The Witten deformation on the product M ×O ∗

In this section,M is acompact oriented even-dimensional K-manifold, equipped with a K-equivariant spin^c bundleS. Let ΦS be the associated moment map onM. Our aim is to compute geometrically the multiplicities of the equivariant indexQK(M,S).

4.5.1. Vanishing theorems

LetH_k be the set of conjugacy classes of the reductive algebras k_ξ, ξ∈k^∗. We denote by S_k the set of conjugacy classes of the semi-simple parts [h,h] of the elements (h)∈H_k.

Recall that an orbit P is an (h)-ancestor of O if P belongs to the Dixmier sheet k^∗_(h) and s(P)=O. Here s(P) is defined as follows: if P=Kµ with kµ=h, then s(P)=

K(µ+o(h)) (see Definition3.5).

Recall that the mapO7!πO:=Q^spin_K (O) is a bijection between the regular admissible orbits and K. Ifb O is a regular admissible orbit, then O^∗:=−O is also admissible and πO^∗=(πO)^∗. If we apply the shifting trick, we see that the multiplicity ofπOinQK(M,S) is equal to

mO= [QK(M,S)⊗(πO)^∗]^K= [QK(M×O^∗,S ⊗SO^∗)]^K. (4.26) Let (k_M) be the generic infinitesimal stabilizer of theK-action onM. In this section, we prove the following vanishing results.

Theorem 4.19. • If ([k_M,k_M])6=([h,h]) for any (h)∈Hk, then QK(M,S) = 0

for any K-equivariant spin^c-bundle S on M.

• Assume that ([kM,kM])=([h,h])for (h)∈Hk. Then m_O= 0

if there is no (h)-ancestor P to O contained in Φ_S(M).

We consider the product M×O^∗ equipped with the spin^c-bundle S ⊗SO^∗. The corresponding moment map is Φ_S⊗SO∗(m, ξ)=Φ_S(m)+ξ. We use the simplified notation Φ_O for Φ_S⊗SO∗,O for the corresponding Kirwan vector field onM×O^∗,Z_O:={O=0}, andd_O for the functiond_S⊗SO∗ onZ_O. Theorem4.19will result from a careful analysis of the function d_O:Z_O!R that was introduced in §4.4. Due to Proposition 4.17 and Corollary4.18, Theorem4.19is a direct consequence of the following theorem.

Theorem 4.20. Let Obe a regular admissible orbit.

• The function d_O is non negative on Z_O.

• If the functiond_O is not strictly positive, then there exists a unique (h)∈Hk such that the following conditions are satisfied:

(1) ([k_M,k_M])=([h,h]).

(2) the orbit O has an(h)-ancestor P contained in Φ_S(M).

Proof. LetP=M×O^∗ and let us compute the function d_O onZ_O. Letm∈M and λ∈O. The pointp=(m,−λ)∈Z_O⊂P if and only Φ_O(p)·p=0. Letβ=Φ_O(p). This means thatβ stabilizesmandλ, and ifµ=Φ_S(m)∈k^∗, thenβ=µ−λ.

We write T_(m,−λ)P=T_mM⊕T_−λO^∗ and, since O^∗ is a regular orbit, we have nTr_T−λO^∗|β|=nTr_k|β|.

We consider aKm-invariant decomposition TmM=k·m⊕Emwhere k·m'k/km, we obtainnTrT_mM|β|=nTrE_m|β|+nTrk|β|−nTrk_m|β|. Thus,

d_O(p) =kβk²+¹₂nTr_T(m,−λ)P|β|−nTrk|β|=kβk²+¹₂nTr_TmM|β|−¹₂nTr_k|β|

=kβk²+¹₂nTr_Em|β|−¹₂nTr_km|β|>kβk²+¹₂nTr_Em|β|−¹₂nTr_kµ|β|.

