}\rangle\geqslant 1$ for some $1\leqslant
r\leqslant n-1$, then $\langle
w_{J}\sigma^{r}(\alpha),\alpha^{\vee}\rangle=0$, which means
$x\overset{(\sigma^{r}(\alpha),n-r)}{\to}x^{\prime}\overset{(\alpha,r)}{\to}x$
with
$x^{\prime}=x-\sigma^{r}(\alpha^{\vee})+\alpha^{\vee}\in\mathcal{S}_{\lambda,b}^{+}$.
Otherwise, $\langle w_{J}\sigma^{i}(\alpha),\mu_{x}\rangle\leqslant 0$ for
$1\leqslant i\leqslant n-1$, which means $\langle
w_{J}(\alpha),\mu_{x}\rangle\geqslant 1$ by Lemma 6.1. So
$x\overset{(\alpha,n)}{\to}x$ and the first statement follows. As
$\mathcal{O}$ is of type I, the second statement follows from Corollary 6.8
and Lemma 7.1 (1) (resp. Lemma 7.1 (2)) if $r\neq n$ (resp. $r=n$). ∎

######  Lemma 7.3.

Suppose $\mathcal{O}$ is of type II. Assume
$\mu_{x^{\prime\prime}}+\vartheta_{\beta}^{\vee}\npreceq\lambda$ for any
$x^{\prime\prime}\in\mathcal{S}_{\lambda,b}^{+}$ and $\beta\in\mathcal{O}$. If
there exist $\gamma\in\mathcal{O}$, $n+1\leqslant r\leqslant 2n-1$, and
$x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such that
$x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime}$, then

(1) $\langle\sigma^{i}(\gamma),\mu_{x}\rangle=0$,
$w_{x}\sigma^{i}(\gamma)=\sigma^{i}(\gamma)$ for $1\leqslant i\neq
r-n\leqslant r-1$;

(2)
$w_{x}\sigma^{r-n}(\gamma)=\sigma^{r-n}(\vartheta_{\gamma}-\sigma^{n}(\gamma))$
and $\langle w_{x}\sigma^{r-n}(\gamma),\mu_{x}\rangle=1$;

(3) $\langle
w_{x}(\vartheta_{\gamma}-\sigma^{n}(\gamma)),\mu_{x}\rangle\geqslant 1$.

Moreover, $gI\sim_{\lambda,b}g\omega I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$, where
$\omega=\gamma^{\vee}+\cdots\sigma^{r-1}(\gamma^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.

###### Proof.

Write
$x^{\prime}=x+\sigma^{r}(\gamma^{\vee})-\sigma^{-r}(\sigma^{r}(\gamma^{\vee}))$.
Then (1), (2) and (3) follow from [30, Lemma 8.2] by using $\sigma^{-1}$
instead of $\sigma$. Let
$\tilde{\theta}=w_{J}\sigma^{r-1}(\gamma)+1\in\tilde{\Phi}^{+}$ and
$\tilde{\vartheta}_{\gamma}=\vartheta_{\gamma}+1\in\tilde{\Phi}^{+}$. By (1)
and (2) we have
$({\tilde{w}}_{x}\sigma)^{i}(\tilde{\theta})=\sigma^{i}(\tilde{\theta})=w_{J}\sigma^{i+r-1}(\gamma)+1$
for $1-n\leqslant i\leqslant 0$, and

$({\tilde{w}}_{x}\sigma)^{i}(\tilde{\theta})=\sigma^{i+n-1}w_{x}^{-1}w_{J}\sigma^{r-n}(\gamma)=\sigma^{i+n-1}w_{J}w_{x}\sigma^{r-n}(\gamma)=w_{J}\sigma^{i+r-1}(\vartheta_{\gamma}-\sigma^{n}(\gamma))$

for $1-r\leqslant i\leqslant-n$. Define
$\textsl{g}=\textsl{g}_{g,-\tilde{\theta},{\tilde{w}}_{x},r}$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$. Then we have

$\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq
IU_{-w_{J}(\tilde{\vartheta}_{\gamma})}{\tilde{w}}_{x}U_{-\sigma(\tilde{\theta})}I\subseteq
I{\tilde{w}}_{x}U_{-\sigma(\tilde{\theta})}I\subseteq I{\rm{Adm}}(\lambda)I,$

where the second inclusion follows from (1) and (3) that $\langle
w_{x}(\vartheta_{\gamma}),\mu_{x}\rangle\geqslant 1$. Thus

$gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=gs_{({\tilde{w}}_{x}\sigma)^{1-r}(\tilde{\theta})}\cdots
s_{({\tilde{w}}_{x}\sigma)^{-1}(\tilde{\theta})}s_{\tilde{\theta}}I=g\omega
u^{-1}I,$

where $u\in W_{0}$ and
$\omega=\gamma^{\vee}+\cdots+\sigma^{r-1}(\gamma^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.
By Corollary 6.8 we have $gI\sim_{\lambda,b}g\omega I$ as desired. ∎

