arXiv:1605.05168v1 [math.RT] 17 May 2016
The Zak transform on strongly proper G-spaces and its
applications
Dominik Jüstel1
Key words. Zak transform, G-spaces, Weil formula, Fourier analysis, Bloch waves,
Poisson summation, trace formula
Abstract
The Zak transform on Rd is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak transform to a class of locally compact G-spaces, where
G is either a locally compact abelian or a second countable unimodular type I group.
This framework unifies previously proposed generalizations of the Zak transform. It is
shown that the Zak transform has invariance properties analog to the classic case and is
a Hilbert space isomorphism between the space of L2 -functions and a direct integral of
Hilbert spaces that is explicitly determined via a Weil formula for G-spaces and a Poisson summation formula for compact subgroups. Moreover, the inversion formula of the
Zak transform is shown to imply a trace formula that generalizes certain versions of the
Arthur-Selberg trace formula. Some applications in physics are outlined.
1Faculty of Mathematics, Technical University of Munich, juestel@ma.tum.de, DJ was partially supported
by the TUM Graduate School. Part of the paper was developped during the author’s stay at the HIM
Trimester Program “Mathematics of Signal Processing” in Bonn.
1
Contents
The Zak transform on strongly proper G-spaces and its applications
1. Introduction
2. The Weil formula for strongly proper G-spaces
2.1. Strongly proper G-spaces
2.2. The Weil formula
2.3. Integration on fundamental domains
3. The Zak transform for abelian actions
3.1. The abelian Zak transform on L1 (X)
3.2. The abelian Zak transform on L2 (X)
4. The Zak transform for non-abelian actions
4.1. The non-abelian Zak transform on L1 (X)
4.2. A Poisson summation formula for quotients by compact groups
4.3. The non-abelian Zak transform on L2 (X)
4.4. The character Zak transform and other alternatives
5. Applications of the Zak transform
5.1. A Bloch-Floquet theorem for group actions
5.2. The Zak transform in radiation design
1
4
5
5
7
13
14
14
19
21
21
25
28
30
32
32
34
Appendix A. Supplementary results
1. A sufficient condition for strong properness
2. Invariant measures on double coset spaces
37
37
41
Bibliography
47
3
4
CONTENTS
1. Introduction
The Zak transform on R has already been known to Gelfand [Gel50] and Weil [Wei64],
and is known under many different names, e.g. the Weil-Brezin map in abstract harmonic
analysis [Bre70], or the kq-representation [Zak67] or Bloch(-Floquet) transform in physics.
It first generated considerable interest, when Zak [Zak67] rediscovered it and applied it
in condensed matter physics as a refinement of the decomposition of electron states into socalled Bloch waves [Blo29]. It also found applications in signal analysis (see e.g. [Jan88]),
in particular as a tool in Gabor analysis, where it is for example used to prove the BalianLow theorem (see [Grö01] for this approach, and [Bal81, Low85] for the original work).
The Zak transform has been generalized to locally compact abelian (lca) groups by
Kaniuth and Kutyniok [KG98] and, by an ingenious construction, to certain non-abelian
locally compact Hausdorff (lcH) groups in [Kut02], where clearly a generalization of Gabor
analysis was the motivation. More recently, a Zak transform on certain semidirect product
groups [AF13] and for actions of lca groups [BHP15, Sal14] were considered.
Here, we propose a generalization of the Zak transform in the spirit of the work by Zak
[Zak67], but without requiring the ‘symmetry’ group to be a subset of the space on which
it acts. Instead, the group acts on a topological space in a suitable way. This approach
allows, for example, to study rotational and helical symmetries in three-dimensional space
in addition to the classic case of translational symmetries, where the group R3 acts on
itself by translations. Along the orbits of the action, the Zak transform decomposes a
function via Fourier analysis on non-abelian groups. In this sense, the Zak transform
might be called a orbit-frequency decomposition.
The construction includes most previous approaches, but does for example not agree
with Kutyniok’s construction in the case of a subgroup of a non-abelian group acting by
translation, illustrating the different needs for different applications. Our construction
will naturally lead to a decomposition of function spaces and operators (e.g. eigenspaces
of the electronic Schrödinger equation and the Hamiltonian) related to a class of noncrystalline structures, called objective structures, that were studied by James in [Jam06]
(see Section 5.1). The Zak transform originally emerged in the author’s work on the design
of electromagnetic radiation for the analysis of molecular structures [FJJ16, JFJ16]. This
application is outlined in Section 5.2
The topological spaces considered in this article are strongly proper G-spaces. These
are lcH spaces X on which an lcH group G acts continuously and properly. In addition,
the orbit space of a strongly proper action is paracompact – a property that is essential to
relate integration on X to the group action. This framework is introduced in Section 2.1.
A sufficient condition for strong properness that is formulated in the language of uniform
spaces is given in the Appendix in Section 1.
When there is a measure on X that is quasi-invariant w.r.t. the strongly proper
action of G (see Definition 2.2), then integration on X can be decomposed into integration
over the group and over the orbit space. This generalization of the classic Weil formula
(Theorem 2.4) for the integration on homogeneous spaces is proved in Section 2.2, and
nicely complements results by Bourbaki [Bou04]. It is also closely related to integration
over fundamental domains of the action (see Section 2.3).
2. THE WEIL FORMULA FOR STRONGLY PROPER G-SPACES
5
The Zak transform on X w.r.t. a strongly proper group action is then defined for the
action of a locally compact abelian (lca) group (Section 3) and for the action of a second
countable unimodular type I group (Section 4). The fact that the introduced transforms
are isometries essentially reduces to an application of the Weil formula and the Plancherel
theorem. In the abelian case, the image of the Zak transform can explicitly be determined
by combining the Poisson summation formula with the Plancherel theorem (Theorem 3.4).
In the case of non-abelian groups, a generalized Poisson summation formula is formulated
that allows to show the analog result (Theorem 4.8).
These results explicitly perform the decomposition of the action on the space of L2 functions into irreducible representations. For abelian actions, functions can be decomposed into invariant functions that are suitably modulated along orbits of the action. For
non-abelian actions, the Zak transform can be viewed as a family of tensor fields on a
fundamental domain of the action. Here the underlying vector space of the tensor field is
not the tangential space at the respective point, but the representation space of an eleb A function can then be decomposed into invariant tensor fields that
ment of the dual G.
are modulated along the orbits by irreducible representations of the group. We call these
equivariant fields Bloch tensor fields. Moreover, invariant differential operators decompose accordingly. In other words, the Zak transform embodies the representation theory
of the action and allows to explicitly determine its decomposition.
In the spirit of Zak’s approach [Zak67], the Zak transform is written as a decomposition
of a function into equivariant measures supported on orbits. This viewpoint reveals the
details of the decomposition into Bloch tensor fields. In Zak’s words, the Bloch fields are
“expressible in infinitely localized Wannier functions” [Zak67].
Finally, in Section 5 some applications are discussed, namely the Bloch decomposition of the electronic eigenspaces of objective structures, and the appearance of the Zak
transform in radiation design (see [FJJ16, JFJ16]).
To not disturb the reading flow, a result on the invariance properties of the measures
constructed for the Weil formula for the important special case of double coset spaces (that
for example include all affine actions on groups) is derived in Section 2 of the Appendix
(Theorem A.8), generalizing work of Liu [Liu65].
Readers that are not interested in the details of the Weil formula can essentially skip
Section 2 and directly proceed to Section 3 after having a look at Definition 2.1, where
a strongly proper action is defined, at Theorem 2.4, where the Weil formula for strongly
proper group actions is found, and at eqs. (2.10) and (2.11), where the measure on the
fundamental domain and its relation to the Weil formula are explained.
2. The Weil formula for strongly proper G-spaces
2.1. Strongly proper G-spaces. The goal of this section is to introduce the setting
of this paper, namely the framework of strongly proper G-spaces. These are topological
spaces on which a topological group G acts in a particularly nice way. The definition of
properness goes back to Palais [Pal60] and was motivated by the attempt to generalize
results for compact group actions to a more general setting. For our purpose, the importance of the stronger condition of strong properness of an action lies in the fact that
the orbit spaces are paracompact. In Section 1 of the Appendix, a sufficient condition for
strong properness is given, which is based on the theory of uniform spaces.
6
CONTENTS
Different authors proposed different inequivalent notions of properness that are tailored to specific needs. In [Bil04], Biller found a way to reformulate and generalize these
definitions in the framework of so-called Cartan actions (a notion that also goes back to
Palais) via properties of the orbit space of the action.
Let G be a topological group acting continuously on a topological space X via the
action ρ : G × X → X. We write ρg (x) := ρ(g, x) for g ∈ G and x ∈ X, and ρG (x) :=
{ρg (x) | g ∈ G} for x ∈ X. Moreover, let ρ\X := {ρG (x) | x ∈ X} be the orbit space,
equipped with the quotient topology, i.e. the final topology w.r.t. the quotient map
πG : X → ρ\X, x 7→ ρG (x). Furthermore, the set GA,B := {g ∈ G | ρg (A) ∩ B 6= ∅}
is called the transporter of the two subsets A, B ⊂ X, and Gx := G{x},{x} denotes the
stabilizer of x ∈ X.
Recall that a topological space is called Hausdorff, if points can be separated by
neighborhoods, called regular, if points and closed sets can be separated by neighborhoods,
and called paracompact, if every open cover has an open locally finite refinement.
Definition 2.1 (Cartan action, (Strongly/Palais-)proper action). Let G be a
topological group acting continuously on a topological space X via the action ρ : G × X →
X.
(i) ρ is called Cartan action, if the stabilizers Gx are compact for all x ∈ X, and for
every x ∈ X and every neighborhood U of Gx there is a neighborhood V of x, s.t.
GV,V ⊆ U.
(ii) ρ is called proper action, if it is a Cartan action and ρ\X is Hausdorff.
(iii) ρ is called Palais-proper action, if it is a Cartan action and ρ\X is regular.
(iv) ρ is called strongly proper action, if it is a Cartan action and ρ\X is paracompact.
Condition (i) is a continuity condition on the map x 7→ Gx . As seen in [Bil04], definitions (i) and (iii) generalize the respective definitions of Palais [Pal60], (ii) is equivalent
to Bourbaki’s definition [Bou89, III, §4.1], and (iv) generalizes the definition of Baum,
Connes, and Higson [BCH94].
When dealing with integration on proper G-spaces, one considers locally compact
Hausdorff (lcH) groups acting on lcH spaces. In this setting, the orbit space is again lcH
(see [Bou04, VII,§2]), so in particular Hausdorff and regular, s.t. (i)-(iii) in Definition 2.1
are equivalent. Moreover, in [Bou89, III, §4.4] it is shown that in this case, properness of
the action is equivalent to compactness of all transporters of compact sets.
Strong properness, i.e. paracompactness of the orbit space, is important to relate
integration on X to integration on ρ\X, as will be seen in Section 2. In [CEM01],
Chabert, Echterhoff, and Meyer showed that all four definitions (i)-(iv) in Definition 2.1
are equivalent when G and X are lcH and second countable. More generally, Biller [Bil04]
showed that the action of a locally compact Lindelöf group on a paracompact lcH space
is strongly proper, if and only if it is proper2 (A topological space is called Lindelöf, if
every open cover has a countable subcover.). A further sufficient condition for strong
properness is given in Section 1 of the Appendix.
2In
fact, Abels [Abe74] conjectured that every Palais-proper action of a connected lcH group on a paracompact space is strongly proper.
2. THE WEIL FORMULA FOR STRONGLY PROPER G-SPACES
7
We give an illustrating example for the different notions of properness. Consider the
action ρ of Z on R2 that is given by
ρn (x) = (2n x1 , 2−n x2 ),
n ∈ Z, x = (x1 , x2 ) ∈ R2 .
b this action is a symplectic transforNote, that interpreting R2 as the phase space R × R,
mation w.r.t. the standard symplectic form ω((x1 , y1 ), (x2 , y2 )) = x1 y2 − x2 y1 .
The action ρ is not a Cartan action, because the stabilizer of the origin (0, 0) is the
whole group, which is not compact. Restricting the action to R2 \ {0} yields a Cartan
action, because the stablizers are then all trivial, and the continuity condition follows
from continuity of the action. However, this action is not proper, because the quotient
space is not Hausdorff. This is easiest seen from the fact that the graph of the action is
not closed (see [Bou89, Prop. 8, I, 8.3]). Alternatively, consider two sets A and B that
are neighborhoods of points on the x-axis and the y-axis, respectively. Then on can show
that the transporter ZA,B is not compact. Now, when further restricting the action to
the set R2 \ {(x, y) | x = 0 or y = 0}, then the action is proper, as the quotient space is
the topological sum of four half-lines. So, this action is also Palais-proper and strongly
proper, what can also be seen with the result of Chabert et al. [CEM01], since Z and R2
are second countable. This example shows that properness is a quite strong condition, as
it already fails for this simple discrete abelian automorphic action. However, it is needed
for the integration theory on G-spaces, as seen in the following section.
2.2. The Weil formula. A Weil formula is a tool to relate integration on a G-space
X to integration on the group G that acts on X. This is achieved by constructing a
suitable measure on the space of orbits.
In the case of X = G being a lcH group with (left) Haar measure µG , and a closed
subgroup H with Haar measure µH acting on G via right translation R : H × G → G,
Rh (g) = gh, h ∈ H, g ∈ G, a classic result is, that there always is a (left G-quasi-invariant)
Radon measure µG/H on the quotient space G/H, s.t. the Weil formula
Z
Z
f · q dµG =
AR f dµG/H , f ∈ Cc (G),
(2.1)
G
G/H
holds, where the orbital mean operator AR : Cc (G) → Cc (G/H) is given by
Z
(AR f )(g) :=
f (gh) dµH (h), f ∈ Cc (G), g ∈ G, h ∈ H
(2.2)
H
(see [Bou04, RS00, Fol95]). Here, q is a continuous and strictly positive function that
satisfies a certain functional equation (and is sometimes called a ρ-function, see [Fol95]).
We will generalize this Weil formula to strongly proper G-spaces under some additional
assumptions. Results on the Weil formula due to Bourbaki (see [Bou04, VII,2]) are nicely
complemented by the following results.
A continuous action ρ of a lcH group G on a lcH space X defines a linear action of G
on the vector space of Borel-measurable functions f on X via
(ρg f )(x) := f (ρ−1
g (x)),
g ∈ G, x ∈ X.
(2.3)
As the action is continuous, equation (2.3) also defines a linear action on the space Cc (X)
of compactly supported continuous functions.
8
CONTENTS
The invariance properties of the measures we will construct are studied in the Appedix
in Section 2 for the special case of double coset spaces and in particular for affine actions.
Since we need the notion of quasi-invariance already in this section, we start with the
following definitions.
Definition 2.2 (Invariant measures). Let X be a lcH G-space on which G acts
continuously via ρ : G × X → X. A Borel measure µ on X is called
(i) ρ-quasi-invariant, if there are continuous functions λg : X → C, g ∈ G, s.t.
Z
Z
ρg f dµ =
f · λg dµ for all f ∈ Cc (X),
X
X
(ii) relatively ρ-invariant, if it is ρ-quasi-invariant, and the functions λg are constant,
(iii) ρ-invariant, if it is relatively ρ-invariant, and λg ≡ 1 for all g ∈ G.
As examples for these properties, consider a Haar measure µG on a locally compact
group G. Then µG is invariant w.r.t. the action of G by left translation, relatively
invariant w.r.t. right translation and quasi-invariant w.r.t. the action of Z2 by inversion:
dµG (hg) = dµG (g),
dµG (gh) = ∆G (h) dµG (g),
dµG (g −1 ) = ∆G (g)−1 dµG (g),
for g, h ∈ G, where ∆G denotes the modular function of G, that is defined by the second
equality. A Haar measure is also relatively invariant with respect to automorphisms.
Let ϕ ∈ Aut(G), then there is a positive number mod(ϕ), called the modulus of ϕ, s.t.
dµG (ϕ(g)) = mod(ϕ) dµG (g) for g ∈ G.
Note that in [Fol95], the notion of quasi-invariance used here is called strong quasiinvariance, while quasi-invariance means mutual absolute continuity of the measures B 7→
µX (ρg (B)), g ∈ G. The approach of this article avoids additional technical issues.
A desirable property of the orbital mean operator (2.2) is to map into the space of
continuous and compactly supported functions on the space of orbits, which allows to
invoke the Riesz representation theorem for the construction of a suitable measure for a
Weil formula. A sufficient condition for this property is properness of the action.
Lemma 2.3 (Orbital mean operator). Let ρ : G × X → X be a proper continuous
action of an lcH group G on an lcH space X. The orbital mean operator Aρ , given by
Z
Aρ f (πρ (x)) :=
ρg f (x) dµG (g), f ∈ Cc (X), x ∈ X.
G
is well-defined and maps a function f ∈ Cc (X) to a function Aρ f ∈ Cc (ρ\X).
