Comptes Rendus

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Comptes Rendus

Mathématique

Alexandre Baldare, Rémi Côme and Victor Nistor

Fredholm conditions for operators invariant with respect to compact Lie group actions

Volume 359, issue 9 (2021), p. 1135-1143 Published online: 3 November 2021 https://doi.org/10.5802/crmath.257

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Creative Commons Attribution4.0International License.

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2021, 359, n9, p. 1135-1143

https://doi.org/10.5802/crmath.257

Operator theory /Théorie des opérateurs

Fredholm conditions for operators invariant with respect to compact Lie group actions

Alexandre Baldare^a, Rémi Côme^band Victor Nistor^b

aInstitut fur Analysis, Welfengarten 1, 30167 Hannover, Germany bUniversité Lorraine, 57000 Metz, France

E-mails:[email protected] (A. Baldare), [email protected] (R. Côme), [email protected] (V. Nistor)

Abstract.LetG be a compact Lie group acting smoothly on a smooth, compact manifold M, letP∈ ψ^m(M;E0,E1) be aG–invariant, classical pseudodifferential operator acting between sections of twoG- equivariant vector bundlesE_i→M,i=0, 1, and letαbe an irreducible representation of the groupG. Then Pinduces a mapπα(P) :H^s(M;E₀)_α→H^s⁻^m(M;E₁)_αbetween theα-isotypical components. We prove that the mapπ_α(P) is Fredholm if, and only if,Pistransversallyα-elliptic, a condition defined in terms of the principal symbol ofPand the action ofGon the vector bundlesE_i.

2020 Mathematics Subject Classification.47A53, 58J40, 57S15, 47L80, 46N20.

Funding.A.B., R.C., and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR) . Manuscript received 7th December 2020, revised 25th March 2021 and 8th July 2021, accepted 10th August 2021.

1. Introduction

We letGbe acompact Lie groupacting smoothly on a smooth,compactriemannian manifold M. There is no loss of generality to assume thatM is endowed with aG-invariant metric. Let ψ^m(M;E0,E1) be the space of orderm, classical pseudodifferential operators acting between sections of two G-equivariant vector bundles E_i →M, i = 0, 1, and letα be an irreducible representation of the groupG. ThenP∈ψ^m(M;E₀,E₁)^Ginduces, byG-invariance, a map

πα(P) :H^s(M;E₀)_α→H^s−m(M;E₁)_α (1) between theα-isotypical components of the corresponding Sobolev spaces of sections. In this short note, we obtain necessary and sufficient conditions forπα(P) to be Fredholm. Fredholm operators are important in many applications to PDEs and geometry. Our result generalizes to the case of compact Lie groups the results of [3, 4], which dealt with finite groups. We assume the reader is familiar with [3] and here we just explain the main differences from the case of finite groups.

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1136 Alexandre Baldare, Rémi Côme and Victor Nistor

1.1. Notation.

In order to state our main result, we need to set up some notation, some of which is classical and some of which was already introduced in [3, 4]. IfGacts onX, we letX^G be the set of fixed points ofG, we letG_x be the stabilizer of a pointx∈X (inG), and, finally, we letX_(H)be the set of points whose stabilizer is conjugated toH. Also, as usual,Gbdenotes the set of equivalence classes of irreducibleG-modules (or representations, agreeing that all our representations are strongly continuous). In general, ifT :V0→V1is aG-equivariant linear map ofG-modules and α∈G, we letb

πα(T) :V_0α→V_1α (2) denote the inducedG-linear map between theα-isotypical components of theG-modulesV_i,i= 0, 1. SincePisG-invariant, its principal symbolσm(P) belongs toC^∞(T^∗Mà{0}; Hom(E₀,E₁))^G, whereπ:T^∗M →M is the cotangent bundle projection. LetG_ξ andG_x denote the isotropy subgroups ofξ∈T_x^∗M andx∈M, as usual. ThenG_ξ ⊂Gx acts linearly on the fibersE0x and E1x. Letgdenote the Lie algebra ofG. Then anyY ∈gdefines a canonical vector fieldYMonM.

