Coherent state triplets and their inner products
Coherent state triplets and their inner products
It is shown that if $\Hb$ is a Hilbert space for a representation of a group $G$, then there are triplets of spaces $({\cal F}_{\Hb}, \Hb, {\cal F}^{\Hb})$, in which ${\cal F}^\Hb$ is a space of coherent state or vector coherent state wave functions and ${\cal F}_\Hb$ is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps ${\cal F}_{\Hb}\to \Hb\to{\cal F}^{\Hb}$ which facilitates the construction of the corresponding inner products. After completion if necessary, the spaces $({\cal F}_{\Hb}, \Hb, {\cal F}^{\Hb})$ become isomorphic Hilbert spaces. It is shown that the inner product for $\Hb$ is often easier to evaluate in ${\cal F}_\Hb$ than in ${\cal F}^\Hb$. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.
math-ph0205034.tex
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\begin{document}
% \draft command makes pacs numbers print
\draft
% repeat the \author\address pair as needed
\title{Coherent state triplets and their inner products}
\author{D.J.~Rowe}
\address{ Department of Physics, University of Toronto,
Toronto, Ontario M5S 1A7, Canada}
\author{J.~Repka}
\address {Department of Mathematics, University of Toronto,
Toronto, Ontario M5S 3G3, Canada}
\date{May 14, 2002} 
%\today
\maketitle
\begin{abstract} It is shown that if $\Hb$ is a Hilbert space for a
representation of a group $G$, then there are triplets of spaces
$({\cal F}_{\Hb}, \Hb, {\cal F}^{\Hb})$, in which ${\cal F}^\Hb$ is a
space of coherent state or vector coherent state wave functions and
${\cal F}_\Hb$ is its dual relative to a conveniently defined measure.
It is  shown also that there is a sequence of maps
${\cal F}_{\Hb}\to \Hb\to{\cal F}^{\Hb}$ which facilitates the construction
of the corresponding inner products.  
After completion if necessary,  the spaces  $({\cal F}_{\Hb}, \Hb,
{\cal F}^{\Hb})$  become isomorphic Hilbert spaces.
It is shown that the inner product for $\Hb$ is 
often easier to evaluate in
${\cal F}_\Hb$ than in  ${\cal F}^\Hb$.
Thus, we obtain integral expressions for the inner
products of coherent state and vector coherent state representations. 
These expressions are equivalent to the algebraic expressions of K-matrix
theory,  but they are frequently more efficient to apply.  
The construction is illustrated by many examples. 

\end{abstract}
% insert suggested PACS numbers in braces on next line
\pacs{02.20.-a, 03.65.Fd}


%\narrowtext
\widetext\date{\today}

\section{Introduction}

Coherent state representations provide a unification of the various
constructions
for inducing representations of groups and Lie algebras.
The  method of induced representations, known originally as the ``Frobenius
method", was introduced into physics by Wigner in two classic papers: in
the first \cite{Wig30} he considered the vibrational spectra of molecules
and, in
the second \cite{Wig39}, he constructed all the irreducible unitary
representations of the Poincar\'e group for positive mass particles.
The theory of induced representations became an important tool in
mathematics and physics following the developments of Mackey \cite{Mackey}.
Other inducing constructions \cite{Knapp} are
given by the Borel-Weil theorem and Harish-Chandra's constructions \cite{HC} of
holomorphic discrete series representations.

Coherent states were first defined by Schr\"odinger \cite{Schr} in 1926 as
minimal
uncertainty wave packets and used to exhibit the classical behavior of
harmonic oscillators within the framework of quantum mechanics.
Such applications has been particularly effective in quantum optics
\cite{glauber}. They were given general group-theoretic definitions by
Klauder 
\cite{klauder}, Perelomov \cite{perelomov},  and Gilmore
\cite{gilmore72}. Coherent state representations were introduced by 
Bargmann
\cite{Barg} and Segal \cite{Segal} and defined more generally by Perelomov
\cite{Perelomov85} and Onofri
\cite{Onofri}. They were subsequently extended to vector-valued
representations \cite{VCS,RRG}. 

It is now known that the inducing constructions are expressed
naturally as coherent state, and vector coherent state, representations
\cite{RR91}. Moreover, the coherent state perspective adds new insights and
has the advantage of being physically intuitive and easy to apply.
In particular, it reveals the correspondence between the unitary representations
of quantum mechanics and the non-unitary representations of classical mechanics
and  is closely related to the methods of geometric quantization  
\cite{BRR}.
Thus, coherent state representation theory is
of considerable practical and pedagogic value.



A scalar coherent state representation is a representation of a group (or Lie
algebra) on a space of complex-valued functions on a coset space.
It is a representation induced from a  one-dimensional representation
of a subgroup.
 The prototype is the well-known Bargmann-Segal
representation \cite{Barg,Segal} of the Heisenberg-Weyl group on a Hilbert
space of entire analytic functions.
A vector coherent state (VCS) representation is a representation of a
group (or Lie algebra) on a space of vector-valued functions.
It is a representation induced from a multi-dimensional representation
of a subgroup.
Such representations have been used widely in the construction of explicit
representations of Lie algebras and Lie groups \cite{HLeBR,RLeBH,LeBR88,LeBR89},
in the construction of  shift tensors \cite{RR95}, and for the computation of
Clebsch-Gordan coefficients for reducing tensor product representations
\cite{RR97,RB00}.



In constructing a coherent state representation, one needs to identify the
 space of wave functions and an appropriate inner product.
When the group is compact or when it is noncompact but the representation
belongs
to the discrete series, there are standard methods for determining a
suitable volume
element relative to which the coherent state wave functions are square
integrable
and the coherent state representation unitary.
The methods are summarized for the holomorphic representations in the book of
Perelomov \cite{Perelomov85} and generalized to VCS representations, by the
methods of Rowe, Rosensteel and Gilmore \cite{RRG}.

Central to the standard construction is the so-called ``resolution of the
identity".
If $T$ denotes a UIR (unitary irreducible representation) 
on a Hilbert space $\Hb$ of a Lie group
$G$ with invariant measure d$\mu$, then, if
$|0\rangle$ is any state in $\Hb$, it follows by Schur's lemma that the
operator
\beq \hat I = \int T^\dagger (g) |0\rangle \langle 0| T(g) \, {\rm d}\mu
(g) \,,\eeq
when defined, is a multiple of the identity.
Thus, with a suitable normalization of the invariant measure,
the inner product of two states of $\Hb$ is expressed
\beq \langle\psi |\varphi\rangle = \int_G \Psi^*(g) \Phi(g) \, {\rm d}\mu
(g) \,,\eeq
where $\Phi$, defined by
\beq \Phi (g) =\langle 0| T(g) |\varphi\rangle \,, \label{eq:1.3}\eeq
is a coherent state wave function for the state $|\varphi\rangle$.
In particular situations, these expressions simplify.
For example,  if $H$
is the isotropy subgroup 
\beq H = \big\{ h\in G \; \big|\; T(h) |0\rangle = |0\rangle \chi(h)
\big\} \,,\eeq 
where $\chi(h)$ is a phase factor, and if $K$ is a
set of coset representives for $H\backslash G$ so that any element $g\in
G$ can be factored $h=hk$, with $h\in H$ and $k\in K$, then the coherent
state wave functions can be restricted to $K$ and their inner products
defined by
\beq \langle\psi |\varphi\rangle = \int_K \Psi^*(k) \Phi(k) \, {\rm d}\mu
(k) \,,\label{eq:5}\eeq 
where d$\mu(k)$ is the measure on $K$ inherited from the invariant
measure on $G$. 



A problem encountered in the practical application of such methods 
is that the integrals are difficult to evaluate 
and in some cases may not converge.
For this reason, an alternative K-matrix theory has been developed
\cite{VCS,RLeBH,RR91}  which  provides numerically 
tractable recursive
algorithms for constructing orthonormal bases for coherent state and VCS
representations.  Such bases, which are needed for the explicit construction of
the matrices of Lie algebra representations, have been determined by K-matrix
methods (reviewed in ref.~\cite{Kmatrix}) for representatives of all
the classical series of Lie algebras
\cite{VCS,HLeBR,RLeBH}. This approach works well for the matrices of Lie
algebra representations. However, for other purposes, particularly when
working at the group level,  it is important to have explicit integral
expressions for inner products. 
Thus,  we consider here integral expressions for the results of K-matrix
theory and show that they are generally easier to derive and use
than the standard  methods.
In this paper, we consider coherent state triplets for scalar
coherent state representations. VCS triplets will be considered in a
following paper. 


The new method makes use of overlap kernels such as
\beq {\cal S}(g_1,g_2)=\langle 0|T(g_1) T^\dagger(g_2) |0\rangle \,.
\label{eq:6}\eeq
Thus, if a state $|\varphi\rangle \in \Hb$ is represented by a wave
function $\varphi$ defined over $G$ such that
\beq |\varphi\rangle = \int T^\dagger (g) |0\rangle \, \varphi(g)\, {\rm
d} v(g) \,,\label{eq:1.5}\eeq
where d$v$ is any convenient measure on the group, then the inner product
of states is given by
\beq \langle\psi |\varphi\rangle = \int\!\!\int \psi^*(g_1)\,
 {\cal S}(g_1,g_2)\, \varphi(g_2)\, {\rm d} v(g_1)\,{\rm d} v(g_2)\,.
\label{eq:1.6}\eeq
Likewise, matrix elements of the representation are given by
\beq \langle\psi |T(g)|\varphi\rangle = \int\!\!\int \psi^*(g_1)\,
 {\cal S}(g_1g,g_2)\, \varphi(g_2)\, {\rm d} v(g_1)\,{\rm d} v(g_2)\,.
\label{eq:1.7}\eeq

The new and old methods are related as follows.
 From Eqs.\ (\ref{eq:1.3}) and (\ref{eq:1.5}), it is seen that the coherent
state wave function $\Phi$ for the state $|\varphi\rangle$ is related to
$\varphi$
by
\beq \Phi(g) = \int {\cal S}(g,g_2)\, \varphi(g_2)\, {\rm d} v(g_2) \,.
\label{eq:1.10}\eeq
Moreover, the inner product of two states $|\psi\rangle$ and
$|\varphi\rangle$ is
given by
\beq \langle\psi |\varphi\rangle = \int\psi^*(g)\,
 \Phi(g)\,{\rm d} v(g)\,.\eeq
Thus, the wave functions $\varphi$ and $\Phi$ are dual to one another
relative to the volume d$v$.
The advantage of using the space of functions dual to the coherent
state wave functions with
inner product given by Eq.\ (\ref{eq:1.6}) is the freedom to choose the
volume d$v$ such that the integrals are easy to define and evaluate.
It will be shown in the following, in a number of representative examples,
that there are natural choices in given situations.

The new techniques greatly facilitate the
use of algebraic and group structures in the solution of physical
problems.  Indeed, they were developed following the discovery, while
computing SU(3)  Clebsch--Gordan coefficients \cite{RB00}, that it is
generally very much easier to compute inner products by means of eqn.\
(\ref{eq:1.6}) than by using the coherent state inner product
(\ref{eq:5}) directly.


\section{Scalar coherent state triplets}

\subsection{Finite-dimensional representations}

Let $T$ be a unitary irrep (irreducible represesentation) of a
real algebraic  Lie group $G$ on a finite dimensional Hilbert space
$\Hb$ with inner product of two vectors
$|\psi\rangle$ and $|\varphi\rangle$ denoted by $\langle
\psi|\varphi\rangle$.   
Assume there is an extension to a representation 
of $G^c$,  the complex extension of $G$, which is compatible with the
natural complex extension of its Lie algebra.
 Let $|0\rangle$ be a fixed vector in $\Hb$ and let
$N$ be a subset of $G^c$, such  that the states
\beq \{T^\dagger(z) |0\rangle\;  ; \;z \in N\} \label{eq:2.1}\eeq
span $\Hb$.
Then, any state $|\psi\rangle \in \Hb$ is uniquely defined by the overlaps
\beq \Psi(z) = \langle  0| T(z) |\psi\rangle \, , \quad z\in N \,
.\label{eq:1.2}\eeq
The complex function $\Psi$ on $N$ is a coherent-state wave
function for the state $|\psi\rangle$ and the space  ${\cal F}^\Hb$ of
such coherent-state wave functions  carries a   representation $\Gamma$ of
$G$ isomorphic to $T$ and defined by
\beq [\Gamma(g) \Psi](z) = \langle 0|T(z) T(g) |\psi\rangle \, , \quad
g\in G \, . \label{eq:2.3}\eeq



For example, if $P\subset G^c$ is the isotropy subgroup
\beq P = \Big\{ p\in G^c \Big| \langle 0|T(p) |\psi\rangle =
\chi(p)\langle 0|\psi\rangle \,,\; \forall |\psi\rangle\in \Hb\Big\} ,\eeq
where $\chi$ is a character of a one--dimensional irrep of $P$, then $N$ 
is often chosen to 
be a set of $P\backslash G^c$ coset representatives or some subgroup
of $G^c$ complementary to $P$ such that any element $g\in G$ can be
factored as $g=pz$ for some $z\in N$ and some
$p\in P$.  


To identify the space ${\cal F}^{\Hb}$ 
of coherent state wave functions, consider first a space ${\cal F} =
{\cal L}^2(N, {\rm d}v)$, defined by a convenient volume element d$v$
such that the map
\beq {\cal F}\to \Hb \, :\,\varphi \to  |\varphi\rangle = \int
{\rm d}v(z)\, T^\dagger (z) |0\rangle \, \varphi(z)  \label{eq:F2H} 
\eeq
is  well-defined and surjective onto $\Hb$.
There is then a sequence of maps  ${\cal F}\to\Hb\to {\cal
F}^{\Hb}$  defined by
\beq \psi \to |\psi\rangle \to \Psi =  \hat{\cal S}\psi \, ,\eeq
with
\beq |\psi\rangle = \int T^\dagger (x)|0\rangle\, \psi(x) \,  {\rm d}
v(x)\eeq 
and
\beq  \Psi(z) =\langle 0 |T(z) |\psi\rangle = \int {\cal S}
(z,x)
\,\psi(x) \, {\rm d}v(x) = [\hat{\cal S}\psi](z) \,,\eeq
where
\beq {\cal S} (z,x) = \langle 0|T(z) T^\dagger (x)
|0\rangle \, . \eeq
Thus, if
\beq {\cal F}_0 = \{ \varphi \in {\cal F}\, |\, \langle \varphi
|\varphi\rangle =0\} \eeq
denotes the kernel of the  map ${\cal F}\to \Hb$  and ${\cal
F}_{\Hb}$ is  the quotient space ${\cal F}/{\cal F}_0$, it
follows that the spaces ${\cal F}_\Hb$ and ${\cal F}^\Hb$ are dual to one
another relative to the volume element d$v$, i.e.,
\beq \langle \varphi |\psi\rangle =
\int \varphi^*(z)\, (\hat{\cal S} \psi)(z) \, {\rm d}v(z)  = 
\int \varphi^*(z)\, \Psi(z) \, {\rm d}v(z) \, .\eeq
It also follows that matrix elements of the representation $T$ are given by
\beq \langle \varphi |T(g) |\psi\rangle =
  \int \varphi^*(z)\, [\Gamma(g) \hat{\cal S}\psi](z)\, {\rm d}v(z)] =
  \int \varphi^*(z)\, [\Gamma(g) \Psi](z)\, {\rm
  d}v(z)\, \label{eq:2.12} 
\eeq
We refer to the triple of spaces $({\cal F}_\Hb, \Hb, {\cal F}^\Hb )$ as
a {\it coherent state triplet\/}.

 

The space ${\cal F}^\Hb$ of coherent state wave functions is clearly
a Hilbert space with respect to an inner product inherited from $\Hb$.
Thus, if the space ${\cal F}_\Hb$ is
equipped with the inner product
\beq (\varphi , \psi) = \langle \varphi |\psi\rangle =
\int\!\!\!\int {\rm d}v(z)\,\varphi^*(z)\, {\cal S}(z,x)\,
\psi(x)\,  {\rm d}v(x) \, ,\eeq
the three spaces $({\cal F}_\Hb, \Hb, {\cal F}^\Hb)$ become
isomorphic to one another. One can then use whichever gives the simplest
expression in any situation.
Examples will be given in the following in which it is easiest to
evaluate inner products in ${\cal F}_\Hb$.
One can then construct an orthonormal basis for ${\cal F}_\Hb$ and
corresponding orthornormal bases for $\Hb$ and ${\cal F}^\Hb$, e.g., if $\{
\psi_\nu\}$ is an orthonormal basis for ${\cal F}_\Hb$, then $\{
\Phi_\nu = \hat {\cal S}\psi_\nu\}$ is an orthonormal basis for ${\cal
F}^\Hb$.

The relationship of the above to K-matrix theory \cite{Kmatrix} is
obtained by noting that, since 
$\hat{\cal S}$ is a positive definite operator on ${\cal F}_{\Hb}$, it
can be factored
\beq \hat {\cal S} =\hat K\hat K^\dagger, \label{eq:1.8}\eeq
and the inner product for ${\cal F}_{\Hb}$  expressed  
\beq  \langle \varphi |\psi\rangle =
\int (\hat K^\dagger\varphi)^*(z)\,  (\hat K^\dagger\psi)(z)
\, {\rm d}v(z) \,  .\eeq
 Thus, if
$\{\psi_\nu\}$ is an orthonormal basis for
${\cal F}_\Hb$, the functions $\{ \varphi_\nu =\hat K^\dagger \psi_\nu\}$
satisfy the equation
\beq
\int \varphi_\mu^*(z)\,  \varphi_\nu(z)
\, {\rm d}v(z) = \delta_{\mu\nu}\,,\eeq
and the corresponding orthonormal coherent state basis is given by
$\{  \Psi_\nu = \hat K\hat K^\dagger\psi_\nu = \hat K \varphi_\nu\}$. 
Hence, if $\{ \varphi_\nu\}$ is an orthonormal basis for ${\cal
F}$, the operator $\hat K$ maps this basis to an orthonormal basis 
$\{\Psi_\nu =  \hat K \varphi_\nu\}$ for
${\cal F}^\Hb$ by annihilating the unwanted states of ${\cal F}$.


The above construction of coherent state wave functions and their
inner products has a powerful and, for practical applications,  very
significant attribute of facilitating the construction of
representations in bases which reduce some desired subgroup $H\subset
G$.   Basically what has to be done is to choose a measure
d$v$ for ${\cal F}$  such that the restriction of
the action of $\Gamma$ to the subgroup $H\subset G$ is unitary on ${\cal
F}$.
Then,  ${\cal S}$ becomes $H$--invariant, in the
sense that
\beq {\cal S} (zh,xh) = \langle 0|T(z)T(h) T^\dagger(h) T^\dagger (x)
|0\rangle = {\cal S} (z,x)\,, \eeq
and $\hat {\cal S}$  block diagonal in a basis for
${\cal F}$ that reduces the unitary representation of $H$.
Specifically, if ${\cal F} = \sum_{\alpha\kappa} {\cal F}^{\alpha\kappa}$
is a decomposition of ${\cal F}$ as a direct sum of irreducible
$H$--invariant subspaces, where $\kappa$ labels an irrep of $H$ and
$\alpha$ distinguishes equivalent irreps, and if
$\{\varphi_{\alpha\kappa\nu}\}$ is an orthonormal basis for
${\cal F}^{\alpha\kappa}$, then
\beq  \langle \varphi_{\alpha\kappa\mu}|\varphi_{\beta\kappa'\nu}\rangle =
\int\!\!\int
\varphi^*_{\alpha\kappa
\mu}(z)
\, {\cal S}(z,x)\,\varphi_{\beta\kappa'\nu}(x) \, {\rm d}v(z)\, {\rm
d}v(x)
 = \delta_{\kappa\kappa'}\delta_{\mu\nu}\,  {\cal
S}_{\alpha\kappa\beta}\,.
\eeq 
It then follows that an orthonormal basis
$\{\varphi_{\alpha\kappa\nu}\}$ for ${\cal F}$ which diagonalizes the
matrix ${\cal S}_\kappa$, i.e., for which
${\cal S}_{\alpha\kappa\beta} = \delta_{\alpha\beta} K^2_{\alpha\kappa}$,
defines a corresponding orthonormal basis $\{
\Psi_{\alpha\kappa\nu} = K_{\alpha\kappa} \varphi_{\alpha\kappa\nu}\}$
of coherent state wave functions.
It is shown explicitly how this is done in the following examples.

