# Configuration Poisson groupoids of flags

Jiang-Hua Lu  Department of Mathematics
The University of Hong Kong
Pokfulam Road
Hong Kong<EMAIL_ADDRESS>,  Victor Mouquin<EMAIL_ADDRESS>and
Shizhuo Yu  School of Mathematical Sciences
Nankai University<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a connected complex semi-simple Lie group and ${\mathcal{B}}$ its
flag variety. For every positive integer $n$, we introduce a Poisson groupoid
over ${\mathcal{B}}^{n}$, called the $n$th total configuration Poisson
groupoid of flags of $G$, which contains a family of Poisson sub-groupoids
whose total spaces are generalized double Bruhat cells and bases generalized
Schubert cells in ${\mathcal{B}}^{n}$. Certain symplectic leaves of these
Poisson sub-groupoids are then shown to be symplectic groupoids over
generalized Schubert cells. We also give explicit descriptions of symplectic
leaves in three series of Poisson varieties associated to $G$.

###### Contents

  1. 1 Introduction and statements of results
    1. 1.1 Introduction
    2. 1.2 Generalized Schubert cells and configuration Poisson groupoids of flags
    3. 1.3 Two isomorphic models of $(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$
    4. 1.4 The third isomorphic model via generalized double Bruhat cells
    5. 1.5 Organization of the paper
    6. 1.6 Notation and basic definitions
    7. 1.7 Acknowledgments
  2. 2 A series of Poisson groupoids associated to Poisson Lie groups
    1. 2.1 Poisson Lie group actions and gauge Poisson groupoids
    2. 2.2 A series of Poisson groupoids
  3. 3 The total configuration Poisson groupoids of flags and isomorphic models
    1. 3.1 The complex semisimple Poisson Lie group $(G,\pi_{\rm st})$
    2. 3.2 The total configuration Poisson groupoids of flags
    3. 3.3 The Poisson groupoid $({\mathcal{C}}_{2n},\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{n},\overline{\pi}_{n})$
    4. 3.4 The Poisson groupoid $(F_{2n}^{o}\times T,\pi_{2n}\bowtie 0)\rightrightarrows(F_{n},\pi_{n})$
  4. 4 Special configuration Poisson groupoids of flags and isomorphic models
    1. 4.1 $T$-leaves
    2. 4.2 The Poisson groupoid $(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$ and two isomorphic models
    3. 4.3 Generalized double Bruhat cells as Poisson groupoids
  5. 5 Configuration symplectic groupoids of flags
    1. 5.1 The symplectic groupoid $(\Lambda^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$
    2. 5.2 Symplectic leaves in $G^{{\bf u},{\bf u}}$ as symplectic groupoids
  6. A Mixed product Poisson structures
    1. A.1 Mixed product Poisson structures
    2. A.2 The Poisson varieties $({\mathcal{C}}_{n},\widehat{\pi}_{n})$ and $({\mathcal{B}}^{n},\overline{\pi}_{n})$ as mixed products
  7. B The Poisson isomorphism $J_{n}$
    1. B.1 The isomorphism $J_{n}$
    2. B.2 Some auxiliary lemmas
    3. B.3 Proof of Theorem B.1
  8. C Generalized double Bruhat cells
    1. C.1 Generalized double Bruhat cells associated to conjugacy classes
    2. C.2 Piece-wise Poisson isomorphisms from $(G_{m,n},\widetilde{\pi}_{m,n})$ to $(\Gamma_{m+n},\widetilde{\pi}_{m+n})$
    3. C.3 Proof of Theorem C.5
  9. D Symplectic leaves of $({\mathcal{O}}^{\bf w}_{e}\times T,\pi_{n}\bowtie 0)$
    1. D.1 Notation
    2. D.2 Description of symplectic leaves of $({\mathcal{O}}^{\bf w}_{e}\times T,\pi_{n}\bowtie 0)$
    3. D.3 Bott-Samelson coordinates and Lusztig toric charts on ${\mathcal{O}}^{\bf w}$
    4. D.4 Proof of Theorem D.7
    5. D.5 Symplectic leaves in generalized double Bruhat cells
    6. D.6 Symplectic leaves of $(\widetilde{F}_{n},\widetilde{\pi}_{n})$

##  1\. Introduction and statements of results

###  1.1. Introduction

Symplectic groupoids, and more generally Poisson groupoids, were introduced by
M. Karasev [14] and A. Weinstein [27, 28] to study singular foliations and
quantizations of Poisson manifolds. A Poisson manifold is said to be
integrable if it is the base of a symplectic groupoid. While not every Poisson
manifold is integrable (see [4] for the obstructions), “natural” Poisson
manifolds are expected to have natural integrations to symplectic groupoids.
When the Poisson manifold is algebraic, one would also want the symplectic
groupoids to be algebraic. It is a fundamental problem of Poisson geometry to
construct explicit and systematic examples of symplectic groupoids, especially
in the category of algebraic Poisson manifolds.

Lie theory provides a rich class of Poisson manifolds: every connected complex
semi-simple Lie group $G$ carries a standard multiplicative Poisson structure
$\pi_{\rm st}$, defined using the choice of a pair $(B,B_{-})$ of opposite
Borel subgroups of $G$ (see $\S$3.1), and many important manifolds in Lie
theory carry Poisson structures closely related to the Poisson Lie group
$(G,\pi_{\rm st})$. Four series of such Poisson manifolds have been introduced
and studied by the first two authors in [17, 18]. Among them are the two
series of quotients of $G^{n}$, denoted respectively as

$\widetilde{F}_{n}=G\times_{B}\times\cdots\times_{B}G\hskip
14.45377pt\mbox{and}\hskip
14.45377ptF_{n}=\widetilde{F}_{n}/B=G\times_{B}\times\cdots\times_{B}G/B,\hskip
14.45377ptn\geq 1.$

Here and for the rest of the paper, we consider the right action of $G^{n}$ on
itself by

$(g_{1},g_{2},\ldots,g_{n})\cdot(h_{1},h_{2},\ldots,h_{n})=(g_{1}h_{1},h_{1}^{-1}g_{2}h_{2},\ldots,h_{n-1}^{-1}g_{n}h_{n}),\;\;h_{j},g_{j}\in
G.$ (1.1)

Then $\widetilde{F}_{n}$ is the quotient of $G^{n}$ by
$B^{n-1}\times\\{e\\}\subset G^{n}$, while $F_{n}$ is the quotient of $G^{n}$
by $B^{n}\subset G^{n}$. By [17, $\S$7.1], the product Poisson structure
$(\pi_{\rm st})^{n}$ on $G^{n}$ projects to well-defined Poisson structures on
$\widetilde{F}_{n}$ and on $F_{n}$, respectively denoted as
$\widetilde{\pi}_{n}$ and $\pi_{n}$. See $\S$2.2 for a more general
construction.

For $Z=\widetilde{F}_{n}$ or $F_{n}$, denote by
$[g_{1},\ldots,g_{n}]_{\scriptscriptstyle Z}$ the image in $Z$ of
$(g_{1},\ldots,g_{n})\in G^{n}$. Let $T=B\cap B_{-}$, a maximal torus of $G$.
Then $T$ acts on $\widetilde{F}_{n}$ and $F_{n}$ respectively by

$\displaystyle t\cdot[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}=[tg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle F}_{n}},$ (1.2)
$\displaystyle
t\cdot[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}}=[tg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}},$
(1.3)

preserving the Poisson structures $\widetilde{\pi}_{n}$ and $\pi_{n}$. A
systematic study of the $T$-orbits of symplectic leaves, or $T$-leaves for
short (see Definition 1.3), of both $(\widetilde{F}_{n},\widetilde{\pi}_{n})$
and $(F_{n},\pi_{n})$ are given in [18]. In particular, it is shown in [18,
Theorem 1.3 and Theorem 1.1] that both
$(\widetilde{F}_{n},\widetilde{\pi}_{n})$ and $(F_{n},\pi_{n})$ have finitely
many $T$-leaves. Setting ${\mathcal{B}}=G/B$, the flag variety of $G$, note
that for each $n$ one has the isomorphism

$F_{n}\longrightarrow{\mathcal{B}}^{n},\;\;[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}\longmapsto({g_{1}}_{\cdot}B,\,(g_{1}g_{2})_{\cdot}B,\,\ldots,\,(g_{1}g_{2}\cdots
g_{n})_{\cdot}B).$

We thus also regard $F_{n}$ as a product of flag varieties. Similarly, one has
the isomorphism

$\widetilde{F}_{n}\longrightarrow{\mathcal{B}}^{n-1}\times
G,\;\;[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}}\longmapsto({g_{1}}_{\cdot}B,\,(g_{1}g_{2})_{\cdot}B,\,\ldots,\,(g_{1}g_{2}\cdots
g_{n-1})_{\cdot}B,\,g_{1}g_{2}\cdots g_{n}).$

As the main results of the paper, we show that for each $n\geq 1$ the Poisson
manifold $(\widetilde{F}_{2n},\widetilde{\pi}_{2n})$ is a Poisson groupoid
over $(F_{n},\pi_{n})$, and that certain $T$-leaves of
$(\widetilde{F}_{2n},\widetilde{\pi}_{2n})$ (resp. symplectic leaves therein)
are Poisson (resp. symplectic) sub-groupoids. We also give several isomorphic
models for the Poisson groupoids, which shed different lights and put them in
different contexts. The isomorphic models are established through Poisson
isomorphisms between $T$-leaves in various $T$-Poisson varieties, the proofs
of which, being technical, are presented in the appendices. In Appendix $\S$D
in particular, we determine symplectic leaves in all the $T$-leaves of three
series of $T$-Poisson varieties including
$(\widetilde{F}_{n},\widetilde{\pi}_{n})$ for all $n$. We remark that while
$T$-leaves of many $T$-Poisson varieties associated to the Poisson Lie group
$(G,\pi_{\rm st})$ have been determined (see, for example, [7, 15, 18]),
describing the symplectic leaves therein is a harder problem and it has been
done only for the case of $(G,\pi_{\rm st})$ itself by M. Kogan and A.
Zelevinsky in [13]. Results in Appendix $\S$D thus constitute a big step
towards a general theory of leaves in $T$-leaves and is thus of independent
interest.

In the rest of the Introduction, we explain our motivation and give more
details of the main results of the paper. See in particular $\S$1.3 on
identifications of the total and base spaces of the Poisson groupoids in this
paper with cluster varieties studied by L. Shen and D. Weng in [25], and with
augmentation varieties of Legendrian links by H. Gao, L. Shen and D. Weng in
[9]. We also point out the recent work [1] by D. Alvarez, which contains a
construction of a Poisson groupoid over $F_{n}$ as the moduli space of flat
$G$-bundles over the disk with decorated boundary. Precise relations between
the Poisson groupoids in [1] and the ones in this paper, and further study on
their dual Poisson groupoids and double symplectic groupoids will be given
elsewhere.

###  1.2. Generalized Schubert cells and configuration Poisson groupoids of
flags

Let $W$ be the Weyl group of $(G,T)$, and recall that the flag variety
${\mathcal{B}}=G/B$ has the decomposition into Schubert cells111In the
literature, $BuB/B$, for $u\in W$, is sometimes called a Bruhat cell in $G/B$.
In this paper, we use the term Schubert cell, reserving the term Bruhat cell
for the sub-manifold $BuB$ in $G$ as suggested by A. Berenstein.
${\mathcal{O}}^{u}:=BuB/B$, where $u\in W$. Similarly, for $n\geq 1$ and ${\bf
u}=(u_{1},\ldots,u_{n})\in W^{n}$, denote the image of
$Bu_{1}B\times\cdots\times Bu_{n}B$ in $F_{n}$ as

${\mathcal{O}}^{\bf u}=(Bu_{1}B)\times_{B}\cdots\times_{B}(Bu_{n}B)/B\subset
F_{n}.$

One has the disjoint union decomposition

$F_{n}=\bigsqcup_{{\bf u}\in W^{n}}{\mathcal{O}}^{\bf u}.$

For ${\bf u}\in W^{n}$, ${\mathcal{O}}^{\bf u}\subset F_{n}$ is called a
generalized Bruhat cell in [5, 18]. In this paper, we will refer to them as
generalized Schubert cells to be consistent with the case when $n=1$. By [18,
Theorem 1.1], each generalized Schubert cell ${\mathcal{O}}^{\bf u}\subset
F_{n}$, being a union of (finitely many) $T$-leaves of $\pi_{n}$, is a Poisson
sub-manifold of $(F_{n},\pi_{n})$. Following [5, 18], the restriction of
$\pi_{n}$ to ${\mathcal{O}}^{\bf u}$, again denoted as $\pi_{n}$, is called
the standard Poisson structure on ${\mathcal{O}}^{\bf u}$.

In this paper, we first give in $\S$2 a general construction of a series of
Poisson groupoids associated to any Poisson Lie group and a closed Poisson Lie
sub-group. The natural Poisson groupoid structure on
$(\widetilde{F}_{2n},\widetilde{\pi}_{2n})$ over $(F_{n},\pi_{n})$ is then a
special case of the general construction (see Theorem 3.2). We introduce the
sub-manifold

$\Gamma_{2n}\stackrel{{\scriptstyle{\rm
def}}}{{=}}\\{[g_{1},g_{2},\ldots,,g_{2n}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}:g_{1}g_{2}\cdots
g_{2n}\in B_{-}\\}$

of $\widetilde{F}_{2n}$, which, by Proposition 3.1 and Theorem 3.2, is a union
of $T$-leaves of $(\widetilde{F}_{2n},\widetilde{\pi}_{2n})$ and a Poisson
sub-groupoid of the Poisson groupoid
$(\widetilde{F}_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$.
We call

$\theta_{\pm}:\;(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n}),\hskip
14.45377ptn\geq 1,$

the $n$th total configuration Poisson groupoid of flags of $G$ or simply the a
total configuration Poisson groupoid, where $\theta_{+}$ and $\theta_{-}$ are
the source and target maps. For each ${\bf u}\in W^{n}$, let

$\Gamma^{({\bf u},{\bf u}^{-1})}\stackrel{{\scriptstyle\rm
def}}{{=}}\theta_{+}^{-1}({\mathcal{O}}^{\bf
u})\cap\theta_{-}^{-1}({\mathcal{O}}^{\bf
u})\rightrightarrows{\mathcal{O}}^{\bf u}$

be the full sub-groupoid of $\Gamma_{2n}\rightrightarrows F_{n}$ over
${\mathcal{O}}^{\bf u}$ (see Definition 1.1) . We show in Theorem 4.4 that
$\Gamma^{({\bf u},{\bf u}^{-1})}$ is a single $T$-leaf of
$(\Gamma_{2n},\widetilde{\pi}_{2n})$. Consequently

$\theta_{+},\theta_{-}:\;\;(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ (1.4)

is an (algebraic) Poisson sub-groupoid of
$(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$.
Furthermore, we show that the symplectic leaf of $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})$ through the section of units of the groupoid
in (1.4) is an (algebraic) symplectic groupoid over $({\mathcal{O}}^{\bf
u},\pi_{n})$. We call $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ a special configuration Poisson groupoid of flags of $G$.

Our interest in generalized Schubert cells ${\mathcal{O}}^{\bf u}$ with their
standard Poisson structures $\pi_{n}$, for ${\bf u}\in W^{n}$, stems from some
of their remarkable features in relation to cluster algebras and to other
Poisson manifolds related to the Poisson Lie group $(G,\pi_{\rm st})$.

First of all, it is shown in [5] that for any ${\bf u}=(u_{1},\ldots,u_{n})\in
W^{n}$, one can use root subgroups of $G$ and reduced decompositions for each
$u_{i}$ to parametrize ${\mathcal{O}}^{\bf u}$ by ${\mathbb{C}}^{l({\bf u})}$,
thus obtaining the so-called Bott-Samelson coordinates
$(z_{1},\ldots,z_{l({\bf u})})$ on ${\mathcal{O}}^{\bf u}$ and a Poisson
bracket $\\{\,,\,\\}_{\bf u}$ on ${\mathbb{C}}[z_{1},z_{2},\ldots,z_{l({\bf
u})}]$. Here $l({\bf u})=l(u_{1})+\cdots+l(u_{n})$ and $l(u_{i})$ is the
length of $u_{i}$. Explicit formulas for $\\{\,,\,\\}_{\bf u}$ are given in
[5, Theorem 5.15] in terms of root strings and structure constants of the Lie
algebra of $G$. In particular, it is shown in [5] that the polynomial Poisson
algebra $({\mathbb{C}}[z_{1},z_{2},\ldots,z_{l({\bf u})}],\\{\,,\,\\}_{\bf
u})$ is a symmetric Poisson CGL extension in the sense of K. Goodearl and M.
Yakimov, a special class of Poisson polynomial algebras introduced and studied
in [11, 12] by the same authors in the context of cluster algebras.

Secondly, generalized Schubert cells with the standard Poisson structures form
basic building blocks for many of the Poisson manifolds associated to the
Poisson Lie group $(G,\pi_{\rm st})$. For example, it is shown in [20] that a
number of Poisson homogeneous spaces $(G/Q,\pi_{{\scriptscriptstyle
G}/{\scriptscriptstyle Q}})$ of the Poisson Lie group $(G,\pi_{\rm st})$,
including $(G,\pi_{\rm st})$ itself and $(G/B,\pi_{1})$, admit so-called Bott-
Samelson atlases which are built out of generalized Schubert cells, and the
Poisson structure $\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$ is
presented as symmetric Poisson CGL extensions in all of the coordinate charts
of the Bott-Samelson atlas. We refer to [20] for more detail.

The explicit and natural Poisson and symplectic groupoids over the generalized
Schubert cells $({\mathcal{O}}^{u},\pi_{n})$ constructed in this paper add
another dimension to this distinguished class of Poisson manifolds. The
Poisson groupoids $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\\!\rightrightarrows\\!({\mathcal{O}}^{\bf
u},\pi_{n})$ are interesting on their own. Indeed, we give three additional
isomorphic models of $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\\!\rightrightarrows\\!({\mathcal{O}}^{\bf
u},\pi_{n})$, each having advantages over the others and putting these Poisson
and symplectic groupoids in different perspectives. We now give more details
on these models.

###  1.3. Two isomorphic models of $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$

Let ${\mathcal{B}}=G/B$ be the flag variety of $G$, and let
${\mathcal{A}}=G/N$ be the decorated flag variety (also known as the basic
affine space) of $G$, where $N$ be the unipotent sub-group of $B$. Let
${\mathcal{A}}^{o}=B_{-}N/N$, an open subvariety of ${\mathcal{A}}$. For
$n\geq 1$, set
${\mathcal{C}}_{2n}={\mathcal{B}}^{2n-1}\times{\mathcal{A}}^{o}$. Under
natural isomorphisms ${\mathcal{C}}^{2n}\to\Gamma_{2n}$ and
$F_{n}\to{\mathcal{B}}^{n}$, the Poisson groupoid
$(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$ becomes
the Poisson groupoid

$({\mathcal{C}}_{2n},\,\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{n},\,\overline{\pi}_{n}),$
(1.5)

see Theorem 3.4 for detail. Correspondingly, for each ${\bf u}\in W^{n}$, the
special configuration Poisson groupoid $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ is then isomorphic to a Poisson sub-groupoid

$({\mathcal{C}}^{({\bf u},{\bf
u}^{-1})},\,\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{\bf
u},\overline{\pi}_{n})$

of the Poisson groupoid in (1.5), where ${\bf
u}^{-1}=(u_{n}^{-1},\ldots,u_{2}^{-1},u_{1}^{-1})\in W^{n}$ if ${\bf
u}=(u_{1},u_{2},\ldots,u_{n})$ (see Corollary 4.5). The variety
${\mathcal{C}}^{({\bf u},{\bf u}^{-1})}\subset{\mathcal{C}}_{2n}$ consists of
sequences of flags with relative Tits distances prescribed by ${\bf u}$ (see
Lemma 4.2) and is an example of a decorated double Bott-Samelson cell
introduced by L. Shen and D. Weng in [25]. We explain in $\S$A.2 that
$({\mathcal{B}}^{n},\overline{\pi}_{n})$ is a mixed product of $n$ copies of
the Poisson variety $({\mathcal{B}},\pi_{1})$ in the sense defined in [17],
and a similar statement holds for the Poisson variety
$({\mathcal{C}}_{2n},\widehat{\pi}_{2n})$. By [25], each ${\mathcal{C}}^{({\bf
u},{\bf u}^{-1})}$ is a Poisson cluster variety when $G$ is of adjoint type.
Relations between the Poisson groupoid structure defined in this paper and the
cluster structure on these varieties defined in [25] will be a very
interesting topic to explore (see [25, Remark 1.10]).

