# Combinatorial geometry of flag domains in $G/B$

Dissertation

zur Erlangung des Doktorgrades
der Naturwissenschaften
an der Fakultät für Mathematik
der

Ruhr-Universität Bochum

vorgelegt von

Faten Said Qarmout
nèe Abu-Shoga

(im
August 2017)

To everyone who wished the best for me.

## Acknowledgement

”Be a scientist, if you can not, be a student, if you can not, love
scientists, if not, don’t hate them”

It is a great honor to thank everyone who supported me in my scientific career
and helped me overcome the obstacles and challenges that have always faced me.
I would like to express my thanks to God Almighty, who has enlightened my life
and made me choose science to be my path.

Undertaking this Ph.D. has been a truly life-changing experience for me and it
would not have been possible to do without the support and guidance that I
received from my supervisors Prof. Dr. Peter Heinzner and Prof. Dr. Alan
Huckleberry. I take the opportunity to send rivers of thanks to them for
giving me the golden opportunity to come to Germany and complete my Ph.D.
study.

I would like to express the deep appreciation of Professor Peter Heinzner for
granting me first knowledge of the theory of Lie group and complex geometry.
Thank you very much for directing many useful and interesting seminars on
transformation groups and for making my time as a graduate student at the Ruhr
University of Bochum very fruitful. I would also like to thank you for your
support and assistance in passing all the challenges that I have encountered
during my studies, and forced me to work hard and to complete my research.

I would like to express the deep appreciation to Professor Alan Huckleberry
for his guidance and encouragement during my studies. He has been actively
interested in my work and has always been available to advise me. Thank you
for forcing me to work more and more and think in different ways. It would
have been very hard for me to understand combinatorial geometry and complete
my research without your observations and advice and your deep knowledge of
cycle spaces.

Besides my advisers, I would like to thank PD. Dr. St phanie Cupit-Foutou who
supported me in mathematical discussions and also fortunately by talking about
things other than my research. Also I would like to thank all my colleagues in
the Transformation Groups seminar.

I gratefully acknowledge the funding received towards my Ph.D. from the
Islamic Development Bank-The Merit Scholarship- during the first $3$ years in
my study. I am also grateful to the funding received from the Ruhr University
of Bochum.

I would also like to thank my parents for their wise advice and sympathetic
ear. They were always with me. I send my thanks to the candles that illuminate
my life, my parents, my brother, my sister, my husband and my children, Ahmed
and Mohammed.

###### Contents

  1. 1 Introduction
    1. 1.1 Preliminaries
    2. 1.2 The structure of the thesis
  2. 2 Basic ingredients
    1. 2.1 Introduction to the flags in $Z$
    2. 2.2 The real form $Sp(2n,\mathbb{R})$
      1. 2.2.1 Preliminaries
      2. 2.2.2 Base points
      3. 2.2.3 Weyl group
      4. 2.2.4 Flag domains and base cycles
    3. 2.3 The real form $SO^{*}(2n)$
      1. 2.3.1 Preliminaries
      2. 2.3.2 Base points
      3. 2.3.3 Weyl groups, flags domains and base cycles
    4. 2.4 The real form $SO(p,q)$
      1. 2.4.1 Preliminaries
      2. 2.4.2 Base points
      3. 2.4.3 Weyl groups, flag domains and base cycles
    5. 2.5 The real form $Sp(2p,2q)$
      1. 2.5.1 Preliminaries
      2. 2.5.2 Base points
      3. 2.5.3 Weyl group, flag domains and base cycles
  3. 3 Fixed Point Theorem
    1. 3.1 Introduction
    2. 3.2 Description of the $B_{I}$-orbits
    3. 3.3 The Moving Lemma and a Fixed Point Theorem
  4. 4 Cycle intersection for $SP(2n,\mathbb{R})$
    1. 4.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$
    2. 4.2 Intersection points of Schubert duality
  5. 5 Cycle intersection for $SO^{*}(2n)$
    1. 5.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$
      1. 5.1.1 Length of the elements of ${\displaystyle S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$
    2. 5.2 Intersection points of Schubert duality
  6. 6 Cycle intersection for $SO(p,q)$
    1. 6.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$
    2. 6.2 Introduction to the combinatorics
    3. 6.3 Intersection points of Schubert duality
  7. 7 Cycle intersection for $SP(2p,2q)$
    1. 7.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$
    2. 7.2 Introduction to the combinatorics
    3. 7.3 Intersection points of Schubert duality

##  Chapter 1 Introduction

###  1.1 Preliminaries

Let $G$ denote a complex semisimple Lie group with real form $G_{0}$ and
$Z:=G/Q$ be a $G$-flag manifold. It is known that $G_{0}$ has only finitely
many orbits in $Z$ and that there is a unique closed orbit $\gamma^{cl}$. The
open $G_{0}$-orbits are referred to as _flag domains_ $D$. Every maximal
compact subgroup $K_{0}$ of $G_{0}$ has a unique orbit $C_{0}$ in $D$ which is
a complex submanifold. This is the unique $K_{0}$-orbit of minimal dimension
in $D$. If $K$ is the corresponding (reductive) subgroup of $G$, then it can
also be characterized as the unique $K$-orbit which is contained in $D$.
Having fixed $K_{0}$, we refer to $C_{0}$ as the associated _base cycle_. For
the above and other background information see ([7] and [15]).

The main goal of our work can be formulated as that of determining the class
of $C_{0}$ in the (topological) cohomology ring of $Z$. If $B$ is any Borel
subgroup of $G$ and $r:=dim_{\mathbb{C}}C_{0}$, this amounts to determining
the intersection numbers of the $r$-codimensional $B$-Schubert varieties $S$
with $C_{0}$. Of course it is possible that this intersection is empty.

The original motivation for this project was another: If $B$ is maximally real
in the sense that it fixes a point in $\gamma^{cl}$, or equivalently if it
contains the $A_{0}N_{0}$-factor of an Iwasawa-decomposition
$G_{0}=K_{0}A_{0}N_{0}$, and $S$ is a $B$-Schubert variety (we refer to such
as Iwasawa-Schubert varieties) of complementary dimension to the cycle
$C_{0}$, then $S\cap D$ is contained in the open $B$-orbit $\mathcal{O}$ in
$S$ and the intersection $\mathcal{O}\cap C_{0}$ is finite and transversal at
each of its points (see [7]). In fact each component $\Sigma$ of
$\mathcal{O}\cap D$ intersects $C_{0}$ in exactly one point $z_{\Sigma}$ with
$\Sigma=A_{0}N_{0}.z_{\Sigma}$. This in turn gives rise to an optimal Barlet-
Koziarz trace-transform from holomorphic functions on $\mathcal{O}\cap D$ to
holomorphic functions on the appropriately defined space $\mathcal{C}(D)$ of
cycles (see [3], [7]). Thus, although the determination of the
$r$-codimensional Iwasawa-Schubert varieties which have non-empty intersection
with $D$, along with their points of intersection with $C_{0}$, is apparently
a problem of a combinatorial nature, such information has complex analytic
significance.

Since $S\cap D\neq\emptyset$ implies that $S\cap C_{0}\neq\emptyset$, the
first goal of this project is to determine which Schubert varieties $S$ have
nonempty intersection with $C_{0}$. After doing so, we then describe this
intersection in the case where $S$ is of complementary dimension to $C_{0}$.
The Schubert varieties are determined by the elements of the Weyl group
$W_{I}$ of a distinguished maximal torus $T_{I}$ in the Borel subgroup $B_{I}$
which fixes a certain base point in $\gamma_{c\ell}$. These Schubert varieties
are denoted by $S_{w}$ where $w\in W_{I}$. Consequently our results in are
formulated in terms of conditions on elements $w$ in the Weyl group $W_{I}$.

In this thesis we make several restrictions. First, we only consider the case
where $G$ is a classical group, i.e., associated to one of the Lie algebras
$A_{n}$, $B_{n}$, $C_{n}$ or $D_{n}$. The case of $A_{n}$ (and all of its real
forms) was handled by Brecan ([4],[5]). Our work here is therefore devoted to
the cases of $B_{n}$, $C_{n}$ and $D_{n}$. Furthermore, we restrict to the
setting where $Z=G/B$ is the $G$-manifold of full flags. This is reflective of
our work with the advantage that technical complications are minimized. The
case of a general $G$-flag manifold $Z=G/Q$ is handled in author paper ([1]).

###  1.2 The structure of the thesis

As indicated above this thesis is devoted to study the combinatorial geometry
of the flag domains in $G/B$ of the real forms $SP(2n,\mathbb{R}),SO^{*}(2n),$
$SO(p,q)$ and $SP(2p,2q)$, where $B$ is Borel subgroup. We study the
intersection between the base cycles of the flag domains and the Schubert
varieties. In the cases of $SP(2n,\mathbb{R})$ and $SO^{*}(2n)$ a remarkable
unicity appears so that an exact description of the Weyl elements can be
given. Analogous to the case of $SU(p,q)$ in the work of Brecan, except for a
few interesting special cases, for $SO(p,q)$ and $SP(2p,2q)$ these elements
are described by algorithms. In all cases the number of intersection points
with $C_{0}$ is explicitly computed.

The work here is organized in six chapters which are briefly described as
follows:

Chapter $2$: In this chapter we explain the formulas for all real forms
$G_{0}$ which arise in this thesis. In each case two base points are
introduced: a base point $F_{I}$ in $\gamma_{c\ell}$ and a base point $F_{S}$
in a certain standard cycle. These two base points are full flags associated
to bases which define the maximal Tori $T_{I}$ and $T_{S}$ respectively. The
Borel subgroup $B_{I}$ and the Weyl group $W_{I}$ are described in terms of
the ordered basis which defines $F_{I}$. Also, we define a special way of
writing the Weyl group elements for the Lie groups of type $B_{n},C_{n}$ and
$D_{n}$ in terms of flags. In each case the flag domains and base cycles are
described by signature invariants.

Chapter $3$: Here we prove two general fixed point theorems:

######  Theorem 1.1.

For every $w\in W$, if the intersection $S_{w}\cap C_{0}$ is non-empty, then
it contains a $T_{S}$-fixed point.

######  Theorem 1.2.

If $S_{w}$ is of complementary dimension to $C_{0}$, then $S_{w}\cap
C_{0}\subset Fix(T_{S})$.

In the following chapters we focus on the goal of understanding the
combinatorial geometry of the topological class $[C_{0}]$ by attempting to
find precises formulas which answer the following question in each of the
cases $B_{n}$, $C_{n}$ and $D_{n}$.

######  Question 1.

What are the conditions on the Weyl elements $w$ which parametrize the
Iwasawa-Schubert Varieties $S_{w}$ with non-empty intersection with the base
cycle $C_{0}$?

######  Question 2.

What are the points of intersection? How many of these points do we have?

Chapter $4$: This chapter is concerned with the case of the real form
$SP(2n,\mathbb{R})$. In this case there is a certain Weyl element, which we
refer to as a ”super generous permutation”. Since we are considering flag
domains in $G/B$, all base cycles have the same dimension. In terms of a super
generous permutation the main result of this chapter can be stated as follows.

######  Theorem 1.3.

If $S_{w}$ is an Iwasawa-Schubert variety of complementary dimension to a base
cycle $C_{0}$ and $S_{w}\cap C_{0}\not=\emptyset$, then $w$ is the super
generous permutation. Furthermore, the intersection $S_{w}\cap C_{0}$ with
every base cycle consists of just one point.

Chapter $5$: This chapter is devoted to study the real form $SO^{*}(2n)$ which
is analogous to that for the case of the real form $SP(2n,\mathbb{R})$. Again
a uniqueness theorem of the above type is valid, but in this case the results
depend on $n$ being even or odd.

Chapter $6$: The real form of this chapter is $SO(p,q)$. The results are
stated in terms of algorithms (See definitions 6.1.1, 6.2.1 and Corollary
6.3.2) ; in fact it seems impossible to avoid this. Also it would seem to be
difficult to find a concrete formula for the intersection points in the
general case. We do however explicitly compute the numbers of intersection
points (see Theorem 6.3.3 and Corollary 6.3.2).

Chapter $7$: This chapter is devoted to the real form $SP(2p,2q)$. Our work
here is similar to that for $SO(p,q)$. Again the results are stated in terms
of algorithms (Corollary 7.3.2, Definition 7.1.1 and Definition 7.2.1) and
again it would seem to be difficult to find a concrete formula for the
intersection points in the general case. We do, however, explicitly compute
the numbers of intersection points (see Theorem 7.3.3).

##  Chapter 2 Basic ingredients

The Lie subgroup $G_{0}$ is a real form of the classical group $G$ which is
assumed to be of type $B_{n}$, $C_{n}$ or $D_{n}$, and $Z$ is in each case the
associated manifold of full flags. In each case we introduce appropriate bases
of the relevant vector space so that $G$, $G_{0}$ and the compact and
reductive groups, $K_{0}$ and $K$, are described in matrix terms which are
useful for our purposes. Regarded as flags, these bases define base points in
$Z$ along with Borel subgroups with distinguished maximal tori. Of particular
importance is the base point $F_{I}$ in the closed $G_{0}$-orbit, because the
Iswasawa-Schubert cells which will be considered are orbits of its isotropy
group $B_{I}$ of points $w(F_{I})=:F_{w}$, where $w$ runs through the Weyl
group $W_{I}$ defined by the distinguished maximal torus $T_{I}$ in $B_{I}$.
Finally, in each of the cases we describe the flag domains $D$ in $Z$ by
signature invariants and the base cycle $C_{0}$ in terms of splittings of $V$.

###  2.1 Introduction to the flags in $Z$

Let $G$ be a semisimple Lie group of type $B_{n},C_{n}$ or $D_{n}$, defined in
each case by a non-degenerate bilinear form $b(v,w)$. A maximally
$b$-isotropic full flag $F$ with respect to $b$ is a sequence of
$(m+1)$-vector spaces

$F=(\\{0\\}=V_{0}\subset V_{1}\subset...\subset V_{m}=\mathbb{C}^{m})$

such that $dim~{}V_{i}=i,~{}for~{}all~{}0\leq i\leq m$ and
$V_{m-i}=V_{i}^{\bot},~{}\text{for}~{}1\leq
i\leq\left\lfloor\frac{m}{2}\right\rfloor$, where $m=2n$ or $2n+1$. In
particular $V_{i}$ is isotropic for $1\leq i\leq n$. The set of all maximally
isotropic flags is denoted by $Z$. The manifold structure of $Z$ arises from
the fact that the Lie group $G$ acts transitively on it with $Z=G/B$ where $B$
is the stabilizer of any particular maximally $b$-isotropic flag in $Z$.

In all cases $\sigma:\mathbb{C}^{2n}\to\mathbb{C}^{2n}$, $u\mapsto\bar{u}$, is
standard complex conjugation and $G_{0}$ is defined to be the fixed set of
$\sigma$,i.e,

$G_{0}=\\{g\in G:\sigma(g)=g\\}.$

The signature of a flag $F=(0\subset V_{1}\subset...\subset V_{m})\in Z$ with
respect to $h$ consists of three sequences

$a:0\leq a_{1}\leq a_{2}\leq...\leq a_{m}$ $b:0\leq b_{1}\leq b_{2}\leq...\leq
b_{m}$ $d:0\leq d_{1}\leq d_{2}\leq...\leq d_{m}$

with $sign(V_{i})=(a_{i},b_{i},d_{i})$ where $a_{i}$ (resp. $b_{i}$) denotes
the dimension of a maximal negative (positive) subspace and $d_{i}$ the
degeneracy $V_{i}$ of the restriction of $h$ to $V_{i}$ and
$a_{i}+b_{i}+d_{i}=i~{}~{}\forall 1\leq i\leq 2n$. The 3-tuple $(a,b,d)$ is
called the signature of the flag $F$. If $d\not=0$, we refer to $F$ as being
non-degenerate and write $sign(F)=(a,b)$.

###  2.2 The real form $Sp(2n,\mathbb{R})$

####  2.2.1 Preliminaries

In order to define the symplectic group $G=Sp(2n,\mathbb{C})$ we introduce
$J_{n}$ to be

$J_{n}=\left(\begin{array}[]{ccccc}0&0&\cdots&0&1\\\ 0&0&\cdots&1&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&1&\cdots&0&0\\\
1&0&\cdots&0&0\end{array}\right)$

and $b(v,w)=v^{t}\left(\begin{array}[]{cc}0&J_{n}\\\
-J_{n}&0\end{array}\right)w.$ Then

$Sp(2n,\mathbb{C}):=\\{g\in
GL(2n,\mathbb{C}):g^{t}\left(\begin{array}[]{cc}0&J_{n}\\\
-J_{n}&0\end{array}\right)g=\left(\begin{array}[]{cc}0&J_{n}\\\
-J_{n}&0\end{array}\right)\\}.$

The Hermitian form $h(v,w)$ of the signature $(n,n)$ is defined by

$h(v,w)=b(v,\sigma(w))=\sum_{i=1}^{n}v_{i}\bar{w}_{i}-\sum_{i=n+1}^{2n}v_{i}\bar{w}_{i}\,$

and the real form $G_{0}=Sp(2n,\mathbb{R})=Sp(2n,\mathbb{C})\cap U(n,n)$.

Here a Cartan involution $\theta$ is given by

$\theta:g\longrightarrow\left(\begin{array}[]{cc}I_{n}&0\\\
0&-I_{n}\end{array}\right)g\left(\begin{array}[]{cc}I_{n}&0\\\
0&-I_{n}\end{array}\right)$

and the maximal compact subgroup associated to it is

$K_{0}=\\{\left(\begin{array}[]{cc}U&0\\\
0&J_{n}(U^{t})^{-1}J_{n}\end{array}\right):U\in U(n,\mathbb{C})\\}\approx
U(n)$

####  2.2.2 Base points

The standard basis $(e_{1},e_{2},...,e_{2n})$ of $\mathbb{C}^{2n}$ defines the
standard maximal torus $T_{S}$ as the subgroup of diagonal matrices, i.e,

$T_{S}=\\{g\in
G_{0}:g=diag(t_{1},t_{2},...,t_{n},-t_{n},...,-t_{2},-t_{1}),t_{i}\in\mathbb{C}\\}$

and the standard Borel subgroup $B$ of upper triangular matrices in
$Sp(2n,\mathbb{R})$.

The standard basis defines a base point $F_{S}$ in $Z$ as the associated flag

$<e_{1}>\subset<e_{1},e_{2}>\subset...\subset<e_{1},...,e_{2n}>$

Reorderings on this basis will determine the flag domains, the base cycles and
the intersection points.

In order to define a base point in $\gamma^{c\ell}$ we introduce the basis

$e_{1}-e_{2n},e_{2}-e_{2n-1}...,e_{n}-e_{n+1},e_{n}+e_{n+1},...,e_{2}+e_{2n-1},e_{1}+e_{2n}$
(2.1)

with $T_{I}$ being the maximal torus of diagonal matrices in $G$ and $B_{I}$
the Iwasawa-Borel subgroup of $G$ which fixes the associated flag $F_{I}$.

####  2.2.3 Weyl group

If $(r_{1},\ldots,r_{n},s_{1},\ldots,s_{n})$ is a basis with
$b(r_{i},r_{j})=0$, $b(s_{i},s_{j})=0$ and $b(r_{i},s_{j})=\delta_{ij}$, then
we regard $V=U\oplus U^{*}$ with $U=Span\\{r_{1},\ldots,r_{n}\\}$ and
$U^{*}=Span\\{s_{1},\ldots,s_{n}\\}$. For such a basis $b$ is in _canonical
form_. If $T$ is the maximal torus in $G$ defined by such a basis, then the
Weyl group $W(T)$ acts on $T$ by certain permutations of the basis. These are
of the form $w=\pi E$ where $\pi$ is an arbitrary permutation of
$(r_{1},\ldots,r_{n})$ and $E$ exchanges an arbitrary number of the $r_{i}$
with the corresponding dual vectors $s_{i}$. If we have a maximally isotropic
flag associated to some permutation of the ordered basis
$(r_{1},\ldots,r_{n},s_{n},\ldots,s_{1})$, then in the first $n$ positions of
the flag the above condition holds: if $r_{i}$ appears, then $s_{i}$ does not,
and vice versa. Now the full flag is determined by the first $n$ positions;
regard it as a permutation of the $r_{i}$ and, if $s_{i}$ occurs instead of an
$r_{i}$, we mark that spot with a minus sign. In this way $W(T)$ is the
semidirect product ${\displaystyle S_{n}\ltimes\mathbb{Z}_{2}^{n}}$ of a
symmetric group and a normal elementary abelian 2-subgroup. If the basis in
question is that in (2.1) above, then we denote this Weyl group by $W_{I}$.
Analogously, if the basis in question is the standard basis
$e_{1},...,e_{2n}$, then we denote this Weyl group by $W_{T_{S}}$.

