A General Simonenko Local Principle and Fredholm Condition for Isotypical Components

Abstract

In this paper, we derive, from a general Simonenko’s local principle, Fredholm criteria for restriction to isotypical components. More precisely, we give a full proof, of the equivariant local principle for restriction to isotypical components of invariant pseudodifferential operators announced in Baldare et al. (Muenster J Math, 2021). Furthermore, we extend this result by relaxing the hypothesis made in the preceding quoted paper.

1 Introduction

This paper is devoted to the proof and an application of a general Simonenko’s local principle to G-invariant operators on closed manifolds. Local principles first appeared in Simonenko’s work [52] and more general forms appeared in [1, 28, 31]. Since then local principles were intensively used to obtain Fredholm condition for singular operators, see for examples [14, 37, 46, 51, 58] and the references therein. As a consequence of the general Simonenko’s local principle, we derive Fredholm conditions for restriction of G-invariant pseudodifferential operators to isotypical components.

Let G be a compact Lie group and denote by $\widehat{G}$ the set of isomorphism classes of irreducible unitary representations of G. If $P : \mathcal H\rightarrow \mathcal H'$ is a G-invariant continuous linear map between Hilbert spaces and $\alpha \in \widehat{G}$, then the operator P induces a well defined continuous linear map between the $\alpha$-isotypical components

$$\begin{aligned} \pi _\alpha (P) : \mathcal H_\alpha \rightarrow \mathcal H'_\alpha . \end{aligned}$$

In this paper, we are interested in the case where P is a pseudodifferential operator acting between sections of two vector bundles.

Assume that our compact Lie group G acts smoothly and isometrically on a compact Riemannian manifold M and on two hermitian vector bundles $E_0$ and $E_1$. Furthermore, let $P : \mathcal C^\infty (M,E_0) \rightarrow \mathcal C^\infty (M,E_1)$ be a G-invariant, classical, order m, pseudodifferential operator on M. Since P is G-invariant, its principal symbol $\sigma _m(P)$ belongs to $\mathcal C^{\infty }(T^*M \smallsetminus \{0\}; {\text {Hom}}(E_0, E_1))^G$. Let $G_\xi$ and $G_x$ denote the isotropy subgroups of $\xi \in T_x^*M$ and $x \in M$, as usual. Then $G_\xi \subset G_x$ acts linearly on the fibers $E_{0x}$ and $E_{1x}$. Following [4], denote by $T^*_GM$ the G-transverse cotangent space, see Eq. (1) and by $S^*_GM := S^*M \cap T^*_GM$ the set of unit covectors in the G-transverse cotangent space $T^*_GM$.

The previous set leads to the definition of G-transversally elliptic operators [4, 54]. Recall that a G-transversally elliptic pseudodifferential operator on M is a G-invariant pseudodifferential operator whose principal symbol becomes invertible when restricted to $T^*_GM\smallsetminus \{0\}$. Since M is compact, we know that this operators are generally not Fredholm due to the lack of full ellipticity. Nevertheless, the, now well known, Atiyah-Singer’s result states that if P is G-transversally elliptic then $\pi _\alpha (P)$ is Fredholm for any $\alpha \in \widehat{G}$, [4, 54]. This allows directly to define an index for G-transversally elliptic operators as an element of the K-homology of $C^*G$, the group $C^*$-algebra of G. Furthermore, with little more work, Atiyah and Singer showed that this index is, in fact, a Ad-invariant distribution on G. See also [6, 8, 33, 35, 36] for related results and [7, 12, 44] for index theorems on G-transversally elliptic operators using equivariant cohomology. The Fredholm property of this restrictions to isotypical component was the starting point for the study carried out in [11].

We now proceed to state the main result studied in this paper but first we need few more notations and definitions from [9,10,11].

Assume M/G connected and let K be a minimal isotropy subgroup of G, see [16, 56]. We shall say that P is transversally $\alpha$-elliptic if for all $\xi \in (S^*_GM)^K$ the linear map

$$\begin{aligned} \sigma _m(P)(\xi ) \otimes {\text {Id}}_{\alpha ^*} : (E_{0\xi } \otimes \alpha ^*)^K \rightarrow (E_{1\xi } \otimes \alpha ^*)^K \end{aligned}$$

is invertible.

One of the main results of [11] states that P is transversally $\alpha$-elliptic if, and only if, $\pi _\alpha (P)$ is Fredholm. Here, we point out that the transversal $\mathbf {1}$-ellipticity is related with transversal ellipticity on (singular) foliations [3, 25]. For G finite, this results were proved before [9, 10].

In the present paper, we recall, in Definition 3.1, the notion of locally $\alpha$-invertible operator at $x\in M$ introduced in [10] and we show in full generality the following result, see Theorem 3.11.

Theorem

Assume that M is a closed, smooth manifold and that G is a compact Lie group acting smoothly on M. Let $P\in \psi ^m(M;E_0,E_1)^G$ and $\alpha \in \widehat{G}$. Then the following are equivalent:

• (1) $\pi _\alpha (P) : H^s(M;E_0)_\alpha \rightarrow H^{s-m}(M;E_1)_\alpha$ is Fredholm for any $s \in \mathbb R$,

• (2) P is transversally $\alpha$-elliptic,

• (3) P is locally $\alpha$-invertible.

Notice that in [10], the equivalence between (1) and (2) was stated without proof under the hypothesis that $\dim G< \dim M$. Moreover, the triple equivalence was then deduced in the case of finite group using the main result of [10]. Here care is taken to state it in full generality and relax the hypothesis $\dim G < \dim M$. This proposition enlightens the results from [9,10,11] in the sense that it explains the local computations done.

The previous theorem, as well as intermediate results in this paper, were obtained during discussions with R. Côme, M. Lesch and V. Nistor.

We point out that the Fredholm conditions obtained in this paper are closely related to the ones in [48], for G-operators, and the ones in [18], for complexes of operators. Fredholm conditions were also investigated in different forms in [15, 49, 50, 55] for boundary problems and in [23, 29, 30, 32, 40,41,42, 45] using techniques of limit operators and also $C^*$-algebras methods. The techniques of limit operators are similar to the one used in [11] to obtain the Fredholm criterion for $\alpha$-transversally elliptic operator, see also Sect. 1.3. Recent developments on singular operators including groupoid and $C^*$-algebras were accomplished in [2, 13, 19,20,21,22, 24, 26, 38, 39].

2 Preliminaries

This section is devoted to background material and results. The reader can find more details in [9,10,11]. The reader familiar with [9,10,11] can skip this section at a first reading.

2.1 Group Actions

Throughout the paper, we let G be a compact Lie group. Assume that G acts on a space X and that $x \in X$, then Gx is the G orbit of x and

$$\begin{aligned} G_x := \{ g \in G \, \vert \ g x = x \} \subset G \end{aligned}$$

the isotropy group of the action at x.

If $H \subset G$ is a subgroup, then $X_{(H)}$ will denote the set of elements of X whose isotropy $G_{x}$ is conjugated to H in G and $G\times _{H} X$ the space

$$\begin{aligned} G \times _H X \, := \,(G \times X)/\sim , \end{aligned}$$

where $(g h,x)\sim (g ,hx)$ for all $g \in G, h\in H$, and $x\in X$.

