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

⎞