International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 1 (2017), 22-31 http://www.etamaths.com PETER-WEYL THEOREM FOR HOMOGENEOUS SPACES OF COMPACT GROUPS ARASH GHAANI FARASHAHI∗ Abstract. This paper presents a structured formalism for a constructive generalization of the Peter- Weyl Theorem over homogeneous spaces of compact groups. Let H be a closed subgroup of a compact group G and µ be the normalized G-invariant measure on the compact left coset space G/H. We then present an abstract TH-version of the Peter-Weyl Theorem for the Hilbert function space L 2(G/H,µ). 1. Introduction The abstract aspects of harmonic analysis over homogeneous spaces of compact non-Abelian groups or precisely left coset (resp. right coset) spaces of non-normal subgroups of compact non-Abelian groups is placed as building blocks for classical harmonic analysis [5, 7], coherent states analysis [8, 11], theoretical and particle physics [1]. Over the last decades, abstract and computational aspects of Plancherel formulas over symmetric spaces have achieved significant popularity in geometric analysis, mathematical physics and scientific computing (computational engineering), see [2–4, 6, 12, 13] and references therein. The Peter-Weyl theorem is a fundamental result in the theory of classical harmonic analysis, apply- ing to compact topological groups that are not necessarily abelian. It was initially proved by Hermann Weyl and Fritz Peter, in the setting of a compact topological groups [15]. The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representations of finite groups, as presented by F. G. Frobenius and Issai Schur, see [1, 9, 10] and classical references therein. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of a compact groups G are dense in the space C(G) of continuous complex-valued func- tions on G, and thus also in the space L2(G) of square-integrable functions. The second part asserts the complete reducibility of unitary representations of G. The final part then asserts that the regular representation of G on L2(G) decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of L2(G). Let G be a compact group and H be a closed subgroup of G. Also, let G/H be the left coset space of H in G and Ĝ/H be the abstract dual space of G/H. Let µ be the normalized G-invariant measure over the homogeneous space G/H with respect to the probability measures of H and G, associated to the Weil’s formula. Then we present a structured formalism for a constructive generalization of the Peter-Weyl Theorem for the Hilbert function space L2(G/H,µ). The paper is organized as follows. Section 2 is devoted to fixing notations and a brief summary on the non-Abelian Fourier analysis of compact groups, general formalism of the Peter-Weyl theorem, and preliminaries and classical results on harmonic analysis of compact homogeneous spaces. Then we present a systematic study of abstract harmonic analysis over the Hilbert function space L2(G/H,µ). In section 4, using the abstract notion of the dual space Ĝ/H of the homogeneous space G/H, we prove that the Hilbert function space L2(G/H,µ) satisfies a canonical decomposition into a direct sum of some closed and mutually orthogonal subspaces. This decomposition coincides with the Peter-Weyl Received 31st June, 2016; accepted 10th August, 2016; published 3rd January, 2017. 