2.1 Notation

Let $\Phi =\Phi (G,T)$ denote the set of roots of T in G. We denote by $\Phi _+$ (resp. $\Phi _-$ ) the set of positive (resp. negative) roots distinguished by B. Let $\Delta $ be the set of simple roots and $\Delta ^\vee $ be the corresponding set of simple coroots. Let $X_*(T)$ be the set of cocharacters, and let $X_*(T)_+$ be the set of dominant cocharacters.

The Iwahori-Weyl group $\tilde W$ is defined as the quotient $N_{G(L)}T(L)/T(\mathcal O)$ . This can be identified with the semi-direct product $W_0\ltimes X_{*}(T)$ , where $W_0$ is the finite Weyl group of G. We denote the projection $\tilde W\rightarrow W_0$ by p. We have a length function $\ell \colon \tilde W\rightarrow \mathbb Z_{\geq 0}$ given as

$$ \begin{align*}\ell(w_0\varpi^{\lambda})=\sum_{\alpha\in \Phi_+, w_0\alpha\in \Phi_-}|\langle \alpha, \lambda\rangle+1|+\sum_{\alpha\in \Phi_+, w_0\alpha\in \Phi_+}|\langle \alpha, \lambda\rangle|,\end{align*} $$

where $w_0\in W_0$ and $\lambda \in X_*(T)$ .

Let $S\subset W_0$ denote the subset of simple reflections, and let $\tilde S\subset \tilde W$ denote the subset of simple affine reflections. We often identify $\Delta $ and S. The affine Weyl group $W_a$ is the subgroup of $\tilde W$ generated by $\tilde S$ . Then we can write the Iwahori-Weyl group as a semi-direct product $\tilde W=W_a\rtimes \Omega $ , where $\Omega \subset \tilde W$ is the subgroup of length $0$ elements. Moreover, $(W_a, \tilde S)$ is a Coxeter system. We denote by $\le $ the Bruhat order on $\tilde W$ . For any $J\subseteq \tilde S$ , let $^J\tilde W$ be the set of minimal length representatives for the cosets in $W_J\backslash \tilde W$ , where $W_J$ denotes the subgroup of $\tilde W$ generated by J.

Let $w\in \tilde W$ . There exists a positive integer k such that $w^k=\varpi ^\lambda $ for some $\lambda \in X_*(T)$ . We set $\nu _w=\lambda /k \in X_*(T)_{\mathbb Q}$ . This is independent of the choice of k.

For $w\in W_a$ , we denote by $\operatorname {\mathrm {supp}}(w)\subseteq \tilde S$ the set of simple affine reflections occurring in every (equivalently, some) reduced expression of w. Note that $\tau \in \Omega $ acts on $\tilde S$ by conjugation. We define the $\sigma $ -support $\operatorname {\mathrm {supp}}_\sigma (w\tau )$ of $w\tau $ as the smallest $\tau \sigma $ -stable subset of $\tilde S$ which contains $\operatorname {\mathrm {supp}}(w)$ .

For $w,w'\in \tilde W$ and $s\in \tilde S$ , we write $w{\xrightarrow {s}}_\sigma w'$ if $w'=sw\sigma (s)$ and $\ell (w')\le \ell (w)$ . We write $w\rightarrow _\sigma w'$ if there is a sequence $w=w_0,w_1,\ldots , w_k=w'$ of elements in $\tilde W$ such that for any i, $w_{i-1}{\xrightarrow {s_i}}_\sigma w_i$ for some $s_i\in S$ . If $w\rightarrow _\sigma w'$ and $w'\rightarrow _\sigma w$ , we write $w\approx _\sigma w'$ .

For $\alpha \in \Phi $ , let $U_\alpha \subseteq G$ denote the corresponding root subgroup. We set

$$ \begin{align*}I=T(\mathcal O)\prod_{\alpha\in \Phi_+}U_{\alpha}(\varpi\mathcal O)\prod_{\beta\in \Phi_-}U_{\beta}(\mathcal O)\subseteq G(L),\end{align*} $$

which is called the standard Iwahori subgroup associated to the triple $T\subset B\subset G$ .

In the case $G=\operatorname {\mathrm {GL}}_n$ , we will use the following description. Let T be the torus of diagonal matrices, and we choose the subgroup of upper triangular matrices B as Borel subgroup. Let $\chi _{ij}$ be the character $T\rightarrow \mathbb G_m$ defined by $\mathrm {diag}(t_1,t_2,\ldots , t_n)\mapsto t_i{t_j}^{-1}$ . Then we have $\Phi =\{\chi _{ij}\mid i\neq j\}$ , $\Phi _+=\{\chi _{ij}\mid i< j\}$ , $\Phi _-=\{\chi _{ij}\mid i> j\}$ and $\Delta =\{\chi _{i,i+1}\mid 1\le i <n\}$ . Through a natural isomorphism $X_*(T)\cong \mathbb Z^n$ , ${X_*(T)}_+$ can be identified with the set $\{(m_1,\ldots , m_n)\in \mathbb Z^n\mid m_1\geq \cdots \geq m_n\}$ . The finite Weyl group is the symmetric group of degree n. Let us write $s_1=(1\ 2), s_2=(2\ 3), \ldots , s_{n-1}=(n-1\ n)$ . Set $s_0=\varpi ^{\chi _{1,n}^{\vee }}(1\ n)$ , where $\chi _{1,n}$ is the unique highest root. Then $S=\{s_1,s_2,\ldots , s_{n-1}\}$ and $\tilde S=S\cup \{s_0\}$ . The Iwahori subgroup $I\subset K$ is the inverse image of the lower triangular matrices under the projection $K\rightarrow G(\overline {\mathbb F}_q)$ induced by $\varpi \mapsto 0$ . Set $\tau ={\begin {pmatrix} 0 & \varpi \\ 1_{n-1} & 0\\ \end {pmatrix}}$ . We often regard $\tau $ as an element of $\tilde W$ , which is a generator of $\Omega \cong \mathbb Z$ . Note that $b\in \operatorname {\mathrm {GL}}_n(L)$ is superbasic if and only if $[b]=[\tau ^m]$ in $B(\operatorname {\mathrm {GL}}_n)$ for some m coprime to n.

2.2 Affine Deligne–Lusztig varieties

For $w\in \tilde W$ and $b\in G(L)$ , the affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety $G(L)/I$ is defined as

$$ \begin{align*}X_w(b)=\{xI\in G(L)/I\mid x^{-1}b\sigma(x)\in IwI\}.\end{align*} $$

For $\mu \in X_*(T)_+$ and $b\in G(L)$ , the affine Deligne–Lusztig variety $X_{\mu }(b)$ in the affine Grassmannian ${\mathcal {G}} r=G(L)/K$ is defined as

$$ \begin{align*}X_{\mu}(b)=\{xK\in {\mathcal{G}} r\mid x^{-1}b\sigma(x)\in K\varpi^{\mu}K\}.\end{align*} $$

The closed affine Deligne–Lusztig variety is the closed reduced $\overline {\mathbb F}_q$ -subscheme of ${\mathcal {G}} r$ defined as

$$ \begin{align*}X_{\preceq\mu}(b)=\bigcup_{\mu'\preceq \mu}X_{\mu'}(b).\end{align*} $$

Left multiplication by $g^{-1}\in G(L)$ induces an isomorphism between $X_\mu (b)$ and $X_\mu (g^{-1}b\sigma (g))$ . Thus, the isomorphism class of the affine Deligne–Lusztig variety only depends on the $\sigma $ -conjugacy class of b. Moreover, we have $X_\mu (b)=X_{\mu +\lambda }(\varpi ^{\lambda }b)$ for each central $\lambda \in X_*(T)$ .

The admissible subset of $\tilde W$ associated to $\mu $ is defined as

$$ \begin{align*}\operatorname{\mathrm{Adm}}(\mu)=\{w\in \tilde W\mid w\le \varpi^{w_0\mu}\ \text{for some}\ w_0\in W_0\}.\end{align*} $$

Note that $\operatorname {\mathrm {Adm}}(\mu ')\subseteq \operatorname {\mathrm {Adm}}(\mu )$ if $\mu '\preceq \mu $ . Indeed if $w\le \varpi ^{w_0\mu '}$ and $\mu '\preceq \mu $ , then $w\le \varpi ^{w_0\mu }$ by [Reference Haines16, Lemma 4.5]. Set ${^S\mathrm {Adm}}(\mu )=\operatorname {\mathrm {Adm}}(\mu )\cap {^S\tilde W}$ . Then by [Reference Görtz and He11, Theorem 3.2.1] (see also [Reference Görtz, He and Rapoport15, Section 2.5]), we have

$$ \begin{align*}X_{\preceq\mu}(b)=\bigsqcup_{w\in{^S\mathrm{Adm}}(\mu)}\pi(X_w(b)),\end{align*} $$

where $\pi \colon G(L)/I\rightarrow G(L)/K$ is the projection. This is the so-called Ekedahl–Oort stratification.

For any $w\in {^S\tilde W}$ , set

2.3 Deligne–Lusztig reduction method

The following Deligne–Lusztig reduction method was established in [Reference Görtz and He10, Corollary 2.5.3].

The following result is proved in [Reference He and Nie22, Theorem 2.10], which allows us to reduce the study of $X_w(b)$ for any w, via the Deligne–Lusztig reduction method, to the study of $X_w(b)$ for w of minimal length in its $\sigma $ -conjugacy class.

