#  UUITP-54/21

Nijenhuis tensor and invariant polynomials

F. Bonechi111INFN Sezione di Firenze, email<EMAIL_ADDRESS>, J.
Qiu 222Matematiska institutionen, Institutionen för fysik och astronomi,
Uppsala Universitet email<EMAIL_ADDRESS>M. Tarlini333INFN Sezione di
Firenze, email<EMAIL_ADDRESS>E. Viviani444INFN Sezione di Firenze
and Dipartimento di Fisica, Università di Firenze, email:
<EMAIL_ADDRESS>
###### Abstract

We discuss the diagonalization problem of the Nijenhuis tensor in a class of
Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We
study its action on the ring of invariant polynomials of a Thimm chain of
subalgebras. The existence of $\phi\,$-minimal representations defines a
suitable basis of invariant polynomials that completely solves the
diagonalization problem. We prove that such representations exist in the
classical cases AIII, BDI, DIII and CI, and do not exist in the exceptional
cases EIII and EVII. We discuss a second general construction that in these
two cases computes partially the spectrum and hints at a different behavior
with respect to the classical cases.

##  1 Introduction

The notion of symplectic groupoid was introduced by A. Weinstein ([16]) with
the problem of quantization of the underlying Poisson manifold in mind. The
basic idea is that a proper quantization must be compatible with the
additional groupoid structure so that the output of the procedure is an
algebra, regarded as the algebra of operators. If the quantization scheme is
given by geometric quantization, both prequantization and polarization should
be compatible with the groupoid structures. In [17] it has been shown that the
natural notion of compatible prequantization is encoded in a central extension
of the symplectic groupoid; moreover, if the groupoid is prequantizable as a
symplectic manifold, then such compatible prequantization always exists and is
unique. In [8], where this approach has been revived, a natural notion of
multiplicative polarization has been introduced. Finding polarizations that
make geometric quantization work is in general highly demanding, so that there
are basically only two big classes of symplectic manifolds where this program
can be completed: cotangent and Kahler manifolds. The corresponding
polarizations are not in general compatible with the groupoid structure so
that this ambitious picture is bit meagre in terms of concrete examples.

In [2] it was shown that a non degenerate symplectic Poisson-Nijenhuis
structures (PN), a particular example of bihamiltonian geometry, is a source
of such multiplicative polarizations. A symplectic $PN$ structure on a smooth
manifold $M$ consists of a symplectic structure $\omega$ and a Poisson
structure $\pi$ such that $\omega^{-1}$ and $\pi$ are compatible (i.e.
$[\pi,\omega^{-1}]=0$). Among the consequences of this property, the Nijenhuis
tensor $N=\pi\circ\omega$ has vanishing torsion and defines a hierarchy of
compatible Poisson structures $N^{n}\circ\omega^{-1},n=0,1,\ldots$. When $N$
has maximal rank, the hamiltonian forms

$\Omega^{1}_{ham}=\\{\alpha\in\Omega^{1}(M),d\alpha=d_{N}\alpha=0\\}\;,$ (1)

where $d_{N}$ is the algebroid differential, define a lagrangian polarization
that is in general very singular and so unfit for standard geometric
quantization. As explained in [12], the PN structure can be integrated to a
multiplicative PN structure $N_{\cal G}$ on the symplectic groupoid ${\cal G}$
integrating any Poisson structure of the hierarchy so that the polarization
defined by $N_{\cal G}$ is multiplicative too. The polarization defined by the
hamiltonian forms is still singular but it allows one to define the
topological groupoid of Lagrangian leaves and, upon regularity assumptions,
the groupoid of Bohr-Sommerfeld leaves. One can then consider its convolution
algebra as the quantization. This procedure was worked out in [3] for
${\mathbb{C}}P_{n}$ where the Bruhat-Poisson structure ([14]) $\pi$ is
compatible with the Fubiny-Study symplectic form $\omega$. The $C^{*}$-algebra
of the groupoid of Bohr-Sommerfeld leaves coincides with the $C^{*}$-algebra
of quantum homogeneous spaces, as shown in [15]. This is a particular case of
a class of $PN$ structures defined on compact hermitian symmetric spaces
$M_{\phi}$, introduced in [10], where $\pi$ is the Bruhat-Poisson structure
and $\omega$ the KKS symplectic form.

Motivated by the problem of quantization, the study of these PN geometries was
started in [4], where the diagonalization of the Nijenhuis tensor was solved
for the classical cases (AIII, BI, DIII, CI). Let $M_{\phi}=K/K_{\phi}$ be a
compact hermitian symmetric space, where $K$ is a compact simple Lie group,
${\mathfrak{k}}$ its Lie algebra and $K_{\phi}\subset K$ the subgroup
integrating ${\mathfrak{k}}_{\phi}\subset{\mathfrak{k}}$ the Lie subalgebra
defined by the non compact root $\phi$. In particular it was proved that $i$)
if $\lambda$ is an eigenvalue of any solution ${\cal M}$ of the matrix
equation, called the master equation,

$Nd{\cal M}=d{\cal M}^{-}{\cal M}+{\cal M}d{\cal M}^{+}+rd{\cal
M}\,,\;\;\;d{\cal M}^{+}+d{\cal M}^{-}=kd{\cal M}\,$ (2)

for some $k,r\in{\mathbb{C}}$ then $\tilde{\lambda}=k\lambda+r$ is an
eigenvalue of $N$; $ii$) the ${\mathfrak{k}}$-moment map evaluated in a
$\phi\,$-minimal representation (Definition 6.1) satisfies the master
equation. Such representations were provided for the classical cases. By a
case by case analysis, it was proved that for each of the classical cases
there exists a chain of subalgebras

${\mathfrak{k}}\supset{\mathfrak{k}}_{1}\supset{\mathfrak{k}}_{2}\ldots\supset{\mathfrak{k}}_{n}=\mathfrak{t}$

where $\mathfrak{t}$ is a Cartan subalgebra, such that the corresponding
minors of the moment map in the $\phi\,$-minimal representation satisfy (2).
We refer to it as Thimm chain of subalgebras ([6]) The exceptional $EIII,EVII$
spaces were not considered.

In this paper we discuss a new approach to this problem. We focus more on the
structure of the Hamiltonian forms $\Omega^{1}_{ham}(M_{\phi})$ rather than
the eigenvalues of $N$ and describe them in terms of invariant polynomials
with respect to the Thimm chain. Notice that Nijenhuis eigenvalues are in
general only continuous functions and their derivatives have singularities.
Since the multiplicative polarization of the groupoid integrates the
hamiltonian forms on $M_{\phi}$, it is important that we describe them in
terms of global variables. Moreover, we start the study of the exceptional
cases that were not discussed in [4].

Let us describe in some details our results. We consider the subcomplex
$(\Omega^{{\mathfrak{k}}_{1}}_{pol},d)$ of the de Rham complex generated by
the invariant polynomials $S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$ of any
subalgebra ${\mathfrak{k}}_{1}\subset{\mathfrak{k}}$ satisfying the
compatibility condition (11) with the complex structure $J$ of
${\mathfrak{k}}$. We compute in Proposition 4.2 the general formula (13) for
$d_{N}p$ where $p\in S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$ and we write
the sufficient condition (15) that implies that $d_{N}p$ is
${\mathfrak{k}}_{1}$-basic. This condition is satisfied if
${\mathfrak{k}}_{1}={\mathfrak{k}}_{\phi}$ for all compact hermitian symmetric
spaces so that $d_{N}p$ is a ${\mathfrak{k}}_{\phi}$-basic form (Proposition
8.3).

We consider Thimm chains of subalgebras satisfying (11) in Section 5. We are
able to prove that the sufficient condition (15) is satisfied for each
subalgebra provided a $\phi\,$-minimal representation of ${\mathfrak{k}}$
exists and we compute explicitly the action of $d_{N}$ on a suitable basis of
invariant polynomials (Theorem 6.3). In particular, in this case
$(\Omega^{{\mathfrak{k}}_{i}}_{pol},d_{N})$ is a subcomplex of the Nijenhuis
complex and the diagonalization is easily obtained (Corollary 6.4). These
$\phi\,$-minimal representations exist for the classical cases: in particular
the diagonalization results of [4] are better understood in this more
conceptual Lie theoretical framework.

We prove also that these representations do not exist for EIII and EVII
(Proposition 7.1) so that Corollary 6.4 does not apply to these cases. We can
still use the general result of Proposition 8.3 that implies that
$d_{N}S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}}$ are
${\mathfrak{k}}_{\phi}$-basic forms, where
${\mathfrak{k}}_{\phi}={\mathfrak{so}}(10)\oplus{\mathfrak{so}}(2)$ for EIII
and ${\mathfrak{k}}_{\phi}={\mathfrak{e}}_{6}\oplus{\mathfrak{so}}(2)$ for
EVII. We give its explicit description in Section 9 and prove that
$(\Omega^{{\mathfrak{k}}_{\phi}}_{pol},d_{N})$ is a subcomplex of the
Nijenhuis complex for EIII. In the case of EVII a new phenomenon occurs that
suggests that we have to consider a polynomial ring of invariants bigger than
$S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}}$ . These results show that
the exceptional cases behave quite differently with respect to the classical
cases and open the way to the complete solution that we plan to address in a
future paper.

##  2 Compact Hermitian symmetric spaces

We fix here notations and basic facts of compact hermitian symmetric spaces
(see [18, 11]. Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let
${\mathfrak{k}}$ be its compact real form. We denote with $(,)$ the Killing
form. Let us fix a compact Cartan subalgebra
$\mathfrak{t}\subset{\mathfrak{k}}$ and let us denote by
$\mathfrak{t}_{\mathbb{C}}$ its complexification. We denote with $\Delta$,
$\Delta^{+}$ and $\Pi$ the roots, a choice of positive roots and the simple
roots, respectively. For each $\alpha\in\Delta$ we denote with
$\mathfrak{g}_{\alpha}$ the root space and we fix a root vector $e_{\alpha}$.
We define $J:\mathfrak{g}\rightarrow\mathfrak{g}$ as

$J(\mathfrak{t})=0\,,\;\;\;\;J(e_{\alpha})=i\ {{\rm
sign}}(\alpha)e_{\alpha}\;.$ (3)

We fix the normalization of the root vectors as
$(e_{\alpha},e_{-\alpha})(\alpha,\alpha)=2$ so that
$[e_{\alpha},e_{-\alpha}]=h_{\alpha}$, where $h_{\alpha}$ is the coroot vector
of $\alpha$.

A simple root $\phi$ is called non compact (see [18]) if the decomposition of
any $\alpha\in\Delta^{+}$ along $\phi$ is either $0$ or $1$. Roots (and
positive roots) are decomposed accordingly as
$\Delta=\Delta_{c}\coprod\Delta_{nc}$ where $\Delta_{c}$ are the compact roots
and $\Delta_{nc}$ the non compact ones. Since the sum of two compact roots, if
it is a root, is compact, then

${\mathfrak{k}}_{\phi}=\mathfrak{t}\dot{+}_{\alpha\in\Delta_{c}^{+}}(\mathfrak{g}_{\alpha}\dot{+}\mathfrak{g}_{-\alpha})\cap{\mathfrak{k}}$
(4)

is a subalgebra. Its center ${\mathfrak{z}}_{\phi}$ is one dimensional and is
generated by $\rho_{\phi}\in\mathfrak{t}$ defined as $\alpha(\rho_{\phi})=0$
for each $\alpha\in\Delta_{c}$ and normalized as $\phi(\rho_{\phi})=i$. We get
the decomposition

${\mathfrak{k}}_{\phi}={\mathfrak{k}}_{\phi}^{\prime}\oplus{\mathfrak{z}}_{\phi}$
(5)

with ${\mathfrak{k}}_{\phi}^{\prime}$ semisimple. We denote with
${\mathfrak{k}}_{\phi}^{\perp}=\dot{+}_{\alpha\in\Delta_{nc}^{+}}(\mathfrak{g}_{\alpha}\dot{+}\mathfrak{g}_{-\alpha})\cap{\mathfrak{k}}$
the orthogonal complement with respect to the Killing form. We have that

$J|_{{\mathfrak{k}}_{\phi}^{\perp}}=[\rho_{\phi},-]\,,\;\;\;(J|_{{\mathfrak{k}}_{\phi}^{\perp}})^{2}=-1\;.$
(6)

If $K$ and $K_{\phi}\subset K$ integrate ${\mathfrak{k}}$ and
${\mathfrak{k}}_{\phi}$, we denote with $M_{\phi}=K/K_{\phi}$ the
corresponding homogeneous space, that is a compact hermitian symmetric space.
After identifying ${\mathfrak{k}}$ with ${\mathfrak{k}}^{*}$, $M_{\phi}$ can
be realized as an adjoint orbit $M_{\phi}=K\rho_{\phi}K^{-1}$. We denote with
$\omega$ the Konstant-Kirillov-Souriau symplectic form and with
$\mu:M_{\phi}\rightarrow{\mathfrak{k}}$ the moment map for the hamiltonian
$K$-action. We will denote with $X^{\sharp}$ the fundamental vector field
associated to $X\in{\mathfrak{k}}$ and with $v_{f}=\omega^{-1}df$ the
hamiltonian vector field of $f\in C^{\infty}(M_{\phi})$.

