Torus quotient of Richardson varieties in Orthogonal and Symplectic Grassmannians

TL;DR

This paper establishes criteria for the existence of semistable points in Richardson varieties within orthogonal and symplectic Grassmannians under torus actions, contributing to geometric invariant theory in these contexts.

Contribution

It provides a new criterion for semistability of Richardson varieties in G/P for groups of types B, C, D, advancing understanding of their geometric invariant theory.

Findings

Criteria for semistability of Richardson varieties established.

Applicable to groups of types B, C, D.

Enhances understanding of torus actions on these varieties.

Abstract

For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.

Equations

$${\mu }^{\mathcal{L}}(x,\lambda ):=-min\{m_i:i=1,\cdots ,r\}$$

$$l_B(\sigma )=\frac{inv(\sigma )+neg(\sigma )}{2},$$

$$l_D(\sigma )=\frac{inv(\sigma )-neg(\sigma )}{2},$$

$$v=\left\{\begin{array}[]{lr}(-(n-1),-(n-3),\ldots,-3,-1,2,4,\ldots,n-2,n),&\text{if n is even}\\ (-(n-1),-(n-3),\ldots,-4,-2,1,3,5,\ldots,n-2,n),&\text{if n is odd}.\\ \end{array}\right.$$

$${\omega }_r=2m({\alpha }_1+{\alpha }_2+\cdots +{\alpha }_r)+\sum _{i=1}^{2m-1}(2m-i){\alpha }_{r+i},2\le r\le n-1.$$

$$w=\left\{\begin{array}[]{lr}(-n,-(n-2),\ldots,-4,-2,1,3,\ldots,n-3,n-1),&\text{if n is even}\\ (-n,-(n-2),\ldots,-3,-1,2,4,\ldots,n-3,n-1),&\text{if n is odd.}\\ \end{array}\right.$$

Full text

Keywords: Schubert variety, Richardson variety, Semi-stable point, Line bundle.

2010 Mathematics Subject Classification: 14F15; 20G05; 22E45.

1. Introduction

For the action of a maximal torus $T$ on the Grassmannian $G_{r,n}$, the GIT quotients have been studied by several authors. In [7] Hausmann and Knutson identified the GIT quotient of the Grassmannian $G_{2,n}$ by the natural action of the maximal torus with the moduli space of polygons in $\mathbb{R}^{3}$ and this GIT quotient can also be realized as the GIT quotient of an $n$-fold product of projective lines by the diagonal action of $PSL(2,\mathbb{C})$. In the symplectic geometry literature these spaces are known as polygon spaces as they parameterize the $n$-sides polygons in $\mathbb{R}^{3}$ with fixed edge length up to rotation. More generally, $G_{r,n}//T$ can be identified with the GIT quotient of $(\mathbb{P}^{r-1})^{n}$ by the diagonal action of $PSL(r,\mathbb{C})$ called the Gelfand-MacPherson correspondence. In [13] and [14] Kapranov studied the Chow quotient of the Gassmannians and he showed that the Grothendieck-Knudsen moduli space $\overline{M}_{0,n}$ of stable $n$-pointed curves of genus zero arises as the Chow quotient of the maximal torus action on the Grassmannian $G_{2,n}$.

Let $G$ be a simply connected semi-simple algebraic group over an algebraically closed field $K$. Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T$. In [8] and [9], the parabolic subgroups $P$ of $G$ containing $B$ are described for which there exists an ample line bundle $\mathcal{L}$ on $G/P$ such that the semistable points $(G/P)^{ss}_{T}(\mathcal{L})$ are the same as the stable points $(G/P)^{s}_{T}(\mathcal{L})$. In [25] Strickland reproved these results.

In [11], when $G$ is of type $A$, $P$ is a maximal parabolic subgroup of $G$ and $\mathcal{L}$ is the ample generator of the Picard group of $G/P$, it is shown that there exists unique minimal Schubert variety $X(w)$ admitting semistable points with respect to $\mathcal{L}$. For other types of classical groups the minimal Schubert varieties admitting semistable points were described in [12] and [21].

A Richardson variety $X_{w}^{v}$ in $G/P$ is the intersection of the Schubert variety $X_{w}$ in $G/P$ with the opposite Schubert variety $X^{v}$ therein. For $G=SL_{n}$ and $P$ a maximal parabolic in $G$ a criterion for the Richardson varieties in $G/P$ to have nonempty semistable locus with respect to an ample line bundle $\mathcal{L}$ on $G/P$ is given in [10]. In this paper, we give a criterion for a Richardson variety in $G/P$ to have nonempty semistable locus with respect to the action of a maximal torus $T$ on $G/P$, where $G$ is of type $B$, $C$ and $D$ and $P$ is a maximal parabolic subgroup in $G$.

The organisation of the paper is as follows. Section 2 consists of preliminary notions and some terminologies from algebraic groups and Geometric invariant theory. Section 3 gives a necessary condition for a Richardson variety to admit a semistable point. In section 4 we give a sufficient condition for a Richardson variety in type $B$ and $C$ to admit a semistable point and in section 5 a sufficient condition is given for type $D$.

2. Preliminaries and notation

In this section, we set up some notation and preliminaries. We refer to [3], [5], [6] and [24] for preliminaries in Lie algebras and algebraic groups. Let $G$ be a semi-simple algebraic group over an algebraically closed field $K$. We fix a maximal torus $T$ of $G$ and a Borel subgroup $B$ of $G$ containing $T$. Let $U$ be the unipotent radical of $B$. Let $N_{G}(T)$ (respectively, $W=N_{G}(T)/T$) be the normalizer of $T$ in $G$ (respectively, the Weyl group of $G$ with respect to $T$). Let $B^{-}$ be the Borel subgroup of $G$ opposite to $B$ determined by $T$. We denote by $R$ the set of roots with respect to $T$ and we denote by $R^{+}$ the set of positive roots with respect to $B$. Let $U_{\alpha}$ denote the one-dimensional $T$-stable subgroup of $G$ corresponding to the root $\alpha$ and we denote $U_{\alpha}^{*}$ by the open set $U_{\alpha}\setminus\{identity\}$. Let $S=\{\alpha_{1},\ldots,\alpha_{l}\}\subseteq R^{+}$ denote the set of simple roots and for a subset $I\subseteq S$ we denote by $P_{I}$ the parabolic subgroup of $G$ generated by $B$ and $\{n_{\alpha}:\alpha\in I^{c}\}$, where $n_{\alpha}$ is a representative of $s_{\alpha}$ in $N_{G}(T)$. Let $W^{I}=\{w\in W:w(\alpha)\in R^{+}\,\,\mbox{for each}\,\,\alpha\in I^{c}\}$ and $W_{I}$ be the subgroup of $W$ generated by the simple reflections $s_{\alpha}$, $\alpha\in I^{c}$. Then every $w\in W$ can be uniquely expressed as $w=w^{I}.w_{I}$, with $w^{I}\in W^{I}$ and $w_{I}\in W_{I}$. Denote by $w_{0}$ the longest element of $W$ with respect to $S$. Let $X(T)$ (respectively, $Y(T)$) denote the group of all characters of $T$ ( respectively, one-parameter subgroups of $T$ ). Let $E_{1}:=X(T)\otimes\mathbb{R}$ and $E_{2}=Y(T)\otimes\mathbb{R}$. Let $\langle.,.\rangle:E_{1}\times E_{2}\rightarrow\mathbb{R}$ be the canonical non-degenerate bilinear form. Let $\{\lambda_{j}:j=1,2,\cdots l\}\subset E_{2}$ be the basis of $E_{2}$ dual to $S$. That is, $\langle\alpha_{i},\lambda_{j}\rangle=\delta_{ij}$ for all $1\leq i,~{}j\leq l$. Let $\bar{C}:=\{\lambda\in E_{2}|\langle\alpha,\lambda\rangle\geq 0\,\,\forall\,\,\alpha\in R^{+}\}$. Note that for each $\alpha\in R$, there is a homomorphism $SL_{2}\xrightarrow{\phi_{\alpha}}G$ (see [2, p.19 ] ). We have $\check{\alpha}:G_{m}\rightarrow G$ defined by $\check{\alpha}(t)=\phi_{\alpha}(\begin{pmatrix}t&0\\ 0&t^{-1}\end{pmatrix})$. We also have $s_{\alpha}(\chi)=\chi-\langle\chi,\check{\alpha}\rangle\alpha$ for all $\alpha\in R$ and $\chi\in E_{1}$. Set $s_{i}=s_{\alpha_{i}}$ for every $i=1,2,\ldots,l$. Let $\{\omega_{i}:i=1,2,\ldots,l\}\subset E_{1}$ be the fundamental weights; i.e. $\langle\omega_{i},\check{\alpha_{j}}\rangle=\delta_{ij}$ for all $i,j=1,2,\ldots,l$.

Let $X_{w}=\overline{BwB/B}$ ( respectively, $X^{v}=\overline{B^{-}vB/B}$ ) denote the Schubert variety corresponding to $w$ (respectively, the opposite Schubert variety corresponding to $v$ ). Let $X_{w}^{v}:=\overline{BwB/B}\cap\overline{B^{-}vB/B}$ denote the Richardson variety corresponding to $v$ and $w$ where $v\leq w$ in the Bruhat order. Such varieties were first considered by Richardson in [22], who shows that such intersections are reduced and irreducible whereas the cell intersection $C_{w}\cap C^{v}$ have been studied by Deodhar [4]. Richardson varieties have shown up in several contexts: such double coset intersections $BwB\cap B^{-}vB$ first appear in [15], [16] and their standard monomial theory is studied in [17] and [2]. We refer to [18] for preliminaries in standard monomial theory.

We recall the definition of the Hilbert-Mumford numerical function and the definition of semistable points from [19]. We refer to [20] for notations in geometric invariant theory.

Let $X$ be a projective variety with an action of a reductive group $G$. A point $x\in X$ is said to be semi-stable with respect to a $G$-linearized line bundle $\mathcal{L}$ if there is a positive integer $m\in\mathbb{N}$, and a $G$-invariant section $s\in H^{0}(X,\mathcal{L}^{m})^{G}$ with $s(x)\neq 0$.

Let $\lambda$ be a one-parameter subgroup of $G$. Let $x\in\mathbb{P}(H^{0}(X,\mathcal{L})^{*})$ and $\hat{x}=\sum_{i=1}^{r}v_{i}$, where each $v_{i}$ is a weight vector of $\lambda$ of weight $m_{i}$. Then the Hilbert-Mumford numerical function is defined by

[TABLE]

Then the Hilbert-Mumford criterion says that $x$ is semistable if and only if $\mu^{\mathcal{L}}(x,\lambda)\geq 0$ for all one parameter subgroup $\lambda$.

We recall the following result from [23] which will be used in section 3.

Lemma 2.1**.**

Let $G$ be a semisimple algebraic group, $T$ be a maximal torus, $B$ be a Borel subgroup of $G$ containing $T$ and $\overline{C}$ be as defined above.

(a) Let $\mathcal{L}$ be a line bundle defined by the character $\chi\in X(T)$. Then if $x\in G/B$ is represented by $bwB$, $b\in B$ and $w\in W$ is represented by an element of $N$ in the Bruhat decompsition of $G$ and $\lambda$ is a one parameter subgroup of $T$ which lies in $\overline{C}$, we have $\mu^{\mathcal{L}}(x,\lambda)=-\langle w(\chi),\lambda\rangle$.

(b) Given any set $S$ of finite number of one parameter subgroup $\lambda$ of $T$, there is an ample line bundle $\mathcal{L}$ on $G/B$ such that $\mu^{\mathcal{L}}(x,\lambda)\neq 0$ for all $x\in G/B$, $\lambda\in S$.

In this paper, we present results for Richardson varieties in the orthogonal and symplectic Grassmannians. For any character $\chi$ of $B$, we denote by $\mathcal{L}_{\chi}$, the line bundle on $G/B$ given by the character $\chi$. We denote by $(X^{v}_{w})_{T}^{ss}(\mathcal{L}_{\chi})$ the semistable points of $X_{w}^{v}$ for the action of $T$ with respect to the line bundle $\mathcal{L}_{\chi}$. Using the notations from [2] we recall the following theorem which is needed in the proofs of the main theorems.

Theorem 2.2** ([2], Proposition 6.).**

Let $\lambda$ be a dominant weight. The restriction to $X^{v}_{w}$ of the $p_{\pi}$, where $v\leq e(\pi)\leq i(\pi)\leq w$ form a basis of $H^{0}(X^{v}_{w},\mathcal{L}_{\lambda})$.

In the rest of this section we recall Bruhat ordering in the Weyl groups of type $B$, $C$ and $D$ and how it is related to the Bruhat order for the symmetric group $S_{n}$.

Bruhat order for type $B_{n}$ or $C_{n}$: We consider $\alpha_{1}$ as special node of Dynkin diagram for type $B$ or $C$. So as a set of generators of Weyl group, we take $S=\{s_{1},s_{2},\ldots,s_{n}\}$, where $s_{1}=(1,-1)$ and $s_{i}=(i-1,i)$ $\forall 2\leq i\leq n$ as in [1].

