Invariant Nijenhuis Tensors and Integrable Geodesic Flows

Invariant Nijenhuis Tensors
and Integrable Geodesic Flows

Konrad LOMPERT † and Andriy PANASYUK ‡

K. Lompert and A. Panasyuk

† Faculty of Mathematics and Information Science, Warsaw University of
Technology,
† ul. Koszykowa 75, 00-662 Warszawa, Poland<EMAIL_ADDRESS>
‡ Faculty of Mathematics and Computer Science, University of Warmia and
Mazury,
‡ ul. Słoneczna 54, 10-710 Olsztyn, Poland<EMAIL_ADDRESS>
Received December 19, 2018, in final form August 02, 2019; Published online
August 07, 2019

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a
reductive Lie group $G$ from the point of view of integrability of a
Hamiltonian system of differential equations with the $G$-invariant
Hamiltonian function on the cotangent bundle $T^{*}(G/K)$. Such a tensor
induces an invariant Poisson tensor $\Pi_{1}$ on $T^{*}(G/K)$, which is
Poisson compatible with the canonical Poisson tensor $\Pi_{T^{*}(G/K)}$. This
Poisson pair can be reduced to the space of $G$-invariant functions on
$T^{*}(G/K)$ and produces a family of Poisson commuting $G$-invariant
functions. We give, in Lie algebraic terms, necessary and sufficient
conditions of the completeness of this family. As an application we prove
Liouville integrability in the class of analytic integrals polynomial in
momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of
compact Lie groups $G$ for two kinds of metrics: the normal metric and new
classes of metrics related to decomposition of $G$ to two subgroups
$G=G_{1}\cdot G_{2}$, where $G/G_{i}$ are symmetric spaces, $K=G_{1}\cap
G_{2}$.

bi-Hamiltonian structures; integrable systems; homogeneous spaces; Lie
algebras; Liouville integrability

37J15; 37J35; 53D25

##  1 Introduction

By Maupertuis’s principle integrability of the geodesic flow of a
(pseudo-)Riemannian metric is a question as old as classical mechanics itself.
In this paper we consider Hamiltonian systems and understand integrability in
the sense of Arnold–Liouville, i.e., as existence of a complete family of
first integrals in involution. The Clairaut theorem on existence of linear
integral for the motion of a free particle on a surface of revolution is
traditionally mentioned as one of the first results on Arnold–Liouville
integrability of geodesic flows. Next classical cases are the Euler top and
geodesics on ellipsoid. In modern mathematical literature one could find many
examples of integrable geodesic flows on homogeneous spaces of Lie groups
starting probably with the papers [15, 30], see also the review [6] and
references therein and later works [7, 12, 16, 17].

The present paper continues this line and develops a new approach for
constructing integrable geodesic flows on homogeneous spaces. Let $G$ be a
reductive Lie group, $K\subset G$ its closed subgroup. The cotangent bundle
$T^{*}(G/K)$ with its canonical Poisson structure $\Pi$ is a phase space of a
Hamiltonian system with the Hamiltonian function equal to the quadratic form
$q$ of an $G$-invariant pseudo-Riemannian metric, which can be constructed as
follows. Let $\langle\,,\,\rangle$ be an $\operatorname{Ad}G$-invariant
symmetric bilinear form on $\mathfrak{g}$, the Lie algebra of $G$. It gives
rise to a bi-invariant metric on $G$, which induces on $G/K$ an $G$-invariant
metric $\langle\,,\,\rangle_{G/K}$ called _normal_. Besides, one can consider
a symmetric $\operatorname{ad}\mathfrak{k}$-invariant linear operator (called
_inertia operator_)
$n_{\mathfrak{k}^{\perp}}\colon\mathfrak{k}^{\perp}\to\mathfrak{k}^{\perp}$,
where $\mathfrak{k}$ is the Lie algebra of $K$ and $\mathfrak{k}^{\perp}$ is
its orthogonal complement in $\mathfrak{g}$ with respect to
$\langle\,,\,\rangle$. It will give rise to an $G$-invariant $(1,1)$-tensor
$N\colon T(G/K)\to T(G/K)$, which is symmetric with respect to
$\langle\,,\,\rangle_{G/K}$, and to another $G$-invariant metric
$\langle\cdot,\cdot\rangle_{N}:=\langle N\cdot,\cdot\rangle_{G/K}$. The
question of integrability of the geodesic flows of both metrics
$\langle\,,\,\rangle_{G/K}$ and $\langle\cdot,\cdot\rangle_{N}$ on $G/K$
consists of finding a family of $\dim(G/K)-1$ independent analytic and
_polynomial in momenta_ functions on $T^{*}(G/K)$ which Poisson commute with
the quadratic form $q$ and with each other. It is known [6, Section 5] that
there are two families $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ of analytic
functions on $T^{*}(G/K)$ that Poisson commute with each other, in which one
can look for desirable integrals. These are the family $\mathcal{F}_{2}$ of
$G$-invariant functions and the family $\mathcal{F}_{1}$ of functions of the
form $\mu_{\rm can}^{*}f$, where $\mu_{\rm can}\colon
T^{*}(G/K)\to\mathfrak{g}^{*}$ is the momentum map corresponding to the
natural Hamiltonian action of $G$ on $T^{*}(G/K)$ and $f$ is an analytic
function on $\mathfrak{g}^{*}$. Obviously $q\in\mathcal{F}_{2}$ and taking a
family $\mathcal{F}$ of commuting polynomials on $\mathfrak{g}^{*}$ (by the
Sadetov theorem [29] there exist _complete_ such families, see also [3]) one
gets the family $\mathcal{A}:=\mu_{\rm can}^{*}(\mathcal{F})$ of integrals of
$q$ polynomial in momenta. Thus the problem now is reduced to the following
one: construct a family $\mathcal{B}\subset\mathcal{F}_{2}$ of commuting
polynomial in momenta integrals of $q$ such that the family
$\mathcal{A}+\mathcal{B}$ is complete.

An approach for constructing such a family $\mathcal{B}$ was proposed in [17].
The homogeneous spaces considered were the coadjoint orbits $\mathcal{O}$ of
$G$. A second $G$-invariant Poisson structure $\Pi_{1}$ was constructed on
$T^{*}(G/K)$ which is compatible with $\Pi$ and the family $\mathcal{B}$ was
the canonical family of functions in involution related with the Poisson pair
$(\Pi^{\prime},\Pi_{1}^{\prime})$ being the reduction of the Poisson pair
$(\Pi,\Pi_{1})$ with respect to the action of $G$. Essential role in the
construction of $\Pi_{1}$ played the Kirillov–Kostant–Suriau symplectic form
$\omega_{\mathcal{O}}$ on $\mathcal{O}$, as
$\Pi_{1}=(\omega+\pi^{*}\omega_{\mathcal{O}})^{-1}$, where $\omega=-\Pi^{-1}$
is the canonical symplectic form on $T^{*}\mathcal{O}$ and $\pi\colon
T^{*}\mathcal{O}\to\mathcal{O}$ is the canonical projection.

In this paper we propose a novel approach for constructing the family
$\mathcal{B}$. Similarly to the case above, we construct a second Poisson
structure $\Pi_{1}$ compatible with $\Pi$, but we use invariant Nijenhuis
$(1,1)$-tensors $N\colon T(G/K)\to T(G/K)$ for this purpose instead, in
particular avoiding the restriction on $G/K$ of being a coadjoint orbit. In
more detail, $\Pi_{1}=\widetilde{N}\circ\Pi$, where $\widetilde{N}$ is the so-
called _cotangent lift_ of $N$, see Definition 4.5. Obviously, an invariant
$(1,1)$-tensor on $G/K$ is determined by a linear operator
$n\colon\mathfrak{g}\to\mathfrak{g}$. We get some Lie algebraic conditions on
this operator which are necessary and sufficient for the so-called
_kroneckerity_ of the Poisson pair $(\Pi^{\prime},\Pi_{1}^{\prime})$ obtained
as the reduction of the pair $(\Pi,\Pi_{1})$ and, as a consequence, of the
completeness of the family $\mathcal{B}$ (and $\mathcal{A}+\mathcal{B}$), see
Theorem 5.1, the main result of this paper, and Theorem 5.4. As an application
we construct two series of invariant Nijenhuis $(1,1)$-tensors on homogeneous
spaces $G_{k}/K_{k}$ of compact simple Lie groups, where $(G_{k},K_{k})$ is
$({\rm SU}(2k),{\rm S}({\rm U}(2k-1)\times{\rm U}(1))\cap{\rm Sp}(k))$ or
$({\rm SO}(2k+2),{\rm SO}(2k+1)\cap{\rm U}(k+1))$, which lead to invariant
metrics with geodesic flow Liouville integrable in the class of integrals
analytic and polynomial in momenta (Theorem 6.2). Besides we prove
integrability of the normal metric on these homogeneous spaces. Below the
content of the paper is discussed in more detail.

In Section 2 we study Lie algebraic conditions on the operator
$n\colon\mathfrak{g}\to\mathfrak{g}$ which guarantee the vanishing of the
Nijenhuis torsion of $N$ (Theorem 2.7) and consider some examples.

A crucial role in our considerations play bi-Hamiltonian (bi-Poisson)
structures, i.e., pencils of Poisson structures generated by pairs of
compatible ones. We devote Section 3 to related notions and preparatory
results which will enable us to study the completeness of families of
functions in involution. Theorem 3.7 gives some criteria of completeness of
the canonical family of $G$-invariant functions related to an action of a Lie
group $G$ on a bi-Poisson manifold $M$ being Hamiltonian with respect to
almost all Poisson structures from the pencil. The theorem requires some
assumptions among which the most significant one says that the action of $G$
on $M$ is _locally free_. This assumption enables to use the so-called
bifurcation lemma and to prove the constancy of rank of the reduced bi-Poisson
structure for almost all values of the parameter, which is a first step for
achieving the kroneckerity.

In Section 4 we study bi-Poisson structures on $T^{*}(G/K)$ generated by
Poisson pairs $(\Pi,\Pi_{1}=\widetilde{N}\circ\Pi)$, where $N$ is a semisimple
invariant Nijenhuis $(1,1)$-tensor. We show that almost all (_generic_)
Poisson structures from the corresponding Poisson pencil are nondegenerate and
calculate the dimensions of the symplectic leaves of the _exceptional_ (not
being generic) Poisson structures (Lemma 4.8). We prove the hamiltonicity of
canonical action of $G$ on $T^{*}(G/K)$ with respect to the generic Poisson
structures, as well as the hamiltonicity of the actions of some subgroups
(stabilizers of the symplectic leaves) on the symplectic leaves of the
exceptional ones. We calculate the corresponding momentum maps (see Lemma 4.9)
as well as these stabilizers (Lemma 4.10).

The main result, Theorem 5.1, which gives necessary and sufficient conditions
for kroneckerity of the reduced Poisson pair $(\Pi^{\prime},\Pi_{1}^{\prime})$
in terms of the _indices_ of the Lie algebra $\mathfrak{g}$ and some its
contractions (see formula (5.3)), is proved in Section 5. As a corollary we
prove Theorem 5.4 stating the complete integrability of the geodesic flow of
the normal metric and the metric with the inertia operator
$n|_{\mathfrak{k}^{\perp}}$ under the assumption that the sufficient
conditions from Theorem 5.1 are satisfied.

In Section 6 we apply the above results to construct examples of metrics with
integrable geodesic flow. The main idea which enables to fit conditions of
Theorem 5.1 is based on the Brailov theorem (see Theorem 6.1) stating equality
of indices of a semisimple Lie algebra and its
$\mathbb{Z}_{2}$-_contractions_. We observe that among the examples of
invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ from Section
2 related to the Onishchik list of decompositions
$\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ of a simple compact Lie
algebra to two subalgebras (Example 2.12) there are two series
$(\mathfrak{g}(k),\mathfrak{g}_{1}(k),\mathfrak{g}_{2}(k))$ in which both the
pairs $(\mathfrak{g}(k),\mathfrak{g}_{1}(k))$ and
$(\mathfrak{g}(k),\mathfrak{g}_{2}(k))$ are _symmetric_ , i.e., by the Brailov
theorem these examples satisfy conditions (5.3) of Theorem 5.1 (the Lie
algebra $\mathfrak{k}$ of the group $K$ is equal
$\mathfrak{g}_{1}(k)\cap\mathfrak{g}_{2}(k)$). In order to apply this theorem
for the proof of complete integrability of the geodesic flow one needs only
ensure that the action of $G$ on $T^{*}(G/K)$ is locally free. This is done in
the proof of Theorem 6.2 stating the complete integrability of the geodesic
flows of the normal metric and the metric with the corresponding inertia
operator.

The explicit formulae for the realizations of Lie algebras $\mathfrak{g}(k)$,
$\mathfrak{g}_{1}(k)$, $\mathfrak{g}_{2}(k)$ for both series as well as for
the corresponding inertia operators are given in Appendix A. There we also
indicate conditions under which these operators (and the corresponding
metrics) are positive definite. We end the paper by concluding remarks
(Section 7) in which we discuss some details of the paper and possible
perspectives.

Fix some notations. We write $P\colon G\to G/K$, $\pi\colon T^{*}M\to M$, and
$p\colon M\to M/G$ for the canonical projections.

All objects in this paper are real analytic or complex analytic. Given a
vector bundle $E$, we write $\Gamma(E)$ for the space of sections of $E$, and
$\mathcal{E}(M)$ will stand for the space of functions on a manifold $M$ (of
the corresponding category).

##  2 Invariant Nijenhuis tensors on homogeneous spaces

######  Definition 2.1.

Let $M$ be a connected manifold. A $(1,1)$-tensor field $N\colon TM\to TM$ is
_a Nijenhuis tensor_ if its Nijenhuis torsion vanishes, i.e., for any vector
fields $X,Y\in\Gamma(TM)$:

$\displaystyle T_{N}(X,Y):=[NX,NY]-N[X,Y]_{N}=0,$

where we put

$\displaystyle[X,Y]_{N}:=[NX,Y]+[X,NY]-N[X,Y].$

Similarly, given any Lie algebra $(\mathfrak{g},[\,,\,])$, a linear operator
$n\colon\mathfrak{g}\to\mathfrak{g}$ is an _algebraic Nijenhuis operator_ if
it satisfies $T_{n}(X,Y):=[nX,nY]-n([nX,Y]+[X,nY]-n[X,Y])=0$ for all vectors
$X,Y\in\mathfrak{g}$ (cf. [9, 13]).

Let an action of a Lie group $G$ on a manifold $M$ be given.

######  Definition 2.2.

We say that a $(1,1)$-tensor field $N\colon TM\to TM$ is $G$-_invariant_ if
for any element of the Lie group $g\in G$, the tensor $N$ commutes with the
tangent map $g_{*}\colon TM\to TM$ to the diffeomorphism $g\colon M\to M$,
i.e., the following diagram is commutative

$\displaystyle\begin{matrix}TM&\stackrel{{\scriptstyle
N}}{{\longrightarrow}}&TM\\\ \downarrow g_{*}&&\downarrow g_{*}\\\
TM&\stackrel{{\scriptstyle N}}{{\longrightarrow}}&TM.\end{matrix}$

A distribution of subspaces $D_{x}\subset T_{x}M$ is $G$-_invariant_ , if for
any $g\in G$ and any $x\in M$ we have

$\displaystyle g_{*,x}(D_{x})=D_{gx}.$

The following lemmas are crucial ingredients in further considerations. Let
$G$ be any Lie group and $K$ a closed Lie subgroup of $G$ (the quotient
$M=G/K$ is then a smooth $G$-manifold).

######  Lemma 2.3.

Let $N\colon T(G/K)\to T(G/K)$ be a semisimple $(1,1)$-tensor and assume that
$N$ is $G$-invariant. Then the eigenvalues of $N$ are constant.

###### Proof.

Since the operator $N$ is $G$-invariant, it follows that its eigenfunctions
are also $G$-invariant, therefore on homogeneous space they are constant
functions. ∎

Given a real manifold $M$, we write $T^{\mathbb{C}}M$ for the complexified
tangent bundle to $M$.

######  Lemma 2.4.

Let $G/K$ be a real homogeneous space. There is a one-to-one correspondence
between $G$-invariant distributions $D\subset T^{\mathbb{C}}(G/K)$ and
subspaces $d\subset\mathfrak{g}^{\mathbb{C}}$ such that
$\mathfrak{k}^{\mathbb{C}}\subset d$ and
$\big{[}\mathfrak{k}^{\mathbb{C}},d\big{]}\subset d$ $($here
$\mathfrak{g},\mathfrak{k}\subset\mathfrak{g}$ are the Lie algebras of the Lie
groups $G,K\subset G)$. An $G$-invariant distribution $D$ is involutive if and
only if the subspace $d$ is a subalgebra in $\mathfrak{g}^{\mathbb{C}}$.
Moreover, $D$ is real, i.e., $D=\overline{D}$, where the bar stands for the
complex conjugation on $T^{\mathbb{C}}(G/K)$, if and only if so is $d$, i.e.,
$d=\bar{d}$, where the bar denotes the complex conjugation in
$\mathfrak{g}^{\mathbb{C}}$ with respect to the real form $\mathfrak{g}$.

###### Proof.

Below we let $P\colon G\to G/K$ to denote the canonical projection. An
invariant distribution $D$ on $G/K$ defines the distribution
$\widehat{D}:=P_{*}^{-1}(D)\subset T^{\mathbb{C}}G$, which by construction is
left $G$-invariant. Indeed, the invariance of $D$, $g_{*,x}(D_{x})=D_{gx}$,
implies $L_{g}$-invariance of $\widehat{D}$ as the commutativity of the
following diagram shows

$\displaystyle\begin{matrix}T_{y}G&\stackrel{{\scriptstyle
L_{g,*}|_{y}}}{{\longrightarrow}}&T_{gy}G\\\ \downarrow P_{*,y}&&\downarrow
P_{*,gy}\\\ T_{x}(G/K)&\stackrel{{\scriptstyle
g_{*,x}}}{{\longrightarrow}}&T_{gx}(G/K);\end{matrix}$

here $L_{g}$ is the left translation by $g$ and $y\in G$ is so that $P(y)=x$.

Moreover, $\widehat{D}$ is right $K$-invariant. To show this observe that,
since $P$ is a surjective submersion, in a vicinity of points $g\in G$ and
$P(g)\in G/K$ there exist local coordinate systems
$(x_{1},\dots,x_{m},y_{1},\dots,y_{k})$ and
$(x^{\prime}_{1},\dots,x^{\prime}_{m})$ respectively such that
$P(x_{1},\dots,x_{m},y_{1},\dots,y_{k})=(x^{\prime}_{1},\dots,x^{\prime}_{m})$,
$x^{\prime}_{i}=x_{i}$, $i=1,\dots,m$. Let
$X_{1}(x^{\prime}),\dots,X_{l}(x^{\prime})$,
$X_{r}(x^{\prime})=X_{r}^{i}(x_{1}^{\prime},\dots,x_{m}^{\prime})\frac{\partial}{\partial
x_{i}^{\prime}}$, be local linearly independent vector fields on $G/K$
generating the distribution $D$. Then the distribution $\widehat{D}$ is
generated by the vector fields
$\widehat{X}_{r}(x)=X_{r}^{i}(x_{1},\dots,x_{m})\frac{\partial}{\partial
x_{i}}$, $r=1,\dots,l$, and $Y_{1},\dots,Y_{k}$, where the last ones are the
fundamental vector fields of the right $K$-action. These last are tangent to
the fibers of $P$, locally can be expressed as combinations of
$\frac{\partial}{\partial y_{j}}$ and vice versa, $\frac{\partial}{\partial
y_{j}}$ can be locally expressed as combinations of $Y_{1},\dots,Y_{k}$.
Obviously, $\big{[}Y_{i},\widehat{X}_{j}\big{]}=f_{ij}^{s}Y_{s}$ for some
functions $f_{ij}^{s}$, which together with the involutivity of the system of
vector fields $\\{Y_{1},\dots,Y_{k}\\}$ gives
$\big{[}Y_{i},\widehat{D}\big{]}\subset\widehat{D}$.

Let $d:=\widehat{D}_{e}\subset T^{\mathbb{C}}_{e}G$, where $e\in G$ is the
neutral element. The left and right $K$-invariance of $\widehat{D}$ implies,
under the identification $T^{\mathbb{C}}_{e}G\cong\mathfrak{g}^{\mathbb{C}}$,
the $\operatorname{Ad}(K)$-invariance of the subspace
$d\subset\mathfrak{g}^{\mathbb{C}}$, or, on the infinitesimal level, its
$\operatorname{ad}(\mathfrak{k})$-invariance:
$\big{[}\mathfrak{k}^{\mathbb{C}},d\big{]}\subset d$.

Now if $D$ is involutive, then so is $\widehat{D}$. Indeed, the systems of
vector fields $\\{X_{j}\\}$ and, consequently,
$\big{\\{}\widehat{X}_{j}\big{\\}}$ are involutive. Hence so is the total
system of vector fields $\big{\\{}Y_{i},\widehat{X}_{j}\big{\\}}$.
Infinitesimally this can be expressed as $[d,d]\subset d$.

Vice versa, let $d\subset\mathfrak{g}^{\mathbb{C}}\cong T^{\mathbb{C}}_{e}G$
be an $\operatorname{ad}\mathfrak{k}$-invariant subspace. Define a
distribution $\widehat{D}\subset T^{\mathbb{C}}G$ by
$\widehat{D}_{g}=L_{g,*}d$. Then $\widehat{D}$ is left $G$-invariant and right
$K$-invariant and descends to a uniquely defined invariant distribution
$D\subset T^{\mathbb{C}}(G/K)$ by means of the complexified tangent map
$P_{*}^{\mathbb{C}}\colon\allowbreak T^{\mathbb{C}}G\to T^{\mathbb{C}}(G/K)$.

If $d\subset\mathfrak{g}^{\mathbb{C}}$ is a subalgebra, then clearly the
distribution $\widehat{D}$ is involutive. Moreover, from the above local
description it follows that the system of vector fields
$\big{\\{}\widehat{X}_{j}\big{\\}}$ is involutive and
$P_{*}\widehat{X}_{j}=X_{j}$, and, as a consequence, so is the system
$\\{X_{j}\\}$. Therefore $D\subset T^{\mathbb{C}}(G/K)$ is involutive.

