ClassicalFourierAnalysisoverHomogeneous Spaces of Compact Groups

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)