Classification of unitary irreducible representation 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 $\mathrm{SL}(3,\mathbb R)$.

It may be worth adding that for complex reductive or semi-simple classical groups (viewed as "real" by restriction of scalars), Gelfand-Naimark nearly-completely classified irreducible unitaries by 1950. (There were some oversights, corrected by Knapp-Stein, for example.) The point is (related to Matt E's points) that the groups themselves have no discrete series, nor do any of the Levi components of parabolics, vastly simplifying the classification.