The last assertion of the lemma is obvious. ∎

######  Lemma 2.5.

Let $D\subset T(G/K)$ be an $G$-invariant integrable distribution on $G/K$
relative to a subalgebra $\mathfrak{h}\subset\mathfrak{g}$,
$\mathfrak{h}\supset\mathfrak{k}$ $($as in Lemma 2.4 but we admit also the
complex analytic case$)$, and let $H\subset G$ be the corresponding subgroup.
Denote by $P\colon G\to G/K$ the canonical projection. Then

  1. $1)$

the leaves of the foliation tangent to $D$ are the projections with respect to
$P$ of the left cosets $gH$, $g\in G$;

  2. $2)$

given $\xi\in\mathfrak{g}$, the fundamental vector field $X_{\xi}$ of the
$G$-action on $G/K$ is tangent to the leaf $P(gH)$ if and only if
$\xi\in\operatorname{Ad}_{g}\mathfrak{h}\subset\mathfrak{g}$.

###### Proof.

Consider the integrable distribution $\widehat{D}$ built in the proof of Lemma
2.4. Then it is easy to see that the foliation tangent to $\widehat{D}$
coincides with the foliation of the left cosets $gH$, $g\in G$. Since
$P_{*}\big{(}\widehat{D}\big{)}=D$, the leaves of the corresponding foliations
are projected on each other by means of $P$, which proves item 1.

The right invariant vector field $\xi_{R}$ on $G$, $\xi_{R}|_{e}=\xi$, is
tangent to $gH$ at the point $gh\in gH$ if and only if $\xi_{R}(gh)\in
T_{gh}(gH)\Leftrightarrow R_{gh,*}(\xi)\in
T_{gh}(gH)=L_{gh,*}\mathfrak{h}\Leftrightarrow\xi\in
R_{(gh)^{-1},*}L_{gh,*}\mathfrak{h}=\operatorname{Ad}_{gh}\mathfrak{h}=\operatorname{Ad}_{g}\mathfrak{h}$
(here $\xi\in\mathfrak{g}\cong T_{e}G$). Hence $X_{\xi}=P_{*}\xi_{R}$ is
tangent to $P(gH)$ if and only if $\xi\in\operatorname{Ad}_{g}\mathfrak{h}$. ∎

######  Lemma 2.6.

Let $N\colon TM\to TM$ be a semisimple $(1,1)$-tensor with constant distinct
eigenvalues $\lambda_{1},\dots,\lambda_{s}\in\mathbb{C}$ $($or
$\lambda_{1},\dots,\lambda_{s}\in\mathbb{R})$ and let $D_{i}\subset
T^{\mathbb{C}}M$ $($or, respectively $D_{i}\subset TM)$ be the
eigendistribution corresponding to $\lambda_{i}$. Then $T_{N}=0$ if and only
if the distributions $D_{i}$ and $D_{i}+D_{j}$ are involutive for any $i$,
$j$.

###### Proof.

Assume $N$ is Nijenhuis. It is easy to see that $T_{N-\lambda I}=T_{N}=0$ for
any $\lambda\in\mathbb{C}$. In particular
$\big{[}\big{(}N^{\mathbb{C}}-\lambda_{i}I\big{)}X,\big{(}N^{\mathbb{C}}-\lambda_{i}I\big{)}Y\big{]}=\big{(}N^{\mathbb{C}}-\lambda_{i}I\big{)}[X,Y]_{N^{\mathbb{C}}-\lambda_{i}I}$
for any vector fields $X$, $Y$ and the image of
$N^{\mathbb{C}}-\lambda_{i}I\colon T^{\mathbb{C}}M\to T^{\mathbb{C}}M$ is an
integrable distribution. As a consequence,
$D_{i}=\bigcap_{k\not=i}\operatorname{im}\big{(}N^{\mathbb{C}}-\lambda_{k}I\big{)}$
and
$D_{i}+D_{j}=\bigcap_{k\not=i,j}\operatorname{im}\big{(}N^{\mathbb{C}}-\lambda_{k}I\big{)}$
are integrable.

Now, let the decomposition $T^{\mathbb{C}}M=D_{1}\oplus\cdots\oplus D_{s}$ be
such that $D_{i}+D_{j}$ are integrable for any $i$, $j$. By the bilinearity of
Nijenhuis torsion tensor it is enough to prove that $T_{N}(x,y)=0$ for
$x\in\Gamma(D_{i})$, $y\in\Gamma(D_{j})$, $1\leq i,j\leq n$:

$\displaystyle T_{N}(x,y)=[Nx,Ny]-N([Nx,y]+[x,Ny])+N^{2}[x,y]$
$\displaystyle\hphantom{T_{N}(x,y)}{}=\lambda_{i}\lambda_{j}([x,y]_{i}+[x,y]_{j})-N(\lambda_{i}([x,y]_{i}+[x,y]_{j})$
$\displaystyle\hphantom{T_{N}(x,y)=}{}+\lambda_{j}([x,y]_{i}+[x,y]_{j}))+N(\lambda_{i}[x,y]_{i}+\lambda_{j}[x,y]_{j})$
$\displaystyle\hphantom{T_{N}(x,y)}{}=\lambda_{i}\lambda_{j}([x,y]_{i}+[x,y]_{j})-\big{(}\lambda_{i}^{2}[x,y]_{i}+\lambda_{i}\lambda_{j}[x,y]_{j}+\lambda_{i}\lambda_{j}[x,y]_{i}+\lambda_{j}^{2}[x,y]_{j}\big{)}$
$\displaystyle\hphantom{T_{N}(x,y)=}{}+(\lambda_{i}^{2}[x,y]_{i}+\lambda_{j}^{2}[x,y]_{j})=0$

(here we denote by $[x,y]_{i}$ the $i$-th component of the element $[x,y]$
with respect to the decomposition above). The proof in the case of real
eigenvalues is analogous. ∎

Let $G$ be any Lie group and $\mathfrak{g}=\operatorname{Lie}(G)$ its Lie
algebra, $K$ a closed Lie subgroup of $G$ and
$\mathfrak{k}=\operatorname{Lie}(K)$.

######  Theorem 2.7.

There is a one-to-one correspondence between

  * $(i)$

$G-$invariant semisimple Nijenhuis $(1,1)$-tensors $N\colon T(G/K)\to T(G/K)$
with the spectrum $\\{\lambda_{1},\dots,\lambda_{s}\\}$, where $\lambda_{i}$
are distinct, $\lambda_{1},\dots,\lambda_{2p}\in\mathbb{C}$,
$\lambda_{i}=\overline{\lambda_{i+p}}$ for $i=1,\dots,p$ and
$\lambda_{2p+1},\dots,\lambda_{s}\in\mathbb{R}$,

and

  * $(ii)$

decompositions
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{g}_{1}+\dots+\mathfrak{g}_{s}$ of
$\mathfrak{g}^{\mathbb{C}}$ to the sum of subspaces such that:

    1. $1)$

$\forall_{i,j\in\\{1,\dots,s\\},i\not=j}$
$\mathfrak{g}_{i}\cap\mathfrak{g}_{j}=\mathfrak{k}^{\mathbb{C}}$;

    2. $2)$

the induced decomposition of the factor space
$\mathfrak{g}^{\mathbb{C}}/\mathfrak{k}^{\mathbb{C}}$ is direct:
$\mathfrak{g}^{\mathbb{C}}/\mathfrak{k}^{\mathbb{C}}=\big{(}\mathfrak{g}_{1}/\mathfrak{k}^{\mathbb{C}}\big{)}\oplus\cdots\oplus\big{(}\mathfrak{g}_{s}/\mathfrak{k}^{\mathbb{C}}\big{)}$;

    3. $3)$

$\forall_{i,j\in\\{1,\dots,s\\}}$ $\mathfrak{g}_{i}+\mathfrak{g}_{j}$ are Lie
subalgebras in $\mathfrak{g}^{\mathbb{C}}$;

    4. $4)$

$\mathfrak{g}_{i}=\overline{\mathfrak{g}_{i+p}}$ for $i=1,\dots,p$ and
$\mathfrak{g}_{j}=\overline{\mathfrak{g}_{j}}$ for $j=2p+1,\dots,s$.

The decomposition $(ii)$ induces the decomposition
$T^{\mathbb{C}}(G/K)=D_{1}\oplus\cdots\oplus D_{s}$ to involutive subbundles,
the corresponding $(1,1)$-tensor $N$ is then given by
$N|_{D_{i}}=\lambda_{i}\operatorname{Id}_{D_{i}}$ and, vice versa, given $N$
as in $(i)$ one constructs the decomposition $(ii)$ by the decomposition
$T^{\mathbb{C}}(G/K)=D_{1}\oplus\cdots\oplus D_{s}$ of $T^{\mathbb{C}}(G/K)$
to the eigendistributions of $N$.

###### Proof.

Let $N$ be an $G$-invariant semisimple Nijenhuis $(1,1)$-tensor on $G/K$ with
the spectrum
$\\{\lambda_{1},\dots,\lambda_{s};\,\lambda_{i}\in\mathbb{C},\,\lambda_{i}\neq\lambda_{j},\text{
for }i\neq j\\}$. From Lemma 2.6 it follows that there is a decomposition
$T^{\mathbb{C}}(G/K)=D_{1}\oplus\cdots\oplus D_{s}$ into integrable
distributions, which, as the eigenspaces of an $G$-invariant tensor, are also
$G$-invariant. By Lemma 2.4 there is a one-to-one correspondence between
$G$-invariant distributions $D_{i}$ and subalgebras $\mathfrak{g}_{i}$
containing $\mathfrak{k}^{\mathbb{C}}$, hence there is a decomposition of
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{g}_{1}+\cdots+\mathfrak{g}_{s}$, such
that $\mathfrak{g}_{i}\cap\mathfrak{g}_{j}=\mathfrak{k}^{\mathbb{C}}$ for any
$i\not=j$. Applying Lemma 2.4 to the sum of distributions $D_{i}+D_{j}$ we see
that it is involutive if and only if $\mathfrak{g}_{i}+\mathfrak{g}_{j}$ is a
subalgebra.

Item 3 follows from the last assertion of Lemma 2.4 and from the obvious fact
that $D_{i}=\overline{D_{i+p}}$ for $i=1,\dots,p$ and $D_{j}=\overline{D_{j}}$
for $j=2p+1,\dots,s$.

The proof in reverse direction follows the same argumentation with the use of
the equivalences in lemmas cited. ∎

Below we present some examples for which decompositions of Lie algebras
mentioned in Theorem 2.7 are given explicitly.

First series of examples come from semisimple algebraic Nijenhuis operators
$n\colon\mathfrak{g}\to\mathfrak{g}$, which are
$\operatorname{ad}\mathfrak{k}$-invariant for some Lie subalgebra
$\mathfrak{k}\subset\mathfrak{g}$, i.e.,
$n\circ\operatorname{ad}k=\operatorname{ad}k\circ n$ for all
${k\in\mathfrak{k}}$. Then by $\operatorname{ad}\mathfrak{k}$-invariance we
can extend it to an invariant Nijenhuis $(1,1)$-tensor $N$ on $G/K$.

In the literature the following two classes of algebraic Nijenhuis operators
are widely known [9, 13, 27]:111The is one more class defined on the full
matrix algebra by $nX=AXB+BAX$, where $A^{2}=B^{2}=I$ [19]. For some
particular cases of the matrices $A$ and $B$ the corresponding operator is
semisimple. However these cases are beyond the scope of this paper. first is
related to a direct decomposition of the algebra $\mathfrak{g}$ to two
subalgebras, second is related to the operator of left multiplication on the
full matrix algebra. Below we consider particular cases of these two classes.

######  Example 2.8.

Let $\mathfrak{g}$ be a semisimple Lie algebra with the root system $R$ with
respect to a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$. Let
$\mathfrak{g}=\mathfrak{h}+\sum\limits_{\alpha\in R}\mathfrak{g}_{\alpha}$ be
the corresponding root decomposition. Choose $R^{+}$ and $R^{-}$ to be sets of
positive and negative roots and let $S\subset\Pi$ be any subset of the set of
positive simple roots. We denote by $[S]$ the set of positive roots generated
by $S$. Consider the decomposition
$\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{p}^{\perp}$, where
$\mathfrak{p}:=\mathfrak{h}+\sum\limits_{\alpha\in
R^{-}}\mathfrak{g}_{\alpha}+\sum\limits_{\alpha\in[S]}\mathfrak{g}_{\alpha}$
is the corresponding parabolic subalgebra and
$\mathfrak{p}^{\perp}=\sum\limits_{\alpha\in
R^{+}\setminus[S]}\mathfrak{g}_{\alpha}$ (the orthogonal complement with
respect to Killing form). Then $\mathfrak{p}^{\perp}$ is obviously a
subalgebra too. The operator $n\colon\mathfrak{g}\to\mathfrak{g}$ defined by
$n|_{\mathfrak{g}_{1}}=\lambda_{1}\operatorname{Id}_{\mathfrak{g}_{1}}$,
$n|_{\mathfrak{g}_{2}}=\lambda_{2}\operatorname{Id}|_{\mathfrak{g}_{2}}$ with
$\mathfrak{g}_{1}=\mathfrak{p}$ and $\mathfrak{g}_{2}=\mathfrak{p}^{\bot}$ and
with arbitrary $\lambda_{1}$, $\lambda_{2}$ is algebraic Nijenhuis (cf. [9,
24]).

One may take $\mathfrak{k}=\mathfrak{p}\cap\mathfrak{p}^{\text{opposite}}$,
where
$\mathfrak{p}^{\text{opposite}}=\mathfrak{h}+\sum\limits_{\alpha\in-[S]\subset
R^{-}}\mathfrak{g}_{\alpha}+\sum\limits_{\alpha\in
R^{+}}\mathfrak{g}_{\alpha}$. Then the operator $n$ will be
$\operatorname{ad}\mathfrak{k}$-invariant and will generate an $G$-invariant
Nijenhuis $(1,1)$-tensor on $G/K$, where $G$, $K\subset G$ are the
corresponding Lie groups. The decomposition of Theorem 2.7 looks as follows:
$\mathfrak{g}_{1}:=\mathfrak{p},\mathfrak{g}_{2}:=\mathfrak{p}^{\text{opposite}}$.
An instance of such a situation for $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{R})$
can be schematically presented as

$\displaystyle\mathfrak{p}=\begin{bmatrix}*&0&0\\\ *&*&*\\\
*&*&*\end{bmatrix},\qquad\mathfrak{p}^{\perp}=\begin{bmatrix}0&*&*\\\ 0&0&0\\\
0&0&0\end{bmatrix},\qquad\mathfrak{p}^{\text{opposite}}=\begin{bmatrix}*&*&*\\\
0&*&*\\\ 0&*&*\end{bmatrix},\qquad\mathfrak{k}=\begin{bmatrix}*&0&0\\\
0&*&*\\\ 0&*&*\end{bmatrix},$

where the corresponding set $S$ consists of the sole root $e_{2}-e_{3}$,
$e_{i}(H)$ being the $i$-th diagonal element of $H\in\mathfrak{h}$.

######  Example 2.9.

Let $\mathfrak{g}=\mathfrak{gl}(n,\mathbb{K})$,
$\mathbb{K}=\mathbb{R},\mathbb{C}$, and consider $n=L_{A}$, the operator of
left multiplication by a matrix $A\in\mathfrak{g}$. Then it is easy to see
that $n$ is an algebraic Nijenhuis operator. Taking
$A=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$,
$\lambda_{i}\not=\lambda_{j}$, $i\not=j$, we get a semisimple operator, whose
eigenspaces $\ker(n-\lambda_{i}\operatorname{Id})$ consist of matrices with
the only nonzero $i$-th row. Obviously, $n$ is
$\operatorname{ad}\mathfrak{k}$-invariant for $\mathfrak{k}=Z(A)$, the
centralizer of $A$, which coincides with the subalgebra of diagonal matrices.
The decomposition of Theorem 2.7 is
$\mathfrak{g}=\sum\limits_{i=1}^{n}\mathfrak{g}_{i}$, where
$\mathfrak{g}_{i}=\ker(n-\lambda_{i}\operatorname{Id})+\mathfrak{k}$ consists
of the matrices having non zero elements at most on the diagonal and $i$-th
row.

The generalization to the case when multiplicities in the spectrum of $A$ are
admitted is straightforward. This example has also an obvious generalization
to the case $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{K})$.

Our next example is quite classical, as this is the complex structure operator
on the adjoint orbits of the compact Lie groups which was intensively studied
in the literature. We adapt the description of this operator to our notations.
An alternative description can be found in [1, Chapter 8.B].

######  Example 2.10.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra,
$\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra,
$\mathfrak{g}=\mathfrak{h}+\sum\limits_{\alpha\in R}\mathfrak{g}_{\alpha}$ the
corresponding root grading. For any $\alpha\in R$ choose
$E_{\alpha}\in\mathfrak{g}_{\alpha}$ such that $\langle
E_{\alpha},E_{-\alpha}\rangle=1$ and put
$H_{\alpha}:=[E_{\alpha},E_{-\alpha}]$. Then
$\mathfrak{u}=\sum\limits_{\alpha\in R^{+}}\mathbb{R}({\rm
i}H_{\alpha})+\sum\limits_{\alpha\in
R^{+}}\mathbb{R}(E_{\alpha}-E_{-\alpha})+\sum\limits_{\alpha\in
R^{+}}\mathbb{R}({\rm i}(E_{\alpha}+E_{-\alpha}))\subset\mathfrak{g}$, where
$R^{+}\subset R$ is a subset of positive roots, is the compact real form of
$\mathfrak{g}$ [11, Theorem 6.3, Chapter III]. By [10, Theorem 1.3, Chapter 6]
the centralizer $Z_{\mathfrak{u}}(a)$ of any element $a\in\mathfrak{u}$ (which
is necessarily semisimple) is of the form
$Z_{\mathfrak{u}}(a)=\sum\limits_{\alpha\in R^{+}}\mathbb{R}({\rm
i}H_{\alpha})+\sum\limits_{\alpha\in[S]}\mathbb{R}(E_{\alpha}-E_{-\alpha})+\sum\limits_{\alpha\in[S]}\mathbb{R}({\rm
i}(E_{\alpha}+E_{-\alpha}))$, where $S\subset R^{+}$ is a subset of the set of
simple positive roots (cf. Example 2.8). Consider the operator
$j\colon\mathfrak{u}^{\perp}\to\mathfrak{u}^{\perp}$, where
$\mathfrak{u}^{\perp}:=\sum\limits_{\alpha\in
R^{+}\setminus[S]}\mathbb{R}(E_{\alpha}-E_{-\alpha})+\sum\limits_{\alpha\in
R^{+}\setminus[S]}\mathbb{R}({\rm i}(E_{\alpha}+E_{-\alpha}))$, given by
$j(E_{\alpha}-E_{-\alpha})={\rm i}(E_{\alpha}+E_{-\alpha})$, $j({\rm
i}(E_{\alpha}+E_{-\alpha}))=-(E_{\alpha}-E_{-\alpha})$. Note that
$j^{\mathbb{C}}(E_{\alpha})={\rm i}E_{\alpha}$,
$j^{\mathbb{C}}(E_{-\alpha})=-{\rm i}E_{-\alpha}$. The eigenspaces
$\mathfrak{g}_{1}^{\prime}:=\sum\limits_{\alpha\in
R^{+}\setminus[S]}\mathbb{C}(E_{\alpha})$ and
$\mathfrak{g}_{2}^{\prime}:=\sum\limits_{\alpha\in
R^{+}\setminus[S]}\mathbb{C}(E_{-\alpha})$ are subalgebras as well as the
subspaces
$\mathfrak{g}_{i}:=\mathfrak{g}_{i}^{\prime}\oplus\mathfrak{k}^{\mathbb{C}}$,
$\mathfrak{k}=Z_{\mathfrak{u}}(a)$. Hence by Theorem 2.7 the operator $j$
induces an invariant integrable almost complex structure on $U/K$, where
$U,K\subset U$ are the Lie groups corresponding to the Lie algebras
$\mathfrak{u}$, $\mathfrak{k}$. We conclude that, although this operator is
not arising from an algebraic Nijenhuis operator, the corresponding
decomposition in fact coincides with that from Example 2.8).

Now we come to a series of examples of different nature.222By this we mean
that they are not necessarily related with an
$\operatorname{ad}\mathfrak{k}$-invariant algebraic Nijenhuis operator on the
Lie algebra $\mathfrak{g}$. For instance, in Example 2.11 there are two ways
to build a compatible with the decomposition
$\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ algebraic Nijenhuis operator
$n\colon\mathfrak{g}\to\mathfrak{g}$:
$n|_{\mathfrak{g}_{1}}=\lambda_{1}\operatorname{Id}_{\mathfrak{g}_{1}}$,
$n|_{\mathfrak{g}_{2}^{\prime}}=\lambda_{2}\operatorname{Id}_{\mathfrak{g}_{2}^{\prime}}$
with $\mathfrak{g}_{2}^{\prime}=\mathfrak{g}_{1}^{\bot}$, or
$n|_{\mathfrak{g}_{2}}=\beta_{2}\operatorname{Id}_{\mathfrak{g}_{2}}$,
$n|_{\mathfrak{g}_{1}^{\prime}}=\beta_{1}\operatorname{Id}_{\mathfrak{g}_{1}^{\prime}}$
with $\mathfrak{g}_{1}^{\prime}=\mathfrak{g}_{2}^{\bot}$. However, in both the
cases the operator in general will not be
$\operatorname{ad}\mathfrak{k}$-invariant. Concerning the decompositions from
the Onishchik list, see Example 2.12, it seems that it is even impossible to
build a compatible Nijenhuis operator for some of them. The decomposition of
Theorem 2.7 will still consist of two components which now need not be
symmetric with respect to the involution interchanging $\mathfrak{g}_{\alpha}$
and $\mathfrak{g}_{-\alpha}$. In other words, any decomposition
$\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ of a Lie algebra
$\mathfrak{g}$ to two subalgebras can be taken into consideration (with
$\mathfrak{k}=\mathfrak{g}_{1}\cap\mathfrak{g}_{2}$). One of possible natural
generalizations of Example 2.8 is considering two “nonsymmetric” parabolic
subalgebras. Their intersection is the so-called _seaweed subalgebra_.