2010 Mathematics Subject Classification. Primary 47A67, 43A85. Key words and phrases. compact group; homogeneous space; G-invariant measure; Peter-Weyl Theorem. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 22 PETER-WEYL THEOREM FOR HOMOGENEOUS SPACES OF COMPACT GROUPS 23 decomposition, when H is a normal subgroup of G. This result can be considered as a generalization of the Peter-Weyl Theorem for homogeneous spaces of compact groups. 2. Preliminaries and Notations Let H be a separable Hilbert space. An operator T ∈ B(H) is called a Hilbert-Schmidt operator if for one, hence for any orthonormal basis {ek} of H we have ∑ k ‖Tek‖ 2 < ∞. The set of all Hilbert-Schmidt operators on H is denoted by HS(H) and for T ∈ HS(H) the Hilbert-Schmidt norm of T is ‖T‖2HS = ∑ k ‖Tek‖ 2. The set HS(H) is a self adjoint two sided ideal in B(H) and if H is finite-dimensional we have HS(Hπ) = B(H). An operator T ∈ B(H) called trace-class, whenever ‖T‖tr = tr[|T |] < ∞, where tr[T] = ∑ k〈Tek,ek〉 and |T| = (TT ∗)1/2, see [14]. Let G be a compact group with the Haar measure dx, H be a closed subgroup of G with the left Haar measure dh. The left coset space G/H is considered as a locally compact homogeneous space that G acts on it from the left and q : G → G/H given by x 7→ q(x) := xH is the surjective canonical mapping. The function space C(G/H) consists of all functions TH(f), where (see Proposition 2.48 of [1]) f ∈C(G) and TH(f)(xH) = ∫ H f(xh)dh. (2.1) Let µ be a Radon measure on G/H and x ∈ G. The translation µx of µ is defined by µx(E) = µ(xE), for Borel subsets E of G/H. The measure µ is called G-invariant if µx = µ, for x ∈ G. If G is compact, the homogeneous space G/H has a G-invariant measure µ, which satisfies the following Weil’s formula, for f ∈ L1(G) (see [1]) ∫ G/H TH(f)(xH)dµ(xH) = ∫ G f(x)dx. (2.2) If µ is a G-invariant measure on the homogeneous space G/H and p ≥ 1, the notation Lp(G/H,µ) stands for the Banach space of all equivalence classes of µ-measurable complex valued functions φ : G/H → C such that ‖φ‖Lp(G/H,µ) < ∞. Each irreducible representation of G is finite dimensional and every unitary representation of G is a direct sum of irreducible representations, see [1, 9]. The set of of all unitary equivalence classes of irreducible unitary representations of G is denoted by Ĝ. This definition of Ĝ is in essential agreement with the classical definition when G is Abelian, since each character of an Abelian group is a one dimensional representation of G. If π is any unitary representation of G, for u,v ∈ Hπ the functions πu,v(x) = 〈π(x)u,v〉 are called matrix elements of π. If {ej} is an orthonormal basis for Hπ, then πij means πei,ej . The notation Eπ is used for the linear span of the matrix elements of π and the notation E is used for the linear span of ⋃ [π]∈ĜEπ. The Peter-Weyl Theorem (see [1, 9]) guarantees that if G is a compact group, E is uniformly dense in C(G), L2(G) = ⊕ [π]∈ĜEπ, and {d −1/2 π πij : i,j = 1...dπ, [π] ∈ Ĝ} is an orthonormal basis for L2(G). Using the Peter-Weyl Theorem, for f ∈ L2(G) we have f = ∑ [π]∈Ĝ dπ∑ i,j=1 cπij(f)πij, (2.3) where cπi,j(f) = dπ〈f,πij〉L2(G). 