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## 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

Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. 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 (for char k = 0, this has been proved earlier); this answers the other

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/T where T is a

maximal torus of G and W the Weyl group.