Admissible Wavelets on Groups and their Homogeneous Spaces

In this note, let G be a locally compact group and H be a compact subgroup of G. We investigate the square integrable representations of homogeneous spaces G/H and admissible wavelets for these representations. Also, we consider the relation between the square integrable representations of locally compact groups and their homogeneous spaces. Moreover, the connection between existence of admissible wavelets for locally compact groups and their homogeneous spaces is described. Mathematics Subject Classification: Primary 43A15; Secondary 43A85, 65T60


Introduction and Preliminaries
For a locally compact group G with left Haar measure λ it is well known that a continuous unitary representation π of G is called square integrable if there exists a non-zero vector ζ in Hilbert space H such that Such a unital vector ζ is called an admissible vector.The square integrable representations on homogeneous spaces that admit G-invariant measure and relatively invariant measure have been studied in [1,5].In this manuscript we investigate the relation between square integrable representations of locally compact group G and its homogeneous space G/H, in which H is compact subgroup of G. To be more precise we need to fix some notations and review some basic concepts.For more details one may refer to [8,6,12,3].
Let G be a locally compact group and H be a closed subgroup of G with left Haar measures λ and λ H , respectively.Consider G/H as a homogeneous space on which G acts from left and the Radon measure μ on G/H is said to be G-invariant if μ x (yH) = μ(yH) for all x, y ∈ G, where μ x is defined by where Δ G , Δ H are the modular functions on G and H, respectively.In this case we have, The Radon measure μ on G/H is called strongly quasi invariant if there is a positive continuous function α of G×G/H such that dμ x (yH) = α(x, yH)dμ(yH), for all x, y ∈ G.If the function α(x, .)reduce to constants, the measure μ is called relatively invariant measure.
A rho-function for the pair (G, H) is defined to be a continuous positive function ρ from G which satisfies In [10] it has been shown that the existence of a homomorphism rho-function for the pair (G, H) is a necessary and sufficient condition for the existence of a relatively invariant measure μ on G/H.
In this paper we introduce square integrable representations of homogeneous space G/H equiped with a relatively invariant measure μ.The main aim in this paper is to compare square integrable representations of locally compact groups and their homogeneous spaces.Finally we describe the relation between admissible wavelets on groups and their homogeneous spaces.

Main Result
Let G be a locally compact group and H be a Hilbert space.We recall square integrable representations of locally compact groups.An irreducible representation π of group G is said to be square integrable if there exists a nonzero element (2.1) The condition (2.1) is known as the admissibility condition for the square integrable representations of G on H.The element ζ ∈ H for which ζ = 1 is called admissible wavelet for the square integrable representation π of group G.The wavelet constant associated to the admissible wavelet is denoted by c ζ and defined as: ( For more details about admissible wavelets on locally compact groups the reader can be consult with [13,9]).
In [5], we have studied the squre integrable representations of homogeneous spaces and admissible wavelets for these representations.Here we define representations of homogeneous spaces which will be needed in the difinition of square integrable representations of homogeneous spaces.A unitary representation of the homogeneous space G/H is a map from G/H into the group U(H), of all unitary operators on some nonzero Hilbert space H, for which the map xH We call c ζ the wavelet constant associated to the admissible wavelet ζ.Note that since H is compact subset of G, the definition (2.2) is well define.
In the sequle we intend to establish a relation between square integrable representations of locally compact groups and their homogeneous spaces.We recall that a unitary representation of homogenous space G/H is in one to one correspondence to a unitary representation of G whose kernel π, which is denoted by N, contains H.We denote by q N , q H , p, the canonical mappings of G onto G/N, of G onto G/H and of N onto N/H.Let λ N and λ H be the left Haar measure on N and H, respectively.Then there exists a G-invariant measure μ N/H on N/H .On the otherhand, let λ be a left Haar measure on G.One can then form a left invariant mesures μ G/N on quotient group G/N and a relatively invarint measure μ G/H on homogeneous space G/H which arises from rho-function ρ of G.It is clear that, the mapping (x,

It is easy to show that
The following lemma shows that the map ω x is proper.

Lemma 2.1. Let E be a compact subset of G/H and K be a compact subset of G. Then ∪ x∈K ω −1
x (E) is relatively compact in N/H.In particular, ∪ x∈K ω −1  x (E) is contained in a compact subset of N/H.Proof.Let F be a compact subset of G such that q H (F ) = E. Let L be the set of n ∈ N such that Kn intersects F .Then L is compact (see [2], chapterIII, 4.5, theorem1).Let n ∈ N, such that p(n) ∈ ∪ x∈K ω −1  x (E).Thus there exists x ∈ K such that ω x (p(n)) ∈ E. i.e. q H (xn) ∈ E and since q x (E) ⊆ p(L).Let M(N/H) and M(G/H) be complex measure spaces on homogeneous spaces N/H and G/H, respectively, as introduced in [3,12].Lemma 2.2 shows that the mapping ω x is proper.Then ω x extends continuously to a map from The function Ψ is continuous and compact support.Morevere by G-invariant measure μ N/H , we have is continuous with compact support, for all ϕ ∈ C c (G/H).
Proof.By (1.1), for ϕ ∈ C c (G/H) we have where ϕ = P f and f ∈ C c (G).Also, (2.5) (ii) suppose that there exists a bounded positive measure μ G/H on homogenous space G/H .Then there exists a bounded positive measure on homogeneous space N/H.
(ii) The function 1 on G/H is μ G/H -integrable.By the part (i), the function 1 on N/H is μ N/H -integrable.Thus μ N/H is bounded.
As before we mention, a unitary representation of homogeneous space G/H define a representation π of G such that kerπ = N contains H. Consider π, the representation of G/N by letting π(xN) = π(x).The following theorem show that the representation of G/H is square integrable if and only if the representation π of quotient group G/N is square integrable.
is said to be invariant with respect to if (xH)M ⊆ M, for all x ∈ G.A unitary representation is said to be irreducible if the only invariant subspaces of H are {0} and H.An irreducible representation of G/H on H, in which G is a locally compact group and H is a compact subgroup of G, is said to be square integrable if there exists a nonzero element ζ ∈ H such that is a relatively invariant measure on G/H which arises from a rho function ρ : G → (0, ∞).If ζ satisfies (2.2), it is called an admissible vector.An admissible vector ζ ∈ H is called an admissible wavelet if ζ = 1.In this case, we define the wavelet constant c ζ as * , for each x, y ∈ G.It is worthwhile to note that this defines a unitary representation π of G in which the subgroup H is contained in the kernel of π.Conversely, any unitary representation π of G which is trivial on H induces a unitary representation of G/H, by letting (xH) = π(x).Moreover, a closed subspace M of H Theorem 2.4.Let be a unitary representation of G/H.The unitary representation of G/H is square integrable if and only if the representation π of quotient group G/N is square integrable.