Proof. For x ∈ X and f ∈ Cc (X), consider the function fx : G → C, g 7→ ρg f (x).
The support of fx can be written as a transporter of two compact sets:
supp(fx ) = Gsupp(f ),{x} = {g ∈ G | ρg (supp(f )) ∩ {x} =
6 ∅}.
Consequently, by properness of the action, the function fx is compactly supported. It
is also continuous by the following identity of set functions: fx−1 = π1 ◦ ρ−1 ◦ f −1 , using
continuity of f and ρ, and the fact that the projection π1 : G × X → G, (g, x) 7→ g, is
open. So, in particular, fx ∈ Cc (G) ⊂ L1 (G), showing that the integral in the definition
of Aρ f converges for every x ∈ X.
2. THE WEIL FORMULA FOR STRONGLY PROPER G-SPACES
9
The function Aρ f is well-defined on ρ\X since the integral is constant on orbits. For
f ∈ Cc (X) and h ∈ G, by left-invariance of µG ,
Z
Z
Z
−1
ρg f (x) dµG (g).
ρg f (ρh (x)) dµG (g) =
f (ρ(hg)−1 (x)) dµG (g) =
G
G
G
It remains to show that Aρ f ∈ Cc (ρ\X). The support supp(Aρ f ) is compact as a closed
subset of the compact set πρ (supp(f )). Continuity of Aρ f follows with properness of the
action as in [Bou04, 2.1, Prop. 1].
We are now in place to state the main theorem of this section. It complements results
of Bourbaki for proper actions and generalizes classic results on quotients G/H.
Theorem 2.4 (Weil formula for strongly proper G-spaces). Let ρ : G ×X → X
be a strongly proper action of an lcH group G on an lcH space X. When there is a ρquasi-invariant Radon measure µX on X with functions λg , s.t. dµX (ρg (x)) = λg dµX (x)
for g ∈ G, x ∈ X, then there is a strictly positive and continuous function q on X that
satisfies
∆G (g)
ρg q(x) =
q(x), g ∈ G, x ∈ X,
(2.4)
λg−1 (x)
and a unique Radon measure µρ\X on ρ\X, s.t.
Z
Z
f · q dµX =
Aρ f dµρ\X , f ∈ Cc (X).
(2.5)
X
ρ\X
In particular, when µX is relatively ρ-invariant with λg ≡ ∆G (g −1 ), then (2.5) is satisfied
with q ≡ 1.
Note that the measure q dµX is realtively ρ-invariant with λg ≡ ∆G (g), since by the
functional equation (2.4) for q,
ρg q(x) dµ(ρg (x)) = ∆G (g)q(x) dµ(x),
g ∈ G, x ∈ X.
This shows that the measure q dµX is the unique measure µ#
ρ\X in [Bou04, Prop. 4, VII,
2.2], that satisfies the Weil formula.
For future reference, we introduce a name for the setting in Theorem 2.4.
Definition 2.5 (Weil G-space). Let G be an lcH group that acts strongly proper on
an lcH space X via the action ρ : G × X → X, and µX a ρ-quasi-invariant measure on
X.
The measure space (X, µ#
ρ\X ) is called a Weil G-space, if there is a measure µρ\X on
ρ\X and a function q on X that satisfies (2.4), s.t. the Weil formula (2.5) holds, and
dµ#
ρ\X (x) = q(x) dµX (x), x ∈ X.
A technical tool that is needed to prove Theorem 2.4, is a so-called Bruhat function.
Definition 2.6 (Bruhat function). Let ρ : G × X → X be a proper continuous
action of a lcH group G on a lcH space X.
A continuous function β on X is called a Bruhat function for ρ, if
(i) Aρ β ≡ 1,
10
CONTENTS
(ii) for every compact set C ⊆ X, the restriction of β to ρG (C) is non-negative and
compactly supported.
Thus, a Bruhat function is a preimage of the constant function 1 under the orbital
mean operator that has nice compactness properties. It will be used to define the function
q in (2.5). The existence of a Bruhat function is guaranteed by the following Lemma that
is in this generality due to Bourbaki and goes back to Bruhat (see [Bru56]).
Lemma 2.7 (Existence of a Bruhat function). Let ρ : G × X → X be a strongly
proper action of an lcH group G on an lcH space X. Then there is a Bruhat function for
ρ.
Proof. A proof can be found in [Bou04, VII, 2.4], Chapter VII, 2.4. We sketch the
proof to show how paracompactness of ρ\X and properness of ρ are used.
Every x ∈ X has an open neighborhood with compact closure, say Ux , by local compactness of X. Now, consider the open cover (πρ (Ux ))x∈X of ρ\X. By paracompactness
of ρ\X, there is an open locally finite refinement of this cover, say (Vi )i∈I with Vi ⊆ ρ\X,
i ∈ I, for some index set I. So, for i ∈ I there are xi ∈ X, s.t. Vi ⊂ πρ (Ui ). Consequently,
setting Wxi := πρ−1 (Vi ) ∩ Uxi , we find that πρ (Wxi ) = Vi . Thus, we have a locally finite
cover (πρ (Wxi ))i∈I of ρ\X with sets Wxi ⊂ X that are open with compact closure by
properness and continuity of ρ.
P Now, for i ∈ I, choose functions ψi ∈ Cc (X), ψi ≥ 0, with supp(ψi ) = Wxi , set ψI :=
i∈I ψi (which is well-defined by local finiteness of the cover and satisfies condition (ii)
of Definition 2.6), and define β := ψI /Aρ ψI , which is well-defined and satisfies conditions
(i) and (ii) of Definition 2.6.
Note that the proofs of this lemma for quotient spaces G/H in [RS00] and [Fol95] use
paracompactness implicitly.
A Bruhat function nicely relates invariant functions on X to functions on ρ\X. Given
a function f ∈ Cc (X), we associate to it the ρ-invariant function fρ on X that is given
by fρ (x) := Aρ f (πρ (x)). Then
Aρ (β · fρ ) = Aρ f.
(2.6)
This shows that in our setting one can associate a function on ρ\X to an invariant function
on X without reference to a specific fundamental domain.
Proof. (of Theorem 2.4) The proof generalizes the arguments of the proofs of the
special case of homogeneous spaces G/H given in [Fol95, RS00].
We start by investigating the functions λg , g ∈ G, that are associated to the ρ-quasiinvariance of µX . Let f ∈ Cc (X) and g1 , g2 ∈ G, then
Z
Z
Z
Z
f · λg1 g2 dµX =
ρg1 g2 f dµX =
ρg2 f · λg1 dµX =
f · ρg2−1 λg1 · λg2 dµX .
X
X
X
X
Since f was arbitrary, this shows that
ρg2 λg1 =
λg1 g2−1
λg2−1
for g1 , g2 ∈ G.
(∗)
2. THE WEIL FORMULA FOR STRONGLY PROPER G-SPACES
11
Now, define the function q : X → (0, ∞) by
Z
λg−1 (x)
q(x) :=
ρg β(x)
dµG (g), x ∈ X,
(2.7)
∆G (g)
G
where β is a Bruhat function for ρ, which exists by Lemma 2.7. The function q satisfies
the functional equation 2.4, since for h ∈ G and x ∈ X,
Z
Z
ρh λg−1 (x)
ρh λg−1 h (x)
ρh q(x) =
ρhg β(x)
dµG (g) =
ρg β(x)
dµG (g)
∆G (g)
∆G (h−1 g)
G
G
Z
λg−1 hh−1 (x)
∆G (h)
(∗)
dµ
(g)
=
q(x).
=
ρg β(x)
G
∆G (h−1 )∆G (g)λh−1 (x)
λh−1 (x)
G
Now, we define a positive linear functional Iρ\X on Cc (ρ\X) that will yield the desired
measure µρ\X via the Riesz representation theorem. For f ∈ Cc (ρ\X), we define
Z
Iρ\X (f ) :=
fe · q dµX , with fe ∈ Cc (X), s.t. Aρ fe = f.
G
We first need to show that this functional is well-defined. For this purpose, associate
to fe ∈ Cc (X) the bounded and continuous ρ-invariant function feρ (x) := Aρ fe(πρ (x)) =
R
ρ fe(x) dµG (g), x ∈ X, and set f := Aρ fe. Now, take a function ϕ ∈ Cc (X), ϕ 6= 0,
G g
ϕ ≥ 0, with ϕρ = 1 on supp(fe). That such a function exists can be seen along the lines
of [Fol95, Lemma 2.47], adapted to the situation, using properness of ρ (see also [Bou04,
VII, 2.1]). When feρ = 0, i.e. when f = 0, then
Z
Z Z
Z
Z
e
e
e
0=
ϕ · fρ · q dµX =
ϕ · ρg f · q dµG (g) dµX =
f·
ρg−1 ϕ · ρg−1 q · λg dµG (g) dµX
X
X G
X
G
Z
Z
Z
Z
−1
∆
(g
)
(2.4)
G
=
q · λg dµG (g) dµX =
fe ·
fe · q ·
ρg−1 ϕ ·
ρg ϕ dµG (g) dµX = Iρ\X (f ),
λg
X
X
G
G
{z
}
|
=ϕρ =1
showing that Iρ\X (f ) = 0. Thus, Iρ\X is well-defined.
By the Riesz
R representation theorem, there is a unique Radon measure µρ\X on ρ\X,
s.t. Iρ\X (f ) = ρ\X f dµρ\X . In particular, for any f ∈ Cc (X) we have
Z
Z
f · q dµX = Iρ\X (Aρ f ) =
Aρ f dµρ\X ,
X
ρ\X
i.e. the Weil formula (2.5) holds.
When µX is relatively ρ-invariant with λg ≡ ∆G (g −1 ), then q ≡ 1 satisfies the functional equation, yielding a measure s.t. (2.5) holds.
A simple example for the Weil formula (2.5) is the case of a closed subgroup H of a
lcH group G acting by left translation Lh (g) := hg. The space of orbits is the quotient
space H\G. The Haar measure µG is L-invariant, s.t. a function q is needed that satisfies
q(h−1 g) = ∆G (h)q(g). This is true for q = ∆−1
G , showing that there is a unique Radon
measure µH\G on H\G, s.t.
Z
Z
Z
−1
−1
f dµG =
f · ∆G dµG =
AL f dµH\G , f ∈ Cc (G).
(2.8)
G
G
H\G
12
CONTENTS
Theorem 2.4 can be generalized to L1 (X, µX ) via classic measure theoretic arguments
(as in [RS00] for quotients G/H). In particular, one can show that every L1 -function on X
is in L1 (G) along almost all orbits. This generalization is sometimes called the extended
Weil formula. Moreover, with a modified orbital mean operator
Z
ρg f
Aρ,q f (πρ (x)) :=
dµG (g), f ∈ L1 (G), x ∈ X,
ρ
q
G g
the following version of (2.5) holds:
Z
Z
f dµX =
X
Aρ,q f dµρ\X ,
f ∈ L1 (X, µX ).
(2.9)
ρ\X
This is sometimes called the (extended) Mackey-Bruhat formula.
The extended Weil formula can be used to study integrability of functions along orbits.
In particular, we get the following results for Lp -functions that will be crucial for the
definition of the Zak transform on L2 (X) in Sections 3 and 4.
Corollary 2.8 (Orbital mean on Lp (X)). Let G be an lcH group that acts strongly
#
p
proper on a Weil G-space (X, µ#
ρ\X ) via ρ : G × X → X. Let f ∈ L (X, µρ\X ) for
1 ≤ p ≤ ∞. Then
(i) The function (πρ (x), g) 7→ ρg f (x) is in Lp ((ρ\X) × G).
(ii) For almost every πρ (x) ∈ ρ\X, the function fx : G → C, g 7→ ρg f (x), is in
Lp (G).
(iii) The orbital mean operator Aρ can be extended to a bounded linear operator Aρ :
p
Lp (X, µ#
ρ\X ) → L (ρ\X) with kAρ kp ≤ 1.
Proof. (i) and (ii): Using the extended Weil formula on the function |f |p ∈ L1 (X, µρ\X # ),
we get
Z
Z
Z
#
p
p
∞ > kf kp =
|f | dµρ\X =
|ρg f (x)|p dµG (g) dµρ\X (πρ (x)),
X
ρ\X
G
R
showing (i) and that πρ (x) 7→ G |fx |p dµG is finite for almost every πρ (x) ∈ ρ\X, i.e. that
(ii) is true.
(iii): Continuing the above calculation using Jensen’s inequality for z 7→ |z|p , we find
Z
Z
p
kf kp =
|ρg f (x)|p dµG (g) dµρ\X (πρ (x))
ρ\X G
p
Z
Z
ρg f (x) dµG (g) dµρ\X (πρ (x)) = kAρ f kpp ,
≥
ρ\X
G
p
justifying the extension of Aρ to an operator Lp (X, µ#
ρ\X ) → L (ρ\X) and showing that
kAρ kp ≤ 1.
Further examples for the Weil formula, namely versions for double coset spaces and
affine actions can be found in Section 2 in the Appendix.
2. THE WEIL FORMULA FOR STRONGLY PROPER G-SPACES
13
2.3. Integration on fundamental domains. The Weil formula can be used to
construct measures on measurable fundamental domains of the action. A fundamental
domain is a set F ⊆ X, s.t. the restriction of πρ to F is a bijection. In other words, it is
a set of representatives of the equivalence relation x ∼ρ y :⇔ πρ (x) = πρ (y).
Unimodularity of G turns out to be necessary to relate integration over F to integration
over ρ\X in a nice way. In particular, it makes sure that the integral of a ρ-invariant
function over different fundamental domains agrees. Thus, for the rest of this section
assume that G is unimodular.
Now, let (X, µ#
ρ\X ) be a Weil G-space. For f ∈ Cc (X), we use the Weil formula twice,
first for the action ρ on X, and then inside the orbital mean integral for the action of Gx
on G by left translation Lgx (g) := gx g, gx ∈ Gx , g ∈ G. Since G is unimodular, the latter
Weil formula is particularly simple (see eq. (2.8)).
Z
Z
Z
#
f dµρ\X =
ρg f dµG (g) dµρ\X
X
ρ\X G
Z
Z
Z
ρgx−1 g f (x) dµGx (gx ) dµGx \G (πL (g)) dµρ\X (πρ (x))
=
ρ\X Gx \G Gx
Z
Z
ρg f (x) dµGx \G (πL (g)) dµρ\X (πρ (x)).
=
µGx (Gx )
Gx \G
ρ\X
The choice of the representatives of the orbits does not influence the volume of the stabilizer, since stabilizers of different points on the same orbit are conjugated via Gρg (x) =
gGx g −1 , g ∈ G. Unimodularity of G yields that the map x 7→ µGx (Gx ) is ρ-invariant. For
convenience, we normalize µGx \G and µGx s.t. µGx (Gx ) = 1 for all x ∈ X.
Now, given a measurable fundamental domain F ⊆ X of ρ, we define a measure µF
on F as the pushforward of µρ\X w.r.t. the map ϕ : ρ\X → F , πρ (x0 ) 7→ x0 .
µF := ϕ∗ µρ\X .
(2.10)
Here, the pushforward measure of a measure µ by a measurable function ψ is defined
by ψ∗ µ(B) := µ(ψ −1 (B)) for measurable sets B. The fact that integrals of ρ-invariant
functions over different fundamental domains agree is clear from the definition.
In summary, we have shown that for every measurable fundamental domain F ,
Z
Z Z
#
f dµρ\X =
(2.11)
ρg f (x0 ) dµGx0 \G (πL (g)) dµF (x0 ), f ∈ L1 (X).
X
F
Gx0 \G
This result agrees with the construction for discrete groups in [Bou04, VII, 2.10]. In this
case, one gets the following additional relation to integration over X w.r.t. µ#
ρ\X . Let
1
f ∈ L (X), and fF,ρ the ρ-invariant extension of the restriction of f to F . Let 1A denote
the characteristic function of the measurable set A, then
Z
Z
Z
X
#
#
f dµρ\X =
1F · fF,ρ dµρ\X =
ρg 1F (x0 )ρg fF,ρ (x0 ) dµF (x0 )
F
X
=
Z
F
f (x0 )
X
πL (g)∈Gx0 \G
F π (g)∈G \G
x0
L
1ρg (F ) (x0 ) dµF (x0 ) =
Z
F
f dµF ,
14
CONTENTS
where it is used that
1ρg (F ) (x0 ) = 1 ⇔ x0 ∈ ρg (F ) ⇔ g ∈ Gx0 ⇔ πL (g) = Gx0 .
For non-discrete G, the above construction doesn’t work when the fundamental domain
is a null set. Instead, in this situation, we can use the measure µF to define a measure δF
on X that allows to integrate a function over the fundamental domain, namely
Z
Z
f dδF :=
f dµF , f ∈ Cc (X).