Let V and W be locally convex spaces and $\mathcal L(V; W)$ be the set of continuous, linear maps $V \rightarrow W$. We let $\mathcal L(V) := \mathcal L(V; V)$. A representation of G on V is a continuous group morphism $G \rightarrow \mathcal L(V)$, where $\mathcal L(V)$ is equipped with the strong operator topology. Said differently, the map $G \times V \rightarrow V$ given by $(g, v) \mapsto gv$ is continuous and $v \mapsto gv$ is linear. We shall also call V a G-module.

For any two G-modules $\mathcal H$ and $\mathcal H'$, we shall denote by

$$\begin{aligned} {\text {Hom}}_{G}(\mathcal H, \mathcal H') \, = \,{\text {Hom}}(\mathcal H, \mathcal H')^G \, = \,\mathcal L(\mathcal H, \mathcal H')^G, \end{aligned}$$

the set of continuous linear maps $T : \mathcal H\rightarrow \mathcal H'$ that commute with the action of G, that is, $T (g v) = g T(v)$ for all $v \in \mathcal H$ and $g\in G$.

Let $\mathcal H$ be a G-module and $\alpha$ an irreducible representation of G. Then $p_\alpha$ will denote the G-invariant projection onto the $\alpha$-isotypical component $\mathcal H_\alpha$ of $\mathcal H$, defined as the largest (closed) G-submodule of $\mathcal H$ that is isomorphic to a multiple of $\alpha$. In other words, $\mathcal H_\alpha$ is the sum of all G-submodules of $\mathcal H$ that are isomorphic to $\alpha$. Notice that $\mathcal H_{\alpha } \simeq \alpha \otimes {\text {Hom}}_{G}(\alpha , \mathcal H)$ and

$$\begin{aligned} \mathcal H_\alpha \, \ne \, 0 \ \Leftrightarrow \ {\text {Hom}}_G(\alpha , \mathcal H) \, \ne \, 0 \ \Leftrightarrow \ {\text {Hom}}_G(\mathcal H, \alpha ) \, \ne \, 0 . \end{aligned}$$

Recall that we denote by $\widehat{G}$ the set of equivalence classes of irreducible unitary representations of G. Let $\chi _\alpha$ be the character of $\alpha \in \widehat{G}$ and $z_\alpha :=\dim \alpha \chi _\alpha \in C^*G$ be the central projection associated in the group $C^*$-algebra $C^*G$ of G. Then $p_\alpha$ is the image of $z_\alpha$ induced by the group action on $\mathcal H$. If $T \in \mathcal L(\mathcal H)^G$ then $T(\mathcal H_\alpha ) \subset \mathcal H_\alpha$ and we let

$$\begin{aligned} \pi _\alpha : \mathcal L(\mathcal H)^G \rightarrow \mathcal L(\mathcal H_\alpha ) \,, \quad \pi _\alpha (T) \, := \,p_\alpha T\vert _{\mathcal H_\alpha }\,, \end{aligned}$$

be the associated morphism. The morphism $\pi _\alpha$ will play an essential role in what follows.

As before, we consider a compact Lie group G and we now assume that G acts by isometries on a closed Riemannian manifold M. Let TM and $T^*M$ be respectively the tangent and cotangent bundle on M and recall that they can be identified using the G-invariant Riemannian metric on M. Let $S^*M$ denote the unit cosphere bundle of M, that is, the set of unit vectors in $T^*M$, as usual. Denote by $\mathfrak {g}$ the Lie algebra of G. Then any $Y \in \mathfrak {g}$ defines as usual the vector field $Y_M$ given by $Y_M(m)=\frac{d}{dt}_{|_{t=0}} e^{tY}\cdot m$. Denote by $\pi : T^*M \rightarrow M$ the canonical projection and let us introduce as in [4] the G-transversal space

$$\begin{aligned} T^*_G M \, := \,\{\xi \in T^*M\ |\ \xi (Y_M(\pi (\xi )))=0, \forall Y\in \mathfrak {g}\}. \end{aligned}$$

(1)

We denote by $T_GM$ the image of $T^*_GM$ in TM obtained using the Riemannian metric. In other words, $T_GM$ is the orthogonal to the orbits in TM. Finally, let $S^*_GM$ be the set of unit covectors in $T^*_GM$, that is $S^*_GM=S^*M \cap T^*_GM$.

Recall that if M/G is connected, there is a minimal isotropy subgroup K such that any isotropy subgroup of G acting on M contains a subgroup conjugated to K and $M_{(K)}$ is an open dense submanifold of M called the principal orbit bundle, see [16, Section IV. 3] and [56, Section I. 5].

2.2 Pseudodifferential Operators

Let G be a compact Lie group acting smoothly by isometries on a compact, Riemannian manifold without boundary M as before. We shall denote by $\psi ^{m}(M; E)$ the space of order m, classical pseudodifferential operators on M. Let $\overline{\psi ^{0}}(M; E)$ and $\overline{\psi ^{-1}}(M; E)$ denote the respective norm closures of $\psi ^{0}(M; E)$ and $\psi ^{-1}(M; E)$. The action of G then extends to a continuous action on $\psi ^{m}(M; E)$, $\overline{\psi ^{0}}(M; E)$, and $\overline{\psi ^{-1}}(M; E)$, see [5] for example. We shall denote by $\mathcal K(\mathcal H)$ the algebra of compact operators acting on a Hilbert space $\mathcal H$. Of course, we have $\overline{\psi ^{-1}}(M; E)= \mathcal K(L^2(M; E))$.

We shall denote, as usual, by $\mathcal C(S^*M; {\text {End}}(E))$ the set of continuous sections of the lift of the vector bundle ${\text {End}}(E) \rightarrow M$ to $S^*M$. We have the following well known exact sequence

$$\begin{aligned} 0 \, \rightarrow \, \mathcal K(L^2(M; E))^G \, \rightarrow \, \overline{\psi ^{0}}(M; E)^G\, {\mathop {-\!\!\!\longrightarrow }\limits ^{\sigma _{0}}}\, \mathcal C(S^*M; {\text {End}}(E))^G \, \rightarrow \, 0 \,. \end{aligned}$$

See, for instance, [9, Corollary 2.7], where references are given.

Recall that a G-invariant classical pseudodifferential operator P of order m is said elliptic if its principal symbol is invertible on $T^* M\smallsetminus \{0\}$ and G-transversally elliptic if its principal symbol is invertible on $T^*_G M\smallsetminus \{0\}$ [4, 5, 44], see Eq. (1) for the definition of $T^*_GM$.

We may now state the classical result of Atiyah and Singer [4, Corollary 2.5].

Theorem 1.1

(Atiyah-Singer [4, 54]) Let P be a G-transversally elliptic operator. Then, for every irreducible representation $\alpha \in \widehat{G}$, $\pi _\alpha (P) : H^s(M; E_0)_\alpha \, \rightarrow \, H^{s-m}(M; E_1)_\alpha ,$ is Fredholm.

Let us recall the following fact which is a direct consequence of the fact that G acts by unitary multiplier on $\mathcal K(\mathcal H)$.

