Fourier analysis on the affine group,
quantization and noncompact Connes geometries

Victor Gayral,1 José M. Gracia-Bondı́a2,3 and Joseph C. Várilly4

1 Laboratoire de Mathématique, Université de Reims Champagne–Ardenne, 51687 Reims, France

2 Departamento de Fı́sica Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain

3 Escuela de Fı́sica, Universidad de Costa Rica, 11501 San José, Costa Rica

4 Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica

J. Noncommut. Geom. 2 (2008), 215–261

Abstract

We find the Stratonovich–Weyl quantizer for the nonunimodular affine group of the line. A
noncommutative product of functions on the half-plane, underlying a noncompact spectral triple
in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the
time-frequency distributions of signal processing. The same construction leads to scalar Fourier
transformations on the affine group, simplifying and extending the Fourier transformation
proposed by Kirillov.

Dedicated to Orietta Protti, who started it all

Contents

1 Introduction 2

2 The orbit method for the group of affine transformations 4

3 An unusual special function 10

4 The Stratonovich–Weyl quantizer for the ‘𝑎𝑥 + 𝑏’ group 12

5 The Moyal twisted product on the half-plane 17

6 Right-covariant quantization 23

7 Fourier–Moyal transformations on the ‘𝑎𝑥 + 𝑏’ group 24

8 Discussion 30

1



9 Spectral triples on the half-plane 33

10 Outlook 36

1 Introduction
The theory of noncompact spin geometries in the sense of Connes [20, 21] or noncompact spectral
triples, broached in [43], was developed in references [38, 40, 67]. The coordinate algebras treated
in [67] have locality properties analogous to those of the commutative case, whereas [38, 40] deal
with truly noncommutative contexts that are essentially flat, respectively the Moyal 2𝑛-planes and
some of their generalizations.

As such, the theory remains underdeveloped. This is partly for want of suitable noncommutative
noncompact spectral triples. Among the myriads of deformations or “star products”, twisted product
algebras with the crucial traciality property distinguish themselves in that the classical integral yields
a faithful tracial state. This is what made the original (Groenewold–)Moyal product [9, 47, 61] so
popular in quantum field theory [32, 72, 78]. Not least, it ensures relevant properties in cyclic
cohomology [22].

Let 𝑋 be a phase space, 𝜇 a convenient measure on it (often the Liouville measure) and H

the Hilbert space associated to (𝑋, 𝜇).1 Denote by 𝛿𝜇 (𝑥, 𝑥′) the reproducing kernel for 𝜇. A
Stratonovich–Weyl quantizer or tracial quantizer for (𝑋, 𝜇,H) is an operator-valued distribution Ω

on 𝑋 , with values in the space of selfadjoint operators on H, spanning a weakly dense subset of
B(H), and verifying

TrΩ(𝑥) = 1, Tr
[
Ω(𝑥)Ω(𝑥′)

]
= 𝛿𝜇 (𝑥, 𝑥′).

Quantizers in this sense, if they exist, are essentially unique. Ownership of a quantizer solves in
principle all quantization problems: quantization of a (sufficiently regular) function or “symbol” 𝑎
on 𝑋 is effected by

𝑎 ↦→
∫
𝑋

𝑎(𝑥)Ω(𝑥) 𝑑𝜇(𝑥) =: 𝑄(𝑎),

and dequantization of an operator 𝐴 ∈ 𝐵(H) is achieved by

𝐴 ↦→ Tr[Ω(·)𝐴] =: 𝑊𝐴 (·). (1.1)

Indeed Ω can just as well be called a dequantizer. It follows that 1H ↦→ 1 by dequantization, and
also

Tr𝑄(𝑎) =
∫
𝑋

𝑎(𝑥) 𝑑𝜇(𝑥).

Moreover, since the set Ω(𝑋) is total, it is clear that

𝑊𝑄(𝑎) (𝑥) = Tr
[(∫

𝑋

𝑎(𝑥′)Ω(𝑥′) 𝑑𝜇(𝑥′)
)
Ω(𝑥)

]
= 𝑎(𝑥),

so 𝑄 and 𝑊 are inverse to one another. In particular, 𝑊𝑄(1) = 1 says that 1 ↦→ 1H by quantization.
Finally, the following relation holds:

Tr[𝑄(𝑎)𝑄(𝑏)] =
∫
𝑋

𝑎(𝑥)𝑏(𝑥) 𝑑𝜇(𝑥). (1.2)

1This paragraph and the next are excerpted from the report by one of us (JMGB) to the Oberwolfach conference on
Dirac Operators and Noncommutative Geometry, in November 2006; see Oberwolfach Reports 3 (2006), pp. 3151–52.

2



This is the tracial property.
Most of the interesting cases occur in the context of group actions; that is to say, there is a

Lie group 𝐺 for which 𝑋 is a symplectic homogeneous 𝐺-space, with 𝜇 then being a 𝐺-relatively
invariant measure on 𝑋 , and 𝐺 acts by a (multiplier) unitary irreducible representation 𝑈 on H. A
quantizer for the data set (𝑋, 𝜇,H, 𝐺,𝑈) satisfies the previous defining equations and is endowed
with the covariance property:

𝑈 (𝑔)Ω(𝑥)𝑈†(𝑔) = Ω(𝑔 ⊲ 𝑥),
for all 𝑔 ∈ 𝐺, 𝑥 ∈ 𝑋 . Orbits of the coadjoint action of 𝐺 on its Lie algebra dual g∗ and symplectic
homogeneous manifolds are essentially the same thing [53, Sect. 1.4], and in this paper we think of
the action denoted by ⊲ above as an instance of the coadjoint action of 𝐺.

A covariant collection as above, but satisfying only∫
𝑋

Ω(𝑥) 𝑑𝜇(𝑥) = 1H and TrΩ(𝑥) = 1,

may be called a semitracial quantizer.

▶ Once in possession of the quantizer, one can in principle immediately construct a twisted product
that will be normalized (its identity being the constant function 1), hermitian (complex conjugation
being the involution), covariant under an appropriate group action, and tracial. We give the details
in the body of the paper.

In summary, the “Stratonovich–Weyl” label here refers neither to general deformations nor to
star products obtained (roughly speaking) by reduction [43], extension [39] or induction from the
original one; but to a restricted category, defined by a precise set of postulates, designed to capture
the main trait behind the success of Moyal’s formalism for Quantum Mechanics; namely, that
quantum and classical expected values should be computed by the same rule. The products thereby
obtained are non-formal and analytically controlled. The quest for quantizers in our sense is richly
rewarding. In this paper we show by example how the theory of covariant tracial quantizers meshes
with, and substantially complements, Kirillov’s method of orbits in representation theory. Its main
end products are the (scalar) Fourier–Moyal kernels on g∗ × 𝐺:

𝔼(𝑥, 𝑔) := Tr
[
Ω(𝑥)𝑈 (𝑔)

]
; 𝔼mod(𝑥, 𝑔) := Tr

[
Ω(𝑥)𝑈 (𝑔)

√
𝑑
]
,

with 𝑑 the formal dimension operator for 𝑈. There is a good case, that we have made before [31]
and we make here again, for these to be the central objects in harmonic analysis. For, as soon as
a tracial quantizer is available, the abstract Plancherel theorem of group Fourier transform theory
becomes a concrete one on the coadjoint orbits. Also, quantizers have an important applied side,
with their relations to wavelets in signal processing and to quantum optics.

Before giving the usual guide to the article, we briefly return to noncompact spectral triples.
It seems natural to look for noncommutative tracial algebras on the surface of constant negative
curvature, whether it be modelled by a hyperboloid in ℝ3, the unit disk, or the Poincaré upper
half-plane Π; we shall focus on the half-plane. Even so, the prescription that the noncommutative
coordinate algebras on Π carry the full SL(2,ℝ) symmetry may be too much to ask, because then
the first-order condition of [20, 21] for the standard Dirac operator on Π cannot be satisfied. This
will be shown in Section 9. One can of course inherit the full symmetry and use the alternative
Dirac operator; or perhaps “deform” suitably the latter.2

2We thank Pierre Bieliavsky for illuminating discussions of these aspects.

3



Meanwhile we concentrate on the smaller ‘𝑎𝑥+𝑏’-type group of symmetry of Π. As mentioned,
there is another compelling motivation for revisiting the Stratonovich–Weyl quantizers: the progress
of photonics [57] allows nowadays a quasi-measurement of the Wigner functions – see [7,71,80] as
theoretical harbingers. This is calling for investigation of systems with solvable group symmetry;
and before tackling them, it helps to visit the case of affine-group symmetry.

▶ In the next section, we remind the reader of the Kirillov method for constructing the unitary
irreducible representations (unirreps) of the ‘𝑎𝑥 + 𝑏’ group, in order to make the exposition self-
contained. We exhibit the corresponding characters and the Duflo–Moore operator for the nontrivial
representations. Section 3 is a modicum of real analysis, eventually needed for establishing the
properties of the affine-group quantizers.

Sections 4 to 8 deal with the main issue. As it happens, the literature on wavelets and time-
frequency distributions already contains the information needed to extract the quantizers on the
half-plane [10, 11]: an impressive example that concrete problem-oriented work can lead to far-
reaching conceptual results. A suitable modification of the classical Weyl quantization rule holds.
We give the quantizers explicitly and verify the key tracial property in Section 4. In the next
section, we find the (left-covariant) twisted product associated to the quantizer, required for spectral
triple theory, and investigate its symmetry properties. Section 6 deals with the right-covariant
counterparts for the quantizer and the twisted product. In Section 7, the foregoing illuminates
harmonic analysis: scalar Fourier–Moyal transformations are found for the ‘𝑎𝑥+𝑏’ group, allowing
us to recover the representation characters and to improve on Kirillov’s Fourier transformation. The
basic results of Fourier analysis, up to and including the Plancherel formula, are shown to hold in
our context, for this nonunimodular case. Section 8 connects and compares our formulation with
others, including Fronsdal’s★-representation program [36] and the approach of a remarkable series
of recent papers [2,3,54] by Ali and coworkers, also inspired by the literature on wavelet transforms.
We take the occasion to set the record straight on the matter of Stratonovich–Weyl quantizers.

Section 9 revisits the noncompact spectral triples over the half-plane that have motivated the
present work; the first-order property for the Dirac operator can now be established. Section 10
gives pointers for further rapprochement of Connes’, Kirillov’s and Moyal’s paradigms inter alia.

It remains to add that the standards of formality in this paper are about the usual ones in
mathematical physics; this saves considerable spacetime. All our arguments are in fact rigorous, as
will be shown in [41].

2 The orbit method for the group of affine transformations
2.1 The coadjoint orbits

The group Aff of orientation-preserving linear transformations of the real line, or affine group for
short, also known as the ‘𝑎𝑥 + 𝑏’ group, is a semidirect product ℝ×+ ⋊ ℝ, with the handy matrix
realization

Aff ≡
{ (
𝑎 𝑏

0 1

)
: 𝑎 > 0, 𝑏 ∈ ℝ

}
. (2.1)

Its Lie algebra is realized by the matrices

aff :=
{
𝑋 =

(
𝑢 𝑣

0 0

)
: (𝑢, 𝑣) ∈ ℝ2

}
, 𝑋 = 𝑢𝑋1 + 𝑣𝑋2,

4



with commutation relation [𝑋1, 𝑋2] = 𝑋2. The group is solvable, since its Lie algebra has the ideal
ℝ𝑋2 = [aff, aff] with aff/ℝ𝑋2 abelian. Recall that a group 𝐺 and its tangent Lie algebra g are
called exponential if the map exp: g→ 𝐺 is a surjective diffeomorphism. Note that

ad 𝑋 =

(
0 −𝑣
0 𝑢

)
, with eigenvalues 0, 𝑢.

By an old result of Dixmier [25], since these eigenvalues are not (nonzero) purely imaginary, the
group is exponential. Of course, this can be seen already from (2.1), since

𝑔(𝑢, 𝑣) := exp(𝑢𝑋1 + 𝑣𝑋2) =
(
𝑒𝑢 𝑣(𝑒𝑢 − 1)/𝑢
0 1

)
. (2.2)

Since tr(ad 𝑋) ≠ 0 in general, the group is not unimodular. Indeed, the right and left Haar
measures 𝑑𝑟𝑔 and 𝑑𝑙𝑔 on Aff are respectively given by

𝑑𝑟 (exp 𝑋) = det
(
𝑒ad 𝑋 − 1

ad 𝑋

)
𝑑𝑋 =

𝑑𝑢 𝑑𝑣

𝜆(−𝑢) =
𝑑𝑎 𝑑𝑏

𝑎
,

𝑑𝑙 (exp 𝑋) = det
(
1 − 𝑒− ad 𝑋

ad 𝑋

)
𝑑𝑋 =

𝑑𝑢 𝑑𝑣

𝜆(𝑢) =
𝑑𝑎 𝑑𝑏

𝑎2 , (2.3)

where 𝑑𝑋 = 𝑑𝑢 𝑑𝑣 and

𝜆(𝑡) :=
𝑡𝑒𝑡

𝑒𝑡 − 1
=

𝑡

1 − 𝑒−𝑡 =
𝑒𝑡/2

sinch(𝑡/2) = 𝑒
𝑡/2 Γ

(
1 + 𝑡

2𝜋𝑖
)
Γ
(
1 − 𝑡

2𝜋𝑖
)
, (2.4)

with sinch 𝑡 := (sinh 𝑡)/𝑡 (and sinch 0 := 1), a well-known nonvanishing even function (so named,
by analogy with the sinus cardinalis of sampling theory). Note that the right Haar measure on Aff
is the product of the Haar measures of ℝ×+ and ℝ; this is a general property for semidirect products.
Neither the left nor right Haar measure coincides with the measure induced on Aff by the Lebesgue
measure on aff. The last equalities on the right hand sides of (2.3) follow from (2.2). Therefore the
modular function Δ(𝑔) := 𝑑𝑙𝑔/𝑑𝑟𝑔 is given by 1/𝑎. We also note for future use the product of the
densities (normalized at 0) of the Haar measures with respect to the Lebesgue measure,

𝑗𝑙 (𝑋) 𝑗𝑟 (𝑋) :=
𝑑𝑙 (exp 𝑋)

𝑑𝑋

𝑑𝑟 (exp 𝑋)
𝑑𝑋

= sinch2(𝑢/2). (2.5)

The adjoint action of Aff on aff, and the contragredient coadjoint action on aff∗, are respectively
given by

Ad 𝑔(𝑋) := 𝑔𝑋𝑔−1 =

(
𝑢 𝑎𝑣 − 𝑏𝑢
0 0

)
; ⟨𝑔 ⊲ 𝐹, 𝑋⟩ := ⟨𝐹,Ad 𝑔−1(𝑋)⟩,

for 𝑋 ∈ aff, 𝐹 ∈ aff∗; we generally use the letter 𝐹 for points in coalgebras g∗. Also, aff∗ can be
realized by matrices

𝐹 = (𝑥, 𝑦) :=
{ (
𝑥 0
𝑦 0

)
: (𝑥, 𝑦) ∈ ℝ2

}
.

5



Thus ⟨𝐹, 𝑋⟩ = tr(𝐹𝑋) = 𝑢𝑥 + 𝑣𝑦, and the coadjoint action is given by

(𝑥, 𝑦) ↦→ 𝑔 ⊲ (𝑥, 𝑦) ≡ Coad 𝑔 (𝑥, 𝑦) =
(
𝑥 + 𝑏𝑦

𝑎
,
𝑦

𝑎

)
.

The orbits of this action are two open half-planes, plus an axis of fixed points. Indeed, if we choose
any point 𝐹 = (𝑥, 0), then the whole group leaves it invariant, whereas all the other points of aff∗
are found in the orbits of 𝐹 = (0, +1) and of 𝐹 = (0,−1). The isotropy group in the last two cases
is trivial, and both orbits, which we denote by O±, are diffeomorphic to the group itself. We think
of O± as Poincaré half-planes, respectively Π and −Π, adopting complex-variable notation when
convenient.

We can describe these “solvmanifolds” by group parameters. If

𝑧(𝑔) := 𝑔 ⊲ ±𝑖 = (±𝑏/𝑎,±1/𝑎), with inverse 𝑧 ↦→ 𝑔𝑧 = (±1/𝑦, 𝑥/𝑦), (2.6)

then of course 𝑔⊲𝑧(𝑔′) = 𝑧(𝑔𝑔′), for 𝑔, 𝑔′ ∈ Aff. One can employ this to transfer the group operation
onto the orbit by 𝑧(𝑔) · 𝑧(𝑔′) := 𝑧(𝑔𝑔′). Explicitly,

(𝑥 + 𝑖𝑦) · (𝑥′ + 𝑖𝑦′) = 𝑥′ ± 𝑥𝑦′ ± 𝑖𝑦𝑦′. (2.7)

Reciprocally, 𝑧 ↦→ 𝑔𝑧 is an isomorphism, namely 𝑔𝑧 𝑔𝑧′ = 𝑔𝑧·𝑧′ .
The invariant symplectic forms 𝜔± on O± are exact in the present case, being clearly given by

𝜔±(𝑧) = 𝑑𝑥 𝑑𝑦/|𝑦 |; note that 𝜔±(𝑧(𝑔)) = 𝑑𝑙𝑔. Darboux coordinates are 𝑞 = 𝑥/𝑦, 𝑝 = |𝑦 |.
In general, we say that a subalgebra h ⊆ g is subordinate to 𝐹 ∈ g∗ if 𝐹 | [h,h] = 0 and the map

𝑋 ↦→ ⟨𝐹, 𝑋⟩ is a one-dimensional representation of h. The entire Lie algebra aff is subordinate to
each (𝑥, 0). Any one-dimensional subalgebra of aff is subordinate to (0, 1) or to (0,−1), but only
the ideal [aff, aff] is Pukánszky, which means that 𝐹 + h⊥ ⊆ O𝐹 : indeed, (𝑥, 0) + (0, 0) = (𝑥, 0)
and (0,±1) + (𝑥′, 0) ∈ O±.

