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 $\mathrm{SL}(3,\mathbb R)$.
Best Answer
If $G$ is a real Lie group, then the collection of isomorphism classes of irreducible unitary reps. of $G$ is known as the unitary dual of $G$, and the problem you ask about, namely of explicitly describing the unitary dual of $G$, is one of the major open problems in the theory.
If $G$ is compact, then any finite-dimensional irreducible representation is unitary (you can take any positive definite inner product on the underlying vector space of the representation, and then average it over $G$), and conversely, any irreducible unitary rep. is finite-dimensional (this is part of the Peter--Weyl theorem).
Of course, the finite dimensional irreps. of $G$ are classified by usual highest weight theory, and so the problem is solved for compact $G$.
For abelian $G$, the unitary irreps. of $G$ are just characters, and so the unitary dual of $G$ coincides with its Pontrjagin dual, for which there is a detailed theory, which is more or less completely understood.
The major open problem is the case when $G$ is semi-simple, but non-compact.
For $GL_n(\mathbb R)$ (which is not semi-simple, I guess, but is so modulo its centre) the classification is complete (it's due to David Vogan). It is known in some other cases as well (e.g. Vogan also treated $G_2$). I think one should be able to deduce the unitary dual for $SL_n(\mathbb R)$ from the case of $GL_n(\mathbb R)$ (and I would guess this is done in the literature).
For arbitary semi-simple groups, Harish--Chandra classified the discrete series representations, which are the unitary irreps. that can be embedded into $L^2(G)$, and (building on Harish--Chandra's results) Knapp and Zuckerman classified the so-called tempered irreps. for any $G$.
There are other results known; some of the relevant names are --- in addition to Vogan --- Adams, Arthur, Barbasch, Schmid and Vilonen. However, as far as I know, for the moment the problem remains open for general semisimple groups.