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 W 1 of W (including all parabolic subgroups), the elements of minimal reflection length in any coset wW 1 are all conjugate, provided w normalises W 1. 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.
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