A GENERALIZATION OF STEINBERG'S CROSS SECTION

摘要

Let G be a connected semisimple algebraic group over an algebraically closed field. Let B, B~- be two opposed Borel subgroups of G with unipotent radicals U, U and let T = B ∩ B~-, a maximal torus of G. Let NT be the normalizer of T in G and let W NT/T be the Weyl group of T. a finite Coxeter group with length function 1. For w ∈ W let u, be a representative of u in NT. The following result is due to Steinberg [St, 8.9] (but the proof in loc.cit. is omitted): if w is a Coxeter element of minimal length in W, then (i) the conjugation action of U on UωU has trivial isotropy groups and (ii) the subset (U ωU ω~1)ω meets any U-orbit on UωU in exactly one point; in particular, (iii) the set of U-orbits on UωU is naturally an affine space of dimension 1(ω).