Quotients by conjugation action, cross-sections, singularities, and representation rings

Let $G$ be a connected semisimple algebraic group over an algebraically closed field $k$. In 1965 R. 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 of $G$ is bijective; this answers Grothendieck's question. In particular, for $char(k)=0$, the converse to Steinberg's theorem holds. The existence of a cross-section in $G$ implies, at least for $char(k)=0$, that the algebra $k[G]^G$ of class functions on $G$ is generated by $rk(G)$ 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 $G$; this answers the other Grothendieck's question.