# Normal forms of $\omega$-Hamiltonian vector fields with symmetries

Patrícia H. Baptistelli, Maria Elenice R. Hernandes
Departamento de Matemática, Universidade de Maringá
Av. Colombo, 5790, 87020-900, Maringá - PR, Brazil 111Email address:
<EMAIL_ADDRESS>(corresponding author<EMAIL_ADDRESS>Eralcilene
Moreira Terezio
Instituto Latino-Americano de Ciências da Vida e da Natureza
Universidade Federal da Integração Latino-Americana
Av. Tancredo Neves, 6731, 85867-900, Foz do Iguaçu - PR, Brazil 222Email
address<EMAIL_ADDRESS>
###### Abstract

In this paper, we present algebraic tools to obtain normal forms of
$\omega$-Hamiltonian vector fields under a semisymplectic action of a Lie
group, by taking into account the symmetries and reversing symmetries of the
vector field. The normal forms resulting from the process preserve the
Hamiltonian condition and the types of symmetries of the original vector
field. Our techniques combine the classical method of normal forms of
Hamiltonian vector fields with the invariant theory of groups.

Keywords: normal forms, $\omega$-Hamiltonian vector field, symmetries,
semisymplectic action

2020 AMS Classification: 37C79, 37J40, 58D19

##  1 Introduction

There are several methods to understand the local qualitative behavior of a
vector field and one of them is the normal form theory. Some authors, as in
[12, 16, 17, 21, 22], use this theory to study limit cycles, families of
periodic orbits, relative equilibria, relative periodic solutions, etc.

The classic way to obtain a normal form of a vector field consists of
performing changes of coordinates around a singular point (namely $x=0$) on a
formal power series of the vector field. More precisely, we consider
perturbations of the identity of the form $x=y+\varphi^{k}(y)$, with $k\geq 2$
and $\varphi^{k}$ a homogeneous polynomial of degree $k$, which preserve the
linear part and allow to eliminate terms of degree $k\geq 2$ of the original
vector field, by resulting in a formally conjugate vector field written in a
more convenient way. For each $k\geq 2$, the truncated equation of degree $k$
of the associated power series is called a normal form of order $k$ of the
vector field.

Over the years, some methods have been developed to provide normal forms as
described above. Belitskii [7, 8] reduces this problem to the computation of
solutions of a partial differential equation. Elphick et al. [14] provide an
algebraic method, in which it is possible to obtain a normal form by choosing
non-linear terms that commute with the action of the one-parameter group

$\mathbf{S}=\overline{\left\\{e^{sL^{T}}:\ s\in{\mathbb{R}}\right\\}},$ (1)

where the bar denotes the closure of the set and $L$ is the linearization of
the vector field at the origin. In this case, the normal form process
introduces additional symmetries into the problem since every vector field is
formally conjugated to a vector field with symmetry group $\mathbf{S}$.
Several works deal with vector fields that present symmetric geometric
configurations, as we can see in [4, 5, 9, 15, 18]. In particular, in the same
way as Elphick et al. [14] and based on algebraic invariant theory,
Baptistelli, Manoel and Zeli present a method ([4, Theorem 4.7]) to obtain
normal forms of a reversible equivariant vector field that preserve the
symmetries of the original vector field.

The class of Hamiltonian vector fields, and consequently Hamiltonian systems,
deserves special attention due to its historical contribution in Mathematics
and its applications in Physics, among others. Given $(V,\omega)$ a symplectic
real vector space of dimension $2n$ and $H:V\to\mathbb{R}$ a smooth function,
the $\omega$-Hamiltonian vector field associated with $H$ can be written as
$X(x)=([\omega]^{-1})^{T}\nabla H(x)$ for all $x\in V$, where $[\omega]$
denotes the matrix of $\omega$ relative to a basis of $V$ (see Definition
2.1). In symmetric Hamiltonian context, several works have contributed in
different aspects, such as [1, 10, 11, 18]. In [18], Montaldi, Roberts and
Stewart deal with Hamiltonian systems on $(\mathbb{R}^{2n},\omega_{0})$, where
$\omega_{0}$ is the canonical bilinear form, under an orthogonal action of a
Lie group that admits an involutory reversible symmetry. In [11], Buzzi and
Teixeira also analyze the dynamics of reversible Hamiltonian vector fields on
$(\mathbb{R}^{2n},\omega_{0})$ with respect to a symplectic involution. More
recently, Alomair and Montaldi [1] study the existence of families of periodic
solutions in two classes of Hamiltonian systems with involutory symmetry (of
types SR and AE according to Table 1). In that paper, the authors consider two
group homomorphisms and classify the symmetries into up four types.

The aim of this work is presenting an algebraic method to obtain normal forms
of $\omega$-Hamiltonian vector fields on $(V,\omega)$ under a semisymplectic
action of a Lie group $\Gamma$ (see Definition 3.1), by taking into account
the symmetries and reversing symmetries of $X$. For this purpose, we assume
the existence of two group homomorphisms
$\sigma_{1},\sigma_{2}:\Gamma\to\mathbb{Z}_{2}$, where $\sigma_{1}$ is related
to semisymplectic action of $\Gamma$ and $\sigma_{2}$ establishes the
symmetries of the vector field. In this context, we consider that the matrix
of the symplectic form $\omega$ may possibly be different from the matrix of
the canonical symplectic form given by

$J=\left[\begin{array}[]{cc}0&I_{n}\\\ -I_{n}&0\end{array}\right],$ (2)

where $I_{n}$ is the identity matrix of order $n$. One of the reasons for this
is that the symmetries of vector fields may not be preserved by
symplectomorphisms. In addition, some works (as [19, 20]) also deal with
Hamiltonian vector fields for matrices of the symplectic form different from
$J$.

The method we present gives us a normal form that preserves the types of
symmetries and Hamiltonian condition of the original vector field. In this
case, there is a computational advantage compared to the method presented in
[4], since a normal form of a Hamiltonian vector field can be obtained by
means of a normal form of the associated Hamiltonian function (see, for
instance, [13]).

This paper is organized as follow: in Section 2 we introduce some notations
and basic notions of Hamiltonian and equivariant contexts. In Section 3 we use
the semisymplectic group to recognize a semisymplectic action on $(V,\omega)$
(Theorem 3.2). In Section 4, we describe the types of symmetry in Hamiltonian
context with respect to a (Lie) group of symmetries $\Gamma$ and its action on
$(V,\omega)$. In Section 5 we present the classic normal form theory of
$\omega$-Hamiltonian vector fields and give a method to compute these normal
forms (Subsection 5.1). The Section 6 deals with the main issue of the paper,
by presenting an algebraic method to obtain normal forms of
$\omega$-Hamiltonian vector fields with symmetries (Theorem 6.3 and Corollary
6.4). We illustrate such a method with two examples (Subsections 6.1 and 6.2)
and we finish the paper with a section of some technical proofs (Section 7).

##  2 Preliminary concepts

###  2.1 Preliminaries on symplectic geometry

A symplectic vector space is a pair $(V,\omega)$, where $V$ is a finite-
dimensional real vector space and $\omega:V\times V\to\mathbb{R}$ is an
alternating and non-degenerate bilinear form. It is possible to prove that the
dimension of $V$ is even and the matrix $[\omega]$ of $\omega$ relative to a
basis of $V$ is a skew-symmetric and invertible matrix. The next definition
presents our main object of study and it can naturally be extended to a
symplectic manifold.

###### Definition 2.1

Let $(V,\omega)$ be a symplectic vector space and $H:V\to\mathbb{R}$ a smooth
function. The $\omega$-Hamiltonian vector field associated with $H$ is the
unique vector field $X_{H}:V\to V$ such that $\omega(X_{H}(x),\cdot)=dH_{x}$,
for all $x\in V$. In other words, $X_{H}$ is written as

$X_{H}(x)=([\omega]^{-1})^{T}\nabla H(x)$

for all $x\in V$, where $T$ denotes the transpose of the matrix and $\nabla H$
is the gradient vector of $H$. The function $H$ is called a Hamiltonian
function.

We intend performing changes of coordinates in an $\omega$-Hamiltonian vector
field by preserving its Hamiltonian structure. Changes of coordinates called
symplectic or antisymplectic satisfy this property (see [6, Theorem 4.2]).

###### Definition 2.2

Let $(V,\omega)$ be a symplectic vector space. A differentiable map $\xi:V\to
V$ is called symplectic if $\xi^{*}\omega=\omega$ and it is called
antisymplectic if $\xi^{*}\omega=-\omega$, where $\xi^{*}\omega$ is the
pullback of $\omega$ by $\xi$.

If $(V,\omega)$ has dimension $2n$, then the matrices
$B\in\mathbb{M}_{2n}(\mathbb{R})$ associated with symplectic linear maps
satisfy the equality $B^{T}[\omega]B=[\omega]$ and are called
$\omega$-symplectic matrices (see [6, Proposition 2.3]). The set of all
$\omega$-symplectic matrices is a Lie group denoted by
$Sp_{\omega}(n;\mathbb{R})$ and called $\omega$-symplectic group. In addition,
matrices $B$ which are associated with antisymplectic linear maps are called
$\omega$-antisymplectic matrices and satisfy $B^{T}[\omega]B=-[\omega]$. The
set of all $\omega$-antisymplectic matrices is denoted by
$Sp_{\omega}^{-1}(n;\mathbb{R})$ and it does not have a group structure.

###### Remark 2.3

We highlight the following properties of the set
$Sp_{\omega}^{-1}(n;\mathbb{R})$: if $B\in Sp_{\omega}^{-1}(n;\mathbb{R})$
then $B^{-1}\in Sp_{\omega}^{-1}(n;\mathbb{R})$; the product of an even number
of elements in $Sp_{\omega}^{-1}(n;\mathbb{R})$ belongs to
$Sp_{\omega}(n;\mathbb{R})$, while the product of an odd number of elements in
$Sp_{\omega}^{-1}(n;\mathbb{R})$ still belongs to
$Sp_{\omega}^{-1}(n;\mathbb{R})$.

In [6] we present a study of the symplectic algebra for an arbitrary
symplectic vector space $(V,\omega)$, by introducing the concept of
$\omega$-symplectic group and characterizing its Lie algebra as a useful tool
in the recognition of $\omega$-Hamiltonian vector fields. More precisely, we
define the $\omega$-semisymplectic group by the disjoint union

$\Omega_{n}=Sp_{\omega}(n;\mathbb{R})\ \dot{\cup}\
Sp_{\omega}^{-1}(n;\mathbb{R}).$ (3)

The Lie algebra of $\Omega_{n}$, which coincides with the Lie algebra of
$Sp_{\omega}(n;\mathbb{R})$, is given by the set

${\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})=\left\\{L\in{\mathbb{M}}_{2n}({\mathbb{R}}):L^{T}[\omega]+[\omega]L=0\right\\}.$
(4)

The elements of this algebra are called $\omega$-Hamiltonian matrices.

###  2.2 Preliminaries on invariant theory

Consider $V$ a finite-dimensional real vector space and $\Gamma$ a Lie group
acting on $V$ by means of a differentiable action $(\gamma,x)\mapsto\gamma x$,
for all $\gamma\in\Gamma$ and $x\in V$. Consider ${\rho}_{\gamma}:V\to V$ the
diffeomorphism induced by this action, given by ${\rho}_{\gamma}(x)=\gamma x$.
If $X:V\to V$ is a vector field, for each $\gamma\in\Gamma$ we denote by
$(\rho_{\gamma})_{*}X:V\to V$ the pushforward of $X$ by ${\rho}_{\gamma}$,
namely $(\rho_{\gamma})_{\ast}X(y)=d(\rho_{\gamma})_{x}(X(x))$ for
$x=\rho_{\gamma}^{-1}(y)$. We say that $X$ is $\Gamma_{\sigma}$-equivariant if
there exists a group homomorphism $\sigma:\Gamma\to\mathbb{Z}_{2}=\\{\pm 1\\}$
satisfying $(\rho_{\gamma})_{*}X=\sigma(\gamma)X,$ that is,

$X(\rho_{\gamma}(x))=\sigma(\gamma)d(\rho_{\gamma})_{x}(X(x))$

for all $\gamma\in\Gamma$ and $x\in V$. When $\sigma$ is trivial, $X$ is
called $\Gamma$-equivariant. An element $\gamma\in\Gamma$ is a symmetry of $X$
if $\gamma\in\ker\sigma$ and it is a reversible symmetry of $X$ if
$\gamma\in\Gamma\backslash\ker\sigma$.

A function $f:V\to\mathbb{R}$ is called $\Gamma_{\sigma}$-invariant if

$f(\rho_{\gamma}(x))=\sigma(\gamma)f(x),$

for all $\gamma\in\Gamma$ and $x\in V$. If $\sigma$ is trivial, then $f$ is
called $\Gamma$-invariant.

Denote by $\mathcal{P}$ and $\overrightarrow{\mathcal{P}}$ the vector spaces
of all polynomial functions $V\to\mathbb{R}$ and polynomial maps $V\to V$,
respectively. We introduce the following notations:

$\begin{array}[]{ccl}{{\mathcal{P}}}(\Gamma)&=&\\{f\in\mathcal{P}:f(\rho_{\gamma}(x))=f(x),\
\forall\gamma\in\Gamma,\ x\in V\\};\\\
{{\mathcal{P}}_{\sigma}}(\Gamma)&=&\\{f\in\mathcal{P}:f(\rho_{\gamma}(x))=\sigma(\gamma)f(x),\
\forall\gamma\in\Gamma,\ x\in V\\};\\\
\overrightarrow{\mathcal{P}}(\Gamma)&=&\\{F\in\overrightarrow{\mathcal{P}}:F(\rho_{\gamma}(x))=\rho_{\gamma}(F(x)),\
\forall\gamma\in\Gamma,\ x\in V\\};\\\
{\overrightarrow{\mathcal{P}}_{\sigma}}(\Gamma)&=&\\{F\in\overrightarrow{\mathcal{P}}:F(\rho_{\gamma}(x))=\sigma(\gamma)\rho_{\gamma}(F(x)),\
\forall\gamma\in\Gamma,\ x\in V\\}.\end{array}$ (5)

The set ${{\mathcal{P}}}(\Gamma)$ has a ring structure and
${{\mathcal{P}}_{\sigma}}(\Gamma)$ is a module over ${{\cal{P}}}(\Gamma)$. If
$\Gamma$ acts linearly on $V$, the sets
${\overrightarrow{\mathcal{P}}}(\Gamma)$ and
${\overrightarrow{\mathcal{P}}_{\sigma}}(\Gamma)$ are also modules over
${{\cal{P}}}(\Gamma)$.

For our purposes, given two groups $\Gamma_{1}$ and $\Gamma_{2}$, an action of
the semidirect product $\Gamma_{1}\rtimes\Gamma_{2}$ on $V$ can be given by
the mapping $(\Gamma_{1}\rtimes\Gamma_{2})\times V\to V$ defined by
$(\gamma_{1},\gamma_{2})x=\gamma_{1}(\gamma_{2}x)$. For $j=1$,$2$, if
$\beta_{j}:\Gamma_{j}\to\mathbb{Z}_{2}$ is a group homomorphism, we define the
homomorphism ${\beta}:\Gamma_{1}\rtimes\Gamma_{2}\to\mathbb{Z}_{2}$ by

${\beta}(\gamma_{1},\gamma_{2})=\beta_{1}(\gamma_{1})\beta_{2}(\gamma_{2}).$
(6)

The following result relates the invariant theory for
$\Gamma_{1}\rtimes\Gamma_{2}$ with the invariant theory for each group,
$\Gamma_{1}$ and $\Gamma_{2}$. This is a natural extension of [4, Proposition
3.2], where the authors consider $\beta_{1}$ in (6) as the trivial
homomorphism.

###### Proposition 2.4

Let $\Gamma_{1}$ and $\Gamma_{2}$ be Lie groups acting linearly on $V$ and
${\beta}$ defined as in (6). Then

  1. 1.

${{\mathcal{P}}}(\Gamma_{1}\rtimes\Gamma_{2})={{\mathcal{P}}}(\Gamma_{1})\cap{{\mathcal{P}}}(\Gamma_{2})$;

  2. 2.

${\overrightarrow{\mathcal{P}}}(\Gamma_{1}\rtimes\Gamma_{2})={\overrightarrow{\mathcal{P}}}(\Gamma_{1})\cap{\overrightarrow{\mathcal{P}}}(\Gamma_{2})$;

  3. 3.

${{\mathcal{P}}_{{\beta}}}(\Gamma_{1}\rtimes{\Gamma_{2}})={{\mathcal{P}}_{\beta_{1}}}({\Gamma}_{1})\cap{{\mathcal{P}}_{\beta_{2}}}({\Gamma}_{2})$;

  4. 4.