Let us consider the non compact real form

$\mathfrak{g}_{0}={\mathfrak{k}}_{\phi}\dot{+}i{\mathfrak{k}}_{\phi}^{\perp}$

of $\mathfrak{g}$ where the Cartan involution $\theta$ is defined as
$\theta|_{{\mathfrak{k}}_{\phi}}={\rm id}$ and
$\theta_{{\mathfrak{k}}_{\phi}^{\perp}}=-{\rm id}$. A $\theta$-stable Cartan
subalgebra ${\mathfrak{h}}$ of $\mathfrak{g}_{0}$ is decomposed as
${\mathfrak{h}}=\mathfrak{t}_{0}\oplus i{\mathfrak{a}}$; it is maximally non
compact when the dimension of ${\mathfrak{a}}$ is as large as possible. A
maximal abelian subalgebra
${\mathfrak{a}}\subset{\mathfrak{k}}_{\phi}^{\perp}$ is unique up to
$K_{\phi}$-conjugation (see [11]). We refer to $\dim{\mathfrak{a}}$ as the
rank of $M_{\phi}$.

Here we list the classification of compact hermitian symmetric spaces.

  * $AIII$)

${\mathfrak{k}}={\mathfrak{su}}(n+1)$ and $\phi=\alpha_{i}$,
${\mathfrak{k}}_{\phi}={\mathfrak{s}}({\mathfrak{u}}(i)\oplus{\mathfrak{u}}(n+1-i))$.
The rank is ${\rm min}\\{i,n+1-i\\}$.

  * $BDI$)

${\mathfrak{k}}={\mathfrak{so}}(n+2)$ and $\phi=\alpha_{1}$,
${\mathfrak{k}}_{\phi}={\mathfrak{so}}(n)\oplus{\mathfrak{so}}(2)$. The rank
is $2$.

  * $DIII$)

${\mathfrak{k}}={\mathfrak{so}}(2n)$ and $\phi=\alpha_{n}$,
${\mathfrak{k}}_{\phi}={\mathfrak{u}}(n)$. The rank is $[n/2]$.

  * $CI$)

${\mathfrak{k}}={\mathfrak{sp}}(n)$, $\phi=\alpha_{n}$ and
${\mathfrak{k}}_{\phi}={\mathfrak{u}}(n)$. The rank is $n$.

  * $EIII$)

${\mathfrak{k}}={\mathfrak{e}}_{6}$, $\phi=\alpha_{6}$ and
${\mathfrak{k}}_{\phi}={\mathfrak{so}}(10)\oplus{\mathfrak{so}}(2)$. The rank
is $2$.

  * $EVII$)

${\mathfrak{k}}={\mathfrak{e}}_{7}$, $\phi=\alpha_{1}$ and
${\mathfrak{k}}_{\phi}={\mathfrak{e}}_{6}\oplus{\mathfrak{so}}(2)$. The rank
is $3$.

We mark in Figure 1 the non compact roots $\phi$ in the Dynkin diagram of the
complexified Lie algebras $\mathfrak{g}$.

$A_{n}$(45,0)(20,0)
,2(-5,0)(5,0),2(-5,0)(5,0),(35,10)(20,0)$\scriptscriptstyle\alpha_{1}$,
$\scriptscriptstyle\alpha_{2}$,$\scriptscriptstyle\alpha_{i}$,
$\scriptscriptstyle\alpha_{n{-}1}$,$\scriptscriptstyle\alpha_{n}$$\scriptstyle\uparrow$

$B_{n}$(45,0)(20,0) ,2(-5,0)(5,0),(35,10)(20,0)$\scriptscriptstyle\alpha_{1}$,
$\scriptscriptstyle\alpha_{2}$,$\scriptscriptstyle\alpha_{n{-}2}$,
$\scriptscriptstyle\alpha_{n{-}1}$,$\scriptscriptstyle\alpha_{n}$$\scriptstyle\uparrow$

$C_{n}$(45,0)(20,0) ,2(-5,0)(5,0),(35,10)(20,0)$\scriptscriptstyle\alpha_{1}$,
$\scriptscriptstyle\alpha_{2}$,$\scriptscriptstyle\alpha_{n{-}2}$,
$\scriptscriptstyle\alpha_{n{-}1}$,$\scriptscriptstyle\alpha_{n}$$\scriptstyle\uparrow$

$D_{n}$(45,0)(20,0) ,2(-5,0)(5,0),(35,10)(20,0)$\scriptscriptstyle\alpha_{1}$,
$\scriptscriptstyle\alpha_{2}$,$\scriptscriptstyle\alpha_{n{-}3}$,
$\scriptscriptstyle\alpha_{n{-}2}$,
$\scriptscriptstyle\alpha_{n{-}1}$$\scriptscriptstyle\alpha_{n}$$\scriptstyle\uparrow$$\scriptstyle\leftarrow$$\scriptstyle\leftarrow$

$E_{6}$(45,0)(20,0) ,,,(35,10)(20,0)$\scriptscriptstyle\alpha_{1}$,
$\scriptscriptstyle\alpha_{2}$,$\hskip 11.0pt\scriptscriptstyle\alpha_{3}$,
$\scriptscriptstyle\alpha_{5}$,$\scriptscriptstyle\alpha_{6}$$\scriptscriptstyle\alpha_{4}$$\scriptstyle\uparrow$$\scriptstyle\uparrow$

$E_{7}$(45,0)(20,0) ,,,,(35,10)(20,0)$\scriptscriptstyle\alpha_{1}$,
$\scriptscriptstyle\alpha_{2}$,$\scriptscriptstyle\alpha_{3}$, $\hskip
8.0pt\scriptscriptstyle\alpha_{4}$,$\scriptscriptstyle\alpha_{6}$,$\scriptscriptstyle\alpha_{7}$$\hskip
18.0pt\scriptscriptstyle\alpha_{5}$$\scriptstyle\uparrow$

Figure 1: Dynkin diagrams with the non compact roots marked.

##  3 Nijenhuis tensor on compact hermitian symmetric spaces

We collect here the basic facts that we need about Poisson-Nijenhuis geometry
(see for instance [2] for more details) and we introduce the examples we are
going to study. For background in Poisson-Lie groups and Poisson homogeneous
spaces see for instance [13]. Let $M$ be a smooth manifold, $\omega$ be a
symplectic form and $\pi$ be a Poisson tensor on $M$ that are compatible, i.e.

$[\omega^{-1},\pi]=0\;.$ (7)

As a consequence the $(1,1)$-tensor $N=\pi\circ\omega$ has vanishing torsion

$T(N)(v_{1},v_{2})=N([Nv_{1},v_{2}]+[v_{1},Nv_{2}]-N[v_{1},v_{2}])-[Nv_{1},Nv_{2}]=0\;.$
(8)

We remark that the PN geometry defined by (7) has been recently generalized in
terms of compatibility between the Nijenhuis tensor and Dirac structures,
together with the lift to Lie groupoids, in [5].

Let $\Omega(M)$ denote the graded ring of differentiable forms and let
$\iota_{N}:\Omega(M)\rightarrow\Omega(M)$ the degree zero derivation defined
as $\iota_{N}(f)=0$ and $\iota_{N}(\nu)=N^{t}\nu$ for all $f\in C^{\infty}(M)$
and $\nu\in\Omega^{1}(M)$. As a consequence of the vanishing of the Nijenhuis
torsion

$d_{N}=[d,\iota_{N}]=d\iota_{N}-\iota_{N}d$ (9)

squares to zero. Remark that $d_{N}f=Ndf$ for $f\in C^{\infty}(M)$. We refer
to the complex $\Omega_{N}=(\Omega(M),d_{N})$ as the Nijenhuis complex. We
have clearly that $[d,d_{N}]=0$; Hamiltonian forms defined in (1) are then
those one forms that are closed with respect to both $d$ and $d_{N}$.

The symplectic form $\omega:TM\rightarrow T^{*}M$ defines an isomorphism of
complexes between $(\Omega(M),d_{N})$ and the complex $(\Gamma(\Lambda
TM),d_{\pi})$ computing the Lichnerowicz Poisson cohomology of $\pi$.

Let $K$ a compact simple Lie group; we denote with ${\mathfrak{k}}$ its Lie
algebra and with $\mathfrak{t}$ a choice of Cartan subalgebra. Let $\phi$ be a
non compact root, ${\mathfrak{k}}_{\phi}\subset{\mathfrak{k}}$ the
corresponding Lie subalgebra and $K_{\phi}\subset K$ the subgroup. Let
$\rho_{\phi}$ be the generator of the one dimensional center
${\mathfrak{z}}(k_{\phi})$ normalized by $\phi(\rho_{\phi})=i$. The
homogeneous space $K/K_{\phi}$ can be realized as the adjoint orbit $M_{\phi}$
of $\rho_{\phi}$, endowing it with the KKS symplectic form $\omega$. The
moment map of the $K$-action is denoted

$\mu:M_{\phi}\rightarrow{\mathfrak{k}}^{*}\equiv{\mathfrak{k}},$

where the identification is done thanks to the Killing form.

The standard Poisson-Lie structure $\pi_{K}$ on $K$ is defined for each $k\in
K$ as

$\pi_{K}(k)=l_{k}(r)-r_{k}(r)$

where $r$ is the classical $r$-matrix

$r=\frac{i}{2}\sum_{\alpha\in\Delta^{+}}e_{\alpha}\wedge e_{-\alpha}\;.$

The subgroup $K_{\phi}$ is a Poisson subgroup and induces a Poisson structure
on $M_{\phi}$ that we denote with $\pi$ and we call the Bruhat-Poisson
structure. The homogeneous $K$-action on $M_{\phi}$ is a Poisson action, i.e.
for each $X\in{\mathfrak{k}}$ we have that

$[X^{\sharp},\pi]=\delta(X)^{\sharp}\;,$

where $\delta:{\mathfrak{k}}\rightarrow\Lambda^{2}{\mathfrak{k}}$ is the
cocycle encoding the dual Lie algebra on ${\mathfrak{k}}^{*}$ (see Thm.2.6 in
[14]).

The compatibility between the KKS symplectic from and the Bruhat-Poisson
structure $\pi$ has been proven in [10].

We skip all details concerning the Bruhat-Poisson structure, that can be found
for instance in [14]. In this paper we will only need the following formula
that has been proven in Theorem 6.1 of [4]

$d_{N}\mu=d\mu-[J(d\mu),\mu]\;.$ (10)

The following Lemma is a consequence of the fact that the $K$-action on
$M_{\phi}$ is a Poisson action. Let ${\mathfrak{k}}_{1}\subset{\mathfrak{k}}$
any Lie subalgebra and let $(\Omega^{{\mathfrak{k}}_{1}},d)$ be the complex of
${\mathfrak{k}}_{1}$ invariant forms.

######  Lemma 3.1.

$(\Omega^{{\mathfrak{k}}_{1}},d_{N})$ is a subcomplex of the Nijenhuis
complex.

Proof. Since the ${\mathfrak{k}}$-action (and then ${\mathfrak{k}}_{1}$) is
hamiltonian, the cochain map $\omega:(\Omega,d_{N})\rightarrow(\Gamma(\Lambda
TM),d_{\pi})$ exchanges the ${\mathfrak{k}}$-actions via Lie derivatives on
forms and multivector fields. So it is enough to prove the statement on the LP
complex. Indeed, for each $X\in{\mathfrak{k}}_{1}$ and $A\in\Gamma(\Lambda
TM)$ such that $L_{X^{\sharp}}(A)=0$ we have

$L_{X^{\sharp}}(d_{\pi}(A))=[X^{\sharp},[\pi,A]]=[[X^{\sharp},\pi],A]=[\delta(X)^{\sharp},A]=0\,.$

∎

##  4 Nijenhuis operator and invariant polynomials

We saw in Lemma 3.1 that invariant forms with respect to ${\mathfrak{k}}$ (and
to any subalgebra) form a subcomplex of the Nijenhuis complex. Here we ask the
same question about the subcomplex generated by invariant polynomials.

We limit ourselves to a class of subalgebras that is compatible with the
complex structure $J$ defined in (3). Let
${\mathfrak{k}}_{1}\subset{\mathfrak{k}}$ be a Lie subalgebra and
$K_{1}\subset K$ be the subgroup integrating it. It is clear that
${\mathfrak{k}}_{1}^{\perp}$ is invariant under the adjoint action of
${\mathfrak{k}}_{1}$. Let us ask now that both ${\mathfrak{k}}_{1}$ and
${\mathfrak{k}}_{1}^{\perp}$ are $J$-invariant and that for each
$X\in{\mathfrak{k}}_{1}$ we have that

$[\operatorname{ad}_{X},J|_{k_{1}^{\perp}}]=0\;.$ (11)

We denote with ${\rm pr}_{{\mathfrak{k}}_{1}}$ and ${\rm
pr}_{{\mathfrak{k}}_{1}^{\perp}}$ the orthogonal projections to
${\mathfrak{k}}_{1}$ and ${\mathfrak{k}}_{1}^{\perp}$ respectively.