The above definitions can be generalized in several ways.
For example,  the condition that $\Hb$ should be irreducible can be relaxed.
The representation $\Gamma$ is then isomorphic to a subrepresentation of $T$.
Moreover, if $|0\rangle$ belongs to an irreducible subspace of $\Hb$, then the
space spanned by $\{ T^\dagger (z) |0\rangle ; z\in N\}$, as well as the
corresponding coherent state representation, is irreducible.






\subsection{Infinite-dimensional representations}

The above construction extends easily to infinite-dimensional
representations. However, although the extension is natural in most situations,
there are technical constraints that must be respected
in the choice of state $|0\rangle$ and subset $N\subset G^c$ to ensure
that the operators $\{ \langle 0|T(z), z\in N\}$ are well defined and
have the properties required to define a coherent state representation.
For simplicity, it will be assumed here that the representation $T$
is irreducible and unitary. 
One concern is that an  infinite-dimensional irrep 
$T$ of a real group $G$ on a Hilbert space $\Hb$ does not
automatically extend to a representation of the complex group $G^c$.
For, whereas a real Lie algebra ${\frak g}$ has a natural extension
to ${\frak g}^c$ by linearity,     
\beq 
T(z \otimes X) = z T(X),  \qquad {\rm for} \quad X \in {\frak g},\,
z \in {\Bbb C}, \label{eq:2B.complexext}
\eeq 
it can happen, for example, that if $\hat X = T(X)$ is the
operator representing an element $X\in {\frak g}^c$, then the action of the
associated group element $e^{\hat X} = T(e^X)$
on a state $|\psi\rangle\in \Hb$
may not converge to a normalizable state $e^{\hat X}|\psi\rangle\in\Hb$.
Nevertheless, as we show by examples in the following, it is frequently
possible to choose a functional $\langle 0|$ and a subset $N\subset G^c$ 
so that the techniques described above can be generalized.  In doing this,  it 
is not possible to expect that the representation $T$ can be
extended so that $N$ acts on all of $\Hb$,  or that the functional
$\langle 0|$  will be normalizable. 
Both these difficulties can be handled by working with a dense subspace 
of $\Hb$.  A good example to keep in mind is the delta function,  which 
is not defined on all of ${\cal L}^2({\Bbb R})$,  but does make sense on 
the dense subspace of continuous functions in ${\cal L}^2({\Bbb R})$.



 Specifically,  what is required is: 
a  dense subspace $\Hb_D\subset \Hb$ of the Hilbert space; 
a subset $N\subset G^c$; 
an  extension of the representation $T$ from $G$ to 
operators $T(z)$, $z \in N$,  which are defined on $\Hb_D$;   
and a linear functional $\langle 0|$ on the space spanned 
by  $\{ T(z) | \psi \rangle :   | \psi \rangle \in \Hb_D,  z \in N \}$.
The extension of $T$ to $N$ must be compatible at the 
level of the Lie algebra with 
eqn.\ (\ref{eq:2B.complexext}) and have 
the property that 
\beq \{ \langle 0| T(z) \,;\, z\in N\} \label{eq:2.28} 
\eeq
are well-defined functionals on $\Hb_D$ and are 
a  {\it sufficient set},  in the sense that any
$|\psi\rangle\in\Hb_D$ is uniquely defined by the overlaps
\beq \Psi(z) = \langle  0| T(z) |\psi\rangle \, , \quad z\in N \,
.\label{eq:1.2b}\eeq
The complex function $\Psi$ on $N$ is then a coherent-state wave
function for the state $|\psi\rangle$.
When the subspace $\Hb_D$ is invariant under
the (real) group action then
the space  ${\cal F}^{\Hb_D}$ of all
such coherent-state wave functions  carries a   representation $\Gamma$ of
$G$, isomorphic to $T$, defined by
\beq [\Gamma(g) \Psi](z) =
\langle 0|T(z) T(g) |\psi\rangle \, , \quad
g\in G \, . \label{eq:2.3b}\eeq
However, sometimes it will be convenient to choose a dense subspace
that is not $G$--invariant.  The group action, defined by eqn.\
(\ref{eq:2.3b}) is then extended to the $G$--invariant completion of
the space of coherent state wave functions relative to the inner product
inherited from $\Hb$. 
 
The final step is to generalize the map (\ref{eq:F2H}).  Since $\langle
0|$  is in general not an element of $\Hb$ (i.e., not
normalizable),  it is necessary to choose  the measure and the space of
functions carefully.   Specifically, 
what is required is: 
a space ${\cal F}$ of functions on $N$ and 
a measure d$v$ on $N$ 
such that, for every $\varphi \in {\cal F}$, the integral 
\beq \int {\rm d}v(z)\, \varphi(z)^* \langle 0|T (z)     \eeq 
converges to an element of 
$\Hb_D^*$, the dual of the dense subspace $\Hb_D$ of $\Hb$ for
which coherent  state wave functions are defined.   
This integral then gives a map
\beq {\cal F} \to \Hb_D^* \, 
:\,\varphi \to  \langle\varphi| = \int
{\rm d}v(z)\, \varphi(z)^* \langle 0|T (z)  \label{eq:2.32}   \eeq
which is  well-defined for all elements of ${\cal F}$.  
We also require that  the image of this map contains $\Hb_D$  and write
${\cal F}_D$ for the space of all elements of ${\cal F}$  which the map
takes to elements of $\Hb_D$.

 
There is now a sequence of maps
${\cal F}_D \to \Hb_D \to {\cal F}^{\Hb_D} $ for which
\beq \psi \to |\psi\rangle \to \Psi = \hat {\cal S}\psi \,,\eeq
with
\beq|\psi\rangle = \int T^\dagger(x) |0\rangle\, \psi(x)\, {\rm d}v(x)
\eeq 
and
\beq \Psi(z) = [\hat {\cal S}\psi](z) = \int {\cal S}(z,x) \psi (x) \,
{\rm d}v(x) \,,
\eeq
where
\beq  {\cal S}(z,x) \psi (x) = \langle 0| T(z) T^\dagger(x)|0\rangle \,.
\eeq

 Let
\beq {\cal F}_0 = \{ \varphi \in {\cal F}\, 
|\, \langle \varphi |\varphi\rangle
= 0 \} \eeq
denote the kernel of the map ${\cal F}\to\Hb^*_D$ 
given in 
(\ref{eq:2.32}), i.e., ${\cal F}_0$ is the set of
all functions in ${\cal F}$ that map to functionals that are zero on all of
$\Hb_D$, and define ${\cal F}_\Hb$ to be the completion of the
space ${\cal F}_D/{\cal F}_0$,  relative to the inner product
\beq (\varphi , \psi) = \langle \varphi |\psi\rangle =
\int\!\!\!\int {\rm d}v(z)\,\varphi^*(z)\, {\cal S}(z,x)\,
\psi(x)\,  {\rm d}v(x) \, ,\label{eq:2.7}\eeq
where
\beq {\cal S} (z,x) = \langle 0|T(z) T^\dagger (x) 
|0\rangle \, . \eeq
Then  ${\cal F}_\Hb$ is a Hilbert space isomorphic to $\Hb$ and, if
${\cal F}^\Hb$ is the completion of the space of coherent state wave
functions relative to the inner product inherited from $\Hb$, it too is
isomorphic to $\Hb$.

Finally observe that, if the map ${\cal F}\to\Hb_D^*$ takes
$\varphi \to \langle\varphi|$  and if a state
$|\psi\rangle\in\Hb_D$ has coherent state wave function
$\Psi\in {\cal F}^{\Hb_D}$, then
\beq \langle \varphi |\psi\rangle = 
\int \varphi^*(z)\, (\hat{\cal S} \psi)(z) \, {\rm d}v(z)  =
\int \varphi(z)^* \Psi(z)\, {\rm d}v(z)\,.\eeq  
It follows that 
${\cal F}_\Hb$ and ${\cal F}^\Hb$ are in duality 
relative to the volume element d$v$.
It also follows that matrix
elements of the representation $T$ are given by
\beq \langle \varphi |T(g) |\psi\rangle =
  \int\varphi^*(z)\, [\Gamma(g) \hat{\cal S}\psi](z)\, {\rm d}v(z)]=
  \int \varphi^*(z)\, [\Gamma(g) \Psi](z)\, {\rm
  d}v(z)\, .\label{eq:2.12b}\eeq
We again refer to the triple of spaces $({\cal
F}_\Hb, \Hb, {\cal F}^\Hb )$ as a {\it coherent state triplet\/}.  



\section{Simple  examples} 


The following well-known examples are developed in some detail as 
prototypes of four classes of application of the triplets method:
holomorphic representations of compact and non-compact semisimple Lie
groups; non-holomorphic representations induced from a non-canonical
subgroup; and Mackey type representations \cite{Mackey} of a
semidirect product group induced from an Abelian normal subgroup.

 \subsection{SU(2)}


Let $\{ J_0, J_+,J_-\}$ with commutation relations
\beq [J_+,J_-] = 2J_0 \, ,\quad [J_0,J_\pm] = \pm J_\pm \, ,\eeq
be a standard basis for the Lie algebra $\frak{su}(2)^c$.
Let $T^j:J_k \to \hat J_k$ be a UIR (unitary irreducible representation)
of SU(2) of angular momentum $j$ on a (finite dimensional)
Hilbert space $\Hb^j$ with (normalized) highest weight state
$|jj\rangle$ satisfying
\beq \hat J_+ |jj\rangle = 0 \, ,\quad \hat J_0  |jj\rangle = j|jj\rangle \, .
\eeq

With fixed state $|0\rangle$ set equal to the highest weight state
$|jj\rangle$, a natural choice of $N\subset {\rm SU(2)}^c$ is the
nilpotent subgroup
\beq N = \{ e^{zJ_+}; z\in \Cb\} \eeq
and a convenient measure d$v$ on $N$ is the Bargmann measure,
relative to which the polynomial functions
 $\{ \varphi_\nu\}$, with $\varphi_\nu (z) = z^\nu / \sqrt{\nu!}$,
satisfy the orthogonality relationship
\beq \int \varphi_\mu^*(z)\, \varphi_\nu(z)\, {\rm d}v(z) = \delta_{\mu\nu} \, .
\eeq
The space ${\cal F} = {\cal L}^2(N,{\rm d}v)$ is then the Bargmann
space
\cite{Barg} of entire analytic functions of $z$ that are square
integrable relative to d$v$.

The overlap kernel, ${\cal S}(z,x)  = 
\langle 0| e^{z\hat J_+} e^{x^*\hat J_-} |0\rangle$,  is evaluated by
noting that almost any SL$(2,\Cb)$ matrix has Gauss
factorization
\beq \pmatrix{a&b\cr c&d\cr} = \pmatrix{1&0\cr x&1\cr}
\pmatrix{a&0\cr 0&a^{-1}} \pmatrix{1&z\cr 0&1\cr}
= e^{xJ_-}\, a^{2J_0}\, e^{zJ_+} \label{eq:Gauss}\eeq
with $z =b/a$ and $x =c/a$.
It follows that,  as elements of SL$(2, \Cb)$,
\beq e^{zJ_+} e^{x^*J_-} = \pmatrix{ 1+zx^* & z\cr x^* & 1\cr}
= e^{\alpha x^* J_-}\, (1+zx^*)^{2J_0}\, e^{\alpha zJ_+} \,,\eeq
with $\alpha = (1+zx^*)^{-1}$.
Thus, the overlap kernel is given by
\beq {\cal S}(z,x) = \langle 0|(1+zx^*)^{2\hat J_0}|0\rangle
=  (1+zx^*)^{2j}  \eeq
and has expansion
\beq {\cal S}(z,x) = \sum_{\nu = 0}^{2j} {(2j)!\over (2j-\nu)! \nu!} \,
(zx^*)^\nu = \sum_{\nu = 0}^{2j} K_\nu^2 \varphi_\nu (z)
\varphi^*_\nu (x) \,,
\eeq
with
\beq K_\nu^2 =  {(2j)!\over (2j-\nu
)!} \, .\eeq


The space ${\cal F}_j$ is now defined as the Hilbert
space of holomorphic functions of
a complex variable $z$ with inner product
\beq \langle\varphi |\psi\rangle  = \int\!\!\!\int \psi^*(z)\,{\cal
S}(z,x)\,\varphi(x)\, {\rm d}v(z)\, {\rm d}v(x) \, .\eeq
With respect to this ${\cal F}_j$  inner product, the overlaps of the
functions $\{\varphi_\nu\}$ are given by
\beq \langle \varphi_\mu|\varphi_\nu\rangle =
 \delta_{\mu\nu}\times  \cases{K_\nu^2 & for $\nu\leq 2j$ \,, \cr
0 & for $\nu > 2j$\,.}\eeq
 Thus, with the substitution $\nu = j-m$ (to conform to
convention),  
and choosing the positive square root of $ K_\nu^2 $,  
an orthonormal basis for ${\cal F}_j$ is given by the set of
holomorphic functions
\beq \left\{\psi_{jm}(z) = {1\over K_{j-m}}\, \varphi_{j-m}(z) =
\sqrt{ (j+m)!\over (2j)! (j-m)!}\, z^{j-m} \, ;\;\; m = -j,
\ldots , j\right\} \, . \label{orthonormalbasis} \eeq

The map ${\cal F}_j \to {\Hb^j}$, in which 
\beq \psi_{jm} \to |jm\rangle = \int
e^{z^*\hat J_-}|0\rangle\, \psi_{jm}(z)\,  {\rm d} v(z)\, , 
\label{firstmap}
\eeq
and the coherent-state map $\Hb^j \to {\cal F}^j$,  given by
\beq |jm\rangle \to \Psi_{jm}(z) =  \langle 0|e^{z\hat J_+}|jm \rangle
= \int \langle 0|e^{z\hat J_+} e^{x^*\hat J_-}|0\rangle \, \psi_{jm}(x) \,
{\rm d} v(x) \, ,  
\label{secondmap} 
\eeq 
are isomorphisms.  
Thus, orthonormal bases for  ${\Hb^j}$ and ${\cal F}^{j}$,
corresponding to the basis 
(\ref{orthonormalbasis}) for ${\cal F}_j$, are given by 
\beqa &\displaystyle\left\{ |jm\rangle = {1\over K_{j-m}}\,
|\varphi_{j-m}\rangle =
\sqrt{ (j+m)!\over (2j)! (j-m)!}\, (\hat
J_-)^{j-m}|0\rangle \, ;\;\; m = -j, \ldots , j\right\} ,&\\
&\displaystyle\left\{\Psi_{jm}(z) = K_{j-m}\, \varphi_{j-m}(z)=
\sqrt{ (2j)!\over  (j+m)!(j-m)!}\, z^{j-m} \, ;\;\;
m = -j,
\ldots , j\right\} ,& \label{eq:4.64}
\eeqa
respectively. 


The coherent state representation $\Gamma^j$ of SU(2) on ${\cal F}^j$, is
defined by
\beq [\Gamma^j(g) \Psi](z) = \langle 0|e^{z\hat J_+} T^j(g) |\psi\rangle
\eeq (cf.\ eqn.\ (\ref{eq:2.3})).
 From the identity
\beq e^{zJ_+} \pmatrix{ a&b\cr -b^*&a^*\cr} =
\pmatrix{ a-zb^* & b+za^* \cr -b^* & a^*\cr}
= e^{-\alpha b^* J_-}\, (a-b^*z)^{2J_0}\, e^{\alpha (b+ a^*z)J_+}
\eeq
with $\alpha = (a-b^*z )^{-1}$, the action of $\Gamma^j (g)$ is determined
to be
\beq [\Gamma^j(g) \Psi](z) = (a-b^*z)^{2j} \Psi (z\cdot g) \,,
\label{eq:4,63}\eeq
with
\beq g = \pmatrix{ a&b\cr -b^*&a^*\cr} \, ,\quad z\cdot g = {b+a^*z\over 
  a-b^*z} \, .\eeq 

Finally, since the inner product can be written in the form
$\langle \psi |\varphi\rangle = \int\psi^*(z) \,\Phi (z)
\, {\rm d}v(z)$, matrix elements
of the representation $T$ are given  by
\beq \langle \psi |T^j(g)|\varphi\rangle = \int
\psi^*(z)\, [\Gamma^j(g)\Phi] (z)\, {\rm d}v(z)\, ,
\label{eq:4.37a}\eeq
where $\varphi$, $|\varphi\rangle$, and $\Phi$ are related by the
${\cal F}_j \to {\Hb^j}\to {\cal F}^j$ maps given by
eqns.~(\ref{firstmap}) and (\ref{secondmap}). 
For example, setting $c=\cos \beta/2$ and $s=\sin\beta/2$ and
\beq g = \pmatrix{c&-s\cr s&c\cr} \, ,\eeq
one obtains, from eqn.\ (\ref{eq:4.37a}),  an
expression for the Wigner function
\beqa
d^j_{mn}(\beta)&=& \langle jm |T^j(g) |jn\rangle \nonumber\\
&=& \int \psi^*_{j-m}(z)\,[\Gamma^j(g)\, \Psi_{j-n}](z)\,
{\rm d}v(z) \nonumber\\
&=& {K_{j-n}\over K_{j-m}}
\int \varphi^*_{j-m}(z)\,[\Gamma^j(g) \varphi_{j-n}](z)\, {\rm d}
v(z) \nonumber\\
&=& {K_{j-n}\over K_{j-m}}
\int \varphi^*_{j-m}(z)\, (c+sz)^{j+n}\, \varphi_{j-n}(-s+cz)\, {\rm d}
v(z)
\, ,
\eeqa
which integrates to the standard result. 



The coherent state representation 
of the $\frak{su}$(2) Lie algebra, defined by
\beq [\Gamma^j (J_k)\Psi](z) = \langle jj| e^{z\hat J_+} \hat J_k
|\psi\rangle \, ,\eeq
yields the known expressions
\beq \Gamma^j (J_+) = {d\over dz} \, ,\quad \Gamma^j(J_0) = j - z {d\over
dz}\, ,\quad \Gamma^j (J_-) = 2jz - z^2{d\over dz}\, .\eeq 
Acting on the orthonormal coherent state basis wave functions of eqn.\
(\ref{eq:4.64}), they give the expected results
\beqa  &\Gamma^j(J_0)\Psi_{jm} = m\Psi_{jm} &\nonumber\\
&\Gamma^j(J_\pm)\Psi_{jm} =
 \sqrt{(j\mp m)(j\pm m+1)}\ \Psi_{j,m\pm 1}\, . &
\eeqa


The above expressions may be compared to those of the standard  holomorphic
representations of SU(2). The latter are identical to the
coherent state representations but specify the inner product for the
Hilbert space ${\cal F}^j$ by
\beq \langle \psi |\varphi\rangle = {2j+1\over \pi}
 \int  {\Psi^*(z)\Phi(z)\over (1+|z|^2)^{2j+2}}\, {\rm d}^2z \, , 
\eeq
where the integral is over the complex plane with 
${\rm d}^2 z = {\rm d}x\,{\rm d}y$ and $z=x+iy$; 
cf.\ ref.\ \cite{Perelomov85} and Appendix A.
Thus, whereas ${\cal F}^j$ is  a space of holomorphic functions
with norm defined by this inner product, the
space ${\cal F}_j$, isomorphic to ${\cal F}^j$, is the space of holomorphic
functions with inner product
\beq \langle \psi |\varphi\rangle = \int\!\!\!\int \psi^*(z)\,
(1+zx^*)^{2j}\,  \varphi(x)\,{\rm d} v(z)\,{\rm d} v(x) \,.\eeq
The latter integrals are easier to evaluate and by regarding 
${\cal F}_j$ and ${\cal F}^j$ as mutual duals, we have the still simpler
hybrid expression
\beq \langle \psi |\varphi\rangle = \int\psi^*(z) \,\Phi (z)
\, {\rm d}v(z)\,.\eeq





\subsection{SU(1,1)}


 

Let $\{ J_0, J_+,J_-\}$ with commutation relations
\beq [J_-,J_+] = 2J_0 \, ,\quad [J_0,J_\pm] = \pm J_\pm \, ,\eeq
be a standard basis for the Lie algebra $\frak{su}(1,1)^c$.
Let $T^\lambda$ be a UIR (unitary irreducible representation) of
SU(1,1) of lowest weight $\lambda$ on an infinite dimensional 
Hilbert space
$\Hb^\lambda$, where $\lambda$ is a positive integer, defined such
that, if $|0\rangle$ is the lowest weight state, then
\beq \hat J_- |0\rangle =0\,,\quad 2\hat J_0 |0\rangle = \lambda
|0\rangle \,, \label{lwaction}\eeq
where $\hat J_k = T^\lambda (J_k)$.

