Induced representations and covariant transform of the affine

Amjad Alghamdi

1. Introduction The affine group is the set Aff := {(a, b): a > 0, b ∈ R} with the group law ∗ defined by:

(1) (a, b) ∗ (a0, b0) = (aa0, ab0 + b), where (a, b), (a0, b0) ∈ Aff. The identity element is e = (1, 0) and the inverse is (a, b)−1 = (a−1, −ba−1). It is also called the ax + b group since for any (a, b) ∈ R+ × R we can define a transformation of the real line Aa,b : R → R such that

Aa,b(x) = (a, b) · x = ax + b, x ∈ R,

Aa,b is called an affine transformation of R, and this action is consistent with the group law (1). The group(Aff, ∗) is isomorphic to the group of all upper triangular 2 × 2 real matrices of the form a b g = (a, b) = , 0 1

with matrix multiplication as the group law. The identity and the inverse elements are 1 0 a−1 −ba−1 I = , g−1 = . 0 1 0 1

The affine group Aff can be decomposed as the semi-direct product Aff = N o A where N is the normal closed given by {(1, b): b ∈ R} which can be identify with R via the correspondence (1, b) ↔ b and the subgroup A = {(a, 0) : a > 0} identify with R+ where (a, 0) ↔ a. [4][2] 1 2

2. Haar measure The affine group Aff with respect to the group law (1)is a locally compact group on which the left (right) Haar measure is respectivly given by [4]: (2) dν(a, b) = a−2dadb,

(3) dµ(a, b) = a−1dadb. Now since dν(a, b) = a−2dadb = a−1dµ(a, b), thus the affine group is non-unimodular group and the function 4(a, b) = a−1 is the modular function of the group.

3. Induced representation of the affine group There are three types of representation of the affine group which can be obtained by the induction from characters of its subgroup: • the left regular representation on the group Aff itself which induced by a character of the subgroup H = {e}. • the co-adjoint representation on the half real lines which in- duced by a character of the subgroup N. • the quasi-regular representation on the real line which induced by a character of the subgroupA. 3.1. The left regular representation. Let H = {e} be the trivial subgroup of the affine group. Then we have the Aff/H = Aff which generates the left regular representation. This representation arises from an action of the Aff group on itself by left translations. Let L2(Aff, dν) be the Hilbert space of all square inte- grable complex-valued functions on Aff with respect to the left Haar measure dν such that: Z 2 2 dadb kF k = |F (a, b)| 2 < ∞. Aff a

Thus the left regular representation of the affine group on L2(Aff, dν) is given by u v − b (4) Λ(a, b)F (u, v) := F ((a, b)−1 ∗ (u, v)) = F , , a a where (u, v) ∈ Aff. This representation is unitary since kΛ(a, b)F kAff = kF kAff for all F ∈ L2(Aff, dν). Also it is reducible since it has closed invariant subspaces for example the Hardy space H2(Aff, dν). 3

3.2. The co-adjoint representation. For the subgroup N = {(1, b): b ∈ R} the homogeneous space X = Aff/N ' A = R+ generates the co-adjoint representation ρ of Aff on L2(R). The natural projection map is given by p : Aff → X and by parametrise the homogenous space X we get the following map,

ep : Aff → R+ where ep(a, b) = a. Then let es : R+ → Aff where es(a) = (a, 0), such that [ep ◦ es](a) = a ⇒ ep ◦ es = I. where I is the identity map. The unique decomposition of any (a, b) ∈ Aff defined by es is of the form, b (a, b) = (a, 0) ∗ 1, . a The space X is a left homogeneous space under the Aff-action defined in terms of ep and es as follow: (a, b): w 7→ (a, b) · w = ep((a, b) ∗ es(w)) = aw, where (a, b) ∈ Aff, w ∈ X and · is the action of Aff on X from the left. Let χτ : N → T be a character of N defined by: 2πibτ (5) χτ (1, b) = e , τ ∈ R, which induces a linear representation of Aff constructed in the Hilbert χτ space L2 (Aff) consisting of the functions Fτ : Aff → C with the prop- erty:

b F (a, b) = χ 1, F (a, 0), τ τ a and the norm Z da kF k2 = |F (a, 0)|2 , τ A a R+ where F ∈ L2(R+). This space is invariant under the left shift of the group Aff .The restriction of the left shift on this space is the left regular representation (4). There are another induced representation constructed on the homo- geneous space by using lifting and pulling map. First introduce the map r : Aff → N such that b r(a, b) = s(a)−1 ∗ (a, b) = (1, ). e a 4

