# What are the finite-dimensional irreducible unitary representations of $E(3)$?

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!

• 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? Commented Jun 11 at 13:08
• 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? Commented Jun 11 at 13:20
• 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. Commented Jun 11 at 14:26