${\overrightarrow{\mathcal{P}}_{{\beta}}}(\Gamma_{1}\rtimes\Gamma_{2})={\overrightarrow{\mathcal{P}}_{\beta_{1}}}({\Gamma}_{1})\cap{\overrightarrow{\mathcal{P}}_{\beta_{2}}}({\Gamma}_{2})$.

##  3 Semisymplectic actions

Semisymplectic actions on a finite-dimensional symplectic vector space are the
natural actions in the study of Hamiltonian systems with symmetries and
reversing symmetries (see, for instance, [1, 18, 23]). For what follows,
$\Gamma$ is a Lie group acting on a $2n$-dimensional symplectic vector space
$(V,\omega)$.

###### Definition 3.1

An action of $\Gamma$ on $(V,\omega)$ is called $\sigma$-semisymplectic if
there exists a group homomorphism $\sigma:\Gamma\to{\mathbb{Z}}_{2}$ such that
the pullback $(\rho_{\gamma})^{*}\omega$ satisfies
$(\rho_{\gamma})^{*}\omega=\sigma(\gamma)\omega$, that is,

$\omega(d(\rho_{\gamma})_{x}\mathbf{u},d(\rho_{\gamma})_{x}\mathbf{v})=\sigma(\gamma)\omega(\mathbf{u},\mathbf{v}),$

for all $\gamma\in\Gamma$ and $x,\mathbf{u},\mathbf{v}\in V$. If $\sigma$ is
the trivial homomorphism, we say that the action of $\Gamma$ on $V$ is
symplectic or $\Gamma$ acts symplectically on $V$.

An element $\gamma\in\ker\sigma$ is called symplectic and
$\gamma\in\Gamma\setminus\ker\sigma$ is called antisymplectic. These
nomenclatures are suitable, since $\gamma\in\ker\sigma$ if and only if
$\rho_{\gamma}$ is a symplectic map and $\gamma\in\Gamma\setminus\ker\sigma$
if and only if $\rho_{\gamma}$ is an antisymplectic map.

If $\sigma$ is surjective, then the quotient group $\Gamma/\ker\sigma$ is
isomorphic to $\mathbb{Z}_{2}$. Thus, given
$\delta\in\Gamma\setminus\ker\sigma$, we have
$\Gamma\setminus\ker\sigma=\delta\ker\sigma$ as the unique non-trivial left-
coset of $\ker\sigma$. Hence

$\Gamma=\ker\sigma\ \dot{\cup}\ \delta\ker\sigma.$

If the action of $\Gamma$ on $V$ is linear, the next result characterizes
subgroups of the $\omega$-symplectic and $\omega$-semisymplectic groups in
terms of properties of this action. For this purpose, we identify the group
$GL(V)$ of invertible linear operators on $V$ with the group $GL(2n)$ of
invertible matrices of order $2n$.

###### Theorem 3.2

Let $\Gamma$ be a Lie group acting linearly on a $2n$-dimensional symplectic
vector space $(V,\omega)$ and $\rho:\Gamma\to GL(2n)$ the representation of
$\Gamma$ on $V$. Then $\rho(\Gamma)$ is a subgroup of $\Omega_{n}$ defined in
(3) if and only if the action of $\Gamma$ on $(V,\omega)$ is
$\sigma$-semisymplectic, for some group homomorphism
$\sigma:\Gamma\to\mathbb{Z}_{2}$. In particular, $\rho(\Gamma)$ is a subgroup
of $Sp_{\omega}(n;{\mathbb{R}})$ if and only if the action of $\Gamma$ on
$(V,\omega)$ is symplectic.

Proof. Suppose $\rho(\Gamma)$ is a subgroup of $\Omega_{n}$ and identify
$\rho(\gamma):=\rho_{\gamma}$ with its matrix, for each $\gamma\in\Gamma$.
Then $\rho_{\gamma}\in Sp_{\omega}(n;{\mathbb{R}})$ or $\rho_{\gamma}\in
Sp_{\omega}^{-1}(n;{\mathbb{R}})$.

By [6, Proposition 2.3], if $\rho(\Gamma)\subset Sp_{\omega}(n;{\mathbb{R}})$,
then $\rho_{\gamma}$ is a symplectic linear operator for all
$\gamma\in\Gamma$, that is, $(\rho_{\gamma})^{*}\omega=\omega$. Thus, the
action of $\Gamma$ on $(V,\omega)$ is symplectic.

If $\rho(\Gamma)\cap Sp_{\omega}^{-1}(n;{\mathbb{R}})\neq\emptyset$, we define
the map $\sigma:\Gamma\to\mathbb{Z}_{2}$ by

$\sigma(\gamma)=\left\\{\begin{array}[]{rl}1,&\text{if}\ \rho_{\gamma}\in
Sp_{\omega}(n;{\mathbb{R}})\\\ -1,&\text{if}\ \rho_{\gamma}\in
Sp_{\omega}^{-1}(n;{\mathbb{R}})\end{array}\right..$

Note that $\sigma$ is a group epimorphism (Remark 2.3). Furthermore,
$\rho_{\gamma}$ is a symplectic map if $\rho_{\gamma}\in
Sp_{\omega}(n;{\mathbb{R}})$, whence
$(\rho_{\gamma})^{*}\omega=\omega=\sigma(\gamma)\omega$. Similarly, it follows
from [6, Proposition 2.3] that $\rho_{\gamma}$ is an antisymplectic map if
$\rho_{\gamma}\in Sp_{\omega}^{-1}(n;{\mathbb{R}})$, whence
$(\rho_{\gamma})^{*}\omega=-\omega=\sigma(\gamma)\omega$.

Conversely, if $\Gamma$ is a Lie group acting linearly and
$\sigma$-semisymplectically on $(V,\omega)$, for some homomorphism
$\sigma:\Gamma\to\mathbb{Z}_{2}$, then $\rho_{\gamma}$ is a symplectic map if
$\gamma\in\ker\sigma$ and antisymplectic if
$\gamma\in\Gamma\setminus\ker\sigma$. If $\gamma\in\ker\sigma$ then
$\rho_{\gamma}\in Sp_{\omega}(n;{\mathbb{R}})$ and if
$\gamma\in\Gamma\setminus\ker\sigma$ then $\rho_{\gamma}\in
Sp_{\omega}^{-1}(n;{\mathbb{R}})$. Hence, $\rho(\Gamma)$ is a subgroup of
$\Omega_{n}$ such that $\rho(\ker\sigma)\subset Sp_{\omega}(n;{\mathbb{R}})$.
In particular, if the action of $\Gamma$ on $(V,\omega)$ is symplectic, we can
consider $\sigma$ as the trivial homomorphism and, therefore,
$\rho_{\gamma}\in Sp_{\omega}(n;{\mathbb{R}})$ for all $\gamma\in\Gamma$.

As we mentioned, an important group in the study of dynamical systems which
presents symmetric geometric configurations is the linear Lie group
$\mathbf{S}$ given in (1). When $[\omega]=J$ as in (2) and
$L\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$, it is known that $\mathbf{S}$
is a subgroup of $Sp_{\omega}(n;\mathbb{R}).$ In particular, the authors in
[18] use this property to obtain some results for Hamiltonian vector fields.
In a more general context, we characterize when $\mathbf{S}$ is a subgroup of
$Sp_{\omega}(n;\mathbb{R})$ in terms of the matrices in
${\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$.

###### Proposition 3.3

Let $(V,\omega)$ be a symplectic vector space and consider the group
$\mathbf{S}$ defined in (1). Then $\mathbf{S}$ is a subgroup of
$Sp_{\omega}(n;\mathbb{R})$ if and only if
$L^{T}\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$.

Proof. We can consider the Lie algebra of $Sp_{\omega}(n;\mathbb{R})$ as

${\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})=\left\\{L\in{\mathbb{M}}_{2n}({\mathbb{R}}):e^{tL}\in
Sp_{\omega}(n;\mathbb{R}),\ \forall t\in{\mathbb{R}}\right\\},$ (7)

since $Sp_{\omega}(n;\mathbb{R})$ is a closed subgroup of $GL(2n)$. If
$\mathbf{S}\subset Sp_{\omega}(n;\mathbb{R})$, then $e^{sL^{T}}\in
Sp_{\omega}(n;\mathbb{R})$, for all $s\in\mathbb{R}$. Thus
$L^{T}\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$ by (7). On the other hand,
if $L^{T}\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$, then $e^{sL^{T}}\in
Sp_{\omega}(n;\mathbb{R})$, for all $s\in\mathbb{R}$, that is,
$\\{e^{sL^{T}}:s\in\mathbb{R}\\}\subset Sp_{\omega}(n;\mathbb{R})$. We claim
that $\mathbf{S}$ is also a subgroup of $Sp_{\omega}(n;\mathbb{R})$. Indeed,
given $B\in{\bf S}$, there exists a sequence
$(B_{m})\subset\\{e^{sL^{T}}:s\in\mathbb{R}\\}$ such that $\lim B_{m}=B$. Then

$B^{T}[\omega]B=(\lim B_{m})^{T}[\omega](\lim
B_{m})=\lim\left(B_{m}^{T}[\omega]B_{m}\right)=\lim[\omega]=[\omega],$

that is, $B\in Sp_{\omega}(n;\mathbb{R})$.

###### Corollary 3.4

Let $(V,\omega)$ be a $2n$-dimensional symplectic vector space such that
$[\omega]^{2}=-I_{2n}$. If $L\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$,
then $\mathbf{S}$ is a subgroup of $Sp_{\omega}(n;\mathbb{R})$.

Proof. Note that if $[\omega]^{2}=-I_{2n}$, then by (4) we have that
${\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})=\\{L\in{\mathbb{M}}_{2n}({\mathbb{R}}):[\omega]L^{T}[\omega]=L\\}$.
In this case, if $L\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$, then
$L^{T}\in{\mathfrak{sp}}_{\omega}(n;{\mathbb{R}})$. Thus, the result is a
consequence of the previous proposition.

###### Example 3.5

Let $(\mathbb{R}^{4},\omega)$ be the symplectic vector space and $L$ in the
definition of $\mathbf{S}$ such that

$[\omega]=\left[\begin{array}[]{cccc}0&1&0&2\\\ -1&0&-1&0\\\ 0&1&0&1\\\
-2&0&-1&0\end{array}\right]\quad\textrm{ and}\quad
L=\left[\begin{array}[]{cccc}-1&1&-1&2\\\ 3&0&4&1\\\ -1&2&0&2\\\
3&1&1&1\end{array}\right].$

Since $L^{T}\notin{\mathfrak{sp}_{\omega}(2,\mathbb{R})}$, Proposition 3.3
ensures that $\mathbf{S}$ is not a subgroup of $Sp_{\omega}(2;\mathbb{R})$.

##  4 Types of symmetry on Hamiltonian context

Consider $\sigma_{1},\sigma_{2}:\Gamma\to{\mathbb{Z}}_{2}$ two group
homomorphisms. Assume that $\Gamma$ acts $\sigma_{1}$-semisympletically on
$(V,\omega)$. In this section, we relate symmetries of a
$\Gamma_{\sigma_{2}}$-equivariant $\omega$-Hamiltonian vector field with
symmetries of its associated Hamiltonian function.

The motivation for this correspondence are some works in the literature. For
instance, in [11, Proposition 3] Buzzi and Teixeira consider reversible
Hamiltonian vector fields under the action of a linear involution $\delta$ and
verify that their associated Hamiltonian functions $H$ are
$(\mathbb{Z}_{2})_{\sigma}$-invariant, with $\sigma(\delta)=-1$. Alomair and
Montaldi [1] extend this result by considering an orthogonal action of a
matrix group $\Gamma$ on $({\mathbb{R}}^{2n},\omega_{0})$ and show that there
are up to four types of symmetry for Hamiltonian vector fields. Theorem 4.2
generalizes these approach, in the sense that we impose neither symplectic
coordinates, nor linearity and orthogonality of the action of $\Gamma$ on
$(V,\omega)$.

From an algebraic point of view, the four types of symmetry $\gamma\in\Gamma$
as presented in [1] are: symplectic equivariant (SE), when
$\sigma_{1}(\gamma)=\sigma_{2}(\gamma)=1$; symplectic reversible (SR), when
$\sigma_{1}(\gamma)=-\sigma_{2}(\gamma)=1$; antisymplectic equivariant (AE),
when $\sigma_{1}(\gamma)=-\sigma_{2}(\gamma)=-1$ and antisymplectic reversible
(AR), when $\sigma_{1}(\gamma)=\sigma_{2}(\gamma)=-1$.

In the next result we denote by $[G:H]$ the index of the subgroup $H$ of $G$
in $G$.

###### Proposition 4.1

Let $\Gamma$ be a Lie group acting on a symplectic vector space $(V,\omega)$
and $\sigma_{1},\sigma_{2}:\Gamma\to{\mathbb{Z}}_{2}$ two distinct
epimorphisms. If the action of $\Gamma$ on $(V,\omega)$ is
$\sigma_{1}$-semisymplectic and $X$ is an $\Gamma_{\sigma_{2}}$-equivariant
$\omega$-Hamiltonian vector field, then $X$ presents all types of symmetry:
SE, SR, AE and AR.

Proof. Clearly the identity of $\Gamma$ is SE. Since $\sigma_{1}$ and
$\sigma_{2}$ are distinct we have $\ker\sigma_{1}\neq\ker\sigma_{2}$. Moreover
$\ker\sigma_{2}\not\subset\ker\sigma_{1}$, otherwise we would have

$2=[\Gamma:\ker\sigma_{2}]=[\Gamma:\ker\sigma_{1}]\cdot[\ker\sigma_{1}:\ker\sigma_{2}]=2\cdot[\ker\sigma_{1}:\ker\sigma_{2}],$

which implies $\ker\sigma_{1}=\ker\sigma_{2}$. Similarly,
$\ker\sigma_{1}\not\subset\ker\sigma_{2}$. Thus there exists $\gamma\in\Gamma$
such that $\gamma\in\ker\sigma_{2}$ and $\gamma\notin\ker\sigma_{1}$, that is,
$\gamma\in\ker\sigma_{2}\cap\left(\Gamma\backslash\ker\sigma_{1}\right)$. In
this case $\gamma$ is AE. There also exists $\delta\in\Gamma$ such that
$\delta\in\ker\sigma_{1}\cap\left(\Gamma\backslash\ker\sigma_{2}\right)$.
Therefore $\delta$ is SR. Hence,
$\sigma_{1}(\gamma\delta)=\sigma_{1}(\gamma)\sigma_{1}(\delta)=-1$ and
$\sigma_{2}(\gamma\delta)=\sigma_{2}(\gamma)\sigma_{2}(\delta)=-1$, that is,
$\gamma\delta$ is AR.

Consider the homomorphism $\sigma_{1}\sigma_{2}:\Gamma\to{\mathbb{Z}}_{2}$
defined by
$(\sigma_{1}\sigma_{2})(\gamma)=\sigma_{1}(\gamma)\sigma_{2}(\gamma)$, for all
$\gamma\in\Gamma$. If $\sigma_{1}$ and $\sigma_{2}$ are non-trivial and
distinct, then the product $\sigma_{1}\sigma_{2}$ is a third epimorphism,
which is distinct from the both ones.

We remark that the hypothesis $H(p)=0$ is not necessary for the reciprocal of
the next theorem.

###### Theorem 4.2

Let $\Gamma$ be a Lie group acting ${\sigma_{1}}$-semisymplectically on a
symplectic vector space $(V,\omega)$ and $X_{H}$ an $\omega$-Hamiltonian
vector field on $V$, with $H:V\to{\mathbb{R}}$ satisfying $H(p)=0$ for some
$p\in V$. If $X_{H}$ is $\Gamma_{\sigma_{2}}$-equivariant, then the
Hamiltonian function $H$ is $\Gamma_{\sigma_{1}\sigma_{2}}$-invariant.
Reciprocally, if $H$ is $\Gamma_{\sigma_{2}}$-invariant, then $X_{H}$ is
$\Gamma_{\sigma_{1}\sigma_{2}}$-equivariant.

