Cross-sections, quotients, and representation rings of semisimple algebraic groups

Let $G$ be a connected semisimple algebraic group over an algebraically closed field $k$. In 1965 Steinberg proved that if $G$ is simply connected, then in $G$ there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary $G$ such a cross-section exists if and only if the universal covering isogeny $\tau\colon \tG\to G$ is bijective; this answers Grothendieck’s question cited in the epigraph. In particular, for $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}k=0$, the converse to Steinberg’s theorem holds. The existence of a cross-section in $G$ implies, at least for $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}k=0$, that the algebra $k[G{]}^G$ of class functions on $G$ is generated by $\mathrm{r}\mathrm{k}G$ elements. We describe, for arbitrary $G$, a minimal generating set of $k[G{]}^G$ and that of the representation ring of $G$ and answer two Grothendieck’s questions on constructing generating sets of $k[G{]}^G$. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary $G$ and the existence of a rational cross-section in $G$ (for $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}k=0$, this has been proved earlier); this answers the other Grothendieck’s question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational $W$-equivariant map $T⇢G\mathrm{/}T$ where $T$ is a maximal torus of $G$ and $W$ the Weyl group.