%%% Date: Wed, 13 Jun 2001 09:53:40 -0500
%%% From: tonthat <tonthat@math.uiowa.edu>
%%% X-Accept-Language: en
%%% To: Ulf Rehmann <rehmann@Mathematik.Uni-Bielefeld.DE>
%%% Subject: [Fwd: Tuong Ton-That's manuscript]
%%% Content-Type: multipart/mixed;
%%%  boundary="------------7103F97F448EB01B61B20797"
%%% X-UIDL: 00bbc9f3ab0494971e13341086ef9cd7
%%% 
%%% This is a multi-part message in MIME format.
%%% --------------7103F97F448EB01B61B20797
%%% Content-Type: text/plain; charset=us-ascii
%%% Content-Transfer-Encoding: 7bit
%%% 
%%% Dear Professor Rehmann,
%%% I would like to submit the attached preprint to the Linear Algebraic
%%% Groups and Related Structures Preprint Server.Thank you very much.
%%% Best regards,
%%% Tuong Ton-That
%%% 
%%% --------------7103F97F448EB01B61B20797
%%% Content-Type: message/rfc822
%%% Content-Transfer-Encoding: 7bit
%%% Content-Disposition: inline
%%% 
%%% MIME-Version: 1.0
%%% 
%%% Received: from math.uiowa.edu (dhcp099.divms.uiowa.edu [128.255.26.202]) by server.divms.uiowa.edu with id RAA18527; Mon, 4 Jun 2001 17:17:12 -0500 (CDT)
%%% Message-ID: <3B1C08E8.33AF8540@math.uiowa.edu>
%%% Date: Mon, 04 Jun 2001 17:17:12 -0500
%%% From: Cymie Wehr <cwehr@math.uiowa.edu>
%%% X-Mailer: Mozilla 4.61 [en] (Win98; I)
%%% X-Accept-Language: en,pdf
%%% MIME-Version: 1.0
%%% To: qforms@marais.math.lsu.edu
%%% CC: Tuong Ton-That <tonthat@math.uiowa.edu>
%%% Subject: Tuong Ton-That's manuscript
%%% Content-Type: multipart/mixed;
%%%  boundary="------------5FF949D3B5479B98428809C7"
%%% X-Mozilla-Status2: 00000000
%%% 
%%% This is a multi-part message in MIME format.
%%% --------------5FF949D3B5479B98428809C7
%%% Content-Type: text/plain; charset=us-ascii
%%% Content-Transfer-Encoding: 7bit
%%% 
%%% The organizer of the Quadratic Forms Conference
%%% 
%%% Tuong Ton-That asked me to send the Tex electronic file copy of his
%%% manuscript, A Generalized Poincare Theorem for Dual Lie Transformation
%%% Groups.
%%% 
%%% Please let me know if you have problems with receiving this electronic
%%% file copy.
%%% 
%%% --
%%% Cymie Wehr, Technical Typist
%%% Department of Mathematics
%%% University of Iowa
%%% Iowa City IA 52240
%%% 
%%% 
%%% --------------5FF949D3B5479B98428809C7
%%% Content-Type: application/x-tex;
%%%  name="generalizedlatex2e.tex"
%%% Content-Transfer-Encoding: 7bit
%%% Content-Disposition: inline;
%%%  filename="generalizedlatex2e.tex"
%%% 
%%% %
%%% %    Documenta-LaTeX.sty (2e)
%%% %
%%% %    Version 0.4
%%% %
%%% %    Copyright (C) 1996-2000 Ulf Rehmann
%%% %    rehmann@mathematik.uni-bielefeld.de
%%% %
%%% % ------------------------------ PROLOGUE -----------------------------
%%% %
%%% % This prologue is to be prepended to your LaTeX file in order to
%%% % give it the layout required by Documenta Mathematica.
% Please edit the \documentclass line below, add \usepackage lines
% according to your needs and
% go to the line starting with \Title below, fill in title,
% authors' names and so on and follow the further instructions there.
% 
% Choose one of the commands \documentclass[]{} or \documentstyle[]{}
% depending on the version of LaTeX you use:
%
%
% If you use LaTeX 2e use the following line (or similar):
\documentclass[10pt,twoside]{article}
% Append here commands \usepackage{...} if you want.
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\numberwithin{equation}{section}
%
% If you use LaTeX 2.09 or earlier use the following line (or similar):
%\documentstyle[10pt,twoside]{article}
%
% Please go to the line starting with \Title below, fill in title,
% authors' names and so on and follow the further instructions there.
%
% The final values of the following 4 uncommented lines will be filled in
% by Doc. Math. after the manuscript has been accepted for publication.
%
\def\YEAR{\year}\newcount\VOL\VOL=\YEAR\advance\VOL by-1995
\def\firstpage{1}\def\lastpage{1000}
\def\received{}\def\revised{}
\def\communicated{}
%

\makeatletter
\def\magnification{\afterassignment\m@g\count@}
\def\m@g{\mag=\count@\hsize6.5truein\vsize8.9truein\dimen\footins8truein}
\makeatother

%%% Choose 10pt/12pt:
%\oddsidemargin1.66cm\evensidemargin1.66cm\voffset1.2cm%10pt
\oddsidemargin1.91cm\evensidemargin1.91cm\voffset1.4cm%10pt
%\magnification1200\oddsidemargin.41cm\evensidemargin.41cm\voffset-.75cm%12pt

%\textwidth12.5cm\textheight19.5cm
\textwidth12.0cm\textheight19.0cm

\font\eightrm=cmr8
\font\caps=cmcsc10                    % Theorem, Lemma etc
\font\ccaps=cmcsc10                   % Sections
\font\Caps=cmcsc10 scaled \magstep1   % Title
\font\scaps=cmcsc8

%
%-----------------headlines-----------------------------------

%\input german.sty      % uncomment if necessary
%\usepackage{german}    % uncomment if necessary

\pagestyle{myheadings}
\pagenumbering{arabic}
\setcounter{page}{\firstpage}
\def\bfseries{\normalsize\caps}

\makeatletter
\setlength\topmargin {14\p@}
\setlength\headsep   {15\p@}  
\setlength\footskip  {25\p@}  
\setlength\parindent {20\p@} 
\@specialpagefalse\headheight=8.5pt
\def\DocMath{}
\renewcommand{\@evenhead}{%
    \ifnum\thepage>\lastpage\rlap{\thepage}\hfill%
    \else\rlap{\thepage}\slshape\leftmark\hfill{\caps\SAuthor}\hfill\fi}%
\renewcommand{\@oddhead}{%
    \ifnum\thepage=\firstpage{\DocMath\hfill\llap{\thepage}}%
    \else{\slshape\rightmark}\hfill{\caps\STitle}\hfill\llap{\thepage}\fi}%
\makeatother