Proof. Suppose
$X_{H}(\rho_{\gamma}(x))=\sigma_{2}(\gamma)d(\rho_{\gamma})_{x}X_{H}(x)$, for
all $\gamma\in\Gamma$ and $x\in V$. Since $X_{H}$ is $\omega$-Hamiltonian, we
have for all $x,\mathbf{v}\in V$ that

$\displaystyle dH_{x}\mathbf{v}$
$\displaystyle=\omega(X_{H}(x),\mathbf{v})=\sigma_{1}(\gamma)\omega(d(\rho_{\gamma})_{x}X_{H}(x),d(\rho_{\gamma})_{x}\mathbf{v})$
$\displaystyle=\sigma_{1}(\gamma)\omega(\sigma_{2}(\gamma)X_{H}(\rho_{\gamma}(x)),d(\rho_{\gamma})_{x}\mathbf{v})=(\sigma_{1}\sigma_{2})(\gamma)dH_{\rho_{\gamma}(x)}d(\rho_{\gamma})_{x}\mathbf{v}$
$\displaystyle=d((\sigma_{1}\sigma_{2})(\gamma)(H\circ\rho_{\gamma}))_{x}\mathbf{v},$

where the second equality holds since the action of $\Gamma$ on $(V,\omega)$
is $\sigma_{1}$-semisymplectic. Thereby
$(\sigma_{1}\sigma_{2})(\gamma)(H\circ\rho_{\gamma})(x)=H(x)+c$, for some
constant $c\in{\mathbb{R}}$, for all $\gamma\in\Gamma$ and $x\in V$. In
particular, by considering $\gamma=1$ the identity of $\Gamma$, we have
$c=H(p)+c=(\sigma_{1}\sigma_{2})(1)H(\rho_{1}(p))=H(p)=0.$ Therefore,
$H(\rho_{\gamma}(x))=(\sigma_{1}\sigma_{2})(\gamma)H(x)$.

Conversely, given $\gamma\in\Gamma$ and $x,\mathbf{v}\in V$, since
$H(\rho_{\gamma}(x))=\sigma_{2}(\gamma)H(x)$, we have
$\sigma_{2}(\gamma)dH_{x}\mathbf{v}=d(H\circ{\rho}_{\gamma})_{x}\mathbf{v}=dH_{{\rho}_{\gamma}(x)}d({\rho}_{\gamma})_{x}\mathbf{v}$.
Thus

$\displaystyle\omega(X_{H}({\rho}_{\gamma}(x)),d({\rho}_{\gamma})_{x}\mathbf{v})$
$\displaystyle=dH_{{\rho}_{\gamma}(x)}d({\rho}_{\gamma})_{x}\mathbf{v}=\sigma_{2}(\gamma)dH_{x}\mathbf{v}$
$\displaystyle=\sigma_{2}(\gamma)\omega(X_{H}(x),\mathbf{v})=\sigma_{2}(\gamma)\sigma_{1}(\gamma)\omega(d({\rho}_{\gamma})_{x}X_{H}(x),d({\rho}_{\gamma})_{x}\mathbf{v})$
$\displaystyle=\omega((\sigma_{1}\sigma_{2})(\gamma)d({\rho}_{\gamma})_{x}X_{H}(x),d({\rho}_{\gamma})_{x}\mathbf{v}),$

where the last equality follows by the bilinearity of $\omega$. Therefore,

$\omega(X_{H}(\rho_{\gamma}(x))-(\sigma_{1}\sigma_{2})(\gamma)d(\rho_{\gamma})_{x}X_{H}(x),d(\rho_{\gamma})_{x}\mathbf{v})=0,$

for all $\gamma\in\Gamma$ and $x,\mathbf{v}\in V$. Since $\omega$ is non-
degenerate and $d(\rho_{\gamma})_{x}$ is an isomorphism, we conclude that
$X_{H}(\rho_{\gamma}(x))=(\sigma_{1}\sigma_{2})(\gamma)d(\rho_{\gamma})_{x}X_{H}(x)$,
for all $\gamma\in\Gamma$ and $x\in V$.

The last two results can be naturally extended to symplectic manifolds. As
consequence of Theorem 4.2 we present the following table, which also appears
in [1] for the case where $\Gamma$ acts orthogonally on
$({\mathbb{R}}^{2n},\omega_{0})$. The signs of $\omega$, $X_{H}$ and $H$
represent, respectively, the signs of $\sigma_{1}(\gamma)$,
$\sigma_{2}(\gamma)$ and $(\sigma_{1}\sigma_{2})(\gamma)$ for
$\gamma\in\Gamma$ satisfying the condition in the first column.

Symmetry | $\omega$ | $X_{H}$ | $H$
---|---|---|---
Symplectic equivariant (SE) | $+1$ | $+1$ | $+1$
Symplectic reversible (SR) | $+1$ | $-1$ | $-1$
Antisymplectic equivariant (AE) | $-1$ | $+1$ | $-1$
Antisymplectic reversible (AR) | $-1$ | $-1$ | $+1$
Table 1: Possible types of symmetry in the Hamiltonian context.

In [10, 18] the authors consider a unique homomorphism
$\sigma:=\sigma_{1}=\sigma_{2}$ so that a symmetric Hamiltonian vector field
admits only symmetries of type SE or AR. The next example presents a
Hamiltonian vector field with the four types of symmetry of Table 1.

###### Example 4.3

Consider $\Gamma={\mathbb{Z}}_{2}^{\tau}\times{\mathbb{Z}}_{2}^{\psi}$ acting
on $(\mathbb{R}^{4},\omega)$ by matrix multiplication, where

$\tau=\left[\begin{array}[]{cccc}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{array}\right],\quad\psi=\left[\begin{array}[]{cccc}-1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{array}\right]\quad\text{and}\quad[\omega]=\left[\begin{array}[]{cccc}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{array}\right].$

Notice that $\psi\in Sp_{\omega}(2;{\mathbb{R}})$ and $\tau\in
Sp_{\omega}^{-1}(2;{\mathbb{R}})$. Then $\Gamma$ is a subgroup of $\Omega_{2}$
and, by Theorem 3.2, its action on $(\mathbb{R}^{4},\omega)$ is
$\sigma_{1}$-semisymplectic so that
$\ker\sigma_{1}=\left\\{(I_{4},I_{4}),(I_{4},\psi)\right\\}$. Given
$\lambda\in\mathbb{R}^{*}$, the $\omega$-Hamiltonian vector field

$X(x_{1},x_{2},x_{3},x_{4})=\lambda(x_{2},-x_{1},x_{4},-x_{3})$

associated with
$H(x_{1},x_{2},x_{3},x_{4})=-\lambda\left(x_{1}x_{3}+x_{2}x_{4}\right)$ is
$\Gamma_{\sigma_{2}}$-equivariant, with
$\ker\sigma_{2}=\left\\{(I_{4},I_{4}),(\tau,\psi)\right\\}$. By Theorem 4.2,
$H$ is $\Gamma_{\sigma_{1}\sigma_{2}}$-invariant, where
$\sigma_{1}\sigma_{2}:\Gamma\to{\mathbb{Z}}_{2}$ is the epimorphism such that
$\ker\sigma_{1}\sigma_{2}=\left\\{(I_{4},I_{4}),(\tau,I_{4})\right\\}$. By
Proposition 4.1, the vector field $X$ presents four types of symmetry:

SE | SR | AE | AR
---|---|---|---
$(I_{4},I_{4})$ | $(I_{4},\psi)$ | $(\tau,\psi)$ | $(\tau,I_{4})$
Table 2: Types of symmetry of
$X(x_{1},x_{2},x_{3},x_{4})=\lambda(x_{2},-x_{1},x_{4},-x_{3})$.

##  5 Normal forms of $\omega$-Hamiltonian vector fields

The aim of this section is to present the normal form theory of
$\omega$-Hamiltonian vector fields on $(V,\omega)$, based on the approach
given in [13] for the canonical symplectic space
$(\mathbb{R}^{2n},\omega_{0})$. Moreover, we establish a relation between the
normal forms of an $\omega$-Hamiltonian vector field and of its associated
Hamiltonian function.

For what follows, we denote respectively by ${\mathcal{P}}^{k}$ and
${\overrightarrow{{\mathcal{P}}}^{k}}$ the vector spaces of the homogeneous
maps of degree $k$ in ${\mathcal{P}}$ and in $\overrightarrow{{\mathcal{P}}}$.
Similarly, ${\mathcal{P}}^{k}(\Gamma)$ and
${\overrightarrow{{\mathcal{P}}}^{k}}(\Gamma)$ are the vector spaces of the
homogeneous maps of degree $k$ in ${\mathcal{P}}(\Gamma)$ and in
$\overrightarrow{{\mathcal{P}}}(\Gamma)$ (see (5)), respectively.

Consider $(V,\omega)$ a symplectic vector space of dimension $2n$ and
$X(x)=([\omega]^{-1})^{T}\nabla H(x)$ an $\omega$-Hamiltonian vector field on
$V$ with an isolated singularity at the origin, that is, $X(0)=0$. Hence we
have $\nabla H(0)=0$ and assume $H(0)=0$. Thus, the Taylor series of $H$
centered at the origin is written as

$H^{2}(x)+H^{3}(x)+\cdots+H^{r}(x)+O(|x|^{r+1}),$ (8)

where $H^{k}\in{\mathcal{P}}^{k}$.

###### Lemma 5.1

The $\omega$-Hamiltonian vector field $X_{H^{2}}$ associated with $H^{2}$
determines the linear part of $X$, that is, $X_{H^{2}}(x)=Lx,$ where
$L=dX_{0}$ is the linearization of $X$ at the origin.

Proof. Since $X(x)=([\omega]^{-1})^{T}\nabla H(x)=(dH_{x}[\omega]^{-1})^{T}$,
we obtain

$L=dX_{0}=(d^{2}H_{0}[\omega]^{-1})^{T}=([\omega]^{-1})^{T}d^{2}H_{0}.$

By definition, we have
$H^{2}(x)=\frac{1}{2}\mathbf{x}^{T}d^{2}H_{0}\mathbf{x}$. Thus $\nabla
H^{2}(x)=d^{2}H_{0}x$ and

$X_{H^{2}}(x)=([\omega]^{-1})^{T}\nabla
H^{2}(x)=([\omega]^{-1})^{T}d^{2}H_{0}x=Lx.$

Based on the classical normal form theory, we want to determine a change of
coordinates $x=\xi(y)$ that preserves the linearization $L=dX_{0}$ and,
naturally, the dynamics of the system $\dot{x}=X(x)$ around the origin. For
each $k\geq 3$,

$X_{H^{2}+\cdots+H^{k}}(x)=X_{H^{2}}(x)+\cdots+X_{H^{k}}(x)=Lx+X_{H^{3}}(x)+\cdots+X_{H^{k}}(x),$

where $X_{H^{k}}=([\omega]^{-1})^{T}\nabla
H^{k}\in{{\overrightarrow{{\mathcal{P}}}^{k-1}}}$. From now on, we assume that

$X(x)\ =Lx+X_{H^{3}}(x)+X_{H^{4}}(x)+\cdots+X_{H^{r}}(x)+O(|x|^{r})$ (9)

and denote $\mu(x)=X_{H^{3}}(x)+X_{H^{4}}(x)+\cdots+X_{H^{r}}(x)+O(|x|^{r})$.
Consider $x=\xi(y)$ a change of coordinates in a neighborhood $U$ of the
origin, that is, $d\xi_{y}$ is invertible with $\xi(0)=0.$ Thus

$\xi^{-1}_{*}X(y)\ =\ d\xi_{y}^{-1}(L\xi(y)+\mu(\xi(y))),$ (10)

with $y\in U$. We denote by $\varphi(y)$ the right side of (10) and by $\nu$
the derivative of $d\xi_{y}^{-1}$ with respect to $y$. Then the Jacobian of
$\varphi$ is given by

$J_{\varphi}(y)=\nu(L\xi(y)+\mu(\xi(y)))+d\xi_{y}^{-1}(Ld\xi_{y}+d\mu_{\xi(y)}d\xi_{y}).$

For $y=0$, we have
$J_{\varphi}(0)=d\xi_{0}^{-1}Ld\xi_{0}+d\xi_{0}^{-1}d\mu_{0}d\xi_{0}=d\xi_{0}^{-1}Ld\xi_{0},$
since $d\mu_{0}=0$. Thus, the linear part of (10) is
$d\xi_{0}^{-1}Ld\xi_{0}y$.

Assume that $\xi$ has the form $\xi(y)=y+O(|y|^{2}),$ with $y\rightarrow 0$.
Then $d\xi_{0}=d\xi_{0}^{-1}={\rm Id}$ whence the linear part of (10) is $Ly$.
Therefore,

$\xi^{-1}_{*}X(y)\ =Ly+g(y),$ (11)

with $y\in U$, where $g(y)$ has only terms of degree greater than or equal to
2.

The idea is determining $\xi$ so that (11) is still an $\omega$-Hamiltonian
vector field. This is possible if $\xi$ is a symplectic diffeomorphism and, in
this case, we obtain a new $\omega$-Hamiltonian vector field associated with
$H\circ\xi$ (see [6, Theorem 4.2]).

###### Proposition 5.2

For each $k\geq 3$, given $\xi^{k}\in{\mathcal{P}}^{k}$, there exist a
neighborhood of the origin $U_{k}\subset V$ and a symplectic diffeomorphism
$\xi:U_{k}\to V$ written as

$\xi(y)=y+X_{\xi^{k}}(y)+O(|y|^{k}),$ (12)

with $y\in U_{k}$, where $X_{\xi^{k}}=([\omega]^{-1})^{T}\nabla\xi^{k}$.

Proof. Consider the dynamic system $\phi$ associated with $Y=X_{\xi^{k}}$.
Hence $\phi(t,y)$ satisfies the initial value problem

$\begin{cases}\dot{\phi}(t,y)=Y(y)\\\ \phi(0,y)=y\end{cases}$ (13)

and it can be expanded in a Taylor series around the origin as

$\phi(t,y)=\phi^{1}(t)y+\phi^{2}(t,y)+\cdots+\phi^{k-1}(t,y)+O(|y|^{k}),$ (14)

where $\phi^{1}:\mathbb{R}\to{\mathbb{M}}_{2n}(\mathbb{R})$,
$\phi^{j}:\mathbb{R}\times V\to V$ and, for all fixed $t$,
$\phi^{j}(t,\cdot)\in{{\overrightarrow{\mathcal{P}}^{j}}}$, $j=2,\ldots,k-1$.
Since $Y$ is a homogeneous polynomial map of degree $k-1$, by replacing (14)
in (13) we get

$\begin{cases}\dot{\phi}^{1}(t)=0\\\ \phi^{1}(0)=I_{2n}\end{cases}$ and
$\begin{cases}\dot{\phi}^{j}(t,y)=0\\\ \phi^{j}(0,y)=0\end{cases}$,

for all $j=2,\ldots,k-2$. Thus, given $2\leq j\leq k-2$, for all
$t\in\mathbb{R}$ in the maximal interval and all $y\in V$ we have
$\phi^{1}(t)=I_{2n}$ and $\phi^{j}(t,y)=0$, since the first line in each IVP
ensures us that $\phi^{1}$ and $\phi^{j}(\cdot,y)$ are constant in $t$.
Moreover

$\begin{cases}\dot{\phi}^{k-1}(t,y)=Y(y)\\\ \phi^{k-1}(0,y)=0\end{cases}.$

Now, by integrating with respect to $t$ and using the initial condition we
have $\phi^{k-1}(t,y)=tY(y)$. Hence $\phi^{k-1}(1,y)=Y(y)=X_{\xi^{k}}(y)$ and
the map $Y_{1}$ defined by $Y_{1}(y)=\phi(1,y)$ is symplectic (see [24,
Proposition V.24]). By (14),

$Y_{1}(y)=y+X_{\xi^{k}}(y)+O(|y|^{k})$

is a symplectic map. Finally take $\xi=Y_{1}$. By inverse mapping theorem,
there exists a neighborhood $U_{k}\subset V$ such that $\xi$ is a local
diffeomorphism.

We now recall some concepts and results about Hamiltonian vector fields. Given
two $C^{\infty}$ functions $F,G:V\to\mathbb{R}$, the Poisson bracket of $F$
and $G$ is a function $\\{F,G\\}:V\to\mathbb{R}$ defined by

$\\{F,G\\}(x)=\omega\left(X_{F}(x),X_{G}(x)\right),$

where $X_{F}$ and $X_{G}$ are $\omega$-Hamiltonian vector fields associated
with $F$ and $G$, respectively. It is possible to prove that
$\\{F,G\\}(x)=\langle\nabla F(x),X_{G}(x)\rangle_{\rm can}$ for all $x\in V$,
where $\langle\cdot,\cdot\rangle_{\rm can}$ is the canonical inner product on
$V$. Moreover

$X_{\\{F,G\\}}=[X_{G},X_{F}],$ (15)

where $[\ ,\ ]$ denotes the Lie bracket of vector fields. For more details,
see [24, Proposition V.28]).