2.2 The Kirillov map and the unirreps

The Kirillov theory asserts the existence of a map 𝐾 : aff∗/Aff → Âff, where Aff acts via Coad,
the space aff∗/Aff is endowed with the (non-Hausdorff, not even 𝑇1) quotient topology and the
unitary dual Âff with its standard Fell topology, determined by the hull-kernel topology on the set
of primitive ideals of 𝐶∗(𝐺) – this matter is well explained in [26, Chap. 3] or [29, Chap. 7]. For
exponential groups, 𝐾 has been proved by Leptin and Ludwig [55] to be bijective and bicontinuous.
(For the similar correspondence between g∗/𝐺 and the set of primitive ideals of the enveloping
algebra U(g), we refer to [58].) It is known that all unirreps for exponential groups are monomial,
that is, induced by an abelian character of some closed subgroup 𝐻. If 𝐻 is the closed subgroup
generated byh subordinate to 𝐹, the Pukánszky condition guarantees that the induced representations

𝐾 [O𝐹] (exp 𝑋) := IndAff
𝐻 𝑈𝐹,𝐻 (exp 𝑋) = IndAff

𝐻 𝑒2𝜋𝑖⟨𝐹,𝑋⟩

are indeed irreducible. In the present case,

𝑈(𝑥,0),Aff (exp 𝑋) = 𝑒2𝜋𝑖𝑥𝑢 and 𝑈±,exp(ℝ𝑋2) (exp 𝑏𝑋2) = 𝑒±2𝜋𝑖𝑏 .

So we obtain in the first place the unitary one-dimensional representations (𝑎, 𝑏) ↦→ 𝑎2𝜋𝑖𝑥 of Aff;
observe that 𝐾 [O(0,0)] is the trivial representation.

6



Now denote 𝑈± := 𝐾 [O±]. The Kirillov scheme “predicts” (this is only of heuristic value) that
the “functional dimension” of𝑈± is 1

2 dimO± = 1; that is,𝑈± can be realized on spaces of functions
of one variable, in such a way that the smooth vectors are smooth functions and the enveloping
algebra 𝑈 (aff) acts by differential operators. The actual induction process leads us to consider the
space of functions 𝑓 on Aff such that

𝑓 (𝑎, 𝑏) = 𝑒±2𝜋𝑖𝑏𝜓(𝑎),

and then, necessarily,

𝑈±(𝑎, 𝑏) 𝑓 (𝑎′, 𝑏′) = 𝑓 (𝑎𝑎′, 𝑏𝑎′ + 𝑏′), or 𝑈±(𝑎, 𝑏)𝜓(𝑎′) = 𝑒±2𝜋𝑖𝑏𝑎′𝜓(𝑎𝑎′).

That is, we may settle on

𝑈±(𝑎, 𝑏)𝜓(𝑟) = 𝑒2𝜋𝑖𝑏𝑟𝜓(𝑎𝑟) = 𝑈±(1, 𝑏)𝑈±(𝑎, 0)𝜓(𝑟),

with 𝑟 > 0 for 𝑈+ and 𝑟 < 0 for 𝑈−. The 𝑈± preserve the space of smooth functions on the
semiaxis, vanishing in some neighbourhood of 𝑟 = 0, and are unitary on the Hilbert spaces
K± := 𝐿2((0,±∞), 𝑟−1 𝑑𝑟). Observe that𝑈+ and𝑈− are mutually dual; this is a general property for
𝐾 [O] and 𝐾 [−O]. (Needless to say, the process of going from O to a putative 𝐾 [O] is not always
so smooth; experience points to the importance of O being spin and of its twisted Dirac operator to
construct 𝐾 [O] – see [23] in this respect.)

The selfadjoint infinitesimal generators for the unirreps are given by

𝑈±(𝑋1)𝜓(𝑟) = −𝑖
𝑑

𝑑𝑢

���
𝑢=0

𝑈±(exp(𝑢, 0))𝜓(𝑟) = −𝑖𝑟 𝜓′(𝑟);

𝑈±(𝑋2)𝜓(𝑟) = −𝑖
𝑑

𝑑𝑣

���
𝑣=0
𝑈±(exp(0, 𝑣))𝜓(𝑟) = 2𝜋𝑟 𝜓(𝑟).

We denote the generators𝑈±(𝑋1), 𝑈±(𝑋2) by 2𝜋𝛽±, 2𝜋 𝑓± respectively. Note 2𝜋𝑖[𝛽±, 𝑓±] = 𝑓±.
Interesting (improper) denizens of K± are the “plane waves” or eigenfunctions of 𝛽±. They are

given by 𝜓𝛽 (𝑟) := 𝑟±2𝜋𝑖𝛽, for 𝛽 real, and constitute a (generalized) orthonormal and complete set.
Complete orthonormal sets within K± are also known, but here we shall not use them.

2.3 Characters of the unirreps and Kirillov’s Fourier transform

Any operator 𝐴 on K± determines an integral kernel, with suitable genuflections to rigour:

𝐴𝜓(𝑟) =:
∫
ℝ±

𝐴(𝑟, 𝑠)𝜓(𝑠) 𝑑𝑠
𝑠
,

so that the operator kernel of𝑈±(𝑎, 𝑏) is

𝑈±(𝑎, 𝑏; 𝑟, 𝑠) = 𝑒2𝜋𝑖𝑏𝑟 𝑠 𝛿(𝑠 − 𝑎𝑟).

Let 𝑓 ∈ D(Aff) ≡ 𝐶∞𝑐 (Aff). Following Kirillov [53, Sect. 4.1], we define the associated
operators𝑈±( 𝑓 ) on K± by

𝑈±( 𝑓 )𝜓(𝑟) :=
∫ ∞

0

∫ ∞

−∞
𝑓 (𝑎, 𝑏) 𝑒2𝜋𝑖𝑏𝑟𝜓(𝑎𝑟) 𝑑𝑎 𝑑𝑏

𝑎
=

∫ ±∞

0

∫ ∞

−∞
𝑓 (𝑠/𝑟, 𝑏) 𝑒2𝜋𝑖𝑏𝑟𝜓(𝑠) 𝑑𝑠𝑑𝑏

𝑠
,

7



with kernels
𝑈±( 𝑓 ; 𝑟, 𝑠) =

∫ ∞

−∞
𝑓 (𝑠/𝑟, 𝑏) 𝑒2𝜋𝑖𝑏𝑟 𝑑𝑏,

where 𝑟 > 0, 𝑠 > 0 or 𝑟 < 0, 𝑠 < 0, respectively. One would naı̈vely expect that 𝑈±( 𝑓 ) be nuclear,
for 𝑓 a test function. This is not the case unless 𝑓 is of zero mean with respect to the second
variable; otherwise 𝑈±( 𝑓 ) is not even compact [52]: 𝐶∗(Aff) is not liminaire. Assume, however,
that this requirement holds; then the traces are given by

Tr𝑈±( 𝑓 ) =
∫ ∞

−∞

∫
ℝ×±

𝑓 (1, 𝑏) 𝑒2𝜋𝑖𝑏𝑟 𝑑𝑏 𝑑𝑟

𝑟

for 𝑟 > 0 or 𝑟 < 0, respectively. By definition, the (generalized) characters 𝜒± of 𝑈± are the
functionals such that

𝜒±( 𝑓 ) = Tr𝑈±( 𝑓 ).
Let 𝑓 := 𝑓 ◦exp. We see that 𝜒±( 𝑓 ) only depends on the values of 𝑓 on [aff, aff]; this is a particular
instance of a property established by Duflo [27]. We can say that

𝜒±(𝑎, 𝑏) = 𝛿(𝑎 − 1)
∫
ℝ×±

𝑒2𝜋𝑖𝑏𝑟 𝑑𝑟

𝑟
, (2.8)

or 𝐹2𝜒±(𝑎, 𝑟) = 𝜃 (±𝑟)
𝑟

𝛿(𝑎−1), where 𝜃 is the Heaviside function, with the obvious caveats for good
definition in these “ultraviolet divergent” expressions.

For simply connected nilpotent groups, which are unimodular and whose Haar measure is the
Lebesgue measure in exponential coordinates, Kirillov postulated and showed the existence of a
unitary transformation matching 𝐿2(𝐺) with 𝐿2(g∗), here denoted 𝔽𝐾 , of the form

𝔽𝐾 [ 𝑓 ] (𝐹) :=
∫
g
𝑓 (𝑋) 𝑒2𝜋𝑖⟨𝐹,𝑋⟩ 𝑑𝑋,

and he established the formula

𝜒𝐾 [O] (exp 𝑋) =
∫
O

𝑒2𝜋𝑖⟨𝐹,𝑋⟩+𝜔, (2.9)

where 𝜔 is the invariant symplectic form on O, so that

Tr𝐾 [O] ( 𝑓 ) :=
∫
g
𝑓 (exp 𝑋) 𝜒𝐾 [O] (exp 𝑋) 𝑑𝑋 =

∫
O

𝔽𝐾 [ 𝑓 ] (𝐹) 𝑑𝜇𝜔 (𝐹). (2.10)

Here 𝜇𝜔 is the Liouville measure on the orbit, given by 𝜔 1
2 dimO/( 12 dimO)! ; also, (2.10) clearly

means that 𝔽𝐾 [𝜒𝐾 [O]] – a tempered distribution defined on g by transposition, transported to 𝐺 by
the exponential map – coincides precisely with that measure. This was one of the earliest triumphs
of the method of orbits. The similarity of (2.9) with the relation between the classical and quantum
partition functions has been pointed out and exploited before: see [70].

For general solvable groups, this will not do. Part of the problem is that the orbit is open and the
invariant symplectic structure is singular at the boundary. Also, it is unclear which measure to use

8



on g, in the nonunimodular case. Kirillov suggests that the recipe (2.9) be replaced by a weighted
version:

𝜒𝐾 [O] (exp 𝑋) = 1
𝑞(𝑋)

∫
O

𝑒2𝜋𝑖⟨𝐹,𝑋⟩+𝜔

and for 𝑞 he chooses ( 𝑗𝑙 𝑗𝑟)1/4. In view of (2.5), for Aff this leads to

𝔽𝐾 [ 𝑓 ] (𝑧) =
∬

ℝ2
exp{2𝜋𝑖(𝑢𝑥 + 𝑣𝑦)} 𝑓

(
𝑒𝑢,

𝑣

𝜆(−𝑢)

)
𝑑𝑢 𝑑𝑣√︁
sinch 𝑢

2
(2.11)

=

∬
Aff

exp{2𝜋𝑖(𝑥 log 𝑎 + 𝑦𝑏𝜆(− log 𝑎))} 𝑓 (𝑎, 𝑏) 𝜆1/4(log 𝑎)𝜆5/4(− log 𝑎) 𝑑𝑎 𝑑𝑏
𝑎

,

for 𝑧 ∈ aff∗. It is necessary and sufficient that 𝔽𝐾 [ 𝑓 ] go to zero on the boundary of the orbits for
𝐾 [O±] ( 𝑓 ) to be nuclear. Then (2.10) is still valid. This is scarcely surprising since, as pointed out
earlier, only the value of 𝜆 at 0 enters the calculation. However, the last formula is certainly not
pretty.

Yet another a priori Fourier map is defined in [3] by a formula of the type

𝔽𝐴𝐹𝐾 [ 𝑓 ] (𝐹) =
√︁
𝜉 𝑗 (𝐹)

∫
g∗

exp{2𝜋𝑖⟨𝐹, 𝑋⟩} 𝑓 (exp 𝑋)
√︁
𝑚(𝑋) 𝑑𝑋. (2.12)

We explain their notation: 𝑗 is an index parametrizing the orbits; 𝜉 𝑗 = 𝑑𝐹/𝑑𝜔 𝑗 ; and 𝑚(𝑋) =
𝑑𝑙 (exp 𝑋)/𝑑𝑋 . Here, for 𝑦 ≠ 0, we take 𝑗 ∈ {±}; 𝜉±(𝑧) = |𝑦 |; and 𝑚(𝑢, 𝑣) = 1/𝜆(𝑢). Hence

𝔽𝐴𝐹𝐾 [ 𝑓 ] (𝑧) =
√︁
|𝑦 |

∬
ℝ2

exp{2𝜋𝑖(𝑢𝑥 + 𝑣𝑦)} 𝑓 (𝑒𝑢, 𝑣/𝜆(−𝑢)) 1√︁
𝜆(𝑢)

𝑑𝑢 𝑑𝑣. (2.13)

By construction, this is an isometry between 𝐿2(Aff, 𝑑𝑙𝑔) and 𝐿2(aff∗, 𝑑𝜔+ ∪ 𝑑𝜔−). However, it
does not give the character.

In this paper we introduce two (left and right) powerful alternative transforms to Kirillov’s
Fourier map, that likewise recover the character, and are closely related to 𝔽𝐴𝐹𝐾 .

2.4 The Duflo–Moore operators

The decomposition of the regular representation of Aff, defined as usual by

Λ(𝑔) 𝑓 (𝑔′) := 𝑓 (𝑔−1𝑔′),

is well known for the affine group: Λ decomposes into a continuous direct sum of representations
equivalent to 𝑈+ ⊕ 𝑈−. More concretely, the Plancherel measure is in our case just the counting
measure on the two-element set {𝑈±}, and there is a unitary map 𝑃, the Plancherel transform:

𝑃 : 𝐿2(𝐺) → HS(K+) ⊕ HS(K−),

with HS(K±) denoting the Hilbert algebras of Hilbert–Schmidt operators on K±, and 𝑃 given by

𝑃 𝑓 := 𝑈+( 𝑓 ) 𝑑1/2
+ ⊕ 𝑈−( 𝑓 ) 𝑑1/2

− , (2.14)

9



where the 𝑑± are positive operators on K± with densely defined inverses, determined (up to a
positive constant) by the semi-invariance relation:

𝑈±(𝑔) 𝑑±𝑈†±(𝑔) = Δ−1(𝑔) 𝑑±. (2.15)

For a general proof of this uniqueness for 𝐺 not unimodular, see [28] and also [4]. Because the𝑈±
are induced from the subgroup {1, 𝑏} belonging to the kernel of Δ, it is easily seen that

𝑑±𝜓(𝑟) = |𝑟 | 𝜓(𝑟),

where we have chosen a convenient normalization. These 𝑑± are the formal dimension operators
as originally defined by Duflo and Moore in [28], although later authors use the phrase “Duflo–
Moore operators” for 𝑑−1/2

± instead. For their theory, one may consult [28] and also its excellent
precursor [74]. The remarkable thing is that the operators𝑈±( 𝑓 ) 𝑑1/2

± (or rather, their closures) are
Hilbert–Schmidt whenever 𝑓 belongs to 𝐿2(𝐺) – actually, our treatment of the harmonic analysis
on Aff in the long Section 7 amounts to an indirect proof of this fact. Then (2.14) holds, and
moreover

𝑃(Λ(𝑔) 𝑓 ) =
(
𝑈+ ⊕ 𝑈−

)
(𝑔) 𝑃 𝑓 .

Also, for 𝑓 in a suitable dense subspace of 𝐿2(𝐺), the operators𝑈 ( 𝑓 ) 𝑑± are nuclear.

3 An unusual special function
Before we plunge into calculating the quantizer, it is convenient to perform a few exercises in real
analysis, to be rewarded with later simplification.

We begin with the function 𝜆 of (2.4). Note that 𝜆(0) = 1 and 𝜆(𝑡) > 0 for all 𝑡 ∈ ℝ, that
𝜆(𝑡) ↓ 0 as 𝑡 → −∞, and that 𝜆(𝑡) ∼ 𝑡 as 𝑡 → +∞ (see Figure 1). It is easy to see that

𝜆(−𝑡) = 𝑒−𝑡𝜆(𝑡), (3.1a)
𝜆(𝑡) − 𝜆(−𝑡) = 𝑡. (3.1b)

These functional equations determine 𝜆 uniquely. It is an analytic function for |𝑡 | < 2𝜋, with
expansion

𝜆(𝑡) =
∑︁
𝑛⩾0
(−1)𝑛𝐵𝑛

𝑡𝑛

𝑛!
; thus 𝜆(−𝑡) =

∑︁
𝑛⩾0

𝐵𝑛
𝑡𝑛

𝑛!
,

where the 𝐵𝑛 are the well-known Bernoulli numbers.
The derivative of 𝜆(𝑡) is

𝜆′(𝑡) = (1 − 𝑒−𝑡)−1 − 𝑡𝑒−𝑡 (1 − 𝑒−𝑡)−2 =
𝜆(𝑡)
𝑡
− 𝜆(𝑡)𝜆(−𝑡)

𝑡

=
𝜆(𝑡)
𝑡
(1 − 𝜆(−𝑡)) = 𝜆(𝑡)

𝑡
(1 + 𝑡 − 𝜆(𝑡)). (3.2)

Thus 𝜆(𝑡) is strictly increasing on ℝ. Since (3.1b) entails 𝜆′(𝑡) + 𝜆′(−𝑡) = 1, we see that
𝜆′(0) = 1

2 . A brief calculation shows that 𝜆′′(𝑡) > 0 for 𝑡 > 0; since 𝜆′′(𝑡) = 𝜆′′(−𝑡), it follows that
𝜆′(𝑡) is increasing (i.e., 𝜆(𝑡) is convex), with 𝜆′(𝑡) ↑ 1 as 𝑡 → +∞, and therefore 𝜆′(𝑡) < 1 for all 𝑡.

