Finite reflection groups in a Euclidean space

150 Views Asked by At

This question is regarding an exercise in Humphrey's Reflection groups and Coxeter groups. Let W be a finite reflection group acting on an n dimensional vector space V and so it naturally acts on $S(V^*)$ via the contragredient action of W on $V^*$. $S(V^*)$ may be identified with polynomial functions on V. The exercise is W has an invariant of degree 2 in $S(V^*)$. How to prove this? Hint is to use the fact that W is a subgroup of orthogonal group. Please help me.