In our particular case there are two maximal tori, the maximal torus $T_{I}$
defined by 2.1 and the maximal torus $T_{S}$ defined by
$(e_{1},e_{2},...,e_{2n})$. Recall that these two maximal tori $T_{I},T_{S}$
are conjugate, and a conjugation induces an isomorphism $\psi$ of the
associated Weyl groups. In this case we define a bijective map
$\psi(\mp(i))=\pm i,$ for all $1\leq i\leq n$ for later use.

####  2.2.4 Flag domains and base cycles

Since in every flag domain $D$ there is a unique closed $K$-orbit, the
associated base cycle $C_{0}$, in order to parametrize the flag domains it is
enough it is enough to parametrize the closed $K$-orbits, (see [7] , $\S 4.2$
and $\S 4.3$). For this we choose a base point $F_{S}\in Fix(T_{S})$. It
follows that the set $\mathcal{C}$ of closed $K$-orbits can be identified with
the Weyl group orbit $W_{G}(T_{S}).F_{S}$ (see [7], Corollary 4.2.4). As we
have observed above, $W_{G}(T_{S})\cong S_{n}\ltimes\mathbb{Z}_{2}^{n}$ where
the $S_{n}$ factor can be identified with $W_{K}(T_{S})$. In fact it is
exactly the stabilizer of the $K.F_{S}$ in $\mathcal{C}$. Consequently
$\mathcal{C}$ can be identified with $\mathbb{Z}_{2}^{n}.F_{S}$. We regard an
element of $\mathbb{Z}_{2}^{n}$ as a vector $\alpha$ of length $n$ consisting
of plus- and minus-signs, and thus $\mathcal{C}$ can be identified with the
set of such vectors. In concrete terms the $K$-orbits (resp. $G_{0}$-orbits )
of base points $(\pm e_{1},\ldots,\pm e_{n})$ are exactly the base cycles
(resp. flag domains) in $Z$. The respective flag domains are denoted by
$D_{\alpha}$.

Observe that $\alpha$ defines a non-degnerate signature $(a,b)$ and that
$D_{\alpha}\subset D_{a,b}:=\\{F:sign(F)=(a,b)\\}$. Conversely, given $F$ with
$sign(F)=(a,b)$ we may choose a basis $(v,w):=(v_{1},\ldots,v_{n},w_{n},\ldots
w_{1})$ which defines $F$ and which has the following properties:

  1. 1.

$b$ is in canonical form.

  2. 2.

$(v,w)$ is $h$-orthogonal.

  3. 3.

$\|v_{j}\|^{2}_{h}=\pm 1$ with $sgn(\|v_{j}\|_{h})=\alpha_{j}$.

  4. 4.

$\|w_{j}\|^{2}_{h}=-\|v_{j}\|^{2}_{h}$.

If $sign(F)=sign(\hat{F})$, then we choose bases $(v,w)$ and
$(\hat{v},\hat{w})$ for $F$ and $\hat{F}$, respectively, and note that the
transformation which takes the one basis to the other is in $G_{0}$. This
argument shows that $D_{a,b}=D_{\alpha}$.

Denote $E_{+}=<e_{1},...,e_{n}>$, and $E_{-}=<e_{n+1},...,e_{2n}>$, where
$\mathbb{C}^{2n}=E_{+}\oplus E_{-}$.

######  Proposition 2.2.1.

For a fixed open orbit $D_{\alpha}$, the base cycle $C_{\alpha}$ is the set

$C_{\alpha}=\\{F\in Z:V_{i}=(V_{i}\cap E^{-})\oplus(V_{i}\cap E^{+}),1\leq
i\leq 2n\\}.$

###### Proof.

Firstly, it was shown above that there is a base point $F_{\alpha}$ with the
splitting condition. Since $K_{0}$ is the product $K_{0}=K_{0}^{+}\times
K_{0}^{-}$, it acts transitively on all such flags. Let $F\in C_{\alpha}$
where $V_{i}^{-}=V_{i}\cap E^{-}$ and $V_{i}^{+}=V_{i}\cap E^{+}$. Define
$F^{+}$ and $F^{-}$ to be the sets of maximal isotopic flags in $E^{+}$ and
$E^{-}$ respectively. Recall that $K_{0}=K_{0}^{+}\times K_{0}^{-}$ where
$K_{0}^{+}$ and $K_{0}^{-}$ act transitively on the sets $F^{+}$ and $F^{-}$
respectively, which implies that $K_{0}$ acts transitively on the set of
maximal isotropic flags obtained by put flags from $F^{+}$ and $F^{-}$ in away
such that the new flag has signature $\alpha$. Hence $C_{\alpha}$ is a complex
manifold. But $K_{0}$ has a unique orbit in $D_{\alpha}$ which is a complex
manifold. Therefor $C_{\alpha}$ is the base cycle. ∎

###  2.3 The real form $SO^{*}(2n)$

####  2.3.1 Preliminaries

Let $G=SO(2n,\mathbb{C})$ be the special orthogonal group defined by the non-
degenerate complex bilinear form
$b:\mathbb{C}^{2n}\longrightarrow\mathbb{C}^{2n}$ where
$b(v,w)=\sum_{i=1}^{2n}v_{2n-i+1}w_{i}$. The real form $G_{0}=SO^{*}(2n)$ is
the subgroup of elements of $G$ which leave invariant the Hermitian form
defined by

$h(v,w)=b(v,\bar{w})=\sum_{i=1}^{n}v_{i}\bar{w}_{i}-\sum_{i=n+1}^{2n}v_{i}\bar{w}_{i}.$

This real form is the fixed point set of the involution
$\tau:SO(2n,\mathbb{C})\longrightarrow SO(2n,\mathbb{C})$ defined by

$\tau(g)=\left(\begin{array}[]{cc}0&J_{n}\\\
J_{n}&0\end{array}\right)\bar{g}^{-t}\left(\begin{array}[]{cc}0&J_{n}\\\
J_{n}&0\end{array}\right),~{}~{}~{}where~{}J_{n}=\left(\begin{array}[]{ccccc}0&0&\cdots&0&1\\\
0&0&\cdots&1&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&1&\cdots&0&0\\\
1&0&\cdots&0&0\end{array}\right).$

Note that $SO^{*}(2n)\cong SO(2n,\mathbb{C})\cap SU(n,n)$.
A Cartan involution $\theta:G_{0}\longrightarrow G_{0}$ is given by

$\theta:g\longrightarrow\left(\begin{array}[]{cc}I_{n}&0\\\
0&-I_{n}\end{array}\right)g\left(\begin{array}[]{cc}I_{n}&0\\\
0&-I_{n}\end{array}\right)$

and the maximal compact subgroup associated to it is

$K_{0}=\\{\left(\begin{array}[]{cc}U&0\\\
0&-J_{n}(U^{t})^{-1}J_{n}\end{array}\right):U\in U(n)\\}\approx U(n)$

####  2.3.2 Base points

By an argument which is completely analogous to that in the case of
$Sp(2n,\mathbb{R})$ the standard basis $(e_{1},e_{2},...,e_{2n})$ of
$\mathbb{C}^{2n}$ defines the standard maximal torus $T_{S}$, i.e,

$T_{S}:=\\{g\in
G_{0}:g=diag(t_{1},t_{2},...,t_{n},-t_{n},...,-t_{2},-t_{1}),t_{i}\in\mathbb{C}\\}$

and the standard Borel subgroup $B$ of upper triangular matrices in
$SO^{*}(2n)$.

The standard basis defines a base point $F_{S}$ in $Z$ as the associated flag

$<e_{1}>\subset<e_{1},e_{2}>\subset...\subset<e_{1},...,e_{2n}>$

Reorderings on this basis will determine the flag domains, the base cycles and
the intersection points.

In order to define a base point in $\gamma^{c\ell}$ we introduce the basis

$e_{1}+ie_{n+1},e_{2}+ie_{n+2}...,e_{n}+ie_{2n},e_{n}-ie_{2n},...,e_{2}-ie_{n+2},e_{1}-ie_{n+1},~{}~{}\text{if~{}n~{}is~{}even};$
(2.2)

or

$e_{1}+ie_{n+1},...,e_{n}+ie_{2n},e_{\frac{n+1}{2}},e_{n+\frac{n+1}{2}},e_{n}-ie_{2n},...,e_{1}-ie_{n+1},~{}~{}\text{if~{}n~{}is~{}odd}.$
(2.3)

with $T_{I}$ being the maximal torus of diagonal matrices in $G$ and $B_{I}$
the Iwasawa-Borel subgroup of $G$ which fixes the associated flag $F_{I}$.

####  2.3.3 Weyl groups, flags domains and base cycles

By arguments which are essentially the same as those in $\S\ref{Weyl}$, if $T$
is a maximal torus defined by a basis of $(r,s)$-type, then
$W(T)=S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ where now the sign-change vector
$\alpha$ has only an even number of minuses. As in the case of
$Sp(2n,\mathbb{R})$, to describe the flag domains and their $K_{0}$-base
cycles we use the Weyl group $W(T_{S})$ of the standard torus $T_{S}$. The
base point for $\alpha$ consisting only of pluses is the flag $F_{S}$ is
defined by the standard basis. Then all base points are $F_{\alpha}:=w(F_{S})$
where $w\in\mathbb{Z}_{2}^{n-1}$ is associated to the sign-change vector
$\alpha$. As before the base cycles and flag domains are the orbits
$K.F_{\alpha}$ and $G_{0}.F_{\alpha}$, respectively, and $\alpha$ defines a
non-degenerate sign with $sign(F_{\alpha})=(a,b)$ and then
$D_{\alpha}=D_{a,b}$.

In our particular case there are two maximal tori, the maximal torus $T_{I}$
defined by (2.2) or (2.3) and the maximal torus $T_{S}$ defined by
$(e_{1},e_{2},...,e_{2n})$. Recall that these two maximal tori $T_{I},T_{S}$
are conjugate, and a conjugation induces an isomorphism $\psi$ of the
associated Weyl groups. In this case we define a bijective map
$\psi(\mp(i))=\pm i,$ for all $1\leq i\leq n-1$ and $\psi(\pm n)=\pm n$. .

Denote $E_{+}=<e_{1},...,e_{n}>$, and $E_{-}=<e_{n+1},...,e_{2n}>$, where
$\mathbb{C}^{2n}=E_{+}\oplus E_{-}$. Again, by an argument which is completely
analogous to Proposition 2.2.1 of the case $SP(2n,\mathbb{R})$, the base cycle
$C_{0}$ of the flag domain $D_{\alpha}$ of $SO^{*}(2n)$ is the set

$C_{0}=\\{F\in Z:V_{i}=(V_{i}\cap E^{-})\oplus(V_{i}\cap E^{+}),1\leq i\leq
2n\\}.$

which is convenient to use the orbit $\mathbb{Z}_{2}^{n-1}.F_{S}$ of the base
point as the set of intersection point of $S_{w}\cap D_{\alpha}$. The
intersection dimensions are determined by $\alpha$ (resp. $(a,b)$). See
section 2.2.4 for more details.

###  2.4 The real form $SO(p,q)$

####  2.4.1 Preliminaries

Consider the semisimple Lie group $G=SO(m,\mathbb{C})$ where $m=2n$ or $2n+1$.
In this case it is convenient to choose the bilinear form $b$ depending on the
$p$ and $q$. If $p$ or $q$ is even, then we choose it in the usual way:

$b(v,w)=-\sum_{i=1}^{q}v_{i}w_{i}+\sum_{i=q+1}^{m}v_{i}w_{i}$

Let $\sigma$ be the complex conjugation $\sigma(v)=\bar{v}$, then the
Hermitian form $h(v,w)$ of signature $(p,q)$ which defines the real form is
defined by

$h(v,w)=b(v,\sigma(w))=-\sum_{i=1}^{q}v_{i}\sigma(w_{i})+\sum_{i=q+1}^{m}v_{i}\sigma(w_{i}).$

If both $p$ and $q$ are odd, then the complex bilinear form is

$b(v,w)=-\sum_{i=1}^{q-1}v_{i}w_{i}+(v_{q}w_{q+1}+v_{q+1}w_{q})+\sum_{i=q+2}^{m}v_{i}w_{i}$

and the Hermitian form $h(v,w)$ is defined by

$h(v,w)=-\sum_{i=1}^{q}v_{i}\bar{w}_{i}+\sum_{i=q+1}^{m}v_{i}\bar{w}_{i}.$

The real form is $G_{0}=SO(p,q)=SO(2n,\mathbb{C})\cap SU(p,q)$.

A Cartan involution
$\theta:\mathfrak{so}(p,q)\longrightarrow\mathfrak{so}(p,q)$ in the Lie
algebra level is given by $\theta(g)=-g^{t}$, and the maximal compact subgroup
associated to it is $K_{0}:=S(O(p)\times O(q))\subset S(U(p)\times U(q))$.

####  2.4.2 Base points

In the following we define bases depending on $p$ and $q$ as well as the
positive and negative spaces which will be essential for understand the base
cycle. The fundamental maximal torus $T_{S}$ is defined in each case by
requiring that each basis vector is a $T_{S}$-eigenvector.

$\bullet$ If $m=2n$ and $q$ is even:

The ordered $b$-isotropic basis is

$\displaystyle
e_{1}+ie_{2},e_{3}+ie_{4},......,e_{2n-1}+ie_{2n},e_{2n-1}-ie_{2n},......,e_{1}-ie_{2},$
(2.4)

where
$E^{-}:=<e_{1}+ie_{2},e_{3}+ie_{4},...,e_{q-1}+ie_{q},e_{1}-ie_{2},e_{3}-ie_{4},...,e_{q-1}-ie_{q}>$
and
$E^{+}:=<e_{q+1}+ie_{q+2},...,e_{2n-1}+ie_{2n},e_{q+1}-ie_{q+2},...,e_{2n-1}-ie_{2n}>$.

$\bullet$ If $m=2n$ and $q$ is odd:

The ordered $b$-isotropic basis is

$\displaystyle
e_{1}+ie_{2},e_{3}+ie_{4},......,e_{2n-1}+ie_{2n},e_{q},e_{q+1},e_{2n-1}-ie_{2n},......,e_{1}-ie_{2},$
(2.5)

where
$E^{-}=<e_{1}+ie_{2},e_{3}+ie_{4},...,e_{q},e_{1}-ie_{2},e_{3}-ie_{4},...,e_{q-1}-ie_{q}>$
and
$E^{+}=<e_{q+1},e_{q+2}+ie_{q+3},...,e_{2n-1}+ie_{2n},e_{q+2}-ie_{q+3},...,e_{2n-1}-ie_{2n}>$.
Note that the vectors $e_{q}$ and $e_{q+1}$ are isotropic and
$b(e_{q},e_{q+1})=-1$. Moreover $h(e_{3},e_{3})=-1$ and $h(e_{4},e_{4})=1$
which means that $e_{q}\in E^{-}$ and $e_{q+1}\in E^{+}$.

$\bullet$ If $m=2n+1$ and $q$ is even:

The ordered $b$-isotropic basis is

$\displaystyle
e_{1}+ie_{2},e_{3}+ie_{4},......,e_{2n}+ie_{2n+1},e_{q+1},e_{2n}-ie_{2n+1},......,e_{1}-ie_{2},$
(2.6)

where
$E^{-}:=<e_{1}+ie_{2},e_{3}+ie_{4},...,e_{q-1}+ie_{q},e_{1}-ie_{2},e_{3}-ie_{4},...,e_{q-1}-ie_{q}>$
and
$E^{+}:=<e_{q+1},e_{q+2}+ie_{q+3},...,e_{2n}+ie_{2n+1},e_{q+2}-ie_{q+3},...,e_{2n}-ie_{2n+1}>$.

$\bullet$ If $m=2n+1$ and $q$ is odd:

The ordered $b$-isotropic basis is

$\displaystyle
e_{1}+ie_{2},e_{3}+ie_{4},......,e_{2n}+ie_{2n+1},e_{q},e_{2n}-ie_{2n+1},......,e_{1}-ie_{2},$
(2.7)

where
$E^{-}=<e_{1}+ie_{2},e_{3}+ie_{4},...,e_{q},e_{1}-ie_{2},e_{3}-ie_{4},...,e_{q-2}-ie_{q-1}>$
and
$E^{+}=<e_{q+1}+ie_{q+2},...,e_{2n}+ie_{2n+1},e_{q+1}-ie_{q+2},...,e_{2n}-ie_{2n+1}>$.

######  Remark 2.4.1.

If $m=2n+1$ we have the vector $e_{q}$ or $e_{q+1}$ in the ordered
$b$-isotropic basis. In this case the vector sits in a fixed position in the
middle of the basis and the terms at the beginning skip over $e_{q}$ or
$e_{q+1}$.

The above bases define the standard maximal Tours $T_{S}$ in each case as the
subgroup of diagonal matrices, i.e,

$T_{S}=\\{g\in
G_{0}:g=diag(t_{1},t_{2},...,t_{n},-t_{n},...,-t_{2},-t_{1}),t_{i}\in\mathbb{C}\\}~{}~{}\text{
if $m=2n$},$

or

$T_{S}=\\{g\in
G_{0}:g=diag(t_{1},t_{2},...,t_{n},0,-t_{n},...,-t_{2},-t_{1}),t_{i}\in\mathbb{C}\\}~{}~{}\text{
if $m=2n+1$}.$

For any of the ordered bases above denote by $F_{S}$ the associated flag in
$Z$. Reorderings on these bases will determine the flag domains, the base
cycles and the intersection points.

Just as in the case of $T_{S}$, the maximal Torus $T_{I}$ is defined to have a
certain basis of eigenvectors which depends on $m$ being odd or even.

  * •

If $m$ is even, then the basis is

$\displaystyle
e_{1}+e_{2q},...,e_{q}+e_{q+1},e_{2q+1}+ie_{2q+2},e_{2q+3}+ie_{2q+4},...,e_{2n-1}+ie_{2n},$
$\displaystyle
e_{2n-1}-ie_{2n},...,e_{2q+3}-ie_{2q+4},e_{2q+1}-ie_{2q+2},e_{q}-e_{q+1},...,e_{1}-e_{2q}.$
(2.8)

  * •

If $m$ is odd, then the basis is

$\displaystyle
e_{1}+e_{2q},e_{2}+e_{2q-1},...,e_{q}+e_{q+1},e_{2q+1},e_{2q+2}+ie_{2q+3},e_{2q+4}+ie_{2q+5},...,e_{2n}+ie_{2n+1},$
$\displaystyle
e_{2q+1},e_{2n}-ie_{2n+1},...,e_{2q+4}-ie_{2q+5},e_{2q+2}-ie_{2q+3},e_{q}-e_{q+1},...,e_{2}-e_{2q-1},e_{1}-e_{2q}~{}~{}~{}$
. (2.9)

and $B_{I}$ the Iwasawa-Borel subgroup of $G$ which fixes the associated flag
$F_{I}$.

####  2.4.3 Weyl groups, flag domains and base cycles

Let us use shorthand notation for the bases used above. By $(r,s)-form$ we
mean a basis $(r_{1},\ldots,r_{n},s_{n},\ldots s_{1})$ where
$b(r_{i},r_{i})=b(s_{i},s_{i})=0$ and $b(r_{i},s_{i})=\delta_{ij}$ for all $i$
and $j$. This occurs in the even-dimensional case. By $(r,t,s)$-form, which
occurs in odd-dimensional case, the $r_{i}$ and $s_{i}$ satisfy the same
conditions and $t$ is a single vector with $b(t,t)=\pm 1$ and
$b(t,r_{i})=b(t,s_{i})=0$ for all $i$. Here we discuss the Weyl groups
$W(T_{S})$ and $W(T_{I})$ by their actions on these bases. If $T$ is either of
these tori, then it stabilizes the spaces spanned by $r_{i}$ and $s_{i}$ for
$i=1,\ldots,n$ for both kinds of bases. In this we regard $T$ as a product
$T=T_{1}\cdot\ldots\cdot T_{n}$, in the second case acting trivially on the
space spanned by $t$. In both cases an arbitrary permutation in $S_{n}$ acting
diagonally on these spaces by $(r_{i},s_{i})\mapsto(r_{\pi(i)},s_{\pi(i)})$
normalizes the $T$-action and is in the given orthogonal group $G$. A simple
reflection $(r_{i},s_{i})\mapsto(s_{i},r_{i})$ for one such $i$, and being the
identity on the other $2$-dimensional spaces, is as usual denoted by a sign
change. Such normalizes $T$, but has negative determinant. In the first case,
this means that only an even number of sign changes is allowed. In the second
case we may couple the sign change with the map $t\mapsto-t$ so that this
slightly modified sign change has positive determinant. Thus in both cases we
have the action of $S_{n}\ltimes\mathbb{Z}_{2}^{n}$ normalizing the
$T$-representation on the basis at hand. This is in fact the action of the
full Weyl group.