\def\TSkip{\bigskip}
\newbox\TheTitle{\obeylines\gdef\GetTitle #1
\ShortTitle  #2
\SubTitle    #3
\Author      #4
\ShortAuthor #5
\EndTitle
{\setbox\TheTitle=\vbox{\baselineskip=20pt\let\par=\cr\obeylines%
\halign{\centerline{\Caps##}\cr\noalign{\medskip}\cr#1\cr}}%
	\copy\TheTitle\TSkip\TSkip%
\def\next{#2}\ifx\next\empty\gdef\STitle{#1}\else\gdef\STitle{#2}\fi%
\def\next{#3}\ifx\next\empty%
    \else\setbox\TheTitle=\vbox{\baselineskip=20pt\let\par=\cr\obeylines%
    \halign{\centerline{\caps##} #3\cr}}\copy\TheTitle\TSkip\TSkip\fi%
%\setbox\TheTitle=\vbox{\let\par=\cr\obeylines%
%\halign{\centerline{\caps##} #4\cr}}\copy\TheTitle\TSkip\TSkip%
\centerline{\caps #4}\TSkip\TSkip%
\def\next{#5}\ifx\next\empty\gdef\SAuthor{#4}\else\gdef\SAuthor{#5}\fi%
\ifx\received\empty\relax
    \else\centerline{\eightrm Received: \received}\fi%
\ifx\revised\empty\TSkip%
    \else\centerline{\eightrm Revised: \revised}\TSkip\fi%
\ifx\communicated\empty\relax
    \else\centerline{\eightrm Communicated by \communicated}\fi\TSkip\TSkip%
\catcode'015=5}}\def\Title{\obeylines\GetTitle}
\def\Abstract{\begingroup\narrower
    \parskip=\medskipamount\parindent=0pt{\caps Abstract. }}
\def\EndAbstract{\par\endgroup\TSkip}

\long\def\MSC#1\EndMSC{\def\arg{#1}\ifx\arg\empty\relax\else
     {\par\narrower\noindent%
     2000 Mathematics Subject Classification: #1\par}\fi}

\long\def\KEY#1\EndKEY{\def\arg{#1}\ifx\arg\empty\relax\else
	{\par\narrower\noindent Keywords and Phrases: #1\par}\fi\TSkip}

\newbox\TheAdd\def\Addresses{\vfill\copy\TheAdd\vfill
    \ifodd\number\lastpage\vfill\eject\phantom{.}\vfill\eject\fi}
{\obeylines\gdef\GetAddress #1
\Address #2 
\Address #3
\Address #4
\EndAddress
{\def\xs{4.3truecm}\parindent=0pt
\setbox0=\vtop{{\obeylines\hsize=\xs#1\par}}\def\next{#2}
\ifx\next\empty % 1 address
     \setbox\TheAdd=\hbox to\hsize{\hfill\copy0\hfill}
\else\setbox1=\vtop{{\obeylines\hsize=\xs#2\par}}\def\next{#3}
\ifx\next\empty % 2 addresses
     \setbox\TheAdd=\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
\else\setbox2=\vtop{{\obeylines\hsize=\xs#3\par}}\def\next{#4}
\ifx\next\empty\ % 3 addresses
     \setbox\TheAdd=\vtop{\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
                \vskip20pt\hbox to\hsize{\hfill\copy2\hfill}}
\else\setbox3=\vtop{{\obeylines\hsize=\xs#4\par}}
     \setbox\TheAdd=\vtop{\hbox to\hsize{\hfill\copy0\hfill\copy1\hfill}
	        \vskip20pt\hbox to\hsize{\hfill\copy2\hfill\copy3\hfill}}
\fi\fi\fi\catcode'015=5}}\gdef\Address{\obeylines\GetAddress}

\hfuzz=0.1pt\tolerance=2000\emergencystretch=20pt\overfullrule=5pt
\begin{document}
%%%%% ------------- fill in your data below this line  -------------------
%%%%%    The following lines \Title ... \EndAddress must ALL be present
%%%%%    and in the given order.
\Title
%%%%%    Put here the title. Line breaks will be recognized. 
A Generalized Poincar\'{e} Theorem 
for Dual Lie Transformation Groups
\ShortTitle A Generalized Poincar\'{e} Theorem 
%%%%%    Running title for odd numbered pages, ONE line, please. 
%%%%%    If none is given, \Title will be used instead.          
\SubTitle   
%%%%%    A possible subtitle goes here.
\Author 
%%%%%    Put here name(s) of authors. Line breaks will be recognized.  
Tuong Ton-That
\ShortAuthor 
%%%%%%   Running title for even numbered pages, ONE line, please. 
%%%%%%   If none is given, \Author will be used instead.          
\EndTitle
\Abstract 
%%%%%    Put here the abstract of your manuscript.
Let $k$ and $n$ be integers such that $k>2n>0$. Let $M$ be the complex
analytic manifold defined by $M=\{x\in\mathbb{C}^{n\times k}:xx^{t}%
=0,\;\operatorname*{rank}$ $(x)=n\}$. Let $G=\mathrm{SO}(k,\mathbb{C})$ and
$G^{\prime}=\mathrm{GL}(n,\mathbb{C})$, then Witt's theorem on quadratic forms
implies that $G$ is a maximal connected Lie group acting transitively on $M$
by right multiplication. Also, $G^{\prime}$ is a maximal connected Lie group
acting freely on $M$ by left multiplication. If $f\in C^{\infty}(M)$, $x\in
M$, $g\in G$, and $g^{\prime}\in G^{\prime}$ define $R(g)f$
(resp.~$L(g^{\prime})f$) by
\[
(R(g)f)(x)=f(xg)\text{\quad and\quad}(L(g)f)(x)=f(g^{-1}x).
\]

If $\mathcal{D}^{\omega}(M)$ denotes the algebra of all analytic differential
operators on $M$ then an element $D\in\mathcal{D}^{\omega}(M)$ is called right
(resp.~left)-invariant if $DR(g)=R(g)D$, $\forall\,g\in G$
(resp.~$DL(g^{\prime})=L(g^{\prime})D,$ $\forall\,g^{\prime}\in G^{\prime}$).

THEOREM: Let $\mathcal{D}_{l}^{\omega}(M)$ (resp.~$\mathcal{D}_{r}^{\omega
}(M)$) denote the subalgebra of $\mathcal{D}^{\omega}(M)$ of all left
(resp.~right)-invariant analytic differential operators on $M$. Let
$\widetilde{\mathcal{U}}(\frak{g})$ (resp.$~\widetilde{\mathcal{U}}%
(\frak{g}^{\prime})$) denote the universal enveloping algebra generated by the
infinitesimal action of $R(g)$ (resp.~$L(g^{\prime})$). Then we have
\[
\mathcal{D}_{l}^{\omega}(M)=\widetilde{\mathcal{U}}(\frak{g})\text{ and
}\mathcal{D}_{r}^{\omega}(M)=\widetilde{\mathcal{U}}(\frak{g}^{\prime
})\text{.}%
\]
Moreover, the commutant of $\mathcal{D}_{l}^{\omega}(M)$ in $\mathcal{D}%
^{\omega}(M)$ is $\mathcal{D}_{r}^{\omega}(M)$, and vice-versa.

This theorem also holds for other types of dual Lie transformation groups
acting on analytic manifolds.

