generators of the symplectic group In Masoud Kamgarpour's paper "Weil Representations" he uses a set of generators for the symplectic group, referring to a book by R. Steinberg which I do not have access to. If it matters at all, I am working in characteristic zero.
After choosing a symplectic basis, the generators can be written
\begin{equation}
\left(
\begin{array}{cc}
A & 0 \newline
0 & (A^t)^{-1}
\end{array} \right), \
\left(
\begin{array}{cc}
I & B \newline
0 & I
\end{array} \right), \ \text{and} \
\left(
\begin{array}{cc}
0 & I \newline
-I & 0
\end{array} \right),
\end{equation}
where $A$ ranges through invertible matrices and $B$ ranges through symmetric matrices. Does anyone know of a reference or an explanation for this, especially a coordinate-free conceptual and/or geometric one?
 A: I think this is essentially a bloc version of LU decompostion (called Bruhat decomposition) : any symplectic bloc upper triangular matrix can be written as a product of the first two (types of matrices), conjugacy by the third gives you bloc lower triangulars. The point being that your symplectic bloc-diagonal matrix are of the of the first matrix you describe. So you can prove this formally by checking Bruhat in GL(2), which is both obvious and geometric (look up flag).
A: I don't know if this precisely answers your question, but a study of generators by symplectic transvections for fields of characteristic $\ne 2$ was carried out by methods using graphs in 
R. Brown and S.P. Humphries, ``Orbits under symplectic transvections I'',  Proc. London Math. Soc. (3) 52 (1986) 517-531.
The main result is:  for a symplectic space $V$ with symplectic form $\cdot$ and a subset $S$ of $V$,  define a graph $G(S)$ with vertex set  $S$ and an edge between $a$ and $b$ if and only if $a\cdot b \ne 0$. Then the transvections corresponding to the elements of $S$ generate the symplectic group of $(V, \cdot)$ if and only if $S$ spans $V$ and $G(S)$ is connected. 
(The immediately following sequel did the more complicated case of characteristic $2$.) 
A: This is done in
Symplectic groups By Onorato Timothy O'Meara
and uses the fact that the symplectic group over a field is generated by symplectic transvections.