As in the cases of $Sp(2n,\mathbb{R})$ and $SO^{*}(2n)$, to describe the flag
domains and their $K_{0}$-base cycles we use the Weyl group $W(T_{S})$ of the
standard torus $T_{S}$. The base point for $\alpha$ is the flag $F_{S}$
defined by the standard basis. Then all base points are $F_{\alpha}:=w(F_{S})$
where $w\in\mathbb{Z}_{2}^{n-1}$ if $m=2n$ and $w\in\mathbb{Z}_{2}^{n}$ if
$m=2n+1$ is associated to the sign-change vector $\alpha$. As before the base
cycles and flag domains are the orbits $K.F_{\alpha}$ and $G_{0}.F_{\alpha}$,
respectively, and $\alpha$ defines a non-degenerate sign with
$sign(F_{\alpha})=(a,b)$ and then $D_{\alpha}=D_{a,b}$.

In our particular case there are two maximal tori, the maximal torus $T_{I}$
and the maximal torus $T_{S}$ which defined above. Recall that these two
maximal tori $T_{I},T_{S}$ are conjugate, and a conjugation induces an
isomorphism $\psi$ of the associated Weyl groups. In this case the bijective
map $\psi$ is described, depending on the case, as follows:

$\bullet$ If $m=2n$ and $q$ is even:

Define $W_{T_{S}}$ to be the Weyl group with respect to the basis (2.4) and
let $W_{I}$ be the Weyl group with respect to the basis • ‣ 6.1. In this case
the bijective map between $W_{I}$ and $W_{T_{S}}$ is $\psi(\mp(2i-1))=\pm
i,\psi(\mp 2i)=\pm(q-i+1)$ if $1\leq i\leq\frac{q}{2}$ and $\psi(\pm i)=\pm i$
if $i>q$.

$\bullet$ If $m=2n$ and $q$ is odd:

Define $W_{T_{S}}$ to be the Weyl group with respect to basis (2.5) and let
$W_{I}$ be the Weyl group with respect to the basis (• ‣ 6.1), then the
bijective map in this case is $\psi:W_{I}\longrightarrow W_{T_{o}}$ defined by
$\psi(\mp(2i-1))=\pm i,\psi(\mp 2i)=\pm(q-i)$ if $1\leq i\leq\frac{q-1}{2}$
and $\psi(\pm i)=\pm(i-1)$ if $i>q$ and $\psi(-q)=-n.~{}$

$\bullet$ If $m=2n+1$ and $q$ is even:

Define $W_{T_{e}}$ to be the Weyl group with respect to the basis (2.6) and
let $W_{I}$ be the Weyl group with respect to the basis (• ‣ 6.1), then we can
define the bijective map between them to be $\psi(\mp(2i-1))=\pm i,\psi(\mp
2i)=\pm(q-i+1)$ if $1\leq i\leq\frac{q}{2}$ and $\psi(\pm i)=\pm i$ if $i>q$.

$\bullet$ If $m=2n+1$ and $q$ is odd:

Define $W_{T_{o}}$ to be the Weyl group with respect to basis (2.7). Let
$W_{I}$ be the Weyl group with respect to the basis (• ‣ 6.1). The bijective
map $\psi$ between $W_{I}$ and $W_{T_{o}}$ can be defined as
$\psi(\mp(2i-1))=\pm i,\psi(\mp 2i)=\pm(q-i)$ if $1\leq i\leq\frac{q-1}{2}$
and $\psi(\pm i)=\pm(i)$ if $i>q$.

Moreover, through an argument quiet similar to Proposition 2.2.1, the base
cycle $C_{\alpha}$ is the set

$C_{\alpha}=\\{F\in Z:V_{i}=(V_{i}\cap E^{-})\oplus(V_{i}\cap E^{+}),1\leq
i\leq m\\}.$

where $E^{-}$ and $E^{+}$ defined in $\S\ref{basepoint}$ and the intersection
dimensions are determined by $\alpha$ (resp. $(a,b)$). The flag domains are
parametrized by the signature of its base cycles.

###  2.5 The real form $Sp(2p,2q)$

####  2.5.1 Preliminaries

Let $G:=Sp(2n,\mathbb{C})$ be the Lie group of complex linear transformations
is defined in $\S\ref{Sp}$. We realize the real form $G_{0}=Sp(2p,2q)$ as a
group of matrices $g\in Sp(2n,\mathbb{C})$ which leave invariant the Hermitian
form $h(v,w)$ of the signature $(2p,2q)$ defined by

$h(v,w)=-\sum_{i=1}^{q}v_{i}\bar{w}_{i}+\sum_{i=q+1}^{p+q}v_{i}\bar{w}_{i}+\sum_{i=p+q+1}^{2p+q}v_{i}\bar{w}_{i}-\sum_{i=p+2q+1}^{2n}v_{i}\bar{w}_{i}.$

such that $Sp(2p,2q)=Sp(2n,\mathbb{C})\cap SU(2p,2q)$. A Cartan involution
$\theta:\mathfrak{sp}(2p,2q)\longrightarrow\mathfrak{sp}(2p,2q)$ in the Lie
algebra level is given by

$\theta(g)=-I_{q,p,p,q}\overline{g}^{t}I_{q,p,p,q},~{}~{}~{}where~{}I_{q,p,p,q}=\left(\begin{array}[]{cccc}-I_{q}&0&0&0\\\
0&I_{p}&0&0\\\ 0&0&I_{p}&0\\\ 0&0&0&-I_{q}\\\ \end{array}\right)$

In particular, the maximal compact subgroup of $Sp(2p,2q)$ is
$K_{0}:=Sp(2q)\times Sp(2p)$

####  2.5.2 Base points

Let $Z$ be the space of all maximally $b$-isotropic full flags. By an argument
which is completely analogous to that in the case of $Sp(2n,\mathbb{R})$, the
standard basis $(e_{1},e_{2},...,e_{2n})$ of $\mathbb{C}^{2n}$ defines a base
point $F_{S}$ in $Z$ as the associated flag

$<e_{1}>\subset<e_{1},e_{2}>\subset...\subset<e_{1},...,e_{2n}>$

Reorderings on this basis will determine the flag domains, the base cycles and
the intersection points.

The standard basis defines the standard maximal torus $T_{S}$, i.e,

$T_{S}:=\\{g\in
G_{0}:g=diag(t_{1},t_{2},...,t_{n},-t_{n},...,-t_{2},-t_{1}),t_{i}\in\mathbb{C}\\}$

and the standard Borel subgroup $B$ of upper triangular matrices in
$Sp(2p,2q)$.

In order to define a base point in $\gamma^{c\ell}$ we introduce the basis

$\displaystyle
e_{1}+e_{2q},e_{2n-2q+1}-e_{2n},...,e_{q}+e_{q+1},e_{2n-q}-e_{2n-q+1},e_{2q+1},e_{2q+2},...,e_{n},e_{n+1},$
$\displaystyle
e_{n+2},...,e_{2n-q},e_{q}-e_{q+1},e_{2n-q}+e_{2n-q+1},...,e_{1}-e_{2q},e_{2n-2q+1}+e_{2n}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(2.10)

with $T_{I}$ being the maximal torus which fixes the basis in the sense of
eigenvectors.

####  2.5.3 Weyl group, flag domains and base cycles

By arguments which are completely analogous to those in $\S\ref{Weyl}$, if $T$
is a maximal torus defined by a basis of $(r,s)$-type, then
$W(T)=S_{n}\ltimes\mathbb{Z}_{2}^{n}$

As in the case of $Sp(2n,\mathbb{R})$, to describe the flag domains and their
$K_{0}$-base cycles we use the Weyl group $W(T_{S})$ of the standard torus
$T_{S}$. The base point for $\alpha$ is the flag $F_{S}$ defined by the
standard basis. Then all base points are $F_{\alpha}:=w(F_{S})$ where
$w\in\mathbb{Z}_{2}^{n}$ is associated to the sign-change vector $\alpha$. As
before the base cycles and flag domains are the orbits $K.F_{\alpha}$ and
$G_{0}.F_{\alpha}$, respectively, and $\alpha$ defines a non-degenerate sign
with $sign(F_{\alpha})=(a,b)$ and then $D_{\alpha}=D_{a,b}$.

The maximal tori $T_{I}$ and $T_{S}$ defined in the previous section are
conjugate, and a conjugation induces an isomorphism $\psi$ of the associated
Weyl groups. In this case the bijective
$\psi(\pm(2i-1))=\pm(2q-i+1),~{}\psi(\pm 2i)=\mp i$ if $1\leq i\leq q$ and
$\psi(\pm i)=\pm i$ if $i>2q$. Note that $-(2q-i+1)=2n-2q-i+2$.

Finally, Define $E^{-}:=<e_{1},...,e_{q},e_{2n-q+1},...,e_{2n}>$ and
$E^{+}:=<e_{q+1},...,e_{2n-q}>$. By the same argument as $\S\ref{flag}$ and
Proposition 2.2.1, for a fixed flag domain $D_{\alpha}$, the base cycle
$C_{\alpha}$ is the set

$C_{\alpha}=\\{F\in Z:V_{i}=(V_{i}\cap E^{-})\oplus(V_{i}\cap E^{+}),1\leq
i\leq 2n\\}.$

As all cases above the intersection dimensions are determined by $\alpha$
(resp. $(a,b)$).

##  Chapter 3 Fixed Point Theorem

###  3.1 Introduction

Here, as throughout our work, we consider a homogeneous $G$-manifold of full
flags $Z$ equipped with the actions of a real form $G_{0}$ as listed in the
previous chapter. The Iwasawa-Borel subgroup $B_{I}$ is the isotropy group at
the flag $F_{I}$ given there by an explicit basis $\mathcal{B}_{I}$ which also
defines the maximal torus $T_{I}$ and its Weyl group $W_{I}$. For $w\in W_{I}$
we study the intersections $S_{w}\cap C_{\alpha}$ of the Schubert cells
$S_{w}=B_{I}.F_{w}$ with any base cycle $C_{\alpha}$, where $F_{w}=w(F_{I})$ .

On the other hand, we have a standard basis $\mathcal{B}_{S}$ which defines a
torus $T_{S}$ and a base point $F_{S}$ in a standard base cycle which when
moved by sign permutations in the Weyl group $W_{S}$ gives base points in all
other base cycles $C_{\alpha}$. Recall that these base cycles are the closed
$K$-orbits which are described by the intersections with $E^{+}$ and $E^{-}$
in each case. All of the above has been defined in Chapter $2$ for all cases.

In this chapter we first give detailed formulas for the orbits $b(F_{I}),b\in
B_{I}$, case by case for all real forms. Then we prove a ”Moving Lemma”(See
the Moving Lemma 3.3.1): Given a point $b.w(F_{I})$ in $C_{\alpha}$, we move
it continously in $C_{\alpha}$ to a $T_{S}$-fixed point by starting with the
basis given for $b.w(F_{I})$ and moving continuously by a curve $\gamma(t)$ of
bases $\mathcal{B}(t)$ which defines flags in $S_{w}\cap C_{\alpha}$. In
general the intersection $S_{w}\cap C_{\alpha}$ may have many connected
components. The moving procedure implies that in each component of $S_{w}\cap
C_{\alpha}$ there is a fixed point. In the case where $S_{w}$ is of
complementary dimension to $C_{\alpha}$ the intersection $S_{w}\cap
C_{\alpha}$ is finite; so the curve $\gamma(t)$ is just a fixed point and the
result states that $S_{w}\cap C_{\alpha}$ consists only of $T_{S}$-fixed
points (See Corollary 3.3.4).

###  3.2 Description of the $B_{I}$-orbits

In the previous chapter in every case we have chosen a flag $F_{I}$ defined by
the basis $(v_{1},\ldots,v_{m}),m=2n$ or $m=2n+1$, which belongs to the
$G_{0}$-closed orbit $\gamma^{cl}$. Given a Weyl element $w\in W$, we consider
the Schubert cell $B_{I}.F_{w}$ and an element $F=b(F_{w})$ in it which is
defined by an ordered basis $(b.v_{w_{1}},\ldots,b.v_{w_{m}})$, $b\in B_{I}$.
If $F\in C_{\alpha}$, then it can be defined by an ordered basis
$(\varepsilon_{1},\ldots,\varepsilon_{m})$ where for all $k$ the vector
$\varepsilon_{k}$ is either in $E^{-}$ or $E^{+}$ where $V=E^{+}\oplus E^{-}$,
and is a linear combination

$\varepsilon_{k}=\sum_{j\leq k}c_{j}b.v_{w_{j}}\,.$

In this case we call the basis a split basis. As a result of the discussion in
the previous section we can write formulas for $b.v_{w_{j}}$ for any element
$b\in B_{I}$ for each real form as follows:

1\. If $G_{0}=SP(2n,\mathbb{R})$,

then

$b.v_{w_{j}}=\eta_{j}(e_{r}+e_{2n-r+1})+\zeta_{j}(e_{r}-e_{2n-r+1})+B_{j}=K_{j}+B_{j},~{}1\leq
r\leq n$

where $B_{j}$ does not involve the basis vectors $e_{r}$ and $e_{2n-r+1}$.

2\. If $G_{0}=SO^{*}(2n)$,

then

$b.v_{w_{j}}=\eta_{j}(e_{r}+ie_{n+r})+\zeta_{j}(e_{r}-ie_{n+r})+B_{j}=K_{j}+B_{j},~{}1\leq
r\leq n$

where $B_{j}$ does not involve the basis vectors $e_{r}$ and $e_{n+r}$.

3\. If $G_{0}=SO(p,q)$,

then

(i)

$b.v_{w_{j}}=\eta_{ij}(e_{r}+e_{2q-r+1})+\zeta_{ij}(e_{r}-e_{2q-r+1})+\eta_{kj}(e_{r+1}+e_{2q-r})+\zeta_{kj}(e_{r+1}-e_{2q-r})+B_{j}\\\
=K_{j}+B_{j},$ where $~{}1\leq r\leq q$ and $\eta_{kj}=\pm i\eta_{ij}$ and
$\zeta_{kj}=\pm i\zeta_{ij}$ and $B_{j}$ does not involve the basis vectors
$e_{r}$, $e_{r+1}$, $e_{2q-r+1}$ and $e_{2q-r}$.

(ii)

$b.v_{w_{j}}=\eta_{j}(e_{m-2r+1}+ie_{m-2r+2})+B_{j}=K_{j}+B_{j},~{}1\leq
r\leq\frac{m-2q}{2}$
where $B_{j}$ does not involve the basis vectors $e_{m-2r+1}$ and
$e_{m-2r+2}$.

(iii)

$b.v_{w_{j}}=\eta_{j}e_{2q+1}+B_{j}=K_{j}+B_{j},$
where $B_{j}$ does not involve the basis vectors $e_{2q+1}$.

4\. If $G_{0}=SP(2p,2q)$,

then

(i)

$b.v_{w_{j}}=\eta_{j}(e_{r}+e_{2q-r+1})+\zeta_{j}(e_{r}-e_{2q-r+1})+B_{j}=K_{j}+B_{j},$
where $~{}1\leq r\leq q$ where $B_{j}$ does not involve the basis vectors
$e_{r}$ and $e_{2q-r+1}$.

(ii)

$b.v_{w_{j}}=\eta_{j}(e_{2n-q+r}+e_{2n-r+1})+\zeta_{j}(e_{2n-q+r}-e_{2n-r+1})+B_{j}=\tilde{K}_{j}+\tilde{B}_{j},$
where $~{}1\leq r\leq q$ where $B_{j}$ does not involve the basis vectors
$e_{2n-q+r}$ and $e_{2n-r+1}$.

(iii)

$b.v_{w_{j}}=\eta_{j}(e_{2q+r})+B_{j}.$ where $~{}1\leq r\leq 2p$ where
$B_{j}$ does not involve the basis vector $e_{2q+r}$.

###  3.3 The Moving Lemma and a Fixed Point Theorem

Assume that the Schubert cell $B_{I}.F_{w}$ has non-empty intersection with
$D_{\alpha}$. This implies that it has a non-empty intersection with the base
cycle $C_{\alpha}$. The flags $F_{0}$ in this cycle can be described by bases
$\mathfrak{B}(F_{0})$ which are split in the sense that their entries
(vectors) are either in $E^{-}$ or $E^{+}$ where $V=E^{-}\oplus E^{+}$.

######  Lemma 3.3.1.

(The Moving Lemma). Given a flag $F_{0}\in B_{I}.F_{I}\cap C_{\alpha}$, there
is a continuous curve of split bases
$\mathcal{F}=\mathfrak{B}(t),t\in[0,t_{0}]$ such that $\mathfrak{B}(t_{0})$
defines the given flag $F_{0}$ and $\mathcal{B}(0)$ defines a $T_{S}$-fixed
point.

###### Proof.

The fixed point $F_{I}$ in $\gamma^{cl}$ is defined by the ordered basis

$(v_{1},v_{2},...,v_{m})$

and $F_{w}=w(F_{I})$. Let $b.F_{w}=:F_{0},b\in B_{I}$, be a given flag in
$B_{I}.F_{w}\cap C_{\alpha}$ and regard it as being given by a basis

$(b.v_{w_{1}},\ldots,b.v_{w_{m}})$

which is given in $\S\ref{S3.2}$. Of course this basis is not necessarily
split. However, the points which belongs to $C_{\alpha}$ are given by split
bases, and we will now construct an associated split basis
$(\varepsilon_{1},\ldots,\varepsilon_{m})$ which belong to $S_{w}\cap
C_{\alpha}$ (term-by-term, recursively). First, we analyze $\varepsilon_{1}$
and describe it in a way that is convenient for our purposes. There are three
possibilities for

$\varepsilon_{1}=b.v_{w_{1}}=\lambda v_{w_{1}}+B_{w_{1}},$

where $B_{w_{1}}=\sum_{j\leq w_{1}}c_{j}v_{j},$ i.e.,

  1. 1.

If $1\leq{w_{1}}\leq q$ 111$q$ comes from the signature $(p.q)$ of the
bilinear form $b$ where $q=n$ in the cases $Sp(2n,\mathbb{R})$ and
$SO^{*}(2n)$. such that the dual vector 222If
$v_{i},\tilde{v}_{i}\in\mathbb{C}^{m}$ and
$b(v_{i},v_{i})=0$,$b(\tilde{v}_{i},\tilde{v}_{i})=0$ and
$b(\tilde{v}_{i},v_{i})=1$ then $\tilde{v}_{i}$ is the dual vector of $v_{i}$.
$v_{m-w_{1}+1}$ is not involved as a term of $B_{w_{1}}$, so there is no
possibility of canceling the term $\lambda v_{w_{1}}$ with $B_{w_{1}}$.

  2. 2.

If $-1\leq{w_{1}}\leq-q$ such that the dual vector $v_{m-w_{1}+1}$ is involved
as a term in $B_{w_{1}}$, so we can find $2\lambda e_{j}$ as combination of
these two vectors.

  3. 3.

If $q+1\leq|w_{1}|\leq\left\lfloor\frac{m}{2}\right\rfloor$, this case appear
for the real forms $G_{0}=SP(p,q)$ and $G_{0}=SO(p,q)$. In this case
$v_{w_{1}}=e_{r}$ for some $r$ if $G_{0}=SP(p,q)$, and $v_{w_{1}}=e_{r}\pm
ie_{r+1}$ for some $r$ or $v_{w_{1}}=e_{q}$ or $v_{w_{1}}=e_{q+1}$ if
$G_{0}=SO(p,q)$.

Note that such combinations in those cases are necessary for the first vector
in the basis $\mathcal{B}(F_{0})$. Now a number of the other vectors $v_{i}$
occur in such pairs and must be combined in the same way. However, the
resulting coefficient is an arbitrary complex number. As a result, if
$\tilde{\lambda}$ is the coefficient of the dual vector $v_{m-w_{1}+1}$, then
the first basis element is

$\varepsilon_{1}=2\lambda
e_{j}+t_{1}e_{j+1}+...+t_{n-j}e_{n},\text{~{}~{}if~{}}\tilde{\lambda}=\lambda$

or

$\varepsilon_{1}=2\lambda
e_{m-j+1}+t_{1}e_{m-j}+...+t_{m-j}e_{n+1}\text{~{}~{}if~{}}\tilde{\lambda}=-\lambda.$

where $m=2n$ or $m=2n+1$. We can not know the values of the coefficients, but
only know that the $t_{i}$ can in principle be any complex number and
$\lambda\neq 0$. Note that the first case $\varepsilon_{1}=b.v_{w_{1}}$ where
$1\leq w_{1}\leq q$ doesn’t happen, because there is no possible linear
combination which produces a split basis element.
Next we construct and describe $\varepsilon_{2}$. Recall that the vector
$b.v_{w_{2}}$ is given by the form

$b.v_{w_{2}}=\lambda v_{w_{2}}+B_{w_{2}},$

where $B_{w_{2}}=\sum_{j\leq w_{2}}c_{j}v_{j}$, but we can modify it by adding
in an appropriate multiple of $\varepsilon_{1}$, i.e.

$\varepsilon_{2}=\lambda v_{w_{2}}+d_{1}\varepsilon_{1}+B_{w_{2}}.$

If we use $d_{1}\varepsilon_{1}$ in the linear combination which defined
$\varepsilon_{2}$, we will call this combination ”external combination”, if
not, we call this combination ”internal combination”. Then there are four
possibilities for $\varepsilon_{2}$ i.e.,

A.

