# A dimension formula of closed affine Deligne-Lusztig varieties of parahoric
level

Arghya Sadhukhan  Department of Mathematics, National University of Singapore,
10 Lower Kent Ridge Drive<EMAIL_ADDRESS>
###### Abstract.

We establish a dimension formula for certain union $X^{G}(\mu,b)_{J}$ of
affine Deligne-Lusztig varieties associated to arbitrary parahoric level
structures of split reductive groups, under certain genericity hypotheses.

###### Key words and phrases:

Affine Deligne-Lusztig variety, Newton stratification, dimension formula,
Kottwitz set, affine Weyl group

###### 2010 Mathematics Subject Classification:

20G25,11G25,20F55

##  1\. Introduction

Let $F$ be a non-archimedean local field with completed maximal unramified
extension of $\breve{F}$ and residue field $\kappa$. Let
$(\mathbf{G},\\{\mu\\},b)$ be a triple where $\mathbf{G}$ is a connected
reductive group over $F$, $\\{\mu\\}$ is a conjugacy class of cocharacters
$\mu:\mathbb{G}_{m,\bar{F}}\rightarrow\mathbf{G}_{\bar{F}}$ and
$[b]\in\mathbf{G}(\breve{F})$. Fix a $F$-rational parahoric level structure
$J$ of $\mathbf{G}$, with corresponding standard parahoric subgroup
$\breve{\mathcal{J}}\subset\mathbf{G}(\breve{F})$. Associated to this data, we
have the following closed affine Deligne-Lusztig variety, cf. [Rap05]

$X^{\mathbf{G}}(\mu,b)_{J}(\kappa):=\\{g\breve{}\mathcal{J}\in\mathbf{G}(\breve{F})/\breve{}\mathcal{J};g^{-1}b\sigma(g)\in\breve{}\mathcal{J}\operatorname{Adm}(\mu)\breve{}\mathcal{J}\\}\subset\mathcal{F}l^{\mathbf{G}}_{J}(\kappa)=\mathbf{G}(\breve{F})/\breve{}\mathcal{J}.$

Here $X^{\mathbf{G}}(\mu,b)_{J}$ is a subscheme, locally of finite type, of
the partial affine flag variety attached to level $J$ in the usual sense in
equal characteristic; in the sense of Zhu [Zhu17] and Bhatt-Scholze [BS17] in
mixed characteristics. Since the underlying topological space of
$\mathcal{F}l^{\mathbf{G}}_{J}$ is Jacobson, any locally closed subscheme is
uniquely determined by its $\kappa$-valued points, hence we freely identify
$X^{\mathbf{G}}(\mu,b)_{J}$ with the above set of its geometric points. These
varieties play an important role in the study of the special fiber of both
Shimura varieties and moduli spaces of shtukas, see [RV14], [Vie18].

By He’s work on the Kottwitz-Rapoport conjecture in [He16a],
$X^{\mathbf{G}}(\mu,b)_{J}\neq\emptyset$ if and only if $[b]$ is neutrally
acceptable, i.e. $[b]\in B(\mathbf{G},\mu)$, which we assume from now on. The
next fundamental question concerns the dimension of these geometric objects.
The first results in this direction was obtained when $\mathbf{G}$ is an
unramified group and $\breve{\mathcal{J}}$ is a hyperspecial subgroup. In this
case, the conjectural dimension formula put forth by Rapoport in [Rap05] was
established in [Gör+06] and [Vie06] for split groups, and in [Ham15] and
[Zhu17] for general quasi-split unramified groups. On the other extreme lies
the case of Iwahori level structure. Here we only have a partial
understanding. For a quasi-split group $\mathbf{G}$ and its Iwahori subgroup
$\breve{\mathcal{I}}$, a dimension formula was first obtained in [HY21] under
certain genericity hypotheses, namely that $\mu$ is superregular and $b$ is
sufficiently small compared to $\mu$; the first of these conditions was
subsequently weakened to $\mu$ being only regular in [Sad23]. Note that by
varying the level structure, we obtain transition maps
$\pi_{J,J^{\prime}}:\mathcal{F}l_{J}\rightarrow\mathcal{F}l_{J^{\prime}}$
whenever $J\subset J^{\prime}$, thus inducing
$\pi_{J,J^{\prime}}:X^{\mathbf{G}}(\mu,b)_{J}\rightarrow
X^{\mathbf{G}}(\mu,b)_{J^{\prime}}$; by [He16a], the latter map is surjective,
hence $\dim X^{\mathbf{G}}(\mu,b)_{J}\geq\dim
X^{\mathbf{G}}(\mu,b)_{J^{\prime}}$.

The purpose of this note is to prove a dimension formula in the case of an
arbitrary parahoric level for split groups that interpolates these two known
formulas.

######  Theorem 1.1.

Suppose that $\mathbf{G}$ is a split group and let $[b]\in B(\mathbf{G},\mu)$.
Then

$\dim
X^{\mathbf{G}}(\mu,b)_{J}=\langle\rho,\underline{\mu}-\nu(b)\rangle-\frac{1}{2}\text{def}(b)+\frac{1}{2}[\\{\ell(w_{0})-\ell_{R}(w_{0})\\}-\\{\ell(w_{J})-\ell_{R}(w_{J})\\}],$
(1.1)

in the following cases:

  1. (i)

$\mu$ is $2$-regular, $\mu\geq\nu(b)+\text{wt}(w_{0},1)$ and $J=\emptyset$ is
the Iwahori level structure.

  2. (ii)

$\mu$ is $4$-regular, $\mu\geq\nu(b)+2\rho^{\vee}+\text{wt}(w_{0},1)$ and $J$
avoids at least one special vertex.

  3. (iii)

$\mu$ is $2\ell(w_{0})+2$-regular,
$\mu\geq\nu(b)+2\rho^{\vee}+\text{wt}(w_{0},1)$ and $J$ is arbitrary.

Here $\ell_{R}(\cdot)$ is the reflection length of elements in finite Weyl
group, and $\text{wt}(\cdot,\cdot)$ is the weight function defined in terms of
its associated quantum Bruhat graph, see section 2 for further details. Note
that in part(i) we only improve [Sad23] by weakening the “gap” condition
$\mu-\nu(b)$ \- which is assumed to be at least $2\rho^{\vee}$ in loc. sit.,
albeit at the cost of introducing some regularity to $\mu$; note that
$\text{wt}(w_{0},1)$ is substantially smaller than $2\rho^{\vee}$, cf. [Sad23,
section 5], e.g. in type $A_{2n}$, we have
$\text{wt}(w_{0},1)=\varpi_{n}^{\vee}+\varpi_{n+1}^{\vee}$.

###  1.1. Strategy

We first discuss the proof of theorem 1.1[(i)]. The assumption $\nu-\nu(b)\geq
2\rho^{\vee}$ essentially enters the proof of [HY21][Theorem 6.1] in asserting
that the KR stratum associated to certain element in $\widetilde{W}$ is top-
dimensional in $X^{\mathbf{G}}(\mu,b)_{\emptyset}$, and the dimension of this
KR stratum is known only under such gap hypothesis from [He21a]. To circumvent
this condition, we simply construct a different top dimensional element (one
in the antidominant chamber) even under the weaker gap hypothesis, by
appealing to the theory of cordial elements in affine Weyl group à la [MV20].

Our basic strategy in proving theorem 1.1[(ii), (iii)] is to adapt the methods
of [HY21] to arbitrary parahoric level. To that end, we note that arguments
involving the EKOR strata in $X^{\mathbf{G}}(\mu,b)_{J}$ \- similar to those
considered in loc. sit. for the KR strata in
$X^{\mathbf{G}}(\mu,b)_{\emptyset}$ \- reduces our problem in section 3 to one
of minimizing distances in the quantum Bruhat graph; more specifically, under
the conditions in the statement of the theorem, determining the dimension is
equivalent to finding

$\min\\{d(x,xw_{0}):x\in~{}^{J}W\\},$

where ${}^{J}W$ is the semi-affine quotient of the finite Weyl group $W$ by
$J$ as introduced in [Sch24]; in case $\breve{}\mathcal{J}$ is contained in a
fixed hyperspecial subgroup, this is the set of minimal length elements in
$W_{J}\backslash W$.

However, explicit construction of an minimizing element is more complicated
when $J\neq\emptyset$. We recall that in the case of Iwahori level structure,
it was established in [HY21] that this minimum is realized in the smaller
subset of elements satisfying $x\leq xw_{0}$; since distance in the quantum
Bruhat graph between two such elements is simply the difference of length, the
previous optimization problem simplifies to finding $\max\\{\ell(x):x\leq
xw_{0}\\}$ \- which is more amenable to an inductive argument. We carry out a
similar argument in section 4 Cartan types $A_{n},B_{n},C_{n}$ when
$J\subset\mathbb{S}$; however this approach does not seem to work nicely in
type $D_{n}$, and in fact the previous observation about the minimum occuring
at an element $x$ with $x<xw_{0}$ is false in general, when $J$ contains an
affine simple reflection.

Hence, we resort to a different method in section 5 to handle the optimization
problem in an uniform manner: by relating the long elements of the finite Weyl
groups $W_{J}$ and $W$. We show that for any subset $J$ of affine simple
roots, one can find a suitable subset $I$ of finite simple roots such that
$w_{0}$ decomposes into product of certain $(~{}^{J}W)^{-1}$-conjugate of
(linearization of) the long element of $w_{J}$ and the long element $w_{I}$ in
a $\ell_{R}$-additive way. This is a proposition of independent interest, and
we deduce it by combining explicit decompositions of $w_{0}$ listed in
[Sad23], with classification of certain involutions in terms of long elements
of parabolic subgroups carried out in [Zib23].

######  Acknowledgement.

The author thanks Xuhua He for his encouragement to work on this project and
Mita Banik for her help with SageMath computations.

