1.1.1. Roots

(a) Let V be a finite dimensional vector space over $\mathbb R$ , $V^*$ the dual space, and let $\Phi \subseteq V^*$ be a (reduced) root system (see, for example, [Reference BehrendBe] or [Reference BourbakiBo, Section VI]).

(b) We denote by $\mathcal {C}=\mathcal {C}_{\Phi }$ the set of all Weyl chambers $C\subseteq V$ of $\Phi $ . For each $C\in \mathcal {C}$ , we denote by $\Phi _{C}\subseteq \Phi $ the set of C-positive roots, by $\Delta _{C}\subseteq \Phi _C$ the set of C-simple roots, and by $\Psi _C\subseteq V^*$ the set of C-fundamental weights.

(c) We set $\widetilde {\Phi }:=\Phi \times \mathbb Z$ and call it the set of affine roots. Every $\widetilde {\alpha }=(\alpha ,n)$ is identified with an affine function $\widetilde {\alpha }:V\to \mathbb R$ , given by the rule $\widetilde {\alpha }(x)=\alpha (x)+n$ . In particular, we identify each root $\alpha \in \Phi $ with affine root $(\alpha ,0)\in \widetilde {\Phi }$ . For a subset $\Phi '\subseteq \Phi $ (resp. a Weyl chamber $C\in \mathcal {C}$ ), we denote by $\widetilde {\Phi '}$ (resp. $\widetilde {\Phi }_C$ ), the set of all $\widetilde {\alpha }=(\alpha ,n)\in \widetilde {\Phi }$ such that $\alpha \in \Phi '$ (resp. $\alpha \in \Phi _C$ ).

(d) Let $W=W_{\Phi }\subseteq \operatorname {Aut} (V)$ be the Weyl group of $\Phi $ , let $\Lambda \subseteq V$ be the subgroup generated by coroots $\{\check {\alpha }\}_{\alpha \in \Phi }$ , and let $\widetilde {W}:=W\ltimes \Lambda $ be the affine Weyl group of $\Phi $ . We will denote by $\pi $ the natural projection $\widetilde {W}\to \widetilde {W}/W=\Lambda $ .

(e) The lattice $\Lambda $ acts on V by translations. Then the group $\widetilde {W}$ acts on V by affine transformations; hence, it acts on $\widetilde {\Phi }$ by the rule $w(\widetilde {\alpha })(x)=\widetilde {\alpha }(w^{-1}(x))$ for all $x\in V$ . In particular, for each $\mu \in \Lambda $ and $(\alpha ,n)\in \widetilde {\Phi }$ , we have $\mu (\alpha ,n)=(\alpha ,n-\langle \alpha ,\mu \rangle )$ .

(f) For each $\widetilde {\alpha }\in \widetilde {\Phi }$ , the affine reflection $s_{\widetilde {\alpha }}$ satisfies $s_{\widetilde {\alpha }}(x)=x-\widetilde {\alpha }(x)\check {\alpha }$ for all $x\in V$ . In particular, for all $(\alpha ,n)\in \widetilde {\Phi }$ , we have equality $s_{\alpha ,n}=(-n\check {\alpha })s_{\alpha }\in \widetilde {W}$ .

(g) For each $\alpha \in \Phi $ , we denote by $\widetilde {W}_{\alpha }\subseteq \widetilde {W}$ the subgroup generated by reflections $s_{\widetilde {\alpha }}$ , with $\widetilde {\alpha }=(\alpha ,n), n\in \mathbb Z$ .

1.1.2. The fundamental Weyl chamber

(a) We fix a Weyl chamber $C_0\in \mathcal {C}$ and denote by $A_0$ the fundamental alcove such that $A_0\subseteq C_0$ and such that $0\in V$ lies in the closure of $A_0$ .

(b) The choice of $C_0$ defines the set of positive roots $\Phi _{>0}=\Phi _{C_0}\subseteq \Phi $ and the set of positive affine roots $\widetilde {\Phi }_{>0}\subseteq \widetilde {\Phi }$ . Explicitly, $\widetilde {\alpha }=(\alpha ,n)\in \widetilde {\Phi }$ is positive if and only if either $n>0$ , or $n=0$ and $\alpha>0$ .

(c) Then $C_0$ defines a set of simple reflection $S\subseteq W$ , and $A_0$ defines a set of simple affine reflections $\widetilde {S}\subseteq \widetilde {W}$ . In particular, a choice of $C_0$ defines length functions and Bruhat orders $\leq $ on both W and $\widetilde {W}$ .

(d) Using $A_0$ , we identify each $w\in \widetilde {W}$ with the corresponding alcove $w(A_0)\subseteq V$ . In particular, we will say that $w\in \widetilde {W}$ belongs to $C\in \mathcal {C}$ , or $w\in C$ , if $w(A_0)\subseteq C$ . Explicitly, this means that $\langle \alpha ,w(A_0)\rangle =\langle w^{-1}(\alpha ),A_0\rangle \geq 0$ for each $\alpha \in \Phi _{C}$ , or, what is the same, $w^{-1}(\Phi _{C})\subseteq \widetilde {\Phi }_{>0}$ .

1.1.3. Fundamental weights

(a) We set $\Psi :=\bigcup _{C\in \mathcal {C}}\Psi _C\subseteq V^*$ . For $\psi \in \Psi $ and $C\in \mathcal {C}$ , we write $C\owns \psi $ , if $\psi \in \Psi _C$ .

(b) Every $\psi \in \Psi $ gives rise to a fundamental coweight $\check {\psi }\in \Lambda _{\mathbb Q}:=\Lambda \otimes _{\mathbb Z}\mathbb Q\subseteq V$ . Namely, $\check {\psi }$ is characterized by condition that for every $C\in \mathcal {C}$ such that $\psi \in \Psi _C$ and every $\alpha \in \Delta _C$ , we have $\langle \alpha ,\check {\psi }\rangle =\langle \psi ,\check {\alpha }\rangle $ . In particular, for every $C\in \mathcal {C}$ , we have $\psi \in \Psi _C$ if and only if $\check {\psi }$ lies in the closure of C.

(c) For every $\psi \in \Psi $ , we denote by $\Phi (\psi )$ (resp. $\Phi ^{\psi }$ ) the set of $\alpha \in \Phi $ such that $\langle \alpha ,\check {\psi }\rangle \geq 0$ (resp. $\langle \alpha ,\check {\psi }\rangle =0$ ). Notice that $\Phi ^{\psi }$ is a root system, and there is a bijection $C\mapsto C^{\psi }$ between Weyl chambers $C\owns \psi $ of $\Phi $ and Weyl chambers of $\Phi ^{\psi }$ . This bijection satisfies the property that $(\Phi ^{\psi })_{C^{\psi }}=\Phi _{C}\cap \Phi ^{\psi }$ . We denote by $W^{\psi }\subseteq W$ and $\widetilde {W}^{\psi }\subseteq \widetilde {W}$ the Weyl group and the affine Weyl group of $\Phi ^{\psi }$ , respectively.

(d) For every $\psi \in \Psi $ , we fix a Weyl chamber $C^{\psi }_0$ of $\Phi ^{\psi }$ . As in Section 1.1.2(b), this choice defines the set of positive affine roots $\widetilde {\Phi }^{\psi }_{>0}\subseteq \widetilde {\Phi }^{\psi }$ , and we denote by $\widetilde {W}_{\psi }\subseteq \widetilde {W}$ the set of all $w\in \widetilde {W}$ such that $w^{-1}(\widetilde {\Phi }^{\psi }_{>0})\subseteq \widetilde {\Phi }_{>0}$ . Then for every $w\in \widetilde {W}$ , there exists a unique decomposition $w=w^{\psi }w_{\psi }$ , where $w^{\psi }\in \widetilde {W}^{\psi }$ and $w_{\psi }\in \widetilde {W}_{\psi }$ (compare, for example, [Reference Bezrukavnikov and VarshavskyBV, Lemma B.1.7(b)]). In other words, $\widetilde {W}_{\psi }\subseteq \widetilde {W}$ is a set of representatives of the set of left cosets $\widetilde {W}^{\psi }\backslash \widetilde {W}$ .

1.1.4. Properties of the Bruhat order

(a) Let $w',w"\in \widetilde {W}$ and $s\in \widetilde {S}$ be such that $w'\leq w"$ . Then we have either $w's\leq w"s$ (resp. $sw'\leq sw"$ ) or $w's\leq w"$ and $w'\leq w"s$ (resp. $sw'\leq w"$ and $w'\leq sw"$ ) or both (see, for example, [Reference Björner and BrentiBB, Proposition 2.2.7]).

(b) Let $w',w"\in \widetilde {W}$ and $s\in \widetilde {S}$ be such that $sw'<w'$ and $sw"<w"$ . Then, by part (a), we have $w'\leq w"$ if and only if $sw'\leq sw"$ .

(c) Let $w,w'$ and $w"$ be elements of $\widetilde {W}$ such that $l(ww')=l(w)+l(w')$ and $ww'\leq ww"$ . Then $w'\leq w"$ . Indeed, if $w=s\in \widetilde {S}$ , then the assertion follows from part (a). The general case follows by induction on $l(w)$ . By a similar argument, if $l(ww")=l(w)+l(w")$ and $w'\leq w"$ , then $ww'\leq ww"$ .

(d) For every $\mu \in \Lambda $ and $u\in W$ , we have $l(u\mu u^{-1})=l(\mu )$ . Indeed, it is enough to show the assertion in the case $u=s=s_{\alpha }$ for a simple root $\alpha $ . In this case, we have $s\mu s=\mu $ , if $\langle \alpha ,\mu \rangle =0$ ; $s\mu>\mu >\mu s$ if $\langle \alpha ,\mu \rangle>0$ ; and $s\mu <\mu <\mu s$ if $\langle \alpha ,\mu \rangle <0$ .