10



•
(0, 1)

𝜆(𝑡)

•
(1, 0)

𝛾(𝑟)

Figure 1: The function 𝜆 and its inverse function 𝛾

Next, let 𝛾(𝑟) denote the inverse function for 𝜆: 𝛾(𝑟) = 𝑡 when 𝑟 = 𝜆(𝑡). It is defined for 𝑟 > 0,
is strictly increasing with 𝛾(1) = 0 and 𝛾′(1) = 2, and 𝛾(𝑟) ↓ −∞ as 𝑟 ↓ 0, while 𝛾(𝑟) ∼ 𝑟 as
𝑟 ↑ +∞. All that is evident on reflecting the graph of 𝑟 = 𝜆(𝑡) about the main diagonal 𝑟 = 𝑡. Of
course, the chain rule and (3.2) give

𝛾′(𝑟) = 1
𝜆′(𝛾(𝑟)) =

𝛾(𝑟)
𝑟 (1 − 𝑟 + 𝛾(𝑟)) .

In particular, 𝛾′(𝑟) > 1 for all 𝑟 > 0.
The special function worthy of our attention is

𝜎(𝑟) := 𝑟 − 𝛾(𝑟), for 𝑟 > 0.

For |𝑟 | < 1 we obtain, by [30, Sect. 2.1] for instance,

𝛾(1 + 𝑟) = 2𝑟 − 2
3
𝑟2 + 4

9
𝑟3 − 44

135
𝑟4 + 104

405
𝑟5 + · · ·

and 𝜎(1 + 𝑟) = 1 − 𝑟 + 2
3
𝑟2 − 4

9
𝑟3 + 44

135
𝑟4 − 104

405
𝑟5 + · · ·

Write 𝑠 = 𝜆(−𝑡) and 𝑟 = 𝜆(𝑡). Then relation (3.1b) shows that

𝑠 = 𝜎(𝑟) ⇐⇒ 𝑟 = 𝜎(𝑠), (3.3)

or, not to put too fine a point on it, 𝜎(𝜎(𝑟)) = 𝑟, that is, 𝜎 is self-inverse.
Note, too, that 𝜎(𝑟) = 𝑟𝑒−𝛾(𝑟) , in view of relation (3.1a). Also, 𝜎(1) = 1 and 𝜎′(1) = −1,

as expected since the graph of 𝜎 must be invariant on reflection in the main diagonal. Now
𝜎′(𝑟) = 1 − 𝛾′(𝑟) < 0 always, so that 𝜎 is a strictly decreasing function. Moreover, 𝜎(𝑟) ∼ −𝛾(𝑟)

11



• (𝑦, 𝑦)

𝜎𝑦 (𝑟)

Figure 2: The self-inverse function 𝜎𝑦 for 𝑦 > 0

as 𝑟 ↓ 0, while 𝜎(𝑟) ∼ 𝑟𝑒−𝑟 as 𝑟 → +∞: the graph of 𝜎 is exponentially asymptotic to both axes in
the first quadrant. Note, as well, from (3.3):

𝛾(𝑟) = −𝛾(𝜎(𝑟)); 𝛾′(𝑟) = −𝛾′(𝜎(𝑟)) 𝜎′(𝑟).

We remark that 𝜎′(𝑟) = 𝑒−𝛾(𝑟) (1 − 𝑟𝛾′(𝑟)) = 𝜎(𝑟) (𝑟−1 − 𝛾′(𝑟)); writing 𝑠 = 𝜎(𝑟) gives

𝑠 − 𝜎′(𝜎(𝑠)) 𝜎(𝑠) = 𝑠 − 𝑠
(

1
𝜎(𝑠) − 𝛾

′(𝜎(𝑠))
)
𝜎(𝑠) = 𝑠𝜎(𝑠)𝛾′(𝜎(𝑠)). (3.4)

It is also useful for our purposes to dilate 𝛾 and 𝜎 by a factor 𝑦 > 0, as follows:

𝛾𝑦 (𝑟) := 𝑦 𝛾(𝑟/𝑦), 𝜎𝑦 (𝑟) := 𝑦 𝜎(𝑟/𝑦).

It is immediate that 𝜎𝑦 (𝜎𝑦 (𝑟)) = 𝑟, so that 𝜎𝑦 also has a graph that is unchanged on reflection in
the diagonal, except that it now crosses the diagonal at (𝑦, 𝑦): see Figure 2. Moreover, the previous
relations between 𝛾 and 𝜎 become

𝜎𝑦 (𝑟) = 𝑟 − 𝛾𝑦 (𝑟), 𝛾𝑦 (𝜎𝑦 (𝑟)) = −𝛾𝑦 (𝑟),
𝜎′𝑦 (𝑟) = 1 − 𝛾′𝑦 (𝑟), 𝛾′𝑦 (𝜎𝑦 (𝑟)) 𝜎′𝑦 (𝑟) = −𝛾′𝑦 (𝑟). (3.5)

4 The Stratonovich–Weyl quantizer for the ‘𝑎𝑥 + 𝑏’ group
4.1 Weyl quantization for the half-plane

Pre-existing work on image processing [10, 11] points the way towards the quantizer: the transfor-
mation from “density matrices” to “Wigner functions” considered in these papers respects reality,
covariance and “unitarity”, which is equivalent to traciality. Consider the Weyl operator,

𝑊±(𝑢, 𝑣) := exp{2𝜋𝑖(𝑢𝛽± + 𝑣 𝑓±)} = 𝑈±(𝑒𝑢, 𝑣(𝑒𝑢 − 1)/𝑢) = 𝑈±(𝑒𝑢, 𝑣𝑒𝑢/𝜆(𝑢)).

Our candidate quantizer is

Ω±(𝑥 + 𝑖𝑦) := |𝑦 |
∬

ℝ2
exp{−2𝜋𝑖(𝑢𝑥 + 𝑣𝑦)}𝑊±(𝑢, 𝑣) 𝑑𝑢 𝑑𝑣. (4.1)

12



It can hardly be simpler! Aside from the factor |𝑦 |, that compensates for the measures used on O±, it
means that the familiar definition from Quantum Mechanics works. The recipe is actually imposed
on us by the heuristic rule that Ω±(𝐹0) should be the quantization of 𝛿(𝐹 − 𝐹0), or the equivalent
remark in [11] that the quantization of a plane wave should be given by the Weyl operator. It stands
to reason that the Weyl operator will play an essential role in an exponential group; one may treat
(4.1) as an Ansatz, and simply prove that it gives the Stratonovich–Weyl quantizer by verifying all
the required properties.

We carry out this verification for the upper half-plane Π. For −Π the argument is identical. It
is no longer worth the trouble to keep always the subscripts ±, so we mostly drop them.

4.2 Identification of the basic “parity” operators

The claim is that we can associate to each symbol 𝑓 (𝑧) on Π an operator 𝐴 on the representation
space K+ ≡ K, and vice versa, by

𝐴 =

∫
Π

Ω(𝑧) 𝑓 (𝑧) 𝑑𝜔(𝑧) =: 𝑄( 𝑓 ), 𝑓 (𝑧) = Tr(Ω(𝑧)𝐴), (4.2)

with the properties required in the Introduction. The quantizer Ω(𝑧) remains to be determined.
With the machinery assembled in the previous section, the computation of Ω is straightforward.

First of all,
𝑊 (𝑢, 𝑣)𝜓(𝑟) = 𝑒2𝜋𝑖𝑣𝑒𝑢𝑟/𝜆(𝑢) 𝜓(𝑒𝑢𝑟),

with operator kernel
𝑊 (𝑢, 𝑣; 𝑟, 𝑠) = 𝑒2𝜋𝑖𝑣𝑠/𝜆(𝑢)𝑠 𝛿(𝑠 − 𝑒𝑢𝑟).

The right hand side of (4.1), applied to 𝜓 ∈ D(0,∞), yields

Ω(𝑧)𝜓(𝑟) = 𝑦
∬

ℝ2
𝑒−2𝜋𝑖(𝑢𝑥+𝑣𝑦−𝑒𝑢𝑟𝑣/𝜆(𝑢)) 𝜓(𝑒𝑢𝑟) 𝑑𝑢 𝑑𝑣

= 𝑦

∫
ℝ

𝑒−2𝜋𝑖𝑢𝑥 𝛿(𝑦 − 𝑟𝑒𝑢/𝜆(𝑢)) 𝜓(𝑒𝑢𝑟) 𝑑𝑢

=
𝑟

𝑦
𝛾′𝑦 (𝑟) 𝑒2𝜋𝑖𝑥𝛾𝑦 (𝑟)/𝑦 𝜓(𝜎𝑦 (𝑟)). (4.3)

For this, consider the diffeomorphism 𝑢 ↦→ 𝑟/𝜆(−𝑢) = 𝑤, whose inverse mapping is 𝑢 = −𝛾(𝑟/𝑤)
with Jacobian 𝐽 (𝑢;𝑤) = 𝑟𝑤−2𝛾′(𝑟/𝑤). The general formula [79, Chap. 1]:

⟨𝑇 (𝑤(𝑢)), 𝜓(𝑢)⟩ = ⟨𝑇 (𝑤), 𝜓(𝑢(𝑤)) |𝐽 (𝑢;𝑤) |⟩

in the present case easily gives∫
ℝ

𝛿(𝑦 − 𝑟𝑒𝑢/𝜆(𝑢)) 𝑒−2𝜋𝑖𝑢𝑥 𝜓(𝑒𝑢𝑟) 𝑑𝑢 =

∫
ℝ

𝛿(𝑦 − 𝑤) 𝑒2𝜋𝑖𝑥𝛾(𝑟/𝑤) 𝜓(𝑟𝑒−𝛾(𝑟/𝑤)) 𝑟𝑤−2𝛾′(𝑟/𝑤) 𝑑𝑤,

yielding (4.3). Similar computations will reappear throughout this paper; often we shall just omit
them. In particular, we have obtained

Ω(𝑖)𝜓(𝑟) = 𝑟𝛾′(𝑟) 𝜓(𝜎(𝑟)).

13



A remarkable event has occurred. For the ordinary Moyal family, the “mother” operator in the
Schrödinger representation is known to be essentially the (Grossmann–Royer) parity operator.
Matters are more involved here, but still Ω(𝑖) is basically given by a kind of reflection; to wit, the
involutive 𝜎 function.

In view of (2.7), covariance requires Ω(𝑧) = 𝑈 (𝑔𝑧)Ω(𝑖)𝑈†(𝑔𝑧), where 𝑈 (𝑔𝑧) = 𝑈 (1/𝑦, 𝑥/𝑦)
with adjoint operator𝑈†(𝑔𝑧) = 𝑈 (𝑦,−𝑥). We leave this direct verification as an exercise.

The kernel of Ω(𝑧) is given by

Ω(𝑧; 𝑟, 𝑠) = 𝑟𝑠
𝑦
𝛾′𝑦 (𝑟) 𝑒2𝜋𝑖𝑥(𝑟−𝑠)/𝑦 𝛿(𝑠 − 𝜎𝑦 (𝑟)). (4.4a)

An alternative form is

Ω(𝑧; 𝑟, 𝑠) = 𝑦 𝑒2𝜋𝑖𝑥 log(𝑟/𝑠) 𝛿

(
𝑦 − 𝑟

𝜆(log(𝑟/𝑠))

)
= 𝑦 𝑒2𝜋𝑖𝑥 log(𝑟/𝑠) 𝛿

(
𝑦 − 𝑟 − 𝑠

log(𝑟/𝑠)

)
. (4.4b)

For the normalization property the operators Ω(𝑧) must be of trace 1, in the distributional sense.
We check this by computing

TrΩ(𝑖) =
∫ ∞

0
Ω(𝑖; 𝑟, 𝑟) 𝑑𝑟

𝑟
=

∫ ∞

0
𝑟𝛾′(𝑟) 𝛿(𝛾(𝑟)) 𝑑𝑟 = 1. (4.5)

The normalization TrΩ(𝑧) = 1 is then automatic, since the𝑈 (𝑔𝑧) are unitary. It can be proved that
the operators 𝑄( 𝑓 ), for 𝑓 any test function on Π, are nuclear [41].

4.3 Selfadjointness of the quantizer operators

Note that the Ω(𝑧) are unbounded operators. They are defined at least on D(0,∞). Indeed, since
𝜎𝑦 is monotonic and smooth, the right hand side of (4.3) lies in D(0,∞) whenever 𝜓 does, and we
see that

∥Ω(𝑧)𝜓∥2 =
1
𝑦2

∫ ∞

0
𝑟𝛾′𝑦 (𝑟)2 |𝜓(𝜎𝑦 (𝑟)) |2 𝑑𝑟

=
1
𝑦2

∫ ∞

0
𝜎𝑦 (𝑠)𝛾′𝑦 (𝜎𝑦 (𝑠))2 |𝜓(𝑠) |2 |𝜎′𝑦 (𝑠) | 𝑑𝑠

=
1
𝑦2

∫ ∞

0
𝑠𝜎𝑦 (𝑠)𝛾′𝑦 (𝑠)𝛾′𝑦 (𝜎𝑦 (𝑠)) |𝜓(𝑠) |2

𝑑𝑠

𝑠
,

where (3.5) has been used. Since 𝑠 ↦→ 𝑠𝛾′𝑦 (𝑠) increases from 1 to +∞ for 0 < 𝑠 < ∞, no bound
of the form ∥Ω(𝑖)𝜓∥ ⩽ 𝐶∥𝜓∥ is possible. We remark that the equivalent operators are (of course)
bounded for compact groups, and for the Heisenberg group. But unboundedness is not unheard of,
as it happens for the Poincaré group [19, Sect. 4].

The Ω(𝑧) are hermitian on the domain D(0,∞). For 𝜙, 𝜓 in this domain, we get

⟨𝜙 | Ω(𝑖)𝜓⟩ =
∫ ∞

0
𝜙(𝑟)Ω(𝑖) 𝜓(𝑟) 𝑑𝑟

𝑟
=

∫ ∞

0
𝜙(𝑟) 𝛾′(𝑟) 𝜓(𝜎(𝑟)) 𝑑𝑟 (4.6)

=

∫ ∞

0
𝜙(𝜎(𝑠)) 𝛾′(𝜎(𝑠)) 𝜓(𝑠) |𝜎′(𝑠) | 𝑑𝑠 =

∫ ∞

0
𝑠𝛾′(𝑠) 𝜙(𝜎(𝑠)) 𝜓(𝑠) 𝑑𝑠

𝑠
= ⟨Ω(𝑖)𝜙 | 𝜓⟩.

14



Likewise, ⟨𝜙 |Ω(𝑧)𝜓⟩ = ⟨Ω(𝑧)𝜙 |𝜓⟩ by covariance, since𝑈†(𝑔𝑧) preserves D(0,∞). We prove that
Ω(𝑧) is closable and identify its closure, and then show selfadjointness. This is easy to do using
the fortunate fact that its square is a multiplication operator, Ω2(𝑧) = 𝑀𝜂2

𝑧
, where the unbounded

positive function 𝜂𝑧 is given by

𝜂2
𝑧 (𝑟) := 𝑟𝑦−2𝜎𝑦 (𝑟) 𝛾′𝑦 (𝑟) 𝛾′𝑦 (𝜎𝑦 (𝑟)).

Notice that 𝜂𝑧 (𝑟) = 𝜂𝑖 (𝑟/𝑦) for 𝑧 ∈ Π. The natural domains for the several Ω(𝑧) are the dense
subspaces 𝐵𝑧 of K defined as

𝐵𝑧 := { 𝜓 ∈ K : 𝜂𝑧𝜓 ∈ K }.
Note that (4.6) remains valid for 𝜙, 𝜓 ∈ 𝐵𝑧 so that Ω(𝑧) is hermitian on this domain.

Proposition 4.1. On the respective domains 𝐵𝑧, the operators Ω(𝑧) are selfadjoint.

Proof. First, take 𝜓 ∈ D(0,∞). One sees at once that indeed Ω2(𝑧)𝜓(𝑟) = 𝜂2
𝑧 (𝑟) 𝜓(𝑟). We already

showed that ∥Ω(𝑧)𝜓∥ = ∥𝜂𝑧𝜓∥, using (4.3); and this continues to hold for all 𝜓 ∈ 𝐵𝑧. Observe that
𝐵𝑧 is complete in the graph norm given by |||𝜓 |||2 := ∥𝜓∥2 + ∥𝜂𝑧𝜓∥2, and that D(0,∞) is dense in 𝐵𝑧
for this norm. Thus Ω(𝑧), with domain 𝐵𝑧, is a closed operator. Clearly DomΩ(𝑧) ⊂ DomΩ(𝑧)†
since Ω(𝑧) is hermitian on 𝐵𝑧.

Note that if 𝜓 ∈ DomΩ(𝑧)†, and if 𝜒𝑛 is the indicator function of the interval [1/𝑛, 𝑛], say, then
𝜒𝑛Ω(𝑧)†𝜓 ∈ 𝐵𝑧 and a routine argument, using the monotone convergence theorem, shows that

∥Ω(𝑧)†𝜓∥ = lim
𝑛→∞
∥𝜒𝑛Ω(𝑧)†𝜓∥ = lim

𝑛→∞
sup
∥𝜙∥=1

��⟨Ω(𝑧) (𝜒𝑛𝜙) | 𝜓⟩�� = · · · = ∥𝜂𝑧𝜓∥,
so that necessarily 𝜓 ∈ 𝐵𝑧. Thus DomΩ(𝑧)† = 𝐵𝑧 = DomΩ(𝑧), as required.