(2.12)
X
F
This definition agrees with special cases for which it is clear what is meant by integration
w.r.t. certain null sets, for example, when a Lie group acts smoothly on a manifold of
higher dimension.
Note that the function fx : G → C, fx (g) := ρg f (x), is constant on cosets Gx g,
so it is in fact well-defined as a function on Gx \G. The orbit ρG (x) is homeomorphic
to the homogeneous space Gx \G, so we can also construct measures µρG (x) := γ∗ µGx \G ,
γ : Gx g 7→ ρg (x), on the orbits of ρ. This analysis shows that when almost all stabilizers
are trivial and G is σ-compact, then the Weil formula actually reduces to Fubini’s theorem.
Viewing the orbits as homogeneous spaces also allows a formulation of the Weil formula
in its form (2.11) as a transform (which should probably be called the Weil transform) on
L2 (X). A function f ∈ L2 (X) is mapped to the function (x0 , πL (g)) 7→ ρg f (x0 ) for x0 ∈ F ,
and πL (g) ∈ Gx0 \G. Applying the extended Weil formula to the function |f |2 ∈ L1 (X)
2
shows
R ⊕ 2 that this transform is a unitary map from L (X) onto the direct integral space
L (Gx0 \G) dµF (x0 ).
F
The question if there is a ‘nice’ measurable fundamental domain is a difficult question
for general actions. As shown in [Bou04, VII, Ex. 12], there is a fundamental domain
whose characteristic function is continuous almost everywhere (or equivalently, whose
boundary is negligable) for a proper continuous action of a countable discrete group.
For lattice subgroups of lcH groups, one can even find relatively compact fundamental
domains as shown in [KG98, Kut02].
The choice of a fundamental domain and the definition (2.10) are important to define
the Zak transform (see Definitions 3.1 and 4.1), since the definition will depend on this
choice.
3. The Zak transform for abelian actions
In this section, the Zak transform for abelian actions will be defined, its main properties will be derived, and different interpretations of its meaning will be discussed. The
mathematical tools used to analyze the transform are the Weil formula that was derived
in Section 2, and Fourier analysis on lca groups. As will turn out, the representation
theoretic decomposition of ρ is fully described by the Zak transform.
3.1. The abelian Zak transform on L1 (X). Before defining the Zak transform
for abelian actions, we recall the main objects and results from Fourier analysis on lca
b of G is the set of characters of G,
groups. Let G be an lca group. The dual group G
which are the continuous homorphisms from G to T := {z ∈ C | |z| = 1}.
b := {χ : G → T | χ continuous, χ(gh) = χ(g)χ(h), g, h ∈ G}.
G
(3.1)
3. THE ZAK TRANSFORM FOR ABELIAN ACTIONS
15
b is an lca group. Given a
With the point-wise product and the compact open topology, G
Haar measure µG of G, the Fourier transform on an lca group is then defined as
Z
b
(3.2)
fb(χ) :=
f (g)χ(g) dµG (g), f ∈ L1 (G), χ ∈ G.
G
The Fourier transform can be inverted as follows. For almost every g ∈ G,
Z
b
f (g) =
fb(χ)χ(g) dµGb (χ), f ∈ L1 (G), fb ∈ L1 (G).
(3.3)
b
G
Another construction that will be used is the so-called reciprocal group HG⊥ of a closed
subgroup H of G.
b | χ(h) = 1 for all h ∈ H},
HG⊥ := {χ ∈ G
(3.4)
which is also called the orthogonal subgroup in [RS00] or the annihilator of H in [HR79]
and generalizes the reciprocal lattice of a lattice subgroup of Rd . The reciprocal group is
a central object in the Poisson summation formula for lca groups, which reads
Z
Z
fbdµHG⊥ , f |H ∈ L1 (H), fb|HG⊥ ∈ L1 (HG⊥ ).
f dµH =
(3.5)
⊥
HG
H
We define the Zak transform of a function as the mean of its modulation by a character
along the orbits.
Definition 3.1 (Zak transform for abelian actions). Let G be an lca group,
#
1
(X, µ#
ρ\X ) a Weil G-space, and f ∈ L (X, µρ\X ). Furthermore, let F be a measurable
fundamental domain of the action ρ. The Zak transform Zρ f of f is defined as
Z
b
Zρ f (x0 , χ) :=
ρg f (x0 )χ(g) dµG (g), x0 ∈ F, χ ∈ G.
(3.6)
G
b by formula (3.6) with x0 ∈ X.
We also define the extended Zak transform ZρX f on X × G
Note, that the transform is well-defined, since an L1 -function is in L1 (G) along µρ\G almost all orbits. This fact justified the extension of the Weil formula to L1 (X).
Example: Consider the discrete abelian group G = AZd , A ∈ GL(d, R), acting on
X = Rd by translation ρx (y) = x + y, x, y ∈ Rd . The Lebesgue measure L on Rd is
translation-invariant, s.t. it can be chosen as the measure µ#
ρ\X := L. The measures on
G and ρ\X that satisfy the Weil formula are then the counting measure and the measure
induced from the restriction of L to a fundamental domain F , respectively. Denoting
b where G′ := 2πA−T Zd
the characters of G by χ(k) (x) := eik·x , x ∈ R3 , k ∈ R3 /G′ ∼
= G,
denotes the reciprocal lattice (which parametrizes the reciprocal group G⊥
R3 ), the resulting
Zak transform Zρ is the classic Zak transform
X
Zρ f (x0 , χ(k) ) :=
f (x0 − v)e−ik·v , x0 ∈ F, k ∈ R3 /G′ , f ∈ L1 (R3 ).
(3.7)
v∈AZd
To simplify notation we chose the notation Zρ for the Zak transform, even though
one has to keep in mind that the definition depends on the choice of fundamental domain
F , so a notation like ZρF would be more precise. The reason for restricting the first
argument to a fundamental domain is the following equivariance property of the extended
Zak transform.
16
CONTENTS
Lemma 3.2 (Invariance of the extended Zak transform). With the assumptions
of Definition 3.1, the extended Zak transform satisfies
b
Z X f (ρ−1 (x0 ), χ) = χ(g)Zρf (x0 , χ), g ∈ G, x0 ∈ F, χ ∈ G.
(3.8)
ρ
g
In particular, the Zak transform vanishes on the set {(x0 , χ) | χ 6∈ (Gx0 )⊥
G }.
Proof. Equation (3.8) is an immediate consequence of the definition (3.6), using leftinvariance of µG .
Now, when x0 ∈ F , and χ 6∈ (Gx0 )⊥
G , then there is a g ∈ Gx0 , s.t. χ(g) 6= 1. In
particular, using eq. (3.8),
Zρ f (x0 , χ) = Zρ f (ρ−1
g (x), χ) = χ(g)Zρ f (x0 , χ),
implying that Zρ f (x0 , χ) = 0.
The invariance property (3.8) shows that the extended Zak transform ZρX f is ρequivariant in the first argument w.r.t. the second argument. So, the operator ZρX
maps to the space of functions with this property. This viewpoint captures the fact that
the extended Zak transform is uniquely determined by the restriction to any fundamental
domain abstractly. However, in applications it is convenient to work with a special choice
of fundamental domain.
The fact that the Zak transform vanishes for (x0 , χ) with χ 6∈ (Gx0 )⊥
G is another
manifestation of the fact that the functions fx0 : g 7→ ρg f (x0 ) are actually functions
⊥
\
∼
on Gx0 \G. Since it can be shown that G
x0 \G = (Gx0 )G , this vanishing property is a
consequence of the Poisson summation formula on G w.r.t. Gx0 . In principle, one could
define the Zak transform of a function as a function on the set {(x0 , χ) | x0 ∈ F, χ ∈
(Gx0 )⊥
G }. For the moment, the idea is to keep the space on which Zρ f lives simple and
capture the details in its properties. These considerations will be made precise in Theorem
3.4.
Note that in the case that X = G is an abelian lcH group with a lattice H acting by
right translation, Definition 3.1 agrees up to the conjugation of the character with the one
given in [KG98]. To see this, one needs to identify a fundamental domain of HG⊥ with the
b
b via the isometry G/H
b ⊥∼
group H
G = H (see e.g. [Fol95, (4.39)]). Also the Zak transform
on semi-direct product groups with the action of a lattice subgroup of the abelian factor
in [AF13] is a special case of our definition. In [BHP15] and [Sal14], the authors propose a
more general representation theoretic approach. They consider a unitary representation σ
P
b for
of a discrete lca group G on L2 (X) and the transform g∈G σg f (x)χ(g), x ∈ X, χ ∈ G
f ∈ L2 (X). This approach is natural in frame theory, but too general for applications that
use the additional structure resulting from the action on the underlying space considered
here.
The definition of the Zak transform can be interpreted in different ways. First, it
may be viewed as the Fourier transform along the orbits of the action. Given a function
f ∈ L1 (X), the Zak transform is given by
Zρ f (x0 , χ) = fc
x0 (χ),
b
x0 ∈ F, χ ∈ G,
(3.9)
where fx0 (g) := ρg f (x0 ), x0 ∈ F , g ∈ G. From this viewpoint, the Zak transform could
be termed an orbit-frequency decomposition.
3. THE ZAK TRANSFORM FOR ABELIAN ACTIONS
17
b Then the
The complementary point of view results from fixing a character χ ∈ G.
χ
X
operator Pρ : f 7→ Zρ f (·, χ) is a so-called symmetry projection, and
ZρX f (x, χ) = Pρχ f (x),
b
x ∈ X, χ ∈ G.
(3.10)
As seen in Lemma 3.2, Pρχ maps f into the space of χ-equivariant functions on X. This
approach goes back to Wigner [Wig31], who used these operators in quantum theory.
The same viewpoint is taken by Bourbaki [Bou89, VII, §2], where instead of characters,
representations on the real multiplicative group are considered.
b
Both viewpoints can be summarized using the Heisenberg group H(G) := G⋉τ (G×T)
of G, where τ [g](χ, z) := (χ, χ(g)z), and the group product is given by
(g, (χ, z)) · (g ′, (χ′ , z ′ )) := (gg ′, τ [g](χ, z) · (χ′ , z ′ )) = (gg ′, (χχ′ , χ(g)zz ′ ))
(3.11)
b and z, z ′ ∈ T. The Heisenberg group is the group theoretic version
for g, g ′ ∈ G, χ, χ′ ∈ G,
of a generalized time-frequency plane or phase space. More precisely, phase space is the
quotient H(G)/T, where T is identified with the center Z(H(G)) = {(e, (1, z)) | z ∈ T} of
H(G). This quotient is in fact isomorphic to the direct product of group and dual group:
b
H(G)/Z(H(G)) ∼
= G × G.
b on L2 (X), that is given by
Now the projective representation ξ of G × G
ξ(g,χ) f (x) := ρg f (x)χ(g),
b
g ∈ G, χ ∈ G.
(3.12)
(3.13)
naturally defines a representation of H(G), namely (up to conjugation of the character)
the classic Schrödinger representation ξ H(G) of the Heisenberg group that is the lift of ξ
b
to the central extension H(G) of G × G:
H(G)
ξ(g,(χ,z)) f (x) := ρg f (x)χ(g)z,
b z ∈ T.
g ∈ G, χ ∈ G,
(3.14)
The Zak transform can be interpreted as a partial orbital mean operator w.r.t. ξ:
Z
b
Zρ f (x0 , χ) =
ξ(g,χ) f (x) dµG (g), x0 ∈ F, χ ∈ G.
(3.15)
G
This viewpoint is the reason for the importance of the classic Zak transform for timefrequency analysis and in particular for Gabor analysis.
Next, we use the Fourier interpretation 3.9 to formulate an inversion formula. We simply use the Fourier inversion formula (3.3) for the second argument of the Zak transform
to get
Z
f (x) =
Zρ f (x0 , χ)χ(g) dµGb (χ), where ρg (x) = x0 ∈ F,
(3.16)
b
G
b for almost all x0 ∈ F .
almost everywhere for functions f ∈ L1 (G) with Zρ f (x0 , ·) ∈ L1 (G)
Note that the right hand side is well-defined, even though the equation x = ρ−1
g (x0 )
′
′
has multiple solutions, namely all (x0 , g g) with g ∈ Gx0 . As seen in Lemma 3.8, the Zak
b that are not constant on stabilizers. Contransform of a function vanishes for all χ ∈ G
∼ \
sequently, choosing a measure µ(Gx0 )⊥G on (Gx0 )⊥
G = Gx0 \G, s.t. the Poisson summation
18
CONTENTS
formula for the subgroup Gx0 of G holds with normalized measures on Gx0 , x0 ∈ F , one
gets the alternative version
Z
(3.17)
Zρ f (x0 , χ)χ(g) dµ(Gx0 )⊥G (χ), where ρg (x) = x0 ∈ F,
f (x) =
(Gx0 )⊥
G
for f and x as above.
To rigorously define the inverse Zak transform, we formulate the condition ρg (x) =
(x )
x0 ∈ F in terms of measures. Consider the point measures δx on F and δx 0 on Gx0 \G
that are given by
(
1, there is a g ∈ G, s.t. ρg (x) ∈ BF ,
δx (BF ) =
(3.18)
0, else,
for Borel-sets BF ⊆ F , and
δx(x0 ) (BGx0 \G ) =
(
1, there is a πL (g) ∈ BGx0 \G , s.t. ρg (x) = x0 ,
0, else,
(3.19)
for Borel-sets BGx0 \G ⊆ Gx0 \G. With these measures we summarize the above considerations in the following result.
Proposition 3.3 (L1 -Zak inversion theorem). Let G be an lca group that acts
strongly proper on a Weil space (X, µ#
ρ\X ), and F a measurable fundamental domain of
the action ρ. The inverse Zak transform Zρ−1 is defined as
Z
Z Z
−1
(3.20)
χ(g) dδx(x0) (πL (g)) dµ(Gx0 )⊥G (χ) dδx (x0 )
h(x0 , χ)
Zρ h(x) :=
F
(Gx0 )⊥
G
Gx0 \G
−1
for h(x0 , ·) ∈ L1 ((Gx0 )⊥
G ) for almost all x0 ∈ F . It satisfies f = Zρ Zρ f point-wise almost
1
1
⊥
everywhere for f ∈ L (G) with Zρ f (x0 , ·) ∈ L ((Gx0 )G ) for almost all x0 ∈ F .
Proof. The proof is a direct consequence of the above discussion, the equations (3.9)
and (3.17), and the L1 -Fourier inversion formula (3.3).
The inversion formula (3.20) can be read as a decomposition of a function into the
R
(x )
measures Gx \G χ(g) dδx 0 on F × Gx0 \G. This is a generalization of Zak’s viewpoint as
0
given in [Zak67, (5), (12)] and will be discussed in more detail below Theorem 3.4.
Example: Let’s reconsider the classic case that the lattice G = AZd , A ∈ GL(d, R),
acts on X = Rd by translation. Since the stabilizers of this action are trivial, we have
b
Gx0 \G = G and (Gx0 )⊥
G = G for all x0 ∈ F , where F is a relatively compact fundamental
(x )
domain of the action. The measure δx 0 on G is simply the Kronecker delta δx0 −x,v .
Consequently, the Zak reconstruction formula reads
Z Z
X
−1
Zρ h(x) =
(3.21)
h(x0 , χ(k) )
eik·v δx0 −x,v dµR3 /G′ (k) dδx (x0 ),
F
R3 /G′
v∈G
b parametrizes
where again G′ = 2πA−T Zd denotes the reciprocal lattice of G, s.t. R3 /G′ ∼
=G
the dual group, and µR3 /G′ is the measure induced by this parametrization. This agrees
3. THE ZAK TRANSFORM FOR ABELIAN ACTIONS
19
with Zak’s inversion formula [Zak67, (5),(12)] up to conjugation of the character. The
constant in Zak’s formula is contained in the normalization of the measure µR3 /G′ .
3.2. The abelian Zak transform on L2 (X). Using Corollary 2.8, we can extend
the Zak transform to L2 (G) and show that it is an isometry. For this purpose, we need the
Plancherel theorem for lca groups which says that the Fourier transform is a Hilbert space
b This can be formulated in a representation theoretic
isomorphism from L2 (G) to L2 (G).
way as follows. The Fourier transform is a Hilbert space isomorphism from L2 (G) to
R⊕
2 b
b = L (G), where the integral is a direct integral of the measurable field of oneb C dµG
G
b act via the irreducible unitary
dimensional Hilbert spaces C on which the characters χ ∈ G
modulation representations Mχ : G → U(C), Mχ (g)z := χ(g)z, where U(C) denotes the
group of continuous unitary operators on C. The Fourier transform intertwines the left
regular representation L (left translation, Lg f (h) := f (g −1h)) of G on L2 (G) with the
R⊕
b This is a precise formulation of
direct integral representation Gb Mχ dµGb (χ) on L2 (G).
the statement that the Fourier transform diagonalizes the translation operator.
The corresponding theorem for the Zak transform is the following.