If $1\leq{w_{2}}\leq q$ such that the dual vector $v_{m-w_{2}+1}$ is not
involved as a term of $B_{w_{2}}$ and $\varepsilon_{1}\neq v_{m-w_{2}+1}$ , so
there is no possibility of canceling the term $\lambda v_{w_{2}}$ with
$B_{w_{2}}$ or with $\varepsilon_{1}$.

B.

If $1\leq{w_{2}}\leq q$ such that the dual vector $v_{m-w_{2}+1}$ is not
involved as a term of $B_{w_{2}}$ but $\varepsilon_{1}=v_{m-w_{2}+1}$ , so we
can find $2\lambda e_{j}$ as combination of $\varepsilon_{1}$ and $v_{w_{2}}$.

C.

If $-1\leq{w_{2}}\leq-q$ such that the dual vector $v_{m-w_{1}+1}$ is involved
as a term in $B_{w_{2}}$, so we can find $2\lambda e_{j}$ as combination of
these two vectors.

D.

If $q+1\leq|w_{2}|\leq\left\lfloor\frac{m}{2}\right\rfloor$, this case appear
for the real forms $G_{0}=SP(p,q)$ and $G_{0}=SO(p,q)$. In this case
$v_{w_{2}}=e_{r}$ for some $r$ if $G_{0}=SP(p,q)$, and $v_{w_{2}}=e_{r}\pm
ie_{r+1}$ for some $r$ or $v_{w_{2}}=e_{q}$ or $v_{w_{2}}=e_{q+1}$ if
$G_{0}=SO(p,q)$.

The combinations in those cases are necessary for the second vector in the
basis $\mathcal{B}(F_{0})$ and all other vectors $\varepsilon_{j},j\geq 2$ .
Now a number of the other vectors $v_{i}$ occur in such pairs and must be
combined in the same way. As a result, if $\tilde{\lambda}$ is the coefficient
of the dual vector $v_{m-w_{2}+1}$, then the second basis element is

$\varepsilon_{2}=2\lambda e_{j}+t_{1}e_{j+1}+...+t_{n-j}e_{n},\text{ if
}\tilde{\lambda}=\lambda,$ $\varepsilon_{2}=2\lambda
e_{m-j+1}+t_{1}e_{m-j}+...+t_{m-j}e_{n+1},\text{ if
}\tilde{\lambda}=-\lambda.$

where $m=2n$ or $m=2n+1$, $t_{i}$ is complex number and $\lambda\neq 0$.

We now assume that $\varepsilon_{1},\ldots,\varepsilon_{j-1}$ have been
constructed as in the case of $\varepsilon_{2}$ depending on
$\tilde{\lambda}$. This means that $\varepsilon_{\ell}=2\lambda
e_{j}+t_{1}e_{j+1}+...+t_{n-j}e_{n},\text{ if }\tilde{\lambda}=\lambda,$ or
$\varepsilon_{\ell}=2\lambda e_{m-j+1}+t_{1}e_{m-j}+...+t_{m-j}e_{n+1},\text{
if }\tilde{\lambda}=-\lambda.$ for $\ell\leq j-1$. Now we will construct
$\varepsilon_{j}$ with these properties. Recall that the vector

$b.v_{w_{j}}=\lambda v_{w_{j}}+B_{w_{j}},$

where $B_{w_{j}}=\sum_{j\leq w_{2}}c_{j}v_{j},$ but we can modify it by adding
in an appropriate linear combination of the $\varepsilon_{i},1\leq i\leq j-1$,
i.e.

$\varepsilon_{j}=\lambda v_{w_{j}}+B_{w_{j}}+\sum_{i<j}d_{i}\varepsilon_{i},$

then there are only three cases for $\varepsilon_{j}$ :

  1. 1.

Some of the coefficients of $B_{j}$ are such that cancellation takes place as
step $C$ above which called internal cancellation.

  2. 2.

There is a linear combination of the previous split basis elements
$\sum_{i<j}d_{i}\varepsilon_{i}$ that cancels one of the terms in expressions
of the type $e_{s}\pm e_{r}$. Similarly as step $B$ above which called
external cancellation..

  3. 3.

The vector $v_{j}$ is equal to $e_{r}$ in the case of the real form $SP(p,q)$
or $v_{j}$ is equal to $e_{r}\pm ie_{r+1}$ or $e_{r}$ in the case of the real
form $SO(p,q)$. Similarly as step $D$ above which is a kind of internal
cancellation..

As a result, if $\tilde{\lambda}$ is the coefficient of the dual vector
$v_{m-w_{1}+1}$, then the $j$-th basis element is

$\varepsilon_{j}=2\lambda
e_{s}+t_{1}e_{s+1}+...+t_{n-s}e_{n},\tilde{\lambda}=\lambda,$

or

$\varepsilon_{j}=2\lambda
e_{m-s+1}+t_{1}e_{m-s}+...+t_{m-s}e_{n+1},\tilde{\lambda}=-\lambda,$

where $m=2n$ or $m=2n+1$ and $t_{i}$ is complex number and $\lambda\neq 0$.

The next claim is we can move continuously from an arbitrary
$\varepsilon$-basis constructed above to a $T_{S}$-fixed point continuously,
i.e., without leaving $C_{\alpha}$. The curve

$\mathfrak{B}(t)=(\varepsilon_{1}(t),\ldots,\varepsilon_{m}(t)),$

which begins at the given basis $(\varepsilon_{1},\ldots,\varepsilon_{m})$ and
ends at a $T_{S}$-fixed bases
$(\varepsilon_{1}(0),\ldots,\varepsilon_{m}(0))$, will be constructed
stepwise. First we show that there is a curve $\varepsilon_{1}(t)$ beginning
at the given vector $\varepsilon_{1}$ and ending at a $T_{S}$-fixed vector
$\varepsilon_{1}(0)$. With this we reach a new base point

$(\varepsilon_{1}(0),\varepsilon_{2},\ldots,\varepsilon_{m}).$

If the vector $\varepsilon_{1}$ in the given base point is $T_{S}$-fixed
point, then the curve $\varepsilon_{1}(t)=\varepsilon_{1}(0)$ for all
$t\in[0,t_{0}]$, i.e a $T_{S}$-fixed point. Then we assume that the given base
point $(\varepsilon_{1},\ldots,\varepsilon_{m})$ has the property that
$\varepsilon_{\ell}$ is $T_{S}$-fixed for $\ell<k$ such that the curve
$\varepsilon_{\ell}(t)=\varepsilon_{\ell}(0),\ell<k$ for all $t\in[0,t_{0}]$
and then construct a curve $\varepsilon_{k}(t)$ beginning at $\varepsilon_{k}$
and ending at a $T_{S}$-fixed vector $\varepsilon_{k}(0)$. This defines a new
base point in $S_{w}\cap C_{\alpha}$ with $\varepsilon_{\ell}$ begin
$T_{S}$-fixed for $\ell\leq k$. As a result we define a sequence of curves
whose composition is the desired continuous curve.

Suppose $F\in S_{w}\cap C_{\alpha}$ is associated to a split basis and not
$T_{S}$-fixed. We may therefore choose an $\varepsilon$-basis with a smallest
index $k$ such that $\varepsilon_{k}$ is not fixed and $\varepsilon_{\ell}$ is
fixed for $\ell<k$.

So we can define the curve such that $\varepsilon_{\ell}$ is $T_{S}$-fixed ,
then we will construct the curve $\varepsilon_{k}(t)$ where $\varepsilon_{k}$
is the first non $T_{S}$-fixed.

It is sufficient to consider the case where $\varepsilon_{k}\in V^{-}$. The
first step is to analyze the $\varepsilon_{k}$, but they are constructed in
such a way that for the movement $b(t)$ we have

$\varepsilon_{k}=(\lambda_{1}e_{r}+\sum_{j\leq
q}a_{j}e_{j})+i(\lambda_{2}e_{r+1}+\sum_{s\leq q}b_{s}e_{s})$

where $a_{r}=0$,$b_{r+1}=0$ and $a_{j}\not=0$, $b_{s}\not=0$ for some other
$j$ and $s$. Furthermore, if $\varepsilon_{\ell}=e_{j}$ for some $\ell<r$,
then $a_{j}=b_{s}=\lambda_{2}=0$, and if $\varepsilon_{\ell}=e_{j}+ie_{j+1}$
for some $\ell<r$, then $a_{j}=b_{s}=0$. In general, if $G_{0}=SO(p,q)$, then
$\lambda_{1}=\pm\lambda_{2}$, and if $G_{0}\neq SO(p,q)$, then
$\lambda_{2}=b_{s}=0$.

If $k=1$ then the curve is

$\varepsilon_{1}(t)=c_{1}b(t).v_{w_{1}}\,.$

It follows that

$\varepsilon_{1}(t)=\varepsilon_{1}+(t-1)K_{1}\,.$

where $K_{1}=0$ corresponds to $e_{r}+ie_{r+1}$ or $e_{r}$ not being involved
in $\varepsilon_{1}$. Note that $\varepsilon_{1}\in V^{-}$ implies that
$\zeta_{1}=0$ and $\tilde{\zeta}_{1}=0$, and $\varepsilon_{1}\in V^{+}$
implies that $\eta_{1}=0$ and $\tilde{\eta}_{1}=0$ in the formulas of section
3.2. Thus $\varepsilon_{1}(t)\in V^{-}$ (resp. $V^{+}$) whenever
$\varepsilon_{1}\in V^{-}$(resp. $V^{+}$). In other words, the image of the
curve $\gamma$ is contained in $S_{w}\cap C_{\alpha}$. Finally, observe that
$\eta_{1}$ is the coefficient of $e_{r}$ in $\varepsilon_{1}$. This has be
scaled to $1$ and as a result the coefficient of $e_{r}$ or $e_{r}+ie_{r+1}$
in $\varepsilon_{1}(t)$ is $t$. Thus

$\varepsilon_{1}(t)=\varepsilon_{1}+(t-1)e_{r},\text{~{}if~{}}G_{0}\neq
SO(p,q),$

or

$\varepsilon_{1}(t)=\varepsilon_{1}+(t-1)(e_{r}+ie_{r+1}),\text{~{}if~{}}G_{0}=SO(p,q).$

In the various cases we then have the following descriptions of
$\varepsilon_{1}(0)$;

$\bullet$ If $G_{0}=SP(2n,\mathbb{R})$,

then

$\varepsilon_{1}(0)=\varepsilon_{1}-e_{r}=[(e_{r}+e_{2n-r+1})]-e_{r}=e_{2n-r+1},~{}1\leq
r\leq n.$

$\bullet$ If $G_{0}=SO^{*}(2n)$,

then

$\varepsilon_{1}(0)=\varepsilon_{1}-e_{r}=[(e_{r}+ie_{n+r})]-e_{r}=ie_{n+r},~{}1\leq
r\leq n,$

or

$\varepsilon_{1}(0)=\varepsilon_{1}-e_{r}=[-i(e_{r-n}+e_{r})]-e_{r}=-ie_{r-n},~{}n+1\leq
r\leq 2n.$

$\bullet$ If $G_{0}=SO(p,q)$,

then

$\varepsilon_{1}(0)=\varepsilon_{1}-(e_{r}\mp ie_{r+1})=(e_{r}\mp
ie_{r+1})+(e_{2q-r}\mp ie_{2q-r+1})-(e_{r}\mp ie_{r+1})=(e_{2q-r}\mp
ie_{2q-r+1}),$

where $~{}1\leq r\leq q.$

$\bullet$ If $G_{0}=SP(2p,2q)$,

then

$\varepsilon_{1}(0)=\varepsilon_{1}-e_{r}=(e_{r}+e_{2q-r+1})-e_{r}=e_{2q-r+1},~{}1\leq
r\leq q.$

Recall that the above non fixed $\varepsilon_{k}$ is
$\varepsilon_{k}=b.v_{w_{k}}=K_{k}+B_{k}$ which defined for all cases in
$\S\ref{S3.2}$ where $B_{k}$ does not involve the terms of $K_{k}$. In the
following we will define a curve $\gamma$ which sent the non fixed point to
$T_{S}$-fixed point . Define $b(t)\in B_{I}$ by
$b(t)(v_{w_{j}})=tK_{j}+B_{j}$. The curve $\gamma$ is defined by
$t\mapsto(b(t)(v_{w_{1}}),\ldots,b(t)(v_{w_{2n}}))$ which is defined by the
corresponding $\varepsilon$-basis,

$\varepsilon_{k}(t)=\sum_{j\leq k}c_{j}b(t).v_{w_{j}}\,.$

It follows that

$\varepsilon_{\kappa}(t)=\varepsilon_{\kappa}+(t-1)\sum_{j\leq\kappa}c_{\kappa
j}K_{j}\,.$

Notice that $\sum_{j\leq\kappa}c_{\kappa j}K_{j}=0$ corresponds to
$e_{r}+ie_{r+1}$ or $e_{r}$ not being involved in $\varepsilon_{\kappa}$.
Therefore $\varepsilon_{\kappa}(t)=\varepsilon_{\kappa}$ for $\kappa<k$. In
general, $\varepsilon_{\kappa}\in V^{-}$ implies that $\sum c_{\kappa
j}\zeta_{j}=0$ and $\sum c_{\kappa j}\tilde{\zeta}_{j}=0$, and
$\varepsilon_{\kappa}\in V^{+}$ implies that $\sum c_{\kappa j}\eta_{j}=0$ and
$\sum c_{\kappa j}\tilde{\eta}_{j}=0$ in the formulas of section 3.2. Thus
$\varepsilon_{\kappa}(t)\in V^{-}$ (resp. $V^{+}$) whenever
$\varepsilon_{\kappa}\in V^{-}$(resp. $V^{+}$). In other words, the image of
the curve $\gamma$ is contained in $S_{w}\cap C_{\alpha}$. Finally, observe
that $\sum c_{\kappa j}\eta_{j}$ is the coefficient of $e_{r}$ in
$\varepsilon_{k}$. This has be scaled to $1$ and as a result the coefficient
of $e_{r}$ or $e_{r}+ie_{r+1}$ in $\varepsilon_{k}(t)$ is $t$. Thus

$\varepsilon_{k}(t)=\varepsilon_{k}+(t-1)e_{r},\text{~{}if~{}}G_{0}\neq
SO(p,q),$

or

$\varepsilon_{k}(t)=\varepsilon_{k}+(t-1)(e_{r}+ie_{r+1}),\text{~{}if~{}}G_{0}=SO(p,q).$

Since at least one $a_{j}\not=0$, and one $b_{j}\not=0$ and
$\varepsilon_{\ell}(t)=\varepsilon_{\ell}$ for $\ell<k$. In the various cases
we then have the following descriptions of $\varepsilon_{k}(0)$;

$\bullet$ If $G_{0}=SP(2n,\mathbb{R})$,

then

$\varepsilon_{k}(0)=\varepsilon_{k}-e_{r}=[(e_{r}+e_{2n-r+1})]-e_{r}=e_{2n-r+1},~{}1\leq
r\leq n.$

$\bullet$ If $G_{0}=SO^{*}(2n)$,

then

$\varepsilon_{k}(0)=\varepsilon_{k}-e_{r}=[(e_{r}+ie_{n+r})]-e_{r}=ie_{n+r},~{}1\leq
r\leq n,$

or

$\varepsilon_{k}(0)=\varepsilon_{k}-e_{r}=[-i(e_{r-n}+e_{r})]-e_{r}=-ie_{r-n},~{}n+1\leq
r\leq 2n.$

$\bullet$ If $G_{0}=SO(p,q)$,

then

$\varepsilon_{k}(0)=\varepsilon_{k}-(e_{r}\mp ie_{r+1})=(e_{r}\mp
ie_{r+1})+(e_{2q-r}\mp ie_{2q-r+1})-(e_{r}\mp ie_{r+1})=(e_{2q-r}\mp
ie_{2q-r+1}),$

$~{}1\leq r\leq q.$

$\bullet$ If $G_{0}=SP(2p,2q)$,

then

$\varepsilon_{k}(0)=\varepsilon_{k}-e_{r}=(e_{r}+e_{2q-r+1})-e_{r}=e_{2q-r+1},~{}1\leq
r\leq q.$

all of the above give us a vector in the standard basis $T_{S}$ which implies
that we move the point to a $T_{S}$-fixed point. The above recursive
construction defines the $\varepsilon(t)_{j}$ as desired and these in turn
define the curve $\mathfrak{B}(t)$ of bases. ∎

The above proof shows that if $v(t)$ is any of the basis vectors
$\varepsilon_{j}(t)$ in $\mathfrak{B}(t)$, then the following hold.

######  Remarks 3.3.2.

  1. 1.

At most one coefficient of $v(t)$ in $\mathfrak{B}(t)$ is restricted not to
vanish (This happens as in the case $C$ where $2\lambda$ arises as a
coefficient!).

  2. 2.

Except for the previous case, all coefficients are not restricted. However,
they may or may not be related, e.g., in the case where an external linear
combination was used.

  3. 3.

In all cases there are limiting vectors when various $t_{i}$ go to $0$ which
are single elements of the standard basis $(e_{1},...,e_{2n})$ except for the
real form $SO(p,q)$ the basis is

$(e_{1}\pm ie_{2},....,e_{m-1}\pm ie_{m})$.

A Schubert cell intersects $C_{\alpha}$ in several connected component
(sometimes only one as the case $Sp(2n,\mathbb{R})$, but sometimes many as the
case $SO(p,q)$). Since the curve given by the Moving Lemma is continuous, it
follows that every such component contains a $T_{S}$-fixed point.

######  Theorem 3.3.3.

(The Fixed Point Theorem). Every component of the intersection $S_{w}\cap
C_{\alpha}$ has a fixed point.

######  Corollary 3.3.4.

If $codim(S_{w})=dim(C_{\alpha})$ then

$S_{w}\cap C_{\alpha}\subset Fix(T_{s})\,.$

###### Proof.

If $S_{w}$ of complementary dimension, then the components of intersection are
isolated points. By using Theorem 3.3.3 every intersection point is a fixed
point.
∎

##  Chapter 4 Cycle intersection for $SP(2n,\mathbb{R})$

###  4.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$

In this chapter the manifold $Z$ under consideration is the homogeneous space
of maximally isotropic full flags of the complex symplectic group
$G=Sp(2n,\mathbb{C})$ equipped with the action of the real form
$G_{0}=Sp(2n,\mathbb{R})$. Recall that $F_{I}$ is the fixed point in the
closed orbit $\gamma^{cl}$ with isotropy group $B_{I}$, i.e., which is
associated to the basis (2.1).

As described in $\S 2.2.3$ the Weyl group $W_{I}$ associated to $T_{I}$ acts
on the basis (2.1) by permutation plus sign change. If $w\in W_{I}$ is
regarded as an element of $G$, then $F_{w}:=w(F_{I})$ and $S_{w}:=B_{I}.F_{w}$
is the associated Iwasawa-Schubert cell. The dimension of the cells
corresponds to the length of the word $w$ in the Weyl group, i.e. if
$F_{w}=w.F_{I}$, then $dim(B.F_{w})=l(w)$.
Below we will describe the Weyl elements which parametrize the Iwasawa-
Schubert cells which have non-empty intersections with flag domains (See
Theorem 4.1.4). For this we call attention to a particular class of Weyl
elements.

######  Definition 4.1.1.

An element $w=(w_{1},w_{2},...,w_{n})\in S_{n}\ltimes\mathbb{Z}_{2}^{n}$ is
called a generous permutation111Recall that we define a special way of
denoting to write the Weyl elements in section $\S 2.2.3$. if $w_{i}<0$ for
all $1\leq i\leq n$.

######  Remark 4.1.2.

When discussing the $B_{I}$-orbit of the base point $F_{I}$ it is important to
explicitly understand the orbits $B_{I}.(e_{i}-e_{2n-i+1})$ and
$B_{I}.(e_{i}+e_{2n-i+1})$. As we explained in $\S 3.2$ we have the following
facts:
If $F_{I}$ and $B_{I}$ are as above, then the orbits of interest are
$B_{I}.(e_{i}\pm e_{2n-i+1})$, $i=1,\ldots,n$. In this case the orbits
$B_{I}.(e_{i}+e_{2n-i+1})$ and $B_{I}.(e_{i}-e_{2n-i+1})$ have points of the
forms

$b(e_{i}+e_{2n-i+1})=\lambda(e_{i}+e_{2n-i+1})+\ldots+b_{n}(e_{n}+e_{n+1})+a_{n}(e_{n}-e_{n+1})+\ldots+a_{1}(e_{1}-e_{2n})$

and

$b(e_{i}-e_{2n-i+1})=\lambda(e_{i}-e_{2n-i+1})+a_{i-1}(e_{i-1}-e_{2n-i})+\ldots+a_{1}(e_{1}-e_{2n})$

respectively, with $\lambda\not=0$. Note that in the above orbits, if $b$ is
chosen appropriately, then we can arrange $b(e_{i}+e_{2n+i-1})=e_{i}$ or
$b(e_{i}+e_{2n+i-1})=e_{2n-i+1}$ for all $i$. This plays a role in the proof
of Theorem 4.1.4.∎

Recall that our base point in the closed orbit $\gamma^{cl}$ is the flag
$F_{I}$ associated to the ordered basis

$e_{1}-e_{2n},e_{2}-e_{2n-1}...,e_{n}-e_{n+1},e_{n}+e_{n+1},...,e_{2}+e_{2n-1},e_{1}+e_{2n}$
(4.1)

and the base point which defined the flag domains and the base cycles is the
flag associated to the ordered basis

$e_{1},e_{2},e_{3},......,e_{2n-1},e_{2n}$ (4.2)

These two bases defined two maximal tori which are conjugate and the
corresponding Weyl groups $W_{I}$ and $W_{T_{S}}$ are isomorphic.
Let $\psi$ be a bijective map between Weyl groups $W_{I}$ and $W_{T_{S}}$
defined by $\psi(\mp(i))=\pm i,$ for all $1\leq i\leq n$, then we have the
following Proposition;

######  Proposition 4.1.3.

If $w$ is a generous permutation, then the flag $F_{\psi(w)}$ belongs to the
orbit $B_{I}(F_{w})$.