Fixed $k\geq 3$, consider a symplectic change of coordinates near the identity
as in (12). For each $y\in U_{k}$, we have $d\xi_{y}={\rm
Id}+d(X_{\xi^{k}})_{y}+O(|y|^{k-1})$, whence $d\xi_{y}^{-1}={\rm
Id}-d(X_{\xi^{k}})_{y}+O(|y|^{k-1})$. So we can rewrite (10) like

$\displaystyle\xi^{-1}_{*}X(y)$ $\displaystyle=$ $\displaystyle
d\xi_{y}^{-1}(L\xi(y)+\mu(\xi(y)))\vspace{0.3cm}$ $\displaystyle=$
$\displaystyle\left({\rm
Id}-d(X_{\xi^{k}})_{y}+O(|y|^{k-1})\right)\left(Ly+LX_{\xi^{k}}(y)+\mu(\xi(y))\right)\vspace{0.3cm}$
$\displaystyle=$ $\displaystyle
Ly+LX_{\xi^{k}}(y)+X_{H^{3}}(y)+\cdots+X_{H^{k}}(y)-d(X_{\xi^{k}})_{y}Ly\vspace{0.1cm}$
$\displaystyle-d(X_{\xi^{k}})_{y}LX_{\xi^{k}}(y)-d(X_{\xi^{k}})_{y}(\mu(\xi(y)))+O(|y|^{k})\vspace{0.3cm}$
$\displaystyle=$ $\displaystyle
Ly+X_{H^{3}}(y)+\cdots+X_{H^{k-1}}(y)\vspace{0.1cm}$
$\displaystyle+\left(X_{H^{k}}(y)-(d(X_{\xi^{k}})_{y}Ly-
LX_{\xi^{k}}(y))\right)+O(|y|^{k}),$

with $y\in U_{k}$. Since $d(X_{\xi^{k}})_{y}L-LX_{\xi^{k}}$ is the Lie bracket
$[L,X_{\xi^{k}}]$, it follows by Lemma 5.1 and by (15) that the term in
parentheses is equal to

$X_{H^{k}}(y)-[X_{H^{2}},X_{\xi^{k}}](y)=X_{H^{k}}(y)-X_{\\{\xi^{k},H^{2}\\}}(y)=X_{H^{k}-\\{\xi^{k},H^{2}\\}}(y).$

Therefore, the vector field (9) is formally conjugated333A vector field $Y$ is
said formally conjugated to $X$ if a power series of $Y$ is conjugated to a
power series of $X$, as formal vector fields. to

$\xi^{-1}_{*}X(y)=Ly+X_{H^{3}}(y)+\cdots+X_{H^{k-1}}(y)+X_{H^{k}-\\{\xi^{k},H^{2}\\}}(y)+O(|y|^{k}),$
(16)

in which the linear part $L$ and the terms of degree less than $k-1$ of (9)
are preserved.

###### Definition 5.3

Given a quadratic form $H^{2}:V\to\mathbb{R}$, for each $k\geq 3$ we define
the linear operator $ad_{H^{2}}^{k}:\ {\mathcal{P}}^{k}\to{\mathcal{P}}^{k}$
by

$ad_{H^{2}}^{k}(\xi^{k})=\\{\xi^{k},H^{2}\\}.$ (17)

For the next theorem, consider ${\mathcal{D}}^{k}$ a complementary subspace of
$ad_{H^{2}}^{k}({\mathcal{P}}^{k})$ in ${\mathcal{P}}^{k}$, that is,

${\mathcal{P}}^{k}=ad_{H^{2}}^{k}({\mathcal{P}}^{k})\oplus{\mathcal{D}}^{k}.$
(18)

###### Theorem 5.4

Let $(V,\omega)$ be a symplectic vector space and $X:V\to V$ an
$\omega$-Hamiltonian vector field such that $X(0)=0$ and $dX_{0}=L$. Suppose
that the Hamiltonian function associated with $X$ admits a Taylor expansion at
the origin as in (8) and consider the decomposition (18), for $k=3,\ldots,r$.
Then $X$ is formally conjugated to the $\omega$-Hamiltonian vector field

$Y(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{r}}(y)+O(|y|^{r})$ (19)

associated with $K=H^{2}+K^{3}+\cdots+K^{r}+O(|y|^{r})$, where
$K^{k}\in{\mathcal{D}}^{k}$ for $k=3,\ldots,r$.

Proof. The idea is to exhibit a sequence of symplectic changes of coordinates
near the identity of the form $x=y+X_{\xi^{k}}(y)+O(|y|^{k}),$ with $y\in
U_{k}$, where ${\xi^{k}}\in{\mathcal{P}}^{k}$ and $U_{k}$ is a neighborhood of
the origin, with $U_{k+1}\subseteq U_{k}$, such that in the new coordinates
the vector field (9) is written as in (19).

By (18), there exist $G^{3}\in ad_{H^{2}}^{3}({\mathcal{P}}^{3})$ and
$K^{3}\in{\mathcal{D}}^{3}$ such that ${H^{3}}=G^{3}+K^{3}$. Hence we can
choose $\xi^{3}\in{\mathcal{P}}^{3}$ so that
$G^{3}=ad_{H^{2}}^{3}(\xi^{3})=\\{\xi^{3},H^{2}\\}$. Consider the symplectic
diffeomorphism given by $\xi(y)=y+X_{\xi^{3}}(y)+O(|y|^{3}),$ with $y\in
U_{3}$, for some neighborhood $U_{3}$ of the origin. Then $X$ is conjugated to
$\xi^{-1}_{*}X$ as in (16), that is,

$\xi^{-1}_{*}X(y)=Ly+X_{H^{3}-ad_{H^{2}}^{3}(\xi^{3})}(y)+O(|y|^{3})=Ly+X_{K^{3}}(y)+O(|y|^{3}).$

Now we proceed by induction on $k$, by remembering that such change of
coordinates do not modify terms of degree less than $k-1$. We assume for
$3\leq k-1<r$ the existence of a sequence of symplectic diffeomorphisms as in
(12) such that $X$ is conjugated to

$\xi^{-1}_{*}X(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{k-1}}(y)+O(|y|^{k-1}),$ (20)

with $y\in U_{k-1}$, where $U_{k-1}\subset U_{k-2}$ is a neighborhood of the
origin and $K^{j}\in{\mathcal{D}}^{j}$ for $j=3,\ldots,k-1$. Since (20) is an
$\omega$-Hamiltonian vector field we have

$\xi^{-1}_{*}X(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{k-1}}(y)+X_{{\overline{H}}^{k}}(y)+X_{{\overline{H}}^{k+1}}(y)+\cdots,$
(21)

where $X_{{\overline{H}}^{k+j}}$ is a homogeneous polynomial map of degree
$k+j-1$ for $j\geq 0$.

Consider $\xi(y)=y+X_{\xi^{k}}(y)+O(|y|^{k})$, with $y\in U_{k}$, for some
neighborhood of the origin $U_{k}\subseteq U_{k-1}$, where we choose
$\xi^{k}\in{\mathcal{P}}^{k}$ such that
${\overline{H}}^{k}=ad_{H^{2}}^{k}(\xi^{k})+K^{k}$ for some
$K^{k}\in{\mathcal{D}}^{k}$. Write $Y(y):=\xi^{-1}_{*}X(y)$. Hence (21) is
given by

$\displaystyle Y(y)$
$\displaystyle=Ly+X_{K^{3}}(y)+\cdots+X_{K^{k-1}}(y)+X_{{\overline{H}}^{k}-ad_{H^{2}}^{k}(\xi^{k})}(y)+O(|y|^{k})$
$\displaystyle=Ly+X_{K^{3}}(y)+\cdots+X_{K^{k-1}}(y)+X_{K^{k}}(y)+O(|y|^{k}),$

with $y\in U_{k}$.

The previous result induces the following definition.

###### Definition 5.5

Given an Hamiltonian function $H:V\to\mathbb{R}$ such that $H(0)=0$ and
$\nabla H(0)={0}$, a normal form of $H$ up to order $r$ is a sum

$K=H^{2}+K^{3}+\cdots+K^{r},$

where $H^{2}$ is the quadratic form associated with the Hessian of $H$ and
$K^{k}\in{{\mathcal{D}}^{k}}$ for $k=3,\ldots,r$. If $X$ is an
$\omega$-Hamiltonian vector field associated with $K$, we say that

$Y(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{r}}(y)$

is a normal form of $X$ up to order $r-1$.

In [7, 8], Belitskii develops a theory of normal forms for arbitrary vector
fields $X:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $X(0)=0$ and $L=dX_{0}$. For
each $k\geq 3$, the author defines the linear operator
$Ad_{L}^{k-1}:{\overrightarrow{{\mathcal{P}}}^{k-1}}\to{\overrightarrow{{\mathcal{P}}}^{k-1}}$
as $Ad_{L}^{k-1}(\xi)=[L,\xi]$ and consider a subspace ${\mathcal{C}}^{k-1}$
in the decomposition

${{\overrightarrow{{\mathcal{P}}}^{k-1}}}=Ad_{L}^{k-1}({{\overrightarrow{{\mathcal{P}}}^{k-1}}})\oplus{\mathcal{C}}^{k-1}.$

It is possible to prove that $X$ is formally conjugated to $Y(y)\
=Ly+g^{2}(y)+\cdots+g^{r}(y)+O(|y|^{r+1}),$ where
$g^{k-1}\in{\mathcal{C}}^{k-1}$ (see [13, Chapter 2, Theorem 1.1]). From [13,
Chapter 2, Theorem 1.7] we can assume ${\mathcal{C}}^{k-1}=\ker
Ad_{L^{T}}^{k-1}$ and by [15, XVI Lemma 5.4] we have

$\ker Ad_{L^{T}}^{k-1}={{\overrightarrow{{\mathcal{P}}}^{k-1}}}(\mathbf{S}),$
(22)

where $\mathbf{S}$ is defined in (1). In the next subsection, we will relate
the normal form of an $\omega$-Hamiltonian vector field $X$ given in (19) with
the obtained by the classical theory developed in [7, 8] for an appropriate
subspace $\mathcal{D}^{k}$.

###  5.1 An algebraic method to determine ${\mathcal{D}}^{k}$

In this subsection, we exhibit a convenient complementary subspace
${\mathcal{D}}^{k}$ as defined in (18). In [18, Theorem 2.1, Lemma 2.3], the
authors consider the canonical symplectic space
$({\mathbb{R}}^{2n},\omega_{0})$ and show that a possible choice for
${\mathcal{D}}^{k}$ is the set ${{\mathcal{P}}^{k}}({\bf S})$ of all
S-invariant homogeneous polynomial functions of degree $k$. For this result,
they use that ${\bf S}$ acts symplectically on $(\mathbb{R}^{2n},\omega_{0})$.
However, this property is not always true on an arbitrary symplectic vector
space $(V,\omega)$ (see Example 3.5 and Theorem 3.2).

From now on, we assume that the action of $\mathbf{S}$ on $(V,\omega)$ is
symplectic. In order to characterize the subspace ${\mathcal{D}}^{k}$, we
introduce an inner product in ${\mathcal{P}}^{k}$: given
$F,G\in{\mathcal{P}}^{k}$ and $X_{F}$, $X_{G}$ their respective
$\omega$-Hamiltonian vector fields, we define

$\ll F,G\gg=\langle X_{F},X_{G}\rangle,$

where $\langle\cdot,\cdot\rangle$ is a convenient inner product in
${{\overrightarrow{{\mathcal{P}}}^{k-1}}}$ such that the equality (22) holds
(see [15, XVI Theorem 5.3]). Notice that a possible choice for
${\mathcal{D}}^{k}$ is the orthogonal complementary subspace of
$ad_{H^{2}}^{k}({{\mathcal{P}}^{k}})$ characterized by
$\ker(ad_{H^{2}}^{k})^{*}$, where $(ad_{H^{2}}^{k})^{*}$ is the adjoint
operator of $ad_{H^{2}}^{k}$. Since ${\bf S}$ acts symplectically on
$(V,\omega)$, we highlight that $Y(x)=L^{T}x$ is an $\omega$-Hamiltonian
vector field (see Proposition 3.3 and [6, Proposition 4.6]), that is, there
exists a quadratic form denoted by $H^{2}_{T}:V\to\mathbb{R}$ such that
$Y(x)=X_{H^{2}_{T}}(x)$. For each $k\geq 3$, it is possible to prove that
$ad_{H^{2}_{T}}^{k}$ is the adjoint operator of $ad_{H^{2}}^{k}$ with respect
to the inner product $\ll\cdot,\cdot\gg$. The proof for this result is similar
to that found in [18, Lemma 2.3] for $(\mathbb{R}^{2n},\omega_{0})$.

Therefore, in order to obtain a normal form up to order $r$ of a Hamiltonian
function $H:V\to\mathbb{R}$, it is enough to determine $\ker
ad_{H^{2}_{T}}^{k}$, $k=3,\ldots,r$. By properties of Poisson bracket and by
Lemma 5.1,

$ad_{H^{2}}^{k}(\xi^{k})(x)=\\{\xi^{k},H^{2}\\}(x)=\langle\nabla\xi^{k}(x),X_{H^{2}}(x)\rangle_{\rm
can}=\langle\nabla\xi^{k}(x),Lx\rangle_{\rm can}.$

Hence we need to solve the partial differential equation
$\langle\nabla\xi^{k}(x),L^{T}x\rangle_{\rm can}=0,$ with $x\in V$. However we
present an alternative algebraic characterization for this kernel. First, we
remark that $F\in\ker ad^{k}_{H^{2}_{T}}$ if and only if
$\\{F,H^{2}_{T}\\}=ad_{H^{2}_{T}}^{k}(F)=0$, which occurs if and only if
$X_{\\{F,H^{2}_{T}\\}}=([\omega]^{-1})^{T}\nabla\\{F,H^{2}_{T}\\}=0$. Hence

$\begin{array}[]{rcl}\ker
ad_{H^{2}_{T}}^{k}&=&\\{F\in{{\mathcal{P}}^{k}}:X_{\\{F,H^{2}_{T}\\}}=0\\}=\\{F\in{{\mathcal{P}}^{k}}:[X_{H^{2}_{T}},X_{F}]=0\\}\vspace{.1cm}\\\
&=&\\{F\in{{\mathcal{P}}^{k}}:X_{F}\in{\overrightarrow{{\mathcal{P}}}^{k-1}}({\bf
S})\\}={\mathcal{P}}^{k}({\bf S}),\end{array}$ (23)

where the third equality follows from (22), since
$Ad_{L^{T}}^{k-1}(X_{F})=[X_{H^{2}_{T}},X_{F}]$. In the last equality we use
that the action of S on $(V,\omega)$ is symplectic and Theorem 4.2, by
considering $\Gamma=\textbf{S}$ and
$\sigma_{1}=\sigma_{2}:\mathbf{S}\to\mathbb{Z}_{2}$ the trivial homomorphism.

Therefore, a good choice for the complementary subspace ${\mathcal{D}}^{k}$ in
(18) is the vector space ${\mathcal{P}}^{k}({\bf S})$, that is,

${\mathcal{P}}^{k}=ad^{k}_{H^{2}}({\mathcal{P}}^{k})\oplus{\mathcal{P}}^{k}({\bf
S}).$ (24)

Thus, in order to describe a normal form up to order $r-1$ of the
$\omega$-Hamiltonian vector field $X_{H}$, it is enough to characterize the
ring ${\mathcal{P}}^{k}({\bf S})$, for $k=3,\ldots,r$. In fact, we show in
Theorem 5.4 that the original vector field is formally conjugated to (19). By
(23), if $\mathcal{D}^{k}={\mathcal{P}}^{k}({\bf S})$, then
$X_{K^{k}}\in{\overrightarrow{{\mathcal{P}}}^{k-1}}({\bf S})$, where
${\overrightarrow{{\mathcal{P}}}^{k-1}}({\bf S})$ is a choice for the
complementary subspace $\mathcal{C}^{k-1}$ according to (22).

Therefore, if $L$ is a non-null matrix, by Theorem 5.4 and by (24), the normal
form of the $\omega$-Hamiltonian vector field $X$ presents symmetries, even if
$X$ does not have any symmetry. Similarly for the normal form of the
Hamiltonian function associated with $X$.

##  6 Normal forms of $\omega$-Hamiltonian vector fields with symmetries

Fixed a compact Lie group $\Gamma$ acting linearly on a finite-dimensional
real vector space, Golubitski, Stewart and Schaeffer [15, XVI Theorem 5.8;
Theorem 5.9] present a normal form of a $\Gamma$-equivariant vector field,
based on method of Elphick et al. [14]. Afterwards, Baptistelli, Manoel and
Zeli [4, Theorem 4.7] generalize such a result in the context of
$\Gamma_{\sigma}$-equivariant vector fields. The union of the classical method
of normal forms in the Hamiltonian context presented in Section 5 with the
algebraic method developed in [4] is the starting point for the study of
normal forms of $\omega$-Hamiltonian vector fields with symmetries taking into
account the semisymplectic action of $\Gamma$ on $(V,\omega)$.