Following [Reference He, Nie and Yu23, Section 3.4], we construct the reduction trees for w by induction on $\ell (w)$ .

The vertices of the trees are elements of $\tilde W$ . We write $x\rightharpoonup y$ if $x,y\in \tilde W$ and there exists $x'\in \tilde W$ and $s\in \tilde S$ such that $x\approx _\sigma x'$ , $\ell (sx'\sigma (s))=\ell (x')-2$ and $y\in \{sx', sx'\sigma (s)\}$ . These are the (oriented) edges of the trees. A reduction tree of w is a tree with these vertices and edges whose unique starting point is w and whose end points are of minimal length in its $\sigma $ -conjugacy class of $\tilde W$ .

The existence of a (not necessarily unique) reduction tree of w can be proved as follows. If w is of minimal length in its $\sigma $ -conjugacy class of $\tilde W$ , then the reduction tree for w consists of a single vertex w and no edges. Assume that w is not of minimal length and that a reduction tree is given for any $z\in \tilde W$ with $\ell (z)<\ell (w)$ . By Theorem 2.4, there exist $w'$ and $s\in \tilde S$ with $w\approx _\sigma w'$ and $\ell (sw'\sigma (s))=\ell (w')-2$ . By our assumption, there exist reduction trees of $sw'$ and $sw'\sigma (s)$ . Then a reduction tree of w consists of the given reduction trees of $sw'$ and $sw'\sigma (s)$ and the edges $w\rightharpoonup sw'$ and $w\rightharpoonup sw'\sigma (s)$ .

Let $\mathcal T$ be a reduction tree of w. Recall that an end point of $\mathcal T$ is a vertex in $\mathcal T$ of minimal length. A reduction path in $\mathcal T$ is a path $\underline p\colon w \rightharpoonup w_1\rightharpoonup \cdots \rightharpoonup w_n$ , where $w_n$ is an end point of $\mathcal T$ . Set $\operatorname {\mathrm {end}}(\underline p)=w_n$ . We say that $x\rightharpoonup y$ is of type I (resp. II) if $\ell (x)-\ell (y)=1$ (resp. $\ell (x)-\ell (y)=2$ ). For any reduction path $\underline p$ , we denote by $\ell _{I}(\underline p)$ (resp. $\ell _{II}(\underline p)$ ) the number of type I (resp. II) edges in $\underline p$ . We write $X_{\underline p}$ for a locally closed subscheme of $X_w(b)$ which is $J_b(F)$ -equivariant universally homeomorphic to an iterated fibration of type $(\ell _{I}(\underline p),\ell _{II}(\underline p))$ over $X_{\operatorname {\mathrm {end}}(\underline p)}(b)$ .

Let $B(\tilde W,\sigma )$ be the set of $\sigma $ -conjugacy classes in $\tilde W$ . Let $\Psi \colon B(\tilde W,\sigma )\rightarrow B(G)$ be the map sending $[w]\in B(\tilde W,\sigma )$ to $[\dot w]\in B(G)$ , where $\dot w\in G(L)$ is a lift of w. It is known that this map is well-defined and surjective, see [Reference He21, Theorem 3.7]. By [Reference He, Nie and Yu23, Proposition 3.9], we have the following description of $X_w(b)$ .

Proposition 2.5 Let $w\in \tilde W$ and $\mathcal T$ be a reduction tree of w. For any $b\in G(L)$ , there exists a decomposition

$$ \begin{align*}X_w(b)=\bigsqcup_{\substack{\underline p\ \text{is a reduction path in}\ \mathcal T;\\ \Psi(\operatorname{\mathrm{end}}(\underline p))=[b]}}X_{\underline p}.\end{align*} $$

In the case that $G=\operatorname {\mathrm {GL}}_n$ and $b=\tau ^m$ with m coprime to n, we can count the number of top irreducible components and rational points of $X_w(b)^0=\{gI\in X_w(b)\mid \kappa (g)=v_L(\det (g))=0\}$ using the reduction tree for w. By [Reference He and Nie22, Proposition 3.5], the $\sigma $ -conjugacy class of $\tau ^m$ in $\tilde W$ is the unique element in $B(\tilde W,\sigma )$ which maps to $[\tau ^m]\in B(G)$ under $\Psi $ . Note also that $\tau ^m$ is the unique minimal length element in its $\sigma $ -conjugacy class. We define a polynomial as

where $\underline p$ runs over all the reduction paths in $\mathcal T$ with $\operatorname {\mathrm {end}}(\underline p)=\tau ^m$ .

Proposition 2.6 Assume that $G=\operatorname {\mathrm {GL}}_n$ and $b=\tau ^m$ with m coprime to n. Let $w\in \tilde W$ and let $\mathcal T$ be a reduction tree of w. Then the number of top irreducible components of $X_w(b)^0$ is equal to the leading coefficient of $F_{w,b}$ (as a polynomial in $\mathbf q-1$ ). Moreover, we have

$$ \begin{align*}|X_w(b)^{0,\sigma}|=F_{w,b}|_{\mathbf q=q}.\end{align*} $$

2.4 The $\mathbb J$ -stratification

For any $g,h\in G(L)$ , let $\mathrm {inv}(g,h)$ denote the relative position, i.e., the unique dominant cocharacter such that $g^{-1}h\in K\varpi ^{\mathrm {inv}(g,h)} K$ . By definition, two elements $gK,hK\in G(L)/K$ lie in the same $\mathbb J$ -stratum if and only if for all $j\in J_b(F)$ , $\mathrm {inv}(j,g)=\mathrm {inv}(j,h)$ . Clearly, this does not depend on the choice of $g,h$ . By [Reference Chen and Viehmann2, Proposition 2.11], the $\mathbb J$ -strata are locally closed in ${\mathcal {G}} r$ . By intersecting each $\mathbb J$ -stratum with $X_\mu (b)$ (resp. $X_{\preceq \mu }(b)$ ), we obtain the $\mathbb J$ -stratification of $X_\mu (b)$ (resp. $X_{\preceq \mu }(b)$ ).

As explained in [Reference Chen and Viehmann2, Remark 2.1], the $\mathbb J$ -stratification heavily depends on the choice of b in its $\sigma $ -conjugacy class. So we need to fix a specific representative to compare the $\mathbb J$ -stratification on $X_\mu (b)$ (or $X_{\preceq \mu }(b)$ ) to another stratification. It is pointed out in loc. cit that if $[b]$ is a basic class in $B(G, \mu )$ , then a reasonable choice of b is the unique length $0$ element $\tau _{\mu }$ . Also, for any $w\in \tilde W$ , the $J_{\dot w}(F)$ -stratification is independent of the choice of a lift in $G(L)$ . See [Reference Görtz9, Lemma 2.5].

In the case where $G=\operatorname {\mathrm {GL}}_n$ and $b=\tau ^m$ with m coprime to n, there is a group-theoretic way to describe the $\mathbb J$ -stratification, which we will call the semi-module stratification. Indeed, by [Reference Chen and Viehmann2, Remark 3.1 and Proposition 3.4], the $\mathbb J$ -stratification on ${\mathcal {G}} r$ coincides with the stratification

$$ \begin{align*}G(L)/K=\bigsqcup_{\lambda\in X_*(T)} I \varpi^\lambda K/K.\end{align*} $$

So in this case, each $\mathbb J$ -stratum of $X_\mu (b)$ (resp. $X_{\preceq \mu }(b)$ ) coincides with $X_\mu ^\lambda (b)$ (resp. $X_{\preceq \mu }^\lambda (b)$ ) for some $\lambda \in X_*(T)$ , where $X_\mu ^\lambda (b)=X_\mu (b)\cap I \varpi ^\lambda K/K$ (resp. $X_{\preceq \mu }^\lambda (b)=X_{\preceq \mu }(b)\cap I \varpi ^\lambda K/K$ ). Set $J_b(F)^0=J_b(F)\cap K=J_b(F)\cap I$ . Note that $\tau X_\mu ^\lambda (b)=X_\mu ^{\tau \lambda }(b)$ and $J_b(F)/J_b(F)^0=\{\tau ^k J_b(F)^0\mid k\in \mathbb Z\}$ . Thus,

$$ \begin{align*}J_b(F)X_\mu^\lambda(b)=\bigsqcup_{k\in \mathbb Z} X_\mu^{\tau^k\lambda}(b)\quad \text{and}\quad J_b(F)X_{\preceq\mu}^\lambda(b)=\bigsqcup_{k\in \mathbb Z} X_{\preceq\mu}^{\tau^k\lambda}(b).\end{align*} $$

See Section 3.1 for the precise definition of (extended) semi-modules. As we will explain in Section 3.2, the set $\{\lambda \in X_*(T)\mid X_\mu ^\lambda (b)\neq \emptyset \}$ can be regarded as semi-modules for $\mu $ . Let $w_{\max }$ be the longest element in $W_0$ . Then we have

$$ \begin{align*}\{\lambda\in X_*(T)\mid X_{-w_{\max}\mu}^\lambda(b^{-1})\neq \emptyset\}=\{-w_{\max}\lambda\in X_*(T)\mid X_\mu^\lambda(b)\neq \emptyset\}.\end{align*} $$

Indeed it is easy to check that the image of $X_\mu ^\lambda (b)$ under the automorphism of ${\mathcal {G}} r$ by $gK\mapsto w_{\max }^t g^{-1}K$ is $X_{-w_{\max }\mu }^{-w_{\max }\lambda }(b^{-1})$ . This gives the description of “dual” semi-modules for $\mu $ .