As in [26] we use a formula for computing the length of $\sigma\in W$ given by

[TABLE]

where $inv(\sigma)=|\{(i,j)\in[-n,n]\backslash\{0\}\times[-n,n]\backslash\{0\}:i<j,\sigma(i)>\sigma(j)\}|$ and $neg(\sigma)=|\{i\in[1,n]:\sigma(i)<0\}|$.

The following result gives a combinatorial characterization of the Bruhat order in $B_{n}$.

Lemma 2.3** ([26], Proposition 2.8).**

Let $\sigma$, $\tau$ $\in W$. Then $\sigma\leq\tau$ in the Bruhat order of $B_{n}$ if and only if $\sigma\leq\tau$ in the Bruhat order of the symmetric group $S_{[-n,n]\backslash\{0\}}$ where $S_{[-n,n]\backslash\{0\}}$ is the permutation group of integers $-n,-(n-1),\ldots,-1,1,\ldots,n-1,n$.

Bruhat order for type $D_{n}$: As above we consider $\alpha_{1}$ as special node for Dynkin diagram for type $D$. For a set of generators of Weyl group we have $S=\{s_{1},s_{2},\ldots,s_{n}\}$, where $s_{1}=(1,-2)(-1,2)$ and $s_{i}=(i-1,i)$ $\forall 2\leq i\leq n$ as in [1].

As in [26] we use a formula for computing the length of $\sigma\in W$ given by

[TABLE]

where $inv(\sigma)$ and $neg(\sigma)$ are as defined above.

The following result gives a combinatorial characterization of the Bruhat order in $D_{n}$.

Lemma 2.4** ([1], Theorem 8.2.8).**

Let $\sigma$, $\tau$ $\in W$. Then $\sigma\leq\tau$ in the Bruhat order of $D_{n}$ if and only if

(i) $\sigma\leq_{B}\tau$ (Bruhat order in type $B$) and

(ii) $\forall a,b\in[1,n]$, if $[-a,a]\times[-b,b]$ is an empty rectangle for both $\sigma$ and $\tau$ and $\sigma[-a-1,b+1]=\tau[-a-1,b+1]$, then $\sigma[-1,b+1]\equiv\tau[-1,b+1]$ (mod $2$) where $\sigma[i,j]=|\{a\in[-n,n]:a\leq i\text{ and }\sigma(a)\geq j\}|$ for $i,j\in[-n,n]$.

3. A necessary condition for admitting semi-stable points

Let $G$ be a simple simply-connected algebraic group and $P_{r}$ be a parabolic subgroup of $G$ corresponding to the simple root $\alpha_{r}$. Let $\mathcal{L}_{r}$ be the line bundle on $G/P_{r}$ corresponding to the fundamental weight $\omega_{r}$. In this section, we provide a criterion for Richardson varieties in $G/P_{r}$ to admit semistable points with respect to $\mathcal{L}_{r}$. This criterion was proved for type $A$ in [10].

Proposition 3.1**.**

Let $G$ be a simple simply connected algebraic group and let $P_{r}$ be the maximal parabolic corresponding to the simple root $\alpha_{r}$. Let $\mathcal{L}_{r}$ be the line bundle on $G/P_{r}$ corresponding to the fundamental weight $\omega_{r}$. Let $v,w\in W^{P_{r}}$. If $(X^{v}_{w})_{T}^{ss}(\mathcal{L}_{r})\neq\emptyset$ then $v(n\omega_{r})\geq 0$ and $w(n\omega_{r})\leq 0$.

Proof.

Let $\chi=n\omega_{r}$. Assume that $(X^{v}_{w})_{T}^{ss}(\mathcal{L}_{\chi})\neq\emptyset$. Let $x\in((BwP_{r}/P_{r})\bigcap(B^{-}vP_{r}/P_{r}))_{T}^{ss}(\mathcal{L}_{\chi})$. Then by Hilbert-Mumford criterion [19, Theorem 2.1], we have $\mu^{\mathcal{L}_{\chi}}(x,\lambda)\geq 0$ for all one parameter subgroups $\lambda$ of $T$. Since $x\in((BwP_{r}/P_{r})\bigcap(B^{-}vP_{r}/P_{r}))_{T}^{ss}(\mathcal{L_{\chi}})$, using [23, Lemma 2.1], we see that $\mu^{\mathcal{L}_{\chi}}(x,\lambda)=-\langle w(\chi),\lambda\rangle$ for every one parameter subgroup $\lambda$ in the fundamental chamber associated to $B$, and $\mu^{\mathcal{L}_{\chi}}(x,\lambda)=\mu^{\mathcal{L}_{\chi}}(w_{0}x,w_{0}\lambda w_{0}^{-1})=-\langle w_{0}v(\chi),w_{0}(\lambda)\rangle=-\langle v(\chi),\lambda\rangle$ for every one parameter subgroup $\lambda$ of $T$ in the Weyl chamber associated to $B^{-}$. Since $x$ is a semistable point, we have $\mu^{\mathcal{L}_{\chi}}(x,\lambda)\geq 0$ for every one parameter subgroup $\lambda$ of $T$. Hence $\langle w(\chi),\lambda\rangle\leq 0$ for all $\lambda$ in the Weyl chamber associated to $B$ and $\langle v(\chi),\lambda\rangle\geq 0$ for all $\lambda$ in the Weyl chamber associated to $B^{-}$. This implies that $w(\chi)\leq 0$ and $v(\chi)\geq 0$. ∎

For $G$ is of type $A$ in [10] it is shown that the above conditions are also sufficient. For type $B,C$ and $D$ the example below shows that the conditions $w(\chi)\leq 0$ and $v(\chi)\geq 0$ are only necessary but not sufficient.

Example: Let $G$ be either of type $B_{4}$ or $C_{4}$ and $\chi=\omega_{3}$. Let $v=(1,2,-3,4)=s_{3}s_{2}s_{1}s_{2}s_{3}$ and $w=(1,4,-3,2)=s_{3}s_{4}s_{2}s_{1}s_{2}s_{3}$. We have $v(\omega_{3})=\alpha_{4}$ and $w(\omega_{3})=-\alpha_{3}$. The sections of $\mathcal{L}_{\chi}$ on $X^{v}_{w}$ are of the form $p_{v}^{m}p_{w}^{n}$ where $m,n\in\mathbb{N}$. But, $mv(\chi)+nw(\chi)\neq 0$ for any $m,n\in\mathbb{N}$. So these sections are not $T$-invariant. So the set $(X^{v}_{w})_{T}^{ss}(\mathcal{L}_{\chi})$ is empty.

If $G$ is of type $D_{4}$ and $\chi=\omega_{3}$, we take $v=(-1,4,-2,3)=s_{4}s_{1}s_{2}s_{3}$ and $w=(-1,2,-4,3)=s_{4}s_{3}s_{1}s_{2}s_{3}$. Here we have $v(\omega_{3})=\alpha_{3}$ and $w(\omega_{3})=-\alpha_{4}$. As in the last paragraph, here also we conlude that the set $(X^{v}_{w})_{T}^{ss}(\mathcal{L}_{\chi})$ is empty.

In order to find a sufficient condition for the Richardson varieties to admit semistable points we first need to classify all $v,w\in W^{P}$ satisfying the above conditions. Since $\chi$ is a dominant weight we have $w_{1}(\chi)\leq w_{2}(\chi)$ for $w_{1}\geq w_{2}$. So we just need to describe all maximal $v$ and minimal $w$ such that $v(\chi)\geq 0$ and $w(\chi)\leq 0$. Note that for $G$ is of type $A$ since all the fundamental weights are minuscule the maximal $v$ and minimal $w$ satisfying the above conditions are unique (see [11]) but for other types this is not the case.

We conclude this section by introducing some notation here:

Notation: For $s,t\in\mathbb{Z}$ such that $s\leq t$ we set $[s,t]=\{s,s+1,\ldots,t\}$. For $p\in\mathbb{N}$ we set $J_{p,[s,t]}=\{(i_{1},i_{2},\ldots i_{p}):i_{k}\in[s,t],\forall k$ and $i_{k+1}-i_{k}\geq 2\}$. For $\underline{i}\in J_{p,[s,t]}$ we set $[\underline{i}]=\{i_{1},i_{2},\ldots,i_{p}\}$ and $-\underline{i}=(-i_{p},-i_{p-1},\ldots,-i_{1})\in J_{p,[-t,-s]}$. For a set $S\subset\mathbb{Z}$, $S\uparrow$ denotes the integers in the set $S$ occurring in increasing order and $S\downarrow$ denotes the integers in the set $S$ occurring in decreasing order.

4. Type B and C

Now for $G$ is of type $B$ and $C$ and for a fundamental weight $\omega_{r}$, we are in a position to describe all the minimal $w\in W^{I_{r}}$ and maximal $v\in W^{I_{r}}$ such that $v(\omega_{r})\geq 0$ and $w(\omega_{r})\leq 0$.

Proposition 4.1**.**

The set of all maximal $v$ in $W^{I_{r}}$ such that $v(\omega_{r})\geq 0$ for type $B_{n}$ and $C_{n}$ are the following:

(i) Let $r=1$. Then

[TABLE]

(ii) Let $2\leq r\leq n-1$ and $(n+1)-r=2m$. For any $\underline{i}=(i_{1},i_{2},\ldots,i_{m})\in J_{m,[2,n]}$, there exists unique maximal $v_{\underline{i}}\in W^{I_{r}}$ such that $v_{\underline{i}}(\omega_{r})=(\sum_{k=1}^{m}\alpha_{i_{k}})$. We have $v_{\underline{i}}=([1,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},\underline{i})$, where $\underline{i^{\prime}}=(i_{1}-1,i_{2}-1,\ldots,i_{m}-1)\in J_{m,[1,n-1]}$.

(iii) Let $2\leq r\leq n-1$ and $(n+1)-r=2m+1$. For any $\underline{i}=(i_{1},i_{2},\ldots,i_{m})\in J_{m,[3,n]}$, there exists unique maximal $v_{\underline{i}}\in W^{I_{r}}$ such that $v_{\underline{i}}(\omega_{r})=(\alpha_{1}+\sum_{k=1}^{m}\alpha_{i_{k}})$ (for $B_{n}$) and $v_{\underline{i}}(\omega_{r})=(\frac{1}{2}\alpha_{1}+\sum_{k=1}^{m}\alpha_{i_{k}})$ (for $C_{n}$). We have $v_{\underline{i}}=([1,n]\backslash\{1,[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},1,\underline{i})$, where $\underline{i^{\prime}}=(i_{1}-1,i_{2}-1,\ldots,i_{m}-1)\in J_{m,[2,n-1]}$.

(iv) Let $r=n$. Then $v=(2,3,4,\ldots,n-1,n,1)$.

We prove this proposition after proving the following lemma.

Lemma 4.2**.**

Let $v$, $v_{\underline{i}}$ are as defined in Proposition 4.1.

(i) Let $r=1$. Then $w>v$ and $l(w)=l(v)+1$ iff $w=s_{k}v$ where $k$ takes the following values \left\{\begin{array}[]{lr}\ k\in\{2,4,6,\ldots,n-2,n\},&{n\,\,is\,\,even}\\ k\in\{1,3,5,\ldots,n-2,n\},&{n\,\,is\,\,odd}.\end{array}\right.

(ii) Let $2\leq r\leq n-1$. Then $w>v_{\underline{i}}$ and $l(w)=l(v_{\underline{i}})+1$ if and only if $w$ is either $s_{\alpha_{i_{t}}}v_{\underline{i}}$ or $s_{\alpha_{i_{t}}+\alpha_{{i_{t}}+1}}v_{\underline{i}}$ for some $i_{t}$ such that $|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{{i_{t}}-1}}v_{\underline{i}}$ for some $i_{t}$ such that $|i_{t}-i_{t-1}|\geq 3$.

(iii) Let $r=n$. Then $w>v$ and $l(w)=l(v)+1$ iff $w=s_{1}v$.

Proof.

We will prove this lemma for case (ii) and $(n+1)-r=2m$. For other cases the proof is similar.

Let $w>v_{\underline{i}}$ and $l(w)=l(v_{\underline{i}})+1$. Then $w=s_{\beta}v_{\underline{i}}$, for some positive root $\beta$ such that height of $\beta$ is less than or equal to $2$. So, $\beta$ is either $\alpha_{j}$ or $\alpha_{j}+\alpha_{j+1}$ for some simple root $\alpha_{j}$.

Case 1: $\beta=\alpha_{j}$.

If $\beta=\alpha_{i_{t}}$ then $s_{\beta}v_{\underline{i}}(\omega_{r})=\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t}}$. Since $s_{\beta}v_{\underline{i}}(\omega_{r})<v_{\underline{i}}(\omega_{r})$ and $\omega_{r}$ is a dominant weight we have $s_{\beta}v_{\underline{i}}>v_{\underline{i}}$. Since $\beta$ is a simple root we have $l(s_{\beta}v_{\underline{i}})=l(v_{\underline{i}})+1$.

