A Generalization of Steinberg’s Cross Section

0.1. 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 l. For w ∈ W let ẇ be a representative of w 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 UẇU has trivial isotropy groups and (ii) the subset (U ∩ ẇU−ẇ−1)ẇ meets any U -orbit on UẇU in exactly one point; in particular, (iii) the set of U -orbits on UẇU is naturally an affine space of dimension l(w). More generally, assuming that w is any elliptic element of W of minimal length in its conjugacy class, it is shown in [L3] that (i) holds and, assuming in addition that G is of classical type, it is shown in [L5] that (iii) holds. In this paper we show for any w as above and any G that (ii) (and hence (iii)) hold; see 3.6(ii) (actually we take ẇ of a special form but then the result holds in general since any representative of w in NT is of the form tẇt−1 for some t ∈ T ). We also prove analogous statements in some twisted cases, involving an automorphism of the root system or a Frobenius map (see Theorem 3.6) and a version over Z of these statements using the results in [L2] on groups over Z. Note that the proof of (ii) given in this paper uses (as does the proof of (i) in [L3]) a result in [GP, 3.2.7] and a weak form of the existence of “good elements” [GM] in an elliptic conjugacy class in W .