Let theG-transverse cotangent space of Mbe given by T_G^∗M :=©

ξ∈T^∗M¯

¯∀Y∈g,〈ξ,Y_M(π(ξ))〉 =0ª

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as in [2]. As usual,S^∗Mdenotes the unit cosphere bundle ofM. LetS_G^∗M:=S^∗M∩T_G^∗Mdenote the set of unit covectors in theG-transverse cotangent spaceT_G^∗M. The groupGacts on {G_ξ|ξ∈ T^∗M} byg·G_ξ:=G_g_ξ=gG_ξg⁻¹. Forρ∈Gb_ξdefineg·ρ∈Gb_g_ξby (g·ρ)(h)=ρ(g⁻¹hg), for allh∈G_g_ξ. The characterization of Fredholm operators can be reduced to each component of the orbit space M/G, and therefore we can and will assumeM/Gto beconnected. Under this hypothesis there exists a minimal isotropy subgroupKsuch that any isotropy subgroup ofGacting onMcontains a subgroup conjugated toKand the set of pointsM_(K)with stabilizer conjugated toKis an open dense submanifoldM_(K)ofMcalled theprincipal orbit bundle of M ,see [9].

1.2. Theα-principal symbol andα-ellipticity LetE=E₀⊕E₁andΩM(E) :=©

(ξ,ρ)∈S_G^∗M×Gb_ξ¯

¯E_ξρ6=0ª . Then σ^Gm(P) : ΩM(E)→ [

(ξ,ρ)∈ΩM(E)

Hom(E₀_ξρ,E₁_ξρ)^G^ξ, σ^Gm(P)(ξ,ρ) :=π_ρ¡

σm(P)(ξ)¢

∈Hom(E₀_ξρ,E₁_ξρ)^G^ξ, ξ∈(S_G^∗M)_x,

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is theG-principal symbolσ^Gm(P) ofP. (Here we have used thatE_x=E_ξfor anyξ∈T_x^∗M, by the definition of the pullbackπ^∗(E) ofE →M toT^∗M.) If AandBare compact groups, ifH is a subgroup of bothAandB, and if HomH(α,β)6=0, thenα∈Abandβ∈Bbare saidH -associated[4].

Let us fix a minimal isotropy groupK⊂GforMand let Ω^α_M(E) :=©

(ξ,ρ)∈ΩM(E)¯

¯∃g∈G,g·ρandαareK-associatedª

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In the definition of the spaceΩ^α_M(E) above, it is implicit that the group elementg∈Gis such that K⊂g·G_ξ.

Definition 1. Theα-principal symbolσ^αm(P)of P isσ^αm(P) :=σ^Gm(P)|_Ω^α_M(E). We shall say that P∈ ψ^m(M;E₀,E₁)^Gistransversallyα-ellipticif itsα-principal symbolσ^αm(P)is invertible everywhere on its domain of definition.

The transversal1-ellipticity is related with transversal ellipticity on (singular) foliations [1,7,8].

We can now formulate our main result.

C. R. Mathématique—2021, 359, n9, 1135-1143

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Theorem 2. Let m∈R, P∈ψ^m(M;E₀,E₁)^Gandα∈G. Thenb πα(P) :H^s(M;E0)_α→H^s−m(M;E1)_α is Fredholm if, and only if P is transversallyα-elliptic.

ForG finite, our main result was proved before [3, 4]. This is the first paper that deals with the non-discrete case. As far as the statement of the result goes, the case non-discrete is different from the discrete case in thatS_G^∗M 6=S^∗M, in general. The proof in the non-discrete case, is, however, significantly different from the one in the discrete case. Extending our results to the case of compact Lie groups is motivated, in part, by questions in Index Theory and also by the recent preprint [6] and the reference therein. The techniques used in this paper to obtain Fredholm conditions are related also to the ones in [16], used forG-opeators, and the ones in [5], used for complexes of operators. See also [8, 12, 14, 15].

We thank Matthias Lesch, Paul-Emile Paradan, and Elmar Schrohe for useful discussions.

2. Background material

This section is devoted to background material and results. The reader can find more details in [3, 4]. We concentrate only on the material that is significantly different from the discrete case, for which we refer to [3]. There is no loss of generality to assume thatM/Gis connected (recall thatGis a compact Lie group acting by isometries on the compact Riemannian manifoldM).

One of the main differences in the non-discrete case is that we need two versions of induction.