Proposition 1.2

We have natural isomorphisms

$$\begin{aligned}&p_\alpha \overline{\psi ^{-1}}(M;E)^G \simeq \pi _\alpha (\overline{\psi ^{-1}}(M;E)^G)\\&= \pi _\alpha (\mathcal K(L^2(M; E))^G) = \mathcal K(L^2(M;E)_\alpha )^G\,, \end{aligned}$$

where the first isomorphism map is simply $\pi _\alpha$ and

$$\begin{aligned} \mathcal K(L^2(M; E))^G = \overline{\psi ^{-1}}(M; E)^G \simeq \oplus _{\alpha \in \widehat{G}}\mathcal K(L^2(M; E)_\alpha )^G\,. \end{aligned}$$

Proof

See, for example, [9, Section 3] for a proof. $\square$

2.3 $\alpha$-transversally Elliptic Operators

Let G be a compact Lie group acting smoothly by isometries on a compact, Riemannian manifold without boundary M as before. Let $P : \mathcal C^\infty (M,E_0) \rightarrow \mathcal C^\infty (M,E_1)$ be a G-invariant pseudodifferential operator. Let $p : M \rightarrow M/G$ be the projection. Let $M/G =\bigsqcup _{i\in I} C_i$ be the decomposition into connected components of M/G. Notice that I is finite and let $K_i \subset G$ be a minimal isotropy group for $M_i:=p^{-1}(C_i)$. Denote by $(S^*_GM_i)^{K_i}$ the subset of $K_i$-invariant elements of $S^*_GM_i$, see Eq. (1).

Definition 1.3

[11] We shall say that $P \in \psi ^m(M; E_0, E_1)^{G}$ is transversally $\alpha$-elliptic if for any $i\in I$, and $\xi \in (S^*_GM_i)^{K_i}$,

$$\begin{aligned} \sigma _m(P)(\xi ) \otimes {\text {Id}}_{\alpha ^*} : (E_{0\xi } \otimes \alpha ^* )^{K_i} \rightarrow (E_{1\xi } \otimes \alpha ^* )^{K_i} \end{aligned}$$

is invertible.

Let us recall the main result of [11], see also [9, 10] for finite groups.

Theorem 1.4

[11] Let $m\in \mathbb R$, $P \in \psi ^m(M; E_0, E_1)^{G}$ and $\alpha \in \widehat{G}$. Then

$$\begin{aligned} \pi _\alpha (P) : H^s(M; E_0)_\alpha \, \rightarrow \, H^{s-m}(M; E_1)_\alpha \end{aligned}$$

is Fredholm if, and only if P is transversally $\alpha$-elliptic.

We now briefly relate Definition 1.3 with the notion of limit operators, see [23, 29, 30, 32, 40,41,42, 45]. In order to simplify notations, let us assume $\alpha =\mathbf {1}$ and M/G connected and let K be the minimal isotropy subgroup. We follow [11]. Let $(x_0 ,\xi ) \in S^*_GM_{(K)}$ and assume $G_{x_0}=K$. Let $U \subset (T_GM)_{x_0} \simeq (T^*_GM)_{x_0}$ be a slice at $x_0$, let $W = G\exp _{x_0}(U) \cong G/K \times U$ be the associated tube around $x_0$, and let

$$\begin{aligned} \eta \in E_{x_0}^{K}\ \text{ and } \ f \in \mathcal C^{\infty }_c(U)\,, \ f(x_0) = 1 \,. \end{aligned}$$

Notice that $(S^*_{K} U)_{x_0} = S^*_{x_0} U$, because $x_0 \in M_{(K)}$ and hence $\xi \in S^*_{x_0} U$. Let us define $s_\eta \in \mathcal C^{\infty }_c(W; E)^G$ and $e_t \in \mathcal C^\infty (W)^G$ by $s_\eta ( g \exp _{x_0} (y) ) \, := \,f(y) g\eta$ and $e_t( g \exp _{x_0} (y)) \, := \,e^{\imath t \langle y, \xi \rangle }$, $t \in \mathbb R$. In other words, they are the functions on W extending the functions $y \mapsto f(y) \eta$ and $y \mapsto e^{\imath t \langle y, \xi \rangle }$ defined on $U \subset T_{x_0}U = (T_KU)_{x_0}$ by G-invariance via $W = G \exp _{x_0}(U)$. Using oscillatory testing techniques, see, for instance [34, 57], the following proposition can be shown, see [11].

Proposition 1.5

Assume that $0 \ne \eta \in E_{x_0}^K$. Then, for every $P \in \psi ^0(M; E)$, we have $\lim _{t \rightarrow \infty } P ( e_t s_\eta ) (x_0) \, = \,\sigma _0(P)(\xi )\eta$. In particular, if $P\in \psi ^0(M; E)^G$, then

$$\begin{aligned} \lim _{t \rightarrow \infty } \pi _{\mathbf {1}}(P) ( e_t s_\eta ) (x_0) \, = \,\sigma _0(P)(\xi )\eta =: \, \pi _{(\xi , \mathbf {1})} \big (\sigma _0(P)\big ) \eta \,. \end{aligned}$$

Remark 1.6

Let $t>0$ and $V_0={\text {Id}}: E_{x_0}^K \rightarrow E_{x_0}^K$. Let $V_t : E_{x_0}^K \rightarrow C^\infty (M,E)^G$ be the map given by $V_t(\eta )=e_t s_\eta$ and let $V_{-t}={\text {ev}}_{x_0} : C^\infty (M,E)^G \rightarrow E_{x_0}^K$ be the evaluation map at $x_0$. Then we have $V_{-t}V_t = V_0 = {\text {Id}}_{E_{x_0}^K}$ and

$$\begin{aligned} \sigma _0(P)(\xi )=\lim \limits _{t\rightarrow \infty } V_{-t}\pi _\mathbf {1}(P)V_t : E^K_{x_0} \rightarrow E^K_{x_0}. \end{aligned}$$

(2)

Equation (2) is similar to the definition of limit operators, see [23, 29, 30, 32, 40,41,42, 45].

3 Simonenko’s General Localization Principle

In this section, we recall the essentials of the usual Simonenko’s localization principle [52], see also [53]. The results of this section are well-know from experts, we shall include proofs for the convenience of the reader. We refer in particular to [45, Chapter 2] and [47, Chapter 2], where more general situations are treated. The general localization principle of this section will be used in the sequel to deduce Fredholm conditions for restriction to isotypical components of invariant operators on closed manifolds.

Throughout this section, we let T be a compact Hausdorff topological space and $\mathcal C(T)$ be the $C^*$-algebra of complex valued continuous functions on T. Let A be a unital $C^*$-algebra and assume that $\mathcal C(T)$ identifies with a unital sub-$C^*$-algebra in A, meaning, in particular, that the image of the unit $1_{\mathcal C(T)}$ of $\mathcal C(T)$ is the unit $1_A$ of A.

Definition 2.1

An element $a \in A$ is said to have the strong Simonenko local property with respect to $\mathcal C(T)$ if, for every $\phi , \psi \in \mathcal C(T)$ with compact disjoint supports, we have $\phi a \psi = 0$.

