Let $G$ be a (connected) reductive group over a field $k$. Then there is a natural functor from the category of representations of $G$ to the category of representations of $G(k)$.


Under which circumstances can one say that this functor is a) full and b) faithful?


Edited for clarity:


Here we think of algebraic representations of $G$ over the field $k$ (so, for example, morphisms of $G$ to $GL(V)$ for $k$-vector spaces $V$, defined over $k$).


Given such a morphism $\phi : G \rightarrow GL(V)$, we take $k$ points to obtain a $k$-linear representation of the group $G(k)$ on the $k$-vector space $V$ (thus a homomorphism $\phi(k) : G(k) \rightarrow GL(V)(k)$).



Points of reductive groups
A bit of Tannakian formalism clarifies the situation. Recall that for every abstract group $\Gamma$ there is a notion of "algebraic hull" $\Gamma^{alg}$ constructed as follows: Consider pairs $(\varphi,H)$ where $H$ is an algebraic group over $k$ and $\varphi:\Gamma\to H(k)$ a homomorphism of groups with Zariski-dense image. Maps between such pairs are defined in the obvious was, and one defines
$$\Gamma^{alg} := \lim\_{(\varphi,H)}H$$
So this is an affine group scheme over $k$. The group $\Gamma^{alg}$ has a universal property, namely that for any algebraic group $H$ over $k$, any map $\Gamma \to H(k)$ factors over $\Gamma^{alg}$. From Tannakian formalism we know:



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> The canonical map $\Gamma\to\Gamma^{alg}$ induces an equivalnence of monoidal categories between the category of finite dimensional algebraic $\Gamma^{alg}$-representations and finite dimensional $k$-representations of $\Gamma$.
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Now, let's apply this to $\Gamma = G(k)$ for a fixed algebraic, not necessarily reductive group $G$ over $k$. The universal property of the algebraic hull yields a canonical morphism $\Gamma^{alg}\to G$ whose image is the Zariski-closure of $\Gamma$ in $G$. We have then results like this:



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> Proposition (Deligne, LNM900, p.139): Let $f: H\to G$ be a morphism of affine group schemes over $k$ and let $\omega$ be the induced functor $Rep\_k(G) \to Rep\_k(H)$ between categories of finite dimensional representations. The morphism $f$ is faithfully flat if and only if $\omega$ is fully faithful and if for every representation $V$ of $G$ every subrepresentation of $\omega(V)$ is isomorphic to the image of a subrepresentation of $V$.
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For instance, faithful flatness of $f$ implies surjectivity of $f$, which is therefore a necessary condition for the induced functor of representations to be fully faithful - this is what's behind Brian's comment. If you assume $G$ to be reductive then the category $Rep(G)$ is semisimple (in characteristic zero) and the last condition in Deligne's proposition can, maybe, be rephrased in a simpler way.


