Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

\documentstyle{era-l}
\def\N{{\Bbb N}}
\def\Z{{\Bbb Z}}
\def\Q{{\Bbb Q}}
\def\R{{\Bbb R}}
\def\C{{\Bbb C}}
\def\c{\mathrm{\bf c}}
\def\S{\mathrm{\bf S}}
\def\H{{\cal H}}
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cQ{{\cal Q}}
\def\cL{{\cal L}}
\def\cW{{\cal W}}
\def\cZ{{\cal Z}}
\def\cR{{\cal R}}
\def\X{{\cal X}}
\def\cI{{\cal I}}
\def\Y{{\cal Y}}
\def\J{{\cal J}}
\def\i{{\imath}}
\def\th{{\delta_{\c}}}
\def\cInf{\mathop{C^{\infty}}\nolimits}
\def\cer{\mathop{\rm cer}\nolimits}
\def\Ad{\mathop{\mathrm Ad}\nolimits}
\def\span{\mathop{\rm span}\nolimits}
\def\Vect{\mathop{\rm Vect}\nolimits}
\def\OB{\mathop{\cal OB}\nolimits}
\def\cC{{\cal C}}
\def\id{\mbox{id}} 
%\renewcommand{\baselinestretch}{1.1}
\newcommand{\SNbr}[2]{\mathrm{\pmb{[}#1,#2\pmb{]}}}
\newcommand{\Pbr}[2]{\{#1,#2\}}
\newcommand{\eqRef}[1]{(\ref{#1})}
\newcommand{\D}[1]{\frac{\partial}{\partial #1}}
\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\pe}[2]{\frac{\partial\epsilon_{#1}}{\partial #2}}
\newcommand{\peMark}[3]{\frac{\partial\epsilon#1_{#2}}{\partial #3}}
\newcommand{\DD}[2]{\frac{\partial}{\partial#1}\wedge\frac{\partial}{\partial#2}}
\newcommand{\DDD}[3]{\frac{\partial}{\partial #1}\wedge\frac{\partial}{\partial #2}\wedge\frac{\partial}{\partial #3}}
\newtheorem{theorem}{Theorem}
\newtheorem{claim}{Claim}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\title{On embedding the $1:1:2$ resonance space in a Poisson manifold}
\author{\'{A}g\'{u}st Sverrir Egilsson}
%\author{{A}g{u}st Sverrir Egilsson}
%\date{\today}
\date{May 8, 1995\inRevisedForm June 2, 1995}
\email{egilsson@@math.berkeley.edu}
\address{University of Iceland, Department of Mathematics,
101 Reykjavik, Iceland.}
\communicated_by{Frances Kirwan}
\subjclass{53}

\begin{document}

\setcounter{page}{48}
\def\currentvolume{1}
\def\currentissue{2}
\def\currentyear{1995}

\begin{abstract}
The Hamiltonian actions of $\S^{1}$ on the symplectic manifold
$\R^{6}$ in the $1:1:-2$ and $1:1:2$ resonances are studied.
Associated to each action is a Hilbert basis of polynomials defining
an embedding of the orbit space into a Euclidean space $V$ and of the
reduced orbit space $J^{-1}(0)/\S^{1}$ into a hyperplane $V_{J}$ of
$V$, where $J$ is the quadratic momentum map for the action. The orbit
space and the reduced orbit space are singular Poisson spaces with
smooth structures determined by the invariant functions.  It is shown
that the Poisson structure on the orbit space, for both the $1:1:2$
and the $1:1:-2$ resonance, cannot be extended to $V$, and that the
Poisson structure on the reduced orbit space $J^{-1}(0)/\S^{1}$ for
the $1:1:-2$ resonance cannot be extended to the hyperplane $V_{J}$.
\end{abstract}

\maketitle

%\tableofcontents

%\clearpage

\section{Introduction}

In this paper we study certain singular Poisson spaces arising from
Hamiltonian actions of Lie groups on symplectic manifolds.  The
singular Poisson spaces are embedded, using Hilbert maps, into
Euclidean spaces and we prove that the singular Poisson structure
cannot be extended to the Euclidean spaces.  This disproves a
conjecture raised by Cushman and Weinstein \cite{SL,WPC}.

For a Hamiltonian action of a Lie group $G$ on a symplectic manifold
$(M,\omega)$ with an equivariant momentum map $J$, Marsden and
Weinstein define in \cite{MW} the reduced orbit space $M_{\mu}$, for a
value $\mu$ in the dual of the Lie algebra of $G$. The space $M_{\mu}$
is the quotient space $J^{-1}(\mu)/G_{\mu}$ where $G_{\mu}$ is the
isotropy group of $\mu$ with respect to the coadjoint action of $G$.
For weakly regular values $\mu$ of $J$, if $G_{\mu}$ acts freely and
properly on the manifold $J^{-1}(\mu)$, $M_{\mu}$ is a manifold and
there is a unique symplectic structure on $M_{\mu}$ which lifts to
$i_{\mu}^{\ast}\omega$ where $i_{\mu}$ is the inclusion map
$i_{\mu}:J^{-1}(\mu)\rightarrow M$.

