Proof of Theorem 5.4

In this subsection we work with a fixedµ∈I(N)∩Λ. For anyε∈g(N)^⊥, we consider the moment map Φ_ε=Φ−µ−ε. We denote by andε the Kirwan vector fields attached to the moment maps Φ₀ and Φ_ε, respectively.

We start with the fundamental lemma.

Lemma5.7. The map ε7![Q_G(N,S,Φ⁻¹_ε (0),Φ_ε)⊗C−µ]^G is constant in a neighbor-hood of zero.

Proof. ChangingS toS ⊗[C−µ], we might as well takeµ=0.

Letr>0 be smallest non-zero critical value ofkΦk², and letU:=Φ⁻¹

ξkξk<¹₂r . Using Remark2.11, we haveU ∩{0=0}=Φ⁻¹(0).

We describe now{ε=0}∩U using a parametrization similar to those introduced in [25,§6].

Letgi, i∈I, be the finite collection of infinitesimal stabilizers for theG-action on the compact set U. LetD be the subset of the collection of subspaces g^⊥_i of g^∗ such that Φ⁻¹(0)∩N^gi6=∅.

Note that D is reduced to I(N) if 0 is a regular value of Φ:N!I(N). If ∆=g^⊥_i belongs toD, andε∈I(N), write the orthogonal decompositionε=ε∆+β∆withε∆∈∆, andβ∆∈gi. Let

Bε={β∆=ε−ε∆: ∆∈ D}

be the set ofβ so obtained.

ε

Figure 7. The pointεand its projectionsε∆.

We denote byZ_ε the zero set of the vector fieldεassociated with Φ_ε. Thus, ifεis sufficiently small (kεk<¹₂r),

Zε∩U= [

β∈B_ε

N^β∩Φ⁻¹_ε (β). (5.37)

With (5.37) in hands, we easily see that t∈[0,1]7!σ(N,S,Φtε)|_U is a homotopy of transversally elliptic symbols onU. Hence they have the same index

QG(U,S,Φ⁻¹(0),Φ) =QG(U,S, Zε∩U,Φ_ε) = X

β∈Bε

QG(N,S,Φ⁻¹_ε (β)∩N^β,Φ_ε).

The lemma will be proved if we check that [QG(N,S,Φ⁻¹_ε (β)X∩N^β,Φ_ε)]^G=0 for any non-zeroβ∈Bε.

We fix now a non-zero β:=β∆∈Bε. The Atiyah–Segal localization formula for the Witten deformation (Remark4.13) gives

QG(N,S,Φ⁻¹_ε (β)∩N^β,Φε) =QG(N^β,dβ(S)⊗Sym(Vβ),Φ⁻¹_ε (β),Φε)

= X

X ⊂N^β

QG(X,dβ(S)|_X⊗Sym(Vβ)|_X,Φ⁻¹_ε (β),Φε),

whereVβ!N^β is the normal bundle of N^β in N and the sum runs over the connected componentsX ofN^βthat intersect Φ⁻¹_ε (β). Like in the proof of Proposition4.17, we will show that the term [QG(N,S,Φ⁻¹_ε (β)∩N^β,Φ_ε)]^Gvanishes by looking to the infinitesimal action ofβ, denoted byL(β), on the fibers of the vector bundle d_β(S)|X⊗Sym(Nβ)|X.

Letn∈Φ⁻¹_ε (β)∩X: we have Φ(n)=β+ε=ε∆, and thereforehΦ(n), βi=hε∆, β∆i=0.

Lemma4.11tell us that L(β)|_db(S)|X=iaId_db(S)|X, where

a=hΦ(n), βi+¹₂nTr_TnN|β|=¹₂nTr_TN|X|β|.

We obtain

1 iL(β)

=¹₂Tr_TN|X(|β|), ondβ(S)|_X,

>0, on Sym(Nβ)|_X.

So we have checked thatL(β)/i>¹2Tr_TN|X(|β|) ondβ(S)|_X⊗Sym(Nβ)|_X.