3. Abstract Harmonic Analysis over Homogeneous Spaces of Compact Groups Throughout this article we assume that H is a closed subgroup of a compact group G with normalized Haar measures dh and dx respectively. We start this section with an extension of the linear map TH : C(G) → C(G/H) for other function spaces related to the homogeneous space G/H. If p = 1, it is easy to check that ‖TH(f)‖L1(G/H,µ) ≤ ‖f‖L1(G). Proposition 3.1. Let H be a closed subgroup of a compact group G. The linear map TH : C(G) → C(G/H) is a uniformly continuous. 24 A. GHAANI FARASHAHI Proof. Let f ∈C(G) and x ∈ G. Then we have |TH(f)(xH)| = ∣∣∣∣ ∫ H f(xh)dh ∣∣∣∣ ≤ ∫ H |f(xh)|dh ≤‖f‖sup (∫ H dh ) = ‖f‖sup, which implies ‖TH(f)‖sup ≤‖f‖sup. � Next we prove that the linear map TH is norm-decreasing in L 2-spaces. Theorem 3.1. Let H be a closed subgroup of a compact group G, µ be the normalized G-invariant measure on G/H associated to the Weil’s formula. The linear map TH : C(G) →C(G/H) has a unique extension to a bounded linear map from L2(G) onto L2(G/H,µ). Proof. We shall show that, each f ∈C(G) satisfies ‖TH(f)‖L2(G/H,µ) ≤‖f‖L2(G). Let f ∈C(G). Using compactness of H and the Weil’s formula we have ‖TH(f)‖2L2(G/H,µ) = ∫ G/H |TH(f)(xH)|2dµ(xH) = ∫ G/H ∣∣∣∣ ∫ H f(xh)dh ∣∣∣∣2 dµ(xH) ≤ ∫ G/H (∫ H |f(xh)|dh )2 dµ(xH) ≤ ∫ G/H ∫ H |f(xh)|2dhdµ(xH) = ∫ G/H ∫ H |f|2(xh)dhdµ(xH) = ∫ G/H TH(|f|2)(xH)dµ(xH) = ∫ G |f(x)|2dx = ‖f‖2L2(G). Thus, we can extend TH to a bounded linear operator from L 2(G) onto L2(G/H,µ), which we still denote it by TH. � Let Jp(G,H) := {f ∈ Lp(G) : TH(f) = 0}. Then, J2(G,H)⊥ is the orthogonal completion of the closed subspace J2(G,H) in L2(G). As an immediate consequence of Proposition 3.1 we deduce the following corollary. Corollary 3.1. Let H be a closed subgroup of a compact group G and µ be a G-invariant measure on G/H. The linear map TH : L 2(G) → L2(G/H,µ) is partial isometric. Proof. Let ϕ ∈ L2(G/H,µ) and ϕq := ϕ◦q. Then, we have ϕq ∈ L2(G). Indeed, ‖ϕq‖2L2(G) = ∫ G |ϕq(x)|2dx = ∫ G/H TH ( |ϕq|2 ) (xH)dµ(xH) = ∫ G/H (∫ H |ϕq(xh)|2dh ) dµ(xH) = ∫ G/H (∫ H |ϕ(xhH)|2dh ) dµ(xH) = ∫ G/H (∫ H |ϕ(xH)|2dh ) dµ(xH) = ∫ G/H |ϕ(xH)|2 (∫ H dh ) dµ(xH) = ∫ G/H |ϕ(xH)|2dµ(xH) = ‖ϕ‖2L2(G/H,µ). PETER-WEYL THEOREM FOR HOMOGENEOUS SPACES OF COMPACT GROUPS 25 Also T∗H(ϕ) = ϕq and THT ∗ H(ϕ) = ϕ. Because using the Weil’s formula, for all f ∈ L 2(G) we achieve 〈T∗H(ϕ),f〉L2(G) = 〈ϕ,TH(f)〉L2(G/H,µ) = ∫ G/H ϕ(xH)TH(f)(xH)dµ(xH) = ∫ G/H ϕ(xH)TH(f)(xH)dµ(xH) = ∫ G/H TH(ϕq.f)(xH)dµ(xH) = ∫ G ϕq(x)f(x)dx = 〈ϕq,f〉L2(G). Now a straightforward calculation implies TH = THT ∗ HTH. Then by Theorem 2.3.3 of [14], TH is a partial isometric operator. � We can conclude the following corollaries as well. Corollary 3.2. Let H be a closed subgroup of a compact group G. Let PJ2(G,H) and PJ2(G,H)⊥ be the orthogonal projections onto the closed subspaces J2(G,H) and J2(G,H)⊥ respectively. Then, for each f ∈ L2(G) and a.e. x ∈ G we have PJ2(G,H)⊥(f)(x) = TH(f)(xH), PJ2(G,H)(f)(x) = f(x) −TH(f)(xH). (3.1) Corollary 3.3. Let H be a compact subgroup of a locally compact group G and µ be a G-invariant measure on G/H. The following statements hold. (1) J2(G,H)⊥ = {ψq : ψ ∈ L2(G/H,µ)}. (2) For all f ∈J2(G,H)⊥ and each h ∈ H we have Rhf = f. (3) For all ψ ∈ L2(G/H,µ) we have ‖ψq‖L2(G) = ‖ψ‖L2(G/H,µ). (4) For all f,g ∈J2(G,H)⊥ we have 〈TH(f),TH(g)〉L2(G/H,µ) = 〈f,g〉L2(G). Remark 3.1. Invoking Corollary 3.3 one can regard L2(G/H,µ) as a closed subspace of L2(G), that is the subspace consists of all f ∈ L2(G) which satisfies Rhf = f for all h ∈ H. Then Theorem 3.1 and Proposition 3.1 guarantees that the linear map TH : L 2(G) → L2(G/H,µ) ⊂ L2(G) is an orthogonal projection. 