The space $M/G$ is assigned a smooth structure $\cInf(M/G)$ by the
$G$-invariant functions on $M$ and $M/G$ inherits a Poisson bracket
from $M$ making $M/G$ a Poisson variety. Even when $\mu\in J(M)$ is
not a weakly regular value the reduced orbit space $M_{\mu}$ has a
smooth structure defined by restricting functions in $\cInf(M/G)$ to
$J^{-1}(\mu)$.  It is proved in \cite{ACG} that the space $M_{\mu}$
inherits, by restriction, the structure of a Poisson variety from
$M/G$. For a compact Lie group $G$ Sjamaar and Lerman show in
\cite{SL} that the reduced orbit space $M_{0}$ is a union of
symplectic manifolds and moreover a stratified symplectic space.

If the Lie group $G$ is compact and the action on $M$ has only
finitely many orbit types then by a theorem of Schwarz \cite{Sz} there
are functions $f_{1},\ldots, f_{n}$ in $\cInf(M/G)$ such that
$F^{\ast}\cInf(\R^{n}) = \cInf(M/G)$ for
$F=(f_{1},\ldots,f_{n})$. This defines an embedding $M/G\rightarrow
\R^{n}$ and in particular an embedding $M_{\mu}\rightarrow \R^{n}$.
If a compact Lie group $G$ acts orthogonally on a Euclidean space $M$
then we can choose the invariants $f_{1},\ldots,f_{n}$ to be
generators of the algebra of invariant polynomials on $M$, in which
case the embedding $F$ is called a Hilbert map for the action and
$f_{1},\ldots,f_{n}$ are said to form a Hilbert basis for the algebra.
It is known for many simple cases, see for example \cite{SL}, that the
Poisson structure on the image of $M/G$ can be extended to all of
$\R^{n}$.  We will, however, outline a proof showing that for the
$1:1:-2$ resonance action and a Hilbert map corresponding to a minimal
homogeneous Hilbert basis, described below, defining the embedding
$M/G\rightarrow \R^{11}$ there exists no smooth Poisson structure on
$\R^{11}$ extending the Poisson structure on the orbit space.
Furthermore we show that the Poisson structure on the singular variety
$M_{0}$ cannot be extended to the hyperplane in $\R^{11}$ induced by
the momentum map for the action.  The nonexistence of Poisson
structures on $\R^{11}$ extending the Poisson structure on the orbit
space of the $1:1:2$ resonance action is then established using
results already obtained for the $1:1:-2$ resonance.
%\clearpage 

\section{The $1:1:-2$ resonance}
\subsection{Preliminaries}
Consider the space $\R^{6}$ as $\C^{3}$ and define an $\S^{1}$ action
\begin{equation*}\label{eq-A04}
\star:\S^{1}\times \R^{6}\rightarrow \R^{6}\mbox{ by }z\star(u_{1},u_{2},u_{3})=(zu_{1},zu_{2},z^{-2}u_{3})
\end{equation*}
where the coordinates on $\R^{6}$ and $\C^{3}$ are related by
$u_{i}=x_{i}+\i y_{i}$ for $i=1,2,3$. Also define coordinates
$v_{i}=x_{i}-\i y_{i}$ on $\C^{3}$ for $i=1,2,3$, see \cite{Mo}.

The existence of a Hilbert basis for the invariant polynomials follows
from a theorem of Hilbert \cite{HW}, but for the action above it is
easily established directly.

A minimal homogeneous generating set for the invariant polynomials in
\begin{equation*}\label{eq-A05}
\R[x,y]=\R[x_{1},y_{1},x_{2},y_{2},x_{3},y_{3}]
\end{equation*}
is given by the polynomials $f_{1},\ldots,f_{11}$, i.e., 
\begin{equation*}\label{eq-A06}
\R[x,y]^{\S^{1}} = \R[f_{1},\ldots,f_{11}],
\end{equation*}
where
\begin{itemize}
\item[] $f_{1}=x_{1}^{2}+y_{1}^{2},~f_{2}=x_{2}^{2}+y_{2}^{2},~f_{3}=x_{3}^{2}+y_{3}^{2},$\medskip
\item[] $f_{4}+\i f_{5} = u_{1}v_{2},~f_{6}+\i f_{7} = u_{1}^{2}u_{3},~f_{8}+\i f_{9} = u_{2}^{2}u_{3}$ and\medskip
\item[] $f_{10}+\i f_{11} = u_{1}u_{2}u_{3}$.\medskip
\end{itemize}
A Hilbert map $F:\R^{6}\rightarrow \R^{11}$ corresponding to the
Hilbert basis $f_{1},\ldots,f_{11}$ is given by
$F=(f_{1},\ldots,f_{11})$.  We will refer to this particular choice of
a Hilbert map for the $1:1:-2$ action as the Hilbert map for the
$1:1:-2$ action in standard form.