\EndAbstract
\MSC 
%%%%%    1991 Mathematics Subject Classification: 
Primary 15A63, 16S32; Secondary 16S30, 14L35.
\EndMSC
\KEY 
%%%%%    Keywords and Phrases:     
\EndKEY
%%%%%    All 4 \Address lines below must be present. To center the last
%%%%%    entry, no empty lines must be between the following \Address
%%%%%    and \EndAddress lines.
\Address 
%%%%%    Address of first Author here
Department of Mathematics
The University of Iowa
Iowa City, IA 52242-1419
USA
tonthat@math.uiowa.edu
\Address
%%%%%    Address of second Author here etc.
\Address
\Address
\EndAddress
%%
%%       Make sure the last tex command in your manuscript
%%       before the first \end{document} is the command  \Addresses
%%
%%---------------------Here the prologue ends---------------------------------
%%--------------------Here the manuscript starts------------------------------
\section{Introduction}

In 1900 H. Poincar\'{e} established the existence of the universal enveloping
algebra of a Lie algebra and proved one of the most fundamental results in the
theory of Lie groups and Lie algebras. This theorem which is valid for a Lie
algebra over an arbitrary field is usually called the Poincar\'{e}%
-Birkhoff-Witt theorem; however for the case of a real or complex Lie algebra
it is entirely due to Poincar\'{e} as shown in \cite{TT-T}.

\begin{theorem}
\label{theorem1.1}$($Poincar\'{e}$)$. Let $G$ be a real or complex Lie group
with Lie algebra $\frak{g}$. Let $\mathcal{U}(\frak{g})$ denote the universal
enveloping algebra of $\frak{g}$. If $\{X_{i}:1\leq i\leq n\}$ is a basis of
$\frak{g}$ then the ordered monomials $1$ and $X_{i_{1}}\cdots X_{i_{s}}%
(s\geq1,i_{1}\leq\cdots\leq i_{s})$ form a basis for $\mathcal{U}(\frak{g})$.
\end{theorem}

Assume that $G$ is a real or complex \textit{connected} Lie group. For each
$g\in G$, the translations $l_{g},r_{g}:G\rightarrow G$ defined by
$l_{g}(x)=gx$ and $r_{g}(x)=xg$, $x\in G$, are analytic diffeomorphisms of $G$
onto itself. Let $\mathcal{D}^{\omega}(G)$ denote the algebra of all analytic
differential operators on $G$. A differential operator $D$ of $\mathcal{D}%
^{\omega}(G)$ is said to be \emph{left} (resp.~\emph{right})-\emph{invariant}
if it is invariant under all left (resp.~right) translations. Let
$[\cdot,\cdot]$ denote the commutator product of $\mathcal{D}^{\omega}(G)$. If
$\mathcal{A}$ is a subalgebra of $\mathcal{D}^{\omega}(G)$ then the
\emph{centralizer} (or \emph{commutant}) of $\mathcal{A}$ in $\mathcal{D}%
^{\omega}(G)$ is defined as the set $\{D^{\prime}\in\mathcal{D}^{\omega
}(G):[D^{\prime},D]=0,\,\forall\,D\in\mathcal{A}\}$, and the \emph{centre} of
$\mathcal{A}$ is defined as the set $\{D^{\prime}\in\mathcal{A}:[D^{\prime
},D]=0,\,\forall\,D\in\mathcal{A}\}$. Then the following can be easily deduced
from the Poincar\'{e} theorem.

\begin{corollary}
\label{corollary1.2}$($To Poincar\'{e} Theorem$)$. If $\mathcal{D}_{l}%
^{\omega}(G)$ $($resp.~$\mathcal{D}_{r}^{\omega}(G))$ denotes the subalgebra
of $\mathcal{D}^{\omega}(G)$ of all left $($resp.~right$)$-invariant analytic
differential operators on $G$ then $\mathcal{D}_{l}^{\omega}(G)$
$($resp.~$\mathcal{D}_{r}^{\omega}(G))$ is isomorphic to the universal
enveloping algebra $\mathcal{U}(\frak{g})$. Moreover, the centralizer of
$\mathcal{D}_{l}^{\omega}(G)$ in $\mathcal{D}^{\omega}(G)$ is $\mathcal{D}%
_{r}^{\omega}(G)$, and vice-versa. Finally, the centres of $\mathcal{D}%
_{l}^{\omega}(G)$ and $\mathcal{D}_{r}^{\omega}(G)$ coincide with
$\mathcal{D}_{l}^{\omega}(G)\cap\mathcal{D}_{r}^{\omega}(G)$.
\end{corollary}

In the context of Lie transformation groups on analytic manifolds the
corollary above can be phrased as follows: Consider $G$ as a $G$%
-transformation group acting on the analytic manifold $M=G$ to the right and
as a $G^{\prime}$-transformation acting on $M$ to the left; then the
subalgebras of all left (resp.~right)-invariant analytic differential
operators on the analytic manifold $M$ are mutual commutants in $\mathcal{D}%
^{\omega}(M)$. We shall generalize this result to dual transformation groups
acting on analytic manifolds. The simplest case with $G=\mathrm{GL}%
(k,\mathbb{C})$, $G^{\prime}=\mathrm{GL}(n,\mathbb{C})$, $M=$ \{$x\in
\mathbb{C}^{n\times k}:x$ of maximum rank\} was considered in \cite{TT5}. In
this article three more cases are considered. They are more intricate and
\emph{Witt's theorems on quadratic forms and skew-symmetric bilinear forms}
play a crucial role in their resolution. The general case will be considered
in a future publication.

\section{A Duality Theorem for Commutants in \mathversion{bold}$\mathcal{D}%
^{\omega}(M)$}

Let $E=\mathbb{C}^{n\times k}$, $G=\mathrm{SO}(k,\mathbb{C})$, $G^{\prime
}=\mathrm{GL}(n,\mathbb{C})$. Then it is clear that $G^{\prime}$
(resp.~$\mathrm{GL}(k,\mathbb{C})$) is the maximum linear group acting on $E$
by left (resp.~right) multiplication. As a subgroup of $\mathrm{GL}%
(k,\mathbb{C})$, $G$ acts on $E$ by right multiplication and leaves the
nondegenerate symmetric bilinear form $(x,y)\rightarrow\operatorname*{tr}%
(xy^{t})$, $x,y\in\mathbb{C}^{n\times k}$, invariant. If $S(E^{\ast})$ is the
symmetric algebra of all polynomial functions on $E$ then the action of $G$ on
$E$ induces an action of $G$ on $S(E^{\ast})$, denoted by $g\cdot p$, for
$g\in G$, $p\in S(E^{\ast})$. We say that $p\in S(E^{\ast})$ is $G$%
-\emph{invariant} if $g\cdot p=p$, for all $g\in G$. The $G$-invariant
polynomial functions form a subalgebra $J(E^{\ast})$ of $S(E^{\ast})$. If
$J_{+}(E^{\ast})$ is the subset of all elements in $J(E^{\ast})$ without
constant term we let $J_{+}(E^{\ast})S(E^{\ast})$ denote the ideal in
$S(E^{\ast})$ generated by $J_{+}(E^{\ast})$. Recall (\cite[Theorem 2.9A]{We})
that $J_{+}(E^{\ast})S(E^{\ast})$ is generated by the $n(n+1)/2$ algebraically
independent polynomials
\begin{equation}
p_{ij}(x)=\sum_{s=1}^{k}x_{is}x_{js},\qquad1\leq i\leq j\leq n,\quad x\in E,
\label{eq2.1}%
\end{equation}
together with the $(k\times k)$ minors of the matrix $x$ (which are $0$ when
$k>n$). If $P$ is the null cone of the common zeros of polynomial functions in
$J_{+}(E^{\ast})S(E^{\ast})$ then by the Hilbert Nullstellensatz the ideal in
$S(E^{\ast})$ of all polynomial functions which vanish on $P$ is $\sqrt
{J_{+}(E^{\ast})S(E^{\ast})}$. By \cite[Theorem 2.1]{D-TT} the ideal
$J_{+}(E^{\ast})S(E^{\ast})$ is prime if and only if $k>2n$, and the scheme
$P$ which is then equal to the set $\{x\in E:xx^{t}=0\}$ is a complete
intersection, with one open dense orbit.