##  2\. Preliminaries

###  2.1. Group-theoretic notations

Recall that $\mathbf{G}$ is a connected reductive quasi-split group over a
non-archimedean local field $F$. Let $\breve{F}$ be the completion of the
maximal unramified extension of $F$ and $\sigma$ be the Frobenius morphism of
$\breve{F}/F$. The residue field of $F$ is a finite field $\mathbb{F}_{q}$ and
the residue field of $\breve{F}$ is the algebraically closed field
$\bar{\mathbb{F}}_{q}$. We use the same symbol $\sigma$ for the induced
Frobenius morphism on $\breve{G}:=\mathbf{G}(\breve{F})$. Let $S$ be a maximal
$\breve{F}$-split torus of $\mathbf{G}$ defined over $F$, which contains a
maximal $F$-split torus. Let $\mathcal{A}$ be the apartment of
$\mathbf{G}_{\breve{F}}$ corresponding to $S_{\breve{F}}$. We fix a
$\sigma$-stable alcove $\mathfrak{a}$ in $\mathcal{A}$, and let
$\breve{}\mathcal{I}\subset\breve{G}$ be the Iwahori subgroup corresponding to
$\mathfrak{a}$. Then $\breve{}\mathcal{I}$ is $\sigma$-stable.

Let $T$ be the centralizer of $S$ in $\mathbf{G}$. Then $T$ is a maximal
torus. We denote by $N$ the normalizer of $T$ in $\mathbf{G}$. The
_Iwahori–Weyl group_ (associated to $S$) is defined as

$\widetilde{W}=N(\breve{F})/T(\breve{F})\cap\breve{}\mathcal{I}.$

We denote by $\ell$ the length function on $\tilde{W}$ determined by the base
alcove $\mathfrak{a}$ and denote by $\tilde{}\mathbb{S}$ the set of simple
reflections in $\tilde{W}$. Let $W_{\text{aff}}$ be the subgroup of
$\widetilde{W}$ generated by $\tilde{}\mathbb{S}$. Then $W_{\text{aff}}$ is an
affine Weyl group. Let $\Omega\subset\widetilde{W}$ be the subgroup of length-
zero elements in $\widetilde{W}$. Then
$\widetilde{W}=W_{\text{aff}}\rtimes\Omega.$ Note that this action of $\Omega$
preserves $\widetilde{\mathbb{S}}$. For any $J\subset\widetilde{\mathbb{S}}$,
let $\widetilde{W_{J}}\subset\widetilde{W}$ be the subgroup generated by the
simple reflections in $J$ and by ${}^{J}\widetilde{W}$ the set of minimal
length elements for the cosets $\widetilde{W}_{J}\backslash\widetilde{W}$.

Let $W=N(\breve{F})/T(\breve{F})$ be the relative Weyl group. We denote by
$\mathbb{S}$ the subset of $\widetilde{\mathbb{S}}$ consisting of simple
reflections generating $W$. We let $\Phi$ (resp. $\Delta$,
$\Delta_{\text{aff}}$) denote the set of roots (resp. simple roots, simple
affine roots). We write $\Gamma$ for $\operatorname{Gal}(\bar{F}/F)$, and
write $\Gamma_{0}$ for the inertia subgroup of $\Gamma$. Then fixing a special
vertex of the base alcove $\mathfrak{a}$, we have the splitting

$\widetilde{W}=X_{*}(T)\rtimes W=\\{t^{\lambda}w;\lambda\in X_{*}(T),w\in
W\\}.$

For an element $x\in\widetilde{W}$ we denote by $\overline{x}$ its projection
to $W$; similarly, for a root $\alpha\in\Delta_{\text{aff}}$ we denote its
spherical part by $\overline{\alpha}$. For an irreducible Weyl group $W$ of
rank $n$, we follow the labeling of roots as in [Bou02] and we usually write
$s_{i}$ instead of $s_{\alpha_{i}}$, where $\Delta=\\{\alpha_{i}:1\leq i\leq
n\\}$. The additional element in $\Delta_{\text{aff}}$ is then the affine
simple root $-\theta+1$, where $\theta$ is the longest root in $\Phi$, and we
denote the corresponding reflection in $\widetilde{\mathbb{S}}$ by $s_{0}$. We
will also sometimes denote by $\alpha_{s}$ the affine simple root
corresponding to $s\in\widetilde{\mathbb{S}}$. For
$J\subset\widetilde{\mathbb{S}}$ we let $w_{J}$ be the longest element of the
corresponding parabolic subgroup and write $w_{0}=w_{\mathbb{S}}$. Let $\rho$
be the dominant weight with $\langle\alpha^{\vee},\rho\rangle=1$ for any
$\alpha\in\Delta$. Let $\\{\varpi_{i}^{\vee}:1\leq i\leq n\\}$ be the set of
fundamental coweights. If $\varpi^{\vee}_{i}$ is minuscule, we denote the
image of $t^{\varpi_{i}^{\vee}}$ under the projection
$\widetilde{W}\rightarrow\Omega$ by $\tau_{i}$; then conjugation by $\tau_{i}$
is a length preserving automorphism of $\widetilde{W}$ which we denote by
$\text{Ad}(\tau_{i})$.

For any dominant coweight $\lambda$ we define
$\text{depth}(\lambda)=\min\\{\langle\alpha,\lambda\rangle:\alpha\in\Delta\\}$.
Given $X\subset\widetilde{W},k\in\mathbb{Z}_{\geq 0}$, we denote by $X_{>k}$
(resp. $X_{\leq k}$) the subset of $X$ whose elements have associated dominant
translation of depth at least $k+1$ (resp. at most $k$); we call such elements
to be $k$-regular.

###  2.2. The $\sigma$-conjugacy classes of $\breve{G}$

We say that two elements $b,b^{\prime}\in\breve{G}$ are $\sigma$-conjugate if
there is some $g\in\breve{G}$ such that $b^{\prime}=gb\sigma(g)^{-1}$. Let
$B(\mathbf{G})$ be the set of $\sigma$-conjugacy classes on $\breve{G}$. By
the work of Kottwitz in [Kot85] and [Kot97], any $\sigma$-conjugacy class
$[b]$ is determined by two invariants:

  * •

The Kottwitz point $\kappa([b])\in\pi_{1}(\mathbf{G})_{\Gamma}$, the set of
$\Gamma$-coinvariants of the Borovoi fundamental group $\pi_{1}(\mathbf{G})$;

  * •

The Newton point
$\nu([b])\in((X_{*}(T)_{\Gamma_{0},\mathbb{Q}})^{+})^{\langle\sigma\rangle}$,
the set of $\langle\sigma\rangle$-invariants of the intersection of
$X_{*}(T)_{\Gamma_{0}}\otimes\mathbb{Q}=X_{*}(T)^{\Gamma_{0}}\otimes\mathbb{Q}$
with the set $X_{*}(T)_{\mathbb{Q}}^{+}$ of dominant elements in
$X_{*}(T)_{\mathbb{Q}}$.

We denote by $\leqslant$ the dominance order on $X_{*}(T)_{\mathbb{Q}}^{+}$,
i.e., for $\nu,\nu^{\prime}\in X_{*}(T)_{\mathbb{Q}}^{+}$, we have
$\nu\leqslant\nu^{\prime}$ if and only if $\nu^{\prime}-\nu$ is a non-negative
(rational) linear combination of positive coroots over $\breve{F}$. The
dominance order on $X_{*}(T)_{\mathbb{Q}}^{+}$ extends to a partial order on
$B(\mathbf{G})$. Namely, for $[b],[b^{\prime}]\in B(\mathbf{G})$, we say that
$[b]\leqslant[b^{\prime}]$ if and only if $\kappa([b])=\kappa([b^{\prime}])$
and $\nu([b])\leqslant\nu([b^{\prime}])$.

####  2.2.1. Neutrally acceptable elements

Let $\\{\mu\\}$ be a conjugacy class of cocharacters over $\bar{F}$. Choose
$\mu$ be a dominant representative of $\\{\mu\\}$ and denote by
$\underline{\mu}$ its image in $X_{*}(T)_{\Gamma_{0}}$. We define the
$\mu$-admissible set

$\operatorname{Adm}(\mu)=\\{w\in\widetilde{W};\ w\leqslant
t^{x(\underline{\mu})}\text{ for some }x\in W\\}.$

Let $\breve{}\mathcal{J}$ be a standard $F$-rational parahoric subgroup of
$\mathbf{G}(\breve{F})$, i.e., a $\sigma$-invariant parahoric subgroup that
contains $\breve{I}$. This corresponds to a spherical type $J$, i.e. a subset
$J\subset\widetilde{\mathbb{S}}$ such that $\widetilde{W}_{J}$ is finite. Set
${}^{J}\text{Adm}(\mu)=\operatorname{Adm}(\mu)\cap~{}^{J}\widetilde{W}$. We
also have the set of neutrally acceptable elements

$B(\mathbf{G},\mu)=\\{[b]\in
B(\mathbf{G})\mid\kappa([b])=\mu^{\natural},\nu([b])\leq\mu\\}.$

Here $\mu^{\natural}$ denotes the common image of $\mu\in\\{\mu\\}$ in
$\pi_{1}(\mathbf{G})_{\sigma}$.

The set $B(\mathbf{G},\mu)$ naturally inherits the partial order from
$B(\mathbf{G})$.

###  2.3. Affine Deligne-Lusztig varieties

The _affine Deligne-Lusztig variety_ associated to $w\in\widetilde{W}$ and
$b\in\breve{G}$ is a locally closed subscheme of the partial affine flag
variety of $\bf G$, locally of finite type over $\overline{\mathbb{F}}_{p}$
and of finite dimension, with the set of geometric points given by

$X_{w}(b)=\\{g\in\breve{G}/\breve{}\mathcal{I}:g^{-1}b\sigma(g)\in\breve{}\mathcal{I}w\breve{}\mathcal{I}\\}.$

If $F$ is of equal characteristic, then by affine flag variety we mean the
“usual” affine flag variety; in the case of mixed characteristic, this notion
should be understood in the sense of perfect schemes, as developed by Zhu
[Zhu17] and by Bhatt and Scholze [BS17].

Let $J$ be a spherical type. We set

$X^{\mathbf{G}}(\mu,b)_{J}=\\{g\breve{}\mathcal{J}\in\breve{G}/\breve{}\mathcal{J};g^{-1}b\sigma(g)\in\breve{}\mathcal{J}\operatorname{Adm}(\mu)\breve{}\mathcal{J}\\}.$

We usually omit the adoration $\mathbf{G}$ from the notation and also write
$X(\mu,b)$ for $X(\mu,b)_{\emptyset}$, which is then an union of affine
Deligne-Lusztig varieties. Settling the Kottwitz-Rapoport conjecture made in
[KR03] and [Rap05] about the non-emptiness pattern for $X(\mu,b)_{J}$, He
proves the following result in [He16].

######  Theorem 2.1.

[He16a, theorem A] $X(\mu,b)_{J}\neq\emptyset$ if and only if $[b]\in
B(\mathbf{G},\mu)$.

###  2.4. Quantum Bruhat graphs

We recall the quantum Bruhat graph introduced by Brenti, Fomin, and Postnikov
in the context of quantum cohomology ring of complex flag variety, see
[BFP99], also [FGP97]. By definition, a _quantum Bruhat graph_ $\Gamma_{\Phi}$
is a directed graph with

  * •

vertices given by the elements of $W$;

  * •

upward edges $w\rightharpoonup ws_{\alpha}$ for some $\alpha\in\Phi^{+}$ with
$\ell(ws_{\alpha})=\ell(w)+1$;

  * •

downward edges $w\rightharpoondown ws_{\alpha}$ for some $\alpha\in\Phi^{+}$
with $\ell(ws_{\alpha})=\ell(w)-\langle 2\rho,\alpha^{\vee}\rangle+1$.

The _weight of a path_ in $\Gamma_{\Phi}$ is defined to be the sum of weights
of the edges in the path. For any $x,y\in W$, we denote by $d(x,y)$ the
minimal length among all paths in $\Gamma_{\Phi}$ from $x$ to $y$. Any path
between $x$ and $y$ affording $d(x,y)$ as its length is called a shortest path
between them. Then the following lemma defines a function $\text{wt}:W\times
W\rightarrow\mathbb{Z}_{\geq 0}\Phi^{\vee}$.

######  Lemma 2.2.

[Pos05, Lemma 1], [BFP99, Lemma 6.7] Let $x,y\in W$.

  1. (1)

There exists a directed path (consisting of possibly both upward and downward
edges) in $\Gamma_{\Phi}$ from $x$ to $y$.

  2. (2)

Any two shortest paths in $\Gamma_{\Phi}$ from $x$ to $y$ have the same
weight, which we denote by $\text{wt}(x,y)$.

  3. (3)

Any path in $\Gamma_{\Phi}$ from $x$ to $y$ has weight
$\geqslant\text{wt}(x,y)$.

######  Lemma 2.3.

[HY21, §4.2], [HN21, §2.5] Let $x,y\in W$. Then

  1. (1)

We have $\langle\text{wt}(x,y),\alpha\rangle\leq 2$, for any simple root
$\alpha$.

  2. (2)

$\ell(y)-\ell(x)=d(x,y)-\langle 2\rho,\text{wt}(x,y)\rangle$.

######  Lemma 2.4.

$d(x,xy)=\ell_{R}(y)$ if $\ell(xy)=\ell(x)-\ell(y)$.

###### Proof.

We have $\text{wt}(x,xy)=\text{wt}(x^{-1}\lhd xy,1)$ by [Sad23][Corollary
3.3], where $\lhd$ denotes the left downward Demazure product, see
[HL15][§2.2]. Thus, $d(x,xy)=\ell(xy)-\ell(x)+\ell(x^{-1}\lhd
xy)+\ell_{R}(x^{-1}\lhd xy)$ by [Sad23][Lemma 8.4] and lemma 2.3[(2)]. Now,
$x^{-1}\lhd xy=y$ if and only if $\ell(y)=\ell(x^{-1})-\ell(xy)$, but in that
case the first equation simplifies to $d(x,xy)=\ell_{R}(y)$. ∎

##  3\. Recollections from previous works

###  3.1. Some standard reductions

Note that if
$\mathbf{G}=\mathbf{G}_{1}\times\mathbf{G}_{2},b=(b_{1},b_{2}),\mu=(\mu_{1},\mu_{2}),\breve{}\mathcal{J}=\breve{}\mathcal{J}_{1}\times\breve{}\mathcal{J}_{2}$,
then by definition $X^{\mathbf{G}}(\mu,b)_{J}\simeq
X^{\mathbf{G}_{1}}(\mu_{1},b_{1})_{J_{1}}\times
X^{\mathbf{G}_{2}}(\mu_{2},b)_{J_{2}}$; similarly, the terms on the RHS of the
dimension formula eq. 1.1 are additive as well. Hence, we may assume from now
on that $\mathbf{G}$ is quasi-simple and split over $F$.

Next, we argue that in certain cases we may assume that the parahoric subgroup
is contained in a hyperspecial subgroup, i.e. $J\subset\mathbb{S}$. To do so,
we need the following observation from [He14][Lemma 4.3].

######  Lemma 3.1.

Let $w\in\widetilde{W},\tau\in\Omega$. Then $X_{w}(b)$ is isomorphic to
$X_{{}^{\tau}w}(b)$.

Note that $s_{i}w\geq w\iff~{}^{\tau}s_{i}~{}^{\tau}w\geq~{}^{\tau}w$, and
thus $\text{Ad}(\tau)(^{J}\widetilde{W})=^{\text{Ad}(\tau)(J)}\widetilde{W}$.
Also, ${}^{\tau}\text{Adm}(\mu)=\text{Adm}(\mu)$, whence

$\text{Ad}(\tau)(^{J}\widetilde{W}\cap\text{Adm}(\mu))=^{\text{Ad}(\tau)(J)}\widetilde{W}\cap\text{Adm}(\mu)=^{\text{Ad}(\tau)(J)}\operatorname{Adm}(\mu).$

It is proved in [GH15][§3.4] (see also [HR17][Theorem 6.21]) that

$\dim X(\mu,b)_{J}=\max\limits_{w\in^{J}\text{Adm}(\mu)}\dim X_{w}(b).$

This comes from a decomposition of $X(\mu,b)_{J}$ that is analogous, in terms
of Shimura varieties, to the decomposition of a Newton stratum into its
intersections with the EKOR strata, cf. [HR17].

In view of lemma 3.1, we then have $\dim X(\mu,b)_{J}=\dim
X(\mu,b)_{\text{Ad}(\tau)(J)}$. Hence, if there exists a simple reflection $s$
corresponding to a special vertex of $\mathfrak{a}$ so that $s\notin J$, we
may use a suitable element $\tau\in\Omega$ so that
$\text{Ad}(\tau)(J)\subset\mathbb{S}$ and then focus on the dimension problem
for this new level structure. For example, this assumption on $J$ is satisfied
for any parahoric level structure in type affine Weyl group of type $A_{n}$,
so here we may assume $J\subset\mathbb{S}$.

###  3.2. Virtual dimension formula

Recall that $w\in\widetilde{W}$ can be written in a unique way as
$w=ut^{\lambda}v$ with $\lambda$ dominant, $u,v\in W$ such that
$t^{\lambda}v\in~{}^{\mathbb{S}}\widetilde{W}$. We then set
$\eta(w)=vu,p_{l}(w)=u$.

Let $\mathbf{J}_{b}$ be the reductive group over $F$ with
$\mathbf{J}_{b}(F)=\\{g\in\breve{G};gb\sigma(g)^{-1}=b\\}.$ Then the _defect_
of $b$ is defined by
$\rm{def}_{\mathbf{G}}(b)=\operatorname{rank}_{F}\mathbf{G}-\operatorname{rank}_{F}\mathbf{J}_{b}$,
where $\operatorname{rank}_{F}$ denotes the $F$-rank.

Following [He14, Section 10.1], we define the _virtual dimension_ for
$X_{w}(b)$ to be

$d_{w}(b)=\frac{1}{2}\big{(}\ell(w)+\ell(\eta(w))-\rm{def}_{\mathbf{G}}(b)-\langle
2\rho,\nu([b])\rangle).$

It is proved in [He14, Corollary 10.4] that $\dim X_{w}(b)\leq d_{w}(b)$
whenever $\kappa(w)=\kappa([b])$.

For $J\subset\widetilde{\mathbb{S}}$, define $W_{J}$ to be the subgroup of $W$
generated by $\\{\overline{s}:s\in J\\}$ and ${}^{J}W:=\\{w\in
W:w^{-1}\overline{\alpha_{s}}\in\Phi^{+},~{}\forall s\in J\\}$; this is
notationally consistent, i.e. when $J\subset\mathbb{S}$, the latter is the set
of minimal length elements in $W_{J}\backslash W$. Given any
$J\subset\widetilde{\mathbb{S}},w\in W$, it is proved in [Sch24][Lemma 3.7]
that we can find $w_{1}\in W_{J},w_{2}\in~{}^{J}W$ so that $w=w_{1}w_{2}$.

######  Lemma 3.2.

Let $\mu$ be dominant regular; then $p_{l}(^{J}\text{Adm}(\mu))=~{}^{J}W$ if
$J\subset\mathbb{S}$, and $p_{l}(^{J}\text{Adm}(\mu)_{>1})=~{}^{J}W$
otherwise.