\subsection{Complex coefficients}
On $\R^{6}$ the standard Poisson bivector field $\varrho$ is given by
\begin{equation*}\label{eq-A07}
\varrho = \DD{x_{1}}{y_{1}}+\DD{x_{2}}{y_{2}}+\DD{x_{3}}{y_{3}}.
\end{equation*}
Denote by $M=\R^{6}$ and by $V=\R^{11}$ the spaces with linear coordinates 
\begin{equation*}\label{eq-A08}
\{x_{1},y_{1},x_{2},y_{2},x_{3},y_{3}\}\mbox{ and }\{l_{1},l_{2},l_{3},R_{1},I_{1},\ldots,R_{4},I_{4}\}
\end{equation*}
respectively and in which $F:M\rightarrow V$ is given by the formula
$F=(f_{1},\ldots,f_{11})$.
\\
Now let $M_{\c}$ and $V_{\c}$ be real vector spaces with linear
coordinates
\begin{equation*}\label{eq-A09}
\{z_{1},w_{1},z_{2},w_{2},z_{3},w_{3}\}\mbox{ and }\{l_{1},l_{2},l_{3},Z_{1},W_{1},\ldots,Z_{4},W_{4}\}
\end{equation*}
respectively.
\medskip

For the Euclidean space $E = \R^{n}$ denote by $\X^{\ast}(E)$ the
$\cInf$-multivector fields on $E$, by $\X^{\ast}[E]$ the multivector
fields on $E$ with polynomial coefficients and by $\X^{\ast}[[E]]$ the
multivector fields on $E$ with formal coefficients.

Let $T$ be the Taylor series operator
\begin{equation*}\label{eq-A27}
T:\X^{\ast}(V)\rightarrow \X^{\ast}[[V]]
\end{equation*}
replacing each coefficient with its Taylor series at $0$.
%\pagebreak\\

Define a $\C$-algebra isomorphism
\begin{equation*}\label{eq-A10}
\kappa : \C \otimes \X^{\ast}[V] \rightarrow \C \otimes \X^{\ast}[V_{\c}]
\end{equation*}
by the formulas 
\begin{itemize}
\item[] $l_{s} \mapsto l_{s},~R_{t} \mapsto \frac{1}{2}(Z_{t}+W_{t})$ and $I_{t} \mapsto \frac{1}{2\i}(Z_{t} - W_{t})$,\medskip
\item[] $\D{l_{s}} \mapsto \D{l_{s}},~\D{R_{t}} \mapsto \D{Z_{t}} + \D{W_{t}}$ and $\D{I_{t}} \mapsto \i (\D{Z_{t}} - \D{W_{t}})$\medskip
\end{itemize}
and similarly define a $\C$-algebra isomorphism
\begin{equation*}\label{eq-A11}
\kappa' : \C \otimes \X^{\ast}[M] \rightarrow \C \otimes \X^{\ast}[M_{\c}]
\end{equation*}
by the formulas 
\begin{itemize}
\item[] $x_{t} \mapsto \frac{1}{2}(z_{t}+w_{t})$ and $y_{t} \mapsto \frac{1}{2\i}(z_{t} - w_{t})$,\medskip
\item[] $\D{x_{t}} \mapsto \D{z_{t}} + \D{w_{t}}$ and $\D{y_{t}} \mapsto \i (\D{z_{t}} - \D{w_{t}})$.\medskip
\end{itemize}
Let
\begin{equation*}\label{eq-A12}
\tau:\R[V^{\ast}]\rightarrow \R[M^{\ast}]
\end{equation*}
be the algebra morphism given by 
\begin{equation*}\label{eq-A13}
\tau(g)=g\circ F
\end{equation*}
and let $\tau_{\c}$ be the $\C$-algebra morphism induced by the diagram
\begin{equation*}\label{eq-6}
\begin{array}{ccc}
\R[V^{\ast}] & \stackrel{\tau}{\longrightarrow} & \R[M^{\ast}]\\
\Big{\downarrow}\kappa&&\Big{\downarrow}\kappa'\\
\C[V_{\c}^{\ast}] & \stackrel{\tau_{\c}}{\longrightarrow} & \C[M_{\c}^{\ast}]\\
\end{array}
\end{equation*}
and use $\tau_{\c}$ to define the complex counterpart of the Hilbert
map
\begin{equation*}\label{eq-A14}
F_{\c}: \C\otimes M_{\c} \rightarrow \C\otimes V_{\c}
\end{equation*}
with complex polynomial coordinate functions defined by 
\begin{equation*}\label{eq-A15}
\tau_{\c}(p)=p\circ F_{\c},
\end{equation*}
furthermore let $F_{\c\ast}$ be the $\C$-linear derivative of
$F_{\c}$.
\medskip

There is a one-to-one relationship between bivector fields $\pi \in
\X^{2}[V]$ which are $F$-related to $\varrho$
\begin{equation*}\label{eq-A16}
F_{\ast}\circ \varrho = \pi\circ F
\end{equation*}
and bivector fields $\pi_{\c} \in \C\otimes \X^{2}[V_{\c}]$ which are
$F_{\c}$-related to $\varrho_{\c}$
\begin{equation*}\label{eq-A17}
F_{\c\ast}\circ\varrho_{\c} = \pi_{\c}\circ F_{\c}
\end{equation*}
and satisfy $\kappa^{-1}(\pi_{\c}) \in \X^{2}[V]$. The relationship is
given by $\pi = \kappa^{-1}(\pi_{\c})$.
%\pagebreak\\