######  Example 2.11.

Let $\mathfrak{g}$ be a semisimple Lie algebra with the root system $R$ with
respect to a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$. Let
$\mathfrak{g}=\mathfrak{h}+\sum\limits_{\alpha\in R}\mathfrak{g}_{\alpha}$ be
the corresponding root decomposition. Choose $R^{+}$ and $R^{-}$ to be sets of
positive and negative roots and let $S,S^{\prime}\subset\Pi$ be any subsets of
the set of positive simple roots. Consider the parabolic subalgebras
$\mathfrak{g}_{1}=\mathfrak{h}+\sum\limits_{\alpha\in
R^{-}}\mathfrak{g}_{\alpha}+\sum\limits_{\alpha\in[S]}\mathfrak{g}_{\alpha}$
and $\mathfrak{g}_{2}=\mathfrak{h}+\sum\limits_{\alpha\in
R^{+}}\mathfrak{g}_{\alpha}+\sum\limits_{\alpha\in-[S^{\prime}]}\mathfrak{g}_{\alpha}$.
An instance of such a situation for $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{R})$
can be schematically presented as

$\displaystyle\mathfrak{g}_{1}=\begin{bmatrix}*&0&0\\\ *&*&*\\\
*&*&*\end{bmatrix},\qquad\mathfrak{g}_{2}=\begin{bmatrix}*&*&*\\\ *&*&*\\\
0&0&*\end{bmatrix},\qquad\mathfrak{k}=\begin{bmatrix}*&0&0\\\ *&*&*\\\
0&0&*\end{bmatrix},$

where the corresponding sets $S$ and $S^{\prime}$ consist of the roots
$e_{2}-e_{3}$ and $e_{1}-e_{2}$ respectively, cf. Example 2.8.

In [20] A.L. Onishchik classified all decompositions
$\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ for compact simple Lie
algebras $\mathfrak{g}$ and we list them below. (In [21] he also gave a
classification of decompositions of reductive Lie algebras $\mathfrak{g}$ to
two subalgebras reductive in $\mathfrak{g}$, but we omit this case here.)

######  Example 2.12.

Let $\mathfrak{g}$ be a compact simple Lie algebra. The following table
presents all pairs of subalgebras $(\mathfrak{g}_{1},\mathfrak{g}_{2})$ such
that $\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ together with possible
embeddings $i^{\prime}\colon\mathfrak{g}_{1}\rightarrow\mathfrak{g}$,
$i^{\prime\prime}\colon\mathfrak{g}_{2}\rightarrow\mathfrak{g}$ up to
conjugations. Below $N$ stands for the trivial representation, $\varphi_{i}$
for the specific representation mentioned in [20] and $T$ for the
1-dimensional Lie algebra.

$\mathfrak{g}$ | $\mathfrak{g}_{1}$ | $i^{\prime}$ | $\mathfrak{g}_{2}$ | $i^{\prime\prime}$ | $\mathfrak{k}=\mathfrak{g}_{1}\cap\mathfrak{g}_{2}$ | restrictions
---|---|---|---|---|---|---
$A_{2n-1}$ | $C_{n}$ | $\varphi_{1}$ | $A_{2n-2}$ | $\varphi_{1}+N$ | $C_{n-1}$ | $n>1$
$A_{2n-1}$ | $C_{n}$ | $\varphi_{1}$ | $A_{2n-2}\oplus T$ | $\varphi_{1}+N$ | $C_{n-1}\oplus T$ | $n>1$
$B_{3}$ | $G_{2}$ | $\varphi_{2}$ | $B_{2}$ | $\varphi_{1}+2N$ | $A_{1}$ |
$B_{3}$ | $G_{2}$ | $\varphi_{2}$ | $B_{2}\oplus T$ | $\varphi_{1}+2N$ | $A_{1}\oplus T$ |
$B_{3}$ | $G_{2}$ | $\varphi_{2}$ | $D_{3}$ | $\varphi_{1}+N$ | $A_{2}$ |
$D_{n+1}$ | $B_{n}$ | $\varphi_{1}+N$ | $A_{n}$ | $\varphi_{1}+\varphi_{n}$ | $A_{n-1}$ | $n>2$
$D_{n+1}$ | $B_{n}$ | $\varphi_{1}+N$ | $A_{n}\oplus T$ | $\varphi_{1}+\varphi_{n}$ | $A_{n-1}\oplus T$ | $n>2$
$D_{2n}$ | $B_{2n-1}$ | $\varphi_{1}+N$ | $C_{n}$ | $\varphi_{1}+\varphi_{1}$ | $C_{n-1}$ | $n>1$
$D_{2n}$ | $B_{2n-1}$ | $\varphi_{1}+N$ | $C_{n}\oplus T$ | $\varphi_{1}+\varphi_{1}$ | $C_{n-1}\oplus T$ | $n>1$
$D_{2n}$ | $B_{2n-1}$ | $\varphi_{1}+N$ | $C_{n}\oplus A_{1}$ | $\varphi_{1}+\varphi_{1}$ | $C_{n-1}\oplus A_{1}$ | $n>1$
$D_{8}$ | $B_{7}$ | $\varphi_{1}+N$ | $B_{4}$ | $\varphi_{4}$ | $B_{3}$ |
$D_{4}$ | $B_{3}$ | $\varphi_{3}$ | $B_{2}$ | $\varphi_{1}+3N$ | $A_{1}$ |
$D_{4}$ | $B_{3}$ | $\varphi_{3}$ | $B_{2}\oplus T$ | $\varphi_{1}+3N$ | $A_{1}\oplus T$ |
$D_{4}$ | $B_{3}$ | $\varphi_{3}$ | $B_{2}\oplus A_{1}$ | $\varphi_{1}+3N$ | $A_{1}\oplus A_{1}$ |
$D_{4}$ | $B_{3}$ | $\varphi_{3}$ | $D_{3}$ | $\varphi_{1}+2N$ | $A_{2}$ |
$D_{4}$ | $B_{3}$ | $\varphi_{3}$ | $D_{3}\oplus T$ | $\varphi_{1}+2N$ | $A_{2}\oplus T$ |
$D_{4}$ | $B_{3}$ | $\varphi_{3}$ | $B_{3}$ | $\varphi_{1}+N$ | $G_{2}$ |

##  3 Bi-Poisson structures, kroneckerity, $\boldsymbol{G}$-invariance,
and complete families of functions in involution

If $M$ is a real or complex analytic manifold, ${\mathcal{E}}(M)$ will stand
for the space of analytic functions on $M$ in the corresponding category. We
will write $\mathbb{K}$ for the corresponding ground field. We recall basic
definitions and concepts related to bi-Poisson structures, their kroneckerity
and invariance (cf. [17]).

We will say that some functions from the set ${\mathcal{E}}(M)$ are
independent at a point $x\in M$ if their differentials are independent at $x$.
For any subset ${\mathcal{F}}\subset{\mathcal{E}}(M)$ denote by
$\operatorname{ddim}_{x}{\mathcal{F}}$ the maximal number of independent
functions from the set ${\mathcal{F}}$ at a point $x\in M$. Put
$\operatorname{ddim}{\mathcal{F}}:=\max\limits_{x\in
M}\operatorname{ddim}_{x}{\mathcal{F}}$.

######  Definition 3.1.

A bivector field (bivector for short) is a skew-symmetric morphism $\Pi\colon
T^{*}M\to TM$. It is called _Poisson_ if the operation
$\\{f,g\\}^{\Pi}:=\Pi(f)g$ is a Lie algebra on $\mathcal{E}(M)$ (here
$\Pi(f):=\Pi(df)$ stands for the _Hamiltonian vector field_ corresponding to
the function $f$). Define $\operatorname{rank}\Pi:=\max\limits_{x\in
M}\dim\Pi(T_{x}^{*}M)$ and $R^{\Pi}:=\\{x\in
M\,|\,\dim\Pi(T_{x}^{*}M)=\operatorname{rank}\Pi\\}$. A function
$f\in\mathcal{E}(U)$ over an open set $U\subset M$ is called a _Casimir
function_ for a Poisson bivector $\Pi$ if $\Pi(f)\equiv 0$. The set of all
Casimir functions for $\Pi$ over $U$ will be denoted by $Z^{\Pi}(U)$ (note
that $Z^{\Pi}(U)$ is the centre of the Lie algebra
$\big{(}\mathcal{E}(U),\\{\,,\,\\}^{\Pi}\big{)}$).

Given a poisson bivector $\Pi$, the generalized distribution
$\Pi(T^{*}M)\subset TM$ (called the _characteristic distribution_ of $\Pi$) is
integrable, the restrictions of $\Pi$ to its leaves are correctly defined
nondegenerate Poisson bivectors and the leaves are called the _symplectic
leaves_ of $\Pi$. In particular the set $R^{\Pi}$ is the union of all the
symplectic leaves of maximal dimension.

######  Definition 3.2.

A set $I\subset\mathcal{E}(U)$ of functions over $U\subset M$ is called
involutive with respect to a Poisson bivector $\Pi$ if $\\{f,g\\}^{\Pi}=0$ for
any $f,g\in I$ (we also say that such functions are _in involution_). An
involutive set is _complete_ with respect to $\Pi$ if there exist
$f_{1},\dots,f_{s}\in I$, where $s=\dim M-\frac{1}{2}\operatorname{rank}\Pi$,
independent at any point from some open dense set $U_{0}\subset U$.

If $I$ is a complete involutive set over $U$, then among $f_{i}$ there are
$\dim M-\operatorname{rank}\Pi$ Casimir functions of $\Pi$. Any such set $I$
is a set of functions constant along a lagrangian foliation of dimension
$\frac{1}{2}\operatorname{rank}\Pi$ defined on an open dense set in any
symplectic leaf of maximal dimension.

######  Definition 3.3.

Two Poisson structures $\Pi_{1}$ and $\Pi_{2}$ on a manifold $M$ is called
_compatible_ if $\Pi_{t}:=t_{1}\Pi_{1}+t_{2}\Pi_{2}$ is a Poisson bivector for
any $t=(t_{1},t_{2})\in\mathbb{K}^{2}$; the whole 2-dimensional family of
Poisson bivectors (in case $\Pi_{1,2}$ are linearly independent)
$\\{\Pi_{t}\\}_{t\in\mathbb{R}^{2}}$ is called a bi-Poisson or a bi-
Hamiltonian structure.

######  Definition 3.4.

A bi-Poisson structure $\\{\Pi_{t}\\}$ on $M$ is Kronecker at a point $x\in M$
if $\operatorname{rank}_{\mathbb{C}}(t_{1}\Pi_{1}+t_{2}\Pi_{2})|_{x}$ is
constant with respect to $(t_{1},t_{2})\in\mathbb{C}^{2}\setminus\\{0\\}$ (in
the real analytic case we consider $(\Pi_{j})_{x}$ as a skew-symmetric
bilinear form on the complexified cotangent space
$(T^{*}_{x}M)^{\mathbb{C}}$). We say that $\\{\Pi_{t}\\}$ is _Kronecker_ if it
is Kronecker at any point of some open dense subset in $M$.

Importance of this notion is explained by the following

######  Theorem 3.5.

Let $\\{\Pi_{t}\\}$ be a Kronecker bi-Poisson structure on $M$. Then for any
open set $U\subset M$ such that $\operatorname{ddim}Z^{\Pi_{t}}(U)=\dim
M-\operatorname{rank}\Pi_{t}$ for any $t$ the set

$\displaystyle
Z^{\\{\Pi_{t}\\}}(U):=\operatorname{Span}\bigg{(}\bigcup_{t\not=0}Z^{\Pi_{t}}(U)\bigg{)}$

is a complete involutive set of functions with respect to any $\Pi_{t}\not=0$
$($see Definition 3.2).

The reader is referred to [2] for the proof. The condition that
$\operatorname{ddim}Z^{\Pi_{t}}(U)=\dim M-\operatorname{rank}\Pi_{t}$ for any
$t$ is always satisfied for any sufficiently small open set $U$ and, in many
cases also for an open and dense set in $M$.

######  Remark 3.6.

Recall that a real analytic submanifold $M$, $\dim_{\mathbb{R}}M=n$, in a
complex manifold ${M}^{c}$, $\dim_{\mathbb{C}}{M}^{c}=n$, is called _maximal
totally real_ if in a neighbourhood of any point in $M$ there exists a
holomorphic coordinate system $z=(z_{1},\dots,z_{n})$, $z_{j}=x_{j}+{\rm
i}y_{j}$, such that $M$ locally is given by the equations $y_{j}=0$. We say
that ${M}^{c}$ is a _complexification_ of $M$ and $M$ is _real form_ of
${M}^{c}$. The holomorphic coordinates as above will be called _adapted_ to
$M$. A complexification exists for any real analytic $M$ [33].

Let $M$ be a real analytic manifold and ${M}^{c}$ its complexification. Any
real analytic tensor $T$ defined on $M$ can be uniquely extended to a
holomorphic tensor ${T}^{c}$ defined in a vicinity of $M$ in ${M}^{c}$ by
extending its coefficients to holomorphic functions and substituting
$\frac{\partial}{\partial x_{j}}$ and ${\rm d}x_{j}$ by
$\frac{\partial}{\partial z_{j}}$ and ${\rm d}z_{j}$ respectively in the
adapted systems of coordinates. Vice versa, if a holomorphic tensor ${T}^{c}$
is given on ${M}^{c}$ such that in the adapted coordinates its coefficients
restricted to $M$ are real, then it is the holomorphic extension of some real
analytic tensor $T$ on $M$. Obviously, if
$\\{\Pi_{t}\\}_{t\in\mathbb{R}^{2}}$, $\Pi_{t}=t_{1}\Pi_{1}+t_{2}\Pi_{2}$, is
a real analytic bi-Poisson structure on $M$, then it is Kronecker at a point
$m\in M$ if and only if so is its holomorphic extension
$\\{{\Pi}^{c}\\}_{t\in\mathbb{C}^{2}}$,
${\Pi}^{c}_{t}=t_{1}{\Pi}^{c}_{1}+t_{2}{\Pi}^{c}_{2}$.

Let $G$ be a Lie group acting on a manifold $M$. Denote by
$\mathcal{E}^{G}(M)$ the space of all $G$-invariant functions from the set
${\mathcal{E}}(M)$. We say that a bi-Poisson structure $\\{\Pi_{t}\\}$ is
$G$-_invariant_ if so is each bi-vector $\Pi_{t}$, $t\in\mathbb{R}^{2}$.

Now we assume that the action of $G$ on $M$ is proper, as for instance is in
the case of any smooth action of a compact Lie group. Fix some isotropy
subgroup $H\subset G$ determining the principal orbit type. In this case the
subset

$\displaystyle M_{H}=\big{\\{}x\in M\colon G_{x}=gHg^{-1}\ \mathrm{for\ some}\
g\in G\big{\\}}$

of $M$, consisting of all orbits $G\cdot x$ in $M$ isomorphic to $G/H$, is an
open and dense subset of $M$ (see [8, Section 2.8 and Theorem 2.8.5]). It is
well known that the orbit space $M^{\prime}_{H}:=M_{H}/G$ is a smooth
manifold. There is a natural identification of spaces $\mathcal{E}^{G}(M_{H})$
and $p^{*}\mathcal{E}(M^{\prime}_{H})$, where $p\colon M_{H}\to
M^{\prime}_{H}=M_{H}/G$ is the canonical projection, in particular
$\operatorname{ddim}_{x}\mathcal{E}^{G}(M)=\operatorname{ddim}\mathcal{E}^{G}(M)$
for $x\in M_{H}$. Moreover, if an $G$-invariant bi-Hamiltonian structure
$\\{\Pi_{t}\\}$ is given on $M$, all the Poisson bivectors $\Pi_{t}|_{M_{H}}$
are projectable with respect to $p$, i.e., there exist a correctly defined bi-
Poisson structure $\\{\Pi^{\prime}_{t}\\}$ on $M_{H}/G$ such that
$\Pi^{\prime}_{t}=p_{*}\Pi_{t}$, and the identification mentioned is a Poisson
map:

$\displaystyle
p^{*}\\{f,g\\}^{\Pi^{\prime}_{t}}=\\{p^{*}f,p^{*}g\\}^{\Pi_{t}},\qquad
f,g\in\mathcal{E}(M^{\prime}_{H}).$

Assuming that the reduced bi-Poisson structure $\\{\Pi^{\prime}_{t}\\}$ is
Kronecker, by Theorem 3.5 for a sufficiently small $U\subset M^{\prime}_{H}$
we get an involutive family of functions

$\displaystyle
Z^{\\{\Pi^{\prime}_{t}\\}}(U):=\operatorname{Span}\bigg{(}\bigcup_{t\not=0}Z^{\Pi^{\prime}_{t}}(U)\bigg{)},$

which is complete with respect to any Poisson structure $\Pi^{\prime}_{t}$. In
some cases the corresponding set of functions
$p^{*}Z^{\\{\Pi^{\prime}_{t}\\}}(U)$ on $p^{-1}(U)\subset M$ which by the
considerations above is involutive with respect to any Poisson bivector
$\Pi_{t}$ can be extended to a complete involutive set of functions. One such
situation is touched in Theorem 3.7 below. This theorem also describes a
method of proving the kroneckerity of the bi-Poisson structure
$\\{\Pi^{\prime}_{t}\\}$ reducing the problem to the calculation of rank of a
finite number of the reduced Poisson structures, which was used in [17, 23].

######  Theorem 3.7.

Retaining the assumptions above assume moreover that

  1. $(a)$

the associated action $\rho\colon\mathfrak{g}\to\Gamma(TM)$ of the Lie algebra
$\mathfrak{g}$ of $G$ on $M$ can be extended to a holomorphic action
$\rho^{c}\colon\mathfrak{g}^{\mathbb{C}}\to\Gamma(T{M}^{c})$ of the
complexification $\mathfrak{g}^{\mathbb{C}}$ of the Lie algebra $\mathfrak{g}$
on some complexification ${M}^{c}$ of $M$ on which a holomorphic extension
$\\{{\Pi}^{c}_{t}\\}$ of $\\{\Pi_{t}\\}$ is defined $($see Remark 3.6) and
$\\{{\Pi}^{c}_{t}\\}$ is $\mathfrak{g}^{\mathbb{C}}$-invariant, i.e., the Lie
derivative $\mathcal{L}_{\rho^{c}(\xi)}{\Pi}^{c}_{t}$ is equal to zero for any
$t\in\mathbb{C}^{2}$ and any $\xi\in\mathfrak{g}^{\mathbb{C}}$; here
$\Gamma(TM)$ stands for the space of real analytic vector fields on $M$ and
$\Gamma(T{M}^{c})$ for the space of holomorphic vector fields on ${M}^{c}$;

  2. $(b)$

the action of $G$ on $M$ is generically locally free, i.e., the stabilizer $H$
corresponding to the principal orbit type is finite; in particular, a generic
stabilizer algebra of the actions $\rho$ and $\rho^{c}$ is trivial;

  3. $(c)$

$\operatorname{codim}\operatorname{Sing}\mathfrak{g}^{*}\geq 2$, where
$\operatorname{Sing}\mathfrak{g}^{*}\subset\mathfrak{g}^{*}$ is the union of
the coadjoint orbits of nonmaximal dimension, i.e.,
$\operatorname{Sing}\mathfrak{g}^{*}=\mathfrak{g}^{*}\setminus
R^{\Pi_{\mathfrak{g}^{*}}}$ for the Lie–Poisson structure
$\Pi_{\mathfrak{g}^{*}}$ on $\mathfrak{g}^{*}$;

  4. $(d)$

for almost all $t$ the bivector ${\Pi}^{c}_{t}$ is nondegenerate and the
action $\rho^{c}$ is Hamiltonian with respect to ${\Pi}^{c}_{t}$, i.e., there
exists a set $E\subset\mathbb{C}^{2}$ being the union of a finite number of
$1$-dimensional linear subspaces $\langle t_{1}\rangle,\dots,\langle
t_{s}\rangle$, a map
$\mu^{c}_{t}\colon{M}^{c}\to\big{(}\mathfrak{g}^{\mathbb{C}}\big{)}^{*}$,
$t\in\mathbb{C}^{2}\setminus E$ $($the so-called momentum map$)$, such that
$\operatorname{rank}{\Pi}^{c}_{t}=\dim M$, $t\in\mathbb{C}^{2}\setminus E$,
and any fundamental vector field $\rho^{c}(\xi)$,
$\xi\in\mathfrak{g}^{\mathbb{C}}$, of this action is a Hamiltonian vector
field ${\Pi}^{c}_{t}\big{(}H_{t}^{\xi}\big{)}$ with the Hamiltonian function
$H_{t}^{\xi}(x)=\langle\mu^{c}_{t}(x),\xi\rangle$ and $\mu^{c}_{t}$ is a
Poisson map from the Poisson manifold $({M}^{c},{\Pi}^{c}_{t})$ to the
Lie–Poisson manifold
$\big{(}\big{(}\mathfrak{g}^{\mathbb{C}}\big{)}^{*},\Pi_{(\mathfrak{g}^{\mathbb{C}})^{*}}\big{)}$;

  5. $(e)$

the restriction $\mu_{t}=\mu^{c}_{t}|_{M}$, $t\in\mathbb{R}^{2}$, takes values
in $\mathfrak{g}^{*}\subset\big{(}\mathfrak{g}^{\mathbb{C}}\big{)}^{*}$; in
particular the action $\rho$ itself is Hamiltonian with respect to any
$\Pi_{t}$, $t\not\in\mathbb{R}^{2}\cap E$:
$\rho(\xi)=\Pi_{t}\big{(}H_{t}^{\xi}|_{M}\big{)}$, $\xi\in\mathfrak{g}$.