Theorem 3.4 (Abelian Zak transform on L2 (X)). Let G be an abelian lcH group
that acts strongly proper on a Weil G-space (X, µ#
ρ\X ) via ρ : G × X → X and F a
measurable fundamental domain of the action ρ. The Zak transform can be extended to a
Hilbert space isomorphism
Z ⊕Z ⊕
2
b
(3.22)
Zρ : L (X) →
C dµ(Gx0 )⊥G dµF (x0 ) =: L2ρ (F × G),
F
(Gx0 )⊥
G
where µF is given by eq. (2.10), and µ(Gx0 )⊥G is normalized, s.t. the Poisson summation
formula (3.5) for Gx0 holds with µGx0 (Gx0 ) = 1 for x0 ∈ F .
The Zak transform intertwines
the action ρ of G on L2 (X) with the left modulation
R⊕R⊕
representation of G on F (Gx )⊥ C dµ(Gx0 )⊥G dµF (x0 ):
0 G
Zρ [ρg f ](x0 , χ) = χ(g)Zρ f (x0 , χ),
g ∈ G, x0 ∈ F, χ ∈ (Gx0 )⊥
G,
(3.23)
for f ∈ L2 (G).
b
The inverse Zak transform Zρ−1 is given by the extension of eq. (3.20) to L2ρ (F × G).
Proof. (of Theorem 3.4) By Corollary 2.8, the functions fx , g 7→ ρg f (x), are in
L2 (G) for almost all πρ (x) ∈ ρ\X, so the Zak transform is well-defined almost everywhere
by eq. (3.9).
We show that the Zak transform is an isometry by combining the Weil formula with
Plancherel’s theorem. For f ∈ L2 (X, µ#
ρ\X ), we use eq. (3.9) to get
Z
Z
Z
Z
Plancherel
2 Weil
2
kf k2 =
|fx | dµG dµρ\X (πρ (x))
=
|fbx |2 dµGb dµρ\X (πρ (x))
b
ρ\X G
ρ\X G
Z
Z
(3.9)
=
|Zρ f (x, χ)|2 dµGb (χ) dµρ\X (πρ (x)).
ρ\X
b
G
20
CONTENTS
R
Now, by Lemma 3.2, the function x 7→ Gb |ZρX f (x, χ)| dµGb (χ) is ρ-invariant, so with eq.
(2.11) for the fundamental domain F (note that G is abelian, so unimodular), we get
Z Z
Z Z
2
X
2
kf k2 =
|Zρ f (x, χ)| dµGb (χ) dµF (x) =
|Zρ f (x0 , χ)|2 dµGb (χ) dµF (x0 ).
F
b
G
F
b
G
Furthermore, using the fact that Zρ f vanishes for χ 6∈ (Gx0 )⊥
G together with the Weil
formula for Gx0 \G and the normalization of µ(Gx0 )⊥G , we get
Z Z
2
|Zρ (x0 , χ)|2 dµ(Gx0 )⊥G (χ) dµF (x0 ) = kZρ f k2L2 (F ×G)
kf k2 =
b .
F
ρ
(Gx0 )⊥
G
b
This shows that the Zak transform is a linear isometry into L2ρ (F × G).
Next, we show that the extension is a Hilbert space isomorphism by explicitly writing
b Consider the equivariant extension hX
down the preimage of a function h ∈ L2ρ (F × G).
b that is given by hX (x, χ) = χ(g)h(x0 , χ) for x ∈ X and χ ∈ (Gx0 )⊥ , where
of h to X × G
G
x0 and g are given by ρg (x) = x0 . Note again, that, since χ ∈ (Gx0 )⊥
G , it is constant on
stabilizer cosets, so the extension is well-defined.
Then |hX (x, χ)|2 is ρ-invariant and the above calculation can be carried out backwards.
The inverse Plancherel-Fourier transform of χ 7→ h(x0 , χ) yields a function in q
h ∈ L2 (F ×
G). It needs to be checked, that this function is of the form (x0 , g) 7→ f (ρ−1
g (x0 )) for some
f ∈ L2 (X) and almost all x0 ∈ F . But this is true, since h(x0 , χ) is only supported on
q
(Gx0 )⊥
G , so h(x0 , ·) is Gx0 -invariant by the Plancherel-theorem for the group Gx0 \G, using
⊥
\
∼
the isomorphism G
x0 \G = (Gx0 )G .
Finally, an application of the Weil formula yields that Zρ−1 h ∈ L2 (X). That this
transform inverts the Zak transform is immediate from eq. (3.9) and Plancherel’s theorem.
The intertwining property (3.23) is shown by direct calculation, using the definition
(3.6) and the translation invariance of the Haar measure.
Using the Hausdorff-Young inequality for lca groups kfbkq ≤ kf kp for 1 < p < 2 and
q = p/(p − 1) and f ∈ L1 ∩ Lp (G), instead of the Parseval identity, one obtains an Lp -Zak
p
q
b
transform Zp : Lp (X, µ#
b ), using the L -version of Corollary 2.8.
ρ\X ) → L (F × G, µF ⊗ µG
We next consider Zak’s approach to the Zak transform on Rd in [Zak67], where he
considers the transform as a decomposition of a function into simultaneous eigenfunctions
of the group of translation operators associated to a lattice and the group of modulations
associated to the reciprocal lattice. In fact, the objects he considers are not functions,
but measures. More precisely, the classic Zak transform decomposes a function into Dirac
combs on orbits of the lattice group that are modulated by characters belonging to a
fundamental domain of the reciprocal lattice (the so-called first Brillouin zone). The
same construction in the general abelian case works as follows.
Let G be an lca group acting strongly proper on the Weil G-space (X, µ#
ρ\X ) via the
action ρ, and let F be a measurable fundamental domain of this action. Now, consider
(x ,χ)
b x0 ∈ F , χ ∈ G,
b that are defined
the locally finite complex Borel measures δG 0 on F × G,
by
Z
(x ,χ)
δG 0 (ϕ) :=
ρg ϕ(x0 )χ(g) dµG(g) = Zρ ϕ(x0 , χ),
G
ϕ ∈ Cc (X).
(3.24)
4. THE ZAK TRANSFORM FOR NON-ABELIAN ACTIONS
21
We call these measures Zak measures. The values of the Zak transform of a function
f ∈ L2 (X) can be considered as the coefficients of the expansion into these measures as
seen by the following calculation. For ϕ ∈ Cc (X),
Z
(x ,χ)
Zρ f (x0 , χ)δG 0 (ϕ) dµF ×Gb (x0 , χ) = hZρ f, Zρ ϕiL2 (F ×G)
b = hf, ϕiL2 (X) = f (ϕ),
b
F ×G
R
where f is interpreted as a functional on Cc (X), namely ϕ 7→ f (ϕ) := X f · ϕ dµX . Here,
we used the isometry property of the Zak transform, and Zρ ϕ(x0 , χ) = Zρ ϕ(x0 , χ). This
a weak inversion theorem in terms of measures. The strong version is Proposition 3.3.
(x ,χ)
b are the analogs of Zak’s
This shows that the Zak measures δG 0 , x0 ∈ F , χ ∈ G,
‘functions’ ψkq in [Zak67, (5)]. Just as these, they are eigenfunctions of the action of G.
For ϕ ∈ Cc (X),
(x ,χ)
(x ,χ)
(x ,χ)
0
−1
ρg δG 0 (ϕ) = δG 0 (ρ−1
(ϕ). (3.25)
g ϕ) = Zρ [ρg ϕ](x0 , χ) = χ(g)Zρ ϕ(x0 , χ) = χ(g)δG
Since the characters of G are not related to some notion of characters on X, the analog
of being an eigenfunction of the modulation action of the reciprocal group is less clear.
However, if we had a complete system of functions on F (or equivalently on ρ\X), we could
extend them to ρ-equivariant functions on X by modulating them along the orbits with a
b So, the ρ-invariant functions would play the role of the reciprocal group.
character χ ∈ G.
Consequently, we would expect that the Zak measures are eigenfunctions of modulation
by any ρ-invariant function. This is true in a trivial way. Let m be a ρ-invariant function
on X. Let the modulation of ϕ ∈ Cc (X) by m be defined as Mm ϕ(x) := ϕ(x)m(x),
x ∈ X. Then
Z
(x0 ,χ)
(x0 ,χ)
(x ,χ)
Mm δG (ϕ) = δG (Mm ϕ) =
ρg ϕ(x0 )ρg m(x0 )χ(g) dµG (g) = m(x0 )δG 0 (ϕ).
G
(3.26)
Before generalizing the results of this section to the non-abelian case, we note that an
b actually is identical to L2 (F × G).
b This
interesting question is, when the space L2ρ (F × G)
is for example the case when for almost every x0 ∈ F , the stabilizer Gx0 is trivial. Then
the vanishing property becomes irrelevant in the inversion formula, and one can invert
b as in the proof of Theorem 3.4.
any function in L2 (F × G)
4. The Zak transform for non-abelian actions
To define a Zak transform for non-abelian groups, we can use eq. (3.9) in a setting
where a Fourier transform is available. Moreover, to prove the analog of Theorem 3.4, we
need a Plancherel formula. A reasonable setting is that of second countable unimodular
type I groups.
4.1. The non-abelian Zak transform on L1 (X). We quickly review the main
results of harmonic analysis on second countable unimodular type I groups. For a detailed
expositions of the theory, see for example [Fol95, Dix82, Lip74, HR70].
b of G is the set of equivalence classes
Let G be a locally compact group. The dual G
of unitary irreducible representations of G.
b = {[σ] | σ : G → U(Hσ ) irreducible},
G
(4.1)
22
CONTENTS
where [σ] denotes the equivalence class w.r.t. the relation of unitary equivalence. Two
unitary representations σ and σ ′ of G are called unitary equivalent, if there is a bounded
linear unitary operator T : Hσ → Hσ′ that intertwines σ and σ ′ , i.e. that satisfies
σ ′ = T σT −1 . To simplify notation, we will omit the brackets for the equivalence classes
throughout the text. The dual is a topological space with the so-called Fell topology (see
e.g. [Fol95, 7.2]).
The dual of a type I group has special properties that are usually formulated in
representation theoretic terms. For second countable groups, there is a way to formulate
b
this property equivalently in terms of the toplogy and the measurable structure of G.
b
Namely, being type I is in this situation equivalent to G being a T0 -space (Kolmogorov
b being standard. The latter means that G
b is
space), and to the Borel σ-algebra on G
measurably isomorphic to a Borel subset of a complete separable metric space [Fol95, Thm.
7.6]. All abelian and compact lcH groups are type I, while a discrete group is type I, if
and only it has a normal abelian subgroup of finite index [Tho64]. The latter fact will
be used in Section 5 to apply the Zak transform to discrete subgroups of the Euclidean
group E(3) of isometries of R3 .
For a second countable unimodular type I group G, the Fourier transform on L1 (G)
is defined by
Z
b
b
f (σ) :=
f (g)σ(g)∗ dµG (g), f ∈ L1 (G), σ ∈ G,
(4.2)
G
b are
where µG is a left Haar measure on G. The single ‘Fourier coefficients’ fb(σ), σ ∈ G,
unitary operators on a Hilbert space Hσ . The Fourier transform defines what is called a
b (see [Fol95, 7.4]). When f ∈ L1 (G)∩L2 (G),
measurable field of operators {fb(σ)}σ∈Gb on G
b where µ b is
then these operators are Hilbert-Schmidt operators for µGb -almost all σ ∈ G,
G
b
the measure on G, s.t. the inverse Fourier transform is given by
Z
b
f (g) =
(4.3)
tr(fb(σ)σ(g)) dµGb (σ), f ∈ L1 (G) fb ∈ L1 (G)
b
G
for almost every g ∈ G (see [Lip74] for more details).
In Section 4.2, a Poisson summation formula for a general compact subgroup H of
a second countable unimodular type I group will be constructed. The following set HG⊥ ,
which we call reciprocal space of H in G, will play a role as that of the reciprocal group
in the abelian case.
b | mult(1, resG
(4.4)
HG⊥ := {σ ∈ G
H (σ)) ≥ 1},
This can be intuitively understood by noting that at least the present invariant components will lead to some ‘resonance’.
Now, we are in place to define the Zak transform on second countable unimodular
type I groups and then prove the analog of Lemma 3.2.
Definition 4.1 (Zak transform for non-abelian actions). Let G be a second
#
1
countable unimodular type I group, (X, µ#
ρ\X ) a Weil G-space, and f ∈ L (X, µρ\X ).
Furthermore, let F be a µX -measurable fundamental domain of the action ρ. The Zak
4. THE ZAK TRANSFORM FOR NON-ABELIAN ACTIONS
transform Zρ f of a function f ∈ L1 (X, µ#
ρ\X ) is defined as
Z
b
Zρ f (x0 , σ) :=
ρg f (x0 )σ(g)∗ dµG (g), x0 ∈ F, σ ∈ G.
23
(4.5)
G
b is given by eq. (4.5) with x0 ∈ X.
Again, the extended Zak transform ZρX on X × G
Note that Definition 4.1 is a natural extension of Definition 3.1 in the abelian case;
the only change is that the characters χ are replaced by the, possibly higher-dimensional,
irreducible unitary representations σ of the group.
b In analogy to Lemma
The Zak transform is a measurable field of operators on F × G.
3.2, we can prove an equivariance property.
Lemma 4.2 (Equivariance of the non-abelian extended Zak transform). With
the assumptions of Definition 4.1, the extended Zak transform satisfies
b
ZρX f (ρ−1
g ∈ G, x0 ∈ F, σ ∈ G.
(4.6)
g (x0 ), σ) = Zρ f (x0 , σ)σ(g),
b | σ 6∈ (Gx0 )⊥ }.
In particular, Zρ f (x0 , σ) vanishes on the set {(x0 , σ) ∈ F × G
G
Proof. Equation (4.6) follows immediately from the definition (4.5), using translation
invariance of µG .
Now, for x0 ∈ F , let σ be not contained in (Gx0 )⊥
G . Every Gx0 -irreducible component of
G
resH (σ) is then non-trivial. For f ∈ L1 (G) and G ∈ Gx0 , using the equivariance property,
Zρ f (x0 , σ) = Zρ f (ρ−1
g (x0 ), σ) = Zρ f (x0 , σ)σ(g),
showing that Zρ f (x0 , σ) vanishes on every Gx0 -invariant subspace by non-triviality of the
respective component of resG
H (σ). Since all components of the representation are nontrivial, Zρ (x0 , σ) = 0.
As in the abelian case, the equivariance property (4.6) can be used to interpret the
b
extended Zak transform as a map into the space of measurable fields of operators on G
that satisfy this property, without the necessity to specify a fundamental domain.
The fact that the Zak transform vanishes away from (Gx0 )⊥
G can be refined as follows.
Even though, σ ∈ (Gx0 )⊥
,
the
Zak
transform
might
vanish
on subspaces of the repreG
b and consider
sentation space Hσ . More precisely, for x0 ∈ F , take any element σ ∈ G
G
resH (σ). Some Gx0 -irreducible components might be non-trivial, showing that Zρ f (x0 , σ)
vanishes on these components. This is part of the content of the generalization of the
Poisson summation formula that is considered in Section 4.2.
First, we consider the analogs of the interpretations (3.9), (3.10), and (3.15). The
Fourier viewpoint is still valid. We have
b
x0 ∈ F, σ ∈ G.
(4.7)
Zρ f (x0 , σ) = fc
x0 (σ),
The same is true for the symmetry-projection viewpoint of Wigner. The operator Pρσ that
is given by
b
ZρX f (x0 , σ) = Pρσ f (x), x ∈ X, σ ∈ G
(4.8)
maps f into the space of σ-equivariant measurable fields of operators on X (see Corollary
5.1). However, we cannot formulate a version of the viewpoint (3.15), since the phase
b is not a group.
space G × G
24
CONTENTS
In analogy to Proposition 3.3, one has the following inversion theorem. Since we
have no Poisson summation formula yet, we also have no suitable measure on (Gx0 )⊥
G to
formulate the analog of eq. (3.17). Thus, we stick to µGb and keep the vanishing property
in mind for the moment. The formula will be made precise in Theorem 4.8.
Proposition 4.3 (Non-abelian L1 -Zak inversion theorem). Let G be a second
countable unimodular type I group that acts strongly proper on a Weil space (X, µ#
ρ\X ),
and F a measurable fundamental domain of the action ρ. The inverse Zak transform Zρ−1
is defined as
!
Z Z
Z
Zρ−1 h(x) :=
F
b
G
tr h(x0 , σ)
Gx0 \G
σ(g) dδx(x0 ) (πL (g))
dµGb (σ) dδx (x0 )
(4.9)
b for almost all x0 ∈ F . It satisfies f = Z −1 Zρ f point-wise almost
for h(x0 , ·) ∈ L1 (G)
ρ
b for almost all x0 ∈ F .
everywhere for f ∈ L1 (G) with Zρ f (x0 , ·) ∈ L1 (G)
Proof. The proof is a direct consequence of the above discussion, the equation (4.7)
and the L1 -Fourier inversion formula (4.3).