Extend the Schouten-Nijenhuis \cite{Nh,Sc} bracket $\SNbr{}{}$ by
$\C$-bilinearity to the spaces $\C\otimes\X^{\ast}[V]$ and
$\C\otimes\X^{\ast}[V_{\c}]$, then the identity
\begin{equation*}\label{eq-A18}
\SNbr{\kappa(X)}{\kappa(Y)} = \kappa(\SNbr{X}{Y})\mbox{ for }X,Y\in \X^{\ast}[V]
\end{equation*}
follows easily.

Let
\begin{equation*}\label{eq-A19}
\varrho_{\c} = \kappa'(\varrho) = -2\i\sum_{n}\DD{z_{n}}{w_{n}}
\end{equation*}
then the commutative diagram below
\begin{equation}\label{eq-7}
\begin{array}{ccc}
(\R[V^{\ast}],\pi) & \stackrel{\tau}{\longrightarrow} &
(\R[M^{\ast}],\varrho)\\
\Big{\downarrow}\kappa&&\Big{\downarrow}\kappa'\\
(\C[V_{\c}^{\ast}],\pi_{\c}) & \stackrel{\tau_{\c}}{\longrightarrow} &
(\C[M_{\c}^{\ast}],\varrho_{\c})
\end{array}
\end{equation}
demonstrates the setting for finding a bivector field $\pi \in
\X^{2}[V]$ which is $F$-related to $\varrho$.

For the above diagram we have
\begin{equation*}\label{eq-2}
\SNbr{\pi}{\pi} = 0 \mbox{ if and only if }\SNbr{\pi_{\c}}{\pi_{\c}} = 0
\end{equation*}
or equivalently \cite{Lz,Nh} that $\pi\in \X^{2}[V]$ is a Poisson
bivector field if and only if $\pi_{\c}=\kappa(\pi)$ is a Poisson
bivector field.

\subsection{Embeddings}\label{E}
The $1:1:-2$ resonance action above has momentum map $J$ given by the
quadratic polynomial
\begin{equation*}\label{eq-A20}
J=\frac{1}{2}(x_{1}^{2}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}-2x_{3}^{2}-2y_{3}^{2}).
\end{equation*}
There exists a unique functional $\J$ in $V^{\ast}$ satisfying
\begin{equation*}\label{eq-A22}
J = \J\circ F
\end{equation*}
and $F$ maps the space $J^{-1}(0)$ into the hyperplane
\begin{equation*}\label{eq-A24}
V_{J}=\J^{-1}(0)\subset V.
\end{equation*}
The smooth structure on the reduced orbit space $J^{-1}(0)/\S^{1}$,
obtained, see \cite{ACG}, by restricting the invariant functions on
$M$ to $J^{-1}(0)$, is described by
\begin{equation*}\label{eq-A25}
\cInf(J^{-1}(0)/\S^{1}) = (\eta\circ F)^{\ast}\cInf(V_{J})|_{J^{-1}(0)}
\end{equation*}
where $\eta$ is the projection
\begin{equation*}\label{eq-A26}
\eta:V\rightarrow V_{J}
\end{equation*}
onto $V_{J}$ along the vector $\J^{\ast} = \frac{1}{2}(l_{1}^{\ast}+l_{2}^{\ast}-2l_{3}^{\ast})$.
%\pagebreak\\

Extending the Poisson structure on $M/\S^{1}$ to all of $V$ is
equivalent to finding a bivector field $\zeta$ in $\X^{2}(V)$
satisfying the conditions
\begin{enumerate}
\item[1a)] $\SNbr{\zeta}{\zeta}=0$ and\medskip
\item[1b)]$\zeta\circ F = F_{\ast}\circ\varrho$.\medskip
\end{enumerate}
Extending the Poisson structure on $J^{-1}(0)/\S^{1}$ to all of $V_{J}$ is equivalent to finding a bivector field $\xi'$ in $\X^{2}(V_{J})$ satisfying
\begin{enumerate}
\item[2a)] $\SNbr{\xi'}{\xi'}=0$ and\medskip
\item[2b)]  $\xi'\circ (\eta\circ F)|_{J^{-1}(0)} = (\eta\circ F)_{\ast}\circ\varrho|_{J^{-1}(0)}$.\medskip
\end{enumerate}
Further a Poisson bivector $\xi'$ in $\X^{2}(V_{J})$, i.e.,
$\SNbr{\xi'}{\xi'}=0$, extending the structure on $J^{-1}(0)/\S^{1}$
can be extended to a Poisson bivector $\xi$ on all of $V$ satisfying
\begin{enumerate}
\item[3a)] $\xi\circ\iota = \iota_{\ast}\circ\xi'$,\medskip
\item[3b)]  $\xi\circ F|_{J^{-1}(0)} = F_{\ast}\circ\varrho|_{J^{-1}(0)}$,\medskip
\item[3c)] $\SNbr{\xi}{\J} = 0$, i.e., $\J$ is Casimir,
\end{enumerate}
and where $\iota$ is the inclusion map $V_{J}\rightarrow V$.

