# Total positivity in twisted product of flag varieties

Huanchen Bao  Department of Mathematics, National University of Singapore,
Singapore<EMAIL_ADDRESS>and  Xuhua He  The Institute of Mathematical
Sciences and Department of Mathematics, The Chinese University of Hong Kong,
Shatin, N.T., Hong Kong SAR, China<EMAIL_ADDRESS>
###### Abstract.

We show that the totally nonnegative part of the twisted product of flag
varieties of a Kac-Moody group admits a cellular decomposition, and the
closure of each cell is a regular CW complex homeomorphic to a closed ball. As
a consequence, we show the totally nonnegative braid variety is a regular CW
complex homeomorphic to a closed ball.

Moreover, we prove that the link of any totally nonnegative double Bruhat cell
in a reductive group is a regular CW complex homeomorphic to a closed ball,
solving an open problem of Fomin and Zelevinsky.

###### Key words and phrases:

flag varieties, Kac-Moody groups, double Bruhat cells, total positivity

###### 2010 Mathematics Subject Classification:

14M15, 20G44, 15B48

##  1\. Introduction

###  1.1. Flag varieties

We first give a quick review of the totally nonnegative flag varieties.

Let $G$ be a connected Kac-Moody group, split over $\mathbb{R}$. Let
$B^{+}=TU^{+}$ be a Borel subgroup of $G$ and $B^{-}=TU^{-}$ be the opposite
Borel subgroup. Let $W$ be the Weyl group of $G$. In [19] and [20], Lusztig
introduced the totally nonnegative monoid

$G_{\geqslant 0}=U^{+}_{\geqslant 0}T_{>0}U^{-}_{\geqslant 0}=U^{-}_{\geqslant
0}T_{>0}U^{+}_{\geqslant 0}.$

Let $\mathcal{B}=G/B^{+}$ be the flag variety. It admits the decomposition
into the open Richardson varieties
$\mathcal{B}=\sqcup\mathring{\mathcal{B}}_{v,w}$, where $v$ and $w$ are
elements in the Weyl group $W$, and the open Richardson variety
$\mathring{\mathcal{B}}_{v,w}$, by definition, is the intersection of the
Schubert cell $B^{+}\dot{w}B^{+}/B^{+}$ and the opposite Schubert cell
$B^{-}\dot{v}B^{+}/B^{+}$. The totally nonnegative flag variety
$\mathcal{B}_{\geqslant 0}$ is defined as the closure of $U^{-}_{\geqslant
0}B^{+}/B^{+}$ in $\mathcal{B}$ with respect to the Hausdorff topology.

The totally nonnegative flag is a “remarkable polyhedral subspace” (cf. [19]).
In particular, we have

  1. (1)

(decomposition into cells) $\mathcal{B}_{\geqslant
0}=\sqcup\mathcal{B}_{v,w,>0}$, where each totally positive stratum
$\mathcal{B}_{v,w,>0}=\mathcal{B}_{\geqslant
0}\cap\mathring{\mathcal{B}}_{v,w}$ is a cell and admits an explicit
parametrization (see [14] and [3]);

  2. (2)

the closure of each totally positive stratum is a union of totally positive
strata (see [24] and [4]);

  3. (3)

the closure of each totally positive stratum is homeomorphic to a closed ball
(see [9] and [4]);

  4. (4)

If $G$ is a connected reductive group, then the flag variety $\mathcal{B}$
admits a natural duality which stabilizes the totally nonnegative part
$\mathcal{B}_{\geqslant 0}$ and sends each totally positive stratum to another
totally positive stratum (see [19] and [21]).

###  1.2. Main result

The main object in this paper is the twisted product of flag varieties

$\mathcal{Z}=G\times^{B^{+}}G\times^{B^{+}}\cdots\times^{B^{+}}G/B^{+}\quad(n\text{
factors}).$

For any sequence $\underline{w}=(w_{1},\cdots,w_{n})$ of elements in $W$, we
define

$\mathring{\mathcal{Z}}_{\underline{w}}=B^{+}\dot{w}_{1}B^{+}\times^{B^{+}}B^{+}\dot{w}_{2}B^{+}\times^{B^{+}}\cdots\times^{B^{+}}B^{+}\dot{w}_{n}B^{+}/B^{+}.$

This is analogous to Schubert cell in the flag $\mathcal{B}$. For any $v\in
W$, we define $\mathring{\mathcal{Z}}^{v}$ as
$m^{-1}(B^{-}\dot{v}B^{+}/B^{+})$, where $m:\mathcal{Z}\mathcal{B}$ is the
convolution product. This is analogous to the opposite Schubert cell in the
flag $\mathcal{B}$. The open Richardson variety of $\mathcal{Z}$ is defined by
$\mathring{\mathcal{Z}}_{v,\underline{w}}=\mathring{\mathcal{Z}}_{\underline{w}}\cap\mathring{\mathcal{Z}}^{v}$.
These open Richardson varieties include as special cases the Bott-Samelson
varieties and the braid varieties.

We define the totally nonnnegative part $\mathcal{Z}_{\geqslant 0}$ of
$\mathcal{Z}$ to be the Hausdorff clsoure of $(U^{-}_{\geqslant
0},U^{-}_{\geqslant 0},\ldots,U^{-}_{\geqslant 0}B^{+}/B^{+})$ in
$\mathcal{Z}$. We set $\mathcal{Z}_{v,\underline{w},>0}=\mathcal{Z}_{\geqslant
0}\cap\mathring{\mathcal{Z}}_{v,\underline{w}}$ and call it a totally positive
stratum  of $\mathcal{Z}$. We have the decomposition $\mathcal{Z}_{\geqslant
0}=\sqcup\mathcal{Z}_{v,\underline{w},>0}$.

The goal of this paper is to show that the totally nonnegative twisted product
of flag is still a “remarkable polyhedral subspace”.

######  Theorem 1.

[Theorem 3.4, Theorem 4.6 $\&$ Proposition 5.7] (1) Each stratum
$\mathcal{Z}_{v,\underline{w},>0}$ is a topological cell and we have an
explicit parameterization of the cell;

(2) The closure of each totally positive stratum in $\mathcal{Z}$ is a union
of some totally positives strata in $\mathcal{Z}$;

(3) The closure of each totally positive stratum in $\mathcal{Z}$ is a regular
CW complex homeomorphic to a closed ball.

(4) If $G$ is a connected reductive group, we have a natural duality on
$\mathcal{Z}_{\geqslant 0}$ generalizing Lusztig’s duality on the totally
nonnegative flag varieties.

###  1.3. Special case: double flag varieties

In this subsection, we consider the special case where $n=2$. In this case,
$\mathcal{Z}$ is isomorphic to the double flag $\mathcal{B}\times\mathcal{B}$.
Under this identification, the open Richardson varieties in $\mathcal{Z}$ are
the intersections of the $B^{+}\times B^{-}$-orbits and the diagonal
$G$-orbits in $\mathcal{B}\times\mathcal{B}$. Below we give an explicit
example of totally nonnegative double flag varieties.

######  Example 1.1.

Let $G=SL_{2}$. In this case $\mathcal{B}\cong\mathbb{P}^{1}$ and
$\mathcal{B}_{\geqslant 0}=[0,\infty]$. The totally nonnegative double flag
variety $\mathcal{Z}_{\geqslant 0}=\\{(a,b);a,b\in[0,\infty],b-a\geqslant
0\\}$. So $\mathcal{Z}_{\geqslant 0}$ is a proper subspace of
$\mathcal{B}_{\geqslant 0}\times\mathcal{B}_{\geqslant 0}$ defined by some
non-trivial inequalities.

The notion of totally nonnegative double flag varieties was introduced by
Webster and Yakimov in [26]. The totally nonnegative double flag variety of
$SL_{2}$ above was the only case understood for many years. The essential
difficulty is to handle the extra non-trivial inequalities arising from the
double flag. There is also a representation-theoretical construction of
$\mathcal{Z}_{\geqslant 0}$ relying on the canonical bases on tensor products
of $G$-modules. The difficulty in studying $\mathcal{Z}_{\geqslant 0}$ versus
$\mathcal{B}_{\geqslant 0}\times\mathcal{B}_{\geqslant 0}$ can also be seen
from the difficulty in constructing (geometrically or categorically) canonical
bases on tensor products of $G$-modules.

As a special case of Theorem 1, we see that the totally nonnegative double
flag variety is a “remarkable polyhedral subspace”. This in particular
verifies a conjecture of Webster and Yakimov in [26].

###  1.4. Special case: double Bruhat cells

The double Bruhat cells of a reductive group are the intersection of the
$B^{+}\times B^{+}$-orbits and the $B^{-}\times B^{-}$-orbits on $G$. They
play an important role in the theory of cluster algebras (see [8]).

It is easy to see that the maximal torus $T$ of $G$ acts freely on each double
Bruhat cell $G^{s,t}$ of $G$. We denote by $L^{s,t}=G^{s,t}/T$ the reduced
double Bruhat cell.

We may identify the quotient space $G/T$ with a subvariety of $\mathcal{X}$
sending a reduced double Bruhat cell $L^{s,t}$ to a stratum
$\mathring{\mathcal{X}}^{u}_{v,w}$ of the double flag variety. Moreover, under
this identification, the image of the totally positive part
$G^{s,t}_{>0}:=G^{s,t}\cap G_{\geqslant 0}$ of the double Bruhat cells
$G^{s,t}$ coincides with the totally positive part
${\mathcal{X}}^{u}_{v,w,>0}$ of the stratum $\mathring{\mathcal{X}}^{u}_{v,w}$
in the double flag variety $\mathcal{X}$. Now as a consequence of Theorem 1,
we obtain

######  Theorem 2 (Theorem 5.5).

The link of each totally positive double Bruhat cell of a connected reductive
group is a regular CW complex homeomorphic to a closed ball.

This solves an open problem of Fomin and Zelevinsky (see [22, Page 9] and [9,
Page 571]). It also generalizes the work of Hersh [13] on the cells in
$U^{-}_{\geqslant 0}$. The special case for $GL_{n}$ was first proved by
Galashin, Karp and Lam in [9] using the embedding of the double Bruhat cells
in $GL_{n}$ into the Grassmannian $Gr(n,2n)$ due to Postnikov and the
regularity of postroid cells in $Gr(n,2n)$. Our approach for $GL_{n}$ is
different from [9].

###  1.5. Special case: braid varieties

A braid $\underline{s}$ is a sequence $(s_{i_{1}},\dots,s_{i_{n}})$ of simple
reflections in $W$. Let $w=s_{i_{1}}\ast\cdots\ast s_{i_{n}}$ be the Demazure
product. The closed (resp. open) braid variety is defined as
$\mathcal{Z}_{w,\underline{s}}$ (resp.
$\mathring{\mathcal{Z}}_{w,\underline{s}}$). When $G$ is a reductive group,
the open braid varieties contain open Richardson varieties as special cases.
As another consequence of Theorem 1, we obtain

######  Theorem 3 (Theorem 5.6).

The totally nonnegative braid variety $\mathcal{Z}_{w,\underline{s},\geqslant
0}$ is regular CW complex homeomorphic to a closed ball.

The cluster algebra structure on the coordinate ring of the open braid variety
$\mathring{\mathcal{Z}}_{w,\underline{s}}$ for reductive groups has recently
been established by Casals, Gorsky, Gorsky, Le, Shen and Simental in [5] and
by Galashin, Lam, Sherman-Bennett, and Speyer in [10]. It is interesting to
compare the total positivity structure with the cluster algebra structure.

The totally nonnegative braid variety $\mathcal{Z}_{w,\underline{s},\geqslant
0}$ also provides a geometric realization of the subword complex considered by
Knutson and Miller in [18, Corollary 3.8].

###  1.6. Our strategy

One key ingredient of our approach is the “thickening” map. Let $G$ be the
Kac-Moody group we start with and $I$ be the set of simple roots. Let
$\tilde{I}=I\sqcup\\{\infty_{1},\infty_{2},\dots,\infty_{n-1}\\}$ be the
thickening of $I$ obtained by adding $n-1$ vertices labeled with $\infty_{1}$,
$\dots$, $\infty_{n-1}$ and the edges $\Leftrightarrow$ between the vertices
$\infty_{i}$ and the vertices in $I$ (see the Figure 1 for $G=SL_{4}$ and
$n=3$). Let $\tilde{G}$ be the corresponding “thickening” Kac-Moody group.

$\circ$$\scriptstyle 1$$\circ$$\scriptstyle 2$$\circ$$\scriptstyle
3$$\circ$$\scriptstyle\infty_{1}$$\circ$$\scriptstyle\infty_{2}$ Figure 1.
$\tilde{G}$ for $SL_{4}$

We construct in Proposition 3.2 a locally trivial
$(\mathbb{C}^{*})^{n-1}$-fibration from a subvariety
$\mathcal{Z}^{\prime}\subset\tilde{\mathcal{B}}$ to $\mathcal{Z}$. The
projection map sends an open Richardson variety in $\tilde{\mathcal{B}}$ to a
stratum $\mathring{\mathcal{Z}}_{v,\underline{w}}$ in $\mathcal{Z}$. Moreover,
the fibration is compatible with the totally positive structure on the twisted
product of flag varieties $\mathcal{Z}$ and on the thickening flag variety
$\tilde{}\mathcal{B}$. We then deduce the local structure on the totally
nonnegative twisted product of flag varieties $\mathcal{Z}_{\geqslant 0}$ from
the local structure on the totally nonnegative flag variety
$\tilde{}\mathcal{B}$ established in [4].

To deduce the global structure of $\mathcal{Z}_{\geqslant 0}$, we need to show
the face poset $\hat{Q}$ of $\mathcal{Z}_{\geqslant 0}$ is combinatorially a
regular CW complex. We show the face poset $\hat{Q}$ is pure, thin, and
shellable in Theorem 4.1. Then the combinatorial regularity follows from a
result of Bjoner in [2].

