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I've heard it said that there are no faithful unitary irreps of the Euclidean group $E_n$ (for $n\geq 2$) that are square-integrable. In other words, if $\pi: E_n \to \mathcal{U}(V)$ is a faithful unitary irrep then we must have $$ \int_{E_n} | \langle \psi, \pi(g) \psi \rangle|^2 dg = \infty$$ for all $\psi \neq 0$ (Haar measure).

  1. Is this statement true, and if yes, how is it proven?
  2. If this is true, how could we decompose a reducible rep on $L^2(E_n)$ in the spirit of the Peter-Weyl Theorem (compact cases) or Pontryagin duality (abelian cases).
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