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A generalization of Steinberg's cross section

Xuhua He, George Lusztig

2012
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Journal of The American Mathematical Society
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Introduction 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 letẇ 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
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... eter 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 takeẇ 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 . 0.2. Let w be an elliptic element of W which has minimal length in its conjugacy class C and let γ be the unipotent class of G attached to C in [L3]. Recall that γ has codimension l(w) in G. As an application of our results we construct (see 4.2(a)) a closed subvariety Σ of G isomorphic to an affine space of dimension l(w) such that Σ ∩ γ is a finite set with a transitive action of a certain finite group whose order is divisible only by the bad primes of G. In the case where C is the Coxeter class, Σ reduces to Steinberg's cross section [St] which intersects the regular unipotent class in G in exactly one element.

doi:10.1090/s0894-0347-2012-00728-0
fatcat:4ek255e7znhztd3g57tbh5ymuu