EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5735 ISSN 1307-5543 – ejpam.com Published by New York Business Global Note on the Affine Group Representations Amjad Saleh Alghamdi1,∗, Taghreed Alqurashi2 1 Department of Mathematics, Umm Al-Qura University, Makkah, Saudi Arabia 2 Department of Mathematics, Al-Baha University, Albaha, Saudi Arabia Abstract. The representation of the affine group is a crucial topic in harmonic analysis. In this paper, we use the wavelet transform to investigate the intertwining operator. This approach helps to compare different unitary representations of the affine group. 2020 Mathematics Subject Classifications: 22D10, 43A32, 22E46, 42C40, 44A10 Key Words and Phrases: Affine group, intertwining operator, unitary representations, wavelet transform, Poisson integral, Laplace transform 1. Introduction The affine group is a non-commutative, locally compact Lie group of the smallest dimensionality, and it is often used to build wavelets. Gelfand and Naimark[7] initially introduced the unitary representations of the affine group. The induced representations of the affine group from a complex character was explained in [4, 5]. Moreover, in [4] they described the intertwining operators related to Hilbert spaces in terms of representations of the affine group. In this paper, we will illustrate the intertwining operators between all three affine representations as follows: • the Poisson integral between the quasi-regular representation on H2(R) and the left regular representation on L2(Aff, dν). • the Laplace transform between the co-adjoint representation on L2(R+) and the left regular representation on L2(Aff, dν). • the Fourier transform between the quasi-regular representation on H2(R) and the co-adjoint representation on L2(R+). ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5735 Email addresses: asmghamdi@uqu.edu.sa (A. Alghamdi), talqorashi@bu.edu.sa (T. Alqurashi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Alghamdi, T. Alqurashi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5735 2 of 7 2. The Affine Group An element of the affine group Aff [4, 5] is denoted by (a, b), where a ∈ R+ and b ∈ R. The group operation on Aff is defined by (a, b) ∗ (a′, b′) = (aa′, ab′ + b), (1) where e = (1, 0) is the identity element and the inverse of (a, b) is given by (a, b)−1 = (a−1,−ba−1). We can decompose the affine group as a semi-direct product Aff = A⋉N . The subgroup N, defined by {(1, b) : b ∈ R}, is a normal subgroup that can be identified with R by mapping (1, b) ↔ b. Moreover, the subgroup A = {(a, 0) : a > 0} is identified with R+ where (a, 0) ↔ a, [8]. The affine group is locally compact. Thus, it has a left Haar measure, which is given as follows: dν(a, b) = a−2dadb, (2) and it is left invariant measure that is dν((a′, b′) ∗ (a, b)) = dν(a, b). In addition, we can obtain a right Haar measure dµ(a, b) = a−1dadb, which is right invariant. Therefore, the affine group is non-unimodular , and the modular function of the group is given by △(a, b) = a−1 [8]. The measure on the subgroup A is the Haar measure da a , and on the subgroup N is the Lebesgue measure db. 3. Unitary representations of the affine Group The affine group has three unitary representations[4, 5] given in the following: • Left regular representation [ Λ(a, b)F ] (x, y) := F ((a, b)−1 ∗ (x, y)) = F ( x a , y − b a ) , (3) where (x, y) ∈ Aff. • Co-adjoint representation on the half real lines [ρ±(a, b)g](x) = √ ae2πibxg(ax), (4) where g ∈ L2(R±, da). • Quasi-regular representation on the real line. The Hilbert space L2(R) with respect to π contains precisely two closed proper invariant subspaces H2(R) and H⊥ 2 (R) such that L2(R) = H2(R)⊕H⊥ 2 (R). A. Alghamdi, T. Alqurashi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5735 3 of 7 Therefore, the quasi-regular representation π is decomposed into two irreducible representations. That is π(a, b) = π+(a, b)⊕ π−(a, b). The operator of representation π+ : H2(R) → H2(R) is given by the following: [π+(a, b)f ](x) = ( 1 a )− 1 2 f ( x− b a ) , (5) where f ∈ H2(R). Also, for the representation π− : H⊥ 2 (R) → H⊥ 2 (R), the operator is given by (5) where f ∈ H⊥ 2 (R). 4. Intertwining Operators In this section, we study the intertwining operators between the unitary representations of the affine group by using the wavelet transform and the induced wavelet transform. Left regular representation on L2(Aff, dν) Quasi-regular representation on H2(R) Co-adjoint representation on L2(R+) Po iss on int eg ra l Fourier transform Fourier transform Laplace transform Figure 1: Intertwining operators between affine group representations 4.1. Wavelet Transform Definition 1. [12] Let V be a Hilbert space with an inner product ⟨., .