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Perturbations of basic Dirac operators on Riemannian foliations

Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.


Introduction
It is well-known that the index of the de Rham operator In [18], the authors expanded the method of Witten deformation and provided the formula for the index in all cases when D is a Dirac-type operator and Z is an admissible bundle map satisfying certain nondegeneracy conditions (similar to the Morse conditions).
The purpose of this paper is to find an expression for the basic index of a transversal Dirac operator over a Riemannian foliation in terms of local quantities associated to the singular set of a foliated bundle map satisfying admissibility and nondegeneracy conditions (see Sections 2.2 and 2.3). We use the method of Witten deformation to achieve localization (see Corollary 4.6). Our main result (Theorem 4.12) is the formula Here, D b is a basic Dirac operator, ℓ is a critical leaf closure for a basic bundle map Z, and x is an arbitrary point of ℓ. The linear maps L ± j (x) are obtained from the linearization of the Clifford form of Z at x (see Section 4.2), the space E λ is the eigenspace associated to the eigenvalue λ, and H x is the infinitesimal holonomy group associated to x. The Hopf index theorem for Riemannian foliations proved in [4] can be easily derived from this formula.
We now briefly explain the setup for the formula; precise definitions are in Section 2.1. The reader may consult the introduction in [11] and Section 3 of [19] for more complete expositions concerning basic Dirac operators, and more information on Riemannian foliations is contained in [24] and [16]. A Dirac operator has the form D = c (e j ) ∇ E e j , where ∇ E is a Clifford connection on a Hermitian vector bundle E over M, {e j } is a local orthonormal frame of T M, and c denotes Clifford multiplication. Suppose now that M has the additional structure of a Riemannian foliation F , i.e. a layering of M by immersed submanifolds (leaves), and a transverse Riemannian metric that is invariant along the leaves. A simple example of this structure is that of the orbits of a compact Lie group action, where all the orbits have the same dimension, and where the invariant transverse metric can be obtained as the average of an arbitrary transverse metric along the orbits (leaves).
On a Riemannian foliation, there exist natural operators called transversal Dirac operators. Consider a Hermitian vector bundle E over M that is a module over the complexified Clifford algebra of the normal bundle NF to the foliation. The formula for the transversal Dirac operator is the same as that of the ordinary Dirac operator, but the sum is merely over a local orthonormal frame of the normal bundle NF ⊂ T M. This operator restricts to an operator on Γ b (M, E, F ), the space of basic sections of E. A section s of E is called basic if it is invariant under parallel translation along the leaves, i.e. if ∇ E X s = 0 for all X ∈ Γ(T F ). The leaf space M/F may be quite singular; these basic sections provide a type of desingularization of the leaf space. On a Riemannian foliation, a transverse Dirac operator maps the space Γ b (M, E, F ) to itself, but it is not necessarily symmetric with respect to the L 2 inner product. A basic Dirac operator D b is the symmetric operator defined as where the sum is over an orthonormal frame of NF and κ b is the basic component of the mean curvature one-form.
The basic index of a basic Dirac operator D b is , where one restricts to the subspace of basic sections of the graded Hermitian bundle E = E + ⊕E − . It turns out that this basic index is a Fredholm index, implying that the dimensions are finite and that the index is stable under perturbations. Using Molino theory (see Section 3), one may show that this index is equivalent to the equivariant index of a certain Dirac operator on an O(n)-manifold.
It is an important problem to express the basic index in geometric terms. We refer to [9,Problem 2.8.9], where this problem was first stated explicitly. One example of an index formula is the Gauss-Bonnett theorem for Riemannian foliations, proved in [6]. Also, in [6], J. Brüning, F. W. Kamber, and Richardson use the equivariant index theorem of [5] to give a geometric formula for the basic index of a basic Dirac operator in terms of Atiyah-Singer type integrands and eta invariants. In [10], A. Gorokhovsky and J. Lott proved a different formula for the basic index of a basic Dirac operator, in the case where all the infinitesimal holonomy groups of the foliation are connected tori and when Molino's commuting sheaf is abelian and has trivial holonomy. In contrast, in our formula the index is a sum of integers, each of which is computed at a single leaf closure.
A related natural question is whether the basic index of a transversally elliptic operator on a Riemannian foliation has similar properties to those of an elliptic operator. Until now, it was not known whether this index is always zero for transversally elliptic differential operators on a Riemannian foliation of odd codimension. Previously, this seemingly elementary fact had only been proved for the basic Euler characteristic (in [12]), and the general case is now proved in Corollary 4.8.