(e) Note that $w\in \widetilde {W}$ belongs to $C_0$ if and only if $l(sw)>l(w)$ for every $s\in S$ . In other words, $\widetilde {W}\cap C_0$ is the set of the shortest representatives of cosets $W\backslash \widetilde {W}$ . In particular, for every $w\in \widetilde {W}\cap C_0$ and $u\in W$ , we have $l(uw)=l(u)+l(w)$ , and for every $u\leq u'$ in W, we have $uw\leq u'w$ .

(f) The characterization of $C_0$ given in part (e) implies that for every $w\in \widetilde {W}\cap C_0$ and $s\in \widetilde {S}$ with $ws<w$ , we have $ws\in C_0$ .

(g) For every $u\in W$ and every $\mu \in \Lambda \cap C_0$ , we have $u\leq \mu $ . Indeed, it is enough to show that $u\leq _R\mu $ (see [Reference Björner and BrentiBB, Definition 3.1.1]). Hence, by [Reference Björner and BrentiBB, Proposition 3.1.3], it is enough to show that for every affine root $\widetilde {\alpha }>0$ such that $u(\widetilde {\alpha })<0$ , we have $\mu (\widetilde {\alpha })<0$ . If $\widetilde {\alpha }=(\alpha ,n)>0$ satisfies $u(\widetilde {\alpha })=(u(\alpha ),n)<0$ , then $n=0$ , and $\alpha>0$ . Hence, $\mu (\widetilde {\alpha })=(\alpha ,-\langle \alpha ,\mu \rangle )<0$ because $\mu \in C_0$ is regular; thus, $\langle \alpha ,\mu \rangle>0$ .

Proof. (a) By induction, it is enough to show that for every element $s\in S$ , we have $sw'<sw"$ . By Section 1.1.4(b), it is enough to show that $w'<sw'$ if and only if $w"<sw"$ . Let $u\in W$ be such that $C=u(C_0)$ . Then it follows from Section 1.1.4(e) that each condition $w'<sw'$ and $w"<sw"$ is equivalent to $u<su$ .

(b) Using part (a) and Section 1.1.4(e), we may assume that $C=C_0$ . If $l(w")-l(w')=1$ , there is nothing to prove, so we can assume that $l(w")-l(w')>1$ . By induction, it is enough to show the existence of $w\in C_0$ such that $w'<w<w"$ .

Choose $s\in \widetilde {S}$ such that $w"s<w"$ . Then $w"s\in C_0$ by Section 1.1.4(f). If $w'<w"s$ , then $w:=w"s$ does the job. If not, then by Section 1.1.4(a) we get $w's<w'$ and $w's<w"s$ . Then by Section 1.1.4(f), we have $w's\in C_0$ , so by induction on $l(w")$ , there exist $w\in C_0$ such that $w's<w<w"s$ .

If $ws<w$ , then it follows from Section 1.1.4(a) that $w'\leq w<w"s$ , contradicting our assumption. Hence, we may assume that $ws>w$ , in which case by Section 1.1.4(a) we have $w'<ws<w"$ ; thus, it is enough to show that $ws\in C_0$ .

Assume that $ws\notin C_0$ . Since $w\in C_0$ , this would imply that there exists a simple root $\alpha $ of $C_0$ such that $ws=s_{\alpha }w$ . Then we have $w'<s_{\alpha }w$ and $w'\in C_0$ and therefore by Section 1.1.4(c) that $w'\leq w<w"s$ , contradicting the assumption.

(c) Using Sections 1.1.4(d),(e), we can assume that $C=C_0$ . Now the proof goes by induction on $l(w)$ . Choose $s\in \widetilde {S}$ such that $ws<w$ . Then $ws\in C_0$ by Section 1.1.4(f); hence, by the induction hypothesis, we have

$$\begin{align*}l(\mu ws)=l(\mu)+l(ws)=l(\mu)+l(w)-1. \end{align*}$$

Thus, it is enough to show that $\mu ws<\mu w$ .

Let $\alpha $ be a simple affine root such that $s=s_{\alpha }$ . Then $\widetilde {\beta }:=w(\alpha )<0$ because $ws<w$ , and we want to show that $\mu (\widetilde {\beta })=\mu w(\alpha )<0$ . Write $\widetilde {\beta }$ in the form $(\beta ,n)$ , where $\beta \in \Phi $ . Then $\mu (\widetilde {\beta })=\widetilde {\beta }-\langle \beta ,\mu \rangle $ , so it remains to show that $\langle \beta ,\mu \rangle \geq 0$ .

Since $\widetilde {\beta }<0$ , we get $n\leq 0$ ; therefore, $w^{-1}(\beta )=\alpha -n>0$ . This implies that $\beta \in \Phi _{C_0}$ because $w\in C_0$ ; hence, $\langle \beta ,\mu \rangle \geq 0$ because $\mu \in C_0$ .

Notation 1.2.1. (a) Let $\widetilde {\alpha }\in \widetilde {\Phi }$ and $w\in \widetilde {W}$ . We say that $s_{\widetilde {\alpha }}w<_{\widetilde {\alpha }}w$ if $w^{-1}(\widetilde {\alpha })>0$ .

(b) Let $\Phi '\subseteq \Phi $ be a subset, and $w',w"\in \widetilde {W}$ . We say that $w"<_{\Phi '} w'$ if there exist affine roots $\widetilde {\alpha }_1,\ldots ,\widetilde {\alpha }_n\in \widetilde {\Phi '}$ such that $s_{\widetilde {\alpha }_i}\ldots s_{\widetilde {\alpha }_1}w'<_{\widetilde {\alpha }_i}s_{\widetilde {\alpha }_{i-1}}\ldots s_{\widetilde {\alpha }_1}w'$ for all i, and $w"=s_{\widetilde {\alpha }_n}\ldots s_{\widetilde {\alpha }_1}w'$ . For $\alpha \in \Phi $ , we write $w"<_{\alpha } w'$ instead of $w"<_{\{\alpha \}} w'$ .

(c) Let $\Phi '\subseteq \Phi $ , and $x',x"\in V$ . We say that $x"<_{\Phi '} x'$ if the difference $x'-x"$ is a positive linear combination of elements $\check {\alpha }$ with $\alpha \in \Phi '$ . For $\alpha \in \Phi $ , we write $x"<_{\alpha } x'$ instead of $x"<_{\{\alpha \}} x'$ .

(d) For each $C\in \mathcal {C}$ , $\psi \in \Psi $ (and $\psi \in C$ ), we write $<_C$ (resp. $<_{\psi }$ , resp. $<_{C^{\psi }}$ ) instead of $<_{\Phi _C}$ (resp. $<_{\Phi (\psi )}$ , resp. $<_{\Phi ^{\psi }(C^{\psi })}$ ).

Proof. (a) Fix $x\in A_0$ . Then $w^{-1}(\widetilde {\alpha })>0$ if and only if $w^{-1}(\widetilde {\alpha })(x)=\widetilde {\alpha }(w(x))>0$ . Thus, $s_{\widetilde {\alpha }}w<_{\widetilde {\alpha }}w$ if and only if $s_{\widetilde {\alpha }}w(x)=w(x)-\widetilde {\alpha }(w(x))\check {\alpha }<_{\alpha }w(x)$ .

(b) Let $r\in \mathbb Z$ such that the affine root $\widetilde {\alpha }=(\alpha ,r)$ satisfies $0<\widetilde {\alpha }(\check {\alpha }w(x))<1$ . Using identity $\widetilde {\alpha }(s_{\widetilde {\alpha }}(\check {\alpha }w(x)))=- \widetilde {\alpha }(\check {\alpha }w(x))$ , we get $0<(\widetilde {\alpha }+1)(s_{\widetilde {\alpha }}(\check {\alpha }w(x)))<1$ . Thus, by the observation of part (a), we have $w=s_{\widetilde {\alpha }+1}s_{\widetilde {\alpha }}(\check {\alpha }w)<_{\widetilde {\alpha }+1}s_{\widetilde {\alpha }}(\check {\alpha }w)<_{\widetilde {\alpha }}\check {\alpha }w$ ; hence, $w<_{\alpha }\check {\alpha }w$ .

(c) The ‘only if’ assertion follows from definitions. To see the ‘if’ assertion, we choose a Weyl chamber $C\ni \psi $ , and let $\alpha _{\psi }\in \Delta _C$ be the simple root, corresponding to $\psi $ . Then the difference $x"-x'$ can be (uniquely) written in the form $\sum _{\alpha \in \Delta _C} c_{\alpha }\check {\alpha }$ with $c_{\alpha }\in \mathbb R$ , and the assumption that $\langle \psi ,x'\rangle \leq \langle \psi ,x"\rangle $ implies that $c_{\alpha _{\psi }}\geq 0$ . Now the assertion follows from the observation that for every $\alpha \in \Delta _C\smallsetminus \{\alpha _{\psi }\}$ , we have $\alpha \in \Phi (\psi )$ and $-\alpha \in \Phi (\psi )$ .

Proof. (a) If $w<_{\alpha }w'$ , then $w\in \widetilde {W}_{\alpha }w'$ (by definition), and $w(x)<_{\alpha }w'(x)$ for all $x\in A_0$ (by Lemma 1.2.2(a)). Conversely, assume that $w=uw'$ with $u\in \widetilde {W}_{\alpha }$ such that $w(x)<_{\alpha }w'(x)$ for all $x\in A_0$ . Then we have either $u=s_{\widetilde {\alpha }}$ or $u=\check {\alpha }^m$ for some $m\in \mathbb Z_{<0}$ . In the first case, we have $w<_{\alpha }w'$ by Lemma 1.2.2(a), while in the second one, we have $w<_{\alpha }w'$ by Lemma 1.2.2(b).

