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We show that the Eigenspace Representations for $\mathbb{R}^{n}$ semidirectwith a finite pseudo-reflection group $K$, which satisfy some generic propertyare equivalent to the induced representations from $\mathbb{R}^{n}$ to$\mathbb{R}^{n} \rtimes K$, which satisfy the same property by Mackey littlegroup method.And the proof of the equivalence is by using matrix coefficientsand invariant theory.As a consequence, these eigenspace representations areirreducible.

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