In Section 2.1, we review properties of foliated vector bundles and Dirac operators over Riemannian foliations. In Sections 2.2 and 2.3, we establish conditions on Z so that the index computation localizes to small neighborhoods of critical leaf closures. The necessary condition for localization is proved in Theorem 2.5. We show in Section 3 how to reduce basic differential operator calculations on a tubular neighborhood of a leaf closure to equivariant differential operator calculations in Euclidean space. In Section 4.1, we define the model operator and prove the localization theorem. We show in Section 4.2 that Z can be deformed to a special form called Clifford form, which is used in our formula for ind b (D b ). We prove the main result (Theorem 4.12) in Section 4.3. Finally, in Section 5, we apply Theorem 4.12 to compute the basic Euler characteristic and basic signature of some foliations. In particular, Example 5.3 shows that localization is possible even when D b Z + ZD b is a first order differential operator.

Perturbing basic Dirac operators
2.1. Preliminaries and Notational Conventions. Let (M, F ) be a closed, connected, smooth manifold M equipped with a smooth foliation F of codimension q. We assume that (M, F ) has additional geometric structure. That is, let Q = T M T F → M be the normal bundle of F , and let g Q be a metric on Q. We assume that g Q is holonomy-invariant, meaning that its Lie derivative in directions tangent to F is zero. A foliation along with such a g Q is called a Riemannian foliation. When (M, F , g Q ) is a Riemannian foliation, one may always choose a metric g on M such that the restriction of g to (T F ) ⊥ agrees with g Q ; such metrics are called bundle-like metrics. Given a bundle-like metric, the leaves of F are locally equidistant. Throughout this paper, we assume that we have chosen a bundlelike metric g on M compatible with g Q . See [20], [16], and [24] for standard facts about Riemannian foliations.
We now recall the definitions (see [13] and [16]) of foliated bundle and basic connection. Let G be a compact Lie group. We say that a principal G-bundle P → (M, F ) is a foliated principal bundle if it is equipped with a foliation F P (the lifted foliation) such that the distribution T F P is invariant under the right action of G, is transversal to the tangent space to the fiber, and projects onto T F . A connection ω on P is called adapted to F P if the associated horizontal distribution contains T F P . An adapted connection ω is called a basic connection if it is basic as a g-valued form on (P, F P ); that is, i X ω = 0 and i X dω = 0 for every X ∈ T F , where i X denotes the interior product with X. Note that in [13] the authors showed that basic connections always exist on a foliated principal bundle over a Riemannian foliation.
Similarly, a complex vector bundle E → (M, F ) of rank k is foliated if E is associated to a foliated principal bundle P → (M, F ) via a representation ρ from G to U (k). Let Ω (M, E) denote the space of forms on M with coefficients in E. If a connection form ω on P is adapted, then we say that an associated covariant derivative operator ∇ E on Ω (M, E) is adapted to the foliated bundle. We say that ∇ E is a basic connection on E if in addition the associated curvature operator ∇ E 2 satisfies i X ∇ E 2 = 0 for every X ∈ T F . Note that ∇ E is basic if ω is basic.
For m ∈ M, let Cl(Q m ) be the complex Clifford algebra associated to the vector space Q and quadratic form g Q . We say that (M, F , g Q ) is transversally spin c if Q is spin c . That is, there exists a complex spinor Cl(Q)-bundle S over M such that, for all m ∈ M, the action of Cl(Q m ) on S m is an irreducible representation of Cl (Q m ). A simple example shows that this condition is necessary even for spin c manifolds; if X is any manifold that is not spin c , then the product foliation X ×X would not be transversally spin c even though X ×X always admits a complex structure and thus is spin c .
Let in addition E → M be a Hermitian Cl(Q) Clifford bundle. We assume that the basic connection ∇ E is compatible with these additional structures.
Let Γ (M, E) denote the space of smooth sections of E over M. Let Γ b (M, E, F ) denote the space of basic sections of E, meaning that

Similarly, a bundle endomorphism A is basic if ∇
End(E) X A = 0 for all X ∈ Γ(T F ). Now we define the transversal Dirac operator, which is similar to the standard Dirac operator but is elliptic only when restricted to directions orthogonal to the tangent bundle T F of the foliation. Formally, the transversal Dirac operator D tr is the composition of the maps where the last map denotes Clifford multiplication, and the operator ∇ E tr is the projection of ∇ E to Γ (Q * ⊗ E). Clifford multiplication by an element v ∈ Q x on the fiber E x is denoted by c (v). Clifford multiplication by cotangent vectors in Q * will use the same notation: → Q x is the metric isomorphism. The transversal Dirac operator fixes the basic sections but is not symmetric on this subspace. By modifying D tr by a bundle map, we obtain a symmetric and essentially self-adjoint operator be the mean curvature of the foliation, where π : T M → Q denotes the projection and {f i } i=1,··· ,p is a local orthonormal frame of T F . Let κ = H ♭ be the corresponding 1-form, so that H = κ ♯ . Let κ b := P b κ be the L 2 -orthogonal projection of κ onto the space of basic forms (see [1], [17]). We now define the basic Dirac operator by where {e i } i=1,··· ,q is a local orthonormal frame of Q. A direct computation shows that D b preserves the basic sections. The operator D b : Γ b (E) → Γ b (E) is transversally elliptic, has discrete spectrum, and is Fredholm ( [9]). The spectrum of D b was shown to depend only on g Q and not on g in [11], in spite of the fact that the mean curvature and L 2 inner product certainly depend on the choice of g.
