Let $X$ and $X^{\prime }$ be compact Riemann surfaces of genus $\ge 3$, and let $G$ and $G^{\prime }$ be nonabelian reductive complex groups. If one component ${ℳ}_G^d\left(X\right)$ of the coarse moduli space for semistable principal $G$–bundles over $X$ is isomorphic to another component ${ℳ}_{G^{\prime }}^{d^{\prime }}\left(X^{\prime }\right)$, then $X$ is isomorphic to $X^{\prime }$.