Are adjoint quotients of connected reductive groups of adjoint type?

Question

Suppose we have a connected reductive group $G$, and consider the adjoint quotient $G^{ad}=G/Z(G)$. Is $G^{ad}$ a group of adjoint type? I'm having difficulty coming up with a counterexample.

Answer 1

I assuming "being of adjoint type" means having trivial center (note that this automatically implies semisimple since quotients of reductive are reductive, and seimsimple=reductive+finite center). This is, in fact, true, but it's less obvious than one might imagine. The key point is that the kernel of the adjoint map $H\to \mathrm{GL}(\mathfrak{h})$ has kernel $Z(H)$ (e.g. see this) for any reductive group $H$. Thus, to show that $H:=G/Z(G)$ has trivial center, it suffices to show that the map $H\to\mathrm{GL}(\mathfrak{h})$ has trivial kernel. The rest is an exercise I leave to you.