A grading on E is induced by the action of the chirality operator γ (in the transverse direction). Recall that if e 1 , ..., e q is an oriented orthonormal basis of Q x , then the chirality operator is multiplication by where k = q/2 if q is even and k = (q + 1) /2 if q is odd. The ±1 eigenspaces of γ determine a grading of E = E + ⊕ E − . This grading is called the natural grading. The other possible gradings are classified in [18, pp. 319ff]. Observe that the chirality operator is a basic bundle map, because for X ∈ Γ(T F ), u ∈ Γ (E), and a local framing (e 1 , ..., e q ) of Q, c(e 1 )...c(∇ E X e j )...c(e q ) = 0, because (M, F , g Q ) is Riemannian and we may choose the local framing so that each e j is a local basic section.
and define the operators D ± s by restricting in the obvious ways. The basic index ind b (D b ) depends only on the homotopy type of the principal transverse symbol of D b and satisfies Thus, we need to study the operator As will be shown later, the leading order behavior of the eigenvalues of this operator as s → ∞ is determined by combinatorial data at the singular set of the operator Z. This "localization" allows one to compute ind b (D b ) in terms of that data. A sufficient condition for localization techniques to work is the requirement that the operator ZD b + D b Z restricts to a bounded operator on the space of smooth basic sections Γ b (M, E, F ). We need the following lemmas.
A foliation F of codimension q is called transversally parallelizable if there exists a global basis of Q consisting of basic vector fields. For any Riemannian foliation F , the lifted foliation F on the orthonormal transverse frame bundle M is always transversally parallelizable (see [16]).
If F is transversally parallelizable and (M, F ) is Riemannian, then the leaf closures are the fibers of a Riemannian submersion for any choice of bundle-like metric. There is a set {s 1 , ..., s k } of basic sections of E such that for every x ∈ M, {s 1 (x) , ..., s k (x)} is a basis of E x . Then every basic section can be written as , so that Xf j for every j and every X ∈ T F . Thus, the functions f j must be basic.
As in the last proof, any section can be written as f j (x) s j (x), and for V basic and tangent to the leaf closures, we have V f j = 0, and Let V be a basic vector field that is tangent to every leaf closure of F , and let V be its horizontal lift in the orthonormal transverse frame bundle M . Observe that horizontal lifts of basic vector fields are basic for the lifted foliation, so that V is a basic vector field. Since the leaf closures of F are principle bundles over the leaf closures of F , V must be tangent to the leaf closures of F. Since M , F is transversally parallelizable, ∇ V is bounded as a linear operator on the space of all basic sections Γ b M , π * E, F . Since for every section  ., e q , e q+1 , ..., e q } of NF near x consisting of basic vector fields such that e 1 , ..., e q is a local transverse orthonormal frame for the leaf closure containing x, and in a neighborhood of x the fields e 1 , ..., e q remain tangent to the leaf closures of F .
Proof. From [16], there exists a local orthonormal frame of NF near x consisting of basic vector fields. It remains to show that the frame may be chosen to be adapted to the leaf closures near x. Let B be a transversal submanifold to the leaf through x. By [16, Ch. 1, 5, App. D], the orbits of the closure of the holonomy pseudogroup acting on B are exactly the intersections of the leaf closures with B, and there exist a smooth family of local isometries that generate the leaf closures near x. This family is at least q-dimensional, where q is the dimension of the leaf closure through x. Choose q one-parameter families of local isometries in the pseudogroup such that their initial velocities generate linearly independent vectors at x. By orthogonalization, we may alter these families so that these velocity vectors e 1 , ..., e q at x are orthonormal. The velocity vectors of the families of isometries naturally extend to a neighborhood of x, and by construction they are basic fields that are tangent to all leaf closures near x and are linearly independent. They are automatically orthonormal on the portion of the leaf closure containing x inside B, and we may orthogonalize over B in a neighborhood of this leaf closure so that e 1 , ..., e q is a local orthonormal set in a neighborhood of x, consisting of basic fields tangent to the leaf closures of F . We may then extend this local frame to a local orthonormal frame of Q near x in B, and then we may extend the frame in a neighborhood of the transversal so that the fields are basic. Thus, the new frame {e 1 , ..., e q , e q+1 , ..., e q } has the desired properties.
The following theorem gives a sufficient condition for the operator Proof. Suppose that the hypothesis holds. Let x ∈ M, and using Lemma 2.4 choose a local orthonormal frame e 1 , ..., e q , e q+1 , ..., e q of NF near x be such that e 1 , ..., e q is a local orthonormal frame for the leaf closure containing x, and the fields e 1 , ..., e q remain tangent to the leaf closures in a neighborhood of x. Then By Lemma 2.3, ∇ E e i is bounded on the space of basic sections for i ≤ q, and thus ZD b + D b Z is a bounded operator on the space of basic sections.
