2.1. The reductive group ${\mathbf G}$ and its Iwahori–Weyl group

Let F be a nonarchimedean local field and $\breve F$ be the completion of the maximal unramified extension of F. We write $\Gamma $ for $\operatorname {\mathrm {Gal}}\left (\overline F/F\right )$ , where $\overline F$ is an algebraic closure of F. We write $\Gamma _0$ for the inertia subgroup of $\Gamma $ . Let t be a uniformiser in F.

Let ${\mathbf G}$ be a connected reductive group over F. Let $\sigma $ be the Frobenius morphism of $\breve F/F$ . We write $\breve G$ for $\mathbf G\left (\breve F\right )$ . We use the same symbol $\sigma $ for the induced Frobenius morphism on $\breve G$ .

We fix a maximal $\breve F$ -split torus S in ${\mathbf G}$ defined over F which contains a maximal F-split torus. Let T be the centraliser of S in ${\mathbf G}$ . Then T is a maximal torus. Let $\mathcal A$ be the apartment of $\mathbf G_{\breve F}$ corresponding to $S_{\breve F}$ . Thus $\mathcal A$ is (noncanonically) isomorphic to $V=X_*(T)_{\Gamma _0} \otimes _{\mathbb Z} \mathbb R$ . The Frobenius $\sigma $ naturally acts on $\mathcal A$ . We fix a $\sigma $ -stable alcove $\mathfrak a$ in $\mathcal A$ , and let $\breve I \subset \breve G$ be the Iwahori subgroup corresponding to $\mathfrak a$ . Thus $\breve I$ is $\sigma $ -stable.

We denote by N the normaliser of T in ${\mathbf G}$ . The relative Weyl group $W_0$ is defined to be $N\left (\breve F\right )/T\left (\breve F\right )$ . The Iwahori–Weyl group (associated to S) is defined as

$$ \begin{align*}\tilde{W}= N\left(\breve F\right)/T\left(\breve F\right) \cap \breve I.\end{align*} $$

For any $w \in \tilde {W}$ , we choose a representative $\dot w$ in $N(L)$ .

We have a natural short exact sequence $0 \rightarrow X_*(T)_{\Gamma _0} \rightarrow \tilde {W} \rightarrow W_0 \rightarrow 0$ . We choose a special vertex of $\mathfrak a$ and represent $\tilde {W}$ as a semidirect product,

$$ \begin{align*}\tilde{W}=X_*(T)_{\Gamma_0} \rtimes W_0=\left\{t^{\lambda} w; \lambda \in X_*(T)_{\Gamma_0}, w \in W_0\right\}.\end{align*} $$

The Iwahori–Weyl group $\tilde {W}$ contains the affine Weyl group $W_a$ as a normal subgroup and we have

$$ \begin{align*} \tilde{W}=W_a \rtimes \Omega, \end{align*} $$

where $\Omega $ is the stabiliser of $\mathfrak a$ . The length function $\ell $ and Bruhat order $\le $ on $W_a$ extend in a natural way to $\tilde {W}$ . The Frobenius $\sigma $ naturally acts on $\tilde {W}$ , in such a way that the subset $\tilde {\mathbb {S}} \subset \tilde {W}$ is stable.

For any $K \subset \tilde {\mathbb {S}}$ , we denote by $W_K$ the subgroup of $\tilde {W}$ generated by $s \in K$ . Let ${}^K \tilde {W}$ (resp., $\tilde {W}^K$ ) be the set of minimal length elements in their cosets in $W_K \backslash \tilde {W}$ (resp., $\tilde {W}/W_K$ ).

Let $\mathbb {S} \subset \tilde {\mathbb {S}}$ be the set of simple reflections of $W_0$ . By convention, the dominant Weyl chamber of V is opposite to the unique Weyl chamber containing $\mathfrak a$ . Let $\Delta $ be the set of relative simple roots determined by the dominant Weyl chamber. For any $s \in \mathbb {S}$ , we denote by $\alpha _s \in \Delta $ the corresponding simple root and $\alpha _s^\vee $ the corresponding simple coroot. We denote by $w_{\mathbb {S}}$ the longest element of $W_0$ .

We define the $\sigma $ -conjugation action on $\breve G$ by $g \cdot _{\sigma } g'=g g' \sigma (g) ^{-1}$ . Let $B(\mathbf G)$ be the set of $\sigma $ -conjugacy classes on $\breve G$ . The classification of the $\sigma $ -conjugacy classes was obtained by Kottwitz in [Reference KottwitzKo85]. Any $\sigma $ -conjugacy class $[b]$ is determined by two invariants:

• • the element $\kappa ([b]) \in \Omega _{\sigma }$ and

• • the Newton point $\nu _b \in \left (\left (X_*(T)_{\Gamma _0, \mathbb {Q}}\right )^+\right )^{\langle \sigma \rangle }$ .

Here $-_{\sigma }$ denotes the $\sigma $ -coinvariants and $\left (X_*(T)_{\Gamma _0, \mathbb {Q}}\right )^+$ denotes the set of dominant elements in $X_*(T)_{\Gamma _0}\otimes \mathbb {Q}=X_*(T)^{\Gamma _0}\otimes \mathbb {Q}$ ; the action of $\sigma $ on $\left (X_*(T)_{\Gamma _0, \mathbb {Q}}\right )/W_0$ is transferred to an action on $\left (X_*(T)_{\mathbb {Q}}\right )^+$ (L-action).

For any $w \in \tilde {W}$ , we write $\kappa (w)$ for $\kappa (\dot w)$ . It is easy to see that $\kappa (w)$ is independent of the choice of the representative w.

We use the convention of Bruhat and Tits that the translation element $t^{\lambda }$ acts by $-\lambda $ on the apartment. In this way, we have $\ell \left (x t^{\lambda }\right )=\ell (x)+\ell \left (t^{\lambda }\right )$ for any $x \in W_0$ and $\lambda $ dominant.