######  Lemma 7.4.

Suppose $\mathcal{O}=\mathcal{O}_{\xi}$ for some $\xi\in C_{\lambda,b,x}$ and
$\mathcal{O}$ is of type II. Assume
$\mu_{x^{\prime\prime}}+\vartheta_{\beta}^{\vee}\npreceq\lambda$ for any
$x^{\prime\prime}\in\mathcal{S}_{\lambda,b}^{+}$ and $\beta\in\mathcal{O}$. If
there do not exist $\gamma\in\mathcal{O}$, $1\leqslant r\leqslant 2n-1$, and
$x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such that
$x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime}$. Then there exists
$\alpha\in\mathcal{O}$ such that

(1) $\langle\sigma^{i}(\alpha),\mu_{x}\rangle=0$,
$w_{x}\sigma^{i}(\alpha)=\sigma^{i}(\alpha)$ for $1\leqslant i\neq n\leqslant
2n-1$;

(2) $w_{x}\sigma^{n}(\alpha)=\vartheta_{\alpha}-\alpha$ and $\langle
w_{J}\sigma^{n}(\alpha),\mu_{x}\rangle=1$;

(3) $\langle w_{x}(\vartheta_{\alpha}),\mu_{x}+\alpha^{\vee}\rangle\geqslant
1$;

(4) $\langle w_{x}(\vartheta_{\alpha}),\mu_{x}\rangle\geqslant 1$.

As a consequence, $\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$.

###### Proof.

The statements (1), (2) and (3) follow from [30, Lemma 8.3 & Lemma 8.4]. Note
that $\vartheta_{\alpha}$ is $J$-anti-dominant. So (4) follows from (1) and
Lemma 6.1. By (3) and (4) we have

$\displaystyle\langle w_{x}(\vartheta_{\alpha}),\mu_{x}\rangle\geqslant
1,\text{ and either }\langle w_{x}(\vartheta_{\alpha}),\mu_{x}\rangle\geqslant
2\text{ or }\langle w_{x}(\vartheta_{\alpha}),\alpha^{\vee}\rangle\geqslant
0.$ (a)

Let $g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ and
$\tilde{\theta}=w_{J}\sigma^{-1}(\alpha)+1\in\tilde{\Phi}^{+}$, and
$\tilde{\vartheta}=w_{J}\sigma^{-1}(\vartheta_{\alpha})+1\in\tilde{\Phi}^{+}$.
By (1) and (2) we have
$({\tilde{w}}_{x}\sigma)^{1-n}(\tilde{\vartheta})=\sigma^{1-n}(\tilde{\vartheta})=w_{J}(\vartheta_{\alpha})$
and

$({\tilde{w}}_{x}\sigma)^{-n}(\tilde{\theta})=\sigma^{-1}w_{x}^{-1}w_{J}\sigma^{-n}(\alpha)=\sigma^{-1}w_{J}w_{x}\sigma^{-n}(\alpha)=w_{J}\sigma^{-1}(\vartheta_{\alpha}-\alpha).$

Define $\textsl{g}:\mathbb{P}^{1}\to G({\breve{F}})/I$ by

$\textsl{g}(z)=gU_{-\tilde{\theta}}(z)\cdots{}^{({\tilde{w}}_{x}\sigma)^{1-n}}U_{-\tilde{\theta}}(z)U_{-\tilde{\vartheta}}(cz^{1+q^{-n}})\cdots{}^{({\tilde{w}}_{x}\sigma)^{1-n}}U_{-\tilde{\vartheta}}(cz^{1+q^{-n}})I,$

where $c\in\mathcal{O}_{\breve{F}}^{\times}$ (as $\Phi$ is simply-laced) such
that

${}^{({\tilde{w}}_{x}\sigma)^{-n}}U_{-\tilde{\theta}}(z)U_{-\tilde{\theta}}(z)U_{-\tilde{\vartheta}}(cz^{1+q^{-n}})=U_{-\tilde{\theta}}(z){}^{({\tilde{w}}_{x}\sigma)^{-n}}U_{-\tilde{\theta}}(z).$

Then by (1) we compute that

$\textsl{g}^{-1}b\sigma(\textsl{g})=U_{-w_{J}(\tilde{\vartheta}_{\alpha})}{\tilde{w}}_{x}U_{-\sigma(\tilde{\theta})}I\subseteq
I{\tilde{w}}_{x}U_{-\sigma(\tilde{\theta})}I\subseteq
I{\rm{Adm}}(\lambda)\sigma I,$

where the first inclusion follows by (a) that
${}^{{\tilde{w}}_{x}^{-1}}U_{-w_{J}(\tilde{\vartheta}_{\alpha})},[{}^{{\tilde{w}}_{x}^{-1}}U_{-w_{J}(\tilde{\vartheta}_{\alpha})},U_{-\sigma(\tilde{\theta})}]\subseteq
I$. Thus

$gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=g(s_{\tilde{\vartheta}}s_{\theta^{\prime}})\cdots\sigma^{1-n}(s_{\tilde{\vartheta}}s_{\theta^{\prime}})I=g\omega_{\mathcal{O}}u^{-1}I,$

where
$\theta^{\prime}=({\tilde{w}}_{x}\sigma)^{1-n}(\tilde{\vartheta})\in\Phi$ and
$u\in W_{0}$. By Corollary 6.8 we have
$gI\sim_{\lambda,b}g\omega_{\mathcal{O}}I$ and
$\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$ as desired. ∎

### 7.2.

Now we have the following result.