###### Proof.

Let $w\in W_{I}$ be a generous permutation, and $F_{w}=w(F_{I})\in\gamma^{cl}$
be the isotropic full flag associated to $w$ such that the first $n$-subspaces
in the flag are

$\\{0\\}\subset<v_{w_{1}}>\subset<v_{w_{1}},v_{w_{2}}>\subset......\subset<v_{w_{1}},...,v_{w_{n}}>$

where $v_{w_{i}}=e_{j}\pm e_{2n-j+1}$. We will show that the orbit
$B_{I}(F_{w})$ contains the flag $F_{\psi(w)}$. For this purpose let
$\tilde{w}\in W_{S}$ be the image of $w$ under the bijective map $\psi$ and
let $Y_{\tilde{w}}$ be the isotropic flag associated to $\tilde{w}$, i.e., the
flag

$Y_{\tilde{w}}:=(\\{0\\}\subset<e_{\tilde{w}_{1}}>\subset<e_{\tilde{w}_{1}},e_{\tilde{w}_{2}}>\subset......\subset<e_{\tilde{w}_{1}},...,e_{\tilde{w}_{n}}>)$

We will show that $Y_{\tilde{w}}=b(F_{w})$ for $b$ appropriately chosen in
$B_{I}$. For this recall that since $w$ is a generous permutation it follows
that $w_{i}<0$ for all $1\leq i\leq n$ , and therefore
$B_{I}.(e_{j}+e_{2n-j+1})$, $j=1,\ldots,n$, are the orbits of interest. By
Remark 4.1.2 the orbit $B_{I}.<e_{i}+e_{2n-i+1}>$ contains the points

$y=b(e_{i}+e_{2n-i+1})=\lambda(e_{i}+e_{2n-i+1})+\ldots+b_{n}(e_{n}+e_{n+1})+a_{n}(e_{n}-e_{n+1})+\ldots+a_{1}(e_{1}-e_{2n})$

where $\lambda\neq 0$. Choose all constants to be $0$ except
$\lambda=a_{i}=\frac{1}{2}$ , then $y=<e_{i}>=<e_{\tilde{w}(i)}>$.
We can repeat this step n-times to have the first half of the flag
$Y_{\tilde{w}}$, then we can extend this flag to maximal $h$-isotropic flag, .
Therefore, the flag $Y_{\tilde{w}}$ is constructed. ∎

######  Theorem 4.1.4.

(Generous Permutation Theorem). The following are equivalent

(i)

$w$ is generous.

(ii)

$B_{I}(F_{w})\cap D_{\alpha}\not=\emptyset$ for some $\alpha$.

(iii)

$B_{I}(F_{w})\cap D_{\alpha}\not=\emptyset$ for every $\alpha$.

Furthermore, under any of these conditions, for every $\alpha$ the
intersection $B_{I}(F_{w})\cap C_{\alpha}$ contains a $T_{S}$ fixed point.

###### Proof.

(iii)$\Rightarrow$ (ii)

Trivial direction.

(ii)$\Rightarrow$ (i)

We assume by contradiction that $w$ is not a generous permutation and will
show that $S_{w}\cap D_{\alpha}=\emptyset,$ for all $\alpha$, i.e. $S_{w}$ has
no $T_{S}$-fixed points222See Corollary 3.3.4. Recall that the complex
bilinear form $b$ has been defined to satisfy the following orthogonality
condition:

$b(e_{i},e_{2n-i+1})=1\ \text{and}\ b(e_{i},e_{k})=0\ \text{for}\
k\not=2n-i+1\,.$

Let $w\in W_{I}$ be a non-generous permutation. This implies that for some
$j\leq n$ the sign of $w_{j}$ is positive. Let $(w_{1},\ldots,w_{2n})$ be the
basis of the $w$-permuted full flag $F_{w}$ and suppose that for some $j\leq
n$ the sign of $w_{j}$ is not negative. Thus for every $g\in B_{I}$

$g(w_{j})=\lambda(e_{k}-e_{2n-k+1})+\sum_{\ell<k}c_{j}(e_{\ell}-e_{2n-\ell+1})\,.$

Observe that $g(w_{j})$ is $h$-isotropic and therefore if $j=1$, then, since
$\lambda\not=0$, it follows that $g(F_{w})\not\in C_{\alpha}$ for all
$\alpha$.

Now suppose that $j>1$ and suppose that $g(F_{w})$ is in some $C_{\alpha}$.
This implies in particular that for every $i<j$ there are linear combinations

$\xi_{i}=g(w_{i})+\sum_{\ell<i}c_{i\ell}g(w(\ell))\,.$

which when expressed in the standard basis contain only vectors from $V^{-}$
or $V^{+}$. For $g(F_{w})$ to be in $C_{\alpha}$ the same must be true for
$g(w_{j})$. This means that there is a linear combination

$\xi_{j}=g(w_{j})+\sum_{i<j}a_{i}\xi_{i}$

which has the same property. Hence, $e_{k}$ or $e_{2n-k+1}$ (or both) must be
among the vectors with non-zero coefficients in the $\xi_{i}$ for $i<k$. If
both appear, e.g., in $\xi_{\kappa}$ and $\xi_{\delta}$, then
$b(\xi_{\kappa},\xi_{\delta})=1$, contrary to the flag $g(F_{w})$ being
maximally $b$-isotropic. If only one appears, e.g., $e_{k}$ in $\xi_{\kappa}$,
then $e_{2n-k+1}$ appears in $\xi_{j}$ and $b(\xi_{\kappa},\xi_{j})=1$ which
is again a contradiction. Hence the intersection is empty and this completes
the proof.

(i)$\Rightarrow$ (iii)

Let $w$ be a generous permutation. Then $w_{i}<0$ for all $1\leq i\leq n$ and
$B_{I}.(e_{j}+e_{2n-j+1})$, $j=1,\ldots,n$, are the orbits of interest. From
Remark 4.1.2 it follows that the orbit $B_{I}.<e_{i}+e_{2n-i+1}>$ contains the
point

$y=b(e_{i}+e_{2n-i+1})=\lambda(e_{i}+e_{2n-i+1})+\ldots+b_{n}(e_{n}+e_{n+1})+a_{n}(e_{n}-e_{n+1})+\ldots+a_{1}(e_{1}-e_{2n})$

where $\lambda\neq 0$. Then we can choose all constants to be $0$ except
$\lambda=a_{i}=\frac{1}{2}$ if the corresponding sign in $\alpha$ is $-$ ,
then $y=<e_{i}>=<e_{\tilde{w}_{i}}>$. If the corresponding sign in $\alpha$ is
$+$ , then we can choose all constants to be $0$ except
$\lambda=-a_{i}=\frac{1}{2}$ and therefore
$y=<e_{2n-i+1}>=<e_{\tilde{w}(i)}>$. Note that the vectors $e_{i}$ and
$e_{2n-i+1}$ are of negative and positive norms, respectively.

In summary, we have constructed a set of $T_{S}$-fixed points in the orbits
$S_{w}$ for $w$ generous with no two belonging to the same flag domain. By
directly checking we see that all possible signatures have been obtained and
therefore every flag domain has non-empty intersection with one such $S_{w}$.

∎

######  Remark 4.1.5.

Assume that we have an intersection point of the Schubert variety $S_{w}$ with
the flag domain $D_{\alpha}$ given by the flag associated to the basis
$(e_{\tilde{w}_{1}},e_{\tilde{w}_{2}},...,e_{\tilde{w}_{n}})$. Therefore this
point belongs to the open orbit $D_{\alpha}$ given by
$\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})$ where $\alpha_{i}=+$ if
$\tilde{w}_{i}>0$ and $\alpha_{i}=-$ if $\tilde{w}(i)<0$. In this case if
$\tilde{w}_{i}<0$, then $e_{\tilde{w}_{i}}=e_{2n-w(i)+1}$.∎

######  Remark 4.1.6.

The basis which defines the flag $F_{I}$ in the closed orbit of
$SP(2n,\mathbb{R})$ is almost the same basis as that in Brecan’s thesis [4]
for the case $SU(p,q)$, and the bases which define the base cycles is the same
in both cases. Thus in principle Brecan’s algorithms and our description for
$Sp(2n,\mathbb{R})$ should be similar. In Brecan’s thesis [4] for the case
$SU(n,2n)$ if the number $2n-i+1$ stays at the left of the number $i$ for all
$1\leq i\leq n$ in the one line notation of the permutation $w$, then the
Schubert variety $S_{w}$ has nonempty intersection with some base cycles. If
we translate this condition to our style of writing the permutations with
negatives, the number $2n-i+1$ will translates to $-i$. So the condition
becomes : If the number $-i$ stays at the left of the number $i$ for all
$1\leq i\leq n$ in the one line notation of the permutation $w$, then the
Schubert variety $S_{w}$ has nonempty intersection with some base cycles.
Recall that in our permutation only $i$ or $-i$ appear in the first half which
we used to write the permutations, see $\S 2.2.3$. ∎

Recall that the base cycle $C_{\alpha}$ is the set

$C_{\alpha}=\\{F\in Z:V_{i}=(V_{i}\cap E^{-})\oplus(V_{i}\cap E^{+}),1\leq
i\leq 2n\\}.$

Since $C_{\alpha}=K/(K\cap B_{z_{0}})$, where $z_{0}$ is an appropriate point
in an open $G_{0}$-orbit [i.e. $G_{0}.z_{0}$ is open in $Z$], the dimension of
$C_{\alpha}$ is $\frac{n(n-1)}{2}$ and Iwasawa-Schubert varieties dual to
$C_{\alpha}$ has dimension $\frac{n(n+1)}{2}$.

The final step in this section is to determine the Schubert varieties of
complementary dimension to that of the base cycle $C_{\alpha}$.

#### The length of the elements of ${\displaystyle
S_{n}\ltimes\mathbb{Z}_{2}^{n}}$

For dimension computations, let us state how can we compute the length of
elements $w\in{\displaystyle S_{n}\ltimes\mathbb{Z}_{2}^{n}}$ relative to our
notation for the Weyl group elements.

######  Lemma 4.1.7.

Fix $w\in{\displaystyle S_{n}\ltimes\mathbb{Z}_{2}^{n}}$, construct an
$\tilde{w}\in{\displaystyle S_{n}\ltimes\mathbb{Z}_{2}^{n}}$ by the following
algorithm:

  1. 1.

Start from left to right in $w$, using simple reflections, place all positive
numbers in $w$ step by step in a sequence of $n-$empty boxes beginning from
the first one in $\tilde{w}$, in the same order as they appeared in $w$.

  2. 2.

From left to right in $w$ replace a negative number with its absolute value in
the $n-$empty boxes starting from right to left.

If $\tilde{w}=(\tilde{w}(1),\tilde{w}(2),...,\tilde{w}(n))$, then define
$L(\tilde{w})=\frac{n^{2}-n}{2}-number~{}of~{}\\{\tilde{w}(i):i<k~{}and~{}\tilde{w}(i)<\tilde{w}(k)\\}$
, and if we have $m$ negative signs in $w$ in the following positions
$j_{1},j_{2},...,j_{m}$, then define $f(w)=\sum_{i=1}^{m}[(n-j_{i})]$.It
follows that the length of $w$ is

$l(w)=L(\tilde{w})+f(w)+m$

###### Proof.

The length of the permutation $w\in S_{n}\ltimes\mathbb{Z}_{2}^{n}$ is the
minimal number of simple reflection which define $w$. To compute this note
that we have two kind of reflections , the first $n-1$ reflections are the
simple reflections in $S_{n}$ and the last one is the reflection which flips
the sign. This means that if we want to flip the sign of $w(j)$, we should
move $w(j)$ from its position to the last position and then flip its sign and
then return it back to its position. More precisely, consider the word
$w_{0}=(123...n)$ and let $w$ be the word where
$w_{j_{i}},w_{j_{2}},...,w_{j_{m}}$ in the positions
$j_{1}<j_{2}<...<j_{m},~{}1\leq j_{i}\leq n$, are negatives. To construct $w$
from $w_{0}$, we first flip the signs for the numbers
$w_{j_{i}},w_{j_{2}},...,w_{j_{m}}$. For this purpose we move each number
,starting from $w_{j_{m}}$, to the last position and then flip the sign of it.
Then we apply the simple reflection to the positive numbers to put them in the
same order as they appear in the word $w$. In this way the sum of all these
movements and flips is exactly $L(\tilde{w})+m$. The last step is to move each
$w_{j_{i}}$ to its original position in $w$ and denote the total number of
these movements by $f(w)$. It then follows that $l(w)=L(\tilde{w})+f(w)+m$. ∎

######  Example 4.1.8.

Let $w=((-12)5(-34)\in S_{5}\ltimes\mathbb{Z}_{2}^{5}$, then by following the
above remark we have

$(-12)5(-34)\Longrightarrow 254(-1-3)\Longrightarrow 254(31)$

so $w=(-12)5(-34)$ and $\tilde{w}=254(31)$, then $L(\tilde{w})=7$ and
$f(w)=5$.
Hence $l(w)=7+5+2=14$

######  Definition 4.1.9.

The element $w_{0}=(-n-(n-1)....-2-1)\in S_{n}\ltimes\mathbb{Z}_{2}^{n}$ is
called Super generous permutation and has length $\frac{n(n+1)}{2}$.

######  Theorem 4.1.10.

There exists a unique Iwasawa Schubert variety $S_{w}$ of dimension
$\frac{n(n+1)}{2}$. It is given by the super generous permutation
$w_{0}=(-n-(n-1)....-2-1)$.

###### Proof.

Define the set $W_{gen}$ to be the set of all generous permutations, i.e. the
set of all permutations ${\displaystyle w\in S_{n}\ltimes\mathbb{Z}_{2}^{n}}$
such that all signs in $w$ are negative. Then by using the above formula to
compute the length of $w$ , it is clear that $f(w)+n=\frac{n(n+1)}{2}$ for all
$w\in W_{gen}$. Thus the shortest element of $W_{gen}$ has $l(\tilde{w})=0$,
i.e. $\tilde{w}$ is the shortest element in $S_{n}$ which is unique and equal
to $(12...n)$. Therefore there is a unique element $w_{0}\in W_{gen}$ with
length $\frac{n(n+1)}{2}$ which is $(-n,\ldots,-1)$.
∎

###  4.2 Intersection points of Schubert duality

Assuming that $w_{0}\in W_{I}$ is the super generous permutation, then
$S_{w_{0}}$ is of complementary dimension to the cycles, and the intersection
$S_{w_{0}}\cap C_{\alpha}$ is in fact only one point which is $T_{S}$-fixed
(see Corollary 4.2.2). It should be a simple matter to compute all such
intersection points. The argument in the case of $Sp_{2n}(\mathbb{R})$ goes as
follows.

Since $w=(-n,\ldots,-1)$, the flag basis corresponding to $w(F_{I})$

$(\hat{v}_{n},\ldots\hat{v}_{1},v_{1},\ldots v_{n})$

where $v_{k}=e_{k}-e_{2n-k+1}$ and $\hat{v}_{k}=e_{k}+e_{2n-k+1}$.

######  Proposition 4.2.1.

Let $w$ be a super generous permutation, and let the $\varepsilon$-basis
$(\varepsilon_{1},\ldots,\varepsilon_{n})$ of $T_{S}$-eigenvectors be the
basis of an intersection point in $S_{w}\cap C_{\alpha}$, then the
$\varepsilon$-basis is given by $\varepsilon_{k}=e_{n-k+1}$ or $e_{n+k-1}$,
depending on the signature $\alpha$.

###### Proof.

This is a consequence of the following:

  1. 1.

$b.\hat{v}_{k}=\lambda_{k}(e_{k}+e_{2n-k+1})+a_{k}(e_{k}-e_{2n-k+1})+B_{k}=K_{k}+B_{k}$

  2. 2.

$\lambda_{k}\not=0$

  3. 3.

The intersection $S_{w}\cap C_{\alpha}$ is a flag defined by
$T_{S}$-eigenvectors.

From the expression for $\hat{v}_{k}$ it is immediate that all of the
possibilities in the statement occur. Furthermore, since the $\lambda_{k}$ are
non-zero, for every $k$ a non-zero contribution from $K_{k}$ occurs in the sum

$\varepsilon_{k}=\sum_{j\leq k}c_{kj}b.v_{w_{j}}\,.$

Since $e_{n-k+1}$ and $e_{n+k-1}$ do not occur in $b.v_{w_{j}}$ for $j<k$, it
follows that $\varepsilon_{k}=K_{k}+B_{k}$ in the standard basis (See $\S
2.2.2$). Finally, since $\varepsilon_{k}$ is a $T_{S}$-eigenvector, it follows
that $\varepsilon_{k}=K_{k}$ and is of the type in the the statement of the
proposition. ∎

As a result, if $w$ is super generous, then the set of all intersection points
of $(S_{w}\cap C_{\alpha})$ can be defined to be the following:

$\text{Supset}(w):=\\{F_{\tilde{w}}:\tilde{w}\in
W\text{and}~{}\tilde{w}\text{~{}obtained~{}by~{}change~{}none,~{}some~{}or~{}all
of~{}the~{}numbers}~{}-i~{}by~{}i\\}$

where $F_{\tilde{w}}$ is the maximally $b$-isotropic flag associated to the
standard basis $\mathbb{C}^{2n}$ given in (4.2). Here $\tilde{w}$ gives us a
point in flag domain and to know the signature of this flag domain just
replace $i$ by $+$ and $-i$ by $-$.

One more remark, the number of flag domains in $G/B$ is $2^{n}$. So as a
results from the above discussion we have the following corollary

######  Corollary 4.2.2.

If $w$ is the super generous permutation in $W$, then

  1. 1.

The total number of intersection points is $2^{n}$, each with different
signature.

  2. 2.

The intersection $S_{w}\cap C_{\alpha}$ is exactly one $T_{S}$ fixed point.

######  Example 4.2.3.

The set of all intersection points of the Schubert variety parametrized by the
element $w=(-3-1-2)$ is
$Supset(w)=\\{(-3-1-2).\tilde{z},(3-1-2).\tilde{z},(-31-2).\tilde{z},(-3-12).\tilde{z},(31-2).\tilde{z},(-312).\tilde{z},(3-12).\tilde{z},(312).\tilde{z}\\}$
. So $S_{w}$ intersects the following flag domains:
$(---),(+--),(-+-),(--+),(++-),(+-+),(++-),(+++)$.

##  Chapter 5 Cycle intersection for $SO^{*}(2n)$

Here we deal with the homogeneous space $Z$ of maximally isotropic full flags
of the complex orthogonal symmetric group $G=SO(2n,\mathbb{C})$ equipped with
the action of the real form $G_{0}=Sp(2n,\mathbb{R})$.

###  5.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$

######  Definition 5.1.1.

An element $w=(w_{1},w_{2},...,w_{n})\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ is
called a dense permutation111Recall that we define a special way of denoting
to write the Weyl elements in section $\S 2.2.3$. if $w_{i}<0$ for all $1\leq
i\leq n$ if $n$ is even, and $w_{i}<0$ for all $1\leq i\leq n-1$ and $w_{n}>0$
if $n$ is odd .

This definition will aid in determining the Weyl elements which parametrize
the Schubert varieties that have nonempty intersection with base cycles of
flag domains which contain $T_{S}$-fixed points.