Remark 2.6. The reader may verify that the converse of the theorem above is also true.  2.3. Proper perturbations of basic Dirac Operators. In this section, we state the nondegeneracy conditions on the perturbation. Note that the perturbations of the basic de Rham operators in [2] and [4] are special cases.
(1) We say that a leaf closure ℓ is critical for Z if Z (x) = 0 for all x ∈ ℓ.
(2) We say that a critical leaf closure ℓ for Z in M is nondegenerate if on any sufficiently small tubular neighborhood U of a critical leaf closure ℓ, there exists a constant c > 0 such that for all α ∈ Γ (U, E) and all y ∈ U, where · y is the pointwise norm on E y and where d (y, ℓ) is the distance from y to the leaf closure ℓ.
a open ball in R q centered at the origin. The corresponding coordinates (x, y) are called the adapted coordinates if each coordinate y j is a locally defined basic function for F , and {(x, y) : y = 0} = ℓ ∩ U. In this case, we say that U is adapted to ℓ.
Since ℓ is an embedded submanifold of M, there exists an adapted neighborhood of every point in ℓ.
Lemma 2.11. A critical leaf closure ℓ of (M, F ) is nondegenerate for a basic bundle map Z if and only if every point of ℓ has an adapted neighborhood U with adapted coordinates (x, y) such that there exist bundle maps Z j for 1 ≤ j ≤ q over U such that Z = j y j Z j on U, and Z is invertible over U \ ℓ.
Proof. Suppose U is an adapted neighborhood of a point of ℓ, a critical leaf closure for Z. Since Z is smooth and vanishes at each (x, 0), Z = j y j Z j for some bundle maps Z j . Let U be the closure of U; we assume that we have chosen U to be small enough so that U is diffeomorphic to a product of compact sets in ℓ and R q . The inequality in the definition of nondegenerate above is equivalent to for every σ ∈ S q−1 , α ∈ Γ (U, E) , (x, y) ∈ U such that α (x,y) = 1. Since the left hand side of the inequality is a continuous function of σ and α over the compact set S q−1 × α ∈ Γ U , E : α (x,y) = 1 for all (x, y) ∈ U , its infimum is attained. It follows that on each such neighborhood, Z is invertible away from ℓ if and only if the inequality holds with c > 0. Compactness of ℓ implies the inequality holds in tubular neighborhoods of ℓ.
be the basic Dirac operator associated to a bundle of graded Clifford modules. Let (2) All critical leaf closures for Z are nondegenerate.
The following lemma shows that only certain ranks of vector bundles may admit proper perturbations.
Lemma 2.14. (in [18])For any positive integer k, there exists a linear map L : The following result shows that nonsingular proper perturbations always exist on Clifford modules over a Riemannian foliation of odd codimension. Proof. Near a point of M, let (e 1 , ..., e q ) be a local basic orthonormal frame of the normal bundle. Let Z = i q(q+1)/2 c (e 1 ) c (e 2 ) ...c (e q ), which is well-defined independent of the choice of orthonormal frame, and it is a basic bundle map from E ± to E ∓ . It satisfies the conditions of Theorem 2.5. Since the adjoint of Z is we have a proper basic perturbation of D b that is globally invertible.

Vector bundles over foliations and compact group actions
In this section, we restrict to a subset of foliated vector bundles over (M, F ). We will discuss F -equivariant vector bundles; most of the interesting cases of geometrically constructed foliated bundles are also F -equivariant.
Let (M, F , g Q ) be a Riemannian foliation with bundle-like metric g, let NF = (T F ) ⊥ ∼ = Q. Let M F denote the space of leaf closures.
Definition 3.1. The bundle E → M is called an F -vector bundle if the following holonomy lifting property is satisfied. For any holonomy diffeomorphism h : U → V associated to a path γ in a leaf from x ∈ D to y ∈ D, where D is a submanifold at x transverse to F and U and V are open subsets of D such that x, y ∈ U ∩ V , h lifts to a bundle isomorphism h : E| U → E| V , and all such maps satisfy Such a vector bundle is often called equivariant with respect to the foliation groupoid, and these vector bundles are examples of foliated vector bundles (see [13]), as described in Section 2.1. They come equipped with a partial connection ∇ E , which may be completed to a basic connection (with the same name). When E is an F -vector bundle, the basic sections are those sections that satisfy h (s x ) = s y for every holonomy diffeomorphism h as above with h (x) = y, i.e. the holonomy-invariant sections. Let Diff b (M, E, F ) denote the space of differential operators that preserve the basic sections, and let Diff * b (M, E, F ) denote the space of their restrictions to Γ b (M, E, F ). In the above, we call an F -vector bundle an Hermitian F -vector bundle if it is equipped with a holonomy-invariant Hermitian inner product.