######  Lemma 4.1.

If ${\mathfrak{k}}_{1}$ satisfies the conditions (11) then, for each
$\xi,\eta\in{\mathfrak{k}}_{1}^{\perp}$ we have that

${\rm pr}_{{\mathfrak{k}}_{1}}([J\xi,\eta]+[\xi,J\eta])=0$

Proof. For each $X\in{\mathfrak{k}}_{1}$ we have that

$(X,[J\xi,\eta]+[\xi,J\eta])=([X,J\xi],\eta)+([X,\xi],J\eta)=(J[X,\xi],\eta)+([X,\xi],J\eta)=0$

where in the second equality we used (11) and in the last one the antisymmetry
of $J$ with respect to the Killing form. ∎

We denote with
$\mu_{{\mathfrak{k}}_{1}}=M_{\phi}\rightarrow{\mathfrak{k}}_{1}$ the moment
map and let $\mu=\mu_{{\mathfrak{k}}_{1}}+\mu_{{\mathfrak{k}}_{1}^{\perp}}$
the decomposition of ${\mathfrak{k}}$-moment map given by
${\mathfrak{k}}={\mathfrak{k}}_{1}\dot{+}{\mathfrak{k}}_{1}^{\perp}$. It is
clear that for each $X\in{\mathfrak{k}}_{1}$

$X^{\sharp}(\mu_{{\mathfrak{k}}_{1}^{\perp}})=[X,\mu_{{\mathfrak{k}}_{1}^{\perp}}]\,,\,\,\,X^{\sharp}(J\mu_{{\mathfrak{k}}_{1}^{\perp}})=[X,J\mu_{{\mathfrak{k}}_{1}^{\perp}}]\,,$
(12)

where the second equality is a consequence of (11).

Let us consider the space ${S}^{r}({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$
of invariant polynomials that are homogeneous of degree $r$ and let
${S}({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}=\oplus_{r\geq
0}{S}^{r}({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}\subset
C^{\infty}({\mathfrak{k}}^{*})$. If
$p\in{S}^{r}({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$, we denote with
$\tilde{p}$ the corresponding $r$-linear application
${\mathfrak{k}}_{1}\otimes\ldots\otimes{\mathfrak{k}}_{1}\rightarrow{\mathbb{C}}$.
In order to avoid cumbersome notations, we use the same symbol $p$ to denote
$\mu^{*}_{{\mathfrak{k}}_{1}}(p)\in C^{\infty}(M_{\phi})$. We denote with
$(\Omega^{{\mathfrak{k}}_{1}}_{b},d)\subset(\Omega^{{\mathfrak{k}}_{1}},d)$
the subcomplex of ${\mathfrak{k}}_{1}$-basic forms and with
$(\Omega^{{\mathfrak{k}}_{1}}_{pol},d)\subset(\Omega^{{\mathfrak{k}}_{1}}_{b},d)$
the subcomplex generated by $S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$. We
would like to study the conditions under which
$\Omega^{{\mathfrak{k}}_{1}}_{pol}$ is a subcomplex of the Nijenhuis complex.
We prove the following preliminary result.

######  Proposition 4.2.

  * $i$)

If ${\mathfrak{k}}_{1}$ satisfies conditions (11) then for each $p\in
S^{r}({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$ we compute

$d_{N}p=dp-r\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\dots,\mu_{{\mathfrak{k}}_{1}},dA_{{\mathfrak{k}}_{1}})\,,$
(13)

where we defined

$A_{{\mathfrak{k}}_{1}}=\frac{1}{2}{\rm
pr}_{{\mathfrak{k}}_{1}}[J\mu_{{\mathfrak{k}}_{1}^{\perp}},\mu_{{\mathfrak{k}}_{1}^{\perp}}]\,.$
(14)

As a consequence $d_{N}p$ is a ${\mathfrak{k}}_{1}$-invariant form.

  * $ii$)

If

$[A_{{\mathfrak{k}}_{1}},\mu_{{\mathfrak{k}}_{1}}]=0$ (15)

then $d_{N}p$ is a ${\mathfrak{k}}_{1}$-basic form.

Proof. We compute from (10)

$d_{N}\mu_{{\mathfrak{k}}_{1}}=d\mu_{{\mathfrak{k}}_{1}}-{\rm
pr}_{{\mathfrak{k}}_{1}}[Jd\mu,\mu]=d\mu_{{\mathfrak{k}}_{1}}-[Jd\mu_{{\mathfrak{k}}_{1}},\mu_{{\mathfrak{k}}_{1}}]-{\rm
pr}_{{\mathfrak{k}}_{1}}[Jd\mu_{{\mathfrak{k}}_{1}^{\perp}},\mu_{{\mathfrak{k}}_{1}^{\perp}}]\;.$

From Lemma 4.1 we see that

${\rm
pr}_{{\mathfrak{k}}_{1}}[Jd\mu_{{\mathfrak{k}}_{1}^{\perp}},\mu_{{\mathfrak{k}}_{1}^{\perp}}]=dA_{{\mathfrak{k}}_{1}}\;.$

As a consequence of (12) we have that for each $X\in{\mathfrak{k}}_{1}$

$X^{\sharp}(A_{{\mathfrak{k}}_{1}})=[X,A_{{\mathfrak{k}}_{1}}]\,.$ (16)

Let $p\in{S}({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$. We compute

$\displaystyle d_{N}p$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{r}\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\dots,d_{N}\mu_{{\mathfrak{k}}_{1}},\ldots,\mu_{{\mathfrak{k}}_{1}})=r\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\dots,\mu_{{\mathfrak{k}}_{1}},d_{N}\mu_{{\mathfrak{k}}_{1}})$
$\displaystyle=$ $\displaystyle
dp-r\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\dots,\mu_{{\mathfrak{k}}_{1}},[Jd\mu_{{\mathfrak{k}}_{1}},\mu_{{\mathfrak{k}}_{1}}]+dA_{{\mathfrak{k}}_{1}})\;.$

By using the invariance of $\tilde{p}$ we get (13).

Since $A_{{\mathfrak{k}}_{1}}$ satisfies (16) and $p$ is an invariant
polynomial, it follows that for each $X\in{\mathfrak{k}}_{1}$

$L_{X^{\sharp}}(d_{N}p)=0\>,$

so that $d_{N}p$ is invariant. Let us assume that
$[\mu_{{\mathfrak{k}}_{1}},A_{{\mathfrak{k}}_{1}}]=0$; from (16) we now
compute

$\iota_{X^{\sharp}}(d_{N}p)=-r\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\dots,\mu_{{\mathfrak{k}}_{1}},[X,A_{{\mathfrak{k}}_{1}}])=-r\sum_{i=1}^{r-1}\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\ldots,[A_{{\mathfrak{k}}_{1}},\mu_{{\mathfrak{k}}_{1}}],\ldots,X)=0.$

∎

######  Corollary 4.3.

Let $q,p\in S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$ then

$\\{q,p\\}_{\pi}=0\;.$ (17)

Let us suppose that (15) holds and let
${\mathfrak{k}}_{2}\subset{\mathfrak{k}}_{1}$. Then (17) holds for each $q\in
S({\mathfrak{k}}_{2})^{{\mathfrak{k}}_{2}}$ and $p\in
S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$.

Proof. Let $v_{q}=\omega^{-1}(dq)$ be the hamiltonian vector field of $q$. We
compute from (13)

$\displaystyle\\{q,p\\}_{\pi}$ $\displaystyle=$
$\displaystyle\iota_{v_{q}}(d_{N}p)=-r\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\ldots,\mu_{{\mathfrak{k}}_{1}},v_{q}(A_{{\mathfrak{k}}_{1}}))=-r\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\ldots,\mu_{{\mathfrak{k}}_{1}},[X_{q},A_{{\mathfrak{k}}_{1}}])$
(18) $\displaystyle=$ $\displaystyle
r\sum_{i=1}^{r-1}\tilde{p}(\mu_{{\mathfrak{k}}_{1}},\ldots[X_{q},\mu_{{\mathfrak{k}}_{1}}],\ldots
A_{{\mathfrak{k}}_{1}})$ (19)

where $X_{q}\in{\mathfrak{k}}_{1}\subset{\mathfrak{k}}$ is defined as

$(Y,X_{q})=\tilde{q}(\mu_{{\mathfrak{k}}_{1}},\ldots,\mu_{{\mathfrak{k}}_{1}},Y)$

for each $Y\in{\mathfrak{k}}_{1}$ so that $X^{\sharp}_{q}=v_{q}$ and the
second step follows from (16). It is now clear that
$[X_{q},\mu_{{\mathfrak{k}}_{1}}]=0$, since for each $Y\in{\mathfrak{k}}_{1}$
we have that

$(Y,[\mu_{{\mathfrak{k}}_{1}},X_{q}])=([Y,\mu_{{\mathfrak{k}}_{1}}],X_{q})=\tilde{q}(\mu_{{\mathfrak{k}}_{1}},\ldots,\mu_{{\mathfrak{k}}_{1}},[Y,\mu_{{\mathfrak{k}}_{1}}])=0$

since $q$ is invariant.

If (15) holds, then by using the invariance of $\tilde{p}$ in the last
equality of the first line of (19), we get that (17) holds also when $q\in
S({\mathfrak{k}}_{2})^{{\mathfrak{k}}_{2}}$ and $p\in
S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$. ∎

##  5 The Thimm chain of subalgebras

We recall here the Thimm method for constructing integrable models (see [6]
for details). Let us consider
${\mathfrak{k}}\supset{\mathfrak{k}}_{1}\supset{\mathfrak{k}}_{2}$ nested Lie
subalgebras and let $q\in S({\mathfrak{k}}_{1})^{{\mathfrak{k}}_{1}}$ and
$p\in S({\mathfrak{k}}_{2})^{{\mathfrak{k}}_{2}}$. Since
$\mu_{{\mathfrak{k}}_{2}}=p_{21}\circ\mu_{{\mathfrak{k}}_{1}}$, where
$p_{21}:{\mathfrak{k}}_{1}^{*}\rightarrow{\mathfrak{k}}_{2}^{*}$ we have that

$\\{\mu^{*}_{{\mathfrak{k}}_{2}}p,\mu^{*}_{{\mathfrak{k}}_{1}}q\\}=\\{\mu^{*}_{{\mathfrak{k}}_{1}}p_{21}^{*}p,\mu^{*}_{{\mathfrak{k}}_{1}}q\\}=\mu_{{\mathfrak{k}}_{1}}^{*}\\{p_{21}^{*}p,q\\}_{{\mathfrak{k}}_{1}^{*}}=0$

since $S^{{\mathfrak{k}}_{1}}({\mathfrak{k}}_{1})\subset
C^{\infty}({\mathfrak{k}}_{1}^{*})$ is the centre of the Poisson algebra. Let
us consider now a chain of Lie subalgebras

${\mathfrak{k}}\equiv{\mathfrak{k}}_{0}\supset{\mathfrak{k}}_{1}\supset\ldots{\mathfrak{k}}_{n}\;$

and let $K_{i}\subset K$ the corresponding subgroups. Let us denote with

$S({\mathfrak{k}}\supset{\mathfrak{k}}_{1}\ldots\supset{\mathfrak{k}}_{n})=\cup_{i}S({\mathfrak{k}}_{i})^{{\mathfrak{k}}_{i}}$
(20)

the union of invariant polynomials of any subalgebra of the chain. By the
above observation, all these polynomials are in involution.

Let us discuss if they are independent and so form an integrable model. The
following two properties are equivalent: $i$) the ring of invariant functions
$C^{\infty}(M)^{{\mathfrak{k}}_{1}}$ is abelian with respect to the Poisson
bracket; $ii$) for each coadjoint orbit $O\subset{\mathfrak{k}}^{*}$ the
symplectic reduction $\mu_{\mathfrak{k}}^{-1}(O)/K$ is a point. If one of the
above condition (and then both) is true then the $K$-action is said to be
multiplicity free ([7]). If for each $K_{i}$\- orbit $O\subset M$, the action
of $K_{i-1}$ on $O$ is multiplicitely free then the hamiltonians in (20)
define an integrable model.

Let us consider now the case $M=M_{\phi}$; we want to see the conditions under
which the hamiltonians (20) commute also with respect to the Poisson structure
$\pi$. Let us choose a non compact root $\phi_{1}$ of ${\mathfrak{k}}$ and let
$\rho_{1}$ the normalized generator of the one dimensional centre
${\mathfrak{z}}_{1}$ of ${\mathfrak{k}}_{\phi_{1}}$. In general this root
$\phi_{1}$ can be different from $\phi$ involved in the definition of the
hermitian symmetric space.