######  Remark 5.1.2.

When discussing the $B_{I}$-orbits of the base point $F_{I}$ it is important
to explicitly understand the orbits $B_{I}.(e_{i}-e_{n+i})$ and
$B_{I}.(e_{i}+e_{n+i})$. If $F_{I}$ and $B_{I}$ are as above, then the orbits
of interest are $B_{I}.(e_{i}\pm e_{n+i})$, $i=1,\ldots,n$ if $n$ is even, and
$B_{I}.(e_{i}\pm e_{n+i})$, $i=1,\ldots,n-1$ and $B_{I}.(e_{\frac{n+1}{2}})$
and $B_{I}.(e_{n+\frac{n+1}{2}})$ if $n$ is odd . In this case the orbits
$B_{I}.(e_{i}+e_{n+i})$ , $B_{I}.(e_{i}-e_{n+i})$, $B_{I}.(e_{\frac{n+1}{2}})$
and $B_{I}.(e_{n+\frac{n+1}{2}})$ have points of the forms

$\bullet$ If $n$ is even:

then for $i=1,\ldots,n$

$b(e_{i}-e_{n+i})=\lambda(e_{i}-e_{n+i-1})+\ldots+b_{n}(e_{n}-e_{2n})+a_{n}(e_{n}+e_{2n})+\ldots+a_{1}(e_{1}+e_{n+1}),$
$b(e_{i}+e_{n+i})=\lambda(e_{i}+e_{n+i})+a_{i-1}(e_{i-1}+e_{n+i-1})+\ldots+a_{1}(e_{1}+e_{n+1}),\text{with}\lambda\not=0.$

$\bullet$ If $n$ is odd:

then for $i=1,\ldots,n-1$

$b(e_{i}-e_{n+i})=\lambda(e_{i}-e_{n+i-1})+\ldots+b_{n-1}(e_{n}-e_{2n})+b_{n}e_{n+\frac{n+1}{2}}+a_{n}e_{\frac{n+1}{2}}+a_{n-1}(e_{n}+e_{2n})+$

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ldots+a_{1}(e_{1}+e_{n+1}),$

$b(e_{i}+e_{n+i})=\lambda(e_{i}+e_{n+i})+a_{i-1}(e_{i-1}+e_{n+i-1})+\ldots+a_{1}(e_{1}+e_{n+1}),$
$b(e_{\frac{n+1}{2}})=\lambda
e_{\frac{n+1}{2}}+a_{n-1}(e_{n}+e_{2n})+\ldots+a_{1}(e_{1}+e_{n+1}),$

and

$b(e_{n+\frac{n+1}{2}})=\lambda
e_{n+\frac{n+1}{2}}+a_{n}e_{\frac{n+1}{2}}+a_{n-1}(e_{n}+e_{2n})+\ldots+a_{1}(e_{1}+e_{n+1}),$

with $\lambda\not=0$.

Note that in the above orbits, if $b$ is chosen appropriately, then we can
arrange $b(e_{i}-e_{n+i})=e_{i}$ or $b(e_{i}-e_{n+i})=e_{n+i}$ for all $i$.
This plays a role in the proof of Theorem 5.1.4. ∎

Depending on $n$ being odd or even, the base point $F_{I}\in\gamma^{c\ell}$ is
that associated to the following ordered basis:

$\bullet$ If $n$ is even:

$e_{1}+ie_{n+1},e_{2}+ie_{n+2}...,e_{n}+ie_{2n},e_{n}-ie_{2n},...,e_{2}-ie_{n+2},e_{1}-ie_{n+1}.$
(5.1)

$\bullet$ If $n$ is odd:

$e_{1}+ie_{n+1},...,e_{n}+ie_{2n},e_{\frac{n+1}{2}},e_{n+\frac{n+1}{2}},e_{n}-ie_{2n},...,e_{1}-ie_{n+1}.$
(5.2)

and the base point which defines the flag domains and the base cycles is the
flag associated to the ordered basis

$e_{1},e_{2},e_{3},......,e_{2n-1},e_{2n}$ (5.3)

These two bases define two maximal tori which are conjugate and the
corresponding Weyl groups $W_{I}$ and $W_{S}$ are isomorphic.
Let $\psi$ be the bijective map between Weyl groups $W_{I}$ and $W_{S}$ which
is defined as follows:

$\bullet$ If $n$ is even:

$\psi(\mp(i))=\pm i,$ for all $1\leq i\leq n$.

$\bullet$ If $n$ is odd:

$\psi(\mp(i))=\pm i,$ for all $1\leq i\leq n-1$ and $\psi(\pm n)=\pm n$.

######  Proposition 5.1.3.

If $w$ is a dense permutation, then the flag $F_{\psi(w)}$ belongs to the
orbit $B_{I}(F_{w})$.

###### Proof.

Without less of generality assume that $n$ is even. Let $w\in W_{I}$ be a
dense permutation, denote the first $n$ subspaces of the associated maximally
isotropic flag $F_{w}$ by

$\\{0\\}\subset<u_{w_{1}}>\subset<u_{w_{1}},u_{w_{2}}>\subset......\subset<u_{w_{1}},...,u_{w_{n}}>$

where $u_{i}=e_{j}\pm e_{n+j}$. We will show that the orbit $B_{I}(F_{w})$
contains the flag $F_{\psi(w)}$. For this purpose let $\tilde{w}\in W_{S}$ be
the image of $w$ under the bijective map $\psi$ and let $Y_{\tilde{w}}$ be the
isotropic flag associated to $\tilde{w}$, i.e., the flag

$Y_{\tilde{w}}:=(\\{0\\}\subset<e_{\tilde{w}_{1}}>\subset<e_{\tilde{w}_{1}},e_{\tilde{w}_{2}}>\subset......\subset<e_{\tilde{w}_{1}},...,e_{\tilde{w}_{n}}>)$

We will show that $Y_{\tilde{w}}=b(F_{w})$ for $b$ appropriately chosen in
$B_{I}$. For this recall that if $w$ is a dense permutation, then $w_{i}<0$
for all $1\leq i\leq n$ , and therefore $B_{I}.(e_{j}-ie_{n+j})$,
$j=1,\ldots,n$, are the orbits of interest. By Remark 5.1.2 the orbit
$B_{I}.<e_{j}-ie_{n+j}>$ contains the points

$y=b(e_{j}-ie_{n+j})=\lambda(e_{j}-ie_{n+j})+\ldots+b_{n}(e_{n}-ie_{2n})+a_{n}(e_{n}+ie_{2n})+\ldots+a_{1}(e_{1}+ie_{2n})$

where $\lambda\neq 0$. Choose all constants to be $0$ except
$\lambda=a_{i}=\frac{1}{2}$ , so that $y=<e_{i}>=<e_{\tilde{w}_{i}}>$.
We repeat this step n-times to construct the first half of the flag
$Y_{\tilde{w}}$, and then extend this flag to maximal $h$-isotropic flag, .
Therefore, the flag $Y_{\tilde{w}}$ is constructed. ∎

######  Theorem 5.1.4.

(Dense Permutation Theorem). The following are equivalent

(i)

$w$ is dense.

(ii)

$B_{I}(F_{w})\cap D_{\alpha}\not=\emptyset$ for some $\alpha$.

(iii)

$B_{I}(F_{w})\cap D_{\alpha}\not=\emptyset$ for every $\alpha$.

Furthermore, under any of these conditions, for every $\alpha$ the
intersection $B_{I}(F_{w})\cap C_{\alpha}$ contains a $T_{S}$-fixed point.

###### Proof.

(iii)$\Rightarrow$ (ii)

Trivial direction.

(ii)$\Rightarrow$ (i)

We handle the case where $n$ is even. The proof for $n$ goes analogously. We
assume by contradiction that $w$ is not a dense permutation and will show that
$S_{w}\cap D_{\alpha}=\emptyset$ for all $\alpha$. In particular, $S_{w}$ has
no $T_{S}$-fixed points222See theorem LABEL:TM1. Recall that the complex
bilinear form $b$ is defined by the following conditions:

$b(e_{i},e_{n+i})=1\ \text{and}\ b(e_{i},e_{k})=0\ \text{for}\ k\not=n+i\,.$

Let $w\in W_{I}$ be a non-dense permutation. Let $(w_{1},\ldots,w_{2n})$ be
the basis of the $w$-permuted full flag $F_{w}$ and suppose that for some
$j\leq n$ the sign of $w_{j}$ is not negative. Thus for every $g\in B_{I}$

$g(w_{j})=\lambda(e_{k}+ie_{n+k})+\sum_{\ell<k}c_{j}(e_{\ell}+ie_{n+\ell})\,.$

Observe that $g(w_{j})$ is $h$-isotropic and therefore if $j=1$, then, since
$\lambda\not=0$, it follows that $g(F_{w})\not\in C_{\alpha}$ for all
$\alpha$.

Now suppose that $j>1$ and suppose that $g(F_{w})$ is in some $C_{\alpha}$.
This implies in particular that for every $i<j$ there are linear combinations

$\varepsilon_{i}=g(w_{i})+\sum_{\ell<i}c_{i\ell}g(w_{\ell})\,.$

which when expressed in the standard basis contain only vectors from $V^{-}$
or $V^{+}$. For $g(F_{w})$ to be in $C_{\alpha}$ the same must be true for
$g(w_{j})$. This means that there is a linear combination

$\varepsilon_{j}=g(w_{j})+\sum_{i<j}a_{i}\varepsilon_{i}$

which has the same property. Hence, $e_{k}$ or $e_{n+k}$ (or both) must be
among the vectors with non-zero coefficients in the $\varepsilon_{i}$ for
$i<k$. If both appear, e.g., in $\varepsilon_{\kappa}$ and
$\varepsilon_{\delta}$, then $b(\varepsilon_{\kappa},\varepsilon_{\delta})=1$,
contrary to the flag $g(F_{w})$ being maximally $b$-isotropic. If only one
appears, e.g., $e_{k}$ in $\varepsilon_{\kappa}$, then $e_{n+k}$ appears in
$\varepsilon_{j}$ and $b(\varepsilon_{\kappa},\varepsilon_{j})=1$ which is
again a contradiction. Hence the intersection is empty and this completes the
proof.

(i)$\Rightarrow$ (iii)

Again, since the two cases are dealt with analogously, we assume here that $n$
is even. Let $w$ be a dense permutation. Then $w_{i}<0$ for all $1\leq i\leq
n$ and $B_{I}.(e_{j}-ie_{n+j})$, $j=1,\ldots,n$, are the orbits of interest.
From Remark 5.1.2 it follows that the orbit $B_{I}.<e_{i}+ie_{n+i}>$ contains
the point

$y=b(e_{j}-ie_{n+j})=\lambda(e_{j}-ie_{n+j})+\ldots+b_{n}(e_{n}-ie_{2n})+a_{n}(e_{n}+ie_{2n})+\ldots+a_{1}(e_{1}+ie_{n+1})$

where $\lambda\neq 0$. Then we can choose all constants to be $0$ except
$\lambda=a_{j}=\frac{1}{2}$. It follows that $y=<e_{j}>=<e_{\tilde{w}_{j}}>$
if the corresponding sign in $\alpha$ is $-$. If the corresponding sign in
$\alpha$ is $+$, then we can choose all constants to be $0$ except
$\lambda=-a_{j}=\frac{1}{2}$ , and $y=<e_{n+j}>=<e_{\tilde{w}_{j}}>$. Note
that the vectors $e_{j}$ and $e_{n+j}$ are of negative and positive norms,
respectively. In summary, we have constructed a set of $T_{S}$-fixed points in
the orbits $S_{w}$ for $w$ dense with no two belonging to the same flag
domain. By directly checking we see that all possible signatures have been
obtained and therefore every flag domain has non-empty intersection with one
such $S_{w}$.

∎

By following the above proof we have the following corollary;

######  Corollary 5.1.5.

If $w$ is a dense permutation, then for all $\alpha$ the intersection
$S_{w}\cap C_{\alpha}$ contains exactly one $T_{S}$-fixed point and in
particular is connected.

######  Remark 5.1.6.

Assume that we have an intersection point given by the flag associated to the
basis $(e_{\tilde{w}_{1}},e_{\tilde{w}_{2}},...,e_{\tilde{w}_{n}})$. Therefore
this point belongs to the open orbit $D_{\alpha}$ given by
$\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})$ where $\alpha_{j}=+$ if
$\tilde{w}_{j}>0$ and $\alpha_{j}=-$ if $\tilde{w}_{j}<0$. In this case if
$\tilde{w}_{j}<0$, then $e_{\tilde{w}_{j}}=e_{n+w_{j}}$.∎

Recall that the complexification of the maximal compact subgroup $K$ of $G$
has a unique compact orbit in each open $G_{0}$-orbits which is a complex
manifold (see [15]) and called the base cycle $C_{0}$. Similarly as the case
$SP(2n,\mathbb{R})$, the base cycle $C_{0}$ of the flag domain $D_{\alpha}$ of
$SO^{*}(2n)$ is the set

$C_{0}=\\{F\in Z:V_{i}=(V_{i}\cap E^{-})\oplus(V_{i}\cap E^{+}),1\leq i\leq
2n\\}.$

where the intersection dimensions are determined by $\alpha$. Since
$C_{0}=K/(K\cap B_{z_{0}})$ where $z_{0}$ is any base point in $C_{\alpha}$,
it follows that the dimension of $C_{0}$ is $\frac{n(n-1)}{2}$ and the Iwasawa
Schubert variety must be of dimension $\frac{n(n-1)}{2}$.

####  5.1.1 Length of the elements of ${\displaystyle
S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$

For dimension computations, let us state how can we compute the length of
elements $w\in{\displaystyle S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$ relative to
our notation for the Weyl group elements.

######  Lemma 5.1.7.

Fix $w\in W$, construct $\tilde{w}\in W$ by the following algorithm:

  1. 1.

Start from left to right in $w$, using simple reflections place all positive
numbers in $w$ step by step in a sequence of $n-$empty boxes beginning from
the first one in $\tilde{w}$, in the same order as they appeared in $w$.

  2. 2.

From left to right in $w$ replace a negative number with its absolute value in
the $n-$empty boxes starting from right to left.

If $\tilde{w}=(\tilde{w}_{1},\tilde{w}_{2},...,\tilde{w}_{n})$, then define
$L(\tilde{w})=\frac{n^{2}-n}{2}-\text{the cardinality of
}\\{\tilde{w}_{j}:j<k~{}and~{}\tilde{w}_{j}<\tilde{w}_{k}\\}$ , and if we have
$2m$ negative signs in $w$ in the following positions
$k_{1},k_{2},...,k_{2m}$, then define
$f(w)=\sum_{j=1}^{m}[(2n-1-k_{2j}-k_{2j-1})]$. It follows that the length of
$w$ is

$l(w)=L(\tilde{w})+f(w)+m$

###### Proof.

The length of the permutation $w\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ is the
minimal number of simple reflections which define $w$. To compute this note
that we have two kinds of reflections , the first $n-1$ reflections are the
simple reflections in $S_{n}$ and the last one is the reflection which flips
the sign. This means that if we want to flip the sign of $w_{j}$, we should
move $w_{j}$ from its position to the last position and then flip its sign and
then return it back to its position. More precisely, consider the word
$w_{0}=(123...n)$ and let $w$ be the word where
$w_{j_{i}},w_{j_{2}},...,w_{j_{m}}$ in the positions
$j_{1}<j_{2}<...<j_{m},~{}1\leq j_{i}\leq n$, are negatives. To construct $w$
from $w_{0}$, we first flip the signs for the numbers
$w_{j_{i}},w_{j_{2}},...,w_{j_{m}}$. For this purpose we move each number
,starting from $w_{j_{m}}$, to the last position and then flip the sign of it.
Then we apply the simple reflection to the positive numbers to put them in the
same order as they appear in the word $w$. In this way the sum of all these
movements and flips is exactly $L(\tilde{w})+m$. The last step is to move each
$w_{j_{i}}$ to its original position in $w$ and denote the total number of
these movements by $f(w)$. It then follows that $l(w)=L(\tilde{w})+f(w)+m$. ∎

######  Example 5.1.8.

Let $w=((-12)5(-34)6\in S_{6}\ltimes\mathbb{Z}_{2}^{5}$, then by following the
above proof we have

$(-12)5(-34)-6\Longrightarrow 2546(-1-3)\Longrightarrow 2546(31)$

so $w=(-12)5(-34)6$ and $\tilde{w}=2546(31)$, then $L(\tilde{w})=9$ and
$f(w)=6$.
Hence $l(w)=9+6+1=16$.

######  Definition 5.1.9.

Let $w=(-n-(n-1)....-2-1)\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ if $n$ is even
and $w=(n-(n-1)....-2-1)\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ if $n$ is odd.
any of these two elements called Super dense and has dimension
$\frac{n(n-1)}{2}$.

As the case of the real form $SP(2n,\mathbb{R})$, by direct calculation of the
dimension of the dense permutations we get the following result.

######  Theorem 5.1.10.

There exists a unique Iwasawa Schubert variety $S_{w}$ of dimension
$\frac{n(n-1)}{2}$. It is given by the super dense permutation
$\tilde{w}=(-n-(n-1)....-2-1)$ if $n$ is even and by
$\tilde{w}=(n-(n-1)....-2-1)$ if $n$ is odd.

###### Proof.

Assume firstly that n is even, then consider the set $W_{den}$ to be the set
of all dense permutations, i.e. the set of all permutations ${\displaystyle
w\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$ such that all signs in $w$ are
negative. Then, using our formula to compute the length of $w$ it is clear
that $f(w)+n=\frac{n(n-1)}{2}$ for all $w\in
S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$. Thus the shortest element of $W_{den}$ has
$l(\tilde{w})=0$, i.e. $\tilde{w}$ is the shortest element in $S_{n}$ which is
unique and equal to $(12...n)$. Therefore there is a unique element $w\in
W_{den}$ with length $\frac{n(n-1)}{2}$ which is $(-n-(n-1)....-2-1)$.

Secondly, let $n$ is odd and let $\tilde{W}_{den}$ to be the set of all dense
permutations, i.e. the set of all permutations ${\displaystyle w\in
S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$ such that all signs in $w$ are negative
except the sign of the entry $n$ is positive. Then, using our formula to
compute the length of $w$, it is clear that $f(w)+n=\frac{n(n-1)}{2}$ for all
$w\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ only if the entry $n$ sitting in the
first box of the permutation. Thus shortest element of $\tilde{W}_{den}$ has
$l(\tilde{w})=0$. Then the shortest element in $S^{n}$ is $\tilde{w}$ which is
unique and equal to $(12...n)$ which implies that we have a unique element
$w\in\tilde{W}_{den}$ with length $\frac{n(n-1)}{2}$ which is
$(n-(n-1)....-2-1)$.
∎

###  5.2 Intersection points of Schubert duality

Assuming that the element $w_{0}\in W_{I}$ is $w_{0}=(-n-(n-1)....-2-1)$ if
$n$ is even and $w_{0}=(n-(n-1)....-2-1)$ if $n$ is odd , we have shown that
$S_{w}$ is of complementary dimension to the cycles, and the intersection
$S_{w}\cap C_{\alpha}$ is in fact only one $T_{S}$-fixed point (See corollary
5.1.5). Here we describe all intersection points. The argument in the case of
$SO^{*}(2n)$ , which is similar to that in the case $SP(2n,\mathbb{R})$, goes
as follows.

Since $w=(-n,\ldots,-1)$, the flag basis corresponding to $w_{0}$

$(\hat{v}_{n},\ldots\hat{v}_{1},v_{1},\ldots v_{n})$

where $v_{k}=e_{k}+ie_{n+k}$ and $\hat{v}_{k}=e_{k}-ie_{n+k}$.

######  Proposition 5.2.1.

If $w$ is a super dense permutation, and the $\varepsilon$-basis
$(\varepsilon_{1},\ldots,\varepsilon_{n})$ of $T_{S}$-eigenvectors defines the
intersection point in $S_{w}\cap C_{\alpha}$, then $\varepsilon_{k}=e_{k}$ or
$e_{n+k}$, depending on the signature $\alpha$.

###### Proof.

This is a consequence of the following:

  1. 1.

$\hat{v}_{k}=\lambda_{k}(e_{k}-e_{n+k})+a_{k}(e_{k}+ie_{n+k})+R_{k}=K_{k}+R_{k}$

  2. 2.

$\lambda_{k}\not=0$

  3. 3.

The intersection $S_{w}\cap C_{\alpha}$ is a flag defined by
$T_{S}$-eigenvectors.

From the expression for $\hat{v}_{k}$ it is immediate that all of the
possibilities in the statement occur. Furthermore, since the $\lambda_{k}$ are
non-zero, for every $k$ a non-zero contribution from $K_{k}$ occurs in the sum

$\varepsilon_{k}=\sum_{j\leq k}c_{kj}b(w(j))\,.$

Since $e_{k}$ and $e_{n+k}$ do not occur in $b(w(j))$ for $j<k$, it follows
that $\varepsilon_{k}=K_{k}+E_{k}$ in the standard basis (See $\S 2.2.2$).
Finally, since $\varepsilon_{k}$ is a $T_{s}$-eigenvector, it follows that
$\varepsilon_{k}=K_{k}$ and is of the type in the the statement of the
proposition. ∎

If $w$ is a super dense permutation, then the set of all intersection points
of $S_{w}$ is denoted by $Supset_{e}(w)$ if n is even and by $Supset_{o}(w)$
if n is odd, where

$Supset_{e}(w):=\\{\tilde{w}(F_{S})\in
W:\tilde{w}~{}obtained~{}by~{}change~{}none,~{}some~{}or~{}all~{}of~{}the~{}numbers~{}-j~{}by~{}j\\}$
$Supset_{o}(w):=\\{\tilde{w}(F_{S})\in
W:\tilde{w}~{}obtained~{}by~{}change~{}none,~{}some~{}or~{}all~{}of~{}the~{}numbers~{}-j~{}by~{}j$
$~{}or~{}n~{}by~{}-n\\},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $F_{S}$ is the maximally $b$-isotropic flag associated to the standard
basis $e_{1},e_{2},...,e_{2n}$. Here $\tilde{w}(F_{S})$ gives us a point in an
open orbit $D_{\alpha}$ and to know this open orbit just replace $i$ by $+$
and $-i$ by $-$.