Gauss factorization of almost any SL$(2,\Cb)$ matrix gives
\beq \pmatrix{a&b\cr c&d\cr} = \pmatrix{1&x\cr 0&1\cr}
\pmatrix{d^{-1}&0\cr 0&d} \pmatrix{1&0\cr z&1\cr}
= e^{-xJ_+}\, (d)^{-2J_0}\, e^{zJ_-} \label{eq:Gauss2} \,,\eeq
with $x=b/d$ and $z=c/d$. 
This implies
that, if $g$ is the SU(1,1) matrix
\beq g =  \pmatrix{a&b\cr b^* &a^* \cr} \quad {\rm with}\;\; aa^* -bb^*
    = 1 \, ,\eeq
then $T^\lambda (g)$ can be expanded
\beq T^\lambda (g) =  e^{-x\hat J_+}\, (a^*)^{-2\hat J_0}\, 
   e^{z\hat J_-}  \eeq
and
\beq \langle 0| T^\lambda (g)|\psi\rangle = (a^*)^{-\lambda}\,
\langle 0| e^{z\hat J_-}|\psi\rangle \,,
\eeq 
with $x=b/a^*$ and $z =b^*/a^*$.
Thus, a natural choice of  
$N\subset {\rm SU(1,1)}^c \cong {\rm SL(2,\Rb)}^c$ is the 
Borel subgroup
\beq N = \{ e^{zJ_-}\,;\, z\in \Cb \}\,.\eeq
With this choice of $N$, the functionals
\beq \{ \langle 0| e^{z\hat J_-} ; z\in \Cb\}
\eeq
are well--defined on the dense subspace $\Hb_D$ of finite
linear combinations of the weight  vectors (the so-called $K$-finite
vectors).  
Thus, a coherent state wave function for a state
$|\psi\rangle\in \Hb_D$ is defined by
\beq \Psi (z) = \langle 0| e^{z\hat J_-} |\psi\rangle \,.\eeq
These coherent state wave functions are polynomials in $z$.

Let ${\cal F}$  denote  the Bargmann Hilbert space 
of holomorphic functions over $\Cb$ with orthonormal (polynomial)
basis $\{\varphi_\nu ; \nu = 0,1,2, \ldots\}$, where
$\varphi_\nu (z)  = z^\nu / \sqrt{\nu !}$.
Now observe that the map 
\beq  \varphi \to \langle \varphi | = \int {\rm d}v(z) \, \varphi(z)^*\,
\langle 0| e^{z\hat J_-}
\eeq 
takes any function $\varphi\in {\cal F}$ to a well--defined functional on
$\Hb_D$. The image of this ${\cal F}\to \Hb_D^*$ map contains $\Hb_D$. 
Moreover ${\cal F}_D\subset {\cal F}$ is identified as the dense subspace
of finite polynomials.
 Thus, if ${\cal F}^{\lambda} = {\cal F}^{\Hb_D}$ is the space of coherent
state wave functions for states in $\Hb_D$, then composition of the maps
${\cal F}_D \to \Hb_D$ and $\Hb_D \to {\cal F}^{\Hb_D}$  gives
the map
\beq \hat{\cal S} \, :\, {\cal F}_D \to {\cal F}^{\Hb_D} \, ; \, \psi \mapsto
\Psi = \hat{\cal S}\psi \, ,\eeq
with
\beq \Psi (z)  = \int  {\cal S} (z,x) \,\psi(x) \, {\rm d}v(x) \, ,\eeq
and
\beq {\cal S}(z,x) = \langle 0|e^{z\hat J_-} e^{x^*\hat J_+}|0
\rangle  \, . \label{eq:Ssu11def}\eeq



\medskip
\noindent {\it Claim:}  The overlap integral ${\cal S}(z,x)$ is
the distribution on ${\cal F}_D$ 
\beq {\cal S}(z,x) =  \sum_\nu^\infty K_\nu^2 \varphi_\nu (z)
\varphi^*_\nu (x)
\,, \label{eq:3.series} \eeq
with
\beq K^2_\nu =  {(\lambda + \nu -1)!\over (\lambda-1)!}\,.\eeq

\medskip
\noindent {\it Proof:}
First observe that, from its definition (\ref{eq:Ssu11def}), 
${\cal S}(z,x)$ is holomorphic in $z$ and $x^*$. 
Thus, it is sufficient to prove the claim for $z$ and $x^*$ lying
in some open domain of $\Cb$.
In fact, the ${\rm SU}(1,1)^c={\rm SL}(2,\Cb)$ expansion
\beqa e^{zJ_-} e^{x^*J_+} &=& \pmatrix{1&0\cr z&1\cr} 
\pmatrix{1 &-x^*\cr 0&1\cr} 
=\pmatrix{ 1&-x^*\cr z&1-zx^*\cr} \label{gausssu11} \\
&=& e^{\alpha x^*J_+} \, (\alpha)^{2J_0} \, e^{\alpha zJ_-} ,
\eeqa
with $\alpha = (1-zx^*)^{-1}$, together with the assumption that
\beq T^\lambda ( e^{\alpha x^*J_+} \, (1-zx^*)^{-2J_0} \, e^{\alpha zJ_-}) =
e^{\alpha x^*\hat J_+} \, (1-zx^*)^{-2\hat J_0} \, e^{\alpha z\hat J_-} \, ,
\label{eq:91}\eeq
 gives the correct result for $|zx^* | < 1$:
\beq {\cal S}(z,x)=  \langle 0| (1-zx^*)^{-2\hat J_0}|0\rangle =
  (1-zx^*)^{-\lambda} = \sum_\nu^\infty K_\nu^2 \varphi_\nu (z)
  \varphi^*_\nu (x)\,, \quad {\rm for}\;\; |zx^*| <1 \,. \label{eq:92}
\eeq 
However, this result could 
be  questioned because 
the representation $T^\lambda$  of SU(1,1) does not in general extend to
all of SL$(2,\Cb)$,  so applying it to elements of 
the complex group is not, in general, justified; an
infinite-dimensional irrep of SL$(2,\Cb)$ does not remain irreducible on
restriction to SU(1,1).
Nevertheless, eqn.\ (\ref{eq:92}) is valid for $|z|^2<1$ and
$|x|^2<1$, because there exist matrices
\beq Z_+(z^*) = \pmatrix{ a^{-1} & az^*\cr 0& a\cr} \,,\quad
Z_-(x) = \pmatrix{ b^{-1} & 0\cr -b^*x& b\cr} \,, \eeq
with $|a|^2 = (1-|z|^2)^{-1}$ and $|b|^2 = (1-|x|^2)^{-1}$,
such that
\beqa &Z_+(z^*) e^{zJ_-} =\pmatrix{ a^{-1} & a^*z^*\cr 0& a\cr}
\pmatrix{1 &0\cr z &1\cr} = \pmatrix{ a^* & a^*z^*\cr az &a\cr} \, ,&\\
&e^{x^*J_+}Z_-(x)  =
\pmatrix{1 &-x^*\cr 0 &1\cr}\pmatrix{ b^{-1} & 0\cr -b^*x& b\cr} 
= \pmatrix{b^* & -bx^*\cr -b^*x&b\cr} \, ,&\eeqa 
are elements of the real Lie group SU(1,1).
Factorization of the product of real SU(1,1) group elements
\beq Z_+(z^*)\, e^{zJ_-}e^{x^*J_+}\, Z_-(x)
=Z_+(z^*)\,e^{\alpha x^* J_+}\, (\alpha)^{2J_0}\, e^{\alpha
zJ_-}\, Z_-(x)\,,\eeq 
with $\alpha = (1-zx^*)^{-1}$, then gives the result (\ref{eq:92})
and confirms the the validity of the claim.
\hfill [QED] 

\medskip




For polynomial functions $\psi$,  $\varphi$,  the 
inner product
\beq \langle \psi |\varphi\rangle = \int\!\!\!\int \psi^*(z)\, {\cal S}(z,x)
\,\varphi (x) \, {\rm d}v(z)\, {\rm d}v(x) 
= \sum_\nu K_\nu^2 \int \psi^*(z)\varphi_\nu(z) \, {\rm d}v(z) 
\int \varphi_\nu^*(x) \varphi (x) \,\, {\rm d}v(x) \,
\label{eq:su11InP}\eeq  
is now well-defined.
In particular, the wave functions $\{ \varphi_\nu\}$ have inner products
$\langle \varphi_\mu |\varphi_\nu\rangle  = \delta_{\mu\nu}
K^2_\nu$ and norms that are  non-vanishing for all integers
$\nu \geq 0$. Thus, the Hilbert space ${\cal F}_\lambda$, isomorphic to 
$\Hb^\lambda$, is  
the completion of the space of holomorphic polynomials with respect
to the  inner product (\ref{eq:su11InP}).     Relative to this inner
product, the normalized functions
\beq \left\{ \psi_{\lambda\nu} (z) ={1\over K_\nu}  \varphi_\nu (z)
= \sqrt{(\lambda-1)!\over (\lambda+\nu -1)! \nu!}\ z^\nu \; ;\;\;
\nu = 0,1,2, \ldots  \right\} \eeq
form an orthonormal basis for ${\cal F}_\lambda$.



The (densely-defined) map ${\cal F}_\lambda \to \Hb^\lambda$, with
\beq \psi_{\lambda\nu} \to |\lambda\nu\rangle = \int
 e^{z^*\hat J_+}|0\rangle \, \psi_{\lambda\nu}(z) \, {\rm d} v(z) \eeq
and the coherent state map $\Hb^\lambda \to {\cal
F}^{\lambda}; |\lambda\nu\rangle \to \Psi_{\lambda\nu}$, with
\beqa \Psi_{\lambda\nu} (z) =  \langle 0|e^{z\hat J_-}|\lambda\nu
\rangle &=& \int \langle 0|e^{z\hat J_-} e^{x^*\hat J_+}|0\rangle \,
\psi_{\lambda\nu} (x)
\, {\rm d} v(x) \nonumber\\
&=& \int {\cal S}(z,x)\, \psi_{\lambda\nu} (x) \, {\rm d}v(x) \,,\eeqa
define corresponding orthonormal bases
\beqa &\displaystyle \left\{|{\lambda\nu}\rangle = {1\over K_\nu}
|\varphi_\nu\rangle =
\sqrt{(\lambda-1)!\over (\lambda+\nu -1)! \nu!}\ (\hat J_+)^\nu
|0 \rangle \;
;\;\;\nu = 0,1,2, \ldots \right\}\, ,&  \label{su11basis} \\
&\displaystyle\left\{ \Psi_{\lambda\nu} (z) = {K_\nu}\varphi_\nu (z) =
\sqrt{(\lambda+\nu -1)!\over (\lambda -1)! \nu!}\ z^\nu \; ;\;\;
\nu = 0,1,2, \ldots  \right\}\, ,& \label{eq:4.104}\eeqa
for $\Hb^\lambda$ and ${\cal F}^\lambda$, respectively.


Proceeding as for SU(2),  
we find that the coherent state representation $\Gamma^\lambda$ of
SU(1,1) acts on
$\Psi (z) \in {\cal F}^\lambda$ by
\beq [\Gamma^\lambda(g) \Psi](z) = (a^*+bz)^{-\lambda} \Psi (z\cdot g)
\,  ,\eeq with
\beq g = \pmatrix{ a&b\cr b^*&a^*\cr} \, ,\quad 
z\cdot g ={b^*+az\over a^*+bz}
\, .\eeq 


Matrices of the representation $\Gamma^\lambda$ are easily evaluated
in the basis (\ref{su11basis}).  
For example, for
\beq g = \pmatrix{ c&s\cr s&c\cr} ,\quad c = \cosh \beta/2 ,\quad
s=\sinh \beta/2 ,\eeq
an SU(1,1) Wigner function is given by
\beqa \langle \lambda\mu| T^\lambda (g) |\lambda \nu\rangle 
 &=&
\int \psi^*_{\lambda \mu} (z)\,[\Gamma^\lambda(g) \Psi_{\lambda \nu}](z)\, {\rm d}
v(z)\nonumber\\  
&=& {K_\nu\over K_\mu} 
\int \varphi^*_\mu(z)\,[\Gamma^\lambda(g) \varphi_\nu](z)\, {\rm d}
v(z)\nonumber\\
&=&\sqrt{(\lambda+\nu -1)!\over (\lambda +\mu -1)! \mu!\nu! }
\int (z^*)^\mu (s+cz)^\nu (c+sz)^{-(\lambda +\nu)}\, {\rm d} v(z)
\nonumber\\
&=&\sqrt{ {\mu!\nu!\over (\lambda +\mu -1)! (\lambda +\nu -1)!}}\
\sum_{n=0}^\mu (-1)^n {(\lambda+\nu+n-1)!\over
(\nu-\mu+n)!(\mu-n)! n!} \nonumber\\
&& \times (\cosh \beta/2)^{\mu-\nu -\lambda -2n}
(\sinh\beta/2)^{\nu-\mu +2n} \,,\eeqa
cf.\ expressions obtained by Dunne \cite{dunne} and Ui \cite{Ui70}. 

Coherent state representations of the $\frak{su}(1,1)$ Lie algebra have
the known expressions
\beq \Gamma^\lambda (J_+) = \lambda z + z^2{d\over dz} \, ,\quad
\Gamma^\lambda(J_0) =\half
\lambda + z {d\over dz}\, ,\quad \Gamma^\lambda (J_-) = {d\over dz}\,
.\eeq 
Acting on the orthonormal basis wave functions of eqn.\
(\ref{eq:4.104}), they give
\beqa  &\Gamma^\lambda(J_0)\Psi_{\lambda\nu} = \half (\lambda +2\nu)
\Psi_{\lambda\nu} &\nonumber\\
&\Gamma^\lambda(J_+)\Psi_{\lambda\nu} =
 \sqrt{(\lambda +\nu)(\nu +1)}\ \Psi_{\lambda,\nu+1}\, , &\\
&\Gamma^\lambda(J_-)\Psi_{\lambda,\nu} =
 \sqrt{(\lambda +\nu -1)\nu }\ \Psi_{\lambda,\nu -1}\, .&\nonumber
\eeqa 



The inner product for ${\cal F}^\lambda$ is known to be given, for a
holomorphic discrete series representation  (for which $\lambda$ is an 
integer greater than 1), by
\beq \langle \psi |\varphi\rangle = {\lambda -1\over \pi}
\int_{|z|<1} {\Psi^*(z)\, \Phi(z)\,
 (1-|z|^2)^{\lambda -2}}\,  {\rm d^2}z  \,
,\label{eq:4.52}\eeq where  $\Psi = \hat {\cal S}\psi$, $\Phi= \hat {\cal
S}\varphi$ and the integral is over the interior of the unit disk
\cite{dunne}. When $\lambda > 1$, this inner product is identical to
that given for  ${\cal F}_\lambda$ by eqn.\ (\ref{eq:su11InP}).
However, the latter is easier to evaluate.
Moreover, unlike the integral of eqn.\ (\ref{eq:4.52}), it is defined
for all non-negative values of $\lambda$ and gives results, not
only for SU(1,1), but also for its universal covering 
group. 



 
\subsection{The semidirect product group $[\Rb^6]$SO(3)}

  
Let $G$ be the semidirect product $[\Rb^6]$SO(3) of an
Abelian normal  subgroup $\Rb^6$, isomorphic to the additive group of real
symmetric 
$3\times 3$ matrices, and the rotation group SO(3). 
Elements of $[\Rb^6]$SO(3) are pairs of group elements, $\{ (Q,\Omega); 
Q\in \Rb^6, \Omega \in {\rm SO(3)}\}$ with product 
\beq (Q,\Omega) \circ (Q',\Omega') = (Q+\Omega
Q'\tilde\Omega,\Omega\Omega') \,  , 
\label{r6so3action} 
\eeq 
where $\tilde \Omega = \Omega^{-1}$ is the transpose of $\Omega$. 
With $Q$ expressed as a sum $Q=\sum_{ij} Q_{ij}\Im_{ij} $, where $\{
\Im_{ij}\}$ is a basis of real symmetric matrices, 
this group is interpreted as the dynamical group of a rotor for
which the corresponding  Lie algebra 
is spanned by moments $\{ \Im_{ij}\}$ 
of an inertia tensor $\Im$  and components  $\{ L_k\}$ of angular
momentum (infinitesimal generators of  SO(3)).  

Let $T$ be a UIR of $G$ on a Hilbert space $\Hb$. The coherent-state
construction of such an irrep mimics  Mackey's 
induction \cite{Mackey}.
The construction starts with a one-dimensional unitary irrep of the
$\Rb^6$ subalgebra defined by a so-called {\it intrinsic inertia tensor}
$\bar\Im$ which assigns numerical values $\bar \Im_{ij}$ to the moments of
inertia
$\Im_{ij}$; i.e.,
\beq  \bar\Im \, : \, \Im_{ij} \to \bar\Im_{ij} = \delta_{ij}\bar\Im_i \,.
\eeq 
A corresponding one-dimensional unitary irrep of the $\Rb^6$ group is
then defined by
\beq Q\to\chi_{\bar \Im}(Q) =e^{i\, {\rm Tr}(Q\bar\Im)} \,. \eeq
Now let 
\beq D= \{ \omega\in {\rm SO(3)} |\omega
\bar\Im \tilde\omega =\bar\Im\}\,, 
\eeq 
with $(\omega\bar\Im\tilde\omega)_{ij} = \sum_{kl}
\omega_{ik}\bar\Im_{kl}\omega_{jl}$, denote the isotropy subgroup of
rotations that leave the intrinsic moments of inertia invariant.  To be
specific, we take the intrinsic moments of inertia
$(\bar\Im_1,\bar\Im_2,\bar\Im_3)$ to be positive  and distinct. Then $D$
is  the discrete group $D_2$  comprising the four rotations
(parameterized in terms of Euler  angles): 
\beq \omega_1 \sim (0,0,0) \, ,\quad \omega_2 \sim (0,\pi,0)\, ,\quad
\omega_3 \sim  (\pi,0,0)  \, ,\quad \omega_4 =
\omega_3\omega_2 \sim  (\pi,\pi,0) \, . \label{eq:D2rots} \eeq 
This group has four irreps,  all of which 
are  one-dimensional.  
Thus, an irrep $\rho_{\bar\Im,\varepsilon}$ of  $[\Rb^6]D_2$ is defined
by 
\beq  
(Q, \omega_i)  \to  \rho_{\bar\Im,\varepsilon}(Q,\omega_i) =
\chi_{\bar\Im}(Q)\, \varepsilon(\omega_i) 
\, ,\quad i = 1, \ldots , 4 , \label{eq:4.96}\eeq  
with
$\varepsilon(\omega_1)=1$, $\varepsilon(\omega_2)= \pm 1$, 
$\varepsilon(\omega_3)=\pm 1$ and 
$\varepsilon(\omega_4) = \varepsilon(\omega_2) \varepsilon(\omega_3)$.  



