Hi All,


I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:


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?



Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group
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
$T^\*(N/M)\to\mathfrak n^\*$.)


I'd say the basic paper on the subject is [this one](http://www.ams.org/mathscinet-getitem?mr=911085) by Corwin, Greenleaf, and Grélaud. It has references to the earlier work by Kirillov himself, and you'll find more in mathscinet's [forward references](http://www.ams.org/mathscinet/search/publications.html?revcit=911085) to reviews citing it. 