The above Poisson bivector field $\xi$ is obtained from $\xi'$ by
defining $\xi$ to be constant along the fibres of the projection $\eta$.

\begin{theorem}\label{tm-1}
Let $F:M\rightarrow V$ be the Hilbert map in standard form for the
$1:1:-2$ resonance action $\star:\S^{1}\times M\rightarrow M$.  Then
the Poisson structure on the singular orbit space $M/\S^{1}$, embedded
into $V$ by $F$, cannot be extended to $V$. Furthermore the Poisson
structure on the singular reduced orbit space $J^{-1}(0)/\S^{1}$
cannot be extended to $V_{J}$.
\end{theorem}\noindent
Before proving the above theorem we establish the lemmas
below.
\medskip

Choose a bivector field $\pi$ in $\X^{2}[V]$ such that
\begin{equation}\label{eq-A28}
\pi\circ F = F_{\ast}\circ\varrho,
\end{equation}
e.g. use diagram \eqRef{eq-7}; the existence of a bivector field
satisfying \eqRef{eq-A28} follows from the definition of a Hilbert
basis.

We can further assume that
\begin{equation*}\label{eq-A29}
\SNbr{\pi}{\J} = 0
\end{equation*}
and using the natural grading in $\X^{2}[V]$ write 
\begin{equation*}\label{eq-A30}
\pi = \pi^{1}+\pi^{2+}
\end{equation*}
where $\pi^{1}$ has coefficients of degree $1$ and $\pi^{2+}$ has
coefficients of degrees $2$ and higher.

Then $\pi^{1}$ is uniquely determined by \eqRef{eq-A28} and is already
a Poisson bivector,
\begin{equation*}\label{eq-A31}
\SNbr{\pi^{1}}{\pi^{1}}=0.
\end{equation*}
Using the Poisson bivector $\pi^{1}$ we define the
$\pi^{1}$-cohomology coboundary operator, see \cite{Lz}, on
$\X^{\ast}(V)$ by
\begin{equation*}\label{eq-A32}
\delta = -\SNbr{\pi^{1}}{\cdot}.
\end{equation*}
Also define $\pi_{\c},\pi^{1}_{\c}$ and $\pi^{2+}_{\c}$ in $\C\otimes\X^{2}[V_{\c}]$ as the images of $\pi,\pi^{1}$ and $\pi^{2+}$ under $\kappa$ respectively, and define $\delta_{\c}$ to be the coboundary operator on $\C\otimes\X^{\ast}(V_{\c})$ given by
\begin{equation*}\label{eq-A33}
\delta_{\c} = -\SNbr{\pi^{1}_{\c}}{\cdot}.
\end{equation*}
Furthermore define $\Gamma$ to be the ideal in $\C[V^{\ast}_{\c}]$
generated by the terms
\begin{equation*}\label{eq-A34}
\J,~l_{3},~Z_{1},~W_{1},~Z_{2},~W_{2},~Z_{3},~W_{3},~Z_{4},~W_{4}^{2}
\end{equation*}
and all the monomials of degree $3$ and higher.
\medskip

For $X\in\C\otimes\X^{3}[V_{\c}]$ denote the coefficient of
$\DDD{Z_{2}}{W_{2}}{W_{4}}$ in $X$ by $X_{\phi}$, i.e.,
\begin{equation*}\label{eq-A35}
X_{\phi} = X(dZ_{2}\wedge dW_{2}\wedge dW_{4}),
\end{equation*}
and for $Y\in \C\otimes\X^{2}[V_{\c}]$ define
$\widetilde{Y}$ by
\begin{equation*}
\widetilde{Y}=Y_{Z_{2}W_{2}}\DD{Z_{2}}{W_{2}}
+Y_{Z_{2}W_{4}}\DD{Z_{2}}{W_{4}}
+Y_{W_{2}W_{4}}\DD{W_{2}}{W_{4}}
\end{equation*} 
where $Y_{Z_{2}W_{2}}$ is the coefficient of $\DD{Z_{2}}{W_{2}}$ in
$Y$.
\begin{lemma}\label{re-1}
Let $\epsilon_{\c}$ be a bivector in $\C\otimes \X^{2}[V_{\c}]$.
Then $\delta_{\c}(\epsilon_{\c})_{\phi} \equiv
-\SNbr{\varphi_{\c}}{\widetilde{\epsilon_{\c}}}_{\phi}\pmod{\Gamma}$
where $\varphi_{\c}$ is the bivector field given by the formula
\begin{equation*}\label{eq-12}
\begin{array}{rl}
\varphi_{\c}&=-2\i(
W_{4}\DD{l_{1}}{W_{4}}
+W_{4}\DD{l_{2}}{W_{4}}
+W_{4}\DD{l_{3}}{W_{4}}\medskip\\&
+(l_{2}-l_{1})\DD{Z_{1}}{W_{1}}
+2W_{4}\DD{Z_{1}}{W_{2}}
+2W_{4}\DD{W_{1}}{W_{3}}).
\end{array} 
\end{equation*}
\end{lemma}\noindent
{\it Proof: } All the coefficients of the basis vectors
$\DD{Z_{2}}{W_{2}},~\DD{Z_{2}}{W_{4}}$ and $\DD{W_{2}}{W_{4}}$ in
$\pi_{\c}^{1}$ and $\varphi_{\c}$ are zero and the bivector field
$\pi_{\c}^{1}-\varphi_{\c}$ is in the ideal generated by
$\Gamma\cdot\X^{2}[V_{\c}]$.  From this and the definition of the
Schouten-Nijenhuis bracket it follows that
$\delta_{\c}(\epsilon_{\c})_{\phi} \equiv
-\SNbr{\varphi_{\c}}{\widetilde{\epsilon_{\c}}}_{\phi}\pmod{\Gamma}$.
$\Box$
\medskip

