comprehensive presentation of the unitary dual of $SO_0(n,1)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:40:28Z http://mathoverflow.net/feeds/question/105387 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105387/comprehensive-presentation-of-the-unitary-dual-of-so-0n-1 comprehensive presentation of the unitary dual of $SO_0(n,1)$ emiliocba 2012-08-24T13:32:20Z 2013-01-13T12:39:39Z <p>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.-.</p> http://mathoverflow.net/questions/105387/comprehensive-presentation-of-the-unitary-dual-of-so-0n-1/105666#105666 Answer by Jim Humphreys for comprehensive presentation of the unitary dual of $SO_0(n,1)$ Jim Humphreys 2012-08-28T00:29:54Z 2012-08-28T00:29:54Z <p>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 <code>$\mathrm{Spin}(n,1)$</code>. One 1974 paper <em>The unitary representations of the generalized Lorentz groups</em> 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. </p> <p>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. </p> http://mathoverflow.net/questions/105387/comprehensive-presentation-of-the-unitary-dual-of-so-0n-1/118806#118806 Answer by Aakumadula for comprehensive presentation of the unitary dual of $SO_0(n,1)$ Aakumadula 2013-01-13T12:39:39Z 2013-01-13T12:39:39Z <p>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. </p>