If $\beta=\alpha_{j},j\neq i_{t}$, then $s_{\beta}v_{\underline{i}}(\omega_{r})\geq v_{\underline{i}}(\omega_{r})$. So $s_{\beta}v_{\underline{i}}\leq v_{\underline{i}}$ in $W^{I_{r}}$, a contradiction.

Case 2: $\beta=\alpha_{j}+\alpha_{j+1}$.

If $j=i_{t}$ and $|i_{t+1}-i_{t}|\geq 3$ then $s_{\beta}v_{\underline{i}}(\omega_{r})=\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t}+1}$. So $s_{\beta}v_{\underline{i}}(\omega_{r})<v_{\underline{i}}(\omega_{r})$ and hence $s_{\beta}v_{\underline{i}}>v_{\underline{i}}$. Now we will show that $l(s_{\beta}v_{\underline{i}})=l(v_{\underline{i}})+1$ for this $\beta$.

Note that $s_{\beta}v_{\underline{i}}=(-\underline{i},\hat{\underline{i^{\prime}}},[-n,-1]\backslash\{[-\underline{i}],[-\hat{\underline{i^{\prime}}}]\}\uparrow,[1,n]\backslash\{[\underline{i}],[\hat{\underline{i^{\prime}}}]\}\uparrow,-\hat{\underline{i^{\prime}}},\underline{i})$ where $\hat{\underline{i^{\prime}}}=(i_{1}-1,i_{2}-1,\ldots,i_{t-1}-1,i_{t}+1,i_{t+1}-1\ldots,i_{m}-1)$. In $v_{\underline{i}}$, the position of $i_{t}+1$ is right to the position of $i_{t}-1$ and left to $i_{t}$ but in $s_{\beta}v_{\underline{i}}$, the position of $i_{t}$ remains unchanged and the positions of $i_{t}+1$ and $i_{t}-1$ are interchanged. Similarly in $v_{\underline{i}}$ the position of $-i_{t}-1$ is right to $-i_{t}$ and left to $-i_{t}+1$ but in $s_{\beta}v_{\underline{i}}$ the position of $-i_{t}$ remains unchanged and the positions of $-i_{t}-1$ and $-i_{t}+1$ are interchanged. So $inv(s_{\beta}v_{\underline{i}})=inv(v_{\underline{i}})+2$. Hence $l(s_{\beta}v_{\underline{i}})=l(v_{\underline{i}})+1$.

If $j=i_{t}$ and $|i_{t+1}-i_{t}|=2$ then since $s_{\beta}v_{\underline{i}}(\omega_{r})=v_{\underline{i}}(\omega_{r})$, we have $s_{\beta}v_{\underline{i}}=v_{\underline{i}}$ in $W^{I_{r}}$, a contradiction.

If $j,j+1\neq i_{t}$ then $s_{\beta}v_{\underline{i}}(\omega_{r})\geq v_{\underline{i}}(\omega_{r})$. So $s_{\beta}v_{\underline{i}}\leq v_{\underline{i}}$ in $W^{I_{r}}$, a contradiction.

Case 3. $\beta=\alpha_{j-1}+\alpha_{j}$. In this case the proof is similar to the previous case.

The converse part is clear from the definition of $v_{\underline{i}}$. ∎

Proof of proposition $4.1$: We prove case (ii). The proofs of other cases are similar.

We prove that for any $\underline{i}\in J_{m,[2,n]}$ there exists $v_{\underline{i}}\in W^{I_{r}}$ such that $v_{\underline{i}}(\omega_{r})=\sum_{k=1}^{m}\alpha_{i_{k}}$.

Note that,

[TABLE]

Now consider the partial order on $J_{m,[2,n]}$, given by $(i_{1},i_{2},\ldots,i_{m})\leq(j_{1},j_{2},\ldots,j_{m})$ if $i_{k}\leq j_{k}$, $\forall k$ and $(i_{1},i_{2},\ldots,i_{m})<(j_{1},j_{2},\ldots,j_{m})$ if $i_{k}\leq j_{k}$ $\forall k$ and $i_{k}<j_{k}$ for some $k$. We will prove the theorem by induction on this order.

For $(j_{1},j_{2},\ldots,j_{m})=(n-(2m-2),n-(2m-4),\ldots,n-2,n)$, we have $v_{\underline{j}}(\omega_{r})=([1,n]\backslash\{[\underline{j}],[\underline{j^{\prime}}]\},\underline{j},\underline{j^{\prime}})(\omega_{r})=\displaystyle\sum_{t=1}^{m}\alpha_{n-2m+2t}$ where $\underline{j}=(j_{1},j_{2},\ldots,j_{m})$ and $\underline{j^{\prime}}=(j_{1}-1,j_{2}-1,\ldots,j_{m}-1)$. For $(i_{1},i_{2},\ldots,i_{m})\in J_{m,[2,n]}$ not maximal, there exists $t$ maximal such that $i_{t}<n-2m+2t$. Now $(i_{1},i_{2},\ldots,i_{t-1},i_{t}+1,i_{t+1},\ldots,i_{m})\in J_{m,[2,n]}$ and $(i_{1},i_{2},\ldots,i_{t-1},i_{t}+1,i_{t+1},\ldots,i_{m})>(i_{1},i_{2},\ldots,i_{m})$. So by induction, there exists $w\in W^{I_{r}}$ such that $w(\omega_{r})=\sum_{k\neq t}\alpha_{i_{k}}+\alpha_{1+i_{t}}$. Now $s_{1+i_{t}}s_{i_{t}}w(\omega_{r})=\sum_{k=1}^{m}\alpha_{i_{k}}$. So for any $(i_{1},i_{2},\ldots,i_{m})\in J_{m,[2,n]}$, there exists $v_{\underline{i}}\in W^{I_{r}}$ such that $v_{\underline{i}}\omega_{r}=\sum_{k=1}^{m}\alpha_{i_{k}}$.

For $\underline{i}\in J_{m,[2,n]}$, if there exists another $u_{\underline{i}}\in W^{I_{r}}$ such that $u_{\underline{i}}(\omega_{r})=v_{\underline{i}}(\omega_{r})$ we have $u_{\underline{i}}=v_{\underline{i}}$ in $W^{I_{r}}$. This gives the uniqueness of $v_{\underline{i}}$.

Now we will prove that the $v_{\underline{i}}$’s in $W^{I_{r}}$ having this property are maximal.

If $v_{\underline{i}}$ is not maximal, then there exists $\beta$ $\in$ $R^{+}$ such that $s_{\beta}v_{\underline{i}}>v_{\underline{i}}$ with $s_{\beta}v_{\underline{i}}(\omega_{r})\geq 0$. We may assume that $l(s_{\beta}v_{\underline{i}})=l(v_{\underline{i}})+1$. So by Lemma 4.2, $\beta$ is either $\alpha_{i_{t}}$ or $\alpha_{i_{t}}+\alpha_{{i_{t}}+1}$ or $\alpha_{i_{t}}+\alpha_{{i_{t}}-1}$.

If $\beta=\alpha_{i_{t}}$, we have $s_{\beta}v_{\underline{i}}(\omega_{r})=\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t}}$ and if $\beta=\alpha_{i_{t}}+\alpha_{i_{t}+1}$, we have $s_{\beta}v_{\underline{i}}(\omega_{r})=\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t+1}},$ for $|i_{t}-i_{t+1}|\geq 3$. Similarly, if $\beta=\alpha_{i_{t}}+\alpha_{i_{t}-1}$, we have $s_{\beta}v_{\underline{i}}(\omega_{r})=\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t-1}},$ for $|i_{t}-i_{t-1}|\geq 3$.

So in all these cases, we see that $s_{\beta}v_{\underline{i}}$ is not greater than 0, a contradiction. Thus all $v_{\underline{i}}$’s are maximal having the property that $v_{\underline{i}}(\omega_{r})\geq 0$.

It remains to show that above listed $v$’s are the only maximal elements having the property that $v(\omega_{r})\geq 0$. Let $\lambda=\sum_{t=1}^{m}\alpha_{i_{t}}$ be in the weight lattice such that $\langle\alpha_{i_{k}},\alpha_{i_{k}+1}\rangle\neq 0$ for some $k$. Let $w\in W^{I_{r}}$ be such that $w(\omega_{r})=\lambda$. Note that $0\leq s_{i_{k}}w(\omega_{r})=\sum_{j\neq k}\alpha_{i_{j}}<\lambda$. Hence $s_{i_{k}}w>w$. This implies that $w$ is not maximal having the property that $w(\omega_{r})\geq 0$.

Proposition 4.3**.**

The set of all minimal $w$ in $W^{I_{r}}$ such that $w(\omega_{r})\leq 0$ for type $B_{n}$ and $C_{n}$ are the following:

(i) Let $r=1$. Then

[TABLE]

(ii) Let $2\leq r\leq n-1$. For $(n+1)-r=2m$, $\underline{i}\in J_{m,[2,n]}$ and $v_{\underline{i}}(\omega_{r})=\sum_{k=1}^{m}\alpha_{i_{k}}$, we have $w_{\underline{i}}=s_{i_{1}}s_{i_{2}}\ldots s_{i_{m}}v_{\underline{i}}=([1,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},\underline{i^{\prime}})$ where $\underline{i^{\prime}}=(i_{1}-1,i_{2}-1,\ldots,i_{m}-1)$. In this case $w_{\underline{i}}(\omega_{r})=-v_{\underline{i}}(\omega_{r})$.

(iii) Let $2\leq r\leq n-1$. For $(n+1)-r=2m+1$, $\underline{i}\in J_{m,[3,n]}$ and $v_{\underline{i}}(\omega_{r})=\alpha_{1}+\sum_{k=1}^{m}\alpha_{i_{k}}$, we have $w_{\underline{i}}=s_{1}s_{i_{1}}s_{i_{2}}\ldots s_{i_{m}}v_{\underline{i}}=([1,n]\backslash\{1,[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-1,\underline{i^{\prime}})$, where $\underline{i^{\prime}}=(i_{1}-1,i_{2}-1,\ldots,i_{m}-1)\in J_{m,[2,n-1]}$. In this case also $w_{\underline{i}}(\omega_{r})=-v_{\underline{i}}(\omega_{r})$.

(iv) For $r=n$, $w=(2,3,\ldots,n-1,n,-1)$.

Proof.

For the proof of minimality of $w$ refer to [12]. ∎

Proposition 4.4**.**

Let $v$, $w$, $v_{\underline{i}}$, and $w_{\underline{j}}$ be as stated in Proposition $4.1$ and Proposition $4.3$ respectively.

(i) For $2\leq r\leq n-1$, $X^{v_{\underline{i}}}_{w_{\underline{j}}}$ is nonempty if and only if $|i_{k}-j_{k}|\leq 1$ $\forall 1\leq k\leq m$.

(ii) For $r=1,n$, $X^{v}_{w}$ is non-empty for any $v$ and $w$.

Proof.

We will prove this lemma for $2\leq r\leq n-1$ and $(n+1)-r=2m$. For other cases the proof is similar.

Let $X^{v_{\underline{i}}}_{w_{\underline{j}}}$ be nonempty. So $v_{\underline{i}}<w_{\underline{j}}$. Since $l(v_{\underline{i}})=l(v_{\underline{j}})$ we have $l(w_{\underline{j}})-l(v_{\underline{i}})=m$. By the repeated application of Lemma 4.2 we have $w_{\underline{j}}=\displaystyle\prod_{\mathbf{card}\{\beta\}=m}s_{\beta}v_{\underline{i}}$, for some $\beta$ such that $\beta$’s is either $\alpha_{i_{t}}$ or $\alpha_{i_{t}}+\alpha_{{i_{t}}+1}$ or $\alpha_{i_{t}}+\alpha_{{i_{t}}-1}$ for some $i_{t}$. So $w_{\underline{j}}(\omega_{r})=-\displaystyle\sum_{{j_{k}}:|i_{k}-j_{k}|\leq 1}\alpha_{j_{k}}$. Hence, $|i_{k}-j_{k}|\leq 1$ $\forall 1\leq k\leq m$.

Conversely, let $|i_{k}-j_{k}|\leq 1$ for all $1\leq k\leq m$. So, $w_{\underline{j}}=\displaystyle\prod_{t\in\{t:i_{t}=j_{t}\}}s_{\alpha_{i_{t}}}$ $\displaystyle\prod_{t\in\{t:j_{t}=i_{t}-1\}\downarrow}s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}$ $\displaystyle\prod_{t\in\{t:j_{t}=i_{t}+1\}\uparrow}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}$ $v_{\underline{i}}$. The arrows $\uparrow$ and $\downarrow$ denote that the reflections in the product are applied in increasing and decreasing order of $t$ respectively. Since $\underline{i},\underline{j}\in J_{m,[2,n]}$ and $|i_{k}-j_{k}|\leq 1$, we see that the sets $\{\bigcup\limits_{\{t:j_{t}=i_{t}+1\}}\{i_{t}-1,i_{t},i_{t}+1\}\}$, $\{\bigcup\limits_{\{t:j_{t}=i_{t}-1\}}\{i_{t}-2,i_{t}-1,i_{t}\}\}$ and $\{\bigcup\limits_{\{t:j_{t}=i_{t}\}}\{i_{t}-1,i_{t}\}\}$ are mutually disjoint.

