Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of characteristic 0 it's equivalent to "reductive algebraic group". Anyway, I'm led to raise a more historical question about how the term originated. The earliest use I've seen in the literature was by my late colleague John Fogarty (who had been a student of Mumford), in his 1969 W.A. Benjamin lecture notes Invariant Theory (Definition 4.6).

Is this the earliest use of the term "linearly reductive"?

In Fogarty's notes and elsewhere, the setting is classical: Fix an algebraically closed field which I'll call $K$, to distinguish it from an arbitrary field $k$. Given a linear algebraic group $G$ over $K$ in the traditional Borel-Chevalley sense, which need not be connected, consider its "rational" representations $G \rightarrow \mathrm{GL}(V)$ with $\dim V < \infty$. Call $G$ linearly reductive if every such representation is completely reducible. (This kind of definition carries over easily to the language of group schemes, but doesn't acquire extra significance.)

The basic motivation for such a condition comes from invariant theory: when $G$ is linearly reductive, all rings of invariants $K[V]^G$ are finitely generated $K$-algebras. After Hilbert's early work and M. Nagata's counterexample to the Hilbert Fourteenth Problem, the papers of Nagata in the 1960s explored the notion of linear reductivity (giving it the name "reductive"). In particular, he found that the only algebraic groups with this property are the reductive ones in the current sense when $\mathrm{char} K = 0$ (i.e., those having trivial unipotent radical, as in the Borel-Tits theory over arbitrary fields), together with the groups whose identity component $G^0$ is a torus while $[G:G^0]$ is prime to $p$ when $\mathrm{char} K = p>0$.

By the way, in his 1965 book Geometric Invariant Theory, Mumford worked just in characteristic 0 and used the term "reductive" as Nagata did to mean "linearly reductive"; the expanded second edition by Mumford and Fogarty in 1982 substituted the latter term. In any case, Mumford had conjectured that in prime characteristic a weaker condition (called by Seshadri and others "geometric reductivity") would hold for reductive groups in any characteristic, ensuring finitely generated rings of invariants. (This is consistent with classical results on finite groups.) It was eventually proved in 1975 by Haboush.


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