Note also that the original domain D(0,∞), being dense in each 𝐵𝑧 for the graph norm, is a
common core for all the Ω(𝑧), which are therefore essentially selfadjoint on that domain. □

A consequence of the operators Ω(𝑧) being selfadjoint (not just formally so) is that they become
observables in the quantum-mechanical sense. For the Stratonovich–Weyl operators in the standard
(Heisenberg covariant) case, see the discussion in [71].

4.4 Traciality

Lemma 4.2. The quantizer is tracial, in the sense that

Tr
(
Ω(𝑤)Ω(𝑧)

)
= 𝑦 𝛿(𝑤 − 𝑧) for all 𝑤, 𝑧 ∈ Π,

where the right hand side is the reproducing kernel for the Hilbert space 𝐿2(Π, 𝑑𝑥 𝑑𝑦/𝑦).

Proof. With 𝑤 = 𝑢 + 𝑖𝑣, 𝑧 = 𝑥 + 𝑖𝑦, we obtain

Tr
(
Ω(𝑤)Ω(𝑧)

)
=

∫ ∞

0

∫ ∞

0
Ω(𝑤; 𝑟, 𝑠)Ω(𝑧; 𝑠, 𝑟) 𝑑𝑟 𝑑𝑠

𝑟𝑠

=

∫ ∞

0

∫ ∞

0

𝑟𝑠

𝑣𝑦
𝛾′

( 𝑠
𝑣

)
𝛾′

( 𝑟
𝑦

)
𝑒−2𝜋𝑖(𝑟−𝑠) (𝑥/𝑦−𝑢/𝑣) 𝛿(𝑟 − 𝜎𝑣 (𝑠)) 𝛿(𝑠 − 𝜎𝑦 (𝑟)) 𝑑𝑟 𝑑𝑠

=

∫ ∞

0

𝑟

𝑣
𝜎

( 𝑟
𝑦

)
𝛾′

(
𝜎𝑦 (𝑟)
𝑣

)
𝛾′

( 𝑟
𝑦

)
𝑒−2𝜋𝑖(𝑥/𝑦−𝑢/𝑣) (𝑟−𝜎𝑦 (𝑟)) 𝛿(𝜎𝑣 (𝜎𝑦 (𝑟)) − 𝑟) 𝑑𝑟.

15



The argument of this last delta vanishes if and only if 𝜎𝑣 (𝑟) = 𝜎𝑦 (𝑟), that is, if and only if 𝑣 = 𝑦.
On setting 𝑓 (𝑣) := 𝜎𝑣 (𝜎𝑦 (𝑟)) − 𝑟, it follows that

𝑓 ′(𝑣)
��
𝑣=𝑦

= 𝜎

(
𝜎

( 𝑟
𝑦

))
− 𝜎′

(
𝜎

( 𝑟
𝑦

))
𝜎

( 𝑟
𝑦

)
=
𝑟

𝑦
− 𝜎′

(
𝜎

( 𝑟
𝑦

))
𝜎

( 𝑟
𝑦

)
=
𝑟

𝑦
𝜎

( 𝑟
𝑦

)
𝛾′

(
𝜎

( 𝑟
𝑦

))
,

using (3.4). Therefore,

𝑟

𝑦
𝜎

( 𝑟
𝑦

)
𝛾′

(
𝜎

( 𝑟
𝑦

))
𝛿(𝑟 − 𝜎𝑣 (𝜎𝑦 (𝑟))) = 𝛿(𝑣 − 𝑦),

and (4.5) follows at once:

Tr
(
Ω(𝑤)Ω(𝑧)

)
= 𝛿(𝑣 − 𝑦)

∫ ∞

0
𝛾′

( 𝑟
𝑦

)
𝑒−2𝜋𝑖(𝑥−𝑢)𝛾(𝑟/𝑦) 𝑑𝑟 = 𝑦 𝛿(𝑥 − 𝑢) 𝛿(𝑣 − 𝑦). □

Corollary 4.3. The maps (4.2) establish an isometric isomorphism between the Hilbert space of
Hilbert–Schmidt operators on K and the Hilbert space 𝐿2(Π, 𝑑𝜔(𝑧)) of square-summable functions
on the upper Poincaré half-plane with the left-invariant measure. □

4.5 Relation with the Wigner functions on the half-plane

Starting from (4.4a), by dequantization we obtain

𝑊𝐴 (𝑧) = Tr(Ω(𝑧)𝐴) =
∫ ∞

0

∫ ∞

0
𝐴(𝑟, 𝑠)Ω(𝑧; 𝑠, 𝑟) 𝑑𝑟

𝑟

𝑑𝑠

𝑠

=
1
𝑦

∫ ∞

0

∫ ∞

0
𝐴(𝑟, 𝑠) 𝛾′(𝑠/𝑦) 𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) 𝛿(𝑟 − 𝜎𝑦 (𝑠)) 𝑑𝑟 𝑑𝑠

=
1
𝑦

∫ ∞

0
𝐴(𝜎𝑦 (𝑠), 𝑠) 𝛾′(𝑠/𝑦) 𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) 𝑑𝑠

=

∫ ∞

−∞
𝐴(𝑦𝜆(𝑢), 𝑦𝜆(−𝑢)) 𝑒−2𝜋𝑖𝑥𝑢 𝑑𝑢. (4.7)

Note how by means of (4.4b) we recover the kernel of 𝐴 from its dequantization:

𝐴(𝑟, 𝑠) =
∫
Π

𝑊𝐴 (𝑧)Ω(𝑧; 𝑟, 𝑠)
𝑑𝑥 𝑑𝑦

𝑦
=

∫ ∞

−∞
𝑊𝐴

(
𝑥,

𝑟 − 𝑠
log(𝑟/𝑠)

)
𝑒2𝜋𝑖𝑥 log(𝑟/𝑠) 𝑑𝑥. (4.8)

In particular, the Wigner function 𝑊𝜓 of a state 𝜓 is simply the expected value of the
Stratonovich–Weyl operator Ω or the dequantization of the projector |𝜓⟩⟨𝜓 |:

𝑊𝜓 (𝑧) := ⟨𝜓 | Ω(𝑧)𝜓⟩ = ⟨Ω(𝑧)𝜓 | 𝜓⟩ = Tr
(
Ω(𝑧) |𝜓⟩⟨𝜓 |

)
.

The operator |𝜓⟩⟨𝜓 | has kernel 𝐴(𝑟, 𝑠) = 𝜓(𝑟)𝜓∗(𝑠). Thus,

𝑊𝜓 (𝑧) =
∫
ℝ

𝑒−2𝜋𝑖𝑥𝑢 𝜓(𝑦𝜆(𝑢)) 𝜓∗(𝑦𝜆(−𝑢)) 𝑑𝑢. (4.9)

Now, for the upper half-plane, a plethora of “affine Wigner functions” were originally constructed
by the Bertrands [10]; see also [59]. One family among several options satisfying good covariance

16



properties (under the ‘𝑎𝑥 + 𝑏’ group and some extensions of it) is distinguished by “unitarity”, that
is, the correspondence |𝜓⟩⟨𝜓 | → 𝑊𝜓 should extend to a unitary isomorphism between Hilbert–
Schmidt operators and 𝐿2(Π, 𝑑𝑥 𝑑𝑦/𝑦). For our purposes, it is enough to check that (4.9) with
our quantizer coincides with their Wigner functions. This is done by inspection of formula (57)
in [59], modulo our conventions for the unirrep 𝐾 [O+], or equation (IV.7) in [11], where Darboux
coordinates on the phase space are used.

For an arbitrary normalized state 𝜓, we remark that
∫
Π
|𝑊𝜓 (𝑧) |2 𝑑𝜔(𝑧) = 1, which seems

curious since 𝜓 need not belong to the domains of all Ω(𝑧). We return to this question in [41].
Geometrical properties of the affine Wigner functions have been much investigated, in regard to
positivity, localization, marginal distributions, interference, etc. On this, we can do little better than
to refer to Flandrin’s articles [33, 34].

4.6 Summary

The strategy outlined in subsection 4.1 has been successful. The outcome is that the deceptively
simple formula for the operator-valued distribution

D(Π) ⊗̂ B(K) ∋ Ω(𝐹) :=
∫
aff
𝑒−2𝜋𝑖⟨𝐹 |𝑋⟩𝑈 (exp 𝑋) 𝑑𝑋, (4.10)

with 𝑑𝑋 the Lebesgue measure, makes sense, and the bounded operators∫
aff∗

𝑎(𝐹)
∫
aff
𝑒−2𝜋𝑖⟨𝐹 |𝑋⟩𝑈 (exp 𝑋) 𝑑𝑋 𝑑𝐹

for 𝑎 a test function on aff∗, supported on Π, are explicitly given. Lest the reader be misled by the
heuristic approach ostensibly taken in subsection 4.1, it must be said that (4.10) recommends itself
because from it covariance of Ω is ensured to hold by an abstract argument. Again, the reader will
have no difficulty in checking it. This point of view had been emphasized in [75].

5 The Moyal twisted product on the half-plane
The Moyal product 𝑓 ★ ℎ of two functions 𝑓 , ℎ on Π is by definition the dequantization of the
operator 𝑄( 𝑓 )𝑄(ℎ); to wit,

𝑓 ★ ℎ(𝑧) := Tr
[
Ω(𝑧)

∫
𝑓 (𝑤)Ω(𝑤) 𝑑𝜔(𝑤)

∫
𝑔(𝑡)Ω(𝑡) 𝑑𝜔(𝑡)

]
=

∬
Π2
𝐾★(𝑧, 𝑤, 𝑡) 𝑓 (𝑤) ℎ(𝑡) 𝑑𝜔(𝑤) 𝑑𝜔(𝑡), (5.1)

where
𝐾★(𝑧, 𝑤, 𝑡) := Tr

(
Ω(𝑧)Ω(𝑤)Ω(𝑡)

)
. (5.2)

The “trikernel” 𝐾★ enjoys left invariance:

Tr
(
Ω(𝑍 · 𝑧)Ω(𝑍 · 𝑤)Ω(𝑍 · 𝑡)

)
= Tr

(
𝑈 (𝑔𝑍 )Ω(𝑧)Ω(𝑤)Ω(𝑡)𝑈†(𝑔𝑍 )

)
= Tr

(
Ω(𝑧)Ω(𝑤)Ω(𝑡)

)
.

17



Using this invariance we can rewrite the Moyal product of two functions – also, using (1.2) and the
cyclicity of the trace-integral, eventually of many distributions [44] – on Π, gifted with the invariant
measure 𝑑𝜔(𝑧) = 𝑑𝑥 𝑑𝑦/𝑦, in the following ways:

𝑓 ★ ℎ(𝑧) =
∬

Π2
𝐾★(𝑧, 𝑤, 𝑡) 𝑓 (𝑤)ℎ(𝑡) 𝑑𝜔(𝑤) 𝑑𝜔(𝑡)

=

∬
Π2
𝐾★(𝑖, 𝑧−1 · 𝑤, 𝑧−1 · 𝑡) 𝑓 (𝑤)ℎ(𝑡) 𝑑𝜔(𝑤) 𝑑𝜔(𝑡)

=

∬
Π2
𝐾★(𝑖, 𝑤, 𝑡) 𝑓 (𝑧 · 𝑤)ℎ(𝑧 · 𝑡) 𝑑𝜔(𝑤) 𝑑𝜔(𝑡).

Let 𝑅𝑤, 𝑅𝑡 denote right multiplication operators (i.e., the right regular action) for the group structure
on Π, and note the elegance of the final “deformation” formula:

𝑓 ★ ℎ =

∬
Π2
𝐾★(𝑖, 𝑤, 𝑡) 𝑅𝑤 𝑓 𝑅𝑡ℎ 𝑑𝜔(𝑤) 𝑑𝜔(𝑡). (5.3)

We do not omit, finally, the tracial identity for our star product:∫
Π

𝑓 ★ ℎ(𝑧) 𝑑𝜔(𝑧) =
∫
Π

𝑓 (𝑧) ℎ(𝑧) 𝑑𝜔(𝑧);

this comes straight from (5.1) on using the properties of the Stratonovich–Weyl quantizer.

5.1 The trikernel for the twisted product and its symmetries

We need the solution 𝜅(𝑦0, 𝑦1, 𝑦2) of the equation

𝑠 = 𝜎𝑦0 (𝜎𝑦1 (𝜎𝑦2 (𝑠))).

Equivalently, it is the solution of

𝜎𝑦0 (𝑠) = 𝜎𝑦1 (𝜎𝑦2 (𝑠)), or 𝜎𝑦2 (𝑠) = 𝜎𝑦1 (𝜎𝑦0 (𝑠)).

Given any (positive) values of 𝑦0, 𝑦1, 𝑦2, there is indeed a unique solution to these equations, for
one of the sides increases monotonically from 0 to ∞, whereas the other decreases from ∞ to 0.
Under the exchange 𝑦0 ↔ 𝑦2, with 𝑦1 held fixed, the value of 𝜅 is unchanged. Moreover, 𝜅 is
a homogeneous function of degree 1, since 𝑐−1 𝜅(𝑐𝑦0, 𝑐𝑦1, 𝑐𝑦2) and 𝜅(𝑦0, 𝑦1, 𝑦2) solve the same
equation. We abbreviate 𝜎012 := 𝜎𝑦0 ◦ 𝜎𝑦1 ◦ 𝜎𝑦2 , whose inverse function is 𝜎210 := 𝜎𝑦2 ◦ 𝜎𝑦1 ◦ 𝜎𝑦0 .

With this in hand, one can proceed to compute the trikernel. After a straightforward, though
tedious, calculation, one obtains

𝐾★(𝑧0, 𝑧1, 𝑧2)

=
𝜅 𝜎𝑦0 (𝜅) 𝜎𝑦2 (𝜅) 𝛾′𝑦2 (𝜅) 𝛾

′
𝑦0 (𝜎𝑦0 (𝜅)) 𝛾′𝑦1 (𝜎𝑦2 (𝜅))

𝑦0𝑦1𝑦2
(
1 − 𝜎′012(𝜅)

) 𝑒
2𝜋𝑖

[
𝑥0
𝑦0
𝛾𝑦0 (𝜅)−

𝑥1
𝑦1
𝛾𝑦1 (𝜎𝑦2 (𝜅))−

𝑥2
𝑦2
𝛾𝑦2 (𝜅)

]
(5.4)

where the dependence on 𝑦0, 𝑦1, 𝑦2 through 𝜅 is understood.

18



The trikernel is of the general form

𝐾★(𝑧0, 𝑧1, 𝑧2) = 𝐴(𝑧0, 𝑧1, 𝑧2) 𝑒2𝜋𝑖𝑆(𝑧0,𝑧1,𝑧2)

for real amplitude 𝐴 : 𝑀3 → ℝ and phase 𝑆 : 𝑀3 → ℝ functions. By its construction, we expect
𝐾★ to have several symmetries. First of all, cyclical symmetry. With 𝜅(𝑦0, 𝑦1, 𝑦2) defined by
𝜅 = 𝜎201(𝜅), with the obvious notation, the same calculation gives

𝐾★(𝑧2, 𝑧0, 𝑧1) (5.5)

=
𝜅 𝜎𝑦2 (𝜅) 𝜎𝑦1 (𝜅) 𝛾′𝑦1 (𝜅) 𝛾

′
𝑦2 (𝜎𝑦2 (𝜅)) 𝛾′𝑦0 (𝜎𝑦1 (𝜅))

𝑦0𝑦1𝑦2
(
1 − 𝜎′201(𝜅)

) 𝑒
2𝜋𝑖

[
𝑥2
𝑦2
𝛾𝑦2 (𝜅)−

𝑥0
𝑦0
𝛾𝑦0 (𝜎𝑦1 (𝜅))−

𝑥1
𝑦1
𝛾𝑦1 (𝜅)

]
.

But 𝜅 is just 𝜎𝑦2 (𝜅); thus

𝛾𝑦2 (𝜅) = −𝛾𝑦2 (𝜅), and also 𝜎𝑦0 (𝜅) = 𝜎𝑦1 (𝜅), implying 𝛾𝑦0 (𝜎𝑦1 (𝜅)) = −𝛾𝑦0 (𝜅),

and one sees at once that the numerator of the fraction in (5.4) and the phase factor coincide with
those of the new formula (5.5). Moreover,

𝜎′201(𝜅) = 𝜎
′
𝑦2 (𝜎𝑦0 (𝜎𝑦1 (𝜅))) 𝜎′𝑦0 (𝜎𝑦1 (𝜅)) 𝜎′𝑦1 (𝜅)

= 𝜎′𝑦2 (𝜅) 𝜎
′
𝑦0 (𝜎𝑦1 (𝜎𝑦2 (𝜅))) 𝜎′𝑦1 (𝜎𝑦2 (𝜅)) = 𝜎′012(𝜅),

and we conclude that, as expected,

𝐾★(𝑧0, 𝑧1, 𝑧2) = 𝐾★(𝑧2, 𝑧0, 𝑧1) = 𝐾★(𝑧1, 𝑧2, 𝑧0).

Next we investigate the switch 𝑧0 ↔ 𝑧2 (with 𝑧1 fixed). We observe that

1
1 − 𝜎′012(𝜅)

=
−𝜎′210(𝜎210(𝜅))

1 − 𝜎′210(𝜎210(𝜅))
=
−𝜎′210(𝜅)

1 − 𝜎′210(𝜅)
,

since 𝜎210(𝜅) = 𝜅. On multiplying the numerator of the fraction occurring in (5.4) by

−𝜎′210(𝜅) = −𝜎
′
𝑦2 (𝜎𝑦1 (𝜎𝑦0 (𝜅))) 𝜎′𝑦1 (𝜎𝑦0 (𝜅)) 𝜎′𝑦0 (𝜅)

and taking (3.5) into account, the whole fraction becomes

𝜅 𝜎𝑦0 (𝜅) 𝜎𝑦2 (𝜅) 𝛾′𝑦2 ((𝜎𝑦2 (𝜅)) 𝛾′𝑦1 (𝜎𝑦0 (𝜅)) 𝛾′𝑦0 (𝜅)
𝑦0𝑦1𝑦2 (1 − 𝜎′210(𝜅))

.