2.5 Length positive elements

We denote by $\delta ^+$ the indicator function of the set of positive roots, i.e.,

$$ \begin{align*}\delta^+\colon \Phi\rightarrow \{0,1\},\quad \alpha \mapsto \begin{cases} 1 & (\alpha\in \Phi_+) \\ 0 & (\alpha\in \Phi_-). \end{cases}\end{align*} $$

Note that any element $w\in \tilde W$ can be written in a unique way as $w=x\varpi ^\mu y$ with $\mu $ dominant, $x,y\in W_0$ such that $\varpi ^\mu y\in {^S\tilde W}$ . We have $p(w)=xy$ and $\ell (w)=\ell (x)+\langle \mu , 2\rho \rangle -\ell (y)$ . We define the set of length positive elements by

$$ \begin{align*}\operatorname{\mathrm{LP}}(w)=\{v\in W_0\mid \langle v\alpha,y^{-1}\mu\rangle+\delta^+(v\alpha)-\delta^+(xyv\alpha)\geq 0\ \text{for all} \alpha\in \Phi_+\}.\end{align*} $$

Then we always have $y^{-1}\in \operatorname {\mathrm {LP}}(w)$ . Indeed y satisfies the condition that $\langle \alpha , \mu \rangle \geq \delta ^+(-y^{-1}\alpha )\ \text {for all} \alpha \in \Phi _+.$ Since $\delta ^+(\alpha )+\delta ^+(-\alpha )=1$ , we have

$$ \begin{align*}\langle y^{-1}\alpha, y^{-1}\mu\rangle+\delta^+(y^{-1}\alpha)-\delta^+(x\alpha)=\langle \alpha,\mu\rangle-\delta^+(-y^{-1}\alpha)+\delta^+(-x\alpha)\geq 0.\end{align*} $$

Lemma 2.8 For any $w=x\varpi ^\mu y\in \tilde W$ as above, we define

Here $\delta ^-$ denotes the indicator function of the set of negative roots. Then we have

$$ \begin{align*}y\operatorname{\mathrm{LP}}(w)=\{r^{-1}\in W_0\mid r(\Phi_+\setminus \Phi_w)\subset \Phi_+\ \text{or equivalently,}\ r^{-1}\Phi_+\subset \Phi_+\cup-\Phi_w\}.\end{align*} $$

Proof Let $r\in W_0$ such that $r(\Phi _+\setminus \Phi _w)\subset \Phi _+$ . Let $\alpha \in \Phi _+$ . If $r^{-1}\alpha \in \Phi _+$ , then we can check that $y^{-1}r^{-1}\in \operatorname {\mathrm {LP}}(w)$ similarly as the case $r=1$ above. If $r^{-1}\alpha \in \Phi _-$ , then we must have $r^{-1}\alpha \in -\Phi _w$ . Since $\delta ^-(-\alpha )=\delta ^+(\alpha )$ , it follows that

$$ \begin{align*} &\langle y^{-1}r^{-1}\alpha,y^{-1}\mu\rangle+\delta^+(y^{-1}r^{-1}\alpha)-\delta^+(xr^{-1}\alpha)\\ =&-(\langle -r^{-1}\alpha, \mu\rangle-\delta^-(-y^{-1}r^{-1}\alpha)+\delta^-(-xr^{-1}\alpha))=0. \end{align*} $$

Thus, $y^{-1}r^{-1}\in \operatorname {\mathrm {LP}}(w)$ . This shows $\{r^{-1}\in W_0\mid r(\Phi _+\setminus \Phi _w)\subset \Phi _+\}\subseteq y\operatorname {\mathrm {LP}}(w)$ .

Let $v\in \operatorname {\mathrm {LP}}(w)$ and let $\alpha \in \Phi _+$ . If $yv\alpha \in \Phi _-$ , then

$$ \begin{align*} \langle-yv\alpha,\mu\rangle-\delta^-(-v\alpha)+\delta^-(-xyv\alpha) =-(\langle v\alpha, y^{-1}\mu\rangle+\delta^+(v\alpha)-\delta^+(xyv\alpha))\le 0. \end{align*} $$

On the other hand, by the characterization of y above, we have

$$ \begin{align*} \langle-yv\alpha,\mu\rangle-\delta^-(-v\alpha)+\delta^-(-xyv\alpha) =\langle-yv\alpha,\mu\rangle-\delta^+(v\alpha)+\delta^+(xyv\alpha)\geq 0. \end{align*} $$

Thus, $\langle -yv\alpha ,\mu \rangle -\delta ^-(-v\alpha )+\delta ^-(-xyv\alpha )=0$ and hence $yv\alpha \in -\Phi _w$ . This shows $y\operatorname {\mathrm {LP}}(w)\subseteq \{r^{-1}\in W_0\mid r(\Phi _+\setminus \Phi _w)\subset \Phi _+\}$ . The proof is finished.

The notion of length positive elements is defined by Schremmer [Reference Schremmer35]. The description of $\operatorname {\mathrm {LP}}(w)$ in Lemma 2.8 is due to Lim [Reference Lim30].

We say that the Dynkin diagram of G is $\sigma $ -connected if it cannot be written as a union of two proper $\sigma $ -stable subdiagrams that are not connected to each other. The following theorem is a refinement of the non-emptiness criterion in [Reference Görtz, He and Nie12], which is conjectured by Lim [Reference Lim30] and proved by Schremmer [Reference Schremmer36, Proposition 5].

In the case $G=\operatorname {\mathrm {GL}}_n$ , there exists a length-preserving automorphism $\varsigma $ of $\tilde W$ defined as

$$ \begin{align*}w_0\varpi^\lambda\mapsto w_{\max}w_0w_{\max}^{-1}\varpi^{-w_{\max}\lambda},\quad w_0\in W_0,\ \lambda\in X_*(T).\end{align*} $$

Note that $\varsigma (\tau ^m)=\tau ^{-m}$ , $\varsigma (s_0)=s_0$ and $\varsigma (s_i)=s_{n-i}$ for $1\le i\le n-1$ . Let $w=x\varpi ^\mu y$ be as above. For any $\alpha \in \Phi _+$ and $v\in \operatorname {\mathrm {LP}}(w)$ , we have

$$ \begin{align*} &\langle \varsigma(v)(-w_{\max}\alpha), \varsigma(y^{-1})(-w_{\max}\mu)\rangle+\delta^+(\varsigma(v)(-w_{\max}\alpha))-\delta^+(\varsigma(xy)\varsigma(v)(-w_{\max}\alpha))\\ &=\langle v\alpha,y^{-1}\mu\rangle+\delta^+(v\alpha)-\delta^+(xyv\alpha)\geq 0. \end{align*} $$

Thus, $\operatorname {\mathrm {LP}}(\varsigma (w))=\varsigma (\operatorname {\mathrm {LP}}(w))=w_{\max }\operatorname {\mathrm {LP}}(w)w_{\max }^{-1}$ . In particular, there exists $v\in \operatorname {\mathrm {LP}}(w)$ such that $v^{-1}p(w)v$ is a Coxeter element if and only if the same is true for $\varsigma (w)$ and $\operatorname {\mathrm {LP}}(\varsigma (w))$ .

3.1 Extended semi-modules

Here we recall the definition of extended semi-modules in a combinatorial way from [Reference Viehmann42]. Note that although we choose the subgroup of upper triangular matrices B as a Borel subgroup in this paper, the fixed Borel subgroup in [Reference Viehmann42] is the subgroup of lower triangular matrices.

For a semi-module A, there exists a unique $\mu '\in \mathbb N^n$ satisfying the following condition: Let $a_0=\min \bar A$ and let inductively $a_i=a_{i-1}+m-\mu '(i)n$ for $i=1,\ldots , n$ . Then $a_0=a_n$ and $\{a_0,a_1,\ldots ,a_{n-1}\}=\bar A$ . We call $\mu '$ the type of A.

For any $\lambda \in X_*(T)$ , we denote by $\lambda _{\operatorname {\mathrm {dom}}}$ the dominant conjugate of $\lambda $ . Let $\mu '$ be the type of a semi-module for $m,n$ . Let $\varphi $ be a function such that $(1)$ and the equation in $(3)$ hold. Then it is easy to check that $(A,\varphi )$ is a cyclic semi-module for $\mu ^{\prime }_{\operatorname {\mathrm {dom}}}$ . In general, the following lemma holds.

Let $e_0,\ldots , e_{n-1}$ be the standard basis of $L^n$ . Then the lattice $\mathcal O^n$ is generated by $e_0,\ldots , e_{n-1}$ . For $i\in \mathbb Z$ , we define $e_i$ by $e_{i+n}=\varpi e_i$ . Note that we have $\tau e_i=e_{i+1}$ for any i. In the sequel, we identify ${\mathcal {G}} r$ and $\{M\subset L^n\ \text {lattice}\}$ by $gK\mapsto g\mathcal O^n$ .