The following lemma follows for example from similar arguments as in [45, Theorem 2.1.6] and [47, Theorem 2.5.6].

Lemma 2.2

The set $B \subset A$ of elements $a\in A$ satisfying the strong Simonenko local property is the set of elements of A commuting with $\mathcal C(T)$.

Proof

We are going to show that the set of elements $a \in A$ with the strong Simonenko local property is a $C^*$-algebra B containing $\mathcal C(T)$ and that every irreducible representation of B restricts to a scalar valued representation on $\mathcal C(T)$, and hence that $\mathcal C(T)$ commutes with B.

Let us show first that B is a sub-$C^*$-algebra of A. Note that B is not empty since $\mathcal C(T) \subset B$. To show that B is a sub-$C^*$-algebra, the only fact that is non-trivial to prove is that $ab \in B$, for all $a,b \in B$. Let $\phi$ and $\psi \in \mathcal C(T)$ with disjoint compact supports and let $\theta$ be a function equal to 1 on $\mathrm {supp}(\psi )$ and 0 on $\mathrm {supp}(\phi )$, which exists by Urysohn’s lemma. Then we have

$$\begin{aligned} \phi ab\psi = \phi a(\theta +1-\theta )b \psi =\phi a \theta b\psi +\phi a(1-\theta )b\psi = 0, \end{aligned}$$

since $\phi a \theta =0$ and $(1-\theta )b\psi =0$.

Let $\pi : B \rightarrow \mathcal {L}(H)$ be an irreducible representation of B. First, let us show that for any $\phi , \psi \in \mathcal C(T)$ with disjoint support, we either have $\pi (\phi )=0$ or $\pi (\psi ) =0$. Indeed we have $\pi (\phi )\pi (a)\pi (\psi )=0$ since $\phi a \psi =0$, for any $a \in B$. Assume that $\pi (\psi ) \ne 0$ then there is $\eta \in H$ such that $\pi (\psi )\eta \ne 0$. Now, $\pi$ is irreducible so we get that the set $\{\pi (a)\pi (\psi )\eta ,\ a \in B \}$ is dense in H. Thus $\pi (\phi )=0$ on a dense subspace of H and so on H.

Assume now that $\pi (\mathcal C(T)) \ne \mathbb C1_H$. Then there exist two distinct characters $\chi _0, \chi _1 \in \mathrm {Sp}(\pi (\mathcal C(T)))$. Denote by $h_\pi : \mathrm {Sp}(\pi (\mathcal C(T))) \rightarrow \mathrm {Sp}(\mathcal C(T))=T$ the injective map adjoint to $\pi$, and choose $\phi , \psi \in \mathcal C(T)$ with disjoint supports such that $\phi (h_\pi (\chi _0)) = 1$ and $\psi (h_\pi (\chi _1)) = 1$. Then $\pi (\phi )(\chi _0) = 1$ and $\pi (\psi )(\chi _1) = 1$, which contradicts the fact that either $\pi (\phi ) = 0$ or $\pi (\psi ) = 0$. $\square$

We now fix notations and hypothesis that will remain valid until the end of this section.

Notationandhypothesis 2.3

As before, let T be a compact Hausdorff topological space and denote by $\mathcal C(T)$ the $C^*$-algebra of continuous functions on T. Let $\mathcal G$ be a Hilbert space and let $\mathcal C(T) \rightarrow \mathcal L(\mathcal G)$ be a non degenerate faithful representation (i.e. $\mathcal C(T)$ identifies with its image in $\mathcal L(\mathcal G)$ and the image of the constant function 1 is ${\text {Id}}\in \mathcal L(\mathcal G)$). Assume that the image of $\mathcal C(T)$ does not intersect $\mathcal K(\mathcal G)\smallsetminus \{0\}$. In other words, we are assuming that $\mathcal C(T)$ identifies with a unital sub-$C^*$-algebra of the Calkin algebra $\mathcal Q(\mathcal G):= \mathcal L(\mathcal G)/\mathcal K(\mathcal G)$. We shall denote by $M_\phi$ the image of a function $\phi \in \mathcal C(T)$ in $\mathcal L(\mathcal G)$ and call it the multiplication operator by $\phi$.

Remark 2.4

If X is a locally compact space and $\mathcal G=L^2(X,\mu )$ then the representation of $\mathcal C_0(X)$ is faithful if and only if $\mu$ is a strictly positive measure, i.e. $\mu (U)>0$ for every open set $U \subset X$. In this case, the only compact operator in $\mathcal C_b(X)$ is zero, where $\mathcal C_b(X)$ denotes the $C^*$-algebra of bounded continuous function, see Lemma 3.9 below for more details.

We shall now turn to the definition of local invertibility. The definition in the present paper and in for example [47, Section 2.5] are at the first reading not the same but they describe the same property by Lemma 2.2. See also [47, Section 2.4.1].

Definition 2.5

An operator $P \in \mathcal L(\mathcal G)$ is said to be locally invertible at $x\in T$ if there exist:

• (i) a neighbourhood $V_x$ of x and

• (ii) operators $Q_1^x$ and $Q_2^x \in \mathcal L(\mathcal G)$

such that, for all $\phi \in \mathcal C_c(V_x)$

$$\begin{aligned} Q_1^xPM_\phi \, = \,M_\phi \, = \,M_\phi PQ_2^x \in \mathcal L(\mathcal G). \end{aligned}$$

The operator P is said to be locally invertible on T if it is locally invertible at any $x\in T$.

Notation 2.6

We let $\Psi _T(\mathcal G) \subset \mathcal L(\mathcal G)$ denote the $C^*$-algebra consisting of all $P \in \mathcal L(\mathcal G)$ such that $M_\phi P M_\psi \in \mathcal K(\mathcal G)$, for all $\phi , \psi \in \mathcal C(T)$ with disjoint support. We let $\mathcal B_T(\mathcal G)$ denote the image of $\Psi _T(\mathcal G)$ in the Calkin algebra $\mathcal Q(\mathcal G)$.

In other words, $\mathcal B_T(\mathcal G)=q(\Psi _T(\mathcal G))$ where $q : \mathcal L(\mathcal G) \rightarrow \mathcal Q(\mathcal G)$ is the canonical projection.

Remark 2.7

We know by Lemma 2.2 that

$$\begin{aligned} \mathcal B_T(\mathcal G)=\{P \in \mathcal {Q}(\mathcal G) \mid M_\phi P =PM_\phi \text { for all } \phi \in \mathcal C(T)\}. \end{aligned}$$

Said differently, $\Psi _T(\mathcal G)$ is the essential commutant of $\mathcal C(T)$, that is

$$\begin{aligned} \Psi _T(\mathcal G)=\{P \in \mathcal L(\mathcal G),\ M_\phi P - PM_\phi \in \mathcal K(\mathcal G),\ \forall \phi \in \mathcal C(T)\}, \end{aligned}$$

see the relation with the work [59]. Moreover, the family of morphisms

$$\begin{aligned} \mathcal B_T(\mathcal G) \rightarrow \mathcal B_T(\mathcal G)/\ker ({\text {ev}}_x)\mathcal B_T(\mathcal G), \end{aligned}$$

$x \in T$ is exhaustive, see [43, Definition 3.1] for the precise definition. This follows from the definition of the central character map, see for example [10, Remark 2.11].

