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 $\mathbb{R}^{n} \rtimes K$, 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.