\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{metaplectic group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{DoubleCover}{Double cover $Mp$ of $Sp$}\dotfill \pageref*{DoubleCover} \linebreak
\noindent\hyperlink{CircleExtension}{Circle extension $Mp^c$ of $Sp$}\dotfill \pageref*{CircleExtension} \linebreak
\noindent\hyperlink{CirclExtensionMUc}{Circle extension $MU^c$ of $U$}\dotfill \pageref*{CirclExtensionMUc} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_the_metalinear_group}{Relation to the metalinear group}\dotfill \pageref*{relation_to_the_metalinear_group} \linebreak
\noindent\hyperlink{NonTrivialityOfExtensions}{(Non-)Triviality of extensions}\dotfill \pageref*{NonTrivialityOfExtensions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{DoubleCover}{}\subsubsection*{{Double cover $Mp$ of $Sp$}}\label{DoubleCover}

For $(V,\omega)$ a [[symplectic vector space]], the \emph{metaplectic group} $Mp(V,\omega)$ is the [[Lie group]] which is [[generalized the|the]] [[universal cover|universal]] [[double cover]] of the [[symplectic group]] $Sp(V,\omega)$.

This has various more explicit presentations. One is by [[quadratic Hamiltonians]]: The metaplectic group is that subgroup of the [[quantomorphism group]] of the [[symplectic manifold]] $(V,\omega)$ whose elements are given by [[paths]] of [[Hamiltonians]] that are [[homogeneously quadratic Hamiltonians]] (due to \hyperlink{Leray81}{Leray 81, section 1.1}, see also \hyperlink{RobbinSalamon93}{Robbin-Salamon 93, sections 9-10}). (The more general subgroup given by possibly inhomogeneous [[quadratic Hamiltonians]] this way is the [[extended affine symplectic group]]. The subgroup given by linear Hamiltonians is the [[Heisenberg group]] $Heis(V,\omega)$.)

\hypertarget{CircleExtension}{}\subsubsection*{{Circle extension $Mp^c$ of $Sp$}}\label{CircleExtension}

There is also a nontrivial [[circle group]]-[[group extension|extension]] of the [[symplectic group]], called $Mp^c$. This is the circle extension [[associated bundle|associated]] to the plain metaplectic group $Mp$ \hyperlink{DoubleCover}{above}, via the canonical [[action]] of $\mathbb{Z}_2$ on $U(1)$ (by [[complex conjugation]]): (\hyperlink{ForgerHess79}{Forger-Hess 79 (2.4)})

\begin{displaymath}
\begin{aligned}
    Mp^c(V,\omega) 
     & \coloneqq
    Mp(V,\omega) \times_{\mathbb{Z}_2} U(1)
    \coloneqq 
    ( Mp(V,\omega) \times U(1) )/\mathbb{Z}_2
  \end{aligned}
\end{displaymath}
where the last line denotes the [[quotient group]] by the [[diagonal action]] of $\mathbb{Z}_2$.

(This is in direct [[analogy]] to the group [[Spin{\tt \symbol{94}}c]] and its relation to [[Spin]].)

Again, this has various more explicit presentations.

The \emph{[[Segal-Shale-Weil representation]]} is the following. By the [[Stone-von Neumann theorem]] there is an essentially unique [[irreducible representation|irreducible]] [[unitary representation]] $W$ of the [[Heisenberg group]] $Heis(V,\omega)$. This being essentially unique implies that for each element $g\in Sp(V,\omega)$ of the [[symplectic group]], there is a unique [[unitary operator]] $U_g$ such that for all $v\in V$

\begin{displaymath}
W(g(v)) = U_g W(v) U^{-1}_g
  \,.
\end{displaymath}
The group $Mp^c$ is the [[subgroup]] of the [[unitary group]] of all such $U_g$ for $g\in Sp(V,\omega)$. The map $U_g \mapsto g$ exhibits this as a [[group extension]] by the [[circle group]]

\begin{displaymath}
U(1)\longrightarrow Mp^c(V,\omega) \longrightarrow Sp(V,\omega)
  \,.
\end{displaymath}
e.g. (\hyperlink{RobinsonRawnsley89}{Robinson-Rawnsley 89, p. 19}, \hyperlink{DerezinskiG&#233;rard13}{Dereziski-G\'e{}rard 13, def. 10.24})

Alternatively, there is again a characterization by [[quadratic Hamiltonians]] (\hyperlink{RobinsonRawnsley89}{Robinson-Rawnsley 89, theorem (2.4)}

\hypertarget{CirclExtensionMUc}{}\subsubsection*{{Circle extension $MU^c$ of $U$}}\label{CirclExtensionMUc}

A [[symplectic vector space]] $(V,\omega)$ has a compatible [[complex structure]] $J$. Write

\begin{displaymath}
U(V,J) \hookrightarrow Sp(V,\omega)
\end{displaymath}
for the corresponding [[unitary group]].

\begin{prop}
\label{MUc}\hypertarget{MUc}{}
The restriction ([[pullback]]) of $Mp^c$ \hyperlink{CircleExtension}{above} to this subgroup is denoted $MU^c$ in (\hyperlink{RobinsonRawnsley89}{Robinson-Rawnsley 89, p. 22})

\begin{displaymath}
\itexarray{
      U(1) &=& U(1)
      \\
      \downarrow && \downarrow
      \\
      MU^c(V,J) &\hookrightarrow& Mp^c(V,\omega)
      \\
      \downarrow && \downarrow
      \\
     U(V,J) &\hookrightarrow& Sp(V,\omega)
  }
\end{displaymath}
\end{prop}
(beware the notational clash with the [[Thom spectrum]] [[MU]], which is unrelated).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_the_metalinear_group}{}\subsubsection*{{Relation to the metalinear group}}\label{relation_to_the_metalinear_group}

