# Galleries for root subsystems

Vladimir Shchigolev  Financial University under the Government of the Russian
Federation, 49 Leningradsky Prospekt, Moscow, Russia
<EMAIL_ADDRESS>
###### Abstract.

We consider projection and lifting of labelled galleries to and from roots
subsystems. Our constructions allow us to construct some topological
embeddings of Bott-Samelson varieties skew equivariant with respect to the
compact torus and order-preserving on the sets of points fixed by it.

##  1\. Introduction

Let $E$ be a Euclidian space and $\Phi\subset E$ be a finite root system. We
require neither that $\Phi$ span $E$ nor that $\Phi$ be crystallographic. A
subset $\Psi\subset\Phi$ is called a root subsystem if it is not empty and is
stable under the reflections with respect to the roots $\alpha\in\Psi$ (in
other words, if $\Psi$ is a root system in its own right). Root subsystems
were objects of extensive study, see for example [W] and [O]. Moreover A.
Borel and J. de Siebental [BdS] established a connection between closed
connected subgroups of compact Lie subgroups of maximal rank and the maximal
closed root subsystems. The aim of this paper is to relate chambers and
galleries for the root systems $\Phi$ and $\Psi$ and apply these results to
Bott-Samelson varieties.

For any root $\alpha\in\Phi$, we denote by $L_{\alpha}$ the hyperplane in $E$
perpendicular to $\alpha$. We consider $E$ with respect to the metric topology
and for any subset $S\subset E$ we denote by $\bar{S}$ its closure. For a
subset $X\subset\Phi$, we denote the connected components of the difference
$E\setminus\bigcup_{\alpha\in X}L_{\alpha}$ by $\mathbf{Ch}_{X}$ and call them
chambers with respect to $X$. We say that a chamber $C\in\mathbf{Ch}_{X}$ is
attached to $L_{\alpha}$ if $\overline{C}\cap L_{\alpha}$ has a nonempty
interior in $L_{\alpha}$. Similarly, two chambers $C,D\in\mathbf{Ch}_{X}$ are
connected through $L_{\alpha}$ if $\overline{C}\cap\overline{D}\cap
L_{\alpha}$ has a nonempty interior in $L_{\alpha}$.

For any chamber $C\in\mathbf{Ch}_{\Phi}$, we denote by $C_{\Psi}$ the unique
chamber of $\mathbf{Ch}_{\Psi}$ containing $C$. Proposition 1, allows us to
make a (labelled) gallery of chambers of $\mathbf{Ch}_{\Psi}$ from a gallery
of chambers of $\mathbf{Ch}_{\Phi}$ in the obvious way, see Section 3.2. We
call this procedure projection.

The transition in the opposite direction, called here lifting, is more
difficult and is not unique. First we have to lift chambers of
$\mathbf{Ch}_{\Psi}$ to chambers of $\mathbf{Ch}_{\Phi}$, see Theorem 4. Then
we prove Theorem 11 for galleries. This result establishes existence and
uniqueness in a weak form.

The projection and lifting procedures applied successively allow us to
construct the so-called $p$-pairs of galleries described by some wall crossing
properties, see Definition 8. We also discuss gluing of galleries produced
this way, see Section 3.5.

Finally, we embed Bott-Somelson varieties into one another in Section 4 as an
application of the theory we develop. Our main result here is Theorem 21,
which asserts the existence of topological embeddings of Bott-Samelson
varieties skew equivariant with respect to the compact torus $K$ and order-
preserving on the set of the $K$-fixed points.

##  2\. Roots subsystems

###  2.1. Definitions and notation

We assume in the sequel the notation of the introduction. We denote by
$\omega_{\alpha}$ the reflection of $E$ through $L_{\alpha}$. The Weyl group
$W$ is the subgroup of $\mathrm{GL}(E)$ generated by all reflections
$\omega_{\alpha}$. We denote its neutral element by $1$. The scalar product on
$E$ is denoted by $(\cdot,\cdot)$. For any subset $X\subset E$, we set
$-X=\\{-x\mathbin{\rm|}x\in X\\}$ and denote by $\mathbb{R}X$ the set of all
$\mathbb{R}$-linear combinations $c_{1}x_{1}+\cdots+c_{n}x_{n}$, where
$c_{1},\ldots,c_{n}\in\mathbb{R}$ and $x_{1},\ldots,x_{n}\in X$. This linear
combination is called positive (respectively, negative) if
$c_{1},\ldots,c_{n}>0$ (respectively, $c_{1},\ldots,c_{n}<0$).

We will generally follow [St] and recall now briefly the main definitions
introduced therein. An ordering of $E$ is a subset $E^{+}\subset E$ closed
under addition and multiplication by positive scalars such that
$E^{+}\cap(-E^{+})=0$ and $E^{+}\cup(-E^{+})=E$. The positive (respectively,
negative) system of $\Phi$ relative to this ordering is the intersection
$\Phi^{+}=\Phi\cap E^{+}$ (respectively, $\Phi^{-}=\Phi\cap(-E^{+})$). The
fundamental chamber of this positive system is

$\\{e\in E\mathbin{\rm|}\forall\alpha\in\Phi^{+}\;(e,\alpha)>0\\}.$

A simple system of $\Phi$ is a subset $X\subset\Phi$ that is linearly
independent and any root of $\Phi$ is either a positive or a negative linear
combination of elements of $X$.

For any chamber $C\in\mathbf{Ch}_{\Phi}$, we denote by $\Phi^{+}(C)$ the
positive system with fundamental chamber $C$, that is,

$\Phi^{+}(C)=\\{\alpha\in\Phi\mathbin{\rm|}\forall e\in C\;(e,\alpha)>0\\}.$

Clearly, $C$ is the fundamental chamber of $\Phi^{+}(C)$.

We also define $\Phi^{-}(C)=-\Phi^{+}(C)$ and choose the (unique) simple
system $\Phi^{s}(C)$ contained in $\Phi^{+}(C)$. These operators are
$W$-invariant:

$\Phi^{+}(wC)=w\Phi^{+}(C),\quad\Phi^{-}(wC)=w\Phi^{-}(C),\quad\Phi^{s}(wC)=w\Phi^{s}(C)$
(1)

for any $w\in W$. We will use the similar notation $\Psi^{+}(C)$,
$\Psi^{-}(C)$ and $\Psi^{s}(C)$ for $C\in\mathbf{Ch}_{\Psi}$.

The proof of the following simple result is left to the reader.

######  Proposition 1.

(i) Let $C$ and $D$ be chambers of $\mathbf{Ch}_{\Phi}$ connected through
$L_{\beta}$ for $\beta\in\Phi\setminus\Psi$. Then $C_{\Psi}=D_{\Psi}$.

(ii) Let $C$ and $D$ be chambers of $\mathbf{Ch}_{\Phi}$ connected through
$L_{\alpha}$ for $\alpha\in\Psi$. Then $C_{\Psi}$ and $D_{\Psi}$ are also
connected through $L_{\alpha}$. Moreover, $C_{\Psi}=D_{\Psi}$ if and only if
$C=D$.

Our favorite root subsystems are given by the following result.

######  Lemma 2.

Let $\Psi$ be a nonempty subset of $\Phi$. The following conditions are
equivalent:

  1. (i)

$\Psi=\mathbb{R}X\cap\Phi$ for some nonempty subset $X\subset\Phi$;

  2. (ii)

$\Psi=\mathbb{R}\Psi\cap\Phi$.

A subset $\Psi$ satisfying these conditions is a root subsystem of $\Phi$.

###### Proof.

Clearly (ii) implies (i). Suppose conversely that (i) is satisfied. As
$\Psi\subset\mathbb{R}X$, we get $\mathbb{R}\Psi\subset\mathbb{R}X$. On the
other hand, $X\subset\Psi$, whence $\mathbb{R}X\subset\mathbb{R}\Psi$. Thus
$\mathbb{R}X=\mathbb{R}\Psi$ and we get (ii). The last claim holds, as both
$\mathbb{R}\Psi$ and $\Phi$ are stable under the reflections with respect to
the roots of $\Psi$. ∎

If the conditions of this lemma hold, we say that $\Psi$ is saturated in
$\Phi$.