Again, the inversion formula can be viewed as a decomposition of a function into
measures as in [Zak67]. However, these measures are operator-valued in the non-abelian
case.
Interestingly, the Zak inversion formula (4.9) is closely related to the Arthur-Selberg
trace formula that goes back to Selberg [Sel56]. More precisely, it implies the following
trace formula for Weil G-spaces.
Corollary 4.4 (Trace formula for Weil G-spaces). Let G be a second countable
unimodular type I group that acts strongly proper on a Weil G-space (X, µ#
ρ\X ), and F a
1
b
measurable fundamental domain of the action ρ. Then for f ∈ L (X) with fb ∈ L1 (G),
Z
f
(G)
(x0 ) dµF (x0 ) =
F
Z
b
G
f (G) (σ) dµGb (σ),
where the integrand on the left (‘geometric’) side is given by
Z
(G)
ρg f (x0 ) dµGx0 \G (πL (g)),
f (x0 ) :=
(4.10)
x0 ∈ F,
Gx0 \G
and the integrand on the right (‘spectral’) side is given by
f (G) (σ) := tr
Z Z
F
Gx0 \G
!
ρg f (x0 )σ(g)∗ dµGx0 \G (πL (g)) dµF (x0 ) ,
b
σ ∈ G.
Proof. The left hand side is an application of the Weil formula to the integral
(see eq. (2.11)).
R
X
f dµρ\G
4. THE ZAK TRANSFORM FOR NON-ABELIAN ACTIONS
25
Applying the inverse Zak transform inside the integral yields
Z
Z
Z Z
f dµρ\G =
tr(Zρ f (x0 , σ)σ(g)) dµGb dµGx0 \G (πL (g)) dµF (x0 )
X
Gx0 \G
F
=
Z
tr
b
G
b
G
Z Z
F
Gx0 \G
Zρ f (x0 , σ)σ(g) dµGx0 \G (πL (g)) dµF (x0 )
using Fubini’s theorem.
!
dµGb ,
In particular, when G is an lcH group, and a discrete subgroup H with compact
quotient H\G acts by conjugation ρh (g) := hgh−1 , then the trace formula (4.10) reduces
to what Arthur calls the ‘unrefined trace formula’ [Art05, (1.3)]. The fundamental domain
F of the action is a set of representatives of the conjugacy classes of H, and the stabilizer
of a point g ∈ G is the so-called centralizer CH (g) that contains all elements of H that
commute with g. This yields the geometric side of the unrefined trace formula. On the
spectral side, the double integral in eq. (4.10) can be combined to an integral over G.
This shows that the trace formula (4.10) generalizes some versions of the Arthur-Selberg
trace formula.
In [Art05], Arthur claims that the difficulty in generalizing the trace formula is due to
non-compact quotients and the resulting non-discreteness on the spectral side. However,
the above discussion and Corollary 4.4 show that actually one can formulate results for
paracompact quotients and even for the action of non-discrete groups. The true difficulty
lies in generalizations to non-proper actions and resulting non-compact stabilizers. In fact,
Arthur’s example that illustrates the problems is a non-proper action. Still, he managed
to prove versions of the trace formula that can handle non-proper actions. The difficulty
of his work illustrates how hard this step is.
The Arthur-Selberg trace formula has found applications in the Langlands program
and in number theory, where it is applied to quotients of the form G(F)\G(AF ), where G
is an algebraic group, F is a number field, and AF is the ring of adeles over F. Corollary
4.4 suggests that the Zak transform might also be of interest in these fields.
4.2. A Poisson summation formula for quotients by compact groups. We
prove an analog of the Poisson summation formula (3.5) for the case that G is a second
countable unimodular type I group, and H is a compact closed subgroup. For this purpose,
we need to develop some kind of harmonic analysis on the quotient space G/H. It will
turn out that we will have to leave classic representation theory, since the objects that
emerge are not unitary operators on some Hilbert space, but linear operators between
different Hilbert spaces. Parts of the reasoning are well-known and related to harmonic
analysis on Gelfand pairs (see e.g. [Die60]).
[ of G/H, before developing the theory. The
We write down the dual object G/H
elements are simply the non-trivial orbital means of the unitary irreducible representations
of G w.r.t. the right action of H.
[ := {AR σ | σ ∈ G}
b
b \ {0Hσ | σ ∈ G},
G/H
(4.11)
where 0Hσ denotes the trivial operator that maps all vectors in Hσ to zero. Note that
these operators are well-defined, because H is compact. To compute these orbital means,
26
CONTENTS
L
L
b and Hσ =
we decompose the restricted representation resG
σ
,
σ
∈
H,
j
j
Hσ =
j
j Hj .
L
(1)
Now define the part of σ that reduces to the trivial representation σ := j,σj ≡1 idHj
L
(1)
which is the identity on Hσ := j,σj ≡1 Hj . Moreover, let σ ⊥ be the restriction to the
non-trivial components collected in Hσ⊥ , s.t.
(1)
resG
⊕ σ⊥,
Hσ = σ
b
Then, for σ ∈ G,
AR σ(gH) =
Z
σ(gh) dµH (h) = σ(g)
H
Hσ = Hσ(1) ⊕ Hσ⊥ .
Z
H
resG
Hσ
dµH = σ(g)
Z
σ (1) ⊕ σ ⊥ dµH
H
= σ(g)(AR σ (1) ⊕ AR σ ⊥ )(eH) = σ(g)(idH(1)
⊕ 0H⊥σ ),
σ
where the last equality results from the Schur orthogonality relations for the irreducible
(1)
representations of H. To simplify notation, we set Pσ := idH(1)
⊕ 0H⊥σ , which is the
σ
(1)
orthogonal projection to Hσ . In particular, we find that AR σ = 0H for σ 6∈ HG⊥ (see
(4.4)), because by definition the restrictions of these representations do not contain the
trivial representation of H. In summary, we found that
[ = {σG/H := σ · P (1) | σ ∈ H ⊥ }.
G/H
σ
G
(4.12)
(1)
In particular, the objects σG/H can be interpreted as linear operators from Hσ to Hσ ,
suggesting a more general approach than representation theory, where the structure of a
space is represented by operators between Hilbert spaces and not by unitary operators on
one Hilbert space.
As Hilbert-Schmidt operators are often identified with tensors in Hσ ⊗Hσ (as antilinear
[ can be considered as tensors
maps from Hσ∗ = Hσ to Hσ ), the operators σG/H ∈ G/H
(1)
(1)
∗
b can be considered as a tensor in
in Hσ ⊗ Hσ . Consequently, σG/H
= (Pσ )∗ σ ∗ , σ ∈ G,
(1)
Hσ ⊗ Hσ .
Next, we define the Fourier transform of a function f ∈ Cc (G/H). For this purpose,
we let fG (g) := f (gH), g ∈ G, be the invariant extension of f to G, and calculate its
b
G-Fourier transform, using the classic Weil formula. For σ ∈ G,
Z
Z
Z
Weil
fc
fG · σ ∗ dµG =
fG (gh)σ(gh)∗ dµH (h) dµG/H (gH)
G (σ) =
G
G/H H
Z
Z
∗
∗
=
f (gH)AR σ (gH) dµG/H (gH) =
f · σG/H
dµG/H .
G/H
G/H
1
This allows to define the Fourier transform on L (G/H) as
Z
∗
[
b
f (σG/H ) :=
f σG/H
dµG/H , f ∈ L1 (G/H), σG/H ∈ G/H.
(4.13)
G/H
⊥
1
In particular, this implies that supp(fc
G ) ⊆ HG for f ∈ L (G/H), s.t. we can define
[ σ 7→ σG/H , which is
µHG⊥ := µGb |HG⊥ . Moreover, consider the map ϕ : HG⊥ → G/H,
obviously surjective. It is also injective, since for σ, σ ′ ∈ HG⊥ of same dimension,
(1)
(1)
ϕ(σ) = ϕ(σ ′ ) ⇔ σ(g) · Pσ(1) = σ ′ (g) · Pσ′ ⇔ (σ ′−1 σ)(g) · Pσ(1) = Pσ′ .
4. THE ZAK TRANSFORM FOR NON-ABELIAN ACTIONS
(1)
27
(1)
Since the spaces Hσ and Hσ′ are non-trivial, this implies that there is at least a onedimensional invariant subspace of σ ′−1 σ, which implies σ ′ = σ by irreducibility. Thus ϕ
[ with the final topology w.r.t. ϕ. Moreover, we define
is bijective and we can endow G/H
[ by µ [ := ϕ∗ µ ⊥ .
a measure on G/H
HG
G/H
This allows us to prove the following inversion theorem for the Fourier transform on
L1 (G/H).
Proposition 4.5 (L1 -Fourier inversion for quotients by compact groups). Let
G be a second countable unimodular type I group, H a compact closed subgroup of G, and
f ∈ L1 (G/H) with fb ∈ L1 (G/H). Then for almost every gH ∈ G/H,
Z
tr fb(σG/H )σG/H (gH) dµG/H
f (gH) =
(4.14)
[ (σG/H ).
[
G/H
In particular, when f is in addition continuous, then eq. (4.14) is true for every gH ∈
G/H.
Proof. Again, we can transfer the calculation to L1 (G) and use the Fourier inversion
formula (3.3) to get the result. Let fG (g) := f (gH), g ∈ G, then for almost every
gH ∈ G/H,
Z
Z
c
tr(fb(σG/H )σ(g)) dµHG⊥ (σ)
f (gH) = fG (g) =
tr(fG (σ)σ(g)) dµGb (σ) =
=
Z
[
G/H
b
G
⊥
HG
tr(fb(σG/H )σG/H (gH)) dµG/H
[ (σG/H ).
When f is continuous, then fG is continuous, so the point-wise inversion formula follows
from the inversion theorem on G.
[
Note, that the inversion formula (4.14) implies a Gelfand-Raikov theorem for G/H.
The elements separate any two points, as shown by the inversion formula for a function in
Cc (G/H) that does not agree on those two points (which exists by the Urysohn lemma,
since G/H is paracompact and thus normal).
Another consequence is the following analog of the Poisson summation formula.
Proposition 4.6 (Poisson summation formula for compact quotients). Let G
be a second countable unimodular type I group, H a compact closed subgroup of G, and
b Then
f ∈ Cc (G) with f |H ∈ L1 (H) and fb ∈ L1 (G).
Z
Z
tr(Pσ(1) · fb(σ)) dµHG⊥ (σ).
(4.15)
f (h) dµH (h) =
H
⊥
HG
Proof. We write the integral over H as the evaluation of the orbital mean at the
neutral element and apply the Fourier inversion formula (4.14).
Z
Z
d
tr(A
f (h) dµH (h) = AR f (eH) =
R f (σG/H )σG/H (gH)) dµG/H (gH)
[
G/H
H
Z
Z
\
\
tr((A
⊥ (σ).
tr((AR f )G (σ)σ(e)) dµHG⊥ (σ) =
=
R f )G (σ)) dµHG
⊥
HG
⊥
HG
28
CONTENTS
We calculate the function in the trace.
Z
Z Z
∗
\
(AR f )G (σ) = (AR f )G (g)σ(g) dµG (g) =
f (gh) dµH (h)σ(g)∗ dµG (g)
G
G H
Z
Z
G unimod.
=
f (g)
σ(gh−1)∗ dµH (h) dµG (g) = Pσ(1) · fb(σ),
G
H
yielding eq. (4.15).
The exact same arguments can be used to prove the analog results for the left quotient
H\G with the reciprocal space
[ = {σH\G := P (1) · σ | σ ∈ H ⊥ }.
H\G
σ
G
(4.16)
(1)
The elements σH\G can also be considered as tensors, namely as elements of Hσ ⊗ Hσ .
The resulting Poisson summation formula is identical to the version for right quotients.
It will be used in Theorem 4.8 for the left quotient of G by a stabilizer of the action.
Lipsman [Lip74] proved a Poisson summation formula for quotients by normal subgroups, s.t. the quotient group is unimodular. The special case that the group is in
addition compact is contained in Proposition 4.6.
The above approach suggest an approach to harmonic analysis on quotient spaces in
more generality. This, however, goes far beyond the purpose of this article and will be
treated elsewhere.
Finally, we formulate the corollary that will allow to prove the analog of Theorem 3.4.
Corollary 4.7 (Plancherel theorem for H-invariant functions). Let G be a
second countable unimodular type I group, and H a compact subgroup. The Planchereltransform
is an isometry from the space of right H-invariant functions onto the space
R⊕
(1)
[ , and an isometry from the space of left H-invariant functions onto
[ Hσ ⊗ Hσ dµG/H
G/H
R⊕
(1)
the space H\G
[.
[ Hσ ⊗ Hσ dµH\G
Proof. By compactness of H, the space of right H-invariant functions is a closed
subspace of L2 (G). In addition, it is isomorphic to L2 (G/H) via f 7→ f ◦ πR . The
extension of the Fourier transform from L1 (G/H) ∩ L2 (G/H) to L2 (G/H) is thus an
R⊕
(1)
isometry, because of Plancherel’s theorem for L2 (G). It maps into G/H
[,
[ Hσ ⊗ Hσ dµG/H
because of eq. (4.13) and the identification of the operators with the respective tensors.
The special form of the tensor spaces reflects the vanishing of the ‘columns’ of the Fourier
coefficients resulting from the Schur orthogonality relations. R
⊕
Surjectivity is shown by embedding this image space into Gb Hσ ⊗ Hσ dµGb (by extension with zero columns), performing the inverse Plancherel transform and using eq.
(4.14).
The analog argument yields the result for the space of left H-invariant functions. 4.3. The non-abelian Zak transform on L2 (X). The L2 -theory of the Fourier
transform is again captured in a Plancherel theorem.
L
For compact groups G, the Peter-Weyl theorem states that L2 (G) = σ∈Gb Eσ , where
Eσ is the translation invariant subspace of L2 (G) that is spanned by the matrix elements
of σ. We can combine left and right translation in the two-sided regular representation T
4. THE ZAK TRANSFORM FOR NON-ABELIAN ACTIONS
29
of G × G on L2 (G) (T(h,h′ ) f (g) :=L
f (h−1 gh′ )), then the Fourier transform intertwines T
and the direct sum representation σ∈Gb (σ ⊗σ)⊕dσ , (dσ = dim(Hσ )), of G×G that acts on
L
the space σ∈Gb (Hσ ⊗ Hσ )⊕dσ by left and right modulation (Mσ : G × G → U(Hσ ⊗ Hσ ),
Mσ (g, h)A := σ(g)Aσ(h)∗ , A ∈ Hσ ⊗ Hσ ), where we identify the Fourier coefficient fb(σ)
with a tensor in Hσ ⊗ Hσ by interpreting the linear operator as an antilinear map from
Hσ∗ = Hσ to Hσ . This can be done, since fb(σ) is a Hilbert-Schmidt operator, and is
natural, when for example calculating matrix coefficients of the representation w.r.t. to
a basis of Hσ and the dual basis of Hσ∗ = Hσ .
The Plancherel theorem for second countable unimodular type I groups [Fol95, Thm.
b s.t.
7.44] says essentially the same. Namely, that there is a unique measure µGb on G,
R
⊕
the Fourier transform (4.2) is a Hilbert space isomorphism betwenn L2 (G) and Gb Hσ ⊗
Hσ dµGb (σ), and that the Fourier inversion formula (4.3) can be extended to L2 (G). As for
compact groups, the Fourier transform intertwines the two-sided regular representation
with the two-sided modulation representation.
Before we can state the analog of Theorem 3.4, we need to give a precise version of
the refinement of the vanishing property of the Zak transform given after Lemma 4.2.
b an irreducible
Let G be a second countable unimodular type I group and σ ∈ G
representation of G. For x0 ∈ F , the stabilizer Gx0 is a compact subgroup of G by
properness of the action.
As we know that the functions fx0 : g 7→ ρg f (x0 ) are left Gx0 -invariant, we can
apply Corollary 4.7 to show that the Plancherel transform maps fx0 to a tensor field in
(1,x )
Hσ 0 ⊗ Hσ on (Gx0 )⊥
G , where the additional index x0 reflects the dependence on the
stabilizer Gx0 .
These considerations allow to formulate the following theorem.
Theorem 4.8 (Non-abelian Zak transform on L2 (X)). Let G be a second countable unimodular type I group that acts strongly proper on a Weil G-space (X, µ#
ρ\X ) via
ρ : G × X → X, and F a measurable fundamental domain of the action ρ. The Zak
transform can be extended to a Hilbert space isomorphism
Z ⊕Z ⊕
#
2
b
Hσ(1,x0 ) ⊗ Hσ dµ(Gx0 )⊥G (σ) dµF (x0 ) =: L2,op
Zρ : L (X, µρ\X ) →
ρ (F × G), (4.17)
F
(Gx0 )⊥
G
where µF is given by eq. (2.10).