Then

  1. $1)$

the set
$U:=M_{H}\setminus\big{(}\bigcup_{t\in\mathbb{R}^{2}}\mu_{t}^{-1}(\operatorname{Sing}\mathfrak{g}^{*})\big{)}$
is an $G$-invariant open dense set in $M_{H}$;

  2. $2)$

the reduced bi-Poisson structure $\\{\Pi^{\prime}_{t}\\}$ on
$M^{\prime}_{H}=M_{H}/G$ is Kronecker at a point $x^{\prime}\in p(U)$ if and
only if

$\displaystyle\operatorname{corank}\Pi^{\prime}_{t_{i}}|_{x^{\prime}}=\operatorname{ind}\mathfrak{g},\qquad
i=1,\dots,s;$

  3. $3)$

if $\\{\Pi^{\prime}_{t}\\}$ is Kronecker and $\mathcal{F}$ stands for any
complete involutive set of polynomial functions on
$(\mathfrak{g}^{*},\Pi_{\mathfrak{g}^{*}})$ $($which exists by the Sadetov
theorem [29]), the set of functions

$\displaystyle\mathcal{I}:=p^{*}\big{(}Z^{\\{\Pi^{\prime}_{t}\\}}(M_{H}^{\prime})\big{)}\bigcup\mu_{t_{0}}^{*}\mathcal{F}$
(3.1)

is complete on $M_{H}$ with respect to any $\Pi_{t_{0}}$, $t_{0}\not\in
E\cap\mathbb{R}^{2}$;

  4. $4)$

moreover,
$p^{*}\big{(}Z^{\\{\Pi^{\prime}_{t}\\}}(p(U))\big{)}=\operatorname{Span}\big{(}\bigcup_{t\not=0}\mu_{t}^{*}\big{(}Z^{\Pi_{\mathfrak{g}^{*}}}(\mu_{t}(U))\big{)}\big{)}$.

Here $\operatorname{ind}\mathfrak{g}$, the index of the Lie algebra
$\mathfrak{g}$, is the codimension of a coadjoint orbit of maximal dimension,
i.e.,
$\operatorname{ind}\mathfrak{g}=\dim\mathfrak{g}-\operatorname{rank}\Pi_{\mathfrak{g}^{*}}$.

###### Proof.

The $G$-invariance of the set $U$ follows from the well-known fact that the
Poisson property of the moment map $\mu_{t}$ is equivalent to its
$G$-equivariance (with respect to the coadjoint action of $G$ on
$\mathfrak{g}^{*}$), which implies the $G$-invariance of
$\mu_{t}^{*}(\operatorname{Sing}\mathfrak{g}^{*})$.

The so-called “bifurcation lemma” says that for any $x\in M_{H}$ the image
$(\mu_{t})_{*}(T_{x}M_{H})\subset\mathfrak{g}^{*}$ coincides with the
annihilator in $\mathfrak{g}^{*}$ of the Lie algebra of the isotropy group
$G^{x}$ of $x$ [22, Proposition 4.5.12]. Since this algebra vanishes by
Assumption (b), $\operatorname{rank}\mu_{t}(x)=\dim\mathfrak{g}^{*}$ and the
image $\mu_{t}(M_{H})$ contains an open subset of $\mathfrak{g}^{*}$. The set
$\operatorname{Sing}\mathfrak{g}^{*}$ is algebraic and its complement in
$\mathfrak{g}^{*}$ is open and dense, hence Assumption (c) guarantees that the
set $U$ is also open and dense.

To prove item 2 observe that for any $t\not=t_{i}$, $i=1,\dots,s$ and any
$x\in M_{H}$, by the holomorphic version of the bifurcation lemma and by a
simple algebraic fact (Lemma 3.8 below)
$\operatorname{corank}(({\Pi}^{c}_{t})^{\prime})_{x}=\operatorname{corank}(\Pi_{(\mathfrak{g}^{\mathbb{C}})^{*}})_{\mu^{c}_{t}(x)}$.
Here $(({\Pi}^{c}_{t})^{\prime})_{x}$ is the restriction of the bivector
$({\Pi}^{c}_{t})_{x}$ treated as a bilinear skew-symmetric form on
$T^{*}_{x}{M}^{c}$ to the annihilator $(T_{x}\mathcal{O})^{\circ}\subset
T^{*}_{x}{M}^{c}$ of the tangent space $T_{x}\mathcal{O}$ to the
$\mathfrak{g}^{\mathbb{C}}$-orbit $\mathcal{O}$ passing through $x$ and it is
known that the space $T_{x}\mathcal{O}$ is the skew-orthogonal complement to
the tangent space through $x$ of the fiber of the moment map $\mu^{c}_{t}$.

Hence, if moreover $x\in U$, then
$\operatorname{corank}_{\mathbb{R}}(\Pi_{t}^{\prime})_{p(x)}=\operatorname{corank}_{\mathbb{C}}(({\Pi}^{c}_{t})^{\prime})_{p(x)}=\operatorname{ind}\mathfrak{g}^{\mathbb{C}}=\operatorname{ind}\mathfrak{g}$.
Therefore the reduced Poisson pencil $\\{\Pi_{t}\\}$ is Kronecker at $p(x)$ if
and only if the corank at $p(x)$ of the reductions $(\Pi_{t_{i}})^{\prime}$ of
the exceptional Poisson structures $\Pi_{t_{i}}$, $i=1,\dots,s$, is equal to
$\operatorname{ind}\mathfrak{g}$.

Item 3 follows from the well known fact that once we have a pair of Poisson
submersions $p_{1}\colon(M,\Pi)\to(M_{1},\Pi_{1})$ and
$p_{2}\colon(M,\Pi)\to(M_{2},\Pi_{2})$ with skew-orthogonal fibers with
respect to $\Pi$ and complete families of functions $\mathcal{F}_{1}$,
$\mathcal{F}_{2}$ on $(M_{1},\Pi_{1})$, $(M_{2},\Pi_{2})$ respectively, the
family $p_{1}^{*}(\mathcal{F}_{1})\cup p_{2}^{*}(\mathcal{F}_{2})$ is complete
on $(M,\Pi)$ [23, Proposition 2.22].

The last item is a consequence of another well known fact that
$p_{1}^{*}\big{(}Z^{\Pi_{1}}\big{)}=p_{2}^{*}\big{(}Z^{\Pi_{2}}\big{)}$ [23,
Corollary 2.19]. ∎

######  Lemma 3.8.

Let $V$ be a vector space over $\mathbb{K}$ and $\omega\colon V\times
V\to\mathbb{K}$ a nondegenerate skew-symmetric bilinear form. Denote by
$\Pi\colon V^{*}\times V^{*}\to\mathbb{K}$ its inverse bivector. Let
$V_{1},V_{2}\subset V$ be two vector subspaces being orthogonal complements of
each other with respect to $\omega$. Then the restrictions of $\Pi$ to the
subspaces $W_{1}:=V_{1}^{\circ}\subset V^{*}$ and $W_{2}:=V_{2}^{\circ}\subset
V^{*}$ have the same coranks.

###### Proof.

Indeed, since $W_{1}$ and $W_{2}$ are mutual orthogonal complements with
respect to $\Pi$, we have $\ker(\Pi|_{W_{1}\times W_{1}})=W_{1}\cap
W_{2}=\ker(\Pi|_{W_{2}\times W_{2}})$. ∎

##  4 Bi-Poisson structures on cotangent bundles
related to Nijenhuis $\boldsymbol{(1,1)}$-tensors

######  Definition 4.1.

Let $Q$ be a manifold and $X\in\Gamma(TQ)$ be a vector field on $Q$. Then the
formula $\widetilde{X}:=\Pi\big{(}\overline{X}\big{)}$, where
$\Pi=\omega^{-1}=\partial_{q}\wedge\partial_{p}$ is the canonical
nondegenerate Poisson structure on $T^{*}Q$ inverse to the canonical
symplectic form $\omega={\rm d}p\wedge{\rm d}q$ and $\overline{X}$ stands for
the linear function on $T^{*}Q$ corresponding to $X$, gives a vector field
$\widetilde{X}\in\Gamma(TT^{*}Q)$ which will be called the cotangent lift of
$X$. The local characterization in the canonical $(q,p)$-coordinates is as
follows: if $X=X^{i}(q)\frac{\partial}{\partial q^{i}}$, then
$\overline{X}=X^{i}(q)p_{i}$ and
$\widetilde{X}=X^{i}(q)\frac{\partial}{\partial q^{i}}-p_{j}\frac{\partial
X^{j}(q)}{\partial q^{i}}\frac{\partial}{\partial p_{i}}$.

######  Remark 4.2.

One can also describe the Hamiltonian function $\overline{X}$ as the
evaluation $\theta\big{(}\widetilde{X}\big{)}$ of the canonical Liouville
1-form $\theta=p_{i}\,{\rm d}q^{i}$ on $\widetilde{X}$.

######  Remark 4.3.

In particular, if a Lie group $G$ with $\operatorname{Lie}(G)=\mathfrak{g}$
acts on a manifold $Q$ and $X_{\xi}$ is the fundamental vector field of this
action corresponding to an element $\xi\in\mathfrak{g}$, then
$\widetilde{X}_{\xi}$ is the corresponding fundamental vector field of the
extended cotangent action of $G$ on $T^{*}Q$.

Let $(M,\omega)$ be a symplectic manifold, $\omega_{1}$ another symplectic
form on $M$. Then $\omega$, $\omega_{1}$ are Poisson compatible (i.e., the
Poisson structures $\Pi:=\omega^{-1}$, $\Pi_{1}:=\omega_{1}^{-1}$ are
compatible) if and only if the $(1,1)$-tensor
$\widetilde{N}:=\Pi_{1}\circ\omega\colon TM\to TM$ is Nijenhuis (cf. [14,
Proposition 7.1]). Assume this is the case. Let
$\Pi^{\lambda}=\Pi_{1}-\lambda\Pi$. We have
$\Pi^{\lambda}=\big{(}\widetilde{N}-\lambda I\big{)}\Pi$, therefore
$\operatorname{im}\Pi^{\lambda}=\operatorname{im}\big{(}\widetilde{N}-\lambda
I\big{)}\Pi=\operatorname{im}\big{(}\widetilde{N}-\lambda I\big{)}$ since
$\Pi$ is nondegenerate. This gives us relation between characteristic
distributions of Poisson structures and eigendistributions of a Nijenhuis
tensor. In particular, we have proved the following lemma.

######  Lemma 4.4.

Retaining the assumptions above assume additionally that $\widetilde{N}$ is
semisimple and has constant eigenvalues $\lambda_{1},\dots,\lambda_{s}$,
$\lambda_{i}\not=\lambda_{j}$, $i\not=j$ $($assumed to be real in the real
category$)$. Let $\widetilde{D}_{i}$, $i=1,\dots,s$, be the eigendistribution
corresponding to the eigenvalue $\lambda_{i}$. Then the foliation
$\mathcal{F}_{i}$ tangent to the distribution
$\sum\limits_{j\not=i}\widetilde{D}_{j}$ $($which is integrable by Lemma 2.6)
coincides with the symplectic foliation of the degenerate Poisson bivector
$\Pi^{\lambda_{i}}$.

The following definition is due to F.-J. Turiel [32].

######  Definition 4.5.

Let $Q$ be a manifold and $K\colon TQ\to TQ$ be a $(1,1)$-tensor. Define its
cotangent lift $\widetilde{K}\colon TM\to TM$, $M:=T^{*}Q$, as follows. Let
$K^{t}\colon T^{*}Q\to T^{*}Q$ be the map transposed to $K$ understood as a
smooth map $M\to M$, let $\omega$ be the canonical symplectic form on
$T^{*}Q$, and let $\omega_{1}:=\big{(}K^{t}\big{)}^{*}\omega$. Put
$\widetilde{K}:=\omega^{-1}\circ\omega_{1}$.

If $\\{q^{i}\\}$ is a system of local coordinates on $Q$ and
$K=K^{i}_{j}(q)\frac{\partial}{\partial q^{i}}\otimes{\rm d}q^{j}$, then in
the corresponding coordinates $\big{(}q^{i},p_{i}\big{)}$ on $T^{*}Q$ we have

$\displaystyle\widetilde{K}=K^{i}_{j}(q)\left(\frac{\partial}{\partial
q^{i}}\otimes{\rm d}q^{j}+\frac{\partial}{\partial p_{j}}\otimes{\rm
d}p_{i}\right)+p_{k}\left(\frac{\partial K^{k}_{j}}{\partial
q^{i}}-\frac{\partial K^{k}_{i}}{\partial
q^{j}}\right)\frac{\partial}{\partial q^{j}}\otimes{\rm d}q^{i}.$

Obviously, if $K$ is a fiberwise invertible $(1,1)$-tensor, then
$\widetilde{K^{-1}}=\widetilde{K}^{-1}$.

######  Lemma 4.6 ([32]).

$T_{K}=0\Longleftrightarrow T_{\widetilde{K}}=0$.

In particular, we have the following statement.

######  Lemma 4.7.

Let $N\colon TQ\to TQ$ be a fiberwise invertible Nijenhuis $(1,1)$-tensor.
Then the pair of bivectors $(\Pi,\Pi_{1})$, where $\Pi:=\omega^{-1}$ is the
canonical Poisson bivector on $T^{*}Q$, $\Pi_{1}:=\widetilde{N}\circ\Pi$, and
$\omega$ is the canonical symplectic form on $T^{*}Q$, is a pair of compatible
Poisson bivectors on $T^{*}Q$.

###### Proof.

Obviously,
$\widetilde{N}\circ\Pi=\widetilde{N}\circ\omega^{-1}=\big{(}\omega\circ\widetilde{N}^{-1}\big{)}^{-1}=\big{(}\omega\circ\widetilde{N^{-1}}\big{)}^{-1}=\big{(}\big{(}(N^{-1})^{t}\big{)}^{*}\omega\big{)}^{-1}=N^{t}_{*}\omega^{-1}=N^{t}_{*}\Pi$,
hence $\Pi_{1}$ is a Poisson bivector. Since $\widetilde{N}$ is a Nijenhuis
tensor, $\Pi$ and $\Pi_{1}$ are compatible. ∎

From now on we assume that $N$ is an invertible semisimple Nijenhuis
$(1,1)$-tensor with constant eigenvalues $\lambda_{1},\dots,\lambda_{s}$,
$\lambda_{i}\not=\lambda_{j}$, $i\not=j$ (which are real in the real category)
of multiplicities $k_{1},\dots,k_{s}$ respectively and let $D_{i}\subset TQ$
to be the eigendistribution corresponding to the eigenvalue $\lambda_{i}$. We
also denote by $\widetilde{D}_{i}\subset TM$ the eigendistribution of the
$(1,1)$-tensor $\widetilde{N}$ corresponding to the eigenvalue $\lambda_{i}$
(of multiplicity $2k_{i}$).

Let $L(N)$ stand for the Lie algebra of vector fields on $Q$ preserving $N$
(i.e., $V\in L(N)$ if and only if $\mathcal{L}_{V}N=0$) and let
$\theta=\theta_{1}+\cdots+\theta_{s}$ be the decomposition of the canonical
Liouville one-form $\theta$ on $M:=T^{*}Q$ related to the decomposition
$TM=\widetilde{D}_{1}\oplus\cdots\oplus\widetilde{D}_{s}$, i.e.,
$\theta_{i}|_{\widetilde{D}_{i}}=\theta|_{\widetilde{D}_{i}}$ and
$\theta_{i}\big{(}\sum\limits_{j\not=i}\widetilde{D}_{j}\big{)}=0$.

######  Lemma 4.8.

Retain the assumptions above. Then the following statements hold.

  1. $1.$

For any $i\in\\{1,\dots,s\\}$ the leaves of the symplectic foliation
$\mathcal{F}_{i}$ of the Poisson bivector
$\Pi^{\lambda_{i}}:=\Pi_{1}-\lambda_{i}\Pi$ are all of the same dimension and
have codimension $2k_{i}$. For any such leaf $F$ its image $\pi(F)$ under the
canonical projection $\pi\colon T^{*}Q\to Q$ is a leaf of the foliation
tangent to the distribution $\check{D}_{i}:=\sum\limits_{j\not=i}D_{j}$
$($which is integrable by Lemma 2.6). The leaf $\pi(F)$ is of codimension
$k_{i}$ in $Q$. For any leaf $F_{0}$ of the foliation tangent to
$\check{D}_{i}$ the set $\pi^{-1}(F_{0})$ is a Poisson submanifold of the
Poisson manifold $\big{(}T^{*}Q,\Pi^{\lambda_{i}}\big{)}$.

  2. $2.$

The vector fields from $L(N)$ tangent to the distribution $D_{i}$ form an
ideal $L_{i}$ of the Lie algebra $L(N)$ and there is a direct decomposition
$L(N)=L_{1}\oplus\cdots\oplus L_{s}$.

  3. $3.$

For any $V\in L(N)$ and $i\in\\{1,\dots,s\\}$ the function
$f^{i}_{V}:=\theta_{i}\big{(}\widetilde{V}\big{)}\in\mathcal{E}(T^{*}Q)$,
where $\widetilde{V}$ is the cotangent lift of $V$ $($see Definition 4.1), is
a Casimir function of the bivector $\Pi^{\lambda_{i}}$. In particular, since
$f^{i}_{V}$ linearly depends on $V\in L(N)$, for any leaf $F$ of the foliation
$\mathcal{F}_{i}$ the formula $V\mapsto\Phi^{i}_{F}(V):=f^{i}_{V}|_{F}$
defines a linear functional on $L(N)$.

  4. $4.$

If $V\in L(N)$ is tangent to the leaf $\pi(F)$, where $F$ is a symplectic leaf
of the Poisson bivector $\Pi^{\lambda_{i}}$, then $\Phi^{i}_{F}(V)=0$.

  5. $5.$

Given any leaf $F_{0}\subset Q$ of the foliation tangent to the distribution
$\check{D}_{i}$, a vector field $V\in L(N)$ is tangent to $F_{0}$ if and only
if $\widetilde{V}$ is tangent to $\pi^{-1}(F_{0})$.

###### Proof.

Recall that $k_{i}$ is the multiplicity of the eigenvalue $\lambda_{i}$,
$i=1,\dots,s$. By Lemma 2.6 in a vicinity of every point on $Q$ there exist a
system of local coordinates
$\big{(}q^{j^{1}_{1}},\dots,q^{j^{1}_{k_{1}}},\dots,q^{j^{s}_{1}},\dots,\allowbreak
q^{j^{s}_{k_{s}}}\big{)}$, where
$\big{(}j^{1}_{1},\dots,j^{1}_{k_{1}},\dots,j^{s}_{1},\dots,j^{s}_{k_{s}}\big{)}=(1,\dots,\dim
Q)$, such that the eigendistribution $D_{i}$ is spanned by the vector fields
$\frac{\partial}{\partial q^{j^{i}_{1}}},\dots,\frac{\partial}{\partial
q^{j^{i}_{k_{i}}}}$. Then
$\widetilde{N}=\sum_{i}\lambda_{i}\big{(}\sum\limits_{n=1}^{k_{i}}\big{(}\frac{\partial}{\partial
q^{j^{i}_{n}}}\otimes{\rm d}q^{j^{i}_{n}}+\frac{\partial}{\partial
p_{j^{i}_{n}}}\otimes{\rm d}p_{j^{i}_{n}}\big{)}\big{)}$ and by Lemma 4.4 the
tangent space to the symplectic foliation of $\Pi^{\lambda_{i}}$ is generated
by the vector fields $\frac{\partial}{\partial q^{l}}$,
$\frac{\partial}{\partial p_{m}}$,
$l,m\not\in\big{\\{}j^{i}_{1},\dots,j^{i}_{k_{i}}\big{\\}}$ (the corresponding
Casimir functions are $q^{j^{i}_{n}}$, $p_{j^{i}_{n}}$, $n=1,\dots,k_{i}$). On
the other hand, the tangent distribution to the leaves of
$\sum\limits_{j\not=i}D_{j}$ is spanned by the vector fields
$\frac{\partial}{\partial q^{l}}$,
$l\not\in\big{\\{}j^{i}_{1},\dots,j^{i}_{k_{i}}\big{\\}}$. This proves the
first assertion of the lemma.

To prove item 2 notice that, given a vector field
$V=V^{l}(q)\frac{\partial}{\partial q^{l}}$, the equality $\mathcal{L}_{V}N=0$
holds if and only if $[V,NX]=N[V,X]$ for any vector field $X$. Substituting
$X=\frac{\partial}{\partial q^{j^{i}_{n}}}$ to the last equality we get
$\lambda_{i}\big{[}V,\frac{\partial}{\partial
q^{j^{i}_{n}}}\big{]}=N\big{[}V,\frac{\partial}{\partial
q^{j^{i}_{n}}}\big{]}$, which means that $\big{[}V,\frac{\partial}{\partial
q^{j^{i}_{n}}}\big{]}$ is an eigenvector of $N$ corresponding to
$\lambda_{i}$. Hence $\big{[}V,\frac{\partial}{\partial q^{j^{i}_{n}}}\big{]}$
is expressed as a linear combination of $\frac{\partial}{\partial
q^{j^{i}_{1}}},\dots,\frac{\partial}{\partial q^{j^{i}_{k_{i}}}}$. In other
words, the coefficients $V^{j^{i}_{1}},\dots,V^{j^{i}_{k_{i}}}$ depend only on
the coordinates $q^{i}:=\big{(}q^{j^{i}_{1}},\dots,q^{j^{i}_{k_{i}}}\big{)}$
for any $i$.

For the proof of item 3 observe that
$\theta_{i}=\sum\limits_{n=1}^{k_{i}}p_{j^{i}_{n}}\,{\rm d}q^{j^{i}_{n}}$ in
the coordinates mentioned and the evaluation $f^{i}_{V}$ of this form on the
cotangent lift

$\displaystyle\widetilde{V}=V^{l}(q)\frac{\partial}{\partial
q^{l}}-p_{l}\frac{\partial V^{l}(q)}{\partial q^{n}}\frac{\partial}{\partial
p_{n}}$ (4.1)

of a vector field $V\in L(N)$ is equal to

$\displaystyle
f^{i}_{V}=\sum_{n=1}^{k_{i}}V^{j^{i}_{n}}\big{(}q^{i}\big{)}p_{j^{i}_{n}}.$

Any leaf $F_{0}$ of the foliation tangent to $\check{D}_{i}$ is given in these
coordinates by the equations $q^{j^{i}_{n}}=c^{j^{i}_{n}}$, $n=1,\dots,k_{i}$,
and any symplectic leaf $F\subset\pi^{-1}(F_{0})$ of the foliation
$\mathcal{F}_{i}$ by the equations $q^{j^{i}_{n}}=c^{j^{i}_{n}}$,
$p_{j^{i}_{n}}=C_{j^{i}_{n}}$, $n=1,\dots,k_{i}$, whose right hand sides are
some constants. This proves item 3.

If a vector field $V\in L(N)$ is tangent to $\pi(F)$, then

$\displaystyle V^{j^{i}_{n}}\big{(}c^{i}\big{)}=0,\qquad n=1,\dots,k_{i},$

where we put $c^{i}:=\big{(}c^{j^{i}_{1}},\dots,c^{j^{i}_{k_{i}}}\big{)}$, in
particular $\Phi^{i}_{F}=f^{i}_{V}|_{F}=0$.

The last item follows easily from formula (4.1). ∎

######  Lemma 4.9.

Retaining the assumptions of the preceding lemma assume that a transitive left
action $\rho\colon\mathfrak{g}\to\Gamma(TQ)$, $\xi\mapsto V_{\xi}$, of a Lie
algebra $\mathfrak{g}$ on $Q$ is given such that $\rho$ preserves $N$, i.e.,
$\rho(\mathfrak{g})\subset L(N)$. Denote by $\tilde{\rho}$ the extended
cotangent action, $\tilde{\rho}(\xi)=\widetilde{\rho(\xi)}$,
$\xi\in\mathfrak{g}$. $($Note that $\rho$ is an antihomomorphism, the map
$V\mapsto\widetilde{V}$ is a homomorphism, hence $\tilde{\rho}$ is an
antihomomorphism, i.e., a left action.$)$ Given a leaf $F$ of the symplectic
foliation $\mathcal{F}_{i}$, $i\in\\{1,\dots,s\\}$, let
$\varphi^{i}_{F}:=\Phi^{i}_{F}\circ\rho\in\mathfrak{g}^{*}$ be the linear
functional induced on $\mathfrak{g}$ by the functional
$\Phi_{F}^{i}\in(L(N))^{*}$ from Lemma 4.8(3) and let $\mathfrak{g}_{F}$ stand
for the stabilizer algebra of $F$, i.e., the set of elements
$\xi\in\mathfrak{g}$ such that $\tilde{\rho}(\xi)$ is tangent to $F$. Let
$p_{i}\colon\Gamma(TQ)\to\Gamma(D_{i})$ be the projection related to the
decomposition $TQ=D_{1}\oplus\cdots\oplus D_{s}$. Then

  1. $1)$

for any $i\in\\{1,\dots,s\\}$ the map $L(N)\to L_{r}$ induced by the
projection $p_{i}$ is a homomorphism of Lie algebras; in particular
$\rho_{i}\colon\mathfrak{g}\to\Gamma(TQ)$, where $\rho_{i}:=p_{i}\circ\rho$,
is a left action of the Lie algebra $\mathfrak{g}$;

  2. $2)$

the action $\tilde{\rho}$ is Hamiltonian with respect to the Poisson structure
$\Pi^{\lambda}$ for any $\lambda\not=\lambda_{i}$, $i=1,\dots,s$ with the
momentum map $\mu_{\lambda}\colon T^{*}Q\to\mathfrak{g}^{*}$ given by

$\displaystyle\langle\mu_{\lambda}(x),\xi\rangle=(\psi(\lambda)^{*}\theta)(\tilde{\rho}(\xi))(x),\qquad\xi\in\mathfrak{g},$
(4.2)

where $\theta$ is the canonical Liouville $1$-form on $T^{*}Q$ and
$\psi(\lambda)$ is the diffeomorphism of $T^{*}Q$ given by $\big{(}(N-\lambda
I)^{t}\big{)}^{-1}$ $($we used the notation $(\cdot)^{t}$ for the transposed
map$)$; equivalently,
$\langle\mu_{\lambda}(x),\xi\rangle=\sum\limits_{i=1}^{s}\frac{f^{i}_{\rho(\xi)}(x)}{\lambda_{i}-\lambda}$,
see Lemma 4.8(3) for the definition of $f^{i}_{V}$; moreover,

$\displaystyle\mu_{\lambda}=\mu_{\rm can}\circ\psi(\lambda),$ (4.3)

where $\mu_{\rm can}$ is the moment map corresponding to the canonical Poisson
bivector $\Pi$;

  3. $3)$

given a leaf $F$ of the symplectic foliation $\mathcal{F}_{i}$, the restricted
action of $\mathfrak{g}_{F}$ on $F$ is Hamiltonian with respect to the
restriction of the Poisson structure $\Pi^{\lambda_{i}}$ to $F$ with the
momentum map $\mu_{\lambda_{i}}^{F}\colon F\to\mathfrak{g}^{*}$ given by
$\big{\langle}\mu_{\lambda_{i}}^{F}(x),\xi\big{\rangle}=(\psi_{\lambda_{i}}^{*}\theta)(\tilde{\rho}(\xi))(x)$,
$\xi\in\mathfrak{g}_{F}$, where $\theta$ is the canonical Liouville $1$-form
on $T^{*}Q$ and $\psi_{\lambda_{i}}$ is the smooth map of $T^{*}Q$ given by
$\psi_{\lambda_{j}}|_{D^{*}_{i}}=\big{(}(N-\lambda_{i}I)^{t}\big{)}^{-1}|_{D^{*}_{i}}$,
$j\not=i$, $\psi_{\lambda_{i}}|_{D^{*}_{i}}=0$; here
$T^{*}Q=D^{*}_{1}\oplus\cdots\oplus D^{*}_{s}$ is the decomposition
corresponding to $TQ=D_{1}\oplus\cdots\oplus D_{s}$;333Here an equivalent
description of the momentum map similar to that from item 2 is also possible:
$\langle\mu_{\lambda_{i}}^{F}(x),\xi\rangle=\sum\limits_{j=1}^{s}\frac{f^{j}_{\rho(\xi)}(x)}{\lambda_{j}-\lambda_{i}}$
(note that the $i$-th term in the sum is correctly defined since
$f^{i}_{\tilde{\rho}(\xi)}$ vanishes for $\xi\in\mathfrak{g}_{F}$, cf. Lemma
4.8(4)).

  4. $4)$

the cotangent extension $\tilde{\rho}_{i}$,
$\tilde{\rho}_{i}(\xi):=\widetilde{\rho_{i}(\xi)}$, of the action $\rho_{i}$
defined in item $1$ is Hamiltonian with respect to the canonical Poisson
bivector $\Pi$ with the momentum map $\nu_{i}\colon
T^{*}Q\to\mathfrak{g}^{*}$,
$\langle\nu_{i}(x),\xi\rangle={f^{i}_{\rho(\xi)}(x)}$;

  5. $5)$

for any leaf $F_{0}\subset Q$ of the foliation tangent to the distribution
$\check{D}_{i}$ its stabilizer algebra $\mathfrak{g}_{F_{0}}$ with respect to
the action $\rho$, i.e., the set of $\xi\in\mathfrak{g}$ such that $\rho(\xi)$
is tangent to $F_{0}$, coincides with the stabilizer algebra of the
submanifold $\pi^{-1}(F_{0})$ with respect to the action $\tilde{\rho}$, i.e.,
the set of $\xi\in\mathfrak{g}$ such that $\tilde{\rho}(\xi)$ is tangent to
$\pi^{-1}(F_{0})$;

  6. $6)$

the following inclusion holds:
$\nu_{i}\big{(}\pi^{-1}(F_{0})\big{)}\subset\mathfrak{g}_{F_{0}}^{\bot}\cong(\mathfrak{g}/\mathfrak{g}_{F_{0}})^{*}$;

  7. $7)$

moreover, the relation $F\mapsto\varphi_{F}^{i}=\nu_{i}|_{F}$ is an
$\mathfrak{g}_{F_{0}}$-equivariant one-to-one correspondence between the
symplectic leaves $F$ of $\Pi^{\lambda_{i}}$ such that $\pi(F)=F_{0}$ and
linear functionals from a $k_{i}$-dimensional linear subspace in
$(\mathfrak{g}/\mathfrak{g}_{F_{0}})^{*}$ $($which in fact coincides with
$(\mathfrak{g}/\mathfrak{g}_{F_{0}})^{*}$, see Lemma 4.10(4)).

  8. $8)$

the stabilizer algebra $\mathfrak{g}_{F}\subset\mathfrak{g}$ of a leaf
$F\subset\pi^{-1}(F_{0})$ with respect to $\tilde{\rho}$ is equal to the
stabilizer algebra of the functional
$\varphi_{F}^{i}\in(\mathfrak{g}/\mathfrak{g}_{F_{0}})^{*}$ with respect to
the action of $\mathfrak{g}_{F_{0}}$.

###### Proof.

The claim of item 1 follows from the fact that each $L_{i}$ is an ideal in
$L(N)$ (see Lemma 4.8(2)).

To prove items 2 and 3 use coordinates from the proof of the previous lemma.
We have the following formulas:
$\Pi=\sum\limits_{i}\sum\limits_{n=1}^{k_{i}}\frac{\partial}{\partial
p_{j^{i}_{n}}}\wedge\frac{\partial}{\partial q^{j^{i}_{n}}}$,
$\Pi_{1}=\sum\limits_{i}\lambda_{i}\big{(}\sum\limits_{n=1}^{k_{i}}\frac{\partial}{\partial
p_{j^{i}_{n}}}\wedge\frac{\partial}{\partial q^{j^{i}_{n}}}\big{)}$,
$\Pi^{\lambda}=\sum\limits_{i}(\lambda_{i}-\lambda)\big{(}\sum\limits_{n=1}^{k_{i}}\frac{\partial}{\partial
p_{j^{i}_{n}}}\wedge\frac{\partial}{\partial q^{j^{i}_{n}}}\big{)}$, and
$\Pi^{\lambda_{j}}=\sum\limits_{i\not=j}(\lambda_{i}-\lambda_{j})\big{(}\sum\limits_{n=1}^{k_{i}}\frac{\partial}{\partial
p_{j^{i}_{n}}}\wedge\frac{\partial}{\partial q^{j^{i}_{n}}}\big{)}$. For the
vector field
$V=V_{\xi}=\sum\limits_{i}\sum\limits_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{\partial}{\partial
q^{j^{i}_{n}}}$ its cotangent lift $\widetilde{V}=\widetilde{V}_{\xi}$ takes
the form

$\displaystyle\widetilde{V}=\sum_{i}\sum_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{\partial}{\partial
q^{j^{i}_{n}}}-\sum_{i}\sum_{n,m=1}^{k_{i}}p_{j^{i}_{n}}\frac{\partial
V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}}{\partial
q^{j^{i}_{m}}}\frac{\partial}{\partial p_{j^{i}_{m}}}$

and is a Hamiltonian vector field with respect to $\Pi$:
$\widetilde{V}=\Pi(H_{\xi})$, where
$H_{\xi}=\theta\big{(}\widetilde{V}\big{)}=\sum\limits_{i}\sum\limits_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}p_{j^{i}_{n}}$
(cf. Remark 4.2). On the other hand, obviously,
$\widetilde{V}=\Pi^{\lambda}(H^{\lambda}_{\xi})$, where we put

$\displaystyle
H^{\lambda}_{\xi}=\sum_{i}\sum_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}p_{j^{i}_{n}}/(\lambda_{i}-\lambda).$

In fact, the functions $H^{\lambda}_{\xi}$ are global and correctly defined
(i.e., they do not depend on the choices of local coordinates), which can be
seen from the equality
$H^{\lambda}_{\xi}=(\psi(\lambda)^{*}\theta)\big{(}\widetilde{V}_{\xi}\big{)}$.
Yet another description of the function $H^{\lambda}_{\xi}$ is as follows:
$H^{\lambda}_{\xi}=\sum_{i}f^{i}_{V_{\xi}}/(\lambda_{i}-\lambda)$ (see Lemma
4.8(3) for the definition of $f^{i}_{V_{\xi}}$).

Using the equality $V_{[\xi,\zeta]}=[V_{\xi},V_{\zeta}]$ we get

$\displaystyle\Pi^{\lambda}\big{(}H^{\lambda}_{\xi}\big{)}H^{\lambda}_{\zeta}-H^{\lambda}_{[\xi,\zeta]}=\sum_{i}\sum_{n,m=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{\partial
V_{\zeta}^{j^{i}_{m}}\big{(}q^{i}\big{)}}{\partial
q^{j^{i}_{n}}}\frac{p_{j^{i}_{m}}}{\lambda_{i}-\lambda}$
$\displaystyle\qquad{}-\sum_{i}\sum_{n,m=1}^{k_{i}}p_{j^{i}_{n}}\frac{\partial
V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}}{\partial
q^{j^{i}_{m}}}V_{\zeta}^{j^{i}_{m}}\big{(}q^{i}\big{)}\frac{1}{\lambda_{i}-\lambda}-\sum_{i}\sum_{n=1}^{k_{i}}V_{[\xi,\zeta]}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{p_{j^{i}_{n}}}{\lambda_{i}-\lambda}=0,$

which proves the hamiltonicity of $\tilde{\rho}$ with respect to
$\Pi^{\lambda}$, $\lambda\not=\lambda_{i}$.

Formula (4.3) is a consequence of (4.2) as $\langle\mu_{\rm
can}(x),\xi\rangle=\theta(\tilde{\rho}(\xi))(x)$.

Now assume that $V=V_{\xi}$ is tangent to the symplectic leaf $F$ given in the
local coordinates by the equations $q^{j^{i}_{n}}=\mathrm{const}$,
$p_{j^{i}_{n}}=\mathrm{const}$, $n=1,\dots,k_{i}$. Then by (4.1) we get

$\displaystyle\widetilde{V}|_{F}=\sum_{l\not=i}\sum_{n=1}^{k_{l}}V_{\xi}^{j^{l}_{n}}\big{(}q^{l}\big{)}\frac{\partial}{\partial
q^{j^{l}_{n}}}-\sum_{l\not=i}\sum_{n,m=1}^{k_{l}}p_{j^{l}_{n}}\frac{\partial
V_{\xi}^{j^{l}_{n}}\big{(}q^{l}\big{)}}{\partial
q^{j^{l}_{m}}}\frac{\partial}{\partial
p_{j^{l}_{m}}}=\Pi^{\lambda_{i}}(H_{\xi})\big{|}_{F},$

where
$H_{\xi}=\sum\limits_{l\not=i}\sum\limits_{n=1}^{k_{l}}V_{\xi}^{j^{l}_{n}}\big{(}q^{l}\big{)}p_{j^{l}_{n}}/(\lambda_{l}-\lambda_{i})$.
The function $H_{\xi}$ is global and correctly defined for any $\xi$ as
$H_{\xi}=\sum\limits_{l\not=i}f^{l}_{V_{\xi}}/(\lambda_{l}-\lambda_{i})$ and

$\displaystyle\Pi^{\lambda_{i}}(H_{\xi})H_{\zeta}-H_{[\xi,\zeta]}=\sum_{l\not=i}\sum_{n,m=1}^{k_{l}}V_{\xi}^{j^{l}_{n}}\big{(}q^{l}\big{)}\frac{\partial
V_{\zeta}^{j^{l}_{m}}\big{(}q^{l}\big{)}}{\partial
q^{j^{l}_{n}}}\frac{p_{j^{l}_{m}}}{\lambda_{l}-\lambda_{i}}$
$\displaystyle\qquad{}-\sum_{l\not=i}\sum_{n,m=1}^{k_{l}}p_{j^{l}_{n}}\frac{\partial
V_{\xi}^{j^{l}_{n}}\big{(}q^{l}\big{)}}{\partial
q^{j^{l}_{m}}}V_{\zeta}^{j^{l}_{m}}\big{(}q^{l}\big{)}\frac{1}{\lambda_{l}-\lambda_{i}}-\sum_{l\not=i}\sum_{n=1}^{k_{l}}V_{[\xi,\zeta]}^{j^{l}_{n}}\big{(}q^{l}\big{)}\frac{p_{j^{l}_{n}}}{\lambda_{l}-\lambda_{i}}=0.$

Since $F$ is a Poisson submanifold with respect to $\Pi^{\lambda_{i}}$, we
have

$\displaystyle\\{H_{\xi}|_{F},H_{\zeta}|_{F}\\}_{\Pi^{\lambda_{j}}|_{F}}=\\{H_{\xi},H_{\zeta}\\}_{\Pi^{\lambda_{j}}}|_{F}=H_{[\xi,\zeta]}|_{F}.$

To prove item 4 notice that, if
$V_{\xi}=\sum\limits_{i}\sum\limits_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{\partial}{\partial
q^{j^{i}_{n}}}$, $\xi\in\mathfrak{g}$, is the fundamental vector field of the
action $\rho$, then
$\rho_{i}(\xi)=\sum\limits_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{\partial}{\partial
q^{j^{i}_{n}}}$ is the fundamental vector field of the action $\rho_{i}$. Its
cotangent lift $\tilde{\rho}_{i}(\xi)$ is a Hamiltonian vector field with
respect to $\Pi$ with the Hamiltonian function
$H_{\xi}^{i}:=f^{i}_{\rho_{i}(\xi)}=f^{i}_{V_{\xi}}=\sum\limits_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}p_{j^{i}_{n}}$.
Now it remains to use the equality
$\rho_{i}([\xi,\zeta])=[\rho_{i}(\xi),\rho_{i}(\zeta)]$, which implies

$\displaystyle\Pi\big{(}H^{i}_{\xi}\big{)}H^{i}_{\zeta}-H^{i}_{[\xi,\zeta]}=\sum_{n,m=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}\frac{\partial
V_{\zeta}^{j^{i}_{m}}\big{(}q^{i}\big{)}}{\partial
q^{j^{i}_{n}}}{p_{j^{i}_{m}}}-\sum_{n,m=1}^{k_{i}}p_{j^{i}_{n}}\frac{\partial
V_{\xi}^{j^{i}_{n}}\big{(}q^{i}\big{)}}{\partial
q^{j^{i}_{m}}}V_{\zeta}^{j^{i}_{m}}\big{(}q^{i}\big{)}$
$\displaystyle\hphantom{\Pi\big{(}H^{i}_{\xi}\big{)}H^{i}_{\zeta}-H^{i}_{[\xi,\zeta]}=}{}-\sum_{n=1}^{k_{i}}V_{[\xi,\zeta]}^{j^{i}_{n}}\big{(}q^{i}\big{)}{p_{j^{i}_{n}}}=0.$

Item 5 follows from Lemma 4.8(5) and item 6 follows from Lemma 4.8(4) in view
of the fact that $\pi^{-1}(F_{0})$ is foliated by the symplectic leaves of the
Poisson bivector $\Pi^{\lambda_{i}}$ (see Lemma 4.8(1)) and from the equality
$\Phi^{i}_{F}(\rho_{i}(\xi)):=f^{i}_{\rho_{i}(\xi)}|_{F}$, where $F$ is any
such leaf.

To prove item 7 first notice that the $\mathfrak{g}_{F_{0}}$-equivariance
follows from $\mathfrak{g}$-equivariance of the moment map $\nu_{i}$. Now
recall (see the proof of Lemma 4.8) that

$\displaystyle\varphi_{F}^{i}(\xi)=\sum_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}c^{i}\big{)}C_{j^{i}_{n}},\qquad\xi\in\mathfrak{g},$

where the constants $c^{i}$, $C_{j^{i}_{n}}$ specify the particular leaf $F$
and $V_{\xi}^{j^{i}_{n}}$ are the coefficients of the fundamental vector field
$\rho(\xi)=V_{\xi}^{l}(q)\frac{\partial}{\partial q^{l}}$.

Now fix a leaf $F_{0}$ of the foliation tangent to a distribution
$\check{D}_{i}$, i.e., fix constants $\big{(}c^{i}\big{)}$. For any
$\xi\in\mathfrak{g}$ we have a linear map444Note that although the range of
constants $c^{i}$ is bounded by that of the local coordinates $q^{i}$, the
constants $C_{j^{i}_{n}}$ can take any value.

$\displaystyle\big{(}C_{j^{i}_{1}},\dots,C_{j^{i}_{k_{i}}}\big{)}\mapsto\sum_{n=1}^{k_{i}}V_{\xi}^{j^{i}_{n}}\big{(}c^{i}\big{)}C_{j^{i}_{n}},$

expressing the correspondence $F\mapsto\varphi_{F}^{i}(\xi)$, where
$k_{i}=\operatorname{corank}\check{D}_{i}$. Thus the claim of item 7 is
equivalent to the nondegeneracy of the following matrix

$\displaystyle\left[\begin{matrix}V_{\xi_{1}}^{j^{i}_{1}}\big{(}c^{i}\big{)}&\cdots&V_{\xi_{1}}^{j^{i}_{k_{i}}}\big{(}c^{i}\big{)}\\\
\vdots&&\vdots\\\
V_{\xi_{k_{i}}}^{j^{i}_{1}}\big{(}c^{i}\big{)}&\cdots&V_{\xi_{k_{i}}}^{j^{i}_{k_{i}}}\big{(}c^{i}\big{)}\end{matrix}\right],$

where $\xi_{1},\dots,\xi_{k_{i}}\in\mathfrak{g}$ are linearly independent
elements not belonging to $\mathfrak{g}_{F_{0}}$. In turn, the nondegeneracy
of this matrix follows from the fact that $\mathfrak{g}$ acts transitively on
$G/K$ and, as a consequence, on the space of leaves of the foliation tangent
to the distribution $\check{D}_{i}$.

Finally the last item follows from item 7. ∎

Now we apply the preceding results to homogeneous spaces. Let $G/K$ be a
homogeneous space and let $N$ be an $G$-invariant semisimple Nijenhuis
$(1,1)$-tensor on $G/K$ with the real spectrum
$\\{\lambda_{1},\dots,\lambda_{s}\\}$. Then by Theorem 2.7 there exists a
decomposition $\mathfrak{g}=\mathfrak{g}_{1}+\cdots+\mathfrak{g}_{s}$ to the
sum of subspaces such that

  1. 1)

$\forall_{i,j\in\\{1,\dots,s\\},i\not=j}$
$\mathfrak{g}_{i}\cap\mathfrak{g}_{j}=\mathfrak{k}$;

  2. 2)

$\forall_{i,j\in\\{1,\dots,s\\}}$ $\mathfrak{g}_{i}+\mathfrak{g}_{j}$ are Lie
subalgebras in $\mathfrak{g}$;

  3. 3)

the decomposition above induces the decomposition
$T(G/K)=D_{1}\oplus\cdots\oplus D_{s}$ to integrable subbundles and
$N|_{D_{i}}=\lambda_{i}\operatorname{Id}_{D_{i}}$.