###  2.2. Lifting

Let $C\in\mathbf{Ch}_{\Phi}$ and $D\in\mathbf{Ch}_{\Psi}$. We say that $C$
lifts $D$ if $\Psi^{s}(D)\subset\Phi^{s}(C)$. Obviously, the last inclusion
implies $\Psi^{+}(D)\subset\Phi^{+}(C)$. The converse is not true, see Example
5.

######  Lemma 3.

If $C$ lifts $D$, then $C_{\Psi}=D$.

###### Proof.

As $C$ and $D$ are the fundamental chambers of $\Phi^{+}(C)$ and
$\Psi^{+}(D)$, respectively and $\Psi^{+}(D)\subset\Phi^{+}(C)$, we get
$C\subset D$. Hence by definition $D=C_{\Psi}$. ∎

######  Theorem 4.

(i) Suppose that $\Psi$ is saturated in $\Phi$. Then for any
$D\in\mathbf{Ch}_{\Psi}$, there exists at least one chamber
$C\in\mathbf{Ch}_{\Phi}$ lifting $D$.

(ii) Suppose that a chamber $C\in\mathbf{Ch}_{\Phi}$ lifts a chamber
$D\in\mathbf{Ch}_{\Psi}$. Then $\Psi$ is saturated in $\Phi$.

###### Proof.

(i) We denote $V=\mathbb{R}\Psi$ for brevity. Let $V^{+}$ be an ordering of
$V$ such that $\Psi^{+}(D)=\Psi^{+}(D\cap V)=\Psi\cap V^{+}$. Let
$e_{1},\ldots,e_{k}$ be a basis of the orthogonal compliment
$V^{\perp}=\\{e\in E\mathbin{\rm|}\forall v\in V(e,v)=0\\}$. We define $E^{+}$
as the set of all sums $c_{1}e_{1}+\cdots+c_{k}e_{k}+v$, where
$c_{1},\ldots,c_{k}\in\mathbb{R}$ and either $c_{1}=\cdots=c_{k}=0$ and $v\in
V^{+}$ or $v\in V$ and there is some $i=1,\ldots,k$ such that
$c_{1}=\cdots=c_{i-1}=0$ and $c_{i}>0$. It is clear that $E^{+}$ is an
ordering of $E$ and $V^{+}=E^{+}\cap V$. Moreover, the following property
holds:

  * ($*$)

A positive linear combination of vectors of $E^{+}$ belongs to $V$ if and only
if each vector belongs to $V^{+}$.

Let $C$ be the fundamental chamber relative to the positive system $\Phi\cap
E^{+}$. Obviously $\Psi^{+}(D)\subset\Phi^{+}(C)=\Phi\cap E^{+}$. We claim
also that $\Psi^{s}(D)\subset\Phi^{s}(C)$. Indeed let $\alpha\in\Psi^{s}(D)$.
Then $\alpha\in\Phi^{+}(C)$ and we have a decomposition
$\alpha=c_{1}\beta_{1}+\cdots+c_{n}\beta_{n}$, where
$\beta_{1},\ldots,\beta_{n}\in\Phi^{s}(C)$ and $c_{1},\ldots,c_{n}>0$. By
($*$), we get $\beta_{1},\ldots,\beta_{n}\in V^{+}$. As $\Psi$ is saturated in
$\Phi$, we also get $\beta_{1},\ldots,\beta_{n}\in\Psi$, whence
$\beta_{1},\ldots,\beta_{n}\in\Psi^{+}(D)$. We get a contradiction unless
$n=1$ and $c_{1}=1$, which proves the claim.

(ii) Let $\Phi^{s}(C)=\\{\alpha_{1},\ldots,\alpha_{n}\\}$, where
$\alpha_{1},\ldots,\alpha_{k}\in\Psi^{s}(D)$,
$\alpha_{k+1},\ldots,\alpha_{n}\notin\Psi^{s}(D)$ and all roots
$\alpha_{1},\ldots,\alpha_{n}$ are pairwise distinct.

Let $\beta\in\mathbb{R}\Psi\cap\Phi^{+}(C)$. There are two $\mathbb{R}$-linear
decompositions $\beta=c_{1}\alpha_{1}+\cdots+c_{k}\alpha_{k}$ and
$\beta=d_{1}\alpha_{1}+\cdots+d_{n}\alpha_{n}$ with $d_{1},\ldots,d_{n}>0$.
Hence $d_{k+1}=\cdots=d_{n}=0$ and $\beta$ is a positive linear combination of
the roots of $\Psi^{s}(D)$. Similarly, if
$\beta\in\mathbb{R}\Psi\cap\Phi^{-}(C)$, then $\beta$ is a negative
combination of the roots of $\Psi^{s}(D)$. Thus $\Psi^{s}(D)$ is a simple
system for both root systems $\mathbb{R}\Psi\cap\Phi$ and $\Psi$. As a simple
system uniquely determines its root system, we get
$\mathbb{R}\Psi\cap\Phi=\Psi$. ∎

######  Example 5.

Let $\Phi$ be the root system of type $A_{2}$ with the simple roots $\alpha$
and $\beta$. We consider the root subsystem $\Psi=\\{-\alpha,\alpha\\}$. For
any root $\gamma\in\Phi$ (respectively $\gamma\in\Psi$), we denote by
$C(\gamma)$ (respectively $D(\gamma)$) the chamber of $\mathbf{Ch}_{\Phi}$
(respectively $\mathbf{Ch}_{\Psi}$) containing $\gamma$. The only chambers of
$\mathbf{Ch}_{\Phi}$ lifting $D(\alpha)$ are $C(\alpha+\beta)$ and
$C(-\beta)$. On the other hand,
$\Psi^{+}(D(\alpha))=\\{\alpha\\}\subset\\{-\beta,\alpha,\alpha+\beta\\}=\Phi^{+}(C(\alpha))$
and
$\Psi^{s}(D(\alpha))=\\{\alpha\\}\not\subset\\{-\beta,\alpha+\beta\\}=\Phi^{s}(C(\alpha))$.

######  Example 6.

Let $\Phi$ be the root system of type $B_{2}$ and $\Psi$ be its subsystem
consisting of the short roots (of type $A_{1}\times A_{1}$). One root of any
simple system of $\Phi$ is long and both roots of any simple system of $\Psi$
are short. Therefore, no chamber of $\mathbf{Ch}_{\Phi}$ lifts a chamber of
$\mathbf{Ch}_{\Psi}$. We have $\mathbb{R}\Psi\cap\Phi\neq\Psi$, that is,
$\Psi$ is not saturated in $\Phi$ in accordance with Theorem 4.

We will need the following additional criterion.

######  Lemma 7.

Let $C,D\in\mathbf{Ch}_{\Phi}$ and $\Psi$ be a root subsystem of $\Phi$. The
chamber $C$ lifts $D_{\Psi}$ if and only if there exists
$X\subset\Phi^{s}(C)\cap\Phi^{+}(D)$ such that $\Psi=\mathbb{R}X\cap\Phi$. If
these conditions hold, then we can take $X=\Psi^{s}(D_{\Psi})$.

###### Proof.

Suppose that $C$ lifts $D_{\Psi}$. Let us take $X=\Psi^{s}(D_{\Psi})$. By our
assumption, we have $X\subset\Phi^{s}(C)$. On the other hand,
$X\subset\Psi^{+}(D_{\Psi})$ and $D\subset D_{\Psi}$. Hence for any $\alpha\in
X$ and $e\in D$, we get $(\alpha,e)>0$. Thus $\alpha\in\Phi^{+}(D)$. As
$\alpha$ was chosen arbitrarily in $X$, we get $X\subset\Phi^{+}(D)$.
Moreover, by Theorem 4(ii), we get
$\mathbb{R}X\cap\Phi=\mathbb{R}\Psi\cap\Phi=\Psi$.