Let $X_\mu (b)^0$ be a $\overline {\mathbb F}_q$ -subscheme of $X_\mu (b)$ defined as $X_\mu (b)^0=\{gK\in X_\mu (b)\mid \kappa (g)=0\}$ . We associate to $M\in X_\mu (b)^0$ an extended semi-module for $\mu $ . Let $v\in L^n$ . Then we can write $v=\sum _{i\in \mathbb Z}[\alpha _i]e_i$ with $\alpha _i\in \overline {\mathbb F}_q$ and $\alpha _i=0$ for sufficiently small i. Here, $[\alpha _i]$ denotes the Teichmüller lift of $\alpha _i$ if $\operatorname {\mathrm {ch}} F=0$ and $[\alpha _i]=\alpha _i$ if $\operatorname {\mathrm {ch}} F>0$ . Let

$$ \begin{align*}\mathcal I\colon L^n\setminus \{0\}\rightarrow \mathbb Z,\quad v\mapsto \min\{i\mid \alpha_i\neq 0\}.\end{align*} $$

For $M\in {\mathcal {G}} r$ , we define the set

$$ \begin{align*}A(M)=\{\mathcal I(v)\mid v\in M\setminus \{0\}\}.\end{align*} $$

It is easy to check that if $M\in X_\mu (b)^0$ , then $A(M)$ is a normalized semi-module for $m,n$ . We also define $\varphi (M)\colon \mathbb Z\rightarrow \mathbb N\cup \{-\infty \}$ by

$$ \begin{align*}a\mapsto \begin{cases} \max\{k\mid \exists v\in M\setminus \{0\} \text{\ with\ }\mathcal I(v)=a, \varpi^{-k}b\sigma(v)\in M\} & (a\in A(M))\\ -\infty & (a\notin A(M)). \end{cases}\end{align*} $$

For an extended semi-module $(A,\varphi )$ for $\mu $ , let

$$ \begin{align*}S_{A,\varphi}=\{M\mid A(M)=A,\varphi(M)=\varphi\}\subset {\mathcal{G}} r.\end{align*} $$

Let $\mathbb A_\mu $ be the set of extended semi-modules for $\mu $ . Set $\mathbb A_\mu ^{\mathrm {top}}= \{(A,\varphi )\in \mathbb A_\mu \mid \dim S_{A,\varphi }=\dim X_\mu (b)\}$ . By Proposition 3.7 below, $J_b(F)\backslash \operatorname {\mathrm {Irr}} X_\mu (b)$ is parametrized by $\mathbb A_\mu ^{\mathrm {top}}$ . In the sequel, we also use the symbol $\mathbb A$ to denote the affine space as usual. We hope our notation will not cause confusions.

For an extended semi-module $(A,\varphi )$ for $\mu $ , let

$$ \begin{align*}\mathcal V(A,\varphi)=\{(a,c)\in A\times A\mid c>a, \varphi(a)>\varphi(c)>\varphi(a-n)\}.\end{align*} $$

Here we briefly describe $U_{A,\varphi }$ and the map $U_{A,\varphi }\rightarrow S_{A,\varphi }$ . For any $x\in \overline {\mathbb F}_q^{|\mathcal V(A,\varphi )|}=\mathbb A^{|\mathcal V(A,\varphi )|}$ , we denote the coordinate of x by $x_{a,c}$ . We associate to every x a set of elements $\{v(a)\in L^n\mid a\in A\}$ which satisfies the following equations.

If $a=\max \bar A$ , then

$$ \begin{align*}v(a)=e_a+\sum_{(a,c)\in \mathcal V(A,\varphi)}[x_{a,c}] v(c).\end{align*} $$

For any other element $a\in \bar A$ , we want

$$ \begin{align*}v(a)=v'+\sum_{(a,c)\in \mathcal V(A,\varphi)}[x_{a,c}] v(c),\end{align*} $$

where $v'=\varpi ^{-\varphi (a')}b\sigma (v(a'))$ for $a'$ being minimal satisfying $a'+m-\varphi (a')n=a$ . For $a\in n+A$ , we want

$$ \begin{align*}v(a)=\varpi v(a-n)+\sum_{(a,c)\in \mathcal V(A,\varphi)}[x_{a,c}] v(c).\end{align*} $$

Here $[x_{a,c}]$ denotes the Teichmüller lift of $x_{a,c}$ if $\operatorname {\mathrm {ch}} F=0$ and $[x_{a,c}]=x_{a,c}$ if $\operatorname {\mathrm {ch}} F>0$ . The set $\{v(a)\in L^n\mid a\in A\}$ is uniquely determined by the equations above. Hence, the map $\mathbb A^{|\mathcal V(A,\varphi )|}\rightarrow {\mathcal {G}} r, x\mapsto \langle v(a)\rangle _{a\in A}$ is well-defined. By applying $\sigma $ on the above equations for x, we can easily check that this map is compatible with the action of $\sigma $ , i.e., maps to $\sigma \langle v(a)\rangle _{a\in A}$ . Let $U_{A,\varphi }$ be the preimage of $S_{A,\varphi }$ under this map. Then $S_{A,\varphi }$ and hence $U_{A,\varphi }$ are stable under $\sigma $ (because $\sigma (b)=b$ ). In particular, we have $|S_{A,\varphi }^\sigma |=|U_{A,\varphi }^\sigma |$ . So if $(A,\varphi )$ is cyclic, then $|S_{A,\varphi }^\sigma |=q^{|\mathcal V(A,\varphi )|}$ . Although not needed in this paper, it is also worth mentioning that if $(A,\varphi )$ is noncyclic, then $S_{A,\varphi }$ is never universally homeomorphic to an affine space.

3.2 The stratification by extended semi-modules

For any $\lambda \in X_*(T)$ , set $A^\lambda =\{(i-1)+\lambda (i)n+k n \mid 1\le i\le n, k\in \mathbb N\}.$ It is easy to check that for a lattice $M\in I\varpi ^\lambda K/K$ , we have $A(M)=A^\lambda $ . Thus, we have the following lemma, which relates the semi-module stratification to the stratification by extended semi-modules.

Lemma 3.9 Let $\lambda \in X_*(T)$ with $\lambda (1)+\cdots +\lambda (n)=0$ . Then $X_\mu ^{\lambda }(b)\neq \emptyset $ if and only if there exists an extended semi-module $(A^\lambda ,\varphi )$ for $\mu $ . If this is the case, we have

$$ \begin{align*}X_\mu^{\lambda}(b)=\bigsqcup_{\varphi}S_{A^\lambda,\varphi},\end{align*} $$

where $\varphi $ runs over all the functions $\mathbb Z\rightarrow \mathbb N\cup \{-\infty \}$ such that the pair of $A^\lambda $ and the function is an extended semi-module for $\mu $ .

For $\lambda \in X_*(T)$ with $X_\mu ^\lambda (b)\neq \emptyset $ , let $1\le i_0\le n$ such that $(i_0-1)+\lambda (i_0)n=\min \overline {A^\lambda }$ . Let $1\le m_0<n$ be the residue of m modulo n, and let $\lambda _{b,\operatorname {\mathrm {dom}}}$ be $((\lfloor \frac {m}{n} \rfloor +1)^{(m_0)},\lfloor \frac {m}{n} \rfloor ^{(n-m_0)})$ . Then

$$ \begin{align*} &(i_0-1)+\lambda(i_0)n+m-(\lambda(i_0)+\lambda_{b,\operatorname{\mathrm{dom}}}(c^m(i_0))-\lambda(c^m(i_0)))n\\ {}&= c^m(i_0)-1+\lambda(c^m(i_0))n\in \overline {A^\lambda}, \end{align*} $$

where $c=s_1\cdots s_{n-1}$ . Repeating the same argument, we can check that the type of $A^\lambda $ is a conjugate of $b\lambda -\lambda =c^m\lambda +\lambda _{b,\operatorname {\mathrm {dom}}}-\lambda $ . By Lemma 3.4, an extended semi-module $(A^\lambda ,\varphi )$ for $\mu $ is cyclic if and only if $b\lambda -\lambda \in W_0\mu $ .

For $\mu =(\mu (1),\ldots , \mu (n-1),0)\in X_*(T)_+$ , set $\mu ^*=(\mu (1), \mu (1)-\mu (n-1),\ldots ,\mu (1)-\mu (2),0)$ and $b^*=\tau ^{n\mu (1)-m}$ . If $(A^\lambda ,\varphi )$ is an extended semi-module for $\mu $ , then there exists $\varphi '\colon \mathbb Z\rightarrow \mathbb N\cup \{-\infty \}$ such that $(A^{-w_{\max }\lambda },\varphi ')$ is an extended semi-module for $\mu ^*$ (see Section 2.4). Clearly, $b\lambda -\lambda \in W_0\mu $ if and only if $b^*(-w_{\max }\lambda )+w_{\max }\lambda \in -W_0\mu ^*$ . Thus, we have the following lemma.

3.3 The minuscule case

In this subsection, we treat the minuscule case. Consider $G^d$ with a Frobenius automorphism $\sigma _{\bullet }$ given by

$$ \begin{align*}(g_1,g_2,\ldots, g_d)\mapsto (g_2,\ldots, g_d, \sigma(g_1)).\end{align*} $$

For $\mu _{\bullet }=(\mu _1, \ldots , \mu _d)\in X_*(T)^d_+$ and $b_{\bullet }=(1,\ldots , 1, b)\in G^d(L)$ with $b\in G(L)$ , we define $X_{\mu _{\bullet }}(b_{\bullet })\subset {\mathcal {G}} r^d=G^d(L)/K^d$ as

$$ \begin{align*}X_{\mu_{\bullet}}(b_{\bullet})=\{x_{\bullet} K^d\in {\mathcal{G}} r^d\mid x_{\bullet}^{-1}b_{\bullet}\sigma_{\bullet}(x_{\bullet})\in K^d\varpi^{\mu_{\bullet}}K^d\}.\end{align*} $$

Let us denote by $\operatorname {\mathrm {Irr}} X_{\mu _{\bullet }}(b_{\bullet })$ the set of irreducible components of $X_{\mu _{\bullet }}(b_{\bullet })$ . Through the identification $J_b(F)\cong J_{b_{\bullet }}(F)$ given by $g\mapsto (g,\ldots ,g)$ , this set is equipped with an action of $J_b(F)$ .