LetH ⊂G be a closed subgroup andV be anH-module, we define, as usual, thecontinuous induced representationby

c₀- Ind^G_H(V) :=C(G,V)^H= ©

f ∈C(G,V)¯

¯f(g h⁻¹)=h f(g)ª

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Assume thatVis a Hilbert space and that the representation is unitary. Then we let thehilbertian induced representation L²- Ind^G_H(V) be the completion ofc₀- Ind^G_H(V) with respect to the induced Hilbert space norm. ThenGacts by left translation onc₀- Ind^G_H(V) andL²- Ind^G_H(V). Frobenius reciprocity holds for both types of induction and that is used in the proof.

Let us summarize two important properties of theG-transversal spacesT_G^∗MandS^∗_GM.

Lemma 3. Let H be a closed subgroup of G and S be a H -manifold. Then T_G^∗(G×HS)∼=G×HT_H^∗S. If M/G is connected with minimal isotropy group K , then S^∗_GM_(K₎is dense in S_G^∗M and T_G^∗M also has K as minimal isotropy group.

Proof. Let us show thatKis a minimal isotropy group forT_G^∗M. We only need to show that there isξ0∈T_G^∗Msuch thatG_ξ₀=K and that, for allξ∈T_G^∗M, a conjugate ofKwill be contained inG_ξ. Since (x, 0)∈T_G^∗Mit follows thatG_(x,0)=K for allx∈M_(K)withG_x=K. Now let (x,ξ)∈T_G^∗Mand identifyT^∗MandT M via the Riemannian metric. We haveG_(x_,_ξ₎=G_(x,_λξ₎, for anyλ∈R^∗since the action ofGis given onT Mby the differential of the action onM. Replacingξbyλξ, we can assume thatξis in the domain of the Riemannian exponential map exp_x :TxM→M. Since the exponential map exp_x identifies (G_x-equivariantly) a small neighborhoodU_x⊂(T_GM)_x with a sliceS_xatx∈M, we get thatG_(x,_ξ₎=G_exp

x(ξ)∩G_x. But the slice theorem also implies thatG_y⊂G_x for anyy ∈S_x thereforeG_(x,_ξ₎=G_exp

x(ξ). Now sinceK is minimal forM, it follows that there is

g∈Gsuch thatg·K⊂G_exp_x_(ξ)=G_(x,ξ).

Let

A_M:=C(S_G^∗M; End(E)) . andA^G_M:=C(S_G^∗M; End(E))^G.

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Remark 4. Recall that a two-sided idealI⊂Aof aC^∗-algebra Ais calledprimitiveif it is the kernel of anon-zero,irreducible∗-representation ofA. By Prim(A) we shall denote the set of primitive ideals ofA, called theprimitive ideal spectrumofA, see [10] for more details. For any σ∈A^G_Mand (ξ,ρ)∈ΩM(E) (see Equation 2), we define

π(ξ,ρ)(σ) :=πρ(σ(ξ))=σ(ξ)|E_ξρ. (7) Then the mapχ:ΩM(E)→Prim(A^G_M) given by χ(ξ,ρ)=ker(π(ξ,ρ)) induces a bijection χ0: ΩM(E)/G→Prim(A^G_M), see [4] for a detailed proof, which appliesmutatis mutandisby replac- ingS^∗M withS_G^∗MandΓwithG. See also [11]. Let us recall briefly the ideas of the proof. The mapC(S^∗_GM)^G →A^G_M given by f 7→ fId_E allows to seeC(S_G^∗M)^G ⊂ A^G_M as a central subalge- bra. Ifπ:A^G_M→L(H) is an irreducible representation ofA^G_M, thenπ(C(S^∗_GM)^G)⊂π(A^G_M)⁰:= {T ∈L(H), Tπ(a)=π(a)T,∀a∈A^G_M}. By Schur’s lemma, it follows thatπ(C(S^∗_GM)^G)=CId_H, becauseπ(Id_E)6=0. Therefore,π|_C(S^∗_GM)^G is a multiple of a character ofC(S^∗_GM)^G which corresponds to an orbitGξ. Let evGξ be the evaluation map at the orbit of ξ. Now notice that End(E_ξ)^G^ξ∼=A^G_M/ ker(ev_G_ξ)A^G_Mand that End(E_ξ)^G^ξ=L

