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.