Let π : M → M denote the bundle of orthonormal transverse frames over (M, F , g Q ). Let π * F be the pullback foliation, which has dimension dim (F ) + dim (O (q)). The connection on M is flat along the leaves, since (M, F , g Q ) is Riemannian. The lifted foliation F is defined on M by horizontally lifting the leaves of F locally. The tangent space of F is the intersection of the tangent space to the pullback foliation and the horizontal space coming from the adapted connection. The lifted foliation F has the same dimension as F and is transversally parallelizable, and this implies that the leaf closure space W = M F is a manifold, called the basic manifold (see [16]). The quotient map p : M → W is a For each Hermitian F -vector bundle π : E → M, we construct a canonically associated Hermitian O (q)-vector bundle π ′ : E ′ → W that has similar geometric and analytic properties, as follows. Specifically, the bundle E ′ over W is defined by , and it is a O (q)-equivariant Hermitian vector bundle of finite rank (see [9, section 2.7]) . It is also true that p * E ′ = π * E, and we have the algebra isomorphisms The isomorphisms are determined by the correspondences Given a differential operator P ∈ Diff * b (M, E, F ), it induces an operator P on Γ (W, where s is the unique section in Γ b (M, E, F ) such that s (w) = π * s| p −1 (w) . The following proposition from [2] gives the important properties of this construction.
. The isomorphism preserves transverse ellipticity, and there is a metric on W so that the isomorphism preserves formal adjoints with respect to the L 2 inner products. Corollary 3.3. The previous proposition is also valid locally. Namely, if M is replaced with an ε-tubular neighborhood T ε of a leaf closure ℓ ⊂ M and W is replaced by a tubular neighborhood Let C ∞ b (M, F ) denote the space of smooth basic functions, and let C ∞ (W ) O(q) denote the space of smooth O (q)-invariant functions. The following proposition is similar to [2, Propositions 3.3 and 3.4], which considers the special case of differential forms. As above, where H is the isotropy subgroup of G at x. Further, the isomorphism puts transversally elliptic operators in in one-to-one correspondence with elliptic operators in In addition, there is a Riemannian metric on B ε such that isomorphism preserves formal adjoints with respect to L 2 inner products.
Proof. First, note that sections in Γ T ′ ε , E ′ | T ′ ε G are in one-to-one correspondence with sections of Γ B ε , E ′ | Bε H . To see this, given a section s ∈ Γ T ′ ε , E ′ | T ′ ε G , the restriction Φ (s) = s| Bε is an element of Γ B ε , E ′ | Bε H . Note that Φ is a linear map that is one-to-one, Let S be a small open ball complementary and transverse to the H-orbits through the identity in G. Let ℓ x = {sx : s ∈ S} for x ∈ B ε . Note that ℓ x ∩ ℓ y = ∅ if x = y and x, y ∈ B ε ; if s 1 x = s 2 y with s 1 , s 2 ∈ S implies s −1 2 s 1 x = y, so that s −1 2 s 1 ∈ H, s 1 ∈ s 2 H, so that s 1 = s 2 since S is transverse to the H-orbits in G. The leaves {ℓ x : x ∈ B ε } form a trivial Riemannian foliation L of a neighborhood N of B ε in T ′ ε . Note that the restriction of to N is a subspace of the space of basic sections of E ′ | N , which is a foliated vector bundle over (N, L). Note that u is a basic section if and only if ∇ E ′ X u = 0 for all leafwise vector fields X.

Given an operator
Then P | Γ(N,E ′ ) is a differential operator that maps Γ b (N, E ′ ), the space of basic sections E ′ over N, to itself, because for g ∈ S, and u ∈ Γ T ′ ε , E ′ | T ′ ε , P (g · u) = g·P (u). We choose a framing of T N adapted to L: (e 1 , ..., e p , e p+1 , ..., e n ), where (e 1 , ..., e p ) is tangent to L and (e p+1 , ..., e n ) is perpendicular to L. Then we may write P | Γ b (N,E ′ ) as a polynomial over C ∞ (N) in the covariant derivatives ∇ E ′ e j , such that where P 1 is a differential operator whose only derivatives are of the form ∇ E ′ e j with j > p.

Here we have used the fact that operators of the form
a are first-order differential operators for a and b coordinate vector fields. Thus the restriction satisfies Thus, P restricted to basic sections may be expressed as a differential operator on the local quotient N L, which is diffeomorphic to B ε . Since P is H-equivariant, the corresponding operator on B ε must be H-equivariant. If we let P ′ be that operator, the proof is complete.
Remark 3.5. If E comes equipped with a grading, a similar grading is induced on E ′ , and the constructions and isomorphisms above preserve those gradings. Let

Localization and consequences
We wish to study the asymptotics of the O (q)-invariant part of the spectrum of H s in the semiclassical limit as s → ∞. The goal of this section is to construct the model operator, whose spectrum approximates the spectrum of H s , under the following conditions: (1) (D ′ ) 2 is a second order, transversally elliptic differential operator with positive definite principal transverse symbol.