######  Proposition 7.5.

Let $\mathcal{O}$ be the $\sigma$-orbit of some element in
$\cup_{x\in\mathcal{S}_{\lambda,b}^{+}}C_{\lambda,b,x}$. Then
$\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$.

###### Proof.

If $\mathcal{O}$ is of type I, the statement follows from Lemma 7.2. If
$\mu_{x^{\prime\prime}}+\vartheta_{\beta}^{\vee}\preceq\lambda$ for some
$x^{\prime\prime}\in\mathcal{S}_{\lambda,b}^{+}$ and $\beta\in\mathcal{O}$,
then we also have
$\omega_{\mathcal{O}}=\omega_{\mathcal{O}_{\vartheta_{\beta}}}\in\mathcal{A}_{\lambda,b}$
since $\mathcal{O}_{\vartheta_{\beta}}$ is of type I. Assume
$\mu_{x^{\prime\prime}}+\vartheta_{\beta}^{\vee}\npreceq\lambda$ for any
$x^{\prime\prime}\in\mathcal{S}_{\lambda,b}^{+}$ and $\beta\in\mathcal{O}$. If
$\mathcal{O}$ is of type III, the statement is proved in §8.2. Suppose
$\mathcal{O}$ is of type II. By Lemma 7.4 we can assume that there exist
$\gamma\in\mathcal{O}$, $1\leqslant r\leqslant 2n-1$, and
$x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such that
$x\overset{(\gamma,r)}{\to}x^{\prime}$, and hence
$x^{\prime}\overset{(\sigma^{r}(\gamma),2n-r)}{\to}x$. If $n+1\leqslant
r\leqslant 2n-1$ (resp. $1\leqslant r\leqslant n$), we have
$gI\sim_{\lambda,b}g\omega I$ by Lemma 7.3 (resp. by Lemma 7.1 (1) & (3)),
where
$\omega=\gamma^{\vee}+\cdots\sigma^{r-1}(\gamma^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.
Similarly, we have
$g\omega\sim_{\lambda,b}g\omega\omega^{\prime}I=g\omega_{\mathcal{O}}I$, where
$\omega^{\prime}=\sigma^{r}(\gamma^{\vee})+\cdots+\sigma^{2n-1}(\gamma^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.
So $gI\sim_{\lambda,b}g\omega_{\mathcal{O}}I$ and
$\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$ as desired. ∎

###### Proof of Proposition 2.5.

First note that $(\mathbb{Z}\Phi^{\vee}/\mathbb{Z}\Phi_{J}^{\vee})^{\sigma}$
is panned by $\omega_{\mathcal{O}}$, where $\mathcal{O}$ ranges over
$\sigma$-orbits of $\SS_{0}$. Let $J\subseteq\SS_{0}^{\prime}\subseteq\SS_{0}$
be such that $\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$ for each
$\sigma$-orbit of $\SS_{0}^{\prime}$. It suffices to show
$\SS_{0}^{\prime}=\SS_{0}$. Assume otherwise. Following the proof of [30,
Proposition 4.3], we can assume that $\Phi$ is simply-laced, and there exist
$\alpha=\sigma^{d}(\alpha)\in\SS_{0}\setminus\SS_{0}^{\prime}$,
$\vartheta=\sigma^{d}(\vartheta)\in\Phi^{+}$ such that
$\vartheta^{\vee}-\alpha^{\vee}\in\mathbb{Z}\Phi_{\SS_{0}^{\prime}}^{\vee}$
and either (b1)
$\vartheta\in\cup_{x\in\mathcal{S}_{\lambda,b}^{+}}C_{\lambda,b,x}$, or (b2)
$x\overset{(\beta,d)}{\to}x^{\prime}$ and
$x\overset{(\vartheta+\beta,d)}{\to}x^{\prime}$ for some
$x\in\mathcal{S}_{\lambda,b}^{+}$ and
$\beta\in\Phi_{\SS_{0}^{\prime}}\setminus\Phi_{J}$ such that
$x^{\prime}=x-\beta^{\vee}+\sigma^{d}(\beta^{\vee})\in\mathcal{S}_{\lambda,b}^{+}$
and $\vartheta+\beta\in\Phi^{+}$.

