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$\DeclareMathOperator\SO{SO}\DeclareMathOperator\ISO{ISO}$I'd like to know if there are any papers that study the following problems:

  • Determine when decomposing the unitary irreps of $\ISO(d,1)$ into those of $\SO(d)$, whether the multiplicities are finite or not.

  • The same problem for $\SO(p)\times \SO(q)\to \ISO(p,q)$.

  • The Poincaré-like groups $\ISO(p,q)$ can be treated as the Inonu-Wigner contractions of the semisimple ones $\SO(p+1,q)$ (the physics picture is for example to take flat limit of symmetric spaces of $\SO(p+1,q)$). The $K$-contents of quasisimple irreps are better understood than those of $\ISO(p,q)$, and it's easy to further decompose $\SO(p+1)$ into $\SO(p)$. Is it possible to understand the Inonu-Wigner contraction at the level of $K$-content?

For the simplest case $\ISO(2,1)$, I know the answer of question 1 is affirmative by explicit calculations, and partially know the results of question 3.

Any suggestions and related references are appreciated!

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    $\begingroup$ What does ISO($p,q$) denote? (I know O($p,q$) and SO($p,q$)) $\endgroup$
    – YCor
    Commented May 16, 2023 at 5:27
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    $\begingroup$ @YCor It's similar to the Poincare group, the semidirect product of $SO(p,q)$ on $\mathbb{R}^{p+q}$. "I" means inhomogeneous. The maximal compact subgroup is the same as $SO(p,q)$, hence $SO(p)\times SO(q)$. $\endgroup$
    – Lacia
    Commented May 16, 2023 at 6:22
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    $\begingroup$ Thank you. Of course I know this group but I was not familiar with the notation. $\endgroup$
    – YCor
    Commented May 16, 2023 at 7:02

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