For the second isomorphic model, consider the open sub-manifold $F_{n}^{o}$ of
$F_{n}$ given by

$F_{n}^{o}=\\{[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}:\;g_{1}g_{2}\cdots g_{n}\in B_{-}B/B\\},$

and for ${\bf w}\in W^{n}$, let

${\mathcal{O}}^{\bf w}_{e}={\mathcal{O}}^{\bf w}\cap
F_{n}^{o}=\\{[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}\in{\mathcal{O}}^{{\bf w}}:g_{1}g_{2}\cdots g_{n}\in B_{-}B/B\\}.$
(1.6)

By [18, Theorem 1.1], ${\mathcal{O}}^{\bf w}_{e}$ is the open $T$-leaf of
$({\mathcal{O}}^{\bf w},\pi_{n})$. The Poisson variety $(F_{n}^{o},\pi_{n})$
has a natural $T$-extension $(F_{n}^{o}\times T,\pi_{n}\bowtie 0)$, whose
$T$-leaves with respect to the diagonal action of $T$ are precisely all the
sub-varieties ${\mathcal{O}}^{\bf w}_{e}\times T$ for ${\bf w}\in W^{n}$. Here
$\pi_{n}\bowtie 0$, as a Poisson structure on $F_{n}^{o}\times T$, is the sum
of the product Poisson structure $(\pi_{n},0)$ and a certain mixed term
defined using the $T$-action on $F_{n}^{o}$ (see (1.9)). We show in Corollary
4.5 that, via a Poisson isomorphism
$J_{2n}:(\Gamma_{2n},\widetilde{\pi}_{2n})\to(F_{2n}^{o}\times
T,\pi_{2n}\bowtie 0)$ and for each ${\bf u}\in W^{n}$, the Poisson groupoid
$(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ is isomorphic to a Poisson groupoid

$({\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times T,\,\pi_{2n}\bowtie
0)\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n}).$ (1.7)

The advantages of the isomorphic model in (1.7) are at least three-fold. First
of all, as $({\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times
T,\pi_{2n}\bowtie 0)$ is the $T$-extension of $({\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}_{e},\pi_{2n})$, we can apply results in [5, 16] on arbitrary
generalized Schubert cells and their $T$-extensions. In particular, the model
in (1.7) allows us to describe all the symplectic leaves of $(\Gamma^{({\bf
u},{\bf u}^{-1})},\widetilde{\pi}_{2n})$, thereby proving that the symplectic
leaf of $(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})$ through the
section of units of the groupoid $\Gamma^{({\bf u},{\bf
u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$ is a symplectic groupoid over
$({\mathcal{O}}^{\bf u},\pi_{n})$.

Secondly, each $({\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times
T,\pi_{2n}\bowtie 0)$ is a (localization of a) Poisson symmetric CGL extension
by [5, Theorem 5.12] and [20, $\S$4.2], so the Goodearl-Yakimov theory in [12]
gives a cluster variety structure on ${\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}_{e}\times T$. The isomorphisms ${\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}_{e}\times T\cong\Gamma^{({\bf u},{\bf
u}^{-1})}\cong{\mathcal{C}}^{({\bf u},{\bf u}^{-1})}$ thus provide tools for
future research on comparing the cluster structure on ${\mathcal{O}}^{({\bf
u},{\bf u}^{-1})}_{e}\times T$ via the Goodearl-Yakimov theory with that on
${\mathcal{C}}^{({\bf u},{\bf u}^{-1})}$ established by L. Shen and D. Weng in
[25].

Thirdly, in their work [9] on Lagrangian fillings of Legendrian links, H. Gao,
L. Shen, and D. Weng show that varieties of the form ${\mathcal{O}}^{\bf
w}_{e}$ for $G=SL_{n}$ are isomorphic to augmentation varieties of certain
positive braid Legendrian links. It would be very interesting to explore
connections between the Poisson groupoid structure on ${\mathcal{O}}^{({\bf
u},{\bf u}^{-1})}_{e}\times T$ in this paper and the results in [9].

###  1.4. The third isomorphic model via generalized double Bruhat cells

For the third isomorphic model for the Poisson groupoid $(\Gamma^{({\bf
u},{\bf u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$, we first explain some background. Recall from [8] that
associated to each pair $u,v\in W$ one has the double Bruhat cell
$G^{u,v}=BuB\cap B_{-}vB_{-}$, and that (see, for example, [13]) the
decomposition $G=\bigsqcup_{u,v\in W}G^{u,v}$ is that of $(G,\pi_{\rm st})$
into $T$-leaves for the $T$-action on $G$ by left translation. The first two
authors proved in [19] that for any $u\in W$, $(G^{u,u},\pi_{\rm st})$ is a
Poisson groupoid over $({\mathcal{O}}^{u},\pi_{1})$, but the groupoid
structure depends on a choice $\dot{u}$ of a representative of $u$ in the
normalizer subgroup $N_{G}(T)$ of $T$ in $G$. Using the description of
symplectic leaves of $\pi_{\rm st}$ in $G^{u,u}$ given by M. Kogan and A.
Zelevinsky in [13], it is then proved in [19] that all symplectic leaves of
$(G^{u,u},\pi_{\rm st})$ are symplectic groupoids over
$({\mathcal{O}}^{u},\pi_{1})$.

Generalizing the decomposition of $G$ into double Bruhat cells, the first two
authors introduced in [18] a Poisson manifold
$(G_{n,n},\widetilde{\pi}_{n,n})$, for each integer $n\geq 1$, and its
decomposition

$G_{n,n}=\bigsqcup_{{\bf u},{\bf v}\in W^{n}}G^{{\bf u},{\bf v}}$

into $T$-leaves, where each $G^{{\bf u},{\bf v}}$ is called a generalized
double Bruhat cell (one has $(G_{1,1},\pi_{1,1})\cong(G,\pi_{\rm st})$). The
second author then showed in [24] that for every ${\bf u}\in W^{n}$, $(G^{{\bf
u},{\bf u}},\widetilde{\pi}_{n,n})$ is a Poisson groupoid over
$({\mathcal{O}}^{\bf u},\pi_{n})$, where again the groupoid structure depends
on a choice $\dot{{\bf u}}=(\dot{u}_{1},\ldots,\dot{u}_{n})\in N_{G}(T)^{n}$
representing ${\bf u}$. The question of whether or not symplectic leaves of
$(G^{{\bf u},{\bf u}},\widetilde{\pi}_{n,n})$ are symplectic groupoids over
$({\mathcal{O}}^{\bf u},\pi_{n})$ was left unanswered in [24], due to the fact
that description of symplectic leaves of $(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})$ was not available.

In this paper, we show that for any ${\bf u}\in W^{n}$, each choice $\dot{{\bf
u}}\in N_{G}(T)^{n}$ representing ${\bf u}$ gives a $T$-equivariant Poisson
embedding

$E_{\dot{{\bf u}}}:\;(G^{{\bf u},{\bf
u}},\,\widetilde{\pi}_{n,n})\;{\hookrightarrow}\;(\Gamma_{2n},\,\widetilde{\pi}_{2n})$
(1.8)

whose image is exactly $\Gamma^{({\bf u},{\bf u}^{-1})}$. The Poisson groupoid
$(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$
defined in [24] using the choice $\dot{{\bf u}}$ is then shown to become
precisely the groupoid $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ via $E_{\dot{{\bf u}}}$. Through the ($\dot{{\bf u}}$-dependent)
Poisson embedding $E_{\dot{{\bf u}}}$, we have thus an intrinsic explanation
on the origin of the Poisson groupoid structures on $G^{{\bf u},{\bf u}}$ as
well as on their dependence on the representatives $\dot{{\bf u}}$. At this
connection, we point out that the construction of the Poisson groupoids
$(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$ in
[24] is based on a general theory on local Poisson groupoids over mixed
product Poisson manifolds and actions by double symplectic groupoids, an
approach completely different from what we use in this paper.

When $n=1$, $\Gamma_{2}=(G/B)\times B_{-}$ is the action groupoid $(G/B)\times
B_{-}\rightrightarrows G/B$ for the action of $B_{-}$ on $G/B$ by left
translation. The fact that $(\Gamma_{2},\widetilde{\pi}_{2})$ is a Poisson
groupoid over $(G/B,\pi_{1})$, and that one has the Poisson embeddings in
(1.8) for $n=1$, are also proved in [19]. Putting these results in [19] (for
$n=1$) and that in [24] (on the Poisson groupoid structures on $G^{{\bf
u},{\bf u}}$) in one unified framework was part of the motivation for this
paper.

###  1.5. Organization of the paper

After a general construction in $\S$2 of a series of Poisson groupoids
associated to any Poisson Lie group and a closed Poisson Lie sub-group, we
turn to the Poisson Lie group $(G,\pi_{\rm st})$ and its Poisson Lie sub-group
$B$ in $\S$3, where we introduce the total configuration Poisson groupoids

$(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n}),\hskip
14.45377ptn\geq 1,$

of flags. In $\S$4, we discuss the Poisson sub-groupoids

$(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n}),\hskip 14.45377pt{\bf u}\in W^{n},\,n\geq 1,$

of $(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$ and
their three isomorphic models. In $\S$5 we prove that the symplectic leaf of
$(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})$ passing through the
section of units of the groupoid $\Gamma^{({\bf u},{\bf
u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$ is a symplectic groupoid over
$({\mathcal{O}}^{\bf u},\pi_{n})$.

The paper contains several extensive appendices. The most technical parts of
the paper are the proofs that various isomorphisms between varieties are in
fact isomorphisms of Poisson varieties, and we present these proofs in the
appendices $\S$A \- $\S$C.

In $\S$A, we show how the various Poisson varieties considered in this paper
are mixed product Poisson varieties as defined in [17].

In $\S$B, we show that the $T$-equivariant isomorphism $J_{n}:\Gamma_{n}\to
F_{n}^{o}\times T$ defined in (3.14) is a Poisson isomorphism from
$(\Gamma_{n},\widetilde{\pi}_{n})$ to $(F_{n}^{o}\times T,\pi_{n}\bowtie 0)$.

Generalizing the case of $m=n$ from [18, $\S$1.4], we introduce in $\S$C a
$T$-Poisson manifold $(G_{m,n},\widetilde{\pi}_{m,n})$ for any pair of
integers $m,n\geq 1$ whose $T$-leaves are shown to be generalized double
Bruhat cells $G^{{\bf u},{\bf v}}$, where $({\bf u},{\bf v})\in W^{m}\times
W^{n}$. The main results of $\S$C are certain explicit $T$-equivariant Poisson
isomorphisms between the single $T$-leaves

$(G^{{\bf u},{\bf
v}},\widetilde{\pi}_{m,n})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\left({\mathcal{O}}^{({\bf
u},{\bf v}^{-1})}_{e}\times T,\,\pi_{m+n}\bowtie 0\right)$

for all $({\bf u},{\bf v})\in W^{m}\times W^{n}$, and similar Poisson
isomorphisms between $T$-leaves of $(\widetilde{F}_{n},\widetilde{\pi}_{n})$
to those of the form $({\mathcal{O}}^{{\bf w}}_{e}\times T,\pi_{n+1}\bowtie
0)$ for ${\bf w}\in W^{n+1}$. These facts illustrate again the role of
generalized Schubert cells (or their $T$-extensions for the examples in this
paper) as building blocks for Poisson varieties associated to the Poisson Lie
group $(G,\pi_{\rm st})$.

In $\S$D we determine the symplectic leaves of $({\mathcal{O}}^{\bf
w}_{e}\times T,\pi_{n}\bowtie 0)$ for any $n\geq 1$ and ${\bf w}\in W^{n}$.
Although only the cases of ${\bf w}=({\bf u},{\bf u}^{-1})$ are needed in the
main text of the paper, the results for arbitrary ${\bf w}$ allow us to
determine the symplectic leaves in all generalized double Bruhat cells
$G^{{\bf u},{\bf v}}$, thereby extending the result of M. Kogan and A.
Zelevinsky in [13] for the case of $u,v\in W$. We in fact describe the
symplectic leaves in all the three series

$(F_{n}^{o}\times T,\,\pi_{n}\bowtie 0),\hskip
14.45377pt(G_{m+n},\,\widetilde{\pi}_{m,n}),\hskip
14.45377pt(\widetilde{F}_{n},\,\widetilde{\pi}_{n}),\hskip 14.45377ptm,n\geq
1,$

of $T$-Poisson varieties. Results in $\S$D provide test stone examples towards
a general theory of symplectic leaves in $T$-leaves to be carried out
elsewhere.

###  1.6. Notation and basic definitions

For a manifold $X$ and $1\leq k\leq\dim X$, let ${\mathfrak{X}}^{k}(X)$ be the
space of all $k$-vector fields on $X$, i.e., all smooth sections of
$\wedge^{k}TX$, where $TX$ is the tangent bundle of $X$. When $X$ is a complex
manifold, $TX$ will stand for the holomorphic tangent bundle and
${\mathfrak{X}}^{k}(X)$ the space of all holomorphic $k$-vector fields on $X$.
If $X$ and $Y$ are manifolds and if $V_{\scriptscriptstyle
X}\in{\mathfrak{X}}^{k}(X)$ and $V_{\scriptscriptstyle
Y}\in{\mathfrak{X}}^{k}(Y)$, let $(V_{\scriptscriptstyle
X},0),(0,V_{\scriptscriptstyle Y})\in{\mathfrak{X}}^{k}(X\times Y)$ be given
by

$(V_{\scriptscriptstyle X},\,0)(x,y)=i_{y}V_{\scriptscriptstyle X}(x)\hskip
14.45377pt\mbox{and}\hskip 14.45377pt(0,\,V_{\scriptscriptstyle
Y})(x,y)=i_{x}V_{\scriptscriptstyle Y}(y),\hskip 14.45377ptx\in X,y\in Y,$

where $i_{y}:X\to X\times Y,\,x^{\prime}\mapsto(x^{\prime},y)$ for
$x^{\prime}\in X$, and $i_{x}:Y\to X\times
Y,\,y^{\prime}\mapsto(x,y^{\prime})$ for $y^{\prime}\in Y$. This convention
also extends to multi-vector fields on $n$-fold product manifolds
$X_{1}\times\cdots\times X_{n}$ for any integer $n\geq 2$. If
$V_{i}\in{\mathfrak{X}}^{k}(X_{i})$ for $i\in[1,n]$, we also write

$\displaystyle(V_{1},V_{2},\ldots,V_{n})$ $\displaystyle=V_{1}\times
V_{2}\times\cdots\times V_{n}$
$\displaystyle=(V_{1},0,\ldots,0)+(0,V_{2},\ldots,0)+\cdots+(0,0,\ldots,V_{n})\in{\mathfrak{X}}^{k}(X_{1}\times
X_{2}\times\cdots\times X_{n}).$

Recall that a Poisson manifold is a pair $(X,\pi)$, where $X$ is a manifold
and $\pi\in{\mathfrak{X}}^{2}(X)$, called a Poisson structure, satisfies
$[\pi,\pi]=0$, where $[\,,\,]$ is the Schouten bracket on
${\mathfrak{X}}^{\bullet}(X)=\oplus_{k}{\mathfrak{X}}^{k}(X)$. For a Poisson
structure $\pi$ on $X$, define

$\pi^{\\#}:\;\;T^{*}X\longrightarrow
TX,\;\;(\pi^{\\#}(\alpha),\;\beta)=\pi(\alpha,\beta),$

where $\alpha$ and $\beta$ are any $1$-forms on $X$. If $X_{1}$ is a Poisson
sub-manifold of $(X,\pi)$, i.e., $X_{1}$ is a sub-manifold of $X$ such that
$\pi(x)\in\wedge^{2}T_{x}X_{1}$ for all $x\in X_{1}$, the restriction of $\pi$
to $X_{1}$ will still be denoted by $\pi$, so that $(X_{1},\pi)$ is a Poisson
manifold.

The identity element of a group is typically denoted as $e$. Let $A$ be any
Lie group with Lie algebra $\mathfrak{a}$. The left and right translations on
$A$ by $a\in A$ will be denoted by $L_{a}$ and $R_{a}$ respectively, and for
any integer $k\geq 1$ and $x\in\wedge^{k}\mathfrak{a}$ and
$\xi\in\wedge^{k}\mathfrak{a}^{*}$, $x^{L}$ and $x^{R}$ (resp. $\xi^{L}$ and
$\xi^{R}$) will respectively denote the left and right invariant $k$-vector
fields (resp. $k$-forms) on $A$ with value $x$ (resp. $\xi$) at $e\in A$. If
$A$ acts on a manifold $Y$ from the right with the action map $\rho:Y\times
A\to Y$, we will also denote by $\rho$ the Lie algebra homomorphism

$\rho:\;\mathfrak{a}\longrightarrow{\mathfrak{X}}^{1}(X),\;\;\rho(x)(y)=\frac{d}{dt}|_{t=0}\rho(y,\exp(tx)),\hskip
14.45377ptx\in\mathfrak{a},\,y\in Y.$

Similarly, if $\lambda:A\times Y\to Y$ is a left action by $A$, one has the
Lie algebra anti-homomorphism

$\lambda:\;\mathfrak{a}\longrightarrow{\mathfrak{X}}^{1}(X),\;\;\lambda(x)(y)=\frac{d}{dt}|_{t=0}\lambda(\exp(tx),y),\hskip
14.45377ptx\in\mathfrak{a},\,y\in Y.$

For clarity and for convenience of the reader, we recall some terminology on
groupoids and Poisson groupoids and refer to [22, 27, 28] for the basics of
the subject.

######  Definition 1.1.

1) A sub-groupoid of a groupoid $\Gamma\rightrightarrows M$ is said to be wide
if it contains the set of units of $\Gamma\rightrightarrows M$.

2) The full sub-groupoid of a groupoid $\Gamma\rightrightarrows M$ over a
subset $M^{\prime}\subset M$ is the intersection
$\theta_{+}^{-1}(M^{\prime})\cap\theta_{-}^{-1}(M^{\prime})$ as a groupoid
over $M^{\prime}$ whose structure maps are the restrictions of those for
$\Gamma\rightrightarrows M$, where $\theta_{+}$ and $\theta_{-}$ are
respectively the source and target maps of $\Gamma\rightrightarrows M$.

######  Definition 1.2.

A Poisson groupoid is a Lie groupoid $\Gamma\rightrightarrows M$ together with
a Poisson structure $\pi$ on $\Gamma$ such that
$\\{(\gamma,\gamma^{\prime},m(\gamma,\gamma^{\prime})):\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime})\\}\subset\Gamma^{3}$
is a coisotropic sub-manifold of $(\Gamma^{3},\pi\times\pi\times(-\pi))$,
where $\theta_{+},\theta_{-}:\Gamma\to M$ are the source and target maps, and

$m:\\{(\gamma,\,\gamma^{\prime})\in\Gamma^{2}:\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime})\\}\longrightarrow\Gamma$

is the partially defined multiplication on $\Gamma$. In such a case, there is
a unique Poisson structure $\pi_{\scriptscriptstyle M}$ on $M$ such that
$\theta_{+}:(\Gamma,\pi)\to(M,\pi_{\scriptscriptstyle M})$ is Poisson and
$\theta_{-}:(\Gamma,\pi)\to(M,\pi_{\scriptscriptstyle M})$ is anti-Poisson,
and we also say that $(\Gamma,\pi)$ is a Poisson groupoid over
$(M,\pi_{\scriptscriptstyle M})$. If in addition $\pi$ is non-degenerate and
$\dim\Gamma=2\dim M$, one says that
$(\Gamma,\pi)\rightrightarrows(M,\pi_{\scriptscriptstyle M})$ is a symplectic
groupoid over $(M,\pi_{\scriptscriptstyle M})$.

A Poisson (resp. symplectic) groupoid
$(\Gamma,\pi)\rightrightarrows(M,\pi_{\scriptscriptstyle M})$ is said to be
(complex) algebraic if both $\Gamma$ and $M$ are smooth algebraic manifolds
over ${\mathbb{C}}$, all structure maps of $\Gamma\rightrightarrows M$ are
smooth algebraic morphisms, and both $\pi$ and $\pi_{\scriptscriptstyle M}$
are algebraic Poisson (resp. symplectic) structures. $\diamond$

We also recall the notion of $\mathbb{T}$-leaves that will be used throughout
the paper.

######  Definition 1.3.

If $\mathbb{T}$ is a torus, by a $\mathbb{T}$-Poisson manifold we mean a
Poisson manifold $(X,{\pi_{\scriptscriptstyle X}})$ with an action of
$\mathbb{T}$ by Poisson isomorphisms. For a $\mathbb{T}$-Poisson manifold
$(X,{\pi_{\scriptscriptstyle X}})$, a $\mathbb{T}$-orbit of symplectic leaves,
or a $\mathbb{T}$-leaf for short, of $(X,{\pi_{\scriptscriptstyle X}})$ is a
sub-manifold $L$ of $X$ of the form

$L=\bigcup_{t\in\mathbb{T}}t\Sigma,$

where $\Sigma$ is a symplectic leaf of $(X,{\pi_{\scriptscriptstyle X}})$, and
the map $\mathbb{T}\times\Sigma\to L,(t,x)\to tx,$ is a submersion.

We now recall a construction from [16, $\S$2.6]. Let $\mathfrak{t}$ be the Lie
algebra of $\mathbb{T}$ and assume that $\langle\,,\,\rangle$ is a symmetric
non-degenerate bilinear form on $\mathfrak{t}$. Let
$\\{h_{i}:i=1,\ldots,r=\dim\mathfrak{t}\\}$ be an orthonormal basis of
$\mathfrak{t}$ with respect to $\langle\,,\,\rangle$. Given a
$\mathbb{T}$-Poisson manifold $(X,{\pi_{\scriptscriptstyle X}})$ with the
$\mathbb{T}$-action $\mathbb{T}\times X\to X$, we then have the Poisson
structure ${\pi_{\scriptscriptstyle X}}\bowtie_{\sigma}0$ on
$X\times\mathbb{T}$ given by

${\pi_{\scriptscriptstyle X}}\bowtie_{\sigma}0=({\pi_{\scriptscriptstyle
X}},\,0)+\sum_{i=1}^{r}(\sigma(h_{i}),0)\wedge(0,h_{i}^{R}),$ (1.9)

where $h_{i}^{R}$ denoted the right (left) invariant vector field on
$\mathbb{T}$ defined by $h_{i}$. We call ${\pi_{\scriptscriptstyle
X}}\bowtie_{\sigma}0$ a $\mathbb{T}$-extension of the Poisson structure
${\pi_{\scriptscriptstyle X}}$, and we call
$(X\times\mathbb{T},{\pi_{\scriptscriptstyle X}}\bowtie_{\sigma}0)$ a
$\mathbb{T}$-extension of the $\mathbb{T}$-Poisson manifold
$(X,{\pi_{\scriptscriptstyle X}})$. It is easy to see that
$(X\times\mathbb{T},{\pi_{\scriptscriptstyle X}}\bowtie 0)$ is a
$\mathbb{T}$-Poisson manifold with respect to the diagonal
$\mathbb{T}$-action. The following fact is proved in [16, Lemma 2.23].

