# Periodic and open classical spin Calogero-Moser chains

Nicolai Reshetikhin  N.R.: YMSC, Tsinghua University, Beijing; Department of
Mathematics, University of California, Berkeley, & Physics Department, St.
Petersburg University<EMAIL_ADDRESS>
###### Abstract.

We construct a class of interacting spin Calogero-Moser type systems. They can
be regarded as a many particle system with spin degrees of freedom and as an
integrable spin chain of Gaudin type. We prove that these Hamiltonian systems
are superintegrable.

## Introduction

### 1

Classical Calogero-Moser (CM) systems were among the first integrable
$N$-particle systems of one dimensional particles [3][25] with the potential
$1/(q_{i}-q_{j})^{2}$. This model was generalized to the potential
$1/sh^{2}(q_{i}-q_{j})$ in [36]. Then it was extended to other root systems
and to elliptic potentials in [28], to a model involving spin degrees of
freedom in [16].

There is an extensive literature on spin versions of CM systems. For example
in [21][22] solutions to equations of motion to the elliptic spin Calogero-
Moser system were related to special elliptic solutions to the matrix KP
hierarchy. The relation to gauge theories were explored in many papers, see
for example [27][15]. A variety of spin CM systems were obtained by L. Feher,
see for example [10][11][12], in particular he derived important examples
related to homogeneous spaces. Two spin CM systems were studied in [18][19].
Integrable chains of relativistic spin CM type systems were studied in [5][2].

Superintegrability of spin CM systems and of spin Ruijsenaars systems was
established in [29]. In [31] the superintegrability of spin CM systems on
homogeneous spaces was established. A family of superintegrable systems on
moduli spaces of flat connections was constructed in [1]. This family includes
systems studied in [5][2]. In these particular case the system is also
Liouville integrable.

In this paper we will describe classical superintegrable system which we call
spin Calogero-Moser(CM) chains. We call them spin CM chains because they
combine features of many particle systems (as in CM systems) and of spin
chains. We distinguish two cases: a periodic chain and an open chain. The
periodic case is the classical version of a quantum integrable system where
joint eigenfunctions of quantum commuting Hamiltonians are trace functions,
see [7]. In this case the spin part of the system reminds a spin chain with
periodic boundary conditions. In case of rank 1 orbits for $\mathfrak{sl}_{n}$
these systems are linearized versions of [5] and [2]. In the open case they
are a classical version of quantum integrable systems constructed in [35][33].
For these systems the spin part of the system is similar to an open spin
chain.

In both cases, i.e. in the periodic and in the open spin Calogero-Moser
chains, the phases space is a stratified symplectic space [23], which, in some
cases have only one stratum and becomes a symplectic manifold.

### 2

Recall that a superintegrable system is the structure on a symplectic manifold
$\mathcal{M}$ that consists of a Poisson manifold $\mathcal{P}$, a Poisson
manifold $\mathcal{B}$ with the trivial Poisson structure (i.e. zero Poisson
tensor) and two surjective Poisson projections

(1) $\mathcal{M}\stackrel{{\scriptstyle
p_{1}}}{{\rightarrow}}\mathcal{P}\stackrel{{\scriptstyle
p_{2}}}{{\rightarrow}}\mathcal{B}$

such that $\dim(\mathcal{M})=\dim(\mathcal{P})+\dim(\mathcal{B})$. For a
superintegrable system a generic fiber of $p_{1}$ is an isotropic submanifolds
of dimension $\dim(\mathcal{B})$ and connected components of a generic fiber
of $p_{2}$ is a disjoint union of symplectic leaves of $\mathcal{P}$. For
details see [26], [30] and references therein. Here we adopt this notion to
the case of stratified symplectic and Poisson spaces in which case $p_{1}$ and
$p_{2}$ are Poisson mapping between stratifies spaces. In this paper the
superintegrability means the balance of dimensions for the big stratum. How
the system behave at smaller strata will be a subject of a separate
publication. In the algebraic case, the appropriate setting is symplectic and
Poisson stacks.

Let $I$ be a Poisson commutative subalgebra of $A=C^{\infty}(\mathcal{M})$
that consists of functions which are constant on fibers of $p_{2}\circ p_{1}$
(the pull-back of functions on $\mathcal{B}$ to functions on $\mathcal{M}$)
and $J$ be the Poisson algebra of functions which are constant on fibers of
$p_{1}$ (the pull-back of functions on $\mathcal{P}$). The condition on
$(\mathcal{M},\mathcal{P},\mathcal{B})$ for being a superintegrable system is
equivalent to the following condition on $I\subset J\subset A$. The Poisson
algebra $A$ has trivial center, $I\subset A$ is a Poisson commutative
subalgebra, such that $J$, its centralizer in $A$ maximal possible Gelfand-
Kirillov dimension for the given Gelfand-Kirillov dimension of $I$.

The Hamiltonian dynamics generated by a function $H\in I$ is called
superintegrable. Any function from $J$ is constant along flow lines of the
vector field generated by $H$ and thus, it is an integral of motion for the
Hamiltonian dynamics generated by $H$. This is why we call elements of the
Poisson commutative subalgebra $I$ Hamiltonians and elements of $J$
conservation laws.

### 3

Throughout this paper $G$ is a split real connected semisimple Lie algebra
with finite center which admits a complexification, and
$\Theta\in\textup{Aut}(G)$ is a Cartan involution. We denote by $K=G^{\Theta}$
the closed subgroup of fixed points of $\Theta$, which is connected and
maximal compact. Let $\theta$ the corresponding Cartan involution111Recall
that an involution $\theta:\mathfrak{g}\to\mathfrak{g}$ is a Cartan involution
when the bilinear from $(-\theta(x),y)$ on $\mathfrak{g}$ is positive
definite. Here $(\cdot,\cdot)$ is the Killing form. of $\mathfrak{g}$, and
$\mathfrak{k}$ the Lie algebra of $K$. The associated Cartan decomposition of
$\mathfrak{g}$ is $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, with
$\mathfrak{p}$ the $(-1)$-eigenspace of $\theta$.

Let $\mathfrak{a}\subset\mathfrak{g}$ be maximally noncompact $\theta$-stable
Cartan subalgebra of $\mathfrak{g}$. Since $\mathfrak{g}$ is split we have
$\mathfrak{a}\subseteq\mathfrak{p}$. On the Lie group level,
$A=\exp(\mathfrak{a})\subset G$ is a maximal real split torus in $G$ and
$H:=Z_{G}(A)$, the centraliser of $A$ in $G$, is a Cartan subgroup in $G$
containing $A$. The exponential map provides an isomorphism
$\mathfrak{a}\overset{\sim}{\longrightarrow}A$, whose inverse we denote by
$\log:A\rightarrow\mathfrak{a}$.

Consider the root decomposition of $\mathfrak{g}$ with respect to the Cartan
subalgebra $\mathfrak{a}$,

$\mathfrak{g}=\mathfrak{a}\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{\alpha}$

where $R\subset\mathfrak{a}^{*}$ is the root system of $\mathfrak{g}$ relative
to $\mathfrak{a}$. Choose $e_{\alpha}\in\mathfrak{g}_{\alpha}$ such that

(2) $\theta(e_{\alpha})=-e_{-\alpha}$

and $(e_{\alpha},e_{-\alpha})=1$ for each $\alpha\in R$, and choose a subset
$R_{+}\subset R$ of positive roots. Let
$W\subset\textup{GL}(\mathfrak{a}^{*})$ be the Weyl group of $R$.

The Weyl group $W$ is isomorphic to $N_{G}(A)/H$, where $N_{G}(A)$ is the
normaliser of $A$ in $G$. Denote by $A_{reg}$ the set of regular elements in
$a\in A$,

$A_{reg}:=\\{a\in A\,\,|\,\,a_{\alpha}:=e^{\alpha(\log(a))}\not=1\hbox{ for
all }\alpha\in R\\}.$

It is the union of all the regular $W$-orbits in $A$. A fundamental domain for
the $W$-action on $A_{reg}$ 222In case of $SL_{n}(\mathbb{R})$ one can take
$A$ to be the diagonal unimodular matrices with positive real entries, and
$A_{reg}$ consists of those with distinct diagonal entries. is the positive
Weyl chamber

$A_{+}=\\{a\in A\,\,|\,\,a_{\alpha}:=e^{\alpha(\log(a))}>1\mbox{ for any
}\alpha\in R_{+}\\}.$

Let $G^{\prime}\subset G$ be the set of elements $g\in G$ which are
$G$-conjugate to some element in $A_{reg}$. The inclusion
$A_{reg}\hookrightarrow G^{\prime}$ induces a bijection
$A_{reg}/W\overset{\sim}{\longrightarrow}G^{\prime}/G$, with $A_{reg}/W$ the
set of $W$-orbits in $A_{reg}$ and $G^{\prime}/G$ the set of $G$-conjugacy
classes in $G^{\prime}$.

The Weyl group $W$ is also isomorphic to $N_{K}(A)/M$, where
$N_{K}(A)=N_{G}(A)\cap K$ and $M=H\cap K$ (note that $M$ is a finite group
since $G$ is split). The inclusion map $A\hookrightarrow G$ induces an
isomorphism $A/W\overset{\sim}{\longrightarrow}K\backslash G/K$. We write
$G_{reg}=KA_{+}K$ for the union of the double $(K,K)$-cosets intersecting
$A_{reg}$.

### 4

The phase space of a periodic spin Calogero-Moser chain corresponding to a
collection $\mathcal{O}=\\{\mathcal{O}_{1},\dots,\mathcal{O}_{n}\\}$ of
coadjoint orbits $\mathcal{O}_{i}\subset\mathfrak{g}^{*}$ is the regular part
of the symplectic leaf $\mathcal{S}(\mathcal{O})$ of the stratified Poisson
space $T^{*}(G^{\times n})/G_{n}$, with the action of the gauge group
$G_{n}=G^{\times n}$ the lift of a twisted conjugation action on $G^{\times
n}$(see section 1.1).333In this paper we assume that all quotients $X/H$ are
GIT quotients. Here we assume that each of $\mathcal{O}_{i}$ is non-trivial,
i.e. $\mathcal{O}_{i}\neq\\{0\\}$ These symplectic leaves are obtained by the
Hamiltonian reduction, as it is described in section 1.1. As a stratified
symplectic space

$\mathcal{S}(\mathcal{O})\simeq\\{(x_{1},\dots,x_{n},g)\in{\mathfrak{g}^{*}}^{\times
n}\times
G\,\,|\,\,x_{1}-\textup{Ad}_{g^{-1}}^{*}x_{n}\in\mathcal{O}_{1},\,\,x_{i}-x_{i-1}\in\mathcal{O}_{i},\,\,i=2,\ldots,n\\}/G,$

see section 1.3.444There are $n$ such natural isomorphisms $\varphi_{j}$
($1\leq j\leq n$), see section 1.2. In the introduction we use $\varphi_{n}$.
Its regular part is defined as the intersection
$\mathcal{S}(\mathcal{O})_{reg}=\mathcal{S}(\mathcal{O})\cap({\mathfrak{g}^{*}}^{\times
n}\times G^{\prime})/G$.555A better way to think about the periodic spin
Calogero-Moser system for a real split simple Lie group $G$ is to define
$\mathcal{S}(\mathcal{O})_{\mathbb{C}}$ is complex algebraic setting and then
to take the corresponding real slice. This will be addressed in another
publication. The regular part has the following structure as a symplectic
manifold,
$\mathcal{S}(\mathcal{O})_{reg}\simeq\bigl{(}\nu_{\mathcal{O}}^{-1}(0)/H\times
T^{*}A_{reg}\bigr{)}/W$, where
$\nu_{\mathcal{O}}:\mathcal{O}_{1}\times\cdots\mathcal{O}_{n}\rightarrow\mathfrak{a}^{*}$
is the moment map for the diagonal coadjoint action of $H$ on
$\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}$ (see section 1.3).

Trivialization of $T^{*}G$ by right translations gives an isomorphism
$T^{*}(G^{\times n})\simeq{\mathfrak{g}^{*}}^{\times n}\times G^{\times n}$
and the Poisson projection $T^{*}(G^{\times
n})/G_{n}\to{(\mathfrak{g}^{*}/G)}^{\times n}$, which is the projection to the
cotangent directions followed by the quotienting with respect to the coadjoint
action of $G^{\times n}$. Poisson commuting Hamiltonians of the periodic spin
Calogero-Moser system are functions on $T^{*}(G^{\times n})/G_{n}$ which are
constant on fibers of this Poisson projection. More precisely, tPoisson
commuting Hamiltonians are such functions restricted to
$\mathcal{S}(\mathcal{O})_{reg}$.

Consider the increasing set of natural numbers $2=d_{1}\leq\cdots\leq d_{r}$
with $r=\textup{rank}(\mathfrak{g})$ and $d_{k}-1$ the exponents of
$\mathfrak{g}$. Let $c_{d_{k}}$ be the nonzero coadjoint invariant functions
on $\mathfrak{g}^{*}$ of degree $d_{k}$, known as Casimir functions. The
function $c_{2}$ is the quadratic Casimir of $\mathfrak{g}$. Let
$H^{(l)}_{d_{k}}$ be the function on $({\mathfrak{g}^{*}/G})^{\times n}$ which
is $c_{d_{k}}$ on the $l$-th factor and constant on all other factors.

Let us denote vectors in $\mathcal{O}_{k}\subset\mathfrak{g}^{*}$ by
$\mu^{(k)}$, its Cartan component by $\mu_{0}^{(k)}$ and set
$\mu_{\alpha}^{(k)}=\mu^{(k)}(e_{-\alpha})$ for $\alpha\in R$. Denote $(p,a)$
points on $T^{*}A\simeq\mathfrak{a}^{*}\times A$. Now let us describe
quadratic Hamiltonians in terms of these variables.

The $n$-th quadratic Hamiltonian is the spin Calogero-Moser Hamiltonian. It
has particularly simple form:

$H^{(n)}_{2}=\frac{1}{2}(p,p)-\sum_{\alpha>0}\frac{\mu_{\alpha}\mu_{-\alpha}}{2\textup{sh}^{2}(q_{\alpha})}$

where we used the parametrization $a_{\alpha}=e^{q_{\alpha}}$ (so
$q_{\alpha}=\alpha(\log(a))$) and
$\mu_{\alpha}=\mu_{\alpha}^{(1)}+\cdots+\mu_{\alpha}^{(n)}$, and
$(\cdot,\cdot)$ is the Euclidean form on $\mathfrak{a}^{*}$ obtained by
dualising the restriction of the Killing form of $\mathfrak{g}$ to
$\mathfrak{a}$.

The differences $D_{k}=H^{(k)}_{2}-H^{(k-1)}_{2}$ for $1<k\leq n$ are
classical analogs of topological Knizhnik-Zamolodchikov-Bernard differential
operators,

(3) $D_{k}=(\mu^{(k)}_{0},p)-\sum_{l=1}^{k-1}r_{lk}+\sum_{l=k+1}^{n}r_{kl}$

where $r_{kl}$ for $k\not=l$ is a classical version of the Felder’s dynamical
$r$-matrix [13],

(4)
$r_{kl}=-\frac{1}{2}(\mu^{(k)}_{0},\mu^{(l)}_{0})+\sum_{\alpha}\frac{\mu^{(k)}_{-\alpha}\mu^{(l)}_{\alpha}}{a_{\alpha}-1}$

and $\sum_{\alpha}$ stands for the sum over all the roots $\alpha\in R$. This
explicit form of $D_{k}$ is derived in section 1.3.

The superintegrability of this system is described in section 1.5. The
projection method for constructing solutions of equations of motion and angle
variables are described in section 1.6.

One can choose $G$ to be a the maximal compact real form of the
complexification $G_{\mathbb{C}}$. In this case the integrable system is
similar, but hyperbolic functions gets replaces by the trigonometric ones. The
structure of the phase space is again a stratified symplectic space. The
superintegrability of the quantum counterpart of such compact case is proven
in [32].

### 5

The phase space of an open Calogero-Moser spin chain is the regular part of a
symplectic leaf of the Poisson manifold $T^{*}(G^{\times n+1})/(K\times
G^{\times n}\times K)$ where the action of the gauge group $K\times G^{\times
n}\times K$ is described in section 2.3, and $K\subset G$ is as above. Such
symplectic leaves are given by the Hamiltonian reduction. They are
parametrized by collections of coadjoint orbits
$\mathcal{O}=\\{\mathcal{O}^{K}_{\ell},\mathcal{O}_{1},\dots,\mathcal{O}_{n},\mathcal{O}^{K}_{r}\\}$
where $\mathcal{O}_{i}\subset\mathfrak{g}^{*}$ and
$\mathcal{O}^{K}_{\ell,r}\subset\mathfrak{k}^{*}\subset\mathfrak{g}^{*}$ are
coadjoint orbits. We assume that none of $\mathcal{O}_{i}$ is trivial, i.e.
$\mathcal{O}_{i}\neq\\{0\\}$.

We denote the corresponding symplectic leaf by $\mathcal{S}(\mathcal{O})$. It
is a stratified symplectic space. Using Cartan decomposition $G=KAK$ and a
”gauge fixing fixing”, we define the regular part
$\mathcal{S}(\mathcal{O})_{reg}$ of $\mathcal{S}(\mathcal{O})$ as the strtatum

$S(\mathcal{O})_{reg}\simeq(T^{*}A_{reg}\times\mathcal{O}^{K}_{1}\times\mathcal{O}_{1}\times\dots\times\mathcal{O}_{n}\times\mathcal{O}^{K}_{2}))/N_{K}(A),$

where on the right we have a natural product symplectic structure.