2.2. Affine Deligne–Lusztig varieties

We have the following generalisation of the Bruhat decomposition:

$$ \begin{align*}\breve G=\sqcup_{w \in \tilde{W}} \breve I \dot w \breve I,\end{align*} $$

due to Iwahori and Matsumoto [Reference Iwahori and MatsumotoIM65] in the split case and to Bruhat and Tits [Reference Bruhat and TitsBT72] in the general case. Let $Fl=\breve G/\breve I$ be the affine flag variety. For any $b \in \breve G$ and $w \in \tilde {W}$ , we define the corresponding affine Deligne–Lusztig variety in the affine flag variety:

$$ \begin{align*}X_w(b)=\left\{g \breve I \in \breve G/\breve I; g ^{-1} b \sigma(g) \in \breve I \dot w \breve I\right\} \subset Fl.\end{align*} $$

In the equal characteristic, $X_w(b)$ is the set of $\bar {\mathbb F}_q$ -points of a scheme [Reference Bhatt and ScholzeBS17].

As discussed in [Reference Görtz, He and NieGHN15, §2], the study of the nonemptiness pattern and dimension formula of affine Deligne–Lusztig varieties for an arbitrary reductive group may be reduced to simple and quasi-split groups over F. From now on, we assume that ${\mathbf G}$ is simple and quasi-split over F. In this case, the $\sigma $ -action on $\tilde {W}$ preserves $W_0$ and $X_*(T)_{\Gamma _0}$ . Moreover, we have $\sigma (\mathbb {S})=\mathbb {S}$ and $\sigma (\Delta )=\Delta $ .

Now we recall the definition of the virtual dimension in [Reference HeHe14, §10.1].

Note that any element $w \in \tilde {W}$ may be written in a unique way as $w=x t^\mu y$ with $\mu $ dominant, $x, y \in W_0$ , such that $t^\mu y \in {}^{\mathbb {S}} \tilde {W}$ . In this case,

(2.1) $$ \begin{align} \ell(w)=\ell(x)+\ell(t^\mu)-\ell(y). \end{align} $$

We set

$$ \begin{align*}\eta_{\sigma}(w) = \sigma^{-1}(y) x.\end{align*} $$

Let $\mathbf J_b$ be the reductive group over F with $\mathbf J_b(F)=\left \{g \in \breve G; g b \sigma (g) ^{-1}=b\right \}.$ The defect of b is defined by $\text {def}(b)=\operatorname {\mathrm {rank}}_F \mathbf G-\operatorname {\mathrm {rank}}_F \mathbf J_b.$ Here for a reductive group $\mathbb H$ defined over F, $\operatorname {\mathrm {rank}}_F$ is the F-rank of the group $\mathbb H$ . Let $\rho $ be the dominant weight with $\left \langle \alpha ^\vee , \rho \right \rangle =1$ for any $\alpha \in \Delta $ . The virtual dimension is defined to be

$$ \begin{align*}d_w(b)=\frac 12 \big( \ell(w) + \ell(\eta_{\sigma}(w)) -\text{def}(b) \big)-\langle\nu_b, \rho\rangle.\end{align*} $$

The following result is proved in [Reference HeHe14, Corollary 10.4] for residually split groups and in [Reference HeHe15, Theorem 2.30] for the general case:

For any $w \in W_0$ , we denote by $\operatorname {\mathrm {supp}}(w) \subset \mathbb {S}$ the set of simple reflections appearing in some (or equivalently, any) reduced expression of w. We set $\operatorname {\mathrm {supp}}_{\sigma }(w)=\cup _{i \in \mathbb {Z}} \sigma ^i(\operatorname {\mathrm {supp}}(w))$ .

For any $w \in \tilde {W}$ , let $\lambda _w$ be the unique dominant coweight such that $w \in W_0 t^{\lambda _w} W_0$ . For any $\lambda \in X_*(T)_{\Gamma _0}$ , we denote by $\lambda ^{\diamondsuit }$ the average of the $\sigma $ -orbit of $\lambda $ . For any $\lambda , \lambda ' \in X_*(T)^+_{\mathbb {Q}}$ , we write $\lambda \ge \lambda '$ if $\lambda -\lambda ' \in \sum _{\alpha \in \Delta } \mathbb {Q}_{\ge 0} \alpha ^\vee $ and write $\lambda \ge _{\mathbb {Z}} \lambda '$ if $\lambda -\lambda ' \in \sum _{\alpha \in \Delta } \mathbb {N} \alpha ^\vee $ . Here $\mathbb {N}$ is the set of natural numbers, that is, the set of nonnegative integers.

A critical strip of the apartment V is the subset $\left \{v; -1<\langle v, \alpha \rangle <0\right \}$ for a given positive root $\alpha $ in the reduced root system associated to the affine Weyl group $W_a$ . We remove all the critical strips from V and call each connected component of the remaining subset of V a shrunken Weyl chamber.

3.1. Minimal length elements

For any $\sigma $ -conjugacy class $\mathcal O$ in $\tilde {W}$ , we denote by $\mathcal O_{\min }$ the set of minimal length elements in $\mathcal O$ . For $w, w' \in \tilde {W}$ and $s \in \tilde {\mathbb {S}}$ , we write $w {\xrightarrow {s}}_{\sigma } w'$ if $w'=s w \sigma (s)$ and $\ell (w') \le \ell (w)$ . We write $w {\rightarrow }_{\sigma } w'$ if there is a sequence $w=w_0, w_1, \dotsc , w_n=w'$ of elements in $\tilde {W}$ such that for any k, $w_{k-1} {\xrightarrow {s}}_{\sigma } w_k$ for some $s \in \tilde {\mathbb {S}}$ . We write $w \approx _{\sigma } w'$ if $w \rightarrow _{\sigma } w'$ and $w' \rightarrow _{\sigma } w$ . It is easy to see that $w \approx _{\sigma } w'$ if $w \rightarrow _{\sigma } w'$ and $\ell (w)=\ell (w')$ .