Let
${\mathfrak{k}}_{1}={\mathfrak{k}}_{\phi_{1}}={\mathfrak{k}}_{\phi_{1}}^{\prime}\oplus{\mathfrak{z}}_{1}$
with ${\mathfrak{k}}_{\phi_{1}}^{\prime}$ being simple. Let $\phi_{2}$ be a
non compact root of ${\mathfrak{k}}_{\phi_{1}}^{\prime}$ with $\rho_{2}$ the
normalized generator of the one dimensional center ${\mathfrak{z}}_{2}$ of
${\mathfrak{k}}_{\phi_{2}}\subset{\mathfrak{k}}_{\phi_{1}}^{\prime}$. Let
${\mathfrak{k}}_{2}={\mathfrak{k}}_{\phi_{2}}\oplus{\mathfrak{z}}_{1}={\mathfrak{k}}_{\phi_{2}}^{\prime}\oplus{\mathfrak{z}}_{2}\oplus{\mathfrak{z}}_{1}$.
We get the decomposition

${\mathfrak{k}}={\mathfrak{k}}_{2}\dot{+}{\mathfrak{k}}_{2}^{\perp}={\mathfrak{k}}_{2}\dot{+}({\mathfrak{k}}_{2}^{\perp}\cap{\mathfrak{k}}_{1})\dot{+}{\mathfrak{k}}_{1}^{\perp}={\mathfrak{k}}_{2}\dot{+}({\mathfrak{k}}_{2}^{\perp}\cap{\mathfrak{k}}_{\phi_{1}}^{\prime})\dot{+}{\mathfrak{k}}_{1}^{\perp}\;,$

where the third equality follows because we included ${\mathfrak{z}}_{1}$ in
the definition of ${\mathfrak{k}}_{2}$. We remark that the perpendicular is
always taken in ${\mathfrak{k}}$. We can iterate the procedure and choose
$\phi_{i}$ non compact root of ${\mathfrak{k}}_{\phi_{i-1}}^{\prime}$ with
$\rho_{i}$ the normalized generator of the centre ${\mathfrak{z}}_{i}$ of
${\mathfrak{k}}_{\phi_{i}}\subset{\mathfrak{k}}_{\phi_{i-1}^{\prime}}$ and
define

${\mathfrak{k}}_{i}={\mathfrak{k}}_{\phi_{i}}\oplus{\mathfrak{z}}_{i-1}\oplus\ldots\oplus{\mathfrak{z}}_{1}\;.$
(21)

We get the $i-th$ decomposition of ${\mathfrak{k}}$

${\mathfrak{k}}={\mathfrak{k}}_{i}\dot{+}({\mathfrak{k}}_{i}^{\perp}\cap{\mathfrak{k}}_{\phi_{i-1}}^{\prime})\dot{+}{\mathfrak{k}}_{i-1}^{\perp}={\mathfrak{k}}_{i}\dot{+}({\mathfrak{k}}_{i}^{\perp}\cap{\mathfrak{k}}_{\phi_{i-1}}^{\prime})\dot{+}({\mathfrak{k}}_{i-1}^{\perp}\cap{\mathfrak{k}}_{\phi_{i-2}}^{\prime})\ldots\dot{+}{\mathfrak{k}}_{1}^{\perp}\;.$
(22)

We remark that $J$ acts on each addend
${\mathfrak{k}}_{i}^{\perp}\cap{\mathfrak{k}}_{\phi_{i-1}^{\prime}}$ of (22)
as $\operatorname{ad}_{\rho_{i}}$.

######  Lemma 5.1.

Every subalgebra ${\mathfrak{k}}_{i}$ satisfies condition (11).

Proof. By construction the Cartan subalgebra $\mathfrak{t}$ is included in
${\mathfrak{k}}_{i}$ for each $i$. Moreover, given any root $\alpha$, either
the root vector $e_{\alpha}\in{\mathfrak{k}}_{i}$ or
$e_{\alpha}\in{\mathfrak{k}}_{i}^{\perp}$. This implies that
${\mathfrak{k}}_{i}$ and ${\mathfrak{k}}_{i}^{\perp}$ are $J$-invariant.
Equation (11) is straightforward if
$X\in\mathfrak{t}\subset{\mathfrak{k}}_{i}$. Let
$X=e_{\alpha}\in{\mathfrak{k}}_{i}$ and
$e_{\beta}\in{\mathfrak{k}}_{i}^{\perp}$ be two root vectors. If
$\alpha+\beta$ is not a root then both terms are zero; if $\alpha+\beta$ is a
root then it is non compact and it will be positive if and only if $\beta$ is
positive so that $[X,J(e_{\beta})]=J[X,e_{\beta}]$. ∎

We will see in the examples that, for each simple compact Lie algebra
${\mathfrak{k}}$, it is possible to find such a chain that ends with
${\mathfrak{k}}_{n}=\mathfrak{t}$ the Cartan subalgebra of ${\mathfrak{k}}$.
Such a choice will not be unique. We call such a chain

${\mathfrak{k}}\supset{\mathfrak{k}}_{1}\supset{\mathfrak{k}}_{2}\ldots\supset{\mathfrak{k}}_{n}=\mathfrak{t}$
(23)

a Thimm chain.

##  6 The $\phi\,$-minimal representations

Let $\phi$ be a non compact root of the simple Lie algebra ${\mathfrak{k}}$
and let $\rho_{\phi}$ the normalized generator of the center ${\mathfrak{z}}$
of ${\mathfrak{k}}_{\phi}$. Let $R_{\Lambda}:{\mathfrak{k}}\rightarrow{\rm
End}(V(\Lambda))$ be an irreducible representation of highest weight
$\Lambda\in\mathfrak{t}^{*}$. It is easy to see that $V(\Lambda)$ decomposes
in eigenspaces of $\rho_{\phi}$ as

$V(\Lambda)=\oplus_{\ell\geq 0}V(\Lambda)_{\ell}$ (24)

where $V(\Lambda)_{\ell}$ are ${\mathfrak{k}}_{\phi}$-invariant,
$\rho_{\phi}|_{V(\Lambda)_{\ell}}=\Lambda_{\phi}-i\,\ell$ with
$\Lambda_{\phi}\equiv\Lambda(\rho_{\phi})$ and
$e_{\alpha}:V(\Lambda)_{\ell}\rightarrow V(\Lambda)_{\ell-1}$ for every non
compact positive root $\alpha$.

######  Definition 6.1.

We say that $R_{\Lambda}$ is $\phi\,$-minimal if the sum in (24) runs over
$\ell=0,1$ and $V(\Lambda)_{0}$ is irreducible. In this case we denote
$V_{+}\equiv V(\Lambda)_{0}$ and $V_{-}\equiv V(\Lambda)_{1}$.

Let now $\phi$ be the noncompact root defining the Hermitian symmetric space
$M_{\phi}$ and let us suppose that $V(\Lambda)$ is $\phi\,$-minimal. Since
$M_{\phi}$ is the adjoint orbit of $\rho_{\phi}$, the moment map
$\mu_{\Lambda}=R_{\Lambda}(\mu)$ in the representation $R_{\Lambda}$ satisfies
then

$(\mu_{\Lambda}-\Lambda_{\phi})(\mu_{\Lambda}-\Lambda_{\phi}+i)=\mu_{\Lambda}^{2}-(2\Lambda_{\phi}-i)\mu_{\Lambda}+\Lambda_{\phi}(\Lambda_{\phi}-i)=0\;.$
(25)

Let us consider now the Thimm chain (23) and let us suppose that $V(\Lambda)$
is also $\phi_{1}$-minimal as $V(\Lambda)=V_{+}^{1}\oplus V_{-}^{1}$. We can
go on along the Thimm chain (23) and let us assume that $V_{+}^{1}$ is
$\phi_{2}$-minimal as $V_{+}^{1}=V_{+}^{2}\oplus V_{-}^{2}$ and so on. At
level $i$, we assume that $V_{+}^{i-1}$ is $\phi_{i}$-minimal so that
$V^{i-1}_{+}=V^{i}_{+}\oplus V^{i}_{-}$. At each level $i$, we get the
decomposition of $V$ into ${\mathfrak{k}}_{i}$-representation as

$V(\Lambda)=W^{i}_{+}\oplus W^{i}_{-}$ (26)

where $W^{i}_{+}=V^{i}_{+}$ and $W^{i}_{-}=V^{i}_{-}\oplus
V^{i-1}_{-}\ldots\oplus V^{1}_{-}$. If these conditions are satisfied we say
that the representation $R_{\Lambda}$ is minimal with respect to the chain
(23).

######  Lemma 6.2.

If the representation $R_{\Lambda}$ is minimal with respect to the Thimm chain
(23) then each $\xi\in{\mathfrak{k}}_{i}^{\perp}$ decomposes with respect to
(26) as

$R_{\Lambda}(\xi)=\left(\begin{array}[]{cc}0&\nu\cr-\nu^{\dagger}&\lambda\end{array}\right)\,,\;\;\;R_{\Lambda}(J(\xi))=\left(\begin{array}[]{cc}0&i\nu\cr
i\nu^{\dagger}&J(\lambda)\end{array}\right)$

Proof. Let us consider the decomposition (26) of $V$ in ${\mathfrak{k}}_{i}$
representations. It is clear that
${\mathfrak{k}}_{i}^{\perp}:W_{+}^{i}\rightarrow W_{-}^{i}$. Indeed if
$\xi\in{\mathfrak{k}}_{i}^{\perp}\cap{\mathfrak{k}}_{\phi_{i-1}}^{\prime}$
then $\xi:W_{+}^{i}=V_{+}^{i}\rightarrow V_{-}^{i}\subset W^{i}_{-}$; if
$\xi\in{\mathfrak{k}}_{i-1}^{\perp}$ then $\xi:W_{+}^{i}=V_{+}^{i}\subset
V_{+}^{i-1}\rightarrow V_{-}^{i-1}\subset W^{i}_{-}$. Let $\alpha$ be a
positive root such that
$e_{\alpha}\in({\mathfrak{k}}_{i}^{\perp}\cap{\mathfrak{k}}_{\phi_{i-1}}^{\prime})\otimes{\mathbb{C}}$,
then $e_{\alpha}|_{W_{+}^{i}}=e_{\alpha}|_{V_{+}^{i}}=0$; if
$e_{\alpha}\in{\mathfrak{k}}_{i-1}^{\perp}\otimes{\mathbb{C}}$ then
$e_{\alpha}|_{W_{+}^{i}}=e_{\alpha}|_{V_{+}^{i}}=(e_{\alpha}|_{V^{i-1}_{+}})|_{V_{+}^{i}}=0$.
∎

Let us consider now the the ring of ${\mathfrak{k}}_{i}$-invariant polynomials
generated by the traces in the $W^{i}_{+}$ representation: let us define

$I_{r}^{(i)}(X)=\frac{i^{r}}{r}\textrm{Tr}_{W^{i}_{+}}(X^{r})\;.$ (27)

######  Theorem 6.3.

We compute

$d_{N}I_{r}^{(i)}=-2i\Lambda_{\phi}dI_{r}^{(i)}+2dI_{r+1}^{(i)}\;.$ (28)

Proof. We then have to compute (14) in the representation $W^{i}_{+}$. If we
write

$R_{\Lambda}(\mu_{{\mathfrak{k}}_{i}})=\left(\begin{array}[]{cc}R_{W^{i}_{+}}(\mu_{{\mathfrak{k}}_{i}})&\nu_{i}\cr-\nu_{i}^{\dagger}&R_{W^{i}_{-}}(\mu_{{\mathfrak{k}}_{i}})+\lambda_{i}\end{array}\right)\,,\;\;\;$

then by using Lemma 6.2 we can compute $A_{{\mathfrak{k}}_{i}}$ defined in
(14) in the representation $W_{+}^{i}$ as

$R_{W^{i}_{+}}(A_{{\mathfrak{k}}_{i}})=-i\nu_{i}\nu_{i}^{\dagger}\;.$

From (25) we find that

$R_{W^{i}_{+}}(A_{{\mathfrak{k}}_{i}})=-iR_{W^{i}_{+}}(\mu_{{\mathfrak{k}}_{i}})^{2}+(2i\Lambda_{\phi}+1)R_{W^{i}_{+}}(\mu_{{\mathfrak{k}}_{i}})-\Lambda_{\phi}(1+i\Lambda_{\phi})\;.$
(29)

Let us now insert it in (13) with $p=I_{r}$ and get

$\displaystyle d_{N}I_{r}^{(i)}$ $\displaystyle=$ $\displaystyle
dI_{r}^{(i)}+r\tilde{I_{r}}^{(i)}\left(\mu_{{\mathfrak{k}}_{i}},\ldots,id(\mu_{{\mathfrak{k}}_{i}}^{2})-(2i\Lambda_{\phi}+1)d\mu_{{\mathfrak{k}}_{i}}\right)$
$\displaystyle=$ $\displaystyle
dI_{r}^{(i)}+i^{r+1}\textrm{Tr}_{W^{i}_{+}}(\mu_{{\mathfrak{k}}_{i}}^{r-1}d(\mu_{{\mathfrak{k}}_{i}}^{2}))-(2i\Lambda_{\phi}+1)dI_{r}^{(i)}\;,$

so that we finally get (28). ∎

From (28) it is now easy to prove the following corollary.

######  Corollary 6.4.

  * $i$)

The subcomplex $(\Omega_{{\mathfrak{k}}_{i}}^{pol},d)$ of the de Rham complex
generated by $S({\mathfrak{k}}_{i})^{{\mathfrak{k}}_{i}}$, defines also a
subcomplex $(\Omega_{{\mathfrak{k}}_{i}},d_{N})$ of the Nijenhuis complex (9);

  * $ii$)

if $\lambda$ is an eigenvalue of the moment map $\mu_{k_{i}}$ then

$\tilde{\lambda}=2i(\lambda-\Lambda_{\phi})$ (30)

is a Nijenhuis eigenvalue.

In the next section we will prove that such minimal representations exist in
the classical symmetric spaces so that the above Corollary reproduces the
diagonalization proved case by case in [4].