Let $\langle\bar\Im,\varepsilon |$ be a functional on a
suitably defined dense subspace $\Hb_D$ of $\Hb$ which {\it picks out} a
particular irrep of the subgroup $[\Rb^6]D_2\subset [\Rb^6]$SO(3) in the
sense that
\beq  \langle\bar\Im,\varepsilon |T(Q,\omega) |\psi\rangle = 
   \chi_{\bar\Im}(Q)\varepsilon(\omega) \langle \bar\Im,\varepsilon 
  |\psi\rangle \, , \quad\forall|\psi\rangle \in 
  \Hb_D\,,\quad\forall \omega \in D_2 
\,.\label{eq:3.131}\eeq
A suitable dense subspace $\Hb_D$ is the SO(3)-invariant space
of
$K$-finite vectors in $\Hb$ comprising all states that are finite linear
combinations of vectors from a multiplicity of  SO(3) irreps. 
A coherent state wave function $\Psi$ for a vector 
$|\psi\rangle\in\Hb_D$ is then defined by 
\beq \Psi(\Omega) = \langle \bar\Im,\varepsilon |\hat R(\Omega)
  |\psi\rangle \,.\eeq 
The space of such wave functions,  ${\cal F}^{\Hb_D}$,  carries a
representation of $[\Rb^6]$SO(3) defined   by 
\beq [\Gamma(Q,\Omega) \Psi ](\Omega')=\langle\bar\Im,\varepsilon
|T(\Omega'Q\tilde\Omega',\Omega'\Omega) |\psi\rangle =  
\chi_{\bar\Im}(\Omega'  Q\tilde\Omega') \Psi(\Omega'\Omega) \, . 
\label{eq:4.98} 
\eeq   
The action of $\Gamma$  
defined by eqn.\ (\ref{eq:4.98}) is unitary on ${\cal L}^2$(SO(3)).  
Thus, we infer from eqn.\ (\ref{eq:3.131}) that the coherent state
representation is realized on  an irreducible subspace of ${\cal
L}^2$(SO(3)) that  satisfies the  constraint 
\beq \Psi (\omega\Omega) = \varepsilon(\omega)\Psi(\Omega) , \quad 
\forall \omega\in D_2\,. \label{eq:R5so3constraint}\eeq  

 

To construct the triplet,  we start with an orthonormal basis for the
space  ${\cal F}={\cal L}^2$(SO(3)) given by functions 
\beq \varphi_{KLM} = \sqrt{2L+1\over 8\pi^2}\, {\cal D}^L_{KM} \, ,\eeq 
where  ${\cal D}^L_{KM}$ is an SO(3) Wigner function. 
Consider the map ${\cal F}^* \to \Hb_D^*$ 
which sends 
\beq
\varphi_{KLM}^* \mapsto\langle KLM | 
= \sqrt{{2L+1\over 8\pi^2}} \int 
 \,  {\rm d}\Omega \,\,{\cal D}^{L}_{KM}(\Omega )^*
\langle \bar\Im, \varepsilon |  R(\Omega) \,.
\label{dualmap}
\eeq
As a functional on $\Hb_D$, $\langle KLM |$ is 
non-zero on the finite-dimensional subspace consisting of 
copies of the SO(3) irrep with angular momentum $L$,  but 
it is zero on the orthogonal complement of this space.  In 
particular,  it can be regarded as a linear functional on a 
finite-dimensional space,  so it must actually be given by 
a vector in that space. 
The inner products of such vectors in $\Hb$ are then expressed
\beqa  \langle K'L'M'|KLM\rangle &=& \delta_{L'L} \delta_{M'M} {2L+1\over
8\pi^2}\int\!\!\int {\cal D}^{L}_{K'M}(\Omega')^* \, \langle
\bar\Im,\varepsilon |\hat
R(\Omega'\Omega^{-1})|\bar\Im,\varepsilon\rangle \, {\cal D}^L_{KM}
(\Omega)\,    {\rm d} \Omega \,{\rm d} \Omega'
\nonumber\\
&=& \delta_{L'L} \delta_{M'M} \int {\cal D}^{L}_{K'K}(\Omega) ^*\, 
\langle \bar\Im,\varepsilon|\hat R(\Omega)|\bar\Im,\varepsilon\rangle  \,
 {\rm d} \Omega \, . \eeqa
Thus, we obtain a densely--defined sequence of maps ${\cal F}_D \to
\Hb_D \to {\cal F}^{\Hb_D}$ in which $\varphi_{KLM} \mapsto |KLM\rangle
\mapsto \Phi_{KLM}$, where  the coherent state wave function
$\Phi_{KLM}$ for the state $|KLM\rangle$ is given by
\beq \Phi_{KLM} (\Omega) = \langle
\bar\Im,\varepsilon|\hat R(\Omega)|KLM\rangle 
= \int {\cal S}(\Omega\tilde\Omega')\, \varphi_{KLM}(\Omega') \, {\rm
d}\Omega'
\eeq
and ${\cal S}$ is the functional on ${\cal F}$ given by 
\beq {\cal S}(\Omega) = \langle \bar\Im,\varepsilon|\hat
R(\Omega)|\bar\Im,\varepsilon\rangle \,.  \label{eq:Srotor}
\eeq
Thus, the inner product for $\Hb$ is expressed in terms of the $\{
|KLM\rangle \}$ basis
\beq \langle K'L'M'|KLM\rangle =\int \varphi_{K'L'M'}^*(\Omega)\, 
\Phi_{KLM} (\Omega)\,    {\rm d} \Omega  \,.
\eeq
Clearly the functions $\varphi_{KLM}$ and $\Phi_{KLM}$ are dual
representations of the state vector $|KLM\rangle$ relative to the measure
d$\Omega$.

The norm of the functional $\langle \bar\Im,\varepsilon |$, and hence
of
${\cal S}$, is arbitrary.
The essential requirement is that the coherent state functions
$\{ \Phi_{KLM}\}$ obtained from all basis functions $\{\varphi_{KLM}\}$
of ${\cal F}$ should satisfy the constraint condition of eqn.\
(\ref{eq:R5so3constraint}).
Thus, ${\cal S}$ must satisfy the equation
\beq {\cal S}(\omega\Omega)= \varepsilon(\omega) {\cal S}(\Omega) \, , 
\quad \forall\,\omega\in {\rm D}_2 \, .\eeq
 From its definition, eqn.\ (\ref{eq:Srotor}), ${\cal S}$ also satisfies
the equation
\beq (\Omega \bar\Im\tilde\Omega) {\cal S}(\Omega)= \bar\Im\, {\cal S}(\Omega) \,
,\quad \forall\, \Omega\in {\rm SO(3)}\, .\eeq 
Thus,  to within an arbitrary norm factor, it has the unique
solution
\beq {\cal S}(\Omega) =  {1\over 4}\sum_{\omega\in {\rm
D}_2}
\varepsilon (\omega) \delta(\Omega,\omega) \,.\label{eq:4.101}\eeq 
With this normalization, 
\beqa \Phi_{KLM} (\Omega) &=& \sqrt{{2L+1\over 8\pi^2}} \int   {\rm d}\Omega' \,\,
{\cal S}(\Omega\tilde\Omega') 
\, {\cal D}^L_{KM}(\Omega' )\,  \nonumber\\ 
&=&  
 \sqrt{{2L+1\over 8\pi^2}}\ {(1+(-1)^K\varepsilon_3)\over 4}
\Big(  {\cal D}^L_{KM}(\Omega) + (-1)^{L+K}\varepsilon_2 {\cal 
D}^L_{-K,M}(\Omega)\Big) \,, 
\eeqa 
where $\varepsilon_2 = \varepsilon(\omega_2)$ and
$\varepsilon_3 = \varepsilon(\omega_3)$.
  

It is seen that the states $\{ |KLM\rangle\}$ 
are not all linearly independent.  
In particular, the wave functions $\Phi_{KLM}$ with 
$(-1)^K\varepsilon(\omega_3)=-1 $ vanish 
as do those with $K=0$ and $(-1)^L\varepsilon_2=-1 $.
Moreover, the wave functions
$\Phi_{\pm  KLM}$ differ by only a phase factor. 
Thus, it is appropriate to restrict to labels $KLM$ with $K \ge 0$ and
non-vanishing wave functions $\Phi_{KLM}$.  
The subspace of ${\cal F}$ spanned by the corresponding restricted basis 
functions $\{\varphi_{KLM}\}$ is then isomorphic, as a vector space,
to the space obtained by factoring out the  kernel of the map ${\cal F}
\to\Hb$. 
  
 
With $|KLM\rangle$ restricted in this way, the overlaps are 
given by 
\beqa \langle K'L'M'|KLM\rangle &=& \int \varphi^*_{K'L'M'}(\Omega)\, 
\Phi_{KLM}(\Omega) \, d\Omega 
\nonumber\\ 
&=& \hf  (1+\delta_{K,0})\delta_{L'L}\, \delta_{M'M} \delta_{K'K} 
\, . \eeqa 
The triplet of Hilbert spaces ${\cal F}_\Hb \sim \Hb \sim {\cal 
F}^\Hb$ is then defined with 
orthornormal bases given, respectively, by 
\beqa & \displaystyle 
\left\{\psi_{KLM}= \sqrt{2L+1\over 4\pi^2 (1+\delta_{K,0})}\, {\cal 
D}^L_{KM}\, ; K \geq 0\right\} , & \nonumber\\ 
&\displaystyle 
\left\{ \sqrt{2\over (1+\delta_{K,0})}\, |KLM\rangle \, ; K \geq 
0\right\} , &\nonumber\\  
&\displaystyle 
\left\{ \Psi_{KLM}=\sqrt{2L+1\over 16\pi^2  (1+\delta_{K,0})}\,  
\left[ {\cal D}^L_{KM} + \varepsilon_2 (-1)^{L+K} {\cal D}^L_{-\!
K,M}\right]     \,   ; K \geq 0 \right\}  ,& 
\eeqa 
with $K$ restricted to even or odd integer values according as  
$\varepsilon_3 = \pm 1$ 
and with $L$ restricted to even or odd integer values when $K=0$ according as 
$\varepsilon_2 = \pm 1$. 

 
 
These wave functions satisfy the orthogonality relations  
\beq \int \psi^*_{KLM}(\Omega) \, \Psi_{K'L'M'} \, {\rm d}\Omega 
= \delta_{LL'} \delta_{KK'} \delta_{MM'} \, .\eeq 
In addition, with the chosen norm for ${\cal S}(\Omega)$, the coherent
state wave functions on their own also satisfy 
\beq \int \Psi^*_{KLM}(\Omega) \, \Psi_{K'L'M'} \, {\rm d}\Omega 
= \delta_{LL'} \delta_{KK'} \delta_{MM'} \, .\eeq 
This is a special case that arises because the appropriate volume element 
for the  Hilbert space of coherent-state wave functions for the rotor
group happens to be the  SO(3)-invariant volume. 
A similar result occurs for the Heisenberg-Weyl group with 
the Bargmann measure. 
It is nevertheless useful to see this special case in the context of the 
general construction which can be applied when the end results are less 
easily anticipated. 
The following derivation of SU(3) irreps in an SO(3) basis is an 
example for which the construction is used with considerable advantage. 
 
 

\subsection{SU(3) $\supset$ SO(3)}

Let $|\lambda\mu\rangle$ denote a highest weight state for an SU(3) irrep 
$T^{(\lambda\mu)}$ of highest weight $(\lambda,\mu)$
with respect to the diagonal Cartan subgroup. 
It is known that the Hilbert space $\Hb^{(\lambda\mu)}$ for such an irrep
is spanned by the set of states generated by rotating the state
$|\lambda\mu\rangle$
with the elements of the subgroup SO(3) $\subset$ SU(3) \cite{JPE}.
Thus, we may take $N$ to be SO(3) and consider the coherent state
representation in
which a vector $|\psi\rangle \in \Hb^{(\lambda\mu)}$ is represented by
a wave function over SO(3) with values
\beq \Psi(\Omega ) = \langle \lambda\mu |\hat R(\Omega) |\psi\rangle \, ,
\quad
\Omega\in
{\rm SO(3)}\, , \label{eq:3.42}\eeq
where $\hat R$ is the restriction of  $T^{(\lambda\mu)}$ to the SO(3)
$\subset$ SU(3) subgroup.

Let ${\cal F}= {\cal L}^2({\rm SO(3)})$ be the square integrable functions
on SO(3)
relative to its invariant measure d$\Omega$.
 The map ${\cal F} \to \Hb^{(\lambda\mu)}$ is then
\beq \varphi \to |\varphi\rangle =
\int  \hat R(\Omega ^{-1}) |\lambda\mu\rangle \,
\varphi(\Omega )\,  {\rm d}\Omega \eeq
and the space ${\cal F}_{(\lambda\mu)}$ isomorphic to
$\Hb^{(\lambda\mu)}$ is defined by the inner product
\beq (\psi,\varphi) = \langle \psi |\varphi\rangle =
\int\!\!\!\int \psi^*(\Omega')\,{\cal S}(\Omega'\tilde\Omega) \,
\varphi(\Omega) \, {\rm d}\Omega'
\, {\rm d}
\Omega \, , \label{eq:3.44}\eeq
where
\beq {\cal S}(\Omega) = \langle \lambda\mu
|\hat R(\Omega)|\lambda\mu\rangle
\, .\eeq


 The overlap kernel ${\cal S}$ is  
easily evaluated by considering a realization of the SU(3) irrep in a
Bargmann space
\cite{Barg} in which the highest weight state $|\lambda\mu\rangle$ is
represented by the wave function $\phi_{\lambda\mu}$ with values
\beq \phi_{\lambda\mu} ({\bf r}, {\bf r}') \propto z^\lambda Y^\mu  
\, ,\eeq 
where $z$ is a component of a complex vector 
${\bf r}\equiv (x,y,z)$ and $Y$ is a component of ${\bf r} \times {\bf
r}'\equiv (X,Y,Z)$. 
Thus, with
\beq [\hat R(\Omega) \phi_{\lambda\mu} ]({\bf r}, {\bf r}') =
\phi_{\lambda\mu} ({\bf r}\Omega, {\bf r}'\Omega) \, , \eeq
and $\Omega \equiv (\alpha, \beta,\gamma)$ parameterized by Euler
angles, ${\cal S}$ is given by \cite{EH}
\beq  {\cal S}(\Omega) =\langle \lambda\mu
|\hat R(\Omega)|\lambda\mu\rangle = (\cos\beta )^\lambda (\cos\alpha
\cos\gamma - \sin\alpha \cos\beta\sin\gamma )^\mu
\, .\eeq 

 From the Bargmann representation,
it is also apparent that the state $|\lambda\mu\rangle$ spans a
one--dimensional representation of the $D_2$ subgroup of SO(3) in which,
in the notations of eqn.\ (\ref{eq:D2rots}),
\beq \hat R(\omega) |\lambda\mu \rangle =
\varepsilon(\omega)|\lambda\mu\rangle \,, \quad \omega \in D_2\,,
\eeq
with
\beq \varepsilon(\omega_2)=(-1)^\lambda \,,\quad
\varepsilon(\omega_3)= (-1)^\mu \,. 
\eeq
Thus, the SU(3) coherent state wave functions satisfy the symmetry
property
\beq \Psi(\omega\Omega ) = \varepsilon(\omega) \Psi(\Omega ) \,, \quad
\omega \in D_2\,. \label{eq:SU3symmetry}
\eeq

According to the Peter-Weyl theorem \cite{PW}, a basis for ${\cal F}= {\cal
L}^2({\rm SO(3)})$ is given by the functions
\beq \varphi_{KLM} = \sqrt{2L+1\over 8\pi^2}\, {\cal D}^L_{KM} \, ,\eeq
where  ${\cal D}^L_{KM}$ is an SO(3) Wigner (matrix coefficient) function.
These functions satisfy
\beq \int \varphi^*_{KLM}(\Omega)\, \varphi_{K'L'M'} (\Omega)
\, {\rm d}\Omega = \delta_{LL'} \delta_{KK'}\delta_{MM'} \, .\eeq
The map ${\cal F} \to\Hb^{(\lambda\mu)}$ then gives
\beq \varphi_{KLM} \to |KLM\rangle =
\sqrt{{2L+1\over 8\pi^2}} \int \hat R(\Omega ^{-1})|\lambda\mu\rangle
\, {\cal D}^L_{KM}(\Omega )\,  {\rm d}\Omega \, .\eeq


A maximal linearly-independent subset of the states $\{ |KLM\rangle\}$
comprises a so-called {\it Elliott basis} for $\Hb^{(\lambda\mu)}$.
 From Eq.\ (\ref{eq:3.44}), the overlaps
of Elliott states are given by
\beqa \langle K'L'M'|KLM\rangle &=& \delta_{L'L} \delta_{M'M} {2L+1\over 8\pi^2}
\int\!\!\!\int {\cal D}^{L}_{K'M}(\Omega')^*\, {\cal
S}(\Omega'\tilde\Omega)\, {\cal D}^L_{KM} (\Omega)\,{\rm d}\Omega' 
\, {\rm d} \Omega \nonumber\\
&=& \delta_{L'L} \delta_{M'M}\int {\cal D}^{L}_{K'K}(\Omega)^* 
\, {\cal S}(\Omega)\,{\rm d} \Omega \, . \eeqa

Let  ${\cal S}^L$ be the SO(3)-invariant 
matrix of overlaps with elements
\beq {\cal S}^L_{K'K} = \langle K'LM|KLM\rangle =
 \int {\cal D}^{L}_{K'K}(\Omega)^* \, {\cal S}(\Omega) \, {\rm d}
\Omega \, .\eeq
 From the symmetry property (\ref{eq:SU3symmetry}),
 it follows that
\beq {\cal S}^L_{K'K} = (-1)^{\mu+K'} {\cal S}^L_{K'K}
= (-1)^{\mu+K} {\cal S}^L_{K'K}
=(-1)^{\lambda+L+K'} {\cal S}^L_{-K',K} 
=(-1)^{\lambda+L+K} {\cal S}^L_{K',-K} \,. \label{eq:Ssymmetry}
\eeq
Thus, ${\cal S}^L_{K'K}$ is zero unless $(-1)^{K'}= (-1)^K =
(-1)^\mu$ and, for $K$ or $K'=0$, unless $(-1)^{\lambda+L} = 1$. 
Then, with $\{ |KLM\rangle\}$
restricted to a maximal linearly-independent set of states with non--zero
norms, the matrix ${\cal S}^L$ is  Hermitian positive definite and can be
expressed as a product
${\cal S}^L = {\cal K}(L){\cal K}^\dagger (L)$; i.e.,
\beq {\cal S}^L_{K'K} = \sum_\gamma {\cal K}_{K'\gamma}(L) {\cal
K}^*_{K\gamma}(L) \, .\eeq
Now, if we define $\bar {\cal K}(L)$ to be the inverse of the matrix
${\cal K}^\dagger(L)$, so that
\beq \sum_{K\geq 0} {\cal K}^*_{K\beta}(L) \bar{\cal K}_{K\alpha}(L)
=
\delta_{\alpha\beta} \,
,\eeq
then the transformed basis states
\beq |\alpha LM\rangle = \sum_K |KLM\rangle \bar {\cal K}_{K\alpha}(L) \, ,
\label{eq:3.54}\eeq
satisfy
\beq \langle \beta L'M'|\alpha LM\rangle = \delta_{LL'} 
\delta_{MM'} \sum_{K'K\gamma} \bar{\cal
K}^*_{K'\beta}(L) {\cal K}_{K'\gamma}(L) {\cal K}^*_{K\gamma}(L) \bar{\cal
K}_{K\alpha}(L) = \delta_{LL'} \delta_{MM'} \delta_{\beta\alpha}
\eeq
and form an orthonormal basis. 