Conversely, suppose that a set $X$ as in the formulation of the lemma exists.
We claim that $X$ is a simple system for $\Psi$. Indeed let
$\Phi^{s}(C)=\\{\alpha_{1},\ldots,\alpha_{n}\\}$, where
$\alpha_{1},\ldots,\alpha_{k}\in X$, $\alpha_{k+1},\ldots,\alpha_{n}\notin X$
and all roots $\alpha_{1},\ldots,\alpha_{n}$ are pairwise distinct. Let
$\beta\in\Psi\cap\Phi^{+}(C)$. There are two $\mathbb{R}$-linear
decompositions $\beta=c_{1}\alpha_{1}+\cdots+c_{k}\alpha_{k}$ and
$\beta=d_{1}\alpha_{1}+\cdots+d_{n}\alpha_{n}$ with $d_{1},\ldots,d_{n}>0$.
Hence $d_{k+1}=\cdots=d_{n}=0$ and we have the positive linear combination
$\beta=d_{1}\alpha_{1}+\cdots+d_{k}\alpha_{k}$. We get a similar negative
linear combination for $\beta\in\Psi\cap\Phi^{-}(C)$. Finally note that the
set $X$ is linearly independent as a subset of the simple system
$\Phi^{s}(C)$.

Let $\alpha\in X$, $e\in D$ and $v\in D_{\Psi}$. As $\alpha\in\Psi$, the
hyperplane $L_{\alpha}$ does not separate the points $e$ and $v$. Hence and
from $(e,\alpha)>0$, we get $(v,\alpha)>0$. Thus we have proved that
$\alpha\in\Psi^{+}(D_{\Psi})$. As $\alpha$ was chosen arbitrarily in $X$, we
get $X\subset\Psi^{+}(D_{\Psi})$. Hence and from the previous paragraph we get
$\Psi^{s}(D_{\Psi})=X\subset\Phi^{s}(C)$. ∎

##  3\. Galleries

###  3.1. Definitions

For any integers $i\leqslant j$, we denote $[i,j]=\\{i,i+1,\ldots,j\\}$,
$[i,j)=\\{i,i+1,\ldots,j-1\\}$, etc. A gallery111It would be more correct to
use the term “labelled gallery”. However we omit the adjective, as we do not
consider here unlabelled galleries. On the other hand, we prefer to keep the
adjective, when we consider combinatorial galleries in Section 4.2. in $\Phi$
is a sequence

$\Gamma=(C_{0},L_{\alpha_{1}},C_{1},L_{\alpha_{2}},\ldots,C_{n-1},L_{\alpha_{n}},C_{n}),$
(2)

where $C_{0},C_{1},\ldots,C_{n}\in\mathbf{Ch}_{\Phi}$ and
$\alpha_{1},\ldots,\alpha_{n}\in\Phi$ are such that the chambers $C_{i-1}$ and
$C_{i}$ are connected through $L_{\alpha_{i}}$ for any $i=1,\ldots,n$.
Clearly, either $C_{i}=\omega_{\alpha_{i}}C_{i-1}$ or $C_{i}=C_{i-1}$. The
number $n$ of walls in $\Gamma$ is called the length of $\Gamma$ and is
denoted by $|\Gamma|$. The chambers $C_{0}$ and $C_{n}$ are called the initial
and the final chambers of $\Gamma$ respectively and the chamber $C_{i}$ is
called the $i$th chamber of $\Gamma$. We also say that $\Gamma$ begins at
$C_{0}$ and ends at $C_{n}$.

For $\Gamma$ given by (2) and integers $i$ and $j$ such that $0\leqslant
i\leqslant j\leqslant n$, we consider the following gallery:

$\Gamma_{[i,j]}=(C_{i},L_{\alpha_{i+1}},C_{i+1},L_{\alpha_{i+2}},\ldots,C_{j-1},L_{\alpha_{j}},C_{j}).$

Moreover, if the final chamber of a gallery $\Gamma$ coincides with the
initial chamber of a gallery $\Delta$, then they can by glued

$\Gamma\cup\Delta=(C_{0},L_{\alpha_{1}},\ldots,C_{n-1},L_{\alpha_{n}},C_{n},L_{\beta_{1}},\ldots,D_{m-1},L_{\beta_{m}},D_{m}),$

where $\Gamma$ is denoted as in (2) and
$\Delta=(C_{n},L_{\beta_{1}},D_{1},L_{\beta_{2}},\ldots,D_{m-1},L_{\beta_{m}},D_{m}\\}$.

We will pay special attention to the sequence of walls
$(L_{\alpha_{1}},L_{\alpha_{2}}\ldots,L_{\alpha_{n}})$ of a gallery $\Gamma$
denoted as in (2). Considering its subsequences, we get the following
definition that is very close to the definition of a $(p,w)$-pair from [S1]
(see Section 3.2 therein or Section 4.4 in this paper).

######  Defintition 8.

Let $\Gamma=(C_{0},L_{\alpha_{1}},\ldots,C_{n-1},L_{\alpha_{n}},C_{n})$ and
$\Delta=(D_{0},L_{\beta_{1}},\ldots,D_{m-1},L_{\beta_{m}},D_{m})$ be galleries
and $p:[1,m]\to[1,n]$ be an increasing embedding. We call $(\Delta,\Gamma)$ a
$p$-pair iff $L_{\alpha_{p(i)}}=L_{\beta_{i}}$ for any $i=1,\ldots,m$.

For any pair of galleries $(\Gamma,\Delta)$ as in this definition, we define
its sign $\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{m})$ by

$\varepsilon_{i}=\left\\{\\!\\!\begin{array}[]{rl}1&\text{ if
}L_{\beta_{i}}\text{does not separate }D_{i}\text{ and }C_{p(i)};\\\
-1&\text{otherwise}.\end{array}\right.$

Similarly, we define the cosign $\mu=(\mu_{1},\ldots,\mu_{m})$ by

$\mu_{i}=\left\\{\\!\\!\begin{array}[]{rl}1&\text{ if }L_{\beta_{i}}\text{does
not separate }C_{i-1}\text{ and }D_{p(i)-1};\\\
-1&\text{otherwise}.\end{array}\right.$

We say that the sign or cosign is positive if all its entries are so.

###  3.2. Projection

Let $\Psi$ be a root subsystem of $\Phi$. For a gallery $\Gamma$ in $\Phi$
denoted as in (2), we denote

$I_{\Psi}(\Gamma)=\\{i\in[1,n]\mathbin{\rm|}\alpha_{i}\in\Psi\\}=\\{i_{1}<i_{2}<\cdots<i_{m}\\}.$

We define

$\Gamma_{\Psi}=((C_{0})_{\Psi},L_{\alpha_{i_{1}}},(C_{i_{1}})_{\Psi},L_{\alpha_{i_{2}}},(C_{i_{2}})_{\Psi},\ldots,(C_{i_{m-1}})_{\Psi},L_{\alpha_{i_{m}}},(C_{i_{m}})_{\Psi}).$

Note that for any $t=1,\ldots,m+1$ and $j\in[i_{t-1},i_{t})$, we get

$(C_{i_{t-1}})_{\Psi}=(C_{j})_{\Psi}.$ (3)

Here and in what follows, we assume $i_{0}=0$. From Proposition 1, we obtain
the following result.

######  Proposition 9.

For any gallery $\Gamma$ in $\Phi$ and any root subsystem $\Psi$ of $\Phi$,
the sequence $\Gamma_{\Psi}$ is a gallery in $\Psi$.

###  3.3. Lifting

We formulate the following definition inspired by Section 2.2.

######  Defintition 10.

A gallery $\Gamma$ in $\Phi$ lifts a gallery $\Delta$ in $\Psi$ if the
sequences of walls of these galleries coincide and the $i$th chamber of
$\Gamma$ lifts the i$th$ chamber of $\Delta$ for any admissible $i$.

We are going to prove the following existence and uniqueness result.

######  Theorem 11.

For any gallery $\Delta$ in $\Psi$ and a chamber $C\in\mathbf{Ch}_{\Phi}$
lifting the $i$th chamber of $\Delta$, there exists a unique gallery in $\Phi$
lifting $\Delta$ whose $i$th chamber is $C$.