######  Lemma 1.4.

With respect to the diagonal $\mathbb{T}$-action on $X\times\mathbb{T}$, the
$\mathbb{T}$-leaves of $(X\times\mathbb{T},{\pi_{\scriptscriptstyle
X}}\bowtie_{\sigma}0)$ are precisely all the sub-manifolds of the form
$L\times\mathbb{T}$, where $L$ is a $\mathbb{T}$-leaf of
$(X,{\pi_{\scriptscriptstyle X}})$.

###  1.7. Acknowledgments

We would like to thank L. Shen and D. Weng for explaining to us their work in
[25]. The research by the first and the third authors in this paper was
partially supported by the RGC of the Hong Kong SAR, China (GRF 17304415 and
GRF 17307718), and the paper was largely written while the second author was a
faculty member at the School of Mathematical Sciences of Shanghai Jiaotong
University. The third author acknowledges support from the NSF China
(1210010907).

##  2\. A series of Poisson groupoids associated to Poisson Lie groups

In this section, we give a construction of a series of Poisson groupoids
associated to any Poisson Lie group and a closed Poisson Lie subgroup.

###  2.1. Poisson Lie group actions and gauge Poisson groupoids

We refer to [3, 6] and especially to [17, $\S$2] for basic facts and sign
conventions on Poisson Lie groups and Lie bialgebras.

Recall first that a Poisson Lie group is a pair $(G,{\pi_{\scriptscriptstyle
G}})$, where $G$ is a Lie group and ${\pi_{\scriptscriptstyle G}}$ a Poisson
bi-vector field on $G$ which is multiplicative in the sense that the group
multiplication $(G\times G,\,{\pi_{\scriptscriptstyle
G}}\times{\pi_{\scriptscriptstyle G}})\rightarrow(G,{\pi_{\scriptscriptstyle
G}})$ is a Poisson map. A right Poisson action of a Poisson Lie group
$(G,{\pi_{\scriptscriptstyle G}})$ on a Poisson manifold
$(X,{\pi_{\scriptscriptstyle X}})$ is, by definition, a Poisson map

$\rho:\;\;(X\times G,\,{\pi_{\scriptscriptstyle
X}}\times{\pi_{\scriptscriptstyle
G}})\longrightarrow(X,\,{\pi_{\scriptscriptstyle X}}),\;\;(x,\,g)\longmapsto
xg,$ (2.10)

which also defines a right Lie group action of $G$ on $X$. Left Poisson
actions of $(G,{\pi_{\scriptscriptstyle G}})$ are defined similarly. Recall
also that a coisotropic subgroup of a Poisson Lie group
$(G,{\pi_{\scriptscriptstyle G}})$ is a Lie subgroup of $G$ which is also a
coisotropic sub-manifold with respect to the Poisson structure
${\pi_{\scriptscriptstyle G}}$. The following fundamental fact is proved in
[26].

######  Lemma 2.1.

Suppose that $\rho$ is a Poisson Lie group action as in (2.10), and suppose
that $Q$ is a coisotropic subgroup of $(G,{\pi_{\scriptscriptstyle G}})$ such
that the restricted action of $Q$ on $X$ is free and the quotient $X/Q$ is a
smooth manifold. Then the Poisson structure ${\pi_{\scriptscriptstyle X}}$
projects to a well-defined Poisson structure on $X/Q$, which will be called
the quotient Poisson structure of ${\pi_{\scriptscriptstyle X}}$.

######  Example 2.2.

Suppose that $(X,{\pi_{\scriptscriptstyle X}})$ and
$(Y,{\pi_{\scriptscriptstyle Y}})$ are Poisson manifolds, both with right
Poisson actions by a Poisson Lie group $(G,{\pi_{\scriptscriptstyle G}})$ and
such that the diagonal action of $G$ on $X\times Y$ is free and the quotient
space $(X\times Y)/G$ is a smooth manifold. Then $(X\times
Y,{\pi_{\scriptscriptstyle X}}\times(-{\pi_{\scriptscriptstyle Y}}))$ has the
right product Poisson action by the product Poisson Lie group $(G\times
G,{\pi_{\scriptscriptstyle G}}\times(-{\pi_{\scriptscriptstyle G}}))$. Since
the diagonal of $G\times G$ is a coisotropic subgroup with respect to
${\pi_{\scriptscriptstyle G}}\times(-{\pi_{\scriptscriptstyle G}})$, the
quotient space $(X\times Y)/G$ has the well-defined quotient Poisson structure
of ${\pi_{\scriptscriptstyle X}}\times(-{\pi_{\scriptscriptstyle Y}})$.
$\diamond$

Suppose now that $X\to X/G$ is a principal $G$-bundle for a Lie group $G$. Let
$(X\times X)/G$ be the quotient space for the diagonal $G$-action on $X\times
X$, and denote elements in $X/G$ by $[x]$ and in $(X\times X)/G$ by
$[x_{1},x_{2}]$, where $x,x_{1},x_{2}\in X$. Recall that the gauge groupoid of
$X\to X/G$ is the manifold $(X\times X)/G$ with the following Lie groupoid
structure over $X/G$:

$\displaystyle\mbox{source map}\;\theta_{+}:\;\;(X\times X)/G\longrightarrow
X/G:\;\;[x_{1},\,x_{2}]\longmapsto[x_{1}],$ $\displaystyle\mbox{target
map}\;\theta_{-}:\;\;(X\times X)/G\longrightarrow
X/G:\;\;[x_{1},\,x_{2}]\longmapsto[x_{2}],$ $\displaystyle\mbox{unit
map}\;\epsilon:\;\;X/G\longrightarrow(X\times X)/G,\;\;[x]\longmapsto[x,\,x],$
$\displaystyle\mbox{inverse map}\;{\rm inv}:\;\;(X\times
X)/G\longrightarrow(X\times X)/G:\;\;[x_{1},\,x_{2}]\longmapsto[x_{2},x_{1}],$
$\displaystyle\mbox{multiplication:}\;\mbox{for}\;\gamma=[x_{1},\,x_{2}]\;\mbox{and}\;\gamma^{\prime}=[x_{3},\,x_{4}]\;\mbox{with}\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime}),\;\gamma\gamma^{\prime}=[x_{1}g,\,x_{4}],$
$\displaystyle\hskip 14.45377pt\hskip 14.45377pt\hskip 14.45377pt\hskip
14.45377pt\hskip 14.45377pt\mbox{where}\;g\in G\;\mbox{is the unique element
such that}\;x_{2}g=x_{3}.$

Assume, in addition, that ${\pi_{\scriptscriptstyle G}}$ is a multiplicative
Poisson structure on $G$, ${\pi_{\scriptscriptstyle X}}$ is a Poisson
structure on $X$ such that the $G$-action on $X$ for the principal bundle
$X\to X/G$ is a right Poisson action of the Poisson Lie group
$(G,\pi_{\scriptscriptstyle G})$ on $(X,{\pi_{\scriptscriptstyle X}})$. Let
$\pi\in{\mathfrak{X}}^{2}((X\times X)/G)$ be the quotient Poisson structure of
${\pi_{\scriptscriptstyle X}}\times(-{\pi_{\scriptscriptstyle X}})$ and
$\pi_{{\scriptscriptstyle X}/{\scriptscriptstyle
G}}\in{\mathfrak{X}}^{2}(X/G)$ the quotient Poisson structure of
${\pi_{\scriptscriptstyle X}}$. The following should be well-known but we have
not been able to find a reference.

######  Lemma 2.3.

With the Poisson structure $\pi$ on $(X\times X)/G$ and
$\pi_{{\scriptscriptstyle X}/{\scriptscriptstyle G}}$ on $X/G$, the gauge
groupoid $(X\times X)/G\rightrightarrows X/G$ is a Poisson groupoid.

###### Proof.

Let $\Gamma=(X\times X)/G$ for notational simplicity. Let
$Y=\\{(x_{1},x_{2},x_{2},x_{4},x_{1},x_{4})\in X^{6}\\}$. It is clear that $Y$
is a coisotropic sub-manifold of $(X^{6},\pi_{\scriptscriptstyle X}^{(6)})$,
where

$\pi_{\scriptscriptstyle X}^{(6)}={\pi_{\scriptscriptstyle
X}}\times(-{\pi_{\scriptscriptstyle X}})\times{\pi_{\scriptscriptstyle
X}}\times(-{\pi_{\scriptscriptstyle X}})\times(-{\pi_{\scriptscriptstyle
X}})\times{\pi_{\scriptscriptstyle X}}\in{\mathfrak{X}}^{2}(X^{6}).$

Let
$Y_{1}=\\{(\gamma,\gamma^{\prime},\gamma\gamma^{\prime})\in\Gamma^{3}:\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime})\\}$,
and let

$J:\;X^{6}\longrightarrow\Gamma^{3},\;\;(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})\longmapsto([x_{1},x_{2}],[x_{3},x_{4}],[x_{5},x_{6}]).$

Then $J(Y)\subset Y_{1}$. Conversely, given
$y_{1}=(\gamma,\gamma^{\prime},\gamma\gamma^{\prime})\in Y_{1}$, let
$\gamma=[x_{1},\,x_{2}]$ and $\gamma^{\prime}=[x_{3},\,x_{4}]$ for some
$x_{1},x_{2},x_{3},x_{4}\in X$ such that $[x_{2}]=[x_{3}]$, and let $g$ be the
unique element in $G$ such that $x_{2}g=x_{3}$. Then
$y=(x_{1}g,x_{2}g,x_{3},x_{4},x_{1}g,x_{4})\in Y$ and $J(y)=y_{1}$. This shows
that $J(Y)=Y_{1}$. Since $J:(X^{6},\pi_{\scriptscriptstyle
X}^{(6)})\to(\Gamma^{3},\pi\times\pi\times(-\pi))$ is Poisson, it follows from
the definition of coisotropic sub-manifolds that $Y_{1}=J(Y)$ is coisotropic
in $(\Gamma^{3},\pi\times\pi\times(-\pi))$. Note also that both $\theta_{+}$
and $\theta_{-}$ are surjective submersions. Thus $(\Gamma,\pi)$ is a Poisson
groupoid. By the definition of the Poisson structure $\pi_{{\scriptscriptstyle
X}/{\scriptscriptstyle G}}$, $(\Gamma,\pi)$ is a Poisson groupoid over
$(X/G,\pi_{{\scriptscriptstyle X}/{\scriptscriptstyle G}})$.

Q.E.D.

We now adapt the constructions of quotient Poisson structures in Example 2.2
and the gauge Poisson groupoids in Lemma 2.3 to a setting suitable for
applications in this paper.

######  Example 2.4.

[17, $\S$7.1] Suppose that $(G,{\pi_{\scriptscriptstyle G}})$ is a Poisson Lie
group, $(X,{\pi_{\scriptscriptstyle X}})$ is Poisson manifold with a free
right Poisson action $(x,g)\mapsto xg$ by $(G,{\pi_{\scriptscriptstyle G}})$,
and $(Y,{\pi_{\scriptscriptstyle Y}})$ is a Poisson manifold with a left
Poisson action $(g,y)\mapsto gy$ by $(G,{\pi_{\scriptscriptstyle G}})$. Then
one has the right Poisson action of the Poisson Lie group $(G\times
G,{\pi_{\scriptscriptstyle G}}\times(-{\pi_{\scriptscriptstyle G}}))$ on
$(X\times Y,\,{\pi_{\scriptscriptstyle X}}\times{\pi_{\scriptscriptstyle Y}})$
given by $(x,y)\cdot(g_{1},g_{2})=(xg_{1},\;g_{2}^{-1}y)$. Denote by
$X\times_{G}Y$ the quotient of $X\times Y$ by the diagonal $G$-action
$(x,\,y)\cdot g=(xg,\,g^{-1}y)$ for $x\in X$, $y\in Y$, and $g\in G$. Assuming
that $X\times_{G}Y$ is a smooth manifold, it then has the well-defined
quotient Poisson structure of ${\pi_{\scriptscriptstyle
X}}\times{\pi_{\scriptscriptstyle Y}}$. $\diamond$

Assume now that $(X,{\pi_{\scriptscriptstyle X}})$ is a Poisson manifold with
a right Poisson action $(x,g)\to xg$ by a Poisson Lie group
$(G,{\pi_{\scriptscriptstyle G}})$ and assume that $X\to X/G$ is a principal
bundle. We make the further assumption that $\kappa:X\to X$ is an anti-Poisson
involution with respect to the Poisson structure ${\pi_{\scriptscriptstyle
X}}$. One then has the unique left action $(g,x)\mapsto gx$ of $G$ on $X$
determined by

$\kappa(g^{-1}x)=\kappa(x)g,\hskip 14.45377ptx\in X,\,g\in G,$

which is a left Poisson action of $(G,{\pi_{\scriptscriptstyle G}})$ on
$(X,{\pi_{\scriptscriptstyle X}})$. Applying the construction in Example 2.4,
one has the quotient space $X\times_{G}X$ of $X\times X$ by the right
$G$-action

$(x_{1},\,x_{2})\cdot g=(x_{1}g,\,g^{-1}x_{2}),\hskip 14.45377ptx_{1},x_{2}\in
X,\,g\in G,$

and the quotient Poisson structure $\pi$ on $X\times_{G}X$ of
${\pi_{\scriptscriptstyle X}}\times{\pi_{\scriptscriptstyle X}}$. Denote again
elements in $X/G$ by $[x]$ and in $X\times_{G}X$ by $[x_{1},x_{2}]$, where
$x,x_{1},x_{2}\in X$, and let again $\pi_{{\scriptscriptstyle
X}/{\scriptscriptstyle G}}\in{\mathfrak{X}}^{2}(X/G)$ be the quotient Poisson
structure of ${\pi_{\scriptscriptstyle X}}$.

######  Lemma 2.5.

With the assumption and notation as above, $(X\times_{G}X,\pi)$ is a Poisson
groupoid over $(X/G,\pi_{{\scriptscriptstyle X}/{\scriptscriptstyle G}})$ with
the following groupoid structure:

$\displaystyle\mbox{source map}\;\theta_{+}:\;\;X\times_{G}X\longrightarrow
X/G:\;\;[x_{1},\,x_{2}]\longmapsto[x_{1}],$ $\displaystyle\mbox{target
map}\;\theta_{-}:\;\;X\times_{G}X\longrightarrow
X/G:\;\;[x_{1},\,x_{2}]\longmapsto[\kappa(x_{2})],$ $\displaystyle\mbox{unit
map}\;\epsilon:\;\;X/G\longrightarrow
X\times_{G}X,\;\;[x]\longmapsto[x,\,\kappa(x)],$ $\displaystyle\mbox{inverse
map}\;{\rm inv}:\;\;X\times_{G}X\longrightarrow
X\times_{G}X:\;\;[x_{1},\,x_{2}]\longmapsto[\kappa(x_{2}),\,\kappa(x_{1})],$
$\displaystyle\mbox{multiplication:}\;\mbox{for}\;\gamma=[x_{1},\,x_{2}]\;\mbox{and}\;\gamma^{\prime}=[x_{3},\,x_{4}]\;\mbox{with}\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime}),\;\gamma\gamma^{\prime}=[x_{1}g,\,x_{4}],$
$\displaystyle\hskip 14.45377pt\hskip 14.45377pt\hskip 14.45377pt\hskip
14.45377pt\hskip 14.45377pt\mbox{where}\;g\in G\;\mbox{is the unique element
such that}\;\kappa(x_{2})g=x_{3}.$

###### Proof.

The map $X\times_{G}X\to(X\times X)/G,[x_{1},x_{2}]\to[x_{1},k(x_{2})]$ is
both an isomorphism of Lie groupoids and an isomorphism of Poisson manifolds,
where $(X\times X)/G$ has the gauge Poisson groupoid for the right Poisson
action $(x,g)\to xg$ as in Lemma 2.3.

Q.E.D.

###  2.2. A series of Poisson groupoids

Assume that $(G,{\pi_{\scriptscriptstyle G}})$ is a Poisson Lie group and that
$Q$ a closed Poisson Lie subgroup. For each integer $n\geq 1$, one then has
the quotient spaces

$X_{n}=\overbrace{G\times_{Q}\cdots\times_{Q}G}^{n}\hskip
14.45377pt\mbox{and}\hskip
14.45377ptY_{n}=\overbrace{G\times_{Q}\cdots\times_{Q}G}^{n}/Q.$ (2.11)

The manifolds $X_{n}$ and $Y_{n}$ can be taken as successive quotient spaces
as in Example 2.4. Consequently, one has the well-defined quotient Poisson
structures on $X_{n}$ and $Y_{n}$, i.e., well-defined projections of the
$n$-fold product Poisson structure $\pi_{\scriptscriptstyle G}^{n}$ on
$G^{n}$, which we will denote respectively as $\pi_{{\scriptscriptstyle
X}_{n}}$ and $\pi_{{\scriptscriptstyle Y}_{n}}$. By the multiplicativity of
${\pi_{\scriptscriptstyle G}}$, one has the following Poisson Lie group
actions

$\displaystyle(G,{\pi_{\scriptscriptstyle
G}})\times(X_{n},\,\pi_{{\scriptscriptstyle
X}_{n}})\longrightarrow(X_{n},\,\pi_{{\scriptscriptstyle
X}_{n}}),\;(g,\,[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
X}_{n}})\longmapsto[gg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle X}_{n}},$
(2.12) $\displaystyle(X_{n},\,\pi_{{\scriptscriptstyle
X}_{n}})\times(G,{\pi_{\scriptscriptstyle
G}})\longrightarrow(X_{n},\,\pi_{{\scriptscriptstyle
X}_{n}}),\;([g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
X}_{n}},\,g)\longmapsto[g_{1},\ldots,g_{n-1},g_{n}g]_{{\scriptscriptstyle
X}_{n}},$ (2.13) $\displaystyle(G,{\pi_{\scriptscriptstyle
G}})\times(Y_{n},\,\pi_{{\scriptscriptstyle
Y}_{n}})\longrightarrow(Y_{n},\,\pi_{{\scriptscriptstyle
Y}_{n}}),\;(g,\,[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
Y}_{n}})\longmapsto[gg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle Y}_{n}}.$
(2.14)

As the inverse map on $G$ is an anti-Poisson involution for
$(G,{\pi_{\scriptscriptstyle G}})$, it follows that

${\bf I}_{{\scriptscriptstyle X}_{n}}:\;(X_{n},\,\pi_{{\scriptscriptstyle
X}_{n}})\longrightarrow(X_{n},\,\pi_{{\scriptscriptstyle
X}_{n}}),\;[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
X}_{n}}\longmapsto[g_{n}^{-1},\ldots,g_{2}^{-1},g_{1}^{-1}]_{{\scriptscriptstyle
X}_{n}}$

is anti-Poisson. As a direct application of Lemma 2.5 by taking $X=X_{n}$ with
the right Poisson action by the Poisson Lie group $(Q,{\pi_{\scriptscriptstyle
G}}|_{\scriptscriptstyle Q})$ given in (2.13) and by taking $\kappa={\bf
I}_{{\scriptscriptstyle X}_{n}}$, one has

######  Theorem 2.6.

For any Poisson Lie group $(G,{\pi_{\scriptscriptstyle G}})$, any closed
Poisson Lie group $Q$ of $(G,{\pi_{\scriptscriptstyle G}})$, and any positive
integer $n$, $(X_{2n},\pi_{{\scriptscriptstyle X}_{2n}})$ is a Poisson
groupoid over $(Y_{n},\pi_{{\scriptscriptstyle Y}_{n}})$ with the following
groupoid structure:

$\displaystyle\mbox{source map}\;\theta_{+}:\;\;X_{2n}\to
Y_{n}:\;\;[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
X}_{2n}}\mapsto[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle Y}_{n}};$
$\displaystyle\mbox{target map}\;\theta_{-}:\;\;X_{2n}\to
Y_{n}:\;\;[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
X}_{2n}}\mapsto[g_{2n}^{-1},\ldots,g_{n+1}^{-1}]_{{\scriptscriptstyle
Y}_{n}};$ $\displaystyle\mbox{unit map}\;\epsilon:\;\;Y_{n}\to
X_{2n},\;\;[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle
Y}_{n}}\mapsto[g_{1},\ldots,g_{n},\,g_{n}^{-1},\ldots,g_{1}^{-1}]_{{\scriptscriptstyle
X}_{2n}};$ $\displaystyle\mbox{inverse map}\;{\rm inv}:\;\;X_{2n}\to
X_{2n}:\;\;[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
X}_{2n}}\mapsto[g_{2n}^{-1},\ldots,g_{1}^{-1}]_{{\scriptscriptstyle X}_{2n}};$
$\displaystyle\mbox{multiplication:}\;\mbox{for}\;\gamma=[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
X}_{2n}}\;\mbox{and}\;\gamma^{\prime}=[g_{1}^{\prime},\ldots,g_{2n}^{\prime}]_{{\scriptscriptstyle
X}_{2n}}\;\mbox{with}\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime}),$
$\displaystyle\hskip 14.45377pt\hskip
14.45377pt\gamma\gamma^{\prime}=[g_{1},\;\ldots,\;g_{n},\;g_{n+1}\cdots
g_{2n}g_{1}^{\prime}\cdots
g_{n}^{\prime}g_{n+1}^{\prime},\;g_{n+2}^{\prime},\ldots,g_{2n}^{\prime}]_{{\scriptscriptstyle
X}_{2n}}.$

##  3\. The total configuration Poisson groupoids of flags and isomorphic
models

We now apply the construction in $\S$2 to the standard complex semisimple Lie
group $(G,\pi_{\rm st})$.

