100 1. Higher order Fourier analysis

group G/G≥s (with the degree s − 1 filtration (G≥i/G≥s)) by the central

vector space G≥s. Thus one can view degree s filtered nilpotent groups as an

s-fold iterated tower of central extensions by finite-dimensional vector spaces

starting from a point; for instance, the Heisenberg group is an extension of

R2

by R.

We thus see that nilpotent filtered Lie groups are generalisations of vec-

tor spaces (which correspond to the degree 1 case). We now turn to filtered

nilmanifolds, which are generalisations of tori. A degree s filtered nilman-

ifold G/Γ = (G/Γ,G•, Γ) is a filtered degree s nilpotent Lie group G•,

together with a discrete subgroup Γ of G, such that all the subgroups G≥i

in the filtration are rational relative to Γ, which means that the subgroup

Γ≥i := Γ ∩ G≥i is a cocompact subgroup of G≥i (i.e., the quotient space

G≥i/Γ≥i is cocompact, or equivalently one can write G≥i = Γ≥i · K≥i for

some compact subset K≥i of G≥i. Note that the subgroups Γ≥i give Γ the

structure of a degree s filtered nilpotent group Γ•.

Exercise 1.6.14. Let G :=

R2

and Γ :=

Z2,

and let α ∈ R. Show that the

subgroup {(x, αx) : x ∈ R} of G is rational relative to Γ if and only if α is

a rational number; this may help explain the terminology “rational”.

By hypothesis, the quotient space G/Γ = G≥0/Γ≥0 is a smooth compact

manifold. The space G≥s/Γ≥s is a compact connected abelian Lie group, and

is thus a torus; the degree s filtered nilmanifold G/Γ can then be viewed as

a principal torus bundle over the degree s− 1 filtered nilmanifold G/(G≥sΓ)

with G≥s/Γ≥s as the structure group; thus one can view degree s filtered

nilmanifolds as an s-fold iterated tower of torus extensions starting from a

point. For instance, the Heisenberg nilmanifold

G/Γ :=

⎛

⎝0

1 R R

1

R⎠

0 0 1

⎞

/

⎛

⎝0

1 Z Z

1

Z⎠

0 0 1

⎞

is an extension of the two-dimensional torus

R2/Z2

by the circle R/Z.

Every torus of some dimension d can be viewed as a unit cube [0,

1]d

with opposite faces glued together; up to measure zero sets, the cube then

serves as a fundamental domain for the nilmanifold. Nilmanifolds can be

viewed the same way, but the gluing can be somewhat “twisted”:

Exercise 1.6.15. Let G/Γ be the Heisenberg nilmanifold. If we abbreviate

[x, y, z] :=

⎛

⎝0

1 x y

1

z⎠

0 0 1

⎞

Γ ∈ G/Γ

for all x, y, z ∈ R, show that for almost all x, y, z, that [x, y, z] has exactly

one representation of the form [a, b, c] with a, b, c ∈ [0, 1], which is given by