For minuscule $\mu _{\bullet }\in X_*(T)^d_+$ and $b_{\bullet }=(1,\ldots ,1,b)\in G^d(L)$ , we define

Here, $X_{\mu _{\bullet }}^{\lambda _{\bullet }}(b_{\bullet })$ denotes $X_{\mu _{\bullet }}(b_{\bullet })\cap I^d\varpi ^{\lambda _{\bullet }} K^d/K^d$ . For $\lambda _{\bullet }, \lambda ^{\prime }_{\bullet }\in \mathcal A_{\mu _{\bullet }}^{\mathrm {top}}$ , we write $\lambda _{\bullet }\sim \lambda ^{\prime }_{\bullet }$ if $\lambda _{\bullet }=\tau ^k \lambda ^{\prime }_{\bullet }=(\tau ^k \lambda ^{\prime }_1,\ldots , \tau ^k \lambda ^{\prime }_d)$ for some $k\in \mathbb Z$ . Let $\mathbb A_{\mu _{\bullet }}^{\mathrm {top}}$ denote the set of equivalence classes with respect to $\sim $ , and let $[\lambda _{\bullet }]\in \mathbb A_{\mu _{\bullet }}^{\mathrm {top}}$ denote the equivalence class represented by $\lambda _{\bullet }\in \mathcal A_{\mu _{\bullet }}^{\mathrm {top}}$ . Then $J_{b}(F)\backslash \operatorname {\mathrm {Irr}} X_{\mu _{\bullet }}(b_{\bullet })$ is parametrized by $\mathbb A_{\mu _{\bullet }}^{\mathrm {top}}$ as follows.

Proposition 3.12 Assume that $\mu _{\bullet }\in X_*(T)_+^d$ is minuscule. Then the map $\lambda _{\bullet }\mapsto \overline {X_{\mu _{\bullet }}^{\lambda _{\bullet }}(b_{\bullet })}$ induces a bijection

$$ \begin{align*}\mathbb A_{\mu_{\bullet}}^{\mathrm{top}}\cong J_{b}(F)\backslash \operatorname{\mathrm{Irr}} X_{\mu_{\bullet}}(b_{\bullet}).\end{align*} $$

We also define

for $1\le j\le \dim X_{\mu _{\bullet }}(b_{\bullet })$ . We can similarly consider the equivalence relation $\sim $ as above. If $d=1$ , then can be identified with (extended) semi-modules for $\mu $ whose corresponding stratum has dimension j, see Lemma 3.4 and Lemma 3.9.

4.1 Crystals and young tableaux

In this subsection, we first recall the definition of $\widehat G$ -crystals from [Reference Xiao and Zhu45, Definition 3.3.1].

Definition 4.1 A (normal) $\widehat G$ -crystal is a finite set $\mathbb B$ , equipped with a weight map $\operatorname {\mathrm {wt}}\colon \mathbb B\rightarrow X_*(T)$ , and operators $\tilde e_\alpha , \tilde f_\alpha \colon \mathbb B\rightarrow \mathbb B\cup \{0\}$ for each $\alpha \in \Delta $ , such that

1. (i) for every $\mathbf {b}\in \mathbb B$ , either $\tilde e_\alpha \mathbf {b}=0$ or $\operatorname {\mathrm {wt}}(\tilde e_\alpha \mathbf {b})=\operatorname {\mathrm {wt}}(\mathbf {b})+\alpha ^\vee $ , and either $\tilde f_\alpha \mathbf {b}=0$ or $\operatorname {\mathrm {wt}}(\tilde f_\alpha \mathbf {b})=\operatorname {\mathrm {wt}}(\mathbf {b})-\alpha ^\vee $ ,

2. (ii) for all $\mathbf {b}, \mathbf {b}'\in \mathbb B$ one has $\mathbf {b}'=\tilde e_\alpha \mathbf {b}$ if and only if $\mathbf {b}=\tilde f_\alpha \mathbf {b}'$ , and

3. (iii) if $\varepsilon _\alpha , \phi _\alpha \colon \mathbb B\rightarrow \mathbb Z,\ \alpha \in \Delta $ are the maps defined by

   $$ \begin{align*} \varepsilon_\alpha(\mathbf{b})=\max\{k\mid \tilde e_\alpha^k\mathbf{b}\neq 0\}\ \ \text{and}\ \ \phi_\alpha(\mathbf{b})=\max\{k\mid \tilde f_\alpha^k\mathbf{b}\neq 0\}, \end{align*} $$

   then $\phi _\alpha (\mathbf {b})-\varepsilon _\alpha (\mathbf {b})=\langle \alpha , \operatorname {\mathrm {wt}}(\mathbf {b})\rangle $ .

For a $\widehat G$ -crystal $\mathbb B$ , let $\mathbb B^*=\{\mathbf {b}^*\mid \mathbf {b}\in \mathbb B\}$ be the dual $\widehat G$ -crystal. Setting $0^*=0$ , the maps are given by

$$ \begin{align*}\operatorname{\mathrm{wt}}(\mathbf{b}^*)=-\operatorname{\mathrm{wt}}(\mathbf{b}),\quad \tilde e_\alpha(\mathbf{b}^*)=(f_\alpha \mathbf{b})^*,\quad \text{and}\quad \tilde f_\alpha(\mathbf{b}^*)=(\tilde e_\alpha \mathbf{b})^*.\end{align*} $$

For $\lambda \in X_*(T)$ , we denote by $\mathbb B(\lambda )$ the set of elements with weight $\lambda $ for $\widehat G$ , called the weight space with weight $\lambda $ for $\widehat G$ . Let $\mathbb B_1$ and $\mathbb B_2$ be two $\widehat G$ -crystals. A morphism $\mathbb B_1\rightarrow \mathbb B_2$ is a map of underlying sets compatible with $\operatorname {\mathrm {wt}},\tilde e_\alpha $ and $\tilde f_\alpha $ .

In the sequel, we write $\tilde e_i$ and $\tilde f_i$ (resp. $\varepsilon _i$ and $\phi _i$ ) instead of $\tilde e_{\chi _{i,i+1}}$ and $\tilde f_{\chi _{i,i+1}}$ (resp. $\varepsilon _{\chi _{i,i+1}}$ and $\phi _{\chi _{i,i+1}}$ ) for simplicity.

We give a realization of $\mathbb B_\mu $ by Young tableaux. This allows us to treat it in a combinatorial way.

Definition 4.3 A Young diagram is a collection of boxes arranged in left-justified rows with a weakly decreasing number of boxes in each row. For a dominant cocharacter $\mu \in X_*(T)_+$ , we denote by $Y_\mu $ the Young diagram having $\mu (i)$ boxes in the ith row. A skew Young diagram is a diagram obtained by removing a smaller Young diagram from a larger one that contains it. For dominant cocharacters $\mu ,\nu \in X_*(T)_+$ with $\nu (i)\le \mu (i)$ , we denote by $Y_{\mu /\nu }$ the skew Young diagram obtained by removing $Y_\nu $ from $Y_\mu $ .

Definition 4.4 A tableau is a (skew) Young diagram filled with numbers, one for each box. A semi-standard tableau is a tableau obtained from a (skew) Young diagram by filling the boxes with the numbers $1,2,\ldots , n$ subject to the conditions

1. (i) the entries in each row are weakly increasing from left to right,

2. (ii) the entries in each column are strictly increasing from top to bottom.

Let $K_{\mu /\nu }(\lambda )$ be the number of all semi-standard tableaux $\mathbf {b}$ of shape $Y_{\mu /\nu }$ such that the number of appearing in $\mathbf {b}$ is $\lambda (i)$ for $1\le i\le n$ . This is sometimes called the Kostka number. In Section 4.3, we need the following well-known result.

We denote by $\mathcal B(Y)$ the set of all semi-standard tableaux of shape Y.

In the sequel, we identify $\mathbb B_\mu $ and $\mathcal B(Y)$ by Theorem 4.6. For a semi-standard tableau $\mathbf {b}\in \mathbb B_\mu $ , let $k_i$ denote the number of i’s appearing in $\mathbf {b}$ . Then the weight map $\operatorname {\mathrm {wt}}$ on $\mathbb B_\mu $ is given by $\operatorname {\mathrm {wt}}(\mathbf {b})=(k_1,\ldots , k_n)$ . The following result is an explicit description of the actions of $\tilde e_i$ and $\tilde f_i$ on $\mathbb B_\mu $ .