Finally, combining the local structure and the combinatorial regularity with
the generalized Poincaré conjecture, we show that the closure of each totally
positive stratum in the twisted product of flag varieties is a regular CW
complex.

It is worth mentioning that even for the totally nonnegative double flag
varieties of reductive groups, our proof still relies on the knowledge of the
totally nonnegative flag of Kac-Moody groups (of non-finite, non-affine type)
in [4].

Acknowledgement:  HB is supported by MOE grants A-0004586-00-00 and
A-0004586-01-00. XH is partially supported by funds connected with Choh-Ming
Chair at CUHK, by Hong Kong RGC grant 14300221 and by Xplorer prize. We thank
George Lusztig for helpful discussions and for the question on §5.1, and we
thank Steven Karp for informing us the background on the open problem of Fomin
and Zelevinsky. We thank Lauren Williams for helpful discussions on
shellability of posets.

##  2\. Flag varieties and their totally nonnegative parts

###  2.1. Kac-Moody groups

A Kac-Moody root datum is a quintuple

$\mathcal{D}=(I,A,X,Y,(\alpha_{i})_{i\in I},(\alpha^{\vee}_{i})_{i\in I}),$

where $I$ is a finite set, $A=(a_{ij})_{i,j\in I}$ is a symmetrizable
generalized Cartan matrix in the sense of [15, §1.1], $X$ is a free
$\mathbb{Z}$-module of finite rank with $\mathbb{Z}$-dual $Y$, and the
elements $\alpha_{i}$ of $X$ and $\alpha^{\vee}_{i}$ of $Y$ such that
$\langle\alpha^{\vee}_{j},\alpha_{i}\rangle=a_{ij}$ for $i,j\in I$.

Let $\Delta^{re}=\\{w(\pm\alpha_{i})\in X\mid i\in I,w\in W\\}\subset X$ be
the set of real roots. Then $\Delta^{re}=\Delta^{re}_{+}\sqcup\Delta^{re}_{-}$
is the union of positive real roots and negative real roots. Let $\mathbf{k}$
be an algebraically closed field. The minimal Kac-Moody group $G$ associated
to the Kac-Moody root datum $\mathcal{D}$ is the group generated by the torus
$T=Y\otimes_{\mathbb{Z}}\mathbf{k}^{\times}$ and the root subgroup
$U_{\alpha}\cong\mathbf{k}$ for each real root $\alpha$, subject to the Tits
relations [25]. Let $U^{+}\subset G$ (resp. $U^{-}\subset G$) be the subgroup
generated by $U_{\alpha}$ for $\alpha\in\Delta^{re}_{+}$ (resp.
$\alpha\in\Delta^{re}_{-}$). Let $B^{\pm}\subset G$ be the Borel subgroup
generated by $T$ and $U^{\pm}$. For each $i\in I$, we fix isomorphisms
$x_{i}:\mathbb{C}U_{\alpha_{i}}$ and $y_{i}:\mathbb{C}U_{-\alpha_{i}}$ such
that the map

$\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}x_{i}(a),\begin{pmatrix}b&0\\\
0&b^{-1}\end{pmatrix}\alpha^{\vee}_{i}(a),\begin{pmatrix}1&0\\\
c&1\end{pmatrix}y_{i}(c)$

defines a homomorphism $SL_{2}G$. The data $(T,B^{+},B^{-},x_{i},y_{i};i\in
I)$ is called a pinning for $G$.

Let $W$ be the Weyl group of $G$. For $i\in I$, we denote by $s_{i}\in W$ the
corresponding simple reflection and write
$\dot{s}_{i}=x_{i}(1)y_{i}(-1)x_{i}(1)\in G$. For any $w\in W$ with reduced
expression $w=s_{i_{1}}\cdots s_{i_{n}}$, we define
$\dot{w}=\dot{s}_{i_{1}}\cdots\dot{s}_{i_{n}}\in G$. It is known that
$\dot{w}$ is independent of the choice of the reduced expressions.

Let $J\subset I$ (not necessarily of finite type). We denote by $P^{+}_{J}$
the standard parabolic subgroup of $G$ corresponding to $J$ and $P^{-}_{J}$
the opposite parabolic subgroup of $G$. Let $L_{J}=P^{+}_{J}\cap P^{-}_{J}$ be
the standard Levi subgroup. Let $W_{J}$ be the subgroup of $W$ generated by
$\\{s_{j}\\}_{j\in J}$. Then $W_{J}$ is the Wey group of $L_{J}$. For any
parabolic subgroup $P$, we denote by $U_{P}$ its unipotent radical. We have
the Levi decomposition $P^{\pm}_{J}=L_{J}\ltimes U_{P^{\pm}_{J}}$.

###  2.2. Flag varieties

We denote by $\mathcal{B}=G/B^{+}$ the (thin) full flag variety [16], equipped
with the ind-variety structure. Let $v,w\in W$. Let
$\mathring{\mathcal{B}}_{w}=B^{+}\dot{w}B^{+}/B^{+}$ be the Schubert cell
corresponding to $w$, $\mathring{\mathcal{B}}^{v}=B^{-}\dot{v}B^{+}/B^{+}$ be
the opposite Schubert cell corresponding to $v$ and
$\mathring{\mathcal{B}}_{v,w}=\mathring{\mathcal{B}}_{w}\cap\mathring{\mathcal{B}}^{v}$
be the open Richardson variety corresponding to the pair $(v,w)$. It follows
from [17] that

(a) $\mathring{\mathcal{B}}_{v,w}\neq\emptyset$ if and only if $v\leqslant w$.
In this case, $\mathring{\mathcal{B}}_{v,w}$ is irreducible of dimension
$\ell(w)-\ell(v)$.

We have the decompositions $\mathcal{B}=\sqcup_{w\in
W}\mathring{\mathcal{B}}_{w}=\sqcup_{v\in
W}\mathring{\mathcal{B}}^{v}=\sqcup_{v\leqslant
w}\mathring{\mathcal{B}}_{v,w}$. Let
$\mathcal{B}_{w},\mathcal{B}^{v},\mathcal{B}_{v,w}$ be the (Zariski) closure
of
$\mathring{\mathcal{B}}_{w},\mathring{\mathcal{B}}^{v},\mathring{\mathcal{B}}_{v,w}$
respectively. By [16, Proposition 7.1.15 & 7.1.21], we have

$\mathcal{B}_{w}=\bigsqcup_{w^{\prime}\leqslant
w}\mathring{\mathcal{B}}_{w^{\prime}},\qquad\mathcal{B}^{v}=\bigsqcup_{v^{\prime}\geqslant
v}\mathring{\mathcal{B}}^{v^{\prime}},\qquad\mathcal{B}_{v,w}=\bigsqcup_{v\leqslant
v^{\prime}\leqslant w^{\prime}\leqslant
w}\mathring{\mathcal{B}}_{v^{\prime},w^{\prime}}.$

###  2.3. Expressions and subexpressions

An expression is a sequence

$\mathbf{w}=(t_{1},t_{2},\ldots,t_{n})$

in $I\sqcup\\{1\\}$. In this case, we write $w=t_{1}t_{2}\cdots t_{n}\in W$,
and we say that $\mathbf{w}$ is an expression of $w$. The expression
$\mathbf{w}$ is called reduced if $n=\ell(w)$. In this case, $t_{i}\in I$ for
all $i$.

A subexpression of $\mathbf{w}$ is a sequence
$\mathbf{w}^{\prime}=(t^{\prime}_{1},t^{\prime}_{2},\ldots,t^{\prime}_{n})$,
where $t^{\prime}_{i}\in\\{t_{i},1\\}$ for all $i$. If $\mathbf{w}$ is an
reduced expression, then the subexpression $\mathbf{w}^{\prime}$ is called
positive if $t^{\prime}_{1}\cdots t^{\prime}_{i-1}<t^{\prime}_{1}\cdots
t^{\prime}_{i-1}t_{i}$ for all $i$.

By [14, Lemma 3.5], for any $w,w^{\prime}\in W$ with $w^{\prime}\leqslant w$
and for any reduced expression $\mathbf{w}$ of $w$, there exists a unique
positive subexpression for $w^{\prime}$ in $\mathbf{w}$. We denote it by
$\mathbf{w}^{\prime}_{+}$.

###  2.4. Totally nonnegative flag varieties

We follow [19] and [21]. Let $U^{+}_{\geqslant 0}$ be the submonoid of $G$
generated by $x_{i}(a)$ for $i\in I$ and $a\in\mathbb{R}_{>0}$ and
$U^{-}_{\geqslant 0}$ be the submonoid of $G$ generated by $y_{i}(a)$ for
$i\in I$ and $a\in\mathbb{R}_{>0}$. Let $T_{>0}$ be the identity component of
$T(\mathbb{R})$. The totally nonnegative monoid $G_{\geqslant 0}$ is defined
to be the submonoid of $G$ generated by $U^{\pm}_{\geqslant 0}$ and $T_{>0}$.
By [21, §2.5], $G_{\geqslant 0}=U^{+}_{\geqslant 0}T_{>0}U^{-}_{\geqslant
0}=U^{-}_{\geqslant 0}T_{>0}U^{+}_{\geqslant 0}$.

Let $\mathcal{B}_{\geqslant 0}=\overline{U^{-}_{\geqslant 0}B^{+}/B^{+}}$ be
the closure of $U^{-}_{\geqslant 0}B^{+}/B^{+}$ in $\mathcal{B}$ with respect
to the Hausdorff topology. For any $v\leqslant w$, let
$\mathcal{B}_{v,w,>0}=\mathring{\mathcal{B}}_{v,w}\bigcap\mathcal{B}_{\geqslant
0}$. Then $\mathcal{B}_{\geqslant 0}=\bigsqcup_{v\leqslant
w}\mathcal{B}_{v,w,>0}$.

Following [14, Definition 5.1], we set

$G_{\bf v_{+},\bf w,>0}=\\{g_{1}g_{2}\cdots g_{n}|g_{j}=\dot{s}_{i_{j}},\text{
if }t_{i_{j}}=s_{i_{j}};\text{ and }g_{j}\in y_{i_{j}}(\mathbb{R}_{>0}),\text{
if }t_{i_{j}}=1\\}.$

Note that the obvious map $\mathbb{R}_{>0}^{\ell(w)-\ell(v)}\rightarrow G_{\bf
v_{+},\bf w,>0}$ is a homeomorphism.

By [14, Theorem 11.3] for reductive groups and [3, Theorem 4.10] for Kac-Moody
groups, we have the following result.

(a) Let $v\leqslant w$. For any reduced expression ${\bf w}$ of $w$, the map
$gg\cdot B^{+}$ gives a homeomorphism $G_{\bf v_{+},\bf
w,>0}\cong\mathcal{B}_{v,w,>0}$. In particular,
$\mathcal{B}_{v,w,>0}\cong\mathbb{R}_{>0}^{\ell(w)-\ell(v)}$ is a topological
cell.

######  Proposition 2.1.

[4, Proposition 5.2] Let $v\leqslant w$. Then

(1) $\mathcal{B}_{v,w,>0}$ is a connected component of
$\mathcal{B}_{v,w}(\mathbb{R})$.

(2) The Hausdorff closure of $\mathcal{B}_{v,w,>0}$ equals $\sqcup_{v\leqslant
v^{\prime}\leqslant w^{\prime}\leqslant
w}\mathcal{B}_{v^{\prime},w^{\prime},>0}$.

For reductive groups, the above results were established by Rietsch in [23,
24].

###  2.5. Twisted product of flag varieties

Let $n\in\mathbb{N}$. Recall the twisted product $\mathcal{Z}_{n}$ of the flag
varieties:

$\mathcal{Z}_{n}=G\times^{B^{+}}G\times^{B^{+}}\cdots\times^{B^{+}}G/B^{+}\quad(n\text{
factors}).$

We often write $\mathcal{Z}=\mathcal{Z}_{n}$ whenever the number of factors is
clear. The isomorphism $G^{n}\rightarrow G^{n}$,
$(g_{1},g_{2},\dots,g_{n})(g_{1},g_{1}g_{2},\dots,g_{1}g_{2}\cdots g_{n})$
induces an isomorphism

$\alpha:\mathcal{Z}_{n}\mathcal{B}^{n},\qquad(g_{1},g_{2},\ldots,g_{n})(g_{1}B^{+},g_{1}g_{2}B^{+},\ldots,g_{1}g_{2}\cdots
g_{n}B^{+}).$

In particular, we have the convolution product

$m:\mathcal{Z}_{n}\mathcal{Z}_{1}=G/B^{+},\qquad(g_{1},\ldots,g_{n})g_{1}g_{2}\cdots
g_{n}B^{+}.$

The stratification on $\mathcal{Z}_{n}$ we are interested in is different from
the stratification on $\mathcal{B}^{n}$ given by the product of
$\mathring{\mathcal{B}}_{-,-}$. Let $\underline{w}=(w_{1},\cdots,w_{n})$ be a
sequence of elements in $W$. Recall the Schubert cell in $\mathcal{Z}_{n}$
associated to $\underline{w}$ is defined to be

$\mathring{\mathcal{Z}}_{n,\underline{w}}=B^{+}\dot{w}_{1}B^{+}\times^{B^{+}}B^{+}\dot{w}_{2}B^{+}\times^{B^{+}}\cdots\times^{B^{+}}B^{+}\dot{w}_{n}B^{+}/B^{+}.$

Also recall the opposite Schubert cell in $\mathcal{Z}_{n}$ associated to
$v\in W$ is defined to be

$\mathring{\mathcal{Z}}^{v}_{n}=\\{x\in\mathcal{Z}_{n};m(x)\in\mathring{\mathcal{B}}^{v}\\}.$

The open Richardson variety of $\mathcal{Z}_{n}$ is defined by
$\mathring{\mathcal{Z}}_{n,v,\underline{w}}=\mathring{\mathcal{Z}}_{n,\underline{w}}\cap\mathring{\mathcal{Z}}^{v}_{n}$.

