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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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2 Answers 2

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.

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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.

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