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 $GL\_n(\mathbb C)\subset GL\_n(A)$.




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**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](https://mathoverflow.net/questions/570/deformation-theory-of-representations-of-an-algebraic-group) $H^{>0}(G,V\otimes V^\*)$, 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.



Are representations of a linearly reductive group discretely parameterized?
I don't think this is literally true. For example, suppose that $G$ is a finite cyclic group of order 2 generated by $s$, suppose that $L$ is an non-trivial 2-torsion invertible $A$-module over a Dedekind ring $A$. Then $L^{\oplus 2}$ is isomorphic to $A^{\oplus 2}$; you can let $G$ act on $L^{\oplus 2}$ with the rule $s(a,b) = (a, -b)$. This gives a family of representations that does not come from $\mathbb C$. You can give a similar example in which $G$ is a torus.


On the other hand, what you want is true Zariski-locally; would that be enough for your needs?


[Edit] Let me add a proof that the statement is Zariski-locally true. The key point is the following: given a locally free sheaf on a scheme $X$ with an action of $G$, the subsheaf of invariants is also locally free. This is easy: if $X = \mathop{\rm Spec} A$ and $E$ corresponds to a projectve $A$-module $M$, then $M^G$ is a direct summand of $M = M^G \oplus M\_G$, because $G$ is linearly reductive (and $M$ is a locally finite representation of $G$, this is standard); and $M\_G$ is also an $A$-submodule, so $M^G$ is a projective $A$-module. Now, let $E$ be a locally free sheaf with an action of $G$. Let $p$ be a rational point of $X$, and $V$ be the representation of $G$ appearing as the fiber $E(p)$ of $E$ at $p$. Consider the locally free sheaf $Hom\_{\mathcal O\_X}(V \otimes \mathcal O\_X, E)$; the sheaf of invariants, that is, the sheaf of $G$-equivariant homomorphisms $V \otimes \mathcal O\_X \to E$, is locally free. At the point $p$ we have a section of its fiber, given by the tautological isomorphism $V \simeq E(p)$; since the sheaf is locally free, this extends to a section in a neighborhood. By further restricting the neighborhood, this sectioin will be an isomorphism, and this concludes the proof.