###  2.6. Monoid structure on $W$

We follow [11, §1]. For any $x,y\in W$, the subset
$\\{xy^{\prime};y^{\prime}\leqslant y\\}$ (resp.
$\\{x^{\prime}y;x^{\prime}\leqslant x\\}$,
$\\{x^{\prime}y^{\prime};x^{\prime}\leqslant x,y^{\prime}\leqslant y\\}$) of
$W$ contains a unique maximal element (with respect to the Bruhat order
$\leqslant$). Moreover, we have

$\max\\{xy^{\prime};y^{\prime}\leqslant
y\\}=\max\\{x^{\prime}y;x^{\prime}\leqslant
x\\}=\max\\{x^{\prime}y^{\prime};x^{\prime}\leqslant x,y^{\prime}\leqslant
y\\}.$

We denote this element by $x\ast y$ and we call it the Demazure product of $x$
and $y$. Moreover, $(W,\ast)$ is a monoid and the Demazure product can be
determined by the following two rules

  * •

$x\ast y=xy$ if $x,y\in W$ such that $\ell(xy)=\ell(x)+\ell(y)$;

  * •

$s\ast w=w$ if $s\in I$, $w\in\tilde{W}$ such that $sw<w$.

We define the maps $m_{\bullet},m_{\ast}:W^{n}W$ by

$\displaystyle m_{\bullet}(w_{1},w_{2},\ldots,w_{n})=w_{1}w_{2}\cdots w_{n},$
$\displaystyle m_{\ast}(w_{1},w_{2},\ldots,w_{n})=w_{1}\ast
w_{2}\ast\cdots\ast w_{n}.$

We have the following result describing the nonemptiness pattern of the
Richardson varieties in $\mathcal{Z}_{n}$.

######  Lemma 2.2.

Let $v\in W$ and $\underline{w}\in W^{n}$. Then
$\mathring{\mathcal{Z}}_{n,v,\underline{w}}\neq\emptyset$ if and only if
$v\leqslant m_{\ast}(\underline{w})$.

###### Proof.

We have $\mathring{\mathcal{B}}_{m_{\ast}(\underline{w})}\subset
m(\mathring{\mathcal{Z}}_{n,\underline{w}})\subset\sqcup_{w^{\prime}\leqslant
m_{\ast}(\underline{w})}\mathring{\mathcal{B}}_{w^{\prime}}$. Now the
statement follows from the equivalence of the following three conditions:

$v\leqslant
m_{\ast}(\underline{w})\quad\Longleftrightarrow\quad\mathring{\mathcal{B}}^{v}\cap\mathring{\mathcal{B}}_{m_{\ast}(\underline{w})}\neq\emptyset\quad\Longleftrightarrow\quad\mathring{\mathcal{B}}^{v}\cap(\sqcup_{w^{\prime}\leqslant
m_{\ast}(\underline{w})}\mathring{\mathcal{B}}_{w^{\prime}})\neq\emptyset.\qed$

##  3\. The thickening map

Let $n\in\mathbb{Z}_{>0}$. We write $\mathcal{Z}=\mathcal{Z}_{n}$ throughout
this section.

###  3.1. The thickening group

We construct the set of simple roots and the associated generalized Cartan
matrix of the thickening Kac-Moody $\tilde{G}$ from the original Kac-Moody
group ${G}$.

The set $\tilde{I}$ of simple roots is the union of $I$ with new vertices
$\\{\infty_{1},\ldots,\infty_{n-1}\\}$. The Dynkin diagram of $\tilde{I}$ is
obtained from the Dynkin diagram of $I$ by adding an edge
$i\Longleftrightarrow\infty_{l}$ for any $i\in I$ and $1\leqslant l\leqslant
n-1$. In other words, the generalized Cartan matrix
$\tilde{A}=(\tilde{a}_{i,j})_{i,j\in\tilde{I}}$ is defined as follows:111In
fact, the argument in this paper works once
$\tilde{a}_{i,\infty_{l}},\tilde{a}_{\infty_{l},i}\neq 0$ for $i\in I$ and
$1\leqslant l\leqslant n-1$.

  * •

$\tilde{a}_{i,j}=a_{i,j}$ for $i,j\in I$;

  * •

$\tilde{a}_{i,\infty_{l}}=\tilde{a}_{\infty_{l},i}=-2$ for $i\in I$ and
$1\leqslant l\leqslant n-1$;

  * •

$\tilde{a}_{\infty_{l},\infty_{l}}=2$ for $1\leqslant l\leqslant n-1$;
$\tilde{a}_{\infty_{l},\infty_{l^{\prime}}}=-2$ for $1\leqslant
l,l^{\prime}\leqslant n-1$ with $l\neq l^{\prime}$.

Let $\tilde{G}$ be the minimal Kac-Moody group of simply connected type
associated to $(\tilde{I},\tilde{A})$ and $\tilde{W}$ be its Weyl group. Let
$\tilde{W}_{I}$ be the parabolic subgroup of $\tilde{W}$ generated by simple
reflections in $I$ and $\tilde{L}_{I}$ be the standard Levi subgroup of the
parabolic subgroup $\tilde{P}^{+}_{I}$ of $\tilde{G}$ that corresponds to $I$.
We have natural identifications

$\displaystyle i:W\tilde{W}_{I}\text{ and }i:G\tilde{L}_{I}.$

We fix a pinning
$(\tilde{T},\tilde{B}^{+},\tilde{B}^{-},x_{i},y_{i};i\in\tilde{I})$ of
$\tilde{G}$ that is compatible with the pinning of $G$ via the identification
$i$. We define the thickening maps

$\displaystyle
th:W^{n}\tilde{W},(w_{1},w_{2},\ldots,w_{n})w_{1}s_{\infty_{1}}w_{2}s_{\infty_{2}}\cdots
s_{\infty_{n-1}}w_{n};$ $\displaystyle th:(\mathbb{C}^{*})^{n-1}\times
G^{n}\tilde{G},(a_{1},\dots,a_{n-1},g_{1},\ldots,g_{n})g_{1}y_{\infty_{1}}(a_{1})g_{2}\cdots
g_{n-1}y_{\infty_{n-1}}(a_{n-1})g_{n}.$

We denote by $\tilde{}\mathcal{B}$ the flag variety of $\tilde{G}$, and denote
by $\mathring{\tilde{}\mathcal{B}}^{-}$, $\mathring{\tilde{}\mathcal{B}}_{-}$,
$\mathring{\tilde{}\mathcal{B}}_{-,-}$ the Schubert cells, the opposite
Schubert cells and the open Richardson varieties of $\tilde{}\mathcal{B}$
respectively.

###  3.2. Positive subexpressions in $W^{n}$

Let $\underline{w}=(w_{1},\dots,w_{n})$ and
$\underline{w^{\prime}}=(w^{\prime}_{1},\dots,w^{\prime}_{n})$ be in $W^{n}$.
We say $\underline{w^{\prime}}\leqslant\underline{w}$ if
$w^{\prime}_{i}\leqslant w_{i}$ in the usual Bruhat order for all $i$.

######  Definition 3.1.

Let $\underline{w},\underline{v}\in W^{n}$ with
$\underline{v}\leqslant\underline{w}$. We say that $\underline{v}$ is positive
in $\underline{w}$ if for any $i$, $v_{1}^{\prime}\leqslant
v_{1},\dots,v^{\prime}_{i-1}\leqslant v_{i-1},v^{\prime}_{i}\leqslant w_{i}$
with $v_{1}^{\prime}\cdots v_{i-1}^{\prime}v^{\prime}_{i}=v_{1}\cdots
v_{i-1}v_{i}$ implies that
$(v_{1}^{\prime},\cdots,v_{i}^{\prime})=(v_{1},\cdots,v_{i})$.

Let $\underline{w}=(w_{1},\dots,w_{n})\in W^{n}$. We choose an expression
$\mathbf{w}_{i}$ for each $w_{i}$. We write
$\underline{\mathbf{w}}=(\mathbf{w}_{1};\mathbf{w}_{2};\dots;\mathbf{w}_{n})$.
We then associate two expressions to it:

  * •

$i(\underline{\mathbf{w}})=(\mathbf{w}_{1},1,\mathbf{w}_{2},1,\dots,1,\mathbf{w}_{n})$,
an expression of $i(m_{\bullet}(\underline{w}))=w_{1}w_{2}\cdots w_{n}$;

  * •

$th(\underline{\mathbf{w}})=(\mathbf{w}_{1},s_{\infty_{1}},\mathbf{w}_{2},\dots,s_{\infty_{n-1}},\mathbf{w}_{n})$,
an expression of $th(\underline{w})=w_{1}s_{\infty_{1}}w_{2}\cdots
s_{\infty_{n-1}}w_{n}$.

It is easy to see that

  * •

for $v,v^{\prime}\in W$, $v\leqslant v^{\prime}$ in $W$ if and only if
$i(v)\leqslant i(v^{\prime})$ in $\tilde{W}$;

  * •

for $\underline{w},\underline{w^{\prime}}\in W^{n}$,
$\underline{w}\leqslant\underline{w^{\prime}}$ if and only if
$th(\underline{w})\leqslant th(\underline{w^{\prime}})$ in $\tilde{W}$;

  * •

for $v\in W$ and $\underline{w}\in W^{n}$, $v\leqslant
m_{\ast}(\underline{w})$ if and only if $i(v)\leqslant th(\underline{w})$.

The following result follow directly from the definition.

(a) Let $\underline{w},\underline{v}\in W^{n}$ with
$\underline{v}\leqslant\underline{w}$. Let $\mathbf{w}_{i}$ be a reduced
expression of $w_{i}$ and $(\mathbf{v}_{i})_{+}$ be the positive subexpression
for $v_{i}$ in $\mathbf{w}_{i}$. Then $\underline{v}$ is positive in
$\underline{w}$ if and only if $i(\underline{\mathbf{v}}_{+})$ is a positive
subexpression of $th(\underline{\mathbf{w}})$, where
$\underline{\mathbf{v}}_{+}=((\mathbf{v}_{1})_{+};(\mathbf{v}_{2})_{+};\dots;(\mathbf{v}_{n})_{+})$
and
$\underline{\mathbf{w}}=(\mathbf{w}_{1};\mathbf{w}_{2};\dots;\mathbf{w}_{n})$
in the sense of §2.3.

(b) Let $\underline{w}\in W^{n}$ and $v\in W$ with $v\leqslant
m_{\ast}(\underline{w})$. Then there exists a unique $\underline{v}$ in
$W^{n}$ with $m_{\bullet}(\underline{v})=v$ and $\underline{v}$ is positive in
$\underline{w}$. In this case, $m_{\ast}(\underline{v})=v$.

###  3.3. The thickening map

Let $\alpha_{\infty_{l}}$ be the simple root corresponding to
$\infty_{l}\in\tilde{I}$ for $1\leqslant l\leqslant n-1$. We write the
subvariety $U_{-\alpha_{\infty_{l}}}-\\{e\\}\subset U_{-\alpha_{\infty_{l}}}$
as $U_{-\alpha_{\infty_{l}}}^{*}$.

######  Proposition 3.2.

Set

$\mathcal{Z}^{\prime}=\tilde{P}^{-}_{I}\cdot\tilde{B}^{+}/\tilde{B}^{+}\cap\tilde{P}^{+}_{I}(\tilde{B}^{+}\dot{s}_{\infty_{1}}\tilde{B}^{+})\tilde{P}^{+}_{I}(\tilde{B}^{+}\dot{s}_{\infty_{2}}\tilde{B}^{+})\cdots(\tilde{B}^{+}\dot{s}_{\infty_{n-1}}\tilde{B}^{+})\tilde{P}^{+}_{I}/\tilde{B}^{+}.$

Then

  1. (1)

$\mathcal{Z}^{\prime}=\bigsqcup_{v\leqslant
m_{\ast}(\underline{w})}\mathring{\tilde{\mathcal{B}}}_{i(v),th(\underline{w})}$
is a locally closed subvariety of $\tilde{}\mathcal{B}$;

  2. (2)

there exists a locally trivial $(\mathbb{C}^{*})^{n-1}$-fibration
$\pi:\mathcal{Z}^{\prime}\mathcal{Z}$;

  3. (3)

the morphism $\pi$ is stratified such that $\pi$ maps
$\mathring{\tilde{\mathcal{B}}}_{i(v),th(\underline{w})}$to
$\mathring{\mathcal{Z}}_{v,\underline{w}}$ via restriction.

###### Proof.

Set
$\mathcal{Z}^{\prime\prime}=\tilde{P}^{+}_{I}(\tilde{P}^{+}_{\infty_{1}})\tilde{P}^{+}_{I}(\tilde{P}^{+}_{\infty_{2}})\cdots(\tilde{P}^{+}_{\infty_{n-1}})\tilde{P}^{+}_{I}/\tilde{B}^{+}$.
Then $\mathcal{Z}^{\prime\prime}$ is closed in $\tilde{}\mathcal{B}$.
Moreover,
$\tilde{P}^{+}_{I}(\tilde{B}^{+}\dot{s}_{\infty_{1}}\tilde{B}^{+})\tilde{P}^{+}_{I}(\tilde{B}^{+}\dot{s}_{\infty_{2}}\tilde{B}^{+})\cdots(\tilde{B}^{+}\dot{s}_{\infty_{n-1}}\tilde{B}^{+})\tilde{P}^{+}_{I}/\tilde{B}^{+}$
is open in $\mathcal{Z}^{\prime\prime}$. As
$\tilde{P}^{-}_{I}\cdot\tilde{B}^{+}/\tilde{B}^{+}$ is open in
$\tilde{}\mathcal{B}$, $\mathcal{Z}^{\prime}$ is locally closed in
$\tilde{}\mathcal{B}$. This shows (1).