Note that $|\mathcal{O}_{\alpha}|=|\mathcal{O}_{\vartheta}|=d$ and
$\omega_{\mathcal{O}_{\alpha}}^{-1}\omega_{\mathcal{O}_{\vartheta}}\in(\mathbb{Z}\Phi_{\SS_{0}^{\prime}}^{\vee}/\mathbb{Z}\Phi_{J}^{\vee})^{\sigma}\subseteq\mathcal{A}_{\lambda,b}$.
If (b1) occurs, then
$\omega_{\mathcal{O}_{\vartheta}}\in\mathcal{A}_{\lambda,b}$ by Proposition
7.5. Hence $\omega_{\mathcal{O}}\in\mathcal{A}_{\lambda,b}$ and
$\alpha\in\SS_{0}^{\prime}$, which is a contradiction. Suppose (b2) occurs.
Let
$\omega=\beta^{\vee}+\cdots+\sigma^{d-1}(\beta^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.
Then
$\omega\omega_{\mathcal{O}^{\prime}}=(\beta+\vartheta)^{\vee}+\cdots+\sigma^{d-1}((\beta+\vartheta)^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.
We claim that

$\displaystyle g\omega
I\sim_{\lambda,b}gI\sim_{\lambda,b}g\omega\omega_{\mathcal{O}_{\vartheta}}I\text{
for }g\in\mathbb{J}_{b,{\tilde{w}}_{x}}.$ (a)

Given (a) we have $g\omega
I\sim_{\lambda,b}g\omega\omega_{\mathcal{O}_{\vartheta}}I$, and hence
$\omega_{\mathcal{O}_{\vartheta}}\in\mathcal{A}_{\lambda,b}$, which is again a
contradiction. Thus $\SS_{0}^{\prime}=\SS_{0}$ as desired.

It remains to show (a). By symmetry, it suffices to show
$gI\sim_{\lambda,b}g\omega I$. By switching $x$ with $x^{\prime}$ we can
assume $\beta\in\Phi^{+}\setminus\Phi_{J}$ and $\beta$ is $J$-anti-dominant
and $J$-minuscule (see [30, Lemma 6.6]). In particular, $\sigma^{d}(\beta)\in
C_{\lambda,b,x}$. If $\mathcal{O}_{\beta}$ is of type I, it follows from Lemma
7.1. If $\mathcal{O}_{\beta}$ is of type III, it follows from Lemma 8.1. If
$\mathcal{O}_{\beta}$ is of type II, by Lemma 5.2 and Lemma 7.1 we have

$\text{either }gI\sim_{\lambda,b}g\omega I\text{ or
}g\omega\sim_{\lambda,b}g\omega\omega^{\prime}I=g\omega_{\mathcal{O}_{\beta}}I\text{
for }g\in\mathbb{J}_{b,{\tilde{w}}_{x}},$

where
$\omega^{\prime}=\sigma^{d}(\beta^{\vee})+\cdots+\sigma^{2d-1}(\beta^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.
Note that $gI\sim_{\lambda,b}g\omega_{\mathcal{O}_{\beta}}I$ by Proposition
7.5. So we always have $gI\sim_{\lambda,b}g\omega I$ as desired. So (a) is
proved. ∎

##  8\. The case that $\sigma$ has order $3d$

We assume that $\sigma$ has order $3d$. Then some/any connected component of
$\SS_{0}$ is of type $D_{4}$.

### 8.1.

Let $\alpha,\beta\in\SS_{0}$ such that $\langle\alpha,\beta^{\vee}\rangle=-1$
and $\beta=\sigma^{d}(\beta)$. Then the subset
$\\{\alpha,\sigma^{d}(\alpha),\sigma^{2d}(\alpha),\beta\\}$ is a connected
component of $\SS_{0}$. Assume $J=\mathcal{O}_{\beta}$.

Let $x,x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$ such that
$x\overset{(\alpha,r)}{\to}x^{\prime}$ for some $J$-anti-dominant root
$\alpha\in\Phi^{+}\setminus\Phi_{J}$ and $1\leqslant r\leqslant 3d-1$. Let
$\omega=\gamma^{\vee}+\cdots+\sigma^{r-1}(\gamma)^{\vee}\in\pi_{1}(M_{J})\cong\Omega_{J}$.

######  Lemma 8.1.

If $1\leqslant r\leqslant d$, then $gI\sim_{\lambda,b}gy^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ and some $y\in
W_{0}^{J}\omega^{-1}W_{J}^{a}$.