The following result is proved in [Reference He and NieHN14, §2]:

3.2. Deligne–Lusztig reduction

Now we recall the ‘reduction’ à la Deligne and Lusztig for affine Deligne–Lusztig varieties (see [Reference Deligne and LusztigDL76, Proof of Theorem 1.6] and [Reference Görtz and HeGH10, Corollary 2.5.3]).

Here, by convention, we set $\dim \emptyset =-\infty $ and $-\infty +n=-\infty $ for any $n \in \mathbb {R}$ .

3.3. The relation $\Rightarrow $

Following [Reference Görtz and HeGH10, Definition 3.1.4], for $w, w' \in \tilde {W}$ we write $w \Rightarrow _{\sigma } w'$ if for any $b \in \breve G$ ,

$$ \begin{align*} \dim X_w(b)-d_w(b) \ge \dim X_{w'}(b)-d_{w'}(b). \end{align*} $$

Again by convention, we set $\dim \emptyset =-\infty $ . If the right-hand side is $-\infty $ , then the inequality holds regardless of the left-hand side. It is also easy to see that the relation is transitive.

Note that by the definition of virtual dimension, $w \Rightarrow _{\sigma } w'$ if and only if for any $b \in \breve G$ with $X_{w'}(b) \neq \emptyset $ , $X_w(b) \neq \emptyset $ , and in this case,

$$ \begin{align*} \dim X_w(b)-\dim X_{w'}(b) \ge \frac{1}{2}\bigl( \ell(w) + \ell(\eta_{\sigma}(w)) -\ell(w')-\ell(\eta_{\sigma}(w'))\bigr). \end{align*} $$

We write $w \Leftrightarrow _{\sigma } w'$ if $w \Rightarrow _{\sigma } w'$ and $w' \Rightarrow _{\sigma } w$ .

3.4. The monoid structure on $\tilde {W}$

By [Reference HeHe09, Lemma 1], for any $w, w' \in \tilde {W}$ the subset $\{u w'; u \le w\}$ of $\tilde {W}$ contains a unique maximal element which we denote by $w \ast w'$ . Moreover, $w \ast w'=\max \{u v; u \le w, v \le w'\}$ . Hence $\ast $ is associative. This gives a monoid structure on $\tilde {W}$ . If $w_1 \le w$ and $w^{\prime }_1 \le w'$ , then $w_1 \ast w^{\prime }_1 \le w \ast w'$ .

4.1. Definition

There is a natural partial ordering $\le $ on $B(\mathbf G)$ defined as follows: Set $[b], [b'] \in B(\mathbf G)$ . Then $[b] \le [b']$ if $\kappa (b)=\kappa (b')$ and $\nu _b \le \nu _{b'}$ .

Now we recall the cordial elements introduced by Milićević and Viehmann in [Reference Milićević and ViehmannMV20].

For any $w \in \tilde {W}$ , there is a unique maximal $\sigma $ -conjugacy class $[b]$ such that $X_w(b) \neq \emptyset $ . We denote this $\sigma $ -conjugacy class by $[b_w]$ . The element w is called cordial if $\dim X_w(b_w)=d_w(b_w)$ . Equivalently, w is cordial if and only if $\ell (w)-\ell (\eta _{\sigma }(w))=\left \langle \nu _{b_w}, 2 \rho \right \rangle -\text {def}(b_w)$ [Reference Milićević and ViehmannMV20, Definition 3.14].

By definition, if $w \Leftrightarrow _{\sigma } w'$ , then w is a cordial element if and only if $w'$ is a cordial element. The following result is proved in [Reference Milićević and ViehmannMV20, Theorem 1.1 & Corollary 3.17]:

It is mentioned in [Reference Milićević and ViehmannMV20] that fully characterising the cordial elements is fairly difficult. In [Reference Milićević and ViehmannMV20, Theorem 1.2], some interesting families of cordial elements are provided. The main result of this section is to provide another family of cordial elements.

4.2. Mazur’s inequality

Recall that ${\mathbf G}$ is quasi-split over F. Let $\breve K \supset \breve I$ be a $\sigma $ -stable special maximal parahoric subgroup of $\breve G$ . The nonemptiness pattern of the affine Deligne–Lusztig varieties in the affine Grassmannian $\breve G/\breve K$ is determined in terms of Mazur’s inequality. This was established by Gashi [Reference GashiGa10, Theorem 1.1] for unramified groups and proved in the general case in [Reference HeHe14, Theorem 7.1]. We may reformulate the result as follows:

4.3. Proof of Theorem 4.2

Set $w \in \tilde {W}$ . By definition, $[b_w]$ is the unique maximal $\sigma $ -conjugacy class that intersects $\breve I \dot w \breve I$ . By [Reference ViehmannVi14, Corollary 5.6], $[b_w]$ is also the unique maximal $\sigma $ -conjugacy class that intersects $\overline {\breve I \dot w \breve I}$ . Since $t^{\lambda } \le x t^{\lambda } \le w_{\mathbb {S}} t^{\lambda }$ , we have

$$ \begin{align*}\breve I \dot t^{\lambda} \breve I \subset \overline{\breve I \dot x \dot t^{\lambda} \breve I} \subset \overline{\breve I \dot w_{\mathbb {S}} \dot t^{\lambda} \breve I}=\overline{\breve K \dot t^{\lambda} \breve K}.\end{align*} $$

By Theorem 4.4, $\left [b_{w_{\mathbb {S}} t^{\lambda }}\right ]=\left [\dot t^{\lambda }\right ]$ . Thus $\left [b_{x t^{\lambda }}\right ]=\left [\dot t^{\lambda }\right ]$ .

