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How to cite: E. Kurniadi, N. Gusriani, and B. Subartini, “Duflo-Moore Operator for The Square-Integrable
Representation of the 2-Dimensional Affine Lie Group”, J. Mat. Mantik, vol. 6, no. 2, pp.114-122, October
2020.
Jurnal Matematika MANTIK
Vol. 6, No. 2, October 2020, pp. 114-122
ISSN: 2527-3159 (print) 2527-3167 (online)
Duflo-Moore Operator for The Square-Integrable Representation
of the 2-Dimensional Affine Lie Group
Edi Kurniadi1, Nurul Gusriani2, Betty Subartini3
1Universitas Padjadjaran Bandung, edi.kurniadi@unpad.ac.id
2Universitas Padjadjaran Bandung, nurul.gusriani@unpad.ac.id
3Universitas Padjadjaran Bandung, betty.subartini@unpad.ac.id
doi: https://doi.org/10.15642/mantik.2020.6.2.114-122
Abstrak. Dalam artikel ini, dipelajari representasi quasi-regular dan representasi unitar tak tereduksi
grup Lie affine Aff +(1) berdimensi dua. Pertama, diberikan bukti lengkap dari hasil kerja Fuhr
tentang transformasi Fourier untuk representasi quasi-regular dari Aff +(1). Kedua, ketika
representasi dari grup Lie affine Aff +(1) adalah square-integrable maka dihitung operator Duflo-
Moore secara langsung tanpa menggunakan transformasi Fourier seperti dalam hasil Fuhr.
Kata kunci: Grup Lie affine; Operator Duflo-Moore; Representasi square-integrable.
Abstract. In this paper, we study the quasi-regular and the irreducible unitary representation of
affine Lie group Aff +(1) of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier
transform of quasi-regular representation of Aff +(1). The second, in such the representation of
affine Lie group Aff +(1) is square-integrable then we compute its Duflo-Moore operator instead
of using Fourier transform as in Führ’s work.
Keywords: Affine Lie group; Duflo-Moore operator; Square-integrable representation.
http://u.lipi.go.id/1458103791
E. Kurniadi, N. Gusriani, B. Subartini
Duflo-Moore Operator for The Square-Interable Representation of 2-Dimensional Affine Lie Group
115
1. Introduction
The current research about square-integrable representations of Lie groups can be
found, for instance in [1] and [2]. In the previous work, the notion of square-integrable
representation of a Lie group associating to wavelet transforms was introduced by
Grossmann, Morlet, and Paul (see [3]). Particularly, they investigated the nice examples
of a square-integrable representation of 𝑎𝑥 + 𝑏- group, known as affine Lie group
Aff(1) as can be seen in [4]. In the other hand, the research about 𝑎𝑥 + 𝑏-groups can also
be found, for instance in [5] and [6].
It is well known that Aff(1) is the exponential solvable Lie group which is non
unimodular group whose Lie algebra of Aff(1) is Frobenius. Other examples are
parabolic subgroups which are Fobenius as well (see [7] and[8]). But we thought that
Grossmann’s work is the best example for young researchers how to understand the
square-integrable representations for case nonunimodular groups which is started from
the Aff(1) Lie group. Moreover, other examples of nonunimodular groups are Lie groups
whose Lie algebras are 4-dimensional real Frobenius Lie algebras. Kurniadi and Ishi [9]
showed that irreducible unitary representations of these Lie groups are square-integrable
representations and they wrote the Duflo-Moore operators in the terms of groups Fourier
transforms.
Many reseachers study affine Lie algebras and the structure of affine for instance we
see some results in [10], [11], [12], [13],[14], [15], [16], and [17].
In the other hand, in easier stage we can also study square-integrable representations
for unimodular Lie groups case. Heisenberg Lie groups of dimension 2𝑛 + 1 and filiform
Lie groups are in these types. In fact, the Duflo-Moore operators for square-integrable
representations of unimodular groups are scalar multiple (see [18]). In current work,
Kurniadi in [19] proved that irreducible-unitary representation of Lie group of 4-
dimensional standard filiform Lie algebra is square-integrable and its Duflo-Moore
operator is scalar multiple of identity which is equal to one.
In this work, we shall give another alternative to compute the Duflo-Moore operator
for square-integrable representation of Aff +(1) by direct computations instead of forming
in group Fourier transform which was written in [18].