I would like to thank an anonymous referee for pointing out to me how to simplify the proof of the next proposition and for the reference [45, Proposition 2.2.3] where a more general situation is treated.

Proposition 2.8

Assume that $P \in \Psi _T(\mathcal G)$ is locally invertible on T. Then P is Fredholm.

Proof

By assumption P is locally invertible on T therefore for any $x \in T$ there are open neighborhood $V_x$ and operators $Q_1^{x}$, $Q_2^x$ such that for all $\phi \in \mathcal C_c(V_x)$,

$$\begin{aligned} Q_1^xPM_\phi \, = \,M_\phi \, = \,M_\phi PQ_2^x \in \mathcal L(\mathcal G). \end{aligned}$$

Since T is compact, there are $x_1, \cdots , x_N$ such that $(V_{x_j})_{j=1}^N$ is a finite open cover of T. Now let $(\phi _j)_{j=1}^{N}$ be a partition of unity subordinated to $(V_j)_{j=1}^N$ then for all $j=1,\cdots ,N$, we have

$$\begin{aligned} Q_1^{x_j}PM_{\phi _j} \, = \,M_{\phi _j} \, = \,M_{\phi _j} PQ_2^{x_j} \in \mathcal L(\mathcal G). \end{aligned}$$

It follows that $Q^1:= \sum _{j=1}^N Q_1^{x_j} M_{\phi _j}$ and $Q^2 := \sum _{j=1}^N M_{\phi _j} Q_2^{x_j}$ are respectively left inverse and right inverse of P modulo compact operators. Indeed, if $[A,B]=AB-BA$ denotes the commutator, we have

$$\begin{aligned} Q_1 P&= \sum _{j=1}^N Q_1^{x_j} M_{\phi _j} P= \sum _{j=1}^N Q_1^{x_j} [M_{\phi _j} , P] + \sum _{j=1}^N Q_1^{x_j} P M_{\phi _j}\\&=\sum _{j=1}^N Q_1^{x_j} [M_{\phi _j} , P] + \sum _{j=1}^N M_{\phi _j}\\&=\sum _{j=1}^N Q_1^{x_j} [M_{\phi _j} , P] + {\text {Id}}, \end{aligned}$$

and similarly

$$\begin{aligned} PQ_2 = \sum _{j=1}^N [P,M_{\phi _j}]Q_2^{x_j} +{\text {Id}}. \end{aligned}$$

Since $P \in \Psi _T(\mathcal G)$, we know from Remark 2.7 that $[P,M_{\phi _j}]$ is compact. Thus, $\sum _{j=1}^N Q_1^{x_j} [M_{\phi _j} , P]$ and $\sum _{j=1}^N [P,M_{\phi _j}]Q_2^{x_j}$ are compact operators and therefore P is Fredholm $\square$

Definition 2.9

We shall say that the representation $\mathcal C(T) \rightarrow \mathcal L(\mathcal G)$ of Notations and Hypothesis 2.3 has the property of strong convergence to 0 if for any $x\in T$ $M_{\chi _V}$ converges strongly to zero, where V runs the set of neighborhoods of x and $\chi _V\in \mathcal C(T,[0,1])$ is equal to 1 on a neighborhood of x, with values in [0, 1] and is supported in V. Said differently, $\mathcal C(T) \rightarrow \mathcal L(\mathcal G)$ as the property of strong convergence to 0 if $\forall x\in T$, $\forall h\in \mathcal G$, $\forall \varepsilon >0$, there is a neighborhood $V'$ of x such that for any neighborhood V of x, if $V\subset V'$ then $\Vert M_{\chi _V}h\Vert <\varepsilon$.

Proposition 2.10

(General Simonenko’s localization principle) Let $P \in \Psi _T(\mathcal G)$. Assume that $\mathcal C(T) \rightarrow \mathcal L(\mathcal G)$ is as in Notation and Hypothesis 2.3 and has the property of strong convergence to 0, see Definition 2.9. Then P is locally invertible on T if, and only if, P is Fredholm.

Proof

The first implication is exactly Proposition 2.8.

Let us prove the opposite implication. That is, let us assume that P is Fredholm and let us prove that P is locally invertible at $x \in T$, where x is fixed, but arbitrary. To this end, let $Q \in \mathcal L(\mathcal G)$ be an inverse modulo $\mathcal {K}(\mathcal G)$ for P, i.e. $PQ={\text {Id}}+K$ and $QP={\text {Id}}+K^\prime$, with $K,K' \in \mathcal K(\mathcal G)$. Using Lemma [27, Proposition 1.3.10], we can assume that $Q\in \Psi _T(\mathcal G)$ if one desires. Let $\chi \in \mathcal C(T)$ be equal to 1 on a neighbourhood $V_x$ of x, with values in [0, 1] and supported in a neighborhood $V_x'$. Let $\phi \in \mathcal C_c(V_x)$ then

$$\begin{aligned} \begin{gathered} M_\phi M_\chi PQM_\chi \, = \,M_\phi M_\chi ^2+M_\phi M_\chi KM_\chi \qquad \text{ and } \\ M_\chi Q P M_\chi M_\phi \, = \,M_\chi ^2M_\phi + M_\chi K^\prime M_\chi M_\phi \,. \end{gathered} \end{aligned}$$

Since $\phi$ is supported in $V_x$, we have $\phi \chi =\phi$ and so

$$\begin{aligned} M_\phi PQ M_\chi \, = \,M_\phi (1+ M_\chi K M_\chi )\quad \text{ and } \quad M_\chi Q P M_\phi \, = \,(1+ M_\chi K^\prime M_\chi )M_\phi \,. \end{aligned}$$

As $V_x'$ becomes small, we have that $M_\chi$ converges strongly to 0 because $\mathcal C(T) \rightarrow \mathcal L(\mathcal G)$ has the property of strong convergence to 0, see Definition 2.9. Since K is compact, we obtain that $\Vert M_\chi K M_\chi \Vert \rightarrow 0$. Thus, by choosing $V_x'$ small enough, we may assume that $\Vert M_\chi K M_\chi \Vert < 1$ and $\Vert M_\chi K' M_\chi \Vert < 1$.

It follows that $(1+ M_\chi K M_\chi )$ and $(1+ M_\chi K^\prime M_\chi )$ are invertible and this implies

$$\begin{aligned} \begin{gathered} M_\phi P\big (Q M_\chi (1+ M_\chi K M_\chi )^{-1}\big ) \, = \,M_\phi \qquad \text{ and }\\ \big ((1+ M_\chi K^\prime M_\chi )^{-1} M_\chi Q\big )PM_\phi \, = \,M_\phi \,, \end{gathered} \end{aligned}$$

that is, P is locally invertible. This completes the second implication, and hence the proof. $\square$

Simonenko’s principle is then [52]:

Proposition 2.11

Let M be a closed manifold, E a hermitian vector bundle on M, $\Psi _M(L^2(M,E))$ be as in 2.6 and let $P \in \Psi _M(L^2(M,E))$. We have that P is locally invertible on M if, and only if, it is Fredholm.