4. Peter-Weyl Theorem for Homogeneous Spaces of Compact Groups For a closed subgroup H of G, define H⊥ := { [π] ∈ Ĝ : π(h) = I for all h ∈ H } , (4.1) If G is Abelian, each closed subgroup H of G is normal and the locally compact group G/H is Abelian and so Ĝ/H is precisely the set of all characters (one dimensional irreducible representations) of G which are constant on H, that is precisely H⊥. If G is a non-Abelian group and H is a closed normal subgroup of G, then the dual space Ĝ/H which is the set of all unitary equivalence classes of unitary representations of G/H, has meaning and it is well-defined. Indeed, G/H is a non-Abelian group. In this case, the map Φ : Ĝ/H → H⊥ defined by σ 7→ Φ(σ) := σ ◦ q is a Borel isomorphism and Ĝ/H = H⊥, see [1]. Thus if H is normal, H⊥ coincides with the classic definitions of the dual space either when G is Abelian or non-Abelian. For a closed subgroup H of G and a continuous unitary representation (π,Hπ) of G, define TπH := ∫ H π(h)dh, (4.2) where the operator valued integral (4.2) is considered in the weak sense. In other words, 〈TπHζ,ξ〉 = ∫ H 〈π(h)ζ,ξ〉dh, for ζ,ξ ∈Hπ. (4.3) 26 A. GHAANI FARASHAHI The function h 7→ 〈π(h)ζ,ξ〉 is bounded and continuous on H. Since H is compact, the right integral is the ordinary integral of a function in L1(H). Hence, TπH defines a bounded linear operator on Hπ with ‖TπH‖≤ 1. Remark 4.1. Let (π,Hπ) be a continuous unitary representation of G with TπH 6= 0. Let (σ,Hσ) be a continuous unitary representation of G such that [π] = [σ]. Let S : Hπ → Hσ be the unitary operator which satisfies σ(x)S = Sπ(x) for all x ∈ G. Then we have STπH = S (∫ H π(h)dh ) = ∫ H Sπ(h)dh = ∫ H σ(h)Sdh = (∫ H σ(h)dh ) S = TσHS, which implies that TσH 6= 0 as well. Thus we deduce that the non-zero property of T π H depends only on [π], that is the unitary equivalence class of (π,Hπ). Let KHπ := {ζ ∈Hπ : π(h)ζ = ζ ∀h ∈ H} . (4.4) Then, KHπ is a closed subspace of Hπ and R(TπH) = K H π , where R(TπH) = {T π Hζ : ζ ∈Hπ}. It is easy to see that [π] ∈ H⊥ if and only if KHπ = Hπ. Proposition 4.1. Let H be a closed subgroup of a compact group G and (π,Hπ) be a continuous unitary representation of G. Then, (1) The operator TπH is an orthogonal projection onto K H π . (2) The operator TπH is unitary if and only if [π] ∈ H ⊥. Proof. (1) Using compactness of H, it can be easily checked that (TπH) ∗ = TπH. As well as we achieve that TπHT π H = (∫ H π(h)dh )(∫ H π(t)dt ) = ∫ H π(h) (∫ H π(t)dt ) dh = ∫ H (∫ H π(h)π(t)dt ) dh = ∫ H (∫ H π(ht)dt ) dh = ∫ H TπHdt = T π H. (2) Since TπH is a projection, the operator T π H is unitary if and only if T π H = I. The operator TH is identity if and only if π(h) = I for all h ∈ H. Thus, TπH is unitary if and only if [π] ∈ H ⊥. � Definition 4.1. Let H be a closed subgroup of a compact