(b) By definition, it is enough to assume that $w=s_{\widetilde {\alpha }}w'<_{\widetilde {\alpha }}w'$ . In this case, the first assertion follows from Lemma 1.2.2(a). Next, since $0\in V$ lies in the closure of $A_0\subseteq V$ , the second one follows from the equality $\pi (w)=w(0)$ .

(c) Assume that $\mu <_{\Phi '}\mu '$ in the sense of Section 1.2.1(c). By our assumption of $\Phi '$ , we may assume that $\mu =\mu '-\check {\alpha }$ for some $\alpha \in \Phi '$ . In this case, it follows from Lemma 1.2.2(b) that $\mu <_{\Phi '}\mu '$ in the sense of Section 1.2.1(b). The converse assertion follows from part (b).

Proof. First, we claim that for every $w',w"\in \widetilde {W}\cap C$ and $\widetilde {\alpha }\in \widetilde {\Phi }$ such that $w'=s_{\widetilde {\alpha }}w"$ , we have $w'<_C w"$ if and only if $w'< w"$ .

Replacing $\widetilde {\alpha }$ by $-\widetilde {\alpha }$ , if necessary, we may assume that $\widetilde {\alpha }=\alpha +n$ with $\alpha \in \Phi _C$ . Then $w'<_C w"$ holds if and only if $w^{\prime \prime -1}(\widetilde {\alpha })>0$ . However, since $w',w"\in C$ , we get that $(w")^{-1}(\alpha )>0$ and $(w')^{-1}(\alpha )>0$ . Since $s_{\widetilde {\alpha }}(\widetilde {\alpha })=-\widetilde {\alpha }$ , we get $s_{\widetilde {\alpha }}(\alpha )=-\alpha -2n$ ; therefore,

$$\begin{align*}(w')^{-1}(\alpha ) =(w")^{-1}s_{\widetilde {\alpha }}(\alpha )=-(w")^{-1}(\alpha )-2n>0. \end{align*}$$

This together with $(w")^{-1}(\alpha )>0$ implies that $n<0$ ; thus, $\widetilde {\alpha }<0$ . Therefore, $w'< w"$ holds if and only if $w^{\prime \prime -1}(\widetilde {\alpha })>0$ .

Now we are ready to show our assertion. Assume that $w'\leq _C w"$ , and we are going to show that for each sufficiently regular $\mu \in \Lambda \cap C$ , we have $\mu w'\leq \mu w"$ . By induction, we can assume that $w'=s_{\widetilde {\alpha }}w"$ for some $\widetilde {\alpha }\in \widetilde {\Phi }$ . Choose $\mu \in \Lambda \cap C$ sufficiently regular so that $\mu w',\mu w"\in C$ . Then $\mu w'\leq _C \mu w"$ (by Remark 1.2.4(a)(i)), and $\mu w'=s_{\mu (\widetilde {\alpha })}\mu w"$ . Hence, by what is shown above, $\mu w'\leq \mu w"$ .

Conversely, assume that for every sufficiently regular element $\mu \in \Lambda \cap C$ , we have $\mu w'\leq \mu w"$ , and we want to show that $w'\leq _C w"$ . Replacing $w'$ and $w"$ by $\mu w'$ and $\mu w"$ , respectively, and using Remark 1.2.4(a)(i), we may assume that $w',w"\in C$ and $w'\leq w"$ . Then using Lemma 1.1.5(b), we may assume in addition that $w'=s_{\widetilde {\alpha }}w"$ . Then, by what is shown above, $w'\leq _C w"$ .

Proof. (a) By Proposition 1.2.5, there exists $\mu \in \Lambda \cap C$ such that $\mu w'\leq \mu w"$ . Since $\mu , w'\in C$ , the assertion follows from Lemma 1.1.5(c) and Section 1.1.4(c).

(b) Using Lemma 1.1.5(c) and Section 1.1.4(c), we conclude that $\mu w'\leq \mu w"$ for every $\mu \in \Lambda \cap C$ . Therefore, we get $w'\leq _C w"$ by Proposition 1.2.5.

(c) follows from parts (a) and (b).

Proof. (a) Since $(\Phi ^{\psi })_{C^{\psi }}\subseteq \Phi _C$ , the ‘if’ assertion is obvious. Conversely, assume that $w"w\leq _C w'w$ . Then there exist affine roots

$$\begin{align*}\widetilde{\beta}_1=(\beta_1,n_1),\ldots,\widetilde{\beta}_r=(\beta_r,n_r)\in\widetilde{\Phi}_C \end{align*}$$

such that $w"w=s_{\widetilde {\beta }_n}\ldots s_{\widetilde {\beta }_1}w'w$ , and $s_{\widetilde {\beta }_i}\ldots s_{\widetilde {\beta }_1}w'w<_{\widetilde {\beta }_i} s_{\widetilde {\beta }_{i-1}}\ldots s_{\widetilde {\beta }_1}w'w$ for all i. Then for every $x\in A_0$ , the difference $w'w(x)-w"w(x)$ is a positive linear combination of the $\check {\beta }_i$ ’s (by Lemma 1.2.2(a)).

Since $w"w\in \widetilde {W}^{\psi }w'w$ , we conclude that $w'w(x)-w"w(x)$ is a linear combination of coroots of $\Phi ^{\psi }$ . Therefore, each $\beta _i$ is a root of $\Phi ^{\psi }$ ; thus, $\beta _i\in (\Phi ^{\psi })_{C^{\psi }}$ . But this implies that $w"w\leq _{C^{\psi }} w'w$ .

(b) By part (a), we have to show that $w"w\leq _{C^{\psi }} w'w$ if and only if $w"\leq _{C^{\psi }} w'$ . Thus, we can assume that $w"=s_{\widetilde {\beta }}w'$ for some $\widetilde {\beta }\in \widetilde {\Phi }^{\psi }$ . In other words, we have to show that $w^{\prime -1}(\widetilde {\beta })\in \widetilde {\Phi }^{\psi }_{>0}$ if and only if $w^{-1}(w^{\prime -1}(\widetilde {\beta }))\in \widetilde {\Phi }_{>0}$ . But this follows from the assumption that $w\in \widetilde {W}_{\psi }$ .

Notation 1.3.3. (a) For $\overline {\mu },\overline {\mu }'\in V^{\mathcal {C}}$ (resp. $\overline {w},\overline {w}'\in \widetilde {W}^{\mathcal {C}}$ ), we will say that $\overline {\mu }\leq \overline {\mu }'$ (resp. $\overline {w}\leq \overline {w}'$ ) if $\mu _C\leq _C\mu ^{\prime }_C$ (resp. $w_C\leq _C w^{\prime }_C$ ) for all $C\in \mathcal {C}$ .

(b) For $\overline {\mu }\in V^{\mathcal {C}}$ , we define by $V^{\leq \overline {\mu }}$ the set of all $x\in V$ such that $x\leq _C\mu _C$ for all $C\in \mathcal {C}$ .

1.3.4. Quasi-admissible tuples in $V^{\mathcal {C}}$ . (a) The set of quasi-admissible tuples in $V^{\mathcal {C}}$ (resp. $\Lambda ^{\mathcal {C}}$ ) can be naturally identified with $\mathbb R^{\Psi }$ (resp. $\mathbb Z^{\Psi }$ ).

Indeed, for each quasi-admissible tuple $\overline {\mu }\in V^{\mathcal {C}}$ and every $\psi \in \Psi $ , the element $\overline {\mu }(\psi ):=\langle \psi ,\mu _C\rangle $ does not depend on $C\owns \psi $ . To see this, we observe that for every pair of Weyl chambers $C,C'\owns \psi $ , there exists $w\in W_{\Phi ^{\psi }}$ such that $C'=w(C)$ . Therefore, $\overline {\mu }$ defines a tuple $\{\overline {\mu }(\psi )\}_{\psi \in \Psi }\in \mathbb R^{\Psi }$ .

Conversely, every tuple $\{\overline {\mu }(\psi )\}\in \mathbb R^{\Psi }$ gives rise to a quasi-admissible tuple $\overline {\mu }\in V^{\mathcal {C}}$ defined by the rule $\mu _C:=\sum _{\alpha _i\in \Delta _C}\overline {\mu }(\psi _i)\check {\alpha }_i$ , where $\psi _i\in \Psi _C$ is the fundamental weight corresponding to $\alpha _i\in \Delta _C$ .

(b) The set of quasi-admissible tuples in $V^{\mathcal {C}}$ (resp. $\Lambda ^{\mathcal {C}}$ ) is a group with respect to the coordinatewise addition in V (resp. $\Lambda $ ). Moreover, the identification of part (a) identifies this group with $\mathbb R^{\Psi }$ (resp. $\mathbb Z^{\Psi }$ ). Also, the set of admissible tuples in $V^{\mathcal {C}}$ (resp. $\Lambda ^{\mathcal {C}}$ ) is a submonoid.

(c) The identification of part (a) preserves coordinatewise ordering. In other words, for every two quasi-admissible tuples $\overline {\mu },\overline {\mu }'\in V^{\mathcal {C}}$ , we have $\mu _C\leq _C\mu ^{\prime }_C$ for all $C\in \mathcal {C}$ if and only if $\overline {\mu }(\psi )\leq \overline {\mu }'(\psi )$ for all $\psi \in \Psi $ . In particular, for every quasi-admissible tuples $\overline {\mu }\in V^{\mathcal {C}}$ , the subset $V^{\leq \overline {\mu }}\subseteq V$ (see Section 1.3.3(b)) consists of all $x\in V$ such that $\langle \psi ,x\rangle \leq \overline {\mu }(\psi )$ for all $\psi \in \Psi $ .