###  3.1. The complex semisimple Poisson Lie group $(G,\pi_{\rm st})$

Let $G$ be a connected complex semi-simple Lie group with Lie algebra
$\mathfrak{g}$. We recall the so-called standard multiplicative Poisson
structure on $G$ and refer the readers to [3, 6, 17, 18] for more detail.

Fix again a pair $(B,B_{-})$ of opposite Borel subgroups of $G$ and let
$T=B\cap B_{-}$, a maximal torus of $G$. Let $N$ and $N-$ the respective
unipotent radicals of $B$ and $B_{-}$. Denote the Lie algebras of
$G,B,B_{-},N,N_{-}$ and $T$ by
$\mathfrak{g},\mathfrak{b},\mathfrak{b}_{-},\mathfrak{n}$ and $\mathfrak{h}$
respectively. Fix also a non-degenerate symmetric invariant bi-linear form
$\langle\,,\,\rangle_{\mathfrak{g}}$ on $\mathfrak{g}$. Let
$\mathfrak{g}=\mathfrak{h}+\sum_{\alpha\in\Delta}\mathfrak{g}_{\alpha}$ be the
root decomposition of $\mathfrak{g}$ with respect to $\mathfrak{h}$, let
$\Delta^{+}\subset\mathfrak{g}^{*}$ be the set of positive roots with respect
to $\mathfrak{b}$, and for each $\alpha\in\Delta^{+}$, let
$E_{\alpha}\in\mathfrak{g}_{\alpha}$ and
$E_{-\alpha}\in\mathfrak{g}_{-\alpha}$ be such that $\langle
E_{\alpha},E_{-\alpha}\rangle_{\mathfrak{g}}=1$. Let
$\\{h_{i}\\}_{i=1}^{\dim\mathfrak{h}}$ be a basis of $\mathfrak{h}$ which is
orthonormal with respect to the restriction of
$\langle\,,\,\rangle_{\mathfrak{g}}$ to $\mathfrak{h}$. Then the element

$r_{\rm st}=\sum_{i=1}^{\dim\mathfrak{h}}h_{i}\otimes
h_{i}+2\sum_{\alpha\in\Delta^{+}}E_{-\alpha}\otimes
E_{\alpha}\in\mathfrak{g}\otimes\mathfrak{g}$ (3.1)

is called the standard $r$-matrix on $\mathfrak{g}$. Let $\Lambda_{\rm
st}\in\wedge^{2}\mathfrak{g}$ be the skew-symmetric part of $r_{\rm st}$,
i.e.,

$\Lambda_{\rm st}=\sum_{\alpha\in\Delta^{+}}(E_{-\alpha}\wedge
E_{\alpha})=\sum_{\alpha\in\Delta^{+}}(E_{-\alpha}\otimes
E_{\alpha}-E_{\alpha}\otimes E_{-\alpha}),$

and let $\pi_{\rm st}$ be the bi-vector field on $G$ defined by (see notation
in $\S$1.6)

$\pi_{\rm st}={\Lambda}_{\rm st}^{L}-{\Lambda}_{\rm st}^{R}.$ (3.2)

Then $\pi_{\rm st}$ is a multiplicative Poisson structure on $G$, and
$(G,\pi_{\rm st})$ is a standard semi-simple Poisson Lie group. It is well-
known (see, for example, [10]) that both $B$ and $B_{-}$ are Poisson Lie
subgroups of $(G,\pi_{\rm st})$. One thus has the Poisson Lie subgroups
$(B,\pi_{\rm st})$ and $(B_{-},\pi_{\rm st})$ of $(G,\pi_{\rm st})$.

###  3.2. The total configuration Poisson groupoids of flags

Continuing with the set-up in $\S$3.1, we can now apply the constructions in
$\S$2.2 to the Poisson Lie group $(G,\pi_{\rm st})$ and its closed Poisson Lie
subgroup $B$. In this particular case, as we have already done in $\S$1.1, we
denote the Poisson spaces $X_{n}$ and $Y_{n}$ in (2.11) respectively as

$\widetilde{F}_{n}=\overbrace{G\times_{B}\cdots\times_{B}G}^{n}\hskip
14.45377pt\mbox{and}\hskip
14.45377ptF_{n}=\overbrace{G\times_{B}\cdots\times_{B}G}^{n}/B,$

and we denote the quotient Poisson structures on $\widetilde{F}_{n}$ and
$F_{n}$ respectively as $\widetilde{\pi}_{n}$ and $\pi_{n}$. Note that we have
Poisson Lie group actions

$\displaystyle\tilde{\lambda}_{n}:\;\;(G,\pi_{\rm
st})\times(\widetilde{F}_{n},\widetilde{\pi}_{n})\longrightarrow(\widetilde{F}_{n},\widetilde{\pi}_{n}),\;\;(g,[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}})\longmapsto[gg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}},$
(3.3) $\displaystyle\lambda_{n}:\;\;(G,\pi_{\rm
st})\times(F_{n},\pi_{n})\longrightarrow(F_{n},\pi_{n}),\;\;(g,[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}})\longmapsto[gg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle F}_{n}}.$
(3.4)

Recall that $T=B\cap B_{-}$, a maximal torus of $G$. Since $\pi_{\rm
st}|_{T}=0$, the restrictions of the actions to $T$ (see (1.2) and (1.3)) make
both $(\widetilde{F}_{n},\widetilde{\pi}_{n})$ and $(F_{n},\pi_{n})$ into
$T$-Poisson manifolds. The multiplicativity of $\pi_{\rm st}$ also implies
that we have the well-defined Poisson map

$\mu_{{\scriptscriptstyle\widetilde{F}}_{n}}:\;(\widetilde{F}_{n},\widetilde{\pi}_{n})\longrightarrow(G,\pi_{\rm
st}),\;\;[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}}\longmapsto
g_{1}g_{2}\cdots g_{n}.$ (3.5)

For ${\bf w}=(w_{1},\ldots,w_{n})\in W^{n}$, set

$B{\bf
w}B=(Bw_{1}B)\times_{B}\cdots\times_{B}(Bw_{n}B)\subset\widetilde{F}_{n},$
(3.6)

the image of $(Bw_{1}B)\times\cdots\times(Bw_{n}B)$ in $\widetilde{F}_{n}$.
One thus has

$\widetilde{F}_{n}=\bigsqcup_{{\bf w}\in W^{m},v\in W}(B{\bf
w}B)\cap\mu_{{\scriptscriptstyle\widetilde{F}}_{n}}^{-1}(B_{-}vB_{-})\hskip
14.45377pt(\mbox{disjoint union}).$ (3.7)

For $w\in W$, let $l(w)$ be the length of $w$. For ${\bf
w}=(w_{1},\ldots,w_{n})\in W^{n}$, let $l({\bf w})=l(w_{1})+\cdots+l(w_{n})$.
The following description of $T$-leaves of
$(\widetilde{F}_{n},\widetilde{\pi}_{n})$ is proved in [18, Theorem 1.3].

######  Proposition 3.1.

1) For any ${\bf w}\in W^{n}$ and $v\in W$, the intersection $(B{\bf
w}B)\cap\mu_{{\scriptscriptstyle\widetilde{F}}_{n}}^{-1}(B_{-}vB_{-})$ is a
non-empty smooth sub-manifold of $\widetilde{F}_{n}$ of dimension $l({\bf
w})+l(v)+\dim T$;

2) The decomposition in (3.7) is that of $\widetilde{F}_{n}$ into the
$T$-leaves of $\widetilde{\pi}_{n}$.

Introduce now

$\Gamma_{n}=\\{[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}}:\,g_{1}g_{2}\cdots
g_{n}\in
B_{-}\\}=\mu_{{\scriptscriptstyle\widetilde{F}}_{n}}^{-1}(B_{-})\subset\widetilde{F}_{n}.$
(3.8)

By Proposition 3.1, $\Gamma_{n}$ is a union of $T$-leaves of
$\widetilde{\pi}_{n}$ and thus a Poisson sub-manifold of
$(\widetilde{F}_{n},\widetilde{\pi}_{n})$. We now apply Theorem 2.6 to the
Poisson Lie group $(G,\pi_{\rm st})$ and its Poisson Lie subgroup $B$.

######  Theorem 3.2.

For any integer $n\geq 1$, $(\widetilde{F}_{2n},\widetilde{\pi}_{2n})$ is a
Poisson groupoid over $(F_{n},\pi_{n})$ with the following groupoid structure:

$\displaystyle\mbox{source map}\;\theta_{+}:\widetilde{F}_{2n}\to
F_{n},\;[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\mapsto[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}};$ $\displaystyle\mbox{target map}\;\theta_{-}:\widetilde{F}_{2n}\to
F_{n},\;[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\mapsto[g_{2n}^{-1},\ldots,g_{n+1}^{-1}]_{{\scriptscriptstyle
F}_{n}};$ $\displaystyle\mbox{unit
map}\;\epsilon:F_{n}\to\widetilde{F}_{2n},\;[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}\mapsto[g_{1},\ldots,g_{n},g_{n}^{-1},\ldots,g_{1}^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}};$
$\displaystyle\mbox{inverse map}\;{\rm
inv}:\widetilde{F}_{2n}\to\widetilde{F}_{2n},\;[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\mapsto[g_{2n}^{-1},\ldots,g_{1}^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}};$
$\displaystyle\mbox{multiplication:}\;\mbox{for}\;\gamma=[g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\;\mbox{and}\;\gamma^{\prime}=[g_{1}^{\prime},\ldots,g_{2n}^{\prime}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\;\mbox{with}\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime}),$
$\displaystyle\hskip 14.45377pt\hskip 14.45377pt\hskip 14.45377pt\hskip
14.45377pt\hskip
14.45377pt\gamma\gamma^{\prime}=[g_{1},\;\ldots,\;g_{n},\;g_{n+1}\cdots
g_{2n}g_{1}^{\prime}\cdots
g_{n}^{\prime}g_{n+1}^{\prime},\;g_{n+2}^{\prime},\,\ldots,\,g_{2n}^{\prime}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}.$

Furthermore, $\Gamma_{2n}\subset\widetilde{F}_{2n}$ is a wide Poisson sub-
groupoid of
$(\widetilde{F}_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$.

###### Proof.

Applying Theorem 2.6 directly to the Poisson Lie group $(G,\pi_{\rm st})$ and
its Poisson Lie subgroup $B$, we see that
$(\widetilde{F}_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$ as
described in Theorem 3.2 is a Poisson groupoid. One also checks directly from
the definitions of the structure maps of $\widetilde{F}_{2n}\rightrightarrows
F_{n}$ that $\Gamma_{2n}$ is a wide (set-theoretical) sub-groupoid of
$\widetilde{F}_{2n}\rightrightarrows F_{n}$ (Definition 1.1). As $\Gamma_{2n}$
is a Poisson sub-manifold of $(\widetilde{F}_{2n},\widetilde{\pi}_{2n})$, it
is a Poisson sub-groupoid of
$(\widetilde{F}_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$ as
long as we prove that $\theta_{+}:\widetilde{F}_{2n}\to F_{n}$ restricts to a
submersion from $\Gamma_{2n}$ to $F_{n}$. To prove this latter statement, we
note the isomorphisms

$\displaystyle\widetilde{F}_{2n}\longrightarrow(G/B)^{2n-1}\times
G,\;\;[g_{1},g_{2},\ldots,g_{2n}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\longmapsto(g_{1}{}_{\cdot}B,g_{1}g_{2}{}_{\cdot}B,\ldots,g_{1}g_{2}\cdots
g_{2n-1}{}_{\cdot}B,g_{1}g_{2}\cdots g_{2n}),$ $\displaystyle
F_{n}\longrightarrow(G/B)^{n},\;\;[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}\longmapsto(g_{1}{}_{\cdot}B,g_{1}g_{2}{}_{\cdot}B,\ldots,g_{1}g_{2}\cdots
g_{n}{}_{\cdot}B),$

under which $\Gamma_{2n}$ is mapped to $(G/B)^{2n-1}\times B_{-}$, and
$\theta_{+}|_{\Gamma_{2n}}:\Gamma_{2n}\to F_{n}$ becomes the projection of
$(G/B)^{2n-1}\times B_{-}$ from the product $(G/B)^{n}$ of the first $n$
factors and is thus a submersion.

Q.E.D.

######  Definition 3.3.

For $n\geq 1$ the Poisson groupoid
$(\Gamma_{2n},\widetilde{\pi}_{2n}))\rightrightarrows(F_{n},\pi_{n})$ in
Theorem 3.2 is called the $n$th total configuration Poisson groupoid of flags
of $G$. $\diamond$

In the next $\S$3.3 and $\S$3.4, we introduce two isomorphic models of
$(\Gamma_{2n},\widetilde{\pi}_{2n}))\rightrightarrows(F_{n},\pi_{n})$.

###  3.3. The Poisson groupoid
$({\mathcal{C}}_{2n},\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{n},\overline{\pi}_{n})$

Recall again that ${\mathcal{B}}=G/B$ is the (full) flag variety of $G$, and
${\mathcal{A}}=G/N$ is the decorated flag variety of $G$, where $N$ is the
unipotent radical of $B$. For an integer $n\geq 1$, recall that we have set

${\mathcal{C}}_{n}={\mathcal{B}}^{n-1}\times{\mathcal{A}}^{o},$ (3.9)

where ${\mathcal{A}}^{o}=B_{-}N/N$ is the open $B_{-}$-orbit in
${\mathcal{A}}=G/N$. Referring to an element $f\in{\mathcal{B}}$ as a flag and
an element $\hat{f}$ in ${\mathcal{A}}$ as a decorated flag, the space
${\mathcal{C}}_{n}$ then consists of all $n$-tuples
$(f_{1},\ldots,f_{n-1},\hat{f}_{n})$, where $f_{1},\ldots,f_{n-1}$ are flags
and $\hat{f}_{n}$ is a decorated flag that is in general position with the
flag $f_{-}$ represented by $B_{-}$.

The spaces ${\mathcal{C}}_{n}$ appeared in [25], where the authors consider
the special cases when $G=G_{\rm sc}$ is simply connected and when $G=G_{\rm
ad}$ is of adjoint type. For any pair of words $(b,d)$ of length $n$ in the
simple reflections in $W$ (called positive braids in [25]), Shen and Weng
introduced certain configuration spaces of flags, denoted as ${\rm
Conf}^{b}_{d}({\mathcal{A}}_{\rm sc})$ and ${\rm
Conf}^{b}_{d}({\mathcal{A}}_{\rm ad})$ and called decorated double Bott-
Samelson cells (see [25, $\S$2.2]), which can be embedded in
${\mathcal{C}}_{n}$ for $G=G_{\rm sc}$ and $G=G_{\rm ad}$ respectively [25,
$\S$2.3] (see Remark 4.3). As one of their main results, Shen and Weng prove
in [25, Theorem 1.1 and Theorem 1.2] that both ${\rm
Conf}^{b}_{d}({\mathcal{A}}_{\rm sc})$ and ${\rm
Conf}^{b}_{d}({\mathcal{A}}_{\rm ad})$ are smooth affine varieties; the
coordinate rings ${\mathcal{O}}({\rm Conf}^{b}_{d}({\mathcal{A}}_{\rm sc}))$
is an upper cluster algebra, ${\mathcal{O}}({\rm
Conf}^{b}_{d}({\mathcal{A}}_{\rm ad}))$ is an upper cluster Poisson algebra,
and the pair $({\rm Conf}^{b}_{d}({\mathcal{A}}_{\rm sc}),{\rm
Conf}^{b}_{d}({\mathcal{A}}_{\rm ad}))$ form a cluster ensemble for which the
Fock-Goncharov cluster duality conjecture holds. See [25] for detail.

For $n\geq 1$, consider now the isomorphisms

$\displaystyle\widehat{\Theta}_{n}:\;\Gamma_{n}\longrightarrow{\mathcal{C}}_{n},\;[g_{1},\cdots,g_{n-1},g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}}\longmapsto(g_{1}{}_{\cdot}B,\ldots,g_{1}\cdots
g_{n-1}{}_{\cdot}B,\,g_{1}\cdots g_{n-1}g_{n}{}_{\cdot}N),$ (3.10)
$\displaystyle\Theta_{n}:\;F_{n}\longrightarrow{\mathcal{B}}^{n},\;[g_{1},\cdots,g_{n-1},g_{n}]_{{\scriptscriptstyle
F}_{n}}\longmapsto(g_{1}{}_{\cdot}B,\ldots,g_{1}\cdots
g_{n-1}{}_{\cdot}B,\,g_{1}\cdots g_{n-1}g_{n}{}_{\cdot}B).$ (3.11)

Under the isomorphisms
$\widehat{\Theta}_{2n}:\Gamma_{2n}\to{\mathcal{C}}_{2n}$ and
$\Theta_{n}:F_{n}\to{\mathcal{B}}^{n}$, one checks directly that the Lie
groupoid $\Gamma_{2n}\rightrightarrows F_{n}$ in Theorem 3.2 becomes the
following Lie groupoid ${\mathcal{C}}_{2n}\rightrightarrows{\mathcal{B}}^{n}$,
where we denote an element in ${\mathcal{A}}^{o}=B_{-}N/N$ as
$b_{-}{}_{\cdot}N$ for a unique $b_{-}\in B_{-}$:

$\displaystyle\mbox{source
map}\;\theta_{+}:{\mathcal{C}}_{2n}\to{\mathcal{B}}^{n}:(f_{1},\ldots,f_{2n-1},b_{-}{}_{\cdot}N)\mapsto(f_{1},\ldots,f_{n});$
$\displaystyle\mbox{target
map}\;\theta_{-}:{\mathcal{C}}_{2n}\to{\mathcal{B}}^{n}:(f_{1},\ldots,f_{2n-1},b_{-}{}_{\cdot}N)\mapsto(b_{-}^{-1}f_{2n-1},\ldots,b_{-}^{-1}f_{n});$
$\displaystyle\mbox{unit
map}\;\epsilon:{\mathcal{B}}^{n}\to{\mathcal{C}}_{2n},(f_{1},\ldots,f_{n})\to(f_{1},\ldots,f_{n},f_{n-1},\ldots,f_{1},e{}_{\cdot}N);$
$\displaystyle\mbox{inverse map}\;{\rm
inv}:{\mathcal{C}}_{2n}\to{\mathcal{C}}_{2n}:(f_{1},\ldots,f_{2n-1},b_{-}{}_{\cdot}N)\mapsto(b_{-}^{-1}f_{2n-1},\ldots,b_{-}^{-1}f_{2},b_{-}^{-1}f_{1},\,b_{-}^{-1}{}_{\cdot}N);$
$\displaystyle\mbox{multiplication:}\;\mbox{for}\;\gamma=(f_{1},\ldots,f_{2n-1},b_{-}{}_{\cdot}N)\;\mbox{and}\;\gamma^{\prime}=(f_{1}^{\prime},\ldots,f_{2n-1}^{\prime},b_{-}^{\prime}{}_{\cdot}N)\;\mbox{with}\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime}),$
$\displaystyle\hskip 14.45377pt\hskip 14.45377pt\hskip 14.45377pt\hskip
14.45377pt\gamma\gamma^{\prime}=(f_{1},\;\ldots,\;f_{n},\;b_{-}f_{n+1}^{\prime},\;\ldots\;b_{-}f_{2n-1}^{\prime},\;b_{-}b_{-}^{\prime}{}_{\cdot}N).$

Define Poisson structures $\widehat{\pi}_{n}$ on ${\mathcal{C}}_{n}$ and
$\overline{\pi}_{n}$ on ${\mathcal{B}}^{n}$ respectively by

$\widehat{\pi}_{n}=\widehat{\Theta}_{n}(\widetilde{\pi}_{n})\hskip
14.45377pt\mbox{and}\hskip 14.45377pt\overline{\pi}_{n}=\Theta_{n}(\pi_{n}).$
(3.12)

We now have the following direct consequence of Theorem 3.2.

######  Theorem 3.4.

For any positive integer $n$, $({\mathcal{C}}_{2n},\widehat{\pi}_{2n})$, with
the groupoid structure as above, is a algebraic Poisson groupoid over
$({\mathcal{B}}^{n},\overline{\pi}_{n})$.

For each $n\geq 1$, we call $\widehat{\pi}_{n}$ (resp. $\overline{\pi}_{n}$)
the standard Poisson structure on ${\mathcal{C}}_{n}$ (resp.
${\mathcal{B}}^{n}$). We prove in Proposition A.6 (see also Corollary A.7)
that $({\mathcal{B}}^{n},\overline{\pi}_{n})$ is a mixed product of $n$ copies
of the Poisson variety $({\mathcal{B}},\pi_{1})$ in the sense defined in [17].
A similar statement for $({\mathcal{C}}_{n},\widehat{\pi}_{n})$ is given in
Remark A.8.