2. Preliminaries
Let Aff +(1) be the 2-dimensional affine Lie group whis is expressed as a semidirect
product of the set of all real numbers ℝ and the set of all positive real numbers ℝ+.
Namely, we can write this group as Aff +(1) ≔ ℝ ⋊ ℝ+. Particularly, in this work we
concentrate to Aff +(1) which is the exponential solvable nonunimodular Lie group. To
make easier in computations we write Aff +(1) in matrix terms. Namely, we have
Aff +(1) ∋ (
𝛼 𝛽
0 1
) , 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ. (1)
Regarding this notations, we denote 𝑔(𝛼, 𝛽) ≔ (
𝛼 𝛽
0 1
), Δ(𝛼) ≔ (
𝛼 0
0 1
), and ∇(𝛽) ≔
(
1 𝛽
0 1
). The Lie algebra of Aff +(1) is denoted by aff(1) whose basis is {𝑒1, 𝑒2}. The
nonzero bracket of aff(1) is given by [𝑒1, 𝑒2] = 𝑒2. The Lie algebra aff(1) is a Frobenius
Lie algebra which has two open coadjoint orbits as follows (see [20]).
Ω±: = {(𝑎, 𝑏) ; 𝑎, 𝑏 ∈ ℝ, ± 𝑏 > 0}. (2)
Jurnal Matematika MANTIK
Volume 6, No. 2, October 2020, pp.114-122
116
The representations of the affine Lie group Aff(1) can be realized on the Hilbert
space of all square-integrable functions L2(ℝ+). Before doing that, let us mention here
some basic notion of representation theory of Lie groups corresponding to our research.
Definition 1 [21]. Let π be a representation of a Lie group G on the carrier space ℋ. π is
said to be irreducible if π has no nontrivial π-invariant subspace ℋ0 in ℋ. Moreover, π
is said to be uintary if for each f ∈ ℋ and each g ∈ G
‖𝜋(𝑔)𝑓‖ = ‖𝑓‖. (3)
Proposition 2 [20]. The irreducible unitary representations of Aff +(1) correponsding to
open coadjoint orbit Ω+ in eqs. (2) in the space L
2(ℝ+) is of the form
𝜋+(𝑔)𝑓(𝑥) = 𝑒
2𝜋𝑖𝛽𝑥 𝑓(𝛼𝑥), (4)
where 𝑔 ≔ 𝑔(𝛼, 𝛽) ∈ Aff +(1) , 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ and 𝑓 ∈ L
2(ℝ+).
Furthermore, the representation of affine Lie group Aff +(1) can be realized as a
quasi-regular representations (see [18]). It is written in the formula as follows.
𝜋(𝑔(𝛼, 𝛽)) = 𝛼
−
1
2𝜓(
𝑥−𝛽
𝛼
), 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ and 𝜓 ∈ L
2(ℝ+). (5)
We are mainly interested in the square-integrable representation. Let 𝜋 be an
irreducible unitary representation of a Lie group 𝐺 realized on the space ℋ and L2(𝐺) be
the space of all square-integrable functions on 𝐺. For vector 𝑓1 ∈ ℋ, we define the
operator on ℋ given by
ℰ𝑓1 : ℋ ∋ 𝑓2 ↦ ℰ𝑓1 𝑓2 ∈ L
2(𝐺). (6)
where ℰ𝑓1 𝑓2(𝑥) = 〈𝑓1|𝜋(𝑥)𝑓2〉.
Definition 3 [22].The irreducible unitary representation π of locally compact topological
group G realized on a space ℋ is said to be square-interable if there exist two vectors
f1, f2 ∈ ℋ − {0} such that
‖ℰ𝑓1 𝑓2‖
2
= 〈𝑓1|𝜋(𝑥)𝑓2〉 = ∫ 𝑓1(𝑔)𝜋(𝑥)𝑓2(𝑔)
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ dμ(𝑔)
𝐺
< +∞. (7)
In the other words, 〈f1|π(x)f2〉 ∈ L
2(G, μG) where μG is a measure on G. Such vectors
which satisfied eqs. (7) are called admissible vectors.
