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Consider Poincare group $ISO(1,d-1)$ in some dimension $d>4$. There are two Casimirs. Let's look at massless one-particle states: the little group is $ISO(d-2)$, and if we restrict to finite dimensional representations, it is actually $SO(d-2)$.

I'm interested in giving a concrete expression for the analogue of helicity in $d=4$, namely generalize the following formula (2.5.42 in Weinberg's book) $$ U(\Lambda) \Psi_{p,\sigma} = N e^{i \sigma \theta} \Psi_{\Lambda p,\sigma} $$ where $N$ is normalization and $\theta=\theta(\Lambda,p)$, to the case $d>4$, where I expect something similar to $e^{i \sigma \theta}$ factor, but involving unitary finite dimensional irreps $D(W)_{\sigma \sigma'}$ of $SO(d-2)$, depending on the little group element $W =W(\Lambda,p) \in SO(d-2)$ (we should fix a Cartan basis, and the irrep is given in terms e.g. of Young diagram), and I'm wondering whether the argument that in $d=4$ restricts $\sigma$ to be half integer (basically by requiring $e^{4\pi i \sigma}=1$) still applies, and how.

By concrete I mean that, say when $\Lambda$ itself is in some maximal 'torus' of $so(1,d-1)$, say rotations in planes $(j,j+1)$, the formula should be simple enough, possibly again just a bunch of phases.

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