\emph{Henceforth we assume that} $k>2n$. Let $M=\{x\in E:xx^{t}%
=0,\;\operatorname*{rank}(x)=n\}$ then obviously $M$ is dense in $P$. Since
$(g^{\prime}x)(g^{\prime}x)^{t}=g^{\prime}(xx^{t})(g^{\prime})^{t}$ it follows
immediately that $G^{\prime}$ is the maximum linear group acting on $M$ by
left multiplication. For $\gamma\in\mathrm{GL}(k,\mathbb{C})$ and $p\in
S(E^{\ast})$ define $R(\gamma)p$ by $(R(\gamma)p)(x)=p(x\gamma)$, then clearly
$\gamma$ leaves $M$, and hence $P$, invariant if and only if $R(\gamma
)p_{ij}\in J_{+}(E^{\ast})S(E^{\ast})$ for all $1\leq i\leq j\leq n$.
Obviously, $R(\gamma)p_{ij}$ are quadratic polynomials, and since the $p_{ij}$
form a basis for the quadratic polynomials in $J_{+}(E^{\ast})S(E^{\ast})$ we
have
\begin{equation}
R(\gamma)p_{ij}=\sum_{r,s}C_{rs}^{ij}p_{rs},\qquad1\leq r\leq s\leq n,
\label{eq2.2}%
\end{equation}
where $C_{rs}^{ij}\in\mathbb{C}$ are constants depending on $\gamma$. For
$1\leq i\leq n$, $1\leq t\leq k$ let $x(i,t)$ denote the element of $E$ which
has the $(i,t)$-entry equal to $1$ and all other entries equal $0$. Then an
easy computation shows that
\[
(R(\gamma)p_{ii})(x(i,t))=\sum_{l=1}^{k}\gamma_{tl}^{2}=\sum_{r,s}C_{rs}%
^{ii}p_{rs}(x(i,t))=C_{ii}^{ii}\text{.}%
\]
It follows that $\sum_{l=1}^{k}\gamma_{tl}^{2}=C\in\mathbb{C}$ for all $t$,
and $i$. Choose $x$ of the form $x(i,t)+x(i,t^{\prime})$ with $t\neq
t^{\prime}$ then we obtain
\begin{align*}
(R(\gamma)p_{ii})(x(i,t)+x(i,t^{\prime}))  &  =\sum_{l=1}^{k}(\gamma
_{tl}+\gamma_{t^{\prime}l})^{2}\rule[-15pt]{0pt}{15pt}\\
&  =\sum_{r,s}C_{rs}^{ii}p_{rs}(x(i,t)+x(i,t^{\prime}))\rule[-15pt]%
{0pt}{15pt}\\
&  =2C_{ii}^{ii}=2C.
\end{align*}
Thus
\begin{align*}
\sum_{l=1}^{k}(\gamma_{tl}+\gamma_{t^{\prime}l})^{2}  &  =\sum_{l=1}^{k}%
\gamma_{tl}^{2}+\sum_{l=1}^{k}\gamma_{t^{\prime}l}^{2}+2\sum_{l=1}^{k}%
\gamma_{tl}\gamma_{t^{\prime}l}^{\prime}=2C\rule[-12pt]{0pt}{12pt}\\
&  =C+C+2\sum_{l=1}^{k}\gamma_{tl}\gamma_{t^{\prime}l}\text{. }%
\end{align*}
It follows that we have the system of equations
\begin{equation}
\sum_{l=1}^{k}\gamma_{tl}^{2}=C,\qquad\sum_{l=1}^{k}\gamma_{tl}\gamma
_{t^{\prime}l}=0\qquad\text{for all }t,t^{\prime}=1,\dots,k,\;t\neq t^{\prime
}, \label{eq2.3}%
\end{equation}
or equivalently, $\gamma^{t}\gamma=CI_{k}$.

Since $(\det(\gamma))^{2}=C^{k}$ and $\gamma$ is invertible it follows that
$C\neq0$. Let $\lambda$ be a fixed square root of $C$ and set $g=\frac
{1}{\lambda}\gamma$, then $g^{t}g=I_{k}$, or $g\in\mathrm{O}(k,\mathbb{C})$.
It follows that the largest group acting on $M$ by right multiplication is
$\mathbb{C}^{\ast}\mathrm{O}(k,\mathbb{C})=\{\lambda g:\lambda\in
\mathbb{C},g\in\mathrm{O}(k,\mathbb{C)}\}$, and $G$ is a maximal connected
linear group acting on $M$ by right multiplication.

By Witt's theorem on symmetric bilinear forms (see, e.g., \cite{Ar} and
\cite[Lemma 2.8]{TT1}) $G$ acts \emph{analytically} and \emph{transitively} on
$M$. More precisely, if $x_{0}\in M$ then $M$ is the $G$-orbit of $x_{0}$, and
if $G_{x_{0}}$ is the stability subgroup at $x_{0}$, then it is easy to verify
that $G_{x_{0}}$ is isomorphic to $\mathrm{SO}(k-n,\mathbb{C})$. Moreover, the
map $G_{x_{0}}g\rightarrow x_{0}g$ is an analytic diffeomorphism of $G_{x_{0}%
}\backslash G$ onto $M$ (see, e.g., \cite[Theorem 2.9.4]{Va}). Thus $M$ is an
analytic manifold of complex dimension $nk-n(n+1)/2$ (this also follows from
\cite[Lemma 2.9]{TT1} and the implicit function theorem for analytic functions
\cite[Theorem 2.1.2]{Ho}).

Now let us show that $G^{\prime}$ acts \emph{freely }on $M$, i.e., the
stability subgroup $G_{x}^{\prime}$ is $\{1_{G^{\prime}}\}$ at every $x\in M$.
Indeed, if $x\in M$ then by the assumption $\operatorname*{rank}(x)=n$ there
exist $n$ columns $x_{i_{1}}\cdots x_{i_{n}},i_{1}<\cdots<i_{n}$, of $x$ such
that the $n\times n$ matrix $x_{n}$ formed by them is invertible. So
$g^{\prime}x=x$ implies that $g^{\prime}x_{n}=x_{n}$ or $g^{\prime}=x_{n}%
x_{n}^{-1}=1_{G^{\prime}}$. Now let us recall the definition of differential
operators on a complex manifold $M$ of dimension $m$ (see, e.g., \cite[Chapter
10]{He}).

