Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by $$E(3)=SO(3)\ltimes T(3)$$ where $T(3)$ is the translation group.

I am looking for a reference classifying all the finite-dimensional irreducible unitary representations of $E(3)$. Several sources (such as "Unitary Representations and Harmonic Analysis: An Introduction", Chapter IV or "Noncommutative Harmonic Analysis" Chapter 5 §4) give an analog classification for $E(2)$ but do not treat the three-dimensional case.

Any help or comment is greatly appreciated!

  • 2
    $\begingroup$ I'm confused. Don't all finite dimensional representations have $T(3)$ in the kernel? And isn't the same also true in two dimensions? $\endgroup$ Commented Jun 11 at 13:08
  • $\begingroup$ Yes that's what i thought. Is it true that all irreps of $E(3)$ come from the irreps of $SO(3)$ by setting $ \pi(g) := \pi(r)$ for $g=rt$ where $r \in SO(3)$ is a rotation and $t \in T(3)$ a translation? Are you aware of any good references for this claim? $\endgroup$ Commented Jun 11 at 13:20
  • $\begingroup$ You really don't need a reference, just restrict to $T(3)$. This group is abelian, so any finite dimensional complex representation has just one dimensional composition factors. A one dimensional representation of $T(3)$ has an associated $2$-space as its kernel, and any $SO(3)$-invariant collection of $2$-spaces is infinite. $\endgroup$ Commented Jun 11 at 14:26


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