⟩ and ρ be a unitary representation of a group G in the space V . Let F : V → C be the functional υ 7→ ⟨υ, υ0⟩ defined by a vector υ0 ∈ V . The vector υ0 is called the mother wavelet. We define a wavelet transform W acting from V to the space L2(G,C) of C-valued functions on G by the formula: W : υ 7→ υ̃(g) = ⟨ρ(g−1)υ, υ0⟩ = ⟨υ, ρ(g)υ0⟩, υ ∈ V, g ∈ G. (6) The collection of the vectors υg = ρ(g)υ0 is called wavelets. A. Alghamdi, T. Alqurashi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5735 4 of 7 Theorem 1. [12] The wavelet transform (6) intertwines the unitary representation ρ and the left regular representation on L2(G,C): Wρ(g) = Λ(g)W. Corollary 1. The Poisson integral for a ∈ R+ and b ∈ R is given by [Pφ](a, b) = 1 2 ∫ R a (x− b)2 + a2 φ(x)dx. (7) It is the wavelet transform that intertwines the quasi-regular representations π± (5) with the left regular representation Λ(a, b) given by (3). Proof. We will prove it for the representation π+ (5) and the result is valid for the representation π−. Let the fiducial operator F : H2(R) → C be the functional φ 7→ ⟨φ,φ0⟩, and the mother wavelet be the conjugate Poisson kernel φ0 = −x π(1+x2) . Then, φ̄0 = 1 π(1+x2) is the Poisson kernel. Hence, the wavelet transform that intertwines the quasi-regular representation with the left regular representation is given as follows: [Wπ+(a, b)φ](x) = ⟨φ, π+(a, b)φ0⟩ = ∫ R φ(x) 1√ a φ̄0 ( x− b a ) dx = 1√ a ∫ R φ(x) 1 π(1 + (x−ba )2) dx = 1√ aπ ∫ R φ(x) a2 a2 + (x− b)2 dx = √ a π ∫ R φ(x) a a2 + (x− b)2 dx = 2 √ a π [Pφ](a, b). Corollary 2. The Laplace transform F (a+ ib) = ∫ R+ f(t)e−2π(a+ib)xdx, a+ ib ∈ C, is the wavelet transform that intertwines the co-adjoint representation of the affine group ρ±χ , with the left regular representation Λ(a, b) given by (3). Proof. It is enough to prove the corollary for the representation ρ+(4). The result works for ρ−. Let the fiducial operator F : L2(R+) → C be the functional f 7→ ⟨f, f0⟩, A. Alghamdi, T. Alqurashi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5735 5 of 7 and the mother wavelet be f0(λ) = e2πζ . Then, the wavelet transform is given as follows: [Wf0ρ + χ (a, b)f ](ζ) = ⟨f, ρ+(a, b)f0⟩ = ∫ R+ f(ζ)ρ+(a, b)f0(ζ)dζ = ∫ R+ f(ζ) √ ae−2πibζf0(aζ)dζ = √ a ∫ R+ f(ζ)e−2iπbζe−2πaζdζ = √ a ∫ R+ f(λ)e−2π(a+ib)ζdζ = √ aF (a+ ib). 4.2. Induced wavelet Transform In this subsection, we will study the wavelet transform that produces functions on a homogeneous space rather than the entire group. Definition 2. [9] Let H be a closed subgroup of the group G, and H is a Hilbert space. For some character χ of H where h ∈ H and ρ is a unitary representation of the group G in the space H there is a mother wavelet υ0 ∈ H such that: ρ(h)υ0 = χ(h)υ0. (8) Then, for any continuous section s : G/H → G the induced wavelet transform is given as follows: Wυ0 : υ 7→ υ̃(x) = ⟨υ, ρ(s(x))υ0⟩, where x ∈ G/H (9) intertwines ρ with the representation ρχ in a certain function space on the homogeneous space G/H induced by the character χ of H. Corollary 3. The induced wavelet transform that intertwines respectively the quasi-regular representations π+ and π−(5) with the co-adjoint representation ρ+ and ρ− (4) is the Fourier transform [Ff ](λ) = f̂(λ) = ∫ ∞ −∞ f(x)e−2πixλdx, λ ∈ R. (10) Proof. For simplicity, it suffices to prove the corollary for the representation π+(5). The same argument is valid for π−. Let the mother wavelet be ψ0(x) = e2πix. It is clear that ψ0 satisfies the following condition: π+(1, b)ψ0 = χ(1, b)ψ0, A. Alghamdi, T. Alqurashi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5735 6 of 7 where χ(1, b) = e2πib is the character of the subgroup N . Let s : R+ → Aff be the continuous section defined as s(a) = (a, 0) where a ∈ R+. Then, for f ∈ H2(R), we calculate the induced wavelet transform as follows: [Wψ0f ](λ) = ⟨f, π+(s(λ))ψ0⟩ = ⟨f, π+(λ, 0)ψ0⟩ = ∫ R f(x)π+(λ, 0)ψ0(x)dx = 1√ λ ∫ R f(x)ψ0 (x λ ) dx = 1√ λ ∫ R f(x)e−2πi x λdx = 1√ λ f̂ ( 1 λ ) , λ ∈ R+. This is the Fourier transform. Next, for the co-adjoint representation ρ+, the mother wavelet ψ0(x) = 1 satisfies the condition ρ+(a, 0)ψ0 = χ(a, 0)ψ0, where χ(a, 0) = a 1 2 is the character of the subgroup A. Let s : R → Aff be the continuous section defined as s(b) = (1, b), where b ∈ R. Then, for g ∈ L2(R+) the induced wavelet transform [Wψ0g](ξ) = ⟨g, ρ+(s(ξ))ψ0⟩ where ξ ∈ R, is the Fourier transform. 5. Conclusion We demonstrated the intertwining operator of the affine group, which allows a connec- tion between the representations while preserving the action of the affine group. We find that the Laplace transform intertwines the co-adjoint representation with the left-regular representation of the affine group. Moreover, the Poisson integral is the intertwining op- erator between the quasi-regular and left-regular representations. Finally, the Fourier transform intertwines the co-adjoint with the quasi-regular representation. References [1] Syed Ali, Jean-Pierre Antoine, and Jean-Pierre Gazeau. 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