Nilpotent Orbits and Commutative Elements Original Research Article

LetWbe a simply laced Coxeter group with generating setS, and letWcdenote the subset consisting of those elements whose reduced expressions have no substrings of the formstsfor any non-commutings, t set membership, variant S. We give a root system characterization ofWc, and in the case whereWcorresponds to a finite Weyl group, show thatWcis a union of Spaltenstein–Springer–Steinberg cells. The latter is valid also for affine Weyl groups of typeA, but not for typeDorE.