One can choose $\pi^{2+}_{\c}$ to be homogeneous of degree $2$ and an example of $\widetilde{\pi^{2+}_{\c}}$ is given by
\begin{equation*}
\widetilde{\pi^{2+}_{\c}} = -2\i(
(l_{1}^{2}+4l_{1}l_{3})\DD{Z_{2}}{W_{2}}
+(l_{1}Z_{1}+2l_{3}Z_{1})\DD{Z_{2}}{W_{4}}).
\end{equation*}
Call an element $\rho$ in $\X^{\ast}[V]$ a relation if
$\rho\circ F = 0$ and define
\begin{equation*}\label{eq-A37}
\Phi^{\ast}[V] = \{\rho'\in \X^{\ast}[V]:\rho'\circ F = 0\}.
\end{equation*}
%\pagebreak\\
\begin{lemma}\label{le-1}
Let $\zeta\in \X^{2}[V]$ be a bivector of the form
\begin{equation*}\label{eq-A36}
\zeta = \pi+\rho+\J\gamma
\end{equation*}
for a relation $\rho$ in $\Phi^{2}[V]$, a bivector $\gamma\in
\X^{2}[V]$ and such that at least one of the two conditions below are
satisfied
\begin{enumerate}
\item[i)] $\SNbr{\zeta}{\J}$ contains no terms of degree $2$,
\item[ii)] $\gamma$ contains no constant terms.
\end{enumerate}
Then $\SNbr{\zeta}{\zeta}$ has nonzero terms of degree $2$.
\end{lemma}\noindent
{\it Proof:} Let $\zeta_{\c} = \kappa(\zeta)$. Using Lemma \ref{re-1}
we calculate that an $\epsilon$ in $\Phi^{2}[V] + \J\X^{2}[V]$ satisfies the formula $\delta_{\c}(\kappa(\epsilon))_{\phi} \equiv 0 \pmod{\Gamma}$.\\
Furthermore the formula
$\SNbr{\pi^{1}_{\c}}{\pi^{2+}_{\c}}_{\phi} \not\equiv 0 \pmod{\Gamma}$ holds.\\
Now we conclude that if either of conditions i) or ii) holds then
\begin{equation*}
\SNbr{\zeta_{\c}}{\zeta_{\c}}_{\phi} \equiv 2\SNbr{\pi^{1}_{\c}}{\pi^{2+}_{\c}}_{\phi} \pmod{\Gamma}
\end{equation*}
and the lemma follows. $\Box$
\begin{lemma}\label{le-2}
Let $\Delta\in\X^{\ast}(V)$ satisfy $\Delta\circ
F|_{J^{-1}(0)}=0$. Then there exists a formal multivector field
$\gamma\in\X^{\ast}[[V]]$ such that $T(\Delta)-\J\gamma$ is a formal
relation, i.e.,
\begin{equation*}\label{eq-A40}
T(\Delta) - \J\cdot\gamma \in
\Phi^{\ast}[[V]]=\{\rho\in\X^{\ast}[[V]]:\rho\circ F = 0\}.
\end{equation*}
\end{lemma}\noindent
This lemma is proved by using the Malgrange-Mather division theorem
\cite{Ma,Mr2}.$\Box$
\bigskip

{\it Outline of the proof of Theorem \ref{tm-1}: } Let $\zeta\in
\X^{2}(V)$ be a bivector field such that $\zeta$ and $\varrho$ are
$F$-related, i.e., $\zeta\circ F = F_{\ast}\circ\varrho$.
 
Now write $T(\zeta) - \pi = \rho' + {\rho'}^{3+}$ where
$\rho'\in\Phi^{2}[V]$ has polynomial coefficients and such that
${\rho'}^{3+}\in\Phi^{2}[[V]]$ contains only terms of degrees greater
than two.

