Arash Ghaani Farashahi

Numerical Harmonic Analysis Group(NuHAG),Faculty Mathematics,University of Vienna,Vienna,Austria.

ClassicalFourierAnalysisoverHomogeneous Spaces of Compact Groups

Arash Ghaani Farashahi?

Numerical Harmonic Analysis Group(NuHAG),Faculty Mathematics,University of Vienna,Vienna,Austria.

.This paper introduces a unif i ed operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups.Let G be a compact group and H be a closed subgroup of G.Let G/H be the left coset space of H in G and μ be the normalized G-invariant measure on G/H associated to the Weil’s formula. Then,we present a generalized abstract framework of Fourier analysis for the Hilbert function space L2(G/H,μ).

Compact group,homogeneous space,dual space,Fourier transform,Plancherel (trace)formula,Peter-Weyl Theorem.

AMS SubjectClassif i cations:20G05,43A85,43A32,43A40,43A90.

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 coherent states analysis[2–4,12],theoretical and particle physics[1,9–11,13].Over the last decades,abstract and computational aspects of Plancherel formulas over symmetric spaces have achieved signi fi cant popularity in geometric analysis,mathematical physics and scienti fi c computing(computational engineering),see[6,7,13–18]and references therein.

Let G be a compact group,H be a closed subgroup of G,and μ be the normalized G-invariant measure on G/H associated to the Weil’s formula.The left coset space G/H is considered as a compact homogeneous space,which G acts on it via the left action. This paper which contains 5 sections,is organized as follows.Section 2 is devoted to fi x notations and preliminaries including a brief summary on Hilbert-Schmidt operators,non-Abelian Fourier analysis over compact groups,and classical results on abstract harmonic analysis over locally compact homogeneous spaces.We present some abstract harmonic analysis aspects of the Hilbert function space L2(G/H,μ),in Section 3.Then we def i ne the abstract notion of dual spacefor the homogeneous space G/H and we will show that this def i nition is precisely the standard dual space for the compact quotient group G/H,when H is a closed normal subgroup of G.We then introduce the def i nition of abstract operator-valued Fourier transform over the Banach function space L1(G/H,μ)and also generalized version of the abstract Plancherel(trace)formula for the Hilbert function space L2(G/H,μ).The paper closes by a presentation of Peter-Weyl Theorem for the Hilbert function space L2(G/H,μ).

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＜ ∞. 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 isThe set HS(H)is a self adjoint two sided ideal in B(H)and if H is f i nite-dimensional we have HS(H)=B(H).An operator T ∈ B(H)is trace-class,whenever=tr[|T|]＜ ∞,if tr[T]=and |T|=[20].

Let G be a compact group with the probability Haar measure dx.Then each irreducible representation of G is f i nite dimensional and every unitary representation of G is a direct sumofirreducible representations,see[1,10].The setof ofall unitary equivalence classes of irreducible unitary representations of G is denoted byThis def i nition ofis in essential agreement with the classical def i nition when G is Abelian,since each characterofanAbelian groupis a onedimensionalrepresentationof G.If π is any unitary representationof G,for ζ,ξ∈Hπthe functions πζ,ξ(x)= 〈π(x)ζ,ξ〉are called matrix elements of π.If{ej}is an orthonormal basis for Hπ,then πijmeansThe notation Eπis used for the linear span of the matrix elements of π and the notation E is used for the linear spanThen Peter-Weyl Theorem[1,10]guarantees that if G is a compact group,E is uniformly dense in C(G),L2(G)=andis an orthonormal basis for L2(G).For f∈ L1(G)and[π ]∈ bG,the Fourier transform of f at π is def i ned in the weak sense as an operator in B(Hπ)by

If π (x)is represented by the matrix(πij(x)) ∈ Cdπ×dπ.Then∈ Cdπ×dπis the matrix with entries given bywhich satisf i es

Let H be a closed subgroup of G with the probability Haar measure dh.The left coset space G/H is considered as a compact homogeneous space that G acts on it from the left and q:G → G/H given by xq(x):=xH is the surjective canonical map.The classical aspects of abstract harmonic analysis on locally compact homogeneous spaces are quite well studiedby several authors,see[5,8,10,11,22]and references therein.If G is compact, each transitive G-space can be considered as a left coset space G/H for some closed subgroup H of G.The function space C(G/H)consists of all functions TH(f),where f∈ C(G) and