Proof

The hypothesis of Proposition 2.10 are satisfied because if $h\in L^2(M,E)$ then $\int _{V} |h|^2 \mathrm {dvol}$ goes to 0 when the volume of V goes to 0, see also Remark 2.4. $\square$

4 Equivariant Local Principle for Closed Manifolds

Let G be a compact Lie group that we assume to act smoothly by isometries on a closed Riemannian manifold M as before. We shall denote, as before, by $\widehat{G}$ the set of isomorphism classes of irreducible unitary representations of G. Let $\mathcal {H} := L^{2}(M,E)$ and let $\mathcal H_\alpha \cong \alpha \otimes (\alpha ^* \otimes \mathcal H)^G$ be the $\alpha$-isotypical component associated to $\alpha \in \widehat{G}$, as in the introduction and Sect. 1.

Any $\phi \in \mathcal C(M)^G$ acts by multiplication on $\mathcal H_\alpha$ and we shall denote also by $M_\phi$ the induced multiplication operator, as in Sect. 2. Furthermore, the representation of $\mathcal C(M/G)=\mathcal C(M)^G$ given by the previous multiplication operator on $\mathcal H$ and $\mathcal H_\alpha$ are non degenerate.

Definition 3.1

We shall say that $P \in \mathcal L(\mathcal H)^{G}$ is locally $\alpha$-invertible at $x\in M$ if $\pi _\alpha (P)$ is locally invertible

at $Gx \in M/G$, see Definition 2.5.

We let $\Psi _M^G(\mathcal H)$ denote the G-invariant elements in the $C^*$-algebra $\Psi _M(\mathcal H)$, which was defined in 2.6, in the previous subsection. More precisely, using Remark 2.7

$$\begin{aligned} \Psi _M^G(\mathcal H)=\{P\in \mathcal L(\mathcal H)^G\ |\ [P,M_\phi ]\in \mathcal K(\mathcal H),\ \forall \phi \in \mathcal C(M)\}. \end{aligned}$$

(3)

Before tackling the Simonenko’s equivariant localization principle, let us first justify our hypothesis with the following simple example.

Example 3.2

Let $M=G$ be our manifold with its standard action by translation. In this case, $T^*_GM=G \times \{0\}$ and then every G-invariant pseudodifferential operator P is G-transversally elliptic. It follows that the restriction $\pi _\alpha (P)$ to any isotypical component is Fredholm. Let us then consider the null operator $0 : L^2(G) \rightarrow 0$. Clearly, the restriction to the isotypical component associated with the trivial representation $\mathbf {1}=\mathbb C$ of G is Fredholm. In other words, $0=\pi _\mathbf {1}(0) : L^2(G)^G=\mathbb C\rightarrow 0$ is Fredholm. But obviously, $0=\pi _\mathbf {1}(0)$ is not locally $\mathbf {1}$-invertible.

This pathological example arises from the fact that there are points $x\in M$ such that the slice at x are discrete (and in fact on the whole space $M=G$ in the previous example). Nevertheless, we can extract such a pathological points using the following interesting fact.

Lemma 3.3

Let X be a not necessarily compact G-manifold without boundary and let $x \in X$ be such that $(T^*_GX)_x=\{0\}$. Then the orbit of x is a union of connected components of X in bijection with the connected component of $G/G_x$.

Proof

Since $(T^*_GX)_x=\{0\}$, we obtain that $S_x=\{x\}$ is the only slice at x. From the slice theorem, we deduce that the orbit $Gx\cong G\times _{G_x} \{x\}=G \times _{G_x} S_x$ is open but it is also compact. Therefore, Gx is a union of connected components of X in bijection with the connected components of $G/G_x\cong Gx$. $\square$

Consider the set of points

$$\begin{aligned} \mathcal P:=\{x \in M, \ (T^*_GM)_x=\{0\}\}. \end{aligned}$$

(4)

Then M is the disjoint union of the closed manifolds $M\smallsetminus \mathcal P$ and $\mathcal P$. Indeed, $\mathcal P$ is a union of clopen orbits and therefore it is also compact because M is. The same argument also implies that $M\smallsetminus \mathcal P$ is a closed submanifold of M.

Remark 3.4

The set $\mathcal P$ will be empty for example in the following cases:

• (1) if M is connected and not reduced to a single orbit,

• (2) if $\dim M > \dim G$,

• (3) in particular, if M/G is an orbifold of dimension $>0$.

In other cases, we can use the following useful result.

Lemma 3.5

Let $\mathcal P=\{x \in M, \ (T^*_GM)_x=\{0\}\}$ be the clopen introduced in Eq. (4). Let $\chi$ be the characteristic function of the clopen $M\smallsetminus \mathcal P$. Let then $M_\chi$ be the multiplication operator by $\chi$. Let $P\in \Psi _M(\mathcal H)$ then $P=P_1 + P_2 +P_3$ with $P_1=M_\chi P M_\chi \in \Psi _{M\smallsetminus \mathcal P}(\mathcal H)$, $P_2=M_{1-\chi } P M_{1-\chi } \in \Psi _\mathcal P(\mathcal H)$ and $P_3=M_{\chi } P M_{1-\chi } +M_{1-\chi } P M_{\chi } \in \mathcal K(\mathcal H)$. Furthermore, if $P\in \Psi _M^G(\mathcal H)$ then $\pi _\alpha (P_2)$ is Fredholm for any $\alpha \in \widehat{G}$ and therefore $\pi _\alpha (P)$ is Fredohlm if, and only if, $\pi _\alpha (P_1)$ is.

Proof

The first part is clear since $M_\chi$ and $M_{1-\chi }$ have disjoint supports. For the second part, decompose $\mathcal P=\sqcup _{i=1}^N \mathcal P_i$ into clopen orbits and let $\phi _i$ be the corresponding characteristic functions then as before we can write $P_2=\sum \nolimits _{i=1}^N M_{\phi _i}P_2 M_{\phi _i} + C$, where $C = \sum _{i\ne j} M_{\phi _i} P_2 M_{\phi _j}$ is compact because the supports of $\phi _i$ and $\phi _j$ are disjoint for $i\ne j$. The previous decomposition is in fact a decomposition into G-invariant operators since $\phi _i$ is G-invariant. Therefore, for every $\alpha \in \widehat{G}$, $\pi _\alpha (P_2)$ is Fredholm if, and only if, $\pi _\alpha (M_{\phi _i}P_2 M_{\phi _i})$ is Fredholm for any i. Now notice that $\mathcal P_i \cong G \times _{G_x} \{x\}$ for some $x \in \mathcal P$ therefore $\forall \alpha \in \widehat{G}$,

$$\begin{aligned} L^2(\mathcal P_i,E\vert _{\mathcal P_i})_\alpha&\cong \alpha \otimes \bigg (\alpha ^* \otimes L^2(\mathcal P_i,E\vert _{\mathcal P_i})\bigg )^G\\&\cong \alpha \otimes \bigg (\alpha ^* \otimes L^2(G \times _{G_x} \{x\},G \times _{G_x} E_x)\bigg )^G\\&\cong \alpha \otimes \bigg ( \alpha ^* \otimes E_x\bigg )^{G_x} \end{aligned}$$

is finite dimensional.