###### Proof.

Existence. Let us denote

$\Delta=(D_{0},L_{\beta_{1}},D_{1},\ldots,L_{\beta_{m}},D_{m})$ (4)

We set $C_{i}=C$, define the chambers $C_{1},\ldots,C_{i-1}$ recursively (from
right to left) by

$C_{j-1}=\left\\{\\!\begin{array}[]{ll}C_{j}&\text{if }D_{j-1}=D_{j};\\\
\omega_{\beta_{j}}C_{j}&\text{otherwise}.\end{array}\right.$

and the chambers $C_{i+1},\ldots,C_{m}$ recursively (from left to right) by

$C_{j+1}=\left\\{\\!\begin{array}[]{ll}C_{j}&\text{if }D_{j+1}=D_{j};\\\
\omega_{\beta_{j+1}}C_{j}&\text{otherwise}.\end{array}\right..$

Then we define
$\Delta^{C}_{i}=(C_{0},L_{\beta_{1}},C_{1},\ldots,L_{\beta_{m}},C_{m})$ and
claim that it is the required gallery.

First we consider the case $i=0$. We will apply induction on $m$. The case
$m=0$ is obvious. Suppose that $m>0$ and that the claim is true for the
truncated gallery
$\widetilde{\Delta}=(D_{0},L_{\beta_{1}},D_{1},\ldots,L_{\beta_{m-1}},D_{m-1})$.
As $D_{m-1}$ is attached to $L_{\beta_{m}}$, we get
$\beta_{m}\in\Psi^{s}(D_{m-1})\subset\Phi^{s}(C_{m-1})$. This proves that
$C_{m-1}$ is attached to $L_{\beta_{m}}$. If $D_{m}=D_{m-1}$, then by
definition $C_{m}=C_{m-1}$ and we are done. If $D_{m}\neq D_{m-1}$, then
$D_{m}=\omega_{\beta_{m}}D_{m-1}$ and by definition
$C_{m}=\omega_{\beta_{m}}C_{m-1}$. Hence
$\Psi^{s}(D_{m})=\omega_{\beta_{m}}\Psi^{s}(D_{m-1})\subset\omega_{\beta_{m}}\Phi^{s}(C_{m-1})=\Phi^{s}(C_{m})$
by (1).

The case $i=m$ can be treated similarly. In the case $0<i<m$, we get
$\Delta^{C}_{i}=(\Delta_{[1,i]})^{C}_{i}\cup(\Delta_{[i,m]})_{0}^{C}$ and the
result follows from the two cases considered above.

Uniqueness. Let
$\Gamma=(C^{\prime}_{1},L_{\beta_{1}},\ldots,C^{\prime}_{m-1},L_{\beta_{m}},C^{\prime}_{m})$
be a gallery in $\Phi$ that lifts $\Delta$ whose $i$th chamber is $C$. We are
going to prove that $\Gamma=\Delta_{i}^{C}$. Here we use the notation of the
previous part, that is, denote $\Delta$ as in (4) and compute recursively the
chambers $C_{1},\ldots,C_{m}$. Suppose that we have already proved that
$C_{j}=C^{\prime}_{j}$ for some $j=i,\ldots,m-1$.

First consider the case $D_{j+1}=D_{j}$. Suppose that $C^{\prime}_{j+1}\neq
C_{j+1}=C_{j}$. Then $C^{\prime}_{j+1}=\omega_{\beta_{j+1}}C_{j}$. Changing
the sign of $\beta_{j+1}$, if necessary, we can assume that
$\beta_{j+1}\in\Psi^{s}(D_{j})$. As $C^{\prime}_{j+1}$ lifts $D_{j+1}$ and
$C_{j}$ lifts $D_{j}$, we get
$\beta_{j+1}\in\Phi^{s}(C_{j})\cap\Phi^{s}(C^{\prime}_{j+1})$. On the other
hand,
$-\beta_{j+1}=\omega_{\beta_{j+1}}\beta_{j+1}\in\omega_{\beta_{j+1}}\Phi^{s}(C_{j})=\Phi^{s}(C^{\prime}_{j+1})$,
which is a contradiction. Thus we have proved that $C^{\prime}_{j+1}=C_{j+1}$.

Finally, we consider the case $D_{j+1}\neq D_{j}$. In this case
$D_{j+1}=\omega_{\beta_{j+1}}D_{j}$. Suppose that $C^{\prime}_{j+1}\neq
C_{j+1}=\omega_{\beta_{j+1}}C_{j}$. In that case $C^{\prime}_{j+1}=C_{j}$.
Changing the sign of $\beta_{j+1}$, if necessary, we can assume that
$\beta_{j+1}\in\Psi^{s}(D_{j})\subset\Phi^{s}(C_{j})$. Then we get
$-\beta_{j+1}=\omega_{\beta_{j+1}}\beta_{j+1}\in\omega_{\beta_{j+1}}\Psi^{s}(D_{j})=\Psi^{s}(D_{j+1})\subset\Psi^{s}(C^{\prime}_{j+1})$.
Therefore, we again have a contradiction
$\beta_{j+1},-\beta_{j+1}\in\Phi^{s}(C_{j})$. Thus we have proved that
$C^{\prime}_{j+1}=C_{j+1}$.

Arguing inductively with the help of the previous two paragraphs, we can prove
that $C^{\prime}_{j}=C_{j}$ for any $j=i,\ldots,m$. The prove of this equality
for $j=1,\ldots,i$ is similar. ∎

######  Remark 12.

We have proved that any lifting of $\Delta$ has the form $\Delta_{i}^{C}$. We
will use this notation throughout the paper. Moreover, we abbreviate
${}^{C}\\!\Delta=\Delta^{C}_{0}$ and $\Delta^{C}=\Delta^{C}_{|C|}$.

######  Remark 13.

It follows from Lemma 3 that $\Gamma_{\Psi}=\Delta$ if $\Gamma$ lifts
$\Delta$.

###  3.4. Constructing $p$-pairs

Let $\Gamma$ be a gallery in $\Phi$. Choosing a saturated root subsystem
$\Psi$ of $\Phi$, we get the gallery $\Gamma_{\Psi}$. Now choosing a chamber
$C\in\mathbf{Ch}_{\Phi}$ that lifts the $i$th chamber of $\Gamma_{\Psi}$, we
get the gallery $(\Gamma_{\Psi})^{C}_{i}$.

######  Theorem 14.

For any gallery $\Gamma$ in $\Phi$ and a gallery $\Delta$ in $\Phi$ lifting
$\Gamma_{\Psi}$, the pair $(\Delta,\Gamma)$ is a $p$-pair of positive sign and
cosign, where $p:[1,|\Gamma_{\Psi}|]\to[1,|\Gamma|]$ is the increasing
embedding with image $I_{\Psi}(\Gamma)$.

###### Proof.

Let
$\Gamma=(C^{\prime}_{0},L_{\alpha_{1}},C^{\prime}_{1},L_{\alpha_{2}},\ldots,C^{\prime}_{n-1},L_{\alpha_{n}},C^{\prime}_{n})$
and $I_{\Psi}(\Gamma)=\\{i_{1}<\cdots<i_{m}\\}$. We have $p(t)=i_{t}$. We
$\Delta=(C_{0},L_{\alpha_{i_{1}}},C_{1},\ldots,L_{\alpha_{i_{m}}},C_{m})$ for
some chambers $C_{0},\ldots,C_{m}\in\mathbf{Ch}_{\Phi}$. Hence we derive the
claim about being a $p$-pair.