###### Proof.

As in the proof Lemma 7.1, we can assume
$x\overset{(\alpha,r)}{\rightarrowtail}x^{\prime}$, and it suffices to show

$U_{-(\alpha+\beta)-1}{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}\subseteq
I{\rm{Adm}}(\lambda)I.$

Assume otherwise. Then $r=d$. Moreover, by Lemma 5.2 (*) we have
$\langle\alpha+\beta,w_{x}\sigma^{d}(\alpha+\beta)^{\vee}\rangle=-1$ (which
implies $\langle\beta,\mu_{x}\rangle=1$ and
$w_{x}\sigma^{d}(\alpha+\beta)=s_{\beta}(\sigma^{d}(\alpha)+\beta)=\sigma^{d}(\alpha)$)
and

$\langle\beta,\mu_{x}\rangle=\langle\alpha+\beta,\mu_{x}\rangle=-\langle\sigma^{d}(\alpha),\mu_{x}\rangle=1,\text{
and }\mu_{x}\pm\delta^{\vee}\preceq\lambda,$

where $\delta=\alpha+\beta+\sigma^{d}(\alpha)$. As $\delta$ is central for
$J=\mathcal{O}_{\beta}$, by Lemma 5.1 (2) we have

$U_{-(\alpha+\beta)-1}{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}\subseteq
IU_{-(\delta+1)}{\tilde{w}}_{x}I\subseteq
I\\{s_{\delta+1}{\tilde{w}}_{x},{\tilde{w}}_{x}\\}I\subseteq
I{\rm{Adm}}(\lambda)I,$

which is a contradiction. ∎

######  Lemma 8.2.

Suppose $2d\leqslant r\leqslant 3d-1$ and the following conditions hold:

(1) $\langle\alpha,\mu_{x}\rangle\geqslant 1$;

(2) if $r=2d$, then $\langle\sigma^{d}(\alpha),\mu_{x}\rangle=0$;

(3) if $2d+1\leqslant r\leqslant 3d-1$, then
$\langle\sigma^{r}(\beta),\mu_{x}\rangle=1$, $\langle\beta,\mu_{x}\rangle=0$,
and $\langle\sigma^{i}(\alpha),\mu_{x}\rangle=0$ for
$i\in\\{r-d,r-2d,d,2d\\}$;

(4) ${\tilde{w}}_{x}\sigma^{i}(\alpha)=\sigma^{i}(\alpha)$ for $1\leqslant
i\leqslant r-1$ with $i\notin\\{r-d,r-2d,d,2d\\}$.

Then we have $gI\sim_{\lambda,b}gy^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ and some $y\in
W_{0}^{J}\omega^{-1}W_{J}^{a}$.

###### Proof.

Let $\tilde{\theta}=\sigma^{r-1}(\alpha+\beta)+1\in\tilde{\Phi}^{+}$. Define
$\textsl{g}=\textsl{g}_{g,-\tilde{\theta},{\tilde{w}}_{x},r}$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$.

Case(1): $r=2d$. By (2) and (4) we have

$\displaystyle\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq\begin{cases}IU_{-(\alpha+\beta+\sigma^{d}(\alpha))-1}{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}I,&\text{
if }\langle\beta,\mu_{x}\rangle=1;\\\
IU_{-(\alpha+\beta)-1}{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}I,&\text{
if }\langle\beta,\mu_{x}\rangle=0;\end{cases}$

By (1) and (2),
$\langle\alpha+\beta,\mu_{x}\rangle=\langle\alpha+\beta+\sigma^{d}(\alpha),\mu_{x}\rangle\geqslant\langle\beta,\mu_{x}\rangle+1$,
which means

$\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}I\subseteq
I{\rm{Adm}}(\lambda)I.$

So $g=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=gsI$, where
$s=\prod_{i=0}^{d-1}s_{\sigma^{i}(\alpha+\beta+\sigma^{d}(\alpha))+1}\prod_{i=0}^{d-1}s_{\sigma^{i}(\alpha)}$
if $\langle\beta,\mu_{x}\rangle=1$, and
$s=\prod_{i=0}^{2d-1}s_{\sigma^{i}(\alpha+\beta)+1}$ if
$\langle\beta,\mu_{x}\rangle=0$.

Case(2): $2d+1\leqslant r\leqslant 3d-1$. Let
$\vartheta=\alpha+\sigma^{d}(\alpha)+\sigma^{2d}(\alpha)+2\beta$. By (3) and
(4),

$\displaystyle\textsl{g}^{-1}b\sigma(\textsl{g})\subseteq
IU_{-\vartheta-1}{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}I\subseteq
I{\tilde{w}}_{x}U_{-\sigma^{r}(\alpha+\beta)-1}I\subseteq
I{\rm{Adm}}(\lambda)I,$

which means $gI=\textsl{g}(0)\sim_{\lambda,b}\textsl{g}(\infty)=gsI$, where

$s=\prod_{i=0}^{r-1}s_{\sigma^{i}(\vartheta)+1}s_{\sigma^{i}(\alpha+\beta)}s_{\sigma^{i+d}(\alpha)}\prod_{i=r}^{d-1}s_{\sigma^{i}(\alpha+\beta+\sigma^{d}(\alpha))+1}s_{\sigma^{i}(\alpha+\beta)}.$

The proof is finished. ∎

The following two lemmas follow from the same construction in Lemma 8.2.

