What's the mathematical background to the representation for Gaussian beams? Background
A general optical system (not necessarily having an axis of 
rotational symmetry) can be represented, for small deviations from a base ray,
by the matrix transfer equation,
$$
\left[\begin{array}{c} n'L'\\ n'M'\\ x'\\ y' \end{array}\right]
=
\left[\begin{array}{cc} B & -A\\ -D & C \end{array}\right]
\left[\begin{array}{c} nL\\ nM\\ x\\ y \end{array}\right]
=
D(g)
\left[\begin{array}{c} nL\\ nM\\ x\\ y \end{array}\right]
$$
where $L,M,N$ are the direction cosines of a ray, $n$ is refractive index,
$x,y$ are coords on a reference plane, $A,B,C,D$ are $2\times 2$ matrices
and a prime (') denotes the image space and unprimed quantities are in the 
object space. The sign convention for the matrix sub-blocks is from 
``The Ray and Wave Theory of Lenses'' by A. Walther. 
The $4\times 4$ transfer matrices $D(g)$ are the defining represention
of the symplectic group $Sp(4,R)$ carried on a 4-d vector
space $V_{4}$ . They obey,
$$
D(g)^{T}
\left[\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right]
D(g)
=
\left[\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right]
$$
where the $0$ is $2\times 2$ and $1$ is the unit $2\times 2$ matrix. Thus
one can say that geometric optics is the study of a finite dimensional
representation of $Sp(4,R)$ and each optical system is a $g\in Sp(4,R)$.
Furthermore, two optical systems $g$ followed by $h$ are equivalent
to the single optical system $hg$. 
It turns out (see Guillemin and 
Sternberg ``Symplectic Techniques in Physics''), provided the optical 
system is lossless, that diffractive 
optics in the Fresnel approximation
is the study of an infinite-dimensional projective representation of $Sp(4,R)$
carried on a Hilbert space.
In this case, the complex amplitude $u(x,y)$ on a reference plane
is a vector $|u\rangle$ and the transfer equation between object
and images reference planes is now,
$$
|u'\rangle=K(g)|u\rangle
$$
where $K(g)$ is a kernel which is actually a projective representation of $Sp(4,R)$
carried on the Hilbert space. Note that, in a book on diffraction, like 
Walther's, the transfer equation would be appear as,
$$
u'(x',y')=\int K(x',y'|x,y)u(x,y)dxdy
$$
and the kernel $K(x'y'|x,y)$ depends on the matrices $A,B,C,D$ via a function
called the eikonal. The details are not important for this question.
Question
If one studies the diffraction of a Gaussian beam,
$$
u(x,y)=\exp{\left\{\frac{i}{2}
\left[\begin{array}{cc} x & y \end{array}\right]\left(\frac{q}{n}\right)^{-1}
\left[
\begin{array}{c}
x \\ y
\end{array}
\right]\right\}}
$$
 by an optical system in the Fresnel approximation, then the output beam is also Gaussian and the relation between the $2\times 2$  input and output complex beam parameter matrices is:
$$
\frac{q'}{n'}=\left(C\frac{q}{n}-D\right)\left(B-A\frac{q}{n}\right)^{-1}
$$
This is another representation (realization) of $Sp(4,R)$ carried on the space
of symmetric $2\times 2$ matrices like $q/n$. It's some sort of conformal
transformation because, for cylindrical lenses, $q/n$ is a complex number and 
$A,B,C,D$  are real numbers, so the above equation is a conformal 
transformation. 
The existence of this representation was a complete surprise for me; it's
clearly related to the 
infinite-dimensional projective representation of $Sp(4,R)$ because the 
derivation proceeds via Fresnel theory, but it's carried on the 3-d space 
of symmetric $2\times 2$ matrices whilst the projective representation has no
irreducible subspaces other that the even and odd parity irreps.   
My question is, ``What is the mathematical background to the representation for 
Gaussian beams?'' 
 A: After some searching, it turns out that mathematicians http://ptmat.fc.ul.pt/~pedro/thesis.pdf know about this stuff. Set up,
$$
Z=X+iY
$$
where $X$ and $Y$ are $m\times m$ real symmetric matrices and $Y$ is 
positive definite. The space of matrices like $Z$ is a generalization of the 
upper half of the complex plane which is called the Siegel Upper Half
Plane ($SUHP$). The conformal realization of $SL(2,R)=Sp(2,R)$ 
on the upper half complex plane generalizes as the realization of the symplectic group $Sp(2m,R)$ carried on the $SUHP$. Let, 
$$
D(g)=
\left[
\begin{array}{cc}
A & B\\
C & D
\end{array}
\right]
$$
be a $2m\times 2m$ matrix in the defining rep of the symplectic group $Sp(2m,R)$. The conformal realization of $Sp(2m,R)$ carried on the $SUHP$ is the map of the $SUHP$ to itself,
$$
\rho_{g}(Z)=\left(AZ+B\right)\left(CZ+D\right)^{-1} 
$$
and it's a realization of the group because,
$$
\rho_{hg}(Z)=\rho_{h}(\rho_{g}(Z))
$$
for $g,h\in Sp(2m,R)$.
It's interesting that the Gaussian beam parameter matrix $q/n$ lives
on the $SUHP$ (actually, the $SLHP$ in the usual way it's defined). However,
I still don't understand the relation between the conformal realization of 
$Sp(2m,R)$ carried on the $SUHP$ and the infinite-dimensional 
projective rep of $Sp(2m,R)$ carried on the Hilbert
space that is Fresnel optics.  
