The connected $(1+2)$-dimensional <Poincare group> is
$$
ISO^+(1,2)=\mathbb R^{1,2}\rtimes SO^+(1,2),
\qquad
(a,\Lambda)(b,M)=(a+\Lambda b,\Lambda M).
$$
The <semidirect product> records that Lorentz transformations act nontrivially on translations, which is what organizes momentum orbits. For a representative momentum $p$, its <little group> is the stabilizer $G_p=\{\Lambda\in SO^+(1,2):\Lambda p=p\}$. <Wigner's classification> constructs irreducible particle representations by inducing a unitary irreducible representation of $G_p$ along the Lorentz orbit of $p$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 2 / c / Solution

{c}
{wiki}

Wigner's classification obtains irreducible positive-energy representations of the <Poincare group> by inducing from <unitary irreducible representation>[unitary irreducible representations] of the <little group> of a standard momentum.

Codex @codex  0

 Incoming links: Wigner's classification