(4.27) In the last inequality, we used km⊂kµ as µ=Φ_S(m). By Proposition 3.14, we have kβk²−¹₂nTrk_µ|β|>0 whenβ=µ−λ, asλis very regular (being regular and admissible), andβ∈kµ∩kλ. Then the first point follows.

Assume now that there exists a pointp=(m,−λ)∈Z_O such thatd_O(p)=0. It implies then thatkβk²=¹₂nTrk_µ|β|andnTrE_m|β|=0. The first equality implies, due to Proposi-tion3.14, thatKµis an admissible orbit such thats(Kµ)=O, and that−β∈o(h)⊂[h,h]^∗. We write −β=%^H. Now we have to explain why the condition nTrE_m|%^H|=0 implies ([kM,kM])=([h,h]). Since Φ_S(m)=µ, we have

(kM)⊂(km)⊂(h). (4.28)

Consider theH-invariant sliceY=Φ⁻¹_S (Uµ) constructed in Proposition3.16. The product KY is an invariant neighborhood of m isomorphic to K×HY. The subspace Em can be taken as the subspace T_mY⊂TmM. Now the conditionnTr_Em|%^H|=0 implies that

%^Hacts trivially on the connected componentY_mofY containingm. ElementsX∈[h,h]

such thatX_Ym=0 form an ideal in [h,h]. Since the ideal generated by%^Hin [h,h] is equal to [h,h], we have proved that [h,h] acts trivially on Y_m. As KY_m is an open subset of M, we get

([h,h])⊂(k_M). (4.29)

With (4.28) and (4.29) we get ([kM,kM])=([h,h]). Finally we have proven that if dO

vanishes at some pointp, then ([k_M,kM])=([h,h]) for some (h)∈Hk, and there exists an admissible orbitKµ⊂k^∗_(h)∩ΦS(M) such thats(Kµ)=O.

4.5.2. Geometric properties

We summarize here some of the geometric properties enjoyed by (M,Φ), with Φ=ΦS, whenQK(M,S) is not zero.

Let (h)∈Hk. We choose a representative h. LetH be the corresponding group and NK(H) be the normalizer ofH in K. Consider the decompositionh=[h,h]⊕z, wherez is the center ofh. Thusz^∗⊂h^∗. Consider the open set

h^∗₀={ξ∈h^∗:kξ⊂h}

ofh^∗. Letz^∗₀=h^∗₀∩z^∗ be the corresponding open subset ofz^∗. We first note the following basic proposition.

Proposition 4.21. Let M be a K-manifold such that ([kM,kM])=([h,h]) and let Φ:M!k^∗ be an equivariant map. Then

• Φ(M)⊂Kz^∗.

• Assume that Y:=Φ⁻¹(h^∗₀)is non-empty;then

(a) Y is a submanifold of M invariant by the action of NK(H), and [H, H] acts trivially on Y.

(b) The image Φ(Y)is contained in z^∗₀.

(c) The open subset KY is diffeomorphic to K×N_K(H)Y.

Proof. Let us prove the first item. Using ourK-invariant inner product, we consider Φ as a map Φ:M!k. The condition on the infinitesimal stabilizer (kM) gives thatM= KM^[H,H]. Ifm∈M^[H,H], the term Φ(m) belongs to the Lie algebra gof the centralizer subgroupG:=ZK([H, H]). But one can easily prove thatzis a Cartan subalgebra ofg:

hence Φ(m) is conjugated to an element ofz. This proves the first item.

If Y is non-empty, the proof that it is a submanifold follows the same line than the proof of Proposition3.16. The setKY is a non-empty open set inM: so onY we have (kM)=(ky) on a dense open subsetY⁰. The condition ([kM,kM])=([h,h]) implies that dim[h,h]=dim[ky,ky] onY⁰, but asky⊂k_Φ(y)⊂h, we conclude that [h,h]=[ky,ky]⊂kyonY⁰: in other words [H, H] acts trivially onY, and [h,h]=[ky,ky] for anyy∈Y. Furthermore, ifξ=Φ(y), then [h,h] acts trivially onξ. Soξis in the center ofh.