The singular set of Z ′ is a union of isolated orbits; each such critical orbit corresponds to a critical leaf closure of Z. We will choose an arbitrary point x on each critical orbit, labeled O x . Then the model operator is a direct sum At each x where Z ′ (x) is singular, the corresponding operators (H s ) ′ from Proposition 3.4 are G x equivariant on the transverse balls B ε,x , where G x is the isotropy subgroup at x. Note that the quadratic condition (3) on Z ′ is equivalent to the condition of nondegeneracy for a leaf closure ℓ that is critical for Z in Definition 2.9.
Let H G s denote the restriction of H s to Γ (W, E ′ ) G . It is known (see [3, p. 12-13]) that the operator H G s has discrete spectrum. Near each critical orbit O x of (Z ′ ) 2 , we choose coordinates x = (x 1 , ..., x m ) for the transverse ball B ε,x such that x corresponds to the origin, N x O x = R m , and the volume form at the origin is dx 1 ∧ ... ∧ dx m . Let H x be the isotropy subgroup at x. We choose a trivialization of E ′ near x. Then A = (D ′ ) 2 , B = D ′ Z ′ +Z ′ D ′ , and C = (Z ′ ) 2 become differential operators with matrix coefficients. We define the model operator where ∇ is the induced connection on E ′ ⊗ E ′ * . It is easy to check that C is independent of the coordinates and connection chosen. Let dg denote the differential of the action of g ∈ H x at x, so we write dg : R m → R m . Let the action of g on R m × E ′ x be defined as Proof. Since H s is equivariant with respect to G for each s > 0, it is easy to show that each of the operators A, B, and C is G-equivariant. Then the principal symbol of A is G-equivariant, and in particular the principal symbol of A at x is H x -invariant. Thus, A is H x -invariant. Next, since B| Γ(W,E ′ ) G is equivariant, its restriction B to x is also. Finally, since C is Gequivariant and the connection is G-equivariant, it follows that C is H x -equivariant.
. This operator is elliptic and essentially self-adjoint, and the operator is bounded below by C 1 + C 2 · |x| 2 1, where C 1 ∈ R and C 2 > 0. Since this bound goes to infinity as x → ±∞, the operator K H x x − (C 1 − 1) 1 has a compact resolvent. Thus, the restriction of K H x x to Γ (R m , E ′ x ) H x also has a compact resolvent.
Remark 4.4. The dimension m above may depend on x, even though this is not obvious from the notation. We define the model operator K G by Clearly, this operator has discrete spectrum.
.. be the eigenvalues of H G s , repeated according to multiplicity, correspond to the orthonormal basis of eigensections ω G 1 (s) , ω G 2 (s) , ... . Let µ G 1 ≤ µ G 2 ≤ ... be the eigenvalues of the model operator K G , repeated according to multiplicity, correspond to the L 2 -orthonormal basis of eigensections φ G 1,s , φ G 1,s , .... Then we have the following result. Theorem 4.5. (Equivariant Localization Theorem)Assume that the singular set of Z is not empty. Then, for each fixed N > 0, there exists C > 0 and s 0 > 0 such that for any s > s 0 and any j ≤ N, λ G j (s) − µ G j ≤ Cs −1/5 . If the singular set of Z is empty, then there is a c > 0 such that for s sufficiently large, λ G 1 (s) ≥ cs. Proof. This proof is a generalization of Theorem 1.1 in [23] to the equivariant setting. We identify the parameter s in our theorem with 1 h in [23]. To obtain an upper bound for the eigenvalues of H G s (or a lower bound on the spectral counting function of H G s ), we use eigensections of the model operator K G to produce test sections for H G s in the Rayleigh quotient. Suppose that ψ is an eigensection of We produce a corresponding element φ ∈ Γ (B ε,x , E ′ ) H x that has support in a small neighborhood B ε,x of x, as follows. Let γ be the unit speed geodesic from x to p ∈ B ε,x , let x p be the geodesic normal coordinates of p, and let P γ : E ′ x → E ′ p denote parallel translation along γ. We define φ (p) = P γ φ (x p ) .
