Let me add a little bit to what has already been written above.

The current framework for quantum mechanics motivates the study of unitary representations of the symmetry groups of the physical system. In the context of four-dimensional relativistic quantum field theory (with some mild assumptions), it follows from a celebrated theorem of Coleman and Mandula that the symmetry group is a direct product of (the universal cover of) the Poincaré group and a compact Lie group. (One can get around this theorem by considering not Lie groups but Lie supergroups, but that's another story.) Let me focus on the Poincaré group. The determination of the (physically relevant) unitary irreducible representations of the Poincaré group is due to Wigner, generalising (though perhaps not consciously) Frobenius's method of induced representations. It was Mackey who extended Wigner's method and placed it firmly in the "right" mathematical context.

The point I would like to make is that approaching the representation theory of the Poincaré group (however one motivates this study) in this fashion naturally makes contact with particle physics.

Induced representations

Let us start with finite groups. Let $G$ be a finite group and $H$ be a subgroup and let $\delta: H \to \mathrm{U}(W)$ be a unitary representation on a finite-dimensional hermitian vector space $W$. Consider the vector space $V$ of functions $f:G \to W$ subject to the equivariance condition $f(gh) = \delta(h^{-1}) f(g)$ for all $g \in G$ and $h \in H$. The homomorphism $\rho:G \to \mathrm{GL}(V)$ defined by $$(\rho(g) f)(g') = f(g^{-1}g')$$ makes $V$ into a representation of $G$. (One has to check that $\rho(g) f \in V$ again.)

Moreover, it is possible to define on $V$ a hermitian structure relative to which $\rho$ is a unitary representation. This is best seen by viewing $V$ in a different light. Let $X= G/H$ be space of left $H$-cosets in $G$. Then $V$ is isomorphic to the vector space of functions $\psi: X \to W$, but not canonically. The isomorphism depends a choice of coset representative $\sigma: X \to G$, a section through the surjection $\pi: G \to X$. Then given $f \in V$ we define $\psi(x)= f(\sigma(x))$. Conversely, given $\psi:X \to G$ we define $f \in V$ by writing $g = \sigma(\pi(g))h(g)$ for some $h(g) \in H$ and declaring $f(g) = \delta(h(g)^{-1}) \psi(\pi(g))$. Then we define the inner product of $f_i \in V$ to be $$ \langle f_1,f_2\rangle_V = \sum_{x\in X} \langle f_1(\sigma(x)), f_2(\sigma(x)) \rangle_W.$$ One can show that this is independent of the coset representative precisely because $W$ is a unitary representation of $H$.

The representation $V$ of $G$ is said to be induced from the representation $W$ of $H$.

Wigner's method

Wigner's method is formally very similar: $G$ is the Poincaré group; that is, the semidirect product $L \ltimes T$, where $L = \mathrm{Spin}(3,1)$ is the spin cover of the Lorentz group and $T$ is the translation ideal. Wigner starts by choosing a character $p$ of $T$, which physically is interpreted as a momentum. A version of Schur's Lemma says that on an irreducible representation of the $G$, all the characters of $T$ which appear share the same minkowskian norm $p^2$. Physically relevant representations have $p^2 = - m^2$, where $m\geq 0$ is the mass.

Let $H < G$ denote the stabilizer of $p$. Wigner induces a unitary representation of the Poincaré group from a finite-dimensional unitary representation of $H$. Now $H$ is non-compact, so such representations are necessarily not faithful. They factor through faithful representations of a group known as Wigner's little group. It is isomorphic to $\mathrm{Spin}(3)$ for $m>0$ and $\mathrm{Spin}(2)$ for $m=0$. Irreducible finite-dimensional representations of the little groups are labelled by integers: the spin (a non-negative integer) for $m>0$ the helicity for $m=0$. So Wigner tells us that to a unitary irreducible representation of the Poincaré group one can associate a mass and a spin/helicity, which are the basic data specifying relativistic particles. But there's more.

The space $G/H$ is the hyperboloid $p^2 = - m^2$ (for a fixed mass $m\geq 0$) in the dual to the Lie algebra of $T$. The vector space carrying the induced representation of $G$ consists of (square-integrable) sections of homogeneous vector bundles over $G/H$ associated to the representation of $H$ from which we induce. Thus this gives naturally a representations on geometric objects defined in the space $G/H$ of momenta. Fourier transforming to Minkowski spacetime we arrive at sections of homogeneous bundles over Minkowski spacetime satisfying (linear) partial differential equations coming from Fourier transforming the condition $p^2 = -m^2$ and the other irreducibility conditions. And the nice surprise is that these partial differential equations are precisely the linearised free field equations for the corresponding particles: the Klein-Gordon, Dirac, Weyl, Maxwell,... equations!