###### Proof.

Let $w\in~{}^{J}\text{Adm}(\mu)$ and write $w=xt^{\lambda}y$ with
$t^{\lambda}y\in~{}^{\mathbb{S}}\widetilde{W}$. For $s_{i}\in\mathbb{S}$,
$s_{i}w>w$ then implies $s_{i}x>x$, thus $x^{-1}\alpha_{i}\in\Phi^{+}$. Now
assume $\text{depth}(\mu)>1$. We have $s_{0}w>w$ if and only if
$w^{-1}(-\theta,1)=(-(xy)^{-1}\theta,1-\langle x\lambda,\theta\rangle)$ is a
positive affine root, i.e., $\langle
x\lambda,\theta\rangle=\langle\lambda,x^{-1}\theta\rangle\leq 1$ or $0$,
depending on $-(xy)^{-1}(\theta)\gtrless 0$. Then we must have
$x^{-1}(-\theta)\in\Phi^{+}$, since otherwise $\text{depth}(\lambda)\leq 1$.

For the converse direction we note $w^{\prime}:=xt^{\mu}w_{0}\leq
t^{w_{0}\mu}$ for any $x\in W$, by regularity of $\mu$. If $s_{i}x>x$ for
$s_{i}\in\mathbb{S}$ then $s_{i}w^{\prime}>w^{\prime}$; also, the argument in
the last paragraph shows $s_{0}w^{\prime}>w^{\prime}$ if
$x^{-1}(-\theta)\in\Phi^{+}$, by the depth hypothesis on $\mu$. Hence,
$w^{\prime}\in~{}^{J}\text{Adm}(\mu)$. ∎

Following [HY21], we define
$d_{{}^{J}\text{Adm}(\mu)}(b):=\max\limits_{w\in^{J}\text{Adm}(\mu)}d_{w}(b)$.
Therefore, $\dim X(\mu,b)_{J}\leq d_{{}^{J}\text{Adm}(\mu)}(b)$.

######  Proposition 3.3.

We have

$d_{{}^{J}\text{Adm}(\mu)}(b)=\langle\rho,\mu-\nu([b])\rangle-\frac{1}{2}\mathrm{def}_{\mathbf{G}}(b)+\frac{1}{2}\ell(w_{0})-\frac{1}{2}\min\\{d_{\Gamma}(x,xw_{0}):x\in~{}^{J}W\\},$

if $\text{depth}(\mu)\geq 4$ in case $J\subset\mathbb{S}$, or if
$\text{depth}(\mu)\geq 2\ell(w_{0})+2$ otherwise.

###### Proof.

The proof scheme here is identical to that in the case of Iwahori level
structure in [HY21][§4.2-4.3]. We only need to explain the depth hypothesis
requirement and justify the index set of the last summand. Note that the proof
in [HY21][Proposition 4.4] was carried out under the assumption
$\text{depth}(\mu)>>0$, which was later relaxed in [Sad23][Proposition 7.1] to
$\mu$ being just regular; this latter improvement in part relied on utilizing
the explicit construction of certain elements in $\text{Adm}(\mu)\cap
Wt^{\mu}W$ coming from [HY21][§5.6]. In the absence of this latter sort of
explicit construction in general111However, this is available when
$J\subset\mathbb{S}$ and $W$ is irreducible of type $A_{n},B_{n}/C_{n}$, by
section 4., we may consider the element
$\widetilde{w}:=x^{\prime}t^{\mu-\text{wt}(x^{\prime},x^{\prime}w_{0})}w_{0}x^{\prime-1}$
for some $x^{\prime}\in~{}^{J}W$; note that
$\mu-\text{wt}(x^{\prime},x^{\prime}w_{0})$ is $2$-regular by lemma 2.3[(1)]
due to our imposed depth hypothesis. Then $\widetilde{w}$ is in
${}^{J}\operatorname{Adm}(\mu)$ by [Sch24][Proposition 4.12] and lemma 3.2,
and we can use this element to replace the arguments in the last paragraph of
[Sad23][Proposition 7.1].

Now let us discuss the index set. If $J\subset\mathbb{S}$, lemma 3.2 does the
job. Assume now that $s_{0}\in J$ and $\langle\alpha_{i},\mu\rangle\geq
2\ell(w_{0})+2$ for all $\alpha_{i}\in\Delta$. Pick $x^{\prime}\in~{}^{J}W$
and consider
$w^{\prime}:=x^{\prime}t^{\mu-\text{wt}(x^{\prime},x^{\prime}w_{0})}w_{0}x^{\prime-1}\in~{}^{J}\operatorname{Adm}(\mu)$.
We argue

(a) if $w\in~{}^{J}\text{Adm}(\mu)_{\leq 1}$, then $\ell(w)<\ell(w^{\prime})$.

Write $w=xt^{\lambda}y$ as before, then $\lambda\leq\mu$; assume that
$\langle\alpha_{j},\lambda\rangle\leq 1$ for some $\alpha_{j}\in\Delta$. Note
that $\ell(w)\leq\langle 2\rho,\lambda\rangle+\ell(w_{0})$ and
$\ell(w^{\prime})=\ell(x^{\prime})+\langle
2\rho,\mu-\text{wt}(x^{\prime},x^{\prime}w_{0})\rangle-\ell(w_{0}x^{\prime-1})=\langle
2\rho,\mu\rangle-d(x^{\prime},x^{\prime}w_{0})$ by lemma 2.3[(2)], thus
$\ell(w^{\prime})\geq\langle 2\rho,\mu\rangle-\ell(w_{0})$. Then

$\ell(w^{\prime})-\ell(w)\geq\langle
2\rho,\mu-\lambda\rangle-2\ell(w_{0})\geq\langle
2\varpi_{i},\mu-\lambda\rangle-2\ell(w_{0})\geq\langle\alpha_{i},\mu-\lambda\rangle-2\ell(w_{0})>0.$

Hence, (a) is proved. This gives $d_{w}(b)<d_{w^{\prime}}(b)$ \- thereby
implying
$d_{{}^{J}\text{Adm}(\mu)}(b)=\max\limits_{w\in^{J}\text{Adm}(\mu)_{>1}}d_{w}(b)$,
and finishing off the argument by appealing to lemma 3.2.

∎

###  3.3. Minimizing distance on the quantum Bruhat graph

We now state the key technical result on quantum Bruhat graph that will be
used in section 6.

######  Theorem 3.4.

Let $J\subset\widetilde{\mathbb{S}}$ be a spherical type. Then
$\min\limits_{x\in^{J}W}d(x,xw_{0})=\ell_{R}(w_{0})+\ell(w_{J})-\ell_{R}(w_{J})$.

Here $\ell_{R}(w_{0})$ (resp. $\ell_{R}(w_{J})$) is the reflection length of
the long element $w_{0}$ in $W$ (resp. $w_{J}$ in the parabolic subgroup
$\widetilde{W}_{J}$ of $\widetilde{W}$), i.e. the smallest number $l$ such
that $w_{0}\in W$ (resp. $w_{J}\in\widetilde{W}_{J}$) can be written as a
product of $l$ reflections in $W$ (resp. in $\widetilde{W}_{J}$), see
[HY21][§5.1] for details. We note that
$\ell_{R}(\overline{w_{J}})=\ell_{R}(w_{J})$.

######  Remark 3.5.

When $J=\emptyset$, i.e. in the case of the Iwahori level, the conclusion of
theorem 3.4 is known from [HY21][Theorem 5.1]. However, the proof in loc. sit.
could be considerably simplified. Indeed, $d(x,xw_{0})\geq\ell_{R}(w_{0})$ for
any $x$, and if $x=w_{0}$ then $d(x,xw_{0})=d(w_{0},1)=\ell_{R}(w_{0})$ by
[Sad23][§5], thereby concluding $\min\limits_{x\in
W}d(x,xw_{0})=\ell_{R}(w_{0})$.

In this subsection we establish the lower bound. This relies on the following
lemma.

######  Lemma 3.6.

$d(x,xw_{0})=\ell(y)+d(x,\overline{y}xw_{0})$ whenever $x\in~{}^{J}W$ and
$y\in\widetilde{W}_{J}$

###### Proof.

We will induct on $\ell(y)$. By lemma 3.2, we may write
$w=xt^{\lambda}z\in~{}^{J}\widetilde{W}$ for some $z\in W$ and cocharacter
$\lambda$ of depth $>>0$.

Clearly, the claim is trivial for $y=1$. Assume now the claim is true for
$y^{\prime}\in\widetilde{W}_{J}$ and let $y=sy^{\prime}>y^{\prime}$, with
$s\in J$; then $sy^{\prime}w>y^{\prime}w$. Note that
$y^{\prime}w=\overline{y^{\prime}}xt^{\lambda^{\prime}}z$, where
$\lambda^{\prime}$ still has depth $>>0$. Arguing as in the proof of lemma
3.2, we thus get $(\overline{y^{\prime}}x)^{-1}(\alpha_{i})\in\Phi^{+}$ if
$s=s_{\alpha_{i}}$ is a finite simple reflection, or
$(\overline{y^{\prime}}x)^{-1}(-\theta)\in\Phi^{+}$ if $s=s_{(-\theta,1)}$ is
the affine simple reflection; note that $x^{-1}\alpha_{i}\in\Phi^{+}$ and
$x^{-1}(-\theta)\in\Phi^{+}$ accordingly in these cases. Thus by
[Len+15][Lemma 7.7],
$d(x,\overline{y^{\prime}}xw_{0})=1+d(x,\overline{sy^{\prime}}xw_{0})$ in
either case, whence

$d(x,xw_{0})=\ell(y^{\prime})+d(x,\overline{y^{\prime}}xw_{0})=\ell(y^{\prime})+1+d(x,\overline{sy^{\prime}}xw_{0})=\ell(y)+d(x,\overline{y}xw_{0}).$

∎

In particular, lemma 3.6 implies
$d(x,xw_{0})=\ell(w_{J})+d(x,\overline{w_{J}}xw_{0})$. Suppose now for
contradiction that there exists $x\in~{}^{J}W$ with

$d(x,xw_{0})<\ell_{R}(w_{0})+\ell(w_{J})-\ell_{R}(w_{J}).$

We then get

$\displaystyle\ell_{R}(w_{0})-\ell_{R}(w_{J})>d(x,\overline{w_{J}}xw_{0})\geq\ell_{R}(x^{-1}\overline{w_{J}}xw_{0})$
$\displaystyle\geq\ell_{R}(w_{0})-\ell_{R}(x^{-1}\overline{w_{J}}x)=\ell_{R}(w_{0})-\ell_{R}(w_{J}),$

a contradiction. Hence,

$\min\limits_{x\in^{J}W}d(x,xw_{0})\geq\ell_{R}(w_{0})+\ell(w_{J})-\ell_{R}(w_{J}).$
(3.1)

##  4\. Explicit constructions for type $A_{n},B_{n}/C_{n}$

In this section, for an irreducible Weyl group $W$ of type $A_{n},B_{n}/C_{n}$
and $J\subset\mathbb{S}$, we set out to prove

(a) there exists $x\in~{}^{J}W$ with $x\leq xw_{0}$ and
$\ell(x)=\frac{1}{2}\\{(\ell(w_{0})-\ell_{R}(w_{0}))-(\ell(w_{J})-\ell_{R}(w_{J}))\\}$.