##  7 Existence of $\phi\,$-minimal representations

Here we discuss the existence of $\phi\,$-minimal representations. We give the
explicit examples for the classical groups and prove that they do not exist in
the exceptional cases.

###  7.1 AIII

Let $M_{\phi}=SU(n)/S(U(k)\times U(n-k))$ and let us consider the root
ordering as in $A_{n-1}$ Dynkin diagram in Figure 1; the non compact root is
$\phi=\alpha_{k}$. In the fundamental representation $f_{\mathfrak{su}(n)}$ of
$\mathfrak{su}(n)$,

$\rho_{\alpha_{k}}=\frac{i}{n}\begin{pmatrix}(n-k)1_{k}&0\\\
0&-k\,1_{n-k}\end{pmatrix},$ (31)

so that $f_{{\mathfrak{su}}(n)}$ is $\alpha_{k}$-minimal and decomposes in
representations of
${\mathfrak{s}}({\mathfrak{u}}(k)\oplus{\mathfrak{u}}(n-k))$ as
$f_{\mathfrak{su}(n)}=V_{+}\oplus V_{-}$, where

$\displaystyle V_{+}=(f_{\mathfrak{u}(k)},0_{\mathfrak{u}(n-k)}),$
$\displaystyle V_{-}=(0_{\mathfrak{u}(k)},f_{\mathfrak{u}(n-k)}),$

with $f_{\mathfrak{u}(k)}$ and $0_{\mathfrak{u}(k)}$ respectively the
fundamental and the trivial representation. The eigenvalue of
$\rho_{\alpha_{k}}$ on $V_{+}$ is then

$\Lambda_{\alpha_{k}}=i\frac{n-k}{n}$ (32)

Now, let us consider the Thimm chain of subalgebras

$\mathfrak{s}{\mathfrak{u}}(n)\supset\mathfrak{s}({\mathfrak{u}}(n-1)\oplus{\mathfrak{u}}(1))\supset...\supset\mathfrak{s}({\mathfrak{u}}(n-i)\oplus{\mathfrak{u}}(1)^{i})\supset...\supset{\mathfrak{u}}(1)^{n-1}.$
(33)

Let us choose $\phi_{1}=\alpha_{n-1}$ so that $f_{{\mathfrak{su}}(n)}$
decomposes with respect to
${\mathfrak{s}}({\mathfrak{u}}(n-1)\oplus{\mathfrak{u}}(1))$ as
$f_{\mathfrak{su}(n)}=V^{1}_{+}\oplus V^{1}_{-}$, with

$V^{1}_{+}=\left(f_{\mathfrak{u}(n-1)},0\right).$

We obtain for the $i$-th step of the chain (33) that $f_{{\mathfrak{su}}(n)}$
decomposes with respect to
${\mathfrak{s}}({\mathfrak{u}}(n-i)\oplus{\mathfrak{u}}(1)^{i})$ as

$f_{{\mathfrak{su}}(n)}=W^{i}_{+}\oplus W^{i}_{-},$

with

$W^{i}_{+}\equiv\Big{(}f_{\mathfrak{u}(n-i)},\underbrace{0,...,0}_{i-\text{times}}\Big{)},$

that is $\phi_{i+1}=\alpha_{n-1-i}\,$-minimal, where $\alpha_{n-1-i}$ is a
root of ${\mathfrak{su}}(n-i)$. Then, the fundamental representation
$f_{\mathfrak{su}(n)}$ of $\mathfrak{su}(n)$ is minimal with respect to the
chain of subalgebras (33). Formula (28) then gives

$d_{N}I_{r}^{(i)}=2\frac{n-k}{n}dI_{r}^{(i)}+2dI_{r+1}^{(i)}.$ (34)

###  7.2 BDI

Let us consider $M_{\phi}=SO(n+2)/SO(n)\times SO(2)$ and the root ordering of
Dynkin diagrams $D_{m}$ and $B_{m}$ in Figure 1, for $n+2=2m$ and $n+2=2m+1$
cases respectively. In these Dynkin diagrams, the non compact root defining
the compact hermitian symmetric space is $\phi\equiv\alpha_{1}$. The
fundamental representation is realized by anti-symmetric real matrices acting
on $\mathbb{R}^{n+2}$. We can select a Cartan subalgebra as spanned by

$\displaystyle n+2=2m+1:\quad$
$\displaystyle\mathfrak{t}=\left\\{\begin{pmatrix}0&0\\\
0&a\otimes\sigma\end{pmatrix},a=\mathrm{diag}(a_{1},...,a_{m}),a_{i}\in\mathbb{R}\right\\}$
$\displaystyle n+2=2m:\quad$
$\displaystyle\mathfrak{t}=\left\\{a\otimes\sigma,\;a=\mathrm{diag}(a_{1},...,a_{m}),a_{i}\in\mathbb{R}\right\\},$

with $\sigma=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$. Then, in both cases,

$\rho_{\phi}=\begin{pmatrix}0_{n}&0\\\ 0&\sigma\end{pmatrix},$ (35)

which clearly has 3 different eingenvalues. So, we can conclude that the
fundamental representation is not $\phi\,$-minimal.

We are going to show that the spin representation is $\phi\,$-minimal. Let us
consider the Clifford algebra $Cl(n+2,\mathbb{R})$, let $z_{i}$ coordinates on
${\mathbb{C}}^{m}$. The gamma matrices act on
$S^{(n+2)}=\wedge{\mathbb{C}}^{m}$ as

$\Gamma_{i}=d\bar{z}_{i}\wedge,\qquad\Gamma_{\bar{i}}=\imath_{\partial\bar{z}_{i}}$

together with $\Gamma_{0}=(-1)^{\rm deg}$ if $n$ is odd. Then, the
$\mathfrak{so}(n+2)$-spin representation is realized as

$S(X)=\frac{1}{8}X_{ij}[\Gamma_{i},\Gamma_{j}],\qquad X\in\mathfrak{so}(n+2).$

In particular,

$S(\rho_{\phi})=i\left(\Gamma_{\bar{m}}\Gamma_{m}-\frac{1}{2}\right).$ (36)

It has two different eingenvalues $\pm i/2$ so that the spin representation is
$\phi\,$-minimal, i.e. $S^{(n+2)}=V_{+}\oplus V_{-}$, where

$V_{+}=\wedge\langle d\bar{z}_{i},i=1,...,m-1\rangle,\qquad
V_{-}=V_{+}(\Lambda)\otimes d\bar{z}_{m},$

which are $(S^{(n)},\pm i/2)$ representation of
$\mathfrak{so}(n)\oplus\mathfrak{so}(2)$ respectively. Easily, one obtains
that

$\Lambda_{\phi}=\frac{i}{2}.$ (37)

Now, let us consider the Thimm chains

$\displaystyle\mathfrak{so}(2m)\supset...\supset{\mathfrak{so}}(2m-2j)\oplus{\mathfrak{so}}(2)^{j}\supset...\supset\mathfrak{so}(2)^{m},$
(38)
$\displaystyle\mathfrak{so}(2m+1)\supset...\supset{\mathfrak{so}}(2m+1-2j)\oplus{\mathfrak{so}}(2)^{j}\supset...\supset\mathfrak{so}(3)\oplus\mathfrak{so}(2)^{m-1}.$

This calculation can be iterated step-by-step along these chains and we get

$W_{+}^{j}=\Big{(}S^{(n+2-2j)},\underbrace{\frac{i}{2},...,\frac{i}{2}}_{j-\text{times}}\Big{)},$

with $S^{(n+2-2j)}$ the spin representation of $\mathfrak{so}(n+2-2j)$.
Moreover, $W_{+}^{j}(\Lambda)$ is $\phi_{j+1}=\alpha_{1}$-minimal, where
$\alpha_{1}$ is the non compact root of ${\mathfrak{so}}(n+2-2j)$. The spin
representation is then minimal with respect to the chain in (38). Then, the
Nijenhuis tensor acts on ${\mathfrak{so}}(2n+2-2j)$-invariant polynomials
$I_{r}^{(j)}$, as defined in (27), as

$d_{N}I_{r}^{(j)}=dI_{r}^{(j)}+2dI_{r+1}^{(j)}.$ (39)

###  7.3 DIII

Let us consider $M_{\phi}=SO(2n)/U(n)$ and the same root ordering of the
$D_{n}$ case in Figure 1. The non compact simple root defining this compact
hermitian symmetric space is $\phi\equiv\alpha_{n}$. In the fundamental
representation, the Lie subalgebra $\mathfrak{u}(n)$ is embedded in
$\mathfrak{so}(2n)$ as

$\mathfrak{u}(n)\ni A+iB\;\longrightarrow\;\begin{pmatrix}A&B\\\
-B&A\end{pmatrix},$

with $A,B\in M_{n}(\mathbb{R})$ such those $A=-A^{t},\,B=B^{t}$. The Cartan
subalgebra can be chosen as

$\mathfrak{t}=\left\\{\begin{pmatrix}0_{n}&a\\\
-a&0_{n}\end{pmatrix},a=\mathrm{diag}(a_{1},...,a_{n}),a_{i}\in\mathbb{R}\right\\}.$

So,

$\rho_{\phi}=\begin{pmatrix}0&\frac{1}{2}1_{n}\\\
-\frac{1}{2}1_{n}&0\end{pmatrix}.$ (40)

Then, the fundamental representation $V={\mathbb{C}}^{2n}$ is
$\phi\,$-minimal, i.e. $V=V_{+}\oplus V_{-}$, where

$V_{+}=\left\langle(a,+ia),a\in\mathbb{C}^{n}\right\rangle=f_{\mathfrak{u}(n)},$

and, clearly,

$\Lambda_{\phi}=\frac{i}{2}.$ (41)

Moreover, if one considers the Thimm chain

$\mathfrak{so}(2n)\supset\mathfrak{u}(n)\supset...\supset\mathfrak{u}(n-i)\oplus\mathfrak{u}(1)^{i}\supset...\supset\mathfrak{u}(1)^{n},$
(42)

then, repeating the analysis for the case AIII one obtains that $V$ decomposes
in representations of ${\mathfrak{u}}(n-i)\oplus{\mathfrak{u}}(1)^{i}$ as
$V=W_{+}^{i}\oplus W_{-}^{i}$, where

$W^{i}_{+}\equiv\Big{(}f_{\mathfrak{u}(n-i)},\underbrace{0,...,0}_{(i-1)-\text{times}}\Big{)},$

and $W_{+}^{i}$ is $\phi_{i}=\alpha_{n-i-1}$-minimal, where $\alpha_{n-i-1}$
is the root of ${\mathfrak{su}}(n-i)$. Then the fundamental representation is
minimal with respect to the whole Thimm chain (42) and one obtains that

$d_{N}I_{r}^{(j)}=dI_{r}^{(j)}+2dI_{r+1}^{(j)}.$ (43)

###  7.4 CI

Let us consider $M_{\phi}=Sp(2n)/U(n)$ with the root ordering as in the
$C_{n}$ case in Figure 1. The non compact simple root defining the compact
hermitian symmetric space CI is $\phi\equiv\alpha_{n}$. In the fundamental
representation,

$\mathfrak{sp}(2n)=\left\\{X(A,B)=\begin{pmatrix}A&B\\\
-B^{t}&-A^{t}\end{pmatrix},A,B\in
M_{n}(\mathbb{C}),A=-A^{t},B=B^{t}\right\\},$

while $\mathfrak{u}(n)=\left\\{X(A,0)\in\mathfrak{sp}(2n)\right\\}$. The
Cartan subalgebra is spanned by matrices in $\mathfrak{u}(n)$ with $A$
diagonal. Then,

$\rho_{\phi}=\begin{pmatrix}\frac{1}{2}1_{n}&0\\\
0&-\frac{1}{2}1_{n}\end{pmatrix}.$ (44)

Then, the fundamental representation $f_{{\mathfrak{sp}}(2n)}$ is
$\phi\,$-minimal, i.e. $f_{{\mathfrak{sp}}(2n)}=V_{+}\oplus V_{-}$, where

$V_{+}=f_{\mathfrak{u}(n)},$

and the eigenvalue of $\rho_{\phi}$ is

$\Lambda_{\phi}=\frac{i}{2}.$ (45)

Now, let us consider the Thimm chain

$\mathfrak{sp}(2n)\supset\mathfrak{u}(n)\supset\mathfrak{u}(n-1)\oplus\mathfrak{u}(1)\supset...\supset\mathfrak{u}(1)^{n}.$
(46)

At step $i+1$, $f_{{\mathfrak{sp}}(2n)}$ decomposes in
${\mathfrak{k}}_{i+1}={\mathfrak{u}}(n-i)\oplus{\mathfrak{u}}(1)^{i}$-representations
as $f_{{\mathfrak{sp}}(2n)}=W_{+}^{i}\oplus W_{-}^{i}$, where

$W^{i}_{+}\equiv\Big{(}f_{\mathfrak{u}(n-i)},\underbrace{0,...,0}_{i-\text{times}}\Big{)},$

so that $W_{+}^{i}$ is $\alpha_{n-i-1}$-minimal, where $\alpha_{n-i-1}$ is a
root of ${\mathfrak{su}}(n-i)$. Then, the fundamental representation
$f_{{\mathfrak{sp}}(2n)}$ is minimal with respect to the Thimm chain in (46).
Finally, one obtains that

$d_{N}I_{r}^{(j)}=dI_{r}^{(j)}+2dI_{r+1}^{(j)}.$ (47)

###  7.5 EIII and EVII

######  Proposition 7.1.

There are no $\phi\,$-minimal representations of $e_{6}$ and $e_{7}$.

