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Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$?

Equivalently, what is a good reference (for mathematicians) for the representation theory of the Lorentz group?

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I don't know about a reference, but here goes. A complex irrep is the complexification of a real irrep iff it's self-dual, and then, if the unique invariant bilinear form is symmetric. (The others are quaternionic, and in particular even-dimensional.) On dominant weights, duality is given by $-w_0$, so for $sl_2$ they're all self-dual. The odd-dimensional ones $V_{2n}$ are obviously real. The 2-d one $V_1$ is obviously quaternionic. So each $V_{2n+1} \leq V_{2n} \otimes V_1$ is quaternionic. Summary: each odd-dimensional irrep is the complexification of a real irrep, and that's all of them. –  Allen Knutson Jun 8 at 18:21
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Oops: my question was essentially the answer for $\mathfrak{sl}_2(\mathbb R)$, not its square $\mathfrak{sl}_2(\mathbb C)$. I should have been tipped off by your mention of Lorentz. –  Allen Knutson Jun 8 at 23:30

1 Answer 1

up vote 9 down vote accepted

The real-linear representations are sums of irreducibles, and the irreducibles are parametrized by pairs of nonnegative integers: see Knapp's §2.3 here.

Explicitly the irreducible parametrized by $(m,n)$ acts on the polynomials $f$ in $(z_1,z_2,\bar z_1,\bar z_2)$ that are homogeneous of degree $m$ in $(z_1,z_2)$ and of degree $n$ in $(\bar z_1,\bar z_2)$, by $$ \biggl(\begin{pmatrix}a&b\\c&d\end{pmatrix}f\biggr) \begin{pmatrix}z_1\\z_2\end{pmatrix}= f\biggl(\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}\begin{pmatrix}z_1\\z_2\end{pmatrix}\biggr). $$

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