Write $P\colon G\to G/K$ and $\pi\colon T^{*}(G/K)\to G/K$ for the canonical
projections.

By the construction from the proof of Lemma 2.5 the eigendistribution $D_{i}$
of $N$ corresponding to the eigenvalue $\lambda_{i}$ is equal
$P_{*}\widehat{D}_{i}$, where $\widehat{D}_{i}$ is the left invariant
distribution on $G$ obtained from the subspace
$\mathfrak{g}_{i}\subset\mathfrak{g}\cong T_{e}G$. In particular, since $\ker
P_{*}$ is the left invariant distribution obtained from the subspace
$\mathfrak{k}\subset\mathfrak{g}_{i}\subset\mathfrak{g}\cong T_{e}G$, the rank
of $D_{i}$, i.e., the multiplicity $k_{i}$ of the eigenvalue $\lambda_{i}$, is
equal to $\dim(\mathfrak{g}_{i}/\mathfrak{k})$.

Denote $\check{\mathfrak{g}}_{i}:=\sum\limits_{j\not=i}\mathfrak{g}_{j}$ (this
is a Lie subalgebra in $\mathfrak{g}$ by condition 2) and let $\check{G}_{i}$
be the corresponding subgroup in $G$. By Lemma 2.5 the leaves of the foliation
integrating the distribution $\check{D}_{i}:=\sum\limits_{j\not=i}D_{j}$ are
the projections with respect to $P$ of the left cosets $g\check{G}_{i}$, $g\in
G$. Let $p_{i}\colon TQ\to D_{i}$ be the projection related to the
decomposition $TQ=D_{1}\oplus\cdots\oplus D_{s}$.

######  Lemma 4.10.

Let $N$ be an invertible Nijenhuis $(1,1)$-tensor on a homogeneous space
$Q=G/K$ satisfying the assumptions above. Let $\Pi$ be the canonical poisson
bivector on $T^{*}(G/K)$ and $\Pi_{1}=\widetilde{N}\circ\Pi$ $($see Lemma
4.7). Then

  1. $1)$

for any symplectic leaf $F$ of the Poisson bivector
$\Pi^{\lambda_{i}}:=\Pi_{1}-\lambda_{i}\Pi$ there exists an element $g\in G$
such that $\pi(F)=P\big{(}g\check{G}_{i}\big{)}$; such element $g$ is unique
modulo right multiplication by $h\in\check{G}_{i}$;

  2. $2)$

the stabilizer algebra $\mathfrak{g}_{\pi(F)}\subset\mathfrak{g}$ of the leaf
$\pi(F)=P\big{(}g\check{G}_{i}\big{)}$ of the foliation tangent to the
distribution $\check{D}_{i}$ with respect to the $G$-action on $G/K$ is equal
to $\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i}$;

  3. $3)$

the stabilizer algebra $\mathfrak{g}_{F}\subset\mathfrak{g}$ of the leaf $F$
with respect to the extended $G$-action on $T^{*}(G/K)$ is equal to the
stabilizer algebra
$\mathfrak{g}^{\varphi_{F}^{i}}\subset\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i}$
of the functional
$\varphi_{F}^{i}\in(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}$
constructed in Lemma 4.9 by means of an action $\rho$, where we specify
$\rho\colon\mathfrak{g}\to\Gamma(T(G/K))$ to be the natural action of the Lie
algebra $\mathfrak{g}$ on $G/K$;

  4. $4)$

if $F_{0}\subset G/K$ is a fixed leaf of the foliation tangent to the
distribution $\check{D}_{i}$, $F_{0}=P\big{(}g\check{G}_{i}\big{)}$ $(g$
fixed$)$, the relation $F\mapsto\varphi_{F}^{i}$ is an
$\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i}$-equivariant one-to-one
correspondence between the symplectic leaves $F$ of $\Pi^{\lambda_{i}}$ such
that $\pi(F)=F_{0}$ and linear functionals from
$(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}$.

###### Proof.

First and second items are consequences of Lemma 2.5 applied to the subalgebra
$\mathfrak{h}=\check{\mathfrak{g}}_{i}$. Item 3 follows from item 2 and Lemma
4.9(8). Item 4 follows from Lemma 4.9(7) since
$k_{i}=\dim\mathfrak{g}_{i}-\dim\mathfrak{k}=\dim(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}$.
∎

##  5 Algebraic criterion of kroneckerity in the case
of a locally free action

The theorem below is the main result of this paper. Let $G$ be a compact Lie
group, $K$ its closed subgroup. Assume that the natural action of $G$ on
$M=T^{*}(G/K)$ is generically locally free, i.e., the stabilizer corresponding
to the principal orbit type is finite. Fix such a stabilizer $H$. In this case
the subset

$\displaystyle M_{H}=\big{\\{}x\in M\colon G_{x}=gHg^{-1}\ \mathrm{for\ some}\
g\in G\big{\\}}$

of $M$, consisting of all orbits $G\cdot x$ in $M$ isomorphic to $G/H$, is an
open and dense subset of $M$ and the orbit space $M^{\prime}_{H}:=M_{H}/G$ is
a smooth manifold (cf. Section 3). Write $p\colon M_{H}\to M_{H}/G$ for the
canonical projection.

######  Theorem 5.1.

Let $N$ be an $G$-invariant invertible555Invertibility can be always achieved
by adding the identity operator, which does not change the corresponding
pencil of operators and the related Poisson pencil. semisimple Nijenhuis
$(1,1)$-tensor on $G/K$ with the real spectrum
$\\{\lambda_{1},\dots,\lambda_{s}\\}$, i.e., $($cf. Theorem 2.7) there exists
a decomposition

$\displaystyle\mathfrak{g}=\mathfrak{g}_{1}+\cdots+\mathfrak{g}_{s}$ (5.1)

to the sum of subspaces such that

  * •

$\forall_{i,j\in\\{1,\dots,s\\},i\not=j}$
$\mathfrak{g}_{i}\cap\mathfrak{g}_{j}=\mathfrak{k}$;

  * •

$\forall_{i,j\in\\{1,\dots,s\\}}$ $\mathfrak{g}_{i}+\mathfrak{g}_{j}$ are Lie
subalgebras in $\mathfrak{g}$;

  * •

the decomposition above induces the decomposition
$T(G/K)=D_{1}\oplus\cdots\oplus D_{s}$ to integrable subbundles and
$N|_{D_{i}}=\lambda_{i}\operatorname{Id}_{D_{i}}$.

Let $(\Pi,\Pi_{1})$ be the Poisson pair consisting of the canonical Poisson
bivector $\Pi$ on $T^{*}(G/K)$ and of the Poisson bivector
$\Pi_{1}=\widetilde{N}\circ\Pi$, where $\widetilde{N}$ is the cotangent lift
of the $(1,1)$-tensor $N$ $($see Definition 4.5 and Lemma 4.7).

Then the bi-Poisson structure generated by the reduced Poisson pair
$(p_{*}\Pi,p_{*}\Pi_{1})$ is Kronecker at any point of the set $p(W)$, where
$W\subset M_{H}$ is the open dense set which will be specified in the proof,
if and only if for any $i=1,\dots,s$

$\displaystyle\exists\,a_{i}\in(\mathfrak{g}/\check{\mathfrak{g}}_{i})^{*}\colon\
\operatorname{ind}\mathfrak{g}^{a_{i}}+\operatorname{codim}_{(\mathfrak{g}/\check{\mathfrak{g}}_{i})^{*}}\mathcal{O}_{a_{i}}=\operatorname{ind}\mathfrak{g},$
(5.2)

where $\check{\mathfrak{g}}_{i}=\sum\limits_{j\not=i}\mathfrak{g}_{i}$ and
$\mathfrak{g}^{a_{i}}$ and $\mathcal{O}_{a_{i}}$ are respectively the
stabilizer algebra and the orbit of the element $a_{i}$ with respect to the
coadjoint action
$\operatorname{ad}^{*}\colon\check{\mathfrak{g}}_{i}\to\mathfrak{gl}(({\mathfrak{g}}/{\check{\mathfrak{g}}_{i}})^{*})$.

Equivalently, condition (5.2) can be written as

$\displaystyle\operatorname{ind}(\check{\mathfrak{g}}_{i}\ltimes(\mathfrak{g}/\check{\mathfrak{g}}_{i}))=\operatorname{ind}\mathfrak{g},$
(5.3)

where the term in the l.h.s. is the semidirect product of the Lie algebra
$\check{\mathfrak{g}}_{i}$ and the vector space
$(\mathfrak{g}/\check{\mathfrak{g}}_{i})$ with respect to the
$\operatorname{ad}$-action.

###### Proof.

We first note that conditions (5.2) and (5.3) are equivalent by Lemma 5.2
below.

Let
$U=M_{H}\setminus\big{(}\bigcup_{\lambda}\mu_{\lambda}^{-1}(\operatorname{Sing}\mathfrak{g}^{*})\big{)}$,
where the moment map $\mu_{\lambda}$ is specified in Lemma 4.9(2).

Observe that all the objects involved admit a natural complexification (cf.
Remark 3.6): the compact Lie groups $G$ and $K$ are imbedded in their
Chevalley complexifications $G^{c}$ and $K^{c}$ and the homogeneous space
$Q=G/K$ is imbedded into the complex homogeneous space $Q^{c}=G^{c}/K^{c}$.
Moreover, the decomposition (5.1) implies the decomposition
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{g}_{1}^{\mathbb{C}}+\cdots+\mathfrak{g}_{s}^{\mathbb{C}}$,
which in turn induces the decomposition $TQ^{c}=D_{1}^{c}\oplus\cdots\oplus
D_{s}^{c}$ of the holomorphic tangent bundle to $Q^{c}$ to complex analytic
involutive distributions and a complex analytic $(1,1)$-tensor $N^{c}$ given
by $N|_{D_{i}^{c}}=\lambda_{i}\operatorname{Id}_{D_{i}^{c}}$. By Lemma 4.9(2)
the assumptions of Theorem 3.7 are satisfied (it is well-known that for
reductive Lie algebras
$\operatorname{codim}\operatorname{Sing}\mathfrak{g}^{*}\geq 3$) and we
conclude that the reduced bi-Poisson structure
$\big{\\{}\big{(}\Pi^{\lambda}\big{)}^{\prime}\big{\\}}$,
$\big{(}\Pi^{\lambda}\big{)}^{\prime}=p_{*}\Pi_{1}-\lambda p_{*}\Pi$, is
Kronecker at a point $x^{\prime}\in p(U)$ if and only if
$\operatorname{corank}p_{*}\Pi^{\lambda_{i}}|_{x^{\prime}}=\operatorname{ind}\mathfrak{g}$,
$i=1,\dots,s$, where $\Pi^{\lambda_{i}}=\Pi_{1}-\lambda_{i}\Pi$ (see Theorem
3.7(2)). Below we express the number
$\operatorname{corank}p_{*}\Pi^{\lambda_{i}}|_{x^{\prime}}$ in equivalent
terms, see formula (5.4).

From Lemma 4.9(3) it follows that the restriction of the action
$\tilde{\rho}\colon\mathfrak{g}\to\Gamma(T(T^{*}G/K))$ to the stabiliser
subalgebra $\mathfrak{g}_{F}$ of any symplectic leaf $F$ of the Poisson
bivector $\Pi^{\lambda_{i}}$ is Hamiltonian with respect to this bivector with
the momentum map $\mu_{\lambda_{i}}^{F}\colon F\to\mathfrak{g}^{*}$. Obviously
the action of $\mathfrak{g}_{F}$ is also locally free. Therefore by the
bifurcation lemma (cf. the proof of Theorem 3.7) the corank of the reduction
$\big{(}\Pi^{\lambda_{i}}|F\big{)}^{\prime}$ of the Poisson structure
restricted to the symplectic leaf, $\Pi^{\lambda_{i}}|F$, at the point
$x^{\prime}=p(x)$, where $x\in F$, is equal to the index of the Lie algebra of
$\mathfrak{g}_{F}$, provided
$\mu_{\lambda_{i}}^{F}(x)\not\in\operatorname{Sing}\mathfrak{g}_{F}$. The
algebraic set $\operatorname{Sing}\mathfrak{g}_{F}$ is nowhere dense in
$\mathfrak{g}_{F}^{*}$ and the set $U_{F}=(F\cap
U)\setminus\big{(}\bigcup_{i=1}^{s}\big{(}\mu^{F}_{\lambda_{i}}\big{)}^{-1}\big{)}(\operatorname{Sing}\mathfrak{g}_{F})$
is an open dense set in $F$ and, moreover, $V=\bigcup_{F}U_{F}$ is open and
dense in $M_{H}$. From now on we will consider only points $x^{\prime}\in
p(V)$.

Obviously,
$\operatorname{corank}_{(M_{H}/G)}\big{(}\Pi^{\lambda_{i}}\big{)}_{x^{\prime}}^{\prime}=\operatorname{corank}_{F/G_{F}}\big{(}\Pi^{\lambda_{i}}|F\big{)}_{x^{\prime}}^{\prime}+\operatorname{codim}_{\mathcal{S}_{i}}G\cdot
F$. Here $G_{F}$ is the subgroup in $G$ corresponding to the subalgebra
$\mathfrak{g}_{F}$, $\mathcal{S}_{i}$ stands for the space of symplectic
leaves of the Poisson bivector $\Pi^{\lambda_{i}}$, on which a natural action
of the group $G$ is induced from the action of $G$ on $M$ due to the
$G$-invariance of $\Pi^{\lambda_{i}}$, and $G\cdot F$ denotes the orbit of the
point $F\in\mathcal{S}_{i}$ with respect to this action. Recall (see Lemma
4.8(1)) that the space $\mathcal{S}_{i}$ is foliated by the submanifolds of
the form $\pi^{-1}(F_{0})$, where $F_{0}\subset G/K$ is a leaf of the
foliation tangent to the distribution
$\check{D}_{i}=\sum\limits_{j\not=i}D_{j}$. Since the group $G$ acts
transitively on $G/K$ and as a consequence on the space of leaves of the
foliation tangent to the distribution $\check{D}_{i}$, we have
$\operatorname{codim}_{\mathcal{S}_{i}}G\cdot
F=\operatorname{codim}_{\mathcal{S}_{i}|\pi^{-1}(\pi(F))}G_{\pi(F)}\cdot F$,
where $G_{\pi(F)}$ is the subgroup corresponding to the subalgebra
$\mathfrak{g}_{\pi(F)}$, i.e., the stabilizer of the submanifold
$\pi^{-1}(\pi(F))$ with respect to the cotangent action (see Lemma 4.9(5)) and
$\mathcal{S}_{i}|\pi^{-1}(\pi(F))$ stands for the submanifold in
$\mathcal{S}_{i}$ of leaves contained in the Poisson submanifold
$\pi^{-1}(\pi(F))$. In view of Lemma 4.9(7), Lemma 4.10(4) and Lemma 4.9(8)
$\mathcal{S}_{i}|\pi^{-1}(\pi(F))$ can be identified with
$(\mathfrak{g}/\mathfrak{g}_{\pi(F)})^{*}$, $G_{\pi(F)}\cdot F$ with
$\mathcal{O}_{\varphi^{i}_{F}}$ and $\mathfrak{g}_{F}$ with
$\mathfrak{g}^{\varphi^{i}_{F}}$, where
$\varphi^{i}_{F}\in(\mathfrak{g}/\mathfrak{g}_{\pi(F)})^{*}$ is the functional
corresponding to $F$ and $\mathfrak{g}^{\varphi^{i}_{F}}$ and
$\mathcal{O}_{\varphi^{i}_{F}}$ are respectively its stabilizer and orbit with
respect to the action of $\mathfrak{g}_{\pi(F)}$ on
$(\mathfrak{g}/\mathfrak{g}_{\pi(F)})^{*}$.

Thus we have proven that
$\operatorname{corank}_{(M_{H}/G)}\big{(}\Pi^{\lambda_{i}}\big{)}_{x^{\prime}}^{\prime}=\operatorname{corank}_{F/G_{F}}\big{(}\Pi^{\lambda_{i}}|F\big{)}_{x^{\prime}}^{\prime}+\operatorname{codim}_{\mathcal{S}_{i}}G\cdot
F=\operatorname{ind}\mathfrak{g}_{F}+\operatorname{codim}_{\mathcal{S}_{i}|\pi^{-1}(\pi(F))}G_{\pi(F)}\cdot
F=\operatorname{ind}\mathfrak{g}^{\varphi^{i}_{F}}+\operatorname{codim}_{(\mathfrak{g}/\mathfrak{g}_{\pi(F)})^{*}}\mathcal{O}_{\varphi^{i}_{F}}$.
Finally, in view of Lemma 4.10(2), we have
$\mathfrak{g}_{\pi(F)}=\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i}$ for some
$g\in G$ and

$\displaystyle\operatorname{corank}\big{(}\Pi^{\lambda_{i}}\big{)}_{x^{\prime}}^{\prime}=\operatorname{ind}\mathfrak{g}^{\varphi^{i}_{F}}+\operatorname{codim}_{(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}}\mathcal{O}_{\varphi^{i}_{F}}.$
(5.4)

We are ready to finish the proof. Assume that
$\big{\\{}\big{(}\Pi^{\lambda}\big{)}^{\prime}\big{\\}}$ is Kronecker at
$x^{\prime}$. Then by Theorem 3.7
$\operatorname{ind}\mathfrak{g}_{\varphi^{i}_{F}}+\operatorname{codim}_{(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}}\mathcal{O}_{\varphi^{i}_{F}}=\operatorname{ind}\mathfrak{g}$,
$i\in\\{1,\dots,s\\}$. Acting by $\operatorname{Ad}_{g^{-1}}$ we will get
condition (5.2).

Vice versa, assume that (5.3) is satisfied. Then by the Raïs formula (see
Lemma 5.2)
$\operatorname{ind}\mathfrak{g}=\operatorname{ind}\mathfrak{g}^{a_{i}}+\operatorname{codim}_{(\mathfrak{g}/\check{\mathfrak{g}}_{i})^{*}}\mathcal{O}_{a_{i}}$
for $a_{i}\in R((\mathfrak{g}/\check{\mathfrak{g}}_{i})^{*})$. Obviously also
$\operatorname{ind}\mathfrak{g}=\operatorname{ind}\mathfrak{g}^{g\cdot
a_{i}}+\operatorname{codim}_{(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}}\mathcal{O}_{g\cdot
a_{i}}$ and ${g\cdot a_{i}}\in
R((\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*})$ for any
$g\in G$, where $g\cdot a_{i}:=\operatorname{Ad}^{*}_{g^{-1}}a_{i}$. Fix $g\in
G$ and let $F_{i}$ be the symplectic leaf of the Poisson bivector
$\Pi^{\lambda_{i}}$ corresponding to the element ${g\cdot a_{i}}$ by Lemma
4.10(4) (with $\pi(F_{i})=P\big{(}\operatorname{Ad}_{g}\check{G}_{i}\big{)}$).
Note that the leaves $F_{i}$ are mutually transversal and
$\sum_{i}\operatorname{codim}F_{i}=\dim M$, thus $\bigcap_{i}F_{i}$ is a
point, say $x$.

Recall (see Lemma 4.9(6), (7) and Lemma 4.10(4)) that the map

$\displaystyle\nu_{i}^{g}:=\nu_{i}|_{\pi^{-1}(\pi(F_{i}))}\colon\
\pi^{-1}(\pi(F_{i}))\to\mathfrak{g}_{\pi(F_{i})}^{\bot}\cong(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}$

is an epimorphism. The set
$\big{(}\nu_{i}^{g}\big{)}^{-1}(R((\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*})$
is an open dense set in $\pi^{-1}(\pi(F_{i}))$ and the set
$W=V\cap\big{(}\bigcup_{g\in
G}\bigcap_{i=1}^{s}\big{(}\nu_{i}^{g}\big{)}^{-1}\big{)}(R((\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}))$
is an open dense set in $M_{H}$.

Taking $a_{i}$ such that $g\cdot
a_{i}\in\nu^{g}_{i}\big{(}W\cap\pi^{-1}(\pi(F_{i}))\big{)}$ for any $i$, we
achieve that $x\in W\subset V$. Formula (5.4) shows that
$\operatorname{ind}\mathfrak{g}^{g\cdot
a_{i}}+\operatorname{codim}_{(\mathfrak{g}/\operatorname{Ad}_{g}\check{\mathfrak{g}}_{i})^{*}}\mathcal{O}_{g\cdot
a_{i}}=\operatorname{corank}p_{*}\Pi^{\lambda_{i}}|_{x^{\prime}}$, where
$x^{\prime}=p(x)$. By Theorem 3.7 we conclude that
$\\{(\Pi^{\lambda})^{\prime}\\}$ is Kronecker at $x^{\prime}$. ∎

######  Lemma 5.2.

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}\subset\mathfrak{g}$ its
Lie subalgebra. Then the condition of existing
$a\in(\mathfrak{g}/\mathfrak{h})^{*}$ such that

$\displaystyle\operatorname{ind}\mathfrak{h}^{a}+\operatorname{codim}_{(\mathfrak{g}/\mathfrak{h})^{*}}\mathcal{O}_{a}=\operatorname{ind}\mathfrak{g},$
(5.5)

where $\mathfrak{h}^{a}$ and $\mathcal{O}_{a}$ are respectively the stabilizer
algebra and the orbit of the element $a$ with respect to the coadjoint action
$\operatorname{ad}^{*}\colon\mathfrak{h}\to\mathfrak{gl}(({\mathfrak{g}}/{\mathfrak{h}})^{*})$,
is equivalent to the following one:

$\displaystyle\operatorname{ind}(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))=\operatorname{ind}\mathfrak{g},$
(5.6)

where the Lie algebra in the l.h.s. is the semidirect product of the Lie
algebra $\mathfrak{h}$ and the vector space $(\mathfrak{g}/\mathfrak{h})$ with
respect to the $\operatorname{ad}$-action. Moreover, if one of this condition
holds, the equality (5.5) holds for any $a$ from the open dense set
$R((\mathfrak{g}/\mathfrak{h})^{*})\subset(\mathfrak{g}/\mathfrak{h})^{*}$,
$R((\mathfrak{g}/\mathfrak{h})^{*}):=\\{\alpha\in(\mathfrak{g}/\mathfrak{h})^{*}\,|\,\exists\,\beta\in(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))^{*}\setminus\operatorname{Sing}((\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))^{*})\colon\alpha=\beta|_{(\mathfrak{g}/\mathfrak{h})}\\}$.