Proof. Let $\Lambda$ be a dominant weight and let $V(\Lambda)$ be the finite
dimensional irreducible representation of $\mathfrak{g}$ with highest weight
$\Lambda$ and highest weight vector $v_{\Lambda}$. If $V(\Lambda)$ is
$\phi\,$-minimal then $e_{-\beta}e_{-\alpha}v_{\Lambda}=0$ for each
$\alpha,\beta\in\Delta_{nc}^{+}$ (otherwise the spectrum of $\rho_{\phi}$
contains the three distinct eigenvalues
$\Lambda_{\phi}=\Lambda(\rho_{\phi}),\Lambda_{\phi}-i,\Lambda_{\phi}-2i$).
Taking $\alpha=\beta$ this means that the string $\\{\Lambda+j\alpha\\}$ of
weights extends at most to $-1\leq j\leq 0$, i.e.

$(\Lambda,\alpha)\in\\{0,1\\}\;\;\;\forall\ \alpha\in\Delta_{nc}^{+}\;,$ (48)

where we assume that $(\alpha,\alpha)=2$ for all roots. Suppose now that
$\Lambda$ satisfies (48) and that $\alpha\in\Delta_{nc}^{+}$ is such that
$\Lambda-\alpha$ is a weight of $V(\Lambda)$. Then the string
$\\{\Lambda-\alpha+j\beta\\}$ extends to $-r\leq j\leq q$ with
$r=q+(\Lambda-\alpha,\beta)$. We are going to show that there exists
$\beta\in\Delta^{+}_{nc}$ such that $(\Lambda-\alpha,\beta)=1$ so that
$r=q+1\geq 1$ and $e_{-\beta}e_{-\alpha}v_{\Lambda}\not=0$. In particular this
happens if there exists such $\beta$ such that $(\alpha,\beta)=0$ and
$(\Lambda,\beta)=1$.

${\mathfrak{e}}_{6}$) Let ${\mathbb{R}}^{8}=\langle\epsilon_{i},i=1,\ldots
8\rangle$. The real Cartan subalgebra
$\mathfrak{t}_{\mathbb{R}}=i\mathfrak{t}$ can be described as the subspace of
${\mathbb{R}}^{8}$ generated by $\epsilon_{i}$, $i=1,\dots,5$ and
$\epsilon=\epsilon_{6}+\epsilon_{7}+\epsilon_{8}$. As in [1], the positive
roots $\Delta^{+}$ are

$\\{\epsilon_{ij}=\epsilon_{i}-\epsilon_{j}\\}_{1\leq
i<j}^{5}\coprod\\{f_{ij}=(\epsilon_{i}+\epsilon_{j})\\}_{1\leq
i<j}^{5}\coprod\\{\frac{1}{2}(\epsilon+\sum_{i=1}^{5}s_{i}\epsilon_{i}),s_{i}=\pm,\Pi_{i}s_{i}=-1\\}\,.$
(49)

The simple roots are

$\Pi=\\{\alpha_{i}=\epsilon_{i,i+1}\\}_{i=1}^{4}\coprod\\{\alpha_{5}=f_{45},\alpha_{6}=\frac{1}{2}(\epsilon-\sum_{i=1}^{5}\epsilon_{i})\\}$
(50)

and the non compact simple root $\phi=\alpha_{6}$. The non compact positive
roots are

$\Delta^{+}_{nc}=\\{\phi+\frac{1}{2}\sum_{i=1}^{5}(1+s_{i})\epsilon_{i},\Pi_{i}s_{i}=-1\\}\;.$
(51)

The weight $\Lambda=\sum_{i=1}^{5}\Lambda_{i}\epsilon_{i}+\Lambda_{0}\epsilon$
is dominant if $(\Lambda,\alpha_{i})=N_{i}$, $i=1,\ldots,6$, with
$N_{i}\in{\mathbb{N}}$. A straightforward computation gives

$\Lambda_{1}=N_{1}+N_{2}+N_{3}+\frac{1}{2}(N_{4}+N_{5}),\,\,\,\,\,\,\Lambda_{2}=N_{2}+N_{3}+\frac{1}{2}(N_{4}+N_{5})\,,$
$\Lambda_{3}=N_{3}+\frac{1}{2}(N_{4}+N_{5})\,,\,\,$
$\Lambda_{4}=\frac{1}{2}(N_{4}+N_{5})\,,\,\,\,\Lambda_{5}=\frac{1}{2}(N_{5}-N_{4})\,,$
$\Lambda_{0}=\frac{1}{3}N_{1}+\frac{2}{3}N_{2}+N_{3}+\frac{1}{2}N_{4}+\frac{5}{6}N_{5}+\frac{2}{3}N_{6}\;.$

Condition (48) means

$(\Lambda,\phi)=N_{6}\in\\{0,1\\},\,\,(\Lambda,\phi+\epsilon_{i}+\epsilon_{j})=N_{6}+\Lambda_{i}+\Lambda_{j}\in\\{0,1\\},$
$(\Lambda,\phi+\epsilon_{i}+\epsilon_{j}+\epsilon_{k}+\epsilon_{l})=N_{6}+\Lambda_{i}+\Lambda_{j}+\Lambda_{k}+\Lambda_{l}\in\\{0,1\\}\;,$

where $i,j,k,l=1,\ldots,5$. Let us consider first $N_{6}=1$. It is not
difficult to verify that the only solution is $N_{i}=0$ for $i\not=6$ so that
only $\Lambda=\frac{2}{3}\epsilon$ satisfies (48). Since $(\Lambda,\alpha)=1$
for each $\alpha\in\Delta_{nc}^{+}$, for every couple $\alpha,\beta$ of
orthogonal non compact positive roots (which exists since the rank is $2$)
$\Lambda-\alpha-\beta$ is a weight.

Let us consider now $N_{6}=0$. Since all couples $\Lambda_{i}+\Lambda_{j}$’s
are non negative integers only one couple can be different from zero; and
since
$\Lambda_{1}\geq\Lambda_{2}\geq\Lambda_{3}\geq\Lambda_{4}\geq\Lambda_{5}$ it
can be only $\Lambda_{1}+\Lambda_{2}$. This means $N_{3}=N_{4}=N_{5}=0$ and
$\Lambda_{1}+\Lambda_{2}=N_{1}+2N_{2}=1$ so that $N_{1}=1$ and $N_{2}=0$. The
solution is then $\Lambda=\epsilon_{1}+\frac{1}{3}\epsilon$. Let us choose now
$\alpha=\frac{1}{2}(\epsilon+\epsilon_{1}-\epsilon_{2}-\epsilon_{3}+\epsilon_{4}-\epsilon_{5})$
and
$\beta=\frac{1}{2}(\epsilon+\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{4}+\epsilon_{5})$
so that $(\Lambda,\alpha)=(\Lambda,\beta)=1$ and $(\alpha,\beta)=0$ so that
$\Lambda-\alpha-\beta$ is a weight.

${\mathfrak{e}}_{7}$) The real Cartan subalgebra
$\mathfrak{t}_{\mathbb{R}}=i\mathfrak{t}$ is realized as the subspace of
${\mathbb{R}}^{8}$ generated by $\epsilon_{i}$, $1\leq i\leq 6$ and
$\epsilon=\epsilon_{7}+\epsilon_{8}$. The simple roots can be chosen as

$\Pi=\\{\alpha_{i}=\epsilon_{i}-\epsilon_{i+1},1\leq i\leq
5;\,\alpha_{6}=\epsilon_{5}+\epsilon_{6},\alpha_{7}=\frac{1}{2}(\epsilon-\sum_{i=1}^{6}\epsilon_{i})\\}\;$

and the non compact simple root $\phi=\alpha_{1}$ [1]. The non compact
positive roots $\Delta^{+}_{nc}$ are

$\\{\epsilon_{1}-\epsilon_{j},\epsilon_{1}+\epsilon_{j};2\leq j\leq
6\\}\coprod\\{\epsilon,\frac{1}{2}(\epsilon_{1}+\epsilon+\sum_{i=2}^{6}s_{i}\epsilon_{i}),s_{i}=\pm
1,\Pi_{i=2}^{6}s_{i}=1\\}\;.$

The weight $\Lambda=\sum_{i=1}^{6}\Lambda_{i}\epsilon_{i}+\Lambda_{0}\epsilon$
is dominant if $(\Lambda,\alpha_{i})=N_{i}\in{\mathbb{N}}$. A straightforward
computation gives

$\Lambda_{1}=N_{1}+N_{2}+N_{3}+N_{4}+\frac{1}{2}(N_{5}+N_{6}),\,\,\,\Lambda_{2}=N_{2}+N_{3}+N_{4}+\frac{1}{2}(N_{5}+N_{6})\,,$
$\Lambda_{3}=N_{3}+N_{4}+\frac{1}{2}(N_{5}+N_{6})\,,\,\,$
$\Lambda_{4}=N_{4}+\frac{1}{2}(N_{5}+N_{6})\,,\,\,\,\Lambda_{5}=\frac{1}{2}(N_{5}+N_{6})\,,\Lambda_{6}=\frac{1}{2}(N_{6}-N_{5})$
$\Lambda_{0}=N_{0}+\frac{1}{2}N_{1}+N_{2}+\frac{3}{2}N_{3}+2N_{4}+N_{5}+\frac{3}{2}N_{6}\;.$

Condition (48) means

$(\Lambda,\epsilon_{1}+\epsilon_{j})=\Lambda_{1}+\Lambda_{j}\in\\{0,1\\},\,\,(\Lambda,\epsilon_{1}-\epsilon_{j})=\Lambda_{1}-\Lambda_{j}\in\\{0,1\\},$
$(\Lambda,\epsilon)=2\Lambda_{0}\in\\{0,1\\},$
$(\Lambda,\frac{1}{2}(\epsilon_{1}+\epsilon+\sum_{i=1}^{6}s_{i}\epsilon_{i}))=\frac{\Lambda_{1}}{2}+\Lambda_{0}+\frac{1}{2}\sum_{i=2}^{6}s_{i}\Lambda_{i}\in\\{0,1\\},\,\,\;.$

Let us choose $N_{1}=(\Lambda,\epsilon_{1}-\epsilon_{2})=0$. Then
$(\Lambda,\epsilon_{1}+\epsilon_{2})=\Lambda_{1}+\Lambda_{2}=2(N_{2}+N_{3}+N_{4})+N_{5}+N_{6}\in\\{0,1\\}$
fixes $N_{2}=N_{3}=N_{4}=0$. Then
$2\Lambda_{0}=2N_{0}+2N_{5}+3N_{6}\in\\{0,1\\}$ implies $N_{0}=N_{5}=N_{6}=0$
so that $\Lambda=0$.

Let us choose $N_{1}=1$. Then $\Lambda_{1}+\Lambda_{2}=1$ implies
$N_{2}=\ldots=N_{6}=0$ and $2\Lambda_{0}=2N_{0}+1=1$ implies $N_{0}=0$. We
then get $\Lambda=\epsilon_{1}+\frac{1}{2}\epsilon$. It is clear that
$(\Lambda,\alpha)=1$ for each non compact positive root $\alpha$, so that
$\Lambda-\alpha-\beta$ is a weight for every couple $\alpha,\beta$ of
orthogonal noncompact positive roots (the rank is $3$). ∎

##  8 Invariant polynomials of ${\mathfrak{k}}_{\phi}$

We are going to prove that, if we choose
${\mathfrak{k}}_{1}={\mathfrak{k}}_{\phi}$ as defined in (4), the condition
(15) is satisfied, so that, from Theorem 4.2, the action of the Nijenhuis
tensor on invariant ${\mathfrak{k}}_{\phi}$-polynomials produces
${\mathfrak{k}}_{\phi}$-basic forms. We remark that
${\mathfrak{k}}_{\phi}={\mathfrak{k}}_{1}$ in the Thimm chain defined in
Section 5 for all cases but $M_{\phi}=Gr(k,n)$ with $1<k<n-1$, i.e.
Grassmanians that are not complex projective spaces.

The result is based on a local parametrization of the moment map around
$\rho_{\phi}$. We refer to Section 2 for notations and basic facts about
compact hermitian symmetric spaces. Let us choose
$P_{\phi}\subset\Delta^{+}_{nc}$ satisfying the following properties:

  * $i$)

for each $\alpha,\beta\in P_{\phi}$, $\alpha-\beta\not\in\Delta$;

  * $ii$)

$P_{\phi}$ is maximal with respect to ($i$).

Since the sum of two positive non compact roots is never a root, condition
$i$) implies that roots in $P_{\phi}$ are all orthogonal and in particular
linearly independent.

Let ${\mathfrak{a}}_{P_{\phi}}\subset{\mathfrak{k}}_{\phi}^{\perp}$ denote the
space spanned by $\\{i(e_{\alpha}+e_{-\alpha}),\alpha\in P_{\phi}\\}$ and
$\mathfrak{t}_{P_{\phi}}=(P_{\phi})^{o}\subset\mathfrak{t}\subset{\mathfrak{k}}_{\phi}$.
We denote ${\mathfrak{a}}_{P_{\phi}}^{\prime}=J({\mathfrak{a}}_{P_{\phi}})$,
that is the space spanned by $\\{e_{\alpha}-e_{-\alpha},\alpha\in
P_{\phi}\\}$.