We have the following isomorphism via the convolution product

$\displaystyle
G\times^{B^{+}}(\tilde{B}^{+}\dot{s}_{\infty_{1}}\tilde{B}^{+})\times^{\tilde{B}^{+}}G\times^{B^{+}}(\tilde{B}^{+}\dot{s}_{\infty_{2}}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\times^{B^{+}}(\tilde{B}^{+}\dot{s}_{\infty_{n-1}}\tilde{B}^{+})\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}$
$\displaystyle\cong\,$ $\displaystyle
G(\tilde{B}^{+}\dot{s}_{\infty_{1}}\tilde{B}^{+})G(\tilde{B}^{+}\dot{s}_{\infty_{2}}\tilde{B}^{+})\cdots(\tilde{B}^{+}\dot{s}_{\infty_{n-1}}\tilde{B}^{+})G/\tilde{B}^{+}$
$\displaystyle\cong\,$
$\displaystyle\tilde{P}^{+}_{I}(\tilde{B}^{+}\dot{s}_{\infty_{1}}\tilde{B}^{+})\tilde{P}^{+}_{I}(\tilde{B}^{+}\dot{s}_{\infty_{2}}\tilde{B}^{+})\cdots(\tilde{B}^{+}\dot{s}_{\infty_{n-1}}\tilde{B}^{+})\tilde{P}^{+}_{I}/\tilde{B}^{+}.$

Here the $B^{+}$-action on $\tilde{B}^{+}\dot{s}_{\infty_{1}}\tilde{B}^{+}$ is
by multiplication, while the action of $\tilde{B}^{+}$ on $G$ is via the
quotient $\tilde{B}^{+}B^{+}$. Any element of $\mathcal{Z}^{\prime}$ is of the
form $g_{1}y_{1}g_{2}y_{2}\cdots y_{n-1}g_{n}\tilde{B}^{+}/\tilde{B}^{+}$,
where $g_{i}\in G$ and
$y_{i}\in\tilde{B}^{+}\dot{s}_{\infty_{i}}\tilde{B}^{+}$. It follows from
$\mathcal{Z}^{\prime}\subset\tilde{P}^{-}_{I}\cdot\tilde{B}^{+}/\tilde{B}^{+}$
that
$y_{i}\in\tilde{B}^{+}\dot{s}_{\infty_{i}}\tilde{B}^{+}\cap\tilde{B}^{-}\tilde{B}^{+}$.
Hence we obtain, via restriction,

$\displaystyle Z^{\prime}\cong$ $\displaystyle
G\times^{B^{+}}(U_{-\alpha_{\infty_{1}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}G\times^{B^{+}}(U_{-\alpha_{\infty_{2}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}$
$\displaystyle\cong$ $\displaystyle
G\times^{B^{+}}\big{(}U_{-\alpha_{\infty_{1}}}^{*}\times
G\big{)}\times^{B^{+}}(U_{-\alpha_{\infty_{2}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\cdots\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}.$

The $B^{+}$-action on $G\times\big{(}U_{-\alpha_{\infty_{1}}}^{*}\times G)$ is
given by $b\cdot(x,y,z)=(xb^{-1},byb^{-1},bz)$. Similar $B^{+}$-actions apply
for the other factors. We hence have a well-defined morphism

$\pi_{1}:\mathcal{Z}^{\prime}\rightarrow
G\times^{B^{+}}G\times^{B^{+}}(U_{-\alpha_{\infty_{2}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\cdots\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}.$
(3.1)

Iterating the construction, we obtain a morphism

$\pi:\mathcal{Z}^{\prime}\rightarrow
G\times^{B^{+}}G\times^{B^{+}}\cdots\times^{B^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}\cong\mathcal{Z}.$

Part (3) follows immediately.

We prove (2). Let $r\in W$. We have

$\displaystyle\mathcal{Z}^{\prime}$
$\displaystyle\supset(\dot{r}U^{-}B^{+})\times^{B^{+}}\big{(}U_{-\alpha_{\infty_{1}}}^{*}\times
G\big{)}\times^{B^{+}}(U_{-\alpha_{\infty_{2}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\cdots\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}$
$\displaystyle\cong\dot{r}U^{-}\times U_{-\alpha_{\infty_{1}}}^{*}\times
G\times^{B^{+}}(U_{-\alpha_{\infty_{2}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\cdots\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}$
$\displaystyle\cong
U_{-\alpha_{\infty_{1}}}^{*}\times(\dot{r}U^{-}B^{+})\times^{B^{+}}G\times^{B^{+}}(U_{-\alpha_{\infty_{2}}}^{*}\tilde{B}^{+})\times^{\tilde{B}^{+}}\cdots\cdots\times^{\tilde{B}^{+}}\tilde{P}^{+}_{I}/\tilde{B}^{+}.$

This shows that $\pi_{1}$ in (3.1) is a locally trivial
$\mathbb{C}^{*}$-fibration. Part (2) follows by iteration. ∎

For any $\underline{r}=(r_{1},\dots,r_{n})\in W^{n}$, the thickening map
$th:(\mathbb{C}^{*})^{n-1}\times G^{n}\tilde{G}$ induces a section of the
fibre bundle $\pi:\mathcal{Z}^{\prime}\rightarrow\mathcal{Z}$ as follows

$\displaystyle\,\,\mathbb{C}^{*}\times(\dot{r}_{1}U^{-})\times\mathbb{C}^{*}\times(\dot{r}_{2}U^{-})\times\cdots\times(\dot{r}_{n}U^{-})/B^{+}\rightarrow\mathcal{Z}^{\prime}.$
(3.2)

The projection $\pi$ in Proposition 3.2 is defined over $\mathbb{R}$, hence
induces projections on the set of $\mathbb{R}$-rational points

$\pi:\mathcal{Z}^{\prime}(\mathbb{R})\rightarrow\mathcal{Z}(\mathbb{R}),\qquad\pi:\tilde{\mathcal{B}}_{i(v),th(\underline{w})}(\mathbb{R})\rightarrow\mathcal{Z}_{v,\underline{w}}(\mathbb{R}).$
(3.3)

We draw some consequences from Proposition 3.2.

######  Corollary 3.3.

Let $v\in W$ and $\underline{w}=(w_{1},w_{2},\dots,w_{n})\in W^{n}$ with
$v\leqslant m_{\ast}(\underline{w})$. Then

  1. (1)

$\mathring{\mathcal{Z}}_{v,\underline{w}}$ is irreducible and of dimension
$\ell(w_{1})+\ell(w_{2})+\dots+\ell(w_{n})-\ell(v)$;

  2. (2)

the Zariski closure of $\mathring{\mathcal{Z}}_{v,\underline{w}}$ equals
$\mathcal{Z}^{v}\cap\mathcal{Z}_{\underline{w}}=\bigsqcup_{v\leqslant
v^{\prime},\underline{w^{\prime}}\leqslant\underline{w}\text{ with
}v^{\prime}\leqslant
m_{\ast}(\underline{w^{\prime}})}\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}}}$.

We may also deduce Lemma 2.2 from Proposition 3.2 and the nonemptiness pattern
of the open Richardson varieties in $\tilde{}\mathcal{B}$.

###  3.4. Total positivity on $\mathcal{Z}$

Let $\mathcal{Z}_{\geqslant 0}$ be the Hausdorff closure of $(U^{-}_{\geqslant
0},U^{-}_{\geqslant 0},\cdots,U^{-}_{\geqslant 0}B^{+}/B^{+})$ in
$\mathcal{Z}$. For any $v\in W$ and $\underline{w}\in W^{n}$ with $v\leqslant
m_{\ast}(\underline{w})$, we define

$\mathcal{Z}_{v,\underline{w},>0}=\mathring{\mathcal{Z}}_{v,\underline{w}}\cap\mathcal{Z}_{\geqslant
0}.$

Now we prove the first main result of this paper.

######  Theorem 3.4.

Let $v\in W$ and $\underline{w}\in W^{n}$ with $v\leqslant
m_{\ast}(\underline{w})$. Then

(1) $\mathcal{Z}_{v,\underline{w},>0}$ is a connected component of
$\mathring{\mathcal{Z}}_{v,\underline{w}}(\mathbb{R})$.

(2) The Hausdorff closure of $\mathcal{Z}_{v,\underline{w},>0}$ equals
$\bigsqcup_{v\leqslant
v^{\prime},\underline{w^{\prime}}\leqslant\underline{w}\text{ with
}v^{\prime}\leqslant
m_{\ast}(\underline{w^{\prime}})}\mathcal{Z}_{v^{\prime},\underline{w^{\prime}},>0}$.

(3) Let $\underline{v}\in W^{n}$ with $m_{\bullet}(\underline{v})=v$ and
$\underline{v}$ being positive in $\underline{w}$. Let $\mathbf{w}_{i}$ be a
reduced expression of $w_{i}$ and $(\mathbf{v}_{i})_{+}$ be the positive
subexpression for $v_{i}$ in $\mathbf{w}_{i}$ for all $i$. Then we have an
isomorphism

$\displaystyle G_{(\mathbf{v}_{1})_{+},\mathbf{w}_{1},>0}\times
G_{(\mathbf{v}_{2})_{+},\mathbf{w}_{2},>0}\times\cdots\times
G_{(\mathbf{v}_{n})_{+},\mathbf{w}_{n},>0}$
$\displaystyle\mathcal{Z}_{v,\underline{w},>0},$
$\displaystyle(g_{1},g_{2},\dots,g_{n})$
$\displaystyle(g_{1},g_{2},\dots,g_{n}B^{+}/B^{+}).$

###### Proof.

By Proposition 2.1, we have

$\displaystyle\overline{U^{-}_{\geqslant
0}y_{\infty_{1}}(\mathbb{R}_{>0})U^{-}_{\geqslant
0}y_{\infty_{2}}(\mathbb{R}_{>0})U^{-}_{\geqslant 0}\dots
y_{\infty_{n-1}}(\mathbb{R}_{>0})U^{-}_{\geqslant
0}\tilde{B}^{+}/\tilde{B}^{+}}$ $\displaystyle=$
$\displaystyle\bigcup_{\underline{w}\in
W^{n}}\tilde{\mathcal{B}}_{th(\underline{w}),\geqslant
0}=\\!\\!\\!\\!\\!\bigsqcup_{\stackrel{{\scriptstyle\tilde{v}\leqslant\tilde{w}\in\tilde{W},}}{{\tilde{w}\leqslant
th({\underline{w}})\text{ for some }\underline{w}\in
W^{n}}}}\\!\\!\\!\\!\\!\tilde{\mathcal{B}}_{\tilde{v},\tilde{w},>0}.$

Therefore we have

$\overline{U^{-}_{\geqslant 0}y_{\infty_{1}}(\mathbb{R}_{>0})U^{-}_{\geqslant
0}\dots y_{\infty_{n-1}}(\mathbb{R}_{>0})U^{-}_{\geqslant
0}\tilde{B}^{+}/\tilde{B}^{+}}\cap\mathcal{Z}^{\prime}=\bigsqcup_{v\leqslant
m_{\ast}(\underline{w})}\tilde{\mathcal{B}}_{i(v),th(\underline{w}),>0}.$
(3.4)

For any subset $S\subset\\{1,\dots,l\\}$, let
$\theta_{S}:\tilde{G}\rightarrow\tilde{G}$ be the unique group automorphism of
$\tilde{G}$ that is identity on $G$, $\tilde{T}$, and maps $x_{\infty_{l}}(a)$
to $x_{\infty_{l}}((-1)^{\delta_{l\in S}}a)$ and $y_{\infty_{l}}(a)$ to
$y_{\infty_{l}}((-1)^{\delta_{l\in S}}a)$ for all $1\leqslant l\leqslant n-1$.
Here $\delta_{l\in S}=1$ if $l\in S$ and $\delta_{l\in S}=0$ if $l\not\in S$.

The automorphism $\theta_{S}$ induces an automorphism on
$\mathcal{Z}^{\prime}$, denoted again by $\theta_{S}$. Applying the
automorphism $\theta_{S}$ to (3.4), we obtain

$\displaystyle\overline{\pi^{-1}(U^{-}_{\geqslant 0},U^{-}_{\geqslant
0},\dots,U^{-}_{\geqslant 0}B^{+}/B^{+})}\cap\mathcal{Z}^{\prime}$
$\displaystyle=$ $\displaystyle\,\,\overline{U^{-}_{\geqslant
0}y_{\infty_{1}}(\mathbb{R}^{\ast})U^{-}_{\geqslant 0}\dots
y_{\infty_{n-1}}(\mathbb{R}^{\ast})U^{-}_{\geqslant
0}\tilde{B}^{+}/\tilde{B}^{+}}\cap\mathcal{Z}^{\prime}$ $\displaystyle=$
$\displaystyle\bigsqcup_{v\leqslant
m_{\ast}(\underline{w})}\Big{(}\bigsqcup_{S\subset\\{1,\dots,l\\}}\theta_{S}\Big{(}\tilde{\mathcal{B}}_{i(v),th(\underline{w}),>0}\Big{)}\Big{)}.$

The projection $\pi:\mathcal{Z}^{\prime}\rightarrow\mathcal{Z}$ is invariant
with respect to the automorphism $\theta_{S}$. We obtain that

$\bigsqcup_{v\leqslant
v^{\prime},\underline{w^{\prime}}\leqslant\underline{w}\text{ with
}v^{\prime}\leqslant
m_{\ast}(\underline{w^{\prime}})}\mathcal{Z}_{v^{\prime},\underline{w^{\prime}},>0}\subset\overline{\mathcal{Z}_{v,\underline{w},>0}}.$

The reverse inclusion follows by Proposition 2.1. Part (2) is proved.