######  Lemma 8.3.

Assume $d+1\leqslant r\leqslant 2d-1$ and the following conditions hold:

(1) $\langle\beta,\mu_{x}\rangle=0$ and
$\langle\sigma^{r}(\beta),\mu_{x}\rangle\in\\{0,1\\}$;

(2)
$\langle\sigma^{d}(\alpha),\mu_{x}\rangle=\langle\sigma^{r-d}(\alpha),\mu_{x}\rangle=0$,
and $\langle\alpha,\mu_{x}\rangle\geqslant 1$;

(3) ${\tilde{w}}_{x}\sigma^{i}(\alpha)=\sigma^{i}(\alpha)$ for $1\leqslant
i\leqslant r-1$ with $i\notin\\{r-d,d\\}$.

Then we have $gI\sim_{\lambda,b}gy^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ and some $y\in W_{0}^{J}\omega
W_{J}^{a}$.

######  Lemma 8.4 ([30, Lemma 8.6]).

Suppose $\langle\beta,\mu_{x}\rangle=1$,
$\langle\sigma^{d}(\alpha),\mu_{x}\rangle=\langle\sigma^{2d}(\alpha),\mu_{x}\rangle=0$,
$\langle\alpha,\mu_{x}\rangle\geqslant-1$, and
${\tilde{w}}_{x}\sigma^{i}(\alpha)=\alpha$ for $i\in\mathbb{Z}\setminus
d\mathbb{Z}$. Then $gI\sim_{\lambda,b}gy^{-1}I$ for
$g\in\mathbb{J}_{b,{\tilde{w}}_{x}}$ and some $y\in
W_{0}\omega_{\mathcal{O}_{\alpha}}^{-1}W_{J}^{a}$. Here
$\omega_{\mathcal{O}_{\alpha}}=\alpha^{\vee}+\cdots\sigma^{3d-1}(\alpha^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.

######  Lemma 8.5.

Let $x_{1},x_{2}\in\mathcal{S}_{\lambda,b}^{+}$,
$\delta=\alpha+\beta+\sigma^{2d}(\alpha)$ and $1\leqslant k\leqslant 3d-1$
such that $x_{1}\overset{(\delta,k)}{\to}x_{2}$. Then we have
$gI\sim_{\lambda,b}gy^{-1}I$ for $g\in\mathbb{J}_{b,{\tilde{w}}_{x_{1}}}$ and
some $y\in W_{0}^{J}\omega^{-1}W_{J}^{a}$. Here
$\omega=\delta^{\vee}+\cdots\sigma^{k-1}(\delta^{\vee})\in\pi_{1}(M_{J})\cong\Omega_{J}$.

###### Proof.

It follows from Lemma 7.2 by noticing that $\mathcal{O}_{\delta}$ is of type
I. ∎

######  Lemma 8.6.

Assume $d+1\leqslant r\leqslant 2d-1$ and the following conditions hold:

(1) $\langle\beta,\mu_{x}\rangle=1$ and
$\langle\sigma^{r}(\beta),\mu_{x}\rangle=0$;

(2) $\langle\sigma^{d}(\alpha),\mu_{x}\rangle=-1$,
$\langle\sigma^{r-d}(\alpha),\mu_{x}\rangle=0$,
$\langle\alpha,\mu_{x}\rangle\leqslant 0$, and
$\langle\sigma^{r}(\alpha),\mu_{x}\rangle\leqslant-1$;

(3) ${\tilde{w}}_{x}\sigma^{i}(\alpha)=\sigma^{i}(\alpha)$ for $1\leqslant
i\leqslant r-1$ with $i\notin\\{r-d,d\\}$

Then we have
$\mathbb{J}_{b,{\tilde{w}}_{x}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.

###### Proof.

Let $\delta=\alpha+\beta+\sigma^{2d}(\alpha)$. Assume
$\mu_{x}-\delta^{\vee}\preceq\lambda$. By (2) we have

$x\overset{(\delta,r)}{\to}x^{\prime\prime}:=x-\delta^{\vee}+\sigma^{r}(\delta^{\vee})\overset{(\sigma^{r-d}(\alpha),3d-r)}{\to}x^{\prime}.$

So
$\mathbb{J}_{b,{\tilde{w}}_{x}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime\prime}}}$
by Lemma 8.5. It suffices to show
$\mathbb{J}_{b,{\tilde{w}}_{x^{\prime\prime}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.
If $\langle\sigma^{r}(\sigma),\mu_{x^{\prime\prime}}\rangle\leqslant-1$, then

$x^{\prime\prime}\overset{(\sigma^{r-d},d)}{\to}x^{\prime\prime}-\sigma^{r-d}(\alpha^{\vee})+\sigma^{r}(\alpha^{\vee})\overset{(\sigma^{r}(\alpha),2d-r)}{\to}x^{\prime},$

and the statement follows from Lemma 8.1 that
$\mathbb{J}_{b,{\tilde{w}}_{x^{\prime\prime}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.
Otherwise, by (2) we have $\langle\sigma^{r}(\alpha),\mu_{x}\rangle=-1$, that
is, $\langle\sigma^{r}(\sigma),\mu_{x^{\prime\prime}}\rangle=0$. the statement
follows from Lemma 8.3 that
$\mathbb{J}_{b,{\tilde{w}}_{x^{\prime\prime}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.
Let $l=\min\\{r+1\leqslant i\leqslant
2d-1;\langle\sigma^{i}(\alpha),\mu_{x}\rangle\neq 0\\}$. If
$\langle\sigma^{l}(\alpha),\mu_{x}\rangle\geqslant 1$, then

