Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:41:39Z http://mathoverflow.net/feeds/question/106068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106068/structure-of-the-unitary-representation-l2n-m-when-n-is-a-nilpotent-lie-gr Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group Valerie 2012-08-31T20:26:42Z 2012-11-28T05:46:46Z <p>Hi All,</p> <p>I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:</p> <p>I am trying to understand the structure (e.g., decomposition) of the unitary representation $L^2(N/M)$ where $N$ is a nilpotent Lie group acting by left translation on this Hilbert space (coming from the invariant measure on N/M). Surprisingly, I am unable to find any suitable references. Does anyone here know where one should look for an answer in the literature?</p> http://mathoverflow.net/questions/106068/structure-of-the-unitary-representation-l2n-m-when-n-is-a-nilpotent-lie-gr/106073#106073 Answer by Francois Ziegler for Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group Francois Ziegler 2012-08-31T21:43:37Z 2012-09-01T15:27:02Z <p>The representation you're looking at is $\mathrm{Ind}_M^N1$ and as such, its decomposition into irreducibles is very well understood using Kirillov's orbit method. (Essentially, the irreducibles that enter correspond to the coadjoint orbits in the image of the moment map <code>$T^*(N/M)\to\mathfrak n^*$</code>.)</p> <p>I'd say the basic paper on the subject is <a href="http://www.ams.org/mathscinet-getitem?mr=911085" rel="nofollow">this one</a> by Corwin, Greenleaf, and Grélaud. It has references to the earlier work by Kirillov himself, and you'll find more in mathscinet's <a href="http://www.ams.org/mathscinet/search/publications.html?revcit=911085" rel="nofollow">forward references</a> to reviews citing it. </p> http://mathoverflow.net/questions/106068/structure-of-the-unitary-representation-l2n-m-when-n-is-a-nilpotent-lie-gr/114733#114733 Answer by Vignon for Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group Vignon 2012-11-28T05:46:46Z 2012-11-28T05:46:46Z <p>Is $N$ or $M$ connected? simply connected? Is $N$ commutative? Ronald Lipsman has settled the decomposition of quasiregular representations of Lie groups in many settings. You will find his papers very readable. They are all available on google. </p>