Next we recall the Weyl group action on crystals. Let $\mathbb B$ be a $\widehat G$ -crystal. For any $1\le i\le n-1$ and $\mathbf {b}\in \mathbb B$ , we set

$$ \begin{align*} s_i\mathbf{b}= \begin{cases} \tilde f_i^{\langle \chi_{i, i+1}, \operatorname{\mathrm{wt}}(\mathbf{b})\rangle}\mathbf{b} & \text{if } \langle \chi_{i, i+1}, \operatorname{\mathrm{wt}}(\mathbf{b})\rangle \geq 0\\ \tilde e_i^{-\langle \chi_{i, i+1}, \operatorname{\mathrm{wt}}(\mathbf{b})\rangle}\mathbf{b} & \text{if } \langle \chi_{i, i+1}, \operatorname{\mathrm{wt}}(\mathbf{b})\rangle \le 0. \end{cases} \end{align*} $$

Then we have the obvious relation

$$ \begin{align*}\operatorname{\mathrm{wt}}(s_i\mathbf{b})=s_i(\operatorname{\mathrm{wt}}(\mathbf{b})).\end{align*} $$

By [Reference Kashiwara26, Theorem 7.2.2], this extends to the action of the Weyl group $W_0$ on $\mathbb B$ , which is compatible with the action on $X_*(T)$ . For example, $w_{\max }\mathbf {b}_{\mu }\in \mathbb B_\mu $ has the lowest weight $w_{\max }\mu $ . It is well-known that the dual of $\mathbb B_\mu $ is isomorphic to $\mathbb B_{-w_{\max }\mu }$ (see for example [Reference Hong and Kang25, Lemma 3.5.2]).

Let $\mathbf {b}\in \mathbb B(\lambda )$ . If $\lambda '$ is a conjugate of $\lambda $ , i.e., there exists $w\in W_0$ such that $\lambda '=w\lambda $ , then we call $w\mathbf {b}$ the conjugate of $\mathbf {b}$ with weight $\lambda '$ . By Lemma 4.8, this does not depend on the choice of w.

Finally we consider the minuscule case. If $\mu \in X_*(T)_+$ is minuscule, then $\operatorname {\mathrm {wt}}\colon \mathbb B_\mu \rightarrow X_*(T)$ gives an identification between $\mathbb B_\mu $ and the set of cocharacters which are conjugate to $\mu $ . Suppose $\mu _{\bullet }=(\mu _1,\ldots ,\mu _d)\in X_*(T)_+^d$ is minuscule. We can also identify with the set of cocharacters in $X_*(T)^d$ which are conjugate to $\mu _{\bullet }$ .

For $1\le k<n$ , let $\omega _k$ be the cocharacter of the form $(1,\ldots ,1,0,\ldots ,0)$ in which $1$ is repeated k times. Assume that each $\mu _i$ is equal to $\omega _{k_i}$ for some $1\le k_i<n$ and $i\le j$ if and only if $k_i\le k_j$ . In the rest of paper, we call such $\mu _{\bullet }$ Far-Eastern. If $\mu _{\bullet }$ is Far-Eastern, then is dominant and its last entry is $0$ . Let $\operatorname {\mathrm {FE}}\colon \mathbb B_{|\mu _{\bullet }|}\rightarrow \mathbb B_{\mu _{\bullet }}^{\widehat G^d}$ be a map defined by decomposing $\mathbf {b}\in \mathbb B_\mu $ into its columns from right to left. We call $\operatorname {\mathrm {FE}}$ the Far-Eastern reading.

4.2 Construction of extended semi-modules

In this subsection, we recall from [Reference Shimada40, Section 4.2] the way of constructing extended semi-modules. See [Reference Shimada40, Section 4.3] for some examples of computation. Let $\mu _{\bullet }\in X_*(T)^d_+$ be a Far-Eastern cocharacter. Set $\mu =|\mu _{\bullet }|$ .

Let $\lambda _b$ denote the cocharacter whose i-th entry is $\lfloor \frac {im}{n}\rfloor -\lfloor \frac {(i-1)m}{n}\rfloor $ . Set $\lambda _b^{\operatorname {\mathrm {op}}}=w_{\max }\lambda _b$ . For any $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ , we denote by $\mathbf {b}^{\operatorname {\mathrm {op}}}$ the conjugate of $\mathbf {b}$ with weight $\lambda _b^{\operatorname {\mathrm {op}}}$ . Let $1\le m_0<n$ be the residue of m modulo n. Note that each entry of $\lambda _b$ is $\lfloor \frac {m}{n} \rfloor $ or $\lfloor \frac {m}{n} \rfloor +1$ and $\lambda _b(i)=\lambda _b(n+1-i)$ for any $2\le i\le n-1$ . Let $i_0=1<i_1<i_2<\cdots <i_{m_0}=n$ be the integers such that $\lambda _b(i_1)=\lambda _b(i_2)=\cdots =\lambda _b(i_{m_0})=\lfloor \frac {m}{n} \rfloor +1$ . Then

$$ \begin{align*}\lambda_b^{\operatorname{\mathrm{op}}}=w_{\max}'\lambda_b,\quad \text{where} w_{\max}'=(s_{i_{m_0-1}}\cdots s_{n-1})\cdots (s_{i_1}\cdots s_{i_2-1})(s_1\cdots s_{i_1-1}).\end{align*} $$

Here $\lambda _b(i)=\lfloor \frac {m}{n} \rfloor $ (resp. $\lambda _b(i+1)=\lfloor \frac {m}{n} \rfloor $ ) if and only if $s_{i-1}s_i\le w_{\max }'$ (resp. $s_is_{i+1}\le w_{\max }'$ ). By Lemma 4.8, it follows that $\mathbf {b}^{\operatorname {\mathrm {op}}}$ can be computed by the action of the Coxeter element $w_{\max }'$ . In this computation, each $s_i$ acts as the action of $\tilde e_i$ because $\lfloor \frac {m}{n} \rfloor -(\lfloor \frac {m}{n} \rfloor +1)=-1$ . Therefore, if we write

$$ \begin{align*}\operatorname{\mathrm{FE}}(\mathbf{b})=(\mathbf{b}_1,\ldots,\mathbf{b}_d)\end{align*} $$

then there exists $(w_1, \ldots , w_{d})\in W_0^d$ such that

$$ \begin{align*}\operatorname{\mathrm{FE}}(\mathbf{b}^{\operatorname{\mathrm{op}}})=(w_1\mathbf{b}_1, \ldots, w_d\mathbf{b}_d)\end{align*} $$

and each simple reflection appears exactly once in some $\operatorname {\mathrm {supp}}(w_j)$ .

Set $w(\mathbf {b})=w_1^{-1}\cdots w_d^{-1}$ and $\Upsilon (\mathbf {b})=\{\upsilon \in W_0\mid \upsilon ^{-1}c^m \upsilon =w(\mathbf {b})\}$ , where $c=s_1s_2\cdots s_{n-1}$ . Clearly $|\Upsilon (\mathbf {b})|=n$ .

For any $\mathbf {b}'\in \mathbb B_{\mu }$ , set

$$ \begin{align*}\xi(\mathbf{b}')=(\varepsilon_1(\mathbf{b}')+\cdots+\varepsilon_{n-1}(\mathbf{b}'),\varepsilon_2(\mathbf{b}')+\cdots+\varepsilon_{n-1}(\mathbf{b}'),\ldots,\varepsilon_{n-1}(\mathbf{b}'),0).\end{align*} $$

Let $\lambda _b^-$ be the anti-dominant conjugate of $\lambda _b$ , and let $\mathbf {b}^-$ be the conjugate of $\mathbf {b}$ with weight $\lambda _b^-$ . For any $\mathbf {b}\in \mathbb B_{\mu }(\lambda _b)$ and $\upsilon \in \Upsilon (\mathbf {b})$ , we define $\xi _{{\bullet }}(\mathbf {b}, \upsilon )\in X_*(T)^d$ by

$$ \begin{align*}\xi_{j}(\mathbf{b}, \upsilon)=\upsilon \xi(\upsilon^{-1}\mathbf{b}^-)+\sum_{1\le j'< j}\upsilon w_1^{-1}\cdots w_{j'-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_{j'})\quad (1\le j\le d).\end{align*} $$

Let $C\in \operatorname {\mathrm {Irr}} X_\mu (b)^0$ . By Proposition 3.7, $C=\overline {S_{A,\varphi }}$ for some $(A,\varphi )\in \mathbb A^{\mathrm {top}}_\mu $ . On the other hand, by Proposition 3.12 and [Reference Nie32, Proposition 3.13], there exists a unique $\lambda _{\bullet }\in \mathcal A_{\mu _{\bullet }}^{\mathrm {top}}$ with $\lambda _1(1)+\cdots +\lambda _1(n)=0$ such that $C=\operatorname {\mathrm {pr}}(\overline {X_{\mu _{\bullet }}^{\lambda _{\bullet }}(b_{\bullet })})$ . Here $\operatorname {\mathrm {pr}}\ \colon\ {\mathcal {G}} r^d\rightarrow {\mathcal {G}} r$ denotes the projection to the first factor. The following theorem is established in [Reference Shimada40, Theorem 4.4] by the author.

This correspondence between $\mathbb A^{\mathrm {top}}_\mu $ and $\mathbb B_\mu (\lambda _b)$ is compatible with the natural bijection in the Chen-Zhu conjecture constructed by Nie in [Reference Nie32].

