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- W2059312480 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." @default.
- W2059312480 created "2016-06-24" @default.
- W2059312480 creator A5052762947 @default.
- W2059312480 date "2011-04-07" @default.
- W2059312480 modified "2023-09-26" @default.
- W2059312480 title "Cross-sections, quotients, and representation rings of semisimple algebraic groups" @default.
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- W2059312480 doi "https://doi.org/10.1007/s00031-011-9137-6" @default.
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