98 1. Higher order Fourier analysis
additively) and G is a nilpotent group. Recall that a group G is nilpotent of
step at most s if the
(s+1)th
group Gs+1 in the lower central series vanishes;
thus, for instance, a group is nilpotent of step at most 1 if and only if it
is abelian. Analogously, let us call a filtered group G• nilpotent of degree
at most s if G≥s+1 vanishes. Note that if G≥0 = G and G• is nilpotent of
degree at most s, then G is nilpotent of step at most s. On the other hand,
the degree of a filtered group can exceed the step; for instance, given an
additive group G and an integer d 1, (G, d) has degree d and step 1.
The step is the traditional measure of nilpotency for groups, but the degree
seems to be a more suitable measure in the filtered group category.
We refer to sequences g : Z G which are polynomial maps from (Z,
1) to G• as polynomial sequences adapted to G•. The space of all such
sequences is denoted Poly(Z G); by the machinery of the previous section,
this is a multiplicative group. These sequences can be described explicitly:
Exercise 1.6.11. Let s 0 be an integer, and let G• be a filtered group
which is nilpotent of degree s. Show that a sequence g : Z G is a polyno-
mial sequence if and only if one has
(1.51) g(n) = g0g1
(n)
1
g2
(n)
2
. . . gs
(n)s
for all n Z and some gi G≥i for i = 0,...,s, where
(n)
i
:=
n(n−1)...(n−i+1)
i!
.
Furthermore, show that the gi are unique. We refer to the g0,...,gs as the
Taylor coefficients of g at the origin.
Exercise 1.6.12. In a degree 2 nilpotent group G, establish the formula
gnhn
=
(gh)n[g,
h]−(n)2
for all g, h G and n Z. This is the first non-trivial case of the Hall-
Petresco formula, a discrete analogue of the Baker-Campbell-Hausdorff for-
mula that expresses the polynomial sequence n gnhn explicitly in the
form (1.51).
Define a nilpotent filtered Lie group of degree s to be a nilpotent
filtered group of degree s, in which G = G≥0 and all of the G≥i are con-
nected, simply connected finite-dimensional Lie groups. A model example
here is the Heisenberg group, which is the degree 2 nilpotent filtered Lie
group
G = G≥0 = G≥1 :=

⎝0
1 R R
1
R⎠
0 0 1
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