Now since $\pi+\rho'$ satisfies condition ii) of Lemma \ref{le-1},
deduce that the Taylor series of $\SNbr{\zeta}{\zeta}$ has nonzero
terms of degree $2$.  Hence $\zeta$ is not Poisson.
\medskip
 
Assume now that there exists a Poisson bivector field $\xi'$ in
$\X^{2}(V_{J})$ extending the Poisson structure of $J^{-1}(0)/\S^{1}$
to all of $V_{J}$.
 
By 3a, 3b and 3c we extend $\xi'$ to a Poisson bivector field $\xi$ on
$V$ in such a way that $\xi$ extends the Poisson structure of
$J^{-1}(0)/\S^{1}$ in $V$ and has $\J$ for a Casimir function.

Use Lemma \ref{le-2} to write
\begin{equation*}\label{eq-A42}
T(\xi) = \pi +\rho + \J\gamma + \rho^{3+} + \J\gamma^{2+}
\end{equation*}
where as before $\rho$ is in $\Phi^{2}[V]$ and $\rho^{3+}$ is in
$\Phi^{2}[[V]]$ and has only coefficients of degrees greater than two,
$\gamma$ is in $\X^{2}[V]$ and $\gamma^{2+}$ is a formal bivector
field with coefficients of degrees greater than one.  Conclude, since
$\pi +\rho + \J\gamma$ satisfies condition i) of Lemma \ref{le-1},
that the Taylor series of $\SNbr{\xi}{\xi}$ has nonzero terms of
degree $2$.  Thus contradicting the existence of the Poisson bivector
field $\xi'$.$\Box$
%\clearpage

\section{The $1:1:2$ resonance}
\subsection{Preliminaries}
To analyze the relationship between the $n_{1}:\cdots:n_{k}$ resonance
and the $\sigma_{1}n_{1}:\cdots:\sigma_{k}n_{k}$ resonance,
$\sigma_{i}\in\{1,-1\}$ for $i=1,\ldots,k$, we define an automorphism,
see \cite{ASE}, $\psi^{\sigma}$ on the space $\C\otimes\X^{\ast}[V]$.
Here we will define $\psi^{\sigma}$ for the $1:1:-2$ resonance and the
$\sigma = (1,1,-1)$ case.

For the $1:1:2$ resonance the Hilbert map $F^{\sigma}:M\rightarrow V$
in standard form, by definition here, is obtained by interchanging
$u_{3}$ and $v_{3}$ in the definitions of the invariants
$f_{1},\ldots, f_{11}$.\\ Let $\sigma =
(\sigma_{1},\sigma_{2},\sigma_{3})=(1,1,-1)$ and define
\begin{equation*}\label{eq-A43}
\omega^{\sigma} : \{l_{1},l_{2},l_{3}, Z_{1}, W_{1}, \ldots, Z_{4}, W_{4}\} \rightarrow \{1,-1,\i,-\i \}
\end{equation*}
by the formula
\begin{equation*}\label{eq-A44}
\omega^{\sigma}_{\gamma} = \i^{\deg_{z_{3}}\tau_{\c}(\gamma)} \cdot \i^{\deg_{w_{3}}\tau_{\c}(\gamma)}.
\end{equation*}
Now, using $\omega^{\sigma}$, define the automorphism
\begin{equation*}\label{eq-A45}
\psi^{\sigma}_{\c} : \C \otimes \X^{\ast}[V_{\c}] \rightarrow \C \otimes \X^{\ast}[V_{\c}]
\end{equation*}
by the formulas
\begin{itemize}
\item[] $l_{s} \mapsto \omega^{\sigma}_{l_{s}}l_{s},~Z_{t} \mapsto \omega^{\sigma}_{Z_{t}}Z_{t}$ and $W_{t} \mapsto \omega^{\sigma}_{W_{t}}W_{t}$,\medskip
\item[] $\D{l_{s}} \mapsto \frac{1}{\omega^{\sigma}_{l_{s}}}\D{l_{s}},~\D{Z_{t}} \mapsto \frac{1}{\omega^{\sigma}_{Z_{t}}}\D{Z_{t}}$ and $\D{W_{t}} \mapsto \frac{1}{\omega^{\sigma}_{W_{t}}}\D{W_{t}}$\medskip
\end{itemize}
and also define
\begin{equation*}\label{eq-A46}
\psi^{\sigma}: \C \otimes \X^{\ast}[V] \rightarrow \C \otimes \X^{\ast}[V]\mbox{ by }\psi^{\sigma} = \kappa^{-1}\circ\psi^{\sigma}_{\c}\circ\kappa.
\end{equation*}
The map $\psi^{\sigma}$ is a Schouten-Nijenhuis morphism defining a
cochain map between the complexes
\begin{equation*}\label{eq-A48}
(\C\otimes\X^{\ast}[V],\delta)
\stackrel{\psi^{\sigma}}{\longrightarrow}(\C\otimes\X^{\ast}[V],\delta^{\sigma})
\end{equation*}
where $\delta = -\SNbr{\pi^{1}}{\cdot}$ and $\delta^{\sigma} =
-\SNbr{\psi^{\sigma}(\pi^{1})}{\cdot}$.
\begin{lemma}\label{le-A1}
Let $\pi'$ be a bivector field in $\X^{2}[V]$. The bivectors $\pi'$
and $\varrho$ are $F$-related if and only if the bivectors
$\psi^{\sigma}(\pi')$ and $\varrho$ are $F^{\sigma}$ related.
\end{lemma}\noindent
{\it Proof: }\cite{ASE}.$\Box$
\medskip