Now $\nu _{b_{x t^{\lambda }}}=\lambda ^{\diamondsuit }$ and $\text {def}\left (b_{x t^{\lambda }}\right )=0$ . Hence

$$ \begin{align*}\ell\left(x t^{\lambda}\right)-\ell\left(\eta_{\sigma}\left(x t^{\lambda}\right)\right)=\ell(x)+\ell\left(t^{\lambda}\right)-\ell(x)=\ell\left(t^{\lambda}\right)=\langle \lambda, 2\rho\rangle=\left\langle \lambda^{\diamondsuit}, 2 \rho\right\rangle.\end{align*} $$

Thus $x t^{\lambda }$ is a cordial element.

4.4. Another family of cordial elements

Set $w \in \tilde {W}$ such that $w \mathfrak a$ is in the dominant Weyl chamber–that is, $w=w_{\mathbb {S}} t^{\lambda } y$ , where $\lambda $ is a dominant coweight and $y \in W_0$ with $t^{\lambda } y \in {}^{\mathbb {S}} \tilde {W}$ . It was proved by Milićević and Viehmann in [Reference Milićević and ViehmannMV20, Theorem 1.2 (a)] that w is also a cordial element.

Now we show that it can also be deduced from Theorem 4.2.

Set $w'=\sigma ^{-1}(y) w_{\mathbb {S}} t^{\lambda }$ . By Theorem 4.2, $w'$ is a cordial element. Note that $\eta _{\sigma }(w')=\eta _{\sigma }(w)=\sigma ^{-1}(y) w_{\mathbb {S}}$ . Moreover, it is easy to see that $w \approx _{\sigma } w'$ . Hence $w \Leftrightarrow _{\sigma } w'$ , and w is also a cordial element.

It is also worth mentioning that not every element of the form $x t^{\lambda }$ is $\approx _{\sigma }$ -equivalent to an element in the dominant Weyl chamber.

5.1. The normalised subtraction

For any dominant coweights $\lambda , \lambda '$ , we denote by $\lambda -_{\text {dom}} \lambda '$ the unique minimal element in the set

$$ \begin{align*}\{\mu'; \mu' \text{ is dominant}, \mu'+\lambda' \ge_{\mathbb {Z}} \lambda\}.\end{align*} $$

It is easy to see that if $\lambda -\lambda '$ is dominant, then $\lambda -_{\text {dom}} \lambda '=\lambda -\lambda '$ . We call $-_{\text {dom}}$ the normalised subtraction. Now we prove some of its properties.

Proof. Note that $\mu -(\lambda -x(\lambda '')) \in \sum _{\alpha \in \Delta } \mathbb {N} \alpha $ , $\lambda ''-x(\lambda '') \in \sum _{\alpha \in \Delta } \mathbb {N} \alpha $ and $\lambda '-\lambda '' \in \sum _{\alpha \in \Delta } \mathbb {N} \alpha $ . Thus

$$ \begin{align*} \mu+\lambda' &=(\mu-\lambda+x(\lambda''))+\lambda-x(\lambda'')+\lambda' \\ &=(\mu-\lambda+x(\lambda''))+(\lambda''-x(\lambda''))+(\lambda'-\lambda'')+\lambda \\ & \ge_{\mathbb {Z}} \lambda.\end{align*} $$

Corollary 5.4. Let $\lambda , \lambda _1, \lambda _2$ be dominant coweights. Then

$$ \begin{align*}(\lambda-_{\text{dom}} \lambda_1)-_{\text{dom}} \lambda_2=\lambda-_{\text{dom}} (\lambda_1+\lambda_2).\end{align*} $$

Proof. Set $\mu _1=(\lambda -_{\text {dom}} \lambda _1)-_{\text {dom}} \lambda _2$ and $\mu _2=\lambda -_{\text {dom}} (\lambda _1+\lambda _2)$ . By definition,

$$ \begin{align*}(\lambda_1+\lambda_2)+\mu_1=\lambda_1+(\lambda_2+\mu_1) \ge_{\mathbb {Z}} \lambda_1+(\lambda-_{\text{dom}} \lambda_1) \ge_{\mathbb {Z}} \lambda.\end{align*} $$

So $\mu _1 \ge _{\mathbb {Z}} \mu _2.$

On the other hand,

$$ \begin{align*}\lambda_1+(\lambda_2+\mu_2)=(\lambda_1+\lambda_2)+\mu_2 \ge_{\mathbb {Z}} \lambda.\end{align*} $$

So by definition, $\lambda _2+\mu _2 \ge _{\mathbb {Z}} \lambda -_{\text {dom}} \lambda _1$ and $\mu _2 \ge _{\mathbb {Z}} \mu _1$ .

5.2. The double flat operator

For any subset J of $\mathbb {S}$ , we denote by $\rho ^\vee _J$ the dominant coweight with

$$ \begin{align*}\left\langle\rho^\vee_J, \alpha_s\right\rangle=\begin{cases} 1 & \text{if } s \in J, \\ 0 & \text{if } s \notin J.\end{cases}\end{align*} $$

Let $\eta ^\vee _J$ be the unique dominant coweight in the $W_0$ -orbit of $-\sigma ^{-1}\left (\rho ^\vee _J\right )$ .

Set $w \in \tilde {W}$ . We write w as $w=x t^{\lambda } y$ with $\lambda $ dominant, $x, y \in W_0$ and $t^{\lambda } y \in {}^{\mathbb {S}} \tilde {W}$ . Let $J=\{s \in \mathbb {S}; {s y<y}\}$ . Since $t^{\lambda } y \in {}^{\mathbb {S}} \tilde {W}$ , we have $\langle \lambda , \alpha _s\rangle>0$ for any $s \in J$ . In particular, $\lambda -\rho ^\vee _J$ is dominant. We set

$$ \begin{align*}\lambda_w^{\flat\mkern-2.4mu\flat}=\left(\lambda-\rho^\vee_J\right)-_{\text{dom}} \eta^\vee_J=\lambda-_{\text{dom}} \left(\rho^\vee_J+\eta^\vee_J\right).\end{align*} $$

The main result of this section is as follows:

Theorem 5.5. Assume that ${\mathbf G}$ is quasi-split over F. Set $w \in \tilde {W}$ such that $w \mathfrak a$ is in a shrunken Weyl chamber. Then there exist a dominant coweight $\gamma $ with $\gamma \ge _{\mathbb {Z}} \lambda _w^{\flat \mkern -2.4mu\flat }$ and $a \in W_0$ with $\operatorname {\mathrm {supp}}_{\sigma }(a) \supset \operatorname {\mathrm {supp}}_{\sigma }(\eta _{\sigma }(w))$ such that

$$ \begin{align*}w \Rightarrow_{\sigma} a t^\gamma.\end{align*} $$

5.3. A convenient notation

Following [Reference Görtz and HeGH10, §2.4], we give a convenient notation for varieties of tuples of elements in $Fl$ . We explain the notation by examples. Let $\mathcal O_w=\left \{\left (g \breve I, g \dot w \breve I\right ); g \in \breve G\right \} \subset Fl\times Fl$ . Then we set

Similarly,

The affine Deligne–Lusztig varieties can be written as

In all these cases, we do not distinguish between the sets given by the conditions on the relative position and the corresponding locally closed sub-ind-schemes of the product of affine flag varieties. The following result is proved in [Reference Görtz and HeGH10, Proposition 2.5.2]:

Proposition 5.6. Set $w, w'\in \tilde {W}$ , and set $w'' \in \{ ww', w \ast w' \}$ . Then the map

is surjective. Moreover, all the fibres have dimension

$$ \begin{align*} \dim\pi^{-1}( ( g, g') ) \ge \begin{cases} \ell(w) + \ell(w') - \ell(w*w') & \text{if } w'' = w \ast w',\\ \frac 12 \bigl( \ell(w) + \ell(w') - \ell(ww') \bigr) & \text{if } w'' = ww'. \end{cases} \end{align*} $$

5.4. Proof of Theorem 5.5

We write w as $w=x t^{\lambda _w} y$ with $x, y \in W_0$ and $t^{\lambda _w} y \in {}^{\mathbb {S}} \tilde {W}$ . Let $J=\{s \in \mathbb {S}; s y<y\}$ and $J'=\left \{s \in \mathbb {S}; s\left (\lambda _w-\rho ^\vee _J\right )=\lambda _w-\rho ^\vee _J\right \}$ . We write $\sigma ^{-1}(y) x$ as $\sigma ^{-1}(y)x=x' z$ for some $x' \in W_0^{J'}$ and $z \in W_{J'}$ . By [Reference Görtz and HeGH10, §2.3], the assumption that $w \mathfrak a$ is in a shrunken Weyl chamber implies that $x \alpha _j<0$ for any $j \in J'$ . In particular,

(5.1) $$ \begin{align} \ell\left(x z ^{-1}\right)=\ell(x)-\ell(z). \end{align} $$

Let $\gamma $ be the unique dominant coweight in the $W_0$ -orbit of $\lambda _w-\rho ^\vee _J+(x') ^{-1} \sigma ^{-1}\left (\rho ^\vee _J\right )$ . By Corollary 5.3, $\gamma \ge _{\mathbb {Z}} \lambda _w^{\flat \mkern -2.4mu\flat }$ . Let $K=\{s \in \mathbb {S}; s(\gamma )=\gamma \}$ and set $y' \in W_0^K$ with $\lambda _w-\rho ^\vee _J+(x') ^{-1} \sigma ^{-1}\left (\rho ^\vee _J\right ) =y'(\gamma )$ .

Set $\alpha>0$ with $(y') ^{-1} \alpha <0$ . Then $\left \langle \gamma , (y') ^{-1} \alpha \right \rangle \le 0$ . On the other hand, if $\left \langle \gamma , (y') ^{-1} \alpha \right \rangle =0$ , then since $y' \in W_0^K$ , we have $\alpha =y' \bigl ((y') ^{-1} \alpha \bigr )<0$ . That is a contradiction. Hence $\langle y'(\gamma ), \alpha \rangle =\left \langle \gamma , (y') ^{-1} \alpha \right \rangle <0$ . Since $\lambda _w-\rho ^\vee _J$ is dominant, $\left \langle \lambda _w-\rho ^\vee _J, \alpha \right \rangle \ge 0$ . Thus $\left \langle (x') ^{-1} \sigma ^{-1}\left (\rho ^\vee _J\right ), \alpha \right \rangle =\left \langle \sigma ^{-1}\left (\rho ^\vee _J\right ), x'(\alpha )\right \rangle <0$ . Since $\sigma ^{-1}\left (\rho ^\vee _J\right )$ is dominant, $x'(\alpha )<0$ . By [Reference Görtz and HeGH10, Lemma 2.6.1], we have

(5.2) $$ \begin{align} \ell(x' y')=\ell(x')-\ell(y'). \end{align} $$

Set $s \in J'$ . Since $x' \in W_0^{J'}$ , we have $\ell (x' s)=\ell (x')+1$ . Thus $\ell (x')-\ell (y')=\ell (x' y')=\ell \bigl ((x' s) (s y') \bigr ) \ge \ell (x' s)-\ell (s y')=\ell (x')+1-\ell (s y')$ . So $\ell (s y') \ge \ell (y')+1$ . Therefore

(5.3) $$ \begin{align} y' \in {}^{J'} W_0. \end{align} $$

In particular, we have

(5.4) $$ \begin{align} \ell\left((y') ^{-1} z\right)=\ell(y')+\ell(z). \end{align} $$

Set $w_1=x z ^{-1} t^{\lambda _w-\rho ^\vee _J} y'$ and $w_2=(y') ^{-1} z t^{\rho ^\vee _J} y$ . Then $w=w_1 w_2$ . By formula (5.3), $t^{\lambda _w-\rho ^\vee _J} y' \in {}^{\mathbb {S}} \tilde {W}$ . By the definition of J, we have $t^{\rho ^\vee _J} y \in {}^{\mathbb {S}} \tilde {W}$ . Hence by equation (2.1), we have