The Zak transform intertwines the action ρ of G on L2 (X) with the modulation representation on Zρ (L2 (X)).
Zρ [ρg f ](x0 , σ) = σ(g)Zρ f (x0 , σ),
b
g ∈ G, x0 ∈ F, σ ∈ G,
for f ∈ L2 (X).
The inverse Zak transform Zρ−1 is given by the extension of
!
Z
Z Z
Zρ−1 h(x) :=
tr h(x0 , σ)
F
(Gx0 )⊥
G
Gx0 \G
σ(g) dδx(x0) (πL (g))
(4.18)
dµ(Gx0 )⊥G (σ) dδx (x0 ).
(4.19)
30
CONTENTS
Proof. The proof is essentially identical to that of Theorem 3.4, using the non-abelian
versions of the used results.
By Corollary 2.8, the functions fx0 are in L2 (G) for almost all x0 ∈ F , so the Zak
transform is well-defined almost everywhere by the L2 -version of eq. (4.7).
The Zak transform is an isometry by combining the Weil formula,
the Plancherel theoR
rem, eq. (2.11) for the fundamental domain F , invariance of x 7→ Gb kZρX (x, σ)kHS dµGb (σ),
and the refined vanishing property. For f ∈ L1 (X) ∩ L2 (X), we get
Z
Z
Z
Z
Plancherel
2 Weil
2
kf k2 =
|fx | dµG dµρ\X (πρ (x))
=
kfbx k2HS dµGb dµρ\X (πρ (x))
b
ρ\X G
ρ\X G
Z
Z
(4.7)
=
kZρ f (x, σ)k2HS dµGb (σ) dµρ\X (πρ (x))
b
ρ\X G
Z Z
(2.11)
=
kZρX (x, σ)k2HS dµGb (σ) dµF (x) = kZρ f k2L2 (F ×G)
b .
F
ρ
b
G
b
Thus, Zρ can be extended to an isometry from L2 (X) into Lρ2,op (F × G).
The proof of surjectivity works exactly as in Theorem 3.4, using Corollary 4.7 for the
left Gx0 -invariant functions fx0 .
The intertwining property (4.18) is again shown by direct computation, using the
definition (4.5) and right invariance of µG , which is due to unimodularity of G.
There is a Hausdorff-Young inequality for second countable unimodular type I groups
(see [Kun58]) to prove an analog result for a Zak transform Zρ : Lp (X) → Zρ (Lp (X)) ⊆
b for p ∈ (1, 2) and q = p/(p − 1).
Lq (F × G)
As in the abelian case (see (3.24)), the reconstruction formula can be interpreted as a
decomposition into measures. These operator-valued Zak measures are
Z
(x0 ,χ)
δG (ϕ) :=
ρg ϕ(x0 )σ(g) dµG (g) = Zρ ϕ(x0 , σ ∗ ), ϕ ∈ Cc (X).
(4.20)
G
They have the analog properties as their abelian counterparts and satisfy a weak inversion
formula.
An important special case of Theorem 4.8 is the case that almost allR stabilizers
⊕R⊕
are trivial. In this case, the image of the Zak transform is the full space F Gb Hσ ⊗
Hσ dµGb (σ) dµF .
4.4. The character Zak transform and other alternatives. The above generalization of the Zak transform is not the only possibility. As already mentioned in the
Introduction, one can easily generalize to general actions of groups on L2 (X). A few further approaches to non-abelian actions will be discussed in this section, since they might
be of interest or have already been considered elsewhere.
One alternative to the Zak transform given in Definition 4.1 uses the characters of the
representations. This approach was already considered by Wigner [Wig31].
b a unitary irreducible representation. The character
Let G be an lcH group and σ ∈ G
χσ : G → C of σ is defined as
χσ (g) := tr(σ(g)),
g ∈ G.
(4.21)
4. THE ZAK TRANSFORM FOR NON-ABELIAN ACTIONS
31
In the case that G is abelian, this agrees with the standard definition in eq. (3.1), as the
irreducible representations are one-dimensional. When G is non-abelian, the characters
are not group homomorphisms and transform in a more complicated way, namely
b g, h ∈ G.
χσ (g −1 h) = tr(σ(g)∗σ(h)) = hσ(h), σ(g)iHS, σ ∈ G,
(4.22)
This is the price you pay for decomposing functions on a non-abelian structure into scalar
functions.
We define the character Zak transform as follows.
Definition 4.9 (Character Zak transform). Let G be a second countable unimodular type I group that acts strongly proper on a Weil G-space (X, µ#
ρ\X ) via ρ : G × X → X,
and F a measurable fundamental domain of the action ρ. The character Zak transform
1
Zρtr f of a function f ∈ L1 (X, µ#
ρ\X ) with fx ∈ L (G) for almost every πρ (x) ∈ ρ\X is
defined as
Z
tr
b
Z f (x0 , σ) :=
ρg f (x0 )χσ (g) dµG (g), x0 ∈ F, σ ∈ G.
(4.23)
ρ
G
b is given by eq. (4.23) with x0 ∈ X.
Again the extended Zak transform Zρtr,X on X × G
The character Zak transform is related to the Zak transform defined in Definition 4.1
by simply taking the trace of the latter:
b
Zρtr f (x0 , σ) = tr(Zρ f (x0 , σ)), x0 ∈ F, σ ∈ G,
(4.24)
where f is as in Definition 4.9. From eqs. (4.22) and (4.24), we can infer the following
transformation law of the extended character Zak transform
∗
b
Zρtr,X f (ρ−1
g ∈ G, x0 ∈ F, σ ∈ G,
(4.25)
g (x0 ), σ) = hσ(g), Zρ (x0 , σ) iHS ,
for f as above. A somewhat subtle consequence is that the function f can only be
reconstructed from the extended character Zak transform Zρtr,X f and not from Zρtr f ,
namely
Z
f (x) =
b
G
Zρtr,X f (x, σ) dµGb (σ),
x ∈ X,
(4.26)
which can be seen from eqs. (4.25) and (4.19). Some information that is lost by taking
the trace of Zρ f can thus be retrieved from the transformation law of Zρtr,X f . More
b are conjugation invariant, and only span the space of
precisely, the characters χσ , σ ∈ G
conjugation invariant L2 -functions (which are called class functions)3.
Definition 4.1 is not the only way of generalizing the Zak transform for abelian groups
and actions.
The Zak transform on lca groups G with a closed subgroup H acting by right transb ⊥ instead of H.
b As already noted, the two
lation is usually defined in terms of G/H
G
objects are isomorphic (via the restriction operator) – a fact that is essential for proving
the Poisson summation formula for lca groups. The relevance of the Zak transform for
sampling theory and Gabor analysis is largely due to its close relation to the Poisson
summation formula (for details, see e.g. [Grö01]). In this spirit, one could try to define
3The
resulting Fourier transform of class functions can be understood in more detail. In the case of a
compact group, the sets of conjugacy classes and of characters, both have the structure of a hypergroup.
These hypergroups are dual to each other (see [BH95], [Las]).
32
CONTENTS
a Zak transform on non-abelian groups based on the restriction of the representations of
X = G to a closed subgroup H that acts on G. These restricted representations are in
general not irreducible, s.t. the resulting transform is not equivalent to our definition.
We will not study this version of a Zak transform in more detail here, mainly due to the
fact that one looses the relation to the Fourier transform on H. To study this transform,
one would need to develop a harmonic analysis on quite general quotients G/H, similar
to that developped in Section 4.2.
b of a non-abelian group
In [Kut02], Kutyniok bypasses the shortcomings of the dual G
b where L is a suitable lca group that plays
G by constructing an injective map G → L,
b In addition, a further group Z and an action τ : G → Aut(L × Z) are
the role of G.
constructed, s.t. the group G ⋉τ (L × Z) can be used as substitute for the Heisenberg
group. Under certain technical conditions, this setting yields a Zak transform with most
features of the abelian transform.
Finally, it should be noted that versions of Plancherel’s theorem exist for more general
situations than second countable unimodular type I groups, possibly allowing to generalize
the versions in this article.
5. Applications of the Zak transform
5.1. A Bloch-Floquet theorem for group actions. As noted in the Introduction,
the classic Zak transform was considered in [Zak67] as a refinement of the decomposition
of electron states in crystals into Bloch waves [Blo29], which are periodic functions that
are suitably modulated. The Bloch decomposition relies on the fact that the partial differential equation under consideration – the non-relativistic Schrödinger equation for a
single electron – is invariant under translation by a crystal lattice vector. In representation
theoretic terms, the classic Zak transform diagonalizes the Hamiltonian. Even earlier, Floquet [Flo83] applied the same argument to decompose ordinary differential equations that
are invariant w.r.t. discrete translation groups in a similar manner. The Zak transforms
(3.6) and (4.5) can be used to prove and refine a generalized Bloch-Floquet theorem.
Corollary 5.1 (Bloch-Floquet theorem for strongly proper actions). Let
(X, µ#
ρ\X ) be a Weil G-space, and F a measurable fundamental domain of the action
ρ. Moreover, let B(X) be a linear space of functions on X that is ρ-invariant (e.g. the
solution space of a differential equation on X or an Lp -space), and on which the Zak
transform is well-defined. Then every function f ∈ B(X) can be decomposed as follows:
b
(i) When G is an lca group, there are ρ-invariant functions BFχ : X → C, χ ∈ G,
s.t.
Z
f (x) =
BFχ (x)χ(g) dµGb (χ), x ∈ X, ρg (x) ∈ F.
(5.1)
b
G
The functions x 7→ BFχ (x)χ(g) are called Bloch waves and are given by
BFχ (x) = Zρ f (x0 , χ),
x ∈ ρG (x0 ).
(5.2)
(ii) When G is a non-abelian second countable unimodular type I group, there are
b on X, s.t.
ρ-invariant tensor fields BFσ : X → Hσ ⊗ Hσ , σ ∈ G,
Z
f (x) =
tr(BFσ (x)σ(g)) dµGb (σ), x ∈ X, ρg (x) ∈ F.
(5.3)
b
G
5. APPLICATIONS OF THE ZAK TRANSFORM
33
The tensor fields x 7→ BFσ (x)σ(g) are called Bloch tensor fields and are given by
BFσ (x) = Zρ f (x0 , σ),
x ∈ ρG (x0 ).
(5.4)
In both cases, when the space B(X) is the solution space of an invariant differential
equation, then the Zak transform intertwines the differential operator with an operator
that acts component-wise on the invariant subspaces.
As an application of 5.1, consider the proper action of a discrete subgroup S of the
Euclidean group E(3) = R3 ⋊ O(3) of isometries of R3 . Molecular structures that are
invariant w.r.t. such a group are called objective structures [Jam06] and are natural
generalizations of crystals. Now, consider a single electron in an objective structure with
symmetry group S. The non-relativistic Schrödinger equation for the state ψ of the
electron is of the form
Hψ := (−∆ + V ) ψ = Eψ,
(5.5)
where E is the energy, ∆ is the Laplacian, and V is the ρ-invariant potential. Let ΨE
be the space of L2 -solutions of eq. (5.5). To apply Corollary 5.1 to the space ΨE , we
need to show that the group S is type I (as a discrete group it is second countable and
unimodular) and that the action is strongly proper. The latter follows from the result
in [CEM01], since S and R3 are both second countable. That S is type I can be seen as
follows.
Let T(S) := {(I|c) ∈ S | c ∈ R3 } be the subgroup of pure translations in S, where we
used the notation (Q|c) for an element of E(3), where Q ∈ O(3) and c ∈ R3 , and the
action on R3 is given by ρ(Q|c) (x) = Qx + c, x ∈ R3 . The group T(S) is always an abelian
normal subgroup of S, since for (I|c) ∈ T(S) and (Q|c′ ) ∈ S,
(Q|c′ )(I|c)(Q|c′)−1 = (I|Qc) ∈ T(S).
Now, when the quotient S/T(S) is finite, then S is type I due to [Tho64], since T(S) is
an abelian normal subgroup of finite index. If S/T(S) is not finite, it needs to contain
a screw displacement by an irrational angle, because these are the only elements of E(3)
that generate infinite groups that don’t contain translations. The only groups of the
considered type that contain such an element are helical groups without translations (as
shown in [DEJ]). Moreover, the subgroup of all screw displacements of a helical group is
an abelian normal subgroup of finite index, completing the proof. In summary we have
shown:
Proposition 5.2 (Discrete subgroups of E(3)). All discrete subgroups of E(3) are
type I.
As a corollary, we get the following.
Corollary 5.3 (Bloch/Zak decomposition of electron states in objective
structures). Every solution ψ ∈ ΨE of eq. (5.5) can be decomposed into Bloch tensor
fields:
Z
ψ(x) =
Sb
where BFσ , σ ∈ Sb is given by
tr(BFσ (x)σ(g)) dµSb(σ),
BFσ (x) = Zρ ψ(x0 , σ),
ρg (x) ∈ F.
x ∈ ρG (x0 ).
(5.6)
(5.7)
34
CONTENTS
The Zak transform intertwines the Hamiltionian H with a component-wise operator on
the invariant subspaces.
Note that the existence of a nice fundamental domain F is guaranteed by [Bou89],
since S is discrete and R3 is σ-compact.
The classic Zak transform is intimately related to energy bands in crystals. A rigorous
derivation of this connection can be found in [RS78]. The reasoning is as follows. The
Zak transform intertwines the Schrödinger operator with a direct integral of operators
that act on the irreducible subspaces of the image of the Zak transform. Now, the single
operators have a discrete spectrum that varies continuously w.r.t. the wave vector in the
Brillouin zone (character of the lattice subgroup). As a consequence, the union of all the
spectra is the discrete union of intervals, which are called energy bands.
The interesting question, if this theory can be generalized to objective structures lies
beyond the scope of this article and will be treated elsewhere.
The analog of the Bloch decomposition for certain discrete abelian and finite symmetries has been studied by Banerjee [Ban11, Ban13]. He used this decomposition to design
algorithms for density functional theory for objective structures.
5.2. The Zak transform in radiation design. The success of classic X-ray crystallography for the analysis of molecular structures is due to the highly structured and
sharply peaked diffraction patterns of crystals. The mathematical reason underlying this
phenomenon is the Poisson summation formula, and the patterns contain information on
the symmetry of the crystal and – up to a phase problem – on the atomic structure of a
fundamental domain.
The theory of radiation design asks the question, whether there is radiation that
produces highly structured and sharply peaked diffraction patterns for structures that
are not crystals. The answer is affirmative for helical structures like carbon nanotubes or
filamentous viruses, where the ‘right’ radiation are so-called twisted X-rays (see [FJJ16,
JFJ16] for details).
For general objective structures, parts of the theory (namely for abelian and compact
groups) have been developped in [Jüs14], where the Zak transform emerges as a tool to
interpret the diffraction patterns. The detailed theory for arbitrary objective structures
will be derived elsewhere.
As a direct application of the Zak transform, we can quickly sketch the line of thought.
According to the diffraction model in [FJJ16], the spatial part of the outgoing field when
a sample with electron density ϕ is illuminated with time-harmonic radiation is essentially
(up to a decay factor and constants) given by the so-called radiation transform of ϕ
Z
−i ωc s0 ·x
R[E0 ]ρ(s0 ) =
P(s⊥
ϕ(x) dx,
(5.8)
0 )E0 (x)e
R3
where E0 is a complex vector field (the spatial part of the incoming electric field of
frequency ω > 0), c is speed of light, s0 ∈ S 2 is the outgoing direction, and P(s⊥
0 ) :=
T
3×3
(I − s0 s0 ) ∈ R
is an orthogonal projection. The model relies on the assumptions of a
weak incoming field, a high frequency, and observation in the far-field.
In classic X-ray crystallography, the incoming field is a plane wave E0 (x) = E(k) (x) :=
ik·x
ne , where k ∈ ωc S 2 , and n ∈ C3 , n · k = 0. The radiation transform integral essentially
5. APPLICATIONS OF THE ZAK TRANSFORM
35
reduces to a Fourier transform of ϕ in this case. Now, assume that ϕ is invariant w.r.t. a
discrete isometry group S ≤ E(3) to model the electron density of an objective structure.
b we can define the symmetry projection P σ E(k) of a
Moreover, assume that given σ ∈ S,
S
(k)
plane wave E in a meaningful way, where the Euclidean group acts on vector fields as
follows. Let g = (Q|c) for Q ∈ O(3) and c ∈ R3 , and E a complex vector field on R3 ,
then
ρg E(x) := QE(QT (x − c)), x ∈ R3 .
