TITLE: Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.