Coherent state wave functions corresponding to the non--orthonormal 
$\{ |KLM\rangle \}$ states are given by
\beq \Phi_{KLM} (\Omega') = [\hat{\cal S}\varphi_{KLM}](\Omega')
= \int{\cal S}(\Omega'\tilde\Omega)\,
\varphi_{KLM}(\Omega) \, {\rm d} \Omega \, .
\eeq
Therefore, the coefficients in the expansion
\beq \Phi_{KLM} (\Omega ) = \sum_{K'} a_{K'} (KL) {\cal D}^L_{K'M}
(\Omega)\, ,\eeq
are given by
\beq  a_{K'} (KL) = {2L+1\over 8\pi^2} \int 
{\cal D}^{L}_{K'M}(\Omega) ^* \, \Phi_{KLM} (\Omega)\, {\rm d}\Omega
= \sqrt{2L+1\over 8\pi^2} \int {\cal D}^{L}_{K'K}(\Omega)^* 
{\cal S}(\Omega)\, {\rm d}\Omega\, .
\eeq
It follows that
\beq \matrix{a_{K'} (KL) =
\displaystyle\sqrt{2L+1\over 8\pi^2}\, {\cal S}^L_{K'K} \, ,\cr
a_{-K'} (KL) = (-1)^{\lambda +L+K'} a_{K'} (KL) \, ,\cr} \eeq
and
\beq \Phi_{KLM}(\Omega) = \sum_{K'\geq 0} \sqrt{2L+1\over 8\pi^2}
\left[ {\cal D}^L_{K'M}(\Omega) + (-1)^{\lambda +L+K'} {\cal
D}^L_{-K'M}(\Omega)
\right]
{1\over 1+\delta_{K'0}}\, {\cal S}^L_{K'K} \, .\eeq
Because of the symmetries of the ${\cal S}$ matrix, given by eqn.\
(\ref{eq:Ssymmetry}), these wave functions vanish unless 
$(-1)^{\mu+K}=1$ and, for $K=0$, unless $(-1)^{\lambda+L}=1$.
Furthermore, the wave functions $\Phi_{\pm KLM}$ differ by only a phase
factor.
Thus, with $L$ and $K$ restricted to a subset of values such that the
matrices ${\cal S}^L$ are positive definite (so that no linear
combinations of the wave functions $\Phi_{KLM}$ vanish), an orthornormal
basis for
${\cal F}_{\lambda\mu}$ is given by wave functions
\beq \psi_{KLM}(\Omega) = \sum_{K\geq 0} \bar {\cal K}_{K\alpha}(L)
\,\varphi_{KLM}(\Omega)  = \sqrt{2L+1\over 8\pi^2} 
\sum_{K\geq 0} \bar {\cal K}_{K\alpha}(L) \,{\cal D}^L_{KM}(\Omega)
\eeq
and a corresponding orthonormal basis of coherent state wave functions is
given by
\beqa \Psi_{\alpha LM}(\Omega) &=&   \sum_{K\geq 0} \bar {\cal
K}_{K\alpha}(L) \, \Phi_{KLM}(\Omega)  \nonumber\\
&=& \sum_{K\geq 0} \sqrt{2L+1\over 8\pi^2}\, {{\cal K}_{K\alpha}(L)\over
1+\delta_{K0}} 
\left[{\cal D}^L_{KM}(\Omega) + (-1)^{\lambda +L+K} {\cal
D}^L_{-KM}(\Omega) \right] \, .\eeqa
These expressions are in agreement with known results
\cite{SU3/SO3,RB00}. An explicit expressions for the overlap kernel
${\cal S}^L_{K'K}$ is given in ref.\ \cite{RB00}. 


For a multiplicity-free representation, the ${\cal K}$ matrices reduce to simple
numbers.
For example, for an irrep of highest weight $(\lambda, \mu =0)$,
\beq |{\cal K}(L)|^2 = \langle LM|LM\rangle = 4\pi^2
\int_0^\pi d^L_{00} (\beta)\, (\cos\beta)^\lambda\, \sin\beta\, {\rm
d}\beta \, .\eeq
The identity
\beq \int_0^\pi d^L_{00} (\beta)\, (\cos\beta)^\lambda\, \sin\beta\, {\rm
d}\beta
 =  \int_0^1 P_L(x)\, x^\lambda\, {\rm d}x =
{\lambda!\over (\lambda-L)!! (\lambda +L+1)!!} \eeq
inferrred from Eq.\ 8.14.15 of Abramowitz and Stegun \cite{AS}, then gives
\beq  |{\cal K}(L)|^2 = 4\pi^2 {\lambda!\over (\lambda-L)!! (\lambda +L+1)!!}
[1+(-1)^{\lambda+L}] \, .\eeq
The corresponding ratios
\beq \left|{{\cal K}(L)\over {\cal K}(L-2)}\right|^2
= {\lambda -L+2\over \lambda +L+1} \eeq
agree with those derived previously \cite{SU3/SO3} by requiring the
representation to be unitary.

The coherent state representation of the su(3) Lie algbra is
relatively simple and given explicitly in ref.\ \cite{SU3/SO3}.  
The action  of a group element $g\in$ SU(3) on a coherent state
wave function $\Psi$, defined by
\beq [\Gamma(g) \Psi] (\Omega) =
\langle \lambda\mu |T^{\lambda\mu}(\Omega g) |\psi\rangle \, , \quad
\Omega\in {\rm SO(3)}\subset {\rm SU(3)}\,,  \eeq
is more complicated. It can be obtained from the generalized
Iwasawa factorization for which
\beq g = Z(g) \omega (g) \,, \eeq
where $Z(g)$ is a lower triangular matrix of the form
\beq Z(g) = \pmatrix{ z_{11}(g) & 0&0\cr 
z_{21}(g) & z_{22}(g) &0\cr
z_{31}(g) & z_{32}(g) & z_{33}(g) \cr}\eeq
and $\omega(g)\in$ SO$(3,\Cb)$ is a complex orthogonal matrix.
It follows that
\beq [\Gamma(g) \Psi] (\Omega) = (z_{11}(\Omega g))^{\lambda+\mu}\,
(z_{22}(\Omega g))^\mu \Psi (\omega(\Omega g))\,,\eeq
where
\beqa & z_{11}^2(g) = \displaystyle\sum_i g_{1i}^2 \,,& \nonumber\\
    & z_{11}^2(g) z_{22}^2(g) = \displaystyle\sum_i g_{1i}^2 \sum_j 
g_{2j}^2 - \sum_i (g_{1i} g_{2i})^2 \,.
\eeqa

The above techniques for constructing orthornormal bases were
used effectively in ref. \cite{RB00} for computing SU(3) Clebsch--Gordan
in an SO(3)--coupled basis.   


\section{Scalar coherent-state representations of semisimple Lie groups}

There are four classes of scalar  holomorphic discrete series
representations of semisimple Lie groups all of which are
expressible as coherent state representations over classical domains.
They are constructed when a coherent-state representation of a semisimple
Lie group $G$
is induced from a one-dimensional irrep $\rho$ of a subgroup $K\subset G$
for which
$G/K$ is a Hermitian symmetric space.
These spaces have been studied in depth by Helgason \cite{Helg}.
The classical domains which we discuss here were identified by Cartan
\cite{Cartan}.
The associated coherent state representations have been summarized by
Hua \cite{Hua} and Perelomov \cite{Perelomov85}. 


Hermitian symmetric spaces \cite{Helg} are easily identified at the Lie 
algebra level. If $G$ is semisimple and $H$ is a reductive 
subgroup of $G$, then 
$G/H$ is a Hermitian symmetric space if the complex extension ${\frak
g}^c$ of the Lie algebra of $G$ has decomposition
\beq {\frak g}^c = {\frak h}^c + {\frak n}_+ + {\frak n}_- \,, \eeq
such that ${\frak g}^c$ and ${\frak h}^c$ have a Cartan subalgebra in 
common, ${\frak n}_\pm$ are Abelian, and
\beq [ C,A] \in {\frak n}_+ \,, \quad [C,B] \in {\frak n}_- \,, \quad
[A,B] \in  {\frak h}^c \,,\eeq
for all $A\in {\frak n}_+$, $B\in {\frak n}_-$, and $C\in {\frak h}^c$.
There are then holomorphic representations of $G$ 
with highest weight states which carry one-dimensional irreps of $H$.
These representations have
simple expressions as scalar coherent state representations.




\subsection{SU$(p+q)$ and SU$(p,q)$}

The Lie algebras of SU$(p+q)$ and SU$(p,q)$ have a common complex extension
${\frak su}(n)^c \sim A_{n-1} = {\frak sl}(n, \Cb)$
with $n=p+q$.
The latter is spanned by traceless linear combinations of the matrices
$\{C_{\alpha\beta}\}$, where $C_{\alpha\beta}$ is an $n\times n$ matrix
with entries
$(C_{\alpha\beta})_{ij} = \delta_{\alpha i} \delta_{ \beta j}$.

Let $T^\sigma$ denote an irrep of $A_{n-1}$,  
with $T^\sigma (C_{\alpha\beta}) = \hat C_{\alpha\beta}$,   
on a Hilbert space 
$\Hb^\sigma$ having highest weight state $|0\rangle$ and highest weight 
$(\sigma \ldots  \sigma ; 0 \ldots 0)$ defined by
\beq \matrix{\hat C_{i\nu} |0\rangle = \hat C_{ij} |0\rangle =\hat C_{\mu
\nu} |0\rangle = 0 \,,\cr
\hat C_{ii} |0\rangle = \sigma |0\rangle \, , \quad\hat C_{\nu
\nu}|0\rangle = 0 \,,\cr} \quad i<j <\mu<\nu \,,
\eeq
where $i$ and $j$ run over the range $1,\ldots, p$ and $\mu$ and $\nu$ run
over the range $ p+1,   \ldots, n = p+q$.
If  $\lambda$ is a positive integer  then,  when $\sigma = +\lambda$, this
irrep integrates to a unitary irrep of SU$(p+q)$; it
integrates to a unitary irrep of 
SU$(p,q)$ when  $\sigma = -\lambda$.
In both cases, the subgroup of
elements that leave the highest weight state $|0\rangle$ invariant (to
within a phase
factor) is  the subgroup of
${\rm U}(p)\times {\rm U}(q)$ matrices
\beq {\rm S[U}(p)\times {\rm U}(q) ] = \Big\{ \pmatrix{ a&0\cr 0&b\cr}
\Big|
\; a \in {\rm U}(p) \,, b\in {\rm U}(q)\,, \det b = [\det a]^{-1}\Big\} .
\eeq
It follows that the highest weight state $|0\rangle$ is a single basis
vector for the
one-dimensional irrep $\rho_\sigma$ of ${\rm S[U}(p)\times {\rm U}(q)]$ for
which
\beq \rho_\sigma \pmatrix{ a&0\cr 0&b\cr} = [\det a]^{\sigma} \, .\eeq

Let $\Hb_D \subseteq \Hb^\sigma$ be the space of ``$K$-finite
vectors'',   that is,  the 
vectors obtained by applying polynomials in the lowering operators 
to the state $|0\rangle$.  
For $G = {\rm SU}(p+q)$,  where $\Hb^\sigma$ is finite-dimensional,  this
will be  all of $\Hb^\sigma$,  but for $G = {\rm SU}(p,q)$,  it is a
dense subspace.  


Gauss factorization of almost any SL$(p+q,\Cb)$ matrix gives
\beq g= \pmatrix{a&b\cr c& d\cr} = \pmatrix{I_p&0\cr x&I_q\cr}
\pmatrix{ a&0\cr 0&d-cz} \pmatrix{ I_p&z\cr 0&I_q\cr}
\eeq
with $z=a^{-1}b$, $x=ca^{-1}$, and with 
$I_p$ the $p \times p$ identity matrix. 
It follows that, for an irrep $T^\sigma$ of $G={\rm SU}(p+q)$ or
$G={\rm SU}(p,q)$ with highest weight state
$|0\rangle$ and highest weight $(\sigma \ldots \sigma ; 0\ldots 0)$,
the coherent state wave functions satisfy
\beq \langle 0| T^\sigma(g)|\psi\rangle =    [\det a]^\sigma\langle 0|
e^{\hat z}|\psi\rangle
\, , \quad g= \pmatrix{a&b\cr c& d\cr} \in G \,,\eeq
where $\hat z = \sum_{i\nu}  z_{i\nu} \hat C_{i\nu}$,
for any $| \psi \rangle \in \Hb_D$. 
 Accordingly,
a suitable choice of $N\subset {\rm SU}(p+q)^c$ is the subgroup of
upper triangular matrices
\beq N = \left\{ \pmatrix{ I_p & z\cr 0 & I_q};  z
\in \Cb^{p \times q} \right\} \, . \eeq 
A coherent state wave function $\Psi$ for a state 
$|\psi\rangle \in \Hb_D$ is 
then
defined by
\beq \Psi(z)  = \langle 0|e^{\hat z}|\psi\rangle \eeq
and is seen to be a polynomial in the matrix coefficients of $z$.  



The coherent state action $\Gamma(g)$ of a 
matrix
\beq g = \pmatrix{a&b\cr c&d\cr} \in G\,,\eeq
defined for $\psi \in \Hb_D$ by
\beq [\Gamma(g) \Psi](z) = \langle 0| e^{\hat z}
T^\sigma(g)|\psi\rangle
\, ,\eeq is determined by the generalized Gauss factorization of
the product
\beq \pmatrix{ I_p & z\cr 0 & I_q} \pmatrix{a&b\cr c&d\cr}
=  \pmatrix{ I_p & 0\cr x & I_q} \pmatrix{\alpha &0\cr 0&\beta\cr}
\pmatrix{ I_p & y\cr 0 & I_q} \,,\eeq 
with
\beq \alpha = a+zc\,,\quad y = (a+zc)^{-1} (b+zd) \,.\eeq
This gives
\beq  \langle 0 |e^{\hat z} T(g)|\psi\rangle = [\det
\alpha]^\sigma
\langle 0|  e^{\hat y} |\psi\rangle
\eeq
and
\beq [\Gamma(g) \Psi](z) =  [\det (a+zc)]^\sigma
\Psi( (a+zc)^{-1} (b+zd))\,.  \label{eq:SUpqaction}\eeq

The space ${\cal F}^{\Hb}$ of coherent state wave functions is
identified in the standard way.
Let ${\cal F}$ be the Bargmann space of entire analytic functions in the
$z$ variables, i.e., ${\cal F}$ is the space   ${\cal L}^2(\Cb^{p\times
q}, {\rm d}v)$,  where ${\rm d}v$ is the Bargmann measure on $\Cb^{p
\times q}$. The map of eqn.\  (\ref{eq:2.32})
\beq{\cal F}^*\to \Hb_D^* \, ;\, 
\varphi^* \mapsto  \langle\varphi| = \int
{\rm d}v(z)\, \varphi^*(z) \langle 0|T^\sigma (z)    
\eeq
then takes a polynomial $\varphi^*\in {\cal F}^*$ to a vector
$\langle 0|\varphi(\hat C)^\dagger$ in $\Hb^* \cong
\Hb$, where
$\varphi(\hat C)$ is the operator obtained by replacing every $z_{i\nu}$
in $\varphi(z)$ by the corresponding operator $\hat C_{i\nu}$.
It follows that the dense subspace ${\cal F}_D$ of polynomials in
${\cal F}$ maps onto the dense subspace $\Hb_D\subset\Hb$ as required.
Thus, the Hilbert space ${\cal F}_{\Hb}$ is a space
of holomorphic functions with inner product
\beq  \langle \varphi|\psi\rangle = \int \varphi^*(z)\, {\cal
S}^\sigma(z,x)\,\psi(x)\, {\rm d}v(z)\, {\rm d}v(x)\,,
\eeq
where ${\cal S}^\sigma(z,x)$ is the overlap kernel
\beq {\cal S}^\sigma(z,x)= \langle 0| e^{\hat z} e^{\hat x^\dagger}
|0\rangle \,.
\eeq 
Let $\{ \psi_n\}$ denote an orthonormal polynomial basis  for ${\cal
F}_{\Hb}$.
An orthonormal basis$\{ |n\rangle \}$ for $\Hb$ is then defined by the map
\beq {\cal F}_D \to \Hb_D \,; \psi_n \mapsto |n\rangle
\eeq
and a corresponding orthonormal basis $\{ \Psi_n\}$ of coherent state wave
functions for ${\cal F}^{\Hb}$ is defined by
\beq \Psi_n(z) = \int {\cal S}^\sigma(z,x)\, \psi_n(x) \, {\rm d}v(x) \,.
\eeq
These bases for ${\cal F}_{\Hb}$ and ${\cal F}^{\Hb}$
are in duality relative to the inner product defined by d$v$, i.e.,
\beq  \langle m|n\rangle = \int \psi_m^*(z) \Psi_n(z) \, {\rm d}v(z) \,,
\eeq
and matrix elements of the representation are given by
\beq \langle m|T(g)|n\rangle = \int \psi_m^*(z) [\Gamma(g)\Psi_n](z) \,
{\rm d}v(z) \,.
\eeq



For a finite-dimensional irrep, the raising and lowering
operators satisfy the Hermitian adjoint relationship
\beq \hat C^\dagger_{i\nu} = \hat C_{\nu i} \, .\eeq
Therefore, the overlap kernel for a unitary irrep of SU$(p+q)$, derived as for
SU(2), is given by
\beq {\cal S}^{\sigma = \lambda}(z,x) = \langle 0|e^{\hat z} e^
{\hat x^\dagger} |0\rangle = [\det (I_p+z x^\dagger)
]^\lambda \,. \label{eq:4.198}
\eeq
For an infinite-dimensional unitary representation of  SU$(p,q)$, for
which
$\hat
C_{i\nu}^\dagger = - \hat C_{\nu i}$, the overlap kernel is calculated 
in a manner analogous to that used to find eqn.\ (\ref{eq:92})  
for $SU(1,1)$;  
the result is given formally by  
\beq {\cal S}^{\sigma = -\lambda}(z,x) = \langle 0|e^{\hat z} e^
{\hat x^\dagger} |0\rangle
=[ \det (I_p-z x^\dagger )]^{-\lambda}\,.
\eeq
This expression does not converge for $|zx^\dagger|\geq 1$.
However, it is well--defined as a distribution over the polynomial
functions of ${\cal F}_D$ with the understanding that 
$\hat {\cal S}^{-\lambda}\psi$ is defined by the integral
\beq [\hat {\cal S}^{-\lambda}\psi](z) = 
\int  {\rm Taylor} \, \bigl( {\cal S}^{-\lambda}(z,x) 
\bigr)\,\psi(x)\, {\rm d}v(x)\,
\eeq
where ${\rm Taylor} \, \bigl( {\cal S}^{-\lambda}(z,x)\bigr)$ is an
expansion of ${\cal S}^{-\lambda}(z,x)$ as a Taylor
series in $z x^\dagger$ and the integration is performed term by term. 

It is useful to note that, for both $G= {\rm SU}(p+q)$ and $G =
{\rm SU}(p,q)$, the above construction leads naturally to irreps
in bases that reduce the subgroup  ${\rm S[U}(p)\times {\rm
U}(q)]\subset G$. 
First observe that the action  on ${\cal F}$ of an element
\beq h = \pmatrix{a&0\cr 0&d\cr} \in {\rm S[U}(p)\times {\rm U}(q)] \,,
\eeq 
given by eq.\ (\ref{eq:SUpqaction}), i.e.,
\beq  [\Gamma (h) \varphi](x) = [\det{a}]^\sigma \varphi(a^\dagger x d)
\,,
\eeq
is unitary.
This follows from the observation that the inner product for ${\cal F}$,
which is expressible in the form
\beq \int \psi^*(x) \varphi(x)\, {\rm d}v(x) = \big[
\psi(\partial/\partial x) \varphi(x) \big]_{x=0} \,,
\eeq 
satisfies the identity
\beq  \int \psi^*(a^\dagger xd) \varphi(a^\dagger xd)\, {\rm d}v(x)
= \int \psi^*(x) \varphi(x)\, {\rm d}v(x) \,,
\eeq
and is manifestly ${\rm S[U}(p)\times {\rm U}(q)]$ invariant.
Thus, if $\{ \varphi_{\kappa\nu}\}$ is an orthonormal 
${\rm S[U}(p)\times {\rm U}(q)]$--coupled basis for ${\cal F}$, then
$\hat {\cal S}^\sigma$ is diagonal and ${\cal S}^\sigma(z,x)$ is
expressible 
\beq {\cal S}^\sigma (z,x) =  \sum_{\kappa\nu} K^2_\kappa\,
\varphi_{\kappa\nu} (z) \varphi^*_{\kappa\nu}(x) \, , \label{eq:203}
\eeq
where
\beq K_{\kappa}^2 = \int\!\!\int \varphi_{\kappa\nu}^*(z) {\cal
S}^\sigma (z,x)\varphi_{\kappa\nu} (x)\,{\rm d}v(z)\,{\rm d}v(x) \,,
\eeq
is independent of $\nu$.