Let μ be a Radon measure on G/H and x ∈ G.The translation μxof μ is def i ned by μx(E)= μ(xE),for all Borel subsets E of G/H.The measure μ is called G-invariant if μx= μ,for all x ∈ G.The homogeneousspace G/H has a normalized G-invariant measure μ,which satisf i es the following Weil’s formula[1,22]

and also the following norm-decreasing formula

3 Abstract harmonic analysis of Hilbert function spaces over homogeneous spaces of compact groups

Throughout this paper we assume that G is a compact group with the probability Haar measure dx,H is a closed subgroup of G with the probability Haar measure dh,and also μ is the normalized G-invariant measure on the homogeneousspace G/H which satisf i es (2.4).

In this section,we present some properties of the Hilbert function space L2(G/H,μ) in the framework of abstract harmonic analysis.

First we shall show that the linear map THhas a unique extensionto a bounded linear map from L2(G)onto L2(G/H,μ).

Theorem 3.1.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.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.Let μ be the normalized G-invariant measure on the homogeneous space G/H which satisf i es(2.4)and f ∈ C(G).Then we claim that

To this end,using compactness of H,we have

Then,by the Weil’s formula,we get

which implies(3.1).Thus,we can extend THto a bounded linear operator from L2(G) onto L2(G/H,μ),which we still denote it by THand satisf i es

Thus,we complete the proof.

Let J2(G,H):={f∈ L2(G):TH(f)=0}and J2(G,H)⊥be the orthogonal completion of the closed subspace J2(G,H)in L2(G).

As an immediate consequence of Theorem 3.1 we deduce the following result.

Proposition 3.1.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 TH:L2(G)→L2(G/H,μ)is a partial isometric linear map.

Proof.Let ? ∈ L2(G/H,μ)and ?q:= ?? q.Then ?q∈ L2(G)with

Indeed,using the Weil’s formula we can write

and since H is compact and dh is a probability measure,we get

forall f∈ L2(G),which implies that(?)= ?q.Nowa straightforwardcalculation shows that TH=THT?HTH.Then by Theorem 2.3.3 of[20],THis a partial isometric operator.

We then can conclude the following corollaries as well.

Corollary 3.1.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

1.PJ2(G,H)⊥(f)(x)=TH(f)(xH).

2.PJ2(G,H)(f)(x)=f(x)?TH(f)(xH).

Corollary 3.2.Let H be a compact 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.J2(G,H)⊥={ψq:ψ ∈ L2(G/H,μ)}.

2.For f∈ J2(G,H)⊥and h ∈ H we have Rhf=f.

3.For ψ ∈ L2(G/H,μ)we have ‖ψq‖L2(G)= ‖ψ‖L2(G/H,μ).

4.For f,g ∈ J2(G,H)⊥we have 〈TH(f),TH(g)〉L2(G/H,μ)= 〈f,g〉L2(G). We f i nish this section by the following remark.

Remark 3.1.Invoking Corollary 3.2one can regard theHilbert functionspace L2(G/H,μ) as a closed linear subspace of the Hilbert function space L2(G),that is the closed linear subspace consists of all f∈ L2(G)which satisf i es Rhf=f for all h ∈ H.Then Theorem 3.1 and Proposition 3.1 guarantees that the bounded linear map

is an orthogonal projection.

4 Abstract trace formulas over homogeneous spaces of compact groups

In this section,we present the abstract notions of dual spaces and Plancherel(trace)formulas over homogeneous spaces of compact groups.

For a closed subgroup H of G,let

Then by def i nition we have

If G is Abelian,each closed subgroup H of G is normal and the compact group G/H is Abelian and sois 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 spacewhich is the set of all unitary equivalence classes of unitary representations of the quotient group G/H, has meaning and it is well-def i ned.Indeed,G/H is a non-Abelian group.In this case,the map Φ→ H⊥def i ned by σΦ(σ):= σ ? q is a Borel isomorphism and=H⊥, see[1,19,23].Thus if H is normal,H⊥coincides with the classic def i nitions of the dual space either when G is Abelian or non-Abelian.