Corollary 4.11 Let $(A,\varphi )\in \mathbb A^{\mathrm {top}}_\mu $ . Let $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ such that $\overline {S_{A,\varphi }}=\operatorname {\mathrm {pr}}(\overline {X_{\mu _{\bullet }}^{\xi _{\bullet }^0(\mathbf {b})}(b_{\bullet })})$ . Then $(A,\varphi )$ is cyclic if and only if

$$ \begin{align*}\sum_{1\le j\le d}w_1^{-1}\cdots w_{j-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_j)\in W_0\mu.\end{align*} $$

Proof By Lemma 3.9, we have $A=A^{\xi _1^0(\mathbf {b})}$ . Recall that $(A,\varphi )$ is cyclic if and only if $b\xi _1^0(\mathbf {b})-\xi _1^0(\mathbf {b})\in W_0\mu $ . Since $b\xi _1^0(\mathbf {b})-\xi _1^0(\mathbf {b})$ is a conjugate of $b\xi _1(\mathbf {b},\upsilon )-\xi _1(\mathbf {b},\upsilon )$ , this is also equivalent to $\upsilon ^{-1}b\xi _1(\mathbf {b},\upsilon )-\upsilon ^{-1}\xi _1(\mathbf {b},\upsilon )\in W_0\mu $ . By Theorem 4.10,

$$ \begin{align*}\upsilon^{-1}b\xi_1(\mathbf{b},\upsilon)-\upsilon^{-1}\xi_1(\mathbf{b},\upsilon)=\sum_{1\le j\le d}w_1^{-1}\cdots w_{j-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_j).\end{align*} $$

This finishes the proof.

We say that an element $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ is cyclic if

Now we give another interpretation of Lemma 3.11. Recall that $\mathbb B_\mu ^*$ is isomorphic to $\mathbb B_{\mu ^*}$ . We denote by $\mathbf {b}^*\in \mathbb B_{\mu ^*}$ the dual of $\mathbf {b}\in \mathbb B_\mu $ . Note that we have $(w\mathbf {b})^*=w\mathbf {b}^*$ for any $w\in W_0$ . So if $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ , then ${\mathbf {b}^{\operatorname {\mathrm {op}}}}^*=w_{\max }\mathbf {b}^*\in \mathbb B_{\mu ^*}(\lambda _{b^*})$ .

Proof Note that if $(\mu _1,\ldots ,\mu _d)$ is Far-Eastern, then $(\mu _d^*,\ldots , \mu _1^*)$ is Far-Eastern. So if we write

$$ \begin{align*}\operatorname{\mathrm{FE}}(\mathbf{b})=\mathbf{b}_1\otimes\cdots \otimes \mathbf{b}_d\quad \text{and}\quad \operatorname{\mathrm{FE}}(\mathbf{b}^{\operatorname{\mathrm{op}}})=w_1\mathbf{b}_1\otimes\cdots \otimes w_d\mathbf{b}_d,\end{align*} $$

in $\mathbb B_{\mu _1}\otimes \cdots \otimes \mathbb B_{\mu _d}$ , then we have

$$ \begin{align*}\operatorname{\mathrm{FE}}(\mathbf{b}^*)=\mathbf{b}_d^*\otimes\cdots \otimes \mathbf{b}_1^*\quad \text{and}\quad \operatorname{\mathrm{FE}}({\mathbf{b}^{\operatorname{\mathrm{op}}}}^*)=w_d\mathbf{b}_d^*\otimes\cdots \otimes w_1\mathbf{b}_1^*,\end{align*} $$

in $\mathbb B_{\mu _d^*}\otimes \cdots \otimes \mathbb B_{\mu _1^*}$ . Thus $w({\mathbf {b}^{\operatorname {\mathrm {op}}}}^*)=w_d\cdots w_1=w(\mathbf {b})^{-1}, \Upsilon ({\mathbf {b}^{\operatorname {\mathrm {op}}}}^*)=\Upsilon (\mathbf {b})$ and

$$ \begin{align*} \lambda({\mathbf{b}^{\operatorname{\mathrm{op}}}}^*)&=\operatorname{\mathrm{wt}}(w_d\mathbf{b}_d^*)+w_d\operatorname{\mathrm{wt}}(w_{d-1}\mathbf{b}_{d-1}^*)+\cdots+w_d\cdots w_2\operatorname{\mathrm{wt}}(w_1\mathbf{b}_1^*)\\ &=-w(\mathbf{b})^{-1}\lambda(\mathbf{b})+(d,\ldots, d), \end{align*} $$

as desired.

4.3 Noncyclic semi-standard tableaux

The goal of this section is to specify the dominant cocharacters $\mu $ such that every $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ is cyclic. Set $d=\mu (1)$ .

Proof First we consider the case $n=3$ . In this case, we have $2\le \mu (2)\le \lfloor \frac {m}{n} \rfloor $ because $\mu (3)=0$ . Let $\mathbf {b}$ be the unique element in $\mathbb B_\mu (\lambda _b)$ whose second row contains exactly one $\fbox{3}$ . Then $w(\mathbf {b})=s_2s_1$ and $s_1\in \operatorname {\mathrm {supp}}(w_{d-\lfloor \frac {m}{n} \rfloor })$ .

Since $2\le \mu (2)\le \lfloor \frac {m}{n} \rfloor $ , we have

$$ \begin{align*}w_1^{-1}\cdots w_{d-\mu(2)}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_{d-\mu(2)+1})=(0,1,1)\quad \text{and}\quad w_1^{-1}\cdots w_{d-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_d)=(1,0,1).\end{align*} $$

Thus $\lambda (\mathbf {b})\notin W_0\mu $ because $\mu (n)=0$ . This proves the case $n=3$ .

In the rest of the proof, we assume that $n\geq 4$ . Let $\lambda $ be a conjugate of $\lambda _b$ such that $(\lambda (1),\lambda (2),\lambda (3))=(\lfloor \frac {m}{n} \rfloor , \lfloor \frac {m}{n} \rfloor +\lfloor \frac {2m_0}{n}\rfloor , \lfloor \frac {m}{n} \rfloor +1)$ and $\lambda (4)\geq \cdots \geq \lambda (n)$ . Set

$$ \begin{align*}\mu_0=\left(3\lfloor \frac{m}{n} \rfloor+\lfloor\frac{2m_0}{n}\rfloor+1-\min\{\mu(2),\lfloor \frac{m}{n} \rfloor\},\min\{\mu(2),\lfloor \frac{m}{n} \rfloor\},0,\ldots,0\right)\in X_*(T)_+,\end{align*} $$

and $\lambda _0=(\lambda (1),\lambda (2),\lambda (3),0,\ldots , 0)\in X_*(T)$ . Note that we have $\mu (1)+\mu (2)\geq 3\lfloor \frac {m}{n} \rfloor +\lfloor \frac {2m_0}{n}\rfloor +1$ . Indeed if $\mu (1)+\mu (2)\le 3\lfloor \frac {m}{n} \rfloor +\lfloor \frac {2m_0}{n}\rfloor $ , then by $\mu (1)\geq 2\lfloor \frac {m}{n} \rfloor +\lfloor \frac {2m_0}{n}\rfloor +1$ , we have $\mu (2)\le \lfloor \frac {m}{n} \rfloor -1$ . This implies $\mu (3)+\cdots +\mu (n-1)\le (n-3)(\lfloor \frac {m}{n} \rfloor -1)$ , or equivalently $3\lfloor \frac {m}{n} \rfloor +n+m_0-3\le \mu (1)+\mu (2)$ , which is a contradiction. Thus $Y_\mu $ contains $Y_{\mu _0}$ .

Let $\mathbf {b}_0$ be the unique element in $\mathbb B_{\mu _0}(\lambda _0)$ whose second row contains exactly one $\fbox{3}$ . We will show that there exists $\mathbf {b}'\in \mathbb B_\mu (\lambda )$ that contains $\mathbf {b}_0$ . It is easy to check that $\mu (n-1)\le \lfloor \frac {m}{n} \rfloor $ and $\mu (n-2)\le \mu _0(1)$ . So each column in $Y_{\mu /\mu _0}$ has at most $n-3$ boxes. By filling each column with the numbers $1,\ldots , n-3$ so that the entries are starting with $1$ and increasing by one from top to bottom, we obtain a skew Young tableau of shape $Y_{\mu /\mu _0}$ . Let $k_i$ be the number of $\fbox{$i$}$ in this tableau. Clearly we have $k_1\geq \cdots \geq k_{n-3}$ .

By $(\lambda (4),\ldots ,\lambda (n))\preceq (k_1,\ldots , k_{n-3})$ and Proposition 4.5, there exists at least one skew Young tableau of shape $Y_{\mu /\mu _0}$ such that the number of $\fbox{$i$}$ is $\lambda (i+3)$ for each $1\le i\le n-3$ . By replacing $1,\ldots , n-3$ by $4,\ldots ,n$ respectively, we obtain a skew Young tableau of shape $Y_{\mu /\mu _0}$ such that the number of $\fbox{$i$}$ is $\lambda (i)$ for each $4\le i\le n$ . Let $\mathbf {b}'$ be the tableau obtained by joining $\mathbf {b}_0$ and this skew tableau. Clearly we have $\mathbf {b}'\in \mathbb B_\mu (\lambda )$ , which shows our claim.