Now we remark thatβ does not act trivially on N, sinceβ belongs to the direction of the subspace I(N)=g^⊥_N: this forces Tr_TN|X(|β|) to be strictly positive. Finally, we see thatL(β)/i>0 on dβ(S)|_X⊗Sym(Nβ)|_X, and then

[QG(X,dβ(S)|_X⊗Sym(Vβ)|_X,Φ⁻¹_ε (β),Φε)]^G= 0.

The Lemma5.7is proved.

The proof of Theorem5.4will be completed with the following lemma.

Lemma5.8. If µ+εis a regular value of Φ:N!I(N), the invariant [QG(N,S,Φ⁻¹_ε (0),Φε)⊗C−µ]^G

is equal to the index Q(Nµ+ε,S_µ+ε^µ ).

We assume thatµ+εis a regular value of Φ:N!I(N): the fiberZ=Φ⁻¹(µ+ε) is a submanifold equipped with a locally free action ofG⁰=G/G_N. LetN_µ+ε:=Z/G⁰ be the corresponding “reduced” space, and letπ:Z!N_µ+εbe the projection map. We have the decomposition

TN|_Z'π^∗(TN_µ+ε)⊕[g⁰_C]. (5.38) For anyν∈Λ∩I(N),S_µ+ε^ν is the spinor bundle onNµ+εdefined by the relation

S|_Z⊗C−ν'π^∗(S_µ+ε^ν )⊗h ^ g⁰_Ci

.

The following result is proved in [28,§2.3].

Proposition 5.9. We have the following equality in R(G):b QG(N,S,Φ⁻¹_ε (0),Φε) = X

ν∈Λ∩I(N)

Q(Nµ+ε,S_µ+ε^ν )Cν.

In particular [QG(N,S,Φ⁻¹_ε (0),Φ_ε)⊗C−µ]^G is equal to Q(Nµ+ε,S_µ+ε^µ ).

5.3. [Q, R]=0

We come back to the setting of a compactK-manifoldM, oriented and of even dimension, that is equipped with aK-spinor bundleS. Let det(S) be its determinant bundle, and let Φ_S be the moment map that is attached to an invariant connection on det(S). We assume that there exists (h)∈H_ksuch that ([kM,kM])=([h,h]). Letzbe the center ofh.

We consider an admissible element µ∈z^∗ such that Kµ=H: the coadjoint orbit P:=Kµis admissible and contained in the Dixmier sheetk^∗_(h). Let

MP:= Φ⁻¹_S (P)/K.

In order to define Q^spin(M_P)∈Z, we proceed as follows.

We follow here the notation of§3.3. LetCbe the connected component of h^∗₀:={ξ∈h^∗:Kξ⊂H}

containingµ. The slice Y_C=Φ⁻¹_S (C) is an H-submanifold of M equipped with an H -spin^c bundleS_YC: the associated moment map is Φ_YC:=Φ_S|_YC−%_C, where%_C is defined by (3.15).

The element ˜µ:=µ−%(µ)=µ−%C belongs to the weight lattice Λ of the torusAH:=

H/[H, H], and the reduced spaceMKµ is equal to (YC)µ˜:={ΦY_C= ˜µ}/AH.

By definition, we take Q^spin(MKµ):=Q^spin((YC)µ˜), where the last term is computed as explained in the previous section. More precisely, let us decompose YC into its con-nected componentsY1, ...,Yr. For eachj, letzj⊂zbe the generic infinitesimal stabilizer relative to theAH-action onYj. Then we take

Q^spin(MP) = Q^spin((YC)µ˜) :=X

j

Q^spin((Yj)µ+ε˜ j), (5.39) whereε_j∈z^⊥_j are generic and small enough.

With this definition of quantization of reduced spaces Q^spin(M_P), we obtain the main theorem of this article, inspired by the [Q, R]=0 theorem of Meinrenken–Sjamaar.

LetM be aK-manifold andSbe aK-equivariant spin^c-bundle overM. Let (h)∈Hk

such that ([k_M,k_M])=([h,h]), and consider the setA((h)) of admissible orbits contained in the Dixmier sheetk^∗_(h).

