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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1$\begingroup$ 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 $\endgroup$– Jim HumphreysAug 24, 2012 at 22:56
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2 Answers
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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.
What you get from any of the sources may be slanted more toward pure mathematics or towards physics, but my understanding is that the details about your special case haven't been essentially improved on since Collingwood's monograph appeared. In any case, that's likely to be the only extended exposition focusing on the real rank one case. The unitary dual is unavoidably complicated to arrive at, even in your special example, so the general theory in the background gets heavy at times.
answered Aug 28, 2012 at 0:29
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$\begingroup$ Dear Professor, thank you for your suggestions. I can see that this area is unavoidably complicated to arrive... I will try to undestand Thm 3 in Thieleker (page 362) and to read the survey in Collingwood. $\endgroup$ Aug 28, 2012 at 17:30
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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 38 (1962),258-262.
answered Jan 13, 2013 at 12:39
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1$\begingroup$ Also see the Math Overflow question mathoverflow.net/questions/84762 $\endgroup$ Jun 26, 2013 at 18:21