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 $\text{SL}_n$.
1 Answer
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 $$ \begin{align} X_{2\epsilon_i} &= E_{i,-i}\\ X_{-2\epsilon_i} &= -E_{-i,i}\\ X_{\epsilon_i-\epsilon_j} &= E_{i,j} - E_{-j,-i}\\ X_{-\epsilon_i+\epsilon_j} &= -E_{j,i} + E_{-i,-j}\\ X_{\epsilon_i+\epsilon_j} &= E_{i,-j} + E_{j,-i}\\ X_{-\epsilon_i-\epsilon_j} &= -E_{-i,j} - E_{-j,i}. \end{align} $$
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$\begingroup$ These are the infinitesimal generators, but sadly I'm a little dense. What is the result of exponentiating these into actual elements of $\text{Sp}_{2n}$? $\endgroup$– SarahAug 22, 2013 at 4:42
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2$\begingroup$ Each $X_\alpha$ has square zero, so $\exp(tX_\alpha)= 1 + tX_\alpha$, no? $\endgroup$ Aug 22, 2013 at 4:53
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$\begingroup$ Oh yeah, of course! Sorry for asking a silly question. Thanks for your help! Somehow I had looked for this info in Bourbaki, but I only consulted chapters 4-6 (I should really buy the other volume someday!). $\endgroup$– SarahAug 22, 2013 at 4:54