On Reflection Length in Reflection Groups

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 subgroups), 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 nature 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 byH the set of F -fixed points of any variety H on which F acts. It is well-known that the G -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 H 1991 Mathematics Subject Classification. 20G40, 20G05. Typeset by AMS-TEX 1 defined over Fq, denote by r(H) its Fq-rank (the dimension of any of its maximal Fq-split tori). Thus r(G) = r(T0) = dim(T0). For any element w ∈W , we say that the rational maximal torus T is w-twisted and write T = Tw if T = gT0g for some g ∈ G such that g−1F (g) = ẇ ∈ NG(T0) where ẇT0 = w ∈ W = NG(T0)/T0. The set {w ∈W | T is w-twisted } is a conjugacy class of W . If Tw is a w-twisted rational maximal torus, then we have (see [L, (5.5)]) r(Tw) = `− n(w) (1.3) where ` = r(G). Now suppose x is a semisimple element of G . The connected centraliser CG(x) is a reductive group defined over Fq and the rational conjugacy of the maximally split maximal tori of CG(x), proved in [BT], has the following simple (but not obvious) consequence for the length function. (1.4) Proposition. Let W be the Weyl group of a connected reductive group G over Fq. Suppose W1 ≤ W is the Weyl group of the connected centraliser of a semisimple element of G (e.g. W1 could be any parabolic subgroup of W – see §2 below). Let w be an element of the normaliser in W of W1. Then any two elements of the coset wW1 which have minimal reflection length in the coset are conjugate in W . We shall explain how (1.4) follows from the work of Borel and Tits in the next section, but our main purpose in this note is to prove an elementary but more general result concerning reflection groups, of which (1.4) is a consequence. (1.5) Theorem. Let W be a finite Coxeter group acting on a Euclidean space V . Let Φ be the corresponding root system, with Π a chosen base of Φ. Let σ be an orthogonal transformation of V such that σΠ = Π. Then, for any element w ∈W , the following conditions are equivalent (1) dim(im(1− σw)) is minimal (2) There is an element x ∈W such that σw = x−1σx. (3) σw stabililises some simple system in Φ. We shall see below (see §2 or (4.4)(1)) that (1.4) follows easily from (1.5), with W (of (1.5)) replaced by W1. Moreover, (1.5) shows that the elements of minimal reflection length in wW1 are actually conjugate by an element of W1. If W1 is any reflection subgroup ofW , the choice of a simple system Π forW determines a length function for W and it is the case (see §4 below) that each coset wW1 contains a unique shortest element with respect to this length function. If w normalises W1, our theorem identifies the conjugacy class of elements of minimal reflection length in wW1 as that of this shortest element in the coset (see (4.2) below).