Duflo-Moore state their results in the following theorem
Theorem 4 [23]. If π is square-integrable representations of locally compact group G
realized on the space ℋ then there exists a positive selfadjoint operator Cπ: ℋ → ℋ
which is called the Duflo-Moore operator such that
a. a vector ψ ∈ ℋ − {0} is admissible if and only if ψ is an element of domain of Cπ.
b. if f1, f2 ∈ ℋ and f3, f4 ∈ Dom (Cπ) then
〈ℰ𝑓1 𝑓3|ℰ𝑓2 𝑓4〉L2(𝐺,𝜇𝐺) = 〈𝑓1
|𝑓2〉ℋ 〈𝐶𝜋𝑓4|𝐶𝜋𝑓3〉ℋ . (8)
E. Kurniadi, N. Gusriani, B. Subartini
Duflo-Moore Operator for The Square-Interable Representation of 2-Dimensional Affine Lie Group
117
2. Methods
In this research we apply the literature reviews method, particularly we focus on
results in [18] and [20]. We obtain the quasi-regular representation of Aff +(1) in Fuhr’s
work and we compute the Fourier transform of its representation to determine the Duflo-
Moore operator. On the other hand, we also obtain the irreducible unitary representation
of Aff +(1) corressponding to open coadjoint orbits and we show that representatation is
square-integrable. Using direct computations, we obtain the Duflo-Moore operator for
that representation.
3. Results and Discussion
Our results and discussion consist of two main part as follows.
3.1 The Duflo-Moore Operator for The Quasi-Regular Representation of 𝐀𝐟𝐟+(𝟏).
The following statement can be deduced from [18] in page 30--31. However, we
give a detail proof for its own interest.
Lemma 5 [18]. The Fourier transform of quasi-regular representation π of Aff +(1) as in
eqs. (5) is of the form
ℱ(𝜋(𝑔(𝛼, 𝛽))𝜓)(𝜉) = 𝛼
1
2𝑒−2𝜋𝑖𝜉𝛽 ℱ𝜓(𝛼𝜉). (9)
Proof.
By direct computation we obtain
ℱ(𝜋(𝑔(𝛼, 𝛽))𝜓)(𝜉) = ∫ 𝑒 −2𝜋𝑖𝜉𝑥 (𝜋(𝑔(𝛼, 𝛽))𝜓)(𝑥) 𝑑𝑥
ℝ
= ∫ 𝑒−2𝜋𝑖𝜉𝑥 𝛼−1/2𝜓 (
𝑥 − 𝛽
𝛼
) 𝑑𝑥
ℝ
= ∫ 𝑒−2𝜋𝑖𝜉(𝛼𝜂+𝛽)𝛼 −1/2𝜓(𝜂) 𝛼 𝑑𝜂
ℝ
( Substituting 𝜂 =
𝑥−𝛽
𝛼
)
= ∫ 𝑒−2𝜋𝑖𝜉(𝛼𝜂)𝑒−2𝜋𝑖𝜉𝛽 𝛼1/2𝜓(𝜂) 𝑑𝜂
ℝ
= ∫ 𝑒−2𝜋𝑖(𝛼𝜉)𝜂𝑒−2𝜋𝑖𝜉𝛽 𝛼1/2𝜓(𝜂) 𝑑𝜂
ℝ
= 𝑒−2𝜋𝑖𝜉𝛽 𝛼1/2 ∫ 𝑒−2𝜋𝑖(𝛼𝜉)𝜂𝜓(𝜂) 𝑑𝜂
ℝ
= 𝑒−2𝜋𝑖𝜉𝛽 𝛼
1
2ℱ𝜓(𝛼𝜉).
∎
Proposition 6 [18]. The Duflo-Moore operator for quasi-regular representation π of
Aff +(1) as in eqs. (5) in the term of Fourier transform can be written as follows.
