Cross-sections, Quotients, and Representation Rings of Semisimple Algebraic Groups

LetG 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 τ : b G → G is bijective. 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] of class functions on G is generated by rkG elements. We describe, for arbitrary G, a minimal generating set of k[G] and that of the representation ring of G and answer two Grothendieck’s questions on constructing the generating sets of k[G]. We prove the existence of a rational cross-section in any G (for char k = 0, this has been proved earlier in [CTKPR]). We also prove that the existence of a rational section of the quotient morphism for G is equivalent to the existence of a rational W -equivariant map T 99K G/T where T is a maximal torus of G and W the Weyl group. We show that both properties hold if the isogeny τ is central.