Let us prove the positivity claim. Let $\mu=(\mu_{1},\ldots,\mu_{m})$ be the
cosign of our pair. As usual, we set $i_{0}=0$. Let us consider any
$t=1,\ldots,m$. The chambers $(C^{\prime}_{i_{t-1}})_{\Psi}$ and $C_{t-1}$ are
the $t-1$th chambers of the galleries $\Gamma_{\Psi}$ and $\Delta$
respectively. By definition, the latter chamber lifts the former one. Hence by
Lemma 3 and (3) (with primes added), we get
$(C_{t-1})_{\Psi}=(C^{\prime}_{i_{t-1}})_{\Psi}=(C^{\prime}_{i_{t}-1})_{\Psi}$.
Therefore $L_{\alpha_{i_{t}}}$ does not separate $C_{t-1}$ and
$C^{\prime}_{i_{t}-1}$, whence $\mu_{t}=1$. Thus we have proved that the
cosign is positive. The positivity of the sign is proved similarly. ∎

###  3.5. Gluing

Let $\Gamma$ be a gallery denoted as in (2). Suppose additionally that we are
given integers $i_{0},\ldots,i_{k}$ such that $0=i_{0}\leqslant
i_{1}\leqslant\cdots\leqslant i_{k}=n$. We choose a number $q=0,\ldots,k-1$
and an arbitrary saturated root subsystem $\Psi_{q}$ of $\Phi$. Then we take
any gallery $\Delta_{q}$ lifting $(\Gamma_{[i_{q},i_{q+1}]})_{\Psi_{q}}$. If
$q<k-1$, then we choose any nonempty subset
$X\subset\Phi^{s}(D)\cap\Phi^{+}(C_{i_{q+1}})$, where $D$ is the final chamber
of $\Delta_{q}$, and consider the saturated root subsystem
$\Psi_{q+1}=\mathbb{R}X\cap\Phi$. Lemma 7 guarantees that $D$ lifts
$(C_{i_{q+1}})_{\Psi_{q+1}}$. Then we define
$\Delta_{q+1}={}^{D}(\Gamma_{i_{q+1},i_{q+2}})_{\Psi_{q+1}}$. Similarly, if
$0<q$, then considering the initial chamber $F$ of $\Delta_{q}$, we can choose
a root subsystem $\Psi_{q-1}=\mathbb{R}Y\cap\Phi$ for a nonempty
$Y\subset\Phi^{s}(F)\cap\Phi^{+}(C_{i_{q}})$ and define the gallery
$\Delta_{q-1}=((\Gamma_{i_{q-1},i_{q}})_{\Psi_{q-1}})^{F}$. Continuing to the
left and to the right, we will construct the galleries
$\Delta_{0},\ldots,\Delta_{q-1},\Delta_{q+1},\ldots,\Delta_{k-1}$. It remains
to glue them all to get the gallery
$\Delta_{0}\cup\Delta_{1}\cup\cdots\cup\Delta_{k-1}$.

######  Remark 15.

In the above construction, we could always take
$X\subset\Psi_{q}^{s}((C_{i_{q+1}})_{\Psi_{q}})$. If
$X=\Psi_{q}^{s}((C_{i_{q+1}})_{\Psi_{q}})$, then we get $\Psi_{q+1}=\Psi_{q}$.
The similar argument applies to $Y$. If we always make the choices with the
equalities just mentioned, then $\Delta$ will lift the gallery
$\Gamma_{\Psi_{q}}$.

By Theorem 14, each pair $(\Delta_{l},\Gamma_{[i_{l},i_{l+1}]})$ is a
$p_{l}$-pair for the corresponding embedding
$p_{l}:[1,|\Delta_{l}|]\to[1,|\Gamma_{[i_{l},i_{l+1}]}|]$. We can glue the
embeddings $p_{0},\ldots,p_{k-1}$ to the embedding

$p:[1,|\Delta_{0}|+|\Delta_{1}|+\cdots+|\Delta_{k-1}|]\to[1,|\Gamma|]$

by $p(j)=p_{l}(j-|\Delta_{0}|-\cdots-|\Delta_{l-1}|)+i_{l}$ for
$|\Delta_{0}|+\cdots+|\Delta_{l-1}|<j\leqslant|\Delta_{0}|+\cdots+|\Delta_{l}|$,
where $i_{0}=0$.

######  Proposition 16.

The pair $(\Delta_{0}\cup\Delta_{1}\cup\cdots\cup\Delta_{k-1},\Gamma)$ is a
$p$-pair of positive sign and cosign.

######  Example 17.

Let us consider examples where projection and lifting applied successively
produce the same gallery. One obvious case is $\Psi=\Phi$. The other example
is the following. Let $\Gamma$ be a gallery denoted as in (2) We write
$\Gamma$ indicating the segments for each root subsystem as follows:

$\Delta=(\underbrace{C_{0},L_{\alpha_{1}},C_{1}}_{\\{\alpha_{1},-\alpha_{1}\\}}\\!\\!\\!\\!\\!\\!\\!\\!\overbrace{\quad\;,L_{\alpha_{2}},C_{2}}^{\\{\alpha_{2},-\alpha_{2}\\}},\ldots,\underbrace{C_{n-1},L_{\alpha_{n}},C_{n}}_{\\{\alpha_{n},-\alpha_{n}\\}}).$

Then we get the decomposition

$\Gamma={}^{C_{0}}(\Gamma_{[0,1]})_{\\{\alpha_{1},-\alpha_{1}\\}}\cup{}^{C_{1}}(\Gamma_{[1,2]})_{\\{\alpha_{2},-\alpha_{2}\\}}\cup\cdots\cup{}^{C_{n-1}}(\Gamma_{[n-1,n]})_{\\{\alpha_{n},-\alpha_{n}\\}}.$

##  4\. Bott-Samelson varieties

###  4.1. Construction

Let $G$ be a semisimple complex algebraic group. We fix a Borel subgroup $B<G$
and a maximal torus $T<B$. We denote by $W$ the Weyl group of $G$ and by
$\prec$ the Bruhat order on it. For any simple reflection $t\in W$, we
consider the minimal parabolic subgroup $P_{t}=B\cup BtB$.

Let $s=(s_{1},\ldots,s_{n})$ be a sequence of simple reflections. The
classical Bott-Samelson variety for $s$ is the quotient variety

$P_{s_{1}}\times\cdots\times P_{s_{n}}/B^{n}$

with the following right action of $B^{n}$:

$(p_{1},p_{2},\ldots,p_{n})(b_{1},b_{2},\ldots,b_{n})=(p_{1}b_{1},b_{1}^{-1}p_{2}b_{2},\ldots,b_{n-1}^{-1}p_{n}b_{n}).$

It is convenient here to redefine Bott-Samelson varieties using only compact
groups. Let $C$, $K$ and $C_{t}$ be the maximal compact subgroups of $G$, $T$
and $P_{t}$, respectively. The compactly defined Bott-Samelson variety for $s$
is the quotient variety

$C_{s_{1}}\times\cdots\times C_{s_{n}}/K^{n}$

under the similar right action of $K^{n}$:

$(c_{1},c_{2},\ldots,c_{n})(k_{1},k_{2},\ldots,k_{n})=(c_{1}k_{1},k_{1}^{-1}c_{2}k_{2},\ldots,k_{n-1}^{-1}c_{n}k_{n}).$

It is well-known [GK] that both varieties defined above are homeomorphic (for
the same sequence of simple reflections). We will denote them by
$\mathrm{BS}(s)$, call them simply the Bott-Samelson variety and always use
only the definition via the compact groups.

###  4.2. Combinatorial galleries

Let $\Phi$ be the root system of $G$, $\Phi^{+}$ be the system of positive
roots defined by the choice of $B$ and $A$ be the fundamental chamber of
$\Phi^{+}$. We define $\Phi^{-}=-\Phi^{+}$. For $\alpha\in\Phi^{+}$, we write
$\alpha>0$ and call this root positive. Similarly, for $\alpha\in\Phi^{-}$, we
write $\alpha<0$ and call this root negative. We say that roots $\alpha$ and
$\beta$ have the same sign if either $\alpha,\beta>0$ or $\alpha,\beta<0$.