Theorem 5.10.

QK(M,S) = X

P∈A((h))

Q^spin(MP) Q^spin_K (P). (5.40)

We end this section by giving yet another criterium for the vanishing ofQK(M,S).

Consider the map Φ_S:M!k^∗. At each pointm∈M, the differentialdmΦ_S defines a linear map TmM!k^∗. Letk^⊥_m⊂k^∗. From the Kostant relations, we see thatdmΦ_S takes value ink^⊥_m.

Proposition5.11. • Suppose that the group Kis abelian and that (kM)=0. Then QK(M,S)6=0 only if the image of ΦS contains an element of the weight lattice in its interior.

• For a general group action, we have QK(M,S)6=0only if the subset CM:={m∈M: Image(dmΦ_S) =k^⊥_m}

has non-empty interior.

Proof. If the group K is abelian, and (kM)=0, then the affine subspace I(M) is equal tok^∗. The first point follows then from Remark5.6.

Let us prove the second point. IfQK(M,S)6=0, then Q^spin(M_P)6=0 for an admis-sible orbitP=Kµof type (h). If we consider the decomposition of the sliceYC=S

jYj

in connected components, then, for some j, Q^spin((Yj)µ+ε˜ _j)6=0 (see (5.39)). Thus, due to Remark5.6, we know that the image of Φ_Yj has non-empty interior inz^⊥_j, and there-fore {y∈Yj:Image(dyΦ_Yj)=z^⊥_j} has non-empty interior in Yj. By definition, zj is the infinitesimal stabilizer of the action ofH/[H, H] onYj. Thus,z^⊥_j ⊂z^∗ is equal to k^⊥_y for genericy∈Yj. It follows that C_Yj:={y∈Yj:Image(dyΦ_Yj)=k^⊥_y} has non-empty interior inYj. Finally, the setKCY_j⊂CM has non-empty interior inM.

Corollary 5.12. If the K-action is non-trivial,we have QK(M,S)=0if (a) ΦS is a constant map,or

(b) the 2-form ΩS is exact.

Proof. If the K-action is non-trivial and ΦS is constant, the set CM=M^K is a closed submanifold of M with empty interior. Due to Proposition 5.11, we must have QK(M,S)=0.

If ΩS=dα, by modifying the connection on det(S) byα, our moment map is constant, and we conclude thatQK(M,S)=0 by the previous case.

Note that Corollary5.12implies the well-known Atiyah–Hirzebruch vanishing theo-rem in the spin case [3], as well as the variant of Hattori [15].

6. Examples: multiplicities and reduced spaces

In this last subsection, we give some simple examples in order to illustrate various features of our result relating multiplicities and reduced spaces. An open question remains even

for a toric manifoldM equipped with a non ample line bundleL. The determinant line bundle of the spinor bundle S:=V

CTM⊗L is equal to det(S):=det_C(TM)⊗L^⊗2. A connection∇ on det(S) determines a moment map Φ_∇:M!t^∗ and a curvature

Ω_∇= 1 2i∇².

The push-forward of Ω_∇ by Φ_∇ does not depend on the choice of the connection:

it is a signed measure, denoted by DH(M,S), and called the Duistermaat–Heckmann measure. The support of DH(M,S), which is contained in Φ_∇(M), is a union of convex polytopes. Can we find ∇ such that the image Φ_∇(M) is exactly the support of the Duistermaat–Heckman measure?

6.1. The reduced space might not be connected

We consider the simplest case of the theory. LetP¹:=P¹(C) be the projective space of (complex) dimension 1. Consider the (ample) line bundleL!P¹, dual of the tautological bundle. It is obtained as quotient of the trivial line bundleC²\{(0,0)}×ConC²\{(0,0)} Note that the holomorphic line bundleL^⊗n is not ample if n60. We have

• QT(M,S(n))=−P−1 line bundle equipped with the representationt⁻¹ onC.

