Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant?

In other words, I want to prove that if $A$ is a commutative $\mathbb C$-algebra (without idempotents) and $\rho:G\to GL_n(A)$ is an algebraic group homomorphism (roughly, a family of representations parameterized by $Spec(A)$), then after conjugating by some element of $GL_n(A)$, the image of $\rho$ is actually contained in $GL_n(\mathbb C)\subset GL_n(A)$.

**Remark 1:** A finite-dimensional representation of $G$ is completely determined by the dimensions of its highest weight spaces. For a long time I thought this "discrete" parameterization somehow proved the result, but it doesn't. For example, nilpotent matrices are "discretely" parameterized by Jordan type, but it's obviously possible to have a flat family of nilpotent matrices in which Jordan type jumps, like $\begin{pmatrix}0&t\\ 0& 0\end{pmatrix}$ over the affine line with coordinate $t$.

**Remark 2:** Another reason I thought this was clear is that deformations of a representation $V$ are controlled by cohomology groups $H^{>0}(G,V\otimes V^*)$, which all vanish when $G$ is linearly reductive. This implies that any *formal* family—any flat family over an Artin ring—of representations has to be constant. However, this isn't enough to say that any family has to be locally constant. For example, deformations of $G$, as a group, are controlled by $H^{>0}(G,Ad)$, which vanish when $G$ is linearly reductive. So any formal family of linearly reductive groups has to be constant, but it is possible to have a flat family of linearly reductive groups which is not locally constant. Specifically, the affine line with a doubled origin is a flat group over the affine line. The fiber over the origin is $\mathbb Z/2$, but all the other fibers are trivial.