Let us consider the non compact real form
$\mathfrak{g}_{0}={\mathfrak{k}}_{\phi}\dot{+}i{\mathfrak{k}}_{\phi}^{\perp}$
of $\mathfrak{g}$. We recall that a Cartan subalgebra ${\mathfrak{h}}$ of
$\mathfrak{g}_{0}$ is maximally non compact if the non compact component
${\mathfrak{h}}\cap i{\mathfrak{k}}_{\phi}^{\perp}$ is a maximal abelian
subalgebra of ${\mathfrak{k}}_{\phi}^{\perp}$.

######  Lemma 8.1.

${\mathfrak{h}}=\mathfrak{t}_{P_{\phi}}\oplus i{\mathfrak{a}}_{P_{\phi}}$ is a
maximally non compact Cartan subalgebra of the non compact real form
$\mathfrak{g}_{0}$.

Proof. From the definitions it follows that
$[\mathfrak{t}_{P_{\phi}},{\mathfrak{a}}_{P_{\phi}}]=0$ and
$[{\mathfrak{a}}_{P_{\phi}},{\mathfrak{a}}_{P_{\phi}}]=0$. Since roots in
$P_{\phi}$ are linearly independent, $\dim\mathfrak{t}_{P_{\phi}}={\rm
rk}(\mathfrak{g})-\sharp P_{\phi}={\rm
rk}(\mathfrak{g})-\dim{\mathfrak{a}}_{P_{\phi}}$. In particular
${\mathfrak{h}}$ is a Cartan subalgebra. In order to prove that the non
compact part ${\mathfrak{a}}_{P_{\phi}}$ is maximal, let us suppose that there
exists
$\xi=\sum_{\alpha\in\Delta_{nc}^{+}}(\xi_{\alpha}e_{\alpha}+\xi_{-\alpha}e_{-\alpha})\in{\mathfrak{k}}_{\phi}^{\perp}$
such that $[\xi,a]=0$ for each $a\in{\mathfrak{a}}_{P_{\phi}}$. Then for each
$\beta\in P_{\phi}$ we have that

$[\xi,e_{\beta}+e_{-\beta}]=(\xi_{\beta}-\xi_{-\beta})[e_{\beta},e_{-\beta}]+\sum_{\alpha\in\Delta_{nc}^{+}\setminus
P_{\phi}}\xi_{\alpha}[e_{\alpha},e_{-\beta}]+\xi_{-\alpha}[e_{-\alpha},e_{\beta}]=0\;\;,$

where $[e_{\beta},e_{-\beta}]\in i\mathfrak{t}$. First of all, we conclude
that $\xi_{\beta}=\xi_{-\beta}$ for $\beta\in P_{\phi}$. Then, let us suppose
that $\xi_{\alpha}\not=0$ for some $\alpha\in\Delta_{nc}^{+}\setminus
P_{\phi}$; the above condition implies that $\alpha-\beta$ is not a root for
all $\beta\in P_{\phi}$, but such root does not exist by maximality of
$P_{\phi}$. We can then conclude that $\xi=\sum_{\beta\in
P_{\phi}}\xi_{\beta}(e_{\beta}+e_{-\beta})\in{\mathfrak{a}}_{P_{\phi}}$. ∎

The compact Cartan is then decomposed as
$\mathfrak{t}=\mathfrak{t}_{P_{\phi}}\oplus\mathfrak{t}_{P_{\phi}}^{\prime}$,
where
$\mathfrak{t}_{P_{\phi}}^{\prime}=\mathfrak{t}_{P_{\phi}}^{\perp}\cap\mathfrak{t}$.
It is well known the $K_{\phi}$-orbits of any maximal abelian subalgebra of
${\mathfrak{k}}_{\phi}^{\perp}$ cover all ${\mathfrak{k}}_{\phi}^{\perp}$ (see
Thm. 6.51 in [11]), i.e.

${\mathfrak{k}}_{\phi}^{\perp}=\operatorname{Ad}_{K_{\phi}}({\mathfrak{a}}_{P_{\phi}})\;.$
(52)

We need to prove the following properties.

######  Lemma 8.2.

  * $i$)

$[{\mathfrak{a}}_{P_{\phi}},{\mathfrak{a}}_{P_{\phi}}^{\prime}]\subset\mathfrak{t}_{P_{\phi}}^{\prime}$;

  * $ii$)

$[\mathfrak{t},{\mathfrak{a}}_{P_{\phi}}]={\mathfrak{a}}_{P_{\phi}}^{\prime}$.

Proof. Let us prove $i$). Let $\alpha,\beta\in P_{\phi}$ and compute

$[i(e_{\alpha}+e_{-\alpha}),e_{\beta}-e_{-\beta}]=-2i\delta_{\alpha,\beta}(e_{\alpha},e_{-\alpha})\tau_{\alpha}\in\mathfrak{t}_{P_{\phi}}^{\prime}$

where $(\tau_{\alpha},H)=\alpha(H)$ for each $H\in\mathfrak{t}$. Point $ii$)
follows because
$[H,i(e_{\alpha}+e_{-\alpha})]=i\alpha(H)(e_{\alpha}-e_{-\alpha})$ for each
$H\in\mathfrak{t}$. ∎

The orthogonal decomposition
${\mathfrak{k}}={\mathfrak{k}}_{\phi}\dot{+}{\mathfrak{k}}_{\phi}^{\perp}$
assures that there exists an open neighborhood of $\rho_{\phi}\in M_{\phi}$
where the moment map $\mu\in C^{\infty}(M_{\phi})\otimes{\mathfrak{k}}$ can be
written as

$\mu=\operatorname{Ad}_{k}\operatorname{Ad}_{e^{\xi}}(\rho_{\phi})$

for $k\in K_{\phi}$ and $\xi\in{\mathfrak{k}}_{\phi}^{\perp}$ so that
$\mu\sim\operatorname{Ad}_{e^{\xi}}(\rho_{\phi})$ where $\sim$ denotes up to
$K_{\phi}$-adjoint action. By using (52) and the fact that $\rho_{\phi}$ is
$K_{\phi}$-invariant we can write

$\mu\sim\operatorname{Ad}_{e^{a}}(\rho_{\phi})\;.$ (53)

######  Proposition 8.3.

We have that

$[\mu_{{\mathfrak{k}}_{\phi}},A_{{\mathfrak{k}}_{\phi}}]=0$

where $A_{{\mathfrak{k}}_{\phi}}$ is defined in (14). As a consequence, for
each $p\in S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}}$, $d_{N}p$ is a
basic ${\mathfrak{k}}_{\phi}$-form.

Proof. Let us write

$\mu\sim\tilde{\rho}+\tilde{\xi}$ (54)

where $\tilde{\rho}\in{\mathfrak{k}}_{\phi}$ and
$\tilde{\xi}\in{\mathfrak{k}}_{\phi}^{\perp}$. From (53) we can write

$\tilde{\rho}=\sum_{k=0}\frac{1}{(2k)!}\operatorname{ad}_{a}^{2k}(\rho_{\phi})\,,\;\;\;\tilde{\xi}=\sum_{k=0}\frac{1}{(2k+1)!}\operatorname{ad}_{a}^{2k+1}(\rho_{\phi})\;.$

It is clear that by using Lemma 8.2 $ii$) we see that
$\operatorname{ad}_{a}(\rho_{\phi})\in{\mathfrak{a}}_{P_{\phi}}^{\prime}$,
from $i$)
$\operatorname{ad}^{2}_{a}(\rho_{\phi})\in\mathfrak{t}_{P_{\phi}}^{\prime}\subset\mathfrak{t}$;
we then show that $\operatorname{ad}^{2k}_{a}(\rho_{\phi})\in\mathfrak{t}$ and
$\operatorname{ad}^{2k+1}_{a}(\rho_{\phi})\in{\mathfrak{a}}_{P_{\phi}}^{\prime}$,
so that $\tilde{\rho}\in\mathfrak{t}$ and
$\tilde{\xi}\in{\mathfrak{a}}_{P_{\phi}}^{\prime}$. By recalling (14), we
compute up to $K_{\phi}$-adjoint action that

$[\mu_{{\mathfrak{k}}_{\phi}},A_{{\mathfrak{k}}_{\phi}}]\sim\frac{1}{2}[\tilde{\rho},[J(\tilde{\xi}),\tilde{\xi}]]\in[\mathfrak{t},[J({\mathfrak{a}}_{P_{\phi}}^{\prime}),{\mathfrak{a}}_{P_{\phi}}^{\prime}]]=[\mathfrak{t},[{\mathfrak{a}}_{P_{\phi}},{\mathfrak{a}}_{P_{\phi}}^{\prime}]]=[\mathfrak{t},\mathfrak{t}]=0\;,$

where we used Lemma 8.2 $i$). ∎

We finally give the following explicit description of the terms appearing in
(54).

######  Corollary 8.4.

Let $P_{\phi}=\\{\alpha_{j}\\}_{j=1}^{\mathrm{rank}M_{\phi}}$,
$\mathfrak{a}_{P_{\phi}}$ spanned by
$X_{\alpha_{j}}=i(e_{\alpha_{j}}+e_{-\alpha_{j}})$. Then, for any
$a=\sum_{i=j}^{\mathrm{rank}M_{\phi}}a_{j}X_{\alpha_{j}}$ where
$a_{j}\in\mathbb{R}$, we compute

$\tilde{\rho}=\rho_{\phi}+\sum_{j=1}^{\mathrm{rank}M_{\phi}}f_{j}\,ih_{\alpha_{j}},\qquad\tilde{\xi}=\sum_{j=1}^{\mathrm{rank}M_{\phi}}g_{j}JX_{\alpha_{j}},$
(55)

where

$f_{j}=\frac{1}{2}(\cos(2a_{j})-1),\qquad g_{j}=-\frac{1}{2}\sin(2a_{j}).$
(56)

Moreover,

$[J\tilde{\xi},\tilde{\xi}]=2\sum_{j=1}^{\mathrm{rank}M_{\phi}}\left[f_{j}+f^{2}_{j}\right]\,ih_{\alpha_{j}}.$
(57)

Proof. First of all, let us remind that, for $\alpha_{j},\alpha_{l}\in
P_{\phi}$, $[ih_{\alpha_{j}},X_{\alpha_{l}}]=2\delta_{jl}\,(JX_{\alpha_{l}})$
and $[X_{\alpha_{j}},JX_{\alpha_{l}}]=2\delta_{jl}\,ih_{\alpha_{l}}$. Then, if
we set

$\operatorname{ad}_{a}^{2k}(\rho_{\phi})=\sum_{j}(f_{j})_{k}\,ih_{\alpha_{j}},\qquad\operatorname{ad}_{a}^{2k+1}(\rho_{\phi})=\sum_{j}(g_{j})_{k}JX_{\alpha_{j}},$

we can find the following recursive relations

$(f_{j})_{k}=-4a_{j}^{2}(f_{j})_{k-1},\quad(g_{j})_{k}=-4a_{j}^{2}(g_{j})_{k-1}$

with $(f_{j})_{1}=-2a_{j}^{2}$ and $(g_{j})_{0}=-a_{j}$. Then, we can write

$(f_{j})_{k}=\frac{(-1)^{k}}{2}(2a_{j})^{2k},\quad(g_{j})_{k}=\frac{(-1)^{k+1}}{2}(2a_{j})^{2k+1},$

and so, putting $f_{j}=\sum_{k=1}^{\infty}\frac{1}{2k!}(f_{j})_{k}$ and
$g_{j}=\sum_{k=0}^{\infty}\frac{1}{(2k+1)!}(g_{j})_{k}$, one obtains (55,56).
The last claim is obtained by a straightforward calculation and by considering
that

$g^{2}_{j}=-f_{j}-f^{2}_{j},$

formula obtained by (56) using some basic goniometric relations. ∎

Thanks to (55) and (57) ${\mathfrak{k}}_{\phi}$\- invariant polynomials can be
expressed as polynomials in the variables $f_{j}$, $j=1\ldots,{\rm
rk}(M_{\phi})$. Let us denote with $p_{n}(f)=\sum_{j}c_{j}f_{j}^{n}$, where
$c_{j}=i(h_{\alpha_{j}},\rho_{\phi})=-2/(\alpha_{j},\alpha_{j})$. Remark that
$(ih_{\alpha_{j}},ih_{\alpha_{r}})=2\delta_{jr}c_{j}$. Let us introduce the
following ${\mathfrak{k}}_{\phi}$-invariant polynomials

$I_{1,0}\equiv(\mu_{{\mathfrak{k}}_{\phi}},\rho_{\phi})=(\rho_{\phi},\rho_{\phi})+p_{1}\,,\;\;I_{2,0}\equiv(\mu_{{\mathfrak{k}}_{\phi}},\mu_{{\mathfrak{k}}_{\phi}})=(\rho_{\phi},\rho_{\phi})+2p_{1}+2p_{2}\,,$
(58)
$I_{0,1}=(\rho_{\phi},A_{{\mathfrak{k}}_{\phi}})=p_{1}+p_{2}=\frac{1}{2}(I_{2,0}-(\rho_{\phi},\rho_{\phi}))\,,\;\;\;I_{1,1}\equiv(\mu_{{\mathfrak{k}}_{\phi}},A_{{\mathfrak{k}}_{\phi}})=p_{1}+3p_{2}+2p_{3}\;.$

By applying (13) we now easily compute

$d_{N}I_{1,0}=d(I_{1,0}-\frac{1}{2}I_{2,0})\,,\;\;\;d_{N}I_{2,0}=d(I_{2,0}-\frac{2}{3}I_{1,0}-\frac{4}{3}I_{1,1})\;.$
(59)

Let us remark that while
$I_{0,1}\in\mu^{*}_{{\mathfrak{k}}_{\phi}}(S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}})$,
it is not obvious that the same is true for $I_{1,1}$ so that (13) is not
enough to compute $d_{N}I_{1,1}$. This property will be analyzed in Section 9
for the exceptional cases $EIII$ and $EVII$.