Since each
$\theta_{S}\Big{(}\tilde{\mathcal{B}}_{i(v),th(\underline{w}),>0}\Big{)}$ is a
connected component in
$\tilde{\mathcal{B}}_{i(v),th(\underline{w})}(\mathbb{R})$, we see that
$\mathcal{Z}_{v,\underline{w},>0}$ is a connected component of
$\mathring{\mathcal{Z}}_{v,\underline{w}}(\mathbb{R})$. This proves part (1).

We prove (3). Since $i(\underline{\mathbf{v}})$ is a positive subexpression of
$th(\underline{\mathbf{w}})$, we have the isomorphism by §2.4 (a)

$G_{i(\underline{\mathbf{v}}),th(\underline{\mathbf{w}}),>0}\xrightarrow{\sim}\mathring{\tilde{}\mathcal{B}}_{i(v),th(\underline{w}),>0}.$

It follows from the definition that

$G_{i(\underline{\mathbf{v}}),th(\underline{\mathbf{w}}),>0}\cong
G_{(\mathbf{v}_{1})_{+},\mathbf{w}_{1},>0}\times
y_{\infty_{1}}(\mathbb{R}_{>0})\times\cdots\times
y_{\infty_{n-1}}(\mathbb{R}_{>0})\times
G_{(\mathbf{v}_{n})_{+},\mathbf{w}_{n},>0}$

By [4, Lemma 5.1], we see that

$\tilde{\mathcal{B}}_{i(v),th(\underline{w}),>0}\subset\dot{r}_{1}U^{-}U^{*}_{-\alpha_{\infty_{1}}}\dot{r}_{2}U^{-}\cdots\dot{r}_{n}U^{-}/\tilde{B}^{+},\text{for
}\underline{r}\in W^{n}\text{ with
}\underline{r}\leqslant\underline{w},v\leqslant m_{\bullet}(\underline{r}).$

The section in (3.2) restricts to an isomorphism

$G_{(\mathbf{v}_{1})_{+},\mathbf{w}_{1},>0}\times
y_{\infty_{1}}(\mathbb{R}_{>0})\times\cdots\times
y_{\infty_{n-1}}(\mathbb{R}_{>0})\times
G_{(\mathbf{v}_{n})_{+},\mathbf{w}_{n},>0}\rightarrow\mathcal{Z}_{v,\underline{w},>0}\times\mathbb{R}^{n-1}_{>0}.$

Note that we have natural $\tilde{T}$-action on both sides and the isomorphism
is $\tilde{T}$-equivariant. Applying the argument in [4, Proposition 4.2]
$n-1$ times, we obtain the desired isomorphism in (3). ∎

##  4\. Regularity theorem

###  4.1. Some properties on posets

Let $(P,\leqslant)$ be a poset. For any $x,y\in P$ with $x\leqslant y$, the
interval from $x$ to $y$ is defined to be $[x,y]=\\{z\in P\mid x\leqslant
z\leqslant y\\}$. A subset $C\subset P$ is called convex if for any $x,y\in
C$, we have $[x,y]\subset C$.

The covering relation is denote by $\gtrdot$. For any $x,y\in P$ with
$x\leqslant y$, a maximal chain from $x$ to $y$ is a finite sequence of
elements $y=w_{0}\gtrdot w_{1}\gtrdot\cdots\gtrdot w_{n}=x$. In general, the
maximal chain may not exists.

The $P$ is called pure if for any $x,y\in P$ with $x\leqslant y$, the maximal
chains from $x$ to $y$ always exist and have the same length. A pure poset $P$
is called thin if every interval of length $2$ has exactly $4$ elements, i.e.
has exactly two elements between $x$ and $y$.

The order complex $\Delta_{ord}(P)$ of a finite poset $P$ is a simplicial
complex whose vertices are the elements of $P$ and whose simplices are the
chains $x_{0}<x_{1}<\cdots<x_{k}$ in $P$. A finite pure poset is called
shellable if its order complex $\Delta_{ord}(P)$ is shellable, that is, its
maximal faces can be order $F_{1},F_{2},\dots,F_{n}$ in such a way that
$F_{k}\cap\big{(}\cup_{i=1}^{k-1}F_{i}\big{)}$ is a nonempty union of maximal
proper faces of $F_{k}$ for $k=2,\dots,n$.

###  4.2. Regular CW complexes

Let $X$ be a Hausdorff space. We call a finite disjoint union
$X=\bigsqcup_{\alpha\in P}X_{\alpha}$ a regular CW complex if it satisfies the
following two properties.

  1. (1)

For each $\alpha\in P$, there exists a homeomorphism from a closed ball to
$\overline{X}_{\alpha}$ mapping the interior of the ball to $X_{\alpha}$.

  2. (2)

For each $\alpha$, there exists $P^{\prime}\subset P$, such that
$\overline{X}_{\alpha}=\bigsqcup_{\beta\in P^{\prime}}X_{\beta}$.

We then have a natural poset $({P},\leqslant)$, where $\beta\leqslant\alpha$
if and only if $X_{\beta}\subset\overline{X}_{\alpha}$. The face poset of $X$
is defined to be the augmented poset $\hat{P}=P\sqcup\\{\hat{0}\\}$, where
$\hat{0}$ is the added unique minimal element of $\hat{P}$.

One of the motivations for the combinatorial properties in §4.1 comes from the
following result in [2, Proposition 2.2].

(a) Let $\hat{0}\neq x\in\hat{P}$. If the the closed interval $[\hat{0},x]$ is
pure, thin, and shellable, then it is the face poset of some regular CW
complex which is homeomorphic to a closed ball.

We also recall the following result from [2] which will be used in Theorem
4.6.

(b) Suppose that $X$ is a regular CW complex with (finite) face poset
$\hat{P}$. If $\hat{P}\sqcup\\{\hat{1}\\}$ (adjoining a maximum $\hat{1}$) is
graded, thin, and shellable, then $X$ is homeomorphic to a sphere of dimension
rank$(P)-1$.

###  4.3. Index set of the stratification of $\mathcal{Z}$

Let

$Q=\\{(v,\underline{w})\in W\times W^{n};v\leqslant
m_{\ast}(\underline{w})\\}.$

The partial order on $Q$ is defined by
$(v^{\prime},\underline{w^{\prime}})\leqslant(v,\underline{w})$ if $v\leqslant
v^{\prime}$ and $\underline{w^{\prime}}\leqslant\underline{w}$. The augmented
poset $\hat{Q}=Q\sqcup\\{\hat{0}\\}$ is the face poset of $\mathcal{Z}$. Now
we state the main result of this section.

######  Theorem 4.1.

The augmented poset $\hat{Q}$ is pure, thin. For any $x\in\hat{Q}$, the
interval $[\hat{0},x]$ is shellable.

###### Proof.

Purity is clear. We define

$\tilde{Q}=\\{(x,y)\in\tilde{W}^{op}\times\tilde{W};x\leqslant y\\}.$

The partial order on $\tilde{Q}$ is defined by
$(x^{\prime},y^{\prime})\leqslant(x,y)$ if $x\leqslant x^{\prime}\leqslant
y^{\prime}\leqslant y$. Define the map

$f:Q\tilde{Q},\qquad(v,\underline{w})(i(v),th(\underline{w})).$

The map $f$ identifies $Q$ with a sub-poset of $\tilde{Q}$. Note that the
image of $f$ is

$\text{Im}(f)=\\{(x,y)\in Q;x\in W,y\in Ws_{\infty_{1}}Ws_{\infty_{2}}W\cdots
s_{\infty_{n-1}}W\\}.$

It is easy to see that $\text{Im}(f)$ is a convex subset of $\tilde{Q}$.

We next apply [7] to the Coxeter group $\tilde{W}\times\tilde{W}$ with respect
to the reflection order $\leqslant_{A}$, where $A$ is the set of reflections
in $\tilde{W}\times\\{e\\}$. It follows from the definition in [7, §1] that
$(x^{\prime},y^{\prime})\leqslant_{A}(x,y)$ if and only if $x\leqslant
x^{\prime}$ and $y^{\prime}\leqslant y$. Hence $\tilde{Q}$ is a convex subset
of $(\tilde{W}\times\tilde{W},\leqslant_{A})$. Therefore by [7, Proposition
2.5], $\tilde{Q}$ is thin, and by [7, Proposition 3.9 with
$J^{\prime}=I=\emptyset$] $\tilde{Q}$ is EL-shellable in the sense of [1,
Definition 2.1 & 2.2]. In particular, any convex subset of $\tilde{Q}$ is EL-
shellable, hence shellable by [1, Theorem 2.3]

We conclude that $Q\cong\text{Im}(f)$ is thin and any convex subset of $Q$ is
shellable. For any $x\in{Q}$, since the convex set $\\{y\in Q|y\leqslant x\\}$
is shellable, the augmented poset set $[\hat{0},x]$ is shellable following the
definition.

It remains to consider the intervals of length $2$ with the minimal element
$\hat{0}$. Suppose that the maximal element in this interval is
$(v,\underline{w})$. Then $\ell(\underline{w})-\ell(v)=1$.

Let $\underline{w}=(w_{1},w_{2},\dots,w_{n})$ and $\mathbf{w}_{i}$ be a
reduced expression of $w_{i}$ for each $i$. We write
$(\mathbf{w}_{1},\mathbf{w}_{2},\dots,\mathbf{w}_{n})$ as
$(s_{1},\ldots,s_{m})$. Then there exist $1\leqslant l\leqslant m$ such that
$v=s_{1}\cdots\hat{s}_{l}\cdots s_{m}$. We show that

(a) $\sharp\\{l;v=s_{1}\cdots\hat{s}_{l}\cdots s_{m}\\}=1$ or $2$.

Suppose that there exists $l_{1}<l_{2}$ such that
$v=s_{1}\cdots\hat{s}_{l_{1}}\cdots s_{m}=s_{1}\cdots\hat{s}_{l_{2}}\cdots
s_{m}$. Then we have $s_{l_{1}}\cdots s_{l_{2}-1}=s_{l_{1}+1}\cdots
s_{l_{2}}$. Thus

$s_{1}\cdots s_{m}=s_{1}\cdots\hat{s}_{l_{1}}\cdots\hat{s}_{l_{2}}\cdots
s_{m}<v.$

Also we have

$\displaystyle v=m_{\ast}(\underline{w})$ $\displaystyle=s_{1}\ast\cdots\ast
s_{l_{1}}\ast\cdots\ast s_{l_{2}}\ast\cdots\ast s_{m}$
$\displaystyle=s_{1}\ast\cdots\ast\hat{s}_{l_{1}}\ast\cdots\ast
s_{l_{2}}\ast\cdots\ast s_{m}.$

Hence if there exists $l_{1}<l_{2}<l_{3}$ such that
$v=s_{1}\cdots\hat{s}_{l_{j}}\cdots s_{m}$ for $j=1,2,3$. Then we have

$\displaystyle v=m_{\ast}(\underline{w})$ $\displaystyle=s_{1}\ast\cdots\ast
s_{l_{1}}\ast\cdots\ast s_{l_{2}}\ast\cdots\ast s_{l_{3}}\ast\cdots\ast s_{m}$
$\displaystyle=s_{1}\ast\cdots\ast\hat{s}_{l_{1}}\ast\cdots\ast
s_{l_{2}}\ast\cdots\ast s_{l_{3}}\ast\cdots\ast s_{m}$
$\displaystyle=s_{1}\ast\cdots\ast\hat{s}_{l_{1}}\ast\cdots\ast\hat{s}_{l_{2}}\ast\cdots\ast
s_{l_{3}}\ast\cdots\ast s_{m}.$

In particular, $\ell(v)\leqslant m-2$. That is a contradiction. (a) is proved.
We now analyze the two cases separately.

Case (1): $\\{l;v=s_{1}\cdots\hat{s}_{l}\cdots s_{m}\\}=\\{l_{1}\\}$.

The expression $(s_{1},\ldots,\hat{s}_{l_{1}},\ldots,s_{m})$ define one
element $\underline{w_{1}}\in W^{n}$ with $\underline{w_{1}}<\underline{w}$.
It is also easy to see that $s_{1}\ast\cdots\ast
s_{m}>v=m_{\ast}(\underline{w_{1}})=s_{1}\ast\cdots\ast\hat{s}_{l_{1}}\ast\cdots\ast
s_{m}$. In this case, there are exactly two elements between $\hat{0}$ and
$(v,\underline{w})$. They are $(v,\underline{w_{1}})$ and
$(m_{\ast}(\underline{w}),\underline{w})$.

Case (2): $\\{l;v=s_{1}\cdots\hat{s}_{l}\cdots s_{m}\\}=\\{l_{1},l_{2}\\}$.

The expressions $(s_{1},\ldots,\hat{s}_{l_{1}},\ldots,s_{m})$ and
$(s_{1},\ldots,\hat{s}_{l_{2}},\ldots,s_{m})$ define two elements
$\underline{w_{1}},\underline{w_{2}}\in W^{n}$ with
$\underline{w_{1}},\underline{w_{2}}<\underline{w}$. In this case, there are
exactly two elements between $\hat{0}$ and $(v,\underline{w})$. They are
$(v,\underline{w_{1}})$ and $(v,\underline{w_{2}})$.

We have now finished the proof of thinness of $\hat{Q}$, hence the theorem. ∎

######  Remark 4.2.

EL-shellability is stronger than shellability for a poset. It is interesting
to see if $\hat{Q}$ is EL-shellable. The special case for $n=1$ was
established by Williams in [27].

