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 $Z(G)$?



Centralizer of a reductive subgroup
$\DeclareMathOperator\GL{GL}$No. Take $H=\GL\_2$ embedded diagonally into $G=\GL\_2\times \GL\_2$ and take
$\rho$ equal to $\mathbb C^2 \otimes (\mathbb C^2)^\*$ with the natural action of $\GL\_2$ on $\mathbb C^2$.