If $(\varphi,U)$ is a local chart on $M$ with $\varphi(p)=(x_{1}%
(p),\dots,x_{m}(p))\in\mathbb{C}^{m}$, $p\in U$, and $f\in C^{\infty}(M)$, set
$f^{\ast}=f\circ\varphi^{-1}$: $\varphi(U)\subset\mathbb{C}^{m}\rightarrow
\mathbb{C}$. Set $\partial_{i}=\partial/\partial x_{i}$ $(1\leq i\leq m)$ and
if $\alpha=(\alpha_{1},\dots,\alpha_{m})$ is an $m$-tuple of indices
$\alpha_{i}\geq0$ we put $D^{(\alpha)}=\partial_{1}^{\alpha_{1}}\cdots
\partial_{m}^{\alpha_{m}}$. Then a linear transformation $D$: $C_{c}^{\infty
}(M)\rightarrow C_{c}^{\infty}(M)$ is called a \textit{differential}
\textit{operator} on $M$ if the following condition is satisfied:

For each $p\in M$ and each chart $(\varphi,U)$, $p\in U$, there exists a
locally finite set of functions $h_{(\alpha)}\in C^{\infty}(U)$ such that for
each $f\in C_{c}^{\infty}(M)$ with support contained in $U$,%
\begin{equation}
\left\{
\begin{array}
[c]{lll}%
\left[  Df\right]  (p)=\underset{(\alpha)}{\sum}h_{\alpha}\left[  D^{(\alpha
)}f^{\ast}\right]  (\varphi(p)) &  & \text{if }p\in U,\rule[-14pt]%
{0pt}{14pt}\\
\left[  Df\right]  (p)=0, &  & \text{if }p\notin U\text{.}%
\end{array}
\right.  \label{eq2.4}%
\end{equation}
If $M$ is a complex analytic manifold then a differential operator $D$ is
called a \emph{holomorphic} or \emph{complex analytic} \emph{differential
operator} if the functions $h_{(\alpha)}$ in Eq. (\ref{eq2.4}) are holomorphic
(or complex analytic).

By Hilbert's fifth problem $G$ (resp.~$G^{\prime}$) can be equipped with an
analytic structure (see, e.g., \cite{M-Z}) so that they act analytically on
$M$. Let $D^{\omega}(M)$ denote the algebra of (complex) analytic differential
operators on $M$.

Now consider a \emph{global} $G$-\emph{transformation group} on an analytic
manifold $M$ (see, e.g., \cite{Pa} or \cite[2.16]{Va}). Let $\varphi$:
$G\times M\rightarrow M$ $((g,x)\rightarrow g\cdot x,$ $g\in G,$ $x\in M)$
denote the global action of $G$ on $M$. For $x\in M$, $f\in C^{\infty}(M)$ we
define $\left(  \Phi(g)f\right)  (x)=f(g^{-1}\cdot x)$. Let $\frak{g}$ denote
the Lie algebra of $G$ and $\mathcal{U}(\frak{g})$ the universal enveloping
algebra of $G$. Then for $X\in\frak{g}$ and $x\in M$ we define
\begin{equation}
d\Phi(X)_{x}(f):=\left(  \frac{d}{dt}f(\exp(-tX)\cdot x)\right)  _{t=0}
\label{eq2.5}%
\end{equation}
for all $f$ defined and $C^{\infty}$ in a neighborhood of $x$. The map
$X\rightarrow d\phi(X)$ is a homomorphism of $\frak{g}$ into the Lie algebra
of analytic vector fields on $M$. Therefore it extends to a homomorphism
$a\rightarrow\widetilde{d\phi}(a)$, $a\in\mathcal{U}(\frak{g})$, of
$\mathcal{U}(\frak{g})$ into the algebra $\mathcal{D}^{\omega}(M)$ of analytic
differential operators on $M$ (see \cite[Lemma 2.16.1]{Va}), where if
$a=X_{1}\cdots X_{r}$ $(X_{i}\in\frak{g})$ then
\begin{equation}
\widetilde{d\phi}(a)_{x}(f)=\left.  \left(  \frac{\partial}{\partial t_{1}%
}\cdots\frac{\partial}{\partial t}f(\exp(-t_{r}X_{r})\cdots\exp(-t_{1}%
X_{1})\cdot x\right)  \right|  _{0}, \label{eq2.6}%
\end{equation}
where the suffix $0$ indicates that the derivatives are taken when
$t_{1}=\cdots=t_{r}=0$. For our problem we consider the cases when
$\Phi(g^{\prime})=L(g^{\prime})$ and $\Phi(g)=R(g)$, where $\left(
L(g^{\prime})f\right)  (x)=f\left(  (g^{\prime})^{-1}x\right)  $ and
$(R(g)f)(x)=f(xg)$ for $g^{\prime}\in G^{\prime}$, $g\in G$, and $x\in M$. Let
$\widetilde{\mathcal{U}}(\frak{g}^{\prime})$ and $\widetilde{\mathcal{U}%
}(\frak{g})$ denote the images of $\mathcal{U}(\frak{g}^{\prime})$ and
$\mathcal{U}(\frak{g})$ under the maps $\widetilde{dL}$ and $\widetilde{dR}$, respectively.

\begin{definition}
\label{definition2.1}A differential operator $D$ of $\mathcal{D}^{\omega}(M)$
is said to be \emph{right} $($resp.~\emph{left}$)$-\emph{invariant} if
$D(R(g)f)=R(g)(Df)$ $($resp.~$D(L(g^{\prime})f)=L(g^{\prime})(Df))$ for all
$g\in G$ $($resp.~$g^{\prime}\in G^{\prime})$, and for all $f\in C^{\infty
}(M)$.
\end{definition}

\begin{theorem}
\label{theorem2.2}Let $\mathcal{D}_{l}^{\omega}(M)$ $($resp.$~\mathcal{D}%
_{r}^{\omega}(M))$ denote the subalgebra of $\mathcal{D}^{\omega}(M)$ of all
left $($resp.~right$)$-invariant analytic differential operators on $M$. Then

\begin{enumerate}
\item [$\mathrm{(i)}$]$\mathcal{D}_{l}^{\omega}(M)=\widetilde{\mathcal{U}%
}(\frak{g})$ and $\mathcal{D}_{r}^{\omega}(M)=\widetilde{\mathcal{U}}%
(\frak{g}^{\prime}),$

\item[$\mathrm{(ii)}$] the commutant of $\mathcal{D}_{l}^{\omega}(M)$ in
$\mathcal{D}^{\omega}(M)$ is $\mathcal{D}_{r}^{\omega}(M)$, and vice-versa.
Moreover, the centres of $\mathcal{D}_{l}^{\omega}(M)$ and $\mathcal{D}%
_{r}^{\omega}(M)$ coincide with the subalgebra $\mathcal{D}_{l}^{\omega}(M)$
$\cap\mathcal{D}_{r}^{\omega}(M)$.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) Let $X\in\frak{g}^{\prime}$, $g\in G$, $x\in M$ and $f\in C^{\infty}(M)$.
Then
\begin{align*}
dL(X)(R(g)f)(x)  &  =\frac{d}{dt}\left(  (R(g)f)(\exp(-tX)x)\right)
_{t=0}\rule[-12pt]{0pt}{12pt}\\
&  =\frac{d}{dt}(f(\exp(-tX)xg))_{t=0},
\end{align*}
while
\begin{align*}
R(g)(dL(X)f)(x)  &  =(dL(X)f)(xg)\rule[-10pt]{0pt}{10pt}\\
&  =\frac{d}{dt}(f(\exp(-tX)xg))_{t=0}.
\end{align*}
Thus any vector field $\tilde{X}=dL(X)\in\widetilde{\mathcal{U}}%
(\frak{g}^{\prime})$ is right-invariant, and it follows immediately that
$\widetilde{\mathcal{U}}(\frak{g}^{\prime})\subset\mathcal{D}_{r}^{\omega}%
(M)$. Similarly we have $\widetilde{\mathcal{U}}(\frak{g})\subset
\mathcal{D}_{l}^{\omega}(M)$. Let us show that $\mathcal{D}_{r}^{\omega
}(M)\subset\widetilde{\mathcal{U}}(\frak{g}^{\prime})$ and $\mathcal{D}%
_{l}^{\omega}(M)\subset\widetilde{\mathcal{U}}(\frak{g})$.