Jurnal Matematika MANTIK
Volume 6, No. 2, October 2020, pp.114-122
118
ℱ(𝐶𝜋𝜓)(𝜉) = 𝜉
−1/2ℱ𝜓(𝜉). (10)
Proof. Let 𝜓1 and 𝜓2 be elements of contiunuous functions space of compact support on
Aff +(1) denoted by 𝐶𝑐(Aff
+(1) ). Using Plancherel’s theorem and Fubini’s theorem
we obtain
∫ |〈𝜓1|𝜋(𝑔(𝛼, 𝛽))𝜓2〉|
2
𝑑𝛼
𝛼2
Aff+(1)
𝑑𝛽 = ∫ |〈ℱ𝜓1|ℱ𝜋(𝑔(𝛼, 𝛽))𝜓2〉|
2
𝑑𝛼
𝛼2
Aff+(1)
𝑑𝛽
= ∫ |∫ ℱ𝜓1(𝜉)𝑒
−2𝜋𝑖𝜉𝛽 𝛼1/2ℱ𝜓̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝛼𝜉)
ℝ
𝑑𝜉|
2
𝑑𝛼
𝛼2
Aff+(1)
𝑑𝛽
= ∫ |∫ ℱ𝜓1(𝜉)𝑒
2𝜋𝑖𝜉𝛽 𝛼1/2ℱ𝜓̅̅ ̅̅ (𝛼𝜉)
ℝ
𝑑𝜉|
2
𝑑𝛼
𝛼2
Aff+(1)
𝑑𝛽
= ∫ ∫ |∫ ℱ𝜓1(𝜉)𝑒
2𝜋𝑖𝜉𝛽ℱ𝜓̅̅ ̅̅ (𝛼𝜉)
ℝ
𝑑𝜉|
2
ℝ+
𝑑𝛼
𝛼
ℝ
𝑑𝛽
= ∫ ∫ |ℱ𝜏𝛼 (−𝛽)|
2
ℝ+
𝑑𝛼
𝛼
ℝ
𝑑𝛽
( 𝜏𝛼 (𝜉) = ℱ𝜓1(𝜉)ℱ𝜓̅̅ ̅̅ (𝛼𝜉) )
= ∫ ∫ |ℱ𝜓1(𝜉)ℱ𝜓̅̅ ̅̅ (𝛼𝜉)|
2
ℝ+
𝑑𝛼
𝛼
ℝ
𝑑𝛽
= ∫ |ℱ𝜓1(𝜉)|
2 { ∫ |ℱ𝜓̅̅ ̅̅ (𝛼𝜉)|
2
ℝ+
𝑑𝛼
𝛼
}
ℝ
𝑑𝜉
= {∫ |ℱ𝜓1(𝜉)|
2𝑑𝜉 } { ∫ |ℱ𝜓̅̅ ̅̅ (𝛼𝜉)|
2
ℝ+
𝑑𝛼
𝛼
}
ℝ
= { ∫ |ℱ𝜓1(𝜉)|
2𝑑𝜉 } {∫ |ℱ𝜓̅̅ ̅̅ (𝛼 ′)|
2
ℝ
𝑑𝛼′
𝛼′
}
ℝ
( 𝛼′ ≔ 𝛼𝜉 ).
= ‖ℱ𝜓1‖
2 {∫ |ℱ𝜓̅̅ ̅̅ (𝛼′)|
2
ℝ
𝑑𝛼′
𝛼′
}.
Thus, from the latter equation we obtain the Duflo-Moore operator is equal to
ℱ(𝐶𝜋𝜓)(𝜉) = 𝜉
−1/2ℱ𝜓(𝜉) as desired.
∎
3.2 The Duflo-Moore Operator for The Irreducible Unitary Representation of
𝐀𝐟𝐟+(𝟏)
This session is the main result. First, we recall that the irreducible unitary
representation of group Aff +(1) in Proposition 2 can be written in the following
proposition
E. Kurniadi, N. Gusriani, B. Subartini
Duflo-Moore Operator for The Square-Interable Representation of 2-Dimensional Affine Lie Group
119
Proposition 7. The irreducible unitary representations of Aff +(1) correponsding to open
coadjoint orbit Ω+ in eqs. (2) in the space L
2(ℝ+) is of the form
𝜋+(Δ(𝛼))𝑓(𝑥) = 𝑓(𝛼𝑥),
𝜋+(∇(𝛽))𝑓(𝑥) = 𝑒
2𝜋𝑖𝛽𝑥 𝑓(𝑥), (11)
where 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ and 𝑓 ∈ L
2(ℝ+).