In other words, the fraction is unchanged by the switch 𝑧0 ↔ 𝑧2. Now note that

𝛾𝑦1 (𝜎𝑦2 (𝜅)) = 𝜎𝑦2 (𝜅) − 𝜎𝑦1 (𝜎𝑦2 (𝜅)) = 𝜎𝑦2 (𝜅) − 𝜎𝑦0 (𝜅) = 𝛾𝑦0 (𝜅) − 𝛾𝑦2 (𝜅).

Using this formula, we can reexpress the phase factor in the trikernel (5.4) as

exp
{
2𝜋𝑖

[( 𝑥0
𝑦0
− 𝑥1
𝑦1

)
𝛾𝑦0 (𝜅) +

( 𝑥1
𝑦1
− 𝑥2
𝑦2

)
𝛾𝑦2 (𝜅)

]}
.

19



This is manifestly skewsymmetric under the exchange 𝑧0 ↔ 𝑧2, with 𝑧1 held fixed. In fine, we have
shown that

𝐾★(𝑧0, 𝑧1, 𝑧2) = 𝐾★(𝑧2, 𝑧1, 𝑧0) = 𝐾★(𝑧0, 𝑧2, 𝑧1) = 𝐾★(𝑧1, 𝑧0, 𝑧2),
where cyclic symmetry has been reinvoked; in particular, this confirms that complex conjugation
is an antilinear involution for the twisted product. Corollary 4.3 can now be read as stating that
(𝐿2(Π, 𝑑𝜔), ★) is a Hilbert algebra.

Finally, we expect 𝐾★(𝑧0, 𝑧1, 𝑧2) = 𝐾★(𝑖, 𝑧−1
0 · 𝑧1, 𝑧

−1
0 · 𝑧2), in view of left invariance. Indeed,

the trikernel is invariant under the transformations

𝑥0 ↦→ 0, 𝑥1 ↦→ 𝑥1 − 𝑥0𝑦1/𝑦0, 𝑥2 ↦→ 𝑥2 − 𝑥0𝑦2/𝑦0,

𝑦0 ↦→ 1, 𝑦1 ↦→ 𝑦1/𝑦0, 𝑦2 ↦→ 𝑦2/𝑦0,

on account of the homogeneity properties of 𝜅.
Inspired by earlier exact results [42, 82], Weinstein [81] developed a heuristic argument for the

construction of trikernels on symplectic symmetric spaces. In this approach, the phase function 𝑆 is
postulated to be an (invariant) oriented symplectic area of a geodesic triangle for which 𝑧0, 𝑧1, 𝑧2 are
the midpoints of the sides, and the amplitude 𝐴 is chosen as to achieve associativity of the twisted
product – implicit in our treatment – and other desirable properties. The idea has been further
developed in [69] – where some caveats are made – and in [12] and [24, § 3.3.5]. It is sometimes
linked to the purported role of reflections in producing quantizers. However, it is known [76] that
reflections do not lead directly to Stratonovich–Weyl quantizers in general; and it is straightforward
to verify that, although it enjoys the same symmetries, our phase function is not the area. The
question deserves further investigation [13].

5.2 The extended covariance group of the twisted product

The ordinary Moyal product on the full plane ℝ2 has a larger covariance group than the original
Heisenberg group of phase-space translations under which it is equivariant; this is the inhomo-
geneous metaplectic group of unitaries 𝑈 (𝑔) such that 𝑄( 𝑓 ) ↦→ 𝑈 (𝑔)𝑄( 𝑓 )𝑈†(𝑔) =: 𝑄( 𝑓 ◦ 𝜑)
implements a diffeomorphism 𝜑 of the plane that normalizes the action of the Heisenberg group.
For the half-plane Π, its analogue will be a Lie group of symplectomorphisms normalizing the
action of Aff. At the infinitesimal level, the generators of this group are given by symbols 𝑓𝑖 such
that the Moyal bracket

[ 𝑓𝑖, ℎ]★ := 2𝜋𝑖( 𝑓𝑖 ★ ℎ − ℎ ★ 𝑓𝑖)
coincides, for arbitrary ℎ, with the Poisson bracket

{ 𝑓𝑖, ℎ}PB = 𝑦

(
𝜕 𝑓𝑖

𝜕𝑥

𝜕ℎ

𝜕𝑦
− 𝜕 𝑓𝑖
𝜕𝑦

𝜕ℎ

𝜕𝑥

)
corresponding to the symplectic 2-form 𝑑𝑥 ∧ 𝑑𝑦/𝑦 on Π. In other words, these 𝑓𝑖 are “distinguished
observables” in the sense of [9].

The (neutral component of) the normalizer of Aff within the group of symplectomorphisms
of Π is easily determined [75]. Any one-parameter subgroup is generated by a Hamiltonian vector

20



field of the form 𝐻 𝑓 = 𝑦
(
𝑓𝑦

𝜕
𝜕𝑥
− 𝑓𝑥 𝜕

𝜕𝑦

)
for some 𝑓 ∈ 𝐶∞(Π). Since the action of aff on Π is

generated by the vector fields 𝑦 𝜕
𝜕𝑥

and 𝑦 𝜕
𝜕𝑦

, we require that[
𝐻 𝑓 , 𝑦

𝜕

𝜕𝑥

]
= −( 𝑓𝑥𝑦 + 𝑓𝑦)𝑦

𝜕

𝜕𝑥
+ 𝑦2 𝑓𝑥𝑥

𝜕

𝜕𝑦
,

[
𝐻 𝑓 , 𝑦

𝜕

𝜕𝑦

]
= −(𝑦 𝑓𝑦𝑦 + 𝑓𝑦)𝑦

𝜕

𝜕𝑥
+ 𝑦2 𝑓𝑥𝑦

𝜕

𝜕𝑦

be linear combinations of 𝑦 𝜕
𝜕𝑥

and 𝑦 𝜕
𝜕𝑦

. This easily entails that

𝑓 (𝑥, 𝑦) = 𝛼𝑥 + 𝛽𝑦 + 𝛾 log 𝑦 + 𝛿

for some constants 𝛼, 𝛽, 𝛾, 𝛿. Ignoring the trivial constant term that does not contribute to 𝐻 𝑓 =

−𝛼𝑦 𝜕
𝜕𝑦
+ 𝛽𝑦 𝜕

𝜕𝑥
+ 𝛾 𝜕

𝜕𝑥
, we obtain a solvable 3-parameter group 𝐺 extending Aff by ℝ. The

appearance of the log function above is related to the existence of ray unirreps of 𝐺 given by

𝑈 (𝑎, 𝑏, 𝑐)𝜓(𝑟) = 𝑒2𝜋𝑖𝑏𝑟𝑟2𝜋𝑖𝑐𝜓(𝑎𝑟).

This covariance group was also found in [10] by a not very different method.
To ascertain that this group is indeed a symmetry group of our twisted product, one must verify

that the three functions 𝑥, 𝑦, log 𝑦 are distinguished observables.

Lemma 5.1. For any smooth function ℎ on Π, the following relations hold:

[𝑥, ℎ]★ = 𝑦
𝜕ℎ

𝜕𝑦
; [𝑦, ℎ]★ = −𝑦 𝜕ℎ

𝜕𝑥
; [log 𝑦, ℎ]★ = −𝜕ℎ

𝜕𝑥
. (5.6)

Proof. We first determine the operator kernels corresponding, via (4.8), to the three basic functions:

𝑄𝑥 (𝑟, 𝑠) =
∫
ℝ

𝑥 𝑒2𝜋𝑖𝑥 log(𝑟/𝑠) 𝑑𝑥 =
1

2𝜋𝑖
𝛿′(log 𝑟 − log 𝑠) = 1

2𝜋𝑖
(
𝑠2 𝛿′(𝑟 − 𝑠) − 𝑠 𝛿(𝑟 − 𝑠)

)
,

𝑄𝑦 (𝑟, 𝑠) =
∫
ℝ

𝑟 − 𝑠
log(𝑟/𝑠) 𝑒

2𝜋𝑖𝑥 log(𝑟/𝑠) 𝑑𝑥 =
𝑟

𝜆(log(𝑟/𝑠)) 𝛿(log 𝑟 − log 𝑠) = 𝑟𝑠

𝜆(log(𝑟/𝑠)) 𝛿(𝑟 − 𝑠),

𝑄log 𝑦 (𝑟, 𝑠) =
∫
ℝ

log
(
𝑟 − 𝑠

log(𝑟/𝑠)

)
𝑒2𝜋𝑖𝑥 log(𝑟/𝑠) 𝑑𝑥 =

𝑟

𝜆(log(𝑟/𝑠)) 𝛿(log 𝑟 − log 𝑠)

=
(
𝑠 log 𝑟 − 𝑠 log𝜆(log(𝑟/𝑠))

)
𝛿(𝑟 − 𝑠). (5.7)

We have written 𝑄𝑥 for 𝑄(𝑥), and similarly for the other operators. If the quantized operator 𝑄(ℎ)
has kernel 𝐵(𝑠, 𝑡), then 𝑄

(
[𝑥, ℎ]★

)
= 2𝜋𝑖[𝑄𝑥 , 𝑄(ℎ)] has kernel

2𝜋𝑖
∫ ∞

0

(
𝑄𝑥 (𝑟, 𝑡) 𝐵(𝑡, 𝑠) − 𝐵(𝑟, 𝑡)𝑄𝑥 (𝑡, 𝑠)

) 𝑑𝑡
𝑡

=

∫ ∞

0

(
𝑡 𝛿′(𝑟 − 𝑡) − 𝛿(𝑟 − 𝑡)

)
𝐵(𝑡, 𝑠) − 𝐵(𝑟, 𝑡)

(
𝑡 𝛿′(𝑡 − 𝑠) − 𝛿(𝑡 − 𝑠)

)
𝑑𝑡

=
𝜕

𝜕𝑡

����
𝑡=𝑟

(−𝑡𝐵(𝑡, 𝑠)) − 𝜕

𝜕𝑡

����
𝑡=𝑠

(−𝑡𝐵(𝑟, 𝑡)) = −𝑟 𝜕𝐵
𝜕𝑟
(𝑟, 𝑠) + 𝑠 𝜕𝐵

𝜕𝑠
(𝑟, 𝑠).

21



On the other hand, (4.7) yields

𝑦
𝜕ℎ

𝜕𝑦
(𝑧) =

∫
ℝ

(
𝑦𝜆(𝑢) 𝜕𝐵

𝜕𝑟

(
𝑦𝜆(𝑢), 𝑦𝜆(−𝑢)

)
+ 𝑦𝜆(−𝑢) 𝜕𝐵

𝜕𝑠

(
𝑦𝜆(𝑢), 𝑦𝜆(−𝑢)

) )
𝑒−2𝜋𝑖𝑥𝑢 𝑑𝑢

=
1
𝑦

∫ ∞

0
𝑟
𝜕𝐵

𝜕𝑟
(𝑟, 𝜎𝑦 (𝑟)) 𝛾′𝑦 (𝑟) 𝑒−2𝜋𝑖𝑥𝛾(𝑟/𝑦) 𝑑𝑟 + 1

𝑦

∫ ∞

0
𝑠
𝜕𝐵

𝜕𝑠
(𝜎𝑦 (𝑠), 𝑠) 𝛾′𝑦 (𝑠) 𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) 𝑑𝑠

=
1
𝑦

∫ ∞

0

∫ ∞

0

(
−𝑟 𝜕𝐵

𝜕𝑟
+ 𝑠 𝜕𝐵

𝜕𝑠

)
(𝑟, 𝑠) 𝛾′𝑦 (𝑠)𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) 𝛿(𝑟 − 𝜎𝑦 (𝑠)) 𝑑𝑟 𝑑𝑠,

and using (4.7) once more we obtain the desired relation:

{𝑥, ℎ}PB = 𝑦
𝜕ℎ

𝜕𝑦
= 𝑊𝑄( [𝑥,ℎ]★) = [𝑥, ℎ]★.

The other cases are simpler. One finds from (5.7) that 𝑄( [𝑦, ℎ]★) = 2𝜋𝑖[𝑄𝑦, 𝑄(ℎ)] has kernel
2𝜋𝑖(𝑟 − 𝑠) 𝐵(𝑟, 𝑠) and that 𝑄( [log 𝑦, ℎ]★) has kernel 2𝜋𝑖(log 𝑟 − log 𝑠) 𝐵(𝑟, 𝑠). Therefore,

[𝑦, ℎ]★(𝑧) =
2𝜋𝑖
𝑦

∫ ∞

0
(𝜎𝑦 (𝑠) − 𝑠) 𝐵(𝜎𝑦 (𝑠), 𝑠) 𝛾′𝑦 (𝑠) 𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) 𝑑𝑠

= −2𝜋𝑖
∫ ∞

0
𝐵(𝜎𝑦 (𝑠), 𝑠) 𝛾′𝑦 (𝑠)

(
𝛾(𝑠/𝑦) 𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) ) 𝑑𝑠

= − 𝜕
𝜕𝑥

∫ ∞

0
𝐵(𝜎𝑦 (𝑠), 𝑠) 𝛾′𝑦 (𝑠) 𝑒2𝜋𝑖𝑥𝛾(𝑠/𝑦) 𝑑𝑠

= − 𝜕
𝜕𝑥
(𝑦ℎ(𝑧)) = −𝑦 𝜕ℎ

𝜕𝑥
(𝑧).

An almost identical calculation, with the relation 𝜎𝑦 (𝑠) − 𝑠 = −𝑦 𝛾(𝑠/𝑦) replaced by the identity
log𝜎𝑦 (𝑠) − log 𝑠 = −𝛾(𝑠/𝑦), yields [log 𝑦, ℎ]★ = −𝜕ℎ/𝜕𝑥. Or we may just remark that [·, ℎ]★ is a
derivation. □

On regarding the functions 𝑥, 𝑦 on phase space as elements of the Lie algebra aff, the first two
equalities of (5.6) show that our twisted product is an aff-invariant ★-quantization in the sense
of [9, 36].

5.3 On the universal enveloping algebra product

By duality and symmetrization, the universal enveloping algebra U(g) of a Lie algebra g can be
realized by an algebra P(g∗) of polynomial functions on g∗. Thus it makes sense to compare the
(restriction to Π of) the product on U(aff) as transferred to P(g∗) with the twisted product; in our
case the correspondence gives 𝑋1 ↦→ 𝑥 ≡ 𝑥1; 𝑋2 ↦→ 𝑦 ≡ 𝑥2. Let us denote by ∗ that transported
product. Its expression is complicated in general, but it is well known that one obtains

𝑥𝑖 ∗ ℎ = 𝑥𝑖ℎ +
∞∑︁
𝑛=1

𝐵𝑛

∑︁
𝑘1...𝑘𝑛

{ad(𝑥𝑘1) · · · ad(𝑥𝑘𝑛)}sym(𝑥𝑖)
𝜕𝑛ℎ

𝜕𝑥𝑘1 · · · 𝜕𝑥𝑘𝑛
; (5.8)

ℎ ∗ 𝑥𝑖 = 𝑥𝑖ℎ +
1
2

∑︁
𝑗

[𝑥 𝑗 , 𝑥𝑖]
𝜕ℎ

𝜕𝑥 𝑗
+
∞∑︁
𝑛=2

𝐵𝑛

∑︁
𝑘1...𝑘𝑛

{ad(𝑥𝑘1) · · · ad(𝑥𝑘𝑛)}sym(𝑥𝑖)
𝜕𝑛ℎ

𝜕𝑥𝑘1 · · · 𝜕𝑥𝑘𝑛
;

22



where the 𝑘 𝑗 take the values 1 or 2 in all possible forms and {· · · }sym means total symmetrization
of the operations inside the curly brackets. The rule makes sense because the Lie products are
supposed known: here [𝑥1, 𝑥2] = 𝑥2. For instance, we see that, if ℎ depends only on the second
variable, then the series terminates, and 𝑥 ∗ ℎ(𝑥, 𝑦) = 𝑥ℎ(𝑦) + 1

2 𝑦ℎ
′(𝑦). The second term of the

series in (5.8) is just 1
2 times the Poisson bracket:∑︁

𝑗

[𝑥 𝑗 , 𝑥]
𝜕ℎ

𝜕𝑥 𝑗
= −𝑦 𝜕ℎ

𝜕𝑦
;

∑︁
𝑗

[𝑥 𝑗 , 𝑦]
𝜕ℎ

𝜕𝑥 𝑗
= 𝑦

𝜕ℎ

𝜕𝑥
.

Thus in particular
𝑥𝑖 ∗ ℎ − ℎ ∗ 𝑥𝑖 = {𝑥, ℎ}PB = [𝑥𝑖, ℎ]★.

A detailed comparison between ∗ and the asymptotic expansion of ★would lengthen this paper too
much; we come back on this matter in [41] and [13].