Let $\mathcal{L}$ denote the Lie algebra of all right-invariant analytic
vector fields on $M$. Then $\mathcal{L}$ is an \emph{involutive}
\emph{analytic system} (see \cite[p.~25]{Va} for the definition), i.e., if $U$
is an open subset of $M$ and $X,Y$ are right-invariant vector fields on $M$
then $[X,Y]$ is (obviously) right-invariant. Then the Global Frobenius Theorem
(see, e.g., \cite[Theorem 1.3.6]{Va}) implies that: given any point of $M$,
there is one and exactly one maximal \emph{integral manifold} $\mathcal{S}$
\emph{of} $\mathcal{L}$ containing that point, i.e., $\mathcal{S}$ is a
connected analytic submanifold of $M$ and for each $y\in\mathcal{S}$,
$\mathcal{L}_{y}$ is the tangent space $T_{y}(\mathcal{S})$. In fact since
$\mathcal{L}$ is an infinitesimal group \cite[Theorem IV, p.~98]{Pa} implies
that $\mathcal{S}$ is the image of a unique connected Lie transformation group
$H$ of $M$. Since $G^{\prime}$ is the largest linear group acting on $M$ by
left multiplication and $dL(\frak{g}^{\prime})\subset\mathcal{L}$ it follows
that $G^{\prime}\subset H$, and hence, $G^{\prime}=H$. It follows that if
$\{X_{1},\dots,X_{n^{2}}\}$ is a basis of $\frak{g}^{\prime}$ then
$\{\tilde{X}_{1},\dots,\tilde{X}_{n^{2}}\}$, where $\tilde{X}_{i}=dL(X_{i})$,
$1\leq i\leq n^{2}$, is a basis for right-invariant analytic vector fields on
$M$. Therefore if $D\in\mathcal{D}_{r}^{\omega}(M)$ then we can find a unique
set of locally finite functions $h_{(\alpha)}\in C^{\omega}(M)$ such that
\begin{equation}
D=\sum_{(\alpha)}h_{(\alpha)}\tilde{X}^{(\alpha)}.\label{eq2.7}%
\end{equation}
Since the $\tilde{X}^{(\alpha)}$ are right-invariant, we have
\begin{equation}
D=D^{r_{g}}=\sum_{(\alpha)}h_{(\alpha)}^{r_{g}}\tilde{X}^{(\alpha)}\qquad(g\in
G).\label{eq2.8}%
\end{equation}
The relations (\ref{eq2.7}) and (\ref{eq2.8}) imply that all the $h_{(\alpha
)}$ are right-invariant, and since $G$ acts transitively on $M$, they must be
all constant. Thus $D\in\widetilde{\mathcal{U}}(\frak{g}^{\prime})$ for all
$D\in\mathcal{D}_{r}^{\omega}(M)$, and hence, $\mathcal{D}_{r}^{\omega
}(M)\subset\widetilde{\mathcal{U}}(\frak{g}^{\prime})$. To show that
$\mathcal{D}_{l}^{\omega}(M)\subset\widetilde{\mathcal{U}}(\frak{g})$ we need
the following

\begin{lemma}
\label{lemma2.3}For each $x\in M$ let $G^{\prime}x=\{g^{\prime}x:g^{\prime}\in
G^{\prime}\}$ denote the orbit of $x$. Let $\mathcal{X}=M/G^{\prime}$ be the
set of all orbits $G^{\prime}x$, $x\in M$, and define $\pi$: $M\rightarrow
\mathcal{X}$ by assigning to each $x\in M$ its orbit $G^{\prime}x$. Then
$(M,\mathcal{X},\pi,G^{\prime})$ is a principal $G^{\prime}$-bundle.
\end{lemma}

\begin{proof}
Define $\gamma$: $M\times G^{\prime}\rightarrow M\times M$ by $\gamma
(x,g^{\prime})=(x,g^{\prime}x)$ and $\Gamma=\gamma(M\times G^{\prime
})=\{(x,g^{\prime}x):x\in M,$ $g\in G\}$. Then since $G^{\prime}$ acts freely
on $M$, $\gamma$ is injective. Now suppose that $\lim_{n\rightarrow\infty
}x_{n}=x$, and $\lim_{n\rightarrow\infty}g_{n}^{\prime}x_{n}=y$ for
$x_{n},x,y\in M$, $g_{n}^{\prime}\in G$. The same argument used in the proof
that $G^{\prime}$ acts freely on $M$ implies that there exists a submatrix
$s[x]$ of $x$ such that $s[x]\in G^{\prime}$. If $s[x_{n}]$ denotes the
corresponding submatrix of $x_{n}$ then clearly $\lim_{n\rightarrow\infty
}s[x_{n}]=s[x]$. So for $n$ sufficiently large we may assume that $s[x_{n}]\in
G^{\prime}$. Then clearly $\lim_{n\rightarrow\infty}s^{-1}[x_{n}]=s^{-1}[x]$
and $\lim_{n\rightarrow\infty}g_{n}^{\prime}s[x_{n}]=s[y]$. By the continuity
of the action of $G^{\prime}$ on $M$ we have $s[y]\in G^{\prime}$. Write
$g_{n}^{\prime}=(g_{n}^{\prime}s[x_{n}])s^{-1}[x_{n}]$ for sufficiently large
$n$ then $\lim_{n\rightarrow\infty}g_{n}^{\prime}=s[y]s^{-1}[x]$. Set
$g^{\prime}=s[y]s^{-1}[x]$, then $g^{\prime}\in G^{\prime}$, $\lim
_{n\rightarrow\infty}g_{n}^{\prime}=g^{\prime}$, and $\lim_{n\rightarrow
\infty}g_{n}^{\prime}x_{n}=g^{\prime}x=y$. Thus $\Gamma$ is closed in $M\times
M$, and $\gamma$ is a homeomorphism of $M\times G$ onto $\Gamma$. Now all the
hypotheses of \cite[Theorem 2.9.10]{Va} are met, and we can conclude that
there exists an analytic structure on $\mathcal{X}$ such that $\pi$ is an
analytic \emph{immersion} (i.e., $(d\pi)_{x}$ is injective for all $x\in M$).
Moreover for each $p\in\mathcal{X}$ we can select an open subset $\mathcal{Y}$
of $\mathcal{X}$ containing $p$ and an analytic diffeomorphism $\xi
_{\mathcal{Y}}=\xi$ of $G^{\prime}\times\mathcal{Y}$ onto $\pi^{-1}%
(\mathcal{Y})$, such that
\[
\xi(h^{\prime}g^{\prime},y)=h^{\prime}\xi(g^{\prime},y)\qquad(g^{\prime
},h^{\prime}\in G,\text{ }y\in\mathcal{Y}).
\]
That is, in other words, $(M,\mathcal{X},\pi,G^{\prime})$ is a principal
$G^{\prime}$-bundle.
\end{proof}

Now let us finish the proof of part (i) of the theorem.