group G. Then we define the dual space of G/H, as the subset of Ĝ which is given by Ĝ/H := { [π] ∈ Ĝ : TπH 6= 0 } = { [π] ∈ Ĝ : ∫ H π(h)dh 6= 0 } . (4.5) Evidently, any closed subgroup H of G satisfies H⊥ ⊂ Ĝ/H. (4.6) Next we shall show that the reverse inclusion of (4.6) holds, if and only if H is a normal subgroup of G. Theorem 4.1. Let H be a closed normal subgroup of a compact group G. Then Ĝ/H = H⊥. PETER-WEYL THEOREM FOR HOMOGENEOUS SPACES OF COMPACT GROUPS 27 Proof. Let H be a closed normal subgroup of a compact group G. It is sufficient to show that Ĝ/H ⊂ H⊥. Let [π] ∈ Ĝ/H be given. Due to the normality of H in G, for all x ∈ G the map τx : H → H given by h 7→ τx(h) := x−1hx belongs to Aut(H) and x−1Hx = H. Invoking compactness of G we have d (τx(h)) = dh, for x ∈ G. Now, for x ∈ G we get∫ H π(h)dh = ∫ xHx−1 π(τx(h))d (τx(h)) = ∫ H π(τx(h))dh = ∫ H π(x)∗π(h)π(x)dh = π(x)∗ (∫ H π(h)dh ) π(x) = π(x)∗TπHπ(x). Therefore π(x)TπH = T π Hπ(x) for x ∈ G, which implies T π H ∈ C(π). Irreducibility of π guarantees that TπH = αI for some non-zero α ∈ C with |α| ≤ 1. Thus, for t ∈ H we can write π(t) = α−1π(t)αI = α−1π(t)TπH = α−1 ∫ H π(th)dh = α−1 ∫ H π(h)dh = α−1TπH = I, which implies [π] ∈ H⊥. � Let (π,Hπ) be a continuous unitary representation of G such that TπH 6= 0. Then the functions πHζ,ξ : G/H → C defined by πHζ,ξ(xH) := 〈π(x)T π Hζ,ξ〉 for xH ∈ G/H, (4.7) for ξ,ζ ∈Hπ are called H-matrix elements of (π,Hπ). For xH ∈ G/H and ζ,ξ ∈Hπ, we have |πHζ,ξ(xH)| = |〈π(x)T π Hζ,ξ〉| ≤ ‖π(x)TπHζ‖‖ξ‖≤‖T π Hζ‖‖ξ‖≤‖ζ‖‖ξ‖. Also we can write πHζ,ξ(xH) = 〈π(x)T π Hζ,ξ〉 = πTπHζ,ξ(x). (4.8) Invoking definition of the linear map TH and also T π H we have TH(πζ,ξ)(xH) = ∫ H πζ,ξ(xh)dh = ∫ H 〈π(xh)ζ,ξ〉dh = ∫ H 〈π(x)π(h)ζ,ξ〉dh = 〈π(x)TπHζ,ξ〉, which implies that TH(πζ,ξ) = π H ζ,ξ. (4.9) Theorem 4.2. Let H be a closed subgroup of a compact group G, µ be the normalized G-invariant measure and (π,Hπ) be a continuous unitary representation of G such that TπH 6= 0. Then (1) The subspace Eπ(G/H) depends on the unitary equivalence class of π. (2) The subspace Eπ(G/H) is a closed left invariant subspace of L1(G/H,µ). 28 A. GHAANI FARASHAHI Proof. (1) Let (σ,Hσ) be a continuous unitary representation of G such that [π] = [σ]. Let S : Hπ → Hσ be the unitary operator which satisfies σ(x)S = Sπ(x) for all x ∈ G. Remark 4.1 guarantees that STπH = T σ HS and also T σ H 6= 0. Thus for x ∈ G and ζ,ξ ∈Hπ we can write πHζ,ξ(xH) = 〈π(x)T π Hζ,ξ〉Hπ = 〈S−1σ(x)STπHζ,ξ〉Hπ = 〈σ(x)STπHζ,Sξ〉Hσ = 〈σ(x)TσHSζ,Sξ〉Hσ = σ H Sζ,Sξ(xH), which implies that Eπ(G/H) = Eσ(G/H). (2) It is straightforward. � If ζ,ξ belongs to an orthonormal basis {ei} for Hπ, H-matrix elements of [π] with respect to an orthonormal basis {ej} changes in the form πHij (xH) = π H ej,ei (xH) = 〈π(x)TπHej,ei〉, for xH ∈ G/H. (4.10) The linear span of the H-matrix elements of a continuous unitary representation (π,Hπ) satisfying TπH 6= 0, is denoted by Eπ(G/H) which is a subspace of C(G/H). Definition 4.2. Let H be a closed subgroup of a compact group G and [π] ∈ Ĝ/H. An ordered orthonormal basis B = {e` : 1 ≤ ` ≤ dπ} of the Hilbert space Hπ is called H-admissible, if it is an extension of an orthonormal basis {e` : 1 ≤ ` ≤ dπ,H} of the closed subspace KHπ , which equivalently