Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $$C_{\mathfrak{g}}(\xi)=\{\eta\in\mathfrak{g}:[\xi,\eta]=0\}, \quad C_G(\xi)=\{g\in G:\mathrm{Ad}_\mathfrak{g}(\xi)=\xi\},$$ and $\pi_0(C_G(\xi))=C_G(\xi)/C_G(\xi)_0$. I would appreciate any references you suspect would give useful structural information. Also, I would welcome any advice and suggestions.

Thanks!