Since $G$ acts analytically and transitively on the analytic manifold $M$
\cite[Lemma 2.9.2]{Va} implies that for each $x\in M$ the map $r$:
$g\rightarrow xg$ $(g\in G)$ is an \emph{analytic submersion} of $G$ onto $M$
(i.e., $(dr)_{g}$ is surjective for all $g\in G$). It follows that if
$\{Y_{1},\dots,Y_{d}\}$ is a basis of $\frak{g}$ then there exists a basis for
analytic vector fields of $M$ of the form $\{\tilde{Y}_{1},\dots,\tilde{Y}%
_{m}\}$, where $m=\dim(M)$, and each $\tilde{Y}_{i}=dR(Y_{j})$ for some $j$,
$1\leq j\leq d$. It follows that every $D\in\mathcal{D}_{l}^{\omega}(M)$ can
be expressed as
\begin{equation}
D=\sum_{(\alpha)}k_{(\alpha)}\tilde{Y}^{(\alpha)},\label{eq2.9}%
\end{equation}
where $\{k_{(\alpha)}\}$ is a set of locally finite analytic functions. Since
the $\tilde{Y}^{(\alpha)}$ are left-invariant, we have%
\begin{equation}
D=D^{l_{g^{\prime}}}=\sum_{(\alpha)}k_{(\alpha)}^{l_{g^{\prime}}}\tilde
{Y}^{(\alpha)}\qquad(g^{\prime}\in G^{\prime}).\label{eq2.10}%
\end{equation}
The relations (\ref{eq2.9}) and (\ref{eq2.10}) imply that all the
$k_{(\alpha)}$ are left-invariant. By Lemma \ref{lemma2.3} a basic open set
$M$ is diffeomorphic to $G^{\prime}\times\mathcal{Y}$ where $\mathcal{Y}$ is
an open subset of $\mathcal{X}$. A typical point in that basic open set is,
for example, of the form $x=(\ast g^{\prime}\ast)\in M$. A function
$k_{(\alpha)}$ that is left-invariant will be independent of the $n^{2}$
variables in the block containing $g^{\prime}$, and since we can let
$g^{\prime}$ occupy any block in the matrix $x$ it follows that $k_{(\alpha)}$
must be constant. Hence $D\in\widetilde{\mathcal{U}}(\frak{g})$, and the proof
of part (i) is completed.

(ii) The proof of part (ii) depends on the following

\begin{lemma}
\label{lemma2.4}Let $D\in\mathcal{D}^{\omega}(M)$ then the following
statements hold.

\begin{enumerate}
\item [$\mathrm{(i)}$]$[dL(X),D]=0$ for all $X\in\frak{g}^{\prime}$ if and
only if $D(L(g^{\prime})=L(g^{\prime})D$ for all $g^{\prime}\in G.$

\item[$\mathrm{(ii)}$] $[dR(Y),D]=0$ for all $Y\in\frak{g}$ if and only if
$D(R(g))=R(g)D$ for all $g\in G$.
\end{enumerate}
\end{lemma}

\begin{proof}
Since both $G^{\prime}$ and $G$ are connected the two statements are similar,
so we will only prove (i). To prove (i) we first consider $g^{\prime
}=g^{\prime}(t)=\exp tX$, $X\in\frak{g}^{\prime}$; then we have
\[
L(g^{\prime})DL((g^{\prime})^{-1})=(\exp(dL(tX))D.
\]
It follows that
\[
L(g^{\prime})D=DL^{\prime}(g)\Longleftrightarrow\lbrack L(X),D]=0.
\]
Since $G^{\prime}$ is connected, $G^{\prime}$ is generated by the image of the
exponential map (cf.~\cite[Cor.~I, p.~6.9]{Go}), i.e., an arbitrary element
$g^{\prime}$ of $G^{\prime}$ can be expressed in the form $g^{\prime}%
=\exp(X_{1})\exp(X_{2})\cdots\exp(X_{r})$, $X_{i}\in\frak{g}^{\prime}$, it
follows from \cite[Prop. 2.10.10]{Na} that
\[
\lbrack dL(X),D]=0,\,\forall\,X\in\frak{g}^{\prime}\Longleftrightarrow
L(g^{\prime})D=DL(g^{\prime}),\,\forall\,g^{\prime}\in G^{\prime}.
\]
\end{proof}

Now part (i) of the theorem and Lemma \ref{lemma2.4} imply that the commutant
of $\mathcal{D}_{l}^{\omega}(M)$ in $\mathcal{D}^{\omega}(M)$ is
$\mathcal{D}_{r}^{\omega}(M)$, and vice-versa. Finally, by definition the
centre of $\mathcal{D}_{l}^{\omega}(M)$ is the subalgebra of elements of
$\mathcal{D}_{l}^{\omega}(M)$ which commute with all elements of
$\mathcal{D}_{l}^{\omega}(M)$. So obviously the centre of $\mathcal{D}%
_{l}^{\omega}(M)$ and similarly the centre of $\mathcal{D}_{r}^{\omega}(M)$
coincide with $\mathcal{D}_{l}^{\omega}(M)\cap\mathcal{D}_{r}^{\omega}(M)$.
\end{proof}

Now let us consider the Lie transformation group $G^{\prime}\times G$ acting
on the analytic manifold $M$, where $G^{\prime}=\mathrm{GL}(n,\mathbb{C})$,
$G=\operatorname*{Sp}(2k,\mathbb{C})$, $M=\{x\in\mathbb{C}^{n\times2k}%
:xs_{k}x^{t}=0,$\ $\operatorname*{rank}(x)=n\}$, $k\geq n$, and
\[
s_{k}=\left[
\begin{array}
[c]{cc}%
0 & -I_{k}\\
I_{k} & 0
\end{array}
\right]
\]
with $I_{k}$ denoting the identity matrix of order $k$.

Recall that $\operatorname*{Sp}(2k,\mathbb{C})$ is the group of all complex
$2k\times2k$ matrices $g$ satisfying $gs_{k}g^{t}=s_{k}$. Then by Witt's
theorem on skew-symmetric bilinear form (see, e.g., \cite{Ar} and \cite[Lemma
1.7]{TT2}) it follows that $G$ acts analytically and transitively by right
multiplication on $M$. Obviously $G^{\prime}$ acts freely by left
multiplication on $M$, and $G$ and $G^{\prime}$ are both connected. Thus we have

\begin{theorem}
\label{theorem2.5}For $k\geq n$ let $G=\operatorname*{Sp}(2k,\mathbb{C})$,
$G^{\prime}=\mathrm{GL}(n,\mathbb{C})$, and $M=\{x\in\mathbb{C}^{n\times
2k}:xs_{k}x^{t}=0,$\ $\operatorname*{rank}(x)=n\}$. Then Theorem
$\ref{theorem2.2}$ holds for this pair of Lie transformation groups acting on
$M$.
\end{theorem}