For a given closed subgroup H of G and also a continuous unitary representation (π,Hπ)of G,def i ne

where the operator valued integral(4.3)is considered in the weak sense.In other words,

Def i nition 4.1.Let H be a compact subgroup of a compact group G.The dual spaceof the left coset space G/H,is def i ned as the subset ofbG given by

Then evidently we have

First we shall present an interesting property of(4.5),when the left coset space G/H has the canonical quotient group structure.

Next theorem shows that the reverse inclusion of(4.6)holds,if H is a normal subgroup of G.

Theorem 4.1.Let H be a closed normal subgroup of a compact group G.Then,

Proof.Let H be a closed normal subgroup of a compact group G.Invoking the inclusion (4.6),it is suff i cient to show that? H⊥.Let[π]∈be given.Due to normality of H in G the map τx:H → H given by hτx(h):=x?1hx belongs to Aut(H)and also we have x?1Hx=H,for all x ∈ G.Let x ∈ G.Then by compactness of G we have d(τx(h))=dh and hence we can write

which implies[π]∈ H⊥.

Let

It is easy to see that[π ]∈ H⊥if and only if=

Then,we can also present the following results.

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,

Proof.(1)Using compactness of H,we have

As well as,we can write

Let ? ∈ L1(G/H,μ)and[π]∈.The Fourier transform of ? at[π]is def i ned as the linear operator

on the Hilbert space Hπ,where for each xH ∈ G/H the notation Γπ(xH)stands for the bounded linear operator def i ned on the Hilbert space Hπby Γπ(xH)= π(x)that is

Then we have

for all ζ,ξ∈ Hπ.Indeed,

Remark 4.1.Let H be a closed normal subgroup of a compact group G and μ be the normalized G-invariant measure over the left coset space G/H associated to the Weil’s formula.Then it is easy to check that μ is a Haar measure of the compact quotient group G/H and by Theorem4.1 wehave=H⊥.Also,foreach ?∈L1(G/H,μ)and[π ]∈ H⊥, we have

Thus,we deduce that the abstract Fourier transform def i ned by(4.8)coincides with the classical Fourier transform over the compact quotient group G/H if H is normal in G.

The operator-valued integral(4.8)is considered in the weak sense.That is

Because,we can write

If ζ,ξ∈ Hπ,then we have

The following propositionpresentsthe canonical connectionofthe abstract Fouriertransform def i ned in(4.8)with the classical Fourier transform(2.1).

Proposition 4.2.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,for ? ∈L1(G/H,μ)and[π]∈,we have

Proof.Using the Weil’s formula and also(4.11),for ζ,ξ∈ Hπ,we can write

which implies(4.12).

In the next theorem we show that the abstract Fourier transform def i ned in(4.8)satisf i es a generalized version of the Plancherel(trace)formula.

Theorem 4.2.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,each ? ∈ L2(G/H,μ)satisf i es the following Plancherel formula;

Proof.Let ? ∈ L2(G/H,μ)be given.If[π ]with[π ]thenwe have TπH=0.Hence, for ζ,ξ ∈ Hπ,we have TH(πζ,ξ)=0.Therefore,we get

Indeed,using the Weil’s formula,for ζ,ξ ∈ Hπwe can write

Using Eqs.(4.12),(4.14),invoking Plancherel formula(2.2),and also Corollary 3.2 we achieve

which implies(4.13).

Remark 4.2.Let H be a closed normal subgroup of a compact group G and μ be the normalized G-invariant measure over the left coset space G/H associated to the Weil’s formula.Then Theorem 4.1 implies that=H⊥and hence the Plancherel(trace) formula(4.13)reads as follows;

for all ? ∈ L2(G/H,μ),where

for all[π]∈ H⊥,see Remark 4.1.

5 Peter-Weyl theorem for homogeneous spaces of compact groups

In this section we present a version of Peter-Weyl Theorem[21]for the Hilbert function space L2(G/H,μ).

Let(π ,Hπ)be a continuous unitary representation of G such that0.Then the functionsG/H → C def i ned by

for ξ,ζ∈ Hπare called H-matrix elements of(π,Hπ). For xH ∈ G/H and ζ,ξ∈ Hπ,we have

Also we can write

Invoking def i nition of the linear map THand alsowe have

which implies that

Theorem 5.1.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 that0.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).

