Automorphisms of connected, reductive groups are well understood: the outer automorphism group is an essentially combinatorial object associated to the root datum. I am trying to understand automorphisms of possibly *disconnected* reductive groups, by which I just mean groups whose identity components are reductive.


As in the connected case, one fixes a pinning and uses it to reduce the study of the full automorphism group to a simpler object. In this case, one winds up studying the subgroup of the disconnected group that fixes the pinning, whose identity component is central in the identity component of the original group.


All that is by way of motivation of the following question: suppose that $G$ is a smooth, affine algebraic group whose identity component is a torus. Do we have any concrete description of the automorphisms of $G$, or at least of its ‘identity component’? I put ‘identity component’ in quotes because I'm not too comfortable with discussing group schemes not of finite type; so let me give a work-around entirely in terms of varieties. The identity component of $G$ acts on $G$ by inner automorphisms. If $\Gamma$ is a smooth, connected, affine algebraic group that acts on $G$, then does the action factor through a map $\Gamma \to G^\circ/\mathrm C\_{G^\circ}(G)$?



Automorphisms of étale-by-torus groups
For any group $\Gamma$ that acts on $G$, $\Gamma$ acts on $G^{\circ}$ and on $G/G^{\circ}$. Since $G/G^{\circ}$ is finite and $G^{\circ}$ is a torus, both their automorphism groups are discrete groups, thus because $\Gamma$ is connected, the action of $\Gamma$ on them is trivial.


Automorphisms of $G$ fixing $G^{\circ}$ and $G/G^{\circ}$ are classified by cocycles in $C^1 ( G/G^{\circ}, G^{\circ})$. Let's do this explicitly so we can make sure this works schematically.


For $f$ such an automorphism of $G$, the function that sends $g \in G$ to $f(g) g^{-1}$ is valued in $G^\circ$ and invariant under right multiplication by $G^{\circ}$, thus defines a function $G /G^{\circ} \to G^{\circ}$, which geometrically gives a map from $\Gamma$ to a power of the torus $G^{\circ}$, whose image is contained in the closed subscheme satisfying the cocycle condition.


The closed subscheme of coboundaries is isomorphic to $G^{\circ}/C\_{G^{\circ}}(G)$, so it suffices to show $\Gamma$ is contained in this subscheme. Since $\Gamma$ is connected, it suffices to show this cohomology group is finite.


Because this cohomology group is a subquotient of a torus, to show it is finite, it suffices to show it is $|G/G^{\circ}|$-torsion. To do this we can use the two maps $ G^\circ \to G^{\circ} \otimes \mathbb Z[G/G^{\circ}]\to G^\circ$, whose composition is multiplication by $|G/G^{\circ}|$, to verify that multiplication by $|G/G^{\circ}|$ factors through a map to a projective $G/G^{\circ}$-module and thus acts as zero on cohomology in nonzero degree, as desired.