It follows that there are no condition for the restriction $\pi _\alpha (P_2)$ to be Fredholm, $\forall \alpha \in \widehat{G}$. In other words, for every $\alpha \in \widehat{G}$, $\pi _\alpha (P)$ is Fredholm if, and only if, $\pi _\alpha (P_1)$ is. $\square$

Remark 3.6

Notice that the previous proof implies that the image of $\mathcal C(M)^G$ in $\mathcal L(\mathcal H_\alpha )$ intersects $\mathcal K(\mathcal H_\alpha )$ when $\mathcal P\ne \emptyset$.

Recall that if $x \in M$ then $W_x \cong G \times _{G_x} U_x$ denotes a tube around x and $U_x$ a slice at x, see [56, Section I. 5]. Moreover, we have a G-equivariant isomorphism of vector bundles $E \cong G \times _{G_x} (U_x \times E_x)$. The next lemma could also be deduced from [17, Corollary 1.5].

Lemma 3.7

Let $\alpha \in \widehat{G}$ and let $\mathcal H_\alpha =L^2(M,E)_\alpha \cong \alpha \otimes L^2(M,E\otimes \alpha ^*)^G$. The subset

$$\begin{aligned} \mathcal N_\alpha :=\big \{x\in M, \ \exists W_x,\ \text {such that } L^2(W_x,E)_\alpha =\{0\}\big \} \end{aligned}$$

(5)

is a G-invariant clopen.

Proof

Replacing E with $E\otimes \alpha$, we see that we can assume that $\alpha$ is the trivial representation and therefore that $L^2(W_x,E)_\alpha = L^2(W_x,E)^G$. Clearly, $\mathcal N_\alpha$ is G-invariant. We shall denote simply $\mathcal N_\alpha$ by $\mathcal N$ is this proof since we consider the trivial representation.

Notice now that

$$\begin{aligned} \mathcal N=\big \{x\in M, \ \forall W_x,\ L^2(W_x,E)^G=\{0\}\big \} \end{aligned}$$

(6)

because if $W_x$ and $W_x'$ are two tubes around $x \in M$ then

$$\begin{aligned} L^2(W_x,E)^{G}&\cong L^2(U_x,E_x)^{G_x}\cong L^2(U_x',E_x)^{G_x}\cong L^2(W_x',E_x)^G. \end{aligned}$$

Let us show that $\mathcal N$ is open. Let $x \in \mathcal N$. By definition, there is $W_x$ such that $L^2(W_x,E)^G =\{0\}$. Let $y \in W_x$ and assume that there is a tube $W_y$ around y such that $L^2(W_y,E)^G\ne \{0\}$. By G-invariance of $W_x$ and $W_y$, we see that we can assume $W_y$ small enough such that $W_y \subset W_x$. But then $\{0\}\ne L^2(W_y,E)^G \subset L^2(W_x,E)^G$ which contradicts the fact that $x \in \mathcal N$. Therefore, $W_x \subset \mathcal N$ and $\mathcal N$ is open.

We now show that $\mathcal N$ is closed. Let $x \in M\smallsetminus \mathcal N$. By Eq. (6), there is $W_x$ such that $L^2(W_x,E)^G=L^2(U_x,E_x)^{G_x}\ne \{0\}$. Let K be a minimal isotropy subgroup for the linear action of $G_x$ on $U_x$, see [56, Section 5]. Notice that K acts trivially on $U_x$ by minimality. Then we have $\{0\} \ne L^2(U_x,E_x)^{G_x} \subset L^2(U_x,E_x)^K=L^2(U_x,E_x^K)$. It follows that there is $v \in E_x^K \smallsetminus \{0\}$. Let $y \in U_x$ and denote by $W_y'\cong G_x \times _{G_y} U_y' \subset U_x$ a tube around y in $U_x$. Denote by $W_{y(K)}'$ the principal orbit bundle of $W_y'$ that is the dense open subset of $W_y'$ given by the points with stabilizer conjugated with K in $G_x$. Each point of $W_{y(K)}'$ has a neighborhood of the form $G_x/K \times V$ with K acting trivially on V. Let $s \in \mathcal C(G_x/K \times V, E_x)$ be given by $s([g],z)=gv$. The section s does not depend on the representative of [g] in $G_x$ because $v \in E_x^K$ and is clearly $G_x$-invariant. If $f\in \mathcal C_c(G_x/K \times V)^{G_x}$ is any compactly supported function then $f s \in \mathcal C_c(G_x/K \times V, E_x)^{G_x} \subset L^2(W_y',E_x)^{G_x}$. Now if $W_y\cong G \times _{G_y} U_y$ is a tube around y in $W_x$ then assuming $U_y'$ small enough we have $G \times _{G_x} W_y' \subset G \times _{G_y} U_y$ and thus $0\ne L^2(W_y',E_x)^{G_x} \cong L^2(G \times _{G_x} W_y',E)^G \subset L^2(W_y,E)^G$. It follows that $y \in M \smallsetminus \mathcal N$ and therefore $W_x \subset M\smallsetminus \mathcal N$. In other words, $M\smallsetminus \mathcal N$ is open. This complete the proof. $\square$

Remark 3.8

Let $\mathcal N_\alpha =\big \{x\in M, \ \exists W_x,\ \text {such that } L^2(W_x,E)_\alpha =\{0\}\big \}$ be the clopen defined in Lemma 3.7. We have $L^2(\mathcal N_\alpha ,E)_\alpha =\{0\}$ and therefore there is no condition on $L^2(\mathcal N_\alpha ,E)_\alpha$ for an operator to be Fredholm. By a discussion similar to the one of Lemma 3.5, we see that $P\in \Psi _M(\mathcal H)^G$ is such that $\pi _\alpha (P)$ is Fredholm if, and only, if $\chi _{M\smallsetminus \mathcal N_\alpha }P \chi _{M\smallsetminus \mathcal N_\alpha }$ is Fredholm, where $\chi _{M\smallsetminus \mathcal N_\alpha }$ denotes the characteristic function of $M\smallsetminus \mathcal N_\alpha$.

Moreover, we see that if $\mathcal N_\alpha$ is not empty then the image of $\mathcal C(\mathcal N_\alpha )^G$ in $\mathcal L(\mathcal H_\alpha )$ is 0, i.e. $M_\phi =0$, $\forall \phi \in \mathcal C(\mathcal N_\alpha )^G$. It follows that the image of $\mathcal C(M)^G$ is the same as the image of $\mathcal C(M\smallsetminus \mathcal N_\alpha )^G$.

