I've been trying to prove this (probably very simple) result that is stated in a paper that I'm reading:
Let $G$ and $H$ be connected semisimple algebraic groups defined over a field $F$, and let $f: G \longrightarrow H$ be a separable isogeny. Then $f$ is a central isogeny.
I'm not seeing why the fact that $\mathrm{ker}(f)$ is smooth and finite implies that it is contained in the center of $G$. Any hints would be much appreciated.
Consider the action of $G$ on $\ker(f)$ by conjugation. We have that $\ker(f)$ is finite and smooth, and $G$ is connected, so the action of $G$ on $\ker(f)$ is trivial. Since this conjugaction action is trivial, we have that $\ker(f)$ is central.
The question is stated in a somewhat confusing way, so a little historical perspective may help. Going back to the Chevalley seminar of 1956-58, where these ideas originate, one deals mainly with connected linear (or affine) algebraic groups over an algebraically closed field. Here Chevalley defines an isogeny to be a surjective morphism $f:G \rightarrow G'$ of algebraic groups having finite kernel. This kernel is automatically in the center of $G$ since $G$ is connected, by Chevalley's elementary argument as Will reminds us. But this is not directly relevant to the more refined notion of "central isogeny".
When $G$ and $G'$ are semisimple as well, the main achievement of the seminar was a classification of possible isogenies, hence of possible isomorphisms. As anon indicates, this has an extra complication in prime characteristic due to the existence there of inseparable isogenies (notably powers of the Frobenius map) and even of unusual isogenies such as that between groups of types $B_\ell, C_\ell$ in characteristic 2. These situations illustrate the role of inseparable isogenies, which only exist in prime characteristic and which show up in the behavior of the differential on Lie algebras (though Chevalley made little use of the Lie algebra).
So the answer to the question in the header is given by Chevalley's arguments. The later generalizations to group schemes facilitate the use of arbitrary ground fields and such but don't essentially change the nature of this question. From the Lie algebra point of view, separability of an isogeny between semisimple groups depends on whether the kernel of the differential is trivial (and is automatic in characteristic 0). This got encoded in the notion of "central isogeny" (ruling out for instance Frobenius maps), which adapts to reductive groups and group schemes. [ADDED: In the concrete setting of linear algebraic groups, probably the best reference for central isogenies is the added $\S 22$ in Borel's second edition, GTM 126, Springer, 1991.]
In nonzero characteristic, the answer is not quite as simple as Will suggests. In the realm of smooth algebraic groups (Borel, Humphreys, Springer), there is an extra condition for an isogeny to be central, namely, that the map on Lie algebras be central. In the realm of algebraic group schemes, the automorphism group of the kernel may be a finite connected group scheme. For example, the automorphism group of $\mathbb{Z}/p\mathbb{Z}$ is $\mu_p$, and a connected algebraic group scheme may have a finite connected group scheme as a quotient. However, a smooth connected algebraic group scheme has no finite quotients, so the statement is correct.
