caa a22 4500 445862335 CHVBK 20180317145447.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s00031-011-9137-6 doi (NATIONALLICENCE)springer-10.1007/s00031-011-9137-6 Popov Vladimir Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, 119991, Moscow, Russia aut Cross-sections, quotients, and representation rings of semisimple algebraic groups [Elektronische Daten] [Vladimir Popov] 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. Springer Science+Business Media, LLC, 2011 Transformation Groups SP Birkhäuser Verlag Boston 16/3(2011-09-01), 827-856 1083-4362 16:3<827 2011 16 31 https://doi.org/10.1007/s00031-011-9137-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00031-011-9137-6 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Popov Vladimir Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, 119991, Moscow, Russia aut NATIONALLICENCE 773 0- Transformation Groups SP Birkhäuser Verlag Boston 16/3(2011-09-01), 827-856 1083-4362 16:3<827 2011 16 31 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer