# Representation Theory of $SL(2,\mathbb R)$

The representation theory regarding the finite-dimensional representations of $SL(2,\mathbb C)$ is well-understood; namely, they all decompose into irreducibles $V_n$, $\dim(V) = n > 0$. Furthermore one knows that $V_n$ is isomorphic to the space of degree $n-1$ homogeneous polynomials in two variables with linear group action. The physical interpretation is also well-known: $V_n$ is the Hilbert space for the internal state space corresponding to the spin degrees of freedom for a particle of spin $(n-1)/2$. Aside from the trivial representation none of these are unitary.

The mathematics of its infinite-dimensional unitary representations is well-understood, though not as well-known. I would like to understand the physical interpretations of these representations.

As a simpler case, I thought to first consider the unitary representations of $SL(2,\mathbb R)$, which are also well-understood. Aside from the trivial one, the unitary representations are all infinite-dimensional and fall into four categories. Of these four, two have discrete parameters while the other two have continuous parameters.

A few years back I traced through the isomorphism of $PSL(2,\mathbb C)$ with the Lorentz group and found that $PSL(2,\mathbb R)$ corresponded to the subgroup of Lorentz transformations fixing the $y$-direction. (I've never seen this mentioned anywhere though so I could be mistaken about this.) Granting this physical interpretation of $SL(2,\mathbb R)$, its unitary, irreducible representations should admit physical interpretations compatible with an interpretation for the representations of the total group $SL(2,\mathbb C)$. Presumably these are internal state spaces for particles which are coming to/from infinity and the parameters have to do with angular momentum or something like that. Does anybody know or have a reference?

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• Vilenkin's, or Vilenkin & Klimyk 's books should be good bets: they have left no stone unturned. – Cosmas Zachos Oct 17 '17 at 15:04