Let us consider a sequence of simple reflections $s=(s_{1},\ldots,s_{n})$. The
elements of the set

$\mathbf{Gal}(s)=\\{(\gamma_{1},\ldots,\gamma_{n})\in
W^{n}\mathbin{\rm|}\gamma_{i}=1\text{ or }\gamma_{i}=s_{i}\\}.$

are called combinatorial galleries. We will use the following notation:

$\gamma^{i}=\gamma_{1}\gamma_{2}\cdots\gamma_{i},\qquad{\boldsymbol{\beta}}_{i}(\gamma)=\gamma^{i}(-\alpha_{i}),\quad\widetilde{\boldsymbol{\beta}}_{i}(\gamma)=\gamma^{i-1}(-\alpha_{i}),$

where $\gamma\in\mathbf{Gal}(s)$ and $s_{i}=\omega_{\alpha_{i}}$. There is the
following one-to-one correspondence between $\bigcup_{s}\mathbf{Gal}(s)$ and
the set of all galleries in $\Phi$ starting at $A$:

$\gamma=(\gamma_{1},\ldots,\gamma_{n})\mapsto\mathop{\mathrm{gal}}(\gamma)=(A,L_{{\boldsymbol{\beta}}_{1}(\gamma)},\gamma^{1}A,L_{{\boldsymbol{\beta}}_{2}(\gamma)},\gamma^{2}A,\ldots,L_{{\boldsymbol{\beta}}_{n}(\gamma)},\gamma^{n}A).$
(5)

To specify the final chamber of this gallery, we denote
${\boldsymbol{\pi}}(\gamma)=\gamma^{n}$. We denote $|\gamma|=n$ and call this
number the length of $\gamma$. Clearly
$|\gamma|=|\mathop{\mathrm{gal}}(\gamma)|$.

There is the folding operator $\mathbf{f}_{i}$ on $\mathbf{Gal}(s)$ that
multiplies the $i$th element of a combinatorial gallery by $s_{i}$. We have

${\boldsymbol{\beta}}_{i}(\mathbf{f}_{j}\gamma)=\left\\{\begin{array}[]{ll}{\boldsymbol{\beta}}_{i}(\gamma)&\text{
if }i<j;\\\
\omega_{{\boldsymbol{\beta}}_{j}(\gamma)}{\boldsymbol{\beta}}_{i}(\gamma)&\text{
if }i\geqslant
j\end{array}\right.,\qquad\tilde{\boldsymbol{\beta}}_{i}(\mathbf{f}_{j}\gamma)=\left\\{\begin{array}[]{ll}\widetilde{\boldsymbol{\beta}}_{i}(\gamma)&\text{
if }i\leqslant j;\\\
\omega_{\widetilde{\boldsymbol{\beta}}_{j}(\gamma)}\widetilde{\boldsymbol{\beta}}_{i}(\gamma)&\text{
if }i>j.\end{array}\right.$ (6)

There is also the double folding operator $\mathbf{f}_{i,j}$, where $i<j$,
that is just equal to the composition $\mathbf{f}_{i}\mathbf{f}_{j}$ and is
applicable to a combinatorial gallery $\gamma$ if and only if
${\boldsymbol{\beta}}_{i}(\gamma)=\pm{\boldsymbol{\beta}}_{j}(\gamma)$. This
operator has the property

${\boldsymbol{\pi}}(\mathbf{f}_{i,j}(\gamma))={\boldsymbol{\pi}}(\gamma),$ (7)

whenever it is applicable to $\gamma$.

The set $\mathbf{Gal}(s)$ is identified with the set $\mathrm{BS}(s)^{K}$ of
the $K$-fixed points of $\mathrm{BS}(s)$ by $\gamma\mapsto\gamma K^{n}$. For
any $K$-equivariant map $\varphi:X\to Y$ between sets acted upon by $K$, we
denote by $\varphi^{K}:X^{K}\to Y^{K}$ the restriction of $\varphi$ to the
sets of $K$-fixed points.

###  4.3. Orders

Following [H], we introduce the strict partial orders $\lhd$ and $<$ on
$\mathbf{Gal}(s)$ by

  1. (i)

$\delta\lhd\gamma$ if and only if $\delta^{i}\prec\gamma^{i}$ for the minimal
index $i$ such that $\delta^{i}\neq\gamma^{i}$.

  2. (ii)

$\delta<\gamma$ if and only if
${\boldsymbol{\pi}}(\delta)\,{=}\,{\boldsymbol{\pi}}(\gamma)$ and
$\delta^{i}\prec\gamma^{i}$ for the maximal index $i$ such that
$\delta^{i}\neq\gamma^{i}$.

This definitions allow reformulations in terms of the operators
${\boldsymbol{\beta}}_{i}$ and $\tilde{\boldsymbol{\beta}}_{i}$ introduced
above.

  1. (i′)

$\delta\lhd\gamma$ if and only if ${\boldsymbol{\beta}}_{i}(\gamma)>0$ for the
minimal index $i$ such that $\delta_{i}\neq\gamma_{i}$.

  2. (ii′)

$\delta\,{<}\,\gamma$ if and only if
${\boldsymbol{\pi}}(\delta)\,{=}\,{\boldsymbol{\pi}}(\gamma)$ and
$\widetilde{\boldsymbol{\beta}}_{i}(\gamma)\;{>}\;0$ for the maximal index $i$
such that $\delta_{i}\;{\neq}\;\gamma_{i}$.

We remember here the following simple fact that we will need in the proof of
Theorem 21.

######  Proposition 18.

Let $x,y\in W$ and $\alpha\in\Phi$. The hyperplane $L_{\alpha}$ does not
separate the chambers $xA$ and $yA$ if and only if the roots $x^{-1}\alpha$
and $y^{-1}\alpha$ have the same sign.

###  4.4. $(p,w)$-pairs

Let $s=(s_{1},\ldots,s_{m})$ and $r=(r_{1},\ldots,r_{n})$ be sequences of
simple reflections, $p:[1,m]\to[1,n]$ be an increasing embedding and $w\in W$.
A pair of galleries $(\gamma,\delta)\in\mathbf{Gal}(s)\times\mathbf{Gal}(r)$
is called a $(p,w)$-pair if

$\delta^{p(i)}r_{p(i)}(\delta^{p(i)})^{-1}=w\gamma^{i}s_{i}(\gamma^{i})^{-1}w^{-1}$

for all $i=1,\ldots,m$. We shall write these equations in either of more
convenient forms

${\boldsymbol{\beta}}_{p(i)}(\delta)=\varepsilon_{i}w{\boldsymbol{\beta}}_{i}(\gamma),\quad\widetilde{\boldsymbol{\beta}}_{p(i)}(\delta)=\mu_{i}w\widetilde{\boldsymbol{\beta}}_{i}(\gamma),$
(8)

where $\varepsilon_{i}=\pm 1$ and $\mu_{i}=\pm 1$. The sequences
$(\varepsilon_{1},\ldots,\varepsilon_{n})$ and $(\mu_{1},\ldots,\mu_{n})$ are
called the sign and the cosign of the $p$-pair $(\gamma,\delta)$.

There is the following connection between the notion of a $(p,w)$-pair and a
$p$-pair (see Definition 8). We leave the proof of the following result to the
reader.

######  Proposition 19.

Let $\gamma\in\mathbf{Gal}(s)$ and $\delta\in\mathbf{Gal}(r)$,
$p:[1,|s|]\to[1,|r|]$ be an increasing embedding and $w\in W$. Then
$(\gamma,\delta)$ is a $(p,w)$-pair if and only if
$(w\mathop{\mathrm{gal}}(\gamma),\mathop{\mathrm{gal}}(\delta))$ is a
$p$-pair. The signs and cosigns of these pairs coincide.

###  4.5. Category $\widetilde{\mathbf{Seq}}$

The objects of this category are the sequences of simple reflections including
the empty one. Each morphism $s=(s_{1},\ldots,s_{m})\to
r=(r_{1},\ldots,r_{n})$ is a triple $(p,w,\varphi)$ such that

  1. (1)

$p:[1,m]\to[1,n]$ is an increasing embedding;

  2. (2)

$w\in W$;

  3. (3)

$\varphi:\mathbf{Gal}(s)\to\mathbf{Gal}(r)$ is a map such that
$(\gamma,\varphi(\gamma))$ is a $(p,w)$-pair for any
$\gamma\in\mathbf{Gal}(s)$ and
$\varphi(\mathbf{f}_{i}\gamma)=\mathbf{f}_{p(i)}\varphi(\gamma)$ for any
$\gamma\in\mathbf{Gal}(s)$ and $i=1,\ldots,m$.

