A new method to prove the irreducibility of the eigenspace representations for Rn semidirect with a finite pseudo-reflection group

We show that the Eigenspace Representations for ${\mathbb{R}}^n$ semidirect with a finite pseudo-reflection group $K$, which satisfy some generic property are equivalent to the induced representations from ${\mathbb{R}}^n$ to ${\mathbb{R}}^n⋊K$, which satisfy the same property by Mackey little group method.And the proof of the equivalence is by using matrix coefficients and invariant theory.As a consequence, these eigenspace representations are irreducible.