###  4.4. Stratified isomorphisms

By [4, Lemma 5.1 & Theorem 5.3], we have the following commutative diagram for
any $u\in\tilde{W}$

${\tilde{\mathcal{B}}_{\geqslant
0}\cap\dot{u}\tilde{U}^{-}\tilde{B}^{+}}$${\bigsqcup_{v\leqslant u\leqslant
w}\tilde{\mathcal{B}}_{v,w,>0}}$${\tilde{\mathcal{B}}_{v,w,>0}}$${\mathring{\tilde{\mathcal{B}}}_{u,\geqslant
0}\times\mathring{\tilde{\mathcal{B}}}^{u}_{\geqslant
0}}$${\bigsqcup_{v\leqslant
u}\tilde{\mathcal{B}}_{v,u,>0}\times\bigsqcup_{u\leqslant
w}\tilde{\mathcal{B}}_{u,w>0}}$${\tilde{\mathcal{B}}_{v,u,>0}\times\tilde{\mathcal{B}}_{u,w>0}.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
(4.1)

Here $\mathring{\tilde{\mathcal{B}}}_{u,\geqslant
0}=\mathring{\tilde{\mathcal{B}}}_{u}\cap\tilde{\mathcal{B}}_{\geqslant 0}$
and $\mathring{\tilde{\mathcal{B}}}^{u}_{\geqslant
0}=\mathring{\tilde{\mathcal{B}}}^{u}\cap\tilde{\mathcal{B}}_{\geqslant 0}$.
We have the following similar result for the twisted product.

######  Proposition 4.3.

Let $(u,\underline{x})\in Q$ be a minimal element. We have the following
commutative diagram

${\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\bigsqcup_{v^{\prime}\leqslant
u,\underline{x}\leqslant\underline{w^{\prime}}\text{ with }v^{\prime}\leqslant
m_{\ast}(\underline{w^{\prime}})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}},>0}}$${\displaystyle\bigsqcup_{v^{\prime}\leqslant
u}\tilde{\mathcal{B}}_{i(v^{\prime}),i(u),>0}\times\bigsqcup_{\underline{x}\leqslant\underline{w^{\prime}}}\tilde{\mathcal{B}}_{th(\underline{x}),th(\underline{w^{\prime}}),>0}}$${\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}},>0}}$${\tilde{\mathcal{B}}_{i(v^{\prime}),i(u),>0}\times\tilde{\mathcal{B}}_{th(\underline{x}),th(\underline{w^{\prime}}),>0}.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

######  Remark 4.4.

By Theorem 3.4, the left hand side is open in $\mathcal{Z}_{\geqslant 0}$. By
Proposition 2.1, the right hand side is locally closed in
$\tilde{\mathcal{B}}\times\tilde{\mathcal{B}}$.

###### Proof.

Let $\underline{x}=(x_{1},\dots,x_{n})$ and
$\tilde{x}=th(\underline{x})\in\tilde{W}$. Note that

$\displaystyle\tilde{\mathcal{B}}_{\geqslant
0}\cap\dot{\tilde{x}}\tilde{U}^{-}\tilde{B}^{+}\cap\mathcal{Z}^{\prime}=\bigsqcup_{v\leqslant
m_{\ast}(\underline{x}),\underline{x}\leqslant\underline{w}}\tilde{\mathcal{B}}_{i(v),th(\underline{w}),>0}$
$\displaystyle\subset\dot{x}_{1}U^{-}U^{*}_{-\alpha_{\infty_{1}}}\dot{x}_{2}U^{-}U^{*}_{-\alpha_{\infty_{2}}}\cdots\dot{x}_{n}U^{-}\tilde{B}^{+}/\tilde{B}^{+}.$

By (3.2), we have a section

$\displaystyle\,\,\prod_{i=1}^{n}U_{-\alpha_{\infty_{i}}}^{*}(\mathbb{R}_{>0})\times\\!\\!\\!\\!\bigsqcup_{v^{\prime}\leqslant
u,\underline{x}\leqslant\underline{w}}\\!\\!\\!\\!\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}},>0}\rightarrow\bigsqcup_{v\leqslant
m_{\ast}(\underline{x}),\underline{x}\leqslant\underline{w}}\tilde{\mathcal{B}}_{i(v),th(\underline{w}),>0}.$

By (4.1), we further have the embedding

$\prod_{i=1}^{n}U_{-\alpha_{\infty_{i}}}^{*}(\mathbb{R}_{>0})\times\\!\\!\\!\\!\bigsqcup_{v^{\prime}\leqslant
u,\underline{x}\leqslant\underline{w}}\\!\\!\\!\\!\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}},>0}\rightarrow\mathring{\tilde{\mathcal{B}}}_{i(u),\geqslant
0}\times\mathring{\tilde{\mathcal{B}}}^{i(u)}_{\geqslant 0}.$

Applying (4.1) again, we obtain that

$\displaystyle\displaystyle\prod_{i=1}^{n}U_{-\alpha_{\infty_{i}}}^{*}(\mathbb{R}_{>0})\times\\!\\!\\!\\!\bigsqcup_{v^{\prime}\leqslant
u,\underline{x}\leqslant\underline{w}}\\!\\!\\!\\!\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}},>0}\rightarrow\mathring{\tilde{\mathcal{B}}}_{i(u),\geqslant
0}\times(\mathring{\tilde{\mathcal{B}}}^{i(u)}_{\geqslant
0}\cap\dot{\tilde{x}}\tilde{U}^{-}\tilde{B}^{+})$ (4.2)
$\displaystyle\rightarrow\mathring{\tilde{\mathcal{B}}}_{i(u),\geqslant
0}\times\tilde{\mathcal{B}}_{i(u),th(\underline{x}),>0}\times\mathring{\tilde{\mathcal{B}}}^{th(\underline{x})}_{\geqslant
0}\cong\mathring{\tilde{\mathcal{B}}}_{i(u),\geqslant
0}\times\mathbb{R}_{>0}^{n-1}\times\mathring{\tilde{\mathcal{B}}}^{th(\underline{x})}_{\geqslant
0}.$

Note that
$\tilde{\mathcal{B}}_{i(u),th(\underline{x}),>0}\cong\mathbb{R}^{n-1}_{>0}$
since $(u,\underline{x})\in Q$ is minimal. The composition in (4.2) is a
$\tilde{T}$-equivariant isomorphism. The proposition follows by applying the
argument in [4, Proposition 4.2] repeatedly. ∎

###  4.5. Regularity theorem

Recall that an $n$-dimension topological manifold with boundary is a Hausdorff
space $X$ such that every point $x\in X$ has an open neighborhood homeomorphic
to either $\mathbb{R}^{n}$, or $\mathbb{R}_{\geqslant
0}\times\mathbb{R}^{n-1}$ mapping $x$ to a point in
$\\{0\\}\times\mathbb{R}^{n-1}$. In the latter case, we say that $x$ is on
$\partial X$, the boundary of $X$.

The following statement can be derived from the generalized Poincaré
conjecture and Brown’s collar theorem. We refer to [9, §3.2] for details.

######  Theorem 4.5.

Let $X$ be a compact $n$-dimensional topological manifold with boundary, such
that $\partial X$ is homeomorphic to an $(n-1)$-dimensional sphere and
$X-\partial X$ is homeomorphic to an $n$-dimensional open ball. Then $X$ is
homeomorphic to an $n$-dimensional closed ball.

Combining Theorem 4.5 with Theorem 4.1, we have the following regularity
theorem.

######  Theorem 4.6.

Let $v\in W$ and $\underline{w}\in W\times W$ with $v\leqslant
m_{\ast}(\underline{w})$. Then $\mathcal{Z}_{v,\underline{w},\geqslant 0}$ is
a regular CW complex homeomorphic to a closed ball.

###### Proof.

It follows from Proposition 4.3 and [4, §5.1] that
$\mathcal{Z}_{v,\underline{w},\geqslant 0}$ is a topological manifold with
boundary

$\partial\mathcal{Z}_{v,\underline{w},\geqslant
0}=\bigsqcup_{(v^{\prime},\underline{w^{\prime}})<(v,\underline{w})\text{ in
}Q}\mathring{\mathcal{Z}}_{v^{\prime},\underline{w^{\prime}},>0}.$

We prove the theorem by induction on $\ell(\underline{w})-\ell(v)$. The
statement is clear when $\ell(\underline{w})-\ell(v)=0$, since
$\mathcal{Z}_{v,\underline{w},\geqslant 0}$ is a single point in this case.

Assume $\partial\mathcal{Z}_{v,\underline{w},\geqslant 0}$ is a regular CW
complex with face poset

$Q^{\prime}=\\{\alpha\in\hat{Q}|\alpha<(v,\underline{w})\\}.$

Adding a maximal element $\hat{1}$, we have
$Q^{\prime}\sqcup\\{\hat{1}\\}\cong[\hat{0},(v,\underline{w})]\subset\hat{Q}$
as posets. By §4.2 (b) and Theorem 4.1,
$\partial\mathcal{Z}_{v,\underline{w},\geqslant 0}$ is a regular CW complex
homeomorphic to a sphere of dimension $\ell(\underline{w})-\ell({v})$. Now the
theorem follows from Theorem 4.5. ∎

##  5\. Applications

In this section, we consider various applications of main results in this
paper.

###  5.1. Double flag varieties

Let $\mathcal{Z}=\mathcal{Z}_{2}=G\times^{B^{+}}G/B^{+}$ in this subsection.

#### 5.1.1.

We have the natural $G\times G$-action on $\mathcal{B}\times\mathcal{B}$. By
§2.2, we have the decomposition of $\mathcal{B}\times\mathcal{B}$ into the
$B^{+}\times B^{-}$-orbits

$\mathcal{B}\times\mathcal{B}=\sqcup_{v,w\in
W}\mathring{\mathcal{B}}_{v}\times\mathring{\mathcal{B}}^{w}.$ (5.1)

Let $G_{\Delta}=\\{(g,g);g\in G\\}\subset G\times G$. Following [26], for
$(u,v,w)\in W^{3}$, we consider the intersection of the $G_{\Delta}$-orbit and
the $B^{+}\times B^{-}$-orbit

$\mathring{\mathcal{X}}^{u}_{v,w}=(G_{\Delta}\cdot(B^{+}/B^{+},\dot{u}B^{+}/B^{+}))\cap(\mathring{\mathcal{B}}_{v}\times\mathring{\mathcal{B}}^{w}).$

We have a decomposition of $\mathcal{B}\times\mathcal{B}$ as $\sqcup_{u,v,w\in
W}\mathring{\mathcal{X}}^{u}_{v,w}$. We denote this stratified space by
$\mathcal{X}$ to emphasize that it is different from the stratified space in
(5.1).

The isomorphism $G\times GG\times G$ defined by
$(g_{1},g_{2})(g_{1},g_{1}g_{2})$ induces an isomorphism of stratified spaces

$\alpha:\mathcal{Z}\mathcal{X},\qquad(g_{1},g_{2}B^{+}/B^{+})(g_{1}B^{+}/B^{+},g_{1}g_{2}B^{+}/B^{+}),$
(5.2)

mapping $\mathring{\mathcal{Z}}_{v,\underline{w}}$ isomorphically to
$\mathring{\mathcal{X}}_{w_{1},v}^{w_{2}}$.

#### 5.1.2.

Since $\mathring{\mathcal{X}}^{u}_{v,w}$ is irreducible by Corollary 3.3 and
(5.2), there is a unique $B^{-}\times B^{+}$-orbit
$\mathring{\mathcal{B}}^{v^{\prime}}\times\mathring{\mathcal{B}}_{w^{\prime}}$
such that
$(\mathring{\mathcal{B}}^{v^{\prime}}\times\mathring{\mathcal{B}}_{w^{\prime}})\cap\mathring{\mathcal{X}}^{u}_{v,w}$
is dense in $\mathring{\mathcal{X}}^{u}_{v,w}$.

Recall the isomorphism $\alpha:\mathcal{Z}\rightarrow\mathcal{X}$ in (5.2). We
define $\mathcal{X}_{\geqslant 0}=\alpha(\mathcal{Z}_{\geqslant 0})$ and
$\mathcal{X}^{w_{2}}_{w_{1},v,>0}=\mathcal{X}^{w_{2}}_{w_{1},v}\cap\mathcal{X}_{\geqslant
0}=\alpha(\mathcal{Z}_{v,(w_{1},w_{2}),>0})$. By [11, §1], for any $x,y\in W$,
the subset $\\{yu\in W|u\leqslant x\\}$ has a unique minimal element. We
denote the unique minimal element by $y\circ_{r}x$.

######  Proposition 5.1.

Let $v^{\prime}=w\circ_{r}u^{-1}$ and $w^{\prime}=v\ast u$. Then

(1)
$\mathcal{X}^{u}_{v,w,>0}\subset\mathring{\mathcal{X}}^{u}_{v,w}\cap(\mathring{\mathcal{B}}^{v^{\prime}}\times\mathring{\mathcal{B}}_{w^{\prime}})$;

(2)
$\mathring{\mathcal{X}}^{u}_{v,w}\cap(\mathring{\mathcal{B}}^{v^{\prime}}\times\mathring{\mathcal{B}}_{w^{\prime}})$
is dense in $\mathring{\mathcal{X}}^{u}_{v,w}$.

######  Remark 5.2.

Part (1) arises from a question Lusztig asked about the totally positive big
cells in the double flag for reductive groups.

###### Proof.

Let $\underline{w}=(w_{1},w_{2})$ be such that $w=w_{1}w_{2}$ and
$\underline{w}$ is a positive subexpression in $(v,u)$. Let ${\mathbf{v}}$
(resp. ${\mathbf{u}}$) be reduced expressions of $v$ (resp. $u$). Let
$({\mathbf{w}_{1}})_{+}$ (resp. $({\mathbf{w}_{2}})_{+}$) be the positive
subexpression of $w_{1}$ (resp. $w_{2}$) in ${\mathbf{v}}$ (resp.
${\mathbf{u}}$). By §3.2 (b), we have $v^{\prime}=w_{1}$. By Theorem 3.4, we
have the isomorphism

$G_{(\mathbf{w}_{1})_{+},\mathbf{v},>0}\times
G_{(\mathbf{w}_{2})_{+},\mathbf{u},>0}\mathcal{X}^{u}_{v,w,>0},\quad(g_{1},g_{2})(g_{1}B^{+}/B^{+},g_{1}g_{2}B^{+}/B^{+}).$

This proves (1).