χτ Then define the lifting map Lχτ : L2(R+) → L2 (Aff) for the subgroup N and its character χτ as follow:

2πi b τ [L f](a, b) := χ (r(a, b))f(p(a, b)) = e a f(a), χτ τ e where f(a) = F (a, 0) is a function on the subgroup A. Then the pulling map is given by

χτ P : L2 (Aff) → L2(R+), [PF ](a) = F (s(a)) = f(a),

such that P ◦ Lχτ = I and Lχτ ◦ P = I. + Therefore, the representation ρχτ : L2(R+) → L2(R+) which is in- duced by the character χτ is given by + ρχτ (a, b) := P ◦ Λ(a, b) ◦ Lχτ , simple calculation with (u, v) ∈ Aff give us the following: + −2πi b τ u [ρ (a, b)f](u) = e u f , χτ a da where f ∈ L2(R+, a ). −1 − 1 −1 By changing the variable t = u , g(t) = t 2 f(t ) we get:

+ √ −2πibτt [ρχτ (a, b)g](t) = ae g(at), where g ∈ L2(R+, da). Now since the subgroup A normalize the subgroup N thus −2πiτ(ab) −2πi(aτ)b χτ (1, ab) = e = e = χaτ (1, b), 1 thus chosen a = |τ| implies that aτ = 1 or −1. Then for aτ = 1 the + representation ρχτ is given by

+ √ −2πibt (6) [ρχ (a, b)g](t) = ae g(at), where g ∈ L2(R+, da). and for aτ = −1 we get

− √ −2πibt (7) [ρχ (a, b)g](t) = ae g(at), where g ∈ L2(R−, da). Hence the direct sum of the two inequivalent irreducible representa- + − tion ρχ and ρχ is given as follow: + − ρχ(a, b) = ρχ (a, b) ⊕ ρχ (a, b), such that

√ −2πibt (8) [ρχ(a, b)g](t) = ae g(at), 5 where g ∈ L2(R, da). Proposition 1. [2] The co-adjoint representation (8) is unitary and irreducible.

Proof. The representation ρχ is unitary since kρχ(a, b)gkA = kgkA. To show irreducibility we will use Schur’s lemma[3]. We will show that for any operator T , permutable with the representation ρχ(a, b), is a multiple of the identity operator. The permutability of T with ρχ(a, b) implies its permutability with the derived representations 1 [dρXA g](u) = ug0(u) + g(u), χ 2

XN [dρχ g](u) = −2πiug(u). But the latter differs from the operator of multiplication by u in a constant factor only. It is known that the only operators in the space of infinitely differentiable finite functions permutable with the operator of multiplication by u, are operators of multiplication by a function (see Lemma pg111 [5]). Therefor T has the form T g(u) = q(u)g(u). To find q(u) we use the permutability of T with √ [ρχ(a, 0)g](u) = ag(au).

The equality T ρχ(a, 0) = ρχ(a, 0)T , mean that √ √ q(u) ag(au) = aq(au)g(au), and so q(u) = q(au). consequently q(u) is a constant. Thus any opera- tor T permutable with all ρχ(a, b), is a multiple of the identity operator and hence the representation ρχ(a, b) is irreducible.  3.3. The quasi-regular representation. Now for the subgroup A = {(a, 0), a > 0} the homogeneous space is X = Aff/A ' N = R then the natural projection map is given by p : Aff → X and by parametrise the homogeneous space X we get the following map,

ep : Aff → R where ep(a, b) = b. Then es : R → Aff where es(b) = (1, b), such that [ep ◦ es](a) = a ⇒ ep ◦ es = I, where I is the identity map. 6

The unique decomposition of any (a, b) ∈ Aff defined by es is of the form, (a, b) = (1, b) ∗ (a, 0). The space X is a left homogeneous space under the Aff-action defined in terms of ep and es as follow: (a, b): x 7→ (a, b) · x = ax + b. where (a, b) ∈ Aff, x ∈ X and · is the action of Aff on X from the left. Let χω : A → T be a character of the subgroup A defined by: iω+ 1 (9) χω(a, 0) = a 2 , ω ∈ R, then this character induced a linear representation of the affine group as follow: χω Let L2 (Aff) be a Hilbert space consisting of the functions Fω : Aff → C with the property:

Fω(a, b) = χω(a, 0)F (1, b), and Z 2 2 kFωkN = |F (1, b)| db, R where F ∈ L2(R+). This space is invariant under the left shift of the affine group. The restriction of the left shift on this space is the left regular representation (4). There are another induced representation constructed on the homo- geneous space by using lifting and pulling map. Introduce the map r : Aff → A given by −1 r(a, b) = es(b) ∗ (a, b) = (a, 0), χω then define the lifting map Lχω : L2(R) → L2 (Aff) for the subgroup A and its character χω as follow:

iω+ 1 [L f](a, b) := χ (r(a, b))f(p(a, b)) = a 2 f(b), χω ω e where f(b) = F (1, b) is a function on the subgroup A. The pulling map is given by

0 χω P : L2 (Aff) → L2(R), [P0F ](b) = F (s(b)) = f(b), 0 0 such that P ◦ Lχω = I and Lχω ◦ P = I.

Therefore, the character χω induced the representation πχω : L2(R) → L2(R) which is given as follow 0 πχω (a, b) := P ◦ Λ(a, b) ◦ Lχω , 7 by simple calculation with (u, v) ∈ Aff we get

iω+ 1 1 2 v − b (10) [π (a, b)f](v) = f , χω a a where f ∈ L2(R).

Now, this representation is unitary since kπχω (a, b)fkN = kfkN and in the following we will show that it is reducible.

Using the Szeg¨oprojection PR : L2(R) → H2(R) (see Theorem + B.1[2])we induce a representation πχω : H2(R) → H2(R) by + 0 πχω (a, b) := P ◦ Λ(a, b) ◦ Lχω ◦ PR, hence we get: iω+ 1 1 2 v − b (11) [π+ (a, b)f](v) = f , χω a a where f ∈ H2(R). And using the complementary projection P ⊥ : L ( ) → H⊥( ) we R 2 R 2 R − ⊥ ⊥ induced the representation πχω : H2 (R) → H2 (R) by π− (a, b) := P0 ◦ Λ(a, b) ◦ L ◦ P ⊥, χω χω R thus iω+ 1 1 2 v − b (12) [π− (a, b)f](v) = f , χω a a ⊥ where f ∈ H2 (R). Therefore, the Hilbert space L2(R) under the action πχω contains ⊥ precisely two closed proper invariant subspaces H2(R) and H2 (R) such that ⊥ L2(R) = H2(R) ⊕ H2 (R),

then the representation πχω is reducible on L2(R) and we can de- compose it into two inequivalent irreducible representations i.e. + − πχω (a, b) = πχω (a, b) ⊕ πχω (a, b). 4. Covariant transform Definition 1. [7] Let ρ be a representation of a group G in a space V and F be an operator acting from V to a space U. We define a ρ covariant transform WF acting from V to the space L(G, U) of U- valued functions on G by the formula: ρ −1 (13) WF : υ 7→ υˆ(g) = F (ρ(g )υ), υ ∈ V, g ∈ G. The operator F called a fiducial operator. 8

Definition 2. [7] Let V be a Hilbert space with an inner product h., .i and ρ be a unitary representation of a group G in the space V . Let F : V → C be the functional υ 7→ hυ, υ0i defined by a vector υ0 ∈ V . The vector υ0 is often called the mother wavelet in areas related to signal processing, the vacume state in the quantum framework. In the set-up, transformation (13) is the well-known expression for a wavelet transform −1 (14) W : υ 7→ υ˜(g) = hρ(g )υ, υ0i = hυ, ρ(g)υ0i, υ ∈ V, g ∈ G.

The family of the vectors υg = ρ(g)υ0 is called wavelets or coherent states. The image of (14) consists of scalar valued functions on G.

