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The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.

Its space of irreducible, continuous unitary representations has been described by Gelfand and Neumark in this 1947 paper.

I am not very familiar with this subject, but in my understanding there are now several proofs to this result. My question is: where can I find a "modern" and accessible presentation of this result?

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Some people around here are almost certain to object if you write 'Naimark' instead of 'Neumark'; see the comments to… – Todd Trimble Oct 19 '11 at 7:34
Thanks Todd. It is now corrected. – Mikael de la Salle Oct 19 '11 at 16:00
up vote 15 down vote accepted

I know of two clean approaches to classifying the unitary irreps of the $ax+b$ group. The first is to write the group as a semidirect product $\mathbb R \ltimes \mathbb R_{>0}$. There is a theory (due chiefly to Mackey) that deals with the representations of semidirect products. In this case, the semidirect is fairly simple, so the description of the unitary irreps you get from "the Mackey machine" is simple too.

The second approach is to realize that the $ax+b$ group is exponential solvable. This means that Kirillov's orbit method applies. Thus the unitary irreps lie in correspondence with coadjoint orbits of the group on its Lie algebra. As a vector space, the Lie algebra of the $ax+b$ group is $\mathbb R^2$, and the coadjoint orbits turn out to be the half-planes $\{y>0\}$ and $\{y<0\}$ and the singletons $\{(x,0)\}$, so the picture isn't very complicated.

A good, modern reference for both approaches is Folland's A Course in Abstract Harmonic Analysis (CRC Press, 1995), sections 6.7 and 7.6, respectively. Folland also describes the Plancherel formula in section 7.6. I should mention though that Folland doesn't prove absolutely everything when it comes to the orbit method, but he does give references.

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Thanks Faisal for your very informative answer, this is answers exactly my question. Folland's book is not in the library I am currently in, I will have a look at Folland's book when I get back home. – Mikael de la Salle Oct 19 '11 at 16:03

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