Similarly to the periodic case, quadratic Hamiltonians can be computed
explicitly in terms of Cartan components $\mu^{(k)}_{0}$ and root coordinates
$\mu^{(k)}_{\alpha}$ of vectors $\mu^{(k)}\in\mathcal{O}_{k}$, coordinates
$\mu^{\prime}_{[\alpha]},\mu^{\prime\prime}_{[\alpha]}$ on
$\mathcal{O}_{\ell}^{K}$ and $\mathcal{O}_{r}^{K}$ respectively (in the basis
elements
$e_{[\alpha]}=e_{-\alpha}-e_{\alpha}\in\mathfrak{k}\subset\mathfrak{g}$ for
$\alpha\in R_{+}$), and $(p,a)\in T^{*}A_{reg}$.

Assuming the gauge fixing $\phi_{n}$ (see section 2.3) we have

$H^{(n)}_{2}=\frac{1}{2}(p,p)+\sum_{\alpha>0}\frac{(a_{\alpha}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+a_{\alpha}(\mu_{\alpha}-\mu_{-\alpha}))(a_{\alpha}^{-1}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+a_{\alpha}^{-1}(\mu_{\alpha}-\mu_{-\alpha}))}{(a_{\alpha}-a_{-\alpha})^{2}}$

For other quadratic Hamiltonians the differences

$D_{k}=H_{2}^{(k)}-H_{2}^{(k-1)}\qquad\quad(1\leq k\leq n)$

are more interesting. They are classical analogs of boundary Knizhnik-
Zamolodchikov-Bernard differential operators [35][33]. We have the following
formula for $D_{k}$:

$D_{k}=(\mu^{(k)}_{0},p)-\sum_{l=1}^{k-1}(r_{lk}+r_{lk}^{\theta_{l}})+(\sum_{\alpha}K_{\alpha}\mu^{(k)}_{-\alpha}-\kappa_{k})+\sum_{l=k+1}^{n}(r_{kl}-r_{kl}^{\theta_{k}}).$

Here $r_{kl}$ for $k\not=l$ now is Felder dynamical $r$-matrix rescaled in
$a\in A_{reg}$,

(5)
$r_{kl}=-\frac{1}{2}(\mu^{(k)}_{0},\mu^{(l)}_{0})+\sum_{\alpha}\frac{\mu^{(k)}_{-\alpha}\mu^{(l)}_{\alpha}}{a_{\alpha}^{2}-1},$

$\theta_{k}$ is the transpose of the Cartan involution acting on $\mu^{(k)}$,

$\kappa_{k}=\frac{1}{2}(\mu^{(k)}_{0},\mu^{(k)}_{0})+\sum_{\alpha}\frac{(\mu^{(k)}_{\alpha})^{2}}{1-a_{\alpha}^{2}}$

and

(6)
$K_{\alpha}=\frac{a_{\alpha}\mu^{\prime}_{[\alpha]}+\mu^{\prime\prime}_{[\alpha]}}{a_{\alpha}-a_{\alpha}^{-1}}.$

The differences $D_{k}=H_{2}^{(k)}-H_{2}^{(k-1)}$ are classical analogs of
boundary KZB operators derived in [35][33]. The superintegrability of open
spin CM chains is proven in section 2.6. The projection method for solving
equations of motion and angle coordinates are described in section 2.7.

### 6

The structure of the paper is as follows. In section 1 we construct periodic
spin CM chains by the Hamiltonian reduction and prove the superintegrability.
In section 1.1 we describe the phase space of such a system. In sections 1.2,
1.3 we describe the regular part of the phase space. Hamiltonians of a
periodic spin CM chain, restricted to the regular part of the phase space are
described in section 1.4. The superintegrability of a periodic spin CM chain
is proven in section 1.5. In section 1.6 solutions to equations of motion are
described algebraically by the projection method, and angle variables are
described. In section 2 we focus on open spin CM chains. In section 2.1 we
describe phase spaces. In section 2.3, 2.4 we describe the regular part of the
phase space. Hamiltonians of an open spin CM chain, restricted to the regular
part of the phase space are described in section 2.5. The superintegrability
of an open spin CM chain is proven in section 2.6. In section 2.7 solutions to
equations of motion are described algebraically by the projection method, and
angle variables are described. In the conclusion (section 3) we discuss some
open problems and describe in details periodic CM spin chain for $SL_{N}$ with
orbits of rank 1. In Appendix A we compare our symplectic leaves with the ones
from [31].

Throughout this paper we will focus on split real semisimple Lie groups.
However, since all constructions are algebraic they extend (with appropriate
modifications) to the complex algebraic case. The non-split real case will be
the subject of a separate publication (see [33] for the quantum case). Another
important real case is when $G$ is compact, which can be deduced from the
complex algebraic case by restriction to a compact real form. The structure of
phase spaces as stratified symplectic spaces will be explored further in [6].

### Acknowledgments

This paper was started as a joint project with Jasper Stokman. The author is
grateful to Jasper for many discussions and for the collaboration on this
paper. He also would like to thank Vladimir Fock, Eva Miranda and Hessel
Postuma for important discussions and remarks and to Zhuo Chen, Kai Jiang and
Husileng Xiao for discussions on stratified symplectic spaces. N.R. want to
thank ITS-ETH for the hospitality, where the bulk of this work was completed.
The work of N.R. was supported by the NSF grant DMS-1902226, by the Dutch
research council (NWO 613.009.126), and by the grant RFBR No. 18-01-00916.

##  1\. Periodic spin Calogero-Moser chains

###  1.1. The phase space as the Hamiltonian reduction

Here we will describe the phase space of a periodic spin Calogero-Moser chain
as a Hamiltonian reduction of $T^{*}(G^{\times n})$. Let us start with the
description of these symplectic spaces.

Consider the manifold $T^{*}(G^{\times n})$ with the standard symplectic
structure. The cotangent bundle over a Lie group can be trivialized by right
translations, which gives an isomorphism of vector bundles

$T^{*}(G^{\times n})\simeq(T^{*}G)^{\times n}\simeq{\mathfrak{g}^{*}}^{\times
n}\times G^{\times n}$

We will choose this trivialization throughout the paper.

The Lie group $G_{n}:=G^{\times n}$ acts naturally on itself by left and right
translations. Lifting these actions to $T^{*}(G^{\times n})$, after the
trivialization of the cotangent bundle, we can write the action by left
translations as:

$h_{L}(x,g)=(Ad_{h_{1}}^{*}(x_{1}),Ad_{h_{2}}^{*}(x_{2})\dots,Ad_{h_{n}}^{*}(x_{n}),h_{1}g_{1},h_{2}g_{2},\dots,h_{n}g_{n})$

and the action by right translations as

$h_{R}(x,g)=(x_{1},\dots,x_{n},g_{1}h_{1}^{-1},\dots,g_{n}h_{n}^{-1})$

Both these actions are Hamiltonian with moment maps

$\mu_{L}(x,g)=(x_{1},x_{2},\dots,x_{n})$

and

$\mu_{R}(x,g)=(-Ad_{g_{1}^{-1}}^{*}(x_{1}),\dots,-Ad_{g_{n}^{-1}}^{*}(x_{n}))$

respectively.

Actions by left and right translations can be twisted by permutations. In
particular, we can twist the action by left translations by a cyclic
permutation. Combining the twisted left action with the non-twisted right
action we obtain the ”gauge action” of $G_{n}$ on $G^{\times n}$666One can
twist both left and right actions by a permutation. This leads to other
superintegrable systems.

$h(g_{1},\dots,g_{n})=(h_{1}g_{1}h_{2}^{-1},h_{2}g_{2}h_{3}^{-1},\dots,h_{n}g_{n}h_{1}^{-1})$

Lifting the twisted conjugation action of $G_{n}$ on $G^{\times n}$ to
$T^{*}(G^{\times n})$ we obtain the ”gauge action” on the cotangent bundle:

(7)
$h(x,g)=(Ad_{h_{1}}^{*}(x_{1}),Ad_{h_{2}}^{*}(x_{2}),\dots,Ad_{h_{n}}^{*}(x_{n}),h_{1}g_{1}h_{2}^{-1},h_{2}g_{2}h_{3}^{-1},\dots,h_{n}g_{n}h_{1}^{-1})$

Because this is the diagonal action for two Hamiltonian actions, the gauge
action is also Hamiltonian with the moment map $\mu:T^{*}(G^{\times
n})\to{\mathfrak{g}^{*}}^{\times n}$:

(8)
$\mu(x,g)=\mu_{L}(x,g)+\mu_{R}^{tw}(x,g)=(x_{1}-Ad_{g_{n}^{-1}}^{*}(x_{n}),x_{2}-Ad_{g_{1}^{-1}}^{*}(x_{1}),\dots,x_{n}-Ad_{g_{n-1}^{-1}}^{*}(x_{n-1}))$

where $\mu_{R}^{tw}$ is the right moment map, twised by cyclic permutation.

Because the gauge action (7) of $G_{n}$ is Hamiltonian, the quotient space
$T^{*}(G^{\times n})/G_{n}$ is a Poisson space.777This space is singular.
Having in mind classical-quantum correspondence we need the algebra of
functions on $T^{*}G^{\times n}/G_{n}$. Thus, by the quotient space we will
always mean the GIT quotient. By definition, functions on $T^{*}(G^{\times
n})/G_{n}$ are $G_{n}$-invariant functions on $T^{*}(G^{\times n})$.
Symplectic leaves of $T^{*}(G^{\times n})/G_{n}$ are given by the Hamiltonian
reduction with respect to the moment map (8). Let
$\mathcal{O}_{1},\dots,\mathcal{O}_{n}$ be coadjoint orbits in
$\mathfrak{g}^{*}$, then the corresponding symplectic leaf in $T^{*}(G^{\times
n})/G_{n}$ is

(9)
$\mathcal{S}(\mathcal{O})=\mu^{-1}(\mathcal{O}_{1}\times\dots\times\mathcal{O}_{n})/G_{n}=\\{(x,g)\in{\mathfrak{g}^{*}}^{\times
n}\times G^{\times
n}|x_{i}-Ad_{g_{i-1}^{-1}}^{*}(x_{i-1})\in\mathcal{O}_{i}\\}/G_{n}$

where $G_{n}$ acts by the gauge transformations (7) and the indices $i$ should
be taken modulo $n$.

On each of these symplectic leaves we will construct a superintegrable system
which we will call a periodic spin Calogero-Moser chain.

###  1.2. The gauge fixing

Let us fix $i\in 1,\dots,n$ and $g=(g_{1},\ldots,g_{n})\in G^{\times n}$. Let
$h\in G_{n}$ such that

$h_{j}=\begin{cases}h_{i}g_{i-1}^{-1}\cdots g_{j+1}^{-1}g_{j}^{-1}\quad&\hbox{
for }\,\,1\leq j<i,\\\ h_{i}g_{i-1}^{-1}\cdots
g_{2}^{-1}g_{1}^{-1}g_{n}^{-1}\cdots g_{j+1}^{-1}g_{j}^{-1}\quad&\hbox{ for
}\,\,i<j\leq n.\end{cases}$

Denote such element of $G_{n}$ by $h_{g}$ (we suppress the dependence on $i$).

It is easy to check that the gauge transformation of $g=(g_{1},\dots,g_{n})$
by the element $h_{g}$ brings it to $(1,\ldots,1,h_{i}(g_{i}g_{i+1}\cdots
g_{n}g_{1}g_{2},\ldots g_{i-1})h_{i}^{-1},1,\ldots,1)$, with the
$i^{\textup{th}}$-entry being the nontrivial entry. This identifies the
$G_{n}$ gauge orbit through $g=(g_{1},\dots,g_{n})$ with the $G$-conjugation
orbit through $g_{1}\cdots g_{n}$. It thus gives an ($i$-independent)
isomorphism

$G^{\times n}/G_{n}\overset{\sim}{\longrightarrow}G/G,$

where $G/G$ denotes the set of conjugacy classes in $G$. On the cotangent
bundles the gauge fixing with gives the isomorphism
$\varphi_{i}:\bigl{(}\mathfrak{g}^{*\times n}\times G^{\times
n}\bigr{)}/G_{n}\overset{\sim}{\longrightarrow}\bigl{(}\mathfrak{g}^{*\times
n}\times G\bigr{)}/G$ mapping the $G_{n}$-orbit $G_{n}(x,g)$ through
$(x,g)\in\mathfrak{g}^{*\times n}\times G^{\times n}$ to the $G$-orbit through

$\bigl{(}Ad_{g_{i}^{-1}\cdots
g_{1}^{-1}}^{*}(x_{1}),\ldots,Ad_{g_{i}^{-1}}^{*}(x_{i}),Ad_{g_{i}^{-1}\cdots
g_{1}^{-1}g_{n}^{-1}\cdots
g_{i+1}^{-1}}^{*}(x_{i+1}),\ldots,Ad_{g_{i}^{-1}\cdots
g_{1}^{-1}g_{n}^{-1}}^{*}(x_{n}),g_{i+1}\cdots g_{n}g_{1}\cdots g_{i}\bigr{)}$

(for $i=n$ this should be read as $\bigl{(}Ad_{g_{n}^{-1}\cdots
g_{1}^{-1}}^{*}(x_{1}),\ldots,Ad_{g_{n}^{-1}}^{*}(x_{n}),g_{1}\cdots
g_{n}\bigr{)}$). Here $G$ is acting diagonally on $\mathfrak{g}^{*\times
n}\times G$ via the coadjoint action on $\mathfrak{g}^{*}$ and the conjugation
action on $G$. From now on we will work with the isomorphism $\varphi_{n}$.

###  1.3. The regular part of the phase space

The image of the symplectic leaf $\mathcal{S}(\mathcal{O})$ under the
isomorphism $\varphi_{n}$ is

$\mathcal{S}(\mathcal{O})=\\{(z_{1},\dots,z_{n},g)\in\mathfrak{g}^{*\times
n}\times
G\,\,|\,\,z_{1}-Ad^{*}_{g^{-1}}z_{n}\in\mathcal{O}_{1},\,\,z_{i}-z_{i-1}\in\mathcal{O}_{i},\,\,\,i=2,\dots,n\\}/G.$

Define the regular part
$\mathcal{S}(\mathcal{O})_{reg}\subset\mathcal{S}(\mathcal{O})$ of the phase
space as $\mathcal{S}(\mathcal{O})\cap(\mathfrak{g}^{*\times n}\times
G^{\prime})/G$. On $\mathcal{S}(\mathcal{O})_{reg}$ we can choose a
representative where $g$ is in the regular part $A_{reg}$ of the real split
torus $A$ in $G$: $g=bab^{-1},z_{i}=\textup{Ad}_{b}^{*}x^{(i)}$ with $a\in
A_{reg}$. Then we have

$\begin{split}\mathcal{S}(\mathcal{O})_{reg}=\\{(x^{(1)},\dots,x^{(n)},a)\in\mathfrak{g}^{*\times
n}\times A_{reg}\,\,|\,\,&x^{(1)}-Ad^{*}_{a^{-1}}x^{(n)}\in\mathcal{O}_{1},\\\
&\,\,\,x^{(i)}-x^{(i-1)}\in\mathcal{O}_{i},\,\,i=2,\dots,n\\}/N_{G}(A).\end{split}$

Identify $\mathfrak{g}^{*}\simeq\mathfrak{g}$ and
$\mathfrak{a}^{*}\simeq\mathfrak{a}$ via the Killing form of $\mathfrak{g}$.
The element $y\in\mathfrak{g}^{*}$ then corresponds to
$y_{0}+\sum_{\alpha}y_{\alpha}e_{\alpha}$, where $y_{0}$ is the element in
$\mathfrak{a}$ corresponding to $y|_{\mathfrak{a}}$ and
$y_{\alpha}=y(e_{-\alpha})$. Let $\mu^{(j)}\in\mathcal{O}_{j}$ be vectors
$\mu^{(1)}=x^{(1)}-Ad^{*}_{a^{-1}}x^{(n)}$ and $\mu^{(i)}=x^{(i)}-x^{(i-1)}$
for $i=2,\dots,n$. For coordinates $x^{(i)}_{\alpha}$ and $\mu^{(i)}_{\alpha}$
of vectors $x^{(i)}$ and $\mu^{(i)}$ we then have

$x_{\alpha}^{(1)}-a_{\alpha}^{-1}x^{(n)}_{\alpha}=\mu^{(1)}_{\alpha},\qquad\quad
x^{(i)}_{\alpha}-x^{(i-1)}_{\alpha}=\mu^{(i)}_{\alpha},\qquad i=2,\dots,n.$

For the Cartan components we have

$x^{(i)}_{0}-x^{(i-1)}_{0}=\mu^{(i)}_{0},\qquad i=1,\dots,n,$

with the index $i$ taken to be modulo $n$.