###  3.4. The Poisson groupoid $(F_{2n}^{o}\times T,\pi_{2n}\bowtie
0)\rightrightarrows(F_{n},\pi_{n})$

For $n\geq 1$, let again

$F_{n}^{o}=\\{[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}:\;g_{1}g_{2}\cdots g_{n}\in B_{-}B/B\\},$ (3.13)

an open sub-manifold of $F_{n}$. Recall that for $g\in B_{-}B$, we write
$g=[g]_{-}[g]_{0}[g]_{+}$ with $[g]_{-}\in N_{-},[g]_{0}\in T$, and
$[g]_{+}\in N$. Set also $[g]_{\geq 0}=[g]_{0}[g]_{+}$ for $g\in B_{-}B$.
Define

$J_{n}:\;\;\Gamma_{n}\longrightarrow F_{n}^{o}\times
T,\;\;J_{n}([g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}})=([g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}},\,[g_{1}g_{2}\cdots g_{n}]_{0}).$ (3.14)

Recall that $T$ acts on $F_{n}$ and $\widetilde{F}_{n}$ by (1.2) and (1.3)
respectively. Let $T$ act on itself by translation and on $F_{n}^{o}\times T$
diagonally. Then $J_{n}$ is $T$-equivariant and that the inverse of $J_{n}$ is
given by

$J_{n}^{-1}([g_{1},\ldots,g_{n-1},g_{n}]_{{\scriptscriptstyle
F}_{n}},t)=[g_{1},\ldots,g_{n-1},g_{n}[g_{1}\cdots g_{n-1}g_{n}]_{\geq
0}^{-1}t]_{{\scriptscriptstyle\widetilde{F}}_{n}}.$ (3.15)

On the other hand, we have the Poisson structure $\pi_{n}\bowtie 0$ on
$F_{n}\times T$ which is the $T$-extension of $\pi_{n}$ with respect to the
$T$-action $\lambda_{n}$ on $F_{n}$ (see $\S$1.6 and (3.4)), namely,

$\pi_{n}\bowtie
0:=(\pi_{n},0)+\sum_{i=1}^{\dim\mathfrak{h}}(\lambda_{n}(h_{i}),0)\wedge(0,h_{i}^{R})\in{\mathfrak{X}}^{2}(F_{n}\times
T),$ (3.16)

where $\\{h_{i}\\}$ is a basis of the Lie algebra $\mathfrak{h}$ of $T$
orthonormal with respect to
$\langle\,,\,\rangle_{\mathfrak{g}}|_{\mathfrak{h}}$, and for
$x\in\mathfrak{h}$, $x^{R}$ is again the right (and left) invariant vector
field on $T$ with value $x$ at the identity element.

######  Theorem 3.5.

For any $n\geq 1$,
$J_{n}:(\Gamma_{n},\,\widetilde{\pi}_{n})\to(F_{n}^{o}\times
T,\,\pi_{n}\bowtie 0)$ is a Poisson isomorphism.

###### Proof.

This is Theorem B.1 in the $\S$B.

Q.E.D.

Using the isomorphism $J_{2n}:\Gamma_{2n}\to F_{2n}^{o}\times T$, we now
transfer the groupoid structure on $\Gamma_{2n}$ to one on $F_{2n}^{o}\times
T$. The following Lemma 3.6 is straightforward to check.

######  Lemma 3.6.

Under the isomorphism $J_{2n}:\Gamma_{2n}\to F_{2n}^{o}\times T$, the Lie
groupoid $\Gamma_{2n}\rightrightarrows F_{n}$ becomes the following Lie
groupoid $F_{2n}^{o}\times T\rightrightarrows F_{n}$:

$\displaystyle\mbox{source map}\;\theta_{+}:\;F_{2n}^{o}\times T\to
F_{n},\;([g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
F}_{2n}},t)\mapsto[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle F}_{n}};$
$\displaystyle\mbox{target map}\;\theta_{-}:\;F_{2n}^{o}\times T\to F_{n},\;$
$\displaystyle\hskip 14.45377pt\hskip 14.45377pt\hskip 14.45377pt\hskip
14.45377pt\hskip 14.45377pt\;\;\;\;([g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
F}_{2n}},t)\mapsto[t^{-1}[g_{1}g_{2}\cdots g_{2n}]_{\geq
0}g_{2n}^{-1},g_{2n-1}^{-1},\ldots,g_{n+1}^{-1}]_{{\scriptscriptstyle
F}_{n}};$ $\displaystyle\mbox{unit map}\;\epsilon:\;F_{n}\to F_{2n}^{o}\times
T,\;[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}\mapsto([g_{1},\ldots,g_{n},g_{n}^{-1},\ldots,g_{1}^{-1}]_{{\scriptscriptstyle
F}_{2n}},e);$ $\displaystyle\mbox{inverse map}\;{\rm inv}:F_{2n}^{o}\times
T\to F_{2n}^{o}\times T:\;\;$ $\displaystyle\hskip 14.45377pt\hskip
14.45377pt\hskip 14.45377pt\hskip 14.45377pt\hskip
14.45377pt\;\;\;\;([g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
F}_{2n}},t)\mapsto(t^{-1}[g_{1}g_{2}\cdots g_{2n}]_{\geq
0}g_{2n}^{-1},g_{2n-1}^{-1},\ldots,g_{1}^{-1}]_{{\scriptscriptstyle
F}_{2n}},\;t^{-1});$
$\displaystyle\mbox{multiplication:}\;\mbox{for}\;\gamma=([g_{1},\ldots,g_{2n}]_{{\scriptscriptstyle
F}_{2n}},t)\;\mbox{and}\;\gamma^{\prime}=([g_{1}^{\prime},\ldots,g_{2n}^{\prime}]_{{\scriptscriptstyle
F}_{2n}},t^{\prime})\;\mbox{with}\;\theta_{-}(\gamma)=\theta_{+}(\gamma^{\prime}),$
$\displaystyle\hskip 14.45377pt\hskip
14.45377pt\gamma\gamma^{\prime}=([g_{1},\;\ldots,\;g_{n},\;g_{n+1}\cdots
g_{2n}[g_{1}g_{2}\cdots g_{2n}]_{\geq 0}^{-1}g_{1}^{\prime}\cdots
g_{n}^{\prime}g_{n+1}^{\prime},\;g_{n+2}^{\prime},\,\ldots,\,g_{2n}^{\prime}]_{{\scriptscriptstyle
F}_{2n}},\,tt^{\prime}).$

By construction, we have now made $(F_{2n}^{o}\times T,\pi_{2n}\bowtie 0)$
into a Poisson groupoid over $(F_{n},\pi_{n})$, as stated in the following
companion of Theorem 3.2 and Theorem 3.4.

######  Theorem 3.7.

For any integer $n\geq 1$, $(F_{2n}^{o}\times T,\pi_{2n}\bowtie 0)$ with the
groupoid structure as above is a Poisson groupoid over $(F_{n},\pi_{n})$.

##  4\. Special configuration Poisson groupoids of flags and isomorphic models

###  4.1. $T$-leaves

In view of the three isomorphic models of the total configuration Poisson
groupoids given in $\S$3.2 \- $\S$3.4, we now look at the $T$-leaves in
$(\Gamma_{n},\widetilde{\pi}_{n})$, $({\mathcal{C}}_{n},\widetilde{\pi}_{n})$,
and $(F_{n}^{o}\times T,\pi_{n}\bowtie 0)$ for any integer $n\geq 1$.

Recall again that $T$ acts on $F_{n}$ and $\widetilde{F}_{n}$ via (1.2) and
(1.3) respectively and on
${\mathcal{C}}_{n}={\mathcal{B}}^{n-1}\times{\mathcal{A}}^{o}$ and on
$F_{n}^{o}\times T$ diagonally. Recall also the $T$-equivariant Poisson
isomorphisms

$\widehat{\Theta}_{n}:\;(\Gamma_{n},\widetilde{\pi}_{n})\longrightarrow({\mathcal{C}}_{n},\widehat{\pi}_{n})\hskip
14.45377pt\mbox{and}\hskip
14.45377ptJ_{n}:\;(\Gamma_{n},\widetilde{\pi}_{n})\longrightarrow(F_{n}^{o}\times
T,\pi_{n}\bowtie 0),$

respectively given in (3.10) and (3.14). For ${\bf w}=(w_{1},\ldots,w_{n})\in
W^{n}$, introduce

$\Gamma^{{\bf w}}=\Gamma_{n}\cap(B{\bf w}B)\subset\Gamma_{n},\hskip
14.45377pt{\mathcal{C}}^{{\bf w}}=\widehat{\Theta}_{n}(\Gamma^{{\bf
w}})\subset{\mathcal{C}}_{n},\hskip 14.45377pt\mbox{and}\hskip
14.45377pt{\mathcal{O}}^{{\bf w}}_{e}\times T\subset F_{n}^{o}\times T,$
(4.17)

where recall that $B{\bf w}B$ is the image of $Bw_{1}B\times\cdots\times
Bw_{n}B$ in $\widetilde{F}_{n}$ and that (see (1.6))

${\mathcal{O}}^{\bf w}_{e}=\\{[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}\in{\mathcal{O}}^{\bf w}:\;g_{1}g_{2}\cdots g_{n}\in
B_{-}B/B\\}\subset{\mathcal{O}}^{\bf w}.$ (4.18)

By the definition of $J_{n}$, one has $J_{n}(\Gamma^{\bf
w})={\mathcal{O}}_{e}^{\bf w}\times T$ for each ${\bf w}\in W^{n}$. It follows
from the decomposition $\widetilde{F}_{n}=\bigsqcup_{{\bf w}\in W^{n}}B{\bf
w}B$ that one has the disjoint unions

$\Gamma_{n}=\bigsqcup_{{\bf w}\in W^{n}}\Gamma^{\bf w},\hskip
14.45377pt{\mathcal{C}}_{n}=\bigsqcup_{{\bf w}\in W^{n}}{\mathcal{C}}^{{\bf
w}},\hskip 14.45377ptF_{n}^{o}\times T=\bigsqcup_{{\bf w}\in
W^{n}}({\mathcal{O}}^{{\bf w}}_{e}\times T).$ (4.19)

######  Proposition 4.1.

For any integer $n\geq 1$, the decompositions in (4.19) are that of the
$T$-leaves of $(\Gamma_{n},\widetilde{\pi}_{n})$,
$({\mathcal{C}}_{n},\widetilde{\pi}_{n})$, and $(F_{n}^{o}\times
T,\pi_{n}\bowtie 0)$. In particular, for any ${\bf w}\in W^{n}$,

$\widehat{\Theta}_{n}:\;(\Gamma^{\bf
w},\widetilde{\pi}_{n})\longrightarrow({\mathcal{C}}^{\bf
w},\widehat{\pi}_{n})\hskip 14.45377pt\mbox{and}\hskip
14.45377ptJ_{n}:\;(\Gamma^{\bf
w},\widetilde{\pi}_{n})\longrightarrow({\mathcal{O}}^{{\bf w}}_{e}\times
T,\pi_{n}\bowtie 0)$

are $T$-equivariant isomorphisms of single $T$-leaves.

###### Proof.

The statement for the $T$-leaf decomposition for
$(\Gamma_{n},\widetilde{\pi}_{n})$ follows Proposition 3.1. The rest of
Proposition 4.1 follows from the fact that both
$\widehat{\Theta}_{n}:(\Gamma_{n},\widetilde{\pi}_{n})\rightarrow({\mathcal{C}}_{n},\widehat{\pi}_{n})$
and $J_{n}:(\Gamma_{n},\widetilde{\pi}_{n})\rightarrow(F_{n}^{o}\times
T,\pi_{n}\bowtie 0)$ are $T$-equivariant Poisson isomorphisms. We also note
that by [18, Theorem 1.1], each ${\mathcal{O}}^{\bf w}_{e}$ is the unique open
$T$-leaf of $({\mathcal{O}}^{\bf w},\pi_{n})$, so the fact that the $T$-leaves
of $(F_{n}^{o}\times T,\pi_{n}\bowtie 0)$ are precisely of the form
${\mathcal{O}}^{\bf w}_{e}\times T$ for ${\bf w}\in W^{n}$ also follows from
Lemma 1.4.

Q.E.D.

We give a description of ${\mathcal{C}}^{{\bf w}}$. Let $G_{\rm diag}$ be the
diagonal of $G\times G$. Recall that for $(f_{1},f_{2})\in{\mathcal{B}}^{2}$,
the Tits distance from $f_{1}$ to $f_{2}$ is defined to be the unique $w\in W$
such that $(f_{1},f_{2})\in G_{\rm diag}(e_{\cdot}B,\,w_{\cdot}B)$, and we
write $f_{1}\stackrel{{\scriptstyle w}}{{\longrightarrow}}f_{2}$. In
particular, $f_{1}$ and $f_{2}$ are said to be in general position if
$f_{1}\stackrel{{\scriptstyle w_{0}}}{{\longrightarrow}}f_{2}$, where $w_{0}$
is the longest element in $W$. Set

$f_{0}=e_{\cdot}B\in{\mathcal{B}}\hskip 14.45377pt\mbox{and}\hskip
14.45377ptf_{-}=w_{0}{}_{\cdot}B\in{\mathcal{B}}.$

For $f=b_{-}{}_{\cdot}N\in{\mathcal{A}}^{o}$, where $b_{-}\in B_{-}$, let
$\bar{f}=b_{-}{}_{\cdot}B\in{\mathcal{B}}$. The following statement is now
clear from the definition of the isomorphism
$\widehat{\Theta}:\Gamma_{n}\to{\mathcal{C}}_{n}$.

######  Lemma 4.2.

For ${\bf w}=(w_{1},w_{2},\ldots,w_{n})\in W^{n}$, the sub-variety
${\mathcal{C}}^{{\bf w}}$ of
${\mathcal{C}}_{n}={\mathcal{B}}^{n-1}\times{\mathcal{A}}^{o}$ consists of all
$(f_{1},f_{2},\ldots,f_{n-1},f_{n})\in{\mathcal{B}}^{n-1}\times{\mathcal{A}}^{o}$
such that

$f_{0}\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}f_{1}\stackrel{{\scriptstyle
w_{2}}}{{\longrightarrow}}f_{2}\longrightarrow\cdots\stackrel{{\scriptstyle
w_{n}}}{{\longrightarrow}}\bar{f}_{n}\stackrel{{\scriptstyle
w_{0}}}{{\longrightarrow}}f_{-}.$

######  Remark 4.3.

For ${\bf w}=(w_{1},\ldots,w_{n})\in W^{n}$, choose any a reduced
decomposition $w_{i}=s_{\alpha_{i,1}}\cdots s_{\alpha_{i,l_{i}}}$ for each
$1\leq i\leq n$, where $l_{i}=l(w_{i})$, and let

$b=(s_{\alpha_{1,1}},\,\ldots,\,s_{\alpha_{1,l_{1}}},\,s_{\alpha_{2,1}},\,\ldots,\,s_{\alpha_{2,l_{2}}},\;\ldots,\;s_{\alpha_{n,1}},\,\ldots,\,s_{\alpha_{n,l_{n}}})\in
W^{l({\bf w})}.$

One then has ${\mathcal{C}}^{\bf
w}\cong{\mathcal{C}}^{b}\subset{\mathcal{C}}_{l({\bf w})}$. In the notation of
[25], ${\mathcal{C}}^{b}={\rm Conf}_{e}^{{b}}({\mathcal{A}})$. $\diamond$

###  4.2. The Poisson groupoid $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ and two isomorphic models

Let now ${\bf u}=(u_{1},\ldots,u_{n})\in W^{n}$ and let ${\bf
u}^{-1}=(u_{n}^{-1},\ldots,u_{1}^{-1})$. Consider

$\Gamma^{({\bf u},{\bf u}^{-1})}=\Gamma_{2n}\cap(B({\bf u},{\bf u}^{-1})B).$

By Proposition 4.1, $\Gamma^{({\bf u},{\bf u}^{-1})}$ is a single $T$-leaf of
$(\Gamma_{2n},\widetilde{\pi}_{2n})$. One checks directly from the definitions
that $\Gamma^{({\bf u},{\bf u}^{-1})}$ is the full sub-groupoid (see
Definition 1.1) of $\Gamma_{2n}\rightrightarrows F_{n}$ over
${\mathcal{O}}^{\bf u}$, i.e.,

$\Gamma^{({\bf u},{\bf u}^{-1})}=\theta_{+}^{-1}({\mathcal{O}}^{\bf
u})\cap\theta_{-}^{-1}({\mathcal{O}}^{\bf u}),$

where $\theta_{+},\theta_{-}:\Gamma_{2n}\to F_{n}$ are the source and target
maps of the groupoid
$(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$.

######  Theorem 4.4.

For any ${\bf u}=(u_{1},u_{2},\ldots,u_{n})\in W^{n}$, $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ is a Poisson sub-groupoid of
$(\Gamma_{2n},\widetilde{\pi}_{2n})\rightrightarrows(F_{n},\pi_{n})$.

###### Proof.

Being a $T$-leaf of $(\Gamma_{2n},\widetilde{\pi}_{2n})$, $\Gamma^{({\bf
u},{\bf u}^{-1})}$ is a Poisson sub-manifold of
$(\Gamma_{2n},\widetilde{\pi}_{2n})$. It remains to show that
$\theta_{+}|_{\Gamma^{({\bf u},{\bf u}^{-1})}}:\Gamma^{({\bf u},{\bf
u}^{-1})}\to{\mathcal{O}}^{\bf u}$ is a surjective submersion. Since
$\Gamma^{({\bf u},{\bf u}^{-1})}$ contains the image of ${\mathcal{O}}^{\bf
u}\subset F_{n}$ under the unit map of $\Gamma_{2n}\rightrightarrows F_{n}$,
$\theta_{+}|_{\Gamma^{({\bf u},{\bf u}^{-1})}}:\Gamma^{({\bf u},{\bf
u}^{-1})}\to{\mathcal{O}}^{\bf u}$ is surjective. Define

$\theta:\;\;{\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}\longrightarrow{\mathcal{O}}^{\bf
u},\;\;[g_{1},g_{2},\ldots,g_{2n}]_{{\scriptscriptstyle
F}_{2n}}\longmapsto[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle F}_{n}}.$

Clearly $\theta$ is a submersion. Under the isomorphism $J_{2n}:\Gamma^{({\bf
u},{\bf u}^{-1})}\to{\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times T$, the
map $\theta_{+}|_{\Gamma^{({\bf u},{\bf u}^{-1})}}:\Gamma^{({\bf u},{\bf
u}^{-1})}\to{\mathcal{O}}^{\bf u}$ becomes the projection
$\theta_{+}^{\prime}:{\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times
T\to{\mathcal{O}}^{\bf u},(q,t)\to\theta(q)$. Since ${\mathcal{O}}^{({\bf
u},{\bf u}^{-1})}_{e}$ is open in ${\mathcal{O}}^{({\bf u},{\bf u}^{-1})}$,
$\theta_{+}^{\prime}$ is a submersion. Thus $\theta_{+}|_{\Gamma^{({\bf
u},{\bf u}^{-1})}}$ is a submersion.

Q.E.D.

Recall from (3.11) the Poisson isomorphism
$\Theta_{n}:(F_{n},\pi_{n})\to({\mathcal{B}}^{n},\overline{\pi}_{n})$. Set

${\mathcal{B}}^{\bf u}=\Theta_{n}({\mathcal{O}}^{\bf
u})=\\{(f_{1},\ldots,f_{n})\in{\mathcal{B}}^{n}:f_{0}\stackrel{{\scriptstyle
u_{1}}}{{\longrightarrow}}f_{1}\stackrel{{\scriptstyle
u_{2}}}{{\longrightarrow}}f_{2}\stackrel{{\scriptstyle
u_{3}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
u_{n}}}{{\longrightarrow}}f_{n}\\},$

so ${\mathcal{B}}^{\bf u}$ is a Poisson sub-manifold of
$({\mathcal{B}}^{n},\overline{\pi}_{n})$. We have the following immediate
consequence of Theorem 4.4.

######  Corollary 4.5.

For any $n\geq 1$ and ${\bf u}\in W^{n}$, $({\mathcal{C}}^{({\bf u},{\bf
u}^{-1})},\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{\bf
u},\overline{\pi}_{n})$ is a Poisson sub-groupoid of
$({\mathcal{C}}_{2n},\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{n},\overline{\pi}_{n})$
in Theorem 3.4 and is isomorphic, via the isomorphisms
$\widehat{\Theta}_{2n}:\Gamma^{({\bf u},{\bf u}^{-1})}\to{\mathcal{C}}^{({\bf
u},{\bf u}^{-1})}$ and $\Theta_{n}:{\mathcal{O}}^{\bf u}\to{\mathcal{B}}^{\bf
u}$, to the Poisson groupoid $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$.

######  Corollary 4.6.

For any $n\geq 1$ and ${\bf u}\in W^{n}$, $({\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}_{e}\times T,\,\pi_{2n}\bowtie 0)\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ is a Poisson sub-groupoid of $(F^{o}_{2n}\times T,\pi_{2n}\bowtie
0)\rightrightarrows(F^{n},\pi_{n})$ in Theorem 3.7 and is isomorphic, via the
isomorphism $J_{2n}:\Gamma^{({\bf u},{\bf u}^{-1})}\to{\mathcal{O}}_{e}^{({\bf
u},{\bf u}^{-1})}$, to the Poisson groupoid $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$.