From now on, $\Gamma$ is a compact Lie group and $\sigma_{1},$
$\sigma_{2}:\Gamma\to{\mathbb{Z}}_{2}$ are two group homomorphisms. We assume
that the action of $\Gamma$ on $(V,\omega)$ is linear and
${\sigma_{1}}$-semisymplectic, and $X:V\to V$ is a
$\Gamma_{\sigma_{2}}$-equivariant $\omega$-Hamiltonian vector field, with
$X(0)=0$. We consider $H:(V,\omega)\to\mathbb{R}$ the Hamiltonian function
associated with $X$ such that $H(0)=0$ and $\nabla H(0)=0$. By Theorem 4.2,
$H$ is $\Gamma_{\sigma_{1}\sigma_{2}}$-invariant. Our goal is presenting a
normal form of $X$ which is also $\Gamma_{\sigma_{2}}$-equivariant and
$\omega$-Hamiltonian.

Let $L=dX_{0}$ be the linearization of $X$ at the origin and $\mathbf{S}$ the
linear Lie group defined in (1). If $\sigma_{2}$ is an epimorphism, we have a
well defined action of ${\bf S}\rtimes\Gamma$ on $V$, namely,

$(e^{sL^{T}},\gamma)x=e^{sL^{T}}(\gamma x)=\gamma
e^{\sigma_{2}(\gamma)sL^{T}}x,$ (25)

as in [4]. If $\sigma_{2}$ is trivial, then the actions of $\Gamma$ and
$\mathbf{S}$ commute, since there exists an inner product on $V$ such that the
action of $\Gamma$ is orthogonal (see [15, XII Proposition 1.3]). In this
case, we also assume the action in (25) by taking $\sigma_{2}(\gamma)=1$ for
all $\gamma\in\Gamma$.

For what follows, we denote
$\mathcal{P}_{\sigma}^{k}(\Gamma):=\mathcal{P}^{k}\cap\mathcal{P}_{\sigma}(\Gamma)$
(see (5)). In order to obtain a normal form as in Theorem 5.4, we present now
a result that provides us a complementary subspace of
$ad_{H^{2}}^{k}(\mathcal{P}_{\sigma_{1}}^{k}(\Gamma))$ in
$\mathcal{P}_{\sigma_{1}\sigma_{2}}^{k}(\Gamma)$, where $H^{2}$ is the
quadratic form of $H$ and $ad_{H^{2}}^{k}$ is defined in (17). A proof of this
result is presented in Section 7.

###### Theorem 6.1

If $\mathbf{S}$ acts symplectically on $(V,\omega)$, we have the decomposition

${\mathcal{P}}^{k}_{\sigma_{1}\sigma_{2}}(\Gamma)=ad_{H^{2}}^{k}({\mathcal{P}}^{k}_{\sigma_{1}}(\Gamma))\oplus{\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right),$ (26)

for each $k\geq 3$, where $\tilde{\sigma}_{j}:{\bf
S}\rtimes\Gamma\to{\mathbb{Z}}_{2}$ is defined by

$\tilde{\sigma}_{j}(e^{sL^{T}},\gamma)=\sigma_{j}(\gamma),$ (27)

with $j=1,2$, for all $s\in\mathbb{R}$ and $\gamma\in\Gamma$.

Our aim is to preserve the $\omega$-Hamiltonian condition and the symmetries
of the original vector field. For this, we consider changes of coordinates
$\xi$ which are symplectic, as those described in (12), and
$\Gamma$-equivariant, that is, $\xi\circ\rho_{\gamma}=\rho_{\gamma}\circ\xi,$
for all $\gamma\in\Gamma$. In this case, the pushforward $\xi^{-1}_{*}X$ is
$\Gamma_{\sigma_{2}}$-equivariant, since

$(\rho_{\gamma})_{*}(\xi^{-1}_{*}X)=(\rho_{\gamma}\circ\xi^{-1})_{*}X=(\xi^{-1}\circ\rho_{\gamma})_{*}X=\xi^{-1}_{*}((\rho_{\gamma})_{*}X)=\sigma_{2}(\gamma)\xi^{-1}_{*}X,$

for all $\gamma\in\Gamma$.

In the next lemma, we present a condition for a symplectic diffeomorphism to
be $\Gamma$-equivariant.

###### Lemma 6.2

Suppose $\Gamma$ acts $\sigma_{1}$-semisymplectically on $(V,\omega)$ and let
$\xi$ be a symplectic diffeomorphism as in (12). If
$\xi^{k}\in{\mathcal{P}}^{k}_{\sigma_{1}}(\Gamma)$, then $\xi$ is
$\Gamma$-equivariant.

Proof. By Theorem 4.2, if the action of $\Gamma$ on $(V,\omega)$ is
$\sigma_{1}$-semisymplectic, then ${\xi^{k}}$ is
$\Gamma_{\sigma_{1}}$-invariant if and only if $X_{\xi^{k}}$ is
$\Gamma$-equivariant. Thus, $(\rho_{\gamma})_{*}X_{\xi^{k}}=X_{\xi^{k}}$ and
since $\rho_{\gamma}:V\to V$ is a diffeomorphism we have

$\rho_{\gamma}\circ(X_{\xi^{k}})_{t}=((\rho_{\gamma})_{*}X_{\xi^{k}})_{t}\circ\rho_{\gamma}=(X_{\xi^{k}})_{t}\circ\rho_{\gamma},$

for all $\gamma\in\Gamma$ and $t\in\mathbb{R}$, where $(X_{\xi^{k}})_{t}$ is
the flow of $X_{\xi^{k}}$. In particular, for $t=1$ we have

$\rho_{\gamma}\circ(X_{\xi^{k}})_{1}=(X_{\xi^{k}})_{1}\circ\rho_{\gamma}.$
(28)

In the proof of Proposition 5.2, we choose $\xi$ as a solution for
$\dot{y}=X_{\xi^{k}}(y)$. More precisely we take $\xi=(X_{\xi^{k}})_{1}$. By
(28) we conclude that $\rho_{\gamma}\circ\xi=\xi\circ\rho_{\gamma}$ for all
$\gamma\in\Gamma$, that is, $\xi$ is $\Gamma$-equivariant.

In the next theorem we present a normal form of an $\omega$-Hamiltonian and
$\Gamma_{\sigma_{2}}$-equivariant vector field $X$ which preserves the
$\omega$-Hamiltonian condition and the symmetries of $X$.

###### Theorem 6.3

Let $\Gamma$ be a compact Lie group and $\sigma_{1},$
$\sigma_{2}:\Gamma\to\mathbb{Z}_{2}$ group homomorphisms. Suppose that the
action of $\Gamma$ on $(V,\omega)$ is $\sigma_{1}$-semisymplectic and consider
$X:V\to V$ an $\omega$-Hamiltonian and $\Gamma_{\sigma_{2}}$-equivariant
vector field such that $X(0)=0$ and $L=dX_{0}$. If $\mathbf{S}$ acts
symplectically on $(V,\omega)$, then $X$ is formally conjugated to

$Y(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{r}}(y)+O(|y|^{r}),$

where
$K^{k}\in{\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)$, $k=3,\ldots,r$, and $\tilde{\sigma}_{j}:{\bf
S}\rtimes\Gamma\to{\mathbb{Z}}_{2}$ is defined in (27).

Proof. Consider $X$ as in (9). By $\Gamma_{\sigma_{2}}$-equivariance of $X$,
we have that $L$ and $X_{H^{k}}$ are also $\Gamma_{\sigma_{2}}$-equivariant
for all $k\geq 3$. Thus,
$X_{H^{k}}\in{\overrightarrow{\mathcal{P}}_{\sigma_{2}}^{k}(\Gamma)}$ and by
Theorem 4.2 we know that
$H^{k}\in{\mathcal{P}_{\sigma_{1}\sigma_{2}}^{k}}(\Gamma)$ for each $k\geq 3$.
From here, the proof of this theorem follows similarly to the proof of the
Theorem 5.4. In the induction hypothesis, we assume that for $3\leq k-1<r$
there exists a sequence of symplectic changes of coordinates which transforms
(9) to the $\omega$-Hamiltonian and $\Gamma_{\sigma_{2}}$-equivariant vector
field

$Y(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{k-1}}(y)+X_{{\overline{H}}^{k}}(y)+X_{{\overline{H}}^{k+1}}(y)+\cdots,$
(29)

where
$K^{j}\in{\mathcal{P}}^{j}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)$ for $j=3,\ldots,k-1$. For each $k\geq 3$ we have
$X_{{\overline{H}}^{k}}\in{\overrightarrow{\mathcal{P}}_{\sigma_{2}}^{k}(\Gamma)}$
and, again by Theorem 4.2,
${\overline{H}}^{k}\in{\mathcal{P}}^{k}_{\sigma_{1}\sigma_{2}}(\Gamma)$.
Hence, by (26), there exist $G^{k}\in
ad_{H^{2}}^{k}(\mathcal{P}^{k}_{\sigma_{1}}(\Gamma))$ and
$K^{k}\in{\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)$ such that ${{\overline{H}}^{k}}=G^{k}+K^{k}$. Let
$\xi^{k}\in{\mathcal{P}}^{k}_{\sigma_{1}}(\Gamma)$ such that
$ad_{H^{2}}^{k}(\xi^{k})=G^{k}$ and consider the symplectic diffeomorphism
$\xi(y)=y+X_{\xi^{k}}(y)+O(|y|^{k})$. It follows from the proof of Theorem 5.4
that (29) is formally conjugated to

$Y(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{k-1}}(y)+X_{K^{k}}(y)+O(|y|^{k}),$

with
$K^{k}\in{\mathcal{P}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}^{k}(\mathbf{S}\rtimes\Gamma)}$.
Since $\xi^{k}\in{\mathcal{P}}^{k}_{\sigma_{1}}(\Gamma)$, from Lemma 6.2 we
have that $\xi$ is $\Gamma$-equivariant. Therefore the new vector field
preserves the symmetries (it is $\Gamma_{\sigma_{2}}$-equivariant) and the
$\omega$-Hamiltonian condition of the original vector field.

Since
$K^{k}\in{\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)\subset{\mathcal{P}}^{k}\left({\bf S}\right)$ for each
$k\geq 3$, the truncated equation

$Y(y)=Ly+X_{K^{3}}(y)+\cdots+X_{K^{r}}(y)$ (30)

is a normal form up to order $r-1$ of $X$. In fact, by (23) we have
$X_{K^{k}}\in{\overrightarrow{{\mathcal{P}}}^{k-1}}(\mathbf{S})$. Moreover,
$X_{K^{k}}\in{\overrightarrow{{\mathcal{P}}}_{\sigma_{2}}^{k-1}}(\Gamma)$,
which implies by item 4 of Proposition 2.4 that
$X_{K^{k}}\in{\overrightarrow{\mathcal{P}}_{\tilde{\sigma}_{2}}^{k-1}}(\mathbf{S}\rtimes\Gamma).$
So, Theorem 6.3 gives us a normal form that preserves the symmetries of the
original vector field. On the other hand, by Lemma 5.1 the Hamiltonian
function associated with the vector field (30) is given by
$K=H^{2}+K^{3}+\cdots+K^{r}$, which is clearly a normal form of the
Hamiltonian function $H$ associated with $X$, since
$K^{k}\in\mathcal{P}^{k}(\mathbf{S})$. Hence we have the following result:

###### Corollary 6.4

Under the conditions of the previous theorem, let $H:V\to\mathbb{R}$ be the
Hamiltonian function associated with the vector field $X:V\to V$ such that
$H(0)=0$ and $\nabla H(0)=0$. Let $H^{2}$ be the quadratic form associated
with the Hessian of $H$. Then

$K=H^{2}+K^{3}+\cdots+K^{r},$

with
$K^{k}\in{\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)$, is a normal form up to order $r$ of $H$ so that
$X_{K}$ is a normal form up to order $r-1$ of the original vector field $X$
that preserves its symmetries.

Hence, in order to obtain a normal form of an $\omega$-Hamiltonian vector
field under the action of a symmetry group $\Gamma$, it is enough to determine
a normal form of the function $H$ according to Corollary 6.4. In this case, we
obtain the generators of degree $k$ of the module
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)$ over the ring ${\cal{P}}(\mathbf{S}\rtimes\Gamma)$, if
$\sigma_{1}\sigma_{2}:\Gamma\to\mathbb{Z}_{2}$ is an epimorphism, or a Hilbert
basis of the ring
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)={\cal{P}}(\mathbf{S}\rtimes\Gamma)$, if
$\sigma_{1}\sigma_{2}:\Gamma\to\mathbb{Z}_{2}$ is the trivial homomorphism. In
practice, we exhibit a general form for the elements of
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)$, by using [2, Theorem 3.1] and [3, Theorem 3.2] if
$\mathbf{S}\rtimes\Gamma$ is compact. When the group $\mathbf{S}\rtimes\Gamma$
is not compact, we can still use tools of invariant theory if
${\cal{P}}(\mathbf{S}\rtimes\Gamma)$ is finitely generated.

In the next two subsections, we apply the algebraic method presented in
Theorem 6.3 (and therefore in Corollary 6.4) in order to obtain a normal form
of two $\omega$-Hamiltonian vector fields.

###  6.1 $\omega$-Hamiltonian $({\mathbb{D}}_{4})_{\sigma}$-equivariant vector
fields on $\mathbb{R}^{2}$

Let $\Gamma=\mathbb{D}_{4}$ be the dihedral group generated by matrices

$R_{\frac{\pi}{2}}=\left[\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right]\quad\textrm{and}\quad\kappa=\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]$

acting on an arbitrary symplectic space $({\mathbb{R}}^{2},\omega)$. Consider

$[\omega]=\left[\begin{array}[]{cc}0&a_{12}\\\
-a_{12}&0\end{array}\right]\quad\textrm{and}\quad
L=\left[\begin{array}[]{cc}0&\lambda\\\ -\lambda&0\end{array}\right],$

for some $a_{12}\neq 0$ and $\lambda\in{\mathbb{R}}^{*}$. We want to determine
a normal form of an $\omega$-Hamiltonian and
$({\mathbb{D}}_{4})_{\sigma}$-equivariant vector field
$X:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ whose linearization at the origin is
equal to $L$, where $\sigma:{\mathbb{D}}_{4}\to{\mathbb{Z}}_{2}$ is the
epimorphism such that $\ker\sigma$ is generated by $R_{\frac{\pi}{2}}$. The
vector field $X$ has Hamiltonian function $H:\mathbb{R}^{2}\to\mathbb{R}$
whose quadratic form is

$H^{2}(x_{1},x_{2})=\dfrac{\lambda
a_{12}}{2}\left(x_{1}^{2}+x_{2}^{2}\right).$

Clearly $R_{\frac{\pi}{2}}\in Sp_{\omega}(1,\mathbb{R})$ and $\kappa\in
Sp_{\omega}^{-1}(1,\mathbb{R})$, that is, $\mathbb{D}_{4}$ is a subgroup of
$\Omega_{1}$ as defined in (3). Moreover, $\ker\sigma\subset
Sp_{\omega}(1,\mathbb{R})$. By Theorem 3.2, $\mathbb{D}_{4}$ acts
$\sigma$-semisymplectically on $({\mathbb{R}}^{2},\omega)$. This is a case
where $\sigma_{1}=\sigma_{2}:=\sigma$ and $X$ presents two types of symmetry
of Table 1: the elements of $\ker\sigma$ are SE and the elements of
${\mathbb{D}}_{4}\backslash\ker\sigma$ are AR.

In order to characterize the linear group $\mathbf{S}$, we observe that
$e^{sL^{T}}$ is the rotation matrix

$R_{\lambda s}=\left[\begin{array}[]{cc}\textrm{cos}\ \left(\lambda
s\right)&-\textrm{sen}\ \left(\lambda s\right)\\\ \textrm{sen}\ \left(\lambda
s\right)&\textrm{cos}\ \left(\lambda s\right)\\\ \end{array}\right],$

whence $\left\\{R_{\lambda s}:\ s\in\mathbb{R}\right\\}$ is isomorphic to
rotation group $SO(2)$. Therefore $\mathbf{S}=\overline{\left\\{R_{\lambda
s}:\ s\in\mathbb{R}\right\\}}\simeq SO(2).$ In this way, by [6, Example 3.1]
and Theorem 3.2 the action of S on $(\mathbb{R}^{2},\omega)$ is symplectic,
even if $a_{12}\neq 1$ (non-symplectic system of coordinates).

