Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensinoal representation of $G$ over $\mathbb{C}$. Assume that $\rho|_H$ is reducible. Then is it always the case that the centralizer $Z_G(H)$ is strictly larger than the center $Z(G)$?

Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\rho|_H$ is reducible. Then is it always the case that the centralizer $Z_G(H)$ is strictly larger than the center $Z(G)$?