Let $\mathbf {b}'\in \mathbb B_\mu (\lambda )$ containing $\mathbf {b}_0$ , and let $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ be the conjugate of $\mathbf {b}'$ . Then $s_2s_1\le w(\mathbf {b})$ and $s_1\in \operatorname {\mathrm {supp}}(w_{d-\lfloor \frac {m}{n} \rfloor })$ . Let $k(\mathbf {b}')$ be the number of $\fbox{4}$ in the second row of $\mathbf {b}'$ . If $k(\mathbf {b}')<\lfloor \frac {m}{n} \rfloor $ , then we have

$$ \begin{align*}(w_1^{-1}\cdots w_{d-\min\{\mu(2),\lfloor \frac{m}{n} \rfloor\}}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_{d-\min\{\mu(2),\lfloor \frac{m}{n} \rfloor\}+1}))(2)=1,\end{align*} $$

and

$$ \begin{align*}(w_1^{-1}\cdots w_{d-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_d))(2)=0.\end{align*} $$

Thus, $\lambda (\mathbf {b})\notin W_0\mu $ and hence $\mathbf {b}$ is noncyclic. If $k(\mathbf {b}')\neq 0$ , then $\lambda (\mathbf {b})(1)=\lfloor \frac {m}{n} \rfloor -1$ .

Assume that $\mu (3)<\lfloor \frac {m}{n} \rfloor -1$ . Then $\mathbf {b}$ is always noncyclic by the above argument.

Assume that $\mu (3)\geq 2$ . Let $\mathbf {b}^{\prime }_1=$ $\fbox{$j$}$ be the leftmost box in the third row of $\mathbf {b}'$ , and let $\mathbf {b}^{\prime }_2=$ $\fbox{$j$'}$ be the box right to $\mathbf {b}^{\prime }_1$ . Clearly $4\le j\le j'$ .

Then in $\mathbf {b}'$ , all $\fbox{$j-$1}$ are in the first or second row. Since the number of $\fbox{$j$}$ in the first or second row is less than $\operatorname {\mathrm {wt}}(\mathbf {b}')(j-1)$ , there exists at least one $\fbox{$j-$1}$ such that there is no box beneath it or the number in the box beneath it is greater than j. So the tableau obtained by replacing $\mathbf {b}^{\prime }_1$ by the rightmost one among such $\fbox{$j-$1}$ is semi-standard. Repeating the same argument, we may assume $j=4$ . Similarly, if $\lfloor \frac {m}{n} \rfloor \geq 3$ , we may also assume $j'=4$ . Indeed if $j'\geq 6$ and the leftmost column in $\mathbf {b}'$ contains $\fbox{$j$'$-$1}$ but does not contain $\fbox{$j$'}$ , we replace $\mathbf {b}^{\prime }_2$ by this $\fbox{$j$'$-$1}$ . In other cases, by $\lfloor \frac {m}{n} \rfloor \geq 3$ , there exists at least one $\fbox{$j$'$-$1}$ such that there is no box beneath it or the number in the box beneath it is greater than $j'$ , and we replace $\mathbf {b}^{\prime }_2$ by the rightmost $\fbox{$j$'$-$1}$ among such $\fbox{$j$'$-$1}$ . Then the obtained tableau is semi-standard. Thus, if $\lfloor \frac {m}{n} \rfloor \geq 3$ , there exists $\mathbf {b}'$ containing $\mathbf {b}_0$ such that $k(\mathbf {b}')<\lfloor \frac {m}{n} \rfloor $ , which is noncyclic by the above argument. If $\lfloor \frac {m}{n} \rfloor =2$ and $n=4$ , then $\mathbf {b}$ is noncyclic because $k(\mathbf {b}')<2$ . If $\lfloor \frac {m}{n} \rfloor =2$ and $n\geq 5$ , we may also assume $j'=4$ and hence $\mathbf {b}$ is noncyclic unless the third row of $\mathbf {b}'$ contains three $\fbox{5}$ . If $\lfloor \frac {m}{n} \rfloor =2, n\geq 5$ and the third row of $\mathbf {b}'$ contains three $\fbox{5}$ , then

$$ \begin{align*}(w_1^{-1}\cdots w_{d-2}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_{d-1}))(4)=1\quad \text{and}\quad (w_1^{-1}\cdots w_{d-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_d))(4)=0.\end{align*} $$

Thus, $\lambda (\mathbf {b})\notin W_0\mu $ and hence $\mathbf {b}$ is noncyclic.

Assume that $\lfloor \frac {m}{n} \rfloor =2$ and $\mu (3)=1$ . By the same argument as above, we may assume that the leftmost column of $\mathbf {b}'$ contains $\fbox{4}$ . So $\mathbf {b}$ is noncyclic when $\lambda (4)=2$ . If $\mu (1)> 5+\lfloor \frac {2m_0}{n}\rfloor $ , we may assume that the first row of $\mathbf {b}'$ also contains $\fbox{4}$ . This can be checked easily as above using $\mu (3)=1$ . Thus, if $\mu (1)> 5+\lfloor \frac {2m_0}{n}\rfloor $ , we obtain a noncyclic $\mathbf {b}$ .

If $\mu (1)= 5+\lfloor \frac {2m_0}{n}\rfloor $ , then we have $n=4$ or $5$ . More precisely, we have

$$ \begin{align*}\mu=(6,4,1,0), (5,5,1,1,0),(6,5,1,1,0),(6,6,1,0,0),\ \text{or}\ (6,6,1,1,0),\end{align*} $$

and $\mathbf {b}'$ contains one of the following smaller Young tableaux when $\lambda (4)=3$ .

We can easily check that $\mathbf {b}$ is noncyclic in every case.

Putting things together, we have proved the lemma.

Proof Let $\lambda $ be a conjugate of $\lambda _b$ such that $(\lambda (1),\lambda (2),\lambda (3))=(\lambda _b(1),\lambda _b(2),\lambda _b(3))$ and $\lambda (4)\geq \cdots \geq \lambda (n)$ . Assume that $(\lambda _b(1),\lambda _b(2),\lambda _b(3))=(1,2,2)$ and $\mu (2)\geq 3$ . Similarly as the proof of Lemma 4.14, we can easily show that there exists $\mathbf {b}'\in \mathbb B_\mu (\lambda )$ containing the following smaller Young tableau.

Let $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ be the conjugate of $\mathbf {b}'$ . If $\mu (3)<2$ , then $\mathbf {b}$ is noncyclic because $\lambda (\mathbf {b})(2)=2$ . If $\mu (3)\geq 2$ , then similarly as the proof of Lemma 4.14, we may assume that the second row of $\mathbf {b}'$ does not contain $\fbox{5}$ . In this case, the conjugate $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ of $\mathbf {b}'$ is noncyclic because

$$ \begin{align*}(w_1^{-1}\cdots w_{d-3}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_{d-2}))(3)=1\quad\text{and}\quad (w_1^{-1}\cdots w_{d-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_d))(3)=0.\end{align*} $$

Assume that $(\lambda _b(1),\lambda _b(2),\lambda _b(3))=(1,2,2)$ and $\mu (2)=2$ . Then there exists $\mathbf {b}'\in \mathbb B_\mu (\lambda )$ containing one of the following smaller Young tableaux.

It is easy to check that the conjugate $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ of $\mathbf {b}'$ is noncyclic.

Assume that $(\lambda _b(1),\lambda _b(2),\lambda _b(3))\neq (1,2,2)$ . Then there exists $\mathbf {b}'\in \mathbb B_\mu (\lambda )$ containing one of the following smaller Young tableaux.

Let $\mathbf {b}\in \mathbb B_\mu (\lambda _b)$ be the conjugate of $\mathbf {b}'$ . Since $\lambda (\mathbf {b})(1)=1$ , $\mathbf {b}$ is noncyclic if $\mu (3)=0$ . If $\mu (3)\geq 2$ , then similarly as the proof of Lemma 4.14, we may assume that the second row of $\mathbf {b}'$ does not contain $\fbox{5}$ . In this case, $\mathbf {b}$ is noncyclic because

$$ \begin{align*}(w_1^{-1}\cdots w_{d-2}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_{d-1}))(3)=1\quad\text{and}\quad (w_1^{-1}\cdots w_{d-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_d))(3)=0.\end{align*} $$

If $\mu (3)=1$ and $\mu (1)>3+\lfloor \frac {2m_0}{n}\rfloor $ , then we may also assume that the second row of $\mathbf {b}'$ does not contain $\fbox{5}$ and hence $\mathbf {b}$ is noncyclic. If $\mu (3)=1$ and $\mu (1)=3+\lfloor \frac {2m_0}{n}\rfloor $ , then we may assume that the leftmost column of $\mathbf {b}'$ contains $\fbox{5}$ . We can easily check that $\mathbf {b}$ is noncyclic by an easy calculation.

This finishes the proof.

Proof Let $1<i_1<i_2<\cdots <i_{m_0}=n$ be the integers such that $\lambda _b(i_1)=\lambda _b(i_2)=\cdots =\lambda _b(i_{m_0})=1$ . Let $\mathbf {b}$ be the Young tableau in $\mathbb B_\mu (\lambda _b)$ obtained by filling $Y_\mu $ with $i_1,\ldots , i_{m_0}$ from top to bottom, starting from the leftmost column.

If (1) holds, then $\mathbf {b}$ is noncyclic because

$$ \begin{align*}\operatorname{\mathrm{wt}}(\mathbf{b}_1)(i_m)=1\quad \text{and}\quad (w_1^{-1}\cdots w_{d-1}^{-1}\operatorname{\mathrm{wt}}(\mathbf{b}_d))(i_m)=0.\end{align*} $$

Let $k=\max \{i\mid \mu (i)\neq 0\}$ . If (2) holds, then the Young tableau $\mathbf c\in \mathbb B_\mu (\lambda _b)$ obtained by replacing ${i_k}$ by ${i_{k+1}}$ in $\mathbf {b}$ is noncyclic because $\lambda (\mathbf c)(i_k)=2$ .

This finishes the proof.