In the first step we see that when we multiply $v_{\underline{i}}$ by $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}$ in increasing order of $t\in\{t:j_{t}=i_{t}+1\}$ then after each multiplication the product is greater than $v_{\underline{i}}$ and the length increases by $1$. Let $t$ be maximal such that $j_{t}=i_{t}+1$. By lemma $4.2$, $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}>v_{\underline{i}}$ and $l(s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}})=l(v_{\underline{i}})+1$.

Let $t$ be such that $i_{t-1}=i_{t}-2$ and $j_{t-1}=i_{t-1}+1$. Note that $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}=(-\underline{i},\hat{\underline{i^{\prime}}},[-n,-1]\backslash\{[-\underline{i}],[-\hat{\underline{i^{\prime}}}]\}\uparrow,[1,n]\backslash\{[\underline{i}],[\hat{\underline{i^{\prime}}}]\}\uparrow,-\hat{\underline{i^{\prime}}},\underline{i})$, where $\hat{\underline{i^{\prime}}}=(i_{1}-1,i_{2}-1,\ldots,i_{t}-3,i_{t}+1,i_{t+1}-1\ldots,i_{m}-1)$ and $s_{\alpha_{i_{t}-2}+\alpha_{i_{t}-1}}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}=(-\underline{i},\hat{\hat{\underline{i}^{\prime}}},[-n,-1]\backslash\{[-\underline{i}],[-\hat{\hat{\underline{i}^{\prime}}}]\}\uparrow,[1,n]\backslash\{[\underline{i}],[\hat{\hat{\underline{i}^{\prime}}}]\}\uparrow,-\hat{\hat{\underline{i}^{\prime}}},\underline{i})$, where $\hat{\hat{\underline{i}^{\prime}}}=(i_{1}-1,i_{2}-1,\ldots,i_{t}-1,i_{t}+1,i_{t+1}-1\ldots,i_{m}-1)$. In $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}$, the position of $i_{t}-3$ is left to the positions of both $i_{t}-1$ and $i_{t}-2$ but in $s_{\alpha_{i_{t}-2}+\alpha_{i_{t}-1}}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}$, the position of $i_{t}-2$ remains unchanged and the positions of $i_{t}-1$ and $i_{t}-3$ are interchanged. Similarly the positions of $-i_{t}+1$ and $-i_{t}+3$ are also interchanged in $s_{\alpha_{i_{t}-2}+\alpha_{i_{t}-1}}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}$. So $inv(s_{\alpha_{i_{t}-2}+\alpha_{i_{t}-1}}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}})=inv(s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}})+2$. Hence $l(s_{\alpha_{i_{t}-2}+\alpha_{i_{t}-1}}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}})=l(s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}})+1$. By lemma $2.3$ we see that $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}<s_{\alpha_{i_{t}-2}+\alpha_{i_{t}-1}}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}$.

Repeating this process we can see that $\displaystyle\prod_{t\in\{t:j_{t}=i_{t}+1\}\uparrow}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}>v_{\underline{i}}$ and the length is increased by the number of reflections multiplied.

Since $\{\bigcup\limits_{\{t:j_{t}=i_{t}+1\}}\{i_{t}-1,i_{t},i_{t}+1\}\},\{\bigcup\limits_{\{t:j_{t}=i_{t}-1\}}\{i_{t}-2,i_{t}-1,i_{t}\}\}$ and $\{\bigcup\limits_{\{t:j_{t}=i_{t}\}}\{i_{t}-1,i_{t}\}\}$ are mutually disjoint, by repeating the above process we see that $\displaystyle\prod_{\{{t:j_{t}=i_{t}}\}}s_{\alpha_{i_{t}}}\prod_{t\in\{t:j_{t}=i_{t}-1\}\downarrow}s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}\\ \prod_{t\in\{t:j_{t}=i_{t}+1\}\uparrow}s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}>v_{\underline{i}}$ and the length is increased by the number of reflections multiplied. So $w_{\underline{j}}>v_{\underline{i}}$ and hence $X^{v_{\underline{i}}}_{w_{\underline{j}}}$ is nonempty. ∎

Remark: Note that $\underline{i}$ denotes the positions of the simple roots with nonzero coefficients in $v_{\underline{i}}(\omega_{r})$ and similarly, $\underline{j}$ denotes the positions of the simple roots with nonzero coefficients in $w_{\underline{j}}(\omega_{r})$.

Theorem 4.5**.**

Let $v$, $w$, $v_{\underline{i}}$ and $w_{\underline{j}}$ be as stated in Proposition $4.1$ and Proposition $4.3$ respectively.

(i) For $2\leq r\leq n-1$, $(X{{}^{v_{\underline{i}}}_{w_{\underline{j}}}})^{ss}_{T}(\mathcal{L}_{r})$ is nonempty if and only if $\underline{i}=\underline{j}$.

(ii) For $r=1$ and $n$, $(X^{v}_{w})^{ss}_{T}$ is non-empty for any $v$ and $w$.

Proof.

We will prove this theorem for case (i). Proof of case (ii) is similar. Let $\underline{i}=\underline{j}$. Then we have $v_{\underline{i}}(\omega_{r})$+$w_{\underline{j}}(\omega_{r})=0$. So $p_{v_{\underline{i}}}p_{w_{\underline{j}}}$ is a non-zero $T$-invariant section of $\mathcal{L}_{r}$ on $G/P_{r}$ which does not vanish identically on $X^{v_{\underline{i}}}_{w_{\underline{j}}}$. Hence, $(X^{v_{\underline{i}}}_{w_{\underline{j}}})^{ss}_{T}(\mathcal{L}_{r})$ is non-empty.

Conversely, if $\underline{i}\neq\underline{j}$, then there exists $t$ such that $j_{t}\neq i_{t}$. Since $X^{v_{\underline{i}}}_{w_{\underline{j}}}\neq\emptyset$, by Proposition $4.4$ we have $j_{t}=i_{t}+1$ or $j_{t}=i_{t}-1$. If $j_{t}=i_{t}+1$ then $w_{\underline{j}}(\omega_{r})$ = $-\sum_{k\neq t}{\alpha_{i_{k}}}-\alpha_{i_{t}+1}$ and if $j_{t}=i_{t}-1$ then $w_{\underline{j}}(\omega_{r})$ = $-\sum_{k\neq t}{\alpha_{i_{k}}}-\alpha_{i_{t}-1}$. Let $u\in W^{I_{r}}$ be such that $v_{\underline{i}}\leq u\leq w_{\underline{j}}$. Then $u$ is of the form $u=(\displaystyle\prod_{\beta}s_{\beta})v_{\underline{i}}$, where $\beta$’s are some positive roots. For $j_{t}=i_{t}+1$ at most one $\beta$ can be $\alpha_{i_{t}}+\alpha_{i_{t}+1}$ and none of the other $\beta$’s can contain $\alpha_{i_{t}}$ or $\alpha_{i_{t}+1}$ as a summand. So in $u(\omega_{r})$, the coefficient of $\alpha_{i_{t}}$ is either zero or one and the coefficient of $\alpha_{i_{t}+1}$ is either zero or $-1$. Similarly for $j_{t}=i_{t}-1$ at most one $\beta$ can be $\alpha_{i_{t}}+\alpha_{i_{t}-1}$ and none of the other $\beta$’s can contain $\alpha_{i_{t}}$ or $\alpha_{i_{t}-1}$ as a summand. So in $u(\omega_{r})$ the coefficient of $\alpha_{i_{t}}$ is either zero or one and the coefficient of $\alpha_{i_{t}-1}$ is either zero or $-1$. For $j_{t}=i_{t}+1$, $u(\omega_{r})$ contains either $\alpha_{i_{t}}$ or $\alpha_{i_{t}+1}$ as a summand and for $j_{t}=i_{t}-1$, $u(\omega_{r})$ contains either $\alpha_{i_{t}}$ or $\alpha_{i_{t}-1}$ as a summand. Hence there does not exist a sequence $v_{\underline{i}}=u_{1}\leq u_{2}\leq\ldots\leq u_{k}=w_{\underline{j}}$ such that $\sum_{l=1}^{k}u_{l}(\omega_{r})=0$ and so we don’t have a non-zero $T$- invariant section of $\mathcal{L}_{r}$ which is not identically zero on $X^{v_{\underline{i}}}_{w_{\underline{j}}}$. So, we conclude that the set $(X_{w_{\underline{j}}}^{v_{\underline{i}}})^{ss}_{T}(\mathcal{L}_{r})$ is empty. ∎

We illustrate Proposition $4.4$ and Theorem $4.5$ with an example.

Example: $B_{5}$, $\omega_{4}=(2,2,2,2,1)$

$\underline{i}$                         $v_{\underline{i}}$                     $v_{\underline{i}}(\omega_{2})$                   $w_{\underline{i}}(\omega_{2})$                   $w_{\underline{i}}$

(2)               $(3,4,5,-1,2)$       $(0,1,0,0,0)$           $(0,-1,0,0,0)$       $(3,4,5,-2,1)$

(3)               $(1,4,5,-2,3)$       $(0,0,1,0,0)$           $(0,0,-1,0,0)$       $(1,4,5,-3,2)$

(4)               $(1,2,5,-3,4)$       $(0,0,0,1,0)$           $(0,0,0,-1,0)$       $(1,2,5,-4,3)$

(5)               $(1,2,3,-4,5)$       $(0,0,0,0,1)$           $(0,0,0,0,-1)$       $(1,2,3,-5,4)$

So from the above observation, $X_{w_{(2)}}^{v_{(2)}}$, $X_{w_{(2)}}^{v_{(3)}}$, $X_{w_{(3)}}^{v_{(2)}}$, $X_{w_{(3)}}^{v_{(3)}}$, $X_{w_{(3)}}^{v_{(4)}}$, $X_{w_{(4)}}^{v_{(3)}}$ and $X_{w_{(4)}}^{v_{(4)}}$, $X_{w_{(4)}}^{v_{(5)}}$, $X_{w_{(5)}}^{v_{(4)}}$, $X_{w_{(5)}}^{v_{(5)}}$ are all non-empty. We have $(X_{w_{(2)}}^{v_{(3)}})^{ss}_{T}(\mathcal{L}_{4})$, $(X_{w_{(3)}}^{v_{(2)}})^{ss}_{T}(\mathcal{L}_{4})$, $(X_{w_{(3)}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{4})$, $(X_{w_{(4)}}^{v_{(3)}})^{ss}_{T}(\mathcal{L}_{4})$, $(X_{w_{(4)}}^{v_{(5)}})^{ss}_{T}(\mathcal{L}_{4})$ and $(X_{w_{(5)}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{4})$ are empty whereas $(X_{w_{(2)}}^{v_{(2)}})^{ss}_{T}(\mathcal{L}_{4})$, $(X_{w_{(3)}}^{v_{(3)}})^{ss}_{T}(\mathcal{L}_{4})$, $(X_{w_{(4)}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{4})$ and $(X_{w_{(5)}}^{v_{(5)}})^{ss}_{T}(\mathcal{L}_{4})$ are non-empty.

5. Type D

As in types $B$ and $C$ here also for a fundamental weight $\omega_{r}$ we describe all the maximal $v\in W^{I_{r}}$ and minimal $w\in W^{I_{r}}$ such that $v(\omega_{r})\geq 0$ and $w(\omega_{r})\leq 0$ and then we use the same techniques to describe $v,w\in W^{I_{r}}$ for which the Richardson variety $X_{w}^{v}$ have nonempty semistable locus for the action of a maximal torus $T$ and with respect to the line bundle $\mathcal{L}_{r}$.

Proposition 5.1**.**

Let $G$ be of type $D_{n}$. Let $v\in W^{I_{r}}$ be maximal such that $v(\omega_{r})\geq 0$. Then the description of $v$ is the following:

(i) For $r=1$,    v(4\omega_{1})=\left\{\begin{array}[]{lr}\ 2\alpha_{2}+2\sum_{i=2}^{\frac{n}{2}}\alpha_{2i},&n\equiv 0(mod\,\,4)\\ 2\alpha_{1}+2\sum_{i=2}^{\frac{n}{2}}\alpha_{2i},&n\equiv 2(mod\,\,4)\\ \alpha_{1}+3\alpha_{2}+2\alpha_{3}+2\sum_{i=2}^{\frac{n-1}{2}}\alpha_{2i+1},&n\equiv 1(mod\,\,4)\\ 3\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\sum_{i=2}^{\frac{n-1}{2}}\alpha_{2i+1},&n\equiv 3(mod\,\,4).\end{array}\right.

where v=\left\{\begin{array}[]{lr}\ (-(n-1),-(n-3),\ldots,-3,-1,2,4,6,\ldots,n-2,n),&n\equiv 0(mod\,\,4)\\ (-(n-1),-(n-3),\ldots,-3,1,2,4,6,\ldots,n-2,n),&n\equiv 2(mod\,\,4)\\ (-(n-1),-(n-3),\ldots,-4,-1,2,3,5,\ldots,n-2,n),&n\equiv 1(mod\,\,4)\\ (-(n-1),-(n-3),\ldots,-4,1,2,3,5,\ldots,n-2,n),&n\equiv 3(mod\,\,4).\end{array}\right.