Remark thatP¹is homogeneous underU(2), so there exists a uniqueU(2)-invariant connection onLn. The corresponding moment map Φ_S(n)is such that

Φ_S(n)([z1, z2]) = (n+1) |z1|²

It is in agreement with our theorem. Indeed all characters occurring inQ_T(M,S(n)) are the integral points in the relative interior ofI_n, and all reduced spaces are points.

−2

−1 0 1 2 3 4

0.2 0.4 0.6 0.8 1

Figure 8. The graph of Φ.

If we consider simply the action ofT onP¹, the choice of connection may vary. In fact, given any smooth functionf onR, we can define a connection∇^f onLn such that the corresponding moment map is

Φ^f_S(n)([z1, z2]) = Φ_S(n)([z1, z2])+f

|z1|²

|z1|²+|z2|²

|z1|²|z2|² (|z1|²+|z2|²)².

Let Ω^f_S(n) be one half of the curvature of (Ln,∇^f); then the Duistermaat–Heckman measure (Φ^f_S(n))_∗Ω^f_S(n)is independent of the choice of the connection∇^f and is equal to the characteristic function ofIn.

Take, for example, n=4 and the constant function f(x)=−15. the corresponding moment map is

Φ([z1, z2]) = 5 |z1|²

|z₁|²+|z₂|²−15 |z1|²|z2|² (|z₁|²+|z₂|²)²−1

2. Figure8is the graph of Φ in terms of

x= |z1|²

|z₁|²+|z₂|²,

varying between 0 and 1. We see that the image of Φ is the interval

−¹³₆,⁹₂

, but the image of the signed measure is still

−¹₂,⁹₂

: so for this choice of connection the image of Φ is larger than the support of the Duistermaat–Heckman measure.

Above the integral points in

−¹³₆,−¹₂

, the reduced space is not connected, it con-sists of two points giving opposite contributions to the index. So our theorem holds.

6.2. The image of the moment map might be non-convex

LetM be the Hirzebruch surface. Represent M as the quotient of U=(C²\{(0,0)})×

(C²\{(0,0)}) by the free action ofC^∗×C^∗ acting by

(u, v)·(z₁, z₂, z₃, z₄) = (uz₁, uz₂, uvz₃, vz₄),

and we denote by [z1, z2, z3, z4]∈M the equivalence class of (z1, z2, z3, z4). The map π: [z1, z2, z3, z4]7![z1, z2] is a fibration ofM onP1(C) with fiberP1(C).

Consider the line bundle L(n1, n2) obtained as quotient of the trivial line bundle U ×ConU by the action

(u, v)·(z1, z2, z3, z4, z) = (uz1, uz2, uvz3, vz4, uⁿ¹vⁿ²z)

for (u, v)∈C^∗×C^∗. The line bundleL(n1, n2) is ample if and only ifn1>n2>0.

We have a canonical action of the group K:=U(2) onM: g·[Z1, Z2]=[gZ1, Z2] for Z1, Z2∈C²\{(0,0)} and the line bundle L(n1, n2) with action g·[Z1, Z2, z]=[gZ1, Z2, z]

isK-equivariant.

We are interested in the (virtual)K-module

H⁰(M,O(n1, n2))−H¹(M,O(n1, n2))+H²(M,O(n1, n2)), whereO(n1, n2) be the sheaf of holomorphic sections ofL(n1, n2).

In this case, it is in fact possible to compute directly individual cohomology groups Hⁱ(M,O(n1, n2)). However, we will describe here only results on the alternate sum and relate them to the moment map.

LetT=U(1)×U(1) be the maximal torus ofK. The set Y :={[z1, z2, z3, z4]∈M:z1= 0}

is aT-invariant complex submanifold ofM (with trivial action of (t1,1)). The map Y −!P¹(C),

[0, z2, z3, z4]7−![z⁻¹₂ z3, z4], is aT-equivariant isomorphism and the map

K×Y−!M, (g, y)7−!g·y,

factorizes through an isomorphismK×_TY'M. ThusM is an induced manifold.