Clearly, φ ∈ Γ (B ε,x , E ′ ). Because the connection on E ′ is G-equivariant, parallel translation commutes with the action of H x , and φ ∈ Γ (B ε,x , E ′ ) H x . Abusing notation, we also denote by φ the corresponding G-invariant section of T ′ ε (and thus of W ) using the isomorphism in Proposition 3.4. This specific trivialization of E ′ produces test sections that can be used as in [23] to obtain the upper bounds for the eigenvalues of H G s . We denote Φ : Γ (R m , E ′ x ) H x → Γ (W, E ′ ) G to be the trivialization φ → φ. We may extend Φ to act on the direct sum of the spaces Γ (R m , E ′ x ) H x (with fixed x in each critical orbit O x ). To obtain a lower bound on the eigenvalues of H G s (or an upper bound on the spectral counting function of H G s ), we proceed exactly as in [23]. The functions in the partition of unity are chosen so that those corresponding to neighborhoods of critical points are radial; then the partition of unity will consist of invariant functions. Next, the IMS localization formula allows us to localize to these small neighborhoods, comparing the operators Φ −1 H G s Φ and K G .
is a basic Dirac operator, and let K G be the model operator constructed above, when the singular set of Z is nonempty. Then When the singular set of Z is empty, Choose any real number r > 0, so that r is strictly less than the least positive number in the union of the spectra of K G + and K G − . Then for any s > 0 we have because D + s is an isomorphism between the eigenspaces of H + s and of H − s corresponding to nonzero eigenvalues. By choosing s sufficiently large in the formula above and applying Theorem 4.5, we obtain Remark 4.7. With the notation of Section 2.1, if Z := Z + , (Z + ) * ∈ Γ (M, End (E + ⊕ E − )) is a smooth basic bundle map that has no critical leaf closures and anticommutes with Clifford multiplication by vectors orthogonal to leaf closures, then the corollary implies that the index of the basic Dirac operator D b must be zero. This is clear for several reasons, for instance There exists c > 0 such that for sufficiently large s, s 2 Z 2 + s (ZD b + D b Z) > cs 2 1 , and the kernel is empty. Proof. Proposition 2.15 implies that there exists a basic perturbation Z that is everywhere invertible.

4.2.
Clifford form of proper perturbations. In this section, we show that in a neighborhood of a critical leaf closure, the operator Z may be continuously deformed so that it has a special form, called Clifford form, near this leaf closure.
Definition 4.9. Suppose that Z is a proper perturbation of D b . Then Z is said to be of Clifford form if near every critical leaf closure ℓ, it has the form γ ⊗ Z † (or in odd ), and in coordinates x k in a disk normal to ℓ, Remark 4.10. If Z is of Clifford form, then near ℓ each Z j anticommutes with each c (∂ k ), and the operators L j = c (∂ j ) Z j commute with each other at the origin.
In the following, we use the terminology stable homotopy of endomorphisms and zeroth order operators in K-theory sense.
. Then Z † is equivariant with respect to the action on W (or W † in the odd case with W = W † ⊕ W † ). Since H acts on U by isometries, then if U is even-dimensional, the equivariant K-theory satisfies the representation ring of H, with generators given by is Clifford multiplication by x ∈ C n/2 and ρ : H → U (V ρ ) is an irreducible unitary representation. In more generality (even in odd dimensions), K H (R n ) is generated by such triples; see [8] for the specific details. Therefore, we may stably homotope Z † in a neighborhood of the origin to a Clifford multiplication-type operator, so that Z † satisfies the conditions in the definition above. The homotopy is performed as follows for the even-dimensional case, and the odd-dimensional case is similar. We consider on a single even-dimensional disk normal U to the critical leaf closure. Restrict Z † to a sphere of radius r in the normal disk, where Z † is nonsingular. It defines an element of K H C n/2 , and thus is in the same class as [S + ⊗ V ρ , S − ⊗ V ρ , c ⊗ ρ] as above. After stabilizing, there exists an equivariant homotopy between the two endomorphisms. Let Z † (τ ) satisfy Z † (r) = Z † , Z † (τ ) = c⊗ρ for τ ≤ 1 2 r, and otherwise Z † (τ ) is the H-equivariant homotopy between the two operators for r ≥ τ ≥ 1 2 r. Then the homotopy between Z and γ ⊗ Z † is Φ t = (1 − t) Z + t γ ⊗ Z † near the critical leaf closure and is constant outside the tubular neighborhood of radius r. This homotopy introduces no additional critical leaf closures by construction.

4.3.
Local calculations and the main theorem. In this section we use Corollary 4.6 to prove the main theorem, Theorem 4.12. Given any proper perturbation Z of D b , we stably deform it to a proper perturbation in Clifford form using Proposition 4.11. We consider , with the local form of the model operator at x = 0 given by with each Z k a constant endomorphism that anticommutes with c (∂ j ) for each j, and such that Z j Z k + Z k Z j = −2δ jk , as described in the previous section. As a consequence, the Hermitian operators L j = c (∂ j ) Z j commute with each other. Let The operators L j can be diagonalized simultaneously. Let v be a common eigenvector; let λ j be the eigenvalue of the operator L j corresponding to v. Letting f be a scalar function of x, we have The section f v is in the kernel of K x if and only if each λ j is negative, and, up to a constant, The kernel of K x is the intersection of the direct sum of eigenspaces E λ (L j ) of L j corresponding to negative eigenvalues. Note that L j maps E + to itself (call the restriction L + j ), so that the The calculation above and Corollary 4.6 imply the following theorem.
where for each critical leaf closure ℓ, where each critical leaf closures ℓ of Z corresponds to a fixed point x on the basic manifold W .