Let us prove that π:K×_NK(H)Y!KY is one-to-one. If y1=ky2, we have ξ1=kξ2

withξi=Φ(yi),i=1,2. As Φ(Y)⊂z^∗₀, the stabilizers of ξ1 andξ2 are both equal toH. It follows thatkbelongs to the normalizer ofH.

The following theorem results directly from Theorem 4.20 and Proposition 4.21.

Indeed, in the case where QK(M,S)6={0}, then ([kM,kM])=([h,h]) for some (h)∈Hk. Furthermore, there exists at least a regular admissible orbitOsuch thatm_O is non-zero, and consequently there exists orbitP ⊂k^∗_(h)∩Φ_S(M).

Theorem 4.22. Let M be a K-manifold and let S be an equivariant spin^c-bundle on M with moment mapΦ_S. Assume that QK(M,S)6={0}. Then

(1) There exists (h)∈Hk such that ([k_M,k_M])=([h,h]).

(2) If zis the center of h, then Φ_S(M)⊂Kz^∗ and the open set Φ⁻¹_S (Kz^∗₀)is non-empty.

(3) The group [H, H]acts trivially on the submanifold Y=Φ⁻¹_S (z^∗₀).

This condition (1) on the K-action is always satisfied in the Hamiltonian setting [21], but not always in the spin setting as can be seen in the following example.

Example 4.23. Forn>2, the sphereSⁿ admits a unique spin structure, equivariant under the action of the groupK:=spin(n+1). The generic stabilizer for theK-action is

isomorphic to the groupKM:=spin(n), and we see that (kM)=([kM,kM]) is not equal to ([h,h]), for any (h)∈Hk.

In the next sections, we will restrict the submanifoldY to a connected component C of h^∗₀. We work with the H-invariant submanifold YC:=Φ⁻¹_S (C)⊂Y: here the open subsetKYC is diffeomorphic toK×HYC.

We follow the notations of §3.3. We denote by q^C the vector space k/h equipped with the complex structure J_C. There exists a uniqueH-equivariant spin^c-bundleS_YC onY_C such that

S|_YC'^

q^C⊗S_YC. (4.30)

At the level of determinant line bundles we have det(SYC)=det(S)|YC⊗C−2%C, and the corresponding moment map satisfies the relation Φ_YC=Φ_S|YC−%C.

We know already that the subgroup [H, H] acts trivially on the submanifoldYC(see Theorem4.22). It acts also trivially on the bundleS_YC since the moment map Φ_YC takes value inz^∗ (see Remark2.8).

4.5.3. Localization on Z_O⁼⁰

LetObe a regular admissible orbit. By Theorem4.20and Corollary4.18, we know that our object of study

m_O= [QK(M×O^∗,S ⊗S_O^∗)]^K is equal to [QK(M×O^∗,S ⊗SO^∗, Z_O⁼⁰,Φ_O)]^K.

Let us give a description of the subsetZ_O⁼⁰ofZ_O⊂M×O^∗, whered_O vanishes. We denote byq:M×O^∗!k^∗⊕k^∗ the map given byq(m, ξ)=(Φ_S(m),−ξ). If µbelongs to a coadjoint orbitP, andξ∈µ+o(kµ), thenP is an ancestor to the orbitOofξ.

Definition 4.24. LetP be a coadjoint orbit.

• Define the following subset ofk^∗⊕k^∗:

R(P) ={(µ, ξ) :µ∈ P andξ∈µ+o(kµ)}.

• DefineZ_O^P=q⁻¹(R(P))⊂M×O^∗.

