TITLE: One-parameter subgroups of symplectic group associated to roots
QUESTION [6 upvotes]: 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 $\text{SL}_n$.

REPLY [4 votes]: 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: they are
$$
\begin{align}
X_{2\epsilon_i} &= E_{i,-i}\\
X_{-2\epsilon_i} &= -E_{-i,i}\\
X_{\epsilon_i-\epsilon_j} &= E_{i,j} - E_{-j,-i}\\
X_{-\epsilon_i+\epsilon_j} &= -E_{j,i} + E_{-i,-j}\\
X_{\epsilon_i+\epsilon_j} &= E_{i,-j} + E_{j,-i}\\
X_{-\epsilon_i-\epsilon_j} &= -E_{-i,j} - E_{-j,i}.
\end{align}
$$