means that dπ,H-first elements of B be an orthogonal basis of KHπ . Let [π] ∈ Ĝ/H and Bπ = {e` : 1 ≤ ` ≤ dπ} be an H-admissible basis for the representation space Hπ. Then, each πi` with 1 ≤ i ≤ dπ and 1 ≤ ` ≤ dπ,H, is a well-defined continuous function over G/H. Let E`π(G/H) be the subspace of C(G/H) spanned by the set B`π := { √ dππi` : 1 ≤ i ≤ dπ}. Proposition 4.2. Let [π] ∈ Ĝ/H, Bπ be an H-admissible basis for the representation space Hπ, and 1 ≤ ` 6= `′ ≤ dπ,H. Then (1) dimE`π(G/H) = dπ and B`π is an orthonormal basis for E`π(G/H). (2) E`π(G/H) is a closed left translation invariant subspace of C(G/H). (3) E` ′ π (G/H) ⊥E`π(G/H). Proof. (1) Let 1 ≤ i, i′ ≤ dπ. Then by Theorem 27.19 of [10] we get 〈πi`,πi′`〉L2(G/H,µ) = 〈πi`,πi′`〉L2(G) = d−1π δii′. Since dimE`π(G/H) ≤ dπ we achieve that B`π is an orthonormal basis for E`π(G/H) and hence dimE`π(G/H) = dπ. (2) It is straightforward. (3) Let 1 ≤ i, i′ ≤ dπ. Applying Theorem 27.19 of [10] we get 〈πi`,πi′`′〉L2(G/H,µ) = 〈πi`,πi′`′〉L2(G) = d−1π δii′δ``′, which completes the proof. � The following theorem shows that H-admissible bases lead to orthogonal decompositions of the subspace Eπ(G/H). Theorem 4.3. Let H be a closed subgroup of a compact group G. Let [π] ∈ Ĝ/H and Bπ = {e`,π : 1 ≤ ` ≤ dπ} be an H-admissible basis for the representation space Hπ. Then Bπ(G/H) := { √ dππi` : 1 ≤ i ≤ dπ, 1 ≤ ` ≤ dπ,H} is an orthonormal basis for the Hilbert space Eπ(G/H) and hence it satisfies the following direct sum decomposition Eπ(G/H) = dπ,H⊕ `=1 E`π(G/H). (4.11) PETER-WEYL THEOREM FOR HOMOGENEOUS SPACES OF COMPACT GROUPS 29 Proof. It is straightforward to check that Bπ(G/H) spans the subspace Eπ(G/H). Then Proposition 4.2 guarantees that Bπ(G/H) is an orthonormal set in Eπ(G/H). Since dimEπ(G/H) ≤ dπ,Hdπ we deduce that it is an orthonormal basis for Eπ(G/H), which automatically implies the decomposition (4.11). � Next proposition lists basic properties of H-matrix elements. Proposition 4.3. Let H be a closed subgroup of a compact group G, µ be the normalized G-invariant measure on G/H, and (π,Hπ) be a continuous unitary representation of G. Then, (1) TπH = 0 if and only if Eπ(G) ⊆J 2(G,H). (2) If TπH 6= 0 then TH(Eπ(G)) = Eπ(G/H) and T ∗ H(Eπ(G/H)) ⊆Eπ(G). (3) Eπ(G) ⊆J2(G,H)⊥ if and only if π(h) = I for all h ∈ H. Then we can prove the following orthogonality relation concerning the functions in E(G/H). Theorem 4.4. Let H be a closed subgroup of a compact group G, µ be a normalized G-invariant measure on G/H and [π] 6= [σ] ∈ Ĝ/H. The closed subspaces Eπ(G/H) and Eσ(G/H) are orthogonal to each other as subspaces of the Hilbert space L2(G/H,µ). Proof. Let ψ ∈ Eπ(G/H) and ϕ ∈ Eσ(G/H). Then we have ψq ∈ Eπ(G) and also ϕq ∈ Eσ(G). Using Proposition 4.3, Corollary 3.3, and Theorem 27.15 of [10], we get 〈ϕ,ψ〉L2(G/H,µ) = 〈ϕq,ψq〉L2(G) = 0. which completes the proof. � We can define E(G/H) := the linear span of ⋃ [π]∈Ĝ/H Eπ(G/H). (4.12) Next theorem presents some analytic aspects of the function space E(G/H). Theorem 4.5. Let H be a closed subgroup of a compact group G and µ be the normalized G-invariant measure on G/H associated to the Weil’s formula. Then, (1) The linear operator TH maps E(G) onto E(G/H). (2) E(G/H) is ‖.‖L2(G/H,µ)-dense in L2(G/H,µ). (3) E(G/H) is ‖.