Solving these equations for $x^{(i)}$ we have

(10)
$\begin{split}x^{(i)}_{\alpha}&=\frac{a_{\alpha}(\mu_{\alpha}^{(1)}+\mu_{\alpha}^{(2)}+\cdots+\mu_{\alpha}^{(i)})+\mu_{\alpha}^{(i+1)}+\mu^{(i+2)}+\cdots+\mu_{\alpha}^{(n)}}{a_{\alpha}-1},\\\
x^{(i)}_{0}&=x^{(1)}_{0}+\mu^{(2)}_{0}+\dots+\mu^{(i)}_{0}=x^{(n)}_{0}-\mu^{(n)}_{0}-\dots-\mu^{(i+1)}_{0}\end{split}$

and we have the constraint

(11) $\mu^{(1)}_{0}+\dots+\mu^{(n)}_{0}=0.$

This gives an isomorphism

(12)
$\mathcal{S}(\mathcal{O})_{reg}\overset{\sim}{\longrightarrow}\bigl{(}\nu_{\mathcal{O}}^{-1}(0)/H\times
T^{*}A_{reg}\bigr{)}/W$

which preserves the natural symplectic structures, where
$\nu_{\mathcal{O}}:\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}\rightarrow\mathfrak{a}^{*}$
is the moment map
$(\mu^{(1)},\ldots,\mu^{(n)})\mapsto(\mu^{(1)}+\cdots+\mu^{(n)})|_{\mathfrak{a}}$
for the diagonal action of $H$ on the product
$\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}$ of coadjoint orbits, and
$W=N_{G}(A)/H$ acts diagonally on $\nu_{\mathcal{O}}^{-1}(0)/H\times
T^{*}A_{reg}$. The isomorphism (12) maps the $N_{G}(A)$-orbit through
$\bigl{(}x^{(1)},\ldots,x^{(n)},a\bigr{)}$ to the $W$-orbit through
$\bigl{(}H(x^{(1)}-Ad_{a^{-1}}^{*}x^{(n)},x^{(2)}-x^{(1)},\ldots,x^{(n)}-x^{(n-1)}),x_{0}^{(n)},a\bigr{)}$,
where we used the trivialisation $T^{*}A_{reg}\simeq\mathfrak{a}\times
A_{reg}$. The inverse maps the $W$-orbit through
$\bigl{(}H(\mu^{(1)},\ldots,\mu^{(n)}),p,a\bigr{)}$ to the $N_{G}(A)$-orbit
through $(x^{(1)},\ldots,x^{(n)},a)$, with

(13)
$x^{(i)}=p-\mu_{0}^{(n)}-\cdots-\mu_{0}^{(i+1)}+\sum_{\alpha}\left(\frac{a_{\alpha}(\mu_{\alpha}^{(1)}+\cdots+\mu_{\alpha}^{(i)})+\mu_{\alpha}^{(i+1)}+\cdots+\mu_{\alpha}^{(n)}}{a_{\alpha}-1}\right)e_{\alpha},$

where we use the identification $\mathfrak{g}\simeq\mathfrak{g}^{*}$ via the
Killing form.

###  1.4. Hamiltonians of a periodic spin CM chain

After the trivialization of the cotangent bundle by translations, we have a
natural projection:

(14) $T^{*}(G^{\times n})\simeq{\mathfrak{g}^{*}}^{\times n}\times G^{\times
n}\to{\mathfrak{g}^{*}}^{\times n}$

which is simply the projection to the first factor. This projection depends on
the trivialization. In this paper we alway assume that we use the
trivialization by right translations. However, the corresponding projection of
quotient spaces

(15) $T^{*}(G^{\times n})/G_{n}\to({\mathfrak{g}^{*}/G})^{\times n}$

does not depend on the trivialization and in this sense is canonical.

The projection (15) is Poisson888One of the reasons for this is that the
equation (14) is the moment map for the left diagonal action of $G^{\times n}$
on the cotangent bundle. with the trivial Poisson structure on
${(\mathfrak{g}^{*}/G)}^{\times n}$. Thus the $G^{\times n}$-invariant
functions on ${\mathfrak{g}^{*}}^{\times n}$ give rise to a Poisson
commutative subalgebra in the algebra of functions on $T^{*}(G^{\times
n})/G_{n}$. The restriction of these functions to the symplectic leaf
$\mathcal{S}(\mathcal{O})$ gives the algebra of Poisson commuting functions on
it. This is the subalgebra of Hamiltonians of the periodic spin Calogero-Moser
chain.

Now let us describe the restriction of the Hamiltonians corresponding to
quadratic Casimir functions

$H_{2}^{(k)}(x,g)=\frac{1}{2}\bigl{(}x^{(k)},x^{(k)}\bigr{)}=\frac{1}{2}\bigl{(}x_{0}^{(k)},x_{0}^{(k)}\bigr{)}+\sum_{\alpha>0}x^{(k)}_{\alpha}x^{(k)}_{-\alpha}\qquad(1\leq
k\leq n)$

to the regular part of $\mathcal{S}(\mathcal{O})$, where
$x=(x^{(1)},\ldots,x^{(n)})\in\mathfrak{g}^{*\times n}$ and $g\in G^{\times
n}$. Consider the functions

$D_{k}=H_{2}^{(k)}-H_{2}^{(k-1)}\qquad\quad(1<k\leq n)$

which we call Knizhnik-Zamolodchikov-Bernard (KZB) Hamiltonians999The proper
name would be constant Knizhnik-Zamolodchikov-Bernard Hamiltonians emphasizing
the fact that they are related to finite dimensional simple Lie algebras, not
to the affine Kac-Moody algebras. See for example references [13][8][9][34]..

######  Theorem 1.

The restriction of the KZB Hamiltonians to $\mathcal{S}(\mathcal{O})_{reg}$
can be written as

(16) $D_{k}=(\mu^{(k)}_{0},p)-\sum_{l=1}^{k-1}r_{lk}+\sum_{l=k+1}^{n}r_{kl}$

where $r_{kl}$ for $k\not=l$ is the classical version of the Felder’s
dynamical $r$-matrix [13]:

(17)
$r_{kl}=-\frac{1}{2}(\mu^{(k)}_{0},\mu^{(l)}_{0})+\sum_{\alpha}\frac{\mu^{(k)}_{-\alpha}\mu^{(l)}_{\alpha}}{a_{\alpha}-1}.$

######  Remark 1.

Note that (17) can also be written as

$r_{kl}=-\frac{1}{2}(\mu^{(k)}_{0},\mu^{(l)}_{0})+\sum_{\alpha>0}\frac{\mu^{(k)}_{-\alpha}\mu^{(l)}_{\alpha}}{a_{\alpha}-1}-\sum_{\alpha>0}\frac{a_{\alpha}\mu^{(k)}_{\alpha}\mu^{(l)}_{-\alpha}}{a_{\alpha}-1}.$

###### Proof.

We need to show that formula (16) gives the expression of $D_{k}$ in terms of
the coordinates on $\mathcal{S}(\mathcal{O})_{reg}$ of
$\bigl{(}\nu_{\mathcal{O}}^{-1}(0)/H\times T^{*}A_{reg}\bigr{)}/W$, obtained
from the isomorphism (12). In particular, let
$\bigl{(}H(\mu^{(1)},\ldots,\mu^{(n)}),p,a\bigr{)}\in\nu_{\mathcal{O}}^{-1}(0)/H\times\mathfrak{a}^{*}\times
A_{reg}$ and let $\bigl{(}(x^{(1)},\ldots,x^{(n)}),(1,\ldots,1,a)\bigr{)}$ be
the corresponding point in $\mathfrak{g}^{*\times n}\times G^{\times n}$, with
$x^{(i)}$ given by (13). Taking into account the relation
$x^{(k)}-x^{(k-1)}=\mu^{(k)}$ between the $x^{(i)}$ and the $\mu^{(j)}$ we
have

$\begin{split}D_{k}&=\bigl{(}\mu^{(k)},x^{(k-1)}\bigr{)}+\frac{1}{2}\bigl{(}\mu^{(k)},\mu^{(k)}\bigr{)}\\\
&=\bigl{(}\mu_{0}^{(k)},x_{0}^{(k-1)}+\frac{1}{2}\mu_{0}^{(k)}\bigr{)}+\sum_{\alpha}x_{\alpha}^{(k-1)}\mu_{-\alpha}^{(k)}+\sum_{\alpha>0}\mu_{\alpha}^{(k)}\mu_{-\alpha}^{(k)}.\end{split}$

Substitute here the expression (10) for $x^{(k-1)}_{\alpha}$ in terms of the
$\mu_{\alpha}^{(j)}$:

$D_{k}=\bigl{(}\mu_{0}^{(k)},x_{0}^{(k-1)}+\frac{1}{2}\mu_{0}^{(k)}\bigr{)}+\sum_{l=1}^{k-1}\sum_{\alpha}\frac{a_{\alpha}\mu_{\alpha}^{(l)}\mu^{(k)}_{-\alpha}}{a_{\alpha}-1}+\sum_{l=k+1}^{n}\sum_{\alpha}\frac{\mu_{\alpha}^{(l)}\mu^{(k)}_{-\alpha}}{a_{\alpha}-1}.$

From here using the identities

$\sum_{\alpha}\frac{a_{\alpha}\mu_{\alpha}^{(l)}\mu^{(k)}_{-\alpha}}{a_{\alpha}-1}=-r_{lk}-\frac{1}{2}(\mu_{0}^{(k)},\mu_{0}^{(l)})$

and

$\sum_{\alpha}\frac{\mu_{\alpha}^{(l)}\mu^{(k)}_{-\alpha}}{a_{\alpha}-1}=r_{kl}+\frac{1}{2}(\mu_{0}^{(k)},\mu_{0}^{(l)})$

we conclude

$D_{k}=\Bigl{(}\mu^{(k)}_{0},x_{0}^{(k-1)}+\frac{1}{2}\sum_{l=k}^{n}\mu_{0}^{(l)}-\frac{1}{2}\sum_{l=1}^{k-1}\mu_{0}^{(l)}\Bigr{)}-\sum_{l=1}^{k-1}r_{lk}+\sum_{l=k+1}^{n}r_{kl}.$

Using $x_{0}^{(k-1)}=p-\mu_{0}^{(n)}-\cdots-\mu_{0}^{(k)}$ (see (13)) and the
constraint (11) we obtain (16). ∎

A particularly simple expression has the quadratic Hamiltonian $H_{2}^{(n)}$
on $\mathcal{S}(\mathcal{O})_{reg}$,

$H_{2}^{(n)}=\frac{1}{2}(p,p)+\sum_{\alpha}\frac{\mu_{\alpha}\mu_{-\alpha}}{(1-a_{\alpha})(1-a_{\alpha}^{-1})}.$

Here $\mu_{\alpha}=\mu^{(1)}_{\alpha}+\dots+\mu^{(n)}_{\alpha}$. Setting
$q_{\alpha}=\alpha(\log(a))$ this formula becomes a familiar formula for the
spin Calogero-Moser Hamiltonian,

(18)
$H_{2}^{(n)}=\frac{1}{2}(p,p)-\sum_{\alpha>0}\frac{\mu_{\alpha}\mu_{-\alpha}}{2sh^{2}(q_{\alpha})}.$

Note that the periodic spin CM chain is the classical version of the dynamical
Knizhnik-Zamolodchikov equation from [8][9].

###  1.5. Periodic spin Calogero-Moser chain as a superintegrable system

Now let us establish the superintegrability of the periodic spin CM chain. For
this we should construct an intermediate Poisson manifold and projections as
in [26][30].

Observe that we have natural Poisson projections:

(19) $T^{*}(G^{\times n})/G_{n}\stackrel{{\scriptstyle
p_{1}}}{{\rightarrow}}\mathcal{P}_{n}\stackrel{{\scriptstyle
p_{2}}}{{\rightarrow}}\mathcal{B}_{n}$

Firstly, $\mathcal{P}_{n}=({\mathfrak{g}^{*}}^{\times
n}\times_{{(\mathfrak{g}^{*}/G)}^{\times n}}{\mathfrak{g}^{*}}^{\times
n})/G_{n}$ with

${\mathfrak{g}^{*}}^{\times n}\times_{{(\mathfrak{g}^{*}/G)}^{\times
n}}{\mathfrak{g}^{*}}^{\times n}:=\\{(x,y)\in{\mathfrak{g}^{*}}^{\times
n}\times{\mathfrak{g}^{*}}^{\times n}|Gy_{i}=-Gx_{i-1}\\},$

where $Gz$ is the coadjoint orbit through $z\in\mathfrak{g}^{*}$ and the
indices $i$ are taken modulo $n$, and $G_{n}$ is acting by

(20)
$g(x,y):=(Ad_{g_{1}}^{*}(x_{1}),\ldots,Ad_{g_{n}}^{*}(x_{n}),Ad_{g_{1}}^{*}(y_{1}),\ldots,Ad_{g_{n}}^{*}(x_{n})).$

The map $p_{1}$ is the map induced from the $G_{n}$-equivariant map
$\mu_{L}\times\mu_{R}^{tw}$. Explicitly, the mapping $p_{1}$ acts as

(21) $\begin{split}p_{1}:G_{n}(x,g)&\mapsto
G_{n}(\mu_{L}(x,g),\mu^{tw}_{R}(x,g))\\\
&=G_{n}(x_{1},x_{2},\ldots,x_{n},-Ad_{g_{n}^{-1}}^{*}(x_{n}),-Ad_{g_{1}^{-1}}(x_{1}),\ldots,-Ad_{g_{n-1}^{-1}}(x_{n-1})).\end{split}$

Secondly,

$\mathcal{B}_{n}={(\mathfrak{g}^{*}/G)}^{\times n}$

and the map $p_{2}$ is the projection to the first factor.

Restricting projection $p_{1}$ to the symplectic leaf
$\mathcal{S}(\mathcal{O})$ (see (9)), we obtain the surjective Poisson
projection

$p_{1,\mathcal{O}}:\mathcal{S}(\mathcal{O})\to\mathcal{P}(\mathcal{O})$

where

$\mathcal{P}(\mathcal{O})=\\{(x,y)\in{\mathfrak{g}^{*}}^{\times
n}\times_{(\mathfrak{g}^{*}/G)^{\times n}}{\mathfrak{g}^{*}}^{\times
n}\,\,|\,\,x_{i}+y_{i}\in\mathcal{O}_{i}\\}/G_{n}\subset\mathcal{P}_{n}$

with the $G_{n}$-action described by (20).

Restricting the second projection $p_{2}$ to $\mathcal{P}(\mathcal{O})$ we
have the Poisson projection

$p_{2,\mathcal{O}}:\mathcal{P}(\mathcal{O})\to\mathcal{B}(\mathcal{O})\subset\mathcal{B}_{n},\qquad
G_{n}(x,y)\mapsto(Gx_{1},\dots,Gx_{n})$

where $\mathcal{B}(\mathcal{O})$ is the image of $p_{2,\mathcal{O}}$. It can
be explicitly described as

$\mathcal{B}(\mathcal{O})=\\{(\mathcal{O}^{(1)},\ldots,\mathcal{O}^{(n)})\in(\mathfrak{g}^{*}/G)^{\times
n}\,\,|\,\,\mathcal{O}_{i}\subseteq\mathcal{O}^{(i)}-\mathcal{O}^{(i-1)}\\},$

with the indices $i$ taken modulo $n$.

######  Lemma 1.

The dimension of $\mathcal{B}(\mathcal{O})$ is $nr$ where $r$ is the rank of
the Lie algebra $\mathfrak{g}$.

###### Proof.

Let $\mathfrak{h}^{*}_{\geq 0}$ be the positive Weyl chamber in the dual space
to Cartan subalgebra of $\mathfrak{g}$. For each generic orbit $\mathcal{O}$
there is a unique representative $y\in\mathcal{O}\cup\mathfrak{h}^{*}{>0}$.
Let $x_{1}$ be such representative of $\mathcal{O}^{(1)}$. Let us describe
orbits $\mathcal{O}^{(2)}$ such that $x_{1}+y_{1}\in\mathcal{O}^{(2)}$ for
some $y_{1}\in\mathcal{O}_{1}$. Assume that $\mathcal{O}^{(1)}$ is very large,
i.e. $||x_{1}||>>1$. Because $\mathcal{O}_{1}$ is compact $||y_{1}||<C_{1}$
for some constant $C_{1}$ determined by the orbit $\mathcal{O}_{1}$. Let
$c^{(i)}_{k}$ be the value of $k$-th Casimir function on the orbit
$\mathcal{O}^{(i)}$.

For $k$-th Casimir function $c_{k}^{(2)}$ we have:

$c_{k}^{(2)}=c_{k}(x_{1}+y_{1})=c_{k}^{(1)}+\sum_{i=1}^{r}\frac{\partial
c_{k}(h)}{\partial h_{i}}|_{h=x_{1}}(y_{1})_{i}+O(y^{2})$

Because the matrix $\frac{\partial c_{k}(h)}{\partial h_{i}}$ is nondegerate
for generic $h$, possible values of the Euclidean vector with components
$c_{k}^{(2)}$ span an $r$-dimensional neighborhood of $\\{c_{k}^{(1)}\\}$.

Repeating this argument for each $\mathcal{O}^{(i)}$ we conclude that each of
$\mathcal{O}_{i}$ is non-zero, $\dim(\mathcal{B}(\mathcal{O}))=nr$.

