# Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group

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?

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Appearently, it is clear what $M$ should be here? For a start with the Mackey Machine/Kirillov orbit method/Clifford theory (see F.Z.'s answer) I suggest Weintraub's book on representation theory for finite groups. This explains the representation theory of semidirect products/group extensions in terms of orbits, stabilizer and (proj.) reps of the stabilizer. –  Marc Palm Nov 28 '12 at 9:06

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 $T^*(N/M)\to\mathfrak n^*$.)
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.