Thus $\psi^{\sigma}$ allows us to apply results for the $1:1:-2$
resonance to the $1:1:2$ resonance case.
\subsection{Embeddings}
\begin{theorem}\label{tm-2}
Let $F^{\sigma}:M\rightarrow V$ be the Hilbert map in standard form
for the $1:1:2$ resonance action $\star:\S^{1}\times M\rightarrow
M$. Then the Poisson structure on the singular orbit space $M/\S^{1}$,
embedded into $V$ by $F^{\sigma}$, cannot be extended to $V$.
\end{theorem}\noindent
{\it Proof:} Follows from applying the morphism $\psi^{\sigma}$ to
results for the $1:1:-2$ resonance. $\Box$
%\pagebreak
\section{Generalizations}
In order to analyze which circle actions induce Poisson embeddings of
the orbit spaces one should study the semigroup structures obtained
from the invariant polynomials. The semigroup structure arises from
complexifying the phase space and the ambient Euclidean space.  A
proper framework for Poisson structures and maps for these semigroups
would be helpful in resolving the general case.  The author is
currently working in this direction.

\section{Acknowledgements}
I would like to thank my advisor, Alan Weinstein, for sharing his
ideas on the subject at hand. I would also like to thank Alexander
Givental for many discussions related to the content of this paper and
Vinay Kathotia and the referee for many helpful comments. Much of the
work presented here was formulated during my visit to the CENTRE
International de Recherche EMILE BOREL in Paris for their symplectic
geometry semester in 1994.  I would like to take this opportunity to
thank them for their hospitality.

Moreover I would like to thank IPC in Iceland for their generous
financial support.  The research described here was partially
supported by a grant from the Thor Thors fund and by DOE grant
DE-FG03-93ER25177.

\begin{thebibliography}{10}

\bibitem{ACG}
J.~Arms, R.~Cushman and M.~Gotay.
\newblock A universal reduction procedure for Hamiltonian group actions.
\newblock In {\em The Geometry of Hamiltonian Systems}. Mathematical Sciences Research Institute Publications 22, Springer-Verlag 1991, 33-51.

\bibitem{ASE} \'{A}g\'{u}st S. Egilsson. 
\newblock On embedding a stratified symplectic space in a smooth Poisson manifold.
\newblock Ph.D. thesis 1995.

\bibitem{Lz}
A.~Lichnerowicz.
\newblock Les vari\'et\'es de Poisson et leurs alg\`ebres de Lie associ\'ees.
\newblock {\em J. Differential Geometry} 12 (1977), 253-300.

\bibitem{Ma}
B.~Malgrange.
\newblock Le th\'eor\`eme de pr\'eparation en g\'eom\'etrie diff\'erentiable.
\newblock {\em S\'eminaire Henri Cartan}, 15e ann\'ee, 1962/63, expos\'es 11-13, 22.

\bibitem{MW} J.~Marsden and A.~Weinstein.
\newblock Reduction of symplectic manifolds with symmetry.
\newblock {\em Reports on Mathematical Physics} 5 (1974), 121-130.

\bibitem{Mr2}
J.N. Mather.
\newblock Stability of $\cInf$ mappings: I. The division theorem.
\newblock {\em Annals of Mathematics} 87 (1968), 89-104.

\bibitem{Mr}
J.N. Mather.
\newblock Differentiable invariants.
\newblock {\em Topology} 16 (1977), 145-155.

\bibitem{Mo}
J.K. Moser.
\newblock Lectures on Hamiltonian systems.
\newblock {\em Memoirs of the American Mathematical Society} 81 (1968).

\bibitem{Nh}
A.~Nijenhuis.
\newblock Jacobi-type identities for bilinear differential concomitants of
  certain tensor fields.
\newblock {\em Indagationes Math} 17 (1955), 390-403.

\bibitem{Sc}
J.A. Schouten.
\newblock On the differential operators of first order in tensor calculus.
\newblock {\em Convengo di Geometria Differenziale 1953}, 1-7.

\bibitem{Sz}
G.W. Schwarz.
\newblock  Smooth functions invariant under the action of a compact Lie group.
\newblock {\em Topology} 14 (1975), 63-68.

\bibitem{SL} R.~Sjamaar and E.~Lerman.
\newblock Stratified symplectic spaces and reduction.
\newblock {\em Annals of Mathematics} 134 (1991), 376-422.

\bibitem{WPC} A.~Weinstein.
\newblock Private communication.

\bibitem{HW}
H.~Weyl.
\newblock {\em The Classical Groups}.
\newblock Princeton University Press, 1946.

\end{thebibliography}

\end{document}