*(ii) For $r=2$, v(4\omega_{2})=\left\{\begin{array}[]{lr}\ 2\alpha_{1}+2\sum_{i=2}^{\frac{n}{2}}\alpha_{2i},&n\equiv 0(mod\,\,4)\\ 2\alpha_{2}+2\sum_{i=2}^{\frac{n}{2}}\alpha_{2i},&n\equiv 2(mod\,\,4)\\ 3\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\sum_{i=2}^{\frac{n-1}{2}}\alpha_{2i+1},&n\equiv 1(mod\,\,4)\\ \alpha_{1}+3\alpha_{2}+2\alpha_{3}+2\sum_{i=2}^{\frac{n-1}{2}}\alpha_{2i+1},&n\equiv 3(mod\,\,4),\end{array}\right. and

where v=\left\{\begin{array}[]{lr}\ ((n-1),-(n-3),\ldots,-3,1,2,4,6,\ldots,n-2,n),&n\equiv 0(mod\,\,4)\\ ((n-1),-(n-3),\ldots,-3,-1,2,4,6,\ldots,n-2,n),&n\equiv 2(mod\,\,4)\\ ((n-1),-(n-3),\ldots,-4,1,2,3,5,\ldots,n-2,n),&n\equiv 1(mod\,\,4)\\ ((n-1),-(n-3),\ldots,-4,-1,2,3,5,\ldots,n-2,n),&n\equiv 3(mod\,\,4).\end{array}\right.

(iii) Let $3\leq r\leq n-1$. For $(n+1)-r=2m$ and for any $\underline{i}=(i_{1},i_{2},\ldots,i_{m})\in J_{m,[1,n]}\backslash Z$, there exists an unique $v_{\underline{i}}\in W^{I_{r}}$ such that $v_{\underline{i}}(\omega_{r})=\displaystyle\sum_{k=1}^{m}\alpha_{i_{k}}$, where $Z=\{(1,3,i_{1},i_{2},\ldots,i_{m-2}):i_{k}\in\{5,\ldots,n-1,n\}$ and $i_{k+1}-i_{k}\geq 2,\forall k\}$.

(a) For $\underline{i}\in J_{m,[3,n]}$, $v_{\underline{i}}(\omega_{r})=\displaystyle\sum_{k=1}^{m}\alpha_{i_{k}}$, v_{\underline{i}}=\left\{\begin{array}[]{lr}\ (-1,[2,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},\underline{i}),&\text{m is odd}\\ (1,[2,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},\underline{i}),&\text{m is even}.\end{array}\right.

*(b) For $\underline{i}\in J_{m-1,[4,n]}$, $v_{1,\underline{i}}(\omega_{r})=\alpha_{1}+\displaystyle\sum_{k=1}^{m-1}\alpha_{i_{k}}$, with

v_{1,\underline{i}}=\left\{\begin{array}[]{lr}\ ([3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},1,2,\underline{i}),&\text{m is odd}\\ (-t,[3,n]\backslash\{t,[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},1,2,\underline{i}),&\text{m is even}.\end{array}\right. where $t=min\{[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\}$.*

*(c) For $\underline{i}\in J_{m-1,[4,n]}$, $v_{2,\underline{i}}(\omega_{r})=\alpha_{2}+\displaystyle\sum_{k=1}^{m-1}\alpha_{i_{k}}$ with

v_{2,\underline{i}}=\left\{\begin{array}[]{lr}\ (-t,[3,n]\backslash\{t,[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},-1,2,\underline{i}),&\text{m is odd}\\ ([3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},-1,2,\underline{i}),&\text{m is even},\end{array}\right. where $t=min\{[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\}.$*

(iv) Let $3\leq r\leq n-1$. For $(n+1)-r=2m+1$ and for any $\underline{i}=(i_{1},i_{2},\ldots,i_{m})\in J_{m,[4,n]}$, there exists unique $v_{\underline{i}}\in W^{I_{r}}$ such that $v_{\underline{i}}(\omega_{r})=\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\displaystyle\sum_{k=1}^{m}\alpha_{i_{k}}$. Also, for any $\underline{i}=(i_{1},i_{2},\ldots,i_{m-1})\in J_{m-1,[5,n]}$, there exists unique $v_{\underline{i},1}\in W^{I_{r}}$ such that $v_{\underline{i},1}(\omega_{r})=\frac{3}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{3}+\displaystyle\sum_{k=1}^{m-1}\alpha_{i_{k}}$ and there exists unique $v_{\underline{i},2}\in W^{I_{r}}$ such that $v_{\underline{i},2}(\omega_{r})=\frac{1}{2}\alpha_{1}+\frac{3}{2}\alpha_{2}+\alpha_{3}+\displaystyle\sum_{k=1}^{m-1}\alpha_{i_{k}}$. We have:

(a) v_{\underline{i}}=\left\{\begin{array}[]{lr}\ (-1,[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},2,\underline{i}),&\text{m is odd}\\ (1,[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},2,\underline{i}),&\text{m is even}.\end{array}\right.

(b) v_{\underline{i},1}=\left\{\begin{array}[]{lr}\ (4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},1,2,3,\underline{i}),&\text{m is odd}\\ (-4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},1,2,3,\underline{i}),&\text{m is even}.\end{array}\right.

(c) v_{\underline{i},2}=\left\{\begin{array}[]{lr}\ (-4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},-1,2,3,\underline{i}),&\text{m is odd}\\ (4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},-1,2,3,\underline{i}),&\text{m is even}.\end{array}\right.

(v) For $r=n$, $v=(1,3,4,5,\ldots,n-1,n,2)$ with $v(\omega_{n})=\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}$.

Proof.

The proof of the proposition is similar to the proofs for type B and C which uses the following crucial lemma. ∎

Lemma 5.2**.**

Let $v,v_{\underline{i}},v_{\underline{i},1},v_{\underline{i},2},v_{1,\underline{i}},v_{2,\underline{i}}$ are as defined in Proposition 5.1.

(i) Let $r=1$. Then $w>v$ and $l(w)=l(v)+1$ iff $w=s_{k}v$ where $k$ takes the following values: \left\{\begin{array}[]{lr}\ k\in\{2,4,6,\ldots,n-2,n\},&{n\equiv 0(mod\,\,4)}\\ k\in\{1,4,6,\ldots,n-2,n\},&{n\equiv 2(mod\,\,4)}\\ k\in\{2,5,7,\ldots,n-2,n\},&{n\equiv 1(mod\,\,4)}\\ k\in\{1,5,7,\ldots,n-2,n\},&{n\equiv 3(mod\,\,4)}.\end{array}\right.

(v) Let $r=2$. Then $w>v$ and $l(w)=l(v)+1$ iff $w=s_{k}v$ where $k$ takes the following values: \left\{\begin{array}[]{lr}\ k\in\{1,4,6,\ldots,n-2,n\},&{n\equiv 0(mod\,\,4)}\\ k\in\{2,4,6,\ldots,n-2,n\},&{n\equiv 2(mod\,\,4)}\\ k\in\{1,5,7,\ldots,n-2,n\},&{n\equiv 1(mod\,\,4)}\\ k\in\{2,5,7,\ldots,n-2,n\},&{n\equiv 3(mod\,\,4)}.\end{array}\right.

(iii) Let $3\leq r\leq n-1$ and $(n+1)-r=2m$. Then,

(a) $w>v_{\underline{i}}$ and $l(w)=l(v_{\underline{i}})+1$ iff $w=s_{\alpha_{i_{t}}}v_{\underline{i}}$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}$ for $i_{t}:|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}v_{\underline{i}}$ for $|i_{t}-i_{t-1}|\geq 3$ or $s_{\alpha_{1}+\alpha_{3}}$ or $s_{{\alpha_{2}}+{\alpha_{3}}}$ for $i_{1}=3$.

(b) $w>v_{1,\underline{i}}$ and $l(w)=l(v_{1,\underline{i}})+1$ iff $w=s_{\alpha_{k}}v_{1,\underline{i}}$ for $k\in\{1,i_{t}\}$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{1,\underline{i}}$ for $i_{t}:|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}v_{1,\underline{i}}$ for $i_{t}:|i_{t}-i_{t-1}|\geq 3$ with $i_{1}\geq 5$ or $s_{\alpha_{1}+\alpha_{3}}$ with $i_{1}\geq 5$.

(c) $w>v_{2,\underline{i}}$ and $l(w)=l(v_{2,\underline{i}})+1$ iff $w=s_{\alpha_{k}}v_{2,\underline{i}}$ for $k\in\{2,i_{t}\}$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{2,\underline{i}}$ for $i_{t}:|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}v_{2,\underline{i}}$ for $i_{t}:|i_{t}-i_{t-1}|\geq 3$ with $i_{1}\geq 5$ or $s_{\alpha_{2}+\alpha_{3}}$ with $i_{1}\geq 5$.

(iv) Let $3\leq r\leq n-1$ and $(n+1)-r=2m+1$. Then,

(a) $w>v_{\underline{i}}$ and $l(w)=l(v_{\underline{i}})+1$ iff $w=s_{\alpha_{k}}v_{\underline{i}}$ for $k\in\{1,2,i_{t}\}$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i}}$ for $i_{t}:|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}v_{\underline{i}}$ for $i_{t}:|i_{t}-i_{t-1}|\geq 3$ with $i_{1}\geq 5$.

(b) $w>v_{\underline{i},1}$ and $l(w)=l(v_{\underline{i},1})+1$ iff $w=s_{\alpha_{k}}v_{\underline{i},1}$ for $k\in\{1,i_{t}\}$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i},1}$ for $i_{t}:|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}v_{\underline{i},1}$ for $i_{t}:|i_{t}-i_{t-1}|\geq 3$ with $i_{1}\geq 6$ or $s_{\alpha_{1}+\alpha_{3}+\alpha_{4}}v_{\underline{i},1}$ with $i_{1}\geq 6$.

(c) $w>v_{\underline{i},2}$ and $l(w)=l(v_{\underline{i},2})+1$ iff $w=s_{\alpha_{k}}v_{\underline{i},2}$ for $k\in\{2,i_{t}\}$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}v_{\underline{i},2}$ for $i_{t}:|i_{t}-i_{t+1}|\geq 3$ or $s_{\alpha_{i_{t}}+\alpha_{i_{t}-1}}v_{\underline{i},2}$ for $i_{t}:|i_{t}-i_{t-1}|\geq 3$ with $i_{1}\geq 6$ or $s_{\alpha_{2}+\alpha_{3}+\alpha_{4}}v_{\underline{i},2}$ for $i_{1}\geq 6$.

(v) Let $r=n$. Then $w>v$ and $l(w)=l(v)+1$ iff $w=s_{1}v$ or $s_{2}v$.

Proof.

We prove this lemma only for case (iv) and part (b). Proofs of the other cases are similar to this case.

Let $w\in W^{I_{r}}$ such that $w>v_{\underline{i},1}$ and $l(w)=l(v_{\underline{i},1})+1$. Then $w=s_{\beta}v_{\underline{i},1}$, with $ht(\beta)\leq 2$ or $\beta$ = $\alpha_{1}+\alpha_{3}+\alpha_{4}$ when $i_{1}\geq 6$. Note that $v_{\underline{i},1}(\omega_{r})=\frac{3}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{3}+\displaystyle\sum_{k=1}^{m-1}\alpha_{i_{k}}$.

Case 1. $\beta=\alpha_{k}$, a simple root.

If $k=i_{t}$, then $s_{\alpha_{i_{t}}}v_{\underline{i},1}(\omega_{r})=\frac{3}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{3}+\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t}}$. Since $s_{\beta}v_{\underline{i},1}(\omega_{r})<v_{\underline{i},1}(\omega_{r})$ and $\omega_{r}$ is a dominant weight we have $s_{\beta}v_{\underline{i},1}>v_{\underline{i},1}$. Since $\beta$ is a simple root so $l(s_{\beta}v_{\underline{i},1})=l(v_{\underline{i},1})+1$.

If $\beta=\alpha_{1}$, then $s_{1}v_{\underline{i},1}(\omega_{r})=-\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{3}+\sum{\alpha_{i_{k}}}$. So $s_{1}v_{\underline{i},1}(\omega_{r})<v_{\underline{i},1}(\omega_{r})$. Hence $s_{1}v_{\underline{i},1}>v_{\underline{i},1}$ with $l(s_{\beta}v_{\underline{i},1})=l(v_{\underline{i},1})+1$.

If $\beta=\alpha_{2}$ or $\alpha_{3}$, then $s_{\beta}v_{\underline{i},1}=v_{\underline{i},1}$ in $W^{I_{r}}$, a contradiction.