(d) From now on, we will not distinguish between a quasi-admissible tuple $\{\mu _C\}_C$ in $V^{\mathcal {C}}$ (resp. $\Lambda ^{\mathcal {C}}$ ) and the corresponding tuple $\{\overline {\mu }(\psi )\}_{\psi }$ in $\mathbb R^{\Psi }$ (resp. $\mathbb Z^{\Psi }$ ). In particular, for every $\psi \in \Psi $ , we denote by $\overline {e}_{\psi }\in \Lambda ^{\mathcal {C}}$ the quasi-admissible tuple, corresponding to the standard vector $\overline {e}_{\psi }\in \mathbb Z^{\Psi }$ , given by the rule $\overline {e}_{\psi }(\psi ')=\delta _{\psi ,\psi '}$ .

Examples 1.3.5. (a) Every $\mu \in C_0\subseteq V$ gives rise to an admissible tuple $\overline {\mu }\in V^{\mathcal {C}}$ defined by the rule $\mu _{u(C_0)}:=u(\mu )$ for all $u\in W$ .

(b) Consider the tuple $\overline {w}_{\operatorname {st}}\in W^{\mathcal {C}}\subseteq \widetilde {W}^{\mathcal {C}}$ , defined by the rule $(w_{\operatorname {st}})_{u(C_0)}=u$ . Then $\overline {w}_{\operatorname {st}}$ is admissible. Indeed, by definition, we have to show that for every $u\in W$ and $\alpha \in \Phi _{u(C_0)}$ , we have $s_{\alpha }u<_{\alpha }u$ ; that is, $u^{-1}(\alpha )>0$ . Since $u^{-1}(\alpha )\in \Phi _{C_0}$ , we are done.

(c) Using Remark 1.2.4(a)(i) and Lemma 1.2.2(b), for every (quasi)-admissible tuples $\overline {\mu }\in \Lambda ^{\mathcal {C}}$ and $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ , the tuple is (quasi)-admissible as well. In particular, for every $\mu \in \Lambda $ and (quasi)-admissible tuple $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ , the tuple $\mu \overline {w}:=\{\mu w_C\}_C$ is (quasi)-admissible.

Notation 1.3.6. Arguing as in Section 1.3.4(a), for each quasi-admissible $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ and every $\psi \in \Psi $ , the class $[w_C]\in \widetilde {W}^{\psi }\backslash \widetilde {W}$ and hence also element $(w_C)_{\psi }\in \widetilde {W}_{\psi }$ (see Section 1.1.3(d)) does not depend on $C\owns \psi $ . We will denote this element by $\overline {w}_{\psi }$ .

Proof. We will only prove the assertion for $\overline {w}$ , while the other case is similar, but easier. Assume first that $\overline {w}$ is admissible, and we want to show that for every two Weyl chambers C and $C'$ , we have $w_{C}\leq _{C'} w_{C'}$ . Using Remark 1.2.4(a)(ii), we may assume that $C'=C_0$ . Let $u\in W$ be such that $C=u(C_0)$ , choose a reduced decomposition $u=s_1\ldots s_n$ of u, and for each $j=1,\ldots ,n$ , we set $u_j:=s_1\ldots s_j$ and $C_j:=u_j(C_0)$ . It is enough to show that $w_{C_{j+1}}\leq _{C_0} w_{C_j}$ for each $j=0,\ldots ,n-1$ .

Let $\alpha _{j+1}\in \Delta _{C_0}$ be such that $s_{j+1}=s_{\alpha _{j+1}}$ . By construction, we obtain that $u_{j+1}=u_j s_{j+1}>u_j$ ; hence, $u_j(\alpha _{j+1})\in \Phi _{C_0}$ . Also since $\alpha _{j+1}\in \Delta _{C_0}$ , we get that $u_j(\alpha _{j+1})\in \Delta _{C_j}$ . Since $C_{j+1}=s_{u_j(\alpha _{j+1})}(C_j)$ , the admissibility assumption implies that $w_{C_{j+1}}\leq _{u_j(\alpha _{j+1})}w_{C_j}$ ; thus, we have $w_{C_{j+1}}\leq _{C_0} w_{C_j}$ because $u_j(\alpha _{j+1})\in \Phi _{C_0}$ .

Conversely, assume that $w_{C}\leq _{C'} w_{C'}$ for all $C,C'\in \mathcal {C}$ . Choose $C\in \mathcal {C}$ , $\alpha \in \Delta _C$ , and set $C'=s_{\alpha }(C)$ . Since $w_{C'}\leq _C w_C$ , there exist a tuple of affine roots $\widetilde {\beta }_1=(\beta _i,n_i),\ldots ,\widetilde {\beta }_r=(\beta _r,n_r)\in \widetilde {\Phi }_{C}$ such that $w_{C'}=s_{\widetilde {\beta }_r}\ldots s_{\widetilde {\beta }_1} w_C$ , and $s_{\widetilde {\beta }_i}\ldots s_{\widetilde {\beta }_1}w_C<_{\widetilde {\beta }_i} s_{\widetilde {\beta }_{i-1}}\ldots s_{\widetilde {\beta }_1}w_C$ for all i. Therefore, for each $x\in A_0$ , the difference $w_C(x)-w_{C'}(x)$ is a positive linear combination of the $\check {\beta }_i$ ’s (by Lemma 1.2.2(a)), and hence a positive linear combination of C-simple coroots.

However, since $w_C\leq _{C'}w_{C'}$ , the difference $w_C(x)-w_{C'}(x)$ is also a negative linear combination of $C'$ -simple coroots. Combining these two statements, we conclude that $w_C(x)-w_{C'}(x)$ has to be a positive multiple of $\check {\alpha }$ . Hence, all the $\beta _i$ ’s have to be $\alpha $ ; thus, $w_{C'}\leq _{\alpha } w_C$ .

Proof. (a) For every $C\in \mathcal {C}$ , we have $\mu _C\leq _{C'}\mu _{C'}$ for every $C'\in \mathcal {C}$ (by Lemma 1.3.7); thus, $\mu _C\in V^{\leq \overline {\mu }}$ . Since subset $V^{\leq \overline {\mu }}\subseteq V$ is convex, this implies that the convex hull of $\{\mu _C\}_{C\in \mathcal {C}}$ is contained in $V^{\leq \overline {\mu }}$ .

To show the opposite inclusion, it suffices to show that for every affine function l on V such that $l(\mu _{C'})\leq 0$ for all $C'\in \mathcal {C}$ and every $x\in V^{\leq \overline {\mu }}$ , we have $l(x)\leq 0$ . Let $\lambda \in V^*$ be the vector part of l and choose $C\in \mathcal {C}$ such that $\alpha \in \overline {C}$ . Since $x\leq _C\mu _C$ , we get $l(\mu _C)-l(x)=\langle \lambda ,\mu _C\rangle -\langle \lambda ,x\rangle \geq 0$ ; therefore, $l(x)\leq l(\mu _C)\leq 0$ .

(b) Since $\langle \psi ,\mu _C\rangle \leq \overline {\mu }(\psi )$ for every $C\in \mathcal {C}$ , it suffices to show that for every $C\in \mathcal {C}$ such that $\langle \psi ,\mu _C\rangle =\overline {\mu }(\psi )$ , we have $\psi \in \Psi _C$ . To show the result, we essentially repeat the first part of the proof of Lemma 1.3.7.

Using Remark 1.2.4(a)(ii), we may assume that $\psi \in \Psi _{C_0}$ , and let $u\in W$ be such that $C=u(C_0)$ . It suffices to show that $u\in W^{\psi }$ . Let $u_j$ , $C_j$ and $\alpha _{j+1}$ be as in the proof of Lemma 1.3.7. Since $\overline {\mu }$ is strictly admissible, we get $\mu _{C_{j+1}}<_{u_j(\alpha _{j+1})}\mu _{C_j}$ for every $j=0,\ldots ,n-1$ . Moreover, since $\psi \in \Psi _{C_0}$ and $u_j(\alpha _{j+1})\in \Phi _{C_0}$ , we conclude that $\langle \psi ,u_j(\alpha _{j+1})\rangle \geq 0$ . So the assumption that $\langle \psi ,\mu _C\rangle =\overline {\mu }(\psi )=\langle \psi ,\mu _{C_0}\rangle $ implies that for every j, we have $\langle \psi ,u_j(\alpha _{j+1})\rangle =0$ ; hence, $u_j(\alpha _{j+1})\in \Phi ^{\psi }$ and thus $s_{u_j(\alpha _{j+1})}\in W^{\psi }$ . Therefore, , as claimed.

(c) is an immediate consequence of parts (a) and (b).

Notation 1.3.9. (a) Let $m\in \mathbb R$ and $C\in \mathcal {C}$ . We say that $\mu \in V$ is $(C,m)$ -regular if $\langle \alpha ,\mu \rangle \geq m$ for every $\alpha \in \Phi _{C}$ . We say that $w\in \widetilde {W}$ is $(C,m)$ -regular if $\pi (w)=w(0)\in \Lambda $ is $(C,m)$ -regular.

(b) Let $m\in \mathbb R$ . We say that a tuple $\overline {\mu }\in V^{\mathcal {C}}$ is m-regular if $\mu _C$ is $(C,m)$ -regular for every $C\in \mathcal {C}$ . We say that a tuple $\overline {\mu }\in V^{\mathcal {C}}$ is regular if it is m-regular for some $m>0$ . A tuple $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ is called m-regular (resp. regular) if $\pi (\overline {w})\in \Lambda ^{\mathcal {C}}\subseteq V^{\mathcal {C}}$ is m-regular (resp. regular).

(c) For every $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ and every $\psi \in \Psi $ , we define $\overline {w}^{\psi }\in (\widetilde {W}^{\psi })^{\mathcal {C}^{\psi }}$ by the rule $(w^{\psi })_{C^{\psi }}=(w_C)^{\psi }$ for each $C\owns \psi $ (see Section 1.1.3).

