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

TL;DR

This paper introduces a new proof technique demonstrating that eigenspace representations for certain semidirect product groups are irreducible, leveraging matrix coefficients and invariant theory.

Contribution

It provides a novel method to prove the irreducibility of eigenspace representations for R^n semidirect with finite pseudo-reflection groups, using Mackey's little group method.

Findings

Eigenspace representations are equivalent to induced representations under certain conditions.

The proof employs matrix coefficients and invariant theory.

Eigenspace representations are shown to be irreducible.

Abstract

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.