By Theorem 6.3, it is enough to describe a generator set for the ring
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}(SO(2)\rtimes\mathbb{D}_{4})={\mathcal{P}}(SO(2)\rtimes\mathbb{D}_{4})$,
since
$\tilde{\sigma}_{j}(R_{\theta},\gamma)=\sigma_{j}(\gamma)=\sigma(\gamma)$, for
$j=1,2$. By [3, Theorem 3.2] and [15, XII Examples 4.1], a Hilbert basis of
the ring ${\mathcal{P}}(SO(2)\rtimes\mathbb{D}_{4})$ is
$\\{u(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}\\}$. Hence, $H$ admits only normal
forms of order $2r$ given by

$K(x_{1},x_{2})=H^{2}(x_{1},x_{2})+K^{4}(x_{1},x_{2})+\cdots+K^{2r}(x_{1},x_{2})=\dfrac{\lambda
a_{12}}{2}(x_{1}^{2}+x_{2}^{2})+\displaystyle\sum_{j=2}^{r}C_{j}(x_{1}^{2}+x_{2}^{2})^{j},$

where $C_{j}\in\mathbb{R}$. Therefore a normal form up to order $2r-1$ of the
$\omega$-Hamiltonian vector field $X$ is given by
$X_{K}(x_{1},x_{2})=([\omega]^{-1})^{T}\nabla K(x_{1},x_{2})$, that is,

$X_{K}(x_{1},x_{2})=\left(\lambda+\displaystyle\frac{2}{a_{12}}\sum_{j=2}^{r}jC_{j}(x_{1}^{2}+x_{2}^{2})^{j-1}\right)(x_{2},-x_{1}).$
(31)

From Theorem 6.3, the normal form (31) preserves the two types of symmetry of
the original vector field: symplectic equivariant and antisymplectic
reversible. Moreover, by Theorem 5.4 (without the action of a group $\Gamma$),
all $\omega$-Hamiltonian vector field on $({\mathbb{R}}^{2},\omega)$ with
linearization equal to $L$ has a normal form up to order $2r-1$ given by (31).
This happens because we can assume
${\mathcal{D}}^{k}={\mathcal{P}^{k}}(SO(2))$, which coincides with
${\mathcal{P}^{k}}(SO(2)\rtimes\mathbb{D}_{4})$, for all $k\geq 1$. In fact,
by Proposition 2.4,

${\mathcal{P}^{k}}(SO(2)\rtimes{\mathbb{D}}_{4})={\mathcal{P}^{k}}(SO(2))\cap{\mathcal{P}^{k}}({\mathbb{D}}_{4})={\mathcal{P}^{k}}(SO(2)),$

since all $SO(2)$-invariant function generated by $u$ is also a
${\mathbb{D}}_{4}$-invariant function. Therefore, every $\omega$-Hamiltonian
vector field on $\mathbb{R}^{2}$ with linearization $L$ is formally conjugated
to an $\omega$-Hamiltonian vector field with SE and AR symmetries.

###  6.2 $\omega$-Hamiltonian
$({\mathbb{Z}}_{2}^{\tau}\times{\mathbb{Z}}_{2}^{\psi})_{\sigma_{2}}$-equivariant
vector fields on $\mathbb{R}^{4}$

In Example 4.3, we consider
$\Gamma={\mathbb{Z}}_{2}^{\tau}\times{\mathbb{Z}}_{2}^{\psi}$ acting
$\sigma_{1}$-semisymplectically on $({\mathbb{R}}^{4},\omega)$ and an
$\omega$-Hamiltonian $\Gamma_{\sigma_{2}}$-equivariant vector field
$X(x_{1},x_{2},x_{3},x_{4})=\lambda(x_{2},-x_{1},x_{4},-x_{3})$, for
$\sigma_{1},\sigma_{2}:\Gamma\to{\mathbb{Z}}_{2}$ such that
$\ker\sigma_{1}=\\{(I_{4},I_{4}),(I_{4},\psi)\\}$ and
$\ker\sigma_{2}=\\{(I_{4},I_{4}),(\tau,\psi)\\}$.

Let $Y:{\mathbb{R}}^{4}\to{\mathbb{R}}^{4}$ be an $\omega$-Hamiltonian and
$\Gamma_{\sigma_{2}}$-equivariant vector field whose linearization at the
origin is $L=dX_{0}$. In Table 2, we list the four types of symmetries of $Y$.
By Example 4.3, the quadratic form of the Hamiltonian function $H$ associated
with $Y$ is given by
$H^{2}(x_{1},x_{2},x_{3},x_{4})=-\lambda\left(x_{1}x_{3}+x_{2}x_{4}\right)$.

In order to determine a normal form for $Y$, we must describe the module
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}({\bf S}\rtimes\Gamma)$
over the ring ${\mathcal{P}}({\bf S}\rtimes\Gamma)$. Since $\lambda i$ is the
only algebraically independent eigenvalue of $L$ and this matrix has zero
nilpotent part, by [15, XVI Proposition 5.7] the group $\mathbf{S}$ is the
circle group $S^{1}$ acting on $\mathbb{R}^{4}$ by

$\theta(x)=(x_{1}\cos\theta-
x_{2}\sin\theta,x_{1}\sin\theta+x_{2}\cos\theta,x_{3}\cos\theta-
x_{4}\sin\theta,x_{3}\sin\theta+x_{4}\cos\theta),$

for all $\theta\in S^{1}$ and $x=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{4}$.
A Hilbert basis for ${\mathcal{P}}(S^{1}\rtimes\Gamma)$ is given by
$\\{u_{1},u_{2},u_{3}^{2},u_{4}^{2}\\}$, where $u_{1}(x)=x_{1}^{2}+x_{2}^{2}$,
$u_{2}(x)=x_{3}^{2}+x_{4}^{2}$, $u_{3}(x)=x_{1}x_{3}+x_{2}x_{4}$ and
$u_{4}(x)=x_{2}x_{3}-x_{1}x_{4}$, and $u_{3}$ is a generator of the module
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}(S^{1}\rtimes\Gamma)$
over ${\mathcal{P}}(S^{1}\rtimes\Gamma)$. For these computations, we use the
tools presented in [3, Theorem 3.2] and [2, Theorem 3.1], respectively. By
Corollary 6.4, a normal form up to order $2r$ of $H$ is given by

$K(x)=H^{2}(x)+\displaystyle\sum_{k=2}^{r}K^{2k}(x),$

with
$K^{2k}\in{\mathcal{{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}}(S^{1}\rtimes\Gamma)$
written as

$K^{2k}(x)=\displaystyle(x_{1}x_{3}+x_{2}x_{4})\sum_{|\alpha|=k-1}C_{\alpha}(x_{1}^{2}+x_{2}^{2})^{j_{1}}(x_{3}^{2}+x_{4}^{2})^{j_{2}}(x_{1}x_{3}+x_{2}x_{4})^{2j_{3}}(x_{2}x_{3}-x_{1}x_{4})^{2j_{4}}$,

for $\alpha=(j_{1},j_{2},2j_{3},2j_{4})$,
$|\alpha|=j_{1}+j_{2}+2j_{3}+2j_{4}$, $C_{\alpha}\in{\mathbb{R}}$ and
$k=2,\ldots,r$. Write

$\displaystyle F_{1}^{k}(x)$
$\displaystyle=\sum_{|\alpha|=k-1}2j_{1}C_{\alpha}u_{1}^{j_{1}-1}(x)u_{2}^{j_{2}}(x)u_{3}^{2j_{3}}(x)u_{4}^{2j_{4}}(x),$
$\displaystyle F_{2}^{k}(x)$
$\displaystyle=\sum_{|\alpha|=k-1}2j_{2}C_{\alpha}u_{1}^{j_{1}}(x)u_{2}^{j_{2}-1}(x)u_{3}^{2j_{3}}(x)u_{4}^{2j_{4}}(x),$
$\displaystyle F_{3}^{k}(x)$
$\displaystyle=\sum_{|\alpha|=k-1}(2j_{3}+1)C_{\alpha}u_{1}^{j_{1}}(x)u_{2}^{j_{2}}(x)u_{3}^{2(j_{3}-1)}(x)u_{4}^{2j_{4}}(x),$
$\displaystyle F_{4}^{k}(x)$
$\displaystyle=\sum_{|\alpha|=k-1}2j_{4}C_{\alpha}u_{1}^{j_{1}}(x)u_{2}^{j_{2}}(x)u_{3}^{2j_{3}}(x)u_{4}^{2(j_{4}-1)}(x)$

for each $k=2,\ldots,r$. Thus

$\displaystyle\dfrac{\partial K^{2k}}{\partial x_{1}}(x)$
$\displaystyle=x_{1}u_{3}(x)F_{1}^{k}(x)+x_{3}u_{3}^{2}(x)F_{3}^{k}(x)-x_{4}u_{3}(x)u_{4}(x)F_{4}^{k}(x),$
$\displaystyle\dfrac{\partial K^{2k}}{\partial x_{2}}(x)$
$\displaystyle=x_{2}u_{3}(x)F_{1}^{k}(x)+x_{4}u_{3}^{2}(x)F_{3}^{k}(x)+x_{3}u_{3}(x)u_{4}(x)F_{4}^{k}(x),$
$\displaystyle\dfrac{\partial K^{2k}}{\partial x_{3}}(x)$
$\displaystyle=x_{3}u_{3}(x)F_{2}^{k}(x)+x_{1}u_{3}^{2}(x)F_{3}^{k}(x)+x_{2}u_{3}(x)u_{4}(x)F_{4}^{k}(x),$
$\displaystyle\dfrac{\partial K^{2k}}{\partial x_{4}}(x)$
$\displaystyle=x_{4}u_{3}(x)F_{2}^{k}(x)+x_{2}u_{3}^{2}(x)F_{3}^{k}(x)-x_{1}u_{3}(x)u_{4}(x)F_{4}^{k}(x).$

Therefore, a normal form up to order $2r-1$ of the $\omega$-Hamiltonian vector
field $Y$ is given by

$X_{K}(x)=Lx+[\omega]\nabla\left(\displaystyle\sum_{k=2}^{r}K^{2k}(x)\right)=(X_{1}(x),X_{2}(x),X_{3}(x),X_{4}(x)),$

where

$\begin{cases}\displaystyle X_{1}(x)=\lambda
x_{2}-x_{4}u_{3}(x)\sum_{k=2}^{r}F_{2}^{k}(x)-x_{2}u_{3}^{2}(x)\sum_{k=2}^{r}F_{3}^{k}(x)+x_{1}u_{3}(x)u_{4}(x)\sum_{k=2}^{r}F_{4}^{k}(x)\\\
\displaystyle X_{2}(x)=-\lambda
x_{1}+x_{3}u_{3}(x)\sum_{k=2}^{r}F_{2}^{k}(x)+x_{1}u_{3}^{2}(x)\sum_{k=2}^{r}F_{3}^{k}(x)+x_{2}u_{3}(x)u_{4}(x)\sum_{k=2}^{r}F_{4}^{k}(x)\\\
\displaystyle X_{3}(x)=\lambda
x_{4}-x_{2}u_{3}(x)\sum_{k=2}^{r}F_{1}^{k}(x)-x_{4}u_{3}^{2}(x)\sum_{k=2}^{r}F_{3}^{k}(x)-x_{3}u_{3}(x)u_{4}(x)\sum_{k=2}^{r}F_{4}^{k}(x)\\\
\displaystyle X_{4}(x)=-\lambda
x_{3}+x_{1}u_{3}(x)\sum_{k=2}^{r}F_{1}^{k}(x)+x_{3}u_{3}^{2}(x)\sum_{k=2}^{r}F_{3}^{k}(x)-x_{4}u_{3}(x)u_{4}(x)\sum_{k=2}^{r}F_{4}^{k}(x)\end{cases}.$

##  7 Characterization of the complementary subspace of
${ad_{H^{2}}^{k}}(\mathcal{P}_{\sigma_{1}}^{k}(\Gamma))$ in
$\mathcal{P}_{\sigma_{1}\sigma_{2}}^{k}(\Gamma)$

In this section, our goal is proving Theorem 6.1, which characterizes a
complementary subspace of
${ad_{H^{2}}^{k}}(\mathcal{P}_{\sigma_{1}}^{k}(\Gamma))$ in
$\mathcal{P}_{\sigma_{1}\sigma_{2}}^{k}(\Gamma)$. We start by considering the
action of $\Gamma$ on $(V,\omega)$, that induces for each $\gamma\in\Gamma$
the linear map $\rho_{\gamma}:V\to V$, $\rho_{\gamma}(x)=\gamma x$. Define the
action ${}^{*}:\Gamma\times\mathcal{P}^{k}\to\mathcal{P}^{k}$ of $\Gamma$ on
$\mathcal{P}^{k}$ as

$\gamma^{*}F=(\rho_{\gamma})^{*}F,$ (32)

where $(\rho_{\gamma})^{*}F$ is the pullback of $F\in\mathcal{P}^{k}$ by
$\rho_{\gamma}$, that is, $\gamma^{*}F(x)=F(\rho_{\gamma}(x))=F(\gamma x),$
for all $\gamma\in\Gamma$ and $x\in V$. It follows from the linearity of the
pullback of differential forms that the action (32) is linear. Besides, if
$L=dX_{0}$, then $L\gamma=\sigma_{2}(\gamma)\gamma L$ for all
$\gamma\in\Gamma$, since
$X\in{\overrightarrow{\mathcal{P}}_{\sigma_{2}}}(\Gamma)$.

###### Lemma 7.1

For each $k\geq 3$, the operator $ad_{H^{2}}^{k}$ is
$\Gamma_{\sigma_{2}}$-equivariant under the action (32), that is,
$ad_{H^{2}}^{k}(\gamma^{*}F)=\sigma_{2}(\gamma)\gamma^{*}ad_{H^{2}}^{k}(F)$,
for all $\gamma\in\Gamma$ and $F\in\mathcal{P}^{k}$.

Proof. For each $x\in V$ and $F\in\mathcal{P}^{k}$, we have
$ad_{H^{2}}^{k}(F)(x)=\\{F,H^{2}\\}(x)=dF_{x}(X_{H^{2}}(x))=dF_{x}Lx,$ where
the last equality follows from Lemma 5.1. Thus, for each $\gamma\in\Gamma$ and
$x\in V$, we have

$\displaystyle ad_{H^{2}}^{k}(\gamma^{*}F)(x)$ $\displaystyle=$ $\displaystyle
d(\gamma^{*}F)_{x}Lx=d(F(\rho_{\gamma}(x)))_{x}Lx=dF_{\rho_{\gamma}(x)}d(\rho_{\gamma})_{x}Lx$
$\displaystyle=$ $\displaystyle dF_{\gamma x}\gamma
Lx=\sigma_{2}(\gamma)dF_{\gamma x}L\gamma
x=\sigma_{2}(\gamma)ad_{H^{2}}^{k}(F)(\gamma x)$ $\displaystyle=$
$\displaystyle\sigma_{2}(\gamma)\gamma^{*}ad_{H^{2}}^{k}(F)(x).$

Next, we introduce some subgroups of $\Gamma$ that will be important in the
proof of Lemmas 7.4-7.7. For group homomorphisms
$\sigma_{1},\sigma_{2}:\Gamma\to\mathbb{Z}_{2}$, fix
$\delta_{1},\delta_{2},\delta_{3}\in\Gamma$ (if they there exist) such that

$\sigma_{j}(\delta_{i})=\begin{cases}\ \ 1,&\textrm{if}\ i=j\\\
-1,&\textrm{if}\ i\neq j\end{cases},$ (33)

for $i=1,2,3$ and $j=1,2$. Denote

$\Gamma_{++}=\ker\sigma_{1}\cap\ker\sigma_{2},\quad\Gamma_{--}=(\Gamma\backslash\ker\sigma_{1})\cap(\Gamma\backslash\ker\sigma_{2}),$
$\Gamma_{-+}=(\Gamma\backslash\ker\sigma_{1})\cap\ker\sigma_{2},\quad\Gamma_{+-}=\ker\sigma_{1}\cap(\Gamma\backslash\ker\sigma_{2}).$

The signs “$+$” and “$-$” in the notations refer to the signs of the range of
their elements by $\sigma_{1}$ and by $\sigma_{2}$, respectively. Then we have
the following result:

###### Proposition 7.2

Let $\sigma_{1},\sigma_{2}:\Gamma\to\mathbb{Z}_{2}$ be group homomorphisms and
$\delta_{1},\delta_{2},\delta_{3}\in\Gamma$ (if they there exist) satisfying
(33).

  1. 1.

If $\sigma_{1}$ and $\sigma_{2}$ are distinct epimorphisms, then $\Gamma_{++}$
is a subgroup of $\Gamma$ of index $4$. Moreover, $\Gamma=\Gamma_{++}\
\dot{\cup}\ \Gamma_{+-}\ \dot{\cup}\ \Gamma_{-+}\ \dot{\cup}\
\Gamma_{--}=\Gamma_{++}\ \dot{\cup}\ \delta_{1}\Gamma_{++}\ \dot{\cup}\
\delta_{2}\Gamma_{++}\ \dot{\cup}\ \delta_{3}\Gamma_{++}.$

  2. 2.