‖sup-dense in C(G/H). Proof. (1) It is straightforward. (2) Let φ ∈ L2(G/H,µ) and also f ∈ L2(G) with TH(f) = φ. Then by ‖.‖L2(G)-density of E(G) in L2(G) we can pick a sequence {fn} in E(G) such that f = ‖.‖L2(G) − limn fn. By Proposition 4.3 we have {TH(fn)}⊆E(G/H). Then continuity of the linear map TH : L2(G) → L2(G/H,µ) implies φ = TH(f) = ‖.‖L2(G/H,µ) − lim n TH(fn), which completes the proof. (3) Invoking uniformly boundedness of TH, uniformly density of E(G) in C(G), and the same argument as used in (1), we get ‖.‖sup-density of E(G/H) in C(G/H). � The following theorem can be considered as an abstract extension of the Peter-Weyl Theorem for homogeneous spaces of compact groups. Theorem 4.6. Let H be a closed subgroup of a compact group G and µ be the normalized G-invariant measure on G/H. The Hilbert space L2(G/H,µ) satisfies the following orthogonality decomposition L2(G/H,µ) = ⊕ [π]∈Ĝ/H Eπ(G/H). (4.13) Proof. Using Peter-Weyl Theorem, Proposition 4.3, and since the bounded linear map TH : L 2(G) → L2(G/H,µ) is surjective we achieve that each ϕ ∈ L2(G/H,µ) has a decomposition to elements of Eπ(G/H) with [π] ∈ Ĝ/H, namely ϕ = ∑ [π]∈Ĝ/H cπϕπ, (4.14) 30 A. GHAANI FARASHAHI with ϕπ ∈ Eπ(G/H) for all [π] ∈ Ĝ/H. Since the subspaces Eπ(G/H) with [π] ∈ Ĝ/H are mutually orthogonal we conclude that decomposition (4.14) is unique for each ϕ, which guarantees (4.13). � We immediately deduce the following corollaries. Corollary 4.1. Let H be a closed subgroup of a compact group G and µ be the normalized G-invariant measure on G/H. For each [π] ∈ Ĝ/H, let Bπ = {e`,π : 1 ≤ ` ≤ dπ} be an H-admissible basis for the representation space Hπ. Then we have the following statements. (1) The Hilbert space L2(G/H,µ) satisfies the following direct sum decomposition L2(G/H,µ) = ⊕ [π]∈Ĝ/H dπ,H⊕ `=1 E`π(G/H), (4.15) (2) The set B(G/H) := {πi` : 1 ≤ i ≤ dπ, 1 ≤ ` ≤ dπ,H} constitutes an orthonormal basis for the Hilbert space L2(G/H,µ). (3) Each ϕ ∈ L2(G/H,µ) decomposes as the following ϕ = ∑ [π]∈Ĝ/H dπ dπ,H∑ `=1 dπ∑ i=1 〈ϕ,πi`〉L2(G/H,µ)πi`, (4.16) where the series is converges in L2(G/H,µ). Remark 4.2. Let H be a closed normal subgroup of a compact group G. Also, let µ be the normalized G-invariant measure over G/H associated to the Weil’s formula. Then G/H is a compact group and the normalized G-invariant measure µ is a Haar measure of the quotient compact group G/H. By Theorem 4.1, we deduce that Ĝ/H = H⊥, and for each [π] ∈ Ĝ/H we get TπH = I and dπ,H = dπ. Thus we obtain L2(G/H) = ⊕ [π]∈H⊥ Eπ(G/H), which precisely coincides with the decomposition associated to applying the Peter-Weyl Theorem to the compact quotient group G/H. References [1] G.B. 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Weyl, Die Vollstndigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann., 97 (1927) 737-755. PETER-WEYL THEOREM FOR HOMOGENEOUS SPACES OF COMPACT GROUPS 31 Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Austria ∗Corresponding author: arash.ghaani.farashahi@univie.ac.at, ghaanifarashahi@hotmail.com 1. Introduction 2. Preliminaries and Notations 3. Abstract Harmonic Analysis over Homogeneous Spaces of Compact Groups 4. Peter-Weyl Theorem for Homogeneous Spaces of Compact Groups References