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

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

2012-08-31T20:26:42Z

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?

Answer by Francois Ziegler for Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group

2012-08-31T21:43:37Z

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 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 to reviews citing it.

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

2012-11-28T05:46:46Z

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.