For any (a, b)∈Z², we denote by Ca,b the 1-dimensional representation ofT asso-ciated with the character (t1, t2)7!t^a₁t^b₂. We denote by {e^∗₁, e^∗₂} the canonical basis of t^∗'R². The Weyl chamber ist^∗_>0={xe^∗₁+ye^∗₂:x>y}. The elementse^∗₁ ande^∗₂ are conju-gated by the Weyl group.

Note that the line bundleL(n₁, n₂), when restricted toY'P¹(C), is isomorphic to L^⊗n2⊗[C0,−n1].

We consider the line bundleL=L(3,2) obtained from the reduction of the trivial line bundleV4

C⁴ with natural action ofC^∗×C^∗. We denoteSM:=V

CTM (resp.SY:=

V

CTY) the spin^c-bundle associated with the complex structure onM (resp.Y).

We denote byϕ:Y![0,1] the map defined by ϕ(y) = |a₁|²

|a1|²+|a2|² ify'[a₁, a₂].

Proposition 6.1. • Let S(n₁, n₂) be the spin bundle S_M⊗L(n₁, n₂) on M. Its determinant line bundle is

Ln₁,n₂= [Cdet]⊗L⊗L(2n1,2n2),

where[Cdet]!M is the trivial U(2)-equivariant line bundle associated with the character det:U(2)!C^∗.

• There exists a connection on Ln1,n2 such that the corresponding moment map Φ_n1,n2:K×TY!k^∗ is defined by

Φn₁,n₂([k, y]) = − n1+³₂

+(n2+1)ϕ(y)

k·e^∗₂+¹₂(e^∗₁+e^∗₂).

Proof. For the second point, we construct aU(2)-invariant connection onLn1,n2 by choosing theT-invariant connection on (Ln1,n2)|_Y having moment map

− n1+³₂

+(n2+1)ϕ(y)

e^∗₂+¹₂(e^∗₁+e^∗₂) under theT-action (see equation (6.41)).

From Proposition6.1, it is not difficult to describe the “Kirwan set”

∆(n1, n2) = Image(Φn₁,n₂)∩t^∗_>0

for all cases of n₁ and n₂. It depends of the signs of n₁+³₂, n₂+1, n₁−n₂+¹₂, that is, as we are working with integers, the signs ofn₁+1, n₂+1 and n₁−n₂. We concentrate

in the case wheren1+1>0 and n2+1>0 (other cases are similarly treated). Then, we

Ifn₁>n₂>0, the curvature of the corresponding connection on Ln₁,n₂=L(2n1+3,2n2+2)

(which is an ample line bundle) is non-degenerate, thus the image is a convex subset of t^∗_>0(in agreement with Kirwan convexity theorem), while forn₂>n₁the image set is not convex.

whereS^k(C²) is the space of complex polynomials onC² homogeneous of degreek.

If n₂>0, we know that Q_T(Y,S_Y⊗L^⊗n2)=Pn2

k=0t^k₂. From the induction formula (3.17) (or direct computation via Cech cohomology!) we obtain the following:

• Ifn1>n2, then

Figure 9. K-Multiplicities forQ_K(8,5).

%^K

Figure 10. K-multiplicities forQ_K(3,6).

Let us check how our theorem works in these cases. First, we notice that we are in a multiplicity free case: all the non-empty reduced spaces are points.

• Consider the case wheren1>n2. We see that the parameter ¹₂,−k−¹₂

belongs to the relative interior of the interval ∆(n1, n2) if and only ifn1−n26k6n1.

• Consider the case wheren2>n1. We see that the parameter (¹₂,−k−¹₂) belongs to the relative interior of

−n1−³₂,0

e^∗₂+¹₂(e^∗₁+e^∗₂) if and only if k6n1. Similarly, the parameter k+³₂,¹₂

belongs to the relative interior of

0, n2−n1−¹₂

e^∗₁+¹₂(e^∗₁+e^∗₂) if and only ifk6n2−n1−2.