Proof. Let aff(1) be a Lie algebra of Aff +(1) whose basis is {𝑒1, 𝑒2}. We consider its
dual space as aff(1)∗ ∋ (
𝑎 ∗
𝑏 ∗
), where 𝑎, 𝑏 ∈ ℝ. Moreover, let (
𝛼 𝛽
0 1
) , 𝛼 ∈ ℝ+, 𝛽 ∈
ℝ be an element of group affine Aff +(1). We shall construct the irreducible unitary
representation of Aff +(1) corresponding to open coadjoint orbit Ω+ ≔ {(𝑎, 𝑏) ; 𝑏 >
0}. To do that, fix a point 𝜏 ≔ 𝑒2
∗ ∈ Ω+ ⊂ aff(1)
∗ as a linear functional. For subalgebra
ℵ ≔ 〈𝑒2〉 we have ℵ has maximal dimension and the value of linear functional 𝜏 on the
commutator [ ℵ, ℵ] is given by 𝜏([ ℵ, ℵ]) = 0. Therefore, ℵ is a polarization in aff(1).
Let ℵ⊥ be the orthogonal subspace. Furthermore, since 𝜏 + ℵ⊥ is contained in Ω+ then
ℵ satisfies Pukanszky condition.
Now we construct a 1-dimensional representation 𝜆𝜏 of Ν ≔ exp ℵ as follows.
𝜆𝜏 (exp 𝑒) ≔ 𝑒
2𝜋𝑖〈𝜏|𝑒〉 = 𝑒2𝜋𝑖𝛽 , 𝑒 ≔ 𝛼𝑒1 + 𝛽𝑒2 , 𝜏 ∈ Ω+. (12)
We identify the coset Aff +(1)/Ν by ℝ+ and we obtain the section given by
𝑠: ℝ+ ∋ 𝑥 ↦ exp 𝑥𝑒1 ∈ Aff
+(1). (13)
To obtain the explicit formula of the representation of Aff +(1) we need to solve what
we called the master equation
𝑠(𝑥)𝑔 = ℎ𝑠(𝑥, 𝑔)𝑠(𝑥𝑔), (𝑥 ∈ ℝ+, 𝑔 ∈ Aff
+(1), ℎ𝑠(𝑥, 𝑔) ∈ Ν ). (14)
Using the basis {𝑒1, 𝑒2 } we solve the following master equations with respect to its basis:
a. (
𝑥 0
0 1
) (
𝛼 0
0 1
) = (
1 𝑢
0 1
) (
𝑦 0
0 1
),
by solving with respect to 𝑢 and 𝑦 we obtain 𝑦 = 𝛼𝑥. Therefore, 𝜋+(Δ(𝛼))𝑓(𝑥) =
𝑓(𝛼𝑥). We mention here that we apply a right action of Aff +(1) in space L2(ℝ+).
b. (
𝑥 0
0 1
) (
1 𝛽
0 1
) = (
1 𝑢
0 1
) (
𝑦 0
0 1
).
In this case, we have 𝑦 = 𝑥 and 𝑢 = 𝛽𝑥. Therefore, 𝜋+(∇(𝛽))𝑓(𝑥) = 𝑒
2𝜋𝑖𝛽𝑥 𝑓(𝑥) as
desired.
∎
In the next section, we shall compute the Duflo-Moore operator for the
representation of Aff +(1) with respect to its right Haar measure. The result of Duflo-
Moore operator for the representation of Aff +(1) with respect left Haar measure can be
found in [24] pages 82-85.
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Volume 6, No. 2, October 2020, pp.114-122
120
Proposition 8. The Duflo-Moore operator for the irreducible unitary representation π+ of
Aff +(1) as written in eqs. (11) is of the form
𝐶𝜋+ 𝑓(Δ(𝑥)) = 𝑥
−1/2𝑓(𝑥), ( 𝑓 ∈ L2(ℝ+), 𝑥 ∈ ℝ+ ) (15)
Proof. Let 𝜗1 and 𝜗2 be elements in 𝐶𝑐(Aff
+(1)). Using the right Haar measure, we
shall compute the integral
∫ |〈𝜗1|𝜋+(∇(𝛽))𝜋+(Δ(𝛼))𝜗2〉L2(ℝ+)|
2
𝑑𝛽
𝑑𝛼
𝛼
Aff+(1)
To do that, first we compute the following inner product.
