Abstract: 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 $$ \tau :\hat{G} \to G $$ 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.

EXPLOR_STEP=$WICRI_ROOT/Wicri/Mathematiques/explor/BourbakiV1/Data/Main/Exploration

HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000314 | SxmlIndent | more

HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 000314 | SxmlIndent | more

{{Explor lien
   |wiki=    Wicri/Mathematiques
   |area=    BourbakiV1
   |flux=    Main
   |étape=   Exploration
   |type=    RBID
   |clé=     ISTEX:BA6AB5650FF78F318A038CC3221967A9466C1CE0
   |texte=   Cross-sections, quotients, and representation rings of semisimple algebraic groups
}}