(5.5) $$ \begin{align} \ell(w)=\ell(x)+\ell\left(t^{\lambda_w}\right)-\ell(y) \end{align} $$

(5.6) $$ \begin{align} \ell(w_1)=\ell\left(x z ^{-1}\right)+\ell\left(t^{\lambda_w-\rho^\vee_J}\right)-\ell(y') \end{align} $$

(5.7) $$ \begin{align} \ell(w_2)=\ell\left((y') ^{-1} z\right)+\ell\left(t^{\rho^\vee_J}\right)-\ell(y). \end{align} $$

By equations (5.1) and (5.4), we have

(5.8) $$ \begin{align} \ell(w_1)+\ell(w_2) &=\ell(x)-\ell(z)+\ell\left(t^{\lambda_w-\rho^\vee_J}\right)-\ell(y')+\ell(y')+\ell(z)+\ell\left(t^{\rho^\vee_J}\right)-\ell(y) \nonumber \\ &=\ell(x)+\ell\left(t^{\lambda_w}\right)-\ell(y)=\ell(w). \end{align} $$

By equations (5.2) and (5.4), we have

(5.9) $$ \begin{align} \ell\left((y') ^{-1} z\right)+\ell(x' y') &=\ell(y')+\ell(z)+\ell(x')-\ell(y')=\ell(x')+\ell(z) \nonumber \\ &=\ell(x'z)=\ell\left(\sigma ^{-1}(y) x\right). \end{align} $$

By equation (5.8) we have

Set

Here the isomorphism follows from equation (5.7).

The map $(g, g_1) \mapsto (g_1, b \sigma (g))$ is a universal homeomorphism from $X_{w}(b)$ to $X_1$ . We have $y \sigma \left (x z ^{-1}\right )=\sigma \left (\sigma ^{-1}(y) x z ^{-1}\right )=\sigma (x')$ and

$$ \begin{align*}t^{\rho^\vee_J} y \sigma(w_1)=t^{\rho^\vee_J} \sigma\left(x' t^{\lambda_w-\rho^\vee_J} y'\right)=\sigma(x') \sigma\left(t^{y'(\gamma)}\right) \sigma(y')=\sigma(x' y' t^\gamma).\end{align*} $$

Let

We have

$$ \begin{align*}\dim(X_w(b))=\dim(X_1) \ge \dim(X_2).\end{align*} $$

Since $\gamma $ is dominant, we have $\ell (\sigma (x' y' t^\gamma ))=\ell (\sigma (x' y'))+\ell (\sigma (t^\gamma ))$ . Set

Let $\pi : X_2 \rightarrow X_3$ be the projection map. Set $N_1=\frac {\ell \left (t^{\rho ^\vee _J} y\right )+\ell \left (w_1\right )-\ell \left (x' y' t^\gamma \right )}{2}$ . By Proposition 5.6, the map $\pi $ is surjective and the dimension of each fibre is larger than or equal to $N_1$ .

We show that

(a) $\dim (X_2) \ge \dim (X_3)+N_1$ .

Now suppose that $\dim (X_2)<\dim (X_3)+N_1$ . Let Z be an irreducible component of $X_3$ with $\dim Z=\dim X_3$ . For each irreducible component Y of $\pi ^{-1} (Z)$ , we construct a closed subscheme $Z_Y$ of Z such that $\dim \left (\pi ^{-1}(z) \cap Y\right )<N_1$ if $z \in Z-Z_Y$ . The construction is as follows.

If $\pi (Y)$ is not dense in Z, then let $Z_Y$ be the closure of $\pi (Y)$ . If $\pi (Y)$ is dense in Z, then the morphism $\pi : Y \rightarrow Z$ is dominant. By [Reference Görtz and WedhornGW10, Corollary 14.116], there exists an open dense subscheme V of Z contained in $\pi (Y)$ such that for any $z \in V$ , we have $\dim \left (\pi ^{-1}(z) \cap Y\right )=\dim (Y)-\dim (Z) \le \dim (X_2)-\dim (X_3)<N_1$ . We set $Z_Y=Z-V$ . This finishes our construction.

Note that in either case, $Z_Y$ is a proper subscheme of Z. Hence $\cup _Y Z_Y \subsetneqq Z$ . Set $z \in Z-\cup _Y Z_Y$ . Then $\dim \left (\pi ^{-1}(z) \cap Y\right )<N_1$ for any irreducible component Y of $\pi ^{-1}(Z)$ . Thus $\dim \left (\pi ^{-1}(z)\right )<N_1$ . That gives a contradiction.

So (a) is proved.

Set $a =\sigma ^{-1}\left ((y')^{-1} z\right ) * (x' y')$ . Then

$$ \begin{align*} \operatorname{\mathrm{supp}}_{\sigma}(a) &=\operatorname{\mathrm{supp}}_{\sigma}(\sigma(a))=\operatorname{\mathrm{supp}}_{\sigma}\left((y') ^{-1} z\right) \cup \operatorname{\mathrm{supp}}_{\sigma}(\sigma(x'y')) \\[4pt] &=\operatorname{\mathrm{supp}}_{\sigma}\left((y') ^{-1} z\right) \cup \operatorname{\mathrm{supp}}_{\sigma}(x'y') \\[4pt] & \supset \operatorname{\mathrm{supp}}_{\sigma}\left(x' y' (y') ^{-1} z\right)=\operatorname{\mathrm{supp}}_{\sigma}(x' z) \\[4pt] &=\operatorname{\mathrm{supp}}_{\sigma}(\eta_{\sigma}(w)).\end{align*} $$

We set

By Proposition 5.6 and the same argument as in (a),

$$ \begin{align*}\dim(X_3) \ge \dim(X_5)+\ell\left((y') ^{-1} z\right)+\ell(x' y')-\ell(a)=\dim(X_5)+\ell(\eta_{\sigma}(w))-\ell(a).\end{align*} $$