$x^{\prime\prime}\overset{(\sigma^{l}(\alpha),2d-1)}{\to}x^{\prime\prime}-\sigma^{l}(\alpha^{\vee})+\sigma^{2d}(\alpha^{\vee})\overset{(\sigma^{r-d}(\alpha),l+d-r)}{\to}x^{\prime},$

and the statement follows from Lemma 8.1 & 8.3. If
$\langle\sigma^{l}(\alpha),\mu_{x}\rangle\leqslant-1$, then

$x^{\prime\prime}\overset{(\sigma^{r-d}(\alpha),k+d-r)}{\to}x^{\prime\prime}-\sigma^{r-d}(\alpha^{\vee})+\sigma^{l}(\alpha^{\vee})\overset{(\sigma^{l}(\alpha),2d-l)}{\to}x^{\prime},$

and the statement also follows from Lemma 8.1 & 8.3.

Now we assume $\mu_{x}-\delta^{\vee}\npreceq\lambda$, which means (as
$\mu_{x}-\alpha^{\vee}-\beta^{\vee}=\mu_{x-\alpha^{\vee}}\preceq\lambda$) that

$\displaystyle\langle\sigma^{2d}(\alpha),\mu_{x}\rangle\leqslant-1.$ (a)

If $\langle\sigma^{r+d}(\alpha),\mu_{x}\rangle\geqslant 1$, then we have

$x\overset{(\sigma^{r-d}(\delta),d)}{\to}x-\sigma^{r+d}(\alpha)^{\vee}+\sigma^{r}(\alpha)^{\vee}\overset{(\sigma^{r+d}(\alpha),2d-r)}{\leftarrow}x^{\prime},$

and the statement follows from Lemma 8.5 and Lemma 8.1. So we assume

$\displaystyle\langle\sigma^{r+d}(\alpha),\mu_{x}\rangle\leqslant 0.$ (b)

By (a), (b), (1), and (2), we have

$\sum_{i\in\\{r-d,r,r+d,0,d,2d\\}}\langle\sigma^{i}(\alpha),{\rm
pr}_{J}(\mu_{x})\rangle<0.$

By Lemma 6.1, there exists $r+1\leqslant k\leqslant 3d-1$ with
$k\notin\\{2d,r+d\\}$ such that

$\displaystyle k=\min\\{r+1\leqslant i\leqslant
3d-1;\langle\sigma^{i}(\alpha),\mu_{x}\rangle\geqslant 1\\}.$ (c)

Suppose $\langle\sigma^{j}(\alpha),\mu_{x}\rangle\leqslant-1$ for some
$r+1\leqslant j\leqslant 3d-1$ with $j\notin\\{2d,k+d,k-d,r+d\\}$. Let

$\displaystyle
z=x-\sigma^{k_{1}}(\delta)^{\vee}+\sigma^{j_{1}}(\delta)^{\vee},\
z^{\prime}=x^{\prime}-\sigma^{k_{1}}(\delta)^{\vee}+\sigma^{j_{1}}(\delta)^{\vee}\in\mathcal{S}_{\lambda,b}^{+},$

where $k_{1}=k+d$ if $k>2d$ and $k_{1}=k$ otherwise, and $j_{1}$ is defined in
the same way. By Lemma 8.5, we have
$\mathbb{J}_{b,{\tilde{w}}_{x}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{z}}$
and
$\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{z^{\prime}}}$.
Moreover, there exist $z_{1},z_{2}\in\mathcal{S}_{\lambda,b}^{+}$ such that

$\displaystyle
z\overset{(\alpha,k-2d)}{\to}z_{1}\overset{(\sigma^{k-2d}(\alpha),2d+r-k)}{\to}z^{\prime}\text{
if }r+d+1\leqslant k\leqslant 3d-1;$ $\displaystyle
z\overset{(\alpha,k-d)}{\to}z_{1}\overset{(\sigma^{k-2d}(\alpha),d+r-k)}{\to}z^{\prime},\text{
if }r+1\leqslant k\leqslant 2d-1;$ $\displaystyle
z\overset{(\alpha,k-2d)}{\to}z_{1}\overset{(\sigma^{k-d}(\alpha),d+r-k)}{\to}z_{2}\overset{(\sigma^{k-2d}(\alpha),d)}{\to}z^{\prime},\text{
if }2d+1\leqslant k\leqslant r+d-1.$

By Lemma 8.1,
$\mathbb{J}_{b,{\tilde{w}}_{z}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{z^{\prime}}}$
and the statement follows. So we can assume

$\displaystyle\langle\sigma^{i}(\alpha),\mu_{x}\rangle=0\text{ for }1\leqslant
i\leqslant k-1\text{ with }i\notin\\{r-d,r,r+d,d,2d\\}.$ (d)

As $\langle\sigma^{r-d}(\alpha),\mu_{x^{\prime}}\rangle=-1$, we have
$y:=x^{\prime}+\sigma^{r-d}(\alpha)^{\vee}-\sigma^{k}(\alpha)^{\vee}\in\mathcal{S}_{\lambda,b}^{+}$.