It is now relatively straightforward to construct an
orthonormal ${\rm S[U}(p)\times {\rm U}(q)]$ basis for the Bargmann space
${\cal F}$ and to  map it to orthonormal bases for ${\cal F}_{\Hb}$ and
${\cal F}^{\Hb}$.
First observe that, in the coherent state representation,  the
U$(p)$ and U$(q)$ operators are given, respectively,  by
\beqa 
\Gamma(C_{ij}) = \displaystyle\sigma\delta_{ij} - \sum_\nu z_{j\nu}
{\partial_{i\nu}}\,, 
\label{eq:4.215}\\
 \Gamma(C_{\mu\nu}) = \displaystyle \sum_i z_{i\mu}
{\partial_{i\nu}} \,,\label{eq:4.216}
\eeqa
where
\beq \partial_{i\nu} = {\partial\over \partial z_{i\nu}} \,. \eeq



To simplify the notation, relabel the $z$ indices 
such that $z_{p+1-i, p+\nu}$ becomes $z_{i\nu}$.
The variables $\{ z_{i\nu}\}$ are then ordered by their weights with
$z_{i\mu}$ being of higher weight than $z_{j\nu}$ if $i<j$ or if $i=j$ and
$\mu<\nu$.  
The space ${\cal F}$ is now seen to be a direct sum of irreducible 
 ${\rm S[U}(p)\times {\rm U}(q)]$ subspaces spanned by homogeneous
polynomials in the $\{ z_{i\nu}\}$ variables.
Consider the polynomial
\beq  \varphi_{\kappa\kappa} (z) = N_\kappa \,Z_1^{\kappa_1-\kappa_2}(z)
Z_2^{\kappa_2-\kappa_3}(z) \cdots Z_m^{\kappa_m}(z) \,,
\eeq
indexed by a sequence $\kappa = (\kappa_1,\kappa_2,\ldots,
\kappa_m)$ of $m$ positive integers, satisfying the inequalities
\beq  \kappa_1 \geq \kappa_2 \geq \cdots \geq \kappa_m \geq 0 \,, 
\quad m\leq \min (p,q) \,,
\eeq
with $(Z_1(z),Z_3(z), Z_3(z),\ldots )$  the sequence of
determinants
\beq Z_1(z) = z_{11} \,, \quad Z_2(z) =
\left| \matrix{z_{11} & z_{12} \cr z_{21} & z_{22} \cr}\right| \,,\quad
 Z_3(z) =\left| \matrix{z_{11} & z_{12} &z_{13}\cr z_{21} & z_{22}& z_{23}
\cr z_{31} & z_{32} & z_{33} \cr}\right|
\,,\quad \ldots \,;
\eeq
$N_\kappa$ is a normalization factor. 
The polynomial $\varphi_{\kappa\kappa} (z)$ is seen from
eqns.~(\ref{eq:4.215}) and (\ref{eq:4.216}) to be of 
${\rm U}(p)\otimes{\rm U}(q)$  weight
\beq (\sigma, \ldots ,\sigma - \kappa_m, \ldots \sigma-\kappa_1) \otimes
(\kappa_1,\ldots , \kappa_m, \ldots , 0) \,.
\eeq
Thus, it is the highest weight polynomial of an irreducible ${\rm
S[U}(p)\times {\rm U}(q)]$ invariant subspace ${\cal F}^\kappa$ of ${\cal
F}$.  Let $\{\varphi_{\kappa\tau}\}$ denote an orthonormal basis for
${\cal F}^\kappa$. 

The norm factor $N_\kappa$ is determined by  the requirement that
\beq \int \varphi^*_{\kappa\kappa}(z)\,
 \varphi_{\kappa\kappa}(z)\,{\rm d}v(z) =
[\varphi^*_{\kappa\kappa}(\nabla)\varphi_{\kappa\kappa}(z)]_{z=0}
=1\,,
\label{eq:209}\eeq
where
\beq \varphi^*_{\kappa\kappa}(\nabla) = N^*_\kappa\
\nabla_1^{\kappa_1-\kappa_2} \nabla_2^{\kappa_2-\kappa_3} \cdots
\nabla_m^{\kappa_m} 
\eeq
with
\beq \nabla_1 = \partial_{11} \,, \quad \nabla_2 =
\left| \matrix{\partial_{11} & \partial_{12} \cr \partial_{21} &
\partial_{22}\cr}\right| \,,\quad
 \nabla_3 =\left| \matrix{\partial_{11} & \partial_{12} &\partial_{13}\cr
\partial_{21} & \partial_{22}& \partial_{23}
\cr \partial_{31} & \partial_{32} &\partial_{33} \cr}\right|
\,,\quad \ldots\,.
\eeq
The overlap integral (\ref{eq:209}) is evaluated by use of the
Capelli identity \cite{Weyl}
\beq  \nabla_n Z_n(z) = \left| \matrix{n+ z_1\cdot\partial_1 & z_1\cdot
\partial_2 &  z_1\cdot\partial_3 &\cdots & \cdots &z_1\cdot \partial_n
\cr z_2\cdot\partial_1& n-1+z_2\cdot \partial_2 &z_2\cdot\partial_3& 
   \cdots& \cdots &z_2\cdot \partial_n \cr
\cdots & \cdots &\cdots &\cdots &\cdots &\cdots \cr
\cdots & \cdots &\cdots &\cdots &\cdots &\cdots \cr
z_n\cdot\partial_1& +z_n\cdot \partial_2 &z_n\cdot\partial_3& 
   \cdots& \cdots &1+z_n\cdot \partial_n \cr}\right| \,,
\eeq
where 
\beq z_i\cdot \partial_j =\sum_\nu z_{i\nu} \partial_{j\nu}
\eeq
and it is understood that a determinant of non--commuting operators is
ordered by columns, e.g.,
\beq \left|\matrix{ A&B\cr C&D\cr}\right| = AD-CB \,. \eeq
Thus, when acting on a highest weight polynomial
$\varphi_{\kappa\kappa}$ for which $\kappa_n\not= 0$ but $\kappa_{n+1}
=0$,
\beq \nabla_n Z_n(z) \varphi_{\kappa\kappa}
= (n+\kappa_1)(n-1+\kappa_2) \cdots (1+\kappa_n) \varphi_{\kappa\kappa}
\,.
\eeq
By repeated use of this identity we obtain the expression for the norm
factors
\beq  N_\kappa^{2} = {\prod_{i<j} (\kappa_i-\kappa_j + j-i) \over 
\prod_{i=1}^m (\kappa_i + m-i)!} \,.
\eeq
It is interesting to note that this norm factor is related to the
dimensionality of an irrep of $S_N$ labelled by a partition $\kappa$ of
$N= \sum_i\kappa_i$ as follows:  
\beq\dim{\kappa} =N!\, N_\kappa^2 \, .  \eeq   







The overlap kernel for SU$(p+q)$, given by eqn.\ (\ref{eq:4.198}),
is
\beq {\cal S}^\lambda (z,x)  = 
 \left| \matrix{1+ z_1\cdot x^*_1 & z_1\cdot x^*_2 & 
z_1\cdot x^*_3 &\cdots & \cdots &z_1\cdot x^*_p
\cr z_2\cdot x^*_1& 1+z_2\cdot x^*_2&z_2\cdot x^*_3& 
   \cdots& \cdots &z_2\cdot x^*_p \cr
\cdots & \cdots &\cdots &\cdots &\cdots &\cdots \cr
\cdots & \cdots &\cdots &\cdots &\cdots &\cdots \cr
z_p\cdot x^*_1& +z_p\cdot x^*_2
&z_p\cdot x^*_3& 
   \cdots& \cdots &1+z_p\cdot x^*_p \cr}\right|^\lambda \,,
\eeq
where
\beq z_i\cdot x^*_j = \sum_\nu z_{i\nu} x^*_{j\nu} \,.
\eeq
Thus, ${\cal S}^\lambda$ is the $\lambda$ power of a sum of diagonal
minors of a determinant, i.e., determinants of submatrices obtained by
deleting some number of rows and corresponding columns. Moreover, since
$\varphi_{\kappa\kappa}$ is a highest weight polynomial,  it follows that
\beq \int{\cal S}^\lambda (z,x) \varphi_{\kappa\kappa} \, {\rm d}v(x)
= \int \Big(1+\sum_{n=1}^{\min (p,q)} \Lambda_n(z,x)\Big)^\lambda
\varphi_{\kappa\kappa} (x) \, {\rm d}v(x)
\label{eq:223}\eeq
where
\beq  \Lambda_n(z,x) = \left| \matrix{z_1\cdot x^*_1 & z_1\cdot
 x^*_2 & \cdots &z_1\cdot x^*_n
\cr 
\cdots & \cdots  &\cdots &\cdots \cr
z_n\cdot x^*_1& +z_n\cdot  x^*_2
& \cdots &z_n\cdot x^*_n \cr}\right|
= Z_n(z) Z_n^*(x) \,.
\eeq
We conclude that
\beq \int{\cal S}^\lambda (z,x) \varphi_{\kappa\kappa} \, {\rm d}v(x)
= \left(\matrix{\lambda \cr (\kappa_1-\kappa_2)
(\kappa_2-\kappa_3)
\ldots \kappa_p  \cr} \right) 
N^{-2}_\kappa\, \varphi_{\kappa\kappa} (z)\,,
\eeq
where $\bigl( \quad \bigr)$ 
is a multinomial coefficient, and  that 
\beq {\cal S}^\lambda (z,x) =  \sum_{\kappa\nu} K^2_\kappa\,
\varphi_{\kappa\nu} (z) \varphi^*_{\kappa\nu}(x) \,.
\eeq
with
\beq K_{\kappa}^2 = {\lambda! \over (\lambda - \kappa_1)!
(\kappa_1-\kappa_2)!  (\kappa_2-\kappa_3)! \ldots \kappa_p! } 
N^{-2}_\kappa
\,.
\eeq

For SU$(p,q)$, instead of eqn.\ (\ref{eq:223}),        we have  
\beq \int{\cal S}^{-\lambda} (z,x)\, \varphi_{\kappa\kappa}(x) \, {\rm
d}v(x) =  \Big(1-\sum_{n=1}^p \Lambda_n(z,x)\Big)^{-\lambda}
\varphi_{\kappa\kappa} (z)\,.
\eeq
Thus, for SU$(p,q)$, ${\cal S}^{-\lambda} (z,x)$ is again of the form
\beq  {\cal S}^{-\lambda} (z,x) =  \sum_{\kappa\nu} K^2_\kappa\,
\varphi_{\kappa\nu} (z) \varphi^*_{\kappa\nu}(x) \,,
\eeq
but now with
\beq K_{\kappa}^2 = {(\lambda + \kappa_1-1)! \over
(\lambda-\kappa_1)!(\kappa_1-\kappa_2)!  (\kappa_2-\kappa_3)! \ldots
\kappa_p! } N^{-2}_\kappa
\,.
\eeq  


\subsection{Sp$(n,\Rb)$ and Sp$(n)$}

The compact symplectic group Sp$(n)$ is isomorphic to the subgroup of
unitary matrices in ${\rm Sp}(n,\Cb)$; it is isomorphic to
\beq {\rm Sp}(n) \cong {\rm Sp}(n,\Cb) \cap {\rm U(2n)} \eeq
which means that an element of Sp$(n)$ is a $2n\times 2n$ matrix of the
form
\beq g = \pmatrix{ \alpha &-\beta\cr \beta^* & \alpha^*\cr} \eeq
with
\beq \alpha \alpha^\dagger +\beta\beta^\dagger = I_n  \, ,\quad
\alpha\tilde \beta = \beta\tilde\alpha \, ,\eeq
where $I_n$ is the $n\times n$ identity matrix and $\tilde \beta$ is
the transpose of $\beta$.
Similarly Sp$(n,\Rb)$ is given by the isomorphism
\beq {\rm Sp}(n,\Rb) \cong{\rm Sp}(n,\Cb) \cap {\rm U(n,n)} \eeq
and an element of Sp$(n,\Rb)$ is a $2n\times 2n$ matrix of the
form
\beq g = \pmatrix{ \alpha &\beta\cr \beta^* & \alpha^*\cr} \eeq
with
\beq \alpha \alpha^\dagger -\beta\beta^\dagger = 
I_n    \, ,\quad
\alpha\tilde \beta = \beta\tilde\alpha \, ,\eeq


The Lie algebras of the groups Sp$(n,\Rb)$ and Sp$(n)$ have a common
complex extension $C_n$.
Let $E_{ij}$ denote the matrix with entries $(E_{ij})_{kl} =
\delta_{ik}\delta_{jl}$.
Then the complex Lie algebra $C_n$ is spanned by the matrices 
\beqa &C_{ij} = \pmatrix{ E_{ij} &0\cr 0& -E_{ji}\cr} , &
\quad i, j = 1,\ldots , n   \\
&A_{ij} = A_{ji} = \pmatrix{ 0& - E_{ij} - E_{ji} \cr 0&0\cr} ,&\quad i,j =
1, \ldots , n, \\
&B_{ij} = B_{ji} = \pmatrix{ 0& 0 \cr E_{ij} + E_{ji} &0\cr} ,&
\quad i, j = 1,\ldots , n \eeqa
which generate the group elements 
\beqa  g(x,y) &=& \exp \sum_{ij}\left[ x_{ij} C_{ij} + y_{ij} A_{ij} +
y^*_{ij} B_{ij} \right]  \quad \in {\rm Sp}(n) \,, \\
g(x,y) &=& \exp \sum_{ij}\left[ x_{ij} C_{ij} + y_{ij} A_{ij} -
y^*_{ij} B_{ij}\right]  \quad \in {\rm Sp}(n,\Rb) \,,
\eeqa
with $x$ skew Hermitian (i.e., $x_{ij} = -x_{ji}^*$). 

Let $T$ denote an irrep of $C_n$ with lowest weight in which the
matrices $A_{ij}$, $B_{ij}$, and $C_{ij}$ are mapped to linear operators 
$\hat A_{ij}$, $\hat B_{ij}$, and $\hat C_{ij}$ on  a Hilbert space
$\Hb$.
We consider an irrep $T$ defined by an integer $\sigma$ for which
the lowest weight state satisfies 
\beq \hat B_{ij} |0\rangle = 0\,,\quad\hat C_{ij} 
|0\rangle = \delta_{ij}\, \sigma |0\rangle \,.\eeq
When $\sigma<0$, $T$ 
integrates to a unitary irrep of Sp$(n)$ for which
\beq \hat A_{ij}^\dagger = -\hat B_{ij} \,,\quad \hat C_{ij}^\dagger = \hat
C_{ji} \,.\eeq
And when $\sigma >0$, it integrates to a unitary irrep of
Sp$(n,\Rb)$  for which
\beq \hat A_{ij}^\dagger = \hat B_{ij} \,,\quad \hat C_{ij}^\dagger = \hat
C_{ji} \,.\eeq
The special case $\sigma =0$ is the trivial identity representation.
Note that one also obtains irreps of $C_n$ when $\sigma$ is a positive
half-odd integer.
These restrict to irreps of sp$(n,\Rb)$ that integrate to unitary irreps of
the metaplectic double cover of the group Sp$(n,\Rb)$.
In all non-trivial cases, the subgroup of elements that leave the
lowest-weight state invariant is U$(n)$. 


As in the previous example,  let $\Hb_D$ be the space of $K$-finite
vectors. 
Let $D$ denote the vector space of complex
symmetric
$n\times n$ matrices and let $N$ be the group 
\beq N = \left\{ \pmatrix{ I_n & 0\cr z & I_n};  
z \in D\right\} \,.\eeq
An element of $N$ is then represented by the operator
$\hat z =  \exp\big[\half\sum_{ij}{z_{ij}\hat B_{ij}}\big]$  
 on $\Hb_D$ and a
coherent state wave function $\Psi$ for a state 
$|\psi\rangle \in \Hb_D$ is defined by   
\beq \Psi(z) = \langle 0|e^{\hat z}|\psi\rangle \, .\eeq
Again, these coherent state wave functions are polynomials in the
matrix coefficients of $z$. 

The coherent state action $\Gamma(g)$ of an Sp$(n,\Rb)$ 
or Sp$(n)$ matrix
\beq g = \pmatrix{\alpha &b\cr\beta^*&\alpha^*\cr},\quad b = \pm \beta
\,,\eeq  
defined by
\beq [\Gamma(g) \Psi](z) = \langle 0| e^{\hat z} T(g) |\psi\rangle \,
,\eeq is determined by the generalized Gauss (Harish-Chandra)
factorization of the product 
\beq \pmatrix{ I & 0\cr z & I}
\pmatrix{\alpha &b\cr\beta^*&\alpha^*\cr} =  \pmatrix{ I & x\cr 0 & I}
\pmatrix{a &0\cr 0&\tilde a^{-1}\cr}
\pmatrix{ I & 0\cr y & I} \,,\quad z\in D \,,\eeq
with
\beq a = (\alpha^\dagger +\tilde b z)^{-1} \,,\quad
x= b(\alpha^* +zb)^{-1} \,,\quad y = (\alpha^* +zb)^{-1} (\beta^* +z\alpha )
\,.\eeq  
This gives
\beq  \langle 0| e^{\hat z}  T(g)|\psi\rangle =
[\det a]^\sigma\langle 0|e^{\hat y}|\psi\rangle
\eeq
and
\beq [\Gamma(g) \Psi](z) =  [\det (\alpha^\dagger +\tilde b
z)]^{-\sigma} \Psi( (\alpha^* +zb)^{-1} (\beta^* +z\alpha ))\,.\eeq


For a unitary irrep of Sp$(n)$, for which
$\hat B_{ij}^\dagger = -\hat A_{ij}$,  we derive the overlap
kernel
\beq {\cal S}^{\sigma = -\lambda}(z,x^\dagger) = \langle 0|
e^{\hat z}e^{{\hat x}^\dagger} |0\rangle = [ \det (I_n+
z x^\dagger )]^\lambda \,,
\eeq
where ${\hat x}^\dagger = -\half\sum_{ij}{x^*_{ij}\hat A_{ij}}$.
The Hilbert space ${\cal F}_\Hb$ is then the space of holomorphic
functions in the matrix coefficients of $z$ with inner product
\beq \langle \varphi|\psi \rangle = \int\!\!\!\int \varphi^*(z) \,
[ \det (I_n+z x^\dagger )]^{\lambda}\, \psi (x) \, {\rm d}v(z)     
\, {\rm d}v(x) \, ,\eeq
where the integral is over $D$, which can be identified with 
the $n(n+1)/2$-dimensional complex plane
and the measure d$v$ is the corresponding Bargmann measure.


For a unitary representation of  Sp$(n,\Rb)$, for which
$\hat B_{ij}^\dagger = \hat A_{ij}$,
\beq {\cal S}^{\sigma = \lambda}(z,x^\dagger) =
\langle 0|e^{\hat z}e^{{\hat x}^\dagger} |0\rangle = [ \det (I_n -
z x^\dagger )]^{-\lambda} \,,
\eeq
when $\det (I_n-z x^\dagger ) >0 $,
and ${\cal F}_\Hb$ is the Hilbert space of holomorphic
functions obtained by completing the space of polynomials with the 
inner product
\beq \langle \varphi|\psi \rangle = \int\!\!\!\int \varphi^*(z) \, \psi
(x)\, {\rm Taylor} \Big([ \det (I_n-z x^\dagger )
]^{-\lambda}\Big) \, {\rm d}v(z)\, {\rm d}v(x) \,,
\eeq
where the integral is again over the whole $n(n+1)/2$-dimensional complex
plane with the understanding that ${\rm Taylor} \, \bigl([ \det (I_n-z
x^\dagger ) ]^{-\lambda}\bigr)$ is an
expansion of $[ \det (I_n-z x^\dagger )]^{-\lambda}$ as a Taylor
series in $z x^\dagger$ and the integration is performed term by term. 