If $k\notin\{1,2,3,i_{t}\}$, then $s_{\beta}v_{\underline{i},1}(\omega_{r})>v_{\underline{i},1}(\omega_{r})$. Hence $s_{\beta}v_{\underline{i},1}<v_{\underline{i},1}$, a contradiction.

Case 2. $\beta=\alpha_{k}+\alpha_{k+1}$, a positive root of height $2$.

If $k=i_{t}$ with $|i_{t}-i_{t+1}|\geq 3$, then $s_{\beta}v_{\underline{i},1}(\omega_{r})=\frac{3}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{3}+\displaystyle\sum_{k\neq t}\alpha_{i_{k}}-\alpha_{i_{t}+1}<v_{\underline{i},1}(\omega_{r})$. So $s_{\beta}v_{\underline{i},1}>v_{\underline{i},1}$. Now we prove that $l(s_{\beta}v_{\underline{i},1})=l(v_{\underline{i},1})+1$.

We will prove this case for $m$ is odd. Note that $s_{\beta}v_{\underline{i},1}=(-\underline{i},-3,-2,-1,\hat{\underline{i^{\prime}}},[-n,-5]\\ \backslash\{[-\underline{i}],[-\hat{\underline{i^{\prime}}}]\}\uparrow,-4,4,[5,n]\backslash\{[\underline{i}],[\hat{\underline{i^{\prime}}}]\}\uparrow,-\hat{\underline{i^{\prime}}},1,2,3,\underline{i})$ where $\hat{\underline{i^{\prime}}}=(i_{1}-1,i_{2}-1,\ldots,i_{t-1}-1,i_{t}+1,i_{t+1}-1\ldots,i_{m}-1)$. In $v_{\underline{i},1}$, the position of $i_{t}+1$ is right to the position of $i_{t}-1$ and left to the position of $i_{t}$ but in $s_{\beta}v_{\underline{i},1}$, the position of $i_{t}$ remains unchanged and the positions of $i_{t}+1$ and $i_{t}-1$ are interchanged. Similarly in $v_{\underline{i},1}$ the position of $-i_{t}-1$ is right to the position of $-i_{t}$ and left to the position of $-i_{t}+1$ but in $s_{\beta}v_{\underline{i},1}$ the position of $-i_{t}$ remains unchanged and the positions of $-i_{t}-1$ and $-i_{t}+1$ are interchanged. So $inv(s_{\beta}v_{\underline{i},1})=inv(v_{\underline{i},1})+2$ and hence $l(s_{\beta}v_{\underline{i},1})=l(v_{\underline{i},1})+1$.

If $k=i_{t}$ with $|i_{t}-i_{t+1}|=2$, then $s_{\beta}v_{\underline{i},1}=v_{\underline{i},1}$ in $W^{I_{r}}$ since $s_{\beta}v_{\underline{i},1}(\omega_{r})=v_{\underline{i},1}(\omega_{r})$, a contradiction.

If $\beta=\alpha_{k}+\alpha_{k+1}$ and $k\notin\{2,3,i_{t}\}$ then since $s_{\beta}v_{\underline{i},1}(\omega_{r})\geq v_{\underline{i},1}(\omega_{r})$ we have $s_{\beta}v_{\underline{i},1}\leq v_{\underline{i},1}$ in $W^{I_{r}}$, a contradiction.

If $k=3$ then $\beta=\alpha_{3}+\alpha_{4}$. So $s_{\beta}v_{\underline{i},1}(\omega_{r})>v_{\underline{i},1}(\omega_{r})$ and hence, $s_{\beta}v_{\underline{i},1}<v_{\underline{i},1}$, a contradiction.

If $k=2$ then $\beta=\alpha_{2}+\alpha_{3}$, then $s_{\beta}v_{\underline{i},1}(\omega_{r})=v_{\underline{i},1}(\omega_{r})$ and hence $s_{\beta}v_{\underline{i},1}=v_{\underline{i},1}$ in $W^{I_{r}}$, a contradiction.

If $j,j+1\neq i_{t}$ then $s_{\beta}v_{\underline{i},1}(\omega_{r})\geq v_{\underline{i},1}(\omega_{r})$ and $s_{\beta}v_{\underline{i},1}\leq v_{\underline{i},1}$ in $W^{I_{r}}$, a contradiction.

Case 3. If $\beta=\alpha_{k-1}+\alpha_{k}$, a positive root of height $2$ the proof is similar to above case.

Case 4. If $\beta=\alpha_{1}+\alpha_{3}$, then $s_{\beta}v_{\underline{i},1}=s_{3}s_{1}v_{\underline{i},1}$. From the reduced expression of $v_{\underline{i},1}$ we see that $l(s_{\beta}v_{\underline{i},1})=l(v_{\underline{i},1})+2$, a contradiction.

Case 5. If $\beta=\alpha_{1}+\alpha_{3}+\alpha_{4}$ with $i_{1}\geq 6$, then $s_{\beta}v_{\underline{i},1}(\omega_{r})=\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}-\alpha_{4}+\sum\alpha_{i_{k}}<v_{\underline{i},1}(\omega_{r})$. So $s_{\beta}v_{\underline{i},1}>v_{\underline{i},1}$. We show that $l(s_{\beta}v_{\underline{i},1})=l(v_{\underline{i},1})+1$.

We will prove this case for $m$ is odd. Note that $s_{\beta}v_{\underline{i}}=(-\underline{i},-3,-2,4,\underline{i^{\prime}},[-n,-5]\backslash\{[-\underline{i}],[-\underline{i^{\prime}}]\}\uparrow,1,-1,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i^{\prime}},-4,2,3,\hat{\underline{i}})$. So $inv(s_{\beta}v_{\underline{i}})=inv(v_{\underline{i}})+4$ and $neg(s_{\beta}v_{\underline{i}})=neg(v_{\underline{i}})+2$. Hence $l(s_{\beta}v_{\underline{i}})=l(v_{\underline{i}})+1$.

Proof of the converse is clear from the definition of maximal $v$ such that $v(\omega_{r})\geq 0$. ∎

Proposition 5.3**.**

Let $G$ be of type $D_{n}$ and let $v,v_{\underline{i}},v_{\underline{i},1},v_{\underline{i},2},v_{2,\underline{i}},v_{1,\underline{i}}$ be as defined in Proposition $5.1$. Let $w\in W^{I_{r}}$ be minimal such that $w(\omega_{r})\leq 0$. Then the description of $w$ is the following:

(i) For $r=1$, w=\left\{\begin{array}[]{lr}\ (-n,-(n-2),\ldots,-4,-2,1,3,5,\ldots,n-3,n-1),&n\equiv 0(mod\,\,4)\\ (-n,-(n-2),\ldots,-4,-2,-1,3,5,\ldots,n-3,n-1),&n\equiv 2(mod\,\,4)\\ (-n,-(n-2),\ldots,-5,-3,-2,-1,4,\ldots,n-3,n-1),&n\equiv 1(mod\,\,4)\\ (-n,-(n-2),\ldots,-5,-3,-2,1,4,\ldots,n-3,n-1),&n\equiv 3(mod\,\,4).\end{array}\right.

(ii) For $r=2$, w=\left\{\begin{array}[]{lr}\ (n,-(n-2),\ldots,-4,-2,-1,3,5,\ldots,n-3,n-1),&n\equiv 0(mod\,\,4)\\ (n,-(n-2),\ldots,-4,-2,1,3,5,\ldots,n-3,n-1),&n\equiv 2(mod\,\,4)\\ (n,-(n-2),\ldots,-3,-2,1,4,6,\ldots,n-3,n-1),&n\equiv 1(mod\,\,4)\\ (n,-(n-2),\ldots,-3,-2,-1,4,6,\ldots,n-3,n-1),&n\equiv 3(mod\,\,4).\end{array}\right.

(iii) Let $3\leq r\leq n-1$ and $(n+1)-r=2m$.

If $\underline{i}\in J_{m,[3,n]}$ then $w_{\underline{i}}=s_{i_{1}}s_{i_{2}}\ldots s_{i_{m}}v_{\underline{i}}$ with $w_{\underline{i}}(\omega_{r})=-v_{\underline{i}}(\omega_{r})$ and

w_{\underline{i}}=\left\{\begin{array}[]{lr}\ (-1,[2,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},\underline{i^{\prime}}),&\text{m is odd}\\ (1,[2,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},\underline{i^{\prime}}),&\text{m is even}.\end{array}\right.**

If $\underline{i}\in J_{m-1,[4,n]}$ then $w_{1,\underline{i}}=s_{1}s_{i_{1}}s_{i_{2}}\ldots s_{i_{m-1}}v_{1,\underline{i}}$ with $w_{1,\underline{i}}(\omega_{r})=-v_{1,\underline{i}}(\omega_{r})$ and

w_{1,\underline{i}}=\left\{\begin{array}[]{lr}\ ([3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-2,-1,\underline{i^{\prime}}),&\text{m is odd}\\ (-t,[3,n]\backslash\{t,[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-2,-1,\underline{i^{\prime}}),&\text{m is even}.\end{array}\right.* where $t=min{[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}}$.*

If $\underline{i}\in J_{m-1,[4,n]}$ then $w_{2,\underline{i}}=s_{2}s_{i_{1}}s_{i_{2}}\ldots s_{i_{m-1}}v_{2,\underline{i}}$ with $w_{2,\underline{i}}(\omega_{r})=-v_{2,\underline{i}}(\omega_{r})$ and

w_{2,\underline{i}}=\left\{\begin{array}[]{lr}\ (-t,[3,n]\backslash\{t,[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-2,1,\underline{i^{\prime}}),&\text{m is odd}\\ ([3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-2,1,\underline{i^{\prime}}),&\text{m is even}.\end{array}\right.* where $t=min{[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}}$.*

(iv) Let $3\leq r\leq n-1$ and $(n+1)-r=2m+1$.

For $\underline{i}\in J_{m,[4,n]}$ and $v_{\underline{i}}(\omega_{r})=\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\sum_{k=1}^{m}\alpha_{i_{k}}$, we have $w_{\underline{i}}=s_{1}s_{2}s_{i_{1}}s_{i_{2}}\ldots s_{i_{m-1}}v_{\underline{i}}$. In this case $w_{\underline{i}}(\omega_{r})=-v_{\underline{i}}(\omega_{r})$ and

w_{\underline{i}}=\left\{\begin{array}[]{lr}\ (1,[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-2,\underline{i^{\prime}}),&\text{m is odd}\\ (-1,[3,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-2,\underline{i^{\prime}}),&\text{m is even}.\end{array}\right.**

For $\underline{i}\in J_{m-1,[5,n]}$ and $v_{\underline{i},1}(\omega_{r})=\displaystyle\frac{3}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{3}+\sum_{k=1}^{m-1}\alpha_{i_{k}}$, we have $w_{\underline{i},2}=s_{2}s_{3}s_{1}s_{i_{1}}s_{i_{2}}\ldots s_{i_{m-1}}v_{\underline{i},1}$.

w_{\underline{i},2}=\left\{\begin{array}[]{lr}\ (4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-3,-2,1,\underline{i^{\prime}}),&\text{m is odd}\\ (-4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-3,-2,1,\underline{i^{\prime}}),&\text{m is even}.\end{array}\right.**

For $\underline{i}\in J_{m-1,[5,n]}$ and $v_{\underline{i},2}(\omega_{r})=\displaystyle\frac{1}{2}\alpha_{1}+\frac{3}{2}\alpha_{2}+\alpha_{3}+\sum_{k=1}^{m-1}\alpha_{i_{k}}$, we have $w_{\underline{i},1}=s_{1}s_{3}s_{2}s_{i_{1}}s_{i_{2}}\ldots s_{i_{m-1}}v_{\underline{i},2}$. In these cases $w_{\underline{i},1}(\omega_{r})=-v_{\underline{i},1}(\omega_{r})$ and $w_{\underline{i},2}(\omega_{r})=-v_{\underline{i},2}(\omega_{r})$ and

w_{\underline{i},1}=\left\{\begin{array}[]{lr}\ (-4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-3,-2,-1,\underline{i^{\prime}}),&\text{m is odd}\\ (4,[5,n]\backslash\{[\underline{i}],[\underline{i^{\prime}}]\}\uparrow,-\underline{i},-3,-2,-1,\underline{i^{\prime}}),&\text{m is even}.\end{array}\right.**

(v) For $r=n$, we have $w=s_{1}s_{2}v$ and in this case $w(\omega_{r})=-v(\omega_{r})$.

Proof.