End(E_ξρ)^G^ξ, where the direct sum is over ρ∈Gb_ξsuch thatρ⊂E_ξ. But now Schur’s lemma for group representations implies that End(E_ξρ)^G^ξ is isomorphic to the simple algebraM_k_ρ(C), wherek_ρis the multiplicity ofρinE_ξ. It follows thatπ corresponds modulo the previous identifications to the projection ontoM_k_ρ(C), for someρ∈Gb_ξ such thatρ⊂E_ξ. In other words, we can associate toπan element (ξ,ρ)∈ΩM(E). The element (ξ,ρ)∈ΩM(E) is not unique (it depends on the pointξ∈Gξ), but it is unique modulo the action ofG. Therefore this associates to ker(π) the elementG(ξ,ρ)∈ΩM(E)/Gand defines our desired bijection (which is even a homeomorphism).

3. Primitive ideals and the proof of the main theorem

Letψ⁰(M,E) (respectivelyψ⁻¹(M,E)) be the norm closure of the algebraψ⁰(M;E) (respectively, ψ⁻¹(M;E)) of classical, order zero (respectively, order−1) pseudodifferential operators onM, a compact manifold. LetKbe our fixed minimal isotropy group, letα∈G, and letb παthe restriction morphism to theα-isotypical componentL²(M;E)_α of L²(M;E), see Equation (2). As for the discrete case, the most important (and technically difficult) part of the proof is the identification of the quotientπα(ψ⁰(M;E)^G)/πα(ψ⁻¹(M;E)^G).

As in [2], the mapC(S^∗M; End(E))^G→πα(ψ⁰(M;E)^G)/πα(ψ⁻¹(M;E)^G) descends to a surjective algebra morphism

Re^α_M:A^G_M:=C(S^∗_GM; End(E))^G→πα(ψ⁰(M,E)^G)/πα(ψ⁻¹(M,E)^G).

Since the map Re^α_M is surjective, the question of determining the quotient algebra π_α(ψ⁰(M,E)^G/π_α(ψ⁻¹(M;E)^G) is equivalent to the question of determining the ideal ker(Re^α_M)⊂A^G_M. In turn, this ideal will be determined by solving the following problem:

Problem 5. Let A^G_M:=C(S_G^∗M; End(E))^G, as before. Identify the closed subset Ξ^α(E) :=Prim(A^G_M/ ker(Re^α_M))⊂Prim(A^G_M) .

We notice that ifP is aG-invariant pseudodifferential operator thenπα(P) is Fredholm if, and only if, the image byRe^α_Mof its principal symbol is invertible, by an equivariant version of Atkinson’s theorem [3,4]. We next determine which ker(π(ξ,ρ)) belongs toΞ^α(E), for (ξ,ρ)∈ΩM(E).

3.1. Calculation on the principal orbit bundle

Let us consider first arbitrary (ξ,ρ)∈ΩM_(K)(E). By tensoring withα^∗, we can assume thatα=1^G∈ G, the trivial representation. (This reduction is justified in Theorem 13.)b

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Remark 6. One of the advantages of assumingα=1=1Gis the following simplification:

Ω¹_M_(K₎(E)= n

(ξ,1G_ξ)¯

¯¯ξ∈S_G^∗M_(K₎,E^G_ξ^ξ6=0o .

Indeed, this is because ifα=1^G, then, in the Equation (5) definingΩ¹_M_(K)(E), we haveK=g·G_ξ, and henceρ=1G_ξandE^G_ξ^ξ6=0.

Letx₀:=π(ξ)∈M_(K). There is no loss of generality to assume thatG_x₀=K. LetU⊂(T_GM)_x₀' (T_G^∗M)x₀be a slice atx0, letW=Gexp_x₀(U)∼=G/K×Ube the associated tube aroundx0, and let

η∈E_x^K₀ and f ∈Cc^∞(U) , f(x₀)=1 . (8) We have (S^∗_KU)x₀=S^∗_x

0U, since we have assumedx₀:=π(ξ)∈M_(K₎. Henceξ∈S^∗_x

0U. We define thens_η∈Cc^∞(W;E)^G ande_t∈C(W)^G bys_η(gexp_x

0(y)) := f(y)gηande_t(gexp_x

0(y)) :=e^{ı t}^〈^y,^ξ〉, t∈R. That is, they are the functions onW extending the functionsy7→f(y)ηandy 7→e^{ı t〈y,}^ξ〉

defined onU⊂T_x₀U=(T_KU)_x₀ byG-invariance viaW =Gexp_x₀(U). Let us notice that, if we let Φ^G_K denote the Frobenius isomorphism, thens_η:=Φ^G_K(fη) andet=Φ^G_K(e^{ı t〈•,ξ〉}). Recall the map χ:ΩM(E)→Prim(A^G_M) of Remark 4. Using oscillatory testing techniques, see, for instance [13,17], and Remark (6), we obtain.