In Figures9and10, we draw the Kirwan subsets oft^∗_>0corresponding to the values (n₁, n₂)=(8,5) and (n₁, n₂)=(3,6). The points on the red line represents the admissible points occurring with multiplicity 1 inQK(n₁, n₂). The points on the blue line represents the admissible points occurring with multiplicity−1 inQK(n₁, n₂).

Consider nowM as aT-manifold. Let Φ^T_n

1,n₂:M!t^∗ be the moment map relative

Figure 11. T-multiplicities forQT(8,5).

Figure 12. T-multiplicities forQT(3,6).

to the action ofT which is the composition of Φn₁,n₂:M!k^∗ with the projectionk^∗!t^∗. Thus, the image of Φ^T_n1,n2 is the convex hull of ∆(n1, n2) and its symmetric image with respect to the diagonal.

Consider first the case where (n1, n2)=(8,5). Thus our determinant bundleL8,5 is ample. The image of the moment map Φ^T_8,5:M!t^∗ is drawn in Figure11. It is a convex polytope with vertices −3,¹₂

, ¹₂,−3

, ¹₂,−9

and −9,¹₂

, the images of the four fixed points [1,0,1,0], [1,0,0,1], [0,1,1,0] and [0,1,0,1].

We now concentrate on the case (n₁, n₂)=(3,6). The line bundle L:=L3,6 is not ample, so that its curvature Ω_L is degenerate, and the Liouville form β_L=Ω_L∧Ω_L is a signed measure onM. Let us draw the Duistermaat–Heckman measure (Φ_L)_∗β_L, a signed measure ont^∗. In red the measure is with value 1, in blue the measure is with value−1 (see Figure12).

We also verify that our theorem is true. Indeed, the representationQT(3,6) is 1+t⁻¹₁ +t⁻¹₂ +t⁻²₁ +t⁻¹₁ t⁻¹₂ +t⁻²₂ +t⁻³₁ +t⁻²₁ t⁻¹₂ +t⁻¹₁ t⁻²₁ +t⁻³₂ −t1t2−t1t²₂−t²₁t2. Theλ∈Z² such thatt^λ occurs in QT(3,6) are the integral points in the interior of the image of Φ_L(M): they have multiplicity±1, and the reduced spaces are points.

In this case, we verify thus that the image of the moment map is exactly the support of the Duitermaat–Heckman measure, however, we do not know if (even in the toric case, and non-ample lines bundles) we can always find a connection with this property.

6.3. The multiplicity of the trivial representation comes from two reduced spaces

ConsiderC⁴with its canonical basis{e₁, ..., e₄}. LetK'SU(3) be the subgroup of SU(4) that fixese₄.

LetT=S(U(1)×U(1)×U(1)) be the maximal torus ofK with Lie algebra t={(x1, x2, x3) :x1+x2+x3= 0}

and Weyl chamber t^∗_>0:={ξ1>ξ2>ξ3:ξ1+ξ2+ξ3=0}. We choose the fundamental roots ω1 and ω2 so that Kω₁=S(U(2)×U(1)) and Kω₂=S(U(1)×U(2)). Recall that ω1, ω2

generates the weight lattice Λ⊂t^∗ so that Λ_>0=Nω1+Nω2. Also note that %=ω1+ω2. For any λ∈Λ_>0+%, we denote by πλ the irreducible representation of K with highest weightλ−%.

LetX={0⊂L1⊂L2⊂C⁴:dimLi=i}be the homogeneous partial flag manifold under the action of SU(4). We have two lines bundles over X: L1(x)=L1 and L2(x)=L2/L1

forx=(L1, L2).

Our object of study is the complex submanifold M={(L1, L2)∈X:Ce4⊂L2}.

The groupK acts onM, and the generic stabilizer of the action is [Kω1, Kω1]'SU(2).

We consider the family of lines bundles

L(a, b) =L^⊗a₁ |M⊗L^⊗−b₂ |M, (a, b)∈N². LetSM:=V

CTM be the spin^c-bundle associated with the complex structure onM. We compute the characters

Q_K(a, b) :=Q_K(M,S_M⊗L(a, b))∈R(K).