Finally, let $p$, $q$, and $k$ be positive integers such that $k\geq\max(p,q)$
and consider $G^{\prime}=\mathrm{GL}(p,\mathbb{C})\times\mathrm{GL}%
(q,\mathbb{C})$, $G=\{(g,g^{\checkmark}):g\in\mathrm{GL}(k,\mathbb{C})$,
$g^{\checkmark}=(g^{-1})^{t}\}\approx\mathrm{GL}(k,\mathbb{C})$, and%
\begin{multline*}
M=\left\{  \left[
\begin{array}
[c]{l}%
x_{1}\\
x_{2}%
\end{array}
\right]  \in\mathbb{C}^{(p+q)\times k}:x_{1}\in\mathbb{C}^{p\times k},\text{
}x_{2}\in\mathbb{C}^{q\times k},\right. \\
\left.  x_{1}x_{2}^{t}=0,\;\operatorname*{rank}(x_{1}%
)=p,\;\operatorname*{rank}(x_{2})=q\rule[-10pt]{0pt}{10pt}\right\}  .
\end{multline*}
Then by Witt's theorem on quadratic forms, \cite[Lemma 1.1]{TT3} and
\cite[Theorem 5.1]{TT4}, it follows that $G$ acts analytically and
transitively on $M$ via the action
\[
\left(  \left[
\begin{array}
[c]{l}%
x_{1}\\
x_{2}%
\end{array}
\right]  ,g\right)  \longrightarrow\left[
\begin{array}
[c]{l}%
x_{1}g\\
x_{2}g^{\checkmark}%
\end{array}
\right]  .
\]
Obviously $G^{\prime}$ acts on $M$ freely via the action
\[
\left(  \left(  g_{1}^{\prime},g_{2}^{\prime}\right)  ,\left[
\begin{array}
[c]{l}%
x_{1}\\
x_{2}%
\end{array}
\right]  \right)  \longrightarrow\left[
\begin{array}
[c]{l}%
g_{1}^{\prime}x_{1}\\
g_{2}^{\prime}x_{2}%
\end{array}
\right]  ,\text{\qquad}g_{1}^{\prime}\in\mathrm{GL}(p,\mathbb{C}),\text{
}g_{2}^{\prime}\in\mathrm{GL}(q,\mathbb{C}).
\]
Moreover, both $G$ and $G^{\prime}$ are connected. Thus we have

\begin{theorem}
\label{theorem2.6}Theorem $\ref{theorem2.2}$ holds for the pair of Lie
transformation groups $G$, $G^{\prime}$ acting on the analytic manifold $M$
described above.
\end{theorem}

\section{Conclusion}

In \cite{TT5} we used the duality theorem for commutants in $\mathcal{D}%
^{\omega}(M)$ with $G=\mathrm{GL}(k,\mathbb{C})$, $G^{\prime}=\mathrm{GL}%
(n,\mathbb{C})$, $M=\{x\in\mathbb{C}^{n\times k}:x$ of maximum rank$\}$ to
find the Casimir invariants of the infinite-dimensional group $\mathrm{GL}%
(\infty,\mathbb{C})$. In turn, a set of generators of these Casimir invariants
determine the irreducible unitary representations of the group $U(\infty)$. We
hope that Theorems \ref{theorem2.2}, \ref{theorem2.5}, and \ref{theorem2.6}
will allow us to find the Casimir invariants of some other
infinite-dimensional groups.

\textbf{Acknowledgements}: The author was supported by a Carver Research
Initiative Grant during the preparation of this paper. He also wishes to thank
Ms. Cymie Wehr for her patience in preparing this manuscript.

\begin{thebibliography}{TT-T}
\bibitem[Ar]{Ar}E. Artin, \emph{Geometric Algebra}, Wiley-Interscience
Publication, John Wiley \& Sons, New York (1988), reprint of the 1957 original.

\bibitem[D-TT]{D-TT}O. Debarre and T. Ton-That, \emph{Representations of
}$\mathrm{SO}(k,\mathbb{C})$ \emph{on harmonic polynomials on a null cone},
Proc.\ Amer.\ Math.\ Soc.\ 112 (1991), 31--44.

\bibitem[Go]{Go}R. Godement, \emph{Introduction \`{a} la Th\'{e}orie des
Groupes de Lie}, 2 vols., Publ.\ Math.\ Univ.\ Paris VII, Universit\'{e} de
Paris VII, Paris, 1982.

\bibitem[He]{He}S. Helgason, \emph{Differential Geometry and Symmetric
Spaces}, Academic Press, New York, 1962.

\bibitem[H\"{o}]{Ho}L. H\"{o}rmander, \emph{An Introduction to Complex
Analysis in Several Variables}, third edition, North-Holland, Amsterdam, 1990.

\bibitem[M-Z]{M-Z}D. Montgomery and L. Zippin, \emph{Topological
Transformation Groups}, Robert E. Krieger, Huntington, NY, 1974, reprint of
the 1955 original.

\bibitem[Na]{Na}R. Narasimhan, \emph{Analysis on Real and Complex Manifolds},
North-Holland Mathematical Library, vol.\ 35, North-Holland, Amsterdam, 1985,
reprint of the 1973 second edition.

\bibitem[Pa]{Pa}R. Palais, \emph{A global formulation of the Lie theory of
transformation groups}, Mem.\ Amer.\ Math.\ Soc.\ 22 (1957).

\bibitem[TT1]{TT1}T. Ton-That, \emph{Lie group representations and harmonic
polynomials of a matrix variable}, Trans.\ Amer.\ Math.\ Soc.\ 216 (1976), 1--46.

\bibitem[TT2]{TT2}T. Ton-That, \emph{Symplectic Stiefel harmonics and
holomorphic representations of symplectic groups},
Trans.\ Amer.\ Math.\ Soc.\ 232\textbf{ }(1977), 265--277.

\bibitem[TT3]{TT3}T. Ton-That, \emph{Sur la d\'{e}composition des produits
tensoriels des repr\'{e}sentations irr\'{e}ductibles de} $\mathrm{GL}%
(k,\mathbf{C})$, J.\ Math.\ Pures Appl. (9)\ 56\textbf{ }(1977)\textbf{, }251--261.

\bibitem[TT4]{TT4}T. Ton-That, \emph{Dual representations and invariant
theory}, Representation Theory and Harmonic Analysis (T. Ton-That, et al.,
eds.), Contemp.\ Math., vol. 191, American Mathematical Society, Providence,
1995, pp.\ 205--221.

\bibitem[TT5]{TT5}T. Ton-That, \emph{Poincar\'{e}-Birkhoff-Witt theorems and
generalized Casimir invariants for some infinite-dimensional Lie groups: I},
J.\ Phys.\ A\ 32\textbf{ }(1999), 5975--5991.

\bibitem[TT-T]{TT-T}T. Ton-That and T.D. Tran, \emph{Poincar\'{e}'s proof of
the so-called Birkhoff-Witt theorem}, Rev.\ Histoire Math.\ 5 (1999), 249--284.

\bibitem[Va]{Va}V.S. Varadarajan, \emph{Lie Groups, Lie Algebras, and Their
Representations}, Graduate Texts in Math., vol. 102, Springer-Verlag, New
York, 1984, reprint of the 1974 original, Prentice-Hall, Englewood Cliffs, NJ.

\bibitem[We]{We}H. Weyl, \emph{The Classical Groups: Their Invariants and
Representations}, second edition, Princeton University Press, Princeton, NJ, 1946.
\end{thebibliography}


%%--------------------Here the manuscript ends--------------------------------
\Addresses
\end{document}

--------------5FF949D3B5479B98428809C7--


--------------7103F97F448EB01B61B20797--