This is analogous to the construction discussed in [HY21][section 5.6] for the
case of $J=\emptyset$. Our argument is based on an downward induction on
$|J|$. If $J=\mathbb{S}$, then $x=1$ satisfies the condition in (a). Let
$J^{\prime}\subset\mathbb{S}$ with $J^{\prime}=J\setminus\\{s_{j}\\}$ for some
$1\leq j\leq n$. For induction, note that

${}^{J^{\prime}}W=\coprod\limits_{i\in^{J^{\prime}}W_{J}}i\cdot~{}^{J}W.$

By inductive hypothesis, there exists $x\in~{}^{J}W$ as asserted in (a). Now,
suppose that there exists $i\in^{J^{\prime}}W_{J}$ such that

  1. (i)

$i<iw_{J}$, and

  2. (ii)

$\ell(i)=\frac{1}{2}\\{(\ell(w_{J})-\ell_{R}(w_{J}))-(\ell(w_{J^{\prime}})-\ell_{R}(w_{J^{\prime}})\\}$.

Upon taking minimal length left coset representatives, the inequality $x\leq
xw_{0}$ gives $x\leq~{}^{J}(xw_{0})=w_{J}xw_{0}$; since
$x,w_{J}xw_{0}\in^{J}W$ and $i,iw_{J}\in W_{J}$, we have
$\ell(ix)=\ell(i)+\ell(x),\ell(iw_{J}\cdot
w_{J}xw_{0})=\ell(iw_{J})+\ell(w_{J}xw_{0})$. We thus obtain $ix\leq
iw_{J}\cdot w_{J}xw_{0}=ixw_{0}$. Furthermore,
$\ell(ix)=\ell(i)+\ell(x)=\frac{1}{2}\\{(\ell(w_{0})-\ell_{R}(w_{0}))-(\ell(w_{J^{\prime}})-\ell_{R}(w_{J^{\prime}}))\\}$.
Therefore, $ix\in~{}^{J^{\prime}}W$ is the desired element for which the lower
bound for the function $yd(y,yw_{0})$ on ${}^{J^{\prime}}W$ is achieved, and
this completes the induction step.

Now we do a case-by-case analysis to give explicit description of such
elements $i\in~{}^{J^{\prime}}W_{J}$. Note that it suffices for this purpose
to reduce to the case where $J$ is irreducible. Since we only consider Weyl
groups of type $A_{n},B_{n}/C_{n}$ in this subsection, the relevant subdiagram
$J$ is also of the same type. To simplify notation, we write
$s_{[a,b]}=s_{a}s_{a+1}\cdots s_{b}$ for $1\leq a<b\leq n$.

####  4.0.1. $J$ is of type $A_{n}$

We need to consider only the case of $1\leq j\leq\frac{\lceil n\rceil}{2}$,
because conjugation by $w_{0}$ is a symmetry of the Dynkin diagram taking $j$
to $n+1-j$. We have
$\ell(w_{J})-\ell_{R}(w_{J})=\frac{n(n+1)}{2}-\lceil\frac{n}{2}\rceil$.

If $j=1$, then $J^{\prime}\simeq A_{n-1}$. In this case, it is already
determined in [HY21] that $i=s_{[1,\lfloor\frac{n}{2}\rfloor]}$ fits the bill.

If $2\leq j\leq\frac{\lceil n\rceil}{2}$, then $J^{\prime}\simeq A_{j-1}\times
A_{n-j}$, and thus

$\ell(w_{J^{\prime}})-\ell_{R}(w_{J^{\prime}})=\frac{j(j-1)}{2}+\frac{(n-j)(n-j+1)}{2}-(\lceil\frac{j-1}{2}\rceil+\lceil\frac{n-j}{2}\rceil).$

Then $i=s_{[j,n]}s_{[j-1,n-1]}\cdots
s_{[j+1-\lfloor\frac{j}{2}\rfloor,n+1-\lfloor\frac{j}{2}\rfloor]}\delta$,
where

$\delta=\begin{cases}&1,\text{ if }j\text{ is even };\\\
&s_{[\lceil\frac{j}{2}\rceil,\lfloor\frac{n}{2}\rfloor]},\text{ if }j\text{ is
odd}.\end{cases}$

It is an easy calculation to check that $\ell(i)$ is equal to
$\frac{1}{2}(nj-j^{2}+j)$ if $j$ is even, and $\frac{1}{2}(nj-j^{2}+j-1)$ if
$j$ is odd.

####  4.0.2. $J$ is of type $B_{n}$ or $C_{n}$

We have $\ell(w_{J})-\ell_{R}(w_{J})=n^{2}-n$. If $j=1$, then
$J^{\prime}\simeq B_{n-1}$ (resp. $C_{n-1}$). In this case, it is already
determined in [HY21] that $i=s_{[1,n-1]}$ fits the bill.

If $2\leq j\leq n$, then $J^{\prime}\simeq A_{j-1}\times B_{n-j}$, and thus

$\ell(w_{J^{\prime}})-\ell_{R}(w_{J^{\prime}})=\frac{j(j-1)}{2}+(n-j)^{2}-(\lceil\frac{j-1}{2}\rceil+n-j).$

Then
$i=\prod\limits_{\kappa=0}^{\lceil\frac{j}{2}\rceil-1}s_{[j-\kappa,n]}\prod\limits_{\eta=0}^{\lceil\frac{j}{2}\rceil-1}s_{[\lceil\frac{j}{2}\rceil-\eta,n-\gamma-2\eta]}$,
where

$\gamma=\begin{cases}&2,\text{ if }j\text{ is even };\\\ &1,\text{ if }j\text{
is odd}.\end{cases}$

An easy calculation verifies that
$\ell(i)=2jn-\frac{3j^{2}+j}{2}+\lceil\frac{j-1}{2}\rceil$.

######  Remark 4.1.

The discussion above shows that

$\min\limits_{x\in~{}^{J}W}d(x,xw_{0})=\min\limits_{x\in~{}^{J}W,x\leq
xw_{0}}d(x,xw_{0})=\ell(w_{0})-2\max\limits_{x\in~{}^{J}W,x\leq
xw_{0}}\ell(x).$

This is in similar spirit to the argument in [HY21]. However, an attempt at
the above sort of explicit construction results into cumbersome formula for
type $D_{n}$. Furthermore, this is false when $s_{0}\in J$; for instance, take
$W$ to be of type $B_{4}$ and $J=\\{s_{0},s_{1},s_{3},s_{4}\\}$, then
$\\{x\in~{}^{J}W:x\leq xw_{0}\\}=\emptyset$.

##  5\. Good decomposition of the long element in finite Weyl groups

In this subsection we analyze certain decomposition of the long element
$w_{0}$ well-suited to our purpose of realizing the minimum asserted in
theorem 3.4.

######  Proposition 5.1.

Let $J\subset\widetilde{\mathbb{S}}$ be a spherical type. Then there exists
$x\in~{}^{J}W$ and $I\subset\mathbb{S}$ such that

  1. (i)

$w_{0}=~{}(^{x^{-1}}\overline{w}_{J})w_{I}$;

  2. (ii)

The above decomposition is $\ell_{R}$-additive, i.e.
$\ell_{R}(w_{0})=\ell_{R}(w_{J})+\ell_{R}(w_{I})$;

  3. (iii)

$d(x,xw_{I})=\ell_{R}(w_{I})$.

Before proceeding with the proof of this proposition, we first establish
theorem 3.4.

###### Proof of theorem 3.4.

For Weyl groups of type $A_{n},B_{n}/C_{n}$, we may use the construction
scheme in section 4 if $J\subset\mathbb{S}$ : note that

$d(x,xw_{0})=\ell(xw_{0})-\ell(x)=\ell_{R}(w_{0})+\ell(w_{J})-\ell_{R}(w_{J})$

for such $x$, thereby proving theorem 3.4 in these cases.

Let $J\subset\widetilde{\mathbb{S}}$ be arbitrary. For the element
$x\in~{}^{J}W$ described in proposition 5.1, we compute

$d(x,\overline{w_{J}}xw_{0})=d(x,x^{x^{-1}}\overline{w_{J}}w_{0})=d(x,xw_{I})=\ell_{R}(w_{I})=\ell_{R}(w_{0})-\ell_{R}(w_{J}).$

Therefore, by lemma 3.6 we have
$d(x,xw_{0})=\ell(w_{J})+\ell_{R}(w_{0})-\ell_{R}(w_{J})$. We are done. ∎

###### Proof of proposition 5.1.

It will be shown in section 5.0.1-section 5.0.8 that there exists $z\in W$ and
$I\subset\mathbb{S}$ such that ${}^{z}(w_{0}w_{I})=\overline{w_{J}}$ and
$\ell_{R}(w_{J})=\ell_{R}(w_{0}w_{I})=\ell_{R}(w_{0})-\ell_{R}(w_{I})$. Let us
now establish the remaining properties in the desiderata.