∎

Now let us describe the fiber
$\mathcal{P}(\mathcal{O};\mathcal{O}^{(1)},\dots,\mathcal{O}^{(n)})$ of
$p_{2,\mathcal{O}}$ over
$(\mathcal{O}^{(1)},\dots,\mathcal{O}^{(n)})\in\mathcal{B}(\mathcal{O})$:

$\begin{split}\mathcal{P}(\mathcal{O};\mathcal{O}^{(1)},\dots,\mathcal{O}^{(n)})&=\\{(x,y)\in{\mathfrak{g}^{*}}^{\times
n}\times{\mathfrak{g}^{*}}^{\times n}|x_{i}+y_{i}\in\mathcal{O}_{i},\ \
x_{i}\in\mathcal{O}^{(i)},y_{i}\in-\mathcal{O}^{(i-1)}\\}/G_{n}\\\
&=\prod_{i=1}^{n}\bigl{\\{}(x_{i},y_{i})\in\mathcal{O}^{(i)}\times-\mathcal{O}^{(i-1)}\,\,|\,\,x_{i}+y_{i}\in\mathcal{O}_{i}\bigr{\\}}/G\end{split}$

with the index $i$ taken modulo $n$ and with $G$ acting by the diagonal
coadjoint action on $\mathcal{O}^{(i)}\times-\mathcal{O}^{(i-1)}$. Set

(22)
$\mathcal{M}(\mathcal{O}^{(1)},\mathcal{O}^{(2)},\mathcal{O}^{(3)})=\\{(x,y,z)\in\mathcal{O}^{(1)}\times\mathcal{O}^{(2)}\times\mathcal{O}^{(3)}\,\,|\,\,x+y+z=0\\}/G,$

with $G$ acting by the diagonal coadjoint action. Then

(23)
$\begin{split}\bigl{\\{}(x_{i},y_{i})\in\mathcal{O}^{(i)}\times-\mathcal{O}^{(i-1)}\,\,|\,\,x_{i}+y_{i}\in\mathcal{O}_{i}\bigr{\\}}/G&\overset{\sim}{\longrightarrow}\mathcal{M}(-\mathcal{O}^{(i)},\mathcal{O}^{(i-1)},\mathcal{O}_{i})\\\
G(x_{i},y_{i})\mapsto G(-x_{i},-y_{i},x_{i}+y_{i}),\end{split}$

and hence we conclude that

(24)
$\mathcal{P}(\mathcal{O};\mathcal{O}^{(1)},\dots,\mathcal{O}^{(n)})\simeq\prod_{i=1}^{n}\mathcal{M}(-\mathcal{O}^{(i)},\mathcal{O}^{(i-1)},\mathcal{O}_{i}),$

with the index $i$ taken modulo $n$.

######  Lemma 2.

Let $\mathcal{O}^{(1)},\mathcal{O}^{(2)}$ be generic, sufficiently large
coadjoint orbits and $\mathcal{O}^{(3)}\neq 0$, then

$\dim(\mathcal{M}(\mathcal{O}^{(1)},\mathcal{O}^{(2)},\mathcal{O}^{(3)}))=\dim(\mathcal{O}^{(3)})-2r.$

###### Proof.

Let $x\in\mathcal{O}^{(1)}$ be the unique representative which lies in the
positive Weyl chamber. Assume this orbit is ”big”, i.e. $||x||>>1$. The
condition $x+y+z=0$ for $y\in\mathcal{O}^{(2)}$ and $z\in\mathcal{O}^{(3)}$
for a large orbit $\mathcal{O}^{(2)}$ means that we have $r$ constraints
$c_{k}(-x-y)=c_{k}^{(3)}$ on $y$. For large orbits $\mathcal{O}^{(1)}$ and
$\mathcal{O}^{(2)}$ these constraints are independent. Taking into account
that we are quotiening by $H$ we have
$\dim(\mathcal{M}(\mathcal{O}^{(1)},\mathcal{O}^{(2)},\mathcal{O}^{(3)})=\dim(\mathcal{O}^{(3)}-2r$.
∎

######  Corollary 1.

Thus the dimension of the fiber is
$\dim(\mathcal{P}(\mathcal{O};\mathcal{O}^{(1)},\dots,\mathcal{O}^{(n)}))=\sum_{i=1}^{n}\dim(\mathcal{O}_{i})-2nr$.

Each of the factors in (24) is the Hamiltonian reduction of the product of the
three coadjoint orbits relative to the moment map of the diagonal coadjoint
$G$-action, and therefore carries a natural symplectic structure. Moduli
spaces $\mathcal{M}(\mathcal{O}^{(1)},\mathcal{O}^{(2)},\mathcal{O}^{(3)})$
and therefore fibers of $p_{2,\mathcal{O}}$ are stratified symplectic spaces.

######  Theorem 2.

The Hamiltonian system generated by any Hamiltonian for the periodic spin CM
chain described in section 1.4 is superintegrable with the superintegrable
structure described by the Poisson maps

$\mathcal{S}(\mathcal{O})\stackrel{{\scriptstyle
p_{1,\mathcal{O}}}}{{\longrightarrow}}\mathcal{P}(\mathcal{O})\stackrel{{\scriptstyle
p_{2,\mathcal{O}}}}{{\longrightarrow}}\mathcal{B}(\mathcal{O})$

as introduced earlier in this section.

Here, as everywhere above, we assume that $\mathcal{O}_{i}\neq\\{0\\}$ for
each $i=1,\dots,n$.

###### Proof.

For $G_{n}(x,y)\in\mathcal{P}(\mathcal{O})$ let $\widetilde{g}_{i}\in G$ such
that $y_{i+1}=-Ad_{\widetilde{g}_{i}^{\,-1}}^{*}(x_{i})$. Then

$p_{1,\mathcal{O}}^{-1}\bigl{(}G_{n}(x,y)\bigr{)}=\\{G_{n}(x,g)\in\mathcal{S}(\mathcal{O})\,\,|\,\,g_{i}\in\widetilde{g}_{i}Z_{G}(y_{i+1})\\}$

(index $i$ taken modulo $n$) which, generically, is isotropic and of dimension
$nr=\dim(\mathcal{B}(\mathcal{O}))$.

It remains to check the balance of dimensions. It follows from (12) that

$\dim(\mathcal{S}(\mathcal{O}))=\sum_{i=1}^{n}\dim(\mathcal{O}_{i}).$

By Remark 2 we have, for generic
$(\mathcal{O}^{(1)},\ldots,\mathcal{O}^{(n)})\in(\mathfrak{g}^{*}/G)^{\times
n}$,

$\begin{split}\dim(\mathcal{P}(\mathcal{O}))&=\dim(\mathcal{B}(\mathcal{O}))+\dim(\mathcal{P}(\mathcal{O};\mathcal{O}^{(1)},\ldots,\mathcal{O}^{(n)}))\\\
&=\dim(\mathcal{B}(\mathcal{O}))+\sum_{i=1}^{n}\dim(\mathcal{O}_{i})-2nr.\end{split}$

Then

$\dim(\mathcal{P}(\mathcal{O}))+\dim(\mathcal{B}(\mathcal{O}))=\sum_{i=1}^{n}\dim(\mathcal{O}_{i})=\dim(\mathcal{S}(\mathcal{O})),$

as desired. ∎

######  Remark 2.

In the compact case, the quantum version of functions on
$\mathcal{M}(\mathcal{O}^{(1)},\mathcal{O}^{(2)},\mathcal{O}^{(3)})$ is the
algebra of endomorphisms $End((V_{\lambda_{1}}\otimes V_{\lambda_{2}}\otimes
V_{\lambda_{3}})^{G})$ of the subspace of $G$-invariant vectors in the tensor
product $V_{\lambda_{1}}\otimes V_{\lambda_{2}}\otimes V_{\lambda_{3}}$ with
$V_{\lambda_{i}}$ the representation corresponding to $\mathcal{O}^{(i)}$.

The quantum version of the algebra of functions on the fiber
$\mathcal{P}(\mathcal{O};\mathcal{O}^{(1)},\dots,\mathcal{O}^{(n)})$ is the
algebra of endomorphisms of the vector space

$\textup{Hom}_{G}(V_{\lambda_{1}},V_{\lambda_{n}}\otimes
V_{1})\otimes\textup{Hom}_{G}(V_{\lambda_{2}},V_{\lambda_{1}}\otimes
V_{2})\otimes\cdots\otimes\textup{Hom}_{G}(V_{\lambda_{n}},V_{\lambda_{n-1}}\otimes
V_{n}).$

Here the orbits $\mathcal{O}_{i}$ correspond to $V_{i}$, and
$\textup{Hom}_{G}(V_{\lambda_{i}},V_{\lambda_{i-1}}\otimes V_{i})$ is the
space of $G$-linear intertwiners $V_{\lambda_{i}}\rightarrow
V_{\lambda_{i-1}}\otimes V_{i}$. For details see [RSQCD1].

###  1.6. Constructing solutions by the projection method and angle variables

For $\mathcal{H}$ a $G$-invariant function on $\mathfrak{g}^{*}$, write
$\mathcal{H}^{(i)}$ for the $G_{n}$-invariant function on $T^{*}(G^{\times
n})$ defined by $\mathcal{H}^{(i)}(x,g):=\mathcal{H}(x_{i})$. The Hamiltonian
flow through $(x,g)\in\mathfrak{g}^{*\times n}\times G^{\times n}$ generated
by $\mathcal{H}^{(i)}$ is

(25)
$(x(t_{i}),g(t_{i}))=(x_{1},\dots,x_{n},g_{1},\dots,g_{i-1},e^{\nabla\mathcal{H}(x_{i})t_{i}}g_{i},g_{i+1}\dots,g_{n})$

where $\nabla\mathcal{H}(x)\in\mathfrak{g}$ is the gradient of $\mathcal{H}$
at $x\in\mathfrak{g}^{*}$, i.e.,

$y(\nabla\mathcal{H}(x))=\frac{d}{dt}\mathcal{H}(x+ty)|_{t=0}$

for all $y\in\mathfrak{g}^{*}$. The projection of such flow line to
$T^{*}(G^{\times n})/G^{\times n}$ and further restricted to
$\mathcal{S}(\mathcal{O})\subset T^{*}(G^{\times n})/G_{n}$ is a flow line of
the Hamiltonian vector field generated by the restriction of
$\mathcal{H}^{(i)}$ to $\mathcal{S}(\mathcal{O})$.

Now let us construct angle variables, i.e., functions on $S(\mathcal{O})$
which evolve linearly with respect to the evolution (25) for each
$i=1,\ldots,n$. Write $\mathfrak{a}_{+}^{*}\subset\mathfrak{g}^{*}$ for the
elements $x\in\mathfrak{g}^{*}$ which vanish on root spaces and satisfy
$(x,\alpha)>0$ for $\alpha\in R_{+}$, where $(\cdot,\cdot)$ is the bilinear
form on $\mathfrak{g}^{*}$ induced by the Killing form. Write
$\mathfrak{g}^{\prime*}$ for the elements in $\mathfrak{g}^{*}$ which are
$G$-conjugate to an element in $\mathfrak{a}_{+}^{*}$, relative to the
coadjoint action.

For $(x,g)\in\mathfrak{g}^{\prime*\times n}\times G^{\times n}$ define
$s_{i}\in G$ by the property $Ad^{*}_{s_{i}}(x_{i})\in\mathfrak{a}_{+}^{*}$.
These elements are defined only up to $s_{i}\mapsto a_{i}s_{i}$ where
$a_{i}\in H$. Gauge transformations $h\in G_{n}$ act by
$(s_{1},\ldots,s_{n})\mapsto(s_{1}h_{1}^{-1},\ldots,s_{n}h_{n}^{-1})$.

Let $G_{\mathbb{C}}$ be a complexification of $G$, which we take to be
connected. Let $H_{\mathbb{C}}\subset G_{\mathbb{C}}$ be the Cartan subgroup
$Z_{G_{\mathbb{C}}}(\mathfrak{h})$, where $\mathfrak{h}$ is the Cartan
subalgebra $\mathfrak{a}\oplus i\mathfrak{a}$ of the Lie algebra
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\oplus i\mathfrak{g}$ of
$G_{\mathbb{C}}$. Then $A\subseteq H\subset H_{\mathbb{C}}$. We identify
$\mathfrak{g}^{*}$ with the real subspace of $\mathfrak{g}_{\mathbb{C}}^{*}$
consisting of the complex linear functionals that take real values on
$\mathfrak{g}$. For finite dimensional $G_{\mathbb{C}}$-representations
$V_{1},\ldots,V_{n}$ choose vector $v_{i}\in V_{i}$ of $H_{\mathbb{C}}$-weight
$\lambda_{i+1}$ and linear functionals $u_{i}^{*}\in V_{i}^{*}$ of
$H_{\mathbb{C}}$-weight $-\lambda_{i}$ (indices $i$ taken modulo
$n$).101010Recall that $G$ acts on dual vectors as
$(gu^{*})(v)=u^{*}(g^{-1}v)$. Define

(26)
$f_{u,v}(x,g)=u_{1}^{*}(s_{1}g_{1}s_{2}^{-1}v_{1})u_{2}^{*}(s_{2}g_{2}s_{3}^{-1}v_{2})\dots
u_{n}^{*}(s_{n}g_{n}s_{1}^{-1}v_{n})$

for $(x,g)\in(\mathfrak{g}^{\prime*}\times G)^{\times n}$. This expression is
well defined (i.e., invariant with respect to transformations $s_{i}\to
a_{i}s_{i}$ with $a_{i}\in H$), and invariant with respect to gauge
transformations. Thus, it defines a function on the subset
$(\mathfrak{g}^{\prime*\times n}\times G^{\times n})/G_{n}$ of
$T^{*}(G^{\times n})/G_{n}$.

From the $G$-invariance of $\mathcal{H}$ we have the identity

$u^{*}_{i}(s_{i}e^{t_{i}\nabla\mathcal{H}(x_{i})}g_{i}s_{i+1}^{-1}v_{i})=e^{t_{i}\lambda_{i}(\nabla\mathcal{H}(y_{i}))}u^{*}_{i}(s_{i}g_{i}s_{i+1}^{-1}v_{i})$

where $y_{i}=Ad_{s_{i}}^{*}(x_{i})\in\mathfrak{a}_{+}^{*}$, and consequently

(27)
$f_{u,v}(x(t_{i}),g(t_{i}))=e^{t_{i}\lambda_{i}(\nabla\mathcal{H}(y_{i}))}f_{u,v}(x,g).$

Logarithms of these functions evolve linearly, and hence they produce angle
variables for the Hamiltonians $\mathcal{H}^{(i)}$ on
$S(\mathcal{O})\cap(\mathfrak{g}^{\prime*\times n}\times G^{\times n})/G_{n}$.

##  2\. Open spin Calogero-Moser chains

Recall from the introduction that $G$ is a split real connected Lie group with
finite center which admits a complexification, and $K\subset G$ is the
subgroup of fixed points of a fixed Cartan involution $\Theta$ of $G$. Recall
furthermore the root space decomposition
$\mathfrak{g}=\mathfrak{a}\oplus\bigoplus_{\alpha>0}(\mathbb{R}e_{\alpha}\oplus\mathbb{R}e_{-\alpha})$
with the Cartan subalgebra $\mathfrak{a}\subset\mathfrak{g}$ and the root
vectors $e_{\alpha}\in\mathfrak{g}_{\alpha}$ such that the infinitesimal
Cartan involution $\theta$ acts as

$\theta(h)=-h,\ \ \theta(e_{\alpha})=-e_{-\alpha}$

for $h\in\mathfrak{a}$ and $\alpha\in R$. We will furthermore normalise the
root vectors in such a way that $(e_{\alpha},e_{-\alpha})=1$, with
$(\cdot,\cdot)$ the Killing form of $\mathfrak{g}$.

To avoid cumbersome notations, we will not always indicate in notations that
we are in the open case. This leads to an overlap of some of the notations
with the ones for the periodic case. For instance, the moment maps, Poisson
spaces and Poisson projections will be denoted in the same way.

###  2.1. The phase space as the Hamiltonian reduction

Consider for $n\geq 0$ the manifold $T^{*}(G^{\times n+1})$ with the standard
symplectic structure. We trivialize the cotangent bundle $T^{*}(G^{\times
n+1})$ by right translations:

(28) $T^{*}(G^{\times n+1})\simeq(T^{*}G)^{\times
n+1}\simeq{\mathfrak{g}^{*}}^{\times n+1}\times G^{\times n+1}$

We have a natural action of $K\times G^{\times n}$ on $G^{\times n+1}$ by left
translations:

$(k_{\ell},h_{1},\dots,h_{n})_{L}(g_{0},g_{1},\dots,g_{n})=(k_{\ell}g_{0},h_{1}g_{1},\dots,h_{n}g_{n})$

This action lifts to the following Hamiltonian action on $T^{*}G^{\times
n+1}$,

$\begin{split}(k_{\ell},h_{1},\dots,h_{n})_{L}(x_{0},&\dots,x_{n},g_{0},g_{1},\dots,g_{n})=\\\
&=(Ad^{*}_{k_{\ell}}(x_{0}),Ad^{*}_{h_{1}}(x_{1}),\dots,Ad^{*}_{h_{n}}(x_{n}),k_{\ell}g_{0},h_{1}g_{1},\dots,h_{n}g_{n})\end{split}$

with the moment map

$\mu_{L}(x,g)=(\pi(x_{0}),x_{1},\dots,x_{n})$

where the projection $\pi:\mathfrak{g}^{*}\rightarrow\mathfrak{k}^{*}$ is the
dual map dual to the embedding $\mathfrak{k}\hookrightarrow\mathfrak{g}$.