###### Proof.

Recall [28, Proposition 1.3(i)] that

$\displaystyle\operatorname{ind}(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))=\operatorname{ind}\mathfrak{h}^{a}+\operatorname{codim}_{(\mathfrak{g}/\mathfrak{h})^{*}}\mathcal{O}_{a}$

for $a\in R((\mathfrak{g}/\mathfrak{h})^{*})$. Hence (5.6) implies (5.5). On
the other hand, for arbitrary $a$ we have
$\operatorname{ind}\mathfrak{h}^{a}+\operatorname{codim}_{(\mathfrak{g}/\mathfrak{h})^{*}}\mathcal{O}_{a}=\operatorname{corank}\Pi_{(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))^{*}}|_{\beta}\geq\operatorname{ind}(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))$,
where
$\beta\in\operatorname{Sing}(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))^{*}$
is any element with $a=\beta|_{(\mathfrak{g}/\mathfrak{h})}$ (cf. [25, Theorem
1.1]) and, moreover, the number
$\operatorname{ind}(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))$ is
bounded below by $\operatorname{ind}\mathfrak{g}$ (since
$\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h})$ is a contraction of
$\mathfrak{g}$). Thus, if
$\operatorname{ind}\mathfrak{h}^{a}+\operatorname{codim}_{(\mathfrak{g}/\mathfrak{h})^{*}}\mathcal{O}_{a}=\operatorname{ind}\mathfrak{g}$
for some $a$, then
$\operatorname{ind}(\mathfrak{h}\ltimes(\mathfrak{g}/\mathfrak{h}))=\operatorname{ind}\mathfrak{g}$.
∎

######  Remark 5.3.

In the case when $K=\\{e\\}$ is the trivial subgroup of the Lie group $G$,
condition (5.2) coincides with the necessary and sufficient condition of
kroneckerity of the Lie–Poisson pencil related to an algebraic Nijenhuis
operator obtained in [24, Theorem 2.5].

######  Theorem 5.4.

Retain the assumptions of Theorem 5.1 and assume that one of the equivalent
conditions (5.2), (5.3) hold. Let $b$ be the $G$-invariant metric on $G/K$,
called normal, induced by some biinvariant metric on $G$, i.e., by an
$\operatorname{Ad}G$-invariant bilinear form $B$ on $\mathfrak{g}$. Then the
$G$-invariant metric $b_{N}$, $b_{N}(x,y):=b((N+N^{*})x,y)$,
$x,y\in\Gamma(T(G/K))$, on $G/K$ corresponding to the symmetric $(1,1)$-tensor
$N+N^{*}$, where $N^{*}$ is the adjoint to $N$ $(1,1)$-tensor,
$b(N^{*}x,y)=b(x,Ny)$, as well as the normal metric itself have completely
integrable geodesic flows in the class of analytic integrals polynomial in
momenta.

###### Proof.

It is well-known that a function of the form $\mu_{\rm can}^{*}f$, where
$\mu_{\rm can}$ is the moment map of the $G$-action on $T^{*}(G/K)$
corresponding to the canonical Poisson bivector $\Pi$ and $f$ is any
polynomial on $\mathfrak{g}^{*}$, is analytic and polynomial in momenta.
Indeed, the analyticity is obvious and the polynomiality can be argued as
follows. If $\xi\in\mathfrak{g}$ is treated as a linear function on
$\mathfrak{g}^{*}$, the function $H_{\xi}=\mu_{\rm can}^{*}\xi$ is the
Hamiltonian function of the corresponding fundamental vector field $V_{\xi}$,
which in turn can be treated as a fiberwise linear function on $T^{*}(G/K)$
(cf. Definition 4.1 and Remark 4.3). Thus, if $f$ is a polynomial in $\xi$,
then $\mu_{\rm can}^{*}f$ is fiberwise polynomial.

By Theorem 3.7(3) the involutive set of functions $\mathcal{I}$ (3.1), where
$\Pi_{t_{0}}=\Pi$, is complete on $T^{*}(G/K)$. We have to prove that the
quadratic forms $q(x):=b(x,x)$ and $q_{N}(x):=b_{N}(x,x)$,
$x\in\Gamma(T^{*}(G/K))$, where we identified $T(G/K)$ with $T^{*}(G/K)$ by
means of $b$, is contained in this set. Let $Q(x)=B(x,x)$ be the quadratic
form of $B$ understood as a Casimir function on $\mathfrak{g}^{*}$ after the
identification of $\mathfrak{g}$ and $\mathfrak{g}^{*}$ by means of $B$. Then
by Theorem 3.7(4) the function $\mu_{\rm can}^{*}Q$ belongs to $\mathcal{I}$.
One can show that in fact $\mu_{\rm can}^{*}Q$ coincides with $q$. Indeed, $b$
belongs to the class of the so-called submersion metrics obtained from the
right-invariant metrics on $G$ by the canonical submersion $G\to G/K$. The
quadratic forms of all the submersion metrics are of the form $\mu_{\rm
can}^{*}f$, where $f$ is the corresponding quadratic polynomial on
$\mathfrak{g}\cong\mathfrak{g}^{*}$ [6, Section 7].

To prove that $q_{N}\in\mathcal{I}$ recall that by Theorem 3.7(4) the set
$\mathcal{I}$ besides the functions $\mu_{\rm can}^{*}\mathcal{F}$ consists of
the functions of the form $\mu_{\lambda}^{*}f$, $\lambda\not=\lambda_{i}$,
$f\in Z^{\Pi}$. On the other hand, $\mu_{\lambda}=\mu_{\rm
can}\circ((N-\lambda I)^{-1})^{t}$ by formula (4.3), hence the functions of
the form $b\big{(}((N-\lambda I)^{-1})^{*}x,\big{(}(N-\lambda
I)^{-1}\big{)}^{*}x\big{)}=b((N-\lambda I)^{-1}\big{(}(N-\lambda
I)^{-1}\big{)}^{*}x,x)$, $x\in\Gamma(T^{*}(G/K))$, belong to $\mathcal{I}$.
Moreover, $\mathcal{I}$ will contain also the coefficients of the Laurent
expansion $b\big{(}(N-\lambda I)^{-1}\big{(}(N-\lambda
I)^{-1}\big{)}^{*}x,x\big{)}=\frac{1}{\lambda^{2}}b(x,x)+\frac{1}{\lambda^{3}}b((N+N^{*})x,x)+\cdots$
corresponding to the expansion $(N-\lambda
I)^{-1}=-\big{(}\frac{1}{\lambda}I+\frac{1}{\lambda^{2}}N+\cdots\big{)}$.

Finally, the functions $\mu_{\lambda}^{*}f$, where $f$ are polynomial Casimir
functions of $\Pi$, are polynomial in momenta (since $\big{(}(N-\lambda
I)^{-1}\big{)}^{t}$ is a fiberwise linear map). ∎

##  6 Applications: two homogeneous spaces
with integrable geodesic flows

In the table from Example 2.12 among the triples
$(\mathfrak{g},\mathfrak{g}_{1},\mathfrak{g}_{2})$ of compact Lie algebras
such that $\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ of one can find two
distinguished from our point of view series: $(A_{2n-1},C_{n},A_{2n-2}\oplus
T)$ and $(D_{n+1},B_{n},A_{n}\oplus T)$. For both of them the pairs
$(\mathfrak{g},\mathfrak{g}_{i})$, $i=1,2$, are symmetric, i.e., the Lie
algebra $\mathfrak{g}_{i}$ is the fixed point set of an automorphism of
$\mathfrak{g}$ of second order (cf. [11, Tables II, III, Section 6, Chapter
X]). In this context we have to mentioned the following reformulation of the
result of Brailov [31, Theorem 5, Section 37].

######  Theorem 6.1.

Let $\mathfrak{g}$ be a semisimple Lie algebra and
$\mathfrak{g}_{0}\subset\mathfrak{g}$ its symmetric subalgebra. Then

$\displaystyle\operatorname{ind}\mathfrak{g}=\operatorname{ind}(\mathfrak{g}_{0}\ltimes(\mathfrak{g}/\mathfrak{g}_{0})),$

where $\mathfrak{g}_{0}\ltimes(\mathfrak{g}/\mathfrak{g}_{0})$ is the so-
called $\mathbb{Z}_{2}$-contraction of $\mathfrak{g}$, i.e., the semidirect
product of the Lie algebra $\mathfrak{g}_{0}$ and the vector space
$\mathfrak{g}/\mathfrak{g}_{0}$ with respect of the natural
$\operatorname{ad}$-representation of $\mathfrak{g}_{0}$ in
$\mathfrak{g}/\mathfrak{g}_{0}$.

In particular, it follows from this result that both the series of
decompositions $\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ mentioned
satisfy condition (5.3) of Theorem 5.1. This allows us to formulate the
following theorem.

######  Theorem 6.2.

Let $G/K$ be one of the following homogeneous spaces:

  1. $(a)$

${\rm SU}(2n)/({\rm S}({\rm U}(2n-1)\times{\rm U}(1))\cap{\rm Sp}(n))$;

  2. $(b)$

${\rm SO}(2n+2)/({\rm SO}(2n+1)\cap{\rm U}(n+1))$.

Then the geodesic flow of

  1. $1)$

the normal metric $b$ on $G/K$ and

  2. $2)$

the $G$-invariant metric $b_{N}$ $($see Theorem 5.4) corresponding to the
$G$-invariant Nijenhuis $(1,1)$-tensor $N$ on $G/K$ with the real spectrum
$\\{\lambda_{1},\lambda_{2}\\}$, $\lambda_{1}\not=\lambda_{2}$,
$\lambda_{i}\not=0$, related to the decomposition
$\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ with
$\mathfrak{k}=\mathfrak{g}_{1}\cap\mathfrak{g}_{2}$ by Theorem 2.7

is completely integrable in the class of analytic integrals polynomial in
momenta.

Here $\mathfrak{g}$ and $\mathfrak{k}$ are the Lie algebras of $G$ and $K$
respectively and the triples of subalgebras
$(\mathfrak{g},\mathfrak{g}_{1},\mathfrak{g}_{2})$ are equal to
$(A_{2n-1},C_{n},A_{2n-2}\oplus T)$ and $(D_{n+1},B_{n},A_{n}\oplus T)$
respectively. The explicit formulae for the embeddings
$\mathfrak{g}_{i}\subset\mathfrak{g}$ as well as the decomposition
$\mathfrak{k}^{\perp}=\mathfrak{k}_{1}\oplus\mathfrak{k}_{2}$ of the
complementary to $\mathfrak{k}$ space corresponding to the decomposition
$\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$ and the “inertia operator”
$n_{\mathfrak{k}^{\perp}}+n_{\mathfrak{k}^{\perp}}^{*}=(N+N^{*})|_{T_{o}(G/K)\cong\mathfrak{k}^{\perp}}$
$($here $o=P(e))$ are listed in Appendix A.

###### Proof.

In view of Theorem 6.1 the result will follow form Theorem 5.4 if we ensure
that the action of $G$ on $T^{*}(G/K)$ is locally free (which is an essential
assumption of Theorem 5.4). Below we prove this fact, which is equivalent to
the fact that the stabilizer $\mathfrak{k}^{E}:=\operatorname{stab}_{\rho}(E)$
of a generic element $E$ in $\mathfrak{k}^{\perp}$ under the isotropy action
$\rho\colon\mathfrak{k}\rightarrow\mathfrak{gl}\big{(}\mathfrak{k}^{\perp}\big{)}$
vanishes; here $\mathfrak{k}^{\perp}$ is the orthogonal complement to
$\mathfrak{k}$ with respect to the (nondegenerate) Killing form on
$\mathfrak{g}$ and we identify isotropy and coisotropy action by means of this
form restricted to $\mathfrak{k}^{\perp}$. In other words, $\mathfrak{k}^{E}$
coincides with $Z_{\mathfrak{k}}(E)$, the centralizer of the element
$E\in\mathfrak{k}^{\perp}$ in $\mathfrak{k}$. In fact, since the function
$\dim\mathfrak{k}^{E}$ is lower semicontinuous, it is enough to show the
existence of an element $E$ with $\mathfrak{k}^{E}=\\{0\\}$. Note that it is
sufficient to show the existence of such an element for the complexified
action which we do below. We list explicit realizations of the
complexifications
$\big{(}\mathfrak{g}^{\mathbb{C}},\mathfrak{g}^{\mathbb{C}}_{1},\mathfrak{g}^{\mathbb{C}}_{2}\big{)}$
for the above mentioned triples
$(\mathfrak{g},\mathfrak{g}_{1},\mathfrak{g}_{2})$ as well as the subspace
$\big{(}\mathfrak{k}^{\mathbb{C}}\big{)}^{\perp}$ complementary to the
subspace
$\mathfrak{k}^{\mathbb{C}}=\mathfrak{g}_{1}^{\mathbb{C}}\cap\mathfrak{g}_{2}^{\mathbb{C}}$
with respect to the Killing form. Besides, we indicate the element
$E\in\big{(}\mathfrak{k}^{\mathbb{C}}\big{)}^{\perp}$ with
$\operatorname{stab}_{\rho}(E)=\\{0\\}$ and outline the proof of the last
equality. We consider separately cases (a) and (b).666We switch to modern
notations and denote the classical Lie algebras by small Gothic letters.

_Case $(a)$:
$(\mathfrak{a}_{2n-1},\mathfrak{c}_{n},\mathfrak{a}_{2n-2}\oplus\mathfrak{t})$._

$\displaystyle\mathfrak{g}^{\mathbb{C}}=\mathfrak{sl}(2n,\mathbb{C}),\qquad\mathfrak{g}_{1}^{\mathbb{C}}=\left\\{\left[\begin{matrix}Z_{1}&Z_{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
Z_{3}&-{Z_{1}}^{T}\end{matrix}\right]\,|\,Z_{2}=Z_{2}^{T},\,Z_{3}=Z_{3}^{T}\right\\}\cong\mathfrak{sp}(n,\mathbb{C}),$
$\displaystyle\mathfrak{g}_{2}^{\mathbb{C}}=\left\\{\left[\begin{matrix}Z&\mathbf{0}\\\
\mathbf{0}^{T}&-t\end{matrix}\right]\,|\,Z\in\mathfrak{gl}(2n-1,\mathbb{C}),t=\operatorname{Tr}Z\right\\}\cong\mathfrak{gl}(2n-1,\mathbb{C}),$
$\displaystyle\mathfrak{k}^{\mathbb{C}}=\left\\{\left[\begin{matrix}\tilde{A}&\mathbf{0}&\tilde{B}&\mathbf{0}\\\
\mathbf{0}^{T}&t&\mathbf{0}^{T}&0\\\
\tilde{C}&\mathbf{0}&-\tilde{A}^{T}&\mathbf{0}\\\
\mathbf{0}^{T}&0&\mathbf{0}^{T}&-t\end{matrix}\right]\,|\,\tilde{A},\tilde{B},\tilde{C}\in\mathfrak{gl}(n-1,\mathbb{C}),\,\tilde{B}=\tilde{B}^{T},\,\tilde{C}=\tilde{C}^{T},\,t\in\mathbb{C}\right\\}$
$\displaystyle\hphantom{\mathfrak{k}^{\mathbb{C}}}{}\cong\mathfrak{sp}(n-1,\mathbb{C})\oplus\mathfrak{t},$
$\displaystyle\big{(}\mathfrak{k}^{\mathbb{C}}\big{)}^{\perp}=\left\\{\left[\begin{array}[]{c|c|c|c}A&\mathbf{v}_{1}&B&\mathbf{v}_{2}\\\
\hline\cr\mathbf{u}_{1}^{T}&a&\mathbf{u}_{2}^{T}&b\tsep{2pt}\\\ \hline\cr
C&\mathbf{v}_{3}&A^{T}&\mathbf{v}_{4}\tsep{2pt}\\\
\hline\cr\mathbf{u}_{3}^{T}&c&\mathbf{u}_{4}^{T}&a\tsep{2pt}\end{array}\right]\,|\,\mathbf{v}_{i},\mathbf{u}_{i}\in\mathbb{C}^{n-1},\,B=-B^{T},\,C=-C^{T},\right.$
$\displaystyle\left.\hphantom{\big{(}\mathfrak{k}^{\mathbb{C}}\big{)}^{\perp}=}{}a=-\operatorname{Tr}A,\,b,c\in\mathbb{C}\vphantom{\begin{array}[]{c|c|c|c}A&\mathbf{v}_{1}&B&\mathbf{v}_{2}\\\
\hline\cr\mathbf{u}_{1}^{T}&a&\mathbf{u}_{2}^{T}&b\tsep{2pt}\\\ \hline\cr
C&\mathbf{v}_{3}&A^{T}&\mathbf{v}_{4}\tsep{2pt}\\\
\hline\cr\mathbf{u}_{3}^{T}&c&\mathbf{u}_{4}^{T}&a\tsep{2pt}\end{array}}\right\\}.$

The isotropy action
$\rho\colon\mathfrak{k}\to\mathfrak{gl}\big{(}\mathfrak{k}^{\perp}\big{)}$ can
be decomposed into direct sum of two invariant subspaces

$\displaystyle
V_{1}:=\left\\{\left[\begin{array}[]{c|c|c|c}A&\mathbf{0}&B&\mathbf{0}\\\
\hline\cr\mathbf{0}^{T}&0&\mathbf{0}^{T}&b\tsep{2pt}\\\ \hline\cr
C&\mathbf{0}&A^{T}&\mathbf{0}\tsep{2pt}\\\
\hline\cr\mathbf{0}^{T}&c&\mathbf{0}^{T}&0\tsep{2pt}\end{array}\right]\right\\},\qquad
V_{2}:=\left\\{\left[\begin{array}[]{c|c|c|c}0&\mathbf{v}_{1}&0&\mathbf{v}_{2}\\\
\hline\cr\mathbf{u}_{1}&0&\mathbf{u}_{2}&0\\\ \hline\cr
0&\mathbf{v}_{3}&0&\mathbf{v}_{4}\\\
\hline\cr\mathbf{u}_{3}&0&\mathbf{u}_{4}&0\end{array}\right]\right\\},$

and a trivial 1-dimensional representation which will be neglected.

Let $\rho\colon\mathfrak{k}\rightarrow\mathfrak{gl}(V_{1}\oplus V_{2})$ be the
coisotropy representation with the invariant subspaces $V_{i}$ and let
$E_{i}\in V_{i}$. Then obviously
$\operatorname{stab}_{\rho}(E_{1}+E_{2})=\operatorname{stab}_{\tilde{\rho}}(E_{2})$,
where $\tilde{\rho}:=\rho|_{\operatorname{stab}_{\rho}(E_{1})}$.

Take the element

$\displaystyle\mathfrak{k}^{\perp}\ni
E_{1}=\left[\begin{array}[]{c|c|c|c}A&0&0&0\\\ \hline\cr 0&0&0&1\\\ \hline\cr
0&0&A^{T}&0\tsep{2pt}\\\ \hline\cr 0&0&0&0\end{array}\right],$

with $A=\left[\begin{smallmatrix}0&1&&\\\ &\ddots&\ddots&\\\ &&&1\\\ &&&0\\\
\end{smallmatrix}\right]$, the standard nilpotent matrix. Then
$\operatorname{stab}_{\rho}(E_{1})$ consists of the matrices of the form

$\displaystyle Y=\left[\begin{array}[]{c|c|c|c}A&0&B&0\\\ \hline\cr 0&0&0&0\\\
\hline\cr C&0&-A^{T}&0\tsep{2pt}\\\ \hline\cr 0&0&0&0\end{array}\right],$

where

$\displaystyle A=\left[\begin{matrix}a_{1}&a_{2}&\dots&a_{n-1}\\\
&a_{1}&\ddots&\vdots\\\ &&\ddots&a_{2}\\\ &&&a_{1}\end{matrix}\right],\qquad
B=\left[\begin{matrix}b_{1}&\dots&b_{n-2}&b_{n-1}\\\
\vdots&\reflectbox{$\ddots$}&b_{n-1}&\\\ b_{n-2}&\reflectbox{$\ddots$}&&\\\
b_{n-1}&&&\end{matrix}\right],$ $\displaystyle
C=\left[\begin{matrix}&&&c_{1}\\\ &&c_{1}&c_{2}\\\
&\reflectbox{$\ddots$}&\reflectbox{$\ddots$}&\vdots\\\
c_{1}&c_{2}&\dots&c_{n-1}\end{matrix}\right].$

Choose $E_{2}\in V_{2}$ with $\mathbf{u}_{1}=\mathbf{v}_{1}^{T}=(1,\dots,1)$
and trivial $\mathbf{v}_{i}$, $\mathbf{u}_{i}$, $i>1$. Then for
$Y\in\operatorname{stab}_{\rho}(E_{1})$ we have

$\displaystyle[E_{2},Y]=\left[\begin{array}[]{c|c|c|c}0&\mathbf{u}&0&0\\\
\hline\cr\mathbf{v}&0&\mathbf{w}_{2}&0\\\ \hline\cr 0&\mathbf{w}_{1}&0&0\\\
\hline\cr 0&0&0&0\end{array}\right],$

where

$\displaystyle\mathbf{u}=\left[\begin{array}[]{l}a_{1}+\dots+a_{n-2}+a_{n-1}\\\
a_{1}+\dots+a_{n-2}\\\ \vdots\\\
a_{1}\end{array}\right],\qquad\mathbf{v}=-\left[\begin{array}[]{l}a_{1}\\\
a_{1}+a_{2}\\\ \vdots\\\ a_{1}+a_{2}+\dots+a_{n-1}\end{array}\right]^{T},$
$\displaystyle\mathbf{w}_{1}=-\left[\begin{array}[]{r}c_{1}+c_{2}+\dots+c_{n-1}\\\
c_{2}+\dots+c_{n-1}\\\ \vdots\\\
c_{n-1}\end{array}\right],\qquad\mathbf{w}_{2}=\left[\begin{array}[]{r}b_{1}+b_{2}+\dots+b_{n-1}\\\
b_{2}+\dots+b_{n-1}\\\ \vdots\\\ b_{n-1}\end{array}\right]^{T}.$

If $Y\in\operatorname{stab}_{\tilde{\rho}}(E_{2})$, then $a_{i}=0$, $b_{i}=0$,
and $c_{i}=0$, which implies $\operatorname{stab}_{\rho}(E)=\\{0\\}$, where
$E:=E_{1}+E_{2}$.