We first show that

(a) ${}^{x}(w_{0}w_{I})=\overline{w_{J}}$ for $x=^{J}z$.

Choose $\zeta\in\widetilde{W}_{J}$ such that
$\overline{\zeta}=z_{J}\in\overline{\widetilde{W}_{J}}=W_{J}$. It suffices to
show that ${}^{\zeta^{-1}}w_{J}=w_{J}$ for such $\zeta\in\widetilde{W}_{J}$,
and hence $\overline{{}^{\zeta^{-1}}w_{J}}=\overline{w}_{J}$, i.e.
${}^{z_{J}^{-1}}\overline{w_{J}}=\overline{w_{J}}$. Suppose otherwise; then
$\ell(^{\zeta^{-1}}w_{J})<\ell(w_{J})$, since
${}^{\zeta^{-1}}w_{J}\in\widetilde{W}_{J}$. By lemma 3.6,

$\displaystyle d(x,xw_{0})$
$\displaystyle=\ell(^{\zeta^{-1}}w_{J})+d(x,(^{z_{J}^{-1}}\overline{w_{J}})xw_{0})=\ell(^{\zeta^{-1}}w_{J})+d(x,x(^{z^{-1}}\overline{w_{J}})w_{0})$
$\displaystyle=\ell(^{\zeta^{-1}}w_{J})+d(x,xw_{I})=\ell(^{\zeta^{-1}}w_{J})+\ell_{R}(w_{I})$
$\displaystyle=\ell(^{\zeta^{-1}}w_{J})+\ell_{R}(w_{0})-\ell_{R}(w_{J})<\ell(w_{J})+\ell_{R}(w_{0})-\ell_{R}(w_{J}).$

By eq. 3.1, this is a contradiction and hence
${}^{z_{J}^{-1}}\overline{w_{J}}=\overline{w_{J}}$.

Hence, we have shown that the set

$\Xi_{J,I}=\\{x\in~{}^{J}W:\text{ (a), (b) of the desiderata is fulfilled for
the pair }(x,I)\\}$

is nonempty.

Next, we claim that

(b) if $x$ is an element of maximal length in $\Xi_{J,I}$, then
$x\Delta_{I}\subset\Phi^{-}$.

Suppose otherwise that $x\alpha_{i}>0$ for some $\alpha_{i}\in\Delta_{I}$;
then $xs_{i}>x$. We show that $(xs_{i},I)\in\Xi_{J}$, contradicting the choice
of $x$. It suffices to check that ${}^{s_{i}}(w_{0}w_{I})=w_{0}w_{I}$. Note
that the long element is central, unless it corresponds to a Dynkin diagram of
type $A_{m}(m\geq 2),D_{2m+1}$ and $E_{6}$. This consideration alone takes
care of most of the cases arising in the following calculations, and we may
easily verify this condition in the remaining cases with the description of
$I$ at hand.

Finally, we note that

(c) $d(x,xw_{I})=\ell_{R}(w_{I})$ for the element found in (b).

For $x\in W$, define
$\text{Inv}(x)=\\{\alpha\in\Phi^{+}:x\alpha\in\Phi^{-}\\}$; then
$\text{Inv}(w_{I})=\Phi_{I}^{+}$. The well-known length formula of product of
elements in the Weyl group

$\ell(xy)=\ell(x)-\ell(y)+2|\text{Inv}(x)^{c}\cap\text{Inv}(y^{-1})|$

gives $\ell(xw_{I})=\ell(x)-\ell(w_{I})$ by (b), and then we deduce (c) from
lemma 2.4.

∎

Let us write $a\sim b$ to denote that $a,b\in W$ are in the same conjugacy
class. Below we explicitly determine the assignment $JI$ via a case-by-case
analysis. For this purpose, we may reduce to the case $J\subset\mathbb{S}$;
indeed, since $\overline{w_{J}}$ is an involution in $W$, there exists
$J^{\prime}\subset\mathbb{S}$ so that $\overline{w_{J}}\sim w_{J^{\prime}}$,
then we may apply the assignment $J^{\prime}I$ to derive the conclusion of
proposition 5.1 in this case. Note that the element $w_{0}w_{I}$ as in the
proposition must be an involution and we utilize below the classification of
conjugacy classes of such involutions as carried out in [Zib23], often without
further reference; the author in loc. sit. considers conjugacy in the
centralizer subgroup $C_{W}(w_{0})$, whereas our situation allows for more
flexibility and we exploit that to ensure the additional properties claimed in
the proposition.

We explain in detail the case of type $D_{n}$, which is the most complicated,
and give only brief sketch of arguments for types $A_{n},B_{n}/C_{n}$. For the
exceptional types, the tables contain the information of $(J,\ell_{R}(w_{J}))$
in the first row and $(I,\ell_{R}(I))$ in the second row and the roles of
$(J,I)$ are symmetric. For brevity, we skip the cases with factors of type
$A_{k}$ with $k\geq 2$; using the fact
$w_{A_{k}}\sim\prod\limits_{\lceil\frac{k}{2}\rceil\text{ copies}}w_{A_{1}}$,
we record the latter cases instead. Also, in type $E_{n}$ we use
$w_{[1,6]}=w_{E_{6}}\sim w_{D_{4}}$. Finally, We have
$\ell_{R}(w_{0}w_{I})=\ell_{R}(w_{0})-\ell_{R}(w_{I})$ by a direct check of
explicit $\ell_{R}$-reduced expression of $w_{0}$.

####  5.0.1. Type $D_{n}$

Here $J\simeq A_{n_{1}}\times\cdots\times A_{n_{r}}\times D_{l}$ where
$n_{1},\cdots,n_{r}>0$, and either $l=0$ or $l\geq 2$. Our convention here is
to consider the largest connected component of $J$ containing $\\{n-1,n\\}$ to
be of type $D_{l}$, if it exists. We write
$J=[i_{1},i_{1}+n_{1}]\cup\cdots\cup[i_{r},i_{k}+n_{r}]\cup[n-l+1,n]$, where
the last interval is present only when $\\{n-1,n\\}\subset J$.

We first show that

(a) $w_{J}\sim(\prod\limits_{k\text{ copies}}w_{A_{1}})\cdot w_{D_{l}},\text{
with }k=\lceil\frac{n_{1}}{2}\rceil+\cdots+\lceil\frac{n_{r}}{2}\rceil.$

Now, for a Weyl group with root system of type $A_{m}$, its longest element
$w_{A_{m}}$ is conjugate to $w_{0,J^{\prime}}$ where
$J^{\prime}=[1,m]\cap\mathbb{Z}_{\text{odd}}$. Thus, for each $1\leq j\leq r$
we can find $x_{j}^{\prime}\in W(A_{n_{j}})$ such that
${}^{x_{j}^{\prime}}w_{0,A_{n_{j}}}=w_{0,J_{j}^{\prime}}$, where
$J_{j}^{\prime}=\\{i_{j},i_{j}+2,\cdots,i_{j}+2(\lceil\frac{n_{j}}{2}\rceil-1)\\}$
for $1\leq j\leq r$. Note that $\text{supp}(x_{i}^{\prime})$ and
$\text{supp}(w_{A_{n_{j}}})$ is orthogonal whenever $i\neq j$; thus setting
$x^{\prime}=x_{1}^{\prime}\cdots x_{k}^{\prime}$, we see that

${}^{x^{\prime}}w_{J}=(\prod\limits_{j=1}^{r}\prod\limits_{\kappa_{j}=0}^{\lceil\frac{n_{j}}{2}\rceil-1}s_{i_{j}+2\kappa_{j}})\cdot
w_{D_{l}}.$

Next, we argue that

(b) the first factor in the RHS above is conjugate to $s_{1}s_{3}\cdots
s_{2k-1}$ by an element whose support is orthogonal to $[n-l+1,n]$.

We work with one connected component of $J$ at a time. Suppose that $i_{1}>1$.
Since
${}^{s_{\alpha_{a-1}+\alpha_{a}}}s_{a}=s_{a-1},^{s_{\alpha_{a-1}+\alpha_{a}}}s_{b}=s_{b}$
whenever $b\geq a+2$ or $b\leq a-3$, we can apply repeated conjugation to get

$\prod\limits_{\kappa_{1}=0}^{\lceil\frac{n_{1}}{2}\rceil-1}s_{i_{1}+2\kappa_{1}}\sim
s_{1}\prod\limits_{\kappa_{1}=1}^{\lceil\frac{n_{1}}{2}\rceil-1}s_{i_{1}+2\kappa_{1}}\sim
s_{1}s_{3}\prod\limits_{\kappa_{1}=2}^{\lceil\frac{n_{1}}{2}\rceil-1}s_{i_{1}+2\kappa_{1}}\sim\cdots\sim
s_{1}s_{3}\cdots s_{\lceil\frac{n_{1}}{2}\rceil}.$

Hence, passing on to a further conjugate, we have that

${}^{x^{\prime\prime}}w_{J}=s_{1}s_{3}\cdots
s_{\lceil\frac{n_{1}}{2}\rceil}(\prod\limits_{j=2}^{r}\prod\limits_{\kappa_{j}=0}^{\lceil\frac{n_{j}}{2}\rceil-1}s_{i_{j}+2\kappa_{j}})\cdot
w_{D_{l}}.$

Let $i_{p}$ be the minimum positive integer such that
$i_{p}-\\{i_{p-1}+2(\lceil\frac{n_{p-1}}{2}\rceil-1)\\}\geq 3$. In this case,
we already have
$\prod\limits_{j=1}^{p-1}\prod\limits_{\kappa_{j}=0}^{\lceil\frac{n_{j}}{2}\rceil-1}s_{i_{j}+2\kappa_{j}}=s_{1}s_{3}\cdots
s_{\lceil\frac{n_{p-1}}{2}\rceil}$. It is now clear that we can run a similar
sequence of conjugation to move the block starting with $s_{i_{p}}$ too,
finishing our argument.

