Representation theory of the Lorentz group

Example

Let $G = \mathrm{SO}^+(1, 3)$ be the restricted Lorentz group: the linear maps of $\R^4$ that preserve $t^2 - x^2 - y^2 - z^2$, have determinant $1$ and preserve the direction of time. Its universal cover is the group SL(2, C) $\mathrm{SL}_2(\C)$, a double cover with kernel $\set{\pm I}$.

For half-integers $m, n \in \set{0, \frac12, 1, \frac32, \dots}$, let $V_{2m}$ be the space of homogeneous polynomials of degree $2m$ in two variables, with $\mathrm{SL}_2(\C)$ acting by $(\pi_{2m}(g) P)(z) = P(g^{-1} z)$. Restricted to SU(2), this is the irreducible representation of dimension $2m + 1$. Put $$\pi_{m,n}(g) = \pi_{2m}(g) \otimes \pi_{2n}(\bar g) \quad \text{on} \quad V_{2m} \otimes V_{2n},$$ where $\bar g$ is the matrix with complex conjugate entries. This is a continuous homomorphism, so it is a representation of $\mathrm{SL}_2(\C)$ of dimension $(2m + 1)(2n + 1)$.

The classification: the $\pi_{m,n}$ are irreducible and pairwise non-isomorphic. View $\mathrm{SL}_2(\C)$ as a real Lie group. Then each irreducible finite-dimensional complex representation of it is isomorphic to exactly one $\pi_{m,n}$. The element $-I$ acts by $(-1)^{2m + 2n}$, so $\pi_{m,n}$ is a representation of $G$ exactly when $m + n$ is an integer.

Check on the Lie algebra: the complexification of $\mathfrak{so}(1, 3)$ is $\mathfrak{sl}_2(\C) \oplus \mathfrak{sl}_2(\C)$. On it, $\pi_{m,n}$ becomes the tensor product of the irreducible representations of highest weights $2m$ and $2n$ of the two factors.