Proof. (a) We will only show the assertion for $\overline {w}$ . Fix $C\in \mathcal {C}$ , let $\alpha \in \Phi _C$ , and set $C'=s_{\alpha }(C)$ . We want to show that $w_{C'}\leq _{\alpha } w_C$ . Since $\overline {w}$ is quasi-admissible, we get $w_{C'}\in \widetilde {W}_{\alpha }w_{C}$ . Therefore, for every $x\in A_0$ , we have $w_{C'}(x)=w_C(x)-a\check {\alpha }$ for some $a\in \mathbb R$ . Since $\overline {w}$ is regular, we conclude $\langle \alpha ,w_C(x)\rangle>0$ and $\langle \alpha ,w_{C'}(x)\rangle <0$ . Thus, $a>0$ , and hence, $w_{C'}<_{\alpha } w_C$ (by Corollary 1.2.3(a)).

(b) Since $\overline {\mu }(\psi )=\langle \psi ,\mu _C\rangle $ for every Weyl chamber $C\owns \psi $ (by definition), we have $\langle \alpha ,\mu _C\rangle>0$ for every $\alpha \in \Delta _C$ (since $\overline {\mu }$ is regular), and $\psi =\sum _{\alpha \in \Delta _C} c_{\alpha }\alpha $ with $c_{\alpha }\geq 0$ for all $\alpha \in \Delta _C$ , the assertion follows.

(c) follows from Lemma 1.2.7(b) and Lemma 1.3.7.

(d) Let $C\owns \psi $ be a Weyl chamber. We have to show that if w is $(C,m+1)$ -regular, and $\psi \in C$ , then $w^{\psi }$ is $(C^{\psi },m)$ -regular.

Notice that if w is $(C,m+1)$ -regular, then for all $\alpha \in \Phi _C$ and $x\in A_0$ , we have $\langle w^{-1}(\alpha ),x\rangle =\langle \alpha ,w(x)\rangle>m$ , or equivalently, $w^{-1}(\alpha )-m\in \widetilde {\Phi }_{>0}$ . Conversely, if $\langle \alpha ,w(x)\rangle>m$ for all $\alpha \in \Phi _C$ , then $\langle \alpha ,w(0)\rangle \geq m$ for all $\alpha \in \Phi _C$ ; thus, w is $(C,m)$ -regular.

Thus, it suffices to show that for every $\alpha \in (\Phi ^{\psi })_{C^{\psi }}$ , we have $w^{-1}(\alpha )-m\in \widetilde {\Phi }_{>0}$ if and only if $(w^{\psi })^{-1}(\alpha )-m\in \widetilde {\Phi }^{\psi }_{>0}$ . Since $w=w^{\psi }w_{\psi }$ , the assertion follows from the fact that $w_{\psi }\in \widetilde {W}_{\psi }$ .

Proof. By Lemma 1.3.10(a), the tuple $\overline {\mu }$ is strictly admissible. Then, by Corollary 1.3.8(c), the intersection of $V^{\leq \overline {\mu }}$ with the set of $x\in V$ such that $\langle \psi ,x\rangle =\overline {\mu }(\psi )$ is equal to the convex hull of $\{\mu _C\}_{C\owns \psi }$ .

Therefore, it is enough to show that for every Weyl chamber $C\owns \psi $ , we have $\langle \alpha ,\mu _C\rangle>0$ . Since tuple $\overline {\mu }$ is regular, it is enough to show that $\langle \alpha ,y\rangle>0$ for some $y\in C\subseteq V^*$ . But this follows from our assumption $\langle \alpha ,\check {\psi }\rangle>0$ together with observation that $\check {\psi }\in \overline {C}$ (see Section 1.1.3(b)).

Proof. First, we claim that there exists an admissible tuple $\overline {\mu }\in \Lambda ^{\mathcal {C}}$ such that $\mu _C^{-1}u_C\leq _C \mu ^{-1}_{C'}u_{C'}$ for every $C,C'\in \mathcal {C}$ . Indeed, $\overline {u}$ is admissible; hence, for each $C\in \mathcal {C}$ , $\alpha \in \Delta _C$ and $x\in A_0$ , we have $u_C(x)-u_{s_{\alpha }(C)}(x)=m_{C,\alpha ,x}\check {\alpha }$ for some constant $m_{C,\alpha ,x}\geq 0$ (use Lemma 1.2.2(a)). Let $m'$ be the supremum of the $m_{C,\alpha ,x}$ ’s, choose $\mu \in C_0\cap \Lambda $ such that $\langle \alpha ,\mu \rangle \geq m'$ for all $\alpha \in \Delta _{C_0}$ , and let $\overline {\mu }\in \Lambda ^{\mathcal {C}}$ be the tuple, corresponding to $\mu $ as in Section 1.3.5(a).

We claim that $\mu _C^{-1}u_C\leq _C \mu ^{-1}_{C'}u_{C'}$ for every $C,C'\in \mathcal {C}$ . Indeed, arguing as in Lemma 1.3.7 word-by-word, it is enough to check that $\mu _C^{-1}u_C\leq _{\alpha } \mu ^{-1}_{C'}u_{C'}$ for all $C\in \mathcal {C},\alpha \in \Delta _C$ and $C'=s_{\alpha }(C)$ . Then by Corollary 1.2.3(a), it is enough to check that $\mu _C^{-1}u_C(x)\leq _{\alpha } \mu ^{-1}_{C'}u_{C'}(x)$ for each $x\in A_0$ . By construction, we have $\mu ^{-1}_{C'}u_{C'}(x)-\mu _C^{-1}u_C(x)= (\langle \alpha ,\mu _C\rangle -m_{C,\alpha ,x})\check {\alpha }$ , so the assertion follows from the fact that $m_{C,\alpha ,x}<m'\leq \langle \alpha ,\mu _C\rangle $ .

Denote m to be the maximum of the $\langle \alpha ,\mu _C\rangle +1$ ’s, taken over all $C\in \mathcal {C}$ and $\alpha \in \Delta _{C}$ . We claim that such an m satisfies the required property.

To see this, we choose any m-regular admissible tuple $\overline {w}$ , and we claim that tuple is admissible. By Section 1.3.5(c), it is quasi-admissible, so by Lemma 1.3.10(a), it is enough to show that it is regular. For every $C\in \mathcal {C}$ , $\alpha \in \Delta _C$ , we have $\langle \alpha ,\mu _C^{-1}\pi (w_C)\rangle = \langle \alpha ,\pi (w_C)\rangle -\langle \alpha ,\mu _C\rangle>0$ because $\langle \alpha ,\pi (w_C)\rangle \geq m$ by m-regularity of $\overline {w}$ , and $\langle \alpha ,\mu _C\rangle \leq m-1$ , by construction.

Let $\mu \in \Lambda $ be such that $\mu u_C\leq _{C^{\psi }} w_C$ for each $C\owns \psi $ , and let $C'\in \mathcal {C}$ be arbitrary. We want to show that $\mu u_{C'}\leq _{C'} w_{C'}$ . Using Remark 1.2.4(a)(ii), it is enough to do it in the case $C'=C_0$ , so using Remark 1.2.4(a)(i), we have to show that $\mu _{C_0}^{-1}\mu u_{C_0}\leq _{C_0}\mu _{C_0}^{-1}w_{C_0}$ .

Choose $u\in W$ of minimal length such that $\psi _0:=u^{-1}(\psi )$ belongs to $\Psi _{C_0}$ , and set $C:=u(C_0)$ . Since is admissible, we conclude from Lemma 1.3.7 that $\mu _{C}^{-1}w_{C}\leq _{C_0}\mu _{C_0}^{-1}w_{C_0}$ , while by our construction, we get $\mu _{C_0}^{-1}u_{C_0}\leq _{C_0} \mu _{C}^{-1}u_{C}$ ; hence, $\mu _{C_0}^{-1}\mu u_{C_0}\leq _{C_0} \mu _{C}^{-1}\mu u_{C}$ (by Remark 1.2.4(a)(i)). Thus, it is enough to show that $\mu _C^{-1}\mu u_C\leq _{C_0} \mu _C^{-1}w_{C}$ , or, equivalently, that $\mu u_C\leq _{C_0} w_{C}$ .

Since $\psi _0\in C_0$ , we get that $\psi \in C$ . Hence, by our assumption, $\mu u_C\leq _{C^{\psi }} w_{C}$ . Therefore, to show that $\mu u_C\leq _{C_0} w_{C}$ , it suffices to check that $(\Phi ^{\psi })_{C^{\psi }}\subseteq \Phi _{C_0}$ .

If $\beta \in (\Phi ^{\psi })_{C^{\psi }}$ , then $u^{-1}(\beta )\in (\Phi ^{\psi _0})_{C_0^{\psi _0}}$ . Since $u\in W$ is an element of minimal length such that $\psi =u(\psi _0)$ , we get that $u((\Phi ^{\psi _0})_{C_0^{\psi _0}})\subseteq \Phi _{C_0}$ . In particular, we have $\beta =u(u^{-1}(\beta ))\in \Phi _{C_0}$ .

Notation 2.1.1. (a) Let k be an algebraically closed field, $K:=k((t))$ the field of Laurent power series over k, and $\mathcal {O}=\mathcal {O}_K=k[[t]]$ the ring of integers of K. For every affine scheme X over $\mathcal {O}$ (resp. K), we denote by $L^+X$ (resp. $LX$ ) the corresponding arc- (resp. loop-) space.

(b) Let G be a semi-simple and simply connected group over k. Fix a maximal torus $T\subseteq G$ , let $\Phi =\Phi (G,T)$ be the root system of $(G,T)$ , let $W=W_{G}$ be the Weyl group of G, and $\widetilde {W}=N_{LG}(LT)/L^+T$ the affine Weyl group of G. Then, in the notation of Section 1.1.1, we have natural isomorphisms $\Lambda \overset {\thicksim }{\to } X_*(T)$ and $W_{\Phi }\overset {\thicksim }{\to } W$ . Moreover, the map $\mu \mapsto \mu (t)$ defines an embedding $\Lambda \hookrightarrow LT$ , which in turn induces isomorphisms of groups $\Lambda \overset {\thicksim }{\to } LT/L^+T$ and $\widetilde {W}_{\Phi }\overset {\thicksim }{\to }\widetilde {W}$ .