For the proof of minimality of $w$ refer to [12]. ∎

Proposition 5.4**.**

Let $v$, $w$, $v_{\underline{i}}$, $w_{\underline{i}}$, $v_{\underline{i},1}$, $w_{\underline{i},1}$, $v_{\underline{i},2}$, $w_{\underline{i},2}$, $v_{1,\underline{i}}$, $w_{1,\underline{i}}$, $v_{2,\underline{i}}$, $w_{2,\underline{i}}$ are as defined in Proposition 5.1 and Proposition 5.3. Let $w\in W^{P_{r}}$ be minimal and $v\in W^{P_{r}}$ be maximal such that $w(\omega_{r})\leq 0$ and $v(\omega_{r})\geq 0$. Then $X_{w}^{v}\neq\emptyset$ iff the pair $(v,w)$ is one of the following:

(i) Let $3\leq r\leq n-1$. For $(n+1)-r=2m$:

(v,w)=\left\{\begin{array}[]{lr}\ (v_{\underline{i}},w_{\underline{j}}),&{s.t.|i_{k}-j_{k}|\leq 1\,\,\forall\,\,1\leq k\leq m}\\ (v_{\underline{i}},w_{1,\underline{j}}),(v_{\underline{i}},w_{2,\underline{j}}),&{s.t.|i_{k+1}-j_{k}|\leq 1\,\,\forall\,\,1\leq k\leq m-1\,\,with\,\,i_{1}=3}\\ (v_{1,\underline{i}},w_{\underline{j}}),(v_{2,\underline{i}},w_{\underline{j}}),&{s.t.|i_{k}-j_{k+1}|\leq 1\,\,\forall\,\,1\leq k\leq m-1\,\,with\,\,j_{1}=3}.\end{array}\right.**

For $(n+1)-r=2m+1$:

(v,w)=\left\{\begin{array}[]{lr}\ (v_{\underline{i}},w_{\underline{j}}),&{s.t.|i_{k}-j_{k}|\leq 1\,\,\forall\,\,1\leq k\leq m}\\ (v_{\underline{i}},w_{\underline{j},1}),(v_{\underline{i}},w_{\underline{j},2})&{s.t.|i_{k+1}-j_{k}|\leq 1\,\,\forall\,\,1\leq k\leq m-1\,\,with\,\,i_{1}=4}\\ (v_{\underline{i},1},w_{\underline{j}}),(v_{\underline{i},2},w_{\underline{j}})&{s.t.|i_{k}-j_{k+1}|\leq 1\,\,\forall\,\,1\leq k\leq m-1\,\,with\,\,j_{1}=4}\\ (v_{\underline{i},1},w_{\underline{j},2}),(v_{\underline{i},2},w_{\underline{j},1})&{s.t.|i_{k}-j_{k}|\leq 1\,\,\forall\,\,1\leq k\leq m-1}.\end{array}\right.**

(ii) For $r=1,2$ or $n$, $X_{w}^{v}\neq\emptyset$ for any $v$ and $w$.

Proof.

Let $X_{w}^{v}$ be non-empty. So, $v\leq w$. We prove this part for case (i), $(n+1)-r$ is odd and $(v,w)=(v_{\underline{i}},w_{\underline{j},1})$. For other cases the proofs are similar. Let $(n+1)-r=2m+1$. We have $v_{\underline{i}}<w_{\underline{j},1}$. Now assume that $|i_{k+1}-j_{k}|\leq 1,\forall 1\leq k\leq m-1$. We need to show that $i_{1}=4$. If not, then $i_{1}>4$. This implies that the coefficient of $\alpha_{4}$ in $v_{\underline{i}}(\omega_{r})$ is zero. Since $l(w_{\underline{j},1})-l(v_{\underline{i}})=m+2$ and $w_{\underline{j},1}>v_{\underline{i}}$ there exists $m+2$ positive roots $\beta$ such that $w_{\underline{j},1}=\displaystyle\prod_{card(\beta)=m+2}s_{\beta}v_{\underline{i}}$. On the other hand we have $v_{\underline{i}}(\omega_{r})=\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+0.\alpha_{3}+0.\alpha_{4}+\sum_{k=1}^{m}\alpha_{i_{k}}$ and $w_{\underline{j},1}(\omega_{r})=-\frac{3}{2}\alpha_{1}-\frac{1}{2}\alpha_{2}-\alpha_{3}-\sum_{k=1}^{m-1}\alpha_{j_{k}}$. So, $w_{\underline{j},1}=[\displaystyle\prod_{t\in\{t:j_{t}=i_{t+1}\}}s_{\alpha_{i_{t+1}}}$ $\displaystyle\prod_{t\in\{t:j_{t}=i_{t+1}-1\}\downarrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}-1}}$ $\displaystyle\prod_{t\in\{t:j_{t}=i_{t+1}+1\}\uparrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}]s_{1}s_{3}s_{4}s_{3}s_{2}(s_{5}s_{6}\ldots s_{i_{1}-1}s_{i_{1}})(s_{4}s_{5}\ldots\\ s_{i_{1}-2}s_{i_{1}-1})v_{\underline{i}}$. Note that the expression in the square bracket contains exactly $m-1$ reflections corresponding to $m-1$ distinct positive roots and it is independent of the word in between this expression and $v_{\underline{i}}$. So we have, $w_{\underline{j},1}=\displaystyle\prod_{card(\beta)>m+2}s_{\beta}v_{\underline{i}}$, a contradiction. So, $i_{1}=4$.

Let $i_{1}=4$. We will show that $v_{\underline{i}}<w_{\underline{j},1}$ implies that $|i_{k+1}-j_{k}|\leq 1,\forall 1\leq k\leq m-1$. Since $l(w_{\underline{j},1})-l(v_{\underline{i}})=m+2$, by Lemma 5.2 we have $w_{\underline{j},1}=\displaystyle(\prod_{\mathbf{card}\{\beta\}=m-1}s_{\beta})s_{1}s_{3}s_{4}s_{3}s_{2}v_{\underline{i}}$ for positive roots $\beta$ such that $\beta$ is either $\alpha_{i_{t}}$ or $\alpha_{i_{t}}+\alpha_{{i_{t}}+1}$ or $\alpha_{i_{t}}+\alpha_{{i_{t}}-1}$ for some $i_{t}$, $2\leq t\leq m$. So $w_{\underline{j},1}(\omega_{r})=-\frac{3}{2}\alpha_{1}-\frac{1}{2}\alpha_{2}-\alpha_{3}\displaystyle-\sum_{{j_{k}}:|i_{k+1}-j_{k}|\leq 1}\alpha_{j_{k}}$. Hence, $|i_{k+1}-j_{k}|\leq 1$ $\forall 1\leq k\leq m-1$.

Conversely, let $v=v_{\underline{i}},w=w_{\underline{j},1}$ with $|i_{k+1}-j_{k}|\leq 1\,\,\forall\,\,1\leq k\leq m-1$ and $i_{1}=4$. Then we need to show that $v_{\underline{i}}<w_{\underline{j},1}$. Note that in this case $w_{\underline{j},1}=\displaystyle\prod_{\{t:j_{t}=i_{t+1}\}}s_{\alpha_{i_{t+1}}}$ $\displaystyle\prod_{\{t:j_{t}=i_{t+1}-1\}\downarrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}-1}}$ $\displaystyle\prod_{\{t:j_{t}=i_{t+1}+1\}\uparrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}$ $(s_{\alpha_{1}})(s_{\alpha_{4}+\alpha_{3}})(s_{\alpha_{2}})$ $v_{\underline{i}}$. We claim that in this product multiplication of each reflection to $v_{\underline{i}}$ amounts to increase the length by one and the product is greater than $v_{\underline{i}}$. Since $\underline{i}\in J_{m,[4,n]}$, $\underline{j}\in J_{m-1,[5,n]}$ and $|i_{k+1}-j_{k}|\leq 1\forall 1\leq k\leq m-1$ with $i_{1}=4$ we observe that the sets $\{\bigcup\limits_{\{t:j_{t}=i_{t+1}+1\}}\{i_{t+1}-1,i_{t+1},i_{t+1}+1\}\}$, $\{\bigcup\limits_{\{t:j_{t}=i_{t+1}-1\}}\{i_{t+1}-2,i_{t+1}-1,i_{t+1}\}\}$ and $\{\bigcup\limits_{\{t:j_{t}=i_{t+1}\}}\{i_{t+1}-1,i_{t+1}\}\}$ are mutually disjoint.

It is easy to see that $s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}>v_{\underline{i}}$ and $l(s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}})=l(v_{\underline{i}})+3$.

Now we claim that $s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}<\displaystyle\prod_{\{t:j_{t}=i_{t+1}+1\}\uparrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}$ and the length is increased by the number of reflections multiplied. Let $t$ be maximal such that $j_{t}=i_{t+1}+1$. Since the sets $\{1,2,3,4\}$ and $\{\cup_{\{t:j_{t}=i_{t+1}+1\}}\{i_{t+1}-1,i_{t+1},i_{t+1}+1\}\}$ are mutually disjoint, from Lemma $5.2$ we see that $s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}<s_{\alpha_{i_{t+1}+i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}$.

Let $t$ be such that $i_{t}=i_{t+1}-2$ and $j_{t-1}=i_{t}+1$. We have $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}\newline s_{\alpha_{2}}v_{\underline{i}}(\omega_{r})=-\frac{3}{2}\alpha_{1}-\frac{1}{2}\alpha_{2}-\alpha_{3}+\displaystyle\sum_{k=2,\neq\{t,t+1\}}^{m}\alpha_{i_{k}}-\alpha_{i_{t+1}+1}-\alpha_{i_{t}+1}$. So $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}\newline s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}(\omega_{r})<s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}(\omega_{r})$. Hence $s_{\alpha_{i_{t}}+\alpha_{i_{t}+1}}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}\\ s_{\alpha_{2}}v_{\underline{i}}>s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}$. From the one line notations of these two elements we see that the length is increasing by $1$.

Repeating this process we conclude that $v_{\underline{i}}<\displaystyle\prod_{t\in\{t:j_{t}=i_{t+1}\}}s_{\alpha_{i_{t+1}}}\prod_{t\in\{t:j_{t}=i_{t+1}-1\}\downarrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}-1}}\newline \prod_{t\in\{t:j_{t}=i_{t+1}+1\}\uparrow}s_{\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}}s_{\alpha_{1}}s_{\alpha_{3}+\alpha_{4}}s_{\alpha_{2}}v_{\underline{i}}$. ∎

Theorem 5.5**.**

Let $G$ be of type $D_{n}$ and let $P_{r}$ be the maximal parabolic subgroup corresponding to the simple root $\alpha_{r}$. Let $\mathcal{L}_{r}$ be the line bundle corresponding to the fundamental weight $\omega_{r}$. Then $(X^{v}_{w})^{ss}_{T}(\mathcal{L}_{r})$ is non-empty if and only if the pair $(v,w)$ is one of the following:

*(i) For $3\leq r\leq n-1$,

$(n+1)-r=2m$:    (v,w)\,\,s.t.\left\{\begin{array}[]{lr}v\leq v_{\underline{i}}\,\,and\,\,w\geq w_{\underline{i}}\\ v\leq v_{1,\underline{i}}\,\,and\,\,w\geq w_{1,\underline{i}}\\ v\leq v_{2,\underline{i}}\,\,and\,\,w\geq w_{2,\underline{i}}.\end{array}\right.*

$(n+1)-r=2m+1$:    (v,w)\,\,s.t.\,\,\left\{\begin{array}[]{lr}v\leq v_{\underline{i}}\,\,and\,\,w\geq w_{\underline{i}}\\ v\leq v_{\underline{i},1}\,\,and\,\,w\geq w_{\underline{i},2}\\ v\leq v_{\underline{i},2}\,\,and\,\,w\geq w_{\underline{i},1}.\end{array}\right.

(ii) For $r=1,2$ and $n$, $(X^{v}_{w})^{ss}_{T}(\mathcal{L}_{r})$ is non-empty for any $v$ and $w$.

where $v$, $w$, $v_{\underline{i}},w_{\underline{i}},v_{\underline{i},1},v_{\underline{i},2},w_{\underline{i},1}$, $w_{\underline{i},2}$, $v_{1,\underline{i}}$, $w_{1,\underline{i}}$,$v_{2,\underline{i}}$ and $w_{2,\underline{i}}$ are as in Proposition 5.3.

Proof.

Now since $X_{w}^{v}\subseteq X_{w^{\prime}}^{v^{\prime}}$ implies $(X^{v}_{w})^{ss}_{T}(\mathcal{L}_{r})\subseteq(X^{v^{\prime}}_{w^{\prime}})^{ss}_{T}(\mathcal{L}_{r})$, we can assume that $v$ is maximal and $w$ is minimal having the property that $v(\omega_{r})\geq 0$ and $w(\omega_{r})\leq 0$. We prove the theorem for case (i) and $(n+1)-r$ is odd. For other cases the proof is similar.

Let $(n+1)-r=2m+1$. For each pair $(v,w)$, we construct a non-zero $T$-invariant section of $\mathcal{L}_{r}$ on $G/P_{r}$ which is not identically zero on $X_{w}^{v}$.

For $(v,w)=(v_{\underline{i}},w_{\underline{i}})$ we have $v_{\underline{i}}(\omega_{r})+w_{\underline{i}}(\omega_{r})=0$. So $p_{v_{\underline{i}}}p_{w_{\underline{i}}}$ is a non-zero $T$-invariant section of $\mathcal{L}_{r}$ on $G/P_{r}$ which is not identically zero on $X_{w}^{v}$.