Definition 3. [6] Let H be a closed subgroup of the group G and υ0 ∈ V such that

(15) ρ(h)υ0 = χ(h)υ0, for some character χ of H where h ∈ H and ρ is a unitary repre- sentation of the group G in the space V . For any continuous section s : G/H → G the map υ 7→ υ˜(x) =υ ˜(s(x)), intertwines ρ with the rep- resentation ρχ in a certain function space on the homogeneous space G/H induced by the character χ of H. We call the map

(16) Wυ0 : υ 7→ υ˜(x) = hυ, ρ(s(x))υ0i, where x ∈ G/H the induced wavelet transform. Example 1. The induced wavelet transform that intertwines the quasi- regular representation π+ (11) for ω = 0 and the co-adjoint represen- + tation ρχ (6) is the Fourier transform. To prove that let the mother 2πix wavelet be ϕ0(x) = e which satisfy the condition (15) as follow + π (1, b)ϕ0 = χ(1, b)ϕ0. Therefore + [Wϕ0 f](λ) = hf, π (s(λ))ϕ0i + = hf, π (λ, 0)ϕ0i Z + = f(x)π (λ, 0)ϕ0(x)dx R 1 Z x = √ f(x)ϕ0( )dx λ R λ Z 1 −2πi x = √ f(x)e λ dx λ R Z = pξ f(x)e−2πξxdx = pξfˆ(ξ). R 9

1 where ξ = λ ∈ A. Hence we can write the co-adjoint representation as the Fourier trans- form of the quasi regular representation (10) √ ˆ −2πibλ ˆ (17) [ρχ(a, b)f](λ) = ae f(aλ), where f ∈ L2(R). Similarly, the induced wavelet transform that intertwines the co- + + adjoint representation ρχ and the the quasi-regular representation π for ω = 0 of the affine group is the Fourier transform. Theorem 1. [7] The covariant transform (13) intertwines ρ and the left regular representation on L(G, U): Wρ(g) = Λ(g)W.

+ Example 2. For the co-adjoint representation of the affine group ρχ −2πλ which is given by (6) let the mother wavelet be υ0(λ) = e then the wavelet transform is given as follow: + + [Wυ0 ρχ (a, b)f](λ) = hf, ρχ (a, b)υ0i Z + = f(λ)ρχ (a, b)υ0(λ)dλ R+ Z √ 2πibλ = f(λ) ae υ0(aλ)dλ R+ √ Z = a f(λ)e2iπbλe−2πaλdλ R+ √ Z √ = a f(λ)e−2π(a−ib)λdλ = aF (a − ib). R+ Hence the intertwining operator between the co-adjoint representation and the left regular is the Laplace transform.

Definition 4. [1][7] The mother wavelet υ0 is an admissible vector if it satisfy the following condition:

Z 2 2 (18) kυe0k = |hυ0, ρ(g)υ0i| dg < ∞. G If the admissible vector exists then the wavelet transform is a map into the square integrable functions with respect to the left Haar measure on G. Theorem 2. [1][7] 10

The quasi-regular representation (10) of the affine group is square in- tegrable representation and any function f such that its Fourier trans- form fˆ(λ) satisfies Z ∞ |fˆ(λ)|2 (19) < ∞, −∞ |λ| is admissible. Proof. By using the Fourier transform of the quasi-regular representa- tion (17) the square integrability of (10) follows from a straightforward calculation: Z Z 2 Z Z Z 2 dadb 1 2iπbλ dadb hf,ˆ ρ(a, b)fˆi = fˆ(λ)a 2 e fˆ(aλ)dλ a2 a2 R R+ R R+ R

Z Z Z Z 0 dadb = ae2iπb(λ−λ )fˆ(λ)fˆ(λ0)fˆ(aλ0)fˆ(aλ)dλdλ0 a2 R R+ R R Z Z Z da = δ(λ − λ0)fˆ(λ)fˆ(λ0)fˆ(aλ0)fˆ(aλ)dλdλ0 a R R+ R Z Z da Z dλ = |fˆ(aλ)|2|fˆ(λ)|2 dλ = kfk2 |fˆ(λ)|2 < ∞. a |λ| R R+ R Therefore, a function f is admissible if R |fˆ(λ)|2 dλ < ∞. R |λ|  Example 3. (The Mexican hat wavelet ) The Mexican hat is defined by the second derivative of a Gaussian func- tion as −x2 d2 −x2 f(x) = (1 − x2)exp( ) = − exp( ), 2 dx 2 its Fourier transform given by √ −ξ2 fˆ(ξ) = 2πξ2exp( ). 2 Therefore Z ∞ |fˆ(ξ)|2 dξ = 2π < ∞, −∞ |ξ| hence f is an admissible vector. Example 4. The Haar mother wavelet function ψ(t) can be described as