Notation 2.1.2. (a) For every $C\in \mathcal {C}$ , let $B_C\subseteq G$ be the Borel subgroup containing T such that $\Phi (B_C,T)=\Phi _C$ , and let $U_C\subseteq B_C$ be the unipotent radical.

(b) Choose $C_0\in \mathcal {C}_{\Phi }$ as in Section 1.1.2, let $T\subseteq B_0=B_{C_0}\subseteq G$ be the corresponding Borel subgroup, let $B_0^{-}\supseteq T$ be the opposite Borel subgroup, and let $I\subseteq L^+G$ be the Iwahori subgroup, defined as the preimage of $B_0^{-}\subseteq G$ under the projection $L^+G\to G$ .

(c) For every $\alpha \in \Phi $ , we have a natural isomorphism $\exp _{\alpha }:\operatorname {Lie} U_{\alpha }\overset {\thicksim }{\to } U_{\alpha }$ . For $\widetilde {\alpha }=(\alpha ,n)\in \widetilde {\Phi }$ , we set $U_{\widetilde {\alpha }}:=\exp _{\alpha }(t^n \operatorname {Lie} U_{\alpha })\subseteq L(U_{\alpha })$ , and $\widetilde {\alpha }':=(-\alpha ,n)$ .

(d) In the conventions of parts (b), (c), we get the equality .

2.1.3. Affine flag varieties

(a) Denote by $\operatorname {Fl}=\operatorname {Fl}_G$ the affine flag variety $LG/I$ of G over k, and by $\operatorname {Gr}=\operatorname {Gr}_G$ the affine Grassmannian $LG/L^+G$ . We have a natural projection $\operatorname {pr}:\operatorname {Fl}\to \operatorname {Gr}$ . Note that both $\operatorname {Fl}$ and $\operatorname {Gr}$ are equipped with an action of the ind-group scheme $LG$ , and that projection $\operatorname {pr}$ is $LG$ -equivariant.

(b) The embedding $N_{LG}(LT)\hookrightarrow LG$ induces embeddings $\widetilde {W}\hookrightarrow \operatorname {Fl}$ and $\Lambda \hookrightarrow \operatorname {Gr}$ , and we identify $\widetilde {W}$ (resp. $\Lambda $ ) with its image in $\operatorname {Fl}$ (resp. $\operatorname {Gr}$ ). Furthermore, both $\operatorname {Fl}$ and $\operatorname {Gr}$ are equipped with the action of $T\subseteq L^+(T)\subseteq LG$ , and these identifications identify $\widetilde {W}$ (resp. $\Lambda $ ) with the locus of T-fixed points $\operatorname {Fl}^T$ (resp. $\operatorname {Gr}^T$ ).

(c) Note that $\operatorname {Fl}$ decompose as a union $\operatorname {Fl}=\bigcup _{w\in \widetilde {W}}Iw$ of I-orbits, and for every $w\in \widetilde {W}$ , we denote by $\operatorname {Fl}^{\leq w}\subseteq \operatorname {Fl}$ the closure of the I-orbit $Iw\subseteq \operatorname {Fl}$ . Then $\operatorname {Fl}^{\leq w}$ is a reduced projective subscheme of $\operatorname {Fl}$ called the affine Schubert variety.

(d) Fix any $C\in \mathcal {C}$ . Then we have decompositions $\operatorname {Fl}=\bigcup _{w\in \widetilde {W}}L(U_C)w$ and $\operatorname {Gr}=\bigcup _{\mu \in \Lambda }L(U_C)\mu $ by $L(U_C)$ -orbits. For every $w\in \widetilde {W}$ (resp. $\mu \in \Lambda $ ), we denote by $\operatorname {Fl}^{\leq _{C}w}\subseteq \operatorname {Fl}$ (resp. $\operatorname {Gr}^{\leq _{C}\mu }\subseteq \operatorname {Gr}$ ) the closure of the $L(U_C)$ -orbit $L(U_C)w\subseteq \operatorname {Fl}$ (resp. $L(U_C)\mu \subseteq \operatorname {Gr}$ ). We also set $\operatorname {Fl}^{\leq ^{\prime }_{C}\mu }:=\operatorname {pr}^{-1}( \operatorname {Gr}^{\leq _{C}\mu })\subseteq \operatorname {Fl}$ . Notice that $\operatorname {Fl}^{\leq _{C}w}$ , $\operatorname {Gr}^{\leq _{C}\mu }$ and $\operatorname {Fl}^{\leq ^{\prime }_{C}w}$ are closed reduced ind-subschemes.

(e) For every tuple $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ (resp. $\overline {\mu }\in \Lambda ^{\mathcal {C}}$ ), we denote by $\operatorname {Fl}^{\leq \overline {w}}$ (resp. $\operatorname {Gr}^{\leq \overline {\mu }}$ ) the reduced intersection $\bigcap _C \operatorname {Fl}^{\leq _C w_C}$ (resp. $\bigcap _C \operatorname {Gr}^{\leq _C \mu _C}$ ), and set $\operatorname {Fl}^{\leq ' \overline {\mu }}:=\operatorname {pr}^{-1}(\operatorname {Gr}^{\leq \overline {\mu }})$ .

Proof. We will show the assertion for $Z\subseteq \operatorname {Fl}$ and $w\in \widetilde {W}$ , while the proof of the second assertion is identical.

Clearly, if $w\in Z$ , then $Z\cap L(U_C)w\neq \emptyset $ . Conversely, let z be an element of $Z\cap L(U_C)w$ , and pick $u\in L(U_C)$ such that $z=uw$ . For any $\nu \in \Lambda =\operatorname {Hom}(\mathbb G_m,T)$ and $a\in \mathbb G_m$ , we have $\nu (a)(z)=(\nu (a)u\nu (a)^{-1})(w)$ because $w\in \operatorname {Fl}$ is T-invariant; hence, $(\nu (a)u\nu (a)^{-1})(w)\in Z$ because Z is T-invariant. Next, for $\nu \in \Lambda \cap C$ , the morphism $a\mapsto (\nu (a)u\nu (a)^{-1})(w):\mathbb G_m\to Z\subseteq \operatorname {Fl}$ extends to the morphism $\mathbb A^1\to \operatorname {Fl}$ , which sends $0$ to w. Since $Z\subseteq \operatorname {Fl}$ is closed, we conclude that $w\in Z$ .

Proof. (a) is a standard.

(b) In the notation of Section 2.1.2(c), for every $\widetilde {\alpha }\in \widetilde {\Phi }$ and $w\in \widetilde {W}$ , we have $wU_{\widetilde {\alpha }}w^{-1}=U_{w(\widetilde {\alpha })}$ and $w(\widetilde {\alpha }')=w(\widetilde {\alpha })'$ . Combining this with Section 2.1.2(d), we see that for every $w\in \widetilde {W}$ , we have

(2.1) $$ \begin{align} Iw=\left(\prod_{\widetilde{\alpha}>0,w^{-1}(\widetilde{\alpha})<0}U_{\widetilde{\alpha}'}\right)w. \end{align} $$

Using formula (2.1), it remains to check that every $\widetilde {\alpha }=(\alpha ,n)>0$ such that $w^{-1}(\widetilde {\alpha })<0$ satisfies $U_{\widetilde {\alpha }'}\subseteq L(U_C)$ ; that is, $-\alpha \in \Phi _C$ . However, $n\geq 0$ because $\widetilde {\alpha }>0$ . Therefore, $w^{-1}(\alpha )=w^{-1}(\widetilde {\alpha })-n<0$ . Thus, $w^{-1}(-\alpha )>0$ ; hence, $-\alpha \in \Phi _C$ because $w\in C$ .

(c) If $Iw\cap L(U_C)w'\neq \emptyset $ , then $\operatorname {Fl}^{\leq w}\cap L(U_C)w'\neq \emptyset $ . Since $\operatorname {Fl}^{\leq w}\subseteq \operatorname {Fl}$ is closed and T-invariant, we get $w'\in \operatorname {Fl}^{\leq w}$ (by Lemma 2.1.4); thus, $w'\leq w$ (by part (a)).

Proof. Assume that $w'\leq _C w"$ . Then by Proposition 1.2.5, there exists $\mu \in \Lambda \cap C$ such that $\mu w'\leq \mu w"$ and $\mu w"\in C$ . Then $\mu w'\in \operatorname {Fl}$ lies in the closure of $I\mu w"\subseteq \operatorname {Fl}$ (by Lemma 2.1.5(a)), and thus in the closure of $L(U_C)\mu w"\subseteq \operatorname {Fl}$ (by Lemma 2.1.5(b)). Since $U_C$ is normalized by T, this implies that $w'\in \operatorname {Fl}$ lies in the closure of $\mu ^{-1} L(U_C)\mu w" =L(U_C)w"\subseteq \operatorname {Fl}$ ; that is, $w'\in \operatorname {Fl}^{\leq _C w"}$ .

Conversely, assume that $w'\in \operatorname {Fl}$ lies in the closure of $L(U_C) w"\subseteq \operatorname {Fl}$ . Then there exists a closed subgroup scheme $U'\subseteq L(U_C)$ such that $w'\in \operatorname {Fl}$ lies in the closure of $U' w"\subseteq \operatorname {Fl}$ . Then $\mu w'\in \operatorname {Fl}$ lies in the closure of $\mu U' w"=(\mu U'\mu ^{-1})\mu w"$ for every $\mu \in \Lambda $ . However, if $\mu \in \Lambda \cap C$ is sufficiently regular, then $\mu U'\mu ^{-1}\subseteq I$ ; thus, $\mu w'\in \operatorname {Fl}$ lies in the closure of $I\mu w"\subseteq \operatorname {Fl}$ . This implies that $\mu w'\leq \mu w"$ (by Lemma 2.1.5(a)); thus, $w'\leq _C w"$ (by Proposition 1.2.5).