Proof.(1)Let(σ,Hσ)be a continuous unitary representation of G such that[π]=[σ].Let S:Hπ→ Hσbe the unitary operator which satisf i es σ(x)S=Sπ(x)for all x ∈ G.Then=S and also0.Thus for x ∈ G and ζ,ξ∈ Hπwe can write

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

The linear span of the H-matrix elements of a continuous unitary representation(π,Hπ) satisfying0,is denoted by Eπ(G/H)which is a subspace of C(G/H).

Def i nition 5.1.Let H be a closed subgroup of a compact group G and[π]∈An orderedorthonormalbasisB={e?:1≤?≤dπ}oftheHilbertspaceHπiscalled H-admissible, if it is an extension of an orthonormal basis{e?:1 ≤ ?≤ dπ,H}of the closed subspacewhich equivalently means that dπ,H-f i rst elements of B be an orthogonal basis of.

Proposition 5.1.Let[π]∈Bπbe an H-admissible basis for the representation space Hπ, and 1 ≤?′≤ dπ,H.Then

(2)It is straightforward.

(3)Let 1 ≤ i,i′≤ dπ.Applying Theorem 27.19 of[11]we get

which completes the proof.

The following theorem shows that H-admissible bases lead to orthogonal decompositions of the subspace Eπ(G/H).

Theorem 5.2.Let H be a closed subgroup of a compact group G.Let[π]∈and Bπ= {e?,π:1 ≤ ?≤ dπ}be an H-admissible basis for the representation space Hπ.Then Bπ(G/H):=is an orthonormal basis for the Hilbert space Eπ(G/H)and hence it satisf i es the following direct sum decomposition

Proof.It is straightforward to check that Bπ(G/H)spans the subspace Eπ(G/H).Then Proposition 5.1 guarantees that Bπ(G/H)is an orthonormal set in Eπ(G/H).Since dimEπ(G/H) ≤ dπ,Hdπwe deduce that it is an orthonormal basis for Eπ(G/H),which automatically implies the decomposition(5.5).

Next proposition lists basic properties of H-matrix elements.

Proposition 5.2.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,

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 5.3.Let H be a closed subgroup of a compact group G,μ be a normalized G-invariant measure on G/H and[π ][σ]∈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 ? ∈ Eσ(G/H).Then we have ψq∈ Eπ(G)and also ?q∈ Eσ(G). Using Proposition 5.2,Corollary 3.2,and Theorem 27.15 of[11],we get

which completes the proof.

We can def i ne

Next theorem presents some analytic aspects of the function space E(G/H).

Theorem 5.4.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 THmaps 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)? limnfn. By Proposition 5.2 we have{TH(fn)} ? E(G/H).Then continuity of the linear map TH: L2(G)→ L2(G/H,μ)implies

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 5.5.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)satisf i es the following orthogonality decomposition

Proof.Using Peter-Weyl Theorem,Proposition 5.2,and since the bounded linear map TH:L2(G) → L2(G/H,μ)is surjective we achieve that each ? ∈ L2(G/H,μ)has a decomposition to elements of Eπ(G/H)withnamely

with ?π∈ Eπ(G/H)for all[π]∈Since the subspaces Eπ(G/H)with[π ]∈are mutually orthogonal we conclude that decomposition(5.8)is unique for each ?,which guarantees(5.7).

We immediately deduce the following corollaries.

Corollary 5.1.Let H be a closed subgroup of a compact group G and μ be the normalizedG-invariant measure on G/H.For each[π]∈let Bπ={e?,π:1 ≤ ?≤ dπ}be an H-admissible basis forthe representationspaceHπ.Thenwe have thefollowing statements.

1.The Hilbert space L2(G/H,μ)satisf i es the following direct sum decomposition

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 ? ∈ L2(G/H,μ)decomposes as the following:

where the series is converges in L2(G/H,μ).

Remark 5.1.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 thatand for eachwe get=I and dπ,H=dπ.Thus we obtain

which precisely coincides with the decomposition associated to applying the Peter-Weyl Theorem to the compact quotient group G/H.

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Received 23 September 2014;Accepted(in revised version)28 July 2016

?Corresponding author.Email addresses:arash.ghaani.farashahi@univie.ac.at,ghaanifarashahi@ hotmail.com(A.Ghaani Farashahi)