6 Right-covariant quantization
In this section the notation 𝑑𝑙𝑧 will be used for 𝑑𝜔(𝑧) = 𝑑𝑥 𝑑𝑦/𝑦, and 𝑑𝑟 𝑧 for 𝑑𝑥 𝑑𝑦/𝑦2. For the
purposes outlined in the introduction, we are actually interested in a right-covariant star product, as
well as the left-covariant one constructed so far. We summarize again our desideratum: a pair of
quantizers Ω𝐿 , Ω𝑅, both acting on K, satisfying

(i) Ω𝐿,𝑅 (𝑧)† = Ω𝐿,𝑅 (𝑧),

(ii) 𝑈 (𝑔𝑧′)Ω𝐿 (𝑧)𝑈†(𝑔𝑧′) = Ω𝐿 (𝑧′ · 𝑧) and 𝑈†(𝑔𝑧′)Ω𝑅 (𝑧)𝑈 (𝑔𝑧′) = Ω𝑅 (𝑧 · 𝑧′),

(iii) TrΩ𝐿,𝑅 (𝑧) = 1,

(iv) Tr
(
Ω𝐿,𝑅 (𝑧)Ω𝐿,𝑅 (𝑧′)

)
= 𝐼𝐿,𝑅 (𝑧, 𝑧′),

with 𝐼𝐿 and 𝐼𝑅 denoting the reproducing kernels for 𝐿2(Π, 𝑑𝑙𝑧) and 𝐿2(Π, 𝑑𝑟 𝑧), respectively. Here
Ω𝐿 ≡ Ω is the quantizer already found. Define

𝑓 (𝑧) := 𝑓 (𝑧−1), with 𝑧−1 = (𝑥 + 𝑖𝑦)−1 := −𝑥/𝑦 + 𝑖/𝑦,

the inverse for the product (2.7). It seems natural to replace the quantization rule (4.2) by

𝐴 =

∫
Π

Ω𝑅 (𝑧) 𝑓 (𝑧) 𝑑𝑟 𝑧 =: 𝑄𝑅 ( 𝑓 ),

when using the right-covariant quantizer. We obtain 𝑄𝑅 ( 𝑓 ) = 𝑄( 𝑓 ) if we declare that

Ω𝑅 (𝑧) ≡ Ω(𝑧−1).

It is not idle to check consistency of this rule with previous use of the coadjoint action and the
diffeomorphism 𝑧 ↦→ 𝑔𝑧. We need to verify that

𝑧 ⊳ 𝑔𝑧′ ≡ (𝑔−1
𝑧′ ⊲ 𝑧

−1)−1 = 𝑧 · 𝑧′. (6.1)

23



Indeed, (
(𝑦′,−𝑥′) ⊲ (−𝑥/𝑦 + 𝑖/𝑦)

)−1
=

(
−𝑥′ − 𝑥𝑦′
𝑦𝑦′

+ 𝑖

𝑦𝑦′

)−1
= 𝑥′ + 𝑥𝑦′ + 𝑖𝑦𝑦′.

The right-covariant quantizer is thus given by the following expression:

Ω𝑅 (𝑧) = 1
𝑦

∬
ℝ2
𝑒2𝜋𝑖(𝑢𝑥/𝑦−𝑣/𝑦)𝑈 (𝑒𝑢, 𝑣/𝜆(−𝑢)) 𝑑𝑢 𝑑𝑣.

It is easy to check consistency of this with Ω(𝑧) = Ω𝑅 (𝑧−1). It is also straightforward to verify,
along the same lines as before, the four properties listed above; in particular,

Tr
(
Ω𝑅 (𝑧)Ω𝑅 (𝑤)

)
= 𝑦2 𝛿(𝑧 − 𝑤).

For the trikernel, one obtains

𝐾𝑅★ (𝑧, 𝑤, 𝑡) = Tr
(
Ω𝑅 (𝑧)Ω𝑅 (𝑤)Ω𝑅 (𝑡)

)
= 𝐾★(𝑧−1, 𝑤−1, 𝑡−1).

This yields the following tautological relation between the twisted products ★ and ★𝑅 :

𝑓 ★𝑅 ℎ(𝑧) =
∫
Π2
𝐾𝑅★ (𝑧, 𝑤, 𝑡) 𝑓 (𝑤) ℎ(𝑡) 𝑑𝑟𝑤 𝑑𝑟 𝑡

=

∫
Π2
𝐾★(𝑧−1, 𝑤−1, 𝑡−1) 𝑓 (𝑤) ℎ(𝑡) 𝑑𝑟𝑤 𝑑𝑟 𝑡

=

∫
Π2
𝐾★(𝑧−1, 𝑤, 𝑡) 𝑓 (𝑤−1) ℎ(𝑡−1) 𝑑𝑙𝑤 𝑑𝑙𝑡 = ( 𝑓 ★ ℎ̆)˘(𝑧),

consistent with
𝑄𝑅 ( 𝑓 ★𝑅 ℎ) = 𝑄𝑅 ( 𝑓 )𝑄𝑅 (ℎ).

We finally register the following formula for the product ★𝑅, similar to (5.3):

𝑓 ★𝑅 ℎ =

∬
Π2
𝐾★(𝑖, 𝑤, 𝑡)𝐿𝑤 𝑓 𝐿𝑡ℎ 𝑑𝑙𝑤 𝑑𝑙𝑡,

and the tracial identity for the ★𝑅 product:∫
Π

𝑓 ★𝑅 ℎ(𝑧) 𝑑𝑟 𝑧 =
∫
Π

𝑓 (𝑧)ℎ(𝑧) 𝑑𝑟 𝑧.

7 Fourier–Moyal transformations on the ‘𝑎𝑥 + 𝑏’ group
7.1 The Fourier–Moyal kernels

Consider the following distribution or kernel:

𝔼(𝑧, 𝑔) ≡ 𝔼𝐿 (𝑧, 𝑔) := Tr
(
Ω±(𝑧)𝑈±(𝑔)

)
= 𝑊𝑈± (𝑔) (𝑧),

for 𝑧 ∈ ±Π, respectively, and 𝑔 ∈ Aff. As a consequence of this definition, we see that 𝔼 is the
symbol for the unirreps:

𝑈±(𝑔) =
∫
±Π

𝔼(𝑧, 𝑔)Ω(𝑧) 𝑑𝜔±(𝑧). (7.1)

24



We compute the kernel, expecting the covariance property

𝔼(ℎ ⊲ 𝑧, ℎ𝑔ℎ−1) = 𝔼(𝑧, 𝑔). (7.2)

It comes as a “nice surprise” that the kernel is a U(1)-valued smooth function. Take ℑ𝑧 > 0. From
equation (4.1), one gets

Ω(𝑧)𝑈 (𝑔) = 𝑦
∬

ℝ2
𝑒−2𝜋𝑖(𝑥𝑢+𝑦𝑣)𝑈 (𝑒𝑢𝑎, 𝑒𝑢 (𝑏 + 𝑣/𝜆(𝑢)) 𝑑𝑢 𝑑𝑣,

and the kernel of this operator is thus

Ω𝑈 (𝑧, 𝑔; 𝑟, 𝑠) = 𝑦
∬

ℝ2
𝑒−2𝜋𝑖(𝑥𝑢+𝑦𝑣) 𝑒2𝜋𝑖𝑒𝑢 (𝑏+𝑣/𝜆(𝑢))𝑟 𝑠 𝛿(𝑠 − 𝑟𝑒𝑢𝑎) 𝑑𝑢 𝑑𝑣

= 𝑦

∬
ℝ2
𝑒−2𝜋𝑖(𝑥𝑢+𝑦𝑣) 𝑒2𝜋𝑖𝑒𝑢 (𝑏+𝑣/𝜆(𝑢))𝑟 𝛿(𝑢 − log(𝑠/𝑟𝑎)) 𝑑𝑢 𝑑𝑣

= 𝑦

∫
ℝ

𝑒−2𝜋𝑖(𝑥 log(𝑠/𝑟𝑎)+𝑦𝑣) 𝑒2𝜋𝑖𝑎−1𝑠(𝑏+𝑣/𝜆(log(𝑠/𝑟𝑎))) 𝑑𝑣

= 𝑦 𝑒−2𝜋𝑖𝑥 log(𝑠/𝑟𝑎) 𝑒2𝜋𝑖𝑏𝑠/𝑎 𝛿
(
𝑦 − 𝑠/𝑎 𝜆(log(𝑠/𝑟𝑎))

)
. (7.3)

Its trace is the desired kernel, explicitly:

𝔼(𝑧, 𝑔) =
∫ ∞

0
Ω𝑈 (𝑧, 𝑔; 𝑟, 𝑟) 𝑑𝑟

𝑟

= 𝑦 𝑒2𝜋𝑖𝑥 log 𝑎
∫ ∞

0
𝑒2𝜋𝑖𝑟𝑏/𝑎 𝛿

(
𝑦 − 𝑟/𝜆(log 𝑎)

) 𝑑𝑟
𝑟

= 𝑦 𝑒2𝜋𝑖𝑥 log 𝑎
∫ ∞

0
𝑒2𝜋𝑖𝑟𝑏/𝑎 𝜆(log 𝑎) 𝛿

(
𝑟 − 𝑦𝜆(log 𝑎)

) 𝑑𝑟
𝑟

= 𝑒2𝜋𝑖(𝑥 log 𝑎+𝑦 𝑏𝜆(log 𝑎)/𝑎) . (7.4)

We then see that 𝔼 is smooth and of modulus 1. We check the coadjoint covariance of this
kernel. With ℎ = (𝑎′, 𝑏′), one indeed finds that

𝔼(𝑥 + 𝑏′𝑦/𝑎′ + 𝑖𝑦/𝑎′; (𝑎, 𝑎′𝑏 + 𝑏′(1 − 𝑎))) = 𝔼(𝑥 + 𝑖𝑦; (𝑎, 𝑏)),

since (𝑎−1 − 1)𝜆(log 𝑎) = − log 𝑎 from the definition (2.4). By computing on the second orbit, the
formula is valid for 𝑦 < 0 as well.

Similarly, the right-covariant quantizer yields another kernel:

𝔼𝑅 (𝑧, 𝑔) := Tr
(
Ω𝑅 (𝑧)𝑈 (𝑔)

)
= 𝔼(𝑧−1, 𝑔),

with the expected covariance property:

𝔼𝑅 (𝑧 ⊳ ℎ, ℎ−1𝑔ℎ) = 𝔼𝑅 (𝑧, 𝑔),

which follows from (7.2), in view of (6.1). Thus we get, explicitly,

𝔼𝑅 (𝑧, 𝑔) = 𝑒2𝜋𝑖(−𝑥 log 𝑎+𝑏𝜆(log 𝑎)/𝑎)/𝑦 .

25



It is enlightening to pass to Lie-algebra coordinates in the 𝔼-functions, for the first group
arguments:

𝔼(𝑧; 𝑢, 𝑏) = 𝑒2𝜋𝑖(𝑢𝑥+𝑏𝜆(−𝑢)𝑦) , 𝔼𝑅 (𝑧; 𝑢, 𝑏) = 𝑒2𝜋𝑖(−𝑢𝑥+𝑏𝜆(−𝑢))/𝑦;

or for both:
𝔼(𝑧; 𝑢, 𝑣) = 𝑒2𝜋𝑖(𝑢𝑥+𝑣𝑦) , 𝔼𝑅 (𝑧; 𝑢, 𝑣) = 𝑒2𝜋𝑖(−𝑢𝑥/𝑦+𝑣/𝑦) .

Note the simplicity of the last result for 𝔼. If we regard 𝑧 as an element of aff∗, then

Tr
(
Ω±(𝑧)𝑈±(exp 𝑋)

)
= exp

(
2𝜋𝑖⟨𝑧 | 𝑋⟩

)
, for all 𝑋 ∈ aff,

has been proved to hold.
In conclusion, the left Fourier–Moyal transformation, denoted 𝔽𝑀 , is given by

𝔽𝑀 [ 𝑓 ] (𝑧) :=
∫

𝔼(𝑧, 𝑔) 𝑓 (𝑔) 𝑑𝑙𝑔

=

∬
aff

exp{2𝜋𝑖(𝑢𝑥 + 𝑣𝑦)} 𝑓 (𝑒𝑢, 𝑣/𝜆(−𝑢)) 𝑑𝑢 𝑑𝑣
𝜆(𝑢)

=

∬
Aff

exp{2𝜋𝑖(𝑥 log 𝑎 + 𝑏𝜆(log 𝑎)𝑦/𝑎)} 𝑓 (𝑎, 𝑏) 𝑑𝑎 𝑑𝑏
𝑎2 .

Compare (2.11). The right Fourier–Moyal transformation 𝔽 𝑟
𝑀

in turn is given by

𝔽 𝑟
𝑀
[ 𝑓 ] (𝑧) :=

∫
𝔼𝑅 (𝑧, 𝑔) 𝑓 (𝑔) 𝑑𝑟𝑔

=

∬
aff

exp{2𝜋𝑖(−𝑢𝑥/𝑦 + 𝑣/𝑦)} 𝑓 (𝑒𝑢, 𝑣/𝜆(−𝑢)) 𝑑𝑢 𝑑𝑣
𝜆(−𝑢)

=

∬
Aff

exp{2𝜋𝑖(−𝑥 log 𝑎/𝑦 + 𝑏𝜆(log 𝑎)/𝑎𝑦)} 𝑓 (𝑎, 𝑏) 𝑑𝑎 𝑑𝑏
𝑎

.

We run a first few checks on these Fourier–Moyal kernels. For 𝑎 = 1, 𝑏 = 0, we recover trivially
TrΩ𝐿,𝑅 (𝑧) = 1. Also, from (7.3) for 𝑔 = 1Aff one gleans without effort the form (4.4) for the kernel
of Ω(𝑧), that was useful to invert the Wigner function in subsection 4.5.

We should be able, as well, to recover the character from the right kernel, say. Indeed, since∫
±Π Ω𝑅 (𝑧) 𝑑𝑟 𝑧 = 1K± , the characters of the representations𝑈± are retrieved from∫

±Π
𝑒−2𝜋𝑖𝑥 log 𝑎/𝑦 𝑒2𝜋𝑖𝑏𝜆(log 𝑎)/𝑎𝑦 𝑑𝑥 𝑑𝑦

𝑦2 =

∫
ℝ×±

𝛿(𝑎 − 1) 𝑒2𝜋𝑖𝑦𝑏 𝑑𝑦

|𝑦 | = 𝜒±(𝑔).

This leads to 𝔽 𝑟
𝑀
[𝜒±] = 𝜔±: the ugly duckling of a character (2.8) turns here into the swan of the

symplectic form. The same holds for 𝔽𝑀 . Also, the following equalities are immediate:

Tr𝑈± [ 𝑓 ] =
∫
±Π

𝔽𝑀 [ 𝑓 ] (𝑧) 𝑑𝑙𝑧 =
∫
±Π

𝔽 𝑟
𝑀
[ 𝑓 ] (𝑧) 𝑑𝑟 𝑧. (7.5)

26



7.2 The modified Fourier–Moyal kernels

Just as the operator Fourier transform needs modification for groups that are not unimodular, we
must redefine our Fourier kernels in order to get a Fourier inversion theorem and a Parseval formula.
Consider now the following distribution or kernel:

𝔼mod(𝑧, 𝑔) := Tr
(
Ω±(𝑧)𝑈±(𝑔) 𝑑1/2

±
)
= 𝑊

𝑈± (𝑔)𝑑1/2
±
(𝑧),

for 𝑧 ∈ ±Π, respectively. Take ℑ𝑧 > 0. From (4.1), we obtain

Ω(𝑧)𝑈 (𝑔) 𝑑1/2 = 𝑦

∬
ℝ2
𝑒−2𝜋𝑖(𝑥𝑢+𝑦𝑣)𝑈 (𝑒𝑢𝑎, 𝑒𝑢 (𝑏 + 𝑣/𝜆(𝑢))𝑀√· 𝑑𝑢 𝑑𝑣,

and the kernel of this operator is thus

Ω𝑈𝑑1/2(𝑧, 𝑔; 𝑟, 𝑠) = 𝑦
∬

ℝ2
𝑒−2𝜋𝑖(𝑥𝑢+𝑦𝑣) 𝑒2𝜋𝑖𝑒𝑢 (𝑏+𝑣/𝜆(𝑢))𝑟 𝑠3/2 𝛿(𝑠 − 𝑟𝑒𝑢𝑎) 𝑑𝑢 𝑑𝑣

= 𝑦

∬
ℝ2
𝑒−2𝜋𝑖(𝑥𝑢+𝑦𝑣) 𝑠1/2 𝑒2𝜋𝑖𝑒𝑢 (𝑏+𝑣/𝜆(𝑢))𝑟 𝛿(𝑢 − log(𝑠/𝑟𝑎)) 𝑑𝑢 𝑑𝑣

= 𝑦

∫
ℝ

𝑒−2𝜋𝑖(𝑥 log(𝑠/𝑟𝑎)+𝑦𝑣) 𝑒2𝜋𝑖𝑎−1𝑠(𝑏+𝑣/𝜆(log(𝑠/𝑟𝑎))) √𝑠 𝑑𝑣

= 𝑦 𝑒−2𝜋𝑖𝑥 log(𝑠/𝑟𝑎) 𝑒2𝜋𝑖𝑏𝑠/𝑎 √𝑠 𝛿
(
𝑦 − 𝑠

𝑎 𝜆(log(𝑠/𝑟𝑎))

)
.

Its trace gives us the desired kernel:

𝔼mod(𝑧, 𝑔) =
∫ ∞

0
Ω𝑈𝑑1/2(𝑧, 𝑔; 𝑟, 𝑟) 𝑑𝑟

𝑟

= 𝑦 𝑒2𝜋𝑖𝑥 log 𝑎
∫ ∞

0
𝑒2𝜋𝑖𝑟𝑏/𝑎 𝛿

(
𝑦 − 𝑟/𝜆(log 𝑎)

) 𝑑𝑟
√
𝑟

= 𝑦 𝑒2𝜋𝑖𝑥 log 𝑎
∫ ∞

0
𝑒2𝜋𝑖𝑟𝑏/𝑎 𝜆(log 𝑎) 𝛿

(
𝑟 − 𝑦𝜆(log 𝑎)

) 𝑑𝑟
√
𝑟

=
√︁
|𝑦 |𝜆(log 𝑎) 𝑒2𝜋𝑖(𝑥 log 𝑎+𝑦 𝑏𝜆(log 𝑎)/𝑎) .