##  Chapter 6 Cycle intersection for $SO(p,q)$

In this chapter we study the case of the real form $SO(p,q)$ of
$SO_{m}(\mathbb{C})$. The general results here are stated in terms of
algorithms (See definitions 6.1.1 and 6.2.1); in fact it seems impossible to
avoid this. In corollary 6.3.2 we give concrete formulas for the intersection
points in $S_{w}\cap C_{\alpha}$ if the intersection non-empty and $S_{w}$ is
of complementary dimension . Also, the number of intersection points with
$C_{0}$ is explicitly computed in Theorem 6.3.3.

###  6.1 Conditions for $S_{w}\cap C_{\alpha}\neq\emptyset$

In the present section we describe the conditions for an element $w$ of the
Weyl group to parametrize Schubert variety with $S_{w}$ must satisfy in order
that $S_{w}\cap D_{\alpha}\not=\emptyset$ for some flag domain $D_{\alpha}$.
As would be expected, a special type of permutation plays a fundamental role.

######  Definition 6.1.1.

An element $w\in W$ is called a harmonic permutation if it satisfies the
following conditions:
If $q$ is even: The number $-(2i-1),1\leq i\leq\frac{q}{2}$, sits in any place
to the left of the number $(2i)$ or $(-2i),1\leq i\leq\frac{q}{2}$, in the one
line notation of the permutation and the order of the numbers $q+i$ or
$-(q+i)$, where $1\leq i\leq p-q$ is arbitrary.
If $q$ is odd: The number $-(2i-1),1\leq i\leq\frac{q-1}{2}$, sits in any
place to the left of the number $(2i),1\leq i\leq\frac{q-1}{2}$, and the
number $-q$ sits in the last position in the one line notation of the
permutation, and the order of the numbers $q+i$ or $-(q+i)$, where $1\leq
i\leq\frac{p-q}{2}$ is arbitrary.

######  Example 6.1.2.

In $SO(4,2)$ the relevant pairs are $(-12)$ and $(-1-2)$. As a result we have
$6$ harmonic permutations. These are: $(-12-3),(-1-32),(-3-12),(-1-23),\\\
(-13-2),(3-1-2)$.

Recall that the fixed point in the closed orbit $\gamma^{cl}$ is the flag
associated to the following ordered basis

  * •

If $m$ is even, then the basis is

$\displaystyle
e_{1}+e_{2q},...,e_{q}+e_{q+1},e_{2q+1}+ie_{2q+2},e_{2q+3}+ie_{2q+4},...,e_{2n-1}+ie_{2n},$
$\displaystyle
e_{2n-1}-ie_{2n},...,e_{2q+3}-ie_{2q+4},e_{2q+1}-ie_{2q+2},e_{q}-e_{q+1},...,e_{1}-e_{2q}.$
(6.1)

  * •

If $m$ is odd, then the basis is

$\displaystyle
e_{1}+e_{2q},e_{2}+e_{2q-1},...,e_{q}+e_{q+1},e_{2q+1},e_{2q+2}+ie_{2q+3},e_{2q+4}+ie_{2q+5},...,e_{2n}+ie_{2n+1},$
$\displaystyle
e_{2q+1},e_{2n}-ie_{2n+1},...,e_{2q+4}-ie_{2q+5},e_{2q+2}-ie_{2q+3},e_{q}-e_{q+1},...,e_{2}-e_{2q-1},e_{1}-e_{2q}~{}~{}~{}$
. (6.2)

######  Remark 6.1.3.

For $v_{i}$ any such basis vector and $b\in B_{I}$ the form of $b.v_{i}$ is
given as follows:

  1. 1.

$b.(e_{i}-e_{2q-i+1})=\lambda(e_{i}-e_{2q-i+1})+\ldots+b_{n}(e_{q}-e_{q+1})+a_{n}(e_{q}+e_{q+1})+\ldots+a_{1}(e_{1}+e_{2q})$,

  2. 2.

$b.(e_{i}+e_{2q-i+1})=\lambda(e_{i}+e_{2q-i+1})+a_{i-1}(e_{i-1}-e_{2q-i})+\ldots+a_{1}(e_{1}-e_{2q})$,

  3. 3.

$b.(e_{2q+2j-1}-ie_{2q+2j})=\lambda(e_{2q+2j-1}-ie_{2q+2j})+\ldots+b_{m}(e_{m-1}-e_{m})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+a_{m}(e_{m-1}+e_{m})+\ldots+a_{1}(e_{1}+e_{2q})$

where $m=2n$ or $2n+1$.

  4. 4.

$b.(e_{2q+2j-1}+ie_{2q+2j})=\lambda(e_{2q+2j-1}+ie_{2q+2j})+a_{i-1}(e_{2q+2j+1}+ie_{2q+2j+2})+\ldots+a_{1}(e_{1}+e_{2q})$,

  5. 5.

$b.(e_{2q+1})=\lambda(e_{2q+1})+a_{n}(e_{2n}+ie_{2n+1})+\ldots+a_{1}(e_{1}+e_{2q})$,
if $m=2n+1$,

with $\lambda\not=0$ in all cases above. Note that in the above orbits, if $b$
is chosen appropriately, then we can arrange all linear combination for every
one of the above vectors to be in the standard basis of $T_{S}$-eigenvectors.
See Theorem 6.1.5.∎

In the following result $\psi:W_{I}\to W_{S}$ denotes the bijective map
between Weyl groups which was introduced in Chapter 2.

######  Proposition 6.1.4.

If $w$ is a a harmonic permutation, then the flag $F_{\psi(w)}$ belongs to the
orbit $B_{I}(F_{w})$.

###### Proof.

We handle the case where $m=2n$. The proof for $m=2n+1$ goes analogously. Let
us first prove the theorem for the case that $q$ is even. For this let $w\in
W_{I}$ be a harmonic permutation and define $F_{w}=w.(F_{I})$ to be the
isotropic full flag associated to $w$. Denote by

$\\{0\\}\subset<u_{w_{1}}>\subset<u_{w_{1}},u_{w_{2}}>\subset......\subset<u_{w_{1}},...,u_{w_{n}}>$

the first $n$ subspaces of $F_{w}=w(F_{I})$, where $u_{w_{i}}$ is a vector
from the basis above. Let $\tilde{w}\in W_{S}$ be the image of $w$ under the
bijective map $\psi$ and let $Y_{\tilde{w}}$ be the isotropic flag associated
to $\tilde{w}$ such that the first half of $Y_{\tilde{w}}$ is

$\\{0\\}\subset<\varepsilon_{\tilde{w}_{1}}>\subset<\varepsilon_{\tilde{w}_{1}},\varepsilon_{\tilde{w}_{2}}>\subset......\subset<\varepsilon_{\tilde{w}_{1}},...,\varepsilon_{\tilde{w}_{n}}>$

Our claim here is that this flag is an intersection point in $B.F_{w}\cap
C_{\alpha}$. To prove this we will construct $b$ with
$b(F_{w})=Y_{\tilde{w}}$. From the definition of harmonic permutation, there
are two possibilities for $w_{1}$: $|w_{1}|=2i-1\leq q$ or $|w_{1}|>q$.
Case 1: If $|w_{1}|>q$. Consequently then $u_{w_{1}}=v_{\tilde{w}_{1}}$, so
the orbit $B_{I}.<u_{w_{1}}>$ contains the point
$<\varepsilon_{\tilde{w}_{1}}>$.
Case 2: If $|w_{1}|=2i-1\leq q$, then we must consider the orbit
$B_{I}.<e_{2\tilde{w}_{1}-1}-e_{2q+w_{1}+1}>$. By Remark (6.1.3) above, the
orbit $B_{I}.<e_{2\tilde{w}_{1}-1}-e_{2q+w_{1}+1}>$ contains points of the
form

$y=<\alpha_{1}(e_{2\tilde{w}_{1}-1}+e_{2q+w_{1}+1})+\alpha_{2}(e_{2\tilde{w}_{1}}+e_{2q+w_{1}})+\alpha_{3}(e_{2\tilde{w}_{1}}-e_{2q+w_{1}})+\alpha_{4}(e_{2\tilde{w}_{1}-1}-e_{2q+w_{1}+1})>$

where $\alpha_{1}=\pm\alpha_{4}$ and $\alpha_{2}=\pm\alpha_{3}$. By taking
$\alpha_{1}=\alpha_{4}=\frac{1}{2}$ and $\alpha_{2}=\alpha_{3}=\frac{1}{2}i$,
it follows that
$y=<e_{2\tilde{w}_{1}-1}+ie_{2\tilde{w}_{1}}>=<v_{\tilde{w}_{1}}>$.

To construct the $j$-vector of $b(v_{w_{j}})$ to obtain the subspace
$V_{\tilde{w}_{j}}$ we must consider three cases:
Case 1: If $|w_{j}|>q$, then $w_{j}=\tilde{w}_{j}$ and
$\varepsilon_{\tilde{w}_{j}}=u_{w_{j}}$, so the orbit $B_{I}.<u_{w_{j}}>$
contains the point $\varepsilon_{\tilde{w}_{j}}$.
Case 2: If $|w_{j}|=2i-1\leq q$, then our job goes through the orbit
$B_{I}.<e_{2\tilde{w}_{j}-1}-e_{2q+w_{j}+1}>$. By using Remark (6.1.3) we see
that the orbit $B_{I}.<e_{2\tilde{w}_{j}-1}-e_{2q+w_{j}+1}>$ contains points
of the form

$y=<\alpha_{1}(e_{2\tilde{w}_{j}-1}+e_{2q+w_{j}+1})+\alpha_{2}(e_{2\tilde{w}_{j}}+e_{2q+w_{j}})$
$+\alpha_{3}(e_{2\tilde{w}_{j}}-e_{2q+w_{j}})+\alpha_{4}(e_{2\tilde{w}_{j}-1}-e_{2q+w_{j}+1})>$

where $\alpha_{1}=\pm\alpha_{4}$ and $\alpha_{2}=\pm\alpha_{3}$. By taking
$\alpha_{1}=\alpha_{4}=\frac{1}{2}$ and $\alpha_{2}=\alpha_{3}=\frac{1}{2}i$,
it follows that
$y=<e_{2\tilde{w}_{j}-1}+ie_{2\tilde{w}_{j}}>=<\varepsilon_{\tilde{w}_{j}}>$.
Therefore $b$ is constructed to obtain the flag

$\\{0\\}\subset<\varepsilon_{\tilde{w}_{1}}>\subset<\varepsilon_{\tilde{w}_{1}},\varepsilon_{\tilde{w}_{2}}>\subset......\subset<\varepsilon_{\tilde{w}_{1}},...,\varepsilon_{\tilde{w}_{j}}>.$

Case 3: If $|w_{j}|=2i$ , then the orbit is
$B_{I}.<e_{|w_{j}|}+e_{|2\tilde{w}_{j}|-1}>$ is relevant. In this case the
points

$y=<\alpha_{1}(e_{|w_{j}|-1}+e_{|2\tilde{w}_{j}|})+\alpha_{2}(e_{|w_{j}|}+e_{|2\tilde{w}_{j}|-1})+\alpha_{3}(e_{|w_{j}|-1}+ie_{|w_{j}|})>$

belong to the orbit $B_{I}.<e_{|w_{j}|}+e_{|2\tilde{w}_{j}|-1}>$. For
$\alpha_{1}=-i,\alpha_{2}=1$ and $\alpha_{3}=i$ we have
$y=<e_{|2\tilde{w}_{j}|-1}+ie_{|2\tilde{w}_{j}|}>$. Therefore, the $j$-vector
of $b$ is constructed in this case as well..
Thus by induction we observe that $b\in B_{I}$ can be constructed with
$b(F_{w})=Y_{\tilde{w}}$.

We complete the proof by handling the case where $q$ is odd. For this point we
can repeat the steps of the proof above for $w_{j}>q$ and $|w_{j}|=2i-1$. For
the case where $|w_{j}|=2i$, we apply the same method as above, only changing
$2|\tilde{w}_{j}|$ by $2(|\hat{w}_{j}|+1)$. If $w_{n}=-q$, then
$\hat{w}_{j}=-n$. In this case the orbit of relevance is
$B_{I}.<e_{q}-e_{q+1}>$. As we see from Remark 6.1.3
$y=\alpha_{1}(e_{q}+e_{q+1})+\alpha_{2}(e_{q}-e_{q+1})$ belongs to this orbit
. The desired result is then obtained by taking
$\alpha_{1}=1,~{}\alpha_{2}=-1$ the point $y=e_{q+1}$ belongs to the orbit
$B_{I}.<e_{q}-e_{q+1}>$. Therefore the element of $B_{I}$ is also constructed
in the case where $q$ is odd.
∎

######  Theorem 6.1.5.

(Harmonic Permutation Theorem). The following are equivalent

(i)

$w$ is harmonic.

(ii)

$B_{I}(F_{w})\cap D_{\alpha}\not=\emptyset$ for some $\alpha$.

Under either of these conditions for every $\alpha$ the nonempty intersection
$B_{I}(F_{w})\cap C_{\alpha}$ contains a $T_{S}$-fixed point.

###### Proof.

(i)$\Rightarrow$ (ii)

Is exactly the statement of Proposition 6.1.4.

(ii)$\Rightarrow$ (i)

Assuming that $w$ is not harmonic permutation, we will show that $S_{w}\cap
D_{\alpha}=\emptyset,$ for all $\alpha$, i.e. $S_{w}$ has no $T_{S}$-fixed
points111See Corollary 3.3.4. Assume to the contrary that there exists $b\in
B_{I}$ such that $b.(F_{w})\in S_{w}\cap C_{\alpha}$. For $b(F_{w})$ to be
fixed, $b$ has to have a certain shape and at each stage where the condition
of harmonicity is violated, we should prove that there is no such $b$. For
this recall that the complex bilinear form $b$ has been defined to satisfy the
following orthogonality condition:

$b(e_{j}-e_{2q-j+1},e_{j}+e_{2q-j+1})=1,~{}1\leq j\leq q\ \text{and}\
b(e_{j}\pm e_{2q-j+1},e_{k}\pm e_{2q-k+1})=0\ \text{for}\ k\not=j\,.$

Since $w$ is not a harmonic permutation, there exist a pair of the form
$(-(2i-1),2i)$ or of the form $(-(2i-1),-2i)$, where $1\leq i\leq\frac{q}{2}$,
such that $\mp 2i$ sits to the left of $-(2i-1)$. Assume that $w_{j}=2i$ is
the first even number which appears in $w$ such that $2i$ sits to the left of
$-(2i-1)$. Then $b$ does in fact yield a $T_{S}$-fixed partial flag,i.e.

$\\{0\\}\subset<\varepsilon_{\tilde{w}_{1}}>\subset<\varepsilon_{\tilde{w}_{1}},\varepsilon_{\tilde{w}_{2}}>\subset......\subset<\varepsilon_{\tilde{w}_{1}},...,v_{\tilde{w}_{j-1}}>$
(6.3)

Now we check that there is no $b\in B_{I}$ so that $b.(v_{w_{j}})$ defines an
extended partial flag which is $T_{S}$-fixed. For this we consider the orbit
$B_{I}.<e_{|w_{j}|}+e_{|2\tilde{w}_{j}|-1}>$ which contains points of the form

$\alpha_{1}(e_{1}+e_{2q})+\alpha_{2}(e_{2}+e_{2q-1})+......+\alpha_{j}(e_{|w_{j}|}+e_{|2\tilde{w}_{j}|-1})$
(6.4)

This is a linear combination of $h$-isotropic vectors for all $\alpha_{i}$.
Recall that the flag of intersection has non $h$-isotropic vectors and to have
one of these vectors in this step which is linearly independent with all
vectors in the flag (6.3) above , we should add the point
$(e_{|w_{j}|-1}+ie_{|w_{j}|})>$ from the flag (6.3) to the linear combination
in (6.4), but this vector is not in the flag (6.3). Thus, as was claimed, no
$b\in B_{I}$ has that property that $b(F_{w})$ is $T_{S}$-fixed.
If $w_{j}=-2i$ is the first even number such that $-2i$ sits to the left of
$-(2i-1)$, then the relevant orbit is
$B_{I}.<e_{|w_{j}|}-e_{|2\tilde{w}_{j}|-1}>$ which contains points of the form

$\alpha_{1}(e_{1}+e_{2q})+...+\alpha_{q}(e_{q}+e_{q+1})+\alpha_{q+1}(e_{q}-e_{q+1})+...+\alpha_{s}(e_{|w_{j}|}-e_{|2\tilde{w}_{j}|-1}).$

To have non-isotropic point which is linearly independent with all points in
the flag (6.3) , we should add any of the points
$(e_{|w_{j}|-1}+ie_{|w_{j}|})>$, $(e_{|w_{j}|-1}-ie_{|w_{j}|})>$,
$(e_{2q-|w_{j}|+1}+ie_{2q-|w_{j}|+2})>$ or
$(e_{2q-|w_{j}|+1}-ie_{2q-|w_{j}|+2})>$ from the flag (6.3) to the linear
combination in (6.4). But the flag (6.3) does not contain any of these
points.So again, for all $b\in B_{I}$ the flag $b(F_{w})$ is not $T_{S}$-fixed
and consequently $S_{\alpha}\cap D_{\alpha}=\emptyset$.

∎

###  6.2 Introduction to the combinatorics

For the remainder of this chapter we only discuss the intersection properties
of the Iwasawa-Schubert cells which are of complementary dimension to $C_{0}$.
Recall that the maximal compact subgroup of $SO(p,q)$ is $K_{0}:=S(O(p)\times
O(q))\subset S(U(p)\times U(q))$. For $E^{+}$ and $E^{-}$ as in $\S 2.4.2$,
the base cycle in the flag domain $D_{a,b}$ is given by

$C_{0}=\\{F\in Z:\text{dim}V_{i}\cap
E^{-}=\sum_{j=1}^{i}a_{j}~{}and~{}\text{dim}V_{i}\cap
E^{+}=\sum_{j=1}^{i}b_{j},1\leq i\leq m\\}.$

If $m=2n$, we have two cases: If q is even, the dimension of $C_{0}$ is
$\frac{p(p-2)}{4}+\frac{q(q-2)}{4}$. If q is odd, then $C_{0}$ has the
dimension $\frac{(p-1)^{2}}{4}+\frac{(q-1)^{2}}{4}.$ If $m=2n+1$, then the
dimension of the base cycle is $\frac{(p-1)^{2}}{4}+\frac{q(q-2)}{4}$ if q is
even and is $\frac{p(p-2)}{4}+\frac{(q-1)^{2}}{4}$ if q is odd. Since we have
restricted to the case where $Z=G/B$ is the manifold of complete flags, it
follows that $\text{dim}~{}S_{w}=\frac{pq}{2}$ if $q$ and $n$ are even, and
$\text{dim}~{}S_{w}=\frac{pq-1}{2}$ if $q$ or $n$ are odd.

######  Definition 6.2.1.

A harmonic permutation $w\in S_{n}\ltimes\mathbb{Z}_{2}^{n-1}$ is called a
perfect harmonic permutation if it is constructed by the following algorithm:

A.

Start with a sequence of $n$ empty boxes which are to be filled in order to
construct $w$.

B.

Consider the pairs $(-(2j-1),2j)$ and $(-(2j-1),-2j)$ for all $1\leq
j\leq\frac{q}{2}$.

C.

If $q$ is even:

  1. 1.

The pairs in step B are $(-1,2),(-1,-2),(-3,4),...,(-(q-1),q),(-(q-1),-q)$.

  2. 2.

Step by step, starting from $(-1,2)$ until $(-(q-1),-q)$. For each $1\leq
j\leq\frac{q}{2}$ we have two pairs of the forms $(-(2j-1),2j)$ and
$(-(2j-1),-2j)$, so choose only one pair of them for each step and omit the
other form the above list.

  3. 3.

If we choose the pair of the form $(-(2j-1),2j)$, place this pair in any box
in $w$ such that the components $-(2j-1)$ and $2j$ of this pair sits as close
as possible to each other.

  4. 4.

If we choose the pair of the form $(-(2j-1),-2j)$, place this pair in any box
in $w$ such that the components $-(2j-1)$ and $-2j$ of this pair sits as close
as possible to each other and all pairs $(-(2i-1),2i)$ or $(-(2i-1),-2i)$ with
$i>j$ are sitting to the left of this pair and the pairs of the form
$(-(2i-1),-2i)$ with $i<j$ are sitting in a decreasing order with respect to
$i$ to the right of $(-(2j-1),-2j)$.