For example, let $G=SO(3)$, let $M=S^2_1 \sqcup S^2_1$ be the disjoint union of two spheres $S^2 \subset \mathbb R^3$ with a trivial action on $S_1^2$ and the induced action from $\mathbb R^3$ on $S_2^2$. Let then $E=M \times \mathbb C^3$ with the natural action on $\mathbb C^3$. Then $L^2(M,E)^G=L^2(S_2^2,\mathbb C^3)^G \ne 0$. Indeed, $(\mathbb C^3)^G=0$ because $\mathbb C^3$ is an irreducible representation of SO(3) therefore $L^2(S_1^2,\mathbb C^3)^G=L^2(S_1^2,(\mathbb C^3)^G)=0$. Moreover, the function $f(x)=x \in \mathbb R^3 \subset \mathbb C^3$ is G-invariant and belongs to $L^2(S_2^2,\mathbb C^3)^G$. Now if $\chi _{S^2_1}$ is the characteristic function of $S_1^2$ then $M_{\chi _{S^2_1}} : L^2(M,E)^G \rightarrow L^2(M,E)^G$ is zero.

Lemma 3.9

Let $\mathcal P$ be the clopen introduced in Eq. (4) and let $\mathcal N_\alpha$ be the clopen introduced in Eq. (5). Let $\alpha \in \widehat{G}$ and let $\mathcal H_\alpha =L^2(M,E)_\alpha \cong \alpha \otimes L^2(M,E\otimes \alpha ^*)^G$. Let $f\in \mathcal C(M\smallsetminus (\mathcal P\cup \mathcal N_\alpha ))^G$ then $M_f \in \mathcal K(\mathcal H_\alpha )$ if, and only, if $f=0$.

Proof

We may assume $\alpha$ to be the trivial representation $\mathbf {1}\in \widehat{G}$. Let $f \in \mathcal C(M\smallsetminus (\mathcal P\cup \mathcal N_\alpha ))^G$ be non zero such that $M_f$ is compact on $\mathcal H_\mathbf {1}=L^2(M,E)^G$. Let then $W_x \cong G \times _{G_x} U_x$ be a tube on which $|f|>\varepsilon >0$. Denote by $\chi _{x}$ the characteristic function of $W_x$ and let $\widetilde{M_f}$ be the restriction of $M_f$ to $L^2(W_x,E)^G\cong L^2(U_x,E_x)^{G_x}$. Then $\widetilde{M_f}$ is invertible with inverse $M_{f\chi _{x}}$. Thus by Banach open mapping theorem $\widetilde{M_f}$ is open but also compact therefore $L^2(U_x,E_x)^{G_x}$ is finite dimensional. Notice that $L^2(U_x,E_x)^{G_x}\ne 0$ since $x \in M\smallsetminus \mathcal N_\alpha$. Let then $s \in L^2(U_x,E_x)^{G_x}$ be non zero. For any $n \in \mathbb N$, we can certainly find $n+1$ disjoint $G_x$-invariant annulus in $U_x$ such that the restriction of s to each of this annulus is non zero because the action is isometric and $\dim U_x >0$ since $x \in M \smallsetminus \mathcal P$. By considering the characteristic functions $\chi _i$ of this $n+1$ annulus, we get $n+1$ linearly independent functions $\chi _i s$, thus a contradiction. $\square$

Recall that we denote by $\mathcal H=L^2(M,E)$. Let $\alpha \in \widehat{G}$, let $P\in \mathcal L(\mathcal H)^G$ and recall that $\pi _\alpha (P) : \mathcal H_\alpha \rightarrow \mathcal H_\alpha$ is the restriction of P to the $\alpha$-isotypical component $\mathcal H_\alpha \cong \alpha \otimes (\alpha ^* \otimes \mathcal H)^G$ of $\mathcal H$.

Proposition 3.10

(Simonenko’s equivariant localization principle) Let M be a closed G-manifold as before. Let $\mathcal P$ be as in Eq. (4) and let $\alpha \in \widehat{G}$. Let $P \in \Psi _M^G(\mathcal H)=\{P\in \mathcal L(\mathcal H)^G | [P,M_\phi ] \in \mathcal K(\mathcal H), \forall \phi \in \mathcal C(M)\}$. Then P is locally $\alpha$-invertible on $M\smallsetminus \mathcal P$ (see Definition 3.1) if, and only if, $\pi _\alpha (P)$ is Fredholm.

Proof

Let $\mathcal N_\alpha$ be as in Eq. (5). Notice that any operator is locally $\alpha$-invertible at $x \in \mathcal N_\alpha$ as operator between the null vector space. Similarly, on $\mathcal N_\alpha$ the operator $\pi _\alpha (P)$ is Fredholm.

By Lemma 3.5 and Remark 3.8, we may replace M with $M\smallsetminus (\mathcal P\cup N)$ and assume that for any $x\in M$, $(T^*_GM)_x$ is not reduce to $\{0\}$ and that there is a tube $W_x$ around x such that $L^2(W_x,E_x)_\alpha \ne \{0\}$. Under this hypothesis, we have that $\mathcal C(M/G) \rightarrow \mathcal L(\mathcal H_\alpha )$ is faithful, non degenerate and does not intersect $\mathcal K(\mathcal H_\alpha )\smallsetminus \{0\}$, see Lemma 3.9 and Notation and Hypothesis 2.3. Moreover, $\mathcal C(M/G) \rightarrow \mathcal L(\mathcal H_\alpha )$ has the property of strong convergence to 0, see Definition 2.9. Indeed, this is equivalent to say that the volume of the slice at x goes to zero when it becomes small. Let us now introduce the $C^*$-algebra $\Psi _{M/G}(\mathcal H_\alpha )$ defined in 2.6. Clearly, $\pi _\alpha (\Psi _M^G(\mathcal H))$ is a sub-$C^*$-algebra of $\Psi _{M/G}(\mathcal H_\alpha )$. Therefore, $\pi _\alpha (P) \in \pi _\alpha (\Psi _M^G(\mathcal H)) \subset \Psi _{M/G}(\mathcal H_\alpha )$ is Fredholm if, and only if, it is locally invertible on M/G. By definition, P is locally $\alpha$-invertible at $x\in M$ if, and only, if $\pi _\alpha (P)$ is locally invertible at $Gx \in M/G$, thus the result follows from Proposition 2.10. $\square$

Theorem 3.11

Let M be a closed G-manifold as before and let $\mathcal P$ be as in Eq. (4). Let $P\in \psi ^m(M;E_0,E_1)^G$ and $\alpha \in \widehat{G}$. Then the following are equivalent:

• (1) $\pi _\alpha (P) : H^s(M;E_0)_\alpha \rightarrow H^{s-m}(M;E_1)_\alpha$ is Fredholm for any $s \in \mathbb R$,

• (2) P is transversally $\alpha$-elliptic (see Definition 1.3),

• (3) P is locally $\alpha$-invertible on $M\smallsetminus \mathcal P$ (see Definition 3.1).

Proof

The first equivalence is given by Theorem 1.4. Now Proposition 3.10 implies that (1) is equivalent to (3). $\square$

In particular, we obtain the following consequence of the localization principle.

Corollary 3.12

Let $P\in \psi ^{m}(M; E)^G$ be a G-transversally elliptic operator, see Sect. 1.2. Then P is locally $\alpha$-invertible on $M\smallsetminus \mathcal P$ for any $\alpha \in \widehat{G}$, as in Definition 3.1.

Proof

Using Theorem 1.1 we obtain that $\pi _\alpha (P)$ is Fredholm. Therefore by Proposition 3.10P is locally $\alpha$-invertible. $\square$