Notice that $\ell (\sigma (a t^\gamma ))=\ell (\sigma (a))+\ell (\sigma (t^\gamma ))$ . Thus the map $(g_1, g_4) \mapsto g_1$ gives an isomorphism $X_5 \cong X_{\sigma \left (a t^\gamma \right )}(b)$ , which is universally homeomorphic to $X_{a t^\gamma }(b)$ . If $X_{a t^\gamma }(b)\ne \emptyset $ , then $X_{\sigma \left (a t^\gamma \right )}(b) \neq \emptyset $ and $X_{w}(b)\ne \emptyset $ . Note that $\ell \left ((y') ^{-1} z\right )+\ell (x' y')=\ell \left (\sigma ^{-1}(y) x\right )=\ell (\eta _{\sigma }(w))$ . Therefore,

$$ \begin{align*} \dim & X_w(b)-\dim X_{a t^\gamma}(b) \\[4pt] & \ge \frac{\ell\left(t^{\rho^\vee_J} y\right)+\ell(w_1)-\ell\left(x' y' t^\gamma\right)}{2}+\ell(\eta_{\sigma}(w))-\ell(a)\\[4pt] &=\frac{\ell\left(t^{\rho^\vee_J} y\right)+\ell\left((y') ^{-1} z\right)+\ell(w_1)-\ell(x' y' t^\gamma)+\ell(x' y')-\ell(a)}{2}+\frac{\ell(\eta_{\sigma}(w))}{2}-\frac{\ell(a)}{2} \\[4pt] & =\frac{\ell(w_2)+\ell(w_1)-\ell(a t^\gamma)}{2}+\frac{\ell(\eta_{\sigma}(w))}{2}-\frac{\ell(a)}{2} \\[4pt] &=\frac{\ell(w)-\ell(a t^\gamma)}{2}+\frac{\ell(\eta_{\sigma}(w))}{2}-\frac{\ell(a)}{2}=d_{w}(b)-d_{a t^\gamma}(b).\end{align*} $$

So $w \Rightarrow _{\sigma } a t^\gamma $ . The theorem is proved.

6.1. The $(J, w, \delta )$ -alcove elements

We recall the alcove elements introduced in [Reference Görtz, Haines, Kottwitz and ReumanGHKR10] for split groups and then generalised to quasi-split groups in [Reference Görtz, He and NieGHN15].

For any $J \subset \mathbb {S}$ with $\sigma (J)=J$ , we denote by $\mathbb M_J \subset \mathbf G$ the standard Levi subgroup corresponding to J and let $\mathbb P_J \supset \mathbb M_J$ be the standard parabolic subgroup. Let $\mathbb U_{\mathbb P_J}$ be the unipotent radical of $\mathbb P_J$ .

Set $J \subset \mathbb {S}$ with $\sigma (J)=J$ and $x \in W_0$ . Set $w\in \tilde {W}$ . We say that w is a $(J, x, \sigma )$ -alcove element if $x ^{-1} w \sigma (x) \in \tilde {W}_J$ and $\mathbb {}^{\dot x} \mathbb U_{\mathbb P_J}\left (\breve F\right ) \cap {}^{\dot w} \breve I \subseteq \mathbb {}^{\dot x} \mathbb U_{\mathbb P_J}\left (\breve F\right ) \cap \breve I$ . The following result is proved in [Reference Görtz, He and NieGHN15, Corollary 3.6.1].Footnote ¹

6.2. The emptiness pattern

Suppose that $w \mathfrak a$ is in a shrunken Weyl chamber and $\left (\lambda _w^{\flat \mkern -2.4mu\flat }\right )^{\diamondsuit } \ge \nu _b$ . We write w as $w=x t^{\lambda } y$ with $x, y \in W_0$ and $t^{\lambda } y \in {}^{\mathbb {S}} \tilde {W}$ . If $\kappa (b) \neq \kappa (w)$ , then $X_w(b)=\emptyset $ .

Now suppose that $\kappa (b)=\kappa (w)$ and $\operatorname {\mathrm {supp}}_{\sigma }\left (\sigma ^{-1}(y) x\right )\neq \mathbb {S}$ . Set $J=\operatorname {\mathrm {supp}}_{\sigma }\left (\sigma ^{-1}(y) x\right )$ . By [Reference Görtz, He and NieGHN15, Lemma 3.6.3], w is a $\left (J, \sigma ^{-1}(y), \sigma \right )$ -alcove element. Set $b' \in [b] \cap \mathbb M_J\left (\breve F\right )$ . We denote by $\nu ^{\mathbb M_J}_{b'}$ the image of $b'$ under the Newton map for $\mathbb M_J$ . Then $\nu ^{\mathbb M_J}_{b'} \in W_0 (\nu _b)$ . Hence $\nu _b-\nu ^{\mathbb M_J}_{b'} \in \sum _{\alpha \in \Delta } \mathbb {Q}_{\ge 0} \alpha ^\vee $ .

By assumption, $\lambda ^{\diamondsuit }-\nu _b \in \sum _{\alpha \in \Delta } \mathbb {Q}_{>0} \alpha ^\vee $ . Thus $\lambda ^{\diamondsuit }-\nu ^{\mathbb M_J}_{b'} \in \sum _{\alpha \in \Delta } \mathbb {Q}_{>0} \alpha ^\vee $ and cannot be written as a linear combination of the coroots in $\mathbb M_J$ . Therefore $\kappa _{\mathbb M_J}\left (\sigma ^{-1} (y) w y ^{-1}\right ) \neq \kappa _{\mathbb M_J}(b')$ . By Theorem 6.3, $X_w(b)=\emptyset $ .