Inside the [[symplectic group]] $Sp(2n, \mathbb{R})$ in dimension $2n$ sits the [[general linear group]] in dimension $n$

\begin{displaymath}
Gl(n,\mathbb{R}) \hookrightarrow Sp(2n,\mathbb{R})
\end{displaymath}
as the subgroup that preserves the standard [[Lagrangian submanifold]] $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$. Restriction of the metaplectic [[group extension]] along this inclusion defines the [[metalinear group]] $Ml(n)$

\begin{displaymath}
\itexarray{
    Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n, \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    Gl(n, \mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R})
  }
  \,.
\end{displaymath}
Hence a [[metaplectic structure]] on a [[symplectic manifold]] induces a [[metalinear structure]] on its [[Lagrangian submanifolds]].

\hypertarget{NonTrivialityOfExtensions}{}\subsubsection*{{(Non-)Triviality of extensions}}\label{NonTrivialityOfExtensions}

\begin{prop}
\label{}\hypertarget{}{}
The extension

\begin{displaymath}
U(1)\to Mp^c(V,\omega) \to Sp(V,\omega)
\end{displaymath}
is nontrivial (does not give a [[split exact sequence]]).

\end{prop}
(\hyperlink{RobinsonRawnsley89}{Robinson-Rawnsley 89, theorem (2.8)})

\begin{prop}
\label{MUcExtensionSplits}\hypertarget{MUcExtensionSplits}{}
The extension

\begin{displaymath}
U(1)\to MU^c(V,J) \to U(V,J)
\end{displaymath}
is trivial (does give a [[split exact sequence]]).

\end{prop}
(\hyperlink{RobinsonRawnsley89}{Robinson-Rawnsley 89, theorem (2.9)})

\begin{cor}
\label{SpStructureLiftsToMpcStructure}\hypertarget{SpStructureLiftsToMpcStructure}{}
Every [[symplectic manifold]] admits a [[metaplectic structure]].

\end{cor}
(\hyperlink{RobinsonRawnsley89}{Robinson-Rawnsley 89, theorem (6.2)})

\begin{proof}
Since the [[unitary group]] $U(V,J)$ is the [[maximal compact subgroup]] of the [[symplectic group]] (see \href{maximal+compact+subgroup#ExamplesForLieGroups}{here}) every $Sp(V,\omega)$-[[principal bundle]] has a [[reduction of the structure group|reduction]] to a $U(V,J)$-principal bundle. By prop. \ref{MUcExtensionSplits} this reduction in turn lifts to a $MU^c(V,J)$-structure. By def. \ref{MUc} this induces an $Mp^c$-structure under inclusion along $MU^c \hookrightarrow Mp^c$.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[metaplectic structure]]


\item [[metaplectic correction (in geometric quantization)]]


\item [[metaplectic representation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Original references include

\begin{itemize}%
\item [[Andre Weil]], \emph{Sur certains groupes d'op\'e{}rateurs unitaires}, Acta Math. 111: 143--211. (1964).


\item M. Kashiwara; [[Michèle Vergne]], \emph{On the Segal-Shale-Weil Representations and Harmonic Polynomials}, Inventiones mathematicae (1978) (\href{https://eudml.org/doc/142517}{EuDML}, \href{http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0044&divID=LOG_0008&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0044%7C&targetFileName=PPN356556735_0044_LOG_0008.pdf&}{pdf})


\item [[Michael Forger]], Harald Hess, \emph{Universal metaplectic structures and geometric quantization}, Comm. Math. Phys. Volume 64, Number 3 (1979), 269-278. (\href{https://projecteuclid.org/euclid.cmp/1103904723}{EUCLID})


\item [[Jean Leray]], \emph{Lagrangian analysis and quantum mechanics}, MIT press 1981 \href{http://www.maths.ed.ac.uk/~aar/papers/leraybook.pdf}{pdf}



\end{itemize}
Further discussion includes

\begin{itemize}%
\item P. L. Robinson, [[John Rawnsley]], \emph{The metaplectic representation, $Mp^c$-structures and geometric quantization}, 1989


\item [[Joel Robbin]], [[Dietmar Salamon]], \emph{Feynman path integrals on phase space and the metaplectic representation} Math. Z. \textbf{221} (1996), no. 2, 307--335, (\href{http://www.ams.org/mathscinet-getitem?mr=98f:58051}{MR98f:58051}, \href{http://dx.doi.org/10.1007/BF02622118}{doi}, [[RobbinSalamonMetaplectic.pdf:file]]), also in [[Dietmar Salamon]] (ed.), \emph{Symplectic Geometry}, LMS Lecture Note series 192 (1993)


\item [[John Rawnsley]], \emph{On the universal covering group of the real symplectic group}, Journal of Geometry and Physics 62 (2012) 2044--2058 (\href{http://www.maths.ed.ac.uk/~aar/papers/rawnsley.pdf}{pdf})


\item [[Michel Cahen]], [[Simone Gutt]], \emph{$Spin^c$, $Mp^c$ and Symplectic Dirac Operators}, Geometric Methods in Physics Trends in Mathematics 2013, pp 13-28 (\href{http://homepages.ulb.ac.be/~sgutt/spinbiel.pdf}{pdf})


\item Jan Dereziski, Christian G\'e{}rard, \emph{Mathematics of Quantization and Quantum Fields}, Cambridge University Press, 2013



\end{itemize}
[[!redirects metaplectic groups]]

[[!redirects Mp]] [[!redirects Mp{\tt \symbol{94}}c]]



\end{document}