### Book chapter

## 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.