Let finally express the above formula on the basis given by the symmetric
polynomials $p_{j}$. We get

$d_{N}p_{1}=-dp_{2}\,,\;\;\;\;d_{N}p_{2}=-\frac{4}{3}dp_{3}\;.$ (60)

It is tempting to state that

$d_{N}f_{i}=-2f_{i}df_{i}\;,i=1,\ldots{\rm rk}(M_{\phi})$ (61)

that actually imply (60). Since in (61) there are two equations, if ${\rm
rk}(M_{\phi})=2$ then (61) and (60) are equivalent. To see what happens when
${\rm rk}(M_{\phi})>2$ it is useful to connect with the results of Theorem 6.3
and Corollary 6.4 for the classical $BCD$ cases. Let us consider a
$\phi\,$-minimal representation $V(\Lambda)=V_{+}\oplus V_{-}$ of
${\mathfrak{k}}$ (see Definition 6.1). We need the following result.

######  Lemma 8.5.

In a $\phi\,$-minimal representation, on $V_{+}$ we have that

$h_{\alpha_{i}}h_{\alpha_{j}}=\delta_{ij}h_{\alpha_{i}}$

for each $\alpha_{i}\in P_{\phi}$.

Proof. Let $\alpha$ be a positive non compact root and let us consider the
${\mathfrak{sl}}(2,{\mathbb{C}})$ subalgebra generated by
$\\{h_{\alpha},e_{\alpha},e_{-\alpha}\\}$. Let $v_{\lambda}\in V_{+}$ be a
common eigenvector of the Cartan subalgebra with weight $\lambda$ such that
$\lambda_{\alpha}=\lambda(h_{\alpha})\not=0$. The eigenvector
$v_{\lambda-\alpha}=e_{-\alpha}v_{\lambda}\in V_{-}$ is non vanishing (indeed
$e_{\alpha}v_{\lambda-\alpha}=h_{\alpha}v_{\lambda}=\lambda_{\alpha}v_{\lambda}$).
Let us consider $W_{\alpha}=\langle v_{\lambda},v_{\lambda-\alpha}\rangle$;
since $e_{\alpha}v_{\lambda}=e_{-\alpha}v_{\lambda-\alpha}=0$, $W_{\alpha}$ is
the two dimensional irreducible representation of
${\mathfrak{sl}}(2,{\mathbb{C}})$. As a consequence $\lambda_{\alpha}=1$. We
can then conclude that the eigenvalues of $h_{\alpha}$ on $V_{+}$ can be $0,1$
(and so $h_{\alpha}$ is idempotent).

Let us denote with $\lambda_{i}=\lambda(h_{\alpha_{i}})$. For each $i\not=j$
we compute

$\lambda_{i}\lambda_{j}=(v_{\lambda},h_{\alpha_{i}}h_{\alpha_{j}}v_{\lambda})=(v_{\lambda},[e_{\alpha_{i}},e_{-\alpha_{i}}][e_{\alpha_{j}},e_{-\alpha_{j}}]v_{\lambda})\;,$

where $(,)$ here denotes the scalar product that makes $V(\Lambda)$ unitary.
Since $\alpha_{i}$ are non compact roots, in a $\phi\,$-minimal representation
$e_{\alpha_{i}}e_{\alpha_{j}}=0$; since $\alpha_{i}\in P_{\phi}$ are mutually
orthogonal then $[e_{\pm\alpha_{i}},e_{\mp\alpha_{j}}]=0$ for $i\not=j$. As a
consequence $\lambda_{i}\lambda_{j}=0$. ∎

We then conclude that the non zero eigenvalues of
$\mu_{{\mathfrak{k}}_{\phi}}$ in the representation $V_{+}$ are
$\\{\Lambda_{\phi}+if_{i}\\}_{i=1}^{{\rm rk}(M_{\phi})}$. By applying the
formula of Corollary 6.4 $ii$) we see that $-2f_{i}$ is a Nijenhuis
eigenvalue, i.e. (61) holds.

##  9 A formula for $S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}}$ on
$EIII$ and $EVII$

We are going to discuss here the exceptional cases. We saw in the previous
section that minimal representations do not exist for ${\mathfrak{e}}_{6}$ and
${\mathfrak{e}}_{7}$ so that we cannot apply Corollary 6.4 to the Thimm chains
discussed in Section 5. For ${\mathfrak{k}}_{\phi}$, that is the first
subalgebra of the chain, we can apply Proposition 8.3: we are going to compute
the explicit form of the subcomplex of the Nijenhuis complex generated by
${\mathfrak{k}}_{\phi}$-invariant polynomials.

###  9.1 EIII

Let us consider $M_{\phi}=E_{6}/SO(10)\times SO(2)$. We are going to give the
explicit formulas for the action of the Nijenhuis tensor on
${\mathfrak{so}}(10)\oplus{\mathfrak{so}}(2)$-invariant polynomials.

By using the description of positive roots and non compact positive roots of
${\mathfrak{e}}_{6}$ given in (49) and (51) respectively, we get that

$P_{\phi}=\left\\{\phi=\frac{1}{2}(\epsilon-\sum_{i=1}^{5}\epsilon_{i}),\psi=\frac{1}{2}(\epsilon+\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}-\epsilon_{5})\right\\}\subset\Delta^{+}_{nc}$
(62)

are orthogonal noncompact roots. Since ${\rm rk}(M_{\phi})=2$, this subset is
maximal. Let ${\mathfrak{a}}_{P_{\phi}}$ be the spaces spanned by
$\\{X_{\phi}=i(e_{\phi}+e_{-\phi}),X_{\psi}=i(e_{\psi}+e_{-\psi})\\}$,
according to Section 8. Thus, formulas (55) and (56) are written as

$\displaystyle\tilde{\rho}=\rho_{\phi}+f_{\phi}\,ih_{\phi}+f_{\psi}\,ih_{\psi},$
(63) $\displaystyle\tilde{\xi}=g_{\phi}JX_{\phi}+g_{\psi}JX_{\psi}.$

Since ${\rm rk}(M_{\phi})=2$ the formulas computed in (59) completely describe
the action of $d_{N}$ on
${\mathfrak{so}}(10)\oplus{\mathfrak{so}}(2)$-invariant polynomials. Indeed
from (62) we have that
$c_{\phi}=i(h_{\alpha_{\phi}},\rho_{\phi})=c_{\psi}=i(h_{\alpha_{\psi}},\rho_{\phi})=-1$
while $(\rho_{\phi},\rho_{\phi})=-\frac{4}{3}$. Then,
$p_{n}(f)=-f_{\phi}^{n}-f_{\psi}^{n}$ and we have the following polynomial
relation

$p_{3}=-\frac{3}{2}p_{1}p_{2}-\frac{1}{2}p_{1}^{3}.$

From (60) we get that

$d_{N}p_{1}=-dp_{2}\,,\;\;\;\;d_{N}p_{2}=\frac{4}{6}d\left(3p_{1}p_{2}+p_{1}^{3}\right)\;$

so that $p_{1}$ and $p_{2}$ generate the subcomplex
$(\Omega^{{\mathfrak{so}}(10)\oplus{\mathfrak{so}}(2)}_{pol},d_{N})$. We can
conclude that

$d_{N}f_{\phi}=-2f_{\phi}df_{\phi}\,,\;\;\;\;d_{N}f_{\psi}=-2f_{\psi}df_{\psi}\,,$

i.e. $-2f_{\phi}$ and $-2f_{\psi}$ are Nijenhuis eigenvalues.

###  9.2 EVII

The case $M_{\phi}=E_{7}/E_{6}\times SO(2)$ presents a new feature. Indeed,
since ${\rm rk}(M_{\phi})=3$ we need three generators to describe
$\mu_{{\mathfrak{k}}_{\phi}}^{*}(S({\mathfrak{k}}_{\phi}))^{{\mathfrak{k}}_{\phi}}$.
Two of them can be $I_{1,0}=(\rho_{\phi},\mu_{{\mathfrak{k}}_{\phi}})$ and
$I_{2,0}=(\mu_{{\mathfrak{k}}_{\phi}},\mu_{{\mathfrak{k}}_{\phi}})$, we have
to look for the third one. The general formulas (59) involve also
$I_{1,1}=(\mu_{{\mathfrak{k}}_{\phi}},A_{{\mathfrak{k}}_{\phi}})$ which, by
looking at the expression given in (59), as a cubic polynomials in the $f$’s,
is clearly independent. The problem is now to understand if
$I_{1,1}\in\mu_{{\mathfrak{k}}_{\phi}}^{*}(S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}})$.
In the previous case of $EIII$, since the rank was $2$, $I_{1,1}$, as a
polynomial of (maximal) degree $3$ in two variables, was clearly generated by
$I_{1,0}$ and $I_{2,0}$. The next independent generator in
$S({\mathfrak{e}}_{6})^{{\mathfrak{e}}_{6}}$ appears in degree $5$; we can
write it as $I_{5,0}=\textrm{Tr}_{V_{\Lambda}}\mu_{{\mathfrak{e}}_{6}}^{5}$
for some representation $V_{\Lambda}$ of ${\mathfrak{e}}_{6}$; it is a
polynomial of degree $5$ in the $f$-variables. By counting the $f$-degree, we
easily conclude that $I_{1,1}$ cannot be polynomially generated by
$\\{I_{1,0},I_{2,0},I_{5,0}\\}$ and $I_{5,0}$ is a polynomial function of
$\\{I_{1,0},I_{2,0},I_{1,1}\\}$. By inverting this relation, $I_{1,1}$ can be
expressed as a non polynomial function of the moment map, i.e.
$I_{1,1}\in\mu_{{\mathfrak{e}}_{6}}^{*}(C^{\infty}({\mathfrak{e}}_{6}^{*})^{{\mathfrak{e}}_{6}})$.
We can then conclude that if we want to preserve the polynomial structure, we
are forced to deal with the bigger ring that includes also polynomials in the
variables $A_{{\mathfrak{e}}_{6}}$. More importantly, the computation of
$d_{N}I_{1,1}$ requires an extension of (13) to this bigger ring of
invariants.

##  10 Conclusions

In this paper we discussed two different approaches to the problem of
diagonalizing the Nijenhuis tensor $N$ on the hermitian symmetric spaces
$M_{\phi}=K/K_{\phi}$. The first one is based on the existence of a special
representation of ${\mathfrak{k}}$ that we call $\phi\,$-minimal: in this case
we can define a Thimm chain of subalgebras
${\mathfrak{k}}\supset{\mathfrak{k}}_{1}\ldots\supset{\mathfrak{k}}_{i}\ldots$
together with the set of generators $I_{r}$ of the ring of invariant
polynomials $S({\mathfrak{k}}_{i})^{{\mathfrak{k}}_{i}}$ that make the
diagonalization problem easily solved.

The second construction is based on a local parametrization around
$\rho_{\phi}\in M_{\phi}$. This method is valid for all cases, including the
exceptional ones, and produces the subset of the spectrum related to the
subalgebra ${\mathfrak{k}}_{\phi}$. It detects a different behavior of the
EVII case: in fact the ring of invariant polynomials
$S({\mathfrak{k}}_{\phi})^{{\mathfrak{k}}_{\phi}}$ is not preserved by the
algebroid differential $d_{N}$ and the bigger ring of polynomials of
$\mu_{{\mathfrak{k}}_{\phi}}$ and $A_{{\mathfrak{k}}_{\phi}}$ must be taken
into consideration. Formula (13) must be then generalized. Moreover,
preliminary computations for the second subalgebra
${\mathfrak{k}}_{2}={\mathfrak{so}}(8)\oplus{\mathfrak{so}}(2)$ in the Thimm
chain of $EIII$ show that the sufficient condition
$[\mu_{{\mathfrak{k}}_{2}},A_{{\mathfrak{k}}_{2}}]=0$ in (15) is not
satisfied. These facts indicate that the cases $EIII$ and $EVII$ have an
exceptional behavior. It is possible that it will be convenient to relate it
to the fact that
$S({\mathfrak{k}})^{\mathfrak{k}}\not=S({\mathfrak{a}}_{{\mathfrak{k}}_{\phi}^{\perp}})^{{\mathfrak{k}}_{\phi}}$
for the exceptional cases proved by Helgason in [9]. This problem will be
addressed in a future publication.

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