Proposition 7. Assume that06=η∈E^K_x₀(recall that K x0=x0). Then, for every P∈ψ⁰(M;E), we havelimt→∞P(ets_η)(x₀)=σ0(P)(ξ)η. In particular, if P∈ψ⁰(M;E)^G, then

tlim→∞π₁(P)(e_ts_η)(x₀)=σ0(P)(ξ)η=:π(ξ,1⁾

¡σ0(P)¢ η. Consequently,ker¡

π(ξ,1⁾

¢∈Ξ¹(E). Equivalently,χ¡

Ω¹_M_(K₎(E)¢

⊂Ξ¹(E).

AboveM_(K), we also have the opposite inclusion.

Theorem 8. We haveχ¡

Ω¹_M_(K₎(E)¢

=Ξ¹(E)∩Prim(A^G_M

(K)), and hence Ξ¹0(E) :=Ξ¹(E)∩Prim(A^G_M

(K))={ker¡ π(ξ,1Gξ)

¢|ξ∈S_G^∗M_(K₎,E^G^ξ

ξ 6=0}.

Proof. Proposition 7 states that χ¡

Ω¹_M_(K₎(E)¢

⊂ Ξ¹(E). This gives hence the inclusion χ¡

Ω¹_M_(K)(E)¢

⊂ Ξ¹₀(E). To prove our result, we need to prove the opposite inclusion. Let χ(ξ,ρ) :=ker(π(ξ,ρ))∈Ξ¹₀(E). By definition, this means:

• (ξ,ρ)∈ΩM(E) andξ∈S_G^∗M_(K₎and, most importantly,

• π(ξ,ρ)|_ker(_R_e¹

M)=0.

We need to prove that (ξ,ρ)∈Ω¹_M_(K)(E). As before, by replacingξwith a suitable conjugategξ, we can assume thatG_ξ=K and therefore we need to prove thatE^K_ξ 6=0 andρ=1K. Sincex:=π(ξ)∈ M_(K), we can replaceMwith a tubeWaroundxwith contractible sliceUon whichEis trivial. We shall prove by contradiction thatE^K_ξ 6=0. Indeed, if 0=E_ξ^K=E_x^K, thenL²(W;E)^G=0, by Frobenius reciprocity, and hence ker(Re¹_W)=A^G_W. This is, however, not possible since ker(Re¹_W)⊂ker(π(ξ,ρ)).

We shall prove, again by contradiction, thatρ=1K. Let us hence assume that1K6=ρ∈Kb. Let p_ρbe the projection onto the isotypical component corresponding toρin End(E_ξ)^K'End(E_x)^K. We havep_ρ6=0, since (ξ,ρ)∈ΩM(E). Recall thatΦ^G_K denotes the induction map. Letf ∈Cc^∞(U) be equal to 1 nearxand extendf p_ρto aG-invariant elementQe:=Φ^G_K(f p_ρ)∈C^∞(M; End(E))^G⊂ ψ⁰(M,E)^G viaW=Gexp(U) with principal symbolσ0(Q)e =:qe=Qe◦π∈A^G_M:=C^∞(M; End(E))^G. LetP∈C^∞(M_(K); End(E))^G⊂C^∞(S_G^∗M_(K₎; End(E))^Gbe the projection defined by

P(y)=p₁_{G y}, y∈M_(K), (9) that is,P(y)∈End(E_y) is the projection onto the vectors fixed byG_y, the stabilizer ofy. It has the property thatPis the identity onL²(M;E)^GandQeP=0∈C^∞(M(K); End(E)). We have

π(ξ,ρ)(q) :e =π(ξ,ρ)

¡σ0(Qe)¢

=f(π(ξ))p_ρ|E_ξρ =p_ρ and π₁(Q) :e =Qe|L²(M;E)^G =QeP|_L²_(M;E₎^G=0 .

