ON REFLECTION LENGTH IN REFLECTION GROUPS

Abstract. Let W be the Weyl group of a connected reductive group over a finite field. It is a consequence of the Borel-Tits rational conjugacy theorem for maximal split tori that for certain reflection subgroups W1 of W (including all parabolic sub-groups), the elements of minimal reflection length in any coset wW1 are all conjugate, provided w normalises W1. We prove a sharper and more general result of this na-ture for any finite Coxeter group. Applications include a fusion result for cosets of reflection subgroups and the counting of rational orbits of a given type in reductive Lie algebras over finite fields. 1. Background and statement of results LetW be a finite Coxeter group acting as a reflection group on the Euclidean space V of dimension `. We refer to [B] for background. (1.1) Definition. The reflection length n(w) of an element w ∈W is the minimal integer n such that w = r1r2...rn, where the ri are reflections in W. The function n(w) is clearly conjugacy invariant and it is well known (see [C1, Lemma 2] or [HL], for example) that for any element w ∈W, we have n(w) = dim(im(1 − w)). (1.2) When W is the Weyl group of a connected reductive group G which is defined and split (see [BT]) over a finite field Fq, the function n(w) arises in the study of rationality properties of tori. For background about the following matters the reader is referred to [L] and the references there. Let F: G → G be the Frobenius endomorphism associated with the Fq-structure on G. We refer to an F-stable subvariety of G as rational and denote byHF the set of F-fixed points of any variety H on which F acts. It is well-known that the GF-conjugacy classes of rational (that is, F-stable) maximal tori of G are parametrised by the conjugacy classes of W. Denote by T0 a fixed maximal torus of G which is split over Fq. For any group