Again,

We notice that K_ω1 corresponds to the subgroup ofK that fixes the lineCe₃. The setY:={(L₁, L₂)∈X:L₂=Ce₃⊕Ce₄}is aK_ω1-invariant complex submanifold ofM such that the mapK×Y3(k, y)7!ky∈M factorizes through an isomorphismK×_Kω

1Y'M.

In particular, the multiplicity ofπ% (the trivial representation) in QK(a, b) is equal to

−2.

We now verify the formula (5.40) in our case. The spin^c-bundle SM is equal to SKω₁⊗K×K_ω1SY. The corresponding determinant line bundle det(SM) satisfies

det(SM) =K×K_ω1C3ω₁⊗K×K_ω1det(SY) =K×K_ω1C2ω₁⊗L^⊗−2₁ . Hence, for the spin^c-bundleSM⊗L(a, b), we have

det(SM⊗L(a, b)) = det(SM)⊗L(a, b)^⊗2=K×K_ω1C(2b+2)ω1⊗L^{⊗2(a+b−1)}₁ .

The line bundle det(S_M⊗L(a, b)) is equipped with a natural holomorphic and Hermitian connection∇. To compute the corresponding moment map Φa,b:M!k^∗, we notice that L1=K×K_ω1L⁻¹, where L!P¹ is the prequantum line bundle over P¹ (equipped with the Fubini–Study symplectic form). If we letϕ:Y'P¹![0,1] be the function defined by

ϕ([z1, z2]) = |z1|²

We know (see Exemple 3.9) that the setA((k_ω1)) is equal to the collection of orbits K ¹₂(1+2n)ω_i

,n∈N,i=1,2, and we haveQ_K K ¹₂ω_i

=0 andQ_K K ¹₂(3+2k)ω_i

= π_kωi+% whenk>0.

If we apply (5.40), we see that πkω₁+% occurs in QK(a, b) only if ¹₂(3+2k)<b+1:

so k∈{0, ..., b−1}. Similarly πjω₂+% occurs in QK(a, b) only if ¹₂(3+2j)<a−2: thus j∈{0, ..., a−4}. For all these cases the corresponding reduced spaces are points and one could check that the corresponding quantizations are all equal to−1 (see (5.36)).

In these cases, two orbitsP_i=K ³₂ω_i

,i=1,2, are the ancestors of the trivial repre-sentation inA((k_ω1)), and the multiplicity of the trivial representation in

QK(M,SM⊗L(a, b)) is equal to

Q^spin(M_P1)+Q^spin(M_P2) =−2.

References

[1] Atiyah, M. F., Elliptic Operators and Compact Groups. Lecture Notes in Mathematics, 401. Springer, Berlin–New York, 1974.

[2] Atiyah, M. F. & Bott, R., The Yang–Mills equations over Riemann surfaces. Philos.

Trans. Roy. Soc. London Ser. A, 308 (1983), 523–615.

[3] Atiyah, M. F. & Hirzebruch, F., Spin-manifolds and group actions, inEssays on Topol-ogy and Related Topics (M´emoires d´edi´es `a Georges de Rham), pp. 18–28. Springer, New York, 1970.

[4] Berline, N., Getzler, E. & Vergne, M., Heat Kernels and Dirac Operators.

Grundlehren Text Editions. Springer, Berlin–Heidelberg, 2004.

[5] Berline, N. & Vergne, M., L’indice ´equivariant des op´erateurs transversalement ellip-tiques.Invent. Math., 124 (1996), 51–101.

[6] Braverman, M., Index theorem for equivariant Dirac operators on noncompact manifolds.

K-Theory, 27 (2002), 61–101.

[7] — The index theory on non-compact manifolds with proper group action.J. Geom. Phys., 98 (2015), 275–284.

[8] Duflo, M., Construction de repr´esentations unitaires d’un groupe de Lie, in Harmonic Analysis and Group Representations, pp. 129–221. Liguori, Naples, 1982.