6.3. Dimension formula

Suppose that $\kappa (w)=\kappa (b)$ and $\operatorname {\mathrm {supp}}_{\sigma }(\eta _{\sigma }(w))=\mathbb {S}$ . By Theorem 5.5, there exist a dominant coweight $\gamma \ge _{\mathbb {Z}} \lambda _w^{\flat \mkern -2.4mu\flat }$ and $a \in W_0$ with $\operatorname {\mathrm {supp}}_{\sigma }(a)=\mathbb {S}$ such that

$$ \begin{align*}w \Rightarrow_{\sigma} a t^\gamma.\end{align*} $$

By our assumption, $\gamma ^{\diamondsuit } \ge \left (\lambda _w^{\flat \mkern -2.4mu\flat }\right )^{\diamondsuit } \ge \nu _b$ . By Theorem 4.2, $\left [\dot t^\gamma \right ]=\left [b_{a t^\gamma }\right ]$ . Since $\kappa (w)=\kappa (t^\gamma )=\kappa (b)$ , we have $[b] \le \left [\dot t^\gamma \right ]$ .

By [Reference HeHe15, Theorem 2.27], $X_{a t^\gamma }(\dot {\tau } ) \neq \emptyset $ , where $\tau \in \Omega $ with $\kappa (w)=\kappa (t^\gamma )=\kappa (\tau )$ . Since $\kappa (w)=\kappa (b)$ , we have $[\dot \tau ] \le [b]$ .

By Theorem 4.2, $a t^\gamma $ is a cordial element. Hence by Theorem 4.1(1), $X_{a t^\gamma }(b) \neq \emptyset $ , and by Theorem 4.1(2), $\dim X_{a t^\gamma }(b)=d_{a t^\gamma }(b)$ .

So by the definition of $\Rightarrow _{\sigma }$ , we have $X_w(b) \neq \emptyset $ and

$$ \begin{align*}\dim X_w(b)-d_w(b) \ge \dim X_{a t^\gamma}(b)-d_{a t^\gamma}(b)=0.\end{align*} $$

Hence $\dim X_w(b) \ge d_w(b)$ . On the other hand, by Theorem 2.1, $\dim X_w(b) \le d_w(b)$ . So $\dim X_w(b)=d_w(b)$ .

6.4. Some remarks on the condition $\left (\lambda _w^{\flat \mkern -2.4mu\flat }\right )^{\diamondsuit } \ge \nu _b$

We first consider the case where $[b]$ is basic. In this case, the condition $\left (\lambda _w^{\flat \mkern -2.4mu\flat }\right )^{\diamondsuit } \ge \nu _b$ follows directly from the condition $\kappa (b)=\kappa (w)$ .

Now we consider nonbasic $[b]$ . Suppose that $\lambda _w^{\diamondsuit } \ge \nu _b+2 \rho ^\vee $ . In this case, although $\lambda _w-2 \rho ^\vee $ may not be dominant, its $\sigma $ -average is dominant and is larger than or equal to $\nu _b$ . By definition, $\lambda _w^{\flat \mkern -2.4mu\flat }-\left (\lambda _w-\rho ^\vee _J-\eta ^\vee _J\right ) \in \sum _{\alpha \in \Delta } \mathbb {Q}_{\ge 0} \alpha ^\vee $ for some J. Note that $2 \rho ^\vee -\rho ^\vee _J-\eta ^\vee _J \in \sum _{\alpha \in \Delta } \mathbb {Q}_{\ge 0} \alpha ^\vee $ . We have $\lambda _w^{\flat \mkern -2.4mu\flat }-\left (\lambda _w-2 \rho ^\vee \right ) \in \sum _{\alpha \in \Delta } \mathbb {Q}_{\ge 0} \alpha ^\vee $ . Hence $\left (\lambda _w^{\flat \mkern -2.4mu\flat }\right )^{\diamondsuit } \ge \lambda _w^{\diamondsuit }-2 \rho ^\vee \ge \nu _b$ . It is also easy to see that $\lambda _w^{\diamondsuit }-\nu _b \in \sum _{\alpha \in \Delta } \mathbb {Q}_{>0} \alpha ^\vee $ .

In particular, if $\lambda _w=n \omega ^\vee $ , where $\omega ^\vee $ is a fundamental coweight and $n \gg 0$ with respect to $[b]$ , then $\lambda _w^{\diamondsuit } \ge \nu _b+2 \rho ^\vee $ , and hence the condition $\left (\lambda _w^{\flat \mkern -2.4mu\flat }\right )^{\diamondsuit } \ge \nu _b$ is satisfied in this case.

6.5. A side remark

By Theorem 4.4, if $X_w(b) \neq \emptyset $ , then $\kappa (b)=\kappa (w)$ and $\nu _b \le \lambda _w^{\diamondsuit }$ .

Set $w \in \tilde {W}$ such that $w \mathfrak a$ is in a shrunken Weyl chamber. If $\operatorname {\mathrm {supp}}_{\sigma }(\eta _{\sigma }(w))=\mathbb {S}$ , then Theorem 6.1 describes the nonemptiness pattern and the dimension formula of $X_w(b)$ for most of the $\sigma $ -conjugacy classes $[b]$ with $\kappa (b)=\kappa (w)$ and $\nu _b \le \lambda _w^{\diamondsuit }$ .

If $\operatorname {\mathrm {supp}}_{\sigma }(\eta _{\sigma }(w))=J \subsetneqq \mathbb {S}$ , then by [Reference Görtz, He and NieGHN15, Lemma 3.6.3] w is a $\left (J, \sigma ^{-1}(y), \sigma \right )$ -alcove element for some $y \in W_0$ . Then the Hodge–Newton decomposition (see [Reference Görtz, Haines, Kottwitz and ReumanGHKR10, Theorem 2.1.4] for the split group and [Reference Görtz, He and NieGHN15, Propositon 2.5.1 & Theorem 3.3.1] in general) reduces the study of $X_w(b)$ to the study of a suitable affine Deligne–Lusztig variety associated to the Levi subgroup $\mathbb M_J$ . One may apply Theorem 6.1 to the latter. In this way, one also obtains an explicit description of the nonemptiness pattern and the dimension formula of $X_w(b)$ for most of the $\sigma $ -conjugacy classes $[b]$ with $\kappa (b)=\kappa (w)$ and $\nu _b \le \lambda _w^{\diamondsuit }$ .