_Case $(b)$:
$(\mathfrak{d}_{n+1},\mathfrak{b}_{n},\mathfrak{a}_{n}\oplus\mathfrak{t})$._

$\displaystyle\mathfrak{g}^{\mathbb{C}}=\left\\{\left[\begin{matrix}Z_{1}&Z_{2}\\\
Z_{3}&-{Z_{1}}^{T}\end{matrix}\right]\,|\,Z_{i}\in\mathfrak{gl}(n+1,\mathbb{C}),\,Z_{2}=-Z_{2}^{T},\,Z_{3}=-Z_{3}^{T}\right\\}\cong\mathfrak{so}(2n+2,\mathbb{C}),$
$\displaystyle\mathfrak{g}_{1}^{\mathbb{C}}=\left\\{\left[\begin{array}[]{c|c|c|c}0&\mathbf{u}^{T}&0&\mathbf{v}^{T}\\\
\hline\cr-\mathbf{v}&W_{1}&-\mathbf{v}&W_{2}\\\ \hline\cr
0&\mathbf{u}^{T}&0&\mathbf{v}^{T}\tsep{2pt}\\\
\hline\cr-\mathbf{u}&W_{3}&-\mathbf{u}&-W_{1}^{T}\tsep{2pt}\end{array}\right]\\!|\,W_{2}=-W^{T}_{2},\,W_{3}=-W_{3}^{T},\,\mathbf{v},\mathbf{u}\in\mathbb{C}^{n}\\!\right\\}\\!\cong\mathfrak{so}(2n+1,\mathbb{C}),$
$\displaystyle\mathfrak{g}_{2}^{\mathbb{C}}=\left\\{\left[\begin{matrix}\tilde{A}&0\\\
0&-\tilde{A}^{T}\end{matrix}\right]\in\mathfrak{g}\,|\,\tilde{A}\in\mathfrak{gl}(n+1,\mathbb{C})\right\\}\cong\mathfrak{gl}(n+1,\mathbb{C}),$
$\displaystyle\mathfrak{k}^{\mathbb{C}}=\left\\{Y:=\left[\begin{array}[]{c|c|c|c}0&\mathbf{0}^{T}&0&\mathbf{0}^{T}\\\
\hline\cr\mathbf{0}&A&\mathbf{0}&0\\\ \hline\cr
0&\mathbf{0}^{T}&0&\mathbf{0}^{T}\tsep{2pt}\\\
\hline\cr\mathbf{0}&0&\mathbf{0}&-{A}^{T}\tsep{2pt}\end{array}\right]\,|\,A\in\mathfrak{gl}(n,\mathbb{C})\right\\}\cong\mathfrak{gl}(n,\mathbb{C}),$
$\displaystyle\big{(}\mathfrak{k}^{\mathbb{C}}\big{)}^{\perp}=\left\\{Z:=\left[\begin{array}[]{c|c}\begin{array}[]{c|c}a&\mathbf{u}^{T}\\\
\hline\cr-\mathbf{v}&0\end{array}&Z_{2}\\\ \hline\cr
Z_{3}&\begin{array}[]{c|c}-a&\mathbf{v}^{T}\tsep{2pt}\\\
\hline\cr-\mathbf{u}&0\end{array}\end{array}\right]\,|\,Z_{2}=-Z_{2}^{T},\,Z_{3}=-Z_{3}^{T},\,\mathbf{u},\mathbf{v}\in\mathbb{C}^{n},\,a\in\mathbb{C}\right\\}.$

For $Y\in\mathfrak{k}^{\mathbb{C}}$ and
$Z\in\big{(}\mathfrak{k}^{\mathbb{C}}\big{)}^{\perp}$ as above with

$\displaystyle Z_{2}=\left[\begin{array}[]{c|c}0&\mathbf{w}^{T}_{2}\\\
\hline\cr-{\mathbf{w}}_{2}&B\end{array}\right],\qquad
Z_{3}=\left[\begin{array}[]{c|c}0&\mathbf{w}^{T}_{3}\\\
\hline\cr-{\mathbf{w}}_{3}&C\end{array}\right],$

where $B$ and $C$ are skew-symmetric $n\times n$ matrices, one has

$\displaystyle[Y,Z]=\left[\begin{array}[]{c|c|c|c}0&-\mathbf{u}^{T}A&0&(A\mathbf{w}_{2})^{T}\\\
\hline\cr A\mathbf{v}&0&-A\mathbf{w}_{2}&AB+BA^{T}\tsep{2pt}\\\ \hline\cr
0&-\mathbf{w}_{3}^{T}A&0&-(A\mathbf{v})^{T}\tsep{2pt}\\\ \hline\cr
A^{T}\mathbf{w}_{3}&-(A^{T}C+CA)&A^{T}\mathbf{u}&0\tsep{2pt}\end{array}\right].$

We will prove the triviality of the stabilizer of the element
$E=\left[\begin{array}[]{c|c}0&J\\\ \hline\cr
J&0\end{array}\right]\in\mathfrak{k}^{\perp}$, where
$J=\left[\begin{smallmatrix}0&1&&\\\ -1&&&\\\ &&\ddots&1\\\
&&-1&0\end{smallmatrix}\right]$. Observe that conditions
$A\mathbf{w}_{2}=0,A^{T}\mathbf{w}_{3}=0$ for
$\mathbf{w}_{2}=\mathbf{w}_{3}=(1,0,\dots,0)^{T}$ imply that $a_{1,i}=0$,
$a_{i,1}=0$, where we put $A=||a_{ij}||_{i,j=1}^{n}$. Thus for any
$Y\in\operatorname{stab}(E)$ the matrix $A$ is of the form
$\left[\begin{array}[]{c|c}0&0\\\ \hline\cr 0&A_{n-1}\end{array}\right]$ where
$A_{n-1}\in\mathfrak{gl}(n-1,\mathbb{C})$. Next, such
$Y\in\operatorname{stab}(E)$ if and only if simultaneously

$\displaystyle AB+BA^{T}=\left[\begin{matrix}0&0&0&0&0&0\\\
0&0&a_{22}&a_{32}&\dots&a_{n,2}\\\ 0&-a_{22}&0&*&*&*\\\ 0&-a_{32}&*&0&*&*\\\
0&\vphantom{\int^{0}}\smash[t]{\vdots}&*&*&&*\\\
0&-a_{n,2}&*&*&*&0\end{matrix}\right]=0$

and

$\displaystyle A^{T}C+CA=\left[\begin{matrix}0&0&0&0&0&0\\\
0&0&a_{22}&a_{23}&\dots&a_{2,n}\\\ 0&-a_{22}&0&*&*&*\\\ 0&-a_{23}&*&0&*&*\\\
0&\vphantom{\int^{0}}\smash[t]{\vdots}&*&*&&*\\\
0&-a_{2,n}&*&*&*&0\end{matrix}\right]=0.$

Therefore $A$ has to be of the form $\left[\begin{array}[]{c|c}0&0\\\
\hline\cr 0&A_{n-2}\end{array}\right]$ with
$A_{n-2}\in\mathfrak{gl}(n-2,\mathbb{C})$. By induction we conclude that
$A=0$, i.e., $\operatorname{stab}(E)=\\{0\\}$. ∎

##  7 Concluding remarks

We would like to note that in the proof of Theorem 5.1 we tried to maximally
accurately indicate the open dense set $W$ such that the reduced bi-Poisson
structure is Kronecker at any point of $p(W)$ (and _is not Kronecker_ in the
complement). This is important from the point of view of study qualitative
analysis of the geodesic flow since outside this set the singularities of the
corresponding lagrangian fibration appear (cf. [4]).

The assumption of compactness of the Lie groups $G$ and $K$ which appeared in
Theorem 5.1 (see also Theorem 3.7) was used in order to guarantee (1) the
existence of complexification of the homogeneous space $G/K$ and as a
consequence of other related objects; (2) the existence of an $G$-invariant
open dense set $M_{0}$ in $M=T^{*}(G/K)$ (the set $M_{H}$) such that the orbit
space $M_{0}/G$ is a smooth manifold. In fact, the assumption of compactness
can be essentially weakened (since conditions (1) and (2) can be achieved for
a wider class of Lie groups) preserving the conclusion of the theorem. We did
not discuss these weaker assumptions as the main application (Theorem 6.2) is
aimed in the class of compact homogeneous spaces.

The assumption that the action of $G$ on $T^{*}(G/K)$ is free, which is
essential in Theorems 3.7 and 5.1, can be bypassed by a special reduction to
smaller groups instead of $G$ and $K$, see [18] and [17].

We would like to mention that a matter of further research is the study of the
cases when the necessary and sufficient conditions (5.3) are not satisfied. In
such cases the canonical commuting set of functions $\mathcal{B}$ related to
the reduced bi-Hamiltonian structure _is not complete_. However, based on the
experience from the study [26] of bi-Hamiltonian structures related with Lie
pencils (hence in fact reductions of $(T^{*}(G/K),\Pi,\Pi_{1})$ with trivial
$K$) one could expect additional symmetries in this case and, as a
consequence, additional Noether integrals. One can ask for algebraic
conditions sufficient for the completeness of the family $\mathcal{B}$
enlarged by these integrals.

Finally, it is worth mentioning that our theory related to the triples
$(\mathfrak{g},\mathfrak{g}_{1},\mathfrak{g}_{2})$ (see Section 6) is very
close to that appearing within the generalized chain method [5, 6]. Note
however, that the assumption of maximality of rank of the symmetric space,
which is essential in [6, Theorem 8.6], is not satisfied for our symmetric
pairs $(\mathfrak{g},\mathfrak{g}_{i})$, i.e., the overlap between the
theories mentioned is minimal (and requires further study).

##  Appendix A Compact real forms of the triples
$\boldsymbol{(\mathfrak{g},\mathfrak{g}_{1},\mathfrak{g}_{2})}$
and inertia operators

Below we list explicit realizations of the compact real forms for the triples
$(\mathfrak{g},\mathfrak{g}_{1},\mathfrak{g}_{2})$ used in Theorem 6.2 as well
as the decompositions of the subspace
$\mathfrak{k}^{\perp}=\mathfrak{k}_{1}\oplus\mathfrak{k}_{2}$ complementary to
the subspace $\mathfrak{k}$ with respect to the Killing form induced by the
decompositions $\mathfrak{g}=\mathfrak{g}_{1}+\mathfrak{g}_{2}$, where
$\mathfrak{k}_{i}=\mathfrak{g}_{i}\cap\mathfrak{k}^{\perp}$, and formulae for
the “inertia operators”
$n_{\mathfrak{k}^{\perp}}+n^{*}_{\mathfrak{k}^{\perp}}\colon\mathfrak{k}^{\perp}\to\mathfrak{k}^{\perp}$
induced by the operator
$n_{\mathfrak{k}^{\perp}}\colon\mathfrak{k}^{\perp}\to\mathfrak{k}^{\perp}$,
$n_{\mathfrak{k}^{\perp}}|_{\mathfrak{k}_{i}}=\lambda_{i}\operatorname{Id}_{\mathfrak{k}_{i}}$.
We also note that in both cases below the inertia operators are positive
definite under the restrictions

$\displaystyle 0<\lambda_{1},\qquad
0<\lambda_{2},\qquad{\frac{\lambda_{1}}{\big{(}\sqrt{2}+1\big{)}^{2}}}<\lambda_{2}<{\frac{\lambda_{1}}{\big{(}\sqrt{2}-1\big{)}^{2}}}.$

_Case
$(\mathfrak{a}_{2n-1},\mathfrak{c}_{n},\mathfrak{a}_{2n-2}\oplus\mathfrak{t})$._

$\displaystyle\mathfrak{g}=\mathfrak{su}(2n)=\big{\\{}A\in\mathfrak{sl}(2n,\mathbb{C})\,|\,A=-\overline{A}^{T}\big{\\}},$
$\displaystyle\mathfrak{g}_{1}^{\prime}=\left\\{\left[\begin{matrix}Z_{1}&Z_{2}\\\
-\overline{Z}_{2}&\overline{Z}_{1}\end{matrix}\right]\,|\,Z_{1}=-\overline{Z}_{1}^{T},\,Z_{2}=Z_{2}^{T}\right\\}\cong\mathfrak{sp}(n),$
$\displaystyle\mathfrak{g}_{2}=\left\\{\left[\begin{matrix}Z&\mathbf{0}\\\
\mathbf{0}^{T}&-t\end{matrix}\right]\,|\,Z\in\mathfrak{u}(2n-1),\,t=\operatorname{Tr}Z\right\\}\cong\mathfrak{su}(2n-1)\oplus\mathfrak{t},$
$\displaystyle\mathfrak{k}=\left\\{\left[\begin{array}[]{c|c|c|c}Z_{1}&\mathbf{0}&Z_{2}&\mathbf{0}\\\
\hline\cr\mathbf{0}^{T}&{\rm i}t&\mathbf{0}^{T}&0\tsep{2pt}\\\
\hline\cr-\overline{Z}_{2}&\mathbf{0}&\overline{Z}_{1}&\mathbf{0}\tsep{2pt}\\\
\hline\cr\mathbf{0}^{T}&0&\mathbf{0}^{T}&-{\rm
i}t\tsep{2pt}\end{array}\right]\,|\,Z_{i}\in\mathfrak{gl}(n-1,\mathbb{C}),\,Z_{1}=-\overline{Z}_{1}^{T},\,Z_{2}=Z_{2}^{T},\,t\in\mathbb{R}\right\\}$
$\displaystyle\hphantom{\mathfrak{k}}{}\cong\mathfrak{sp}(n-1)\oplus\mathfrak{t},$
$\displaystyle\mathfrak{k}^{\perp}=\left\\{X:=\left[\begin{array}[]{c|c|c|c}Z_{1}&\mathbf{v}&Z_{2}&\mathbf{u}\\\
\hline\cr-\overline{\mathbf{v}}^{T}&a&\mathbf{u_{1}}^{T}&z\tsep{2pt}\\\
\hline\cr\overline{Z}_{2}&-\overline{\mathbf{u}}_{1}&-\overline{Z}_{1}&\mathbf{v}_{1}\tsep{2pt}\\\
\hline\cr-\overline{\mathbf{u}}^{T}&-\overline{z}&-\overline{\mathbf{v}}_{1}^{T}&a\tsep{2pt}\end{array}\right]\,|\,Z_{1}=-\overline{Z}_{1}^{T},\,Z_{2}=-Z_{2}^{T},\,\mathbf{v},\mathbf{u},\mathbf{v}_{1},\mathbf{u}_{1}\in\mathbb{C}^{n-1},\right.$
$\displaystyle\left.\hphantom{\mathfrak{k}^{\perp}=}{}\vphantom{\left[\begin{array}[]{c|c|c|c}Z_{1}&\mathbf{v}&Z_{2}&\mathbf{u}\\\
\hline\cr-\overline{\mathbf{v}}^{T}&a&\mathbf{u_{1}}^{T}&z\tsep{2pt}\\\
\hline\cr\overline{Z}_{2}&-\overline{\mathbf{u}}_{1}&-\overline{Z}_{1}&\mathbf{v}_{1}\tsep{2pt}\\\
\hline\cr-\overline{\mathbf{u}}^{T}&-\overline{z}&-\overline{\mathbf{v}}_{1}^{T}&a\tsep{2pt}\end{array}\right]}{}z\in\mathbb{C},\,a=-\operatorname{Tr}Z_{1}\right\\},$
$\displaystyle\mathfrak{k}_{1}=\left\\{\left[\begin{array}[]{c|c|c|c}0&\mathbf{v}&0&\mathbf{u}\\\
\hline\cr-\overline{\mathbf{v}}^{T}&0&\mathbf{u}^{T}&z\tsep{2pt}\\\ \hline\cr
0&-\overline{\mathbf{u}}&0&\overline{\mathbf{v}}\\\
\hline\cr-\overline{\mathbf{u}}^{T}&-\overline{z}&-\mathbf{v}^{T}&0\tsep{2pt}\end{array}\right]\right\\},\\!\\!\qquad\mathfrak{k}_{2}=\left\\{\left[\begin{array}[]{c|c|c|c}Z_{1}&\mathbf{v}_{1}&Z_{2}&0\\\
\hline\cr-\overline{\mathbf{v}}_{1}^{T}&a&\mathbf{u}_{1}^{T}&0\tsep{2pt}\\\
\hline\cr\overline{Z}_{2}&-\overline{\mathbf{u}}_{1}&-\overline{Z}_{1}&0\tsep{2pt}\\\
\hline\cr 0&0&0&a\\\ \end{array}\right]|\,a=-\operatorname{Tr}Z_{1}\right\\},$
$\displaystyle\frac{1}{2}\big{(}n_{\mathfrak{k}^{\perp}}+n_{\mathfrak{k}^{\perp}}^{*}\big{)}X=\left[\begin{array}[]{c|c|c|c}\lambda_{2}Z_{1}&\begin{array}[]{c}\lambda_{2}\mathbf{v}\\\
+\frac{(\lambda_{1}-\lambda_{2})}{2}\overline{\mathbf{v}}_{1}\end{array}&\lambda_{2}Z_{2}&\begin{array}[]{c}\lambda_{1}\mathbf{u}\\\
+\frac{(\lambda_{1}-\lambda_{2})}{2}{\mathbf{u}}_{1}\end{array}\\\
\hline\cr\begin{array}[]{c}-\lambda_{2}\overline{\mathbf{v}}^{T}\\\
-\frac{(\lambda_{1}-\lambda_{2})}{2}{\mathbf{v}}_{1}^{T}\end{array}&\lambda_{2}a&\begin{array}[]{c}\lambda_{2}\mathbf{u_{1}}^{T}\\\
+\frac{(\lambda_{1}-\lambda_{2})}{2}\mathbf{u}^{T}\end{array}&\lambda_{1}z\\\
\hline\cr\lambda_{2}\overline{Z}_{2}&\begin{array}[]{c}-\lambda_{2}\overline{\mathbf{u}}_{1}\\\
-\frac{(\lambda_{1}-\lambda_{2})}{2}\overline{\mathbf{u}}\end{array}&-\lambda_{2}\overline{Z}_{1}&\begin{array}[]{c}\lambda_{1}\mathbf{v}_{1}\\\
+\frac{(\lambda_{1}-\lambda_{2})}{2}\overline{\mathbf{v}}\end{array}\\\
\hline\cr\begin{array}[]{c}-\lambda_{1}\overline{\mathbf{u}}^{T}\\\
-\frac{(\lambda_{1}-\lambda_{2})}{2}\overline{\mathbf{u}}_{1}^{T}\end{array}&-\lambda_{1}\overline{z}&\begin{array}[]{c}-\lambda_{1}\overline{\mathbf{v}}_{1}^{T}\\\
-\frac{(\lambda_{1}-\lambda_{2})}{2}\mathbf{v}^{T}\end{array}&\lambda_{2}a\end{array}\right].$

_Case
$(\mathfrak{d}_{n+1},\mathfrak{b}_{n},\mathfrak{a}_{n}\oplus\mathfrak{t})$_,
cf. [11, solution to Exercise B.3, Chapter VI].

$\displaystyle\mathfrak{g}=\left\\{\left[\begin{matrix}W&Z\\\
\overline{Z}&\overline{W}\end{matrix}\right]\,|\,Z,W\in\mathfrak{gl}(n+1,\mathbb{C}),\,W=-\overline{W}^{T},\,Z=-Z^{T}\right\\}\cong\mathfrak{so}(2n+2,\mathbb{R}),$
$\displaystyle\mathfrak{g}_{1}=\left\\{\left[\begin{array}[]{c|c|c|c}0&\mathbf{u}^{T}&0&\overline{\mathbf{u}}^{T}\\\
\hline\cr-\overline{\mathbf{u}}&W_{1}&-\overline{\mathbf{u}}&Z_{1}\\\
\hline\cr 0&\mathbf{u}^{T}&0&\overline{\mathbf{u}}^{T}\tsep{2pt}\\\
\hline\cr-\mathbf{u}&\overline{Z}_{1}&-\mathbf{u}&\overline{W}_{1}\tsep{2pt}\end{array}\right]\,|\,Z_{1},W_{1}\in\mathfrak{gl}(n,\mathbb{C}),\,Z_{1}=-Z_{1}^{T},\,W_{1}=-\overline{W}_{1}^{T},\,\mathbf{u}\in\mathbb{C}^{n}\right\\}$
$\displaystyle\hphantom{\mathfrak{g}_{1}}{}\cong\mathfrak{so}(2n+1,\mathbb{R}),$
$\displaystyle\mathfrak{g}_{2}=\left\\{\left[\begin{matrix}A&0\\\
0&-A^{T}\end{matrix}\right]\,|\,A\in\mathfrak{u}(n+1)\right\\}\cong\mathfrak{u}(n+1),$
$\displaystyle\mathfrak{k}=\left\\{\left[\begin{array}[]{c|c|c|c}0&\mathbf{0}^{T}&0&\mathbf{0}^{T}\\\
\hline\cr\mathbf{0}&W_{1}&\mathbf{0}&0\\\ \hline\cr
0&\mathbf{0}^{T}&0&\mathbf{0}^{T}\tsep{2pt}\\\
\hline\cr\mathbf{0}&0&\mathbf{0}&\overline{W}_{1}\tsep{2pt}\end{array}\right]\,|\,W_{1}=-\overline{W}_{1}^{T}\right\\}\cong\mathfrak{u}(n),$
$\displaystyle\mathfrak{k}^{\perp}=\left\\{X:=\left[\begin{array}[]{c|c|c|c}{\rm
i}a&\mathbf{u}^{T}&0&\overline{\mathbf{v}}^{T}\\\
\hline\cr-\overline{\mathbf{u}}&0&-\overline{\mathbf{v}}&Z_{1}\\\ \hline\cr
0&\mathbf{v}^{T}&-{\rm i}a&\overline{\mathbf{u}}^{T}\tsep{2pt}\\\