If $\sigma_{1}$ is trivial and $\sigma_{2}$ is an epimorphism, then
$\Gamma_{++}=\ker\sigma_{2}$ is a subgroup of $\Gamma$ of index $2$ and
$\Gamma=\Gamma_{++}\ \dot{\cup}\ \Gamma_{+-}=\Gamma_{++}\ \dot{\cup}\
\delta_{1}\Gamma_{++}.$

  3. 3.

If $\sigma_{1}$ is an epimorphism and $\sigma_{2}$ is trivial, then
$\Gamma_{++}=\ker\sigma_{1}$ is a subgroup of $\Gamma$ of index $2$ and
$\Gamma=\Gamma_{++}\ \dot{\cup}\ \Gamma_{-+}=\Gamma_{++}\ \dot{\cup}\
\delta_{2}\Gamma_{++}.$

  4. 4.

If $\sigma:=\sigma_{1}=\sigma_{2}$ is an epimorphism, then
$\Gamma_{++}=\ker\sigma$ is a subgroup of $\Gamma$ of index $2$ and we can
write $\Gamma=\Gamma_{++}\ \dot{\cup}\ \Gamma_{--}=\Gamma_{++}\ \dot{\cup}\
\delta_{3}\Gamma_{++}.$

  5. 5.

If $\sigma_{1}=\sigma_{2}$ is the trivial homomorphism, then
$\Gamma=\Gamma_{++}$.

Proof. We only prove the first item, since the other ones follow similarly. If
$\sigma_{1}$ and $\sigma_{2}$ are distinct group epimorphisms, then by the
proof of Proposition 4.1 there exist $\delta_{1}$, $\delta_{2}$ and
$\delta_{3}=\delta_{1}\delta_{2}$ in $\Gamma$ satisfying (33), that is,
$\delta_{1}\in\Gamma_{+-}$, $\delta_{2}\in\Gamma_{-+}$ and
$\delta_{3}\in\Gamma_{--}$. We now prove that $\Gamma_{++}$ is a subgroup of
$\Gamma$ of index $[\Gamma:\Gamma_{++}]=4$. In fact, consider the restricted
map $\sigma_{1}\mid_{\ker\sigma_{2}}:\ker\sigma_{2}\to\mathbb{Z}_{2}$. Clearly
$\ker(\sigma_{1}\mid_{\ker\sigma_{2}})\subset\ker\sigma_{1}\cap\ker\sigma_{2}=\Gamma_{++}$.
On the other hand, if $\gamma\in\Gamma_{++}$ then
$\gamma\in\ker\sigma_{1}\cap\ker\sigma_{2}$. Thus
$\sigma_{1}\mid_{\ker\sigma_{2}}(\gamma)$ is well defined and

$\sigma_{1}\mid_{\ker\sigma_{2}}(\gamma)=\sigma_{1}(\gamma)=1,$

which implies that $\Gamma_{++}\subset\ker(\sigma_{1}\mid_{\ker\sigma_{2}})$.
Therefore, $\Gamma_{++}=\ker\left(\sigma_{1}\mid_{\ker\sigma_{2}}\right)$ is a
subgroup of $\ker\sigma_{2}$ of index $2$. Hence

$[\Gamma:\Gamma_{++}]=[\Gamma:\ker\sigma_{2}]\cdot[\ker\sigma_{2}:\Gamma_{++}]=4.$

In this case, the left-cosets of $\Gamma_{++}$ in $\Gamma$ are $\Gamma_{++}$,
$\Gamma_{+-}=\delta_{1}\Gamma_{++}$, $\Gamma_{-+}=\delta_{2}\Gamma_{++}$ and
$\Gamma_{--}=\delta_{3}\Gamma_{++}$, which completes the proof.

An important tool in representation theory of groups is the Haar integral, an
invariant form of integration by translation of elements of a Lie group. Based
on the approach presented in [15, XVI Theorem 5.8; Theorem 5.9] and [4,
Theorem 4.7], we define a projection of the ring $\mathcal{P}^{k}(\mathbf{S})$
of all S-invariant polynomial functions onto the module
${\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}^{k}(\mathbf{S}\rtimes\Gamma)$
(Lemma 7.7), where the action of $\mathbf{S}$ on $(V,\omega)$ is symplectic
and $\tilde{\sigma}_{j}:\mathbf{S}\rtimes\Gamma\to\mathbb{Z}_{2}$ is defined
as in (27).

Suppose $\Gamma_{++}=\ker\sigma_{1}\cap\ker\sigma_{2}$ is a linear Lie group.
In this case, $\Gamma_{++}$ is closed in $\Gamma$ and, therefore, compact.
This hypothesis is important in the computation of the normalized Haar
integral over $\Gamma_{++}$.

###### Definition 7.3

Consider the action ${}^{*}:\Gamma\times\mathcal{P}^{k}\to\mathcal{P}^{k}$
defined in (32). For each $k\geq 3$, define the maps
$\overline{\pi},\pi:{\mathcal{P}^{k}}\to{\mathcal{P}^{k}}$ as

$\overline{\pi}(F)=\displaystyle\frac{1}{[\Gamma:\Gamma_{++}]}\left(\int_{\tau\in\Gamma_{++}}\tau^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau)^{*}F\right)$

and

$\pi(F)=\displaystyle\frac{1}{[\Gamma:\Gamma_{++}]}\left(\int_{\tau\in\Gamma_{++}}\tau^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau)^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau)^{*}F\right),$

where $\displaystyle\int_{\tau\in\Gamma_{++}}$ is the normalized Haar integral
over $\Gamma_{++}$. We assume
$\displaystyle\int_{\tau\in\Gamma_{++}}(\delta_{i}\tau)^{*}F=0$ if
$\delta_{i}$ does not exist, $i=1,2,3$.

###### Lemma 7.4

For each $k\geq 3$, the map
$\overline{\pi}:{\mathcal{P}^{k}}\to{\mathcal{P}_{\sigma_{1}}^{k}}(\Gamma)$ is
an idempotent linear projection, that is, $\overline{\pi}^{2}=\overline{\pi}$.

Proof. Since the pullback and the Haar integral are linear, we have that
$\overline{\pi}$ is linear. For the same reason, if the degree of $F$ is $k$,
then the degree of $\overline{\pi}(F)$ is also $k$. We want to show that
$\overline{\pi}(F)\in{\mathcal{P}_{\sigma_{1}}^{k}}(\Gamma)$, that is,
$\gamma^{*}\overline{\pi}(F)=\sigma_{1}(\gamma)\overline{\pi}(F)$, for any
$\gamma\in\Gamma$ and $F\in\mathcal{P}^{k}$. Based on Proposition 7.2, we
prove the case where $\sigma_{1}$ and $\sigma_{2}$ are distinct epimorphisms,
that is, $[\Gamma:\Gamma_{++}]=4$. The other cases follow similarly.

Given $\gamma\in\Gamma$,

$\begin{array}[]{rcl}\gamma^{*}\overline{\pi}(F)&=&\gamma^{*}\left(\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}\tau^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau)^{*}F\right)\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}\gamma^{*}\tau^{*}F+\int_{\tau\in\Gamma_{++}}\gamma^{*}(\delta_{1}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}\gamma^{*}(\delta_{2}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}\gamma^{*}(\delta_{3}\tau)^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}(\tau\gamma)^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau\gamma)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau\gamma)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau\gamma)^{*}F\right),\end{array}$

where in the second equality we use the continuity of the pullback and in the
third equality we use that $(\tau\gamma)^{*}F=\gamma^{*}\tau^{*}F$ for all
$\tau,\gamma\in\Gamma$ and $F\in\mathcal{P}^{k}$.

If $\gamma\in\Gamma_{++}$, then $\tau\gamma\in\Gamma_{++}$. Since the Haar
integral is invariant by elements of $\Gamma_{++}$ we have that
$\gamma^{*}\overline{\pi}(F)=\overline{\pi}(F)=\sigma_{1}(\gamma)\overline{\pi}(F)$.

If $\gamma\in\Gamma_{+-}=\delta_{1}\Gamma_{++}$, then there exists
$\bar{\gamma}\in\Gamma_{++}$ such that $\gamma=\delta_{1}\bar{\gamma}$. Note
that $\tau\delta_{1}\in\Gamma_{+-}=\delta_{1}\Gamma_{++}$,
$\delta_{3}\delta_{1}\in\Gamma_{-+}=\delta_{2}\Gamma_{++}$ and
$\delta_{2}\delta_{1}\in\Gamma_{--}=\delta_{3}\Gamma_{++}$, because
$\sigma_{j}(\tau\delta_{1})=\sigma_{j}(\delta_{1})$,
$\sigma_{j}(\delta_{3}\delta_{1})=-\sigma_{j}(\delta_{1})=\sigma_{j}(\delta_{2})$
and $\sigma_{j}(\delta_{2}\delta_{1})=-1=\sigma_{j}(\delta_{3})$, for all
$j=1,2$. Hence we conclude that there exist
$\bar{\gamma},\bar{\tau},\bar{\delta}_{2},\bar{\delta}_{3}\in\Gamma_{++}$ such
that $\gamma=\delta_{1}\bar{\gamma},$ $\tau\delta_{1}=\delta_{1}\bar{\tau},$
$\delta_{3}\delta_{1}=\delta_{2}\bar{\delta}_{2}$ and
$\delta_{2}\delta_{1}=\delta_{3}\bar{\delta}_{3}.$ In this case, by the
invariance of the Haar integral by $\Gamma_{++}$ we have

$\begin{array}[]{rcl}\gamma^{*}\overline{\pi}(F)&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}(\tau\delta_{1}\bar{\gamma})^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau\delta_{1}\bar{\gamma})^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau\delta_{1}\bar{\gamma})^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau\delta_{1}\bar{\gamma})^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{1}\bar{\tau}\bar{\gamma})^{*}F+\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{1}^{2}\bar{\tau}\bar{\gamma})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{2}\delta_{1}\bar{\tau}\bar{\gamma})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{3}\delta_{1}\bar{\tau}\bar{\gamma})^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{1}\bar{\tau})^{*}F+\int_{\bar{\tau}\in\Gamma_{++}}\bar{\tau}^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{3}\bar{\delta}_{3}\bar{\tau})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{2}\bar{\delta}_{2}\bar{\tau})^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{1}\bar{\tau})^{*}F+\int_{\bar{\tau}\in\Gamma_{++}}\bar{\tau}^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{3}\bar{\tau})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{2}\bar{\tau})^{*}F\right)\vspace{0.15cm}\\\
&=&\overline{\pi}(F)=\sigma_{1}(\gamma)\overline{\pi}(F).\end{array}$

If $\gamma\in\Gamma_{-+}=\delta_{2}\Gamma_{++}$, since
$\tau\delta_{2}\in\Gamma_{-+}=\delta_{2}\Gamma_{++}$,
$\delta_{3}\delta_{2}\in\Gamma_{+-}=\delta_{1}\Gamma_{++}$ and
$\delta_{1}\delta_{2}\in\Gamma_{--}=\delta_{3}\Gamma_{++}$, then there exist
$\bar{\gamma},\bar{\tau},\bar{\delta}_{1},\bar{\delta}_{3}\in\Gamma_{++}$ such
that $\gamma=\delta_{2}\bar{\gamma},$ $\tau\delta_{2}=\delta_{2}\bar{\tau},$
$\delta_{3}\delta_{2}=\delta_{1}\bar{\delta}_{1}$ and
$\delta_{1}\delta_{2}=\delta_{3}\bar{\delta}_{3}.$ Again, by the invariance of
the Haar integral by $\Gamma_{++}$, we have similarly to the previous case
that

$\begin{array}[]{rcl}\gamma^{*}\overline{\pi}(F)&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}(\tau\delta_{2}\bar{\gamma})^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau\delta_{2}\bar{\gamma})^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau\delta_{2}\bar{\gamma})^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau\delta_{2}\bar{\gamma})^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{2}\bar{\tau})^{*}F+\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{3}\bar{\tau})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}\bar{\tau}^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{1}\bar{\tau})^{*}F\right)\vspace{0.15cm}\\\
&=&-\overline{\pi}(F)=\sigma_{1}(\gamma)\overline{\pi}(F).\end{array}$

If $\gamma\in\Gamma_{--}=\delta_{3}\Gamma_{++}$, since
$\tau\delta_{3}\in\Gamma_{--}=\delta_{3}\Gamma_{++}$,
$\delta_{2}\delta_{3}\in\Gamma_{+-}=\delta_{1}\Gamma_{++}$ and
$\delta_{1}\delta_{3}\in\Gamma_{-+}=\delta_{2}\Gamma_{++}$, then there exist
$\bar{\gamma},\bar{\tau},\bar{\delta}_{1},\bar{\delta}_{2}\in\Gamma_{++}$ such
that $\gamma=\delta_{3}\bar{\gamma},$ $\tau\delta_{3}=\delta_{3}\bar{\tau},$
$\delta_{2}\delta_{3}=\delta_{1}\bar{\delta}_{1}$ and
$\delta_{1}\delta_{3}=\delta_{2}\bar{\delta}.$ Hence we obtain

$\begin{array}[]{rcl}\gamma^{*}\overline{\pi}(F)&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}(\tau\delta_{3}\bar{\gamma})^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau\delta_{3}\bar{\gamma})^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau\delta_{3}\bar{\gamma})^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau\delta_{3}\bar{\gamma})^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{3}\bar{\tau})^{*}F+\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{2}\bar{\tau})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}(\delta_{1}\bar{\tau})^{*}F-\int_{\bar{\tau}\in\Gamma_{++}}\bar{\tau}^{*}F\right)\vspace{0.15cm}\\\
&=&-\overline{\pi}(F)=\sigma_{1}(\gamma)\overline{\pi}(F).\end{array}$

Therefore, in each case
$\overline{\pi}(F)\in\mathcal{P}_{\sigma_{1}}^{k}(\Gamma)$. Since
$F\in\mathcal{P}^{k}$ is arbitrary, we conclude that
$\overline{\pi}(\mathcal{P}^{k})\subset\mathcal{P}_{\sigma_{1}}^{k}(\Gamma)$.
Reciprocally, given a function $F\in\mathcal{P}_{\sigma_{1}}^{k}(\Gamma)$, we
have by (32) that $\gamma^{*}F(x)=F(\gamma x)=\sigma_{1}(\gamma)F(x)$ for all
$\gamma\in\Gamma$ and $x\in V$. Then

$\begin{array}[]{rcl}\overline{\pi}(F)&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}\tau^{*}F+\int_{\tau\in\Gamma_{++}}(\delta_{1}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{2}\tau)^{*}F-\int_{\tau\in\Gamma_{++}}(\delta_{3}\tau)^{*}F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}\sigma_{1}(\tau)F+\int_{\tau\in\Gamma_{++}}\sigma_{1}(\delta_{1}\tau)F-\int_{\tau\in\Gamma_{++}}\sigma_{1}(\delta_{2}\tau)F-\int_{\tau\in\Gamma_{++}}\sigma_{1}(\delta_{3}\tau)F\right)\vspace{0.15cm}\\\
&=&\displaystyle\frac{1}{4}\left(\int_{\tau\in\Gamma_{++}}F+\int_{\tau\in\Gamma_{++}}F-\int_{\tau\in\Gamma_{++}}(-F)-\int_{\tau\in\Gamma_{++}}(-F)\right)\vspace{0.15cm}\\\
&=&F\displaystyle\int_{\tau\in\Gamma_{++}}1=F,\end{array}$

where in the third equality we have the Haar integral of the constant function
$f(\gamma)=F$. Thus $F\in\overline{\pi}(\mathcal{P}^{k})$ and we conclude that
$\overline{\pi}(\mathcal{P}^{k})=\mathcal{P}_{\sigma_{1}}^{k}(\Gamma)$.
Moreover $\overline{\pi}\mid_{\mathcal{P}_{\sigma_{1}}^{k}(\Gamma)}={\rm Id}$.
Therefore, $\overline{\pi}$ is an idempotent projection.

The proof of the next lemma is similar to the previous one.

###### Lemma 7.5

For each $k\geq 3$, the map
$\pi:{\mathcal{P}^{k}}\to{\mathcal{P}_{\sigma_{1}\sigma_{2}}^{k}}(\Gamma)$ is
an idempotent linear projection.