( 1 1 0 ≤ t < 2 , 1 ψ(t) = −1 2 ≤ t < 1, 0 otherwise 11

Its Fourier transform given as follow:

(1 − e−iπwt)2 ψˆ(w) = , 2iπw we can see that it satisfy the admissibility condition (19). Example 5. The quasi-regular representation of the affine group on Lp(R) is given by 1 1 p x − b (20) [π (a, b)f](x) = f . p a a

Consider the operators F± : Lp(R) → C defined by: 1 Z f(x) F±(f) = dx. πi R i ± x 1 1 In L2(R) we note that F+(f) = hf, ci, where c(x) = πi i+x . Comput- −λ ing the Fourier transform cˆ(λ) = χ(0,∞)(λ)e , we see that cˆ ∈ H2(R). Moreover, cˆ does not satisfy admissibility condition (19). In Lp(R) the covariant transform is given by 1 a p Z f(x) fe(a, b) = F (π((a, b)−1)f) = dx, πi R x − (b + ia) this is the Cauchy integral from Lp(R) to the space of functions fe(a, b) − 1 such that a p fe(a, b) is in the Hardy space on the upper/lower half- 2 Hp(R±).

−x2 Example 6. The Gaussian function f(x) = e 2 in L2(R) is not ad- missible vector for the quasi regular representation. Proposition 2. [7] Let G be a and ρ be a representation of G in a space V . Let [Wf](g) = F (ρ(g−1f) be a covariant transform de- fined by a fiducial operator F : V → U. Then the right shift [Wf](gg0) by g0 is the covariant transform [W0f](g) = F 0(ρ(g−1)f), defined by the fiducial operator F 0 = F ◦ ρ(g−1). In other words the covariant transform intertwines right shifts R(g): f(h) → f(gh) on the group G with the associated action −1 ρB(g): F 7→ F ◦ ρ(g ), on fiducial operators:

R(g) ◦ WF = WρB (g)F , g ∈ G. 12

Corollary 1. [7] Let a fiducial operator F be a null-solution, AF=0, P Xj for the operator A = j ajdρB , where Xj ∈ g and aj are constants . −1 Then the covariant transform [WF ](g) = F (ρ(g )f) for any f satisfies

X Xj D(WF f) = 0 where D = ajL . j Here,LXj are the left invariant fields(Lie derivatives) on G correspond- ing to Xj. Example 7. Consider the quasi-regular representation of the affine 1 0 0 1 group (20) with p = 1. Let X = and X = be A 0 0 N 0 0 the basis of the g of the affine group. They generate one -parameter of g: et 0 1 t a(t) = and n(t) = , 0 1 0 1 then the derived representations are: [dπXA f](x) = −f(x) − xf 0(x), [dπXN f](x) = −f 0(x). The corresponding left invariant vector fields on the affine group are:

XA XN L = a∂a, L = a∂b. 1 The mother wavelet x+i is a null solution of the operator d −dπXA − idπXN = I + (x + i) . dx Therefore, the image of the covaiant transform with the fiducial opera- tor 1 Z f(x) F+(f) = dx, πi R i − x consists of the null solutions to the operator

XA XN −L + iL = ia(∂b + i∂a), ∂ ∂ that is in the essence of Cauchy-Riemann operator ∂z = ∂x +i ∂y in the upper half-plane. Example 8. A step in a different direction is a consideration of non- linear operators. For the affine (20) define Fm to be a homogeneous(but non linear) functional V → R+: 1 Z 1 (21) Fm(f) = |f(x)|dx. 2 −1 13

Hence the covariant transform is: (22) 1     Z 1 p Z b+a m 1 −b 1 1 a p [W f](a, b) = Fm πp , f = a f(ax + b) dx = |f(x)|dx. a a 2 −1 2a b−a

Example 9. The fiducial functional Fm is a null solution of the fol- lowing functional equation: 1 1 1 1 1 1 F − F ◦ π , − F ◦ π , − = 0 m 2 m ∞ 2 2 2 m ∞ 2 2 Consequently, the image of wavelet transform Wm (22) consists of func- tions which solve the equation :  1 1 1 1 1 1 I − R , − R , − f = 0. 2 2 2 2 2 2 14

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