Case(1): $r+1\leqslant k\leqslant 2d-1$. Then

$x\overset{(\sigma^{k}(\delta),r-k)}{\to}x-\sigma^{k}(\delta)^{\vee}+\sigma^{r}(\delta)^{\vee}\overset{(\alpha,k-d)}{\to}y\overset{(\sigma^{r-d}(\alpha),k-r+d)}{\to}x^{\prime}.$

By Lemma 8.1, it suffices to show
$\mathbb{J}_{b,{\tilde{w}}_{y}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.
If $\langle\sigma^{r}(\alpha),\mu_{x}\rangle\leqslant-2$, that is,
$\langle\sigma^{r}(\alpha),\mu_{y}\rangle\leqslant-1$, it follows from that

$y\overset{(\sigma^{r-d}(\alpha),d)}{\to}x^{\prime}+\sigma^{r}(\alpha)^{\vee}-\sigma^{k}(\alpha)^{\vee}\overset{(\sigma^{r}(\alpha),k-r)}{\to}x^{\prime}.$

Otherwise, we have $\langle\sigma^{r}(\alpha),\mu_{x}\rangle=-1$ by (2), that
is, $\langle\sigma^{r}(\alpha),\mu_{y}\rangle=0$. Then the statement follows
from Lemma 8.3.

Case(2): $2d+1\leqslant k\leqslant 3d-1$. Then we have

$x\overset{(\sigma^{k+d}(\delta),r-k-d)}{\to}x-\sigma^{k+d}(\delta)^{\vee}+\sigma^{r}(\delta)^{\vee}\overset{(\alpha,k-2d)}{\to}y\overset{(\sigma^{r-d}(\alpha),k-r+d)}{\to}x^{\prime}.$

Again, it suffices to show
$\mathbb{J}_{b,{\tilde{w}}_{y}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$.
If $k\leqslant r+d-1$, it follows similarly as in Case(1). Otherwise, it
follows from that

$\displaystyle
y\overset{(\sigma^{k-d}(\alpha),r+2d-k)}{\to}y-\sigma^{k-d}(\alpha)^{\vee}+\sigma^{r+d}(\alpha)^{\vee}\overset{(\sigma^{k-2d}(\alpha),d)}{\to}y-\sigma^{k-2d}(\alpha)^{\vee}+\sigma^{r+d}(\alpha)^{\vee}$
$\displaystyle\
\overset{(\sigma^{r-d}(\alpha),k-r-d)}{\to}y-\sigma^{r-d}(\alpha)^{\vee}+\sigma^{r+d}(\alpha)^{\vee}\overset{(\sigma^{r+d}(\alpha),k-r-d)}{\to}x^{\prime},$

where the first arrow follows from (b) that
$\langle\sigma^{r+d}(\alpha),\mu_{y}\rangle=\langle\sigma^{r+d}(\alpha),\mu_{x}\rangle-1\leqslant-1$.

∎

### 8.2.

Now we finish the proofs for the case that $\sigma$ has order $3d$.

###### Proof of Proposition 2.3.

Let $x,x^{\prime}\in\mathcal{S}_{\lambda,b}^{+}$. To show
$\mathbb{J}_{b,{\tilde{w}}_{x}}\sim_{\lambda,b}\mathbb{J}_{b,{\tilde{w}}_{x^{\prime}}}$,
by Proposition 5.4 we can assume
$x\overset{(\gamma,r)}{\rightarrowtail}x^{\prime}$ for some $1\leqslant
r\leqslant 2d-1$ and $\gamma\in\Phi^{+}\setminus\Phi_{J}$ with $\gamma^{\vee}$
is $J$-anti-dominant and $J$-minuscule. In particular, $\sigma^{r}(\gamma)\in
C_{\lambda,b,x}$. If $\mathcal{O}_{\gamma}$ is of type I, the statement
follows from Lemma 7.1 and Corollary 4.6. Otherwise, we can assume
$J=\mathcal{O}_{\beta}$ and $\gamma=\alpha$ as in §8.1. If $1\leqslant
r\leqslant d$, the statement follows from Lemma 8.1 and Corollary 4.6.
Otherwise, by the proof of [30, Proposition 6.8], either Lemma 8.3 or Lemma
8.6 applies. So the statement also follows. ∎

###### Proof of Proposition 7.5.

As $\mathcal{O}$ is of type III, we can assume
$\mathcal{O}=\mathcal{O}_{\alpha}$ and $J=\mathcal{O}_{\beta}$, where