We have written |𝑦 | for 𝑦, so the formula remains valid on the second orbit. We then see that 𝔼mod

is no longer of modulus 1. We check the coadjoint variation of this kernel, and find that

𝔼mod(𝑧, 𝑔) = Δ1/2(ℎ) 𝔼mod(ℎ ⊲ 𝑧, ℎ𝑔ℎ−1).

In conclusion, we make the new definition

𝔽mod
𝑀
[ 𝑓 ] (𝑧) :=

∫
𝔼mod(𝑧, 𝑔) 𝑓 (𝑔) 𝑑𝑙𝑔

=
√︁
|𝑦 |

∬
aff

exp{2𝜋𝑖(𝑢𝑥 + 𝑣𝑦)} 𝑓 (𝑒𝑢, 𝑣/𝜆(−𝑢)) 𝑑𝑢 𝑑𝑣√︁
𝜆(𝑢)

.

This is seen to coincide with the 𝔽𝐴𝐹𝐾 transform of [3]: compare (2.13).

27



▶ Summarizing, the situation is as follows: the formulas analogous to (7.5) are true as well for the
Fourier–Kirillov transform, as pointed out in Section 2; that is of course well known, and happens
for the good reason that the character is concentrated at 𝑎 = 1 (i.e., at the subgroup generated by
the Pukánszky subalgebra subordinate to the maximal orbits). On the other hand, for the inversion
and Plancherel theorems to be perfect analogues of the ordinary case, we also need a modified
Fourier–Moyal transform. This “fact of life” reflects the non-unimodularity of the group. The
relevant results are established in the next subsection.

7.3 The basic theorems of Fourier analysis

Theorem 7.1. The left Fourier–Moyal kernel and transformation enjoy the following properties:

• Hermiticity: complex conjugation gives 𝔼(𝑧, 𝑔) = 𝔼(𝑧, 𝑔−1).

• Covariance: 𝔼(𝑧, 𝑔) = 𝔼(ℎ ⊲ 𝑧, ℎ𝑔ℎ−1) for all ℎ ∈ Aff.

• Character formula:
∫
±Π

𝔼(𝑧, 𝑔) 𝑑𝜔±(𝑧) = Tr𝑈±(𝑔).

• Convolution theorem: 𝔽𝑀 [ 𝑓 ∗ ℎ] = 𝔽𝑀 [ 𝑓 ] ★ 𝔽𝑀 [ℎ].

Analogous properties hold for the right Fourier–Moyal kernel and map.

Proof. The first property is obvious from the definition and the selfadjointness of Ω(𝑧). The second
and third properties have already been established. The fourth is easy: note first that

𝔼(·, 𝑔) ★𝔼(·, 𝑔′) = 𝔼(·, 𝑔𝑔′) (7.6)

on account of (7.1) and (5.2). Therefore,

𝔽𝑀 [ 𝑓 ∗ ℎ] =
∬

Aff ×Aff
𝔼(·, 𝑔𝑔′) 𝑓 (𝑔) ℎ(𝑔′) 𝑑𝑙𝑔 𝑑𝑙𝑔′ = 𝔽𝑀 [ 𝑓 ] ★ 𝔽𝑀 [ℎ] . □

Analogous properties hold for the modified kernel, as follows.

Theorem 7.2. The modified Fourier–Moyal kernel and transformation enjoy the following proper-
ties:

• Modified hermiticity: 𝔼mod(𝑧, 𝑔) = Δ−1/2(𝑔) 𝔼mod(𝑧, 𝑔−1).

• Modified covariance: 𝔼mod(𝑧, 𝑔) = Δ1/2(ℎ) 𝔼mod(ℎ ⊲ 𝑧, ℎ𝑔ℎ−1) for all ℎ ∈ Aff.

• Modified convolution: for all 𝑓 ∈ 𝐿1(Aff, 𝑑𝑙𝑔) and ℎ ∈ 𝐿1 ∩ 𝐿2(Aff, 𝑑𝑙𝑔),

𝔽mod
𝑀
( 𝑓 ∗ ℎ) = 𝔽𝑀 [ 𝑓 ] ★ 𝔽mod

𝑀
[ℎ] = 𝔽mod

𝑀
[ 𝑓 ] ★ 𝔽𝑀 [Δ−1/2ℎ] .

Proof. The first two are elementary. For the third, first notice that

𝔽𝑀 [ 𝑓 ] (𝑧) =
∫

Aff
Tr

(
Ω(𝑧)𝑈±(𝑔)

)
𝑓 (𝑔) 𝑑𝑙𝑔 = Tr

(
Ω(𝑧)𝑈±( 𝑓 )

)
= 𝑊𝑈± ( 𝑓 ) (𝑧)

28



for 𝑧 ∈ ±Π, when 𝑓 ∈ 𝐿1(Aff, 𝑑𝑙𝑔). On the other hand, when ℎ ∈ 𝐿1 ∩ 𝐿2(Aff, 𝑑𝑙𝑔), the relation

𝑈±(ℎ) 𝑑1/2
± = 𝑑

1/2
± 𝑈±(Δ−1/2ℎ)

follows from selfadjointness of 𝑑 and the semi-invariance relation (2.15), integrated over the group.
More precisely, this equality on the domain of 𝑑1/2

± extends to the operator closures, which are
everywhere defined and Hilbert–Schmidt on K [3, 28]. Thus, over the orbits ±Π we obtain

𝔽mod
𝑀
( 𝑓 ∗ ℎ) = 𝑊

𝑈± ( 𝑓 )𝑈± (ℎ)𝑑1/2
±

= 𝑊
𝑈± ( 𝑓 ) ★𝑊𝑈± (ℎ)𝑑1/2

±
= 𝔽𝑀 [ 𝑓 ] ★ 𝔽mod

𝑀
[ℎ]

= 𝑊
𝑈± ( 𝑓 )𝑑1/2

± 𝑈± (Δ−1/2ℎ) = 𝑊𝑈± ( 𝑓 )𝑑1/2
±
★𝑊

𝑈± (Δ−1/2ℎ) = 𝔽mod
𝑀
[ 𝑓 ] ★ 𝔽𝑀 [Δ−1/2ℎ] . □

The modifications make the nontrivial theorems of Fourier analysis available: namely, the
analogue of the Schur orthogonality relations for compact groups (the𝑈± are discrete-series repre-
sentations), the Fourier inversion theorem and the Plancherel–Parseval unitarity relation. (See [31]
for the compact semisimple case and an application of it.) To state them, we extend the measure
𝜔 = 𝜔+ ∪ 𝜔− from Π ∪ −Π to aff∗ by declaring the complement ℝ to be a nullset.

Theorem 7.3. These additional properties hold for the modified Fourier–Moyal kernel and trans-
formation:

• Orthogonality:
∫

Aff
𝔼mod(𝑧, 𝑔) 𝔼mod(𝑤, 𝑔) 𝑑𝑙𝑔 = |𝑦 | 𝛿(𝑧 − 𝑤) for 𝑧, 𝑤 ∈ ±Π.

• Inversion theorem: 𝑓 (𝑔) =
∫
aff∗

𝔼mod(𝑧, 𝑔) 𝔽mod
𝑀
[ 𝑓 ] (𝑧) 𝑑𝜔(𝑧) whenever 𝑓 ∈ D(Aff).

• Plancherel formula:
∫
aff∗

��𝔽mod
𝑀
[ 𝑓 ] (𝑧)

��2 𝑑𝜔(𝑧) = ∫
Aff
| 𝑓 (𝑔) |2 𝑑𝑙𝑔whenever 𝑓 ∈ 𝐿2(Aff, 𝑑𝑙𝑔).

Proof. For the Plancherel formula, it is enough to show the relation for 𝑓 ∈ D(Aff); the extension
to 𝐿2(Aff, 𝑑𝑙𝑔) by unitarity is immediate.

Take 𝑧 = 𝑥 + 𝑖𝑦, 𝑤 = 𝑢 + 𝑖𝑣 in Π; the case of 𝑧, 𝑤 ∈ −Π is similar. The orthogonality relation is
straightforward:∫

Aff
𝔼mod(𝑧, 𝑔) 𝔼mod(𝑤, 𝑔) 𝑑𝑙𝑔 =

∫
Aff

√
𝑦𝑣 𝜆(log 𝑎) 𝑒2𝜋𝑖((𝑥−𝑢) log 𝑎+𝑏(𝑣−𝑦)𝜆(log 𝑎)/𝑎) 𝑑𝑎 𝑑𝑏

𝑎2

=

∫ ∞

0

√
𝑦𝑣 𝑒2𝜋𝑖(𝑥−𝑢) log 𝑎 𝛿(𝑦 − 𝑣) 𝑑𝑎

𝑎
= |𝑦 | 𝛿(𝑥 − 𝑢) 𝛿(𝑦 − 𝑣).

The result obviously implies that

Ω(𝑧) =
∫

Aff
𝔼mod(𝑧, 𝑔)𝑈 (𝑔) 𝑑1/2 𝑑𝑙𝑔,

where the ± signs have been omitted.

29



For the inversion formula, take 𝑓 ∈ D(Aff); then∫
aff∗

𝔼mod(𝑧, 𝑔) 𝔽mod
𝑀
[ 𝑓 ] (𝑧) 𝑑𝜔(𝑧) =

∫
aff∗

∫
Aff

𝔼mod(𝑧, 𝑔) 𝔼mod(𝑧, 𝑔′) 𝑓 (𝑔′) 𝑑𝑙𝑔′ 𝑑𝜔(𝑧)

=

∫
Π∪−Π

|𝑦 |
∫

Aff

√︁
𝜆(log 𝑎)

√︁
𝜆(log 𝑎′) 𝑒−2𝜋𝑖(𝑥(log 𝑎−log 𝑎′)+𝑦(𝑏𝜆(log 𝑎)/𝑎−𝑏′𝜆(log 𝑎′)/𝑎′))

× 𝑓 (𝑎′, 𝑏′) 𝑑𝑎
′ 𝑑𝑏′

𝑎′2
𝑑𝑥 𝑑𝑦

𝑦

=

∫
ℝ

∫
Aff
𝜆(log 𝑎′) 𝛿(𝑎 − 𝑎′) 𝑒−2𝜋𝑖𝑦(𝑏−𝑏′)𝜆(log 𝑎′)/𝑎′ 𝑓 (𝑎′, 𝑏′) 𝑑𝑎

′ 𝑑𝑏′

𝑎′
𝑑𝑦

=

∫
ℝ

∫
ℝ

𝜆(log 𝑎)
𝑎

𝑒−2𝜋𝑖𝑦(𝑏−𝑏′)𝜆(log 𝑎)/𝑎 𝑓 (𝑎, 𝑏′) 𝑑𝑏′ 𝑑𝑦 = 𝑓 (𝑎, 𝑏).

We note the agreeable similarity between the Fourier transformation and cotransformation.
Finally, the Plancherel relation follows directly from the inversion theorem. Again we take

𝑓 ∈ D(Aff), for simplicity:∫
aff∗

��𝔽mod
𝑀
[ 𝑓 ] (𝑧)

��2 𝑑𝜔(𝑧) = ∫
aff∗

∫
Aff

𝑓 (𝑔) 𝔼mod(𝑧, 𝑔) 𝔽mod
𝑀
[ 𝑓 ] (𝑧) 𝑑𝑙𝑔 𝑑𝜔(𝑧)

=

∫
Aff

𝑓 (𝑔) 𝑓 (𝑔) 𝑑𝑙𝑔.

We invite the reader to make a direct proof of this; it proceeds along the lines of the inversion
formula, and is even shorter. □

Recall that in the case of a compact semisimple group, the Plancherel measure 𝑑𝜔(𝑧) is supported
on the integral coadjoint orbits O 𝑗 , on each of which the formal dimension 𝑑 𝑗 is a constant. By
taking 𝔽mod

𝑀
(𝑧, 𝑔) := 𝑑

1/2
𝑗

𝔽𝑀 (𝑧, 𝑔) when 𝑧 ∈ O 𝑗 , one recovers the usual aspect of the Plancherel
formula for ∥ 𝑓 ∥2 as a weighted sum of integrals over these orbits [31]. The role of the Duflo–Moore
operator as a formal dimension operator is transparent in our context.

Using Moore’s concept of reduced character [60], defined for all 𝑓 ∈ D(Aff), one can establish
a character property for 𝔽mod

𝑀
. We forgo this. Finally, the right Fourier–Moyal kernel and trans-

formation may be modified in the same way, leading to altogether analogous harmonic analysis
properties, mutatis mutandis.

8 Discussion
8.1 The Fronsdal program and differential equations for the Fourier–Moyal kernel

In the terminology of [36], our 𝔼-function is a★-representation, that is, it satisfies equation (7.6). In
view of the first part of Theorem 7.1, we have a symmetric★-representation (here called hermitian)
in the sense of that reference. Such ★-representations are intrinsic objects on coadjoint orbit,
introduced by Fronsdal as a (putative) lifting to the group level of the ★-exponentials of [9],
which play a fundamental role in the theory of star products. They fulfil systems of differential
equations. Concretely, Fronsdal’s generic proposal for the ★-representation kernel is the locally

30



given ★-exponential:

𝔼𝐹 (𝐹, 𝑔) = exp★[2𝜋𝑖𝑋] (𝐹) :=
∞∑︁
𝑛=0

(2𝜋𝑖𝑋)★𝑛 (𝐹)
𝑛!

, if 𝑔 = 𝑒𝑋 . (8.1)

The coefficient 2𝜋 thrown in here is convenient, given our definitions. Good treatments of the
★-exponential are given by Arnal [5] and Gutt [48]. One readily sees that this object satisfies
formally the equation of a ★-representation:

𝔼𝐹 (·, 𝑔) ★𝔼𝐹 (·, 𝑔′) = 𝔼𝐹 (·, 𝑔𝑔′). (8.2)

From the covariance relation (8.2) one derives ordinary PDE for this type of ★-representation
kernel, that may be sufficient to determine it under favourable circumstances. Substituting 𝑒𝑡𝑋 for
𝑔 and 𝑔′, for any 𝑋 ∈ g one obtains by differentiation of (8.2) at the formal level,

[𝑋,𝔼𝐹]★ = [𝑟 (𝑋) − 𝑙 (𝑋)]𝔼𝐹 , (8.3)

with 𝑙 (𝑋), 𝑟 (𝑋) respectively the corresponding left- and right-invariant vector fields. We proceed
now directly on aff and use its standard basis; then 𝑋1 ≡ 𝑥, 𝑋2 ≡ 𝑦. Thus because of the invariance
formulas (5.6), there must hold in our case:

𝑦
𝜕

𝜕𝑦
𝔼(𝑥, 𝑦; 𝑎, 𝑏) =

[
𝑟 (𝑋1) − 𝑙 (𝑋1)

]
𝔼(𝑥, 𝑦; 𝑎, 𝑏) = 𝑏 𝜕

𝜕𝑏
𝔼(𝑥, 𝑦; 𝑎, 𝑏),

𝑦
𝜕

𝜕𝑥
𝔼(𝑥, 𝑦; 𝑎, 𝑏) =

[
𝑙 (𝑋2) − 𝑟 (𝑋2)

]
𝔼(𝑥, 𝑦; 𝑎, 𝑏) = (𝑎 − 1) 𝜕

𝜕𝑏
𝔼(𝑥, 𝑦; 𝑎, 𝑏). (8.4)

This is the fundamental Fronsdal differential system for Aff. Direct inspection of the explicit
form (7.4) of our 𝔼 shows that these equations are indeed fulfilled. We already saw in subsection 7.3
that the analogue of (8.2) is satisfied by our 𝔼 as well.

Following the Fronsdal program, invariant affine ★-quantization was studied in [51]; the latter
is the oldest work on quantization based on the affine group of which we are aware. Equations (8.4)
coincide with equations (2.9) of [51], when allowance is made for a slightly different definition of
the affine group multiplication. Of course, our focus in this paper is on the tracial property rather
than general covariance. Thus we did obtain a distinguished solution.

8.2 Relation with the formalism of Ali et al

In [2, 3, 54], taken in the context of the affine group of the line, the Wigner functions are indirectly
defined as the images of the map

HS(K+) ⊕ HS(K−) → 𝐿2(aff∗, 𝑑𝜔+ ∪ 𝑑𝜔−),

obtained by composition of the inverse Plancherel transformation and the 𝔽𝐴𝐹𝐾-transformation
already given in subsection 2.3:

𝑊 [𝐴] = 𝔽𝐴𝐹𝐾 [𝑃−1(𝐴)] .
These authors furthermore propose “formal Wigner operators”𝑊 (𝐹) via the property

Tr(𝐴𝑊 (𝐹)) := 𝑊 [𝐴] (𝐹).

31



In view of the results in subsection 7.3, it is clear that the dequantization 𝑊𝐴 in (1.1) and in
subsection 4.5 is the same as𝑊 [𝐴], and the formal Wigner operators𝑊 (𝐹) are just the Stratonovich–
Weyl (de)quantizers Ω(𝐹), which are not merely formal at all, and have been explicitly calculated
in this paper. It is remarkable that the integral expression (49) for𝑊 (𝐹) in the first reference in [2]
makes manifest use of the Duflo–Moore operators, whereas ours does not; they however coincide.