######  Definition 4.7.

For ${\bf u}\in W^{n}$, we refer to either of the three isomorphic Poisson
groupoids

$(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n}),\;\;\;\;({\mathcal{C}}^{({\bf u},{\bf
u}^{-1})},\widehat{\pi}_{2n})\rightrightarrows({\mathcal{B}}^{\bf
u},\overline{\pi}_{n}),\;\;\;\;({\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}_{e}\times T,\,\pi_{2n}\bowtie 0)\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$

as a special configuration Poisson groupoid (of flags of $G$). $\diamond$

The special configuration Poisson groupoids have a simple set-theoretical
description. We first set up some notation. Recall that $N_{G}(T)\subset G$ is
the normalizer of $T$ in $G$. For $u\in W$, let $uT\subset N_{G}(T)$ be the
set of representatives of $u$ in $N_{G}(T)$. For $\dot{u}\in uT$, set

$C_{\dot{u}}=N\dot{u}\cap\dot{u}N_{-}$ (4.20)

It is well-known (see, for example, [8, Proposition 2.9]) that the maps

$C_{\dot{u}}\times B\longrightarrow BuB,\,(c,b)\longmapsto cb\hskip
14.45377pt\mbox{and}\hskip 14.45377ptB\times C_{\dot{u}}\longrightarrow
Bu^{-1}B,\,(b,c)\longmapsto bc^{-1},$ (4.21)

are both isomorphisms. For ${\bf u}=(u_{1},\ldots,u_{n})\in W^{n}$, let ${\bf
u}T^{n}=u_{1}T\times\cdots\times u_{n}T\subset N_{G}(T)^{n}$ and call any
$\dot{{\bf u}}=(\dot{u}_{1},\ldots,\dot{u}_{n})\in{\bf u}T^{n}$ a
representative of ${\bf u}$. For $\dot{{\bf
u}}=(\dot{u}_{1},\ldots,\dot{u}_{n})\in{\bf u}T^{n}$, set

$C_{\dot{{\bf u}}}=C_{\dot{u}_{1}}\times C_{\dot{u}_{2}}\times\cdots\times
C_{\dot{u}_{n}}.$ (4.22)

One then has the isomorphisms

$\displaystyle C_{\dot{{\bf u}}}\times B\longrightarrow B{\bf
u}B,\;\;(c_{1},\,c_{2},\,\ldots,\,c_{n},\,b)\longmapsto[c_{1},\cdots
c_{n-1},c_{n}b]_{{\scriptscriptstyle\widetilde{F}}_{n}},$ (4.23)
$\displaystyle C_{\dot{{\bf u}}}\longrightarrow{\mathcal{O}}^{\bf u}=B{\bf
u}B/B,\;\;(c_{1},c_{2},\ldots,c_{n})\longmapsto[c_{1},c_{2},\ldots,c_{n}]_{{\scriptscriptstyle
F}_{n}},$ (4.24) $\displaystyle B\times C_{\dot{{\bf u}}}\longrightarrow B{\bf
u}^{-1}B,\;\;(b,c_{1}^{\prime},\ldots,c_{n-1}^{\prime},c_{n}^{\prime})\longmapsto[b(c_{n}^{\prime})^{-1},\,(c_{n-1}^{\prime})^{-1},\,\ldots,\,(c_{1}^{\prime})^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{n}}.$
(4.25)

For $c=(c_{1},c_{2},\ldots,c_{n})\in C_{\dot{{\bf u}}}$, set
$[c]_{{\scriptscriptstyle F}_{n}}=[c_{1},\ldots,c_{n}]_{{\scriptscriptstyle
F}_{n}}$ and $\underline{c}=c_{1}c_{2}\cdots c_{n}\in G$. Define

${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}=\\{(c,b,b_{-},c^{\prime}):c,c^{\prime}\in C_{\dot{{\bf u}}},\,b\in
B,\,b_{-}\in B_{-},\,\underline{c}\,b=b_{-}\underline{c^{\prime}}\\}\subset
C_{\dot{{\bf u}}}\times B\times B_{-}\times C_{\dot{{\bf u}}}.$ (4.26)

We then have the isomorphism ${{\mathcal{I}}}_{\dot{{\bf
u}}}:{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf u}}}\to\Gamma^{({\bf u},{\bf
u}^{-1})}$ given by

${{\mathcal{I}}}_{\dot{{\bf
u}}}((c_{1},\ldots,c_{n}),\,b,\,b_{-},(c_{1}^{\prime},\ldots,c_{n}^{\prime}))=[c_{1},\ldots,c_{n-1},c_{n}b,\,(c_{n}^{\prime})^{-1},(c_{n-1}^{\prime})^{-1},\ldots,(c_{1}^{\prime})^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}.$

######  Proposition 4.8.

For any ${\bf u}\in W^{n}$ and $\dot{{\bf u}}\in{\bf u}T^{n}$,
${{\mathcal{I}}}_{\dot{{\bf u}}}$ is an isomorphism of Lie groupoids over
${\mathcal{O}}^{\bf u}$, where the groupoid $\Gamma^{({\bf u},{\bf
u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$ is given as in Theorem 4.4,
and ${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$ is the groupoid defined in [24] as
follows:

$\displaystyle\mbox{source map}\;\theta_{+}:\;{\mathcal{G}}^{\dot{{\bf
u}},\dot{{\bf u}}}\longrightarrow{\mathcal{O}}^{\bf
u},\;\;(c,\,b,\,b_{-},\,c^{\prime})\longmapsto[c]_{{\scriptscriptstyle
F}_{n}},$ $\displaystyle\mbox{target
map}\;\theta_{-}:\;{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\longrightarrow{\mathcal{O}}^{\bf
u},\;\;(c,\,b,\,b_{-},\,c^{\prime})\longmapsto[c^{\prime}]_{{\scriptscriptstyle
F}_{n}},$ $\displaystyle\mbox{unit map}\;\epsilon:\;{\mathcal{O}}^{\bf
u}\longrightarrow{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}},\;\;[c]_{{\scriptscriptstyle F}_{n}}\longmapsto(c,\,e,\,e,\,c),$
$\displaystyle\mbox{inverse map}\;{\rm inv}:\;{\mathcal{G}}^{\dot{{\bf
u}},\dot{{\bf u}}}\longrightarrow{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}},\;\;(c,\,b,\,b_{-},\,c^{\prime})\longmapsto(c^{\prime},\,b^{-1},\,b_{-}^{-1},\,c),$
$\displaystyle\mbox{multiplication}:(c,\,b,\,b_{-},\,c^{\prime})(c^{\prime},\,b^{\prime},\,b_{-}^{\prime},\,c^{\prime\prime})=(c,\,bb^{\prime},\,b_{-}b_{-}^{\prime},\,c^{\prime\prime}).$

###### Proof.

By (4.23) and (4.25), ${{\mathcal{I}}}_{\dot{{\bf u}}}$ is an isomorphism of
varieties. The fact that ${{\mathcal{I}}}_{\dot{{\bf u}}}$ is an isomorphism
from the groupoid ${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$ to $\Gamma^{({\bf u},{\bf
u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$ follows directly from the
definitions of the two groupoids.

Q.E.D.

######  Remark 4.9.

Under the isomorphism $C_{\dot{{\bf u}}}\to{\mathcal{O}}^{\bf
u},c\mapsto[c]_{{\scriptscriptstyle F}_{n}}$, one can regard
${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$ as a groupoid
${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf u}}}\rightrightarrows C_{\dot{{\bf
u}}}$, and as a such, it is a sub-groupoid of the direct product

$C_{\dot{{\bf u}}}\times B\times B_{-}\times C_{\dot{{\bf
u}}}\rightrightarrows C_{\dot{{\bf u}}}$

of two groupoids: the pair groupoid $C_{\dot{{\bf u}}}\times C_{\dot{{\bf
u}}}\rightrightarrows C_{\dot{{\bf u}}}$ and the direct product group $B\times
B_{-}$ as a groupoid over the one point space, where we identify

$C_{\dot{{\bf u}}}\times B\times B_{-}\times C_{\dot{{\bf u}}}\longrightarrow
C_{\dot{{\bf u}}}\times C_{\dot{{\bf u}}}\times B\times
B_{-},\;\;(c,\,b,\,b_{-},\,c^{\prime})\longmapsto(c,\,c^{\prime},\,b,\,b_{-}).$

###  4.3. Generalized double Bruhat cells as Poisson groupoids

We now give the third isomorphic model of the special configuration Poisson
groupoids using generalized double Bruhat cells. For $n\geq 1$, recall the
right action of $G^{n}$ on itself given in (1.1). Let

$\widetilde{F}_{-n}=G\times_{B_{-}}\cdots\times_{B_{-}}G,$

the quotient of $G^{n}$ by $B_{-}^{n-1}\times\\{e\\}\subset G^{n}$, and let
$\widetilde{\pi}_{-n}$ be the Poisson structure on $\widetilde{F}_{-n}$ that
is the (well-defined) projection of the Poisson structure $(\pi_{\rm st})^{n}$
on $G^{n}$. For $(g_{1},g_{2},\ldots,g_{n})\in G^{n}$, denote its image in
$\widetilde{F}_{-n}$ as
$[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{-n}}$. For any
integers $m,n\geq 1$, we introduce in $\S$C.1 a Poisson structure
$\widetilde{\pi}_{m,n}$ on $\widetilde{F}_{m}\times\widetilde{F}_{-n}$, which
is the sum of the product Poisson structure
$(\widetilde{\pi}_{m},\widetilde{\pi}_{-n})$ with a certain mixed term. See
Definition C.1. The Poisson structure $\widetilde{\pi}_{m,n}$ is $T$-invariant
under the $T$-action on $\widetilde{F}_{m}\times\widetilde{F}_{-n}$ given by

$t\cdot([g_{1},g_{2},\ldots,g_{m}]_{{\scriptscriptstyle\widetilde{F}}_{m}},[k_{1},k_{2},\ldots,k_{n}]_{{\scriptscriptstyle\widetilde{F}}_{-n}})=([tg_{1},g_{2},\ldots,g_{m}]_{{\scriptscriptstyle\widetilde{F}}_{m}},[tk_{1},k_{2},\ldots,k_{n}]_{{\scriptscriptstyle\widetilde{F}}_{-n}}).$
(4.27)

Introduce the sub-manifold

$G_{m,n}=\\{([g_{1},g_{2},\ldots,g_{m}]_{{\scriptscriptstyle\widetilde{F}}_{m}},[k_{1},k_{2},\ldots,k_{n}]_{{\scriptscriptstyle\widetilde{F}}_{-n}}):g_{1}g_{2}\cdots
g_{n}=k_{1}k_{2}\cdots
k_{n}\\}\subset\widetilde{F}_{m}\times\widetilde{F}_{-n}.$ (4.28)

For ${\bf u}=(u_{1},\ldots,u_{m})\in W^{m}$ and ${\bf
v}=(v_{1},\ldots,v_{n})\in W^{n}$, let again $B{\bf u}B$ be the image of
$(Bu_{1}B)\times\cdots\times(Bu_{m}B)$ in $\widetilde{F}_{m}$, let $B_{-}{\bf
v}B_{-}\subset\widetilde{F}_{-n}$ be the image of
$(B_{-}v_{1}B_{-})\times\cdots\times(B_{-}v_{n}B_{-})$ in
$\widetilde{F}_{-n}$, and set

$G^{{\bf u},{\bf v}}=G_{m,n}\cap(B{\bf u}B\times B_{-}{\bf v}B_{-})\subset
G_{m,n}.$ (4.29)

It follows from the Bruhat decomposition of $G$ that one has the disjoint
union

$G_{m,n}=\bigsqcup_{({\bf u},{\bf v})\in W^{m}\times W^{n}}G^{{\bf u},{\bf
v}}.$ (4.30)

We prove in Corollary C.4 that $G_{m,n}$ is a $T$-invariant Poisson sub-
manifold of
$(\widetilde{F}_{m}\times\widetilde{F}_{-n},\widetilde{\pi}_{m,n})$, and that
(4.30) is the decomposition of $(G_{m,n},\widetilde{\pi}_{m,n})$ into its
$T$-leaves. Generalizing the case of

$(G_{1,1},\widetilde{\pi}_{1,1})\cong(G,\pi_{\rm st}),$

we call $G^{{\bf u},{\bf v}}$, for any ${\bf u}\in W^{m}$ and ${\bf v}\in
W^{n}$, a generalized double Bruhat cell. When $m=n$, the Poisson manifold
$(\widetilde{F}_{n}\times\widetilde{F}_{-n},\widetilde{\pi}_{n,n})$, as well
as the generalized double Bruhat cells $G^{{\bf u},{\bf v}}$ for ${\bf u},{\bf
v}\in W^{n}$, were introduced in [18, $\S$1.4]. We also note that for any
$m,n$, the projections
$(\widetilde{F}_{m}\times\widetilde{F}_{-n},\widetilde{\pi}_{m,n})\to(\widetilde{F}_{m},\widetilde{\pi}_{m})$
and
$(\widetilde{F}_{m}\times\widetilde{F}_{-n},\widetilde{\pi}_{m,n})\to(\widetilde{F}_{-n},\widetilde{\pi}_{-n})$
to the factors are Poisson. It follows in particular that for any $m\geq 1$,
the map

$(G_{m,1},\;\widetilde{\pi}_{m,1})\longrightarrow(\widetilde{F}_{m},\widetilde{\pi}_{m}),\;([g_{1},g_{1},\ldots,g_{m}]_{{\scriptscriptstyle\widetilde{F}}_{m}},g_{1}g_{2}\cdots
g_{m})\longmapsto[g_{1},g_{1},\ldots,g_{m}]_{{\scriptscriptstyle\widetilde{F}}_{m}},$
(4.31)

is a Poisson isomorphism.

Let now ${\bf u}=(u_{1},\ldots,u_{n})\in W^{n}$, and consider the generalized
double Bruhat cell $G^{{\bf u},{\bf u}}$. Fix any $\dot{{\bf
u}}=(\dot{u}_{1},\ldots,\dot{u}_{n})\in{\bf u}T^{n}$. As
$B_{-}uB_{-}=B_{-}C_{\dot{u}}$ is a direct product decomposition for any $u\in
W$ and any $\dot{u}\in uT$, one has the isomorphism

$B_{-}\times C_{\dot{{\bf u}}}\longrightarrow B_{-}{\bf
u}B_{-},\;\;(b_{-},\,c_{1}^{\prime},\,c_{2}^{\prime},\,\ldots,\,c_{n}^{\prime})\longmapsto[b_{-}c_{1}^{\prime},\,c_{2}^{\prime},\,\ldots,\,c_{n}^{\prime}]_{{\scriptscriptstyle\widetilde{F}}_{-n}}.$
(4.32)

Recalling the definition of ${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\subset C_{\dot{{\bf u}}}\times B\times B_{-}\times C_{\dot{{\bf u}}}$,
one then has the isomorphism

${\mathcal{J}}_{\dot{{\bf u}}}:\;{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\longrightarrow G^{{\bf u},{\bf
u}},\;(c,b,b_{-},c^{\prime})\longmapsto([c_{1},\ldots,c_{n-1},c_{n}b]_{{\scriptscriptstyle\widetilde{F}}_{n}},[b_{-}c_{1}^{\prime},c_{2}^{\prime},\ldots,c_{n}^{\prime}]_{{\scriptscriptstyle\widetilde{F}}_{-n}}),$
(4.33)

where $c=(c_{1},\ldots,c_{n-1},c_{n})$,
$c^{\prime}=(c_{1}^{\prime},c_{2}^{\prime},\ldots,c_{n}^{\prime})$, and
$(c,b,b_{-},c^{\prime})\in{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf u}}}$.

######  Definition 4.10.

[24] For each choice $\dot{{\bf u}}\in{\bf u}T^{n}$, define the groupoid
$G^{{\bf u},{\bf u}}\rightrightarrows{\mathcal{O}}^{\bf u}$ such that

${\mathcal{J}}_{\dot{{\bf u}}}:\;{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\longrightarrow G^{{\bf u},{\bf u}}$

is a groupoid isomorphism from the groupoid ${\mathcal{G}}^{\dot{{\bf
u}},\dot{{\bf u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$ in Proposition 4.8
to $G^{{\bf u},{\bf u}}\rightrightarrows{\mathcal{O}}^{\bf u}$.

Using the parametrization ${\mathcal{J}}_{\dot{{\bf
u}}}:{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf u}}}\to G^{{\bf u},{\bf u}}$, we
can now define

$\displaystyle E_{\dot{{\bf u}}}:\;G^{{\bf u},{\bf
u}}\longrightarrow\Gamma_{2n},\;\;$
$\displaystyle([c_{1},\ldots,c_{n-1},c_{n}b]_{{\scriptscriptstyle\widetilde{F}}_{n}},[b_{-}c_{1}^{\prime},c_{2}^{\prime},\ldots,c_{n}^{\prime}]_{{\scriptscriptstyle\widetilde{F}}_{-n}})$
(4.34)
$\displaystyle\longmapsto[c_{1},\ldots,c_{n-1},c_{n}b,\;(c_{n}^{\prime})^{-1},\ldots,(c_{2}^{\prime})^{-1},(c_{1}^{\prime})^{-1}]_{\widetilde{F}_{2n}},$

where
$((c_{1},\ldots,c_{n}),b,b_{-},(c_{1}^{\prime},\ldots,c_{n}^{\prime}))\in{\mathcal{G}}^{\dot{{\bf
u}},\dot{{\bf u}}}$.

######  Theorem 4.11.

For each $\dot{{\bf u}}\in{\bf u}T^{n}$, the map $E_{\dot{{\bf u}}}$ is a
$T$-equivariant Poisson embedding of $(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})$ into $(\Gamma_{2n},\widetilde{\pi}_{2n})$ and
gives a Poisson isomorphism

$E_{\dot{{\bf u}}}:\;\;(G^{{\bf u},{\bf
u}},\;\widetilde{\pi}_{n,n})\longrightarrow(\Gamma^{({\bf u},{\bf
u}^{-1})},\;\widetilde{\pi}_{2n})\subset(\Gamma_{2n},\widetilde{\pi}_{2n}).$

Consequently, with the groupoid structure defined as in Definition 4.10,
$(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$ is
a Poisson groupoid, and $E_{\dot{{\bf u}}}$ is an isomorphism from the Poisson
groupoid $(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$ to
the Poisson groupoid $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$.

###### Proof.

The fact that $E_{\dot{{\bf u}}}$ is a Poisson isomorphism onto
$(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})$ is a special case of
Corollary C.7 by taking ${\bf u}={\bf v}$ and $E_{\dot{{\bf u}}}=E_{{\bf
u},\dot{{\bf u}}}$. That $E_{\dot{{\bf u}}}$ is a groupoid isomorphism follows
from the definition of the groupoid $G^{{\bf u},{\bf
u}}\rightrightarrows{\mathcal{O}}^{\bf u}$ and Proposition 4.8.

Q.E.D.

######  Remark 4.12.

We emphasize that the groupoid structure on $G^{{\bf u},{\bf u}}$ over
${\mathcal{O}}^{\bf u}$ depends on the choice of the representative $\dot{{\bf
u}}\in{\bf u}T^{n}$. If $\hat{{\bf u}}=(\hat{u}_{1},\ldots,\hat{u}_{n})\in{\bf
u}T^{n}$ is another such choice, and if $t\in T$ is such that
$\hat{u}_{1}\cdots\hat{u}_{n}=t\dot{u}_{1}\cdots\dot{u}_{n}$, then
$E_{\dot{{\bf u}}}=E_{\hat{{\bf u}}}\circ r_{t}$, where $r_{t}:G^{{\bf u},{\bf
u}}\to G^{{\bf u},{\bf u}}$ is given by

$r_{t}([g_{1},\ldots,g_{n-1},g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{m}},[k_{1},\ldots,k_{n-1},k_{n}]_{{\scriptscriptstyle\widetilde{F}}_{-n}})=([g_{1},\ldots,g_{n-1},g_{n}t]_{{\scriptscriptstyle\widetilde{F}}_{m}},[k_{1},\ldots,k_{n-1},k_{n}t]_{{\scriptscriptstyle\widetilde{F}}_{-n}}),$

Thus $r_{t}:G^{{\bf u},{\bf u}}\to G^{{\bf u},{\bf u}}$ defines a groupoid
isomorphism from the groupoid structure on $G^{{\bf u},{\bf u}}$ defined by
$\dot{{\bf u}}$ to that defined by $\hat{{\bf u}}$. $\diamond$

##  5\. Configuration symplectic groupoids of flags

In this section, we assume that $G$ is connected and simply connected.
Symplectic leaves of $({\mathcal{O}}^{\bf w}_{e}\times T,\pi_{n}\bowtie 0)$,
for any $n\geq 1$ and ${\bf w}\in W^{n}$, are determined in $\S$D, where we
also give a complete description of the symplectic leaves of all the three
series

$(F_{n}^{o}\times T,\,\pi_{n}\bowtie 0),\hskip
14.45377pt(G_{m+n},\,\widetilde{\pi}_{m,n}),\hskip
14.45377pt(\widetilde{F}_{n},\,\widetilde{\pi}_{n}),\hskip 14.45377ptm,n\geq
1,$

of $T$-Poisson varieties. See $\S$D for details. For any $n\geq 1$ and ${\bf
u}\in W^{n}$, we show in this section that all the units of the groupoid
$\Gamma^{({\bf u},{\bf u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$ are
contained in a single symplectic leaf, denoted as $\Lambda^{({\bf u},{\bf
u}^{-1})}$, of $(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})$, and
that $\Lambda^{({\bf u},{\bf u}^{-1})}$ is a Lie sub-groupoid of
$(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$, obtaining thus a symplectic groupoid

$(\Lambda^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n}).$

Using Theorem 4.11, we then show that every symplectic leaf of $(G^{{\bf
u},{\bf u}},\widetilde{\pi}_{n,n})$ is symplectic groupoid over
$({\mathcal{O}}^{\bf u},\pi_{n})$.