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Therefore,Re¹_M(q)e =π1(Q)e +π1(ψ⁻¹(M;E))=0 butπ(ξ,ρ)(Φ^G_K(f p_ρ))6=0, which contradicts our assumption thatπ(ξ,ρ)|_ker(_R_e1

M)=0. Henceρ=1^K^.

3.2. Calculation for singular(ξ,ρ)

We now consider (ξ,ρ)∈ΩM(E) such thatx:=π(ξ)∉M_(K₎. A lot of work in this subsection will be devoted to finding the right definitions of theqeandQeused in the previous subsection and still satisfying the relationsπ(ξ,ρ)(q)e =p_ρ,qe=σ0(Qe), but, this time,PQe=0.

3.2.1. The symbolq and a distinguished neighborhood ofe (ξ,ρ).

We first consider the construction ofq. We could constructe qeto satisfyπ(ξ,ρ)(q)e =p_ρusing a tubular neighborhood ofξinS^∗M, but we needqeto satisfy a few other properties in order to be able to constructQein the next subsection. AssumeK⊂G_ξ. Let us then define succesively

F_ξ:={β∈Gb_ξ|β⊂E_ξandβ^K6=0}, p_K:= X

β∈F_ξ

p_β, and e_ξ_,K :=Id_E_ξ−p_K. (10)

(In particular,pK∈End(E_ξ)^G^ξ.) To construct the rightq, we restrict to a tubular neighoborhoode ofx :=π(ξ). Let thenH be a closed subgroup ofG and letV be a vector space on which H acts by isometries. LetW =G×HV and assume thatE=G×H(V×E⁰) for someH-moduleE⁰. Let (ξ,ρ)∈ΩW(E) be our fixed element. Ifπ(ξ)=:x=[e, 0]∈G×HV =W, thenξ=[e, 0,v^∗]∈ S^∗W ∼=G×H(V×S^∗_xW), withv ∈S^∗_xW. We have that H acts onS^∗_xW. We construct then qe first on the sphereS^∗_xW to be H-invariant and to coincide withe_ξ_,K 6=0 atξby using anH- tubular neighborhood ofv^∗inS^∗_xW and call this functionp. The main properties ofpare thus thatp∈C^∞(S^∗_xW)^H andp(v^∗)=e_ξ,K. Then we extendptoV×S^∗_xW to be constant in theV- direction. Since this extension (still calledp) isH-invariant, we can further extendptoS^∗W ∼= G×H(V×S^∗_xW) to beG-invariant. Finally,qe:=f p∈Cc^∞

¡S^∗W; End(E)¢G

, where f ∈Cc^∞(W)^G is such thatf(x)=1, which defines also an element ofCc^∞

¡S^∗M; End(E)¢G

if the support of f is small. Recall the projectionP∈C^∞¡

M_(K); End(E)¢G

,P(y)=p₁_{G y}, of Equation (9).

Lemma 9. Let(ξ,ρ)∈ΩM(E)with K⊂G_ξandρ^K=0. Then (i) π(ξ,ρ)(q)e =p_ρ6=0.

(ii) π(ξ,1Gξ)(q)e =0, for allξ∈S^∗M_(K). (iii) Pqe=qeP=0on S^∗M_(K).

(iv) Let Vq˜={(ζ,β)∈ΩM(E)|π(ζ,β)(q)e 6=0}. Thenχ(Vq˜)is an open neighbourhood ofχ(ξ,ρ)in Prim(A^G_M)andχ(Vq˜)∩Ξ¹₀(E)= ;. In particular,χ(ξ,ρ)∉Ξ¹₀(E).

Proof. (i). Let us notice first that p_ρ ∈End(E_ξ)^G^ξ = End(E⁰)^G^ξ is non zero because (ξ,ρ) ∈ ΩM(E) implies thatρ⊂E_ξ, by definition. Let us prove (i). Sinceρ^K =0 we get thatρ∉F_ξ (see Equation 10). Therefore, for anyρ⁰∈F_ξ,p_ρ0p_ρ=0. This gives that

π(ξ,ρ)(q) :e =q(eξ)p_ρ =e_ξ_,Kp_ρ=p_ρ− X

ρ⁰∈F_ξ

p_ρ0p_ρ=p_ρ =Id_E_ξρ∈End(E_ξρ).