[9] Duistermaat, J. J.,The Heat Kernel Lefschetz Fixed Point Formula for the Spin-cDirac Operator. Progress in Nonlinear Differential Equations and their Applications, 18.

Birkh¨auser, Boston, MA, 1996.

[10] Grossberg, M. D. & Karshon, Y., Bott towers, complete integrability, and the extended character of representations.Duke Math. J., 76 (1994), 23–58.

[11] — Equivariant index and the moment map for completely integrable torus actions.Adv.

Math., 133 (1998), 185–223.

[12] Guillemin, V., Reduced phase spaces and Riemann–Roch, inLie Theory and Geometry, Progr. Math., 123, pp. 305–334. Birkh¨auser, Boston, MA, 1994.

[13] Guillemin, V. & Sternberg, S., Geometric quantization and multiplicities of group representations.Invent. Math., 67 (1982), 515–538.

[14] — A normal form for the moment map, inDifferential Geometric Methods in Mathematical Physics (Jerusalem, 1982), Math. Phys. Stud., 6, pp. 161–175. Reidel, Dordrecht, 1984.

[15] Hattori, A., Spin^c-structures andS¹-actions.Invent. Math., 48 (1978), 7–31.

[16] Hochs, P. & Mathai, V., Quantising proper actions on Spin^c-manifolds.Asian J. Math., 21 (2017), 631–686.

[17] Hochs, P. & Song, Y., Equivariant indices of Spin^c-Dirac operators for proper moment maps.Duke Math. J., 166 (2017), 1125–1178.

[18] Jeffrey, L. C. & Kirwan, F. C., Localization and the quantization conjecture.Topology, 36 (1997), 647–693.

[19] Karshon, Y. & Tolman, S., The moment map and line bundles over presymplectic toric manifolds.J. Differential Geom., 38 (1993), 465–484.

[20] Kirwan, F. C.,Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathe-matical Notes, 31. Princeton University Press, Princeton, NJ, 1984.

[21] Lerman, E., Meinrenken, E., Tolman, S. & Woodward, C., Nonabelian convexity by symplectic cuts.Topology, 37 (1998), 245–259.

[22] Ma, X. & Zhang, W., Geometric quantization for proper moment maps: the Vergne conjecture.Acta Math., 212 (2014), 11–57.

[23] Meinrenken, E., Symplectic surgery and the Spin^c-Dirac operator. Adv. Math., 134 (1998), 240–277.

[24] Meinrenken, E. & Sjamaar, R., Singular reduction and quantization. Topology, 38 (1999), 699–762.

[25] Paradan, P.-E., Formules de localisation en cohomologie equivariante.Compositio Math., 117 (1999), 243–293.

[26] — Localization of the Riemann–Roch character.J. Funct. Anal., 187 (2001), 442–509.

[27] Paradan, P.-E. & Vergne, M., Index of transversally elliptic operators.Ast´erisque, 328 (2009), 297–338.

[28] — Witten non abelian localization for equivariant K-theory and the [Q, R]=0 theorem.

To appear inMem. Amer. Math. Soc.

[29] — Admissible coadjoint orbits for compact lie groups. To appear inTransform. Groups.

[30] Cannas da Silva, A., Karshon, Y. & Tolman, S., Quantization of presymplectic man-ifolds and circle actions.Trans. Amer. Math. Soc., 352 (2000), 525–552.

[31] Sjamaar, R., Holomorphic slices, symplectic reduction and multiplicities of representa-tions.Ann. of Math., 141 (1995), 87–129.

[32] Teleman, C., The quantization conjecture revisited.Ann. of Math., 152 (2000), 1–43.

[33] Tian, Y. & Zhang, W., An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg.Invent. Math., 132 (1998), 229–259.

[34] Vergne, M., Multiplicities formula for geometric quantization. I, II. Duke Math. J., 82 (1996), 143–179, 181–194.

[35] — Equivariant index formulas for orbifolds.Duke Math. J., 82 (1996), 637–652.

[36] Witten, E., Two-dimensional gauge theories revisited.J. Geom. Phys., 9 (1992), 303–368.

Paul-Emile Paradan

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