Examples
The perturbations Z in this section are already in Clifford form at the critical leaf closures.
The leaf closures for |z| < 1 are two dimensional, and the closures corresponding to the poles (z = ±1) are one dimensional. In the natural metric, the foliation is Riemannian. We wish to calculate the basic Euler characteristic of this foliation by using our theorem.
be the exterior derivative restricted to basic forms, and let δ b be the L 2 -adjoint of d b . The operator D b = d b + δ b acting on even degree basic forms is called the basic de Rham operator. The basic Laplacian is ∆ b = (d b + δ b ) 2 , and its kernel consists of basic harmonic forms. Since the Hodge theorem is valid for Riemannian foliations, the standard argument shows that the index of D b is the basic Euler characteristic is the basis cohomology group (see [16], [20]).
For any bundle-like metric, the basic de Rham operator is where κ b is the mean curvature one-form and α = (α∧) * for one-forms α. Let the perturbation be Z = dz ∧ +dz . The reader may verify that Z anticommutes with the principal symbol of D b ; as explained in Section 2.2, this implies that D b Z +ZD b is zeroth order. The critical leaf closures correspond to the poles z = ±1. In the coordinates x, y near the poles, Z = xZ 1 + yZ 2 , where At the north pole z = +1, Similarly, at the south pole ( z = −1 ), Here E + = Λ even T * 0 R 2 , E − = Λ odd T * 0 R 2 , and the vector subspaces found are O (2)-invariant, so that We now directly calculate the Euler characteristic of this foliation. Since the foliation is taut, the standard Poincare duality works [14] [15] , and Example 5.2. We will compute the basic Euler characteristic of the Carrière example from [7] in the 3-dimensional case. Let A be a matrix in SL 2 (Z) with tr (A) > 2. We denote respectively by V 1 and V 2 the eigenvectors associated with the eigenvalues λ and 1 λ of A with λ > 1 irrational. Let the hyperbolic torus T 3 A be the quotient of T 2 × R by the equivalence relation which identifies (m, t) to (A(m), t + 1). The flow generated by the vector field V 1 can be made into a Riemannian foliation with a bundle-like metric g, as follows. Let (x, y, t) denote the local coordinates in the V 1 , V 2 , and R directions, respectively, and let g = λ 2t dx 2 + λ −2t dy 2 + dt 2 .
The operator Z is a proper perturbation, as in the previous example. The critical leaf closures are those points of T 3 A corresponding to t = 1 4 , 3 4 . The bundle map has the local form Z (t) = −2π t − 1 4 (dt ∧ +dt ) near t = 1 4 and Z (t) = 2π t − 3 4 (dt ∧ +dt ) near t = 3 4 on the one-dimensional orthogonal disk, and so that , and thus There is no holonomy, so in both cases H x is trivial. We compute the eigenvalues of A , F ∼ = 0, which verifies our computation. Example 5.3. (Localization when D b Z + ZD b is a first order operator.) We will examine a transverse signature operator for a one-dimensional foliation. The manifold is the suspension of a Z-action on CP 2 . Specifically, let (θ 0 , θ 1 , θ 2 ) ∈ T 3 = S 1 × S 1 × S 1 be a generator of a discrete dense subgroup of T 3 , and let φ ([z 0 , z 1 , z 2 ]) = (θ 0 , θ 1 , θ 2 ) · [z 0 , z 1 , z 2 ] = [exp (iθ 0 ) z 0 , exp (iθ 1 ) z 1 , exp (iθ 2 ) z 2 ] for all [z 0 , z 1 , z 2 ] ∈ CP 2 . The suspension is M = CP 2 × R ∼, where (z, t) ∼ (φ (z) , t + 1) for all z ∈ CP 2 , t ∈ R. The t-parameter curves form a Riemannian foliation of M with a natural bundle-like metric. Next, let D b be the transverse signature operator, the pullback of the signature operator on CP 2 via the local projections π 1 : U → CP 2 for U open in M.We may identify the bundle Q with T CP 2 . Then c : Q * → End (Λ * Q * ) is the standard Clifford action by cotangent vectors, so that the basic Dirac operator on M is the pullback the operator D b = d + d * on T 3 -invariant forms on CP 2 . The grading on the bundle E = Λ * Q * is given by the standard signature involution operator on forms on CP 2 . Next, let Z = ic (V * ), where V is the infinitesimal generator of the S 1 action t → [exp (itα 0 ) z 0 , exp (itα 1 ) z 1 , exp (itα 2 ) z 2 ] on CP 2 ; we assume that α 0 , α 1 , α 2 are linearly independent over Q. The reader may check that D b Z + ZD b is a first order differential operator but is bounded on the basic forms.