Reductive and linearly reductive complex algebraic groups

Let $G$ be a complex affine algebraic group (Classical complex affine algebraic actions and rational modules).

Unipotent subgroups. A closed subgroup $U\subseteq G$ is unipotent if every non-zero finite-dimensional rational $U$-module has a non-zero $U$-fixed vector. This fixed-vector condition is the characterization used in Brion's Example 1.22 (printed p. 8), and it is the only form of the notion used in this pair. Equivalently, by the Lie–Kolchin theorem, $U$ is unipotent in the standard sense that it admits no non-trivial rational characters and all its elements are unipotent; this equivalence is recorded as a sourced parenthetical companion to the definition (Brion Example 1.22 cites Lie–Kolchin) and is not used as a supplier anywhere in this pair, so no edge to the higher-order unipotent/solvable page is introduced.

Reductive and linearly reductive groups. The group $G$ is reductive if it has no non-trivial closed normal unipotent subgroup, and linearly reductive if every finite-dimensional rational $G$-module is completely reducible, i.e. a direct sum of simple $G$-submodules (A completely reducible representation as a finite direct sum of irreducible subrepresentations, Semisimple modules as direct sums of simple modules).

Over $\mathbf{C}$ the two notions coincide: every complex reductive affine algebraic group is linearly reductive, and conversely a linearly reductive $G$ has no non-trivial closed normal unipotent subgroup. Both implications, together with the identity-component reduction below, are proved in Complete reducibility and the Reynolds operator for a complex reductive group ↗; this definition only records them for consumers of that theorem.

Both notions depend only on the identity component, in the sense that $G$ is reductive if and only if $G^{\circ }$ is, and $G$ is linearly reductive if and only if $G^{\circ }$ is; the passage from $G^{\circ }$ to the finite component group for both notions is carried out in the same theorem.

• Positive characteristic. The equivalence is false in characteristic $p>0$ and must never be extended: ${\mathrm{S}\mathrm{L}}_2$ in characteristic $2$ has non-semisimple representations, and a linear algebraic group over a field of characteristic $p\mathrm{\ne }0$ is linearly reductive if and only if its identity component is of multiplicative type and $p$ does not divide the component index (Milne Definition 12.52, Example 12.55 and Remark 12.56). No statement in this pair is made over a field other than $\mathbf{C}$.
• Scope. No notion of geometric reductivity is introduced. The fixed-vector definition of unipotence is Brion's Example 1.22 convention; the standard-sense equivalence quoted above is not consumed by any proof in this pair, and the proofs that do need unipotence of a constructed subgroup re-establish it from the fixed-vector condition.
• Choice. This definition and its transcriptions of Brion's definitions use no Axiom of Choice.