It's probably most natural to consider this as a question about the (rational) representations of Frobenius kernels, in the spirit of Jantzen's book Representations of Algebraic Groups (Chapter II.3). Given a connected, simply connected semisimple group $G$, the irreducible representations of its Frobenius kernel $G_r$ are parametrized naturally by $p^r$ of the highest weights for $G$ relative to a fixed maximal torus. Only the zero weight corresponds to a 1-dimensional representation (i.e., character of $G_r$) because $G$ is semisimple.
It's probably most natural to consider this as a question about the (rational) representations of Frobenius kernels, in the spirit of Jantzen's book Representations of Algebraic Groups. Given a connected semisimple group $G$, the irreducible representations of its Frobenius kernel $G_r$ are parametrized naturally by $p^r$ of the highest weights for $G$ relative to a fixed maximal torus. Only the zero weight corresponds to a 1-dimensional representation (i.e., character of $G_r$) because $G$ is semisimple.