(5.9)
σ (k)
For example, this is possible for compact groups S. The resulting object PS E can be
considered as a field of elements of the tensor product Hσ ⊗ Hσ ⊗ R3 . By a formal calculation we can show that the radiation transform reduces to an integral over a fundamental
domain F of the action of S on R3 . For this purpose we assume that the Zak transform
Zρ E(k) (x0 , σ) := PSσ E(k) (x)σ(g), ρg (x) = x0 ∈ F , is well-defined and has the usual properties. Then, using the Weil formula and the equivariance of the Zak transform, writing
e(ℓ) (x) := eiℓ·x for ℓ ∈ R3 ,
Z
ω
σ (k)
⊥
PSσ E(k) (x)e−i c s0 ·x ϕ(x) dx
R[PS E ]ϕ(s0 ) = P(s0 )
3
ZR Z
3
−i ωc s0 ·ρ−1
g (x0 )
ρg ϕ(x0 ) dµS (g) dµF (x0 )
= P(s⊥
ZρR E(k) (ρ−1
0)
g (x0 ), σ)e
F S
Z Z
ω
⊥
= P(s0 )
Zρ E(k) (x0 , σ)σ(g)ρg (e(− c s0 ) · ϕ)(x0 ) dµS (g) dµF (x0 )
ZF S
ω
= P(s⊥
Zρ E(k) (x0 , σ)Zρ (e( c s0 ) · ϕ)(x0 , σ) dµF (x0 )
0)
F
Now, using the isometry property of the Zak transform, integration over Sb yields
Z
(k)
( ωc s0 )
(k) ( ωc s0 )
R[PSσ E(k) ]ϕ(s0 ) dµSb(σ) = P(s⊥
· ϕ)i = P(s⊥
· ϕi
0 )hZρ E , Zρ (e
0 )hE , e
Sb
= P(s⊥
b ωc s0 − k),
0 )nϕ(
where the scalar products are defined component-wise. This is again essentially a Fourier
coefficient of ϕ. In particular, if one could measure the radiation transform w.r.t. this
b one could reconstruct the density ϕ. Howsymmetry-adapted radiation for every σ ∈ S,
ever, in diffraction experiments one can only measure the intensity of the outgoing radiation that is essentially given by the norm kR[PSσ E(k) ]ϕ(s0 )k22 of the radiation transform.
This is similar to the phase problem, but it’s a combined phase-and-orientation problem
for a complex vector.
APPENDIX A
Supplementary results
1. A sufficient condition for strong properness
In the following we give a sufficient condition for strong properness of a proper action
of an lcH group on a paracompact lcH space, using the theory of uniform spaces.
The definition of uniform spaces goes back to André Weil (see [Wei37]) and is a
general framework that allows to define uniform notions, like uniform continuity and
completeness, by introducing a concept of ‘nearness’. Many textbooks are available that
treat different aspects of the theory, e.g. the chapters in [Bou89,Kel55,Eng75,Pag78] and
the textbooks [Isb64, RD81, Jam90, How95, Jam99, Pac12].
Consider a set X and two relations R, S ⊆ X × X. Then inversion and composition
of relations are defined as follows:
R−1 := {(y, x) | (x, y) ∈ R}, and R◦S := {(x, z) | (x, y) ∈ S, (y, z) ∈ R for some y ∈ X}.
The identity relation (or diagonal) is denoted by idX := {(x, x) | x ∈ X}.
Definition A.1 (Uniform space). Let X be a set and U a non-empty family of sets
E ⊆ X × X. The pair (X, U) is called a uniform space, if for every E ∈ U
(i) idX ⊆ E,
(ii) E −1 ∈ U,
(iii) there is a F ∈ U, s.t. F ◦ F ⊆ E,
(iv) if F ∈ U, then E ∩ F ∈ U,
(v) if E ⊆ F ⊆ X × X, then F ∈ U.
The set U is called uniformity, the elements of U are called entourages. A subset B of U,
s.t. every E ∈ U contains an element B ∈ B is called a base of the uniformity.
In other words, a uniformity is a neighborhood filter of the diagonal idX that is compatible with inversion and composition. It can be shown that a family B of subsets of
X × X is a base for some uniformity U, if and only if it satisfies (i)-(iii), and the intersection of two members contains a member (see [Kel55, Thm. 6.2]). In this case we write
U = U(B) and call U the uniformity generated by B. A uniform structure U on X defines
a unique topology T (U) on X, s.t. the neighborhood filter at x ∈ X is given by the sets
Ex := {y ∈ X | (x, y) ∈ E},
E∈U
(see [Bou89, Prop. I, p. 171]). We say that a uniformity U on a topological space (X, T )
is compatible with the topology, if T (U) = T . We also write EA := {y ∈ X | (a, y) ∈
A for some a ∈ A} for A ⊆ X.
LcH groups are illustrative examples of uniform spaces. There are two different natural
uniform structures that are compatible with the topology of an lcH group G, namely the
37
38
A. SUPPLEMENTARY RESULTS
left and right uniformities that are generated by the entourages
E L (U) := {(g, g ′) ∈ G × G | g ′ ∈ gU},
and E R (U) := {(g, g ′) ∈ G × G | g ′ ∈ Ug},
respectively, where U runs through all neighborhoods of the neutral element e ∈ G.
These are not the only uniformities compatible with the topology. In [RD81], two further
uniformities that are symmetric w.r.t. left and right multiplication are studied in detail
– the upper and lower uniformities, that are generated by
E L∨R (U) := {(g, g ′) ∈ G × G | g ′ ∈ gU ∩ Ug},
E L∧R (U) := E L (U) ◦ E R (U),
respecively, where U again runs through all neighborhoods of e. These four bases of
uniformities on G satisfy
(E L (U))g = gU, (E R (U))g = Ug, (E L∨R (U))g = gU ∩Ug, (E L∧R (U))g = UgU, g ∈ G.
Paracompactnes of a locally compact space is equivalent to the existence of a special
uniformity, enabling us to transport paracompactness to the quotient by transporting this
special uniformity.
Proposition A.2 (Uniformly locally compact spaces). Let (X, T ) be a topological space. Then the following statements are equivalent:
(i) X is a uniformly locally compact (ulc) space. I. e. there is a uniformity U on
X that is compatible with T and contains an entourage E c , s.t. the sets Exc are
compact for all x ∈ X.
(ii) The space (X, T ) is locally compact and paracompact.
(iii) The space (X, T ) is the topological sum of a family (Xj )j∈J of σ-compact spaces.
Proof. (ii) ⇔(iii) is a classic result of Bourbaki [Bou89, Thm. 5, p. 96].
(i)⇔(ii) is sketched in [Kel55, Ex. 6.T], and carried out in detail in [Pag71] (via the
path (i)⇒(iii)⇒(ii)⇒(i)).
The definition (i) of ulc spaces is a natural compatibility condition on the locally
compact topology and the uniform structure. Ulc spaces are fairly general, as e.g. all
locally compact metrizable spaces, all lcH groups, all quotients of an lcH group by a
closed subgroup, and also all (semi-)hypergroups (see [Las] or [BH95]) are ulc spaces.
In summary, to this point, we assumed paracompactness to have a uniform structure
compatible with local compactness and properness to transport the topological structure
to the space of orbits. We need one further property to transport the uniform structure.
Consider an equivalence relation R on X and the quotient map πR . As shown in
[RD81, Jam90], the images of the entourages under πR form a base of a uniformity on the
quotient X/R, when the following property, called weak compatibility, holds. For every
entourage E of X, there is an entourage F of X, s.t. F ◦ R ◦ F ⊆ R ◦ E ◦ R. For a group
action ρ and the corresponding equivalence relation, this property can be formulated as
follows.
Definition A.3 (Weakly compatible action). Let (X, U) be a uniform space. A
continuous action ρ is said to be weakly compatible with U, if for every entourage E ∈ U
there is an entourage F ∈ U, s.t.
(x, ρg (y)), (y, z) ∈ F
⇒
there are g1 , g2 ∈ G, s.t. (ρg1 (x), ρg2 (z)) ∈ E.
(A.1)
1. A SUFFICIENT CONDITION FOR STRONG PROPERNESS
39
A stronger condition, named compatibility [Jam90, Def. 6.2], is that for every E ∈ U
there is an F ∈ U, s.t. ρG (Fx ) ⊆ EρG (x) for all x ∈ X. Since the topology is generated by
U, uniform equicontinuity of the family (ρg )g∈G is a sufficient condition for compatibility.
There are compatible actions that are not uniformly equicontinuous as we will see below
(see [RD81] or [Jam90] for further considerations concerning compatibility).
Weak compatibility of a proper action ρ of a lcH group and the uniformity on a ulc
space X implies paracompactness of the orbit space:
Lemma A.4 (Strong properness of weakly compatible proper actions). Let
G be an lcH group acting properly on a ulc space X. When the action is weakly compatible
with the uniformity on X, then the orbit space ρ\X with the quotient topology is a ulc
space with the quotient uniform structure
Uρ\X := U(πρ × πρ (UX )),
(A.2)
where πρ × πρ : E 7→ {(πρ (x), πρ (y)) | (x, y) ∈ E}. In particular, ρ\X is paracompact, i.e.
ρ is strongly proper.
Proof. By the definition of weak compatibility , the sets (πρ × πρ )(E), E ∈ U are a
base of a uniformity on ρ\X (see [RD81, Jam90]), which shows that Uρ\X is a uniformity.
As πρ is open, the topology generated by Uρ\X is the quotient topology (see [Jam90], p.
41).
Since X is ulc, there is an entourage E c ∈ UX with Exc compact for every x ∈ X
by Proposition A.2 (i). Now, πρ × πρ (E c ) ∈ Uρ\X is an entourage in ρ\X with (πρ ×
πρ (E c ))πρ (x) = πρ (Exc ) compact for every πρ (x) ∈ ρ\X by continuity of πρ , showing that
ρ\X is a ulc space. Paracompactness is immediate from Proposition A.2 (ii).
When the action is not only weakly compatible, but compatible, then the canonical
map is uniformly continuous and uniformly open as shown in [Jam90, Prop. 2.15].
We clearify the different assumptions, by considering a few examples. First, the action
of Z on R by translation ρn (x) := x + n is compatible, and thus weakly compatible,
with the uniform structure associated to the Euclidean metric (generated by the sets
E ε = {(x, y) ∈ R × R | |x − y| < ε} for ǫ > 0). In contrast, the action ρen (x) := 2n · x
is not compatible with this uniform structure, since the action makes the sets ρen (Fx )
arbitrarily large, while the sets Eρg (x) are just translates of Ex . Still, the action ρe is
weakly compatible. It suffices to show (A.1) for a base of the uniformity, so consider
E = E ε vor some ε > 0. Let (x, ρen (y)), (y, z) ∈ F = E δ for x, y, z ∈ R, n ∈ Z, and δ < 2ε ,
i.e. |x − 2n y|, |y − z| < δ. For n1 , n2 ∈ Z, we find that
|2n1 x − 2n2 z| = 2n1 |x − 2n2 −n1 y| ≤ 2n1 (|x − 2n y| + |2n y − 2n2 −n1 z|).
Choosing n1 = n2 − n yields |2n1 x − 2n2 z| < 2n2 (2−n + 1)δ, s.t. with n2 = 0 for n ≥ 0
and n2 = n for n < 0, the choice δ < 2ε is sufficient for (e
ρn1 (x), ρn2 (z)) ∈ E ε . This
shows that compatibility is too strong to even include simple scaling actions, while weak
compatibility is general enough. Note, that the action ρe is not Cartan as G{0} = Z is
not compact. The quotient ρe\X is not Hausdorff, as the graph of the action is not closed
(see [Bou89], I, 8.3, Prop. 8). However, the space R× := R \ {0} is a proper ulc G-space
with this action.
40
A. SUPPLEMENTARY RESULTS
There are several important actions for the case X = G an lcH group. Let H be a
closed subgroup of G acting continuously on G by right translation R : H × G → G,
Rh (g) := gh−1, h ∈ H, g ∈ G. G is a ulc space with respect to the left and right uniform
structures UL and UR . It is easily seen that R is compatible with the right uniform
structure. Consequently, G is a ulc H-space with respect to the right uniform structure,
and, using Lemma A.4, the orbit space R\G := {πR (g) | g ∈ G} is a ulc space. Since
R\G is identical to the quotient space G/H := {gH | g ∈ G}, it follows that G/H is a
ulc space. In particular, G/H is paracompact w.r.t. the quotient topology. Note that the
R
uniform structure on G/H is the right quotient uniform structure UG/H
that is generated
by the entourages
R
EG/H
(U) := {(gH, g ′H) ∈ G/H × G/H | g ′ ∈ Ug}.
The action by left translation Lh (g) := hg might not be weakly compatible with the
right uniform structure. This motivates the following observations. The left and right
(and upper and lower) uniformities of a lcH group G agree, if and only if for every
neighborhood U ⊆ G of the neutral element e ∈ G there is a neighborhood V of e, s.t.
gV g −1 ⊆ U for all g ∈ G. Groups with this property are called SIN groups (meaning
groups with Small Invariant Neighborhoods), since it is equivalent to the existence of a
conjugation invariant local basis at e (as shown in [Itz76]). Examples are all abelian, all
compact, and all discrete groups. Every SIN group is unimodular, while the converse is
generally not true (e.g. SL(n, R))1. The action of H by right translations is compatible
with the left uniformity if the SIN property is satisfied for all h ∈ H. We intoduce the
following notions.
Definition A.5 ((Weakly) H-SIN group). Let G be a lcH group and H a closed
subgroup of G. G is called H-SIN (or group with small H-invariant neighborhoods), if for
every neighborhood U of the neutral element e ∈ G, there is a neighborhood V of e, s.t.
hV h−1 ⊆ U
for all h ∈ H.
(A.3)
The group G is called weakly H-SIN, if for every neighborhood U of the neutral element
e ∈ G, there is a neighborhood V of e, s.t. for every h ∈ H there is a h′ ∈ H, s.t.
h′ V (h′ )−1 , s′ s−1 V s(s′ )−1 ⊆ U.
(A.4)
When G is weakly H-SIN, then the actions of H by left and right translation are
weakly compatible with the left and right (and upper and lower) uniformities, as can be
seen by considering eq. (A.3) for these special cases. We will generalize this concept to
the action of general automorphism groups in Section 2 in the Appendix, Definition A.9.
When H is compact, then G is always H-SIN as follows from [HM13, Prop. 1.11].
Note also that when G is H-SIN, then H is SIN.
Now, let K be a further closed subgroup of G that acts continuously on G/H by left
translation L : K × G/H → G/H, Lk (πR (g)) := πR (kg) for k ∈ K and g ∈ G. When G is
weakly K-SIN, this action is weakly compatible with the right quotient uniform structure
on G/H, and when G is weakly H-SIN, this action is weakly compatible with the left
1The
properties abelianity, being SIN, and unimodularity can be viewed as different levels of conjugation
invariance, namely at the levels of singletons, of local bases of the topology, and of the Haar measure.
2. INVARIANT MEASURES ON DOUBLE COSET SPACES
41
quotient uniform structure on G/H. Hence, in either case, the space of orbits L\(G/H)
that, as a topological space, is identical to the double coset space
K\G/H := {KgH | g ∈ G}
is a ulc space with the quotient topology. This example extends the list of ulc spaces
given after Proposition A.2.
To address the general problem of paracompactness of quotients, one could ask for the
existence of a uniformity on X that is compatible with its topology, while at the same
time being weakly compatible with the action. However, this idea lies beyond the scope
of this paper.
2. Invariant measures on double coset spaces
In this section we study double coset spaces in further detail. In particular, we investigate invariance properties of the measures that satisfy the Weil formula (2.5), generalizing
work of T. S. Liu [Liu65].
Let G be a lcH group and K, H closed subgroups of G and consider the normalizers
NG (H) := {g ∈ G | gH = Hg} and NG (K) := {g ∈ G | gK = Kg}
of H and K in G. The product NG (K) × NG (H) acts continuously in a well-defined way
on the double coset space K\G/H via left and right translation T : (NG (K) × NG (H)) ×
K\G/H → K\G/H, T(n,m) (KgH) := Kngm−1 H for n ∈ NG (K), m ∈ NG (H), and
KgH ∈ K\G/H. The normal subgroup K × H of NG (K) × NG (H) acts trivially, so
we can factor it out to obtain an action of the lcH group (K\NG (K)) × (NG (H)/H) by
restricting T . We simplify notation by introducing the names MK := K\NG (K) and
M H := NG (H)/H. In the case that G is weakly NG (H)- or weakly NG (K)-SIN, the orbit
space
MK \(K\G/H)/M H := {πT (KgH) | KgH ∈ K\G/H}
is a ulc space by Lemma A.4. The action of MK × M H is natural when considering
invariance properties of measures on double coset spaces. Note that a normalizer NG (H)
carries a natural semidirect product structure NG (H) = H⋊ϕ M H with ϕ : M H → Aut(H)
given by conjugation ϕ[m](h) := mhm−1 ∈ H. This shows that MK \(K\G/H)/M H is
uniformly homeomorphic to the double coset space NG (K)\G/NG (H).