We now prove (2). Let $a,b\in W$ be such that
$\mathring{\mathcal{X}}^{u}_{v,w}\cap(\mathring{\mathcal{B}}^{a}\times\mathring{\mathcal{B}}_{b})\neq\emptyset$.
Thanks to (1), it suffices to show $v^{\prime}\leqslant a$ and $b\leqslant
w^{\prime}$.

We first have

$G_{\Delta}\cdot(B^{+}/B^{+},\dot{u}B^{+}/B^{+})\cap(\mathring{\mathcal{B}}^{a}\times\mathring{\mathcal{B}}^{w})\neq\emptyset.$

Hence
$(B^{-}\dot{w}B^{+}/B^{+})\cap(\dot{a}B^{+}\dot{u}B^{+}/B^{+})\neq\emptyset$.
Note that $\dot{a}B^{+}\dot{u}B^{+}/B^{+}\subset\sqcup_{x\leqslant a\ast
u}B^{+}xB^{+}/B^{+}$. Hence $w\leqslant a\ast u$, or equivalently
$w\circ_{r}u^{-1}\leqslant a$ by a proof similar to [12, Lemma 4.3].

Similarly, we have

$G_{\Delta}\cdot(B^{+}/B^{+},\dot{u}B^{+}/B^{+})\cap(\mathring{\mathcal{B}}_{v}\times\mathring{\mathcal{B}}_{b})\neq\emptyset.$

Hence
$(B^{+}\dot{b}B^{+}/B^{+})\cap(\dot{v}B^{+}\dot{u}B^{+}/B^{+})\neq\emptyset$.
Hence $b\leqslant v\ast u$. ∎

#### 5.1.3.

We assume that $G$ is a connected reductive group. Let $w_{0}$ be the longest
element of the finite Weyl group $W$. By [19, Theorem 8.7], we have
$U^{-}_{>0}B^{+}/B^{+}=U^{+}_{>0}\dot{w}_{0}B^{+}/B^{+}$. Thus

$\alpha(U^{-}_{>0},U^{-}_{>0}B^{+}/B^{+})=\alpha(U^{-}_{>0},U^{+}_{>0}\dot{w}_{0}B^{+}/B^{+})=(G_{>0})_{\Delta}\cdot(B^{+}/B^{+},\dot{w}_{0}B^{+}/B^{+}).$

We may naturally identify $\mathcal{X}$ with $G/B^{+}\times G/B^{-}$. Then
$\mathcal{X}_{\geqslant 0}$ is identified with the Hausdorff closure of
$(G_{>0})_{\Delta}\cdot(B^{+}/B^{+},B^{-}/B^{-})$ in $G/B^{+}\times G/B^{-}$.
The later space was considered in [26].

Now we may reformulate Theorem 3.4 for $\mathcal{Z}_{\geqslant 0}$ as

(a) The subspace $\mathcal{X}_{\geqslant 0}$ of $\mathcal{X}$ gives a totally
positive structure for the stratified space
$\mathcal{X}=\bigsqcup\mathcal{X}^{u}_{v,w}$.

In particular, we verify a conjecture of Webster and Yakimov [26, Conjecture
1].

###  5.2. Links on double Bruhat cells

We still write $\mathcal{Z}=\mathcal{Z}_{2}$ in this subsection.

#### 5.2.1.

We define the link of the cell
$\mathcal{Z}_{v^{\prime},\underline{w}^{\prime},>0}$ in the space
$\mathcal{Z}_{v,\underline{w},\geqslant 0}$ for any
$(v^{\prime},\underline{w}^{\prime})<(v,\underline{w})\in Q$. Our definition
of links is similar to [9, §3.1].

We first define a closed subspace of $\mathcal{Z}_{\geqslant 0}$ as follows

${\rm
Star}_{\geqslant(v^{\prime},\underline{w}^{\prime})}=\bigsqcup_{(v^{\prime},\underline{w}^{\prime})\leqslant(v^{\prime\prime},\underline{w}^{\prime\prime})\in
Q}\mathcal{Z}_{(v^{\prime\prime},\underline{w}^{\prime\prime}),>0}.$

Let $(u,\underline{u_{1}})\in Q$ be a minimal element such that
$(u,\underline{u_{1}})\leqslant(v^{\prime},\underline{w}^{\prime})$. By
Proposition 4.3, we have the following isomorphism via restriction

$\phi:{\rm
Star}_{\geqslant(v^{\prime},\underline{w}^{\prime})}\rightarrow\bigsqcup_{v^{\prime\prime}\leqslant
v^{\prime}}\tilde{\mathcal{B}}_{i(v^{\prime\prime}),i(u),>0}\times\bigsqcup_{\underline{w}^{\prime}\leqslant\underline{w}^{\prime\prime}}\tilde{\mathcal{B}}_{th(\underline{u_{1}}),th(\underline{w}^{\prime\prime}),>0}$

Applying (4.1) again, we obtain the isomorphism

${\rm
Star}_{\geqslant(i(v^{\prime}),\underline{w}^{\prime})}\rightarrow\tilde{\mathcal{B}}_{i(v^{\prime}),i(u),>0}\times\tilde{\mathcal{B}}_{th(\underline{u_{1}}),th(\underline{w}^{\prime}),>0}\times\bigsqcup_{v^{\prime\prime}\leqslant
v^{\prime}}\tilde{\mathcal{B}}_{i(v^{\prime\prime}),i(v^{\prime}),>0}\times\bigsqcup_{\underline{w}^{\prime}\leqslant\underline{w}^{\prime\prime}}\tilde{\mathcal{B}}_{th(\underline{w}^{\prime}),th(\underline{w}^{\prime\prime}),>0}.$
(5.3)

We recall some representation theoretical notations. Denote by
$\tilde{X}^{++}$ the set of regular dominant weights of $\tilde{G}$. For
$\lambda\in\tilde{X}^{++}$, we denote by $V^{\lambda}$ the highest weight
simple $\tilde{G}$-module over $\mathbb{C}$ with highest weight $\lambda$. We
denote the highest weight vector by $\eta_{\lambda}$. We denote by
$V^{\lambda}(\mathbb{R})$ the $\mathbb{R}$-subspace of $V^{\lambda}$ spanned
by the canonical basis. We equip
$V^{\lambda,\lambda}(\mathbb{R})=V^{\lambda}(\mathbb{R})\otimes_{\mathbb{R}}V^{\lambda}(\mathbb{R})$
with the standard Euclidean norm with respect to the tensor product of
canonical bases, denoted by $||\cdot||$.

We continue from (5.3). We consider the projection

$\displaystyle\tilde{\mathcal{B}}_{i(v^{\prime}),i(u),>0}\times\tilde{\mathcal{B}}_{th(\underline{u_{1}}),th(\underline{w}^{\prime}),>0}\times\bigsqcup_{v^{\prime\prime}\leqslant
v^{\prime}}\tilde{\mathcal{B}}_{i(v^{\prime\prime}),i(v^{\prime}),>0}\times\bigsqcup_{\underline{w}^{\prime}\leqslant\underline{w}^{\prime\prime}}\tilde{\mathcal{B}}_{th(\underline{w}^{\prime}),th(\underline{w}^{\prime\prime}),>0}$
(5.4) $\displaystyle\rightarrow$
$\displaystyle\bigsqcup_{v^{\prime\prime}\leqslant
v^{\prime}}\tilde{\mathcal{B}}_{i(v^{\prime\prime}),i(v^{\prime}),>0}\times\bigsqcup_{\underline{w}^{\prime}\leqslant\underline{w}^{\prime\prime}}\tilde{\mathcal{B}}_{th(\underline{w}^{\prime}),th(\underline{w}^{\prime\prime}),>0}.$

For any $\tilde{v},\tilde{w}\in\tilde{W}$, we have a natural embedding

$\displaystyle\pi^{\prime}:\mathring{\tilde{\mathcal{B}}}^{\tilde{w}}(\mathbb{R})\times\mathring{\tilde{\mathcal{B}}}_{\tilde{v}}(\mathbb{R})$
$\displaystyle\xrightarrow{((\dot{\tilde{w}})^{-1},(\dot{\tilde{v}})^{-1})\cdot-}\tilde{U}^{-}(\mathbb{R})\tilde{B}^{+}/\tilde{B}^{+}\times\tilde{U}^{-}(\mathbb{R})\tilde{B}^{+}/\tilde{B}^{+}$
(5.5) $\displaystyle\xrightarrow{(u_{1},u_{2})u_{1}\eta_{\lambda}\otimes
u_{2}\eta_{\lambda}}V^{\lambda,\lambda}(\mathbb{R}).$

We take $\tilde{v}=i(v^{\prime})$ and $\tilde{w}=th(\underline{w^{\prime}})$
in (5.5), and denote the composition of (5.3), (5.4) and (5.5) by $\pi:{\rm
Star}_{\geqslant(v^{\prime},\underline{w}^{\prime})}\rightarrow
V^{\lambda,\lambda}(\mathbb{R})$.

For any
$(v^{\prime},\underline{w}^{\prime})<(v^{\prime\prime},\underline{w}^{\prime\prime})\leqslant(v,\underline{w})\in
Q$, we define the link space as

$\displaystyle
Lk_{(v^{\prime},\underline{w}^{\prime})}(\mathcal{Z}_{v,\underline{w},\geqslant
0})$ $\displaystyle=\pi({\rm
Star}_{\geqslant(v^{\prime},\underline{w}^{\prime})}\cap\mathcal{Z}_{v,\underline{w},\geqslant
0})\bigcap\\{x\in
V^{\lambda,\lambda}(\mathbb{R})|\,||x-\eta_{\lambda}\otimes\eta_{\lambda}||=1\\};$
$\displaystyle
Lk_{(v^{\prime},\underline{w}^{\prime}),(v^{\prime\prime},\underline{w}^{\prime\prime}),>0}$
$\displaystyle=Lk_{(v^{\prime},\underline{w}^{\prime})}(\mathcal{Z}_{v,\underline{w},\geqslant
0})\cap\pi({\rm
Star}_{\geqslant(v^{\prime},\underline{w}^{\prime})}\cap\mathcal{Z}_{v^{\prime\prime},\underline{w}^{\prime\prime},>0}).$

We hence have a stratified space

$Lk_{(v^{\prime},\underline{w}^{\prime})}(\mathcal{Z}_{v,\underline{w},\geqslant
0})=\bigsqcup_{(v^{\prime},\underline{w}^{\prime})<(v^{\prime\prime},\underline{w}^{\prime\prime})\leqslant(v,\underline{w})\in
Q}Lk_{(v^{\prime},\underline{w}^{\prime}),(v^{\prime\prime},\underline{w}^{\prime\prime}),>0}.$

It follows from the construction that the definition of
$Lk_{(v^{\prime},\underline{w}^{\prime})}(\mathcal{Z}_{v,\underline{w},\geqslant
0})$ is independent of the choice of the minimal element
$(u,\underline{u_{1}})\in Q$ and the regular dominant weight $\lambda$, up to
a stratified isomorphism.

######  Proposition 5.3.

Let $(v^{\prime},\underline{w}^{\prime})<(v,\underline{w})\in Q$. Then
$Lk_{(v^{\prime},\underline{w}^{\prime})}(\mathcal{Z}_{v,\underline{w},\geqslant
0})$ is a regular CW complex homeomorphic to a closed ball.

###### Proof.

Let

$\displaystyle S=\\{x\in
V^{\lambda,\lambda}(\mathbb{R})|\,||x-\eta_{\lambda}\otimes\eta_{\lambda}||=1\\},\quad
R=\tilde{\mathcal{B}}^{i(v)}_{\geqslant
0}\times\tilde{\mathcal{B}}_{th(\underline{w}),\geqslant
0}\subset\tilde{\mathcal{B}}_{\geqslant 0}\times\tilde{\mathcal{B}}_{\geqslant
0},$ $\displaystyle T^{\prime}=\bigsqcup_{v^{\prime\prime}\leqslant
v^{\prime}}\tilde{\mathcal{B}}_{i(v^{\prime\prime}),i(v^{\prime}),>0}\times\bigsqcup_{\underline{w}^{\prime}\leqslant\underline{w}^{\prime\prime}}\tilde{\mathcal{B}}_{th(\underline{w}^{\prime}),th(\underline{w}^{\prime\prime}),>0}\subset\tilde{\mathcal{B}}_{\geqslant
0}\times\tilde{\mathcal{B}}_{\geqslant 0},\quad T=T^{\prime}\cap R.$

It follows from the construction that
$Lk_{(v^{\prime},\underline{w}^{\prime}),(v,\underline{w}),\geqslant 0}$ is
isomorphic to $\pi^{\prime}(T)\cap S$ as stratified space. The space
$\pi^{\prime}(T)\cap S$ is a special case of links considered in [4, §4]
related with the flag variety of the Kac-Moody group
$\tilde{G}\times\tilde{G}$.

By [4, Proposition 4.4], $\pi^{\prime}(T)\cap S$ is a regular CW complex
homeomorphic to a closed ball. The proposition follows now. ∎

#### 5.2.2.

We assume $G$ is a connected reductive group. We denote by $w_{0}$ the longest
element in the Weyl group of $G$.

