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Lecture Notes on the Affine Group Representations

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Lecture Notes on the Affine Group Representations Induced representations and covariant transform of the affine group Amjad Alghamdi 1. Introduction The affine group is the set Aff := f(a; b): a > 0; b 2 Rg with the group law ∗ defined by: (1) (a; b) ∗ (a0; b0) = (aa0; ab0 + b); where (a; b); (a0; b0) 2 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) 2 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 2 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 subgroup given by f(1; b): b 2 Rg which can be identify with R via the correspondence (1; b) $ b and the subgroup A = f(a; 0) : a > 0g 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 = feg. • 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 = feg be the trivial subgroup of the affine group. Then we have the homogeneous space 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 = jF (a; b)j 2 < 1: 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) 2 Aff: This representation is unitary since kΛ(a; b)F kAff = kF kAff for all F 2 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 = f(1; b): b 2 Rg 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) 2 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) 2 Aff, w 2 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 2 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 = jF (a; 0)j2 ; τ A a R+ where F 2 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) 2 Aff give us the following: + −2πi b τ u [ρ (a; b)f](u) = e u f ; χτ a da where f 2 L2(R+; a ): −1 − 1 −1 By changing the variable t = u , g(t) = t 2 f(t ) we get: + p −2πibτt [ρχτ (a; b)g](t) = ae g(at); where g 2 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 = jτj implies that aτ = 1 or −1. Then for aτ = 1 the + representation ρχτ is given by + p −2πibt (6) [ρχ (a; b)g](t) = ae g(at); where g 2 L2(R+; da): and for aτ = −1 we get − p −2πibt (7) [ρχ (a; b)g](t) = ae g(at); where g 2 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 p −2πibt (8) [ρχ(a; b)g](t) = ae g(at); 5 where g 2 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 p [ρχ(a; 0)g](u) = ag(au): The equality T ρχ(a; 0) = ρχ(a; 0)T , mean that p p 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 = f(a; 0); a > 0g 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) 2 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) 2 Aff, x 2 X and · is the action of Aff on X from the left. Let χ! : A ! T be a character of the subgroup A defined by: i!+ 1 (9) χ!(a; 0) = a 2 ;! 2 R; then this character induced a linear representation of the affine group as follow: χ! Let L2 (Aff) be a Hilbert space consisting of the functions F! : Aff ! C with the property: F!(a; b) = χ!(a; 0)F (1; b); and Z 2 2 kF!kN = jF (1; b)j db; R where F 2 L2(R+).
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