  5. 5.

Once a pair is placed its position can be ignored so that the places at the
immediate left and right of this pair become adjacent.

  6. 6.

After all pairs are placed, the remaining numbers $\pm(q+i)$ for $1\leq
i\leq\frac{p-q}{2}$ are placed in the remaining spots in the strictly
increasing order with respect to $|w(i)|$ such that all number $\pm(q+i)$ for
$1\leq i\leq\frac{p-q}{2}$ are sitting to the left of the pairs
$(-(2i-1),-2i)~{}~{}1\leq i\leq\frac{q}{2}$ . If the number of negative signs
in all pairs from steps 2 and 3 is even, then all numbers of the reminder
numbers are positive, and if the number of negative signs is odd, then all
numbers of the reminder numbers are positive except the number $n$ is
negative.

D.

If $q$ is odd:

  1. 1.

The pairs in step B are $(-1,2),(-3,4),(-5,6),...,(-(q-2),q-1),-q$.

  2. 2.

In the last box of $w$ put the number $-q$.

  3. 3.

Step by step, starting from $(-1,2)$ until $(-(q-2),q-1)$, choose a pair and
place this pair in any box of the first $n-1$ boxes in $w$ such that the
components $-(2j-1)$ and $2j$ of this pair sit as close as possible to each
other. This means that once a pair is placed it can be ignored so that the
places at the immediate left and right of this pair become adjacent.

  4. 4.

After all pairs are placed, the numbers $\pm(q+i)$ for $1\leq
i\leq\frac{p-q}{2}$ are placed in the remaining spots in the stricly
increasing order with respect to $|w(i)|$ . If the number of negative signs in
all pairs from step 1 and 2 is even, then all numbers of the reminder are
positive, and if the number of negative signs is odd then all remaining
numbers of the reminder are positive expect the number $n$ is negative.

######  Remark 6.2.2.

If $q$ is odd, then the signature of the flag domain $D_{\alpha}$ has $+$ in
the last position.∎

######  Remark 6.2.3.

If $W=S_{n}\ltimes\mathbb{Z}_{2}^{n}$, then a perfect harmonic permutation
consists only of pairs of the form $(-(2j-1),2j)$ for all $1\leq
j\leq\frac{q}{2}$. It is constructed as above, in particular such that the
sign of $n$ is $+$.∎

######  Example 6.2.4.

If $p=6,~{}q=4$, then the perfect harmonic permutations are:
$(-12)(-34)5,(-12)5(-34),5(-12)(-34),(-3~{}4)(-1~{}2)5,(-3~{}4)5(-1~{}2),5(-3~{}4)(-1~{}2),\\\
(-12)-5(-3-4),-5(-12)(-3-4),-5(-3-4)(-12),(-34)-5(-1-2),-5(-34)(-1-2),\\\
5(-3-4)(-1-2),(-3-1~{}2~{}4)5,5(-3-1~{}2~{}4),-5(-3-12-4)$.

######  Example 6.2.5.

If $p=10,~{}q=6$, then the element $78(-5~{}6)(-3-4)(-1~{}2)$ is a perfect
harmonic permutation while the element $78(-3-4)(-1~{}2)(-5~{}6)$ is not a
perfect harmonic permutation.

Recall that the dimension of the cells corresponds to the length of the word
$w$ in the Weyl group, i.e. if $F_{w}=w.F_{I}$, then $dim(B.F_{w})=l(w)$. So
if we want to discuss the dimension of the Schubert cell it is enough to
discuss the length of Weyl elements.

######  Proposition 6.2.6.

Every perfect harmonic permutation $w\in{\displaystyle
S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$ has dimension $\frac{pq}{2}$ if $q$ is even
and $\frac{pq-1}{2}$ if $q$ is odd.

###### Proof.

Given a perfect harmonic permutation $w$ we consider three cases.These depend
on which of the pairs $(-1~{}2),(-3~{}4),(-1-2)$ or $(-3-4)$ is contained in
$w$. In each of these cases our proof goes by induction on the dimension of
the flag manifold. Without loss of generality let $p>q\geq 6$ because if
$q\leq 6$ then the permutaion has only one or tow pairs and the proof become
trivial.

Case 1: The permutation $w$ contains the pairs $(-1~{}2)$ and $(-3~{}4)$.
First, remove the pairs $(-1~{}2)$ and $(-3~{}4)$ from $w$ to have a new
permutation $v$ consisting of the numbers $\\{5,6,...,n\\}$. Define a function
$f:\\{5,...,n\\}\longrightarrow\\{1,...,n-4\\}$ by $f(i)=i-4$ for all $1\leq
i\leq n-4$. This is a bijective map which sends $v$ to
$\hat{w}\in{\displaystyle S_{n-4}\ltimes\mathbb{Z}_{2}^{n-5}}$. Note that
$\hat{w}$ is a perfect harmonic permutation in ${\displaystyle
S_{n-4}\ltimes\mathbb{Z}_{2}^{n-5}}$. Thus by the induction assumption
$l(\hat{w})=\frac{(p-4)(q-4)}{2}$. Since $v=f^{-1}(\hat{w})$ it follows that v
has the same length as $\hat{w}$. So we put the numbers $1234$ to the left of
$v$ to have an element $\tilde{w}\in{\displaystyle
S_{n}\ltimes\mathbb{Z}_{2}^{n-1}}$ with length $\frac{(p-4)(q-4)}{2}$. To
split the sign of 1 and 3 (i.e., to change the postive sign to negative sign)
we add $n-4+1$ to $\frac{(p-4)(q-4)}{2}$ to send $3$ to the last position and
add $n-4+2$ to send 1 to the position before the last position. Consequently,
we have the element $(24)v(13)$ with length
$\frac{(p-4)(q-4)}{2}+n-4+1+n-4+2=\frac{(p-4)(q-4)}{2}+2n-5$ and it follows
that the element $(24)v(-3-1)$ has length $\frac{(p-4)(q-4)}{2}+2n-4$. We then
return to the original $w$ and remove only the pair $(-1~{}2)$. Then we define
$g$ to be the number of positions to the left of $(-34)$. As a result we have
$n-4-g$ positions to left of $(-3~{}4)$ and $-3$ should cross $n-4-g+1$
positions to end up in the last position and $4$ should cross $g$ boxes to end
up in the first position.

Finally we return to the original $w$ and define $h$ to be the number of
positions to the left of $(-1~{}2)$ and $n-2-f$ to the right. In this
situation $-1$ must cross $n-2-h+1$ positions to end up in the last position
and $2$ must cross $g$ positions to end up in the first position. It follows
that the length of $w$ is
$\frac{(p-4)(q-4)}{2}+2n-4+n-4-g+1+g+n-2-h+1+h=\frac{pq}{2}$.
If $q$ is odd, then the length of $\hat{w}\in{\displaystyle
S_{n-4}\ltimes\mathbb{Z}_{2}^{n-5}}$ is $l(\hat{w})=\frac{(p-4)(q-4)-1}{2}$.
After applying the same steps as those above it follows that
$l(w)=\frac{pq-1}{2}$

Case 2: The permutation $w$ contains the pair $(-1~{}-2)$.
Note that this case only occurs if q is even. Since $w$ is a perfect harmonic
permutation, the pair $(-1-2)$ sits in the last 2 positions of $w$. Remove the
pair $(-1-2)$ from $w$ to obtain $\hat{v}$. Since $q$ is even, a similar
argument to that above shows that $l(\hat{v})=\frac{(p-2)(q-2)}{2}$ because q
is even . Put the pair $(1~{}2)$ to the left of $\hat{v}$. It follows that
$\tilde{v}=(1~{}2)\hat{v}$ has the same length as $\hat{v}$. To split the sign
of 1 and 2, each of them must cross $n-2$ positions. Having made this move we
then apply $2$ reflection to obtain the pair $(-1-2)$ in the last two
positions. Hence the length of $w$ become
$l(w)=\frac{(p-2)(q-2)}{2}+2(n-2)+2=\frac{pq}{2}$.

Case 3: The permutation $w$ contains the pairs $(-1~{}2)$ and $(-3~{}-4)$.
Note that this case appear only if q is even. In this case these two pairs
appear in $w$ in the following forms: $(-3-4-1~{}2)$ or $(-3-1~{}2-4)$ or the
pair $(-3-4)$ sits in the last two positions of $w$. If the pair $(-3-4)$ sits
in the last two positions of $w$, then the argument goes as in Case 2 above.
If we have the form $(-3-1~{}2-4)$, then we must add $n-4$ to
$\frac{(p-2)(q-2)}{2}$ to put 3 in the last position and add $n-2$ to
$\frac{(p-2)(q-2)}{2}$ to put 4 in the last position. Then we must add 1 to
split the signs and 3 to sent $-3$ to its position in the original $w$, then
the length of $w$ become $\frac{(p-2)(q-2)}{2}+(n-4)+(n-2)+1+3=\frac{pq}{2}$ .
If we have the form $(-3-4-1~{}2)$, then we must add $n-4$ to
$\frac{(p-2)(q-2)}{2}$ to put 3 in the last position and add $n-1$ to
$\frac{(p-2)(q-2)}{2}$ to put 4 in the position before the last position. Then
we must add 1 to split the signs, 3 to send $-3$ to its position in the
original $w$ and 1 to send $-4$ to its position in the origenal $w$. It
follows that the length of $w$ is
$\frac{(p-2)(q-2)}{2}+(n-4)+(n-3)+1+3+1=\frac{pq}{2}$.
∎

######  Proposition 6.2.7.

Every perfect harmonic permutation $w\in{\displaystyle
S_{n}\ltimes\mathbb{Z}_{2}^{n}}$ has dimension $\frac{pq}{2}$.

###### Proof.

The argument goes exactly along the lines as that for the above Proposition.
Here it is in fact simpler, because only the pairs $(-1~{}2)$ and $(-3~{}4)$
appear. ∎

###  6.3 Intersection points of Schubert duality

let $w\in W_{I}$ be a perfect harmonic permutation, in particular so that
$S_{w}$ is of complementary dimension to the cycles. Recall that in this case
either $S_{w}\cap C_{\alpha}=\emptyset$ or is pointwise $T_{S}$-fixed (see
Theorem 3.3.3 and corollary 3.3.4). The main goal in this section is to
compute all such intersection points. The argument in the case of $SO(p,q)$ is
carried out by means of algorithms. Nevertheless we are able to provide
formulas for the cardinality of $S_{w}\cap C_{\alpha}$ when it is non-zero and
the total number of cycles $C_{\alpha}$ for which this intersection is non-
empty (see Theorem 6.3.3).

######  Proposition 6.3.1.

If $w$ is a perfect harmonic permutation so that $B_{I}.F_{w}$ intersects a
cycle $C_{\alpha}$ at a point given by the $\varepsilon$-basis
$(\varepsilon_{1},\ldots,\varepsilon_{m})$ of $T_{S}$-eigenvectors, then for
any such eigenvector $\varepsilon_{k}$ it follows that the $\varepsilon$-basis
is given by $\varepsilon_{k}=e_{2j-1}+ie_{2j}$ or $e_{2j-1}-ie_{2j}$ if $q$
and $p$ are even , and $\varepsilon_{k}=e_{2j-1}+ie_{2j}$ , $e_{2j-1}-ie_{2j}$
, $e_{q}$ , $e_{q+1}$ or $e_{2q+1}$ if $q$ or $p$ is odd, depending on the
dimension $m$ and the signature $\alpha$.

###### Proof.

This is a consequence of the following:

  1. 1.

$w$ is a perfect harmonic permutation and the flag basis is that of
$w(F_{I})$.

  2. 2.

We have the following cases

$\displaystyle\bullet~{}b.v_{j}$ $\displaystyle=$
$\displaystyle\eta_{j}(e_{r}+e_{2q-r+1})+\zeta_{j}(e_{r}-e_{2q-r+1})+\tilde{\eta}_{j}(e_{r+1}+e_{2q-r})+\tilde{\zeta}_{j}(e_{r+1}-e_{2q-r})+B_{j}$
$\displaystyle=$ $\displaystyle K_{j}+B_{j},.$

$\bullet~{}b.v_{j}=\eta_{j}(e_{2q+r}+ie_{2q+r+1})+B_{j}=K_{j}+B_{j},$

where $\tilde{\eta}_{j}=\pm i\eta_{j}$ and $\tilde{\zeta}=\pm i\zeta_{j}$ and
$B_{j}$ does not involve the basis vectors $e_{r}$, $e_{r+1}$, $e_{2q-r+1}$
and $e_{2q-r}$.

$\bullet~{}b.v_{j}=\eta_{j}(e_{2q+1})+B_{j}=K_{j}+B_{j},~{}\text{if}~{}m=2n+1$

  3. 3.

$\eta_{j}\not=0$ and $\tilde{\eta}_{j}\neq 0$.

  4. 4.

The intersection $S_{w}\cap C_{\alpha}$ is a flag defined by
$T_{S}$-eigenvectors.

From the expression for $v_{j}$ it is immediate that all of the possibilities
in the statement occur. Furthermore, since the $\eta_{j}\not=0$ and
$\tilde{\eta}_{j}\neq 0$ are non-zero, for every $j$ a non-zero contribution
from $K_{j}$ occurs in the sum

$v_{j}=\sum_{k\leq j}c_{kj}b.v_{w_{k}}\,.$

Since $e_{2j-1}+ie_{2j}$ and $e_{2j-1}-ie_{2j}$ and $e_{q}$ or $e_{q+1}$ do
not occur in $b.v_{w_{k}}$ for $k<j$, it follows that
$\varepsilon_{j}=K_{j}+E_{j}$ in the standard basis. Finally, since
$\varepsilon_{j}$ is a $T_{S}$-eigenvector, it follows that
$\varepsilon_{j}=K_{j}$ and is of the type in the the statement of the
proposition. ∎

As a result of the above Proposition the following corollary gives us all
intersection points of $S_{w}\cap C_{\alpha}$:

######  Corollary 6.3.2.

Let $D_{\alpha}$ be a flag domain parametrized by a sequence $\alpha$ and
$w\in W_{I}$ be a perfect harmonic permutation such that $S_{w}\cap
D_{\alpha}\neq\phi$. Then the following algorithm produces us all intersection
points of $S_{w}\cap C_{\alpha}$:

If $q$ is even:

  * •

Consider a copy of $\alpha$ denoted by $\beta$.

  * •

For any pair $(-(2j-1),2j),1\leq j\leq\frac{q}{2},$ in $w$ if the
corresponding signature of it in $\beta$ is $+-$ then replace this $+-$ in
$\beta$ by

$<e_{2q-2j+1}+ie_{2q-2j+2},e_{2j-1}-ie_{2j}>\text{ or
}<e_{2q-2j+1}-ie_{2q-2j+2},e_{2j-1}+ie_{2j}>,$

and if the corresponding signature of it in $\beta$ is $-+$ then replace this
$-+$ in $\beta$ by

$<e_{2j-1}+ie_{2j},e_{2q-2j+1}-ie_{2q-2j+2}>\text{ or
}<e_{2j-1}-ie_{2j},e_{2q-2j+1}+ie_{2q-2j+2}>$

  * •

For any pair $(-(2j-1),-2j),1\leq j\leq\frac{q}{2},$ in $w$ if the
corresponding signature of it in $\beta$ is $+-$ then replace this $+-$ in
$\beta$ by

$<e_{2q-2j+1}+ie_{2q-2j+2},e_{2j-1}+ie_{2j}>\text{ or
}<e_{2q-2j+1}-ie_{2q-2j+2},e_{2j-1}-ie_{2j}>,$

and if the corresponding signature of it in $\beta$ is $-+$ then replace this
$-+$ in $\beta$ by

$<e_{2j-1}+ie_{2j},e_{2q-2j+1}+ie_{2q-2j+2}>\text{ or
}<e_{2j-1}-ie_{2j},e_{2q-2j+1}-ie_{2q-2j+2}>$

  * •

For the remaining numbers, for each $q+1\leq j\leq n-1$ replace the
corresponding $+$ in $\beta$ by $(e_{2j-1}+ie_{2j})$ and for $\pm n$ replace
the corresponding $+$ by $(e_{2n-1}\pm ie_{2n})$.

If $q$ is odd:

  * •

For any pair $(-(2j-1),2j),1\leq j\leq\frac{q-1}{2}$, in $w$ if the
corresponding signature of it in $\beta$ is $+-$ then replace this $+-$ in
$\beta$ by

$<e_{2q-2j+1}+ie_{2q-2j+2},e_{2j-1}-ie_{2j}>\text{ or
}<e_{2q-2j+1}-ie_{2q-2j+2},e_{2j-1}+ie_{2j}>,$

and if the corresponding signature of it in $\beta$ is $-+$ then replace this
$-+$ in $\beta$ by

$<e_{2j-1}+ie_{2j},e_{2q-2j+1}-ie_{2q-2j+2}>\text{ or
}<e_{2j-1}-ie_{2j},e_{2q-2j+1}+ie_{2q-2j+2}>$

  * •

For the number $-q$ replace the corresponding $+$ in $\beta$ by $e_{q+1}$.

  * •

For the remaining numbers, for each $q+1\leq j\leq n-1$, replace the
corresponding $+$ in $\beta$ by $(e_{2j-1}+ie_{2j})$ and for $\pm n$ replace
the corresponding $+$ by $(e_{2n-1}\pm ie_{2n})$.

Each point obtained from this algorithm is a point of the intersection of
$S_{w}\cap D_{\alpha}$ .

######  Theorem 6.3.3.

A Schubert variety $S_{w}$ which is parametrized by a perfect harmonic
permutation $w$ intersects $2^{\frac{q}{2}}$ flag domains if $q$ is even and
intersects the base cycles of these flag domains in $2^{q}$ points. If $q$ is
odd, it intersects $2^{\frac{q-1}{2}}$ flag domains and intersects the base
cycles of these flag domains in $2^{q-1}$ points.

###### Proof.

Let $w\in W_{I}$ be a perfect harmonic permutation. We first show that if $q$
is even, then $S_{w}\cap C_{\alpha}$ consists of $2^{q}$ points and $2^{q-1}$
points if q is odd. Since $w$ is a perfect harmonic permutation, we have two
cases. In the first case if $w$ contains the pair $(-1~{}2)$ then the pair
$(-1~{}2)$ sits inside consecutive boxes in $w$. The goal here is to show that
there is exactly $4$-possibilities for this pair which can be completed to
maximal isotropic flag. We begin by considering the $B_{I}$-orbit of
$<e_{1}-e_{2q}>$. By Remark 6.1.3 the elements in this orbit have the form

$\beta_{1}(e_{1}+e_{2q})+....+\beta_{2n}(e_{1}-e_{2q}).$

The question is how many 1-dimensional subspaces (spanned by vectors in
Proposition 6.3.1) do we have such that these subspaces can be complete to
maximal isotropic flag? To compute this number we denote by $v_{1}$ the vector
we have from the first step which spans the 1-dimensional subspace, in the
second step we consider the orbit $B_{I}.<e_{2}+e_{2q-1}>$, which has points
of the form

$\beta_{1}(e_{1}+e_{2q})+\beta_{2}(e_{2}+e_{2q-1}).$

The $2$-dimension subspace in the flag is spanned by linear combinations of
the form

$v_{2}=\beta_{1}(e_{1}+e_{2q})+\beta_{2}(e_{2}+e_{2q-1})+\gamma v_{1}.$

Note that $v_{2}$ should be in $E^{+}$ or $E^{-}$ and of the form stated in
proposition 6.3.1, and therefore $v_{1}$ should contain the terms $e_{1}$ and
$e_{2}$ or the terms $e_{2q-1}$ and $e_{2q}$. Thus we have the following 4
possibilities of $v_{1}$ as follows:
$e_{1}-ie_{2},e_{1}+ie_{2},e_{2q-1}+ie_{2q},e_{2q-1}-ie_{2q}$.
If $v_{1}=e_{1}\mp ie_{2}$, then for $\beta_{1}=\pm i,~{}\beta_{2}=1,$ and
$~{}\gamma=\mp i$, the vector $v_{2}$ is $v_{2}=e_{2q-1}\pm ie_{2q}$, so the
2-dimensional subspace corresponding to the pair $(-1~{}2)$ is spanned by

$<e_{1}\mp ie_{2},e_{2q-1}\pm ie_{2q}>.$

If $v_{1}=e_{2q-1}\mp ie_{2q}$, then for $\beta_{1}=1,~{}\beta_{2}=\pm i,$ and
$~{}\gamma=\mp i$, it follows that $v_{2}$ is $v_{2}=e_{1}\pm ie_{2}$. Thus
the 2-dimensional subspace corresponding to the pair $(-1~{}2)$ is spanned by

$<e_{2q-1}\mp ie_{2q},e_{1}\pm ie_{2},>.$

Having constructed the $2$-dimensional space we ignore the pair $(-12)$ in $w$
and repeat the same steps for the next pairs step by step, So if $w$ contains