Similarly, the action of $G^{\times n}\times K$ on $G^{\times n+1}$ by right
translations

$(h_{1},\dots,h_{n},k_{r})_{R}(g_{0},g_{1},\dots,g_{n})=(g_{0}h_{1}^{-1},g_{1}h_{2}^{-1},\dots,g_{n-1}h_{n}^{-1},g_{n}k_{r}^{-1})$

lifts to the following Hamiltonian action on $T^{*}G^{\times n+1}$,

$\begin{split}(h_{1},\dots,h_{n},k_{r})_{R}(x_{0},&\dots,x_{n},g_{0},g_{1},\dots,g_{n})\\\
&=(x_{0},x_{1},\dots,x_{n},g_{0}h_{1}^{-1},g_{1}h_{2}^{-1},\dots,g_{n-1}h_{n}^{-1},g_{n}k_{r}^{-1}),\end{split}$

with the moment map

$\mu_{R}(x,g)=(-Ad^{*}_{g_{0}^{-1}}(x_{0}),-Ad^{*}_{g_{1}^{-1}}(x_{1}),\dots,-Ad^{*}_{g_{n-1}^{-1}}(x_{n-1}),-\pi(Ad^{*}_{g_{n}^{-1}}(x_{n}))).$

As a result, the group $G_{n,K}:=K\times G^{\times n}\times K$ acts on
$T^{*}(G^{\times n+1})$ as

(29)
$\begin{split}(k_{\ell},h_{1},\dots,&h_{n},k_{r})(x_{0},\dots,x_{n},g_{0},\dots,g_{n})=\\\
=&(Ad^{*}_{k_{\ell}}(x_{0}),Ad^{*}_{h_{1}}(x_{1}),\dots,Ad^{*}_{h_{n}}(x_{n}),k_{\ell}g_{0}h_{1}^{-1},h_{1}g_{1}h_{2}^{-1},\dots,h_{n}g_{n}k_{r}^{-1})\end{split}$

with $k_{\ell},k_{r}\in K$ and $h_{i}\in G$. This action is Hamiltonian with
the moment map $\mu:T^{*}(G^{\times
n})\to\mathfrak{k}^{*}\times{\mathfrak{g}^{*}}^{\times
n}\times\mathfrak{k}^{*}$ given by

(30)
$\begin{split}\mu((&x_{0},\dots,x_{n},g_{0},\dots,g_{n})=(\mu_{L}(x,g),0)+(0,\mu_{R}(x,g))=\\\
&=(\pi(x_{0}),x_{1}-Ad^{*}_{g_{0}^{-1}}(x_{0}),x_{2}-Ad^{*}_{g_{1}^{-1}}(x_{1}),\dots,x_{n}-Ad^{*}_{g_{n-1}^{-1}}(x_{n-1}),-\pi(Ad^{*}_{g_{n}^{-1}}(x_{n}))).\end{split}$

For $n=0$ this is the $K\times K$ action
$(k_{\ell},k_{r})(x,g)=(Ad^{*}_{k_{\ell}}(x),k_{\ell}gk_{r}^{-1})$ on
$T^{*}G$, with the moment map $(x,g)\mapsto(\pi(x),-\pi(Ad_{g^{-1}}(x)))$. It
is easy to check explicitly that this moment map intertwines the action of
$G_{n,K}$ on $T^{*}(G^{\times n+1})$ given by (29) with its diagonal coadjoint
action on
$\mathfrak{k}^{*}\times{\mathfrak{g}^{*}}^{\times(n-1)}\times\mathfrak{k}^{*}$.

Because the action of $G_{n,K}$ on $T^{*}(G^{\times n+1})$ is Hamiltonian, the
space $T^{*}(G^{\times n+1})/G_{n,K}$111111Recall that here and in everywhere
else in this paper $X/H$ means the GIT quotient for a Lie group $H$ action on
a manifold $X$. is Poisson with symplectic leaves being given by the
Hamiltonian reduction with respect to the moment map (30). Let
$\mathcal{O}=(\mathcal{O}_{\ell}^{K}\times\mathcal{O}_{1}\times\dots,\mathcal{O}_{n}\times\mathcal{O}^{K}_{r})$
with $\mathcal{O}_{i}\subset\mathfrak{g}^{*}$ coadjoint $G$-orbits and
$\mathcal{O}_{\ell}^{K},\mathcal{O}_{r}^{K}\subset\mathfrak{k}^{*}$ coadjoint
$K$-orbits, then the corresponding symplectic leaf in $T^{*}(G^{\times
n+1})/G_{n,K}$ is

(31) $\begin{split}\mathcal{S}&(\mathcal{O})=\mu^{-1}(\mathcal{O})/G_{n,K}\\\
&=\bigl{\\{}(x_{0},\dots,x_{n},g_{0},\dots,g_{n})\in{\mathfrak{g}^{*}}^{\times
n+1}\times G^{\times
n+1}\,\,|\,\,\pi(x_{0})\in\mathcal{O}^{K}_{\ell},\,-\pi(Ad_{g_{n}^{-1}}^{*}(x_{n}))\in\mathcal{O}^{K}_{r},\\\
&\qquad\qquad\qquad\qquad\qquad\quad
x_{1}-Ad^{*}_{g_{0}^{-1}}(x_{0})\in\mathcal{O}_{1},\dots,x_{n}-Ad^{*}_{g_{n-1}^{-1}}(x_{n-1})\in\mathcal{O}_{n}\bigr{\\}}/G_{n,K}.\end{split}$

Each symplectic leaf $\S(\mathcal{O})$ is a stratified symplectic space and is
the phase space for the corresponding open spin Calogero-Moser chain. We will
describe the largest stratum of $\mathcal{S}(\mathcal{O})$ later.

###  2.2. The Hamiltonians of the open spin Calogero-Moser chain

After the trivialization (28) of $T^{*}(G^{\times n+1})$ by right translations
we have a natural Poisson projection $T^{*}(G^{\times
n+1})\to{\mathfrak{g}^{*}}^{\times n+1}$ to the first factor. It is
$G_{n,K}$-invariant with the following action of $G_{n,K}$ on
${\mathfrak{g}^{*}}^{\times n+1}$

$(k_{\ell},h_{1},\dots,h_{n},k_{r}):(x_{0},x_{1},\dots,x_{n})\mapsto(Ad^{*}_{k_{\ell}}(x_{0}),Ad^{*}_{h_{1}}(x_{1}),\dots,Ad^{*}_{h_{n}}(x_{n})).$

This gives rise to the projection

$p:T^{*}(G^{\times n+1})/G_{n,K}\to(\mathfrak{g}^{*}/G)^{\times n+1}$

which is Poisson because it is the composition of natural Poisson projections

(32) $T^{*}(G^{\times n+1})/G_{n,K}\to({\mathfrak{g}^{*}}^{\times
n+1})/G_{n,K}=\mathfrak{g}^{*}/K\times(\mathfrak{g}^{*}/G)^{\times
n}\to(\mathfrak{g}^{*}/G)^{\times n+1}.$

Here the Poisson structure on the right is trivial (the Poisson tensor is
zero). The last projection is a consequence of the embedding $K\hookrightarrow
G$.

Restricting this projection to the symplectic leaf $\mathcal{S}(\mathcal{O})$
we have the Poisson projection

(33)
$p_{\mathcal{O}}:\mathcal{S}(\mathcal{O})\to\mathcal{B}(\mathcal{O}),\quad
G_{n,K}(x_{0},\dots,x_{n},g_{0},\dots,g_{n})\mapsto(Gx_{0},\dots,Gx_{n})$

where $\mathcal{B}(\mathcal{O})\subset(\mathfrak{g}^{*}/G)^{\times n+1}$ is,
by definition, the image of $\mathcal{S}(\mathcal{O})$. It can be described
explicitly from the description of $\mathcal{S}(\mathcal{O})$ as

(34)
$\begin{split}\mathcal{B}(\mathcal{O})=\bigl{\\{}(\mathcal{O}^{(0)},\dots,\mathcal{O}^{(n)})\in(\mathfrak{g}^{*}/G)^{\times
n+1}&\,\,|\,\,\mathcal{O}_{\ell}^{K}\subseteq\pi(\mathcal{O}^{(0)}),\mathcal{O}_{r}^{K}\subseteq-\pi(\mathcal{O}^{(n)}),\\\
&\mathcal{O}_{1}\subseteq\mathcal{O}^{(1)}-\mathcal{O}^{(0)},\ldots,\mathcal{O}_{n}\subseteq\mathcal{O}^{(n)}-\mathcal{O}^{(n-1)}\bigr{\\}}.\end{split}$

Hamiltonians of the open spin Calogero-Moser system are pull back $p^{*}$ of
functions on ${(\mathfrak{g}^{*}/G)}^{\times n+1}$ restricted to
$\mathcal{S}(\mathcal{O})$.

The subalgebra of Hamiltonians is a Poisson commuting subalgebra. Quadratic
Hamiltonians are given by Casimir functions. We will compute the radial
components of the quadratic Casimirs explicitly in section 2.5. Hamiltonians
are constant on fibers of the projection $p_{\mathcal{O}}$.

###  2.3. The gauge fixing

Fix $i\in\\{0,\ldots,n\\}$. For $g=(g_{0},\dots,g_{n})\in G^{\times n+1}$ and
$k_{\ell},k_{r}\in K$ define $h\in G^{\times n}$ (depending on
$i,g,k_{\ell},k_{r}$) by

$h_{j}=\begin{cases}k_{\ell}g_{0}g_{1}\cdots g_{j-1}\qquad&\hbox{ if
}\,\,j\leq i,\\\ k_{r}g_{n}^{-1}g_{n-1}^{-1}\cdots g_{j}^{-1}\qquad&\hbox{ if
}\,\,j>i.\end{cases}$

Then $(k_{\ell},h_{1},\dots,h_{n},k_{r})$ acts on $g\in G^{\times n+1}$ as

$g\mapsto(1,\dots,1,k_{\ell}g_{0}g_{1}\dots g_{n}k_{r}^{-1},1,\dots,1),$

Here the nontrivial entry is at the position $i$. This gives an
$i$-independent isomorphism

(35) $G^{\times n+1}/G_{n,K}\to K\backslash G/K,\qquad
G_{n,K}(g_{0},\dots,g_{n})\mapsto Kg_{0}g_{1}\dots g_{n}K.$

For the cotangent bundles the gauge fixing gives an isomorphism

(36) $\phi_{i}:\bigl{(}\mathfrak{g}^{*\times n+1}\times G^{\times
n+1}\bigr{)}/G_{n,K}\overset{\sim}{\rightarrow}K\backslash({\mathfrak{g}^{*}}^{\times{n+1}}\times
G)/K$

mapping the $G_{n,K}$-orbit through $(x,g)\in\mathfrak{g}^{*\times n+1}\times
G^{\times n+1}$ to the double $K$-coset through

$\bigl{(}x_{0},Ad_{g_{0}}^{*}(x_{1}),\ldots,Ad_{g_{0}\cdots
g_{i-1}}^{*}(x_{i}),Ad_{g_{n}^{-1}\cdots
g_{i+1}^{-1}}^{*}(x_{i+1}),\ldots,Ad_{g_{n}^{-1}}^{*}(x_{n}),g_{0}g_{1}\cdots
g_{n}\bigr{)}$

For example for $i=n$ this expression is
$\bigl{(}x_{0},Ad_{g_{0}}^{*}(x_{1}),\ldots,Ad_{g_{0}\cdots
g_{n-1}}^{*}(x_{n}),g_{0}g_{1}\cdots g_{n}\bigr{)}$. In (36) the double
$K$-cosets in the codomain of $\phi_{i}$ are taken relative to the
$i$-dependent $K\times K$-action

$(k_{\ell},k_{r})(x_{0},\ldots,x_{n},g)=(Ad_{k_{\ell}}^{*}(x_{0}),\cdots,Ad_{k_{\ell}}^{*}(x_{i}),Ad_{k_{r}}^{*}(x_{i+1}),\ldots,Ad_{k_{r}}^{*}(x_{n}),k_{\ell}gk_{r}^{-1})$

on ${\mathfrak{g}^{*}}^{\times n+1}\times G$.

Now we can describe the symplectic leaf $\mathcal{S}(\mathcal{O})$ as a
subvariety in $K\backslash({\mathfrak{g}^{*}}^{\times{n+1}}\times G)/K$ though
the isomorphism $\varphi_{n}$ as

(37)
$\begin{split}\mathcal{S}(\mathcal{O})=K\backslash\bigl{\\{}(y_{0},y_{1},\ldots,y_{n},g)\in\mathfrak{g}^{*\times
n+1}\times
G&\,\,|\,\,\pi(y_{0})\in\mathcal{O}_{\ell}^{K},-\pi\bigl{(}Ad_{g^{-1}}^{*}(y_{n}))\in\mathcal{O}_{r}^{K},\\\
&\quad\,\,\,y_{1}-y_{0}\in\mathcal{O}_{1},\ldots,y_{n}-y_{n-1}\in\mathcal{O}_{n}\bigr{\\}}/K.\end{split}$

Note that, as in the periodic case, $\mathcal{S}(\mathcal{O})$ is a symplectic
stratified space. From now on we will focus mostly on the largest stratum
$\mathcal{S}(\mathcal{O})_{reg}$.

###  2.4. The regular part of the symplectic leaf $\mathcal{S}(\mathcal{O})$

We use the gauge fixing isomorphism $\varphi_{n}$ in the remainder of the
text. We now use $K\backslash G/K\simeq A/W$ with $W=N_{K}(A)/M$ the Weyl
group of $G$ (see subsection §3 of the introduction) to describe the regular
part of the symplectic leaf $\mathcal{S}(\mathcal{O})$ in radial coordinates.

Define the regular part of the phase space $\mathcal{S}(\mathcal{O})$ (see
(37)) as

$\mathcal{S}(\mathcal{O})_{reg}=\mathcal{S}(\mathcal{O})\cap
K\backslash(\mathfrak{g}^{*\times n+1}\times G_{reg})/K.$

The regular part
$\mathcal{S}(\mathcal{O})_{reg}\subset\mathcal{S}(\mathcal{O})$ is its largest
stratum of the stratified symplectic space $\mathcal{S}(\mathcal{O})$.

We can then choose a representative of
$K(y_{0},\ldots,y_{n},g)K\in\mathcal{S}(\mathcal{O})_{reg}$ with the
$G$-component in $A_{reg}$ by writing $g=k_{\ell}ak_{r}^{-1}$ and
$y_{i}=Ad_{k_{\ell}}^{*}(x^{(i)})$ with $k_{\ell},k_{r}\in K$ and $a\in
A_{reg}$. It follows that

$\begin{split}\mathcal{S}(\mathcal{O})_{reg}\simeq\bigl{\\{}(x^{(0)},\dots,x^{(n)},a)\in\mathfrak{g}^{*\times
n+1}\times&A_{reg}\,\,|\,\,\pi(x^{(0)})\in\mathcal{O}^{K}_{\ell},-\pi(Ad_{a^{-1}}^{*}(x^{(n)}))\in\mathcal{O}^{K}_{r},\\\
&\,\,x^{(1)}-x^{(0)}\in\mathcal{O}_{1},\ldots,x^{(n)}-x^{(n-1)}\in\mathcal{O}_{n}\bigr{\\}}/N_{K}(A).\end{split}$

Here $N_{K}(A)$ acts diagonally on $\mathfrak{g}^{*\times n+1}\times A_{reg}$
via the coadjoint action on $\mathfrak{g}^{*}$ and the conjugation action on
$A_{reg}$.121212We use here the fact that
$k_{\ell}ak_{r}^{-1}=k_{\ell}^{\prime}a^{\prime}k_{r}^{\prime-1}$ for
$k_{\ell},k_{\ell}^{\prime},k_{r},k_{r}^{\prime}\in K$ and $a,a^{\prime}\in
A_{reg}$ is implying that
$k_{\ell}^{-1}k_{\ell}^{\prime}=k_{r}^{-1}k_{r}^{\prime}\in N_{K}(A)$, cf.,
e.g., [20, §VII.3]. This essentially follows from the global Cartan
decomposition of $G$. We can now also divide out the action of $M=Z_{K}(A)$,
to obtain the isomorphism

$\begin{split}\mathcal{S}(\mathcal{O})_{reg}\simeq\bigl{\\{}(M(x^{(0)},\dots,x^{(n)}),a)\in\mathfrak{g}^{*\times
n+1}/M\times&A_{reg}\,\,|\,\,\pi(x^{(0)})\in\mathcal{O}^{K}_{\ell},-\pi(Ad_{a^{-1}}^{*}(x^{(n)}))\in\mathcal{O}^{K}_{r},\\\
&\,\,x^{(1)}-x^{(0)}\in\mathcal{O}_{1},\ldots,x^{(n)}-x^{(n-1)}\in\mathcal{O}_{n}\bigr{\\}}/W,\end{split}$

where $M$ acts by the diagonal coadjoint action on $\mathfrak{g}^{*\times
n+1}$ and $W$ acts diagonally on the space $\mathfrak{g}^{*\times n+1}/M\times
A_{reg}$ in the natural way.

Recall that we identified $\mathfrak{g}\simeq\mathfrak{g}^{*}$ and
$\mathfrak{a}\simeq\mathfrak{a}^{*}$ via the Killing form, so that
$x\in\mathfrak{g}^{*}$ corresponds to
$x_{0}+\sum_{\alpha}x_{\alpha}e_{\alpha}$ with $x_{0}$ the element in
$\mathfrak{a}$ corresponding to $x|_{\mathfrak{a}}\in\mathfrak{a}^{*}$ and
$x_{\alpha}=x(e_{-\alpha})$. Denote by $x_{0}^{(k)},x_{\alpha}^{(k)}$ the
components of vectors $x\in\mathfrak{g}^{*\times n+1}$ from the $k$-th factor
of ${\mathfrak{g}^{*}}^{n+1}$, and $\mu_{0}^{(k)},\mu_{\alpha}^{(k)}$ the
components of $\mu\in\mathcal{O}_{k}$. For $y\in\mathfrak{k}^{*}$ we write
$y_{[\alpha]}=y(e_{-\alpha}-e_{\alpha})$, so that
$y_{[\alpha]}=-y_{[-\alpha]}$.