Proposition 4.25. Assume that M is a K-manifold with ([k_M,kM])=([h,h]). Let S be a K-equivariant spin^c-bundle over M with moment map Φ_S. Let O be a regular admissible coadjoint. Then

Z_O⁼⁰=G

P

Z_O^P,

where the disjoint union is over the set of (h)-ancestors to O. Furthermore, if P=Kµ a (h)-ancestor to O, the set Z_O^P is equal to

K(Φ⁻¹_S (µ)×{−µ+o(kµ)^∗}). (4.31) Proof. In the proof of Theorem4.20, we have seen that, ifd_O(m,−λ)=0, then the element µ=Φ_S(m) is such that (k_µ)=(h) and λ=β+µ with β∈o(k_µ). So Kµ is an (h) ancestor ofOandq(m,−λ)∈F

PZ_O^P. This proves the inclusionZ_O⁼⁰⊂F

PZ_O^P.

Conversely, take now (m,−ξ)∈Z_O^P and define µ=Φ_S(m). SoKµis an (h) ancestor ofOandξ=µ+β withβ∈o(k_µ). ByK-invariance, we may assume thatµ∈z^∗₀, som∈Y.

We haveT_mM=k/k_m⊕T_mY. So

d_O(m,−ξ) =kβk²−¹₂nTr_km|β|+¹₂nTr_TmY|β|.

Sinceβ∈o(h)⊂[h,h] acts trivially onY by Proposition4.21, we have d_O(m,−ξ) =k%^Hk²−1

2nTrk_m|%^H|.

But as [h,h]⊂km⊂h, and then ¹₂nTr_km|%^H|=¹₂nTr_h|%^H|=k%^Hk², finally d_O(m,−ξ)=0.

If Kµis an ancestor of the regular admissible orbitO, thenµ+o(k_µ) is aK_µ-orbit contained inO. It follows that the set

Z_O^P:={(m, ξ)∈M×O^∗: ΦS(m)∈ P and −ξ∈ΦS(m)+o(kΦ_S(m))}

is equal to (4.31).

At this stage we have proved that

mO=X

P

m^P_O, (4.32)

where the sum runs over the (h)-ancestor ofOand

m^P_O= [QK(M×O^∗,S ⊗S_O^∗, Z_O^P,Φ_O)]^K.

In the next section we will go into the computation of the terms m^P_O. We finish this section with the following important fact.

Proposition 4.26. Each individual term m^P_O is independent of the choice of the moment map Φ_S.

Proof. Let Φ^t_S, t∈[0,1] be a continuous family of moment maps for S. This gives a family Φ^t_O(m, ξ):=Φ^t_S(m)+ξforS ⊗S_O^∗.

Let ^t_O be the Kirwan vector field associated with Φ^t_O, and let Z_O(t):={_O^t=0}.

Recall thatZ_O⁼⁰(t)=F

PZ_O^P(t), where the union runs over the (h)-ancestors ofO, and Z_O^P(t) =K((Φ^t_S)⁻¹(µ)×{−µ+o(kµ)^∗}),

ifP=Kµ.

We denote simply byσ^t the symbolσ(M×O^∗,S ⊗S_O^∗,Φ^t_O). For any t∈[0,1], we consider the quantityQ^P_O(t)∈R(K) which is the equivariant index ofb σ^t|U_t, whereUt is a (small) neighborhood ofZ_O^P(t) such thatUt∩Z_O(t)=Z_O^P(t).

Let us prove that the multiplicity m^P_O(t)=[Q^P_O(t)]^K is independent oft. It is suffi-cient to prove that t7![Q^P_O(t)]^K is locally constant: let us show that it is constant in a neighborhood of 0. We follow the same line of proof that the proof of the independence of the connection of the local pieceQK(M,S,Φ⁻¹_S (O)∩ZS) ofQK(M,S) in [28].

LetU0be a neighborhood ofZ_O^P(0) such that

U₀∩Z_O(0) =Z_O^P(0). (4.33)

The vector fieldO⁰ does not vanish on∂U₀: there existsε>0 so thatO^t does not vanish on∂U₀fort∈[0, ε]. The familyσ^t|U0,t∈[0, ε] is then a homotopy of transversally elliptic symbols: hence they have the same equivariant index.