By [S1, Lemma 3], it suffices that $(\gamma,\varphi(\gamma))$ be a
$(p,w)$-pair just for one $\gamma\in\mathbf{Gal}(s)$. The (co)sign of
$(p,w,\varphi)$ is defined to be the (co)sign of each pair
$(\gamma,\varphi(\gamma))$. By [S1, Lemma 2] and the similar argument for
$\widetilde{\boldsymbol{\beta}}_{k}$ this definition does not depend on the
choice of $\gamma$. The composition law for the triples is
$(p^{\prime},w^{\prime},\varphi^{\prime})(p,w,\varphi)=(p^{\prime}p,w^{\prime}w,\varphi^{\prime}\varphi)$.

###  4.6. Embeddings

The category $\widetilde{\mathbf{Seq}}$ yields the following result on
embeddings.

######  Proposition 20 ([S2, Lemma 13]).

For any morphism $(p,w,\varphi):s\to r$ of the category
$\widetilde{\mathbf{Seq}}$, there exists a topological embedding
$\iota:\mathrm{BS}(s)\to\mathrm{BS}(r)$ such that $\iota^{K}=\varphi$ and
$\iota(ka)=wkw^{-1}a$ for any $k\in K$ and $a\in\mathrm{BS}(s)$.

Constructing $p$-pairs with the help of Theorem 14, we ge the following
result.

######  Theorem 21.

Let $r$ be a sequence of simple reflections, $\gamma\in\mathbf{Gal}(r)$ and
$\Psi$ be a saturated root subsystem of $\Phi$. Let $\Delta$ be any gallery
lifting $\mathop{\mathrm{gal}}(\gamma)_{\Psi}$, $w\in W$ be such that
$w^{-1}\Delta$ begins at $A$ and $s$ be the sequence of simple reflections
such that $\mathop{\mathrm{gal}}^{-1}(w^{-1}\Delta)\in\mathbf{Gal}(s)$. Then
there exists a topological embedding $\iota:\mathrm{BS}(s)\to\mathrm{BS}(r)$
mapping $\mathop{\mathrm{gal}}^{-1}(w^{-1}\Delta)$ to $\gamma$ such that
$\iota(ka)=wkw^{-1}a$ for any $k\in K$, $a\in\mathrm{BS}(s)$ and $\iota^{K}$
preserves both orders $\lhd$ and $<$.

###### Proof.

We denote $\mu=\mathop{\mathrm{gal}}^{-1}(w^{-1}\Delta)$ for brevity and
consider the map $p:[1,|s|]\to[1,|r|]$, which is the increasing imbedding with
image $I_{\Psi}(\mathop{\mathrm{gal}}(\gamma))$. By Theorem 14, the pair
$(\Delta,\mathop{\mathrm{gal}}(\gamma))$ is a $p$-pair of positive sigh and
cosign. As

$(\Delta,\mathop{\mathrm{gal}}(\gamma))=(ww^{-1}\Delta,\mathop{\mathrm{gal}}(\gamma))=(w\mathop{\mathrm{gal}}(\mu),\mathop{\mathrm{gal}}(\gamma)),$

we obtain from Proposition 19 that $(\mu,\gamma)$ is a $(p,w)$-pair of
positive sign and cosign. By [S1, Lemma 3], we obtain a map
$\varphi:\mathbf{Gal}(s)\to\mathbf{Gal}(r)$ such that $\varphi(\mu)=\gamma$
and $(p,w,\varphi):s\to r$ is a morphism in $\widetilde{\mathbf{Seq}}$.
Applying Proposition 20, we obtain the required map $\iota$.

It remains to prove that $\varphi=\iota_{K}$ preserves both orders $\lhd$ and
$<$. Let $\lambda,\rho\in\mathbf{Gal}(s)$ be such that $\lambda\lhd\rho$. Let
$i$ be the minimal index such that $\lambda_{i}\neq\rho_{i}$. Then there exist
indices $i_{1},\ldots,i_{m}>i$ such that
$\rho=\mathbf{f}_{i}\mathbf{f}_{i_{1}}\cdots\mathbf{f}_{i_{k}}\lambda$.
Applying property (3) of the definition of $\widetilde{\mathbf{Seq}}$, we get
$\varphi(\rho)=\mathbf{f}_{p(i)}\mathbf{f}_{p(i_{1})}\cdots\mathbf{f}_{p(i_{k})}\varphi(\lambda)$.
As $p$ is increasing, we get $p(i_{1}),\ldots,p(i_{k})>p(i)$, which proves
that $\varphi(\lambda)_{p(i)}\neq\varphi(\rho)_{p(i)}$ but
$\varphi(\lambda)_{j}=\varphi(\rho)_{j}$ for any $1\leqslant j<p(i)$.

Remember that $p(j)\in I_{\Psi}(\mathop{\mathrm{gal}}(\gamma))$ for any
$j\in[1,|s|]$ by the definition of $p$. Hence from (5), we get

${\boldsymbol{\beta}}_{p(j)}(\gamma)\in\Psi.$ (9)

Now we consider a representation
$\rho=\mathbf{f}_{j_{m}}\cdots\mathbf{f}_{j_{1}}\mu$. Applying $\varphi$ to
both sides, we get by part (3) of the definition of $\widetilde{\mathbf{Seq}}$
that $\varphi(\rho)=\mathbf{f}_{p(j_{m})}\cdots\mathbf{f}_{p(j_{1})}\gamma$.
We are going to prove by induction on $m$ that
${\boldsymbol{\beta}}_{p(j)}(\varphi(\rho))\in\Psi$ for any $j\in[1,|s|]$. The
case $m=0$ follows from (9). Suppose that $m>0$ and the claim holds for $m-1$
folding operators. By the first formula of (6) and the inductive hypothesis,
we get for some $\varepsilon\in\\{0,1\\}$ that

${\boldsymbol{\beta}}_{p(j)}(\varphi(\rho))={\boldsymbol{\beta}}_{p(j)}(\mathbf{f}_{p(j_{m})}\cdots\mathbf{f}_{p(j_{1})}\gamma)=\omega_{{\boldsymbol{\beta}}_{p(j_{m})}(\mathbf{f}_{p(j_{m-1})}\cdots\mathbf{f}_{p(j_{1})}\gamma)}^{\varepsilon}{\boldsymbol{\beta}}_{p(j)}(\mathbf{f}_{p(j_{m-1})}\cdots\mathbf{f}_{p(j_{1})}\gamma)\in\Psi.$

By the first formula of (8) and the positivity of the sign, we get
${\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))=w{\boldsymbol{\beta}}_{i}(\rho)$.
As $\Delta$ lifts $\mathop{\mathrm{gal}}(\gamma)_{\Psi}$, the initial chamber
$wA$ of $\Delta$ lifts the initial chamber $A_{\Psi}$ of
$\mathop{\mathrm{gal}}(\gamma)_{\Psi}$. Hence by Lemma 3, we get
$(wA)_{\Psi}=A_{\Psi}$, whence no wall $L_{\beta}$ for $\beta\in\Psi$
separates $A$ and $wA$. In particular the wall
$L_{{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))}$ does not separate these
chambers. By Proposition 18, we get that the roots
${\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))$ and
$w^{-1}{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))={\boldsymbol{\beta}}_{i}(\rho)$
have the same sign. However, as $\lambda\lhd\rho$, the latter root is positive
by (i′) from Section 4.3. Hence ${\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))>0$
and $\varphi(\lambda)\lhd\varphi(\rho)$ again by (i′).

The proof of the fact that $\iota^{K}$ preserves $<$ is very similar. Indeed,
let $\lambda,\rho\in\mathbf{Gal}(s)$ be such that $\lambda<\rho$. Let $i$ be
the maximal index such that $\lambda_{i}\neq\rho_{i}$. Similarly to the above
arguments, we get $\varphi(\lambda)_{j}=\varphi(\rho)_{j}$ for any
$p(i)<j\leqslant|s|$. As
${\boldsymbol{\pi}}(\lambda)={\boldsymbol{\pi}}(\rho)$, we get
$\rho=\mathbf{f}_{i_{1},j_{1}}\cdots\mathbf{f}_{i_{k},j_{k}}\lambda$ for some
sequence of the double folding operators. Applying property (3) of the
definition of $\widetilde{\mathbf{Seq}}$, we get
$\varphi(\rho)=\mathbf{f}_{p(i_{1}),p(j_{1})}\cdots\mathbf{f}_{p(i_{k}),p(j_{k})}\varphi(\lambda)$.
Hence ${\boldsymbol{\pi}}(\varphi(\rho))={\boldsymbol{\pi}}(\varphi(\lambda))$
by (7).

Just as for the case of $\lhd$, we can prove that
$\widetilde{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))\in\Psi$. By the second
formula of (8) and the positivity of the cosign, we get
$\widetilde{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))=w\widetilde{\boldsymbol{\beta}}_{i}(\rho)$.
As
$L_{\tilde{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))}=L_{{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))}$
and we proved above that this wall does not separate $A$ and $wA$, we get by
Proposition 18 that the roots
$\widetilde{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))$ and
$w^{-1}\widetilde{\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))=\widetilde{\boldsymbol{\beta}}_{i}(\rho)$
have the same sign. However, as $\lambda<\rho$, the latter root is positive by
(ii′) from Section 4.3. Hence ${\boldsymbol{\beta}}_{p(i)}(\varphi(\rho))>0$
and $\varphi(\lambda)<\varphi(\rho)$ again by (ii′). ∎

######  Remark 22.

Proposition 20 allows us to construct embeddings of Bott-Samelson varieties
for lifings $\Delta$ different from that used in Theorem 21, for example, for
several such liftings glued together as in Example 2 below. Wether the orders
$\lhd$ and $<$ are preserved is an open question.

###  4.7. Example 2

Let $\Phi$ be the root system of type $A_{3}$ with the following Dynkin
diagram:

$\scriptstyle\alpha_{1}$$\scriptstyle\alpha_{2}$$\scriptstyle\alpha_{3}$

The simple reflections are $\omega_{i}=\omega_{\alpha_{i}}$. Let us take
$r=(\omega_{2},\omega_{3},\omega_{2},\omega_{1},\omega_{2},\omega_{1},\omega_{3},\omega_{1})$
and
$\gamma=(1,\omega_{3},1,\omega_{1},\omega_{2},\omega_{1},1,\omega_{1})\in\mathbf{Gal}(r)$
and consider the gallery $\Gamma=\mathop{\mathrm{gal}}(\gamma)$. We draw
$\Gamma$ indicating the segments where the root subsystems $\Psi_{1}$ and
$\Psi_{2}$ are applied as follows:

$\Gamma=(\overbrace{A,L_{\alpha_{2}},A,L_{\alpha_{3}},\omega_{3}A,L_{\alpha_{2}+\alpha_{3}},\omega_{3}A,L_{\alpha_{1}},\omega_{3}\omega_{1}A,L_{\alpha_{1}+\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{1}\omega_{2}A}^{\Psi_{1}},\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{\qquad\qquad}_{\Psi_{2}}\\\
\underbrace{L_{\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{1}\omega_{2}\omega_{1}A,L_{\alpha_{1}+\alpha_{2}},\omega_{3}\omega_{1}\omega_{2}\omega_{1}A,L_{\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{1}\omega_{2}A}_{\Psi_{2}}).$

Here as usual $A$ denotes the fundamental chamber. We are going to project,
lift and glue according to the general plan described in Section 3.5. During
this process, we will choose the root systems $\Psi_{1}$ and $\Psi_{2}$.

First, we take
$\Psi_{1}=\mathbb{R}\\{\alpha_{1}+\alpha_{2},\alpha_{3}\\}\cap\Phi$. Then we
get $\Psi_{1}^{s}(A_{\Psi_{1}})=\\{\alpha_{1}+\alpha_{2},\alpha_{3}\\}$ and

$(\Gamma_{[0,5]})_{\Psi_{1}}=(A_{\Psi_{1}},L_{\alpha_{3}},\omega_{3}A_{\Psi_{1}},L_{\alpha_{1}+\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{2}A_{\Psi_{1}}).$

Now we are going to lift this gallery. In order to do it, we need lift its
initial chamber $A_{\Psi_{1}}$. There are two possible liftings $\omega_{1}A$
and $\omega_{2}\omega_{3}A$. Let us take the first one. We get

${}^{\omega_{1}A}(\Gamma_{[0,5]})_{\Psi_{1}}=(\omega_{1}A,L_{\alpha_{3}},\omega_{1}\omega_{3}A,L_{\alpha_{1}+\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}A).$

We have
$\Phi^{s}(\omega_{1}\omega_{3}\omega_{2}A)=\\{\alpha_{2}+\alpha_{3},\alpha_{1}+\alpha_{2},-\alpha_{1}-\alpha_{2}-\alpha_{3}\\}$.
Thus we can take
$\Psi_{2}=\mathbb{R}\\{\alpha_{2}+\alpha_{3}\\}\cap\Phi=\\{\alpha_{2}+\alpha_{3},-\alpha_{2}-\alpha_{3}\\}$.
We get

$(\Gamma_{[5,8]})_{\Psi_{2}}=(\omega_{1}\omega_{3}\omega_{2}A_{\Psi_{2}},L_{\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}\omega_{1}A_{\Psi_{2}},L_{\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}A_{\Psi_{2}}).$

Then we lift

${}^{\omega_{1}\omega_{3}\omega_{2}A}(\Gamma_{[5,8]})_{\Psi_{2}}=(\omega_{1}\omega_{3}\omega_{2}A,L_{\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}\omega_{1}A,L_{\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}A).$

Finally we glue both liftings and get the gallery

$\Delta={}^{\omega_{1}A}(\Gamma_{[0,5]})_{\Psi_{1}}\cup{}^{\omega_{3}\omega_{2}\omega_{1}A}(\Gamma_{[5,8]})_{\Psi_{2}}\\\
=(\omega_{1}A,L_{\alpha_{3}},\omega_{1}\omega_{3}A,L_{\alpha_{1}+\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}A,L_{\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}\omega_{1}A,L_{\alpha_{2}+\alpha_{3}},\omega_{1}\omega_{3}\omega_{2}A).$

Hence it is clear that $(\Delta,\Gamma)$ is a $p$-pair, where
$p:[1,4]\to\\{2,5,6,8\\}$ is the increasing bijection. Multiplying by
$\omega_{1}$, we get the gallery starting at the fundamental chamber

$\omega_{1}\Delta=(A,L_{\alpha_{3}},\omega_{3}A,L_{\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{2}A,L_{\alpha_{1}+\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{2}\omega_{1}A,L_{\alpha_{1}+\alpha_{2}+\alpha_{3}},\omega_{3}\omega_{2}A).$

We get the combinatorial gallery
$\delta=\mathop{\mathrm{gal}}^{-1}(\omega_{1}\Delta)=(\omega_{3},\omega_{2},\omega_{1},\omega_{1})\in\mathbf{Gal}(s)$
for $s=(\omega_{3},\omega_{2},\omega_{1},\omega_{1})$. The reader can easily
check that $(\delta,\gamma)$ is a $(p,\omega_{1})$-pair of positive sign and
cosign. By Proposition 20, we get an embedding

$\iota:\mathrm{BS}(\omega_{3},\omega_{2},\omega_{1},\omega_{1})\hookrightarrow\mathrm{BS}(\omega_{2},\omega_{3},\omega_{2},\omega_{1},\omega_{2},\omega_{1},\omega_{3},\omega_{1}),$

such that $\iota(ka)=\omega_{1}k\omega_{1}^{-1}a$ for any $k\in K$ and
$a\in\mathrm{BS}(s)$ and

$\iota(\omega_{3}^{\varepsilon_{1}},\omega_{2}^{\varepsilon_{2}},\omega_{1}^{\varepsilon_{3}},\omega_{1}^{\varepsilon_{4}})=(1,\omega_{3}^{\varepsilon_{1}},1,\omega_{1},\omega_{2}^{\varepsilon_{2}},\omega_{1}^{\varepsilon_{3}},1,\omega_{1}^{\varepsilon_{4}})$

for any
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\in\\{0,1\\}$.
Here we used the fact that $(p,\omega_{1},\iota^{K})$ is a morphism in
$\widetilde{\mathbf{Seq}}$.

## References

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