###### Lemma 7.6

For each $k\geq 3$, we have
$\pi(ad_{H^{2}}^{k}(\mathcal{P}^{k}))=ad_{H^{2}}^{k}(\overline{\pi}(\mathcal{P}^{k}))=ad_{H^{2}}^{k}(\mathcal{P}_{\sigma_{1}}^{k}(\Gamma)).$

Proof. As before, we prove the case where $\sigma_{1}$ and $\sigma_{2}$ are
distinct epimorphisms. The other cases are similar. Moreover, to simplify the
notation, we will omit $\tau\in\Gamma_{++}$ in the Haar integral over
$\Gamma_{++}$.

Given $F\in\mathcal{P}^{k}$, the linearity of the operator $ad_{H^{2}}^{k}$
ensures that

$\int ad_{H^{2}}^{k}(\tau^{*}F)=ad_{H^{2}}^{k}\left(\int\tau^{*}F\right).$

Furthermore, by Lemma 7.1, we have
$\gamma^{*}ad_{H^{2}}^{k}(F)=\sigma_{2}(\gamma)ad_{H^{2}}^{k}(\gamma^{*}F)$
for all $\gamma\in\Gamma$. Since $\sigma_{1}$ and $\sigma_{2}$ are distinct,
there exist $\delta_{1},\delta_{2},\delta_{3}\in\Gamma$ satisfying (33). For
all $\tau\in\Gamma_{++}$, we have
$\sigma_{2}(\delta_{1}\tau)=\sigma_{2}(\delta_{3}\tau)=-\sigma_{2}(\delta_{2}\tau)=-\sigma_{2}(\tau)=-1$.
Moreover, $[\Gamma:\Gamma_{++}]=4$. So, for all $F\in\mathcal{P}^{k}$,

$\displaystyle\pi(ad_{H^{2}}^{k}(F))$
$\displaystyle=\frac{1}{4}\left(\int\tau^{*}ad_{H^{2}}^{k}(F)-\int(\delta_{1}\tau)^{*}ad_{H^{2}}^{k}(F)-\displaystyle\int(\delta_{2}\tau)^{*}ad_{H^{2}}^{k}(F)+\int(\delta_{3}\tau)^{*}ad_{H^{2}}^{k}(F)\right)$
$\displaystyle=\displaystyle\frac{1}{4}\left(\int\sigma_{2}(\tau)ad_{H^{2}}^{k}(\tau^{*}F)-\int\sigma_{2}(\delta_{1}\tau)ad_{H^{2}}^{k}((\delta_{1}\tau)^{*}F)\right.$
$\displaystyle\left.\displaystyle\quad-\int\sigma_{2}(\delta_{2}\tau)ad_{H^{2}}^{k}((\delta_{2}\tau)^{*}F)+\int\sigma_{2}(\delta_{3}\tau)ad_{H^{2}}^{k}((\delta_{3}\tau)^{*}F)\right)$
$\displaystyle=\displaystyle\frac{1}{4}\left(\int
ad_{H^{2}}^{k}(\tau^{*}F)+\int ad_{H^{2}}^{k}((\delta_{1}\tau)^{*}F)-\int
ad_{H^{2}}^{k}((\delta_{2}\tau)^{*}F)-\int
ad_{H^{2}}^{k}((\delta_{3}\tau)^{*}F)\right)$
$\displaystyle=ad_{H^{2}}^{k}(\overline{\pi}(F)).$

Thus we obtain
$\pi(ad_{H^{2}}^{k}(\mathcal{P}^{k}))=ad_{H^{2}}^{k}(\overline{\pi}(\mathcal{P}^{k}))$.
By Lemma 7.4, we have
$\pi(ad_{H^{2}}^{k}(\mathcal{P}^{k}))=ad_{H^{2}}^{k}(\mathcal{P}_{\sigma_{1}}^{k}(\Gamma))$,
since $\overline{\pi}$ is surjective.

For the next result, we consider again
$\tilde{\sigma}_{j}:\mathbf{S}\rtimes\Gamma\to\mathbb{Z}_{2}$ as in (27), with
$j=1,2$. We define in (6) a group homomorphism on semidirect product
$\Gamma_{1}\rtimes\Gamma_{2}$. In our case,
$\tilde{\sigma}_{j}(e^{sL^{T}},\gamma)=\beta_{j1}(e^{sL^{T}})\beta_{j2}(\gamma)$,
where $\beta_{j1}:\mathbf{S}\to\mathbb{Z}_{2}$ is the trivial homomorphism and
$\beta_{j2}:\Gamma\to\mathbb{Z}_{2}$ coincides with $\sigma_{j}$. Furthermore,

$\displaystyle(\tilde{\sigma}_{1}\tilde{\sigma}_{2})(e^{sL^{T}},\gamma)=\tilde{\sigma}_{1}(e^{sL^{T}},\gamma)\tilde{\sigma}_{2}(e^{sL^{T}},\gamma)=(\sigma_{1}\sigma_{2})(\gamma),$

for all $s\in\mathbb{R}$ and $\gamma\in\Gamma$.

###### Lemma 7.7

For each $k\geq 3$, we have
$\pi(\mathcal{P}^{k}(\mathbf{S}))=\mathcal{P}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}^{k}(\mathbf{S}\rtimes\Gamma)$.

Proof. By item 3 of Proposition 2.4,

${{\mathcal{P}}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}}(\mathbf{S}\rtimes{\Gamma})={{\mathcal{P}}}(\mathbf{S})\cap{{\mathcal{P}}_{\sigma_{1}\sigma_{2}}}({\Gamma}).$
(34)

Given
$F\in\mathcal{P}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}^{k}(\mathbf{S}\rtimes\Gamma)$,
it follows that $F\in\mathcal{P}^{k}(\mathbf{S})$ and
$F\in\mathcal{P}_{{\sigma}_{1}{\sigma}_{2}}^{k}(\Gamma)$. From proof of Lemma
7.5 we have $F=\pi(F)\in\pi\left(\mathcal{P}^{k}(\mathbf{S})\right)$. Thus
$\mathcal{P}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}^{k}(\mathbf{S}\rtimes\Gamma)\subset\pi(\mathcal{P}^{k}(\mathbf{S}))$.

On the other hand, given $F\in\mathcal{P}^{k}(\mathbf{S})$, we want to show
that
$\pi(F)\in\mathcal{P}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}^{k}(\mathbf{S}\rtimes\Gamma)$,
that is,

$\pi(F)((e^{sL^{T}},\gamma)x)=(\tilde{\sigma}_{1}\tilde{\sigma}_{2})(e^{sL^{T}},\gamma)\pi(F)(x)=(\sigma_{1}\sigma_{2})(\gamma)\pi(F)(x),$

for all $\gamma\in\Gamma,\ x\in V$. By (25), it is enough to show that

$\left(\gamma
e^{\sigma_{2}(\gamma)sL^{T}}\right)^{*}\pi(F)=(\sigma_{1}\sigma_{2})(\gamma)\pi(F),$
(35)

for all $\gamma\in\Gamma$ and $s\in\mathbb{R}$, where ∗ is the action of
$\mathbf{S}\rtimes\Gamma$ on $\mathcal{P}^{k}$ defined as in (32). By (25),
$e^{\sigma_{2}(\gamma)sL^{T}}=\gamma e^{sL^{T}}\gamma^{-1}$ and, in
particular, for all $\gamma\in\ker\sigma_{2}$ we have $\gamma
e^{sL^{T}}=e^{sL^{T}}\gamma$. Thus if $\gamma\in\Gamma$, $s\in\mathbb{R}$ and
$x\in V$, then

$\displaystyle(\gamma e^{\sigma_{2}(\gamma)sL^{T}})^{*}F(x)$
$\displaystyle=F(\gamma
e^{\sigma_{2}(\gamma)sL^{T}}x)=F(\gamma^{2}e^{sL^{T}}\gamma^{-1}x)$
$\displaystyle=F(e^{sL^{T}}\gamma^{2}\gamma^{-1}x)=F(\gamma x)$
$\displaystyle=\gamma^{*}F(x),$ (36)

where the third and fourth equalities follow since
$\gamma^{2}\in\ker\sigma_{2}$ and $F\in\mathcal{P}^{k}(\mathbf{S})$,
respectively. Thus, for all $\bar{\gamma}\in\Gamma$ and $x\in V$, we have

$\displaystyle(\bar{\gamma}\gamma e^{\sigma_{2}(\gamma)sL^{T}})^{*}F(x)$
$\displaystyle=(\gamma
e^{\sigma_{2}(\gamma)sL^{T}})^{*}((\bar{\gamma})^{*}F)(x)=(\gamma
e^{\sigma_{2}(\gamma)sL^{T}})^{*}F(\bar{\gamma}x)$
$\displaystyle=\gamma^{*}F(\bar{\gamma}x)=\gamma^{*}(\bar{\gamma})^{*}F(x)=(\bar{\gamma}\gamma)^{*}F(x),$

where the third equality follows from (7).

As before, we present a proof only when $\sigma_{1}$ and $\sigma_{2}$ are
distinct epimorphisms and omit $\tau\in\Gamma_{++}$ in the Haar integral. In
this case, for all $\gamma\in\Gamma$ and $s\in\mathbb{R}$, we have

$\displaystyle\left(\gamma e^{\sigma_{2}(\gamma)sL^{T}}\right)^{*}\pi(F)$
$\displaystyle=\left(\gamma
e^{\sigma_{2}(\gamma)sL^{T}}\right)^{*}\left(\frac{1}{4}\left(\int\tau^{*}F-\int(\delta_{1}\tau)^{*}F-\int(\delta_{2}\tau)^{*}F+\int(\delta_{3}\tau)^{*}F\right)\right)$
$\displaystyle=\frac{1}{4}\left(\int(\tau\gamma
e^{\sigma_{2}(\gamma)sL^{T}})^{*}F-\int(\delta_{1}\tau\gamma
e^{\sigma_{2}(\gamma)sL^{T}})^{*}F\right.$
$\displaystyle\left.\quad-\int(\delta_{2}\tau\gamma
e^{\sigma_{2}(\gamma)sL^{T}})^{*}F+\int(\delta_{3}\tau\gamma
e^{\sigma_{2}(\gamma)sL^{T}})^{*}F\right)$
$\displaystyle=\frac{1}{4}\left(\int(\tau\gamma)^{*}F-\int(\delta_{1}\tau\gamma)^{*}F-\int(\delta_{2}\tau\gamma)^{*}F+\int(\delta_{3}\tau\gamma)^{*}F\right)$
$\displaystyle=\gamma^{*}\pi(F)=(\sigma_{1}\sigma_{2})(\gamma)\pi(F),$

where the last equality follows from Lemma 7.5. Therefore, the equation (35)
is valid.

We are now able to present a proof for Theorem 6.1.

Proof of Theorem 6.1: Since $\pi:\mathcal{P}^{k}\to\mathcal{P}^{k}$ is linear,
we can apply it to both sides of decomposition (24) in order to get

$\pi(\mathcal{P}^{k})=\pi(ad_{H^{2}}^{k}(\mathcal{P}^{k}))+\pi(\mathcal{P}^{k}(\mathbf{S})).$

From Lemmas 7.5, 7.6 and 7.7, respectively, we have

${\mathcal{P}}^{k}_{\sigma_{1}\sigma_{2}}(\Gamma)=ad_{H^{2}}^{k}({\mathcal{P}}^{k}_{\sigma_{1}}(\Gamma))+{\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right),$

with $\tilde{\sigma}_{j}$ as defined in (27), $j=1,2$. From (34) we know that
${\mathcal{P}}^{k}_{\tilde{\sigma}_{1}\tilde{\sigma}_{2}}\left({\bf
S}\rtimes\Gamma\right)={\mathcal{P}}^{k}(\mathbf{S})\cap{\mathcal{P}}_{\sigma_{1}\sigma_{2}}^{k}(\Gamma)$
and by decomposition (24) we have that
$ad_{H^{2}}^{k}({\mathcal{P}}^{k})\cap{\mathcal{P}}^{k}(\mathbf{S})=\left\\{0\right\\}$,
which completes the proof.

Acknowledgments. Eralcilene Moreira Terezio was financed in part by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance 001. The research of Patrícia Hernandes Baptistelli was supported by
FAPESP, grant 2019/21181-0.

## References

  * [1] Alomair, R., Montaldi, J., Periodic orbits in Hamiltonian systems with involutory symmetries. Journal of Dynamics and Differential Equations 29 (4), 1283-1307 (2017).
  * [2] Antoneli, F., Baptistelli, P. H., Dias, A. P. S., Manoel, M., Invariant theory and reversible-equivariant vector fields. Journal of Pure and Applied Algebra 213 (5), 649-663 (2009).
  * [3] Baptistelli, P. H., Manoel, M., Invariants and relative invariants under compact Lie groups. Journal of Pure and Applied Algebra 217 (12), 2213-2220 (2013).
  * [4] Baptistelli, P. H., Manoel, M., Zeli, I. O., Normal form theory for reversible equivariant vector fields. Bulletin of the Brazilian Mathematical Society, New Series 47 (3), 935-954 (2016).
  * [5] Baptistelli, P. H., Manoel, M., Zeli, I. O., Normal form of bireversible vector fields. Bulletin des Sciences Mathématiques 154, 102-126 (2019).
  * [6] Baptistelli, P. H., Hernandes, M. E., Terezio, E. M., $\omega$-Symplectic algebra and Hamiltonian vector fields. arXiv:2106.14355v2.
  * [7] Belitskii, G. R., Equivalence and normal forms of germs of smooth mappings. Russian Mathematical Surveys 33 (1), 107-177 (1978).
  * [8] Belitskii, G. R., Normal forms in relation to the filtering action of a group. Trudy Moskovskogo Matematicheskogo Obshchestva 40, 3-46 (1979).
  * [9] Broer, H., Formal normal forms theorems for vector fields and some consequences for bifurcations in the volume preserving case. In: Rand, D., Young, L.-S. (eds.) Dynamical Systems and Turbulence, Warwick 1980\. Lecture Notes in Mathematics 898, 54-74. Springer, Conventry (1981).
  * [10] Buono, P-L, Laurent-Polz, F., Montaldi, J., Symmetric Hamiltonian bifurcations. In: Montaldi, J., Ratiu, T. (eds.) Geometric Mechanics and Symmetry: The Peyresq Lectures. London Mathematical Society. Lecture Note Series 306, 357-402. Cambridge University Press, Cambridge (2005).
  * [11] Buzzi, C. A., Teixeira, M. A., Time-reversible Hamiltonian vector fields with symplectic symmetries. Journal of Dynamics and Differential Equations 16 (2), 559-574 (2004).
  * [12] Buzzi, C. A., Roberto L. A., Teixeira M. A., Branching of periodic orbits in reversible Hamiltonian systems. In: Manoel, M., Fuster, M.C.R., Wall, C.T.C. (eds.) Real and Complex Singularities. London Math. Soc. Lecture Note Ser. 380, 46-70. Cambridge University Press, Cambridge (2010).
  * [13] Chow, S.-N., Li, C., Wang, D., Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994).
  * [14] Elphick, C., Tirapegui, E., Brachet, M.E., Coullet, P., Iooss, G., A simple global characterization for normal forms of singular vector fields. Physica D: Nonlinear Phenomena 29, 95-127 (1987).
  * [15] Golubitsky, M., Stewart, I., Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, Vol. II. Appl. Math. Sci. 69, Springer-Verlag, New York (1987).
  * [16] Lima, M. F. S., Teixeira, M. A., Families of periodic orbits in resonant reversible systems. Bulletin of the Brazilian Mathematical Society, New Series 40 (4), 511-527 (2009).
  * [17] Martins, R. M., Formal equivalence between normal forms of reversible and Hamiltonian dynamical systems. Communications on Pure and Applied Analysis 13 (2), 703-713 (2013).
  * [18] Montaldi, J., Roberts, M., Stewart, I., Existence of nonlinear normal modes of symmetric hamiltonian systems. Nonlinearity 3 (3), 695-730 (1990).
  * [19] Robinson, R. C., Lectures on Hamiltonian Systems. Monografias de Matemática bf 7. IMPA, Guanabara (1971).
  * [20] Sevryuk, M. B., Reversible Systems. Lectures Notes in Mathematics 1211. Springer, Berlin (1986).
  * [21] Teixeira, M. A., Martins, R. M., Reversible-equivariant systems and matricial equations. Anais da Academia Brasileira de Ciências 83 (2), 375-390 (2011).
  * [22] Teixeira, M. A., Mereu, A., Reversibility and branching of periodic orbits. Discrete and Continuous Dynamical Systems 33 (3), 1177-1199 (2012).
  * [23] Wulff, C., Roberts, M., Hamiltonian systems near relative periodic orbits. SIAM Journal on Applied Dynamical Systems 1 (1), 1-43 (2002).
  * [24] Zehnder, E., Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities. EMS Textbooks in Mathematics 11. European Mathematical Society, Freiburg (2010).