Lemma4.27. For small t,we have

U0∩Z_O⁼⁰(t) =Z_O^P(t).

Indeed, by Proposition 4.25, Z_O⁼⁰(t) projects by the first projection Φ^t_S:M×O^∗! M!k^∗ to a finite union of coadjoint orbits (the (h)-ancestors toO), andZ_O^P(0) projects onP. So, for smallt,U0∩Z_O⁼⁰(t) is the subset Z_O^P(t) ofZ_O⁼⁰(t) projecting onP.

So, for small t, we have the decomposition U0∩Z_O(t)=Z_O^P(t)∪Zt, where Zt is a component contained inZ_O^>0(t). Finally, for smallt, we have

Q^P_O(0) = Index_K(σ⁰|U0) = Index_K(σ^t|U0) =Q^P_O(t)+QK(M×O^∗,S ⊗SO^∗, Z_t,Φ^t_O).

As [QK(M×O^∗,S ⊗SO^∗, Zt,Φ^t_O)]^K=0 by Proposition4.17, the proof of Proposition4.26 is completed.

4.5.4. Computation of m^P_O In this section we compute

m^P_O:= [QK(M×O^∗,S ⊗S_O^∗, Z_O^P,Φ_O)]^K for an (h)-ancestorP ofO.

Let C be a connected component of h^∗₀ that intersects the orbitP. With the help of Proposition 4.15, we will reduce the computation of m^P_O to a similar computation where the groupK acting onM is replaced with the torusA_H=H/[H, H] acting on the sliceY_C.

Let us recall the geometric property of the orbits P and O that we need for our computation. Letµ∈P ∩C be such that P=Kµ with Kµ=H. Let λ∈µ+o(h). Since P is an ancestor of the regular admissible orbit O, we have O=Kλ and the weight λ−%(λ)=µ−%(µ)∈z^∗ defines a character ofH (see Lemma 3.8).

Then Z_O^P is equal to K(Φ⁻¹_S (µ)×{−µ+o(h)^∗}). Here o(h) is the%-orbit for H, so o(h)=o(h)^∗ and Q^spin_H (o(h)^∗) is the trivial representation ofH.

Let YC=Φ⁻¹_S (C) be the slice relative to the connected component C (see §4.5.2).

ThusKYC is an open neighborhood of Φ⁻¹_S (P) in M diffeomorphic withK×HYC. We see that

Z_O^P⊂(K×HYC)×O^∗'^j K×H(YC×O^∗), where the isomorphismj is defined by ([k;y],[k⁰])7![k; (y,[k⁻¹k⁰])].

We consider theH-manifoldN⁰:=YC×o(h)^∗ and theK-manifold N=K×H(B_q×N⁰) =K×H(B_q×YC×o(h)^∗), whereBq is a small open ball inq, centered at 0 and H-invariant.

Consider the map Ψ: (X, ξ)7!exp(X)(−µ+ξ), fromq×o(h)^∗ into O^∗. For any ξ∈

o(h)^∗, its differential at (0, ξ) is the isomorphism (X, Y)7!−µad(X)−ξad(Y) from q⊕h/hξ intok/kξ. Hence, whenBqis small enough, the map Ψ defines a diffeomorphism betweenBq×o(h)^∗ into anH-invariant neighborhood of theH-orbit−µ+o(h)^∗.

Hence a K-invariant neighborhood of Z_O^P inM×O^∗ is diffeomorphic to N. Under this isomorphism, the equivariant map ΦO=ΦS+iO^∗ defines a map Φ onN. Fork∈K, X∈Bq,y∈YC andξ∈o(h)^∗, we have

Φ([k;X, y, ξ]) =k(Φ_S(y)+exp(X)(−µ+ξ)).