〈𝜗1|𝜋+(∇(𝛽))𝜋+(Δ(𝛼))𝜗2〉L2(ℝ+) = ∫ 𝜗1(𝑥)𝜋+(∇(𝛽))𝜋+(Δ(𝛼))𝜗2
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅
ℝ+
(𝑥)
𝑑𝑥
𝑥
= ∫ 𝜗1(𝑥)𝜋+(Δ(𝛼))𝑒
2𝜋𝑖𝛽𝑥 𝜗2
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅
ℝ+
(𝑥)
𝑑𝑥
𝑥
= ∫ 𝑒 −2𝜋𝑖𝛽𝑥 𝜗1(𝑥)𝜋+(Δ(𝛼))𝜗2
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅
ℝ+
(𝑥)
𝑑𝑥
𝑥
.
Using Plancherel’s theorem we have
∫ |〈𝜗1|𝜋+(∇(𝛽))𝜋+(Δ(𝛼))𝜗2〉L2(ℝ+)|
2
𝑑𝛽 =
ℝ
∫ |𝑒−2𝜋𝑖𝛽𝑥 𝜗1(𝑥)𝜋+(Δ(𝛼))𝜗2
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑥)|
2
ℝ+
𝑑𝑥
𝑥2
= ∫ |𝜗1(𝑥)𝜋+(Δ(𝛼))𝜗2
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑥)|
2
ℝ+
𝑑𝑥
𝑥2
= ∫ |𝜗1(𝑥)𝜗2̅̅ ̅(𝛼𝑥)|
2
ℝ+
𝑑𝑥
𝑥2
.
Therefore, using Fubini’s theorem we obtain
∫ |〈𝜗1|𝜋+(∇(𝛽))𝜋+(Δ(𝛼))𝜗2〉L2(ℝ+)|
2
𝑑𝛽
𝑑𝛼
𝛼
= ∫ |𝜗1(𝑥)|
2 { ∫ |𝜗2(𝛼𝑥)|
2
𝑑𝛼
𝛼
ℝ+
}
ℝ+Aff+(1)
𝑑𝑥
𝑥2
= ∫ |𝜗1(𝑥)|
2
𝑑𝑥
𝑥2
{ ∫ |𝜗2(𝛼
′)|2
𝑑𝛼′
𝛼′
ℝ+
}
ℝ+
(𝛼′ ≔ 𝛼𝑥)
= ∫ |𝑥 −1/2𝜗1(𝑥)|
2 𝑑𝑥
𝑥
{ ∫ |𝜗2(𝛼
′)|2
𝑑𝛼′
𝛼′
ℝ+
}
ℝ+
= ∫ |𝑥−1/2𝜗1(𝑥)|
2 𝑑𝑥
𝑥
ℝ+
. ‖𝜗2‖
2.
E. Kurniadi, N. Gusriani, B. Subartini
Duflo-Moore Operator for The Square-Interable Representation of 2-Dimensional Affine Lie Group
121
Therefore, The Duflo-Moore operator for the irreducible unitary representation of
Aff +(1) as written in eqs. (11) is of the form 𝐶𝜋+ 𝑓(Δ(𝑥)) = 𝑥
−1/2𝑓(𝑥) as desired.
∎
4. Conclusions
The Duflo-Moore operator for the representations of Aff +(1) in this paper is
considered in two cases. The first case, it is for the quasi-regular representation and
written in the term of Fourier transform. Namely, we obtain ℱ(𝐶𝜋𝜓)(𝜉) = 𝜉
−1/2ℱ𝜓(𝜉)
(see [18]). The second case, the Duflo-Moore operator is considered for irreducible
unitary representation with respect to its right Haar measure and we have 𝐶𝜋+ 𝑓(Δ(𝑥)) =
𝑥−1/2𝑓(𝑥). On the other hand, the Duflo-Moore operator for a square-intergarble
representation of Aff +(1) with respect to its left Haar measure can be seen in [24] pages
82-85.
It is more interesting to compute the Duflo-Moore operator for the representation of
higher dimension of affine Lie groups.
5. Acknowledgement
The first author is very grateful to Professor Hideyuki Ishi from Osaka City
University Japan for generous support and guidance to the first author to study
representation theory of Lie groups during his study. We also thank to the University of
Padjadjaran who has funded the work through Riset Percepatan Lektor Kepala (RPLK)
year 2020 with contract number 1427/UN6.3.1/LT/2020.
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