In both cases if the map
${\cal F}_\Hb\to \Hb$ sends $\varphi \to |\varphi\rangle$ and the map
$\Hb\to {\cal F}^\Hb$ sends $|\psi\rangle \to \Psi$, then the inner
product of the two states is the overlap
\beq  \langle \varphi|\psi\rangle = \int \varphi^*(z) \Psi(z) \, {\rm
d}v(z)\,.
\eeq
Thus, as usual, the spaces ${\cal F}_\Hb$ and ${\cal F}^\Hb$ are in
duality with respect to the measure d$v$ and  matrix elements of the
representation are given by
\beq  \langle \varphi|T(g)|\psi\rangle = \int \varphi^*(z)
[\Gamma(g)\Psi](z) \, {\rm d}v(z)\,.
\eeq


An orthonormal U$(n)$ basis $\{
\varphi_{\kappa\nu}\}$ for the Bargmann space ${\cal F}$ can  be
constructed following the techniques given in section IVA albeit using a
generalization of the Capelli identity
\cite{LeB&R}. 
Similarly, by an expansion of the overlap kernels as in eqn.\
(\ref{eq:203}), one can determine the renormalization factors $\{
K_\kappa\}$ which map this basis for ${\cal F}$ to orthonormal bases
$\{ \psi_{\kappa\nu} = K_\kappa^{-1} \varphi_{\kappa\nu}\}$ and
$\{ \Psi_{\kappa\nu}= K_\kappa \varphi_{\kappa\nu}\}$ for ${\cal F}_\Hb$
and ${\cal F}^\Hb$, respectively. 

\subsection{SO$(2n)$ and SO*$(2n)$}


The real orthogonal group SO$(2n)$ is isomorphic to the subgroup
of unitary matrices in the  complex orthogonal group  SO$(2n,\Cb)$;
\beq {\rm SO}(2n) \cong {\rm SO} (2n,\Cb) \cap {\rm U}(2n)
\,.\eeq 
Thus, if  SO$(2n,\Cb)$ is realized as the group of matrices
\beq {\rm SO}(2n,\Cb) =\{ M\in {\rm SL}(2n,\Cb)\, |\,
MJ\tilde M=J
\} \,,\eeq where 
\beq J = \pmatrix { 0&I_n\cr I_n&0\cr} \,,\eeq 
then an element $g\in {\rm SO}(2n)$ is a $2n\times 2n$ matrix of
the form
\beq g = \pmatrix{ \alpha &\beta\cr \beta^*&\alpha^*\cr}\eeq
with
\beq \alpha\alpha^\dagger + \beta\beta^\dagger = I_n  \,,\quad
\alpha\tilde\beta + \beta\tilde\alpha = 0\,.\eeq
Similarly, SO*$(2n)$ is given by the isomorphism
\beq {\rm SO}^*(2n) \cong {\rm SO} (2n,\Cb) \cap {\rm U}(n,n)
\eeq 
and an element of SO*$(2n)$ is a $2n\times 2n$ matrix of the
form
\beq g = \pmatrix{\alpha &\beta\cr -\beta^*&\alpha^*\cr}\eeq
with
\beq \alpha\alpha^\dagger - \beta\beta^\dagger = I_n \,,\quad
\alpha\tilde\beta + \beta\tilde\alpha = 0\,.\eeq



The Lie algebras of the groups SO$(2n)$ and SO*$(2n)$ have
common complex extension $D_n$.
Let $E_{ij}$ denote the matrix with entries $(E_{ij})_{kl} =
\delta_{ik}\delta_{jl}$.
Then the complex Lie algebra $D_n$ is spanned by the matrices 
\beqa & C_{ij} = \pmatrix{ E_{ij} &0\cr 0&-E_{ji}\cr}\,,&
\quad i, j = 1,\ldots , n \\
&A_{ij} = \pmatrix{0& E_{ij}-E_{ji}\cr 0&0\cr} = -A_{ji}\,,&
\quad i, j = 1,\ldots , n ,\\
&B_{ij} = \pmatrix{ 0&0\cr E_{ij}-E_{ji} &0\cr} = -B_{ji} \,,&
\quad i, j = 1,\ldots , n, \eeqa
which generate the group elements
\beqa  g(x,y) &=& \exp \sum_{ij}\left[ x_{ij} C_{ij} + y_{ij} A_{ij} +
y^*_{ij} B_{ij}\right]  \quad \in {\rm SO}(2n) \,, \\
g(x,y) &=& \exp \sum_{ij}\left[ x_{ij} C_{ij} + y_{ij} A_{ij} -
y^*_{ij} B_{ij}\right] \quad \in {\rm SO}^*(2n) \,,
\eeqa
with $x$ skew Hermitian (i.e., $x_{ij} = -x_{ji}^*$). 

Let $T$ denote an irrep of $D_n$ with lowest weight in which the
matrices $A_{ij}$, $B_{ij}$, and $C_{ij}$ are mapped to linear operators 
$\hat A_{ij}$, $\hat B_{ij}$, and $\hat C_{ij}$ on  a Hilbert space
$\Hb$.
We consider an irrep $T$ defined by an integer $\sigma$ for which
the lowest weight state satisfies 
\beqa & \hat B_{ij} |0\rangle = 0\,,&\nonumber\\
&\hat C_{ij} |0\rangle = \delta_{ij} \sigma |0\rangle \,. 
& \eeqa
For $\sigma \leq 0$ this $D_n$ irrep integrates to a unitary irrep of
SO$(2n)$ for which
\beq \hat A^\dagger_{ij} = -\hat B_{ij} \,,\quad \hat C^\dagger_{ij} =
\hat C_{ji} \,.\eeq
For $\sigma>0$ it integrates to a unitary irrep of SO*$(2n)$ and
\beq \hat A^\dagger_{ij} = \hat B_{ij} \,,\quad \hat C^\dagger_{ij} =
\hat C_{ji} \,.\eeq
The special case $\sigma =0$ is the trivial identity representation.
One also obtains spinor irreps when $\sigma$ is a half-odd integer.
In all non-trivial cases, the subgroup of elements that leave the
lowest-weight state invariant is U$(n)$.

Let $\Hb_D$ be the subspace of $K$-finite vectors in $\Hb$.  
Let $D$ denote the vector space of complex  symmetric
$n\times n$ matrices and let $N$ be the group of lower
triangular matrices 
\beq N = \left\{ \pmatrix{ I_n & 0\cr z & I_n};  
z \in D\right\} \,.\eeq
An element of $N$ is then represented by the operator
$e^{\hat z}$ with $\hat z =  \half\sum_{ij}{z_{ij}\hat B_{ij}}$ 
 on $\Hb_D$ and a coherent state wave
function $\Psi$ for a state 
$|\psi\rangle \in \Hb_D$ is defined by 
\beq \Psi(z) = \langle 0|e^{\hat z}|\psi\rangle \, .\eeq
Again, these coherent state wave functions are polynomials in the
matrix coefficients of $z$.



The coherent state action $\Gamma(g)$ of an SO$(2n)$ or
SO*$(2n)$ matrix
\beq g = \pmatrix{ \alpha &\beta\cr b^* &\alpha^*\cr}\,,\quad b= \pm
\beta\,,\eeq
defined by
\beq [\Gamma (g) \Psi ](z) = \langle 0| e^{\hat z} T(g)
|\psi\rangle \,,\eeq is determined by the generalized Gauss
(Harish-Chandra) factorization of the product
\beq \pmatrix{ I&0\cr z&I\cr}
\pmatrix{ \alpha &\beta\cr b^* &\alpha^*\cr} =
\pmatrix{ I&x\cr 0&I\cr}
\pmatrix{ a&0\cr 0&\tilde a^{-1}\cr}
\pmatrix{I&0\cr y&I\cr} , \quad z\in D\,,\eeq
with
\beq x=\beta (\alpha^*+z\beta)^{-1}\,,\quad
a= (\alpha^\dagger -\tilde\beta z)^{-1}\,,\quad
y = (\alpha^*+z\beta)^{-1} (b^*+z\alpha )\,.\eeq 
This gives
\beq \langle 0| e^{\hat z} T(g)
|\psi\rangle = [\det a]^\sigma \langle 0| e^{\hat y} |\psi\rangle
\eeq
and
\beq [\Gamma(g) \Psi ](z) = [\det (\alpha^\dagger -\tilde \beta
z)]^{-\sigma}
\Psi ((\alpha^* +z\beta)^{-1} (b^*+z\alpha))\,.\eeq

For a unitary irrep of SO$(2n)$, the overlap kernel is
\beq {\cal S}^{\sigma = -\lambda} (z,x^\dagger ) = \langle 0| e^{\hf\sum
z_{ij} \hat B_{ij}} e^{-\hf\sum x^*_{ij} \hat A_{ij} } |0\rangle = [\det
(I+z x^\dagger )]^\lambda \eeq
and the inner product is defined by
\beq \langle \psi |\varphi \rangle = \int\!\!\!\int
\psi^* (z)
\, [\det (I+x^\dagger z) ]^\lambda \, \varphi (x) \, {\rm d} v (z)\, {\rm
d} v(x)
\,,\eeq where the integral is over $D$,  which can be 
identified with the $n(n-1)/2$-dimensional complex
plane, 
and the measure d$v$ is the corresponding Bargmann measure.
For a unitary representation of SO*$(2n)$,
\beq {\cal S}^{\sigma = \lambda} (z,x^\dagger) =
 \langle 0| e^{\hf\sum z_{ij} \hat B_{ij}}
e^{\hf\sum x^*_{ij} \hat A_{ij}} |0\rangle = [\det
(I-zx^\dagger) ]^{-\lambda} \,,\eeq
when $\det(I-zx^\dagger)>0$, and the inner product is
\beq \langle \psi |\varphi \rangle = \int\!\!\!\int \psi^* (z)
\,  {\rm Taylor} \Big([\det (I-x^\dagger z) ]^{-\lambda} \Big) \,
\varphi (x) \, {\rm d} v (z)\,  {\rm d} v(x)\,,\eeq 
where the integral is again over the $n(n-1)/2$-dimensional complex
plane with the understanding that ${\rm Taylor} \, \bigl([ \det (I_n-z
x^\dagger ) ]^{-\lambda}\bigr)$ is an
expansion of $[ \det (I_n-z x^\dagger )]^{-\lambda}$ as a Taylor
series in $z x^\dagger$ and the integration is performed term by term. 


\subsection{SO$(p+2)$ and SO$(p,2)$}

The groups SO$(p+2)$ and SO$(p,2)$ are defined by
\beqa {\rm SO}(p+2) = &\left\{ M\in {\rm SL}(p+2,\Rb)\,
\Big|\, M J^+_{p+2}\tilde M= J^+_{p+2} \right\} &\\
{\rm SO}(p,2) = &\left\{ M\in {\rm SL}(p+2,\Rb)\, \Big|\, M
J^-_{p+2} \tilde M = J^-_{p+2} \right\} \,,&
\eeqa
where
\beq J^\pm_{p+2} = \pmatrix{ \pm I_2 & 0\cr  0& I_p\cr} \,; \eeq
$I_p$ and $I_2$ are, respectively, the $p\times p$ and $2\times 2$ identity
matrices.

Let
\beq L_{\alpha\beta}= -i (E_{\alpha\beta}-E_{\beta\alpha}) \,,\quad
\alpha,\beta =
1, \ldots , p+2 ,
\eeq
 denote the infinitesimal generators (angular momenta)
 of SO$(p+2)$; they satisfy the  commutation relations
\beq [L_{\alpha\beta},L_{\gamma\delta}] = -i
(\delta_{\beta\gamma} L_{\alpha\delta} - \delta_{\beta\delta} L_{\alpha\gamma}
+\delta_{\alpha\delta}L_{\beta\gamma} - \delta_{\alpha\gamma}L_{\beta\delta})\,.
\eeq
The common complex extension of the Lie algebras so$(p+2)$ and
so$(p,2)$ is then spanned by the matrices
\beq \matrix{ L_{12}  && \text{infinitesimal generator of
SO(2)} ,\cr L_{\alpha\beta} && \alpha,\beta =3,
\ldots , p+2 ,\cr A_\alpha = L_{1\alpha} + iL_{2\alpha} && \alpha = 3,
\ldots , p+2 ,\cr B_\alpha =  L_{1\alpha} - iL_{2\alpha} && \alpha = 3,
\ldots , p+2 .\cr}
\eeq
For example, $L_{12}$ is the matrix of block form
\beq L_{12} = \pmatrix{ l_{12} & 0\cr
0&0 
\cr} , \quad l_{12} = \pmatrix{0&-i\cr i &0\cr} . \eeq
Group elements are then expressed in terms of this basis by
\beqa  g(\varphi,\theta,x) &=& \exp i\Big[ \varphi L_{12}
+\sum_{\alpha\beta}\theta_{\alpha\beta}L_{\alpha\beta}
+ \sum_\alpha(x_\alpha A_\alpha + x_\alpha^*B_\alpha )\Big]  \quad
\in {\rm SO}(p+2) \,, \\ 
g(\varphi,\theta,x) &=& \exp i\Big[ \varphi L_{12}
+\sum_{\alpha\beta}\theta_{\alpha\beta}L_{\alpha\beta}
+ \sum_\alpha(x_\alpha A_\alpha - x_\alpha^*B_\alpha )\Big] \quad \in
{\rm SO}(p,2) \,,
\eeqa
where $\varphi$ is real, $\theta_{\alpha\beta}$ is real and antisymmetric,
and $x_\alpha$ is complex. 
 
By Gauss factorization, an SO$(p+2)$ or
SO$(p,2)$ matrix $g$ is expressed as a product
\beq g = e^{z\cdot A} \pmatrix{a & 0\cr 0& d\cr} e^{x^*\cdot B} \,,\eeq
with $a \in {\rm SO}(2,\Cb)$ and $d\in {\rm SO}(p,\Cb)$,
where  $z = (z_3,\ldots , z_{p+2})$  and $x^* = (x^*_3, \ldots ,
x^*_{p+2})$ are $p$-component complex vectors, and
\beq z\cdot A = \sum_\alpha z_\alpha A_\alpha\,, \quad
x^*\cdot B = \sum_\alpha x^*_\alpha B_\alpha \,. \eeq
Therefore, as matrices,
\beqa z\cdot A = \pmatrix{ 0& ez \cr -\tilde z \tilde e & 0\cr} ,&\quad&
e^{z\cdot A} = \pmatrix{ I_2 - \hf (z\cdot z) e \tilde e &ez\cr
-\tilde z\tilde e & I_p \cr} ,\\
x^*\cdot B = \pmatrix{ 0& -e^*x^* \cr x^\dagger e^\dagger & 0\cr} ,&\quad&
e^{x^*\cdot B} = \pmatrix{I_2- \hf (x^*\cdot x^*) e^* e^\dagger
&-e^*x^*\cr x^\dagger e^\dagger & I_p \cr} ,
\eeqa
where $\tilde z$ denotes the transpose
(column vector) of $z$, and $e$, $e^*$,
$\tilde e$, and $e^\dagger$ are the vectors
\beq e=\pmatrix{-i\cr 1} , \quad  e^* =\pmatrix{i\cr 1} , \quad
\tilde e=(-i,1) \,, \quad e^\dagger = (i,1) \,. \eeq
The matrix $a$ is expressible as a function of a complex parameter
$\phi$
in the form
\beq a(\phi)= e^{\phi L_{12}} = \pmatrix{ \cosh\phi &-i\sinh\phi \cr
i\sinh\phi &\cosh\phi\cr} .\eeq
It has the useful property that
\beq e^\dagger a(\phi) = e^\phi e^\dagger \,,\quad a(\phi) e = e^\phi
e\,.\label{eq:5.249}\eeq




We now consider a representation $T$ with highest weight state
$|0\rangle$, which maps $L_{\alpha\beta} \to \hat
L_{\alpha\beta}$, etc., and which, for integer values of $\sigma$, satisfies the
equations
\beq\hat A_\alpha |0\rangle = 0 \,, \quad
\hat L_{12} |0\rangle=\sigma |0\rangle \,, \quad
\hat L_{\alpha\beta}|0\rangle=0 \,.\eeq
For $\sigma\geq 0$, this irrep is finite dimensional
and integrates to a unitary irrep of
SO$(p+2)$ for which
\beq \hat A^\dagger_\alpha =\hat B_\alpha \,.\eeq
For $\sigma <0$, it is infinite dimensional 
and integrates to a unitary irrep of
SO$(p,2)$ with
\beq \hat A^\dagger_\alpha = - \hat B_\alpha \,.\eeq
The special case $\sigma =0$ is the trivial identity representation.
The highest weight state
$|0\rangle$ is seen to span a 1-dimensional irrep of the subgroup    
${\rm SO}(2, \Cb ) \times {\rm SO}(p , \Cb )$ with
\beq T\pmatrix{ a(\phi) & 0\cr 0& d\cr} |0\rangle = 
e^{\phi\hat L_{12} }|0\rangle  = e^{\sigma\phi} |0\rangle
\,.\eeq


A vector $|\psi\rangle$ in the space of $K$-finite vectors 
has holomorphic coherent state wave function
\beq \Psi (z) = \langle 0| e^{z\cdot \hat A} |\psi\rangle \,.\eeq
The action of a group element on coherent state wave functions is obtained by
considering the actions of each of the factors in the Gauss factorization.
One obtains immediately
\beq [\Gamma (e^{z\cdot A}) \Psi ](z') =\langle 0| e^{z'\cdot \hat A}
e^{z\cdot \hat A} |\psi\rangle = \Psi (z'+z) \,.\label{eq:5.255}\eeq
And, from the identity
\beq e^{z\cdot \hat A} \pmatrix{ a(\phi) & 0\cr 0& d\cr}=
\pmatrix{ a(\phi) & 0\cr 0& d\cr} e^{(e^{-\phi}zd)\cdot \hat A} \,,\eeq
it follows that
\beq \left[ [\Gamma \pmatrix{ a(\phi) & 0\cr 0& d\cr} \Psi
\right](z) =e^{\sigma\phi} \Psi (e^{-\phi}zd) \,.\eeq

The action
\beq [\Gamma (e^{x^*\cdot B}) \Psi ](z) =\langle 0| e^{z\cdot \hat A}
e^{x^*\cdot B} |\psi\rangle \label{eq:5.257}\eeq
is determined by Gauss factorization of the product
\beq  e^{z\cdot A}e^{x^*\cdot B}
= e^{X^*\cdot B} \pmatrix{a(\phi) & 0\cr 0& d\cr}  e^{Z\cdot A} \,.\eeq
This $(p+2)\times(p+2)$ matrix identity is equivalent to the four
matrix equations
\beqa &&[ I_2-\hf (X^*\cdot X^*) e^* e^\dagger ] a
[I_2-\hf (Z\cdot Z) e \tilde e] + (X^*d\tilde Z) e^* \tilde e \nonumber\\
&& \qquad\qquad =
[I_2-\hf (z\cdot z) e \tilde e] [I_2-\hf (x^*\cdot x^*) e^* e^\dagger ]
+ (z\cdot x^*) e e^\dagger \,,\label{eq:I}\\
&& e^\phi eZ - e^\phi (X^*\cdot X^*) e^* Z - e^* X^* d = ez-e^*x^* + (z\cdot z)
ex^* \,,\label{eq:II} \\
&& e^\phi X^\dagger e^\dagger - e^\phi (Z\cdot Z) X^\dagger \tilde e - d\tilde Z
\tilde e = x^\dagger e^\dagger - \tilde z \tilde e + (x^*\cdot x^*) \tilde z
e^\dagger \,,\label{eq:III}\\
&&2e^\phi X^\dagger Z + d = I_p + 2\tilde z x^\dagger \,.\label {IV}
\eeqa
where we have used eqn.\ (\ref{eq:5.249}).
The $2\times 2$ matrix equation (\ref{eq:I}) is equivalent to the
four equations 
\beqa &e^\phi = 1 + (x^*\cdot x^*) (z\cdot z) + 2 (x^*\cdot z) \,,&
\label{eq:V}\\
&e^\phi (Z\cdot Z) = z\cdot z \,,&\\
&e^\phi (X^*\cdot X^*) = x^*\cdot x^* \,,&\\
&e^\phi X^* dZ = x^*\cdot z \,, &
\eeqa
and the $2\times p$ matrix eqn.\ (\ref{eq:II}) is equivalent to the
two equations
\beqa & e^\phi Z + e^\phi (X^*\cdot X^*)Z + X^*d = z+x^*+(z\cdot z) x^*
\,, &\\
  & e^\phi Z -  e^\phi (X^*\cdot X^*)Z - X^*d = z-x^*+(z\cdot z) x^*
\,. &
\eeqa
Thus, we obtain 
\beq Z = e^{-\phi} [ z+(z\cdot z) x^*] \,.\eeq
These results imply that
\beq [\Gamma(e^{x^*\cdot B)} \Psi](z) = e^{\sigma\phi} \Psi (e^{-\phi}
[z+(z\cdot
z)x^*]) \label{eq:5.268}\eeq
with $e^\phi$ given by eqn.\ (\ref{eq:V}).