Under this isomorphism, the setZ_O^P becomes

Z:=K×_H({0_q}×Φ⁻¹_S (µ)×o(h)^∗)⊂Z_Φ.

The map Φ:N!k^∗, when restricted to N⁰, is the H-equivariant map Φ⁰:N⁰!h^∗ defined by Φ⁰(y, ξ)=Φ_S(y)−µ+ξ. Furthermore, if Bq is small enough, Φ([1;X, y, ξ]) belongs toh^∗if and onlyX=0. AsX∈q, we also see that

(Φ([1;X, y, ξ]), X) = (Φ_S(y), X)+(exp(X)(−µ+ξ), X) = (Φ_S(y)−µ+ξ, X) = 0 for all (X, y, ξ)∈Bq×YC×o(h)^∗. Conditions (4.23) are satisfied. Then Proposition4.15 tells us that and the corresponding moment map is Φ^P_Y

C:=Φ_S|_YC−µ.

sinceλ−%(λ)=µ−%(µ)∈z^∗ defines a character ofH (see Lemma3.8).

As the spinor bundleSq is equal to the productV q^C⊗V

We consider now theH-manifoldYC equipped with the spin^c-bundleS_Y^P_C. Let Q_H(Y_C,S_Y^P

C,{0})∈R(Hb ) (4.34)

be the equivariant index localized on the compact component{Φ^P_Y

C=0}={Φ_S=µ}⊂YC. LetAH be the torus H/[H, H]. Since [H, H] acts trivially on (YC,S_Y^P

C), we may also define the localized indexQA_H(YC,S_Y^P

C,{0})∈R(Ab H).

We can now prove the main result of this section.

Theorem 4.29. The multiplicity m^P_O is equal to

[QH(YC,S_Y^P_C,{0})]^H= [QA_H(YC,S_Y^P_C,{0})]^AH.

Proof. LetZ⁰:=Φ⁻¹_S (µ)×o(h)^∗. The characterQ_H(N⁰,S⁰, Z⁰,Φ⁰)∈R(Hb ) is equal to the equivariant index of σ(N⁰,S⁰,Φ⁰)|_U, where U ⊂N⁰ is an invariant open subset such that U ∩Z_Φ⁰=Z⁰. For (y, ξ)∈N⁰=Y_C×o(h)^∗ and (v, η)∈T_(y,ξ)N⁰, the endomorphism σ(N⁰,S⁰,Φ⁰)|_(y,ξ)(v, η) is equal to

c1(v+(ΦS(y)−µ+ξ)·y)⊗Id_So(h)∗|ξ+ε1⊗c2(η+(ΦS(y)−µ+ξ)·ξ).

Here c1 acts on S_Y^P

C|y, c2 acts on S_o(h)∗|ξ and ε1 is the canonical grading operator on S_Y^P

C|y.

Since o(h)^∗ is compact, we can replace the termc2(η+(Φ_S(y)−µ+ξ)·ξ) simply by c2(η). As [H, H] acts trivially on YC, andξ∈[h,h], the vector fieldy7!(Φ_S(y)−µ+ξ)·y is equal toy7!(Φ_S(y)−µ)·y. Thus our symbol is homotopic to the symbol

c₁(v+(Φ_S(y)−µ)·y)⊗IdS_o(h)|ξ+ε₁⊗c2(η).

This last expression is the product symbol of the H-transversally elliptic symbol c1(v+(Φ_S(y)−µ)·y) on YC and of the elliptic symbol c2(η) on o(h)^∗. The equivariant indices multiply under the product (as one is elliptic) ([1],[27]).

Now the H-equivariant index of c2(η) acting on S_o(h)^∗ is the trivial representa-tion of H. Thus we obtain our theorem. We also have to remark that the identity [QH(YC,S_Y^P

C,{0})]^H=[QA_H(YC,S_Y^P

C,{0})]^AHfollows from the fact that [H, H] acts triv-ially on (YC,S_Y^P

C).

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