# comprehensive presentation of the unitary dual of $SO_0(n,1)$

The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO_0(n,1)$. I know the paper of Baldoni Silva-Barbasch ("The unitary spectrum for real rank one", Invent. math. 72) and the reference therein, but I would like to have a more comprehensive presentation, for example in a book. Thanks in advance.-.

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Have you tried more recent work of Barbasch or Vogan? They are certainly experts in this area and could be contacted by email at Cornell or MIT resp. There is also the incomplete online Atlas of Lie Groups: liegroups.org listing its organizers. A book I don't have at hand might also be worth looking at, though it's long out of print (and the author has left this subject): David H. Collingwood, Representations of rank one Lie groups. Research Notes in Mathematics, 137. Pitman (Advanced Publishing Program), Boston, MA, 1985. ISBN 0-273-08697-9 – Jim Humphreys Aug 24 '12 at 22:56

Though the notes by Collingwood aren't aimed directly at the unitarity question for groups of real rank one, his survey 9.1 toward the end fills in some of the ideas with a lot of references to the literature up to then. In this treatment your group occurs in the framework of its 2-fold universal covering $\mathrm{Spin}(n,1)$. One 1974 paper The unitary representations of the generalized Lorentz groups by Ernest Thieleker is cited by Collingwood but is not included in the references of Baldoni-Silva and Barbasch, which lists instead a slightly earlier one. The later paper in Trans. Amer. Math. Soc. 199, 327-367, is freely available online at www.ams.org/journals/ and might (or might not) help to fill in more details.
I am told that the first person who classified all the unitary reps of $O(n,1)$ is T.Hirai: On Irreducible Representations of the Lorentz Group of $n$-th order, Proc.Japan Acad {\bf 38} (1962),258-262.