Submission: 2009, Aug 6
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 of G is bijective. 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 of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of this algebra and that of the representation ring of G and we answer two Grothendieck's questions on constructing the generating sets of this algebra. We prove the existence of a rational cross-section in any G (for char k=0, this has been proved earlier in LAGRS preprint no. 320 (2009)). We also prove that the existence of a rational section of the quotient morphism for G is equivalent to the existence of a rational W-equivariant map of T to G/T where T is a maximal torus of G and W the Weyl group. We show that both properties hold if the universal covering isogeny of G is central.
2000 Mathematics Subject Classification: 14M99, 14L30, 14R20, 14L24, 17B45
Keywords and Phrases: Semisimple algebraic group, conjugating action, cross-section, quotient, representation ring
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