Denote by $w_{k,l}:=w_{I}$ the longest element for the sub-diagram
$I=\\{1,3,\cdots,2k-1,n-l+1,n-l+2,\cdots n\\}$222In [Zib23], this element is
denoted by $c_{k,l-1}$.. So far, we have established $w_{J}\sim w_{k,l}$.

Note that $2k-1\leq n-2$, unless $n$ is even and $J\sim A_{n-1}$, i.e.
$J=\mathbb{S}\setminus\\{n-1\\}$ or $J=\mathbb{S}\setminus\\{n\\}$. Let us
exclude these cases for now and consider the element $w_{I}$ defined as

$w_{I}=\begin{cases}&w_{k,0},\text{ if }n=2k+l,\text{ or }n-(2k+l)=1\text{ and
}l\text{ is even };\\\ &w_{k,n-2k-(l-1)},\text{ if }n-(2k+l)\geq 1\text{ and
}l\text{ is odd};\\\ &w_{k,n-2k-l},\text{ otherwise}.\end{cases}$

We have $\ell_{R}(w_{0}w_{I})=\ell_{R}(w_{0})-\ell_{R}(w_{I})$, by the
explicit decomposition of $w_{0}$ in [Sad23][§5].

  * •

In the first case, $w_{I}$ is a $\ell_{R}$-subword of $w_{0}$ since all simple
reflections $s_{i}$ such that $i\leq n-2$ and $i$ is odd occurs in a
$\ell_{R}$-reduced expression of $w_{0}$.

  * •

In the remaining cases, the second index in the subscript of the definition of
$w_{I}$ has same parity as that of $n$, and thus comparing the explicit
expression of long element of Weyl groups of type $D_{l}$ (supported on
$\\{n-l+1,n-l+2,\cdots n\\}$) and $D_{n}$, we see that the former is a
$\ell_{R}$-subword of the latter.

Furthermore, $w_{I}w_{0}=w_{0}w_{I}$ and hence $w_{I}w_{0}$ is an involution.
It follows directly from [Zib23] that $w_{0}w_{I}\sim w_{J}$ in the first two
cases; for the third case, we note $w_{J}=w_{k,l}\sim w_{k,l+1}\sim
w_{0}w_{k,n-2k-l}=w_{0}w_{I}$.

Now we deal with the cases where $n$ is even and $J\sim A_{n-1}$. Here

  * •

$w_{0}w_{\mathbb{S}\setminus\\{n-1\\}}\sim
w_{\\{1,3,\cdots,n-3,n\\}},w_{0}w_{\mathbb{S}\setminus\\{n\\}}\sim
w_{\\{1,3,\cdots,n-3,n-1\\}}$, if $n=0$ mod $4$;

  * •

$w_{0}w_{\mathbb{S}\setminus\\{n-1\\}}\sim
w_{\\{1,3,\cdots,n-3,n-1\\}},w_{0}w_{\mathbb{S}\setminus\\{n\\}}\sim
w_{\\{1,3,\cdots,n-3,n\\}}$, if $n=2$ mod $4$.

####  5.0.2. Type $A_{n}$

Here $J\simeq A_{n_{1}}\times\cdots\times A_{n_{r}}$. Then
$w_{J}\sim(\prod\limits_{p\text{ copies }}w_{A_{1}})$ with
$p=\sum\limits_{i=1}^{r}\lceil\frac{n_{i}}{2}\rceil$, which is further
conjugate to either
$w_{(\lceil\frac{n}{2}\rceil-p,\lceil\frac{n}{2}\rceil+p)}$ or
$w_{(\lceil\frac{n}{2}\rceil-p,\lceil\frac{n}{2}\rceil+p]}$, depending on
whether $n$ is odd or even. Hence, setting $I=[p+1,n-p]$ we obtain $w_{J}\sim
w_{0}w_{I}$.

####  5.0.3. Type $B_{n}/C_{n}$

Our argument here for type $C_{n}$ directly adapts to type $B_{n}$. Here
$J\simeq A_{n_{1}}\times\cdots\times A_{n_{r}}\times C_{l}$, with either $l=0$
or $l\geq 2$. Then $w_{J}\sim(\prod\limits_{p\text{ copies
}}w_{A_{1}})w_{B_{l}}$, with
$p=\sum\limits_{i=1}^{r}\lceil\frac{n_{i}}{2}\rceil$. Hence, setting
$I=\\{1,3,\cdots,2p-1\\}\coprod\\{2p+l+1,\cdots,n\\}$ we obtain $w_{J}\sim
w_{0}w_{I}$.

####  5.0.4. Type $E_{6}$

$(\emptyset,0)$ | $(A_{1},1)$ | $(A_{1}^{2},2)$ | $(A_{1}^{3},3)$ | $(D_{4}\text{ or }D_{5},4)$
---|---|---|---|---
$(\Delta,4)$ | $(\Delta\setminus\\{2\\}\sim A_{5},3)$ | $(\\{3,4,5\\}\sim A_{3},2)$ | $(\\{4\\}\sim A_{1},1)$ | $(\emptyset,0)$

####  5.0.5. Type $E_{7}$

$(\emptyset,0)$ | $(A_{1},1)$ | $(A_{1}^{2},2)$ | $(A_{1}^{3}\text{ except }\\{2,5,7\\},3)$ | $(\\{2,5,7\\},3)$
---|---|---|---|---
$(\Delta,7)$ | $(\Delta\setminus\\{1\\}\sim D_{6},6)$ | $(\Delta\setminus\\{1,6\\}\sim D_{4}\times A_{1},5)$ | $(\\{1,2,5,7\\}\sim A_{1}^{4},4)$ | $(\\{2,3,4,5\\}\sim D_{4},4)$

####  5.0.6. Type $E_{8}$

$(\emptyset,0)$ | $(A_{1},1)$ | $(A_{1}^{2},2)$ | $(A_{1}^{3},3)$ | $(A_{1}^{4},4)$ | $(D_{4},4)$
---|---|---|---|---|---
$(\Delta,8)$ | $(\Delta\setminus\\{8\\}\sim E_{7},7)$ | $(\Delta\setminus\\{1,8\\}\sim D_{6},6)$ | $(\\{1,2,5,8\\}\sim A_{1}^{4},4)$ | $(\\{1,4,6,8\\}\sim A_{1}^{4},4)$ | $(D_{4},4)$

####  5.0.7. Type $F_{4}$

$(\emptyset,0)$ | $(\\{1\\}\text{ or }\\{2\\},1)$ | $(\\{3\\}\text{ or }\\{4\\},1)$ | $(A_{1}^{2},2)$ | $(\\{2,3\\}\sim C_{2},2)$
---|---|---|---|---
$(\Delta,4)$ | $(\Delta\setminus\\{1\\}\sim C_{3},3)$ | $(\Delta\setminus\\{4\\}\sim C_{3},3)$ | $(A_{1}^{2},2)$ | $(\\{2,3\\}\sim C_{2},2)$

####  5.0.8. Type $G_{2}$

$(\emptyset,0)$ | $(\\{1\\},1)$ | $(\\{2\\},1)$
---|---|---
$(\Delta,2)$ | $(\\{1\\},1)$ | $(\\{2\\},1)$

##  6\. Proof of the main theorem

We first discuss the case of Iwahori level structure. The proof of this part
is different in spirit from the argument in [HY21], whereas the proof of the
remaining parts is a direct adaptation of the proof scheme in loc. sit.

Basically, the restriction $\mu^{\diamond}\geq\nu(b)+2\rho^{\vee}$ enters in
the proof of [HY21][Theorem 6.1] because the dimension for the affine Deligne-
Lusztig variety associated to $w:=xt^{\mu}w_{0}x^{-1}$ \- the specific element
chosen in loc. sit. during the proof - can be asserted to be equal to
$d_{w}(b)$ by [He21] only under that stated restriction. Note that it is then
shown for this element, we have $d_{w}(b)=d_{\text{Adm}(\mu)}(b)$ \- by which
one concludes that $\dim X(\mu,b)=d_{\text{Adm}(\mu)}(b)$. In other words, for
this particular choice of $w$, the associated affine Deligne-Lusztig variety
has a known dimension by design, and also makes it to the top dimensional
component of $X(\mu,b)$.

###### Proof of theorem 1.1(i).

Instead of working with the element $w$ as above, let us consider the element
$w^{\prime}:=w_{0}t^{\mu-\text{wt}(w_{0},1)}$. We claim that

(a) $w^{\prime}\in\text{Adm}(\mu)$.

Indeed, proceeding as in the proof of [Sad23][Proposition 4.4] and utilizing
the decomposition of $w_{0}$ in section 5 of loc. sit., we have

$t^{\mu-\text{wt}(w_{0},1)}\leq t^{\mu}w_{0},\text{~{}and
hence~{}}w^{\prime}\leq w_{0}t^{\mu}w_{0}.$

This last inequality needs depth at least $2$ to ensure that
$\mu-\text{wt}(w_{0},1)$, as well the coweights appearing in the intermediate
steps (arising from application of the proof technique in loc. sit.), are
dominant so that the proof is valid here. Alternatively, we may apply
[Sch24][Proposition 4.12]; either way, we have $w^{\prime}\in\text{Adm}(\mu)$.

Now, we claim that

(b) $d_{w^{\prime}}(b)=d_{\text{Adm}(\mu)}(b)$.

To that end, let us compute

$\displaystyle d_{w^{\prime}}(b)$
$\displaystyle=\frac{1}{2}\\{\ell(w^{\prime})+\ell(w_{0})-\text{def}(b)-\langle
2\rho,\nu([b])\rangle\\}$ $\displaystyle=\frac{1}{2}\\{\ell(w_{0})+\langle
2\rho,\mu-\text{wt}(w_{0},1)\rangle+\ell(w_{0})-\text{def}(b)-\langle
2\rho,\nu([b])\rangle\\}$ $\displaystyle=\frac{1}{2}\\{\langle
2\rho,\mu-\nu(b)\rangle-\text{def}(b)\\}+\\{\ell(w_{0})-\langle\rho,\text{wt}(w_{0},1)\rangle\\}.$

Note that lemma 2.3[2] gives $\langle
2\rho,\text{wt}(w_{0},1)\rangle=\ell(w_{0})+d(w_{0},1)$, but the explicit
decomposition of $w_{0}$ shows that $d(w_{0},1)=\ell_{R}(w_{0})$. Combining,
we have
$\ell(w_{0})-\langle\rho,\text{wt}(w_{0},1)\rangle=\ell(w_{0})-\frac{1}{2}(\ell(w_{0})+\ell_{R}(w_{0}))=\frac{1}{2}\\{\ell(w_{0})-\ell_{R}(w_{0})\\}$.

Finally, we have that

(c) $\text{dim}X_{w^{\prime}}(b)=d_{w^{\prime}}(b)$, whenever
$X_{w^{\prime}}(b)\neq\emptyset$.

By [MV20, Theorem 1.2] every element in the antidominant chamber is cordial.
Therefore $w^{\prime}$ is cordial, and then above claim follows from [MV20,
Corollary 3.17].

Finally, note further that the basic element in $B(\mathbf{G})$ satisfying
$\kappa(b)=\kappa(w^{\prime})$ is indeed the minimal element of
$B(\mathbf{G})_{w^{\prime}}$ by [GHN19][Theorem B]. Hence,
$B(\mathbf{G})_{w^{\prime}}=\\{[b]:[b]\leq[b_{w^{\prime}}]\\}$. By [He21a,
Theorem 4.2], we have $\nu([b_{w^{\prime}}])=\mu-\text{wt}(w_{0},1)$. This in
turn enforces the condition $\mu\geq\nu([b])+\text{wt}(w_{0},1)$ in order to
ensure that $X_{w^{\prime}}(b)\neq\emptyset$. We are done. ∎

###### Proof of theorem 1.1(ii),(iii).

Note that $\dim X(\mu,b)_{J}$ is bounded above by RHS of eq. 1.1 by
proposition 3.3 and theorem 3.4.

Let us now consider the element $w:=xt^{\mu-\text{wt}(x,xw_{0})}w_{0}x^{-1}$,
where $x\in~{}^{J}W$ is as found in proposition 5.1. By lemma 3.2,
$w\in~{}^{J}\widetilde{W}$ and by [Sch24][Proposition 4.12],
$w\in\operatorname{Adm}(\mu)$. Since

$\mu-\text{wt}(x,xw_{0})-\nu(b)\geq\mu-\text{wt}(w_{0},1)-\nu(b)\geq
2\rho^{\vee},$

we have $\dim X_{w}(b)=d_{w}(b)$ by [He21a][Theorem 1.1]. By a straightforward
computation,
$d_{w}(b)=\langle\rho,\underline{\mu}-\nu(b)\rangle-\frac{1}{2}\text{def}(b)+\frac{1}{2}\\{\ell(w_{0})-d(x,xw_{0})\\}$.
We are now done by theorem 3.4.

∎

######  Remark 6.1.

Since we may use the explicit construction for $J\subset S$ in type
$A_{n},B_{n},C_{n}$ \- where we have $\text{wt}(x,xw_{0})=0$ for the element
$x$ found in section 4 \- we may modify the proof of theorem 1.1[(ii)]
accordingly so that it holds under the assumption
$\mu\geq\nu(b)+2\rho^{\vee}$.

One could ask if we could emulate the proof technique of theorem 1.1[(i)] for
arbitrary parahoric levels and weaken the gap $\mu-\nu(b)$ further. Here one
obstacle is that there may not exist any cordial element
$w\in~{}^{J}\operatorname{Adm}(\mu)$ for which $\dim
X_{w}(b)=d_{{}^{J\operatorname{Adm}(\mu)}}(b)$.

## References

  * [BFP99] Francesco Brenti, Sergey Fomin and Alexander Postnikov  “Mixed Bruhat operators and Yang-Baxter equations for Weyl groups”  In _Internat. Math. Res. Notices_ , 1999, pp. 419–441
  * [Bou02] Nicolas Bourbaki  “Lie groups and Lie algebras. Chapters 4–6” Translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin)  Springer-Verlag, Berlin, 2002, pp. xii+300
  * [BS17] Bhargav Bhatt and Peter Scholze  “Projectivity of the Witt vector affine Grassmannian”  In _Invent. Math._ 209.2, 2017, pp. 329–423
  * [FGP97] Sergey Fomin, Sergei Gelfand and Alexander Postnikov  “Quantum Schubert polynomials”  In _J. Amer. Math. Soc._ 10.3, 1997, pp. 565–596
  * [GH15] Ulrich Görtz and Xuhua He  “Basic loci of Coxeter type in Shimura varieties”  In _Camb. J. Math._ 3.3, 2015, pp. 323–353
  * [GHN19] Ulrich Görtz, Xuhua He and Sian Nie  “Fully Hodge-Newton decomposable Shimura varieties”  In _Peking Math. J._ 2.2, 2019, pp. 99–154
  * [Gör+06] Ulrich Görtz, Thomas J. Haines, Robert E. Kottwitz and Daniel C. Reuman  “Dimensions of some affine Deligne-Lusztig varieties”  In _Annales scientifiques de l’École Normale Supérieure_ Ser. 4, 39.3 Elsevier, 2006, pp. 467–511
  * [Ham15] Paul Hamacher  “The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian”  In _Int. Math. Res. Not. IMRN_ , 2015, pp. 12804–12839
  * [He14] Xuhua He  “Geometric and homological properties of affine Deligne-Lusztig varieties”  In _Ann. of Math. (2)_ 179.1, 2014, pp. 367–404
  * [He16] Xuhua He  “Hecke algebras and $p$-adic groups”  In _Current developments in mathematics 2015_ Int. Press, Somerville, MA, 2016, pp. 73–135
  * [He16a] Xuhua He  “Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties”  In _Ann. Sci. Éc. Norm. Supér. (4)_ 49.5, 2016, pp. 1125–1141
  * [He21] Xuhua He  “Affine Deligne-Lusztig varieties associated with generic Newton points”  arXiv, 2021  URL: https://arxiv.org/abs/2107.14461
  * [He21a] Xuhua He  “Cordial elements and dimensions of affine Deligne–Lusztig varieties”  In _Forum of Mathematics, Pi_ 9, 2021, pp. e9
  * [HL15] Xuhua He and Thomas Lam  “Projected Richardson varieties and affine Schubert varieties”  In _Ann. Inst. Fourier (Grenoble)_ 65.6, 2015, pp. 2385–2412
  * [HN21] Xuhua He and Sian Nie  “Demazure product of the affine Weyl groups”  arXiv, 2021  URL: https://arxiv.org/abs/2112.06376
  * [HR17] X. He and M. Rapoport  “Stratifications in the reduction of Shimura varieties”  In _Manuscripta Math._ 152.3-4, 2017, pp. 317–343
  * [HY21] Xuhua He and Qingchao Yu  “Dimension formula for the affine Deligne-Lusztig variety $X(\mu,b)$”  In _Math. Ann._ 379.3-4, 2021, pp. 1747–1765
  * [Kot85] Robert E. Kottwitz  “Isocrystals with additional structure”  In _Compositio Math._ 56.2, 1985, pp. 201–220
  * [Kot97] Robert E. Kottwitz  “Isocrystals with additional structure. II”  In _Compositio Math._ 109.3, 1997, pp. 255–339
  * [KR03] R. Kottwitz and M. Rapoport  “On the existence of $F$-crystals”  In _Comment. Math. Helv._ 78.1, 2003, pp. 153–184
  * [Len+15] Cristian Lenart et al.  “A uniform model for Kirillov–Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph”  In _International Mathematics Research Notices_ 2015.7 Oxford University Press, 2015, pp. 1848–1901
  * [MV20] Elizabeth Milićević and Eva Viehmann  “Generic Newton points and the Newton poset in Iwahori-double cosets”  In _Forum Math. Sigma_ 8, 2020, pp. Paper No. e5018
  * [Pos05] Alexander Postnikov  “Quantum Bruhat graph and Schubert polynomials”  In _Proc. Amer. Math. Soc._ 133.3, 2005, pp. 699–709
  * [Rap05] Michael Rapoport  “A guide to the reduction modulo $p$ of Shimura varieties” Automorphic forms. I  In _Astérisque_ , 2005, pp. 271–318
  * [RV14] Michael Rapoport and Eva Viehmann  “Towards a theory of local Shimura varieties”  In _Münster J. Math._ 7.1, 2014, pp. 273–326
  * [Sad23] Arghya Sadhukhan  “Affine Deligne-Lusztig varieties and quantum Bruhat graph”  In _Math. Z._ 303.1, 2023, pp. Paper No. 2134
  * [Sch24] Felix Schremmer  “Affine Bruhat order and Demazure products”  In _Forum of Mathematics, Sigma_ 12, 2024, pp. e53
  * [Vie06] Eva Viehmann  “The dimension of some affine Deligne-Lusztig varieties”  In _Annales scientifiques de l’École Normale Supérieure_ Ser. 4, 39.3 Elsevier, 2006, pp. 513–526
  * [Vie18] Eva Viehmann  “Moduli spaces of local ${\bf G}$-shtukas”  In _Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures_ World Sci. Publ., Hackensack, NJ, 2018, pp. 1425–1445
  * [Zhu17] Xinwen Zhu  “Affine Grassmannians and the geometric Satake in mixed characteristic”  In _Ann. of Math. (2)_ 185.2, 2017, pp. 403–492
  * [Zib23] Marcus Zibrowius  “Dual involutions in finite Coxeter groups”  In _Communications in Algebra_ Taylor & Francis, 2023, pp. 1–27