(ii). In view of the support ofqeand itsG-invariance, this relation reduces toe_ξ_,Kp₁_K =0 onV (see Equation 10). The relatione_ξ_,Kp₁_K =0 is valid becauseE_ξ^K =L

ρ⁰∈F_ξ(E_ξρ0)^K and therefore p_Kp₁_K =p₁_K.

(iii). The relation (iii) is just another formulation of (ii) sinceP(y)=p₁_K onV(K).

(iv). It is standard that the set χ(Vq˜) is open, see for example [3]. By Theorem 8, Ξ¹₀(E|W)=

and then (ii) implies (iv).

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3.2.2. Density ofΞ¹₀inΞ¹

Lemma 10. Assume that(ξ,ρ)∈ΩM(E)is such thatχ(ξ,ρ)∉Ξ¹₀(E)and that K⊂G_ξ. Then (i) ρ^K=0, and hence, in particular,χ(Ω¹_M(E))=Ξ¹₀(E)=χ(Ω¹_M_(K)(E))and

(ii) there isQe∈ψ⁰(M;E)^Gsuch thatσ0(Q)e =q ande π₁(Q)e =0(i.e.,Qe=0on L²(M;E)^G).

Proof. (i). We already showed in Theorem 8 thatΞ¹₀(E)=χ(Ω¹_M_(K₎(E)). Since ker(π(ξ,ρ)) is maxi- mal (becauseA^G_M/ ker(π(ξ,ρ))'End(E_ξρ)^G^ξ'M_k(C), due to the fact thatρ∈Gb_ξ), the assumption thatχ(ξ,ρ)∉Ξ¹₀(E) implies that there isσ∈A^G_Msuch thatπ(ξ,ρ)(σ)=Id_E_ξρ and the associated neighbourhoodV_σ=©

(η,ρ⁰)∈ΩM(E)¯

¯π(η,ρ⁰)(σ)6=0ª

does not intersectΞ¹₀(E). Henceπ(ζ,1⁾(σ)=0 for all (ζ,1⁾∈Ω¹_M_(K₎(E). Let us assume, by contradiction, thatρ^K 6=0 and letζn ∈S_G^∗M_(K)be a sequence converging toξ, which exists by Lemma 3. Let us then writeσ(ζn)|E_ξρ =σ(ζn)|E_ξρ^K ⊕ σ(ζn)|(E^K_ξρ)^⊥, forn large enough. Since (E_ξρ)^K ⊂E_ξ^K andσ(ζn)|E^K_ξ =π(ζn,1⁾(σ)=0, we obtain that Id_E_ξρ =π(ξ,ρ)(σ)=limσ(ζn)|E_ξρ vanishes on (E_ξρ)^K 6=0, which is a contradiction. The relation χ(Ω¹_M(E))=Ξ¹₀(E) follows by combining what we have just proved with Lemma 9 (iv). The rest follows fromΞ¹₀(E)=χ¡

Ω¹_M_(K)(E)¢ .

(ii). Since the problem is local, we can assume that M =W :=G×HV, as in the previous subsection. Using the compactness ofG/H, let us cover it with finitely many contractible local chartsD_i and let (ϕ²_i) be a partition of unity subordinated to this covering {D_i}. Letκ:W := G×HV →G/H be the projection and Y_i :=κ⁻¹(D_i). ThenY_i is a finite covering of W and E|Yi 'Y_i×E_i⁰is trivial. Letχi∈Cc^∞(D_i) be such thatχiϕi =ϕi. We extend trivially the partition of unityφi and the functionsχi toW by replacing, for instance,φiwithφi◦κ. EachY_i identifies with an open subset of a vector spaceZi. Then letψi∈C^∞(Z_i^∗) be such thatψ(0)=0 if|η| <1/2 andψ(η)=1 whenever|η| ≥1. For any order zero classical symbola∈S⁰_cl(Y_i×Y_i×Z_i^∗) onY_iand anys∈Cc^∞(Yi,E), we let

Op(a)s(y) :=(2π)^−dimW Z

Z_i^∗

Z

Zi

e^ı(y⁻^z)·ηa(y,z,η)s(z)dzdη.