###  5.1. The symplectic groupoid $(\Lambda^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$

Assume that $G$ is connected and simply connected. Let $X^{*}(T)$ be the
character lattice of $T$. For $\lambda\in X^{*}(T)$ and $t\in T$, write
$t^{\lambda}$ for the value of $\lambda$ at $t$. Let $\Phi_{0}\subset
X^{*}(T)$ be the set of simple roots determined by the choice of $B$, and let
$\\{\omega_{\alpha}:\alpha\in\Phi_{0}\\}\subset X^{*}(T)$ be the corresponding
set of fundamental weights. For $u\in W$ and $t\in T$, we also set
$t^{u}=\dot{u}^{-1}t\dot{u}$ using any $\dot{u}\in uT$.

Let now ${\bf u}=(u_{1},\ldots,u_{n})\in W^{n}$ be arbitrary. Define

${\widetilde{T}^{{\bf u}}}=\\{t\in
T:t^{\omega_{\alpha}}=1,\,\forall\,\alpha\in{\rm supp}^{o}({\bf u})\\},$

where ${\rm supp}^{o}({\bf
u})=\\{\alpha\in\Phi_{0}:u_{i}\omega_{\alpha}=\omega_{\alpha},\,\forall\,i\in[1,n]\\}$.
Choose any $\dot{{\bf u}}=(\dot{u}_{1},\ldots,\dot{u}_{n})\in{\bf u}T^{n}$,
and let again $C_{\dot{{\bf u}}}=C_{\dot{u}_{1}}\times\cdots\times
C_{\dot{u}_{n}}$. Let $u=u_{1}u_{2}\cdots u_{n}\in W$. Using the isomorphism
${{\mathcal{I}}}_{\dot{{\bf u}}}:{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\to\Gamma^{({\bf u},{\bf u}^{-1})}$ to write an element in $\Gamma^{({\bf
u},{\bf u}^{-1})}$ uniquely as

$\gamma=[c_{1},\ldots,c_{n-1},c_{n}b,\,(c_{n}^{\prime})^{-1},(c_{n-1}^{\prime})^{-1},\ldots,(c_{1}^{\prime})^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}},$
(5.1)

where $(c_{1},\ldots,c_{n}),(c_{1}^{\prime},\ldots,c_{n}^{\prime})\in
C_{\dot{{\bf u}}}$, $b\in B,b_{-}\in B_{-}$, and $c_{1}\cdots
c_{n-1}c_{n}\,b=b_{-}c_{1}^{\prime}c_{2}^{\prime}\cdots c_{n}^{\prime}$, we
define the sub-variety $\Lambda^{({\bf u},{\bf u}^{-1})}$ of $\Gamma^{({\bf
u},{\bf u}^{-1})}$ by

$\Lambda^{({\bf u},{\bf u}^{-1})}=\\{\gamma\in\Gamma^{({\bf u},{\bf
u}^{-1})}\,\mbox{as in
\eqref{eq:gamma-1}}:\;[b]_{0}[b_{-}]_{0}^{u}=e,\,[b]_{0}\in\widetilde{T}^{{\bf
u}}\\}.$

Note that $\Lambda^{({\bf u},{\bf u}^{-1})}$ contains all the units of the
groupoid $\Gamma^{({\bf u},{\bf u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf
u}$, and that

${\mathcal{I}}_{\dot{{\bf u}}}^{-1}(\Lambda^{({\bf u},{\bf
u}^{-1})})={\mathcal{S}}^{\dot{{\bf u}},\dot{{\bf
u}}}\stackrel{{\scriptstyle\rm
def}}{{=}}\\{(c,b,b_{-},c^{\prime})\in{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}:\,[b]_{0}[b_{-}]_{0}^{u}=e,\,[b]_{0}\in\widetilde{T}^{{\bf u}}\\}.$ (5.2)

Note that ${\mathcal{S}}^{\dot{{\bf u}},\dot{{\bf u}}}$ is a wide sub-groupoid
of ${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$. It thus follows from Proposition
4.8 that $\Lambda^{({\bf u},{\bf u}^{-1})}$ is a wide sub-groupoid of
$\Gamma^{({\bf u},{\bf u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$.

######  Theorem 5.1.

For any ${\bf u}\in W^{n}$, $\Lambda^{({\bf u},{\bf u}^{-1})}$ is a symplectic
leaf of $(\Gamma^{({\bf u},{\bf u}^{-1})},\widetilde{\pi}_{2n})$ and is a Lie
sub-groupoid of $\Gamma^{({\bf u},{\bf
u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$. Consequently,
$(\Lambda^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})\rightrightarrows({\mathcal{O}}^{\bf
u},\pi_{n})$ is a symplectic groupoid.

###### Proof.

Symplectic leaves of $(\Gamma^{\bf w},\widetilde{\pi}_{n})$ for any $n\geq 1$
and any ${\bf w}\in W^{n}$ are described in Theorem D.16, and the case of
${\bf w}=({\bf u},{\bf u}^{-1})\in W^{2n}$ is given in Example D.18. More
concretely, for $\gamma\in\Gamma^{({\bf u},{\bf u}^{-1})}$ in (5.1), let
$(c_{n+1},\ldots,c_{2n})\in C_{\dot{u}_{n}^{-1}}\times\cdots\times
C_{\dot{u}_{1}^{-1}}$ and $b_{n},\ldots,b_{1}\in B$ be such that

$b(c_{n}^{\prime})^{-1}=c_{n+1}b_{n},\;\;b_{n}(c_{n-1}^{\prime})^{-1}=c_{n+2}b_{n-1},\;\ldots,\;\;b_{2}(c_{1}^{\prime})^{-1}=c_{2n}b_{1}.$

Then
$\gamma=[c_{1},\ldots,c_{n-1},c_{n},\,c_{n+1},\ldots,c_{2n-1},c_{2n}b_{1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}$
and $[b_{1}]_{0}=[b]_{0}^{u^{-1}}$. Let

$\dot{{\bf
w}}=(\dot{u}_{1},\ldots,\dot{u}_{n},\dot{u}_{n}^{-1},\ldots,\dot{u}_{1}^{-1})\in{\bf
w}T^{2n}.$

We then have the alternative description of $\Lambda^{({\bf u},{\bf
u}^{-1})}\subset\Gamma^{({\bf u},{\bf u}^{-1})}$ as consisting of all

$\gamma=[c_{1},\ldots,c_{n-1},c_{n},\,c_{n+1},\ldots,c_{2n-1},c_{2n}b_{1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\in\Gamma^{({\bf
u},{\bf u}^{-1})},$

where $(c_{1},\ldots,c_{n},c_{n+1},\ldots,c_{2n})\in C_{\dot{{\bf w}}}$,
$b_{1}\in B$, and $b_{-}=c_{1},\cdots c_{n}c_{n+1}\cdots c_{2n}b_{1}\in B_{-}$
are such that $[b_{-}]_{0}\in{\widetilde{T}^{{\bf u}}}$ and
$[b_{1}]_{0}[b_{-}]_{0}=e$. By Example D.18, $\Lambda^{({\bf u},{\bf
u}^{-1})}$ is the symplectic leaf of $(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})$ passing through the point

$[{\bf u},{\bf
u}^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\,\stackrel{{\scriptstyle\rm
def}}{{=}}\,[\dot{u}_{1},\ldots,\dot{u}_{n},\,\dot{u}_{n}^{-1},\ldots,\dot{u}_{1}^{-1}]_{{\scriptscriptstyle\widetilde{F}}_{2n}}\in\Gamma^{({\bf
u},{\bf u}^{-1})}.$

We already know that $\Lambda^{({\bf u},{\bf u}^{-1})}$ is a sub-groupoid of
$\Gamma^{({\bf u},{\bf u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$. To
show that $\Lambda^{({\bf u},{\bf u}^{-1})}$ is a Lie sub-groupoid of
$\Gamma^{({\bf u},{\bf u}^{-1})}\rightrightarrows{\mathcal{O}}^{\bf u}$, it
remains to show that the source map $\theta_{+}:\Gamma^{({\bf u},{\bf
u}^{-1})}\to{\mathcal{O}}^{\bf u}$ restricts to a surjective submersion
$\theta:=\theta_{+}|_{\Lambda^{({\bf u},{\bf u}^{-1})}}:\Lambda^{({\bf u},{\bf
u}^{-1})}\to{\mathcal{O}}^{\bf u}$. To this end, let

$\Sigma^{({\bf u},{\bf u}^{-1})}=J_{2n}(\Lambda^{({\bf u},{\bf
u}^{-1})})\subset{\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times T.$

Then $\Sigma^{({\bf u},{\bf u}^{-1})}$ is the symplectic leaf of
$\pi_{2n}\bowtie 0$ in ${\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times T$
through the point

$([\dot{u}_{1},\ldots,\dot{u}_{n},\,\dot{u}_{n}^{-1},\ldots,\dot{u}_{1}^{-1}]_{{\scriptscriptstyle
F}_{2n}},\;e)\in{\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times T.$

By Example D.15, $\Sigma^{({\bf u},{\bf u}^{-1})}=\Sigma^{\dot{{\bf w}}}$,
consisting of all $([c_{1},\ldots,c_{2n}]_{{\scriptscriptstyle
F}_{2n}},t)\in{\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e}\times T$, where
$t\in{\widetilde{T}^{{\bf u}}}$ and $(c_{1},\ldots,c_{2n})\in C_{\dot{{\bf
w}}}$ such that

$c_{1}c_{2}\cdots c_{2n}\in B_{-}B\hskip 14.45377pt\mbox{and}\hskip
14.45377ptt^{2}=[c_{1},c_{2},\ldots,c_{2n}]_{0}.$

We also know from Example D.15 that the projection $P:\Sigma^{({\bf u},{\bf
u}^{-1})}\to{\mathcal{O}}^{({\bf u},{\bf u}^{-1})}_{e},(q,t)\mapsto q,$ is a
$2^{|{\rm supp}({\bf u})|}$-to-$1$ covering map, where ${\rm supp}({\bf
u})=\Phi_{0}\backslash{\rm supp}^{o}({\bf u})$. Let

$p:\;{\mathcal{O}}^{({\bf u},{\bf
u}^{-1})}_{e}\longrightarrow{\mathcal{O}}^{\bf
u},\;[g_{1},\ldots,g_{n},g_{n+1},\ldots,g_{2n}]_{{\scriptscriptstyle
F}_{2n}}\longmapsto[g_{1},\ldots,g_{n}]_{{\scriptscriptstyle F}_{n}}.$

Since $p$ is a submersion, one sees that $\theta=p\circ P\circ
J_{2n}:\Lambda^{({\bf u},{\bf u}^{-1})}\to{\mathcal{O}}^{\bf u}$ is a
submersion.

Q.E.D.

###  5.2. Symplectic leaves in $G^{{\bf u},{\bf u}}$ as symplectic groupoids

Let ${\bf u}\in W^{n}$ and consider the Poisson manifold $(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})$. As $(G^{{\bf u},{\bf u}},\widetilde{\pi}_{n,n})$
is a single $T$-leaf, every symplectic leaf of $(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})$ passes through the point $(\dot{{\bf u}},\dot{{\bf
u}})\in G^{{\bf u},{\bf u}}$ for some $\dot{{\bf u}}\in{\bf u}T^{n}$. Let
$\dot{{\bf u}}\in{\bf u}T^{n}$, and let $S^{\dot{{\bf u}},\dot{{\bf u}}}$ be
the symplectic leaf of $(G^{{\bf u},{\bf u}},\widetilde{\pi}_{n,n})$ through
$(\dot{{\bf u}},\dot{{\bf u}})$. Using the parametrization
${\mathcal{J}}_{\dot{{\bf u}}}:{\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightarrow G^{{\bf u},{\bf u}}$ in (4.33), we write every ${\bf g}\in
G^{{\bf u},{\bf u}}$ uniquely as

${\bf
g}=([c_{1},\ldots,c_{n-1},c_{n}b]_{{\scriptscriptstyle\widetilde{F}}_{n}},[b_{-}c_{1}^{\prime},c_{2}^{\prime},\ldots,c_{n}^{\prime}]_{{\scriptscriptstyle\widetilde{F}}_{-n}}).$
(5.3)

where
$((c_{1},\ldots,c_{n}),b,b_{-},(c_{1}^{\prime},\ldots,c_{n}^{\prime}))\in{\mathcal{G}}^{\dot{{\bf
u}},\dot{{\bf u}}}$. Recall from (5.2) the wide sub-groupoid
${\mathcal{S}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$ of the groupoid
${\mathcal{G}}^{\dot{{\bf u}},\dot{{\bf
u}}}\rightrightarrows{\mathcal{O}}^{\bf u}$. Recall also from Theorem 4.11 the
Poisson isomorphism $E_{\dot{{\bf u}}}:(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})\to(\Gamma^{({\bf u},{\bf
u}^{-1})},\widetilde{\pi}_{2n})$. The following Theorem 5.2 is now a direct
consequence of Theorem 4.11 and Theorem 5.1.

######  Theorem 5.2.

The symplectic leaf $S^{\dot{{\bf u}},\dot{{\bf u}}}$ of $(G^{{\bf u},{\bf
u}},\widetilde{\pi}_{n,n})$ through $(\dot{{\bf u}},\dot{{\bf u}})$ is given
by

$\displaystyle S^{\dot{{\bf u}},\dot{{\bf u}}}=E_{\dot{{\bf
u}}}^{-1}(\Lambda^{{\bf u},{\bf u}^{-1}})=\\{{\bf g}\in G^{{\bf u},{\bf
u}}\,\mbox{as in
\eqref{eq:bfg-0}}:\;[b]_{0}[b_{-}]_{0}^{u}=e,\,[b]_{0}\in\widetilde{T}^{{\bf
u}}\\}.$

Equip $S^{\dot{{\bf u}},\dot{{\bf u}}}$ with the structure of a Lie groupoid
over ${\mathcal{O}}^{\bf u}$ via the isomorphism ${\mathcal{J}}_{\dot{{\bf
u}}}:{\mathcal{S}}^{\dot{{\bf u}},\dot{{\bf u}}}\to S^{\dot{{\bf u}},\dot{{\bf
u}}}$. Then $(S^{\dot{{\bf u}},\dot{{\bf
u}}},\widetilde{\pi}_{n,n})\rightrightarrows({\mathcal{O}}^{\bf u},\pi_{n})$
is a symplectic groupoid.

##  Appendix A Mixed product Poisson structures

In $\S$A.1, we recall from [17] the construction of mixed product Poisson
structures using Poisson Lie group actions. We then show in $\S$A.2 that the
Poisson varieties $({\mathcal{C}}_{n},\widehat{\pi}_{n})$ and
$({\mathcal{B}}^{n},\overline{\pi}_{n})$ defined in $\S$3.3 are all mixed
product Poisson varieties.

###  A.1. Mixed product Poisson structures

The following definition was introduced in [17].

######  Definition A.1.

Given two manifolds $Y_{1}$ and $Y_{2}$, by a mixed product Poisson structure
on the product manifold $Y_{1}\times Y_{2}$ we mean a Poisson bivector field
$\pi$ on $Y_{1}\times Y_{2}$ that projects to well-defined Poisson structures
on $Y_{1}$ and $Y_{2}$. Given $Y_{i}$, $1\leq i\leq n$, where $n\geq 2$, a
Poisson structure $\pi$ on the product manifold $Y=Y_{1}\times\cdots\times
Y_{n}$ is said to be a mixed product if the projection of $\pi$ to
$Y_{i}\times Y_{j}$ is a well-defined mixed product Poisson structure on
$Y_{i}\times Y_{j}$ for any $1\leq i<j\leq n$, and in this case, we also call
the pair $(Y_{1}\times\cdots\times Y_{n},\pi)$ a mixed product Poisson
manifold. $\diamond$

Assume now that $((G,{\pi_{\scriptscriptstyle
G}}),(G^{*},{\pi_{{\scriptscriptstyle G}^{*}}}))$ is any pair of dual Poisson
Lie groups, i.e. the Lie bialgebras $(\mathfrak{g},\delta_{\mathfrak{g}})$ and
$(\mathfrak{g}^{*},\delta_{\mathfrak{g}^{*}})$ are dual to each other, where
recall that $\delta_{\mathfrak{g}}:\mathfrak{g}\to\wedge^{2}\mathfrak{g}$ and
$\delta_{\mathfrak{g}^{*}}:\mathfrak{g}^{*}\to\wedge^{2}\mathfrak{g}^{*}$ are
respectively the linearizations of ${\pi_{\scriptscriptstyle G}}$ and
${\pi_{{\scriptscriptstyle G}^{*}}}$ at the identity elements of $G$ and
$G^{*}$. Suppose that $(X,{\pi_{\scriptscriptstyle X}})$ and
$(Y,{\pi_{\scriptscriptstyle Y}})$ are two Poisson manifolds, with respective
right and left Poisson actions

$\rho:\;(X,{\pi_{\scriptscriptstyle X}})\times(G^{*},{\pi_{{\scriptscriptstyle
G}^{*}}})\longrightarrow(X,{\pi_{\scriptscriptstyle X}})\hskip
14.45377pt\mbox{and}\hskip 14.45377pt\lambda:\;(G,{\pi_{\scriptscriptstyle
G}})\times(Y,{\pi_{\scriptscriptstyle
Y}})\longrightarrow(Y,{\pi_{\scriptscriptstyle Y}})$ (A.1)

by the Poisson Lie groups $(G^{*},\pi_{{\scriptscriptstyle G}^{*}})$ and
$(G,\pi_{\scriptscriptstyle G})$. Let
$\rho:\mathfrak{g}^{*}\to{\mathfrak{X}}^{1}(X)$ and
$\lambda:\mathfrak{g}\to{\mathfrak{X}}^{1}(Y)$ be the corresponding Lie
algebra homomorphism and Lie algebra anti-homomorphism (see notation in
$\S$1.6). Then $\rho$ and $\lambda$ are respectively right and left Lie
bialgebra actions (see [17, $\S$2.5]). Define the bi-vector field

${\pi_{\scriptscriptstyle X}}\times_{(\rho,\lambda)}{\pi_{\scriptscriptstyle
Y}}=({\pi_{\scriptscriptstyle X}},0)+(0,{\pi_{\scriptscriptstyle
Y}})-\sum_{i=1}^{n}(\rho(\xi_{i}^{*}),\,0)\wedge(0,\;\lambda(\xi_{i}))$ (A.2)

on $X\times Y$, where $\\{\xi_{i}\\}_{i=1,\ldots,n}$ is any basis for
$\mathfrak{g}$ and $\\{\xi_{i}^{*}\\}_{i=1,\ldots,n}$ its dual basis for
$\mathfrak{g}^{*}$. Note that the definition of ${\pi_{\scriptscriptstyle
X}}\times_{(\rho,\lambda)}{\pi_{\scriptscriptstyle Y}}$ only uses the Lie
algebra actions $\rho$ and $\lambda$.

######  Lemma-Definition A.2.

[17, Proposition 4.2] For any pair $(\rho,\lambda)$ of Poisson Lie group
actions in (A.1), the bi-vector field ${\pi_{\scriptscriptstyle
X}}\times_{(\rho,\lambda)}{\pi_{\scriptscriptstyle Y}}$ on $X\times Y$ is
Poisson, and it is called the mixed product of ${\pi_{\scriptscriptstyle X}}$
and ${\pi_{\scriptscriptstyle Y}}$ associated to the pair $(\rho,\lambda)$.

######  Example A.3.

Let $(\rho,\lambda)$ be a pair of Poisson Lie group actions as in (A.1).
Equipping $G^{*}\times Y$ with the Poisson structure
$\pi=\pi_{{\scriptscriptstyle G}^{*}}\times_{(\rho_{{\scriptscriptstyle
G}^{*}},\lambda)}{\pi_{\scriptscriptstyle Y}}$, where
$\rho_{{\scriptscriptstyle G}^{*}}$ is the right action action of $G^{*}$ on
itself by right translation, one then has the left Poisson Lie group action

$\lambda^{\prime}:\;\;(G^{*},\;\pi_{{\scriptscriptstyle
G}^{*}})\times(G^{*}\times Y,\;\pi)\longrightarrow(G^{*}\times
Y,\;\pi),\;(a,\,(a^{\prime},y))\longmapsto(aa^{\prime},y),\hskip
14.45377pta,a^{\prime}\in G^{*},\,y\in Y.$

Let $X\times_{G^{*}}(G^{*}\times Y)$ is the quotient of $X\times(G^{*}\times
Y)$ by the right diagonal action of $G^{*}$. The Poisson structure
${\pi_{\scriptscriptstyle X}}\times\pi$ on $X\times(G^{*}\times Y)$ then
projects to a well-defined Poisson structure on $X\times_{G^{*}}(G^{*}\times
Y)$, which we denote as $\pi^{\prime}$. On the other hand, one has the
isomorphism

$\psi:\;\;X\times_{G^{*}}(G^{*}\times Y)\longrightarrow X\times
Y,\;[x,\,(a,y)]\longmapsto(xa,\,y),\;\hskip 14.45377ptx\in X,\,a\in
G^{*},\,y\in Y.$

By an argument similar to that used in the proof of [16, Lemma A.1], one sees
that

$\psi:(X\times_{G^{*}}(G^{*}\times Y),\;\pi^{\prime})\longrightarrow(X\times
Y,\;{\pi_{\scriptscriptstyle
X}}\times_{(\rho,\lambda)}{\pi_{\scriptscriptstyle Y}})$

is a Poisson map. $\diamond$

Consider now the standard complex semi-simple Lie group $(G,\pi_{\rm st})$,
and we keep the notation set up in $\S$3.1. Define the non-degenerate bilinear
pairing $\langle\,,\,\rangle_{(\mathfrak{b},\mathfrak{b}_{-})}$ between
$\mathfrak{b}$ and $\mathfrak{b}_{-}$ by

$\langle
x_{-}+x_{0},\;\,y_{+}+y_{0}\rangle_{(\mathfrak{b}_{-},\mathfrak{b})}=\frac{1}{2}\langle
x_{-},\;y_{+}\rangle_{\mathfrak{g}}+\langle
x_{0},\;y_{0}\rangle_{\mathfrak{g}},\hskip
14.45377ptx_{-}\in\mathfrak{n}_{-},\,\,x_{0},\,y_{0}\in\mathfrak{h},\,y_{+}\in\mathfrak{n}_{.}$
(A.3)

For a basis $\\{x_{i}\\}$ for $\mathfrak{b}_{-}$ and dual basis $\\{x^{i}\\}$
for $\mathfrak{b}$ under
$\langle\,,\,\rangle_{(\mathfrak{b}_{-},\mathfrak{b})}$, we choose (see
$\S$3.1)

$\displaystyle\\{x_{i}\\}_{i=1}^{\dim\mathfrak{b}_{-}}=\\{h_{i}:1\leq
i\leq\dim\mathfrak{h}\\}\cup\\{2E_{-\alpha}:\alpha\in\Delta^{+}\\},$ (A.4)
$\displaystyle\\{x^{i}\\}_{i=1}^{\dim\mathfrak{b}}=\\{h_{i}:1\leq
i\leq\dim\mathfrak{h}\\}\cup\\{E_{\alpha}:\alpha\in\Delta^{+}\\}.$ (A.5)

We now give three pairs of Poisson Lie groups to be used to form mixed Poisson
products.