Consider

$\begin{split}T(\mathcal{O})_{reg}=\bigl{\\{}(x^{(0)},\dots,x^{(n)},a)\in\mathfrak{g}^{*\times
n+1}\times
A_{reg}\,\,|\,\,&\pi(x^{(0)})\in\mathcal{O}^{K}_{\ell},-\pi(Ad_{a^{-1}}^{*}(x^{(n)}))\in\mathcal{O}^{K}_{r},\\\
&x^{(1)}-x^{(0)}\in\mathcal{O}_{1},\ldots,x^{(n)}-x^{(n-1)}\in\mathcal{O}_{n}\bigr{\\}}.\end{split}$

Clearly $S(\mathcal{O})_{reg}=T(\mathcal{O})_{reg}/N_{K}(A)$.

For $(x^{(0)},\ldots,x^{(n)},a)\in T(\mathcal{O})_{reg}$ write
$\mu^{\prime}=\pi(x^{(0)})\in\mathcal{O}_{\ell}^{K}$,
$\mu^{\prime\prime}=-\pi(Ad_{a^{-1}}^{*}(x^{(n)}))\in\mathcal{O}_{r}^{K}$ and
$\mu^{(i)}=x^{(i)}-x^{(i-1)}\in\mathcal{O}_{i}$ for $i=1,\ldots,n$. The Cartan
components of $x^{(k)}$ and their root coordinates then satisfy

(38)
$\begin{split}x_{\alpha}^{(0)}-x_{-\alpha}^{(0)}&=\mu_{[\alpha]}^{\prime},\qquad\quad
a_{\alpha}x_{-\alpha}^{(n)}-a_{\alpha}^{-1}x_{\alpha}^{(n)}=\mu_{[\alpha]}^{\prime\prime},\\\
x^{(i)}_{\alpha}-x^{(i-1)}_{\alpha}&=\mu_{\alpha}^{(i)},\qquad\quad
x_{0}^{(i)}-x_{0}^{(i-1)}=\mu_{0}^{(i)}\end{split}$

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

It is easy to solve the equations for Cartan parts $x_{0}^{(i)}$ ($0<i<n$) in
terms of Catran components of $x^{(0)},x^{(n)}$ and $\mu^{(j)}$,

(39)
$x^{(i)}_{0}=\mu^{(i)}_{0}+\dots+\mu^{(1)}_{0}+x^{(0)}_{0}=x^{(n)}_{0}-\mu^{(n)}_{0}-\dots-\mu^{(i+1)}_{0}$

######  Proposition 1.

The following identities hold for $\alpha\in R$ and $k=0,1,\ldots,n$:

(40)
$x^{(k)}_{\alpha}=K_{\alpha}+\sum_{l=1}^{k}\frac{a_{\alpha}\mu^{(l)}_{\alpha}-a_{\alpha}\mu^{(l)}_{-\alpha}}{a_{\alpha}-a_{\alpha}^{-1}}+\sum_{l=k+1}^{n}\frac{a_{\alpha}^{-1}\mu^{(l)}_{\alpha}-a_{\alpha}\mu^{(l)}_{-\alpha}}{a_{\alpha}-a_{\alpha}^{-1}}$

where

(41)
$K_{\alpha}=\frac{a_{\alpha}\mu_{[\alpha]}^{\prime}+\mu^{\prime\prime}_{[\alpha]}}{a_{\alpha}-a_{\alpha}^{-1}}.$

###### Proof.

Denote

(42) $\mu=\mu^{(1)}+\dots+\mu^{(n)}.$

Note that $x^{(n)}-x^{(0)}=\mu$.

Fix $\beta\in R_{+}$. For $x^{(1)}_{\pm\beta}$ and $x^{(n)}_{\pm\beta}$ the
formula $x^{(n)}-x^{(0)}=\mu$ implies

$x^{(n)}_{\beta}-x^{(0)}_{\beta}=\mu_{\beta},\ \
x^{(n)}_{-\beta}-x^{(0)}_{-\beta}=\mu_{-\beta}.$

Combined with the first line of (38) we end up with four linear equations in
$x^{(0)}_{\beta},x^{(0)}_{-\beta},x^{(n)}_{\beta},x^{(n)}_{-\beta}$ which, by
the assumption that $a$ is regular, are uniquely solved by

(43)
$\begin{split}x^{(0)}_{\alpha}&=\frac{a_{\alpha}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+(a_{\alpha}^{-1}\mu_{\alpha}-a_{\alpha}\mu_{-\alpha})}{a_{\alpha}-a_{\alpha}^{-1}},\\\
x^{(n)}_{\alpha}&=\frac{a_{\alpha}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+(a_{\alpha}\mu_{\alpha}-a_{\alpha}\mu_{-\alpha})}{a_{\alpha}-a_{\alpha}^{-1}}\\\
\end{split}$

for $\alpha=\beta,-\beta$ (here we used that $a_{-\beta}=a_{\beta}^{-1}$,
$\mu_{[-\beta]}^{\prime}=-\mu_{[\beta]}^{\prime}$ and
$\mu^{\prime\prime}_{[-\beta]}=-\mu^{\prime\prime}_{[\beta]}$).

By the second line of (38) we then obtain

$x^{(k)}_{\alpha}=\frac{a_{\alpha}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+(a_{\alpha}^{-1}\mu_{\alpha}-a_{\alpha}\mu_{-\alpha})+(a_{\alpha}-a_{\alpha}^{-1})(\mu_{\alpha}^{(1)}+\cdots+\mu_{\alpha}^{(k)})}{a_{\alpha}-a_{\alpha}^{-1}}$

for $k=0,1,\ldots,n$. Substituting (42) it is now easy to see that this is
exactly what we wanted to prove. ∎

The proposition and (39) give an isomorphism

(44)
$S(\mathcal{O})_{reg}\simeq\bigl{(}(\mathcal{O}_{\ell}^{K}\times\mathcal{O}_{1}\times\dots\times\mathcal{O}_{n}\times\mathcal{O}_{r}^{K})/M\times
T^{*}A_{reg}\bigr{)}/W,$

mapping $N_{K}(A)(x^{(0)},\ldots,x^{(n)},a)$ to the $W$-orbit of
$(M(\mu^{\prime},\mu^{(1)},\ldots,\mu^{(n)},\mu^{\prime\prime}),x_{0}^{(n)},a)$,
which preserves the natural symplectic structures. Here the finite discrete
group $M=Z_{K}(A)\subset K$ acts diagonally via the coadjoint action, and
$W=N_{K}(A)/M$ acts diagonally. The quantum version of this isomorphism is
described in [35].

###  2.5. Quadratic Hamiltonians of open spin Calogero-Moser chain on the
regular part of the phase space

In this section we compute the restriction of the Hamiltonian corresponding to
the quadratic Casimir function on $\mathfrak{g}^{*}$,

$H_{2}^{(k)}(x,g)=\frac{1}{2}(x^{(k)},x^{(k)})=\frac{1}{2}(x^{(k)}_{0},x^{(k)}_{0})+\sum_{\alpha>0}x^{(k)}_{\alpha}x^{(k)}_{-\alpha}$

to the regular part of $\mathcal{S}(\mathcal{O})$ (see (31)) for
$k=0,\ldots,n$, where $(x,g)\in\mathfrak{g}^{*\times n+1}\times G^{\times
n+1}$. Here $(\cdot,\cdot)$ is the Killing form and
$x_{\alpha}^{(i)},x^{(i)}_{0}$ are the components of $x^{(i)}$ which were
computed in the previous section on the regular part of the phase space.

We first consider the differences, which we will call the boundary Knizhnik-
Zamolodchikov-Bernard (bKZB) Hamiltonians,

$D_{k}=H_{2}^{(k)}-H_{2}^{(k-1)}\qquad\quad(1\leq k\leq n).$

######  Theorem 3.

For the bKZB Hamiltonians we have the following formula:

(45)
$D_{k}=(\mu^{(k)}_{0},x^{(n)}_{0})-\sum_{l=1}^{k-1}(r_{lk}+r_{lk}^{\theta_{l}})+(\sum_{\alpha}K_{\alpha}\mu^{(k)}_{-\alpha}-\kappa_{k})+\sum_{l=k+1}^{n}(r_{kl}-r_{kl}^{\theta_{k}}).$

Here $r_{kl}$ for $k\not=l$ is Felder’s rescaled dynamical $r$-matrix

(46)
$r_{kl}=-\frac{1}{2}(\mu^{(k)}_{0},\mu^{(l)}_{0})+\sum_{\alpha}\frac{\mu^{(k)}_{-\alpha}\mu^{(l)}_{\alpha}}{a_{\alpha}^{2}-1},$

$\theta_{k}$ is the transpose of the Chevalley involution $\theta$ acting on
$\mu^{(k)}$,

$\kappa_{k}=\frac{1}{2}(\mu^{(k)}_{0},\mu^{(k)}_{0})+\sum_{\alpha}\frac{({\mu^{(k)}_{\alpha}})^{2}}{1-a_{\alpha}^{2}}$

is the core quadratic classical dynamical $k$-matrix and $K_{\alpha}$ is given
by (41).

###### Proof.

The first step of the proof is the same as in the proof of Theorem 1,
resulting in the expression

(47)
$D_{k}=(\mu_{0}^{(k)},x_{0}^{(k-1)}+\frac{1}{2}\mu_{0}^{(k)})+\sum_{\alpha}x_{\alpha}^{(k-1)}\mu_{-\alpha}^{(k)}+\sum_{\alpha>0}\mu_{\alpha}^{(k)}\mu_{-\alpha}^{(k)}.$

Now let us the formula (40) for $x^{(k-1)}_{\alpha}$,

(48)
$\begin{split}\sum_{\alpha}x^{(k-1)}_{\alpha}\mu^{(k)}_{-\alpha}&=\sum_{l=1}^{k-1}\sum_{\alpha}\frac{a_{\alpha}\mu^{(l)}_{\alpha}\mu_{-\alpha}^{(k)}-a_{\alpha}\mu^{(l)}_{-\alpha}\mu_{-\alpha}^{(k)}}{a_{\alpha}-a_{\alpha}^{-1}}\\\
&+\sum_{\alpha}K_{\alpha}\mu^{(k)}_{-\alpha}+\sum_{\alpha}\frac{a_{\alpha}^{-1}\mu^{(k)}_{\alpha}\mu_{-\alpha}^{(k)}-a_{\alpha}\mu^{(k)}_{-\alpha}\mu_{-\alpha}^{(k)}}{a_{\alpha}-a_{\alpha}^{-1}}\\\
&+\sum_{l=k+1}^{n}\sum_{\alpha}\frac{a_{\alpha}^{-1}\mu_{-\alpha}^{(k)}\mu^{(l)}_{\alpha}-a_{\alpha}\mu_{-\alpha}^{(k)}\mu^{(l)}_{-\alpha}}{a_{\alpha}-a_{\alpha}^{-1}}.\end{split}$

We express the different terms in the right hand side of (48) in terms of the
dynamical $r$-matrix and $k$-matrix.

Note first that

$r_{kl}^{\theta_{k}}=\frac{1}{2}(\mu_{0}^{(k)},\mu_{0}^{(l)})-\sum_{\alpha}\frac{\mu_{\alpha}^{(k)}\mu_{\alpha}^{(l)}}{a_{\alpha}^{2}-1}=r_{lk}^{\theta_{l}}.$

Then the terms in the right hand side of (48) with $l$ strictly smaller than
$k$ can be rewritten as

$\sum_{\alpha}\frac{a_{\alpha}\mu^{(l)}_{\alpha}\mu_{-\alpha}^{(k)}-a_{\alpha}\mu^{(l)}_{-\alpha}\mu_{-\alpha}^{(k)}}{a_{\alpha}-a_{\alpha}^{-1}}=-(r_{lk}+r_{lk}^{\theta_{l}})$

while the terms in the right hand side of (48) with $l$ strictly larger than
$k$ reduce to

$\sum_{\alpha}\frac{a_{\alpha}^{-1}\mu_{-\alpha}^{(k)}\mu^{(l)}_{\alpha}-a_{\alpha}\mu_{-\alpha}^{(k)}\mu^{(l)}_{-\alpha}}{a_{\alpha}-a_{\alpha}^{-1}}=(\mu_{0}^{(k)},\mu_{0}^{(l)})+(r_{kl}-r_{kl}^{\theta_{k}}).$

Finally, for the middle term in (48) a direct computation shows that

$\sum_{\alpha}\frac{a_{\alpha}^{-1}\mu^{(k)}_{\alpha}\mu_{-\alpha}^{(k)}-a_{\alpha}\mu^{(k)}_{-\alpha}\mu_{-\alpha}^{(k)}}{a_{\alpha}-a_{\alpha}^{-1}}=\frac{1}{2}(\mu_{0}^{(k)},\mu_{0}^{(k)})-\sum_{\alpha>0}\mu_{\alpha}^{(k)}\mu_{-\alpha}^{(k)}-\kappa_{k}.$

Substitute these formulas in (48), then the resulting formula (47) for $D_{k}$
becomes

$\begin{split}D_{k}&=(\mu_{0}^{(k)},x_{0}^{(k-1)}+\mu_{0}^{(k)}+\mu_{0}^{(k+1)}+\cdots+\mu_{0}^{(n)})\\\
&-\sum_{l=1}^{k-1}(r_{lk}+r_{lk}^{\theta_{l}})+(\sum_{\alpha}K_{\alpha}\mu^{(k)}_{-\alpha}-\kappa_{k})+\sum_{l=k+1}^{n}(r_{kl}-r_{kl}^{\theta_{k}}).\end{split}$

By (39) this reduces to the formula (45). ∎

The quantum versions of the boundary KZB Hamiltonians in the present context
were obtained in [35, §6]. It was extended to the case of non-split real
semisimple Lie groups $G$ in [33].

For the Hamiltonian $H^{(n)}_{2}$ we obtain by (43) the expression

$H^{(n)}_{2}=\frac{1}{2}(p,p)+\sum_{\alpha>0}\frac{(a_{\alpha}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+a_{\alpha}(\mu_{\alpha}-\mu_{-\alpha}))(a_{\alpha}^{-1}\mu_{[\alpha]}^{\prime}+\mu_{[\alpha]}^{\prime\prime}+a_{\alpha}^{-1}(\mu_{\alpha}-\mu_{-\alpha}))}{(a_{\alpha}-a_{-\alpha})^{2}}$

on $\mathcal{S}(\mathcal{O})_{reg}$, where $\mu=\mu^{(1)}+\cdots+\mu^{(n)}$.
Here we use notation $p=x^{(n)}_{0}$ for the cotangent vectors to $A_{reg}$ in
formula (44). Note that the potential term only depends on the restrictions
$\pi(\mu^{(i)})$ of $\mu^{(i)}\in\mathfrak{g}^{*}$ to $\mathfrak{k}$, since
$\mu_{\alpha}-\mu_{-\alpha}=-(\pi(\mu))_{[\alpha]}$. The radial component of
the quantum quadratic Hamiltonian in the current open context was obtained in
[35, §6].

###  2.6. The superintegrability of the open spin CM chain

In this section we will prove that Poisson commutative subalgebra of
Hamiltonians constructed in section 2.2 defines a superintegrable system. Fix
$\mathcal{O}=(\mathcal{O}_{\ell}^{K},\mathcal{O}_{1},\ldots,\mathcal{O}_{n},\mathcal{O}_{r}^{K})\in(\mathfrak{k}^{*}/K\times(\mathfrak{g}^{*}/G)^{\times
n}\times\mathfrak{k}^{*}/K$. We willl construct Poisson projections

(49) $S(\mathcal{O})\stackrel{{\scriptstyle
p_{1,\mathcal{O}}}}{{\longrightarrow}}\mathcal{P}(\mathcal{O})\stackrel{{\scriptstyle
p_{2,\mathcal{O}}}}{{\longrightarrow}}\mathcal{B}(\mathcal{O})$

such that $p_{\mathcal{O}}=p_{2,\mathcal{O}}\circ p_{1,\mathcal{O}}$ (see
(33)), satisfying the desired properties.

Let $(\mathfrak{k}^{*}\times\mathfrak{g}^{*\times
n})\times_{(\mathfrak{g}^{*}/G)^{\times n+1}}(\mathfrak{g}^{*\times
n}\times\mathfrak{k}^{*})$ be the subset of
$(\mathfrak{k}^{*}\times\mathfrak{g}^{*\times n})\times(\mathfrak{g}^{*\times
n}\times\mathfrak{k}^{*})$ consisting of elements
$(z_{\ell},x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z_{r})$ satisfying

$z_{\ell}\in-\pi(Gy_{1}),\quad x_{i}\in-Gy_{i+1}\,\,(1\leq i<n),\quad
z_{r}\in-\pi(Gx_{n}).$

The gauge group $G_{n,K}$ acts on
$(\mathfrak{k}^{*}\times\mathfrak{g}^{*\times
n})\times_{(\mathfrak{g}^{*}/G)^{\times n+1}}(\mathfrak{g}^{*\times
n}\times\mathfrak{k}^{*})$ by

(50)
$\begin{split}(k_{\ell},h_{1},\dots,h_{n},k_{r})&(z_{\ell},x_{1},\dots,x_{n},y_{1},\dots,y_{n},z_{r})=\\\
&=(Ad^{*}_{k_{\ell}}z_{\ell},Ad^{*}_{h_{1}}x_{1},\dots,Ad^{*}_{h_{n}}x_{n},Ad^{*}_{h_{1}}y_{1},\dots,Ad^{*}_{h_{n}}y_{n},Ad^{*}_{k_{r}}z_{r}).\end{split}$

Consider the resulting Poisson space

$\mathcal{P}=\bigl{(}(\mathfrak{k}^{*}\times\mathfrak{g}^{*\times
n})\times_{(\mathfrak{g}^{*}/G)^{\times n+1}}(\mathfrak{g}^{*\times
n}\times\mathfrak{k}^{*})\bigr{)}/G_{n,K}$

and define the Poisson map

$p_{1}:T^{*}(G^{\times n+1})/G_{n,K}\rightarrow\mathcal{P}$

by

$\begin{split}p_{1}\bigl{(}G_{n,K}(x,g)\bigr{)}&=G_{n,K}\bigl{(}\mu_{L}(x,g),\mu_{R}(x,g)\bigr{)}\\\
&=G_{n,K}\bigl{(}\pi(x_{0}),x_{1},\ldots,x_{n},-Ad_{g_{0}^{-1}}^{*}(x_{0}),\ldots,-Ad_{g_{n-1}^{-1}}^{*}(x_{n-1}),-\pi(Ad_{g_{n}^{-1}}^{*}(x_{n}))\bigr{)}.\end{split}$

Here $(x,g)=(x_{0},\ldots,x_{n},g_{0},\ldots,g_{n})\in\mathfrak{g}^{*\times
n+1}\times G^{\times n+1}\simeq T^{*}(G^{\times n+1})$. Define the Poisson
projection

$p_{2}:\mathcal{P}\to(\mathfrak{g}^{*}/G)^{\times n+1}$

by

$p_{2}\bigl{(}G_{n,K}(z_{\ell},x_{1},\dots,x_{n},y_{1},\dots,y_{n},z_{r})\bigr{)}=(-Gy_{1},Gx_{1},\dots,Gx_{n}),$

with the trivial Poisson structure on the target space.

