This is a very well-written paper. Apart from the interesting rationality results proved, there are several comments and examples interspersed which add to the understanding of the topic (even for nonexperts like me).
For a connected, reductive affine algebraic group \(G\) over a field \(k\), J.-P. Serre defined a \(k\)-subgroup \(H\) to be \(G\)-completely reducible (abbreviated as \(G\)-cr.) over \(k\), if each parabolic subgroup \(P\) defined over \(k\), and containing \(H\), contains a Levi subgroup over \(k\) which itself contains \(H\). In fact, the authors can deal with groups which are not connected by replacing parabolic subgroups and Levi subgroups by what the authors call Richardson parabolic subgroups (R-parabolics corresponding to multiplicative one parameter subgroups as defined below) and their Levi subgroups.
Using the notions introduced and rationality results proved in this paper, the authors deduce: If \(k_1\) is a separable extension of \(k\), and \(H\) is \(G\)-cr. over \(k_1\), then \(H\) is \(G\)-cr. over \(k\).
Recently, the first three authors have also proved the converse. Together, these results provide an affirmative answer to a question raised by Serre.
Another problem addressed by the authors is the so-called center conjecture of Tits, which has been recently proved by Mühlherr & Tits. A more general version of the conjecture is answered in the special case of a contractible, fixed point subcomplex of the spherical Tits building of \(G\) over \(k\), using the techniques of the paper.
The authors introduce a new notion which proves the key to the study of rationality properties of algebraic group actions on affine varieties. Let \(V\) be an affine variety defined over \(k\) and let \(G\) be a reductive (possibly not connected) affine algebraic group over \(k\) which acts on \(V\) over \(k\). For \(v\in V\) (not necessarily in \(V(k)\)), the \(G(k)\)-orbit of \(v\) is said to be cocharacter-closed over \(k\) if, for any multiplicative one parameter subgroup \(\lambda\) of \(G\) over \(k\) for which the morphism \(a\mapsto\lambda(a)v\) extends to \(0\) (that is, \(\lim_{a\to 0}\lambda(a)v\) exists), this limit belongs to \(G(k)v\).
If either \(k\) is perfect or \(G\) is connected, the authors show that this property is equivalent to the property that the above limit is in the orbit under \(R_u(P_\lambda)(k)\) where \(P_\lambda\) is the Richardson parabolic subgroup \[ P_\lambda:=\{g\in G:\lim_{a\to 0}\lambda(a)g\lambda(a)^{-1}\text{ exists}\}. \] There are many other applications of this notion; we describe only one below.
For a subgroup \(H\) of \(G\) and some embedding \(G\to\mathrm{GL}_n\), a tuple \(\mathbf h\in H^r\) is said to be ‘generic’ if its image under the embedding generates the associative subalgebra of \(M_n\) spanned by \(H\).
The authors prove: Let \(G\) be connected. Let \(H\) be a subgroup and \(\mathbf h\in H^r\) a generic tuple of \(H\). Then, \(H\) is \(G\)-cr. over \(k\) if and only if \(G(k)h\) is cocharacter-closed over \(k\).