Strong Lefschetz elements of the coinvariant rings of finite Coxeter   groups
  For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups. 
Toshiaki Maeno, Yasuhide Numata and Akihito Wachi