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So the Heisenberg nilmanifold is an addition rule on triples $(a,b,c) + (x,y,z) \equiv (a+x, b+y + m\; xc ,c+z)$. This rule is associative and $$ n(a,b,c) = \left(na, nb + m\frac{n(n-1)}{2}ac, nc\right) $$ However, we are going to quotient by those triples whose coordinates are integers. $$H = (\mathbb{R},\mathbb{R},\mathbb{R}) \mod (\mathbb{Z},\mathbb{Z},\mathbb{Z}) $$ In this relation the quotient should be (can someone help me shrink these fractions please?) $$n(a,b,c) \equiv \left( \{ n a\},\left\{ nb + m \frac{n(n-1)}{2}ac - m\;\; na\lfloor nc\rfloor \right\} , \{nc \} \right)$$ I use additive notation since if $n = 0$, this is just the 3-torus $\mathbb{T}^3$ and it really looks like addition.

In this way it is a circle bundle over the torus (Euler class = n in this case, for $\mathbb{T}^3$, Euler class is $0$.). It can also be thought of the mapping cylinder of the linear map, $ \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) $

These come in handy when reading the papers by Tao, Green and Ziegler on Gowers's norms, where they look for patterns in the primes by correlating with various sequences mod 1. $\leftarrow$ If you can help me understand that paper would be nice

I would like to know how functions $f:H \to \mathbb{R}$, i.e. in $f \in L^2(H)$ decomposes into a "Fourier-like series" (maybe the term is spectral decomposition).

If this were just the 3-torus ($n=0$), the modes would be $e^{i(mx+ny+pz)}$, but these Heisenberg groups are not quite abelian.

If I recall... the starting point if to observe the middle copy of the circle, $(0, \mathbb{T}, 0)$ is in the center of the group and then you can write induced representations...

  • $(a,b,c) \mapsto e^{i(ma + nc)}$ with integers $m, n \in \mathbb{Z}$ is a two-dimensional family...
  • Wikipedia suggests you can act on functions, $(a,b,c)f(x) = e^{i(cx + mb)}f(x+m a)$.

If this were over $H(\mathbb{Z}_p)$ (with $\mathbb{Z}_p =\{ e^{2\pi i k /p}: k = 0 \dots, p-1 \}$ ) instead of $H(\mathbb{T})$ we'd have $p^2$ 1-dimensional representations and $p-1$ of $p$-dimensional representations. Then $p^2\cdot 1 + (p-1)\cdot p^2 = p^3 = |H(\mathbb{Z}_p)|$, so we have all of them. Then let $p \to \infty$ (doesn't have to be prime) and $\mathbb{Z}_p \to S^1$.

I would expect the representations of $H$ to form some kind of 3-dimensional lattice (if $n=0$ representations of $\mathbb{T}^3$ are indexed by $\mathbb{Z}^3$. and then I still don't know of they make up all of $L^2(H)$

Finally, I'm worried I'm not getting the 2-step nilcharacters mentioned in Tao Green Ziegler's papers such as $e^{n^2 x}$ or $e^{n \sqrt{2} \lfloor n \sqrt{3} \rfloor}$ .

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    $\begingroup$ @ John: As I understand, what you call $H$ is the quotient of the Heisenberg group $Heis(\mathbb{R})$ over the reals, by the discrete Heisenberg group $Heis(\mathbb{Z})$. Please pay attention to the fact that, since $Heis(\mathbb{Z})$ is NOT a normal subgroup, this quotient makes sense only as a homogeneous space, and carries no group structure! If I understand correctly, you want to decompose $L^2(H)$ into irreducible representations of $Heis(\mathbb{R})$; except for 1-dimensional characters, these are given by the Stone-von Neumann theorem. See en.wikipedia.org/wiki/Heisenberg_group $\endgroup$
    – Alain Valette
    Nov 30, 2011 at 19:51
  • $\begingroup$ PS: I'm sure that this spectral decomposition has been worked out somewhere, I just don't know the reference. $\endgroup$
    – Alain Valette
    Nov 30, 2011 at 19:53
  • $\begingroup$ I'm sure it is too... actually it's more like $Heis(S^1)$ since it's a bundle over the torus $(S^1)^2$. $\endgroup$
    – john mangual
    Dec 1, 2011 at 0:13

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You can find the explicit decomposition of the representation of the Heisenberg group on $L^2$ of the Heisenberg nilmanifold in Invent. Math. 78 (1984), 101-112 for example, together with references.

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  • $\begingroup$ I'm surprised this leads to Atiyah-Singer index theorem. I was just asking simple number theory question. $\endgroup$
    – john mangual
    Dec 1, 2011 at 0:22
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You might find what you need in Starkov's "Dynamical systems on homogeneous spaces", in the chapter "Spectral theory of homgoeneous flows". Apparently he analyzes the case of an ergodic flow on a nilmanifold in general and shows how the decomposition works in particular for the Heisenberg nilmanifold, which is the case you are interested in. Link.

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Fourier analysis on the Heisenberg group is a large subject, and googling "Fourier analysis on the heisenberg group" will give many references. Here is one:

Neil Lyall: The Heisenberg Group Fourier Transform

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  • $\begingroup$ This is Heisenberg group over $\mathbb{R}$ and $\mathbb{R}^n$ but yes, this kind of thing. $\endgroup$
    – john mangual
    Nov 30, 2011 at 18:41

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