Proof. (a) By Lemma 1.3.7 and Proposition 2.1.6, a tuple $\overline {w}\in \widetilde {W}^{\mathcal {C}}$ is admissible if and only if $w_{C}\in \operatorname {Fl}^{\leq \overline {w}}=\bigcap _{C'\in \mathcal {C}}\operatorname {Fl}^{\leq _{C'} w_{C'}}$ for each $C\in \mathcal {C}$ . Since $\operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}$ is closed and T-invariant, the assertion now follows from Lemma 2.1.4.

(b) The ‘if’ assertion follows from Proposition 2.1.6. Conversely, if $\operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}^{\leq \overline {u}}$ , then $w_C\in \operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}^{\leq \overline {u}}$ (as in part (a)); hence, $w_C\in \operatorname {Fl}^{\leq _C u_C}$ for every C. Therefore, $w_C\leq _C u_C$ by Proposition 2.1.6.

(c) Let z be any element of $\operatorname {Fl}^{\leq \overline {w}}$ , let $u\in \widetilde {W}$ be such that $z\in Iu$ , and let $C\in \mathcal {C}$ be such that $u\in C$ . We want to show that $u\leq w_C$ , and thus $z\in \operatorname {Fl}^{\leq w_C}$ .

By Lemma 2.1.5(b), we get $z\in Iu\subseteq L(U_C)u$ . However, we have $z\in \operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}^{\leq _C w_C}$ . Therefore, by Proposition 2.1.6, we get that $u\leq _C w_C$ , which by Corollary 1.2.6(a) implies that $u\leq w_C$ .

(d) By construction, $z\in \operatorname {Fl}^{\leq \overline {u}}\cap L(U_C)u_C$ for all $C\in \mathcal {C}$ ; hence, $\overline {u}$ is admissible by part (a). Since $z\in L(U_C)u_C\cap Z$ , we get $u_C\in Z$ by Lemma 2.1.4.

(e) follows immediately from part (d).

(f) It is enough to show that for every $C\in \mathcal {C}$ , the preimage $\operatorname {pr}^{-1}(\operatorname {Gr}^{\leq _C\mu _C})$ equals $\operatorname {Fl}^{\leq _C (\mu _C (w_{\operatorname {st}})_C)}$ . Using Proposition 2.1.6, we have to check that for every $\mu \in \Lambda $ and $u\in W$ , we have $\mu \leq _C \mu _C$ if and only if $\mu u\leq _C\mu _C (w_{\operatorname {st}})_C$ .

The ‘only if’ assertion follows from Corollary 1.2.3(b). Conversely, if $\mu \leq _C \mu _C$ , then $\mu u\leq _C \mu _C u$ by Lemma 1.2.2(b). So by Remark 1.2.4(a)(i), it is enough to show that $u\leq _C (w_{\operatorname {st}})_C$ . Since $\overline {w}_{\operatorname {st}}$ is admissible and $u=(w_{\operatorname {st}})_{u(C_0)}$ , the assertion follows from Lemma 1.3.7.

2.2.1. Let $m\in \mathbb N$ . Recall that $w\in \widetilde {W}$ is called m-regular if $\pi (w)\in \Lambda $ is m-regular; that is, we have $|\langle \alpha ,\pi (w)\rangle |\geq m$ for all $\alpha \in \Phi $ . For each $w\in \widetilde {W}$ , we denote by $\widetilde {W}^{\leq w}$ the set of $w'\in \widetilde {W}$ such that $w'\leq w$ .

Proof. (a) Denote by $X(w)$ the closed ind-subscheme $\bigcap _C(\bigcup _{w'\leq w} \operatorname {Fl}^{\leq _C w'})\subseteq \operatorname {Fl}$ (compare Corollary 2.1.7(c)), and we claim that $X(w)$ equals $\operatorname {Fl}^{\leq w}$ . Indeed, the inclusion $\operatorname {Fl}^{\leq w}\subseteq X(w)$ follows from Corollary 2.1.7(e), while the opposite inclusion $X(w)\subseteq \operatorname {Fl}^{\leq w}$ follows from identity $X(w)=\bigcup _{\overline {w}'\in (\widetilde {W}^{\leq w})^{\mathcal {C}}}\operatorname {Fl}^{\leq \overline {w}'}$ and Corollary 2.1.7(c).

Next, since $\operatorname {Fl}^{\leq w}=\bigcup _{\overline {w}'\in (\widetilde {W}^{\leq w})^{\mathcal {C}}}\operatorname {Fl}^{\leq \overline {w}'}$ is irreducible, there exists a tuple $\overline {w}=\{w_C\}_C\in (\widetilde {W}^{\leq w})^{\mathcal {C}}$ such that $\operatorname {Fl}^{\leq w}=\operatorname {Fl}^{\leq \overline {w}}$ .

Then for each $w'\leq w$ and $C\in \mathcal {C}$ , we have $w'\in \operatorname {Fl}^{\leq w}\subseteq \operatorname {Fl}^{\leq _C w_C}$ . Thus, by Proposition 2.1.6, we have $w'\leq _C w_C$ ; that is, $w_C$ is the biggest element of $\widetilde {W}^{\leq w}$ with respect to ordering $\leq _C$ . In particular, for every other Weyl chamber $C'$ , we have $w_{C'}\leq _C w_C$ . Thus, by Lemma 1.3.7, we conclude that $\overline {w}$ is admissible.

The uniqueness of $\overline {w}$ follows immediately from Corollary 2.1.7(b).

(b) Choose any $\mu \in \Lambda \cap C_0$ , and let r be the maximum of the $2\langle \psi ,\mu \rangle $ ’s, taken over $\psi \in \Psi _{C_0}$ . We claim this r satisfies the required property; that is, for every $m\in \mathbb N$ and every $(m+r)$ -regular $w\in \widetilde {W}$ , the tuple $\overline {w}$ is m-regular. In other words, we claim that $w_{u(C_0)}$ is $(u(C_0),m)$ -regular or, equivalently, that $u^{-1}w_{u(C_0)}$ is $(C_0,m)$ -regular for all $u\in W$ .

Claim 2.2.3. Let $w\in \widetilde {W}$ , and let $\overline {w}:=\{w_C\}_{C\in \mathcal {C}}$ be the tuple from Theorem 2.2.2(a).

(a) If $w=w_0w_+$ , where $w_0\in W$ is the longest element, and $w_+\in \widetilde {W}\cap C_0$ , then $w_{u(C_0)}=uw_+$ for all $u\in W$ .

(b) If $w\in Ww_+$ with $w_+\in \widetilde {W}\cap C_0$ , then for all $u\in W$ , we have inequalities

$$\begin{align*}\mu^{-1}w_+\leq_{C_0}u^{-1}w_{u(C_0)}\leq_{C_0}w_+. \end{align*}$$

Proof. (a) Fix $u\in W$ . We will show that $w\leq w_0u^{-1} w_{u(C_0)}\leq w$ , which will imply that $w_0u^{-1} w_{u(C_0)}=w_0w_+$ , and thus $w_{u(C_0)}=uw_+$ .

Since $u\leq w_0$ , we get $uw_+\leq w_0w_+=w$ (use Section 1.1.4(e)). Therefore, by the characterization of $\overline {w}$ , given Theorem 2.2.2(a), we get $uw_+\leq _{u(C_0)} w_{u(C_0)}$ . Hence, $w=w_0w_+\leq _{w_0(C_0)}w_0u^{-1} w_{u(C_0)}$ (by Remark 1.2.4(a)(ii)); thus, $w\leq w_0u^{-1} w_{u(C_0)}$ (by Corollary 1.2.6(a)). However, $w_{u(C_0)}\leq w=w_0w_+$ ; thus, using Section 1.1.4(e), we conclude that $w_0u^{-1} w_{u(C_0)}\leq w$ .

(b) By Remark 1.2.4(a)(ii), it is enough to show that

$$\begin{align*}u\mu^{-1}w_+\leq_{u(C_0)}w_{u(C_0)}\leq_{u(C_0)}uw_+. \end{align*}$$

Consider element $w':=w_0w_+$ . Then $w\leq w'$ (use Section 1.1.4(e)), and thus, we have $w_{u(C_0)}\leq w'$ . Hence, $w_{u(C_0)}\leq _{u(C_0)} w^{\prime }_{u(C_0)}$ (by the characterization of $w^{\prime }_{u(C_0)}$ , given in Theorem 2.2.2(a)); thus, $w_{u(C_0)}\leq _{u(C_0)}uw_+$ (by part (a)). To show the other inequality, it is enough to show that $u\mu ^{-1}w_+\leq w$ . Since w and hence also $w_+$ is $(m+r)$ -regular, our definition of r implies that $\mu ^{-1}w_+\in \widetilde {W}\cap C_0$ . Since $u\leq \mu $ (by Section 1.1.4(g)), and $l(w_+)=l(\mu )+l(\mu ^{-1}w_+)$ (by Lemma 1.1.5(c)), we conclude from Section 1.1.4(c) and part (e) that $u(\mu ^{-1}w_+)\leq w_+\leq w$ .

Let us come back to the proof of the Theorem. By Claim 2.2.3(b) and Corollary 1.2.3(b), we have

$$\begin{align*}\mu^{-1}\pi(w_+)\leq_{C_0}\pi(u^{-1}w_{u(C_0)})\leq_{C_0}\pi(w_+). \end{align*}$$

Hence, we have $\pi (u^{-1}w_{u(C_0)})=\pi (w_+)- \sum _{\alpha \in \Delta _{C_0}}m_{\alpha }\check {\alpha }$ , such that $0\leq m_{\alpha }\leq \langle \psi _{\alpha },\mu \rangle $ , where $\psi _{\alpha }\in \Psi _{C_0}$ is the fundamental weight corresponding to $\alpha $ for each $\alpha \in \Delta _{C_0}$ . In particular, for each $\alpha \in \Delta _{C_0}$ , we have

$$\begin{align*}\langle\alpha,\pi(u^{-1}w_{u(C_0)})\rangle\geq \langle\alpha,\pi(w_+)\rangle-2m_{\alpha}\geq (m+r)-r=m \end{align*}$$

because $w_+$ is $(m+r)$ -regular, and $2m_{\alpha }\leq 2\langle \psi _{\alpha },\mu \rangle \leq r$ .

