I'm having trouble sorting out some basic definitions concerning Chevalley groups. The groups I'm interested in are the simply connected groups of type $C\_n$, so the groups $\text{Sp}\_{2n}$. The roots in $C\_n$ are $\{\pm 2 \epsilon\_i \text{ $|$ } 1 \leq i \leq n\} \cup \{\pm \epsilon\_i \pm \epsilon\_j \text{ $|$ } 1 \leq i < j \leq n\}$. Associated to every root in $C\_n$ is a one-parameter subgroup of $\text{Sp}\_{2n}$. Can someone tell me the actual matrices generating these one-parameter subgroups? Or give me a reference discussing this in a simple and concrete way? All the sources I've consulted only do things concretely for $\text{SL}\_n$.



One-parameter subgroups of symplectic group associated to roots
The actual matrices depend of course on the particular alternating bilinear form you use to define $\mathrm{Sp}\_{2n}$. A standard choice is to use the form whose matrix relative to a basis $(e\_1,\dots,e\_n,e\_{-n},\dots,e\_{-1})$ is
$$
J=
\begin{pmatrix}
& & & & & 1\\
& & & & \cdots\\
& & & 1\\
& & -1\\
& \cdots\\
-1
\end{pmatrix}.
$$
This has the advantage that 1º) the diagonal matrices in $\mathfrak g$ make a Cartan subalgebra, with basis the matrices $H\_i=E\_{i,i}-E\_{-i,-i}$ where the $E\_{i,j}$ are the standard matrix units, $(E\_{i,j})\_{kl}=\delta\_{ik}\delta\_{jl}$. And 2º) the infinitesimal generators of the root subgroups that you ask for are listed in [Bourbaki's tables](http://books.google.com/books?id=oCO0xOzNLhAC&pg=PA205): they are
$$
\begin{align}
X\_{2\epsilon\_i} &= E\_{i,-i}\\
X\_{-2\epsilon\_i} &= -E\_{-i,i}\\
X\_{\epsilon\_i-\epsilon\_j} &= E\_{i,j} - E\_{-j,-i}\\
X\_{-\epsilon\_i+\epsilon\_j} &= -E\_{j,i} + E\_{-i,-j}\\
X\_{\epsilon\_i+\epsilon\_j} &= E\_{i,-j} + E\_{j,-i}\\
X\_{-\epsilon\_i-\epsilon\_j} &= -E\_{-i,j} - E\_{-j,i}.
\end{align}
$$