Nevertheless, the difference between our axiomatic approach and the treatment based on the
Plancherel transform and the a priori Fourier transformation (2.12) is not moot. Only the coadjoint
orbits have an interpretation as elementary physical systems (see [19] in this respect); the coalgebra
g∗ by itself is an empty vessel. Now, the second definition raises the problem of the eventual
indecomposability of 𝑊 [𝐴] on the coadjoint orbits; or, on account of the Kirillov map, on the
unirreps. This indecomposability actually happens [3]. Moreover the definition of 𝔽𝐴𝐹𝐾 will not
do for compact groups; and the character formula à la Kirillov is lost with it, anyway. It seems
preferable to accept that the Stratonovich–Weyl quantizer generally determines the correct scalar
Fourier transform, rather than the other way around, and that for a non-unimodular group there are
four such pertinent objects: 𝔽𝑀 , 𝔽mod

𝑀
and 𝔽 𝑟

𝑀
, 𝔽mod,𝑟

𝑀 .
In other words, our approach is geared to fit better with Kirillov theory. It has an obvious

drawback, in that no one knows precisely for which categories of groups and unirreps do quantizers
exist (for non-type-I groups there is no hope whatsoever). We give an aperçu of the question in the
following subsection, through the story of Stratonovich–Weyl operators so far.

To conclude, note that our Weyl Ansatz for 𝑄( 𝑓 ) can be rewritten in the form

𝑄( 𝑓 ) =
∫
aff
𝐹 [ 𝑓 ] (𝑢, 𝑣)𝑈±(𝑒𝑢, 𝑣𝑒𝑢/𝜆(𝑢)) 𝑑𝑢 𝑑𝑣,

where 𝐹 is the ordinary Fourier transformation between functions on aff∗ and on aff, and to use it,
we extend 𝑓 by zero on the complement of Π. Thus, the quantization prescription is not unrelated to
the proposal of Manchon [56] for Weyl quantization of solvable Lie groups – which however ignores
the issue of supports within coadjoint orbits, needed to establish boundedness or compactness of
the quantized operators.

8.3 Setting the record straight

The concept of Stratonovich–Weyl quantizer was introduced in the late eighties [45,77] by two of us.
In [46, Sect. 3.5] we reported that the name had not caught on, and called them “Moyal quantizers”
instead. But the concept itself certainly did catch on, and beyond [15], which inaugurated a wealth
of applications, we find it in [2, 54] under the name “Wigner operators”. Lest that nomenclature
be misread as a priority claim, it seems wise to revert to form. We still speak here of Moyal-type
quantization for tracial quantization, and of Fourier–Moyal kernels and transformations.

The main motivation for the early works was to extend the remit of phase-space Quantum
Mechanics. In particular, tracial twisted products covariant under SU(2), for dealing with spinning
particles, were developed in full detail, including applications, in [77]. There we were elaborating on
old work by Stratonovich [73] – who should thus be credited with introducing the “fuzzy sphere” –
and were unaware of another precedent [1]. An equivalent version of the SU(2)-Stratonovich–Weyl
quantizer, simpler than our original expression, is given in [49].

The Stratonovich–Weyl quantizer appropriate to deal with relativistic particles [19] was devel-
oped shortly after [45, 77]. Indeed, the prevalence of the Heisenberg groups in quantization is an

32



artifact. From the physical viewpoint, the coadjoint orbit for the 7-dimensional Heisenberg group
makes its appearance as a direct factor of the splitting group G̃al of the covering group of the Galilei
group [17], that linearizes its multiplier representations. Thus the restriction of the quantizer for
G̃al to the flat part of the orbits renders the standard Moyal quantizer [45]; for an explicit calculation
showing the multiplier Galilean covariance of the ordinary Moyal framework, peruse [65]. All this
often goes unremarked.

Stratonovich–Weyl quantizers exist for all compact Lie groups. This was shown in principle
in [31] by the time-honoured method of interpolating between the “active” and “passive” symbols
associated to semitracial quantizers. Then in the nineties, apparently unaware of that work, N. V.
Pedersen introduced a similar set of postulates, and proved the existence of Stratonovich–Weyl
quantizers, for nilpotent Lie groups [64]. See also [37, Sect. 4.5] on this matter. Prior to all that, the
Unterbergers [76] had shown by the interpolation method the existence of a Moyal-type quantization
for the discrete-series representations of SL(2,ℝ). In this regard, we wish to mention [66] as well.
(To our knowledge, however, no one has been able to exhibit explicitly the Stratonovich–Weyl
quantizers for this case.) These older examples and the work of Ali and coworkers indicate that
Stratonovich–Weyl quantizers exist for large classes of semi-direct product groups. The time seems
ripe for a renewed assault on Moyal-type quantization and scalar group Fourier transforms covariant
under larger classes of solvable and reductive groups.

Fourier–Moyal kernels are arguably even more important than Stratonovich–Weyl quantizers,
because of their crucial role in harmonic analysis. They seem destined to complete Kirillov theory.
For years, the abstract nature of expansions of functions on Lie groups in terms of equivalence
classes of unitary transformations has been a source of some dissatisfaction [50]. However, one
still finds the Plancherel measure usually realized on 𝐺, rather than on g∗. For compact group
symmetry, we demonstrated in [31] how the Fourier–Moyal transformation solves the problem of
giving a formulation of harmonic analysis parallel to standard Fourier analysis. This section and
the previous one show a wider applicability of its method; and, although here we have opted for
concrete proofs, there is a good chance, in view of the Leptin–Ludwig theorem, that similar results
are valid for all exponential groups. To finish, we should mention that a bit earlier – see [6] and
references therein – a concept of scalar “adapted Fourier transform” had been proposed; because of
covariance trouble it actually does not seem to be all that well adapted to the context.

9 Spectral triples on the half-plane
We turn at last to noncommutative spectral triples. The upper half-planeΠ is a model for the simplest
hyperbolic geometry, living on a Riemannian surface with negative constant scalar curvature. It
may be regarded as a homogeneous space of the group SL(2,ℝ), acting by Möbius transformations
𝑧 ↦→ (𝑎𝑧 + 𝑏)/(𝑐𝑧 + 𝑑) on Π.

Writing a typical element of the Iwasawa decomposition SL(2,ℝ) = 𝐴𝑁𝐾 as

𝑔 = 𝑎𝑡𝑛𝑠𝑘𝜃 =

(
𝑒𝑡/2 0
0 𝑒−𝑡/2

) (
1 𝑠

0 1

) (
cos 1

2𝜃 sin 1
2𝜃

− sin 1
2𝜃 cos 1

2𝜃

)
,

the compact subgroup 𝐾 = SO(2) fixes 𝑖 and thus Π ≈ SL(2,ℝ)/SO(2) is a principal homogeneous
space for the subgroup 𝐴𝑁 . The orbits of 𝐴 are half-lines emanating from 0, and the orbits of 𝑁
are horizontal lines: see Figure 3.

33



•𝑖

Figure 3: Orbits of 𝐴 and 𝑁 for the half-plane

Under the identification

𝐴𝑁 ∋ 𝑎𝑡𝑛𝑠 ←→ 𝑎𝑡𝑛𝑠 · 𝑖 = 𝑎𝑡 · (𝑖 + 𝑠) = 𝑠𝑒𝑡 + 𝑖𝑒𝑡 ≡ 𝑥 + 𝑖𝑦 = 𝑧 ∈ Π, (9.1)

we may transfer the group operation of 𝐴𝑁 to Π by letting

(𝑥 + 𝑖𝑦) • (𝑥′ + 𝑖𝑦′) := 𝑥 + 𝑥′𝑦 + 𝑖𝑦𝑦′.

On the other hand, we may identify 𝐴𝑁 with Aff by the group isomorphism

𝑎𝑡 𝑛𝑠 ↦−→
(
𝑒𝑡 −𝑠𝑒𝑡
0 1

)
. (9.2)

Observe, however, that the product • induced by the (left) Möbius action of Aff on Π is opposite to
that of (2.7), induced by the (left) coadjoint action of Aff on Π. This is to be expected, since (9.1)
and the identification (9.2) leads to (𝑎, 𝑏) = (𝑦,−𝑥), which is 𝑔−1

𝑧 with the definition (2.6).
The spinor bundle 𝑆 → Π over the Poincaré half-plane has rank two and is the direct sum of

two trivial line bundles, since Π is contractible. Let H0 := 𝐿2(Π, 𝑦−2 𝑑𝑥 𝑑𝑦) and let H := H0 ⊕H0
be the Hilbert space of spinors. The Dirac operator is /𝐷 := −𝑖(𝜎+ ∇𝑆𝐸+ + 𝜎− ∇

𝑆
𝐸−
), using the

isotropic basis 𝐸+ := 2𝑦 𝜕𝑧 = 𝑦(𝜕𝑥 − 𝑖 𝜕𝑦), 𝐸− := 2𝑦 𝜕𝑧 = 𝑦(𝜕𝑥 + 𝑖 𝜕𝑦), and the spin connection
∇𝑆
𝐸𝜌

:= 𝐸𝜌 − 1
4 Γ̂

𝛽
𝜌𝛼 𝜎

𝛼𝜎𝛽 is determined by the Christoffel symbols of the Levi-Civita connection,
for any zweibein {𝐸𝜌}. The Levi-Civita connection is canonical here because Π is a symmetric
space [68]. Standard formulas [46, 62] then yield

/𝐷 = −𝑖
(

0 2𝑦 𝜕𝑧 + 𝑖
2

2𝑦 𝜕𝑧 − 𝑖
2 0

)
.

Following Palmer et al [63], one can find a representation 𝜏 of SL(2,ℝ) on H under which /𝐷
is invariant. It can be written in the form

[𝜏(𝑔)𝜓] (𝑧) := 𝑢(𝑔, 𝑧) 𝜓(𝑔−1 · 𝑧),

34



where the factor 𝑢 is of the form

𝑢(𝑔, 𝑧) :=
(
𝑣(𝑔−1, 𝑧)1/2 0

0 𝑣(𝑔−1, 𝑧)−1/2

)
, where 𝑣(𝑔, 𝑧) :=

𝑐𝑧 + 𝑑
𝑐𝑧 + 𝑑 for 𝑔 =

(
𝑎 𝑏

𝑐 𝑑

)
,

whenever the square root of 𝑧 ↦→ 𝑣(𝑔, 𝑧) can be chosen smoothly, e.g., for 𝑔 lying in suitable
one-parameter subgroups. A set of infinitesimal generators 𝐹0, 𝐹1, 𝐹2 representing sl(2,ℝ) is found
to be

𝐹0 = −1
2 (1 + 𝑧

2) 𝜕𝑧 − 1
2 (1 − 𝑧

2) 𝜕𝑧 + 1
4 (𝑧 − 𝑧) 𝜎3,

𝐹1 = −(𝑧 𝜕𝑧 + 𝑧 𝜕𝑧) = −(𝑥 𝜕𝑥 + 𝑦 𝜕𝑦), 𝐹0 + 𝐹2 = −𝜕𝑥 .

The invariance of /𝐷 follows directly from the relations [𝐹𝑗 , /𝐷] = 0. For instance, [𝐹1, /𝐷] vanishes
because [𝑥 𝜕𝑥 + 𝑦 𝜕𝑦, 𝑦 𝜕𝑥 ∓ 𝑖𝑦 𝜕𝑦] = 0. One can observe that the components of /𝐷 are left-invariant
differential operators on Π, regarded as a group. This is why they commute with the fundamental
vector fields 𝐹1 and 𝐹0 + 𝐹2, which are of course right-invariant [18].

▶ Now let a suitably chosen algebra of functions on Π, under the twisted product, act diagonally
on spinors. This defines on Π a noncommutative operator module in the sense of [8]. That is to
say, there is a noncommutative algebra (A, ★) involutively represented by bounded operators on a
Hilbert space, and a self-adjoint operator 𝐷 on the same Hilbert space, unbounded in the present
case, whose domain is preserved by the action of A.

We shall now show that the basic pre-condition for a spectral triple holds, to wit, the commutator
of 𝐷 with the twisted multiplication by elements of A is bounded.

The expression (5.3) may be applied componentwise when the function ℎ is replaced by the
two-spinor 𝜙, namely:

𝑅𝑤𝜙(𝑧) =
(
𝜙1(𝑧 · 𝑤)
𝜙2(𝑧 · 𝑤)

)
=

(
𝜙1(𝑤 • 𝑧)
𝜙2(𝑤 • 𝑧)

)
=: 𝐿•𝑤𝜙(𝑧),

with an obvious notation. Now, because the factor 𝑢(𝑔, 𝑧) is trivial for 𝑔 ∈ 𝐴𝑁 , the invariance of /𝐷
under all 𝜏(𝑔) allows us to conclude that 𝐿•𝑤 /𝐷 = /𝐷𝐿•𝑤, and thus

/𝐷 ( 𝑓 ★ 𝜙) (𝑧) − 𝑓 ★ /𝐷𝜙(𝑧) =
∬

Π2
𝐾★(𝑖, 𝑤, 𝑡)

( /𝐷𝑧 (𝐿•𝑤 𝑓 (𝑧) 𝐿•𝑡 𝜙(𝑧)) − 𝐿•𝑤 𝑓 (𝑧) /𝐷𝑧 𝐿
•
𝑡 𝜙(𝑧)

)
𝑑𝑙𝑤 𝑑𝑙𝑡

=

∬
Π2
𝐾★(𝑖, 𝑤, 𝑡) 𝐿•𝑤 /𝐷 𝑓 (𝑧) 𝐿•𝑡 𝜙(𝑧) 𝑑𝑙𝑤 𝑑𝑙𝑡 = /𝐷 𝑓 ★ 𝜙(𝑧),

where the second equality uses the invariance of the Dirac operator under the (restriction to 𝐴𝑁

of) the SL(2,ℝ)-action by Möbius transformations on its orbit Π. Thus the Leibniz rule is valid
for the action of /𝐷 on the (left-covariant for the coadjoint action, right-covariant for the Möbius
action) twisted product. In particular, if 𝐿★( 𝑓 ) denotes the operator of left twisted multiplication,
then [ /𝐷, 𝐿★( 𝑓 )] = 𝐿★( /𝐷 𝑓 ) is bounded whenever 𝑓 has a bounded exterior derivative. Moreover,
arguing as in [38], we also get Connes’ first-order condition [21] from the associativity and complex
conjugation properties of the twisted product. Needless to say, from here to showing or disproving
that the Moyal half-plane in our sense is a noncommutative geometry, there is still some way to
travel.

35



Some reflection on what has been achieved – and what has not – is in order. The half-plane
carries a natural symmetry, namely the left Möbius action of SL(2,ℝ). However, in order to
preserve the Leibniz rule, we have been led to an algebra which is right-invariant, rather than
left-invariant, under (the restriction to the subgroup 𝐴𝑁 of) that action. To our knowledge, this was
first stated in [14], which moreover contains a beautiful study of the differential equations a general
covariant trikernel must satisfy. If one insists on having left-invariance of the algebra under the
Möbius action, one can bring into play instead the Dirac operator associated to the right-invariant
metric on 𝐴𝑁 , which is given by

𝐷\ = −𝑖
(

0 (𝑥 − 𝑖) 𝜕𝑥 + 𝑦 𝜕𝑦 + 1
2

(𝑥 + 𝑖) 𝜕𝑥 + 𝑦 𝜕𝑦 + 1
2 0

)
.

Alternatively, one might try to deform the Dirac operator itself.

10 Outlook
We conclude with a brief review of possible ramifications for our work in this paper.

• Harmonic analysis by way of the scalar Fourier kernels can of course be pursued much further,
around standard lines. For instance, the third assertion in Theorem 7.2 remains true when 𝑓

is a bounded measure. In general, one uses the power 𝑑1/𝑝
± of the dimension operators for 𝐿𝑝-

Fourier analysis. Some matters of rigour – see the remark at the end of subsection 4.5 – will
be treated separately [41]. More to the point, the role of the Fourier–Moyal transformation
in relating Wildberger’s group class and coadjoint orbit hypergroups [83], arguably in the
spirit of [35], is an appealing subject of research. See, with regard to Wildberger’s theory, the
remarks in [53, Sect. 6.4].

• The issue of SL(2,ℝ) symmetry for a satisfactory star product is not ended. It should be
obvious that 𝐾𝑅★ can be extended to define semitracial, SL(2,ℝ)-covariant star products on the
half-plane. That should allow a fresh attack on the determination of the Stratonovich–Weyl
quantizers for this group [13]. A solution would bring the prize of a suitable and strong
definition of noncommutative Riemann surfaces.

• There seems to be no obstruction to the generalization of our method for constructing star
products on 𝐴𝑁-symmetric spaces, on the basis of the Iwasawa decomposition. For complex
groups, this leads naturally to Manin triples – see for instance [16].3

Acknowledgments

Most of this work was done at Universidad Complutense of Madrid (UCM), at a period when the
second named author was a staff member there. We are thankful for helpful discussions to Paolo
Aniello, Pierre Bieliavsky, Alain Connes, Bruno Iochum, Fedele Lizzi, Giuseppe Marmo, Patrick
Várilly, Jasson Vindas, and Patrizia Vitale. Special thanks are due to André Unterberger, who
shared with us his unpublished manuscript [75], after the first version of our article was written. VG
thanks the UCM for hospitality. JMG-B acknowledges partial support from CICyT, Spain, through

3Patrizia Vitale pointed this out to us.

36



the grant FIS2005–02309, and is grateful to Università di Napoli Federico II for warm hospitality.
JCV is grateful to UCM and to Banco Santander for a Visitante Distinguido fellowship, and thanks
the Universidad de Zaragoza and SISSA, Trieste, for friendly hospitality. Support for JCV from the
Universidad de Costa Rica is acknowledged. We also thank the referee, whose comments helped to
improve the final manuscript.

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41