The restriction of the Poisson projection $p_{1}$ to the symplectic leaf
$S(\mathcal{O})\subset T^{*}(G^{\times n+1})/G_{n,K}$ (see (31)) gives the
Poisson projection

$p_{1,\mathcal{O}}:S(\mathcal{O})\to\mathcal{P}(\mathcal{O})$

where

$\mathcal{P}(\mathcal{O})=(\mu_{L}\times\mu_{R})(\mu^{-1}(\mathcal{O}))/G_{n,K},$

or, more explicitly,

(51)
$\begin{split}\mathcal{P}(\mathcal{O})=\bigl{\\{}(z_{\ell},x_{1},\ldots,x_{n},&y_{1},\ldots,y_{n},z_{r})\in(\mathfrak{k}^{*}\times\mathfrak{g}^{*\times
n})\times_{(\mathfrak{g}^{*}/G)^{\times n+1}}(\mathfrak{g}^{*\times
n}\times\mathfrak{k}^{*})\,\,|\,\,\\\
&z_{\ell}\in\mathcal{O}_{\ell}^{K},\,x_{1}+y_{1}\in\mathcal{O}_{1},\ldots,x_{n}+y_{n}\in\mathcal{O}_{n},\,z_{r}\in\mathcal{O}_{r}^{K}\bigr{\\}}/G_{n,K}.\end{split}$

The generic fibers of this mapping are isotropic submanifolds in
$\mathcal{S}(\mathcal{O})$.

The restriction of the Poisson projection $p_{2}$ to
$\mathcal{P}(\mathcal{O})$ gives a surjective Poisson projection

$p_{2,\mathcal{O}}:\mathcal{P}(\mathcal{O})\to\mathcal{B}(\mathcal{O}),$

with $\mathcal{B}(\mathcal{O})$ given by (34). Clearly, the composition of
$p_{2,\mathcal{O}}\circ
p_{1,\mathcal{O}}:\mathcal{S}(\mathcal{O})\rightarrow\mathcal{B}(\mathcal{O})$
is the projection $p_{\mathcal{O}}$ as given by (33).

Now let us describe fibers of $p_{2,\mathcal{O}}$ are symplectic leaves of
$\mathcal{P}(\mathcal{O})$.

######  Lemma 3.

We have he following symplectomorphysm

(52)
$p_{2,\mathcal{O}}^{-1}((\mathcal{O}^{(0)},\dots,\mathcal{O}^{(n)}))\overset{\sim}{\longrightarrow}\mathcal{M}(-\mathcal{O}^{(0)},\mathcal{O}_{\ell}^{K})\times\prod_{i=1}^{n}\mathcal{M}(-\mathcal{O}^{(i)},\mathcal{O}^{(i-1)},\mathcal{O}_{i})\times\mathcal{M}(\mathcal{O}^{(n)},\mathcal{O}_{r}^{K})$

where symplectic spaces
$\mathcal{M}(-\mathcal{O}^{(i)},\mathcal{O}^{(i-1)},\mathcal{O}_{i})$ are
defined in (22) and

$\mathcal{M}(\mathcal{O}^{\prime},\mathcal{O}^{K})=\\{(x,z)\in\mathcal{O}^{\prime}\times\mathcal{O}^{K}\,\,|\,\,\pi(x)+z=0\\}/K$

Here $\mathcal{O}\subset\mathfrak{g}^{*}$ is a $G$-coadjoint orbit and
$\mathcal{O}^{K}\subset\mathfrak{k}^{*}$ is a $K$-coadjoint orbit. It has a
natural symplectic structure because it is the Hamiltonian reduction of
$\mathcal{O}\times\mathcal{O}^{K}$ with respect to the Hamiltonian diagonal
action of $K$.

###### Proof.

Let $(\mathcal{O}^{(0)},\ldots,\mathcal{O}^{(n)})\in\mathcal{B}(\mathcal{O})$.
By a direct computation, the fiber
$p_{2,\mathcal{O}}^{-1}((\mathcal{O}^{(0)},\dots,\mathcal{O}^{(n)}))$ consists
of the $G_{n,K}$-orbits in $\mathcal{P}$ with representatives

$(z_{\ell},x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z_{r})\in(\mathfrak{k}^{*}\times\mathfrak{g}^{*\times
n})\times(\mathfrak{g}^{*\times n}\times\mathfrak{k}^{*})$

satisfying the following conditions

$\begin{matrix}&z_{\ell}\in\mathcal{O}_{\ell}^{K}\cap\pi(\mathcal{O}^{(0)}),\quad&x_{i}+y_{i}\in\mathcal{O}_{i}\quad(1\leq
i\leq n),\,\,&z_{r}\in\mathcal{O}_{r}^{K}\cap-\pi(\mathcal{O}^{(n)}),\\\
&-y_{1}\in\mathcal{O}^{(0)},\quad&x_{i}\in\mathcal{O}^{(i)},-y_{i+1}\mathcal{O}^{(i)}\in\,(1\leq
i\leq n-1),\quad&x_{n}\in\mathcal{O}^{(n)}.\end{matrix}$

Using this explicit description of the fiber, we can write it as a direct
product of symplectic spaces. The isomorphism (23) for symplectic spaces
$\mathcal{M}(\mathcal{O}^{(1)},\mathcal{O}^{(2)},\mathcal{O}^{(3)})$ defined
by (22) gives factors
$\mathcal{M}(-\mathcal{O}^{(i)},\mathcal{O}^{(i-1)},\mathcal{O}_{i})$. The
space

$\mathcal{M}(\mathcal{O}^{\prime},\mathcal{O}^{K})=\\{(x,z)\in\mathcal{O}^{\prime}\times\mathcal{O}^{K}\,\,|\,\,\pi(x)+z=0\\}/K$

with $K$ acting by the diagonal coadjoint action is symplectic because, see
above. The isomorphism

$\mathcal{M}(\mathcal{O}^{\prime},\mathcal{O}^{K})\overset{\sim}{\longrightarrow}(\mathcal{O}^{\prime}\cap\pi^{-1}(\mathcal{O}^{K}))/K,$

completes the proof. The isomorphism maps
$G_{n,K}(z_{\ell},x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z_{r})\in
p_{2,\mathcal{O}}^{-1}((\mathcal{O}^{(0)},\dots,\mathcal{O}^{(n)}))$ to

$(K(\widetilde{y}_{1},z_{\ell}),G(-x_{1},-y_{1},x_{1}+y_{1}),\ldots,G(-x_{n},-y_{n},x_{n}+y_{n}),K(\widetilde{x}_{n},z_{r})),$

with $\widetilde{y}_{1}\in Gy_{1}=-\mathcal{O}^{(0)}$ such that
$-\pi(\widetilde{y}_{1})=z_{\ell}$ and $\widetilde{x}_{n}\in
Gx_{n}=\mathcal{O}^{(n)}$ such that $-\pi(\widetilde{x}_{n})=z_{r}$. ∎

Note that for generic $\mathcal{O}^{\prime}$, the symplectic space
$\mathcal{M}(\mathcal{O}^{\prime},\mathcal{O}^{K})$ is of dimension
$\dim(\mathcal{O}^{K})$.

######  Remark 3.

In the compact case, the algebra of function on the fiber of
$p_{2,\mathcal{O}}$ has the algebra of endomorphisms of the vector space

$\textup{Hom}_{K}(V_{\lambda_{0}},U_{\nu_{\ell}})\otimes\textup{Hom}_{G}(V_{\lambda_{1}},V_{\lambda_{0}}\otimes
V_{\mu_{1}})\otimes\cdots\otimes\textup{Hom}_{G}(V_{\lambda_{n}},V_{\lambda_{n-1}}\otimes
V_{\mu_{n}})\otimes\textup{Hom}_{K}(U_{\nu_{r}},V_{\lambda_{n}})$

as natural quantization, where the finite dimensional $G$-representation
$V_{\lambda_{i}}$ (resp. $V_{\mu_{i}}$) corresponds to $\mathcal{O}^{(i)}$
(resp. $\mathcal{O}_{i}$) and the $K$-representation $U_{\nu_{\ell}}$ (resp.
$U_{\nu_{r}}$) corresponds to $\mathcal{O}_{\ell}^{K}$ (resp.
$\mathcal{O}_{r}^{K}$), compare with Remark 2 in the cyclic case. For details
see [35] (which treats the noncompact case) and [RSQCD2].

######  Lemma 4.

Dimensions of spaces $\mathcal{B}(\mathcal{O})$ and $\mathcal{P}(\mathcal{O})$
are

$\dim(\mathcal{B}(\mathcal{O}))=(n+1)r,\ \
\dim(\mathcal{P}(\mathcal{O}))=\dim(\mathcal{O})-2nr$

where we define $\dim(\mathcal{O})$ as
$\dim(\mathcal{O}_{\ell}^{K})+\sum_{i=1}^{n}\dim(\mathcal{O}_{i})+\dim(\mathcal{O}_{r}^{K})$.

###### Proof.

The proof of the dimension formula for $\mathcal{B}(\mathcal{O})$ is
completely similar to the periodic case. It is enough to consider large
orbits. For the dimension of $\mathcal{P}(\mathcal{O})$ we have:

$\dim(\mathcal{P}(\mathcal{O}))=\dim(\mathcal{B}(\mathcal{O}))+\dim\bigl{(}p_{2,\mathcal{O}}^{-1}((\mathcal{O}^{(0)},\ldots,\mathcal{O}^{(n)}))$

and by (52) the dimension of
$p_{2,\mathcal{O}}^{-1}((\mathcal{O}^{(0)},\ldots,\mathcal{O}^{(n)}))$ is
equal to

$\begin{split}\dim(\mathcal{M}(-\mathcal{O}^{(0)},\mathcal{O}_{\ell}^{K}))&+\sum_{i=1}^{n}\dim(\mathcal{M}(-\mathcal{O}^{(i)},\mathcal{O}^{(i-1)},\mathcal{O}_{i}))+\dim(\mathcal{M}(\mathcal{O}^{(n)},\mathcal{O}_{r}^{K}))=\\\
&=\dim(\mathcal{O}_{\ell}^{K})+\sum_{i=1}^{n}(\dim(\mathcal{O}_{i})-2r)+\dim(\mathcal{O}_{r}^{K})=\dim(\mathcal{O})-2nr.\end{split}$

This finishes the proof. ∎

We now have the following main result of this section.

######  Theorem 4.

The Hamiltonian system generated by any Hamiltonian for the open spin CM chain
described in section 2.2 is superintegrable with the superintegrable structure
described by the surjective Poisson maps

$\mathcal{S}(\mathcal{O})\stackrel{{\scriptstyle
p_{1,\mathcal{O}}}}{{\longrightarrow}}\mathcal{P}(\mathcal{O})\stackrel{{\scriptstyle
p_{2,\mathcal{O}}}}{{\longrightarrow}}\mathcal{B}(\mathcal{O})$

as introduced earlier in this section.

Recall that $\mathcal{O}_{i}\neq\\{0\\}$ for all $i=0,1,\dots,n$.

###### Proof.

We already verified most of the conditions. What remains to show is the
matching of dimensions,

(53)
$\dim(\mathcal{S}(\mathcal{O}))=\dim(\mathcal{P}(\mathcal{O}))+\dim(\mathcal{B}(\mathcal{O})).$

For the collection
$\mathcal{O}=(\mathcal{O}_{\ell}^{K},\mathcal{O}_{1},\dots,\mathcal{O}_{n},\mathcal{O}_{r}^{K})$
of coadjoint orbits we write $\dim(\mathcal{O})$ for the sum of the dimensions
of the coadjoint orbits.

For the dimension of the symplectic leaf $\mathcal{S}(\mathcal{O})$ we have,
using (44),

$\dim(\mathcal{S}(\mathcal{O}))=2r+\dim(\mathcal{O}),$

with $r$ the rank of $\mathfrak{g}$. For $\mathcal{P}(\mathcal{O})$ we obtain
for generic
$(\mathcal{O}^{(0)},\ldots,\mathcal{O}^{(n)})\in\mathcal{B}(\mathcal{O})$,

Because

$\dim(\mathcal{B}(\mathcal{O}))=(n+1)r,$

we have

$\begin{split}\dim(\mathcal{P}(\mathcal{O}))+\dim(\mathcal{B}(\mathcal{O}))&=\dim(\mathcal{O})-2nr+2\dim(\mathcal{B}(\mathcal{O}))\\\
&=\dim(\mathcal{O})+2r\\\ &=\dim(\mathcal{S}(\mathcal{O})),\end{split}$

as desired.

∎

###  2.7. Constructing solutions by the projection method and angle variables

Let $\mathcal{H}$ be a $G$-invariant function on $\mathfrak{g}^{*}$ and
$\mathcal{H}^{(i)}$ for $i=0,\ldots,n$ the associated $G_{n}$-invariant
function $(x,g)\mapsto\mathcal{H}(x_{i})$ on $T^{*}(G^{\times n+1})$ (cf.
section 1.6). The Hamiltonian flow generated by $\mathcal{H}^{(i)}$ on
$T^{*}(G^{\times n+1})\simeq\mathfrak{g}^{*\times n+1}\times G^{\times n+1}$
was already described in section 1.6. The flow line passing through $(x,g)$ at
$t=0$ is

(54)
$(x(t_{i}),g(t_{i}))=(x_{0},\dots,x_{n},g_{0},\dots,g_{i-1},e^{\nabla\mathcal{H}(x_{i})t_{i}}g_{i},g_{i+1}\dots,g_{n}).$

The corresponding Hamiltonian flow on the symplectic leaf
$\mathcal{S}(\mathcal{O})\subset T^{*}(G^{\times n+1})/G_{n,K}$ is obtained by
projecting the flow (54) to $T^{*}(G^{\times n+1})/G_{n,K}$ and restricting it
to $\mathcal{S}(\mathcal{O})$. Thus, we reformulated the problem of solving
nonlinear differential equations of motion for open spin Calogero-Moser chains
to a problem of linear algebra. This is a version of the original projection
method which goes back to earlier papers on Calogero-Moser type systems [28].

Now let us describe angle variables for this integrable dynamics. We use the
notations from section 1.6. Fix $(x,g)\in\mathfrak{g}^{\prime*\times
n+1}\times G^{\times n+1}$. For $i=1,\dots,n$ define elements $s_{i}\in G$ by
the condition $Ad^{*}_{s_{i}}(x_{i})\in\mathfrak{a}^{*}_{+}$. As in the
periodic case (see section 1.6) it is defines up to $s_{i}\mapsto a_{i}s_{i}$
with $a_{i}\in H\subset G$, where $H\subset G$ is the Cartan subgroup
containing $A$. Define $s_{0}\in K$ such that
$Ad_{s_{0}}^{*}(x_{0})|_{\mathfrak{p}}\in\mathfrak{a}_{+}^{*}$ (where we view
$\mathfrak{a}_{+}^{*}$ now as subset of $\mathfrak{p}^{*}$ in the natural
manner). The element $s_{0}$ is defined up to $s_{0}\mapsto ms_{0}$ with $m\in
M=Z_{K}(A)$. Similarly, we define $s_{n+1}\in K$ such that
$Ad_{s_{n+1}}^{*}(x_{n})|_{\mathfrak{p}}\in\mathfrak{a}_{+}^{*}$.

Choose finite dimensional representations $V_{0},V_{1},\dots,V_{n}$ of
$G_{\mathbb{C}}$, $H_{\mathbb{C}}$-weight vectors $v_{i}\in V_{i}$ of weight
$\lambda_{i+1}$ for $0\leq i<n$ and $H_{\mathbb{C}}$-weight vectors
$u_{j}^{*}\in V_{j}^{*}$ of weight $-\lambda_{j}$ for $0\leq j\leq n$.
Finally, we choose $M$-invariant vectors $u_{0}^{*}\in V_{0}^{*}$ and
$v_{n}\in V_{n}$ (i.e., $mu_{0}^{*}=u_{0}^{*}$ and $mv_{n}=v_{n}$ for all
$m\in M$). Define

(55)
$f_{u,v}(x,g)=u^{*}_{0}(s_{0}g_{0}s^{-1}_{1}v_{0})u^{*}_{1}(s_{1}g_{1}s^{-1}_{2}v_{1})\cdots
u_{n-1}^{*}(s_{n-1}g_{n-1}s_{n}^{-1}v_{n-1})u_{n}^{*}(s_{n}g_{n}s_{n+1}^{-1}v_{n}).$

It is an easy check that $f_{u,v}(x,g)$ is a well defined $G_{n,K}$-invariant
function on $\mathfrak{g}^{\prime*\times n+1}\times G^{\times n+1}$.