We shall use this construction fora_i(y,z,η) :=χi(y)ψi(η)qe³ y,_|η|^η´

χi(z) to obtainQ_i :=Op(a_i), where we have regarded (y,η)∈Yi×Z_i^∗=T^∗Yi ⊂T^∗W. Next we defineQ :=P

iϕiQiϕi and Qe:=Av(Q) :=R

GgQ g⁻¹d g. ThenQeis an order zero,G-invariant pseudodifferential operator on W with principal symbolsqebecause the average map commutes with the principal symbol map.

Let us prove now thatQe|L²(W,E)^G =0. By density, it is sufficient to show thatQ s(y)e =0 for y∈W_(K)ands∈Cc^∞(W,E)^G. By Lemma 9 (iii), we have fory∈W_(K₎∩Y_i,z∈Y_i, andη∈Z_i^∗that Pa_i(y,z,η)=0. Thus for anys∈Cc^∞(Y_i,E) andy∈Y_i∩W_(K), we obtain thatP(y)Q_i(s)(y)=0. Thus we get thatPQ(s)=0 onL²(W;E)=L²(W(K);E). Using the average map, we obtain

0=£ Av¡

PQ(s)¢¤

(y)=P(y)£ Av(Q)s¤

(y)=P(y)(Q s)(ye ), ∀y∈W(K).

It follows thatPQe=0 onL²(W;E). SinceQe(L²(W;E)^G)⊂L²(W;E)^GandPacts as the identity on L²(W;E)^G, we obtain thatQe(L²(W;E)^G)=PQ(Le ²(W;E)^G)=0.

Theorem 11. The setΞ¹(E) :=Prim(A^G_M/ ker(Re¹_M))⊂Prim(A^G_M)associated to the idealker(Re¹_M) of A^G_M:=C(S_G^∗M; End(E))^G is the closure inPrim(A^G_M)of the setΞ¹₀(E) :=Ξ¹(E)∩Prim(A^G_M

(K)).

Proof. SinceΞ¹(E) is closed, it is enough to show that, if ker(π(ξ,ρ))∉Ξ¹₀(E), then ker(π(ξ,ρ))∉ Ξ¹(E). We may assume thatK⊂G_ξandM=G×HV as before. From Lemma 10 (i), we know that ρ^K =0. Furthermore Lemma 10 (ii) gives that there isQe∈ψ⁰(M;E)^G such thatσ0(Q)e =qeand π1(Q)e =0 onL²(M;E)^G. Therefore,σ0(Q)e =qe∈ker(Re¹_M). Since,π(ξ,ρ)(q)e 6=0 by Lemma 9 (i),π(ξ,ρ)

does not vanish on ker(Re¹_M), which means that ker(π(ξ,ρ))∉Ξ¹(E).

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Combining the previous results, we obtain the desired determination of the set Ξ¹(E) and hence of the kernel of the morphismRe¹_M.

Proposition 12. We haveΞ¹(E)=Ξ¹₀(E)=χ(Ω¹_M(E)), and hence χ0:Ω¹_M(E)/G →Ξ¹(E)is a bijection.

The form of the main result given in the introduction is then obtained from the following sequence of equivalences (which was referred to as “reduction toα=1”). The last equivalence of the following theorem was shown for finite groups in [3, Proposition 5.9].

Theorem 13. The following statements are equivalent:

(i) πα(P)is Fredholm;

(ii) π1(P⊗Id_α∗)is Fredholm;

(iii) P⊗Id_α∗is transversally1^-elliptic;

(iv) P is transversallyα-elliptic;

(v) For allξ∈(S^∗_GM)^Kthe linear map

σm(P)(x,ξ)⊗Id_α∗: (E_x⊗α^∗)^K→(E_x⊗α^∗)^K is an isomorphism.

Sketch of the proof. The equivalence of (i) and (ii) is an elementary, direct operator theory argument. The equivalence of (ii) and (iii) is the difficult part of the statement and it is what we have proved for the most part of this note. The equivalence of (iii) and (iv) is an elementary, direct representation theoretic argument. Finally, (iii)⇒(v) by the definition of1-transverse ellipticity.

Leth∈G_ξbe such thath·K⊂G_ξ. Thenh⁻¹will mapE_ξtoE_h−1ξandE_ξ^h^·^KtoE^K

h⁻¹ξ. The invertibility of theG-invariant symbolσm(P) on E^h_ξ^·^K (and hence on each E_ξβ withβ^h^·^K 6=0) will follow therefore from its invertibility onE^K

h⁻¹ξ. Thus (v)⇒(iii).

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