Notation 2.3.1. Fix $\psi \in \Psi $ .

(a) Denote by $P_{\psi }\supseteq T$ the parabolic subgroup of G such that $\Phi (P_{\psi },T)=\Phi (\psi )$ (see Section 1.1.3(c)), by $M_{\psi }\supseteq T$ the Levi subgroup of $P_{\psi }$ , by $U_{\psi }\subseteq P_{\psi }$ the unipotent radical, by $M^{\operatorname {sc}}_{\psi }$ the simply connected covering of the derived (=commutator) group of $M_{\psi }$ . Let $P_{\psi }\to M_{\psi }$ be the natural projection, and set $P^{\operatorname {sc}}_{\psi }:=P_{\psi }\times _{M_{\psi }}M^{\operatorname {sc}}_{\psi }$ .

(b) Note that we have a natural homomorphism $P^{\operatorname {sc}}_{\psi }\to P_{\psi }\subseteq G$ ; thus, the loop group $L(P^{\operatorname {sc}}_{\psi })$ acts on $\operatorname {Fl}$ . For every $w\in \widetilde {W}$ , we denote by $\operatorname {Fl}^{\leq _{\psi }w}\subseteq \operatorname {Fl}$ the closure of the $L(P^{\operatorname {sc}}_{\psi })$ -orbit $L(P^{\operatorname {sc}}_{\psi })w\subseteq \operatorname {Fl}$ . For every $\mu \in \Lambda $ , we denote by $\operatorname {Gr}^{\leq _{\psi }\mu }\subseteq \operatorname {Gr}$ the closure of the $L(P^{\operatorname {sc}}_{\psi })$ -orbit $L(P^{\operatorname {sc}}_{\psi })\mu \subseteq \operatorname {Gr}$ .

Proof. (a) Assume first that $w'\leq _{\psi } w"$ , and we want to prove that $L(P^{\operatorname {sc}}_{\psi })w'\subseteq \operatorname {Fl}$ is contained in the closure of $L(P^{\operatorname {sc}}_{\psi })w"\subseteq \operatorname {Fl}$ . By definition, we can assume that $w'=s_{\widetilde {\beta }}w"<_{\widetilde {\beta }}w"$ , where $\widetilde {\beta }=(\beta ,m)$ , and $\langle \beta ,\psi \rangle \geq 0$ . Then there exists a Weyl chamber $C\owns \psi $ such that $\beta \in \Phi _{C}$ . Then $w'<_C w"$ ; hence, by Proposition 2.1.6, $w'$ lies in the closure of $L(U_C)w"\subseteq \operatorname {Fl}$ . Since $L(U_C)\subseteq L(P^{\operatorname {sc}}_{\psi })$ , the assertion follows.

Conversely, assume that $w'$ belongs to the closure of $L(P^{\operatorname {sc}}_{\psi })w"\subseteq \operatorname {Fl}$ . Choose any Weyl chamber $C\owns \psi $ . Then $L(P^{\operatorname {sc}}_{\psi })w"$ is a union of orbits $\bigcup _{w\in \widetilde {W}^{\psi }}L(U_C)ww"$ . Therefore, $w'$ belongs to the closure of $L(U_C)ww"\subseteq \operatorname {Fl}$ for some $w\in \widetilde {W}^{\psi }$ . Hence, by Proposition 2.1.6, we get $w'\leq _C ww"$ , and thus $w'\leq _{\psi }ww"$ . However, since $w\in \widetilde {W}^{\psi }$ , we also get $ww"\leq _{\psi } w"$ .

(b) Choose any Weyl chamber $C\owns \psi $ . Then $w_C\in \widetilde {W}^{\psi }\overline {w}_{\psi }$ ; hence, we have $\overline {w}_{\psi }\leq _{\psi } u$ if and only if $w_C\leq _{\psi } u$ .

Assume first that $w_C\leq _{\psi } u$ . Then by part (a) we have $\operatorname {Fl}^{\leq _{\psi } w_C}\subseteq \operatorname {Fl}^{\leq _{\psi } u}$ . However, we always have inclusions $\operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}^{\leq _C w_C}\subseteq \operatorname {Fl}^{\leq _{\psi } w_C}$ , which imply that $\operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}^{\leq _{\psi } u}$ . Conversely, since $\overline {w}$ is admissible, we get $w_C\in \operatorname {Fl}^{\leq \overline {w}}$ by Lemma 1.3.7. Therefore, if $\operatorname {Fl}^{\leq \overline {w}}\subseteq \operatorname {Fl}^{\leq _{\psi } u}$ , we get $w_C\in \operatorname {Fl}^{\leq _{\psi } u}$ . Hence, by part (a), we have $w_C\leq _{\psi } u$ .

Proof. (a) Using equality $\operatorname {pr}^{-1}(\operatorname {Gr}^{\leq _{\psi }\mu "})=\bigcup _{w\in W} \operatorname {Fl}^{\leq _{\psi }\mu "w}$ , we see that $\mu '\in \operatorname {Gr}^{\leq _{\psi } \mu "}$ if and only if $\mu '\in \operatorname {Fl}^{\leq _{\psi } \mu " w}$ for some $w\in W$ . Hence, by Lemma 2.3.2(a), this happens if and only if we have $\mu '\leq _{\psi }\mu "w$ for some $w\in W$ . Since $\pi (\mu "w)=\mu "$ , it thus follows from Corollary 1.2.3(b),(c) and Section 1.2.4(c) that this happens if and only if $\mu '\leq _{\psi }\mu "$ in the sense of Section 1.2.1(c). Now the assertion follows from Lemma 1.2.2(c).

(b) follows immediately from part (a).

(c) Notice that for every $C\in \mathcal {C}$ and $\psi \in \Psi _C$ , the inclusion $U_C\subseteq P^{\operatorname {sc}}_{\psi }$ implies the inclusion $\operatorname {Gr}^{\leq _{C}\mu _C}\subseteq \operatorname {Gr}^{\leq _{\psi }\mu _C}=\operatorname {Gr}^{\leq _{\psi }\overline {\mu }(\psi )}$ , from which the inclusion ‘ $\subseteq $ ’ follows.

Conversely, for every $y\in \bigcap _{\psi \in \Psi }\operatorname {Gr}^{\leq _{\psi }\overline {\mu }(\psi )}$ and $C\in \mathcal {C}$ , let $\nu \in \Lambda $ be such that $y\in L(U_C)\nu $ , and we want to show that $\nu \leq _C\mu _C$ . Since the ind-subscheme $\bigcap _{\psi \in \Psi }\operatorname {Gr}^{\leq _{\psi }\overline {\mu }(\psi )}\subseteq \operatorname {Gr}$ is closed and T-invariant, it follows from Lemma 2.1.4 that $\nu \in \bigcap _{\psi \in \Psi }\operatorname {Gr}^{\leq _{\psi }\overline {\mu }(\psi )}$ . Hence, by part (a), we have $\langle \psi ,\nu \rangle \leq \overline {\mu }(\psi )$ for each $\psi \in \Psi $ , from which inequality $\nu \leq _C\mu _C$ follows from Section 1.3.4(c).

Proof. We denote by Z the reduced intersection $\operatorname {Fl}^{\leq \overline {w}'}\cap \operatorname {Fl}^{\leq \overline {w}"}$ in the case (a), and $\operatorname {Fl}^{\leq \overline {w}}\cap \operatorname {Fl}^{\leq _{\psi } u}$ in the case (b). Then, by Corollary 2.1.7(c), in both cases, Z is a closed T-invariant subscheme of $\operatorname {Fl}$ of finite type; thus, the intersection $\widetilde {W}\cap Z$ is finite.

By Corollary 2.1.7(d), each $z\in Z$ defines an admissible tuple $\overline {u}=\overline {u}(z)\in \widetilde {W}^{\mathcal {C}}$ satisfying $u_C\in \widetilde {W}\cap Z$ for each $C\in \mathcal {C}$ . It follows that the set of tuples $\{\overline {u}(z)\}_{z\in Z}$ is finite, so it will suffice to show the equality

$$ \begin{align*} Z=\bigcup_{z\in Z} \operatorname{Fl}^{\leq \overline{u}(z)}. \end{align*} $$

One inclusion follows from the fact that $z\in \operatorname {Fl}^{\leq \overline {u}(z)}$ for every $z\in Z$ . To show the converse, it is enough to show that if $\overline {u}\in \widetilde {W}^{\mathcal {C}}$ satisfies $u_C\in Z$ for all $C\in \mathcal {C}$ , then $\operatorname {Fl}^{\leq \overline {u}}\subseteq Z$ . Using definition of Z, it remains to show the corresponding assertion in the cases $Z=\operatorname {Fl}^{\leq \overline {w}}$ and $Z=\operatorname {Fl}^{\leq _{\psi }u}$ . In the first case, we have $u_C\in \operatorname {Fl}^{\leq _C w_C}$ ; hence, $\operatorname {Fl}^{\leq _C u_C}\subseteq \operatorname {Fl}^{\leq _C w_C}$ for all $C\in \mathcal {C}$ , and thus, $\operatorname {Fl}^{\leq \overline {u}}\subseteq \operatorname {Fl}^{\leq \overline {w}} $ . In the second case, the assertion follows from Lemma 2.3.2(b).