Similarly as in the periodic case (see section 1.6) we then have for
$i=1,\ldots,n$,

(56)
$f_{u,v}(x(t_{i}),g(t_{i}))=e^{t_{i}\lambda_{i}(\nabla\mathcal{H}(y_{i}))}f_{u,v}(x,g)$

with $y_{i}=Ad_{s_{i}}^{*}(x_{i})\in\mathfrak{a}_{+}^{*}$. Logarithms of these
functions thus evolve linearly, and hence give rise to angle variables for the
Hamiltonians $\mathcal{H}^{(i)}$ on
$\mathcal{S}(\mathcal{O})\cap(\mathfrak{g}^{\prime*\times n+1}\times G^{\times
n+1})/G_{n,K}$.

For $i=0$ we need to restrict further to $(x,g)\in\mathfrak{g}^{\prime*\times
n+1}\times G^{\times n+1}$ with $x_{0}\in\mathfrak{p}$, and assume that
$u_{0}^{*}\in V_{0}^{*}$ is not only $M$-invariant but also a
$H_{\mathbb{C}}$-weight vector, say of weight $-\lambda_{0}$. In this case
$\operatorname{Ad}_{s_{0}}^{*}(x_{0})=y_{0}\in\mathfrak{a}_{+}^{*}$ and hence

$u^{*}_{0}(s_{0}e^{t_{0}\nabla\mathcal{H}(x_{0})}g_{0}s_{1}^{-1}v_{0})=e^{t_{0}\lambda_{0}(\mathcal{H}(y_{0}))}u^{*}_{0}(s_{0}g_{0}s_{1}^{-1}v_{0}).$

As a consequence (56) then also holds true for $i=0$, and the logarithm of
$f_{u,v}(x,g)$ becomes a linear functions of time $t_{0}$.

##  3\. A Liouville integrable example of a periodic spin Calogero-Moser
example for orbits of rank $1$.

### 3.1.

Let us briefly discuss a particular case of periodic spin CM chain
corresponding to $G=SL_{N}(\mathbb{R})$ with rank one orbits
$\mathcal{O}_{k}$. This case is related to the original paper [16] where spin
CM systems were first introduced.

Take $\mathfrak{a}\subset\mathfrak{sl}_{N}$ the Cartan subalgebra consisting
of diagonal matrices, and denote the roots by
$\\{\epsilon_{i}-\epsilon_{j}\\}_{i\not=j}\subset\mathfrak{a}^{*}$ with
$\epsilon_{i}\in\mathfrak{a}^{*}$ the linear functional picking out the
$i^{th}$ diagonal entry. We identify $\mathfrak{sl}_{N}$ with its dual via the
Killing form $(x,y)=2N\,\textup{Tr}(xy)$. Then for
$p\in\mathfrak{a}^{*}\simeq\mathfrak{a}$ we have
$(p,p)=2N\sum_{i=1}^{N}p_{i}^{2}$, with $p_{i}$ the $i^{th}$ diagonal entry of
the diagonal matrix $p$. For
$y\in\mathfrak{sl}_{N}^{*}\simeq\mathfrak{sl}_{N}$ and $i\not=j$ we have
$y_{\epsilon_{i}-\epsilon_{j}}=\sqrt{2N}y_{ij}$, with $y_{ij}$ the
$(i,j)^{th}$ entry of the matrix $y$.

For $\xi\in\mathbb{R}$ set

$\mathcal{O}^{(\xi)}=\Bigl{\\{}x-\frac{\xi}{N}\textup{id}_{N}\,\,\,|\,\,\,x\textup{
is a rank one $N\times N$ matrix with }\,\,\textup{Tr}(x)=\xi\,\Bigr{\\}}.$

Then $\mathcal{O}^{(\xi)}$ is a coadjoint orbit in
$\mathfrak{sl}_{N}\simeq\mathfrak{sl}_{N}^{*}$ of dimension $2(N-1)$.

Viewing elements in $\mathbb{R}^{N}$ as column vectors, we have a natural
mapping

(57)
$\bigl{\\{}(a,b)\in\mathbb{R}^{N}\times\mathbb{R}^{N}\,\,|\,\,a^{t}b=\xi\bigr{\\}}/\mathbb{R}^{\times}\overset{\sim}{\longrightarrow}\mathcal{O}^{(\xi)},\qquad\mathbb{R}^{\times}(a,b)\mapsto
ba^{t}-\frac{\xi}{N}\textup{id}_{N}$

where $\lambda\in\mathbb{R}^{\times}$ acts by $(a,b)\mapsto(\lambda
a,\lambda^{-1}b)$ and $a^{t}$ is the transpose of $a\in\mathbb{R}^{N}$.
Because of the rank one condition, this is an isomorphism. It is easy to check
that this is symplomorphism, with the Poisson brackets of the coordinate
functions $a_{i}$ and $b_{j}$ of $(a,b)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$
given by

$\\{b_{i},a_{j}\\}=\delta_{ij},\ \ \\{a_{i},a_{j}\\}=0=\\{b_{i},b_{j}\\}=0.$

The value of the quadratic Casimir function
$y\mapsto\frac{(y,y)}{2N}=\sum_{i,j=1}^{N}y_{ij}y_{ji}$ on
$\mathcal{O}^{(\xi)}$ is easily computed using (57):

$\sum_{i,j=1}^{N}\mu_{ij}\mu_{ji}=\xi^{2}\Big{(}1-\frac{1}{N}\Bigr{)},\qquad\qquad\mu\in\mathcal{O}^{(\xi)}.$

Here we use notations $\mu=y|_{\mathcal{O}^{(\xi)}}$.

### 3.2.

The quadratic $n$-th Hamiltonian $H_{2}^{(n)}$ in radial coordinates, rescaled
by a factor $2N$, then is

(58)
$H_{2}=\frac{1}{2}\sum_{i=1}^{N}p_{i}^{2}-\sum_{i<j}\frac{\mu_{ij}\mu_{ji}}{2\textup{sh}^{2}(q_{i}-q_{j})}$

(see (18)), where $q_{i}=\epsilon_{i}(\log(a))$ and

$\mu_{ij}=\sum_{k=1}^{n}\mu_{ij}^{(k)}.$

We now consider the Hamiltonian (58) on
$\mathcal{S}(\mathcal{O})_{reg}\simeq(\nu_{\mathcal{O}}^{-1}(0)/H\times
T^{*}A_{reg})/W$ (see (12)) with
$\mathcal{O}=(\mathcal{O}^{(\xi_{1})},\ldots,\mathcal{O}^{(\xi_{n})})$ a
collection of $n$ rank one orbits. Here $H\subset\textup{SL}_{N}(\mathbb{R})$
is the Cartan subgroup of diagonal matrices and
$\nu_{\mathcal{O}}^{-1}(0)\subset\mathcal{O}^{(\xi_{1})}\times\cdots\times\mathcal{O}^{(\xi_{n})}$
consists of the $n$-tuple of rank one matrices

(59)
$(\mu^{(1)},\ldots,\mu^{(n)})=\Bigl{(}b^{(1)}a^{(1)t}-\frac{\xi_{1}}{N}\textup{id}_{N},\ldots,b^{(N)}a^{(N)t}-\frac{\xi_{N}}{N}\textup{id}_{N}\Bigr{)}$

where the diagonal action of $h\in H$ is given by $a_{i}^{(k)}\to
h_{i}a_{i}^{(k)}$, $b_{j}^{(k)}\to h_{j}^{-1}b_{j}^{(k)}$, and vectors
$a^{(k)},b^{(k)}\in\mathbb{R}^{N}$ satisfy the relations

(60) $\sum_{i=1}^{N}a_{i}^{(k)}b_{i}^{(k)}=\xi_{k}\quad(1\leq k\leq
n),\qquad\sum_{k=1}^{n}a_{i}^{(k)}b_{i}^{(k)}=\frac{\bm{\xi}}{N}\quad(1\leq
i\leq N).$

Here $\bm{\xi}=\sum_{k=1}^{n}\xi_{k}$. In other words,
$\mu^{(k)}_{ij}=b_{i}^{(k)}a_{j}^{(k)}-\delta_{ij}\xi_{k}/N$, where
$b_{i}^{(k)},a_{j}^{(k)}$ are as above.

In terms of the variables $a^{(k)}$ and $b^{(k)}$ the Hamiltonian $H_{2}$ on
$\mathcal{S}(\mathcal{O})_{reg}$ can be rewritten as

(61)
$H_{2}=\frac{1}{2}\sum_{i=1}^{N}p_{i}^{2}-\sum_{i<j}\frac{\sum_{k,\ell=1}^{n}b_{i}^{(k)}a_{j}^{(k)}b_{j}^{(\ell)}a_{i}^{(\ell)}}{2\textup{sh}^{2}(q_{i}-q_{j})}.$

### 3.3.

Here we will rewrite the Hamiltonian (61) in terms of variables attached to
each $q_{i}$. It is natural to think of these variable as spin variables
attached to a one dimensional particle with the position $q_{i}$. They are
defined as follows.

For $\xi\in\mathbb{R}$ denote by $\widetilde{\mathcal{O}}^{(\xi)}$ the rank
one coadjoint $\textup{SL}_{n}(\mathbb{R})$-orbit defined as

$\widetilde{\mathcal{O}}^{(\xi)}=\Bigl{\\{}x-\frac{\xi}{n}\textup{id}_{n}\,\,\,|\,\,\,x\textup{
is a rank one $n\times n$ matrix with}\,\,\textup{Tr}(x)=\xi\,\Bigr{\\}},$

and set

$\widetilde{\mathcal{O}}:=\underbrace{(\widetilde{\mathcal{O}}^{(\bm{\xi}/N)},\ldots,\widetilde{\mathcal{O}}^{(\bm{\xi}/N)})}_{N}.$

The coadjoint action of the Cartan subgroup
$\widetilde{H}\subset\textup{SL}_{n}(\mathbb{R}$) is Hamiltonian and gives the
moment map

$\widetilde{\nu}_{\widetilde{\mathcal{O}}}:\underbrace{\widetilde{\mathcal{O}}^{(\bm{\xi}/N)}\times\cdots\times\widetilde{\mathcal{O}}^{(\bm{\xi}/N)}}_{N}\rightarrow\widetilde{\mathfrak{a}}^{*},\qquad(g_{1},\ldots,g_{N})\mapsto\bigl{(}g_{1}+\cdots+g_{N})_{0}$

where $\widetilde{\mathfrak{a}}=\textup{Lie}(\widetilde{H})$.

Finally, consider the traceless diagonal $n\times n$-matrix

$t_{\underline{\xi}}=\textup{diag}\Bigl{(}\xi_{1}-\frac{\bm{\xi}}{n},\ldots,\xi_{n}-\frac{\bm{\xi}}{n}\Bigr{)}\in\widetilde{\mathfrak{a}}.$

From the above we immediately have the following statement.

######  Lemma 5.

We the following isomorphism of $2(n-1)(N-1)$-dimensional symplectic varieties

$\begin{split}\nu_{\mathcal{O}}^{-1}(0)/H&\overset{\sim}{\longrightarrow}\widetilde{\nu}_{\widetilde{\mathcal{O}}}^{-1}(t_{\underline{\xi}})/\widetilde{H},\\\
H(\mu^{(1)},\ldots,\mu^{(n)})&\mapsto\widetilde{H}(g^{(1)},\ldots,g^{(N)}),\end{split}$

Here $\mu^{(k)}_{ij}=b_{i}^{(k)}a_{j}^{(k)}-\delta_{ij}\xi_{k}/N$ and the
local spin variables $g^{(i)}_{k\ell}$ are

$g_{k\ell}^{(i)}=b_{i}^{(k)}a_{i}^{(\ell)}-\delta_{k\ell}\frac{\bm{\xi}}{Nn}.$

It is easy to check that if $i\neq j$ the following identity holds:

(62) $\sum_{k,\ell=1}^{n}g^{(i)}_{k\ell}g_{\ell
k}^{(j)}=\mu_{ij}\mu_{ji}-\frac{\bm{\xi}^{2}}{N^{2}n},$

Thus, we can rewrite the Hamiltonian (58) in terms of spin variables from
$\widetilde{\nu}_{\widetilde{\mathcal{O}}}^{-1}(t_{\underline{\xi}})/\widetilde{H}\times
T^{*}A_{reg}$ as

(63)
$H_{2}=\frac{1}{2}\sum_{i=1}^{N}p_{i}^{2}+\sum_{i<j}\frac{\textup{Tr}(g^{(i)}g^{(j)})+\frac{\bm{\xi}^{2}}{N^{2}n}}{2\textup{sh}^{2}(q_{i}-q_{j})}.$

This Hamiltonian describes $N$ classical particles each carrying a ”spin” from
a rank one coadjoint orbit in $\mathfrak{sl}_{n}^{*}$ with the Casimir value
given by

(64)
$\sum_{\alpha,\beta=1}^{n}(g_{i})_{\beta}^{\alpha}(g_{i})_{\alpha}^{\beta}=\frac{c^{2}}{n^{2}}(1-\frac{1}{N})$

The system is Liouville integrable since we constructed $n(N-1)$ integrals for
the periodic spin chain earlier (see the proof of Theorem 2).

Integrable system described above are closely related [16] and [21].

### 3.4.

This project, together with results of [1], is the first step towards
constructing superintegrable systems on moduli spaces of flat connections on a
surface where on part of the boundary the gauge group $G$ is constrained to
$K$. When the boundary gauge group is not constrained, corresponding
integrable systems are described in [1]. We expect that such moduli spaces
have the structure of a cluster variety similar to the one described in [14].
It would be interesting to to extend the construction of spin CM chains to the
elliptic case as it was done for $N=1$ in [21].

##  Appendix A Comparison with the $n=2$ case from [31]

Consider the periodic spin CM chain from section 1 for $n=2$. The symplectic
leaves of $T^{*}(G^{\times 2})/G_{2}$ are then

(65)
$\mathcal{S}(\mathcal{O}_{1},\mathcal{O}_{2})=\\{(x_{1},x_{2},g_{1},g_{2})\,\,|\,\,x_{1}-Ad^{*}_{g_{2}^{-1}}(x_{2})\in\mathcal{O}_{1},\
\ x_{2}-Ad^{*}_{g_{1}^{-1}}(x_{1})\in\mathcal{O}_{2}\\}/G_{2}$

where $\mathcal{O}_{1},\mathcal{O}_{2}$ are coadjoint orbits in
$\mathfrak{g}^{*}$, relative to the gauge action

$(h_{1},h_{2})(x_{1},x_{2},g_{1},g_{2})=(Ad_{h_{1}}^{*}(x_{1}),Ad_{h_{2}}^{*}(x_{2}),h_{1}g_{1}h_{2}^{-1},h_{2}g_{2}h_{1}^{-1}).$

In [31, §3 & App. C] the following Hamiltonian action of $G_{2}$ on
$T^{*}(G^{\times 2})$ is considered,

(66)
$(h_{1},h_{2})_{*}(x_{1},x_{2},g_{1},g_{2})=(Ad_{h_{1}}^{*}(x_{1}),Ad_{h_{1}}^{*}(x_{2}),h_{1}g_{1}h_{2}^{-1},h_{1}g_{2}h_{2}^{-1}),$

with corresponding moment map $\mu_{*}:T^{*}(G^{\times
2})\rightarrow\mathfrak{g}^{*\times 2}$ given by

$\mu_{*}(x_{1},x_{2},g_{1},g_{2})=(x_{1}+x_{2},-Ad_{g_{1}^{-1}}^{*}(x_{1})-Ad_{g_{2}^{-1}}^{*}(x_{2})).$

The corresponding symplectic leaves are

$\begin{split}\mathcal{S}_{*}(\mathcal{O}_{1},\mathcal{O}_{2})&=\mu_{*}^{-1}(\mathcal{O}_{1},\mathcal{O}_{2})/G_{2}\\\
&=\bigl{\\{}(x_{1},x_{2},g_{1},g_{2})\,\,|\,\,x_{1}+x_{2}\in\mathcal{O}_{1},\,\,-Ad_{g_{1}^{-1}}^{*}(x_{1})-Ad_{g_{2}^{-1}}(x_{2})\in\mathcal{O}_{2}\bigr{\\}}/G_{2},\end{split}$

with the gauge group $G_{2}$ now acting by (66). These symplectic leaves were
used in [31]. They are related to the symplectic leaves
$\mathcal{S}(\mathcal{O}_{1},\mathcal{O}_{2})$ in the following way.

Consider the map $\psi:T^{*}(G^{\times 2})\rightarrow T^{*}(G^{\times 2})$,
defined by

$\psi(x_{1},x_{2},g_{1},g_{2})=(-x_{1},Ad_{g_{1}}^{*}(x_{2}),g_{1},g_{1}g_{2}g_{1}).$

Then $\psi$ is $G_{2}$-equivariant,

$\psi((h_{1},h_{2})(x_{1},x_{2},g_{1},g_{2}))=(h_{1},h_{2})_{*}\psi(x_{1},x_{2},g_{1},g_{2}),$

and the resulting map on the $G_{2}$-orbits restricts to an isomorphism

$\mathcal{S}(\mathcal{O}_{1},\mathcal{O}_{2})\overset{\sim}{\longrightarrow}\mathcal{S}_{*}(\mathcal{O}_{2},\mathcal{O}_{1}).$

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