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Transformation Groups

, Volume 16, Issue 3, pp 827–856 | Cite as

Cross-sections, quotients, and representation rings of semisimple algebraic groups

  • Vladimir L. Popov
Article

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 TG/T where T is a maximal torus of G and W the Weyl group.

Keywords

Algebraic Group Toric Variety Maximal Torus Rational Section Representation Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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