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I recently learnt that one can explicitly classify the unitary irreducible representations of $\mathrm{SL}(2,\mathbb R)$. In the end one has a list of all these representations given by explicit formulas.

Now I wonder for which other Lie groups such an explicit classification is possible as well.

I know that it is possible for $\mathrm{SO}(2)$ (and maybe for all compact Lie groups? and for all abelian ones?). I am more interested in whether it is possible for more "complicated" Lie groups such as $\mathrm{SL}(3,\mathbb R)$.

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