Suppose that I am over an algebraically closed field of char $0$, and $G$ is a simply connected semisimple group.
For a dominant weight $\lambda$, there is an irreducible representation $W_{\lambda}$ of highest weight $\lambda$.
The center $Z(G)$ acts by scalars on $W_{\lambda}$.
Is the converse true? If $g \in G$ acts on $W_{\lambda}$ by scalar multiplication, then $g$ is central? How can I see this?
Sorry if this is a totally obvious question.
EDIT: added simply connected... Thanks!
If $g$ acts by scalar multiplication, then its image in $\text{GL}(W_{\lambda})$ is central. This need not imply that $g$ itself is central; for example, $G$ could be a product of two simple groups, $W_{\lambda}$ could be an irrep of one of them, and $g$ could live in the other. But it's true if $W_{\lambda}$ is a faithful representation.