Now, when K × H acts strongly proper via T : (K × H) × G → G, T(k,h) (g) := kgh−1,
we are in the setting of Theorem 2.4, i.e. there is a function q : G → (0, ∞) and a Radon
measure µK\G/H , s.t. the Weil formula (2.5) holds.
As already noted, the group NG (K)×NG (H) acts in a well-defined manner on K\G/H
via T : ((n, m), KgH) 7→ Kngm−1 H. We denote both this action and the action on G
by T . To understand the behaviour of the measure µK\G/H w.r.t. this action, we first
need to understand the interplay of the action and the orbital mean operator. Recall that
MK := K\NG (K) and M H := NG (H)/H. We start with the following lemma.
Lemma A.6 (Conjugation and Haar measure). Let G be a lcH group, and H
a closed subgroup. Consider the action of the normalizer NG (H) on H by conjugation,
42
A. SUPPLEMENTARY RESULTS
conj : NG (H) × H → H, conjm (h) := mhm−1 . Then the Haar measure µH of H satisfies
dµH (mhm−1 ) =
∆M H (mH)
∆MH (Hm)
dµH (h) =
dµH (h),
∆NG (H) (m)
∆NG (H) (m)
m ∈ NG (H), h ∈ H.
(A.1)
Proof. For every m ∈ NG (H), conjm is an automorphism of H. As shown in [Bou04],
VII, 2.7, Cor. to Prop. 11, the modulus mod(conjm ) is of the desired form in the case
that H acts from the right, yielding the first equality.
We show the second equality directly, using the Mackey-Bruhat version (2.9) of Theorem 2.4 for the quotient H\NG (H). Since H is a normal subgroup of NG (H), the Weil
formula is satisfied with q = ∆−1
NG (H) (see (2.8)) and µMH the Haar measure of MH . So,
for f ∈ Cc (NG (H)) and m ∈ NG (H),
Z
Z
Z
−1
∆NG (H) (m )
f dµNG (H) =
conjm f dµNG (H) =
AL,q conjm f dµMH
NG (H)
NG (H)
MH
Z Z
=
f ((m−1 h−1 m)(m−1 nm))∆NG (H) (h−1 n) dµH (h) dµMH (Hn)
ZMH ZH
=
f (h−1 n)∆NG (H) ((mh−1 m−1 )(mnm−1 )) dµm (h) dµMH (Hm(Hn)Hm−1 )
MH H
Z
−1
f dµNG (H) ,
= c(m)∆MH (Hm )
NG (H)
where we used that ∆NG (H) is conjugation invariant. By continuity, c(m) =
all m ∈ NG (H), yielding the second equation of (A.1).
∆M H (Hm)
∆NG (H) (m)
for
Lemma A.6 yields the following result on the interplay of orbital mean and the action
T.
Corollary A.7 (Commutation relation for orbital mean operator and action
of NG (K) × NG (H)). Let G be a lcH goup, K, H two closed subgroups of G. Let T denote
both the actions of NG (K)×NG (H) on G and on K\G/H, and assume that the restriction
of T to K × H acts properly. Then
AT ◦ T(n,m) = cT(n,m) · T(n,m) ◦ AT ,
where cT(n,m) =
n ∈ NG (K), m ∈ NG (H),
(A.2)
∆MK (Kn)∆M H (mH)
.
∆NG (K) (n)∆NG (H) (m)
Proof. Using Lemma A.6, for n ∈ NG (K), m ∈ NG (H), g ∈ G, and f ∈ Cc (G),
Z
AT ◦ T(n,m) f (πT (g)) =
T(n,m) f (k −1 gh) dµK×H (k, h)
K×H
Z Z
=
f ((n−1 k −1 n)(n−1 gm)(m−1 hm) dµH (h) dµK (k) = cT(n,m) · T(n,m) ◦ AT f (πT (g)).
K
H
Now we can show the following result on the invariance properties of measures on
double coset spaces.
2. INVARIANT MEASURES ON DOUBLE COSET SPACES
43
Theorem A.8 (Invariant measures on double coset spaces). Let G be a lcH
goup, K, H two closed subgroups of G. Let T denote both the actions of NG (K) × NG (H)
on G and on K\G/H. Assume that the restriction of T to K × H acts strongly proper.
(i) There always is a T -quasi-invariant measure µK\G/H on K\G/H with
λ(n,m) (KgH) =
cT(n,m)−1
T(n,m)−1 q(g)
∆G (m)
q(g)
·
(A.3)
for n ∈ NG (K),m ∈ NG (H), g ∈ G, with cT(n,m) defined as in (A.2) and q as in
Theorem 2.4.
(ii) The measure in (i) is relatively T -invariant, if and only if q is relatively T invariant (w.r.t. the action of NG (K) × NG (H)).
(iii) The measure in (i) is T -invariant, if and only if q satisfies
q(ngm−1) = cT(n,m) · ∆G (m) · q(g)
(A.4)
for n ∈ NG (K),m ∈ NG (H), and g ∈ G.
Proof. (i) First, we note that the Haar measure µG on G is relatively T -invariant
with dµG (ngm−1 ) = ∆G (m−1 ) dµG (g) for n ∈ NG (K), m ∈ NG (H), and g ∈ G. So,
we can apply Theorem 2.4 for the action of K × H. In particular, there is a function
q : G → (0, ∞) and a measure µK\G/H , s.t. (2.5) is satisfied. Using the commutation
realtion (A.2) and the Weil formula, we find that for f ∈ Cc (G), n ∈ NG (K), and
m ∈ NG (H),
Z
Z
T
AT ◦ T(n,m) f dµK\G/H
T(n,m) ◦ AT f dµK\G/H = c(n,m)−1
K\G/H
K\G/H
Z
Z
T
T
= c(n,m)−1
T(n,m) f · q dµG = c(n,m)−1
f · T(n,m)−1 q · ∆G (m−1 ) dµG
G
G
cT(n,m)−1 Z
cT(n,m)−1 Z
T(n,m)−1 q
T(n,m)−1 q
f·
· q dµG =
AT f ·
dµK\G/H .
=
∆G (m) G
q
∆G (m) K\G/H
q
T
−1 q
By the functional equation of q, (2.4), the function g 7→ (n,m)
is invariant w.r.t the
q
action of K × H, so it can be pulled out of the orbital mean operator, yielding
!
Z
Z
cT(n,m)−1 T(n,m)−1 q
dµK\G/H ,
T(n,m) ◦ AT f dµK\G/H =
AT f ·
·
∆G (m)
q
K\G/H
K\G/H
proving T -quasi-invariance of µK\G/H .
(ii) and (iii) are simply reformulations of relative T -invariance, and T -invariance, respectively.
This theorem generalizes results obtained in [Liu65] for the case of a compact group
K with the normalizer NG (K) acting from the left.
As an example, we will study the action of an affine group S ≤ Aff(G) = G ⋊ Aut(G)
on an lcH group G.
44
A. SUPPLEMENTARY RESULTS
Let G be an lcH group. First, we consider the case that an lcH automorphism group
S ≤ Aut(G) acts continuously on G via ρ : S × G → G. Since the stabilizers for automorphism groups are trivial except for the neutral element e where Se = S, compactness
of S is a necessary condition for properness of the action. We will investigate conditions
for weak compatibility (see (A.1)) of ρ with the left and right uniformities on G.
In the case of the left uniformity, given an open neighborhood U of e ∈ G, we need
a neighborhood V of e, s.t. when ρs (y) ∈ xV and z ∈ yV , then there are s1 , s2 ∈ S, s.t.
ρs2 (z) ∈ ρs1 (x)U. Using the fact that the functions ρs , s ∈ S, are automorphisms, the
assumptions yield that
ρs2 (z) ∈ ρs2 s−1 (x) · ρs2 s−1 (V ) · ρs2 (V ),
s.t. when setting s1 := s2 s−1 , we need for every s ∈ S an s2 ∈ S, s.t. ρs2 s−1 (V ) · ρs2 (V ) ⊆
U. Analogously, when considering the right uniformity, one needs ρs2 (V ) · ρs2 s−1 (V ) ⊆ U.
Similar expressions are obtained when considering the upper and lower uniformities. This
observation motivates the following generalization of Definition A.5.
Definition A.9 (Weakly ρ-SIN group). Let G be a lcH group, and S ≤ Aut(G)
lcH, acting continuously via ρ : S × G → G. G is called ρ-SIN, if for every neighborhood
U of the neutral element e ∈ G, there is a neighborhood V of e, s.t.
ρs (V ) ⊆ U
for all s ∈ S.
(A.5)
The group G is called weakly ρ-SIN, if for every neighborhood U of the neutral element
e ∈ G, there is a neighborhood V of e, s.t. for every s ∈ S there is a s′ ∈ S, s.t.
ρs′ (V ), ρs′ s−1 (V ) ⊆ U.
(A.6)
When S ≤ Inn(G) is a group of inner automorphisms, the concepts reduce to the
ones in Definition A.5. Again, when S is compact, then G is always ρ-SIN as follows
from [HM13, Prop. 1.11].
As a direct corollary of the above considerations and Lemma A.4, we get
Proposition A.10 (Weakly ρ-SIN groups and weak compatibility). Let G be
a lcH group, and S ≤ Aut(G) a compact automorphism group, acting continuously via
ρ : S × G → G. Then the space ρ\G is a proper ulc S-space, and therefore paracompact.
Proof. Given a neighborhood U of e, choose a neighborhood V ′ of e with V ′ · V ′ ⊆ U.
As G is weakly ρ-SIN, there is a neighborhood V of e, s.t. for every s ∈ S there is a
s′ ∈ S, s.t. ρs′ (V ), ρs′ s−1 (V ) ⊆ V ′ . In particular,
ρs2 s−1 (V ) · ρs2 (V ), ρs2 (V ) · ρs2 s−1 (V ) ⊆ V ′ · V ′ ⊆ U,
showing weak compatibility with both the left and right uniformities, as seen above (weak
compatibility with the upper and lower uniformities follows analogously).
That ρ\G is a proper ulc S-space is an immediate consequence of Lemma A.4.
As an example for a weakly ρ-SIN space, reconsider the action ρen (x) := 2n · x, n ∈ Z,
x ∈ R× (see Section 2.1). This action is obviously by automorphisms, so by Proposition
′
A.10, we need for every ε > 0 a δ > 0, s.t. for all n ∈ Z there is a n′ ∈ Z, s.t. 2n · Bδ (0),
′
′
′
2n −n · Bδ (0) ⊆ Bε (0). In other words, we need 2n δ < ε and 2n −n δ < ε. For δ < 2ε , this
2. INVARIANT MEASURES ON DOUBLE COSET SPACES
45
is satisfied for n′ = 0 whenever n ≥ 0, and n′ = n for n < 0, showing that R is weakly
ρe-SIN and ρe\R× is paracompact.
We can write the space of orbits of an automorphic action as a double coset space via
the following construction. Let G be a lcH group, and ρ a continuous action of an lcH
automorphism group S ≤ Aut(G). The two groups can be combined to form a semidirect
product G ⋊ρ S with group operation
(g1 , s1 ) · (g2 , s2 ) := (g1 · ρs1 (g2 ), s1 · s2 ), g1 , g2 ∈ G, s1 , s2 ∈ S.
(A.7)
It is easily seen that G is homeomorphic to the quotient (G ⋊ρ S)/({eG } × S) via g 7→
(g, eS ) · ({eG } × S), where eG and eS are the respective neutral elements. Moreover, the
action of {eG } × S on this quotient from the left becomes the action of S on G via this
identification, since (eG , s) · (g, eS ) = (ρs (g), s) ∈ (ρs (g), eS ) · ({eG } × S). In other words,
the double coset space ({eG } × S)\(G ⋊ρ S)/({eG } × S) is homeomorphic to the space
ρ\G of orbits of the action ρ. From now on we will write S\(G ⋊ S)/S to abbreviate this
double coset space.
More generally, consider the affine group Aff(G) := G ⋊ Aut(G). Every s ∈ Aff(G)
can be split into its translational part gs and its automorphic part ϕs , i.e. s = (gs , ϕs ).
As action of Aut(G) on G consider the left action L(gs ,ϕs) g := gs ϕs (g). We can write
the space of orbits of the action of a lcH affine group S ≤ Aff(G) as a double coset
space S\Aff(G)/Aut(G). Paracompactness of the quotient can, for example, be shown
by finding a lcH group A ≤ Aut(G), s.t. S ≤ G ⋊ A and G ⋊ A is weakly S-SIN. In
particular, we can then apply the Weil formula (Theorem 2.4) and derive the invariance
property of the corresponding measure on the space of orbits via Theorem A.8.
First, we study the structure of affine groups. Consider an affine group S ≤ Aut(G)
and the projection π2 : (gs , ϕs ) 7→ ϕs to the automorphic component of an element of S.
The image of π2 is called the isogonal point group IP(S) of S:
IP(S) = {ϕs ∈ Aut(G) | (gs, ϕs ) ∈ S for some gs ∈ G}.
(A.8)
The set IP(S) is a subgroup of Aut(S), but not necessarily a subgroup of S. Moreover,
π2 is a group homomorphism by eq. (A.7), s.t. the kernel T(S),
T(S) = {gs ∈ G | (gs, idG ) ∈ S},
(A.9)
called the translational subgroup of S, is a normal subgroup of S, and IP(S) ∼
= S/T(S) by
the first isomorphism theorem. In other words, the isogonal point group parametrizes a
translational fundamental domain of the group S 2. Consequently, applying the classic Weil
formula (2.1), and denoting by modG (ϕ) the positive number that satisfies dµG (ϕ(g)) =
modG (ϕ) dµG (g) for ϕ ∈ Aut(G), we obtain
Z
Z
Z
f (gs · t, ϕs )
f dµS =
dµT(S) (t) dµIP(S) (ϕs ), f ∈ Cc (S),
(A.10)
S
IP(S) T(S) modT(S) (ϕs )
where gs ∈ G, s.t. (gs , ϕs ) ∈ S, choosing the representatives (gs , ϕs ) of S/T(S) and using
that (gs , ϕs ) · (t, idG ) = (gs · ϕs (t), ϕs ). Note that the topology on IP(S) is the quotient
topology of S/T(S) and not the relative topology w.r.t. Aut(G).
This yields a particularly nice Weil formula for proper affine actions.
2For
example, in structural biology, the translational fundamental domain of a filamentous virus that has
helical symmetry (e.g. tobacco mosaic virus) is called a helical repeat of the virus.
46
A. SUPPLEMENTARY RESULTS
Corollary A.11 (Weil formula for affine actions). Let G be an lcH group, and
ρ : S × G → G a strongly proper continuous affine action of an lcH group S ≤ Aff(G).
Then there is a continuous function q : G → (0, ∞), that satisfies
ρs q(g) = ∆S (s)mod(ϕs )q(g),
s = (gs , ϕs ) ∈ S, g ∈ G,
and a unique Radon measure µρ\G on ρ\G, s.t. for f ∈ Cc (G),
Z
Z Z
Z
ρ(gs ·t,ϕs ) f
f · q dµG =
dµT(S) (t) dµIP(S) (ϕs ) dµρ\G ,
G
ρ\G IP(S) T(S) modT(S) (ϕs )
(A.11)
(A.12)
where gs ∈ G, s.t. (gs , ϕs ) ∈ S.
Moreover, the measure µρ\G is quasi-invariant w.r.t. the action of NG⋊IP(S) (S) ×
NG⋊IP(S) (IP(S)) as given in Theorem A.8.
Proof. Eq. (A.11) is the special case of eq. (2.4) with the Haar measure µG , which
is relatively Aut(G)-invariant.
Eq. (A.12) is a direct consequence of Theorem 2.4 and eq. (A.10).
For the invariance properties of µρ\G , the quotient space is identified with the double
coset space S\(G⋊IP(S))/IP(S), as discussed above. The properties follow with Theorem
A.8, using that S ≤ G ⋊ IP(S), and that IP(S) is a lcH group since S is lcH.
A nice example for the affine Weil formula (A.12) is the case of a closed unimodular
subgroup S of the Euclidean group E(3) := R3 ⋊ O(3) of isometries of R3 acting on R3 .
Since modR3 (ϕ) = 1 for ϕ ∈ O(3), the constant function q ≡ 1 satisfies eq. (A.11), s.t.
there is a unique Radon measure on ρ\R3 with
Z
Z
Z
Z
f (x) dx =
ρ(cQ +c,Q)f (x) dµT(S) (c) dµIP(S) (Q) dµρ\R3 (πρ (x))
R3
ρ\R3
IP(S)
T(S)
for f ∈ L1 (R3 ), where cQ ∈ R3 , s.t. (cQ , Q) ∈ S.
Acknowledgement. The author thanks Gero Friesecke, Rupert Lasser, and Arash
Ghaani Farashahi for helpful discussions on various topics associated to this work, and
the organizers of the HIM Trimester Program “Mathematics of Signal Processing” for
inviting me and for their hospitality.
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