For any $v,w\in W$, we define the double Bruhat cell
$G^{v,w}=B^{+}\dot{w}B^{+}\cap B^{-}\dot{v}B^{-}$. Let $\pi:G\rightarrow
G/T=L$ be the quotient map. We define the reduced double Bruhat cells as
$L^{v,w}=\pi(G^{v,w})$. By [26, Proposition 2.1], we have the stratified
embedding

$f:L\rightarrow\mathcal{B}\times\mathcal{B}\cong\mathcal{Z},\quad
gT(gB^{+}/B^{+},g\dot{w}_{0}B^{+}/B^{+})(g,\dot{w}_{0}B^{+}/B^{+}),$

mapping $L^{u,v}$ isomorphically to
$\mathring{\mathcal{Z}}_{{ww_{0}},(v,w_{0})}$ We further define $L_{\geqslant
0}=\pi(G_{\geqslant 0})$. It is clear that $L_{\geqslant
0}=\sqcup_{v,w}L^{v,w}_{>0}$ with $L^{v,w}_{>0}=L^{v,w}\cap L_{\geqslant 0}$.

Let ${\mathbf{v}}=(s_{i_{1}},\cdots,s_{i_{m}})$ and
${\mathbf{w}}=(s_{j_{1}},\cdots,s_{j_{n}})$ be reduced expressions of $v$ and
$w$, respectively. It follows by [19] that the map

$\displaystyle\mathbb{R}_{>0}^{m+n}$ $\displaystyle\longrightarrow
L^{v,w}_{>0},$ (5.6) $\displaystyle(a_{1},\dots,a_{n},b_{1},\dots,b_{m})$
$\displaystyle y_{j_{1}}(a_{1})\cdots y_{j_{n}}(a_{n})x_{i_{1}}(b_{1})\cdots
x_{i_{m}}(b_{m}).$

is an isomorphism.

######  Lemma 5.4.

We have $L_{\geqslant 0}=\overline{\pi(G_{>0})}$ in $L$.

###### Proof.

Since $\overline{G_{>0}}=G_{\geqslant 0}$, we have $L_{\geqslant
0}\subset\overline{\pi(G_{>0})}$. We have the following commutative diagram

${L_{\geqslant
0}}$${\sqcup_{v,w}L^{v,w}_{>0}}$${L^{v,w}_{>0}}$${\overline{\pi(G_{>0})}}$${\overline{\mathcal{Z}_{e,(w_{0},w_{0}),>0}}\cap
L=\sqcup_{v,w}\mathcal{Z}_{vw_{0},(w,w_{0}),>0}}$${\mathcal{Z}_{vw_{0},(w,w_{0}),>0}.}$

We claim $L^{v,w}_{>0}\rightarrow\mathcal{Z}_{vw_{0},(w,w_{0}),>0}$ is
surjective, hence an isomorphism. Let
${\mathbf{v}}=(s_{i_{1}},\cdots,s_{i_{m}})$ and
${\mathbf{w}}=(s_{j_{1}},\cdots,s_{j_{n}})$ be reduced expressions of $v$ and
$w$, respectively. We consider the morphism following (5.6)

$\mathbb{R}_{>0}^{m+n}\longrightarrow
L^{v,w}_{>0}\longrightarrow\mathcal{Z}_{vw_{0},(w,w_{0}),>0}$

mapping $(a_{1},\dots,a_{n},b_{1},\dots,b_{m})$ to $(y_{j_{1}}(a_{1})\cdots
y_{j_{n}}(a_{n}),x_{i_{1}}(b_{1})\cdots
x_{i_{m}}(b_{m})\dot{w}_{0}B^{+}/B^{+})$. By [19, Theorem 8.7], we have
$U^{+}_{>0}\dot{w}_{0}B^{+}/B^{+}=U^{-}_{>0}B^{+}/B^{+}$. Hence the image of
$\mathbb{R}_{>0}^{m+n}$ is the entire $\mathcal{Z}_{vw_{0},(w,w_{0}),>0}$ by
Theorem 3.4. This finishes the lemma. ∎

Via the embedding $L_{\geqslant 0}\rightarrow\mathcal{Z}_{\geqslant 0}$, we
can define the link of $L^{v,w}_{>0}$ in $L_{\geqslant 0}$ as

$Lk_{(v,w)}(L_{\geqslant 0})=Lk_{(v,(w,w_{0}))}(\mathcal{Z}_{\geqslant
0}),\quad\text{for any }v,w\in W\text{ such that }(v,w)\neq(w_{0},w_{0}).$

By Proposition 5.3, we have the following theorem, solving an open problem of
Fomin and Zelevinsky.

######  Theorem 5.5.

Let $v,w\in W$ such that $(v,w)\neq(w_{0},w_{0})$. The link
$Lk_{(v,w)}(L_{\geqslant 0})$ of $L^{v,w}_{>0}$ in $L_{\geqslant 0}$ is a
regular CW complex homeomorphic to a closed ball.

###  5.3. Braid varieties

Let $\underline{s}=(s_{i_{1}},\dots,s_{i_{n}})\in W^{n}$ be such that all
$s_{i_{j}}$ are simple reflections. Let $w=m_{\ast}(\underline{s})$. We define
the open braid variety associated to $\underline{s}$ as

$\\{(g_{1},g_{2},\dots,g_{n})\in
B^{+}\dot{s}_{i_{1}}B^{+}\times^{B^{+}}\cdots\times^{B^{+}}B^{+}\dot{s}_{i_{n}}B^{+}/B^{+}|g_{1}g_{2}\cdots
g_{n}/B^{+}=\dot{w}B^{+}/B^{+}\\}.$

Note that $B^{+}\dot{s}_{i_{1}}B^{+}\cdots
B^{+}\dot{s}_{i_{n}}B^{+}/B^{+}\subset\overline{B^{+}\dot{w}B^{+}/B^{+}}$ and
$\overline{B^{+}\dot{w}B^{+}/B^{+}}\cap
B^{-}\dot{w}B^{+}/B^{+}=\dot{w}B^{+}/B^{+}$. So the open braid variety is
exactly $\mathring{\mathcal{Z}}_{w,\underline{s}}$. Hence the closure is
${\mathcal{Z}}_{w,\underline{s}}$.

Let ${Q}_{w,\underline{s}}=\\{(w,\underline{s^{\prime}})\in W\times
W^{n}|m_{\ast}(\underline{s^{\prime}})=w,\underline{s^{\prime}}\leqslant\underline{s}\\}$
be the face poset of $\mathcal{Z}_{w,\underline{s}}$ and let
$\hat{Q}_{w,\underline{s}}$ be the augmented face poset. Note that Theorem 4.1
implies that

(a) The poset $\hat{Q}_{m_{\ast}(\underline{s}),\underline{s}}$ is graded,
thin, and shellable.

Let $Q^{\prime}=\\{\underline{s^{\prime}}\in
W^{n}|\underline{s^{\prime}}<\underline{s},m_{\ast}(\underline{s^{\prime}})=w\\}$.
Note that
$Q^{\prime}\cong\\{(w,\underline{s^{\prime}})\in\hat{Q}_{w,\underline{s}}|\underline{s^{\prime}}\neq\underline{s}\\}$.
The (dual) order complex of $Q^{\prime}$ is the subword complex considered in
[18]. So (a) recovers part of [18, Theorem 2.5].

Applying Theorem 4.6 to the closed braid variety, we obtain the following
theorem.

######  Theorem 5.6.

The totally nonnegative braid variety
$\mathcal{Z}_{m_{\ast}(\underline{s}),\underline{s},\geqslant 0}$ is a regular
CW complex homeomorphic to a closed ball.

This theorem provides a geometric realization of the subword complex [18,
Corollary 3.8].

On the other hand, the cluster algebra structure on the coordinate ring of the
open braid variety $\mathring{\mathcal{Z}}_{w,\underline{s}}$ has recently
been established in [5, 10] for a reductive group $G$. It is interesting to
compare the total positivity structure with the cluster algebra structure.

###  5.4. The duality map

In this subsection, we assume that $G$ is a connected reductive group. We
write $\mathcal{Z}=\mathcal{Z}_{n}$.

Let $\iota$ be the involutive group automorphism on $G$ which fixes $T$ and
sends $x_{i}(a)$ to $x_{i}(-a)$, $y_{i}(a)$ to $y_{i}(-a)$ for $i\in I$ and
$a\in\mathbb{C}$. Then $\iota$ preserves the Borel subgroup $B^{+}$ and
induces an involution on $\mathcal{B}$.

We define

$\Phi:\mathcal{B}\mathcal{B},\qquad
gB^{+}/B^{+}\iota(\dot{w}^{-1}_{0}g)B^{+}/B^{+}.$

It is easy to see that $\Phi$ is an involution on $\mathcal{B}$ and for any
$v\leqslant w$ in $W$, we have
$\iota(\mathring{\mathcal{B}}_{v,w})=\mathring{\mathcal{B}}_{v,w}$ and
$\Phi(\mathring{\mathcal{B}}_{v,w})=\mathring{\mathcal{B}}_{w_{0}w,w_{0}v}$.
Since
$\iota(\dot{w}^{-1}_{0}U^{-}_{>0})B^{+}/B^{+}=U^{+}_{>0}\dot{w}_{0}B^{+}/B^{+}=U^{-}_{>0}B^{+}/B^{+}$
by [19, Theorem 8.7], $\Phi$ restricts to an isomorphism on
$\mathcal{B}_{\geqslant 0}$.

We define the involutions

$\displaystyle\Phi:\mathcal{B}^{n}\mathcal{B}^{n},\quad(h_{1}B^{+}/B^{+},\dots,h_{n}B^{+}/B^{+})(\Phi(h_{n}B^{+}/B^{+}),\dots,\Phi(h_{1}B^{+}/B^{+})),$
$\displaystyle\Phi:\mathcal{Z}\mathcal{Z},\quad(g_{1},g_{2},\dots,g_{n})(\iota(\dot{w}^{-1}_{0}g_{1}g_{2}\cdots
g_{n}),\iota(g_{n}^{-1}),\dots,\iota(g_{2}^{-1})).$

Recall the isomorphism $\alpha:\mathcal{Z}\rightarrow\mathcal{B}^{n}$. Then we
have the commutative diagram

$\textstyle{\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathcal{X}.}$

######  Proposition 5.7.

We have $\Phi(\mathcal{Z}_{\geqslant 0})=\mathcal{Z}_{\geqslant 0}$.

###### Proof.

Note that $U^{-}_{>0}$ is Hausdorff dense in $U^{-}_{\geqslant 0}$. Thus
$\mathcal{Z}_{\geqslant 0}$ is the Hausdorff closure of
$(U^{-}_{>0},\dots,U^{-}_{>0})$ in $\mathcal{Z}$. By definition,
$\iota(U^{-}_{>0})^{-1}=U^{-}_{>0}$.

Let $g_{1},g_{2},\dots,g_{n}\in U^{-}_{>0}$. By [9, Lemma 5.6], we have
$\dot{w}^{-1}_{0}g_{1}=g^{\prime}_{1}b_{1}\text{ for some }g^{\prime}_{1}\in
U^{-},b_{1}\in B^{+}_{>0}$. We write $b_{1}g_{2}=g_{2}^{\prime}b_{2}$ for some
$g^{\prime}_{2}\in U^{-}_{>0}$, $b_{2}\in B^{+}_{>0}$. We similarly obtain
$g^{\prime}_{i}\in U^{-}_{>0}$, $b_{i}\in B^{+}_{>0}$ for all $2\leqslant
i\leqslant n$.

Then $\iota(g_{1}^{\prime}\cdots
g^{\prime}_{n})B^{+}=\iota(\dot{w}^{-1}_{0}g_{1}\cdots
g_{n})B^{+}=\iota(\dot{w}^{-1}_{0}g_{1}g_{2}\dot{w}_{0})\dot{w}_{0}B^{+}\subset
U^{+}_{>0}\dot{w}_{0}B^{+}$. Hence $\iota(g_{1}^{\prime}\cdots
g^{\prime}_{n})B^{+}\in U^{-}_{>0}B^{+}$ by [9, Lemma 5.6]. It follows that
$\iota(g_{1}^{\prime}\cdots g^{\prime}_{n})\in U^{-}_{>0}$ since
$\iota(g_{1}^{\prime}\cdots g^{\prime}_{n})\in U^{-}$. Therefore we have

$\displaystyle\,(\iota(\dot{w}^{-1}_{0}g_{1}g_{2}\cdots
g_{n}),\iota(g_{n}^{-1}),\dots,\iota(g_{2}^{-1}))$ $\displaystyle=$
$\displaystyle\,(\iota(g^{\prime}_{1}\cdots,g^{\prime}_{n}),\iota(g^{\prime}_{n})^{-1},\dots,\iota(g^{\prime}_{2})^{-1})\in(U^{-}_{>0},U_{>0}^{-},\dots,U^{-}_{>0})\subset\mathcal{Z}_{\geqslant
0}.$

Hence $\Phi(\mathcal{Z}_{\geqslant 0})\subset(\mathcal{Z}_{\geqslant 0})$.
Since $\Phi$ is an isomorphism, we must have $\Phi(\mathcal{Z}_{\geqslant
0})=(\mathcal{Z}_{\geqslant 0})$. This finishes the proof. ∎

For any $v\in W$ and $\underline{w}=(w_{1},w_{2},\dots,w_{n})\in W^{n}$ with
$v\leqslant m_{\ast}(\underline{w})$, it follows by direct computation and the
above proposition that

$\Phi(\mathring{\mathcal{Z}}_{v,(w_{1},w_{2},\dots,w_{n})})=\mathring{\mathcal{Z}}_{w_{0}w_{1},(w_{0}v,w_{n}^{-1},\dots,w_{2}^{-1})},\quad\Phi(\mathcal{Z}_{v,(w_{1},w_{2}),>0})={\mathcal{Z}}_{w_{0}w_{1},(w_{0}v,w_{2}^{-1}),>0}.$

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