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

Let $G$ be a connected semisimple algebraic group over an algebraically closed field $\Bbbk$. The celebrated Steinberg’s theorem of 1965 claims 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 address the problem of the existence of such a cross-section in arbitrary $G$. The existence of a crosssection in $G$ implies, at least for char $\Bbbk$ = 0, that the algebra $\Bbbk[G]^G$ of class functions on G is generated by $rk(G)$ elements. We describe, for arbitrary G, a minimal generating set of $\Bbbk[G]^G$ and that of the representation ring of G and answer two Grothendieck’s questions on constructing generating sets of $\Bbbk[G]^G$. We also address the problem of the existence of a rational (i.e., local) cross-section in any $G$.