######  Example A.4.

With the non-degenerate pairing
$\langle\,,\,\rangle_{(\mathfrak{b}_{-},\mathfrak{b})}$ between Lie algebras
$\mathfrak{b}_{-}$ of $B_{-}$ and $\mathfrak{b}$ of $B$ given in (A.3), we
have the first pair of dual Poisson Lie groups $((B_{-},\pi_{\rm
st}),\;(B,-\pi_{\rm st}))$. Under the same pairing
$\langle\,,\,\rangle_{(\mathfrak{b}_{-},\mathfrak{b})}$, the Poisson Lie
groups $(B_{-}^{\rm op},\pi_{\rm st})$ and $(B,\pi_{\rm st})$ are also dual to
each other, where $B_{-}^{\rm op}$ is the manifold $B_{-}$ with the group
structure opposite to that on $B_{-}$. A third pair of dual Poisson Lie groups
is $((B_{-},-\pi_{\rm st}),(B,\pi_{\rm st}))$, with the pairing between the
Lie algebras $\mathfrak{b}_{-}$ and $\mathfrak{b}$ given by
$-\langle\,,\,\rangle_{(\mathfrak{b}_{-},\mathfrak{b})}$. Finally we also have
the pair of dual (direct product) Poisson Lie groups

$(A,\pi_{{\scriptscriptstyle A}})=(B_{-},\pi_{\rm st})\times(B_{-}^{\rm
op},\pi_{\rm st}),\hskip 14.45377pt(A^{*},\pi_{{\scriptscriptstyle
A}^{*}})=(B,-\pi_{\rm st})\times(B,\pi_{\rm st}),$

where the non-degenerate pairing between the Lie algebras
$\mathfrak{a}=\mathfrak{b}_{-}\oplus\mathfrak{b}_{-}$ and
$\mathfrak{a}^{*}=\mathfrak{b}\oplus\mathfrak{b}$ is the direct sum of the
pairing $\langle\,,\,\rangle_{(\mathfrak{b}_{-},\mathfrak{b})}$ with itself.
$\diamond$

######  Example A.5.

Consider the maximal torus $T$ of $G$ with the zero Poisson structure as a
Poisson Lie group. By identifying the Lie algebra $\mathfrak{h}$ of $T$ with
$\mathfrak{h}^{*}$ using the bilinear form
$\langle\,,\,\rangle_{\mathfrak{g}}|_{\mathfrak{h}}$, $T$ is also a dual
Poisson Lie group of itself. Let $(X,{\pi_{\scriptscriptstyle X}})$ be a
$T$-Poisson manifold with $T$-action $\sigma:T\times X\to X$. The Poisson
structure ${\pi_{\scriptscriptstyle X}}\bowtie_{\sigma}0$ on $X\times T$ given
in (1.9) is then the mixed product Poisson structure defined using $(T,0)$ as
a Poisson Lie group. $\diamond$

###  A.2. The Poisson varieties $({\mathcal{C}}_{n},\widehat{\pi}_{n})$ and
$({\mathcal{B}}^{n},\overline{\pi}_{n})$ as mixed products

Consider now the complex semi-simple Poisson Lie group $(G,\pi_{\rm st})$, and
let the notation be as in $\S$3.1. Let $Q$ be any closed coisotropic subgroup
of $(G,\pi_{\rm st})$, i.e., $Q$ is a closed Lie subgroup of $G$ which is also
a coisotropic with respect to the Poisson structure $\pi_{\rm st}$. For an
integer $n\geq 1$, let

$F_{n,{\scriptscriptstyle
Q}}=\overbrace{G\times_{B}\cdots\times_{B}G}^{n}/Q=\widetilde{F}_{n}/Q,$

and denoted by $\pi_{n,{\scriptscriptstyle Q}}$ the Poisson structure on
$F_{n,{\scriptscriptstyle Q}}$ that is the quotient of the Poisson structure
$(\pi_{\rm st})^{n}$ on $G^{n}$. When $n=1$, the quotient Poisson structure on
$G/Q$ will be denoted as $\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle
Q}}$. Note that when $Q=\\{e\\}$ and when $Q=B$, we respectively have
$\pi_{n,{\scriptscriptstyle Q}}=\widetilde{\pi}_{n}$ and
$\pi_{n,{\scriptscriptstyle Q}}=\pi_{n}$. In particular, the flag variety
${\mathcal{B}}:=G/B$ has the Poisson structure
$\pi_{1}=\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle B}}$, and the
decorated flag variety ${\mathcal{A}}=G/N$ has the Poisson structure
$\widehat{\pi}_{1}:=\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle N}}$.

Returning to the case of an arbitrary closed coisotropic Lie subgroup $Q$ of
$(G,\pi_{\rm st})$, set

${\mathcal{Q}}_{n}={\mathcal{B}}^{n-1}\times(G/Q),\hskip 14.45377ptn\geq 1.$

One then has the isomorphism

$\Theta_{n,{\scriptscriptstyle Q}}:\;F_{n,{\scriptscriptstyle
Q}}\to{\mathcal{Q}}_{n},\;[g_{1},\cdots,g_{n-1},g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}}\longmapsto(g_{1}{}_{\cdot}B,\ldots,g_{1}\cdots
g_{n-1}{}_{\cdot}B,g_{1}\cdots g_{n-1}g_{n}{}_{\cdot}Q).$

Define the Poisson structure $\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}$ on
${\mathcal{Q}}_{n}$ by

$\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}\,=\,\Theta_{n,{\scriptscriptstyle
Q}}(\pi_{n,{\scriptscriptstyle Q}}).$

Our goal now is to express the Poisson manifold
$({\mathcal{Q}}_{n},\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}})$ as a mixed
product of $n-1$ copies of $({\mathcal{B}},\pi_{1})$ with
$(G/Q,\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle Q}})$.

Let $\lambda_{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$ be the left
Poisson action of $(G,\pi_{\rm st})$ on $(G/Q,\pi_{{\scriptscriptstyle
G}/{\scriptscriptstyle Q}})$ given by translation, i.e,

$\lambda_{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}:\;(G,\pi_{\rm
st})\times(G/Q,\,\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle
Q}})\longrightarrow(G/Q,\,\pi_{{\scriptscriptstyle G}/{\scriptscriptstyle
Q}}),\;(g,\,g_{1}{}_{\cdot}Q)\longmapsto gg_{1}{}_{\cdot}Q.$

Recall the bases $\\{x_{i}\\}_{i=1}^{\dim\mathfrak{b}_{-}}$ of
$\mathfrak{b}_{-}$ and $\\{x^{i}\\}_{i=1}^{\dim\mathfrak{b}_{-}}$ of
$\mathfrak{b}$ in (A.4). For $1\leq j<k\leq n-1$, define
$\mu_{j,k}^{(n-1)}\in{\mathfrak{X}}^{2}({\mathcal{B}}^{n-1})$ by (see notation
in $\S$1.6)

$\mu_{j,k}^{(n-1)}=\sum_{i=1}^{\dim\mathfrak{b}_{-}}(0,\;\ldots,\;0,\;\underset{j^{\text{th}}\,\text{entry}}{\lambda_{{\scriptscriptstyle
G}/{\scriptscriptstyle
B}}(x^{i})},\;0,\;\ldots,\;0)\wedge(0,\;\ldots,\;0,\;\underset{k^{\text{th}}\,\text{entry}}{\lambda_{{\scriptscriptstyle
G}/{\scriptscriptstyle B}}(x_{i})},\;0,\;\ldots,\;0),$

and for $1\leq j\leq n-1$, define $\mu_{j}^{(n,{\scriptscriptstyle
Q})}\in{\mathfrak{X}}^{2}({\mathcal{Q}}_{n})$ by

$\mu_{j}^{(n,{\scriptscriptstyle
Q})}=\sum_{i=1}^{\dim\mathfrak{b}_{-}}(0,\;\ldots,\;0,\;\underset{j^{\text{th}}\,\text{entry}}{\lambda_{{\scriptscriptstyle
G}/{\scriptscriptstyle
B}}(x^{i})},\;0,\;\ldots,\;0)\wedge(0,\;\ldots,\;0,\;\lambda_{{\scriptscriptstyle
G}/{\scriptscriptstyle Q}}(x_{i})).$

######  Proposition A.6.

For any closed coisotropic subgroup $Q$ of $(G,\pi_{\rm st})$, one has

$\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}=(\pi_{1},\,\ldots,\,\pi_{1},\,\pi_{{\scriptscriptstyle
G}/{\scriptscriptstyle Q}})+\sum_{1\leq j<k\leq
n-1}(\mu_{j,k}^{(n-1)},\;0)+\sum_{j=1}^{n-1}\mu_{j}^{(n,{\scriptscriptstyle
Q})}.$ (A.6)

###### Proof.

Note first that the standard $r$-matrix $r_{\rm st}$ on $\mathfrak{g}$ given
in (3.1) is given by

$r_{\rm st}=\sum_{i=1}^{\dim\mathfrak{b}_{-}}x_{i}\otimes
x^{i}\in\in\mathfrak{g}\otimes\mathfrak{g}.$

The case of $Q=\\{e\\}$ follows from the proof of [17, Proposition 8.1] by
setting, in the notation of [17, Proposition 8.1], $Q_{+}=B$ and $r=r_{\rm
st}$. The case for arbitrary $Q$ then follows from the following commutative
diagram of Poisson manifolds

${(F_{n,\\{e\\}},\,\pi_{n,\\{e\\}})}$${({\mathcal{B}}^{n-1}\times
G,\,\pi_{{\scriptscriptstyle{\mathcal{G}}_{n}}})}$${(F_{n,{\scriptscriptstyle
Q}},\,\pi_{n,{\scriptscriptstyle
Q}})\;}$${\;\;({\mathcal{Q}}_{n},\,\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}),}$$\scriptstyle{\Theta_{n,\\{e\\}}}$$\scriptstyle{\Theta_{n,{\scriptscriptstyle
Q}}}$

where the second vertical arrow is ${\rm
Id}_{{\scriptscriptstyle{\mathcal{B}}^{n-1}}}\times p_{{\scriptscriptstyle
G}/{\scriptscriptstyle Q}}$, with $p_{{\scriptscriptstyle
G}/{\scriptscriptstyle Q}}:G\to G/Q$ the projection.

Q.E.D.

For the special case of $Q=B$, let
$\overline{\pi}_{n}=\Theta_{n,{\scriptscriptstyle B}}(\pi_{n})$.

######  Corollary A.7.

For any $n\geq 1$, the Poisson structure $\overline{\pi}_{n}$ on
${\mathcal{B}}^{n}$ is given by

$\overline{\pi}_{n}=(\pi_{1},\pi_{1},\ldots,\pi_{1})+\sum_{1\leq j<k\leq
n}\mu_{jk}^{(n)},$ (A.7)

where for $1\leq j<k\leq n$,
$\mu_{j,k}^{(n)}\in{\mathfrak{X}}^{2}({\mathcal{B}}^{n})$ is given by

$\mu_{j,k}^{(n)}=\sum_{i=1}^{\dim\mathfrak{b}_{-}}(0,\;\ldots,\;0,\;\underset{j^{\text{th}}\,\text{entry}}{\lambda_{{\scriptscriptstyle
G}/{\scriptscriptstyle
B}}(x^{i})},\;0,\;\ldots,\;0)\wedge(0,\;\ldots,\;0,\;\underset{k^{\text{th}}\,\text{entry}}{\lambda_{{\scriptscriptstyle
G}/{\scriptscriptstyle B}}(x_{i})},\;0,\;\ldots,\;0).$

######  Remark A.8.

By (A.6), $\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}$ is a mixed product
Poisson structure on ${\mathcal{Q}}_{n}$ in the sense of Definition A.1. The
Poisson structure $\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}$ on
${\mathcal{Q}}_{n}={\mathcal{B}}^{n-1}\times G/Q$ can also be written as two
fold mixed products: consider again the pair of dual Poisson Lie groups
$((B_{-},\pi_{\rm st}),(B,-\pi_{\rm st}))$ from Example A.4. For $1\leq k\leq
n-1$, define the right Poisson action $\tilde{\rho}_{k}$ of $(B,-\pi_{\rm
st})$ on ${\mathcal{B}}^{k}$ by

$\displaystyle\tilde{\rho}_{k}:$
$\displaystyle\;({\mathcal{B}}^{k},\overline{\pi}_{k})\times(B,-\pi_{\rm
st})\longrightarrow({\mathcal{B}}^{k},\overline{\pi}_{k}),$
$\displaystyle\;((f_{1},f_{2},\ldots,f_{k}),\,b)\longmapsto(b^{-1}f_{1},\,b^{-1}f_{2},\ldots,b^{-1}f_{k}),$

and let $\lambda_{n-k,{\scriptscriptstyle Q}}$ be the left Poisson action of
$(B_{-},\pi_{\rm st})$ on
$({\mathcal{Q}}_{n-k},\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n-k}}})$ given by

$\displaystyle\lambda_{n-k,{\scriptscriptstyle Q}}:$
$\displaystyle\;(B_{-},\pi_{\rm
st})\times({\mathcal{Q}}_{n-k},\,\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n-k}}})\longrightarrow({\mathcal{Q}}_{n-k},\,\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n-k}}}),$
$\displaystyle\;(b_{-},\,(f_{1},\,\ldots,\,f_{n-k-1},\,g_{\cdot}Q))\longmapsto(b_{-}f_{1},\,\ldots,\,b_{-}f_{n-k-1},\,b_{-}g_{\cdot}Q)).$

It then follows from Proposition A.6 that

$\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n}}}=\overline{\pi}_{k}\times_{(\tilde{\rho}_{k},\lambda_{n-k,{\scriptscriptstyle
Q}})}\pi_{{\scriptscriptstyle{\mathcal{Q}}_{n-k}}}.$

See [17, Remark 6.10] for the general setting. In particular, setting
$\widehat{\pi}_{n}=\pi_{n,{\scriptscriptstyle Q}}$ for $Q=N$, the Poisson
structure $\widehat{\pi}_{n}$ on ${\mathcal{B}}^{n-1}\times{\mathcal{A}}$ then
satisfies

$\widehat{\pi}_{n}=\overline{\pi}_{k}\times_{(\tilde{\rho}_{k},\lambda_{n-k,{\scriptscriptstyle
N}})}\widehat{\pi}_{n-k},\hskip 14.45377pt1\leq k\leq n-1.$ (A.8)

$\diamond$

##  Appendix B The Poisson isomorphism $J_{n}$

###  B.1. The isomorphism $J_{n}$

For $n\geq 1$, recall the Poisson action

$\lambda_{n}:\;(G,\pi_{\rm
st})\times(F_{n},\pi_{n})\longrightarrow(F_{n},\pi_{n}),\;\;\lambda_{n}(g,[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}})=[gg_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle F}_{n}},$

which in particular makes $(F_{n},\pi_{n})$ into a $T$-Poisson manifold. By
(1.9), one has the $T$-extension $(F_{n}\times
T,\,\pi_{n}\bowtie_{\lambda_{n}}0)$ of $(F_{n},\pi_{n})$. For notational
simplicity, we set

$\pi_{n}\bowtie
0=\pi_{n}\bowtie_{\lambda_{n}}0=(\pi_{n},0)+\sum_{i=1}^{\dim\mathfrak{h}}(\lambda_{n}(h_{i}),0)\wedge(0,h_{i}^{R})\in{\mathfrak{X}}^{2}(F_{n}\times
T).$

Recall that the open sub-manifold $F_{n}^{o}$ of $F_{n}$ is defined as

$F_{n}^{o}=\\{[g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}:g_{1}g_{2}\cdots g_{n}\in B_{-}\\},$

and that we have the $T$-equivariant isomorphism $J_{n}:\Gamma_{n}\to
F_{n}^{o}\times T$ (see (3.14)) given by

$J_{n}([g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle\widetilde{F}}_{n}})=([g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}},\,[g_{1}g_{2}\cdots g_{n}]_{0}),$

where $T$ acts on $F_{n}^{o}\times T$ diagonally. In this section we prove the
following fact also stated as Theorem 3.5.

######  Theorem B.1.

For any $n\geq 1$,
$J_{n}:(\Gamma_{n},\,\widetilde{\pi}_{n})\to(F_{n}^{o}\times
T,\,\pi_{n}\bowtie 0)$ is a Poisson isomorphism.

###  B.2. Some auxiliary lemmas

Consider the pair of dual Poisson Lie groups $((B_{-},\pi_{\rm
st}),(B,-\pi_{\rm st}))$ from Example A.4. The left Poisson action
$\lambda_{n}$ of $(G,\pi_{\rm st})$ on $(F_{n},\pi_{n})$ restricts to a left
Poisson action of $(B_{-},\pi_{\rm st})$ on $(F_{n},\pi_{n})$, still denoted
by $\lambda_{n}$. One also has the induced right Poisson action of
$(B,-\pi_{\rm st})$ on $(F_{n},\pi_{n})$ by

$\displaystyle\rho_{n}:\;$ $\displaystyle(F_{n},\,\pi_{n})\times(B,-\pi_{\rm
st})\longrightarrow(F_{n},\pi_{n}),$ (B.1)
$\displaystyle([g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}},\,b)\longmapsto[b^{-1}g_{1},g_{2},\ldots,g_{n}]_{{\scriptscriptstyle
F}_{n}}.$

Recall that $\lambda_{n}$ also denotes the induced Lie algebra anti-
homomorphism $\lambda_{n}:\mathfrak{g}\to{\mathfrak{X}}^{1}(F_{n})$, and
recall that for $x\in\mathfrak{g}$, $x^{R}$ is the right invariant vector
field on $G$ with value $x$ at the identity element $e$. For every sub-
manifold $S$ of $G$ that is invariant under left translation by elements in
$B_{-}$, we have the Lie algebra anti-homomorphism

$\lambda_{\mathfrak{b}_{-}}:\mathfrak{b}_{-}\longrightarrow{\mathfrak{X}}^{1}(S),\;\;\lambda_{\mathfrak{b}_{-}}(x)=x^{R}|_{{\scriptscriptstyle
S}}.$

We further simplify notation as follows.

######  Notation B.2.

If $(X,{\pi_{\scriptscriptstyle X}})$ is a Poisson manifold with a right
Poisson action

$\rho:\;(X,{\pi_{\scriptscriptstyle X}})\times(B,-\pi_{\rm
st})\longrightarrow(X,{\pi_{\scriptscriptstyle X}}),\;\;(x,b)\longmapsto xb,$
(B.2)

of the Poisson Lie group $(B,-\pi_{\rm st})$, we set

$\displaystyle{\pi_{\scriptscriptstyle X}}\bowtie_{\rho}\pi_{\rm st}$
$\displaystyle={\pi_{\scriptscriptstyle
X}}\times_{(\rho,\lambda_{\mathfrak{b}_{-}})}\pi_{\rm
st}=({\pi_{\scriptscriptstyle X}},0)+(0,\pi_{\rm
st})-\sum_{i=1}^{\dim\mathfrak{b}_{-}}(\rho(x^{i}),0)\wedge(0,x_{i}^{R})\in{\mathfrak{X}}^{2}(X\times
G),$ $\displaystyle{\pi_{\scriptscriptstyle X}}\bowtie_{\rho}\pi_{n}$
$\displaystyle={\pi_{\scriptscriptstyle X}}\times_{(\rho,\lambda_{n})}\pi_{\rm
st}=({\pi_{\scriptscriptstyle
X}},0)+(0,\pi_{n})-\sum_{i=1}^{\dim\mathfrak{b}_{-}}(\rho(x^{i}),0)\wedge(0,\lambda_{n}(x_{i}))\in{\mathfrak{X}}^{2}(X\times
F_{n})$