Eqns.\ (\ref{eq:5.255}), (\ref{eq:5.257}), and (\ref{eq:5.268}) are all
special cases of the general expression
\beq \left[ \Gamma (g) \Psi \right] (z) = e^{\sigma\phi (g)}
\Psi(zg) \,,\eeq
where, for
\beq g =   \pmatrix{A&B\cr C&D\cr} ,\eeq
\beqa &e^\phi (g) = \hf (e^\dagger - z\cdot z\, \tilde e) Ae + zCe \,,& \\
&zg = e^{-\phi}
\left\{ \hf (e^\dagger -z\cdot z\, \tilde e )B +zD\right\} \,.&\eeqa

For $\sigma = \lambda > 0$ (unitary irrep of O$(p+2)$), the overlap integral
\beq {\cal S}^\lambda (z,x) = \langle 0| e^{z\cdot A} e^{x^*\cdot B}
|0\rangle
\eeq is determined, from eqns.\ (\ref{eq:5.257}) and (\ref{eq:5.268}), to be
given by
\beq {\cal S}^\lambda(z,x) = [ 1+ (z\cdot z)(x^*\cdot x^*) + 2 z\cdot
x^*]^\lambda
\,.\eeq
This expression is well-defined for all $z$ and $x^*$ in the $p$-dimensional
complex plane.
For $\sigma = -\lambda <0$ (unitary irrep of O$(p,2)$),
\beqa {\cal S}^{-\lambda} (z,x) &=& \langle 0| e^{z\cdot A} e^{-x^*\cdot
B} |0\rangle \nonumber\\
&=& [ 1+ (z\cdot z)(x^*\cdot x^*) - 2 z\cdot x^*]^{- \lambda} 
\label{eq:4.342}\eeqa
for all $z$ and $x^*$ in the domain
\beq 
D = \left\{ z\in \Cb^p\, \big|\, 
1+|z\cdot z|^2 - 2 z^*\cdot z > 0 \right\} \,. 
\eeq
Because ${\cal S}^{-\lambda} (z,x)$ is holomorphic in the components
of $z$ and $x^\dagger$, it is now defined by eqn.~(\ref{eq:4.342}) for all
$z$ and $x$ by the Taylor series expansion
\beq  {\cal S}^{-\lambda} (z,x) = {\rm Taylor} [ 1+ (z\cdot z)(x^*\cdot
x^*) - 2 z\cdot x^*]^{- \lambda} \,,
\eeq  
 as in (\ref{eq:3.series}) for $SU(1,1)$. 
The integral expression given
below for the inner product then makes sense for all polynomials;
for any polynomials
$\psi$, $\varphi$,  only finitely many terms in the series make non-zero
contributions.  

In both cases, the inner products are given by
\beq \langle \psi |\varphi\rangle = \int \!\!\!\int \psi^*(z)\, {\cal S}^\sigma
(z,x)\, \varphi(x)\, {\rm d} v(z)\, {\rm d} v(x)\, ,\eeq where the integrals 
are with respect to the Bargmann measure 
over the whole $p$-dimensional complex plane. 



\section{Concluding remarks}

The above examples show that the coherent state method is very versatile 
and gives all the standard irreps  induced from one-dimensional subgroups.
Moreover, it does so in a way that considerably simplifies  the evaluation
of inner products.
Such simplification is invaluable in practical applications of the 
inducing construction, e.g., to problems in physics.
For example, the analysis of section IVC has recently been used effectively for
the calculation of SU(3) Clebsch-Gordan coefficients in an SO(3) basis
\cite{RB00}; coefficients that are used widely, for example, in nuclear
physics.

The above techniques become even more powerful and much more general in their
extension to the vector coherent state induction of irreps from
multi-dimensional irreps of a subgroup.
Applications of the triplets method to VCS representations will be given
in a sequel to the present paper.


\begin{appendix} 

\section{Inner products for standard holomorphic representations}

Let $T$ be an irrep of SU(2) on a Hilbert space $\Hb$ with highest
weight state $|0\rangle$. 
The standard method of finding the inner product of
coherent state wave functions 
for this representation \cite{Perelomov85} makes use of the
resolution of the identity
\beq \hat I = \int_{G_0} T^\dagger (g) |0\rangle \langle 0| T(g) \, {\rm
d}\mu (g) \,
,\eeq
where d$\mu$ is a right-invariant measure for the group $G$.
Substituting the Gauss factorization for an SU(2) matrix
\beq \pmatrix{a&-b^*\cr b&a^*\cr} = \pmatrix{1&0\cr b/a &1\cr}
\pmatrix{ a&0\cr 0&1/a\cr} \pmatrix{1&-b^*/a\cr 0&1\cr}
= e^{{b\over a} J_-} a^{2J_0} e^{{-b^*\over a} J_+} \, ,\eeq
and setting $z = -b^*/a$ with the constraint $|a|^2 = (1+|z|^2)^{-1}$, we then
have
\beq \hat I = \int T^\dagger (z) |0\rangle \langle 0| T(z) \,
(1+|z|^2)^{-2j}\, {\rm d}\sigma (z) \,
,\eeq
where d$\sigma (z)$ is the (suitably normalized) right $G$-invariant
measure for $N = \{ e^{zJ+}; z\in \Cb\}$.
Taking matrix elements between states of  $\Hb$
gives the inner product
\beq \langle \psi |\varphi\rangle = \int \Psi^*(z)\Phi(z) (1+|z|^2)^{-2j}\,
{\rm d}\sigma (z) \, ,\eeq
in terms of coherent state wave functions.
The SU(2)-invariant measure d$\sigma$ for $N$ is determined by considering
the right translations of $N$ by SU(2).
The appropriate SU(2)$:N\to N; z\mapsto z\cdot g$ action is obtained by
observing that
\beq  \pmatrix{1&z\cr 0&1\cr}\pmatrix{a&-b^*\cr b&a^*\cr}
= \pmatrix{1&0\cr {b\over bz+a} & 1\cr}
\pmatrix{bz+a & 0\cr 0 & {1\over bz+a}\cr}
\pmatrix{1& {a^*z-b^*\over bz+a} \cr 0 & 1\cr}
\eeq
which implies that
\beq  z\cdot g = {a^*z-b^*\over bz+a} \, .\eeq
By computing the Jacobian of the $z\to z\cdot g$ transformation, one  determines
 the invariant volume element
\beq {\rm d}\sigma (z) \propto (1+|z|^2)^{-2} {\rm d}x\, {\rm d} y \eeq
with $z= x+iy$; the factor $(1+|z|^2)^{-2}$ which gives an 
SU(2)-invariant volume
 in terms of the corresponding Euclidean volume is known as a Bergman kernel
\cite{Hua}. Finally, the normalization constant is fixed by requiring the
coherent-state wave function for the highest weight state to be normalized
to unity.
Thus, one obtains the inner product
\beq \langle \psi |\varphi\rangle = {2j+1\over \pi} \int\!\!\!\int
{\Psi^*(z)\Phi(z) \over (1+|z|^2)^{2j+2}}\,  {\rm d}x \, {\rm d}y \, .\eeq
 \end{appendix}

\begin{references}  

\bibitem {Wig30} E. Wigner, ``Uber die elastischen Eigenschwingungen
  symmetrischer  Systeme", Nachr. Ges. Wiss. G\"ottingen Math.-Phys. Kl.
  (1930) 133-146.
\bibitem{Wig39} E.~P.~Wigner, {\it Ann.\ Math.}\ {\bf 40} (1939) 149-204.
\bibitem{Mackey}  G.~W.~Mackey, Annals of Math.\ {\bf55}, 101 (1952);
    {\bf 58}, 193 (1952);
    ``Induced representations of groups and quantum mechanics",
                         (Benjamin, New York, 1969).
\bibitem{Knapp} A.~W.~Knapp, ``Representation theory of  semi-simple
   groups", Princeton Univ.\ Press N.J., 1986.
\bibitem{HC}  Harish-chandra, Am.\ J.\ Math.\ {\bf 78}, 1, 564 (1956).
\bibitem{Schr} E. Schr\"odinger, ``Der Stetige Ubergang von der Mikro-zur
  Makromechanik", Naturwissenschaften {\bf 14}, 664-666 (1926).
\bibitem{glauber} R.~Y.~Glauber, Phys.\ Rev.\ {\bf 131}, 2766
   (1963).
\bibitem{klauder} 
  J. R. Klauder, {\it J.\ Math.\ Phys.} {\bf 4}, 1058 (1963).
\bibitem{perelomov}
  A. Perelomov, {\it Commun.\ Math.\ Phys.} {\bf 26}, 222 (1972).
\bibitem{gilmore72}
  R. Gilmore, {\it Ann.\ Phys.\ (N.Y.)} {\bf 74}, 391 (1972).
\bibitem{Barg} V.~Bargmann, Commun.\  Pure Appl.\  Math.\ 
   {\bf 14}, 187 (1961).
\bibitem{Segal} I.\ Segal, ``Mathematical problems of relativistic
   physics", (Providence, Rhode Island, 1963).
\bibitem{Perelomov85}  A.~Perelomov, ``Generalized coherent states and
 their applications",  Springer (1985).
\bibitem{Onofri} E.~Onofri, J.~Math.\ Phys.\ {\bf 16}, 1087 (1975).
\bibitem{VCS} D.J.~Rowe, J.\ Math.\  Phys. {\bf 25}, 2662 (1984);
  D.J.~Rowe, G.~Rosensteel and R.~Carr,   J.\ Phys.\ A:
  Math.\ Gen. {\bf 17}, L399 (1984).
\bibitem{RRG} D.J.~Rowe, G.~Rosensteel, and
  R.~Gilmore,  J.\ Math.\ Phys. {\bf 26}, 2787 (1985).
\bibitem{RR91} D.J.~Rowe and J.~Repka,   J.\ Math.\ Phys. {\bf 32}, 2614
  (1991).
\bibitem{BRR} S.\ Bartlett, D.J.\ Rowe, and J.\ Repka, ``Coherent state
representations as geometric quantization", (in preparation).
\bibitem{HLeBR} K.T.\ Hecht, R.\ Le Blanc, and D.J.\ Rowe, J.\ Phys.\
  A: Math.\ Gen.\ {\bf 20}, 2241 (1987).
\bibitem{RLeBH} D.J.\ Rowe, R.\ Le Blanc, and K.T.\ Hecht, J.\ Math.\
  Phys.\ {\bf 29}, 287 (1988).
\bibitem{LeBR88}  R.\ Le Blanc, and D.J.\ Rowe,  J.\ Math.\
  Phys.\ {\bf 29}, 758, 767 (1988).
\bibitem{LeBR89} R.\ Le Blanc, and D.J.\ Rowe,  J.\ Math.\
  Phys.\ {\bf 30}, 1415 (1989); {\bf 31}, 14 (1990);
  C.\ Quesne, J.\ Phys.\ A: Math.\ Gen.\ {\bf 23}, 5383, 5411 (1990).
\bibitem{RR95} D.J.~Rowe and J.~Repka,   J.\ Math.\ Phys. {\bf 36}, 2008
  (1995).
\bibitem{RR97} D.J.~Rowe and J.~Repka,   J.\ Math.\ Phys. {\bf 38}, 4363
  (1997).
\bibitem{RB00} D.J.~Rowe and C.\ Bahri,  J.\ Math.\ Phys. {\bf 41},
  1520 (2000). 
\bibitem{Kmatrix} D.J.~Rowe, J.\ Math. Phys.\ {\bf 35},
 860 (1995).
\bibitem{dunne} S.A.~Dunne, J.\ Math. Phys.\ {\bf 10},
 860 (1969).
\bibitem{Ui70} H.\ Ui, Prog.\ Theor.\ Phys.\ {\bf 44}, 687 (1970).
\bibitem{JPE}  J.~P.~Elliott, Proc.\ Roy.\ Soc.\ {\bf A245} (1958) 128,
   562.
\bibitem{EH}  J.~P.~Elliott and M.~Harvey,  Proc.\ Roy.\ Soc.\
{\bf A272},  557 (1963).
\bibitem{PW} F.\ Peter and H.\ Weyl, Math.\ Ann.\ {\bf 97}, 737 (1927).
\bibitem{AS} {\it Handbook of Mathematical Functions\/}, ed.\ M.
  Abramowitz and I. Stegun (Dover, New York, 1965.
\bibitem{SU3/SO3} D.J.~Rowe, R.~Le Blanc and J.~Repka, J.\ Phys.\
    A: Math.\ Gen.\ {\bf 22}, L309 (1989);
    D.J.~Rowe,  M.G.~Vassanji and J.~Carvalho, Nucl.\ Phys.\ {\bf A504},
   76 (1989).
\bibitem{Cartan} E. Cartan, Abh.\ Math.\ Sem.\ Univ.\ Hamburg {\bf 11}, 116
   (1935).
\bibitem{Hua} L.-K. Hua, ``Harmonic analysis of functions of several
  complex variables in classical domains", (Am.\ Math.\ Soc., Providence,
  Rhode Island, 1963).
\bibitem{Helg}  S. Helgason, ``Differential Geometry, Lie Groups and
   Symmetric Spaces", (Academic Press, New York 1978).
\bibitem{Weyl}  H.~Weyl ``The Classical Groups, Their Invariants and
Representations" (Princeton Univ.\ Press, Princeton, N.J.)
\bibitem{LeB&R} R.~Le Blanc and D.J.~Rowe, J.~Phys.\ A: Math.\ Gen.\ {\bf
20} L

\end{references} 

\end{document}

math-ph0205034.extracted_bib
\bibitem {Wig30} E. Wigner, ``Uber die elastischen Eigenschwingungen
  symmetrischer  Systeme", Nachr. Ges. Wiss. G\"ottingen Math.-Phys. Kl.
  (1930) 133-146.

\bibitem{Wig39} E.~P.~Wigner, {\it Ann.\ Math.}\ {\bf 40} (1939) 149-204.

\bibitem{Mackey}  G.~W.~Mackey, Annals of Math.\ {\bf55}, 101 (1952);
    {\bf 58}, 193 (1952);
    ``Induced representations of groups and quantum mechanics",
                         (Benjamin, New York, 1969).

\bibitem{Knapp} A.~W.~Knapp, ``Representation theory of  semi-simple
   groups", Princeton Univ.\ Press N.J., 1986.

\bibitem{HC}  Harish-chandra, Am.\ J.\ Math.\ {\bf 78}, 1, 564 (1956).

\bibitem{Schr} E. Schr\"odinger, ``Der Stetige Ubergang von der Mikro-zur
  Makromechanik", Naturwissenschaften {\bf 14}, 664-666 (1926).

\bibitem{glauber} R.~Y.~Glauber, Phys.\ Rev.\ {\bf 131}, 2766
   (1963).

\bibitem{klauder} 
  J. R. Klauder, {\it J.\ Math.\ Phys.} {\bf 4}, 1058 (1963).

\bibitem{perelomov}
  A. Perelomov, {\it Commun.\ Math.\ Phys.} {\bf 26}, 222 (1972).

\bibitem{gilmore72}
  R. Gilmore, {\it Ann.\ Phys.\ (N.Y.)} {\bf 74}, 391 (1972).

\bibitem{Barg} V.~Bargmann, Commun.\  Pure Appl.\  Math.\ 
   {\bf 14}, 187 (1961).

\bibitem{Segal} I.\ Segal, ``Mathematical problems of relativistic
   physics", (Providence, Rhode Island, 1963).

\bibitem{Perelomov85}  A.~Perelomov, ``Generalized coherent states and
 their applications",  Springer (1985).

\bibitem{Onofri} E.~Onofri, J.~Math.\ Phys.\ {\bf 16}, 1087 (1975).

\bibitem{VCS} D.J.~Rowe, J.\ Math.\  Phys. {\bf 25}, 2662 (1984);
  D.J.~Rowe, G.~Rosensteel and R.~Carr,   J.\ Phys.\ A:
  Math.\ Gen. {\bf 17}, L399 (1984).

\bibitem{RRG} D.J.~Rowe, G.~Rosensteel, and
  R.~Gilmore,  J.\ Math.\ Phys. {\bf 26}, 2787 (1985).

\bibitem{RR91} D.J.~Rowe and J.~Repka,   J.\ Math.\ Phys. {\bf 32}, 2614
  (1991).

\bibitem{BRR} S.\ Bartlett, D.J.\ Rowe, and J.\ Repka, ``Coherent state
representations as geometric quantization", (in preparation).

\bibitem{HLeBR} K.T.\ Hecht, R.\ Le Blanc, and D.J.\ Rowe, J.\ Phys.\
  A: Math.\ Gen.\ {\bf 20}, 2241 (1987).

\bibitem{RLeBH} D.J.\ Rowe, R.\ Le Blanc, and K.T.\ Hecht, J.\ Math.\
  Phys.\ {\bf 29}, 287 (1988).

\bibitem{LeBR88}  R.\ Le Blanc, and D.J.\ Rowe,  J.\ Math.\
  Phys.\ {\bf 29}, 758, 767 (1988).

\bibitem{LeBR89} R.\ Le Blanc, and D.J.\ Rowe,  J.\ Math.\
  Phys.\ {\bf 30}, 1415 (1989); {\bf 31}, 14 (1990);
  C.\ Quesne, J.\ Phys.\ A: Math.\ Gen.\ {\bf 23}, 5383, 5411 (1990).

\bibitem{RR95} D.J.~Rowe and J.~Repka,   J.\ Math.\ Phys. {\bf 36}, 2008
  (1995).

\bibitem{RR97} D.J.~Rowe and J.~Repka,   J.\ Math.\ Phys. {\bf 38}, 4363
  (1997).

\bibitem{RB00} D.J.~Rowe and C.\ Bahri,  J.\ Math.\ Phys. {\bf 41},
  1520 (2000). 

\bibitem{Kmatrix} D.J.~Rowe, J.\ Math. Phys.\ {\bf 35},
 860 (1995).

\bibitem{dunne} S.A.~Dunne, J.\ Math. Phys.\ {\bf 10},
 860 (1969).

\bibitem{Ui70} H.\ Ui, Prog.\ Theor.\ Phys.\ {\bf 44}, 687 (1970).

\bibitem{JPE}  J.~P.~Elliott, Proc.\ Roy.\ Soc.\ {\bf A245} (1958) 128,
   562.

\bibitem{EH}  J.~P.~Elliott and M.~Harvey,  Proc.\ Roy.\ Soc.\
{\bf A272},  557 (1963).

\bibitem{PW} F.\ Peter and H.\ Weyl, Math.\ Ann.\ {\bf 97}, 737 (1927).

\bibitem{AS} {\it Handbook of Mathematical Functions\/}, ed.\ M.
  Abramowitz and I. Stegun (Dover, New York, 1965.

\bibitem{SU3/SO3} D.J.~Rowe, R.~Le Blanc and J.~Repka, J.\ Phys.\
    A: Math.\ Gen.\ {\bf 22}, L309 (1989);
    D.J.~Rowe,  M.G.~Vassanji and J.~Carvalho, Nucl.\ Phys.\ {\bf A504},
   76 (1989).

\bibitem{Cartan} E. Cartan, Abh.\ Math.\ Sem.\ Univ.\ Hamburg {\bf 11}, 116
   (1935).

\bibitem{Hua} L.-K. Hua, ``Harmonic analysis of functions of several
  complex variables in classical domains", (Am.\ Math.\ Soc., Providence,
  Rhode Island, 1963).

\bibitem{Helg}  S. Helgason, ``Differential Geometry, Lie Groups and
   Symmetric Spaces", (Academic Press, New York 1978).

\bibitem{Weyl}  H.~Weyl ``The Classical Groups, Their Invariants and
Representations" (Princeton Univ.\ Press, Princeton, N.J.)

\bibitem{LeB&R} R.~Le Blanc and D.J.~Rowe, J.~Phys.\ A: Math.\ Gen.\ {\bf
20} L