For $(v,w)=(v_{\underline{i},1},w_{\underline{i},2})$, we consider the sequence $v_{\underline{i},1}\leq s_{1}v_{\underline{i},1}\leq s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}s_{1}v_{\underline{i},1}\\ \leq s_{2}s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}s_{1}v_{\underline{i},1}=w_{\underline{i},2}$. We have $v_{\underline{i},1}(\omega_{r})+s_{1}v_{\underline{i},1}(\omega_{r})+s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}s_{1}\\ v_{\underline{i},1}(\omega_{r})+w_{\underline{i},2}(\omega_{r})=0$ and so $p_{v_{\underline{i},1}}p_{s_{1}v_{\underline{i},1}}p_{s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}s_{1}v_{\underline{i},1}}p_{w_{\underline{i},2}}$ is a non-zero $T$-invariant section of $\mathcal{L}_{r}$ on $G/P_{r}$ which is not identically zero on $X_{w}^{v}$.

For $(v,w)=(v_{\underline{i},2},w_{\underline{i},1})$, we consider the sequence $v_{\underline{i},2}\leq s_{2}v_{\underline{i},2}\leq s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}\\ s_{2}v_{\underline{i},2}\leq s_{1}s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}s_{2}v_{\underline{i},2}=w_{\underline{i},1}$. We have $v_{\underline{i},2}(\omega_{r})+s_{2}v_{\underline{i},2}(\omega_{r})+s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}\\ s_{2}v_{\underline{i},2}(\omega_{r})+w_{\underline{i},1}(\omega_{r})=0$ and so $p_{v_{\underline{i},2}}p_{s_{2}v_{\underline{i},2}}p_{s_{\alpha_{i_{1}}}s_{\alpha_{i_{2}}}\ldots s_{\alpha_{i_{m-1}}}s_{3}s_{2}v_{\underline{i},2}}p_{w_{\underline{i},1}}$ is a non-zero $T$-invariant section of $\mathcal{L}_{r}$ on $G/P_{r}$ which does not vanish identically zero on $X_{w}^{v}$.

So, in all these cases we conclude that $(X{{}^{v}_{w}})^{ss}_{T}(\mathcal{L}_{r})\neq\emptyset$.

Conversely, let $(X^{v}_{w})^{ss}_{T}(\mathcal{L}_{r})$ be non-empty.

Let $(v,w)=(v_{\underline{i},1},w_{\underline{j},2})$. If $\underline{i}\neq\underline{j}$, then there exists $t$ such that $j_{t}\neq i_{t}$. Since $X^{v_{\underline{i},1}}_{w_{\underline{j},2}}\neq\emptyset$, by proposition 5.4 we have either $j_{t}=i_{t}+1$ or $j_{t}=i_{t}-1$. If $j_{t}=i_{t}+1$ then $w_{\underline{j},2}(\omega_{r})$ = $-\frac{1}{2}\alpha_{1}-\frac{3}{2}\alpha_{2}-\alpha_{3}-\sum_{k=1,\neq t}^{m-1}{\alpha_{i_{k}}}-\alpha_{i_{t}+1}$ and if $j_{t}=i_{t}-1$ then $w_{\underline{j},2}(\omega_{r})$ = $-\frac{1}{2}\alpha_{1}-\frac{3}{2}\alpha_{2}-\alpha_{3}-\sum_{k=1,\neq t}^{m-1}{\alpha_{i_{k}}}-\alpha_{i_{t}-1}$. Let $u\in W^{I_{r}}$ be such that $v_{\underline{i},1}\leq u\leq w_{\underline{j},2}$. Then $u$ is of the form $u=(\displaystyle\prod_{\beta}s_{\beta})v_{\underline{i},1}$, where $\beta$’s are some positive roots. For $j_{t}=i_{t}+1$ at most one $\beta$ can be $\alpha_{i_{t}}+\alpha_{i_{t}+1}$ and none of the other $\beta$’s contain $\alpha_{i_{t}}$ or $\alpha_{i_{t}+1}$ as a summand. So in $u(\omega_{r})$ the coefficient of $\alpha_{i_{t}}$ is either zero or one and the coefficient of $\alpha_{i_{t}+1}$ is either zero or $-1$. Similarly for $j_{t}=i_{t}-1$ at most one $\beta$ can be $\alpha_{i_{t}}+\alpha_{i_{t}-1}$ and none of the other $\beta$’s contain $\alpha_{i_{t}}$ or $\alpha_{i_{t}-1}$ as a summand. So in $u(\omega_{r})$ the coefficient of $\alpha_{i_{t}}$ is either zero or one and the coefficient of $\alpha_{i_{t}-1}$ is either zero or $-1$. For $j_{t}=i_{t}+1$, $u(\omega_{r})$ contains either $\alpha_{i_{t}}$ or $\alpha_{i_{t}+1}$ as a summand and for $j_{t}=i_{t}-1$, $u(\omega_{r})$ contains either $\alpha_{i_{t}}$ or $\alpha_{i_{t}-1}$ as a summand. So, there does not exist any sequence $v_{\underline{i},1}=u_{1}\leq u_{2}\leq\ldots\leq u_{k}=w_{\underline{j},2}$ such that $\sum_{l=1}^{k}u_{l}(\omega_{r})=0$. So we don’t have a nonzero $T$-invariant section which is not identically zero on $X_{w}^{v}$.

Let $(v,w)=(v_{\underline{i}},w_{\underline{j},1})$ where $\underline{i}=(4,i_{2},\ldots,i_{m})$ and $\underline{j}=(i_{2},\ldots,i_{m})$. Then $v_{\underline{i}}(\omega_{r})$ = $\frac{1}{2}\alpha_{1}+\frac{1}{2}\alpha_{2}+\alpha_{4}+\sum_{k=2}^{m}{\alpha_{i_{k}}}$ and $w_{\underline{j},1}(\omega_{r})$ = $-\frac{3}{2}\alpha_{1}-\frac{1}{2}\alpha_{2}-\alpha_{3}-\sum_{k=2}^{m}{\alpha_{i_{k}}}$. Then any $u\in W^{I_{r}}$ such that $v_{\underline{i}}\leq u\leq w_{\underline{j},1}$ is of the form $u=(\displaystyle\prod_{\beta}s_{\beta})v_{\underline{i}}$, where $\beta$’s are some positive roots. At most one $\beta$ can be $\alpha_{3}+\alpha_{4}$ and none of the other $\beta$’s can contain $\alpha_{3}$ or $\alpha_{4}$ as a summand. So, the coefficient of $\alpha_{4}$ in $u(\omega_{r})$ is either zero or one and the coefficient of $\alpha_{3}$ in $u(\omega_{r})$ is either zero or $-1$. So for any such $u$, $u(\omega_{r})$ contains either $\alpha_{3}$ or $\alpha_{4}$ as a summand. So, in this case also there is no non zero $T$-invariant section which is not identically zero on $X_{w}^{v}$.

Let $(v,w)=(v_{\underline{i}},w_{\underline{j},1})$ with $i_{1}=4$. If $(i_{2},i_{3},\ldots,i_{m})\neq\underline{j}$, then there exists $t$ such that $j_{t}\neq i_{t+1}$. Since $X^{v_{\underline{i}}}_{w_{\underline{j},1}}\neq\emptyset$, by proposition 5.4 we have $j_{t}=i_{t+1}+1$ or $j_{t}=i_{t+1}-1$. If $j_{t}=i_{t+1}+1$ then $w_{\underline{j},1}(\omega_{r})$ = $-\frac{3}{2}\alpha_{1}-\frac{1}{2}\alpha_{2}-\alpha_{3}-\sum_{k=1,\neq t}^{m-1}{\alpha_{i_{k+1}}}-\alpha_{i_{t+1}+1}$ and if $j_{t}=i_{t+1}-1$ then $w_{\underline{j},1}(\omega_{r})$ = $-\frac{3}{2}\alpha_{1}-\frac{1}{2}\alpha_{2}-\alpha_{3}-\sum_{k=1,\neq t}^{m-1}{\alpha_{i_{k+1}}}-\alpha_{i_{t+1}-1}$. Then any $u\in W^{I_{r}}$ such that $v_{\underline{i}}\leq u\leq w_{\underline{j},1}$ is of the form $u=(\displaystyle\prod_{\beta}s_{\beta})v_{\underline{i}}$, where $\beta$’s are some positive roots. For $j_{t}=i_{t+1}+1$ at most one $\beta$ can be $\alpha_{i_{t+1}}+\alpha_{i_{t+1}+1}$ and none of the other $\beta$’s contain $\alpha_{i_{t+1}}$ or $\alpha_{i_{t+1}+1}$ as a summand. So, in $u(\omega_{r})$ the coefficient of $\alpha_{i_{t+1}}$ is either zero or one and the coefficient of $\alpha_{i_{t+1}+1}$ is either zero or $-1$. Similarly for $j_{t}=i_{t+1}-1$ at most one $\beta$ can be $\alpha_{i_{t+1}}+\alpha_{i_{t+1}-1}$ and none of the other $\beta$’s contain $\alpha_{i_{t+1}}$ or $\alpha_{i_{t+1}-1}$ as a summand. So, in $u(\omega_{r})$ the coefficient of $\alpha_{i_{t+1}}$ is either zero or one and the coefficient of $\alpha_{i_{t+1}-1}$ is either zero or $-1$. For $j_{t}=i_{t+1}+1$, $u(\omega_{r})$ contains either $\alpha_{i_{t+1}}$ or $\alpha_{i_{t+1}+1}$ as a summand and for $j_{t}=i_{t+1}-1$, $u(\omega_{r})$ contains either $\alpha_{i_{t+1}}$ or $\alpha_{i_{t+1}-1}$ as a summand. So, like in previous cases here also we don’t have a non zero $T$-invariant section which is not identically zero on $X_{w}^{v}$.

For the pair $(v,w)=(v_{\underline{i}},w_{\underline{j}})$ with $\underline{i}\neq\underline{j}$ the proof is similar as in the cases in type $B$ and $C$. ∎

We illustrate Proposition 5.4 and Theorem 5.5 with an example.

Example: $D_{5}$, $\omega_{3}=(\frac{3}{2},\frac{3}{2},3,2,1)$

$v_{\underline{i}}$                   $v_{\underline{i}}(\omega_{3})$                   $w_{\underline{i}}(\omega_{3})$                        $w_{\underline{i}}$

$v_{(4)}$ = $(-1,5,-3,2,4)$       $(\frac{1}{2},\frac{1}{2},0,1,0)$       $(-\frac{1}{2},-\frac{1}{2},0,-1,0)$       $(1,5,-4,-2,3)=w_{(4)}$

$v_{(5)}$ = $(-1,3,-4,2,5)$       $(\frac{1}{2},\frac{1}{2},0,0,1)$       $(-\frac{1}{2},-\frac{1}{2},0,0,-1)$       $(1,3,-5,-2,4)=w_{(5)}$

$v_{1}$ = $(4,5,1,2,3)$               $(\frac{3}{2},\frac{1}{2},1,0,0)$       $(-\frac{1}{2},-\frac{3}{2},-1,0,0)$       $(4,5,-3,-2,1)=w_{2}$

$v_{2}$ = $(-4,5,-1,2,3)$         $(\frac{1}{2},\frac{3}{2},1,0,0)$       $(-\frac{3}{2},-\frac{1}{2},-1,0,0)$      $(-4,5,-3,-2,-1)=w_{1}$

So from the above observation, $X_{w_{(4)}}^{v_{(4)}}$, $X_{w_{(5)}}^{v_{(4)}}$, $X_{w_{(4)}}^{v_{(5)}}$, $X_{w_{(5)}}^{v_{(5)}}$, $X_{w_{1}}^{v_{(4)}}$, $X_{w_{2}}^{v_{(4)}}$, $X_{w_{(4)}}^{v_{1}}$, $X_{w_{(4)}}^{v_{2}}$, $X_{w_{2}}^{v_{1}}$ and $X_{w_{1}}^{v_{2}}$ are all non-empty. We have $(X_{w_{(4)}}^{v_{(5)}})^{ss}_{T}(\mathcal{L}_{3})$, $(X_{w_{(5)}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{3})$, $(X_{w_{1}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{3})$, $(X_{w_{(4)}}^{v_{1}})^{ss}_{T}(\mathcal{L}_{3})$, $(X_{w_{(4)}}^{v_{2}})^{ss}_{T}(\mathcal{L}_{3})$, and $(X_{w_{2}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{3})$ are empty whereas $(X_{w_{(4)}}^{v_{(4)}})^{ss}_{T}(\mathcal{L}_{3})$, $(X_{w_{(5)}}^{v_{(5)}})^{ss}_{T}(\mathcal{L}_{3})$, $(X_{w_{1}}^{v_{2}})^{ss}_{T}(\mathcal{L}_{3})$ and $(X_{w_{2}}^{v_{1}})^{ss}_{T}(\mathcal{L}_{3})$ are non-empty.